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Pool-riffle dynamics in mountain streams : implications for maintenance, formation and equilibrium Chartrand, Shawn M. 2017

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Pool-riffle dynamics in mountain streams: implications formaintenance, formation and equilibriumbyShawn M. ChartrandBA, Environmental Geology, Case Western Reserve University, 1995MS, Geological Sciences, Case Western Reserve University, 1997A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Geography)The University of British Columbia(Vancouver)July 2017c© Shawn M. Chartrand, 2017AbstractIt is common for mountain riverbeds to exhibit a repetitive pattern of topographic lows andhighs known respectively as pools and riffles. Pool-riffle structures are ecologically importantbecause salmon rely on them for birth, growth and regeneration, and they are physically im-portant because pool-riffles are observed across diverse landscape settings. A common phys-ical characteristic of pool-riffles is that pool spacing is proportional to channel width, for lon-gitudinal bed slopes that vary by two-orders of magnitude. Furthermore, field, numericaland laboratory based studies observe that pools are colocated with points of channel narrow-ing, and riffles with points of widening. What is not known, however, is how downstreamchanges of channel width give rise to, and maintain pool-riffles. The goal of my thesis is toaddress this knowledge gap, and to specifically build physical understanding for the observedspatial correlation between channel width and pool-riffle architecture. I use field work, lab-oratory experiments and theory to address this goal. In Chapter 2 I apply non-parametricstatistics and self-organizing maps to understand the spatial and temporal character of rif-fle bed surface texture spanning 11 different sediment mobilizing floods, and conclude thatfrequent texture adjustment is part of the maintenance process for pool-riffles which exhibittopographic stationarity. I build from this finding in Chapters 3, 4 and 5 with laboratory exper-iments designed to investigate how pool-riffles form and evolve along variable width channelreaches. In Chapter 4 I conclude that pool-riffle formation is physically driven by two com-peting timescales which reflect the tendency to build riverbed topography through sedimentdeposition, vs. the tendency to destroy topography through net particle entrainment. I cap-ture these timescales in a mathematical model I develop using theory with physical scaling.In Chapter 5 I show that the (dis)equilibrium state of pool-riffle evolution is quantitatively de-scribed by a competition between two rates which reflect the temporal adjustment of riverbedtopography and riverbed surface texture. I conclude that equilibrium, or comparability be-tween the rates of topographic and sediment texture adjustment, is most likely to occur whenoverall sediment mobility and grain size sorting are relatively high.iiLay SummaryMountain streams commonly display a riverbed shape that has a repetitive pattern of topo-graphic lows and highs known respectively as pools and riffles. Visually, pools appear asrelatively deep portions of a river, with slow water velocities, and riffles appear as compara-tively shallow portions, with more rapid water velocities. Pool-riffles are ecologically impor-tant because salmon rely on them for birth, growth and regeneration, and they are physicallyimportant because pool-riffles are observed across diverse landscape settings. Despite theirimportance, the scientific community lacks a clear explanation for pool-riffle formation. Thisresearch shows that pool-riffles develop in response to how channel width and water velocitychange moving in the downstream direction, reflecting a tendency to either build or destoryriverbed topography. We demonstrate our finding with a mathematical model motivated byexperimental observations, and built using a combination of theory and physical scaling.iiiPrefaceThis thesis is original work completed by Shawn Chartrand. Guidance was given by the super-visory committee, and laboratory assistance was provided by Rick Ketler, Carles Ferrer-Boix,and Ryan Buchanan.This thesis includes one manuscript, and three complementary Chapters that will be submit-ted for publication as two or more manuscripts. The published manuscript is presented inChapter 2. Chapter 3, Chapter 4 and Chapter 5 are the complementary Chapters.A version of the work in Chapter 2 is published in Water Resources Research Chartrand et al.(2015). The co-authors are Marwan Hassan and Valentina Radic´. I am responsible for devel-oping the field sampling program, implementation, and analysis of all field data presentedin Chapter 2, except use of self-organizing maps (SOMs), which was completed by ValentinaRadic´. I completed a majority of the writing presented in Chapter 2. Marwan Hassan andValentina Radic´ provided editorial review of the manuscript prior to publication.Chartrand, S. M., M. A. Hassan, and V. Radic´ (2015), Pool-riffle sedimentation and surface tex-ture trends in a gravel bed stream, Water Resources Research, 51, 9127-9140.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Symbols and Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi1 Introduction and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 River size at the local scale . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Evidence and knowledge gaps of the physical connection between chan-nel width variation and pool-riffle architecture . . . . . . . . . . . . . . . 41.3 Making sense of coupling between channel width and bed architecture . . . . . 62 Pool-riffle sedimentation and surface texture trends in a gravel bed stream . . . . 102.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Study site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Data collection and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.1 Riffle texture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.2 V* and pool cross-section surveys . . . . . . . . . . . . . . . . . . . . . . 23v2.4.3 Bedload transport measurements and modeling . . . . . . . . . . . . . . 242.4.4 Bed surface sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.5.1 Riffle texture adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.5.2 V* and cross-section surveys . . . . . . . . . . . . . . . . . . . . . . . . . 322.5.3 Sediment transport and bed surface sampling . . . . . . . . . . . . . . . 332.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.6.1 Riffle texture dynamics and spatial organization . . . . . . . . . . . . . . 342.6.2 Pool-riffle sedimentation coupling . . . . . . . . . . . . . . . . . . . . . . 372.7 Concluding remarks and next steps . . . . . . . . . . . . . . . . . . . . . . . . . . 382.8 Details of SOM methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 Experimental setup and measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 413.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Laboratory experiment and methods . . . . . . . . . . . . . . . . . . . . . . . . . 423.2.1 Setup and construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2.2 Experimental design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2.3 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2.4 Experimental measurements and processing . . . . . . . . . . . . . . . . 493.2.5 Bed surface grain size distributions . . . . . . . . . . . . . . . . . . . . . 533.2.6 Manual water and bed surface profiles . . . . . . . . . . . . . . . . . . . 543.2.7 Flow depth, flow area and average streamwise velocity . . . . . . . . . . 554 Morphodynamics of a width-variable gravel-bed stream: new insights on pool-riffle formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.3.1 Identifying general response regimes with sediment flux, mean bed to-pography and bed sediment texture . . . . . . . . . . . . . . . . . . . . . 594.3.2 Topographic response: channel-wide and longitudinal profile develop-ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.3.3 Effects of initial conditions on topographic responses . . . . . . . . . . . 704.3.4 Summary of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.4 Physically linking channel width changes to topographic response . . . . . . . 714.4.1 Downstream changes in flow speed and mobility . . . . . . . . . . . . . 714.4.2 Downstream changes in channel width and bed slope . . . . . . . . . . . 734.4.3 Theory for the local channel profile . . . . . . . . . . . . . . . . . . . . . 764.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.5.1 Predicting local channel slope along variable-width channels . . . . . . 80vi4.5.2 Maintenance of bed topography along variable-width channels: supportfor an emerging view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.5.3 Development of pool-riffles along variable width channels . . . . . . . . 854.5.4 General implications of unique profiles for sediment transport theory . 864.6 Conclusions and next steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875 Morphodynamic evolution of a width-variable gravel-bed stream: a battle be-tween local topography and grain size texture . . . . . . . . . . . . . . . . . . . . . . 895.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.3 Morphodynamic evolution metrics at the scale of a channel width . . . . . . . . 925.3.1 Problem set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.3.2 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.3.3 Nondimensional Exner and Hirano equations . . . . . . . . . . . . . . . 975.3.4 Dimensionless channel response number: Ne . . . . . . . . . . . . . . . . 995.3.5 Calculations of δ2, Ub, Up and Ne . . . . . . . . . . . . . . . . . . . . . . . 1005.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.4.1 Topographic and sediment texture response numbers: Nt and Np . . . . 1055.4.2 Sediment texture δ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.4.3 Channel response number: Ne . . . . . . . . . . . . . . . . . . . . . . . . 1075.4.4 Results summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.5.1 Local contributions to equilibrium conditions during pool-riffle and rough-ened channel development and maintenance . . . . . . . . . . . . . . . . 1085.5.2 Local and channel response numbers: a new view of fluvial equilibrium 1095.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.2 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118A Numerical channel evolution model description . . . . . . . . . . . . . . . . . . . . 131A.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131A.2 1D nonuniform hydrodynamic model . . . . . . . . . . . . . . . . . . . . . . . . 132A.3 Mixed grain sediment transport model . . . . . . . . . . . . . . . . . . . . . . . . 133A.4 Channel evolution model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135A.5 Grain sorting model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135A.6 Model set-up and boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 136A.7 Model application to experiment PRE1 . . . . . . . . . . . . . . . . . . . . . . . . 137viiB Probabilistic friction angle model of particle mobility . . . . . . . . . . . . . . . . . 138B.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138B.2 Simplifying assumption of the present model . . . . . . . . . . . . . . . . . . . . 139B.3 Solution for τc with the p simplification . . . . . . . . . . . . . . . . . . . . . . . 140B.4 Sample simulation inputs and results . . . . . . . . . . . . . . . . . . . . . . . . 142B.5 Friction angle model availability and citation . . . . . . . . . . . . . . . . . . . . 144C Sediment transport as a rarefied phenomenon . . . . . . . . . . . . . . . . . . . . . . 145viiiList of TablesTable 2.1 Characteristics of the sampling program . . . . . . . . . . . . . . . . . . . . . 14Table 2.2 McNemar sequential event transect test results . . . . . . . . . . . . . . . . . 19Table 3.1 Experimental details for PRE1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Table 4.1 Experimental details for PRE1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Table 4.2 Values of downstream width change between subsampling locations . . . . 65Table 4.3 Values of average channel bed slope for initial and steady-state conditions . 67Table 4.4 Mean values of Ux/Ux and τ∗/τ∗re f for subsampling locations . . . . . . . . . 74Table 6.1 Mean values of normalized residual depth ηˆx for subsampling locations . . 115ixList of FiguresFigure 1.1 Overview of links between watershed scale processes and channel morphol-ogy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Figure 1.2 Aerial photograph of a mountain river segment in Southwest Iceland. . . . 3Figure 1.3 Graphic of East Creek showing pool-riffle pairs and sequences with a pho-tograph of one pair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Figure 2.1 Majors Creek project site. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Figure 2.2 Majors Creek grain size distribution curves. . . . . . . . . . . . . . . . . . . 15Figure 2.3 Record of mean daily flow for the study period. . . . . . . . . . . . . . . . . 16Figure 2.4 Riffle sediment texture data organization for non-parametric tests. . . . . . 20Figure 2.5 Schematic illustration of SOM implementation. . . . . . . . . . . . . . . . . . 22Figure 2.6 Dimensionless bed load transport rates for Majors Creek. . . . . . . . . . . . 25Figure 2.7 Riffle texture point count results. . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 2.8 Spatial observations of riffle texture. . . . . . . . . . . . . . . . . . . . . . . . 29Figure 2.9 Spatial observations of riffle texture change. . . . . . . . . . . . . . . . . . . 30Figure 2.10 SOM grid of characteristic temporal patterns of riffle texture. . . . . . . . . 31Figure 2.11 Spatial distribution of the three characteristic riffle texture temporal patterns. 31Figure 2.12 V* and residual pool and sediment volume change of the pool. . . . . . . . 32Figure 2.13 Repeat cross-sectional surveys at the riffle head. . . . . . . . . . . . . . . . . 33Figure 2.14 Cumulative exceedence curves for streamflow and bedload sediment. . . . 34Figure 2.15 Test of the Buffington and Montgomery (1999a) shear stress-grain size model. 35Figure 2.16 V* and fine sediment fraction change in the pool. . . . . . . . . . . . . . . . 37Figure 3.1 Overview graphic of the experimental setup and field stream reach. . . . . 43Figure 3.2 PRE1 cumulative grain size distributions. . . . . . . . . . . . . . . . . . . . . 44Figure 3.3 PRE1 initial bed topography conditions. . . . . . . . . . . . . . . . . . . . . . 48Figure 3.4 PRE1 details of water and sediment supply. . . . . . . . . . . . . . . . . . . . 49Figure 3.5 Example sub-sampled DEM and photograph pair for station 10000 mm,elapsed time 2150 minutes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Figure 4.1 Downstream pool spacing as a function of the local channel width. . . . . . 58xFigure 4.2 PRE1 morphodynamics summary: water and sediment supply, sedimentflux, longitudinal mean bed topography, and geometric mean and charac-teristic coarse grain sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Figure 4.3 Summary panel of PRE1 topographic responses illustrated with DEMs. . . 64Figure 4.4 Summary panel of longitudinal profiles with zero-crossing residuals. . . . . 66Figure 4.5 Summary panel of SS box-and-whiskers plots. . . . . . . . . . . . . . . . . . 68Figure 4.6 Profile traces of average topographic response for two steady-state cases. . 69Figure 4.7 Normalized topographic profiles of the six steady-state cases. . . . . . . . . 70Figure 4.8 Average steady-state topography with downstream changes in flow speedand particle mobility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Figure 4.9 Average channel bed slope as a function of the downstream change in chan-nel width and flow speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Figure 4.10 Downstream changes in local mean flow speed for associated changes inchannel width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Figure 4.11 Prediction of local SS channel slope across the range of channel width con-ditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 4.12 Summary of Λ of Equation 4.11 vs. Γ for PRE1. . . . . . . . . . . . . . . . . . 83Figure 5.1 Conceptual illustration of how topographic and sediment texture filters work. 93Figure 5.2 Example values of δ2 of Equation 5.14 for differing values of fus, fb, and fes. 99Figure 5.3 Example bed sediment texture data used to build stratigraphic column atstation 10000 mm for te = 19, 50 and 110 minutes. . . . . . . . . . . . . . . . 102Figure 5.4 Summary of Qs f , Nt, Np, δ2 and Ne for PRE1. . . . . . . . . . . . . . . . . . . 106Figure 6.1 Observations of residual depths for PRE1 determined from zero-crossingprofiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114Figure 6.2 Concept mountain streambed architecture regime diagram. . . . . . . . . . 115Figure 6.3 Photograph of an inverted pool-riffle channel segment along the BridgeRiver, near Camoo Creek Road, BC, Canada. . . . . . . . . . . . . . . . . . . 116Figure A.1 Profile comparison of experimental and simulation outcomes for te = 2150minutes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Figure B.1 Schematic view of friction angle based mobilization problem as defined byKirchner et al. (1990). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139Figure B.2 Sample grain size distribution for the friction angle model. . . . . . . . . . . 142Figure B.3 Simulated grain mobility conditions using the friction angle model. . . . . . 143Figure C.1 Measured sediment flux from the flume. . . . . . . . . . . . . . . . . . . . . 145xiList of Symbols and AcronymsRoman SymbolsSymbol Definition Units[Ap]HSV HSV color of a sampled pixelbased on 1 of 12 colors used topaint sediment for experiments(-)[Ap]RGB RGB color of a sampled pixelbased on 1 of 12 colors used topaint sediment for experiments(-)A flow area L2bw Wilcock-Crowe sediment transportfunction fitting exponent1C f dimensionaless bed resistancecoefficient1Cn Courant stability number 1c1 image scale factor in thestreamwise direction(-)c2 image scale factor in the crossstream direction(-)Dc characteristic grain size LDg geometric mean grain size LxiiDg f geometric mean grain size of thesediment flux at the outletLDgs geometric mean grain size of thesediment supplyLDi grain size of percentile i LDsm mean grain size of the bed surface L; mmDg′ ratio of Dgs/Dg f 1D90 f 90th percentile grain size of thesediment flux at the outletLD90s 90th percentile grain size of thesediment supplyLD90′ ratio of D90s/D90 f 1DEM digital elevation model;topographic map(-)d cross-sectionally average waterdepthLFi proportion of grain size class i ofthe bed surface used in Equation2.101Fs total proportion of sand-sizedsediments of the bed surface usedin Equation 2.91F(x) step function for SOM Equation2.8C(-)Fr Froude number: U2x/gLc 1xiiiFrr Froude number ratio 1fa volume probability density ofgrain size class ψ on the bedsurface, or of the active layer1f ∗a normalized volume probabilitydensity of grain size class ψ on thebed surface, or of the active layer1fb volume probability density ofgrain size class ψ in the localbedload supply1fc characteristic volume probabilitydensity of grain size class ψ1fes volume probability density of ofgrain size class ψ of the exchangesurface at the lower boundary ofthe active layer(-)fs volume probability density ofgrain size class ψ in the local bedsubstrate, which contributes to fes1fus volume probability density ofgrain size class ψ in the long termaverage sediment supply1g acceleration due to gravity onEarthL · t−2g′ measure of the relative bedstrength: g[(ρs/ρw)− 1]1h neighborhood or kernal functionused in SOM Equation 2.8A(-)xivHSV image hue, saturation and value(brightness) color values(-)ij grid points of the mapping domainfor SOM Equation 2.8A(-)ik grid point of interest withapplication of the learning rateparameter for SOM Equation 2.8A(-)K total number of grain size fractionsused in Equation 5.19(-)k constant between 1 and 2, heretaken to have a value of 2, used todetermine active layer length scale(-)ks measure of local bed roughness:nkD901L forward difference length scaleused in Equation 4.1LLa active layer length scale: kD90 LLr length scale ratio 1Ne dimensionless channel responsenumber1Np dimensionless bed sedimentparticle number1Nt dimensionless bed topographynumber1n learning rate parameter used inSOM Equation 2.8A(-)xvnk roughness scaling factor 1px pixel (-)Q Cochran non-parametric teststatistic1Qr flow rate ratio 1Qs f sediment flux per unit channelwidthL2 · t−1Qss sediment supply per unit channelwidthL2 · t−1Qw water supply per unit channelwidthL2 · t−1q∗b dimensionless sediment flux forgrain size class i1qb total sediment flux per unitchannel widthL2 · t−1qbi sediment flux for grain size class iper unit channel widthL2 · t−1qw unit flow rate: Qw/w L2 · t−1qψ sediment flux for grain size class ψper unit channel widthL2 · t−1R relative density: R = [(ρs/ρw)− 1] 1RGB image red, blue and green colorvalues(-)S elevation gradient 1xviSlocal local elevation gradientdetermined between subsamplinglocations1SS steady-state (-)Sinuosity ratio of channel length to valleylength1sb grain size fraction sample standarddeviation of the local bedloadsupply1ses grain size fraction sample standarddeviation of the local exchangesurface1sus grain size fraction sample standarddeviation of the upstream bedloadsupply1t time tta activation time tte experimental elapse time tto dimensionless experimental time 1tr bed response to steady-state period (-)tt bed response to developing flowperiod(-)t∗ dimensionless time 1Ub bed speed: rate of channeltopography changeL · t−1xviiUc characteristic velocity L · t−1Up particle speed: rate of sedimentparticle size changeL · t−1Ux Cross-sectionally averageddownstream flow velocityL · t−1Uxr Cross-sectionally averageddownstream flow velocity ratio1u(z) vertical velocity L · t−1u∗ shear velocity L · t−1V∗ dimensionless pool sedimentstorage1w channel width: measured from topof bank to top of bankLwo normalized channel width: w/w′ 1w′ reach average channel width:average of widths measured over alengthscale of many channelwidthsLw average channel width: average ofwidths measured over asubsampling region withlengthscale of 320 mmLW∗i Wilcock-Crowe dimensionlesssediment transport function1X real world coordinate in thestreamwise directionLxviiiXo real world coordinate of the origin Lx streamwise location Lx∗ dimensionless streamwise location Lxˆ image coordinate in streamwisedistanceLxo image center coordinate LY real world coordinate in the crossstream directionLYo real world coordinate of the origin Lyˆ image coordinate in cross streamdistanceLyo image center coordinate Lz elevation of grain top used in thefriction angle based mobilitymodelLzo elevation of the lowest grainelevation exposed to the oncomingflow used in the friction anglebased mobility modelLzj initial data vectors used in SOM (-)Greek Symbolsα width to depth ratio: w/d 1αr dimensionless sediment transportcoefficient1xixβ partitioning coefficient that rangesfrom 0 to 11Γ downstream change in channelwidth1∆ denotes a difference of somevariable or quantity(-)∆w(x)∆L forward difference equation whereL is the differencing length scale,and w is differencing quantity1δ1 ( fb − fes)/ε expresses thedissimilarity between thefractional composition of the localbedload supply and the localexchange surface(-)δ2 expresses the dissimilaritybetween the fractional compositionof the three grain size populationswhich contribute to bed sedimenttexture: fus/( fb − fes)(-)ε solid fraction in the bed: 1− φ (-)η channel bed elevation Lη∗ dimensionless channel bedelevation1κ van Karmen’s constant, hereassumed to have a value of 0.4071Λ(g′)0.5D1.5cεUc Lc 1ρ′ relative density: (ρs/ρw)− 1 1xxρs sediment density m · L−3ρw water density m · L−3σg geometric standard deviation Lσv δ1/La (-)T time scale of local bed elevationadjustmenttτ shear stress m · L−1 · t−2τ∗ dimensionless Shields stresscondition1τ∗c critical or threshold dimensionlessShields stress1τˆ∗ normalized dimensionless Shieldsstress1τ∗re f reference dimensionless Shieldsstress1τc50 threshold shear stress for the 50thpercentile grain sizem · L−1 · t−2τ∗c50 dimensionless threshold shearstress for the 50th percentile grainsize1τri reference threshold shear stress forgrain size im · L−1 · t−2τrm reference threshold shear stress forthe mean grain size of the bedsurface im · L−1 · t−2xxiτrs50 reference threshold shear stress for50th percentile grain size of the bedsurfacem · L−1 · t−2φ porosity of bed sedimentsdetermined as volume of the voidsand taken here as a constant: 0.40(1)X length scale of local bed slopeadjustmentLχ numerical constant (-)ψ log2 grain size scale: log2D = Dpsi (-)xxiiAcknowledgmentsSeven years ago, Emma and I hatched a plan to move to Vancouver. A new adventure, aimedat providing me the privilege of sitting around and thinking all day, about problems thatmaybe only a handful of other people in the world care about. This move was an attemptto get back in touch with my curiosity for how planet Earth works. In the end, this has beenan utterly life changing journey, touched by many people. Here I endeavor to thank all of you.In the Fall 2010, I met Marwan Hassan outside his flume lab of the old Ponderosa Build-ing. I was prepared to beg, plead really, for a chance to work with him and learn about hisexperimental craft. Instead, I was met with an immediate openness and acceptance, which inmy experience, is a rare trait amongst us humans, let alone academic researchers. We took thisjourney together, as world citizens, and in the process you have taught me to stand on my owntwo feet, and chart my own path through the world of learning. More importantly, there weretimes when you stood back and let me make mistakes, no matter how much time you knew itwould cost me, because you knew it was the only way I would learn. And learning, for us, iswhat matters. Thank you does not begin to express how lucky I feel to have your mentorshipand support over the last 5 years, and friendship. Here’s to many more years of fun together,venturing into questions which at present, may only be a faint glimmer of thought.In 1999, I met Mark Jellinek when I was working at Marmot Mountain Works, in Berkeley,CA. Mark has a mass to his personality that makes our own sun jealous. As a result, I immedi-ately went into orbit. We have shared much together over the years, but to the present point,you have shaped me for the scientist I am. I have pile, upon pile of papers, with commentsscribbled in red, blue, black, green, really all the colors (guess you got bored with just one),which together, is testament to the craft and precision you have opened to, and taught me.Many of my thoughts and ideas live in my head. Thanks to your untraining, and then retrain-ing, that will change. To all the late nights, and early mornings reading my half-baked, poorlycommunicated ideas - thanks for your unwavering support, mentorship and acceptance ofthis challenge. Despite all of this, I still will not share the maple-peacan scone I saved fromyesterday (ever!). And since the first time you crushed my hand in the Marmot entrance, theinitial orbiting response has grown into something more a kin to a binary star system. But, Iwill always stand in awe.In the Fall 2012 I had the privilege to meet Carles Ferrer-Boix. Your friendship and supportthrough my work have been absolutely invaluable. You have taught me much, and havexxiiiplayed an instrumental role in development of all my seemingly crazy ideas, and I assure you,Marwan and Mark thank you. Your patience and supportive teaching in working throughdifficult ideas is something I will strive to carry forward. Sometime soon, may we enjoy a niceCatalonian Cava on the southern beaches of Spain, and chart a path to venture into some new,and exciting problem. Here’s to the road ahead Carles.To my lab mates from both Marwan’s lab, and the mjcj-research group. Carles, Leo, David,Katie, Ashley, Elli, Maria, Tobias (thanks for the hours working with me trying to figure out thecomposite photograph!), Matteo, Marco, Emma, Yinlue, Kevin, Alex, Claudia, Ilana; David,Reka, Andreas, Anna G., Anna M, Thomas, Yoshi (thanks for the EOSC 352 night time home-work jams!), Natasha, Georgia, okay, this is getting ridiculous. I am the luckiest person I knowto have shared and learned from each of you over the past five years - thanks. The mostcherished moments go to David, Reka and Andreas though, for having taught me about thephysics of Guinness, the physics of planetary bow shocks (just about the coolest thing ever),and how to plan an asteroid mining mission. To Sir Christian, thanks for teaching me aboutthe ways of the force. You have equipped me to venture into the unknown. To David Fur-bish, thanks for support over these past two years. You have given me the courage to take thenext step. To Olav Slaymaker, you have and always will be my philosophical guru. To JohnBuffington, thanks for your support and careful review of my PhD research proposal. Yourcomments and thoughts had a significant role in how I took the proposal forward. And toTom Dunne, thanks for your careful review of my thesis.To my Mother, Father and sister Suzanne, thank you for your love and support throughit all. You cultivated my curiosity for science and nature, and encouraged me to attend Caseand pursue my passions, which among all other things has shaped almost every aspect of myadult life. It all began high up in those trees in the woods, waving in the wind, and withthe first chemistry set you bought me, for which the local ant and rock population sufferedendlessly. Here’s to all the nights sitting around the table drinking tea and eating pinwheelcookies, entertaining all my questions. All my love and thanks.Many other people have helped and supported me along the way. To Peter Whiting, yougave me a chance and my start as a scientist, nurturing me through those early years. I amforever grateful for your kindness, and setting me on a path that has evolved into an endlesscuriosity for rivers, and how they work. Thanks Peter. To Barry Hecht, you taught me thevalue of the big picture, and folding together the many aspects of Earth Science to shape thestory of place. You are one of a kind, and I am beyond lucky to have learned from you. Thanksfor everything, most of all, room to grow as an Earth Scientist, under your ever watchful eye.To my colleagues at Balance, thanks for the opportunities to venture into the world of reshap-ing how rivers look. It is one we will learn from for many years to come, and it would not havebeen possible without your engagement, help and support. Here’s to many more sundowns,working with an operator to place that last piece of streamwood. To Dr. A, you were the hook,line and sinker that got me into Earth Science. I will never forget your enthusiasm for teachingxxivand life. To my SIESD NCED crew, thanks for a wonderful 10 days in 2016! Here’s to newfriendships and collaborations.The work presented in this thesis was thankfully funded by the University of British ColumbiaFour-Year Fellowship, the National Science and Engineering Research Council of Canada Alexan-der Graham Bell Three-Year Research Fellowship, the Mitacs Accelerate Program Internship,and the Canada Foundation of Innovation. I thank these funding agencies for their generoussupport of my research, which included painting 6 tons of rock. The unexpected consequenceof which, was that the experimental channel looked as if it were filled with peanut M&Ms,explaining why it took so long to complete my work, and why Aidan and Mira visited the lab,frequently. Ever excited to work with me.To all the rivers that I have gazed upon, measured, prodded and otherwise sat next to.Thanks for sharing some of your secrets...Rivers and streams, far and wide,Change their shape, and their rise,Ask why is this so, and you will see,It is for how size changes, simple as can be.xxvDedicationThis journey is dedicated to the fun-hoggin grissies: Emma, Aidan and Mira. Life is an adven-ture of many journeys. I could not, and would not have done this one without each of you.Your smiles, giggles, big time wrestling on the bed, walks in the woods for hot chocolate, helpshoveling sediment in the Lab, putting up with my late nights running experiments, or writ-ing code, your curiosity for those big math problems that had an unexciting answer of zero,and your absolute love and support saw me through it all. This milestone is as much yours asit is mine. Congratulatons! I can’t wait for our next journey. All my love and gratitude.xxviChapter 1Introduction and motivationOverview of Links Between Watershed Scale Processes and Channel MorphologyLandscape Settingmountains, valleySediment Supplylow, highHydrologymagnitude, frequencyChannel Morphologysize, shape, slopedeterminesdistributesSub-aerial Shapeuniformtopographic variationBed Sediment Sizefine, coarsewell sorted, patchyHydrodynamicsuniform, non-uniformstrength of mixingscales scalesscalesreflectsscalesTemporal and Spatial Scales> 102 km2> 100 km2> 102 m2> 100 m2> 106 yr> 102 yr> 101 yr> 100 yrFigure 1.1: Overview of links between watershed scale processes and channel morphol-ogy. The landscape setting scales basin contributions of water and sediment, whichtogether determines channel morphology as one moves down a river basin. Chan-nel morphology in turn reflects local bed shape, and scales bed sediment texture andhydrodynamics, which evolve in feedbacks which tend to reinforce local responsesin the absence of significant disturbances. Approximate temporal and spatial scalesof these attributes are provided at the left. Figure motivated by Church and Jones(1982); Church (2006); Hassan et al. (2008)11.1. Overview1.1 OverviewThe landscape setting, its relief, geographic location and diversity scales the supplies of waterand sediment delivered to channel networks (Figure 1.1). Streamflow, in turn and over manyyears distributes and transports the sediment supply throughout the basin. This drives anddetermines the development of channel morphology, which over length scales of many chan-nel widths comprises a river’s overall shape, size, and longitudinal slope. The broad characterof channel morphology is described at the local scale of a channel width by the shape of thebed, the spatial distribution of sediment grain sizes on the bed, which we define as bed surfacetexture, and the flow dynamics, which is characterized by the spatial character of the veloc-ity field. For time scales of many floods, and upstream supplies which vary around somelong-term average, the shape of the bed, the bed surface texture and the velocity field interactthrough feedbacks which lead to local conditions of sediment continuity (Church and Ferguson,2015).Within this context, we understand that changes to the upstream water and sediment sup-plies through large floods or landslides can cause changes to channel morphology throughoutthe downstream basin. When this occurs, four basic morphologic responses are possible (Fig-ure 1.1): a change of channel position, size, steepness, or bed surface roughness. A changeof channel position occurs through meandering of the whole channel, or of the primary flowpath, called the thalweg, and through complete channel relocation by break out and avulsion.A change of channel size occurs through bank erosion, and through mass movement encroach-ment, leading to partial channel blockage. A change of river bed steepness occurs throughsediment deposition, or net bed material entrainment. A change of bed surface texture oc-curs through preferential entrainment of particle size fractions present on the bed surface, orthrough deposition of particle size fractions present within the sediment supply, but which oc-cur in differing concentrations on the bed surface. Predicting how a particular channel reachof many widths in length will respond is difficult, because the four responses are coupled byfeedbacks, and these feedbacks trigger additional responses, which occur over differing timescales.I simplify the problem in two ways. First, we consider rivers within mountain settings,where lateral mobility is constrained by banks and valley walls sufficiently strong to drivedevelopment of river reaches that are relatively straight, or which exhibit minor amounts ofcurvature. Second, we recognize that rivers tend to express sizes which:1. Reflect the supply of water and sediment delivered by floods of moderate magnitude(Wolman and Miller, 1960; Leopold and Maddock, 1953; Leopold et al., 1964; Emmett, 1999;Whiting et al., 1999; Emmett and Wolman, 2001), or a series of moderate floods (Pickup andRieger, 1979), conditioned over times scales of at least 101 to 102 years, depending onbasin size (Howard, 1982); and2. Are constrained by local conditions such as the occurrence of bedrock outcrops, landslide21.2. Motivationdeposits, and mature stands of riparian vegetation.This perspective immediately focuses the problem to one whereby rivers exhibit a downstreamvariation of size, and given the links shown in Figure 1.1, floods drive adjustments to lo-cal channel steepness, bed surface texture, and flow structure, for which the adjustments aremodulated by local width conditions (Bolla Pittaluga et al., 2014). My interest, therefore, is withriver size, and my thesis specifically examines variations of river size at the local scale, andhow the properties of size variation mechanically drives riverbed shape, sediment texture andflow character.scale: 2200 meters across imageFigure 1.2: Aerial photograph of a mountain river segment in Southwest Iceland, withflow direction from image right to left. The photograph illustrates channel widthvariations that range in length from 1 to 5 times the local average width, driven bythe occurrence of bedrock, landsilde deposits and lateral bar deposits. Image source:Google Earth.1.2 Motivation1.2.1 River size at the local scaleRivers change size at length scales that range from a few channel widths (Figure 1.2), consid-ered the local scale, to that of the full river. Size change at the largest scale for rivers withinnon-arid climatic zones reflects increasing downstream contributions of water (Leopold andMaddock, 1953), but changes at the scale of a few channel widths are due to local processesand properties, for example the occurrence of bedrock, riparian tree mortality, or landslidedeposits (Figure 1.2). We understand that channel width influences the expression of chan-nel bed architecture, which includes bed shape, or topography, and the bed surface texture(Richards, 1976a; Keller and Melhorn, 1978; Lisle, 1979; Montgomery and Buffington, 1997; Lisle andHilton, 1999; Chartrand and Whiting, 2000; Chin, 2002; Church, 2006; Church and Zimmermann,2007; Chin, 2002; Chartrand et al., 2011). But how and why does width matter for channel bedarchitecture?Understanding width-bed coupling at the local scale is important because each are basic31.2. Motivationelements of fluvial landscapes, but in particular:a. Width scales how much sediment is stored at various points of a river system. Therefore,width regulates the redistribution of sediment down a river, including attenuation ofsediment signals from mass movements, earthquakes or forest fires;b. Width scales the footprint and diversity of habitats available to aquatic organisms. There-fore, width and its variation sets the foundation of how streams support aquatic organ-isms, and as a result, width is a key element to the provision of ecosystem services byrivers and streams; andc. The periodic character of meanders, pool-riffles and step-pools are each described bychannel width Leopold et al. (1964); Richards (1976a); Keller and Melhorn (1978); Chin (2002).Therefore, width is one of the basic properties of rivers which helps to explain theirphysical character.Furthermore, the size of rivers reveals information about the local physical character of a riversegment, as well as the history of how the physical character has shaped responses driven byupstream conditions.1.2.2 Evidence and knowledge gaps of the physical connection between channelwidth variation and pool-riffle architecturePool-riffles are perhaps the most common channel architecture of mountain streams, wherethey occur in pairs (Carling and Wood, 1994), and often in sequences of many pairs. The pe-riodic nature of pool-riffles highlights the idea that an underlying, and spatially consistentmechanism is responsible for formation and maintenance (Figure 1.3). A viable explanationstems from the observation and experimental replication of pool colocation with channel andvalley segments that are narrowing, or are relatively narrow (Richards, 1976a; Dolan et al., 1978;Carling, 1991; Clifford, 1993a; Lisle, 1986; Sear, 1996; Montgomery et al., 1995; Montgomery andBuffington, 1997; Thompson et al., 1998, 1999; Repetto et al., 2002; MacWilliams et al., 2006; Har-rison and Keller, 2007; Wilkinson et al., 2008; Thompson and McCarrick, 2010; White et al., 2010;de Almeida and Rodrı´guez, 2012; Venditti et al., 2014; Nelson et al., 2015), and riffle colocationwith segments that are widening, or are relatively wide (Richards, 1976a; Sear, 1996; Carling,1991; Montgomery and Buffington, 1997; Repetto et al., 2002; Wilkinson et al., 2008; White et al.,2010; de Almeida and Rodrı´guez, 2012; Nelson et al., 2015). One- and multi-dimensional numeri-cal models built to simulate specific field cases also reproduce the spatial association of poolsand riffles with relatively narrow and wide channel segments (Thompson et al., 1998; Bookeret al., 2001; Cao et al., 2003; MacWilliams et al., 2006; Harrison and Keller, 2007; de Almeida andRodrı´guez, 2011, 2012) (See Appendix A). Therefore, spatial correlations between width andpool-riffle architecture are suggested by multiple lines of evidence.Yalin (1971) proposes that channel spanning, macroturbulent eddies with overturning lengthscales comparable to the flow depth are responsible for riffle spacing along straight channel41.2. MotivationFigure 1.3: Graphic of East Creek showing pool-riffle pairs and sequences with a photo-graph of one pair. The overview map is from Papangelakis and Hassan (2016), andthe photograph shows a pool-riffle pair mid-way down RP1 during a relatively lowwinter flow condition. The photograph perspective is looking downstream. Photo-graph by Shawn Chartrand.segments with a meandering thalweg. Carling and Orr (2000) interpret Yalin’s hypothesis asrelated to the length scale over which these eddies deliver sufficient momentum to the chan-nel bed to entrain sediment and dig a pool. The theoretical lower limiting case for the eddylength scale is approximately three channel widths (Yalin, 1971; Carling and Orr, 2000), or halfthe average meander wavelength (Richards, 1976a); the central tendency of the eddy lengthscale based on field data is six channel widths (Richards, 1978; Carling and Orr, 2000). But pool-51.3. Making sense of coupling between channel width and bed architectureriffle architecture is not necessarily limited to large-scale meanders, or a meandering thalweg.Lisle (1986) found that stationary pools and associated riffles were colocated with stream sideobstructions and bedrock outcrops, often at channel bends.Clifford (1993a) combines the above works and proposes a systematic 3-step process ofpool-riffle formation, that is both probabilistic and autogenic in nature, and consists of a pooldigging phase, followed by maturation and autogenic (i.e driven by emergent local conditions)phases. Phase one is probabilistic in nature, and begins with the random occurrence of a localobstruction to the flow, which stimulates pool construction. The obstruction drives lateral flowconvergence into the developing pool area due to local width narrowing, which establishes astreamwise or cross-stream gradient in velocity sufficient to entrain bed sediments for pooldevelopment. Entrained bed sediments are transported downstream, or away from the pool,and deposited to form a riffle. At the end of phase one, an upstream-downstream pool-rifflepair has formed, and the obstruction which drove development persists. During phase 2, thepool-riffle pair matures, an upstream riffle develops, and the nucleating obstruction eitherpersists, in the case of bedrock, etc., or is removed. The upstream riffle takes shape due to thepresence of the pool, and associated development of a streamwise gradient in velocity, andparticle drag, which shapes the upstream bed to slope into the pool. At the end of phase 2, apool-riffle unit consisting of a riffle-pool-riffle morphologic feature emerges. During phase 3,the autogenic process begins and further pool-riffle pair creation occurs due to local flow andsediment transport perturbations driven by the initial pool-riffle unit.Published field measurements, as well as physical experiments provide constraints on theoverall width variation needed to drive pool-riffle development and maintenance. Fieldworkconducted by Lisle (1986) suggests that an obstruction extend at least 30% of the channel widthto develop a channel-spanning pool. Wilkinson et al. (2008) show by contrast that width en-croachment as small as 16% may be sufficient to promote pool development in an experimen-tal setting. Yet, flume experiments conducted by Thompson and McCarrick (2010) demonstratepool and downstream riffle development for a width reduction of 40%, and in the most recentcase, Nelson et al. (2015) produce sequences of pool-riffles along a sinusoidal shaped channelthat has a downstream width variation of 40%. Furthermore, 1-D numerical model resultsof specific field cases suggest that roughly 50% may be necessary (Carling and Wood, 1994;de Almeida and Rodrı´guez, 2011, 2012). In summary, the published results show that pool-rifflesform and are maintained when downstream total width change varies from 15 to 50%.1.3 Making sense of coupling between channel width and bedarchitectureMy primary objective is to examine coupling between local variations of channel width, andchannel bed shape and texture. Through this objective I address the questions of how andwhy width matters, and in the process build understanding of channel bed architecture ex-pression, for the natural conditions of mountain stream settings. I use a combination of field61.3. Making sense of coupling between channel width and bed architectureand experimental work to address my primary objective. The field study addresses next stepsmotivated by Nelson et al. (2009) and Hodge et al. (2013), and the experimental study buildsdirectly from Thompson et al. (1998); de Almeida and Rodrı´guez (2011, 2012); MacVicar and Rennie(2012) and Nelson et al. (2015), with the key focus that natural channels exhibit non-uniformchanges in downstream channel width (e.g. Richards, 1976a; Thompson et al., 1999; Harrison andKeller, 2007; Thompson and McCarrick, 2010; de Almeida and Rodrı´guez, 2012; Nelson et al., 2015).I address the primary thesis objective through four specific questions and knowledge gaps:A. How variable is riffle surface texture in time and space due to natural variations of waterand sediment supply? (Chapter 2)B. What mechanisms give rise to, as well as modulate, the spatial organization of pools andriffles along rivers? (Chapter 4)C. Are all pool-riffles created by a similar set of processes, and does channel bed constitu-tion predetermine a particular outcome? (Chapter 4)D. Which physical processes shape the (dis)equilibrium conditions of gravel-bed streams?