UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Geometrical constraints on the formation and melt of ridged sea ice Amundrud, Trisha Lynne 2005

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_2004-994228.pdf [ 9.86MB ]
Metadata
JSON: 831-1.0052754.json
JSON-LD: 831-1.0052754-ld.json
RDF/XML (Pretty): 831-1.0052754-rdf.xml
RDF/JSON: 831-1.0052754-rdf.json
Turtle: 831-1.0052754-turtle.txt
N-Triples: 831-1.0052754-rdf-ntriples.txt
Original Record: 831-1.0052754-source.json
Full Text
831-1.0052754-fulltext.txt
Citation
831-1.0052754.ris

Full Text

GEOMETRICAL CONSTRAINTS ON THE FORMATION AND MELT OF RIDGED SEA ICE by Trisha Lynne Amundrud B.Sc. Hons., Physical Science, University of  Guelph, Canada, 1998 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF EARTH AND OCEAN SCIENCES We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 2004 © Trisha Lynne Amundrud, 2004 The Arctic ice pack consists of  flat  level ice, open water, and large ridge structures. During winter, ice thickens and is compacted into ridges, increasing the Arctic ice volume. In summer, ridging is accompanied by ice melt processes, which act to decrease ice volume. Current ice-atmosphere-ocean models cannot reproduce the evolution of  the ridged ice fraction,  suggesting that ridging or melt may be inappropriately parameterized. To increase our understanding of ridged ice evolution, this thesis investigates the factors  that constrain the ridging and melt processes. rat-A unique ice draft  distribution model is developed to simulate ice evolution in the Beaufort  Sea, allowing direct comparison with observations of  ice draft  by moored sonar. Conventional ridging algorithms used in a 24-day simulation were found  to overestimate the amount of  very thick ice. Observations of  level ice reveals that 75% of  all ice floes  are too small to create ridges of maximum draft.  In addition, observed ridges have cusp-shaped keels with concave flanks, containing less thick ice and increased amounts of  thinner ice than the triangular shaped keels assumed by most models. Including the observed constraints into the redistribution model produces ridged ice in agreement with observations, confirming  the importance of  the geometrical constraints to the creation of  ridged ice. During the melt season, simulations of  ice ablation in the Beaufort  Sea indicate that level ice melt processes cannot reproduce the observed enhanced melt rates of  ridged ice. A semi-quantitative model for  internal melt due to flow  through the porous keel is developed and an enhanced internal melt rate estimated. The rate of  melting within the porous structure of  the ridge keel is up to an order of  magnitude greater than the rate of  melting at the surface  of  level ice floes.  Including the internal melt within the ice draft  distribution model can reproduce the enhanced melt of  ridged ice and is thus essential for  the accurate simulation of  the evolution of  ridged ice. Similar to the geometric constraints on ice ridging, the internal geometry of  ridge keels plays a large role in the annual evolution of  the thickest sea ice. TABLE OF CONTENTS: ABSTRACT II LIST OF TABLES VI LIST OF FIGURES VII PREFACE XVI ACKNOWLEDGEMENTS & DEDICATION XVII SECTION A : OVERVIEW AND INTRODUCTION 1 1. GENERAL INTRODUCTION 1 1.1. Motivation 1 1.2. Thermodynamics of  sea ice 3 1.3. Ridge morphology 5 1.4. Mechanics of  ridge formation  8 1.5. Winter ridge evolution 12 1.6. Spring and summer ridge ablation 14 1.7. Research objectives 17 2. SEA ICE REDISTRIBUTION MODELS 20 2.1. General overview of  statistical descriptions 20 2.2. Ice draft  redistribution models 21 2.3. Ice area evolution models 24 SECTION B : THE WINTER REDISTRIBUTION OF SEA ICE IN THE BEAUFORT 26 3. APPLICATION TO THE BEAUFORT SEA 26 3.1. Geographic location 26 3.2. Available data from moorings 29 3.3. Natural experiments 32 4. A REGIONAL ICE DRAFT REDISTRIBUTION MODEL 37 4.1. General model 37 4.2. Thermodynamic ice growth 38 4.3. Ice motion: divergence and ridging 42 4.4. Observational errors 45 SECTION C: GEOMETRIC CONSTRAINTS ON ICE RIDGING 48 5. CONTRIBUTION OF ICE FEATURES TO ICE EVOLUTION 48 5.1. Maximum keel draft  48 5.2. Floe size and thickness 51 5.3. Characteristic floe  length in buckling 54 5.4. Truncation of  ridge building via limits on ice available 56 5.5. Keel shape 59 6. DISCUSSION 64 6.1. Simulation of  pack-ice development during 1991-1992 64 6.2. An idealized simulation of  pack-ice evolution in winter 66 6.3. Convergent motion and ridging 68 6.4. Keel shape 69 6.5. Importance of  divergence on the local characteristics of  the ice pack 71 6.6. Stress levels in the ice pack : 75 6.7. Time and distance scales 76 6.8. Model limitations 77 7. CONCLUSIONS: GEOMETRIC FACTORS INFLUENCING RIDGE FORMATION 79 SECTION D: SUMMER RIDGE EVOLUTION 81 8. ADAPTING THE ICE DRAFT REDISTRIBUTION MODEL TO SUMMER MELT 81 8.1. Winter 2000 81 8.2. Level ice melt 83 8.3. Model predictions 86 8.4. Evidence for  enhanced melting 90 9. SUMMER RIDGE ABLATION PROCESSES 94 9.1. The influence  of  internal ridge geometry on ice melt 94 9.2. Under-ice ocean characteristics 96 9.3. Porous flow  through the ice 101 9.4. Pressure distribution on keel 106 9.5. Pore velocity .'. 109 9.6. Heat transfer  to a porous media I l l 9.7. Melt of  ridged sea ice 113 10. MELT WITHIN A SINGLE KEEL 119 10.1. Model sensitivity 119 10.2. Sensitivity of  melt to parameters for  porous flow  119 10.3. Sensitivity of  melt to parameters for  heat transfer  122 10.4. Stratification  and stability: 2-d components of  fluid  flow..  123 10.5. Sensitivity of  melt to block thickness and porosity 124 10.6. Keel width and shape 127 10.7. Consolidation due to melt 128 11. INTERNAL MELT OF A DISTRIBUTION OF RLDGE KEELS 130 11.1. Keel statistics 1 130 11.2. Keel width 130 11.3. Keel shape 133 11.4. Melt of  a population of  ridged ice 134 11.5. Enhanced melt in redistribution models 138 11.6. Ability for  internal melt to account for  enhanced melt rates 140 12. CONCLUSIONS: GEOMETRIC FACTORS INFLUENCING RIDGE MELT 143 SECTION E: THESIS SUMMARY 145 13. CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH 145 13.1. Conclusion 145 13.2. Future directions 146 13.3. Implications of  research 147 REFERENCES ' * 150 APPENDICES 158 Appendix A Variables and Parameters 158 Appendix B Hydraulic radius for  spheres and rectangles .'. 162 Appendix C Derivation of  keel width 165 Table 1-1: Ice growth rates for  central Arctic pack ice used as model input by Thorndike et al (1975).. 4 Table 1-2: Common ridge statistical ratios reported in Timco and Burden (1997) from a survey of 112 first  year ridges and 64 multi year ridges. Italicized values are reported in Melling et al., (1993) from the literature. Of  interest is the ratio of  keel to sail geometries with keels taller and wider than sails by an approximate factor  of  four  8 Table 9-1: Mean currents observed with depths during the SHEBA mast experiment in summer 1998 99 Table 9-2: Mean ocean temperature over a ten-day period for  depths 3m to 10m 100 Table 9-3: Porous flow  parameters. 'Englelund (1953) data as reported in Burcharth and Andersen (1995). 2Diameter is given as D50, the diameter, for  which 50% of  the sample is smaller than by mass. 3Three different  rectangular irregular samples with their results are listed 105 Table 9-4: Experimental drag coefficients  for  a laboratory model of  a keel in a stratified  flow (Cummins et al., 1994). The drag coefficient  decreases as the Froude number, and hence the current velocity, increases 108 Table 10-1: Experimental drag coefficients  for  a laboratory model of  a keel in a stratified  flow (Cummins et al., 1994) 122 LIST OF FIGURES Figure 1-1: View of  ice cover in the Beaufort  Sea. The intersection of  two floes  to form a ridge structure is clearly visible across the upper left  side of  the photo. Additional structures are also present. Courtesy of  H. Melling 6 Figure 1-2: Schematic of  the dominant processes acting on ridges during winter and summer. Thermodynamic melt is indicated in red, thermodynamic growth in blue, and mechanical forcing  in grey (movement of  ice) and purple (movement of  water) 9 Figure 1 -3: Keel schematic showing trapezoidal (solid line) and triangular (dashed line) cross sections 11 Figure 1-4: Typical temperature and salinity water profiles  with depth [dbar] from the Beaufort Sea in winter and summer seasons. The typical changes in the mixed layer depths are evident here with the summer mixed layer being much shallower, and the change in density greater, than in the winter. Data provided by H. Melling 15 Figure 2-1: The statistical representation of  the probability density distribution of  sea ice draft (fraction  per metre). Data are from the end of  December, 1991 in the coastal Beaufort  Sea. The level ice peaks at 1.2 and 0.6 m are visible as strong signals in the distribution. Open water is a much smaller peak. The exponential tail of  ridged ice stretches from 1.5 m to 15 m, at which point no further  observations of  ridged ice were made. Data courtesy H. Melling 20 Figure 2-2: Schematic of  the effects  of  mechanical and thermal forcing  on the probability density function  of  ice draft.  Level ice peak is at 1.8 m and indicates the proportion the ice pack that is level ice of  this thickness. Figure courtesy of  H. Melling 23 Figure 2-3: Form of  mechanical ridging function,  y/. Level ice is removed by the participation function  and redistributed through the transfer  function.  Open water is created to conserve area 23 Figure 3-1: The Beaufort  Sea area of  the Canadian Basin. Mooring sites in the Beaufort  Sea and the land-fast  ice edge in January and February 1998 (from Regional Ice Chart, CIS). Ice data were acquired at Sites 1 and 2 during 1997-1998 and at Site 3 during 1991-1992. All four sites are located north of  the land fast  ice and the recurring flaw lead at the fast  ice edge... 27 Figure 3-2: Partial ice chart for  the Western Canadian Arctic on August 30th, 1999 from the Canadian Ice Service. The dotted area indicates the open water with less than 10% ice concentration while areas closer to the shore contains bergy water where total ice concentration is less than 10% cover 28 Figure 3-3: February 1st 1998 ice cover. The land fast  ice is visible as area 'E' in this plot, the offshore  ice in section 'X' is assumed to be homogeneous in an alongshore direction. The flaw lead is at the boundary between areas 'E' and 'X' but is not visible in the ice chart. Partial Ice Chart, Canadian Ice Service 32 Figure 3-4: Eulerian displacement vector illustrating ice drift  at Site 2 from December 28th, 1997 to February 1st, 1998. Stars indicate relative positions of  moorings on December 28th. Squares mark the endpoints of  track segments used to calculate distributions of  draft  34 Figure 3-5: Evolution of  ice thickness during the winter of  1997-1998 at site 2. Contours showing the discontinuity in level ice thickness. Progressively darker shaded regions contain 1, 5, 10 and 30% of  the distribution (heavy black line is the 10% contour). The break in the seasonal level-ice mode is evident at Day 65 of  1998. Later level-ice modes track ice formed at other locations or at times after  freeze  up 35 Figure 3-6: Eulerian versus Lagrangian ice perspectives. Contours showing the discontinuity in level ice thickness. Progressively darker shaded regions contain 1, 5, 10 and 30% of  the distribution (heavy black line is the 10% contour). The inconsistent thermal growth of  the level ice peak in a section of  1991-1992 data indicates that care must be taken in drawing conclusions from natural experiments 36 Figure 4-1: The Lipscomb (2001) remapping model algorithm (adapted from Lipscomb, 2001) applied to an assumed distribution function  (note y-axis is the probability distribution of  ice [m"']). The initial distribution is indicated in the upper left  panel by the light grey bar. With a constant growth rate that does not change with ice thickness, the initial distribution grows to 0.5m and diffuses  slightly. With the original Thorndike algorithm (upper right panel), the distribution spreads out over a range from 0 to lm in draft.  Using a growth rate that is dependent on ice thickness the Lipscomb remapping model grows ice to 0.25m in a sharp peak (lower left  panel). The Thorndike algorithm grows ice to this height, but introduces significant  diffusion  (lower right panel) 39 Figure 4-2: Observed and modeled probability density of  draft  (fraction  per decimetre) for  ice growing thermodynamically during the winter of  1997 at site 2. (a) Growth from conditions initialized using observations on 17 October, (b) Growth from an ice-free  sea surface  at the observed time of  freeze-up.  The model indicates more level ice than observations because mechanical redistribution is not active in these simulations 40 Figure 4-3: Results of  the Thorndike/Hibler and Lipscomb thermal redistribution schemes (Lipscomb, 2001) on the final  probability density distribution of  sea-ice (fraction  per decimetre), (a) Thorndike/Hibler approach, (b) Lipscomb approach 41 Figure 4-4: Modelled draft  distribution and observed distributions on November 26th, 1991 (fraction  per decimetre). The model does an excellent job recreating the level ice depth and shows little excess diffusion.  Ice greater than the level ice peak is not present in the model distribution as rafting  and ridging of  ice was not permitted in this run 42 Figure 4-5: The power spectral density for  the north-south ice velocity data at site 2 during the winter of  1997-1998. The semi-diurnal tidal peak is clearly visible. The horizontal line represents the noise level associated with the sampling error of  1 cm s"1 46 Figure 4-6: Influence  of  24 (heavy solid line) and 6 (heavy dashed line) filters  on the north-south ice velocity data from site 2. Lower panel is an expanded section of  upper panel to clearly show daily oscillations in data 47 Figure 5-1: Probability density of  draft  (fraction  per decimeter) from a model run with customary assumptions that all ridge keels are built to a draft  of  20 m1/2 hm and have triangular shape, for  site 2 in 1998. Relative to observations, far  too much ice of  draft  exceeding 10 m is created. Thin lines are confidence  bounds representing +/- one standard sampling error based on degrees of  freedom 49 Figure 5-2: Empirical relationship between the draft  of  large keels and that of  level ice adjacent to them, based on observations of  seasonal pack in the Beaufort  Sea during the 1990s. The curve indicates truncation of  keel development at a draft  of  20 m1/2 hul 50 Figure 5-3: Cumulative distribution of.floe  size (fractionper  floe  size) in the Beaufort  Sea during 1997-1998 at Sites 1 and 2. All floes  larger than 560m are capable of  forming  a ridge of maximum size. Inset: probability density function  of  floe  size (fraction  per metre on a logarithmic scafe)  : 53 Figure 5-4: Probability density of  the logarithm of  floe  size (in m) for  all (0-1.3 m) drafts  of  level ice (solid line). The log-normal curve (dotted line) has a mean of  4.95 and a standard deviation of  1.7. The shaded area encompasses probability densities of  floe  size of  specific draft  by 0.1 m increment in the range 0-1.3 m. The gray line is the mean probability density of  size for  floes"  of  1.3-2.0 m draft.  The truncation of  data at a 10 m floe  size reflects  the minimum scale used in the definition  of  level ice. Figure 5-5: Critical buckling stress (in MPa) for  a 0.5 m thick level ice floe,  indicating the expected critical buckling stress for  level ice (5.3.1). Visible is the minimum buckling stress Figure 5-6: Flow chart illustrating the decision process for  deciding the value of  the floe  size to Figure 5-7: Mean probability density of  draft  (fraction  per decimeter) for  ridged ice calculated with various geometrical constraints on ice ridging. Model is run from the 9th January to the 1st of  February 1998 at site 2. Comparative curves are shown smoothed with a 5-point running average for  runs with (a) uniform volume distribution and a constant truncation (b) uniform volume distribution and the floe-availability  constraint, (c) exponential (6-m scale) volume distribution and a constant truncation, and (d) exponential (6-m scale) volume distribution and the floe-availability  constraint. Thin lines indicate the variation between 500 simulations using the floe  availability constraint indicated as one standard deviation. Similar variation is found  for  curve (b) 59 Figure 5-8: Probability density of  draft  (fraction  per 0.5m) within 20 keels of  12.5+0.5 m maximum draft.  Small circles represent individual keels and large circles show average values. An exponential curve with 6.0 m e-foldmg scale fits  the average, (b) Cross-sectional shapes (cusped and triangular) corresponding to negative exponential and constant probability density of  draft  within keels. . 61 Figure 5-9: Difference  in ridged ice volume between the modeled and observed density of  ice volume at each draft  (volume fraction  of  ice greater than 9 m thick). Model is run from the 9th January to the 1st February 1998 at site 2. Comparative curves are shown smoothed with a 5-point running average for  runs with (a) uniform volume distribution and a constant truncation (b) uniform volume distribution and the floe-availability  constraint, (c) exponential (6 m scale) volume distribution and a constant truncation, and (d) exponential (6 m scale) volume distribution and the floe-availability  constraint. Combining the exponential volume distribution with the floe-availability  constraint, curve (d), produces the best ice draft  distribution at both thin and thick ridged ice drafts.  Shaded region indicates confidence bounds on the observed distribution representing +/- one standard sampling error based on degrees of  freedom 63 (for  floes  larger than Z c , (5.3.4)) given as the solid horizontal line. 56 be used in the truncation constraint (previous page). 59 Figure 6-1: The ice motion across mooring site 3 during the winter 1991-1992 is plotted as accumulated displacement. The natural experiment from days 330 to 348 of  1991 when the ice pack remained in the vicinity of  the mooring, is used for  the verification  of  the thermal growth model. The trajectory of  ice drift  over the mooring during this period is shown in the inset (plus sign indicates start of  ice motion event) 64 Figure 6-2: Probability density of  ice draft  (fraction  per decimeter) using the floe-availability constraint and assuming an exponential keel shape to simulate pack-ice development in the winter of  1991-1992. The trajectory of  ice drift  over the mooring is shown in the inset (arrow heads indicate direction) 66 Figure 6-3: Probability density of  ice draft  (fraction  per decimeter) from simulation of  idealized pack-ice development over 90 days, assuming various combinations of  the truncation model and transfer  functions.  Best-fit  exponential curves with e-folding  scales of  4.81 and 3.11m are found  for  the floe  availability model with triangular and exponential keel shape functions respectively; for  exponential keels with a constant truncation the e-folding  scale is 3.89 m.68 Figure 6-4: Schematic of  ridge shapes illustrating the varying distribution of  ice with draft.  The exponential roll-off  (heavy solid line) places far  less ice at the thickest keel drafts  than the triangular keel shape assumed by Thorndike et al., (1975) or the trapezoidal keel shape of Hopkins (1996a) 70 Figure 6-5: Motion of  ice over sites 2 and 1 during event A in 1998. Upper panel displays the northward motion of  ice across the moorings from days 10 to 47. The IDR model simulates the evolution of  the distribution of  ice from days 10 to 32. Crosses indicate the temporal start point of  the ice profiles  used to construct the statistically independent density distributions of  draft.  Circles indicate distributions plotted in figure  below. The lower panel displays the divergence [day"1] calculated as the velocity difference  between the sites divided by the distance (68 km) 72 Figure 6-6: Probability density of  draft  distributions (fraction  per decimeter) for  sites 2 and 1 during and after  convergent-divergent event A. Days (of  1998) listed represent the start date of  the data profile  used to create the distribution 73 Figure 6-7: Probability density draft  (fraction  per decimeter) of  ice evolving from the 9th of January to the 1st of  February. The divergence in the model is equal to the total divergence observed between sites 1 and 2, and ridging is assumed to create open water in the same area as the observed ice. The impact of  the divergence on the ridged ice is seen in the inset figure. While the total amount of  ridged ice decreases (area conservation), the shape of  the ridged ice does not change 74 Figure 6-8: Onshore and offshore  convergence travelled by ice during the winter of  1997/1998. Upper panel demonstrates the convergence (km) travelled in one hour. Lower panels illustrate the convergence (km) travelled in one direction before  the motion changes direction. Lower panels are identical except for  vertical scale. The convergence travelled exceeds 560 m for  all but one convergent (and therefore  ridging) event 77 Figure 8-1: North - South motion of  ice over mooring site 2 during the first  half  of  the year 2000. Observed ice draft  distributions are centred at points indicated by circles on the path. Initially, during January and February, ice moves south compacting against the coast before remaining relatively stationary in the north-south direction (thin lines with arrows indicate range of  motion). Melt begins around day 150 arid four  ice motion events are identified where the ice moves in an onshore-offshore  pattern (indicated by lines double ended with arrows).... 82 Figure 8-2: Probability density of  ice draft  (fraction  per decimeter) for  the results of  the IDR model at site 2 during the first  50 days of  2000. Note that the model (heavy solid line) reproduces the level ice growth of  the final  ice viewed at site 2 (heavy dashed line) and the thickest ridged ice at site 1 (thin solid line), 15 km south of  the final  location of  the modelled ice distribution 83 Figure 8-3: Ice and snow thickness observed at Cape Parry from 1959 to 1992. The decrease in snow cover can be seen in early May, preceding the decrease in ice thickness by two to four weeks. Error bars indicate +/- one standard deviation from the mean (large circles). These raw data are from the Canadian Ice Service Archives, http://ice-glaces.ec.gc.ca/App/WsvPageDsp.cfm?ID=210&Lamg=eng.  (Figure courtesy of  Humfrey Melling.) 85 Figure 8-4: Observed level ice draft  (dominant mode) and quadratic best-fit  curve used to estimate level ice melt rates. D is the day number of  year 2000. Inset figure  shows the increasing melt rates (as negative growth rates) as the season progresses 86 Figure 8-5: Probability density of  draft  (fraction  per decimetre) for  the modelled change in the distribution of  ridged ice during summer melt from days 154 to 173. Allowing ridging to occur, the model predicts an increase in thick ice due to ridging (heavy solid line). Without ridging the model predicts a decrease in ridged ice density expected from observed level ice melt rates independent of  draft  (thin solid line). The observed thick ice shows evidence for an enhanced melt process, with density at large drafts  decreasing faster  than predicted (dashed line)..... 89 Figure 8-6: Probability density of  draft  (fraction  per decimetre) for  the modelled change in the distribution of  ridged ice during summer melt from days 192 to 207. Similar to the results for  event A, the amount of  ridged ice is greatly overestimated by the model (heavy solid line). The model was run again without ridging (thin solid line) and the melt of  ridged ice is clearly underestimated by the model 90 Figure 8-7: Determining ablation rates from the cumulative density distribution. Here the cumulative density distribution is defined  to reach unity for  the thinnest ice (open water), as the thickest ice is least prominent. The ablation rate can be found  by subtracting the level ice thicknesses (Ah= hi - h2) associated with a chosen value of  the cumulative density and dividing Ah by the time separating the two distributions 91 Figure 8-8: Observed ablation rates as a function  of  draft  during four  ice motion events. Level ice melt for  these time periods is indicated by the dotted line of  the same pattern. Both level ice melt rates and ridged ice ablation rates increase throughout the melt season 92 Figure 8-9: Observed melt for  event D (thick line) with error (thin lines) representing the maximum errors using one standard deviation from the cumulative density distributions... 93 Figure 9-1: Components of  oceanic velocity relative to level ice, 13-18th April 1989 in the Beaufort  Sea. Level ice depth is 1.5 m, water depth is 54.6 m, and ice is moving westward. The direction and magnitude of  the velocity remain relatively constant below the ice, with a mean speed of  6.9 cm s"1. Data courtesy of  H. Melling 97 Figure 9-2: Velocity measured during turbulence measurements for  the SHEBA program 98 Figure 9-3: Mean temperature and salinity profiles  from automatic CTD profiles  collected during the SHEBA program in June and July 1998. Profiles  show mean values over ten days, centred at days 155, 165, 175, 185, 195, 205 100 Figure 9-4: Keel schematic showing the definition  of  axes used in the theory. The oceanic velocity relative to the ice, U, and the porous flow  through the ice, u, are in the direction of the ice motion x 102 Figure 9-5: Distribution of  pressure gradient along the flow  direction on a bed form such as a sand dune at the bottom of  a channel 109 Figure 9-6: Distribution of  pressure (9.5.4), pressure gradient (9.5.7), and pore velocity (9.5.2) through a 9 m keel with block thicknesses of  0.5 m and a ocean current of  0.1 m s"1 Ill Figure 9-7: Top: Internal heat flux  through keel showing the decrease in heat transfer  across the keel. Bottom: Change in the water temperature through the keel due to the heat lost to melt and the influx  of  fresh  water at the freezing  point. At deeper drafts  the water flows  faster (from left  to right on figure,  see Figure 9-6) moving further  through the keel before  the water reaches the freezing  temperature. Keel porosity is 0.3 and block thickness is 0.5 m 116 Figure 9-8: Melt rates from porous flow  through a keel 9 m (heavy black line). Thin black lines show the melt rates upstream and downstream of  the keel crest. Expected melt from level ice is shown as the thin dashed line, observed melt rates are shown as a thick dash-dot line. The melt rate from a small 4 m keel is shown for  comparison (heavy dashed line). Keel porosity is 0.3 and block thickness is 0.5 m for  both keels 117 Figure 9-9: Top: Porous flow  velocity due to melt through the keel. Increase in melt is very small compared with porous velocities that can exceed 20 mm s"1 (Figure 9-6). Keel porosity is 0.3 and block thickness is 0.5 m 118 Figure 10-1: Variations in calculated maximum pore velocity for  various coefficients  a and J3.120 Figure 10-2: Increase in pore velocity due to influx  of  melt water 124 Figure 10-3: Density anomaly (from 1000 kg m"3) through the keel due to the influx  of  fresh water and heat loss. Density varies by only 0.004% throughout the keel and pore water is stably stratified  : 124 Figure 10-4: Variation in melt rates with porosity for  a 9m keel with block thickness 0.5m.... 125 Figure 10-5: Variation in melt rate with block thickness for  a 9 m keel with a porosity of  0.3.. 127 Figure 11-1: The ratio, R, of  keel extent to keel draft  for  keels observed during event A (+) and event D (x) 131 Figure 11-2: schematic of  keel extent as seen by the sonar and true keel width, W 132 Figure 11-3: Mean e-folding  scales for  keels observed in 2000 during the melt season spanning events A and D (circles) and during the entire winter 1999/2000 (diamonds). Standard deviations of  the e-folding  scales are plotted as error bars. Observed e-folding  scales from the 1997 data (section 5.5) are plotted as filled  squares. The linear best-fit  line takes the form  A=0.50H  -0.32 m over the entire winter and spring seasons and match observations well 134 Figure 11-4: Melt rates within ridged ice during event D. The enhanced melt rate due to the internal melt (heavy black line) increases the melt rate from the level ice melt (thin dashed line) to match the observed melt rates (thin line). The probability density distribution of ridged ice (fraction  per decimetre) is indicated by the heavy dashed line 136 Figure 11-5: Distributions of  the maximum draft  of  ridged ice keels (heavy line) and rafted  ice (thin line) during event D (fractions  per decimetre). The weighted distribution of  ridged ice keels is shown as the dashed line and represents the proportion of  thick ice at that draft  that is in porous ridges 137 Figure 11-6: Melt rates within ridged ice during event A. The enhanced melt rate due to the internal melt (heavy black line) increases the melt rate from the level ice melt (thin dashed line) to represent most of  the observed melt rates (thin line). The probability density distribution of  ridged ice (fraction  per decimetre) is indicated by the heavy dashed line... 138 Figure 11-7: Probability density of  draft  (fraction  per decimetre) for  the internal melt model predicted distribution of  ridged ice during summer melt from days 154 to 173. Allowing ridging to occur, the model predicts an increase in thick ice due to ridging (thick solid line). Without ridging (thin solid line) the model predicts a decrease in ridged ice density in agreement with observed final  distribution (dashed line) 139 Figure 11-8: Probability density of  draft  (fraction  per decimetre) for  the internal melt model predicted distribution of  ridged ice during summer melt from days 192 to 207 of  2000. When ridging is included, the enhanced melt from porous flow  (heavy solid line) reduces the production of  ridged ice when compared to the model with only level ice melt (thin dashed line) 140 Figure 11-9: Probability density of  ice draft  (fraction  per decimeter) for  the internal melt model predicted distribution of  ridged ice during summer melt from days 192 to 207 of  2000. Including the enhanced melt from porous flow  (heavy solid line) with a assumed freezing temperature departure of  0.4 degrees reproduces most of  the features  of  the observed distribution, except for  an over-prediction of  ice within drafts  of  4-8 m 142 Portions of  Chapters III-VI have been published and presented as: T.L. Amundrud, H. Melling, and R. G. Ingram. Geometrical constraints on the evolution of ridged sea ice. Journal of  Geophysical Research (Oceans), vol. 109, C06005, doi: 10.1029/2003JC002251, June 2004 T.L. Bellchamber-Amundrud, H. Melling, and R. G. Ingram. Modelling the evolution of  draft distribution in the sea ice pack of  the Beaufort  Sea. Ice in the Environment: Proceedings of  the 16th IAHR International Symposium on Ice, Dunedin, NZ, pp. 243-250. 2002. There are many people who have supported me during this research. My primary supervisor, Grant Ingram, has provided much needed support. Grant, your enthusiasm has helped me to enjoy this process and you've been an excellent role model and mentor! Committee members Garry Clarke and Susan Allen have been wonderful.  Garry has made me feel  welcome in his stimulating glacier research group and Susan's care in working through the thesis in the early stages has improved this work greatly. Three individuals at the Institute of  Ocean Sciences deserve credit for  their assistance and support. My co-supervisor, Humfrey  Melling, has been an invaluable resource and a questioning mind throughout this research. David Riedel performed  the initial data analysis on the ice profile data. His efforts  let me sleep well at nights knowing that all errors are mine alone. While at sea on the icebreaker Laurier in September 2002, Peter Gamble's instruction in maintaining the field equipment was wonderful  and his enthusiasm and words of  wisdom kept me happy. I would also like to acknowledge financial  support from NSERC, Fisheries and Oceans Canada, and UBC. My parents, Neil and Judy, have always encouraged me to pursue my research and offered financial  support during my undergraduate degree when I took a low paying summer research position to gain experience. Your commitment to me, and your belief  in my abilities has been essential to my success. My grandparents, Murray and Dorothy, have also been wonderful cheerleaders, always taking an interest and supporting my goals. My in-laws, Sue and Jim, and various other extended family  members make a great support system. Dr. Gerry McKeon deserves credit for  getting me excited about physics in the very beginning, and taking an interest ever since (and solving equation 9.7.5) > . I want to especially thank my husband Cory, who has been there every step of  the way through this research: listening, suggesting ideas, and learning far  more about ice than you'd ever planned. The past few  years would be nowhere near as bright without you. Thank you! This thesis is dedicated to two strong and loving women who helped me become the person I am: For Vera Arnold, nee Callaghan, (1920-2004). Who taught me to see beauty in everything (and taught me how to drink whiskey). and For Audrey Bellchamber, nee Everitt (1913 -1993). Who listened, believed, and taught me to be strong. Thank you both for  the love and laughter. SECTION A: OVERVIEW AND INTRODUCTION 1. GENERAL INTRODUCTION 1.1. Motivation The pack ice of  the Arctic Ocean stretches over its polar seas in a thick, complex structure of ridges, leads, and flat  level ice. Acting as an insulator between the cold Arctic atmosphere and warmer ocean water, the ever-changing arrangement of  the pack ice regulates the heat flux between the atmosphere and ocean and thus influences  the local and global heat budget. On timescales of  days, gradients in the forces  acting on moving pack ice cause floes  to collide, raft over one another, and sometimes buckle and fail  on impact. Ridges form as the ice fragments from sustained local failure  are pushed into heaps along the line of  interaction. These pressure ridges have been found  to contain 68-73% of  the total ice mass in the Beaufort  Sea (Melling and Riedel, 1995) and are both an important component in the Arctic ice volume and a significant hazard for  shipping and offshore  structures. As ridges form in the Arctic Ocean, the ice motion also opens leads and larger open water areas called polynyas that exist throughout the Arctic during the winter and spring seasons. Created by wind forcing,  and water current drag, or thermal ocean heating, these areas are prevalent in thinner and marginal sea ice zones, and comprise 1% of  the total pack area (Smith et al., 1990). Heat exchange in these open water areas is two orders of  magnitude greater than the typical surrounding ice and snow, and thus are large contributors to regional heat budgets (Smith et al., 1990). In addition, the refreezing  of  these open water areas produces thin ice during midwinter and spring seasons that is easily ridged between thicker, older ice. The complex arrangement of  ice and open water in the Arctic influences  the global climate in addition to the local heat budget. While the ice cover is an important influence  on global climate, it is itself  is one of  the most sensitive factors  responding to climate change with complex and indirect responses to global warming (Melling, 2002). Anticipated increased heat in the Arctic atmosphere could lead to decreasing amounts of  multi year sea ice, and perhaps even a complete loss of  the Arctic multi year ice pack. Millennial timescales in sedimentary records of  large circulation changes (Darby et al., 2001) indicate that temperature increases of  up to 5 degrees C have occurred over the past few  thousand years in the Arctic. It is essential to be able to model and understand the ice-ocean-atmosphere system accurately, so as to make reliable predictions of the response of  the ice pack to projected climate changes similar in magnitude to those of  the recent geological past. Complicating the task of  predicting ice cover changes in coastal regions is the large interannual variability and the influence  of  topography; the large number of  islands contributes to the formation  and movement of  the ice pack. Recent work (Melling, 2002) in the Canadian Arctic Archipelago has found  that the heavily ridged ice of  the region is prevented from entering the Beaufort  Sea by ice bridges between islands. Melling (2002) speculates that warming conditions thus may not bring uniformly  lighter ice conditions to the Beaufort,  as warming could decrease ice bridge blocking and flush  heavily ridged ice into the Beaufort  Sea. In a region of  Canada where shipping is already highly dependent on ice conditions, the difference  between a decrease in ice cover and an increase in ridged ice is of  utmost importance. The most useful  tool scientists have for  ice prediction is often  computer modelling. Based on the observed conditions, a model should be able to forecast  the future  ice cover if  the processes affecting  climate are properly represented. Yet, current models show greatly varying results and often-erroneous  ice cover characteristics (Proshutinsky et al.,2001). Model runs with current conditions will often  underestimate the sea ice extent and thickness and overestimate melt (Cattle and Crossley, 1995). Others are biased towards larger thicknesses of  ice (Flato and Hibler, 1995) leading to excess formation  and slow degradation of  thick ice ridges. The thermal growth and melt of  flat  level ice has been extensively researched and successfully  modelled (ie. Maykut, 1986, Melling and Riedel, 1996a). Thus, the failure  of  models to evolve ice appropriately can be attributed to errors in the modelling of  the processes controlling the formation  and evolution of the thickest ice in ridges and rubble fields  and suggests these processes must be further investigated. While many physical components are incorporated into any ice-ocean-atmosphere model, the mechanical and thermal processes that affect  the ridged ice require specific  considerations. Thermal processes are often  simplified  under the assumption that simple heat balance calculations can account for  changes in ice thickness. Yet simple heat balance models have failed  to reproduce the ablation rates of  the thickest, ridged sea ice (Flato and Hibler, 1995; Proshutinsky et al.,2001) leading to overestimates of  ice draft  in climate models. Since the ridges created by mechanical forces  are complex three-dimensional structures, perhaps the enhanced melt rates of  ridged ice are due to the interaction of  thermal processes with these complex geometries. In concert with thermally driven freeze  and melt processes; wind and ocean current forcing  keeps the ice in motion, creating ridges and giving the ice pack its complex thickness distribution structure (Flato and Hibler, 1995). The mechanical processes acting to form a single ridge have been extensively studied due to the requirements of  human activities in the north. Engineering design of  ships, pipelines, and offshore  structures must account for  the unique constraints of  the ice cover. This has motivated ridge research (Timco and Burden, 1997), leading to an understanding of  the probable ridge distribution in the central Arctic ice pack (Thorndike et al.,1975; Lowry and Wadhams, 1979; and Mock et al.,1972) and ridge formation  from level ice (e.g. Hopkins and Hibler, 1991). Applications of  this knowledge of  individual ridging processes to the thickness distribution for  sea ice over regional scales should allow for  more accurate representation of  ridged ice in thickness redistribution models. For ice-ocean-atmosphere models to correctly predict the distribution of  sea ice, the mechanical and thermal processes that act on a ridge throughout the year must be better understood and appropriately modelled. In the following  sections, the processes that create and maintain ridged sea ice will be reviewed and areas where our knowledge of  the processes is limited identified. 