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Diss. ETH No. 17874 Mechanics and Thermodynamics of Polythermal Glaciers A dissertation submitted to the ETH ZURICH for the degree of Doctor of Sciences presented by ANDREAS ASCHWANDEN Dipl. Natw. ETH born 27. Dezember 1977 citizen of Seelisberg - Switzerland accepted on the recommendation of Prof. Dr. H. Blatter, examiner Prof. Dr. C. Schär, co-examiner Prof. Dr. P. Jansson, co-examiner 2008
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Mechanics and Thermodynamics of Polythermal Glaciers · Mechanics and Thermodynamics of Polythermal Glaciers A dissertation submitted to the ETH ZURICH for the degree of Doctor of

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Page 1: Mechanics and Thermodynamics of Polythermal Glaciers · Mechanics and Thermodynamics of Polythermal Glaciers A dissertation submitted to the ETH ZURICH for the degree of Doctor of

Diss. ETH No. 17874

Mechanics and Thermodynamicsof

Polythermal Glaciers

A dissertation submitted to theETH ZURICH

for the degree ofDoctor of Sciences

presented byANDREAS ASCHWANDEN

Dipl. Natw. ETHborn 27. Dezember 1977

citizen of Seelisberg - Switzerland

accepted on the recommendation ofProf. Dr. H. Blatter, examiner

Prof. Dr. C. Schär, co-examinerProf. Dr. P. Jansson, co-examiner

2008

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Page 3: Mechanics and Thermodynamics of Polythermal Glaciers · Mechanics and Thermodynamics of Polythermal Glaciers A dissertation submitted to the ETH ZURICH for the degree of Doctor of

Perfection is achieved,not when there is nothing more to add,but when there is nothing left to take away.

Antoine de Saint-Exupéry

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ZusammenfassungPolytherme Gletscher bestehen aus kaltem Eis (Eis unterhalb des Druckschmelzpunk-tes) und temperiertem Eis (Eis am Druckschmelzpunkt).Die Viskosität von Eis ist abhängig von der Temperatur des kalten Eises und vom

Wassergehalt des temperierten Eises. Kaltes und temperiertes Eis werden durch eineinterne freie Übergangsfläche getrennt. Kaltes Eis enthält kein flüssiges Wasser, und derWärmefluss in kaltem Eis wird durch die Fouriergleichung beschrieben. TemperiertesEis hingegen enthält einen kleinen Anteil an flüssigem Wasser. Über den Wasserflussin temperiertem Eis ist wenig bekannt, vorgeschlagen wurden Darcy- oder Fick’scheDiffusion. Da die Diffusion aber ohne dies relativ klein ist, wird sie in vielen Fällenvernachlässigt.Für eine umfassende Fliessmodellierung polythermer Gletscher ist es wichtig, die

Temperaturverteilung des kalten Eises sowie den Wassergehalt des temperierten Eiseszu kennen. Eine von vier möglichen Quellen für flüssiges Wasser in temperiertem Eisist Schmelzwasserproduktion durch Reibungswärme, in vielen arktischen Gletschernsogar die einzige. Sie kann abgeschätzt werden, indem die Reibungswärme entlangvon Trajektorien integriert wird. In einer Fallstudie über den Storglaciären, Schweden,wurde ein Wassergehalt von bis zu 10 Gramm Wasser pro Kilogramm Mischung auf-grund von Reibungswärme im Zungenbereich berechnet. Für einen solchen Wert istder sogenannte “rate factor” mehr als dreimal so gross als für wasserfreies Eis. Diesist ein Grund, warum Schmelzwasserprodukton durch Reibungswärme wichtig für dieModellierung temperierter und polythermer Gletscher ist.Auch in Eisschilden gibt es temperierte basale Zonen, die Wasser enthalten. Aktuelle

Eisschildmodelle können in zwei Kategorien unterteilt werden: (1) “Kalteis-Modelle”,welche die Temperatur auf den Druckschmelzpunkt limitieren und den Einfluss desflüssigen Wassers auf das Eisfliessen ignorieren; und (2) echte polytherme Model-le. Solche Modelle lösen die Fourier-Gleichung für kaltes Eis und eine Advektions-Produktions-gleichung für den Wassergehalt im temperierten Eis zusammen mit denSprungbedinungen und den kinematischen Bedingungen am Übergang. Für die nu-merische Behandlung von Phasenübergängen existieren sogenannte “front-tracking”Methoden und Enthalpie-Methoden, wovon letztere relativ leicht zu implementierensind. Wird der Wärmefluss mit Hilfe des Enthalpiegradienten ausgedrückt, so nenntman dies Enthalpiegradientenmethode. Ein mathematisches Modell, welches auf einerEnthalpiegradientmethode basiert, wird für die Behandlung der Thermodynamik vonpolythermen Gletschern entwickelt. Es wird gezeigt, dass diese Methode gleichwertigzur oben erwähnten Trajektorienmethode ist, falls der Feuchtestrom gegen Null geht.Die vielseitige Anwendbarkeit dieser Methode wird anhand einiger Beispiele kalter,temperierter und polythermer Situationen demonstriert.

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AbstractPolythermal glaciers and ice sheets contain both cold ice (i.e. below the pressuremelting point) and temperate ice (i.e. at the pressure melting point).The viscosity of cold ice and temperate ice depends on temperature and on water

content, respectively. Cold and temperate ice are separated by the cold-temperatetransition surface, which is an internal free surface of phase change and thus a ther-modynamical discontiuitiy. Cold ice is free of liquid water, and the heat flux in cold icecan be expressed by Fourier’s law. On the other hand, temperate ice is characterizedby existence of liquid water in the ice matrix. Heat flux in temperate ice is difficultto constitute, Darcy-type and Fick-type diffusion have been proposed. Moisture dif-fusivity is, however, assumed to be small, and moisture diffusion is therefore oftenneglected.To comprehensively model the flow of a polythermal ice mass, it is essential to

know the temperature distribution in the cold part and the spatial variation in thewater content in the temperate part. One of four possible sources for liquid water intemperate ice is melting due to strain heating. Strain heating is the only source ofliquid water in high arctic glaciers with a temperate basal zone in the lower tongue areabeneath otherwise cold ice. Meltwater accumulation can be estimated by integratingstrain heating along trajectories. In a case study on Storglaciären, a small polythermalglacier in northern Sweden, values reach more than 10 grams water per kilogram ice-water mixture in the lowest parts of the temperate domain. For this water content,the rate factor is more than three times higher than for water-free ice, and therefore,water production by strain heating is important for the modeling of temperate andpolythermal glaciers.In ice sheets, temperate zones can be present at the bed, resulting in temperate

ice containing liquid water. Present-day ice sheet models can be divided into twocategories: (1) “cold-ice method” ice sheet models which limit temperatures to thepressure-melting point and ignore the influence of liquid water in temperate ice onice flow; and (2) truly polythermal ice sheet models. Such models solve the Fourierequation for the cold parts and a moisture advection-production equation for thetemperate parts together with jump conditions and kinematic conditions at the cold-temperate transition surface. To handle phase changes, front-tracking methods andenthalpy methods are commonly used. It is proposed to model polythermal glaciersand ice sheets by an enthalpy method because enthalpy methods are known for theirease of implementation. Heat flux is expressed in terms of the gradient of enthalpy.This technique is called “enthalpy gradient method”. Based on the enthalpy gradientmethod, a mathematical model for the thermodynamics of polythermal glaciers isdeveloped. It is shown that, in the limit of vanishing moisture diffusion, the enthalpygradient method is equivalent to the trajectory integration method. The versatility ofthe enthalpy gradient method is demonstrated by application to cold, temperate and

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polythermal test cases. The use of an enthalpy method removes the cold-temperatetransition surface from the list of discontinuities in polythermal glaciers and ice sheets.

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Contents

1 Introduction 11.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Thermal Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Aim of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Numerical Simulations of Isothermal Glacier Flow 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.2 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . 92.2.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Scale Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6.1 Horizontal Surface Velocities . . . . . . . . . . . . . . . . . . . . 172.6.2 Effective Strain Rates . . . . . . . . . . . . . . . . . . . . . . . 17

2.7 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Meltwater Production due to Strain Heating in Storglaciären, Sweden 273.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.1 First Order Ice Flow Model . . . . . . . . . . . . . . . . . . . . 293.2.2 Trajectory Model . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.3 Strain Heating and Water Content . . . . . . . . . . . . . . . . 32

3.3 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.5 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Modeling Polythermal Glaciers 394.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.1 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2.2 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . 414.2.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 424.2.4 Enthalpy Gradient Method . . . . . . . . . . . . . . . . . . . . . 434.2.5 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

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4.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3.1 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . 474.3.2 Storglaciären . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3.3 Ice flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3.4 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3.5 Role of advection and diffusion . . . . . . . . . . . . . . . . . . 514.3.6 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.5 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 55

5 An Enthalpy Method for Glaciers and Ice Sheets 595.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2.1 Ice flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2.2 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2.3 Enthalpy Gradient Method . . . . . . . . . . . . . . . . . . . . . 63

5.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.3.1 Ice flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.3.2 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.4.1 Uncoupled Experiments . . . . . . . . . . . . . . . . . . . . . . 665.4.2 Coupled Experiments . . . . . . . . . . . . . . . . . . . . . . . . 67

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6 Outlook 75

References 77

Acknowledgements 85

Curriculum Vitae 87

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1 IntroductionFluid dynamics of glaciers are a complex interplay of climate, the geological andgeomorphological setting, hydrology and hydraulics on different scales. The non-Newtonian strain-softening behavior of ice is similar to that of many metals near themelting point (e.g. Norton, 1929). The viscosity of ice depends, among other variables,on temperature and on water content in cold and in temperate ice, respectively. Tocomprehensively model glaciers and ice sheets, knowledge of their thermal structureis important.

1.1 ThermodynamicsThe term polythermal glacier was first introduced by Fowler and Larson (1978). Poly-thermal glaciers consist of both temperate ice (i.e. ice at the melting point) and coldice (i.e. ice below the melting point). This definition is not rigorous, as reconized byLliboutry (1971), since the melting temperature depends on stresses, the interfacialfree energies, the salt content of the liquid phase and on the nature of defects in thesolid. Temperate ice is characterized by the occurrence of liquid water within the ice.He suggested the following definition: “Temperate ice is ice which contains within ita liquid phase and which is in local equilibrium with it.” The permeability of temper-ate ice was already discussed in the eighteenth century by, for example, L. Agassiz,F. A. Forel or the Schlagintweit brothers, see Lliboutry (1971, and references therein)for an account. For the movement of liquid water in temperate ice, Fick-type diffusion(Hutter, 1982) and Darcy-type (Fowler, 1984) diffusion were propsed. Lliboutry andDuval (1985) attempted to quantify the effect of the water content on the viscosity.Four different sources are assumed to be the origin of liquid water in the ice matrix

in glaciers (Paterson, 1971; Lliboutry, 1976):

1. water trapped in ice as water-filled pores close-off at the firn-ice transition in theaccumulation area

2. melting due to strain heating

3. water entering the glacier at the surface

4. adjustment of the pressure melting point due to changes in overburden pressureheats or cools temperate ice. The energy is balanced by freezing or melting ofliquid water within the ice

Methods to estimate liquid water content can be divided into thermodynamical meth-ods, remote sensing and numerical modeling. In-situ calorimetry is a thermodynamicalmethod and can be used to determine the absolute water content at the cold-temperate

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2 1 Introduction

transition surface (CTS) if the temperature gradient at, and the migration rate of, theCTS are known (Pettersson et al., 2004). The liquid water content relative to a ref-erence point can be estimated from radio back-scatter power (Hamran et al., 1996).Radio back-scatter techniques, which belong to the group of remote sensing methods,were applied, e.g., by Moore et al. (1999), Murray et al. (2000) and Pettersson et al.(2004). The content of liquid water can differ remarkably between different glaciers,estimates range from 0 % to 9 %; an overview of water content determinations for var-ious glaciers is given in Pettersson et al. (2004). The contribution of melting due tostrain heating can be estimated by (1) integrating strain heating along trajectoriesor by (2) solving a partial differential equation for the liquid water content. Method(1), which is numerically less critical since stable numerical integration schemes ex-ist, is applied in chapter 3 to Storglaciären, northern Sweden. Method (2) is used inchapter 4; and a comparison shows the equivalence of (1) and (2) in the limit of zerodiffusion.Cold ice, on the other hand, is free of liquid water. Heat flux in cold ice can be

described by the Fourier law and the temperature-depence of the viscosity is wellestablished (e.g. Paterson, 1994; Smith and Morland, 1981).In a polythermal glacier or ice sheet, the CTS represents a surface of phase change.

Water arriving at the CTS freezes and the released energy is transported into thecold ice through a nonvanishing temperature gradient (Greve, 1997b). Hutter (1982)presented a mathematical model of polythermal glaciers and ice sheets. Cold ice istreated as a non-linear viscous, heat conducting fluid, while temperate ice is regardedas a binary mixture of ice and water using two balance laws for mass but only onefor momentum and energy. Boundary surfaces are treated as non-material surfaces ofdiscontinuity where finite jumps may occur. However, only sensible heat flux is con-sidered in the energy balance equation. Fowler (1984) used two momentum equationsand constituted the moisture flux in temperate ice with Darcy’s law. Hutter (1993)then suggested to treat temperate ice in polythermal glaciers and in wholly temperateglaciers differently. The amount of water generated in polythermal glaciers is likely tobe small as strain heating and geothermal heat are the only sources. Water is expectedto move along grain boundaries rather than in large cracks. Thus the momentum ofwater is about that of ice; and a Fick-type diffusive model is appropriate. The watercontent in fully temperate glaciers is relatively large, hence the velocity of the intersti-tial water exceeds that of the ice. Consequently, considering the momentum balanceof water is important. Temperate ice is therefore assumed to be a porous media forwhich Darcy’s law is more applicable. Greve (1995) includes latent heat flux in theenthalpy balance. A comprehensive overview can be found in Greve (1997b).Currently, prognostic ice-sheet models of various complexity (e.g. Huybrechts, 1990;

Saito et al., 2003) are used to investigate ongoing processes in, and to predict thefuture behavior of Greenland and Antarctica. These models can be divided into twogroups: (1) “Cold-ice method” ice sheet models limit temperatures to the meltingpoint, thus locally violating energy conservation; and (2) truly polythermal ice sheetmodels. The “simulation code for polythermal ice sheets” (SICOPOLIS) (Greve, 1995,1997a,b) is the only truly polythermal ice sheet model. It solves an advection-diffusion-production equation for temperature in cold ice and an advection-production equationfor the water content in temperate ice, together with the jump conditions and the

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1.2 Thermal Structures 3

kinematic condition at the CTS. A front-tracking method is used to handle the CTSas a moving boundary.

1.2 Thermal StructuresBlatter and Hutter (1991) identified five different thermal structures in polythermalglaciers, and Blatter (1991) added a sixth. Glaciers with a thermal layering as inFigures 1.1a and 1.1b are found in cold regions such as the Canadian arctic (Blatterand Kappenberger, 1988). The bulk of ice is cold except for a temperate layer nearthe bed which exists due to melt water resulting from strain heating. This type shallbe referred to as Canadian-type in the following. Figure 1.1c illustrates a glacierwhere the lower part of the accumulation area experiences high melting in spring andsummer. This meltwater infiltrates the firn pack. Refreezing at the transition to thecold firn produces latent heat that warms up the firn until it becomes temperate andallows for formation of temperate ice within an otherwise cold glacier. Glaciers of thetype shown in Figure 1.1d can be found in the European Alps at high altitudes andon small Greenlandic ice caps (Haeberli, 1976). Figure 1.1e illustrates a polythermalstructure commonly found on Svalbard and on the eastern side of the Scandinavianmountains (Pettersson, 2004, and references therein). This type is sometimes calledScandinavian-type or Svalbard-type. High net ablation at lower altitudes can turn thecold ice into temperate ice, while the cold layer can still exist higher up (Figure 1.1f).Polythermal glaciers can be found mainly at high latitudes (Hutter et al., 1988),

but also at high altitudes in the European Alps (Haeberli, 1976) and in China (Mao-huan, 1990). Borehole measurements by Lüthi et al. (2002) reveal temperate basalpatches beneath otherwise cold ice in Jakobshavn Isbræ, west Greenland. Evidencefor temperate ice in Antarctica is given by the existence of sublacial lakes (Siegertet al., 2005). The EPICA (European Project for Ice Coring in Antarctica) stoppeddrilling at Dome C about 15 m above bedrock because seismic soundings suggested thepresence of meltwater (Parrenin et al., 2007). Also close to the bedrock, Motoyama(2007) found unusual frozen water chips and interpreted them as water that leakedinto the borehole and had frozen in the drill.

1.3 Aim of ThesisMy work aims at furthering our understanding of thermodynamical processes in poly-thermal glaciers by means of mathematical models and numerical simulations. Themain goal is to develop a mathematical model which describes the thermodynamicsof cold, polythermal and temperate glaciers and ice sheets adequately yet avoidinginternal free surfaces.