(Chapter 5)In Chapter 2, I address question A through a 3-year field-based study of pool-riffle textureand sedimentation dynamics, along a gravel-bed stream located in coastal California, U.S.I demonstrate that riffle texture change is spatially organized across 11 sediment transport-ing events, including a 20-year flood, and that the pool-riffle pair responds to the upstreamsediment supply in a dissimilar manner for 10 of the 11 transport events. Riffle texture mea-surements occur through a fixed sampling grid of five transects, where texture is determinedat 160–180 locations, depending on riffle footprint. Pool sedimentation measurements occurby quantifying the sediment storage volume relative to the available sediment storage volumeof the upstream pool. Cochran’s Q and McNemar’s tests indicate that riffle sediment surfacetexture is spatially and temporally varied across each sampling transect, with distinct finingand coarsening trends. This result supports and motivates use of self-organizing maps tocharacterize the spatial and temporal character of riffle texture. Self-organizing maps (SOM)is a type of machine learning which uses unsupervised learning algorithms. Application ofSOM to pool-riffle sediment texture shows that the study riffle responds to sediment supplyevents in a spatially organized way, with different temporal trends carried preferentially overspecific riffle areas. Texture response occurs in a manner that is disconnected from the up-stream pool, except for the largest flood, which triggers a fining trend of both pool and riffle.The work demonstrates that riffles in approximate topographic equilibrium with upstreamwater and sediment supplies, modulate supply changes with texture responses that are spa-tially organized. This result has implications for pool-riffle maintenance, because organizedtexture responses are up and till now, an unidentified part of riffle maintenance processes. Fora majority of floods, maintenance occurs in pool and riffle along differing trajectories, despite71.3. Making sense of coupling between channel width and bed architectureneighboring proximities. This highlights the localized nature of sediment transport, withincontinually disturbed gravel-bed river systems.In Chapter 3, I provide the details of pool-riffle experiment 1 (PRE1), conducted at the Uni-versity of British Columbia, Canada. I use the experiments to address Questions B–D, withresults presented in Chapters 4 and 5. Experiments occurred in an 18-m long flume whichre-circulates water but not sediment. Experimental set-up was guided by a gravel-bed streamreach located near Maple Ridge, B.C., Canada. The experiment uses three different flow rates,ranging from an approximate 2-year, to a 10-year flood. Sediment supply is at or near thetheoretical capacity. The key characteristic of the experimental channel is downstream vary-ing width, with downstream gradients that range from (-0.26)–(+0.18). The experiment uses apoorly-sorted grain size distribution that ranges from 0.5–32 mm, with a geometric mean sizeof 7.3 mm and a geometric standard deviation of 2.5. The experiment consists of two phases,an initial and repeat phase. During the experiments I collect (a) 1 mm resolution topographicmaps (DEM) of the entire channel; (b) composite photographs of the entire channel, mappedto the same coordinate system of the DEMs; (c) 1 Hz sediment flux; and (d) manual measure-ments of longitudinal bed topography and water surface elevation. This data set facilitatesexamination of questions B–D. The experiments produce pool, riffle and roughened channelfeatures, which persist across the full range of external supplies.In Chapter 4, I address questions B and C with PRE1 experimental data, theory and scal-ing to demonstrate that bed topography expression is systematically organized across the fullrange of experimental width conditions. Experimental data indicates that pools develop fordownstream channel width gradients less than -0.10, riffles for width gradients ∆w(x) · ∆Lgreater than +0.10, and roughened channel features for width gradients in between -0.10 and+0.10. Topographic diversity is higher when transport conditions are closer to threshold, buttopographic relief is higher for increasing transport conditions. This suggests that the effectof channel width variation changes with water and sediment supply, becoming stronger assupplies increase. Results from PRE1 and two other studies demonstrate that the local bedslope Slocal exhibits a systematic trend for downstream width gradients that range from (-0.30)–(+0.30), and for larger-scale reach averaged bed slopes that vary by 1 order of magnitude. Thisresults motivates development and use of a mathematical model to examine the coupling be-tween the local bed slope, and the downstream channel width gradient. The mathematicalmodel shows that the local bed slope response is driven by Λ, which is a ratio of velocities thatrepresents slope production as a balance between the characteristic spreading time scale ofbed sediments, to the forcing time scale that quantifies the magnitude of local momentum fluximparted to the bed surface. Characteristically large spreading time scales drive pool develop-ment, small time scales give rise to riffles, and intermediate conditions lead to roughened chan-nel segments. These results are described reasonably well by the mathematical model, whichgiven the range of larger-scale reach average slopes, suggests a scale invariant response, thatmay be a generalized attribute of river segments governed by downstream changes to channel81.3. Making sense of coupling between channel width and bed architecturewidth.In Chapter 5, I address question D with PRE1 experimental data, theory and scaling todemonstrate that fluvial equilibrium conditions Ne are more readily achieved under increas-ing supplies of water and sediment, when the rate of local topographic Nt and bed surface tex-ture Np change are comparable. I present a new local view of fluvial equilibrium using state-ments of mass conservation for the bulk bed, and the particles that compose the bed, scaledto quantities that represent the time, length and dynamical properties of gravel-bed mountainstreams. My perspective is motivated by an idea that local bed topography and bed sedimentsurface texture act as filters to incoming supplies of sediment. The topographic and texture fil-ters change the upstream sediment supply Qss through deposition (topographic filter), or viasize-preferential entrainment (texture filter), or both. Specifically, and in the simplest case, sed-iment particles in motion entering a local bed region exhibit 1 of 2 outcomes. The particles caneither continue in transport downstream, or come to rest. The outcome is determined by thetopographic filter, which scales the average local downstream velocity magnitude Ux, whichin turn scales the average particle drag. Particles resting on the bed surface also exhibit 1 of 2outcomes. The particles can either remain at rest, or can be entrained by the flow. The outcomeis determined by the texture filter, which scales the local bed surface roughness, which in turnscales the mobility of particles resting on the bed surface. The resultant equilibrium statementNe quantifies the topographic and texture filters as a ratio of velocities, that represents the rateof change of local bed topography, and bed surface composition. Accordingly, fluvial equi-librium is achieved when the ratio of velocities is O(1). The work demonstrates that for thePRE1 conditions, rates of topographic adjustment are dominant and protracted when particlemobility conditions are relatively close to critical, and that Ne trends toward equilibrium, butdoes not achieve it. At higher relative mobility conditions, however, rates of topographic andbed surface composition change are rapid, and exhibit comparable values after an initial re-sponse period that is relatively short in duration. Hence, fluvial equilibrium is more readilyachieved. My view of equilibrium is one of the few attempts to build a formal definition forequilibrium in gravel-bed rivers. It is also flexible in terms of the choices that can be maderegarding appropriate scales, and as a result it may be useful beyond the present application.9Chapter 2Pool-riffle sedimentation and surfacetexture trends in a gravel bed stream2.1 SummaryA 3-year field campaign was completed to investigate spatial and temporal variability of sed-imentation trends for a single pool-riffle pair located in the Santa Cruz Mountains, Califor-nia. Our measurements represent a range of hydrologic conditions over eleven sediment mo-bilizing events. Two different statistical methods were used to explore riffle sedimentation.Cochran’s Q and McNemar’s non-parametric tests (one method) indicate that riffle sedimentsurface texture was spatially and temporally varied at the transect level. For McNemar’s test,variation was significant at p < 0.05, with several trends evident, including strong riffle fin-ing triggered by a 20-year flood event. A nonlinear, empirical orthogonal function methodknown as Self-organizing maps (SOMs; the second method) shows that riffle sediment surfacetexture is well described by two characteristic temporal signals, and one transitional signal atthe sampling node level. SOM mapping to each sampling node clearly shows riffle sedimentsurface texture change was spatially organized over the eleven sediment mobilizing events.Observations of pool sediment storage indicate that the pool-riffle pair exhibited a coupledsedimentation response (i.e. similar texture trends between pool and riffle) following the 20-year flood. The coupled response was characterized by a trend toward overall sedimentationconditions that were similar to those measured at the beginning of the study. The reportedtexture trends may be of interest to salmonid habitat studies that examine factors contributingto successful vs. unsuccessful fry emergence.2.2 IntroductionMountain streams exhibit a diversity of bed sediment textures that manifest at a range ofscales. Bed sediment texture is defined as the general spatial distribution of grain sizes on astreambed (the focus of this study) (Venditti et al., 2012), or the detailed distribution of bed102.2. Introductionsediments into patches distinguished by grain size and sorting (Paola and Seal, 1995; Buffingtonand Montgomery, 1999a). The spatial and temporal character of bed sediment texture is drivenby localized patterns of sediment transport, which continually responds to variations of up-stream sediment supply (Parker and Klingeman, 1982; Dietrich et al., 1989, 2005; Lisle et al., 1993;Buffington and Montgomery, 1999b; Hassan and Church, 2000; Nelson et al., 2009), spatial patternsof local grain size organization (Paola and Seal, 1995; Church et al., 1998; Hassan and Church,2000), as well as spatial patterns of local bed topography (Dietrich and Smith, 1984; Dietrich andWhiting, 1989; Clayton and Pitlick, 2007; Nelson et al., 2009, 2010). In sum these works providefor clear process linkages to bed sediment texture development and response. But relativelyfew field-based studies of bed sediment texture variability over many hydrograph events orfield seasons have been completed (Jackson and Beschta, 1982; Lisle and Madej, 1992; Clifford,1993b; Lisle and Hilton, 1999; Dietrich et al., 2005).Spatial and temporal variation in bed sediment texture has important implications for theavailability of habitats suitable for salmonids (Kondolf and Wolman, 1993; Kondolf , 2000), suc-cessful emergence of salmonid fry from streambed sediments and processes that maintainpool-riffles sequences (e.g. Hodge et al., 2013). Since publication of the pool-riffle velocity-reversal maintenance hypothesis (Keller, 1971), subsequent studies have shown that pool-rifflepairs are maintained through a combination of at least several mechanisms that operate andinteract over a range of temporal and spatial scales (Lisle, 1979; Clifford and Richards, 1992; Clif-ford, 1993b; Sear, 1996; Thompson et al., 1999; Carling and Orr, 2000; MacWilliams et al., 2006;Thompson, 2010; White et al., 2010; Sawyer et al., 2010; de Almeida and Rodrı´guez, 2011; Caaman˜oet al., 2012; Hodge et al., 2013). Pool-tail and riffle-crest sediment structuring enhances the rela-tive stability of a pool-riffle pair. Structuring occurs through in situ grain vibration and shortparticle movements (Sear, 1996; MacVicar and Roy, 2011; MacVicar and Best, 2013), and in somecases jetting of fines into pool tail and riffle crest sediments (Hodge et al., 2013), resulting in amortaring effect. Pool depth and volume is maintained over multi-year and longer timescalesvia flow convergence into a pool (Thompson and McCarrick, 2010), driven by topographic steer-ing of flow by bars (MacWilliams et al., 2006) or by changes in valley width (Sawyer et al., 2010;White et al., 2010), or due to the effects of drag along pool channel margins (MacVicar and Best,2013). Pool-riffle form is further maintained during flood hydrographs by riffle crest growth,and an associated upstream pool backwatering, which can limit pool degradation (de Almeidaand Rodrı´guez, 2011), and perhaps promote pool sedimentation. These local interactions how-ever can be disrupted by downstream effects, and shift the balance of response, promotingpool erosion and riffle stability (Pasternack et al., 2008; de Almeida and Rodrı´guez, 2011). Lastly,the spatial and temporal character of sediment sorting (bed sediment texture) over a pool-rifflepair may play a role in maintenance by limiting or moderating effects of upstream backwa-tering, or channel bed evolution – hydrodynamic feedbacks in general (Clifford, 1993b; Sear,1996; de Almeida and Rodrı´guez, 2011; Hodge et al., 2013). Little regarding the last mechanism,however, is known.112.3. Study siteThere is a need to explore how pool-riffle texture responds to discrete floods, and how thecumulative texture effect is carried over multiple flood seasons. This study addresses the firstneed. Our research was guided by two questions: (a) How variable is general riffle surfacegrain-size texture in time and space as a result of natural variations in sediment supply andstreamflow discharge? (b) Does a pool-riffle pair exhibit equivalent sedimentation as a resultof a sediment-mobilizing flood? Furthermore, are sedimentation responses consistent throughtime? Question (b) could be interpreted or conceptualized a few different ways. To be clear, wespecifically asked whether the pool filled (an increase of pool sediment storage) and the rifflefined, or the pool scoured (depletion of pool sediment storage) and the riffle coarsened. Eithercase would qualify as equivalent sedimentation, which we term sedimentation coupling. Wenote that one could also describe equivalent sedimentation as pool erosion and riffle deposi-tion, with the depositional material comprised of the eroded pool substrate. Whereas the latterdescription has a process-basis, we use the former description because it is compatible withaggradation or degradation; the latter description is not.The study research questions were addressed through a combination of field measure-ments of riffle texture, pool sedimentation (V*), cross-sectional geometry and bedload sed-iment transport, sediment transport numerical modeling, and statistical analysis, for a se-quence of 11 floods that occurred over a 3-year study period. We hypothesized that:1. The study measurement strategy would be sufficient to characterize spatial and temporaltrends of pool-riffle texture adjustment; and2. The pool-riffle pair would exhibit sedimentation coupling.2.3 Study siteMajors Creek is a mountain stream that drains the western slopes of the Santa Cruz Mountains(Figure 2.1) from a contributing area of 9.2 km2, ranging in elevation from 120 to 560 m in itsheadwaters. Mean annual precipitation ranges from 660 mm (coast) to 1067 mm (headwaters)Rantz (1971). The study site constitutes a pool-riffle pair located in an old-growth redwood(Sequoia sempervirens ‘Adpressa’) forest preserve managed by California State Parks. A 3.5 mdam is located 450 m downstream of the study site. We installed a continuously recordingstream gage (gage) within the subject pool (Figure 2.1), 7 years prior to this study.122.3. Study site(a) (b)stream gagepool X-SriffleFigure 2.1: (A) Photograph shows the pool and riffle pair discussed within this paper,view looking upstream. Solid line indicates location of associated monitoring cross-section. Tape strung over the riffle illustrates the five surface texture sampling tran-sects. Location of stream gage shown in the mid-left part of the image. Photographtaken in October 2012 during low flow. (B) Location map of study site.Present land uses are mostly rural residential, rangeland and public lands with no majorland disturbance in the past 50-years or more. Upstream of the study site the watershed isunderlain by two primary geologic units: (1) the southwest-dipping Lompico sandstone ofmiddle Miocene age; and (2) Cretaceous-aged crystalline basement rocks typed as granite andadamellite (Leo, 1961; Clark, 1966; Brabb, 1989). Smaller areas underlain by the upper MioceneSanta Margarita sandstone also occur. The Lompico and Santa Margarita sandstones producesand-size sediments, whereas the crystalline basement rocks can produce sand- to cobble-sizesediments. Sediment contributions from these geologic materials are reflected in the mediangrain size of the sampled bedload (D50 = 0.6 mm) and streambed substrate (D50 = 7 mm)(Figure 2.2).Bankfull discharge at the study site is estimated as 5.70 m3/s. Bankfull channel widthvaries from 6 to 8 m, and bankfull depth is approximately 0.75 to 1 m. Bankfull discharge hasan estimated recurrence interval of 2 years, and is associated with well-defined channel-bankslope breaks within the vicinity of the study site. The recurrence interval estimate is based ona HEC-SSP 2.0 flood-frequency analysis (Log-Pearson Type III distribution) completed with 15years of available peak flow data (gage records supplemented with USGS Station #11161570).The mean annual flow for the study period was 0.11 m3/s (Figure 2.3). During the studyperiod several near bankfull flows occurred, an estimated 20-year flood occurred on March 26,2011 (24.64 m3/s), and a 5-year flood occurred on December 19, 2010 (12.03 m3/s) (Table 2.1;Figure 2.3). Flood recurrence intervals were computed with HEC-SSP 2.0. On March 2, 2011a relatively small tanoak (Notholithocarpus densiflorus) (diameter ∼ 30 cm) fell from the streamright bank (looking downstream) and through the pool longitudinally. The tree was removed132.3. Study siteTable 2.1: Characteristics of the sampling programMean Daily FlowPeak FlowPeak Average Bed StressTotal Flow VolumeBedload Dischargem 3 /s m 3 /s Pa 106 cubic meterstonsTexture Sampling Events (sampling events)-E1: January 14-15, 20100.29 3.11 7.84 0.30 4.30E2: January 27-28, 20100.24 5.07 8.76 0.27 5.51E3: February 10-11, 20100.22 4.45 9.81 0.48 9.13E4: March 10, 20100.06 1.38 5.47 1.34 1.01E5: December 13 and 15, 20100.52 12.03 11.22 0.98 36.13E6: January 5-6, 20110.40 24.64 14.28 3.10 121.47E7: April 5-6, 20110.09 n/a 5.59 1.57 1.11E8: November 4 and 9, 20110.05 1.53 5.44 0.41 0.65E9: Janaury 30-31, 20120.16 9.54 10.16 0.85 22.49E10: April 3-4, 20120.14 3.00 7.65 0.25 2.29E11: April 24-25, 2012Total: 204.09Notes1. Bedload discharge equals total for the indicated time period, and reflects results of sedimen transport modeling. Of the 204.09 tons of bedload computed for the study period, 163.9 tons was sand-sized, and 40.2 tons was gravel-sized up to 32 mm.2. Peak average bed-stress was computed using the mean daily flow corresponding to the peak flow date.Bed-stress was computed in the sediment flux model using a gaging-derived empirical relationship for water depth.Bed stress was computed as: ρ w gd'S, where d' is the section-average water depth and S is the reach-average bed slope. Intra-sampling PeriodsApril 6-November 3, 2011January 29-February 10, 2010January 16-27, 2010April 5-April 24, 2012November 5, 2011-January 30, 2012February 1-April 3, 2012February 12-March 9, 2010March 11-December 12, 2010December 14, 2010 -January 4, 2011January 6-April 4, 2011Peak flood of period 3/26/11from the stream in late April 2011 and had minor effects on measured conditions during thestudy, as evidenced by widespread fine bed texture upstream and downstream of the studysite following the March 26th flood.A pronounced eastward bend in the stream is located 8Wb (bankfull widths) upstream ofthe study site, with another eastward trending bend located more than 10Wb downstream. Atall wood jam was located about 30Wb upstream of the study site. The pool-riffle pair measuresabout 2Wb longitudinally, along a reach of stream with an average bed slope of 0.4-0.5%, anda bankfull (and higher) water surface slope of roughly 0.3%, based on two field surveys ofhigh-water marks. The reach is relatively confined.The pool-riffle pair was chosen for study because it occurs along a near straight reach ofstream which changed little in character during the 7 years prior to the start of the study.142.4. Data collection and analysisFigure 2.2: Cumulative percent finer data for 19 bedload samples collected immediatelyupstream of the subject riffle, and for 2 bed surface bulk samples collected just down-stream of the subject riffle.As a result it was hypothesized that the pool-riffle pair is a quasi-stable attribute of MajorsCreek. We also monitored other pool-riffle pairs. Some of them however were affected bysignificant wood accumulations during the study period whereas others were influenced bydam operations. The pool-riffle pair was visually identified during low-flow as a backwaterfeature (pool) of a largely emergent sand-gravel-cobble deposit (riffle).2.4 Data collection and analysisThis section details the data collected during the study, and the analytical methods used toaddress the research questions. Data collection occurred from January 2010 through April2012 (Table 2.1). Measurement dates (sampling events) bracket storm periods during whichmeasurable volumes of bedload transport occurred (Table 2.1), and measurement dates werecharacterized by relatively low winter flows during which little to no bedload transport oc-curred. As a result it is important to note that our measurements of riffle texture and poolfine sediment storage reflect the net effect of sediment transport during the intra-samplinghigh-flow periods. Measured data were analyzed with a particularly broad set of analyticalmethods because we have attempted to explore how event-based riffle textures manifestedtemporally, and in relation to the magnitude of local sediment transport fluxes and pool sed-imentation conditions. As a result, the following discussion is quite detailed. The section isorganized by data type, and for each data type proceeds by first discussing data collectionmethods, followed by how the data were analyzed.152.4. Data collection and analysisFigure 2.3: Record of mean daily streamflow for the study period, Majors Creek immedi-ately upstream of monitored pool-riffle pair, Santa Cruz County, CA. Dates of bedtexture, V*, cross-sectional geometry and bedload sampling events are indicated bythe ovals and diamonds, respectively. Dashed line represents the mean annual flowfor the study period (0.11 m3/s), and the dashed-dot line represents the sedimentmobilizing flow (0.25 cms; cf. Figure 2.14 this paper.)2.4.1 Riffle textureData collectionRiffle surface texture was characterized according to an adaptation of the Sampling Frame andTemplate (SFT) and Environmental Monitoring and Assessment Program (EMAP) protocols(Bunte et al., 2009). Adaptation of the SFT protocol involved constructing a fixed samplinggrid over the subject riffle, coupled with use of grain size classes as specified with the EMAPprotocol. The SFT protocol was developed to reduce operator variability and bias from particleselection and size measurement, common to different methodologies, e.g., the more familiarWolman pebble count (Wolman, 1954). The adapted SFT protocol was suitable for the presentstudy because it permitted bed texture characterization from approximately the same locationduring each sampling event, deemed necessary in order to compare temporal trends of riffletexture. Grain size calls at each sampling node over the 11 sampling events were generallynon-ambiguous, and reflected the local (∼ several cm2) texture character.Riffle extents were established at the start of the study period and a total of five cross-sections were established to construct fixed sampling transects (Figure 2.1). The cross-sectionendpoints were established with rebar and metal tags, located at what we determined to bethe bankfull elevation. One cross-section each was placed at the downstream and upstreamextent of the riffle (4.6 m apart), and the remaining three cross-sections were located 0.9 m,162.4. Data collection and analysis1.8 m and 3.0 m from the downstream-most section. Over the course of the study period theupstream and downstream extents of the riffle varied < 1 m from that observed at the time ofcross-section establishment.Sampling event set-up involved pulling one tape through all cross-section endpoints (Fig-ure 2.1) by fixing the start of the tape to the left end of the downstream most transect, permit-ting confirmation of general sampling set-up consistency. The sampling grid was defined by afixed event-based sampling interval along all five transects. The sampling interval for the firstsampling event was 18 cm, for the second was 24 cm and for the remaining 9 events was 23cm. Grid spacing was chosen in order to minimize sampling the largest bed sediment grainmore than once (Church et al., 1987). Node zero along each transect was consistently placed atthe right-bank margin. Riffle texture characterization was conducted by placement of a fine-tipped rod at each sampling node, and recording whether the grain was organic, sand, gravel,cobble, boulder, or bedrock (a modest adaptation of the EMAP classes), according to a simplegrain size score from 0 (organic) to 5 (bedrock), incremented by a value of one.Quantitative analysis is limited by the texture sample sizes (n = 161 to 180), associatedspatial densities and lack of actual grain lengths. As a result we did not utilize, for example,the statistical methods detailed and used by Nelson et al. (2012). We have however used acombination of appropriate statistical analyses to characterize texture trends, and learn howto improve study methods.Non-parametric statisticsTwo non-parametric tests were used to evaluate similarity and dissimilarity of temporal riffletexture trends for each sampling transect during the study period. Similarity of trends wouldimply spatially homogeneous texture conditions, for example replacement of a gravel-sizedgrain with a gravel-sized grain at some specific location (cf. Dietrich et al., 2005), and perhapsconsistent sediment supply in terms of grain-size distribution (GSD). Dissimilarity would im-ply spatially variable texture conditions, and the possibility of either consistent or dynamicsediment supply in terms of GSD, as well as expanding, contracting or migrating sedimentpatches.We used non-parametrics to evaluate texture trends at the sampling transect scale, ratherthan for the entire riffle, for example, because the riffle texture data spatial resolution reportedhere is best suited for transect consideration. A denser sampling of bed surface sedimentslends itself to consideration of texture trends at the entire riffle scale, as reported, for example,by Nelson et al. (2012). Non-parametric tests are appropriate because there is no reason toassume residuals normality for riffle texture (Paola and Seal, 1995), and because Aberle andNikora (2006) previously observed non-Gaussian distributions for armored laboratory gravelbeds.Texture trends along each transect were evaluated using Cochran’s test for related obser-vations and the McNemar test for significance of change (Conover, 1980). Cochran’s test was172.4. Data collection and analysisused for each transect for all sampling events (r×11 matrix – r sampling locations and 11 sam-pling events), and McNemar test was used for each transect for sequential sampling events(2×2 contingency table). Cochran’s test evaluates general transect-based texture trends overthe whole study period, whereas McNemar evaluates transect-based trends on an event byevent basis, permitting identification of floods which resulted in similar vs. dissimilar tex-ture conditions. Event by event testing with McNemar was predicated on Cochran test resultswhich indicated dissimilar bed texture for any given transect over the 11 sampling events.Both tests assume nominal or binary data types, however, Cochran’s further assumes thatsampling locations were randomly selected from the populations of all possible locations, andMcNemar that the repeat measurement pairs are mutually independent, consistent in princi-ple with Wiberg and Smith (1987); Kirchner et al. (1990) and Buffington et al. (1992), and in spiritwith Schmeeckle and Nelson (2003) and Furbish et al. (2012). Our texture data set was convertedto a nominal scale of two classes by specifying sand and finer grains zeros (fine) and graveland coarser grains ones (coarse).The Cochran test statistic is defined as (Conover, 1980):Q = c (c− 1) ∑cj=1(Cj − Nc)2∑ri=1 Ri (c− Ri)(2.1)The parameter c is the total number of sampling events (degrees of freedom), Cj is the totalcolumn-wise score for each sampling event (the random variable subject to the Cochran test;simply the sum of zeros and ones for any given sampling event), r is the total number ofsampling locations along each transect (Table 2), Ri is the total row-wise score for each sam-pling location and N is the grand total score across all sampling sites and events (Figure 2.4).The variable N/c reflects the estimate of E(Cj), and the coefficient c(c-1) and the denominatorreflects the estimate of Var(Cj); Equation 2.1 can therefore be seen as approximating the dis-tribution of Cj with the chi-squared distribution and c-1 degrees of freedom (Conover, 1980).The zero-hypothesis, Ho, is that all floods are equally effective (i.e. generated similar bed tex-tures), and that for each sampling location, the probability of a flood yielding a coarse grainis independent of which flood is considered. The distribution of Q is set by assuming that thenumber of sampling locations along all transects is large, for the present study ranging from29 to 35 locations. If Q is larger than the 1-α (0.05) quantile of a chi-square random variablewith (c-1) degrees of freedom, Ho is rejected. The MATLAB cochraneqtest.m script was usedfor the analysis.The McNemar contingency table is constructed with the four possible pair values for eachpaired sampling event. If X1 is the nominal grain size at location L1 for the first event and Y1 isthe nominal grain size at location L1 for the second of the paired events, the four possible pairvalues for X1 and Y1 at L1 are (0,0), (0,1), (1,0), and (1,1) (Figure 2.4). For each paired samplingevent (e.g. E2–E3), the number of pairs of Xi and Yi which satisfy each of the four possible pair182.4. Data collection and analysisTable 2.2: McNemar sequential event transect test resultsTransect 1 Transect 2 Transect 3 Transect 4 Transect 5n = 31-35 n = 32-35 n = 30-35 n = 32-35 n = 29-35- - - - -f f fcf f f f fcc cf  Significant change in transect texture (p  < 0.05)fc1. See Table 2.1 for the sampling event dates.2. Event is short for Sampling Event, as presented in the text.3. McNemar test was not performed for the event 1-event 2 sequence because texture sampling occurred with different sampling intervals. Notes  indicates more gravel and coarser grains countedEvent 5 - Event 6Event 6 - Event 7Event 7 - Event 8Event 8 - Event 9Legend  indicates more sand and smaller grains counted  No significant change in transect textureEvent 10 - Event 11Paired Texture Sampling EventsEvent 1 - Event 2Event 2 - Event 3Event 3 - Event 4Event 4 - Event 5Event 9 - Event 10values is summed and used to build the 2×2 contingency table (Conover, 1980):ac = ∑i=ri=1 pairs (Xi = 0, Yi = 0) bc = ∑i=ri=1 pairs (Xi = 0, Yi = 1)cc = ∑i=ri=1 pairs (Xi = 1, Yi = 0) dc = ∑i=ri=1 pairs (Xi = 1, Yi = 1)(2.2)As above, r is the number of sampling locations along any particular transect for paired sam-pling events (Table 2). The Ho specifies that the marginal probability between row and columnoutcomes is equal (i.e. P(ac) + P(cc) = P(ac) + P(bc)orP(cc) + P(dc) = P(bc) + P(dc)), notingthat the marginal probabilities associated with ac and dc cancel and Ho becomesP(cc) = P(bc)(Conover, 1980). If Ho is rejected it can be recognized that bed texture along any given transectchanged significantly between paired sampling events, and either from finer to coarser, or viceversa (i.e. there is an imbalance between cc and bc and P(cc) 6= P(bc)) . For the present study, acontingency table, Equation 2.2, was built for each sampling event sequence (e.g. E2–E3), and192.4. Data collection and analysisi = 1j = 1     2     3     4     5     sampling events0     1     0     0     1     .      .      .      .      .      1  i = 2 1     0     0     1     1     .      .      .      .      .      0 transect sampling location1     1     0     1     0     .      .      .      .      .      1i = 3....1     1     0     0     0     .      .      .      .      .      00     0     0     1     0     .      .      .      .      .      1i = 4i = 5. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . j = c..... . . . . N0     1     1     0     1     .      .      .      .      .      