1.2. Thermodynamics of  sea ice Thermodynamic sea ice growth or melt is determined by the balance of  heat fluxes  between the air, ocean, and ice. The rate of  ice growth is a function  of  the longwave and shortwave radiation, conductive and oceanic heat fluxes,  optical properties of  the ice and snow surfaces,  and the turbulent heat exchange. In addition, these parameters are functions  of  the cloud cover, open water area, wind forcing,  and the current thickness of  ice in the area (Maykut, 1986). Of  all of these factors,  in winter, when the water temperature is close to freezing,  the temperature of  the air often  becomes the controlling variable. The simplest models of  initial sea ice growth start with an estimate based on Stefan's  Law, relating the number of  cumulative freezing  days, 6,  to the ice thickness (Maykut, 1985). This yields a relationship of  the form: (1.2.1) h2 + Ah=BO where A and B are constants varying with location and snow cover (Maykut, 1986) and h is the thickness of  the sea ice. Nearshore melt can be modelled in a similar fashion  as a linear relationship between the accumulated days with temperature greater than freezing  (Maykut, 1986). From the above relationship (1.2.1) the estimate of  ice growth, dh/dt,  will be of  the form: (1.2.2)  dh/dt  = f(l/h)  + other terms suggesting that the rate of  observed ice growth will be inversely proportional to the ice thickness. The atmospheric temperatures are much lower than the oceanic temperatures during the winter season. The resulting heat flux  from the water to the atmosphere causes ice formation.  As ice forms,  it begins to act as an insulating layer inhibiting ice growth by reducing the heat flux  from the water to the atmosphere, which for  drafts  from 0-0.4m is 1-2 orders of  magnitude greater than the heat input from thicker perennial ice (Hibler, 1980). In addition, initial ice growth often occurs at higher rates due to turbulent wind mixing (Smith et al., 1990) in the upper centimetres of  the water column. Frazil ice crystals can be mixed below the surface  increasing growth rates. Wind also advects the newly formed  frazil  ice to the edge of  an open water area, thus allowing more frazil  ice to form and enhancing the ice growth rates (Cavalieri and Martin, 1994). Ice thickness Rate of  ice growth 0 cm 13 cm day"1 50 cm 2 cm day"1 100 cm 0.8 cm day"1 400 cm <0.1 cm day"1 Table 1-1: Ice growth rates for  central Arctic pack ice used as model input by Thorndike et al (1975). A heat balance model (Cavalieri and Martin, 1994) of  ice formation  accounts for  all significant input of  heat into the ice from the ocean and atmosphere: (1-2.3) F t = F t + F,  - Fb  - F  +F v ' 1 n e t 1 t ^ 1 L 1 B 1 s ^ 1 oceanic where the components on the right hand side are the turbulent heat flux,  the longwave radiation down and up, the shortwave radiation down (Cavalieri and Martin, 1994), and the oceanic heat flux  (Melling and Riedel, 1996a) respectively. In this model, snow is allowed to absorb a fraction of  the temperature difference  between the atmosphere and the ice, reducing the temperature gradient and conductive heat flux  but is not modelled explicitly (Melling and Riedel, 1996a). From this, the freezing  growth rate of  ice becomes: (1.2.4)  dh/dt = F„ e/piLi where L[ is the latent heat of  fusion  and />, is the ice density (Cavalieri and Martin, 1994). The individual components of  the heat flux  utilize measured meteorological data to estimate sea ice growth. The observationally based model of  Cavalieri and Martin (1994) is easily implemented and has been adapted to allow for  possible snow cover and applied successfully  to ice growth in the Beaufort  Sea (Melling and Riedel, 1996a). Most current ice growth models should contain the caveat that they are only accurate for  shallow sea ice drafts  less than one metre in thickness (Maykut, 1985). At drafts  larger than this, influences  of  porosity and non-uniformity  in the ice may become more important. For most purposes, these estimates of  sea ice growth are adequate as sea ice at 0-0.4m grows an order of magnitude faster  than thicker ice (Maykut, 1985). This growth rate ensures that the greatest contribution of  thermal cooling to ridge formation  is in the development of  thin ice available for ridging rather than thermal growth of  previously ridged ice. 1.3. Ridge morphology In an aerial view, the ridge sail protrudes above the level ice surface  (Figure 1-1) while the greater mass of  the ridge keel extends below into the water. Ridges often  appear in linear sections joined by complex corners or bends with corner regions exhibiting greater accumulations of  ice than linear sections (Bowen and Topham, 1996). Some locations may exhibit other ridge configurations  such as the honeycomb appearance observed in the Weddell Sea (Granberg and Lepparanta, 1999) where large spatial variations in ridge configuration  are not correlated to the distance from the ice edge. In the Beaufort  Sea the arrangement of  ridges has been observed to form orientated failure  patterns (Hibler and Schulson, 2000). Using a sea ice model, Hibler and Schulson (2000) found  ice flaws  formed  along preferred  failure  lines intersecting at angles of  25-30 degrees with a natural preference  for  flaws  to form in an alternating manner as damage propagated. Full modelling of  ridge orientation becomes complicated further  by land-ice interactions and as such, ridge orientations are often  treated statistically or ignored in ice models. Typical to all ridges, the keel underneath the sail is both wider and taller to achieve isostatic balance. This balance occurs over scales of  20 to 100 m across a ridge (Bowen and Topham, 1996; Melling et al., 1993) as the force  generated by the sail's ice area can exceeds the buoyant force  created by the area of  the keel directly underneath the sail and some proportion of  the sail gravitational force  is thus carried the wider flanks  of  the keel. In addition, the keel must balance the additional gravitation force  of  any snow cover. Figure 1-1: View of  ice cover in the Beaufort  Sea. The intersection of  two floes  to form a ridge structure is clearly visible across the upper left  side of  the photo. Additional structures are also present. Courtesy of  H. Melling. Observations of  ridge structure often  focus  on the ridge sail topography due to the relative ease of data collection above the waterline. This is a questionable approach for  quantifying  ice volume, as the undersea area and volume of  ridged ice is considerably greater than that of  the upper surface.  Multi year sails are often  eroded with sail to keel thicknesses decreasing to approximately 3A of  their original values (Timco and Burden, 1997) and can be as smooth as level ice (Granberg and Lepparanta, 1992). Further complications include snow drifts  that pile up at ridges similar to drifts  at a snow fence  making discernment of  ridge shapes difficult  (Granberg and Lepparanta, 1992). Field studies have provided rough generalizations of  ridge geometry, although individual ridges vary greatly. Averaged ridge ratios and geometries displayed in Table 1-2 illustrate the sail-keel thickness and width ratios needed to satisfy  isostacy. The idealized triangle keel geometry assumed in most ice ridging models (e.g. Thorndike et al., 1975; Flato and Hibler, 1995) is not observed except in larger ridges (Lepparanta and Hakala, 1992). Multi year ridges generally exhibit broader and more rectangular keels (Timco and Burden, 1997) than first  year ice, differences  attributed to seasonal degradation processes (see chapters 1.5 and 1.6). Due to the mechanisms of  ridge formation,  most ridges begin with a loose porous block structure where the draft  is a multiple of  block thicknesses (Melling et al.,1993). These structures consolidate through freeze  and melt processes as the winter progresses (Rigby and Hanson, 1976) although this consolidation layer is highly variable (Timco and Burden, 1997) and freeze-melt processes are not understood for  porous structures. Some estimates of  first  year ridge porosity have ranged in the literature from 20-45 % (Bowen and Topham, 1996) with reported averages of 30% (Timco and Burden, 1997). Multi year ridges, due to ridged ice evolution processes (discussed in chapters 1.5 and 1.6), generally have very low porosities. The porosity also varies within the vertical structure of  a ridge, ;with higher porosity in the keel than the sail attributed to lower relative gravitational packing forces  (Lepparanta and Hakala, 1992; Melling et al., 1993). Ordered identical spheres will pack to a porosity of  25%, this comparison indicates the loose nature of  the typical ridge structure with porosity 30% (Lepparanta and Hakala, 1992). The strength of  sea ice is obviously influenced  by the ridge structures.. For engineering design, the load applied to a mechanical structure will be a function  of  the ridge thickness, porosity/consolidation, and horizontal ridge extent (Tom Brown, University of  Calgary, personal communication). On larger scales, models of  ridge formation  consider the strength of  level ice, which must fracture  and crumble to form ridges. Compressive strength of  the ice is a function  of the amount of  thin ice and its thickness (Richter-Menge and Elder, 1998). Ice strength decreases with increasing scale, such that lab-scale strengths are approximately 2 orders of  magnitude greater than mesoscale strengths (Dempsey et al., 1999). The tendency for  the tensile strength in-situ strength of  ice to be less than results from  lab testing is attributed to pre-existing cracks in the ice (Richter-Menge and Elder, 1998). Ridge ratios First year ridges Multi year ridges Keel to sail height 3.95 + 0.12 4.5-5 3.17 + 0.08 3.1 -3.3 Keel width to height 3.91 +0.16 Keel width to sail width 4.4 3.2 Keel area to sail area 7.96 + 0.33 Sail angle 32.9 + 9.2° 25° 20° Keel angle 26.6+13.4° 33° Porosity -30% Very low Table 1-2: Common ridge statistical ratios reported in Timco and Burden (1997) from  a survey of 112 first  year ridges and 64 multi year ridges. Italicized values are reported in Melling et al., (1993) from  the literature. Of  interest is the ratio of  keel to sail geometries with keels taller and wider than sails by an approximate factor  of  four. 1.4. Mechanics of  ridge formation The ridge geometries and keel drafts  of  the previous sections are not static, but constantly evolving. The mechanical and thermal processes acting on the ice vary seasonally to provide an annual cycle in ridge evolution. This balance of  processes can be simplified  to a picture where the formation  of  ridges occurs mainly in winter, and ridge degradation primarily in summer (Figure 1-2). It is expected that with a solid understanding of  winter and summer, the shoulder season sea ice development can be modelled as a combination of  known mechanisms. For simplicity, this thesis will address mechanical ridge formation  in winter and summer ridge ablation. During formation,  the convergence and divergence of  the pack generate both ridge and open water formation.  In the ice covered and land-bounded Arctic Ocean, divergence and convergence are essentially one process, as divergence in one region causes convergence in another. Winds and currents continue to open leads and polynyas throughout the winter season where new ice forms  and can then be compressed by thicker pack ice to form  pressure ridges (Smith et al., 1990). Winter: Directions of ice movement driven by external forcing leading to ridge creation. I Ice growtff" Open water Ice growth Surface mixed layer Summer: Atmospheric heat flux Open Atmospheric heat flux water Melt from atmospheric heat flux added to surface water Erosion from enhanced flow past keel. Possible hydraulic flow Surface mixed layer Figure 1-2: Schematic of  the dominant processes acting on ridges during winter and summer. Thermodynamic melt is indicated in red, thermodynamic growth in blue, and mechanical forcing  in grey (movement of  ice) and purple (movement of  water). The contributions of  mechanical forces  to ridge formation  have been extensively modelled through particle simulations. Initial work by Parmerter and Coon (1972) dealt with the compression of  a rubble-filled  lead by an ice sheet that would break under flexure  with limiting assumptions restricting the repose angles of  sail and keel. The central pile was found  to grow until a maximum height was reached at which point it began to grow laterally. The maximum sail height of  the modelled keels was 5 to 6 times the thickness of  the level ice consumed in ridging. This direct linear relation contrasts with the observed relationship of  16 m'/2 hU 2 found  by Melling et al., (1993) in the Beaufort  Sea and is at the upper limit of  the range reported by Lepparanta and Hakala (1992) of  hm to h\ With the increasing capacities of  numerical models, more recent work has improved the particle simulations. Models have gained realistic block definitions  and inter-particle forces  with frictional  and elastic contacts (Hopkins et al., 1991) and varying initial conditions (Hopkins and Hibler, 1991). Differing  ice types involved in the ridge formation  have been investigated including the compression of  thin ice floes  (Hopkins, 1994) or rubble filled  leads with both polygonal and disc shaped particles (Hopkins et al., 1991). A coherent picture of  the stages of  pressure ridging can be developed (Hopkins, 1998) based on the results of  various particle simulations. Initially, the compression of  thin ice against a thicker floe  initiates ridge formation.  The keel appears to function  as the support for  the sail portion of the ridge with growth exhibiting cyclic alterations of  sail and keel development (Hopkins, 1994). With continuous driving ice velocities, the sail reaches a maximum height first  (Hopkins, 1996a) after  which the keel grows to its maximum draft.'  At this final  stage, growth continues in the horizontal direction until all thin ice has been consumed. Finally, the ridge evolution continues with a compression of  the ridge as characterized by a thick rubble field  (Hopkins, 1998). The final  draft  reached by the keel in these models agrees with the observed relationships of  Melling et al., (1993) and produces porous structures matching observed porosity estimates. Of  interest, several features  and controlling factors  of  ridge development are apparent from particle simulations that produce ridge geometries and porosities consistent with observations (Hopkins, 1998). While the thickness of  lead ice determines the ridge width and thickness (Hopkins, 1994), the ice sheet velocity (within observed ranges) does not have a significant  effect on the resulting ridge structure (Hopkins, 1998). In all simulations, the ridge profiles  appear to have a trapezoidal keel shape (Figure 1-3), in agreement with observations (Lepparanta and Hakala, 1992) and in contrast to the triangular shapes of  larger scale models (e.g. Thorndike et al., 1975). The ridges formed  from  an intact frozen  lead require much more energy for  the potential energy of  the constructed ridge than a similar ridge constructed out of  rubble blocks (Hopkins and Hibler, 1991). This is likely due to the excess energy required to break the blocks, although as the ratio of  work to potential energy is approximately 15 or 20 to 1, most of  the energy is consumed by frictional  processes regardless of  the initial ice conditions (Hopkins and Hibler, 1991; Hopkins, 1994). Keel cross section [m] Figure 1-3: Keel schematic showing trapezoidal (solid line) and triangular (dashed line) cross sections. Convergent ice motion can lead to rafting  of  sea ice as well as ridging. Rafting  involves the movement of  one ice sheet over another, and in contrast to ridging, produces low porosity structures. Babko et al (2002) identify  two types of  rafting  processes; simple rafting  where one sheet slides atop another, and finger  rafting,  where the ice sheets fracture  perpendicular to the ice edge and form  fingers  of  overthrusts and underthrusts. In the Weddell Sea, rafting  of  thin sea ice has been observed to increase ice growth substantially (Lange et al., 1989). Waves break and raft the newly formed  ice resulting in thicker rafted  ice areas and regions of  open water. These regions of  open water are then able to freeze  at the open water rate leading to an increased volume of  ice production. A conventional-view of,  rafting  holds that only the thinnest ice is available to raft.  Parmerter and Coon (1972) restrict rafting  to ice thinner than 0.17 m while Babko et al., (2002) suggest a higher cut-off  value of  0.75 m where ridging takes place only for  ice thicker than 0.75 m. These simplifications  do not truly represent the ice conditions observed. Ice over 2 m in thickness has been observed to raft,  facilitated  by breaking at the edges and lubrication from  brine drainage in the upper sheet or by a rubble field  of  thin ice which can serve as a ramp for  the thicker ice sheet (Babko et al., 2002). Rafting  and ridging are equally possible for  medium thicknesses of  ice and there exists no range at which either should be excluded. No guidelines are currently in place for determining the initial conditions leading to rafting  although it is suggested (Babko et al., 2002) that rafting  is the preferred  form  of  interaction between floes  of  uniform  thickness and ridging occurs, as simulated (Hopkins, 1994), from  the interaction of  one thick and one thinner floe. While the standard ice thickness evolution models (Thorndike et al., 1975; Flato and Hibler, 1995) do not include a rafting  mechanism, newer ice models are beginning to include a rafting component (Haapala, 2000) within their mechanical processes. This will likely improve the accuracy of  modelled ice within the rafted  draft  range (1 - 5 m). The accuracy of  the thick ice found  in the largest ridges produced by the ridging models has yet to be addressed and is the focus  of  this thesis. 1.5. Winter ridge evolution Once a ridge is formed,  it continues to evolve through the growth of  a consolidated layer, packing and lateral erosion, the deterioration of  the unconsolidated rubble, and ice melt (Lepparanta et al.,1995; H(|>yland, 2002). Of  these, the separate contributions of  the processes of  packing, erosion, and melt are unclear as they are thought to occur in parallel and are discussed in chapter 1.6. The process of  consolidation is in comparison, well understood. After  a ridge is formed,  the rubble will begin to consolidate by freezing  from  the water surface  downwards (Bowen and Topham, 1996) until there exists a frozen  nucleus at the water level connecting with the level ice as a continuous ice field  (Lepparanta and Hakala, 1992). The consolidation process can be further  separated into two phases, an initial phase where heat energy is redistributed between water and the surrounding ice blocks, and the main consolidated layer formation  phase (H())yland, 2002). Models of  this initial consolidation phase indicate that a purely conductive heat transfer  is too slow (H(|>yland, 2002) and thus some flow  of  water through the ridge structure is expected. This is supported by observations of  consolidation processes in level ice where fine-grained  brine drainage networks (Cole and Shapiro, 1998) having porosities of  up to 10% are developed. Combining this with a ridge structure's macro porosity averaging 30%, the total porosity of  a ridge on formation  is estimated to be 40% (H(|>yland, 2002). It is this high porosity that we believe causes the quick heat energy transfer  throughout a ridge structure and this porosity is essential for  transferring  heat to the ice during the melt season as well (see sections D and E). Observations by divers at the start of  the spring melt found  that the ridge keels were porous to bubbles from  the divers equipment, suggesting a permeable ridge structure (Lepparanta and Hakala, 1992). Once the heat energy has been redistributed, the consolidation layer begins to form  as the atmospheric freezing  temperatures are transferred  through the ice via conductive heat flux (H(|>yland, 2002). As the consolidated layer grows, its growth is determined by the thermal history rather than the surface  heat balance (Maykut, 1986) due to the time needed for temperature changes to penetrate through the already consolidated ice. The growth of  the consolidated layer proceeds much faster  than level ice growth, as would be expected for  a structure which is not forming  a complete sheet of  new ice, but merely freezing  the gaps between ice blocks. Analytical models predict growth rates approximately twice that of  level ice (Lepparanta and Hakala, 1992; H<|>yland, 2002; Veitch et al., 1991). Comparing the observed consolidated layer depth to expected values; Lepparanta et al., (1995) observed consolidated layer depths in agreement with analytical estimates while H(j)yland and L<|>set (1999) and Veitch et al., (1991) observe shallower than predicted consolidated layers, perhaps due to incorrect porosity assumptions, insulation by the ridge, or two and three-dimensional effects. Observational studies by Lepparanta et al (1995) and H<|)yland and Lcfiset  (1999) document the complete degradation of  seasonal pack ice ridges in the Baltic Sea and in a Spitsbergen fjord,  and describe the evolution of  the consolidated layer in the ridges. From early February to mid April the consolidated layer of  ice grows downward as water pockets freeze  and unifying  the loose porous blocks. This leads to a reduction in total keel porosity from  28 to 18% (Lepparanta et al., 1995) and along with mechanical arrangements in block packing, begins to decreases the high porosity of  a first  year ridge. Due to the formation  of  the consolidated layer, the porosity of  the ridge is now not uniform.  Consolidated ice may have porosity of  only 3.4% (Hoyland and Loset, 1999) while the porosity of  the unconsolidated keel remains high. Observations have shown either no change in the unconsolidated keel porosity after  consolidation (Lepparanta et al., 1995) or minimal decrease (Hoyland and Loset, 1999) and thus suggest that a constant porosity in the loose, unconsolidated blocks is an adequate assumption for  most modelling approaches (Hoyland, 2002). This implies, that while the consolidated layer of  a ridge is resistant to erosion processes, the loose and unconsolidated keel may be susceptible to mechanical erosion. 1.6. Spring and summer ridge ablation The third process to affect  ridge evolution, after  formation  and consolidation, is ablation from thermal melt and mechanical erosion. These processes are most visible in summer months when overall ridge draft  decreases. Melt in summer occurs through solar heating on the ice surface leading to melt ponds and through heat added to the water column in areas of  open water. For melt occurring in the winter, any heat must come from  below the mixed layer where sufficient heat may be entrained upwards (Figure 1-2). In Arctic waters, the temperature in winter can increase with depth due to atmospheric heat loss (Figure 1-4) although the temperature anomaly above freezing  is very small when compared with the summer oceanic temperatures. Heat budget models similar to those discussed for  ice growth can be derived for  the melt of  level ice (Lepparanta et al., 1995) where the dominant direction for  melt is vertical rather than horizontal ablation. This follows  as lateral melt rates are the same order of  magnitude as vertical rates such that most of  the change in volume is due to changes in thickness ratherthan area (Bjork, 1992). The main body of  work examining ridge degradation and summer evolution has been observational and often  focused  on seasonal ice. For example, Lepparanta et al (1995), Lepparanta and Hakala (1992), Veitch et al., (1991), and H<(>yland and L(j)set (1999) observe the complete degradation of  seasonal pack ice ridges in the Baltic Sea and a Spitsbergen fjord. Although the reported work focused  mainly on the consolidation processes, some observations on ridge melt were made. Lepparanta and Hakala (1992) reported that at the start of  the spring melt season the keel blocks were well rounded and easily collected by divers. Lepparanta et al., (1995) also reported loose keel blocks although they found  no evidence of  erosion. A summer melt cycle, including the transition to multi year ice, was observed by Rigby and Hanson (1976). They observed a strong depth dependent melt rate with deep draft  features showing higher loss rates than shallower portions of  the keel. Shallower portions also showed more resistance to drilling, indicating harder ice, and it is suggested that the deeper parts of  the keel may shield the shallower portions from  thermal melt and mechanical erosion. Consolidation in the first  year ridge was evident with voids in the ridge volume becoming slush filled  as the summer progressed, leading to multi year ridges of  lower porosity and lower melt rates than first year ridged ice. Of  importance, the general ablation rate for  the ridge keel that is several times that of  the level ice indicating that ridge specific  processes are influential. Winter Temperature Profiles Summer Temperature Profiles Figure 1 -4: Typical temperature and salinity water profiles  with depth [dbar] from  the Beaufort  Sea in winter and summer seasons. The typical changes in the mixed layer depths are evident here with the summer mixed layer being much shallower, and the change in density greater, than in the winter. Data provided by H. Melling. Thermal melt in a ridge is more complicated than in level ice due to its structure of  level ice blocks that are interspersed with large voids that will affect  drainage, temperature conduction through the ice, and mechanical circulation. No complete models of  ridged ice melt exist, although simplified  models have investigated the importance of  brine pockets and channels (Bitz, 1999) and geometry (Schramm et al., 2000). on melt rates. A complete understanding of  ridge melt is of  great importance to ice melt estimates but models consistently fail  to reproduce the observed (Rigby and Hanson, 1976) higher melt rates (Flato and Hibler, 1995; Schramm et al., 2000). With ridged ice containing up to 68-73% of  the total ice mass of  some regions of  the Arctic (Melling and Riedel, 1996a), an improved understanding of  their melt processes is essential. Why is the melt rate higher in keels than level ice? In agreement with Rigby and Hanson (1976), Flato and Hibler (1995) found  that a lack of  preferential  thick ice melt made their model over predict the thick ice in their distribution. Schramm et al., (2000) suggest that factors  such as increased surface  area due to the sloping sides and accelerated flow  around the keel increase the turbulent heat flux  and contribute to thermal melt. Their work models 1-D and 2-D heat conduction effects  in ridged ice to determine if  purely thermodynamical processes can explain melt rates. By assuming 2-D heat conduction (which better approximates reality than a 1-D model) they find  that increased surface  area and internal heat conduction can account for  only a small portion of  the observed enhanced melt rates. The increased heat flux  to the keel from accelerated flow  velocity past a keel structure could produce melt rates 3 times that of  the surrounding level ice, which may explain part of  the enhanced melt rate observed (Rigby and Hanson, 1976) and could work in tandem with mechanical erosion and the influence  of  porosity on the melt processes. It seems likely that the porous nature of  first  year ridges will greatly aid in the conduction of  heat through a ridge and thus increase the melt rate although no provision for porosity exists in current models (Schramm et al., 2000). The observed contributions of  mechanical erosion to ridge degradation is contradictory; Lepparanta et al., (1995) observe little mechanical erosion in winter, Rigby and Hanson (1976) report mechanical erosion causing high draft  loss in summer. Samples of  winter keel blocks in the Baltic (Lepparanta and Hakala, 1992) show loose, eroded, and porous blocks. Rigby and Hanson (1976) observe similar loose blocks throughout the entire summer and estimate that erosion of these blocks accounts for  losses of  up to half  of  the original thickness of  a first  year ridge. An explanation for  the lack of  erosion reported by Lepparanta et al., (1995) could stem from  regional differences.  Rigby and Hanson (1976) observed a ridge in a highly salinity and temperature-stratified  region (similar to Figure 1-4). Lepparanta et al., (1995) observed ridges in the Baltic Sea, where different  stratification  regimes could reduce the potential for  mechanical erosion. Support for  seasonal variations in erosion lies in the reduction of  mixed layer depth (Figure 1-4) in summer. Solar heating both increases the temperature and freshens  the water through ice melt, which increases stratification.  If  the water column is simplified  to a two-layer flow,  the decrease in depth of  the upper layer can enhance flow  past a keel due to conservation of  volume. Enhanced velocity of  flow  will not only increase heat conduction (Schramm et al., 2000) but may trigger hydraulic jumps and internal waves. Similar to the two-layer flows  over mountains or fjord  sills (Farmer and Denton, 1985), the upper 100m of  the Arctic forms  a two-layer system flowing  past a ridge keel where the keel surface slopes are on the same order of  magnitude as mountain slopes (Pite et al., 1995). Laboratory experiments (Pite et al., 1995) predict the existence of  a supercritical lee jet formed  from  a hydraulic jump downstream of  a keel with a forward  propagating bore. Numerical simulations (Cummins et al., 1995) support laboratory experiments and predict variations in hydraulic flow including soliton formation  upstream of  keels for  steep-sided ridges (Cummins, 1995). Predicted hydraulic jumps and solitons occur for  Froude values in the range of  0.25 to 1.35 (Cummins, 1995; Cummins et al., 1994, and Pite et al., 1995) which, using the density variations calculated from  the profiles  in Figure 1-4, can occur under common keel drafts  and flow  speeds. The flows  past the keel are tidally driven, and as such, produce a cyclic pattern of  upstream bores, jumps, and soliton formation  (Cummins, 1995). Ongoing observational work (Marsden, personal communication with R.G. Ingram) is attempting to observe these hydraulic features  in two-layer flow  past ridges in the Canadian Arctic Archipelago. Oceanographic situations vary from numerical and laboratory experiments in that the Arctic ice pack, as they consist of  multiple ridge keels, such that hydraulic flow  past one keel will likely interact with the next ridge feature.  Thus predicting the effects  of  enhanced velocity and hydraulic flow  on ridge melt becomes complex, yet the resulting increase in turbulent and potential energy to the water column has the potential to increase erosion and melt. In summary, while the thermal melt of  level ice is well understood, the processes of  mechanical erosion, thermal melt with flow  through a porous structure, and enhanced velocity past a keel on the ablation rates of  ridges is largely unknown. 1.7. Research objectives The above reviews of  keel structure, formation,  consolidation, and ablation processes highlight what is known about ridged ice formation  and evolution. Previous work has investigated many of the processes that contribute to the evolution of  an individual sea ice ridge, and yet the compilation and assimilation of  these processes into larger scale models is incomplete. The failure  of  current models to produce accurate and inter-model consistent ice cover characteristics (Proshutinsky et al., 2001) indicates that the mechanical and thermal processes that act on a ridge need more accurate representation in models of  sea ice evolution. The goals of  this thesis are therefore: a. To determine why sea ice models are misrepresenting the amount of  thick ridged ice. b. To increase our understanding of  the physical processes of  ridging and melt (and constraints on those processes) that act on thick ridged ice. c. To develop more accurate sea ice evolution algorithms for  the evolution of  the thickest ice. These objectives require a unique approach to sea ice modelling. While most models evolve ice on either basin (e.g. Hibler, 1980) or individual ridge (e:g. Hopkins, 1998) scales, this research will model the evolution of  mesoscale parcels of  ice (0(50km)). This unique scale will allow large area-averaged ice pack characteristics to be compared with individual ridged ice features. Using an ice draft  redistribution model adapted to short (-15-30 days) natural experiments in the Beaufort  Sea, the ridging and melt processes evolving ridged ice can be isolated and studied. This research can be separated into three sections. In section B this research sets out to evaluate currently used sea ice ridging and melt algorithms through the following  steps: 1. Adapt the commonly used ice redistribution model (Thorndike et al., 1975) to evolve ice undergoing onshore convergence in the Beaufort  Sea during the winter of  1997-1998. 2. Compare model results with observations of  ice draft  from  moored subsea sonars. 3. Identify  the discrepancy between modelled and observed ridged ice distributions during winter when ridged ice is being produced. The tendency for  models to create excess thick ridge ice is then explored in Section C. By combining model output with observed ice draft  profiles,  new insights into limitations of  ridge sizes and shapes can be found.  This portion of  the research can be summarized as the steps: 4. Use observations of  ice pack geometrical characteristics to determine geometric constraints on the formation  of  ridged ice. 5. Adapt ridging algorithms to include constraints on ridge size and shapes, allowing for improved simulation of  ice evolution. 6. Validate the importance of  these geometrical constrains through application of  the model to a different  year (the winter of  1991-1992) and to idealized distributions. In the latter part of  the thesis (Section D), the enhanced melt of  ridged ice is investigated. Before sea ice models can reproduce the evolution of  ridged ice, we must understand the process or processes that cause the observed enhanced melt rates in thick ice. Using the ice draft redistribution model, estimates of  the enhanced melt can be made and will lead to the development of  a semi-quantitative model for  porous melt within ridge keels. The steps taken to develop this model are: 7. Adapt the winter model to the melt season and identify  enhanced ablation rates within the thick, ridged ice. 8. Develop a semi-quantitative model for  porous melt within ridge keels. 