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UBC Theses and Dissertations

A multicomponent coupled model of glacier hydrology Flowers, Gwenn Elizabeth 2000

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A B S T R A C T Multiple lines of evidence suggest a causal link between subglacial hydrology and phe-nomena such as fast-flowing ice. This evidence includes a measured correlation between water under alpine glaciers and their motion, the presence of saturated sediment beneath Antarctic ice streams, and geologic signatures of enhanced paleo-ice flow over deformable substrates. The complexity of the glacier bed as a three-component mixture presents an obstacle to unraveling these conundra. Inadequate representations of hydrology, in part, prevent us from closing the gap between empirical descriptions and a comprehensive consistent framework for understanding the dynamics of glacierized systems. I have developed a distributed numerical model that solves equations governing glacier surface runoff, englacial water transport, subglacial drainage, and subsurface groundwater flow. Ablation and precipitation drive the surface model through a temperature-index pa-rameterization. Water is permitted to flow over and off the glacier, or to the bed through a system of crevasses, pipes, and fractures. A macroporous sediment horizon transports subglacial water to the ice margin or to an underlying aquifer. Governing equations are derived from the law of mass conservation and are expressed as a balance between the internal redistribution of water and external sources. Each of the four model compo-nents is represented as a two-dimensional, vertically-integrated layer that communicates with its neighbors through water exchange. Stacked together, these layers approximate a three-dimensional system. I tailor the model to Trapridge Glacier, where digital maps of the surface and bed have been derived from ice-penetrating radar data. Observations of subglacial water pressure provide additional constraints on model parameters and a basis for comparison of simulations with real data. Three classical idealizations of glacier geometry are used for simple model experi-ments. Equilibrium tests emphasize geometric controls on hydrology, while time-dependent simulations disclose where and how the input forcing is manipulated by the system. Results of sensitivity tests show good qualitative agreement with the glaciological lore. Using Trapridge Glacier geometry and meteorological observations, I investigate hydro-logy on sub-hourly, diurnal, and seasonal timescales. Examples from 1990, 1995, and 1997 collectively substantiate the merit of the model in a variety of situations. u Table of Contents A B S T R A C T ii List of Tables vii List of Figures viii Preface xii Acknowledgements xiii Chapters 1 1 I N T R O D U C T I O N 1 1.1 Preamble 1 1.2 Water in glacial systems 3 1.2.1 Storage 3 1.2.2 Chemical and geological relevance 3 1.2.3 Ice-dynamical relevance 4 1.3 Model representation of glacier hydrology 6 1.3.1 Evolution of subglacial theory and models 6 1.3.2 Distributed deterministic models 7 1.4 Trapridge Glacier study site and project 8 1.4.1 Physical environment 8 1.4.2 Scientific context 9 1.5 Thesis structure and objectives 10 2 T R A P R I D G E G L A C I E R DEMS 12 2.1 Data collection and preparation 13 2.1.1 Instrumentation and field methods 14 2.1.2 Picking direct and reflected arrivals 15 2.2 Data interpolation 16 2.2.1 Data preconditioning 17 2.2.2 Geostatistical analysis 18 iii 2.2.3 Omnidirectional semivariograms 20 2.2.4 Directional semivariograms 21 2.2.5 Kriging and post-processing 22 2.3 Interpolation results 23 2.4 Derived geometric quantities 26 2.4.1 Piezometric surface 26 2.4.2 Terrain analysis 28 2.5 Borehole connection record 31 2.6 Summary 33 3 M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 35 3.1 Conceptual model 35 3.2 Surface ablation and runoff 37 3.3 Englacial water transport and storage 42 3.4 Drainage through a subglacial water sheet 47 3.5 Subsurface groundwater flow 50 3.6 Coupling exchange terms 53 3.7 Summary of equations and boundary conditions 55 4 P A R A M E T E R I Z A T I O N OF SUBGRID PROPERTIES 57 4.1 Macroporous flow layer thickness 57 4.1.1 Borehole drilling observations: data selection and assumptions . . 59 4.1.2 Statistical method and results 61 4.1.3 Hydrological implications 64 4.2 Hydraulic conductivity 68 4.2.1 Topographic autocorrelation 71 4.2.2 Topographic power 78 4.2.3 Derived relationship for Ks(hs) 80 4.3 Summary , 81 5 N U M E R I C A L M E T H O D 83 5.1 Model domain and coordinate system 83 5.1.1 Discretization 84 5.2 Generalized form of discrete equations 87 5.2.1 Boundary and initial conditions 90 5.3 Solution Algorithm 91 5.3.1 Ablation routine 92 5.3.2 Dynamics loop 93 iv 5.3.3 Linearization: Newton-Raphson iteration 93 5.3.4 Solving the linear system 98 5.4 Summary and practical implementation 99 6 A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 100 6.1 Glacier geometry 100 6.2 Steady state test description 102 6.2.1 Reference model parameters 102 6.3 Steady state results 107 6.3.1 Terrain 1: inclined glacier bed 107 6.3.2 Terrain 2: parabolic inclined glacier bed 115 6.3.3 Terrain 3: undulating inclined glacier bed 120 6.3.4 Synopsis of results 124 6.3.5 Determination of subglacial drainage structure 125 6.4 Transient test description 131 6.4.1 Additional parameters for time-dependent investigations 131 6.5 Transient results 132 6.5.1 Reference model 133 6.5.2 Model sensitivities 136 6.5.3 Geometric effects 152 6.6 Summary 155 7 A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 157 7.1 Implementation of the ablation-runoff model 157 7.1.1 Geometric data requirements and calculations 157 7.1.2 Determination of melt 159 7.1.3 Additional requirements 162 7.2 Parameter refinement 163 7.2.1 Initial model 165 7.2.2 Comparisons with data 170 7.2.3 Final reference model 171 7.3 Hydraulic release events: rapid disturbances in the water system 174 7.3.1 1990 Event synopsis and interpretation 176 7.3.2 Model adaptations and inputs 177 7.3.3 Results and discussion 179 7.3.4 Model sensitivities 183 7.3.5 Summary of release event simulation 185 7.4 Seasonal transitions in the drainage system 185 7.4.1 Evidence from Trapridge Glacier 186 v 7.4.2 Modelling approach 189 7.4.3 Results and discussion 189 7.4.4 Summary of seasonal transitions 206 7.5 Summary and conclusions 209 7.5.1 Model performance 209 7.5.2 Trapridge hydrology 209 8 C O N C L U S I O N S , R E C O M M E N D A T I O N S , A N D O U T L O O K 212 8.1 Summary of work 212 8.2 Model evaluation and future priorities 214 8.2.1 Development of a channelized hydrology 215 8.3 Outlook 216 References 218 Appendices 236 A Derivation of kriging weights 236 B Input parameters for J C B L O K 238 C Discrete equations for multicomponent hydrology 240 C.l Surface runoff 240 C.2 Englacial storage and transport 242 C.3 Subglacial drainage 244 C.4 Groundwater flow 247 C.5 Coupling exchange terms 249 D Calculating ablation variables 252 E Trapridge Glacier parameter refinement 254 E . l Quantification of mismatch between observed and simulated records . . . 254 E.2 Results: simulated response to selected model parameters 255 E.3 Performance comparison and ranking 260 F Temporally-varying hydraulic connections, days 140-185, 1995 263 vi List of Tables 2.1 Radar data collected on Trapridge Glacier, 1994-1997 13 2.2 Omnidirectional semivariogram parameters 20 2.3 Directional semivariogram parameters 21 4.1 Characteristics of low-pass Gaussian filters 73 4.2 Attributes of filtered topography 77 4.3 Attributes of amplitude-adjusted topography 78 5.1 Grid-position classification of input, output, and initial fields 85 5.2 Grid-position classification of internal fields and functions 86 6.1 Physical constants and numerical parameters for reference model 103 6.2 Reference model parameters 104 6.3 Intercomparison of steady state results for synthetic topography 125 6.4 Summary of subglacial pressure state 127 6.5 Additional reference model parameters for transient tests 131 6.6 Sensitivity of various quantities to Ka and Kl 146 6.7 Characteristics of decaying and delayed subglacial signals 155 7.1 Initial model parameters for Trapridge Glacier 164 7.2 Numerical parameters for Trapridge Glacier investigations 165 B.l Parameters used in the block-kriging routine JCBLOK 239 D. l Parameters and variables used in Figure D.l 252 E. l Selected initial model parameters for Trapridge Glacier 255 E.2 Trapridge Glacier parameter refinement tests and results 1 256 E.3 Trapridge Glacier parameter refinement test and results 2 257 E.4 Parameter refinement test rankings 262 vu List of Figures 2.1 Trapridge Glacier data coverage, 1994-1997 14 2.2 Geometry and components of radar data collection equipment 14 2.3 Sample radar traces 15 2.4 Histograms of raw and transformed data 17 2.5 Two common idealized variogram models 19 2.6 Experimental semivariograms plotted with best-fit models 20 2.7 Directional anisotropy determined from semivariogram ranges 22 2.8 Interpolated ice-surface elevation obtained by kriging 23 2.9 Comparison of photograph and interpolated ice surface near the terminus 24 2.10 Interpolated bed topography obtained by kriging 24 2.11 Contoured maps of interpolated data sets 25 2.12 Contours of equal hydraulic potential for / = 0.5 and / = 1 27 2.13 Upstream area distributions governed by hydraulic potential 30 2.14 Comparison of borehole connection probability and upstream area . . . . 32 3.1 Conceptual model of glacier hydrology in perspective and cross-section . 36 3.2 Ablation hierarchy for snow, superimposed ice, and glacier ice 39 3.3 Idealized morphology of englacial voids 42 3.4 Porosity adjustments in the saturated subglacial sediment sheet 48 4.1 Rough bed element with and without water 58 4.2 Provenance and distribution of borehole depth data 59 4.3 Results of borehole inclinometry, 1997-98 60 4.4 Porous sediments overlying a rough bed 62 4.5 Probability density and cumulative distribution functions of \SZB\ . . . . 62 4.6 Maximum water depth h*max vs. anomaly amplitude \SZB\) areal-fraction coverage of water Af vs. maximum depth hsmax 65 4.7 Empirical and analytical relationships between h" and h*max 66 4.8 Proposed relationships between p", Af, and h" 67 4.9 Subgrid water distribution in connected and unconnected elements . . . . 69 4.10 Profiles of damped and filtered random topography 70 4.11 Distributions of upstream area in order of increasing total upstream area 71 4.12 Gaussian filter characteristics: normalization, shape, and power 74 4.13 Amplitudes and power spectra for filtered and unfiltered topography . . . 75 vm 4.14 Total upstream area for spatial-correlation tests 77 4.15 Total upstream area for variable-power tests 79 4.16 Proposed relationship for Ks(h°) 81 5.1 Coordinate system and discretization of the model domain 83 5.2 Flow diagram of main program 91 5.3 Flow diagram of ablation routine 92 5.4 Flow diagram of solver including Newton-Raphson iteration 97 6.1 Terrain 1: glacier on an inclined plane 101 6.2 Terrain 2: glacier in a U-shaped valley 101 6.3 Terrain 3: glacier on an undulating bed 102 6.4 Characteristics of surface runoff and englacial storage for Terrain 1 . . . 108 6.5 Characteristics of the subglacial system for Terrain 1 109 6.6 Characteristics of the subsurface system for Terrain 1 . . 110 6.7 Subterraneous water erupts into a proglacial sediment cauldron I l l 6.8 Piezometric and exchange profiles for Terrain 1 with frozen margin . . . 113 6.9 Schematic representation of major drainage routes 114 6.10 Equilibrium water budget and transfer rates for the coupled system . . . 115 6.11 Characteristics of surface runoff and englacial storage for Terrain 2 . . . 116 6.12 Characteristics of the subglacial system for Terrain 2 117 6.13 Characteristics of the subsurface system for Terrain 2 118 6.14 Characteristics of subsurface system with no boundary recharge 119 6.15 Characteristics of surface runoff and englacial storage for Terrain 3 . . . 120 6.16 Characteristics of the subglacial system for Terrain 3 121 6.17 Diversion of subglacial water away from steepest slope 122 6.18 Characteristics of the subsurface system for Terrain 3 123 6.19 Equilibrium distributions of subglacial water pressure 126 6.20 Cumulative distributions of equilibrium subglacial flotation fraction . . . 128 6.21 Comparison of modelled fluxes and hydraulic potential calculations . . . 129 6.22 Temperature forcing and response for transient reference model 133 6.23 Intercomponent phase diagrams 134 6.24 Daily surface air temperature, melt rate, and water pressure 135 6.25 Diurnal cycling of subglacial water pressure and water exchange . . . . . . 136 6.26 Sensitivity of key quantities to rr:e 137 6.27 Sensitivity of key quantities to r e :* 139 6.28 Subglacial behaviour induced by differences in conductivity 140 6.29 Selected quantities affected by subglacial hydraulic conductivity 141 6.30 Discrete implementation of a variable pressure axis 144 6.31 Model simulations using the hydraulic geometry of Fig. 6.30 145 ix 6.32 Sensitivity of </>s:a to hydraulic conductivity of the aquifer Ka 147 6.33 Sensitivity of selected variables to Kl and Ka 148 6.34 Illustration and quantification of time lags due to Kl 149 6.35 Geometry of periglacial permafrost and its effect on <f>":a 150 6.36 Effects of nonuniform aquifer thickness 151 6.37 Englacial component of the water budget for various systems 152 6.38 Time slices of subglacial water pressure for sparse source model 153 6.39 Water exchange profiles and propagation of the diurnal pressure wave . . 154 7.1 Topography surrounding Trapridge Glacier 158 7.2 Digital representations of Trapridge Glacier slope and aspect . . . . . . . 159 7.3 Spatially variable solar radiation incident on Trapridge Glacier 160 7.4 Cumulative surface melt calculated for days 184-207, 1997 161 7.5 Digital representation of Trapridge Glacier crevasse distribution 163 7.6 Equilibrium surface/englacial characteristics of Trapridge hydrology . . . 166 7.7 Equilibrium subglacial characteristics of Trapridge Glacier hydrology . . 167 7.8 Equilibrium characteristics of the groundwater system 168 7.9 Equilibrium profiles of intercomponent water exchange 169 7.10 Sensor locations and records used in parameter optimization 170 7.11 Final reference model comparison with observed pressure records 171 7.12 Time of daily maximum pressure for observed and modelled records . . . 172 7.13 Calculated surface melt rate and modelled subglacial water pressure . . . 174 7.14 Hydraulic release event taxonomy 175 7.15 1990 hydromechanical event: observed surface and subglacial conditions . 176 7.16 Event interpretation by Stone 177 7.17 Release event model adaptations 178 7.18 Location of 1990 field site 180 7.19 Results: modelled evolution of multicomponent hydrology 181 7.20 Results: modelled subglacial and proglacial response 182 7.21 Sensitivity of modelled subglacial behaviour to selected parameters . . . 183 7.22 Seasonal transitions: Trapridge Glacier, fall, 1997 187 7.23 Seasonal transitions: Trapridge Glacier, spring, 1995 188 7.24 Modelled spring transition, 1997: surface conditions 190 7.25 Calculated distribution of snow, superimposed ice, and stored water . . . 191 7.26 Modelled evolution of surface hydrology, days 150-158, 1997 193 7.27 Simulated timeseries of water production, delivery, runoff, and storage . . 195 7.28 Modelled subglacial conditions: days 154 and 158, 1997 197 7.29 Changes in subglacial water pressure and storage 199 7.30 Modelled subsurface conditions: days 154 and 158, 1997 200 7.31 Modelled autumn transition, 1997: surface conditions 202 x 7.32 Simulated timeseries of exchange rate and pressure, days 245-285, 1997 . 204 7.33 Modelled subglacial conditions: days 250 and 285, 1997 205 7.34 Simulated subglacial water pressure at five locations 208 8.1 Initial results from the conduit-coupled model . . . 216 D. l Detailed flow diagram of ablation calculation 253 E. l Simulated timeseries for parameter refinement tests 259 E. 2 Bar chart comparison of 6M, £B, £A, and ep for parameter refinement tests 261 F. l Connected-unconnected transitions: days 140-185, 1995 264 x i Preface Chapter 2 has been published in the Journal of Glaciology [Flowers and Clarke, 1999] with several changes to the text. Figures 2.1-2.6, 2.8, and 2.10-2.14 are reprinted from the Journal of Glaciology with permission of the International Glaciological Society (IGS). The contents of Chapter 7 that pertain to hydraulic release events, along with some information from Chapters 3-5, were presented at the International Symposium on the Verification of Cryospheric Models, Zurich, Switzerland, 16-20 August, 1999. The ac-companying paper is in press and will appear in the Annals of Glaciology, volume 31. Figures 3.1, 7.15, 7.18-7.21 are reprinted from the Annals of Glaciology with permission of the IGS. Other aspects of Chapter 7 have been presented at the Annual Scientific Meeting of the Canadian Geophysical Union, Banff, Alberta (9-13 May, 1999) and the American Geophysical Union Fall Meeting, San Francisco, California (6-10 December, 1998). Chapter 4 was presented at the 1998 Northwest Glaciology Meeting, 16-17 Octo-ber, 1998, Seattle, Washington. X l l Acknowledgements In my second year, I received email from my supervisor, Garry Clarke, who was on sabbatical leave in Cambridge. During one weekend, he had framed, typed, and tested the equations that I would spend the next four years programming to form the backbone of my thesis. It is this vigourous fountain of ideas that I most appreciated while I had the pleasure of working with Garry. I am also exceptionally thankful for the four field seasons I spent with him at Trapridge Glacier. Garry has been gracious to me at every turn, looking out for my success and well-being in ways that sometimes compromised his own interests. For all of his support I am sincerely grateful. My other committee members, Bruce Buffett, Susan Allen, and Roger Beckie, each deserve recognition for their encouragement and consultation over the years, and for their diligent efforts to improve this thesis. Regine Hock from the Climate Impacts Research Centre, Kiruna, Sweden, has been very generous to me by donating software and support for implementing her ablation model. I have enjoyed being part of the UBC Glaciology Group, both in Vancouver and the Yukon. I was fortunate to have Dave Hildes and Jeff Kavanaugh as field companions during four summers. Both Jeff and Dave helped collect the data presented in these pages. Thanks to Dave, in particular, for his many hours towing the radar around with me. The Geophysics Building is a notoriously hard place to work because it is occupied by such an amiable group of people. Of those who have passed through, Kevin Kingdon, Mathieu Dumberry, Barry Zelt, Dave Baird, Chris Wijns, and Ken Matson especially brightened my time here. Kris Innanen, Dave Hildes, Yuval, Phil Hammer, Stephane Rondenay, Len Pasion, Kim Welford, Colin Farquharson, and Eldad Haber make this a very hard place to leave. I am indebted to my friends Sharon Webster and Karen Goodman for supporting me from afar and for visits every year. Shawn Marshall has been a source of joy and strength throughout, and has also contributed his expertise to my work. I am grateful to my parents, not only for beginnings, but for the steady support they have always offered. My graduate studies were funded through the Natural Sciences and Engineering Re-search Council (NSERC) of Canada, UBC Graduate Fellowships, and support from UBC Earth and Ocean Sciences. I would also like to acknowledge the Canadian Geophysical Union for a financial award and for its effort to recognize student contributions to re-search. Digital maps for some of this work were donated by the Geological Survey of Canada, and Parks Canada kindly permits us to return to Trapridge Glacier every year. The Arctic Institute of North America and Kluane Lake Research Station (KLRS) pro-vided logistical support. Special thanks to the KLRS staff for making the base camp a memorable and enjoyable destination. xm Chapter 1 I N T R O D U C T I O N "... large bodies of water formed underneath, or within the glaciers (either on account of the interior heat of the earth, or from other causes), and at length acquired irresistible power, tore the glaciers from their mooring on the land, and swept them over every obstacle into the sea ... " - Mark Twain quoting Mr. Whymper on the events near Mt. Katla, Iceland, 1721 in A Tramp Abroad The foregoing account bespeaks the authority of water in glacial environments, known to the inhabitants of Iceland as a regular and destructive force of Nature. The controls on this "irresistible power" that tears glaciers from their mooring are far from completely resolved, and remain a source of vitality in glaciological research. Toward a better un-derstanding of the role that water plays in these glaciological dramas, I propose a model that describes the production, storage, and transport of water in glacial systems. 1.1 Preamble Because it operates conspicuously before the eyes of the observer, hydrology was one of the first aspects of glaciology to capture attention. Unlike viscous ice-deformation, for example, water meandering over the glacier surface and pouring down vertical shafts could be casually observed. These vertical shafts, called moulins, exist on some glaciers that experience significant surface ablation. Walking over the summer surface of such a glacier, one would observe that runoff derived from melted snow and ice collects in discrete channels. These channels escort water either off the glacier, or into one of its portals. Most alpine glaciers, being bound by the valleys they excavate, are flanked by ice-marginal streams which eventually coalesce. In addition to supraglacial streams, slush swamps and lakes may exist on the glacier surface. The system that stores and transports water over the ice surface, which includes snow and firn (transitional between snow and ice), is the supraglacial component of glacier hydrology. While certain aspects of glacier hydrology are straightforward to observe, our under-standing of it remains somewhat blurred at the point that water disappears from view. 1 Chapter 1. I N T R O D U C T I O N 2 Glacier plumbing provides convenient, though not comprehensive, scientific access to in-formation about the ice interior and bed. Dye-tracing has been used extensively and successfully to determine the speed and pattern of water flow through glaciers, thereby extracting constraints on drainage morphology and evolution [e.g., Kamb et al, 1985; Hooke et al., 1988; Seaberg et al., 1988; Willis et al., 1990; Fountain, 1992; Sharp et al., 1993; Hock and Hooke, 1993; Nienow et al., 1996]. Geophysical methods, primarily ground-penetrating radar and seismic reflection, are also being honed to extract hydro-logical information from glaciers [e.g, Blankenship et ah, 1986,1987; Hamran et al., 1996; Murray et al, 1997; Moore et al, 1999; Nolan and Echelmeyer, 1999a; Nolan and Echel-meyer, 1999b]. "Morphology" is used in the general sense to describe structure, specifically that of the drainage network operating in or under a glacier. A distinction is drawn between englacial and subglacial components of the drainage system, because the former is a two-phase (ice-water) mixture, but the latter is potentially more complicated. Very little is known about englacial drainage [see Fountain and Walder, 1998 for a review]. Bulk glacier ice is relatively impermeable [Raymond and Harrison, 1975], so water that reaches the bed must do so either through pipes or cracks. Theory predicts that pipes through temperate glaciers should be oriented steeply toward the bed, perpendicular to lines of equal hydraulic potential [Shreve, 1972]. Observations suggest, however, that these conduits are sometimes closer to horizontal [e.g. Fountain and Walder, 1998]. Peering over the glacier terminus, one may occasionally observe water being ejected through englacial openings some distance off the ground. The subglacial component of the drainage system is unique. While temperate glaciers exist at the melting point throughout, non-temperate glaciers that are sufficiently thick will attain their pressure-reduced melting point at the bed. In some situations, regelation due to non-hydrostatic stress around rocky protuberances is an important mechanism for glacier motion [Kamb, 1970]. Heat, in excess of that required to maintain basal ice at the melting point, contributes to water production. Geothermal, frictional (due to glacier sliding over the bed), and strain heating (in basal ice) are possible sources. In glaciers that are fed from the surface, basally-derived melt is usually small (millimetres to centimetres per year) compared to melt derived from the surface (decimetres to metres per year). While there are many possible subglacial drainage morphologies, suffice it for now to group them into two categories: channelized and distributed, or fast and slow, re-spectively. This bifurcation is important, because the dependence of water pressure on discharge is distinct for each group. In a fast system, restricted to ice-walled conduits, water pressure decreases with discharge [Rothlisberger, 1972]. Consequently, large tun-nels usurp smaller ones, resulting in an arborescent network feeding a few main arteries. Water pressure in a slow system increases with discharge and the result is self-stabilizing, provided the source is sufficiently small. The slow designation embraces most morpho-logies, including broad canals, water films, linked cavities, and macroporous sediments. Chapter 1. I N T R O D U C T I O N 3 These are described in more detail with references later in this chapter and in Chapter 3. Subglacial discharge is characteristically turbid, at least in summer. Sediment-bearing water can often be observed emanating from the glacier margin in either a channelized or distributed format. Aside from exiting directly off the glacier surface, through englacial tunnels or crevasses, and from the ice-bed contact, water may also escape through the subsurface if the glacier rests on a permeable substrate. Drainage through this fourth component of the system is difficult to study in the field because its physical boundaries are often poorly-defined. In some cases, however, buried aquifers are entirely responsible for evacuating subglacial water [Stone, 1993; Haldorsen et al, 1996]. 1.2 Water in glacial systems In glacial environments, water and ice are mutually influential. Glaciers store water, both solid and liquid, while the chemical, geological, biological, and dynamical processes of a glacierized basin are affected by the presence of water. 1.2.1 Storage According to van der Veen [1999, p. 2], 86.9% of the world's freshwater is frozen in glaciers and ice sheets combined. Liquid water, though usually less abundant, is impounded by glaciers in ice-dammed, proglacial, subglacial, and supraglacial lakes, in englacial and subglacial cavities, and in subglacial sediments. Stored water is episodically released, as runoff [see Fountain and Tangborn, 1985 or Rothlisberger and Lang, 1987 for reviews], or in catastrophic floods [e.g., Mathews, 1973; Nye, 1976; Clarke and Matthews, 1981; Clarke, 1982; Bjornsson, 1992; Fowler, 1999; Skidmore and Sharp, 1999]. Floods of this sort on a massive scale have been inferred from geologic evidence on the continent [e.g. glacial Lake Missoula, Clarke et al, 1984b] and in the ocean [e.g. glacial Lakes Agassiz and Ojibway, Barber et al., 1999]. 1.2.2 Chemical and geological relevance Chemical and geological signatures of glacierized (presently ice-inhabited) or glaciated (once ice-inhabited) regions are distinctively affected by hydrology [see Benn and Evans, 1998, Ch. 3]. Moving water transports erosion products and chemical species away from their source areas and deposits or precipitates them downstream. Therefore, proglacial stream hydrochemistry provides an important indicator of subglacial water quality and the geologic environment [e.g., Sharp et al, 1999a]. In addition to revealing geologic composition of the substrate, suspended and dissolved sediment loads can be quantified Chapter 1. I N T R O D U C T I O N 4 to estimate erosion rate [e.g., Humphrey and Raymond, 1994]. Solute concentration is known to be affected by the nature of the drainage system itself [e.g., Richards et al, 1996; Tranter et al, 1996; Anderson et al, 1999; Skidmore and Sharp, 1999], exhibiting characteristic diurnal [e.g., Collins, 1995] and seasonal [e.g., Collins, 1981; Brown et al, 1996] variability. Chemical analyses of recently deglaciated terrain can be used to decipher the prevailing hydrological conditions when the area was ice-covered [e.g., Hallet, 1979b]. Even biological information is forthcoming, thanks to the preservation of ancient lifeforms in subglacial lakes such as Lake Vostok, Antarctica [see Bell and Karl, 1998], and a mounting awareness of subglacial biogeochemical processes in general [e.g., Sharp et al, 1999a]. By thermal and mechanical weathering, water sculpts the ice it encounters to create shafts, tunnels, and cavities. The action of water, applied directly to a glacier substrate, erodes channels in sediment and bedrock. Geologic legacies of subglacial drainage exist in many forms, including (1) sinuous sediment ridges, or "eskers", thought to be relict tunnel deposits [see Hambrey, 1994], (2) channels incised in bedrock [Nye, 1973], and (3) sheet-like deposits of fine-grained till, suggestive of canal drainage over a deformable bed [e.g., Clark and Walder, 1994]. By facilitating subglacial erosion and debris transport [e.g., Boulton et al, 1974; Hooke, 1991; Iverson, 1991; Humphrey and Raymond, 1994; Alley et al, 1997], water influences the geologic imprint made by glaciers on the landscape. 1.2.3 Ice-dynamical relevance In ways that are perhaps more subtle, water influences glacier dynamics. Glaciers move by a combination of internal creep, basal slip, and substrate deformation [Alley, 1989b]. Water plays a facilitating role in all three. In polycrystalline ice, liquid water assists grain movement and hence overall deformation [Barnes et al, 1971]. At the glacier bed, high-pressure water distributed over large areas reduces friction and promotes basal slid-ing [e.g., Iken et al, 1983; Iken and Bindschadler, 1986; Iken and Truffer, 1997]. The existence of this relationship was already recognized in the late 18th century [Saussure, 1779-96 in Clarke, 1987c], but not incorporated into glaciological theory until much later [e.g., Lliboutry, 1968; Weertman, 1979; Kamb, 1970; Iken, 1981]. The precise mechanism involved and the controlling hydrological variable (e.g., water volume, pressure, pressure fluctuation) remain contentious issues [Iken et al, 1983; Fischer and Clarke, 1997; Harbor et al, 1997]. For a glacier resting on unconsolidated sediments, pressurized porewater can weaken the sediment and lead to pervasive (viscous) deformation [e.g., Boulton and Hindmarsh, 1987] or plastic failure in thin deforming layers [e.g., Iverson et al, 1998; Kavanaugh, 2000; Tulaczyk et al, 2000a]. The strength of this effect, but not its exis-tence, depends on till rheology. If any of these mechanisms (ice creep, basal drag, bed deformation) proceeds unstably, a glacier is said to be "surging". Chapter 1. I N T R O D U C T I O N 5 Fast glacier flow A small fraction of the world's ice masses demonstrate a dynamical instability known as surging (for glaciers) or streaming (for riverine ice-sheet outlets). Surging is an alternation between fast and slow modes of flow, where long periods (~10-100a) of steady ice-accumulation are punctuated by bursts of rapid motion lasting ~1-10 a. These episodes transport a vast quantity of ice below its equilibrium position, whereupon it wastes away. Ice streams, which are fast flowing sectors of an ice sheet flanked by comparatively stagnant ice, transport mass to an ice shelf or directly to the ocean where it calves away. Like surging glaciers, ice streams have been identified in both states of activity (on) and quiescence (off). For example, buried shear margins suggest that ice stream C, West Antarctica, switched off sometime in the last 250 years [e.g., Whillans, 1987]. Surging glaciers are found only in limited geographical pockets around the world, including Yukon, arctic Canada, Alaska, Iceland, Svalbard, East Greenland, Asia, and the Andes [Paterson, 1994]. Ice streams are found both in Greenland and Antarctica, and their existence has been inferred in the Laurentide Ice Sheet which covered much of North America during the last ice age [see Dyke and Prest, 1987]. While they occupy only small area! fractions of an ice sheet, their behaviour has far-reaching consequences. For example, Heinrich Events (episodic deposition of detrital carbonate in the North Atlantic [Heinrich, 1988]) may have been the result of a periodically-activated ice stream eman-ating from Hudson Strait [Broecker et al, 1992; MacAyeal, 1993; Alley and MacAyeal, 1994; Broecker, 1994; Clarke et al, 1999]. One of the most interesting aspects of surging is that it appears to be internally-regulated by individual glaciers, rather than externally-driven. This has coaxed glaciolo-gists into detailed examinations of the glacier bed in search of a "triggering" mechanism [see Raymond, 1987]. A landmark study of the 1982-83 surge of Variegated Glacier in Alaska [Kamb et al, 1985] resulted in one model for the cause and mechanics of surging [Kamb, 1987]. According to this theory, water is trapped during a surge in a poorly-drained basal cavity network, thereby facilitating sliding. Surge termination coincides with the establishment of a channelized (fast) drainage network that flushes subglacially-stored water. While this theory has done much to clarify the role of water in ice-flow instability, fast-flow over unconsolidated sediments [e.g. Clark, 1995] (such as those de-tected beneath ice stream B, Antarctica [Alley et al, 1986; Blankenship et al, 1986,1987]) remains a perplexing issue. Whether ordinary or catastrophic, glacier fluctuations are of interest especially where civilization establishes itself precariously in harm's way. Today, global sea-level rise is discussed in the daily newspapers, as glaciers undergo historically unprecedented retreat. In Europe during the Little Ice Age (~1550-1850), farmers spoke of evil spells that caused the glaciers to advance, burying their churches and menacing their fields with great torrents of water. Chapter 1. I N T R O D U C T I O N 6 "The expansion of the Grindelwald glaciers was so great in the 1770s that the parishioners called in a monk to exorcise them; but the monk, ... not knowing whether the advance was called by God or the Devil, declined to act." [Richter, 1891 in Ladurie, 1971] Lacking a lucid understanding of the internal dynamics of glacial systems and how they relate to external variables, we are still divining answers. 1.3 Model representation of glacier hydrology Considering the importance attached to water in glacial systems, its representation in glaciological models remains primitive. For basin-scale studies, glaciologists have relied on existing hydrological techniques to model surface runoff and groundwater flow. The problem of transport through glaciers and at the ice-bed interface is unique, however, and has garnered special attention. 1.3.1 Evolution of subglacial theory and models Theoretical underpinnings of today's subglacial hydrology modelling date back to the early 1970's. Papers by Rothlisberger [1972], Shreve [1972], and Nye [1973] propound water flow in ice-walled conduits (R-channels), along grain boundaries, and in bedrock channels (N-channels), respectively. Of these theoretical constructs, observational evi-dence favours R-channel-like conduits most frequently. Hence Rothlisberger theory has received more subsequent attention and refinement. Weertman [1972] evaluated the ability of R-channels to collect subglacial water, and concluded that mm-thick water films would coexist with channels. Analysis of water flow in these films predicted that films thicker than several mm are unstable [Walder, 1982]. That is, viscous heating increases with film thickness, potentially causing run-away growth. Despite the predicted collapse of thick water films, Weertman and Birch-field [1983b] argued that development of an incipient conduit network requires a more substantial water supply, namely from the surface. Walder [1986] and Kamb [1987] introduced the idea of connected subglacial cavities as a viable drainage morphology. Unlike R-channels, water is transported slowly in a cavity network and water pressure increases with discharge. This conception of drainage has been used to explain a myriad of glaciological observations [e.g., Willis et al, 1990; Iken et al., 1996]. These idealizations—channels, sheets, and cavities—were developed for rigid (sediment-free) glacier beds. As agents of erosion, glaciers are unlikely to rest on clean substrates. Deformable or "soft-bed" hydrology was introduced [Shoemaker, 1986; Clarke 1987b; Shoemaker and Leung, 1987] by adopting principles from groundwater Chapter 1. I N T R O D U C T I O N 7 hydrology. The concomitant and electrifying discovery of deformable material beneath ice stream B [Alley et al, 1986, 1987; Blankenship et al, 1986] precipitated a surge of interest in the rheological properties of subglacial sediment [Blankenship et al, 1986; Alley et al, 1987; Boulton and Hindmarsh, 1987; Clarke 1987b; Blake, 1992] and its implications for ice motion [Alley, 1989c; Alley et al, 1986, 1987]. Subsequent developments have addressed more complex drainage morphologies, such as canals incised in basal till [Fowler and Walder, 1993]. Increasing realism in the rep-resentation of subglacial drainage comes at the expense of simple geometry and well-established constitutive laws. While qualitative analyses can modify or combine these idealizations to explain observations, quantitative models are challenged by the require-ment that simple and exact mathematical formulations adequately capture reality. Alley [1996] made a step toward overcoming this difficulty in a one-dimensional numerical model based on the theory of Weertman [1972], with channel drainage emulated by the boundary conditions. His contribution portends the need for two-dimensional physically-based models that can accommodate the full suite of drainage mechanisms operative in diverse glaciological situations and settings. 1.3.2 Distributed deterministic models Lumped and stochastic models have been used extensively in the practical problem of predicting discharge from glacierized basins [see Fountain and Tangborn, 1985 or Lang, 1986 for a review]. Physically-based lumped models, as designed and demonstrated by Clarke [1996], are excellent tools for isolating and understanding the basic mechanisms of water flow through glaciers. For application to specific two- and three-dimensional problems, where the objective is shifted toward understanding processes rather than predicting outcomes, the didactic value of either lumped or stochastic methods is limited. Budd and Jenssen [1987] devised the first distributed model of subglacial drainage and applied it to West Antarctica. Assuming water flows in a film, they used melt rates from an ice-sheet model to compute steady-state distributions of film thickness and water velocity. Deterministic distributed models that include surficial, englacial, and subglacial components (as required for many glaciers) are still, to some extent, in gestation. This is partially due to the difficulty of representing drainage over a large area through discrete elements (e.g., channel networks). Darcy flow through porous subglacial sediments is an exception. Drawing on the more mature science of groundwater hydrology, two-dimensional transport has been modelled in subglacial aquifers by Boulton et al [1995], Piotrowski [1997], and van Weert et al [1997]. The first attempt at multicomponent modelling treated the glacier itself as a porous medium [Campbell and Rasmussen, 1973]. In 1998, Arnold et al. published a physically-based, distributed model of glacier hy-drology. As the first and only of its kind, it defines the state of the art. Three independent submodels handle surface ablation, meltwater routing, and englacial/subglacial conduit Chapter 1. I N T R O D U C T I O N 8 flow. Ablation is computed with a rigorous energy balance scheme developed by Arnold [1996]. Meltwater is directed into moulins or off the glacier, according to surface catch-ment structure as defined by a digital elevation model. Englacial and subglacial water is assumed to travel in a predefined network of conduits, and pipe flow equations are solved with software for modelling flow through storm sewers (developed by the United States Environmental Protection Agency). Predictions of outlet discharge, subglacial wa-ter pressure, and water velocity compare favourably with observations from Haut Glacier d'Arolla, Switzerland, during the melt season. 1.4 Trapridge Glacier study site and project 1.4.1 Physical environment Perched on the northern rim of Kluane National Park, Yukon, Canada (lat. 61°14' N, Ion. 140°20' W), Trapridge Glacier is a small (~4km long) outlet of the Mt. Wood ice complex. As a surge-type glacier, it keeps company with an anomalously high concentration of others in the St. Elias Mountains [Post, 1969]. Trapridge last surged sometime between 1941 and 1949 and has since undergone dramatic thermal and geometrical evolution back to its pre-surge state [Clarke and Blake, 1991]. The glacier spans an elevation range of ~2250-2800m above sea-level (a.s.l.), [Stone, 1993] with an equilibrium-line altitude (ELA) between 2400-2450 m a.s.l. The ELA divides the accumulation zone (where net mass is gained) from the ablation zone (where net mass is lost) and is an indicator of glacier health. Mean ice-thickness over the lowermost 2.5 km of the glacier is 70 m. Ice in the study area (approximate elevation 2360 m) has undergone progressive thinning from ~70m to ~60m over the last decade. Trapridge has a subpolar thermal regime, meaning that the ice is below its melting point except near the base. A surplus of basal heat enables subglacial water to exist, thereby facilitating sliding [Iken and Bindschadler, 1986]. Defying the global trend of retreat, the glacier terminus is stable or slightly advancing. Annual advances in the study area are on the order of 30 m and at least ~90% of this motion is accounted for by a combination of sliding and bed deformation [Blake, 1992]. A layer of deformable saturated sediment, 0.3-0.5 m thick, underbeds the glacier in this area [Blake, 1992]. Sedimentary studies of the area in front of the glacier reveal a hydrostratigraphy consisting of buried layers of sand and gravel (transmissive groundwater aquifer) capped by a comparatively impermeable till aquitard [Stone, 1993]. Plumbing at Trapridge Glacier is particularly interesting, owing to a combination of the glacier thermal regime and local meteorology. Water pressure sensors, emplaced in boreholes drilled to the bed, often record large diurnal fluctuations during the summer Chapter 1. I N T R O D U C T I O N 9 in response to surface meltwater. In winter, records from different instruments are gen-erally uncorrelated. Unlike many temperate glaciers (at the melting point throughout), Trapridge has no moulins and boreholes refreeze within 24 h, so inwash must enter the glacier through crevasses. Because sediments are frozen at the glacier margin, no visible hydraulic outlets (e.g., channels) develop. This oddity prompted investigation, eventually revealing dye-labelled subglacial water in groundwater icings down valley [Stone, 1993]. Except during extreme events, when floodwaters breach the thermal dam, subglacial water escapes through subsurface aquifers [Stone, 1993]. 1.4.2 Scientific context Written records of the Steele Creek drainage basin, in which Trapridge lies, date back to 1936 when W. A. Wood carried out geographical expeditions near the mountain of his namesake [VFtW, 1936]. It was not until 1969, however, that Trapridge was occupied for scientific investigation. From 1969-1989, the geometry and thermal structure of the glacier were studied as it continued to recover from its surge in the 1940s. Until hot-water drilling was developed in the 1960s and 70s and data loggers came into widespread use in the 1980s, scientific information was limited in both quality and quantity. Data acquisi-tion has since become much easier, and direct observations of subglacial hydromechanical conditions have been recorded year-round at Trapridge Glacier since 1988. Development of novel measurement devices has been an integral part of the Trapridge Glacier Project during the last decade. Sensors used to measure subglacial water tur-bidity and electrical conductivity [Stone et al, 1993], "ploughing" of subglacial sediment [Fischer and Clarke, 1994], till deformation [Blake et al, 1992], and glacier sliding [Blake et al, 1994] were designed for this project and are now being deployed in other glaciers [e.g., Porter et al, 1997; Fischer et al, 1998]. With the harvest of information made pos-sible by these and other sensors, process-scale studies at Trapridge Glacier have resulted in seminal contributions pertaining to subglacial hydrology [Murray and Clarke, 1995] and till mechanics [Clarke, 1987b]. Trapridge investigations have largely concentrated on a small area of the glacier bed, while the basin-scale context for these observations has been left unexplored. A dearth of information pertaining to basin-scale meteorology and glacier mass-balance precludes a controlled characterization of basin hydrology, as was undertaken at Haut Glacier d'Arolla, Switzerland [e.g., Sharp et al, 1993; Richards et al, 1996; Arnold et al, 1998; Nienow et al, 1998]. However, the collective value of long-term, in situ observations of dynamics at the glacier bed renders Trapridge suitable, if not extraordinary, for the purpose of relating basin-scale hydrology to process-scale subglacial phenomena. Fur-thermore, being subpolar, soft-bedded, and prone to surge, its study offers potential relevance to present-day Antarctic ice streams [e.g., Blankenship et al, 1986; Alley et al, 1989c; Engelhardt et al, 1990] and some inferred paleo-ice-sheet outlets [e.g., Dyke and Chapter 1. I N T R O D U C T I O N 10 Prest, 1987; MacAyeal, 1993; Clark et al, 1996]. 1.5 Thesis structure and objectives This thesis advances an integrated approach to modelling glacier hydrology, drawing heavily on information from Trapridge Glacier. My objective is (1) to articulate the theoretical and numerical anatomy of such a model and (2) to illustrate the success of this framework for simulating glacier hydrology in real and idealized situations. Much of the groundwork for this study was laid by Dan Stone [1993] in his thesis entitled "Char-acterization of the basal hydraulic system of a surge-type glacier: Trapridge Glacier, 1989-92". I have taken a significantly different approach than that of Arnold et al [1998], and the result fulfills a need not entirely overlapping. Physically-based equations gov-ern water transport in surficial, englacial, subglacial (ice-bed contact), and subsurface systems. For each of these components, governing equations are expressed in terms of vertically-integrated variables and are discretized on a two-dimensional finite-difference mesh. The system is synchronously coupled, with source terms that allow dynamic water exchange between components. Spatially- and temporally-variable forcing derives from a temperature-index parameterization of ablation [Hock, 1999]. Aside from the inclusion of subsurface groundwater flow in this model, the most im-portant departure from the work of Arnold et al is the treatment of subglacial drainage. Inspired by the situation at Trapridge Glacier, I model distributed flow through a mac-roporous sediment horizon (slow) rather than a network of conduits (fast). These two conceptualizations are equally limiting, because fast and slow systems can exist simul-taneously [e.g., Hubbard et al, 1995; Raymond et al, 1995] and can be interconverted during seasonal transitions [e.g., Nienow, 1998] and surge episodes [Kamb, 1987]. How-ever, the one I have chosen is preferentially amenable to soft-bedded glaciers such as Trapridge, and has greater potential relevance to ice streams [see Alley, 1996; Tulaczyk et al, 2000b]. Chapter 2 attends to the rudimentary requirements of a distributed model, namely constructing digital elevation models of the glacier surface and bed as inputs. I describe the collection, processing, and interpolation of a four-year suite of ice-penetrating radar data from Trapridge Glacier. Hydrological quantities, such as fluid potential and up-stream area, are estimated from the derived glacier map and form a basis for speculation into the distribution and drainage of subglacial water. Multicomponent hydrology theory, as conceived for this project, is expounded in Chapter 3. I begin with a conceptual model that defines system geometry and distin-guishes each drainage component by its physical attributes. The presentation of governing equations draws attention to mathematical parallels between components. Constitutive Chapter 1. I N T R O D U C T I O N 11 laws required for closure are derived in Chapter 4. These are separated from the basic theory because I view them as customized and replaceable. Chapter 5 translates the contents of 3 and 4 into a numerical model and outlines a method of solution. Chapters 6 and 7 turn to model applications, examining contrived and actual glac-iers in that order. Basic model behaviour is elucidated through equilibrium and time-dependent tests on simple geometry (Ch. 6). Sensitivity studies are conducted with an effort to relate the results to observations from several different glaciers. Chapter 7 in-troduces the full model capabilities for Trapridge Glacier simulations. Field data serve as model input, guide parameterizations, and provide model "calibration" and performance standards. Hydrology is modelled on three key timescales (sub-hourly, diurnal, seasonal) and qualitatively compared to instrument records. As the first mesoscale model of any kind applied to Trapridge, I restrict model tests to previously-studied phenomena and make only oblique attempts at data interpretation. In my estimation, Chapters 3 and 6 comprise the major contributions of this work. Together they represent what is freshest and most transferable to future problems. Chap-ters 2, 4, and 5 apply existing methods to ancillary aspects of the problem. Chapter 7 provides a welcome respite from idealized parabolic glaciers and may be of interest to those who use instrumentation to study subglacial hydrology. In their 1998 review paper, under "Directions for future research", Fountain and Walder write: "Glaciologists need to adopt a holistic perspective in studying glac-ier hydrology. Indeed, although we have written separately about near-surface, englacial, and subglacial water flow, the three phe-nomena are obviously coupled." ... "A key issue that needs more thorough investigation is how the various components of the glacial drainage system interact in space and time." These statements embody the authors' evaluation of scientific deficiency in the field of glacier hydrology, and summarize the mission that this thesis partially undertakes. Left deliberately ambiguous, the studies to which Fountain and Walder refer must comprise both theoretical and experimental components. Accordingly, and to the best of my ability, I have endeavored to weave these perspectives into a project whose theoretical elements are firmly rooted in reality. Chapter 2 T R A P R I D G E G L A C I E R DEMS Measurements of ice thickness and surface elevation are prerequisite to many glaciological investigations. A variety of techniques has been developed for interpretation of these data, including means of constructing regularly gridded digital elevation models (DEMs) for use in numerical studies. Here I present a simple yet statistically sound method for processing ice-penetrating radar data and describe a technique for interpolating these data onto a regular grid. Digital elevation models generated for Trapridge Glacier are used to derive geometric quantities that give preliminary insights into the underlying basin-scale hydrological system. The simple geometric analysis presented in this chapter suggests that at low water pressures a dendritic drainage network exists. A uniaxial morphology evolves from this as water pressure approaches flotation. Several realizations of these predictions are compared to hydraulic connection probabilities based on borehole drilling. This cursory hydrological analysis serves as a prelude to the modelling that is the main subject of this thesis. Ice-penetrating radar has been used extensively in glaciology, most commonly to de-termine ice thickness (see Bogorodskyet al. [1985] for an overview), as well as for studying englacial and ice core stratigraphy [e.g., Bamber, 1989; Jacobel et al, 1993; Siegert et al, 1998], basal and thermal conditions [e.g., Holmlund, 1993; Bjdrnsson et al, 1996; Urat-suka et al, 1996], and ice fabric or the characteristics of ice-crystal structure [e.g., Fujita and Mae, 1993]. Interpretation of these data is usually the principal priority, so inves-tigators have adapted seismic software or other packaged programs for data processing. These methods are often successful, but using such tools without proper knowledge of the underlying statistics can lead to erroneous results from which it is possible to draw incorrect conclusions. Therefore, an attractive alternative is to use a simple kriging al-gorithm, a statistically based interpolation method that uses weighted spatial functions to compute unknown variable estimates [e.g., Journel and Huijbregts, 1978]. Kriging re-quires geostatistical analysis which quantifies diagnostic characteristics of the data and allows incorporation of a priori knowledge into the interpolation. Such an approach, commonly used in mineral exploration and hydrogeology, is amenable to glaciological ap-plications. The method I adopt is applied to spatially irregular data generally collected along transects, but isolated radar depth soundings can be readily included. My interest in gridded ice-thickness and surface-elevation data arises from the need for accurate inputs to a basin-scale hydrological model of Trapridge Glacier. Knowledge 12 Chapter 2. T R A P R I D G E G L A C I E R DEMS 13 of basin geometry and ice distribution is essential for determining gradients in hydraulic potential—the primary driving force for water flow. In addition, I aim to use these DEMs as predictive maps to identify preferential flow paths and possible areas of water storage. Hydrological drainage features have been inferred directly from geometrical calculations using DEMs [Sharp et al, 1993], but this assumes that drainage is largely controlled by geometric rather than physical factors. To check this, I calculate hydrological proxies from the DEMs obtained for Trapridge Glacier and compare them to field observations of borehole connectivity. 