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Experimental investigations of step-pool channel formation and stability Zimmermann, Andre Eric 2009

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Experimental investigations of step-pool channel formation and stability  by  André Eric Zimmermann B.Sc., The University of British Columbia, 1999 M.Sc., McGill University, 2003  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  In  THE FACULTY OF GRADUATE STUDIES (Geography) THE UNIVERSITY OF BRITISH COLUMBIA  (Vancouver)  March 2009  © André Eric Zimmermann, 2009  1 Abstract The stability of steep streams with step-pool and cascade morphologies cannot be assessed using knowledge developed from lowland streams due to the structured nature of headwater streams. Thus there is a need for experimental studies examining the stability of such channels. The structuring of these channels occurs as a result of boulders and cobbles jamming across the width of the channel and the more typical pattern of armouring and imbrication, which results from a relatively low rate of sediment supply. To conduct such a study, new experimental techniques were developed and an artificial stream channel (flume) was designed and built. Channel width, bed grain size and channel gradient were varied and step-pool bedforms were created and subsequently destroyed. The variables governing the dimensions, frequency and form of step-pools were observed to be channel slope, bed grain size and channel width. Video records show that on occasion groups of larger stones moved together as a coherent group, forming a line of stones that locked together across the width of the channel. The failure of a step most frequently occurred when the downstream scour pool undermined the step-forming stones and was often associated with headward migrating instabilities. With smaller jamming ratios (channel width/D84steps — the diameter at which 84% of the step stones are smaller than) stable beds persisted at larger Shields numbers (ratio of shear stress to grain size), confirming that such channels do gain stability by having grains jam across their width. The failure of the bed was shown to be a stochastic process with nearly half of the failures occurring within the first minute following an increase in discharge, while 26% of the failures did not start to occur until tens of minutes or more after the flow was increased. Detailed bed morphology, channel grain size and flow velocity measurements suggest that a dimensionless hydraulic geometry approach is the best method of predicting flow velocities in headwater channels. The bed stability criterion, in combination with a dimensionless hydraulic geometry approach, provides a means of assessing the stability of mountain stream channels.  ii  2 Table of contents 1  Abstract................................................................................................................................... ii  2  Table of contents ................................................................................................................... iii  3  List of tables .......................................................................................................................... vi  4  List of figures....................................................................................................................... viii  5  List of videos ....................................................................................................................... xvi  6  List of symbols ................................................................................................................... xvii  7  List of equations .................................................................................................................. xix  8  Acknowledgements............................................................................................................. xxii  1  Introduction ............................................................................................................................ 1 1.1 Literature review............................................................................................................. 2 1.1.1 The step-pool form ..................................................................................................... 2 1.1.2 Formation of step-pool channel units ......................................................................... 8 1.1.3 Channel stability ....................................................................................................... 12 1.1.4 Flow resistance ......................................................................................................... 15 1.2 Research objectives ...................................................................................................... 16  2  Methods ................................................................................................................................ 19 2.1 Flume design................................................................................................................. 19 2.2 Instrumentation ............................................................................................................. 20 2.2.1 Grain movement ....................................................................................................... 21 2.2.2 Water and bed surface elevations ............................................................................. 22 2.2.3 Flow velocities and water depth ............................................................................... 27 2.2.4 Bedload transport...................................................................................................... 33 2.2.5 Discharge .................................................................................................................. 34 2.2.6 Flume grain size mix ................................................................................................ 34 2.2.7 Measuring surface grain size .................................................................................... 36 2.2.8 Channel width........................................................................................................... 38 2.2.9 Bank roughness......................................................................................................... 38 2.3 Error analysis ................................................................................................................ 42  3  Two innovative techniques ................................................................................................... 43 3.1 Identifying step-pool units............................................................................................ 43 3.1.1 Field, channel classification and analysis techniques............................................... 45 3.1.1.1 Field survey methods........................................................................................ 45 3.1.1.2 Assessing variability among step-pool researchers .......................................... 46 3.1.1.3 Introducing the scale-free rule-based classification algorithm......................... 48 3.1.1.4 Determining average long profile from bed scans............................................ 50 3.1.1.5 Other published methods of classifying step-pool channels............................. 52 3.1.2 Classification results................................................................................................. 53 3.1.2.1 Respondents’ results ......................................................................................... 53 iii  3.1.2.2 Results from applying the scale-free rule-based classification scheme............ 58 3.1.2.3 Other published methods of classifying step-pool channels............................. 63 3.1.3 Discussion................................................................................................................. 65 3.1.4 Summary................................................................................................................... 66 3.2 Measuring sediment transport rates and grain size....................................................... 67 3.2.1 Equipment................................................................................................................. 67 3.2.2 Design considerations............................................................................................... 69 3.2.3 Image acquisition...................................................................................................... 70 3.2.4 Data analysis............................................................................................................. 72 3.2.5 Calibration of GSD................................................................................................... 78 3.2.6 Validation tests ......................................................................................................... 80 3.2.7 Summary................................................................................................................... 81 4  Experimental idealism .......................................................................................................... 82 4.1 Preconceived notions .................................................................................................... 82 4.2 Froude scale modeling of step-pool streams ................................................................ 83 4.2.1 Prototype 1: Shatford Creek ..................................................................................... 84 4.2.2 Prototype 2: Rio Cordon........................................................................................... 91 4.2.3 Summary comments regarding Froude scaling ........................................................ 92 4.3 Trying to make step-pool channels............................................................................... 92 4.3.1 Determining flow duration ....................................................................................... 93 4.3.2 Determining flow increase........................................................................................ 93 4.3.3 Determining feed rate ............................................................................................... 94 4.3.4 Adjusting flow rate for changes in flume slope and bed grain size.......................... 96 4.4 Success! Step-pools, among other bed features, are formed ........................................ 98 4.4.1 Wedge dynamics..................................................................................................... 106 4.5 Summary thoughts ...................................................................................................... 107  5  The flow, the bed and the transport: Interactions in a steep flume..................................... 108 5.1 Step-pool form and frequency .................................................................................... 108 5.1.1 Controls on step-pool frequency............................................................................. 109 5.1.2 Step dimensions...................................................................................................... 113 5.1.3 Pool dimensions...................................................................................................... 116 5.1.4 Step-pool shape parameters .................................................................................... 118 5.1.5 Summary of controls on step-pool form and frequency ......................................... 119 5.2 Flow properties ........................................................................................................... 120 5.2.1 Froude number........................................................................................................ 120 5.2.2 Width to depth ratio................................................................................................ 122 5.3 Sediment transport and its effect on channel morphology ......................................... 125 5.3.1 Patterns of sediment transport ................................................................................ 126 5.3.2 Patterns of instabilities............................................................................................ 128 5.3.2.1 Headward migrating instabilities.................................................................... 129 5.3.2.2 Splays and cutoffs........................................................................................... 130 5.3.2.3 Sheet flow ....................................................................................................... 130 5.3.3 Role of sediment transport in forming step-pool channels..................................... 131 5.4 Variability among identical runs ................................................................................ 135 5.5 Effect of flow conditioning......................................................................................... 141  6  Step-pool stability: Testing the jammed state hypothesis................................................... 144 iv  6.1 Theoretical background .............................................................................................. 144 6.2 Experimental data ....................................................................................................... 147 6.3 Classifying bed changes ............................................................................................. 147 6.4 Results ........................................................................................................................ 150 6.4.1 Stable bed states...................................................................................................... 150 6.4.2 Effect of varying experimental design and run duration on stability ..................... 156 6.4.3 Differentiating unstable and stable beds................................................................. 156 6.4.4 Replicate experiments............................................................................................. 159 6.4.5 Effect of sediment supply on stable bed configurations......................................... 160 6.4.6 Comparing results with results from other studies ................................................. 168 6.5 Discussion................................................................................................................... 173 6.5.1 Stochastic influences on stability ........................................................................... 173 6.5.2 Influence of bank roughness................................................................................... 175 6.5.3 Comparing different means of assessing shear stress............................................. 177 6.5.4 Revisiting shear stress approaches to entrainment ................................................. 182 7  Water velocities and flow resistance .................................................................................. 183 7.1 Introduction ................................................................................................................ 183 7.2 Experimental overview and data selection ................................................................. 187 7.3 Results ........................................................................................................................ 187 7.3.1 Hydraulic geometry observations........................................................................... 187 7.3.2 Patterns of flow velocity during feed, armouring and hydraulic geometry runs .... 199 7.3.3 Flow resistance functions ....................................................................................... 200 7.4 Discussion................................................................................................................... 208 7.4.1 Resistance partitioning ........................................................................................... 210  8  Conclusions ........................................................................................................................ 213 8.1 Conditions leading to step-pool formation and destabilization .................................. 213 8.2 Effect of history on step-pool channels ...................................................................... 214 8.3 Jamming plot .............................................................................................................. 215 8.4 Flow resistance in steep channels ............................................................................... 215  9  Bibliography ....................................................................................................................... 217  10  Appendix 1: Wall roughness corrections............................................................................ 227  11  Appendix 2: Compendium.................................................................................................. 230  v  3 List of tables Table 1. Range of flume slopes used in past mobile bed steep stream experiments. ................... 12 Table 2. Prototype and model dimensions.................................................................................... 20 Table 3. Metrics measured during experiments and method used to measure their value. .......... 21 Table 4. Functional regression results for depth back calculated using 9 different measures of velocity compared to depth measured using water surface and bed surface scans. ....... 32 Table 5. Error associated with measuring grain size .................................................................... 37 Table 6. Summary of experiments................................................................................................ 39 Table 7. Method used to estimate standard error for each metric directly measured................... 42 Table 8. Error propagation relations............................................................................................. 42 Table 9. Classification results of long profiles distributed to respondents................................... 53 Table 10. Agreement between respondents and field classification............................................. 57 Table 11. Parameters used in rule-based classification algorithm................................................ 58 Table 12. Number of step-pools that were identified in the field, by the respondents and using the classification algorithm................................................................................................. 59 Table 13. Number of step-pools identified using a range of methods applied to the entire length of four channels. ........................................................................................................... 61 Table 14. Equipment and costs of GSD setup .............................................................................. 68 Table 15. Grain size estimation statistics based on calibration and validation results. ................ 79  vi  Table 16. Results of paired t-test comparisons of Dx calculated using the GSD and sieved grain size distribution............................................................................................................. 80 Table 17. Standardized experimental procedure used to armour bed for an experiment with the fine sediment mix and a flume gradient of eight percent. ............................................ 95 Table 18. Range of bed conditions observed during 32 experiments......................................... 109 Table 19. Coefficient of variation from a range of flow and bed property measurements for data from last run of two sets of replicate experiments...................................................... 141 Table 20. Grain size and bedload transport rate for each bed disturbance type. ........................ 149 Table 21. Other studies used to compare results from this study with. ...................................... 170 Table 22. Dimensionless hydraulic geometry regression coefficients for each hydraulic geometry run............................................................................................................................... 196 Table 23. Flow resistance relations and their precision and error .............................................. 206 Table 24. Wall correction results based on Einstein (1941) and Vanoni and Brooks (1957) .... 229  vii  4 List of figures Figure 1. Step-pool bedforms in Deeks Creek, British Columbia.................................................. 3 Figure 2. Steep stream channel morphologies................................................................................ 4 Figure 3. Illustrations with definitions of channel types and commonly measured step-pool metrics ............................................................................................................................. 5 Figure 4. Proposed projection of Shields stability criterion for step-pool channels..................... 17 Figure 5. Video image from overhead camera. ............................................................................ 21 Figure 6. Two bed scans of flume ................................................................................................ 22 Figure 7. Image illustrating calibration template for scanner....................................................... 23 Figure 8. Long profile of bed and water surface. ......................................................................... 25 Figure 9. Filtered water surface scan............................................................................................ 26 Figure 10. Tipping bucket used to inject salt solution.................................................................. 27 Figure 11. Schematic illustrating layout of conductivity probe circuitry..................................... 28 Figure 12. Nine different measures of mean water velocity related to discharge for Experiment 16 during a hydraulic geometry run with a stable bed. ............................................... 30 Figure 13. Relations between depth back calculated using continuity for 9 different measures of water velocity and average depth measured with water surface and bed surface scans. ..................................................................................................................................... 31 Figure 14. Relation between peak in conductivity pulse velocity and harmonic mean derived velocity ........................................................................................................................ 32  viii  Figure 15. Mean velocities of neutral density wax particles compared to mean salt injection velocities...................................................................................................................... 33 Figure 16. Grain size distributions of initial flume mix, surface distribution of a number of steppool creeks (a) and distribution of sediment mixes used in a number of other steepstream experimental studies (b). .................................................................................. 35 Figure 17. a) Base of bank edge along Shatford Creek, b) deviation from bank position as a function of bank distance, c) frequency distribution of deviations from bank position. ..................................................................................................................................... 41 Figure 18. Long profiles distributed for classification. ................................................................ 47 Figure 19. Example of the variation in how different respondents might classify a step-pool unit.. ..................................................................................................................................... 50 Figure 20. a) Scans from two flume runs and b) long profiles ..................................................... 51 Figure 21. Variability associated with location of a) end of pool, b) start of pool, and c) top of step recorded by each respondent.. .............................................................................. 54 Figure 22. Variability associated with a) length of pool and b) step drop height recorded by each respondent.................................................................................................................... 55 Figure 23. Entire profile of Tees Creek with field, rule-based, and minimum step slope classification results illustrated.................................................................................... 60 Figure 24. Entire profile of Giveout Creek with field, rule-based, and minimum step slope classification results illustrated.................................................................................... 60 Figure 25. Results of sensitivity analysis performed by varying each of the parameters independently............................................................................................................... 62 Figure 26. Conceptual perspective drawing of the flume outlet and GSD................................... 68 Figure 27. Examples of particle velocity as a function of position across the light table during low (a) and high (b) flow runs, area (c) and minor axis (d) ........................................ 71 ix  Figure 28. GSD image. ................................................................................................................. 72 Figure 29. Schematic diagram illustrating steps in image analysis. ............................................. 73 Figure 30. Example of sampling window for three sequential images......................................... 75 Figure 31. a) Relation between area of particle measured on light table and weight of particle and b) ellipsoid minor axis measured on light table and measured b axis using callipers. 77 Figure 32. Box plot illustrating error in GSD-estimated sediment transport rate by grain size class based on (a) 13 calibration samples and (b) 8 validation samples after calibration coefficients were applied. ............................................................................................ 79 Figure 33. Plan view map and long profile of study section along Shatford Creek used as the initial prototype for the flume...................................................................................... 85 Figure 34. Shatford Creek at 3000 L/s.......................................................................................... 88 Figure 35. Scaled grain size distribution of Shatford Creek and grain size distribution from Experiment 2 run 8.. .................................................................................................... 89 Figure 36. Long profile of bed from Experiment 2 at end of run 8.............................................. 90 Figure 37. Sediment transport as a function of time for initial portions of three experiments where initial flow rate was much greater than the critical flow to mobilize bed......... 93 Figure 38. Response of sediment transport records to addition of sediment feed for three portions of Experiment 2. .......................................................................................................... 95 Figure 39. Flow, feed, sediment transport and mean water velocity from first 1500 minutes of Experiment 4................................................................................................................ 96 Figure 40. Maximum stable bed gradient as a function of stream power (a, b)and Shields ratio (c, d) for fine sediment experiments. ........................................................................... 97 Figure 41. Images from Experiments 1, 4, 6 and 26.. ................................................................ 100 Figure 42. Images from Experiments 10, 16, 21 and 25. ........................................................... 101 x  Figure 43. Images from Experiment 26...................................................................................... 102 Figure 44. Images from Experiments 23, 28 and 29.. ................................................................ 103 Figure 45. Images from Experiments 30 and 32. ....................................................................... 104 Figure 46. Relation between number of step-pools per bankfull width and bed slope .............. 110 Figure 47. D84/w and D84step/w as a function of bed slope. ......................................................... 111 Figure 48. Fraction of bed composed of step-pools as a function of the D84/w ratio ................. 112 Figure 49. Relation between step height and drop height. ......................................................... 114 Figure 50. Relation between step shape (step length/drop height) and D84/w............................ 116 Figure 51. H / L / S (a) and H / L s / S (b) as a function of channel slope ................................. 118 Figure 52. Froude number as a function of time since experiment started for (a) all runs after at least 60 minutes of flow over the bed and (b) only portion of time just before run ended.......................................................................................................................... 121 Figure 53. For each experiment the Froude number is plotted as a function of time................ 123 Figure 54. Histogram illustrating water surface width/mean depth ratio for all experiments.... 124 Figure 55. Relation between hydraulic radius and flow depth as a function of width ............... 124 Figure 56. Sediment transport record from the end of Experiment 22 when a 9.2 L/s flow was run for a cumulative duration of twenty hours.. ........................................................ 126 Figure 57. (a) Average grain size of bedload leaving the flume and (b) sediment transport rate during the first 93 minutes of Experiment 1.............................................................. 128 Figure 58. Step-pool formed between 8th and 10th hour of Experiment 16. ............................... 132 Figure 59. Sediment transport record from first 24 minutes of Experiment 9 ........................... 134 Figure 60. Time until sediment transport rate increased following an increase in flow. ........... 135 xi  Figure 61. Cumulative sediment yield for replicate portion of experiments.............................. 137 Figure 62. Flow properties as a function of time for the portion of Experiments 5, 6, 7 and 13 until one hour after the last feed ................................................................................ 138 Figure 63. Number of step-pools, bed slope, standard deviation of bed, D50, D84 and D84step as a function of time for the portion of Experiments 5, 6, 7 and 13 until one hour after the last feed...................................................................................................................... 139 Figure 64. Flow properties as a function of time for the portion of Experiments 16, 20 and 22 until three hours after the second flow increase following the last feed. .................. 140 Figure 65 Number of step-pools, bed slope, standard deviation of bed, D50, D84 and D84step as a function of time for portion of Experiments 16, 20 and 22 up until three hours after the second flow increase following the last feed............................................................. 140 Figure 66. Flow properties as a function of discharge for Experiments 5, 6, 7 and 13. ........... 142 Figure 67. Bed properties as a function of discharge for Experiments 5, 6, 7 and 13. ............. 143 Figure 68. Box plot illustrating range of unit sediment transport rates associated with each channel change class.................................................................................................. 149 Figure 69. D50 of bed (a) and D84 of steps (b) as a function of DuBoys derived shear stress. ... 151 Figure 70. Maximum DuBoys shear stress sustained on the bed as a function of jamming ratio (w/D84steps).................................................................................................................. 152 Figure 71. Maximum sustained Shields ratio as a function of jamming ratio based on D84 of bed surface and D84 of steps.. ........................................................................................... 153 Figure 72. Jamming plot for stable beds for all jamming ratios (a) and jamming ratios ≤ 6 (b).154 Figure 73. Relation between predicted Shields ratio using Eq. 26 and observed Shields ratio.. 155 Figure 74. Probability of bed being unstable as a function of the Shields ratio and jamming ratio. ................................................................................................................................... 157  xii  Figure 75. Distinguishing stable and unstable beds based on depth-slope derived shear stress estimates and mean velocity.. .................................................................................... 159 Figure 76. Jamming data from Experiments 5, 6, 7 and 13........................................................ 160 Figure 77. Photographs of bed during Experiment 20 when fine feed was added while discharge was held constant....................................................................................................... 162 Figure 78. Response of surface grain size distribution to (a) sediment feed while discharge was held constant and (b) constant discharge with no sediment feed............................... 163 Figure 79. Jamming plot illustrating effect of feeding fine sediment......................................... 166 Figure 80. Formation of scour hole during the fine sediment feed run of Experiment 19. ........ 168 Figure 81. Map of East Creek showing edge of mobile sediment and edge of water line during flood........................................................................................................................... 169 Figure 82. Data from other studies presented with jamming results from this study................. 172 Figure 83. Stones structuring along banks.................................................................................. 176 Figure 84. Jamming plots illustrating relative bank roughness determined using the channel width (a) and the D50 of the sediment mix (b). .......................................................... 176 Figure 85. Water surface and bed surface elevations averaged across the cross-section from runs before and after flow was increased 20%.. ................................................................ 178 Figure 86. Froude numbers along channel prior to and after flow was increased 20%. ............ 179 Figure 87. Comparison of median along-channel Froude number and salt tracer-continuity derived Froude number.............................................................................................. 180 Figure 88. Change in continuity derived flow depth and mean flow velocity following a 20% increase in flow.......................................................................................................... 181 Figure 89. Histogram illustrating number of velocity measurements for each unique combination of discharge, bed morphology and grain size. ........................................................... 188 xiii  Figure 90. Mean velocity as a function of discharge for 20 hydraulic geometry runs.. ............. 189 Figure 91. Individual hydraulic geometry relations.. ................................................................. 190 Figure 92. Darcy-Weisbach friction factor (f) as a function of relative depth for data from 22 hydraulic geometry experiments.. ............................................................................. 191 Figure 93. Standard deviation about mean long profile based on bed scan as a function of the D84 of the bed............................................................................................................. 192 Figure 94. Darcy-Weisbach friction factor (f) as a function of mean flow depth (y) normalized by the standard deviation of the bed (σ) for data from 22 hydraulic geometry runs. ..... 192 Figure 95. Relation between velocity and discharge normalized by the standard deviation (σ) of the bed and the D84 of the surface for hydraulic geometry runs (a and b) and all runs (c and d).. ....................................................................................................................... 193 Figure 96. Dimensionless hydraulic geometry relations for 22 experiments for data from hydraulic geometry runs during which bed was stable.............................................. 194 Figure 97. Residuals about dimensionless hydraulic geometry relations for individual hydraulic geometry runs. ........................................................................................................... 197 Figure 98. Relation between exponent (α’) in dimensionless hydraulic geometry relation and flow and bed characteristics for 22 hydraulic geometry runs with stable beds. ........ 198 Figure 99. Relation between constant (c’) in dimensionless hydraulic geometry relation and flow and bed characteristics for 22 experiments with hydraulic geometry data................ 198 Figure 100. Velocity as a function of discharge for 20 experiments.......................................... 201 Figure 101. Dimensionless hydraulic geometry relations based on standard deviation of bed elevations (σ) for 20 experiments. ........................................................................... 202 Figure 102. Dimensionless hydraulic geometry relations based on D84 for 20 experiments. .... 203  xiv  Figure 103. Plot illustrating residuals from prediction of velocity using the Darcy-Weisbach loglaw of the wall approach as a function of bed and flow properties. ........................ 208 Figure 104. Plot illustrating residuals from prediction of velocity using a dimensionless hydraulic geometry approach based on unit discharge, D84 and bed slope. ............ 209 Figure 105 Darcy-Weisbach friction factor as a function of time.. ............................................ 212  xv  5 List of videos Video 1. Flow characteristics at 1.7 L/s. ...................................................................................... 88 Video 2. Illustrates flow of 1.7 L/s over a bed with a grain size distribution similar to the scaled distribution of Shatford Creek ........................................................................................ 90 Video 3. Flow at 61 L/s over fine bed at medium width from last run during Experiment 6.. . .. 99 Video 4. Large jump formed during Experiment 25. . .............................................................. 105 Video 5. First 93 minutes of Experiment 1 ................................................................................ 127 Video 6. A headward migrating instability starts nine seconds into the video and eventually travels the entire length of the flume and degrades bed to flume floor. ....................... 129 Video 7. Overhead record of sediment transport for first hour of Experiment 8. ...................... 129 Video 8. Formation of a meandering channel, bars and the passage of waves of sediment during Experiment 9. ............................................................................................................. 130 Video 9. Transport of sheets of sediment during 18% gradient run with fine mix and new bed. ...................................................................................................................................... 131 Video 10. Jamming of large white stone and neighbouring stones against bank during Experiment 16. ............................................................................................................. 133 The videos have been encoded with the Microsoft Windows Video CODEC 9 and can be played with Windows Media Player among other software. They can be found on the DVD that is included with the thesis (hardcopy) and are embedded in the PDF file.  xvi  6 List of symbols a aaxis A Axs AC b baxis c C dr dt D D84 D84step D90 Dmax DC e f_sp f Fr g ĝ H k l L Ls Lsp m m madj n n nw p P Per Q Qs q q* qs R  Constant in hydraulic geometry relation Length of a axis of grain (mm) Channel cross sectional area (m2) Cross sectional area of grain measured on light table (mm2) Alternating current Exponent in hydraulic geometry relation Length of b axis of grain (mm) Constant in dimensionless hydraulic geometry relation (Eq. 4) Chézy resistance coefficient (m1/2/s) Residual depth; crest of pool elevation less minimum elevation in pool (m) (Figure 3) Critical depth at step crest (m) (Figure 3) Grain size (m) Grain size at which 84% of the sediment is finer (m) Grain size at which 84% of the sediment in the steps is finer (m) Grain size at which 90% of the sediment is finer (m) Largest grain size present (m) Direct current Is the root of the natural logarithm Fraction of bed composed of step-pools Darcy-Weisbach friction factor Froude number (Eq. 7) Acceleration due to gravity (9.81 m/s2) Logit value from logistic regression Step height = ys (m) (Figure 3) Nikuradse roughness height (m) Generic roughness length (m) Step to step distance (m) (Figure 3) Pool length (m) (Figure 3) Length of step and pool (m) (Figure 3) Exponent in hydraulic geometry relation Slope in linear regression relation Slope for functional regression relation Sample size Manning n resistance coefficient (m1/3/s) Manning n resistance coefficient associated with wall friction (m1/3/s) Statistical p-value Wetted perimeter (m) Perimeter of grain measured on light table (mm) Water discharge (m3/s) Sediment discharge (m3/s) Unit water discharge (m2/s) = Q/w Dimensionless unit discharge (q/(gl3))1/2 Unit sediment discharge (g/s/m) Hydraulic radius = A/P (m) xvii  Re r2 S S.E. SL ta u* v v* w wb Wt Xminor y ymid ytotal ys z α β κ θ θ LW θc ψ ρs ρ )  π  σ τ τLW τc’ τ’’ τc τ’ τf τs τθ μ ω ωc  Reynolds number (Eq. 8) Coefficient of determination Bed slope (m/m) (Figure 3) Standard error Step-length (mm) (Figure 3) Student t value Shear velocity = (τ/ρ)0.5 =(gRS)0.5 (m/s) Mean water velocity (m/s) Dimensionless velocity = v/(gl)1/2 Channel width (m) Mean bankfull width (m) Weight of grain (g) Minor axis measured by LabView VisionTM (mm) Measure of flow depth (m) Flow depth measured in middle portion of flume (m) Flow depth measured across full width of flume (m) Scour depth = H (m) (Figure 3) Drop height (m) (Figure 3) Greek symbols Exponent for q in dimensionless hydraulic geometry relation (Eq. 4) Exponent for S in dimensionless hydraulic geometry relation (Eq. 4) Von Kármán constant (0.4) Shields number (τ/ τc) determined with DuBoys shear stress estimate Shields number (τ/ τc) determined with law of the wall shear stress estimate Critical Shields number (0.045) Phi units = log base 2 Density of sediment (used 2650) kg/m3 Density of water (used 1000) kg/m3 Probability of bed being unstable from logistic regression Standard deviation (same units as measure) Applied shear stress (Pa) determined with DuBoys shear stress estimate Applied shear stress (Pa) determined with law of the wall shear stress estimate Shear stress required to move a grain on a loose poorly sorted bed (Pa) Residual shear stress (Pa) Critical shear stress for entrainment (Pa) Shear stress consumed by grain resistance (Pa) Shear stress consumed by form resistance (Pa) Shear stress consumed by spill resistance (Pa) Shear stress in excess of τc’ required to move grains in a jammed structured bed (Pa) Dynamic viscosity Unit stream power (Watts/m2) Critical unit stream power (Watts/m2)  xviii  7 List of equations 1  2  τ c = τ c '+(τ θ + τ f + τ s ) v = C RS =  C=  3  4  5  6  8 gRS R S = f n 1 6  8g R = f n  14  16  17  17  v gy Re=vyρ/μ = 0.986 y total + 0.003  19  madj = m r  20 26 29  ± t aσ 1 + 1 / n  36 2  Per 2A P 2A + xs − er 2 − xs xminor = 2 π π 2π 2π 1.50 Wt = 0.00106 Axs baxis = 1.19 X min or − 1.05 ⎛  15  17  Fr =  2  13  15  τ ( ρ s − ρ ) gD50 τ θ ( ρ s − ρ ) gD50 = 0.045 θc  y mid  4  ρs ⎜  Axs  ⎜ 6 3 π 10 ⎜⎜ a axis baxis ⎝ # SP = 1.8 + 1.2 log(S ) w  Wt =  15  15  θ=  11 12  1/2  α ⎡⎛ ⎞ β⎤ v q ⎟ S ⎥ = c ⎢⎜ ⎥ ⎢⎜ gl 3 ⎟ gl ⎠ ⎦ ⎣⎝  7 8 9 10  14  2/3  ⎞ ⎟ ⎟ ⎟⎟ ⎠  73 76 76  3 2  ⎛D ⎞ f _ sp = 1.29 + 1.00 log⎜ 84 ⎟ ⎝ w ⎠  77  110 112  xix  18 19  20 21 22  23 24  25  26  27 28 29 30  z ⎛D ⎞ = 0.19 − 0.44 log⎜ 84 ⎟ D84 ⎝ w ⎠ H ⎛D ⎞ = 0.56 − 0.50 log⎜ 84 ⎟ D84 ⎝ w ⎠  114 114  SL w ⎛D ⎞ = 0.27 − 2.4 log⎜ 84 ⎟ − 0.035 z y ⎝ w ⎠ d r = 0.0044 + 0.33D84  115 117  dr # SP w = 0.61 − 0.20 − 0.014 D84 w y Ls = k = −4.8 − 13.6 Log ( S ) dr 2.3 ⎛ 30θ c R Di ⎞ ωc = ρ (θ c RgDi )3 / 2 log⎜ ⎟ κ ⎝ eS k ⎠ ω ρgSq = ω c 2.3 ⎛ 30θ c R Di ⎞ ρ (θ c RgDi )3 / 2 log⎜ ⎟ κ ⎝ eS k ⎠ ⎛ w ⎞ θ ⎟ = 6.7⎜ ⎜D ⎟ θc ⎝ 84 step ⎠  118 145  145  −0.57  154  ⎛ w ⎞ ⎛ ) ⎟ + 7.8 log⎜ θ g (unstable) = −6.2 + 1.7 log⎜ ⎜θ ⎜D ⎟ ⎝ c ⎝ 84 step ⎠ ) g ( unstable ) e ) π (unstable) = ) 1 + e g (unstable ) ⎛D⎞ v ∝ gyS ⎜ ⎟ ⎝d ⎠  117  ⎞ ⎟⎟ ⎠  157 157  −1  1 ⎛⎜ qD50 ⎞⎟ y= C1 ⎜⎝ gS ⎟⎠  171 2 5  171  1  31  1 D 6 n = 50 = 0.047 D50 6 21.1  184  1  32  8 ⎛ R ⎞6 = a1 ⎜ ⎟ f ⎝D⎠  184  33  v 2.303 ⎛ 12.2 y ⎞ = log⎜ ⎟ κ u* ⎝ k ⎠  184  34  8 2.303 ⎛ 12.2 y ⎞ = log⎜ ⎟ κ f ⎝ k ⎠  184  xx  u* = (gRS)0.5  35  8 ⎛ y⎞ = a⎜ ⎟ f ⎝k⎠  36  184  b  185  1  37 38  1 = n  2.303 g R 6  κ v* = a 1-m S  ⎛ 12.2 y ⎞ log⎜ ⎟ ⎝ k ⎠ 1- m 2  185 185  q *m 1  8 ⎛ y ⎞6 = a1 ⎜ ⎟ f ⎝k⎠  39  40  41  8 = f  42  vres  186  8 ⎛ y⎞ = a2 ⎜ ⎟ f ⎝k⎠ ⎛ y⎞ a1a2 ⎜ ⎟ ⎝k⎠  186  5  2 2⎛ y ⎞3 a1 + a2 ⎜ ⎟ ⎝k⎠ = v pred − vmeas  186  204  43  δvres = δv pred + δvmeas + δvmean _ bias  204  44  δvmean _ bias = pred − obs  205  3/ 2  45 46 47 48 49  ⎛ vn ⎞ Rw = ⎜ w ⎟ ⎝ S⎠ Ab = A − Aw  ⎛ 2R ⎞ ⎛ wy − 2 yRw ⎞ Rb = Ab / w = ⎜ ⎟ = y ⎜1 − w ⎟ w ⎠ w ⎝ ⎠ ⎝ 2 v 8 gA 8 gAb 8 gAw = = = S fp f b pb f w pw  fb = f +  2y ( f − fw ) w  227 227 228 228  228  xxi  8 Acknowledgements This project would not have happened without the support of numerous people. I would like to thank everyone who has supported and guided my research as well as those who have kept me grounded in the real world. Michael Church and Marwan Hassan were terrific supervisors; they provided leadership, encouragement, assistance and freedom. With their support and critical thinking we were able to design and complete some interesting experiments. The jammed state concept, which is tested in this thesis, has been an idea of Michael Church’s for some time. Dan Moore and Rob Millar, who were also on my committee, helped guide and support the research; their input was most appreciated. During the setup of the flume and while running the experiments, I had a terrific series of assistants including Christina Lovatt, Robert Taylor, João Sarmiento, Jennifer Wardle, Kathleen Macdonald, Dylan Heden-Nicely, Rob O’Connor and Christiaan Iacoe. These are the people who painted the stones, sieved the grains, weighted the bedload, surveyed the streams and ran many of the experiments. Bonnie Smith was instrumental in the design of the laser scanner and Jon Tunnicliffe recommended LabViewTM, which proved to be immensely useful. Fellow graduate students including Joshua Caulkins, David Luzi, Jeff Carpenter, Joanna Reid and Nira Salant helped me shovel out the Civil Engineering sump, celebrate successes and were always there for moral and academic support. Françoise Bigillon and Phillippe Frey hosted me during visits to laboratories in Lyon and Grenoble, France, where the original GSD was developed and is in use. I also benefited from a visit to the ETH laboratory in Zurich, Switzerland, where I was hosted by Roman Weichert. Ivan Lui helped design the flume and bed scanner while Harold Schrempp and the other technicians in Civil Engineering provided useful technical guidance on a range of instrumentation challenges. The support of Violeta Martin in the hydraulics laboratory at Civil Engineering was much appreciated and instrumental. I have also benefited from many interesting and fruitful conversations with Brett Eaton.  xxii  I want to specifically thank authors who have included descriptive thoughts and insight in their writing and spent an extra bit of time explaining how they think. I have found detailed descriptions of ideas and the logic used by others to be very useful for my own understanding and formulation of new ideas. I have tried to emulate this writing style. Conversations and the exchange of ideas with Francesco Comiti have been particularly useful. Many step-pool investigators also contributed time and their thoughts to this project. These individuals include Ellen Wohl, Ashley Perkins, Joanna C. Curran, Mark Fonstad, Stephen Bird, Peter Molnar, Francesco Comiti and one anonymous researcher who participated in the channel classification exercise. In addition, Steve Bird and the Research Branch of the British Columbia Ministry of Forests provided the 2002 long profile data for the step-pool channel classification algorithm. The research was supported by Natural Sciences and Engineering Research Council of Canada (NSERC) strategic and discovery grants (awarded to Dr. Michael Church). I was supported by a NSERC Canadian Graduate Scholarship, a Province of British Columbia/Ministry of Postsecondary Education Pacific Century Graduate Scholarship and a University of British Columbia Graduate Fellowship, Ph.D. Tuition Fee Award and the Walter W Jeffrey Memorial Scholarship. Lastly, my family deserves a special thank you as they have always been a source of love, inspiration and encouragement. My dad, Erich Zimmermann, volunteered his craftsmanship and enthusiasm to help me set up the flume and pour concrete for the holding tank. My mom, Shirley Howdle, has always been there to ensure that I still get out skiing and play in the mountains. My wonderful friend and lifetime companion, Katie Graham, continually encourages me to complete excellent work and has given me the time to putter away on my projects while helping me explore the wider world and have fun in the process. I thank her dearly and love all that we share together. To all of you, thank you for being there.  xxiii  1 Introduction Steep streams with step-pool and cascade morphologies are common features of mountainous landscapes and there is considerable interest in the formation and stability of such channels (Chin and Wohl, 2005; Curran, 2007; Weichert et al., 2008). The formation of step-pool structures remains contested (Curran, 2007) while the stability of such channels has rarely been assessed (Church and Zimmermann, 2007). Observing the formation and failure of such morphologies is extremely difficult as the channels generally develop over prolonged periods of time and fail rarely. Thus to improve our understanding of the formation and destruction of such channels laboratory and field studies are necessary. This thesis describes the formation of step-pool features and presents a bed stability criterion based on observations in experimental steep channels. During the experiments hydraulic geometry and flow resistance data were collected in order to assess how best to characterize flow resistance for steep streams. A new flume was built and outfitted with novel instruments to monitor changes in the bed, sediment transport and water velocities. The thesis consists of eight chapters and a 54 page Compendium (Appendix 2) that includes figures and tables that summarize the data collected during each of the experiments. The present chapter provides a brief introduction to step-pool research and the study objectives are defined. A more detailed literature review and the development of the jammed state model were published in Church and Zimmermann (2007). The jammed state hypothesis asserts that streams that are narrow relative to the size of stones on their bed (small ratio of width to grain size) will be more stable. In Chapter 2 the flume and methods used during the laboratory experiments are described. Chapter 3 presents two methods developed during the thesis. More detailed versions of these methods have been published (Zimmermann et al., 2008a; 2008b). The first method describes a means to objectively identify step-pool bedforms from digital elevation models while the second method describes a refined means of measuring sediment transport rates using high speed video data acquisition and image possessing algorithms. A narrative chapter follows that discusses the successes and failures of the experimental design and makes recommendations for future endeavours. It includes a discussion of Froude scaling step-pool channels and the 1  experimental procedure that was developed to form step-pool bedforms. Chapter 4 is included as the experimental design used during these experiments was itself experimental. Clear documentation of what was done, why it was done, and whether in retrospect the chosen approach had merit will be particularly useful for other researchers who want to perform similar experiments. The next three chapters present the main results from the experiments. Chapter 5 examines step-pool form, flow properties, sediment transport conditions and bed failure patterns. Sediment transport, channel form and flow data are then integrated in Chapter 6 to develop a stability criterion for steep streams based on the jammed state concept. In Chapter 7 flow roughness in steep streams is examined and methods of predicting water depth and velocity based on channel form are assessed. Conclusions and recommendations are presented in Chapter 8.  1.1 Literature review Headwater streams comprise 60- 80% of the cumulative length of river networks (Benda et al., 2005) and in mountainous environments are commonly comprised of cascade morphologies with step-pool bed features (Grant et al., 1990). Interest in these channels arises both because they pose a significant risk to people when they destabilize and flood downstream communities with water and sediment, and because of their visual appeal (Figure 1). While researchers have been studying the morphology, hydraulics and formation of such channels since the early 1960s (Peterson and Mohanty, 1960; Judd, 1963) they remain poorly understood. Much of the contemporary research continues to examine what can be perceived as relatively basic questions such as how step-pools form and means of classifying and segregating the channel types (Nickolotsky and Pavlowsky, 2006; Weichert, 2006; Curran, 2007).  1.1.1 The step-pool form Step-pool units are channel forms composed of alternating channel-spanning ribs (steps) and pools (Figure 1; Figure 2a, b, Figure 3b, c) with tumbling flow (Peterson and Mohanty, 1960) that oscillates between subcritical in the pool and supercritical over the step (Hayward, 1980; Grant et al., 1990; Montgomery and Buffington, 1997). Step-pool bedforms are an expression of bed structuring. Bed structuring occurs during extended periods of time during which the bed is partially mobile and the supply of sediment is relatively low. Less stable grains move and are rearranged, forming more stable aggregated bed elements that result in a wide variety of 2  bedforms including imbricated stones, stone-lines, stone-cells, stone-clusters and finally steppools (Hassan et al., 2008). Individual grains and grain structures such as stone-lines, stone-cells and clusters make up subunits with lengths that are a fraction of the width of the channel. An assemblage of subunits forms channel unit features such as pools, bars, steps and runs that have length scales on the order of a channel width. A series of channel units form a reach, which has a length that is typically tens to hundreds of channel widths (Grant et al., 1990).  