9. Using assumptions of  keel geometries and ocean characteristics based on observations, calculate expected internal melt rates and evaluate the ability of  internal melt to account for  observed, enhanced ablation rates. The results of  the thesis will demonstrate that current knowledge of  ridged ice evolution is often limited to the evolution of  a single idealized ridge. By extending our scope to look at ridge specific  processes and their impact on the distribution of  ridged ice, we will be able to build on current knowledge and develop a better understanding of  the entire sea ice pack. 2. SEA ICE REDISTRIBUTION MODELS 2.1. General overview of  statistical descriptions In addition to characteristics of  individual ridges, it is useful  .to look at the statistical descriptions of  a region of  sea ice that consists of  ridges, level ice, and open water. On regional or basin scales, it is efficient  to look at a statistical representation of  sea ice rather than a description of each individual feature.  One common representation used is the probability distribution of  sea ice draft  or (interchangeably) thickness (Figure 2-1) where the level ice is represented in peaks at 1.2 and 0.6 m and thicker ridged ice is apparent in the tail of  the distribution. Visible in Figure 2-1 is the often-observed  (Melling and Riedel, 1996a; Melling and Riedel, 1995) exponential tail of ridged ice in the draft  distribution. Draft  [m] Figure 2-1: The statistical representation of  the probability density distribution of  sea ice draft (fraction  per metre). Data are from  the end of  December, 1991 in the coastal Beaufort  Sea. The level ice peaks at 1.2 and 0.6 m are visible as strong signals in the distribution. Open water is a much smaller peak. The exponential tail of  ridged ice stretches from  1.5 m to 15 m, at which point no further  observations of  ridged ice were made. Data courtesy H. Melling. In addition to the distribution of  sea ice thickness, the spatial distribution of  sea ice provides information  about the horizontal structure of  the sea ice pack. Analogous to ridge drafts,  the spatial ridge distribution though the ice pack off  Alaska has been described with an exponential probability distribution P(x)=p exp(-p x) (Mock et al., 1972) where x is the horizontal distance. This and other statistical models for  the distribution of  pressure ridges (Lowry and Wadhams, 1979; Hibler et al., 1974) begin with an assumption of  randomness. Yet significant  directional anisotropy in ridge orientation has been found  and coastal interaction may cause regions of  higher ridging (Tucker et al., 1979; Hibler and Schulson, 2000). To avoid the assumption of  randomness, Rothrock and Thorndike (1980) propose using more general properties of  the ice profile  such as an autocorrelation function. To describe the regional characteristics of  ice roughness, the fractal  dimension of  sea ice can be employed. The slope of  the autocorrelation function  of  ice draft  can be seen to increase as the lag decreases, indicating that the underside surface  of  the ice has a fractal  dimension (Melling et al., 1993; Rothrock and Thorndike, 1980) and thus is a rough surface.  The reported dimensions for sea ice fall  clearly in the range of  a rough profile  with the exception of  Rothrock and Thorndike (1980) who find  smooth, non-fractal  ice. Fractal dimensions found  in the Beaufort  Sea are remarkably constant (1.57 + 0.06, Melling et al., 1993; 1.50 ± 0.05 for  first  year ice and 1.52 ± 0.02 for  multi year ice, Melling and Riedel, 1995; and 1.44 ± 0.064, Bowen and Topham, 1996) and indicate a rougher profile  than terrestrial landforms  (Melling et al., 1993). Overland et al., (1995) suggests the use of  the fractal  dimension of  sea ice as a state variable representing the aggregate properties of  the pack. Physically, the fractal  dimension is a measure of  the surface  area to volume ratio of  the ice and is high, due to the loose block structure of  a ridge. This influences both surface-exchange  processes and creates a unique habitat for  substrate-dependent biology. 2.2. Ice draft  redistribution models Using a probability distribution of  sea ice draft  to characterize the ice pack in a region allows the representation of  ridge formation  as an evolution of  the thickness or draft  distribution probability density function,  g(h)  (Thorndike et al., 1975). While Thorndike et al., (1975) and later researchers model ice thickness, draft  and thickness may be used interchangeably and for  the purposes of  comparison with observed data, this thesis will refer  to ice draft  redistribution models. If  we consider the distribution shown in Figure 2-2, during ridge formation  ice is converging and diverging in different  areas. This is seen in the distribution as a shift  of  ice to larger drafts,  as more ridged ice is created (convergence), and as the creation of  open water to conserve volume (divergence). The thermal effects  of  melting and freezing  shift  the distribution to lower and higher drafts,  respectively. The evolution of  the thickness distribution function,  g(h), was first  formulated  by Thorndike et al., (1975) who represented the advective, thermodynamic, and ridging processes in the following  form: (2.2.1) -§- = -^'(Mg)-—(Jg)  + y/ at oh The first  term on the right hand side represents divergent motion with u being the horizontal velocity of  the ice. The second term represents thermal growth by the thermal growth function,/ and finally,  the third term represents the mechanical ridging. Thorndike et al., (1975) assumed a static form  for/and  a complex ridging function,  y/, which handled both the creation of  new water through divergence and the transfer  of  level ice to thick ice through ridging (Figure 2-3). This function,  y/, depends on the strain rate along with the assumption that the thinnest 15% of  ice is ridged into structures five  times the level ice thickness. A modified  version of  the thickness redistribution equation was developed by Hibler (1980). In this redistribution model, a lateral melt term, F L, is added and the ice growth function,/,  is represented as a heat budget that contains an ocean boundary layer model with a fixed  oceanic mixed layer depth. This formulation  is expanded to include snow cover in the ocean basin scale model of  sea ice dynamics and thickness (Flato and Hibler, 1995). Both formulations  of  the ridging function,  y/, were improved to create maximum deformed  ice thickness that scales as the square root of  level ice, hm, in agreement with Melling et al., (1993) and Hopkins (1998), although the restriction that only the thinnest 15% of  ice may ridge is maintained. E TO a> O ta O C7SJ O C=J T H E R M O D Y N A M I C S Freezing > <-Leads from divergence < Open Water M elting M E C H A N I C S Ridges from convergence Thermo-dynamic mode Deformed i c e T i m r r r A 6 B Thickness (m) —i— 10 —i 12 Figure 2-2: Schematic of  the effects  of  mechanical and thermal forcing  on the probability density function  of  ice draft.  Level ice peak is at 1.8 m and indicates the proportion the ice pack that is level ice of  this thickness. Figure courtesy of  H. Melling. Participation Function: Ice removed Figure 2-3: Form of  mechanical ridging function,  y/. Level ice is removed by the participation function  and redistributed through the transfer  function.  Open water is created to conserve area. Hopkins (1996b) compared the thickness redistribution function  (Thorndike et al., 1975) with the redistribution of  thickness with the ridge formation  processes of  Hopkins (1996a) in a mesoscale model. The initial ice pack consists of  polygonal parcels of  ice that can interact through the elastic-plastic contact model driven by specified  strain rates. This produces ice pack thickness distributions over time that are found  to be in reasonable agreement to the thickness redistribution predicted by Thorndike et al., (1975) although only a single distribution was calculated in Hopkins'(1996b) study. One limitation to either the Thorndike et al., (1975) or Hibler (1980) approach lies in the details of  the ridging term in (2.2.1). The assumption that only the thinnest 15% may ridge makes the form  of  ^ non-linear; assuming a cut-off  height above which ice cannot ridge, such as the limit of first  year level ice growth, can remove this non-linearity. Thorndike (1992) formulated  ^as a product of  the strain. The thermodynamic growth is represented as a mean annual forcing  which produces thickness distributions that require several years to reach equilibrium. The resulting thickness distributions produced by this simplified  approach resemble those of  the more complicated formulations.  Further work by Thorndike (2000) reaffirms  that a simpler form  of  y/ will reproduce the general distribution and explains the negative exponential form  of  the thickness distribution as the result of  a repetition of  thin ice piling on thicker ice with decreasing probability for  increasing thickness. 2.3. Ice area evolution models Ice draft  redistribution (IDR) models are not the only types of  sea ice evolution models currently used by researchers. A comprehensive review by Savage (2001) for  the Canadian Ice Service focuses  on a different  class of  sea ice models referred  to here as ice area evolution (IAE) models. In brief,  while IDR models follow  the evolution of  the sea ice thickness distribution, an IAE model tracks the evolution of  the mean thickness and area fraction  of  several distinct ice types. Both model types contain information  about the mean thickness of  ice and amount of  ridged ice, but only the redistribution model includes the information  about the shape and size of  the created ridges. The premise of  an IAE model, that the mean thickness of  the ice is tracked rather than the entire thickness distribution, simplifies  the computational requirements. Initial models (e.g. Shulkes 1995) tracked the mean thickness and area fraction  of  both level and ridged ice categories providing information  on the concentration of  ice-covered waters in a region (or the area percentage covered with ice) and the amount of  ridged ice. Further enhancements led to the five-category model of  Haapala (2000) that tracks the mean thickness and area fractions  of  lead, level, rubble, rafted,  and ridged ice categories. The Canadian Ice Service and other sea ice forecast  communities are interested in predicting ice thickness and concentration for  shipping routes and thus much of  the detail in the form  of  the EDR model is not of  use to those communities. For this reason, IAE models have been incorporated in the Canadian Ice Service predictive model. For other sea ice research and applications, the physical shape of  the ridged ice redistribution can be of  importance. In climate models, the distribution of  ridged ice into different  thickness categories may affect  the melt rates of  ridged sea ice, the drag of  ridged ice on currents and winds, and the strength of  the ridged ice components. Predictions of  extreme keel events for  offshore  construction also rely on accurate thickness distributions of  ridged ice. It is necessary for  these reasons that an understanding of  the physical processes affecting  the redistribution of  sea ice is obtained and included in Arctic ice evolution models suggesting that EDR models may be more appropriate. Savage (2001) noted that the underlying physics in redistribution models based on Thorndike et al (1975) has changed little in the past 25 years. He states that they "contain a significant  amount of empiricism in the formulation  of  the redistribution mechanics that is both arbitrary to some degree, and difficult  to verify"  (Savage, 2001). Some researchers have used this as reasoning to exclude a physical representation of  the formation  of  ridges altogether in favour  of  the simpler IAE models. While this approximation may be adequate for  some purposes, I suggest that a physical representation of  ridge formation  is necessary in order to gain an estimate of  the distribution of  the thickness of  ridged ice, to understand the physical factors  influencing  sea ice thickness, and to appropriately represent the ridged ice formed  through a season. Through the development of  a regional ice draft  redistribution model, a new regional scale evaluation of  the underlying physics affecting  the draft  distribution of  sea ice has the potential to update IDR models for  more accurate application to a variety of  research problems. SECTION B: THE WINTER REDISTRIBUTION OF SEA ICE IN THE BEAUFORT 3. APPLICATION TO THE BEAUFORT SEA 3.1. Geographic location The Beaufort  Sea (Figure 3-1) is an excellent location for  studies of  the redistribution of  coastal sea ice. Normal circulation in the region is anti-cyclonic and the sea is bounded by land and land fast  ice to the south and east. In average years, the sea is covered with first  and multi year ice except for  August and September (Figure 3-2) when marginal seas offshore  become ice free  due to easterly winds in early summer (Melling and Riedel, 1996b). This allows for  the development of  new level ice at the time of  freeze  up providing a natural experimental situation for  the tracking of  the development of  first  year level ice to ridged multi year ice. Further, during the winter, a large broad flaw  leads open at the edge of  the land fast  ice due to storm events (Melling and Riedel, 1995; Melling and Riedel, 1996) which forms  new level ice fields  at varying times through the season. Ice motion in,this area is largely divergent (Colony and Thorndike, 1984) such that new ice formed  in the flaw  lead should move offshore.  Perhaps due to the offshore divergence and continued ridging, ice tends to be more heavily ridged further  offshore  (Melling' and Riedel, 1996). Exceptions to the normal summer ice circulation have occurred periodically (eight times from 1954-1991, Melling and Riedel, 1996b) where seasonal anomalies in atmospheric circulation have led to northerly winds reversing the anti-cyclonic circulation of  the Beaufort  Sea and closing the open water. This causes larger ice concentrations in the summer months and potential shipping difficulties  (Melling and Riedel, 1996b). This change in circulation pattern also allows ice import from  the northwest coast of  the Canadian Arctic Archipelago, which is a region subject to high ridging, and thus increases the largest draft  of  ridge features  observed in these years. Observations of  ice in the Arctic show large inter-annual variability (Venegas and Mysak, 2000), which is often  attributed to the ten-year period of  the Arctic Oscillation (Flato, 1995). Of regional interest, some of  the largest variations are often  observed and predicted (Flato, 1995) in the Canadian waters of  the Beaufort  Sea with great impact on shipping and supply delivery for northern communities (Melling and Riedel, 1996b). Figure 3-1: The Beaufort  Sea area of  the Canadian Basin. Mooring sites in the Beaufort  Sea and the land-fast  ice edge in January and February 1998 (from  Regional Ice Chart, CIS). Ice data were acquired at Sites 1 and 2 during 1997-1998 and at Site 3 during 1991-1992. All four  sites are located north of  the land fast  ice and the recurring flaw  lead at the fast  ice edge. REGIONAL ICE ANALYSIS Western Arctic ANALYSE REG IO NALE DE GLACE L'Ouest deUArctique 30 AUG/AOU 1999 7 ^eVD/ ^ I T T 7 SOW- / / V 145V 70N a 8 1 7 4. 1 9 7 4. 2 8 7 4. H / I w C 7. 7 4. F / T \ pii 1 5 4 1 7 4. 7 4. w i i H ^ 3 7 i 1 7 4. v Q 7 M / ^ T X 3 7 9 1 7 4. 7 4. 7. 7 4. « TUKTCTtAJCTUK 4. KZ7 2 5 . 7 4 . . 4 5 7 4. R / 9 + \ • INUVIK K 0 7 V / ^ N 9 1 2 2 7 4. 7 4. WZTX x / £ \ W 1 9 . 7 4 . . Y / J ^ X 2 8 . 7 4 . . 140W 135W TEMPERATURE (°C» (Past 7 days) TEMPERATURE <°c> (7 demiersjoura) Ftoint Barrow Inuvik Kugluktuk Cambridge Ba MEAN MOYENNE NORMAL NORMALE 2.5 M 5.8 8 .4 5 .8 6 .9 4 . 2 4 . 6 KUC 130W 125W 120W Figure 3-2: Partial ice chart for  the Western Canadian Arctic on August 30th, 1999 from  the Canadian Ice Service. The dotted area indicates the open water with less than 10% ice concentration while areas closer to the shore contains bergy water where total ice concentration is less than 10% cover. 3.2. Available data from moorings To determine the evolution of  sea ice on a scale of  hundreds of  kilometres, a tool such as undersea sonar offers  general profiles  of  the thickness of  the sea ice and ridge distributions for  a region. Upward looking sonar on nuclear submarines has been utilized for  several decades. This technique allows for  spatial coverage of  a region but requires multiple trips for  temporal evolution and can be limited by shallow waters and political boundaries. To complement submarine data, upward looking sonars moored to the sea floor  can provide ice draft  profiles. This combines temporal and spatial coverage with the stationary sonar viewing new areas of  ice due to pack movement. While this dependence on natural ice motion provides irregular spatial sampling intervals, the long time records allow for  temporal evolution of  a region's level and ridged ice distribution. Central to the ability to measure spatial distributions is the ice velocity, which allows for  the mapping of  draft  values into a pseudo-spatial co-ordinate. It also provides valuable information  on the recent kinematic history of  the pack, particularly in relation to place of  origin and interactions with the flaw  lead and stamukhi zone to the south and east of  the mooring sites. This is obtained with an acoustic Doppler current profiler  (ADCP) used in tandem with an ice profiling  sonar (IPS) to provide velocity and thickness data as described in detail in Melling et al., (1995) and Melling and Riedel (1994). The moored IPS approach provides a better representation of  the thickness distribution than other experimental techniques. Aircraft  or satellite mounted sensors record only the profiles  of  above water ice, which consists of  only 10% of  the actual volume of  ice (from  buoyancy). The low sail to keel height and width ratios (Table 1 -2) and the possibility for  snowdrifts  to conceal the true thickness distribution also suggest that measurements of  the thickness distribution must take an "upwards-looking" approach. The method of  electromagnetic induction, where the difference  in conductivity between air and seawater can be utilized to determine ice thickness, does provide a "downward-looking" approach (Melling et al., 1995). Yet electromagnetic induction requires surveys by small planes, which is a costly and time-consuming way of  obtaining the evolution of the sea ice draft  distribution through a season. A moored IPS transmits an acoustic pulse upwards and identifies  the echo associated with the bottom of  the ice or air-water interface.  The IPS field  of  view from  50-m depth is approximately 0.8 m across (Melling, 1998b), small enough to have negligible effect  in 'blurring' the topography of  keels that are tens of  metres in width. The sonar detects the envelope of  first returns, which in general defines  the outline of  a volume that may be as little as 70% ice (Melling et al., 1993). The travel time between the pulse and echo combined with data from  tilt, pressure, and temperature sensors determines the draft  of  the ice. Challenges in determining the draft  of the ice and extensive details of  this instrument can be found  in Melling and Riedel (1994). The accuracy of  the draft  of  sea ice determined by an upwards-looking IPS is sufficient  to discern the changing draft  of  sea ice. The varying speed of  sound in water throughout the year, due to seasonal changes in the water properties (Figure 1-4), can introduce errors in the calculated thickness of  up to 0.2 m. However, calibration relative to open water data decreases this error to ± 0.02 m. The total error in the draft  determination is approximately ± 0.05 m (Melling et al., 1995). Confidence  limits for  draft  increase under compact ice conditions, whereas open ice conditions have a detrimental impact on confidence  limits for  velocity. As the ice evolution model in this thesis has a resolution of  10 cm, this accuracy is sufficient  to provide excellent data for  both model initial conditions and validation. Measuring the velocity and direction in a water column is the often-used  application of commercial ADCPs. The ice-water interface  is a very strong target for  the sonar similar to the signals obtained from  the seafloor  in ship mounted ADCP applications. The nominal accuracy of the Doppler velocities obtained in the data used in this thesis is ± 0.7 cm s"1 (Melling and Riedel, 1994). While the resolution and nominal precision of  the ADCP data is given as 0.25 cm s 'and 0.7 cm s" 1 respectively, additional error sources must be considered. The magnitude of  the error introduced into the measured speed by the assumption of  a constant speed of  sound in the water column above the mooring is less than 0.5% and thus not a concern. The dominant source of uncertainty in the velocity determination is due to the error in measuring the Doppler shift,  which was estimated as 0.01 m s"1 in each component for  the ADCP used. Achieving this level of accuracy in the field  was not always possible due to lowered signal-to-noise ratios, and an estimated error of  ± 1 cm s"1 is reasonable. Errors in the direction of  the ice motion are also quantifiable.  Due to the location of  the moorings relatively near the magnetic north pole, the ADCP compass must be specially calibrated in a high latitude nonmagnetic enclosure (at Tuktoyaktuk, NWT; Melling et al., 1995), which reduces the error in the ADCP heading to a random error of  ± 2 degrees. Additionally, sampling errors in each component of  velocity contribute additional uncertainty independent of  speed, leading to larger errors in low-speed drift  (+ 8 degrees at ice speeds of  0.05 m s"1; Melling et al., 1995). An eleven-year data set of  ice draft  profiles  in the Beaufort  Sea from  moored subsea ice profiling and acoustic Doppler sonar is available and provides a rich data resource. Processing of  the data was completed by David Riedel and Humfrey  Melling (IOS, DFO, Sidney, BC) and provided to T. Amundrud for  this thesis. A full  discussion of  the methods used for  data processing can be found  in the technical report by Melling and Riedel (1994). Of  note to this research is the distinction that the ice evolution is sampled in time rather than space. For a thickness distribution to be constructed, the ice draft  from  the IPS must first  be resampled in the spatial domain using the ADCP velocity record. Sufficiently  long sections of  the draft  profile  to allow for  statistical reliability can then be used to produce ice draft  distributions (Melling and Riedel, 1994). The determination of  the processes involved with summer ice degradation may only be partially elucidated by the draft  profile  record. The profiles  provide no data on the consolidated layer depth, the ridge porosity, and characteristics of  the water column below the ice. This limits the ability of  the data to identify  mechanical erosion and repacking events that respectively remove ice mass or conserve mass while decreasing porosity. While the eleven-year data set provides much information  about the sea ice draft  at one location in the Beaufort  Sea, the natural movement of  ice past the moorings can lead to little insight on the evolution of  a Lagrangian parcel (on the order of  100 km) of  ice. Draft  distributions through the Beaufort  Sea are not homogeneous (as observed in the data). Ice can be formed  locally in leads and polynyas, which introduces differing  level ice types into a small parcel of  ice. It becomes important to thus identify  instances where a Lagrangian parcel of  ice is viewed multiple times by the mooring such that the data provides a natural experiment where the temporal evolution of  the ice can be determined. 3.3. Natural experiments Wintertime ice drift  in the southeastern Beaufort  Sea follows  a repeating sequence with three phases. In the first  phase the pack may be almost or completely motionless for  prolonged periods. Ice draft  measured during these periods may be used to evaluate freezing  algorithms in an IDR model. The first  phase ends when strong easterly winds force  the ice pack to the west-north-west, a heading with an offshore  component in the area of  study (Figure 3-1). Westward movement opens a wide flaw  lead at the edge of  fast  ice (which is aligned in an east-west direction, Figure 3-3) and narrow leads within the drifting  pack. Because open areas freeze  rapidly under wintertime conditions, expanses of  young weak ice form  quickly. Northwesterly winds initiate the third phase in the sequence, pushing thick seasonal and perennial ice southward toward the coast to the accompaniment of  heavy ridging of  younger ice. ySfN w H ^ J / j o \ P N/J0\ 9 1 7 4. T P s / £ \ r n r 7 4. V K/roX [ T V f T T " o/ih B 4 74. y  IJ 13 7 4.; .4 ey 1 9 7 4. 4 5 7 1. l/l J 9_ T T ] \5~B7 p Q/ToN R/1o\ T T 7 4. W 9 1 7 4. V6  V v /3"^ w/TtS x / a A Y^Tx i 4 3 3 7 4 1, 6 3 1 7 4 1.| •2 6 2J 7 V 1 1 7 ( 7 4 1 W 7; W Figure 3-3: February 1st 1998 ice cover. The land fast  ice is visible as area 'E' in this plot, the offshore  ice in section 'X' is assumed to be homogeneous in an alongshore direction. The flaw  lead is at the boundary between areas 'E' and 'X' but is not visible in the ice chart. Partial Ice Chart, Canadian Ice Service. During the two active phases, an area of  pack ice might possibly be driven landward then seaward along the same trajectory such that the same ice could be measured by sonar before  and after  the building of  ridges. However, in general, our analysis is reliant upon the banded structure of  the ice field  that is established by the cyclic opening and closing of  the flaw  lead (Melling, 1998a). The nascent band is the flaw  lead itself,  a strip 10-20 km in width and 200 km in length, from  the moorings at 134° W to Cape Bathurst in the northeast. The lead freezes  over to form  an alongshore band of  uniform  ice, typically located inshore of  the bands from  previous openings, which may have been ridged during prior events of  convergence (Melling, 1998a). With the saw-toothed movement of  the pack (onshore and offshore  motion superimposed on the general westward drift,  Figure 3-4), ice moving back over the mooring was formed  further  to the east, but within the same alongshore band as that initially viewed. We assume that the properties of  ice bands are homogeneous in the alongshore direction, by virtue of  identical age and deformation history. The critical 200-km length of  the flaw  lead in conjunction with westward average ice drift  imposes an upper limit to the interval over which the assumption of  alongshore homogeneity is valid. There were three events during the winter of  1997-98 when an ice band was compressed against the land-fast  ice, ridged and then blown offshore  to be re-observed by the sonar at Site 2. This data set is particularly suited for  verifying  the mechanical ridging portion of  the ice draft redistribution model because a separate data set is available from  the mooring at site 1 located 70 km to the south (Figure 3-1). The first  of  these events occurred during January and early February 1998, and ice drift  is indicated on Figure 3-4. Ice draft  distributions were calculated from  observations accumulated over approximately 50 km of  ice at irregular intervals in time. The difference  in north-south ice speed between the sites is the local north-south convergent rate of  strain in the pack. Because the land-fast  ice is close to the moorings and has an east-west orientation, the principal stress causing ice failure  will be that resulting from  north-south convergent strain. Ice movements in the east-west direction are not tightly constrained by land-fast  ice or coastline. The observed thermal growth of  level-ice modes provides guidance in the selection of  events for analysis of  ice draft  redistribution. At Site 2 during the winter of  1997-1998 the sonar tracked progressive growth of  level ice only until the 65th day of  1998 (6 March; Figure 3-5). Because our thermal growth model matches the observations until this date (Bellchamber-Amundrud, et al., 2002), we are confident  that the assumption of  alongshore homogeneity is valid. The subsequent thinning of  level ice despite winter conditions indicates that a different  population of  ice has drifted  over the mooring. Among the three events of  high strain rate in 1997-1998, only the first ice motion event from  days 9 to 32 of  1998 (Figure 3-5, 9th January to the 1stFebruary) occurred while level ice was thickening as predicted via numerical simulation. January 1998 Ice Movement Past Site 2 [km] WEST < > EAST Figure 3-4: Eulerian displacement vector illustrating ice drift  at Site 2 from  December 28th, 1997 to February 1st, 1998. Stars indicate relative positions of  moorings on December 28th. Squares mark the endpoints of  track segments used to calculate distributions of  draft. Figure 3-5 clearly illustrates that the ice seen by the mooring throughout the winter is not all of the same age. How valid is it at any time to assume that the ice passing the mooring over a period of  time is homogeneous and has grown under the same thermal conditions? Figure 3-6 shows the evolution of  the probability density distribution from  days 300 to days 461 during the winter 1991-1992. Similar to the winter of  1997-1998 (Figure 3-5) the inconsistent draft  of  the level ice peak after  day 380 of  1991-1992 is visible in this plot of  the evolution of  draft  distribution. Further discussion of  the applicability of  the data for  comparison and validation of  the modelled ice evolution will be discussed in chapter 4.2. Contours of the distribution of ice draft Ice Draft  [m] Figure 3-5: Evolution of  ice thickness during the winter of  1997-1998 at site 2. Contours showing the discontinuity in level ice thickness. Progressively darker shaded regions contain 1,5, 10 and 30% of  the distribution (heavy black line is the 10% contour). The break in the seasonal level-ice mode is evident at Day 65 of  1998. Later level-ice modes track ice formed  at other locations or at times after  freeze  up. Contours of the distribution of ice draft Draft  [m] Figure 3-6: Eulerian versus Lagrangian ice perspectives. Contours showing the discontinuity in level ice thickness. Progressively darker shaded regions contain 1,5, 10 and 30% of  the distribution (heavy black line is the 10% contour). The inconsistent thermal growth of  the level ice peak in a section of  1991-1992 data indicates that care must be taken in drawing conclusions from  natural experiments. 4. A REGIONAL ICE DRAFT REDISTRIBUTION MODEL 4.1. General model We begin with the general equation for  the evolution of  the thickness distribution, g(h) (Thorndike et. al., 1975): (4.1.1) J = + ^ ot oh Here, the first  term accounts for  divergence of  ice, the second for  the thermal growth of  the ice, and the third for  mechanical redistribution. Equation (4.1.1) evolves the thickness distribution, however, embedded within the terms on the right hand side are conservation of  ice mass and volume equations. Thorndike (1992) has expressed (4.1.1) in discrete form,  where the thickness distribution is defined  as the distribution of  ice within bins of  equal increment in thickness, dh,  represented by the vector G. The evolution of  G with each time step, At, is expressed as: (4.1.2) G t + X & G t = ^ ~ d i v I + F + ^ G t + d i v S where divl,  F,  and ¥ are the matrices representing divergence, thermal growth, and mechanical ridging, respectively. The time step At has been set to one hour, an approximation of  the time required to build a small ridge. At a typical rate of  strain (3% per day) 85 m of  ice is consumed per hour, as the separation of  moorings is 68 km. This is roughly the extent of  the smallest floe participating in ridging (see chapter 5.3). A ridge may therefore  be created in a single time step, but most ridges will require the accumulation of  strain over several such steps. Model results are insensitive to increase of  the time step to as long as 6 hours, which is the expected time for  a ridge of  maximum size to be created. The matrix F,  which controls the thermal evolution of  ice thickness, is discussed in section 4.2. The matrix / is the identity matrix, which allows the thickness distribution to respond to the scalar value for  the rate of  divergence, div  [s"1]. In addition, a source vector, S, creates open water to conserve area with convergent motion. The mechanical redistribution and divergence of  the thickness distribution are discussed in section 4.3. 4.2. Thermodynamic ice growth The change in ice thickness via thermal forcing  is determined by the growth rate, f(h),  which varies with time. For each time step in the IDR model, f(h)  is calculated using the ice growth algorithms developed by Cavalieri and Martin (1994) and adapted to the observed increase in level ice draft  in the Beaufort  Sea by Melling and Riedel (1996) and forced  by atmospheric data observed on the coast at Tuktoyaktuk (Canadian Climate Centre, Environment Canada). For the winter of  1997-1998, the growth rate is calculated assuming no oceanic heat flux  and allowing the snow to accommodate 30% of  the temperature difference  between the ocean and the atmosphere. As the thermal conductivity of  snow is 6-8 times smaller than the thermal conductivity of  ice (Makshtas, 1998), the assumed snow thickness is less than 7% of  the ice thickness, or 7 cm for  1 m ice. Recent work (Lipscomb, 2001) has focused  on the need for  a new thermal growth algorithm that does not display the diffusive  qualities (Figure 4-1) of  traditional thermal growth redistribution methods (the Thorndike/Hibler approach; Thorndike et al., 1975; Thorndike, 1992; Hibler, 1980; and Flato and Hibler, 1995). Lipscomb's (2001) new remapping algorithm employs an additional tracked mean draft  variable to reduce numerical diffusion  of  the sea ice draft  distribution. Comparing the traditional Thorndike/Hibler thermal redistribution with the thermal remapping in a local redistribution model run with no mechanical forcing  illustrates the effectiveness  of  the remapping algorithm in reducing diffusion  of  distribution peaks over short time scales. 5 4 growth 3 rate constant 2 1 0 Remapping model algorithm initial distribution • i j j l i B i i m i i 0.5 Original Thorndike algorithm MBP 0.5 5 4 growth rate a 3 function of 2 thickness 1 0.5 Draft [m] 0.5 Draft [m] Figure 4-1: The Lipscomb (2001) remapping model algorithm (adapted from  Lipscomb, 2001) applied to an assumed distribution function  (note y-axis is the probability distribution of  ice [m1]). The initial distribution is indicated in the upper left  panel by the light grey bar. With a constant growth rate that does not change with ice thickness, the initial distribution grows to 0.5m and diffuses  slightly. With the original Thorndike algorithm (upper right panel), the distribution spreads out over a range from  0 to lm in draft.  Using a growth rate that is dependent on ice thickness the Lipscomb remapping model grows ice to 0.25m in a sharp peak (lower left  panel). The Thorndike algorithm grows ice to this height, but introduces significant  diffusion  (lower right panel). The accurate observations-of  ice draft  from  the Beaufort  Sea provide an excellent opportunity to evaluate the Lipscomb remapping algorithm using the IDR model. The mechanical forcing  within the model is switched off  by setting the strain rate to zero. Results from  a run of  the model for  17 October to 5 November 1997; when the rate of  strain in the pack was weak, demonstrate the accuracy of  the remapping algorithm (Figure 4-2a). A second evaluation ran from  freeze-up  on 27 September to 20 November 1997. The observed and modeled ice-draft  distributions have the same level ice peak (Figure 4-2b). Since the model was run without mechanical forcing  to isolate thermal processes, there is no ridged ice in the modeled distribution at thickness greater than the 0.4-m level-ice peak. The obvious skill in simulating ice growth through freezing  permits the change of  the distribution of  ice thickness associated with ridging to be isolated and independently simulated. rj, (a) level ice growth (b) ice formation draft  (m) draft  (m) Figure 4-2: Observed and modeled probability density of  draft  (fraction  per decimetre) for  ice growing thermodynamically during the winter of  1997 at site 2. (a) Growth from  conditions initialized using observations on 17 October, (b) Growth from  an ice-free  sea surface  at the observed time of  freeze-up.  The model indicates more level ice than observations because mechanical redistribution is not active in these simulations. Further evaluations of  the ability of  the thermodynamic component of  the model are carried out using draft  data from  1991. Figure 4-3 illustrates the improved performance  of  the Lipscomb scheme compared to the Thorndike/Hibler algorithm. Slight differences  in the location of observed and modelled level ice peaks at day 348 may indicate small errors in the thermal growth rates or may be due to the comparison of  Eulerian draft  data with Lagrangian model output. Non-uniform  growth of  level ice throughout the winter season can be observed in the sonar data indicating that the modal ice passing oyer the sonar may have been formed  under slightly different  conditions. Probability density of ice draft: days 330 to 348 (a) Thorndike/Hibler 0.6 1 2 draft [m] n a (b) Lipscomb remapping 1 2 draft [m] Figure 4-3: Results of  the Thorndike/Hibler and Lipscomb thermal redistribution schemes (Lipscomb, 2001) on the final  probability density distribution of  sea-ice (fraction  per decimetre), (a) Thorndike/Hibler approach, (b) Lipscomb approach. Applying the redistribution model to the 1991 freeze-up  season, the model is run from  October 9th to November 26th beginning with open water and allowing divergence to continue to generate open water for  new ice creation throughout the period. The 9th of  October is chosen as the start of the simulation as there were no significant  periods of  possible ice growth observed in the meteorological forcing  data from  Tuktoyuktuk before  this date. The observed and modelled sea ice draft  distributions of  November 26th match very closely (Figure 4-4). A slightly lower draft  is observed for  some of  the level ice in the modelled distribution. This may be due to numerical diffusion  or indicate that the thinner level ice is preferentially  ridged. 