2.1 Data collection and preparation Table 2.1: Number of radar traverses and soundings collected from 199Jf-1991 on Trapridge Glacier. Year Number of traverses Number of soundings 1994 12 354 1995 46 549 1996 19 556 1997 41 277 Total 118 1736 Ground-based radar data were collected on Trapridge Glacier from 1994 to 1997, providing significant area! coverage of the upper and lower basins. Together, these basins represent the entire ablation zone and a fraction of the accumulation zone. Spatial coverage is limited to safely accessible areas where a fixed survey location is visible and where the environment is conducive to high-quality sounding data. These factors preclude regularly spaced data, and yield a paucity of data in regions that are heavily crevassed (e.g., icefalls) and in marginal zones of very thin ice. In addition to surface coordinate survey data associated with radar transects, a variety of other survey data has been collected that can be used to map the glacier surface. These include locations and elevations of a fixed glacier-wide array of velocity stakes, a 250 m x 250 m strain grid, data loggers, boreholes, and longitudinal glacier profiles. These data sources together with the radar survey from 1994-1997 yield a total of ~ 1700 measurements from which to reconstruct an average ice-surface map. Over the same period, 1736 radar soundings were collected (Table 2.1) that are used to reconstruct the corresponding four-year average ice-thickness distribution. Figure 2.1 shows independent projections of areal coverage of ice-surface and ice-thickness (radar) data. The horizontal and vertical axes are parallel to easting and northing coordinates, respectively, and the dominant ice-flow direction is from west to east. Chapter 2. T R A P R I D G E G L A C I E R DEMS 14 Figure 2.1: Areal projections of data coverage on Trapridge Glacier for 1994-1997. (a) Surface eleva-tion survey data. Notable gaps occur coincident with an icefall separating upper (west) and lower (east) basins, and in the southwest corner where steep and crevassed terrain prohibits travel. The density of data in the centre of the ablation zone (east) is due to surveys of data loggers, boreholes, and a strain grid in our study area, (b) Radar data. Gaps are the same as those explained for ice-surface data. The origin (southwest corner) is located at 534003E, 67871S2N in Universal Transverse Mercator (UTM) coordinates. North is shown as upward in this plot and is the same in those that follow unless otherwise indicated. 2.1.1 Instrumentation and field methods 4m SURVEY TARGET 12 MHz RECEIVING END 25m HP PALMTOP COMPUTER SCOPE METER TRANSMITTING END MINIATURE HIGH-POWER IMPULSE TRANSMITTER Figure 2.2: Geometry and components of the data collection equipment. The transmitting end comprises the miniature high-power impulse transmitter and power supply between a full antenna of four-metre half-length (eight-metre total length). Separated from the transmitter by 25 m, the receiving end (a half-antenna) is connected to a digital scopemeter and HP palmtop computer. The sounding location is assumed to be vertically beneath the midpoint of the two antennas, hence the position of the survey target. Chapter 2. T R A P R I D G E G L A C I E R DEMS 15 We use a high-power impulse transmitter [Narod and Clarke, 1994] in conjunction with a resistively loaded transmitting antenna of four-metre half-length, producing a centre frequency of 12 MHz. Geometry of the equipment during data collection is shown in Figure 2.2. The direct air wave is used to trigger the receiver, and data are transmitted via an optical cable to a battery-powered Fluke Model 97 Scopemeter. A serial interface connects the oscilloscope to an HP200LX palmtop computer that controls data collection, processing, and storage. For ease of travel, antennas are oriented parallel to the propagation direction of the air wave and to the given transect. Both the transmitting and receiving antennas are towed on a rope by one person, such that the antenna spacing is fixed. A second person controls measurement spacing with the help of distance-markers on the tow-rope. The elevation of the equipment is assumed to be uniform and equal to the ice-elevation beneath the survey target. In accessible areas of the glacier, we attempt to collect continuous lines of data, generally transverse to the direction of ice flow to ensure good lateral resolution. Where possible, spacing between soundings is either 12.5 or 25.0 m, with surface elevation survey data collected less frequently (usually every 50.0 m). It is implicitly assumed that surface elevation gradients are smaller than elevation gradients at the bed so that surface coordinate data can be linearly interpolated between survey points. 2.1.2 Picking direct and reflected arrivals Time (ns) Figure 2.3: Sample radar traces from three locations showing the direct air wave (d) and bed reflections (r). Traces A to C are presented in order of decreasing data quality and increasing uncertainty in arrival picks, demonstrating the need to quantify pick confidence in the analysis. Englacially reflected wave arrivals (e) are common in areas of complex geometry, in which case I use the signal-to-noise ratio or travel-time comparisons with neighbouring points to determine which arrival is associated with the bed. Chapter 2. T R A P R I D G E G L A C I E R DEMS 16 Figure 2.3 shows a series of typical radar sounding traces as observed on the HP palmtop computer, with the direct and reflected waves indicated. I attempt to pick the arrival time of the first detectable energy, therefore the incidence of the wave as opposed to the peak. This strategy avoids the potential error introduced with data of variable amplitude. The results of this processing are travel times for the direct and reflected waves for each radar sounding. Ice thickness is readily computed from the travel-time difference assuming a homogeneous ice velocity of 170m//s-1, an air velocity of 300 m /xs_1, and a simple parallel-planar geometry of the ice and bed in the measurement vicinity. Data migration is unnecessary for Trapridge, because steeply dipping valley walls, usually the hallmark of glaciated terrain, are not an obvious feature of this basin. Migration techniques have been developed to handle this problem [e.g., Welch et al, 1998] common to many other alpine glaciers. For each radar trace, data quality is assessed and recorded by quantifying two at-tributes; the first pertains to the amplitude of the direct arrival and the second to the time-domain clarity of the reflection. The reflection index is quantitatively related to the arrival-time uncertainty, expressed as a number of oscilloscope pixels. Depending on oscilloscope parameters, each pixel represents the travel time through 0.68 to 1.36 m of ice, so an uncertainty of one pixel can introduce errors in the final ice thickness estimate on the order of 0.35 to 0.70 m. This index is thus a confidence factor for the pick that I exploit to preferentially value high-quality data in the interpolation. 2.2 Data interpolation The preparation I have just described yields two sets of irregularly spaced data from which I construct full arrays of interpolated data that adequately represent the whole glacier surface and ice-thickness distribution. From these, the underlying bed topography can be determined by subtraction. A variety of methods elaborated in various software is available for data interpolation. Selection of a method depends on the nature of the data and the desired outcome. Salient considerations are: whether data should be honoured exactly, how data weighting should vary with distance, and the size of the data set. The method I adopt, kriging with prior geostatistical analysis, is relatively simple and has proven useful in a variety of other geoscience applications [e.g., Piotrowski et al, 1996; Von Steiger et al, 1996; Persicani, 1995]. Kriging is a linear, unbiased, least-squares interpolation procedure that uses weighted functions of spatial autocovariance to compute variable estimates. It does so by mini-mizing the variance of an error function, expressed in terms of kriging weights, with a Lagrangian multiplier [Carr, 1995]. Kriging requires a priori knowledge of the statistical spatial characteristics of the data as revealed in semivariogram analysis. It is unbiased Chapter 2. T R A P R I D G E G L A C I E R DEMS 17 in that the data mean and distribution remain undisturbed. Kriging is an exact interpo-lator; therefore, it perfectly honours data at any of the specified interpolation locations, an attribute that may require spatial averaging as a preconditioning step. Data trans-formations are often necessary if spatial trends exist in the raw data, because kriging operates on the assumption that expected values are not a function of absolute spatial position [Carr, 1995]. A detailed explanation and derivation of kriging weights is given by Carr [1995] and summarized in Appendix A. 2.2.1 Data preconditioning Because the data are dense in some areas and sparse in others, I combine measurements that occur within the same 5 m grid cell. For the ice-thickness data, I compute a weighted average where each datum is favoured according to its reflection quality index. In some cases, very poor data are excluded if they are spatially redundant. For the ice-surface elevation data set, all values are weighted equally since there is no processing step that introduces additional subjective errors. Elevation bins Transformed bins Ice thickness bins Transformed bins Figure 2.4: Histograms of raw and transformed data, (a) Raw ice-surface elevation z s . (b) Transformed ice-surface elevation£Zs. (c) Raw ice-thickness hi. (d) Transformed ice-thickness^- The bimodal nature of the raw distributions in (a) and (c) reflects distinct environmental and geometric factors associated with the upper and lower basins. Both data sets under consideration, ice-surface elevation and ice thickness, are stat-istically corrupted by spatial trends that are apparent when either variable is plotted as a Chapter 2. T R A P R I D G E G L A C I E R DEMS 18 function of a:, the easting coordinate (approximately down-flow). Spatial trends in this di-rection are anticipated due to ice-surface slope and glacier thinning toward the terminus, but in practice, these trends must be empirically identified. Fortunately, in both cases, the variation with x is approximately linear. To precondition the raw surface-elevation data zs, I search for a best-fit line which, when subtracted from the data, results in a mean approximately equal to zero. Let the transformed surface-elevation data be de-noted £ Z s . The result of removing this trend from the data is revealed in Figures 2.4a and 2.4b, where comparative histograms of z s and £ z > qualitatively demonstrate that the transformed variable is distributed about zero. A two-step transformation is applied to precondition the ice-thickness data. To clus-ter the data more effectively about a line, I take the natural logarithm of ice thickness h\. This step also ensures that the interpolation produces positive estimates of ice thickness. In the second step, a linear trend is removed from the logarithmic data to obtain the transformed variable Figures 2.4c and 2.4d show histograms of the raw and trans-formed data, demonstrating that this two-step preconditioning eliminates the original bimodality. 2.2.2 Geostatistical analysis To quantify the spatial characteristics of these data sets, I use a well-known tool of geo-statistical analysis, the experimental semivariogram. A variogram 2^{h) characterizes the spatial variability, or autocovariance, of a quantity Y as a function of lag h (sepa-ration) [Journel and Huijbregts, 1978; Carr, 1995]. Following Cam, the semivariogram *y(h) = 2~f(h)l2 is defined mathematically as = 2^ - * W ) ] 8 , (2-1) »=i where N is the number of data points under consideration and Y(xi+h) is the notation for a variable that is a distance h away from Y(x{), so h = \xi+h — X{\. The semivariogram, presented here for a single variable type Y, is a measure of similarity within the data set and relies on the fact that similarity is inversely proportional to distance. Formally, this assumption is known as the intrinsic hypothesis [Carr, 1995] which states that variable similarity is exclusively a function of lag, not of absolute spatial location. The utility of this analysis is in identifying the appropriate semivariogram model and quantifying its statistical parameters. Among the most common variogram models are those called Gaussian and spherical. These are shown in Figure 2.5 along with visual explanations of their corresponding parameters. The equation of an idealized Gaussian Chapter 2. T R A P R I D G E G L A C I E R DEMS 19 1(h) i GAUSSIAN MODEL i silt /CO , £5 i nugget "1 a h V(h) I SPHERICAL MODEL 1 Sill ' CD 1 -1 CO 1 i nugget r ! b h Figure 2.5: Two common idealized variogram models: (a) Gaussian and (b) spherical. The increase of f(h) for small values ofh indicates spatial correlation of the data at small lags, diminishing as separation between data pairs increases. Parameters germane to the interpolation are indicated for both models. The sill relates to the statistical variance of the data, the nugget is an indication of data errors, and the range delimits the maximum lag at which correlation is significant. In order to represent the same geostatistical property, range is defined differently for the models shown in (a) and (b). semivariogram is [Journel and Huijbregts, 1978] j(h) = CO + C 1 I h " 1 - exp h > 0 (2.2) with 7(0) = 0 where CO, C and a are the standard representations of the statistical para-meters. The "nugget" CO is the value of the semivariogram when the lag is extrapolated back to zero. Ideally CO = 0 because each data point should correlate exactly with itself, so a nonzero nugget value is indicative of random noise or spatial autocorrelation on a scale smaller than the minimum lag. Parameter C = sill — CO where the sill is denned as the constant value that j(h) approaches at large lags and is related to the statistical variance of the data. For the Gaussian model, the "range" a, or transitional correlation lag, is determined by a = a '/•v/3, where 7(a') = 0.95(sill). An idealized spherical model is written as [David, 1977] CO + C = sill, 0 < h < a h > a (2.3) with 7(0) = 0, CO and C as previously described, and a defined as the lag at which the semivariogram attains the sill value. For spherical models the range is determined graphically rather than computed. Chapter 2. T R A P R I D G E G L A C I E R DEMS 20 2.2.3 Omnidirectional semivariograms 250 1(h) 1(h) 0 200 400 600 800 h (metres) 200 600 h (metres) Figure 2.6: Experimental semivariograms (triangles) plotted with best-fit models (lines), (a) £Zs. A Gaussian model provides the best fit to this semivariogram. (b) The experimental calculation and model are only plotted for lags less than 600 m due to an insufficient number of ice-thickness data pairs at large lags. A spherical model produces the best fit to the data. Both experimental semivariograms are generated with a minimum lag of 40 m. Semivariograms that utilize indiscriminately oriented data pairs are described as "om-nidirectional" , and are the standard type. I compute omnidirectional semivariograms for data sets £Z s and ^ using a routine provided by Carr [1995]. These are shown in Figure 2.6. Visual inspection leads to the choice of a Gaussian model for ^ and a spherical model for ^ n which are plotted along with the experimental data. Gaussian models are indicative of data with relatively deterministic spatial variations [Carr, 1995], more plausible for £Z|> than ^ since £ 2 s represents a smoother and simpler surface than Ultimately, the statistical properties of these surfaces relate to the processes that form them. Parameter values used to generate optimum semivariogram models for the data are listed in Table 2.2. Table 2.2: Geostatistical parameters extracted from experimental omnidirectional semivariogram ana-lyses of£Zs and £hl Parameter in Model Gaussian Spherical CO 5.0 0.0050 siU 215 0.0285 c 210 0.0235 a' 550 m — a 318 m 300 m Chapter 2. T R A P R I D G E G L A C I E R DEMS 21 2.2.4 Directional semivariograms Directional semivariograms use data subsets, chosen according to data-pair orientation, to provide additional insight into spatial autocorrelation of the data set as a whole. Results can reveal the imperfection of simple data transformations in removing small-scale trends, and anisotropy of the transformed data can then be quantified as a further constraint on the interpolation. I compute directional semivariograms for eight angles for each of the two data sets. As expected from the omnidirectional results, nearly all of the directional semivariograms for £z„ are fitted well with Gaussian models, and models for £hi are unanimously spherical. The parameters that result from fitting models to these semivariograms are listed in Table 2.3. Table 2.3: Geostatistical parameters extracted from directional semivariogram analyses o/<fZs and (For a linear model, C is defined as the slope of the semivariogram) Data set Azimuth Model CO sill G a' (m) a (m) u 0° Gaussian 5.0 150 145 275 159 22.5° Gaussian 5.0 175 170 350 202 45° Gaussian 5.0 170 165 400 231 67.5° linear 0.0 — 0.36 — — 90° Gaussian 0.0 195 195 500 289 112.5° Gaussian 5.0 240 235 600 346 135° Gaussian 5.0 250 245 600 346 157.5° Gaussian 5.0 150 145 400 231 0° spherical 0.0025 0.0275 0.0250 300 22.5° spherical 0.0050 0.0290 0.0240 — 275 45° spherical 0.0070 0.0300 0.0230 — 300 67.5° spherical 0.0040 0.0225 0.0185 — 150 90° spherical 0.0060 0.0225 0.0165 — 225 112.5° spherical 0.0050 0.0305 0.0255 — 325 135° spherical 0.0070 0.0300 0.0230 — 375 157.5° spherical 0.0025 0.0250 0.0225 — 300 Of direct importance is the comparison of ranges determined from the directional calculations. This gives the orientation and magnitude of spatial autocorrelation aniso-tropy. If vectors with orientations parallel to the individual semivariogram angles, and magnitudes proportional to the corresponding ranges, are plotted with common mid-points, this anisotropy is revealed as an ellipse (Figure 2.7). From the associated angles and ranges for £ Z s , an ellipse striking approximately northwest-southeast is definitive. It Chapter 2. T R A P R I D G E G L A C I E R DEMS 22 500 m Figure 2.7: Directional anisotropy in the two data sets (£hi and £Zs) determined from ranges of the directional semivariograms. The glacier margin is shown for orientation. Ranges are listed in Table 2.3. has an aspect ratio of 1.8, meaning that the range is nearly twice as great along its major axis than perpendicular to it. The directional ranges for ^ vary from 150 to 375 m, and the best-fit ellipse yields an anisotropy angle of 45° west of north with an aspect ratio of 1.4. This orientation is very close to that for £ Z l , so residual spatial trends are similarly present in both preconditioned data sets. Having quantified the orientation and mag-nitude of spatial anisotropy, the required geostatistical analysis is complete, and these constraints can now be applied to the interpolation. 2.2.5 Kriging and post-processing A published routine [Carr, 1995] is used to krige both data sets with the input inform-ation discussed above. Parameters used in the interpolation are listed and explained in Appendix B. Interpolated distributions of zs and hi are recovered from the raw kriging results by reversing the transformations performed initially. All ice-thickness values are positive owing to the logarithmic data transformation. Results obtained by kriging are relatively insensitive to exact geostatistical input parameters, but degenerate quickly if the wrong variogram model or random parameters are chosen. Because much of the Trapridge Glacier terminus is an ice cliff, rather than a gently thinning tongue, I suppress information about the ice extent (the position of its bound-ary) in the interpolation. Furthermore, introducing values of zero ice-thickness disrupts the data distribution and necessitates more complex preconditioning. Following this ap-proach, the glacier margin must be artificially imposed on the interpolated data set. I combine survey data and aerial photography to define the position of the margin, and set ice thickness to zero beyond it. Chapter 2. T R A P R I D G E G L A C I E R DEMS 23 2.3 Interpolation results Figure 2.8: Interpolated ice-surface elevation obtained by kriging. The ice margin has been imposed a posteriori. Prominent features of the interpolated ice-surface annotated in the text are (1) steep slopes representing icefalls, (2) surface depressions that capture supraglacial meltwater, and (3) bulges at the glacier terminus. The results after post-processing are shown in Figures 2.8-2.11. Note that the entire glacier margin is precipitous. While this feature is unrealistic except at the terminus (Fig. 2.9), it is difficult to avoid, and not necessarily problematic for the intended applications. The geometry of Trapridge and its surrounding terrain frequently interrupts line-of-sight to the survey station from the glacier margin. Therefore, data are lacking where the ice tapers to zero thickness. Areas of thin ice are further underrepresented in the original data set because the reflected wave often becomes indistinguishable from the tail of the direct wave. The purpose of constructing these interpolated data sets is for use in hydrological models, and because most of the glaciologically-relevant activity occurs within the interior reaches of the glacier, the marginal cliffs do not present a problem. Locations of features in the interpolated data sets that have been observationally or geophysically verified are indicated by numbers in Figures 2.8 and 2.10. The interpolated ice-surface is naturally smooth, representing icefalls poorly due to a lack of data in diffi-cult terrain. However, appreciably steep slopes are recovered at the four major icefalls, identifiable in Figure 2.8 (locations labelled 1). The first of these areas is on the southern edge where the ice margin extends beyond the grid. The second is the slope from upper to lower basins on the north side; this is an icefall that occurs as the glacier rides over a Chapter 2. T R A P R I D G E G L A C I E R DEMS 24 Figure 2.9: Comparison of a photograph of the Trapridge Glacier terminus (Mt. Wood in the back-ground) with the interpolated ice surface rotated into a similar perspective. Only the lower glacier is shown on the right, because the interpolation does not include the mountains in the photograph. severe bedrock ridge (look ahead to Fig. 2.10, location 2). Two more icefalls are iden-tifiable in the upper basin in Figure 2.8. Two important surface depressions (locations labelled 2) that collect meltwater in summer are visible, and the prominent terminal bulges (locations labelled 3) separated by a surface trough are successfully recovered at the eastern margin. Figure 2.10: Interpolated bed topography obtained by kriging. Features of the interpolated bed model are (1) bedrock ridges, (2) a large bedrock outcrop and (3) overdeepened troughs where ice-surface depressions are observed. Chapter 2. T R A P R I D G E G L A C I E R DEMS 25 Referencing the corresponding location on the image of bed topography (Fig. 2.10), these ridges are apparent (locations labelled 1). The steep ridge on the northern edge of the grid (location labelled 2) does in fact crop out as such, and the over-deepened troughs behind and adjacent to it (locations labelled 3) result in a depressed ice surface and a series of supraglacial ponds. Ice thickness is contoured in Figure 2.11b along with ice-surface elevation (Fig. 2.11a) and bed topography (Fig. 2.11c). Estimates of ice thickness are in excellent agreement with measurements of borehole depths in our study area. Anomalously thick ice recovered near the southern margin has been previously identified by airborne radar [B. B. Narod, personal communication, 1997]. 500m 2000 , metres above A , sea level Figure 2.11: Contoured maps of interpolated data sets, (a) Ice-surface elevation, (b) Ice thickness. Values of interpolated ice thickness are in excellent agreement with measurements of borehole depths where they are available, (c) Bed topography. The upper-basin bedrock ridge mentioned in the text is visible just below the letter Y. Chapter 2. T R A P R I D G E G L A C I E R DEMS 26 2.4 Derived geometric quantities In the simplest case, subglacial drainage might be almost exclusively controlled by geo-metric influences of the ice and bed. In general this explanation does not suffice, owing to the complexities of subglacial geology and drainage morphology. Nevertheless, using digital elevation models alone, insight can be gained into a glacier's predisposition to different drainage regimes. This approach has been used successfully to explain experi-mental dye-tracing results [e.g., Sharp et al, 1993]. I refer to derived geometric quantities as those calculated solely from the interpolated data to distinguish them from physical model results. Two such quantities are piezometric surface (hydraulic potential) and a terrain metric called upstream area. I compute these for several hydrological situations and compare the results to field data. 2.4.1 Piezometric surface Water moves in response to potential gradients, so a useful hydrological predictor is a map of the piezometric or potential surface. Such a map cannot be constructed without knowledge of the presence and pressure of water everywhere, so it is often assumed that water pressure balances the ice overburden pressure. This is an acceptable approxima-tion for winter when an efficient subglacial drainage system is not in place. It is well established from borehole water-pressure measurements that daily excursions in excess of flotation occur during the summer, but this condition cannot be maintained across the entire glacier bed. Therefore, I consider two situations: p w = O.bpi and p w = pi, where pw is water pressure and pi is the ice overburden pressure. These are characteristic limits of diurnal summer water pressure variations. Total hydraulic potential if> is the sum of pressure potential and elevation potential [Shreve, 1972], i> = Pw + Pv, g z\>, (2.4) where pw = 1000 kgm - 3 is the density of water, g = 9.81 m s - 2 is the acceleration due to gravity, and Zb is bed elevation. The water pressure in terms of the ice overburden pressure is pw = /pig hi, where / = pw/pi is the notation fraction, pi = 917 kgm - 3 is the density of ice, and hi is ice thickness. Thus we have i> = f pig h + pwgzh. (2.5) Figure 2.12 shows the results of this calculation for / = 0.5 and / = 1, visualized as Chapter 2. T R A P R I D G E G L A C I E R DEMS 27 contours of equal hydraulic potential. Contours that appear beyond the glacier margin are expressions of elevation potential only. Figure 2.12: Contours of equal hydraulic potential, (a) f — 0.5. (b) f = 1. Both cases are realistic for the melt season. Water moves perpendicular to equipdtential lines and is channeled into areas where contours are diverted to the west. Obvious outlets for water are the terminal reaches of the central and southern ice lobes. The box indicates our field study area. Both assumptions yield similar results, with / — 0.5 producing slightly more struc-ture. This reflects the greater influence of bed topography, which varies over shorter spatial scales. Best seen for / = 1, the strongest suggestions of water channelization at the bed (indicated as contours diverted to the west) are from the upper basin into the transitional region, and near the terminus. For both / = 0.5 and / = 1, the northwest-southeast trending potential low emerging from the upper basin appears to relieve most of the upper catchment. This potential trough continues into the lower basin, but be-comes much less distinct. Water flow to the glacier terminus is likely divided beneath the centre and southern ice lobes. A more distributed drainage network is expected for / = 0.5 than for / = 1 due to the slightly more complicated structure of its potential distribution. Chapter 2. T R A P R I D G E G L A C I E R DEMS 28 2.4.2 Terrain analysis Quantitative characterization of terrain has long been used in geomorphology to study river basins, and is explained by Zevenbergen and Thorne [1987]. Glaciological application of terrain analysis is not unprecedented: Sharp et al. [1993] computed upstream area distributions for Haut Glacier d'Arolla in a study that has already been mentioned; Marshall et al. [1996] used aspects of terrain analysis to determine controls on fast-flowing ice; and Bahr and Peckham [1996] investigated the viability of describing glacier networks using statistical topology models. For the present purpose, the most relevant terrain characteristic described by Zeven-bergen and Thorne is upstream area. For a particular grid cell, this quantity is the sum of all grid-cell areas that are upstream and connected. In a landscape model with known water thicknesses in each cell, computing upstream area would allow total runoff volume passing through any cell to be ascertained directly. I adapt the meaning of "upstream" by using hydraulic potential differences, rather than elevation gradients in this analysis. Much labour has gone into creating rules to compute upstream area accurately [e.g., Tarboton, 1997; Costa-Cabral and Burges, 1994]. Algorithms of varying sophistication have been developed for these calculations, each attended by certain drawbacks related to grid-dependence, artificial dispersion, computational memory requirements, and numer-ical complexity. The simplest method, referred to as D8 or steepest descent [Tarboton, 1997], uses a transfer rule where one grid cell gives its own area plus its upstream area to a single neighbour. This is a numerically simple calculation but has an obvious grid bias. In a glaciological context, D8 might be useful for representing a network of discrete conduits, although it still suffers from limited flow orientation possibilities. Quinn et al. [1991] improved on this by introducing a multiple-direction method that incorporates weighted area transfer, such that all downstream neighbours receive area in proportion to their relative gradients. This method largely alleviates the grid bias encountered us-ing D8, but introduces numerical dispersion and performs poorly at boundaries. Despite this, Costa-Cabral and Burges [1994] advocate this method over the steepest descent algorithm. A clever way of approximating continuous flow direction possibilities, called Doo, was developed by Tarboton [1997]. With this approach each grid cell is divided into triangular facets, and upstream area is partitioned between two receiving cells according to the angle of steepest descent. This method performs similarly to the multiple-direction method with reduced dispersion. Tarboton's results demonstrate this point: the drainage morphology derived from both is similar, but images resulting from Doo calculations are sharper. In one comparative test however, Doo failed to reproduce proper symmetry that the multiple-direction method captured [Tarboton, 1997]. I choose the multiple-direction method as a satisfactory compromise that clearly out-performs D8 and is substantially easier to program than Doo. Because I am interested Chapter 2. T R A P R I D G E G L A C I E R DEMS 29 in qualitative drainage morphology rather than quantitative discharge calculations, this method is adequate. I have performed the same calculations using the D8 method and find that my conclusions remain unaltered, despite the fact that I believe it to be an inferior representation of a hydrauhcally-diffusive glacier bed. I compute upstream area for a range of flotation fractions by repeated sweeps in four directions, allowing areas to cascade down from grid cells of high potential to those of low potential. Results for pw/pi = 0.5 and pw/pi = 1 are shown in Figure 2.13. DEMs with square 20 m grid cells are used in both cases. Additional tests were done with DEMs of 5 and 10 m resolution to elucidate any grid sensitivity in this method. Spatial patterns obtained were visually indistinguishable in all three cases, demonstrating the robustness of this method under the working grid geometry and resolution. In describing these results, I refer to water transport and drainage patterns as in-terpreted from the realizations of upstream area. Upstream area distributions represent estimates of hydraulic catchment. Because the formation of a subglacial drainage system depends on water flux, areas with large catchments have a higher probability of being included in the drainage network. Factors including the spatial distribution of surface sources play a role in drainage development that is unaccounted for in this analysis. I do not use terms that imply quantitative knowledge, such as discharge, since one cannot in general compute flow volumes. This is possible only if a constitutive equation is invoked to relate water-equivalent thickness to subglacial water pressure, a relationship that necessarily depends on the subglacial medium but is not in the spirit of the present analysis. In Figure 2.13, brightness indicates high values of upstream area, thus hydraulically favourable pathways. Results are presented as logarithms of the original calculations to de-emphasize the dominant downslope trend that obscures details. Shaded areas are hydraulically resistive. For pw/pi = 0.5 (Fig. 2.13a), ice and bed topography both exert strong controls over the predicted drainage morphology. The resulting system is primarily distributed, with dendritic branches developing where water is diverted around small-scale obstacles at the bed. In the absence of ice, the bed is naturally predisposed to unconfined distributed drainage, rather than channelization. One notable exception to this is the confluence suggested at the eastern margin, where a major outlet develops in a depression between the two bedrock ridges. Note also the drainage discontinuity between upper (west) and lower (east) basins. Potential ponding is indicated by bright pixels obstructed by dark barriers. Such an area is visible in the upper basin where water is collected from several tributaries but has no obvious outlet. These areas have potential glaciological significance in terms of water storage. For the case of full flotation (Fig. 2.13b), the upstream area distribution changes significantly. Water is channelized beneath the ice, and diverted to the south to avoid a bedrock bump as it exits the upper basin. This axis then bends northeast back into the central lower basin and continues through the study area. Most importantly, it is Chapter 2. T R A P R I D G E G L A C I E R DEMS 30 1.5 2.0 2.5 3.0 3.5 4.0 4.5 1.5 2.0 2.5 3.0 3.5 4.0 4.5 log(upstream area) 500m A Figure 2.13: Upstream, area distributions governed by hydraulic potential, (a) p^/pi = 0.5. (b) Pvr/pi = 1- Plots are presented as logarithms of upstream area to remove the dominant downslope trend. Bright areas represent preferential water flow paths or areas of high connection probability. The box indicating our study area is detailed in Fig. 2.11). the ice-surface slope that is sufficient to drive water from the overdeepened upper basin across the hydraulic barrier that existed at low water pressure. Qualitative differences between upstream area distributions for / — 0.5 and f — 1 are in agreement with my speculations based on hydraulic potential analysis (Fig. 2.12). Low water pressures result in drainage that is poorly confined, while high water pressures lead to a more channelized morphology. As water enters the lower basin where bed and surface slopes are mild, channelization is interrupted by two areas of distributed flow for / = 1. The first of these diverts some water to the southern ice lobe. This water travels in interconnected branches before converging again into a single channel that exits the glacier. The second hydraulic interruption occurs further down-flow just below the study area. Tributaries eventually reconverge between the central bedrock ridges before exiting the glacier. These results could be substantiated or refuted if there were subglacial outlets where water discharge Chapter 2. T R A P R I D G E G L A C I E R DEMS 31 could be measured, but because sediments at the margin of Trapridge Glacier are frozen, there is no observational evidence of subglacial drainage issuing from the ice-bed contact. We infer that discharge generally occurs through the subsurface where it is difficult to detect. Due to the purely geometric nature of this analysis, it is not applicable to all sub-glacial environments. I begin with a simple assumption to characterize the hydrological state of the bed, but do not address the mechanisms that may have brought about its development. Many processes in an open system are likely to invalidate the assumption of a uniform pressure flotation fraction. For example, englacial water routing can deliver large volumes of surface melt to the bed, often transporting it over significant horizon-tal distances [Fountain and Walder, 1998]. For a system where englacial transport of surface water is sufficient to maintain spatial pressure gradients at the bed, this type of analysis may be inappropriate. Furthermore, while Figure 2.13b suggests a uniaxial drainage morphology and identifies the area ripe for conduit development, this analysis takes no account of detailed subglacial physics such as is appropriate for Rothlisberger channels, Nye channels, or canals as proposed by Walder and Fowler [1994]. The interac-tion of a subglacial water sheet and conduits would preclude the possibility of sustaining uniform water pressures at the bed over short time scales. The channel suggested in Figure 2.13b is 60-80 m wide and is interpreted as a preferential flow path through per-meable sediments, rather than a conduit incised in rock or ice. In essence, this analysis provides a means of isolating the hydrological effects of ice surface and bed geometry. These influences are not responsible for the true complexity of the drainage network, but nevertheless modulate its development. Despite these caveats, I use upstream area as a proxy for hydraulic connection prob-ability, and attempt to predict changes in connection probability as a function of water pressure. Supposing that water were injected over the entire glacier bed, the upstream area distribution gives some indication as to the flow through each gridcell. This must be positively correlated with drainage system development, and hence, with connection probability. Spatial and temporal changes in connectivity have been documented at Trapridge Glacier by Murray and Clarke [1995]. They identified regions that are hydraulically connected during periods of high pressure, but become isolated during periods of low pressure. Borehole drilling records can be used to characterize the true connection prob-ability of the glacier bed as a function of position. This provides a basis for comparison with the predicted probability. 2.5 Borehole connection record Hot-water drilling has proceeded at Trapridge Glacier since 1980, and a log has been Chapter 2. T R A P R I D G E G L A C I E R DEMS 32 -10.65 10.45 10.25 •0.05 connection probability 3.50 4.50 1 3.25 4.00 3.00 3.50 2.75 3.00 2.50 2.50 100m log(upstream area) Figure 2.14: Comparison of borehole connection probability and upstream area distributions for the study area, (a) Drilling observations from 1989 to 1997 used to derive connection probabilities, (b) Connection probability distribution for 20 m grid cells obtained by kriging. The box indicates the area of highest connection probability predicted from borehole data, and appears in (c) and (d) for comparison, (c) Upstream area distribution for pw/'pi = 0.5. (d) Upstream area distribution for pw/'pi = 1. Note the difference in grayscales for (c) and (d). maintained to record observations that include the connection status of each hole. A connected hole is one that taps into a detectable subglacial drainage system. When a connection is established, borehole water level drops rapidly because the pressure in the drainage system is much lower than the pressure created by drill water backed up in the borehole. Unconnected holes do not drain or drain too slowly to be observed, indicating that they are hydrauhcally-isolated from their surroundings. In some cases, holes that are unconnected at the time of drilling become connected later on. I use drill log information from boreholes drilled between 1989 and 1997 to construct an average connection probability map of the study area. Figure 2.14a maps the bore-hole locations in the study area and indicates the connection status of each. From the intermingling of connected and unconnected holes, the hydraulic heterogeneity of the bed can be appreciated. For each connected borehole I assign a connection probability of one, and for each unconnected hole a probability of zero. Within each 20 m gridcell, the arithmetic mean of all data occurring in that cell is computed to represent a gridscale probability. From these data I generate an interpolated connection probability map (Fig. 2.14b) using the kriging method described earlier. For comparison with terrain analysis results, I choose 20m grid cells. This low res-olution obscures the sharp heterogeneity of Figure 2.14a, but indicates a region of high connection probability (bright) in the centre of the study area. This could be due to Chapter 2. T R A P R I D G E G L A C I E R DEMS 33 factors other than ice and bed topography, such as laterally extensive cracks near the base of the ice or a continuous layer of permeable sediment. Not surprisingly, neither upstream area calculation predicts the detailed connection probability very well. Bearing in mind that the study area is the only location where we have successfully drilled connected holes on Trapridge Glacier, the fact that most of the water is predicted to be routed through this area (especially in the case Pv,/pi = 1) is remarkable in itself (refer to Figure 2.13). The connection "hot spot" in the centre is slightly better predicted by the upstream area distribution for p^/pi — 1 than for any other distribution from p^/pi — 0.1-0.9, including pw/pi = 0.5 shown here. Grayscales in Figures 2.14c and 2.14d are chosen to maximize contrast within each panel. This belies the fact that the high water-pressure regime focuses a greater flow fraction through the study area. Absolute values of upstream area in the study region and in the hot spot are much higher for p^/pi = 1 than for pw/pi = 0.5. Because our drilling records are based on observations made during the day in mid melt-season, it is sensible that p w/pi = 1 comes closest to capturing this connection feature. 2.6 Summary I have presented a statistically reliable method for interpolating and extrapolating spa-tially irregular data onto a grid that is useful for numerical models. I emphasize the importance of carefully performing the geostatistical analysis before the interpolation, and caution against uninformed use of kriging or other interpolation packages. The re-ward for doing the statistics is the ability to provide custom information about each data set to the interpolator, and hence obtain confident estimates of surface and bed topography. Predictions based on derived geometric quantities lend insight into the basin-scale drainage structure of the glacier (both metrics, piezometric surface and upstream area, predict preferential drainage through the study site), but are unable to rigorously account for detailed variations on scales relevant to predicting borehole instrument responses. Therefore, some features of the drainage clearly arise from topographic gradients of the ice and bed, but other effects, such as sediment permeability and distribution, can exert an equally strong influence. With DEMs now available for Trapridge Glacier, one new data set is added to the small collection of complete geometric data available for alpine glaciers, which includes among others, DEMs for South Cascade Glacier [Fountain and Jacobel, 1997], Haut Glacier d'Arolla [Sharp et al, 1993], and Storglaciaxen [Bjornsson, 1981]. Complete data sets are sparse despite the long scientific history of alpine glaciology, yet if accompanied by surface velocity data, they provide important constraints for ice-flow modelling such as that undertaken by Blatter [1995], Albrecht and Blatter [in preparation], and Hubbard Chapter 2. T R A P R I D G E G L A C I E R DEMS 34 et al. [1998]. These models can make significant contributions to our understanding of stress distribution in ice and the importance of various glacier flow mechanisms. Detailed understanding of the hydraulic connectivity of the glacier bed, and therefore of instrument signals, awaits a model that includes subglacial physics. Chapters 3-5 describe such a model, and Chapter 7 outlines its application to Trapridge Glacier. Chapter 3 M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y Hydrological behaviour at the glacier bed expresses the complexities of a larger system forced by conditions above, within, and beneath the ice. Distribution and movement of subglacial water are widely-recognized controls on glacier and ice-sheet dynamics [e.g., Walder, 1982; Bindschadler, 1983; Clarke et al, 1984a; Raymond, 1987; Alley, 1989a; Al-ley et al., 1994]. Consequently, numerous theories have been developed to describe water flow at the base of a glacier (see Paterson [1994], pp.103-131 for a review). Interactions between the glacier bed and its surroundings can be difficult to characterize, yet bear direct relevance to some of the most captivating glaciological phenomena: hydromechan-ical disturbances and outburst floods, seasonal glacier-climate interactions, and ice-flow instabilities. While this is widely appreciated, few attempts have been made to include these interactions in models of subglacial hydrology. Therefore, existing models suffer from an inability to objectively predict the subglacial response to surficial and englacial changes. The gap in theory that describes component interactions makes it necessary to prescribe driving forces and invoke simple assumptions about the transmission of these forces to the bed. Toward an improved and integrated description of glacier hydrology, I develop a theo-retical model that comprises four components to represent surficial, englacial, subglacial, and subsurface water transport. Stacked as horizontal layers (Figure 3.1), the system is stitched together by intercomponent water exchange. This approach is reminiscent of a similar, but lumped, representation of glacier hydraulics [Clarke, 1996], appealing in its simplicity but limited by a lack of spatial information. The present formulation is two-dimensional vertically-integrated and time-dependent, thus permitting the evolution of spatially-distributed variables. 3.1 Conceptual model A description of glacier hydrology that is mathematically and computationally re-alizable requires tremendous simplification. I begin by writing independent governing equations for each of the four constituents illustrated in Figure 3.1. Ablated snow and ice feed a surface runoff system (Figure 3.1, system 1) which delivers water to the glacier interior via encounters with crevasses and moulins. Excess water runs off supraglacially. 