Figure 1. Step-pool bedforms in Deeks Creek, British Columbia.  Steps may consist of cobble or boulder chains, woody debris –often channel-spanning logs – or bedrock. In this thesis, I focus attention on cobble-boulder step-pools as they are likely to exhibit the most common, repeating and potentially organized steps. Relative roughness (D/y, where D is a diameter representative of the larger clasts and y is flow depth) is near 1 at or near bankfull flow. Montgomery and Buffington (1997) quote a range of 0.3 < D84/y < 0.8, (D84 is the size of grain at which 84% of the sample is finer) while Comiti et al. (2006) observed that the D84/R was 0.85 for a flow event about 80% of bankfull (R is the hydraulic radius; the quotient of channel cross-sectional area and the wetted perimeter). In both French Pete Creek, Oregon (Grant et al., 1990), and Shatford Creek, British Columbia (Zimmermann and Church, 2001), relative roughness remained greater than one at calculated or measured high flows. In East Creek, British Columbia, during an event that was the 3rd largest in 36 years, D84/y was 1.55, in  3  part due to overbank flooding that resulted in a relatively small average depth (0.33 m) despite centre of channel mean depths on the order of 0.65 m.  Figure 2. Steep stream channel morphologies. a) A step-pool unit at East Creek, in the British Columbia Coast Range, during flood (Q = 2.2 m3/s, the 3rd largest recorded flow in 36 years of record; b) same location at low flow (Q ≈ 0.01 m3/s). While wood is present in the channel, the steps are composed entirely of boulders. c) A series of step-pool channel units in a cascade reach dominated by high sediment supply located in Kluane National Park, Yukon. d) A cascade channel unit in Giveout Creek, in the interior of British Columbia; e) Stone-lines in a rapid unit of the Lainbach, Germany.  Whether step-pools constitute a distinct channel type at the reach scale or simply represent a channel morphological unit has been debated. Grant et al. (1990) suggested that they are a channel unit phenomenon with a characteristic scale on the order of one bankfull width set 4  within a larger cascade channel type defined at the reach scale. In contrast, working only at the reach scale, Montgomery and Buffington (1997) suggested that step-pools represent a channel type distinct from the cascade (Figure 2d) and “plane bed” (or rapid; Figure 2e; Figure 3a) channel morphologies. They noted that the cascade channel type may include boulder ribs, but that they do not span the channel width. As the boulder ribs do not span the channel they are distinct from the step-pool morphology, which is comprised of channel spanning steps and pools. In Figure 2c a series of channel unit scale steps and pools are clearly visible in a cascade channel. These units do not span the channel width and it is likely that they are regularly rearranged during sediment transporting flood events. Subunit steps and pools like these are not examined herein, but are thought to have characteristics similar to their channel spanning cousins during low to modest flood events.  Figure 3. Illustrations with definitions of channel types and commonly measured step-pool metrics that are as follows: L, step to step spacing; Lsp, length of step-pool; Ls, scour length; H = ys, step height or scour depth; z, drop height; dr, residual depth; dt, critical depth; S, channel slope.  5  Individual step-pool units can occur immediately downstream of each other (Figure 3b) or with a section of channel between the downstream end of one pool and the top of the next step (Figure 3c). This intermediate section of channel is generically referred to as the tread, but may exhibit a run/rapid or cascade morphology. Extended treads may be considered equivalent to a run. Early researchers were primarily concerned with the step-to-step spacing (L); thus, the identification of the tread section of the channel was not emphasized. Interest in the step-to-step spacing likely arose because antidunes were considered a possible step-pool formation mechanism (Whittaker and Jaeggi, 1982) and step-to-step spacing is similar to the commonly used pool-riffle spacing or wavelength (Robert, 2003). As alternative step-pool formation mechanisms have been introduced (e.g. Zimmermann and Church, 2001; Curran, 2007) and interest has become more focused on the scour dimensions in pools (Comiti et al., 2005), the tread portion of the channel has increasingly been recognized. An awareness of the tread necessitates recognition that the channel is composed of more than steps and pools and makes the definition of a step-pool reach at the channel scale difficult. How much of the bed must be step-pools to qualify as a step-pool morphology? The Grant et al. (1990) classification scheme, which identifies step-pools as a channel unit phenomenon, appears most appropriate. The identification of steps and pools is a major challenge and different methods and metrics have been presented in the literature (Nickolotsky and Pavlowsky, 2006). Wooldridge and Hickin (2002) investigated four means of classifying boulder step-pool and cascade stream channels, including visual identification, bed level-crossings about the mean gradient, bed elevation differencing and power spectrum analysis. They found that visual identification most consistently recognized the geometry of and classified the individual bedforms. Milzow et al. (2006) have since developed a step identification technique that classifies step-pool sections based on the occurrence of a critical slope (e.g. > 15%) followed by a low gradient section (pool). The use of a critical step gradient is developed further in Chapter 3 where a new means of objectively identifying step-pool channels is presented. Individual step-pool units have been reported on gradients greater than 2° or about 4% (Whittaker and Jaeggi, 1982; Chin, 1989; Grant et al., 1990; Montgomery and Buffington, 1997) and continuous step-pool morphology on gradients greater than about 4o or 7%, where structural reinforcement of individual clasts becomes necessary to maintain bed stability (Church, 2002). On lower gradients, boulder and cobble ribs (e.g. Figure 2e and Figure 3a) grade into rapid steps 6  (Hayward, 1980) or stone lines/stone cells (Church et al., 1998). Herein these features are seen to be distinct from step-pools as their relative roughness is generally less than 1.0, they lack channel-spanning pools and occur at the subunit scale (Hassan et al., 2008). Furthermore, steppools are considered to occur if a channel-spanning transition to supercritical flow at the top of the step and a turbulent plunge pool downstream occur (see Figure 2a). Conversely, stone lines are drowned at moderate flows and channel-spanning hydraulic jumps are not present. The criterion of hydraulic function when classifying step-pools has not, however, been universally recognized. Weichert et al. (2008) did not consider the hydraulic function of the steps, particularly the importance of water plunging over the steps at high flows. They suggest that step-pools are self-similar to, and cannot be distinguished from, stone structures occurring at the subunit scale. This example highlights the challenge of interpreting what other researchers mean by a step-pool bedform and the need for clear consistent criteria. Some investigators (e.g. Abrahams et al., 1995; Aberle and Smart, 2003) have suggested 3% as the threshold gradient above which continuous step-pools are found. However, Comiti (2003) and Comiti and Lenzi (2006) noted that, in flume experiments with a 3% slope, antidunes formed that did not break and tumbling flow did not occur; conversely at slopes greater than 4.5% the ribs caused a hydraulic jump to occur and flow conditions resembled those found in step-pools during flood. In the field, within a short reach many steep channels alternate between clearly defined steppools and the less distinctive cascade or rapid-type morphologies described by Montgomery and Buffington (1997). The switching between channel types may occur as a result of local variations in slope associated with the accumulation of sediment behind large steps or the accumulation of material from lateral sediment sources. The discrimination of a lower threshold gradient for steps to form depends both on the relative size of the channel and the step-forming clasts as well as one’s definition of a step-pool unit. I am unaware of any investigation that has critically examined the maximum slope at which steppools occur. Grant et al. (1990) show 40% as the upper limit of their observations of steps. Figure 6 in Wohl and Grodek (1994) shows a compilation of their own data with those of Hayward (1980) and Grant et al. (1990) and suggests that there is no further systematic reduction in pool length on gradients above 20%, when length averages about one metre. Further increase 7  in gradient is entirely accommodated by increasing step height. At 20% gradient, pool plus tread length is 5 times the drop height. One must wonder whether pool length is no longer reduced because at higher gradients there is no longer room for pool formation so that one is dealing rather with a series of drops and sills. Furthermore, above a certain gradient sediment movement in the channel is likely to be dominated by colluvial mass wasting processes, including debris flows, and step-pools will not persist, but they may occur between debris flow events. Debris flows have been observed to initiate in mountain canyons with slopes greater than 27% (Takahashi, 1981) suggesting an upper limit to persistent step-pools. Since at least the 1960s it has been recognized that any assessment of riffle-pool channel shape needs to consider the size of the channel (Leopold et al., 1964). As a result it is common practice to scale dimensions such as the riffle-pool wavelength by measures of channel size (e.g. bankfull width). Nevertheless, few studies examining step-pool form have scaled the dimensions to account for changes in channel size or grain size. As an example, it has been common practice to plot step-pool wavelength as a function of slope without scaling for stream width (see Chin and Wohl, 2005). While a reasonably strong relation exists between wavelength and slope, it may only be casual. Stream width generally decreases with increasing bed slope, and thus a relation between wavelength and channel width would also be expected. I am unaware of any step-pool dimension studies to date that have considered which variables account for the characteristic size of the channel (e.g. width and grain size) before attempting to examine what controls their form.  1.1.2 Formation of step-pool channel units Some of the first research on step-pools was concerned with how step-pools form (Judd, 1963) and this continues to be a topic of much research (Lee, 1998; Curran and Wilcock, 2005; Curran, 2007; Weichert et al., 2008). While it is generally accepted that step height is governed by the grain size of the step-forming clasts (Judd, 1963; McDonald and Day, 1978; Whittaker and Jaeggi, 1982; Allen, 1983; Wohl et al., 1997; Lee, 1998; Chin, 1999; Chartand and Whiting, 2000), the controls on step length remain widely debated. Herein a brief description of formation models and recent research is provided; for a more detailed discussion of formation mechanisms and a historical accounts of step-pool formation theory see Church and Zimmermann (2007) and Curran (2007).  8  Central to many step formation theories is that large, immobile clasts form anchor points against which other stones become imbricated and stop moving. The deposition of these grains yields an initial step. Two general schools of thought have emerged regarding the controls on the formation of additional steps. Following hydraulic tradition, step-pools have been supposed to be controlled by the flow field (Whittaker and Jaeggi, 1982; Grant, 1994; Comiti et al., 2005) but, more recently, it has been claimed that the chance location of steps depends on both the flow and the location of keystones within the channel (Lee, 1998; Zimmermann and Church, 2001; Curran and Wilcock, 2005; Curran, 2007). The delivery of keystones is thought to be random and governed by factors such as the location of erratics following deglaciation, hillslope inputs and bank failures. The existence of a keystone itself does not guarantee a step, but rather it must be located in a position where water is forced over the stone causing a step and downstream scour. A keystone in the bottom of a pool will not generate a step. Milzow et al. (2006) examined the morphology of a 17% gradient step-pool cascade and observed that step spacing was related to the height of the upstream step, suggesting that a strong feedback exists between step height, pool scour and step-pool morphology. Curran (2007) discussed three step-pool formation types — rough bed, exhumation, and dune — which were responsible for 63, 25 and 12% of all step-forming events during her experiments, respectively. Rough bed step formation, which could be interpret to be akin to the jamming and structuring of coarse grains, occurs when a large step-forming grain (keystone) is deposited and other stones stabilize against or around this stone leading to the formation of a channel forming step. Exhumation step formation occurs when overlying sediments are scoured exposing a preexisting step or step-forming stone. Dune step formation occurs when grains smaller than the step-forming stones accumulate and then a step-forming grain is deposited on the upstream side of these grains. Subsequently the water surface is forced into phase with this accumulation of stones and the feature grows into a channel spanning dune, yielding a step. These dunes were observed to have sediment depositing on the lee side of a the bed feature while erosion was occurring on the stoss side of the feature, but no slip face was observed nor did the bedforms migrate (Curran, 2007). Curran termed these features dunes although the water surface was in phase with the bed. In phase flow conditions are typically associated with antidune features rather than dune features (Robert, 2003).  9  Using a 15 cm wide flume, Curran (2007) demonstrated that the step formation process influences the length of the exclusion zone, which is the section downstream of a step, including the pool, where another step does not occur. Curran (2007) found that if a step formed through the rough bed process the exclusion zone was on average 32 cm long. If the step formed through exhumation, the exclusion zone averaged 40 cm. Finally steps were most closely spaced if they were formed through the dune process, which had an exclusion zone of 20 cm. Curran (2007) does not indicate if these exclusions zones were statistically different from each other. The dune process was more common at high flows and high sediment transport rates. Factors that are believed to influence the formation of step-pool channels include stream power (product of velocity and flow depth; a measure of flow stage), the grain size distribution of the sediment mix (Tatsuzawa et al., 1999), sediment supply rate and the jamming ratio (Church and Zimmermann, 2007). The jamming ratio is the ratio between channel width and grain size; it indexes how many stones are required to span the width of the channel — fewer stones are thought to be more stable. Evidence that grain size and stream power/transport stage are important comes from an extensive set of flume experiments by Rosport (1994), Rosport and Dittrich (1995), and Koll (2002). They showed that high flows wash out steps and create a morphology similar to riffle-pools with shorter, irregular barriers. These seem similar to the chute and pool structures described by Simons et al. (1965) for sand bed experiments. Maxwell (2000) performed similar experiments that showed that bed stability also depends on bed slope. Using sketches of the bed structure, Weichert et al. (2004) demonstrated that, during steep flume experiments, the type of bedform present changes as stream power increases. They also observed more frequent channel spanning steps for narrow channels and steeper slopes. Evidence that not only the size of the large grains, but also the entire grain size distribution, affects the formation of steps comes from Tatsuzawa et al. (1999), who performed flume experiments with three sediment mixtures. All mixtures had the same Dmax but two of them had a greater proportion of large stones than the third. They found that step-pools, which they described as anchored antidunes formed when the two mixtures with a larger proportion of coarse material were used, but not when the third mixture was used. High sediment transport rates are also thought to discourage the formation of step-pools. When the largest stone in Curran’s (reported under the name Crowe, 2002) experiments was 45 mm, 10  rather than 64 mm, steps did not form. Yet the jamming ratios (w/D84) from eight other studies that reportedly created steps were smaller suggesting steps should have formed. The main difference between Curran’s study and the other studies was the sediment transport rate. Curran ran constant feed at rates between 110 and 1000 g/s/m while the other studies ran anywhere from no feed to feed at a rate of 170 g/s/m. In Curran’s experiments, even at the lowest flow rates (0.031 - 0.043 m2/s), all sizes in the mixture were readily transported and it is likely that this mobile bed prohibited steps from forming. Curran’s experiments highlight a potential discrepancy between how supposed step-pool channels have been formed during flume experiments and how they may form in nature. Most of the experiments attempting to recreate step-pool channels have used high transport rates, often under equilibrium transport conditions with a completely mobile bed. These conditions may not be akin to those observed in nature. Chin and Phillips (2006) suggested that in cases where steppools are known to have developed after disturbances the steps were not simply a relic of the disturbance event, but rather developed over a number of years. Likewise, Lenzi (2001) noted that the morphology of the steps and pools developed over time after a large debris flood disturbed the Rio Cordon in 1994. Another example that illustrates the importance of partial transport for the formation of steps include the progressive development of step-pools in rehabilitated road crossings (Madej, 2001). During partial transport conditions the finer, more mobile, material is removed from the channel and the large stones move rarely and adjust into more stable locations. Many flume studies (Whittaker and Jaeggi, 1982; Lee, 1998; Tatsuzawa et al., 1999; Koll, 2002; Weichert et al., 2004; Comiti et al., 2007a; in review) have not fed sediment into the flume, but simply degraded the material that was initially in the flume, which may reproduce the formation mechanism observed in some systems (e.g. degrading road crossings), but does not account for the episodic delivery of sediment that is common in headwater streams (Benda et al., 2005). During degradation experiments, the amount the larger stones can be rearranged during partial transport conditions without the flume floor being exposed is limited as there are only a few large stones available compared to feed experiments that have large grains in the feed. A number of authors have noted that the formation of steps takes on different behaviour for bed slopes greater than about 7%. Whittaker and Jaeggi (1982) observed that, below slopes of about 11  7.5%, antidunes remained mobile, while at greater slopes the antidunes were quickly anchored by large stones. Likewise Koll et al. (2000) observed that the self stabilizing processes for runs with bed slopes less than 4% differed from the those observed at slopes greater than 8%. In light of these observations it is noteworthy that Curran’s experiments at slopes ranging from 3.5-5.2% with sub-45 mm sediment produced only mobile, antidune-like features, whilst steps were produced on gradients between 5.0 and 8.3% by adding 45-64 mm sediment. Among all the step-pool experiments with which I am familiar (see Table 1), only the experiments discussed by Whittaker and Jaeggi covered the entire range of gradients on which step-pools appear to exist. It appears that distinct “step” forming mechanisms may occur on moderate gradients compared to truly steep ones, with the former having been chiefly investigated in the experiments reported to date, while the latter appear to depend more strictly on grain congestion and transport intensity. Table 1. Range of flume slopes used in past mobile bed steep stream experiments.  Study Lee Koll Maxwell  Slope (%) 5.6 to 6.7 7.5 to 10 3 to 7  Curran/Crowe Weichert Tatsuzawa et al. Rosport  3.5 to 8.3 3.5 to 9 2.5, 5 and 10 2, 4, 8 and 10 Whittaker and Jaeggi 1 to 24  Reference (Lee, 1998; Lee and Ferguson, 2002) (Koll et al., 2000; Koll and Dittrich, 2001; Koll, 2002) (Maxwell, 2000; Maxwell and Papanicolaou, 2001; Maxwell et al., 2001) (Crowe, 2002; Curran and Wilcock, 2005; Curran, 2007) (Weichert, 2006; Weichert et al., 2008) (Tatsuzawa et al., 1999) (Rosport, 1994; Rosport and Dittrich, 1995; Rosport, 1997; Rosport, 1998) (Whittaker and Jaeggi, 1982)  1.1.3 Channel stability While a number of studies have examined the formation of step-pools few have examined the stability and mechanisms by which beds fail. In early work Chin (1994) used Costa’s (1983) formula based on paleoflood reconstruction to predict the mobility of step-forming stones. Costa’s (1983) approach remains untested and does not consider additional factors affecting step stability such as the diameter and abundance of large stones relative to the channel width (Grant et al., 1990), bed structuring or sediment supply.  12  Grant et al. (1990) suggested that boulder jams form steps, which Church and Zimmermann (2007) suggested may be an example of jammed state physics (Cates et al., 1998; To et al., 2001; Bocquet et al., 2002; De-Song et al., 2003). Jammed state theory suggests that grains moving together down-gradient can jam if the width of the opening in a constriction reaches a critical width/grain size ratio (between 2 and 5 (To et al., 2001)), and that the probability of a jam occurring depends on the angle of the inclined surface as well as the thickness of the grain flow (Bocquet et al., 2002). In addition, below a critical particle velocity, jamming interactions do not occur. Above the critical velocity jamming interactions do occur and the movement of particles is inhibited in a non-linear manner (De-Song et al., 2003). The jammed state concept appears to have merit for step-pool streams. Curran and Wilcock (2005) illustrated that particle interaction form steps and the channel width/grain size ratio can control the development of steppool structures (Grant et al., 1990). However, inasmuch as their concept arises in the physics of granular flows, some of the assumptions of Bocquet et al. (2002) and De-Song et al. (2003) may not hold for step-pool channels. In particular, the movement of the keystones is discontinuous whereas they envision a continuous movement of grains covering the entire channel width with no fluid medium (see Pouliquen (1999) and De-Song et al. (2003) for a description of the experimental setup). In lower gradient streams, bed stability has been shown to be related to the development of surface structures, which depend on low rates of sediment supply (Church et al., 1998; Hassan and Church, 2000). In step-pool channels it is likely that the rate and texture of the sediment supply also regulates the development of the channel. Lenzi (2001) observed that high sediment supply reduced the number and morphological extent of step-pools (H/L was reduced; H is step height and L is step length), likely by having relatively coarse material bury and disturb the steps. High sediment transport rates, particularly of coarse material, may also regulate the likelihood of jamming. Low rates of sediment supply may affect the amount of scour downstream of steps and bed stability. As an indirect means of investigating step-pool stability, Comiti (2003) investigated scour downstream of sills in the absence of sediment supply to determine how much scour could occur. Curran and Wilcock (2005) showed that downstream scour destroyed over 75% of all steps. There is clearly the potential that a low rate of sediment supply can promote scour and destabilize steps. 13  Koll and Dittrich (2001) did a series of experiments in which they fed sediment into the flume to examine the effect of sediment flux on bed stability. Stability was defined in two ways: the bed was considered to be unstable if the surface stones were mobilized or, alternatively, if the grain size distribution of the material leaving the flume increased following the introduction of feed. An increase in the grain size distribution occurred when the surface armour broke. Using these criteria, an increase in feed rate was observed to destabilize the steps. These observations may be a result of pools infilling with an associated decrease in the form/spill resistance. A reduction in form and/or spill resistance would result in an increase in flow velocities and higher grain shear stresses. Upon considering Crowe’s, Comiti’s and Koll’s results, an apparent contradiction emerges. If steps are in fact destroyed due to downstream scour (Curran and Wilcock, 2005), step destruction most likely occurs during low sediment transport conditions as high sediment transport rates can limit scour depths (Whittaker, 1987; Comiti, 2003). Conversely, low sediment supply promotes bed structuring and increases the stress required to mobilize the bed, suggesting low supply rates should promote stability. Koll’s results support this trend as high sediment transport rates were shown to destabilize steps. There is a clear need for an improved understanding of how step stability is affected by sediment transport and if high and/or low sediment supply leads to step instabilities. Depending on flow history and conditions, both high and low sediment transport rates may destabilize the bed. The mobility of grains in step-pools can, at least conceptually, be assessed through the partitioning of the critical shear stress to entrain a grain (τc) into 4 components.  τ c = τ c '+(τ θ + τ f + τ s )  (1)  where τc’ is referred to as grain shear stress and is the shear stress required to move the mean grain size (D50) in a mix on a loose, poorly sorted bed. τc’ = 0.045(ρs-ρ)gD50 (Komar, 1996) where ρs and ρ are the density of sediment and water respectively. τθ is the shear stress applied to the grain in excess of τc’ that is required to mobilize a stone in a jammed, structured bed. Together τc’ and τθ encompass the force acting on the grain (τ’). τf is the shear stress consumed by form resistance and τs is the shear stress consumed by spill resistance. At present there is no means of predicting τθ, τf and τs independently. Whether these shear stresses are actually additive  14  remains unclear (Wilcox et al., 2006). If the applied shear stress (τ) exceeds the critical shear stress (τc) the grain is expected to be mobile.  1.1.4 Flow resistance Along with an interest in the formation and stability of step-pool streams, there has been considerable interest in flow resistance in steep streams. The interest in flow resistance arises because of the need to estimate flow velocities from channel morphology and flow stage data (either discharge or depth) since flow velocities, particularly during flood events, are almost never measured directly. Classical approaches to flow resistance relate velocity averaged across the channel and through the flow depth (v) to channel slope (S) and flow depth (y) or, more appropriately, hydraulic radius (R). These include the Chézy (C), Darcy-Weisbach (f) and Manning (n) resistance formulations, 8 gRS R 2/3 S 1/2 = f n  v = C RS =  (2)  each with their respective coefficients. The resistance coefficients are interchangeable via 1  8g R 6 = C= f n  (3).  As an alternative approach Aberle and Smart (2003) generalized a hydraulic geometry approach (v ∝ qm) and proposed α ⎡⎛ ⎞ β⎤ v q ⎟ S ⎥ = c ⎢⎜ 3 ⎟ ⎜ ⎢ ⎥ gl gl ⎠ ⎣⎝ ⎦  (4),  where l is a roughness length that can be taken to be a grain size or metric such as the standard deviation of bed elevation (σ). While much research has been completed on what is the best way of characterizing a particular roughness coefficient (e.g. what predicts n or f best), few researchers have examined which approach of characterizing resistance works best across a range of relative depths, slopes and channel forms.  15  1.2 Research objectives Despite over four decades of research examining steep channels there remain a number of active and important research questions. These include defining and developing a means of objectively identifying step-pool channels, understanding the process of bed structuring and bed failure mechanisms in headwater channels, means of predicting flow resistance and the effect of sediment transport on bed stability. The purpose of my research is to assess what conditions lead to step-pool formation and what conditions lead to bed destabilization. To address these questions, my project can be broken down into three specific research questions that are based on the issues raised in the preceding review. Question 1: What governs the occurrence and form of step-pool channels? Question 2: Does channel stability in steep streams reflect jammed state physics; in particular,  can stability be predicted based on the applied stress (e.g. shear stress) and jamming ratio? Question 3: How can flow resistance in steep channels best be assessed in order to predict  flow depths during extreme events and hence assess the applied force (e.g. shear stress) on the bed? To address these three research questions, experiments were conducted in a flume and flow rates, sediment supply, the grain size of the bulk mix, channel width and flume slope were varied. The experiments started by scaling a prototype stream and then the slope, width and grain size were changed to move away from this initial scaled experiment in an effort to investigate the limits of step-pool channel existence. Once a limit was established, subsequent experiments that could have been conducted by further perturbing the same variable (e.g. increases in slope) were not conducted. These subsequent experiments were not expected to produce stable channels, either because very large flow rates would have been required to move the bed or because the bed would have immediately failed. These limits are examined in detail in Section 5.1.1. The approach used to assess stability is based the Shields function  16  θ=  τ ( ρ s − ρ ) gD50  (5)  which relates shear stress (τ) to the submerged weight of the stone. For gravel bed rivers with a mixed grain size distribution, in the absence of form roughness and a structured bed, the bed is predicted to mobilize when a Shields number in the range of 0.045 is attained (Miller et al., 1977; Wilcock and Southard, 1988; Komar, 1996; Church, 2006). Using this reference value, a Shields ratio can be defined that is the ratio between the applied Shields number and the critical Shields number:  τ θ ( ρ s − ρ ) gD50 = Shields ratio = 0.045 θc  (6).  Other Shields numbers could be used and a range of values above and below 0.045 have been proposed (Komar, 1996), but the result is no different as the change in Shields number simply results in a constant adjustment. Shear stress can be determined in a number of ways; herein DuBoys’ (1879) approach is used most often. DuBoys (1879) proposed that τ = ρgRS where ρ is water density, g is gravity, R is hydraulic radius and S is energy gradient (which for this study will be considered equivalent to the mean bed gradient over the length of the flume).  Figure 4. Proposed projection of Shields stability criterion for step-pool channels.  17  The Shields number along with sediment concentration and the jamming ratio can be incorporated into a three dimensional surface (Figure 4) that accounts for the additional stability that is believed to occur in steep channels due to jamming and sediment supply. During my experiments the focus will be on the Shields ratio-jamming ratio relation and only cursory observations about sediment concentration (Qs/Q) will be made. Additional research will be necessary to gain a better understanding of the role sediment concentration has on the stability of such channels. To examine bed stability τ will be used rather than τg, the latter would be preferred, but there is currently no means of predicting how much shear stress is actually consumed by form and spill resistance. In order to investigate if form and shear resistance can be related to any morphological variables and thereby account for τf and τs, the residual variability about the relation between the Shields number and the jamming ratio will be compared with measures of channel form and flow resistance (e.g. f). I expect that experiments that have more form and spill roughness will remain stable at higher shear stresses than those that do not. Thus there may be some remaining structure in the data that can be explained by measures such as step steepness or bed standard deviation. Alternatively the jamming ratio may correlate with the development of form and spill resistance and it will not be possible to detect any residual structure in the data. To address the research hypotheses, I performed experiments in a flume that was approximately Froude scaled to natural step-pool streams based on field measurements in Shatford Creek, East Creek and the Rio Cordon; step-pool streams located near Penticton, British Columbia, within the University of British Columbia Malcolm Knapp Research Forest and in the Dolomites near Cortina d’Ampezzo, respectively. A flume-based study was used as it is not possible to observe infrequent events that form and destabilize step-pool streams in nature, nor is it possible to vary sediment feed or monitor sediment transport in natural systems to the degree that is necessary to examine my research questions. The flume experiments differed from other step-pool flume studies in that rough walls were used to simulate bank conditions that occur in natural step-pool streams. Rough walls were thought to be important for the formation of boulder jams and steps. Flume results, particularly those examining flow conditions, sediment transport, grain size, and channel form were compared with field measurements from the prototype streams.  18  2 Methods To examine the evolution and destruction of steep channels a new flume was built and a series of instruments were developed or adapted to monitor channel change, flow conditions and bedload transport. Herein a detailed description of the flume is presented and most of the instrumentation is described on account of the experimental nature of the research. Two observational techniques that were developed as part of the thesis, the identification of step-pools and monitoring of sediment transport using a video camera, are described in the subsequent chapter.  2.1 Flume design To conduct the experiments a flume was constructed in the Hydraulics Laboratory in the Civil Engineering Department at UBC. The flume channel is five meters long, has a maximum width of 83 cm and a maximum depth of 70 cm. Flume slope can be adjusted from 0 to 18%. Depending on the experiment, mean channel width was varied from 25 to 51 cm. At the beginning of each experiment the flume was filled with sediment to a depth of approximately 30 cm. Maximum flow depths were on the order of 25 cm. Approximately, the first and last 50 cm of the flume are affected by inflow and outflow conditions, leaving a four metre section of bed where measurements can be made. The inlet and outlet adjustment lengths were short because step-pools create highly disturbed local flow conditions. The median step to step spacing was 52 cm. To determine channel dimensions in the flume, Froude scaling was employed. Froude scaling assumes that gravity drives the processes and the dimensions were adjusted to ensure that the Froude number, Fr =  v gy  (7),  is the same for the prototype and model. According to Froude scaling, lengths are proportional, velocities scale to the ½ power and discharge scales to the 5/2 power (Henderson, 1966). As  19  Reynolds number scaling (which scales for viscous effects) is incommensurable with Froude scaling, the Reynolds number (Re), Re = vyρ/μ  (8),  differs between the prototype and model (Table 2). In Eq. 8, μ is the viscosity of water. Provided the Reynolds number stays in the turbulent regime (> 2000; Henderson, 1966), the generally held belief is that the reduction in the Reynolds number should have little effect on how well the model represents the prototype. Table 2 gives basic dimensions of East Creek (11.4 % gradient) and Shatford Creek (7.8 % gradient) and the corresponding dimensions for a 1:10 and 1:20 scaling of the prototypes in the flume. These streams were used as prototypes because I have observed both in modest to large flood events and collected detailed channel morphology, flow, sediment transport and bed surface grain size data at both sites.  Shatford Creek  East Creek  Table 2. Prototype and model dimensions.  Parameter Width (m) Step spacing (m) Dmax bed surface (mm) D50 bed surface (mm) Discharge (L/s) event freq ≈ 12 yr Velocity (m/s) event freq ≈ 12 yr Reynolds number, event freq ≈ 12 yr Width (m) Step spacing (m) Bankfull water depth (m) Mean depth at bankfull (m) Dmax (mm) D50 of steps surface (mm) Discharge (L/s) event freq = 1.5 yr Velocity (m/s) event freq = 1.5 yr Duration of a “long event” (hr) Duration of a “short event” (hr) Reynolds number, event freq = 1.5 yr  Prototype 5.7 6.6 1536 200 2183 0.88 250000 8.7 5 1 0.4 1750 600 3000 1 48 6 260000  1:10 scale 0.57 0.66 153.6 20 6.90 0.28 12000 0.87 0.5 0.1 0.04 175 60 9.49 0.32 15.2 1.9 13000  1:20 scale 0.285 0.33 76.8 10 1.22 0.20 4300 0.435 0.25 0.05 0.02 87.5 30 1.68 0.22 10.7 1.3 4400  2.2 Instrumentation During background research on step-pool stability it became apparent that more detailed data on flow velocities, channel roughness, grain movement, bed and water surface elevations and bedload movement were necessary in order to improve our understanding of step stability. These 20  data are needed as channel resistance is intricately tied to the structuring of the bed, and is a key control on channel stability. For this reason a number of innovative technologies that make use of high speed video acquisition were adopted to provide information on channel conditions during step-pool formation and destruction. Table 3. Metrics measured during experiments and method used to measure their value. Measurement Methods used to measure Water (ρ) and sediment (ρs) Taken to be constant (1000, 2650 kg/m3 respectively) density Discharge (Q) Thermo Polysonics DCT 1088 Transit Time Flow Meter Velocity (v) Conductivity probes and a tipping bucket injecting slugs of salt water Mean width (w) Fixed using bank inserts DEM of bed and banks Laser scan of bed (1 mm resolution) Water depth (y) Discharge divided by width and velocity, checked using laser scans of water surface and subsequent scan of bed Sediment supply (Qs) Variable speed conveyor belt Sediment transport out of Light table with high resolution camera flume (Qs) Grain size (D) Digital photos of bed surface composed of colour coded grains Bed slope (S) Best-fit least squares linear regression of long profiles extracted from DEM of bed Grain stability Overhead video and bedload output at end of flume Step stability DEM and overhead video Channel stability DEM  2.2.1 Grain movement  Figure 5. Video image from overhead camera (Experiment 30, coarse mix, wide flume).  To monitor the movement of grains over the length of the flume a Prosilica EC 1380 Firewire video camera, in conjunction with an 8 mm lens, was mounted on the ceiling of the laboratory approximately four metres above the flume (Figure 5). Between 15 and 60 images were  21  captured per second, enabling the largest stones in the flume to be easily tracked and the timing and mechanism of step formation and destruction to be determined.  2.2.2 Water and bed surface elevations A laser profiler was designed to scan the entire bed of the flume. The profiler uses a Prosilica EC1280 (1280 x 1040 pixels) Firewire video camera mounted on a stepping motor driven cart to record a cross-section profile of the flume with a resolution of 1 mm. The stepping motor moves the cart in 2 mm increments and the cross-sections are stacked together to construct a Digital Elevation Model (DEM) 4.13 m long that covers the width of the flume (Figure 6). Each scan takes about 35 minutes to complete. Where the laser beam is hidden from the camera behind stones on the bed or behind bank protrusions, no data are collected. Five repeat scans of the same bed established that the standard deviation (σ) of the bed measurements is ± 1.9 mm and that on average 89% of the points differ by less than 5 mm between repeat scans. This variability is insignificant considering that the largest stones in the fine mix were 90 mm and the D50 is about 18 mm.  Figure 6. Two bed scans of flume, upper scan is from coarse bed experiment while lower is from fine bed experiments.  22  The calibration template used to spatially register the camera’s view in the plane of the laser (see Figure 7) is composed of a series of dots separated by 19 mm in both the x and y direction. The template is mounted on a thin piece of plexiglass and is arranged such that the laser beam shines through the length of the plexiglass sheet. The motor used to drive the scanner is an Applied Motion Products HT17-068 and is connected to a 1:100 Carson 17EP100-L gear box. The drive circuit consists of a rack and pinion setup that drives the cart from the left bank side of the flume. As a result there is about 7 mm of lag on the non-drive side when changing directions. Repeat movements over the entire length of the flume reveal that, over the 4.1 m that are scanned (1,079,300 steps), the scanner is usually displaced about 50 steps from home. This difference is insignificant considering there are 250.7 steps per millimetre. Each time the scanner returns to home, its position is reset to zero. Home is located at the downstream end of the flume where the bed scans begin and is determined using a SR17CJ6 limit device manufactured by Honeywell. The device has an accuracy of ± 0.025 mm.  Figure 7. Image illustrating calibration template for scanner.  23  The motor is controlled by a PCI mounted National Instruments PCI-7332 motion control board that is connected to a Universal Motion Interface 7764 device. The motion control board sends signals to the stepping motor controller/amplifier (Applied Motion Products 2035), which is powered by a Cosel UAW250S-24 power supply. The pulses that actually generate steps are sent from the amplifier to the motor over cables that are lubricated to tolerate a large amount of movement without breaking down. Typically the motor runs at 1500-2000 steps per second and accelerates at 100 rps/s or about 400 steps/s2. The scan begins by loading the calibration template and performing a perspective calibration on the template and storing the calibration information for future use. Next the scanner moves to home, and an image is taken with the laser off. Then a signal is sent to a National Instruments USB 6008 data acquisition device to output 5 volts on an analog output channel. The 5 volt signal triggers a transistor that connects the ground terminal of the laser to ground and the laser turns on. With the laser on a second image is taken with the video camera. The no laser image is then subtracted from the laser image producing an image where just the laser beam’s intersection with the bed is visible. The calibration information stored previously is then applied to the subtracted image. For each vertical column in the image with a pixel value greater than 30 the maximum pixel value and its location are recorded. The querying of the image creates a set of points with x and y pixel coordinates, which are then converted to true x and y coordinates using the calibration information. It is necessary to query the calibration template data for the location of each pixel in real space as calibrating the image does not produce an image with pixels of equal size in real space. Next for each integer millimetre value across the channel (i.e. horizontal position 1, 2, 3…..639, 640 mm) the vertical elevation is interpolated based on the two adjacent points. If both of the adjacent points are valid numbers, linear interpolation is used. If only one of the adjacent points is valid, then only that value is used, if neither point is a valid number, a “NA” value is assigned. This re-sampling provides data that are consistent between scans and makes it possible to store only the elevation data, as the horizontal position is fixed at 1 mm. Once this interpolation is completed a 1 dimensional array consisting of elevation data for each millimetre across the channel is written to a text file. Next, the laser is turned off and the motion cart is sent to the next cross-section, located 2 mm upstream and the entire process is repeated. The elevation data from the 2nd cross-section are then appended to the previous 1 dimensional array creating a two 24  dimensional array where the dimensions are horizontal position and cross-section number. At the end of the scan an array that can be plotted creating a DEM is produced (Figure 6) from which long profiles (Figure 8) and other information can be gathered.  Figure 8. Long profile of bed and water surface.  The water surface can be scanned using the same system; however, more noise is introduced and the data need to be filtered in a supervised manner. To filter the data a program was developed that incorporates user specified elevation accept/reject cut points, a moving window that spatially averages the data, as well as an elevation offset. To determine a mean water surface elevation the top 20 % of the data points within a moving window were selected and the mean and standard deviation of these data points was determined. The window size was specified based on how noisy the data was and ranged in size from 25 to 61 mm. If the coefficient of variation of the top 20% of data points was greater than 0.2 and the highest data point was more than one standard deviation from the mean, the highest data point was removed. This continued until either the highest data point was less than one standard deviation from the mean or the coefficient of variation of the remaining data points was less than 0.2. Once a stable mean of the selected data points was achieved, the mean of these data points was assigned to the point in the center of the moving window. Through a process of trial and error the size of the window, threshold elevation for the cut points and vertical offset were adjusted until the filtered surface corresponded with the water surface. The true water surface was clearly visible in the raw data. Using only the top 20% of the data points removed reflections off the bed. Likewise, by removing points that were more than one standard deviation above the mean, points that were caused by the laser reflecting off spray were removed. Figure 9 illustrates a long profile from a  25  water surface scan. Clearly, noise in the raw data exists, but a reasonable mean water surface profile is evident. The solid line indicates the result from the filtering procedure.  Figure 9. Filtered water surface scan. Points are raw data and line is filtered surface.  When filtering the water surface scans it was easiest to simply select the middle portion of the flume that did not contain any bank protrusions. To investigate if using only this central portion of the channel rather than the bank to bank water surface had an effect on the measurement of mean depth, the depth measured using the central portion of the channel was compared with the depth measured using data gathered from the entire width of the flume. Using a 3.696 m length of channel and 46 water surface scans the mean depth based on the middle portion of the channel (ymid) was regressed against depth determined using the entire width of the flume (ytotal). The regression yielded y mid = 0.986 y total + 0.003 , r2 = 0.99, p < 0.001, n = 46, units are metres  (9).  While the slope in Eq. 9 is significantly different than the 1:1 line, and the intercept is significantly different from zero, the differences are minimal. Depths in the entire data set (ytotal) range from 21 to 253 mm, which yields an error of +2.7 mm (12%) for the shallowest depths and -0.5 mm for the largest depths. These errors are likely smaller than the errors associated with measuring shallow depths as at shallow depths the water surface scans were consistently more difficult to process and less precise. The errors at shallow depths occur as the laser is more frequently reflected off the bed rather than the water surface and it was difficult to identify what is the bed and what is the water surface in the scan data, particularly along the banks. Removing the constant in the regression produced a slope of 1.011, suggesting that the depth derived using the full width is shallower than the mid-channel derived depth; the opposite of what was observed when a constant was included in the regression. In summary, while there are some 26  minor differences introduced by using only the mid-channel depth, the mid-channel depth is considered to adequately reflect the true depth and will be used for all subsequent analysis.  2.2.3 Flow velocities and water depth Average flow velocities were determined using a tipping bucket (Figure 10) that injects approximately 250 ml of dilute salt solution into the upstream end of the flume every minute or so. The time of injection was recorded and the pulse of salt was monitored with two conductivity probes located approximately one metre and four metres from the top of the flume. Figure 11 illustrates the conductivity probe circuits. The probes are powered by an AD698 signal conditioner chip that is supported by an isolated 19.3 V AC power supply. To prevent the two probes from interfering with each other one signal conditioning chip was used and two single pole, double throw relays were used to switch between the two probes. The chip generates an AC signal which was sent to one of the probes and the voltage drop across the probe was read on the same chip and converted to a DC 0-5 Volt output. The relay was then flipped and after a 30 ms delay the voltage drop on the second probe was recorded. Ten samples were taken at a rate of 1500 samples per second and averaged. In order to reduce the number of times the relays were switched, the relays were only switched once for each sampling period. Thus the first sensor was sampled, then relay was switched, then a 30 ms delay occurred, then the second sensor was sampled. During the next sampling period (100 ms after the first sample) the second sensor was sampled again, the relay was switched, a 30 ms delay occurred, and then the first sensor was sampled again.  Figure 10. Tipping bucket used to inject salt solution  27  Conductivity was thus monitored at 10 Hz and the time for the centroid and peak of the pulse to travel between the two probes was determined. Using the switch on the tipping bucket the time from when the tipping bucket flipped till when the peak and centroid were detected on both probes was also measured.  Figure 11. Schematic illustrating layout of conductivity probe circuitry.  To improve the accuracy of tipping bucket injection time, using the average of a number of runs and high speed video that recorded when the salt pulse plunged into the water, the time at which the tipping bucket switch recorded the tipping bucket flipping was adjusted to match the time the salt solution was observed to be injected into the water. Adjusting the time the tipping bucket flipped removed any bias introduced due to the switch being located on one side of the tipping bucket and the time required for the water to drain out of the tipping bucket and into the flume (bias was 0.9 and 0.5 s depending on flip direction). The time for the salt solution to travel to the probes and between the probes, along with the distance between the probes, provided six measures of the average water velocity through the experimental section of the flume. In addition to these six measures of velocity, three other velocity measurements were used based on the recommendation of MacMurray (1985). These methods utilized the cumulative probability function derived from the conductivity pulses to determine the harmonic mean travel time and in turn the mean water velocity. Waldon (2004) notes that the harmonic mean provides an 28  unbiased measure of mean water velocity while the peak in the pulse over estimates the velocity and the centroid underestimates the mean velocity. To illustrate the nature of the velocity data, hydraulic geometry data collected while the bed was stable are plotted in Figure 12. While these data are only a small portion of all the data collected during this study (225 of 56,000 salt injections), the rest of the data show similar behaviour. The hydraulic geometry relations clearly differ depending on the technique used to calculate the velocity despite all of the data coming from the same slugs of salt water. The amount of variability about the best-fit relations is also substantial and varies between methods. The variability suggests that some methods of calculating velocity are more precise than others. Which of the nine velocity measurements was most accurate was not initially known, but was determined by comparing back calculated depths with measured depths. To assess which velocity measurement is the most accurate and precise the mean flow depth predicted using continuity (y = Q/vw where Q is discharge) and each of the nine velocity measurements was compared with the mean water depth determined using the water surface and bed surface scans. Figure 13 illustrates that not all of the velocity measurement techniques predict the same flow depth and that the variability between methods is considerable. Table 4 records the regression coefficients and their standard error for the nine regression models plotted in Figure 13. As there is uncertainty in both the x and y variables the slope and intercept in the regression equations were adjusted to produce reduced major axis solutions using madj = m r  (10).  