0.6 0.5 0.4 0.3 0.2 0.1 Probability density of ice draft 0 0.5 observed distribution modelled distribution i j j initial distribution is open water y | ice growth at day 330 1.5 draft  [m] 2.5 Figure 4-4: Modelled draft  distribution and observed distributions on November 26th, 1991 (fraction per decimetre). The model does an excellent job recreating the level ice depth and shows little excess diffusion.  Ice greater than the level ice peak is not present in the model distribution as rafting and ridging of  ice was not permitted in this run. 4.3. Ice motion: divergence and ridging We have adapted the ice thickness redistribution model of  Thorndike (1992) to represent the ice draft  observations in the Beaufort  Sea. As the sonar was moored close to the edge of  land-fast  ice and because the pack is structured in shore-parallel bands of  homogeneous ice, we assume that the component of  motion perpendicular to the fast-ice  edge is the dominant influence  on the ice draft  distribution. Ridging occurs in a two-dimensional world. Onshore displacements of  the pack are convergent and re-distribute ice into ridges. Offshore  displacements are divergent and create leads. The rate of  convergent strain was calculated from  the difference  in the offshore  component of velocities measured at site 2 and site 1, about 70 km to the south. Because variance at frequencies exceeding 4 cycles per day was dominated by observational error, the time series of  velocity were smoothed using a Butterworth filter  with 6-h cut-off  (see chapter 4.4). Episodes of  prolonged compression and dilation of  the ice cover were identified  within a time series of  uni-axial strain rate. The divergent ice motion was incorporated into the model through the terms -divl  and divS, where div  is the divergent strain rate, set to zero during compressive motion. Observations of  open water by the ice-profiling  sonar at site 2 provided an independent measure of  recent local divergence. For the modelled ice motion event, the local measure of  divergence was much smaller than the regional estimate (about 10%). We conclude that divergence was not uniformly  distributed between the sites, but was concentrated at a large lead observed at site 1. The same scaling was not necessary during convergence, since the large lead moved rapidly south of  site 1 where new ridging within it was no longer incorporated in the convergence calculated from  velocities at sites 1 and 2. This is discussed further  in chapter 6.5. Compressive strain causes mechanical deformation.  The matrix acts to distribute ice into ridges in proportion to the compressive strain rate, £ by taking ice from  bin j and putting it into bin i. This matrix takes the form: (4.3.1) x ¥ 0 j = a e (4.3.2) = (4.3.3) %j = pb(J)s,  for  2(j-1)  + 1 <i <j3(j) Equation (4.3.2) is the participation term and acts to remove level ice from  the categories participating in ridging. The participating ice, of  draft  h, is denoted in the matrix by the subscript j, where h=(j-l)h*.  h* is the bin size or draft  increment between elements of  G. A 0.1-m value is used in this model, as the 95% confidence  in the observations allows draft  increments of  this size in the observed distribution. In a Eulerian perspective, an ice-free  area equivalent to the area of ice "lost" to ridging must appear during convergence. The constant 'a' conserves area. On short time scales, 'a' is set to zero, implying that ice-free area created by local convergence appears somewhere else, likely far  to the north, and is not measured by the sonar." In this approach, we do not conserve area, but instead re-normalize the thickness distribution at each time step. The ice redistribution operator, is applied over all participating level ice thicknesses, j, via a participation function  that describes which level ice thicknesses participate in ridging. The participation function  divides the amount of  strain that is accommodated by ridging among the thickness bins judged to be level ice. Ice-free  leads are closed first.  Thorndike et al., (1975) and Hibler (1980) allow the thinnest 15% of  ice to participate in ridging. They use a weighting coefficient  that decreases linearly with increasing thickness, in order to ridge thinner ice preferentially.  Thorndike (1992) argues that the appropriate range for  participation is all ice thinner than the thermodynamic equilibrium. This assumption is used in the IDR model, implying that all level first-year  ice is available for  ridging. Thin ice is more likely to fail  than thick ice since the buckling strength of  a floating  plate increases as the square of  its thickness (see chapter 5.3). To approximate this sharply rising disincentive to ridging thicker floes,  a participation function  inversely proportional to thickness squared is applied to the strain rate s over the range of  ice draft  participating in ridging. The level ice "lost" during convergent motion is redistributed into thicker ice according to a transfer  function,  b(j),  where j is the index of  the thickness bin wherein ridged ice is created. Traditionally b(j)  is a constant, moving an equal area of  ice into each ridged ice category. This implies that ice is piled into ridges of  triangular cross-section and maximum possible draft (Figure 1-3). (Thorndike, 1992). The value of  b(j)  is chosen to conserve ice volume: (4.3.4) b(j) 7 - 1 PU) I < <•=20-0+1 New ridges are poorly consolidated in most cases and have a high fraction  (30%) of  ice-free  void space (e.g. Lepparanta et al., 1995). We multiply b(j)  by a fixed  factor,/?,  so that the volume occupied by the ridged ice exceeds the volume of  ice within the ridge. This factor  facilitates  the comparison of  model results to observations acquired by sonar that delineates the envelope of first  returns from  ridges. The inclusion of  porosity does not change the geometries of  the keels or the distribution of  ridged ice, affecting  only the proportion of  ridged to level ice. A further  component of  (4.3.3) is the range over which the transfer  function  is applied, here written as 2(j-\)+\  < i < fi(j).  Ice is ridged into a range of  drafts  that extends from  double the draft of  participating level ice (rafting)  to some maximum possible ridge draft.  Thorndike has defined the maximum to be a fixed  multiple, k, of  the participating level-ice thickness (Thorndike, et al., 1975; Thorndike, 1992) where k is the truncation factor.  Hibler (1980) used a maximum thickness that is the product of  the truncation factor  and the square root of  the thickness of  participating level ice. Initially, we use the DDR model to examine the effect  of  a constant truncation factor, setting the maximum keel draft  to 20 times the square root of  the level ice participating such that P(j)  = 20 -1 )/h*  (where h* is the bin size, here 0.1 m, and j= 1 is the open water category). The transfer  function  and truncation factor  control the shapes and dimensions of  ridges and thereby are essential to a model's ability to reproduce observed thickness distributions for  ridged ice. Failure of  a model to replicate observations may imply incorrect formulation  of  these algorithms. 4.4. Observational errors Observational errors in the probability draft  distribution are calculated as outlined in Rothrock (1986) and depend on the decorrelation distance of  the ice profile,  which for  all ridged ice can be found  to be approximately 5 m, or 5 samples as the ice profile  is in 1 m spatial increments. Thus the number of  independent samples of  ice, JV  (h),  within a profile  is l/5th the observed number of samples of  ice at that draft  range, h. Following from  Rothrock (1986) and Melling and Riedel (1995), the percent error on the distribution for  ridged ice is (4.4.1) % error a -gm g(h)L!  5 1/2 where L is the total number of  samples in the ice profile.  For ridged ice, as (\-g(h))  ~1 and g(h)L/5  ~ JV(h),  (4.4.1) can be estimated as JV' 1'2. The velocities used in the model to calculate the strain rate are smoothed using a Butterworth filter  with 6-h cut-off.  The power spectrum for  the raw ice motion data at site 2 is shown in Figure 4-5. Visible in the spectrum is the semi-diurnal tidal signal. The noise due to the sampling error 1 cm s"1) of  the Doppler ADCP translates into a noise level in the power spectral density (Emery and Thomson, 1998) of  approximately 5.28 x 104 cm2 s"1 cycles"1, which the spectrum crosses at approximately 4 cycles per day (~ 4.7 x 10"5 cycles s"1). While the noise level would suggest using a six-hour filter,  the validity of  including tide-driven ice motion could also be questioned. Running the model with either a 6-hour or 24-hour cut-off  for  the Butterworth filter (Figure 4-6) makes no visible difference  to the results, as the average strain over the period of  the simulations does not change. The tidal oscillations are thus included in the model. Total error in the strain rate, before  smoothing velocities, is 0.02 day"1. This signal is smaller than the strain rate at most times (except during periods of  no motion) and, as velocities are smoothed before  the strain is calculated, the error in the strain rate is expected to be small. Additionally, in the model, the mechanical forcing  is a cumulative process, such that random errors in the strain and divergence should be smoothed further  by temporal integration. 10 10 T Power spectral density [cm s~ cycles" ] for the ice velocity data for site 2. ,2 10 ,8 10 ,6 10 ,4 10 2 10 ,0 0 2 4 6 8 10 12 14 16 18 Cycles per day Figure 4-5: The power spectral density for  the north-south ice velocity data at site 2 during the winter of  1997-1998. The semi-diurnal tidal peak is clearly visible. The horizontal line represents the noise level associated with the sampling error of  1 cms'1. Influence of filters on ice motion Day of 1998 Figure 4-6: Influence  of  24 (heavy solid line) and 6 (heavy dashed line) filters  on the north-south ice velocity data from  site 2. Lower panel is an expanded section of  upper panel to clearly show daily oscillations in data. SECTION C: GEOMETRIC CONSTRAINTS ON ICE RIDGING 5. CONTRIBUTION OF ICE FEATURES TO ICE EVOLUTION 5.1. Maximum keel draft The IDR model was run for  the period 9 January to 1 February 1998 using Thorndike's transfer function  with b(j)  (4.3.3) a constant (Thorndike et. al., 1975; Hibler, 1989, Bitz et al., 2001, and Babko et al., 2002). This algorithm clearly apportions far  too much ice into the thick end of  the distribution (Figure 5-1) and too little into the 3-5m draft  range. Confidence  bounds on the constructed distribution from  the observed ice pack represent one standard deviation due to sampling error and are calculated as described in chapter 4.4. A constant transfer  function  places an equal amount of  ridged ice in each bin up to the maximum. This comparison of  model results with observations suggests that the constant values used for  the transfer  function  and truncation factor  embody inappropriate assumptions about keel shape and size. Here we explore the assumption concerning maximum keel draft  as a source of  the overestimate in the fraction  of  thick ridged ice. There is a statistically significant  truncation of  the commonly observed exponential roll-off  in the occurrence of  ridged ice at large draft  (Melling and Riedel, 1996a). The truncation has been ascribed to the finite  strength of  the level ice from  which ridges are built (Melling and Riedel, 1996a; Hopkins, 1998). Hopkins (1998) asserts that the maximum height of  a ridge sail is reached when the force  needed to push another ice block beyond the top of  the ridge exceeds the force that will buckle the level ice next to it. At this point a section of  level ice will fail  and contribute fragments  either to widen the existing ridge or to begin construction of  a new one. Observations provide circumstantial evidence in support of  Hopkins' (1998) argument (Melling and Riedel, 1996a). Adding recent data from  almost a decade of  ice-draft  observations in the Beaufort  Sea, a relationship between the maximum observed draft  for  ridges and the draft  of  level ice contributing to them can be established with some confidence.  Figure 5-2 shows the draft  of the deepest keel observed within each of  approximately one thousand 50-km topographic transects versus the draft  of  the level ice adjacent to it. Note that because only the deepest keel of each transect has been plotted, most of  the keel population is not represented. A plausible upper bound to the keel draft  is 20 m 1 / 2 hyi, where h is the draft  of  the adjacent level ice. Hibler (1980) and Hopkins (1994, 1998) proposed a similar truncation curve via geometrical arguments and via numerical simulations of  ridging, respectively. Figure 5-1: Probability density of  draft  (fraction  per decimeter) from  a model run with customary assumptions that all ridge keels are built to a draft  of  20 m1/2 h m and have triangular shape, for  site 2 in 1998. Relative to observations, far  too much ice of  draft  exceeding 10 m is created. Thin lines are confidence  bounds representing +/- one standard sampling error based on degrees of  freedom. Maximum keel heights [m] observed relative to level ice Figure 5-2: Empirical relationship between the draft  of  large keels and that of  level ice adjacent to them, based on observations of  seasonal pack in the Beaufort  Sea during the 1990s. The curve indicates truncation of  keel development at a draft  of  20 m"2 A1'2. Less than 4% of  the data in Figure 5-2 lie above the truncation curve. These points could be associated with pinnacles on keels or could result from  a mistaken identification  of  ice in a re-frozen  crack near the ridge as material used in building it. They could perhaps be associated with ridges built from  stronger multi year ice, although care was taken to disregard such features  for this study of  ridging in seasonal pack ice. Many of  even the biggest keels fall  below or to the right of  the curve representing the maximum. There are four  possible explanations for  such a shortfall.  The ridge may be old, so that the adjacent level ice has grown thicker since it was fractured  to build the ridge. All available thin ice may have been incorporated in the ridge, leaving only nearby thicker ice that is too strong to fail. The keel may have lost draft  since formation  through mechanical collapse or melting. Finally, some constraint may have prevented the keel from  ever growing to maximum draft. 5.2. Floe size and thickness A ridge will develop to maximum size only if  several conditions are met. First, there must be sufficient  contiguous level ice to provide raw material for  the ridge. Second, the stress developed by the action of  wind and current must be sufficient  to cause structural failure  of  the ice. Third, the critical stress must last long enough to complete building the ridge. In this section we explore the implication of  finite  floe  size as a constraint on keel size. Rapid sampling by sonar in this study has provided fully  resolved topographic profiles  of  pack ice. The shapes and dimensions of  keels and of  level floes,  along the direction of  drift,  can be readily determined. Level ice floes  were identified  using the criteria that the floes  must be greater than 10 m in extent and vary by less than ±0.25 m in draft  (Melling and Riedel, 1996a). Floes are delineated by expanses of  appreciably thinner (refrozen  or open leads) or thicker ice (ridges). Adjacent floes  separated by less than 1 m with drafts  within one standard deviation were merged, since a small piece of  rubble separating floes  is not expected to hinder ridging. Ice floes  of  draft exceeding the maximum growth for  first-year  ice at the time observed were not counted. In a final selection, only those floes  sandwiched between thicker level ice capable of  transmitting a critical stress for  buckling were included. Ensembles of  thin floes  of  differing  draft,  surrounded by thicker ice were considered a single expanse of  ice with draft  equal to the average over the ensemble. If  a ridge keel has a triangular cross-section of  width, W,  equal to four  times the draft,  H,  (Timco and Burden, 1997), the extent of  level ice needed to construct a ridge of  maximum draft  can be calculated by conserving cross-sectional area. In two dimensions, the area balance between floes of  length^ and draft  h and a keel of  draft  H  and porosity p will be: (5.2.1) — " f Lh=\HW{\-p)  = 2H 2{\-p) Substituting the relationship H=  20 m U 2 h v l into (5.2.1), the minimum size of  floe  necessary to build a solid keel of  maximum draft  is 800 m, independent of  level-ice draft.  However, with a porosity factor  of  30%, typical of  new ridges (Timco and Burden, 1997), at least 560-m of  level ice is incorporated into a keel of  maximum draft.  Continued ridging of  a wider floe  would increase the keel's width, but not its draft  (Hopkins, 1998). Based on floe-size  data from  Sites 1 and 2 in 1997-1998 (Figure 5-3), approximately 25% of  the floes  are sufficiently  large to form  keels equal to the maximum size H=  20 m1/2 hV 2 controlled by buckling failure.  Thus it is not a shortage of  large expanses of  level ice that cause the biggest keels to fall  short of  maximum draft. On the other hand, 75% of  the floes  are too small to build a ridge of  maximum size. Thus conventional redistribution models, which consider the supply of  level ice to be unlimited, place too much ice in the categories of  deepest draft.  It follows  that the draft  of  keels created by ridging in models should be constrained by the level ice available. When re-arranged, (5.2.1) takes the form: (5.2.2) H  = , 1/2 JL 2(1 - p ) hl/ 2 Which is analogous to the equation for  the maximum possible draft  of  ridge keels, H=20 m1/2 hU 2. We thus propose constraints on maximum keel draft  dependent on both the 20 m1/2/z1/2 restriction from  ice strength and the (f L /2(l-p)) m hV 2 restriction from  floe  size. A truncated lognormal function  provides a good fit  to the histogram of  observed floe  size (Figure 5-4) constructed from  the floe  sizes in Figure 5-3. Since our definition  of  level ice excludes floes less than 10 m across, the truncation value is 2.3. The histogram of  floe  size does not vary with draft  over the range 1-1.3 m. The histogram also appears to be invariant for  thicker floes, although it is sparsely populated by the data available. Cumulative Probability Distribution of Floe Size floe size [m] Figure 5-3: Cumulative distribution of  floe  size (fraction  per floe  size) in the Beaufort  Sea during 1997-1998 at Sites 1 and 2. All floes  larger than 560m are capable of  forming  a ridge of  maximum size. Inset: probability density function  of  floe  size (fraction  per metre on a logarithmic scale). Probability Distribution of the logarithm of floesize Figure 5-4: Probability density of  the logarithm of  floe  size (in m) for  all (0-1.3 m) drafts  of  level ice (solid line). The log-normal curve (dotted line) has a mean of  4.95 and a standard deviation of 1.7. The shaded area encompasses probability densities of  floe  size of  specific  draft  by 0.1 m increment in the range 0-1.3 m. The gray line is the mean probability density of  size for  floes  of  1.3-2.0 m draft.  The truncation of  data at a 10 m floe  size reflects  the minimum scale used in the definition  of  level ice. 5.3. Characteristic floe  length in buckling The mechanical failure  of  an ice floe  in buckling is commonly modeled as the failure  of  an elastic plate (ice) floating  on an elastic foundation  (seawater) where buoyant forces  are analogous to the elastic restoring force  of  a simple spring (Sanderson, 1988). In ridging, we assume that the ice floe  is a rectangular plate and that the force  is applied evenly over one edge of  the plate. The bending energy of  the plate and the energy stored in the buoyant foundation  can be calculated (Den Hartog, 1952) and an expression for  the critical buckling stress derived: (5.3.1) o , = E7T 2(m 2+\)2 tf  , pwg fI 12(1 -n2)m2 fl m Here E is Young's modulus, n is Poisson's ratio, p* is the density of  water, g is the gravitational acceleration (to avoid confusion  with the probability distribution, g(h)),  and m is a mode number (m  > 1). The minimum buckling stress for  a fixed  thickness occurs where da/df L = 0. The corresponding floe  length is: (5.3.2) r 4 _ EK2(m2 +1)2 , 3 J l "12(1 -n2)pj The smallest size satisfying  (5.3.2) occurs for  m=1 when the floe  buckles as a single arc. This is the characteristic length Lc, (5.3.3) 4 En 1}? 12(1 -nl)pwg !/4 Buckling of  floes  smaller than Lc is unlikely because the buckling stress varies in this range as the inverse square of  floe  size. The minimum buckling stress, evaluated using (5.3.2) in (5.3.1) decreases with mode number. cr = 2K ( 2 m +1 (5.3.4) The lowest stress corresponds to the highest integral mode number satisfying  m2 <2-(f [/L c)2-l. Equation (5.3.4) is equivalent to the buckling force  for  a beam derived by Kovacs and Sodhi (1980) and used in two-dimensional simulations by Hopkins (1998). Equation (5.3.3) is greater than past derivations of  the characteristic length (Hopkins, 1998) by a factor  of[47i 2/(l - n2)]1'4 due to the formulation  of  buckling of  the floe  as a plate rather than a beam (Kovacs and Sodhi, 1980; Hopkins, 1998). Since the width of  the ice floe  is expected to add stability, the inclusion of  this factor  is appropriate. The characteristic lengths for  various thicknesses of  ice can be calculated using estimates for Young's modulus and Poisson's ratio. Poisson's ratio for  ice is often  given as 0.3 (Hopkins, 1998, Fox and Squire, 1991). Estimates of  Young's modulus range from  1 GPa to greater than 6 GPa (Hopkins, 1998; Fox and Squire, 1991; Dempsey et al., 1999). Sea ice is a complex, multiphase material near its melting point and traditional strength parameters such as E and n vary with temperature and salt content such that single definitive  values for  E and n cannot be assigned. For purposes of  approximation, Young's Modulus is assumed to be 2 Gpa, as found  by Fox et al., (2001) in Antarctic sea ice. Using these parameters, the critical buckling stress (5.3.1) for  a level ice floe  can be plotted for  various floe  sizes and mode numbers (Figure 5-5) illustrating the minimum buckling stress (5.3.4) needed to buckle the ice. ice thickness: 0.5 m 300 400 500 floe size [m] 600 700 800 Figure 5-5: Critical buckling stress (in MPa) for  a 0.5 m thick level ice floe,  indicating the expected critical buckling stress for  level ice (5.3.1). Visible is the minimum buckling stress (for  floes  larger than Lc, (5.3.4)) given as the solid horizontal line. 5.4. Truncation of  ridge building via limits on ice available We incorporate the limitation on raw material for  ridge building into the EDR model via a stochastic approach. Initially, for  each time step and for  each bin of  thickness that participates in ridging, the model randomly samples a floe  size from  the empirical lognormal distribution. The selected size is compared with the characteristic length for  buckling for  this thickness category. The selected floe  size is not used if  it is less than the characteristic length for  a particular bin; replacement values are selected randomly from  the floe-size  distribution until a value large enough to buckle is obtained. Ridging continues with this floe  size in subsequent time steps until the entire extent of  the floe has been consumed. At this point, a new floe  size is chosen using the above criteria, and ridging begins using this new floe  size for  a truncation constraint. This process is outlined in the flow chart in Figure 5-6. The randomly selected floe  size is mapped to a value for  the truncation constant, /?, where and H=j3h 1/ 2. Values of  p from  (5.4.1) that are greater than 20 m1/2 are set equal to 20 m1/2 to reflect  the constraint of  ice strength on keel draft. Use of  a keel-draft  truncation factor  based on floe  size within the DDR model decreases the fraction  of  ice in the deep ridged categories. Results are displayed in Figure 5-7. The inclusion of the random selection of  floe  size introduces some variance between model runs. This is shown in Figure 5-7 on line d as one standard deviation from  the mean after  500 simulations. The implementation of  the floe  size constraint on ridge building begins to decrease the amount of  the deepest ridged ice for  drafts  exceeding 1 lm (Figure 5-7; line b). The improved concurrence of these modeling results with observations suggests that finite  size of  level floes  may indeed influence  the draft  of  large keels that develop in first-year  pack ice. (5.4.1) How big a ridge to build? For each timestep, enter loop to determine the size of the ridge Input: floesize and iceleft (both zero at start of simulation) \ 7 4 Calculate the amount of ice consumed by ridging in this time step. Subtract from the iceleft to get the remaining ice A End of ridge building algorithm Figure 5-6: Flow chart illustrating the decision process for  deciding the value of  the floe  size to be used in the truncation constraint (previous page). Probability density of ice draft 10 5 ' J 1 1 : 1 3 6 9 12 15 18 Draft  [m] Figure 5-7: Mean probability density of  draft  (fraction  per decimeter) for  ridged ice calculated with various geometrical constraints on ice ridging. Model is run from  the 9th January to the 1st of February 1998 at site 2. Comparative curves are shown smoothed with a 5-point running average for  runs with (a) uniform  volume distribution and a constant truncation (b) uniform  volume distribution and the floe-availability  constraint, (c) exponential (6-m scale) volume distribution and a constant truncation, and (d) exponential (6-m scale) volume distribution and the floe-availability constraint. Thin lines indicate the variation between 500 simulations using the floe  availability constraint indicated as one standard deviation. Similar variation is found  for  curve (b). 5.5. Keel shape lee-thickness redistribution models commonly assume a constant value for  the transfer  function. This assumes that the cross-sectional shapes of  ridge keels and sails are triangular. Melling and Riedel (1995) noted that the empirical exponential form  both of  the probability density of  ice draft  and of  the occurrence frequency  for  keel draft  has implications for  the average cross-sectional shape of  keels. Assuming that width of  a keel is proportional to its maximum draft  and that its shape is independent of  draft,  then observations indicate that the probability density of draft  within keels must on average also be exponential. Melling and Riedel (1995) found  the average distribution of  ice within 21 keels exceeding 20 m draft  to be exponential with a 15-m e-folding  scale. However, the shape of  individual keels is extremely variable (Melling and Riedel, 1995; Timco and Burden, 1997; and Lepparanta and Hakala, 1992). In an IDR model, the keel shape for  features  of  all sizes is required, not just for  the very large keels. To extend the shape analysis of  Melling and Riedel (1995), keels surrounded by level ice were identified  in the ice-draft  profile  for  days 330 of  1997 to day 59 of  1998. Keels were identified  as ridged ice greater than 0.25 m thicker than the adjacent level ice and with maximum drafts  deeper than three times level ice draft  (to exclude rafted  ice). An average keel shape was defined  by searching data for  all keels within five  ranges of  maximum draft:  5.0±0.5, 7.5±0.5, 10.0±0.5, 12.5±0.5 and 15±0.5 m. The populations of  the categories were 578, 202, 64, 20, and 9 keels, respectively. Exponential curves fitted  to the average draft  distribution for  each size range have mean e-folding  scales of  2.7, 3.9, 4.3, 6.0, and 6.5 m respectively. The large variation in the shape of  individual keels is visible in Figure 5-8; the variation in the draft  distribution for individual keels is more than an order of  magnitude about the mean. However, on average, within a ridge, ice of  deepest draft  is less common than that of  shallowest draft  by as much as an order of magnitude for  deep keels. The fraction  of  ice in the deepest bin of  a 10 m keel is 58% lower than the fraction  in the same bin if  ice is uniformly  distributed. Draft  [m] (b) Keel cross section [m] Figure 5-8: Probability density of  draft  (fraction  per 0.5m) within 20 keels of  12.5+0.5 m maximum draft.  Small circles represent individual keels and large circles show average values. An exponential curve with 6.0 m e-folding  scale fits  the average, (b) Cross-sectional shapes (cusped and triangular) corresponding to negative exponential and constant probability density of  draft  within keels. In our stochastic IDR model the approximation of  ice distribution within a ridge as exponential with characteristic scale X=6  m for  all keel drafts  is appropriate to our present purpose. The e-folding  scales found  may suggest including a depth dependent truncation factor;  however for simplicity, a constant value of  6m is used. This value is representative of  the e-folding  scales for large keels and is easily implemented. The transfer  coefficient  in (4.3.3) becomes: (5.5.1) b(j)  = b0e-z/ A where b0 is a constant to conserve volume and z is the draft  in the range 2h to /3(h),  where h is the level ice and J3  is the truncation point. With this transfer  coefficient  (cusp-shaped keels) and a constant truncation value, the IDR model produces an ice-draft  distribution in agreement with the observations up to 15-m draft,  but does not match the distribution of  the thickest ice (Figure 5-7; line c) similar to the results using the floe  availability truncation. With (5.5.1) and the floe-availability truncation algorithm (5.4.1), the IDR model reproduces the observed ridged ice distribution within observational error bounds. In addition, the underestimate in ridged ice within the 4-6 m range is decreased as the constraints place more ice in shallower drafts  (Figure 5-7; line d). Figure 5-9 shows the percent difference  in volume fraction  of  thick ice between the modeled and the observed data. The constant keel truncation and triangular shape results in a 55.5% overestimate of  ice volume for  ice greater than 9 m. Note that the apparent equivalency of  all models for  ice thinner than 12 m in draft  is due to the short duration of  the simulation. Except for the thickest ice, the initial distribution of  ice, rather than the newly ridged portion dominates the distributions. Using an exponential keel shape alone, and with a floe  availability truncation, decreases the discrepancy by 75% for  each case; to a 12.5% overestimate and a 12.5% underestimate, respectively. Over just 23 days, using geometrical constraints, significantly  less ridged ice area and volume (2.8% and 8.8% respectively) are placed in drafts  exceeding 9 m. Figure 5-9: Difference  in ridged ice volume between the modeled and observed density of  ice volume at each draft  (volume fraction  of  ice greater than 9 m thick). Model is run from  the 9th January to the 1st February 1998 at site 2. Comparative curves are shown smoothed with a 5-point running average for  runs with (a) uniform  volume distribution and a constant truncation (b) uniform volume distribution and the floe-availability  constraint, (c) exponential (6 m scale) volume distribution and a constant truncation, and (d) exponential (6 m scale) volume distribution and the floe-availability  constraint. Combining the exponential volume distribution with the floe-availability constraint, curve (d), produces the best ice draft  distribution at both thin and thick ridged ice drafts.  Shaded region indicates confidence  bounds on the observed distribution representing +/-one standard sampling error based on degrees of  freedom. 6. DISCUSSION 6.1. Simulation of  pack-ice development during 1991-1992 Conditions of  ice drift  suited to simulation using the IDR model were encountered also during the winter of  1991-92 (see Figure 3-6). Ice draft  and movement were observed at Site 3 to the southeast of  Site 2 (Figure 6-1). Beginning on day 324 of  1991, the ice moved shoreward and then seaward in a circle that returned it to the same location on Day 362 (Figure 6-1). Under this unique circumstance, the assumption of  along-shore homogeneity in pa:ck ice properties was not required. 1991-1992 ice Movement WEST < > EAST [km] Figure 6-1: The ice motion across mooring site 3 during the winter 1991-1992 is plotted as accumulated displacement. The natural experiment from  days 330 to 348 of  1991 when the ice pack remained in the vicinity of  the mooring, is used for  the verification  of  the thermal growth model. The trajectory of  ice drift  over the mooring during this period is shown in the inset (plus sign indicates start of  ice motion event). For this event, observations of  ice motion were not available shoreward of  Site 3, where changes to the distribution of  the ridged ice occurred during the 38-day period. However, because Site 3 was only 29 km from  the edge of  land-fast  ice (Regional Ice Chart, Canadian Ice Service), the average strain rate of  ice within the zone was computed from  the velocity at Site 3, assuming zero motion at the edge of  the fast  ice. The 1991 simulation used the same parameters as that for  1997, except that the snow cover in 1991 was set to account for  only 5% of  the temperature difference between the ocean and the atmosphere, to match observed level ice growth. As before,  the calculated divergence was scaled by 1/10th to reflect  local estimates of  opening based on observations of  new ice. This scaling implies that most of  the strain has been accommodated by opening of  a flaw  lead at the edge of  land-fast  ice to the south of  the sonar mooring site (Melling, 1998a). Results are presented in Figure 6-2. The model result on 28 December is a good replica of  the observations. The model truncates the distribution at 17 m draft,  beyond the deepest observed draft.  The difference  represents an absence in the transect of  only 10 m of  ice profile  in this draft range. This loss may be explained by the statistics of  sampling. Figure 6-2: Probability density of  ice draft  (fraction  per decimeter) using the floe-availability constraint and assuming an exponential keel shape to simulate pack-ice development in the winter of  1991-1992. The trajectory of  ice drift  over the mooring is shown in the inset (arrow heads indicate direction). 6.2. An idealized simulation of  pack-ice evolution in winter We now present an idealized simulation of  the development of  seasonal ice over a four-month period, without observational initialization or verification.  An initially ice-free  sea surface develops ice as a result of  atmospheric forcing  observed at Tuktoyaktuk during 1997-1998. The ice is compacted at a constant rate of  1% per day. Results from  the IDR model, with keel-draft  truncation based on floe  availability (5.4.1) and constant (triangular shaped) and exponential keel draft  distributions, are displayed 90 days after initiation (mid January) in Figure 6-3. The combination of  the floe  size constraint and cusped keel shape produce a distribution with significantly  less deep ridged ice. Ice deeper than 10 m covers only 48% of  the area and contains only 45% of  the volume of  the corresponding ice produced by the unconstrained keel and truncation algorithm. This represents a significant  portion of  the ridged ice, with 20.5% of  the volume of  ridged ice exceeding 10 m for  the floe  size constraint and exponential keel shape compared with 49.4% of  the ice exceeding 10 m for  the unconstrained case. Because all approaches create the same volume of  ridged ice, the area of  ridged ice is greater when using the floe  size and keel shape constraints and the average draft  of  deformed  ice is less. Such differences  may have important implications for  the simulation of  pack ice in climate models. After  90 days, the floe  size constraint and cusped keel shape have generated an exponential distribution for  draft  between 3.0 and 18 m, with a 3.11 m e-folding  scale. The impact of  the maximum-draft  constraint, H=20 m1/2 hU 2, is visible as steps in distribution above 10 m representing the maximum keel size (20 m1/2 hV 2) that could be formed  from  the growing level ice draft.  The e-folding  scale is similar to observations previously reported from  the Beaufort  Sea in mid-January (Melling and Riedel, 1995; 1996a). The exponential distribution of  ice in keels and the floe-size  constraint cannot generate realistic draft  distributions for  ice when used alone and must be used in tandem. Figure 6-3: Probability density of  ice draft  (fraction  per decimeter) from  simulation of  idealized pack-ice development over 90 days, assuming various combinations of  the truncation model and transfer  functions.  