1.4 OutlineThis thesis comprises six chapters. Each of chapters 2 – 5 is an independent manuscriptand may therefore contain overlapping information; for example repetition of field and

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4 1 Introduction

d

c

b

aa

b

c

ELA

d

e

f

temperate cold

Figure 1.1: Schematic view of different polythermal structures based on Blatter and Hutter(1991) and modified by Pettersson (2004). The gray color indicates temperate ice andthe equilibrium line altitude (ELA) is indicated with a line. Courtesy of R. Pettersson.

constitutive equations is unavoidable. The manuscripts are ordered thematically andnot chronologially. Conclusions are drawn in each chapter individually and an outlookis given in chapter 6. The references to all manuscripts are presented in summary atthe end of the thesis.

Chapter 2: Numerical Simulations of Isothermal Glacier Fow The goal is twofold.First, the performance of the finite element code used to obtain numerical simulationsin later chapters is tested; and second, the Stokes equations and approximations ofdifferent complexity to it are compared qualitatively.

Chapter 3: Meltwater Production due to Strain Heating in Storglaciären, Swe-den Published under the same title in Journal Geophysical Research with co-authorH. Blatter. One of four possible sources for liquid water in temperate ice is meltingdue to strain heating. This source can be estimated by modeling. Calculated valuesreach more than 10 grams of water per kilogram ice-water mixture. For this moisturecontent, the rate factor is three times higher than for water-free ice. This indicatesthat water production by strain heating is important for the modeling of temperateice.

Chapter 4: Mathematical Model and Numerical Simulation of Polythermal GlaciersSubmitted to Journal Geophysical Research with co-author H. Blatter. In this chap-ter, a novel mathematical model to solve the enthalpy equation in cold and temperateice is presented. The enthalpy function is regularized at the pressure melting point

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1.4 Outline 5

by a brine pocket parametrization known from sea ice modeling. From the simulateddistribution of enthalpy, the temperature distribution and the moisture content in coldand temperate ice, respectively, are then obtained.

Chapter 5: An Enthalpy Method for Glaciers and Ice Sheets This chapter is adraft of work in progress. The mathematical model presented in chapter 4 is simplifiedand refined. The enthalpy function and its relation to temperature and water contentis reinterpreted such that no regularization is required. Basal sliding is introducedthrough a linear sliding law. Thermo-mechanically coupled simulations are presentedand the applicability of the method to cold, temperate and polythermal conditions isdemonstrated.

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6 1 Introduction

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2 Numerical Simulations ofIsothermal Glacier Flow

AbstractDifferent models for glacier flow problems were evaluated and Comsol Multiphysicswas selected to obtain numerical solutions. Numerical simulations based on the fullStokes equations and two approximations to them were then compared for longitudinalprofiles of Haut Glacier d’Arolla, Griesgletscher and Storglaciären. The focus lies insetting up Comsol Multiphysics for glacier flow problems rather than in glacier flowitself. Results are thus discussed only qualitatively. For large aspect ratios (verticalextent of the ice mass divided by the horizontal extent), the shallow ice approximationwas found to deviate significantly from the full Stokes solution and from the first orderapproximation.

2.1 IntroductionNumerical simulations of glacier flow are important in many glaciological applications.Questions involving past, present and future evolution of glaciers and ice sheets (Huy-brechts, 1994; Greve, 1997a; Albrecht et al., 2000; Schneeberger et al., 2001), stabilityanalysis of (avalanching) glaciers to assess their risk potential (Pralong et al., 2003)or dating ice cores (Dansgaard and Johnsen, 1969; Parrenin et al., 2004), all rely onnumerical simulations of glacier flow. Recent dramatic changes in flow velocity of icestreams, most prominently Jakobshavn Isbræ, west Greenland (Joughin et al., 2004)or Helheim Glacier, east Greenland (Howat et al., 2005), require flow modeling toinvestigate possible causes of the observed changes.Bueler et al. (2005) note that our level of confidence in numerical simulations de-

pends on two questions: “Are we solving the correct equations?” and “Are we solvingthe equations correctly?” and add that the terms ‘validation’ and ‘verification’ arestandard in the computational fluid dynamics (CFD) literature. To this end, Wes-seling (2001, p. 560 – 561) provides the following informal definition: “Two types oferrors may be distinguished: modeling errors and numerical errors. Modeling errorsarise from not solving the right equations. Numerical errors arise from not solvingthe equations right. The assessment of modeling errors is called validation, whereasthe assessment of numerical errors is called verification. . . Validation makes sense onlyafter verification, otherwise agreement between measured and computed results maywell be fortuitous.”It is therefore crucial to carefully evaluate and test available glacier flow models.

Numerical glacier flow models may approximate the model domain by using finite

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8 2 Numerical Simulations of Isothermal Glacier Flow

differences (FD), finite elements (FE) and finite volumes (FV), and they may solveeither the Stokes equations (FS) or approximations of different complexity. During thecourse of this work, models and codes listed in Table 2.1 were evaluated. The FD codeglacier flow was written in 2000 by H. Blatter in Fortran 77, updated by A. Aschwandenin 2004 and later translated into C++. Glacier flow was used in chapter 3 to obtainnumerical simulations, and FELib has been successfully used for simulations of glacierflow by Picasso et al. (2003) and Kirner (2007). Nevertheless, both models are limitedin their flexibility, and adaption to other problems than what they are built for issomewhat cumbersome. Comsol Multiphysics (Comsol, 2007), which is based on theFinite Element Method, is capable of solving most kinds of (coupled) partial differentialequations. Ready-made application modes for many often-used problems in fields suchas fluid dynamics, chemical engineering, structural analysis and the like reduce hand-coding substantially. This flexibilty, a Matlab-like syntax and the existence of couplingvariables1 made Comsol Multiphysics the first choice to obtain numerical simulationsin this chapter and in chapters 4 and 5.CFD codes such as Comsol Multiphysics provide the user with benchmark models

that compare numerical solutions with analytical solutions. In the following chapter,numerical simulations based on the Stokes equations and different approximations toit are compared, and the limitations of the approximations are discussed. In a strictsense, validating means comparing to measurements. In this work, however, the Stokesequations serve as a benchmark against which, in a less strict sense, the approximationsare validated.For this kind of study, a longitudinal plane-flow approximation is used to demon-

strate the limits of the approximations used, and the equations are thus written forthe two-dimensional case only.

Table 2.1: Evaluated Models and Programs. FS: full Stokes equations, FOA: first orderapproximation.

Name Method Physics Author/VendorComsol Multiphysics FE FS Comsol AB, Sweden (Comsol, 2007)FELib FE FOA Group J. Rappaz, EPF Lausanneglacierflow FD FOA H. Blatter, ETH Zurich

2.2 Mathematical Model

2.2.1 Field Equations

1Coupling variables are extremely powerful in their ability to make the value and the exact Jacobianof an expression available nonlocally. They are not only useful for modeling coupled problems —one can also use them solely for postprocessing and visualization purposes (Comsol, 2007).

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2.2 Mathematical Model 9

Glacier ice is generally assumed to be a viscous, incompressible, heat-conductiongnon-Newtonian fluid which obeys the Stokes equations,

∇ ·v = 0, (2.1)∇ ·T = −ρg, (2.2)

where ∇ · is the divergence operator, v is the velocity, T is the Chauchy stress tensor,ρ is the density, and g is the acceleration due to gravity. Glacier ice flow is quasi-stationary and thus the acceleration term can be neglected. For incompressible media,the Cauchy stress tensor can be split into a deviatoric part and an isotropic part:

T = T′ − pI, (2.3)

where T′ and I are the deviatoric stress tensor and the unit tensor, respectively; andp = −1

3tr (T) is the isotropic pressure.

2.2.2 Constitutive EquationTo close the system, constitutive equations are required. In the case of isothermal ice,only an equation which relates stress and deformation is required. In glaciology, it iscommon to describe how ice deforms if stress is applied with the following formulation,

D = AF (σeff)T′, (2.4)

where D = 12

(∇v +∇vT

)is the strain rate tensor. The effective stress, σeff , is defined

as

σeff =√

IIT′ =√

12tr (T′ ·T′), (2.5)

where IIT′ is the second invariant of T′. The rate factor A is scalar factor whichdepends on quantities such as temperature, water content, ice crystal orientation,impurities and further quantities. The rate factor is often used as a tuning parameterto simulate velocities in close agreement with measured velocites. F = F (σeff) is thefluidity (creep response function), for which both Glen (1955) and Steinemann (1958)obtained a power-law type relationship:

F (σeff) = σn−1eff . (2.6)

This relationship is known as Norton’s law (Norton, 1929) in metallurgy. The choiceof n is still a matter of debate, but n = 3 is widely used.In standard fluid dynamics literature, it is more common to express the stress field

in terms of the strain field,T′ = 2ηD, (2.7)

where η is the effective viscosity, and thus

η = 12Aσn−1

eff. (2.8)

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10 2 Numerical Simulations of Isothermal Glacier Flow

η can then be written asη = 1

2 A− 1n ε

1−nn

eff , (2.9)

where εeff is the effective strain rate,

εeff =√

IIε =√

12tr (D ·D), (2.10)

and IIε is the second invariant of D. As a consequence, the viscosity becomes infiniteat vanishing effective stress if n > 1. To avoid this, a regularization has been proposed(Meier, 1958),

F (σeff) = σn−1eff + σn−1

0 , (2.11)

where σ0 is a small number. The regularized viscosity,

η = 12A

(σn−1

eff + σn−10

) , (2.12)

now has a finite limit for vanishing effective stress,

limσeff→0

η = 12Aσn−1

0. (2.13)

For n > 1 no analytical inversion is possible, and the viscosity is implicitly defined asa polynomial equation. Nevertheless, by adding a small number ε2

0 to IIε,

εeff =√

IIε + ε20, (2.14)

a regularization similar to equation (2.11) can be achieved. ε0 can be regarded as thesmallest strain rate that can be resolved.

2.2.3 Boundary ConditionsAt the glacier surface tangential stress vanishes and the resulting boundary conditionbecomes

T ·n = −pairn, (2.15)

where pair is the atmospheric pressure and n is the outward unit normal vector.Either a basal velocity, vb, or a given sliding law can be prescribed. If no-slip con-

ditions are assumed, then vb ≡ 0. In case of a sliding law, the basal shear traction, τb,and the bed-parallel sliding velocity, vb, are functionally related through F (τb, vb) = 0.

2.3 Scale AnalysisLargely independent of size, basal stress hardly exceeds one bar in glaciers. This is aresult of the power law stress-strain rate relation making the ice softer rapidly as theeffective stress exceeds one bar. The mean basal shear traction may be approximatedby the mean basal driving stress, τb ∝ H sinα, where H is the mean thickness of the

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2.3 Scale Analysis 11

ice and α is the mean inclination of the ice surface. The aspect ratio is ε = [H]/[L] =[W ]/[U ], where [L] and [H] are the horizontal and vertical extents of an ice mass,and [U ] and [W ] are the typical horizontal and vertical components of the ice velocity,respectively. For large ice sheets, average inclination of the surface slope is of theorder of the aspect ratio and thus α ∝ ε. For smaller glaciers, [H] may be larger thanH, thus underestimating the mean surface slope. With the assumption of a giveninvariant basal stress and assuming [H] = H,

[L] ∝ [H]2. (2.16)

As a consequence, the aspect ratio is inversely proportional to the vertical extent, orcorrespondingly inversely proportional to the square root of the horizontal extent,

ε ∝ 1[H]

, ε ∝√

1[L]

. (2.17)

For glaciers with similar shape, the thickness of the ice only grows with the squareroot of the horizontal extent. For a slab of ice, [U ] ∝ H, thus

[U ] ∝ 1ε, (2.18)

and [W ] is independent of ε. This leads to a hierarchy of approximations. FollowingBlatter (1995), the scaled mass conservation is

∂u

∂x+ ε

∂w

∂z= 0, (2.19)

and, correspondingly, the scaled momentum balance reads as

ε∂

∂x

(4 η∂u

∂x

)+ ∂

∂zη

(∂u

∂z+ ε2

∂w

∂x

)= ρg

∂S

∂x−ε2 ∂

2

∂x2

(∫ S

z

(∂u

∂z+ ε2

∂w

∂x

)dz′), (2.20)

where S = S(x) is the elevation of the upper free surface. The scaled constitutiveequations are

ε∂u

∂x= 1

2 η σxx, (2.21)

and∂u

∂z+ ε2

∂w

∂x= 1

ησxz, (2.22)

where σij are the components of T′. Finally, the scaled boundary condition at theupper free surface is:1− ε2

(∂S

∂x

)2σxz∣∣∣∣

S− ε

(2 ∂S∂x

)σxx

∣∣∣∣S

= 0. (2.23)

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12 2 Numerical Simulations of Isothermal Glacier Flow

Considering only zeroth-order terms in equations (2.20) to (2.23) is commonly re-ferred to as the Shallow Ice Approximation (SIA), wherein ice flow due to deformationis only determined by vertical shearing; see e.g. Hutter et al. (1981). Higher OrderApproximations include terms of at least order ε. The so-called First Order Approx-imation (FOA) developed by Blatter (1995) is now a quasi-standard for glacier flowmodels (e.g. Hubbard et al., 1998, 2000; Pattyn, 2002; Breuer et al., 2006) and becomesincreasingly popular in ice sheet modeling (e.g. Saito et al., 2003). A recent study byPattyn et al. (2008) compares different higher order and full Stokes models. Colingeand Rappaz (1999) proved existence and uniqueness of numerical solutions based onthe FOA. Hindmarsh (2004) compares approximations to the Stokes equations, andlists and describes all commonly used approximations in glaciology.Let Ω be the glacier domain and ΓC its border, where the subscript C = B, S denotes

bed and surface. The three solutions are formulated as follows.

FSThe FS can be written as:

∇ · (−pI + 2ηD) = −ρg in Ω∇ ·v = 0 in Ω

(−pI + 2ηD) ·n = pairn on ΓSv = vb on ΓB

. (2.24)

FOAThe FOA can be written as:

4 ∂∂x

(η ∂u∂x

)+ ∂

∂z

(η ∂u∂z

)= ρg ∂S

∂xin Ω

∂u∂x

+ ∂w∂z

= 0 in Ω∂u∂z− 2 ∂S

∂x∂u∂x

= 0 on ΓSu = ub on ΓB

. (2.25)

SIAThe SIA can be written as:

∂∂z

(η ∂u∂z

)= ρg ∂S

∂xin Ω

∂u∂x

+ ∂w∂z

= 0 in Ω∂u∂z

= 0 on ΓSu = ub on ΓB

. (2.26)

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2.4 Model Setup 13

Effective strain ratesThe corresponding effective strain rates εeff are:

εFSeff =

√12

((∂u∂x

)2+(∂w∂z

)2+ 1

2

(∂u∂z

+ ∂w∂x

)2),

εFOAeff =

√12

((∂u∂x

)2+ 1

2

(∂u∂z

)2),

εSIAeff =

√12

(12

(∂u∂z

)2).

(2.27)

2.4 Model SetupThe focus of this study lies in setting up and testing Comsol Multiphysics for glacierflow problems rather than in glacier flow itself and results are thus discussed morequalitatively than quantitatively. Glaciers with different aspect ratios ε, but compara-ble geometry and basal stress are compared to qualitativly assess the limitations of theFOA and the SIA. To this end, the horizontal and vertical extents of the test geome-tries are stretched by factors Λ and

√Λ, respectively, thus changing the aspect ratio

by a factor 1/√

Λ, Λ ∈ 0.25, 0.5, 1, 2, 5, 10, 100 (Figure 2.1). Longitudinal profiles ofGriesgletscher (Switzerland), Haut Glacier d’Arolla (Switzerland) and Storglaciären(Sweden) serve as test cases (Figure 2.2). Haut Glacier d’Arolla and Griesgletscherare temperate and Storglaciären is polythermal, but for the purpose of this study, thechoice of no-slip conditions at the glacier bed can be justified.Scaling factors for glaciers tested and model parameters used in this study are listed

in Table 2.2 and Table 2.3, respectively. By using a constant rate factor, A = A0, Ais only a multiplier of the velocity field, and thus, the actual numerical value of A0 isnot important because the velocity field willl be normalized later.

Table 2.2: Scaling Factors and Corresponding Aspect Ratios for Tested Glaciers.

Λε 0.25 0.50 1 2 5 10 100Haut Glacier d’Arolla 0.28 0.20 0.14 0.10 0.063 0.044 0.014Griesgletscher 0.33 0.23 0.16 0.11 0.073 0.051 0.016Storglaciären 0.30 0.20 0.15 0.10 0.067 0.046 0.015

2.5 Numerical SimulationsNumerical simulations are obtained with the commercial program package Comsol Mul-tiphysics. Quadratic and linear Lagrange elements are used for velocity and pressure,

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14 2 Numerical Simulations of Isothermal Glacier Flow

H

L

ΛL

√ΛH

Figure 2.1: Example of how glacier geometries are scaled. Horizontal and vertical extent arestretched by Λ and

√Λ, respectively, and Λ = 2.

Table 2.3: Physical Constants and Parameters Used in this Study.