r is the correlation coefficient (Mark and Church, 1977). Once the adjusted slope was determined, the line was positioned through the bivariate mean to determine the y-intercept. Only the tipping bucket to downstream probe harmonic mean velocity was not significantly different from the 1:1 line. In general, the tipping bucket to downstream conductivity probe data produced better results than the other methods. The harmonic mean derived depth is associated with more variance than the peak derived velocities (Figure 13). While processing the data it was also observed that the harmonic mean derived velocities were more likely to produce unreasonable velocities (greater than 300 cm/s or negative). These data were removed. Additional filtering was also performed by comparing the harmonic mean velocity with the  29  tipping bucket to the peak in conductivity pulse at the downstream probe derived velocity data. In particular only velocity measurements falling within the grey region of Figure 14 were included in the subsequent analysis.  Figure 12. Nine different measures of mean water velocity related to discharge for Experiment 16 during a hydraulic geometry run with a stable bed. Bottom figure shows best-fit relations plotted in top 9 illustrations on a single plot to illustrate variability.  30  Figure 13. Relations between mean flow depth back calculated using continuity for 9 different measures of mean water velocity and average depth measured with water surface and bed surface scans.  The processing of the data also revealed that the uncertainty associated with picking a peak in the record, and even more so the centroid and harmonic mean of a pulse was greater than the uncertainty associated with determining the time the tipping bucket flipped and the exact time the salt was injected into the water. The reduced precision associated with probe derived data may explain why the velocities measured using two probes are more variable than those based on the tipping bucket and a single probe.  31  Table 4. Functional regression results for depth back calculated using nine different measures of velocity compared to depth measured using water surface and bed surface scans. Standard error (SE) of slope and intercept based on least squares regression are also given. Bold numbers indicate slope is not significantly different from the 1:1 line (n = 194-196).  Velocity method Tipping to centroid upstream Tipping to peak upstream Tipping to centroid downstream Tipping to peak downstream Upstream to downstream centroid Upstream to downstream peak Tipping to upstream harmonic mean Tipping to downstream harmonic mean Upstream to downstream harmonic mean  r2 0.77 0.83 0.87 0.90 0.58 0.70  Slope 1.43 0.96 1.06 0.83 1.08 0.86  SE of slope 0.05 0.03 0.03 0.02 0.05 0.03  Intercept 0.0042 0.0077 0.0124 0.0129 -0.0108 0.0016  SE of intercept 0.006 0.004 0.003 0.002 0.006 0.004  0.75  1.09  0.04  0.0040  0.005  0.80  0.99  0.03  0.0035  0.004  0.54  1.22  0.06  -0.0299  0.008  Figure 14. Relation between peak in conductivity pulse velocity and harmonic mean derived velocity. n = 43,683 velocity measurements. Data falling outside of grey envelope were removed from subsequent analysis.  As a third means of investigating water velocities a number of wax particles that were 1.5 to 3 cm in diameter were made. They were weighted with sand so that they would be neutrally buoyant. These particles were then dropped in the flume at the tipping bucket location and  32  tracked with the overhead video. The time at which the particles passed the conductivity probes was recorded and their velocity determined. Figure 15 illustrates the mean velocities for the wax particles for 12 runs as well as the salt dilution derived velocities for the same runs. The wax particles have substantially larger mean velocities at low flows, probably because they are too big to flow through the openings between stones near the bed where the flow velocities would be expected to be lower. As the relative roughness (Dmax/y) is quite high (1.5 on average, 3.3 maximum, 0.53 minimum) most of the flow is around and between large grains. As a result the wax particle velocities are likely not representative of the mean flow velocity, nevertheless they can provide detailed local velocity information over steps and pools. Overall the tipping bucket to downstream probe harmonic mean travel time derived velocity is most accurate and was used in all subsequent analysis.  Figure 15. Mean velocities of neutral density wax particles compared to mean salt injection velocities for runs during Experiments 15, 16 and 18. Bed varied between runs. The number of particles per run (data point) ranged from 14 to 39 and averaged 30.  2.2.4 Bedload transport As a means of acquiring detailed temporal sediment transport information while also reducing the need to sieve samples a light table was attached to the end of the flume and a video camera was used to record the shadow of stones passing over the light table. The algorithms and data collection techniques were developed specifically for this research and are explained in detail in Chapter 3.  33  2.2.5 Discharge Discharge was initially measured using a sonic Versa-flow sensor mounted on a straight section of the 20 cm diameter pipe that supplies water to the flume. It became apparent that the flow meter was malfunctioning and it was subsequently replaced with a Thermo Polysonics DCT 1088 Transit Time Flow Meter. Flow was recorded along with conductivity at 10 Hz on a National Instruments 6008 USB data acquisition device. The internal averaging on the DCT 1088 was reduced to one second from thirty seconds to increase the temporal precision of the record. Maximum discharge rates were limited by the outflow tank, which could accommodate 85 L/s. Most days before a run started the amount of water flowing through the bed just prior to the initiation of wide spread surface flow was measured and subsequently subtracted from the measured discharge. The subtraction of subsurface flow was standardized only after Experiment 22.  2.2.6 Flume grain size mix The range of grain size distributions used during previous studies is illustrated in Figure 16. There is about an order of magnitude of variation in the mean size used and substantial variation in the grain size spread. The figure also illustrates a range of grain size distributions from steppool streams and it is evident that the mean grain sizes used during flume experiments have tended to have a wider range than the distribution of surveyed step-pool streams. For this study, with the exception of Experiment 1, all of the experiments were conducted using either a ‘fine’ or ‘coarse’ mix (see Figure 16) that were modelled after Shatford Creek and the Rio Cordon, respectively. Neither mix contained sediment finer than 2 mm. The finer mix was designed to contain sediment approximate 1/20th the size of the Shatford Creek bed material. Ideally a bulk sample of the subsurface sediment in Shatford Creek would have been used to determine the grain size distribution for the flume; however, no bulk sample data were available and collecting a bulk sample in such coarse substrate would have required heavy equipment and considerable cost. Instead the surface distribution of East Creek, which is noticeably finer than Shatford Creek, was used as a first approximation of the Shatford Creek subsurface distribution. East Creek is a less energetic, smaller step-pool stream and it was supposed that its’ surface grain size distribution might match Shatford Creek’s subsurface distribution. As East Creek does not have any of the large boulders that are present in Shatford 34  Figure 16. Grain size distributions of initial flume mix, surface distribution of a number of step-pool creeks (a) and distribution of sediment mixes used in a number of other steepstream experimental studies (b).  Creek, a few larger grain size classes were added to the East Creek derived distribution. After the first experiment, the proportion of sediment less than 40 mm associated with the East Creek surface grain size mix was redistributed among the finest four grain size classes (>2, 2.8, 4 and 35  5.6 mm classes). By redistributing the finer than 40 mm sediment the D50 was kept the same as the scaled D50 of East Creek (9.8 mm) despite truncating the sample at 40 mm (2 mm scaled). For the coarse mix a 1:8 scaled distribution of the surface grains size distribution of the Rio Cordon was used. This distribution is the same as that used by Francesco Comiti (Comiti et al., 2007a; in review), with the exception that my sample was truncated at 2 mm, which removed 2.5% of the sediment. The D50 of this mix was 22.8 mm. For both mixes, each ½ phi grain size class above 11 mm was painted a different colour that was intended to correspond with a distinct grey scale value in black and white digital images.  2.2.7 Measuring surface grain size To determine the surface grain size distribution in the flume digital images were collected from approximately 80 cm above the bed. The images were loaded into a custom LabView program and a user identified the colour, and therefore size, of each stone located at 63 randomly located points in each image. The location of the points was determined by randomly assigning a start location for a rectangular grid of points that was made up of a series of points separated by more than the largest grain size. To cover the bed, seven digital images were used, yielding a sample equivalent to a Wolman count (Wolman, 1954) with more than 300 stones per sample. To investigate the precision of the technique the same set of seven images was re-analyzed 16 times by three different users (four times by one user, six times by the other two). While there were significant differences between users in the mean D10, D16, D25, D50, and D75 (ANOVA, p<0.05, phi units (φ)), the mean difference was at most 0.35 φ. Therefore, between user differences were not considered to be important. The standard error of the 16 samples ranged between 0.04 φ for the D25 and 0.0025 φ for the D95 with a tendency to decrease as the coarse end is approached (Table 5). A reduction in the standard error at the coarse end was also observed by Rice and Church (1996). The standard errors are smaller than the values found by Rice and Church (1996) based on sub-sampling traditional Wolman counts collected from two rivers. Prediction intervals are given in Table 5 and can be used to predict the precision that should be expected for the rest of the grain size measurements. The prediction interval for individual samples was calculated with ± t aσ 1 + 1 / n  (11) 36  where ta is the critical Student t-value (p = 0.05) and σ is the standard deviation. A similar analysis was performed using four sets of photos from Experiment 22 of a bed that hardly changed between runs due to low sediment transport rates and similar results were obtained. Table 5. Error associated with measuring grain size. Error was determined by repeated processing of seven images (n = 16).  Standard error (SE) (φ) Prediction interval (PI) for single sample +/- (φ) SE (φ) PI for single sample +/- (φ) SE (φ) PI for single sample +/- (φ) SE (φ) PI for single sample +/- (φ)  Unit type D10 Entire 0.0068 channel Entire 0.0148 channel  D16  D25  D50  D75  D84  D90  D95  0.0173  0.0378 0.0112 0.0033 0.0033 0.0049 0.0025  0.0383  0.0835 0.0247 0.0071 0.0074 0.0108 0.0055  Steps  0.694  0.650  0.598  0.261  0.025  0.016  0.063  0.145  Steps  1.524  1.427  1.313  0.574  0.054  0.034  0.139  0.318  Pools  0.0052  0.0134  0.0314 0.0125 0.0092 0.0067 0.0039 0.0020  Pools  0.0115  0.0294  0.0690 0.0274 0.0201 0.0146 0.0085 0.0044  Treads  0.0176  0.0451  0.0596 0.0133 0.0042 0.0058 0.0079 0.0064  Treads  0.0387  0.0990  0.1308 0.0292 0.0091 0.0127 0.0173 0.0141  In total 307 sets of photos were processed and their mean grain size determined. During the processing of the images the longitudinal position of each sampling point was recorded. Once the channel was classified into step, pool and tread bed morphological units, the location of each grain size sample was coded to a channel type and the grain size distribution of the steps, pools and tread channel morphological units was determined. To determine the error associated with the grain size distribution of each channel type the data from the same 16 repeat grain size samples were evaluated. The bed that was repeatedly sampled had five step-pool units. Based on the 16 replicate samples, on average 380 (σ = 9) stones were identified during each grain size sample. Twenty-nine of these points were associated with steps (σ = 6), 133 (σ = 6) were associated with pools and the rest (218, σ = 6) were associated with treads. As steps tend to occupy less of the channel compared to the other units, they were sampled less frequently. The prediction interval for the steps is considerably larger than for the other channel units, especially for the finer grain size classes. The increase in 37  uncertainty is because relatively few photo points fell within steps (Table 5). The analysis in subsequent chapters frequently utilizes the D84 of the steps, which has the lowest error range of the step grain size metrics (Table 5). The D84 was used as it was shown to consistently explain the most variance in step-pool and flow resistance metrics. The D50 of the steps was associated with an uncertainty of ± 0.5 φ, which is a substantial amount of uncertainty. The more sample points that exist for each channel unit, and the closer the percentile (D50, D84 etc.) is to the mode of the grain size population, the more precise the grain size sample will be.  2.2.8 Channel width For Experiments 1 through 3 the banks were inclined at an angle of 70° and as a result the channel width changed with discharge and as bed aggradation or degradation occurred. For these experiments the initial bed width was 0.371 m and water surface widths were back calculated based on measured depth (calculated using continuity), width at the bed surface determined using the bed scans, and the angle of the banks. For the other experiments the mean width was predetermined and was 0.245, 0.359 or 0.511 m. A value of 0.359 was used as this scaled to the width of Shatford Creek. The other widths were designed to be 2/3 this width and 3/2 this width (+/- a bit to accommodate the width of a sheet of plywood). Table 6 summarizes the conditions of each experiment.  2.2.9 Bank roughness While past experiments have all used smooth vertical banks, it was believed that to adequately model the jamming of large stones against the banks, rough banks scaled to a natural stream channel should be used. To determine and scale bank roughness the location of the base of the bank was surveyed in East, Shatford and Giveout Creeks. These three creeks are located on the coast, in the Okanagan and in the West Kootenay regions of British Columbia, respectively. All of the deviations caused by boulders, bends, wood and bars were included. As such the surveys provide a detailed description of bank roughness (Figure 17a). Once the survey was completed, using linear or polynomial regression relations a best-fit line was fit to sections of bank and the deviations perpendicular to this line were calculated (Figure 17b). Determining the deviations about the mean bank position provides a description of the roughness along the bank channel. A first order Butterworth high pass filter with a window length of 20 m was then used to filter the data, a histogram of the distribution of deviations was created (Figure 17c) and outlying bank 38  Table 6. Summary of experiments.  Experiment number  Flume slope (%)  Mean width (m)  D50 of mix (mm)  Bank configuration  Additional notes  Experiment duration (hr)  Maximum discharge (L/s)  1  8  0.371  13.1  Angled and rough  Old flow meter- Discharge reading no good.  15.0  Not known  2  8  0.371  9.8  Angled and rough  Fed 340 kg of sediment, high flow to start.  24.3  25.4  3  8  0.371  9.8  Angled and rough  Fed 150 kg of sediment, high flow to start.  14.7  14.2  4  8  0.359  9.8  Rough  Ran feed to balance degradation, fed 430 kg of sediment.  35.7  42.8  5  8  0.359  9.8  Rough  Replicate 1 of proposed design, 1 hr increments throughout.  20.1  50.7  6  8  0.359  9.8  Rough  Replicate 2 of proposed design, 1 hr increments throughout.  21.4  61.6  37.6  51.2  7  8  0.359  9.8  Rough  One hour runs when feed was not being added and 3 hr runs after last feed. Feed rate was reduced to 12 g/s for last three feeds (first feed was 7 g/s). This required feeding to occur over 3 hr for last two feeds.  8  18  0.359  9.8  Rough  Flow not scaled by slope, material removed quickly.  4.0  6.8  9  18  0.359  9.8  Rough  Design standardized, 1 hr until last feed, then 3 hr after that. Flows scaled by stream power.  15.6  8.6  39  10 11 12 13 14 15 16 17 18  Flume slope (%) 3 11 14 8 8 18 14 11 3  Mean width (m) 0.359 0.359 0.359 0.359 0.245 0.245 0.245 0.245 0.245  D50 of mix (mm) 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8 9.8  19  14  0.245  22.8  Rough  20 21  14 18  0.245 0.245  9.8 22.8  Rough Rough  22  14  0.245  9.8  Rough  23 24 25 26 27 28 29 30 31 32  14 8 8 8 3 8 18 14 14 18  0.245 0.245 0.245 0.511 0.511 0.511 0.511 0.511 0.359 0.359  22.8 9.8 22.8 9.8 9.8 22.8 22.8 22.8 22.8 22.8  Smooth Smooth Smooth Rough Rough Rough Rough Rough Rough Rough  Experiment number  Bank configuration  Additional notes  Rough Rough Rough Rough Rough Rough Rough Rough Rough Added fine feed at end to see if bed could be broken, flume full so couldn’t increase Q Added fine feed rather than increasing flows Ran at constant Q for 20 hr rather than increase flow  32.5 29.6 21.7 39.9 43.7 21.2 27.4 38.5 32.8  Maximum discharge (L/s) 76.5 21.0 13.6 50.6 43.7 7.7 12.2 26.5 55.1  22.2  22.4  34.2 36.7  10.0 43.4  46.1  11.7  33.0 27.7 31.7 31.0 25.4 15.5 21.7 28.4 35.7 26.5  50.3 20.8 69.0 54.3 81.1 82.2 53.6 82.3 84.2 37.9  Experiment duration (hr)  40  roughness elements were identified. Any roughness lengths falling outside the 95% confidence interval were removed. A Butterworth high pass filter was used as this filter separated the high frequency bank roughness deviations from the remaining low frequency deviations associated with channel beds.  Figure 17. a) Base of bank edge along Shatford Creek, b) deviation from mean bank position as a function of bank distance, c) frequency distribution of deviations from mean bank position.  41  In the flume plywood slabs were attached to the banks in a manner that reproduced a 1:20 scaled model of randomly selected sections of Shatford Creek. Despite Giveout Creek being narrower than Shatford Creek (5 m vs. 6 m at base of bank) the deviations about the mean bank positions were nearly identical (standard deviation = 0.426 versus 0.434 m respectively). Therefore, a second configuration of bank roughness based on Giveout Creek was not used. However, three experiments were completed at the narrowest flume width with straight banks (see Table 6).  2.3 Error analysis With the measurement of any metric there is always some uncertainty. Depending on the metric, error was estimated based on the variability of replicate measures (e.g. velocity), standard error (e.g. bed slope), or instrument error (e.g. discharge). Table 7 summarizes the method used to estimate the standard error associated with each metric that was directly measured. To determine the error associated with all of the calculated metrics standard error propagation relations were used (Table 8). Table 7. Method used to estimate standard error for each metric directly measured.  Metric Discharge Width Slope Standard deviation of bed Grain size Velocity  Method used to estimate standard error Instrument error (5%) Instrument error (2%) Standard error based on mean squared error from best-fit regression Instrument error (1%) Prediction interval for single sample given in Table 5 Standard error based on standard deviation of replicate measurements  Table 8. Error propagation relations.  Relation among variables B = cx B = x+y-z B = log(x) B = xy/z B = xa  Calculation of error δB = cδx δB = δx + δy + δz δx δB = x  δB  2  ⎛ δx ⎞ ⎛ δy ⎞ ⎛ δz ⎞ = ⎜ ⎟ + ⎜⎜ ⎟⎟ + ⎜ ⎟ B ⎝ x⎠ ⎝ y⎠ ⎝ z⎠ δB δx =a x B 2  2  42  3 Two innovative techniques To examine the formation and failure of step-pools two important methodological challenges needed to be addressed. First, an objective means of identifying a step, pool or tread section of channel was needed and second, a means of recording the temporal variability and grain size of the sediment being transported out of the flume was sought. The essential details of both of these methods are described in Sections 3.1 and 3.2. Revised, more detailed versions of these chapter sections have been published (Zimmermann et al., 2008b; 2008a).  3.1 Identifying step-pool units To determine the dimensions of step-pool units, long profile surveys are commonly completed and the steps and pools are identified in the field. The surveys are usually comprised of breaks in slope and points that identify important channel features (Zimmermann and Church, 2001; Milzow et al., 2006; Nickolotsky and Pavlowsky, 2006), but may alternatively be comprised of survey points collected using a fixed sampling interval (Wooldridge and Hickin, 2002). Subsequently, metrics such as the step length, step height, residual pool depth, and pool length are calculated for each step-pool unit using the survey data. Some researchers (e.g., Chin, 1999; Wooldridge and Hickin, 2002; Milzow et al., 2006) have attempted to identify the location of steps and pools using only long profile data. Thus the traditional approach includes two important subjective components. First, the surveyor must decide where to run the long profile and put the rod to capture the overall channel morphology, and second, the step-pool unit must be identified, either in the field or afterwards. As the study of step-pools has progressed, authors (Wohl and Grodek, 1994; Zimmermann and Church, 2001; Chin and Wohl, 2005; Curran and Wilcock, 2005; Church and Zimmermann, 2007) have incorporated data from other studies without apparently considering the subjectivity associated with identifying the step-pool channels. As Nickolotsky and Pavlowsky (2006) illustrated, the variability introduced by different operators describing step-pools in different ways can lead to important differences in step-pool metrics (pool length, step height etc). Objective, standardized means of identifying step-pool unit breaks would overcome this challenge.  43  While a few objective means of classifying step-pools using long profile surveys have been presented in the literature (Wooldridge and Hickin, 2002; Milzow et al., 2006), these techniques have not been widely tested. Furthermore, they overcome only the subjectivity associated with deciding where the steps and pools are, based on a long profile; they do not suggest how to overcome the subjectivity associated with surveying the long profile data. Wooldridge and Hickin (2002) concluded that subjective field-based classification works best. Traditionally, survey points are chosen so that the data represent significant breaks in the channel morphology. The data do not reproduce the actual details of the channel as this would take too much time. In recent years however, technological advances that include low-level aerial photos, tracking total stations, and tripod-mounted laser scanners have made it possible to objectively record the shape of the channel. As a result, producing detailed digital elevation models (DEMs) that record the morphology of the channel is now possible, but objective means to classify the channel units using these DEMs do not exist. The objectives of this section are as follows: i. to show that subjective classification leads to considerable ambiguity as to what is a step-pool and what is not and that different “experts” classify the same long profile in substantially different ways; ii. to show that a scale-free, rule-based classification scheme can delineate step-pool units based on long profiles in a manner equivalent to the modal response of “experts” and can identify most of the step-pool units observed in the field; iii. to demonstrate that objectively classifying a step-pool channel based on subjectively collected long profile survey data yields results similar to classifying the channel in the field; and iv. to illustrate that the scale-free, rule-based classification scheme performs better than four objective methods previously presented in the literature: minimum slope (Milzow et al., 2006), zero crossing, bed elevation differencing, and spectral analysis (Wooldridge and Hickin, 2002).  44  3.1.1 Field, channel classification and analysis techniques  3.1.1.1 Field survey methods Two different crews completed field surveys at 13 steep channels in British Columbia. The surveys were completed using either a level-tape survey or a total station survey. The spacing of field survey points was either systematically every 20 cm or every few metres at breaks in slope. With the exception of Deeks Creek, where four short reaches (25-80 m in length; 3.5-10.4 bankfull widths) were visited, between 19.4 and 50.1 bankfull widths of channel were surveyed (50 to 362 m of channel). Channel units were identified when they spanned the channel and a dominant hydraulic function appeared to exist. Steps and pools were always identified as a pair, and in order for a step to be identified the step must have spanned the channel and during low to moderate flood events water must plunge off the step into the pool downstream as an impinging jet (see Church and Zimmermann, 2007, after Wu and Rajaratnam, 1996, 1998). An impinging jet requires subcritical flow conditions at the reach scale and the step height to be similar to flow depth during floods. At high flows, whether the flow conditions change to the subcritical surface jet regime or the supercritical nappe, transitional or skimming/rapid regime (see Church and Zimmermann, 2007) is an unresolved research question. In order to identify a step-pool unit, it was necessary to assess whether, during high flows, a step-pool unit visible at low flows would still persist and whether new units would become evident at higher flows. Other studies considering only low flow features may not put the same emphasis on the hydraulic function of the channel units at high flows and may establish different classification methods. The emphasis on hydraulics is driven by the notion that water plunging off the steps scours the bed, forming the pools, and plays a major role in controlling the dissipation of energy in mountain streams. Thus, the field surveys were intended to identify steps and their downstream pools and to characterize the bed roughness for flow resistance considerations (a topic of much research, see Marston, 1982; Abrahams et al., 1995; Maxwell and Papanicolaou, 2001; Lee and Ferguson, 2002; Aberle and Smart, 2003; Wilcox et al., 2006; Comiti et al., 2007b). The step pools considered here are distinct from micro cascade (Grant et al., 1990) features that may include drops, pools, and steps that do not span the channel. None of the survey data was collected with the idea that the profiles would subsequently be used to  45  objectively identify step-pool units; rather the data were initially collected in order to characterize the channel slope and step-pool dimensions.  3.1.1.2 Assessing variability among step-pool researchers To assess how much variability there is in the way different step-pool researchers classify a channel given a long profile, four long profiles were sent to 15 step-pool researchers. I explicitly sought people who had experience classifying step-pool channels by reviewing the step-pool literature and e-mailing the researchers directly. For each profile, lengths of channel representing six to eight bankfull widths were provided (Figure 18) as this length was sufficiently long to characterize a number of step-pool units while still being short enough that the details were not obscured on a sheet of letter-sized paper (21 x 28 cm). An initial set of trial long profiles was created with no vertical exaggeration and distributed to two researchers. It was observed that these were particularly difficult to classify and many of the step-pools that were identified in the field were not readily evident. For this reason vertical exaggeration was adjusted by eye in order to make the bedforms appear reasonable. Determining the vertical exaggeration is a subjective adjustment that was observed to have an effect on classification results, further supporting the need for an objective means of classifying the bed. Each respondent was given the following instructions: I seek to analyze how a number of researchers, who have previous experience classifying step-pool channels, classify the same long profiles. Thus, I have prepared four long profiles that vary in slope from 5 to 15% and in width from 0.47 m to 7.2 m. Three profiles were taken from field surveys and one is the result of a flume experiment. The density of field survey points is either systematically every twenty centimetres or every few metres at breaks in slope. In the case of the flume data, there is a point every two millimetres. To help guide you in this exercise I ask that you consider a step-pool bedform to be a single entity with two sub-sections, the step and the pool. Thus one cannot have a step without a pool and vice-versa. I also ask that you consider the step to be upstream of the pool as the pool is considered to be a morphology caused by water plunging off a step. A step-pool unit is considered to be a channel spanning bedform that has water plunging off the step into the pool at low to modest discharges. Whether the plunge persists at high flows remains an active research question (Church and Zimmermann, 2007). It may be useful to estimate the low flow water profile and thereby identify the pools. 46  Figure 18. Long profiles distributed for classification: a) Carnation Creek, b) Tees Creek, c) Giveout Creek, and d) flume.  47  On each profile please identify where you believe the pool starts, the pool ends/step starts, the step ends and where 'other' sections of channel exist. While the 'other' sections of the channel can be classified as a range of morphologies (including riffle-pool, cascade, run, glide and tread), I ask that you simply classify these as 'other' (or you may simply leave them unlabelled).” In total, 11 researchers returned classified long profiles that were then compared to the fieldbased classification results. For the long profile from flume data, no field-based classification results exist.  3.1.1.3 Introducing the scale-free rule-based classification algorithm In order to objectively classify step-pool units, a classification algorithm must be scale free, and the scheme must have a set of rules that can be applied across a range of channels. Furthermore, the algorithm should work regardless of the spacing between survey points; provided the spacing is sufficiently close to characterize the channel morphology. Attempts to develop such a scheme began with Milzow et al. (2006), who defined a step if a minimum gradient is exceeded. This approach was tried, but did not identify pools consistently. Thus a classification algorithm was adopted that looked for the pool first and then attempted to determine if a step was upstream. The approach was programmed in LabViewTM, but is a simple set of rules that could be easily programmed in other languages. The classification scheme utilizes six scale-free parameters. These include the minimum step length, minimum pool length, minimum residual depth, minimum drop height, minimum step slope, and maximum step length. All of the parameters, except minimum step slope, were normalized by the mean bankfull width (wb) of the reach. Bankfull width was chosen as the scale variable as it is likely to be measured in most studies and provides a reasonably reliable estimate of the scale of the channel. Residual depth (dr) is the difference between the elevation of the bed at the lip of the pool and the deepest point in the pool (Figure 3). The minimum step gradient was determined by calculating the average gradient of the channel and then adding a fixed number of degrees to the mean gradient to determine the minimum step gradient. Thus, minimum step slope was set by the overall channel gradient. The bankfull water line was located in the field using the break in slope at the top of the banks, where evident; otherwise the location of the maximum flood, determined by examining the trash-line, was used. The flume width (w) was scaled to be between the bankfull width and base of bank width measured at 48  Shatford Creek. For the majority of the scaled experiments (excluding Experiments 1-3) the mean flume width was 93.5% of the bankfull width hence the flume width was multiplied by 1.068 to attain the bankfull width. Zimmermann et al. (2008b) used a multiplier of 1.33 as it was thought that the base of bank width was used in the flume experiments, and the base of bank width was 75% narrower than the bankfull width. Herein this mistake has been corrected and a multiplier of 1.068 is used. This results in one more step-pool being identified with the algorithm in the portion of channel reviewed by the respondents. On average using the correct width results in 18% or 1.3 more step-pools units over the length of the flume. The increase in the number of step-pools identified when the smaller multiplier is used is similar in magnitude to the uncertainty associated with the classification algorithm. The algorithm begins at the downstream end of the survey data and moves upstream by first determining the location of the end of a pool. A survey point is coded to be a potential end of pool point if it is the highest point in elevation among all points upstream that fall within a predefined minimum pool length (which is some fraction of the bankfull width of the channel). If this is the case, the pool is extended upstream until a point is reached where the next upstream point is higher than the end of pool. A new bed elevation point is then interpolated such that it has the same elevation as the end of the pool but lies between the point that is just higher than the end of pool and the downstream point that is lower than the end of pool elevation. This interpolated point becomes the start of pool (= end of step). The residual depth is calculated at this time; and should it be less than the minimum required residual depth, all the survey data points making up this potential pool are classified as “other” and the search for a pool continues, starting at the upstream end of the abandoned pool. If the residual depth is greater than the minimum required residual depth, the survey points upstream are analyzed to determine whether or not a step exists. To determine if a step exists, sufficient upstream survey points are grouped together such that both the minimum step height and minimum step length are exceeded and then a linear best-fit relation is determined. If the slope exceeds the critical step slope, a step is identified. If the slope of the best-fit line does not exceed the critical slope, then the next upstream survey point is added to the set of survey points and the slope recalculated. The addition of upstream survey points continues until either the slope exceeds the critical slope or the maximum step length is reached. If the maximum step length is reached, the section between the start of pool and the 49  end of pool is classified as “other” and the search for a pool starts again at the start of pool survey point. To determine the top of step, the slope and r 2 are computed for the fitted line that first exceeds the critical slope and all subsequent lines that can be created by sequentially adding an additional upstream survey point until the maximum step length is reached. If the initial best-fit line is composed of only two points, an r2 of 0.5 is used rather than 1. The upstream-most point among the data points associated with the line that has the largest r2 and a slope greater than the critical slope is classified as the top of step. By using the line with the largest r2, the most continuous steep section of the bed is classified as the step. The search for the next pool begins at the top of step survey point. The rule-based algorithm results in definitions of step height (H), pool length (Ls), residual depth (dr), and step-to-step spacing (L) that correspond with the definitions illustrated in Figure 3 and Figure 19. The method also explicitly allows for the identification of an ‘other’ bed unit and does not require the entire bed to be classified as either a step or a pool.  Figure 19. Example of the variation in how different respondents might classify a step-pool unit. EP, SP, and TS refer to end of pool, start of pool, and top of step, respectively. The rule-based classification scheme results are indicated with bold type.  3.1.1.4 Determining average long profile from bed scans Traditionally, detailed surveys of channel morphology that record the entire surface structure of the channel have been prohibitively time consuming. Instead, a surveyor can best represent the  50  channel by choosing to locate survey points at major breaks in slope and at other defining locations. However, with the advent of laser scanners and tracking total stations, collecting detailed survey data is now possible, making it feasible to objectively record the morphology of the channel.  Figure 20. a) Scans from two flume runs. Coarse mix scan is from a run with a flume slope of 14%; fine mix scan is from a run with a slope of 8%. b) Long profiles are generated by averaging 233 points on each cross section. Critical dimensions used with the classification algorithm are also shown scaled to the fine mix bed scan.  For the flume derived data used in this chapter average long profiles (two of which are illustrated in Figure 20) were constructed by averaging the bed elevation across the channel for each cross section using the data from the middle 233 mm of the channel. As the horizontal resolution is 1  51  mm, averaging across the cross section results in an average composed of 233 bed elevation sample points.  3.1.1.5 Other published methods of classifying step-pool channels To examine whether the rule-based, scale-free classification scheme works more effectively than previously published techniques, four published techniques were also examined: (1) minimum slope, (2) zero-crossing, (3) bedform differencing and (4) power spectral analysis. A brief description of each technique is given here; the reader can consult Milzow et al. (2006) and Wooldridge and Hickin (2002) for the details. It is important to emphasize that the techniques were developed for a particular combination of stream size and survey density and that I am applying them outside of their original range. Furthermore, all of these techniques classify the entire channel as step or pool rather than step, pool, or “other.” Using the minimum slope technique (1), a step is identified if the slope between two points is greater than a critical slope of 0.45 (24º) (Milzow et al., 2006). If successive points are all greater than the critical slope, then they are all part of the same step. Milzow et al. (2006) specified that all steps must be less than a critical height (1.2 m) to discourage bedrock steps from being classified as alluvial steps, and they noted that the method should be applied only to steep streams where large pools do not dominate the morphology. I normalized their critical height by the width (6 m) of the Vogelbach, the stream they surveyed (Milzow, 2004), to enable their criterion to be applied to my surveys. The zero-crossing technique (2) works by classifying all points above a best-fit relation as steps and points below as pools (Wooldridge and Hickin, 2002). With bedform differencing (3), the cumulative departure from the previous bedform must exceed a specified value for a new bedform to be identified. The value Wooldridge and Hickin (2002) found to work best for steppools was 0.31 m. To make the threshold parameter scale free, I normalized 0.31 by the width of the cobble-boulder portion of Wooldridge and Hickin’s Mosquito Creek (7 m) to define a critical bedform height. Wooldridge and Hickin (2002) noted that this technique remains rather subjective, as a threshold value must be specified that is greater than the microscale bed roughness but smaller than the height of the bedforms. Power spectral analysis (4) is carried out by Fourier transforming detrended data that have a fixed sampling interval, and the dominant  52  revealed wavelength is considered to be the step-pool wavelength. Thus a step-pool frequency is predicted, but the locations of the step-pools units are not identified.  3.1.2 Classification results  3.1.2.1 Respondents’ results The results from the 11 respondents who replied are broken into three analyses. First, the differences among respondents in how they classified step-pool channels will be presented; second, the results from the respondents are compared with what was classified in the field; and third, the rule-based classification scheme is compared with the results from the respondents and what was classified in the field.  3.1.2.1.1 Differences among respondents Table 9. Classification results of long profiles distributed to respondents  Number of units identified in field  Tees Creek 3  Giveout Creek 8  Results from respondents: Mean number of step-pools 4.4 11.6 Standard deviation 1.6 3.8 Maximum number of step-pools 7 18 Minimum number of step-pools 2 6 Median number of step-pools 4 11 Number more than one-half the respondents identified 3 13  Carnation Creek 5  Flume N/A  1.6 1.3 5 0 1  5.9 2.6 12 3 6  1  4  Results from scale-free, rule-based classification algorithm: Number of step-pools 3 9 1 (3a) a  5  Three were identified with the unfiltered data, see text for details.  Table 9 shows the range of responses in terms of the number of step-pool units that were classified for each of the four profiles. The variability among step-pool researchers was quite high. To further examine how much variability there was between respondents, how each respondent classified each survey point (step, pool, or other) was recorded and then I determined how many of the respondents on average agreed with the modal response. On average the modal response was supported by 81, 64, and 89% of the respondents for Tees, Giveout, and Carnation  53  Creeks, respectively. Hence, on average between 60 and 90% of the respondents supported the average choice of the group. Respondents who identified more step-pool units in one profile also tended to identify more units in other profiles, implying that individuals were using “consistent criteria” between channels.  Figure 21. Variability associated with location of a) end of pool, b) start of pool, and c) top of step recorded by each respondent. Data are from step-pool units identified by more than half the respondents that were also identified by the algorithm. Sample size for each box plot varies between 10 and 17 and averaged 14. Classification breaks are compared to the breaks identified with the algorithm and are scaled using the bankfull width of the channel. The dashed lines indicate error estimates associated with measuring the location of the transition between channel unit types on a sheet of letter-sized paper (21 x 28 cm) using a ruler (error estimated to be 1 mm). The median of the respondents is also plotted to indicate the result of the average respondent relative to the algorithm. Positive numbers indicate respondent results are upstream of the classification algorithm results.  Upon examining the classified long profiles, differences became evident not only in the number of step-pool units that different researchers classified, but also in the extent of the units. Figure 21 illustrates the variability associated with the location of the end of pool, start of pool, and top 54  of step for each respondent as well as the variability of the median respondent. Figure 22 illustrates the variability associated with pool length and step drop height.  Figure 22. Variability associated with a) length of pool and b) step drop height recorded by each respondent. Data are from step-pool units identified by more than half the respondents that were also identified by the algorithm. Sample size for each box plot varies between 9 and 16 and averaged 14. Results are relative to the pool length and step height determined using the algorithm. The dashed lines indicate error estimates associated with measuring the transition between channel unit types on a sheet of lettersized paper (21 x 28 cm) using a ruler (error estimated to be 2 mm). The median of the respondents is also plotted to indicate the result of the average respondent relative to the algorithm.  It needs to be emphasized that both Figure 21 and Figure 22 are based on data extracted from the classification results faxed to me from the respondents on letter size paper (21 x 28 cm) and some of the variability is certainly associated with measuring where respondents identified the breaks in channel morphology. On account of this uncertainty, the average error associated with measuring where the respondents identified changes in channel type has been estimated and is indicated in Figure 21 and Figure 22 using dashed lines. Respondent 10 did not mark the exact location where each unit started and ended; hence no data are plotted for respondent 10. Respondent 8 tended not to identify “other” sections of channel. In a few cases, respondent 7 did not mark the top of step in a clear, consistent manner, and for a few units the end of pool was estimated for respondent 2.  55  Figure 21 and Figure 22 show that individual respondents are not always consistent and that the variance between different respondents is not equal. A number of the respondents on occasion labelled the end of pool as being some way up the reverse sloping portion of the pool (EP2), but not at the downstream crest of the pool (EP1); however, no general bias is evident (Figure 21a). There was considerable variation in where the start of pool was located. Some respondents marked the start of pool as being near the lowest point in the pool where there was a change in slope (Figure 19: position SP1); others labelled the top of the pool as being where the end of pool elevation projected itself onto the upstream profile (SP2). Figure 21b shows that respondent 4 consistently located the start of pool farther downstream than the other respondents, while respondent 11 consistently located the start of pool farther upstream than the other respondents. The location of the top of the step also varied among the respondents. Some located the top of the step as being at the transition from the steep slope to the relatively lower gradient section of channel upstream (TS1), others considered some of the lower gradient section upstream of the step crest to also be part of the step (TS2), and yet others included much or even all of the lower gradient section upstream of the steep portion of the step as being part of the step (TS3). Figure 21c suggests respondent 4 consistently located the top of step farther downstream than the other respondents, while respondents 7 and 8 consistently related the top of step farther upstream than the other respondents. The variability associated with the different respondents in terms of where a unit starts, ends, and goes through the transition from being a step to a pool results in differences in metrics such as the step length, pool length, step height, and drop height — all commonly calculated metrics in step-pool studies. Figure 22 illustrates the variability associated with pool length and step drop height for each respondent as well as the median result from all respondents. Figure 22a demonstrates that the pools identified by respondents 3 and 4 tended to be shorter than the median response of all respondents. Figure 22b illustrates that respondent 11 identified steps that were consistently about one-half as tall as those identified by the other respondents. Clearly step-pool researchers do not classify step-pool channels identically.  56  3.1.2.1.2 Differences between respondents and field results Table 9 shows that for Tees and Giveout Creeks there is a good agreement between the number of units observed in the field and the median number of units identified by the respondents. In Carnation Creek, the results are not as good, probably because the plot that was provided to the respondents had a median filter (n = 5) run through the data. The filtering replaced the elevation of the survey point with the median value of its elevation and the elevation of its four neighbours (two on either side), which resulted in much of the variation in the profile being removed. Interestingly, the one respondent who always classified more units than anyone else did identify all five units that were identified in the field. Based on the three field profiles, no significant difference was found between the expected number of step-pools (based on the field surveys) and the mean and median number classified by the respondents (χ2 statistic, p > 0.05, df = 2). To further assess how well the respondents identified the step-pool units that were identified in the field, the classification assigned to each survey point in the field was compared with the modal response of 10 of the respondents; the results are presented in Table 10. If there was no agreement between what was classified in the field and what was classified by the respondents and each survey point could be considered independent, pure chance would suggest that 33% of the survey data points would be classified the same. Depending on the stream, between 49 and 81% of the survey points were classified the same (Table 10). The likelihood that the modal response of the respondents matched the classification assigned in the field increases as the percentage of respondents who agreed on how a particular point should be classified increased. Table 10. Agreement between respondents and field classification  Tees Creek Giveout Creek; all survey points Giveout Creek; only survey points agreed by more than one-half of respondents Carnation Creek  Percent of survey points classified by respondents that agree with field classification 81% 49% 55% 70%  57  3.1.2.2 Results from applying the scale-free rule-based classification scheme The scale-free classification algorithm described earlier was applied to the long profiles presented in Figure 18 and the results were compared to the modal response of the respondents and to the field classification results. Through a trial and error process, the parameters for each rule were adjusted until there appeared to be a reasonable match between what looked like a step-pool unit and what was classified as a step-pool unit in the field. Initial values were based on the dimensions and hydraulic characteristics of step-pools. Subsequently, the parameters were adjusted a few times to reduce the difference between the number of step-pool units observed in the field and the number identified using the algorithm-based surveys from all 13 streams. No formal optimization was, however, performed as no correct classification exists. Final parameter values are listed in Table 11 and are illustrated in Figure 20. While any one criterion on its own seems sufficiently small to enable small cascade features to potentially be classified as a step-pool, the metrics act together to prevent micro-cascade (Grant et al., 1990) features from being identified as step-pool units. No other method of classifying step-pool channels uses a combination of rules that individually define all parts of the unit. Table 11. Parameters used in rule-based classification algorithm  Maximum step length (fraction of wb) Minimum step length (fraction of wb) Minimum pool length (fraction of wb) Minimum residual depth (fraction of wb) Minimum drop height between top of step and end of pool (fraction of wb) Minimum step slope for a step to be identified  2 0.0225 0.10 0.0023 0.025 mean slope + 10º  The number of respondents who identified each step-pool identified by the classification algorithm is summarized in Table 12. The data show that there is a good agreement between the respondent classification results and the algorithm classification results. In the case of Giveout Creek, three of the four units not identified by the algorithm had drop heights less than the minimum drop height. The fourth unit was not steep enough to qualify as a step using the algorithm. For the flume data, the one step-pool unit not identified by the majority of the respondents was identified by 4 of the 11 respondents. Figure 21 and Figure 22 also illustrate that the location of the end of pool, start of pool, top of step, pool length, and step drop height  58  determined with the classification algorithm match the median classification result from the respondents. Table 12. Number of step-pools that were identified in the field, by the respondents and using the classification algorithm.  Tees Giveout Creek Creek Results from sub-section of reach Number of step-pools identified by more than 50% of the respondents that were also identified by classification algorithm 3/3 9/13 Number of step-pools identified by classification algorithm that were also identified by more than 50% of the respondents 3/3 9/9 Results from entire reach Number of step-pools identified in the field that were also identified by classification scheme 11/18 27/39 Number of step-pools identified by classification scheme that were also identified in the field 11/12 27/36  Carnation Creek  Flume  1/1  4/4  1/1  5/4  21/34 21/25  When the field classification results are compared to the results from the classification scheme, the correspondence is not as strong, but still reasonable (Tables 9, 12, and 13). Figure 23 illustrates the surveyed section of Tees Creek and shows the step-pool units that were identified by the majority of the respondents, by the algorithm, and in the field. Likewise, Figure 24 illustrates the long profile from Giveout Creek and the classification results. Summary numbers illustrating how many step-pools that were identified using one technique were also identified using a second technique are presented in Table 12. For Carnation Creek, when the classification scheme was run on the same length of channel presented to the respondents (60 m as opposed to the 360 m summarized in Table 12); but using the unfiltered data, the scheme identified three of the five step-pool units that were identified in the field. The algorithm predicted the channel type identified in the field for 80% of the Tees Creek survey points, 53% of the Giveout Creek survey points, and 59% of the Carnation Creek survey points. Figure 23 illustrates that sections of the channel may look like a step-pool and be classified as a step-pool using the rule-based approach (e.g., 115 m in the Tees Creek profile), but not be classified as a step-pool unit in the field. The opposite also occurs; sections of channel can be  59  Figure 23. Entire profile of Tees Creek with field, rule-based, and minimum step slope classification results illustrated.  