0  i = rXi YiFigure 2.4: Schematic organization of the riffle bed sediment texture data for use withthe Cochran non-parametric test. Data organization was the same for all 5 samplingtransects and 11 sampling events, corresponding to the Cochran test statistic – Equa-tion 2.1 (Conover, 1980). The variable N is the total of the row-wise or column-wisescores, or the total number of ones in the table. The terms Xi and Yi correspondto the McNemar test and represent any sampling node for two sequential samplingevents – Equation 2.2 (Conover, 1980).sample proportions (probabilities) were calculated as:P1 = (ac+cc)(ac+cc)+(bc+dc)P2 = (ac+bc)(ac+cc)+(bc+dc)(2.3)It is of general note that the greater the difference between P1 and P2, the more likely Ho willbe rejected for any paired sampling event. The McNemar test statistic T was computed as bcbecause the sum bc + cc was small in all paired events (≤ 20). However, the exact distributionof T is the binomial probability function for size n = bc + cc, p = 0.5 [Conover, 1980], and i =the maximum of bc andcc:P(X ≤ x) =n∑i(ni)pi (1− p)n−i . (2.4)If the binomial probability P(X≤ x) is smaller than α(0.05), the Ho is rejected. The McNemarex-test.m MATLAB script developed by Trujillo-Ortiz et al. (2004) was used for the analysis.202.4. Data collection and analysisSOM methodSelf-organizing maps (SOMs) are a common type of unsupervised artificial neural networkparticularly adept at pattern recognition and classification, and in many respects SOMs areanalogous to more traditional forms of cluster analysis. Kohonen (2001) offers an explanationof the development and details of the SOMs algorithm which by now has been used in awide range of disciplines (e.g. Kaski et al., 1998; Oja et al., 2002). Whereas SOMs method hasbeen commonly used as a clustering tool, it may under perform simpler techniques such asK-means clustering (Bishop, 2006). The value of SOMs therefore lies in application as a discretenonlinear Empirical Orthogonal Function (EOF) method (Cherkassky and Mulier, 1998), ratherthan for clustering. Additional details on the SOMs method are found in Section 2.8, andFigure 2.5 illustrates a conceptual diagram of the SOM’s steps. In this section, we introducethe main features of the SOMs method by comparing it with the more commonly known EOFmethod.The principles behind SOMs and EOF method are similar in their ability to reduce multi-dimensional data into a smaller set of characteristic modes (spatial and/or temporal patterns).To achieve this goal, the SOMs method performs a nonlinear projection from the input dataspace to a set of units (neural network nodes) on a 2-D grid (Figure 2.5). Each characteristicpattern identified by the SOMs method can be viewed as an EOF (spatial pattern or eigenvec-tor) of a particular mode. Whereas temporal information in the EOF method is individuallyexpressed for each mode as a time series of principal components (where the components in-dicate the strength of the particular spatial pattern at a given time), the SOMs patterns areenumerated according to their position in the 2-D grid so the temporal information is given bythe time series of the pattern numbers (i.e. time series of the 2-D grid node locations; Figure2.5). The essential feature of the 2-D grid to which the patterns are mapped is that the neigh-boring units on the 2-D grid represent similar patterns, while dissimilar patterns are mappedonto units farther apart.Our goal is to use the SOMs method to identify characteristic temporal patterns (signals)in the classified bed texture data, discussed above. For the present case, a temporal patternis basically the classified bed texture data (consisting of 2 classes) from any given samplingpoint i plotted for each sampling event as a series (Figure 2.5 - result of pre-processing). Af-ter removing missing values, our data set generated 154 temporal patterns (154 consistentlysampling locations on the subject riffle), or series, of 11 events in length (11 sampling events)(Figure 2.5). These 154 temporal patterns serve as the input data (vectors) to SOMs training(Figure 2.5). SOMs training results in populating the 2-D grid discussed above, which consistsof the most characteristic temporal patterns in the bed texture data (Figure 2.5).212.4. Data collection and analysisFigure 2.5: Schematic illustration of SOMs implementation on our dataset, modified andadapted from Richardson et al. [2003] for the present study. The main steps withSOM implementation are (1) pre-processing, (2) training, and (3) post-processing.222.4. Data collection and analysisInitial estimates of the SOMs 2-D grid size were identified prior to SOMs analysis by com-pleting Principal Components Analysis (PCA) on the bed texture data. PCA results indicatedthat a majority of classified bed texture data variance was explained by the first several PCAmodes. As a result, the initial SOMs 2-D grid size testing began with several nodes. Further-more, SOMs training was completed using several different numbers of units in the 2-D grid,with different parameter sets (e.g. neighborhood function and radius, type of training, initial-ization of weight vectors and number of iterations). As in Radic´ and Clarke (2011), we find thatthe sensitivity of pattern recognition in the SOMs training to the choice of the parameters issmall (i.e. similar patterns in resulting SOMs are produced with varying parameter values)2.4.2 V* and pool cross-section surveysTrends of pool erosion and sedimentation were evaluated within the pool immediately up-stream of the riffle (Figure 2.1) using the V* methodology developed by Lisle and Hilton (1992),and with repeat elevation surveys conducted along one established cross-section (Figure 2.1).The V* methodology uses a repeatable grid-based pattern of depth measurements in a poolto compute the decimal fraction of a pool that is filled with fine sediment (Lisle and Hilton,1992). For example, a V* value of 0.63 means that 63% of the residual pool volume is filledwith fine sediment, noting that the residual pool volume is an explicit function of the down-stream riffle crest elevation. Measurements of V* made during this study were not necessarilyrestricted to measuring volumes of just fine sediment filling the pool. Rod (V* rod) penetrationwas achieved for coarser substrate (the V* rod is stainless steel construction of one-half inchin diameter with a hexagonal section, 0.05 feet etched measurement intervals, and an approx-imately 45-degree tip at one end). Our measurements therefore provide a fuller estimate oftotal pool fill change through the study period. Repeat V* measurements over time permit anobjective evaluation of pool sedimentation trends (Lisle and Hilton, 1992; Hilton and Lisle, 1993).V* was characterized with at least 100 sampling points per event. At each pool samplinglocation we measured the water depth, and then pushed the V* rod into the bed until rod ad-vancement was refused, subsequently recording the refusal depth (i.e. sedimentation depth)(Lisle and Hilton, 1992). Consistent with methodological guidance (Hilton and Lisle, 1993), allV* measurements were made at relatively low discharges when streamflow was clear. Flowvariability between V* measurements over the 3-year study period was minor. V* values werecomputed for each event according to:V∗ =total sediment volumetotal residual pool volume(2.5)Note that the total residual pool volume is the sum of the total sediment and water volumesituated below the downstream controlling riffle crest elevation (Lisle and Hilton, 1992).A cross-section used to measure the controlling riffle crest elevation and track local bedchange was located at the pool tail. The cross-section was surveyed with a laser level at ap-232.4. Data collection and analysisproximately 60 cm intervals within the channel, and at slope breaks along the banks. All bedelevations were recorded to the nearest 2 mm, and the elevation surveys and data reductionwere conducted in accordance with the methods described by Harrelson et al. (1994). It wasnot possible to establish more cross-sections due to time constraints during sampling events.Repeat longitudinal profiles were collected, but were of a resolution not useful to the presentefforts.2.4.3 Bedload transport measurements and modelingData collectionBedload sediment transport was measured with a 76-mm Helley-Smith sampler that was fit-ted with a 0.25 mm mesh collection bag. Samples were collected according to the single equalwidth increment method (Edwards and Glysson, 1988). Bedload particles larger than 16 mm(and smaller than the opening) may be underrepresented by the Helley-Smith sampler due toan apparent trapping efficiency threshold (Emmett, 1980). We therefore consider our measure-ments representative for particles in the range of 0.5 to 16 mm. Bedload was sampled at fourto ten locations across measurement transects for a duration of 30 to 60 seconds per vertical,dependent upon stage variability. Repeat bed load measurements were made for roughly one-quarter of the visits, and bed load samples were weighed, dried, weighed again and sieved at1 phi intervals.Flow measurements accompanied each bedload sediment measurement and were com-pleted with bucket-wheel velocimeters and wading rod. Water level stage was recorded beforeand after each flow measurement. Bedload measurements began two years before the start ofthe present study in January 2008. Bedload measurements were restricted to flows less than2.5 m3/s, or close to one order of magnitude less than the peak flood of the study period. Bed-load measurements primarily reflect Phase 1 (primarily sand-sized grains transported over astable and coarser riffle surface), and to a lesser extent Phase 2 (Phase 1 transported substrateplus mobilization of some fraction of the coarser riffle surface substrate), transport conditionsJackson and Beschta (1982) (Figure 2.2), or Stage 1 and Stage 2 as detailed by Hassan et al. (2005).However, sediment transport modeling (discussed next) indicates that Phase 2 transport con-ditions occurred during most, if not all, of the floods observed through the study period. Theflow related sampling limitation relative to the study goal of examining pool-riffle bed texturetrends was addressed heuristically through construction of a sediment transport flux modelfor the gaging station location.242.4. Data collection and analysisFigure 2.6: Dimensionless bed load transport rates for Majors Creek. The Adjusted W-Creflects sand and gravel component curves joined into one continuous curve, fol-lowing the approach outlined by Wilcock (2001); each component curve was fit tothe data using sum of squares minimization. Bedload samples collected immedi-ately upstream of the subject riffle from December 2008 through April 2012.Sediment transport modelingFractional bedload flux modeling followed the approach developed by Wilcock (2001) whereby,in this case, the Wilcock-Crowe (W-C) function (Wilcock and Crowe, 2003) was calibrated tobedload measurements collected at the gage (Figure 2.6). The Wilcock-Crowe function wasselected because the bulk bed samples indicate roughly 33% sand composition (Figure 2.2).The dimensionless W-C function for any grain class i is defined as:W∗i =0.002φ7.5 φ < 1.3514(1−0.894φ0.5)4.5φ ≥ 1.35(2.6)The parameter φ is a stress ratio (τ/τri) of which τ is the average bed stress and τri is theaverage mobilizing reference stress for any size class i of the bed material. The average bedstress was computed as γRhS where γ is the unit weight of water, Rh is the hydraulic radius,computed with a field-measurement based empirical relationship, and S is the average bedslope in the downstream direction, assumed to approximate the water surface slope over manychannel widths. The average mobilizing reference stress is dependent upon a hiding function(Di/D50) and the average mobilizing reference stress for the mean grain size of the surface252.4. Data collection and analysismaterial(τrs50) is defined as:τriτrs50=(DiD50)bw, (2.7)where Di represents the mean diameter of each size class, and the exponent bw is a fittingparameter dependent on Di/Dsm:bw =0.671+ exp(1.5− DiDsm) , (2.8)where Dsm is the mean grain size of the bed surface material. Wilcock and Crowe (2003) demon-strated that the average mobilizing reference stress for the mean size class of the surface ma-terial (τrm) is dependent upon the surface sand content:τrm = (ρs − ρw) gDsm [0.021+ 0.015 exp (−20Fs)] , (2.9)where ρs is sediment density, here assumed to be 2.65 g/cm3, ρw is the density of fresh water,Fs is the percentage of sand in the surface material and g is gravitational acceleration. Thedimensional transport rate for any grain size class i is computed following an Einstein-typeflux:qbi =W∗i Fiu3∗[(ρs/ρw)− 1] g , (2.10)where qbi is the fractional sediment flux for grain size class i, Fi is the percent of grain sizeclass i present in the bed surface mixture, u3∗ is the shear velocity, computed as(τ/ρw)0.5. Cal-ibration of the W-C function was achieved through adjustments to τri for the sand-sized andgravel-sized and coarser fractions, respectively (i.e. computed values of τrifor sand-sized andgravel-sized and coarser fractions was multiplied by an adjustment factor). The τri adjust-ment factor for sand-sized fractions was 1.48 and for gravel-sized and coarser fractions was2.10. The ultimate calibration fits were determined by sum of squares minimization. The cal-ibrated, composite sediment rating curve shown in Figure 2.6 was developed after Ashworthand Ferguson (1989) and Whiting and King (2003) for comparative purposes. The dimension-less bed load transport rating curve shown in Figure 2.6 is defined by the dimensionless shearstress τ∗ (Shields stress) and the dimensionless Einstein bed load flux q∗. The Shields stressfor grain fraction i can be defined as (Shields, 1936):τ∗i =τ(ρs − ρw)gDi . (2.11)The Einstein bed load flux for grain fraction i can be defined as [Parker, 2008]:q∗i =qbiF−1i{[(ρs/ρw)− 1] gDi}0.5 Di (2.12)262.5. ResultsThe quantity qbi was computed with Equation Bed surface samplingTwo streambed samples were collected in June 2008 in close proximity to the pool-riffle pair inorder to assess differences between bedload samples and general composition of the streambedsurface. The two streambed samples were prepared from three separate sub-samples collectedalong two sampling transects, respectively. Sub-samples were collected using a 130 mm diam-eter sampler driven roughly 150 mm into the streambed. One-hundred and fifty mm approx-imates two times the active layer thickness (Hirano, 1971) according to:La = naD90 whereLa isthe active layer thickness, nais a constant of order 1 to 2 (Parker, 2008) and D90 was measured tobe 55 to 60 mm (Figure 2.2). The transect sub-samples were emptied onto a plastic tarp, thor-oughly mixed, and sampled to produce the two streambed samples, respectively. The sampleswere then weighed, dried, weighed and sieved at one-phi intervals.2.5 Results2.5.1 Riffle texture adjustmentCumulative point count basedFigure 2.7 provides summary results for the eleven riffle texture sampling events accordingto the grain size score 0 to 5. For E1 to E6 riffle texture exhibited a relatively consistent grainsize composition of which a majority was gravel-sized substrate - gravel 61%, sand 22% andcobble 17% (Figure 2.7). Riffle texture shifted to a sand-sized substrate majority following theMarch 26, 2011 flood (E7: Figure 2.7) - sand 48%, gravel 42% and cobble 10% (Figure 2.7). Thesandier riffle composition persisted through E8, after which texture began to trend back to agravel dominated condition. The distribution of grain sizes for the last two sampling events isquite similar to the initial condition (E1: Figure 2.7)Non-parametric statisticsCochran’s Q test results indicate that transect-based bed textures over the eleven bed-mobilizingevents were different, and that the eleven floods did not result in effectively similar transect-based textures. Cochran’s Q for Transect 1 yielded a p-value of 2.17× 10−4 (n=31), for Transect2 a p-value of 8.97 × 10−3 (n=32), for Transect 3 a p-value of 1.36 × 10−7 (n=30), for Transect4 a p-value of 4.59 × 10−11 (n=32) and for Transect 5 a p-value of 2.37 × 10−9 (n=29), all wellbelow α = 0.01.McNemar test results indicate that all transects exhibited at least one texture shift throughthe study period (Table 2: p-value < 0.05). A texture shift occurred at each transect fromsampling event 6 to event 7, presumably due to a large flood that occurred between these272.5. Resultsn = 176 162 163 160 177 174 161 180 171 169 170Figure 2.7: Riffle texture point count results for 11 sampling events from January 2010through April 2012, expressed as percent stacked bars by grain size class.two events (Table 2.1 and Figure 2.2). In addition, Transects 1 and 5 are characterized byone additional texture shift, Transect 2 and 4 by two additional shifts, and Transect 3 by threeadditional shifts (Table 2). Transect texture shifts along relatively finer or coarser texture trendsvaried in time (Table 2). For example, some transects exhibited trends from finer to coarser,and so on (Transects 2 and 3), whereas others exhibited only finer texture shifts (Transects 1and 5), or a combination of both (Transect 4). On the other hand, spatial texture trends weresimilar when changes are considered with respect to event sequence (Table 2). For example,Transects 3 to 5 showed a fining texture trend from sampling event 2 to 3, all transects showeda fining trend from sampling event 6 to 7, and Transect 2 and 3 showed a coarsening trendfrom sampling event 9 to 10.The McNemar test results can be compared qualitatively against spatial representations ofthe riffle texture point count data. Figure 2.8 illustrates the point count data spatially. The pointcount data have been classified as sand-sized or gravel-sized and coarser. Figure 2.9 illustratesriffle texture point count change spatially. Riffle texture change was specified as nodes whichchanged from sand-sized to gravel-sized and coarser, and vice versa. Figure 2.8 shows thatwith the exception of E7 to E9, riffle texture was generally coarse through the riffle center andfine along the riffle margins. Figure 2.9 highlights this point and is particularly useful becauseit provides a visual summary of how riffle texture changed from event to event.SOM methodAfter testing several different sizes of the 2-D grid (i.e. the number of patterns), we settlewith a 2×3 SOM that displays six temporal patterns across all sampling points and samplingevents (Figure 2.10). The most common temporal patterns are 5 and 6 (each associated with26% of the sampling points), whereas the least common is pattern 2 (associated with 6% ofthe sampling points). General interpretation of the six temporal patterns (Figure 2.10) basedon their visual similarity and dissimilarity, results in grouping of patterns into three clusters(Figure 2.10). The first cluster is defined by patterns 1 and 2, the second cluster by pattern 3282.5. ResultsSampling Event 3Sampling Event 2Sampling Event 1Sampling Event 6Sampling Event 5Sampling Event 4Sampling Event 9Sampling Event 8Sampling Event 7Sampling Event 11Sampling Event 10 right bankleft bankflow directionLongitudinal Station (m)Transverse Station (m)sand-sized grain gravel-sized and coarser grainFigure 2.8: Spatial observations of riffle texture based on classifying the point countsshown in Figure 2.7 as either sand-sized or gravel-sized and coarser. Note thatthe sampling interval for E1 was slightly different from that used in all subsequentcounts. As a result E1 data will not be used in additional non-parametric analyses.and the third cluster by patterns 4-6 (Figure 2.10). We consider the second cluster to reflecttransitional temporal conditions to those characterized by clusters one and three, because itcontains some features of the first and third clusters. Specifically, the second cluster reflectsthe first cluster for E1:E5 and reflects the third cluster for E6:E11. All three clusters exhibit bedtexture fining for E6 to E7, providing the only consistent response among the three clusters.The clusters exhibit differences at the beginning of the study period with clusters 1 and 2characterized by fining through E4 and cluster 3 showing initial coarsening followed by littlechange through E6. All three clusters exhibit coarsening from E8 to E9 or E9 to E10, followedby slight to little fining between the last two sampling events.We associate each temporal signal in the 2×3 SOM with a pattern from each samplingpoint, based on the minimum root mean square distance between the observed temporal sig-292.5. Resultsflow directionEvent 4 - Event 5Event 3 - Event 4Event 2 - Event 3Event 7 - Event 8Event 6 - Event 7Event 5 - Event 6Event 10 - Event 11Event 9 - Event 10Event 8 - Event 9right bankleft bankLongitudinal Station (m)Transverse Station (m)change to sand-sized grain change to gravel-sized and coarser grainFigure 2.9: Observations of riffle texture change for sequential sampling events, com-puted with data shown in Figure 2.8 as a shift from either sand-sized to gravel-sizedand coarser grains, or gravel-sized and coarser grains to sand-sized grains. Sam-pling sequence E1 to E2 has been omitted because the sampling stationing for E1was slightly different for each subsequent event.nal at the point, and the temporal signal in the SOM. To show which of the SOM patterns oc-curs for each sampling point, we plot the original grid of longitudinal and transverse samplingpoints and display the cluster number for each point (Figure 2.11). Notably, the third clusterplots as the dominant temporal response through the riffle core into the left bank, whereascluster one prevails along the right bank riffle margin and cluster 2 shows spotty occurrencethrough the riffle core and along the left bank (Figure 2.11).302.5. ResultsFigure 2.10: Final 2×3 SOM (2-D grid) showing six characteristic temporal patterns ofbed texture classes. The temporal pattern index number and number of points(f, in %) associated with the pattern are given above each plot. The patterns aregrouped into three clusters, represented here by different colors: cluster one ingold (consisting of pattern index numbers 1 and 2), cluster 2 in light brown (patternindex number 3) and cluster three in dark brown (pattern index numbers 4, 5 and6). The number of sampling points along each transect is set by the minimumnumber of sampling points n for each transect range provided in Table 2.Figure 2.11: Spatial distribution of the three clustered bed texture temporal signals shownin Figure 2.10. The mapped color for each temporal signal is consistent with thatused in Figure 2.10: gold (temporal patterns 1 and 2), light brown (pattern 3) anddark brown (patterns 4, 5 and 6). The top of the plot represents the right bankriffle margin. For clarity the results have been collapsed in space, i.e. the clusterindexes (1:3) are presented as a uniform grid of sampling points. No numericalinterpolation has been performed between sampling points.312.5. Results2.5.2 V* and cross-section surveysV* conditions basically exhibit two states, those within the range of 0.55 to 0.70, punctuatedby one event duration periods of lower sedimentation, within the range 0.36 to 0.40 (Figure2.12A). The relatively lower V* values of 0.36 and 0.40 were measured on E3 and E10, respec-tively (Figure 2.12A: Table 2.1). The median V* value for the events is 0.61 (standard deviationis 0.12). Figure 2.13 shows results from repeat cross-sectional surveys completed at the tail ofthe pool, and head of the riffle (Figure 2.1). The data illustrate little net change to the bed orthe banks at the section location, despite the occurrence of two relatively large flood events(Table 2.1).Figure 2.12: (A) Pool V* results for 11 sampling events from January 2010 through April2012. (B) Sampling event to event normalized residual pool and sediment volumechange. The residual pool and sediment volume change was computed by sub-tracting former normalized volume values from later volume values, e.g. E3 nor-malized volume values less E2 normalized volume values. Residual volumes havebeen normalized by the median residual pool and sediment volumes, respectively.See Hilton and Lisle [1992; Fig. 1 therein] for graphical depictions of the residual vol-umes. Errors bars in (A) represent the method repeatability as suggested by Hiltonand Lisle (1993).322.5. Resultsright bankleft bankFigure 2.13: Repeat cross-sectional surveys completed at the transition from the pool tothe riffle. High-water mark (HWM) for the flood of record during the study periodis shown. Figure view is looking downstream. The stream banks are populatedwith mature redwood and California Bay-Laurel trees. Cross-section location notedin Figure Sediment transport and bed surface samplingSampled bedload consists predominately of sand-sized grains, and in a few cases, grains inthe 16 to 32 mm and the 32 to 64 mm size classes were caught in the sampler (Figure 2.2).In all cases particles > 16 mm represented less than 3-percent of the total sample mass. Grainsize composition of bed load measurements was generally consistent through the study period(Figure 2.2) and noticeably finer than the bed surface composition. Bedload samples consistedof 80-percent or more sand-sized grains by mass, whereas the two surface samples consistedof 40- and 36-percent sand-sized grains by mass. The median grain diameter of bed load mea-surements ranged from 0.36 to 0.90 mm (medium to coarse sand-size fractions, respectively),whereas the median grain diameter of the surface samples ranged from 6.0 to 9.2 mm (fine tomedium pebble-size fractions, respectively or gravels) (Figure 2.2). The relatively low captureof bedload particles > 16 mm during the course of this study either indicates sampler perfor-mance consistency with the findings of Emmett [1980], or reflects little to no transport of thesesizes at the time bedload measurements were made.Dimensionless bedload flux by grain class predominately increases with the dimensionlessstress (Figure 2.6). Exceptions to this overall trend include two bedload measurements madeon December 30, 2010 , following occurrence of an estimated 5-year flood (December 19th) and11 days of continuous elevated streamflows, including a second significant peak on December29th. The December 30th measurement points show as plotting to the left of the other mea-surements for any particular grain size class. The lowest fractional transport rates shown inFigure 2.6 were measured on April 5, 2010, which corresponds to the lowest measured waterand bedload discharge during the study, 0.3 m3/s and 0.7 tons/day, respectively. Figure 2.14provides a summary of fractional bedload transport modeling. Bedload transport computedover the three winter seasons was estimated as 204.09 tons in total (Table 2.1), and to have332.6. Discussionoccurred during just 11% of the study period, or due to 52% of the total water discharge (Fig-ure 2.14). Appreciable sand-sized transport picked up at about 0.25 cms whereas appreciablegravel-size and coarser transport picked up at about 0.75 cms (Figure 2.14). Over the flows forwhich gravel and coarser transport was non-existent to quite minor, measurable volumes ofsand-sized sediment (∼20% of sand-sized total) was transported (Figure 2.14).Figure 2.14: Accumulative percent of time, streamflow and bedload sediment, sand-sizedand gravel-sized and coarser, for the 3-season study period. Bedload discharge forthe 3-season period was computed with the adjusted WC curve shown in Figure2.6. Analysis after Emmett [1999].2.6 Discussion2.6.1 Riffle texture dynamics and spatial organizationThe McNemar and SOM clusters 1 and 2 provide for a generally consistent interpretation ofriffle texture temporal responses during the study period. The temporal responses were cycli-cal and characterized by initial riffle texture fining, followed by coarsening, fining, coarsening,and then with slight fining between the last two events (Figures 2.8 and 2.9). The overall domi-nant texture response was a clear fining across much of the riffle surface, apparently due to the20-year flood that occurred between sampling events 6 and 7 (Figures 2.8 and 2.9; Table 2: p <0.05). A fining response driven by relatively large reach-average bed stresses is consistent, atfirst order, with theory developed by Buffington and Montgomery (1999a) (Figure 2.15), notingthat construction of Figure 2.15 and the discussion below reflects a general test of their theory,as we are limited by the lack of explicit grain size information for the point-count data.342.6. DiscussionFigure 2.15: Predictions of reach-average median grain size vs. reach-average bed stressas developed by Buffington and Montgomery (1999a). Median grain size predictionsfall along the dashed light lines, which represent equilibrium unit bedload trans-port rates (kg-m−1s−1), and the bold black line which represents the estimated com-petent median grain size for a given average bed stress (i.e. Shields’ condition). Thedot-dashed black line reflects the transition from sand-sized grains to gravel-sizedand coarser grains (denoted as gravel+). Bar labels reflect texture changes for sam-pling event sequences which exhibited significant riffle texture shifts (see Table 2).Texture shifts have been plotted as departures from sand-gravel and coarser transi-tion line depending on whether the riffle in general fined or coarsened. The reach-average stress values reflect peak flood conditions in between each sampling event(see Table 2.1), and as computed with the sediment flux model using a gaging-derived empirical function for water depth. The maximum estimated mobile grainsize for each intra-sampling event peak is provided in Table 2.1. Figure re-drawnfrom Buffington and Montgomery (1999a)Figure 2.15 illustrates event sequence changes in percent sand-sized and gravel-sized andcoarser substrate for event sequences which exhibited a significant change in texture (Table2.2). Specifically, E6–E7 bracket the March 26, 2011 flood, which to some degree drove a 29-percent upward shift in measurement of sand-sized grains on the riffle bed surface. Samplingevent 6 occurred on January 5, 2011 and sampling event 7 occurred on April 5, 2011, the firstday post peak flood when conditions were suitable for measurements. Given the time spreadfrom flood peak to E7, it is not possible to explicitly attribute the texture shift to conditionsduring the peak. Nonetheless, the overall fining response is consistent with Buffington andMontgomery (1999a).The fining response for E6–E7 (Figure 2.15) could also be attributed to a relatively high up-stream supply of fine sediment (Dietrich et al., 1989; Lisle et al., 1993; Paola and Seal, 1995; Nelsonet al., 2009), or due to conditions of high bed stress relative to the critical stress (Paola and Seal,1995). The sediment transport flux model indicates that 95 tons of sand-sized grains were352.6. Discussiontransported in between E6–E7, as compared to 26 tons for gravel-sized and coarser grains.Though it is not clear if the relative upstream supply of fines was higher for the storm pe-riod between E6–E7, as compared to other sequential events, it is clear that the peak studyperiod flood provided average bed stress conditions well above the critical sand and graveland coarser stresses, respectively (Table 2.1). As a result, peak flood conditions of March 26,2011 were likely an important aspect of the measured E6:E7 texture response.The remaining texture responses of McNemar and SOM clusters 1 and 2 reflect a range offlood flows and sediment fluxes that vary by up to a factor four (Table 2.1), including floods ofannual, bankfull and roughly twice-bankfull magnitude. The spatial responses for fining andcoarsening were distributed across the riffle transects, and were not restricted to the channelmargins or the channel center, suggesting widespread bedload movement during the floods(Figure 2.8). Figure 2.15 indicates that the texture responses of E8:9, E4:5, E10:11, E2:3 (sam-pling events; Table 2.1) are consistent with Buffington and Montgomery (1999a). The texture re-sponse of E9–E10 (Figure 2.15) on the other hand are not well described by the reach-averagestress—average D50 model.The coarsening response of E9–E10 is comparably strong amongst E2–E3, E4–E5, E8–E9and E10–E11, as it is a consistent result for McNemar and SOM clusters 1-3. It seems possiblethat E9–E10 coarsening reflects riffle texture recovery from effects of the 20-year flood. Recov-ery could be explained by Jackson and Beschta (1982), who provide that of all the storage sitesfor sediments, riffles may respond most actively as a result of spatially non-uniform bedloadtransport under Phase 2 conditions (i.e. when coarse riffle surface sediments have been mo-bilized). Notably, peak flood conditions in between E9 and E10 were near to twice bankfull.This implies that bedload transport spatial non-uniformity can have a strong influence on rif-fle texture trends in streams with somewhat subdued bed topography, consistent in spirit with(Nelson et al., 2010).Despite the apparent variation of riffle texture temporal response given by McNemar andSOM clusters 1-3, Figure 2.11 indicates that the responses were spatially organized during thestudy period. Following from the preceding discussion, McNemar and SOM clusters 1 and2 comprise one aspect of the spatial organization whereas SOM cluster 3 the other (Figure2.11). This observation bridges the temporal and spatial, and indicates order in a set of resultswhich at first pass appear to be disordered (cf. Figure 2.8). Furthermore, the full complementof results suggest that gravel-bed streams may characteristically respond to moderate andsmaller sized floods through texture adjustment which is spatially and temporally variable.Large floods, however, appear to drive texture adjustments which are spatially and temporallyconsistent, for a period of time scaled by magnitude of disturbance generated by the flood.This time scale will vary from stream to stream.362.6. Discussion2.6.2 Pool-riffle sedimentation couplingWe hypothesized sedimentation coupling between the pool-riffle pair for which coupling re-flects similar temporal trends of pool filling or scouring and pool/riffle texture fining or coars-ening, respectively. Figure 2.16 indicates two sequences of coupled sedimentation trends inthe pool-riffle, perhaps one sequence of a partially coupled response and several sequences ofdecoupled responses. The E10–E9 change index shows coarsening of sediments in the pooland riffle, as well as a somewhat large evacuation of stored pool sediments (Figure 2.16). Onthe other hand the E11–E10 change index shows fining of sediments in the pool and riffle, aswell as a swing back to increased storage of sediments in the pool (Figure 2.16). These resultsindicate coupling of sediment transport process between the pool and riffle, as pool deple-tion suggests general coarsening and pool filling general fining of pool sediments (Lisle andHilton, 1992). Notably, E10–E9 pool depletion and the E11–E10 pool filling each occurred witha concurrent increase in the residual pool volume (Figures 2.12B). We are unaware of previousstudies that reported sedimentation coupling for associated pool-riffle morphologic units.relative riffleand pool coarseningrelative riffleand pool fining1.*: big changeV*: no changepool depletionpool fillFine Fraction Point Count Index of Change E3-E2E4-E3E5-E4E6-E5E7-E6E8-E7E9- E8E11-E10E10-E9Figure 2.16: Sequential sampling period change of normalized riffle and pool fine fraction(sand-sized sediments) point count and V* (top of plot). Riffle and pool fine frac-tion point count data were normalized by the sum of point counts for gravel- andcoarser-sized substrate for each sampling event. The fine fraction index of changewas computed by subtracting former normalized fine fraction point counts fromlater fine fraction point counts, e.g. E3 normalized fine fraction less E2 normalizedfine fraction. V* change plotted as relative change for illustration purposes only.Note that normalized riffle and pool fine fraction change for the first sequence offloods is lacking due to a pool fine fraction data gap for E1.The E3–E2 change index illustrates fining of both the pool and riffle, yet a somewhat large372.7. Concluding remarks and next stepsdepletion of pool sediments (Figure 2.16). In this case pool sediment depletion was character-ized by a large decrease in the residual pool volume (Figure 2.12B), indicating the pool hadless storage volume. The E9–E8 change index shows pool and riffle texture coarsening (Figure2.16), and little change in pool sediment volume storage (Figure 2.16). Residual pool volumeremained generally unchanged during this period (Figure 2.12B). Lastly, the E6–E5 changeindex illustrates minor pool-riffle coarsening, with little change in pool sediment volume stor-age. These event sequences suggest partial sedimentation coupling as the texture responsestrended similarly, yet the volume of pool sediment storage diverged or was unchanged.Decoupled texture trends in the pool-riffle occurred for E4–E3, E5–E4, E7–E6 and E8–E7(Figure 2.16). Decoupled sedimentation trends between pool-riffle pairs reflects spatially non-uniform sediment transport, as previously reported by Jackson and Beschta (1982). Further-more, generally de-coupled sediment transport processes at the scale of a pool-riffle pair isconsistent with that of Sear [1996], most of the pool-riffle maintenance theory (i.e. contrastinghydrodynamics from pool to riffle) and consistent in principle with the numerical results ofde Almeida and Rodrı´guez (2011), who reported that pool-riffle pairs (and sequences) can ex-hibit maintenance coupling via sediment transport processes characterized by bedload ratesand fractional content that differ from pool to riffle ((de Almeida and Rodrı´guez, 2011): Figure2.11 therein]. Whereas decoupled sedimentation trends appear to be related to processes ofpool-riffle maintenance, it is unclear what role, if any, coupled sedimentation trends have forpool-riffles. Building from Jackson and Beschta (1982), periods of sedimentation coupling mustreflect conditions of equal mobility (Parker and Klingeman, 1982) and uniformity of sedimenttransport process. As a result, our limited data set implies that periods of equal mobility, man-ifested over bedform pair length scales, is an important aspect of channel response to largefloods.2.7 Concluding remarks and next stepsThe field-sampling approach permitted quantification of the spatial and temporal characteris-tics of riffle texture over the course of the study. In particular, SOM mapping of riffle texturetrends to each riffle sampling node indicates a spatially and temporally organized process ofriffle texture adjustment. Notably, the McNemar non-parametric test results were consistentwith the EOF temporal trends of riffle texture. Riffle texture trends associated with significantchange compare well with the reach-average bed stress - reach-average D50 model of Buff-ington and Montgomery (1999a). For all but one event sequence (E9–E10), the Buffington andMontgomery (1999a) response model captured both the direction and magnitude of texture ad-justment in response to sediment mobilizing floods. The response trajectory of event sequenceE9–E10 can be explained by Jackson and Beschta (1982)’s observations of spatially non-uniformsediment transport processes over a pool-riffle reach in the Oregon coast range. There mayalso, however, be other explanations.Evaluation of local sedimentation trends for the pool-riffle pair with V* and the riffle tex-382.8. Details of SOM methodsture data suggest a period of coupling between the bedform units following the 20-year flood,consistent with previous work (Dietrich et al., 1989; Lisle and Hilton, 1992). Demonstrationof coupling for the present case was however tenuous, and would benefit from a more con-trolled environment. Therefore, we have used the present work to design flume experimentswhich we hypothesize will support a novel analysis of how sedimentation and surface sedi-ment texture dynamics between pool and riffle factor for pool-riffle maintenance. Beyond themaintenance realm, these issues are particularly important for salmonids, because they relyon pool-riffles for habitat over a wide range of their life cycle.We believe the riffle-texture aspects of the field-measurement strategy and analytical ap-proach holds promise for identification and characterization of bed surface structures and sed-iment patches, as well as exploration of local sediment sorting processes. Further applicationof this program would benefit from a denser sampling of riffle sediments, including georef-erenced photographs of sufficient resolution for more detailed analysis of surface sedimentsdistribution.2.8 Details of SOM methodsSOMs method, introduced by Kohonen (1982, 2001), approximates a dataset in multidimen-sional space by a flexible grid (typically of 1 or 2 dimensions) of cluster centers. For a 2-Drectangular grid, let the grid points or units be symbolized as ij = (l, m), where l and m takeon integer values, i.e. l = 1, . . . , L; m =1, . . . , M; and j =1, . . . , L×M. To initialize the train-ing process, an EOF method is usually performed on the dataset, and the grid ij is mappedto zj(0) (i.e. initial vectors in the data space) lying on the plane spanned by the two leadingEOFs (or eigenvectors). As training proceeds, the initially flat 2-D surface of zj(0) is bent tofit the data. The original SOM is trained in a flow-through manner, i.e. observations (inputvectors) are presented one at a time during training, though algorithms for batch training arealso possible. In flow-through training, there are two steps to be iterated, starting with n = 1:(1) at the n-th iteration, select an observation (input vector) x(n) from the data space, and findamong the points zj(n - 1), the one with the closest Euclidean distance to x(n). Call this closestneighbor zk(n), and the corresponding unit ik(n) the best matching unit (BMU); (2) letzj (n) = zj (n− 1) + η h(∣∣∣∣ij − ik (n)∣∣∣∣2) [x (n)− zj (n− 1)] (2.8A)for which n is the learning rate parameter and h is a neighborhood or kernel function. Theneighborhood function gives more weight to the grid points ij near ik (n), than those far away,an example being a Gaussian drop-off with distance. Note that the distances between neigh-bors are computed for the fixed grid points (ij = (l, m)), not for their corresponding positionszj(n) in the data space. Typically, as n increases, the learning rate is decreased gradually fromthe initial value of 1 towards 0, while the width of the neighborhood function is also graduallynarrowed.392.8. Details of SOM methodsWe use the MATLAB SOM Toolbox (Vesanto et al., 2000) where four types of neighborhoodfunctions are available: ‘bubble’, ‘gaussian’, ‘cutgauss’ and ‘ep’ (or Epanechikov function).Following the guidelines from Liu et al. (2006) we use the ‘bubble’ neighborhood functionwhere h in 2.8A is:h = F(σn − djk)(2.8B)The parameter o´n is the neighborhood radius at n-th iteration, djk is the distance between theunits ij and ik on the 2-D grid, and F is a step function:F(x) ={0 if x < 01 if x ≥ 0 (2.8C)(see Vesanto et al. (2000) for the geometries of these neighborhood functions). The neighbor-hood radius o´nis either constant or linearly decreasing between specified initial and final val-ues.Two quantitative measures (Kohonen, 2001) are used to help determine how many unitsone should use in SOMs, knowing that data underfitting will occur with too few units andoverfitting with too many units. The first measure is the average quantization error (QE)which is the average distance between each data point (input vector) x and zk of its BMU.The second measure is the topographic error (TE) which gives the percentage of data vectorsfor which the first BMU and the second BMU are not neighboring units. Smaller QE and TEvalues indicate better mapping quality. By increasing the number of units, QE can be furtherdecreased; however, TE will eventually rise, indicating that an excessive number of units isused.Our set of final parameters in MATLAB SOM Toolbox consists of the following: hexagonallattice, sheet SOM shape, linearly initialized weights, bubble neighborhood radii of 2 and 1,and batch training performed over 200 iterations.40Chapter 3Experimental setup and measurements3.1 IntroductionA total of four experiments were conducted to complement the fieldwork focus of Chapter 2.In Chapter 2, we report that a pool-riffle pair was maintained across 11 sediment mobilizingfloods, including an approximate 20–year flood. The pool-riffle pair exhibits minor amounts ofnet topographic adjustment across all floods, however, the riffle in particular responds to floodevents through spatially organized bed sediment texture changes. This finding highlights thatpool-riffles in topographic steady-state process flood supplies through changes in bed surfacetexture and roughness. We design our experiments to further examine sediment texture re-sponses of pool-riffle structures to upstream supply forcing, and to develop enhanced data setsfor use with our application of self-organizing maps to characterize sediment texture trends.However, we also specifically design our experiments to examine topographic responses ofpool-riffle structures that are trending to a steady-state condition, as our field case of Chapter2 exhibits statistically-steady behavior. This latter objective forms the main complementarypart to the fieldwork focus of Chapter 2.Here, we provide a summary of the experimental design for pool-riffle experiment 1 (PRE1),as well as details for data collection and processing. In general, we test how channel widthvariations and morphodynamics are coupled, defined by bed topography and sediment tex-ture responses. Experimental water supply ranges from 1.2–2 times the threshold to mobilizebed sediments, on average, and sediment supply is at or near the theoretical capacity for eachflow condition. Our experimental design addresses the following three guiding questions:1. How does bed topography respond to non-uniform downstream changes in