Symbol Variable or Constant Name Value UnitA rate factor 2.22 · 10−24 Pa−n s−1

ε0 viscosity regularization 10−13 s−1

n exponent of the flow law 3 -pair atmospheric pressure 105 Paρ densitiy of ice 900 kg m−3

ub horizontal basal velocity 0 m s−1

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2.5 Numerical Simulations 15

Figure 2.2: Geomtries of Haut Glacier d’Arolla (upper panel), Griesgletscher (middle panel)and Storglaciären (lower panel).

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16 2 Numerical Simulations of Isothermal Glacier Flow

respectively. This choice of elements satisfies the inf-sup condition and thus no furtherstabilization is required. Unstructured triangular elements with a maximum elementsize of 20

√Λ are used. An advantage of Comsol Multiphysics is its ability to solve (cou-

pled or uncoupled) partial differential equations simultaneously. Instead of settingup three independent models (FS, FOA and SIA), all equations are implemented inthe same model. This facilitates post-processing considerably but increases memoryusage.To solve non-linear, stationary problems, Comsol Multiphysics uses an affine variant

form of the damped Newton method (Deuflhard, 1974). The discrete form of theequations can be written as

f(U) = 0, (2.28)where f(U) is the residual vector and U is the solution vector. Starting with an initialguess U0, the software forms the linearized model using U0 as the linearization point.It solves the discretized form of the linearized model,

f ′(U0)δU = −f(U0), (2.29)

for the Newton step, δU , using the selected linear system solver (f ′(U0) is the Jacobianmatrix). It then computes the new iteration,

U1 = U0 + λδU, (2.30)

where λ (0 ≤ λ ≤ 1) is the under-relaxation factor. Next the modified Newtoncorrection estimates the error, E, for the new iteration, U1, by solving

f ′(U0)E = −f(U1). (2.31)

If the relative error corresponding to E is larger than the relative error in the previousiteration, the code reduces the damping factor λ and recomputes U1. This algorithmrepeats the damping-factor reduction until the relative error is less than in the previousiteration or until the damping factor underflows the minimum damping factor. When ithas taken a successful step U1, the algorithm proceeds with the next Newton iteration.A value of λ = 1 results in Newton’s method, which converges quadratically if the

initial guess U0 is sufficiently close to a solution. In order to enlarge the domainof attraction, the solver chooses the damping factors judiciously. Nevertheless, thesuccess of a nonlinear solver depends heavily on a carefully selected initial guess.For this work, however, a constant initial value of 0 for all variables is sufficient forconvergence.The nonlinear iterations terminate when the following convergence criterion is sat-

isfied: Let U be the current approximation to the true solution vector, and let E bethe estimated error in this vector. The software stops the iterations when the relativetolerance exceeds the relative error computed as the weighted Euclidean norm,

err =

√√√√ 1N

N∑i=1

(‖Ei‖Wi

)2

. (2.32)

Here N is the number of degrees of freedom (DOF) and Wi = max(‖Ui‖, Si), where Siis the average of ‖Uj‖ for all DOFs j having the same name as DOF i times 0.1.

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2.6 Results 17

2.6 ResultsIn the following section, simulated horizontal surface velocities and simulated effectivestrain rates are presented.

2.6.1 Horizontal Surface VelocitiesTo facilitate comparisons, all axes and velocities are normalized. FS and FOA velocitiescan be normalized together (subscript t) or independently (subscript i). We then definethe corresponding normalized horizontal velocity field u as

uιt(Λ) = uι(Λ)max [uFS(Λ), uFOA(Λ)] , ι = FS,FOA (2.33)

anduιi(Λ) = uι(Λ)

max [uι(Λ)] , ι = FS,FOA. (2.34)

The SIA is normalized independently. The advantage of the applied scaling is thatthe SIA solution retains its shape, simulated velocities for scaling factors Λ 6= 1 aremultiplied by

√Λ relative to velocities for Λ = 1.

Figure 2.3 shows normalized horizontal surface velocites ut (left column) and ui

(right column) for Haut Glacier d’Arolla (upper panel), Griesgletscher (middle panel)and Storglaciären (lower panel) for selected Λ.

2.6.2 Effective Strain RatesεSIA

eff contains zeroth order strain rates only, εFOAeff additionally includes first order strain

rates. During postprocessing, however, it is possible to calculate “full” effective strainrates ˙ειeff (ι = FOA, SIA) for the corresponding approximations,

˙ειeff =

√√√√√12

(∂uι∂x

)2

+(∂wι

∂z

)2

+ 12

(∂uι

∂z+ ∂wι

∂x

)2. (2.35)

An upper limit for effective strain rates can be estimated from

εeff = Aσneff . (2.36)

Stresses in glaciers are of the order of 105 Pa, a typical value of the rate factor is10−16 Pa−3 a−1 and n = 3. These values yield a typical effective strain rate of 0.1 a−1.Stresses exceeding 3 · 105 Pa would result in strain rates larger than 1 a−1. Strain ratessignificantly larger than 1 a−1 are assumed to be physically unrealistic.Figure 2.4 shows effective FS strain rates for Haut Glacier d’Arolla (a), Gries-

gletscher (b) and Storglaciären (c). In Figures 2.5 and 2.6 FOA and SIA effectivestrain rates are presented for Haut Glacier d’Arolla (a,b), Griesgletscher (c,d) andStorglaciären (e,f). Plots a), c) and e) show εeff while b), d) and f) show ˙εeff . Dif-ferences between εeff and ˙εeff are shown in the left columns of Figure 2.7 (FOA) andFigure 2.8 (SIA), respectively, together with differences between full Stokes strain ratesand ˙εeff in the right columns.

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18 2 Numerical Simulations of Isothermal Glacier Flow

Haut Glacier d’Arolla

Griesgletscher

Storglaciären

Figure 2.3: Normalized horizontal surface velocities ut (left column) and ui (right column)for selected scaling factors Λ.

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2.6 Results 19

Figure 2.4: Effective FS strain rates εeff for Haut Glacier d’Arolla (a), Griesgletscher (b)and Storglaciären (c) for Λ = 1. Values are in one per year.

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20 2 Numerical Simulations of Isothermal Glacier Flow

Figure 2.5: Effective strain rates εeff (left column) and ˙εeff (right column) for FOA for HautGlacier d’Arolla (a,b), Griesgletscher (c,d) and Storglaciären (e,f) for Λ = 1. Values arein one per year.

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2.6 Results 21

Figure 2.6: Effective strain rates εeff (left column) and ˙εeff (right column) for SIA for HautGlacier d’Arolla (a,b), Griesgletscher (c,d) and Storglaciären (e,f) for Λ = 1. Values arein one per year.

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22 2 Numerical Simulations of Isothermal Glacier Flow

Figure 2.7: Differences in effective strain rates εFOAeff − ˙εFOA

eff (left column) and εFSeff − ˙εFOA

eff(right column) for Haut Glacier d’Arolla (a,b), Griesgletscher (c,d) and Storglaciären(e,f) for Λ = 1. Values are in one per year.

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2.6 Results 23

Figure 2.8: Differences in effective strain rates εSIAeff − ˙εSIA

eff (left column) and εFSeff − ˙εSIA

eff (rightcolumn) for Haut Glacier d’Arolla (a,b), Griesgletscher (c,d) and Storglaciären (e,f) forΛ = 1. Values are in one per year.

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24 2 Numerical Simulations of Isothermal Glacier Flow

2.7 Discussion and Conclusions

The FS effective strain rate contains derivatives of the vertical velocity, which arenot included in the FOA effective strain rate. In general, these additional strainrate components result in a higher effective strain rate and, consequently, in highervelocities. This effect is illustrated in the left column in Figure 2.3. The area ofthe steep ice fall of Griesgletscher shows the largest differences between FS and FOAhorizontal surface velocities, while differences are smallest for Haut Glacier d’Arolla.This can be partly explained by the differences in glacier geometries. Nevertheless inpractice, one would tend to choose the rate factor such that simulated and observedsurface velocities match as closely as possible. Thus independent normalization bettervisualizes the performance of the FOA. Figure 2.3 indicates that for Λ = 1, the shapeof the FS and the FOA closely match for Haut Glacier d’Arolla while Griesgletscherand, to a lesser degree, Storglaciären show non-negligible discrepancies. For all threetested glaciers, differences between FS and FOA become smaller with increasing Λ.For Λ ≥ 10, the shapes of FS and FOA are already close to SIA. This indicates thatfor Λ→∞ both FS and FOA converge to the SIA solution as expected from theory.SIA full effective strain rates ˙εSIA

eff are higher than 1 a−1 over large areas (Figure 2.6,right column). Differences between εFS

eff and εSIAeff (Figure 2.8) are about an order of

magnitude higher than differences between εFSeff and εFOA

eff (Figure 2.7).Longitudinal coupling is important in areas where large gradients occur. The

smoothing effect of normal stresses is visible in all plots in Figure 2.3. But eventhe FOA underestimates the velocity speed-up in the ice fall of Griesgletscher (Fig-ure 2.3, left plot, middle panel) for Λ = 1, thus indicating that derivatives of thevertical velocity are important.This analysis shows that for aspect ratios ≥ 0.1, the SIA velocity fields differ sub-

stantially from FS and FOA velocity fields, thus making the SIA a questionable choice.Nevertheless, in some cases, such as the cirque-type Glacier de Saint-Sorlin, France,SIA can still produce reasonable results (Le Meur and Vincent, 2003), especially ifone is interested in volume changes and measured surface velocities are not included.Leysinger Vieli and Gudmundsson (2004) simulated the reaction of alpine glaciers toshifts in the equilibrium line altitude by using a FS and a SIA model. They concludedthat, at least in the absence of significant basal motion, there is no need to includehorizontal stresses. Alpine glaciers are, with few exceptions, temperate and thus basalsliding occurs, at least at moderate slip ratios (ratio between mean sliding velocity andmean deformational velocity) of about 1 (Gudmundsson et al., 1999; Gudmundsson,2003). Increased basal sliding leads to enhanced spatial transmission of stress gradi-ents (Gudmundsson et al., 1999) for which the SIA does not account. This suggeststhat differences between the FS and the SIA may become larger for increasing slipratios (Gudmundsson, 2003).The possibility to solve coupled and uncoupled partial differential equations si-

multaneously with Comsol Multiphysics simplifies the model setup as well as post-processing significantly. Comsol Multiphysics shows good performance for two-dimen-sional, isothermal glacier flow problems and is therefore used in chapters 4 and 5, andadditionally for the author’s contribution to Pattyn et al. (2008).

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2.7 Discussion and Conclusions 25

Acknowledgements Stimulating discussions with H. Blatter improved the manuscriptconsiderably. Thank is due to A. Wiacek for careful proofreading. This work is sup-ported by the Swiss National Science Foundation, grant no. 200020-115881.

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26 2 Numerical Simulations of Isothermal Glacier Flow

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3 Meltwater Production due to StrainHeating in Storglaciären, Sweden

Andy Aschwanden and Heinz BlatterInstitute for Atmospheric and Climate Science

ETH Zurich, Switzerland

Abstract. Storglaciären, northern Sweden, is temperate in most parts exceptfor a cold surface layer in the ablation zone. One of four possible sources forliquid water in temperate ice is melting due to strain heating. Velocity fieldsare calculated with an ice flow model, so that calculated and observed surfacevelocities agree. Melt water accumulation is computed by integrating strainheating along trajectories starting at the surface in the accumulation areaand ending at the cold-temperate transition surface in the ablation zone. Thedistribution of moisture content due to strain heating alone is mapped ina longitudinal section of Storglaciären. Values reach more than 10 gramsof water per kilogram ice-water mixture in the lowest parts of the temperatedomain. For this moisture content, the rate factor is more than 3 times higherthan for water-free ice, and therefore, water production by strain heating isimportant for the modeling of temperate and polythermal glaciers.

Citation: Aschwanden, A., and H. Blatter (2005), Meltwater production dueto strain heating in Storglaciären, Sweden. J. Geophys. Res., 110, F04024,doi: 10.1029/2005JF000328

3.1 IntroductionFlow properties of temperate ice depend on the content of moisture in the ice matrix;however, only one study has attempted to quantify this relation (Lliboutry and Duval,1985). In order to comprehensively model the flow of a temperate or polythermalglacier, it is essential to know the spatial variation in the water content in the tem-perate part of the glacier and the temperature distribution in the cold part. This isparticularly true near the bed in the ablation area where shear rates are high and thus,the impact of the water content on the flow behavior is expected to be high.Temperate ice is sometimes defined as ice at the pressure melting point. This may

be an acceptable approximation in the case of pure ice. However, if salts are present,as at Hansbreen, Spitsbergen (Jania et al., 1996), the definition of and the distinction

27

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28 3 Meltwater Production due to Strain Heating in Storglaciären, Sweden

between temperate and cold ice becomes more diffuse (Paterson, 1971). The thermo-dynamic properties of a ternary mixture of ice, water and salts define a temperaturerange where a change in heat content leads to a change both in temperature and inbrine content (Untersteiner, 1961; Ono, 1967). In this work, ice is treated as temper-ate if a change in heat content leads to a change in liquid water content alone, and isconsidered cold if a change in heat content leads to a temperature change alone. ForStorglaciären, northern Sweden, this assumption seems to be justified by a good cor-respondence between the depth of radar echoes and the depth of the seemingly sharptransition between isothermal temperate ice and cold ice with a non-zero vertical tem-perature gradient, as observed by direct temperature measurements (Pettersson et al.,2004).There are four different potential sources for liquid water in the ice matrix in glaciers

(Paterson, 1971; Lliboutry, 1976): (1) water trapped in the ice as water-filled poresclose-off at the firn-ice transition in the accumulation area; (2) water entering theglacier through cracks and crevasses at the ice surface in the ablation area; (3) changesin the pressure melting point due to changes in the lithostatic pressure; and (4) meltingdue to energy dissipation by internal friction, henceforth referred to as strain heating.Sources (3) and (4) can be estimated by modeling, whereas sources (1) and (2) areextremely difficult to quantify due to complex microphysics and drainage processes.During the melt season, meltwater percolates through the firn and forms an aquifer

with a thickness of about 5m on top of the firn-ice transition in the accumulation area.This transition is assumed to be impermeable (Schneider, 1999). The water-saturatedzone contrasts with the low water content of the ice beneath the firn-ice transition(Schneider, 2000). Freezing of water is negligibly small within this permanently tem-perate zone immediately above and below the firn-ice transition. This indicates anefficient drainage mechanism, either laterally or via crevasses. This makes an estimateof the amount of entrapped water extremely difficult.Water entering the glacier through moulins and crevasses at the ice surface is as-

sumed to remain in and drain through the conduits and thus would not contribute tothe moisture distributed within the ice matrix (Lliboutry, 1971; Raymond and Har-rison, 1975). Fountain et al. (2005), however, in a study on Storglaciären, found anetwork of water filled planar cracks in the temperate ice. These cracks form an in-terconnected pressurized hydraulic system. There is thus a question of the extent towhich water is exchanged between the ice matrix and this drainage system (P. Jansson,personal communication, March 2005). This is closely connected to the question ofmoisture diffusion within the ice matrix, which is not quantifiable to date. However,moisture diffusion is assumed to be negligibly small since the moisture gradients aregenerally small. This may not be the case close to the cracks in the ice if the pressurein the hydraulic system of the cracks is different to the water pressure in the ice matrix.Changes in hydrostatic pressure contribute 0.007 grams of water per kilogram ice-

water mixture per meter difference in depth (gwkg−1m−1) (Pettersson et al., 2004),which makes this contribution of water negligibly small everywhere in the temperatepart of the glacier. Thus, water entrapment in the firn zone and strain heating seemto be the significant sources of moisture in the ice. With an ice flow model, strainheating can be computed and integration along particle trajectories then yields theaccumulated liquid water generated by this process.

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3.2 Methods 29

Strain heating is the only source of liquid water in high arctic glaciers with a tem-perate basal zone in the lower tongue area beneath otherwise cold ice (Fowler andLarson, 1978; Blatter and Hutter, 1991). In temperate and Scandinavian type poly-thermal glaciers, where most ice is temperate except for a cold surface layer in theablation area (Holmlund and Eriksson, 1989; Jansson, 1997; Pettersson et al., 2003),strain heating is one of the significant sources for liquid water in the ice matrix to-gether with water entrapped in the firn of the accumulation area (Vallon et al., 1976;Pettersson et al., 2004).This study compares the contributions of strain heating and water entrapment on

Storglaciären (6755′ N, 1835′ E), a small Scandinavian type polythermal glacier lo-cated on the eastern side of the Kebnekaise massif in northern Sweden (Figure 3.1).The glacier is 3.2 km long from its head at 1730 m a.s.l. to its terminus at 1120 m a.s.l.and it has a total surface area of 3 km2. The accumulation area consists of two cirques,a larger one to the north contributing the major portion of the flow, and a smaller oneto the south. The glacier has an average ice thickness of 95m, with a maximum depthof 250m in the upper part of the ablation area. The average thickness of the cold layeris 31m, with a maximum thickness of 65m along the southern margin. The thicknessdecreases towards the equilibrium line. The surface layer in the accumulation area isonly cold seasonally. In the accumulation area, latent heat due to surface melting isstored in the firn pack, whereas it is lost through surface run-off in the ablation area(Paterson, 1972; Pettersson, 2004, and references therein).Pettersson et al. (2003) mapped the thickness of the cold surface layer and found

a substantial thinning between 1989 and 2001. Pettersson et al. (2004) determined thewater content at the cold-temperate transition surface (CTS) using radar-backscatteredpower calibrated at three thermistor strings with an in-situ calorimetric method. Thepresent work is based on the findings of Pettersson et al. (2003) and Pettersson et al.(2004) and presents a model study of liquid water generated by strain heating in Stor-glaciären. In the next section, the applied flow and trajectory models are described; inSection 3 the data sources, the applied boundary conditions and the tuning procedureare explained; and in Section 4 results are presented.