35 Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 36 Figure 3.1: Conceptual model of glacier hydrology, (a) Perspective diagram illustrating the four coupled systems: (1) surface ablation and runoff, (2) englacial storage and transport, (3) drainage through a porous subglacial sheet, and (4) subsurface groundwater flow. Layers surrounding the subglacial sheet are enlarged in the inset, and arrows indicate possible waterways, (b) Longitudinal slice through the conceptual model. Elevations noted on the side are used in the text: zs=ice surface, zg —glacier bed, zu=upper boundary of aquifer, zi=lower boundary of aquifer. Not shown is the elevation of the saturated horizon in the aquifer z w . The englacial system (Figure 3.1, system 2) routes surface melt to the bed, where it joins water that is subglacially-derived. Beneath the ice, drainage takes place in a porous sed-iment sheet (Figure 3.1, system 3) resting on a till cap or aquitard. Possible subsurface flow is accommodated in a groundwater aquifer separating the till cap from bedrock. (Figure 3.1, system 4). The existence of a low-permeability till layer (the cap) is inspired by the sedimentary stratigraphy in front of Trapridge Glacier. Because glaciers are agents of erosion, many are underlain by a similar layer of low-permeability material. Neighbouring systems are synchronously coupled through exchange terms (f> that rep-resent water transfer from one layer to another. This is the crux of the formulation that facilitates a two-dimensional approximation of a three-dimensional problem. Geometry of the conceptual model, the subsurface stratigraphy in particular, is inspired by Trapridge Glacier. However, the model structure can easily be modified. Alternative governing physics for any of the components can be inserted within the framework presented here, providing the equations describe a continuous system and can be discretely represented. One aspiration of this work is to provide a context for interpreting hydrological behavior as recorded by subglacial sensors. Toward this end, the treatment of individual compo-nents is simple and economical. More exhaustive studies have focused on the details of each component (surface ablation is a good example), from which I endeavour to extract the essentials. Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 37 Throughout the text, superscripts are used to identify variables belonging to a partic-ular system as follows: surface runoff (r), englacial storage and transport (e), subglacial sheet (s), and groundwater aquifer (a). Subscripts are generally reserved for quantities common to more than one system (e.g., ice density pi, glacier bed elevation z#). In the following sections, I outline the basic theory for each modelled system. Governing equa-tions are based on water balance and are written analogously for each layer. I reserve the discussion of model parameters for Chapter 6. 3.2 Surface ablation and runoff Ablation Melting of snow or ice is the result of a complex energy balance that includes contribu-tions from net radiation and turbulent heat fluxes [e.g., Kuhn, 1987]. These quantities vary over short spatial scales, making is difficult to accurately determine melt rates with a small number of measurements [Arnold et al., 1996]. Additional uncertainties arise due to the potential importance of complicated boundary layer processes in the turbulent heat transfer [Hock and Holmgren, 1996]. Despite the complexity of these processes, practical advances have been made in computing ablation of snow and ice with degree-day (or more generally, temperature-index) methods [e.g., Braithwaite and Olesen, 1989]. This approach assumes a proportionality between ablation and the sum of positive temper-atures ( > 0°C), and has become the standard for models that require quantification of melt [e.g., Huybrechts and T'Siobbel, 1995; Marshall and Clarke, 1999a]. Significant effort has been made to relate temperature-index calculations to energy-balance [e.g., Braithwaite, 1995; Arendt, 1997; Hock, 1998], resulting in refinements to the method for ice sheets [Reeh, 1991] and for alpine glaciers. Most recently, Hock [1999] has developed a spatially-distributed temperature-index method that includes the effects of potential direct solar radiation. This adaptation produces substantial improvements over the classical degree-day method in reproducing the high-frequency character of discharge hydrographs from glacierized basins. By con-sidering the effects of topographic shading, slope, and aspect, calculated ablation rates are made spatially variable [Cazorzi and Fontana, 1996]. In principle, this allows a direct comparison between modelled and measured glacier mass balance. Following Hock, the melt rate M of snow or ice is determined by J T (MF + a s n o w / i c e i) T > 0 M = < (3.1) 0 T < 0 , Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 38 where T is air temperature, MF and <zsnow/ice a r e melt- and radiation factors respectively, and I is potential direct solar radiation. Hock computes this quantity as r T ( Rm\ , \P0 COS z) a ( 0 c>\ I = h I 1 ipa cos 0, (3.2) with solar constant I0 = 1368 W m - 2 [Frohlich, 1993], Rm/R the mean over instantaneous Sun-Earth distance, clear-sky transmissivity tj)a = 0.75, P/Po the atmospheric pressure relative to sea level, Z the local zenith angle, and 6 the solar incidence angle. Following Gamier and Ohmura [1968], incidence angle is computed as a function of slope, aspect, solar azimuth, and zenith angle. Cloudy conditions are not accounted for in this analysis. Hock [1999] attempted to capture the deviation from clear-sky conditions in one version of her model by including measurements of global radiation in the calibration of radiation factors. The ratio of measured global radiation to theoretical direct radiation fluctuated between 0.2 and 1.2, confirming that atmospheric effects are important in the determination of incident solar flux. However, no measurable improvement was made in the prediction of ablation or glacier discharge with this version of the model [Hock, 1999]. This fact does not diminish the importance of cloudy conditions in the radiation budget; rather, it suggests that changing a linear calibration factor cannot improve the representation of this effect. Radiation is set to zero during times when the surrounding terrain casts a shadow on the glacier surface. In the absence of site-specific mass balance data, I assume melt and radiation factors are spatially and temporally uniform. To compute ablation then only requires a DEM of the glacier and its surroundings, parameters MF, a s n o w , and alce, a timeseries of air temperature at a known elevation, and a value for surface lapse rate. Following Reeh [1991], modelled snowmelt initially refreezes as superimposed ice (Fig-ure 3.2). This is presumed to occur instantaneously and accounts for some, but not all, of the energy lost to melting snow or ice that is destined to refreeze before leaving the system. Namely, it requires the first portion of the snowpack to be melted twice. Along with in situ water storage, refreezing effectively delays the seasonal onset of runoff in keeping with observations [e.g., Fountain, 1989; Nienow et al, 1998]. For the lack of a thermal model, diurnal refreezing of surface water is neglected. When modelled superimposed ice comprises a certain water-equivalent fraction of the original snowpack (PMAX), refreezing terminates and subsequently-generated melt is stored as liquid within the snow (Fig. 3.2). Although PMAX = 0.6 is usually taken as constant ([Reeh, 1991]), I follow the approach of Woodward et al [1997] who suggest that more realistic estimates of PMAX can be obtained with little additional effort. Based on the premise that the maximum thickness of superimposed ice D™"x is limited by near-surface glacier temperature (assumed to be equal to the mean annual air temperature, Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 39 1 snow 1 ice S L I C E < PMAX s.i. Ice „ P M A X snow(t=0) snow(t=0) super-imposed ice snow-pack storage source to runoff till (1) melted snow refreezes as superimposed ice (2) meltwater stored in snowpack when PMAX reached (3) snowpack saturates (4) stored water released as snow ablates (5) superimposed ice ablates (6) glacier ice ablates • H • 55a snow superimposed glacier water stored Ice Ice In snowpack Figure 3.2: Ablation hierarchy for snow, superimposed ice (s.i.), and glacier ice. (a) Flowchart showing possible ablation pathways from snow and glacier ice to melt available for runoff, (b) Stages of ablation from a snow-covered surface to bare glacier ice. MAAT) [Ward and Orvig, 1953], Woodward et al. derive the empirical relationship T\TYICLX ice -0.69 x MAAT + 0.0096 (3.3) for D%? in cm and MAAT in °C. PMAX is then calculated as D%?x/D°now, where D°snow is the water-equivalent depth of snow in cm at the beginning of the melt season. Runoff over the glacier surface commences when the snowpack saturates, as determ-ined by snowpack porosity. Water is released from storage with further ablation of the snow. Because I assume that water is uniformly distributed within the saturated snow column, the amount of water released depends only on snowpack porosity. Where snow is present in the model, it retards meltwater delivery to the glacier interior as parameterized Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 40 by (3.4) where 0r:e(O) represents the water exchange rate between the ice surface and interior in the absence of snow, s is the areally-averaged snow depth expressed in water-equivalence, and sc is a critical depth required to reduce the exchange coupling by 1/e. Physically-based modelling of water percolation through the snowpack would require knowledge of additional snowpack parameters [e.g., Colbeck and Anderson, 1982; Sturm and Holmgren, 1993]. Parameter sc is meant to embody the delaying effects of percolation. The term <f>r:e(0) describes vertical water exchange and can be a source or sink to the surface runoff system. It usually depends on the depth of runoff and a time constant, as shown later in Equation (3.51). Horizontal transport of water within the snow is neglected as an additional computa-tional cost, as is hydrology of the firn layer. Detailed studies of water transport through snow, firn, and ice suggest a potentially significant role of firn in water storage [e.g., Fountain, 1989; Schneider, 1999]. However, discharge through firn is much lower than discharge through snow, which is lower still than discharge through the body of a temper-ate glacier via moulins [e.g., Baker et al, 1982; Fountain, 1993; Hock and Noetzli, 1997]. Therefore, simulations of subglacial behaviour should not be dramatically compromised for glaciers that experience high surface melt rates. In these situations, water delivery to the glacier interior is accomplished primarily by supraglacial streams, whose discharges dwarf the flow in near-surface aquifers. Models that aim to reproduce proglacial stream hydrographs or changes in glacier water storage through seasonal cycles should certainly include a firn layer. Runoff State-of-the-art runoff models can be empirical, statistical, or physically-based. I elect a physically-based (deterministic) approach for consistency with the rest of the model. Overland flow literature offers two basic starting points: conservation of mass and mo-mentum [e.g., Hong and Mostaghimi, 1997] or kinematic wave theory [e.g, Kuchment et al, 1996]. Both have distinct drawbacks. The former requires simultaneous solution of three variables (flow depth and two horizontal velocities), and the latter allows the possibility of shock waves. I therefore opt for a much simpler approach adapted from Marshall and Clarke [1999b] that alleviates these difficulties. Their work uses linear diffusion to route continental sur-face water to the oceans. Water flux is related to gradients in hydraulic head by a constant diffusivity. In my formulation, runoff routing is governed by the local meltwater runoff Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 41 depth hr(x,y,t) and spatial gradients in runoff depth and elevation. This introduces the appealing feature that runoff flux tends to zero as the source water is depleted. Volume conservation for an incompressible fluid is honoured according to f + S = M - " : - ' ( 3 - 5 ) where M is given by Equation 3.1, and <f>r:e is source/sink term arising from mass ex-change with the underlying englacial storage system. The source/sink term (f>T'a represents exchange with the groundwater system in the glacier forefield (the area in front of the glacier), effectively accounting for infiltration. Thus <j>r:e — 0 where hi — 0 and (jf:a — 0 where hi > 0. Water exchange terms are discussed in more detail at the end of this chapter. The discharge per unit width (hereafter referred to as flux) QT- is computed as Q r = _ ( 3 6 pw g dxk with j,k — (1,2) for two horizontal spatial dimensions, water density py, — 1000kgm-3, and gravitational acceleration g = 9.81ms -2. Total fluid potential tpr in (3.6) is the sum of pressure and elevation potentials above a datum zs = 0, written as i>T = f + P^gzs- (3.7) For surface water, pr = pw ghr'. . For a two-dimensional system, the constant of proportionality Kr^k in Equation (3.6) is a rank-two tensor. I make the simple assumption that Kr-k is isotropic, such that Kjk = Kr 8jk- This assumption pervades the forthcoming developments. JCJfe cannot be interpreted as hydraulic conductivity of a porous medium, nor does it describe the physics of supraglacial runoff. Rather, it is used as a numerical convenience to regulate the rate of meltwater transport. For a glacier surface, the value of Kr-k depends on the combined ease of water transport in firn and in supraglacial channels, weighted by the relative abundance of each. It could be made to evolve in time if the development itself of the runoff network is deemed important to the problem. Aside from the source term M, runoff routing as governed by Equation 3.5 is independent of the snow conditions on the glacier surface. As meltwater channels are observed to forge their way through remnant snow patches, Kjk is not affected by the presence of snow. Accordingly, runoff depth hr does not include water stored in the snowpack. Owing to the head ward migration of the Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 42 snowline during the melt season, only unusual glacier surface geometry would give rise to situations where substantial runoff is captured by a downstream snow aquifer. 3.3 Englacial water transport and storage Water transport within the ice is an important component of glacial drainage [e.g., Hooke et al., 1988] and introduces nonlinearity into subglacial plumbing [Clarke, 1996]. Of all the systems, this is perhaps the most difficult to probe, and hence has received compar-atively less attention [e.g., Holrnlund, 1988]. Previous modelling attempts have assumed instantaneous transfer of surface water to the bed [e.g., Arnold, 1998], but this is some-times contradicted in Nature. Observations on a number of glaciers show that daily subglacial water pressure maxima substantially lag the maximum surface melt [e.g., Ray-mond et al, 1995; Stone, 1993]. Moreover, observations of englacial tunnels have revealed near-horizontal orientations [Fountain and Walder, 1998], demonstrating an opportunity for horizontal migration of water within the ice. I attempt an improved representation of englacial hydrology in this model by including various storage morphologies (crevasses and moulins), and by allowing transport between them via a system of cracks. Figure 3.3: Idealized morphology of englacial voids with dimensions labelled as in the text, (a) Vertical pipe or moulin. (b) Surface crevasse. Parameter fi scales crevasse width at the base relative to width at the surface, (c) Basal crevasse. Figure 3.3 presents three idealized englacial void geometries: (1) vertical pipes or moulins, (2) surface crevasses, and (3) basal crevasses. Each morphology implies a unique relationship between water volume and pressure. The volume v\ of a cylinder, assumed to extend from the surface to bed is v\ - TV r\ hi, (3.8) Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 43 where r\ is the radius and hi is ice thickness. For simplicity, I assume that top-sided crevasses also penetrate to the bed. This is unrealistic according to empirical evidence [Paterson, 1994, p. 187], but its practical consequences are small. An alternative formu-lation assumes crevasses are limited to a particular depth within the ice, introducing an additional unknown quantity. I therefore write the volume V2 of a surface crevasse as V2 = ^w2l2hi(f2 + 1), (3-9) where w2 is crevasse width, l2 is crevasse length, and f2 is a geometric factor that scales the area in contact with the glacier bed to the area exposed at the surface. A choice of f2 = 1 simplifies this geometry to a vertical slot. For basal crevasses, I arbitrarily assume a height of hi/2, thus volume V3 is computed as V3 = ^w3l3hi, (3.10) where w3 and Z3 are the crevasse dimensions at the bed. Total void volume Vr(x,y) is then given by Vr = Y, J Ni(x,y)Vi(x,y)dS, (3.11) i s where Ni(x,y) is the number density of storage morphology i (crevasse, moulin, etc.) per unit area, Vi(x,y) is its corresponding volume given by (3.8), (3.9), or (3.10), and S is area. The relative volumetric contribution of morphology i can be expressed as a fraction 7;(a:,y) = • In practice, it is simplest to prescribe iVj and Vy, and partition void volume among the possible morphologies by adjusting 7 .^ Then, consistent dimensions (e.g., w2, h) for each void type can be determined. To reduce the number of free parameters, one might assume that N\ = N2 = N3 = 1 per model grid cell, and that l2 and ls are equal to the length or width of the cell. This simplification is reasonable for cells on the order of 20-40 m, but becomes problematic as cells become large. For example, if individual cells are large enough to contain entire crevasse fields, local englacial storage cannot be satisfactorily represented by a single idealized crevasse. Using the relationship for hydrostatic pressure, the water pressure at the base of a crevasse or moulin is p = p„ g h, where pw is water density, g is gravitational acceleration, and h is the depth of water in the storage element. Different storage elements have distinct hypsometric curves, giving rise to unique relationships between water pressure and water Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 44 volume. The englacial water pressure pi as a function of water volume V™ in each void type is Pi = P* 9 V? 7T r; (3.12a) P2 Pa Pv, 9 hi f2 ( l - / 2 ) p^ghi -1 + W1 + hi w2 h fi 4 V? hi wz h V™ < Vs. (3.12b) (3.12c) Equations (3.12b) and (3.12c) are positive solutions to the quadratic equations V^fa) and V3w(pz), respectively. Equation (3.12b) has a singularity at / 2 = 1. In this case, the surface crevasse is a vertical slot and the appropriate relationship between pressure and volume is Z>2 Pv, 9 V? w2 h (3.13) An alternative to (3.12c) is required when the uncompressed water volume V$ in a basal crevasse exceeds the void volume V3. Assuming additional water is accommodated by compressing the liquid rather than distending the crevasse walls, Pa 9h | v r 3V3 (3.14) where 8 is the compressibility of water and V3 is given by (3.10). In the interest of computational efficiency, pressures for each morphology in the stor-age system are not tracked independently. Rather, I assume that englacial water pressure can be represented by a single value for a particular location and time. Water is then partitioned among the elements such that pa = p2 = Pz = Pe, where pe(x, y, t) is the bulk pressure of the englacial storage system. To obtain a relationship between pe and the total volume of water in storage Ve(x,y,t), I solve (3.12a-c) for V^w, V^, and V^w and substitute the results into v e = ^ 1 v 7 + JV2V7 + JV3V3w, (3.15) Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 45 with pi = p2 = p3 = pe. For the case f2^l and V™ < V3, this yields Pv,ghi N2 w2 l2 (1 - / 2) - 2N3 w3l3 - (iVi 7T r\ + N2 w2 l2 f2 + N3 w313) 2 Ve + A/(JV17Tr\ + N2 w2 l2 f2 + N3 w3 l3)2 + -i—(N2 w2 l2 (1 -f2)-2 N3 w313) hi (3.16) For the case f2 — 1 and Vgw < V3, this relationship must be recomputed using (3.13) rather than (3.12b) for P2{V2) resulting in P 2 N3 w313 (Ni 7T r 2 + N2 w2 h + N3 w313) yj(JVi 7T r\ + N2 w2 l2 + N3 w313)2 -4 Ve N3 w3 l3 (3.17) When Vgw > V3 (3.14) is used in place of (3.12c) to derive p = P^ghi N2w2 h(l-f2) - (^NiTvr\ + N2w2l2f2 + ^pv,gBN3w3l3h^j + iVi 7r r\ + N2 w2 l2f2 + -pv,gBN3 w313 h^j K ^ W - M (N3w3l3hi-l-P„g8 N3 w313 hi - 4 V 1 hi V * (3.18) for f2 ^ 1 and Pe = Pv,g hi 4 y e AT ^— + pw g 3 N3w3l3 - N3 w313 4:N1Tvrl + 4N2w2l2 + py,gBN3 w313 hi (3.19) for / 2 = 1. Where a single storage morphology exists, the relationship pe(Ve) is described by the appropriate choice of (3.12), (3.13), or (3.14). Where multiple morphologies are Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 46 present, pe(Ve) is described by (3.16), (3.17), (3.18), or (3.19). As outlined above, this system allows for meltwater to reach the bed through surface crevasses and moulins, and provides additional vertical storage in basal crevasses. To accommodate horizontal transport of water within the ice, I envision a system of cracks or joints, which invites a fractured-medium approach. While englacial tunnels may be the dominant delivery conduits in some cases, the difficulty of modelling the physics of their formation and operation, especially in a continuum framework, warrants an alternative idea. Cracks have been observed during borehole drilling at Trapridge Glacier and have occasionally played an important role in subglacial sensor communication. Previous hydrological models have also treated the glacier itself as a porous medium [Campbell and Rasmussen, 1973]. I proceed by denning an areally-averaged englacial water volume he(x,y,t) related to Ve as Ve = J he dS. (3.20) With this, the water balance equation analogous to (3.5) is -dr + -dx- = <f> -+ - ( 3 - 2 1 ) where the right hand side contains source/sink terms arising from exchange with the sur-face runoff system ((f>re) and underlying subglacial sheet {<j>e"). The vertically-averaged discharge is expressed in Darcian terms as Pv,g dxk where fluid potential •0e = pe +pw g ZB, and ZB is the glacier bed elevation. Transmissivity TJk is equal to Ke-k hi, where Ke-k is hydraulic conductivity and hi is the local ice thickness. I use hi as an effective aquifer thickness, under the assumption that it is large enough to function as a Darcy continuum. According to Snow [1968], the hydraulic conductivity K of a fractured aquifer with uniform planar joints or cracks can be written as K = ^^f, (3.23) p 12 Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 47 where /x is the dynamic viscosity of water, NL is a line-density of cracks (with dimensions of inverse-length), and b is crack aperture. The effective porosity ne of such a system is NL b, or in terms of (3.23), For the present purposes, I specify crack aperture b and an isotropic conductivity, again assuming K?k = Ke Sjk- With these values, an effective porosity can be calculated ac-cording to (3.24). The corresponding volume of englacial joints, t S z b zs J J nedz dS, (3.25) is added to the volume computed in (3.11). In areas lacking bulk storage bodies, total englacial void volume is derived entirely from internal joints, thus Vr = Vjoint- This situation requires an alternative expression for englacial water pressure as a function of volume. I let pressure increase linearly with water volume: Pe=pigh (^j, (3.26) with ice density pi = 910kgm~3, and the ice overburden pressure pig hi introduced for scaling. 3.4 Drainage through a subglacial water sheet Numerous idealized drainage structures have been proposed to describe water flow at the base of a glacier, including ice-walled conduits [Rothlisberger, 1972; Shreve, 1972], bedrock channels [Nye, 1976], water films and sheets [Weertman, 1972; Walder, 1982; Weertman and Birchfield, 1983a], linked cavities [Walder, 1986; Kamb, 1987], soft-sediment canals [Walder and Fowler, 1994], porous sediment sheets [Clarke, 1996], and standard aquifers [Shoemaker, 1986; Shoemaker and Leung, 1987] Most glaciers develop multiple drainage morphologies over time, if not simultaneously [e.g., Willis et al., 1990; Stone and Clarke, 1996; Nienow et al., 1998]. Transitions between various states (e.g., low-pressure Rothlisberger networks to high-pressure linked cavities) are implicated in ac-tivating ice-dynamical instabilities [e.g., Kamb et al., 1985; Clarke, 1987a; Echelmeyer et Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 48 al., 1987; Fowler, 1987a; Kamb, 1987; Raymond, 1987; Harrison et al., 1994; Bjornsson, 1998]. At the risk of limiting model applicability, I confine my treatment of subglacial drainage to a porous sheet composed of water and sediment. Trapridge Glacier field studies motivate this approach with extensive evidence for a thin permeable layer un-derlying the ice [e.g., Blake, 1992; Stone and Clarke, 1993; Fischer and Clarke, 1994]. An additional incentive is that this morphology lends itself to continuum mathematics, while enabling a wide range of hydrological behaviour. I imagine this layer to be a con-tinuous saturated horizon separating the ice and bed, capable of porosity adjustments in response to changes in water flux as illustrated in Figure 3.4. This is a common prop-erty of dilatant materials, of which till is a well-known example [e.g., Clarke, 1987b]. A temperate glacier bed is assumed, although frozen conditions can be accommodated by introducing a thermal switch. Determination of frozen versus temperate bed areas would require implementation of a thermal model. a ice AH = A/i Figure 3.4: Porosity adjustments in the saturated subglacial sediment sheet. Porosity (dark shading) is greatly exaggerated, (a) Thickness of the saturated horizon Hs at low water flux, (b) Sheet porosity increases in response to increased flux. Changes in H' and h" are identical. Define the subglacial sheet water thickness h"(x,y,t) as an areally-averaged water volume, hs = | Jn'H> dS> (3.27) where S is area, n"(x,y,t) is the porosity of the sheet, and H"(x,y,t) is the combined thickness of the binary (sediment-water) mixture. The water balance is 1 dt dxj (3.28) Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 49 where b" is a source term that includes basal melting due to geothermal heat and glacier sliding friction, (jf's is the exchange with internal storage (as in Eqn. 3.21), and <f>t:a is exchange with the underlying aquifer. A coupled bed thermal model would allow free determination of b , and additional possibilities such as freeze-on to basal ice. Assuming a temperate glacier bed, I approxi-mate basal melt rate as b> = 9°±2L, (3.29) piL where QG and QF are geothermal and frictional heat fluxes, respectively, and L = 3.34 x 105 J kg - 1 is the latent heat of fusion of ice. Geothermal heat flux depends on geographic location. Frictional heat flux can be estimated in terms of basal shear stress Tbj and basal-ice velocity Vj relative to a deep stationary horizon: QF = TbjVj. (3.30) Shear stress in turn can be approximated as Tbj — PiQ^i sinctsj, (3.31) where as j is the glacier surface slope [Paterson, 1994]. In cases where geothermal activity is high (e.g., Iceland) or where basal melt dominates surface water delivery to the bed, the calculation of b merits careful attention. In many instances however, b3 is negligible compared to other sources. As in Darcian flow, I write the vertically-integrated water flux in (3.28) as Q._ K°kh°dr^ 3 pv, g dxk' with ij>s = p" + pv, g ZB- Note a deviation from the standard definition of Q"- in (3.32), where I have written transmissivity as Kjkh*, rather than the usual K9-kH*. Because the horizon under consideration is thin and h* can tend toward zero, having Q'j oc h* appropriately ensures that Q"- —> 0 as h* —» 0. Hubbard et al. [1995] and Stone [1993] propose that effective hydraulic conductivity can be locally enhanced by the washing away of fine sediments, and have used this idea to explain observed nonlinearity in subglacial discharge. I attempt to capture this property by allowing hydraulic conductivity to vary (3.32) Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 50 in space and time as a function of subglacial water volume. Mathematical closure requires a relationship between subglacial water pressure p" and sheet thickness h*. Chapter 4 attends to defining p"(hs) and parameterizing K^k(h") using data from Trapridge Glacier. 3.5 Subsurface groundwater flow Flow in subglacial aquifers has been documented in several places [e.g., Sigurdsson, 1990; Stone, 1993], and in some cases is the primary outlet for subglacial water [e.g., Stone, 1993; Hcddorsen et al., 1996]. Thus it has received substantial attention from theoret-ical and modelling perspectives [e.g., Shoemaker, 1986; Boulton et al., 1995; Piotrowski, 1997; van Weert et al., 1997]. Studies have focused primarily on paleo ice-sheets that involve large areas and long time scales, thus lending themselves to equilibrium assump-tions. Layered multi-aquifer systems have also been modelled, but to date, have not been coupled to a time-dependent model of drainage at the ice-bed interface. Adopting the subsurface stratigraphy illustrated in Figure 3.1, I consider an aquifer subparallel to the glacier bed and separated from it by a low-permeability till cap, or aquitard. While this geometry is patterned after the sedimentary stratigraphy exposed by streamcuts in the Trapridge Glacier forefield, it is effectively a system of interbedded aquifers and aquitards. An existing mathematical formulation readily enables a descrip-tion of two-dimensional flow in the transmissive unit with water exchange across the resistive cap [e.g., Bredehoeft and Pinder, 1970; Chorley and Frind, 1978]. Both sat-urated and unsaturated modes are permitted in the aquifer. In the saturated regime, the dependence of water density on pressure is included. Let zjXx,y) be the lower boundary of the aquifer and zw(x,y,t) be the elevation of the saturated horizon. The mass of water in the aquifer ma(x,y,t) can be expressed in terms of porosity na(x,y,t) and water density pa(x,y,t) as (3.33) S zL Density is a function of pressure pa and obeys the equation of state pa = p0 exp[6(pa -p0)], (3.34) where p0 is the density at a reference pressure p0, and 3 = 5.04 1 0 Pa 1 is the compress-ibility of water. The reference pressure is usually taken to be atmospheric (po ~ 0) where Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 51 Po = Pw, hence (3.34) reduces to pa = pw exp(3pa). (3.35) I make the usual assumption that horizontal spatial variations in pa are small compared to changes in time. Spatial derivatives of pa are thus neglected and Zyi I na pa dz = na p" (zw — ZL). (3.36) Water thickness in the aquifer ha(x,y,t) is na (zw — ZT,), S O (3.33) can be written as ma = J p a hadS. s (3.37) Changes in mass are assumed to occur exclusively through water exchange with the overlying subglacial sheet (where hi > 0) and the periglacial runoff system (where AT. = 0). The global mass balance expression that applies is dma ~dT J pa (<t>s:a + 4>r:a) dS, (3.38) where <J>s:a is the vertical flux across the aquitard beneath the glacier, and <f>r'a is the vertical flux across the aquitard in the non-glacierized area. For a particular spatial location, only one of <f>*:a or <^r:o is nonzero. To compute from (3.37), I apply Reynold's transport theorem to get dma _ d ~dT ~ dt dS. (3.39) Equating the right hand sides of (3.39) and (3.38), J [ ^ ( Pa ha ) + ^ ( v* pah*) dS = J pa (ra + <f>r:a) dS, (3.40) Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 52 and in its local form, Upaha)+dVj{ v ? p a h a ) = z p a {4>s:a + r ' a ) • ( 3 - 4 1 ) Variable v" is the Darcy velocity, also known as Darcy flux q° in groundwater hydrology [Freeze and Cherry, 1979]. Thus one can define Qa- =v"ha = qjha, where 9 ? = - ^ , (3.42) pvg dxk with fluid potential tpa = pa + pw g zi. Recalling the assumption that <C (3.41) can be rewritten as f) dOa-m { p a h a ) + paix~ = p a { r ' a + r a ) • ( 3 ' 4 3 ) Because appears only as a spatial derivative, I have taken the liberty of using pw rather than pa in the expressions for q" and tpa. Dividing (3.43) through by pa and differentiating the first term, the final balance equation emerges: These equations have been developed to govern both saturated and unsaturated flow in the aquifer. In either case, one of the time derivatives in (3.44) vanishes. When the aquifer is unsaturated the water table is a free-surface, so ^Sr = 0. Neglecting aquifer expansion for the moment, saturation implies that fti 0. The transition between saturated and unsaturated states is another source of numerical stiffness that I seek to minimize in choosing a constitutive law pa(ha). For the unsaturated case, pa = pw9ha ha<nada, (3.45) where da = z\j — zi, is the aquifer thickness, and in the standard diction, ha is identically hydraulic head. When the aquifer is saturated, compressibility effects become important. Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 53 For vertical stress on an aquifer, aquifer compressibility a a can be expressed in terms of porosity and pressure as Q . = 1 ( 3 . 4 6 ) (1 - na) dpa In terms of changes in the aquifer thickness itself, (3.46) is equivalently expressed as [Freeze and Cherry, 1979, p. 57] ^ = aa da. (3.47) dpa I approximate (3.47) to estimate the change in pressure 8pa as a function of the change in aquifer thickness 8da, 8p° = (3.48) aa da Since changes in aquifer thickness are assumed to be equivalent to changes in water thickness 8ha, 6p« = - ^ L (3.49) aa da This impHes a relationship ha -nada , Pa = P*9ha+ a a d a (3-50) for ha >nada. 3.6 Coupling exchange terms Source/sink terms </>r:e, <jf:t, <f>s:a, and <j)r:a appearing in (3.5), (3.21), (3.28), and (3.44), represent the rate of water transport between adjacent systems. By parameterizing water flow in the vertical dimension, the problem is effectively reduced from three dimensions to two, simplifying the mathematics and reducing the computational requirements. The imbedded assumption is either that the areal extent of each system is much greater than its vertical thickness, or that the forces driving water flow are dominantly horizontal. Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 54 Both statements are true for all but the glacier interior. The exchange functions are cast in terms of dependent variables hr, he, h", and ha, such that transport rates evolve simultaneously with the remainder of the system. Positive values of exchange always represent downward movement of water in accordance with gravity. Surface-englacial exchange <f>r:e(x,y,t) is expressed in one of two ways, depending on the vacancy of englacial storage. In the usual case, when stored water volume is less than the void volume (VE < VT), <f>r:e is proportional to the local runoff depth hr and (r r : e ) _ 1 , where r r : e is a time constant. This simple function depends only on hr, because supraglacial runoff is blind to the value of he when crevasses and moulins are underfull. In the special case where storage is filled to capacity (or is predicted to be overfilled), englacial water is permitted to spill out on the glacier surface. These two cases are expressed as where he is related to VE in (3.20), VT is the total englacial void volume in S, and % r : e represents the coupling strength of the two systems. In general, % r : e = 1 where surface crevasses or moulins are present, and zero elsewhere. Recall that coupling between the glacier surface and interior is also affected by the presence of snow according to (3.4). Over extended periods of time, englacial and subglacial systems will converge toward a common operating pressure. However, interesting subglacial behaviour often results from adjustments made over minutes to days, hence the distinction between pe and p" is retained. Englacial-subglacial exchange is formulated in terms of the pressure differential pe - p°, coupling strength X e : s , and a time constant r " as = X * : ' — ^ " J**) • (3-52) Pv, 9 T For (j>e:s, x e a > 0 in the presence of any bulk storage type. For englacial cracks alone, x e : s = o. The till cap regulates exchange between the subglacial sheet and groundwater system. An effective time constant for this process is defined by the aquitard thickness dt(x,y) = zs{x,y) — zu(x,y) divided by its vertical hydraulic conductivity Kt(x, y). With attention VE < VT VE > VT, (3.51) Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 55 to the saturation level of the aquifer, <f)':a is written separately for two cases as ha > na da ha < nada, (3.53) where pa replaces py, in the first case, and pa g dt represents the driving potential due to elevation differences between the two systems. For the purpose of calculating <f>s:a, fluid pressure at the top surface of the aquifer is required. Thus the first term in Equation La „a Ja (3.50) is dropped and pa = a a ^ a • ^ n *ke unsaturated regime (ha < na da), the top surface of the aquifer remains unpressurized, so for the purposes of calculating sheet-groundwater exchange, pa is set to zero as shown. Similarly, where hi — 0, exchange between surface runoff and the groundwater system is 4>r:a = { Xra-^JtW-Va) + Pa9dt] ha > na da ha < na da. (3.54) 3.7 Summary of equations and boundary conditions Equations (3.5), (3.21), (3.28), and (3.44) describe a water balance in each of the four model layers and can be solved simultaneously for the dependent variables hT, he, h3, and ha. Ancillary relationships required by the basic governing equations include ex-pressions for horizontal water flux and fluid potential, the equation of state for water, and constitutive laws for pressure as a function of areally-distributed water thicknesses. Intercomponent coupling is accomplished in Equations (3.51)-(3.54), which govern freely-determined water exchange throughout the system. Ideally, this system is solved on a grid that exceeds the horizontal ice extent. In this case, implementation of boundary conditions is straightforward. As discussed in Chapter 2, DEMs for Trapridge Glacier only encompass the ice margin in the ablation area. On three sides of the grid, ice extends beyond the area covered by the radar mapping campaign, leaving the problem of developing boundary conditions appropriate for both glacierized and non-glacierized cells. Chapter 3. M U L T I C O M P O N E N T H Y D R O L O G Y T H E O R Y 56 At the surface, the forcing is entirely encapsulated in M(x,y,t) (Eqn. (3.1)), the melt rate of snow or ice. Although it has not been explicitly included, precipitation as rain would also constitute a direct forcing. I assume that impermeable bedrock underlies the aquifer, such that the bottom boundary is closed. In principle, either Neumann or Dirichlet conditions could be applied to the lateral boundaries of any of the systems. However, there are few instances when nonzero Neumann conditions would be preferable since the known sources reside largely at the surface. Surface runoff and groundwater systems extend over the entire model domain, while englacial and subglacial components are defined only in glacierized areas. Surface water requires special treatment at the boundaries. Runoff should be permitted to exit the model domain unimpeded, provided that the domain is not a confined basin. Rather than specify a value of hr and the boundary, I pad the outside of the domain with a topographic trough in which runoff is collected, and prescribe a no-flow boundary outside the trough. This achieves the desired result of a passive boundary and has been used in other surface hydrology models (personal communication from S. J. Marshall). The implementation of this condition is detailed in Chapter 5. Conditions on the internal and subglacial systems are imposed at the glacier margin where pressure is atmospheric. Thus pe = 0\hl=o and p s = 0|/iI=o. For the special case that the ice terminates in a proglacial lake, the boundary condition is pe = p" — py, g hiake\hi=o, where hiake is lake depth. Where ice transgresses the model boundary, pe and p" are nonzero. This typically occurs at the upstream boundaries, so one might assume that a reservoir is available to maintain constant pressure in each system. Thus, pe(xo,yo,t) — pl(x0,yo) and ps(x0,yo,t) = Po(x,y), where (xo,yo) are the set of coordinates defining the boundary. Alternatively, no-flow conditions can be prescribed if the domain extends beyond the region affected by an active drainage network. Specifically, if the boundaries are in the accumulation zone, where surface water is unlikely to play an active role, very low transmissivity would be expected within the ice and at the bed. This boundary condition becomes a problem-dependent choice. For Trapridge, the upstream boundaries are above the equilibrium line, so no-flow conditions are most realistic. A similar argument is used in determining conditions on the aquifer, except that they are not as directly related to surface melt. If the aquifer operates on a long enough timescale, it might be reliably saturated at the glacierized boundaries from infiltration of geothermal melt. On the other hand, if the grid extends to steep mountain terrain, the boundary condition is likely ha = 0 or no-flow. For nonglacierized cells, ha(xo, yo, t) — h%(x0,yo) is prescribed. Since nonglacierized cells are typically downstream from the glacier, no-flow conditions are inappropriate. For Trapridge I prescribe no-flow across the top, h" = na da at the lateral boundaries, and ha < na da downstream. Chapter 5 delves into the details of how these governing equations and boundary conditions are implemented on a useful numerical grid. I reserve Chapter 4 for describing a method of parameterizing K^k(h3) and p3(hs) based on subgrid topographic attributes. Chapter 4 P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S In exchange for spatial breadth, most discrete models suffer from the ills of sacrificing process representation on scales below the model resolution. These processes are im-portant because they give rise to phenomena described by the governing equations. For example, in groundwater hydrology the extent of the vadose (unsaturated) zone depends on the relationship between capillary pressure and saturation, which can vary over short distances [Desbarats, 1995]. Ice-sheet models require treatment of surface mass-balance, a quantity subject to variable precipitation patterns and the effects of mesoscale topog-raphy [e.g., Marshall and Clarke, 1999a]. To ameliorate this situation, variables are often parameterized in terms of subgrid quantities, implicitly giving voice to processes not included in the model. This chapter describes an attempt to capture the influence of small-scale glacier bed topography on subgrid hydrology through two key variables. Closure of the coupled equations introduced in Chapter 3 requires that subglacial water pressure p* and hydraulic conductivity K° be expressed as functions of water sheet thickness h". Both ps and K" are expected to vary on subgrid scales (20-40 m). For reasons outlined in the forthcoming text, I assume that these quantities are fundamentally affected by glacier bed topography, and that gridscale parameterizations can be improved by including this. The first half of the chapter uses data from Trapridge Glacier to define a relationship between p" and h", and the second half attends to the parameterization of K* by considering synthetically-generated topography. 4.1 Macroporous flow layer thickness In a typical aquifer, water pressure is a linear function of hydraulic head, fluid density, and gravity. Although Darcian terminology was adopted in Chapter 3 to describe the macroporous horizon, several features distinguish this layer from a groundwater aquifer. Most importantly, it sustains water pressures sufficient to compensate the ice-overburden, in a layer that is only centimetres to decimetres thick. Spatial and temporal changes in horizon thickness are facilitated by dynamic adjustments in porosity. These characterist-ics point to nonlinearity in the relationship ps(h"). Except during extreme events [e.g., Kavanaugh and Clarke, in press], subglacial water pressures recorded at Trapridge Glac-ier rarely exceed ~150% of the ice overburden pressure. The rule for p*(A*) should be 57 Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 58 consistent with this observation. Furthermore, p" must increase more rapidly than h* for hs > h*., where h"c is the critical sheet-water thickness corresponding to ice notation. Failure to choose a rule with this property produces unrealistic heterogeneity in the dis-tribution of h". For the same reason, a poorly-chosen constitutive law can lead to stiffness in the model equations. Figure 4.1: (a) Subgrid topography without water. The soft-bed model assumes that sediments (not shown) cover the rough bed element, effectively smoothing the surface in contact with ice. (b) Subgrid topography with water. Black represents the areal coverage of saturated sediment (sediment itself not shown). Quantification of A* depends on subgrid topography. Assuming a soft-bed model, porous sediments would infill depressions, effectively smoothing an otherwise rough bed element. These sediments would create a more uniform contact with the overlying ice, resulting in a more uniform distribution of stress. Hydrological variables such as h* depend on sediment porosity and distribution, the latter being related to bed topography. Figure 4.1 illustrates a rough bed element with and without water. The porous sediment layer assumed to be deposited on the bed is not shown. Dark shading in Figure 4.1b represents the areal extent of saturated sediment for an arbitrary saturation datum. The volume of porewater required to pressurize the bed element is clearly related to subgrid topography. Using measurements of borehole length and ice-surface elevation, I attempt to char-acterize the topographic variability of the glacier bed. From this, it is possible to extract a gridscale value of h"c and derive a plausible form for pa(h"). Throughout the text, the word "subgrid" will be used to denote areas less than ~1600m2, corresponding to a square 40 m cell. Chapter 4. P A R A M E T E R I Z A T I O N OF SUB GRID P R O P E R T I E S 59 4.1.1 Borehole drilling observations: data selection and assumptions b " -1" " j y . "Kit • « » * Sfc-iV" • • * tit. • 50m Figure 4.2: Provenance and distribution of borehole depth data, (a) Trapridge Glacier bed (gray mesh) with ice skeleton, showing study area location, (b) Locations of selected boreholes drilled in the central study area from 1985-1997. The search radius used for this study r = 20 \ / 2 m is indicated by the circle. Extensive radar surveys have provided good spatial coverage of Trapridge Glacier, en-abling reconstructions of ice-surface and bed elevation as described in Chapter 2. Coarse sample spacing, however, precludes the analysis of subgrid topography. Thus, any sub-grid information must be acquired from borehole drilling or instrument studies which are densely spaced (Fig. 4.2). Boreholes, through which sensors access the glacier bed, have accurately surveyed surface coordinates. An estimate of borehole depth is made by measuring the length of hose required to drill the hole. Assuming a vertical profile, the bed elevation below the borehole can be determined by subtraction. The continuum framework used to describe subglacial hydrology in Chapter 3 assumes a pervasive subglacial sediment sheet. Hot-water drilling is known to excavate sediment from a hemispherical cavity beneath the borehole, disturbing the bed down to several decimeters [Blake et al, 1992]. In many places this would penetrate the entire sediment horizon. For lack of more detailed information, I assume that variations in measured borehole length, when corrected for surface elevation, can be interpreted as variations in bed elevation. By dealing with bed characteristics, which are more persistent than those of the ice (e.g., erosion rates are much smaller than ablation rates), data can be assembled from many different years. I utilize drilling observations from 1985-97 to characterize topographic variability of the bed, and extend this to relate p* and h*. Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S Sources of error 60 Figure 4.3: Results of borehole inclinometry, 1997-98. In all panels, results from lowering (solid lines) and raising (dashed lines) the inclinometer are shown together so that there are two traces per hole. Borehole surface coordinates are adjusted to (0,0). (a) Projection of borehole trajectories on the Easting axis, approximately aligned with ice flow, (b) Projection of borehole trajectories on the Northing axis, (c) Areal projection of borehole trajectories. Ice-flow direction is approximately from left to right, (d) Histogram of differences between borehole length and ice depth. Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 61 The assumption that boreholes are vertical is flawed. Horizontal displacements at the bed are often decimetres to metres compared to the position of the collar. For this reason, inclinometry is used to map true borehole trajectories. Figure 4.3a-c plots vertical and horizontal projections of the boreholes drilled and inclinometered from 1997-98. Incli-nometry data are available only for selected boreholes, so all depth measurements cannot be rigorously corrected. However, horizontal displacements are somewhat systematic (Fig. 4.3c), depending on the particulars of the drill tip and the experience of the drill-master. The Trapridge Glacier Drillmaster consistently bores holes that bend upglacier with depth, and whose ends are misaligned by 0.4-8.0 m. Horizontal displacement is not critical for this analysis so long as the variance of disparities between borehole length and ice depth is small. To proceed, I assume that there is a constant offset between borehole length and ice depth, such that it need not actually be calculated. Differences in borehole length are taken to represent differences in ice depth. From the limited self-consistent inclinometry data available (1997-98), discrepancies between borehole length and local ice depth are not strictly systematic. Figure 4.3d shows the distribution of the difference between these two quantities. Ideally, this distribution would approximate a delta function. While the distribution is sharply peaked, there is a non-negligible amount of energy in the tail. Ninety percent of the total energy is contained in the peak below 0.1m on the abcissa. Eighty percent of the energy lies in the interval 0.025-0.05 m. At best then, an error of 0.05 m could be assigned to the computed elevation anomalies. Considering the data quality, the significance of this analysis remains debatable. However, I present it as a method that could be used to derive p"(ha) with data of higher quality. An excellent dataset was collected in 1999 with improved drilling equipment, but must be augmented in order to repeat this analysis. 