Best-fit  exponential curves with e-folding  scales of  4.81 and 3.11m are found  for the floe  availability model with triangular and exponential keel shape functions  respectively; for exponential keels with a constant truncation the e-folding  scale is 3.89 m. 6.3. Convergent motion and ridging In our present simulations of  ice draft  redistribution, the onshore compression of  pack ice between mooring sites is assumed to be the sole contributor to ice ridging. Confirmation  of  the validity of  this assumption can be made by comparing the amount of  level ice consumed by ridging during the simulation with the change in level ice area from  initial to final  observation. With reference  to Figure 5-1, the model "consumes" approximately 27% of  the area of  level ice and the observations indicate a 25% decrease in level-ice area. The level-ice area lost to shoreward motion is adequate to generate the observed increase in ridged ice volume, without requiring appreciable ice ridging from  alongshore compression or shear. 6.4. Keel shape Runs of  the DDR model with an exponential distribution of  cross-section versus draft  within keels (6-m scale length), but with no constraint on keel draft  from  floe  size, do reproduce the observed change in ice-draft  distribution over tens of  days. However, runs over longer periods create a draft  distribution with too much thick ice: after  90 days the e-folding  scale over drafts  of  3-18 m is 3.89 m (Figure 6-3). The e-folding  scale representative of  smaller drafts  is larger because the modeled draft  distribution without floe-size  constraint tends to the 6 m e-folding  scale typical of keels, far  in excess of  the 2-3 m scale typically observed (Melling and Riedel, 1995; 1996a). The use of  an e-folding  scale of  6m for  creating ridges less than 10m in maximum draft  may seem inappropriate based on observed mean e-folding  scales, however 40% of  all 5m keels have e-folding  scales that exceed 5m indicating that this assumption may be acceptable for  smaller keels. In addition, running the model with a keel shape e-folding  scale of  3m does not visibly change the modelled distribution within the 3-10 m draft  range and thus using a constant value seems a reasonable approximation. Simulations of  ridging by other researchers, without restriction on the amount of  level ice, build ridges with a trapezoidal shape (Hopkins, 1996a). The draft  distribution for  these ridges may have more ice at the maximum draft  than in the thinner keel flanks,  in contrast to the observed exponential roll-off  with draft  in real ice fields  (Figure 6-4). The floe-availability  constraint and cusped keel shape constraint, which limit the draft  and width of  keels, create draft  distributions in agreement with observations. Keel cross section [m] 0 E" -5 Q. -15 0 10 20 30 40 50 Figure 6-4: Schematic of  ridge shapes illustrating the varying distribution of  ice with draft.  The exponential roll-off  (heavy solid line) places far  less ice at the thickest keel drafts  than the triangular keel shape assumed by Thorndike et al., (1975) or the trapezoidal keel shape of  Hopkins (1996a). The observed e-folding  scales of  the 5, 7.5, 10, 12.5, and 15 m keels increase with draft  (as 2.5, 3.8, 4.3, 6.7, and 5.8 m respectively. The draft-dependent  increase in A, was not included in the IDR model for  simplicity, and the constant value of  6 m was chosen to ensure that the distribution of  ice in the thickest keels would be compatible with observations. What would be the impact of  varying X  with h? For keels with smaller drafts,  more ice would be placed in the thinner keel edges, thus increasing the distribution of  ice in the smaller draft  ranges (<10 m). However, during the simulations in this thesis, as the distribution of  ridged ice is on short timescales and the amount of  ridged ice already at these drafts  is appreciable, the difference in draft  when X  was decreased was not visible. In contrast, for  the ice thicker than 10 m, changing X  will greatly affect  the visible amount of  the thickest ice as the ridging in the model makes up a large proportion of  the ice at those drafts. The assumption is also made that the e-folding  scale will not change seasonally. As the distribution of  level ice flow  size is not seen to vary with thickness, and thus season, we assume that keels are created with similar constraints on level ice availability throughout the season. Yet a 5 m keel in the early winter is likely to have been created by a large expanse of  thin ice according to (5.2.2), as thicker ice is not present in the seasonal Beaufort  Sea. However, in midwinter, floe  size availability (Figure 5-3) suggests that a 5 m keel is more likely created by the available thicker ice, as limited by floe  extent. The importance of  level ice extent on keel shape is not known. To investigate this, a model similar to the particle model of  keel formation  developed by Hopkins (e.g. 1998) could explore the evolution of  keel shapes with level ice consumption during ridging. However, such work is beyond the scope of  this thesis. The seasonal variation in keel shape may be due to observational limitations. Ice draft  profiles used to identify  keels may not separate adjacent keels that do not have thinner level ice or open water between them. Extended edges of  keels may consist of  smaller ridges formed  at a later time from  ice abutting the large keels, features  not resolved by the keel criteria. Thus a keel initially formed  with an e-folding  scale of  6 m could be observed to have a greater e-folding  scale after subsequent ridging events. Finally, it must be noted again that the variability in e-folding  scales between individual keels far exceeds the variability in average e-folding  scales with draft.  Keel shapes vary by orders of magnitude, and the e-folding  scale used in the EDR model is a parameterization of  the average ridged ice distribution, not a representation of  the shape of  an individual keel. 6.5. Importance of  divergence on the local characteristics of  the ice pack In developing and adapting the IDR model to the mooring data in the Beaufort  Sea, two changes in the formation  of  open water are made to the model from  the standard equations. In (4.1.2), open water is created directly in response to the divergence of  the ice pack. In addition, open water is created in ridging to balance the loss of  ice area due to convergence. Both of  these open water terms are necessary to balance the area of  ice lost during ridging and divergence. For mooring data, such as the 1997/1998 ice profile  record at site 2, both these sources of  open water are not visible in the probability draft  distributions during the onshore-offshore  motion event A and thus are excluded or reduced in the model. During event A, the ice first  moves southwards compacting against the coast over both sites 1 and 2 (Figure 6-5). While the ice is moving south, the relative motion between the mooring sites is both divergent and then convergent in the first  10 days. According to (4.1.2), open water should be created through divergence from  days 11 to 13, and then" created to balance area lost through ridging during days 14 to 16. Yet in the distribution observed between day 25 and day 27, no open water or newly grown level ice is observed at site 2 (Figure 6-6). At site 1, over 80% of  all ice observed is less than 0.2 m in thickness. At this time of  year, ice will grow to 20 cm in less than 5 days (calculated from  thermal growth model of  Melling and Riedel, 1996). This indicates that the new level ice observed on day 25 at site 1 (Figure 6-6) was created after  day 18 and this is not from  the earlier divergent motion around day 12-13 (Figure 6-5). Figure 6-5: Motion of  ice over sites 2 and 1 during event A in 1998. Upper panel displays the northward motion of  ice across the moorings from  days 10 to 47. The IDR model simulates the evolution of  the distribution of  ice from  days 10 to 32. Crosses indicate the temporal start point of the ice profiles  used to construct the statistically independent density distributions of  draft.  Circles indicate distributions plotted in figure  below. The lower panel displays the divergence [day"1] calculated as the velocity difference  between the sites divided by the distance (68 km). Where is the open water created during divergence and ridging? While ridging occurs locally at moorings, divergence may not. Riding is assumed to occur at many locations during compression, as supported by the success Of the floe  availability constraint for  ridging that creates multiple ridges of  varying sizes. Divergence may occur by leads opening in more localized regions. If  the moorings do not observe these leads because they lie between the sites, then no record will appear in the data. 0.5 0 i 0.5 0 ( 0.5 0 0.5' 0 0.5< Probability density distribution of draft 0.5 site 2 day 10 f\ 0 1 2 3 site 2 : day 24 _ A 0 1 2 3 site 2 : day 32 _ y v _ 0 1 2 3 site 2 day 37 / V 0 1 2 3 site 2 / V day 47 0 ( 0.5 0 ( 0.5 0 0.5* 0 I 0.5 J ^ ^ site 1 day 10 0 1 2 3 I site 1 day 24 0 1 2 3 A. site 1 W day 31 0 1 2 3 A site 1 : day 37 0 1 2 3 / site 1 day 47 1 2 draft  [m] 1 2 draft  [m] Figure 6-6: Probability density of  draft  distributions (fraction  per decimeter) for  sites 2 and 1 during and after  convergent-divergent event A. Days (of  1998) listed represent the start date of  the data profile  used to create the distribution. The second divergent motion even between days 24 to 29 (Figure 6-5) may create open water at the site 1 mooring. The event accounts for  a 12% expansion of  the ice between sites 1 and 2 over 3.1 days. Although no evidence of  this open water is seen in any distributions at site 2, or in distributions after  the divergence event at site 1, the open water and new ice at day 24 may be created by this divergence event (Figure 6-6). The distribution of  ice starting at day 24 is constructed from  a profile  stretching to the end of  day 27. During days 25 and 26, the ice is undergoing divergence and opening water, which may be observed at the mooring site. If  that lead is situated over site 1, it could explain the large area of  open water or thin 10 cm ice covering 80% of  the ice area profiled. 0.25 0.2 0.15 0.1 0.05 0 0 1 2 3 4 draft  [ml Figure 6-7: Probability density draft  (fraction  per decimeter) of  ice evolving from  the 9th of  January to the 1st of  February. The divergence in the model is equal to the total divergence observed between sites 1 and 2, and ridging is assumed to create open water in the same area as the observed ice. The impact of  the divergence on the ridged ice is seen in the inset figure.  While the total amount of  ridged ice decreases (area conservation), the shape of  the ridged ice does not change. probability density: linear plot — 9th January 1st February — model: floe availability constraint probability density: log plot 1st February — Model: 1/10th divergence Model: full divergence 5 10 15 . draft  [m] The origin of  the open water and thin ice observed between days 24 and 27 at site 1 cannot be confirmed;  however, it is clear that open water caused by divergence or ridging is not seen at site 2. For this reason, the IDR model adapted to evolve the distribution of  ice at site 2 during event A does not include all of  the observed divergence between the sites. For the IDR model the divergence is set to one-tenth the total amount and is an estimate of  the divergence that creates open water locally to the ice observed at site 2. This 1/10th fraction  prevents excess open water from  being created and produces thin ice in agreement with observations. Running the model with the total amount of  divergence and allowing ridging to create open water produces far  too much thin level ice (Figure 6-7). While the assumption that the divergence seen by the sonar at site 2 is only 10% of  the total is arbitrary, for  the purposes of  this research, it is unimportant. Increasing or decreasing the percentage of  divergence assumed to be observed at site 2 would increase and decrease the amount of  thin level ice, and inversely impact the amount of  ridged ice. However the shape of  the ridged ice is determined by the ridging algorithms in the matrix and is generally unaffected  by the divergence (Figure 6-7). Large changes in the divergence would have the potential to change the shape of  the ridged ice distribution; as large amounts of  thin ice would lead to smaller ridges being created. This is not apparent in changing the divergence fraction  by a factor  of  10 on short timescales of  weeks. 6.6. Stress levels in the ice pack. With the distribution of  floe  sizes established and the critical stress for  ridging found  to be independent of  floe  size once the characteristic buckling length is exceeded, the possibility for ridging to be limited by stress levels in the ice pack is considered. The determination of  ice stress within a region is not easily obtained. While the patterns of  stress observed at different  sites show temporal correspondence, the magnitudes of  stress varies greatly between sites (Richter-Menge et al., 2002). In addition, thermal ice stresses are often  on the same order of  magnitude as mechanical stress (Richter-Menge et al., 2002) and stress measurements vary greatly between sites on a single floe  (Richter-Menge and Elder, 1998). As such, no observations are available of the distribution of  ice stress through a region and the estimates are from  limited observations. Observations of  the stress during a ridging event in a large, thick (3.5 m) floe  were made in the spring of  1989 (Comfort  and Ritch, 1989). From these data, a mean background stress can be estimated from  the stress levels observed immediately after  a ridging event. A mean background stress of  20 kPa was observed after  widely varying residual stresses from  the experimental set-up were subtracted from  the sensors. Peak ice stresses during the event varied greatly with the location of  the sensors. The maximum stress recorded by any sensor was 240 kPa, and remained at 150 kPa for  approximately 30 minutes at one site. The shear stress was, on average, one third of  the maximum stress. A similar mean background stress of  20 kPa was observed late in the season during the SHEBA and SIMI experiments in the Beaufort  Sea north of  Alaska (Richter-Menge et al., 2002). Baseline stress returned to zero between events before  mid-November, after  which a gradual increase in baseline stress was observed. This was attributed to the continual increase of  compressive strength of  the pack due to mechanical thickening of  the ice. During ridging events much higher maximum stresses are recorded, reaching values of  100-200 kPa (Richter-Menge and Elder, 1998). Commonly reported mean stress levels during ridging events are reported in the ranges of 45 to 50 kPa (Comfort  and Ritch, 1989) and 30-60 kPa (Richter-Menge et al., 2002). For stress to be a limiting factor  on ice ridging, the stress must be less than the critical buckling stress ((5.3.4), Figure 5-5) for  the ice floe.  An estimate of  this using the values of  2 GPa for Young's modulus and 0.3 for  Poisson's ratio is 5 MPa. So is ridging limited by buckling stresses? The maximum stress levels recorded in a floe  are almost an order of  magnitude smaller than the predicted critical stress. Yet ridges are common features  in sea ice, and it seems likely that the high stresses needed to instigate ridging are localized in thin ice, not suitable for  direct measurements of  stress and existing only for  brief  periods of  time before  being dissipated at ridge initiation. To ascertain if  ridging is limited by stress levels, a series of  observations of  stress in new thin ice would be needed. 6.7. Time and distance scales The possibility exists that ridge development could be limited by the time scale of  convergence, or by the amount of  strain during a coherent motion. This seems unlikely when we look at Figure 6-8. For a ridge to be formed  with a maximum truncation factor  of  20 m1/2, just 560 m of  ice must be consumed. As we can see in the Figure 6-8, convergent motion events exceed the 560 m limit before  motion changes direction. 0.2 0 -0.2 -0.4 Dec Jan Feb Mar Total convergence [km] 40 20 Dec Jan Feb Mar Total convergence [km] (scale 0-5 km) Dec Jan Feb Mar Day of 1997/1998 Figure 6-8: Onshore and offshore  convergence travelled by ice during the winter of  1997/1998. Upper panel demonstrates the convergence (km) travelled in one hour. Lower panels illustrate the convergence (km) travelled in one direction before  the motion changes direction. Lower panels are identical except for  vertical scale. The convergence travelled exceeds 560 m for  all but one convergent (and therefore  ridging) event. 6.8. Model limitations The incorporation of  a cusped keel shape and floe  availability into the redistribution algorithm provides a better simulation of  the amount of  deep ridged ice but does little to improve replication of  the bulge of  ice exceeding the exponential distribution in the 3-5 m draft  range (visible in Figure 5-7). Perhaps the bulge is an indication that the ice viewed at the end of  this event may have come from  a population different  from  that used to initialize the model. Convergence in one hour [km] , d This bulge between 3-5 m might also be an indication of  ice rafting  in response to pack-ice compression, a process not incorporated within our model. By including rafting,  Babko et al., (2002) were able to improve agreement between model and observations for  ice drafts  in this range. Further work evaluating the combined effects  of  rafting,  floe  availability constraints and keel shape may be more definitive  in determining the mechanical redistributor over the entire ridged ice distribution. 7. CONCLUSIONS: GEOMETRIC FACTORS INFLUENCING RIDGE FORMATION An ice-draft  redistribution (IDR) model has been developed to simulate the development of seasonal pack ice observed by a moored ice-profiling  sonar in the Beaufort  Sea. Deformation episodes selected carefully  from  the data record, which spans more than a decade in this area, are natural "experiments" on ice-draft  redistribution. These observations provide a unique opportunity to evaluate thermodynamic and mechanical redistribution algorithms. The IDR model uses thermal growth algorithms that utilize a remapping approach (Lipscomb, 2001). These simulations track the observed growth of  drifting  level ice without exhibiting numerical diffusion.  The only prior evaluations of  ice-growth algorithms have been based on observations of  land-fast  ice in shallow water or multi year sea ice. However, the IDR model creates too much ice in the thickest draft  categories when using current algorithms for  ice-thickness redistribution through ridging (Thorndike et al., 1975; Hibler, 1980). Such algorithms assume an unlimited supply of  ice for  ridging. Our analysis of  the lognormal size distribution of level floes  in the Beaufort  Sea indicates that large expanses of  ice are in short supply. Three quarters of  the level floes  observed in the Beaufort  Sea are too small to form  ridges of  the maximum draft  possible for  their thickness. Our IDR model uses a random selection of  floes  from  the empirical size distribution to determine the draft  of  the keel to be built next. Minimum ridge size is constrained by the characteristic length of  a floating  plate in buckling. The floe  availability model is a simple geometric constraint on ridge formation.  It improves the capability of  the model to reproduce the distributions of ridged ice, suggesting that floe  size is an important constraint on ice ridging. The cross-sectional shape of  keels also has a significant  impact on the draft  distribution for  ridged ice. Although underwater shape of  observed keels varies widely, the average distribution of  draft for  larger keels can be represented as an exponential with a 6-m e-folding  scale, truncated at maximum keel draft.  Replacement of  the usual uniform  draft  distribution in keels (triangular keel shape) with an exponential form  improved the model's simulation of  deep ridged ice. Idealized simulations indicate that over longer timescales the use of  cusped keel shape is not in itself sufficient  to replicate observed distributions of  ridged ice. The combined implementation of  the floe-availability  and the exponential-keel-shape constraints in ice-draft  redistribution produces the best results. Using this approach, the volume of  thick ridged ice is 8.8 % (for  draft  exceeding 9 m) over a 23-day "natural" experiment and 20.5 % (for draft  exceeding 10 m) over a 90-day idealized experiment. These differences  will have an impact on the strength of  the pack ice and on its response to thermodynamic forcing  via consolidation and melting. SECTION D: SUMMER RIDGE EVOLUTION 8. ADAPTING THE ICE DRAFT REDISTRIBUTION MODEL TO SUMMER MELT 8.1. Winter 2000 The ice draft  redistribution model developed in the Sections B and C of  the thesis can provide valuable insight into summer ridge ablation processes when adapted to the melt season. Suitable ice motion events where the ice remained in the region of  the moorings through June and July of 2000 suggest that the model be compared with observations from  that year. To develop confidence  in the suitability of  the IDR model for  the year 2000, the model was first  adapted to the midwinter season (January-February). During the first  50 days of  2000, the ice moved south approximately 55 km (Figure 8-1). Subsequent to this movement, no offshore  ice motion was found,  so the ice was not re-viewed by the sonar. Instead, after  ice ridging, the model results are compared to the observed distributions at both site 1 and site 2, as the initial ice surveyed now lies in between the two sites. The winter 2000 model is similar to the 1997-1998 winter model with differences  only in the divergence fraction.  During mid-winter, the divergence is not set to l/10th the observed divergence, as was done for  previous modelling cases. This is due to the simulated ice motion event. Both the 1997-1998 and 1991-1992 simulations of  events included significant  onshore and offshore  motions, where the ice returned to the initial offshore  distance. The year 2000 motion event moves ice in the onshore direction, without a significant  offshore  divergence that could create a large open water lead. As offshore  divergence was not present, the assumption that divergence opens a large flaw  lead, which was not visible to the sonar, is invalid. The small divergent motion events superimposed on the convergent motion may open up water throughout the ice pack. The divergence fraction  is thus set to 1 on the assumption that no single lead dominates the divergence. The results of  the winter ridging model in the year 2000 are shown in Figure 8-2. The modelled level ice thickness lies between the observed level ice thicknesses at sites 1 and 2 suggesting that thermal ice growth is accurate. The ridged ice thickness matches the ridged ice distribution at site 1,15 km south of  the final  location of  the modelled ice distribution. The agreement between modelled and observed distributions of  level and ridged ice for  days 1-50 of  the year 2000 shows that the model is applicable to ice motion events observed during the winter season. , Time [day of 2000] Figure 8-1: North - South motion of  ice over mooring site 2 during the first  half  of  the year 2000. Observed ice draft  distributions are centred at points indicated by circles on the path. Initially, during January and February, ice moves south compacting against the coast before  remaining relatively stationary in the north-south direction (thin lines with arrows indicate range of  motion). Melt begins around day 150 and four  ice motion events are identified  where the ice moves in an onshore-offshore  pattern (indicated by lines double ended with arrows). Draft  [m] Figure 8-2: Probability density of  ice draft  (fraction  per decimeter) for  the results of  the IDR model at site 2 during the first  50 days of  2000. Note that the model (heavy solid line) reproduces the level ice growth of  the final  ice viewed at site 2 (heavy dashed line) and the thickest ridged ice at site 1 (thin solid line), 15 km south of  the final  location of  the modelled ice distribution 8.2. Level ice melt The thermodynamic ice growth model used to predict winter ice growth (Section 4.2) is not relevant to summer melt conditions. Unlike growth, which occurs only at the ice-water interface, melt also occurs at the ice-air interface.  This surface  melt, along with the changing snow cover characteristics, introduces seasonal variability to the surface  albedo due to the varied upper ice surface.  Attempts to parameterize the albedo and the sensible heat flux  have been addressed in detail by other researchers, (e.g. Perovitch et al., 2002; Curry et al., 2001; Flato and Brown, 1996; and Dumas et al., 2003). The pattern of  annual variation in ice and snow thickness can be observed in landfast  ice measurements at Cape Parry (70.15° N, 123.33° W), as shown in Figure 8-3. Data are from  the Canadian Ice Service Archives and span the years 1959 to 1992. In an average year, snow begins to melt at the start of  May and has disappeared by July. Ice melt onset lags snow melt by approximately 2 to 4 weeks, beginning in the latter half  of  June, in agreement with observed level ice draft  melt which began around day 150 (May 29th) of  2000 (Figure 8-4). While snow depth and melt can vary regionally, we expect that during ice motion events A to D (Figure 8-1) the snow cover insulating the ice is decreasing in a similar manner to Figure 8-3. Along with the insulating properties of  snow, the albedo of  the snow-ice surface  begins to vary due to wet snow and an increasing number of  melt ponds, which have low albedo and act to absorb more shortwave radiation than snow or ice. Extensive research during the SHEBA experiment compared the results of  eight albedo parameterizations of  varying complexity with the observed SHEBA averages (Curry et al., 2001). Curry et al., (2001) note that a "surface  albedo parameterization with more complex dependence on surface  features  can give a degraded simulation of  surface  albedo if  the simulation of  surface  features  is unknown." Of  all eight parameterizations, a constant albedo parameterization of  0.75 and 0.5 for  snow and ice, respectively (Parkinson and Washington, 1979) was found  to be one of  the more accurate. This is similar to the results of  Dumas et al., (2003) who found  that a constant albedo of  0.55 for  melting ice was the best at reproducing the evolution of  ice thickness in the central Arctic. Moored sonars provide no information  about the surface  features  (melt ponds, snow cover) of  ice, thus allowing level ice melt to be modelled with no more accuracy than the simplest parameterizations. Cape Parry (Franklin Bay) 1959- 1992 1 1 1 1 1 1 1 r Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Figure 8-3: Ice and snow thickness observed at Cape Parry from  1959 to 1992. The decrease in snow cover can be seen in early May, preceding the decrease in ice thickness by two to four  weeks. Error bars indicate +/- one standard deviation from  the mean (large circles). These raw data are from the Canadian Ice Service Archives, http://ice-glaces.ec.gc.ca/App/WsvPageDsp.cfm?ID=210&Lamg=eng.  (Figure courtesy ofHumfrey  Melling.) Fortunately, accurate modelling of  level ice melt is not essential for  the investigation of  processes enhancing ridged ice melt that are addressed in this thesis. Observed level ice melt rates can be used as draft  independent melt rates for  all ice thicknesses and provide an upper bound on the thermodynamic melt of  thick ice without considering ridge specific  processes. The observed relation between level ice draft  and time can be approximated with a quadratic and is shown in Figure 8-4. Melt rates are then approximated as a linear function  (Figure 8-4, inset) with slope and intercept -0.0004 m day"1 and 0.044 m day"2. Observed Level Ice Draft  [m] Day of 2000 Figure 8-4: Observed level ice draft  (dominant mode) and quadratic best-fit  curve used to estimate level ice melt rates. D is the day number of  year 2000. Inset figure  shows the increasing melt rates (as negative growth rates) as the season progresses. 8.3. Model predictions During the year 2000 melt season, four  potential ice motion events are identified  (Figure 8-1) where the same ice, with the assumption of  alongshore homogeneity, is viewed multiple times by the sonar. In order to allow convergent strain forces  to be estimated, the motion of  the ice must be first  onshore and then offshore  where strain is calculated from  onshore velocities. Events A and D are preferable  for  modelling and bracket the beginning and end of  the melt season. Event B is less suitable; as the ice parcel moves offshore  of  the mooring and accurate compressive strain information  is not available during this time. Event C is excluded due to the short time scale of the motion, the initial and final  observed draft  distributions are constructed from  adjacent sections of  the ice profile  reducing confidence  in the observed ice evolution. The maximum draft  of  ridges created during the melt season will be lower than the draft  of  ridges created from  the same thickness of  level ice during the winter season. Sea ice strength is a function  of  the brine volume within the ice. As the temperature approaches the freezing  point, the brine volume will increase as saline ice pockets begin to melt, and the sea ice strength will decrease. This has implications for  the maximum draft  of  ridges, which during the winter is set at 20 m l / 2 hvl, and represents the point during ridge creation when the force  needed to push another ice block beyond the current ridge draft  exceeds the force  needed to buckle adjacent level ice (Hopkins, 1998). Decreased level ice strength will cause the maximum ridge draft  to be reached at a lower multiple of  hm. Estimates for  the appropriate expression for  the maximum keel draft  are very difficult  to obtain. Ice strength estimates are consistently found  to be lower in the field  than in small laboratory samples (Dempsey et al., 1999) and the temperature of  the ice during the events simulated here is not known. During event A, at the start of  the melt season, air temperatures had recently surpassed freezing  and ice strength would be starting to decrease. In event D, in early July, air temperatures had been above freezing  for  over a month. Initial estimates for  keel truncations can be found  by looking at the maximum draft  of  level ice observed during these melt periods, and using the dominant mode of  level ice during these events to estimate the relationship between keel draft  and level ice. For event A, a maximum keel draft  of  15.2 m is observed, with level ice draft  at 1.8 m, indicating an upper bound on keel draft  may be H=  11 m1/2 hU 2. For event D, a maximum keel draft  of  10.2 m is observed, with level ice draft  at 1.3 m, indicating an upper bound on keel draft  of  H=  9 m l / 2 hV 2. The largest observed ice keels may be remnants of  keels formed  earlier in the winter season, such that these maximum keel drafts  that can be created during the events are overestimates. During events A and D we can be reasonably confident  that no ice is created thicker than predicted by these truncation points based on observations. Further insight into the maximum drafts  of  level ice can be found  from  ice strength measurements (Johnston and Frederking, 2001) conducted using borehole jack tests in the Canadian Eastern Arctic Archipelago (71° 14.7 N, 97° 09.2 W) in first  year sea ice. From the 14th of  May to the 11th of  June, 2001, the ice strength at a depth of  30 cm decreased by over 50%, from  22 to 10 MPa. At larger depths, similar decreases occurred. An extended data set from  2000 shows further decreases in ice strength to mid July. By day 200 of  year 2000, ice strength at all depths was less than 5MPa, with only 25% of  the ice strength in the upper layer during May. These values suggest that the ice strength during events A and D will be considerably less than the mid-winter strength. It appears that reasonable choices for  the relationship between maximum keel draft  and level ice for  events A and D are 11 m1/2 h1'2 and 9 m1/2 hV 2, respectively. During the simulations of  events A and D, (Figure 8-1), the divergence is set to l/10th the observed values (see discussion in Chapter 5) and the models are run with and without ridging. Results are plotted in Figure 8-5 and Figure 8-6. Observed and modelled level ice peaks match, as ice melt is fixed  at observed rates. Large discrepancies between the observed (dashed) and modelled (heavy and thin solid lines) distributions of  thick ice are evident. Including ridging and melt processes, the modelled distribution of  thick ice increases for  both ice motion events indicating that ridging dominates over melt driven changes. However, observed distributions of  thick ice show a decrease in density up to an order of  magnitude greater than the decrease predicted by melt alone (thin solid line). It is evident that level ice melt is insufficient  to account for  the observed melt of  level ice even if  no ridging is occurring and that contributions of ablation processes specific  to ridged ice must be important. draft  [m] Figure 8-5: Probability density of  draft  (fraction  per decimetre) for  the modelled change in the distribution of  ridged ice during summer melt from  days 154 to 173. Allowing ridging to occur, the model predicts an increase in thick ice due to ridging (heavy solid line). Without ridging the model predicts a decrease in ridged ice density expected from  observed level ice melt rates independent of draft  (thin solid line). The observed thick ice shows evidence for  an enhanced melt process, with density at large drafts  decreasing faster  than predicted (dashed line). Figure 8-6: Probability density of  draft  (fraction  per decimetre) for  the modelled change in the distribution of  ridged ice during summer melt from  days 192 to 207. Similar to the results for  event A, the amount of  ridged ice is greatly overestimated by the model (heavy solid line). The model was run again without ridging (thin solid line) and the melt of  ridged ice is clearly underestimated by the model. 8.4. Evidence for  enhanced melting Enhanced ablation rates of  ridged ice in the draft  distributions can be quantified  from  the shift  in the cumulative draft  density distributions, which are defined  as all ice thicker than a draft  value (in contrast to conventional definitions  of  cumulative distributions). For example, over event D, from  days 192 to 207, the cumulative distributions shift  to the left,  indicating that ice is ablating (Figure 8-7). cumulative density distribution (/m) draft  [m] Figure 8-7: Determining ablation rates from  the cumulative density distribution. Here the cumulative density distribution is defined  to reach unity for  the thinnest ice (open water), as the thickest ice is least prominent. The ablation rate can be found  by subtracting the level ice thicknesses (Ah= hi - h2) associated with a chosen value of  the cumulative density and dividing Ah by the time separating the two distributions. A lower bound for  the ablation of  ridged ice can be computed from  observations, assuming that no new ridge building has occurred. Real melt rates will be considerably higher, as ridging acts to increase the distribution of  ridged ice (Figure 8-5, Figure 8-6). However, as the truncation for maximum keel draft  due to ice strength is unknown, ridging cannot be estimated with enough accuracy to predict true melt rates. Therefore,  we use the observed ablation rates for  the ice motion events (Figure 8-1) where the same ice is viewed consecutively as a lower bound. Values are plotted against draft  in comparison for  level ice melt rates in Figure 8-8. The increase in melt with draft  is visible in Figure 8-8. During event D (heavy dashed line), from days 192 to 207, the level ice melt is approximately 2-3 cm/day, while the observed ablation reaches up to 15 cm/day. Clearly, the melt rates needed to reproduce observed ablation are far greater than accounted for  by one-dimensional thermodynamic level ice melt models. Observed Ablation Rates [m day"1] Draft  [m] Figure 8-8: Observed ablation rates as a function  of  draft  during four  ice motion events. Level ice melt for  these time periods is indicated by the dotted line of  the same pattern. Both level ice melt rates and ridged ice ablation rates increase throughout the melt season. Errors in the observed melt are calculated similar to section 4.4. Following Rothrock (1986) and Melling and Riedel (1995), the uncertainty in the cumulative distribution can be found  as: (8.4.1) <j(h)  = LIS Where L is the total number of  samples in the ice profile,  the summation is over H to h, and the decorrelation distance for  ridged ice is approximately 5 samples (see section 4.4). Using this estimate for  the error in the cumulative density profiles,  the maximum error on the observed melt can be found  (Figure 8-9). This error is a generous overestimate, as the error in the initial and final  cumulative density profiles  should be independent, and thus should scale in quadrature. However, this error estimate demonstrates that observed melt rates are, at all times, much greater than the predicted melt rates of  level ice. Observed Melt Rates [m day"1] Figure 8-9: Observed melt for  event D (thick line) with error (thin lines) representing the maximum errors using one standard deviation from  the cumulative density distributions. 