3.2 Methods

3.2.1 First Order Ice Flow Model

The ice flow model used in this study is a two dimensional, first order glacier flowmodel (Blatter, 1995; Colinge and Rappaz, 1999; Pattyn, 2002). The model calculateshorizontal and vertical velocity components, u and w in a Cartesian coordinate system(x, z). The model was run with 50 gridpoints in the x-direction and 100 gridpointsin the z-direction, where x is horizontal and oriented downglacier and z is verticallyupward. For purposes of flow calculations in this study, the glacier is assumed to betemperate. This is the case except for the thin surficial cold layer in the ablation area.Tests were made using a rate factor that depends on temperature. However, results(not shown) reveal that the influence of this temperature dependence is negligibly

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30 3 Meltwater Production due to Strain Heating in Storglaciären, Sweden

surface contour intervals (50m)

base contour intervals (20m)

kinematic center line

thermistor strings CTS 1-3

1km

N

1600

1600

1500

1400

1300

CTS1 CTS2 CTS3

Figure 3.1: Map of surface and bed topography of Storglaciären. Solid and dotted lines indi-cate surface elevation (50m contour interval) and bed elevation (20m contour interval),respectively. The dashed-dotted line is the kinematic center line (KCL), and dots markpositions of thermistor strings.

small except near the cold glacier terminus. Hence, the assumption of an entirelytemperate glacier is justified.The first order force balance is (Blatter, 1995)

2 ∂σxx∂x

+ ∂σxz∂z

= ρgdSdx , (3.1)

where σxx and σxz are the components of the deviatoric stress tensor Σ = T− 12 tr (T) I,

T and I are the Cauchy stress tensor and the unit tensor, respectively, g is the accel-eration due to gravity, and S = S(x) is the elevation of the upper free surface. Therheology of ice is described with a relation of the form

∂u

∂x= F σxx, (3.2)

∂u

∂z= 2F σxz. (3.3)

Using the following notation

G = 1F, Gx = ∂G

∂x, Gz = ∂G

∂z, (3.4)

and substituting equations (3.2) and (3.3) into equation (3.1) yields

G

(2 ∂

2u

∂x2 + 12∂2u

∂z2

)+ 2Gx

∂u

∂x+ 1

2 Gz∂u

∂z= ρg

dSdx . (3.5)

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3.2 Methods 31

The fluidity F is a function of the effective stress σ(II) = 12 tr

(ΣT ·Σ

),

F = F (σ(II)) = F (σxx, σxz) = F [σxx(F ), σxz(F )], (3.6)which can be written in the form of a non-linear equation for F . In the case of amodified Glen’s flow law (Blatter, 1995) with flow law exponent n = 3,

F = A[σ2xx + σ2

xz + σ20

]= 1

2µ, (3.7)

where A and µ are the rate factor and the viscosity, respectively, and σ20 is proportional

to the inverse of the viscosity at vanishing effective stress. The resulting equation thenbecomes

F 3 − F 2Aσ20 − A

(∂u∂x

)2

+ 14

(∂u

∂z

)2 = 0. (3.8)

The boundary condition at the surface is (Blatter, 1995)

σxz,S − 2 dSdx σxx,S = 0. (3.9)

With equations (3.2) and (3.3), equation (3.9) becomes:(∂u

∂z

)S

− 4 dSdx

(∂u

∂x

)S

= 0. (3.10)

Either a basal velocity ub = ub(x) or a given sliding law can be prescribed (Colingeand Blatter, 1998). If no-slip conditions are assumed, then ub = wb = 0. In case ofa sliding law, the basal shear traction τb and the sliding velocity vb are functionallyrelated through F (τb, ub) = 0.

3.2.2 Trajectory ModelGiven is the velocity field v = v(r), where r = −→OP points to a given location P. Thestarting point of a particle is P0 with the location vector r0 = −−→OP0 at the time t0.The trajectory of the particle follows the differential equation

drdt = v. (3.11)

A numerical solution of the differential equation (3.11) uses a forward Euler iterationscheme, sometimes called Petterssen iteration (Seibert, 1993) with an initial step

r1 = r0 + ∆tv(r0) (3.12)and iteration steps

ri+1 = r0 + ∆t2 [v(r0) + v(ri)] , (3.13)

for i = 1, . . . , N . If N = 1, this scheme is called a predictor corrector scheme. ThePetterssen iteration scheme is a fixed point iteration scheme of second order accuracy,which converges towards a fixed point rf = −→OPf ,

rf = r0 + ∆t2 [v0 + v(rf )] . (3.14)

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32 3 Meltwater Production due to Strain Heating in Storglaciären, Sweden

3.2.3 Strain Heating and Water ContentThe density of energy dissipation due to strain, Q, is given by (Greve, 1997b)

Q = tr (Σ ·D) = 2Fσ2xx + 2Fσ2

xz, (3.15)

where D is the strain rate tensor. The liquid water content ω is the mass fraction ofliquid water in the ice-water mixture and thus defined as

ω := ρwρ, (3.16)

where ρw is the density of water and ρ is the density of the ice-water mixture. Themeltwater production due to strain heating can thus be calculated using

ω = ω0 +∫ B

A

Q

ρLdt, (3.17)

where L is the latent heat of melting.The ice flow model computes the strain heating Q = Q(i, j) at each grid point (i, j)

with i = 1, . . . , Nx and j = 1, . . . , Nz. Then, Q is interpolated using splines at allpoints Q(l) for l = 1, . . . , Nt, where Nt is number of trajectory points.The total amount of water produced due to strain heating is obtained through

summation over all discrete points along the trajectory,

ωcts = ω0 + 1ρL

Nt∑l=1

Q(l − 1) +Q(l)2 ∆t(l). (3.18)

3.3 Model SetupThe following data sets were used (provided by P. Jansson and R. Pettersson): (1) bedtopography of Herzfeld et al. (1993); (2) surface topography of Holmlund (1996); (3)CTS topography of Pettersson et al. (2003); (4) velocity measurements between 2001and 2002 (Jansson and Pettersson, unpublished data); and (5) velocity measurementsin 1983 of Hooke et al. (1989). In this study, a kinematic center line (KCL) close tothe one described in Hanson and Hooke (1994) and Jansson (1997) was used for themodeling study (Figure 3.1).A reliable velocity field is required to obtain reliable computations of trajectories

and strain heating. The desirable way to compute the velocity field involves solvingthe force balance (equation 3.5) and the constitutive equations (3.8) with the surfaceboundary condition (equation 3.9) and a sliding parameterization. However, basal slid-ing still constitutes the grand unsolved problem in glaciology, and coefficients occurringin such a sliding law are virtually unknown. For this reason, a different approach ischosen by matching the surface velocities, which are constrained by observations, bytuning the rate factor in the flow law and basal velocities. With given boundary con-ditions, the force balance equation together with the constitutive equation constitutesa well posed problem with a unique solution (Colinge and Rappaz, 1999).

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3.3 Model Setup 33

0 500 1000 1500 2000 2500 3000 35000

5

10

15

20

25

30

distance from bergschrund [m]

velo

city

[m/a

]

annual mean 2001−2002annual mean 1983 (Hooke 1989), outside kclannual mean 1983 (Hooke 1989), on kclsummer 1983 peak (Jansson 1997)average winter (Jansson 1997)simulated, scenario H2simulated, scenario L2

Figure 3.2: Observed and simulated surface velocities and applied basal sliding velocities.Dashed line and dotted line indicate scenario H2 and L2, respectively. Values in metersper year.

Because the velocity data available are sparse, an assimilation approach was chosen.Surface velocity data from observations between 2001 and 2002 were only available inthe ablation zone at altitudes below 1400m a.s.l., mainly because at higher elevations,the glacier surface is steeper and heavily crevassed and the proximity of steep rockwalls shade the satellite signal used for differential GPS measurements (Jansson andPettersson, unpublished data). The annual mean surface velocity between 2001 and2002, interpolated along the chosen kinematic center line (KCL), is depicted in Fig-ure 3.2. Measured mean annual horizontal velocities of 1983 (Hooke et al., 1989) arerepresented in Figure 3.2 with crosses where measuring points are close to the KCL andwith asterisks where measuring points are too far away. Average winter velocity andsummer 1983 peak (Jansson, 1997, Figure 5) are marked with squares and diamonds,respectively.The bed topography of Storglaciären is well known from radar profiles (Herzfeld

et al., 1993), but information about basal sliding is sparse (Hooke et al., 1992) andhigh spatial and temporal variability makes extrapolation from measured points ques-tionable. Starting with the basal velocity presented in Jansson (1997), the basalvelocity was tuned until a reasonable match with the pattern of the observed surfacevelocity was found. The summer 1983 peak data serve as an upper limit for surfacevelocities. Annual mean velocities and average winter velocities are very similar, atleast in the ablation zone.The rate factor A is assumed to depend on physical properties of the ice such

as temperature, pressure, water content, crystal size and orientation, impurities and

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34 3 Meltwater Production due to Strain Heating in Storglaciären, Sweden

density (Paterson, 1994). In the two-dimensional case, a shape factor (Nye, 1965) isoften used in order to account for the valley shape. For isothermal glaciers, the ratefactor is a constant multiplier of the velocity field. Correspondingly, the shape factoris a multiplier of the velocity field, since we assume it to be constant for the entireglacier. Thus, they can be combined to one tuning parameter, B = f A, which is againa multiplier of the velocity field.The general pattern of sliding velocities presented by Jansson (1997) reflect the

surface velocities, with peak values in the lower part of the accumulation zone and inthe vicinity of the riegel. For tuning the model to match observed surface velocities,several tuning parameters are available: a) the rate factor in the flow law for ice, b) theshape factor correcting for the valley shape, and c) an enhancement factor for a givendistribution of sliding velocities. Several criteria served to judge the model results:a) reasonable fit of modeled and observed surface velocity, and b) realistic gradientsin the computed velocity field. Unrealistic patterns in the prescribed basal velocitiesresulted in unrealistically large gradients and consequently, the iteration scheme of theflow model performed badly.

3.4 ResultsIn an earlier study of Storglaciären, Albrecht et al. (2000) found the best agreement be-tween observed and modeled surface velocities by using a rate factor of 0.07 a−1bar−3.To account for the plane-strain approximation, four shape factors and two enhance-ment factors for sliding conditions (1 for high and 0.7 for low sliding) were combined foreight situations (Table 3.1). The high sliding condition was used to match the highersurface velocities of 2001/2002, the low sliding condition matched the lower surfacevelocities of 1983. Best agreement with observed surface velocities was achieved usinga shape factor of 0.40 (scenarios L2 and H2), results are depicted in Figures 3.2 and3.3.Trajectories with time steps of one day and of one year were calculated forward

and then backward starting at the obtained end point. The distance ∆x between thestarting point of the forward and the end point of the backward trajectory increaseswith increasing trajectory lengths; however, the ratio between ∆x and the total lengthl of a flow line is nearly constant, 10−3 for a time step of one year and 10−5 for atime step of one day. In all cases, a time step of one day was used for strain heatingcalculation. Figure 3.4 shows the calculated trajectories.22 backward trajectories were calculated with starting points evenly distributed

along the CTS between the upper end of the cold layer and the point where it reachesthe bed. For scenario H2, the length of the shortest flow line is 659 m with a traveltime of 25 a, the longest is 2924 m and has a travel time of 209 a. Values for the otherscenarios are comparable. Trajectory calculations close to the bed become unreliable,and thus the computed water content also becomes less accurate towards the base ofthe glacier.Figure 3.5 shows the calculated contribution to the water content in the glacier due

to strain heating. Scenarios L4 and H1 indicate upper and lower limits, respectively.

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3.4 Results 35

Table 3.1: Description of tested scenarios together with the computed moisture accumulateddue to strain heating at the three positions of thermistor measurements. Shape factorvalues are unitless and water content values are given in grams of water per kilogramice-water mixture.

Scenario shape factor f sliding ωCTS1 ωCTS2 ωCTS3

L1 0.36 low 0.41 0.45 0.48H1 0.36 high 0.40 0.44 0.47L2 0.40 low 0.42 0.46 0.49H2 0.40 high 0.40 0.44 0.47L3 0.43 low 0.42 0.46 0.49H3 0.43 high 0.40 0.44 0.47L4 0.46 low 0.42 0.46 0.50H4 0.46 high 0.41 0.45 0.48

0 500 1000 1500 2000 2500 3000 35001100

1200

1300

1400

1500

1600

1700

distance from bergschrund [m]

altit

ude

[m a

.s.l.

]

5 10 15 20 25

Figure 3.3: Simulated horizontal velocities for the H2 scenario. Values in meters per year.

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36 3 Meltwater Production due to Strain Heating in Storglaciären, Sweden

0 500 1000 1500 2000 2500 3000 35001100

1200

1300

1400

1500

1600

1700

distance from bergschrund [m]

altit

ude

[m a

.s.l.

]

Figure 3.4: x-z slice along the kinematic center line of Storglaciären. The shading colorindicates temperate ice. The dashed line indicates the position of the transition surfacebetween the cold and temperate layer. Calculated backward trajectories from scenarioH2 start at the CTS (dashed line) and are based on a 1 day time step. The pointsindicate a 10 year time step. The region with dense trajectories is an artifact of thesampling of the starting points evenly distributed along the CTS.

The kinematic center line misses the three thermistor strings by less than 30 m, whichis close enough for a comparison with measured water content.The water content due to strain heating in the vicinity of the thermistor strings does

not differ substantially between the eight scenarios. All values lie between 0.4 and 0.5gwkg−1 (Table 3.1). This indicates that meltwater production by strain heating is fairlyrobust to velocity changes, as long as the sliding pattern is not changed substantially.Pettersson et al. (2004) obtained a total water content at the three thermistor stringsCTS1, CTS2 and CTS3 of 7.5± 0.6, 7.9± 0.7 and 5.8± 0.8 gwkg−1, respectively, andestimated a contribution of 1 gwkg−1 from strain heating. This rough estimate is ofthe same order of magnitude as the values presented in this study, though it was basedon several simplifying assumptions.

3.5 Discussion and ConclusionsThe accuracy of the computation of moisture production by strain heating dependson the accuracy of the computed velocity field. The limiting process for the accuracyis the basal sliding. Near the base the results thus become unreliable. Besides thesmall contribution to the moisture content due to changes in lithostatic pressure, thesignificant contribution due to strain heating seems best quantifiable.The contribution of strain heating to the observed water content at the CTS is less

than 10 % in the neighborhood of the thermistor strings of Pettersson et al. (2004).However, the relative importance of strain induced moisture to total moisture contentincreases further downglacier. Near the lower end of the CTS, this moisture exceeds 10gwkg−1, thus reaching a magnitude where drainage may become significant. Seemingly,an upper limit for water content in ice exists, above which the percolation of water

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3.5 Discussion and Conclusions 37

0

5

10

15

20

25 ω

cts

[g

w/k

g]

ωcts

H2

ωcts

L2

ωcts

L4

ωcts

H1

total water content

0 500 1000 1500 2000 2500 3000 35001100

1200

1300

1400

1500

1600

1700

distance from bergschrund [m]

altitu

de

[m

a.s

.l.]

0.1

0.5

12 4

812

Figure 3.5: Distribution of water content [gwkg−1] due to strain heating alone, and (inset)water content along CTS. Solid thick lines indicate the approximate position of thethree thermistor strings. The asterisks denoting the total water content are data byPettersson et al. (2004).