4.1.2 Statistical method and results Within an area about each borehole drilled to the glacier bed between 1985 and 1997, I compute bed-elevation differences between the central borehole and each of its neigh-bours. The radius of this neighbourhood is r = 20\/2 m = 28.3 m (Fig. 4.2b), the distance between the centre and corner of a 40 m pixel. For each data subset centred on a particular borehole, there are spatial trends arising from bed slope. These trends are removed by subtracting best-fit lines in the easting and northing directions. The resulting data represent topographic anomalies 6ZB, rather than absolute differences in elevation. These anomalies are assumed to represent the shape of the bed surface over which a mantle of sediment is draped (Fig. 4.4). The number of neighbouring boreholes in each subset varies from 0 to 229, and those with less than two neighbours are excluded. Of the remaining boreholes, each is double counted. These data pairs are not redundant because different spatial trends are removed from each subset. There are 552 eligible boreholes in all and 29,417 borehole-neighbour Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 62 Ice Figure 4.4: Conceptual diagram of porous sediments overlying a rough bed. Vertical scale is greatly ex-aggerated. The size and distribution of topographic anomalies (6ZQ, white arrow) is assumed to represent the thickness and distribution of the sediment layer (black arrow). If the layer were entirely saturated, the maximum water thickness h'max would correspond to the size of the black arrow scaled by the sediment porosity. pairs, resulting in 2 x 29,417 = 58,834 data. I use the entire data set simultaneously to quantify bed-topographic variability, which is assumed to be independent of position. To map out variability as a function of spatial location, one would retain a distinction between data subsets. Figure 4.5: Probability density and cumulative distribution functions of \6ZQ\. (a) Histogram of \6ZB\ (dots) with an analytical approximation (line, Eqn. (4-1)) multiplied by the number of data Nj. (b) Cumulative distribution function of\6z-a\ (Eqn. (4-5)). After removing spatial trends to distill the anomalies, I consider absolute values of SZ-Q for the remainder of this analysis. A histogram of all |<5ZB| (Fig. 4.5a) has a maximum near 0.1m and declines to approximately zero when |<5ZB| « 2 m. The data can be Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 63 analytically described by a modified Maxwell distribution written as f(\SzB\) = co [ \j~ 4/2 (10 |«5ZB| + c2)2 exp (10|^B| + c2)2 (4.1) for \SZB\ measured in metres, c\ = 0.0325, c2 = 7 (c2 distribution), and CQ a normalization factor such that 0 for a standard Maxwell oo / f(u) du = 1. (4.2) In this case, c0 = 1 + c2 \ / ¥ G X P H I ) - G r f (C2 = 1.51. (4.3) In this appUcation the choice of a Maxwell distribution is motivated by convenience, rather than a meaningful representation of physical processes. By satisfying Equation (4.2) and having f(\SzB\) > 0, (4.1) can be defined as a probability density function. The probability that a particular value of \SZB\ lies between \SZB\I and |<5ZB|2 is \6zb\2 P(\SZB\I < \SzB\ < \SzB\2)= J f(u) du. (4.4) I**BI To obtain the cumulative distribution function P(\8ZB\), (4.1) is integrated over 0 < |<5ZB| < co- Integrating by parts according to the formula / uv'dx = uv — J vu'dx with , the u co V ^ l M1 0 \SZB\ + c2) and v' = C l (10 \SzB\ + c2) exp [ - § (10 \SzB\ + c2f Chapter 4. P A R A M E T E R I Z A T I O N OF SUB GRID P R O P E R T I E S 64 result is P(\SZB\) = co y -y/^f (10 \SZB\ + c 2 ) exp (10 \Sz3\ + c2f + erf [ y ^ ( 1 0 \SzB\ + c 2 ) ] + c3. (4.5) The constant of integration is c3 = 1 — Co — —0.51 such that P(0) — 0 and ^(oo) = 1. Probabihties can be computed directly from Equation (4.5) as P {\8ZB\\ < \8z-&\ < | ^ B | 2 ) = P ( |£ZB| 2 ) — P (|^B|I)- This distribution is plotted in Figure 4.5b and suggests that almost all topographic anomalies are less than 1 m in amplitude, highlighting the importance of uncertainties in ice depth. 4.1.3 Hydrological implications The fact that Trapridge Glacier rests on a soft-sediment bed is well-established [e.g., Clarke, 1987b; Blake, 1992; Fischer and Clarke, 1994; Kavanaugh and Clarke, submit-ted]. In order to relate subgrid elevation anomalies (Fig. 4.5a) to the hydrology of the macroporous sediment horizon, I assume that variations in horizon thickness are dis-tributed as in Fig. 4.5a. This is equivalent to assuming that sediment infills cavities otherwise existing at the bed (Fig. 4.4). In reality, sediment patches and open cav-ities probably coexist, but the tendency would be toward deposition in the cavities. For simplicity, I assume a uniform water-surface elevation in a given bed element. The cu-mulative distribution function defined by Equation (4.5) can then be translated directly into a relationship between the areal fraction of saturated sediment Af and the maxi-mum water depth in a cell hsmax, given a horizon porosity n3 (Fig. 4.6). This value of n" represents a time-averaged gridscale porosity, aligned with the assumption that porosity adjusts to changes in water supply. Because the distribution of anomalies is fixed, each value of hsmax uniquely defines a volume of water. Expressed in terms of the anomaly \SZB\, hamax — n'\8z-s\, and Equation (4.5) can be rewritten as Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 65 a | / — — j - — H — Figure 4.6: (a) Relationship between maximum water thickness and anomaly amplitude |«5ZB| for a sediment porosity ns = 0.4. (b) Areal fraction of saturated sediment Af as a function of maximum water thickness. I choose n" = 0.4 for fully dilated sediment, such as that imaged beneath ice stream B, Antarctica [Blankenship, 1987]. The function Af(hsmax) (Fig. 4.6b) is identical to P(|<5zB|) (Fig. 4.5b) with the independent variable scaled by ns. Therefore, when hanax- = 0.4 m, nearly all subgrid obstacles are drowned and Af « 1. While hsmax is directly related to \SZB\, it is not a conserved quantity. A mapping is required from hsmax to h*, the areally-averaged thickness of water and one of four dependent variables in the multicomponent model. Suppose the complete assemblage of \SZ-B\ data represents variability in a single gridcell and that each datum applies to a unit area of the cell. For a particular value of h,' , h* h" is the sum of water volume contributions from each unit area with \SZB\ < ™£X, normalized by the total area. Sweeping over a range of A£,aa!, an empirical relationship ha(hmax) c a n k e generated as shown in Figure 4.7a. To proceed analytically, h" is calculated in terms of the original variable \SZB\ as I**b| n* f u f(u) du hs(\SzB\) = ^ (4.7) / /(«) du o where f(u) is as in (4.1) and from (4.2), the denominator is unity. The analytical result Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 66 0.16 0.12 w 0.08 •C 0.04 0J a ; y j / r \ —-A- —\—-\— 0.2 0.4 0.6 0.8 hS (m) max1 ' 0 0.2 0.4 0.6 0.8 hS (m) Figure 4.7: Mapping to h3. (a) Empirical relationship for h' (fcj^). (b) Analytical approximation (Eqn. (4.10)). of (4.7) is ha(\6zB\) = - ^ ^ v ^ c 7 ( ( 1 0 | / 5 z B | ) 2 + 10c2\SzB\ + l) (10 \SzB\ + c2f + c 2 v ^ e r f (IO^ZBI + C Z ) with c4 such that hs(0) = 0, yielding + c4n*, c4 c0 2 -y/2 ci exp C l Written in terms of h*max, (4.8) becomes c0 n° exp 7T ^ 1 0 + 10 c2 + 1 Th + c2 y/ir erf '£i I io + C 2 n + c4n*. (4.8) + Clc2^ erf [y^cT y ] ) = 14.4. • (4.9) (4.10) Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 67 Calculations of hs(h*nas^) based on Eqn. (4.10) (Fig. 4.7b) agree reasonably well with the empirical results (Fig. 4.7a). Relationships have now been established for Af(hsmax) (4.6) and hs{hsmax) (4.10), so Af and hs are connected parametrically. Neither function can be inverted for h*max, so an analytical form of Af(h") remains elusive. The final goal—to derive a physically-based rule for ps(h")—requires a connection between Af and p". I propose something of the form p° = piA^, where the ice-overburden pressure pi = pig hi and 77 > 0. This function is plotted for 77 =1-10 in Figure 4.8a, and the corresponding curves for p°{h") are shown in Figure 4.8b. If basal stress is unequally distributed over a bed element (e.g., concentrated on a few obstacles), one would expect that a significant areal fraction of the sediment layer must be saturated before it becomes pressurized. Furthermore, pressure should increase rapidly once the saturated horizon is as high as these obstacles, such that stress is redistributed onto the saturated matrix. I choose 77 = 10 to reflect these ideas, although any value of 77 > 5 would behave similarly. 0.04 0.08 0.12 0.16 hS(m) 0.04 0.08 0.12 0.16 Figure 4.8: Proposed relationships between p", Af, and h*. (a) p"(Af). Water pressure is expressed as a fraction of the ice-overburden pressure, (b) ps(/is). (c) ps(hs) for r] — 10 (solid line) shown with an analytical approximation (dashed line, Eqn. 4-11). Independent of the choice of 77, this analysis suggests that the critical value of sheet water thickness h"c — 0.147 m. This can be seen as the convergence point in Figure 4.8b 8 where ^ = 1. Because h"c represents water only, the corresponding areally-averaged ho-rizon thickness (water plus sediment) is h*/na — 0.368 m. This value agrees reasonably well with Blake's [1992] estimate of 0.30-0.50m for the depth of sediment deformation beneath Trapridge. Numerical modelling by Kavanaugh [2000] independently predicts deformation from 0.08-0.77 m depth with 0.37 m yielding the best fit to the data. Obser-vational evidence from numerous sensors suggests that the top 10-20 cm of this layer are hydrologically active. It is conceivable that water resides in troughs much longer than it does near the surface where it is most vulnerable to being purged. This "active" horizon probably differs from the bulk sediment layer in porosity and permeability, and can be Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 68 accommodated in part by allowing hydraulic conductivity to vary with h*. The derived relationship for p"(h") can be approximated by the analytical expression * • = * ( £ ) ' . ( « • » ) where the power | is chosen to give the best fit to the derived values. Figure 4.8c shows this function (dashed line) plotted with the empirical relationship with n = 10 (solid line). When h° = hsc, the porespace is saturated and p" = pi. Pressures in excess of flotation correspond to hs > hsc and are accompanied by dilatation of the sediment layer. According to (4.11), a pressure increase from pi to 1.5pi is achieved by an addition of 0.018 m of water, resulting in a saturated matrix expansion from 0.367 to 0.385 m. This corresponds to a change in porosity from 0.4 to 0.428. Making use of Freeze and Cherry's [1979] expression for aquifer compression, it is possible to estimate the compressibility ac of a layer governed by Equation (4.11) as b Spa v ' For a fractional aquifer-expansion 5b/b = 0.018 m/0.368 m = 0.049 and pressure change 5p = 1.5pi - pi = 0.5pigh = 269.9kPa (for g = 9.81ms -2, pi = 917kgm"3, and a nominal ice thickness of 60 m), the relationship in Eqn. (4.12) implies a compressibility of 1.8 x 10 - 7 P a - 1 . This values lies within the large range of compressibilities determined for a layer of this thickness by Stone [1993] from inversion of borehole connection-drainage data (a c « 0.5-5 x 10~8Pa_ 1) and slug test data (ac » 0.6-1 x 10~ 3Pa _ 1). 4.2 Hydraulic conductivity Transitions in the subglacial system between hydraulically-connected and unconnected states can signal a reorganization of the drainage network [Stone and Clarke, 1996]. In the process, lines of hydraulic communication are established or disrupted between different patches of the glacier bed [e.g., Murray and Clarke, 1995]. These patches can be as small as decimetres in size. Subgrid topography probably controls the patterns of water movement and storage on this scale prior to a gridscale connection, effectively determining the ease and speed of establishing such a connection. Figure 4.9 illustrates this point schematically. Two glacier-bed elements have comparable fractions covered with water (black), but the spatial distribution of water implies a difference in hydrology. Water forms isolated ponds in one case (Fig. 4.9a) and interconnected branches in the Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 69 Figure 4.9: Differences in subgrid water distribution for cells with comparable areal coverage. Black indicates water, (a) Unconnected element, (b) Connected element. other (Fig. 4.9b). A discrete model is blind to these potentially important differences in subgrid water distribution, making parameterizations essential. The effective hydraulic conductivity of a gridcell will vary depending on how water is distributed in that cell (Fig. 4.9), even if sediment properties are homogeneous. There-fore, the question of gridscale conductivity becomes one of connectivity. If a relationship can be established between the areally-averaged water thickness h" and connectivity, it may indicate how the gridscale magnitude of hydraulic conductivity changes with h'. For simplicity, I assume that conductivity is isotropic and that it varies between scalar minimum and maximum values that depend on factors other than hs, such as lithology. This problem is somewhat different than the classic problem of "upscaling" hydraulic conductivity in porous media. Upscaling refers to the process of determining a represen-tative value or distribution of values for a volume composed of heterogeneous materials [e.g., Rubin and Gomez-Hernandez, 1990; Durlofsky, 1991; Dykaar and Kitanidis, 1992; Indelman and Dag an, 1993; Sanchez-Vila et al., 1995]. Subgrid hydraulic communication depends on at least two attributes of glacier-bed topography: the amplitude of asperities and their spatial arrangement (Figure 4.10). In the diction of spectral analysis, the former attribute represents the total energy or power. The latter pertains to spatial correlation. Reducing topographic amplitude represents a decrease in total power (Fig. 4.10a) and results in a geometry more conducive to water flow. This process preserves the relative spatial arrangement of obstacles. Spatial correl-ation can be altered independently by filtering the distribution. Low-pass filtering leads to a more smoothly-varying surface, while preserving the original range of amplitudes (Fig. 4.10b). For topography, decreasing the total power or increasing spatial correlation should enhance hydraulic connectivity. As an experimental platform, I choose a 40 x 40 m area, representative of a single gridcell. This cell is divided into 100 x 100 square pixels each with an area of 0.16 m 2. A collection of 1002 elevation anomalies, with a mean equal to h'/n* — 0.368 m and a Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 70 10 20 30 40 50 60 70 80 90 100 Position (m) 10 20 30 40 50 60 70 80 90 100 Position (m) Figure 4.10: Profiles of randomly-generated topography altered in amplitude and spectral content, (a) Reduced amplitude topography (bold line) contains less total power compared to the original (fine line), (b) Low-pass filtering increases spatial correlation (bold line) compared to the original (fine line). distribution resembling that in Fig. 4.5a, is randomly arranged such that each pixel is assigned a value. This topography is superimposed on a bed slope of 6°. A layer of ice is added with a uniform surface slope of 7°, and mean thickness equal to 60 m, the Trapridge Glacier average. Upstream described in Chapter 2, is used as a proxy for connectivity. It is computed in the familiar way according to fluid potential gradients, and is summed over all pixels so that each distribution is characterized by a single quantity. Figure 4.11 shows three distributions presented in order of increasing total upstream area to illustrate its correlation with drainage network development. The prescribed increase in water pressure from 20-70% of flotation produces a drainage evolution from isolated rivulets (Fig. 4.11a) to a well-connected network (Fig. 4.11c). Eventually, at water pressures approaching flotation, total upstream area declines because flowpath tortuosity decreases. At that point, total upstream area ceases to be a good indicator of connectivity. Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 71 XUA = 1.45 x 10 3 m 2 LUA = 3.33x10 3m 2 EUA = 7.87x10 3m 2 Figure 4.11: Distributions of upstream area in order of increasing total upstream area ^2 UA. All are shown as log(UA). (a) j£ = 0.2. (b) j£ = 0.5. (c) j£ = 0.7. 4.2.1 Topographic autocorrelation Autocorrelation is a measure of self-similarity as a function of lag or displacement. Calcu-lating this property can be visualized as overlaying a copy of the field upon itself, moving the two relative to each other, and quantifying the difference between them at each po-sition. The autocorrelation of a signal with a white spectrum is zero except at zero lag [Robinson and Treitel, 1980], indicating that there is no significant spatial correlation in the distribution. Other fields have an autocorrelation maximum of finite width whose shape and dimensions are interpretable. Filter design To isolate the effects of correlation on subgrid hydrology, a suite of distributions is gen-erated by low-pass filtering the original field if>(x,y). One way to do this is to convolve ip(x,y) with a filter function g(x,y). The convolution is expressed in terms of lags x' and y' as [Brace-well, 1986] oo oo ip(x,y)**g(x,y)= J J ib(x',y')g(x - x',y - y')dx dy . (4.13) In practice it is often simpler to filter in Fourier space. It can be shown that the Fourier transform of the convolution is equivalent to the product of the transform of the filter and the transform of the field [Kanasewich, 1981]. Mathematically this can be written Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 72 as ky) = fc„) (4.14) where upper case variables represent the Fourier transforms of their lower case counter-parts and bold variables have been filtered. The result of applying the inverse Fourier transform to &(kx, ky) is to obtain the final filtered result, xb(x,y), identical to the con-volution tp(x,y) * *g(x,y). Both ip(x,y) and g(x,y) are digital functions with identical sampling intervals, so they can be written as tp(nx, ny) and g(nx, ny). In order to compute the Fourier transform, both functions are padded with zeros such that the number of rows and columns is doubled. The two-dimensional discrete Fourier transform of ip(nx,ny) is then computed as *(*.,*>)= E E ^ ^ e x p f - 2 " ^ - ^ " ^ ) x n , = l n , = l V i V * 7 (4.15) "*{ N," By carefully constructing a low-pass or band-pass filter, spatial correlation in the data can be adjusted without compromising total power. Such a filter must be designed to suppress ringing (constructive interference) at both ends of the spectrum. Designing effective filtering "windows" is the subject of an entire body of literature. For the present purposes, I choose a Gaussian filter for its functional simplicity. Centred at zero in the spectral domain, a Gaussian acts as a low-pass filter, while centred elsewhere it becomes a band-pass filter. The standard two-dimensional Gaussian used in statistics is [Bracewell, 1986] ^ ) = 2 ^ e X P ( - R - | J ) - ( 4 ' 1 6 ) I modify this function by replacing o~xo~y with an independently adjustable normalization factor fi2 to conserve total power in the filtered signal. This requirement is sufficient to preserve the maximum autocorrelation amplitude of the filtered field tb(x,y). The filter function can now be written 9 ^ y ) = ^ ™ v ( - h x - h y ) - ( 4 - 1 7 ) Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 73 Empirically-determined values of ft are tabulated with other filter characteristics in Table 4.1. Table 4.1: Characteristics of low-pass Gaussian filters. Maximum filter amplitude /(0, 0) is controlled by fi, and serves to compensate changes in filter width (o~x, cry) to preserve total filter power PF(k*,ky)-Filter number ft 5(0,0) PF(kx,ky) 1 0.225 0.225 0.261 2.34 3.00 2 0.275 0.275 0.337 1.40 3.06 3 0.325 0.325 0.393 1.03 3.02 4 0.375 0.375 0.434 0.845 3.01 5 0.425 0.425 0.467 0.730 3.00 6 0.475 0.475 0.496 0.647 2.99 7 0.525 0.525 0.523 0.582 2.97 8 0.575 0.575 0.549 0.528 2.95 9 0.625 0.625 0.574 0.483 2.92 10 0.725 0.725 0.620 0.414 2.91 11 0.825 0.825 0.662 0.363 2.91 Increasing the filter standard deviation in the spatial domain restricts its spectral width, or reduces its frequency pass-band. Therefore, short-wavelength topography can be progressively excluded by increasing ax and o~y for a low-pass filter. Due to the requirement that the filter and topography have identical sampling intervals, there is a lower limit placed oh filter width. Namely, it must be large enough that the fixed discretization interval sufficiently samples the function. At low sampling intervals, such as are employed here, the Gaussian filter approximates a trapezoid. Spatial representations of filters 1-11 are shown in Figure 4.12b. Power spectra of their transform pairs are shown in Figure 4.12c. These profiles are plotted for positive wave numbers along the diagonal kx — ky. The total one-sided power of each filter PF(kx,ky)i as recorded in Table 4.1, is the volume of a quadrant of the solid of rotation of these profiles about the kx = ky = 0 axis. Results The transformed filtered quantity &(kx, ky) can be computed according to (4.14) as *&(kx, ky) — ^(kx,ky)G(kx,ky), where G(kx,ky) is the transformed filter function and ^(kx, ky) is the transformed variable. The energy density or power spectrum P*(kx, ky) Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 74 Figure 4.12: Gaussian filter characteristics: normalization, shape, and power, (a) Normalization parameter Cl as a function of standard deviation a. (h) Filter shapes for a range of a. (c) Profiles of one-sided power spectra PF(k,.,kv) for kx — ky. of the filtered function is related to the square modulus of &(kx, ky) as * < * • • * > > - £ s r & ( 4 1 8 ) where NxAx, Ny Ay = 40 m are the dimension of the test element. To recover the filtered field ip(nx,ny), the inverse Fourier transform is applied to ^(kx, ky), = w w t i : *CM exp x /27ri(A: y -l)(n y -iy Results of this filtering process for filters 4, 7, and 11 are compared to unfiltered topog-raphy in Figure 4.13. As the filter width a is increased, long-wavelength topography is visibly enhanced. Below each image, the one-sided power spectrum for a profile kx = ky is shown. Note that power is progressively redistributed toward low wave numbers. Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 75 Wave number Wave number G = 0.825 0 I 20m I I 1 m -0.5 0 10 20 30 40 50 Wave number 10.5 m I-0.5 10 20 30 40 Wave number 50 Figure 4.13: Amplitude distributions and profiles of power spectra for filtered and unfiltered topography. Power is computed according to (4-18) and shown for kx = ky. Note the difference in vertical scales between power spectra, (a) Unfiltered. (b) Filter 4- (c) Filter 7. (d) Filter 11. Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 76 The normalized autocorrelation of rj)(x,y) is computed with the intent of assigning a characteristic spatial parameter to each new topographic field. The autocorrelation 4>A(X, y) is the convolution of tj}(x, y) with itself [Bracewell, 1986, p. 40], so the normalized autocorrelation is oo oo / / ^(x', y') tl>{x -x',y- y') dx' dy' M*, V) = oo • (4-20) / / *b{x',y')1>{x',y')dx'dy' In discrete form, £ £ ^ n>y) ^ _ n<, n y _ ^) n'x n'y <t>A{nx,ny) = ,—.x ,—7T • ( 4 - 2 1 ) The autocorrelation matrix has dimensions {2NX — 1) x (2Ny — 1) because it is calcu-lated over lags from -(Nx - l)Ax to (Nx - l)Ax and -{Ny - l)Ay to (7Yy - l)Ay. The maximum correlation coefficient is in the center of the matrix, corresponding to zero lag [Kanasewich, 1981]. As self-similarity in the topography increases, the cross-sectional area or width of this central maximum generally increases. A measure of this width W can be used to quantify the spatial correlation of each distribution. I use the full width at half maximum amplitude (FWHM) of a cross-section through the centre of the function. As in some other cases, this metric is preferable to the "equivalent width" (the width of a rectangle containing the total area of the function and having a height equal to the autocorrelation maximum) [Bracewell, 1986]. Values of W are compiled in Table 4.2, along with the total power of filtered and unfiltered fields. As expected, autocorrelation increases (as suggested by increases in W) with restriction of the filter bandwidth. Power remains constant within one percent, demonstrating that the filters were adequately designed. Effects of autocorrelation on upstream area Results of the upstream area calculations for selected tests in Table 4.2 are presented in Figure 4.14. Total upstream area is shown on a logarithmic scale as a function of the prescribed subglacial water pressure. All of the curves are alike in character, sug-gesting a similar progression from unconnected to connected states (see Fig. 4.11). The lowest curve (1) corresponds to the filtered field with the most high-frequency content. Successive curves represent increasingly long-wavelength topography with greater spatial correlation. Although it is blurred by the logarithmic scale, there appears to be a water Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 77 Table 4.2: Attributes of filtered topography. P*(kx,ky) is computed according to 4.18, W is the FWHM of the autocorrelation function, and total upstream area is reported for water pressure equal to 50% of flotation. Topographic Filter W Total upstream area (m2) field a (m) p'/pi = 0.5 0.225 5.03 xlO 3 0.82 9.08 xlO 2 V>2 0.275 5.03 xlO3 0.82 9.68 xlO 2 ^3 0.325 5.06 xlO3 0.90 1.00 xlO 3 *4 0.375 5.03 xlO3 0.96 1.14 xlO 3 *5 0.425 5.03 xlO3 1.08 1.41 xlO 3 V>6 0.475 5.03 xlO3 1.17 1.48 xlO 3 tb7 0.525 5.03 xlO 3 1.30 2.17 xlO 3 *l>8 0.575 5.03 xlO 3 1.39 2.50 xlO 3 ^9 0.625 5.03 xlO 3 1.53 3.33 xlO 3 1^0 0.725 5.03 xlO3 1.64 4.56 xlO 3 0.825 5.03 xlO 3 2.04 6.41 xlO 3 pressure threshold for rapid accumulation of upstream area. This threshold is most pro-nounced for the lowermost curves, and occurs at lower water pressure as the correlation length of the topography increases. — 105[ , , , . . . . . , CM Figure 4.14: Selected results from spatial-correlation tests (Table 4.2). Total upstream area is shown as a function of subglacial water pressure. Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 78 When the curves attain their maximum value the drainage configuration is responding almost exclusively to ice-surface slope. All curves attain the same maximum value which is reassuring since the ice surface is identical in each case. The position of this maxi-mum migrates to lower water pressures with progressive filtering. Total upstream area differs significantly between tests at low water pressure, because spatial correlation in the topography affords a predisposition to hydraulic communication. While connections may be entirely absent for p"/pi = 0.1, more contiguous ponding occurs for distributions 8-11 than for the others. As mentioned earlier, total upstream area eventually decreases when the tortuosity of individual waterways is reduced and the implied direction of flow becomes aligned with surface slope. This transition occurs between 70-90% of flotation (Fig. 4.14) and represents a breakdown in the analysis. Were one to identify a total up-stream area threshold for gridscale connectivity, it would remain challenging to associate that with a single value of p"/pi given its sensitivity to spatial correlation. 4.2.2 Topographic power Method Table 4.3: Attributes of amplitude-adjusted topography. PyR(kx,ky) is the total power contained in the distribution, and total upstream area is reported for water pressure equal to 50% of ice-flotation. Topographic Amplitude PyR(kx, ky) Relative power Total upstream area (m2) field scaling (R) R2 P*/PL = 0.5 1 1.53 xlO 4 1 0.388 xlO 3 0.9 1.24 xlO 4 0.81 0.404 xlO 3 0.8 9.78 xlO 3 0.64 0.424 xlO 3 ipR, 0.7 7.50 xlO 3 0.49 0.476 xlO3 i>R6 0.6 5.50 xlO3 0.36 0.572 xlO 3 TpRt, 0.5 3.81 xlO 3 0.25 0.780 xlO 3 i>R* 0.4 2.45 xlO 3 0.16 1.37 xlO3 0.3 1.38 xlO 3 0.09 4.56 xlO 3 i>R2 0.2 6.13 xlO2 0.04 10.6 xlO 3 For this test I generate a 100 x 100 array of normally-distributed elevations, and scale the amplitudes to create realizations of variable total power. These values of power bracket those in the previous test (Table 4.2), so the disposition to hydraulic connectivity can be evaluated for bed patches spanning the roughness characteristics of Trapridge Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 79 Glacier. For an amplitude reduction factor R, the power spectrum of is P<j>R(kx, ky) — NxAx NyAy' (4.22) expressed in terms of the Fourier-transformed variable ^(kx, ky). Power scales as R2. The normalized autocorrelation is the Fourier transform of the power spectrum [Bracewell, 1986], so ^(0,0) could equivalently be used to quantify the relative total power. Table 4.3 records the imposed amplitude scaling and the resulting change in total power for nine different fields. Results: effects of power on upstream area I Figure 4.15: Results of variable-power tests (Table 4-3)- Total upstream area is shown as a function of subglacial water pressure. Upstream area calculations are repeated for the fields listed in Table 4.3 and results are plotted in Figure 4.15. As expected, this family of curves shows that reducing total power promotes the development of a drainage network. The distribution with the largest range of amplitudes (ipR^) has the lowest total upstream area for all values of p"/pi- In form and magnitude, these curves bear a qualitative resemblance to those in Figure 4.14. However, the transition to high values of upstream area is more clearly a threshold Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 80 phenomenon in this case. Most of the curves have a common maximum, as well as minimum, suggesting that amplitude reduction is less important than spatial correlation for establishing connections at low water pressure. Similarly, higher values oip'/pi are required (1.0-1.1) than in the previous experiment before subglacial drainage aligns with the surface slope, indicated by the downturn in total upstream area. 4.2.3 Derived relationship for K"(h') To relate the variables of interest, I assume that total upstream area is a reasonable gauge of subgrid connectivity, and that connectivity can be used as a proxy for the effective cell-averaged conductivity. Given these assumptions, Figures 4.14 and 4.15 suggest that the gridscale variation of K* with water pressure is highly nonlinear. Specifically, K" should have the same logarithmic functional form as total upstream area. If the cell acts like a hydraulic resistor of fixed length, its resistance decreases as more area becomes hydraulically-activated. The gridscale value of conductivity is inversely proportional to resistance and should therefore scale with the active area. A constitutive relation of the following form satisfies this requirement: log(A-) = - {\og{K'max) - log ( /CV B ) ) t an - 1 where ka modulates the abruptness of the transition and kb determines its position. Figure 4.16 plots (4.23) for a range of ka and and compares this calculation for ka — 15 and kb = 0.85 to the derived results. Equation (4.11) is used to relate h' and p*. Values of K^in = 4 x 10 _ 4 ms _ 1 and K^ax = 8 x 10 _ 3 ms - 1 are chosen arbitrarily for the purposes of this illustration. For specific model applications, K^ax and K^in would be tailored to the study site. The bold line in Figure 4.16c represents my preferred parameterization of K* as a function of subglacial water pressure. Its form mimics the general character of the upstream area variations obtained from both experiments (shaded) and approximately bisects their region of overlap (darkly shaded). An isolated analysis of total topographic power would recommend a later and steeper transition from minimum to maximum conductivity (between curves 5 and 6 in Fig. 4.15), introducing a greater hindrance to drainage network development. Excluding this analysis would imply slightly less nonlinearity in K*. While the choice of K"(h3) is not quantitatively tied to terrain analysis, its suggested form is unambiguous. Had the upstream area relationships been disparate between tests, Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 81 Figure 4.16: Proposed relationship for K"(h$) as described by Eqn. (4-23). K^ax = 8x 10 - 3 m s and Kmin = 4 x 1 0 - 4 m s - 1 are arbitrary, (a) Equation (4-23) with transition steepness adjusted. k\, = 0.9 and ka =5-60. (b) Equation (4-23) with transition onset adjusted. ka = 20 and kb =0.8-0.975. (c) Morphological comparison between Equation (4-23) and upstream area results. Results from Fig. 4-H (fine dashed lines) and Fig. 4-15 (fine solid lines) are shown together and the domain they encompass is shaded. The region of overlap is darkly shaded. Ordinate values on the left apply to upstream area. Proposed form of K"(hs) (ka = 15, fcf, = 0.85^  is shown in bold. Ordinate values on the right apply to hydraulic conductivity. some more rigorous approach to combining them would be warranted. Given the dearth of detailed spatial data for most of Trapridge Glacier, this analysis must rely in part on studies of synthetic topography. Hence the results are somewhat generic. Optimistically however, they may help account for glacier-wide variability not portrayed in data collected from the study site. 4.3 Summary Parametric relationships for subglacial water pressure and hydraulic conductivity can be derived using a few basic constructs of statistical analysis and signal processing. Some prior consideration of subgrid attributes (elevation) and processes (water flow over a rough glacier-bed) directed these studies toward the effects of variable topography. In principle, densely-spaced borehole-length data can provide a valuable constraint on topographic variability. If the physical picture of soft-bed hydrology as proposed is correct, these data provide a reasonable basis for relating subglacial water pressure p' to water-sheet thickness h*. Equation 4.11, describing an exponential increase in water Chapter 4. P A R A M E T E R I Z A T I O N OF SUBGRID P R O P E R T I E S 82 pressure with h", is the result of this exercise. From the same data, a critical water-thickness of hi = 0.147 m was extracted. While uncertainty in the data is too large to make significant claims hinging on the nature of this relationship or the precise value of hsc, making use of this information is deemed worthwhile. Hopes of obtaining a realistic expression for hydraulic conductivity as a function of h" are somewhat more tenuous due to the lack of available constraints. Assuming the most important factors controlling connectivity on small scales have been identified, namely topographic amplitude and spatial correlation, the results robustly suggest a smooth step transition between unconnected and connected states. Spatial correlation appears to be slightly more influential during the early stages of drainage evolution. Both parameters play out equally when the bed is subjected to water pressures comparable to the hydrostatic value. Interestingly, both parts of this study suggest nonlinearity in the subgrid hydraulic response to changes in water volume. Equations (4.11) and (4.23) provide a means of objectively incorporating this into the model. Chapter 5 N U M E R I C A L M E T H O D As presented in Chapter 3, the mathematical description of glacier hydrology comprises four analogous sets of equations, one per system. Therefore the numerical exposition of each set of equations is conveniently similar. I begin this chapter by describing the model grid, leading to a generalized numerical formulation of the problem using standard finite-difference techniques. A Newton-Krylov iterative procedure is used to obtain sim-ultaneous solutions to this set of nonlinear coupled equations. 5.1 Model domain and coordinate system Figure 5.1: Coordinate system and discretization of the model domain, (a) Orientation of coordinate axes relative to ice-flow direction. Grid alignment with geographical directions as shown is specific to Trapridge Glacier, (b) Multicomponent discretization. Gridcell (i,j) is horizontally aligned between systems. Governing equations do not apply in shaded areas, (c) Plan view of the staggered grid. Dots represent cell-centred nodes and diamonds mark the staggered interfaces. Diamonds are replaced by labels surrounding node (i,j). Designed with glacier-scale problems in mind, the model uses a rectangular grid ref-erenced by Cartesian coordinates. Ideally, the entire body of ice is encompassed by the grid, leaving the boundary ice-free. For experiments with time-dependent ice geometry, the grid should exceed the maximum ice-extent. Although this is not a requirement, 83 Chapter 5. N U M E R I C A L M E T H O D 84 it is often realizable for ice caps and cirque glaciers and it simplifies the implementa-tion of boundary conditions. Individual outlet glaciers fed by larger ice-complexes (e.g., Trapridge Glacier) present a problem due to sheer size or lack of information pertaining to the ice complex. When possible, the grid is oriented with one axis parallel to the primary direction of ice-flow. This helps minimize the negative effects of bias arising from grid orientation. I use a right-handed coordinate system with x 6 [0, Lx] roughly parallel to this longi-tudinal axis and positive downglacier, and y £ [0, Ly] positive toward glacier-left as in Figure 5.1a. The vertical coordinate z is positive upward with its origin at sea level. For Trapridge Glacier, x and y are aligned with Easting (E) and Northing (N) axes, respect-ively (Fig. 5.1a), with the origin (x,y) = (0,0) in the southwest corner at (534003.017E, 6787131.585N). The grid extends to (536563.017E, 6788251.585N), so Lx = 2560 m and Ly = 1120 m. Only the eastern boundary is completely ice-free. 5.1.1 Discretization A suitable finite-difference mesh is generated by partitioning the domain into nx x ny rectangular cells, with nodes positioned at cell-centres. For a uniform mesh, the dimen-sions of each cell are Ax = ^ by Ay = -=^ -. The working cellsize for Trapridge Glacier is Aa; = Ay = 40 m, resulting in nx = 64, ny = 28, and nx x ny = 1792 nodes per system. This includes a padded border, two pixels in width, to accommodate surface runoff boundary conditions. Each of the four model components is discretized identically (Fig. 5.1b), producing a system of size naya = 4 x 1792 = 7168. In practice, the number of non-trivial simul-taneous equations is less than 7168 because englacial and subglacial transport equations are not solved in ice-free cells (e.g., shaded regions in Fig. 5.1b). A discretized swatch of each layer is shown in the same image to illustrate gridcell alignment between systems. Each node in the system shares its horizontal coordinates with up to three other nodes, distinguished by their vertical offsets. Discrete variables are identified by spatial indices and system superscripts (r, e, s, a) as in Chapter 3. Index i € [l,nx] represents the ^-coordinate and j £ [1, wy] the y-coordinate (Fig. 5.1b). I employ a staggered grid as described by Patanhar [1980] such that, in general, scalar fields are computed at cell centres and vector fields are computed across interfaces (Fig. 5.1c). Tables 5.1 and 5.2 group input/output fields and internal functions according to positional status as centred or staggered. This staggered scheme obviates the difficulty of satisfying continuity with unrealistic velocity (or flux) distrib-utions [Patankar, 1980], because gradients of scalars apply to interfaces and divergence of vectors applies to nodes. With this particular system of equations, a staggered grid is prerequisite to numerical stability. Chapter 5. N U M E R I C A L M E T H O D 85 Each vector quantity is represented by two variables in the model, one for each or-thogonal direction. Variables aligned with x are computed at interfaces (i + hij) and 1 (i — iy, j), producing arrays of size (nx + 1) x ny. Similarly, variables aligned with y are - . 1 . 1 computed at (i, j + )^ and (i, j — )^ for arrays of size nx x (ny + 1). Cell-centred variables have nx x ny array elements. Table 5.1: Grid-position classification of input, output, and initial fields as centred or staggered. Centred Staggered: hydraulic conductivity Input fields Ice-surface elevation zs(x,y) Bed elevation zs(x,y) Air temperature T(x,y,t) Precipitation P(x, y, t) Geothermal heat flux Qa(x,y) Sliding velocity vx(x, y, t), vy(x, y, t) Englacial void volume Vr(x,y) Volume of englacial void types jk(x,y) Number density of void types Nk(x,y) Till cap thickness dt(x,y) Aquifer porosity na(x,y) Aquifer thickness da(x,y) Aquifer compressibility aa(x,y) Runoff: Krx(x, y), Ky(x, y) Englacial: K^(x,y), Key(x,y) Subglacial: K*xmax(x, y), K°max(x,y), KXmin(X,y), Kymin(X,y) Aquifer: K^(x,y), K^(x,y) Initial fields Runoff depth hr(x, y, 0) Englacial water thickness he(x,y,0) Subglacial water thickness/pressure h*(x,y,0) or p*(x,y,0) Water thickness/pressure in aquifer ha{x,y,0) orpa(x,t/,0) Output fields Surface melt rate M(x,y,t) Snow depth Dsnow(x,y,t) Superimposed ice thickness Dice(x,y,t) Runoff depth hr(x, y, t) Englacial water thickness he(x,y,t) Subglacial water thickness h"(x,y,t) Water thickness in aquifer ha(x,y,t) Chapter 5. N U M E R I C A L M E T H O D 86 Table 5.2: Grid-position classification of internal fields and functions as centred or staggered. Centred Staggered Internal fields and functions Ice thickness hi(x,y) Fluid density in aquifer pa(*,y,t) Fluid pressure pr(x,y,t) Pe{x,y,t) p3(x,y,t) pa(x,y,t) Fluid potential i(>r(x,y,t) i>e(x,y,t) 4>a(x,y,t) Water exchange rate <f>r:e(x,y,t) <t>T-a{*,y,t) Flux divergence y,t) + 8y ^ ' y,t) »,<) + y,t) y,t) + 'dy { x y,t) y,t) + y,t) Englacial transmissivity T;(x,y),T;{x,y) Subglacial conductivity K'(x,y,t), K'(x,y,t) Water flux Qli*,y,t), Qry(x,y,t) Ql(x,y,t), Qey(x,y,t) Qsx(x,y,t), Qy(x,y,t) QZ(x,y,t), Qay(x,y,t) Potential gradient dip* Chapter 5. N U M E R I C A L M E T H O D 87 5.2 Generalized form of discrete equations The structural similarity of equations governing runoff, englacial, subglacial, and sub-surface model components permits a concise and generalized presentation of the discrete equations. Appendix C contains the discrete equations written out specifically for each component. I use centred-difference approximations for spatial derivatives and forward differences in time. This leads to a discretization error of 0[A£ + (Aa:)2] [Anderson et al, 1984]. Employing the Crank-Nicolson method to partition dependent variables between implicit (future time) and explicit (present time) values affords a reduction in the time-discretization error from 0[At] to 0[(At)2] [Fletcher, 1991]. This partitioning is applied to all but the surface runoff system which is solved explicitly. Integer m is used as a time index such that hm represents the variable h (a generalized water thickness) at time t and hm+1 represents h at t + A i . Define a new variable h* = 0hm+1 + (1 — 0)hm with which to compute /^ -dependent functions in the water balance equation. The range of 9 is 0 < 0 < 1. In the limit that 6 = 0, the formulation becomes fully explicit, and for 8 = 1, fully implicit. Crank-Nicolson stipulates that 8 = 0.5, in which case h* = hm+2. An explicit scheme requires very few iterations to converge, but is prone to large errors and hence instability, if time steps are not sufficiently small. Implicit schemes are typically more stable but less accurate, and require more iterations to obtain a solution. The Crank-Nicolson method advantageously combines these schemes. Only first-order discrete derivatives are required for the equations presented in Chap-ters 3-4. Fortunately, second-order accuracy is obtained by evaluating discrete derivatives of interface variables at nodes. Therefore, blending derivatives of centred and staggered variables does not produce order-of-accuracy inconsistencies. Dirichlet boundary condi-tions make consistency between inner and outer nodes straightforward by allowing an identical formulation of the discrete derivatives. For a generalized cell-centred scalar variable C(x>3/)> the discrete Cartesian gradient Evaluated at the four interfaces defining cell (Fig. 5.1c), the gradient fields are dx (t+i-,j) dx 1 Ax (C(i,j) - C(i-u)) (5.1a) and (5.1b) Chapter 5. N U M E R I C A L M E T H O D 88 For a vector quantity u(x,y), separated into component variables ux(x,y) and uy(x,y), the divergence can be approximated at the (i,j)th node as duk dxk Approximations of the form (5.1) and (5.2) are employed in the discretization of model governing equations that follows. Balance equations written in Chapter 3 have the general form m + dx^ = ^ ( 5 - 3 ) where h is an areally-averaged water thickness, Qu is water flux, and (f> represents sources and sinks. The balance equation for subsurface (aquifer) water is a notable exception, with additional terms arising from the time-dependence of fluid density. However, the same principles for discretizing apply and details are provided in Appendix C. Approx-imating the time derivative for the (i,j)th node as Oh 8t the discrete form of (5.3) is Xt (hTiS " + ~Kx (c?«(*+^-) ~ Q'V-hJ)) + ~ty {Qy^+\) " Qyi^-\)) = (5.5) Variables with asterisks are computed using h*, the time-blended value of water thickness. Multiplying both sides of (5.5) by At, (5.6) Chapter 5. N U M E R I C A L M E T H O D 89 Source/sink terms <t>\ij) typically reference a cell-centred variable in the local system o r (^* i)) a n < ^ o n e fr°m a n adjacent system directly above or below. The generalized water flux is written Qk = ~^,klg ~^~) where subscripts k and I are used for Einstein's notation to avoid confusion with discrete coordinates i and j. Fluid potential has the form i\> = p + pw g z, where p is fluid pressure and z is an elevation relative to some datum. As noted in Chapter 3, K\,i is simplified to K Ski for all applications presented here. In this case, flux can be expressed as Kh di> Pwg dxk' (5.7) where hydraulic conductivity is a vector that can be discretized in the familiar way. In discrete notation, the fluxes contributing to cell are Pw9 K* x(i-j,j) Pwg (5.8a) and O* K* Pwg K y{i<i~2) Pwg (5.8b) with ^(ij) — P(i,j) + Pw gz(i,j)- Asterisks are applied to hydraulic conductivity for the case that it is a calculated function of h*, as for the subglacial system. The first set of square brackets in each expression in (5.8) contains the averaged representation of h*, and the second set encloses the discrete fluid potential gradient centred on the appropriate interface. The final generalized balance equation is obtained by substituting (5.8) into (5.6) and Chapter 5. N U M E R I C A L M E T H O D 90 collecting constants: hTiS ~ kTi,J) + Jp~^g {Kxf { _ K<i+\J) (k(i+hj) + h(i,j)) (^ (i+U) ~ = ^ % ) -All of the actual balance equations differ from (5.9) in the number of source terms. In addition, the englacial water balance substitutes TX^+LJ) for K*^i+^ ^ ^i^li+ij) (^*,j))> and so forth. Appendix C contains the balance equations written out specifically for each system. 5.2.1 Boundary and initial conditions Dirichlet boundary conditions are applied to each of the four systems. Surface runoff is collected in a trough at the edge of the model domain that is two cells in width and depressed ten to several tens of metres, depending on the scale of the topography. The amount of this depression is a sensitive issue, as it determines the fluid potential gradient that drives water out of the domain. The trough must be sufficiently deep to trap the water it encounters, but not so deep that it acts as a potential vacuum. An impermeable boundary is imposed in the trough by forcing ^ = Kr^_^ = Kr^ 2^ = Ky(.n _r) = 0, so that an arbitrary boundary condition on hT^ .j, hT(nxt:y ^ (:,i)> and h\-n^) does not interfere with water collection. At each timestep, the trough is "emptied" by resetting hr^2 = hr^nx_x = hr^. ^ = h^. ny_^ = 0 after recording the volume of trapped water. Boundary conditions for the englacial and subglacial transport systems are imposed in ice-free cells bordering the glacier, wherever the ice does not transgress the edge of the model domain. Where hj^j) — 0, = h'^ = 0. Elsewhere h'^ and hs{^ may take on nonzero values. These conditions can be expressed equivalently in terms of water pressures p^ ^ and ^. The code does not presently accommodate Neumann conditions. Chapter 5. N U M E R I C A L M E T H O D 91 Known values of ,y h"nx h*. ^, and A" or water pressure at the same locations, are prescribed as conditions on the aquifer. In general, boundary values are nonzero. For certain applications, an impermeable barrier is imposed along part of the domain perimeter. This usually applies to upstream ice-occupied cells. Initial conditions can be specified in terms of the dependent variables hr, he, hs, and ha or in terms of the respective water pressures. For time-dependent investigations, starting fields are provided by equilibrium model runs with the appropriate parameters. 5.3 Solution Algorithm The coupled system of equations defined by (C.3), (C.10), (C.18), and (C.23) requires a method for time-stepping and a solution at each time interval. Fig. 5.2 presents the basic numerical algorithm which comprises four nested loops. Working inward these are the: (1) ablation routine, (2) dynamics loop, (3) Newton-Raphson iteration, and (4) Krylov solution to the linear system. M a i n Figure 5.2: Flow diagram of main program. Chapter 5. N U M E R I C A L M E T H O D 92 The first two items are time loops, with the ablation timestep (minutes to hours) assumed to exceed the dynamics timestep (seconds to minutes). The last two items are nested iterations used to obtain a solution at each time interval. Both are contained within the step labelled "Solve system: hr ,he,h",ha" in Figure 5.2, and the Newton it-eration is illustrated separately in Figure 5.4. Before entering the outermost loop, all parameters and input fields including the problem geometry (Table 5.1) are loaded. In practice, many of the input fields such as geothermal heat flux QG and aquifer compress-ibility aa are treated as spatially-uniform and constant in time. 5.3.1 Ablation routine Ablation routine Read variable / [Hock, 1999] | Read precipitation| \Loopover i| Return to min\ Figure 5.3: Flow diagram of ablation routine. Along with precipitation, ablation constitutes the surface forcing and is independent of other internal variables (e.g., hT, he). The ablation routine requires timeseries of air temperature T and precipitation P as inputs, and delivers surface melt rate M, snow depth Dsnow, and superimposed ice thickness Dice as outputs (Table 5.1). A typical timestep of one hour, adopted from Hock [1999], establishes the ablation routine as the Chapter 5. N U M E R I C A L M E T H O D 93 outermost loop. Figure 5.3 presents this routine schematically and includes the possibility of a prescribed, rather than calculated, source rate M. This option is useful for equilib-rium tests and for experiments with synthetic topography (see Ch. 6). Temperature-index calculations are performed inside the box labelled "Calculate M, Danow, Dice" and are detailed with a separate flow diagram in Appendix D. 5.3.2 Dynamics loop The inner sequence in Figure 5.2 comprises the dynamics loop and main body of the numerical model. This cycle is executed once every timestep, where the timestep is per-mitted to vary during an integration. Imbalances in the initial conditions often lead to predicted rates of water exchange that cannot be stably accommodated using a fixed stepsize. For most applications, an exponential increase in the timestep to a prescribed maximum value circumnavigates this problem. Other sources of strife affecting con-vergence, including high-frequency forcing, are handled by an encoded option for auto-matic timestep adjustment. Applied to linear diffusion (the heat equation), Crank-Nicolson is unconditionally sta-ble [Anderson et al, 1984], providing no a priori suggestion for a maximum timestep. In practice, smaller timesteps than necessary for convergence are often warranted to reduce the number of Newton iterations, and hence the total model run-time. For Trapridge applications the dynamical timestep ranges from 30-600 s. In addition to factors such as hydraulic conductivity and intercomponent coupling strength, topography is an import-ant consideration in choosing a maximum timestep. Specifically, complex bed topography often requires a timestep reduction. 