channelwidth, which introduces flow accelerations and corresponding variations in shear stress?2. Do width induced topographic responses persist across mobility conditions which rangefrom 2–4 times the threshold condition? Is there hysteresis in this response: will similarresponses emerge for repeat experiments conducted with pre-conditioned bed topogra-phy and bed sediment texture?413.2. Laboratory experiment and methods3. What time-varying or steady features of bed topography and sediment texture indi-cate an equilibrium between upstream supplies of water and sediment and mountainstreams?In Chapters 4 and 5 we present results from PRE1, and use theory to frame and discuss im-plications of the results for pool-riffle formation, maintenance and equilibrium. Whereas wedid use the characteristics of a field site to layout and calibrate the experiments (discussedin Section 3.2.1), the experiments, results and findings are generally applicable to mountainstreams.3.2 Laboratory experiment and methods3.2.1 Setup and constructionPRE1 was conducted at the BioGeoMorphic eXperimental Laboratory (BGMX Lab) at the Uni-versity of British Columbia, Vancouver, Canada (Figures 3.1a and 3.1b). The experimentalflume measures 16 m in total length and 1 m in width, and we use 15 m to conduct the ex-periments. Water recirculates through the flume via a pump, but sediment does not (Figure3.1). We introduce water to the upstream channel boundary through a series of stacked 5 cm∅ plastic pipes, collectively called the flow normalizer. The normalizer is 1000 mm in length,or roughly 2w′ in length, where w′ is the average channel width, and we use it to establish aninitially uniform flow. We introduce sediment to the flume via a speed-controlled conveyor,which dumps particles into a mixing chamber we call the randomizer. The randomizer con-sists of a vertical shaft with alternating cross-bars spanning the width of the shaft. As particlesfall through the mixing chamber, their pathways are interrupted by the cross-bars, which flingsthe particles along random trajectories, providing a spatially-random distribution of sedimentfall points on the inlet flume bed. The randomizer action provides a spatially and tempo-rally uniform inlet boundary condition, which did not require manual adjustment during theexperiments due to pile construction.The flume outlet elevation is fixed, and the downstream-most 1.0 m of channel consists ofstraight channel walls. We chose this outlet configuration to provide controlled conditions forwater and sediment leaving the flume, which pass through the particle imaging light box forflux measurement (discussed in Section 3.2.4). Figures 3.1a and 3.1b show that the experimen-tal channel consists of downstream varying channel width, which for simplicity, reflects theaverage downstream width condition between inflection points along a field site (discussednext). We achieve the experimental width conditions by constructing a channel inside theflume with rough-faced veneer-grade D plywood, which has a surface roughness that variesfrom 1-4 mm, or roughly 0.15 to 0.60 times the geometric mean grain size of the experimentalgrain size mixture (discussed next).423.2. Laboratory experiment and methodssediment feed randomizerExperimental Channel Stationing (mm)water supply normalizerwater recirculation pipereturn water pumpmeasurement carthi resolution cameraLED lightssediment catch basketwater diverter and pressure platewater tankexperimental water level in tankexperimental channel or flumesediment feed conveyorExperimental Set-up(a)(b)Channel Station (mm)(c)Sub-sampling locations320 mm320 mmw = 0.7 mw = 2.1 mFigure 3.1: Overview and images of the experimental setup and field stream reach. (a)Schematic illustration of the of the experimental setup, including an overhead viewof the experimental channel, showing the downstream width variation and subsam-pling locations indicated by red boxes. (b) Photograph of the experimental chan-nel. The photograph view is looking upstream from station 1000 mm. Photographtaken at experimental time 2150 minutes. (c) Photograph of the field channel EastCreek. The photograph view is looking downstream at the equivalent of experimen-tal channel segment 7500 mm to 5000 mm.3.2.2 Experimental designExperimental scaling was guided by a 75-meter reach of East Creek, University of BritishColumbia Malcolm Knapp Research Forest, located 1.5 hours east of the University. East Creekis a small gravel-bedded mountain stream (Figure 3.1c). The field reach was chosen becauseit exhibits pool-riffles and roughened channel segments, with a reach-average bed slope (S) of433.2. Laboratory experiment and methods0.015. Following Henderson (1966), the geometric scale ratio for our 15 m experimental channelis Lr = 5, where the subscript r indicates the field:model length ratio (Parker et al., 2003). Theexperimental channel width and grain size distribution follows Lr (Figures 3.1 and 3.2), andthe experimental channel slope equals that of the field site. The model grain size distributionranges from 0.5–32 mm, with a geometric mean size of 7.3 mm (Dg), a characteristic coarsegrain size of 21.3 mm (D90: for which 90% of all particles are smaller), and a geometric stan-dard deviation of 2.5 (σg) (Figure 3.2). We paint each grain size fraction a unique color to aidwith analysis.Figure 3.2: Cumulative grain size distribution for the experiment and the field reach alongEast Creek, with grain size in mm shown on the x-axis, and cumulative percent fineron the y-axis. The experimental distribution was scaled according to the geometricratio Lr = 5.The ratio of the maximum to minimum width along the experimental channel is≈ 2.1, andwidth variation provides a range of downstream width gradients from (-0.26)–(+0.18). As aresult, experimental conditions are characterized by segments with minor width gradients, tosegments with strong positive and negative gradients (Figure 3.1). The average channel widthof the experimental channel is: w′ = 547 mm, (∑w)/n, where w is the local width, and forcontext, the characteristic coarse grain size scales as 18 to 36w. The minimum channel widthalong the channel is 370 mm (station 8150 mm), and maximum width is 785 mm (station 9960mm).We apply Froude scaling to determine the water supply flow rates for our experiments,which requires (Henderson, 1966):Frr = 1 (3.1)where Frr is the Froude number field:model ratio, with ratio indicated by the subscript r, andFr = Ux/(gLc). We rearrange Equation 3.1 and solve for the field:model velocity ratio:Ux,r = (grLr)0.5 ∝ L0.5r (3.2)443.2. Laboratory experiment and methodswhere Ux,r is the downstream cross-sectionally averaged flow velocity field:model ratio, andgr is the gravitational acceleration field:model ratio. Equation 3.2 provides a direct link withgeometric scaling of the experiment, and provides the basis of experimental flow scaling:Qr = Ux,rL2r ∝ L2.5r (3.3)Our field-estimate of the bankfull flow is 2.3 to 2.5 m3·s−1. Following Equation 3.3, the lowestexperimental flow is 42 liters per second (l·s−1), and was supplemented with higher flows of60 and 80 (l·s−1) (Table 3.1), which equate to flood magnitudes of roughly 5–, and 10–yearrecurrence intervals, respectively. The magnitude of flows used during PRE1 suggests mo-bile conditions for the entire experimental grain size distribution, reflected by (τ/τre f ) valueswhich approach 2.0 (Table 4.1) (Wilcock and McArdell, 1997). The variable τ is the average bedstress, calculated as τ = ρwCdU2x, and τre f is the reference critical mobility stress for the bedsurface median particle diameter D50. The variable ρw is the density of water, taken as 1.0g·cm−3, and Cd is a dimensionless drag coefficient, generally taken as a constant within openchannel flows when Re > 102−4. The average Reynolds number for fully developed and hy-drostatic flows is Re = (Qd)/(Aν), where d is the cross-sectionally average water depth, Ais the average flow area, and ν is the kinematic viscosity of water at 15◦C (approximate tem-perature of water used in the experiments). Reynolds numbers for PRE1 > 105. The Shieldsequation expresses the critical stress for the (D50) of all sediment particles on the bed surface(Shields, 1936):τ∗c50 =τc50ρ′gD50(3.4)where ρ′ = [(ρs/ρw) − 1], ρs is the density of sediment, taken as 2.65 g·cm3, and g is theacceleration due to gravity on Earth. To close Equation 3.4, the critical dimensional stress τc50is calculated by assuming a reference dimensionless critical stress (τ∗re f ) for the D50, which herewe use a value of 0.030 for gravel mixtures (Parker, 2007; Buffington and Montgomery, 1997). Inreality however, (τ∗re f ) is described by a probability distribution of possible values (e.g. Wibergand Smith, 1987; Kirchner et al., 1990; Buffington et al., 1992) (Appendix B).We chose sediment supply rates through iterative 1D numerical modeling of channel evo-lution, which facilitates examination of how theoretical capacity bedload transport varies alongthe length of the experimental channel due to changes in width and particle drag. We use twodifferent transport functions for the simulations, in order to capture a larger range of possibleexperimental conditions:1. The Wong and Parker (2006) corrected Meyer-Peter and Mu¨ller function (MPMf); and2. The mixed grain size Wilcock-Crowe (Wilcock and Crowe, 2003) function (WCf).The WCf, in particular, is suitable for our experimental grain size distribution, due to the 10%by mass, sand-sized composition of our mixture (Wilcock and Crowe, 2003) (Figure 3.2). Weidentify preliminary sediment supply rates by solving MPMf and WCf for453.2. Laboratory experiment and methodsTable 3.1: Experimental details for PRE1PRE1 Interval Elapsed Time Flow Sediment Feed DEM/Photo(-) (min) (ls−1) (kgm−1) (-)0 0 - - yes1 19 42 0.50 yes2 50 42 0.50 yes3 110 42 0.50 yes4 230 42 0.50 yes5 470 42 0.50 yes6 710 42 0.50 yes7 950 42 0.50 yes8 1190 42 0.50 yes9 1430 42 0.50 yes10 1670 42 0.50 yes11 1910 42 0.50 yes12* 2150 42 0.50 yes13 2225 60 0.80 yes14* 2390 60 0.80 yes15 2450 80 1.00 yes16* 2570 80 1.00 yes17 2630 80 1.00 yes18 2870 42 0.50 yes19 3110 42 0.50 yes20 3350 42 0.50 yes21 3590 42 0.50 yes22 3830 42 0.50 yes23 4070 42 0.50 yes24* 4310 42 0.50 yes25 4385 60 0.80 yes26* 4550 60 0.80 yes27 4610 80 1.00 yes28 4730 80 1.00 yes29 4790 80 1.00 yesaAsterisk denotes achievement of mass equilibrium between feed and flux.bThe repeat phase of PRE1 began at elapse time 2630 minutes.cThe elapse time indicates the end time for the specified experimental interval.cDEM stands for digital elevation model of the experimental channel.a. A uniform channel width, equivalent to w′;b. A uniform longitudinal slope of 0.015;c. The three experimental flow rates 42, 60 and 80 l·s−1;d. A uniform longitudinal flow depth approximated as the normal depth; and463.2. Laboratory experiment and methodse. Sediment textures based on the experimental grain size distribution (Figure 3.2).We use the preliminary sediment supply rates for model simulations to assess topographicprofile development. Preliminary sediment supply rates vary from 0.1 to 1.5 kilograms perminute (kg·m−1). Model run durations are 20–40 simulation hours in total, which is sufficientto achieve steady-state conditions, which we define by the following two criteria (these criteriawill be discussed further in Chapter 5):1. ∂η/∂t → 0: Rate of changes in bed elevation goes to zero everywhere in the modeldomain; and2. ∂D50/∂t → 0: Rate of change of the median grain size on the bed surface goes to zeroeverywhere in the model domain.Steady-state simulation results are necessary so we can evaluate whether projected bed pro-files are compatible with the experimental setup, and specifically to avoid supply rates whichcause profile lowering to the flume floor. These two points represent our evaluative criteriafor the simulations. Interaction of the flow with the flume floor would introduce an addi-tional forcing condition to bed profile and sediment texture development. Introduction of anadditional forcing makes it difficult to understand whether width, or floor-related effects areresponsible for experimental outcomes.Simulations show that capacity bedload transport varies along the experimental channelby a factor of 2–100, for an initially uniform sloping bed surface (S = 0.015), subject to the threeexperimental flow rates. After modeling several supply rates within the range of transport val-ues for each flow, we simplified the problem, and chose the spatially-averaged capacity trans-port of the MPMf and WCf which met our evaluative criteria above. The selected sedimentsupply rates for each flow are: 0.5, 0.80 and 1.0 kilograms per minute (kg·m−1), respectively(Table 3.1). One drawback to this approach is that the difference between the spatial averageand true transport capacity increases with flow rate. As a result, sediment supply rates for60 and 80 (l·s−1) were less than capacity by some small fraction. We use each selected bed-load supply rate for trial experiments to confirm the morphodynamic modeling. Here we usemorphodynamics to refer specifically to bed topography and sediment texture adjustments.See Appendix A for model details and sample results for a simulation reflecting experimentalconditions at 42 l·s− Experimental procedureWe start PRE1 from a smoothed-bed, uniform slope condition (Figure 3.3). The profile inFigure 3.3 has the channel station in mm on the x-axis, and bed elevation in mm on the y-axis.We also provide a DEM (digital elevation model) at the top of Figure 3.3, for reference to thefull channel topography, as well as the width layout. Prior to smoothing, the full thicknessof sediment in the flume was thoroughly mixed to establish a random size distribution, and473.2. Laboratory experiment and methods Elevation (mm)Experimental Channel Topography: t = 0 minutes0 minutes0 2000 4000 6000 8000 10000 12000 14000 16000Channel Station (mm)Figure 3.3: PRE1 initial bed topography shown as a longitudinal profile of the averageelevation along the center 50 mm of the experimental channel from station 1500 to15500 mm, and as a DEM (digital elevation model of the experimental channel) at thetop. The perceptually uniform Polarmap colormap was used to map bed elevationsin the remove textural heterogeneity related to previous trial runs (Figure 3.1b). Figure 3.4 showsthe experimental water supply rate in l·s−1 vs. time in minutes (a), and the sediment supplyrate in kg·m−1 (b). PRE1 consists of an initial and repeat phase (Figure 3.4b). The initial phaseextends from te = 0 minutes, where te is elapse time, to 2630 minutes, and the repeat phaseextends from te = 2630 minutes, to 4790 minutes (Table 3.1; Figure 3.4b). Flow and sedimentsupply continue at constant values until total sediment flux approximates the sediment supplyrate, and in all cases the fractional flux was comparable to the fractional supply as determinedby the Two-sample Kolmogorov-Smirnov test of continuous distribution similarity (Massey,1951). Specifically, water and sediment supply varies over the course of PRE1 as follows (Table3.1; Figure 3.4b):1. For the initial phase:(a) water supply was 42 l·s−1, and sediment supply was 0.5 kg·m−1 from te = 0 to 2150minutes;(b) water supply was 60 l·s−1, and sediment supply was 0.8 kg·m−1 from te = 2150 to2390 minutes;(c) water supply was 80 l·s−1, and sediment supply was 1.0 kg·m−1 from te = 2390 to2630 minutes;2. For the repeat phase:(a) water supply was 42 l·s−1, and sediment supply was 0.5 kg·m−1 from te = 2630 to4310 minutes;(b) water supply was 60 l·s−1, and sediment supply was 0.8 kg·m−1 from te = 4310 to4550 minutes;483.2. Laboratory experiment and methods(c) water supply was 80 l·s−1, and sediment supply was 1.0 kg·m−1 from te = 4550 to4790 minutes;We use a ramping up and down period of 4-5 minutes each time the water supply is raisedand lowered to and from the experimental flows of 42, 60 and 80 l·s−1. The repeat phase(Figure 3.4b) began from the prevailing channel topographic and bed surface sediment sortingconditions established by the end of the initial phase (Figure 3.4b).80 Experimental Water Supply Rate (l/s)6042(a)(b)repeat phaseinitial phaseSediment Supply Rate (kg/m)Elapsed Time (minutes)te: elapsed time (minutes)Figure 3.4: PRE1 details of water and sediment supply. (a) Timing and rate of flow duringPRE1. (b) Timing and supply of sediment during PRE1. The first sequence of 42,60 and 80 l·s−1 water supply (and associated sediment feed) constitutes the initialexperimental phase. The second flow and sediment feed sequence was the repeatphase. The vertical shaded areas reflect flow rates of 60 and 80 l·s−1, respectively.3.2.4 Experimental measurements and processingTo address the general experimental questions presented in Section 3.1, we make direct mea-surements of bed topography, sediment flux from the channel, and water depth along the493.2. Laboratory experiment and methodschannel, and we make indirect measurements of bed sediment texture Measurements aremade for data collection intervals that range from 19 to 240 minutes, with intervals during thefirst 240 minutes following a gemeotric progression starting from time 19 minutes (4 minuteramp up + 15 minute run interval) (Table 3.1). We use a geometric progression in order tobetter evaluate morphodynamic evolution during the first 2 hours of PRE1. The maximumdata collection interval is 240 minutes. Practical considerations dictated that data collectionintervals during the 60 and 80 l·s−1 flows were set by the time it took for substantial bed to-pography change to occur, ranging from 60 to 165 minutes.Bedload fluxWe use a light table to measure bedload flux and enforce mass conservation (Zimmermannet al., 2008). The light table system uses an overhead camera to measure particle positions ina water column 2 to 3 cm thick. The particles and water pass over a positively-sloping semi-transparent lexan base, which is back lit by a constant-voltage LED panel light measuring610 mm2. Images of the silhouetted particles are captured at 15-20 Hz with an Allied VisionTechnology GX2300 CCD camera. The camera uses a Kowa Optimed 16 mm 4/3” megapixelLM16XC lens, which was selected specifically for the GX2300 sensor resolution, and imag-ing distance of the setup. Images are processed with LabViewTM code to compute the time-averaged flux for all grain size classes >2 mm at a temporal resolution of 1Hz (Zimmermannet al., 2008). The particle imaging setup went through extensive validation trials followingZimmermann et al. (2008). To independently evaluate PRE1 light table data, we hung a meshbasket at the downstream end of the light table to catch all flux from the experimental channel,which was weighed, then sub-sampled and sieved for comparison.Bed surface topographyWe periodically stop flow to measure bed topography with a camera-laser setup mounted toan automated cart system. We subsequently use these data to produce DEMs with a spatialresolution of 1 mm. In total, we collected thirty DEMs during PRE1 (Table 3.1). The camera isan Allied Vision Technology Prosilica GC with a Kowa 15 mm 4/3” megapixel lens. We mountthe camera at the downstream center of the measurement cart, looking upstream at an angleof roughly 15-degrees from horizontal. We mount a 5 mW 100 deg fan angle green line laser atthe upstream center of the measurement cart, with the lens plane oriented parallel to the bedof the channel. As the cart moves upstream along the experimental channel, photographs ofthe laser illuminated bed are taken. We compare these photographs with a vertical elevationmodel of the flume to produce the DEMs. The vertical elevation is prepared by imaging adot-matrix board placed in the vertical plane of the laser.We use a 3-step algorithm to process each DEM:1. We clip DEM margins to the experimental channel width;503.2. Laboratory experiment and methods2. We filter anomalously high elevation values; and3. We fill DEM holes.Anomalously high elevation values are caused by laser reflection off of boundaries. Highvalues were iteratively identified by plotting the full distribution of elevations as a cd f , andsubsequently removing values that occur at the extreme lower end of the cd f . We then vi-sually inspect a draft DEM, and the filter threshold is adjusted if point elevations within theexperimental channel were deleted, or if anomalously high values remain.DEM holes occur due to camera view obstruction by large grains, and along short channelsegments which exhibit strongly positive topographic gradients (from the reference frame ofthe camera view). We fill DEM holes assuming the local average elevation within immediatelyadjacent unaffected locations along the same longitudinal coordinate. After some trials, weset the lateral search neighborhood to be 30 grid points, which equates to a length scale of 15mm. Our decision to correct DEM holes with lateral elevations, as opposed to a neighborhoodof elevations, is consistent with the physical cause of the camera blind spots. After DEMprocessing was complete, we clip the DEM to longitudinal stations 1500 and 15500 mm, and11 DEM locations are sub-sampled for further analysis. Sub-sampling locations are shownin Figure 3.1a, are located at 1000 mm increments (≈ 2w′), and measure 320x320 mm2. Weuse one last processing step for each subsampled DEM, and discus it next with the compositephotograph.Composite photographs of bed surfaceWe produce a composite image of the experimental channel with a Canon D60 camera andCanon EF 17-40 mm f4.0 lens, fixed to 40 mm to minimize distortion and maximize imageresolution. We mount the Canon camera at the upstream end of the cart, and upstream fromthe laser we use to collect the DEM. Individual photographs are captured at a sensor resolu-tion of 1920Wx1280H px2. Based on lens construction and camera sensor size, we step themeasurement cart at increments of 10-25 mm, depending on topographic relief, and we cropresultant images to the cart step distance and line each sub-image up edge to edge to producethe composite image. Composite image resolution is approximately 1–2 mm, with image in-formation density ranging from 2.1–2.6 px/mm, or 4.4-27.0 px/sediment particle, dependingon distance between channel bed surface and lens body, and lateral distance from image cen-ter. The upper end of the pixels per particle range exceeds the 20 pixels per particle goal putforth by Zimmermann et al. (2008). We use external LED overhead lights to provide consistentlighting conditions across all composite photographs.We apply a 2-step process to the composite channel photographs to develop the final work-ing image.1. We line up each photograph to the same longitudinal coordinate system as the DEM.This was accomplished by applying a length scale offset based on the distance between513.2. Laboratory experiment and methodsthe first downstream row of photographic pixels and the first row of DEM data. Thealignment operation has a resolution of roughly ±1 mm at its best.2. We clip the photograph to longitudinal stations 1500 and 15500 mm, which facilitatesassociated photographs and DEMs to be overlaid, within the resolution constraint justnoted.As with the DEMs, photographs are subsampled at 11 locations (Figure 3.1a), producing im-ages with physical dimensions to match the subsampled DEMs. From visual inspection, thecomposite photographs show a minor amount of radial lens distortion at the image corners,and as a result, composite images were not corrected for distortion.We apply one last post-processing step to produce subsampled composite images andDEMs which map to the same coordinates. The step involves use of two image mappingstatements, assuming a coplanar perspective between the camera lens and the channel bedHugemann (2010):X− X0 = [c1(xˆ− x0)]z (3.5)Y−Y0 = [c2(yˆ− y0)]z (3.6)where real world coordinates are denoted by X and Y, and the origin defined by the position(X0, Y0), which is co-located with the image center coordinates (x0, y0). Image coordinates aredenoted by xˆ and yˆ, which reflect estimates of position within the image. We use Equations3.5 and 3.6 plus the image resolution (fixed for PRE1) to build image coordinate mappingdatabases of the channel bed for the full range of elevations observed during PRE1. This isanalogous to the distortion-correction mapping database of Hugemann (2010).We produce the image coordinate maps with photographs of a checkerboard (uniformcheck dimensions: 1x1 cm) placed horizontally within the widest flume location, and at threeelevations beginning at the floor of the channel, and 30 and 60 cm above the floor. The totalnumber of checks (Cn) and pixels (Pxn) from image center to edge were counted and recordedfor each of the three photographs. The ratio Pxn/Cn provides a scaling of px/mm from imagecenter to image edge for the photographed elevations, and linear regression provides a con-tinuous scaling over elevations 0–60 cm (variable [c2]z of Equation 3.6. At each subsamplinglocation and moving downstream to upstream through each row of image pixels, we producea subsampled image with the following 2 steps:1. We use the DEM to identify the elevation-specific image coordinate map to query; and2. We locate the lateral image coordinates (xˆ and yˆ) which measure ±180 mm from imagecenter by inverting each ratio Pxn/Cn, multiplying by 1 px and summing.We use the DEM resolution to prepare the subsampled DEM, and then numerically enhanceit by linearly interpolating the DEM to match the resolution of each subsampled image. For523.2. Laboratory experiment and methodsreference, we provide Figure 3.5, which illustrates a sub-sampled DEM and photograph pairfor station 10000 mm, elapsed time 2150 minutes (Table 3.1).Figure 3.5: Example sub-sampled DEM and photograph pair for station 10000 mm,elapsed time 2150 minutes. The DEM is on the left hand side of the image, and thephotograph on the right. The sub-sampled DEM and photograph measure 320x320mm2. Affects from the hole filling procedure are evident within the DEM, near thecenter left, and with respect to the larger, oblong sediment particle that has its majoraxis oriented in the vertical plane of the DEM.3.2.5 Bed surface grain size distributionsWe analyze subsampled images for grain size statistics using a semi-automated MATLAB R©script. The script identifies grain sizes based on the painted color of each grain size classin the experimental mixture. The script begins by loading an RGB and HSV color databasefor each color used in the mixture. We build the color databases by randomly querying therespective color values of grains in several different subsampled images for grain size classesdown to the Wentworth 2 mm gravel/sand threshold. Fifty grains from each grain size classwere used to build the size-specific color databases. The script then proceeds to establish afixed sampling grid of 100 points over the subsampled image, and moves point to point usingbuilt-in MATLAB R© image analysis functions to identify the RGB and HSV color values in a3x3 px2 area Ap.A positive color detect was met if:[Ap]RGB = [Ap]HSV (3.7)533.2. Laboratory experiment and methodsIn the case of a positive color detect, the script increments the associated grain size classcounter by 1, and uses the geometric mean grain size of the grain size class in the statisticscalculations. If [Ap]RGB 6= [Ap]HSV , the user is prompted to identify the color of the sam-pled grain. All grains ≤ 2 mm were counted as 2 mm in diameter, effectively lumping thesegrains into the sand size classes. Validation of script results occurred by comparing the semi-automated results to manual counts for 5 different images. The only notable difference be-tween the semi-automated and manual counts occurred for cases when the sampling nodewas located at the intersection of several grains.The grain count procedure has an inherent error of approximately 5% due to sieve inaccu-racies of the bulk sediments prior to painting. The sieve inaccuracies resulted in inclusion ofsmaller grains within the next larger grain size class. This error was constrained to grains <16 mm in diameter.3.2.6 Manual water and bed surface profilesWe measure centerline longitudinal channel bed and water surface profiles with a point gauge.Profile measurement points were spaced 250 mm apart, providing 58 measurement locationsfor each profile. Profile collection took 12 minutes to complete, and in total, eighty-four man-ual profile measurements were completed in between the higher resolution automated DEMmeasurements. The tip of the point gauge was outfitted with a 3 cm ∅ stainless steel washerso that it sat on the bed surface during bed measurements, and so that it was easier to sightalong the water surface during those measurements. Water surface measurements made withthe point gauge occurred over a period of about 10 seconds in order to capture a local quasi-average condition. Resolution of the bed and water surface measurements was ±1 mm basedon a visual estimate of reading variation of the point gauge scale.All thirty DEMs were associated with a manual water surface profile, which was collectedimmediately prior to collecting the DEMs. Twenty-seven of the thirty DEMs were also as-sociated with a manual bed surface profile. We use the manual bed surface profiles to alignthe water surface profiles within the same coordinate system of the DEM. We shift manually-collected channel bed profiles until alignment is achieved with a profile computed from anassociated DEM. After some trials, the profile correction was determined to be 29 mm basedon minimizing the sum of square differences between manual and DEM profiles. After verticalalignment, we smooth manual water surface profiles with an upstream/downstream movingaverage algorithm, and then interpolate the profiles to the DEM resolution. We use movingaverage window lengths of 2, 4 and 6 observation locations around the smoothing point. Thefinal smoothing window length was chosen based on minimization of sum of squares dif-ferences between the non-smoothed and smoothed profiles, which was generally met with awindow length of 2 or 4 observation locations. We smoothed the water surface profiles toremove the rapidly varying character of some segments of profiles.543.2. Laboratory experiment and methods3.2.7 Flow depth, flow area and average streamwise velocityWe use the DEMs and filtered and smoothed water surface profiles to determine the cross-sectionally averaged water depth (d), wetted flow area (A), and the downstream-orientedflow velocity (Ux) at every longitudinal station j of the DEM:dj =k=n∑k=1(zj − ηj,k)n(3.8)Aj = wjd (3.9)U j =QwAj(3.10)where zj is the smoothed water surface elevation for longitudinal station j, ηj,k is the DEM bedsurface elevation for longitudinal station j, and transversal station k, n is the total number oftransversal stations, w is the experimental channel width at j, and Qw is the flow rate. We cal-culate the local Froude number at location j: Frj = qw,j/(g0.5d1.5j ), where qw,j = Q/wj, and thelocal Reynolds number at j: Rej = Qdj/Ajν. The local channel bed and water surface slopes,Sη and Sw respectively, were computed from the DEMs, and the smoothed water surface pro-files. Side wall corrections were not made because the experimental channel walls providedsome roughness, and because results presented here have focused on downstream differencesin experimental conditions, as opposed to the magnitude of the values themselves.55Chapter 4Morphodynamics of a width-variablegravel-bed stream: new insights onpool-riffle formation4.1 SummaryPool-riffles occur along gravel-bed mountain streams which exhibit downstream variationsin channel width, such that pools are observed along segments of narrowing, and riffles atplaces of widening. Despite recognizing spatial correlations between channel width and bedtopography, we lack a widely accepted mechanistic explanation of the correlation (See Section1.2.2). We address this knowledge gap and build from existing work with systematic exper-imental evaluation of bed topography evolution and development within a channel that ex-hibits downstream width gradients ranging from -0.26 (narrowing) to +0.18 (widening). Ourexperiments reliably produce pools, riffles and roughened channel segments, which persist forsediment mobility conditions that varied on average by a factor 2–4 above the threshold valuenecessary to mobilize the experimental sediment mixture. Our results show that topographicresponses are coupled to changes in channel width, which drives flows to accelerate or decel-erate, for narrowing and widening, respectively. We characterize and understand theoreticallythis coupling in terms of a mathematical model which describes topography as directly depen-dent on spatial variations in the bulk flow speed, and inversely dependent on channel widthand bed surface sediment mobility. The model suggests that a negative feedback betweenbulk flow speed variations and particle mobility drives channel evolution to states that tend toeliminate, or greatly diminish spatial differences in bedload transport. We show that amongreaches of similar grain size, it is possible to project topographic responses knowing nothingmore than how channel width changes downstream, regardless of mean channel slope.564.2. Introduction4.2 IntroductionPool-riffles are generic structures of river bed architecture within bedload dominated systemsof gravel (2–64 mm) to cobble (64–256 mm) composition. Pools are topographic lows, relatedto a local tendency for net particle entrainment, where local is a length scale of 1 to 2 reach av-erage channel widths (w′). Riffles, by contrast, are topographic highs, reflecting the tendencyfor net particle deposition. Pool-riffles are observed across a broad range of natural conditionsfrom mountain headwaters to valley lowland settings, straight to meandering river reaches(Leopold and Wolman, 1957), and for mean longitudinal bed slopes ranging from ≈ 0.0001 to0.03 (Leopold and Wolman, 1957; Leopold et al., 1964; Church and Jones, 1982; Montgomery andBuffington, 1997; Chartrand and Whiting, 2000; Buffington et al., 2002; Hassan et al., 2008). Theprevalence of pool-riffles throughout river systems highlights that the necessary formativeconditions are common to many different parts of the fluvial landscape.Figure 4.1 illustrates that pool spacing along relatively straight pool-riffle channels scaleswith the local channel width (w), and is independent of channel steepness. Data from six pre-vious studies suggest that mean pool spacing is 2–8w′ (Yalin, 1971; Richards, 1978; Keller andMelhorn, 1978; Carling and Orr, 2000), for mean bed slopes that vary by 2 orders of magnitude.Figure 4.1 also shows that pool spacing along channels with large roughness elements suchas wood and boulders scales as 0–2w′ (Montgomery et al., 1995; Beechie and Sibley, 1997), andalong meandering channel segments which commonly exhibit bars scales as 10–14w′ (Richards,1976b). One expectation for the data in Figure 4.1 is that the spacing is related to eddies scalingin size with the water depth d, which deliver sufficient momentum flux to the bed to disaggre-gate it, entrain particles, and drive pool construction (Yang, 1971; Richards, 1976a; Carling andOrr, 2000). However, to our knowledge, no studies demonstrate any direct link between depthscaled eddies and pool-riffle formation. As a result, Figure 4.1 highlights key knowledge gaps:1. Despite the evidence that pool-riffles colocate with narrow and wide channel segmentsalong straight channel segments, respectively, it is not clear how or why channel widthmight enter into a mechanistic scaling for pool spacing.2. We do not know why rivers with generally similar water depth conditions (∼ 1 m) atformative discharge, exhibit such a wide range of pool spacings.3. The distinctive pool spacing for roughness dominated, straight, and meandering riversegments has not been explained.The knowledge gaps motivate two critical questions, which are the focus of this Chapter:A. Figure 4.1 suggests that channel width exerts a strong control over pool spacing, butwhat in particular about channel width matters for pool spacing expression?B. Figure 4.1 includes data from many different river systems, each at differing stages ofdevelopment in response to landscape construction and flood events. Do initial or inher-574.2. IntroductionLog10 Mean Longitudinal Bed Slope8w2w(0.00032) (0.01)(0.0032) (0.032)(0.001)meander wavelength domain ~ (10-14) wlarge wood/isolated boulder driven pool wavelength domain ~< 2 w?Figure 4.1: Downstream pool spacing as a function of the local channel width (w) for pool-riffles along relatively straight reaches. Data are colored according to the log10 meanlongitudinal bed slope. Two gray dashed lines suggest approximate limiting casesof 2 and 8w′; solid darker gray line is MacVicar and Best (2013)’s experimental re-sult of (3–4)w for recovery of flow into and out of a fixed pool, implying a poolwavelength of (6–8)w. Values in parentheses above the slope colorbar are the equiv-alent fractional bed slope, defined as the change in bed elevation over some stream-wise distance of many channel widths in length. Meander wavelength domain perRichards (1976b), and the large wood/boulder driven pool wavelength domain perMontgomery et al. (1995) and Beechie and Sibley (1997). Plotted data from Leopold andWolman (1957); Keller and Melhorn (1978); Montgomery et al. (1995); Sear (1996); Carlingand Orr (2000); Thompson (2001). Only S data used from Leopold and Wolman (1957),and only PR data used from Montgomery and Buffington (1997). The perceptuallyuniform Polarmap colormap was used for slope magnitude.ited topographic and sediment texture conditions predetermine a particular outcome,and contribute to the variance illustrated in Figure 4.1?We address these two questions with scaled laboratory experiments (PRE1: pool-riffle ex-periment 1) and theory, guided by three objectives. First, we examine how bed topographyand bed sediment texture conditions (i.e. morphodynamics) evolve from initial to steady-state conditions along a variable width channel. In all cases initial conditions are far fromsteady-state. Second, we use experimental results to characterize the extent to which morpho-dynamic evolution depends on downstream changes in channel width. Third, we use repeatexperiments to explore the potentially significant effects of hysteresis: does the history of bedevolution condition shape the character of its response in space and time?We hypothesize that bed constitution and the potential for pool-riffle formation adjusts584.3. Resultsto downstream flow accelerations and dynamic pressure variations related to changes in thecross-sectionally averaged velocity driven by channel width differences. We further hypothe-size that the magnitude of these effects on the morphodynamic response, defined as a partic-ular topographic and sediment texture outcome, will scale with the extent of width variation.A primary conclusion of our work is that the downstream channel width gradient is a generalpredictor of topographic response along stream reaches likely to exhibit pool-riffle architec-ture. We also identify restrictive conditions leading to pool digging and riffle construction,and show that bed history may not affect the general response pattern (i.e. the mean longitu-dinal bed slope), but that it does lead to unique responses.4.3 Results4.3.1 Identifying general response regimes with sediment flux, mean bedtopography and bed sediment textureIn Figure 4.2 we show temporal variability of the PRE1 sediment flux (Qs f ), normalized meanbed elevation (η′), normalized geometric mean bed surface grain size (D′g = Dgs/Dg f ), andthe normalized characteristic coarse bed surface grain size (D′90 = D90s/D90 f ), expressed forthe dimensionless time to, defined as the ratio of the elapsed time (te) to the activation time (ta:explained below). For D′g and D′90, the subscript s stands for the bed surface, and the subscriptf stands for the upstream sediment supply. We also show the associated supplies of water(Qw) and sediment (Qss) for context, and plotted quantities are given in Table 4.1. Figure 4.2illustrates that (Qs f ), (η′), and to a lesser extent mean D′g and D′90, vary systematically throughPRE1, and we use these systematic trends to establish four characteristic response regimesfor PRE1. The response regimes capture the morphodynamics of PRE1, and therefore help toexplain how the variable-width experimental channel responds to upstream supplies of waterand sediment. Overall, PRE1 has an initial, and a repeat phase (Figure 4.2a), which extendfrom to = 0–23.9, and to = 23.9–43.5, respectively. The characteristic response regimes are:1. Bed response to the start of an experimental phase (Activation Time: ta): PRE1 has astart-up response we term the activation time ta. The activation time represents the ini-tial sediment redistribution along the experimental channel, which occurs due to therelatively high beginning sediment mobility. High sediment mobility provides a rapidincrease in Qs f during ta to a peak, after which the rate of Qs f change abruptly decreases.The bed evolution η′ during ta exhibits a small positive rate of change. Average D′g andD′90 follow the Qs f trend, and each grain size show rapid increase in size to a peak beforeabruptly changing. ta extends from to = 0 to 1. As a result, the activation time onlyoccurs during the initial phase.2. Bed response to developing flow (Transient Time: tt): The transient period tt has flow de-veloping in response to topographic pattern construction along the experimental chan-594.3. Results0 550 1100 1650 2200 2750 3850 4400 4950330080 Experimental Water Supply Rate (l/s)6042(a)(b)sediment supplysediment fluxrepeat phaseSediment flux and supply (kg/m)steady-stateta tt tr tr 6.5tr tr tr1.51.00.5Normalized average elevation(-)activation time(c)initial phasesteady-state0rifflepoolroughened(d) (e) Mean Dg Mean D90activation time activation timeta tt tr 1tr 6.5tr 1tr 1tr ta tt tr 1tr 6.5tr 1tr 1trto: elapsed time/activation time (-)to: elapsed time/activation time (-) to: elapsed time/activation time (-)te: elapsed time (minutes)Figure 4.2: PRE1 morphodynamics summary: PRE1 morphodynamics summary: waterand sediment supply, sediment flux, longitudinal mean bed topography, and ge-ometric mean and characteristic coarse grain sizes vs. the dimensionless time to,defined as the ratio of elapse time te to activation time ta. (a) Water supply rate(Qw) (l·s−1). (b) Sediment supply rate and flux (Qss and Qs f , respectively) (kg·m−1).