9. SUMMER RIDGE ABLATION PROCESSES 9.1. The influence  of  internal ridge geometry on ice melt The observed enhanced melt rates of  ridged ice (Figure 8-8) suggest that additional melt processes must work in tandem with level ice melt to ablate ridges. Several processes for enhanced melt have been proposed, such as increased surface  area and enhanced flow  past keels; common to each of  these processes is the oceanic source of  heat. Observational evidence supports this hypothesis; for  example, Melling (2002) attributes observations of  limited amounts of  very thick ice within the Canadian Northern Arctic Archipelago (northeast of  the region studied in this thesis) to melting from  oceanic heat sources, rather than local surface  atmospheric processes. Schramm et al., (2000) used numerical models to investigate the importance of  increased keel surface  area to melt rates. The sloping sides of  the keel were found  to enhance thermal melt of ridged ice as much as three times that of  level ice, but only for  unrealistically narrow keels. This increase is insufficient  to account for  the present observations. An additional potential contributor to enhanced melt is accelerated flow  in the upper ocean past ridge keels. The motion resembles a two-layer flow  past an obstacle (Lawrence, 1993) and can be represented with hydraulic terminology. Using lab models designed to reproduce oceanic conditions, Pite et al., (1995) found  enhanced flows  past keels as predicted by hydraulic theory, with the flow characteristics dependent on mixed layer depth and upstream flow  velocity. The enhanced flow has the potential to increase the turbulent heat transfer  to the keel. However, quantifying  this heat transfer  is difficult. It is likely that the increased turbulent heat flux  generated by the two-layer flow  past keels is a significant  contributor to ice melt, however, results from  the simulations of  Skyllingstad et al. (2003) suggest that this may not account for  all of  the observed melt. Simulations of  flow underneath ridge keels were compared with observations made during the SHEBA experiment by Skyllingstad et al., (2003). They found  that keels caused both an increase in velocity and heat transfer  near or at the keel. Modelling the interaction of  the 0.5 m fresh  layer and a keel extending 0.54 m deeper than the surrounding level ice showed heat flux  values increased at the keel crest, to greater than 80 Wm"2, during July 1998. Unfortunately,  no large keels were simulated during this time period. During the March 1998, the impact of  a 10.8 m keel on the turbulent heat flux  was simulated. Increased across-keel velocities at the keel crest of  0.5 m s"1 represented a significant  increase on the average velocity of  0.1 m s"1. Turbulent heat flux  within the water column (Skyllingstad et al., 2003; Figure 7) indicates that for  a freezing  temperature departure of  0.05 upstream heat flux  reached values of  120 W m"2, whereas downstream values reached only 40 W m"2, leading to average values of  80 W m ~2. As temperature scales linearly with heat flux,  increasing the temperature departure to the expected July values of  0.2 0 (see 9.2) implies that heat flux  could reach 320 W m"2. If  all of  that heat were used to melt ice, the melt rates would be approximately 9 cm day"1. Figure 8-8 suggests that events in July (days 183 to 213 of  2000) will have net ablation rates much higher than 9 cm day"1, with total melt rates that are greater than 20 cm day"1. It seems likely that turbulent heat flux  is not the sole contributor to ridged ice melt and another process may play an equal or larger role. We propose that an additional process contributing to enhanced melt of  ridge keels is melt within the porous keel structure due to internal flow.  Ice keels are constructed of  blocks of  ice that form porous structures with irregular envelopes. Fractal dimensions of  the underside of  first  year sea ice (2.57 + 0.06, Melling et al., 1993; 2.50 ± 0.05, Melling and Riedel, 1995, and 2.44 ± 0.064, Bowen and Topham, 1996) indicate that the envelope around a keel is rough with a high surface area to volume ratio. As undersea sonars measure only the envelope of  the keel and do not recognize interstitial voids in the keel structure that are shadowed by lower blocks, the actual surface  area to volume ratio is even higher than this estimate and the fractal  surface  is continued throughout the keel. This fractal  geometry is observed as porosity, which in first  year sea ice ranges from  20-45% (Bowen and Topham, 1996) with a mean of  approximately 30% (Timco and Burden, 1997). Studies of  vertical ridge structure indicated that the porosity is highest in the keel sections, likely due to the lower relative gravitational packing forces  in ridge keels compared with ridge sails (Lepparanta and Hakala, 1992; Melling et al., 1993). Measurements on Baltic first  year ridges indicate that there exists a porous, non-consolidated keel bottom during the melt season (Hoyland and L(|)set, 1999) with the consolidated layer expected to extend to 1.3 - 1.7 times the depth of  the level ice. For large Beaufort  Sea keels, this implies that most ridged ice has a porous structure. During the winter of  1990, the evolution of  a first  year ridge in the Baltic Sea (Lepparanta et al., 1995) was monitored until its break-up in early May. The keel depth decreased from  5.28 m to 3.12 m over three visits spanning the period from  the 4th of  February to the 19th of  April. During this time, the consolidated layer increased from  0.52 + 0.19 m to 1.02 + 0.31 m, reaching a final thickness slightly over 1.75 times the level ice thickness. While the overall ridge porosity decreased from  28% to 17% over the course of  the two months, the porosity of  the unconsolidated portion did not decrease and the change in porosity was due to the increasingly large volume fraction  of  consolidated ice. Porosity below the consolidated layer remained relatively constant at 31%, 29% and 32% for  each of  the three visits (Lepparanta et al., 1995). Immediately under the consolidated layer there was some evidence for  decreasing porosity (to a minimum of  20%) due to packing rearrangements* but porosity was found  to increase in the lower half  of  the keel (Lepparanta et al., 1995). Therefore,  we can assume that porous ice is found  in the lower regions of  the keel throughout the entire melt season. With such relatively high porosities in the lower portions of  ridge keels, oceanic water may flow through the keels and transport heat to the central portions of  thick, ridged ice. The relative rapid percolation of  warm water "short circuits" the conductive transfer  modelled by Schramm et al., (2000). A conceptual model is developed in chapters 9.3 to 9.7 to estimate the potential melt rates due to internal flow.  We explore the potential of  the combination of  the fractal  surface  of the keels and the penetration of  warm oceanic water into the keel structures to account for enhanced melt observed in ridged sea ice. 9.2. Under-ice ocean characteristics The potential for  melt within porous ridged ice depends on the physical characteristics of  the upper layer of  the ocean. For melt to occur, ice motion relative to ocean currents must be strong enough to induce flow  through the porous keel and the ocean must be warm enough to melt the ice. While no upper ocean data are available from  the Beaufort  Sea during the 2000 melt season, comparisons with observations from  previous years will be used to provide estimates for  current speed and temperature. Measurements taken from  the 13th to 18th of  April, 1989 in the Beaufort  Sea Shelf  using an ADCP attached to the ice (data courtesy H. Melling) provide average velocities with depth. The ice drifted  at average velocities of  0.13 to 0.17 m s"1 in a westward direction within the regions bounded by 70.5 - 71.1 °N and 130.9 - 133.4 °W. This region includes site 3 (Figure 3-1). Ocean current speeds relative to the ice show mean speeds of  0.069 m s"1, without significant variations in speed or direction with depth to at least 45m (Figure 9-1). There will be a reduction of  ocean speeds at depths less than 10 m due to a non-slip boundary condition at the ice-ocean surface.  However, for  simplicity, the current velocity relative to the ice is assumed constant with depth as a first  approximation. Figure 9-1: Components of  oceanic velocity relative to level ice, 13-18th April 1989 in the Beaufort Sea. Level ice depth is 1.5 m, water depth is 54.6 m, and ice is moving westward. The direction and magnitude of  the velocity remain relatively constant below the ice, with a mean speed of  6.9 cm s"1. Data courtesy of  H. Melling. Deep ocean data from  1998 are available as part of  the SHEBA (Surface  Heat Budget of  the Arctic Ocean) project (SHEBA data were provided by the SHEBA Project Office,  Univ. of Washington, http://sheba.apl.washington.edu/). During the spring of  1998, data were collected at the SHEBA ice floe  located within the area 74.6 - 80.5 ° N and 143.9-168.1 ° W. As part of  that project; an oceanographic mast experiment was installed to measure the mean and turbulent ocean currents beneath the SHEBA ice floe.  Instrument clusters were mounted on a pole 4 m apart below the ice surface.  The mean velocity from  the 5+1 and 9+1 m clusters provides a time dependent profile  of  the oceanic current speed through the melt season (Figure 9-2). Mean currents in April 1998 at 6 and 10 m depth were 0.12+0.02 m s"1 and 0.10+0.03 m s"1 respectively, higher than 1989 values (Figure 9-1). However, the current meters did not record currents below 0.05 m s"1, increasing the apparent mean value. Measured ocean speed [m s" ] 0.4 ~ 5 m depth 0.3 f • i Q1 1 1 1 I I g 150 160 170 180 190 200 210 220 TD (D 8. 0.4 r 9 m depth 0.3-Q I L_ I I I I 150 160 170 180 190 200 210 Day of 1998 Figure 9-2: Velocity measured during turbulence measurements for  the SHEBA program Values of  mean and standard deviations for  the currents during the melt season are shown in Table 9-1. In June, the depth of  the instruments was raised approximately 2m (at day 176). No data were collected during most of  July, from  days 185 to 210, due to biofouling  and low currents that prevented data collection. The increased velocities at the end of  July did not remain at that magnitude during August (Table 9-1) and may have resulted from  a storm event during that three-day period of  data collection. Month Depth (m) Mean current (m s"1) Standard deviation (m s"1) June 4-6 0.102 0.028 8-10 0.106 0.034 July 4 0.166 0.035 8 0.218 0.045 Aug 3 0.135 0.033 7 0.144 0.054 Table 9-1: Mean currents observed with depths during the SHEBA mast experiment in summer 1998. The time series in Figure 9-2 suggest that the maximum expected oceanic current will be ~ 0.25-0.3 m s"1, with more common speeds at 0.10 m s"1. We assume that the mean current of  0.10 m s"1 will be aligned in a random direction to the keel axis. The component of  velocity in the across keel direction will then be the average of  0.10 cos(O)  m s"1 over the range of  angles from  0 to an angle close to 7t/2 radians. For this range, the mean of  cos(0) is 0.63 and the average velocity perpendicular to the keel is 0.063 ras'1. Ocean temperature and salinity profiles  are also available from  the SHEBA dataset. An automated CTD winch recorded profiles  throughout the 1998 melt season. The ten-day mean temperature and salinity profiles  during June and July, spanning days 150 to 210 of  1998, are plotted in Figure 9-3. It is expected that the upper 15 m of  the water column under the ice will interact with ridge keels observed during this time. This layer appears to be homogeneous, with the mixed layer getting shallower through the melt season. For the salinity profiles,  an upper fresh  layer generated by ice melt begins to appear only after  day 200. Temperature is largely independent of  depth in the upper 15 m, with values increasing with time. Mean 10-day Salinity profiles Temperature profiles 31.5 Salinity 10 15 20 25 \ 0 1 2 % standard deviation, Temperature -1.7 -1.6 -1.5 -1.4 Temperature (degrees C) Figure 9-3: Mean temperature and salinity profiles  from  automatic CTD profiles  collected during the SHEBA program in June and July 1998. Profiles  show mean values over ten days, centred at days 155, 165, 175, 185, 195, 205. Midpoint (day of  1998) 155 165 175 185 195 205 Mean temperature 3-10 m [°C] -1.66 -1.64 -1.63 -1.59 -1.55 -1.50 Table 9-2: Mean ocean temperature over a ten-day period for  depths 3m to 10m The freezing  point for  seawater with a salinity of  ~ 31.5 (over depths of  5-15 m) is approximately -1.73 °C. For all temperature profiles,  the mean departure from  this freezing  point is sufficient  to cause melt (Table 9-2). For the purposes of  model development, the temperature anomaly above freezing  is set at 0.18°C, similar to the observed mean temperature anomaly on day 195 in 1998. 9.3. Porous flow  through the ice The fractal  envelope of  ridged ice is the external manifestation  a porous internal structure. Flow through porous media is well studied in groundwater and engineering literature and results can be adapted to the geometries of  ridged ice. Consider the schematic of  a ridge shown in Figure 9-4. Across-ridge flow  is in the direction x . As the orientation of  the sonar survey line to the keels varies, the cross section viewed by the mooring sonar is generally significantly  wider than the keel width. Further discussion of  ridge cross-sections can be found  in Section 11.1. Within the keel, the average direction of  flow  is assumed to travel in a net x direction and is similar to flow through any porous media. To estimate porous flow,  Darcy's Law is used to describe the discharge velocity through a fluid  filled  medium. Where V  is the discharge flow  velocity, K  is the constant of  hydraulic conductivity [m s"1], and I H is the hydraulic gradient, given the subscript 'W to differentiate  it from  the identity matrix in section B, and related to the pressure gradient as (Batchelor, 1967, pp 224): Here, P is pressure, p*, is the density of  seawater, and g is the gravitational acceleration. Darcy's Law is valid when the Reynolds number, Re = uL/v,  is less than 1. The characteristic velocity within the pores is related to the discharge velocity, u=V/p  where p is the porosity, L is the length scale of  the pores, and vis the kinematic viscosity of  seawater, 1.8xl0"6 m2 s"1. For most groundwater systems, Re <1, and Darcy's Law is sufficient  to predict porous flow.  Ice keels, assuming pore length scales on the order of  1-20 cm (see Appendix B) and velocity scales on the order of  0.005-0.025 m s"1 (see section 9.5) will have Reynolds numbers ranging from  25-3000, indicating non-Darcy flow  is occurring. (9.3.1) (9.3.2) Pw8 C* Figure 9-4: Keel schematic showing the definition  of  axes used in the theory. The oceanic velocity relative to the ice, U, and the porous flow  through the ice, u, are in the direction of  the ice motion x . An extension to Darcy's Law for  slightly turbulent flows,  the Forchheimer equation, includes an inertial component (Burcharth and Andersen, 1995). (9.3.3) I H  =K~ iV  + K I  \V\V where Kt [s2 m"2] is an inertial coefficient. A physical understanding of  (9.3.3) can be obtained by examining the 1-dimensional steady flow case of  the Navier-Stokes equation (Burcharth and Andersen, 1995). Consider the horizontal Navier-Stokes equation in one dimension for  a non-rotating system (Gill, 1982) (9.3.4) Du 1 dP  _2 = + vV  u Dt p dx Where u is the velocity, p is the fluid  density, and v is the kinematic viscosity. As the flow  is steady, we drop the time dependent terms. Rewriting (9.3.4) in terms of  the hydraulic gradient (9.3.2): 2 1 dP  _ u du  v d  u pg dx  g dx  g dx 2 The first  term on the right hand side is an inertial term while the second is a viscous term. Following Burcharth and Andersen (1995), we can relate (9.3.5) to the Forchheimer equation (9.3.3) by defining  characteristic length (L) and velocity (u)  scales for  the fluid  through the pores. This suggests an equation of  the form: v u „ 1 u2 (9.3.6) Ih = a — ~ 2 + P ~ , g L2 g L Here u is the pore velocity in the x direction, related to the discharge velocity, Vas  u=V/p.  The characteristic pore length, L, is defined  as the hydraulic radius, RH,  the ratio of  pore volume to pore surface  area. Burcharth and Andersen (1995) derive RH  for  spherical granules. In the case of rectangular blocks of  thickness h\ RH  = (9.3.7) JtH <\-p)r Full derivations of  the hydraulic radius for  spheres and blocks are in Appendix B. The factor  y is a geometrical factor  related to the shape of  the blocks forming  the keel and is approximated as 4.45 (see Appendix B). The ratio of  the porosity, p, to (1-p)  is the ratio of  the fluid  to solid fractions. For our keels, the porosity in the lower portion of  the keel is assumed constant at 0.3 (Lepparanta et al., 1995) and the hydraulic radius becomes RH  =0.096 h. When the keel is constructed out of 10 cm thick ice, the hydraulic radius will be 0.96 cm, and on a larger scale, for  1 m thick ice, the hydraulic radius will be approximately 9.6 cm. Substituting RH  for  L in (9.3.6) we find  that u and I H  are dependent on v, g, and RH,  along with the parameters, a and /?. No analytical derivation of  these last two parameters exists, but they are expected to include information  about the tortuosity, or interconnectedness of  the pores (Burcharth and Andersen, 1995). The relationship between the hydraulic conductivity and the velocity becomes: r av P i (9.3.8) 7 / / = T T W + i T " gRH z §RH In this form,  (9.3.8) differs  from  the derived Forchheimer equation in Burcharth and Andersen (1995) by the factors  of  / and y, which are included in RH.  Burcharth and Andersen (1995) include these factors  (62 and 6 for  spheres, from  the d16 dependence in RH)  in their parameters (here indicated as a'  and /?'). Care must be taken to remember these factors  when estimating the discharge velocity using published values a'  and /?' for  a and fi  in (9.3.8). (9.3.9) a = = 36 6 A selection of  values for  a'  and /?' found  in literature are summarized in Table 9-3. The coefficients  in (9.3.3) are often  reported, rather than the coefficients  a and /?. In these cases, a' and J3'  are calculated using the relations (Burcharth and Andersen, 1995; Wong et al., 1985): (9.3.10) K - , K j - — — p gh p gh For rough samples, h is chosen as the diameter at which 50% of  the sample, by mass, has a smaller diameter, D50. The uncertainty in this, along with a lack of  information  about the geometry of  the grains makes determining the ideal values for  a and J3  difficult.  Table 9-3 illustrates the variation in coefficients  found  experimentally. A first  approach at determining the flow  through a keel suggests that, based on experimental results with irregular shaped particles, a'and /?' will vary from  400-1300 and 0.5-2, respectively. Adjusting these to the derivation of  the Forchheimer equation for  rectangular blocks, a and /? will range from  11-37, and 0.08-0.34, respectively. Using literature values for  K 1 and K,in  (9.3.3) is not desirable as the porosity of  the non-spherical samples is much greater than the porosity of the ice keel. 73 1. cs OH a u m -c a. cn >n os cfl D — os c/i U E J3 D, w O o e O r-<L> o B t- sd • OS c B SO o oo r-vo OS o (N > Tt ^ © © CN © <n OS •—' t/i a> ^ S: o Q. C/> © C3 oo CN a> s "3 X 6 >o • o o m CN Os xl" © SO © rn T3 <D OS -a c 3 •a O e _3 P <o 00 c/5 c '3 LH O oo 3 S CN CN W 3 too I ! !s W) e <U ca B 3 u OS a Re ts a> cs a. M « 00 c W I I «s i e feb-§ « — oo oo 1 2 X *Tj o 2 L, § M  Q " " 3 ^ —- &0T3 — — D § T3 A S O J3 3 O 3 O B B B 6 <1> a. cd -C cn Ja o JS CM 3 00 Q II „ a> I I S £ 11 B o 2 S Jc tS £ Q 3 toO e 03 c/l so .S <U CO fc .3 SO 03 en 'E .5 « ^ 3 8 E a. CN II o O V T & m CN o CN ' as oo o o II O V 7 ^ I SO © o o II ° " . CN ^ o so <o SO <D CO 03 cn (3 OJ <1> o o +-» 00 00 CL o. u 3 o 3 o Table 9-3: Porous flow  parameters. 'Englelund (1953) data s reported in Burcharth and Andersen (1995). 2Diameter is given as D50, the diameter, for  which 50% of  the sample is smaller than by mass. 3Three different  rectangular irregular samples with their results are listed. The large variations in a and /? between experiments may be due to sensitivity to different packing arrangements (Kells, 1993) along with porosity, surface  roughness, Reynolds number of pore velocity, particle size, and other factors  (Burcharth and Andersen, 1995). For this reason, without experimental results specific  to ice keels, the appropriate coefficients  are unknown. A sensitivity study for  the variation in pore velocity with a and (i follows  in section 10.2. 9.4. Pressure distribution on keel To determine the discharge velocity through the keel, the pressure gradient across the keel must be known. Unlike the groundwater case, the pressure gradient is determined by the velocity component of  the dynamic head, the upper ocean velocity, U, relative to the ice. Bernoulli's theorem for  an incompressible fluid  states that energy is conserved along streamlines and defines a constant Q (often  called H, see Batchelor, 1967; section 3.5), where (9.4.1) Q = -\u\2 + EI +-+gz 21 I  p In this equation, Ej is the internal energy of  the fluid  per unit mass and Q is the total energy per unit mass. Q is constant in a frictionless,  non-conducting fluid  with a steady pressure distribution. In a groundwater situation, the pressure gradient is normally driven by the change in the height of the water column, z, as the dynamic head from  the velocity component is far  smaller. In rivers or oceanic conditions it is the change in velocity, the dynamic head, which changes the pressure gradient. Holding Q constant and changing the velocity (U ) will change the corresponding pressure aligned horizontally and normal to the keel axis, dP/dx  (Cummins et al., 1994): (9.4.2) aP*^p\uf The change in pressure from  the upstream to downstream obstacle face  can be related to the velocity as: ,„ . _. _ Pupstream ?downstream (9.4.3) <~D=-(l/2)p  \u\2 Where CD is the drag coefficient  (Vittal et al., 1977). Determining the drag coefficient,  CD, is straightforward  for  an obstacle in a single-layer homogeneous flow.  Pite et al., (1995) found  that ice keel models in a homogeneous fluid  had constant drag coefficients  of  0.62 and 0.51, depending on keel shape. Lab experiments measuring drag coefficients  of  river sediment bedforms  showed similar results for  the drag on triangular impermeable obstacles with shallow slopes (30°). In rivers, the drag coefficients,  CD, are found  to vary with the ratio of  the bed form  height to stream depth, d]ayer, as CD ~ (H/d, ayerf s where H is the obstacle height (Thibodeaux and Boyle, 1987). Larger obstacles, where H/di ayer exceeds 0.35, may be related as CD ~ (H/d, ayerf 2 (Shen et al., 1990; Elliot and Brooks 1997). Reported values for  the drag coefficients  range from  0.15 to 0.4 (Vittal et al., 1977). For an ice keel floating  on the ocean, the upper-layer depth is analogous to the river depth, but the buoyancy difference  between the upper and lower ocean layers has a reduced gravity three orders of  magnitude less than the air-water interface.  The relative ease of  displacement of  the internal pycnocline in flow  past a keel suggests that values for  drag coefficients  from  non-stratified systems cannot be used here. Pite et al., (1995) used laboratory models with two-layer flows  to estimate the force  on the ice keels. Scaling the laboratory simulation to ratios of  the ice keel to upper-layer depth observed in the Arctic, Pite et al., (1995) found  that the drag coefficient  varied with the upstream Froude number of  the flow,  which ranged from  0.1 to 0.7 under typical Arctic conditions. Maximum values for  the drag coefficient  of  approximately 3.8 were observed at Froude numbers of-0.3-0.5. Drag coefficients  decreased as the Froude number increased. Keel draft  and upper layer water depths were not varied. Drag coefficients  from  a single model keel are shown in Table 9-4. In this experiment the layer and keel depths were kept constant (keel depth is one half  of  upper-layer depth) so variations in the Froude number were solely due to variations in flow  velocity (Piteet al., 1995). To effectively  estimate the drag coefficient  for  an ice keel, the undisturbed internal Froude number, F0, of  the system should be known. Here the Froude number is defined  following  the notation of  Pite et al., (1995): U  , dxd2 (9.4.4) F 0 = —j===; h0 = • 1 2 ° d \ + d 2 Where U  is the current velocity remote from  the keel, g'  is the reduced gravity ~ 0.005 m s"2 under typical Arctic conditions, and d; and d2 are the upper and lower layer depths. For the keels at site 2 in the Beaufort  Sea, we estimated the upper and lower layer depths as 15 and 55 m, respectively (using Figure 1-4 to estimate mixed layer depth). With an oceanic velocity of  0.063 m s"1, the Froude number is approximately 0.26. For peak flows,  exceeding 0.3 m s"1, the Froude number will exceed 0.8. Observed keel depths in summer are On average less than 10 m deep, such that the keel depth is less than half  the upper-layer depth, similar to Pite et al., (1995). Large keels will increase the velocity past the keel such that the full  range of  drag coefficients  from  0.2 to 3.8 may be possible. The sensitivity of  heat transfer  and pore velocity to the variations in the drag coefficient,  CD, will be explored further  in Chapter 10.2. The drag coefficient  is estimated as 3.5 for  the purposes of this investigation to match the expected Froude number, 0.26, at oceanic velocities of  0.063 m s"1. F 0 0.23 0.34 0.44 0.53 0.69 0.84 1.01 1.23 1.64 cD 3.5 3.8 3.5 3.0 1.4 0.77 0.46 0.31 0.20 Table 9-4: Experimental drag coefficients  for  a laboratory model of  a keel in a stratified  flow (Cummins et al., 1994). The drag coefficient  decreases as the Froude number, and hence the current velocity, increases. The drag coefficient  gives information  about the total pressure difference  between the upstream and downstream faces  of  the ice keel, but does not provide information  about the distribution of pressure across the upstream face.  Pressure measurements from  river bedform  models indicate low pressure at the upstream end and crest of  the obstacle with a maximum pressure partway along the upstream face  and a constant, low pressure along the downstream face  (Figure 9-5; Vittal et al., 1977). We assume that the shape of  the pressure distribution along the upstream face  of  a ridge keel is similar. . flow  direction ; • upstream pressure N^ bed form anomaly L^^^K Figure 9-5: Distribution of  pressure gradient along the flow  direction on a bed form  such as a sand dune at the bottom of  a channel. 9.5. Pore velocity Once the hydraulic pressure gradient is known, (9.3.8) can be solved for  the pore velocity, u (z). Equation (9.3.8) can be defined  in terms of  the hydraulic radius, RH,  as: _ , -\dP  B 2 av (9.5.1) — = ^—uz + -u p dx  RH  R H In the simplistic model where the free-stream  velocity is assumed constant with depth, the dependence on z reflects  the fact  that the keel width varies with depth, and thus dP/dx  is a function  of  depth; solving (9.5.1) for  u gives: (9.5.2) = + 2PRH  "\ARH 2P2 PP dx We expect that the pressure anomaly distribution is a function  of  the bed form  shape (Figure 9-5) and reaches a maximum value along the upstream obstacle face  before  falling  back towards zero at the keel crest (Vittal et al., 1977). The total pressure distribution must satisfy: H  | (9.5.3) lAP(z)dz  = -(H-h)C DpU 2 h 1 If  we assume a parabolic shape for  the pressure distribution, the simplest shape to satisfy  the constraints at the keel boundaries, then the pressure drop across the keel (Figure 9-6) can be written as: 2 (9.5.4) AP{z)  = 3 C o p U ^ ((H  + h)z-z2-Hh) 0H-hf The pressure gradient as a function  of  depth depends on the width of  the keel. An exponential distribution of  ice draft  within a keel can be expressed as: (9.5.5) b(z)  = b0e~z'\ where b0 is a constant to conserve volume and X  is the e-folding  scale. If  we assume that the keel has a simple smooth shape, the keel width as a function  of  depth can be written as (see Appendix C): W( e-H/A_ e-z/A) ( 9-5-6> = -H!X  -MX e —e The pressure gradient across the keel (Figure 9-6) is then: dP  AP(z) (9.5.7) ~T = ~~7T dx  0)(z)  • Using (9.5.7) we can solve for  the pore velocity through the keel using (9.5.2). As the keel width must be finite  at the maximum draft  to be consistent with observations, in calculations for  the discrete model (9.5.7) we set H=H+Ah,  where Ah is the bin size in the internal melt model, here 0.1 m. We assume a porosity of  0.3, seawater density of  1024 kg m"3, kinematic viscosity of  1.8 x 10"6 m2 s"1, and a hydraulic radius given by (9.3.7). The ocean current velocity relative to the ice drift  is held constant at 0.1 m s"1, the temperature anomaly above freezing  is 0.18°C, and coefficients  a and [3  are set at 20 and 0.11, respectively, based on results from  van Gent (1995) for  rectangular grains with a porosity of  0.39. This study was chosen as it has both the closest porosity to ice keels and a range of  Reynolds numbers in agreement with the expected Reynolds numbers in our application. The calculations are made for  a keel 9 m in draft,  with block thicknesses of  0.5 m and a width to draft  ratio of  6.8, and an e-folding  scale of  4.2 m. These values are representative of  the average keel shapes observed (Chapter 10). The impact of  assuming these average geometries is discussed in Sections 10.5 and 10.6. Ice keels are assumed to be randomly oriented relative to the direction of  current, so that a representative flow  perpendicular to the keel is 0.063 m s ( s e e Section 9.2). The 9 m keel is chosen as it represents the largest keel observed during event D (Figure 8-1). The pore velocity increases with depth to a maximum velocity of  27.6 mm s"1 (Figure 9-6). The pore velocity at the keel draft  is set to zero, as the block thickness within the keel, 0.5 m, is greater than the bin size used to calculate the pore velocity, 0.1 m, such that no porous flow  is expected at the maximum draft. Pressure drop Pressure gradient across keel [Pa] across keel [Pa m"1] Pore Velocity, [mm s"1] Figure 9-6: Distribution of  pressure (9.5.4), pressure gradient (9.5.7), and pore velocity (9.5.2) through a 9 m keel with block thicknesses of  0.5 m and a ocean current of  0.1 m s"1. 9.6. Heat transfer  to a porous media Analogous to the flow  through tubes, once we know the keel discharge velocity, we can estimate the heat transfer  from  the fluid  to the ice. For a pipe, the rate of  heat transfer  from  the walls to the fluid  can be expressed as (Bird et al., 1960): (9.6.1) 2 = ^ A T , where K  is the thermal conductivity, D is the pipe diameter, Nu  is the dimensionless Nusselt number, AT  is the temperature difference  between the fluid  and the tube wall, and A is the area of contact. Extension of  this theory to porous materials such as packed beds or ice keels is achieved by replacing the pipe area, A, with the surface  area of  the pores, AP. Essentially, the system can be envisioned as a network of  connecting pipes, and Bird et al., (1960) noted the application of (9.6.1) to non-circular cross sections for  turbulent flow  can be accomplished by allowing D to be 4RH,  four  times the mean hydraulic radius. We judge the flow  through the keel to be close to turbulent for  the purposes of  this discussion. From Appendix B, the surface  area of  the blocks, Ap, is given as: (9.6.2) AP = 2 N  (i\b 2 +b]+b2)h2, where N  is the number of  blocks in a volume of  ridged ice, V K, and is given by: (9.6.3) N = * Z P H J L bXb 2h3 Substituting in for  N,  and recalling that RH  is defined  asp h /(y(l-p)),  the surface  area of  the blocks for  a keel volume V K=dx  dy  dz  is given by: 2(l-p)(b lb2+bl+b2)vK  _ H-p)rV K  pdxdydz <9M> A f = b{b 2h h % Equation (9.6.1) can be written as the heat flux,  Qm/dx,  to the keel ice from  the water along the x direction as (Bird et al., 1960): ( 9 6 5 ) Qm _ NUK  pdydz dx  4RH 2 T 1 V Where Qm is the heat generating melt, and T x is the temperature above freezing  of  the water at x. A new parameter in (9.6.1) and (9-6.5) is the Nusselt number, Nu.  For fluids  where the variation in viscosity is negligible (such as seawater, whose viscosity varies on the order of  1% per degree temperature; Batchelor, 1967), turbulent Nusselt numbers are given as (Bird et al., 1960; Knudsen and Katz, 1958): where Pr is the ratio of  the molecular diffusivity  of  momentum to the molecular diffusivity  of heat and is called the Prandtl number (Knudsen and Katz, 1958). The Prandtl number is estimated as 13.6 (Holland and Jenkins, 1999). The Reynolds number defined  here uses the diameter of  the porous system, 4RH, as its length scale (Bird et al., 1960). The turbulent form  of  the Nusselt number, (9.6.6), is applicable for  Reynolds numbers greater than 20,000. Here, with a peak velocity of  24 mm s"1 and Rf l=0.05 (for  h=0.5), the Reynolds number is approximately 2500, less than the fully  turbulent regime, but greater than laminar Reynolds numbers. As no simple parameterization of  the Nusselt number for  the transition region between laminar and turbulent flow  exists, and the nature of  the flow  path through the porous media would intuitively increase turbulence, the turbulent Nusselt number is used as a best approximation. 9.7. Melt of  ridged sea ice For melting ice, this heat transfer  model is complicated by the facts  that not only will the temperature change due to the heat flux,  but the melt will also result in an increasing volume of water. Considering the ice-water system as a packed bed box model, the conservation of  volume flux  through the ice pore can be written as: (9.6.6) Nu  = 0.023Re0"8 P r 1 / 3 or (9.7.1) and can be rewritten as: (9.7.2) NUKD 1/3 X.  , ux = -JT X  dx  + u0^ A p i L i R H £ o where ua is the initial velocity of  the flow,  x is the along tube distance, L\ is the latent heat of fusion  for  ice, 334000 J kg"1, p is the density of  sea ice, 920 kg m"3, and /? is the density of seawater, 1024 kg/m3. Note that the subscripts o,m,x indicate input, melt, or along channel flow  in the x direction respectively, and do not indicate a derivative. This notation is used for  the remainder of  the thesis. Similarly, for  conservation of  heat flux,  Q, the equation Q0-Q,„=Qx becomes: m 2/3 7 7 NUK  pdydz x, 1 _ ?/3 . > (9 7 3) T 0pu0cp p dydz  \T X  dx  = T xpuxc„ p dydz 4RH  0 which reduces to: * 4RH 2pcp (9.7.4) ]T xdx=  TnT^o uo ~Tx ux) 0 Nu  K  p where cp is the heat capacity of  seawater, 4218 J kg"1 K"1. Using (9.7.2) for  ux in (9.7.4), we find the equation for  the heat transfer  through a packed bed is: 2 x pcp x 4RH  pcpu0 (9.7.5) )Tx  dx  + —— T x \T X  dx  = 77^  (T 0 - T x) 0 PiLi o Nu/cp 1/ 3 where T x(0)=T o. This is a complex differential  equation of  the form: dx  dx  dx  p + ay where y = JT xdx The solution for  (9.7.6) can be written in terms of  the Lambert W function  (Corless et al., 1996) as follows: W e*=_zaprs_ (9.7.7) W + rto) a(y~PT 0) -x-apT 0 m + aT 0) and  W  = /3(\  + aT 0) To solve for  T x (9.7.7) can be solved for  y. This was done using the built-in Matlab/Maple solver for  the Lambert W function  (Corless et al., 1996). Once T x is known, the heat flux  from  the water to the keel can be found  from  (9.6.5). Using the above equations for  a hypothetical keel 9 m in draft,  60.8 m wide, and constructed out of  blocks 0.5 m thick, produces internal heat flux  and water temperatures as shown in Figure 9-7. From the heat flux,  the enhanced melt rates for  ridged ice can be calculated as the summation of the internal melt within the keel divided by (1 -p)xn, to account for  the decrease in pore volume, assuming that repacking maintains a constant porosity. Figure 9-8 shows the internal melt rates compared with observed level ice melt rates. The internal melt rate for  ice within a 9 m keel is slightly less than the observed ridged ice melt at 9 m from  event D (days 192 to 207), over all keels, but much greater than the level ice melt alone, suggesting that internal melt may be an important contributor to the enhanced melt of  ridged sea ice. Level ice melt during event D, 23.6 mm day"1, is less than the contribution from  internal melt for  all ridged ice greater than 4 m. The melt rate increases with ridge draft,  consistent with the observations of  Rigby and Hanson (1976). Note that the modelled rates of  melt are tied to a plausible guess for  the value of  T 0 (0.18 K). The actual value is not known. Internal Heat Flux, Wm" -20 -10 0 10 20 water temperature anomaly above freezing [K] -20 -10 0 10 horizontal distance across keel Figure 9-7: Top: Internal heat flux  through keel showing the decrease in heat transfer  across the keel. Bottom: Change in the water temperature through the keel due to the heat lost to melt and the influx  of  fresh  water at the freezing  point. At deeper drafts  the water flows  faster  (from  left  to right on figure,  see Figure 9-6) moving further  through the keel before  the water reaches the freezing temperature. Keel porosity is 0.3 and block thickness is 0.5 m. Melt Rate, [mm day"1] Figure 9-8: Melt rates from  porous flow  through a keel 9 m (heavy black line). Thin black lines show the melt rates upstream and downstream of  the keel crest. Expected melt from  level ice is shown as the thin dashed line, observed melt rates are shown as a thick dash-dot line. The melt rate from  a small 4 m keel is shown for  comparison (heavy dashed line). Keel porosity is 0.3 and block thickness is 0.5 m for  both keels. The bulge in observed melt in the range of  3 to 6 m of  draft  can be explained as a feature  of  the keel population. The internal melt rates from  a 4 m keel constructed of  0.5 m thick blocks are shown on Figure 9-8 (heavy dashed line). A keel population with many keels in the size range of 3-6 m may lead to the observed bulge. A more accurate comparison of  observed and internal melt rates will be discussed in chapter 11 when the melt rates for  keels are combined with the keel population to develop a statistical ridged ice internal melt rate. Equation (9.7.5) is complicated by the inclusion of  the melt water to the porous flow.  