5.3.3 Linearization: Newton-Raphson iteration The most common approach to nonlinear problems is to linearize, so that one of the host of methods developed for linear systems can be applied. Newton's method, an iterative application of the Taylor series approximation, provides the most efficient means of accomplishing the linearization [personal communication from E. Haber). This method, combined with a Krylov subspace solver for the linear system, is used here and comes under the umbrella of Newton-Krylov procedures. At each timestep, one seeks a solution to the equation F(T) = A(T)T = 0, (5.10) where F(T) represents a system of equations in the vector T, generated by applying the Chapter 5. N U M E R I C A L M E T H O D 94 nonlinear matrix operator A(T) to the variable. At a particular iteration n, this equation is not perfectly satisfied with the vector of solution estimates T n , thus F(Tn) = A(Tn)Tn = R(Tn), (5.11) where R(Tn) is the vector residual. Expand F(Tn+1) in a Taylor series about the variable T n and truncate the series after the first-order (linear) term such that F(Tn+1) sa F(Tn) + ^ „ ^ ^ ( T n + 1 - T n ) . (5.12) The derivative dF(Tn)/8Tn+1 is the Jacobian matrix Jn which quantifies the sensitivity of the function to changes in the solution estimate. Assuming that F(Tn+1) — F(Tn) is exactly the residual Rn, Rn = jn ( T n + 1 _ ynj (5.13a) or JnATn = Rn, (5.13b) where A T n = T n + 1 — T n . In terms of the updated solution vector, T n + 1 = T n + (jnj-lRn (5.14) This is the Newton-Raphson iteration, which continues until R(Tn) is sufficiently small and T" is declared an acceptable solution. For a single-component system (e.g., isolated subglacial horizon), T is a vectorized representation of the unknown array h^^ and R is the result of substituting T into the vectorized balance equation (5.9) with terms rearranged to one side. The Jacobian is the derivative of the residual with respect to the implicit unknown h-™^y La this case the residual vector is of length nxny and the Jacobian is nxny x nxny but sparse. I calculate Jacobian entries as analytical derivatives of the balance equations, although it is also possible to approximate the Jacobian using simple variable perturbations. Chapter 5. N U M E R I C A L M E T H O D 95 For the multicomponent system, I obtain simultaneous solutions to the balance equa-tions by including them together in the linearization just described. Partitioning of the solution vector allows unknowns to be stacked such that T ( l :nxnv) rh™+1 ( l : n x , l-.ny) ^ (nxny-\-X : 2nxny) ( l : n x l:ny) (2nxny-\-l : 3nxTij,) (l:nx l:ny) f-(&nxny+l:4nxny) — ah™+1 ( l : n x ,l:ny) where the system superscripts r, e, s, and a precede the variables for legibility. Identical partitioning is applied to the residual vector so that the appropriate balance equation is represented in each position. Thus, the first nxny terms in T represent the unknown runoff depth rhm+1 and the first nxny terms in R represent Eqn. (C.3). By quadrupling of the number of unknowns, T and R grow to length n3ys = Anxny and J is nsy3 x nsys. Because the systems are interdependent via water exchange, each finite-difference stencil contains seven points, five within the horizontal plane and two outside it. For example, h'(iJ) has dependencies on h'{i+1J), hs{i_1:j), A ^ j + 1 ) , h»{iJ_iy hfa, and ha{ij). This gives rise to a Jacobian matrix with a maximum of seven nonzero bands. Two of these are separated from the diagonal by an integer multiple of nxny columns, representing dependencies on other systems. Jacobian elements total nays x nsys 5.14 x 10r in a coupled Trapridge Glacier simulation. With a maximum of seven nonzero bands (less than 7 x nsys = 50,176 nonzero elements), Jacobian sparsity lends itself to indexed vector storage. I employ the method of Press [1992] to load the Jacobian directly as a single vector, with an accompanying index vector of identical size. Terminating the cycle I use a reduction of the L 2 norm of the residual vector (misfit) below a predefined tolerance to halt the iteration. For the coupled system, a misfit must be calculated for each system and independent criteria satisfied. Segmentation of the residual enables individual convergence criteria to be imposed. For example, convergence of the englacial system Chapter 5. N U M E R I C A L M E T H O D 96 occurs when 2nxriy Y , R 2 < i o / e > (5-16) k=nxTty-\-l where the L 2 norm depends on system size but is accommodated in the calculation of tole. Tolerances are designed to control the total solution error for each system at the end of a complete integration. They are computed as (5.17) where (tf — U) is the total integration time, overbars represent spatial averages, and hr, he, hs, and ha are characteristic variable values. The final mean solution error for any variable amounts to ~0.5% of its characteristic value. For a 14-day Trapridge simulation with At = 120 s, tolr = 1.8 x 10"5m, tole = 1.2 x 10- l om, tol* = 2.6 x 10-6m, and tola = 2.1 x 10 - 5 m of water. Upper bounds are placed on these parameters to guard against unreasonably high tolerances resulting from large timesteps or short integration times. Because they depend on timestep, tolerances must be recomputed for each new value of At (Fig. 5.2). Newton's method demonstrates quadratic convergence when the system is close to the solution, such that the Taylor series approximation is valid. For a well-conditioned system, convergence typically occurs in 2-5 iterations. The bulk of computational time is spent within the Newton iteration calculating the Jacobian and solving Equation (5.13b). Convergence is hindered when vigorous water exchange occurs between systems and when dependent variables approach zero. In the first case, numerical oscillations occa-sionally arise in the form of water being pushed back and forth between adjacent layers. A timestep reduction usually alleviates this problem. In the second case, there is a risk of obtaining negative values of water thickness as water becomes scarce. No attempt has been made to enforce positivity in the selection of a solver, as negative solutions are Chapter 5. N U M E R I C A L M E T H O D 97 relatively rare. However, preventative measures are taken to avoid this situation. By calculating where the demand for water is likely to exceed supply, the equations govern-ing outgoing flux from these cells can be effectively modified before the system is solved. While this active interference is successful in discouraging negative solutions, it reliably stifles convergence. This becomes especially problematic in areas where elevation gradi-ents make significant contributions to the driving potential, because these gradients do not diminish with waning water supply. Solver Initialize variables | Construct residual vector fl£ Set failure flag] Figure 5.4: Flow diagram of solver including Newton-Raphson iteration. Figure 5.4 summarizes the practical implementation of Newton's method within the solver. At each timestep, the most recent solution is used as the initial solution guess, equivalent to setting A T n = 1 = 0. Improvements in convergence are negligible for nonzero values of A T " = 1 . The residual vector and individual misfits are calculated before the iteration begins in order to avoid computing the Jacobian if the initial guess provides a satisfactory solution to the equations. If the misfit for any system exceeds its tolerance, a Newton iteration is executed. Jacobian terms are computed and loaded piece-wise into a vector. The linear system Jn A T " = Rn is then solved for the solution step vector A T n . Variables are updated with this result, and residual and misfits recalculated. Iteration continues until the misfit reaches a prescribed tolerance level or until a fixed number of iterations is exceeded. In the latter case, failure is noted and the integration proceeds. Chapter 5. N U M E R I C A L M E T H O D 98 5.3.4 Solving the linear system As written in (5.14), a direct determination of T n + 1 requires that the Jacobian be in-verted. This is an expensive computation. The alternative is to solve (5.13b), which is already in the form handled by Krylov subspace methods [Saad, 1996]. Among these methods are conjugate gradient (CG), bi-conjugate gradient (Bi-CG), preconditioned bi-conjugate gradient (PBCG), and generalized minimum residual (GMRES). This class of methods, which works efficiently for large systems [Barrett et al., 1995], has replaced stationary routines such as Jacobi, Gauss-Seidel, and successive overrelaxation (SOR) [Kelley, 1995]. Stationary methods compute variable estimates that are independent of the iteration history. While equations governing diffusive processes often spawn symmetric systems of al-gebraic equations, the coupling terms in this model produce asymmetry which restricts the selection of solvers. Ordinary CG applies only to symmetric, positive-definite sys-tems while Bi-CG and GMRES accommodate nonsymmetric, nonpositive-definite sys-tems [Greenbaum, 1997]. In simple tests of computational accuracy and efficiency on a linear diffusion problem, Bi-CG outperformed GMRES [Flowers, unpublished manu-script] . Preconditioning further extends the utility of this method to embrace some ill-condi-tioned matrices. Using the Jacobian diagonal as the preconditioner is straightforward and requires no additional memory storage with the sparse packing technique I employ. For systems with a wide range of variable values (many orders of magnitude), condi-tioning must be undertaken with more rigour. Coupled systems are especially prone to ill-conditioning, because multiple variable types are represented in the same mat-rix. Fortunately, characteristic values of hr, he, h3, and ha are sufficiently similar that ill-conditioning does not arise. Problematic variable disparity in these systems can be handled by nondimensionalization or scaling [personal communication from C. G. Far-quharson]. Bi-CG generates search-direction vectors that are functions of the residual. Along with the Jacobian, these are used to compute improved estimates of the solution. Search-direction vectors are mutually orthogonal, thus the entire solution space can be approxim-ated in nsys iterations. I implement PBCG to solve Equation (5.13b) using the Numerical Recipes routine linbcg [Press, 1992]. In most cases, convergence is monotonic and requires 3-30 iterations. The convergence criterion for this routine is dimensionless, so a uniform tolerance is applied to all systems. Stipulating that this be independently satisfied for each component of the model required substantial alteration of linbcg. Chapter 5. N U M E R I C A L M E T H O D 99 5.4 Summary and practical implementation Finite difference conventions are applied to discretize the model domain and cast the governing equations in a compatible manner. The algorithm I have developed iterates toward a solution in two stages at each timestep. The first stage approximates the prob-lem as a linear system (Eqns. (5.11)-(5.12)) and the second solves the resulting algebraic equation (Eqn. (5.13b)) for an improvement to the solution estimate. Simultaneous so-lutions for different model layers are enabled by stacking up to four systems of equations together (Eqns. (C.3), (CIO), (C.18), (C.23)). The numerical strategy described in this chapter was adopted after performing tests on a similar but much simpler set of equations. Order of accuracy in the discretization was confirmed with this system, as were accuracy improvements from spatial and tem-poral grid refinement. The Newton-Krylov iterative solution procedure was selected in a comparative study of five methods where accuracy and economy were quantified for each. Overall, the strategy appears remarkably successful, with the exception of the surface runoff system which must be solved explicitly to achieve reasonable execution times. All code is written in Fortran 90 and compiled using Cray™ software on a Spark Ultra 5 with Solaris operating system. Makefiles are used to handle compilation effi-ciently for the UNIX-based directory structure. Fifteen directories house a total of 86 subroutines excluding F90 modules and libraries. Runtime memory requirements are modest, approximately 47 MBytes for a typical fully-coupled Trapridge Glacier simula-tion. Program runtime varies dramatically depending on the number of coupled systems. For the simulation above, the pace is approximately 25-32 model hours per hour on a Spark Ultra 5 with 360MHz clock speed. Runtimes are substantially less for problems with simple geometry, such as are the focus of Chapter 6. Chapter 6 A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y Glacier geometry exerts a first-order control on the movement of water in its vicinity: surface slope influences the initial incision of supraglacial melt water channels into the ice; water from these channels is gathered by ice-marginal streams whose course is dictated by the glacier perimeter; englacial and subglacial flow trajectories are controlled by the combined influence of ice-thickness gradients and bed topography. The success of most simple glacier hydrology models hinges on the superior explanatory power of geometric effects over material properties or detailed physical processes. In some cases, this tenet is correct. To evaluate the importance of ice and land surface geometry in determining glacier drainage structure, I apply the coupled model described in Chapters 3-5 to imaginary glaciers. These glaciers are idealizations comprising a parabolic ice mass resting on different beds. Though greatly simplified, the bed configurations I choose are derived from those observed in Nature, and each is associated with at least one glacier whose hydrology has been well studied. To evaluate the merit of geometric methods for determining subglacial drainage structure, as introduced in Chapter 2, I compute the steady state hydrology in response to a constant surface melt rate for each idealized glacier. I then explore the behaviour of the entire coupled system in response to time-dependent forcing. Model parameters that strongly influence the system response are identified and explored in the context of sensitivity tests. Model validation, as is practiced in meteorology, is unfortunately beyond our grasp with the present dearth of direct observations. Moreover, as state-of-the-art numerical modelling of glacier hydrology lags that of ice dynamics, formal model intercomparison projects [e.g., Huybrechts et al, 1996] are a thing of the future. To the best of my ability I attempt model evaluation, rather than validation, by (1) choosing reference model parameters grounded in the literature, (2) using sensitivity analyses to investigate plausible system behaviour through a range of physical parameters, and (3) critiquing model results in the context of observations. 6.1 Glacier geometry Figures 6.1-6.3 show parabolic glaciers resting on three idealized beds: an inclined 100 Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 101 Figure 6.1: Terrain 1: parabolic ice mass resting on an inclined plane. Figure 6.2: Terrain 2: parabolic ice mass in an inclined U-shaped valley. plane, an inclined trough, and an inclined series of undulations perpendicular to ice flow along the centreline. The ice is meant to represent the ablation area of a small alpine glacier that is approximately 2000 m in length, 1000 m in width, and 200 m thick at the upstream boundary. In reality, bed topography would have some influence on ice-surface structure. For example, bedrock ridges may produce bulges at the surface. However, for the purposes of intercomparing geometric results, the glacier surfaces are designed to be congruent in each example. By imposing uniform ice surface structures on different beds, the glacier footprint changes in each case. In the text that follows, these three geometrical ice-bed combinations (Figures 6.1-6.3) will be referred to as Terrains 1, 2, and 3, respectively. Terrain 1 represents the simplest bed configuration, and is similar to Trapridge Glacier which flows unconfined in much of the ablation zone. Terrain 2 represents the classical U-shaped valley revealed in the wake of retreating glaciers. Haut Glacier d'Arolla and Unteraargletcher, Switzerland, are good examples of glaciers that occupy the U-shaped valleys they have excavated. Overdeepenings in the Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 102 Figure 6.3: Terrain 3: parabolic ice mass overlying inclined undulations. glacier bed, as designed in Terrain 3, are believed to govern the hydrology of glaciers such as Storglaciaren, Sweden. 6.2 Steady state test description For each terrain, I compute the hydrological steady state in response to a prescribed forcing, using parameters listed in Tables 6.1 and 6.2. These results define a reference model for each geometry that will serve as a basis of comparison throughout the chapter. The equilibrium state of the system is a function of the source rate (in the form of surface melt water generation), boundary conditions, and a myriad of model parameters, all of which are applied identically to each terrain. I impose no-flow conditions at the head of the glacier in both the subglacial sediment sheet and the groundwater aquifer. This ensures that the steady state is primarily a result of surface forcing, rather than poorly-known boundary conditions. Throughout the discussion I use the term subglacial specifically to describe the region in contact with, or just below, the ice. I use subsurface to refer to buried sediment layers that are not directly exposed to the base of the glacier (e.g., the till cap and groundwater aquifer). In the context of groundwater flow, I use the term unsaturated to indicate the existence of an unsaturated zone, not to imply a complete absence of water. 6.2.1 Reference model parameters Physical constants and numerical parameters that apply to all steady state tests are presented in Table 6.1. Physical and geometric parameters are listed in Table 6.2. In some cases, I have carried out detailed investigations of parameter space, comparing model results and Trapridge Glacier sensor records, to identify appropriate reference Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 103 values. Table 6.1: Physical constants and numerical parameters used in the reference model Parameter Value Description Physical constants: 1000 kgm- 3 Density of water at 0 °C PI 917 kgm"3 Density of ice 9 9.81ms-2 Gravitational acceleration 8 5.04 x 10- 1 0Pa- x Compressibility of water L 3.34 x 105 J kg- 1 Latent heat of fusion of water Numerical parameters: LX 2120m Model domain, longitudinal 1240 m Model domain, transverse nx 53 Number of grid cells in x ny 31 Number of grid cells in y Ax 40 m Longitudinal grid spacing Ay 40 m Transverse grid spacing At 120-300 s Maximum time step Ablation and runoff I assume a constant surface melt rate of 1.25 x 10 _ 3 mh- 1 water-equivalent (w.e.). This source rate is applied in every model grid cell and is roughly comparable to melting 3 cm of ice per day. Using a degree-day factor optimized for Storglaciaxen by Hock [1999], this melt rate corresponds to a constant air temperature of 6.8°C. The hydraulic conductivity for surface runoff i f is not used as a true material and fluid property, but as a convenient means of regulating the rate of supraglacial water transport. I choose a reference value of KT = 0.1ms - 1 based on Trapridge Glacier data comparisons. Stone [1993] obtained values of conductivity even greater than 0.1ms - 1 for thin (0.001-0.01 m) subglacial "aquifers", in which flow is expected to be slower than on the surface. Parameter r r : e is introduced to regulate the rate of water exchange between the glacier surface and interior. Physically, it can be interpreted as a subgrid transit time for runoff in a crevasse- (or moulin-) bearing grid cell. This naturally depends on the density of entry portals in a gridcell and the prevailing surface conditions. For example, surface water may percolate through firn before debouching into a channel. Velocities in the near surface are much slower than in an open channel, so r r : e must account for the time delay resulting from a composite travel path. Realistically, this quantity would evolve over the course of the meltseason. For simplicity, I take a constant value of r r : e = 1200 s. Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 104 Table 6.2: Reference model parameters. Symbols are consistent with those used in Chapters 3~4 where superscripts denote one of four model components (r=runoff, e=englacial, s=subglacial, a=groundwater aquifer) Parameter Value Description Ablation/runoff: M l ^ x l O - ' m h - 1 Surface melt rate Rr 0.1ms-1 Effective hydraulic conductivity Tr:e 1200 s Time constant for water exchange xr:e 1 Runoff-englacial coupling strength xr:a 1 Runoff-groundwater coupling strength Englacial storage/transport: VT VI 7I lxl0~ 4 Maximum storage volume fraction 1 Relative abundance of moulins 72 0 Relative abundance of surface crevasses 73 0 Relative abundance of basal crevasses b 2.0xl0- 5m Englacial crack aperture Ke l .OxlO-^ms- 1 Englacial hydraulic conductivity Te:s 7200 s Time constant for water exchange xe:s 1 Englacial-subglacial coupling strength Subglacial sediment sheet: hc 0.10m Critical sheet water thickness Ks • min lx lO^ms" 1 Minimum hydraulic conductivity K° max l x l O ^ m s - 1 Maximum hydraulic conductivity V 7/2 Exponent in p"(ha) relationship V 50 ma"1 Average glacier sliding velocity QG 0.07 Wm" 2 Geothermal heat flux* xs:a 1 Subglacial-aquifer coupling strength Till cap properties: d1 1.0m Thickness K* l .OxlO^ms- 1 Hydraulic conductivity Groundwater aquifer: da 3.0m Thickness Ka l .OxlO^ms- 1 Hydraulic conductivity na 0.4 Porosity a 1.0XlO"8 Pa"1 Compressibility t Cook [1973] Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 105 Englacial storage and transport Compared to surface and subglacial hydrologic processes, relatively little is known about englacial storage and transport. This is reflected in the paucity of field studies from which relevant quantitative information can be extracted. However, recent applications of ground-penetrating radar to map glacier internal structures are rapidly pushing back this barrier [e.g., Murray et al, 1997; Moore et al, 1999]. In these basic tests, crevasses are neglected in favour of moulins (vertical pipes) as englacial storage elements. Table 6.2 reports the relative abundance of each element as a fraction. For simplicity, moulins are present in every model grid cell. I assume that these elements comprise 0.01% of the glacier volume, expressed as the ratio Vr/Vi = 10 - 4. For a 40m x 40m model grid cell, these geometric specifications translate into pipes of diameter 0.45 m. Small values of englacial crack aperture b and hydraulic conductivity Ke reflect the glaciological consensus that percolation of water through the ice matrix is negligible compared to transport through a few developed passageways [Fountain and Walder, 1998]. The parameters I choose translate into an exceptionally conservative volume of void space compared to that estimated from borehole video [Copland et al, 1997]. Englacial transit is responsible for temporal delays and spatial rerouting of water. By representing the ice as a fractured medium, I permit lateral transport of water between its injection point and final destination. The time constant r e :* accounts for tardiness in transfer between the englacial network and the bed. A number of empirical studies have quantified the time lag between peak ablation and maximum glacier outlet discharge (e.g., 2-4 hours for Fels and Black Rapids Glaciers, Alaska [Raymond et al, 1995]), but ambiguity remains in the travel-time partitioning between englacial and subglacial transport. For a distributed (e.g., unchannelized) subglacial drainage network, englacial transport is probably fast compared to flow at the glacier bed. Typical delays between solar noon and subglacial water pressure maxima in the Trapridge Glacier study area are 4-8 hours. With this information, I assign re:s=7200s, assuming the total time lag will be substantially augmented during transit through the subglacial sheet. Subglacial sediment sheet Critical sheet water thickness hc represents the areaily averaged depth of water required to saturate the sediment horizon. It is thus a combined estimate of the total sediment layer thickness and its porosity. I derive a value of hc = 0.147 m in Chapter 4 for Trapridge Glacier from estimates of subgrid topographic variability and sediment distri-bution. This value is a product of horizon thickness equal to 0.368 m and a porosity of 0.4. Blake [1992] estimates 0.30 m as the maximum depth of sediment deformation be-neath Trapridge Glacier, echoing the suggestion that this layer is a few decimetres thick. Beneath Storglaciaren, Sweden, penetrometer tests indicate the presence of a permeable Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 106 layer at least 0.20-0.35m thick [Hooke et al, 1997], and beneath Blue Glacier, Alaska, Engelhardt et al. [1978] observed a sediment layer O.lm-thick. Porter et al. [1997] report that Bakaninbreen, Svalbard, overlies a soft sediment layer 1-3 m in depth. Till 4-7 m thick has been confirmed from seismic soundings [Nolan and Echelmeyer, 1999b] and direct sampling of Black Rapids Glacier, Alaska [Truffer et al., 1999]. Seismic soundings of ice stream B show several metres of saturated subglacial sediment [Blankenship et ah, 1986]. The depth of subglacial sediment varies spatially and from glacier to glacier. I choose a reference sediment thickness of 0.25 m. Porosity of subglacial materials typically varies from about 0.2-0.4 [Paterson, 1994]-Blankenship et al. [1987] estimate a till porosity of ~0.4 beneath ice stream B from seismic travel times. More recent results of soil mechanics tests by Tulaczyk et al. [2000a] suggest values between 0.25 and 0.35. A typical porosity for dilatant materials is 0.4 [Kamb, 1991] with an increase expected during deformation [Boulton et al., 1974]. A value of 0.4 is also obtained by direct sampling from beneath Black Rapids Glacier, Alaska [Truffer et al, 1999] Because sediment deformation is well-documented beneath soft-bedded glaciers, I choose 0.4 as a reference porosity. Thus the reference value of hc is 0.25m x 0.4 = 0.1m. Hydraulic conductivity estimates for subglacial materials have been compiled from numerous laboratory, in-situ, and model experiments [e.g., Stone, 1993; Fountain, 1994; Iverson et al, 1994; Hubbard et al, 1995; Waddington and Clarke, 1995; Iken et al, 1996; Fischer et al, 1998]. Values range from 10 - 9 to 1ms - 1 , depending on the lithology and physical properties of the material. Data can also be affected by spatial sampling of the glacier bed and by the conditions under which they are collected, often leading to disparate (though possibly correct) estimates of hydraulic conductivity for the same material. For example, Trapridge subglacial sediment has been assigned conductivities ranging from 1ms - 1 [Stone, 1993} to 10 _ 9 ms _ 1 [Waddington and Clarke, 1995]. I will elaborate on this point in Chapter 7. For now I choose a smaller range of conductivities, 10~4 to lO^ms- 1 . Sliding velocities vary from zero to several kilometres per year for glaciers in surge [Clarke, 1987a]. Disentangling the contributions to surface motion from basal sediment deformation, basal sliding, and internal ice deformation is a tricky business. However, for some glaciers surface velocity can be roughly equated to sliding velocity. Typical values of sliding velocity (for warm-based glaciers) from in situ measurements and observationally-based calculations are a few to tens of metres per year (Black Rapids Glacier, 100 m a - 1 [Raymond et al, 1995]; Trapridge Glacier, 33ma - 1 [Blake, 1992]). I choose a reference value of 50 m a - 1 . Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 107 Subsurface Aquifer and aquitard properties are modelled after the subsurface hydrostratigraphy of Trapridge Glacier. Stone [1993] estimates a range of layer thicknesses for the till cap and groundwater aquifer and describes their respective compositions. Hydraulic conductivity estimates for the till cap Kl and aquifer Ka are taken from the following ranges given by Smart and Clarke [unpublished manuscript]: Kl = 10 - 1 2 -10 - 7 ms - 1 and Ka = 10~5-10 - 3 ms - 1 . I have assigned aquifer porosity and compressibility according to Freeze and Cherry [1979] as appropriate for the sandy gravel described by Stone. 6.3 Steady state results In order to illustrate the basic configuration and operation of a multicomponent hydro-logical system, I present results for each terrain independently. Differences imparted by geometry are discussed collectively at the end, and a comparison is made between deter-ministic modelling and the hydraulic potential method for predicting subglacial drainage structure. 6.3.1 Terrain 1: inclined glacier bed Figure 6.4 summarizes the key features of equilibrium surface runoff and englacial storage for Terrain 1. The runoff system routes meltwater over the glacier surface and then over land in the glacier forefield. Supraglacial water that reaches the ice margin either infiltrates the aquifer or flows out of the basin. Artesian fountains, when present in the glacier forefield, also contribute to basin runoff. Figure 6.4a shows equilibrium flow vectors for the surface runoff system. Water flux is much higher in the unglacierized area due to greater runoff depth hr (vector length in Figure 6.4a is proportional to flux magnitude, rather than velocity). This reflects the subglacial and groundwater sources to the forefield. Despite the presence of moulins in every model grid-cell, they are escaped by a sub-stantial fraction of the total surface melt. The global rate of surface melt generation is 0.44m 3s - 1, while the global rate of water supply to the subglacial sheet is 0.16m3s_ 1. Aside from a negligible amount of englacial water escape, this means that ~65% of the source water never encounters the glacier bed. The time constant rr'e partially controls this statistic by regulating surface-englacial water exchange. For an infinitesimal value of r r : e , all surface water would be drawn into the glacier. In reality, supraglacial streams and moulins have a causative relationship such that stream water is drawn into the moulins it forms. Because I do not account for this genesis and development in the model, wa-ter capture by moulins (contradicting large-scale surface slope) can be accomplished by adjustments in r r : e . Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 108 Water that finds its way into the englacial network of cracks and moulins can be transported directly to the bed or can migrate horizontally within the ice. Figures 6.4b and 6.4c show longitudinal and transverse profiles of englacially stored water. In this example, there is exactly one moulin per cell. If one thinks of the glacier itself as an unconfined aquifer, then the surface defined by connecting the height of the water columns approximates the water table, or the boundary between saturated and unsaturated zones (no partially-saturated zone is considered in the model). More accurate delineation of the water table would require knowledge of water storage partitioning between bulk elements (moulins) and englacial fractures. However, the fracture void volume is insignificant compared to the bulk storage volume, so making this distinction leads to visually identical results. Runoff flux Longitudinal distance (m) Transverse distance (m) max = 2.6x10"4m2 s"1 Figure 6.4: Equilibrium characteristics of surface runoff and englacial storage for Terrain 1. (a) Runoff flow vectors. Arrow length is •proportional to flux magnitude. Flux is highest in the unglacierized area due to greater runoff depth rather than high velocities, (b) Depth of englacially stored water in moulins along centreline (every second moulin shown for clarity), (c) Depth of englacially stored water along a midglacier transect. Ice-surface and bed profiles are shown for context in (b) and (c). Figure 6.5 encapsulates the subglacial equilibrium response to the prescribed surface conditions. Subglacial sheet flux is shown as a vector field in Figure 6.5a. Flow direction is misaligned with bed slope as water is steered away from the high pressure interior toward the margin. This feature arises from the imbedded assumption that subglacial water pressure and ice overburden pressure are related [e.g., Shreve, 1972]. In the model, a parcel of subglacial water is pressurized by an amount that depends on the ice overburden. Profiles of piezometric surface or hydraulic head (Figures 6.5a and 6.5b) illustrate the combined gravitational and fluid pressure potentials that drive flow in the sheet. For a subglacial drainage system, it also represents the height to which water would rise in a Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 109 Subglacial sheet flux max = 1.1x10'4m2 s"1 400 800 1200 1600 2000 Longitudinal distance (m) 200 400 600 800 1000 1200 Transverse distance (m) Figure 6.5: Equilibrium characteristics of the subglacial system for Terrain 1. (a) Subglacial sheet flow vectors, (b) Centreline profile of piezometric surface, (c) Piezometric surface along midglacier transect. borehole drilled to the glacier bed. By analogy to groundwater flow, this is only valid for a confined layer where flow is restricted to the horizontal. Piezometric surface ift is computed as [Freeze and Cherry, 1979] Pv>9 (6.1) where p w is water pressure, / 9 W is water density, g is gravitational acceleration, and ZB is bed elevation. Note in Figure 6.5 that piezometric surface is equal to bed elevation at the glacier margin, where water pressure is atmospheric. For a drainage network with subaerial outlets, this is a required condition. Mismatched boundary pressures are only possible with a hydraulic barrier at the glacier margin. The surface ablation rate I impose is representative of high melt-season conditions for a mid-latitude glacier. Considering this, the piezometric profiles suggest a remarkably low operational pressure for the sheet. By comparison, the piezometric profile for a floating glacier would be ~ 9/10 of the ice thickness above the bed, as defined by the density ratio pi/py,. Note the similarity between profiles in Figure 6.5 and the water columns in Figure 6.4. The piezometric surface of the subglacial system (which can be likened to a confined aquifer) and the water table in the ice are almost identical. Small differences in these quantities drive water flow between the two systems. Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 110 . f f r a * 1 mm i t H-4r 3 2 1 0 -1 -2 -3 -4 x10 Water exchange (m s*1) Groundwater flux max = 1.1x10"5m2 s"1 400 800 1200 1600 2000 Longitudinal distance (m) 200 400 600 800 1000 1200 Transverse distance (m) 1> 1.225 | 1.205 n 3 1.200 « W 1.195 400 800 1200 1600 2000 Longitudinal distance (m) 200 400 600 800 1000 1200 Transverse distance (m) Figure 6.6: Equilibrium characteristics of the groundwater aquifer and its exchange with the sheet for Terrain 1. (a) Water exchange rate <j>>,a. Positive values indicate aquifer recharge and negative values indicate upwelling of groundwater to the glacier bed. The transition between positive and negative exchange is delineated in white, (b) Longitudinal profiles of <j>s:a from the centreline (C) and toward the glacier margin (A). Transect positions are labelled in (a) and the same sign convention for exchange rate applies, (c) Transverse profiles of (f>$',a as labelled in (a). Transect H is in front of the glacier margin, (d) Groundwater flow vectors, (e) Longitudinal profiles of aquifer saturated thickness, expressed as the equivalent depth of water. Aquifer saturation occurs at a thickness of 1.2m indicated by the dashed line. The transition between glacierized and unglacierized areas can be clearly identified as an abrupt change in slope, (f) Transverse saturated thickness profiles. Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 111 Similarly, flow between the subglacial sheet and groundwater aquifer is driven by pressure differences in the two systems and by elevation potential across the intervening aquitard. Figure 6.6 shows both positive and negative rates of water exchange cp':a, corresponding to aquifer recharge and discharge, respectively. I use the term recharge, rather than infiltration, to designate positive values of exchange, because I assume that water transported across the aquitard immediately joins the saturated horizon. Transects labelled in Figure 6.6a are shown in Figures 6.6b and 6.6c. In this case, the glacier boundary coincides with exchange minima. Groundwater upwelling begins under the glacier where very thin ice and low subglacial water pressures provide little resistance to artesian flow. Rates of upwelling decline almost linearly with distance from the glacier. Profile H is in front of the glacier margin, where exchange rates are low but water is still being pumped to the surface. Vigorous artesian flow has been observed in the proglacial area of one of Kluane National Park's large outlet glaciers (personal communication from M. L. Skidmore). Figure 6.7 documents this phenomenon, showing subterraneous water emerging from a cauldron into a proglacial river. At Trapridge Glacier, aufeis (groundwater icing) has been observed in the forefield, demonstrating that groundwater is expelled subaerially. This ice contained trace amounts of Rhodamine WT dye that had been poured into glacier boreholes [Stone, 1993]. Figure 6.7: Subterraneous water erupts into a proglacial sediment cauldron in Kluane National Park, Yukon, Canada. Photograph courtesy of M. L. Skidmore. Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 112 Due to the sheet's demands on the aquifer, groundwater flux lines are diverted away from the glacier interior as shown in Figure 6.6d. Flow of this form is driven by the conditions illustrated in Figures 6.6e and 6.6f. In the absence of ice, uniformly sloping topography would produce a groundwater flux field with no lateral component. While the aquifer operates under the entire model domain, the glacier's signature is unmistakable in the profiles of saturated thickness (Figs. 6.6e and 6.6f). I define saturated thickness as the equivalent depth of water in the aquifer. Given the reference values of aquifer thickness (da = 3 m) and porosity (na = 0.4), a fully saturated aquifer corresponds to a saturated thickness of 1.2 m. According to Figures 6.6e and 6.6f, unsaturated conditions are only obtained at and beyond the glacier margin. Beneath the ice interior, supersaturation persists. Despite the fact that aquifer discharge is about an order of magnitude less than discharge from the ice-bed interface (0.0246 cf. 0.278 m 3 s - 1 ) , the groundwater system contributes to the relatively low operational pressure of the sheet. Increasing the value of da extends the depth of the unsaturated zone outside the glacier margin, but is otherwise unable to relieve high pressure conditions beneath the sheet. Where the aquifer is unsat-urated, groundwater flow is driven primarily by topography, rather than fluid pressure. For small bed slopes, leakage from the sheet into the aquifer accumulates, driving the aquifer toward saturation in equilibrium. Upon saturation, pressure escalates rapidly with any further addition of water, and the resulting rise in fluid potential stimulates horizontal flow. Figure 6.8 illustrates some limitations of the groundwater system in response to escalating demands from the sheet. In this case, the glacier margin is frozen to the underlying sediment. Transport of water across this "thermal dam" is controlled by the hydraulic conductivity of frozen soil (taken to be 1 x 10 - 1 2 ms _ 1 [Williams and Smith, 1989]). Under these conditions, equilibrium ice-marginal discharge decreases from 0.278 m 3 s - 1 to 0.129 m 3 s _ 1 , and aquifer discharge across the model boundary increases from 0.0246 m 3 s _ 1 to 0.0447 m 3 s - 1. Piezometric surface profiles (Figures 6.8a and 6.8b) are much higher than those in Figures 6.5b and 6.5c, suggesting that groundwater transport is too languorous to depres-surize the sheet. At the glacier margin, the piezometric profile is above the ice surface, evincing artesian water pressure at the bed. The resulting demand on the aquifer can be appreciated by comparing profiles of </>s:a in Figures 6.6b and 6.8c. For a frozen glacier margin (Fig. 6.8c), exchange rates are high and sustained along the length of the glacier. Upwelling of groundwater beyond the terminus (negative values of 4>8'a in Figure 6.8c) is correspondingly more intense. A similar exchange pattern is established at the ice-bed interface. As shown in Figure 6.8d, values of cff'" are positive beneath the glacier interior, indicating flow toward the bed. Approaching the frozen zone, <ff's becomes large and negative over a short distance. In this interval, high sheet pressure forces water back into storage. It is ultimately Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 113 expelled from storage onto the ice surface, where it rejoins the runoff system. Longitudinal distance (m) Transverse distance (m) i Figure 6.8: Profiles of piezometric surface and water exchange for Terrain 1 with a frozen margin, (a) Centreline profile of piezometric surface. The piezometric profile exceeds ice surface elevation at the glacier terminus, indicating artesian subglacial water pressure, (b) Midglacier transect of piezometric surface, (c) Longitudinal profiles of sheet-groundwater exchange <j>ixa. Dashed sections contain no val-ues, (d) Longitudinal profiles of englacial-subglacial exchange (f>e:>. Profiles end at the glacier margin. Transect positions for (c) and (d) are shown in Figure 6.6. Skidmore and Sharp [1999] have observed this phenomenon on John Evans Glacier, Nunavut, Canada, during three consecutive field seasons. Surface crevasses feed water to the bed where it is forwarded downglacier and stored. This water is released in a cyclic sequence that begins with the eruption of an artesian fountain on the glacier surface. Concomitant upwelling through subglacial sediments is observed in the proglacial area (in front of the glacier). The release cycle culminates in an outburst flood of channelized subglacial water that breaches the thermal dam at the glacier margin. Sketches in Figure 6.9 contrast the evacuation plan for water when the glacier margin is frozen (Fig. 6.8) and unfrozen (Figures 6.4-6.6). The diagrams are a qualitative Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 114 interpretation of model results. For an unfrozen glacier margin (Figure 6.9a), the flow of water between systems is primarily in accordance with gravity. Englacial drainage is negligible compared to drainage on the surface, at the bed, and through the aquifer. A frozen margin (Figure 6.9b) creates hydraulic resistance that diverts subglacial water back to the surface, and intensifies groundwater upwelling in the forefield. Figure 6.9: Schematic representation of major drainage routes, (a) Unfrozen glacier margin, (b) Frozen glacier margin. The total volume of water stored in each of the four systems (surface runoff, englacial, subglacial, aquifer) is shown in Figure 6.10 for unfrozen conditions at the margin. Quan-tities Vr and Vlost are components of a common runoff system; Vr is the volume of water stored on the glacier surface, while Vlo,t is the volume of water that exits the basin. The relative storage capacity of each system is largely controlled by a few model parameters: total englacial void volume VT, critical sheet water thickness hc, and aquifer thickness da and porosity na. Global rates of water transfer between systems are also presented in Figure 6.10. The source term (j)melt is the prescribed surface melt rate. Differences in adjacent exchange rates are proportional to the volume of water lost through the intermediate model layer. For example, the difference between <f>melt and <j>r:e represents supraglacial runoff, and the difference between <f>e:3 and <f>":a is due to discharge from the subglacial sheet. Quantities (f>r:e and </>e:* are virtually identical because very little water escapes the system directly through englacial cracks. Uniform exchange rates equal to the melt rate would imply that all water is ultimately destined for the aquifer. In the absence of a surface source, pressure in the englacial and subglacial systems would converge toward a common value, eliminating the need to independently monitor both systems. Such a reduction would produce substantial computational savings and is probably warranted for large systems (e.g., hydrologically coupled ice-dynamics models), especially where surface melt is intermittent or absent. As it is, the closely matched Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 115 Total water volume (m 3 ) Figure 6.10: Equilibrium water budget and transfer rates for the coupled system. (Left) Loga-rithmic chart of water storage partitioning between each of the modelled systems (r=runoff, e=englacial, s=subglacial, a=aquifer). VT is the runoff stored on the glacier surface and Vlost is the runoff that exits the basin. (Centre) Diagram of layered model structure and exchange interfaces. (Right) Rates of global water volume transfer between systems. $melt %$ the prescribed surface melt rate. values of <f>r:e and <j)e:s in this case suggest that the englacial system could be represented as a time delay, rather than an independently modelled component. 6.3.2 Terrain 2: parabolic inclined glacier bed U-shaped valleys are the classic signature of glaciation on the landscape. As is frequently observed, supraglacial water is delivered to one of two ice-marginal streams, and runoff is confined between the glacier and its valley wall (Fig. 6.11a). These streams coalesce into a single channel at the glacier terminus. Flow vectors in Figure 6.11a have several dis-tinct orientations that reflect the digital representation of the glacier perimeter. Arrows pointing toward the interior occur where the glacier narrows by a single model grid cell; at these points, water flows in so as to remain in the groove between the ice and valley wall. Equilibrium characteristics of the englacial system are shown in Figures 6.11b and 6.11c. The longitudinal profile in Figure 6.11b suggests that considerably more water is stored in this case than in the case for Terrain 1 (see Fig. 6.4b). At the head of the glacier, moulins are water-filled up to an elevation of ~200m compared to ~170m in Figure 6.4b. The transverse profile (Figure 6.11c) manifests a similar increase in storage. Subglacial equilibrium conditions are summarized in Figure 6.12. Figure 6.12a shows water moving uphill as subglacial flow vectors diverge from the centreline. This drainage pattern reveals the controlling influence of ice, and can be understood by first rewriting Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 116 300 5 100 Runoff flux max = 1.7x10"3m2 s'1 400 800 1200 1600 2000 Longitudinal distance (m) 200 400 600 800 1000 1200 Transverse distance (m) Figure 6.11: Equilibrium characteristics of surface runoff and englacial storage for Terrain 2. (a) Runoff flow vectors. Abrupt changes in flow direction are caused by the digital representation of the glacier margin, (b) Depth of englacially stored water in moulins along centreline (every second moulin shown for clarity), (c) Depth of englacially stored water along a midglacier transect. Ice-surface and bed profiles are shown for context in (b) and (c). hydraulic potential as ip = Pw + Pv,gzB. (6.2) Assuming p w is a constant fraction / of the ice overburden pressure, p w = / p i g hi, where hi = zs — ZB- Then "0 = f Pi9(zs ~ ZB) + py,gzB, (6.3) and by factoring, V> = pig [fzs + f ) z B pi (6.4) The potential gradient can now be written Pi (6.5) Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 117 assuming spatial changes in p\ and pm are small. For / = 1 (flotation), this reduces to Paterson's Equation 6, Chapter 6 [1994]. According to this simple hydraulic potential analysis, when / = j ^ y f ~ 0.55, surface and bed slopes equally determine the total fluid potential, and hence, the direction of subglacial water flow. For Terrain 2, surface slopes exceed bed slopes (see Fig. 6.12b,c), and / > 0.55 over approximately 60% of the bed. Under these conditions, water pressure is high enough to overcome basal topography. Pater son [1994] uses a similar analysis and a generalization of glacier geometry to explain why hydraulic flow lines are generally convergent in the accumulation area (where valley walls are steep and the ice surface is flat) and divergent in the ablation area (where the ice surface is parabolic). Subglacial sheet flux Longitudinal distance (m) Transverse distance (m) max = 1.8x10"4m2 s"1 Figure 6.12: Equilibrium characteristics of the subglacial system for Terrain 2. (a) Subglacial sheet flow vectors. Irregularities occur where one grid cell has two interfaces through which water can escape the glacier, (b) Centreline profile of piezometric surface, (c) Piezometric surface along midglacier transect. Piezometric surface profiles for the subglacial system accompany the flow vectors in Figure 6.12. A direct comparison between Terrains 1 and 2 (Figs. 6.5 and 6.12) shows that the U-shaped valley glacier operates at a higher subglacial water pressure. Maximum piezometric surface elevation for Terrains 1 and 2 (where bed elevations are identical) is ~165 and 200 m, respectively. This discrepancy is intuitive considering the difficulty of water escaping from a laterally-confined basin. Injection of groundwater along the entire glacier bed contributes to subglacial water retention for Terrain 2, as shown in Figures 6.13a-c. The highest rates of water injection occur at the glacier margin, where subglacial water pressure is low. Exchange rates taper toward the centreline and there is a concomitant reduction in aquifer flux (marked by a decrease in vector length in Figure 6.13d). A similar result was obtained by Tulaczyk et al. [2000b] when evaluating plausible aquifer drainage beneath ice stream B, Antarctica. Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 118 Their modelling affirms that groundwater should be directed upward toward the base of the ice, due to positive concavity in the stream bed profile. Water exchange (m s - 1) max = 5.5x10'5m2 s'1 Figure 6.13: Equilibrium characteristics of the groundwater aquifer and its exchange with the sheet for Terrain 2. (a) Water exchange rate <j}s-a. Ubiquitously negative values result from terrain curvature as explained in the text, (b) Longitudinal profiles of <j>"'a from the centreline (C) and toward the glacier margin (A). Transect positions are labelled in (a) and the same sign convention for exchange rate applies, (c) Transverse profiles of 4>s:a as labelled in (a). Transect H is in front of the glacier margin, (d) Groundwater flow vectors, (e) Longitudinal profiles of aquifer saturated thickness, expressed as the equivalent depth of water, (f) Transverse saturated thickness profiles. Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 119 Groundwater flow trajectories, as determined by vector orientations in Figure 6.13d, are convergent toward the centreline. Identical boundary conditions are used in all of the geometric tests; for the aquifer, ha = 1.20 m along the lateral boundaries, ha = 1.08 m at the downstream edge, and no flow is permitted across the top. Because the lateral model boundaries in Terrain 2 are elevated relative to the glacier bed, water seeps into the aquifer from both sides. These conditions clearly modulate aquifer behaviour and its interaction with the subglacial system. Due to terrain geometry and the additional source of water, higher saturated thicknesses are obtained for Terrain 2 than for Terrain 1, as shown in Figures 6.13e and 6.13f. Saturated thickness still increases abruptly beneath the ice, even though water is not being collected from the sheet. While the aquifer fails to provide an outlet for subglacial water, it does not dramat-ically alter the subglacial pressure regime. Model tests with an impermeable horizon below the sheet produce piezometric profiles similar to those in Figures 6.13b and 6.13c. In other tests, boundary conditions on the aquifer are reformulated to allow no flow ex-cept across the downstream border. In this case, groundwater that is distal from the glacier drains away without being replenished, and the aquifer gradually ceases to feed the sheet. Figure 6.14: Transient characteristics of the Terrain 2 groundwater system with no recharge from the lateral boundaries, (a) Lateral profiles of saturated thickness (cf. Figure 6.13f). Saturated thickness drops rapidly to zero toward the margins (not shown), (b) Distribution of <f>":a (cf. Figure 6.13a). The glacier perimeter is delineated as a fine white line, and the large white gap marks the transition between positive and negative <J>s:a. (c) Lateral profiles of <t>,:a. Figure 6.14a shows profiles of saturated thickness for an aquifer with no-flow side boundaries 180 days after being completely saturated. Due to the steep valley walls, the aquifer has drained entirely near its lateral margins. Compared to the equilibrium Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 120 case with transmissive side margins, this amounts to a 25% difference in total ground-water volume and a 50% decrease in maximum aquifer flux. Figures 6.14b and 6.14c illustrate the retreat of the aquifer's influence on the glacier bed, expressed in terms of (j)s'a. Compared to Figures 6.13a-c, where exchange is unanimously negative, <j>s:a is now greater than zero over more than 40% of the glacier bed. The most noticeable impact on subglacial hydrology is an 8% reduction in ice-marginal discharge. Despite differences in the subsurface equilibrium for Terrains 1 and 2, the water budget remains largely unchanged (see Fig. 6.10 for Terrain 1). This confirms that the physical properties and dimensions of each system (as opposed to geometry) control the steady state water partitioning between them. Intercomponent exchange rates are comparable for both terrains, except the sign of <j>"''a is reversed for Terrain 2. Like the water budget, exchange is also governed by material properties such as hydraulic conductivity. Thus, geometric influences are second-order. Topography can only affect exchange by perturbing the equilibrium pressure distribution. 6.3.3 Terrain 3: undulating inclined glacier bed a Runoff flux Longitudinal distance (m) Transverse distance (m) max = 6.3x10"4m2 s"1 Figure 6.15: Equilibrium characteristics of surface runoff and englacial storage for Terrain 3. (a) Runoff flow vectors, (b) Depth of englacially stored water in moulins along centreline (every second moulin shown for clarity), (c) Depth of englacially stored water along a midglacier transect. Ice-surface and bed profiles are shown for context in (b) and (c). Runoff trajectories inferred from modelled surface flux over Terrain 3 (Figure 6.15a) are strictly confined by basin topography. Supraglacial melt near the ice margin is quickly escorted to the local topographic trough, where it bends 90° to exit the basin. Only water originating near the terminus is destined for the glacier forefield. Profiles of englacial storage in Figure 6.15b hint at the structure of the bed, with storage maxima occurring Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 121 above bedrock overdeepenings. The maximum depth to water from the surface in Figure 6.15c is similar to that for Terrain 1 (Figure 6.4c), despite a 25 m difference in bed elevation. The geometry idealized as Terrain 3 is known to have unique impacts on subglacial hydrology, as suggested by extensive studies of Storglaciaxen, Sweden [e.g., Hooke and Pohjola, 1994; Seaberg et al, 1988]. Model results in Figure 6.16 illustrate this point. Flow vectors in Figure 6.16a show that subglacial water is purged through topographic trenches. Flow is convergent in these troughs and divergent elsewhere, resulting in an unusual subglacial catchment structure. Piezometric profiles in Figures 6.16b and 6.16c resemble the englacial storage distribution (Figure 6.15), with subglacial water accumu-lating in the depressions. Subglacial sheet flux Longitudinal distance (m) Transverse distance (m) max = 1.7x10"4m2 s"1 Figure 6.16: Equilibrium characteristics of the subglacial system for Terrain 3. (a) Subglacial sheet flow vectors. Flow is concentrated toward the two bedrock trenches. Shading delineates catchment areas for two hypothetical outlet streams L and R. (b) Piezometric surface profile along centreline, (c) Piezometric surface along midglacier transect. Due to the combined effects of bed topography and ice thickness, subglacial catch-ment structure for Terrain 3 is unique. Figure 6.16a shows shaded catchment areas for two hypothetical streams on the left (L) and right (R) sides of the glacier. Two major streams emerging from Storglaciaren, Sweden, are positioned similarly with one near the terminus and one slightly upstream. The catchment area of stream L is bounded by the first bedrock ridge. Water just upstream from this ridge flows upglacier into the adjacent bedrock trough. Stream R derives its water from further upglacier. This simple ob-servation illustrates how bed topography can produce upstream-downstream catchment partitioning for two outlets near the terminus. Without this type of bed topography, both streams would tap into water from the upper region of the glacier. One of the Stor-glaciaren streams (similar to L) is known to collect water from the ablation area, while Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 122 the other (similar to R) taps into the accumulation area [Hock and Hooke, 1993]. The one fed by the accumulation area is slightly upstream and carries approximately twice the discharge. This source area partitioning imparts distinct hydrochemical characteristics to each stream. Figure 6.17: Diversion of subglacial water away from steepest bed slope, (a) Diversion of water around a bedrock bowl as suggested for Storglacidren by Hock and Hooke [1993]. (b) Modelled diversion of water away from the maximum bed slope. Other observations reported by Hock and Hooke [1993] indicate that subglacial water conduits are diverted around a bedrock overdeepening near the terminus. Similar be-haviour is suggested by the model where water is diverted away from the line of steepest descent into bedrock troughs. If the troughs were bowl-shaped instead, one would ex-pect these flow lines to reconverge as described for Storglaciaren. Figure 6.17 illustrates this concept for a bedrock bowl, along with modelled flow vectors. This flow feature is uniquely subglacial, and can be contrasted to runoff where flow direction is almost exclusively determined by slope. Topographic variations influence groundwater flow directly through aquifer geometry and indirectly through spatial patterns of recharge. Figure 6.18a shows both positive and negative values of equilibrium water exchange between the sheet and aquifer. Upwelling of groundwater ((f>s:a < 0) occurs where bedrock depressions intersect the glacier margin. This flow reversal is permitted by a combination of concave topography and moderate to low water pressures in the sheet. Water injection to the aquifer ((f>":a > 0) occurs primarily over the upstream half of the glacier bed, with a maximum just upstream of the first bedrock ridge. Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 123 400 800 1200 1600 2000 Longitudinal distance (m) 200 400 600 800 1000 1200 Transverse distance (m) Water exchange (m s"1) CO (0 o c o •a o 13 i-D TO to D f G r 1 n 1. . : Groundwater flux max = 2.6x10'5m2 s'1 400 800 1200 1600 2000 Longitudinal distance (m) 200 400 600 800 1000 1200 Transverse distance (m) Figure 6.18: Equilibrium characteristics of the groundwater aquifer and its exchange with the sheet for Terrain 3. (a) Water exchange rate <f>s:a. Negative values occur in four zones coincident with bedrock troughs, (b) Longitudinal •profiles of 4>s'a from the centreline (C) and toward the glacier margin (A). Transect positions are labelled in (a) and the same sign convention for exchange rate applies, (c) Trans-verse profiles of <j>"a as labelled in (a). Transect H is in front of the glacier margin, (d) Groundwater flow vectors, (e) Longitudinal profiles of aquifer saturated thickness expressed as the equivalent depth of water, (f) Transverse saturated thickness profiles. Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 124 Longitudinal exchange profiles in Figure 6.18b reveal a complexity unique to this terrain. Cusps in the profiles mark the boundary between saturated and unsaturated zones of the aquifer. Along the centreline, positive exchange extends beyond the glacier itself, indicating contributions from proglacial runoff. Moving away from the centreline, upwelling becomes increasingly vigorous in the lower trough (profile A). Due to the efficacy of water evacuation at the glacier bed, near-zero infiltration rates are found along the unglacierized bedrock crest and in the forefield (profiles E and H in Figure 6.18c). Glacier bed geometry emulates the bedrock, such that elevation gradients are iden-tical in the sheet and aquifer. Figure 6.18d shows strong convergence of groundwater in the bedrock hollows. Pressure gradients are sufficient to drive water over the uppermost ridge, but the lower ridge constitutes a true flow divide. Longitudinal profiles of ha (Fig-ure 6.18e) contain sharp boundaries near ridge crests where the aquifer is unsaturated. Transverse profiles (Figure 6.18f) show predominantly saturated conditions except near the glacier terminus (profile G). 6.3.4 Synopsis of results Table 6.3 summarizes equilibrium diagnostics for each terrain. Local diagnostics are quantities at a test point on the centreline 1000 m from the head of the glacier (i, j = (26,16)). For Terrain 3, this point occurs just downstream of the central bedrock ridge where the ice is 130 m thick. The ice is 140 m thick at this location for Terrains 1 and 2. Values of runoff depth hr, source rate to the glacier interior (jf:e, and source rate to the sheet (f>e:* attest to uniform glacier surface conditions and subglacial forcing at this location. The most significant local difference between terrains is the sheet-aquifer exchange <f>":a. Global diagnostics are glacier-wide averages, maxima, or descriptive characteristics. Quantities Vr, Ve, Vs, and Va represent the volume of water stored in each system and together comprise the water budget. Storage varies much more between systems (e.g., Ve vs. Va) than it does for different terrains (e.g., Vs for Terrain 1 vs. Vs for Terrain 2), confirming the idea that the water budget partitioning is largely a function of layer properties and dimensions. Areal fraction of (j>s:a > 0 gives an indication as to whether the sheet is forcing the aquifer or vice versa. Areal fraction of ha > 1.2 m represents the horizontal extent of a completely saturated aquifer. Terrain 2 is clearly the outlier for aquifer behaviour, with the highest areal fraction of ha > 1.2 m, and (f>s:a < 0 over the entire glacier bed. Ice-marginal discharge for all three cases is less than typical summertime measurements which are in the range of 1-6 m 3 s _ 1 [e.g., Willis et al, 1990; Anderson et al 1999]. This quantity depends on glacier size, and mean summer values as low as 0.023-1.01 m 3 s - 1 have been measured by Fountain [1992] for three streams at South Cascade Glacier, Washington. Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 125 Table 6.3: Intercomparison of steady state results. Local diagnostics are water thicknesses h and exchange rates <j> at a centrally-located test point. Superscripts denote individual systems as usual. Global diagnostics V are volumes of water stored in each system. VT is limited to runoff on the glacier itself. Qmax is the maximum modelled flux in a particular system. A ^ ,.;« > o is the areal fraction of the glacier bed over which the aquifer is recharged from above. Ah*> 1.2 m quantifies the horizontal areal fraction of aquifer saturation. Qmargin is the ice-marginal discharge from the subglacial sheet. Local diagnostics Terrain 1 Terrain 2 Terrain 3 hr (m) 3.615 x lO"4 3.615 x lO"4 3.615 x 10"4 (j>r'e (ms"1) 3.012 x 10~7 3.012 x lO"7 3.012 x lO" 7 he (m) 5.373 x l O - 3 7.574 x 10~3 4.321 x 10"3 <j>e:s (ms-1) 3.011 x 10"7 3.012 x 10"7 3.012 x 10"7 h° (m) 0.07797 0.08601 0.07477 4>*-a (ms - 1) 3.012 x lO"8 -2.396 x lO"8 2.706 x lO"8 ha (m) 1.216 1.234 1.213 Global diagnostics Vr (m3) 384.7 600.9 379.6 Ve (m3) 5.211 x 103 6.350 x 103 5.578 x 103 Vs (m3) 9.327 x 104 9.527 x 104 8.985 x 104 Va (m3) 3.112 x 106 3.131 x 106 2.288 x 106 Q e m a x K s - 1 ) 1.620 x lO"7 2.466 x 10-7 1.330 x lO"7 1.087 x 10~4 1.827 x lO"4 1.706 x lO"4 Q a m a x K s " 1 ) 1.083 x lO"5 5.471 x 10~5 2.587 x lO"5 A^f.a > 0 0.943 0 0.857 ^ / i « > 1 . 2 m 0.467 0.812 0.592 Qmargin (m S ) 0.278 0.340 0.282 6.3.5 Determination of subglacial drainage structure Above the process scale (a few to tens of metres) determination of subglacial drainage structure usually relies on glacier geometry as explained in Chapter 2. The most import-ant, and potentially flawed, assumption in hydraulic potential analysis is that subglacial water pressure is either zero, or balances the ice overburden pressure. However, this analysis sometimes produces a realistic drainage configuration [e.g., Sharp et al, 1993]. For the purpose of larger (e.g., ice-sheet) models that include a water system, hydro-logy should be encapsulated in as few parameters as possible. Hence parameters must be chosen carefully. For example, the pressure balance assumption leads to spatially-variable absolute water pressure, but spatially-uniform effective pressure equal to zero. Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 126 Most ice dynamics and till mechanics models that use hydrological quantities to para-meterize basal shear stress or sliding, choose effective pressure as the diagnostic variable [e.g., Fowler, 1987b; Alley, 1989b]. In the simplest case then, spatial variability is lost. For ice and till flow models, both the mean pressure and the spatial distribution of water are important. Surging or streaming criteria, used to switch ice-flow governing physics in a model, include high mean water pressure at the bed [e.g., Marshall and Clarke, 1997]. Spatial distributions of water pressure may be used to determine where surging outlets occur, or to differentiate basal sticky- and slippery spots [e.g., Fischer, 1995]. Therefore, both aspects are relevant in evaluating representations of basal hydrology. In this sec-tion, I present several ways of summarizing the subglacial equilibrium for each terrain and compare model results with geometrically derived drainage structures. Equilibrium pressure regimes Terrain 1 Terrain 2 0.45 10.65 Terrain 3 0.45 Figure 6.19: Equilibrium distributions of subglacial water pressure expressed as a fraction of ice flota-tion, (a) Terrain 1. (b) Terrain 2. (c) Terrain 3. Figure 6.19 presents equilibrium distributions of subglacial water pressure as a fraction of ice overburden pressure. I refer to this ratio as the water pressure flotation fraction Pfi = p"/pi, where pi = pigh\. Like effective pressure, flotation fraction usually conveys a more glaciologically meaningful result than absolute water pressure. Under the prescribed conditions, water pressures are neither uniform, nor close to flotation (Figure 6.19). The respective greyscales show that the Terrain 2 subglacial system operates at a higher flotation fraction than the others. Its central maximum is ~0.65 compared to ~0.45 for Terrains 1 and 3, with a mean 0.50 compared to 0.36-0.37 (see Table 6.4). This higher Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 127 operational pressure is primarily due to the confining landscape, with some contribution from the groundwater system. For each geometry, the maximum water pressure is located toward the centre of the glacier, rather than at the head, due to the no-flow boundary condition across the top. The global maximum flotation fraction for Terrain 2 actually occurs at the glacier margin. This is primarily a result of thin ice (reducing pi), rather than high absolute water pressure. Such a result cautions against using a single metric to diagnose the hydrological state of the bed. Table 6.4: Summary of subglacial pressure state. Absolute pressure is expressed as water column height above the bed. Triangular brackets denote mean value, and (26,16) are the i,j coordinates of the test point. Terrain 1 Terrain 2 Terrain 3 Maximum pfi 0.44 m 0.85m 0.49 m Maximum ps 71.4m 105 m 101m (Pfi) 0.37 0.50 0.36 (Ps) 41.2m 53.4 m 45.0 m P//(26,16) 0.42 0.59 0.37 p'(26,16) 53.7m 75.7m 43.2 m Table 6.4 compiles a few statistics pertaining to Figure 6.19, expressed as both flot-ation fraction and absolute water pressure. Both quantities are useful in evaluating the effects of topography, except in the case of maximum pfi as explained. Figure 6.20 plots cumulative distributions of equilibrium flotation fraction for Ter-rains 1-3. This is more useful than a single parameter for evaluating the influence of geometry on subglacial hydrology, because it includes spatial information. Taken as in-dicators of the hydrological state of the bed, these curves suggest a surprising similarity between Terrains 1 and 3. Both operate at low pressure with minimal spatial variation in comparison to Terrain 2. What these curves fail to summarize, which may be im-portant for ice-dynamics, is the spatial contiguity of high pressure water. For example, the separated pockets of high water pressure in Terrain 3 will inspire an ice-dynamical response distinct from that in Terrain 1, despite their similar distribution functions (Fig. 6.20). Nevertheless, this type of statistical information may be of use in ice models where simple parameterizations of subgrid hydrology are required. While geometry has a conspicuous influence on hydrology, its effects alone are not sufficient to drive the subglacial system into a highly pressurized state such as would be required for fast flow of ice. Subglacial hydraulic properties and the thermal state of the margin (frozen versus unfrozen) are more important in determining the overall operating pressure of the sheet. Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 128 1.0 0 0.2 0.4 0.6 0.8 1.0 Areal fraction of the glacier bed Figure 6.20: Cumulative distributions of equilibrium subglacial flotation fraction. Labels 1-3 indicate terrain. Implications for flow As discussed in Chapter 2, inferences about glacier drainage structure are often derived from hydraulic potential analysis. This method estimates the fluid potential gradient based on glacier surface and bed geometry. A discrete form of Equation (6.5) can be used to generate hydraulic gradient vector distributions for qualitative comparisons with modelled flux fields. Absolute vector magnitudes can not be compared between modelled flux (in units of m 2 s _ 1) and calculated hydraulic potential gradient (in units of Pain"1). For quantitative analysis, the potential method requires additional assumptions about hydraulic conductivity and flow depth. Therefore, comparisons are restricted to flow orientation and spatial distribution of flow strength. Figure 6.21 presents this comparison between modelled- and geometrically derived drainage structures. For the potential method, two different values of flotation fraction / are used in Equation (6.5). In Figure 6.21, Column b, / = 1 is assumed. In Column c> / = {Vfi)i s o a s to uniformly scale down the contribution of water pressure relative to bed topography. Values of (pfi) are taken from Table 6.4. By comparing modelled distributions to those obtained by the potential method, one can evaluate the necessity of including physics to determine equilibrium flow fields. By selecting two values of / , one can differentiate the effects of excluding physics from the effects of assuming a particular water pressure distribution. Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 129 Figure 6.21: Comparison of modelled subglacial fluxes (Column a) and flow fields obtained by the hydraulic potential method (Columns b and c). Each row corresponds to a single terrain. In Column b, f = 1 is assumed. In Column c, f = (pfi). Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 130 For Terrain 1, the drainage configurations suggested by modelled- and geometrically derived results are similar (Figure 6.21, Row 1). Variations in the value of / , used to compute hydraulic potential, yield nearly identical results owing to the constant bed slope. If the hydraulic potential method is reliable in this case, the flow field contains little information about water pressure. The most significant qualitative difference between the two approaches is the enhanced boundary flow predicted by the potential method. Abrupt changes in potential from ice-covered to ice-free areas give rise to this effect. According to the model, interior fluxes are comparatively amplified because they depend on sheet water thickness, which increases toward the glacier interior. Low modelled fluxes at the glacier terminus correspondingly reflect a lack of water supply. Although geometry strongly favours water expulsion from this point, most water has already been evacuated through the side margins. Aside from these differences at the boundary, the physical model and potential method suggest kindred patterns of subglacial drainage for Terrain 1. For Terrain 2, the potential method with / = 1 (Figure 6.21, b2) predicts a large-scale flow configuration similar to that for Terrain 1 (Figure 6.21, bl). Ice-surface slopes are again responsible for the augmented boundary flow. When the flotation fraction is adjusted to (pfi) = 0.5 (Figure 6.21, c2), flow divergence decreases. Moving headward, trajectories are increasingly parallel, even becoming convergent near the margins. Recall the calculated transition value of / = 0.55 where ice- and bed slopes equally determine flow direction. At (pfi) = 0.5, water flows downhill even where bed slopes are slightly less than ice-surface slopes. At slightly lower flotation levels, drainage structure predicted by the potential method for Terrain 2 begins to change dramatically. At / = 0.4, flow is convergent toward the centreline beneath the upper half of the glacier and divergent elsewhere. At / = 0.3, flow is convergent along the entire length of the bed. Thus for Terrain 2, some knowledge of basal water pressure is required to uniquely determine drainage structure with the hydraulic potential method. A similar ambiguity is encountered for Terrain 3, as seen in Figure 6.21, Row 3. Both fields derived by the potential method are dominated by flow at the glacier margin. Using f — 1, the flow field is conspicuously unaffected by bed topography, with little variation in magnitude and no evidence of (upstream) flow reversals. This rules out the possibility of unusual catchment partitioning as discussed for Storglaciaren. In contrast, / = (pfi) = 0.36 produces a drainage pattern much more congruent with the modelled result. Flow reversals occur at ridge crests, contributing to convergence in the bedrock depressions. Furthermore, flow magnitude varies widely over the glacier bed. This example affirms that the subglacial pressure baseline is critical in deriving drainage geometry. Some of these examples suggest that hydraulic potential analysis gives as realistic a portrayal of subglacial drainage structure as deterministic modelling. In saying this, I assume that modelling is the superior approach, albeit much less expedient. With no knowledge of the subglacial water pressure regime, a consensus on flow structure from Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 131 hydraulic potential analysis is impossible, especially in the presence of substantial bed topography. For poorly-drained glaciers, it is plausible that / ~ 1. For those that drain from the ice margin, / < 1 is more likely. Therefore, even without in-situ subglacial observations, hydraulic potential analysis can be refined to discriminate between these two situations. It is worth mentioning that the assumption of proportionality between water and ice pressures is uncertain. It is imbedded in both the model and potential analysis as I have framed it. To my knowledge, there is not enough field data to either justify or refute this idea. 6.4 Transient test description While equilibrium hydrology can be explored with minimal effort using the hydraulic potential method, time-dependent investigations require a physical model. Alpine glacier hydrology varies most notably on diurnal, seasonal, and interannual timescales. On daily and annual timescales, the forcing is quasi-periodic and relatively well-known. On interannual time scales, hydrology is affected by mass balance and other quantities that are assumed to be constant in this model. On sub-diurnal timescales, dynamical processes such as ice-cracking and unsteady glacier motion become important. To illustrate transient hydrological behaviour as portrayed by the model, I focus on the diurnal timescale. Sinusoidally-varying surface air temperature, adjusted for elev-ation, is applied to the equilibrium hydrology for Terrain 1. Boundary conditions and reference model parameters remain the same as in the steady state tests. Several addi-tional parameters required for the surface ablation model are listed in Table 6.5. 6.4.1 Additional parameters for time-dependent investigations Table 6.5: Additional reference model parameters for transient tests Parameter Value Description Source Ablation/runoff: DDF 4.4mmh- l 0 C- 1 Degree-day factor Hock [1999] LR 0.0065 ^ m - 1 Atmospheric lapse rate Arnold et al. [1996] For the purposes of these tests, I do not employ the full capabilities of Hock's temperature-index ablation model. Her model is designed for a geographically located glacier and therefore requires special information including a DEM of the landscape sur-rounding the glacier. I neglect insolation because it depends on latitude and topographic shading. In this case, Hock's ablation model reduces to the classical temperature-index Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 132 method with a single adjustable parameter. Melt rate M can be written in terms of the temperature T as f J - (DDF) T T>0 M = < N m (6.6) I 0 T < 0, where Nm is the number of ablation model time steps per day and DDF is the degree-day factor for ice [Hock, 1999]. In the absence of snow, calculated melt immediately contributes to surface runoff. Details of the ablation-runoff model are elaborated in Chapter 7 when real topography is introduced. Hock calibrates a classical degree-day model to determine DDFs for snow and ice. I use this DDF for ice (see Table 6.5) for consistency with subsequent calculations where I reintroduce Hock's ablation model in its entirety. Atmospheric lapse rate LR, or the change in air temperature with elevation, is taken from estimates of surface lapse rate found in glaciological literature. Surface lapse rate differs from free-air lapse rate as their names imply. For temperature-index ablation models, surface lapse rate is the appropriate quantity, but care must taken to determine whether air- or ice temperatures are required. Most models (including Hock's) use an empirical relationship between air temperature and melt. Lapse rate depends on the radiative energy balance at the surface, and is therefore affected by various properties of the ice (e.g., heat capacity and albedo) and external environmental conditions (e.g., wind speed). For this reason, lapse rate estimates made in glacial environments are most appropriate for this study. I consider values obtained from studies of ice masses that experience surface melt [e.g., Reeh, 1991; Arnold et al., 1996; Arendt, 1997; Anderson et al, 1999], because I am interested in glaciers whose drainage regimes are controlled by surface water. Estimates from the authors above range from 0.0057 to 0.0070 " C m - 1 , and I have chosen a value of 0.0065 ° C m _ 1 as listed in Table 6.5. Lapse rates calculated for high elevation areas or ice accumulation zones tend to be higher. 6.5 Transient results Results of transient tests are presented as timeseries of individual variables and as spatial distributions for particular time slices. This should lend insight into the model repre-sentation of field observables (timeseries from a few sparse points), while providing a larger context for these simulated measurements. Unless otherwise noted, timeseries are extracted from the test point (i,j) = (26,16). Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 133 6.5.1 Reference model Time (days) Time (days) Figure 6.22: Temperature forcing and intercomponent exchange response for the transient reference model, (a) Surface air temperature at the test point. Temperatures at other locations are lapsed according to the rate reported in Table 6.5. (b) Resulting rate of water exchange between systems: <f>T:e (fine line), <j>e'-s (dashed line), <j>s:a (bold line). Prescribed surface air temperature for the test point is shown in Figure 6.22, along with the resulting local variations in water exchange. An initialization period of seven days is used to dampen trends in the response before the five cycles that are shown. The prescribed forcing represents extreme summer conditions with daily maxima exceeding 14°C and freezing occurring overnight. The rate at which water enters the glacier (4>r:e) is almost matched by the rate at which it is delivered to the bed (</>e:*), as can be seen by comparing fine and dashed lines in Figure 6.22b. In contrast, aquifer recharge (</>*:a) is substantially damped and lags by a few hours. Though the similarity between 4>r:e and <jf:s is partially a function of the time constants r r : e and Te:s, it persists over large variations in r e : s . This suggests that the surface forcing itself strongly controls water delivery to the glacier bed. Flow to the aquifer is regulated by physical properties of the intervening aquitard, and is not, therefore, subject to arbitrary time constants. Delays between the surface and various layers of the system, as implied by Figure 6.22b, are revealed in phase diagrams in Figure 6.23. Phase diagrams for two sinusoids are presented for visual comparison (Fig. 6.23a). In Figures 6.23b-e, delays are expressed relative to the solar forcing (temperature), normalized as T — T • T = ' _ 7 , • (6.7) -* max 1 min The dependent variables (hr, he, h*, ha) are normalized in a similar fashion. Approxim-ate lags between solar noon (time of temperature maximum) and each system are: 1 h Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 134 Figure 6.23: Phase diagrams. All variables are normalized following Equation 6.7. (a) Reference diagrams for two sinusoids with relative phases labelled on top of each panel, (b) Temperature and runoff depth hr. (c) Temperature and englacially stored water he. (d) Temperature and subglacial sheet water thickness h". (e) Temperature and groundwater thickness ha. (runoff), 4.8h (englacial), 4.8h (subglacial), and 10.6h (groundwater). Of these, only surface runoff and subglacial lags have been quantified in the field. These delays vary widely between glaciers, between seasons, and between locations on the same glacier. Sensors beneath Trapridge Glacier record surface-subglacial lags of 4-8 h, bracketing the untuned modelled value of 4.8 h. Figure 6.24 encapsulates the salient surface and subglacial features of a single daily cycle. Air temperature beginning at midnight is shown in Figure 6.24a for locations U, M, and L labelled in Figure 6.24b. The spatial distribution of mean daily surface melt rate, as calculated by the temperature-index method, is shown in Figure 6.24b. Over the length of the glacier, the lapse rate produces a range of daily melt rates from 23 to 31mm. Subglacial pressure cycles for three locations are shown together in Figure 6.24c, illustrating differences in amplitude and phase. Diurnal pressure waves migrate headward from the glacier terminus (Fig. 6.24c). At high elevation, melting commences later in the day than it does at low elevation, even though daily temperature maxima occur simultaneously. The resulting delays are discernible between curves L, M , and U in Figure 6.24c. Figure 6.24c also demonstrates that amplitude and offset are determined by the com-bined effects of melt water supply and retention. Location M operates at a higher Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 135 Melt rate (mm day"1) Figure 6.24: Daily surface air temperature, melt rate, and subglacial water pressure, (a) Temperature at locations in the upper (U), middle (M), and lower (L) sections of the glacier, as labelled in (b). (b) Spatial distribution of mean daily surface melt rate, (c) Daily subglacial water pressure variations for points U, M, and L. pressure than U because the local surface melt rate is higher and its supraglacial and subglacial catchment areas are bigger. Location M operates at a higher pressure than L because water is more effectively retained further from the glacier margin. The system as a whole operates at a higher pressure in response to time-dependent forcing than it does in steady state. For a constant melt rate of 30 mm day - 1, equilibrium flotation fraction at the test point is 0.37 (Table 6.4), as compared to 0.43 in the present case. Subglacial water pressure and profiles of water exchange at intervals throughout a diurnal cycle are shown in Figure 6.25. Maximum water pressure, as indicated by bright-ness, occurs glacier-wide between 1600 and 2000 h as expected from the timeseries in Figure 6.24c. The magnitude of spatial and temporal pressure variations, ~15% of flota-tion, is small compared to variations observed in the field (30-60% of flotation). This is an inherent shortcoming of a homogeneous and hydraulically connected subglacial drainage model. The plumbing beneath real glaciers is extremely heterogeneous, with connected and unconnected areas interspersed like patchwork on subgrid scales. Melt water that forces its way through this limited labyrinth creates high pressure in the connected re-gions, leaving the remaining ground undisturbed. Modelling such a system as uniformly connected is a decent first approximation, but behavioural extremes are compromised. Exchange profiles in Figure 6.25 coincide with the glacier centreline and each panel represents a snapshot in time. According to this figure, spatial variations in water ex-change are small compared to temporal variations. Individual profiles move up and down over the course of a day with little appreciable change in shape. Moreover, from 800 h to 2000 h, a hierarchy of exchange is preserved such that cj>r:e > <j>e:3 > </>*:a. In contrast, Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 136 10.45 0.40 0.35 400 h 800 h 1200 h 1600 h 2000 h '0.30 Water pressure (fraction of flotation) 2400 h , 9 400 h ~ 6 M 4 X *" 2 UJ X -<t>e -4>s 800 h 1200 h 1600 h 2000 h 2400 h 0 1000 2000 0 1000 2000 0 1000 2000 0 1000 2000 0 1000 2000 0 1000 2000 Distance (m) Distance (m) Distance (m) Distance (m) Distance (m) Distance (m) Figure 6.25: Spatial distribution of subglacial water pressure and profiles of water exchange through a diurnal cycle. Exchange profiles coincide with the glacier centreline and distance is measured from the head of the glacier. Shown are 4>T'e (fine line), <j>e's (dashed line), and <j>s:a (bold line). at 400 h and 2400 h when surface melt is small, (ff:e < (f>e:a over some fraction of the glacier bed. During most of the day, transport from the glacier bed to the aquifer (f>':a is the lowest and least variable of the exchange terms. As expected from the modelled delay between air temperature and subglacial water pressure (Fig. 6.24), maximum wa-ter input <jf:e occurs well before maximum subglacial water pressure (1200 h compared to 1600-2000 h, according to Figure 6.25). 6.5.2 Model sensitivities Using a range of model parameters, I aim to chart a suite of hydrological possibilities, some of which are inspired by field studies reported in the literature. I reserve parameter optimization for Chapter 7 when model results are scrutinized against Trapridge Glac-ier sensor records. Except for the parameter under investigation, sensitivity tests are conducted using the reference model as defined in Tables 6.1, 6.2, and 6.5. Terrain 1 is the testing ground unless otherwise noted, and sinusoidal air temperature variations are applied as in the reference case. Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 137 Water exchange time constants Simple lumped models of glacier hydrology define various water reservoirs (e.g., snow, firn, and ice) and associated residence time constants to explain observed delays between solar noon (peak melt) and maximum daily discharge [e.g., Hock and Noezli, 1997]. Dis-tributed models require a slightly different approach. I use time constants in the para-meterization of water exchange between systems, and the physical transport equations objectively control the delay incurred within systems. Time constants r r : e and r e : s govern water transfer from the glacier surface to the interior, and from the interior to the bed, re-spectively. The time constant governing subglacial-groundwater exchange is determined by physical properties of the aquitard and will be explored later. Surface-englacial exchange time constant rr:e X r : e (hours) X r : e < n o u r s ) Figure 6.26: Sensitivity of key quantities to rr:e, plotted as functions ofrT:e. (a) Mean subglacial water pressure over five days at the test point, (b) Peak-to-peak subglacial pressure wave amplitude at the test point, (c) Time lag between solar noon (peak melt) and the subglacial pressure maximum at the test point, (d) Supraglacial and subglacial discharge. Subglacial discharge is the mean total sheet discharge from the glacier margin, and supraglacial discharge is the mean total surface runoff. Figure 6.26 illustrates the sensitivity of the subglacial pressure regime and water budget to r r : e . Each point represents a 10-day model integration, of which the last five days are used in this analysis. Subglacial water pressure mean, oscillation amplitude, and time lag Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 138 are recorded at the test point. (Figs. 6.26a-c, respectively). Both amplitude and mean pressure decay with r r : e (Figs. 6.26a,b). By increasing the time constant, less surface water encounters each moulin per unit time (e.g., (f>r:e decreases), reducing the amount that ultimately reaches the glacier bed. Figure 6.26c shows the progressively increasing time lag between solar noon and the daily subglacial pressure maximum. Total time lag between temperature and runoff from the glacier basin is a combination of delays introduced within each system and at exchange interfaces. For reference, r e :* = 2h in these tests. Figures 6.26b and 6.26c together show that r r : e regulates both phase and amplitude, as we would expect by analogy to electrical circuits. Changing r p : e also has implications for global discharge partitioning, because r r : e affects hydraulic coupling between the glacier surface and interior. Other factors, such as crevasse density or the presence of snow, also affect surface-englacial coupling. Mean total supra- and subglacial discharge are plotted together in Figure 6.26d. The subglacial component represents water exiting the system from the ice margin, and the supraglacial component is surface runoff. Figure 6.26d demonstrates that in response to r r : e , a change in either discharge component is mirrored by the other. As surface exchange abates (with increasing r r : e ) , melt water is shunted supraglacially to the forefield (or proglacial area), rather than to the glacier bed. When r r : e ~ lh , these two pathways handle equal discharge. Ultimately, supra- and subglacial waters rejoin. It is this melange that is monitored at proglacial stream sites, where hydrochemical analysis can be used to study subglacial weathering processes [e.g., Anderson et al., 1999]. In order to make accurate estimates of subglacial solute or sediment concentration, one must know the relative proportions of supra- and subglacially derived water in the stream. This is especially important for making quantitative estimates of erosion rate. Englacial-subglacial exchange time constant r e : a Figure 6.27 suggests that the model is categorically insensitive to re :", the time constant for exchange across the ice-bed interface. The same diagnostic quantities are shown as in Figure 6.26: subglacial water pressure mean, oscillation amplitude, time lag, and mean sub- and supraglacial discharges. For each of the model tests presented in Figure 6.27, the reference value of r r : e = 0.333 h is used. Discharge partitioning (Fig. 6.27d) is impervious to re:s, because it depends primarily on coupling between the glacier surface and interior. Short of water piling up in moulins and spilling out the top, supraglacial runoff is blind to conditions within and beneath the ice. Surprisingly, conditions under the glacier are equally unperturbed by changes in re:s, as shown in Figures 6.27a-c. Deconstructing the exchange parameterizations <f>r:e and (f)e:s will disclose an explanation. When crevasses and moulins are filling, <f>r:e oc hr/rr:e, where hr is the local runoff depth. As previously discussed, large r r : e hinders meltwater apportionment to the glacier interior. In a closed system, this would cause a buildup of hr, in turn driving more Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 139 3 f (o .2 £TS a. ra * 5 RJ O IB O 0.46 0.42 0.38 0.34 ** * » * * 0.16 c o = o § 1 0.12 a> 5 | 1 0.08 f ° ** 0.04, 6.5 | 6.0 a) ra | 5.5 5.0, T e : S (hours) _ 0.40 'in « 0.30 E. o> S> 0.20 a £ O (0 5 0.10 supraglacial -*—*—* » subglacial o o o o X e : s (hours) Figure 6.27: Sensitivity of key quantities to r e : 5 , plotted as functions of Te:". Vertical scales are the same as those in Fig. 6.26 (a) Mean subglacial water pressure over five days at the test point, (b) Peak-to-peak subglacial pressure wave amplitude at the test point, (c) Time lag between solar noon (peak melt) and subglacial pressure maximum at the test point, (d) Supraglacial and subglacial discharge. vigorous exchange. However, the glacier surface is an open system, and surplus water is whisked away as runoff. Thus, the value of <f>r:e is subject to r p : e . At the ice-bed contact, <j>e's oc (pe — p s ) /r e : s . Increasing r e : s should therefore impede water flow to the sheet. Instead, increasing re:s first drives up pe, because the englacial storage system is laterally confined except for a poorly-connected system of cracks. Pressure in turn drives (jf'*, thereby completing a negative feedback cycle that bolsters (jf'" against changes in r e :*. While this explanation reconciles model results and process parameterizations, it also contends with some aspect of reality. Time constants represent delays caused by com-plicated or unknown processes. Meandering of supraglacial meltwater sloughs and hy-draulic cracking beneath water-filled crevasses are two such examples of complex physical processes. Because the glacier surface is hydraulically unconfined as discussed, meltwa-ter hindrances at the surface manifest themselves subglacially. For example, a dearth of portals into the glacier would curtail water supply to the bed. Inside the ice, hy-draulic pathways are limited and temporary delays can be incurred. Crevasses that are weakly-connected to the bed tend to fill, until the ice ruptures and a better connection is established. Delays are further reduced with the maturation of the englacial drainage network, wrought by the action of water on conduit walls. In this context, transport to Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 140 the bed is ultimately controlled by water at the surface, and the system's sensitivity to rr:e is meaningful. Results in Figures 6.26-6.27 provide a crude picture of what we might better know if these complex processes could be modelled explicitly. Subglacial hydraulic conductivity As outlined in Chapter 4, modelled spatial and temporal variations in hydraulic conduc-tivity are determined by the dependent variable h" and parameters Kmin and Kmax. To illuminate the effects of conductivity, independent of its functional form, I conduct model tests where Ks = Kmin = Kmax. Therefore, hydraulic conductivity is homogeneous, iso-tropic, and constant in time. By varying conductivity over several orders of magnitude, a full range of physically plausible situations can be explored. Hydraulic conductivity is the most important quantity for determining transient subglacial behaviour. It not only dictates the mean water pressure, but controls the magnitude and nature of pressure excursions. Discharge from the glacier margin, as well as the water budget, are strongly determined by properties of the subglacial system. Figure 6.28: End-members of subglacial behaviour induced by differences in hydraulic conductivity, (a) <j>e:s at the test point for K = 10 - 1 ms'1. (b) <f>e:* at the test point for K = 10 - 7 m* - 1. (c) Subglacial water pressure at the test point for a K = 10 - 1 ms~x. (d) Subglacial water pressure at the test point for K = l<r7 ms"1. Figure 6.28 illustrates end-member models of subglacial behaviour controlled by hydraulic conductivity. Figures 6.28a and 6.28b show modelled timeseries of <f>e:* for K — 10_ 1 and 10 _ 7 ms _ 1 , respectively. As the source rate, <f>e:s represents the forc-ing applied directly to the sheet. Figures 6.28c and 6.28d show the resulting subglacial hydraulic response. For the highly conductive sheet (Fig. 6.28c), a smoothly varying Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 141 input evokes a smoothly varying response with a mean pressure less than 10% of the ice-overburden pressure. The resistive sheet (Fig. 6.28d), however, fails to meet the demands imposed by (f>e:s. Basal pressures develop in excess of flotation, causing water to back up into englacial voids. A striking change in system behaviour occurs when the voids are completely full (Fig. 6.28d). During these times, a constant rate of water exchange persists (Fig. 6.28b, interval A), such that englacial storage volume remains fixed. Con-sequently, subglacial water pressure attains a stable maximum (Fig. 6.28d, interval A). When surface melt subsides, crevasses drain slightly. The source rate (f>e'a drops below the baseline value, and there is an accompanying reduction in subglacial water pressure (Figs. 6.28b and 6.28d, interval B). When surface melt recommences, crevasses rapidly refill and the exchange rate increases temporarily (interval C). Pressure quickly reattains its maximum value. 10"8 10"6 10"* 10"z 10" in m E o Ul fc. a JC O (A 10"° 10"° 10"* 10"' 10" Hydraulic conductivity (m s"1) Hydraulic conductivity (m s"1) Figure 6.29: Selected quantities affected by subglacial hydraulic conductivity, expressed as a function of conductivity from 10~7 to 10 - 1 m s _ 1 . The dashed lines partition two modes of behaviour as identified in Figure 6.28. (a) Peak-to-peak amplitude of subglacial pressure oscillations at the test point, (b) Mean subglacial water pressure at the test point, (c) Mean total discharge through the sheet at the glacier margin, (d) Mean total volume of water stored within and beneath the glacier (plotted independently). Given reference values of the degree-day factor DDF and total englacial storage ca-pacity VT/VI, the transition between end-member models described above occurs at K — 10 - 3 ms - 1 . I do not suggest that this absolute value of conductivity is significant. Rather, I propose that the existence of such a transition is significant, and demonstrate Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 142 that it can occur at physically plausible values of conductivity. Figure 6.29 presents this transition as expressed by four hydrological quantities. The subglacial pressure mean and oscillation amplitude are shown as a function of hydraulic conductivity in Figures 6.29a and 6.29b. Dashed lines partition the two behavioural modes as shown in Figure 6.28. To the left of this line, englacial storage is completely full for some time each day. The duration of full storage increases as the subglacial sheet becomes more resistive. For a highly conductive sheet, oscillation amplitude decreases with conductivity because more resistance is required for substantial periodic pressure build-up. Amplitude also diminishes as K —> 0, in part, because pressure becomes pinned at a maximum limited by the ice thickness (Fig. 6.28d). This is compounded by inefficient drainage. A dramatic change in the subglacial pressure regime, indicated by the mean pressure (Fig. 6.29b), occurs concomitantly with the behavioural switch. Those systems in which englacial voids fill up (left of the dashed line) operate at or above ice flotation levels, while others (right of the dashed line) operate well below. Discharge from the glacier margin is naturally affected by the hydrological state of the bed, as illustrated in Figure 6.29c. Near-zero discharge levels occur in the resistive sheet, but climb smoothly as conductivity increases. High discharge from the glacier margin flushes water from storage as shown in Figure 6.29d. The mean volume of both sub- and englacially stored water declines as the sheet becomes transmissive. Relatively constant mean storage levels are maintained in the resistive systems (left of the dashed line), differing only because crevasses remain full for different lengths of time each day. Without changing the porosity or thickness of the subglacial sediment sheet, its capability of storing water can be reduced by a factor of three due to changes in hydraulic conductivity. For K = 10 - 1 m s - 1 , the subglacial sheet is so transmissive that englacial cavities are nearly devoid of water. These simple tests illustrate that the model is very sensitive to subglacial hydraulic conductivity. Depending on other quantities, such as surface melt rate or englacial stor-age capacity, different behavioural modes can be excited by spatial or temporal changes in conductivity. In the example presented in Figures 6.28-6.29, mode switching is con-tingent upon storage being filled. Glacier surging is one situation in which hydraulic properties of the glacier bed change in time and crevasses (and surface potholes) have been observed to be filled with water [e.g., Kamb et al., 1985]. In-situ observations of surging glaciers show that subglacial pressures are higher and less variable than those observed during quiescent periods. Water is impounded beneath the glacier during a surge, because no efficient drainage system exists. The development of a drainage sys-tem deactivates the surge and breaks the hydraulic seal at the glacier margin. If efficient and impaired drainage structures can be effectively represented by different values of subglacial hydraulic conductivity, these sensitivity tests demonstrate a hydrologically motivated instability akin to surging. Dramatic changes in both discharge and stored water, as shown in Figures 6.29c and 6.29d, have been observed during surge onset and termination [e.g., Kamb et al., 1985]. Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 143 Spatial variability A good example of spatially heterogeneous subglacial hydraulic conductivity is given by Hubbard et al. [1995] for Haut Glacier d'Arolla, Switzerland. Instrument studies of the glacier bed reveal a "variable pressure axis" (VPA) that experiences extreme daily pressure fluctuations. These fluctuations are attenuated with distance from the VPA and cannot be identified beyond ~70m. Hubbard et al. suggest a conceptual model where the VPA is centred on a conduit that delivers water to the adjacent subglacial sediment layer during the day. Their data indicate a diurnally reversing hydraulic gradient that is responsible for driving water back toward the channel at night. This cycle can be explained by a hydraulic conductivity gradient in the subglacial sediment, where K = 3 x 10 - 4 m s _ 1 near the channel and K = 5 x 10 - 7 m s _ 1 distal from the channel. Preferential washing away of fine sediment in the channel vicinity could explain such a gradient. To describe the variation of hydraulic diffusivity D (calculated from their data) with distance from the channel, Hubbard et al. suggest the following relationship: D = 43.5 exp(-0.091 x), (6.8) where x is distance perpendicular to the VPA. Multiplying this function (given in m 2 s _ 1) by specific storage Ss yields a relationship for hydraulic conductivity K. To demonstrate some glaciological realism, I attempt to model the scenario just described using Terrain 2 (U-shaped valley) and a heterogeneous conductivity field. To mimic the VPA, I insert moulins along the glacier centreline where I set Kx = 1 x 10 _ 2 ms _ 1 . Taking a value of 6.4 x 10 _ 6 m _ 1 for specific storage Ss [Murray and Clarke, 1995], I compute discrete cell-interface values of hydraulic conductivity as a func-tion of distance from the glacier centerline, according to the prescription of Hubbard et al. Outside of the VPA, I assume conductivity is isotropic and only a function of distance. For cells further than 70 m from the channel, the conductivity is homogeneous (and iso-tropic) and equal to 5 x 10~ 7ms - 1. Figure 6.30 illustrates my numerical implementation of this conductivity distribution. While I have made no attempt to introduce other information pertaining to Haut Glacier d'Arolla (e.g., melt rates), results presented in Figure 6.31 bear a qualitative resemblance to the observations of Hubbard et al. First, the diurnal pressure wave prop-agates at least 40, but no more than 80 m from the VPA. Hydraulic head in the VPA, and in three cells that are 40, 80, and 120 m away, is plotted in Figure 6.31a. The spa-tial arrangement of these test points is shown in Figure 6.31b. Oscillations in the VPA are largest due to the direct input of surface melt water. The adjacent cell experiences damped and delayed variations about a similar pressure baseline. At 80 m and beyond, hydraulic head is high and stable. In the observed records from Arolla, hydraulic head Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 144 Variable pressure axis Hydraulic conductivity (m s'1) * - > i - » 1.0 x10"2 4.5 x10' 5 7.3 x10' 6 1.2 x10"6 b 1 • i I 1 • I i 4 i • 1 i 4 i • 1 i i I Figure 6.30: Discrete implementation of a variable pressure axis (VPA) inspired by Haut Glacier d'Arolla, Switzerland [Hubbard et al., 1995]. (a) VPA in the context of Terrain 2. (b) Locations and magnitudes of hydraulic conductivity imposed across grid-cell interfaces. The VPA itself is assigned an arbitrarily low conductivity to mimic the presence of a conduit, (c) Legend: conductivity magnitudes. is also high and stable far from the channel, but its value is exceeded by daily maxima near the channel. Hydraulic head gradient between adjacent nodes in Figure 6.31b is plotted in Figure 6.31c. Positive values of hydraulic gradient indicate water outflow from the channel and vice versa. Gradients between the channel and the nearby sediment sheet demonstrate clear diurnal reversals. Outflow occurs after midday and inflow begins around midnight. This timing is similar to that observed near the channel beneath Arolla. Further from the channel, between 40 and 80 m, the gradient magnitude varies diurnally but flow is always toward the VPA. Finally, between 80 and 120 m, the gradient is nearly zero, implying little or no flow. In the data presented by Hubbard et al., hydraulic gradients decline similarly toward zero with distance from the channel, but all show diurnal reversals. While the results of Hubbard et al. are more spatially dense, and therefore more informative, the model succeeds in capturing the two most salient features of this sit-uation: attenuation of the diurnal wave within 70 m from the VPA, and a diurnally reversing hydraulic gradient. The magnitude of hydraulic head and gradient variations is much greater in the observed data, due in part to the arbitrary ablation rate I have imposed. Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 145 VPA b 1 2 3 • • i i • / / 0 0.4 0.8 1.2 1.6 2.0 Time (Days) 0 0.4 0.8 1.2 1.6 Time (Days) Figure 6.31: Model simulations using the hydraulic geometry of Figure 6.30. (a) Hydraulic head (•pressure head plus elevation) for the VPA and three adjacent grid cells, (h) Geometry of test points relative to the VPA. Each point is separated by 40m. (c) Hydraulic gradient computed between each of the four nodes in (b). Flow reversals are suggested by temporal changes in the sign of the gradient. Subsurface properties Due to communication between the glacier bed and groundwater aquifer, changes in subsurface properties should have implications for basal drainage. Poorly constrained characteristics of both the till cap and aquifer include thickness (d* and da respectively) and hydraulic conductivity (K* and Ka). Changes in either property of the aquifer (da, Ka) will have a similar effect on the system. For example, by increasing thickness or conductivity, the transport capacity of the aquifer increases. In contrast, changes in till cap thickness and conductivity work against each other. Increasing the till cap thickness retards the transmission of water, while increasing its conductivity hastens it. Uncertainty in conductivity (both Ka and K*) spans a much greater range—up to several orders of magnitude—than does uncertainty in layer thickness. Therefore, I vary conductivity rather than thickness to yield the most illustrative results. From these tests, the effects of varying layer thickness can be qualitatively surmised. Table 6.6 summarizes the results of independently varying Ka from 5 x 10~r m s _ 1 to 1 x 10 - 3 m s _ 1 and Kl from 1 x 10~12 m s _ 1 to 1 x 10 - 6 ms _ 1 . Values of Ka and Kl are constrained by the requirement that Ka > 100 Kt. This limitation exists because the model neglects horizontal transport in the till cap [Chorley and Frind, 1978]. Table 6.6 attests to the modest impact of varying aquifer conductivity over a large range of values. While the total groundwater discharge across the model boundary changes dramatically (Column 2), the saturated areal fraction (Column 3) remains essen-tially constant around 90%. Incidentally, this value is approximately double that for the equilibrium case (refer to Table 6.3). Five-day mean subglacial water pressure at the test point (Column 4) varies by about 5% and total discharge from the ice margin (Column Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 146 Table 6.6: Sensitivity of various quantities to Ka and K* (Column 1). Asterisks denote reference model values. (Column 2) Total aquifer discharge across the boundary at midnight. (Column 3) Areal fraction of saturated aquifer (ha > 1.2 m). (Column 4) Mean subglacial water pressure (reported as a flotation fraction) over five days at the test point. (Column 5) Mean total ice-marginal discharge (e.g., from the subglacial sheet) over five days. (Column 6) Mean water exchange between sheet and aquifer (4>s:a) over five days at the test point. Ka(ms *) Aquifer discharge (xlO-'m's- 1) Areal fraction ha > 1.2m (pfi(26,16)) Discharge (m^"1) <^»(26,16)> (x lO^ms - 1 ) 5 x lO"7 0.0424 0.899 0.434 0.311 0.510 1 x lO"6 0.0843 0.899 0.434 0.311 0.937 5 x 10-6 0.873 0.899 0.433 0.308 4.37 1 x 1CT5 2.36 0.899 0.431 0.305 7.96 5 x 1(T5 14.8 0.900 0.425 0.292 22.2 1 x 1CT4* 24.6 0.899 0.421 0.283 30.3 5 x 1CT4 57.8 0.887 0.414 0.267 45.7 1 x 1CT3 89.5 0.883 0.412 0.269 48.9 1 x 10~12 7.31 0.891 0.435 0.312 0.0539 1 x lCT 1 1 7.53 0.891 0.434 0.312 0.536 1 x io - 1 0 9.80 0.892 0.432 0.307 5.05 i x icr 9* 24.6 0.899 0.421 0.283 30.3 1 x 1(T8 24.2 0.900 0.402 0.247 72.2 l x i cr 7 8.13 0.900 0.396 0.202 76.3 1 x 10"6 5.34 0.899 0.397 0.165 75.1 5) decreases by 15% as Ka goes from 5 x 10 - 7 to 1 x 10 _ 3 ms _ 1 . This change in sheet discharge occurs because the aquifer plays an increasingly important role in collecting subglacial water. Column 6 documents this progression in terms of (f>*:a, averaged over five days at the test point. Exchange increases by two orders of magnitude as the aquifer becomes more transmissive. While the mean exchange is always positive at the test point, Figure 6.32 shows that diurnal reversals occur for a low conductivity aquifer. Modelled timeseries of <f>g:a are shown in Figure 6.32a for selected Ka. Amplitude and phase of <f)s:a are impervious to Ka, while the baseline is affected in a nonlinear fashion. For Ka > 10 _ 5 ms - 1 , 4>s:a > 0 and water is perpetually pumped into the aquifer. When Ka < 10~6, diurnal reversals in the exchange cycle inject water back to the glacier bed at night. Longitudinal exchange profiles at midnight are plotted in Figure 6.32b for values of Ka reported in Table 6.6. Exchange decreases substantially toward the glacier terminus for Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 147 w £ o o c o X A A A A A A V V l i K a = 10'5ms1 /r, = 10"6ms"1 2 4 6 8 Time (Days) 10 (0 E CO I o 0) o> c (B £ O x UJ ! 5 ^ 1 — e •—fal *-T 1" -• ^7 ^8 400 800 1200 1600 2000 Longitudinal distance (m) Figure 6.32: Sensitivity of <j>g:a to hydraulic conductivity of the aquifer Ka. (a) Timeseries of <j>*:a at the test point: Test point location is indicated by the vertical dashed line in (b). '(b) Longitudinal profiles of <j>s:a at midnight for different values of Ka (in ms'1) as follows: (1) 1 x 10 - 3 ; (2) 5 x 10~4; (3) 1 x lO" 4 ; (4) 5 x lO" 5 ; (5) 1 x K T 5 ; (6) 5 x lO" 6 ; (7) 1 x 1(T 6; (8) 5 x 1(T 7. more transmissive aquifers (profiles 1-4), otherwise remaining relatively constant (profiles 5-8). Profiles 7 and 8 represent resistive aquifers in which diurnal reversals occur, except at the glacier terminus. Although </»*:° is strongly affected by Ka, the sheet competently buffers itself against changes that may have been caused by <f>":a. From a modelling perspective, it is welcome news that small uncertainties in subsurface parameters should not cloud our picture of basal hydrology. Table 6.6 shows that changes in till cap conductivity (K*) produce similar trends, except in the case of aquifer discharge. Figure 6.33 plots four of the diagnostic variables as functions of K* and Ka together. Mean subglacial water pressure (Fig. 6.33a) and mean total ice-marginal discharge (Fig. 6.33b) decline with increases in either Kl or Ka. For individual quantities, the results are nonunique. For example, the mean subglacial flotation fraction is 0.43 for more than one combination of Kl and Ka. However, am-biguity can be resolved by inspecting more than one variable. Changes in Kl generally elicit a larger range of subglacial responses (cf. changes in Ka) due to direct coupling between the till cap and subglacial sheet. Figure 6.33c demonstrates that water exchange is reduced by hydraulic resistance in the till cap (low values of K*). Both (<£ s :a(26,16)) and (pfi(26,16)) (Figs. 6.33a and 6.33c) change rapidly over a restricted range of Kl and Ka. These transitions are centred on the reference values K* — 10~ 9ms _ 1 and Ka - 10 _ 4 ms _ 1 . For very low till cap conductivity Kl, the aquifer is effectively isolated from diurnal oscillations in the sheet. For high Kl, pressure pulses are transmitted through the exchange term (f>":a. In Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 148 0.44 10~10 10"8 10"6 10^  10"12 10"10 10~8 10"6 10~ 0.10 10"12 10 1 0 10"8 10"6 10"4 Hydraulic conductivity (m s"1) 10"12 10"10 10"8 10"6 10"" Hydraulic conductivity (m s"1) Figure 6.33: Sensitivity of selected variables to K* (asterisks) and Ka (open circles). Values are tabulated in Table 6.6. (a) Mean subglacial water pressure over 5 days at the test point, (b) Mean total subglacial discharge from the ice margin over 5 days, (c) Mean water exchange rate <j>s:a over 5 days at the test point, (d) Total discharge from the aquifer boundary at midnight. the transition region, <f>s:a oscillates about a lower baseline than in the high conductivity case. These three regimes (low, transitional, and high Kl) can be loosely likened to an overdamped, underdamped, and undamped oscillator, respectively. Total discharge from the aquifer boundary at midnight is shown in Figure 6.33d, as a function of Kl and Ka. While aquifer discharge escalates rapidly with Ka, it changes only slightly through the whole range of K*, reaching a maximum between Kl = 10~9 and 10 - 8 ms - 1 . Its initial increase, beginning around Kl = 10 - l o ms _ 1 , can be attributed to more efficient water collection from the sheet. Subsequent accumulation of water in the aquifer precipitates a pressure buildup, that eventually drives artesian flow in the glacier forefield (through the exchange term <j>r'a). Although total groundwater transport increases with K*, water intercepted by the proglacial runoff system accounts for the reduction in aquifer boundary discharge, as shown for large K* (Fig: 6.33d). If this forefield outlet were unavailable, aquifer boundary discharge for = 10 - 6 m s _ 1 would be 0.0333 rather than 0.00534m3 s - 1 . Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 149 o 4-1 5 o E <s 0.4 0.8 1.2 1.6 Time (Days) 2.0 so .!= JZ £ w <0 C •O O >. 2 o5 •= °? (/> 12 10 8 6 4 2 0 b \ ha *- * 0 s ; a ^ \ 10" 10" 10 -7 10" K* (m s-1) Figure 6.34: Illustration and quantification of time lags due to Kl. (a) Groundwater thickness ha at the test point for selected values of Kl (in ms'1): (1) 10~6; (2) 10~7; (3) 10~8; (4) 10~9. Dots identify the same daily maximum on each curve, (b) Delay of 4>>:a and ha with respect to solar noon, as a function of K*. Till cap conductivity and thickness act in conjunction as a time constant for water transfer from the sheet to aquifer. Together they control the phase of <£s:a and ha relative to the forcing. Figure 6.34a illustrates this effect with timeseries of ha at the test point for various values oi Kl > 10 - 9 ms _ 1 . Below 10 _ 9 ms _ 1 , diurnal variations in the aquifer are absent. Note the change in offset between curves, as well as the amplitude damping and phase. Time lags relative to solar noon for (f>s:a and ha are quantified in Figure 6.34b. Groundwater thickness ha lags (j>s:a by a constant ~6 hours, while the lag with respect to solar noon decreases with Kt. These additional effects distinguish Kl from Ka, as can be confirmed by referring back to Figure 6.32 where phase and amplitude of <f>":a remain constant with changes in Ka. Spatial variability Studies of glacierized and glaciated areas have concluded that subglacial aquifers are capable of evacuating 25-100% of all water at the glacier bed [e.g., Sigurdsson, 1990; Boulton et al, 1995; Haldorsen et al, 1996; Piotrowski, 1997]. In general, this body of work involves two-dimensional (vertical) ice flow-line models applied to large ice masses, where subsurface hydrostratigraphy is relatively well-known (e.g., European paleo ice-sheets). Workers agree that subsurface geometry and the presence of permafrost play commanding roles in the efficiency and topology of large-scale glacial drainage. Inferences are made about drainage at the glacier bed itself, but this component has not been modelled in conjunction with subsurface aquifers. With this in mind and the tools in hand, I present two simple alternatives to the homogeneous reference model aquifer. The first is partially invaded by permafrost, and the second has a variable thickness defined Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 150 by irregular bedrock topography. Permafrost in periglacial areas impedes drainage because freezing reduces sediment conductivity [Williams and Smith, 1991]. In this example, I assume frozen ground except beneath the glacier, as illustrated in Figure 6.35a. Permafrost affects both the till cap and aquifer in the periglacial region. Where it is frozen, I assign an isotropic conductivity of 10 - 1 2 ms _ 1 to the till cap and 10 _ 1 0 ms _ 1 to the aquifer [Williams and Smith, 1991]. This ring of permafrost has a less startling effect on the subglacial system than it would under a poorly drained ice sheet. Under ice sheets, high water pressures are expected because drainage is impeded by low slopes and large distances between interior points and the ice margin. This implies that subsurface drainage might play a significant role in evacuating basal water. At the glacier-scale, drainage along the ice-bed interface is much easier, hence the subsurface layers are less important. Discharge from the glacier margin undergoes a 30% increase from 0.238 to 0.308 m 3 s - 1 , while discharge loss from the aquifer is reduced from 2.46 x 10 - 2 to 2.46 x 10 - 8 m 3 s _ 1 . Despite these differences, mean subglacial water pressure changes by less than 2%. At the test point, leakage flux <f>s:a into the aquifer is reduced by a factor of 3. Closer to the glacier margin, >^*:a is permanently reversed. The horizontal position of this tran-sition from (f>s:a > 0 to <f>>:a < 0 fluctuates by 360 m with the daily melt cycle. Figure 6.35b shows longitudinal profiles of (f>':a at 0600 and 1700 h, demonstrating this lateral migration of the transition line. At 1700 h, peak flow to the aquifer (near the head of the glacier) and peak flow back to the bed (near the terminus) are approximately equal in magnitude. At 0600 h, the profile is shifted upglacier such that flow to the aquifer is reduced and flow to the bed is enhanced. Figure 6.35: Geometry of periglacial permafrost and its effect on <f>s'a. (a) Model geometry showing frozen and unfrozen zones of the till cap and aquifer, (b) Exchange profiles (<$>*'a) along the glacier centreline for 0600 and 1700 h. Positive values indicate water flow toward the aquifer. Vertical lines mark the transitions from downward to upward flow across the till cap. The glacier extends the entire length of the abcissa. Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 151 This model test suggests that small, well-drained glaciers are not prone to the hydro-logical cycles of large ice-sheets in permafrost areas [e.g., Cutler et al., in press], where a buildup of basal water (due to inadequate drainage through permafrost) culminates in an outburst flood. Factors that contribute to water accumulation under ice sheets include low surface and bed slopes, spatially extensive ice, and thermal damming of the ice margin. Like the effects of permafrost, effects of aquifer geometry on subglacial drainage are also small, despite noticeable changes in groundwater flow patterns and water exchange with the sheet. Figure 6.36 summarizes these changes for an aquifer with variable bottom topography. In this test, aquifer thickness varies smoothly between about 1-6 m, while total aquifer volume remains the same as in the reference case. The centreline profile of saturated thickness (ha/na) is shown in Figure 6.36a with glacier bed and bedrock elevations. To simplify the figure, bed slope has been removed. Along this profile, the saturated horizon approximates the base of the till cap, indicating complete saturation of the aquifer. The areal extent of complete saturation (ha > 1.2 m) considering the whole domain is around 90%, as in the reference case. Given that surface melt and total aquifer volume are the same as in the reference model, this result is not surprising. Figure 6.36b shows that variable aquifer thickness distorts the water exchange profile. However, mean values of exchange, subglacial sheet pressure, and discharge from the ice margin are nearly identical to reference model values. Longitudinal distance (m) ng = 1.60 x 10"5 m2s'1 Q9 =l.09xi0"5m2s-1 max max Figure 6.36: Effects of nonuniform aquifer thickness, (a) Saturated thickness of the aquifer in the context of glacier bed and bedrock elevations. Bed slope has been removed for clarity, (b) Centreline profile of<j>s:a. (c) Groundwater flow vectors in the nonuniform aquifer, (d) Groundwater flow vectors in the reference model. Groundwater flow patterns change markedly with aquifer thickness variations. Fig-ures 6.36c and 6.36d compare vector fluxes for this case to the reference model. Figure Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 152 6.36c shows preferential flow in the vicinity of bedrock depressions and higher maximum fluxes than in the reference case. The overlying glacier and its source water provide a di-ligent driving force that prevents groundwater flow reversals. Despite spatial differences in groundwater flux, total discharge from the aquifer is almost identical in Figures 6.36c and 6.36d. 6.5.3 Geometric effects Several special cases outside of systematic parameter variation warrant mention in the context of model sensitivities. These arise from geometric considerations, including the shape of englacial voids and nonuniformly distributed surface sources. Englacial void space Depending on the particulars of englacial void geometry, systems dominated by crevasses may behave identically to those dominated by moulins. However, this is not generally true. Figure 6.37 shows the englacial component of the water budget for three different storage morphologies: (1) moulins, (2) slightly-tapered crevasses, and (3) wedge-like crevasses. Each morphology has the same volume. Both types of crevasses are 40 m long, but taper is increased in case (3) by setting f2 = 0.05, instead of 0.5. Parameter f2 scales the crevasse area exposed to the bed relative to its area open at the glacier surface. This change is clearly expressed in the water budget, but hardly evident in the subglacial response (not shown). Approximately the same volume of water reaches the bed in all three cases, and differences in the amplitude of subglacial pressure oscillations are small (< 10%). 6000 10001 0 2 3 Time (days) 4 5 Figure 6.37: Englacial component of the water budget for systems comprised of (1) moulins, (2) slightly-tapered crevasses, and (3) wedge-like crevasses. Total englacial void volume is the same in each case. Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 153 The coupled hydrological system strives for potential equilibrium by adjusting its exchange terms. In this case, the volume of englacially stored water changes to establish the optimum pressure balance. Between each of the four model layers, exchange is driven by differences in water pressure. Changes in volume produce larger changes in pressure when a crevasse tapers with depth. Consequently, as shown in Figure 6.37, wedge-like crevasses store the smallest amount of water. This example illustrates that englacial void geometry, not just volume, is important in quantifying water storage for systems where coupling exists between surface, ice, and bed. For glaciers prone to outburst floods, the source water is often dynamically isolated from the rest of the system until flood initiation. Subglacial response to sparse sources Up to this point, surface melt water has been allowed to enter the ice in any model grid-cell, simulating a glacier uniformly riddled with holes. More realistically, points of entry are unevenly distributed across the glacier surface. In this test, moulins occupy three discrete bands across the glacier as indicated in the background of Figure 6.38. The subglacial effects of sparsely distributed sources are evident in Figure 6.38. During periods of high melt (1200 and 1600 h), subglacial water pressure most directly reflects the surface structure, and spatial gradients in water pressure are most dramatic. At night, when the melt rate is low, pressure diffuses both upstream and downstream away from the source. Diffusion of the pressure wave introduces a time lag for areas not directly connected to the surface. For example, by comparing brightnesses in Figure 6.38, one can identify that the daily pressure maximum downglacier from the source bands occurs around 400 h. 400 h 800 h 1200 h 1600 h 2000 h 2400 h Water pressure (fraction of flotation) Figure 6.38: Time slices of subglacial water pressure for the reference model with sparsely distributed sources. Moulins are present in three bands coincident with the stripes in the background. Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 154 Compared to the case of uniformly distributed moulins, source sparsity results in an increase of the mean supraglacial basin runoff from 493 to 1230 m 3 s _ 1, and a correspond-ing decrease in mean ice-marginal discharge from 0.284 to 0.0853 m 3 s _ 1 . These changes reflect the diminished probability of encounter between surface melt water and moulins. 7 16 (A E N 12 o E 8 c i £ O UJ 0 • • — 0 I • flj \ V 0.4 0.3 0.2 0.1 0 s:a c o i o «+-o c o 1 (8 3 CO U) a> 400 800 1200 1600 2000 Longitudinal distance (m) Time (days) Figure 6.39: Intercomponent water exchange patterns and propagation of the diurnal pressure wave, (a) Centreline exchange profiles at 1600 h. <f>r'-e and <j>e:> are indistinguishable on this scale. <j>a:a is plotted on a magnified scale, labelled on the right. Shaded areas indicate moulin locations, (b) Location of moulins (shaded bands) and test points 1-5. (c) Modelled timeseries of subglacial water pressure for points 1-5. Note the migration of the pressure maximum to later times and the waveform evolution from location 1 to 5. Water transfer is instructive in trying to understand the connection between source sparsity and subglacial pressure patterns. Figure 6.39a displays profiles of water transfer along the glacier centreline for each exchange interface. Shaded bands indicate moulin locations. As expected, (f>T'e is highest at the upstream end of each band where runoff first encounters an opening. This pattern is closely matched by <f>e:*, indicating negligible horizontal transport within the ice. Leakage to the groundwater system {(j>"'a) is shown for comparison on a different vertical scale. Its smeared profile suggests that water is redistributed at the glacier bed before being conveyed to the aquifer. Modelled pressure records from points labelled in Figure 6.39b are plotted together in Figure 6.39c. Location 1 occurs in the lowermost source band. Horizontal distances between each point and the nearest source are recorded in Table 6.7, along with response attributes shown in Figure 6.39c. Simple estimates of the pressure wave migration veloc-ity made by dividing source distance by time lag are also reported in Table 6.7. Velocities decline nonlinearly with distance from the source. In Figure 6.39c, note the migration of diurnal pressure maxima to later times as distance from the source increases. Beyond 440 m (location 5), oscillations are completely suppressed. This time lag is accompanied by a breakdown in symmetry between rising and falling limbs of the pressure wave. Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 155 Table 6.7: Characteristics of decaying and delayed subglacial signals. Positions are labelled in Figure 6.39 and distance is relative to position 1 where moulins are present. Implied velocity is calculated as distance divided by time lag. Position Distance Relative amplitude Time lag Implied velocity 1 0 1.0 0 -2 120 m 0.54 2.4h 1.4 x lO"2 ms- 1 3 280 m 0.23 8.5 h 9.1 x lO^ms" 1 4 360 m 0.046 22.5 h 4.4 x l O ^ m s - 1 5 440 m 0 - -6.6 Summary Using reference model parameters from Trapridge Glacier and other field studies, I have shown that the theoretical and numerical model described in Chapters 3-5 successfully mirrors reality when applied to idealized glaciers. Ice-surface and bed geometry strongly dictate equilibrium drainage structure as realized in these tests, but are secondary to the physical properties of each model layer in determining the global water budget. As the water source, the glacier itself uniquely influences basin hydrology, regardless of its particular geometry. In all cases, the groundwater aquifer remains saturated be-neath the ice, with artesian pressures obtained in the glacier forefield. In steady state, discharge from the subglacial sheet is by far the most important avenue for water evacu-ation. Perturbations to profiles of equilibrium pressure and exchange with a frozen glacier margin reflect this. Transport of surface runoff operates independently of subglacial and subsurface conditions, except where they act as additional sources. Comparison of subglacial drainage structure derived from modelled results versus hydraulic potential analysis, as introduced in Chapter 2, demonstrates that a compli-cated model is unwarranted in selected situations. In general, the hydraulic potential method overpredicts relative flow strength at the ice margin, and breaks down when bed topography is significant if subglacial water pressure is assumed equal to flotation. Some knowledge of baseline water pressure or implementation of a statistical distribution of water pressure is a possible improvement for mechanical models of ice and till that incorporate parameterizations of hydrology. Simple transient tests, focused on Terrain 1 with diurnally varying surface input, more lucidly illustrate intercomponent relationships. Time lags between the glacier sur-face and individual modelled systems (using reference model parameters) are in line with values reported in the literature or obtained from Trapridge Glacier instrument records. Parameters that elicit notable sensitivities include the time constant for surface-englacial exchange rr:e and subglacial hydraulic conductivity Ks. Meltwater destination is deter-mined by r r : e , as it regulates the coupling between the glacier surface and what lies Chapter 6. A P P L I C A T I O N T O S Y N T H E T I C G E O M E T R Y 156 beneath. Ultimately, this would affect records of turbidity, electrical conductivity, and solute concentration in a proglacial stream. Hydraulic conductivity can induce differ-ent drainage modes reminiscent of surging and quiescent behaviour. Spatial variations in conductivity can be introduced to stimulate diurnally-reversing flow direction as observed by Hubbard et al. [1995] beneath Haut Glacier d'Arolla, Switzerland. Glacier bed hydrology seems largely impervious to subsurface parameters, but this, as the next chapter will reveal, is uniquely due to the permeable ice margin providing a ready hydraulic outlet. A frozen margin, along with additional complexities associated with Trapridge Glacier, will be introduced in Chapter 7 where direct data comparisons are possible. Results presented in Chapter 6 however, exhibit a qualitative similarity to observed and hypothesized glaciohydraulic phenomena. Thus the model can be evaluated based on its ability to reproduce observed behaviour and can be utilized to test the viability of hypotheses inspired by those observations. Together, these suggest that a great deal of glaciological exploration is possible even without real topography. Chapter 7 A P P L I C A T I O N T O T R A P R I D G E G L A C I E R With its history of intensive instrumentation, Trapridge Glacier is an ideal testing ground for a hydrological model. Data collected over the last decade elevate accountability by providing a quantitative means of scrutinizing model results. The first part of this chapter attends to refining the reference model for Trapridge Glacier by comparing simulated subglacial signals to actual instrument records. The second part turns to an investigation of glacier hydrology on two key timescales. Equally inspired by usual and unusual sensor records, I look at rapid hydrological disturbances (previously studied by Stone [1993]) and seasonal transitions in the drainage system. I begin by introducing the remainder of the surface melt and transport model. 7.1 Implementation of the ablation—runoff model Real topography and geographic location allow me to reintroduce the full ablation model as described in Chapter 3. In addition, knowledge of the glacier surface merits a cus-tomized representation of crevasse distribution for the runoff model. Data requirements, calculations, and input parameters for these tasks are outlined below. 7.1.1 Geometric data requirements and calculations In order to calculate potential direct solar radiation, digital topographic information is required for the glacier and its surroundings. In particular, a DEM extending to the sky-line (as perceived from any point on the glacier) is needed to compute shading. Chapter 2 discusses the construction of high-resolution DEMs for the glacier itself. The Geological Survey of Canada (GSC), in cooperation with Parks Canada, has prepared a DEM of the surrounding area with 400 m grid spacing. Figure 7.1 presents this information in perspective and contour. To splice the two datasets, I interpolate the large DEM onto a 40 m grid and insert the radar-derived DEM for the glacier. For the following applications, I use Trapridge Glacier DEMs with 40mx40m grid cells. Uncertainty in the rendering of surrounding terrain is unimportant, so long as the skyline is accurately portrayed. However, it is imperative to retain topographic detail in the area of interest. Contour mismatch arising from the 157 Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 158 Figure 7.1: Topography surrounding Trapridge Glacier interpolated from the 400 m x 400 m GSC DEM. (a) Topographic mesh. Inset shows study area location, (h) Topographic contour map. Contour interval is 120 m. splice is evident in Figure 7.1b, but does not adversely affect radiation calculations. Mts. Wood and Steele (Fig. 7.1) with their connecting ridge define our skyline to the south and west. To the north and east, the skyline extends to the edge of the DEM, except where the prominent Steele valley exits the basin. Along this narrow corridor, the line of sight extends an additional 35 km. Using the composite DEM (Fig. 7.1), digital models of slope and aspect were com-puted by Regine Hock at the Climate Impacts Research Centre, Kiruna, Sweden. I used her code to compute radiation from these models. Figure 7.2a shows slope contoured for pixels within the model domain. In this image, ice flow is generally from top to bottom. The glacier margin is exceptionally bright because it is steep on all sides. Figure 7.2b shows the glacier illuminated from the east (bottom of image), based on the digital rep-resentation of aspect. Brightness represents the horizontal projection of the slope normal in degrees away from east, such that slopes facing east are light and slopes facing west are dark. Figure 7.2c illuminates the glacier from the north (right) in a similar fashion. Using the information summarized in Figs. 7.1 and 7.2, potential direct clear-sky insolation can be calculated from Equation (3.2). Then, without any additional meteo-rological quantities, spatially variable radiation can be estimated for any time. Figure 7.3a shows radiation incident on Trapridge Glacier at 1400 h on four different days in 1997. Radiation at four different times during day 200, 1997 (July 19th), is shown in Figure 7.3b. The changing strike of the radiation pattern from 1200 h to 1500 h (Fig. 7.3b) discloses the Sun's path from southwest to northwest. Computer disk space places a practical limit on the temporal resolution of radiation calculations. For each time (hour, day, year) a data set of size N = nx x ny must be Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 159 stored. For the applications presented in this chapter, I have computed spatially variable radiation for each hour of the day, every five days, from day 140 to 280. Thus, the same diurnal cycle is used for five consecutive days. (° from East) (° from North) Figure 7.2: Digital representations of slope and aspect for the model domain. Ice flow is generally from top to bottom. The glacier margin is discernible in the lower half of each panel, (a) Slope magnitude, (b) Aspect expressed in degrees away from east. Bright areas have eastern exposure, (c) Aspect expressed in degrees away from north. Bright areas have northern exposure. 7.1.2 Determination of melt To compute ablation now requires parameters MF and asnow/iCe and a timeseries of air temperature. In principle, estimation of these parameters requires little more than a tape measure. Ideally, discharge records from the glacier terminus and direct measurements of snow/ice ablation and density are used together to optimize MF and a s n o w / i c e . A com-bination of factors makes this optimization process unworkable for Trapridge. Discharge from the glacier is not neatly confined to one or a few outlet channels. Rather, it occurs through the subsurface and in several braided streams whose courses are ephemeral. For these reasons, proglacial discharge monitored at two stream sites accounts for only a fraction of the total basin runoff. Consequently, it cannot be equated to observed mass loss from the glacier. Velocity poles on the glacier surface serve as ablation stakes and are measured at least once per year. In order to use this information, it must be accompanied by a record of the nature of the surface (snow vs. ice) and its density at the time of each measurement, Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 160 and by a record of precipitation during that time interval. Precipitation is monitored for only a brief period during the summer, thus precluding the possibility of parameter calibration using annual measurements. Were these measurements available, uncertainty in its variation with elevation would remain. Precipitation measurements confined to the melt season would suffice, provided there were two for each stake so that a difference could be calculated. Unfortunately, repeated measurements are not sufficiently abundant to make this a viable option. Furthermore, pole height measurements are potentially ridden with errors due to the tendency of poles to lean as they melt out. Figure 7.3: Spatially variable solar radiation incident on Trapridge Glacier, computed according to Equation 3.2 using digital models of slope, aspect, and elevation, (a) Radiation at 1400 h on four days in 1997. (b) Radiation at four times during day 200, 1997. Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 161 Given this list of impediments to parameter calibration, I adopt the parameters op-timized by Hock for Storglaciaren, and scale them by a constant factor. This assumes that the relative importance of insolation (compared to temperature) is the same for Trapridge and Storglaciaren, as is the relative ease of melting ice compared to snow. Given that insolation is computed independently for each location, with attention to latitude, elevation, slope, aspect, and shading, the first assumption does not present an obvious problem. The second assumption may be compromised by differing properties of the snowpack. However, since (1) snowpack temperature is the most important property in relating insolation to actual melt, and (2) ice surface and snow temperatures should be correlated, then the relative difficulty of melting ice versus snow should remain fairly con-stant from glacier to glacier. For example, the mean annual air temperature of Trapridge Glacier is ~ —6°C [Clarke et al., 1984a; Clarke and Blake, 1991] and of Storglaciaren is —3.9°C [Hock, 1999]. Therefore, both snow and ice will require slightly more energy to produce the same volume of melt water for Trapridge as compared to Storglaciaren. Figure 7.4: Cumulative surface melt calculated for days 184-207, 1997 with scaled values of MF and ^ s n o w / i c e * Using 1997 air temperature as recorded by several data loggers in the study area, I scale MF and a s n o w / i c e to produce approximately 0.917 mw.e. total melt from days 184-207. Assuming there is no snow present, this corresponds to one metre of ablated ice {pi = 917kgm -3). During the 1997 field season (days 184-207) stakes supporting a fixed-rope from the glacier margin to the study area melted out approximately one metre. While this fixed line extends for about 500 m, it maintains an elevation comparable to the study area where the data loggers are located. Constraining ablation in this way yields a scaling factor for the melt parameters of 0.667, such that MF — 1.2mmd_ 1 ° C _ 1 , asnow = 4.0 x l O - W W ^ m m h - ^ C - 1 , and a i c e = 5.33 x 10~4 m 2 W" 1 mmh"1 o C " 1 . Figure 7.4 shows the spatial distribution of cumulative surface melt for days 184-207, 1997 as calculated with these scaled parameters. Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 162 The scaling factor 0.667 is low, in part, because the temperature timeseries I use is ob-tained from a data logger housed in a poorly-ventilated enclosure. In comparative studies of ventilated and unventilated temperature sensors, it has been shown that radiational heating can introduce errors greater than 8°C in the unventilated measurements [Hock, 1993]. Moreover, these errors fluctuate with radiation and wind speed in a way that eludes a simple correction formula. Hence the scaling factor 0.667 does not accurately reflect environmental differences between Storglaciaren and Trapridge. I have attempted, with little success, to correct unventilated air temperature time-series using ventilated temperature measurements from the camp meteorological station. Ventilated timeseries are short and intermittent, and reflect a different surface energy balance because they are collected over glacier moraine, rather than over ice. Henceforth I will use uncorrected air temperatures from the study area data loggers, except in the case of the 1990 release event. For this, air temperature from the meteorological station is used because temperatures recorded on the glacier in 1990 differ between loggers by up to 10°C. To err on the side of caution, I scale the melt parameters by the same amount (0.667) as in the unventilated case. For investigations using 1997 temperature data, the particular logger I choose is in nearly perfect agreement with 17 out of 21 total loggers. 7.1.3 Additional requirements Subglacial sensors show that hydraulic conditions at a particular location are clearly affected by proximity to a melt water source [e.g., Kavanaugh and Clarke, submitted]. Therefore, the spatial distribution of englacial cavities fed by the ice surface (and deliver-ing water to the bed) is an important requirement. Mapping accurate locations of every crevasse would be perilous and time consuming. Alternatively, I approximate crevasse distribution by analyzing curvature of the digital ice-surface model, assuming convexity as a proxy for strain, and thus a reasonable indicator of crevasse habitat. The outcome of this approach, shown in Figure 7.5, compares favourably with aerial photographs. This method works particularly well for Trapridge Glacier because surface crevasses comprise the majority of exposed englacial storage elements; there are no moulins, and boreholes freeze over within 24 hours of drilling. A proper representation of glacier surface conditions requires precipitation, as well as surface air temperature. Unfortunately, direct measurements of precipitation are made for only a few weeks each summer, leaving us with a dearth of information pertain-ing to winter accumulation. The study area probably receives less than one metre of accumulation during the winter, judging from the rarity of logger failure due to solar panels (perched above the glacier surface) being buried by snow. This provides an upper bound for precipitation estimates and is in accordance with mesoscale climatology of the northwest St. Elias Range. Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 163 Figure 7.5: Digital representation of crevasse distribution. Solid cells host crevasses, providing englacial access for surface runoff. Locations of the 1997 instrument study site and model test points are labelled. Precipitation is measured during the summer with a tipping bucket at the camp me-teorological station. Discrimination between rain and snow is unnecessary for Trapridge, because almost all precipitation falls as snow. Conveniently, the tipping bucket measures precipitation in water-equivalent units directly. I construct input timeseries of precipit-ation by combining tipping-bucket data with a randomly sampled synthetic function. I constrain the total annual precipitation and assume a cosine function for its shape. 7.2 Parameter refinement The availability of real data from Trapridge Glacier permits an objectively guided para-meter selection. Model results generated with a suite of different parameters are evaluated based on quantitative comparisons with data collected in 1997. From these comparisons I extract a final reference model. Table 7.1 defines an initial model for Trapridge investig-ations, showing substantial crossover with the reference model used in Chapter 6. Notable exceptions are K m i n = 2.5 x 1CT2 ms"1, K m a x = 5.0 x K T 2 ms"1, Ka = 5.0 x l O ^ m s - 1 , and the introduction of a s n o w / j c e and Kf, the hydraulic conductivity of frozen soil. Table 7.2 lists the accompanying numerical parameters. Surface water is permitted to enter the glacier only through crevasses as pictured in Figure 7.5, and the ice margin is treated as frozen by applying Kf to the outer interfaces of glacierized grid cells. As in the tests with simple geometry, an impermeable upstream boundary is prescribed in the aquifer. Where the edge of the grid contains ice, impermeable boundaries are also applied to the subglacial sheet. Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 164 Table 7.1: Initial model parameters for Trapridge Glacier. Symbols are consistent with those used in Chapters 3~4 where superscripts denote one of four model components (r—runoff, e=englacial, s=subglacial, a=groundwater aquifer) Parameter Value Description A blation/runoff: LR 0.0065 ^ m - 1 Atmospheric lapse rate MF l .SOxlO^ms-^C- 1 Melt factor a s n o w l . l lx lO-^rrr'W^ms-- i o C - i Radiation factor, snow a i c e l ^ x l O - ^ n ^ W ^ m s -- l o C - l Radiation factor, ice Kr 0.1ms-1 Effective hydraulic conductivity rT:e 1200 s Time constant for water exchange xr:e i Runoff-englacial coupling strength xT:a i Runoff-groundwater coupling strength Englacial storage/transport: ^ 1 X 1 0 " 4 7i 0 Maximum storage volume fraction Relative abundance of moulins 72 1 Relative abundance of surface crevasses 73 0 Relative abundance of basal crevasses h 0.5 Crevasse tapering factor b 2.0xl0- 5m Englacial crack aperture Ke LOxlO-^ms" 1 Englacial hydraulic conductivity r e : s 7200 s Time constant for water exchange xe:a o.i Englacial-subglacial coupling strength Subglacial sediment sheet: hc 0.147 m Critical sheet water thickness Kmin 2.5xl0- 2ms- 1 Minimum hydraulic conductivity Kmax S.OxlO^ms"1 Maximum hydraulic conductivity Kf l .OxlO-^ms- 1 Hydraulic conductivity of frozen soil v 30 m a - 1 Average glacier sliding velocity QG 0.07Wm-2 Geothermal heat flux x3:a i Subglacial-aquifer coupling strength Till cap properties: d* 1.0 m Thickness K* LOxlO^ms" 1 Hydraulic conductivity Groundwater aquifer: da 3.0 m Thickness Ka S-OxlO^ms- 1 Hydraulic conductivity na 0.4 Porosity a l .OxlO^Pa" 1 Compressibility Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 165 Table 7.2: Numerical parameters for Trapridge Glacier investigations. Parameter Value Description Lx 2560 m Ly 1120 m nx 64 ny 28 Ax 40 m Ay 40 m At 120-300 s Model domain, longitudinal Model domain, transverse Number of grid cells in x Number of grid cells in y Longitudinal grid spacing Transverse grid spacing Maximum time step 7.2.1 Initial model An equilibrium model consistent with Tables 7.1 and 7.2 is generated with a constant surface melt rate of 1.25 x 10 _ 3 mh _ 1 , as in Chapter 6. Figures 7.6-7.8 display various fields and profiles representative of this model and highlight several important features foreshadowed by the synthetic tests. Flow of melt water over the glacier surface, as suggested in Figure 7.6a, forms several dominant arteries. These arteries develop in crevasse-free regions of the glacier surface, creating a pattern reminiscent of the upstream area distribution in Figure 2.12b. Corre-spondence between upstream area distributions, based on subglacial hydraulic potential, and surface runoff morphology has also been noted by Nienow et al. [1998]. Several features of the runoff configuration in Fig. 7.6a are substantiated by qualitative field observations. Where runoff appears to converge on the central and southern ice lobes, supraglacial waterfalls are observed in summer. At the weakly implied exit point along the northern margin (marked by an arrow in Fig. 7.6a) an ice-marginal stream originates. Surface water flow through the study area and slightly upstream to the southwest can be broadly confirmed by the presence of supraglacial swamps. While model results re-semble the general observed morphology of surface runoff in the ablation area, the active drainage implied in the upper basin (upper half of Fig. 7.6a) is unrealistic. Perennial snow stifles the development of a mature supraglacial runoff network in the upper basin. The location and abundance of stored water along transects intersecting test point 1 are shown in Figures 7.6b and 7.6c. In both figures, the elevation datum is arbitrary but consistent, and a 7° slope has been removed from the longitudinal profile (Fig. 7.6b). For clarity of presentation, Figures 7.6b and 7.6c are vertically exaggerated 5.7 and 4.9 times, respectively. Profiles will be presented in an identical fashion hereafter. Vertical bars indicate the hydraulic head of englacially stored water. Gaps in storage along the profiles reflect the sparsity of crevasses. Note the general downglacier increase in storage (Fig. Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 166 7.6b) and that the water table eventually exceeds the glacier surface. Water of subglacial origin has been observed to erupt near the terminus, as suggested by this result. Square root runoff flux max = V 2.3x10'3m2 s"1 Figure 7.6: Equilibrium characteristics of surface runoff and englacial storage for Trapridge Glacier. Profiles intersect test point 1 (Fig. 7.5). (a) Runoff flow vectors plotted as the square root of runoff flux. Cross-hairs mark profile locations. Arrow indicates the origin of an ice-marginal stream, (b) Longitudinal profile of englacially stored water, expressed as an equivalent height above the bed. Surface and bed topography are shown with a T slope removed, and the elevation datum is arbitrary. Vertical exaggeration is 5.7. (c) Transverse profile of englacially stored water. Vertical exaggeration is 4-9- The location of test point 1 (Fig. 7.5) is indicated by the vertical line in both profiles. Equilibrium subglacial water pressure for the initial model is shown in Figure 7.7a. The frozen periphery produces distinctively high pressure at the glacier terminus, in contrast to the mild conditions expected for a temperate margin. Aside from at the terminus, areas of high pressure generally correspond to bedrock bowls. The upper basin provides a good example of this where the longitudinal profile crosses a trough between 600-800 m from the head of the glacier (Fig. 7.7b). Bedrock obstacles prevent water accumulation, in some cases completely drying up. Dark patches in the upper basin reveal these areas (Fig. 7.7a), which may be overpredicted by the model due to the impermeable upstream boundary. The most notable area with thin water coverage in the Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 167 lower basin is directly west of the study site (marked by the profile intersection). While it may be coincidental, prospecting for hydraulically unconnected boreholes has led the Trapridge Glacier Drillmaster west of the study area year after year. 60 50 40 30 20 10 0 500 1000 1500 2000 Longitudinal distance (m) 2500 C o ns > LU 140 120 100 80 60 40 C I i 200 400 600 800 1000 Transverse distance (m) 1200 500 m ^ ». Pressure (m) Figure 7.7: Equilibrium characteristics of the Trapridge Glacier subglacial system, (a) Subglacial water pressure expressed in metres, (b) Longitudinal profile of piezometric surface (dashed line). Flotation level is indicated by the dotted line in both (b) and (c). (c) Transverse profile of piezometric surface. As expected from synthetic tests in Chapter 6, profiles of piezometric surface (Figs. 7.7b and 7.7c) closely track the water table (Figs. 7.6b and 7.6c). In this case, englacial profiles are discontinuous, yet give a reasonable indication of the underlying piezometric structure. Dashed lines in Figs. 7.7b and 7.7c are longitudinal and transverse profiles of piezometric surface, shown relative to the ice flotation level (dotted lines). The transverse profile is exceptionally uniform, while the longitudinal profile samples hydraulic extremes: dry bedrock bumps near the head of the glacier and artesian pressures near the terminus. Water distribution in the subsurface aquifer mirrors water pressure in the sheet, as illustrated in Figure 7.8a. Saturated thickness is shown in plan view and profile, ac-companied by groundwater flow vectors. Figure 7.8a demonstrates the highest values of saturated thickness where accumulated sheet water pushes its way into the subsurface, namely in the upper basin bedrock trough. As shown in Chapter 6, this is a combined result of aquifer topography (which is subparallel to the glacier bed) and the distribution of source water. Saturation levels are slightly elevated near the glacier toe, but vary Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 168 smoothly between glacierized and unglacierized terrain owing to the continuity of the aquifer across this boundary. Profiles of saturated thickness delineated in Figure 7.8a are plotted in Figures 7.8b and 7.8c. The intersection of lines B and F marks the study site (test point 1). All profiles are scaled to focus on supersaturated zones ( > 1.20 m). Despite the fact that values of Ka and Kl are substantially greater than in the synthetic reference model, supersaturation persists in most of the aquifer. The saturated areal fraction in this case is 71%. Values that are off-scale in the profiles are close to zero, representing zones where the aquifer is nearly or completely unsaturated. These conditions persist exclusively in the upper basin and are primarily due to lack of source water. Figure 7.8: Equilibrium characteristics in the aquifer, (a) Saturated thickness, (b) Longitudinal profiles of saturated thickness. Values that are off-scale are close to zero. Profile positions are labelled in (a) for both (b) and (c). (c) Transverse profiles of saturated thickness. The study site coincides with the intersection of profiles B and F. (d) Groundwater flow vectors. Pockets of low saturation are highlighted by groundwater flow vectors in Figure 7.8d. Beneath most of the ablation zone groundwater flow patterns are smooth and uninter-esting, except at the terminus where flux reaches its maximum. Enhanced transport beneath the margin as shown in Figure 7.8d reiterates the fact that the aquifer is slave to the sheet, especially when drainage from the ice-bed interface is impeded by permafrost. Such clear delineation of the ice boundary is a direct result of intercomponent water exchange patterns. Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 169 Profiles of exchange between englacial, subglacial, and groundwater systems (Fig. 7.9) link Figures 7.6-7.8. Vertical bars represent englacial-subglacial exchange (f>e:a and solid lines show exchange between the aquifer and sheet (4>s:a, beneath the ice) or aquifer and runoff system {(j)r:a, beyond the ice). Dashed lines serve to visually connect <^ *:a and <f)r:a. Profiles of the ice surface and bed are shown for context as before. Exchange between the glacier interior and bed (4>e:s) is almost exclusively downward (positive), except near the terminus (Fig. 7.9a) where artesian pressures are obtained in the sheet (Fig. 7.7b). Source rate to the sheet <^ e:* is high in the central ablation zone, with a maximum just below the study site (vertical line). Best seen beyond the glacier terminus (Fig. 7.9a) and southern (left) margin (Fig. 7.9b), negative values of 4>r:a indicate upwelling from the aquifer to the glacier forefield. This water is escorted by the runoff system out of the basin. E b < X 2 0) 0 c O -2 .C X UJ 500 1000 1500 2000 Longitudinal distance (m) 2500 E 2.0 ID 1.5 b r- 1.0 X -—' 0.5 a> at 0 c ha -0.5 xc -1.0 UJ 200 400 600 800 1000 Transverse distance (m) Figure 7.9: Equilibrium profiles of intercomponent water exchange (<j>e:s, <j>s:a, and <j>T:a). (a) Longi-tudinal profiles of 4>e:l1 (vertical bars), 4>s:a (fine line), and </>r:a (bold line). Ice and bed topography are shown with slope removed, (b) Transverse profiles of <j>e:s (vertical bars), <j>s:a (fine line), and <j>r:a (bold line). Exchange between the sheet and aquifer is shown on the same scale for comparison. It is dominated by sheet water loss at the ice boundary, where values of <t>*'a exceed the maximum <j)e:> and dwarf variations beneath the glacier interior. This pattern of exchange is responsible for the aquifer flow vector distribution in Figure 7.8d. Interior values of t /> 5 : a exhibit spatial reversals in the exchange direction, a feature unique to complicated terrain. Careful inspection of Figure 7.9a shows that (f>s:a undulates about zero along most of the profile. Bed topography is directly responsible for these reversals, such that negative values of 4>s:a occur in the lee of topographic obstacles. This general result was explained for the U-shaped valley glacier in Chapter 6. With the starting model summarized in Figures 7.6-7.9, time-dependent tests are carried out in order to select a reference model for Trapridge Glacier. Field data collected in 1997 provide an objective basis of comparison. Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 170 7.2.2 Comparisons with data In 1997, a dense array of 16 subglacial pressure transducers was deployed in order to probe spatial correlation in the water system. Sensor spacing is ~ 5m, roughly aligned with the direction of ice flow (Figure 7.10a). Two Campbell CR10 data loggers fixed with multiplexing panels handle data from eight instruments each. The upstream half of the array was in operation for almost three years after installation. Data from most of these sensors were exceptionally coherent during the 1997 melt season, as shown in Figure 7.10b. Based on borehole drilling observations, this particular season was characterized by abundant cracking in the lowermost 10 m of ice and widespread subglacial connectivity. The result is a remarkably homogeneous picture of melt season hydrology in the study area. While such spatial homogeneity is unusual, it presents the best possible opportunity for some form of model calibration. 1760 1764 1768 1772 1776 1780 Easting (- 534000 m) S 40 (0 190 194 198 Day in 1997 202 Figure 7.10: Locations and records of Trapridge Glacier sensors used in model parameter optimization, (a) Dense array of 16 pressure transducers installed in 1997. Asterisks indicate surface locations of boreholes in which instruments were installed. Open circles mark data loggers, (b) Subglacial water pressure as recorded by sensors labelled in (a) for days 190-204, 1997. Records from 97P04 and 97P34 are not shown. Water pressure fluctuates about a baseline of 45.1m (~ 78% of flotation), with peak-to-peak amplitudes ranging from 10-25 m. Phases are coherent between sensors, but vary somewhat systematically over time by as much as 8 h. Fourteen of sixteen sensors are represented in Fig. 7.10b, with 97P13 being the visible outlier. The record from this Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 171 sensor, conforming to the others only during periods of high pressure, is indicative of diurnal switching between hydraulically-connected and unconnected modes [Murray and Clarke, 1995; Kavanaugh and Clarke, submitted]. In its entirety, the 1997 pressure array occupies less than one model grid-cell. To facilitate quantitative comparisons between modelled and observed records, I stack the 14 traces in Figure 7.10b into a single master timeseries with a sampling interval of one hour. Results from the numerical cell that overlaps this region (1997 study site, test point 1 in Fig. 7.5) can now be compared directly to the data. I use the period from day 190-204 to make these comparisons. Beginning with the initial model defined in Tables 7.1 and 7.2, a total of 33 diagnostic tests were performed, varying seven parameters independently and several in combina-tion. Model tests are initiated from equilibrium beginning at day 184, six days before the comparison interval (day 190-204), and are forced with a temperature timeseries from logger #3557. Precipitation is negligible during this period. Each test required 14-19 hours of computer time on a 360 MHz Sun Ultra 5 workstation. Results were recorded hourly to facilitate direct comparisons with the master timeseries. Model evaluation is based on three metrics that quantify simulation error with respect to mean pressure and diurnal oscillation amplitude and phase. Appendix E details these tests and the outcome of comparisons with data. 7.2.3 Final reference model '?90 192 194 196 198 200 202 204 Day in 1997 Figure 7.11: Comparison between observed (solid) and modelled (dashed) pressure records for final reference model. The best overall performance was achieved by adjusting two initial model parameters. Effective conductivity for runoff Kr was increased from 0.1 to 0.15ms -1, and the time constant for surface-englacial exchange r p : e was reduced by half to 600 s. Figure 7.11 Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 172 compares observed and simulated subglacial water pressures. In addition to enabling parameter refinement, these tests demonstrate that the basic model successfully emulates orthodox melt season behaviour. Mean water pressure, recovered by the model within 20 cm, is largely a function of subsurface conductivities (if*, Ka) in the presence of a frozen glacier margin. The most striking differences between observed and modelled records are in the amplitude and phase of diurnal oscillations. More subtle differences he in the value of daily pressure minima and in the shape of the cycles. Except for days 196 and 203, modelled amplitudes are too small. This shortcoming, ubiquitous in the simulations, betrays the limitations of a simple continuum model. In reality, subglacial hydraulic pathways are finite, owing to heterogeneity at the glacier bed. Surface melt is probably routed over a small fraction of the bed, driving maximum pres-sures higher than in the simulations. Excavation and sculpting of the drainage system itself allows very low pressures at night. An interesting discussion of this phenomenon is given by Kavanaugh and Clarke [submitted]. Subgrid channelization of supraglacial run-off would also contribute to high-amplitude fluctuations, becoming an important source of discrepancy where the channelization is misaligned with the large-scale ice-surface slope. Phase lag between modelled and observed records is conspicuous on most days (Fig. 7.11). Uncertainty in the entry point for surface water, as obtained from the derived crevasse map, and its englacial routing could easily be responsible for this. Parameters that yield improved phase correspondence reliably fail in other categories (see Appendix E). Delayed peaks are not of particular concern considering the unusual abundance of englacial fractures observed in 1997. These fractures may have expedited water trans-mission through the system in a way that is anomalous compared to other years. a •o a B 2400 -2300 i o 2200 \ A ,f • \ ' \ v 2100 s 0 _o' / \ —\ 7\ / 2000 / 1900 1800 1700 • \ * 1600 V 190 1 92 1 94 1 96 1 98 200 202 204 Day in 1997 Figure 7.12: Time of daily maximum pressure for observed (solid) and modelled (dashed) records. Figure 7.12 plots the time of daily maximum pressure for simulated and observed records during the test interval. A decisive increase in coherence between the two traces Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 173 occurs at day 197. Because the model neglects ice and bed dynamics, it produces a comparatively simple convolution of the surface forcing in the basal pressure response. Therefore, the similarity in phase evolution after day 197 marks a simplification in the real-system response to surface air temperature. These two modes of behaviour are not evident in the master timeseries itself, but can be identified with the help of the model. However, the cause of this transition remains speculative. Sensors were installed from day 188-189, so little information is available to diagnose the state of the system before day 190. It snowed several times in the days prior to 190, possibly resulting in a volume reduction of the active drainage network. This could be consistent with the observed prematurity of daily pressure maxima. Model results were unchanged by including this small amount of precipitation, pointing to complex changes at the bed as the likely explanation. This example suggests that model shortcomings are potentially useful as an interpretive tool for real data. Shape and depth of the observed diurnal pressure minima are affected by dynamic evolution of the drainage system which is not accounted for in the model. Simulated pressure changes directly represent changes in water volume, while observed pressure changes reflect the relative volumes of water and the drainage system. Observed pressure is also affected by the distribution (and redistribution) of basal stress. Kavanaugh and Clarke [submitted] have noted the tendency for pressure minima to rise with cooler weather, pointing to the role of meltwater in opening and maintaining a high-capacity drainage network. If surface input decreases sufficiently, the subglacial system heals itself and portions become disconnected from the primary drainage network. Inflections on the falling limbs of observed pressure cycles (Fig. 7.11) suggest that this process of collapse is approached nightly given low enough water pressure. Sensor 97P13 (Fig. 7.10) resides in an area with a high disconnection-threshold and disconnects decisively during periods of low pressure. Given the limitations inherent in a model that neglects ice dynamics and thermal conditions at the bed, the correspondence between modelled and observed records as presented in Figure 7.11 is considered very good. Pressure maxima follow a reasonable progression in time, and the system properly demonstrates a memory of its priming on a scale of several days. The latter point can be appreciated in Figure 7.13 where calculated surface melt rate is plotted with subglacial water pressure. Results from a reverse-time integration (performed by reading meteorological timeseries backwards) is plotted along with the reference model in Figure 7.13b to elucidate the influence of multi-day melt history. Note that the reference model pressure peak at day 196 is directly preceded by a day of high melt, but follows a low-melt cycle the day before. This peak shows excellent agreement with the observations (Fig. 7.11), and illustrates that a period of low melt allows the system to drain itself over several days. This idea is reinforced by the reverse-time sequence shown in Figure 7.13b. It is offset in time to align peaks in water pressure resulting from a common melt cycle. Low frequency differences in these Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 174 two curves represent the effect of system priming, since each has inherited a different history. Hysteresis in the daily cycle is evident as asymmetry in the rising and falling limbs of subglacial pressure waves. Figure 7.13: Calculated surface melt rate and modelled subglacial water pressure, (a) Surface melt rate at test point 2. (b) Subglacial water pressure at test point 1 as obtained from the reference model (bold line) and by reverse-time integration (fine line). The reverse-time test is offset to align peaks arising from the same melt pulse. With a reference model that successfully compares with subglacial instrument records during the melt season, I proceed to investigate more complex situations of interest to glacier hydrologists. 7.3 Hydraulic release events: rapid disturbances in the water system Outbursts of subglacial water have been observed and documented beneath numerous alpine glaciers. Such events tend to occur in spring and are thus attributed to an inabil-ity of the winter subglacial drainage system (characterized by high water pressure and low capacity) to accommodate a sudden and profuse influx of surface meltwater. Prior to a release event, prevailing hydrologic conditions are often attended by bursts of glac-ier motion, and the release itself precipitates the restoration of summer plumbing that damps or terminates surface acceleration. The events bear witness to the importance of interactions between surface melt, runoff, englacial water storage, and internal routing in addition to subglacial drainage morphology. Hydraulic release events associated with glaciers have received significant attention from both experimental and theoretical perspectives. Such disturbances and subsequent Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 175 morphological adjustments are known to have ice-dynamical manifestations and feed-backs, for example, ice surface uplift and velocity changes [Iken and others, 1983], with possible relationships to large-scale instabilities [Kamb and Engelhardt, 1987]. These broadly-defined events arrive in a variety of forms. Some are associated with the de-velopment and collapse of subglacial tunnels, exemplified by Iceland's jokulhlaups [e.g. Bjornsson, 1992] (Fig. 7.14a). Others, more common for non-temperate glaciers, involve hydraulic rupture of a thermal dam at the glacier margin [e.g. Stone, 1993; Skidmore and Sharp, in press] (Fig. 7.14b). Figure 7.14: Types of subglacial hydraulic release events, (a) Water impounded in subglacial- or glacier-dammed lakes is released through a tunnel, (b) High-pressure distributed subglacial drainage ruptures a thermal dam at the glacier margin. Nye [1976], Spring and Hutter [1981], and Fowler [1999] among others have theoret-ically described outburst floods of the first type, which typically issue from lakes either subglacial [e.g. Bjornsson, 1992] or glacier-dammed [e.g. Clarke, 1986] and tend to be cyclic in nature, because they are dependent on recharging a storage reservoir. Little modelling work however has been devoted to floods of the second type. These may also be periodic, but occur in heterogeneous flow environments that are difficult to characterize and depend on complex interactions between surface, ice and bed. Release events are most commonly documented by proglacial stream hydrochemistry and suspended sediment analysis, complemented by supraglacial observations [e.g. An-derson et al., in press; Skidmore and Sharp, in press; Humphrey and Raymond, 1994], but rarely have they been recorded subglacially. Trapridge Glacier offers this unique op-portunity. Sensors that operate year-round demonstrate strong seasonality in subglacial hydromechanical behaviour and have recorded anomalous adjustments sometimes correl-ated with outbursts from the glacier margin. Two events in particular have registered subglacial responses over tens to hundreds of metres. The first took place in 1990 and Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 176 was observed and interpreted by Stone [1993] and Stone and Clarke [1996]; the most re-cent occurred in 1995 [Kavanaugh and Clarke, submitted]. Here I revisit the 1990 event equipped with a numerical model to test the hypotheses of Stone [1993]. This doubles as an opportunity to evaluate the capability of the model to simulate rapid transients in the subglacial system. 7.3.1 1990 Event synopsis and interpretation Figure 7.15: Observed surface and subglacial conditions surrounding the 1990 hydromechanical event, (a) Air temperature, (b) Selected subglacial water pressure records, (c) Detail of Fig. 7.15b bracketing the event. The inset (after Stone [1993]) indicates relative sensor positions. According to Stone [1993], the 1990 event struck both mechanical and hydrological sensors within an area of 5000 m 2 just before midnight on day 203 (Fig. 7.15). Its onset was marked by an abrupt reduction in subglacial water pressure according to most, but not all, participating pressure sensors, and proceeded in a southwesterly direction. Then commenced an equally abrupt pressure rise to superflotation values, followed by a general decline over the next several days. Superimposed on this decline are diurnal cycles indicating that formerly unconnected sensors became connected with each other and Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 177 with the surface. Connections themselves are not rare beneath Trapridge, but they are commonly confined to spatial scales less than a few tens of metres [Murray and Clarke, 1995]. Days prior to the event saw high surface melt, judging from elevated air temperatures (Fig. 7.15a), culminating in a seasonal temperature maximum on the afternoon of day 203, and accompanied by no significant precipitation. Proglacial stream stage registered a premature daily maximum the day after the event, followed by a burst of dye that had been injected subglacially in 1985. Except under unusual circumstances, Trapridge proglacial streams carry little or no subglacial water because the frozen glacier margin constitutes a hydraulic barrier. In light of these and other lines of evidence, Stone [1993] attributed this temporary alteration of subglacial drainage structure to a sudden crevasse drainage northwest of the study area (Fig. 7.16). Such a large water pulse impinging on an inefficient drainage system would, lacking an escape conduit, hydraulically lift the glacier (as observed else-where [e.g. Iken and others, 1983]), thus reducing subglacial effective pressure (manifest as the initial pressure drop recorded by most instruments). Thereafter, hydraulic gradi-ents would drive water rapidly from the source area to these newly created regions of low potential, breaching hydraulic barriers and delivering the pulse observed in Figures 7.15b and 7.15c. Resulting and relatively widespread hydraulic connections are maintained for several days before the source water is depleted. Records of subglacial turbidity and con-ductivity tell a similar story, and argue against other mechanisms such as downstream rupture or subglacial cavity formation. A more comprehensive discussion is given by Stone [1993]. Figure 7.16: Event interpretation [Stone, 1993]. (a) Intense surface melt feeds crevasses. Borehole sensors are initially isolated, (b) Crevasse drains rapidly resulting in hydraulic uplift of the glacier, (c) Hydraulic barriers are breached connecting borehole sensors. 7.3.2 M o d e l adaptations and inputs Release events are intrinsically idiosyncratic, because they represent a failure of the existing drainage system. This model has been developed to study conventional soft-bed hydrology; thus, it requires two important modifications in order to simulate release Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 178 events. First it must accommodate both hydrauhcally-connected and isolated behaviour (Fig. 7.17a,b), and allow spatial and temporal transitions between these two states. Secondly, hydraulic uplift of the glacier must be parameterized (Fig. 7.17c). Over grid scales (20-40 m), areally-averaged thickness of the subglacial water sheet h" must exceed some finite value before hydraulic barriers are breached and communication is established with adjacent regions. This threshold is a function of subgrid attributes such as topographic irregularity and sediment distribution, as discussed in Chapter 4. To represent the connection process, I introduce a switch in the model whereby sheet conductivity is nonzero only when the saturated horizon is sufficiently thick. To initialize the model, I assume that all cells are unconnected except those with a direct surface coupling, where a more mature drainage structure is expected. / Ks=0 ? 1 Ks= Ks(hs) i c e ^ _ _ J ? < (x-•V) subglacial void Figure 7.17: Model adaptations, (a) Plan view and schematic cross-section of a hydraulically uncon-nected cell (hs < hc). Ks = 0 on all sides of the cell which acts as an open circuit, (h) Plan view and schematic cross-section of a hydraulically connected cell (hs > hc). K3 = Ks(h") on all sides of the cell, closing the circuit, (c) Hydraulic jacking initiated at (x,y) and experienced distal from the source at (x',y'), according to Eqn. 7.1. Ice-dynamical feedbacks induce a rich variety of subglacial hydrological responses, yet without an ice dynamics model, interactions between ice and bed cannot be freely determined. Consequently, hydraulic uplift must be parameterized based on hydrological and geometric variables alone. To do this I assume that uplift of the ice creates a subglacial volume increase that can be approximated by a modified two-dimensional Gaussian centred on the initiation point. The volume change at location (x',y') can then Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 179 be expressed as AV{x\ y') = a V(x, y) ^ exp ( - ) > P™ > Pi (7.1) where a is a dimensionless scaling factor, and the remaining unprimed variables refer to the origin of the uplift (x,y): V is subglacial volume, pw is subglacial water pressure and pi is ice overburden pressure. Values of <rx, cry and a must be empirically determined; based on the inferred spatial extent of two hydromechanical events at Trapridge Glacier, I take crx = o-y = hi(x,y) where hi is ice thickness. For simplicity, I assume the change described by Equation (7.1) propagates instantaneously from points where subglacial water pressure exceeds ice flotation pressure. The interval from day 200 to 207 (July 19-26), 1990 is examined in this study because it brackets the event of interest on the eve of day 204. Input time series of air temperature and precipitation from this period are required to force the surface hydrology model; both variables are monitored at the field site meteorological station. Precipitation is applied uniformly over the model domain. The model timestep is reduced to 30 s in order to resolve the event. 7.3.3 Results and discussion To model the 1990 event and test the interpretation of Stone [1993], I prescribe the opening of a valve cock that restricts flow between crevasses and the glacier bed. Other event initiation experiments led to the conclusion that its onset must have been extremely rapid—caused by a switch rather than any semi-continuous process. In Nature this could be a rupture in basal ice connecting englacial storage to the bed, but whatever the cause, it must have been a coupled hydromechanical process. The hydrological context of the event is summarized in Figure 7.19, with attention to individual components and their interactions. Modelled timeseries are taken from the study area, highlighted in Figure 7.18. Figure 7.19a shows the areally-averaged depth of ice melted near a feeder crevasse (c in Fig. 7.18) as computed by the distributed ablation model. Global volume evolution of englacially-stored water is chronicled in Figure 7.19b, and demonstrates its sensitivity to the diurnal melt cycle. Prior to the event, local storage was filled to capacity. The drainage episode itself, marked by a swift reduction in stored water volume (ca. day 204), depletes storage by approximately 50%, both globally and locally. Further drawdown is prevented by resistance from the overpressurized subglacial sheet. The pulse of released water delivered to the sheet (Fig. 7.19c) overshadows all subse-quent englacial-subglacial exchange, and the resulting rise in sheet pressure (Fig. 7.19d) Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 180 Figure 7.18: Location of 1990 field site (lightly shaded) in the context of the model grid and computed crevasse map. Modelled results in Figure 7.19 correspond to crevassed cell "c". Those in Figure 7.20 are from adjacent labelled cells. Subscripts indicate their relative positions (e.g. SW = southwest). triggers hydraulic uplift of the surrounding ice, allowing the disturbance to cascade into neighbouring regions. Formerly-latent areas of the bed are activated, thereby improving hydraulic connectivity, stimulating water transport and facilitating widespread subglacial depressurization. The efficacy of this process maintains low pressures even in areas per-petually charged by the surface, as shown in Figure 7.19d after day 204. As a hydraulic buffer, the underlying groundwater aquifer helps dissipate the impact of the flood by accepting a large influx of subglacial water (Fig. 7.19e). Before the event, poorly-connected, high-pressure conditions at the bed give rise to steady seepage into the aquifer, whereas reduced sheet pressures following the event produce an exchange reversal and groundwater upwelling ensues. An aquifer that is too efficient or well-coupled to the glacier bed can prevent hydrological events altogether by expediting the removal of excess water. Trapridge Glacier subsurface hydraulic characteristics (e.g. stratigraphic layer thicknesses and conductivities) define a system that provides enough resistance to allow hydromechanical events, yet in mid-meltseason suffices to evacuate the daily input of surface melt. Modelled subglacial reaction, distal from the source, is shown in Figure 7.20 over several days (Fig. 7.20a) and over several hours (Fig. 7.20b) surrounding the event. These results are extracted from the study area identified in Figure 7.18, and are shown together in Figure 7.20 to convey the spatial variability of response. Direct comparison of model output and instrument records is imprudent in this case. At least two of the records shown in Figure 7.15 (which lack a perfect harmony themselves) were collected from an area smaller than one gridcell, for which there is a single modelled result. However, Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 181 ~ 5.0 E o X X to E 2.5 0 1.0 £ 0.5 u (0 r 4.0 E ?= 2.0 n rx o 2 c o (S E CO is 0 1.5 1.0 0.5 4.0 2.0 C 0 £ to -2.0 a Surface melt rate Global englacial storage Water delivery to subglacial system Subglacial water pressure Subglacial - groundwater exchange 198 200 202 204 206 Day in 1990 Figure 7.19: Modelled evolution of multicomponent hydrology through the event: results (a) and (c)-(e) are from crevassed location "c" (Fig. 7.18), northwest of the 1990 field site, (a) Local surface melt rate M (water equivalent), (h) Global volume of englacially-stored water relative to maximum capacity. Prior to the drainage event, local storage was full, (c) Rate of water transport from the englacial system to the underlying sheet <fre:s. (d) Resulting subglacial water pressure p". (e) Water exchange rate between sheet and groundwater aquifer <f)>:a. Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 182 Figures 7.20a and 7.20b demonstrate that the qualitative features of the event have been captured. The disturbance appears as a spike in the records and marks the transition between a stable, isolated system (characterized by high, steady water pressures) and a connected system that now interacts with its surroundings and with the glacier surface (indicated by diurnal cycling of water pressure). Figure 7.20: Modelled subglacial and proglacial response to an englacial drainage event, (a) Subglacial water pressure over several days surrounding the event. Each trace represents a spatial location over-lapping the 1990 instrument study area (Fig. 7.18). (b) Detail of Figure 7.20a. (c) Subglacial discharge from the glacier margin. Note the near zero ambient discharge preceding the event. A closer look at the event onset (Fig. 7.20b) confirms that it begins with an initial pressure drop due to uplift of the ice, rather than evacuation of water, followed by a pulse that decays over a matter of hours. The linear pressure decline at the onset of the modelled event derives from the assumption of instantaneous hydraulic jacking. Figure 7.20c verifies that the modelled effects of this event ultimately propagate to the glacier margin, culminating in a hydraulic outburst. Preceding this, subglacial discharge is but a small steady leak, owing to the largely disconnected glacier bed and loss to the groundwater aquifer. Afterward, elevated and diurnally-varying discharge persists as the system remains connected to cope with the inundation of surface water. Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 183 7.3.4 Model sensitivities To characterize the variability of results presented in Figures 7.20a and 7.20b I investigate the model sensitivity to three key parameters: the uplift scaling factor a used in Equation (7.1), englacial crevasse volume and crevasse location relative to the study area. Figure 7.21 summarizes results of these tests for a point PNW (Fig. 7.18). | 1.2 3 * 1.0 (0 c 8.2 £15 0.8 o t.0.6 i a i — o=1.11 x lO" 4 — 0=1.67x10^ - - a = 2.0x10- 4 203.94 203.98 204.02 204.06 203.94 203.98 204.02 204.06 ?> a Q_ CO o 1.2 1.0 0.8 0.6 -up, N -up, S - down, N - down, S 203.94 203.98 204.02 Day in 1990 204.06 Figure 7.21: Sensitivity of modelled subglacial behaviour to selected parameters. All results are for a common test point P^w (Fig. 7.18). (a) Variation with uplift scaling factor a (Eqn. 7.1). (b) Variation with relative crevasse volume. Crevasses were full prior to drainage in each case. Absolute crevasse volume is computed as 10~* Vc V\, where V\ is ice volume in the appropriate cell, (c) Detail of Figure 7.21b. (d) Effects of event source location. Labels indicate source position relative to the study area (up=upstream, down—downstream, N=north, S—south). Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 184 The value of a controls scaling between hydraulic uplift and water pressure (Eqn. (7.1)). This relationship dictates the timing and magnitude of response near, and far from, the uplift inception point. As shown in Figure 7.21a, a conservative range of a elicits reaction times spanning the variance observed in the field data. Uplift is minimized for small values of a; therefore, higher pressures are ultimately attained, because a fixed volume of source water is forced through a constricted system. The absolute magnitude of a is not meaningful in itself, as it depends on prescribed initial estimates of subglacial volume. Changes due to small variations in a however, emphasize the system's sensitivity to volume perturbations. Figures 7.21b and 7.21c illustrate the surprisingly modest importance of relative crevasse volume. Bounds can be placed on englacial storage capacity by scrutinizing subglacial diurnal signals, as in Figure 7.21b; for the case of maximum crevasse volume, diurnal variations are noticeably subdued. Crevasses too large to experience substantial pressure excursions in response to daily surface melt provide an unrealistically weak dri-ving force for the sheet. Furthermore, sudden drainage of these crevasses gives rise to modelled pressures significantly higher than those observed (Fig. 7.21c). The accom-panying cases shown in Figure 7.21c, testing crevasses of small and moderate size, both produce acceptable results in light of the data. Adopting parameters a = 1.67 x 10 - 4 and dimensionless crevasse volume Vc = 10, I investigate the effects of event source location. For each case presented in Figure 7.21d, connections between crevasses and the glacier bed were restricted to quadrants either upstream or downstream and either north or south of the study area. These results appear wildly dissimilar and exhibit some noteworthy features. Drainage in the northern downstream quadrant is predictably reminiscent of a downstream rupture: monotonic pressure reduction is initiated as connections propagate upglacier. Stone [1993] ruled out this mechanism in his 1990 event interpretation because it fails to account for the high pressure pulse observed after the initial decline. By comparison, downstream drainage on the south side produces a small but late pressure rise. In this case, the relative proximity of downstream crevasses to the south enables flow upglacier to the study area (see Figure 7.18 for crevasse locations relative to the test point). Both upstream drainage tests yield results closer in character to the data. Pressure maxima exceed ice flotation values, although arrival of the pulse is delayed as the sep-aration between source and study area increases. Of the four tests, drainage from the northern upstream sector best reflects the form and timing of the observed records. These results have a debatable relationship to the data as crevasse locations have been crudely predicted for modelling purposes and do not necessarily honour the glacier geometry of 1990. Yet if anything, they favour event initiation northwest of the instrumented area as inferred by Stone [1993]. Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 185 7.3.5 Summary of release event simulation With extremely simple adaptations of the distributed glacier hydrology model, it is able to mimic the signature of a hydromechanical event as recorded by subglacial sensors. Given the changing glacier geometry (between the 1990 event and 1997 creation of surface and bed maps), the coarse resolution of the model (compared to individual sensor spacing) and the homogeneous theoretical treatment of a heterogeneous environment, the resem-blance of modelled to observed subglacial behaviour is remarkable. This correspondence attests to the robust nature of a soft-bed hydrological response to an englacial drainage episode. Within the limits imposed by the required averaging of physical properties and processes, these numerical results substantiate Stone's [1993] conceptual interpretation of the 1990 event. Application of this integrated modelling approach to similar events on other glaciers could prove interesting given the importance attributed to particulars of glacier surface geography and subglacial geometry in dictating water storage and re-lease. Additionally, incorporation of model tracers would be useful in Unking proglacial dye-tracing observations and the subglacial story. 7.4 Seasonal transitions in the drainage system Seasonal transitions provide one of the most useful contexts in which to study the fun-damentals of glacier hydrology. They offer insight into the development, maintenance, and collapse of various drainage modes, and they showcase complex interactions between basal hydraulics and mechanics. The dependence of glacier dynamics on these different hydrological modes accounts for a vast portion of interest in their study. Links between ice motion and subglacial water properties have been deciphered from a combination of theory [e.g., Kamb, 1970; Iken, 1981; Bindschadler, 1983; Fowler, 1987b; Alley, 1989b] and observation [e.g., Iken et al, 1983; Iken and Bindschadler, 1986; Fischer and Clarke, 1997; Harbor et al, 1997; Kavanaugh and Clarke, submitted], while important quantities such as subglacial storage capacity and drainage system volume have been estimated from field measurements [e.g., Hock and Hooke, 1993; Nienow et al, 1998; Anderson et al, 1999]. Straddling a seasonal divide affords a perspective of extremes, and in many cases, is more informative on subglacial behaviour than the seasons themselves. Glaciologically speaking, winter occupies most of the year and summer fills the gap in between. Summer is normally synonymous with the meltseason and lasts 2-4 months. Spring denotes the characteristically abrupt introduction of summer, a transition at-tended by mechanical and hydrological disruptions. Certain aspects of this transition are similar to the progression of a release event. Fall is usually defined by the permanent cessation of diurnal pressure cycling and a gradual reduction of drainage system volume. Field studies provide us with some consensus on the signatures of these transitions. Most work has focused on the spring, owing to its appealing parallels with surge initiation and Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 186 shutdown albeit on a much shorter timescale. Spring initiates the development of a connected and efficient subglacial drainage sys-tem from the isolated and torpid one of winter [Stone, 1993]. Prior to "spring events", subglacial/englacial water storage augments in response to surface input [e.g., Iken et al, 1983; Fountain, 1994; Anderson et al, 1999; Skidmore and Sharp, 1999]. Event precurs-ors include high subglacial water pressure, uplift of the glacier itself, increased horizontal ice-motion, and elevated solute concentrations in proglacial streams. Stored water is typically released during or after the event. Transition timing is thought to depend significantly on snow cover [Fountain, 1996], either related to sat-uration of the snow/firn aquifer or retreat of the snowline itself. Revelation of moulins or other surface portals in the wake of the snowline has been observationally correlated with the onset of subglacial summer [Nienow et al., 1998]. Hydraulic barriers formed of superimposed ice play a potentially important role in the early season, especially for non-temperate glaciers [Hodgkins, 1997; Skidmore and Sharp, 1999]. If the drainage dis-ruption takes seed, morphological evolution of the internal and subglacial systems leads to an increasingly efficient transit network [Seaberg et al., 1988; Hock and Hooke, 1993] and observable hydraulic connection between formerly isolated patches of the glacier bed [Murray and Clarke, 1995; Stone and Clarke, 1996]. Summer can end as hastily as it begins. Higher water pressures are observed despite the dwindling input [Fountain, 1994], proglacial discharge decreases [Hock and Hooke, 1993], and subglacial diurnal variations eventually cease as hydraulic connections are severed. Late-season melt events can jolt the system from its pending slumber [Raymond et al, 1995], effectively delaying the onset of winter. Both transition types have been recorded by sensors operating year-round at Trapridge Glacier, furnishing an inspiration and guideline for exploring their context. While ne-glect of detailed physical processes compromises the model's ability to simulate point-measurements, I attempt to characterize both transition types more comprehensively than is possible with sparse observations. Specifically, I address transition timing, changes in the water budget, and subglacial seasonal signatures. 7.4.1 Evidence from Trapridge Glacier Six pressure records from the array installed in 1997 (Fig. 7.10a) are shown in Figure 7.22 to illustrate the progression from summer to winter. The transition to an unconnected state, marked by disparity between instruments records, is clearly demonstrated in this suite of data. While the moment of permanent isolation occurs at different times for different sensors, it takes place rapidly for any individual sensor. In most instances, the decisive termination of diurnal fluctuations corresponds to the connected-unconnected transition, suggesting that high water fluxes are required to maintain connections in these areas. This explanation is corroborated by the highest record in Fig. 7.22b which Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 187 reconnects during a peak in pressure on day 248, and by other studies that have recorded the same phenomenon [e.g., Murray and Clarke, 1994; Gordon et al, 1998]. In one case presented here, adjacent sensors remain quasi-connected even after diurnal variations cease. Evidence for this is in the two lowest records (Fig. 7.22b) that track each other until approximately day 265. Figure 7.22: 1997 Trapridge Glacier transition to winter as recorded by 6 pressure transducers (97P11, 97P12, 97P13, 97P14, 97P15, 97P17) from the array shown in Fig. 7.10a. (a) Selected water pressure records for July 19, 1997 to April 9, 1998. (b) Detail of transition shown from September 2 to October 12, 1997. In contrast to the rapidity of the connected-unconnected transitions, subsequent in-creases in pressure continue throughout the winter. Several months are required be-fore winter water pressures exceed the maximum summer values. These two different timescales portend more than one process at work. It has become customary to general-ize winter water pressures as higher than summer water pressures, but this is not always the case (compare Figs. (7.22) and (7.23)). This is another reminder of the heterogeneity and complexity of the glacier bed. Figure (7.22a) illustrates that the spatial variation in Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 188 winter water pressures over a small area (~15m x 15 m) is comparable to the range of diurnal fluctuations in summer. Examples of the spring transition are comparatively scarce because it typically occurs before the field season begins (July 1), if at all. Therefore, it must be captured by sensors emplaced in previous years, some of which have failed by that point or have advected into dormant areas of the bed. The latter describes the fate of the 1997 pressure array. Summer of 1998 was discernible in the records but characterized by complex, aperiodic variations. 0 20 40 60 80 100 120 140 160 180 Day in 1995 Figure 7.23: 1995 Trapridge Glacier transition to summer as recorded by water pressure transducers 94P09 (bold line) and 94P02 (fine line). 94P02 fails on day 162 during a hydromechanical event, and the 94P09 record has been corrected for a concomitant calibration shift. Figure 7.23 chronicles the 1995 onset of summer as recorded by two instruments approximately 30 m apart. Both records have been used in a study by Kavanaugh and Clarke [submitted] documenting a series of three successive hydromechanical events. The first occurred on day 162 where 94P02 fails (fine line, Fig. 7.23), presumably due to an extreme pressure pulse. A correction has been applied to the 94P09 record (bold line) to rectify a calibration shift experienced during this event [Kavanaugh and Clarke, in press]. These records show synchronous and intermittent connected behaviour, with uncon-nected intervals initially as long or longer than periods of connection. Appendix F dis-cusses this in more detail. With the onset of diurnal cycling, both instruments detect mean pressures that are greater than or equal to mean unconnected pressures, reflecting the low capacity of the winter drainage system. Pressure minima are increasingly de-pressed from day 130-170 as the drainage network is excavated by the action of water. Nienow et al. [1998] suggest a trio of mechanisms to explain this development: melting of basal ice, erosion of the bed, and cavity growth due to enhanced glacier sliding. For Trapridge Glacier, winnowing of fines from the subglacial sediment matrix probably plays Chapter 7. A P P L I C A T I O N T O T R A P R I D G E G L A C I E R 189 an important role as well. 7.4.2 Modelling approach Due to the number and complexity of additional processes involved in seasonal tran-sitions, I opt not to parameterize them, but rather to apply the continuum model as is. Reasonable parameterizations would require some knowledge of the deviatoric stress field and spatial distributions of sliding velocity and bed deformation. Coupling with a glacier-dynamics model [e.g., Gudmundsson, 1999] would permit a better evaluation of these quantities. However, worthwhile insights can be gained from the continuum model alone, as many basin-scale features of seasonal transitions are still extricable. For both spring and fall examples, I use 1997 meteorological inputs. Air temperatures are recorded by data loggers in the study area. Snowfall is prescribed for the study area elevation in water-equivalent (w.e.) units. A gradient of 0.0082 cm snow w.e. per metre of elevation gain is applied to estimate the spatial distribution of snow. I obtain this number from the empirical relationship Dsnow — 0.0082z + 10.48 given by Woodward et al. [1997] to describe the w.e. snow depth as a function of elevation for the beginning of the melt season at John Evans Glacier, Nunavut, Canada. Simulated timeseries presented in the following section are taken from a crevassed cell in the study area. Reference model parameters are applied except in the case of hydraulic conductivity. I reduce the minimum conductivity to 10~7 ms - 1 (in accord with the estimates of Fountain [1994] and Hubbard et al. [1995]) and