Figure 24. Entire profile of Giveout Creek with field, rule-based, and minimum step slope classification results illustrated.  60  classified as a step-pool in the field, but no evidence for the existence of a step-pool unit can be seen in the one-dimensional long profile. The variation might be related to judgments made during the field observations or the limitations of a one-dimensional long profile. A unit could be missed during the long profile survey or the profile could record a step that was a microrather than meso-scale channel-spanning bed feature. Field surveyors may also key on morphological features other than those corresponding with long profile survey metrics when classifying step-pool units. Table 13. Number of step-pools identified using a range of methods applied to the entire length of four channels.  Classification method Field Scale-free, rule-based algorithm Minimum slopea Bedform differencingb Power spectrumb Optimized orderb rd Zero3 -order polynomial crossing 2nd-order polynomial 1st-order polynomial a  Tees Creek 18 12 18 37 NA 60c 53 27 6  Giveout Creek 39 36 202 89 7 44d 44 40 31  Carnation Creek 34 25 37 62 NA 151e 114 116 78  Flume N/A 7 216 52 3 19d 50 56 37  Technique from Milzow et al., (2006); b Technique from Wooldridge and Hickin (2002); c sixth-order; d fourth-  order; e Eighth-order  3.1.2.2.1 Sensitivity analysis In order to investigate the sensitivity of the rule-based classification scheme to the choice of a particular parameter, sensitivity analyses were performed in which each of the six parameters was varied between one-third of its optimum value (Table 11) and three times its optimum value while all of the other parameters were held constant. The maximum step-length was found to have no effect on the number of step-pools identified and is simply a calculation stop point during the running of the algorithm. Figure 25 illustrates that the rule-based classification technique is relatively insensitive to the residual pool depth and the minimum step length, somewhat sensitive to minimum slope, and quite sensitive to the minimum pool length and minimum drop height used to define a step. The latter two values are important in defining whether flow is plunging into a pool or is simply flowing smoothly over a bar feature into a 61  downstream pool. A steeper section of bed with a larger drop height is more likely to be a step with plunging flow. In Figure 25, the number of step-pool units identified in the field is also plotted. Apparently increasing or decreasing one of the parameters may increase the correspondence between the algorithm results and the field results for one stream, but decreases the agreement for others. To test the sensitivity of the algorithm to the r2 value used if the step is composed of only two survey data points, the assigned r2 value was varied between 0 and 1 and found to have no effect on the number of step-pools identified.  Figure 25. Results of sensitivity analysis performed by varying each of the parameters independently. Dashed vertical line indicates the optimum parameter value. Dashed horizontal lines indicate the number of units observed in the field for Giveout (G), Carnation (C) and Tees (T) Creeks. Minimum step length, minimum residual depth, minimum drop height, and minimum pool length have been normalized by bankfull width.  3.1.2.2.2 Validation analysis During the analysis, a validation series of tests was attempted using 13 long profiles collected by two different field crews. The first eight long profiles (2002 series) were collected in the same manner as the Tees and Carnation profiles (survey points at breaks in slope and every bankfull width), while the second set (2006 series) was collected in the same manner as the Giveout Creek long profile (every 20 cm). 62  For the eight 2002 streams, on average the algorithm identified 43% fewer step-pools than were identified in the field (range was 11-65%). The comparison is thought to be poor because the surveys were not designed to be used to subsequently classify step-pools and the surveys were not particularly dense. Not all of the important morphological features were necessarily identified. Where pools were recorded as occurring in the field, but were not identified by the algorithm, a residual depth was often not associated with the pool. Less frequently the units identified in the field were visible in the long profile, but were small and thus not identified by the algorithm. The 2006 series of profiles consists of four short reaches (25-80 m long) of Deeks Creek and one 220-m-long reach of Shatford Creek (see Figure 33). These data yield more favourable validation results. At Deeks Creek for reaches 1 through 4 and at Shatford Creek, 11, 6, 9, 2, and 14 step-pool units were identified during the field survey, respectively. The algorithm identified 8, 8, 7, 2, and 21 units, respectively. The algorithm results are thought to have corresponded better with what was observed in the field because the survey data were collected at a finer spatial resolution. The relatively large discrepancy with the Shatford Creek data (33%) is difficult to explain; on a number of occasions the long profile shows what clearly looks like a step-pool yet it was classified as “other” in the field. A fundamental problem with attempting to validate the classification algorithm is that no objective field method exists. Hence no classification is correct. Even experts were shown to produce widely varying results (Section 3.1.2.1.1). Thus, while the two sets of surveys produce different validation results, one cannot readily be considered more accurate than the other. It should also be emphasized that the purpose of collecting the profiles was initially to record bed slope and step-pool dimensions; only after the data were collected was using the profiles to objectively classify step-pool channels considered.  3.1.2.3 Other published methods of classifying step-pool channels The four published classification techniques, with the exception of the power spectrum technique, were applied to the entire length of the three field data sets and the one flume data set. The power spectrum technique was applied only to Giveout Creek and the flume data as only these surveys had a fixed sampling frequency. In both cases, the sampling frequency was less  63  than the D50 and both micro- and macroform roughness was measured. Wooldridge and Hickin (2002) used a sampling frequency slightly greater than the D50. For both the zero-crossing and the power spectrum technique, a best-fit line needs to be fitted through the data. Using the method proposed by Wooldridge and Hickin (2002), which is based on recommendations by Chayes (1970), I increased the order of filtered polynomial until the pvalue calculated for the F-statistic associated with the additional factor exceeded 0.01. A pvalue > 0.01 indicates that a higher order polynomial regression function does not substantially reduce the residual variance that is associated with a function of one less order. Applying this criterion to the flume, Giveout, Carnation, and Tees Creeks data, fourth-, eighth-, sixth- and fourth-order polynomial relations were fit, respectively. Table 13 illustrates the number of step-pool units identified by the four published techniques, the number identified in the field, and the number identified using the rule-based classification scheme. In two cases, the slope-based criterion of Milzow et al. (2006) came close to identifying the same number of units as identified in the field. However, the same technique substantially over-classified the two surveys with a high density of survey points. The overclassification occurs because the minimum step slope technique was designed to be applied to surveys that recorded only major breaks in slope and not the minor details of the bed. In contrast, Giveout Creek was surveyed every 20 cm along the bed of the channel and thus a boulder could be surveyed many times. Using this approach the minimum slope technique could result in the steep section of the boulder being classified as a step. The proposed rule-based method limits the chances of a steep section of a boulder being classified as a step as the minimum pool length, step length, residual depth, and step height work together to ensure that only mesoscale bed features (Hassan et al., 2008) are identified. Figure 23 and Figure 24 illustrate the results from the slope classification technique for Tees and Giveout Creeks. In the case of Tees Creek, the agreement between the field- and the slope-based techniques is acceptable; however, in the case of Giveout Creek, some boulders clearly are being classified as steps. To investigate the effect of survey density, the Giveout Creek data were subjectively subsampled to include only the survey points that would likely be recorded if the bed had been surveyed in a manner that identified only major breaks in slope. The subsampling was achieved by utilizing 64  the field step-pool classification results to identify survey points most likely to be associated with major breaks in slope and resulted in a survey in which survey points were located on average every 1.4 m. When the minimum slope criterion was applied to these data, 44 step-pool units were identified, which is similar to the number of units identified in the field. Both the zero-crossing technique and bedform differencing technique substantially over classify the long profiles (Table 13); they identified more than twice as many step-pools as were identified in the field. If lower order polynomials are used, the number of step-pools identified using the zero-crossing method is generally reduced, but a poor agreement remains between the number of step-pools identified in the field and the number predicted by this technique (Table 13). The power spectrum analysis of Giveout Creek indicates a clear peak at 29 m, and the flume data have a peak at 1.37 m. Both of these values are well above the actual step-pool spacing, and as a result, the number of step-pool units is under-predicted using power spectrum analysis (Table 13).  3.1.3 Discussion The comparison between the field-derived classification and the results from the respondents illustrates that the capacity of long profiles to fully capture what is observed in the field is imperfect but, on the whole, most of the units observed in the field are also visible in the long profile data. Considerable variability among respondents was also evident. The application of four published channel classification techniques shows that these techniques do not work well in all cases. They all classify the entire channel as either steps or pools and are unable to classify an “other” category to describe reaches of plane bed or cascade morphologies. They can also be sensitive to the density of survey points and the size of the channel. The critical slope technique used by Milzow et al. (2006) was found to work well for data collected at the correct spatial density, but poorly if dense survey data were used. The proposed rulebased method incorporates the minimum slope idea of Milzow et al. (2006) while overcoming the survey density limitations. The power spectrum and zero-crossing techniques imply that the location of step-pools is systematic and regular; yet Figure 23 and Figure 24 show no obvious regularity, and mechanisms of step-pool formation are likely to be linked to keystone location and to exhibit a degree of randomness (Zimmermann and Church, 2001).  65  The scale-free, rule-based algorithm was explicitly designed to be applied to data collected over a range of survey intensities and to work on both flume and field channels. Thus to test and develop the technique, long profiles were selected to encompass a wide range of long profile survey densities, from a point every 0.006 bankfull widths to survey points only at major breaks in slope (≈ 0.5 wb). Over this range, the technique was shown to work reasonably well. While there are undoubtedly some step-pool units that will be misclassified, the fact that the method will always classify a long profile the same way is felt to outweigh this drawback. If a long profile is classified by an individual, the results are no more precise; yet they are likely to be affected by the biases of the individual, making the results more difficult to interpret than those coming from the application of the rule-based algorithm. Furthermore, results from different researchers applying the same algorithm are directly comparable, while subjective field or long profile classification results from different researchers are unlikely to produce the same result and their data are best not compared. While completing the validation and calibration portion of the analysis, it became evident that there is as much uncertainty regarding how correct the field classification results are and how well a single long profile can characterize the bed as there is regarding how well the algorithm works. A formal calibration was not completed as there is no means of creating a “correct” classification of a step-pool channel. In some respects, the median classification result of the respondents may be the most correct classification result; in other respects, the field-based classification results may be more correct.  3.1.4 Summary It was demonstrated that the variability among step-pool researchers as to where and how a steppool channel unit is identified is substantial. In recognition of this subjectivity a series of scalefree geometric rules were developed. These rules include a minimum step length (2.25% of wb), minimum pool length (10% of wb), minimum residual depth (0.023% of wb), minimum drop height (2.5% of wb), and minimum step slope (10º greater than the mean bed slope). The rules were shown to perform as well as the median response of 11 step-pool researchers and to perform better than other published techniques. Thus the rules were used to identify step-pool units in all 319 bed scans.  66  3.2 Measuring sediment transport rates and grain size Observing the size, volume and timing of bedload exiting a flume has generally been difficult and usually consists of weighing sediment that exits the flume at set intervals either by swapping buckets at the outlet or by using a recording weighing device. Typically, to determine the grain size distribution of the bedload the sediment is subsequently dried and sieved. The temporal resolution of such data is generally limited to thirty second periods or longer and the time needed to dry and sieve the samples is considerable, often limiting the frequency with which experiments can be completed. Philippe Frey at Cemagref in France developed the first system that tracked bedload with a video camera (Frey et al., 2003) and coined the name GSD (Grain size distribution and Solid Discharge). With their system 640 x 480 pixel images were collected in bursts (e.g. they collected fifteen images per second for two seconds every eight seconds). The images were then post-processed using an algorithm to size and enumerate the particles. To determine a sediment transport rate a mean particle velocity is required, which they determined by manually matching particles in sequential images. By utilizing the concept presented by Frey et al. (2003), advances in video data acquisition hardware and processing algorithms that are built into LabView TM software, I improved the GSD system. With the new system, described herein, video images are continuously captured and particle velocity measurement has been automated, allowing the bedload transport rate and grain size distribution of the bedload to be measured every second. In addition, owing to the higher resolution cameras that can now be utilized, a wider range of grain sizes can be accommodated. The technology provides a significant improvement over the traditional bucket/sieving approach. The objective of this chapter section is to present the improved GSD.  3.2.1 Equipment The GSD (Figure 26) costs about $7900 CDN to assemble and is primarily composed of a video camera, computer, software and a light table (Table 14). The video camera I utilized is a Pulnix 1402CL Cameralink TM with an 8-bit greyscale resolution of 1392 x 1024 pixels and a maximum frame rate of 30 fps.  67  Figure 26. Conceptual perspective drawing of the flume outlet and GSD illustrating the screen and valves used to separate some of the water from the bedload along with a photo of the light table. Only two of four valves are shown for clarity. Sketch is not to scale; light table is 60 cm wide. Table 14. Equipment and costs of GSD setup in Canadian dollars  1392 x 1024 Pulnix CCD Cameralink video camera (4 times the resolution of most video cameras) National Instrument Cameralink video acquisition card Suitable computer (anything with SATA RAID 0 configured hard drives) Hard drives (70 GB per hour of experiment, process overnight and delete, depends on need for continuous run time) LabView Software with Vision module Light table and additional hardware (light table is lit by eighteen 10 W 12V halogen bulbs powered by discarded computer power supplies) Total cost  $3000 $1000 $1000 1500 GB cost $200 $2000 (Academic pricing) $500 $7900  The width of the light table needs to be appropriate for the size of particles in transport and the resolution of the camera. As I was interested in tracking grains between 2 mm and 181 mm a light table 60 cm wide, yielding 0.44 mm pixels, was used. In theory 0.44 mm pixels should have yielded at least 12 pixels per 2 mm particle; however, a 2 mm grain consisted of about 7 pixels due to light diffusing around the particle. Seven pixels per particle is lower than the approximate minimum of 20 pixels per particle that is suggested in order to adequately identify 68  particles: the finest grains were observed to be under-sampled. With a 20 pixel round particle, the average diameter of the particle is 5 pixels; thus an additional pixel would add 20% to the apparent diameter of the particle. If the particles are composed of even fewer pixels, the addition or removal of a single pixel has an even larger effect on the size of the measured particle, and this should be avoided. A higher resolution camera enables a larger range in grain sizes to be monitored. I found that the opaqueness of white plexiglass can vary considerably and affects the results. While the identical type of plexiglass was purchased for both the prototype and final version of the light table, the second sheet was considerably more transparent and individual lights were plainly visible through the plexiglass. To overcome this limitation, a sheet of Mylar was pasted to the underside of the plexiglass. Sheets of copper are used in the transition from the flume to the light table since copper is quite malleable and waterproof joints can easily be soldered together with a propane torch. The choice of lights to be used within the light table is very important as it is critical that the lighting not vary with time. Any lighting that was connected directly to wall 120V AC had a 1-2 Hz flicker that made it difficult to process the images. To avoid this problem eighteen - 12V 10W DC halogen lamps were installed in the light box. The power for the lights is supplied from abandoned computer power supplies, which provide a stable source of 12V power. In total 6 power supplies are used, each one connected to three halogen light bulbs. The images were saved to a SATA RAID 0 disk array consisting of two hard drives. The disk array can continuously record 29 1.4 MB 8-bit tiff images per second. Under normal operating conditions, I captured 15 to 20 frames per second. While it is possible to process the images while they are being collected, I found that when there are more than about 20 particles per image, the processing algorithm slows down causing the frame rate to drop. Thus the images are saved to file and after the experiment is finished they are processed and subsequently deleted. Permanently saving the primary images requires unreasonably large storage volumes.  3.2.2 Design considerations Considerable attention needs to be paid to the design of the flume outlet and light table. Based on my experience the slope of the light table should be adjustable from a few percent to twenty  69  five percent and particles should be kept in motion at a velocity of about 1.0-1.5 m/s by varying the slope of the light table and the amount of water flowing over it. These particle velocities require supercritical flow which will ensure that all the particles travel at the same velocity and do not stall on the light table. I found that to accommodate high flows (greater than about 25 L/s for my setup) some of the water coming out of the flume needed to be routed away from the light table. To route water off the light table I installed a 2 mm screen in the last 30 cm of the flume and added four 10 cm PVC pipes to the end of the flume. These pipes have valves that were adjusted to regulate how much water flows through the screen, bypassing the table (Figure 26). Ideally about 12-20 L/s of water flow over the light table. This arrangement works for flows up to about 50 L/s or about 90 L/s/m on a 0.6 m wide light table. Higher flows are not effectively managed and large standing waves form on the light table causing some wave details to be classified as particles. When operating the flume at flows of a few litres per second with all of the valves closed it was observed that the 2 mm screen was sufficiently rough to stall sediment and it was necessary to cover the screen with a smooth sheet of metal. When the operating width of the flume was narrowed to 25 cm and water diverged at the flume outlet standing waves formed along the edge of the light table as the water reflected off the walls along the edge of the light table. As a result particles were propelled towards the sides of the table. Ideally particles are primarily moved down the centre of the light table to avoid wall drag effects (Figure 27b); thus to reduce the lateral movement of particles, metal guides mounted parallel with the flume were installed at the flume outlet.  3.2.3 Image acquisition The target frame rate was usually specified to be 15 fps and could be increased to 20 fps if particles were moving especially fast and the same particle could not easily be tracked visually on the screen during acquisition. The actual frame acquisition rate was determined by the computer and not set to a fixed interval. It usually ran at the target frame rate but, on occasion, the frame rate dropped when other processes tied up computer memory. Each image file was named using its acquisition time with 1/1000 of a second precision. To account for variations in light intensity across the light table (as is evident in Figure 28), an image was collected of the light table prior to each run and all the subsequent images were subtracted from this image. Capturing a background image at the start of each run controls for 70  Figure 27. Examples of particle velocity as a function of position across the light table during low (a, 3.8 L/s) and high (b, 17.4L/s) flow runs, area (c) and minor axis (d). Each plot illustrates 32000 particle velocity measurements. These were made during 16 minute and 60 minute long experiments for plots a) and b) respectively.  any changes in plexiglass opaqueness and bulb aging. I found that the sizes of the stones that the algorithm recorded were quite sensitive to light intensity. In particular, if the lights become too dim, the particles are less obvious and the outer edges of the particles were not recorded. While it is possible to saturate the entire image, causing the image to appear entirely white, saturating the image was found to cause the smaller particles not to be detected. Experience suggests that the electronic gain on the camera should be set so that the brightest areas in the image are not quite saturated (pixel value should be just under 255). It was also found that configuring the video image acquisition algorithm to adjust the gain so that the mean pixel value (averaged over 71  200-2000 images) was maintained within a narrow range (197-203) helped account for lighting changes caused by varying water depths and changes in room lighting. To assign dimensions to the pixels an image was taken of a calibration template that consisted of a grid of points. A perspective calibration was applied and the calibration coefficients were then applied to all subsequent images.  Figure 28. GSD image: variation in lighting across the light table is evident in larger image. The effect of subtraction, applying a threshold and filtering on a particle is illustrated in three small images, each of which has the same scale. The small images illustrate the effect of particle dilation.  3.2.4 Data analysis In order to develop the particle detection and segmentation algorithm a series of trial runs was completed during which the threshold for particle detection, filter selected and number of dilations were modified and the results evaluated. The goal was to have a fixed procedure that worked reasonably well across a range of flow conditions and, if necessary, correct for any introduced biases through a subsequent calibration procedure. The main challenge faced was finding a procedure that worked acceptably at high flows when waves and bubbles were common, but also worked at low flows when the images were nearly perfect. Figure 29 illustrates the steps used in the procedure that was finally adopted. Each image was subtracted from the background image and the calibration information applied. Then a 3 x 3 number 15 gradient kernel (Perwitt Kernel; National Instruments, 2005) was applied to 72  Figure 29. Schematic diagram illustrating steps in image analysis.  emphasize particle edges. Next the image was converted into a binary image using a threshold range of 150 to 255 to define particles (pixel values range from 0-255). A fixed threshold was applied rather than a floating threshold as it was found that a floating threshold tends to identify bubbles and waves at high flows and had difficulty dealing with both high and low sediment transport rates. Subsequently any particles touching the edge of the image were removed. The outer edge of the particles, which are lighter due to light diffusing around the stones, was often excluded when the threshold was applied since the threshold was set relatively high to reduce the chance that waves and bubbles were identified as grains. To correct for this bias, two dilations of the particles were applied; each dilation adds one layer of pixels around each grain. Dilating the particles also helped smooth the particles, which can become somewhat blocky after the threshold was applied (see Figure 28). The dilation caused some particles to become larger than they actually were, which was corrected during the calibration procedure. Next, any holes in the particles were filled. The particles must be larger than 10 mm2 in projected area and smaller than 25,000 mm2 to be included (that is, having grain diameter between about 3.5 mm and 180 mm). The minimum of 10 mm2 removes some of the smallest particles, but was required to stop small bubbles from being enumerated. Finally the x and y coordinates of each particle, its area and equivalent ellipse minor axis were recorded and these data, along with the time the image was recorded, were saved to a text file. The equivalent ellipse minor axis was computed in LabviewTM using 2  2  Per 2A P 2A xminor = + xs − er 2 − xs 2 π π 2π 2π  (12), 73  where Xminor is the minor axis, Per is the perimeter and Axs is the cross sectional area of the particle (National Instruments, 2005). Increasing the threshold from 150 to 180 in 10 unit increments showed that the total amount of material measured with the GSD decreases as the threshold is increased, but the grain size distribution did not change markedly. In order to calculate the velocity of the particles, images with fewer than 11 particles were selected. The five largest grains greater than 4 mm in diameter that have their center separated from other stones by a minimum of 30 mm in both the x and y directions, and were further than 5% of the width of the light table from the edges of the table were located. Their location and size were recorded. These particles were matched with particles that met the same conditions from the subsequent image and, if a further set of conditions was met, a match was considered to occur and particle velocity was determined using the distance between the centres of each particle and time elapsed between images. The additional conditions that must be met for a particle pair to be considered a match are as follows:  •  The lateral position on the light table of the potentially matching particle must differ by less than 5% of the table width.  •  The potentially matching particle must have moved far enough for a minimum velocity of 100 mm/s to occur. The use of a minimum velocity prevents different particles that occupy the same place on both images from being counted as a pair.  •  The potentially matching particle must not have moved so much that the calculated velocity will be more than 2500 mm/s.  •  Based on a log base 2 scale, the diameter of the potentially matching particle must not differ by more than 10% between the two images.  Next the data were summarized by calculating the running average particle velocity using a sampling window of thirty seconds (± 15 s). A thirty second sampling window constitutes an average from about 450 images (i.e. 30 x 15/s). If fewer than ten velocity measurements existed 74  within a thirty second window, the sampling window was enlarged in two second increments until at least 10 velocity measurements were encountered. The average velocity in conjunction with the time between images was then used to define a portion of each image that records a unique slice of time during which particles were sliding over the table. Figure 30 illustrates the sampling window for three sequential images. As the time between images changes, the window size was changed to accommodate the amount of time that passed between the images. Grains with their centres in the sampling window were included in the final bedload transport data. Application of this procedure to successive images provides a continuous record of bedload transport with no overlap despite having images that capture the same grains multiple times. The window was offset from the upper and lower edges of the image to ensure that large stones were not under-sampled in comparison with smaller stones as a result of them being more likely to have been touching the edge of the image and thus removed.  Figure 30. Example of sampling window for three sequential images. Dark grey indicates portion of each image that were excluded to prevent grains near the edges from being sampled. Light grey indicates the sampling window for each image. A particle velocity of 1200 mm/s has been assumed for demonstration purposes. The elapsed time for the third image was determined using the time the fourth image (not shown) was captured.  On occasion, within any single second the time between images may be sufficiently long that the prescribed sampling window is taller than the portion of the image that can be sampled. When this happened the total transport for the second was increased based on the amount of time that was missed and the amount of transport that was recorded during the second. Typically less than 1% of a run was missed. If frames were missing for more than one second, the average transport rate for the previous ten seconds of the run was used and the data were flagged as extrapolated.  75  There was the potential for up to 5 grain velocity measurements per image or about 3000 velocity measurements per thirty second sampling period. Generally there were a few hundred velocity measurements per thirty second sampling period. A large sample size was sought as the variability in velocity measurements tends to be considerable. Typical velocities were on the order of 1 m/s with a standard deviation of ± 0.4 m/s. Provided discharge rates remain constant, the variation in velocity over the duration of an experiment was minimal, and dwarfed by the variability in velocity that occurs within any one 30 s sampling period. Scatter plots illustrate that there was no systematic change in particle velocity with either particle area or minor axis (Figure 27c and d). Furthermore velocities tend to be consistent across the width of the light table (Figure 27a). A decrease in velocities can be observed along the walls at high flow rates (Figure 27b) and the water and light table should be managed to avoid this. The velocity decrease was compensated for as the slower moving particles were used along with the faster moving particles to calculate the thirty second running mean velocity. If during a particular sampling period more particles move along the walls where their velocity is reduced, the mean velocity will be reduced, decreasing the sampling window and thereby compensating for the reduced particle velocities. Remaining biases introduced due to variable velocities across the width of the table at high flows are considered to be minimal in comparison to other challenges introduced at high flows such as bubbles being classified as particles and waves forming over particles that distort their dimensions. For these reasons one average velocity from a thirty second period was applied across the light table. Particle velocities were generally observed to increase as flow rates increased. As the video records the area (Axs) and the equivalent ellipsoid minor axis (Xminor) of the particles a conversion to weight based on area and b-axis is required. Based on trial runs that involved sliding stones of known dimensions (measured with callipers) and weights across the light table the weight (Wt) in grams and b-axis (baxis) of each grain in millimetres were calculated using the following equations, which are the best-fit linear regression relations through the data plotted in Figure 31. Wt = 0.00106 A1xs.50 ; r2 = 0.95, p < 0.001, n = 199  (13)  baxis = 1.19 X min or − 1.05 ; r2 = 0.95, p < 0.001 , n = 458  (14)  76  The weight versus area relation is based on grains larger than 11 mm and is shown in Figure 31a to be reasonably precise. The standard error range for the constant is 0.00119-0.00095, while the standard error range on the exponent is ± 0.01. The standard error of the estimate is 0.119. The relation between b axes is more variable (with standard errors of ± 0.52 and ± 0.013 for the constant and intercept respectively and a standard error of the estimate of 8.45), particularly for larger grains. The error associated with the large grains occurs as a result of the stones rolling down the light table. As the particles roll, the projected minor axis that is visible in the images varies between the b and c axes. Thus the amount of variability observed when measuring the minor axis depends on the difference between the c and b axes, which is greater for large grains.  Figure 31. a) Relation between area of particle measured on light table and weight of particle and b) ellipsoid minor axis measured on light table and measured b axis using callipers. The linear regression relations and 95% confidence intervals are shown.  If one assumes that the ratio of the a to b axes is the same as the ratio between the b and c axes (which was the case for the 40 stones used to calibrate the GSD; Student t-test, H0 mean not different from 0, p = 0.16) and the formulas for the volume and area of an ellipsoid are utilized it can be shown that  ⎛ 4 ρ s ⎜ Axs ⎜ Wt = 6 3 π 10 ⎜⎜ a axis baxis ⎝  3  ⎞2 ⎟ ⎟ ⎟⎟ ⎠  (15),  77  where ρs is the density of sediment, assumed to be 2650 kg/m3. Eq. 15 reduces to Wt = 0.00123Axs3/2 using a axis / baxis = 1.38, which was the mean value for the 40 stones that had their a, b and c axis measured and were used to calibrate the vision system. Relation 13, developed using the video camera, is not significantly different from the semi-theoretical relation (15). These results confirm that video particle analysis can reasonably measure the shape and weight of grains. The video tracking as described yields un-calibrated grain size specific bedload transport information every second.  3.2.5 Calibration of GSD To calibrate the GSD the material caught in the stilling basin at the end of the flume was compared with the cumulative weight and grain size distribution of the material that was recorded with the GSD. In total 13 calibration samples were sieved and 15 had their total weight recorded. The GSD on average estimated 39% more sediment to be transported than was directly measured (95% confidence interval = 30% - 48%). The over-prediction of sediment in transport is thought to be due to dilated grains appearing larger than they actually were, which affects small grains more than larger grains, and due to the occasional bubble or wave being interpreted as a stone. There is also some uncertainty introduced by the regressions between area and weight and the conversion of observed minor axis to grain b-axis. The insert in Figure 32a illustrates the grain size distributions of the 13 samples used to calibrate the GSD. The mean D50 of the samples (based on weight by sieve) was 7.9 mm. The D50 ranged between 4.9 and 12.4 mm while the material in transport ranged from 2 to 64 mm. The GSD design used during these experiments can accommodate grains up to 181 mm, but was not calibrated for such large grains as stones larger than 64 mm never exited the flume during the calibration runs. Figure 32a and Table 15 compare the amount of sediment in each class actually measured after sieving the sample with the amount measured with the GSD. A systematic bias exists, which may in part be due to the difference between the calliper and sieved results. In all cases, the difference were significant (one sample t-test, α = 0.05). Grains in the 2-2.8 mm and 11-16 mm grain size classes were under-represented, while grains in the 2.8-11 mm grain size classes and grains large than 16 mm were over-represented.  78  Table 15. Grain size estimation statistics based on calibration and validation results.  Grain size class (mm) 2-2.8 2.8-4 4-5.6 5.6-8 8-11 11-16 16-22 22- 32 32-45 45-64 2-2.8 2.8-4 4-5.6 5.6-8 8-11 11-16 16-22 22- 32 32-45 45-64  Calibration results Mean S.D. (%) p-value # runs Mean # grains Maximum & difference (%) minimum # grains -61 8 <0.001 13 46293 114806-4685 79 32 <0.001 13 119125 416747-12141 126 28 <0.001 13 63238 246913-5702 80 25 <0.001 13 18441 73384-1524 30 21 <0.001 13 4397 17193-368 -21 28 0.02 13 1127 3614-67 13 16 0.01 13 866 2844-61 115 96 <0.001 13 229 720-16 110 93 0.002 11 48 130-1 261 217 0.03 6 3 12-0 Validation results after applying calibration constants 2.70 15.5 0.60 8 54373 176298-6208 -12.41 14.9 0.05 8 136416 447728-13151 -12.15 20.3 0.14 8 71351 250-859-4708 -2.23 26.9 0.82 8 21793 75192-1019 -3.87 29.8 0.72 8 5009 17119-148 -8.34 23.5 0.35 8 1147 3856-31 23.40 30.6 0.07 8 1001 3347-24 26.45 50.8 0.18 8 228 845-5 13.50 66.9 0.68 5 25 93-0 1 3-0  Figure 32. Box plot illustrating error in GSD-estimated sediment transport rate by grain size class based on (a) 13 calibration samples and (b) 8 validation samples after calibration coefficients were applied. The insert in (a) illustrates the grain size distributions of the 13 samples used in calibration tests of the GSD.  79  Based on paired sample t-tests (data in phi units, α = 0.05) the predicted D50 and D75 of the 13 calibration samples did not differ from what was measured by sieving. Conversely, there were statistically significant differences in the tails of the grain size distribution (D16, D25, D84, D90 & D95). The mean difference was consistently less than half a phi class for all Dx quantities (Table 16). Table 16. Results of paired t-test comparisons of Dx calculated using the GSD and sieved grain size distribution for the calibration samples (n = 13) and validation samples (n = 8).  Calibration results Dx  D16 D25 D50 D75 D84 D90 D95  Mean increase in Dx (ψ) 0.30 0.24 -0.09 0.06 0.15 0.20 0.17  SD (ψ)  p-value  0.08 0.13 0.27 0.26 0.14 0.06 0.08  <0.001 <0.001 0.25 0.4 .002 <0.001 <0.001  Validation results after applying calibration constants Mean increase in Dx SD (ψ) p-value (ψ) -0.01 0.12 0.80 -0.01 0.21 0.93 0.07 0.18 0.32 0.05 0.15 0.37 0.06 0.13 0.20 0.01 0.13 0.83 -0.05 0.20 0.49  For each grain size class the mean percent difference (Table 15) between the GSD and the sieved result was used as a calibration coefficient and applied to all subsequent GSD data. After applying these coefficients to the calibration samples the total weight predicted by the GSD for the 13 calibration samples was on average 0.1% under predicted and had a standard deviation of ± 12%.  3.2.6 Validation tests Based on 206 validation samples the percent difference between the mass measured with the GSD after applying the calibration coefficients and the actual weight of the sediment averaged only four percent. The percent difference had a standard deviation of 42% and was not significantly different from zero (p = 0.14). Of the 206 validation samples, 8 were sieved to validate the GSD’s ability to measure the grain size distribution. During the validation tests material up to 45 mm was transported. Figure 32b and Table 15 illustrate that the errors were fairly well distributed about all of the grain size classes. Thus the calibration coefficients remove most of the grain size specific error. The values of the D16, D25, D50, D75, D84, D90 and D95 determined with the GSD and by sieving show no statistically significant difference (Table 80  16, Dx in phi units, validation and calibration results combined, n = 22, p > 0.05). These results imply that following calibration even if the GSD does not predict the correct total weight of the sample, it does predict the correct grain size distribution. Additional analysis showed that the total error was not related to discharge. The predicted sample weight can easily be corrected by calculating the ratio between the bucket weight of the sample and the calibrated GSD derived weight and applying this ratio to all grain size classes.  3.2.7 Summary The GSD provides an effective means to monitor the solid discharge and grain size distribution of sediment leaving the flume. It also eliminates the need to dry and sieve samples and thereby dramatically increases the efficiency with which experiments can be run. The flume and GSD were often run for 6-8 hours a day every day of the week. The GSD provides high temporal resolution bedload transport data and has already produced some impressive results (Recking, 2006). The technique is however limited in the range of grain sizes that can be monitored and the flow rates that can be accommodated. Each grain should be composed of at least 20 pixels to be adequately imaged. The finest particles were composed of only 7 pixels, and as a result were under-sampled. My observations also show that the transport of grains out of the flume can be spatially heterogeneous, necessitating that the entire light table be imaged rather than just a subsection of the table.  81  4 Experimental idealism Unlike the other chapters in this thesis, this chapter is purposely written in a narrative style. The reason is two fold. First, when I started to think about doing experiments on step-pools, I wanted to know what other researchers observed to work and not work; in addition to knowing what they found. Second, when discussing the formation and destruction of step-pools and the controls on their development, some observations are strictly qualitative and it is useful to distinguish the qualitative interpretations from the quantitative data that are detailed in subsequent chapters. In this chapter I discuss how I arrived at the methods used to make step-pools and detail observations from the experiments. I also present the range of bedforms and flow conditions that were encountered. Where appropriate, supporting data will be presented. I hope the reader will find my observations interesting, gain some insight into how I approached the modelling of steep streams, and how they could improve the experimental design should they choose to perform similar experiments.  4.1 Preconceived notions Prior to this project I was a field geomorphologist, who had dabbled with some numerical sediment routing. I came to the idea of physical modeling with excitement; I would be able to control the outcome and observe how things worked as I could vary the governing variables. However, I also had a serious concern that step-pools had not and could not be reproduced in flumes. The later concern came out of my appraisal of other studies coupled with my observations from step-pool streams in flood. The sense I had from reading the literature was that many of the previous researchers who had made step-pools had constructed armoured channels with linear stone clusters, but I did not see convincing evidence that water was actually plunging over stable steps into pools. I was uncertain, and to some degree remain uncertain, whether during past studies rapid (or plane-bed) morphologies were more commonly developed than step-pool structures. In some studies (e.g. Whittaker and Jaeggi, 1982; Curran, 2007), my concern arose from the mobility and high sediment transport rates that occurred during the experimental runs (see Section 1.1.2). These transport rates do not agree with my observations 82  of natural step-pool channels. The step-pool channels that I have visited in the field and studied during large flood events have retained stable steps with mobile populations of sands, gravels and cobbles moving through the channel (East Creek, see Figure 2 and Shatford Creek, see Zimmermann and Church (2001)). It is visually obvious that the channels are extremely armoured and that the large boulders are locked together and have been in place for some time. I was, and remain, unsure if flume experiments that are completed with a mobile population of large stones during the entire run produce the same kind of steps and behaviour as beds that are well structured and have large stones which move only rarely. My sense at the beginning of the project was that steps form or, at the very least, are structurally modified during periods of high flow when sediment transport rates are not particularly high. During these conditions the bed is scoured and structured (Chin and Phillips, 2006). For this reason, past studies that ran high flows and high feed rates and observed mobile bedforms that persisted for relatively short periods of time do not agree with my notion of how sediment transport behaves in step-pool channels. I consider the features produced in such studies to either represent early stages of step-pool formation, or to be mobile bedforms akin to dunes or antidunes. Curran (2007) in fact notes that dunes were present in her experiments, especially at high sediment transport rates. None of her runs were conducted without sediment feed; thus it is unclear if the beds stabilized and structured in the manner they would have had there been some periods when there was no sediment feed. There is little doubt in my mind that at sufficiently high transport rates dunes or antidunes can form and migrate along the channel; what I am less certain of is whether these have anything to do with the step-pools we observe in nature. Fundamentally I believed that to form step-pools you need periods of low sediment transport when the bed can structure along with periods of high sediment transport when mobile grains can jam and structure. With these rudimentary ideas about the formation of step-pools I set out to try to see if step-pools could be made and sustained in a flume.  4.2 Froude scale modeling of step-pool streams The initial configuration and flow rates for the flume experiments were based on a 1:20 Froude scaled model of Shatford Creek. Once these experiments were completed flume width and gradient were changed to fill out the parameter space selected for study without scaling the model to any particular prototype. When switching to a coarse sediment mix a 1:8 scaled 83  version of the surface grain size distribution from the Rio Cordon was used. I used this grain size distribution in the hope that the model could be compared to the Rio Cordon observations and because it replicates the approach used by Comiti et al. (2007a). Early during the experiments I was sceptical that Froude scaling reproduced the observed behaviour and was particularly concerned about the apparent reduction in turbulence intensity, which I attribute to the reduction in the Reynolds number (see Table 2). In the next few paragraphs I critically examine how well the model represented the two prototypes. These two prototypes were used as detailed flow, bed and grain size information was available for these streams during flood flows.  4.2.1 Prototype 1: Shatford Creek The initial configuration of the flume was based on a 1:20 Froude scale model of the flows in Shatford Creek (Figure 33), a step-pool stream near Penticton BC. The flows were scaled, rather than the sediment flux, since variable low sediment transport rates and stable channel geometries are expected in such streams. The scaling is based on a 220 m long section of channel that has a mean gradient of 7.8% based on a total station survey completed in 2006. The reach includes reaches 3 and 4 from Zimmermann and Church (2001) that were surveyed in 1998. Based on field observations and the survey data plotted in Figure 33, little changed between 1998 and 2006. During the same period the largest instantaneous peak flow recorded at the Water Survey of Canada gauge (08NM037), located eight kilometres downstream of the study site, had a return interval of five years (based on 30 yr of record). The return interval data suggest that Shatford Creek will not destabilize during a flow of Q5. During the 2006 survey a 957 stone pebble count (Wolman, 1954) of the surface grain size distribution of Shatford Creek was completed and the stones were binned into step (D50 = 630 mm, 157 stones), pool (D50 = 150 mm, 219 stones) and tread (D50 = 253 mm, 616 stones) channel units. The reach average D50 was 270 mm (see Figure 16 for grain size distribution). In most flume studies the sub-surface grain size distribution of the prototype is scaled and used as the bulk sediment grain size distribution in the flume. This approach was not possible since there is no obvious means of determining the sub-surface grain size distribution in a step-pool stream on account of the large boulders that are present, nor is it likely that the subsurface grain size distribution represents the channel forming grain size distribution on account of the armoured and structured nature of the channel. This knowledge provided a quandary: what  84  distribution should be used to achieve an armoured bed grain size distribution similar to the one observed in Shatford Creek. I knew that the distribution needed to be finer than the surface grain size distribution of Shatford Creek and decided to try using the surface grain size distribution of East Creek (Figure 16). This distribution is finer and I was hoping that a mix composed of the East Creek grain size distribution would coarsen to the surface distribution of Shatford Creek.  Figure 33. Plan view map and long profile of study section along Shatford Creek used as the initial prototype for the flume. 1998 long profile surveys of reaches 3 and 4 from Zimmermann and Church (2001) are also shown.  In order to restrict the sediment used in the experiments to gravel sizes that were not readily suspended and that could be tracked with the GSD, only material larger than 2 mm was used in 85  the flume. The inclusion of material only larger than 2 mm meant that the material in the field smaller than 40 mm was not included, which amounted to 14% of the surface sediments. Initially the grain size distribution was truncated at 2 mm and a mix with a mean grain size of 13.8 mm was used for preliminary experiments and Experiment 1. I observed that the scaled bed was becoming too coarse and attributed the coarsening to the increase in the D50 that resulted from truncating the distribution. East Creek has a scaled D50 of 9.8 mm, yet the truncated mix had a D50 of 13.8 mm. To avoid skewing the grain size distribution, the 14% of sediment that was finer than 2 mm was “added back” into the mix by increasing the proportion of sediment in the finest four ½ φ grain size fractions (2-8 mm). The addition of fine material resulted in a mix with a D50 of 9.8 mm. While detailed observations of sediment transport at Shatford Creek do not exist, we do know that a flood with a return interval of 1.5 years at Shatford Creek has a discharge of 3000 L/s, which scales to a discharge in the flume of 1.68 L/s. During such a flood, which was observed in May of 1998, stones could be heard clunking down the channel and fines were mobile but the channel forming stones did not move (Zimmermann and Church, 2001). This observation suggests that grains up to about 64-90 mm were mobile. These grains scale to 3.2 to 4.5 mm and provide me with a rough idea of the size of grains that should be in transport when the scaled flow rate is 1.7 L/s. To determine the scaled width of the flume, both the mean width at the base of the banks and the bankfull width are relevant. The bankfull width of Shatford Creek was determined to be 7.8 m based on the surveys in 1998 and the base of bank width was 5.9 m based on the survey in 2006. A reasonable mean channel width is some average between these two: a width of 0.371 (7.4 m scaled) was used for Experiments 1-3 and a width of 0.359 (7.3 m) was used for the subsequent scaled experiments (Exp 4-7 & 13). Initially the banks were angled at 70° to reproduce bank slopes that are present in the field, but I abandoned angled banks after Experiment 3 as angled banks introduced additional undesirable complications. With the angled banks it was difficult to get a sense of the stability of the bed since bed degradation narrowed the channel’s width and increased the jamming ratio. Degradation also caused the flow depth to increase as the channel’s cross-section narrowed, resulting in an increase in the applied shear stress. Thus both of the independent variables used 86  to assess the jamming concept varied during the experiments. Furthermore, with angled banks I had to continuously monitor width as it changed during experiments. Width was also not constant down the channel as it depended on the amount of degradation that occurred locally. When trying to match the grain size distribution in the flume to that of Shatford Creek I quickly observed that the fine to middle portion of the grain size distribution tended to become coarser than the scaled Shatford Creek distribution. This trend persisted through all of the experiments. In most of the runs the D50 was greater than the scaled D50 of Shatford Creek, though the coarse end of the grain size distribution matched the target distribution. During some runs the D50 did match, but the fine end of the distribution was still much too coarse. I feel that the mismatch at the fine end of the grain size distribution is not a serious concern as one could always add fine feed, thereby fining the finer end of the distribution and creating a matching grain size distribution. I think it is most important to match the coarse end of the grain size distributions since the coarse end creates the bed morphology. Nevertheless, the lack of fines certainly affects the mobility of the finer sediment in the flume. During a number of runs after an armoured bed was formed the flows were reduced to the scaled 1.7 L/s and observed. Video 1 and Figure 34 compare flow characteristics from a model run that had the same scaled discharge as a photo taken of Shatford Creek. The flow in the flume appears too tranquil, which may be a result of the reduction in the Reynolds number, but may also simply be a result of looking at a smaller system. I do not know if the mind can accurately compare the turbulence properties of systems at different scales. To perform a more robust comparison between the flume runs and Shatford Creek, all of the runs with the same flume slope and scaled width as Shatford Creek were examined to see which run had a coarse bed grain size distribution that resembles Shatford Creek. To determine which run’s grain size distribution was most similar to the scaled Shatford Creek grain size distribution the root mean square (RMS) error was calculating using the percent finer data for the 90, 64, 45 and 32 mm grain size classes. The bed from Experiment 2 run 8 was found to have a coarse grain size distribution that was the third most similar to that of Shatford Creek with an RMS of 1.8% compared to 1.2 and 1.3% for beds formed at the end of two other runs. While these other two runs had grain size distributions that were more similar to the scaled Shatford Creek distribution, no low flows (near 1.7 L/s) were run on those beds. Hence these two runs could not 87  be used to assess how well the flume results scale back to Shatford Creek. Figure 35 illustrates both the bed from the end of run 8 on November 16th, 2006 and the scaled distribution of Shatford Creek. Clearly the coarse ends of the distributions are quite similar.  