(c) Longitudinal normalized mean bed topography (η′), calculated as the ratio ofthe time-specific mean bed elevation for all subsampling locations, to the mean bedelevation across all subsampling locations and observation times. (d) Normalizedgeometric mean grain size (Dg), calculated as the ratio of the bed surface Dg to thesupply Dg. (e) Normalized characteristic coarse grain size (D90), calculated as theratio of the bed surface D90 to the supply D90. Activation (ta), transient (tt) andresponse (tr) periods indicated at the top of (b), (d) and (e).604.3. Resultsnel. During the initial tt, Qs f remains relatively consistent until to = 15.2, after whichQs f rises rapidly to the supply rate, and remains consistent with the Qss through the endof the initial tt (to = 19.5). η′ rises at a uniform rate during the initial tt until to = 15.2,after which topography is steady. D′g increases during tt to a peak at to = 13.9, afterwhich it drops through the end of the initial tt (to = 19.5). D′90, on the other hand re-mains steady through the initial tt. Following recovery from the initial period high flowsequence from to = 19.5–23.9, the repeat phase transient responses are similar to thosefor the initial phase, with one exception. D90′ steadily decreases during the repeat phasett, recovering a value at to = 39.2 which is roughly equivalent to the corresponding ini-tial phase value at to = 19.5. The duration of the initial phase tt is 18.2, and the repeatphase is 15.3.3. Steady-state (SS): We define steady-state by two criteria, similar to that used for numeri-cal simulations (Chapter 3.2.2). First, the extent to which mean topography is statisticallystationary. Second, Qss ≈ Qs f . The second criteria holds for the total mass, and grainsize specific (fractional) masses. Steady-state occurs at to = 19.5 and 39.2, after extendedtime of topographic and flux steadiness (Table 4.1; Figure 4.2). Steady-state also occursat the end of each response to steady-state (tr) periods (discussed next), at to = 21.7, 23.9,41.4 and 43.5.4. Response of steady-state beds to supply regime changes (Bed Response Time: tr): Thesteady-state bed response period characterizes how a supply regime change is expressedby the channel from a SS. Step increases in supply during both the initial and repeatphases drives initially rapid increases in Qs f and D90′ , and sharp decreases in η′ andD′g. After these early responses, and for both the initial and repeat phases, Qs f abruptlychanges and exhibits rapid rates of decrease, η′ continues to steadily decline, Dg′ alsoabruptly changes and exhibits increases, and D90′ steadily rises through the end of the trperiods. At the end of the tr periods, topographic and D90′ adjustments are quasi-steady,Qs f settles to the supply rate, and Dg′ is still responding. The first response to steady-state period is used as the characteristic tr time for PRE1. Notably, the second, fourth andfifth tr periods are roughly equivalent to the characteristic tr time. The third tr period,however, is equivalent to 6.5tr.Across the response regimes, Dg′ displays varied adjustments for the pool, riffle and rough-ened channel structures, with no obvious feature-specific trends. However, Dg′ does exhibit aweak overall increasing trend through PRE1. By contrast, D90′ responses are more consistentbetween pool, riffle and roughened channel structures. This suggests that D′g is more respon-sive than D90′ to the PRE1 conditions. Last, it is notable that across PRE1, the Dg measuredalong the channel is generally coarser than that of the supply, whereas the D90 is finer than thesupply.614.3. ResultsWe now use the four characteristics response regimes to present results shown in Figures4.3 to Topographic response: channel-wide and longitudinal profile developmentFigure 4.3 illustrates channel bed topography for thirteen observation times during PRE1. Cor-responding flow rate is indicated within each topographic map (DEM), and at the side of eachDEM we provide the observation time as well as the response regime. We also show the sub-sampling locations in the DEM for te = 0 minutes for reference. The top four DEMs showtopographic development through the entirety of the ta, the fifth through seventh DEMs showtopographic development for different points within the initial tt, and the bottom six DEMsillustrate topography for each SS condition: t=2150, 2390, 2630, 4310, 4550 and 4790 minutes(Table 4.1).Stations within segments of strong widening and narrowing (3600, 10000 and 8000 mm,respectively; Table 4.2) exhibit rapid topographic development during ta (Figure 4.3), which ingeneral has relatively high sediment mobility conditions (Table 4.1), demonstrated by the largerate of Qs f increase to a peak value of 0.064 kg·m−1 by the end of the ta (Figure 4.2). At station8000 mm, channel width has the strongest negative downstream width difference: ∆w(x) =-0.25 (Table 4.2), where we calculate fractional width change as the forward difference in adownstream moving reference frame:∆w(x) =w(x + L)− w(x)∆L(4.1)where w is the channel width at longitudinal station x, and ∆L is the forward difference lengthscale between subsampling locations, ∆L = 1000 mm, or roughly 2w′ (Figure 4.3). The rela-tively large width reduction at 8000 mm correlates with a pool, and by the end of the ta (te =110 minutes), this pool is well developed. The relatively large width increases at 10000 and3600 mm correlates with riffles, and the initial style of topographic construction at these twostations differed during the ta (Figure 4.3). At the downstream location, topography is builtvia progressive deposition of sediments over the entire riffle surface. At the upstream loca-tion, by contrast, topography is built by migrating fronts of bedload sediment. Each locationcorresponds to relatively large positive downstream width changes: ∆w(x) = 0.19 and 0.17,respectively (Table 4.2). Channel segments for which ∆w(x) = O(0) (Table 4.2) exhibit mutedtopographic responses during the ta, relative to the narrow and wide zones.Through the early part of the initial tt, differences observed between stations 10000 and3600 mm diminish, and topographic construction continues at stations 10000 and 3600 mm byincremental deposition of bedload. Subtle topographic development, relative to the narrowand wide zones, also continues during the tt at channel segments for which ∆w(x) = O(0).However, periodic topographic waves are evident at te = 470 minutes and 2150 minutes (endof the initial tt) from station 13000 to 11000 mm (Figure 4.3). These features occur with a624.3. ResultsTable 4.1: Experimental details for PRE1PRE1 Interval to te Qw (Qss) (Qs f ) (η′) Dg′ D90′ τ/τre f(-) (-) (minutes) (l·s−1) (kg·m−1) (kg·m−1) (-) (-) (-) (-)0 0 0 - - - 0.90 0.675 0.820 -1 0.2 19 42 0.50 0.009 0.88 0.849 0.983 1.492 0.5 50 42 0.50 0.027 0.88 1.014 1.013 1.473 1.0 110 42 0.50 0.064 0.89 0.977 1.009 1.484 2.1 230 42 0.50 0.048 0.90 1.007 1.008 1.445 4.3 470 42 0.50 0.027 0.94 1.024 0.996 1.536 6.5 710 42 0.50 0.050 0.97 1.047 0.925 1.597 8.6 950 42 0.50 0.030 0.97 1.074 0.974 1.548 10.8 1190 42 0.50 0.016 1.01 1.148 1.003 1.579 13.0 1430 42 0.50 0.099 1.06 1.299 1.012 1.5410a 15.2 1670 42 0.50 0.058 1.10 1.209 0.985 1.6310b 16.3 1790 42 0.50 0.660 - - - -11a 17.4 1910 42 0.50 0.437 1.07 1.149 0.994 1.5611b 18.5 2030 42 0.50 0.226 - - - -12* 19.5 2150 42 0.50 0.393 1.09 1.202 1.059 1.5613a 19.8 2180 60 0.80 4.284 - - - -13b 20 2195 60 0.80 7.113 - - - -13c 20.2 2225 60 0.80 3.670 1.01 0.991 1.052 1.7514a 20.6 2270 60 0.80 2.296 - - - -14b* 21.7 2390 60 0.80 0.917 0.98 1.143 1.146 1.6615a 21.9 2405 80 1.00 3.848 - - - -15b 22.1 2429 80 1.00 3.840 - - - -15c 22.3 2450 80 1.00 3.303 0.96 1.102 1.185 2.0016a 22.5 2480 80 1.00 2.554 - - - -16b 23 2525 80 1.00 1.336 - - - -16c* 23.4 2570 80 1.00 0.927 0.94 1.294 1.198 1.7817 23.9 2630 80 1.00 1.067 0.94 1.302 1.219 1.7718 26.1 2870 42 0.50 0.033 0.96 1.330 1.190 1.2619 28.3 3110 42 0.50 0.016 1.00 1.309 1.126 1.4520 30.5 3350 42 0.50 0.013 1.04 1.306 1.101 1.5221 32.6 3590 42 0.50 0.018 1.08 1.352 1.100 1.5022 34.8 3830 42 0.50 0.022 1.12 1.330 1.063 1.6423 37 4070 42 0.50 0.143 1.15 1.248 1.032 1.6624* 39.2 4310 42 0.50 0.509 1.14 1.132 0.999 1.6825a 39.4 4336 60 0.80 4.488 - - - -25b 39.6 4351 60 0.80 9.622 - - - -25c 39.7 4370 60 0.80 5.479 - - - -25d 39.9 4385 60 0.80 4.527 1.04 1.131 1.170 1.7126a 40.3 4430 60 0.80 2.436 - - - -26b* 41.4 4550 60 0.80 0.595 1.03 1.225 1.171 1.6227a 41.5 4565 80 1.00 2.231 - - - -27b 41.7 4589 80 1.00 6.212 - - - -27c 41.9 4610 80 1.00 5.062 0.96 1.082 1.210 1.7528a 42.2 4640 80 1.00 2.069 - - - -28b 42.6 4685 80 1.00 1.132 - - - -28c 43 4730 80 1.00 0.892 0.97 1.374 1.195 1.9129 43.5 4790 80 1.00 0.600 0.97 1.158 1.242 1.741. Asterisk denotes achievement of mass equilibrium between feed and flux.2. The elapse time indicates the end time for the specified experimental interval.3. to defined as ratio te/ta, where te is the elapse time and ta is the activation time.4. Sediment flux is the mean flux for the observational interval.5. η′ is the normalized mean bed elevation for the 11 subsampling locations.6. τ/τre f is the mean for the 11 subsampling locations.7. τre f is the reference stress associated with τ∗re f = 0.030 = τ/[(ρs − ρw)gDi].7. The repeat phase of PRE1 began at elapse time 2630 minutes.634.3. ResultsExperimental Channel Station (mm)1Relative Channel Bed Elevation (mm)0 minutes19 minutes50 minutes110 minutes230 minutes2150 minutes2630 minutes4310 minutes4790 minutes2390 minutes4550 minutes42 ls-142 ls-142 ls-142 ls-142 ls-160 ls-180 ls-142 ls-160 ls-180 ls-142 ls-142 ls-1470 minutes710 minutesChannel Station (mm)tattSSSubsampling locationsFigure 4.3: Summary panel of topographic responses observed during PRE1. Topo-graphic responses provided for elapsed time 0, 19, 50, 110, 230, 470, 710, 2150, 2390,2630, 4310, 4550 and 4790 minutes; steady-state topography at times 2150, 2390,2630, 4310, 4550 and 4790 minutes (Table 4.1). At the side of each DEM we providethe elapse time, and within each DEM we indicate the flow rate for the precedingexperimental interval. We show the subsampling locations for reference with thered boxes in the te = 0 min. DEM. DEM coloring based on the perceptually uniformVirdis colormap.spacing ≈ w′, and are comparable to D90 high, where D90 is the 90th percentile grain size ofthe experimental distribution (Figure 3.2). By the end of the initial phase tt, the overall spatialpattern of topography evident by te = 110 minutes remains.Steady-state topography has a few characteristic patterns, depending on flow and sedi-ment supply rates (Figure 4.3). SS at te = 2150 and 4310 minutes (42 l·s−1) has riffles centered644.3. ResultsTable 4.2: Values of downstream width change between subsampling locationsBounding Subsampling Locations3000:4000 4000:5000 5000:6000 6000:7000 7000:8000 8000:9000 9000:10000 10000:11000 11000:12000 12000:130000.187 -0.056 0.0051 0.089 0.059 -0.252 -0.107 0.178 -0.065 0.0721. Downstream width change calculated with Equation 4.1 for length scale h = 1000 mmat stations 9800 and 3600 mm, a pool centered at 8000 mm, and roughened channel segmentselsewhere. Upstream of the pool, topographic magnitude is high relative to downstream ofthe pool, and generally ranges from 200 to 300 mm. Downstream of the pool, topographicmagnitude is relatively low, and generally ranges from 140 to 240 mm. At each subsequent SS,riffles and pools persist at the same stations, albeit at increasingly lower relative elevations,and an additional pool emerges at station 15000 mm. This pool is most evident in SS at te =2630 and 4790 minutes.In order to better characterize the statistically steady conditions just described, we supple-ment the SS DEMs with corresponding longitudinal profiles shown in Figure 4.47. We alsoprovide the initial condition profile and DEM for context, and the elapse time and associatedflow rate is given to the right of each profile (Table 4.1). We determine SS profile residuals withthe zero-crossing method (Melton, 1962; Richards, 1976a), which provides one way to qualita-tively distinguish pools, riffles and roughened channel segments (cf. Carling and Orr, 2000).Furthermore, we project the residuals back onto the SS slopes, to show the scale of bed struc-tures relative to the overall relief. We distinguish pools as negative residual departures fromthe detrended profile for length scales of ∼ w′, and denote these with the letter P. We distin-guish riffles as positive residual departures, again for ∼ w′, and denote these with the letterR. Roughened channel segments have minor residual departures that fluctuate around thedetrended profile, and we denote these with the letters Ro.Figure 4.4 highlights that pools are colocated at points of narrowing, riffles at points ofwidening, and roughened channel segments where width changes are negligible. In general,the prevalence of pools at SS increases with increasing flow and sediment supply rates, indi-cated by the magnitude and downstream extent of the pool depth shown for SS at 42 vs. 60and vs. 80 l·s−1, at stations 8000 and 15000 mm. The growth of a pool at station 4000 mmfor increasing supply rates further demonstrates the dependence of pool prevalence on exter-nal supply conditions. The SS profiles reveal that topographic relief increases with flow andsediment supply rates, but that channel-average longitudinal bed slopes decrease (Table 4.3).PRE1 began from an initial slope of 0.015, steepened to 0.0191 at SS 2150 minutes, decreased to0.0162 at SS 2390 minutes, and decreased yet more so to 0.0138 at SS 2630 minutes (Table 4.3).We observe a similar progression of steepening and relaxing for the repeat phase. Compari-son of the profiles shown in Figure 4.4 with results of Figure 4.2 reveals that Qs f remains atrelatively low values for long durations during profile construction (tt), and rises significantly654.3. Resultszero crossing lineExperimental Channel Elevation (mm)Experimental Channel Topography: te = 0 minutesPRiRoRiRiRiRiRiRiRiRiRiRiRiPPPPPP PPPPPPPPPRoRoRoRoRoRoRoRoRo0 minutes42 ls-12150 minutes60 ls-12390 minutes80 ls-12630 minutes42 ls-14310 minutes60 ls-14550 minutes80 ls-14790 minutes0 2000 4000 6000 8000 10000 12000 14000 16000Channel Station (mm)Figure 4.4: Identification of pool-riffle structures with the zero-crossing method. DEM ofthe channel at to =0 shown at top for reference. Zero-crossing profiles ((Richards,1976a)) projected onto the experimental channel slope at t=0, 2150, 2390, 2630, 4310,4550 and 4790 minutes elapsed time. These times reflect steady-state conditions forthe PRE1 flow and sediment supply rates (see Table 4.1). Profiles are computed forthe center 100 mm of each corresponding DEM. We indicate the general topographicresponse for each steady-state case with the abbreviations P (pool), Ri (riffle), and Ro(roughened channel). te and Qw for the preceding experimental interval is given tothe right of each profile. The zero-crossing line is represented by the light dashedline, specifically called out for t=2150 minutes.and then peaks during profile relaxation (tr). A peak Qs f response occurs for each SS supplychange, except following the third SS at te = 23.9, after which Qs f drops continuously beforeleveling off, and the eventually rising, as the topographic profile is restored.664.3. ResultsTable 4.3: Values of average channel bed slope for initial and steady-state conditionste = 0 min. 2150 min. 2390 min. 2630 min. 4310 min. 4550 min. 4790 min.0.015 0.0191 0.0162 0.0138 0.0186 0.0156 0.01411. Average channel slopes calculated with profiles shown in Figure 4.4.To characterize how SS topographic conditions are organized by downstream changes inchannel width, we augment the SS profiles shown in Figure 4.4 with box-and-whiskers plotsof SS normalized local mean bed elevation (η/η) vs. normalized local width (w/w′) providedin Figure 4.5. Local refers to the specific value of η or w at longitudinal station x, and η isthe associated channel average SS bed elevation. We use the results of Figure 4.5 to charac-terize the width-specific conditions which give rise to topographic expression at the SSs, andcast topographic development tendencies in terms of a balance between the entrainment anddepositional responses, expressed through wo = w/w′.The entrainment response governs SS at wo . 0.90. These relative widths are narrow, andassociated with topographic responses η/η which tend to values of 0.80 or less, indicatingthat the entrainment response is larger than the depositional one. These conditions lead topools. On the other hand, the depositional response governs SS at wo & 1.10. These relativewidths are large, and associated with topographic responses η/η which tend to values of 1.20or more, indicating that the depositional response is larger than the entrainment one. Theseconditions lead to riffles. Last, the tendency for entrainment and deposition to balance iscaptured qualitatively across all SSs for 0.90 & wo . 1.10. These relative widths are similarto the mean width, and are associated with topographic responses η/η which tend to rangebetween 0.90 to 1.10. These conditions lead to roughened channel segments.The magnitude of topographic diversity for any given value of wo is relatively large forsmaller values of the mobility conditions τ/τre f , where τ is the shear stress and τre f is a refer-ence stress for threshold of motion conditions, and diminishes with increasing mobility (Table4.1; Figure 4.5). This result is shown Figure 4.5 by the increasing range of values between thelower and upper quartiles, and the increasing magnitude of associated whisker lengths fordecreasing mobility conditions, and especially for intermediate responses. On the other hand,topographic relief is relatively large for increasing values of τ/τre f , and diminishes with de-creasing mobility (Table 4.1). This result is reflected by the range of η/η values shown in thebox plots (Figure 4.5), as well as the departure of the distribution of η/η values from the 1:1line, which diminishes for increasing values of τ/τre f .We use topographic profiles to explore the response trends shown in Figure 4.5 in moredetail, and in particular to emphasize the spatial character of the data through the normalizedaxes. Figure 4.6 illustrates profile traces for SS at te = 2150 and 2630 minutes. The x- and y-axesof Figure 4.6 are the same as those for Figure 4.5, but in Figure 4.6 we show the mean relativeelevation of each longitudinal position along the experimental channel, expressed according674.3. Results1.000.681.401. range1:1 line1.20.80.642 lps - t2150 min80 lps - t2630 min60 lps - t2390 min42 lps - t4310 min60 lps - t4550 min80 lps - t4790 minEntrainment ResponseDepositional ResponseIntermediateResponseFigure 4.5: Summary panel of SS topographic responses observed during PRE1 illustratedwith box-and-whiskers plots. The physical nature of responses is provided at thetop of the panel, and mobility condition τ/τre f is provided to the right. The termw is the local channel width; w′ = 547 mm and is the mean channel width of theexperimental channel, η is the local elevation provided by the DEM, and η is themean elevation of the experimental channel for each SS condition. The y-axis rangeis consistent for all six the specified range of colors.The two SS profiles indicate that similar values of wo can generate differing topographicresponses. The magnitude of dissimilarity between the wo associated responses depends onthe mobility condition τ/τre f , with lower mobilities driving more accentuated differences (Ta-ble 4.1). The lines and station call outs shown along topographic trace segments correspond tospecific pool, riffle and roughened channel structures within the experimental channel (Figure684.3. ResultsChannel Station (mm) lps - t2630 min9958sta8138sta3498sta3998sta10100sta11000sta42 lps - t2150 min9958sta3498sta8138sta 3998sta10100sta11000sta15,00014,00013,00012,00011,00010,0009,0008,0007,0003,0002,0005,0006,0004,0001.01.0Figure 4.6: Profile traces of topographic response for steady-state conditions at t=2150 and2630 minutes. Same axes as used in Figure 4.5. Colorbar denotes location along ex-perimental channel (see Figures 4.3 and 4.4). The individual elevation points alongthe profile trace reflect the data represented by each corresponding box from Fig-ure 4.5. The arrows indicate the downstream direction along the profile traces. Theperceptually uniform Polarmap colormap was used for location mapping.4.4). Comparison of Figures 4.3 and 4.6 indicates that the traces exhibit a topographic hystere-sis type response for both SS cases. Pool-riffle type structures can develop in a way that yieldsriffle-pool (station 9958 to 8138 mm: widening to narrowing), or pool-riffle (station 3998 to3498 mm: narrowing to widening), in an upstream/downstream moving reference frame, re-gardless of τ/τre f . The profile traces also illustrate that on average, the local bed slope departsfrom the mean longitudinal slope for lower mobility conditions τ/τre f , and tends to the meanlongitudinal slope as the mobility condition τ/τre f increases. Local here refers to bed slopesover length scales of 1–5w′.694.3. Results4.3.3 Effects of initial conditions on topographic responsesWe use the same data for SS conditions shown in Figures 4.5 and 4.6 to examine the effectsof initial conditions on longitudinal topographic response. Figure 4.7 shows normalized localmean bed elevation (η/η) vs. normalized channel station (x/w′), where x is channel station.We provide the dimensional channel station at the top of the plot for reference, a DEM forcontext on how width changes, and the profiles were filtered using a moving average windowlength l = 100 nodes, or 200 mm.t2150t4310t2390t4550t2630t479042 ls-160 ls-180 ls-10 2000 4000 6000 8000 10000 12000 14000 16000Channel Station (mm)Experimental Channel Topography: te = 0 minutesflowFigure 4.7: Steady-state normalized topographic profiles for PRE1. x is the channel sta-tion in mm, and x/w is the normalized channel station. As with Figures 4.5 and4.6, local mean elevation η has been normalized by the associated profile mean ele-vation η. Average, filtered profiles computed for center 50 mm of the experimentalchannel at t= 2150, 2390, 2630, 4310, 4550 and 4790 minutes, and were filtered witha moving average window length l = 100 nodes, or 200 mm. Periodic bedload orsediment waves discussed in Section 4.3.2 are evident in several of the profiles, andin particular for te = 4310 minutes, between stations 22–26.The SS profiles are organized into three populations, and the nature of profile organizationchanges from upstream to downstream, and generally reflects topographic relief. For exam-ple, from x/w = 16–20, the 80 l·s−1 conditions exhibit the largest η/η values, and the 42 l·s−1conditions exhibit the smallest. In contrast, from x/w = 13–16 this organization is flipped,and the 42 l·s−1 conditions exhibit the largest η/η values, and the 80 l·s−1 conditions exhibitthe smallest. Normalized profiles for the 60 l·s−1 conditions are consistently in between thoseof the 42 and 80 l·s−1 cases. Comparison of the profiles with the DEM suggests that the topo-704.4. Physically linking channel width changes to topographic responsegraphic responses are shifted slightly downstream from the widest and narrowest points, bya length scale of roughly 5D90.The SS profiles are visually comparable in shape and pattern from stations 3–20, as sug-gested by consistent riffle occurrence from x/w′ = 3–6 and 16–20, and pool occurrence fromx/w′ = 13–16. In contrast to pattern similarity, each SS profile reflects a qualitatively uniquemorphodynamic response to the external supply conditions, as paired profiles for the 42, 60and 80 l·s−1 cases are visually different, and profiles across all supply conditions are different.The primary visual difference is magnitude of response for any given station x/w′, character-ized by the vertical offset between associated SS profiles.4.3.4 Summary of main resultsPRE1 produced pool-riffle and roughened channel structures that were persistent across mo-bility conditions (τ/τre f ), which on average were greater than 2. Pools were colocated withpoints of width narrowing, where wo . 0.90, riffles with points of widening, where wo & 1.10,and roughened channel beds were expressed along segments where width change was con-strained to the range 0.90 & /wo . 1.10. The characteristic coarse grain size D90′ was notablysimilar between morphologic structures, regardless of width condition. The topographic andsediment texture of pools, riffles and roughened channel beds develop rapidly during thestart-up, or activation time ta, and evolve more slowly thereafter during the transient periodtt, as a SS condition is approached. Topographic and sediment texture perturbations awayfrom steady-state (tr) are of short duration, roughly 2− 2.5to under increased supplies of wa-ter and sediment. The effect of inherited bed states does not precondition the outcome, as ini-tial and repeat SS bed morphologies exhibit consistent spatial patterns of topography and bedslope. However, all six SS bed profiles are unique, exhibiting different absolute topographicresponses, and different D50 and D90 characteristic bed surface grain sizes. The combined re-sults suggest that SS bed topography is coupled to downstream changes in channel width, butsediment texture for PRE1 does not show a clear spatial correlation with width, such as poolswith finer sediment textures than riffles (cf. Lisle, 1979; Hodge et al., 2013). In the next section wepresent and develop a physical explanation for our observations of bed topography-channelwidth coupling.4.4 Physically linking channel width changes to topographicresponse4.4.1 Downstream changes in flow speed and mobilityPools, riffles and roughened channel structures are reliably produced by PRE1, and there is aspatial association between pools and riffles, and segments of channel narrowing and widen-ing. But how does narrowing and widening mechanically lead to pools and riffles? We expect714.4. Physically linking channel width changes to topographic responsefrom flow continuity that mean flow velocity, and hence particle mobility will increase forchannel narrowing, and decrease for widening. We demonstrate this expectation in Figure 4.8,where we show the average DEM (i.e. topographic response) for all six SS conditions, alongwith the normalized mean downstream change in (a) flow speed (Uˆx) and (b) particle mobil-ity (τˆ∗). We determine downstream changes in flow speed and mobility with Equation 4.1,and normalize by the mean speed for all subsampling locations and observation times (Table4.1), and by the reference mobility value τ∗re f = 0.030, respectively. We use Equation 3.10 todetermine the mean flow speed, and the Manning-Strickler formulation of the Shields equa-tion (Parker, 2007, 2008) to determine particle mobility (τ∗), expressed for the uniform flowcondition as (Parker, 2007):τˆ∗ =[(k0.33s q2wα2r g)0.30 ( S0.70ρ′D90)(τ∗re f)−1]x,t0 :tn, (4.2)where ks = nkD90 is a measure of local bed roughness in absence of bedforms, nk = 2 (Parker,2008), qw = Qw/ws, where ws is the mean width for each subsampling location, the constantαr = 8.1 (Parker, 1991), and S is the local channel bed surface slope, and local here is thedistance between subsampling locations. Values of S < 0 were set to S = 0.001.Figure 4.8a shows that flow speed declines in segments of channel widening (warmer col-ors), increases along segments of narrowing (colder colors), and has negligible variation instraight segments where downstream width changes are small (neutral colors). The spatialpattern of flow speed change correlates with the spatial pattern of SS bed topography, sug-gesting a mechanistic link. We observe pools where flows accelerate, reflecting net particleentrainment, riffles where flows decelerate, reflecting net particle deposition, and roughenedchannel segments where flow speed change is negligible (Figure 4.8a; cf. Figure 4.4). How-ever, flow speed change shown at station 4500 mm is declining, yet the bed topography has arelatively low elevation (cf. Figure 4.4).Figure 4.8b shows that particle mobility declines in segments of channel widening (warmercolors), increases along segments of narrowing (colder colors), and has negligible variationin straight segments where downstream width change is minor (neutral colors). However,there is one notable departure from these general spatial correlations. Station 7500 mm showsrelatively low topography, but a strong depositional prediction (bright red circle). The primaryfactor driving this discrepancy in Equation 4.2 is the relatively large decrease in bed slope S,moving from station 8500 mm to station 7500 mm. Figure 4.8a also shows that flow speedchanges very little from station 8500 to 7500 mm. So, whereas the momentum flux decreasesbetween stations 8500 and 7500 mm, the flow speed remains relatively high, thus favoringparticle entrainment conditions, as captured by the DEM.The relationship between spatial patterns of flow speed change and particle mobility, to SSbed topography is consistent with field measurements of riffles located at points of widening,where flow decelerates, and pools located at points narrowing, where flow accelerates, within724.4. Physically linking channel width changes to topographic responseMobility Change (-)EntrainmentDeposition-1 1AccelerationDeceleration-0.04 0.04Flow Speed Change (-)300250200150Bed Elevation (mm)(a)(b)Figure 4.8: Average steady-state topography related to downstream changes in (a) nor-malized cross-sectionally averaged flow speed, and (b) normalized sediment mobil-ity. Average topography determined from the six SS conditions (Table 4.1). Changein downstream flow speed and mobility determined with Equation 4.1 for all sub-sampling locations, averaged across observational times 1-29 (Table 4.1). Changesare plotted mid-way between subsampling locations. Flow speed normalized bythe mean flow speed for all subsampling locations and PRE1 observation times, andmobility normalized by a τ∗re f value of 0.030. The perceptually uniform Polarmapcolormap was used to show flow speed and particle mobility changealluvial (MacVicar and Roy, 2007) and bedrock river reaches (Venditti et al., 2014). Furthermore,the coupling of downstream changes in channel width, flow speed change, particle mobilityand bed topography shown in Figure 4.8, is consistent with theory and field measurements(Furbish, 1998; Furbish et al., 1998). As a result, Figure 4.8 motivates the hypothesis that localchanges in the longitudinal bed slope correlates with downstream changes of channel widthw and mean flow velocity Uˆx.4.4.2 Downstream changes in channel width and bed slopeIn Figure 4.9a we plot the mean channel bed slope Slocal versus the associated downstreamchange in width: ∆w(x) · ∆L−1. We determine local bed slope as the difference in mean bedelevation between subsampling locations using the same form of Equation 4.1, where we av-erage the bed topography within each subsampling location. We determine mean bed slopesfrom the associated values for observations 1–29 (Table 3.1), and the bed slope error bars arethe sample standard deviation across all observations. As with Figure 4.8a, we show the down-stream change in Ux with the circle colors: warm colors for decelerating flow, cold colors for734.4. Physically linking channel width changes to topographic responseTable 4.4: Mean values of Ux/Ux and τ∗/τ∗re f for subsampling locationsSubsampling Locations3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 130000.939 0.980 0.990 1.010 1.056 1.064 0.979 0.951 1.004 0.988 1.0400.150 1.674 0.962 0.321 0.546 3.892 1.120 0.367 1.315 0.476 0.9901. Ux calculated from continuity, and is the cross-sectional average velocity.2. Ux averaged across all times.3. Ux is the mean for all subsampling locations for all times.4. τ∗ calculated with Equation 4.2.5. τ∗re f set to 0.030 (Buffington and Montgomery, 1997).6. Mean values determined from 30 values for each subsampling location.accelerating flow, and neutral colors for negligible flow speed change. Last, the circle size forPRE1 data reflects the average mobility condition τ∗/τ∗re f for observations 1–29 (Table 4.4).In Figure 4.9b we supplement our experimental data with appropriate steady-state results re-ported by de Almeida and Rodrı´guez (2012) and Nelson et al. (2015). de Almeida and Rodrı´guez(2012) provides numerical simulations of the Bear River, AK, U.S. (star symbol; data sourceis Figure 2 of de Almeida and Rodrı´guez (2012)), and Nelson et al. (2015) provides experimentsguided by the physical characteristics of the middle reach of the Elwha River, WA, U.S. (di-amond symbol; data source is Run 1, Figure 6 of (Nelson et al., 2015)). For deAlmeida andNelson’s data points, we use P to indicate pool and R for riffle.Figure 4.9a indicates that as channel segments increasingly narrow, local bed slope steep-ens in the downstream direction (positive values of Slocal), as channel segments increasinglywiden, slopes steepen in the upstream direction (negative values of Slocal), and for segmentswhich exhibit little change in width, local bed slopes respond with negligible downstream orupstream topographic gradients (values close to zero). In a downstream moving referenceframe, Figure 4.9 highlights that pools are favored for relatively large, negative changes indownstream width, for which the bulk flow is accelerating, and τ∗ >> τ∗re f . Riffle type de-posits are more likely for relatively large, positive changes in downstream width, for whichthe bulk flow is decelerating, and τ∗ << τ∗re f . Grain roughness dominated beds for negligiblechanges in width, positive or negative, for which the downstream bulk flow speed change isminor, and τ∗ ≈ τ∗re f (Table 4.4; Figure 4.8).Figure 4.9b illustrates that local bed slopes from PRE1, de Almeida and Rodrı´guez (2012) andNelson et al. (2015) exhibit a systematic response across the full range of downstream widthchange, from (-0.30)–(+0.30). This result is particularly important because the overall reach-average bed slope of de Almeida and Rodrı´guez (2011, 2012)’s simulation (and field site) is 0.002,and the reach-average bed slope of Nelson et al. (2015)’s experimental channel Run 1 (and field744.4. Physically linking channel width changes to topographic responseRiffles (R) at 10,000 and 3,000 mm0.04 -0.04DecelerationAccelerationFlow Speed Change (-)Pool Regime Riffle RegimeLocal Narrowing Local WideningPool (P) at 8,000 mm(a)(b)PPRRRSlope leading into pool (P) at 8,000 mm5 0.21Figure 4.9: (a) Average channel bed slope as a function of the downstream change in chan-nel width and flow speed. We determine bed slope as the forward difference usingEquation 4.1 and then take the mean and sample standard deviation (error bars) forobservations 1–29 (Table 3.1). Flow speed change is depicted by color, and the meanmobility condition τ∗/τ∗re f is indicated by circle size (Table 4.4). (b) Same plot as (a),but we add corresponding data from de Almeida and Rodrı´guez (2012) (stars: Figure2 therein), and Nelson et al. (2015) (diamonds: mean of data from Run 1 in Figure6 therein). For deAlmeida and Nelson’s data, we use P to indicate pool, and R toindicate is 0.007. These bed slopes stand in contrast to the range of our SS experimental channelbed slopes of 0.014 to 0.019 (Table 4.3), highlighting consistency between bed slope responseand width change across almost one order of magnitude of overall reach-average bed slope.The ordered expression of Slocal across the width change domain of Figure 4.9 motivates thehypothesis that the local bed slope can be predicted with information about nearby changesin width and/or flow speed.754.4. Physically linking channel width changes to topographic response4.4.3 Theory for the local channel profileOur results show a two-way coupling between the flow, the bed and general particle mobility,which is driven by downstream changes in channel width (Figures 4.8a and b). Changes inwidth lead to conditions whereby the flow loses speed at segments of channel widening, favor-ing particle deposition, gains speed at segments of narrowing, favoring particle entrainment,and remains relatively uniform where width change is minor, for which particle depositionand entrainment are roughly balanced. Therefore, the results of Figures 4.8a and 4.8b mo-tivates development of a mathematical model which predicts the local channel profile, andwhich is dependent upon how flow speed changes, because particle mobility is dependentupon the fluid drag. We build from Snow and Slingerland (1987); Duro´ et al. (2016); Bolla Pit-taluga et al. (2014) and Ferrer-Boix et al. (2016), and begin our analysis with four assumptions:(1) statistical steady-state conditions, as defined in Chapter 3.2.2 by the requirements that therates of bed elevation, and bed surface sediment texture change of the median grain size eachapproach zero; (2) characteristic grain sizes are spatially uniform, ∂Di/∂x = 0, (3) channelbanks change position at rates much less than those of bed elevation and bed surface sedi-ment texture; and (4) a channel reach of at least 10–20w′ in length has a well-defined averagebed surface slope.With these assumptions, channel profile construction is governed by bed sediment massconservation. Accordingly, the Exner equation (Exner, 1925) in one-dimension is:∂η∂t= −1ε∂qb∂x, (4.3)where η is the channel bed elevation, t is time, the solid fraction in the bed is ε = (1− φ),where φ = 0.4 is the volume-averaged streambed porosity of the active layer La = kDc (Hirano,1971), where k is constant between 1 and 2 (Parker, 2008), and Dc is a characteristic grain size,generally taken as D90, qb is the total bedload transport rate per unit channel width, and xis spatial location in the streamwise orientation. We write Equation 4.3 in expanded formwith the dimensionless Einstein bedload number (Einstein, 1950), expressed for the sum of allbedload fractions following Parker (2007):q∗b =qb(Rg)0.5D1.5c, (4.4)where q∗b is the dimensionless unit bedload transport rate, R is the relative density of sediment:[(ρs/ρw)− 1], for which ρs = 2.65 g·cm−3 is the density of sediment, ρw = 1.0 g·cm−3 is thedensity of water, and Dc is a characteristic grain size. Combining Equations 4.3 and 4.4, andnondimensionalizing with x ≈ Lc, T = ∆η/(∂η/∂t) and X = ∆η/(∂η/∂x), where the scale Lcis defined below, we obtain a dimensionless form of the Exner equation, written in terms ofthe topographic gradient:∂η∂x≈ −Λ∂q∗b∂x∗, (4.5)764.4. Physically linking channel width changes to topographic responsewhere g′ = Rg andΛ =(g′)0.5D1.5cεUcLc. (4.6)Λ characterizes development of the local channel profile in terms of a competition betweentwo time scales: (1) the time scale for topographic spreading, or relaxation over the length Lc,and (2) the time scale for eddy overturning (Yalin, 1971; Carling and Orr, 2000), or bed forcing,which scales the dynamic pressure force imparted at Dc, expressed as:Λ ≡ tspreadingtforcing=((g′Dc)0.5εLc)︸ ︷︷ ︸spreading(DcUc)︸ ︷︷ ︸forcing. (4.7)The spreading time scale changes considerably depending on channel bed packing ε, andfor loosely packed beds this time scale is relatively small, and for tightly packed beds, thetime scale is relatively large. This suggests that as the solid fraction of the bed becomes in-creasingly small, the bed material is more responsive to the flow. As a result, the spreadingtime scale is a measure of bed resistance. More generally though, the spreading time scale isconceptually understood as similar to how honey spreads when poured onto a flat surface, forwhich spreading is driven by gravity acting on the height of the initial honey pile, and resistedby honey’s viscosity. In the present case, Equation 4.7 indicates that the driving gravitationalforce is resisted by the degree of bed packing ε, and that the magnitude of the spreading timescale is a function of how Dc scales relatve to Lc. On this last point it is useful to recognize thatincreases or decreases in bed topography magnitude scale as the characteristic grain sizes Dcthat are locally participating in the adjustment response.The characteristic velocity Uc could be taken to be the rate at which a disturbance or re-sponse propagates downstream along a stream bed (e.g. Stecca et al., 2014; Juez et al., 2016,and citations therein), or the speed with which the bed changes vertical position. Here weassume Uc is governed by the mechanical coupling of the flow to the bed, and thus specifythat Uc ≈ u∗ (u∗ is the shear velocity), and use the Manning-Strickler resistance formulationof the shear velocity:u∗ =√(k0.33s q2wα2r)0.30(gS)0.70 (4.8)Shear velocity is a reasonable choice for Uc because it captures the rate at which shear andmomentum flux are delivered to the top of the bed, which, in turn governs the transport mag-nitude. Equation 4.8 is particularly appropriate because it reflects how flow intensity (qw)changes in the downstream direction, which scales the flow speed change, and hence the mo-bility condition. We take the characteristic length scale Lc to be the channel width, becausewidth inversely scales the cross-sectionally average flow velocity. Figures 4.8 and 4.9 showthat this sets up a spatial variation in flow speed and particle mobility, which correlates with774.4. Physically linking channel width changes to topographic responsespatial patterns of bed topography. With Uc and Lc defined, Equation 4.5 becomes:∂η∂x≈ −Λ∂q∗b∂x∗≈ − (g′)0.5D1.5cεu∗w∂q∗b∂x∗., (4.9)which has two unknowns, Dc and q∗b . We specify Dc as the sediment supply D90 grain size,which means D is treated as spatially uniform and therefore constant. This choice underscoresthe earlier noted assumption of morphodynamic equilibrium, and recognition that the D90has a strong influence on rates of particle mobilization and transport (Schneider et al., 2016;MacKenzie and Eaton, 2017; Masteller and Finnegan, 2017).The dimensionless bedload transport q∗b requires discussion. This parameter can be deter-mined, for example, with the Wong and Parker (2006) corrected form of the Meyer-Peter andMu¨ller bedload transport relation (Meyer-Peter and Muller, 1948): q∗b = 3.97(τ∗− τ∗re f )1.5, whichstates that transport intensity is a non-linear function of the excess Shields stress (Shields, 1936).However, Figure 4.8b illustrates that anomalous Shields stress conditions relative to bed to-pography exist for locations 7000, 5000 and 4000 mm for PRE1. We therefore propose a scalingof q∗b based on Figure 