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38 3 Meltwater Production due to Strain Heating in Storglaciären, Sweden

becomes dominant. The saturation level may depend on ice properties but also on thestate of straining. This seems to be confirmed by observations of a drop in moisturecontent close to the bed at Falljökull in Iceland (Murray et al., 2000) and on the upperplateau of the Vallée Blanche in the Massif du Mont-Blanc (Vallon et al., 1976).Lliboutry and Duval (1985) have attempted to quantify the relationship between

the flow properties of temperate ice and its water content. A change of 10 gwkg−1 inthe water content changes the rate factor by a factor of about three. Thus, the effectof water in the ice is largest near the bed in the ablation area, where the accumulatedwater due to strain heating is highest. This effect is even enhanced by the fact thatmost of the shearing occurs near the bed and therefore, a change in the flow propertiesof ice near the bed has the largest influence on the overall movement of the ice. Inthis respect, ice flow models incorporating the dependence of the rate factor on themoisture content, together with calculations of the moisture distribution, are similarto models incorporating thermo-mechanical coupling for cold ice masses. This will bean important step in the future modeling of polythermal glaciers.The combination of measured moisture content at the CTS, such as presented by

Pettersson et al. (2004), and modeled moisture accumulated by strain heating opensthe possibility of estimating the moisture entrapped in the firn of the accumulationarea. However, this is only true if no other significant process influences the liquidwater content in the ice matrix. The existence of a dense drainage system throughintraglacial cracks (Fountain et al., 2005) suggests the possibility of additional sourcesand sinks of moisture.The difference between the high liquid water content in the top firn layers of the

accumulation zone (Schneider, 1999) and the comparably small amount trapped in theice of the accumulation zone indicates an efficient drainage mechanism. If the moisturetrapped in the firn is limited by saturation, this would reduce the importance of meltin the firn zone and thus reduce the climatic contribution to the liquid water content ofthe temperate ice, and consequently, to the thickness and dynamics of the cold layer.Together with the in situ calorimetric determination of moisture content at the CTS

and the areal extrapolation with radar backscatter (Pettersson et al., 2004), the pro-posed computation of moisture generated by strain heating provides the possibilityof computing the 3-dimensional moisture distribution in the temperate ice of Scan-dianvian type polythermal glaciers. Measurements of the distribution of liquid waterin a glacier, for example by borehole radar, may provide data to validate the modelcomputations, and as a consequence, to further constrain other sources and sinks ofmoisture and possibly, basal sliding.

Acknowledgments The authors thank P. Jansson and R. Pettersson for providingdata. Their comments on an earlier version of the manuscript helped to improve thepaper substantially. Valuable comments from R. LeB. Hooke and an anonymous re-viewer are highly acknowledged. We thank T. Ewen for correcting our english writing.A.A. thanks C. Schär and A. Ohmura for funding.

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4 Mathematical Modeling andNumerical Simulation ofPolythermal Glaciers

Andy Aschwanden and Heinz BlatterInstitute for Atmospheric and Climate Science

ETH Zurich, Switzerland

Abstract. A mathematical model for polythermal glaciers and ice sheets ispresented. The enthalpy balance equation is solved in cold and temperateice together using an enthalpy gradient method. To obtain a relationshipbetween enthalpy, temperature and water content, we apply a brine pocketparametrization scheme known from sea ice modeling. The proposed enthalpyformulation offers two advantages: (1) the discontinuity at the cold-temperatetransition surface is avoided; and (2) no treatment of the transition as aninternal free boundary is required. Fourier’s law and Fick-type diffusion areassumed for sensible heat flux in cold ice and latent heat flux in temperateice, respectively. The method is tested on Storglaciären, northern Sweden.Numerical simulations are carried out with a commercial finite element code.A sensitivity study reveals a wide range of applicability and defines the limitsof the method. Realistic temperature and moisture fields are obtained over alarge range of parameters.

Citation: Aschwanden, A., and H. Blatter (2009), Mathematical modelingand numerical simulation of polythermal glaciers. J. Geophys. Res., doi:10.1029/2008JF001028, in press

4.1 IntroductionPolythermal glaciers consist of cold ice (ice below the pressure melting point) andtemperate ice (ice at the pressure melting point). Temperate ice is characterized bythe existence of a small percentage of liquid water in the ice matrix. The contentof liquid water in temperate ice is defined as the mass fraction of water in the ice-water mixture ω = mw/m, where mw and m are the mass of water and the mass ofthe mixture, respectively. The liquid water content varies spatially and temporallyin a glacier, but is generally less than 3 %, and, additionally, the maximum liquid

39

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40 4 Modeling Polythermal Glaciers

water content differs from glacier to glacier (cf. Pettersson et al., 2003, and referencestherein). Higher values may be found in fully temperate glaciers. Several studies(Vallon et al., 1976; Murray et al., 2000) indicate that an upper limit for the watercontent exists above which the percolation of water becomes important.A first mathematical model for polythermal glaciers and ice sheets was presented by

Fowler and Larson (1978). Hutter (1982) introduced mixture theory for the descrip-tion of temperate parts and Fowler (1984) discussed the role of salts in ice. Hutteret al. (1988) applied a reduced plane flow model and computed water content at thetransition between cold and temperate ice for different moisture diffusivities. A com-prehensive overview is given in Greve (1997b).Polythermal glaciers and ice caps are common at high latitudes, e.g. in the Canadian

Arctic (Blatter, 1987; Blatter and Kappenberger, 1988), in Svalbard (Jania et al.,1996) or in Scandinavia (Holmlund and Eriksson, 1989) but can also be found at highaltitudes in the Alps (Haefeli, 1963; Haeberli, 1976). In ice sheets, temperate zonescan be present at the bed. Many present-day ice sheet models, however, ignore theeffects of melting, and thus locally violate energy conservation. This “cold-ice method”overestimates the thickness and volume of temperate basal layers, as demonstrated byGreve (1997a) for the Greenland ice sheet. Breuer et al. (2006) modified a three-dimensional “cold-ice method” model to assess the influence of temperate ice in KingGeorge Island ice cap. Zwinger et al. (2007) used variational inequalities to imposethe constraint that the ice temperature is limited by the melting point. The only trulypolythermal ice sheet model SICOPOLIS (Greve, 1995, 1997b,a) solves the Fourierequation for the cold parts and a moisture advection-production equation for thetemperate parts together with the jump conditions and kinematic conditions at thefree cold-temperate transition surface (CTS).Numerical approaches to handle phase changes may be divided into two classes:

front-tracking methods and enthalpy methods (EM) (Nedjar, 2002). SICOPOLIS im-plements a front-tracking method treating the CTS as a moving boundary. This eitherrequires deforming grids or transformed coordinate systems, of which the latter is im-plemented in SICOPOLIS. Both deforming grids and transformed coordinate systemsare somewhat cumbersome to implement.In this paper, we propose a different strategy to model polythermal glaciers by us-

ing an EM which solves the enthalpy equation for the cold and temperate domainstogether. Enthalpy methods have been used for more than three decades (e.g. Sham-sundar and Sparrow, 1975; Voller and Cross, 1981; White, 1982; Voller et al., 1987;Elliott, 1987) and are recommended by many authors owing to their ease of imple-mentation (Nedjar, 2002, and references therein). In the standard enthalpy methodthe heat flux is expressed in terms of the temperature gradient. On the other hand, ifheat flux is expressed in terms of the gradient in enthalpy, this technique is called the“enthalpy gradient method” (EGM) (Pham, 1995). Using an EM requires a functionalrelationship between enthalpy, temperature and water content. To this end, we havechosen to apply a regularization of the enthalpy function known from sea ice modeling.In the next section, field equations, constitutive equations and boundary conditions

for polythermal glaciers are introduced together with the applied regularization. Nu-merical simulations are described in section 4.3, and results are presented in section 4.4.We discuss our results and summarize our conclusions in section 4.5.

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4.2 Mathematical Model 41

4.2 Mathematical Model4.2.1 Field EquationsGlacier ice is generally assumed to be an incompressible, viscous, heat-conductingnon-Newtonian fluid which obeys the Stokes equations,

∇ ·v = 0, (4.1)

∇ ·T = −ρg, (4.2)and a balance equation for specific inner energy (Greve, 1997b), u,

ρu = −∇ ·q +Q. (4.3)∇ · is the divergence operator, v is the velocity, T is the Cauchy stress tensor, ρ isthe density of ice, g is the acceleration due to gravity. u is the total derivative of u, qis the energy flux and Q is an internal heat source. The specific energy has SI unitsJ kg−1, and the terms in equation (4.3) have SI units J m−2 s−1 (Moran and Shapiro,2000). The stress tensor can be split into an isotropic and a deviatoric part,

T = −pI + T′ = −pI + 2ηD, (4.4)

where p is the pressure, I and T′ are the identity and the deviatoric stress tensor,respectively, η is the effective viscosity and D = 1/2

(∇v +∇vT

)is the strain rate

tensor.

4.2.2 Constitutive EquationsThe set of field equations (4.1) to (4.3) must be closed by constitutive equations forthe viscosity η and the energy flux q.

Ice flow

The nonlinear rheology of glacier ice is generally expressed by a power law (Glen, 1955;Steinemann, 1958), relating viscosity η and effective strain rate εeff ,

η = 12A− 1n ε

1−nn

eff , (4.5)

where A is the rate factor and n is the flow law exponent. The effective strain rateis

εeff =√

IIε =√

12tr (D ·D), (4.6)

where IIε is the second invariant of D.The rate factor of temperate ice depends on the water content; however, only one

experimental study by Duval (1977) attempted to quantify this relationship. Weassume a constant rate factor because in this way a thermomechanically coupled modelis avoided as the energy balance equation is decoupled from the Stokes equations.

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42 4 Modeling Polythermal Glaciers

Thermodynamics

The specific enthalpy, H, is commonly defined as (e.g. Moran and Shapiro, 2000):

H = u+ p/ρ. (4.7)

If a material is heated under constant pressure, when there are no volume changes,the enthalpy represents the inner energy (Alexiades and Solomon, 1993). Because theterm “enthalpy method” is widely used in the computational fluid dynamics literature,we will refer to enthalpy instead of inner energy in the remainder of this paper andrewrite equation (4.3) as

ρH = −∇ ·q +Q. (4.8)

Ice is defined as cold if a change in enthalpy leads to a change in temperature alone,thus

δH = c δT, (4.9)

where c is the heat capacity under constant pressure. The energy flux in cold ice, q,can be described by Fourier’s law,

q = −k∇T, (4.10)

where k is the thermal conductivity.Ice is defined as temperate if a change in enthalpy leads to a change in liquid water

content alone, thus

δH = L δω, (4.11)

where L is the latent heat of fusion. Temperate ice is at the local melting pointTm, although the possibly small dependencies of the melting point on pressure, air-saturation level of the ice and stresses (Harrison, 1972; Kamb, 1972) are neglected inthis study. As a consequence the energy flux in temperate ice, q, is expressed as

q = Lj, (4.12)

where j is the diffusive moisture flux (Hutter, 1982; Greve, 1997b). Little is knownabout the moisture flux in temperate ice, though Fick-type (Hutter, 1982) or Darcy-type (Fowler, 1984) diffusion have been proposed. A general formulation for the dif-fusive moisture flux in temperate ice may consider water content, its spatial gradient,deformation and gravity (Hutter, 1982).

4.2.3 Boundary conditionsIce flow

At the glacier surface tangential stress vanishes and the resulting boundary conditionbecomes

T ·n = pairn, (4.13)

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4.2 Mathematical Model 43

where pair is the atmospheric pressure and n is the outward unit normal vector.Either a basal velocity vb or a given sliding law can be prescribed. If no-slip condi-

tions are assumed, then vb ≡ 0. In case of a sliding law, the basal shear traction τ band the sliding velocity vb are functionally related through F (τ b,vb) = 0.

Thermodynamics

The thermodynamic boundary conditions at the glacier surface are defined by theenergy balance at the surface. However energy fluxes at the surface are the resultof a complex climatology and therefore, it is common practice to prescribe Dirichletconditions for cold and temperate ice, T = TS and ω = ωS, respectively. At the coldbed, the geothermal heat flux, qgeo, enters the ice, thus a Neumann condition is applied,n ·q = qgeo. At the temperate bed, two cases have to be distinguished, either the iceis also temperate above the bed or the ice is cold immediately above the bed. In thefirst case, all available heat (geothermal and frictional heat) is used for basal melt,thus n ·q = 0. In the latter case, part of the heat flux may enter the ice. However, toproperly treat this case, the thermal boundary of the domain must be lowered deepenough into the lithosphere below the glacier.

4.2.4 Enthalpy Gradient MethodFor the temperature range in consideration, the dependence of thermodynamic quan-tities on temperature is small (e.g. Paterson, 1994). It is thus justified to assumeheat capacity and thermal conductivity to be constant. To solve the enthalpy balanceequation (4.8), we express the energy flux q in terms of enthalpy. In cold ice, thegradient of equation (4.9) gives

∇T = 1c∇H, (4.14)

and introducing equation (4.14) into equation (4.10) yields the sensible heat flux incold ice as a function of enthalpy,

q = −kc∇H. (4.15)

As indicated by the name, the “enthalpy gradient method” assumes a enthalpy gradient-driven diffusion and is thus not compatible with Darcy-type diffusion. Therefore, aFick-type moisture diffusion in temperate ice is assumed and the diffusive moistureflux is then given by

j = −ν∇ω, (4.16)where ν is a moisture diffusivity (Hutter, 1982). In temperate ice, the gradient ofequation (4.11) gives

∇ω = 1L∇H. (4.17)

Introducing equations (4.16) and (4.17) into equation (4.12) yields the latent heat fluxin temperate ice as a function of enthalpy,

q = −ν∇H. (4.18)

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44 4 Modeling Polythermal Glaciers

By defining the diffusivity κ as

κ =

ν/ρ temperate icek/(ρc) cold ice

, (4.19)

thusq = −ρκ∇H, (4.20)

equation (4.8) can be rewritten as

ρ

(∂H

∂t+ v · ∇H

)= ∇ · (ρκ∇H) +Q, (4.21)

for both cold and temperate ice. The first term is the local rate of change of enthalpy,the second term is enthalpy advection and the third term is enthalpy diffusion. Q =tr (D ·T′) is enthalpy production due to strain heating. In this study, we focus on thethermodynamical steady state, thus the enthalpy balance equation reduces to

ρv · ∇H = ∇ · (ρκ∇H) +Q. (4.22)

4.2.5 RegularizationThe enthalpy function of pure water is discontinuous at the pressure melting point. Toconvert between enthalpy, temperature and water content, a function relating thesethree quantities is required. We apply a regularization of the enthalpy as a function oftemperature to obtain such a relationship. The regularization is applied firstly to theboundary conditions and, secondly, is used to derive temperature and water contentfrom the calculated enthalpy field. Temperature and water content of pure water asa function of enthalpy are shown in Figures 4.1a and 4.1b, respectively, together withthe applied regularization.In this work we use a one-sided regularization inspired by the brine pocket parametriza-

tion scheme (BPP) known from sea ice modeling (Untersteiner, 1961; Ono, 1967;Maykut and Untersteiner, 1971; Bitz and Lipscomb, 1999; Huwald et al., 2005). Thebrine pocket parameterization accounts for the temperature and salinity dependenceof the ice properties. The matrix of ice is assumed to consist of freshwater ice and acomplex system of cavities between the ice crystals, mostly along the contact lines andpoints of three and four ice crystals, respectively. These cavities are considered to befilled with a brine solution in thermal equilibrium with the ice. The thermodynamicproperties of a mixture of ice and brine define a temperature range in which a changein enthalpy leads to a change both in temperature and in brine content. We definethe bulk salinity Si of the ice as the specific salinity,

Si = ms

mi +mb, (4.23)

where ms, mi and mb are the mass of salt, ice and brine, respectively. The specificsalinity of the brine solution is defined by

Sb = ms

mb. (4.24)

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4.2 Mathematical Model 45

b)

H

T

H

Tm

HlHs

a)

0

1

Figure 4.1: Temperature-enthalpy relationship for pure water (solid line) and the appliedregularization (dotted line) (a) and water content-enthalpy relationship (solid line) andthe applied regularization (dotted line) (b). Tm is the pressure melting point, and Hsand Hl are the enthalpy of pure ice and pure water at Tm, respectively.

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46 4 Modeling Polythermal Glaciers

Furthermore, the specific mass fraction of brine solution in the ice is defined by

ωb = mb

mi +mb. (4.25)

At equilibrium, the ice temperature, Ti, must be equal to the temperature of theliquid brine pockets, Tb; and thus T = Tb = Ti. If this were not the case, the brinepocket would grow or reduce its size thereby adjusting its salinity in such a way thatTb = Ti. The freezing temperature Tb is a function of the salinity Sb, which in linearapproximation is

Tb(Sb)− T0 = −µSb, (4.26)where µ is an empirical constant and T0 = 273.15 K at atmospheric pressure. Withequation (4.25) and the definitions of Si and Sb in equations (4.23) and (4.24), wearrive at

ωb = − µSi

T − T0= − α

T − T0. (4.27)

We take Si as a constant and, therefore, µSi is henceforth replaced by the regularizationparameter α for convenience. The specific enthalpy H of the ice-brine mixture of massm = mi +mb can be written as

H = 1m

− Lmi + ci (T − T0) mi + cb (T − T0) mb

, (4.28)

where ci and cb are the heat capacity of ice and brine at T0, respectively. The heatcapacity of brine is assumed to be that of pure liquid water at the same temperaturesince the assumed bulk salinity is extremely small. Introducing equations (4.25) and(4.27) into equation (4.28) gives:

H = −L(

1 + α

T − T0

)+ ci (T − T0)

+ (ci − cb)α. (4.29)

Solving equation (4.29) for the temperature T , we obtain two solutions

T± = 12ci

H + L− (ci − cb)α

±√

(H + L− (ci − cb)α)2 − 4ciLα

+ T0. (4.30)

The minus sign yields the physically meaningful solution. For small salt contents,mw ≈ mb, thus ω ≈ ωb, and the water content is then obtained from equation (4.27),

ω = − α

T − T0. (4.31)

As a consequence of the application of the BPP, the position of the CTS is notsharply defined anymore. To find the position of the CTS we use the facts that at

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4.3 Numerical Simulations 47

the CTS temperature must be at the melting point, T = Tm, and the water contentmust be zero, ω = 0 (Hutter, 1982). To this end we introduce equation (4.31) intoequation (4.29), which yields

H = −L (1− ω) + ci (T − T0) (1− ω)+cb (T − T0)ω. (4.32)

In the limits of T → Tm and ω → 0 if the CTS is approached from the cold andtemperate sides, respectively, we get H = −L at the CTS from equation (4.32). Thisdefines the position of the CTS. We therefore redefine

cold ice if H < −L,cold-temperate transition surface if H = −L,temperate ice if H > −L.