Video 1. Flow characteristics at 1.7 L/s.  Figure 34. Shatford Creek at 3000 L/s; equivalent to the discharge shown in Video 1.  88  Figure 35. Scaled grain size distribution of Shatford Creek and grain size distribution from Experiment 2 run 8. The prediction error interval for a single sample is plotted for the Experiment 2 data. The 1:8 scaled distribution of the Rio Cordon is also shown as well as all of the grain size distributions from Experiment 30. The grain size distribution from run 2 of Experiment 19 is also shown as it is most similar to the scaled distribution from the Rio Cordon.  Prior to run 8 during Experiment 2 the bed had sustained 410 minutes of bed structuring flows with a maximum discharge of 12.5 L/s. This duration scales to 31 hr of flow at a maximum discharge of 22 m3/s. Based on the annual instantaneous maximum discharge record at Shatford Creek such an event scales to an event with a recurrence interval on the order of 550 years. So while the bed has received a relatively short period of armouring, the maximum flow the bed withstood is likely to occur rarely. Video 2 illustrates a steady flow of about 1.7 L/s over the bed that was formed at the end of run 8 during Experiment 2. During this flow the velocity was 0.20 m/s (n = 1), which scales to 0.89 m/s. In comparison, in 1998 flow velocities of 0.9-1.1 m/s were recorded at the prototype section of channel during the median annual flood (Zimmermann  89  and Church, 2001). Thus there is a good correspondence between the velocity measured in the flume and the velocity measured in the field. In terms of the number of step-pools, during the 2006 survey 14 step-pools were identified over a 170 m section of channel (excluding the section affected by an old log bridge). Using a width of 7.4 m this step-pool spacing corresponds with a step every 1.6 widths. In comparison, in the flume 8 step-pool units formed, each of which is identified in Figure 36, yielding a step-pool every 1.4 widths. Hence, step-pools were ever-so-slightly less common in the field than they were in the flume. The difference should not be considered significant as there is inherently some uncertainty associated with the classification of step-pools (see Chapter 3). The mean bed gradient of the run was 7% and, therefore, slightly less than the prototype gradient of 7.8%.  Video 2. Illustrates flow of 1.7 L/s over a bed with a grain size distribution similar to the scaled distribution of Shatford Creek (from Experiment 2).  Figure 36. Long profile of bed from Experiment 2 at end of run 8. Pools indicated. Long profile is based on averaging bed elevations across each cross section.  In summary, despite my scepticism about Froude scaling and the belief that the reduction in the Reynolds number may cause the model results to have characteristics distinctly different than the 90  prototype, run 8 from Experiment 2 appeared to be a reasonable representation of Shatford Creek. The comparison is limited, however, as sediment transport data do not exist.  4.2.2 Prototype 2: Rio Cordon To conduct experiments with a coarser grain size distribution I decided to model the Rio Cordon surface grain size distribution as this distribution was being used in similar experiments (see Comiti et al., 2007a; in review) and because the Rio Cordon is an extensively studied steep stream in the Italian Alps that has a long history of sediment transport measurements (Lenzi et al., 1999; Lenzi, 2001; Lenzi et al., 2004). A 1:8 scaled model of the surface grain size distribution of the Rio Cordon was used (Figure 35). The distribution matched that used by Comiti et al., (2007a) with the exception that sediment finer than 2 mm (2.5%) was removed. 2.5% of the sediment in their mix was between 0.2 and 2 mm. The Rio Cordon has a mean gradient of 14% through the prototype reach; hence the flume was set at a slope of 14%. Comiti et al. (2007a; in review) used a flume width of 0.46 m while Experiment 30, which was a model version of the Rio Cordon, had a flume width of 0.511 m. These flume widths scale to 3.68 and 4.09 m respectively. The section of the Rio Cordon used as the prototype has a mean width of four metres (Comiti et al., in review). In Figure 35 the grain size distributions from the 5 runs that occurred during Experiment 30 are plotted along with the scaled distribution from the Rio Cordon. The bed prior to any flows being run is evidently quite similar to that of the Rio Cordon, except it is too fine at the coarse end. The fining at the coarse end likely occurred because fine material hid some of the larger stones. Once water was run over the bed it was observed to coarsen dramatically and in the late stages of the experiment, when the bed would ideally be most similar to the distribution of the Rio Cordon, the D84 from the flume runs was about 120 mm while the idealized D84 was 63 mm. Thus the flume had become much coarser than the Rio Cordon. While not surprising — the bed is expected to coarsen while armouring and step-pools are being formed — it was disappointing. The model runs cannot be compared with the Rio Cordon as the bed became too coarse. Thus while sediment transport data from the Rio Cordon do exist, no comparable model was created. Clearly using the surface grain size distribution for the bulk mix does not work.  91  4.2.3 Summary comments regarding Froude scaling The observations and analysis of the runs during which I attempted to construct Froude scaled models of Shatford Creek and the Rio Cordon illustrate that getting the grain size distribution in the flume to match that of the prototype is not straightforward. The feedback between sediment feed, bed stability, surface grain size distribution and bed slope makes it difficult to form a bed that is appropriately armoured and has the correct structures/bedforms as well as the correct bed gradient and surface grain size distribution. During the runs that were completed, a perfectly scaled model was never created, but the beds that were most similar did produce flow characteristics and bedforms that agreed with field observations. I believe that, to improve the match between the desired model and the trial bed, fine sediment should be fed onto a bed that has approximately the correct grain size distribution at the coarse end, but is too coarse at the fine end. In addition, if sediment transport information is available, sediment should be fed into the flume at a rate and grain size informed by field data. Having no sediment supply rate and grain size distribution data for the material coming into the Shatford Creek study reach limited the degree to which the scaling could be assessed. While sediment transport data exist for the Rio Cordon, it was not utilized as a model with a similar surface grain size distribution was not created. The decision to use the surface grain size distribution of the Rio Cordon as the bulk mix for the flume experiments was an error as it resulted in a model bed with a grain size distribution considerably coarser than the scaled Rio Cordon.  4.3 Trying to make step-pool channels To form step-pool channels a recipe of flows and feed rates was required that was expected to produce the most realistic series of step-pool channels. To develop a recipe I conducted a series of experiments during which the flow rate was increased in steps and the feed rate was adjusted. Unlike lowland streams, the bed of step-pool streams is generally stable and mobile populations of gravels and sands move over the bed (Zimmermann and Church, 2001). By increasing discharge in steps while adding feed at some of the discharge increments I hoped to emulate the natural pattern of step-pool formation.  92  4.3.1 Determining flow duration The first question that was addressed was how long does it take for the bed to adjust to new flow conditions? Figure 37 shows the sediment transport record from three runs (beginning of Experiments 1, 2 and 8) when the flume was filled with sediment and an exceptionally large flow was applied to the bed. The entire bed surface was mobilized but within an hour the majority of the sediment in transport had stopped moving and the bed stabilized. This bed response showed that the bed can adjust to new applied discharges within an hour.  Figure 37. Sediment transport as a function of time for initial portions of three experiments where initial flow rate was much greater than the critical flow to mobilize bed.  4.3.2 Determining flow increase Preliminary experiments illustrate that increasing flow 20% per step was sufficient to cause a marked change in sediment mobility but did not always cause wholesale destruction of the bed. In some cases the transport rate barely responded, while in other cases substantial increases in 93  transport occurred nearly instantaneously and widespread degradation occurred (this behaviour will be explored in greater detail in Sections 5.3.3 and 6.5.1). I considered this behaviour to be ideal for the formation and testing of step-pool channels since it indicates that the flow increases were sufficient to have an effect, but not so much that the armour was consistently destroyed. The variable nature of the response is characteristic of stream beds experiencing stochastic behaviour.  4.3.3 Determining feed rate The next issue that was examined was the response time to an addition of feed. A set of experiments was run at constant discharge and after the bed had structured and the sediment transport rate slowed I began to feed material into the flume. The feeding occurred for about an hour and the response of the sediment transport rate to the step increase in feed was observed. Figure 38 illustrates that the response to feed is not instantaneous. It takes between 15 and 50 minutes for the feed material to cause a response in the transport of material out of the flume. These response times correspond with the stabilization time and suggest a characteristic transit time on the order of one metre in ten minutes. The faster the feed rate, the longer it took for the sediment transport rate to completely respond. Within an hour the transport rate at the end of the flume had always responded. The maximum feed rate used during these runs was similar (115 g/s/m) to the fastest feed rate used to form step-pools during the main experiments (108 g/s/m). These results thus suggest that feeding sediment for an hour ensured the bed had responded to the addition of feed. Based on the feed and flow response each flow increment or sediment feeding event was run for at least an hour to ensure that the bed had sufficiently adjusted to the applied discharge and feed conditions. Experiment 4 was used to determine the feed rates for the rest of the experiments. The flume was filled with sediment and run at an initial flow that was just sufficient to mobilize the smaller grains and initiate bed structuring. After an hour the flow was increased 20% and held constant for another hour before being increased again. If the bed began to degrade below its initial profile, feed was added until the bed aggraded back to its original profile. After feeding, the discharge was held constant for at least an hour before being increased. After nine increases in flow and five sessions of feed the bed was well armoured and structured. The amount of feed that was used during this experiment was then used to guide the selection of feed rate for all  94  Figure 38. Response of sediment transport records to addition of sediment feed for three portions of Experiment 2. Discharge was 9 L/s for all three runs.  subsequent experiments. Figure 39 illustrates the history of sediment transport, flows and feed rates for Experiment 4 while Table 17 summarizes the flow and feed rates used during the subsequent experiments that had a bed gradient of eight percent and during which the fine grain size mix was used. For experiments completed with a different grain size mix or at different slopes the discharge used during the experiments was modified. Table 17. Standardized experimental procedure used to armour bed for an experiment with the fine sediment mix and a flume gradient of 8%.  Flow increment Flow rate (L/s/m) Feed rate (g/s/m)  1 9.8  2 12  3 15 43  4 19  5 23 216  6 28  7 34 301  8 41  9 50 301  10+ previous Q + 20%  95  Figure 39. Flow, feed, sediment transport and mean water velocity from first 1500 minutes of Experiment 4. Dashed line indicates flow rate, grey boxes illustrate feed rate.  4.3.4 Adjusting flow rate for changes in flume slope and bed grain size To adjust for changes in gradient between experiments, I wanted a means of adjusting the flows such that all of the beds would break after approximately the same duration of flow. The hope was that each bed would be subjected to the same relative force during each discharge increment and regardless of slope they would all break after approximately the same number of runs. Such behaviour would ensure that steep and low gradient experiments would run for the same duration. Two possible means of scaling for changes in bed slope exist and I had to choose one. I could either decide to keep the stream power constant or I could decide to keep the Shields ratio constant. To keep the Shields ratio constant the flow depth needs to be estimated prior to the run whereas stream power is simply the product of unit discharge and bed slope. With a stream power approach flow roughness does not need to be considered and for this reason I decided to use a stream power approach. Using a stream power approach meant that if the flume gradient was doubled the discharge was halved. The red data points in Figure 40 illustrate the bed gradient for the last stable bed (i.e. the bed that withstood the greatest force) in each experiment as a function of both unit stream power and the Shields ratio. If stream power adequately scaled for the effect of bed gradient we would see the red data points plot vertically above each other. Instead there is a clear trend: at higher gradients the maximum stream power the bed withstood was lower. These results show that scaling the flows by stream power did not work. It was also observed that during the steeper experiments the bed broke earlier than during the lower gradient experiments. 96  Figure 40. Maximum stable bed gradient as a function of stream power (a, b) and Shields ratio (c, d) for fine sediment experiments. Lines connect sequential experimental runs. Red points are last stable point before bed failed.  The last stable gradient as a function of the Shields ratio for two different widths of channel is indicated by the red data points in Figure 40c and d. For neither channel width is there an obvious trend in the Shields ratio data implying that, regardless of bed gradient, the channel can withstand about the same applied shear stress before failing. The data from runs with different  97  widths are believed to plot separately because of the jamming phenomenon. Figure 40 suggests that despite the experiments not being conducted in a manner whereby a constant Shields ratio was maintained, when the data from the runs are plotted after the experiments were completed, we see that the last stable bed sustained about the same Shields ratio. Clearly the channel’s slope, grain size and roughness adjust to maintain an approximately constant Shields ratio. How the system goes about adjusting these three parameters to attain a constant Shields ratio is unclear. In the future, to conduct experiments at a range of slopes, it would appear best to use a constant Shields ratio approach. Such an approach will require flow roughness to be predicted, itself not an easy task, along with an estimate of how the channel will coarsen and structure. Flow resistance relations presented in Chapter 7 will help, as will grain size evolution data from experiments like these. To adjust for changes in the mean size of the sediment mix, discharge was scaled by q2/q1 = (D1/D2)-3/2, which holds the applied Shields number constant and utilizes the Manning-Strickler flow resistance relation. Had a log-law of the wall, Darcy-Weisbach friction factor type of approach been used a different relation between unit discharge and grain size would have resulted. Once the bed was formed, flows were increased by 20% and run for three hours to provide ample time for the bed to armour. At the end of the three hours, if scour had not caused the base of the flume to become visible, the flow was increased again. Runs continued until either the base of the flume was visible or a flow rate of 85 L/s, the highest flow that could be safely run in the flume, was reached.  4.4 Success! Step-pools, among other bed features, are formed Based on the bed scans and the video along with the observations during the runs I eventually became convinced that during some of the runs step-pools were formed; however, this conclusion was not immediate. During Experiment 1 and the preliminary experiments it seemed that step-pools formed relatively readily (see Compendium). Then to account for the bias introduced by truncating the grain size distribution a finer mix was employed for the subsequent experiments and step-pools became less obvious. In later experiments that used the coarse mix 98  modeled after the Rio Cordon, step-pools were much more evident (see Compendium). With the finer mix, step-pool units appear to be less uniformly spaced and cascade or rapid sections of channel appeared more common. Furthermore, at the widest widths the channel tended to migrate and form bar deposits. In general it was found that looking up the channel during low water flows provided the most convincing visual evidence that step-pools had in fact been created (e.g. Figure 41). Figures 41-45 and Videos 3 and 4 illustrate the range of bedforms and flow conditions observed during the experiments. There was clearly a wide range of conditions observed, from stones jammed across the channel width forming drops to sinuous bars with interconnecting rapid channel types. Furthermore, a wide range of local hydraulic conditions was observed. These included clear free jumps, submerged jumps, cascade tumbling flow and smooth flow over rapid channel types. These are the main flow types that I have observed in headwater streams. Steps that were formed varied between those composed of one large grain and a couple of key smaller grains that formed an interlocking line of grains against the walls (e.g. Figure 44) to steps formed by a large number of stones arranged in a line spanning the channel (e.g. Figure 43).  Video 3. Flow at 61 L/s over fine bed at medium width from last run during Experiment 6. Rhodamine dye was injected with salt solution. Viewed looking upstream.  Reviewing the images in Figures 41-45 and the bed scans in the compendium emphasizes just how hard it is to decide what is a step-pool and what is not; some are evident, but many are less clear. For this reason the classification technique developed in Chapter 3 was used exclusively. 99  Figure 41. Images from Experiments 1, 4, 6 and 26. Small pools visible in (e) are also clearly visible in 5th bed scan (see Compendium). All images are viewed looking upstream.  100  Figure 42. Images from Experiments 10, 16, 21 and 25. a), b), e) and f) are viewed looking upstream. In c) flow is from left to right and d) is viewed looking downstream.  101  Figure 43. Images from Experiment 26. a), b), and d) are viewed looking upstream. In c) flow is from left to right.  102  Figure 44. Images from Experiments 23, 28 and 29. All images are viewed looking upstream.  103  Figure 45. Images from Experiments 30 and 32. a), d), and f) are viewed looking upstream. In b), c) and e) flow is from left to right.  104  Specific details of the geometry and frequency of step-pools related to characteristics such as channel width, slope and grain size are presented in Chapter 5, while general comments related to formation of step-pools, observations from running the experiments and the perceived effectiveness of the recipe used to form the bed are discussed below.  Video 4. Large jump formed during Experiment 25. Viewed looking upstream.  The Compendium illustrates the 314 bed scans that were completed as part of the experiments and the evolution of the bed. At the end of the compendium five scans from a series of runs conducted to model Fitzsimmons Creek in Whistler, BC (Zimmermann, 2008) are also included to illustrate the nature of the channel that forms when the channel is very wide and fine, but at similar slopes to some of the other experiments. For this experiment the bed material was especially fine (D50 = 2.5 mm) and the flume was set to its full width (83 cm) at a slope of 6%. Flows and feed were selected to form a structured channel similar to Fitzsimmons Creek and thus did not follow the same procedure as the main experiments. The top half of the flume after the last Fitzsimmons run can be compared to the step-pool experiments. Along with the wide channel and fine grain size the Compendium illustrates that the bed is lower in the middle of the flume and individual large grains are easily visible in the scan. Unlike the other experiments the coarse stones are not arranged in cells, lines or steps, but are more uniformly spaced as they might be on a bar. If we compare the last bed from the Fitzsimmons experiment with the one from the end of Experiment 26, which had nearly the identical slope and was the widest of the step-pool experiments, clear differences are evident. The last bed from Experiment 26 does not have many channel spanning step-pools (the algorithm identified three), but local accumulations 105  of large stones and small pool like sections are visible. Similar structures are visible in Figure 41e, which is from on earlier run during Experiment 26. Bar features and low sinuosity channels were evident in the experiments with the fine mix and a medium to wide channel width, particularly at steeper slopes. These migrating channels were typically drowned out by the second increase in flow and as a result they were rarely preserved in the scans. One preserved example can be seen in the second scan from Experiment 9 (see Compendium). A sinuous channel can clearly be seen in the middle to upper portion of the flume. Another example can be seen in the top left image in Figure 43, which shows the channel moving from the right bank to the left bank. The upstream bar on the right side of the channel (left side of image) was not mobilized until the flow rate was increased. These sinuous channels at steep slopes are believed to be similar to those observed during Roman Weichert’s (Weichert et al., 2004; Weichert, 2006) research on steep streams.  4.4.1 Wedge dynamics During the feeding of sediment a coarse wedge of material 20-40 cm long frequently developed downstream of the feeder outlet, creating a local slope that was a few percent greater than the bed slope (e.g. Figure 44). Occasionally the wedge would fail, resulting in a release of water and sediment downstream. This process appeared similar to what would be expected for a point source of sediment such as the toe of a landslide. Its influence on the stability and structuring of the channel downstream remains unknown. In retrospect I would like to have run an experiment with no feed in which all the sediment that would have been fed was placed in the flume as a wedge and the channel degraded into this wedge as flows were increased. I suspect step-pools would still form as the wedge was degraded, but do wonder if the same end state would have been achieved. Does the random supply of larger stones from a feeder influence how the bed develops? If the channel has the capacity to degrade into its own bed to entrain new sediment will it attain the same level of stability as it would if the sediment was supplied from upstream? I would guess that the supply of sediment from the feeder promotes stability as the stones that remain on the surface were deposited during conditions when an applied stress existed (shear stress) and should be more stable than those that were placed in a wedge in the absence of an applied stress. A problem with this line of reasoning is that there would be fewer stones in the wedge formed from feeding 106  as some of the fed stones would not find stable states, and would have moved on, while all the stones would be in the wedge if the material was placed in the wedge before hand. A wedge degrading experiment could resolve this question.  4.5 Summary thoughts During the experiments a wide range of bedforms and flow conditions were observed and convincing step-pools formed during some runs, but not during all runs. This variability provides a rich data set that will make it possible to assess what controls the frequency and form of step-pool channels and their stability; which is the focus of the next two chapters respectively. The experimental approach, based on feeding sediment in pulses while increasing discharge in 20 % increments, is unable to clarify the importance of flow duration and feed rate. This challenge is examined in greater detail in the subsequent chapters, however, even then it remains unresolved. Is feed absolutely necessary to form the step-pools I observed? Can flow rates be ramped up quicker and the same conditions observed? There also remains some question in my mind as to how well Froude scaling really works.  107  5 The flow, the bed and the transport: Interactions in a steep flume The objective of this chapter is to characterize the bed morphologies, flow conditions and pattern of sediment transport during the experiments. As part of this analysis the factors governing the occurrence and form of step-pool channels, the first research question, will be examined.  5.1 Step-pool form and frequency During the different experiments, rapid and cascade channel types were formed. These were composed of steps, stone cells, stone lines, runs, pools and bars. While all of these bedforms occurred and would be expected in natural channels, the analysis here is focused on factors that affect the frequency and morphology of step-pool channel units. The dimensions of step-pools have been the focus of many studies and represent one of the earliest aspects of step-pool streams to be studied (Judd, 1963). Chin and Wohl (2005) provide a recent detailed review of the factors that correlate with the occurrence of step-pools and their dimensions. These factors include channel slope, channel width, grain size, flow magnitude and duration and bank roughness. In their review Chin and Wohl emphasize the challenges of multiple cross-correlated terms, differences in classification techniques and sample size. This study overcomes the latter challenge, offers a resolution of the second and provides detailed flow and bed data that can address the cross-correlation challenge to some degree. Table 18 illustrates that a wide range of bed configurations was observed during the experiments, making the data excellent for detailed analysis of step-pool frequency and form. Thus the data are explored to determine if the steppool units formed during the experiments can improve our understanding of the variables that govern step-pool formation. Where step-pool channel units did not occur, the channel was classified as ‘other’ and consisted of features such as cascades, rapid or riffle-pool bedforms. Unlike past studies, during this analysis particular attention is paid to using dimensionless descriptors of step-pool frequency and step-pool dimensions. Thus each independent and dependent variable will be scaled by a variable or group of variables that have the same dimensions. During previous studies, dimensionless groupings have not been used and as a 108  result differences in scale associated with different streams could have biased past findings. As an example, step-pool spacing has frequently been related to slope, but not standardized in any manner. In the past data from small and large streams have been grouped together even though step-pool spacing is certain to scale with channel size. A second significant difference between this analysis and past studies is that in this study treads or ‘other’ bed morphologies are recognized to occur between consecutive step-pool channel units while previous studies have considered the channel to be formed only of steps and pools. Table 18. Range of bed conditions observed during 32 experiments. Portions of bed not composed of step-pools were classified as other.  Number of step-pools Fraction of the bed covered with step-pools (%) Width/D84step ratio (the jamming ratio) D84 of the steps (mm) D84 of bed surface (mm)  Minimum 0 2 1.6 9.6 6.7  Maximum 18 85 60 156 143  Average 7.4 44 7.2 64 51  n 319 297 312 298 312  Beds at the end of runs that lasted at least an hour during which no sediment was fed were exclusively used for this analysis. After an hour the beds had stabilized (see Section 4.3.1) and thus the beds represent stable end-state morphologies. Multiple bed scans from the same experiment but sequentially increasing peak flows were included. In general, sequential bed scans differed as each bed changed in order to accommodate the increase in flow (see Compendium). During the analysis, each bed scan was considered to be independent, despite some sequential bed scans being serially correlated.  5.1.1 Controls on step-pool frequency To explore why step-pools formed more readily during some runs compared to others supervised forward step-wise regression techniques were used. Since the number of step-pool units observed during each experiment was strongly dependant on channel width (r2 = 0.36, p < 0.001, n = 231; other variables examined included D50, D84, D84step, but were inferior) and channel width is a common scaling variable that correlates with discharge and basin area, both of which have been shown to be important in other step-pool studies (Judd, 1963), the number of step-  109  pools observed in the flume over the length of the bed scan (4.128 m) was normalized by the width of the water surface. Based on forward step-wise regression that included S, D50/w, D84/w, D84step/w, D50/D50mix D84/D50mix, D84step/D50mix and the logarithm of S, the number of step-pools per bankfull width was found to be best explained by the logarithm of bed slope (Figure 46) yielding the following functional relation for channels with rough banks: # SP = 1.8 + 1.2 log(S ) , r2 = 0.50, n = 226, p < 0.001 w  (16).  Figure 46. Relation between number of step-pools per bankfull width and bed slope for rough (solid line) and smooth (dashed line) banks. Functional regression relations are shown.  D50/w, D84/w, D84step/w were included in the regression model to account for grain size of the bed relative to the width of the channel and are the inverse of the jamming ratio, while D50/D50mix D84/D50mix, D84step/D50mix were included in the regression model to account for the surface coarsening and structuring associated with the development of the structures. The stepwise regression results showed that neither of these grain size related metrics explained the remaining variability in the data (p > 0.1). If slope was not included in the model, the logarithm of D84/w 110  ratio was found to be related to the number of step-pools per bankfull width, but less of the variance was explained (r2 = 0.26 vs. 0.50). The ability to substitute D84/w for bed slope in the model arises because these two variables are correlated. In Figure 47 we see that the inverse of the jamming ratio (D84step/w) and the D84/w ratio correlate with bed slope. This correlation likely exists as shallow slopes at small jamming ratios were not investigated during the experiments as very large flow rates would have been required to move the bed. Likewise, steep slopes at large jamming ratios were not studied as the beds would have immediately failed. In part the physics underlying this relation dictated the experiments that were run. Similar controls likely also exist in nature; wide channels with fine beds are unlikely to exist on steep slopes as the bed would quickly degrade, and flat, narrow, coarse channels are unlikely to exist, except possibly as lag deposits, as flows capable of moving large stones into and through low gradient narrow channels are unlikely to occur. If they were to occur, they would most likely erode the banks. The correlation between the jamming ratio and bed slope influences much of the analysis in the subsequent sections.  Figure 47. D84/w and D84step/w as a function of bed slope.  Rough walled experiments were observed to have significantly more step-pool channel units per bankfull width compared to the smooth walled experiments (ANCOVA, p < 0.001). Thus the main factors influencing the number of step-pools were found to be width, channel slope, and bank roughness (as illustrated in Figure 46). The importance of gradient is well established in the literature (Judd, 1963), but scaling for changes in channel size with width and the importance of bank roughness are new. Figure 46 also illustrates that while smooth and rough walls affect the number of step-pools, both bank types are associated with similar amounts of scatter. The 111  figure also suggests that on average, with rough banks, step-pool bedforms do not occur below a channel slope of 3 to 4%. Next the fraction of the channel length composed of step-pools (f_sp) was investigated using the same series of independent variables (S, D50/w, D84/w, D84step/w, D50/D50mix, D84/D50mix, D84step/D50mix and the Logarithm of D50/w, D84/w, D84step/w, and S) and stepwise forward regression. As the number of step-pools per bankfull width increased, so did the fraction of the bed composed of step-pools. Regression analysis showed that the fraction of the bed composed of step-pools is more strongly related to the logarithm of D84/w ratio (r2 = 0.56) than the logarithm of the bed slope (r2 = 0.23) suggesting the D84/w ratio is the dominant factor. Like the number of step-pools, during experiments with rough banks a greater fraction of the bed was composed of step-pools compared to smooth bank experiments (Figure 48, ANCOVA, p < 0.001). No other variables were found to influence the fraction of the bed composed of steppools (p > 0.07). Functional regression (see Section 2.2.3) yielded  ⎛D ⎞ f _ sp = 1.29 + 1.00 log⎜ 84 ⎟ , r2 = 0.55, n = 224, p < 0.001 ⎝ w ⎠  (17).  Figure 48. Fraction of bed composed of step-pools as a function of the D84/w ratio. Best-fit functional regression relation for rough banks is shown.  112  The large amount of scatter displayed in Figure 48 suggests that for a channel with a D84/w ratio of 0.20, anywhere from 40-80% of the channel may be step-pools. While some of this variability is certainly associated with measurement uncertainty, much of it is attributable to real variation in the formation of step-pool structures and the multiple stable states that can occur. Long armoured cascade channels may occur that remain stable during some runs, while other runs may be associated with more step-pool structures. The smooth banks are associated with similar variability, suggesting the variability is not simply a response to the rough banks but more endemic to step-pool channels in general. On average the channel was observed to be composed of more than 50% step-pools when the bed slope exceeded 10%. This observation suggests that step-pool dominated streams occur at gradients greater than 10%, somewhat larger than the 7% minimum for ‘step-pool’ streams suggested in the introduction (Section 1.1.1).  5.1.2 Step dimensions Step height can be measured in at least two distinct ways (see Figure 3 for two methods discussed here, for other methods see Nickolotsky and Pavlowsky (2006)). One can measure the drop height (z), which is the elevation from the crest of the step to the elevation of the lip of the pool or one can measure the elevation from the crest of the step to the deepest point in the pool (step height: H). While both measures are likely to be influenced by scour in the pool, not just the formation of the step by stones structuring across the channel, the later is more sensitive to scour in the pool. Both are examined here for completeness: traditionally only H has been examined. While they are highly correlated, step height can be more than twice the drop height (Figure 49). For some time step height has been known to correlate with grain size, and the results from this study confirm this belief, D50, D84, D50mix and D84step all correlate with step height and drop height as does the standard deviation of bed elevation. For step height (H) the D84 of the surface explains more of the variance than any other metric (r2 = 0.73; p < 0.001, n = 224) while for drop height (z) the D50 and D84 of the bed surface explained the same amount of variance (r2 = 0.57 and 0.56, respectively, n = 224). On account of the desire to use a single grain size to scale both measures of step height, the D84 was used to scale drop height and step height.  113  Figure 49. Relation between step height and drop height.  Forward stepwise regression that utilized S, D50/w, D84/w, D84step/w, D50/D50mix, D84/D50mix, D84step/D50mix, #Step-pools/w, Fr, y/D84, f_sp, w/y, logarithm of S, D50/w, D84/w and D84step/w showed that the logarithm of D84/w best explained the variance in both step metrics (z/D84, r2 = 0.23; H/D84, r2 = 0.16; p < 0.001). With the H/D84 data, D84/w performed equally well. Since the jamming ratio correlates with the number of step-pools per unit width and bed slope, variables related to these metrics were removed from the step-wise regression (S, D50/w, # Step-pools/w, D84step/w and their logarithms). For z/D84 and H/D84, D50/D50mix was the next variable to enter the regression; however, the p-value calculated for the F-statistic was greater than 0.01 and thus this term was not added to the model. To avoid the model being over-parameterized a more restrictive p-value of 0.01 was used. Thus the final regression models for rough bank experiments are: z ⎛D ⎞ = 0.19 − 0.44 log⎜ 84 ⎟ ; r2 = 0.23, p < 0.001, n = 224 D84 ⎝ w ⎠  (18)  H ⎛D ⎞ = 0.56 − 0.50 log⎜ 84 ⎟ ; r2 = 0.15, p < 0.001, n = 224 D84 ⎝ w ⎠  (19).  The coefficients for the  D84 terms are not statistically different while the constants are w  significantly different. The correlation coefficients for both relations are relatively poor, with only 23% or less of the variability in the data being explained, which further emphasizes the 114  variability associated with step-pool bedforms. Rough banks were also associated with larger steps (p = 0.058) and drop heights (p = .048, ANCOVA). The commonly reported relation between step-height and bed slope (Chin and Wohl, 2005) is supported by these data as bed slope is statistically related to z/D84 and H/D84 (r2 = 0.04 and 0.11, respectively); however, the logarithm of D84/w explains more of the variance. This finding may result from the D84 occurring on both sides of the relation, or it may represent step height being more closely tied to the jamming ratio than bed slope. Step length (SL) was found to be positively related to drop height (SL = 1.8z+0.013, r2 = 0.49, p < 0.001, n = 228) and was therefore scaled by drop height. Other variables that were included in the step-wise linear regression included H, Ls, dr, D84step, D50, D84, D50mix, w, σbed and log(σbed), but these variables explained less of the variance. The observation that drop height is the best predictor of step length coupled with step height/ grain size relation presented earlier suggests grain size determines the drop height as there is a limit to the height grains can be stacked without a step failing. Drop height then determines step length. This inferred causal pattern is opposite to what is expected for lower gradient streams where riffle-pool wavelength may dictate riffle heights, and emphasizes that fundamentally different processes govern the dimensions of riffle-pool channel units compared to step-pool channel units. To determine an average scaled step length, the length of each step was scaled by the drop height of the step and then an average of all the steps observed in each run was determined. The average scaled step length was then related to S, D50/w, D84/w, D84step/w, D50/D50mix, D84/D50mix, D84step/D50mix, #Step-pools/w, w/y, logarithm of S, D50/w, D84/w and D84step/w in a forward stepwise regression. The logarithm of D84/w was most strongly related to the scaled step-length. On account of the cross-correlation discussed earlier, bed slope was removed from the forward regression along with other variables closely related to D84/w. The width to depth ratio was the next and final variable to enter the multiple regression yielding SL w ⎛D ⎞ = 0.27 − 2.4 log⎜ 84 ⎟ − 0.035 ; r2 = 0.44, p < 0.001, n = 214 z y ⎝ w ⎠  (20).  For this relation the standardize coefficients were -0.73 and -0.16 for D84/w and w/y respectively. This step shape function (1/step slope) implies that steps become steeper as D84/w increases, 115  which correlates with bed slope. Steps formed during smooth walled experiments were significantly steeper than those formed during the rough walled experiments (ANCOVA, p < 0.001; Figure 50).  Figure 50. Relation between step shape (step length/drop height) and D84/w. For the rough bank data the w/y ratio is indicated by the size of the data points, which varies continuously along a linear scale. Rough bank data are grey circles while smooth bank data are black squares. Best-fit line does not include w/y term and is dashed for the smooth walled experiments.  5.1.3 Pool dimensions Working independently Comiti, Curran and Milzow have made considerable progress towards our understanding of the formation of pools and, in particular, the importance of step height and flow magnitude in shaping the dimensions of scour pools (Comiti, 2003; Milzow, 2004; Comiti et al., 2005; Curran and Wilcock, 2005; Milzow et al., 2006). Scour depth has been related to the height of the step and critical flow depth while the length of the pool depends on the formation mechanism, jet dynamics and scour depth. These patterns among others are examined using the data from this study. Unlike Comiti, who indexed scour using step height (ys = H), residual depth (dr) is used here. This decision was made to avoid the serial-correlation that occurs as a result of step height including drop height, which is commonly used to predict scour (Church and Zimmermann, 2007). Average scour depths and average pool lengths for each run are used for this analysis. To convert residual depth to a dimensionless grouping three basic types of parameters are available, the grain size of the bed (e.g. D84), flow depth data (e.g. critical depth dt) and step data 116  (e.g. z). Forward linear regression that examined residual depth and included z, dt, D84step, D84pool, D50, D84, D50mix, w, Ls, SL, σbed and log(σbed) demonstrates that the D84 of the bed surface was the first parameter to enter the regression. Thus residual depth (dr) was scaled by the D84 of the bed. d r = 0.0044 + 0.33D84 ; r2 = 0.65, p < 0.001, n = 224  (21)  Scaled residual depth was then related to S, D50/w, D84/w, D84step/w, D50/D50mix, D84/D50mix, D84step/D50mix, #Step-pools/w, z/D84, w/y, fraction of bed composed of step-pools (f_sp), logarithm of S, D50/w, D84/w and D84step/w in a forward step-wise regression. The #step-pools/w was the first term to enter the relation followed by the w/y ratio. The p-value associated with the F-statistic for the remaining variables was greater than 0.01; hence no additional variables were added to the model. Thus the final pool scour model for rough banks is dr w # SP = 0.61 − 0.20 − 0.014 D84 w y  ; r2 = 0.20, p < 0.001, n = 214  (22).  For this relation the standardize coefficients were -0.32 and -0.28 for #SP/w and w/y respectively. Smooth banks had statistically shallower dimensionless pools (dr/D84) than rough banks (ANCOVA p <0.001). The relation suggests that as the frequency of step-pools increases the scour depth is reduced. This observation is akin to the interference phenomenon observed by Comiti (2003). He observed that sills located a close distance downstream of a pool could limit the amount of scour that occurred in the upstream pool. Drop height and critical flow depth do explain some of the variance in residual depth, which has been shown in the past (Comiti, 2003; Comiti et al., 2005); however, in this study grain size was a better predictor of residual depth. In Comiti’s research the surface grain size distribution was typically not measured and fixed sills were used. Consequently the surface grain size distribution did not dictate the step dimensions. Forward stepwise regression that included S, dt, z, SL, D84step, D84pool, D50, D84, D50mix, w, σbed and log(σbed) revealed that residual depth (dr) explains the greatest proportion of the scatter in pool length (Ls, 38%). In contrast, previous studies that examined rill head cuts (Bennett, 1999) and pools downstream of sills (Gaudio et al., 2000) found that scour pools were not self similar (pool depth could not be predicted based on pool length). On account of the correlation that was 117  observed, pool length was normalized by residual depth yielding Comiti’s k factor (see Comiti, 2003; Comiti et al., 2005). While Comiti et al. (2005) found that k varied between 6 and 8 for flume studies, these data reveal a mean value of 9.5 and a standard deviation of 3.3. Regression results that included S, D50/w, D84/w, D84step/w, D50/D50mix, D84/D50mix, D84step/D50mix, SL/z, w/y, f_sp, logarithm of S, D50/w, D84/w and D84step/w suggest that the variance in Ls/dr was best explained by the logarithm of bed slope. For rough banks the best-fit relation is Ls = k = −4.8 − 13.6 Log ( S ) ; r2 = 0.56, p < 0.001, n = 228 dr  (23).  The p-value associated with the F-statistic for the remaining variables was greater than 0.01; hence no additional variables were added. ANCOVA revealed that relative pool length (Ls/dr) was shorter for smooth bank experiments (p = 0.01). The importance of slope in determining the relative length of the pools is likely related to the jet impinging angle, which is known to influence pool shape (Bormann and Julien, 1991). Steeper streams are likely to have a relatively steeper jet moving the location of maximum scour upstream and shortening the pool. While pool length was statistically related to drop height, which has previously been observed by others (Lenzi and Comiti, 2003; Milzow et al., 2006), for the presented data residual depth and bed slope explain more of the variance in pool length.  5.1.4 Step-pool shape parameters  Figure 51. H / L / S (a) and H / L s / S (b) as a function of channel slope for data from all the bed scans. Best fit linear regression relation is shown in (a).  118  The classic relation describing step-pool shape is the H/L/S ratio introduced by Abrahams et al. (1995), which was initially believed to range between 1 and 2 for step-pool channels that maximize flow resistance. It has subsequently been shown to depend on slope and is no longer considered to have a narrowly defined range (Church and Zimmermann, 2007). Data from all the bed scans completed during these experiments confirm this observation (Figure 51a). If the relation is plotted using pool scour length (Ls) rather than step-pool spacing (L) the relation does not obviously depend on slope and mean values do range between 1 and 2 (Figure 51b). We know from (23) that the scour length normalized by residual depth is negatively related to bed slope, and it can also be shown that the drop height (z) and residual depth (dr) are positively related to bed slope. Therefore, both pool and step dimensions are known to vary as a function of channel slope, which may explain why there is no trend in the plot of H / L s / S (Figure 51b). The dimensions of step-pool units adjust to a constant value and are thus self-similar. If it is these adjustments that are responsible for the absence of a trend in Figure 51b, the presence of a negative trend in Figure 51a should not be interpreted as being due to changes in step-pool shape, but rather an increase in the frequency of step-pools as slope increases and a reduction in the length of treads (Figure 46 and Eq. 16). As the frequency of step-pools increases the proportion of the spacing between step-pools (L) that is composed of pools (Ls) increases and the length of the tread decreases. Under these conditions L approaches Ls and H / L / S approaches H / L s / S for steep slopes.  5.1.5 Summary of controls on step-pool form and frequency The preceding empirical analysis has highlighted that the frequency of step-pools per bankfull width increases as the channel becomes steeper, and that the fraction of channel composed of step-pools decreases with increases in the jamming ratio (w/D). Since lower jamming ratios are also associated with steeper beds this finding suggests that small jamming ratios promote step formation, which is required for steep channels to remain stable. Step-height and step drop height were observed to be strongly related to D84 of the bed. Step-height normalized by grain size was then found to be negatively related to the logarithm of the inverse of the jamming ratio (D84/w); suggesting large jamming ratios produce relatively taller steps. Step length was found to depend on drop height and in turn, step length normalized by drop height was positively related to the logarithm of the jamming ratio and negatively related to the width to depth ratio. This result suggests that as the jamming ratio decreases, steps get longer. The residual depth of 119  the pools was found to depend strongly on the D84 of the bed, while residual depth normalized by grain size was related to the w/y ratio, and the number of step-pools per unit channel width. Finally pool length was strongly related to the residual depth, which yields the ratio k that is slope dependent. The analysis reveals that the fundamental variables correlating with the dimensions and form of steep channels are channel slope, bed grain size and channel width. Both channel slope and bed grain size are largely controlled by geology while channel width scales with flow magnitude or drainage area. Channel slope and bed grain size, particularly over long time scales, are influenced by flow magnitude; however, the availability of large boulders and location of erosion resistant features that can control bed gradient are important at almost all time scales. The commonly cited and interpreted H / L / S relation originally proposed by Abrahams et al. (1995) was found to be essentially correct if pool length (Ls) is used in place of step-to-step spacing (L). It is suggested that the trend observed between H / L / S and bed slope can be attributed to changes in the frequency of steps as a function of slope, a factor not originally considered by Abrahams et al. (1995).  5.2 Flow properties The following section explores flow properties such as the Froude number and the width to depth ratio during the experiments to inform the reader about the range of flow conditions observed.  5.2.1 Froude number The Froude number (Eq. 7) is a particularly interesting parameter for step-pool streams as the flow alternates between supercritical (Fr > 1) flow over steps and subcritical (Fr < 1) flow through the pools. In addition, step-pools have been hypothesized to form due to antidunes (Whittaker and Jaeggi, 1982), which occur during supercritical flow conditions (Simons and Richardson, 1960). The Froude numbers that occurred at least one hour after the experiments started are plotted as a function of time in Figure 52. Only data for which more than one velocity measurement exists are plotted. In addition, data from Experiment 1 are not included as the discharge readings were unreliable. The data illustrated in the plot are exclusively from runs  120  completed at the highest discharge thus far run during the experiments. Therefore the Froude numbers are from ‘channel forming’ flows when the bed is near the limit of stability.  Figure 52. Froude number as a function of time since experiment started for (a) all runs after at least sixty minutes of flow over the bed and (b) only portion of time just before run ended. Error bars illustrate the uncertainty associated with the Froude number based on uncertainty associated with Q, w and standard error of the velocity measurements. Data from the Fitzsimmons experiment and Experiment 1 are not included.  