For a 9 m keel with blocks 0.5 m thick, the increase in flow  velocity due to melt is much smaller than the porous flow.  Figure 9-9 shows that the addition of  melt water to the porous flow  through the keel adds only a maximum of  0.007 mm s 1 to the flow  at a depth of  3.2 m. The ratio between the input from  melt to the initial porous flow  is less than 0.1 %, and the assumption that ua-ux is justified for  the internal melt model. Because the influx  of  melt water is small (Figure 9-9) and can be neglected, ux—u0 and (9.7.4) can be solved directly to give a solution for  the temperature: Xr T Ape RH2U q (9.7.8) \T xdx=—  — (  T 0-T x)t 0 NuKp or, NUK  p1 (9.7.9) T x=r oe 4 P u „ C p RH 2 -30 -20 -10 0 10 20 30 horizontal distance across keel Increase in porous velocity due to melt [mm/s] Figure 9-9: Top: Porous flow  velocity due to melt through the keel. Increase in melt is very small compared with porous velocities that can exceed 20 mm s"' (Figure 9-6). Keel porosity is 0.3 and block thickness is 0.5 m. 10. MELT WITHIN A SINGLE KEEL 10.1. Model sensitivity The internal melt model developed in chapter 9 is an initial estimate that combines estimated keel and ocean parameters, empirical formulas,  and simplifying  assumptions. To calculate the porous flow  we used estimates for  the Forchheimer coefficients  from  similar, but not identical, physical systems. For the estimation of  the pressure gradient we relied on laboratory tests to estimate the drag coefficient  across the keel. To calculate heat transfer,  we used a form  of  the Nusselt number applicable to fully  developed turbulent flow,  although Reynolds numbers indicate that internal flow  will be in the transition region between laminar and turbulent. Finally, we assume that the porous flow  can be approximated with a one-dimensional model, although melt may introduce buoyancy-driven vertical flow  within the keel. All of  these assumptions and estimations introduce some uncertainty into the calculated melt rates. In this chapter we investigate the consequences of  these assumptions on the predicted melt rates and show that, while melt rates may vary, the ability of  the internal flow  to melt ridged ice at melt rates comparable or greater than level ice melt is maintained. 10.2. Sensitivity of  melt to parameters for  porous flow To find  the pore velocity through the keel, the Forchheimer equation (9.5.2) utilizes the parameters a and /?. The appropriate values for  a and P for  this system are unknown. A review of  the literature and personal communications with civil engineers reveal that no research has been conducted for  flows  with similar Reynolds numbers and material with similar block dimension ratios. We therefore  chose to use values for  a and P that represent a similar system of irregular shapes with appropriate Reynolds numbers (115-1150, van Gent, 1995). However the porosity for  this system was higher (0.39) than the average porosity of  ridged ice (0.3, Timco and Burden, 1997) and the particles were much smaller with a median (by mass) dimension of  61 mm. To determine the uncertainty in flow  velocities with a and P, we compared flow  estimates over a range of  published coefficients  (Table 9-3). In the literature, a and p are found  to range from  11-37, and 0.08-0.34, respectively. Calculating the velocity through the keel for  this range of coefficients,  increasing or and /? will decrease the pore velocity (Figure 10-1). The maximum velocity changes by a factor  of  2.2, from  27 mm s"' to 12 mm s"1 for  a 9 m keel with 0.5 m level ice blocks and an ocean current velocity of  0.1 m s"1. The velocity through the keel at 5 m depth varies by a factor  of  2.2 for  the same keel. Variations in Velocity with Alpha and Beta [m s"1] Alpha Figure 10-1: Variations in calculated maximum pore velocity for  various coefficients  a and /?. Further confidence  in our choice of  a and /? is found  when we calculate the porous flow  velocity using values for  or and /? that corresponds to a packed bed of  spheres with a porosity of  0.3, similar to ice (Table 9-3; Hall et al., 1995). From values for  a'  and /?' we find  that a= 19 and P = 0.09 using (9.3.9). This introduces a 9% change in the porous velocity (Figure 10-1), only a small variation from  the velocity found  using van Gent's (1995) values. As the values of  Hall et al., (1995) correspond to a system with an appropriate porosity, and the values of  van Gent (1995) correspond to irregular shaped particles that may better represent the blocks of  ice within a keel, it seems reasonable to assume that true values for  a and /? will not introduce large errors into the porous velocity. As the equation for  water temperature (9.7.9) depends on velocity as an exponential, and the e-folding  scale of  that negative exponential varies as uJNu,  the e-folding  scale for  heat transfer  will vary with the pore velocity to the l/5th, u0vs. However, since the rate of  heat transfer  from  the water to the ice varies as u's (9.6.5), increases in velocity will correspond almost directly to increased melt rates. Thus, over the entire range of  Forchheimer coefficients,  the internal melt rates vary by approximately a factor  of  2 and will not effect  the basic capacity for  internal melt to contribute to the enhanced melt of  ridged ice. A further  variation in the pore velocity is introduced by the influx  of  melt water which acts to increase pore velocity. The Forchheimer equation is derived for  a constant porous flow. Increasing that flow  volume will decrease the pressure gradient, dP/dx,  and thus decrease the pore velocity according to (9.5.2). As the influx  of  melt water will increase the flow  by less than 1% (Figure 9-9), the Forchheimer equation (9.3.6) can be written by expressing the pore velocity, ux, as a perturbation from  the initial flow  u=u0+u': 1 dP  _ u0+u' „(u 0+u')2 (10.2.1) ~ ^ = a v 2~ + / 3 P Sx rh2 RH The adjustment in the pressure gradient due to u' can then be written as: 1 dP'  u'  „nuQu' (10.2.2) P & RH 2 RH and as u' is less than 1% of  u0, and all the terms on the RHS are less than 1% of  their undisturbed values, the decrease in the pressure gradient is minimal. Finally, as the velocity depends on the square root of  the pressure gradient (see (9.5.2)), u is not extremely sensitive to small variations in dP/dx  and the impact on the melt rate will be negligible. The final  parameter impacting the pore velocity is the choice of  the drag coefficient,  CD=3.5. The impact of  the drag coefficient  on the pressure gradient is linear (9.5.4) and the impact on the pore velocity, obtained from  (9.5.2), is CD1/ 2 . Over the range of  coefficients  (3.8 to 1.4; Cummins et al., 1994) associated with Froude numbers typically found  in the Arctic (0.1 to 0.7; Pite et al., 1995), the impact on the pore velocity is thus less than a factor  of  (3.8/1.4)"2, or 1.65. The change in pressure over the whole keel is proportional to 'Ap CDU 2 (9.5.4). Using the laboratory results of  Pite et al., (1995) reported in Cummins et al., (1994), the impact of  various drag coefficients  on the pressure gradient can be found  by computing F a2 CD (Table 10-1), as the Froude number, F 0, is directly related to U  (9.4.4). For all Froude numbers greater than 0.3, the increase in the drag coefficients  corresponds to a decrease in the Froude number (and ocean current velocity) leading to a constant ratio of  F 02 CD ~0.5. An immediate application of  this result is the impact of  storm events on the melt rates. Following from  Table 10-1, we see that once the ocean current increases such that the Froude number exceeds -0.3, the pressure gradient will approach a constant value approximately 250% greater than when the Froude number is 0.23 (Table 10-1). This increases the pore velocity by a maximum factor  2.51/2, or 1.6 (9.5.2). Storm events may then increase the rate of  heat transfer  to the ice by a factor  of  u4'5, (9.6.5), and thus have the capacity to increase melt rates over short time scales by a factor  of  approximately 1.5. However, the drag coefficients  in Table 10-1 are based on one data set where keel drafts  are kept constant at 50% of  the mixed layer drafts.  In the Arctic, keels will be of  varying shapes, sizes, and drafts  and further  research about the drag coefficients  for  the wide variety of  naturally occurring keels is needed to increase confidence  in the drag coefficient. Fo 0.23 0.34 0.44 0.53 0.69 0.84 1.01 1.23 1.64 CD 3.5 3.8 3.5 3.0 1.4 0.77 0.46 0.31 0.20 F0 CD 0.19 0.44 0.68 0.84 0.67 0.54 0.47 0.47 0.54 Table 10-1: Experimental drag coefficients  for  a laboratory model of  a keel in a stratified  flow (Cummins et al., 1994). 10.3. Sensitivity of  melt to parameters for  heat transfer With a known porous velocity through the keel, the heat transfer  from  the water to the ice can be calculated. This heat flux  is dependent on the Nusselt number, Nu,  which influences  both the temperature of  the pore water given by (9.7.9) and the heat flux  to the ice given by (9.6.5). The Nusselt number used to calculate heat transfer  is designed for  turbulent flows  (Bird et al., 1960); however, Reynolds numbers for  a flow  through a keel indicate that the flow  may not be fully turbulent. The transition region from  laminar to turbulent flow  is not well understood, as it is usually avoided in experimental design for  engineering research (Bird et al., 1960). However, a comparison with laminar Nusselt numbers for  heat transfer  in tubes is useful.  Here, the laminar Nusselt number is Nu=\.%6(Re  Pr (4R H/l)) m (Bird et. al., 1960), where / is a length scale we approximate as the keel width. The laminar Nusselt number is within a factor  of  2 of  the turbulent Nusselt number for  a 9 m keel with block thickness of  0.5 m, indicating that errors introduced by the choice of  Nusselt number do not influence  the general ability of  a porous melt model to contribute significantly  to the enhanced melt of  ridged ice. Without further  information  about the turbulent characteristics of  the flow,  no further  refinement  of  the Nusselt number is possible. 10.4. Stratification  and stability: 2-d components of  fluid  flow For simplicity, the Forchheimer model used here to predict fluid  flow  is 1-dimensional. This assumption implies that no significant  flow  in the vertical direction is present. This would be incorrect if  the change in density of  the pore water, due to the influx  of  fresh  water, was large enough to drive vertical flow.  However, the fresh  water influx  is very small compared to the porous flow,  less than 0.2 % (Figure 10-2), and the change in pore water density through the keel (Figure 10-3) is both very small, and stably stratified,  hence the assumption of  horizontal flow  is valid. While the fresh  melt water may not influence  flow  within the keel, the volume of  fresh  water produced by internal melt is not insignificant.  Integrating the flux  of  melt water over the keel, the melt produces 3.5 m3 of  fresh  water per day, exceeding the fresh  water production from  an equal area of  level ice. The discharge flow  from  the keel is not fresh  however, but well mixed with the flow  of  ocean water through the keel and thus would not create a distinct fresh  layer under the ice, but would increase stratification  within the upper water column. Increase in porous velocity Porous velocity Percent increase due to melt [mm s"1] [mm s"1] in velocity Figure 10-2: Increase in pore velocity due to influx  of  melt water. Figure 10-3: Density anomaly (from  1000 kg m3) through the keel due to the influx  of  fresh  water and heat loss. Density varies by only 0.004% throughout the keel and pore water is stably stratified. -30 -20 -10 0 10 20 30 horizontal distance across keel 25.27 25.26 25.25 25.24 Density anomaly through keel [kg m"3] 10.5. Sensitivity of  melt to block thickness and porosity The internal keel geometrical properties of  block thickness and porosity will impact the internal melt rates. Larger blocks lead to increased pore sizes and thus an increased hydraulic radius, RH  , which appears in the equations for  pore velocity and heat flux  (see (9.5.2) and (9.7.9)). Increased porosity will increase the hydraulic radius. As no information  on block sizes or porosity can be gathered from  the sonar data, we investigate the effects  of  variations in geometry on melt rate using a sensitivity analysis. Ridge porosity is reported to range between 20 to 45% (Bowen and Topham, 1996). We investigate the impact of  varying porosity for  a 9 m keel with block thickness set at 0.5 m. Results for  porosities from  0.1 to 0.5 can be seen in Figure 10-4. As porosity increases, melt rates increase due to an increased hydraulic radius and heat transfer.  However, increasing porosity will also decrease the surface  area, as (9.6.4) can be rearranged as AP=(\-p)yV KJh  using (9.3.7) where p is the porosity, ^is a geometrical factor  (see Appendix B), and V K is the keel volume. As porosity increases past 0.4, the increase in melt rate with porosity slows, while at porosity values in excess of  0.5, the melt rates begin to decrease due to the loss of  surface  area for  heat transfer  to occur (not shown in figure).  Melt rates are thus maximized at porosities between 0.4 and 0.5, at the upper bound of  observed values. Melt rate with varying porosity [m day"1] draft  [m] Figure 10-4: Variation in melt rates with porosity for  a 9m keel with block thickness 0.5m. The relationship between block thickness and keel draft  during ridge formation  is known to vary with the distribution of  floe  size (recall (5.2.2)). (10.5.1) H  = fL 2(1 -p) 1/2 hV 2 Unfortunately,  this relationship cannot be extended to provide information  about the block thickness within keels observed by the sonar. While immediately following  formation, surrounding level ice may indicate the thickness of  the blocks within the keel, subsequent growth of  the level ice means that during the winter, block thickness will be less than the thickness of  the surrounding level ice. Further, the floe  size constraint also suggests that all level ice within a floe may be consumed during ridging, leaving only thicker level ice adjacent to the keel. While some keels created from  very thick first  year ice no doubt exist, these keels are expected to be rare, as they are only formed  during the end of  the winter when ice is thickest and when no thinner ice is available to be ridged. The main impact on melt is expected to come from  the more common keels with moderate block thickness. All keels observed during events A and D could conceivably be formed  from  ice 0.5 m thick (based on a truncation of  H=20  m1/2 hU 2) indicating that a block size estimate of  0.5 m may be reasonable. The impact of  different  block thicknesses on melt is plotted in Figure 10-5. Block thickness is varied from  0.1 to 1.2 m to represent the range of  blocks expected within keels. Melt rates increase with decreasing block sizes, as surface  area increases (from  (9.6.4)) and RH  decreases, until approximately h-0.25 m. At this point, melt rates are maximized and being to decrease, as the e-folding  scale for  the heat transfer  (9.7.9) is dependent on RH 2. As h decreases past 0.25 m, RH 2 becomes very small and the e-folding  scale (9.7.9) decreases such that the melt is restricted to an upstream portion of  the keel and the averaged melt with draft  decreases. For the purpose of  illustrating the potential for  internal melt to account for  the enhanced melt of ridged ice, we fix  the porosity at 0.3 to match observations (Timco and Burden, 1997) and the block thickness at 0.5 m as a best guess. Further field  studies of  the distribution of  ridge porosity and block thicknesses, especially of  the numerous small 3-5 m keels would be useful  in making more accurate internal keel geometry estimates. 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 Figure 10-5: Variation in melt rate with block thickness for  a 9 m keel with a porosity of  0.3. 10.6. Keel width and shape For the 1 -d Forchheimer model used to describe porous flow,  the keel shape and width parameters affect  the melt in similar ways. Increasing the width of  a keel at a depth, via the total keel width or via the shape parameter, will decrease the pressure gradient and decrease the porous velocity given by (9.5.2). As keel widths increase and the pore velocity decreases, the e-folding scale for  the heat transfer  given by (9.7.9), will decrease, causing melt to occur mainly on the upstream side of  the keel. As the melt rate is a statistical measure of  the average melt for  all ice of that draft,  the overall melt rate will decrease. Note that in the real world, as no one side of  the keel will consistently be upstream of  the ocean currents, melt should be symmetric. Keel width and shape parameters may not be separable from  keel block thickness. Recall from section 5.2 that keel depth is a function  of  floe  size. Keels that exceed the average width are likely to have formed  from  larger floes  based on .the results of  keel simulations (Hopkins, 1998). These keels are thus formed  when the limiting truncation between //and h" 2 approaches the mechanical limit of  20 mI/ 2 (section 5.1). Then, wider keels may have blocks of  smaller thickness than narrower keels. As the block thickness is such a dominant factor  in determining melt rates, the errors introduced by uncertainties in keel width are secondary to concerns of  block thickness. 10.7. Consolidation due to melt The keel modelled in chapter 9 is assumed to have a porous structure that extends up to a level ice thickness equal to the block thicknesses that make up the ridge. This is the structure of  a newly constructed ridge, before  consolidation processes can increase the solid core of  ice at the water level of  a ridge. As consolidated ridged ice is predicted to have growth rates approximately twice that of  level ice (Lepparanta and Hakala, 1992; Hoyland, 2002; Veitch et al., 1991), the porous structure of  a keel will extend from  the maximum draft  to the bottom of  the consolidated portion of  unknown thickness (as the keel age is unknown). However, as the heat flux  is low in the upper portions of  the keel (Figure 9-7), the overall melt rates within ridged ice will not be affected greatly by the loss of  melt in a consolidated portion and hence the errors in melt rates are assumed to be small. Multi year ridge keels are consolidated with very low porosities. Models and observations of melt demonstrate that this consolidation is unlikely to occur during the winter season as the consolidated layer growth rate is insufficient  to extend throughout the entire keel in one winter season. Rigby and Hanson (1976) observed that voids in the ridge keel became more slush filled as the summer progressed, allowing for  consolidation early in the next winter season. Internal melt due to porous flow  may explain the consolidation process by allowing the keel to melt internally and eventually repack the now soft  and partially melted blocks into a consolidated shape. For the purposes of  model development, we have assumed this repacking would not occur during the events in early June and July simulated here, but itcould occur later in the summer, consistent with observations of  Lepparanta et al., (1995) who found  that porosity did not change during melt in the lower portions of  the keel in the initial part of  the melt season in the Baltic Sea (4th February to 19th April). Further, the model assumes that the block shapes within the keel are retained throughout the summer. As the melt season progresses, the block size and orientation may change, impacting the porous flow  and heat transfer.  The internal melt model may then be most applicable during the early melt season, and further  investigation of  the evolution of  the internal keel geometries due to melt is suggested. 11. INTERNAL MELT OF A DISTRIBUTION OF RIDGE KEELS 11.1. Keel statistics To determine the characteristics of  the keel population during the melt season, keels are identified within the ice draft  profile  using the level ice criteria of  Melling and Riedel (1995). In these criteria, ridges are identified  as regions of  ice surrounded by level ice floes,  which themselves are defined  as greater than 10 m in extent and varying over only 0.25 m in thickness. For the purpose of  an internal melt model, we add one more constraint to the definition  of  ridges. The definition of  Melling and Riedel (1995) included rafted  ice as a ridge if  the extent of  the raft  was less than 10 m. To exclude rafted  ice, which is not porous and thus cannot participate in internal melt, and to exclude small rubble structures where porous melt theory may not apply, the maximum draft  of the keels must be at least three times the minimum draft  of  the surrounding level ice. Further, the edge of  a keel must have a draft  that exceeds the level ice draft  by 0.25 m. This excludes small level ice floes  (less than 10 m) from  being counted in keel flanks. Attempts to parameterize the keel width, shape, and block thickness as functions  of  keel depth should not be interpreted as demonstrating the dependence of  these parameters on keel depth. Rather, the purpose of  sections 11.2 and 11.3 is to allow the identification  of  average keel geometry for  use in constructing a melt model. The causes of  the relationships between geometrical variables and keel depth are beyond the scope of  this thesis. 11.2. Keel width Timco and Burden (1997) report keel widths to be approximately 4 times the keel draft,  a value already used in this thesis to estimate the floe  size needed to build a ridge of  maximum draft. However, this ratio is poorly constrained, since no information  is given on the types of  keels measured or the criteria for  measurement. A tendency to study linear ridges could bias the keel width measurement. The ratio of  the keel extent seen by the sonar to the keel draft,  R, in the ice profile  data during event A in 1999/2000 has a mean value of  12.7 and a standard deviation of  10.2 (Figure 11-1). Ratios of  extent to draft  are slightly higher in event D, perhaps due to a reduction of  keel drafts due to melt. For model development, we want to assume an average keel shape, and the ratio of keel extent to keel draft  is fixed  at i?=13. This gives results similar to those of  Melling and Riedel (1995) who found  keel extent to be approximately 10 times keel draft  for  population of  21 large keels. Ratio of keel extent to keel draft 1 4 0 1 ' ' + event A x x event D 1 2 0 - -1 0 0 - * 8 0 -+ X X X * x 6 0 -t o - " , ; » * „ « 2 0 - ^ * 0 1 — 0 5 10 15 Keel draft  [m] Figure 11-1: The ratio, R, of  keel extent to keel draft  for  keels observed during event A (+) and event D (x). Assuming that keels drift  past the sonar with a random orientation, the true keel width, W,  can be found  from  JF<l/cos (6)  >= <R> H where the angled brackets indicate the average over the range of  angles from  0 to 7i/2 radians (Figure 11-2). As the cosine of  tt/2 is 0, if  the sonar did approach the keel at this angle, the sonar would track along the axis of  the keel and the keel extent seen by the sonar would be infinite  if  the ridge was infinite  in length. Assuming then that the maximum keel length along the keel axis is not more than 8 times the keel width, the range of  angles is then 0 to tan~'(8). Over this range, the mean of  l/cos(#) is 1.92 and the true keel width works out to + event A x event D + Xx* + X * xX * ? * + * + i be 0.52RH.  If  the keel length is 8 times the keel width, then for  an 8m keel, the keel width is 53 m and the maximum expected keel extent is 430 m. Figure 11-2: schematic of  keel extent as seen by the sonar and true keel, width, W. The observed keel width to keel draft  ratio is larger than the ratio of  4 times keel draft  found  by Timco and Burden (1997) from  field  measurements. This discrepancy likely stems from  the inclusion of  irregular shaped ridges in our keel population, including corner regions where the accumulation of  rubble is greater than on linear sections (Bowen and Topham, 1996). For a melt model to work it should be based on the mean shape of  all ridged ice, not just the simple linear sections of  ridges, and we therefore  chose to use a keel width to keel draft  ratio of  0.52R. The use of  the factor  of  4 (Timco and Burden, 1997) in calculating the floe  size consumed in ridging (section 5.2) may still be appropriate. Comer regions or non-linear ridges are the result of complex ridging interactions and are not yet well understood. If  these wider ridged regions are created from  the collision of  more than two plates, then the consumed level ice from  multiple plates may be close to the constraint given by (5.2.2). 11.3. Keel shape The average distribution of  ice draft  within keels was found  to have an exponential form  in Chapter 5.5 with an e-folding  scale A. For five  different  keel sizes, A was shown to increase with H,  with keels at 5, 7.5, 10, 12.5, and 15 m having e-folding  scales of  approximately 2.7, 3.9, 4.3, 6.0, and 6.5 m, respectively. For simplicity, and due to the extreme variations in keel shape, with variations in shape exceeding the variations in e-folding  scale with depth (Figure 5-8), a constant value of  A^ =6 m was chosen. For determining the shape of  the keels used in a conceptual model of internal melt, we want to define  the average-shaped keel. While these extreme variations in keel shape for  individual keels are far  greater than the variations in mean keel shape parameters for varying drafts,  some trends can be observed when a large range of  keels are studied. The data analysis of  keel shapes was repeated for  two data sets collected in 2000: all keels within the winter of  1999/2000 and all keels within the spring melt season (day 150 to day 220 of  2000). Keels were binned with a + 0.5 m range and the average keel shape (from  the mean keel draft distribution) for  each data set was calculated. From the mean keel shape, the e-folding  scale and the standard deviation of  that scale were found.  The variation in e-folding  scale with maximum keel draft  is seen in Figure 11-3 where the keel shape e-folding  scale is estimated as a linear best-fit  line of  the form  A=0.50H-0.32  m, for  H>  1 m. The increased error bars for  the e-folding  scale at large keel draft  reflect  the smaller sample size. While 1012 keels between 3 to 4 m in draft  were observed during the 1999/2000 winter and spring seasons, only 107 keels between 8 -9 m and 8 keels between 13-14 m were observed in the same period. The e-folding  scales found  during 1997/1998 (Figure 11-3, filled  squares) show the same pattern of  a linear increase with draft,  especially for  keels less than 5 m. Melling and Riedel (1995) found  that the mean e-folding  scale for  21 keels with drafts  exceeding 20 m was 15 m. Extrapolation of  the best-fit  line in Figure 11-3 gives an e-folding  scale of  12.2 m for  a 25 m keel, compatible within sampling error to the value found  by Melling and Riedel (1995). A linear best-fit  can be used to estimate the average shape of  a keel, and thereby predict the average melt over a population. However, the extreme variation in the shape of  observed keels means that individual features  will melt in greatly different  ways. e-folding scale [m] keel draft  [m] Figure 11-3: Mean e-folding  scales for  keels observed in 2000 during the melt season spanning events A and D (circles) and during the entire winter 1999/2000 (diamonds). Standard deviations of the e-folding  scales are plotted as error bars. Observed e-folding  scales from  the 1997 data (section 5.5) are plotted as filled  squares. The linear best-fit  line takes the form  A,=0.50H  -0.32 m over the entire winter and spring seasons and match observations well. 11.4. Melt of  a population of  ridged ice Using the keel statistics in sections 11.1 to 11.3, the enhanced melt rates for  keels of  average sizes can be calculated (Figure 11-4). The total melt rate is then a weighted average of  these values using the distribution of  keel drafts  for  ridged ice. A standard definition  of  ridge keels (Melling and Riedel, 1995) includes all features  surrounded by level ice of  lesser draft.  This definition  includes rafted  ice that is less than 10 m in extent. The internal melt model will apply only to ridged ice, as internal flow  through rafted  ice cannot occur. To restrict the melt model, the distributions of  ridged and rafted  ice, g,i(fy  and gra(H)> respectively, and the number of  occurrences of  each is found,  rjri and rjra. Here rafted,  or non-porous, ice is defined  as ice where the maximum draft  is less than three times the level ice thickness. The distribution of  ridged ice is then weighted as: (11.4.1) gri'(H)  = gri(H)- V r i 8 r i i H } Vri  gri(H)  + tlra  gra( H) The resulting distribution of  ridged ice draft  is shown in Figure 11-5. The sum of  the distribution is less than one, approximately 0.75, as 25% of  all thick ice is classed as rafted.  The criteria that ridged ice must reach drafts  greater than 3 times the level ice is a crude one; undoubtedly some small ridges exist that are surrounded by thick level ice. However, this criteria does serve as a good first  approximation to exclude ice that is rafted  or features  that consist of  rubbled ice of large block thickness relative to draft  and thus unlikely to support porous flow.  Including all ice that is not classified  as level ice would be incorrect, although test simulations including these rafted  ice features  did not vary the melt rates significantly. Figure 11-4 shows the modelled enhanced melt rates for  event D. The expected internal melt is much greater than the observed level ice melt and the summation of  internal and level ice melt can be seen to reproduce the observed melt for  all but ice within the range 6-8 m. This discrepancy may be due to ridging of  ice into this 6-8 m range, thus decreasing the observed melt. Ridged ice melt rates [m day"1]: event D Figure 11-4: Melt rates within ridged ice during event D. The enhanced melt rate due to the internal melt (heavy black line) increases the melt rate from  the level ice melt (thin dashed line) to match the observed melt rates (thin line). The probability density distribution of  ridged ice (fraction  per decimetre) is indicated by the heavy dashed line. Probability density of keel draft Figure 11-5: Distributions of  the maximum draft  of  ridged ice keels (heavy line) and rafted  ice (thin line) during event D (fractions  per decimetre). The weighted distribution of  ridged ice keels is shown as the dashed line and represents the proportion of  thick ice at that draft  that is in porous ridges. The enhanced melt of  a population of  ice during event A (Figure 11 -6) is close to the observed melt. To model event A (days 154 to 173), the water temperature was set at 0.09 degrees above freezing.  This is based on the water temperature observed during the SHEBA project at day 165 during the winter of  1998 (Figure 9-3, SHEBA data). However, conditions at SHEBA two years earlier may vary greatly from  conditions at the mooring site 2 in the year 2000. 0.12 0.1 0.08 0.06 0.04 0.02 0 Figure 11-6: Melt rates within ridged ice during event A. The enhanced melt rate due to the internal melt (heavy black line) increases the melt rate from  the level ice melt (thin dashed line) to represent most of  the observed melt rates (thin line). The probability density distribution of  ridged ice (fraction  per decimetre) is indicated by the heavy dashed line. 11.5. Enhanced melt in redistribution models To explore the abilities of  internal melt to contribute to the observed enhanced melt of  ridged ice, the internal melt rates found  in Figure 11-4 and Figure 11-6 are added to the observed level ice melt and the ice draft  redistribution model is run over events A and D with and without ridging. Results are displayed in Figure 11-7 and Figure 11-8 and show the ability of  an internal melt model to decrease the amount of  ridged ice significantly.  In Figure 11-7 we see that when ridging is excluded, the model does an excellent job of  reproducing the observed distribution of  ridged ice, as expected from  the melt rates of  Figure 11-6. With ridging, the model cannot melt enough ice and is still producing new thick ridged ice. Ridged ice melt rates [m day ]: event A internal melt l l l — level ice melt observed melt — distribution of ridge keels -J  / J / 1 ' — , s * i i N draft  [ml Figure 11-7: Probability density of  draft  (fraction  per decimetre) for  the internal melt model predicted distribution of  ridged ice during summer melt from  days 154 to 173. Allowing ridging to occur, the model predicts an increase in thick ice due to ridging (thick solid line). Without ridging (thin solid line) the model predicts a decrease in ridged ice density in agreement with observed final distribution (dashed line). Figure 11-8 shows the distribution of  ridged ice when the internal melt model is run over ice motion event D. Visible in these results is the ability of  the internal melt model to melt significant  amounts of  ice such that, even with ridging (thick line), the model shows a decrease in ridged ice at all depths less than 5 m. At larger drafts,  ridging still dominates over the predicted internal and level ice melt, resulting in an increase in ridged ice rather than the observed decrease in ridged ice. The model results with ridging and level ice melt (from  Figure 8-7) are plotted as the thin dashed line, showing the marked improvement of  the enhanced melt model. Model results without ridging show melt in agreement with observed melt, but are of  limited value without the inclusion of  ridging. draft  [m] Figure 11-8: Probability density of  draft  (fraction  per decimetre) for  the internal melt model predicted distribution of  ridged ice during summer melt from  days 192 to 207 of  2000. When ridging is included, the enhanced melt from  porous flow  (heavy solid line) reduces the production of ridged ice when compared to the model with only level ice melt (thin dashed line). 11.6. Ability for  internal melt to account for  enhanced melt rates The results of  Figure 11-7 and Figure 11-8 demonstrate the importance of  internal melt processes to the enhanced melt rates of  ridged ice. However, the inability of  the model to melt the thickest ice suggests that the strength of  porous melt is being underestimated, which may be caused by the choice of  water temperatures. The water temperature departure from  freezing  may be too low (as SHEBA data may not be relevant to the ocean conditions at site 2 in the year 2000), such that melt rates of  ridged ice are significantly  higher than those found  in section 11.4. The turbulent heat transfer  to the water column due to increased mixing at the interface  of  the two-layer flow  may increase the water temperature. As the large keels extend over a significant fraction  of  the upper layer of  the water column, the hydraulic flow  is amplified  for  these keels, increasing the temperature of  the upper layer of  the ocean. The recent work of  Skyllingstad et al., (2003) demonstrated the ability of  ice keels to increase turbulent velocities. This may increase the temperature of  the water column and thus increase melt rates. The estimated freezing  temperature departures of  0.18 and 0.09 degrees for  events D and A, respectively, are estimates based on data from  another year at a different  location. A temperature profile  from  the 3rd of  June, 1987 at the edge of  the landfast  ice in the Beaufort  Sea shows a freezing  temperature departure of  0.35-0.7 degrees, indicating that a temperature difference  of  0.4 degrees between the water and ice is not unreasonable, and that the earlier choices of  0.09 and 0.18 degrees may be too low. Using 0.4 degrees as the freezing  temperature departure for  event D, with a maximum ridge constraint of  9 m1/2 hV 2, greatly increases the melt of  the ridged ice and produces modelled results in agreement with observed ablation (Figure 11-9). Figure 11-9: Probability density of  ice draft  (fraction  per decimeter) for  the internal melt model predicted distribution of  ridged ice during summer melt from  days 192 to 207 of  2000. Including the enhanced melt from  porous flow  (heavy solid line) with a assumed freezing  temperature departure of  0.4 degrees reproduces most of  the features  of  the observed distribution, except for  an over-prediction of  ice within .drafts  of  4-8 m. The internal melt model proposed here, and used in section 11.6, is an initial approach with many potential unknowns. While we have chosen reasonable values for  the ocean currents, keel statistics, Nusselt number, and Forchheimer coefficients,  these are estimates and contribute uncertainty to the melt rates. However, the goal of  this initial investigation was to demonstrate the potential for  internal melt to contribute to the observed enhanced melt of  ridged ice. The results clearly show that internal melt can make a significant  contribution to enhanced melt and should be investigated further. 