4.8a:q∗b ≈ U∗x ≈ −Ux(g′d)0.5, (4.10)where Equation 4.10 is a form of Froude number, the square of which expresses a balancebetween the kinetic energy available in the velocity field, and the potential energy stored in thebed topography, which is a measure of the relative bed strength. Here g′ = g[(ρs/ρw)− 1], andd is cross-sectionally averaged water depth. Use of g′ in the nondimensionalization requiresthat the bed be viewed as a granular gravity current, rather than a solid boundary. We assumethat changes in the relative strength occur over distances which scale as d, which complementsthe assertion of a flow timescale that scales as t ≈ Lc/Uc.Our choice of Ux nondimensionalization is motivated by the way in which the bed re-sponds to water flow down a channel characterized by downstream changes in width (Figures4.8 and 4.9). The bed responds by either building topography and storing potential energy(PE), or destroying topography to a magnitude commensurate with the kinetic energy (KE)extracted from the velocity field to do the work of mobilizing the bed.Normalization of Ux by (g′d)0.5 yields U∗x values that are 0(1), ranging from 0.70–1.06.Multiplying U∗x by -1 is necessary because the relative magnitude of U∗x is reversed to thatof Ux, because of the effect of water depth. Introducing -1, however, preserves the spatialcharacter of Ux as observed during PRE1 (Figure 4.8). Furthermore, scaling of q∗b in termsof Ux is supported by Figure 4.10, which illustrates that downstream changes in the cross-sectionally averaged velocity are inversely correlated with variations in width, which relatesto the relative mobility condition. There is scatter amongst the data, but the trend is clearand expected (Thompson et al., 1998; MacVicar and Roy, 2007; Thompson and McCarrick, 2010;de Almeida and Rodrı´guez, 2012; MacVicar and Rennie, 2012).784.5. Discussion1:1Figure 4.10: Downstream changes in local mean flow speed for associated changes inchannel width. Changes computed as forward differences with Equation 4.1. PRE1observations indicate that mean downstream changes in flow speed for 42, 60 and80 l·s−1 were inversely correlated with downstream changes in width.With our proposed scaling of q∗b , Equation 4.9 is written in the final form used herein:∂η∂x≈ Λ∂U∗x∂x∗≈ (g′)0.5D1.590εu∗w∂U∗x∂x∗≡ Slocal (4.11)To calculate Slocal from experimental data, and over the range of local width changes, we as-sume steady flow and steady-state topographic profile conditions (Figures 4.3 and 4.4), forwhich bed sediments have sorted to an approximately consistent spatial grain size distribu-tion, defined by the sediment supply. Figure 4.10 permits us to constrain the calculation of slocalbased on the range of observed Ux for the steady-state conditions, and associate the resultingvalues of Slocal with the range of experimental changes in local channel width. Furthermore,w and computed values of u∗ used in Equation 4.11 are averaged between the subsamplinglocations (Figure 4.2), and S in the shear velocity calculation was specified as the initial meanflume slope of 0.015. Use of the initial mean flume slope is appropriate because we are inter-ested in how spatial variations in particle mobility, set by the initial slope, unit discharge, localmean fluid velocity and bed roughness, gives rise to a local slope response.4.5 DiscussionThe combined results of Sections 4.3 and 4.4 raise several questions which require further dis-cussion. First, Figure 4.8 illustrates that local topographic gradients from PRE1, de Almeidaand Rodrı´guez (2012) and Nelson et al. (2015) are systematically expressed across the range of794.5. Discussiondownstream width change −0.3 : +0.3. For PRE1 results, we further observe that flow speedand particle mobility changes inversely with variations of channel width. Therefore, we revisita more focused Question 1 presented within the Introduction: How specifically does channelwidth matter for pool-riffle development? Second, Figure 4.4 illustrates that longitudinal to-pographic gradients are similar across a range of water and sediment supply conditions thatvary by a factor 2, with the lowest experimental flows simulating the bankfull flow. In partic-ular, what does persistence of pool, riffle and roughened channel bed structures at the largestflow and sediment supply rates suggest for pool-riffle maintenance with respect to these con-ditions? Third, results presented in Figures 4.3 and 4.4 indicate that pool-riffles are created byat least two different processes along variable width channels. What are these processes andwhy is it important to identify them? Last, the six SS topographic profiles of Figures 4.4 and 4.7exhibit overall consistent patterns, indicating that pools, riffles and roughened bed structuresare spatially anchored by changes in width. However, Figure 4.7 reveals that each SS profileis unique. We address Question 2 presented in the Introduction, and ask how this finding isimportant for that sediment transport theory which is built from a probabilistic perspective,versus a deterministic one?.4.5.1 Predicting local channel slope along variable-width channelsWe developed Figures 4.8 and 4.9 to help explain how and why channel width matters forpool-riffle development. Results from these figures motivated development of our mathemat-ical model for the local topographic gradient Slocal (Equation 4.11), which we plot in Figure4.11 with our experimental observations of Slocal , plus those of de Almeida and Rodrı´guez (2012)and Nelson et al. (2015). As before, we plot these quantities vs. the downstream change inchannel width, which to simplify discussion we define as Γ. We indicate flow speed change asbefore, based on the specified colors, and the size of circles for the PRE1 results indicates themobility condition magnitude τ∗/τ∗re f (Table 4.4).On the secondary axis we show the downstream change in the average local width to depthratio: ∆α(x) (square symbols). The dashed light gray line is a linear best fit to ∆α(x), as weknow of no theory which describes how the width to depth ratio changes along variable widthchannel segments. The linear best fit has a coefficient value of 22. We determine ∆α(x) as thedifference between subsampling locations, and we take averages of ∆α(x) from the associatedvalues for observations 1–29 (Table 3.1). The error bars are the sample standard deviationacross all observations.We frame the results shown in Figure 4.11 by identifying regimes in terms of Γ–Slocal–∆αparameter space, building from our previous explanations. Figure 4.11 suggests that relativelystraight channel segments of variable downstream width exhibit three regime spaces for Γ andSlocal , which based on the combined results of PRE1, de Almeida and Rodrı´guez (2012) and Nelson804.5. DiscussionPool Regime Riffle RegimeLocal Narrowing Local WideningMean conditions: Eq. 4.115 0.210.04 -0.04DecelerationAccelerationFlow Speed Change (-)Entrainment DepositionalUniformFigure 4.11: Prediction of local SS channel slope across the range of channel width con-ditions using Equation 4.11, plotted with Slocal for PRE1, de Almeida and Rodrı´guez(2012), and Nelson et al. (2015) (circles, stars and diamonds as in Figure 4.9, respec-tively). The predicted Slocal curve reflects the mean of PRE1 water supply condi-tions, and is shown with the dark, solid curve. On the secondary axis we show theaverage downstream change in the width to depth ratio α. The average of ∆α(x)is taken for the associated values of observations 1–29, and error bars are the stan-dard deviation over the observation range. The light gray dashed line is a linearbest fit to ∆α(x) with a coefficient value of al. (2015), we define as:Slope Regimes=Entrainment Regime: Γ < −0.1 and Slocal > 0.2Uniform Regime: − 0.1 < Γ < 0.10.02 > Slocal > −0.1Depositional Regime: Γ > 0.1 and Slocal < −0.1Pool development defines the entrainment regime along relatively straight channel reaches,driven by downstream flow speed changes ∆Ux ·∆L−1 that are increasing, and mobility con-ditions τ∗/τ∗re f which are well above threshold conditions. The uniform regime defines rough-ened channel development, driven by downstream flow speed changes that are relatively mi-nor, and mobility conditions that are near the threshold condition. Riffle construction defines814.5. Discussionthe depositional regime, driven by downstream flow speed changes that are decreasing, andmobility conditions that are well below the threshold condition (cf. Table 4.4).The datasets of Figure 4.11 indicate that the spaces defined by (Γ < 0,Slocal < 0), as wellas (Γ > 0,Slocal > 0) may not be physically possible at the local scale, for relatively straightchannel segments of gravel composition. However, at the basin scale, the (+Γ,+Slocal) regimegives rise to the upward-concave river profile of graded channel conditions, whereby the chan-nel slope evolves to transport the prevailing basin sediment supply, given the associated wa-ter supply conditions (Sternberg, 1875; Gilbert, 1877; Leopold and Maddock, 1953; Langbein andLeopold, 1964). Last, the (−Γ,−Slocal) regime is driven by geologic controls that are relativelydecoupled from the flow-bed coupling conditions described here, with one example being abedrock-controlled channel segment due to normal faulting, which exhibits narrowing (Ouchi,1985; Schumm et al., 2002).Associated with the Γ–Slocal regimes, Γ and ∆α exhibit two regimes, we define as:Width/Depth Regimes=Depth Regime: Γ < −0.1 and α < −2Uniform Regime: − 0.1 < Γ < 0.12 > α > −2Width Regime: Γ > 0.1 and α > 2The depth regime is characterized by water depths that are increasing relative to channelwidth. As a result, flows are comparatively deep and increasing in speed, delivering moremomentum flux to the bed, and it is for such conditions that pools develop. The uniformregime is characterized by comparable changes in width and depth. As a result, flows areapproximately uniform, and it is for such conditions that roughened channel segments de-velop. The width regime is characterized by channel widths that are increasing relative towater depth. As a result, flow are comparatively shallow and decreasing in speed, and it is forsuch conditions that riffles develop.Equation 4.11 indicates that local slope construction depends on the magnitude of Λ, andthe sign and magnitude of ∂U∗x/∂x∗. The influence of Λ is governed by how the magnitude oftspreading compares to t f orcing, noting that local channel width drives the magnitude of tspreading,and local shear velocity drives t f orcing. In Figure 4.12 we show Λ vs. Γ. We observe that Λvaries inversely with Γ, and decreases monotonically from 0.30 to 0.25 as Γ increases from 0 to0.20. As a result, Λ has a particularly strong affect on Slocal under narrowing width conditions,compared to cases for which width is widening (cf. Figures 4.11 and 4.12).Additionally, Figure 4.11 shows that for the narrowest and widest width conditions, Slocalis roughly twice as steep leading into pools, as it is for riffles. Figure 4.12 explains that thesteeper pool entrance slopes are due to how Λ varies against Γ, with values in the pool regimethat are roughly twice as large as values in the riffle regime. From our definition of Λ inEquation 4.7, we therefore understand that pools develop when tspreading is characteristically824.5. DiscussionPool Regime Riffle RegimeFigure 4.12: Summary of Λ of Equation 4.11 vs. Γ for PRE1. Λ varies inversely with Γ,and ranges by roughly a factor 2 across the PRE1 width conditions.large relative to t f orcing, indicating that pool development is governed by momentum fluxdelivery to the bed, which drives particle entrainment. On the other hand, we understandthat riffles develop when tspreading is characteristically small relative to t f orcing, indicating thatriffle development is governed by topographic spreading and growth as a result of reducedmomentum flux delivery to the bed.But why, specifically, are pool entrance slopes roughly twice as steep as those of riffles, for(-0.30 < Γ < -0.10) and (0.10 < Γ < 0.30), respectively? From our choice of physical scales,Equation 4.11 shows that the value of Λ depends on the local shear velocity u∗ and width w,owing to our choice of Dc = D90us , where D90us is a constant and is the 90th percentile sizeclass of the upstream supply. For increasing values of Γ, the shear velocity decreases, and as< Γ → 0.30, the product (u∗w) tends to a constant value, and ∂U∗x/∂x∗ approaches a negativelimiting value. It follows then that the magnitude of these values, expressed through Equation4.11, yield riffle entrance slopes between (-0.02)–(-0.03), or about half that of pools, and for theassociated ranges of Γ noted at the beginning of the paragraph.As a closing remark, the dynamics of slope construction just discussed suggest that theconversion and storage of kinetic as potential energy at locations of sediment deposition andchannel widening, has an upper limiting condition near Γ = 0.30 (Figure 4.11) and for channelreaches that exhibit SS bed profiles. This makes sense because SS profiles ultimately providethe conditions necessary to transport the upstream supply of sediment through segments ofpositive and adverse bed slopes, where adverse slopes are the limiting transport cases. Bycontrast though, Figure 4.11 suggests that a limiting condition for pools occurs for some valueof Γ < -0.30, indicating that the release and conversion of potential to kinetic energy fromwide to narrow segments is not as readily limited, as the reverse case. Which taken together,implies that river flows dig holes that are comparatively deeper than deposits are tall. Last, it isimportant to recognize that the mobility conditions for pool vs. riffle of are similar magnitude834.5. Discussionrelative to the threshold condition, 5 vs. 0.2 (Figures 4.11 and 4.12), yet yield the disparateSlocal responses just discussed.Figure 4.11 illustrates that ∆α(x) varies linearly over the range of PRE1 width conditions,such that:∆α(x) ∝∆w(x)∆L(4.12)Because of a coupling between the flow and the bed, ∆α(x) is a summary of the morphody-namic processes that led to net adjustment of bed topography across the range of imposedwidth changes. Net adjustment of bed topography to external conditions is known to con-verge toward states for which the local divergence of bedload flux goes to zero (Equation4.3; (Bolla Pittaluga et al., 2014)). It follows then that ∆α(x) reflects the tendency to balancelocal fluid momentum and resulting solid fluxes. The balancing occurs over a characteristiclength scale reflected in the∆α(x)−−∆w(x) proportionality, Equation 4.12, which, noting that∆α = ∆w/∆d, we find the length scale is of order:∆x ≈ χ∆d, (4.13)where d is the local cross-sectionally averaged flow depth, and χ is a constant, which for PRE1has a value of 22. For the pool at station 8000 mm, ∆d has an average value of 4.0 cm acrossobservations 1–29, whereas for the riffles at stations 4000 and 10000 mm, the average valuesare -2.5 and -1.6 cm, respectively. This suggests that channel width narrowing drives a meanflow response which manifests over length scales that are roughly twice as long compared tothat for riffles, from ∼ 2w′ for pools vs. ∼ w′ for riffles. These mean flow relaxation lengthscales are reflected in both tspreading and t f orcing, because as the relaxation length gets bigger,as in the case of pools, we expect tspreading to get bigger, and we expect t f orcing to get smaller,and vice versa for riffles, as shown in Figure 4.12. Therefore, Equation 4.13 provides the ba-sic information needed to understand how water depth responds to downstream changes inchannel width, for a range of upstream water and sediment supply conditions.4.5.2 Maintenance of bed topography along variable-width channels: support foran emerging viewWe suggest that the combined results of Figures 4.3 through 4.11 provide evidence that thecombination of PRE1 flows are important for topographic expression, and by extension mor-phodynamics in natural streams. Experimental support for our proposal consists of two parts.First, pool-riffle and roughened channel persistence across all experimental water and sedi-ment supply conditions suggests that morphologic response is reinforced across the range ofsupply conditions. Second, increasing topographic relief for lower overall longitudinal gra-dients, and vice versa suggests that different supply magnitudes maintain channel form indifferent, but equally important ways (Figures 4.4 and Table 4.3).Our perspective is consistent with Pickup and Rieger (1979); Parker et al. (2003); Bolla Pit-844.5. Discussiontaluga et al. (2014) and Brown and Pasternack (2017)’s interpretation that the full distributionof flows under the present day hydrology is important for channel morphology, as raised by(Ferrer-Boix et al., 2016). More importantly, however, recognizing the importance of the full hy-drologic regime in channel form maintenance builds immediate bridges with ecology, and inparticular with the field of environmental flows, and the natural flow paradigm (NFP) concept(e.g. Poff et al., 1997; Acreman et al., 2014b). The NFP reflects the view that the entire flow regimeconsisting of droughts, floods of all size, annual low flows, etc. are critical to the support ofriverine processes and ecological communities. NFP may seem at odds with the perspectivethat bankfull, or the effective flood is the most important flow for mountain stream morpho-logic maintenance (Wolman and Miller, 1960; Emmett, 1999; Whiting et al., 1999). The bankfullor effective flow perspective is based on quantifying the flood magnitude that moves the mostbedload sediment over long periods of time. Since alluvial channels are built by sedimenttransport, it follows that the bankfull or equivalent flow maintains river form or shape. De-spite hydroclimatological variation in the frequency of bankfull or effective flows (Williams,1978), the morphologic basis of bankfull is a critical aspect of geomorphology (Phillips andJerolmack, 2016).To bridge the apparent gap between concepts underpinning views of environmental andbankfull flows, we suggest that results presented here coupled with supporting work by Pickupand Rieger (1979); Parker et al. (2003); Bolla Pittaluga et al. (2014) and Brown and Pasternack (2017)highlights that larger floods build the framework, or foundation skeleton of gravel-bed moun-tain streams, and that smaller, more frequent floods fill out the skeleton (Figure 4.4), whileretaining the shape or morphology of the skeleton (Figure 4.7). The filling out process evolvesaccording to the sequence and magnitude of floods, which work collectively to enhance mor-phologic diversity (Figures 4.5 and 4.6), and build the riverine palette from which measurableecosystem services are realized (Acreman et al., 2014a, Figure 1 therein). Accordingly and overlong periods of time, the bankfull or effective flow would be the most important element ofthe flows which fill out the skeleton.4.5.3 Development of pool-riffles along variable width channelsFigures 4.3 and 4.4 illustrate that two types of pool-riffle structures formed within the exper-imental channel. The first type, referred to as entrainment-driven pool-riffles, occur alongchannel segments with downstream width variations that proceed from relatively wide seg-ments to narrower ones. The riffle-pool from station 10000 to 6000 mm of the experimentalchannel reflects an entrainment-driven feature (Figures 4.3 and 4.4). The second type, referredto as depositional-driven pool-riffles, is the sequential inverse of the entrainment-driven type,whereby width is organized to proceed from relatively narrow segments to wider ones. Thepool-riffle from station 12000 to 10000, and 5000 to 3000 mm reflects a depositional-drivenfeature (Figure 4.3).The names of each pool-riffle type convey the processes responsible for formation. In the854.5. Discussionfirst case, entrainment-driven riffle-pools form due to the downstream release of KE storedwithin the upstream riffle, which drives net particle entrainment downstream of the riffle,and pool formation. Depositional-driven pool-riffles form due to downstream storage of KE,which manifests as locally elevated water surface elevations. Channel segments immediatelyupstream of the points of widening, and locally high water surface elevations, are affected bythis downstream condition, leading to increased water depths, and passive pool formation.Figure 4.4, in particular, illustrates this condition, and also shows that the upstream pool bedslopes are similar, or slightly steeper than the overall longitudinal bed slope. Indicating thatnet particle entrainment has a minor role in pool-riffle formation under depositionally-drivenprocesses. de Almeida and Rodrı´guez (2011) also reports the prevalence of backwater-controlledpool-riffles for their Bear River, AR, U.S. simulation reach.It is important to identify the different processes which give rise to pool-riffles for at leasttwo reasons. First, formative hypotheses must account for the development mechanisms re-quired to explain the observations in Figures 4.3, 4.4 and 4.9. The second practical reason isthat river restoration practitioners should be aware that different design approaches will yieldpool-riffles, but that the associated structures will exhibit differing characteristics.4.5.4 General implications of unique profiles for sediment transport theoryThe paired topographic profiles for 42, 60 and 80 l·s−1 shown in Figure 4.7 probably reflectsjust a few of the many possible SS topographic states that would otherwise result from thesame external conditions. This reality means that the emergence of bed topography within thePRE1 experimental channel is best described by a probability distribution of n possible states(steady-state shapes), conceptually reflecting the idea of microstates as discussed by Furbishet al. (2016), and in line with the probabilistic nature of sediment mobility (Wiberg and Smith,1987; Kirchner et al., 1990; Hassan et al., 1991, e.g.), sediment transport (e.g. Einstein, 1950; Furbishet al., 2012; Ancey and Heyman, 2014), and turbulence. This outcome is not consistent, however,with expectations built from Equations 4.3 and 4.4, which imply a uniform outcome for thesame supplies of water and sediment. So, is it possible to reconcile the probabilistic behaviorshown in Figure 4.7 with the uniform basis of Equations 4.3 and 4.4? We suggest that theemergence of non-unique topographic profiles for similar upstream supply conditions offersa potential link between particle scale probabilistic transport theory (Furbish et al., 2012; Anceyand Heyman, 2014; Furbish et al., 2016), and manifestation of these processes at larger scales.To motivate future work, we provide the following example to illustrate a possible link.The assumptions underpinning Equation 4.11 preclude a probabilistic perspective, but thatdoes not constrain solutions to unique outcomes. The dimensionless downstream mean ve-locity gradient ∂U∗x/∂x∗ responds to the local topographic and surface texture conditions, overlength scales of 1–2w′. If SS topography is described by a probability distribution of possiblestates, ∂U∗x/∂x∗ will correspondingly vary, and drive non-unique outcomes for the same up-stream supply conditions. Furthermore, relaxing the assumption of a spatially fixed D90 of864.6. Conclusions and next stepsEquation 4.11, would introduce more local variability into the problem, which would rein-force the tendency for non-unique topographic responses, and would better reflect grain sizevariability as shown in Figures 4.2d and 4.2e. The particle effect would diminish, however, forcases where natural channels have time periods of likely 101 − 103 years, or more to respondand evolve to uniform upstream conditions, depending on initial conditions relative to thoseof SS (Howard, 1982).4.6 Conclusions and next stepsMotivated by previous observations that pool-riffles are colocated with segments of channelnarrowing and widening, respectively, we use scaled laboratory experiments and theory toexamine how and why downstream channel width variations give rise to these bed structures,and under conditions common to natural streams. Our experiments produce pool-riffle, androughened channel morphologic structures across flow and sediment supply rates that vary bya factor 2. Pools occur where the downstream change in width ∆w(x) ·∆L−1 < -0.10, riffles oc-cur where ∆w(x) ·∆L−1 > +0.10, and roughened channel beds where−0.10 < ∆w(x) ·∆L−1 <+0.10. These general threshold conditions are consistent with data from numerical simula-tions (de Almeida and Rodrı´guez, 2012) and experiments (Nelson et al., 2015), and also highlightthat relatively straight channel segments constrained by −0.10 < ∆hw(x)h−1 < +0.10 areunlikely to develop pool-riffle pairs, unless they are driven by some other external condi-tion which leads to relatively large spatial differences in sediment transport. Furthermore,pool-riffle formation is the result of at least two different processes: entrainment-driven anddepositional driven. Which one ultimately governs local conditions depends on the spatial or-ganization of channel width. Therefore, along relatively straight channel segments, the spatialorganization of channel width drives the general topographic response.We show that local topographic gradients Slocal are systematically expressed across therange of downstream width change (-0.30)–(+0.30), and for reach-average bed slopes that varyby one order of magnitude. This finding points out that we can determine the general behav-ior of Slocal by knowing nothing more than how channel width changes in the downstreamdirection, and over length scales of 1–2 average widths. We examine specific controls on Slocalorganization with a 1D mathematical model developed from statements of mass conservation,bedload transport, and scaling arguments supported by our experimental measurements. Ourmodel is motivated by the observation that bed topography and width change are coupledthrough downstream variations of mean flow speed, but the model indicates that the physicsgoverning this coupling outcome is expressed through the parameter Λ. Λ expresses the rela-tive importance of a relaxation vs. the forcing times scale. Pools emerge when the forcing timescale is characteristically small relative to the relaxation time scale, and riffles emerge whenthe difference between the two time scales decreases by a factor 2 or more. Furthermore, theexpression of Slocal over the range of ∆w(x) ·∆L−1 presented in Figures 4.9 and 4.11 is condi-tioned by the spreading timescale, as it is a measure of bed resistance for alluvial channels.874.6. Conclusions and next stepsOur experiments idealize natural gravel-bed streams as ones with fixed banks, high enoughto contain relatively large floods. Whereas the height of channel walls in PRE1 were highenough to contain all flows, so that the width conditions were controlled across all upstreamsupplies, we do not suggest that our work or results reflect bedrock canyon reaches. As such,it is helpful to contextualize our work as framing an end member case where local widthis the dominant driving mechanism of bed topography expression. Notably, de Almeida andRodrı´guez (2011) and Brown and Pasternack (2017) offer numerical and field-case results for theopposite end member case where the nature of channel width control relaxes to give way toother driving mechanisms, which are discharge dependent. Fruitful next steps include exam-ination of topographic responses under variable upstream sediment supply conditions, withpulses of differing texture, and hydrographs, with an emphasis on whether it is possible tobreak the width control and evolve toward a completely different topographic response, in-cluding the occurrence of lateral bars. We also suggest that our results can frame the basis fora unified pool-riffle formative hypothesis within mountain stream settings.88Chapter 5Morphodynamic evolution of awidth-variable gravel-bed stream: abattle between local topography andgrain size texture5.1 SummaryStatistical steady-state is commonly defined as mass continuity of bedload sediment overchannel reaches of many channel widths in length, or longer. Proposals for equilibrium condi-tions commonly carry on from this steady-state definition by stating that under conditions ofmass continuity, rivers express a longitudinal bed profile which varies around some long-termstable pattern. But this larger-scale view of equilibrium neglects the local physical processesthat give rise to the stationary profiles, and we lack a formal definition of equilibrium basedon these processes. We address this need and use mass conservation statements for the bulkriverbed, and the sediment particles which comprise the riverbed to define two new dimen-sionless numbers which quantify the rates of bed topography and bed sediment texture ad-justment to upstream water and sediment supplies, for which sediment texture is defined bythe local spatial distribution of grain sizes for areas that scale as (w′)2. We hypothesize that alocal equivalence of these rates defines fluvial equilibrium, which can be scaled up to reachesof many channel widths with supporting information on the spatial distribution of these rates.Our equilibrium definition depends on only three quantities: a topographic adjustment ve-locity, a particle composition adjustment velocity, and a term which quantifies the degree ofdifference between the fractional composition of the local bedload supply and the sedimentsstored in the bed substrate, in relation to the fractional composition of the long-term averagesediment supply. We apply our new view to experimental data from pool-riffle experiment 1,and find that equilibrium conditions are achieved for relatively high bed sediment mobilities.895.2. Introduction5.2 IntroductionMountainous rivers flow through channels that are remarkable for their spatial complexity.Water moves over steps of various size, accelerates through narrowings, slows at deep poolsand becomes complex at bends. Depositing and mobilizing sediments along the way. Overrelatively long times, the inherent richness of river systems can give way to some measure oforder, typified by river longitudinal profiles that settle to a statistically steady-state condition(SS) (Howard, 1982; Ahnert, 1994), which we define by two criteria. First, steadiness of averagetopography, and second, that the upstream sediment supply Qss is approximately equal to thedownstream sediment flux at the outlet Qs f (see Figure 4.2). The second criteria holds for boththe total mass of the sediment mixture, and the fractional mass of each grain size.Statistically steady profiles are characterized by a natural downstream progression of chan-nel bedform geometry and topography, and associated bed sediment grain size distributionsor textures (Montgomery and Buffington, 1997). Closer to the headwaters, channels are steep,and exhibit boulder (> 256 mm) bed stepped reaches of many channel widths in length. Asdrainage area increases, channels are more moderately sloped, with undulating cobble andgravel (2–256 mm) bed reaches. Closer to the terminus, or out into the lowland plains, chan-nels are gently sloped, and exhibit meandering forms composed of sand (> 2 mm) coveredbeds (e.g. Leopold and Wolman, 1957; Montgomery and Buffington, 1997; Church, 2006).In Chapter 4 we demonstrate that adjustments of channel bed topography and grain sizedistribution are coupled to local variations of channel size via changes in flow speed and par-ticle mobility condition, expressed through the parameter Λ. Local width variations, whichwe define by length scales of a few average widths, are important because they modulate thetotal mass and fractional composition of sediments transported to downstream reaches, prin-cipally due to sediment storage within relatively wide channel segments, and depending onthe extent to which sediment is mobilized (cf. Chapter 4) (Furbish et al., 1998; Bolla Pittalugaet al., 2014; Ferrer-Boix et al., 2016). As a result, SS at the reach and larger scale is conditioned bythe cumulative time scales necessary for channel profiles to develop in response to local widthvariations (Howard, 1982; Paola et al., 1992; Ahnert, 1994; Bolla Pittaluga et al., 2014). Here, webuild from our findings of Chapter 4, and suggest that the coupling between channel bedformgeometry, surface texture, and channel width is also important for SS conditions, and providesa way to define ”fluvial equilibrium”.Fluvial equilibrium is a useful, but often times confusing concept with a long history.Howard (1982) defines equilibrium as a temporally invariant functional relationship betweensystem inputs and outputs, which in the present case includes the total and fractional massesof all sediment sizes within Qss and Qs f . Ahnert (1994) elaborates on Howard’s proposal, andstates that equilibrium is an equivalence of the rates of processes acting on sediment supply,which drives erosion and deposition, and ultimately gives rise to sediment mass continuity.Ahnert’s proposal builds directly from Domenico Guglielmini (Guglielmini, 1697; Chorley et al.,2009), who postulated in 1697 that ”streams erode or build up their beds until an equilibrium905.2. Introductionis reached between force and resistance” (Chorley et al., 2009, page 84). Du Buat (du Buat, 1786),and then Gilbert (Gilbert, 1877) continued with this view, and Gilbert surmised that equilib-rium is tied to an equality of action, which can be understood as comparability between thecapacity to do work to the streambed, and the resistance offered by the bulk bed. Despite thefrequent use of equilibrium within fluvial studies, we are unaware of a formalized definitionbuilt from Gilbert (1877)’s, Howard (1982)’s, or Ahnert (1994)’s proposals.Here we use Ahnert (1994)’s general proposal, and define equilibrium as an equivalencebetween the rates of constructing local mean bed topography and slope, and bed surface sed-iment texture or roughness, as determined by the bed surface local grain size distribution, forareas that scale as (w′)2 (Venditti et al., 2012; Chartrand et al., 2015). Disequilibrium occurs whenthese rates are not equivalent. We assume that the topographic and texture rates are muchlarger than those of channel bank, or channel position change, and we further assume on thebasis of results presented in Chapter 4, that equilibrium reflects a balance between momentumflux delivery to the bed, and the strength of the bed itself. We are therefore focused on mechan-ical equilibrium, and we quantify the rates of topographic and texture change, at maximumspatial scales of roughly w′, because this is the approximate minimum scale at which channelwidth change drives a morphologic response, absent external forcing by wood or boulders(see Figure 4.1). Our definition of fluvial equilibrium is distinct from conventional views ofstatistical steady-state in two ways:a. Statistical steady-state is defined by mean bed elevation steadiness, and sediment massbalance. Equilibrium is defined by local rates of constructing topography and bed sur-face roughnessb. Statistical steady-state is usually evaluated over length scales of many average channelwidths, whereas equilibrium is determined at scales that measure of maximum of w′.We build our statement of fluvial equilibrium from mass balance expressions and scalingarguments, and our work was motivated by three important questions:1. Can we build a definition of equilibrium that captures the physics discussed in Chapter4, and which is easily tested with experimental, or field data?2. Can equilibrium conditions be determined, or reliably inferred from the appropriatemass balance statements?3. For a given set of water and sediment supplies, will the same balance of forces be ex-pressed by 1 profile/grain size bed pattern (Parker and Wilcock, 1993; Church and Fergu-son, 2015), or a suite of possible patterns as suggested by Figure 4.7? This question alsoaddresses the related issue of whether equilibrium is achieved in generally the samemanner for a set of supply conditions?915.3. Morphodynamic evolution metrics at the scale of a channel widthWe hypothesize that riverbed response to imposed flow and sediment supply conditionsat the local scale of w′ is governed by two filters which either drive the tendency to build to-pography, or the tendency to entrain sediment particles resting on the bed surface (Figure 5.1–discussed in detail within Section 5.3.1). The proposed filters conceptually reflect the physicalprocesses which drive sediment deposition and entrainment. Consequently, at equilibriumthese filters have equivalent gains and local sediment continuity is achieved. Accordingly, wealso hypothesize that equilibrium is expressed through statements of total and fractional massconservation of the riverbed, consistent in principle with Ahnert (1994).We test our hypotheses with flume experiments conducted within a variable-width chan-nel, the details of which are reviewed in Chapter 3. A principal finding of this chapter isthat equilibrium condition is dependent upon comparability of topographic and grain sizeadjustment rates, and similarity between three populations of sediment particles that set lo-cal responses: the local bed subsurface, the local sediment supply, and the long-term averageupstream sediment supply. The grain sizes term is rate limiting, and ultimately governs equi-librium time scales, which expands upon existing topographic-focused ideas (Howard, 1982;Ahnert, 1994), and basin-scale theory (Paola et al., 1992).5.3 Morphodynamic evolution metrics at the scale of a channelwidth5.3.1 Problem set-upAt the local scale of a few channel widths, rivers respond to supply fluctuations of wateror sediment through adjustments of channel size, streamwise topographic profile, and bedsurface texture. Along relatively straight mountain streams, adjustments of channel size areneglected, because topographic profile and texture adjustment rates are relatively much larger,and changes in size are intermittent. Our focus therefore is with adjustments of the streamwisetopographic profile, and bed surface texture, which force disequilibrium conditions by alteringlocal rates of sediment transport for time scales of at least a few flood events (cf. Chapter 2). Weconceptualize this view in Figure 5.1a, where we illustrate the physical processes of sedimentsupply modification in terms of depositional and entrainment filters. As we will show, theaction of these filters modulates channel evolution toward fluvial equilibrium.Figure 5.1a shows that local sediment supply Qss has a magnitude and composition, whichfor disequilibrium conditions, are modified by deposition and/or entrainment. The degree ofmodification sets the properties of the local sediment flux Qs f . For a control volume coveringa unit area of the bed, incoming sediment grains transported along the streambed can eithersettle to the bed, or remain in motion and continue downstream, and additional grains can beentrained, or not. The tendency for any of these outcomes depends on the local profile andbed texture conditions, which in turn reflects departure from the associated equilibrium.The depositional filter (Figure 5.1a) is a result of feedbacks between the local bed topogra-925.3. Morphodynamic evolution metrics at the scale of a channel widthInputs Ouputs Depositional Filter Entrainment FilterChannel EvolutionQsswater supplysubsurface(fss)mixingentrainment gain set by localbed surface texture conditionsx1x2x3coordinate systemsurface (fa)depositional gain set bylocal topographic conditions150 200 250 300 mmhighlowac�ve layer thicknessexchange surfainput signal filteringinterac�onsQwF(Di)320 mmsupply compositionsediment supplywater exportexport compositionsediment exportQsfQwF(Di)Bed/Channel Topography Bed Sediment TexturesourcesinkControl Volumewater surfaceDefinition of Variableswidth: winput outputQb, F(Di)(a)(b) (c)bedload (fb)Figure 5.1: Conceptual illustration of how topographic and sediment texture filters workto drive channel evolution. (a) The top panel illustrates that incoming sedimentsupply Qs f , transported by some supply of water Qw, is acted upon by topographicand sediment texture filters to give rise to the outgoing sediment flux Qs f . Both themagnitude and composition of the incoming supply can change. Images of the DEMand photograph of the bed surface provided for illustration of the concepts. (b) Thecontrol volume for the problem illustrates that the filter actions lead to the channelbed being a sink or source of sediment particles. Filtering the Qss by depositionmeans the channel bed is a sink for particles, and filtering by entrainment meansit is a particle source. (c) Definition of variables used for derivation of the localtopographic and particle response numbers, Nt and Np respectively.phy and the flow. Over bed areas of roughly (w′)2, bed topography scales the average down-stream flow velocity Ux, due to flow continuity and coupling with the local channel widthcondition (see Chapter 4) . Ux, and in particular the downstream change in Ux in turn scalesthe average momentum flux imparted to the bed, and therefore the tendency to deposit sed-iment in motion or not. As we show in Chapter 4, under conditions where local flow speeddecreases, the tendency for deposition is relatively high, where flow speed increases, the ten-dency for sediment grains in motion to continue downstream is relatively high, and whereflow speed change is negligible, either outcome is possible (e.g. Hoey and Ferguson, 1994; Parker,2008).The entrainment filter (Figure 5.1a) is a function of the local sediment texture, which sets935.3. Morphodynamic evolution metrics at the scale of a channel widththe average mobility condition for any grain resting on the bed surface. Mobility is commonlyexpressed (1) as a function of relative grain size, which captures the degree to which smallergrain sizes are sheltered from the flow by larger grains (e.g. Ashida and Michiue, 1972; Parker,1990), (2) based on the content of sand sized grains in the bed surface, which reflects near-bedvelocity structure (Wilcock and McArdell, 1993; Wilcock and Crowe, 2003), and (3) based on howgrains rest on the bed surface, which captures the relative difficulty of pivoting a grain out ofa pocket (Wiberg and Smith, 1987; Kirchner et al., 1990; Buffington et al., 1992). As one example,bed surfaces that are relatively rough, with many grain sizes present, will preferentially