Finally, the BPP is applied to the thermodynamic boundary conditions. Let Ω bethe glacier domain and ΓC its border, where the subscript C = B, S denotes bed andsurface. At the glacier surface, enthalpy is prescribed in terms of temperature (cold ice,c) or water content (temperate ice, t). The transformed surface boundary conditionsfor enthalpy then read:

H =Hc(T ) on ΓS,cHt(ω) on ΓS,t

,

with

Hc(T ) = −L(

1 + α

TS − T0

)+ ci (TS − T0)

+ (ci − cb)α, (4.33)Ht(ω) = −L (1− ωS)− ci

α

ωS+ (ci − cb)α, (4.34)

where TS and ωS are the temperature and the water content at the surface, respec-tively. Equation (4.34) is obtained by introducing equation (4.31) into (4.33). Thetransformed boundary conditions at the bed read:

n ·q = qgeo if H < −Ln ·q = 0 if H ≥ −L . (4.35)

4.3 Numerical Simulations4.3.1 Finite Element MethodNumerical solutions based on the Finite Element Method (e.g. Braess, 2007) are ob-tained using the commercial program package Comsol Multiphysics (www.comsol.com).The glacier domain is approximated by an unstructured triangular mesh. QuadraticLagrange elements are used for velocity and enthalpy, and linear Lagrange elements areused for pressure; this is Comsol’s default setting. Values of constants and parametersused in this study are listed in Table 4.1.

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48 4 Modeling Polythermal Glaciers

Table 4.1: Physical Constants and Parameters Used in the Study

Variable or Constant Name Value UnitA rate factor 2.22 · 10−24 Pa−n s−1

cb specific heat of brine at Tm 4.17 · 103 J kg−1 K−1

ci specific heat of ice at Tm 2.008 · 103 J kg−1 K−1

ε0 viscosity regularization 10−13 s−1

f shape factor 0.4 -ki thermal conductivity ice 2.22 W m−1 K−1

L latent heat of fusion 3.34 · 105 J kg−1

n exponent of the flow law 3 -pair atmospheric pressure 105 Paqgeo geothermal heat flux 0.042 W m−2

ρ densitiy of ice 900 kg m−3

4.3.2 StorglaciärenStorglaciären (6755′ N, 1835′ E), northern Sweden, was selected as test glacier. Itis a small polythermal glacier located on the eastern side of the Kebnekaise massif innorthern Sweden (Figure 4.2). Storglaciären has a Scandinavian-type (sometimes alsocalled Svalbard-type) thermal structure where most of the ice is temperate, except fora cold surface layer in the ablation zone. The average thickness of the cold layer is31 m, with a maximum thickness of 65 m along the southern margin. The thicknessdecreases towards the equilibrium line. The thermal structure of the glacier is knownfrom radar echo soundings (Holmlund and Eriksson, 1989; Pettersson et al., 2003).Using an in-situ calorimetric method, Pettersson et al. (2004) determined absolutevalues of the water content at the CTS at three thermistor string locations.A plane flow approximation for Storglaciären along the kinematic center line is used.

Figure 4.2a shows a map of bed and surface topography and the kinematic center line.Figure 4.2b shows the longitudinal cross section at the kinematic center line, theregions of cold and temperate ice, and the position of the cold-temperate transitionsurface as measured by Pettersson et al. (2003).

4.3.3 Ice flowAschwanden and Blatter (2005) calculated horizontal and vertical velocities for Stor-glaciären along the kinematic center line using a longitudinal stress approximation(first order approximation, FOA) developed by Blatter (1995). The current studysolves the full Stokes (FS) equations, but it uses the same parameters (Table 4.1) andbasal velocities (Figure 4.3) as for the H2 case in Aschwanden and Blatter (2005). Inthe two-dimensional case, a shape factor f (Nye, 1965) accounts for the valley shape.The FOA and FS velocity fields are sufficiently similiar (Figure 4.3) to compare thecorresponding computed temperature and moisture distributions using the trajectory

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4.3 Numerical Simulations 49

1km

N

1600

1600

1500 14

00

1300

surface contour intervals (50m)base contour intervals (20m)kinematic center line

cold-temperate transition surface

a)

b)

Figure 4.2: (a) Map of surface and bed topography of Storglaciären (adapted from Aschwan-den and Blatter, 2005). Solid and dotted lines indicate surface elevation (50 m contourinterval) and bed elevation (20 m contour interval), respectively. The dashed-dottedline is the kinematic center line. (b) Longitudinal cross-section at the kinematic cen-ter line. The gray color indicates cold ice and the dashed line is the position of thecold-temperate transition surface as measured by Pettersson et al. (2003).

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50 4 Modeling Polythermal Glaciers

0 500 1000 1500 2000 2500 3000 35000

10

20

30

distance from bergschrund [m]

velo

city

[m a−1

]

Aschwanden & Blatter (2005)this studybasal velocity

Figure 4.3: Velocities from Aschwanden and Blatter (2005) (dashed line, first order approxi-mation) and this study (solid line, full Stokes). Dotted line is the horizontal componentof the imposed basal velocity. Values are in meters per year.

0 500 1000 1500 2000 2500 3000 35001100

1200

1300

1400

1500

1600

1700

distance from bergschrund [m]

altit

ude

[m a

.s.l.

]

0.5

24 8 12

Figure 4.4: Meltwater production due to strain heating alone. Comparison between fullStokes (solid lines) and first order approximation (dotted lines) velocity field. Valuesare in grams water per kilogram mixture.

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4.3 Numerical Simulations 51

integration method (TIM) explained in Aschwanden and Blatter (2005), see Figure 4.4.Trajectories starting at the temperate surface and ending at the CTS are calculatedand moisture due to strain heating is integrated along these trajectories.A basal velocity vb is prescribed at the glacier base. The dotted line in Figure 4.3

is the horizontal component ub. By assuming that there is no basal melt or freeze, wehave vb ·n = 0 and thus the vertical component is wb = −ub ·nx/nz, where nx and nzare the horizontal and the vertical component of n, respectively.To avoid infinite viscosity at low effective stress, a small number ε0 is added to IIε

in equation (4.6) thus the effective strain rate becomes:

εeff =√

IIε + ε20. (4.36)

ε0 can be regarded as the smallest strain rate that can be resolved. Our choice,ε0 = 10−13 s−1, results in an error in velocity calculation of less than 5 mm a−1 overan ice thickness of 100 m, which is below measurement errors.

4.3.4 ThermodynamicsThe diffusivity is a step function with a step at the CTS from ki/(ρci) to ν/ρ, where kiand ci are thermal conductivity and heat capacity of ice at Tm, respectively. This step isimplemented with a smoothed Heaviside function with a continuous first derivative (apiecewise polynomial of degree three), spreading the step over a range of 1500 J kg−1.The transition between cold and temperate surface conditions is gradual and hence,

a smoothed Heaviside function with a smoothing width of 70 m was applied. The tran-sition occurs at a distance of approximately 1670 m from the bergschrund (Petterssonet al., 2003).A smoothed Heaviside function is applied to spread the thermal transition from

temperate to cold ice at the bed over an enthalpy range of 100 J kg−1.The steady-state enthalpy problem is then defined by the following equations:

ρv · ∇H = ∇ · (ρκ∇H) +Q in ΩH = Ht(ω) on ΓS,cH = Hc(T ) on ΓS,t

n ·q = 0 on ΓB,tn ·q = qgeo on ΓB,c

. (4.37)

The nonlinear system of equations (4.37) is solved by Comsol Multiphysics using anaffine invariant form of the damped Newton method (Deuflhard, 1974), which is theprograms’ default solver for stationary nonlinear problems. From the simulated en-thalpy distribution, temperature and moisture content are then obtained from equa-tions (4.30) and (4.31).

4.3.5 Role of advection and diffusionThe dimensionless Peclet number is a measure of the relative importance of advectionto diffusion; the higher the Peclet number, the more important is advection. Valueslarger than one indicate that the problem is advection-dominated. The global Peclet

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52 4 Modeling Polythermal Glaciers

Table 4.2: Range of Values Used for Sensitivity Analysis. Values Typed in Bold Face AreUsed if not Stated Otherwise.

Variable Description Range Unit

α regularization parameter 10−20, 10−19, . . . ,10−12, . . . 10−0 KθS cold surface temperature −7,−5,−3,−1 CωS temperate surface water content 0, 2, 4, . . . , 20 gw kg−1

ν moisture diffusivity 10−2, 7.5 · 10−3, 5 · 10−3,2.5 · 10−3, 10−3, 7.5 · 10−4,5 ·10−4, 2.5 · 10−4, 10−4 kg m−1 s−1

h maximum element size 5,10, 20, 40 m

number, Pe = U L/D, can be estimated from typical values for velocity U , length scaleL and diffusivity D. Inserting appropriate values for Storglaciären (U = 10 m a−1, L =103 m, D = 10−6 m2 s−1) yields a value of about 300. This indicates that the transportproblem defined by equation (4.37) is expected to be dominated by advection. Thelocal Peclet number Peh, expressed as local quantities of a mesh element, is definedas

Peh =∣∣∣∣∣vhκ

∣∣∣∣∣ , (4.38)

where h is the length of the longest edge of an element. The local Peclet numberindicates areas where either advection or diffusion is the dominant process.

4.3.6 Numerical experimentsA sensitivity study is carried out to assess the relative importance of selected modelparameters. The regularization parameter α, the moisture diffusivity ν, water contentat the temperate surface ωS, temperature at the cold surface TS and maximum ele-ment size h are tested to probe the limits of the enthalpy gradient method. A set ofparameters is chosen as a control run ctrl, the values of which are typed in bold face inTable 4.2, and are used in all simulations unless stated otherwise. Due to the BPP itis not possible to prescribe a temperate surface water content ωS = 0 gw kg−1, insteada small number 10−2 gw kg−1 is used.

4.4 ResultsSimulated enthalpy, derived temperature, derived water content, and local Peclet num-ber for the ctrl run are shown in Figure 4.5. For convenience, temperature is givenin degree Celsius, θ = T − T0. Figure 4.5b reveals that advection is the dominanttransport process in most of the temperate ice, while diffusion prevails in cold ice andin temperate ice near the bergschrund. The mean thickness of the cold surface layeris 38 m.To investigate the influence of mesh size on the solution, both the Stokes equations

and the enthalpy equation are solved on meshes with an maximum element size of

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4.4 Results 53

Figure 4.5: Enthalpy (a), local Peclet number (b), temperature (c), and water content (d)for the control run ctrl. The dashed line marks the position of the calculated CTS.Values are in joule per kilogram (enthalpy), unitless (Peclet number), degree Celsius(temperature), and grams water per kilogram mixture (water content).

5 m, 10 m, 20 m and 40 m. Results (not shown) indicate that the size of the triangularelements does not significantly affect the calculated position of the CTS.Little is known about the possible range of the moisture diffusion coefficient. Applied

values are in the range 10−2−10−6 kg m−1s−1 (Hutter et al., 1988); Greve (1995, 1997b)assumes a vanishing diffusive moisture flux, although, for numerical stability, usesν = 10−6 kg m−1s−1. In this study, the range of moisture diffusivities for which stablenumerical solutions could be obtained is 10−2 − 10−4 kg m−1s−1. Figure 4.6 showsCTS positions for selected moisture diffusivities. For moisture diffusivities smallerthan 5 · 10−3 kg m−1s−1, the position of the CTS does not vary significantly.The viability of the brine pocket parameterization strongly depends on an appro-

priate choice of the regularization parameter α. Here, we explore the parameter rangefor which the BPP produces reliable results. Runs were performed from α = 10−20 to10−1 C in steps of factor ten. Minimum and maximum temperatures are given at thecold surface temperature and by the melting point, respectively. Due to the BPP, themelting point can only be reached within a finite limit. Minimum moisture contentoccurs at the temperate surface but maximum water content depends on strain heat-ing. Figure 4.7 displays absolute values of minimum temperature and maximum watercontent as a function of the regularization parameter. Absolute values of minimumtemperature are constant from α = 10−20 to 10−5 C and then start to increase. Themaximum water content is stable between α = 10−17 and 10−5 C.The mean thickness of the cold surface layer Hc decreases with both increasing

water content at the temperate surface (Figure 4.8a) and increasing temperature atthe cold surface (Figure 4.8b). Hc decreases from 32 m for ωS = 2 gw kg−1 to 16 m for

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54 4 Modeling Polythermal Glaciers

Figure 4.6: Position of the CTS for different moisture diffusivities ν.

Figure 4.7: Range of applicability of the brine pocket parametrization scheme. Absolutevalues of minimum temperature and maximum water content are good indicators.

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4.5 Discussion and Conclusions 55

Figure 4.8: Position of the CTS for selected temperate surface water contents ωS (a) andcold surface temperatures θS (b).

ωS = 20 gw kg−1 and from 42 m for a surface temperature θS = −7 C to 20 m for θS =−1 C. The decrease of the thickness of the cold surface layer with increasing watercontent and temperature at the surface is consistent with the findings of Petterssonet al. (2007) who investigated the sensitivity of Storglaciären’s cold surface layer todifferent forcing parameters.

4.5 Discussion and ConclusionsThe temperature and moisture fields (Figure 4.5c,d) derived from the calculated en-thalpy distribution (Figure 4.5a) clearly demonstrate that the enthalpy gradient method(EGM) is capable of simulating a Scandinavian-type thermal structure. It is not theaim of this study to compare simulated and measured CTS positions but to demon-strate the applicability of the EGM to the Scandinavian-type thermal structure.The system of equations (4.37) define an advection-diffusion-production problem

which is parabolic, but becomes hyperbolic if diffusion is neglected. Very small nu-

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56 4 Modeling Polythermal Glaciers

500 1000 1500 2000 2500 3000

1200

1300

1400

1500

1600

distance from bergschrund [m]

altit

ude

[m a

.s.l.

]

0.5

24 8 12

Figure 4.9: Content of liquid water simulated by TIM (solid line) and EGM (dotted line).Dashed line indicates position of the CTS as calculated by the EGM. A moisture dif-fusivity of ν = 10−4 kg m−1 s−1 was used for EGM. Both TIM and EGM use the FSvelocity field. Values are in grams water per kilogram mixture.

merical values for the diffusion coefficient result in an advection-dominated transportproblem which is very nearly ill-conditioned. We address the advection-dominationthrough a fine global grid, though local grid refinement with target of roughly constanta posteriori values for Peh might be favorable. In this case, for decreasing moisturediffusivities, the maximum element size must decrease as well. The solver did not con-verge for moisture diffusivities smaller than 10−4 kg m−1 s−1. This may be attributedto a high local Peclet number but not to non-Newtonian viscosity effects because theenthalpy equation was solved independently from the Stokes equations. However ourresults are not conclusive.The moisture field computed with the EGM, using the smallest moisture diffusivity

for which results could be obtained, ν = 10−4 kg m−1 s−1, coincides well with the resultof the trajectory integration method (TIM) (Figure 4.9). Both fields were computedwith the same full Stokes velocity field. This suggests that for vanishing moisturediffusion the EGM should be equivalent to the TIM, though we have no mathematicalproof for this. Thus, the EGM offers a viable alternative to trajectory integration.However, the trajectory integration method is only applicable if the thermal structureis known a-priori, e.g. from measurements.The regularization scheme applies the brine pocket parameterization scheme used

in sea ice models. The regularization parameter α depends on the bulk salinity Siand defines the degree of regularization. Values of α less than 10−15 C are too closeto machine precision and produce numerical artifacts. Values larger than 10−5 Cinfluence the resulting fields in a physically unrealistic way. This may be related toour choice of ωS = 10−5 kgw kg−1. As a result of applying the BPP, zero water contentand a temperature of zero degrees Celsius are undefined, and hence, a small non-zero

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4.5 Discussion and Conclusions 57

value must be prescribed instead. The regularization parameter should then be anorder of magnitude smaller than ωS. Temperature as a function of the regularizationparameter is stable over almost the whole tested range except for the largest valuesof α. The moisture content is more sensitive to the choice of α than the temperaturefield. Remarkably, for ωS = 10−5 kgw kg−1, the moisture content and temperaturefields are invariant over almost ten orders of magnitude of α, a fact that supports theapplication of the regularization method.We have demonstrated the feasibility of the enthalpy gradient method for 2-dimen-

sional steady state Scandinavian-type polythermal glaciers. This is a first step to acomprehensive thermodynamical model of glaciers and ice sheets, and points the wayto future model developments: (1) testing the method for different polythermal struc-tures in glaciers, (2) for transient enthalpy fields, (3) for thermomechanically coupledsimulations, and (4) for 3-dimensional situations. The enthalpy gradient method solvesa field equation of a similar mathematical form as the Fourier equation for tempera-ture, which is applied in most ice sheet models. This is an important advantage for theinclusion of the enthalpy gradient method in existing ice sheet models, and it allowsus to convert “cold-ice method” ice sheet models into polythermal ice sheet modelsand to couple both moisture content and temperature fields to rheological propertiesof the ice with relatively minor modifications. Additionally, the corresponding type ofpolythermal structure should result uniquely from the imposed boundary conditions,thus, no extra handling of different cases and tracking of the cold temperate-transitionsurface should be required with the enthalpy gradient method.