12. CONCLUSIONS: GEOMETRIC FACTORS INFLUENCING RIDGE MELT Section D has developed and applied a semi-quantitative model for  the enhanced melt of  ridged ice. First, the inability of  level ice melt processes to ablate thick ice is established using the ice draft  redistribution model of  Sections B and C. Second, a semi-quantitative model for  internal ice melt in ridge keels is developed. Finally, the internal melt model is included in the ice draft redistribution model to demonstrate that internal melt processes ablate thick ridged ice in agreement with observations. Initially, during the melt season, the model is constrained to melt level ice at a rate equal to the observed value. This provides an upper bound on the contribution of  surface  melt to the melt of ridged ice and allows for  comparison between the observed and modelled melt of  thick ice. The ridging model truncates the maximum draft  of  new ridges at lower drafts  than in the winter season to represent the decreasing ice strength. Model results from  two ice motion events during the melt season show an increase in the distribution of  ridged ice, as ridging dominates melt processes, rather than the observed enhanced ablation rates. The inability of  the IDR model to reproduce the evolution of  ridged ice confirms  that ridged ice melt cannot be represented by 1-dimensional level ice melt. The underwater region of  a ridge, consisting of  interconnected ice blocks, is a porous media with a large surface  area. The flow  of  warm oceanic water through the porous keel is suggested as a mechanism for  enhanced melt. Pore velocities are estimated by adapting flow  models from groundwater and engineering literature to the geometries of  ridged ice. Driving the flow  is the oceanic current, which generates pressure gradients across the keel. Porous flows  through a 9 m keel are predicted to reach maximum velocities of  27 mm s"1, for  average ocean currents of  0.1 m s"1. Heat transfer  from  the warm oceanic water to the ridged ice is modelled as the heat transfer  from a fluid  to a packed bed. Integrated melt rates, assuming repacking maintains a constant porosity, reach melt rates almost five  times the level ice melt rate (for  a 9 m keel), supporting the role of internal melt as a contributor to enhanced melt. Estimates for  keel shape, width, and the distribution of  maximum draft  are found  from  the ice draft  profiles.  These parameters are combined with assumed porosity, ice block thickness, ocean current speed, and ocean temperature to calculate melt rates for  a distribution of  ridged ice. The enhanced internal melt rates are of  the same order of  magnitude and shape as the observed ablation rates calculated for  two ice motion events we studied, suggesting that internal melt within the porous keel can explain the observed ablation rates. Including internal melt in the IDR model improved the simulated draft  distributions of  the melting thick ice. Increasing the freezing  temperature departure of  the ocean to a reasonable value of  0.4 degrees above freezing,  the model reproduces the ablation of  ridged ice with only small variations from observed values. It is not unreasonable to assume that better estimates of  ocean temperature, ice strength, and internal keel geometry will further  improve the internal melt model and allow for more accurate redistribution models of  enhanced ridged ice melt. Notwithstanding the uncertainties introduced in the internal melt model through the use of average geometries or model parameters, the internal melt of  ridged sea ice can be seen to explain the observed enhanced melt of  ridged ice. The agreement between modelled and observed melt suggests that we have identified  and quantified  a major contributor to melt in ridged ice. SECTION E: THESIS SUMMARY 13. CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH 13.1. Conclusion The ice pack of  the Arctic Ocean is a complex structure of  ridges, leads, and level ice, of  which the large ridged features  are a significant  contributor to both ice area and volume. An attempt to understand the evolution of  sea ice on seasonal, annual, or long-term timescales thus requires understanding of  ridge-specific  evolution processes. In this thesis the inability of  current sea ice redistribution models to reproduce the evolution of  ridged ice during winter and summer melt was demonstrated using a unique regional ice draft  redistribution model that can be compared directly to observed drafts.  Conventional parameterizations of  ice draft  redistribution during ridging and melt cannot replicate observations and suggest that ridge building and melt processes are not understood. This thesis utilizes observed ice draft  profiles,  ice draft  redistribution model results, and conceptual models to explore geometrical constraints on ridging and melt. During ridging, the maximum possible draft  of  ridge keel is constrained by ice strength. However, most keels do not grow to this maximum draft  and are instead constrained by the extent of  level ice available for  ridging. Observations of  floe  size indicate that 75% of  all floes  are too small to form  ridges of  maximum draft.  This suggests that approximately 75% of  all ridges are size-limited by the level ice available for  their creation. The inclusion of  this floe  size constraint in ridging models is thus necessary to limit ridged ice drafts. Further, the shape of  observed ridges differs  from  the conventional assumption of  a triangular shaped keel. Ridge keels, on average, have a cusp-shape, with broad shoulders and steep narrow crests. The distribution of  ice within a keel is exponential, with e-folding  scales linearly increasing with maximum drafts.  This shape constraint, along with the small ridges produced due to the floe  size constraint, acts to place less ice in the thickest drafts  and therefore  increase the ridged ice area. The porous structure of  a first-year  ridge keel permits percolation of  seawater through the keel. During the melt season, when upper-layer ocean temperatures exceed the freezing  point, this flow of  ocean water through the keel transports heat deep into the structure, causing internal melt. A conceptual model for  internal melt is developed. Melt rates are found  to be dependent on the external keel shape and the geometries of  the blocks within the keels, their thickness and packing structure. Melt rates for  populations of  ridged ice are calculated and found  to be more than five  times the melt rates of  level ice, which is often  assumed to be the sole contributor to ice melt. Reasonable choices for  keel and ocean parameters demonstrate that internal melt processes are capable of explaining the observed enhanced melt of  ridged ice. While the internal melt model developed should be viewed as a first  approach, the enhanced melt of  ridged ice has, for  the first  time, been estimated, and a physical process has been used to explain the observed enhanced melt rates. Both ridging and ridge melt are found  to be dependent on the geometric properties of  the ice pack. Including these geometrical factors  in ridge building and ablation models will produce more accurate estimations of  ice evolution. Our understanding of  ridge ice processes has been greatly increased by the knowledge that floe  size, keel shape, and internal keel geometries will influence  the evolution of  ridged ice throughout the year. 13.2. Future directions The dependence of  the ridge sea ice on the geometrical characteristics of  both level ice and ridged ice has not been previously investigated. As such, future  research is needed to increase our understanding of  the interconnections between such geometrical factors  as keel shape, level ice extent, block thickness, and their evolution throughout the year. The evolution of  keel shape during ridge creation is unknown, but may be related to the extent of level ice consumed in ridging. Future research using particle models (e.g. Hopkins, 1996a) to explore the evolution of  keel shape with level ice consumption would be of  great value. Additionally, field  observations of  keel shape, draft,  and internal geometries could be used to investigate the potential relationship between level ice extent consumed in ridging and keel shape. However, this dataset would require a great effort  to collect. To improve estimates of  internal melt, the accuracy of  the estimated keel shape, internal geometry, ocean temperature, and ocean velocity should be established. Ideally, a field  study of  a first  year ridge and surrounding oceanic conditions would be conducted, with tracers used to estimate porous flow  and observations of  porosity, block thickness, and ablation rates from borehole sampling used to verify  model results. However, such a dedicated study may not be feasible  due to funding  and logistical constraints. Observations of  under-ice oceanic conditions during the spring of  2004 from  the CASES program, not yet available, may provide further guidance on appropriate oceanic temperatures during the melt season in the seasonal sea ice zone of  the Beaufort  Sea. Similar to the research suggested above, field  observations of  the block thickness and external shape parameters for  a wide variety of  keels would allow for  more accurate estimates of  the block thickness. As such a study would be labour extensive and expensive, particle simulations similar to those of  Hopkins (1996a) could be used to suggest block thicknesses for  observed keel shapes and draft  statistics, assuming such a relationship is observed. Finally, melt rates are calculated based on average keel shapes. However, most keels are not average. Instead of  the idealized cusp-shaped keel, with sides that increase continuously to a maximum draft,  ridges may contain multiple draft  maxima (peaks) connected by regions of  lower draft  (troughs). The pressure gradients across these irregular structures will differ  greatly from  the idealized ridges, and simple estimates for  the porous flow  are not possible. To investigate the impact of  irregular keel shapes, a two-dimensional model of  the interaction of  a keel with the upper layer of  the ocean would be needed. This model could also be used to investigate the contributions of  hydraulic flow  to internal melt by quantifying  the turbulent heat transfer  from below the pycnocline to the upper layer. 13.3. Implications of  research Conventional models of  pack ice evolution have focused  on basin-scale forcing  and parameterized ridging without consideration of  small-scale features  of  the ice pack. Here, in both winter and summer season, the evolution of  the large scale pack ice is shown to be dependent on the characteristics of  individual ice features.  Total ice volume in the Arctic will depend on proportion of  ridged ice, whose creation depends on the distribution of  floe  sizes and keel shapes on scales of  10 to 1000 m. The evolution of  the distribution of  ridged ice throughout the melt season depends on the internal geometries of  the ice keels, on scales of  10 cm to 1 m. In order to understand the basin-scale features  of  the ice pack, we must therefore  understand the small-scale processes of  ridging and melt that act to create and modify  the ice cover. The results of  this research are directly relevant to the challenges of  Arctic engineering. For offshore  development and exploration, it is important to be able to reliably estimate both the maximum draft  of  ridges, and the probability of  encountering a ridge of  that maximum draft. With the floe  size constraint, fewer  ridges of  maximum draft  will be encountered, perhaps easing design requirements. For navigation, the strength of  the sea ice, which is related to ice draft,  is a limiting factor.  The floe  size and keel shape constraints on ridging imply that more ridged ice area is created. This increased ridged area is due to the preferential  creation of  smaller ridges, with ice area to volume ratios greater than those of  larger ridges, by these constraints. As ridged ice impedes navigation, not including these geometric constraints in models could lead to overestimates of  navigability. However, the internal melt of  ridged ice suggests that ridged ice may be considerably weakened by melt, easing navigation as the melt season progresses. Offshore  engineering and ship design, along with predictions of  ice cover for  transportation, are thus reliant on the geometric properties of  the ice pack. An understanding of  the evolution of  ridged ice is essential for  predicting changes in sea ice area and thickness due to climate change. At present, the ice cover buffers  the ocean from  extreme variations in atmospheric temperature with thicker ice often  lasting throughout the summer season and thus reducing heat uptake by the ocean in summer months. Climate change is often thought to lead to a decreasing ice cover, as ice melts thermodynamically. We suggest that ridging may be a dominant process in the response of  sea ice to climate change. Potential increased storm activity, associated with high latitude climate variability, would cause increased ridging, and thus impact the ice extent. The floe  availability constraint predicts an increase in ice of  moderate drafts,  rather than ice with very large keels that occupy little area. Increased ridging may therefore  create a larger area of  ice with drafts  too thick to melt during the short summer season. Thus, ice extent may not decrease as rapidly as predicted by thermodynamic models. The internal melt model suggests a mechanism for  consolidation of  ridged ice to solid multi year keels, although the enhanced melt rate also suggests that many smaller keels could ablate completely during the melt season. The adaptation of  present climate models to include realistic ridging and enhanced melt will allow us to better estimate the influence  of  ridged ice on the Arctic heat budget. Parameterizations of  the floe  size constraint on ridging and the internal melt rate for  ridged ice are important corrections to the sea ice models currently used in climate change predictions. The adoption of  these concepts into large-scale redistribution models will allow for  increased confidence  in sea ice simulations. In addition, the results of  this thesis have yielded new information  about the processes of  ridge creation and how ridge melt contributes to ridged ice evolution. This provides us with a new awareness of  the influence  of  small-scale geometric features  on the large-scale characteristics of  pack ice. REFERENCES 1. Amundrud, T. L., H. Melling, and R. G. Ingram, Geometrical constraints on the evolution of  ridged sea ice, J. Geophys. Res., 109, C06005, doi:10.1029/2003JC002251. 2004 2. Babko, O., D.A. Rothrock, and G.A. Maykut, Role of  rafting  in the mechanical redistribution of  sea ice thickness, J. Geophys. Res., 107(C8), doi: 10.1029/1999JC000190, 2002. 3. Batchelor, G.K. An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge, UK. 1967 (reprinted 2002). 615 pages. 4. Bird R.B., W.E. Stewart, and E.N. Lightfoot,  Transport Phenomena. John Wiley, NY, 1960. 5. Bellchamber-Amundrud, T., G. Ingram, and H.M. Melling, Modelling the evolution of draft  distribution in the sea ice pack of  the Beaufort  Sea, in Ice in the Environment, vol. 2, edited by V. Squire and P. Langhorne, University of  Otago, Dunedin, 243-250, 2002. 6. Bitz, C.M., and W.H. Lipscomb, An energy conserving model of  sea ice. J. Geophys. Res., 104(C7), 15669-15677, 1999. 7. Bitz, C.M., M.M. Holland, A.J. Weaver, and M. Eby, Simulating the ice-thickness distribution in a coupled climate model, J. Geophys. Res., 106(C2), 2441-2463, 1999JC000113, 2001. 8. Bjork, G. On the response of  the equilibrium thickness distribution of  sea ice to ice export, mechanical deformation,  and thermal forcing  with application to the Arctic Ocean. J. Geophys. Res., 97(C7), 11287-11298, 1992 9. Bowen, R.G. and D.R. Topham, A study of  the morphology of  a discontinuous section of  a first  year.Arctic pressure ridge: Cold Regions Science and Technology, 24, 83-100,1996. 10. Burcharth, H.F., and O.H. Andersen, On the one-dimensional steady and unsteady porous flow  equations. Coastal Engineering, 24, 233-257, 1995. 11. Cattle, H. and J. Crossley, Modelling Arctic Climate Change, Phil. Trans. R. Soc. London, 352, 201-213, 1995. 12. Cavalieri, D.J., and S. Martin, The contribution of  Alaskan, Siberian, and Canadian coastal polynyas to the cold halocline layer of  the Arctic Ocean, J. Geophys. Res., 99(C9), 18343-18362, 1994. 13. Cole, D.M. and L.H. Shapiro, Observations of  brine drainage networks and microstructure of  first-year  sea ice. J. Geophys. Res., 103 (C10), 21739-21750, 1998. 14. Colony, R. and A.S. Thorndike, An estimate of  the mean field  of  arctic sea ice motion. J. Geophys. Res., 89 (C6) 10623-10629, 1984. 15. Comfort,  G. and R. Ritch, Field Measurements of  Pack Ice Stresses, Vol I & II., Report prepared for  Marine Works, A&ES, by Fleet Technology Ltd., 1989. 16. Corless, R.M., G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey,  and D. E. Knuth, 'On the Lambert W Function, Advances in Computational Mathematics, 5, 329-359, 1996. 17. Cummins, P.F. Numerical simulation of  upstream bores and solitons in a two-layer flow past an obstacle. J. Physical Oceanography, 25, 1504-1515, 1995. 18. Cummins P.F., D.R. Topham, and H.D. Pite. Simulated and experimental two-layer flows past isolated two dimensional obstacles. Fluid Dynamics Research, 14, 105-119. 1994. 19. Curry, J.A., J.L. Schramm, D.K., Perovitch, and J.O. Pinto, Applications of  SHEBA/FIRE data to evaluation of  snow/ice albedo parameterizations. J. Geophys. Res., 106 (D14), 15345-15355,2001. 20. Darby, D., J. Bischof,  G. Cutter, A. de Vernal, C. Hillaire-Marcel, G. Dwyer, J. McManus, L. Osterman, L., Polyak, and R. Poore, New record shows pronounced changes in Arctic Ocean circulation and climate. EOS Transactions, 82(49), 601, 607, 2001 21. Dempsey, J.P., R.M. Adamson, and S.V. Mulmule, Scale effects  on the in-situ tensile strength and fracture  of  ice. Part II: First-year sea ice at Resolute, N.W.T., International Journal of  Fracture, 95, 347-366, 1999.. 22. Den Hartog, J.P., Advanced Strength of  Materials. McGraw-Hill Book Company, Inc. New York, 379pp., 1952. 23. Dumas, J.A., G.M. Flato, and A.J. .Weaver, The impact of  varying atmospheric forcing  on the thickness of  Arctic multi year sea ice, Geophys. Res. Lett., 30(18), 1918, doi:10.1029/2003GL017433, 2003. 24. Elliott, A.H., and N.H. Brooks, Transfer  of  nonsorbing solutes to a streambed with bed forms:  Theory. Water Resources Research, 33(1), 123-136, 1997. 25. Emery, W.J., and R.E. Thomson, Data Analysis Methods in Physical Oceanography. Pergamon. New York, 634pp., 1998. 26. Englelund, F.A., On the laminar and turbulent flows  of  ground water through homogeneous sand. Danish Academy of  Technical Sciences. 1953. 27. Farmer, D.M., and R.A. Denton, Hydraulic control of  flow  over the sill in observatory inlet. J. Geophys. Res., 90 (NC5), 9051-9068, 1985. 28. Flato, G.M., Spatial and temporal variability of  Arctic ice thickness, Ann. of  Glaciol., 21, 323-329,1995. 29. Flato, G.M. and R.D. Brown, Variability and climate sensitivity of  landfast  Arctic sea ice, J. Geophys. Res., 101(C10), 25767-25777, 1996. 30. Flato, G.M. and W.D. Hibler III, Ridging and Strength in modelling the thickness distribution of  arctic sea ice, J. Geophys. Res., 100(C9), 18611-18626. 1995. 31. Fox, C. and V.A. Squire, Coupling between the ocean and an ice shelf,  Ann. of  Glaciol., 15, 101-108, 1991. 32. Fox, C., T. Haskell, and H. Chung, Dynamic, in situ measurement of  sea ice characteristic length, Ann. of  Glaciol., 33, 339-344, 2001. 33. Gill, A. E., Atmosphere-Ocean Dynamics. Academic Press, San Diego, California,  662pp., 1982 34. Granberg, H.R., and M. Lepparanta, Observations of  sea ice ridging in the Weddell Sea J. Geophys. Res., 104(C11), 25735-25745, 1992. 35. Haapala, J. On the modelling of  ice thickness redistribution. J. Glaciol. 46,427-437. 2000 36. Hall, K.R., G.M. Smith, and D.J. Turcke, Comparison of  oscillatory and stationary flow through porous media. Coastal Engineering, 24, 217-232, 1995. 37. Hibler, W.D., EI, Modelling a variable thickness sea ice cover, Mon. Weather. Rev., 108, 1943-1973, 1980. 38. Hibler, W.D. HI and E.M. Schulson, On Modelling the anisotropic failure  and flow  of flawed  sea ice. J. Geophys. Res.. 105(C7) 17105-17120, 2000. 39. Hibler, W.D. m, S.J. Mock, and W.B. Tucker m, Classification  and Variation of  Sea Ice ridging in the western arctic basin. J. Geophys. Res., 79 (18), 2735-2743, 1974. 40. Holland, D.M., and A. Jenkins, Modelling thermodynamic ice-ocean interactions at the base of  an ice shelf.  J. Phys. Oceanogr., 29, 1787-1800. 1999. 41. Hopkins, M, On the ridging of  intact lead ice J. Geophys. Res., 99(C8), 16351-16360, 1994. 42. Hopkins, M. The effects  of  individual ridging events on the ice thickness distribution in the Arctic ice pack, Cold Reg. Sci. Technol., 24, 75-82, 1996a. 43. Hopkins, M.A., On the mesoscale interaction of  lead ice and floes.  J. Geophys. Res., 101 (C8), 18315-18326, 1996b. 44. Hopkins, M., Four stages of  pressure ridging, J. Geophys. Res., 103(C10), 21883-21891, 1998. 45. Hopkins, M. A. and Hibler, On the ridging of  a thin sheet of  lead ice. Annals of  Glaciology, 15,81-86. 1991. 46. Hopkins M.A., W.D. Hibler, III and G.M. Flato, On the numerical simulation of  the sea ice ridging process. J. Geophys. Res., 96, 4809-4820, 1991 47. Hoyland, K.V., Simulations of  the consolidation process in first  year sea ice ridges, Cold Reg. Sci. Technol., 34, 143-158, 2002 48. Hoyland K.V. and S. L(j)set, Measurements of  temperature distribution, consolidation, and morphology of  a first-year  sea ice ridge. Cold Reg. Sci. Technol., 29, 59-74, 1999. 49. Johnston, M., and R. Frederking, Decay of  first-year  sea ice: A second season of  field measurements, 2001. Technical Report, HYD-TR-066, National Research Council, Canada, 2001. 50. Kells, J.A., Spatially varied flow  over rockfill  embankments. Can. J. Civil Eng., 20 (5): 820-827. 1993. 51. Knudsen, J. G. and D.L. Katz., Fluid Dynamics and Heat Transfer.  McGraw-Hill Book Company, New York, 576 pgs, 1958. 52. Kovacs, A., and D. S. Sodhi, Shore ice pile-up and ride up: field  observations, models, theoretical analyses, Cold Reg. Sci. Technol., 2, 209-288, 1980. 53. Lange, M.A., S.F. Ackley, P. Wadhams, G.S. Dieckmass, and H. Eicken, Development of sea ice in the Weddell Sea, Ann. Glaciol., 12, 92-96,1989. 54. Lawrence, G.A., The hydraulics of  steady two-layer flow  over a fixed  obstacle. J. Fluid. Mech, 254, 605-633, 1993. 55. Lepparanta, M., and R. Hakala, The structure and strength of  first  year ice ridges in the Baltic Sea, Cold Reg. Sci. Technol., 20, 295-311, 1992. 56. Lepparanta, M., M. Lensu, P. Kosloff,  and B. Veitch, The life  story of  a first-year  sea ice ridge, Cold Reg. Sci. Technol., 23 279-290, 1995. 57. Lipscomb, W.H., Remapping the thickness distribution in sea ice models, J. Geophys. Res., 106(C7), 13989-14000, 2001. 58. Lowry, R.T. and P. Wadhams, On the statistical distribution of  pressure ridges in sea ice. J. Geophys. Res., 84(C5), 2487-2494, 1979. 59. Makshtas, A.P., Thermodynamics of  sea ice, in Physics of  Ice-Covered Seas, vol. 1, ed. M. Lepparanta, Helsinki University Printing House, Helsinki, pp. 289-304, 1998. 60. Maykut, G.A., The ice environment. In "Sea Ice Biota", R. A. Horner (editor). CRC Press Inc., Florida. 21-82, 1985. 61. Maykut, G.A., The surface  heat and mass balance, in The Geophysics of  Sea Ice, NATO Sci. Ser., Ser B, vol 146, ed. N. Untersteiner, chap. 5, pp. 395-463, Springer-Verlag, New York, 1986. 62. Melling, H. Detection of  features  in first-year  pack ice by synthetic aperture radar (SAR), International Journal of  Remote Sensing, 19(6), 1223-1249, 1998a. 63. Melling, H. Sound scattering by sea ice: Aspects relevant to ice-draft  profiling  by sonar, Journal of  Atmospheric and Oceanic Technology, 15, 1023-1033, 1998b. 64. Melling, H. Sea ice of  the northern Canadian Arctic Archipelago. J. Geophys. Res., 107(C11), 3171, doi: 10.1029/2001JC001102, 2002. 65. Melling, H., and D.A. Riedel, Draft  and Movement of  Pack Ice in the Beaufort  Sea, April 1991-April 1992, Can. Tech. Rep. Hydrogr. Ocean Sci., No. 162, 109 pp. 1994. 66. Melling, H., and D.A. Riedel, The underside topography of  sea ice over the continental shelf  of  the Beaufort  Sea in the winter of  1990, J. Geophys. Res., 100(C7), 13641-13653, 1995. 67. Melling, H., and D.A. Riedel, Development of  seasonal pack ice in the Beaufort  Sea during the winter of  1991-1992: A view from  below, J. Geophys. Res., 101(C5), 11975-11991, 1996a. 68. Melling, H., and D.A. Riedel, The thickness and ridging of  pack ice causing difficult shipping conditions in the Beaufort  Sea, Summer 1991, Atmosphere-Ocean, 34(3), 457-487,1996b. 69. Melling, H., D.R. Topham, and D.A. Riedel, Topography of  the upper and lower surfaces of  ten hectares of  deformed  sea ice. Cold Regions Science and Technology, 21, 349-369, 1993. 70. Melling, H., P. H. Johnston, and D.A. Riedel, Measurements of  the underside topography of  sea ice by moored subsea sonar, J. Atmos. Oceanic Technol., 12, 589-602, 1995. 71. Mock, S.J., A.D. Hartwell, and W.D. Hibler III, Spatial aspects of  pressure ridge statistics. J. Geophys. Res., 77(33), 5945-5953, 1972. 72. Overland, J.E., B.A. Walter, T.B. Curtin and P. Turet, Hierarchy and sea ice mechanics: A case study from  the Beaufort  Sea. J. Geophys. Res., 100(C3), 4559-4572, 1995. 73. Parkinson, C.L., and W.M. Washington, A large-scale numerical model of  sea ice, J. Geophys. Res., 84, 311-337, 1979. 74. Parmerter, R.R. and M.D Coon, Model of  pressure ridge formation  in sea ice, J. Geophys. Res., 77(33), 6565-6575, 1972. 75. Perovitch, D.K., T. C. Grenfell,  B. Light, and P.V. Hobbs, Seasonal evolution of  the albedo of  multi year Arctic sea ice, J. Geophys. Res., 107(C10), 8044, do:10.1029/2000JC000438, 2002. 76. Pite, H.D., D.R.Topham, and B.J. van Hardenberg, Lab measurements of  the drag force  on a family  of  two-dimensional ice keel models in a 2-layer flow.  J. Phys.Oceanogr. 25(12) 3008-3031, 1995. 77. Proshutinsky, A., M. Steele, J. Zhang, G. Holloway, N. Steiner, S. Haakinen, D. Holland, R. Gerdes, C. Koeberle, M. Karcher, M. Johnson, W. Maslowski, W Hibler, and J. Wang, Multinational effort  studies differences  among Arctic ocean models, Eos Trans. AGU, 82(51), 637,643-644, 2001. 78. Richter-Menge, J.A. and B.C. Elder, Characteristics of  pack ice stress in the Alaskan Beaufort  Sea J. Geophys. Res., 103(C10), 21817-21829, 1998. 79. Richter-Menge, J., B. Elder, K. Claffey,  J. Overland, and S. Salo, In situ sea ice stressed in the western Arctic during the winter of  2001-2002, in Ice in the Environment, vol. 2, edited by V. Squire and P. Langhorne, University of  Otago, Dunedin, 423- 430, 2002. 80. Rigby, F.A. and A. Hanson, Evolution of  a large arctic pressure ridge. AIDJEX Bull. 34, 43-71, 1976 81. Rothrock, D.A., Ice thickness distribution - Measurement and theory, in The Geophysics of Sea Ice, NATO Sci. Ser., Ser B, vol 146, ed. N. Untersteiner, chap. 8, pp. 551-575, Springer-Verlag, New York, 1986. 82. Rothrock D.A. and A.S. Thorndike, Geometric properties of  the underside of  sea ice. J. Geophys. Res., 85(C7), 3955-3963, 1980. 83. Sanderson, T.J.O., Ice Mechanics: Risks to Offshore  Structures, Graham and Trotman, London, 253 pp., 1988. 84. Savage, S. B., Review of  Sea-Ice Thickness Redistribution Models. Prepared for  the Canadian Ice Service. Contract KM149-1-85-018, Technical Report 2001-02, 2001. 85. Sayed, M. and R.M. Frederking, Measurements of  ridge sails in the Beaufort  Sea. Can. J. Civ. Eng. 16, 16-21. 1989. 86. Schramm, J.L., Flato, G.M., and Curry, J.A., Toward the modelling of  enhanced basal melting in ridge keels, J. Geophys. Res., 105(C6), 14081-14092, 2000. 87. Shulkes, R.M., A note on the evolution equations for  the area fraction  and the thickness of a floating  ice cover, J. Geophys. Res., 100(C3), 5021-5024, 1995. 88. Shen, H.W., H.M., Fehlman, and C. Mendoza, Bed form  resistance in open channel flows.  J Hydraulic Engineering, 116(6), 799-815, 1990. 89. Skyllingstad, E.D., C.A. Paulson, W.S. Pegau, M.G. McPhee, and T. Stanton, Effects  of keels on ice bottom turbulence exchange, J. Geophys. Res., 108(C12), 3372, doi: 10.1029/2002JC001488, 2003. 90. Smith, S.D., R.D. Muench, and C.H. Pease, Polynyas and Leads: An overview of  physical processes and environment, J. Geophys. Res., 95(C6) 9461-9479, 1990. 91. Thibodeaux, D. J. and J. D. Boyle. Bedform-generated  convective transport in bottom sediment. Nature, 325(22), 341- 343, 1987. 92. Thorndike, A.S., Estimates of  sea ice thickness distribution using observations and theory, J. Geophys. Res., 97(C8), 12601-12605, 1992. 93. Thorndike A.S., Sea ice thickness as a stochastic process, J. Geophys. Res., 105(C1), 1311-1313,2000. 94. Thorndike, A.S., D.A. Rothrock, G.A. Maykut, and R. Colony, The thickness distribution of  sea ice, J. Geophys. Res., 80(33), 4501-4513, 1975. 95. Timco, G.W. and R.P. Burden, An analysis of  the shapes of  sea ice ridges, Cold Reg. Sci. Technol., 25, 65-77, 1997. 96. Tucker, W.B. IE, W. F. Weeks, and M. Frank, Sea Ice ridging over the Alaskan continental shelf  J. Geophys. Res., 4885-4897, 1979. 97. van Gent, M.R.A., Porous flow  through rubble-mounded material, J. Waterw. Port Coastal Ocean Eng., 3, 176-181, 1995. 98. Veitch, B., Kujala, P., Kosloff,  P., and Lepparanta, M., Field measurements of  the thermodynamics of  an ice ridge, M-l 14 Laboratory of  Naval Architecture and Marine Engineering, i - vi, 1 - 33 1991. 99. Venegas, S.A. and L.A. Mysak, Is there a dominant timescale of  natural climate variability in the Arctic?, J. Clim., 13, 3412-3434, 2000 100. Vittal, N., K.G. Ranga Raju, and R.J. Garde. Resistance of  two dimensional triangular roughness. J Hydraulic Research, 15, 19-35, 1977. 101. Wong, J., S. Beltaos, and B.G. Krishnappan, Seepage flow  through simulated grounded ice jam, Can. J. Civil, Eng., 12, 926-929, 1985. APPENDICES Appendix A VARIABLES AND PARAMETERS Symbol Meaning Units a Conservation of  area constant A, B Constants related to thermal ice growth [] b(i),  bQ Conservation of  volume transfer  coefficients b/,  b2 Ice block length scales m cd Average drag coefficient [] cP Local pressure coefficient [] cp heat capacity of  water 4218Jkg1 d Diameter of  sphere m dh d2 depths of  upper and lower layers of  the ocean m D pipe diameter m div Rate of  divergence % day1 dlayer Depth of  oceanic mixed layer m E Young's modulus Pa E, Internal energy of  a fluid  per unit mass 2 -2 m s f Ice thermal growth rate m day-1 F Thermal growth matrix operator F, Heat flux  within the keel Wm"2 F B Longwave heat flux  (up) Wm"2 F L Longwave heat flux  (down) Wm"2 fL  ... Floe extent m Fnet Heat flux Wm"2 F 0 Froude number of  flow [] F 1 oceanic Heat flux  from  ocean Wm"2 F s Shortwave heat flux Wm"2 F, Turbulent heat flux Wm"2 G Vector probability distribution of  thickness, £G=1; m"1 g Gravitational acceleration constant m s"2 g(h) Probability distribution of  ice thickness/draft m'1 gri(H) Probability distribution of  maximum ridge draft m1 gra(H) Probability distribution of  maximum rafted  ice draft m h draft  of  sea ice m Ah change in draft  associated with melt m hl ,h2 draft  of  ice at value of  the cumulative density distribution H Keel draft m h* Bin size m I Identity matrix [] I Hydraulic gradient [] K Constant of  hydraulic conductivity m s"1 K.j Turbulent Forchheimer coefficient s2m"2 L total number of  samples in the ice profile [] Li Latent heat of  fusion,  ice J kg"1 i 1> 13 Arbitrary length scales m Lc Characteristic buckling length m LP Characteristic pore length scale m m Buckling mode number [] N Number of  blocks in V M [] N t Number of  tubes in a section of  keel volume [] n Poisson's ratio [] JV Number of  independent samples in ice profile [] Vri Number of  rafted  keels [] lira Number of  ridged keels [] Nu Nusselt number [] P Porosity % P Pressure Pa Qm Heat flux  of  water due to melt W Qo Heat flux  of  water at entrance to keel w Q* Heat flux  of  water at distance, x, along tube w r ratio of  porous fraction,  p, to solid fraction,  1-p. [] R ratio of  keel extent to keel width [] Re Reynolds number [] RH Hydraulic Radius [m] 5 Open water source vector, Si=l; Sn=0; [] s(z) segment of  keel with a draft,  z m t Time, s or day AT Temperature difference  between fluid  and pipe K T 0 Temperature of  ocean above freezing K T x(x) Temperature above freezing  (K) of  porous flow K u Ice velocity m s-1 u, u0 Characteristic pore velocity m s"1 ux(x) pore velocity across the keel m s"1 U Characteristic free  stream velocity m s"1 V Discharge flow  velocity - m s"1 V B Volume of  a block m3 V M Volume of  material m3 Vp Volume of  pores in material m3 V T Volume containing one tube m3 v K Volume of  keel, similar to V M m3 Vf m Volume flux  of  water due to melt m V Vfo Volume flux  of  water at entrance to keel m V Vf x Volume flux  of  water at distance, x, along tube m V w Width of  keel m W k(z) Width of  keel as a function  of  z and X m £ Convergent strain rate % day"1 Q Bernoulli's constant of  conserved energy m2 s"2 Tv Mechanical ridging matrix operator with indices i,j a',/3' Forchheimer coefficients [] a,/3 Forchheimer coefficients [] m w ) Truncation factors m K Thermal conductivity of  water 0.56 J m"1 s-'K"1 X Characteristic e-folding  scale of  exponential distribution of  draft m within a keel V Viscosity of  water, 1.8 x 10"6 m2 s"1 e Number of  cumulative freezing  days days p> Density of  ice kg m"3 p»,p Density of  water kg m"3 a Buckling stress Pa co(z) keel width at draft,  z m ¥ Mechanical ridging operator £ generic constant used for  a placeholder. [] Appendix B HYDRAULIC RADIUS FOR SPHERES AND RECTANGLES For flow  through a granular medium, the hydraulic radius can be defined  as the ratio of  the pore volume to the pore surface  area. Typically, in hydro-geological systems, the solid particles are approximated by spheres and the hydraulic radius is given as RH= [p/(l-p)]d/6, where d is the sphere diameter and p is the porosity (Burcharth and Andersen, 1995). For spherical particles, V M is the volume of  the media and Vp is the volume of  the pores: (13.3.1) V p=pV u Pore volume can also be defined  as a function  of  the number of  particles, N, when the volume of each particle is given by the volume of  a sphere, Vsphere=7id3/6, where d is the sphere diameter. toP (13.3.2) V P = V M-N o Solving (13.3.1) and (13.3.2) for  N, the number of  particles: (13.3.3) n = V m - V P = WMQ~P) nd3/ 6 ®/3 The hydraulic radius, RH, can then be found  as the ratio of  the pore volume to the pore surface area, which is the same as the particle surface  area, Nnd 2. (13.3.4) R - PV M  = PVM  ri " Nnd 2 nd 2 e-V M{\  -p) RH  = J-P, The hydraulic radius of  a triangular ice keel can be derived in a similar fashion.  The volume of the keel is V K and that of  the pores, VP. (13.3.5) vP = PvK The volume of  the pores is again a function  of  the number of  blocks. Letting the dimensions of the block be multiples bi and  b2 of  the thickness, h, the volume and surface  area of  a block are given by: (13.3.6) V B=blb2h3 (13.3.7) AB =(b lb2+bl+b2)lh2 And pore volume and pore surface  are (same as block surface  area) are: (13.3.8) VP = V K - N b M ? Ap = 2N(b\b2+  ^  +b2)h2 Again solving for  the number of  blocks, n: (1 - p ) V K (13.3.9) N  = - bxb2h3 The hydraulic radius is then the ratio of  pore volume to pore surface  area: (13.3.10) RH= BH 2N{b xb2+bx+b1)h2 / \ P l-pJ bfah 2(b\bi  + +&2) The factor  [p/(l-p)] is the ratio of  the porous fraction  to the solid fraction.  Representing the block geometry, the factor  y is introduced and equation (13.3.10) becomes: (13.3.11) RH  =—-—— (1  -P)r From research looking at block sizes in ice keels in the Beaufort  Sea, we can estimate the hydraulic radius; RH. Block structure data is available from  a detailed study of  landfast  ridge sails near 70° Ns 133° -137° W by Sayed and Frederking (1989) found  that the ratios of  block length and width to block thickness were 2.33 and 1.26 respectively. Substituting these ratios for  b] and b2, y is approximately 4.45. At first  glance, this value for  RH appears larger as the hydraulic radius for  a medium constructed of  spheres. It should be recognized though that the thickness, h, is the smallest of  the three dimensions: height, length, and width; while the diameter is the only dimension. An average diameter for  a block of  sides h x 2.3h x 1.3h would be 1.43h, making the hydraulic radius for blocks proportional to (1/6.4) of  the average dimension, compared to value of  l/6thfor  spheres. For a block thickness of  1 m and porosity of  30%, the hydraulic radius is approximately 10cm, for  a block thickness of  0.1m the hydraulic radius is 1 cm. Appendix C DERIVATION OF KEEL WIDTH The observed averaged distribution of  ice within the keel can be approximated by a negative exponential with an e-folding  scale of  6 m (see Section 5.5). Here, z is the draft  of  the ice along the envelope of  the keel, and h is the level ice draft. (13.3.12) b(i)  = b0e~zlX Assuming that the keel has a smooth cusp-shape (averaged shape), the relative length of  the segment, s(z), at draft  z to the segment at draft  h is: (13.3.13) ^ = zlX e s(h) =~ And the absolute length of  any segment is: (13.3.14) 5 (z)  = Ce{ h~z)n We know the total width of  the keel, W, from  the ice profiles  (see section 11), which allows us to solve for  the constant in (13.3.14) as the total width of  the keel is the sum of  all the segments. W (13.3.15) £ = - h h/A.  f  -z/;I e \e H The width of  the keel at any draft,  z, is then: l e " * ' * z W I 6 —z/A\ (13-3.16) ~h H , x h U H 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0052754/manifest

Comment

Related Items