The data illustrate that reach average Froude numbers were generally well below the critical value of 1. They averaged 0.65 for all runs (n = 747) and 0.66 (n = 208) for sections of the record just prior to the flow stopping when the bed was most stable. The data from the last 121  stable run, earlier runs and the bed after it degraded to the flume floor are each plotted using a different color/symbol combination. There appears to be no systematic shift in the Froude number just before the bed fails suggesting the Froude number cannot be used to characterize bed stability. The average uncertainty was 4.9%, which is considerably less than the mean difference between the harmonic mean derived Froude number (which is illustrated in the figure) and the peak in the conductivity pulse derived Froude number. The peak derived Froude number averaged 0.78 and was on average 18% greater. The plot also illustrates data from different bank configurations and suggests bank type does not have a substantial effect on the Froude number. To better assess the trajectory of Froude numbers during individual experiments, the average Froude number for each flow level that was the highest flow thus far is plotted in Figure 53. There is no systematic increase or decrease in the Froude number with time. Eleven experiments show a statistically significant (slope term in linear regression does not equal zero, p < 0.05) increase in the Froude number as flows are increased and the bed adjusts, while four other experiments show a clear pattern of decreasing Froude number with time. During 17 of the experiments, the Froude number did not change significantly as the experiments progressed. In summary, the Froude number was consistently less than 1 during these experiments and did not increase over the course of the experiments.  5.2.2 Width to depth ratio The width to depth ratio indicates the aspect ratio of the channel and is a useful indicator of the amount of bed as opposed to banks to which the flow is exposed. For these experiments the width to depth ratio averaged 6.6, which is relatively low. The width to depth ratio varied between 0.97 and 190 and for most runs was less than 10 (Figure 54). The relatively low width to depth ratios observed during these experiments, which are also common in natural step-pool channels, necessitates the use of hydraulic radius rather than depth in hydraulic and flow resistance calculations. Figure 55 illustrates that for narrow channels the hydraulic radius can be less than half the flow depth. The importance of using hydraulic radius in streams with such small width to depth ratios cannot be overstated and bears heavily on the discussion of scaling flows for changes in slope and grain size that was presented in Section 4.3.4 and wall resistance corrections discussed in Section 6.1. 122  Figure 53. For each experiment the Froude number is plotted as a function of time. Data are from runs that were the largest discharge the bed had thus far experienced. Best-fit linear regression relation is shown if slope is significantly different from zero (p < 0.05).  123  Figure 54. Histogram illustrating water surface width/mean depth ratio for all experiments. Note x-axis has been logarithmically scaled.  Figure 55. Relation between hydraulic radius and flow depth as a function of width (w). Ellipses are experimental data.  124  5.3 Sediment transport and its effect on channel morphology During the experiments it quickly became clear that sediment transport patterns could vary considerably, in part due to well recognized responses to flow increases and sediment supply, but also in response to the formation of sediment transport instabilities. These instabilities and the general pattern of sediment transport were observed to dictate the formation of step-pool structures. This observation was evident in both the video images collected with the overhead camera and data from the GSD. These data also illustrated that step-forming grains did not move regularly, but rather their movement occurred stochastically. On occasion the bed would mobilize immediately following an increase in flow, while on other occasions the bed remained relatively stable for tens of minutes or more before any significant sediment movement began. The focus of the following section is to describe and illustrate the different sediment transport patterns and describe the types of instabilities that were observed using overhead video and GSD results. Particular effort will be made to illustrate how the bed is governed by the chance mobility of keystones. Subsequently examples will be presented that illustrate how the instabilities interrelate with the step-pool form. DuBoys (1879, cited in Graf, 1984) described sediment transport as a series of layers of grains moving over one another with the top layer moving faster than the lower layers. Subsequently Einstein (1937; 1950) proposed that bedload transport is comprised of individual grains rolling, sliding or hopping over the bed so that grain entrainment and travel distances should be considered using a probabilistic approach. Jackson and Beschta (1982) refined this transport pattern and introduced a two phase model of sediment transport. Fine sediment moves over a stable bed during phase one, while bars, riffles and pools migrate during phase two. Ashworth and Ferguson (1989) subsequently illustrated that at low transport rates partial mobility can occur whereby smaller grains are more mobile than larger grains and the relative mobility of each grain size class depends on flow stage. At high flow stages transport may approach a state of equal mobility whereby all grain sizes move equal distances and in equal proportion to their occurrence in the subsurface. Subsequently Warburton (1992) described a conceptual three phase model that expands upon the two phase model of Jackson and Beschta and incorporates Ashworth and Ferguson’s observations. According to Warburton (1992), phase 1 transport occurs when individual fine grains that are mostly sand move over a stable armour layer. During phase 2 transport conditions all grain sizes may move sporadically and the fine sediment may be 125  generally mobile, but the surface structures remain intact. Phase 3 is noted to occur when all exposed grains are mobile and there is a rapid exchange of sediment between the bedload and bed surface. During these conditions scour and fill are widespread and few remaining patches of stable grains exist.  5.3.1 Patterns of sediment transport During this research, while developing the structured beds and observing sediment transport, all three of Warburton’s sediment transport phases were observed. Qualitative examples of each phase of sediment transport, as they relate to steep channels, are presented below. Phase 1 sediment transport was observed to occur between instabilities and frequently during the latter portion of each run when the bed had stabilized. Data from Experiment 22 after the 1100th minute (4th bed scan) illustrates an example of phase 1 sediment transport. At the end of the experiment, before flows were increased transport rates were low (circa 2400 minutes in the Compendium plots and 1100 minutes in Figure 56). During this period the difference maps illustrated in the Compendium show no change in the surface structures. This period of the experiment is typical of phase 1 transport.  Figure 56. Sediment transport record from the end of Experiment 22 when a 9.2 L/s flow was run for a cumulative duration of twenty hours (1200 min). A hydraulic geometry run was done at 180 minutes and the flow was stopped and restarted eight times to characterize the bed changes. The beginning of this plot corresponds with 1100 minutes in the Compendium.  126  Phase 2 transport is illustrated during the later portion of run 1, Experiment 1, which is shown in Video 5. During phase 2 transport conditions the sediment in transport is much finer than the bulk mix in the flume and the largest stones are rarely mobile. This transport behaviour is illustrated in the later portion of Figure 57, which is a plot of the average Dmax, D84, D50 and D16 from one second samples of sediment transport averaged over thirty seconds (as opposed to the Dmax, D84, D50 and D16 of all the material that left the flume during a 30 second period). After twenty minutes the grain size in transport is indeed much finer than the bulk mix in the flume while the video shows that large stones are still occasionally mobile.  Video 5. First 93 minutes of Experiment 1 recorded at a rate of 30 seconds in one second. Flow rate is 15.8 L/s.  The beginning of Video 5 illustrates phase 3 sediment transport conditions, which are characterized by high sediment transport rates. Individual large grains are clearly mobile. This pattern of transport continues for the first ten minutes of the run. In Figure 57 we see that during the first ten minutes the grain size of the material in transport was much coarser than later in the experiments and similar to the bulk mix. During the ninety minutes that are illustrated in Figure 57 the largest stone that left the flume was a 64-90 mm grain and in total twenty-nine 45-64 mm grains left the flume. In Video 5 we also see that all of the largest grains moved during the first twenty minutes of the run. The video and figure illustrate that the largest grains are not regularly in transport, but are indeed mobile. This transport pattern is considered phase 3 transport as all grain sizes are mobile at some point in time and the transport pattern is much more continuous than the phase 2 transport conditions that occur later in Video 5. Like phase 1 and 2, during phase 3 transport conditions the largest grains are less mobile than their smaller counterparts suggesting that in these steep systems, even during phase 3 conditions, equal mobility does not occur. Warburton (1992) also observed that the sediment transport rate was not steady during phase 3 transport, but rather occurred in pulses as steps and structures in the channel upstream failed. Particularly high sediment transport rates during phase 3 transport conditions may be  127  akin to debris flood conditions that have been observed in nature. Hungr et al. (2001) defined a debris flood as a “very rapid, surging flow of water, heavily charged with debris, in a steep channel”. During a debris flood, sediment is propelled forward due to the tractive forces of the water and they can be distinguished from flood events by the large amount of sediment they move.  Figure 57. (a) Average grain size of bedload leaving the flume and (b) sediment transport rate during the first 93 minutes of Experiment 1. On the right hand y-axis of (a) the grain size of the bulk sediment mix in flume is also noted.  5.3.2 Patterns of instabilities During the experiments it was apparent that the occurrence of each of the transport phases was not simply a result of flow and bed conditions, but also depended on the occurrence of bed instabilities. Three of the most common instabilities that dictated sediment transport conditions and the evolution of the bed are presented below.  128  5.3.2.1 Headward migrating instabilities The dominant step-pool failure mechanism was observed to be headward migrating instabilities. These could start near the downstream end of the flume and take tens of minutes to travel the length of the flume, or only a matter of seconds. Video 6 and data from Experiment 16 document the occurrence of one headward migrating instability. In the video thirty minutes is compressed into thirty seconds. The step-pool feature near the top of the flume formed during a run with a peak flow of 8.2 L/s (see Experiment 16 in the Compendium) and persisted through a three hour run with a peak flow of 10 L/s. When the flow was subsequently brought up to 12.3 L/s, the bed slowly had a few minor changes and then, nine minutes into the run, a section of bed immediately upstream of the flume outlet was eroded causing a headward migrating instability to form. This instability travelled to the downstream end of a scour pool that formed about ¾ of the way up the flume. The channel then remained stable for about ten minutes, before the step collapsed and the keystones forming the step were entrained. The entrainment of the keystones released all of the upstream sediment and the base of the flume became visible.  Video 6. A headward migrating instability starts nine seconds (equivalent to nine minutes) into the video and eventually travels the entire length of the flume and degrades bed to flume floor. Video is from end of Experiment 16.  Video 7. Overhead record of sediment transport for first hour of Experiment 8.  Other examples of upstream migrating instabilities are evident during the late stages of Video 7, which illustrates the first hour of Experiment 8, the same portion of time for which sediment transport data are plotted in Figure 37c. The viewing rate is 1 minute in one second. The erosion of sediment downstream of a group of grains resulted in the upstream grains sliding and  129  jamming together. This pattern of coarse grains grouping together, moving as a group and subsequently jamming together can be seen in many of the video records from the experiments.  5.3.2.2 Splays and cutoffs During steep experiments with the fine bulk sediment mix, splays and cutoffs were observed. The first run during Experiment 9 was conducted with just enough water for surface flow to occur, which was about 1.4 L/s. The first part of Video 8 illustrates the splays and cutoffs that formed. Locally, a channel quickly developed through incision into the bed and a high rate of sediment transport followed. The transport rates exceeded the capacity of the small channel resulting in sediment being deposited in splays. These deposits reduced the local bed slope and promoted lateral migration of the channel upstream of the splay. These instabilities tended to start downstream and migrate upstream as the splays formed, promoting upstream lateral migration and in turn another splay and further upstream lateral migration. The video images also illustrate that the large grains often move as a group rather than individually. If one large grain moved there was an increased likelihood that the neighbouring large grains also moved. Clearly the movement of grains in this video is not strictly fluvial, but represents some combination of gravitational and fluvial processes. In contrast, the experiments at 18% slope with the coarse sediment mix did not produce this behaviour.  Video 8. Formation of a meandering channel, bars and the passage of waves of sediment during Experiment 9. Flume gradient was 18% and flow rate was 1.4 L/s.  5.3.2.3 Sheet flow The most active form of sediment transport observed was the sheet flows that occurred during the beginning of Experiment 8. This experiment was run at a gradient of 18% with the same flow rates as the experiments with a gradient of 8%. As soon as the discharge was increased sheets of grains began to be transported down the flume (Video 9). Most of the smaller grains in transport moved with the flow, while the larger grains tended to bind and move along in spurts.  130  The large white 90-128 mm stone can be seen to slide when it was undermined, which is the process by which most large stones were moved during all of the experiments. The video is taken shortly after transport started and as a result the top portion of the flume had already begun to structure and transport rates were decreasing. All of the sediment originates from the bed as there is no sediment feed. The multiple layers of mobile sediment are one of the features used to distinguish this pattern from some of the other transport patterns observed.  Video 9. Transport of sheets of sediment during 18% gradient run with fine mix and new bed. The discharge was 3.4 L/s.  Figure 37c illustrates the sediment transport record for the first sixty minutes of this run. Video 9 covers the time from 1.05 minutes to 2.14 minutes, which is the period with the highest sediment transport rates. Transport rates were estimated to be 1500 g/s, while the water discharge was about 3.5 L/s. Thus the total discharge was about 4.1 L/s, of which 13% was sediment. This flow would be considered hyperconcentrated based on Jakob and Jordan (2001) not Costa (1988). Since there was no fine sediment in the mix (< 2 mm), the flow differs from traditional hyperconcentrated flows and might be better described as a concentrated grain flow.  5.3.3 Role of sediment transport in forming step-pool channels The preceding sections illustrate the range of transport patterns and instabilities that occurred during the experiments; the following section examines and summarizes how the sediment transport phases and instabilities influence the formation of step-pools and bed structures. The first observation is that there is a clear tendency for larger stones to move down channel until they touch the banks. Reviewing the scans from Experiment 16 (see Compendium) illustrates that during the first seven to eight hours of the experiment the larger stones tended to roll until they touch a bank at which point they often stopped moving. The large stones anchored against the banks then become the keystones associated with step development. As an example, after the 10th hour of Experiment 16 a step-pool structure formed due to a large white stone on the left bank. This structure is illustrated with both 4.4 L/s of water flowing over it and dry in 131  Figure 58. The large white stone can be seen to be making contact with the left bank in the photo (red X in photo). By the end of 12th hour of the experiment this structure broke and was replaced with a cascade structure (see Compendium). Subsequently near the end of the experiment a different keystone formed a scour hole in nearly the identical location. This scour pool appears to be contingent on a large triangular shaped stone that moves from the centre of the flume to the left bank where it can be seen to rest against a bank protrusion in the 15th bed scan.  Figure 58. Step-pool formed between 8th and 10th hour of Experiment 16.  A second observation is that during phase three transport, groups of large stones start to form relatively quickly. Then during succeeding phase 2 transport conditions the subtle movement of these groups of large grains into more stable arrangements occurs. In part this adjustment is accomplished by the removal of fine sediment around the large grains promoting the sliding of large grains into each other and the dynamic formation of steps. Video 10 illustrates an example of this process. A large white stone and adjacent larger stones are undermined and as a result they slide and lock up when the large white stone makes contact with the left bank. It is likely that steps that form through this dynamic sliding/locking mechanism are more stable than steps formed through the passive deposition of individual stones one at a time. The increase in  132  stability is hypothesized to occur as stopping a group of stones in this sliding/locking fashion is likely to produce a lattice of force chains or contact points that span the channel, rather than individual contact points that are more likely to develop when stones are deposited one at a time.  Video 10. Jamming of large white stone and neighbouring stones against bank during Experiment 16. Flow rate = 4.6 L/s with feed. Large stones around the large white stone arrived at site by individually rolling into position. Frame rate equals capture rate.  A third observation during the runs was that the sediment transport rate was not steady and was indicative of a stochastic process. Even when the flow rate was kept constant the transport conditions would change between phases, particularly between phase 1 and 2. As an example, Figure 57 illustrates that clear spikes in transport occur, and these are mirrored by increases in the grain size of the transported sediment. Rather than changing little by little to a more stable state the bed was observed to adjust in spurts. In Figure 56 we see that, even after twenty hours of flow, transport rates remained variable. The bed scans for these same data show that after the fourth hour of flow the movement of large stones is no longer evident, but scour and fill can still be detected in the bed scans. In particular the 8th difference map (which follows eight hours of constant flow) shows that scour occurred in one of the pools just upstream of the mid point of the flume. This scour occurs after a one hour run during which almost no sediment transport occurred. Even during prolonged periods of constant flow the transport rate was observed to be unsteady. Spikes in transport can be seen to occur during the last sixteen hours of Experiment 22, after the flow rate had already been held constant for four hours and only 3.6 kg of sediment transport occurred (see Compendium). The absence of any significant bed changes during these sixteen hours suggests that phase 1 transport conditions were occurring. The detection of spikes in sediment transport and relating them to changes in the flume was difficult as the GSD was immediately influenced by what was happening at the downstream end of the flume and did not always record what was occurring further upstream. As an example, at the beginning of Video 8 high rates of sediment transport can be seen along much of the flume,  133  yet the GSD record, illustrated in Figure 59, shows a relatively low mean transport rate of 30 g/s at the beginning of the run. A peak transport rate of around 1500 g/s did occur, but not until 18 minutes into the run when one of the splays emptied onto the screen at the end of the flume and many grains slid across the light table. The movement of the splay onto the screen can be seen near the end of the video. While the splays and cutoffs make this run exceptional, it was generally observed that if the bed failures are closer to the end of the flume the peaks in sediment transport are more pronounced. Furthermore, the large stones associated with bed failures are more likely to exit the flume and be recorded if failures occurred near the end of the flume. In some cases bed failures occurred at the upstream end of the flume, but the GSD recorded only a small amount of the disturbance as the majority of the sediment was redistributed within the channel.  Figure 59. Sediment transport record from first 24 minutes of Experiment 9; corresponds with the duration of Video 8. No feed, discharge = 1.4 L/s throughout run.  In terms of our overall understanding of sediment transport in steep streams the importance of both the headward migrating instabilities and the stochastic nature of bed entrainment must be emphasised. It was observed that during constant flow and feed conditions the bed could be stable at one moment and unstable at another moment, with the transition being dictated by a combination of stochastic processes and headward migrating instabilities. In particular, if a headward migrating instability is occurring in a downstream portion of the flume there is a 134  strong likelihood that the bed upstream will be disturbed. In contrast, the entrainment of a patch of the bed is driven by stochastic processes and cannot be directly predicted. The range in times that it took after an increase in flow for sediment transport rates to increase is illustrated in Figure 60. The distribution is strongly skewed, with an immediate increase in sediment transport being observed during the first minute 46% of the time, an average response time of 9.2 minutes and a median response time of 1 minute. During twenty-six percent of the runs the bed did not break during the first ten minutes after a flow increase. On one occasion the bed did not break until two hours and sixteen minutes into the run.  Figure 60. Time until sediment transport rate increased following an increase in flow.  5.4 Variability among identical runs To examine the variability associated with the development of the step-pool channels, two sets of replicated experiments were completed. The first set consisted of portions of Experiments 5, 6, 7 and 13 and was completed at a medium width (0.359 m), with a flume slope of 8% and the fine grain size mix (see Table 6). The runs from the start of Experiments 5, 6 and 13 until sixty minutes after the end of the last feed were identical and all followed the standard experimental design used during the majority of the experiments. Likewise runs during the first 300 minutes of Experiment 7 were identical to the other three experiments. After 300 minutes Experiment 7 was no longer identical as the last three feeds were run at a slower rate than the other sediment  135  feeding runs. The second set of replicate experiments comprised Experiments 16, 20 and 22. These experiments were identical until three hours after the last feed had finished and were completed with a narrow channel width (0.254 m), flume slope of 14% and the fine grain size mix (see Table 6). To examine the variability between replicate runs the cumulative sediment yield for each experiment is plotted in Figure 61. For the set of experiments at a steeper slope and narrower channel (Experiments 16, 20 and 22) there is more variability in the cumulative sediment yield compared to the wider, less steep experiments (Experiments 5, 6, 7, and 13). Part of the variability observed in Figure 61a is associated with small differences in hyporheic flow that resulted in different amounts of water flowing above the bed, but this effect only persisted for the first 240 minutes. The difference in variability between the narrow and medium width experiments is possibly associated with the narrower experiments (Figure 61a) having twice as many step-pools as the wider experiments. More step-pools would be expected to produce a less regular pattern of sediment transport as transport rates are contingent on the particular configuration of the keystones and headward migrating instabilities, rather than the less stochastic process of unconstrained sediment entrainment. The observation that the narrower, steeper experiments and wider less steep experiments transported similar amounts of sediment over the first 930 minutes (Figure 61) of the experiments is initially surprising. Two counteracting factors may be at work and cancelling each other out. On the one hand, the narrower experiments would be expected to be more stable due to an increase in the propensity for steps to form, which would be expected to cause less sediment to be evacuated out of the flume. On the other hand, it was illustrated in Section 4.3.4 that steeper experiments are less stable; thus, the steeper experiments would be expected to move more material. Since the steeper experiments are also narrower, the effect of channel slope appears to be compensated for by the effect of the channel width. Along with sediment transport data a number of metrics related to the bed and flow conditions were recorded and can be used to examine the variability associated with replicate experiments. Figure 62 indicates that, during Experiments 5, 6, 7 and 13 until one hour after the last feed, the mean water velocity and back calculated mean depth and mean hydraulic radius were similar for all four experiments, but not without some variability. These data include only conditions just 136  Figure 61. Cumulative sediment yield for replicate portion of experiments. Experiment 7 had slower feed rates than Experiments 5, 6 and 13, but was otherwise the same. Lines from a) are replicated in b) as light grey lines.  137  Figure 62. Flow properties as a function of time for the portion of Experiments 5, 6, 7 and 13 until one hour after the last feed. Circles indicate common flow rate for which data can be directly compared. Height of black box indicates median standard error associated with each metric.  prior to the bed being scanned when the bed was relatively stable and do not include the portion of the experiment when the beds were actively adjusting to feed and increases in discharge. During Experiment 13 the Froude number was larger than during the three other experiments. This increase in Froude number is a result of a slightly faster mean velocity and resulting shallower mean depth (Figure 62a and b). At least some of the variability between experiments is attributed to differences in channel form rather than to measurement error. The median standard error is indicated in each plot by the height of a black box and it is evident that the measurement error is frequently smaller than the difference between the replicate runs. Bed conditions during Experiments 5, 6, 7 and 13 are shown in Figure 63 to vary between the experiments. Like the previous figure the variation between replicate experiments is greater than the uncertainty associated with the measurements and represents real differences in channel form. The flow properties and bed conditions for Experiments 16, 20 and 22 up until three hours after the 2nd flow increase following the last feed are shown in Figure 64 and Figure 65. During  138  Experiments 20 and 22 the initial flow rates were not adjusted correctly for hyporheic flow and thus the measured velocities were especially slow and could not be measured. Beyond the first few hours the three experiments are similar with respect to most of the metrics. One noticeable exception is Experiment 16, which is 1-2% less steep than the other experiments after most of the runs. None of the other metrics (grain size, roughness etc) were obviously affected by the lower bed slope.  Figure 63. Number of step-pools, bed slope, standard deviation of bed, D50, D84 and D84step as a function of time for the portion of Experiments 5, 6, 7 and 13 until one hour after the last feed. Circles indicate common flow rate for which data can be directly compared.  To characterize the variability in bed and flow conditions that might typically occur during replicate experiments data from the last run that was part of the replicated portion of each experiment were used to determine the coefficient of variation for a range of flow and bed parameters. These coefficients of variation are presented in Table 19 and suggest that many metrics associated with the replicate experiments vary between 5 and 20%. The DarcyWeisbach friction factor (f) was particularly variable, with coefficients of variation on the order of 35%. In contrast, grain size, particularly the D84 of the bed surface, varied relatively little with coefficients of variation of approximately 4.5%. Clearly flow resistance, which will be examined in detail in Chapter 7, is unlikely to be explained by grain size alone. In summary, the 139  Figure 64. Flow properties as a function of time for the portion of Experiments 16, 20 and 22 until three hours after the second flow increase following the last feed. Height of black box indicates median standard error associated with each metric.  Figure 65 Number of step-pools, bed slope, standard deviation of bed, D50, D84 and D84step as a function of time for portion of Experiments 16, 20 and 22 up until three hours after the second flow increase following the last feed.  140  analysis of replicate experiments illustrates that some variability in pattern, flow and bed conditions is inherent with these channels. The plots also illustrate that with time the bed coarsened, roughness increased and bed slopes were reduced. Table 19. Coefficient of variation from a range of flow and bed property measurements for data from last run of two sets of replicate experiments. Bold numbers illustrate particularly variable metrics.  Experiments 5, 6, 7 & 13 Experiments 16, 20, 22 Experiments 5, 6, 7 & 13 Experiments 16, 20, 22  q  V  y  R  Fr  f  Slope  # Step-pools  2.4%  12.4%  12.5%  8.8%  19%  29%  3.2%  16.7%  4.9%  11.7%  16.2%  10.2%  19%  42.7%  9.5%  17.8%  D50  D84  D84step  D84pool  D84tread  Ls  dr  z  13.8%  5.1%  22.8%  18.9%  7.1%  20.1%  5.5%  15.6%  16.2%  4.2%  3.5%  5.4%  13.3%  21%  26%  25%  5.5 Effect of flow conditioning During early preliminary experiments and Experiments 1 through 3 it was observed that the bed took between twenty minutes and one hour to stabilize; hence it was decided that the runs should have a duration of at least one hour (Section 4.3.1). Runs prior to the last feed were generally run for an hour, while runs after the last feed were, for the most part, conducted for 3 hours. To examine how flow duration and these choices in run duration affected the experimental outcome two types of flow duration experiments were completed. 1. After the last feed during Experiments 5 and 6 flow rates were held constant for only one hour before the flow was increased again. In contrast, during Experiments 7 and 13 the standard experimental approach was used whereby flow rates were held constant for three hours before being increased. 2. During Experiment 7 for the last three feeds (starting at 300 minutes) a feed rate of 35 g/m/s was used while all other experiments had feed rates of 75, 105 & 105 g/m/s for the same three runs, respectively. While the feed rate was different, the same amount of  141  sediment was fed during Experiment 7 and the other experiments; hence the feeding occurred over a longer period of time. In Figures 66 and 67 flow and bed properties during Experiments 5, 6, 7 and 13 are illustrated and a dashed vertical line is used to indicate when Experiments 7 and 13 began to be run for three hours at a constant discharge rather than only one hour. In neither plot can any bed or flow characteristics be seen to differ between the three hour runs and the one hour runs. Any differences between the experiments that are evident appear to exist prior to the experimental procedures diverging from each other. This observation suggests that letting the bed armour for only one hour had essentially the same affect on the evolution of the bed as letting it armour for three hours. A more detailed investigation of the effective of flow duration on bed stability, potentially the most sensitive factor to run duration, is examined in the subsequent chapter.  Figure 66. Flow properties as a function of discharge for Experiments 5, 6, 7 and 13. Vertical dashed line indicates time when Experiments 7 and 13 began to be run at a constant discharge for three hours rather than one hour.  To evaluate if changing the feed rate had an effect on the evolution of the bed the cumulative sediment transport record up to the end of the last feed was evaluated (Figure 61b) as were flow and bed properties (Figures 62, 63, 66 and 67). The cumulative sediment transport record shows that during Experiment 7, despite the lower feed rate, the experiment evacuated more sediment 142  from the flume than Experiments 5, 6 and 13. However, overlaying the plots from Experiments 16, 20 and 22 (Figure 61b) reveals that Experiment 7 falls within the variability associated with these other replicate experiments suggesting the observed differences may be within the range of natural variability. The flow and bed properties during the portion of Experiment 7 that had the slower feed rates (after 300 minutes; 8 L/s < Q < 16 L/s) were similar to what was observed during Experiments 5, 6 and 13 (Figures 62, 63, 66 and 67). Thus in summary it appears that, despite the differences in feed rates, Experiment 7 did not behave noticeably different than experiments 5, 6 and 13.  Figure 67. Bed properties as a function of discharge for Experiments 5, 6, 7 and 13. Vertical dashed line indicates time when Experiments 7 and 13 began to be run at a constant discharge for three hours rather than one hour.  While this analysis on the effect of flow conditions on the bed state cannot be considered comprehensive, and tests that examine how shorter run times affect bed stability are needed, the two tests that were performed did not show any indication that flow duration affected the bed’s response to an increase in flow. In retrospect, it would have been helpful to run some of the runs for much shorter durations and see how limiting the time the bed had to structure affected the stability of the bed. The limited data that are available do not show that run duration has an effect; however, I suspect that very short duration experiments would produce different results. The effect of run duration will be examined further in the next chapter, which develops a jammed state model that incorporates the main bed stability parameters into a single model. 143  6 Step-pool stability: Testing the jammed state hypothesis The purpose of this chapter is to examine the second research question: is channel stability inversely related to the jamming ratio (channel width/grain size) and can stability be predicted based on the applied stress (e.g. shear stress) and jamming ratio?  6.1 Theoretical background To assess bed mobility in a gravel bed river the mean grain size of the bed surface (D50) is commonly related to the strength of the flow. Flow strength can be characterized using stream power (unit; ω = ρgSq, or total; ω = ρgSQ) or a measure of shear stress (τ). Stream power based approaches use discharge directly and thus do not consider the flow depth/velocity relation. Many approaches to shear stress exist; herein DuBoys’ approach is generally used. DuBoys’ (1879) approach assumes uniform flow conditions and requires knowledge about the flow depth and therefore the relation between velocity and depth (see Sections 1.1.3 and 1.1.4 for further details). The uniform flow assumption is invalid for most experimental runs completed during this study since hydraulic jumps and flow transitions are common; hence, there is interest in alternative approaches. Alternative approaches include law of the wall, (e.g. Wiberg and Smith (1991)), Reynolds stress and drag force approaches. These approaches also have their own limitations. The law of the walls approach is predicated on a smooth velocity profile with slower velocities at depth and faster velocities near the surface, yet observations of velocity profiles in step-pool channels have shown that velocity profiles can deviate substantially from this idealized model (Lee, 1998; Wohl and Thompson, 2000). The Reynolds stress approach is most applicable at point locations as it relies on measuring velocity deviations in the x and z directions and it is unclear how to spatially average such point specific data to determine reach scale bed stability. Finally, with the drag force approach, the appropriate bed cross-sectional area that the flow is exposed to is difficult to estimate, and I am unaware of any means of assessing the drag coefficient for the small relative depths observed during these experiments. On account of these difficulties the DuBoys approach was chosen to estimate reach average shear stress values, despite its own  144  problems. Analyses using the law of the wall and drag force approaches were completed and will be discussed briefly. The classical approach to stability is the Shields’ function (Eq. 5), which relates shear stress to the submerged weight of the stone. Classically the bed is predicted to mobilize when the Shields  τ θ ( ρ s − ρ ) gD50 = ) exceeds 1. Based on the stream power approach ratio (Eq. 6; 0.045 θc developed by Bagnold (1977; 1980) the critical stream power for entrainment is not only related to grain size, but also depends on flow properties (Martin and Church, 2000) and the bed state. By bed state I am referring to the observation that bed structuring and grain arrangement influence the mobility of the bed (Church, 1978; Church et al., 1998; Hassan et al., 2008). The critical stream power relation developed by Ferguson (2005), which incorporates the law of the wall and parameterizes critical entrainment using a Shields number approach (θc) is  ωc =  2.3  κ  ⎛ 30θ c R Di ⎞ ⎟ ⎝ eS k ⎠  ρ (θ c RgDi )3 / 2 log⎜  (24)  Where e is the root of the natural logarithms and k is the roughness length, which is commonly some multiple of grain size. The only difference between this approach and Bagnold’s (1980) original approach is that depth is not specified directly in the relative roughness portion of the relation (log(y/k)), but rather is determined indirectly through the use of the Shields parameter. Bagnold’s original approach utilized the Shields parameter to determine the critical shear stress to mobilize the bed, but not flow depth. Like the shear stress approach, a stream power ratio can be used to assess the stability of the bed. Stream power ratio =  ω ρgSq = ω c 2 .3 ⎛ 30θ c R Di ⎞ ρ (θ c RgDi )3 / 2 log⎜ ⎟ κ ⎝ eS k ⎠  (25)  When comparing Eq. 6 and Eq. 25 some noticeable differences are evident. With the shear stress based approach (6), an increase in the hydraulic radius (R; closely related to flow depth), will increase the Shields ratio as the shear stress approach addresses flow roughness and the relation between flow depth and velocity in the numerator portion of the relation through 145  DuBoys’ depth-slope relation. With the stream power approach the relation between flow depth and velocity is in the denominator portion of the function through the law of the wall relation and hydraulic radius is both in the numerator (indirectly through q) and denominator. Ferguson (2005) emphasizes that the stream power approach should yield more convenient results as it does not require knowledge of flow depth to calculate stream power. However, hydraulic radius and a roughness length (k) are required to calculate the critical stream power to entrain particles on the bed. In steep streams the prediction of the relevant roughness length is particularly tenuous (Ferguson, 2007), especially for small relative depths. A fixed multiple of a grain size is almost certainly not appropriate and the entire concept requires further investigation. Stream power is an attractive means of characterizing bed stability since flow resistance does not need to be determined to calculate stream power. There is, however, no means of characterizing the critical stream power to entrain a grain without invoking a flow roughness dependent parameter (e.g. y or R) and the shear stress required to entrain a grain. One would need a function predicting the power necessary to entrain a grain; however, stream power is a measure of the rate at which work is being done, not a measure of force. Thus there is no reason that a well defined minimum stream power to entrain a grain should exist. In addition, my experiments illustrate that the critical stream power for entrainment is slope dependent (while DuBoys derived shear stress is not; see Section 4.3.4), which adds additional complexity to the use of a stream power model. For these reasons only shear stress based approaches are investigated as a means of predicting bed stability. While previous studies have predicted the stability of step-pool channels using classical sediment entrainment functions (e.g. Chin, 1994), this approach does not appear realistic as it suggests step-pool bedforms should be much more mobile than they actually are (Zimmermann and Church, 2001). The additional stability is believed to come from two sources. First, the bed can imbricate and form structures such as stone cells and stone lines, and second, the jamming of stones against each other and across the width of the channel, forming force chains, can help stabilize the bed. The Shields number can be treated as a representative of the bed state and increased to account for imbrication and structures such as stone cells and stone lines, while the jamming ratio can be used to assess the increase in stability associated with step structures forming across the width of the channel.  146  Wall corrections are typically applied to laboratory data as flume walls are smoother than stream banks. The smooth banks are thought to reduce the bank friction and increase the stress on the bed more than it would be in a natural channel (Vanoni, 1975). Two common wall correction approaches were initially attempted, that of Einstein (1941) and of Vanoni and Brooks (1957). In Appendix 1 example calculations and the problems with both approaches are described in detail; suffice to say here that applying either technique to smooth or rough wall data results in the unreasonable outcome that the bed hydraulic radii can be predicted to be more than twice the total hydraulic radii. These large hydraulic radii result in wall corrected bed shear stresses being much greater than total shear stresses. In particular, mean wall corrected hydraulic radii are predicted to be similar to mean flow depths, which implies that the wetted perimeter associated with the banks is removed when the wall correction is applied. During my experiments the wetted perimeter associated with the banks was on average 30% of the total wetted perimeter (see Section 5.2.2); hence the banks, smooth or rough, contribute significantly to the total resistance. Existing wall corrections appear to over correct for the effect of smooth walls. On account of this observation, wall corrections have not been applied as it appears they are not applicable to channels that have small width to depth ratios. There is an apparent need for a refined wall correction approach.  6.2 Experimental data To select data for the stability analysis the record of flow velocities and sediment transport rates collected before and after each bed scan was reviewed and depending on the transport rate, flow velocities and bed changes a channel stability class was assigned. The data from the first hour of each experiment and Experiment 1 were excluded. Using the selected records average water velocities were then determined and used to calculate flow properties such as water depth and hydraulic radius. To be included in the analysis more than one measurement of water velocity was required.  6.3 Classifying bed changes During the experiments a wide range of responses to an increase in flow were observed. During some runs the bed surface barely changed, while during other runs the bed was completely transformed. In order to examine channel stability these changes needed to be classified and  147  characterized so that data from stable runs could be compared with data from unstable runs. To examine channel changes difference maps were created by subtracting the latter bed scan from the prior (see Compendium for all the difference maps). These led to the development of four classes of stability. Class 0 data were selected by reviewing the sediment transport and flow velocity record from runs with no feed and selecting data from the end of runs during which the sediment transport rate was low and velocities remained stable. These data came from periods of time when the bed had ample time to structure (see Section 4.3.1 for details on response time of channel). For these data bed properties such as bed slope and grain size collected following the run were used to assess the bed state during the run and the bed was considered to be stable. The other three classes all relate to changes in the bed surface that were observed to have occurred over the course of a run. These three classes bear similarities to the three phases of sediment transport discussed in Section 5.3.1, but differ in that they describe changes over a period of time along the entire length of the flume, while the phases describe conditions at a moment in time and space. Multiple phases of sediment transport can, and frequently did, contribute to the disturbance class. These three classes correspond with three categories of change: (I) bed structure underwent little to no change, (II) bed structure modified or (III) bed structure destroyed. Specifically, Class I disturbances consisted of the movement of small grains and isolated large grains but channel spanning patches of scour and/or fill did not occur. Class II disturbances consisted of isolated areas of scour or fill that may have locally formed new steps and pools, but wide spread changes did not occur. Finally, Class III disturbances occurred when there were widespread changes throughout the length of the flume and multiple areas of scour and/or fill could be identified. Three runs during Experiment 23 illustrate Class I, II and III disturbances. The changes that occurred between the 4th and 5th scan (4th difference map; see Compendium) consisted of small patches of scour, mostly near the downstream end of the flume, and were classified as Class I disturbances. During the next run the flow was increase from 27 to 32 L/s and the observed changes were classified as a Class II disturbance since a scour hole clearly developed. Two runs later, the flow was increased from 32 to 39 L/s and widespread degradation occurred (difference  148  map 7; changes occurred between scan 7 and 8). This degradation was classified as a Class III disturbance. The end of each of these runs, when the bed was stable, represents Class 0 beds. Table 20. Grain size and bedload transport rate for each bed disturbance type.  II: Patches  Mean σ Max Min  I: Fine sediment and individual grains 26 1.8 1.7 7.3 0.1  73 8.2 4.9 28 1.0  III: Widespread change 46 35 58 351 3.2  Mean  2.6  8.1  12.9  Mean  0.48  0.51  0.60  Bed disturbance class Sample size Average sediment transport rate (g/s/m) Time until significant sediment transport began (min) D50 bedload/D50 flume mix  Figure 68. Box plot illustrating range of unit sediment transport rates associated with each channel change class.  For the Class I, II and III disturbances the following observations were recorded: the amount of sediment exiting the flume over the duration of the run, time until sediment transport rate increased following a flow increase, and the grain size distributions of the sediment leaving the flume. Table 20 and Figure 68 suggest that the bedload transport data support the qualitative classification scheme. Class III disturbances were associated with significantly more sediment transport than Class II disturbances, which were associated with significantly more sediment transport than Class I disturbances (ANOVA post-hoc Tukey tests reveal p < 0.001 using logarithmically transformed data). Class III disturbances on average took longer to respond to an increase in discharge than Class II, which took longer than Class I; however, the differences were not statistically significant (ANOVA, p = 0.27). While the D50 of the transported material was on average larger for Class III disturbances, the differences were not statistically significant 149  (ANOVA). If, however, a single Class I disturbance from run 12 Experiment 22 that mobilized two 90-128 mm grains during an otherwise stable run is removed from the analysis, Class I disturbances were found to move significantly finer sediment than Class II and III disturbances. The two large grains associated with this outlier contributed 371 g of the 820 g moved during a nine hour run that had an exceptionally low transport rate. The two large grains that moved during this unique low transport rate run strongly biased the grain size distribution of the transported material. For each run that was associated with an increase in discharge, velocity data from the beginning of the run, along with the bed disturbance class of the run and the bed state prior to the start of the run provided data on how the beds responded to an increase in flow. These data, along with data from the end of runs when the bed was stable, are used to develop the bed failure criterion for step-pool channels.  6.4 Results The results are broken into two main sections: first I determine what conditions lead to the formation of stable beds and second I examine when these stable beds break. Subsequent sections examine replicate experiments, the effect of feed and compare my data with data from the literature. For the stable bed analysis the data that were analyzed included data from the end of runs when the bed was stable (Class 0 events) that were conducted at the largest flow the bed had thus far been exposed to and Class I disturbances. For the bed failure analysis, data from hydraulic geometry runs along with data from the runs during which the bed broke (Class II and III) were added to the analysis. The hydraulic geometry data were included as they help distinguish stable and unstable beds. For the majority of the analysis depth-slope (DuBoys) derived estimates of shear stress were used.  6.4.1 Stable bed states The most basic bed stability relation is between grain size and shear stress. As the grain size of the bed increases the shear stress necessary to mobilize the bed also increases. Data from flow conditions that were the largest run thus far during which the bed was stable bed (Class 0 and I) are plotted in Figure 69. The commonly observed relation between mean bed grain size (D50) and shear stress is clearly evident. Data from rough vertical bank experiments plot similar to 150  data from the rough angled experiments, while data from the smooth bank experiments plot above the other data. A similar relation is also evident in Figure 69b, which plots the D84 of the steps as a function of the maximum applied shear stress. This relation is illustrated as the D84 of the steps is thought to represent the grain size of the sediment increasing the stability of the bed by jamming across the channel width.  Figure 69. D50 of bed (a) and D84 of steps (b) as a function of DuBoys derived shear stress.  To directly examine if additional stability is introduced by stones jamming across the channel, in Figure 70 the maximum applied shear stress is plotted as a function of the jamming ratio. The data define an envelope above which stable beds did not occur as the shear stresses would exceed the limit of stability. At small jamming ratios (large grain size to width ratios) the maximum shear stresses attained were much greater than those attained at large jamming ratios. The data from experiments with smooth walls are also plotted and are shown to have a similar trend but plot below the rough bank data. This result suggests that bank roughness increases the stability of the bed. Shear stress is related to grain size (Figure 69) and as a result the relation in Figure 70 is not simply a plot of the effect of the jamming ratio on the maximum shear stress at which the bed remains stable, but also incorporates the relation between shear stress and grain size. To control for differences in grain size between runs the Shields ratio (Eq. 5) is plotted against the jamming ratio in Figure 71. To determine the Shields ratio the D50 of the surface sediment along the 151  entire flume was used. In Figure 71a the ratio between the channel width and D84 of the entire bed is plotted as this plot best illustrates the experimental design and the six combinations of channel width and grain size that were used (See Table 6 for complete summary). The jamming ratio, calculated with the D84 of the steps, is plotted in Figure 71b and used for the subsequent jamming analysis. Comparing Figure 71a and b illustrates that the relation between the Shields ratio and the jamming ratio is slightly more continuous if grain size is indexed with the D84 of the steps instead of the mean D84 of the bed surface. For this reason and because it is the step stones that are thought to jam across the channel, the D84 of the steps is used for all subsequent analysis.  