Acknowldegments We thank J. Rappaz and M. Picasso for stimulating discussions.Comments from, and proofreading by, J. Brown improved the manuscript consid-erably. We also thank M. Church (Editor) and G. Hamilton (Associate Editor) forhelpful comments and suggestions. Thorough reviews from E. Bueler, R. Greve and ananonymous reviewer, which helped to clarify the manuscript substantially, are highlyacknowledged. This work is supported by the Swiss National Science Foundation,grant nos. 200020-115881 and 200021-107480.

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58 4 Modeling Polythermal Glaciers

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5 An Enthalpy Method for Glaciersand Ice Sheets

AbstractIn this manuscript, the mathematical model presented in chapter 4 is refined andsimplified. Enthalpy is assumed to be equivalent to temperature and water contentin cold and temperate ice, respectively. Basal sliding is introduced through a linearsliding law and first thermo-mechanically coupled and uncoupled simulations of coldand temperate glaciers are presented. It is concluded that the model is applicable fordifferent thermal structures of glaciers and ice sheets. The presented enthalpy methodis a suitable replacement for the temperature equation used in “cold-ice method” icesheet models. Substantial improvement is expected if applied to small ice caps and icesheets, where significant parts of temperate ice exist. Note that this is still a workingdraft.

5.1 IntroductionPolythermal conditions in ice sheets are found in Greenland (Lüthi et al., 2002) andAntarctica (Siegert et al., 2005; Motoyama, 2007; Parrenin et al., 2007). Present-dayice sheet models can be divided into two groups: (1) “Cold-ice method” ice sheet mod-els limit temperatures to the melting point, thus locally violating energy conservation;and (2) truly polythermal ice sheet models. SICOPOLIS (Greve, 1997b), so far theonly member of the second group, uses a front-tracking method to handle the inter-nal free cold-temperate transition surface (CTS). Here we apply an enthalpy method(EM) (Nedjar, 2002) to solve the enthalpy equation in both cold and temperate ice,thus avoiding front-tracking. The power of this thermodynamical framework lies in itsability to simulate the thermal structure of a glacier or ice sheet for given boundaryconditions. The EM is applicable to cold, temperate and polythermal glaciers and icesheets.Enthalpy is a continuous field. However, due to phase changes, temperature and

water content as functions of enthalpy are not differentiable at the pressure meltingpoint. In chapter 4, a brine pocket parametrization scheme (Huwald et al., 2005)was applied to ensure differentiability of temperature and water content, and enthalpywas considered as a function of temperature and water content. We now attempt toreinterpret this relationship by defining temperature and water content as functions ofenthalpy. In this respect, enthalpy is the fundamental quantity. It uniquely determinestemperature below and above the pressure melting point and water content at thepressure melting point.

59

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60 5 An Enthalpy Method for Glaciers and Ice Sheets

5.2 Mathematical model5.2.1 Ice flowField equations

Glacier ice is generally assumed to be an viscous, incompressible, heat-conductingnon-Newtonian fluid which obeys the Stokes equations,

∇ ·v = 0, (5.1)∇ ·T = −ρg, (5.2)

where ∇ · is the divergence operator, v is the velocity, T = −pI+2ηD is the Chauchystress tensor, ρ is the density, g is the acceleration due to gravity, I is the identitytensor, p is the pressure, η is the effective viscosity and D = 1/2

(∇v +∇vT

)is the

strain rate tensor.

Constitutive equation

The nonlinear rheology of glacier ice can be expressed by a power law (Glen, 1955;Steinemann, 1958), relating viscosity η and effective strain rate εeff :

η = 12A− 1n ε

1−nn

eff , (5.3)

where A is the rate factor and n is the flow law exponent. The effective strain rate is

εeff =√

IIε =√

12tr (D ·D), (5.4)

where is the IIε the second invariant of D.

Boundary conditions

At the glacier surface tangential stress vanishes and the resulting boundary conditionbecomes

T ·n = −pairn, (5.5)where pair is the atmospheric pressure and n is the outward unit normal vector. Either abasal sliding velocity vb or a given sliding law can be prescribed. If no-slip conditionsare assumed, then vb ≡ 0. In case of a sliding law, the basal shear traction τb =t · (T ·n) and the bed-parallel sliding velocity vb are functionally related throughF (τb, vb) = 0. A Weertman-type sliding law reads

vb =

0 cold ice,−Cb

τpbpqb

temperate ice, (5.6)

where Cb is a sliding coefficient, pb is the basal overburden pressure, p and q arebasal sliding exponents and t is the outer tangential vector. A definition of cold andtemperate ice is given in section 5.2.2.

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5.2 Mathematical model 61

5.2.2 ThermodynamicsField equations

The balance equation for specific inner energy (Greve, 1997a), u, reads

ρu = −∇ ·q +Q, (5.7)

where u is the total derivative of u, q is the energy flux and Q is an internal heatsource. The specific energy has SI units J kg−1, and the terms in equation (5.7) haveSI units J m−2 s−1 (Moran and Shapiro, 2000). The specific enthalpy, H, is commonlydefined as (e.g. Moran and Shapiro, 2000):

H = u+ p/ρ. (5.8)

If a material is heated under constant pressure, when there are no volume changes,the enthalpy represents the inner energy (Alexiades and Solomon, 1993). Because theterm “enthalpy method” is widely used in the computational fluid dynamics literature,we will refer to enthalpy instead of inner energy in the remainder of this paper andrewrite equation (5.7) as

ρH = −∇ ·q +Q. (5.9)

Constitutive equations

The enthalpy of pure ice assumed to be uniquely related to temperature T = T (H, p)and water content ω = ω(H, p) in their respective domains, where ω is defined as themass fraction of water in the ice-water mixture ω = mw/m, where mw and m arethe mass of water and the mass of the mixture, respectively. The total energy fluxq = qs + ql is expressed as

q = qs + ql = −k∇T + Lj, (5.10)

where qs and ql are sensible and latent heat flux, k is the thermal conductivity of themixture, and j is the diffusive moisture flux (Hutter, 1982). Little is known about themoisture flux in temperate ice, Fick-type (Hutter, 1982) or Darcy-type (Fowler, 1984)diffusion have been proposed.Hs = Hs(p) and Hl = Hl(p) are the enthalpy of pure ice and pure water at the

pressure melting temperature Tm = Tm(p). We initialize Hl at p0 with Hl(p0) = 0, andintegrate from p0 to p:

Hl =∫ p

p0c(p′)Tm(p′) dp′, (5.11)

where cw is the specific heat capacity of pure water at Tm. To first order, we get

Hl = cwTm(p) (5.12)

andHs = −L+Hl(p). (5.13)

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62 5 An Enthalpy Method for Glaciers and Ice Sheets

cold temperate liquid

T′

H

Tm

HlHs

1

Figure 5.1: Temperature-enthalpy relationship for pure water (solid line) and water content-enthalpy relationship (dotted line). T ′ is the temperature relative to the pressure meltingpoint Tm; and Hs and Hl are the enthalpy of pure ice and pure water at Tm, respectively.Blue and red colors indicate cold and temperate ice, respectively; and violet indicatesliquid water.

We then define

cold iceT = 1

c(H −Hs) + Tm

ω = 0

if H < Hs,

CTST = Tm

ω = 0

if H = Hs,

temperate iceT = Tm

ω = H−HsL

if Hs < H < Hl,

liquid waterT = 1

c(H +Hl) + Tm

ω = 1

if H ≥ Hl,

(5.14)

where c = c(T ) is the heat capacity of the mixture. Measured (Pettersson et al., 2004,and references therein) water contents in temperate ice are generally less than 3 %and thus, liquid water (H ≥ Hl) is not considered in this work. A schematic plot oftemperature and water content as a function of enthalpy is shown in Figure 5.1.

Boundary conditions

At the glacier surface S = S(x, y), a Dirichlet condition, H = HS, under the con-straints,

HS = Hs + c (TS − Tm) if H < Hs,HS = Hs if H = Hs,HS = Hs + LωS if Hs < H < Hl,

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5.2 Mathematical model 63

is applied, where TS and ωS are the temperature at the cold surface and the watercontent at the temperate surface, respectively. At a cold glacier bed, all geothermalheat qgeo enters the ice, thus n ·q = n ·qgeo. At a temperate bed, two cases can bedistinguished: (1) if the ice above the bed is temperate, all heat (including frictionalheat) is used to melt ice, hence n ·q = 0; and (2) if the ice is cold above the bed, afraction γ, 0 < γ < 1, of the geothermal heat may enter the ice, thus n ·q = γn ·qgeo.To properly treat this case, however, the thermal boundary of the domain must belowered into the lithosphere. We thus consider case (1) only:

n ·q = n ·qgeo if H < Hs,n ·q = 0 if Hs ≤ H < Hl.

5.2.3 Enthalpy Gradient MethodTo solve the enthalpy balance equation (5.9), we express the energy flux q in terms ofenthalpy. In cold ice, the gradient of equation (5.14) gives

∇T (H, p) = 1c

(∇H −∇Hs(p)

)+∇

(1c

)(H −Hs(p)

)+∇Tm(p)

= 1c

(∇H − ∂Hs(p)

∂p∇p

)−(

1c2

∂c∂T∇T

)(H −Hs(p)

)+∂Tm(p)

∂p∇p.

(5.15)

As already mentioned before,Hs and Tm vary only weakly with pressure, thus ∂Hs/∂p =∂Tm/∂p = 0. We then set c′ = ∂c/∂T and obtain

∇T = 1c+ c′(T − Tm)∇H. (5.16)

Introducing equation (5.16) into equation (5.10) yields the sensible heat flux in coldice as a function of enthalpy,

qs = − k

c+ c′(T − Tm)∇H. (5.17)

Sensible heat flux is the only heat flux in cold ice, hence q = qs.A general formulation for the diffusive moisture flux in temperate ice may consider

water content, its spatial gradient, deformation and gravity (Hutter, 1982). However,the testing of different constitutive equations for the moisture flux is beyond the scopeof this study. Therefore, Fick-type moisture diffusion in temperate ice is assumed andthe diffusive moisture flux is then given by

j = −ν∇ω, (5.18)

where ν is a moisture diffusivity (Hutter, 1982). The gradient of equation (5.14) fortemperate ice gives

∇ω = 1L∇H, (5.19)

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64 5 An Enthalpy Method for Glaciers and Ice Sheets

where again is assumed that Hs varies only weakly with pressure. Introducing equa-tions (5.18) and (5.19) into equation (5.10) yields the latent heat flux in temperate iceas a function of enthalpy,

ql = −ν∇H. (5.20)The sensible heat flux in temperate ice,

qs = −c∇Tm, (5.21)

arises due to small temperature variations of the melting point. This contributionshall be ignored henceforth, thus q = ql. By defining κ as

κ =

νρ

temperate icek

ρ(c+c′(T−Tm)) cold ice,(5.22)

thusq = −ρκ∇H, (5.23)

and therefore, equation (5.9) can be rewritten as

ρ

(∂H

∂t+ v · ∇H

)= ρ∇ · (κ∇H) +Q, (5.24)

for both cold and temperate ice. The first term is the local rate of change of enthalpy,the second term is enthalpy advection and the third term is enthalpy diffusion. Q =tr (D ·T′) is enthalpy production due to strain heating. In this study, we focus on thethermodynamical steady state, thus the enthalpy balance equation reduces to

ρv · ∇H = ρ∇ · (κ∇H) +Q. (5.25)

5.3 Numerical SimulationsNumerical solutions based on the Finite Element Method are obtained using the com-mercial program package Comsol Multiphysics (www.comsol.com). For an account onthe Finite Element Method, we refer to, e.g., Braess (2007). The glacier domain isapproximated by an unstructured triangular mesh with a maximum element size of25 m. The mesh is not shown because the used element sizes are too small. QuadraticLagrange elements are used for velocity and enthalpy, and linear Lagrange elements areused for pressure; this is Comsol’s default setting. Values of constants and parametersused in this study are listed in Table 5.1.

5.3.1 Ice flowFor soft, deformable beds (p, q) = (1, 0) is suggested, thus resulting in a linear slidinglaw,

vb =

0 if H < Hs,

−Cbτb if Hs ≤ H < Hl.(5.26)

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5.3 Numerical Simulations 65

Table 5.1: Physical Constants and Parameters Used in the Study

Variable or Constant Name Value Unit

β Clausius-Clapeyron constant 7.53 · 10−8 K Pa−1

c specific heat at Tm 2.008 · 103 J kg−1 K−1

ε0 viscosity regularization 10−13 s−1

k thermal conductivity at Tm 2.22 W m−1 KL latent heat of fusion 3.34 · 105 J kg−1

n exponent of the flow law 3 -ν moisture diffusivity 5 · 10−4 kg m−1 s−1

pair atmospheric pressure 105 Paq⊥geo geothermal heat flux 0.042 W m−2

ρ densitiy of ice 900 kg m−3

The rate factor may be written as A = A0A(ω, T ′), where T ′ = T − Tm is the temper-ature relative to Tm. For the cold ice rate factor, AT ′ , we use a double exponential fitderived by Smith and Morland (1981),

AT ′ =

0.9316 exp (0.32769T ′) + 0.0686 exp (0.07205T ′) T ′ ≥ −7.65C0.7242 exp (0.59784T ′) + 0.3438 exp (0.14747T ′) T ′ < −7.65C

, (5.27)

and for the temperate ice rate factor, Aω, according to Lliboutry and Duval (1985),

Aω = (1 + 1.84ω), (5.28)

which is valid for ω < 1%. In the absence of other constraints, Greve (1995) limits thewater content to this value by using a drainage function. In this work equation (5.28)is applied also to water contents larger than 1 %. The transition at T ′ = −7.65 C isspread over a temperature range of 0.1 C using a smoothed Heaviside function with acontinuous first derivative (a piecewise polynomial of degree three). To avoid infiniteviscosity at low effective stress, a small number ε2

0 is added to IIε in equation (5.4)thus the effective strain rate becomes:

εeff =√

IIε + ε20. (5.29)

ε0 can be regarded as the smallest strain rate that can be resolved.Let Ω be the glacier domain and ΓC its border, where the subscript C = B, S denotes

bed and surface. The steady-state Stokes problem is then defined by the followingequations:

∇ ·v = 0 in Ω∇ ·T = −ρg in ΩT ·n = −pairn on ΓS

vb =

0 if H < Hs

−Cbτb if Hs ≤ H < Hlon ΓB

. (5.30)

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66 5 An Enthalpy Method for Glaciers and Ice Sheets

5.3.2 ThermodynamicsTm can be approximated by Tm = −βρg(S − z), where β is the Clausius-Clapeyronconstant. Thermodynamical properties such as c and k are, in general, functions oftemperature, but are taken constant in this work. The diffusivity κ thus reduces to:

κ =

ν/ρ temperate icek/ρc cold ice

. (5.31)

The diffusivity is a step function with a step at the CTS from k/(ρc) to ν/ρ. Thisstep is implemented with a smoothed Heaviside function, spreading the step over arange of 1500 J kg−1. A smoothed Heaviside function is applied to spread the thermaltransition from temperate to cold ice at the bed over an enthalpy range of 100 J kg−1.We set q⊥geo = n ·qgeo and thus n ·q = q⊥geo. The steady-state enthalpy problem is thendefined by the following equations:

ρv · ∇H = ∇ · (ρκ∇H) +Q in ΩH = HS on ΓS

n ·q = 0 on ΓB,tn ·q = q⊥geo on ΓB,c

, (5.32)

where the subscripts t and c indicate temperate and cold bed, respectively.The coupled nonlinear system of equations (5.30) and (5.32) is solved using an

affine invariant form of the damped Newton method (Deuflhard, 1974). From thesimulated enthalpy distribution, temperature and moisture content are then obtainedfrom equation (5.14).

5.4 ExperimentsTo demonstrate the applicability of the mathematical model to cold, temperate andpolythermal glaciers, three test cases are designed. To this end, longitudinal profilesof three different glaciers (Haut Glacier d’Arolla, Griesgletscher and Laika Glacier)are selected instead of artificial geometries.

5.4.1 Uncoupled ExperimentsExperiment UP

We first simulate the temperature and water content fields in an uncoupled run witha constant rate factor, A = A0 = 2 · 10−24 Pa−3 s−1, and no-slip conditions at the bed.No-slip conditions in temperate ice might be questionable but it is the aim of thisexperiment to show that the EM is capable to simulate different thermal structures,and not to simulate velocities, temperatures and water contents in agreement withmeasurements. Thermo-mechanically coupled simulations of polythermal glaciers in-cluding basal sliding in temperate ice are beyond the scope of this study. A longitudinalprofile of Haut Glacier d’Arolla serves as test geometry.