entrainlarger grains, or grains which sit relatively high in the flow, above the bed surface (Wiberg andSmith, 1987; Kirchner et al., 1990; Buffington et al., 1992) (also see Appendix B). Furthermore, thedepositional and entrainment filters also interact, because filtering by each occurs at differentrates, and this triggers additional possible responses.Figure 5.1b shows the local control volume for our filtering problem, and Figure 5.1c pro-vides definition of variables which will be used below when we derive our statement of fluvialequilibrium. In the problem definition of Figure 5.1b, the upstream water supply Qw is unaf-fected by filtering, and the channel bed is either a sink, or source of sediment. The channel bedacts as a sink due to affects of the depositional filter, and acts as a source due to affects of theentrainment filter.We have set up the local equilibrium problem in terms of depositional and entrainmentfilters, which we link to the local bed topography, and bed surface texture, respectively. Theselinks offer a natural basis from which to derive our equilibrium statement, which we completewith mass conservation statements for the bulk riverbed, and the particles which make up thebed. We make these choices because these conservation statements can quantify the respectivefiltering responses over relatively short time scales, ideally that are less than the duration of aflood event. Furthermore, the mass conservation statements embody information concerningthe dynamics of how each process responds to upstream water and sediment supply forcing(e.g., Hoey and Ferguson, 1994; Paola and Voller, 2005; Parker, 2008; Stecca et al., 2014), and this isthe information needed to examine equilibrium conditions. To build understanding in the nextsection, we characterize bed and particle related dynamics with new dimensionless quantitieswe term the topographic, particle and channel response numbers (Nt, Np and Ne), respectively.Nt and Np are derived by nondimensionalizing the respective mass conservation statements,with choice of specific length, time, velocity and flux scales. Ne is defined as the ratio Nt/Np.The next section presents our derivation of each number.945.3. Morphodynamic evolution metrics at the scale of a channel width5.3.2 Mass conservationExner equation: mass conservation of the riverbedThe general statement of riverbed mass conservation in one dimension is written (Exner, 1925;Paola and Voller, 2005):∂η∂t= −1ε∂qb∂x, (5.1)where η is channel bed elevation at position x, t is time, ε = (1− φ), where φ is the volume-averaged streambed porosity (assumed to be spatially uniform and set as 0.40) of the activelayer La = kDc (Hirano, 1971), where k is constant between 1 and 2 (Parker, 2008), and Dc isa characteristic grain size, generally taken as D90, which is the local grain size for which 90%of bed material within the active layer is smaller, and qb is streamwise bedload flux per unitw. Equation 5.1 expresses mass conservation for riverbeds as a balance between temporalchanges of local bed topography and the net flux of bedload Qs f .Hirano equation: mass conservation of riverbed grain sizesThe particle mass conservation statement in one dimension (Hirano, 1971), is written followingthe derivation by Juez et al. (2016):fes∂∂t(η − La) + ∂∂t( faLa) = −1ε∂qψ∂x, (5.2)where qψ is streamwise sediment flux per unit w for grain size ψ = log2D and D is a grainsize in mm, fes is the volume probability density of ψ at the exchange surface (Viparelli et al.,2010) (Figure 5.1c), and fa is volume probability density of ψ within La, or as expressed at thebed surface (Figure 5.1c), assuming a constant particle density. Based on the two-layer modelof the bed subsurface (Figure 5.1c) (Hirano, 1971), the active layer composition defines whichparticle size classes ψ can be entrained into bedload. Consequently, the probability distribu-tion of fa defines the composition or roughness of the bed surface. Equation 5.2 expressesmass conservation of particles comprising the bed as a balance between temporal changes tothe exchange surface position and the La thickness (Figure 5.1c), to the net fractional flux ofbedload (i.e. flux of each grain size fraction ψ).Equations 5.1 and 5.2 are linked via the definition of qb (Parker et al., 2000):fb = qψ/qb, (5.3)where fb is the volume probability of ψ within the local bedload (Figure 5.1c), and qb = ∑ qψacross all grain size classes. The exchange surface composition fes regulates solid fractionalfluxes between the local bed substrate ( fs) and active layer ( fa) (Figure 5.1c), and fes depends955.3. Morphodynamic evolution metrics at the scale of a channel widthon whether the local bed surface builds or lowers, determined as (Hoey and Ferguson, 1994):fes = fs if ∂η/∂t < 0β fa + (1− β) fb if ∂η/∂t > 0,(5.4)where β is a partitioning coefficient that ranges in value from 0 to 1. During phases of bedsurface lowering, or net particle entrainment, fes is composed of the bed substrate. Duringphases of bed surface heightening, or net particle deposition, fes is a linear combination of thebed substrate and active layer compositions, assuming α 6= 0. It is important to recognize thatEquation 5.2 simplifies to Equation 5.1 for a uniform mixture of bed sediment.Our goal is to express Equation 5.2 in a form where qb is a function of fes, fa, fb, and La. Wetherefore apply Equations 5.1 and 5.3 to Equation 5.2, and rearrange to obtain:∂∂t( faLa)− fes ∂La∂t= −1ε[∂∂x( fbqb)− fes ∂qb∂x](5.5)The last term of Equation 5.5 can be simplified by assuming fb is constant, and moreover thatfb sets the rate at which the bedload transport gradient ∂ fb/∂x changes in x, because fb sets thefractional composition of qb, and the fractional composition changes based on the makeup ofthe bed surface fa. Therefore, fb acts on the bedload flux gradient (Tritton, 1988), and Equation5.5 becomes:∂∂t( faLa)− fes ∂La∂t= −[( fb − fes)ε∂qb∂x](5.6)The first term of Equation 5.6 can be expanded with the product rule, and Equation 5.6 canbe restated, grouping like terms and defining δ1 = ( fb − fes)/ε:La∂ fa∂t+ ( fa − fes)∂La∂t= −δ1 ∂qb∂x(5.7)Because |La| >> |( fa − fes)|, and defining σv = δ1/La [1/L], Equation 5.7 is simplifies to:∂ fa∂t≈ −σv ∂qb∂x(5.8)Equation 5.8 is a kinematic wave equation, and states that the time rate of change of fa dependson the flux of qb, which is modulated by σv. The inverse of σv is a sorting length scale (Folk,1966) for a given δ1. Depending on the magnitude of La, the length scale is characteristicallysmall for comparable bedload and substrate compositions, and large for dissimilar composi-tions. Small sorting length scales implies that evolution of the local bed surface compositionwill occur relatively rapidly, as compared to longer length scales.965.3. Morphodynamic evolution metrics at the scale of a channel width5.3.3 Nondimensional Exner and Hirano equationsNondimensional ExnerWe nondimensionalize Equation 4.3 assuming η ∼ Lc, t ∼ (Lc/Uc), qb ∼ qbc , and x ∼ Lc:Uc∂η∗∂t∗≈ − qbcεLc∂q∗b∂x∗, (5.9)where qbc is a characteristic bedload flux per unit w, and Lc, and Uc are a characteristic lengthscale, and velocity or speed. We next multiply Equation 5.9 by 1/Uc to obtain:∂η∗∂t∗≈ − qbcεLcUc︸ ︷︷ ︸1/Nt∂q∗b∂x∗≈ − 1Nt∂q∗b∂x∗. (5.10)Here, Nt is the dimensionless topographic response number and is a ratio of two veloci-ties: Uc · (Lc/qbc). Or based on the parameter Λ from Chapter 4, a ratio of two time scales:(Uc/Lc · (L2c /qbc). To further characterize Nt, we must specify reasonable choices for qbc , Lc,and Uc. We set qbc as the local sediment supply Qss, determined at maximum spatial scales ofw′. Sediment transport theory commonly uses La to approximate the bed depth which partic-ipates in sediment transport, and thus La is a measure of a local channel bed response lengthscale. Consequently, Lc = La. Since Nt describes the time rate of change of bed topography,the characteristic velocity Uc is defined as the rate at which the local bed surface changes it’svertical position, Ub, because this defines the responsiveness of the bed to upstream supplies.With these definitions, we state Nt as:Nt(t) ≈ εLaUbqbc. (5.11)Physically, Nt(t) expresses the tendency to build or consume local bed topography oversome time period t, which depends on the magnitude and sign of Ub and qbc. The magnitude ofNt(t) depends on how the two velocities Ub and (La/qbc) compare, noting in particular that thebed speed Ub is governed by the degree of local topographic departure from an equilibrium,as will be shown below. When the velocity ratio is relatively large and positive, Ub drivesthe local bed elevation to increase through deposition, and when it is negative, Ub drives thelocal bed elevation to decrease through entrainment When the velocity ratio is relatively small,positive or negative, bed elevation change is negligible, and the local sediment flux Qs f totalmass is close to, or equivalent with the upstream sediment supply Qss. Last, based on ourassumption that qb ∼ qbc , the term ∂q∗b /∂x∗ < will have a value that generally ranges from 0–2or 3. Here, however, we are not concerned with this term.975.3. Morphodynamic evolution metrics at the scale of a channel widthNondimensional HiranoWe determine the particle response number Np in an analogous way to Nt, and nondimen-sionalize Equation 5.8 with the assumptions stated above, plus our assumption that fa ∼ fc:fcUcLc∂ f ∗a∂t∗≈ −σvqbcLc∂q∗b∂x∗, (5.12)where fc is a characteristic volume probability density of grain size class ψ. Whereas fa isdimensionless, defining f ∗a in terms of a characteristic fractional content fc is useful becauseit ultimately permits incorporation of a third grain size population into the problem, which isnecessary in order to represent all grain size sources that drive sediment texture adjustmentat the local scale, as we will discuss below. We next multiply Equation 5.12 by Lc/( fcUc), andobtain:∂ f ∗a∂t∗≈ − σvqbcfcUc︸ ︷︷ ︸1/Np∂q∗b∂x∗≈ − 1Np∂q∗b∂x∗. (5.13)Here, Np is the dimensionless particle response number, which is a ratio of two velocities:Uc · (Lc/qbc), modified by the degree of similarity between the fractional content of the localbedload, and the exchange surface at the base of the active layer (Figure 5.1c). In order tofurther characterize Nt, must specify reasonable choices for qbc , fc, and Uc. As above, weset qbc as the local sediment supply Qss, determined at maximum spatial scales of w′. For thecharacteristic fractional content fc we step away from the local scale, and define it as the basin-scale average upstream bedload supply composition fus. Use of a non-local quantity here isreasonable because over long periods of time (≈> 101− 102 years), fus sets the composition ofchannel reaches many w in length. As a result, local bed areas are ultimately adjusting to theupstream supply compositions over relatively long times.The characteristic speed Uc is defined in the context of streambed texture, which we specifyas the rate at which a characteristic grain size changes: Up. The characteristic grain size couldbe the D50, D90, or some combination of grain sizes, but here we quantify Up in terms of theD90 because of the earlier noted role it has for sediment mobility and transport. With thesedefinitions, and recalling that σv = δ1/La and δ1 = ( fa − fes)/ε, we recast Np as:Np(t) ≈(fusfb − fes)εLaUpqbc≈ δ2 εLaUpqbc, (5.14)where δ2 = fus/( fb − fes), which expresses the tendency for the local fractional compositionto fine or coarsen, depending on the sign of ( fb − fes).Physically, Np(t) expresses the tendency for bed surface texture to change over some timeperiod t, which depends on the sign of Up and qbc . The relative magnitude of Np(t) dependson how the two velocities δ2εUp and (La/qbc) compare, noting that the particle speed Up isgoverned by the degree of local texture departure from an equilibrium, which is quantified985.3. Morphodynamic evolution metrics at the scale of a channel widthby δ2. When the velocity ratio is relatively large and positive, Up drives the local bed surfacefractional composition of a given size class ψ to increase through deposition, and when itis negative, Up drives the local bed surface fractional composition of a given size class ψ todecrease through entrainment When the velocity ratio is relatively small, positive or negative,fractional composition change is negligible, and the local sediment flux Qs f fractional mass isclose to, or equivalent with the upstream sediment supply Qss fractional mass.Last, we highlight that δ2 has a strong affect on the value of the particle velocity δ2εUp.As an example, Figures 5.2a–5.2c show that δ2 generally ranges from (-20:+20) for a steadyupstream supply composition value fus = 0.3, and fb = 0, 0.5 and 1.0, and fes ∈ [0.05, 0.95].Notably, the largest values of δ2 occur when fb and fes approach equivalence (Figures 5.2b and5.2c). This is important because when fb and fes are near equivalence for an extended durationof time, local fractional composition conditions are likely approaching an equilibrium.fb = 1.0fb = 0f b = 0.5f b = 0.5fus = 0.3fb = 0fb = 1.0fb = 0.5fus = 0.3fus = 0.3(a) (b)Figure 5.2: Example values of δ2 of Equation 5.14 vs. (a) fb and (b) fb − fes, for parametervalues fus=0.3, fb= 0, 0.5 and 1.0, and fes ∈ [0.01, 0.99]. δ2 is asymptotic across theentire range of plausible values for the fractional variables fus, fb and fes.5.3.4 Dimensionless channel response number: NeThe hypotheses we presented within Section 5.2 suggests that equilibrium occurs where:∂η∗/∂t∗ ≈ ∂ f ∗a /∂t∗ (5.15)Therefore, we combine Equations 5.11 and 5.14 to introduce a channel response number:Ne(t) =(NtNp)=(1δ2)UbUp(5.16)Ne expresses a balance between the bed and particle velocities, Ub and δ2Up, respectively, andbuilds directly from Ahnert (1994)’s equilibrium proposal, for which we view Ub as the rate995.3. Morphodynamic evolution metrics at the scale of a channel widthof topographic adjustment, and δ2Up as the rate of texture adjustment. More specifically, Ubquantifies the time rate of change of the local bed elevation, and δ2Up quantifies the timerate of change of a local bed surface grain size fraction. Therefore, Ne(t) is a ratio of the twoadjustment rates which determine how local river segments change in response to upstreamsupplies of water and sediment.Since we are specifically interested in whether the time rate of change of topography, ortexture governs the local Ne condition, and given that both velocities can take positive ornegative values, we write Equation 5.16 as an absolute value:Ne ≈∣∣∣∣( 1δ2)UbUp∣∣∣∣ . (5.17)Equation 5.17 highlights that we expect equilibrium conditions where Ne ≈ O(1), which,importantly, is a function of the choices we made to assign scales to Uc and Lc.5.3.5 Calculations of δ2, Ub, Up and NeTo determine Ne, we calculate δ2, Ub and Up for all subsampling locations xj ∈ [4000:1000:13,000],and all observation times tn 1–29 (see Table 3.1), with data from pool-riffle experiment 1(PRE1). Development of the data we use in the calculations is presented in Chapter 3. Thespecific data sets we use are local bed topography derived from the digital elevation models,the local grain size distributions derived from the composite photographs, and the upstreambedload sediment boundary conditions. We now step through calculation of each quantity.Calculation of δ2Instead of focusing on a particular size class, such as the D50, or a combination of size classes,we determine δ2 by assuming that changes in the fractional composition of each grain sizepopulation fus, fb and fes, scales according to changes in the standard deviation of each popu-lation distribution, which contributes to ∂ f ∗a /∂t∗. Therefore, we calculate δ2 at a given locationxj and observation time tn as:δ2,xj,tn ≈sus(sb − ses) , (5.18)where s is the sample standard deviation for grain size fractions of the experimental sedimentsupply (sus), the local bedload supply (sb) and the local exchange surface (ses), for subsamplinglocation xj and for observational time tn. Note that because sb is the local bedload supply, itis determined from subsampling location xj−1, in a downstream moving reference frame. Wecalculate the sample standard deviation of the grain size fractions as:sxj,tn =√√√√ 1KK∑i=1( fi,ψ − f )2xj,tn , (5.19)1005.3. Morphodynamic evolution metrics at the scale of a channel widthwhere sxj,tn is the weighted sample standard deviation at subsampling location xj and time tn,K is the number of grain size fractions, fi,ψ is the fractional content for the experimental sup-ply, near-local bedload supply or the local exchange surface for grain size class ψ, and f is theaverage fractional content across all grain size classes. We choose to calculate δ2 with standarddeviations of the grain size populations, as opposed to a characteristic grain size (e.g. D50or D90), because s reflects how the grain size population for the local bedload and exchangesurface varies in time, as opposed to an explicit size class, which may change independentlyfrom the population. As a result, use of s to evaluate δ2 provides a complete and straightfor-ward test of dissimilarity between the three grain size populations. sus was constant in thecalculations.We do not have a direct measure of sb from PRE1 experimental data. We can howeverestimate sb based on quantifying the local bedload supply distribution fb at xj−1 and tn, whichdepends on how the composition changes from tn−1 to tn:fb,ψ,xj,tn = 0 if fb,ψ,xj−1,tn > fb,ψ,xj−1,tn−1fb,ψ,xj−1,tn−1 + ( fb,ψ,xj−1,tn−1 − fb,ψ,xj−1,tn) if fb,ψ,xj−1,tn < fb,ψ,xj−1,tn−1 (5.20)Equation 5.20 states that the local bedload fractions for any size class ψ, at subsampling loca-tion j, and for time tn:1. Goes to zero if the fractional composition of any size class ψ is enriched at xj−1 betweentn−1 and tn; and2. Is a linear combination of (a) the composition at xj−1 at tn−1 plus (b) the difference be-tween compositions from tn−1 to tn if the fractional composition is depleted.In the latter case, we assume that the bed surface supplies the additional fractional composi-tion, consistent with how we define the control volume (Figure 5.1b). After fb,ψ,xj,tn is calcu-lated for all grain size classes with Equation 5.20, the sum of fb,ψ,xj,tn may have a fractionalvalue less than 1, because of loss of fractional content. It is therefore necessary to renormalizethe fractions such that for N total grain size classes:N∑i=1fb = 1. (5.21)We then use Equations 5.19 and 5.21 with the values for fb to calculate sb.We do not have a direct measure of ses from PRE1 experimental data, but we can estimateit using the times series of bed surface sample standard deviations, sa, which permits us toconstruct a basic stratigraphic column at each subsampling location xj (Figure 5.3). The strati-graphic columns are time dependent, and account for cycles of deposition and entrainment,which means our approach is consistent with theory (Viparelli et al., 2010), given the resolutionof our data.1015.3. Morphodynamic evolution metrics at the scale of a channel widthElevation: 194.23 mmElevation: 202.10 mmElevation: 209.36 mm Station 10000 mm, te = 110 minutes Station 10000 mm, te = 50 minutes Station 10000 mm, te = 19 minutesFigure 5.3: Example bed sediment texture data used to build stratigraphic column at sta-tion 10000 mm for te = 19, 50 and 110 minutes. Note that these three times cor-respond to sequential periods of sediment deposition, and hence bed elevation in-creases. This temporal behavior corresponds to rule 1 discussed on page 102.We begin determination of ses by associating each value of sa with its corresponding relativeaverage bed elevation η, resulting in pairs of values at each subsampling location xj, and foreach time tn: (sa, η)xj,tn . At the end of this process we have 29 value pairs for each subsamplinglocation, and are ready to determine ses. Figure 5.1c indicates that the exchange surface liesat the interface between the subsurface and the active layer. Therefore, we use four differentrules to assign a value to ses at each subsampling location xj and for each observation time tn:1. For sequential observations of deposition during the initial experimental phase (see Fig-1025.3. Morphodynamic evolution metrics at the scale of a channel widthures 3.4 and 4.2) at any xj, the value of ses for time tn is assigned the value of sa for timetn−1. This means that the exchange surface distribution at tn is assumed to reflect the re-cently buried active layer distribution. An example of this rule is subsampling location10000 mm, which is shown as an example in Figure 5.3.2. For sequential observations of entrainment during the initial experimental phase at anyxj, the value of ses for time tn is assigned the value of sus, as long as the average bedelevation at time tn is lower than it was at time tn−1. This means that the exchangesurface distribution at tn is assumed to reflect the bed subsurface material that has notbeen reworked by the flow. This distribution matches the upstream supply distribution.An example of this rule is subsampling location 8000 mm3. For cycles of deposition and entrainment during the initial experimental phase at anyxj, the value of ses is assigned the value of sa for the nearest, preceding elevation that ishigher than the elevation at tn. This assumes that during the best matching precedingtime, the bed was built by a distribution that was uniform in time over the depositionaltime interval.4. For the repeat experimental phase (see Figures 3.4 and 4.2), stratigraphy constructedduring the initial phase is replaced by newly constructed stratigraphy as the bed subsur-face is reworked. Otherwise, we apply the previous 3 ses assignment rules depending onaverage bed elevations dynamics.We then determine δ2 with the values for sus, sb and ses using Equation 5.18.Calculation of UbWe determine the rate of topographic adjustment Ub for each subsampling location and obser-vational time as:Ub ≈η¯xj,tn − η¯xj,tn−1tn − tn−1 ≈∆η∆t∣∣∣∣tn :tn−1, (5.22)where η is the local average bed elevation at t = n and t = n− 1.Calculation of UpWe determine the corresponding grain size adjustment rate Up as:Up ≈D90us − D90xj , tntn − tinit ≈∆D90∆t∣∣∣∣tn :tr, (5.23)where D90us is drawn from the experimental sediment supply, and tinit is the starting time foreach flow sequence of PRE1 (Table 3.1; Figure 5.4b). Therefore, tn− tinit is a cumulative time foreach flow sequence. Equation 5.23 reflects the fact that the local bed surface texture ultimatelyadjusts to the composition of the upstream sediment supply as equilibrium is approached.1035.4. ResultsCalculation of NeWith Equations 5.18, 5.22 and 5.23, we approximate the channel response number for PRE1 as:Ne ≈∣∣∣∣∣(sb − sessus)tn∆ηtn :tn−1∆D90,tn :tinit∆ttn−tinit∆ttn−tn−1∣∣∣∣∣ (5.24)As applied here, Equation 5.24 states that the local channel response is the product of threeratios:1. The first ratio quantifies the magnitude of difference between local grain size distribu-tions and the upstream supply distribution;2. The second ratio quantifies the magnitude of difference between the speed of topo-graphic and surface roughness adjustment; and3. The third ratio quantifies the difference between the surface roughness adjustment timescale, and the time increment of observation.From the three ratios, it is evident that as equilibrium is approached, the grain size and speedratios decline in magnitude, and the time scale ratio increases.5.4 ResultsFigure 5.4 illustrates evolution of the sediment flux Qs f (b), Nt (c), Np (d), δ2 (e) and Ne (f) forPRE1. Each quantity is plotted against the dimensionless time to, defined as the ratio of theelapsed time te to the activation time ta. In the top panel (a) we show the supply of water (Qw),and the light and darker gray vertical fill areas in panels (b)–(f) indicate when supplies change.Within panel (b), we show the times associated with the initial and a repeat experimentalphases, which extend from to = 0–23.9, and to = 23.9–43.5, respectively; at the top of panel (b),we indicate the associated response regimes as either the start-up (ta), transient (tt), steady-state or response to steady-state (tr) periods. The character of each regime was presented inSection 4.3.1, and here we provide a summary recap of each regime, and follow that withdescription according to Nt, Np, δ2 and Ne. For Nt (c), Np (d), δ2 (e) and Ne (f), we show themean as well as 10th and 90th percentile value trends; for Ne (f) we also provide the medianand geometric mean value trends.The start-up period defines the beginning of PRE1, running from to = 0 to to = 1, termedthe activation time (ta) (Figure 5.4b). ta characterizes initial redistribution of sediment alongthe experimental channel in response to coupling between downstream changes in (1) widthand (2) average flow speed. The transient regime (tt) reflects the adjustment of bed topogra-phy and sediment texture to the upstream supplies, leading to the first steady-state, definedabove in Section 5.2 as average topographic steadiness, and mass balance. The initial phase(tt) extends from to = 1–19.5. A total of six SSs occur during PRE1, at to = 19.5, 21.7, 23.9, 39.2,41.4 and 43.5. The response to steady-state tr captures how the channel responds to changes1045.4. Resultsin upstream water and sediment supplies from a SS condition. A total of five tr periods occurduring PRE1 (Figure 5.4b).5.4.1 Topographic and sediment texture response numbers: Nt and NpNt drops one-order of magnitude during the activation time from a high of≈ 10−2, and by theend settles into an approximately steady, to a slight decreasing condition through the initialphase transient period (Figure 5.4c). Just before steady-state at to =19.5, Nt drops to a low ofroughly 10−3.2 (mean response). Nt rises abruptly to a peak during the first response to a SSperiod, and then drops almost as rapidly to the second steady-state condition. Again, Nt risesduring the second response to SS period, but not as rapidly as during the first, reaches a peak,and then drops to a value similar in magnitude to that at the end of the first response period,marking the third steady-state. The 10th and 90th percentile responses are consistent withthe mean response during the initial phase, and bound a response range that spans roughlyone-order of magnitude. The repeat phase behavior of Nt is similar to that of the initial phase,excluding the start-up period, including the strong narrowing of overall response about half-way through, similar SS values, and within a comparable value range. The difference is that10th percentile response departs from the mean at to = 28 (Figure 5.4c).Np drops more than one-order of magnitude during the activation time from a high of or-der 10−2, and continues to steadily drop during the initial phase transient period to a low of10−3.2 (Figure 5.4d), marking conditions at the first SS (mean response). Np rises strongly to apeak during the first response to SS period, and then drops to the second steady-state condi-tion. Again, Np rises during the second response to SS period, reaches a peak, and then dropsto the third steady-state, and to a value almost one-order of magnitude higher compared tothat of the first SS. The 10th and 90th percentile responses are consistent with the mean re-sponse during the initial phase, and reflect a tight variational range of 1.5 to 2Np. The repeatphase behavior of Np is similar to that of the initial phase, excluding the start-up period, in-cluding comparable SS values. A key difference is that the repeat phase first response to SSperiod ends (to = 39.2) at a value about 2Np higher than the value at the first SS (Figure 5.4d).5.4.2 Sediment texture δ2δ2 mean is unresponsive during the activation period, maintaining a value of roughly 100(Figure 5.4e). The 10th and 90th percentile values drop and increase, however. δ2 rises quicklyat the beginning of the initial phase transient period, reaching a peak of 100.8 at to=2.2 (meanvalue). Following this peak δ2 drops to a value of 100, where it remains until the first SS.Relatively small increases occur after the first and second SSs, with equally minor decreasesto the next SS conditions. The repeat phase behavior is similar to the initial phase during thefirst response to SS, but exhibits larger responses during subsequent response to SS periods,comparatively After the fourth SS, δ2 rises rapidly to a value of 100.7, then falls to a value atthe fifth SS consistent with the second SS. After the fifth SS, δ2 rises rapidly to a value of 101,1055.4. Results80 Experimental Water Supply Rate (l/s)6042(a)(b)activation timesediment supplysediment fluxrepeat phaseSediment flux and feed (kg/m)0 550 1100 1650 2200 2750 3850 4400 49503300NtNp(e)steady-stateta tt tr tr 6.5tr tr tr(c)(d)90th percentile (dashed)mean (solid)10th percentile (dashed)equilibriumequilibriummean (solid)median (dashed)geometric mean (marker)101100103102101Ne(e)(f)initial phase10-1to: elapsed time/activation time (-)te: elapsed time (minutes)100Figure 5.4: Summary of Qs f , Nt, Np, δ2 and Ne for PRE1, vs. the dimensionless time to,defined as the ratio of elapsed time te to the activation time ta. Nt determined with5.11, Np and δ2 with 5.14, and Ne with Equation 5.24. (a) Water supply rate. (b)Sediment supply rate and flux. The initial experimental phase occurred from di-mensionless time 0 to roughly 24, and the repeat phase from 24 to 43. Activation(ta) and transient (tt) periods indicated at the top of (b). Steady-state occurred on sixseparate occasions during PRE1, as indicated by the position of the vertical dashedlines. Evolution of (c) Nt, (d) Np, (e) δ2 and (f) Ne during PRE1. Evolution con-sisted of start-up, transient, steady-state and response to steady-state phases. Seetext for explanation of each phase. The solid lines in (c), (d) and (e) are the meansfor 29 observational times across PRE1. The dashed lines below and above the solidlines shown in (c), (d) and (e) illustrate the 10th and 90th percentile responses, re-spectively. The mean (solid line), median (dashed line) and geometric mean (linewith marker) responses of Ne. The range of Ne responses between the 10th and 90thpercentile values is also provided as the red shaded region.1065.4. Resultsone order of magnitude higher than that following the second SS, and then falls again to avalue similar to the third SS. The 10th and 90th percentiles generally cover a value of range ofone-half order of magnitude during the initial and repeat phases (Figure 5.4e).5.4.3 Channel response number: NeBehavior of the channel response number Ne is generally in contrast to Nt and Np, and issensitive to δ2, when it is large relative to the ratio (Nt/Np). Ne begins the activation periodwith a mean value of 100.3, and ends with a mean value of approximately 100.1. From thestart of the initial phase transient period, Ne rises steadily until to=8.6, drops somewhat untilto=10.8, and then increases to a peak mean value of 23.0 at to=17.4, with a correspondingmedian value of 21.9, and geometric mean of 20. Following this peak, Ne declines steadilyto the end of the initial phase transient period, and occurrence of the first SS at to = 19.2,achieving a mean value of 5.3, a median of 6.7 and geometric mean of 3.6.Ne continues to decline through the first response to SS period, and has a mean value of2.4 at the second steady-state, a median of 1.6 and a geometric mean of 1.5. The decliningtrend continues to to=22.3, achieving a low during the initial phase with a mean value of1.7, and a median and geometric mean of 1.0. Ne then rises thereafter through most of thesecond response to SS period of the initial phase, with a mean value of 2.5 at the third SS,a median of 2.0 and a geometric mean of 2.0. The repeat phase exhibits generally similarbehavior compared to the initial phase, rising to the middle of the first response to SS at to =32.6, and falling thereafter to a repeat phase low at the fifth SS, with values of O(1). Thechannel response number at the last SS achieves a slightly higher value to that for the 3rdSS, 4.5 vs. 2.5, respectively. More specifically though, there are a few differences betweenthe initial and repeat phases. Ne peaks earlier during the first response to SS period of therepeat phase, as compared to the initial phase transient period, and Ne responds during thelast response to SS of the repeat phase with an up–down–up cycle, which is not exhibitedduring the second response to SS.The 10th and 90th percentile value range generally approaches or slightly exceeds one-order of magnitude during most of PRE1, and rises and falls with the mean values (Figure 5.4f).Two episodes of greater than one order of magnitude value range occurs during PRE1, oneeach during the initial and repeat phases: to = 10.8–13.0 and to = 28.3–30.5. We suggest thatequilibrium conditions occur around the second, and fifth and sixth SSs, during which timeNe ≈ O(1). Furthermore, steadily declining Ne values leading up to both equilibrium casesindicates conditions throughout the experimental channel were evolving toward equivalencebetween the topographic and texture velocities, Ub and δ2Up, respectively.5.4.4 Results summaryIn summary, results show that Nt and Np respond in a consistent way during PRE1, evolvingfrom highs at to = 0, to lows at the first steady-state. Nt and Np are responsive to flow and sed-1075.5. Discussioniment supply increases, generally showing rapid rises to peaks immediately following supplychanges, with steady decline thereafter to the next steady-state. Nt and Np show overall con-sistency between the initial and repeat phases. δ2 on the other hand responds along generallydifferent trajectories compared to Nt and Np. Immediately following the activation period, δ2peaks, falls rapidly, and then maintains basically a uniform response until the fourth SS, atwhich point it exhibits an up–down–up–down cycle that ranges over one order of magnitude.Ne integrates these three signals into a generally predictable response pattern, rising tohighs during the relatively long first transient and third SS response periods, and thereaftertrending toward O(1) values, which suggests equilibrium conditions during the subsequentSS response periods.5.5 DiscussionThe combined results of Figure 5.4 indicates that our mass balance derived topographic, par-ticle and channel response numbers for river segments of approximate length w′, Equations5.11, 5.14 and 5.24, respectively, provide reasonable results and may be of use in a broadercontext. Specifically, the results highlight two questions: First, how do the bed and particle ve-locities, Ub and δ2Up, respectively, contribute to equilibrium conditions during developmentof the pool-riffle and roughened channel structures discussed in Chapter 4? And second, howdoes our view of fluvial equilibrium in terms of Ne help clarify equilibrium and distinguish itfrom SS?5.5.1 Local contributions to equilibrium conditions during pool-riffle androughened channel development and maintenanceFigure 5.4f shows that the time rate of change of topography governs pool-riffle and rough-ened channel development for PRE1 during the initial phase transient period tt, and the repeatphase first response to steady-state period 6.5tr. During these times, the average mobility con-dition τ/τre f is characteristically low, ranging from 1.20–1.60 (Table 4.1), where τ is the averagebed stress, calculated as τ = ρwCdU2x, and τre f is the reference critical mobility stress for thebed surface median particle diameter D50 (see Chapter 3.2.2 for complete description). Whentopography governs the equilibrium condition Ne, Ub ranges from a factor 2–10 times largerthan Up, given that δ2 is generally uniform with an approximate value of 1 throughout (Fig-ures 5.4c, 5.4d and 5.4e). Hence, at lower relative mobility conditions, the primary adjustmentresponse to pool-riffle and roughened channel development is through the construction oftopography, and bed surface texture, or roughness, plays a less important role.A different dynamic, however, is evident at higher relative mobility conditions τ/τre f (Ta-ble 4.1), following the SSs that result from the initial phase transient period tt, and the repeatphase first response period 6.5tr (Figure 5.4). During the first and second SS response peri-ods, the bed and particle velocities approach equivalence, and maintain a ratio of roughly 3.0.At to = 22.3, however, a mean value of 1.7 is observed, which from the mean trend of the1085.5. Discussionpreceding data, suggests that conditions are evolving to equilibrium. The mean trend of δ2throughout this time generally fluctuates around a value of 1.0, highlighting that adjustmentsof the grain size fractions which contribute to the bed surface texture fa remains uniform, anddoes not affect Ne.During the fourth and fifth SS response periods, the bed and particle velocities again ap-proach equivalence, but for these times δ2 has a large effect on the equilibrium conditions Ne.Figures 5.4c, 5.4d and 5.4e show that when topographic and particle velocities spike, δ2 alsospikes, because the local bedload supply and exchange surface sample standard deviations arecomparable. When the topographic and particle velocities drop, δ2 does as well, and tends tovalues of 1.0. It is during these relaxations that Ne also approaches a value of 1.0, indicatingthat equilibrium is achieved for the mean trend.These findings are important, because in contrast to the lower mobility conditions, athigher mobility the adjustment response to pool-riffle and roughened channel maintenanceis shared equally between topography and bed surface texture. Specifically, texture adjust-ment occurs primarily through bed surface coarsening, with an increase in the D90 grain sizethroughout much of the experimental channel for the higher mobility periods (see Figure 4.2).This response is expected within channels of gravel and cobble composition, due to preferen-tial entrainment of finer grain sizes to balance mobility over the entire distribution (e.g. Parkerand Klingeman, 1982), or when the upstream supply of sediment is reduced (e.g. Dietrich et al.,1989). In the present case, both of these effects play a role.5.5.2 Local and channel response numbers: a new view of fluvial equilibriumWe conceptualize the local processes that contribute to equilibrium conditions along relativelystraight mountain streams as depositional and entrainment filters, which alter the upstreamsediment supply by driving local storage or depletion of sediment particles, until flow andmobility conditions support mass continuity (Figure 5.1). Because the depositional filter re-lates to the local bed topography, and the entrainment filter to the local bed surface texture,we constructed our view of fluvial equilibrium from mass conservation considerations for thebulk riverbed, and the particle fractions making up the riverbed. Our construct is consistentwith Ahnert (1994)’s focus on mass budgets, and also builds from Howard (1982)’s emphasison equilibrium, or grade, reflected in mutual adjustments of the channel gradient and flowcharacteristics.From scaling η, t, qb, x and fa, we propose that local equilibrium and disequilibrium is de-scribed by the ratio of the bed and particle velocities, Ub and δ2Up, respectively. Our proposalbuilds on the existing view that equilibrium is a problem of sediment continuity, expressedby longitudinal profiles that are just capable of transporting the upstream sediment supply,known as the graded river profile (Gilbert, 1877; Mackin, 1948). Whereas we use this view tobuild Equation 5.24, we augment it to account for textural adjustments, or sorting processes,which matter at the local scale. Recognizing the importance of texture and sorting to fluvial1095.6. Conclusionsequilibrium complements the analogous emphasis detailed by Paola and Seal (1995) for theproblem of downstream sediment fining.Taken together, the results presented in Figures 4.8, 4.11 and 5.4 show that the occurrenceof equilibrium defined by Equation 5.24 is characterized by channel width conditions thatvaries by a factor 2, a grain size distribution that spans 0.5–32 mm, and for local bed slopesSlocal that range on average from adverse downstream values of 3%, to positive downstreamvalues of almost 6%. These varied morphodynamic conditions are a solid test of our formalequilibrium statement, defined as Ne = Ub/δ2Up. More importantly, our new view may offera link to help bridge established work on equilibrium at larger spatial scales Sternberg (1875);Gilbert (1877); Mackin (1948); Langbein and Leopold (1964); Snow and Slingerland (1987); Paolaet al. (1992); Bolla Pittaluga et al. (2014); Blom et al. (2016), with that focused on local conditions(Ferrer-Boix et al., 2016). First, our framework can be incorporated into numerical models toevaluate equilibrium at a wide range of spatial and temporal scales, and for an equally broadset of driving conditions. Second, it is possible that our approach can yield estimates of thetime scale needed for a river reach of many channel widths to reach equilibrium. Although,specifying this time scale may require different choices for the characteristic scales, the exercisewould yield an outcome that augments the basin-scale equilibrium times presented by Howard(1982); Paola et al. (1992).5.6 ConclusionsWe use theory and scaling arguments to demonstrate that the ratio of the bed and sedimentparticle velocities, Ub and δ2Up, defines fluvial (dis)equilibrium conditions, and that equi-librium, specifically, is achieved when the velocity ratio is O(1). For PRE1, this occurs forrelatively high sediment mobility conditions, when the rate of bed surface coarsening is com-parable to the rate of topographic adjustment, and notably does not include a significant affectfrom the local sediment sorting state δ2. This finding suggests an important point which con-nects with findings from Chapter 4: equilibrium along channel reaches of variable width canoccur under conditions of spatially non-uniform bed surface sediment texture. This high-lights the intrinsic link between local bed topography and surface roughness, which adjustsand emerges in response to locally-driven flow velocity conditions. Hence, equilibrium is notrestricted to spatially uniform bed surface sediment distributions.Our results further indicate that for lower sediment mobilities, local adjustments of bed to-pography govern the (dis)equilibrium condition, and that this affect continues after mass con-tinuity is achieved. Notably, the post-continuity period of topographic influence on (dis)equilibriumoccurs for time scales that are at least 30% of the time necessary for continuity to emerge. Thisfinding motivates an important expectation: watersheds with rainfall dominated hydrologywill process water and/or sediment supply perturbations relatively quickly, vs. watersheds ofsnow-melt driven hydrology, for which equilibrium time scales are comparatively long. Thelag between mass continuity and equilibrium also highlights a key characteristic of fluvial1105.6. Conclusionsprocesses, which is obscured by defining statistical steady-state in terms of continuity. Aftercontinuity is achieved, processes that build bed shapes and topography, as well as bed sed-iment textures, are still at work, further adjusting to local conditions, albeit, at smaller andsmaller rates, as (dis)equilibrium conditions trend toward a velocity ratio of O(1).Last, PRE1 sediment flux Qs f exhibits consistent rates of relaxation toward continuity fol-lowing four separate episodes of water and sediment supply increases. General rate consis-tency suggests and e-fold time of to ≈ 0.5 or less, or approximately 1 hour under experimentalconditions. Using that the e-fold time is the time necessary for a perturbation to diminish by∼ 37% of its initial value (Slingerland and Kump, 2011), the supply perturbations of PRE1 arecompletely processed within to ≈ 2.0–2.5, or 6 to 8 hours under high mobility experimentalconditions. With a time scaling of Lr/Ur ≈ 2.5, which is based on Froude scaling discussedin Chapter 3, we obtain an approximate perturbation relaxation time scale of order 20 hoursfor high mobility conditions, and natural cases of comparable dimension and character to theEast Creek field site. In Chapter 2 we found that bed sediment texture conditions of a pool-riffle pair along a small mountain stream in the Santa Cruz Mountains, CA, U.S., requiredthree floods to recover from a sediment texture disturbance driven by an approximate 20-yearflood (Table 2.4). Sediment mobilization durations for typical rainfall hydrographs along thissmall mountain stream are less than 24 hours, highlighting reasonable consistency betweenexperimental and field conditions.111Chapter 6Concluding remarksThe aim of my thesis was to build understanding of how pool-riffles form and are maintainedalong channel segments of variable width. I was specifically interested in characterizing themechanical coupling between width conditions, local bed topography and bed surface texture.Accordingly, one of my main goals was to build a physical explanation for the widely reportedspatial correlations between channel width and pools at points of narrowing, and riffles atpoints of widening. In this chapter I provide a summary of the main findings from my work,and I discuss future directions motivated by these findings.6.1 SummaryChapter 3 presents the use of non-parametric statistics and self-organizing maps (SOM) to ex-amine and characterize riffle texture trends across 11 sediment mobilizing floods. The subjectpool-riffle has occurred in its present form and location for at least 14 years, and therefore thework provides insight on pool-riffle maintenance through riffle texture adjustment. I showthat net riffle head elevation is stationary during the study period, and that riffle texture re-sponds to each flood, but generally maintains a coarse riffle center of gravels (2–64 mm) andcobbles (64–256 mm), and finer lateral margins of sand (<2 mm), with one exception. Thelargest flood drove fining of much of the riffle surface, as demonstrated by the McNemarcontingency test completed for each sampling transect, and the SOMs. Riffle fining persistedfor two subsequent floods, and the third event saw texture conditions recover to those mea-sured at the beginning of the field study. I also show that pool sediment storage responds toeach flood, exhibiting a range of conditions. The major result and finding of this study is thatnon-parametric statistics and SOM can be used to characterize the role of riffle texture adjust-ment for pool-riffle maintenance. Topographically stationary pool-riffles are maintained, inpart, through riffle textural adjustments which occur at a frequency equal to that of sedimentmobilizing floods. Furthermore, textural adjustments are spatially and temporally organized,suggesting that sediment transport patterns are conditioned at the scale of a pool-riffle pair.In Chapter 4, I build from the Chapter 3 field study and present experimental and theoret-1126.2. Future directionsical results on pool-riffle formation along a variable width channel. The experiments producepool, riffle and roughened channel bed structures, which I show to colocate with channel seg-ments which are narrowing and widening, and which exhibit negligible width change, respec-tively. Continuity shows that flows accelerate to construct pools, decelerate to construct riffles,and generally uniform flows yield roughened channels. I demonstrate that local channel bedslopes for my experiments, one additional experiment and a numerical study are organizedalong a systematic trend across the range of width gradients from (-0.30)–(+0.30). Notably, thereach-average bed slopes for the three data sets vary by one order of magnitude. These resultsmotivated development of a mathematical model which indicates that slope construction iscontrolled by the ratio of two velocities. The velocity ratio Λ represents two characteristictime scales: the time scale for bed surface sediments to spread a length scale L under gravity,vs. the time scale of momentum flux delivery to the bed surface. Application of the model tothe local bed slope data sets reveals that pools form when the spreading time scale is charac-teristically large, and riffles form when it is small. Each formative condition is associated witha suggested threshold width gradient.In Chapter 5, I present a new definition of fluvial equilibrium based on mass conservationstatements for the bulk riverbed, and the particle size fractions which comprise the riverbed.The final equilibrium statement is obtained by nondimensionalizing the conservation state-ments and specifying scales for elevation, time, length, bedload transport and bed surfacegrain size fractions. Similar to Chapter 4, I find that fluvial equilibrium conditions are de-scribed by a ratio of two velocities. The velocity ratio Ne, or equilibrium condition, repre-sents the speed at which the bed surface increases or decreases in height, and the speed atwhich bed surface grain sizes change fractional composition, through grain size populationfining or coarsening. Application of Ne to my experimental data shows that these two speedsmore readily exhibit comparable values, and hence the local channel is in equilibrium, whensediment mobility conditions are relatively high. This result is consistent with findings fromChapter 3, which highlights the importance of riffle texture adjustment for pool-riffle mainte-nance. Last, Ne permits variable bed surface grain sizes between pools, riffles and roughenedchannel segments at equilibrium, which, foregoing the equilibrium aspect, is consistent withmany observational data sets of natural streams Lisle (1979); de Almeida and Rodrı´guez (2011);Caaman˜o et al. (2012); Hodge et al. (2013); Papangelakis and Hassan (2016).6.2 Future directionsApplication to river restoration designThe results of Chapter 4 have implications for river restoration design strategies and ap-proaches. Specifically, Figure 4.11 suggests that it is possible to determine the average residuallength scale of pools, riffles and roughened channel segments for variable width channels, rel-ative to the reach-average longitudinal bed slope. Having a simple method to predict the aver-1136.2. Future directionsage residual length scale will benefit restoration design, because it helps to reduce uncertaintyinvolving the average bed profile conditions of a particular design concept. The common wayto achieve this presently is through numerical simulations, which are costly and time intensiveto perform. Furthermore, a simulation is restricted to a particular design concept, as opposedto a range of possible conditions.AccelerationDeceleration-0.04 0.04Flow Speed Change (-)RifflePoolPool RegimeRiffleRegimeFlow(a)(b) (c)Figure 6.1: Observations of residual depths for PRE1 determined from zero-crossing pro-files. (a) Average bed topography for the six PRE1 steady-state conditions with theaverage change in downstream flow speed, depicted by the circle colors. Flow speedchange plotted half-way between bounding differencing stations. (b) Residual depthfor 10 locations shown in (a), determined from the zero-crossing profiles computedfor observations 1–29 (Table 3.1). Error bars are the sample standard deviation forthe observations. (c) Photography of a pool-riffle along East Creek, near MapleRidge, BC. Pool-riffle location spatially correlated with narrow and wide channelsegments, respectively.In Figure 6.1, I show the average downstream change in the residual length scale ∆η vs.the downstream change in width, determined for subsampling locations of PRE1. The residuallength scale was determined from the average zero-crossing trend line for PRE1, and residuallengths are normalized by the average total elevation loss across the PRE1 experimental chan-nel. In Table 6.1 I provide the normalized residual lengths for each subsampling location ofPRE1. Figure 6.1 shows that the normalized downstream change in residual length varies di-rectly with the downstream change in channel width, suggesting the residual lengths can be1146.2. Future directionsTable 6.1: Mean values of normalized residual depth ηˆx for subsampling locationsSubsampling Locations3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 130000.1047 -0.0298 0.0140 0.0086 -0.0765 -0.1361 0.1108 0.1197 0.0537 0.0780 0.03471. ηˆx calculated from observations 1–29 for PRE12. ηˆx normalized by the average total elevation loss across the PRE1 experimental channel.4. ηˆx calculated from the associated average zero-crossing trend line for PRE1.projected for a given width condition. The normalized change in residual length scales rangesfrom -25% within the pool at station 8000 mm, to roughly +10% in the riffles at station 10000and 3000 mm. My goal is to use the theory presented in Chapter 4 to develop a straightforwarddesign methodology for pool-riffles of variable width channels.0 11nnBed TimescalePar�cle Interac�ons IncreasesBedload Transport Intensity Increases Pool Development Surface Texture Adjustment Riffle/BarDevelopment ?Fluid TimescalePar�cle TimescaleFluid TimescaleFigure 6.2: Concept mountain streambed architecture regime diagram. The x-axis is theratio of the flow to the particle time scales and the y-axis is the ratio of the flow tothe bed time scales.Identifying a mountain streambed architecture regime diagramEquations 4.7 and 5.16 highlight that the dynamics of pool-riffle formation, and equilibriumcondition are expressed through characteristic time scales, which describe how the bed, parti-cles and the flow participate in development of channel bed architecture. Furthermore, fromFigures 4.11, 4.12 and 5.4 we understand that pools, riffles and roughened channel structuresexhibit a likely range of time scales that are particular to each channel bed response. As a1156.2. Future directionsresult, we hypothesize that the channel bed, particles and flow time scales can be used todevelop of mountain stream bed architecture regime diagram. A concept of the diagram isshown in Figure 6.2, where I plot the ratio of the flow to the particle time scales on the x-axis,and the ratio of the flow to the bed time scales on the y-axis. My hypothesis proposes that:1. Riffles occur when the flow time scale is small relative to the particle or bed time scales.This regime is characterized by minor net bedload transport intensities of partial mo-bility, or Stage II or Phase II transport conditions per Ashworth and Ferguson (1989) andHassan et al. (2005), respectively.2. Roughened channel segments with patches, clusters and surface structuring occur whenthe flow and particle time scales are comparable, but the flow time scale remains smallrelative to the bed time scale. Therefore, particle-particle interactions are relatively large,but bedload intensity remains in the partial-mobility condition.3. Pools occur when the flow, particle and bed time scales are comparable, providing rel-atively high particle-particle interactions, and high rates of bedload transports, or StageIII or Phase III transport conditions per Ashworth and Ferguson (1989) and Hassan et al.(2005), respectively.I have developed flow time scales from the Navier-Stokes Equation, with the next step todetermine particle and bed time scales from the work of Chapters 4 and 5.Figure 6.3: Photograph of an inverted pool-riffle channel segment along the Bridge River,near Camoo Creek Road, BC, Canada. A landslide during the summer 2015 deliv-ered coarse sediment and willow trees to the channel, narrowing the channel at thedelivery point by 60%, and resulting in an upstream backwatered pool, and rifflethrough the landslide deposit.Identifying a unified pool-riffle formation hypothesisThe work of Chapter 4 lays the groundwork to develop a unified pool-riffle formation hy-pothesis, and on which accounts for both the entrainment- and depositionally-driven forma-tive mechanisms. Furthermore, the formative hypothesis would need to account for the not1166.2. Future directionsuncommon observation of pools at points of widening, and riffles at points of narrowing, asshown in Figure 6.3, which I call inverted pool-riffle topography. 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Hassan (2008), Video-based gravel transport mea-surements with a flume mounted light table, Earth Surface Processes and Landforms, 33(14),2285–2296.130Appendix ANumerical channel evolution modeldescriptionA.1 SummaryA de-coupled numerical model of channel bed evolution was developed following Parker(2007) and Wu (2008). The model consists of four components, which together describe thebasic physical processes governing channel evolution:Step 1: 1D non-uniform hydrodynamics (GVF Solution to Saint Venant Equations);Step 2: Mixed grain sediment transport (Wilcock-Crowe, 2003);Step 3: Diffusive bed evolution (Exner Equation);Step 4: Bed sediment sorting and properties (Hirano Type Exner Equation).The model uses a finite differences scheme and bed evolution is solved sequentially over a1D domain in the order presented above, with each step completed prior to moving to thenext step. At the end of the four steps the model advances to the next time step and thecalculations are completed again. Step 1 determines the shear velocity u∗ and the sedimentparticle mobility conditions τ∗. Step 2 determines the sediment transport rate using u∗ andτ∗. Step 3 determines the change in bed elevation over the 1D domain based on the sedimentflux gradient in x. Step 4 determines the fractional composition of the bed surface and fluxat each node based on the mass balance of each size fraction, and from this information thebed roughness is determined. Steps 1 through 4 provide the conditions for the next time step.The model is considered an approximate formulation of channel evolution, useful to exploreand simulate the problem in a manner consistent with the assumptions used to develop thegoverning equations for each of the four model components. Principal aspects of channelevolution not addressed by the model include:131A.2. 1D nonuniform hydrodynamic model1. Flow patterns characterized by lateral and/or vertical motions;2. Flow properties associated with time independent turbulent motions;3. Bedload sediment transport patterns defined by lateral motions;4. Interactions of flow with a rough channel bank;5. Bed stratigraphy for cycles of erosion and deposition;A.2 1D nonuniform hydrodynamic modelFollowing Parker (2007), the hydrodynamic model solves the 1D mass conservation and steady,nonuniform Saint-Venant equations for incompressible flow of constant density. The 1D watermass conservation statement is:∂Uxd∂x= 0, (A.1)where Ux is the downstream average flow velocity, d is the average water depth and x isdistance along the channel. The 1D momentum conservation statement is:∂U2xd∂x= −0.5g∂d2∂x− gd∂η∂x− C f U2x , (A.2)where g is the acceleration of gravity on Earth, η is the channel bed elevation and C f is adimensionless bed resistance coefficient. Equation A.2 is simplified with Equation A.1, and byintegrating Equation A.1, which yields Ud = qw = constant (unit flow rate), and:∂U∂x=qwd∂d∂x(A.3)Applying Equation A.1 to A.2 provides:Ux∂Ux∂x= −0.5g ∂d∂x− gd∂η∂x− C f U2xd(A.4)Applying Equation A.3 to A.4 and rearranging provides the backwater solution to the Saint-Venant Equations:∂d∂x=S− S f1− Fr2 , (A.5)where S = −∂η/∂x (channel bed slope), S f = C f Fr2 (friction slope or slope of the total energyline) and Fr2 = (q2w/gd3) = (U2x/gd) (Fr is the Froude number). For uniform flows, S = S f .Equation A.5 is a first-order differential equation requiring one boundary condition in d. Useof the backwater solution assumes that (1) the flow is gradually varying in x (i.e. the lengthscale of the pressure term (Lx) is therefore assumed to satisfy H/Lx << 1), and hence thepressure force can be approximated as hydrostatic, with the channel bed surface defining the132A.3. Mixed grain sediment transport modellocal datum; and (2) the viscous stress is negligible, but that a friction force acts at the bed–water interface on the overlying fluid through a drag relation: C f U2x .Equation A.5 is solved numerically using the Standard Step Method (STM), with a conver-gence tolerance of 0.0008 m. The backwater solution does not adequately simulate water depthalong pool segments with adverse channel bed slopes. I address this issue with a simple nu-merical approximation that I call the reduced complexity adverse slope algorithm (RCASA).RCASA locates segments along the numerical domain which exhibit adverse bed slopes, andthen it identifies the downstream bed elevation for each segment where the bed slope changessign. This location is the controlling location, and the bed elevation here determines the resid-ual water depth in the upstream pool until the controlling bed elevation is exceeded at someupstream location x. The water depth through the pool is approximated as the normal depth(ho) at the controlling location plus an incremental depth based on an estimated pool watersurface slope. The normal depth is determined with the Manning-Strickler formulation:ho = k 13s q2wα2r gS 310 , (A.6)where αr is a dimensionless quantity assigned a value of 8.1 (Parker, 2008), ks = nkD90 is ameasure of bed roughness using the 90th percentile grain size class, and nk = 2 (Parker, 2008).The pool water surface slope is set by observed associated slopes from pool-riffle experiment1 (PRE1), which for 42 ls−1 ranged from 0.5–1%. Lastly, Equation A.5 is limited by cases whenthe Froude number≥ 1. To address this shortcoming, water depths for Froude numbers higherthan∼0.9995 are approximated by the quasi-normal momentum balance (Cui and Parker, 1997),for which S = S f .A.3 Mixed grain sediment transport modelThe model uses the Wilcock and Crowe (2003) sediment transport function to determine frac-tional rates of bedload transport for the PRE1 bed surface sediment mixture. The dimension-less W-C function for any grain class i is defined as:W∗i =0.002φ7.5 φ < 1.3514(1−0.894φ0.5)4.5φ ≥ 1.35,(A.7)where φ is a stress ratio(τ/τrψ)of which τ is the average bed stress and τrψ is the averagemobilizing reference stress for any size class ψ of the bed material, where ψ = log2(Di) andDi represents the mean diameter of each grain size class according to the Wentworth scalefor half-ψ increments. The average bed stress was determined with the drag equation given133A.3. Mixed grain sediment transport modelabove, and C f was determined using the Manning-Strickler formulation:C f =1α2r(dks)− 13, (A.8)where αr is a dimensionless quantity assigned a value of 8.1 (Parker, 2008), d is the averageflow depth determined with the backwater solution discussed above, ks = nkD90 is a measureof bed roughness using the 90th percentile grain size class, and nk = 2 (Parker, 2008). Theaverage mobilizing reference stress is dependent upon a hiding function (Di/D50) and theaverage mobilizing reference stress for the mean grain size of the surface material(τrs50) isdefined as:τriτrs50=(DiD50)bw, (A.9)where the exponent bw is a fitting parameter dependent on Di/Dsm:bw =0.671+ exp(1.5− DiDsm) , (A.10)and Dsm is the mean grain size of the bed surface material. Wilcock and Crowe (2003) [2003]demonstrated that the average mobilizing reference stress for the mean size class of the surfacematerial (τrm) is dependent upon the surface sand content:τrm = (ρs − ρw) gDsm [0.021+ 0.015 exp (−20Fs)] , (A.11)where ρs is sediment density, here assumed to be 2.65 g/cm3, ρw is the density of fresh water,here assumed to be 1.00 g/cm3 and Fs is the percentage of sand in the surface material. Thedimensional transport rate for any grain size class i is computed following an Einstein-typeflux:qbψ =W∗i fau3∗Rg, (A.12)where qbψ is the fractional sediment flux for grain size class ψ fa is the volume probabilityof ψ within in the bed surface mixture, u3∗ is the shear velocity, computed with the Manning-Strickler formulation:u2∗ =(k0.33s q2wα2r)0.30(gS)0.70 , (A.13)R is the relative density of sediment (ρs/ρw)− 1.134A.4. Channel evolution modelA.4 Channel evolution modelChannel bed evolution is solved with the 1D Exner equation of mass conservation (Exner,1925):∂η∂t= −1ε∂qb∂x, (A.14)where t is time, ε is the solid fraction of the bed, and qb is the total sediment transport flux.The total sediment fluxes needed to compute the change in bed elevation at each spatial nodecomes from the previous step, and for all but the upstream and downstream nodes, spatialgradients of sediment flux in x are determined as central differences. Equation A.14 has, forexample, a diffusion type analytical solution which can be found by assuming (1) steady, uni-form flow and (2) that sediment transport can be approximated as a simple power law functionof average flow velocity, yielding (Soni et al., 1980; Gill, 1983a,b):∂η∂t=bqb3ε∂2η∂x2, (A.15)Provided that the simulations seek projections of channel bed longitudinal profiles, we specifythe starting channel bed profile as an appropriate initial condition, and since Equation A.15is second order, we need two boundary conditions: (1) sediment supply rate at the upstreammodel boundary and (2) a channel bed elevation at the downstream model outlet.A.5 Grain sorting modelThe grain sorting model is based on the active layer concept developed by Hirano (1971), asapplied by Parker (1991), and as further developed by Viparelli et al. (2010), to the problemof bed surface sediment sorting and mass conservation of the various grain sizes present onthe bed surface along a channel profile. The active layer is a relatively thin layer of surficialsediments that are conceptualized to participate in bedload transport, as well as bed evolution.Notably, the active layer concept is applicable only for Stage 1 and 2 transport conditions(Hassan et al., 2005). As noted above, the active layer length scale is commonly estimated asthe 90th percentile grain class of the bed surface times a constant, which ranges from a valueof 1 to 2 (Parker, 2008). The active layer grain size distribution changes due to erosion anddeposition based on fractional bedload transport capacity. The model presently tracks thechanging composition of active layer sediments due to erosion and subsequent deposition.Cycles of erosion and deposition can be handled, but presently are not.The grain sorting model for a unit width of stream bed is computed as:ε[La∂ fa∂t+ ( fa − fes)δLaδt]= −∂qbψ∂x+ fesδqbδx, (A.16)where La is the active layer thickness and fes is the volume probability of ψ within the activelayer/bed substrate interface. The left hand side of Equation A.16 represents the mass of bed135A.6. Model set-up and boundary conditionssediments within a control volume, and the right hand side reflects the net mass inflow rate ofsediment. Exchange of sediments between the active layer and sediments below is controlledby the exchange fraction parameter:fes = fss|z=η−La for bed erosionβ fa + (1− β)pbi for bed deposition,(A.17)where fss is the volume probability of ψ within the bed substrate and β is a partitioning coef-ficient assigned a value of 0.5.A.6 Model set-up and boundary conditionsThe channel bed evolution model is computed following a finite difference numerical scheme.Determination of the spatial and time steps is subject to the Courant stability parameter (Cn)where an estimated mean flow velocity, and the model spatial and temporal differences areused to compute the projected model stability:Cn =u¯∆t∆x(A.18)Courant stability values less than 1 are sought for finite difference numerical models. Theestimated mean flow velocity used in Equation A.18 is computed with the normal flow depthapproximation.As described above, the model requires three boundary conditions in total. One at theupstream end and two at the downstream end of the model domain. The upstream boundarycondition is set as the sediment supply rate for all times t:qb(xo, t) = qbo , (A.19)where xo is the upstream most node in the model domain. The fractional flux qbψ is determinedfrom qb based on the fractional proportion of each grain size class ψ in the supply mixtureThe two downstream boundary conditions for the flume simulations are set as (1) the watersurface elevation at the downstream most computational node for all times t, determined withthe normal depth approximation (ho):h(xL, t) = ho, (A.20)and, (2) the channel bed elevation at the downstream most computational node for all times t,determined by the elevation at the initial time (ηo):η(xL, t) = ηo, (A.21)136A.7. Model application to experiment PRE1where L is the length of the computational domain from upstream to downstream.A.7 Model application to experiment PRE1Figure A.1 compares steady-state longitudinal profiles for PRE1, and channel evolution modelapplication to PRE1, at te = 2150 minutes. We also show the initial profiles, and the initialexperimental DEM for reference. The experimental and simulation profiles exhibit similarmorphologic responses to the upstream supplies of water and sediment, with pool, riffle androughened channel features located in a consistent manner along the channel. The depths ofpools, heights of riffles, and topographic magnitudes of roughened channel segments are alsosimilar. Notably, the simulation profile informed design of PRE1, as discussed in Section 3.2.2.The simulation was run for a spatial step of 10 cm, a time step of 0.1 seconds, and with aRCASA pool water surface slope of crossing lineChannel Elevation (mm)Experimental Channel Topography: te = 0 minutesPRiRoRiP0 minutes42 ls-12150 minutes0 2000 4000 6000 8000 10000 12000 14000 16000Channel Station (mm)42 ls-12150 minutesSimulated Channel TopographyExperimental Channel TopographyPRiRoRiPsimulation initial profileexperimental initial profile(a)(b)(c)Figure A.1: Profile comparison of experimental and simulation outcomes for te = 2150minutes. At the top we show the channel layout and experimental DEM at te =0 minutes for reference. (a) The initial simulation (dashed line) and experimentalprofiles (solid lines). (b) Experimental profile at steady-state te = 2150 minutes,with zero-crossing negative residuals colored for reference. (c) Simulation profile atsteady-state te = 2150 minutes (simulation duration), with zero-crossing negativeresiduals colored for reference. See Section 4.3.2 for information concerning thezero-crossing method.137Appendix BProbabilistic friction angle model ofparticle mobilityB.1 MotivationMobilization of a grain resting on or within the channel bed surface will occur when the down-stream and upward directed forces acting on the grain exceed the forces acting to keep thegrain at rest. The resisting forces acting to keep the grain at rest are Fg and Ff , the gravitationaland resisting frictional forces, respectively (Wiberg and Smith, 1987). The driving forces actingto mobilize the grain are Fd, Fl and Fb, the drag, lift and buoyancy forces, respectively (Wibergand Smith, 1987). From this context, grain mobilization is generally approximated with theShields stress criterion (τ∗c ) (Shields, 1936), which reflects the ratio of driving to resisting forcesfor uniform or non-uniform flow conditions, formulated in the most simplified form possible:τ∗c =τ(ρs − ρw)gDi , (B.1)where τ is the dimensional shear stress, ρs is sediment density, here assumed to be 2.65 g/cm3,ρw is the density of fresh water, here assumed to be 1.00 g/cm3, Di represents the mean di-ameter of each grain size class according to the Wentworth scale for half-ψ increments, andψ = log2(Di). The driving force τ is commonly taken to be equivalent to ρwgdS, where g is theacceleration of gravity on Earth, d is the average water depth and S is the channel bed slope.The resisting force is taken to be the gravitational force acting on the submerged weight of asediment grain.Application of Equation B.1 yields singular values of the Shields stress for any grain sizeDi, or a range of values if the driving force is assumed to vary, achieved by varying the meanflow depth by some specified amount. Either approach will provide Shields stress conditionsthat are constrained over a narrow range. This stands in contrast to the view that sedimenttransport is a probabilistic phenomenon (Hassan et al., 1991; Furbish et al., 2012; Ancey and Hey-138B.2. Simplifying assumption of the present modelman, 2014; Furbish et al., 2016), for which sediment particles on the bed surface have mobilityconditions described by a distribution of possible values (Wiberg and Smith, 1987; Kirchner et al.,1990; Buffington et al., 1992).We address these differences with a numerical model of the particle mobility problem aspresented by Wiberg and Smith (1987); Kirchner et al. (1990) and Buffington et al. (1992). Themodel builds a randomly constructed virtual 1D river bed from a specified grain size distri-bution. Sediment particles from the same distribution are randomly placed along the virtualriverbed, but grain placement is restricted to known locations of grain–grain intersection onthe virtual riverbed. The geometric characteristics of each placed grain are determined, pro-viding measurement of the grain friction angle, which determines the relative mobility of allplaced grains and which is used to determine τ∗c of Equation B.1. The goal of the model is touse a simple mechanics-based approach to examine particle mobility probability distributionsfor poorly sorted gravel-bed rivers. The specific goal is to examine how/if morphodynamicpredictions are affected by the distributions of possible mobility states, as opposed to deter-ministic mobility approximations, such as those associated with Equation B.1. Here I describethe details of the model, and provide some results to demonstrate the model capabilities, andcharacteristics of simulated τ∗c distributions.B.2 Simplifying assumption of the present modelFigure B.1: Schematic view of friction angle based mobilization problem as defined byKirchner et al. (1990). Grain projection (e) is determined by the local average bedelevation, where local is defined by a length scale equivalent to the D84 percentilegrain size. Grain exposure (e) defines the upstream face length scale exposed tothe oncoming fluid flow, determined as the distance from the top of the upstreamneighboring grain to the top of the grain of interest. e can have a value of 0 if thegrain of interest is sheltered by the upstream neighboring grain. The friction angleφ is determined by how the grain of interest sits on the supporting grains.Kirchner et al. (1990) derived a particle force balance solution by expanding Equation B.1into a form for which τ∗c of any particular grain on the bed surface is a function of grain139B.3. Solution for τc with the p simplificationprojection (p), grain exposure (e) and the grain friction angle φ (Figure B.1). Grain projectionscales the magnitude of the drag force: Fd = 0.5ρwCd(0.5As)U2xz, because p sets the datumfrom which the vertical velocity profile is computed, where Cd is the drag coefficient, Uxz isthe vertically-averaged velocity impinging on the upstream exposed face of the grain and Asis the total exposed area of the grain to the surrounding fluid. Kirchner et al. (1990) specifiesthe datum for calculation of the vertical velocity as the nearby upstream/downstream averagebed elevation (yaverage1,2,3,4 : Figure B.1), where the averaging length scale in both directions isequivalent to the D84 percentile grain size on the bed surface. Buffington et al. (1992) used thisset-up to calculate τc values for frozen bed surface samples from Wildcat Creek, Berkeley, CA,U.S. Inclusion of p by Kirchner et al. (1990) in Fd assumes that the flow field feels the riverbedat the specified datum, which requires the flow to be attached and developing from the datumelevation. I simplify the problem by assuming that grain exposure adequately approximatesthe vertical length scale over which the upstream flow field develops to a relatively roughbed surface. This simplification reduces the fluid mechanical assertions implied by use of theprojection length scale.B.3 Solution for τc with the p simplificationKirchner et al. (1990) expands Equation B.1 by accounting for Fd, Fl and the grain friction angleφ:16(ρs − ρw)gpiD3 = Fw = Fdtan φ + Fl , (B.2)where Fw is the weight force, or the immersed weight of a sediment particle of diameter D. Asindicated above, Fd is (Wiberg and Smith, 1987):Fd = 0.5ρwCd(0.5As)〈u2(z)〉, (B.3)where U2xz is rewritten as〈u2(z)〉. The vertical velocity profile is a logarithmic function of thedistance above the reference datum (Kirchner et al., 1990):u(z) =(τρw)0.5κ−1 ln(z + zozo), (B.4)where κ is van Karmen’s constant, here assumed to have a value of 0.407, z is the elevation ofthe grain top exposed to the flow field (ytop: Figure B.1) and zo is the lowest elevation of thegrain exposed to the oncoming flow. The distance z− zo defines the grain exposure e (FigureB.1). The logarithmic term is abbreviated as f (z), and combining Equations B.3 and B.4 gives:Fd = 0.5Cd(0.5As)(τκ−2) f (z)2 (B.5)Fl is a function of the scaled velocity difference (scaled by the lift coefficient) acting over theexposed surface area of a grain from the grain top to the lowest elevation of the grain exposed140B.3. Solution for τc with the p simplificationto the oncoming flow (Wiberg and Smith, 1987):Fl = 0.5ρwCl As(u2t − u2b), (B.6)where u2t is the velocity at the grain top and u2b is the velocity at the lowest elevation of thegrain exposed to oncoming flow. Similar to Fd, the Fl velocity difference term is a function ofthe velocity profile, also assumed to be logarithmic in form, providing Fl as:Fl = 0.5Cl As(τκ−2)[ f (zt)2 − f (zb)2] (B.7)The drag coefficient Cd is taken to have a value 0.40, and the lift coefficient Cl a value of 0.20(Wiberg and Smith, 1985). The grain friction angle φ is calculated as:φ = sin−1d1d2, (B.8)where d1 is the horizontal distance between the overlying grain center, and the downstreamneighboring grain center, and d2 is the straight line distance between the overlying grain cen-ter coordinates, and the downstream neighboring grain center coordinates (Figure B.1). As anote, the downstream neighboring grain lies partially beneath the overlying grain, for whichthe mobility condition is sought. With values for Cd and Cl , a method to determine φ, andEquations B.5 and B.7, all that remains is to specify how to determine As. The surface area of agrain resting on particles beneath is determined through revolution about the axis parallel tothe bed surface (Figure B.1):As =∫ ba2pi(r2 − x2)√(r2r2 − x2)dx, (B.9)where the limits of integration are taken as a = the lowest grain elevation exposed to oncomingflow (point 2 of Figure B.1), which is taken to have a relative elevation of 0 (equivalent to zo),and b = the grain top elevation (yitop of Figure B.1). With the integration limits specified,Equation B.9 simplifies to:As =∫ yitop02pirdx, (B.10)and integrating with respect to x and applying the limits yields:, As = 2pire (B.11)where e is taken as the grain exposure, defined above as the difference between the grain topelevation and the lowest elevation of the grain exposed to the oncoming flow, defined by theelevation of the neighboring upstream grain top (Figure B.1). As indicated in Equation B.5, thedrag force is a function of As normal to the flow, so As is multiplied by 0.5. The lift force on141B.4. Sample simulation inputs and resultsthe other hand is a function of the entire exposed surface area As.Combining Equations B.2, B.5, B.7 and B.11, rearranging, and solving for τ = τc yields arevised form of the force balance describing a spherical grain resting on the channel bed:τcr = (ρs − ρw)gpiD36·[0.5Cd tan−1 θκ−2pire · f (z)2 + Clκ−2pire · ( f (zt)2 − f (zb)2)]−1(B.12)With the definition of zo, the velocity difference quantity on the right hand side of EquationB.12 simplifies to f (z)2. Finally, results from Equation B.12 are used in Equation B.1 to deter-mine the Shields stress τ∗c , provided that e > 0. If e ≤ 0, τ∗c is not computed.B.4 Sample simulation inputs and resultsThe model begins by the user specifying the grain size distribution, which is assumed to benormally distributed grain size sample, described by a mean size and standard deviation (Fig-ure B.2). The model uses the specified distribution to construct the virtual riverbed, whichFigure B.2: Sample grain size distribution for the friction angle model. The illustrateddistribution matches the PRE1 grain size mixture.generally consists of 5000–10000 grains. Grain sizes are randomly chosen from the distribu-tion, and placed sequentially along the riverbed. Grains from the same distribution are thenrandomly placed along the riverbed, and the grain friction φ is determined with EquationB.8, as the center coordinates of every grain on the virtual riverbed, and those placed on the142B.4. Sample simulation inputs and resultsriverbed are stored by the model. Typically, 10–20% of all placed grains are discarded for every5000 placed grains because e ≤ 0. Figure B.3 illustrates sample results from applying EquationB.12 to the grain size distribution shown in Figure B.2, for the steps just discussed. Figure B.3Figure B.3: Simulated grain mobility conditions using the friction angle model. The topplot illustrates measured friction angle φ vs. the relative grain diameter Di/D50, col-ored based on the critical Shields stress value τ∗cr. The bottom plot shows the 10th–90th percentile values for the critical Shields stress value τ∗cr vs. the average graindiameter for each grain size class of the simulated distribution. The gray shadedregion highlights the typical value range of 0.03–0.06 for τ∗cr (cf. Buffington and Mont-gomery, 1997). Results shown with the perceptually uniform Viridis colormap.shows that simulated grain mobility is systematically distributed according to grain size forpoorly sorted gravel mixtures, like that of PRE1 (Figure B.2). Smaller grains tend to exhibithigher critical mobility conditions, as shown with the yellow colors of the top panel in FigureB.3, and the magnitude of associated values for the 10th–90th percentile results of the lower143B.5. Friction angle model availability and citationpanel. On the other hand, larger grains tend to exhibit lower critical mobility conditions, asshown with the blue colors of the top panel in Figure B.3, and the magnitude of associatedvalues for the 10th–90th percentile results of the lower panel. For the simulated grain sizedistribution, the lower panel of results also indicates that grain size classes within the centerof the distribution are characteristically less mobile than the grain size classes on either end ofthe distribution. This result is common to simulation outcomes for mixtures with geometricmean sizes greater than roughly 5–6 mm, with standard deviations greater than roughly 0.5mm, and normal distribution limits of five times the standard deviation. The result indicatesthat there is some optimal hiding or packing problem, particular to grain size classes that lienear the mode of the distribution.B.5 Friction angle model availability and citationThe model was coded in MATLAB R©, and has been partially translated to Python. The model isfreely available on my GitHub site: The model is licensed with a MIT license, and is citable with the following digital objectidentifier: doi: 10.5281/zenodo.250114, which can be found here: CSediment transport as a rarefiedphenomenonmean proportion for interval 2150-2225 minutesFigure C.1: Measured sediment flux from the flume. The plot illustrates sediment fluxas an approximate proportion of the total grains on the bed surface participatingin transport, for a bed area measuring 547x547 mm2 and assuming the bed is com-posed of 8 mm diameter grains (the average channel width of experiment pool-riffle1 (PRE1) and the grain size mixture D50, respectively). Sediment flux at end time2150 minutes reflects steady-state conditions for an approximate bankfull flow, andsediment flux at end time 2225 minutes reflects adjustment of the 2150 minutessteady-state conditions to an increase in flow and sediment feed such that averagesediment mobility was 2x the threshold condition of τ∗c ≈ 0.030. Sediment flux dur-ing the interval 2150-2225 minutes was the highest measured during PRE1, yet withon average only 12% of the sediment particles on the bed surface participating intransport. For the prior interval the average proportion participating in transportwas well below 1% (yellow line at bottom of plot).Figure C.1 shows that the approximate proportion of sediment particles on the bed surface145that participate in sediment transport ranges, on average, from less than 1% to roughly 12%for experiment pool-riffle 1 (PRE1). This range reflects steady-state topographic conditions(end time t=2150 min.), and a significant perturbation to the upstream boundary conditionsthat gave rise to the t=2150 min. steady-state (end time t=2225 min.). This set of results isone demonstration that sediment transport is a rarefied phenomenon (Furbish et al., 2016). SeeChapters 3 and 4 for more details regarding the experimental conditions.146


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