Figure 70. Maximum DuBoys shear stress sustained on the bed as a function of jamming ratio (w/D84steps). Envelopes encompassing smooth and rough wall experiments are fit by eye.  The jamming plot in Figure 71b clearly indicates that at large jamming ratios the Shields ratio remains more or less constant, but is quite variable, then as the jamming ratio decreases the Shields ratio increases. It appears that above a jamming ratio of six the stability of the bed does not increase due to particles jamming across the channel. This observation suggests force chains spanning the width of the channel have a maximum width of about eight to ten grains. The Shields ratio is the preferred means of assessing stability as it predicts mobility based on the grain size of the bed. A Shields number of 0.045 is used here for reference purpose, but other 152  values could also be used and would simply linearly shift the data. Shields ratios for the structured beds that are plotted in Figure 71 are expected to plot above 1 on account of the armouring and structuring of the bed that occurred during the stabilization of the bed.  Figure 71. Maximum sustained Shields ratio as a function of jamming ratio based on D84 of bed surface and D84 of steps. Vertical dashed line at jamming ratio of 6 indicates where jamming ratio appears to begin to affect stability. Each symbol shape corresponds with a particular w/D50mix ratio.  A major uncertainty when examining the data plots is how much of the variability can be attributed to measurement error. To examine this uncertainty, the error associated with each metric used to plot the data was estimated and the cumulative standard errors for each data point are plotted in Figure 72. See Section 2.3 for error calculation details. The mean error associated with the jamming ratio and the Shields ratio was ± 0.29 and 0.28 respectively. As is evident in Figure 72, relative to the range in observed values, there is more uncertainty associated with the Shields ratio than with the jamming ratio. The majority of the uncertainty associated with the Shields ratio originates from the measurement of mean water velocity, which is used to determine flow depth, hydraulic radius and shear stress.  153  Figure 72. Jamming plot for stable beds for all jamming ratios (a) and jamming ratios ≤ 6 (b). The error bars represent the standard error associated with each measurement.  To investigate if some of the scatter in Figure 72 can be related to bed or flow metrics from the experiments, a non-linear regression relation was fit to all of the stable bed data from experiments with rough banks and jamming ratios less than 6. The best-fit non-linear functional regression relation had the form ⎛ w ⎞ θ ⎟ = 6 .7 ⎜ ⎜D ⎟ θc ⎝ 84 step ⎠  −0.57  ; r2 = 0.27, p < 0.001, n = 116  (26)  The residuals from this regression were plotted against a range of bed, flow and grain size parameters to investigate if any of the remaining scatter could be explained. The variables included z, z/D84, average H, maximum H, dr/D84, L, Ls, maximum dr, average dr, dr/yc, average step length, fraction of bed step-pools, # step-pools, # step-pools/w, bank roughness/w, σbed, y/D84, Lsp, L/w, ω, yc, f, n, C, Fr, A, w/y and the start time of each run. The fraction of the bed composed of step-pools explained the most variance (F statistic = 8.5, p = 0.004), but inspection of this relation revealed that the relation was particularly poor. For this reason no additional parameters were added to the stability function and Eq. 26 is the final form. To examine what variables may explain the variation in Shields ratio for jamming ratios greater than 6 a forward step-wise regression was performed using the same variables as above. While a few variables were statistically related to the Shields ratio, inspection of these relations indicated 154  that the relations existed because of three outliers, once these were removed, their was no clear relation between the Shields ratio and the flow or bed state parameters listed in the previous paragraph. To illustrate the amount of variability that this jamming model (Eq. 26) does not explain the predicted Shields ratio for stable beds is plotted against the observed Shields ratio in Figure 73. Error bars associated with the Shields ratio and regression are also included.  Figure 73. Relation between predicted Shields ratio using Eq. 26 and observed Shields ratio. In (a) standard errors associated with measurement of Shields ratio are illustrated. In (b) the plotting location of non-standard experiments is shown and in (c) data from runs before and after the last feed are shown.  155  6.4.2 Effect of varying experimental design and run duration on stability The data included in the analysis thus far are from experiments that both followed the standard experimental design and those that did not. In addition, data from the beginning of the experiments as well as the end, when the bed had experienced more stabilizing flows, have been included. To examine the belief that beds experiencing longer durations of flow are more stable, ANCOVA was performed with logarithmically transformed Shields ratio and w/D84step. Data from standard experimental runs had similar Shields ratios to those from non-standard runs (p = 0.8, n = 116, Figure 73b). The runs that were conducted after the last feed finished were also similar to those conducted before the last feed had finished (p = 0.34, n = 116, Figure 73c). Bank roughness was the only factor that was shown to have a substantial effect on the Shields ratio (ANCOVA, p < 0.001). Beds from smooth banks never remained stable at Shields ratios as high as those with rough banks (Figure 72). This result suggests that bank roughness has a much stronger effect on bed stability than the particular flow and feed history. Further investigations are warranted in which the bank roughness is varied, backed up by field surveys that assess the natural range and controls on bank roughness.  6.4.3 Differentiating unstable and stable beds In the preceding section conditions affecting the maximum stable Shields ratio associated with stable beds were considered; now differentiating stable and unstable beds is considered using data collected after stable beds were exposed to an increase in flow. Unstable beds were classified as the beds that showed signs of isolated pockets of scour and/or fill (Class II, see Section 6.3 for details) or widespread scour and fill (Class III). Hence unstable runs were the runs that had a step failure, which may or may not have propagated. Beds that underwent minor changes (Class I) were grouped with the Class 0 stable beds from the end of runs. Along with this stability information, bed properties before the flow was increased and flow conditions observed once the target flow rate was achieved were used to assess what conditions were associated with channel failures. To extend the parameter space of stable beds, data from hydraulic geometry runs during which the bed was stable were also included. Using these data, logistic regression was performed to predict the likelihood of a bed being unstable for a particular combination of θ/θc and w/D84steps on logarithmically transformed data yielding  156  ⎛ w ) g (unstable) = −6.2 + 1.7 log⎜ ⎜D ⎝ 84 step  ⎞ ⎛ ⎟ + 7.8 log⎜ θ ⎜θ ⎟ ⎝ c ⎠  ⎞ ⎟⎟ ; p < 0.001, n = 489 ⎠  (27)  where ĝ is the logit function (Steinberg and Colla, 2007). The predicted probability of the bed ) being unstable ( π ) is obtained using )  e g (unstable ) π (unstable) = ) 1 + e g ( unstable ) )  (28).  The logistic regression correctly classified 82% of the stable beds (n = 395) and 24% of the unstable beds (n = 94) with an overall classification success of 71% and is illustrated in Figure 74. This relation applies only to the data collected during these experiments and field testing of the relations must be completed before it is applied to other channels.  Figure 74. Probability of bed being unstable as a function of the Shields ratio and jamming ratio.  While the discrimination between stable and unstable beds using the DuBoys derived Shields ratio (plotted in Figure 74) is statistically significant, the discrimination of stable and unstable data is not easily accomplished by eye. To investigate if a better discrimination was possible, 157  each pair of stable and subsequently unstable beds was examined. Examining each pair of data points led to the observation that hydraulic radius did not always increase in response to an increase in flow. This result is counterintuitive and led to a more detailed examination of what factors might better distinguish stable and unstable beds. To examine if other parameters are better predictors of bed stability a large number of parameters were tested in a forward step-wise logistic regression. The parameters examined in the step-wise regression included τ, θ/θc, ω, v, S, Q, q, yc, # step-pools, D50, D84, D84step, D84pool, z, z/D84, H, H/D84, step length, step length/w, Ls, Ls/w, Lsp, Lsp/w, dr, max dr, max H, A, y, R, f, Fr, v/(gσ)0.5, v/(gD84)0.5, v/(gD50)0.5, v/(gD84step)0.5, σ, (gRS)0.5, y/D84, y/σ, v/u*, D50/w, D84/w, D84step/w, w/D84, w/y, fraction bed step-pools, θ_D84/θc, θ_D84step/θc, Re, τLW, θLW/θc and the logarithms of v, S, Q, q, f, v/(gD84)0.5, v/(gD84)0.5, v/(gD50)0.5, v/(gD84_step)0.5, y/D84, y/σ, v/u*, # step-pools, ω, w/D84step, D50/w, D84/w and D84step/w. θ_D84/θc and θ_D84step/θc are identical to the Shields ratio used earlier, but rather than considering the mobility of the D50 as was done earlier, the D84 of the bed and steps was used for these two parameters, respectively. τLW refers to shear stress calculated using the Wiberg and Smith (1991) law of the wall approach. With the Wiberg and Smith approach a Shields ratio was also calculated (θLW/θc). Based on all of the rough bank data (n = 395, 88 unstable), the variable most able to distinguish stable from unstable beds was mean velocity (v), followed by log v, θLW/θc, v/(gD50)0.5, v/(gD84)0.5, log(v/(gD50)0.5) and log(v/(gD84)0.5) in descending order. Thus it was observed that mean flow velocity compared to depth-slope derived estimates of shear stress can more readily distinguish stable from unstable beds. The depth slope derived estimates of shear stress are shown in Figure 75 to be nearly the same for both stable and unstable runs while the mean velocities differ. Interestingly, there is also no clear relation between shear stress and velocity for runs with the same bulk mix (Figure 75). While velocity is a better discriminant of stable and unstable beds, it and estimated shear stress values derived using mean velocity (either Wiberg and Smith law of wall or drag force approaches) failed to illustrate the increase in bed stability associated with smaller jamming ratios. The Wiberg and Smith and drag force based shear stress values did not correlate with the jamming ratio, suggesting jamming did not affect stability, while the DuBoys shear stress estimates showed that below a jamming ratio of six the bed stability was increased as the 158  jamming ratio decreased (Figure 71). The later is the correct result as experimental runs that were identical in every manner except channel width always withstood larger unit discharges when the channel was narrower. While a DuBoys based approach is less able to distinguish stable from unstable beds compared to velocity the DuBoys approach is the most reasonable approach overall. The reason velocity is a better predictor of stability is examined in greater detail in Section 6.5.3.  Figure 75. Distinguishing stable and unstable beds based on depth-slope derived shear stress estimates and mean velocity. Stable bed data are from runs that were completed at the highest discharge thus far run during the experiments and thus represent critically stable bed states. The ellipses enclose the 0.67 probability of the sample means.  6.4.4 Replicate experiments In Section 5.4 the variability in flow and bed metrics that occurred during replicate experiments was explored in some detail. In general the replicate experiments were shown to produce similar, but not identical, bed and flow conditions. Herein, it is worth revisiting these replicate experiments to evaluate how the threshold for bed stability varies between replicate experiments and if variability between replicate experiments exceeds the measurement uncertainties. Data from the replicate experiments completed at an 8% slope that were collected at the end of each run just prior to flow being stopped are presented in Figure 76 (Experiments 5, 6, 7, and 13; see Section 5.4 for details of the replicates). During Experiment 7 bed scans were completed after one hour and three hours and as a result two data points are presented for each run. Along with  159  the data points, standard errors associated with each data point are indicated as are polygons that illustrate the range of conditions that were observed.  Figure 76. Jamming data from Experiments 5, 6, 7 and 13.  Bed stability data from the four replicates are shown to vary, and while much of the variability can be explained by the standard errors associated with the data points, some of the variability cannot (Figure 76). This additional variability is inferred to arise because of real differences in how the beds evolve and, in particular, the grain size of the steps.  6.4.5 Effect of sediment supply on stable bed configurations In the Introduction (Section 1.1.3) it was noted that, depending on flow history and bed conditions, both high and low sediment transport rates were hypothesized to affect step stability. Step destruction was observed herein and by Curran (2007) to occur due to downstream scour, which would be expected to be enhanced during low sediment supply conditions as high transport rates can limit scour depths (Whittaker, 1987; Comiti, 2003). Conversely, low sediment supply promotes bed structuring and is supposed to increase the stress required to mobilize the bed, which suggests high supply rates should reduce stability. Koll and Dittrich’s (1998) results support this suggestion as high sediment transport rates were shown to destabilize steps. In addition, step-pools were observed to be destabilized in the Rio Cordon in response to a sharp increase in sediment supply (Lenzi, 2001). While a complete series of experiments on the effect of sediment supply on step-pool stability is beyond the scope of this thesis, in part because feed duration, texture, and rate are likely all important factors affecting how sediment 160  supply affects a bed, two experiments were performed to explicitly test the effect of supply on stability. During Experiment 20, following the last of the four feeds after the 2nd increase in flow, rather than increasing discharge, sediment was fed into the channel and the response monitored. The experimental procedure is illustrated in the Compendium and consisted of three runs with sediment feed. After each feeding run, and again after an hour of no sediment feed, the flow was stopped to scan and photograph the bed. Feed rates were 2.2, 11 and 28 g/s (8.7, 43 and 110 g/s/m). The fine feed grain size distribution (Figure 16) that was used was a 1:20 scaled representation of the mobile cobbles and gravels observed in Shatford Creek (Zimmermann and Church, 2001). The initial feeding of sediment produced little change in the bed surface; only a few sections of aggradation are evident in the difference maps (see Compendium). One of these patches is visible in Figure 77 and is encircled by a red oval. Despite little evidence of morphological change along the bed, a small reduction in the fine end of the grain size distribution is evident (Figure 78a). Prior to the feed starting, the mean velocity was 55 cm/s, while at the end of the feed run when conditions had stabilized, the velocity had increased significantly to 62 cm/s (ANOVA, p < 0.001). Following each feed, clear water flows were run for an hour. During the first clear water run that followed the first feeding run, the water velocities decreased significantly from 62 cm/s to 56 cm/s (linear regression, velocity as a function of time, p = 0.001). Accompanying this decrease in velocity was an increase in the grain size of the bed at the fine end of the grain size distribution (Figure 78). A section of the bed was also observed to break loose in the middle portion of the flume (7th difference map in Compendium). During the second feeding of sediment no change in the velocity was evident until after 25 minutes of feeding when the bed broke and significant degradation of the upper 1/3 of the flume occurred (see Compendium). This degradation resulted in the bed slope being reduced from 12.4% to 10.7%. In Figure 77 it is evident that some of the larger keystones remained stable (e.g. large white on right bank) while others were mobilized. Furthermore the bed was covered with more fine sediment, which was picked up by the grain size distribution as a reduction in the D50, D25 and D16 (Figure 78). Following an hour of clear water flows the fine end of the grain size distribution coarsened while the water velocity increased slightly from 54 to 57 cm/s. The 161  third feed, like the other two feed runs, resulted in a slightly finer fine end of the grain size distribution (Figure 78), but no appreciable change in the mean water velocity. During the final hour when no sediment was fed into the flume the water velocity slowed to 51 cm/s. At the end of this run the top portion of the flume floor was visible.  Figure 77. Photographs of bed during Experiment 20 when fine feed was added while discharge was held constant. Red oval indicates an area where deposition of fine feed material occurred and illustrates the size of the feed material. Additional areas of fine sediment deposition can be seen in subsequent images. Section is from mid flume to ¾ of the way towards the top of the flume.  162  Figure 78. Response of surface grain size distribution to (a) sediment feed while discharge was held constant and (b) constant discharge with no sediment feed.  To assess whether the degradation observed during Experiment 20 was due to the introduction of feed or simply prolonged run times, Experiment 22 was run to mimic Experiment 20; however, during Experiment 22 no fine sediment was fed into the flume. During Experiment 22 over a seven hour period at a constant discharge the mean velocity decreased from 64 to 52 cm/s, then returned to 64 cm/s and stayed constant for the remaining 10 hours. These changes in velocities were larger than the changes observed during the Experiment 20 fine sediment feeding runs. The difference maps included in the Compendium show little change in the bed during the constant flow runs. A small amount of scour and fill, which was similar to the amount observed during the first fine feed in Experiment 20, was observed during the fourth hour at 9.2 L/s. This scour coincided with the first fine sediment feeding run during Experiment 20, implying that the changes observed during the first fine feed of Experiment 20 could have occurred even if no feeding had occurred. Beyond the fourth hour at 9.2 L/s Experiments 20 and 22 were observed to take on distinct trajectories. During Experiment 20 the fine end of the surface grain size distribution fined after each feed, especially after the first feed, while during Experiment 22 the  163  fine end of the distribution remained more or less constant (Figure 78). Interestingly, in both experiments the coarse end appears to have coarsened slightly, perhaps reflected a slow armouring of the bed (Figure 78). During Experiment 22, by the 16th hour at 9.2 L/s the mean sediment transport rate was reduced to 0.002 g/s whereas the last run during Experiment 20 had a transport rate of 0.015 g/s. The major difference between Experiments 20 and 22 is that during Experiment 22 after 20 hours at a flow rate of 9.2 L/s the bed did not break and the bed proceeded to a stable state with a bed slope of 11.4%. Conversely, after nine hours at the same flow rate with three pulses of fine sediment feed, Experiment 20 degraded to a bed slope of 10% and broke through the armour to reveal the flume floor. The flume floor was not revealed in Experiment 22 until the third hour after the flow rate was increased to 11 L/s. After only an hour at 11 L/s the flume floor was not visible. Interestingly, Experiment 16, which had the same experimental setup, except that the flow was increased 20% after three hours, did not degrade to the flume floor until a flow of 12.4 L/s was run over the bed. The bed was, however, reduced to a gradient of 10.3% after three hours at the 10 L/s flow rate. This observation suggests that the differences between Experiment 20 and 22 could have occurred as a result of the structures that formed and bed conditions that occurred, not as a result of feed: during the first three hours at 10 L/s Experiment 16 degraded to a slope (10.3%) almost as low as that attained at the end of the third fine feed run during Experiment 20 (10.0%). The preceding analysis is based on when the flume floor became visible, which is a poor means of judging stability. During Experiment 22 the flume floor was visible when the bed slope was 11%, but during Experiment 16 it was not visible when the bed slope was 10.3%. The reason for this apparent discrepancy is that the bed at the downstream end of the flume can influence the amount of degradation that can be accommodated without exposing the flume floor. The downstream portion of the flume during the last few runs of Experiment 16 is clearly higher in elevation than the downstream portion of the flume during Experiment 22. This difference in elevation is likely the result of large stones forming structures just upstream of the outlet of the flume. These structures are visible in the scans (see Compendium). For this reason, more formal analysis using the Shields ratio/jammed state approach is necessary to assess the effect of sediment supply on bed stability.  164  As an improved means of assessing the effect of feed, data from Experiments 16, 19, 20 and 22 are plotted in Figure 79 as a function of the Shields ratio and jamming ratio. This approach accounts for flow roughness (through a measurement of hydraulic radius), bed slope, the jamming ratio, and small differences in discharge between the replicate runs making it possible to directly compare the experiments. The data from Experiments 16, 20 and 22 had the same history of sediment feed, flow rates, flow duration, grain size mix and flume slope until three hours after the last feed; thus these data can be directly compared. In the figure the first data points from Experiments 16, 20 and 22 are from the beginning of a run just after the flow was increased to 10, 9.8 and 9.2 L/s, respectively. Figure 79 illustrates that these three ‘identical’ data points have different Shields and jamming ratios even before fine sediment was fed into the flume during Experiment 20. The variability observed among these three data points is not associated with one single variable, nor do the independent variables that influence the Shields ratio interact in a systematic way. For Experiments 16, 20 and 22 the Shields ratio was 2.5, 3.6 and 3.6, respectively. For the same runs the jamming ratio was 4.6, 4.4 and 4.6 and the probability of the bed failing was 0.12, 0.31 and 0.33, respectively. While some of the differences in the Shields ratio can be attributed to the differences in flow resistance (the hydraulic radius was 4.4, 4.9 and 4.0 cm respectively) and differences in the D50 (18-23 mm), bed slope (10.2-12.4%) had the largest influence. Experiment 22 had the highest Shields ratio (3.6) yet it had the smallest hydraulic radius (4.0 cm) and was the second steepest (11.8%). The high Shields ratio can be attributed to the small D50 of the bed surface (18 mm). Conversely, Experiment 16 had the smallest Shields ratio, which can be attributed to the D50 and lowest bed slope, but not to the largest hydraulic radius. The differences between the experiments may result from the random location of the larger grains in the flume and/or be the result of stochastic behaviour during the runs. Moving beyond the period when Experiments 16, 20 and 22 had identical flow histories it is possible to assess the effect of feeding fine sediment (Experiment 20) on bed stability by following how each run progressed. During Experiment 20 the bed was scanned immediately after the feeding of sediment ended, and as such, the bed was not necessarily stable prior to each bed scan. To illustrate if the bed was stable, beds that underwent Class II or III disturbances after they were scanned are plotted as unstable in Figure 79 while those that experienced only Class I disturbances are plotted as stable.  165  Figure 79. Jamming plot illustrating effect of feeding fine sediment. The probability of the bed failing based on the logistic model is also illustrated with grey lines.  During Experiment 20 the first feeding run had a small effect on the jamming ratio while the Shields ratio remained nearly constant. Despite the addition of feed the probability of the bed failing actually decreased from 0.32 to 0.21. The recovery from the feeding led to a large Shields ratio and a larger probability of failing (0.34), largely because the D50 fined from 22 to 20 mm. This fining was unexpected as the bed was expected to coarsen and stabilize rather than become less stable. During the second feeding run the bed degraded from a slope of 12.4 to 10.7%, but this reduction in slope was compensated for by a substantial fining of the bed (20 to 13 mm) resulting in a Shields ratio substantially larger than before the feed started and a probability of bed failure of 0.67. The large Shields ratio can be attributed almost entirely to the reduction in the D50 as there was only a minor change in the hydraulic radius (0.8 mm). During the run that followed the second feed, sediment was mobilized and the bed armoured again and the D50 coarsened to 21 mm, yielding a 0.25 probability of failure. During the third feed a large reduction in the surface grain size distribution was not observed. Instead the steps were rearranged, resulting in coarser steps, a smaller jamming ratio and a 0.19 probability of bed failure. Following the end of the third feed the bed was initially very mobile, resulting in the removal of fine sediment and destruction of a few steps. 166  In the case of Experiment 22 the data from the end of the first three hours (2nd data point) has a 0.33 probability of failing and during the 4th hour the bed degraded from 11.8 to 11.4% slope and coarsened from a D50 of 18 to 22 mm, resulting in a new Shields ratio that had a 0.14 probability of failure (4th data point, Figure 79). During the subsequent thirteen hours at a constant discharge the probability of the bed failing remained between 0.13 and 0.27 (error bars overlap). Experiment 19 did not follow the standard experimental procedure since when it became time to add the fourth coarse feed during the building of the step-pool structure, the bed was already at a slope of 20% and there was insufficient room in the flume to add more sediment. Instead the flow was increased to 22 L/s as per normal and run for three hours, but no sediment was fed into the flume (see Compendium for complete details). Subsequently the intent was to add the fine feed sediment mix at a high feed rate for an hour at discharge of 22 L/s. The feed was added but, by mistake, at a discharge of 19 L/s; nevertheless, similar results likely would have occurred. During the feeding of fine material in Experiment 19 the fines could be seen in suspension despite their relatively coarse size (see Figure 16 for distribution), and the fed sediment moved directly through the flume. Of the 260 kg of sediment fed into the flume, 245 kg was evacuated during the run and only minor changes in the bed could be detected (see Compendium difference maps). It was evident that the fines were easily mobilized through the channel and the structure of the channel changed little. The high Shields ratio in Figure 79 reflects the fining of the surface grain size from 48 to 36 mm due to the feed. Upon the continuation of flow the grain size distribution coarsened to 47 mm suggesting the feed affected the grain size distribution of only the surface grains, but not the structure of the channel. This suggestion is confirmed by the bed scans, which show the bed prior to the start of the feed as being similar to the bed one hour after the feed had been stopped. In contrast, the scan immediately after the end of the feed does show fewer pockets in the bed and a smoother bed, especially in the pools. The only difference between the bed before the start of the feed and the bed afterwards is the formation of a small scour pool about 2/3 of the way up the flume, which is illustrated in the upper third of Figure 80. Comparing the first and second scans it is clear that grains were entrained and sediment was scoured from downstream of a large stone that was jammed across the channel. In the same figure we see that the scour pool downstream has undergone some changes but the adjacent steps remain essentially unchanged. Clearly relatively large stones can move through the channel without destabilizing well formed step structures. 167  Figure 80. Formation of scour hole during the fine sediment feed run of Experiment 19. Uppermost scan is prior to fine feed, middle scan is immediately after feed and bottom scan is after 1 hr of clear water flows after fine feed. Flow is from right to left. Red arrow indicates new scour hole.  In summary, Experiments 19 and 20 illustrate that over an already stable bed feeding fine sediment will reduce the surface grain size distribution, which temporarily increases the Shields ratio but, as the fed material is removed, the surface coarsens again and the Shields ratio returns to levels similar to other experiments that were not exposed to the feeding of fine sediment. Thus my fine feed experiments failed to show that the addition of fine sediment destabilized the larger grains. It would appear that, if the sediment being supplied to a step-pool channel is sufficiently fine, it can be moved over a stable bed without destabilizing the structures.  6.4.6 Comparing results with results from other studies Over the last few decades a number of researchers have examined step-pool streams using both flume and field observations (see Table 21). An initial attempt to incorporate these data into a stability plot was undertaken by Church and Zimmermann (2007). These same data will now be compared with the results from this study. In order to make this comparison, some important differences among the studies had to be overcome. One challenge was the differences amongst the 13 studies in channel widths that were measured. In Figure 81 a section of East Creek is shown. The 2.2 m3/s water line (3rd largest flow in 36 years of record) is noticeably wider (7.55 m) than the edge of the mobile substrate (4.47 m) as a result of over-bank flooding. To construct the original jamming plot in Church and Zimmermann (2007) it was suggested that the flume width used during previous experiments 168  with smooth walls was likely most similar to the narrower portions of the channel, rather than the average bankfull width or average base of bank width. As a result Church and Zimmermann (2007) divided the bankfull width determined during field studies by 1.5 in an attempt to make field widths equivalent to the width of smooth walled flume experiments. A multiplier of 1.5 was used as this multiplier was the difference between the average minimum width of the steppool units (4 m) and bankfull width at East Creek. From a grain jamming perspective smooth walled flumes were considered to represent conditions similar to the minimum width of the channel, rather than the average width of the channel.  Figure 81. Map of East Creek showing edge of mobile sediment and edge of water line during flood.  For the experiments completed during this thesis, rough walls were commonly used and the average width of the rough walled experiments can be scaled directly to Shatford Creek. The average width of the scaled experiments was 6.5% narrower than the bankfull width of Shatford Creek (see Section 4.2.1). Results from other studies that measured bankfull widths were thus scaled to the flume width used during this study by multiplying the observed bankfull width by 0.935. For the smooth walled flume experiments an approach that is different than that of Church and Zimmermann was adopted. Rather than trying to estimate what the smooth walled width corresponds to in nature, the flume width is used as the relevant channel width, but the data from smooth walled experiments are coded and identified as being unique.  169  Table 21. Other studies used to compare results from this study with.  Study  Data type  East Creek (authors’ data) Lenzi et al. (1999) and Lenzi (2001; 2004) Bathurst et al. (1982) Curran (2002) and Curran and Wilcock (2005) Grant and Mizuyama (1991) and Grant (1994) Koll (2002)  Lee (1998) and Lee and Ferguson (2002) Tatsuzawa et al. (1999) Weichert et al. (2004) and R. Weichert, personal communication, 2005 Whittaker and Jaeggi (1982)  Field  Grain size reference Bed  Grain size method Wolman  Width method Mapping active channel Measured over step crests  Slope method  Field  Steps  Wolman of 60-100 stones  Flume  Bed  Wolman  Flume width  Flume slope  Flume  Bulk mix  Bulk mix  Flume width  Flume slope  Flume  Bed  Surface end of run  Flume width  Flume slope  Flume  Bed  Bulk mix  Flume width  Flume  Steps  Wolman  Fiber tape bankfull width  From laser profile of 2.4 m long section Level survey  Flume  Bulk  Surface bulk  Flume width  Flume slope  Flume  Maximum size in mix  Bulk mix  Laser profile of bed  Flume  Bed  Bulk mix  Flume, for experiments with alternate bars, wetted width used Flume width  Total station survey Total station survey  Flume slope  For Curran’s (2002), Koll’s (2002) and Weichert et al.’s (2004)] experiments Dmax of the bulk mix was used in place of D84 of the surface material. For Whittaker and Jaeggi (1982) surface grain size data were available for a few runs and were similar to the bulk mixture. Hence bulk D84 was converted to D84 of the steps by multiplying the reported D84 by 1.4.  A second major difference among past studies was how grain size was measured. In general, but not exclusively, flume studies have tended to report the grain size of the bulk mixture, while  170  reports from field studies present the grain size of either the step-forming grains or the bed surface material. In an attempt to rectify the different grain size measurements, where the surface of the entire stream was sampled, the D84 was multiplied by 1.4 to estimate the D84 of the step-forming stones. A multiplier of 1.4 was used since Wolman grain size samples of East Creek, Shatford Creek and the most step-pool dominated reach of Deeks Creek indicated that the D84 of the step-forming stones was approximately 1.4, 1.42 and 1.38 times the D84 of the entire bed surface, respectively. In the studies of Lee (1998), Crowe (2002) and Weichert et al. (2004), only the D50 of the bulk material was reported; these values were multiplied by 1.9 to approximate the D50 of the surface as this multiplier was the average armouring ratio observed during my experiments. This correction is crude. In order to determine the total applied shear stress flow depth must be known, yet depth was rarely measured. For this reason where depth was not known the Chézy equation adjusted by a measure of relative roughness (Church and Zimmermann, 2007) ⎛D⎞ v ∝ gyS ⎜ ⎟ ⎝d ⎠  −1  (29),  was used along with flow continuity (q = vy) to determine depth via 2  1 ⎛⎜ qD50 ⎞⎟ 5 y= C1 ⎜⎝ gS ⎟⎠  (30).  Here C1 is a constant that is on the order of 1 (it was found to be 1.06 for the Rosport-Koll data; Aberle and Smart, 2003). Some of the scatter evident in Figure 82 is certainly a result of error introduced when trying to standardize the data and the need to estimate certain parameters. As an example, all of the field data with Shields ratios greater than 6.8 comes from the Rio Cordon (Lenzi, 2001; Lenzi et al., 2004) and flow depths at flood stage were estimated. For the Weichert (2006) and Whittaker and Jaeggi (1982) experiments only the flume slope was given and the amount of degradation is not known. In addition, surface grain size distributions were not available and the coarsening of the bed was estimated. Because of challenges like these, the data from these other studies should not be considered absolute. The intent in plotting these other data is three fold: (1) to show that 171  the results from other studies do not systematically differ from my experimental results, (2) to show that my results fill in the parameter space and better confine the jamming relation than the data from other studies and (3) to motivate those conducting experiments and field studies to make measurements that can be compared.  Figure 82. Data from other studies (see Table 21) presented with jamming results from this study. Data from other studies has been adjusted to correspond with metrics used in this study.  Stable bed states from the other flume experiments are predicted to plot below the rough bank data from this study as others have always used smooth walls. The orange triangular data points that plot vertically at a jamming ratio of 14 are all from Bathurst et al.’s (1982) flume experiments that were completed with high sediment feed rates that produced a range of antidune features. Thus the absence of step-pool features at low Shields ratios is likely due to the high sediment feed rates and the absences of an armour layer. Likewise, the triangles and diamonds at jamming ratios of 3.3 and 2.3 and Shields ratios less than 2 are from Curran’s (2007) smooth walled flume experiments that were also completed with a continuous high rate of sediment supply that had the same texture as the bulk mix. The smooth walls coupled with the feeding conditions are likely to have prevented steps from stabilizing. For Curran’s experiments, the steps were classified as unstable for plotting purpose as videos from her experiments showed 172  steps that formed and then disappeared in a matter of minutes as well as a high throughput of sediment. During Curran’s experiments sediment transport rates varied between 110 and 1250 g/s/m while runs from my experiments that had Class III bed failures had an average transport rates of 34 g/s/m and stable beds (class I changes) had an average transport rate of 1.8 g/s/m (Table 20). In my experience, high sediment transport rates are associated with unstable steppool structures.  6.5 Discussion 6.5.1 Stochastic influences on stability At the outset of this study it was envisioned that with precise measures of flow velocity, bed state, grain size and channel changes along with a standardized experimental procedure a jamming plot would be developed with relatively little error. Much of the variability observed in previous jamming plots (Church and Zimmermann, 2007) was thought to be due to data quality issues that ranged from differences in experimental procedure to poor or infrequent measures of bed and flow properties. It was also believed that a more precise stability function would emerge with the quality of data that was recorded during these flume experiments. While a relatively more precise jamming plot was developed (Figure 74), the scatter in the stability plots was considerably more than was expected. At first each ‘outlier’ was examined and questioned, but it soon became apparent that such variability is intrinsic to these systems. Even replicate experiments showed considerable variability (Figure 76 and Figure 79). The source of the variability is attributed to the individual histories of how the structures develop and the stochastic nature of sediment transport. The particular history of how certain steps were formed is believed to have had an effect as some structures are apt to be more stable than others due to the manner in which the grains were deposited and how the force chains developed. For example, steps formed by having grains accumulate one at a time may be less stable than steps formed as a result of a group of grains sliding together and locking across the channel (Section 5.3.2.1 and Video 10). While sediment transport as a stochastic process is not new (Einstein, 1937), at the beginning of the project I thought the stochastic nature of sediment transport would affect the timing and pattern of grain movement, but not the overall stability of the channel. It was, however, demonstrated that the stochastic nature of sediment transport affects how the  173  channel aggraded and degraded. This observation suggests that channel stability needs to be assessed using a probabilistic model rather than a deterministic model. Three examples of stochastic sediment transport behaviour include: (1) a wide range in response times following an increase in flow, (2) mobility of large grains after many hours of constant flow accompanied by low sediment transport rates, and (3) the wide variation in the nature of channel changes that occurred following a 20% increase in flow. Each of these is discussed below. 1. If sediment transport was deterministic a common response to an increase in flow would be expected to occur with a characteristic response time; however, despite a common experimental design and flows consistently being increased 20%, the response time of the bed to a flow increase varied substantially (Figure 60). Among the 103 beds exposed to an increase in flow, 29 of them did not have significant sediment transport in the first 10 minutes following the flow increase, and in one case significant sediment transport did not start until more than two hours after the flow increased. 2. The observation that, during Experiment 22 between the 13th and 20th hour at a constant discharge, two 90-128 mm grains were mobilized that contributed 371 g of the 820 g of sediment transported over a seven hour period also suggests transport conditions were stochastic. In a deterministic system I would expect the largest material to move early on, and transport rates to smoothly decrease along with a decrease in the size of the mobile sediment as time progressed. The transport of large grains after many hours of constant flow during which low sediment transport rates developed would not be expected to occur. 3. Finally, the observation that following a flow increase during some runs the bed hardly changed while during other runs substantial changes occurred suggests that a stochastic process is occurring. Despite the flow rate being consistently increased 20%, Class I bed disturbances (minor changes) occurred 12% of the time. In a deterministic system we would expect the same type of bed changes for all runs. The jamming plots may help illustrate why this stochastic behaviour is occurring. The beds that experienced only Class I disturbances, which was essentially no change at all, were shown on average to plot with the stable data points and separate from the Class II and III disturbances. 174  Presumably, during the development of the channel the bed must have failed during a previous run and as a result of that failure a new bed state was attained that was not just critically stable, but rather capable of withstanding a flow increase without degrading, hence no substantial change occurred despite the 20 % increase in flow. This behaviour suggests the bed does not rearrange itself to attain a state that is just at the limit of stability but rather can jump to a new state that is capable of withstanding future flow increases. It is the stochastic nature of sediment entrainment along with the ‘jumpy’ nature of the channel response that is thought to yield the variability in the jamming plots. Most surprising is that the variability among runs that were identical is similar to the variability caused by varying the feed rate and history of flows. While very long run times were not assessed, the variability between runs was larger than the effect flow duration had on stability (Figure 73). When this research project began the assumption was that history would play a significant role. It was however, observed that while history played a role, it did not dictate the outcome. Data from Experiments 2 and 3 were not expected to plot among the data from the other runs as the bed was not exposed to a careful series of runs that slowly ramped up flows. Instead a flow rate much greater than the minimum bed mobilizing discharge was applied, yet the data from the stable beds that did form plotted among the data from the other experiments (Figure 71).  6.5.2 Influence of bank roughness One parameter that was changed during the runs that had a significant effect on the jamming plots was bank roughness. Despite the variability associated with identical experiments, smooth and rough walled experiments clearly plotted separately. The rough walls were observed to contribute to the structuring of the channel, even during the wide experiments. The large stones in the flume could be seen to frequently stop against bank protrusions and form cellular structures (e.g. Figure 83). These structures increased the stability of the rough bank experiments relative to smooth walled experiments. The importance of rough banks begs the question: when assessing stability how can bank roughness be measured in the field and compared with the results from these experiments? Are the banks of the stream as rough as the rough banks in this experiment? Rougher? Smoother? How much does it matter?  175  Bank roughness scaled by channel width (Figure 84a, σbanks/w) and the D50 of the sediment mix (Figure 84b, σbanks/D50) varied during the experiments. Although bank roughness was fixed, grain size and width did vary. While the plots are not an ideal test of the effect of bank roughness, since varying bank roughness at a fixed width and grain size would be preferred, they  Figure 83. Stones structuring along banks (from Experiment 27). Flow is from right to left. Two structures are visible, one in the upper left and another in the lower right. The downstream stones in each structure were typically deposited at or near the tip of the bank protrusion and additional stones were then deposited behind these stones.  are useful indicators of how bank roughness might affect stability. ANCOVA analysis revealed that the Shields ratio of the σbanks/w = 0.17 experiments was statistically greater than the Shields ratio for the σbanks/w = 0.24 group of experiments (p = 0.029) but not the σbanks/w = 0.36 group of experiments (p = 0.095). There was also no significant difference between the σbanks/w = 0.24 and σbanks/w = 0.36 group of experiments (p = 0.88). These data, in addition to the plot in Figure 84, suggest that bank roughness scaled by channel width had no appreciable effect on bed  Figure 84. Jamming plots illustrating relative bank roughness determined using the channel width (a) and the D50 of the sediment mix (b).  176  stability. ANCOVA based on the σbanks/D50 data suggest the relatively rougher banks were more stable (p< 0.001). This analysis is, however, hampered by the fact that the experiments with the relatively rough banks were also the experiments with the smallest jamming ratios (Figure 84). Without a second series of experiments that fixes channel width and/or grain size while varying bank roughness, it is not possible to fully appreciate how bank roughness influences the stability plots. Figure 84 is as much a plot showing relative bank roughness as it is a plot showing the effect of channel width and grain size. Interestingly, Giveout Creek was observed to have banks that are equally as rough as Shatford Creek, yet Giveout Creek is 17% narrower.  6.5.3 Comparing different means of assessing shear stress The logistical regression results examining what predicts when a bed will fail demonstrated that shear stresses values based on the mean velocity (Wiberg and Smith or drag force based approaches) were better able to distinguish stable and unstable beds compared to DuBoys depthslope derived shear stress values. The reasons underlying this observation are not entirely clear but may be related to the highly non-uniform flow conditions observed during these experiments or the relatively small change in depth-slope estimated shear stress that follows an increase in flow. Both of these factors are examined in greater detail below by examining how flow conditions responded to a 20% increase in flow. To illustrate the nature of the flow conditions and the effect of increasing discharge by 20% has on the flow six sets of water surface scans are shown in Figure 85. Each of the plots illustrates a long profile of the water and bed surface from before and after the discharge was increased. The scans are from runs during which Class I changes only occurred, i.e., the bed changed little in response to the flow increase. The scans clearly demonstrate that even during the runs with the lowest slope (4.5%) the water surface undulates and flow conditions are not uniform. A clearer understanding of flow conditions can be gained by examining the variation in the Froude number along the channel. For each cross-section (spaced 2 mm) the width of the channel is known; therefore, the local Froude numbers can be determined and are illustrated in Figure 86. The Froude numbers clearly varied along the length of the channel (Figure 86) and for the most part were subcritical. Most scans did, however, have a portion of supercritical flow. With the exception of Experiment 23 the water surface was influenced by both the bed surface elevation and variation in width. Experiment 23 was conducted with smooth walls, yet the variation in 177  Froude numbers is similar to the other experiments. This observation suggests the walls alone were not responsible for the variation in observed Froude numbers.  Figure 85. Water surface and bed surface elevations averaged across the cross-section from runs before and after flow was increased 20%. Flow data are for the run after the flow was increased. There is no vertical exaggeration. The shaded boxes refer to sections of the channel where water depths did not increase following an increase in discharge.  178  Figure 86. Froude numbers along channel prior to and after flow was increased 20%.  On account of the spatial heterogeneity I became interested in how the salt tracer derived estimates of Froude number compared with the spatially averaged Froude number based on the water surface scans. In Figure 87 data from all the water surface scans are plotted and it is clear that while there is no general bias, the median Froude number over the length of the channel was  179  only weakly related to the continuity derived Froude number. This result suggests that the spatially averaged values do not directly correspond with the along channel integrated results attained using a salt tracer.  Figure 87. Comparison of median along-channel Froude number and salt tracer-continuity derived Froude number.  To further examine how the flow responded to an increase in discharge the observed changes in flow depth and velocity following a 20% increase in flow are plotted in Figure 88. If uniform flow conditions were occurring, one would expect the data to plot along a line of constant Froude number, or slightly off the line in response to slight changes in Froude number that the increase in flow would produce. Instead we see the data cover a wide range of conditions and do not follow any of the constant Froude number relations. This behaviour occurs despite continuity-derived Froude numbers typically being in the 0.7 range and on average increasing by 0.06 with a standard deviation of 0.11 following a 20% increase in flow. The failure of the data to follow the theoretical Froude number relations, which are based on uniform flow assumptions, is believed to occur as a result of the non-uniform flow conditions. It appears that tracer velocity measurements provide an integrated velocity that does not correspond with the spatially averaged velocity. To illustrate why this observation is thought to occur data from the second to last run of Experiment 32 during which the increase in flow did not break the bed are examined. Based on the water surface scans, following the increase in discharge the along channel average flow depth increased 1.5 cm and mean velocity increased 4 cm/s. In contrast the salt tracer derived velocity increased 24 cm/s yielding a 2.5 cm reduction in  180  continuity derived depth (the data in question is the lowest, furthest right, stable data point in Figure 88). To examine why this reduction in continuity derived depth might happen three sections of these water surface scans are highlighted in Figure 85. At each of these sections the flow depth was observed not to increase following the flow increase. A constant depth implies that a dramatic increase in velocity must have occurred locally. I suspect that the tracer derived velocity measurements are positively biased by these local substantial increases in velocity resulting in a salt-tracer velocity much faster than the spatially averaged velocity. If true, this bias would further explain why tracer derived velocities are a better predictor of bed stability; the beds tend to destabilize due to scour in pools, which is where the water surface scans illustrate that flow depths may not increase following an increase in discharge. Hence the tracer velocity measurements may be more sensitive to increases in flow velocities occurring in the pool area, specifically where the increases in velocity are most likely to destabilize the bed.  Figure 88. Change in continuity derived flow depth and mean flow velocity following a 20% increase in flow. Lines illustrate expected response for a range of Froude numbers.  An additional reason that flow velocity may be a better predictor of stability than shear stress is that following an increase in discharge, flow velocities almost always increased while flow depths, used to determine hydraulic radius and depth-slope derived shear stress did not. Furthermore, the coefficient of variation for velocity increases was 87% while flow depths had a 181  coefficient of variation of 180%. If both velocity and depth-slope derived shear stress had the same underlying affect on stability, the differences in the coefficient of variation alone would result in velocity being a stronger predictor of stability. To summarize, flow velocity may be a better predictor than shear stress of when stable beds break due to the non-uniform flow conditions, which can result in velocity increases and depth reductions following an increase in discharge.  