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5.4 Experiments 67

Figure 5.2: Experiment UP (Haut Glacier d’Arolla). Simulated horizontal (a) and vertical(b) velocity field for a constant rate factor A = A0 = 2 · 10−24 Pa−3 s−1 and no-slipconditions at the glacier bed. Values are in meters per year.

Simulated horizontal and vertical velocities are shown in Figures 5.2a and 5.2b,respectively.Temperature and water content are simulated for surface temperatures TS ∈ -6,

-4, -2, 0, TS in C, and results are presented in Figure 5.3. For this choice of surfacetemperatures, cold, polythermal and temperate conditions are simulated. For thechosen rate factor, a surface temperature TS of −6 C leads to an entirely cold glacier,while TS = −4 and −2 C have a temperate basal layer. The dashed line indicates theposition of the CTS. A surface temperature of TS = 0 C eventually results in a fullytemperate glacier. For an entirely cold glacier, the water content is zero everywhere(Figure 5.3b) and, correspondingly, the temperature relative to the pressure meltingpoint is zero everywhere in a wholly temperate glacier (Figure 5.3h), thus no contourplots could be drawn. No-slip conditions everywhere at the glacier bed leads to largevalues for the water content. Blatter and Hutter (1991), for example, chose basalsliding in temperate areas as small as possible but large enough to prevent too high amoisture content.

5.4.2 Coupled ExperimentsExperiment CT

To simulate a temperate glacier, a longitudinal profile of Griesgletscher, Switzerland,serves as test geometry. Two cases are investigated: (1) no-slip conditions at theglacier bed (Experiment CT1); and (2) slip conditions at the glacier bed using a slidingcoefficient Cb = 1.25 · 10−12 m s−1 Pa−1 (Experiment CT2). For case (2) both at thebergschrund and at the glacier tongue, no-slip conditions are imposed to improveconvergence, and the transition from no-slip to sliding is spread over a distance of200 m with a smoothed Heaviside function. In both cases A0 = 0.7 · 10−24 Pa−3 s−1 isused.

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68 5 An Enthalpy Method for Glaciers and Ice Sheets

Figure 5.3: Experiment UP (Haut Glacier d’Arolla). Temperature relative to the pressuremelting point T ′ (left column) and water content ω (right column) for TS = −6 C(a,b), −4 C (c,d), −2 C (e,f) and 0 C (g,h) using a constant rate factor A = A0 =2 · 10−24 Pa−3 s−1 and no-slip conditions at the bed. The dashed line indicates theposition of the cold-temperate transition surface. Values are in degree Celsius (temper-ature), and grams water per kilogram mixture (water content).

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5.4 Experiments 69

Figure 5.4 (no-slip) and Figure 5.5 (slip) show horizontal velocity at the surface(solid line) and at the base (dotted line), temperature (b), and water content (c). Thetemperature T is equal to Tm, and the temperature relative to the pressure meltingpoint is zero everywhere. Visual comparison of Figure 5.4c and Figure 5.5c showshigher water content for the no-slip case.

Experiment CL

Laika ice cap (unofficial name) is a small icefield with a surface area of about 10 km2

on Coburg Island, Canadian Arctic Archipelago. The ice cap rests on the relatively flatnorth-eastern side of an almost circular shaped hill. The largest of three outlet glaciersis Laika Glacier (Blatter and Kappenberger, 1988). Temperature measurements in fivebore holes by Blatter and Kappenberger (1988) revealed a temperate basal layer. Blat-ter and Hutter (1991) simulated the temperature distribution of Laika Glacier usinga shallow ice approximation velocity field and a constant rate factor. Prescribed tem-peratures at the surfaces are derived and interpolated from measurements by Blatterand Kappenberger (1988) (Figure 5.7b). Their calculations, however, yielded a whollycold glacier, which led to the conclusion that the polythermal state is a remnant ofearlier conditions. Huss et al. (2008) analyzed geometrical changes based on field data,remote sensing and mass balance modelling. Observed geometries of Laika Glacier areavailable for the years 1959 and 1971, of which the first one is used in this work.A0 = 1.2 · 10−25 Pa−3 s−1 is used, and no-slip conditions at the bed and a surfacetemperature distribution given in Figure 5.7b are applied as boundary conditions.Simulated horizontal and vertical velocities are presented in Figures 5.6a and 5.6b,

respectively. Prescribed surface temperature distribution and temperature relative tothe melting point are given in Figures 5.7a, 5.7b and 5.7c, respectively. Near thebergschrund, slightly negative horizontal velocities, |u| −10−2 m a−1, are obtained.This is a result of the negative surface inclination, however, a significant influence onthe temperature distribution further downglacier is not expected. The calculations ofBlatter and Hutter (1991) are based on the similar, but thinner 1971 geometry. Never-theless the simulation presented here, which is based on the 1959 geometry, also yieldsa fully cold glaciers with highest temperatures of about −2.6 C. This corroboratesthe conclusion of Blatter and Hutter (1991) that the polythermal structure of Laikaglacier is only a remnant of an earlier climate.

Experiment CP

A thermo-mechanically coupled experiment for a polythermal structure including basalsliding in temperate ice is set up. Basal sliding is implemented in the model by addinga thin soft layer at the glacier base with a constant viscosity (Vieli et al., 2000). Thelayer has a constant thickness d = 50 m. Viscosity, layer thickness and the slidingcoefficient are related through Cb = d/η, where η is the viscosity of the soft layer. No-slip conditions are implemented using a relatively high viscosity, ηc = 1016 Pa s, andto simulate sliding in temperate ice, ηt = 1013 Pa s is chosen. A smoothed Heavisidefunction is applied to spread the thermal transition from temperate to cold ice at the

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70 5 An Enthalpy Method for Glaciers and Ice Sheets

Figure 5.4: Experiment CT1 (Griesgletscher). Example of a temperate glacier without slid-ing. Simulated horizontal velocity at the surface (solid line) and at the base (dottedline) (a), temperature (b) and water content (c). Temperature T is equal to the pressuremelting point Tm. Values are in meters per year (velocity), degree Celsius (temperature),and grams water per kilogram mixture (water content).

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5.4 Experiments 71

Figure 5.5: Experiment CT2 (Griesgletscher). Example of a temperate glacier with sliding.Simulated horizontal velocity at the surface (solid line) and at the base (dotted line) (a),temperature (b) and water content (c). Temperature T is equal to the pressure meltingpoint Tm. Values are in meters per year (velocity), degree Celsius (temperature), andgrams water per kilogram mixture (water content).

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72 5 An Enthalpy Method for Glaciers and Ice Sheets

Figure 5.6: Experiment CL (Laika Glacier). Simulated horizontal (a) and vertical (b) veloc-ity field. Values are in meters per year.

Figure 5.7: Experiment CL (Laika Glacier). Prescribed surface temperature TS (a) adaptedfrom Blatter and Hutter (1991), and simulated temperature relative to the pressuremelting point T ′ (b). Values are in degree Celsius (temperature).

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5.5 Conclusions 73

Figure 5.8: Experiment CP. Simulated horizontal velocity at the surface (solid line) and atthe base (dotted line) (a), temperature relative to the pressure melting point (b) andwater content (c). Values are in meters per year (velocity), degree Celsius (temperature),and grams water per kilogram mixture (water content).

bed over an enthalpy range of 250 J kg−1. A0 = 5 · 10−24 Pa−3 s−1 and TS = −4 Care used.Figure 5.8 shows horizontal surface and basal velocity (a), temperature relative to

the melting point (b), and water content (c).

5.5 ConclusionsThe presented steady-state solutions of uncoupled and coupled simulations demon-strate the performance of the enthalpy method. The mathematical model is capableto simulate the relevant thermal structures. This formulation is more compact thanthe brine pocket parametrization applied in chapter 4 and more physical than the“cold-ice method” used in current ice sheet models. It nevertheless leads to the sametype of advection-diffusion-production problem as the temperature equation in the

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74 5 An Enthalpy Method for Glaciers and Ice Sheets

“cold-ice method” models which would facilitate the implementation. For these rea-sons, the proposed method could be a replacement for the temperature equation usedin present-day “cold-ice method” ice sheet models.

Acknowledgements Stimulating discussions with H. Blatter and J. Brown improvedthe manuscript substantially. Help from J. Brown and M. Lüthi with the implementa-tion of the slip boundary condition is highly acknowledged. Thank is due to M. Hussfor providing the digitized Laika geometry and R. Greve for providing information onSICOPOLIS. This work is supported by the Swiss National Science Foundation, grantnos. 200020-115881 and 200021-107480.

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6 OutlookOur current generation of prognostic ice-sheet models is not able to adequately cap-ture rapid and non-linear responses of the polar ice masses to environmental changes,such as dramatic increase in ice discharge following weakening and disintegration ofbuttressing ice shelves and ice tongues. While the Intergovernmental Panel on ClimateChange acknowledged the potential importance of ice-dynamical effects, processes thatcould render the Greenland and Antarctic ice sheets more vulnerable to future warm-ing are not incorporated into forecasts of future sea-level rise, primarily because oflimited understanding of the processes involved. Due to these facts, recently ques-tions were raised within the glaciological community concerning the future of ice-sheetmodeling. The community agrees on the need for a “next-generation ice-sheet model”.Workshops have been organized to gather ideas and to formulate the requirements. Itseems clear that, to further our understanding, multi-disciplinary efforts are requiredthat involve (1) field campaigns; (2) advances in remote sensing techniques and (3) innumerical analysis; and (4) novel theoretical developments:(1) Field experiments targeted at ice dynamics and thermodynamics.

(2) Advances in remote sensing techniques help to better constrain boundary con-ditions (e.g. bed and surface topography, surface velocities, geothermal heatflux).

(3) The development of the next generation of ice-sheet models faces several chal-lenges. Model needs to go beyond the “shallow ice paradigm” and include non-linear processes. Key challenges are: (a) the development of prognostic higher-order and full Stokes models which are parallelized from scratch requires closecollaboration with fields such as mathematics and computational fluid dynamics;and (b) nearly mass-conserving numerical methods to treat free surfaces whichcan handle also topological changes are import to increase our confidence inprognostic simulations.

(4) Novel theoretical developments should address topics including, but not limitedto, calving, ice-shelf and grounding-line instability.

The developed enthalpy method contributes to (4). It combines a rigorous representa-tion of the thermodynamics with the ease of implementation. It is much more physicalthan the “cold-ice method” and is thus particularly well suited to replace the temper-ature equation used in current ice sheet models. Nevertheless, some points need to beaddressed before the enthalpy method is ready to be implemented in ice-sheet models:• Improvement of the sliding implementation for polythermal cases. Results pre-

sented for the UP experiment are preliminary, and only a linear sliding law hasbeen implemented. Thorough testing and sensitivity studies are crucial.

75

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76 6 Outlook

• Numerical simulations of the Greenland ice sheet with SICOPOLIS show thatstrain heating produces large amounts of meltwater and thus, a drainage functionto limit the water content is physically reasonable (R. Greve, personal commu-nication, May 2008). Either such a drainage function or a Darcy-type diffusionfor water contents above a given threshold should be implemented. Such anextension offers the possibility to further investigate one of the many roles waterplays in ice dynamics.

• Development of a transient model. This requires to treat the glacier surfaceas a free surface. To this end, Comsol Multiphysics offers two approaches: (1)moving meshes based on an arbitrary Langrangian Eulerian setting; and (2) alevel-set method which treats the glacier and the surrounding air as a two-phaseflow problem. Method (1) has the advantage that it is relatively easy to couplewith the enthalpy balance equation, but would require a lot of hand-coding.Method (2) was successfully applied by Pralong et al. (2003). However, couplingthe level-set equation and the enthalpy balance equation (which is only definedin the glacier domain) is a non-trivial task. Another weakness of the originallevel-set formulation is that mass is not conserved. However, modified level-setformulations, which conserve mass much better than the original formulation,exist. Such a method is implemented in Comsol Multiphysics. With such a modelat hand, studies how a changing climate affects the thermal structure of a glacierbecome possible. For example, how much increase in annual air temperature isrequired that a cold glacier becomes polythermal?

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Acknowledgments

During the course of my doctoral studies, I was supported by many individuals:

First of all, my deepest thanks go to my supervisor Heinz Blatter, who initiated theproject. He guided me through my PhD and answered patiently my countless ques-tions. With his scientific curiosity, he always asked the right questions when I wasstuck.

I still remember the day I first met Peter Jansson at Kiruna airport, Sweden, in April2004. Since then, Peter gave me the change to do field work on Storglaciären, provideddata and shared his ideas on glaciology, on LATEX and on science with me. For this,and for being my external examiner, I’d like to thank him very much.

Martin Funk was a member of my PhD committee and generously provided logisticalsupport for field work. Martin Lüthi’s thoughtful comments and suggestions helpedto solve modeling problems. Jed Brown carefully read parts of this thesis and madehelpful comments on mathematical formulations. I thank Andreas Bauder, JérômeFaillettaz, Matthias Huss, Patrick Riesen, Fabian Walter and Mauro Werder for stim-ulating discussions and great days in the field.

As a member of Atsumu Ohmura’s group for several years, I enjoyed his full supportand many stimulating discussions, along with a pleasant working atmosphere.

I’m grateful to Christoph Schär that he accepted to be co-examiner.

Jaques Rappaz and Marco Picasso (EPF Lausanne) proposed to solve the enthalpybalance equation and thus led the foundation for this thesis.

Sven Friedel (Femlab GmbH) and the support group of Comsol Multiphysics kindlyprovided help to deal with vagaries and pitfalls of Comsol Multiphysics.

I’d like to thank my office mate Christian Ruckstuhl not only for scientific discussions,solving Matlab and LATEX-problems, but also for great mountaineering adventures andhaving fun in Seattle. Shuba Pandey and Harald Rieder, my office mates during thefinal stage of this thesis, had to cope with my ups and downs; thanks for being patientwith me. The 4 o’clock self-help group is responsible for sugar jolts, cakes, cookies and(both serious and funny) discussions on virtually everything. And, most important, alot of laughter and happiness.

Cornelius Senn, together with Edi Im Hof and Hans-Jörg Frei, developed miniatur-ized inclinometer probes which were deployed in a borehole near the glacier tongue of

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Storglaciären. Conny’s enthusiasm and his talent to improvise helped to successfullyinstall the inclinometer system in 2007. Thanks very much to Barbara who was partof Team ETH 2008. Unfortunately, this work is not part of the thesis. I’d also liketo thank Karl Schroff for so many things (for guaranteeing continuous coffee supply,fixing mod-cons or repairing a flat tyre, just to name a few).

Many thanks to Jed Brown, Marc Chiacchio, Simon Lloyd, Harald Rieder and AldonaWiacek for proofreading parts of this work.

Donald E. Knuth started developing TEX in 1977, which was later supplemented bythe LATEX macros. Without LATEX, I might have had to use something else to type mythesis. Almost unthinkable.

Thanks is due to Tim Oke for introducing me to scientific research; you were the sparkwho lit the fire.

My family has always been my greatest and most important backup. Their supportcannot be appreciated too much. Too many friends to name them all who put upwith me during my PhD studies. But I’ll never forget field work, climbing and skiingsessions with Ruschle, Luca and Simon.

This work is dedicated to Boris Müller who left us too early; I will always rememberyou.

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Curriculum Vitae

Personal Information

Name: Andreas AschwandenDate of Birth: 27.12.1977Place of Birth: Altdorf, SwitzerlandCitizen of: Seelisberg (UR)

Education

06/2005 – 06/2008 PhD student at the Institute for Atmospheric andClimate Science, ETH Zurich, Switzerland

11/2004 – 02/2005 Research assistant at the Institute for Atmospheric andClimate Science, ETH Zurich, Switzerland

10/1999 – 10/2004 Diploma student, Departement of Earth Science,ETH Zurich, Switzerland. Diploma inClimatology, Glaciology, Hydrology and Atmospheric Chemistry

10/1997 – 04/1999 Studies in Physics, Departement of Physics,ETH Zurich, Switzerland

08/1990 – 06/1997 Kantonale Mittelschule Uri, Altdorf, SwitzerlandMatura Typus C

PublicationsAschwanden, A. and H. Blatter (2009). Mathematical modeling and numerical simula-tion of polythermal glaciers. J. Geophs. Res., doi: 10.1029/2008JF001028, in press.Pattyn, F., L. Perichon, A. Aschwanden, B. Breuer, B. D. Smith, O. Gagliardini, G. H.Gudmundsson, R. Hindmarsh, A. Hubbard, J. V. Johnson, T. Kleiner, Y. Konovalov,C. Martin, A. J. Payne, D. Pollard, S. Price, M. Rückamp, F. Saito, O. Souček, S.Sugiyama, and T. Zwinger (2008). Benchmark experiments for higher-order and fullstokes ice sheet models (ISMIP-HOM). Submitted to The Cryosphere , 2(1), 111 – 151.

Aschwanden, A. and H. Blatter (2005). Meltwater production due to strain heating inStorglaciären, Sweden. J. Geophs. Res., Vol. 110, F04024, doi:10.1029/2005JF000328.

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