6.5.4 Revisiting shear stress approaches to entrainment Non-uniform flow conditions occurred during all of the runs and thus none of the reach scale means of assessing shear stress is strictly valid; however, more appropriate means of assessing shear stress at the reach scale do not currently exist. The development of an appropriate means of assessing shear stress during non-uniform flow conditions is needed. One means of assessing this question would be to examine what size sediment was mobile during the runs and back calculating the shear stress. The challenge will be to develop a technique that works at the reach scale, where the information is most useful, while capturing the nature of the non-uniform flow conditions. A more detailed Reynolds shear stress approach based on the use of an ADV or similar instrument should be examined carefully as the approach is plagued by two problems. First, the measurements are difficult due to the aeration of the water (Wilcox and Wohl, 2006b) and, second, the stability of a step depends not only on the local shear stress, but also the reach scale disturbances, especially headward migrating instabilities (Section 5.3.2.1). The last point is key: to assess stability one should consider both individual steps and the potential for headward migrating instabilities to form downstream. In this Chapter the jamming function was shown to provide an effective means of characterizing many of the factors affecting the stability of mountain stream channels. To apply the jamming function the grain size of the channel and steps, the channel width and slope and the flow depth during the event in question needs to be known. All of these parameters can easily be measured in the field with the exception of flow depth. To assess flow depth or determine the discharge associated with a specific Shields ratio an estimate of the flow resistance is required. There is no standard approach to flow resistance and most existing approaches are based on low gradient streams. In the next Chapter experiences from these experiments will be used to help develop an approach to estimating flow roughness in steep streams.  182  7 Water velocities and flow resistance 7.1 Introduction To assess the stability of a bed it is generally necessary to estimate depth and discharge during flood stage; depth is necessary to assess the applied shear stress using the uniform flow approximation, and discharge is necessary to have an idea about the return frequency of the flood in question. Due to the rarity of gauging stations a measurement of both water depth and discharge during a flood event is rarely available; rather one of them may be known, either from field measurements (depth) or from regional flood frequency analysis (discharge). To determine the other parameter velocity must be estimated. Velocity is almost never directly available. To estimate velocity channel properties such as bed slope, grain size and roughness are commonly measured and velocity is predicted based on the measured depth or discharge. Traditional approaches from lowland rivers have been shown not to work particularly well (Scheuerlein, 1973) as they are predicated on the dominance of grain resistance, yet the tumbling nature of step-pool channels and their particular morphology dramatically increases flow resistance. Early work was concentrated on what step spacing maximized flow resistance (Abrahams et al., 1995) and more recent work (Aberle and Smart, 2003; Wilcox et al., 2006; Wilcox and Wohl, 2006a; Comiti et al., 2007b; Ferguson, 2007) has found that, in general, only 10-30% of the energy that is dissipated can be attributed directly to the grains based on traditional predictions of grain resistance. In stylized experiments (stone “steps” placed as transverse strips across a flume) Canovaro et al. (2004) showed that form-induced resistance amounted to as much as 80% of the total. To estimate flow resistance two approaches have traditionally been used. In the first approach the grain size of the bed raised to the 1/6 power of D (in metres) is related to the Manning’s n via  183  1 1 D 6 n = 50 = 0.047 D50 6 21.1  (31),  where the D50 is the median bed surface grain size (in metres). This relation was developed by Strickler in 1923 and is based on uniform sand (Simons and Senturk, 1992). Meyer-Peter and Muller suggested a constant of 0.038 (Simons and Senturk, 1992) and that the D90 in metres be used instead. Church et al. (1990) suggested the constant is 0.039 and the D50 should be used. Combining (Eq. 31) and (Eq. 3) yields 1  8 ⎛ R ⎞6 = a1 ⎜ ⎟ f ⎝D⎠  (32),  where a1 incorporates g and the constant. With the second approach the logarithmic law of the wall, v 2.303 ⎛ 12.2 y ⎞ = log⎜ ⎟ κ u* ⎝ k ⎠  (33)  combined with (Eq. 2) yields 8 2.303 ⎛ 12.2 y ⎞ = log⎜ ⎟ f κ ⎝ k ⎠  (34).  where κ is the Von Kármán constant (≈0.4), u* is the shear velocity u* = (gRS)0.5  (35),  and k is the Nikuradse roughness height, which was originally equated with the D50, but is more commonly expressed as some multiple of the D84 (Millar, 1999; Ferguson, 2007). While over rough beds the velocity profile deviates substantially from a lognormal profile, Ferguson (2007) asserted that a number of studies (e.g. Wiberg and Smith, 1991) have shown that a log-law still holds reasonably well with k scaled to some roughness height. For narrow ranges of y/k, (34) can be approximated by  184  8 ⎛ y⎞ = a⎜ ⎟ f ⎝k⎠  b  (36),  with b depending on the value of y/k. Equation 36 is then simply the general form of (32), with the substitution of mean flow depth (y) for hydraulic radius (R), suggesting that the Strickler approach (32) to estimating roughness can yield a functional form similar to the Keulegan-type logarithmic relation. The Manning relation can be presented in a form similar to (34) whereby n depends on relative roughness yielding 1 6  1 2.303 g R ⎛ 12.2 y ⎞ = log⎜ ⎟ κ n ⎝ k ⎠  (37).  Aberle and Smart (2003) generalized a hydraulic geometry approach (v ∝ qm) to proposed dimensionless groupings  α ⎡⎛ ⎞ β⎤ v q ⎟ S ⎥ = c ⎢⎜ 3 ⎟ ⎜ ⎢ ⎥ gl gl ⎠ ⎣⎝ ⎦  (Eq. 4) where l is a roughness length. Using  D84 as the roughness parameter, Comiti et al. (2007b) found no significant dependence on slope and a constant (c) of 0.92 with an exponent (α) of 0.66. Ferguson (2007) remarked that if the roughness lengths in (36) and (4) are equated (k = l) and y = q/v, (36) can be rearranged to yield v* = a 1-m S  1- m 2  q *m  (38),  where v* is dimensionless velocity (= v/(gl)0.5) and q* is dimensionless discharge (= q/(gl3)0.5). This solution is arrived at by adopting the definition of the Darcy-Weisbach friction factor (f = 8gRS/v2); from (2). By adopting the definition of the Darcy-Weisbach friction factor the relation between slope and discharge is fixed and therefore the relation between α and ß. Hence the dimensionless components in (4) are no longer independent. Rickenmann (1991) and Aberle and Smart (2003) found that m is 0.6, which yields b = 1 in (36). When Aberle and Smart (2003) examined individual beds for which they had hydraulic geometry relations, the exponent (m) ranged from 0.54 to 0.86 and increased with mean bed slope. Ferguson (2007) suggested that m = 0.6 is one end member of the range of power 185  functions that may occur in the form of (38). The other end member is achieved using the Strickler friction law, which yields b = 1/6 or m = 0.4 and a slope exponent of 0.3, and is suggested to represent relatively deep flows (Ferguson, 2007). Aberle and Smart (2003), however, suggest the upper limit of m to be 1, which is arrived at by assuming no increase in cross-sectional area with increasing discharge. Clearly m should be somewhat less than 1. Using Ferguson’s upper and lower bounds, for deep flows 1  8 ⎛ y ⎞6 = a1 ⎜ ⎟ f ⎝k⎠  (39)  should hold with typical values of a1 being approximately 7-8. For shallow flows 8 ⎛ y⎞ = a2 ⎜ ⎟ f ⎝k⎠  (40)  should hold with typical values of a2 being 1-4 (Ferguson, 2007). Ferguson (2007) went on to suggest that these two functions represent friction components that can act additively and represent grain and form resistance in a manner similar to that introduced previously (e.g. Millar, 1999). By adding the components  8 = f  ⎛ y⎞ a1a2 ⎜ ⎟ ⎝k⎠ 2 2⎛ y ⎞ a1 + a2 ⎜ ⎟ ⎝k⎠  5 3  (41)  is achieved (Ferguson, 2007). Equation (41) provides a smooth resistance function that depends on relative roughness and captures the effect of form elements being drowned out at high relative flow depths. Other attempts to partition form and grain roughness have not included a relative roughness dependency. The forgoing discussion has introduced the classic resistance functions and presented a number of recently published means of characterizing flow resistance. What has not been stressed above, or in the literature, is that the different approaches integrate the governing variables in fundamentally different ways. The Chézy and Darcy-Weisbach functions suggest that velocity is 186  related to slope and hydraulic radius to the half power (v ∝  RS ), and the Manning approach  considers velocity to be related to R to the 2/3 power rather than the half power (v ∝ R2/3S1/2). In contrast, the dimensionless hydraulic geometry approach considers velocity to be related to discharge with a variable exponent and directly incorporates a measure of roughness (e.g. D84 or σ), but not slope. It can include additional components such as slope, step height/length ratio and the number of steps. While much research has focussed on what is the best way of characterizing a particular roughness coefficient (e.g., what predicts n or f best), few researchers have examined which means of characterizing resistance works best across a range of relative depths, slopes and channel forms. The purpose of this chapter is to examine the third research question, how can flow resistance in steep channels be best assessed in order to predict flow velocities?  7.2 Experimental overview and data selection Over the course of the experiments 58,000 slugs of salt water were injected into the channel to provide an estimate of mean water velocity (see Section 2.2.3 for details on velocity measurement techniques). Of these, 12,460 could be related to a stable bed scan for which the bed surface morphology and grain size distribution were known. Excluding samples with just one velocity measurement, there were in total 424 unique grain size, bed morphology and flow rate combinations. For each of these the number of velocity measurements used to compute an average flow velocity varied between 2 and 73 and averaged 20. The majority of the mean velocities were comprised of 10-40 measurements of velocity (Figure 89). The samples with just one velocity measurement were not used during the analysis as they were often from portions of the experiment when the discharge was being increased or decreased.  7.3 Results 7.3.1 Hydraulic geometry observations Among the 32 experiments there were 27 hydraulic geometry runs on stable beds at a range of bed slopes, grain sizes and channel widths. These runs provide the highest quality data to examine the relation between flow velocity and discharge as the bed was stable and the data came from experiments that cover a wide range of slopes, grain sizes and flow rates. An ideal flow resistance model would fit each of the individual hydraulic geometry runs well and by 187  incorporating bed characteristics collapse the 27 hydraulic geometry runs into a single function. The degree to which the different flow resistance models (Manning, Darcy-Weisbach and dimensionless hydraulic-geometry) achieve this goal is examined below.  Figure 89. Histogram illustrating number of velocity measurements for each unique combination of discharge, bed morphology and grain size.  Figure 90 illustrates hydraulic geometry relations (v = aQb) for Experiments 4 through 32. It is evident that no single hydraulic geometry relation exists and there is considerable variability among the 20 runs. The error bars illustrate the standard error associated with each measurement of discharge and velocity. The coefficient of variation among the velocity measurements ranges between 9 and 0.3% and averages 1.3%. The individual hydraulic geometry relations clearly have modest differences in slopes and significant differences in their y-intercepts. Some of the individual hydraulic geometry runs, which are illustrated in Figure 91, appear to have a slight curvilinear trend and thus may not follow a perfect power relation. Interestingly Experiments 23, 24 and 25 are among the most linear; these were the three experiments completed with smooth banks. Differences between experiments are likely the result of differences in grain size, surface structures and bedforms. To investigate how well traditional measures of grain resistance quantify the observed flow resistance, the Darcy-Weisbach friction factor was calculated and is plotted in Figure 92 against  188  relative depth (y/D84; D84 of bed surface is used) for each hydraulic geometry run from the main set of experiments. The figure clearly illustrates that the Darcy-Weisbach friction factor is not related only to relative roughness, as individual hydraulic geometry runs plot separately. The data are seen to roughly follow lines that fall between the two limit relations suggested by Ferguson (2007) (Eqs. 39 and 40; (8/f)1/2 = a1(y/k)1/6 and (8/f)1/2 = a2(y/k), respectively), but are generally most similar to the latter.  Figure 90. Mean velocity as a function of discharge for 20 hydraulic geometry runs. Error bars indicated standard errors and are for the most part smaller than the symbols.  If typical values suggested by Ferguson (2007) are used in the Darcy-Weisbach-Strickler type relation (e.g. (8/f)1/2 = 8.2(y/D84)1/6), the relation plots above the extent of the data in the figure. The data generally support Ferguson’s suggestion that such resistance functions change from a power relation at small relative depths to a Strickler type relation at large relative depths. Some of the scatter in Figure 92 can easily be attributed to the D84 not fully characterizing the roughness of the channel. Additional factors, such as form and spill resistance have long been known to be important. For these reasons the standard deviation (σ) of bed elevations about the mean long profile has been proposed as a more representative roughness length. As Figure 93 illustrates the D84 and σ are only poorly correlated, suggesting the standard deviation may improve the characterization of channel roughness. The differences between the D84 and the  189  standard deviation are apt to arise because the standard deviation is based on an actual measure of bed height, while the D84 is not, as well as differences in the physical arrangement of the bed due to armouring, structuring and the presence of bedforms.  Figure 91. Individual hydraulic geometry relations. Replicate Experiments 16, 20 and 22 are plotted together, all other experiments were unique.  To investigate if standard deviation of the bed can be used in place of grain size to improve the characterization of flow resistance the relation between the Darcy-Weisbach friction factor and depth scaled by standard deviation of the bed was examined and is illustrated in Figure 94. The scatter between the hydraulic geometry runs is similar to the scatter observed when the D84 is used, or even greater. The plot shows no evidence that the standard deviation of the bed improves the fit. One problem with the standard deviation of the bed is that it does not contain any information about the spatial arrangement of the roughness. Consider a hypothetical  190  situation where a channel is composed of either large blocks or small cubes. The channel with large infrequent blocks can have the same standard deviation as the channel with smaller more frequent cubes, yet large blocks are apt to produce more tumbling flow while small cubes are more likely to result in skimming flow.  Figure 92. Darcy-Weisbach friction factor (f) as a function of relative depth for data from 22 hydraulic geometry experiments. Manning-Strickler type resistance functions are indicated by the dashed lines while power function relations are indicated by the solid lines. Equations for plotted lines are supplied to compare with results from previous studies.  An alternative means of investigating resistance to flow is to examine the multiplier and exponent in dimensionless hydraulic geometry relations such as those illustrated in Figure 95. In Figure 95 unit discharge and mean velocity have been scaled by (a) the standard deviation of the bed and (b) the D84 to develop dimensionless variables. The data in both plots exhibits less scatter than the Darcy-Weisbach approach. This observation is particularly evident for discharge and velocity normalized by the D84 (Figure 95b). In Figure 95a and b it is evident that the individual experimental runs have a steeper dimensionless hydraulic geometry relation than the data in general.  191  Figure 93. Standard deviation about mean long profile based on bed scan as a function of the D84 of the bed at the time of the scan.  Figure 94. Darcy-Weisbach friction factor (f) as a function of mean flow depth (y) normalized by the standard deviation of the bed (σ) for data from 22 hydraulic geometry runs.  192  Figure 95. Relation between velocity and discharge normalized by the standard deviation (σ) of the bed and the D84 of the surface for hydraulic geometry runs (a and b) and all runs (c and d). Solid line indicates non-linear regression model while the dashed line indicates the functional relation (Mark and Church, 1977).  193  Figure 96. Dimensionless hydraulic geometry relations for 22 experiments for data from hydraulic geometry runs during which bed was stable. Replicate Experiments 16, 20 and 22 are plotted together.  194  Compared to the Darcy-Weisbach approach (Figure 94), dimensionless hydraulic geometry relations shown in Figure 95a and b collapse the hydraulic geometry data much more successfully. These data are, however, highly selected as they are all from experiments with no sediment transport and vertical banks. Figure 95c and d illustrate that similar dimensionless hydraulic geometry relations hold if data from all the experiments are used, including runs with sediment feed and sloping banks. The data in Figure 95c and d appear to be slightly curvilinear in the log-log plots, suggesting that a power function may not be the ideal flow resistance model. In particular, the data consistently lie below the line at low dimensionless discharges. This pattern could occur as a result of remaining structure in the data that cannot be explained by the normalization of unit discharge alone. Dimensionless hydraulic geometry data from each unique hydraulic geometry run are plotted separately in Figure 96 and the constant and exponent of the best-fit non-linear regression model for each of these unique hydraulic geometry runs are given in Table 22. The mean relation derived from using all of the hydraulic geometry runs has an exponent of 0.36, while the exponents of the individual runs range from 0.35 to 0.79 and average 0.55 (Table 22). A power function appears to be a good fit for the individual runs. For reference sake, recall that Ferguson (2007) suggested that the exponent should range from 0.4 to 0.6 as the relative depth goes from deep to shallow. The exponents in the dimensionless hydraulic geometry relations are generally lower than those observed in other studies (Aberle and Smart, 2003; Comiti et al., 2007b; Ferguson, 2007). To examine Ferguson’s suggestion that the exponent in the power function decreases with increasing relative depth, the residuals about each power function are plotted in Figure 97 as a function of relative depth to investigate if systematic structure exists within the residuals. If the exponent in the power function was non-stationary, but rather decreased with increases in relative depth, we would see negative residuals for slow and fast velocities and positive residuals for median velocities. Out of 21 hydraulic geometry relations, 5 had significant quadratic relations (95% confidence interval for quadratic slope exponent did not overlap with zero) that supported the expected pattern of residuals. These 5 experiments did not have flow (e.g. relative depth, width/depth ratio, Froude number) or bed characteristics (e.g. number of step-pools, step height, grain size) significantly different (p>0.05, t-tests) from the experiments that do not fit the pattern suggested by Ferguson. Furthermore, careful review of each plot in Figure 97 reveals 195  Table 22. Dimensionless hydraulic geometry regression coefficients (v* = cq*α) for each hydraulic geometry run. Standard deviation of the bed was used to estimate the roughness length.  Experiment All hydraulic geometry runs Exp 2-1 Exp 2-2 Exp 4 Exp 10 Exp 11 Exp 13 Exp 14 Exp 16 Exp 17 Exp 18 Exp 19 Exp 20 Exp 21 Exp 22 Exp 23 Exp 24 Exp 25 Exp 26 Exp 27 Exp 30 Exp 31 Exp 32  n  α  α’2  ASE1-α  c  c’2  ASE1-c  r2  205 5 2 14 11 9 9 11 11 8 11 7 10 9 7 10 10 10 10 10 10 11 10  0.35 0.47 0.45 0.62 0.62 0.54 0.62 0.48 0.59 0.54 0.51 0.35 0.55 0.79 0.59 0.71 0.60 0.56 0.44 0.51 0.56 0.42 0.59  0.39 0.47 0.45 0.62 0.65 0.55 0.63 0.48 0.60 0.54 0.51 0.38 0.56 0.80 0.61 0.71 0.60 0.59 0.44 0.54 0.56 0.47 0.61  0.01 0.01  0.63 0.61 0.60 0.37 0.30 0.55 0.41 0.62 0.60 0.69 0.44 0.55 0.55 0.56 0.57 0.49 0.40 0.46 0.49 0.27 0.43 0.60 0.58  0.57 1.26 1.13 1.03 1.17 1.13 1.10 0.98 0.89 0.97 1.14 0.72 0.92 1.00 1.05 1.15 1.10 0.98 1.24 1.26 1.00 0.78 0.83  0.02 0.01  0.80 1.00 1.00 0.99 0.93 0.99 0.97 0.99 0.97 0.98 0.99 0.85 1.00 0.96 0.94 0.98 0.99 0.91 0.97 0.91 0.99 0.82 0.92  0.02 0.06 0.02 0.05 0.01 0.04 0.03 0.02 0.07 0.02 0.07 0.07 0.04 0.02 0.08 0.03 0.06 0.02 0.07 0.07  1  Asymtotic Standard Error for non-linear regression coefficients.  2  Functional regression slope and intercept coefficients.  0.02 0.06 0.02 0.05 0.01 0.02 0.02 0.02 0.02 0.01 0.02 0.04 0.02 0.05 0.05 0.03 0.05 0.01 0.04 0.02  that the fit in some of the experiments relies heavily on one or two data points and the range of relative depths is small. The preceding analysis suggests that for individual hydraulic geometry data, a dimensionless hydraulic geometry approach is reasonable. The next challenge is to examine how the individual runs differ from each other and if the variation can be predicted and explained. This question is explored in Figure 98 by plotting the variation in the functional regression exponent (α’) from the 22 individual dimensionless hydraulic geometry relations as a function of 15 measures of channel shape and flow characteristics. Linear regression analysis indicates that there were no statistically significant relations between the channel or flow metrics and the exponent in the 196  hydraulic geometry relations. Notably the exponents in the relations do not appear to depend on relative depth, calculated using either the D84 or the standard deviation of the bed (Figure 98n and o). This observation suggests that the exponent in a dimensionless hydraulic geometry relation is not affected by relative depth; however, the range in relative depths is fairly small. Following Aberle and Smart (2003) the exponent was also plotted against bed slope (Figure 98f), but unlike Aberle and Smart who observed a general trend of increasing exponent with increasing bed slope and substantially greater variance with increasing bed slope, no trend is evident in these data.  Figure 97. Residuals about dimensionless hydraulic geometry relations for individual hydraulic geometry runs. Line is shown if quadratic form is statistically significant.  197  Figure 98. Relation between exponent (α’) in dimensionless hydraulic geometry relation and flow and bed characteristics for 22 hydraulic geometry runs with stable beds.  Figure 99. Relation between constant (c’) in dimensionless hydraulic geometry relation and flow and bed characteristics for 22 experiments with hydraulic geometry data. Functional regression constant is used. Best-fit line with 95% confidence interval is shown if slope is statistically different than zero.  198  Between each hydraulic geometry run the functional regression exponent (α’) varied little compared to the constant (c’, Figure 95a and b and Figure 97). To investigate if the variance in the constant could be related to bed or flow parameters, Figure 99 was produced and regression analysis performed. A number of measures of bed characteristics and flow parameters correlate with the constants from the different experiments. The most significant relations are for relative depth, particularly normalized by the standard deviation of the bed. In part the strength of this relation may be a result of standard deviation being used to normalize both velocity and discharge since a spurious correlation is possible. Pool length, Froude number and bed slope also correlate with the constant but these relations were not as significant. Figure 99 suggests that relative depth, bed slope or the number of step-pools may have the most influence on the constant in dimensionless hydraulic geometry relations. If relative depth is used to predict the constant, an iterative solution is required since determining the depth requires knowledge about the flow velocity, which is unknown. In summary, a power function dimensionless hydraulic geometry relation fits the data from individual hydraulic geometry runs well. While there is some variation in the exponent, it is not obviously related to any bed or flow condition. In contrast, the constant in the power function negatively correlates with relative depth, the number of step-pools, bed slope, Froude number and the average pool length. Relative depth and the number of step-pools appear to be the most significant terms.  7.3.2 Patterns of flow velocity during feed, armouring and hydraulic geometry runs The data from hydraulic geometry runs, completed during stable bed conditions, clearly illustrate that resistance functions based on dimensionless hydraulic geometry relations produce reasonably precise models of flow velocity variation with discharge when the bed is stable. Data from runs with mobile beds and sediment feed will now be examined to evaluate how effective dimensionless hydraulic geometry relations are when the bed is undergoing changes. Mean velocity as a function of discharge is illustrated in Figure 100 for runs completed at the start of the experiment (S), during feeding (F), bed armouring (N) and hydraulic geometry (H) investigations. Thus, with the exception of the hydraulic geometry data, the bed was mobile between successive data points. The plot shows that velocities generally start high when the bed 199  is smooth and fine, then increase modestly as feed is added, the bed armours and discharge increases. During the hydraulic geometry investigations velocities are seen to change as a function of discharge more rapidly than they did during the mobile bed portion of the experiments. For the same 20 experiments Figure 101 and Figure 102 illustrate the relation between dimensionless velocity and dimensionless discharge normalized by the standard deviation of bed elevations and the D84, respectively. These figures illustrate that dimensionless hydraulic geometry relations can collapse the data from individual experiments onto a common line even though the bed is actively changing slope, grain size and form. As an example, Experiment 18 shows a clear pattern of initially high velocity at low discharges (50 cm/s at 6.1 L/s; Figure 100) when the experiment was started but after a series of runs the velocity was 32 cm/s at the same discharge. When these data are plotted in dimensionless form, the velocity at the beginning of the run and after the bed was armoured fall on a similar line (Figure 101 and Figure 102). While the data do not always collapse onto a common line, the data from many of the experiments do appear remarkably linear. Visually it appears that neither roughness metric performs better. In three of the plots the standard deviation of the bed appears to perform better than the D84 (e.g. Exp. 25), while for three other plots it appears that the D84 performs better. The data from all of the experiments are plotted in Figure 95, which shows that the addition of mobile beds, smooth banks, rough angled banks and channel armouring does not appreciably change the overall appearance of the dimensionless hydraulic geometry relation. The residuals about the best fit function show a slight pattern of negative residuals at the extreme values and positive residuals for mean values. While the pattern is supported by a significant quadratic function, it is particularly weak.  7.3.3 Flow resistance functions To examine how well the flow velocity data could be explained by a resistance relation and compare six flow resistance models, data from experiments with rough vertical banks that were collected at the end of runs when velocities had stabilized and no sediment was being fed into the flume were selected. The selected portions of the experiments were on average 24 minutes long, ranging from one minute to 74 minutes. The data came from 380 runs. The flow resistance models that were evaluated include: 200  Figure 100. Velocity as a function of discharge for 20 experiments illustrating data from the start of the run (S), hydraulic geometry runs when bed was stable (H), mobile runs when bed may have moved (N) and runs during which sediment was fed into the flume (normal feed, F; fine feed, f). The solid line connects sequential runs. The red data point indicates the first run with data and the blue point illustrates the last run.  201  Figure 101. Dimensionless hydraulic geometry relations based on standard deviation of bed elevations (σ) for 20 experiments illustrating data from the start of the run (S), hydraulic geometry runs when bed was stable (H), mobile runs when bed may have moved (N) and runs during which sediment was fed into the flume (normal feed, F; fine feed, f). The solid line connects sequential runs. The red data point indicates the first run with data and the blue point illustrates the last run.  202  Figure 102. Dimensionless hydraulic geometry relations based on D84 for 20 experiments illustrating data from the start of the run (S), hydraulic geometry runs when bed was stable (H), mobile runs when bed may have moved (N) and runs during which sediment was fed into the flume (normal feed, F; fine feed, f). The solid line connects sequential runs. The red data point indicates the first run with data and the blue point illustrates the last run.  203  1: A Manning velocity formula with an integrated log-law of the wall approach (Eq. 37), 2: Three Darcy-Weisbach resistance models, one with the Manning-Strickler approach (Eq. 32) and two with the integrated log-law of the wall approach (Eq. 34) using either the D84 or σ, and 3: Dimensionless hydraulic geometry relations that utilize the D84 or σ to characterize roughness (Eq. 4 without the slope term). For the Darcy-Weisbach and Manning log-law approaches both the D84 of the bed and the standard deviation of the bed were used to characterize roughness (k) and both slope and intercept coefficients were determined using the data (Eqs. 34 and 37). Thus the relations were fully calibrated with the data from this study. Traditionally, using the log-law of the wall approach no calibration coefficients would be included and the relations would be as they are presented in Equations 34 and 37. Once the best-fit model was determined the model was used to predict velocity and the observed and predicted values were compared. Some of the scatter in the predicted versus observed velocity plots is due to measurement uncertainty associated both with the measurement of velocity and the measurement of the variables used to predict velocity. Hence, a means was sought to characterize this uncertainty and determine if the scatter in the plots could be explained solely based on the error associated with the data used to measure and predict velocity. The approach follows that of Gomez and Church (1989), but an additional term that accounts for the error associated with the variables used to predict velocity was added. For each flow velocity model the difference between the predicted velocity (vpred) and the measured velocity (vmeas) yields the residual velocity (vres) vres = v pred − vmeas  (42)  The cumulative predicted error associated with the residual (δvres) includes three terms  δvres = δv pred + δvmeas + δvmean _ bias  (43)  where δvmean _ bias is the mean bias  204  δvmean _ bias = pred − obs ,  (44)  associated with the regression of predicted velocity versus observed velocity, δvmeas is the uncertainty associated with the measurement of velocity (see Section 2.3) and δv pred is the uncertainty associated with the predicted velocity. δv pred is predicted by combining the error terms associated with each metric that is included in the velocity model (see Section 2.3). If the cumulative predicted error (δvres, Eq. 43) was greater than the velocity residual (vres, Eq. 42) no unexplained error was considered to exist. Such a result implies that the difference between the predicted velocity and measured velocity can fully be explained by the uncertainty in the data. If the velocity residual (vres) is greater than the cumulative error then unexplained error exists. The root mean square of the unexplained error gives an estimate of the amount of unexplained error associated with the velocity predictions and provides a means of directly comparing models that have as input metrics with different amounts of uncertainty. As part of the analysis the unit-discharge-based models were also translated into their depth format, and depth rather than discharge was used to predict velocity. This transformation arranges the dimensionless hydraulic geometry relation in the same form as the Manning and Darcy-Weisbach models. Comparing the standard error from the residuals after the errors associated with each metric have been removed provides a fair comparison of the velocity models as the error associated with each metric used in the model is accounted for. The r2 of the original regression should not be used to compare models as some of the models include the same terms on both sides of the expression. Each flow resistance model, along with its respective regression coefficients and information about the observed versus predicted results, is presented in Table 23. The standard error (SE) for the observed versus predicted data is included as is the standard error of the unexplained error. The unexplained standard error associated with the Darcy-Weisbach-Manning-Strickler approach is surprisingly low (0.11 cm/s), but this model is shown to be a poor fit as the r2 of the predicted versus observed data is only 0.11. In addition the model departs strongly from the 1:1 line: despite the low standard error of the unexplained residuals, this is a poor model. Excluding this model, of the traditional approaches the Darcy-Weisbach with integrated log-law of the wall  205  Table 23. Flow resistance relations and their precision and error (n = 380)  Equation type  Manning- law of the wall with D84 for k Manning- law of the wall with σ for k Darcy-Weisbach: Manning-Strickler for roughness Darcy-Weisbachlaw of the wall with D84 for k Darcy-Weisbachlaw of the wall with σ for k Dimensionless hydraulic geometry with D84 for roughness Dimensionless hydraulic geometry with σ for roughness Dimensionless hydraulic geometry with D84 for roughness and a slope component Dimensionless hydraulic geometry with σ for roughness and a slope component 1  Best-fit linear relations  1 6  ⎛ 7.5 y ⎞ 1 7.2 R ⎟⎟ log⎜⎜ = n D κ ⎝ 84 ⎠  r2  Velocity predicted vs. measured Mean SE SE r2 bias unexpla ined (cm/s)  0.20  0.89  0.331  -1.0  0.38  0.20  0.95  0.281  -0.6  0.40  0.26  0.92  0.112  2.7  0.11  0.47  0.77  0.463  1.3  0.28  0.40  0.84  0.353  0.9  0.35  0.45  0.743  1.1  0.14  1  1 6.0 R 6 ⎛ 4.8 y ⎞ log⎜ = ⎟ n κ ⎝ σ ⎠ 1  ⎛ R ⎞ ⎛ 8 ⎞2 ⎟⎟ ⎜⎜ ⎟⎟ = 2.5⎜⎜ D ⎝f ⎠ 84 ⎝ ⎠  1/ 6  1  ⎛ 8 ⎞ 2 1.6 ⎛ 2.8 y ⎞ ⎟⎟ ⎜⎜ ⎟⎟ = log⎜⎜ κ D ⎝f⎠ 84 ⎝ ⎠ 1  ⎛ 8 ⎞ 2 1.2 ⎛ 2.5 y ⎞ ⎜⎜ ⎟⎟ = log⎜ ⎟ κ ⎝ σ ⎠ ⎝f ⎠ 0.41 ⎡⎛ ⎞ ⎤ v q ⎜ ⎟ ⎥ = 0.70⎢ 3 ⎟ ⎢⎜ ⎥ gD84 ⎢⎣⎝ gD84 ⎠ ⎥⎦  0.80  v = 0.70g0.29q0.41D84-0.12 or v = 0.55g0.5y0.70D84-0.20 0.37 ⎡⎛ ⎞ ⎤ q v ⎟ ⎥ = 0.65⎢⎜ 3 ⎟ ⎜ ⎢ ⎥ gσ 0.81 ⎢⎣⎝ gσ ⎠ ⎥⎦ 0.31 0.37 -0.056 or v = 0.65g q σ v = 0.55g0.5y0.59σ-0.0890 0 .48 ⎡⎛ ⎤ ⎞ v q 0 .20 ⎥ ⎜ ⎟ ⎢ = 1 .12 S ⎢⎜ gD 3 ⎟ ⎥ gD 84 84 ⎠ ⎣⎢⎝ ⎦⎥ 0.83 v = 1.12g0.26q0.48D84-0.23S0.20 or v = 1.24g0.5y0.94D84-0.44S0.38 0 .48 ⎡⎛ ⎤ ⎞ v q ⎟ S 0.28 ⎥ = 1 .08 ⎢⎜ ⎢⎜ g σ 3 ⎟ ⎥ gσ ⎠ ⎣⎝ ⎦ 0.26 0.48 -0.23 0.28 v = 1.08g q σ S or v = 1.16g0.5y0.92σ-0.42S0.54  0.84  Based on flow depth (y) 0.81  0.38  2.8  0.22  0.45  0.723  -1.0  0.24  Based on flow depth (y) 0.74  0.36  -0.9  0.35  0.42  0.783  1.9  0.10  Based on flow depth (y) 0.89  0.40  4.9  0.17  0.44  0.753  -0.3  0.19  Based on flow depth (y) 0.89  0.34  0.8  0.33  non-linear regression, 2 linear regression, 3 functional regression used.  206  approach using the D84 to characterize roughness is the most precise, with a standard error of 0.28 cm/s and a relatively large correlation between the predicted and observed velocities (0.46). The dimensionless hydraulic geometry approach, using the unit discharge based formula and the D84 of the bed surface or the standard deviation of the bed performed better than the traditional flow resistance models. When bed slope was added to these models the correlation between the predicted and observed velocities increased further and the standard error of the unexplained error was further reduced. Overall the best model appears to be the hydraulic geometry model that incorporates bed slope and D84 to characterize bed roughness. The predicted versus observed velocities for this model have an r2 of 0.78 and the standard error of the unexplained residuals was 0.1 cm/s. When the dimensionless hydraulic geometry relations are in the depth based format, instead of unit discharge, they perform less well. In particular, the correlation between predicted and observed velocities decreases substantially (r2 = 0.4 for best performing model) and the mean bias can be substantial (4.9 cm/s). The SE of the unexplained residuals may still be small, but this result is in part due to the large mean bias that was subtracted from each residual to determine the unexplained residuals. The relation between a range of channel and flow parameters and the residuals from the DarcyWeisbach log-law of the wall and dimensionless hydraulic geometry approaches are explored in Figures 103 and 104, respectively. Figure 103 illustrates that the residuals from the DarcyWeisbach approach correlate with the width to depth ratio, cross section area and relative depth. There may also be weak correlations with other parameters. For the same data set, the residuals from the dimensionless hydraulic geometry relation also correlate with the width/depth ratio, relative depth and cross sectional area. These are all parameters that depend indirectly on flow depth and would require a step-wise solution if they were to be incorporated into a flow resistance model. The magnitude of the residuals is clearly larger for the Darcy-Weisbach approach than the dimensionless hydraulic geometry approach, reflecting the precision of the models. If the depth form of the dimensionless hydraulic geometry relation is used, the residuals display a similar pattern except there is more scatter. The plots of the residuals, like the analysis of the hydraulic geometry data earlier, suggest that the width/depth ratio and relative depth are additional factors that influence the flow resistance 207  relations. Correction parameters can be estimated from the plots and used to improve the models if desired. The width/depth ratio, relative depth and cross-sectional area represent the distribution of stresses on the channel wall. The importance of these metrics highlights the importance of channel shape on flow resistance.  Figure 103. Plot illustrating residuals from prediction of velocity using the Darcy-Weisbach log-law of the wall approach as a function of bed and flow properties.  7.4 Discussion There are a number of means by which the flow resistance equations can be evaluated. These include the mean bias, correlation coefficient between observed and predicted velocities, the overall standard error of the residuals and the standard error of the remaining unexplained variance. The largest r2 between observed and predicted data along with the smallest standard errors were associated with the dimensionless hydraulic geometry relation that incorporated D84, q and S suggesting that this model is best overall (Table 23). In some instances only a measure 208  of flow depth will be available, not unit discharge (q) and a flow depth based model must be used. In such situations the choice of models is not as obvious since the Darcy-Weisbach law of the wall model based on y/D84 produced results similar to the dimensionless hydraulic geometry relations. In particular the depth based dimensionless hydraulic geometry relation that incorporated S and D84 had a smaller unexplained standard error, but also a larger mean bias and lower r2 than its’ Darcy-Weisbach counterpart.  Figure 104. Plot illustrating residuals from prediction of velocity using a dimensionless hydraulic geometry approach based on unit discharge, D84 and bed slope.  For the dimensionless hydraulic geometry relations the standard errors of the depth-based predictions of velocity were greater than those of the unit-discharge-based methods. This finding was true for both the standard error of the residuals and the standard error of the unexplained residuals. The reason for this observation is likely that unit discharge is a  209  compound dimension metric (length and time) while depth is only a single dimension metric (length) and unit discharge ‘contains’ velocity while depth only roughly correlates with velocity. Overall a dimensionless hydraulic geometry model appears to be the most suitable means of characterizing flow resistance in steep channels, especially one based on the D84 of the bed and bed slope. Interestingly, the incorporation of bed slope into the dimensionless hydraulic geometry model only modestly improves the performance of the model, yet bed slope (or more specifically energy gradient) appears in all other flow resistance models. Much like Wilcock’s (2001) suggestion that bedload transport predictions can be drastically improved by having even one measurement to calibrate a bedload equation, the prediction of velocity at a site using a dimensionless hydraulic geometry approach can be dramatically improved with the use of one measurement of velocity. One measurement would greatly reduce the uncertainty in the y-intercept, which is associated with most of the variance in these models. Recall that the individual hydraulic geometry runs had exponents that were more similar than their intercepts (Table 22).  7.4.1 Resistance partitioning Resistance partitioning has been sought as a means of examining flow resistance in steep streams and was originally envisioned as a possible approach in this study (Sections 1.1.3 and 1.1.4) as detailed grain and bed morphology data would be available; however, when it came to applying the concept it was realized that the concept is not appropriate for steep shallow streams. To explain it is useful to first examine the history of the concept and then present the specific challenges that steep streams present. The concept of resistance partitioning originates in large lowland rivers where the grains are a small fraction of the flow depth and there is a poor relation between the grain size on the bed and the form of the channel (Einstein and Barbarossa, 1952). In particular the formation of dunes in sand bed rivers necessitate the inclusion of an additional roughness parameter as river beds can evolve from a smooth plane-bed form to a dune bed without a change in grain size but with a change in the depth-velocity-discharge relation (Vanoni and Brooks, 1957; Simons and Richardson, 1960; Henderson, 1966).  210  Owing to the success of the concept that dunes add resistance that grain size cannot account for and the presence of bedforms in gravel bed streams, resistance partitioning in gravel/cobble bed streams has been a popular topic of research. The hope has been that solutions for challenges such as bars, woody debris and steps and pools could be achieved through this approach (Wilcox et al., 2006; Wilcox and Wohl, 2006a). It has been shown that as the flows become shallow the ‘grain’ roughness needs to be modified as relatively shallow flows increase the apparent grain roughness (Bathurst, 1985; Wilcox et al., 2006). While this approach has worked to some degree, Wilcox et al. (2006) showed that the additive approach to resistance in steep streams with wood does not behave as expected. Nevertheless, when I began the experiments I hoped that by using the bed at the beginning of each experiment as a measure of ‘grain’ resistance I would be able to gain a better understanding of partitioning and address some of the challenges associated with flow resistance in steep streams. With each experiment the initial bed was smoothed by shovelling the sediment around and washing the bed surface quickly with a water hose. Then the bed was scanned and the grain size distribution of the bed was determined. These beds were seen to provide a perfect opportunity to assess the effect of structuring and bedforms on water velocities as they had no form or structure. It was thought that by using the data from the first twenty minutes or so of each experiment it would be possible to examine how well traditional measurements of grain resistance correspond with the amount actually observed. It was observed that the resistance coefficient from these first few minutes were similar to coefficients observed after the bed structured (Figure 105), suggesting that ‘grain resistance’ may be responsible for most of the resistance in all the runs. This finding was certainly not the expected result, as grain resistance has often been considered to be a small portion of the total resistance (Aberle and Smart, 2003; Wilcox et al., 2006; Wilcox and Wohl, 2006a; Comiti et al., 2007b; Ferguson, 2007) and the result lead to a more critical assessment of the concept. After considering the history of the concept it was concluded that calculating ‘grain’ resistance for shallow flows does not make sense. At low relative depths the water is bending around the grains and there no established velocity profile. Using the traditional concept of grain roughness, the ‘grains’ would best be described by the dimples and edges of the individual grains, not their actual size. Their actual size forms the structure of the channel around which the water bends and spills. This phenomenon has traditionally been considered form roughness 211  (e.g. water bending around dunes, logs etc.). Thus partitioning was abandoned in favour of a dimensionless hydraulic geometry approach.  Figure 105. Darcy-Weisbach friction factor as a function of time. Friction factor during first twenty minutes of each run is highlighted with solid black circle. Numbers in upper right corner of each plot refer to experiment from which date are taken.  A dimensionless hydraulic geometry relation was favoured as it was shown that a single scaling function can extend from shallow to relatively deep flows without any serious deviation from the overall trend. This finding suggests that a total flow resistance approach can be applied with some success in steep streams. Unlike large lowland streams, grain size is a proxy for channel form in many steep streams, which may in part explain why the dimensionless hydraulic geometry approach works as well as it does. The influence of wood in such channels remains an unresolved flow resistance question.  212  8 Conclusions To examine the stability of mountain streams new experimental methods were developed that provided detailed observations of step-pool channel formation, failure and flow resistance. Using these technologies the conditions that lead to step-pool formation and destabilization were examined. The research focused on three specific questions. Question 1: What governs the occurrence and form of step-pool channels? Question 2: Is channel stability inversely related to the jamming ratio (channel width/grain size  ratio) and can it be predicted based on the applied stress (e.g. shear stress)? Question 3: How can flow resistance in steep channels be best assessed in order to predict  flow depths during extreme events and assess the applied force (e.g. shear stress) on the bed? Concluding comments about the formation and destruction of step-pool channels and each research question are summarized below.  8.1 Conditions leading to step-pool formation and destabilization The experimental observations showed that when the bed was fed sediment and discharge was slowly increased, stable step-pool channels were developed. Step-pool formation was shown to be contingent on a relatively small jamming ratio (w/D84step) with an observable increase in bed stability for jamming ratios less than 6. On average step-pools did not occur on slopes less than 3% and step-pools were observed to compose 50% or more of the channel when the slope exceeded 10%. Keystones were observed to be the anchors around which other large stones were deposited leading to the formation of steps. Some steps were formed when a group of large stones was mobilized together and slid as a group downstream a short way before locking together in a force chain across the channel. Pools were generally observed to form due to scour caused by water flowing over the step stones upstream.  213  The step-pool morphology data suggest the main governing variables on step-pool form are channel slope, bed grain size and channel width. In particular, the frequency of step-pools was related to bed slope and/or the jamming ratio (which are correlated in this study) while the height of steps was related to grain size. The residual depth in pools was found to depend mostly on grain size while the length of the pools was related to the residual depth. The commonly cited and interpreted H/L/S relation originally proposed by Abrahams et al. (1995) was found to be essentially correct if pool length (Ls) is used in place of step-to-step spacing (L). It is suggested that the trend observed between H/L/S and bed slope can be attributed to changes in the frequency of steps as a function of slope, a factor not considered by Abrahams et al.’s earlier work. The destruction of step-pools was observed to be a stochastic process. Following an increase in discharge a substantial increase in sediment transport was observed during the first minute 46% of the time; however, during 25% of the runs an increase did not occur during the first ten minutes of the run. On other occasions it took tens of minutes to a couple of hours for the bed to break. The most common bed failure mechanism was upstream migrating instabilities. Once a section of bed destabilized there was a strong tendency for a head scarp to form and the instability to move upstream. Thus the stability of a particular section of bed depended both on the stochastic process of bed entrainment and the occurrence of headward migrating instabilities that could destabilize the steps.  8.2 Effect of history on step-pool channels While beds that experienced a longer duration of flows were observed to be slightly more stable, the effect of duration was not as substantial as was expected. Runs from Experiments 2 and 3, which were completed by initially having a flow rate much greater than the critical rate for sediment entrainment, attained a stable state similar to the state observed for the later experiments that were completed by slowly increasing flows. The substantial variability due to the stochastic nature of bed failures may have masked the effect of varying the history of flows. The limited number of replicate experiments that were completed showed that variability among replicate experiments can be substantial. While studies that have replicated flume experiments are rare in the literature, on account of the variability observed in this study, it is suggested that  214  conducting replicate experiments should become more common. A study that replicates the same experiment ten or so times may be particularly insightful.  8.3 Jamming plot The largest Shields ratio (Eq. 6) at which stable bed states could exist in the flume was shown to be related to the jamming ratio (Eq. 26). Using the data from stable and unstable beds a logistic regression model was developed that incorporates the Shields ratio and jamming ratio to predict the probability of a stable bed becoming unstable (Eq. 27). Rough banks were consistently shown to produce more stable beds. The stable bed model (Eq. 26) and bed failure models (Eq. 27) provide an important means of assessing over what parameter space stable beds can form and under what conditions stable beds will fail. The models incorporate bed and step grain size, channel slope, hydraulic radius and channel width. To scale for changes in flume slope during the experiments discharge was set to maintain a constant stream power with the hope that all the experiments would fail after approximately the same duration of flows. It was shown, however, that steeper beds failed earlier than less steep beds. For experiments with the same channel width and grain size the Shields ratio the bed could withstand was shown not to vary with slope suggesting experiments can be scaled by the Shields ratio. Scaling flows using the Shields ratio would require the flow depth and bed slope during the run to be predicted, which in turn requires knowledge about the grain size and the use of a flow resistance relation. To some degree this information can be inferred with the jamming plot, but further investigations are required to examine how grain size evolves during experiments and is modified by the feeding of sediment. Such an experimental procedure would require a more complex approach, but would provide an interesting test of whether the evolution of the bed can be predicted. The original concept that changes in flume slope could be scaled by stream power was overly simplistic.  8.4 Flow resistance in steep channels Overall a dimensionless hydraulic geometry model appears to be most suitable means of characterizing flow resistance in steep channels, especially one based on the D84 of the bed and bed slope. Additional factors that were shown to influence the water velocity include the width/depth ratio, relative depth and wetted cross-sectional area. These are all metrics that 215  describe the amount and distribution of stress along the walls. In some instances only a measure of flow depth will be available, not unit discharge (q), and a flow depth based model must be chosen. In such situations the choice of models is not as obvious. 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