Water Content of Glacier Ice at the Bed of ...

Water Content of Glacier Ice at the Bed of Engabreen, Norway, from the Solution of a One-Dimensional Stefan Problem Denis Cohen Department of Geology and Geophysics University of Minnesota 310 Pillsbury Dr. S.E. Minneapolis, MN 55455 December 16, 1998

Contents 1 Introduction

1

2 Field Experiments

1

2.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Numerical Solutions of Stefan Problem 3.1 3.2 3.3 3.4 3.5 3.6

Hypothesis . . . . . . . . . . . . . . . . . . . . . . Mathematical Formulation . . . . . . . . . . . . . . Dimensionless Equations . . . . . . . . . . . . . . . Coordinate Transformation . . . . . . . . . . . . . Numerical Algorithm: Variable Time-Step Method Testing of the Algorithm . . . . . . . . . . . . . . .

4 Results and Discussion

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3 5 6 6 7 10

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2 FIELD EXPERIMENTS

1

1 Introduction Calorimetric methods have long been used to determine the liquid content of temperate ice [Vallon et al., 1976; Duval, 1977]. However, as noted by Hutter et al. [1990], these methods are not well adapted to in situ measurements as they require the extraction of ice cores. Zryd [1991] proposed an in situ method and tested it at Findelengltscher, Switzerland, in an ice tunnel near the snout, a few meters beneath the surface. The method is based on monitoring the outward propagation of a cold front generated by a cold source inserted in the ice. Assuming temperate ice to be a homogeneous pure ice-water mixture, the propagation front is a sharp discontinuity which separates a region of cold ice from a region of temperate ice of uniform physical properties. The propagation speed is directly related to the water content in the temperate region and to the initial and boundary conditions at the cold source and at the propagating front. Hence, the water content can be calculated by comparing the time of travel of the cold front to a numerical solution obtained from solving a Stefan problem. In order to measure the water content of basal ice at Engabreen, an apparatus similar to that of Zryd was designed and tested in three experiments at Engabreen, in ice tunnels near the door (see MSI report Rheology of Basal Ice at Engabreen, Norway; Part I: Field Measurements). The cold wave was generated by circulating a liquid refrigerant through concentric copper tubes with the use of a small electric pump. The refrigerant was cooled by a ice/water/salt bath. The passage of the cold front was measured by thermistors encapsulated in a copper housing and inserted in the basal ice. The water content was determined by comparison with a one-dimensional Stefan problem.

2 Field Experiments 2.1 Apparatus Figure 1 shows the closed refrigeration circuit. The cold source is constructed out of two concentric copper tubes with dierent diameters, the largest one closed on one end, welded together to allow counter-current ow of the refrigerant. The refrigerant, ethylene glycol diluted with 50% water, is cooled while passing through thin copper coils immersed in a ice/water/salt bath. The refrigerant is circulated between the cold bath and the cold source through polyethylene tubing by a small 12 Volt pump. Two thermistors are glued to the wall of the inner copper tube in the cold source to monitor the cold source temperature as a function of time. Another thermistor also records the temperature of the cold bath. Receiving thermistors, encapsulated in a copper housing attached to a Te on rod, are used to measure the temperature in the ice and to record the passage of the cold front. The cold source and the thermistor housings are 12.7 mm in diameter. This set up corrects some of the problems encountered by Zryd, namely: 1. Poor contact between the receiving thermistors and ice. 2. Open circuit for refrigerant. 3. Temperature of cold source not monitored.

WATER CONTENT OF GLACIER ICE

2

Pump Copper coil Ice/Water/Salt bath

CR10

1111111111111111111111 0000000000000000000000 Ice 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 Cold source 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 Te on cylinder 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 Copper housing 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 Thermistor 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111 0000000000000000000000 1111111111111111111111

Figure 1: Refrigeration circuit.

3 NUMERICAL SOLUTIONS OF STEFAN PROBLEM

3

2.2 Experiments Three experiments were performed at Engabreen in November 1997 ice in ice tunnels melted with a hot water jet. Two of the experiments were in the clean, cloudy ice, and one was in the lower sediment-rich basal ice. Prior to each experiment, a fresh ice face was made by removing 30 cm of ice with an electric jack hammer. Then, four holes 5{10 cm apart, one 50 cm deep for the cold source, the other three 35 cm deep for the receiving thermistors, were drilled in the ice. These two steps lasted about one hour. The cold source and the Te on rods with the receiving thermistors were then inserted in the holes, the pump was started, and the temperatures were recorded on a CR10 logger for one to two hours. A few diculties were encountered during the experiments. 1. It was dicult to drill holes parallel to one another. The positions of the receiving thermistors were measured at the ends of the experiments by melting out the ice around them. 2. Some receiving thermistors did not function or did not record any change in temperature. 3. The temperature of the refrigerant increased with time. This was not a great problem since this can be incorporated in the numerical model. The temperature of the cold source was tted to either a straight line or a cubic polynomial. 4. During the experiment in the sediment-rich ice very close to the bed, the ice was extremely soft and contained numerous small water pockets. Table 1 gives details of each experiment. Figure 2 shows typical temperature records from the receiving thermistors. A couple of comments regarding the temperature records are in order. First, the initial temperature, upon insertion of the receiving thermistors in the ice, is not always 0 o C. The thermistors were calibrated to within 0:01 oC. Nevertheless, in Experiment 1, the initial temperature is ;0:07 oC. The ice being at equilibrium with the water phase, the temperatures measured show the eect of ice pressure and solutes in the ice. Second, the temperature record does not show a clear passage of the cold front. Instead, one can distinguish two distinct zones (except for T-28, Experiment 3): a slowly falling temperature followed by a more rapid decrease. The transition between the two takes place over a time span of several minutes, and is associated with the passage of the cold wave.

3 Numerical Solutions of Stefan Problem 3.1 Hypothesis Consider an ice-water mixture initially at the melting temperature m . At time = 0, a cold source at temperature c < m is inserted in the mixture. The cold source is a metallic object whose temperature is a known function of time. A phase-boundary is created by insertion of the cold source, and this boundary travels away from the source. This phase boundary separates a region of cold ice from a region of temperate ice. In order to obtain a tractable mathematical problem, the following assumptions are made:


4

Cold source Temperature Distance from Experiment (t) in o C cold source [mm] 1 ;12:424 + 0:0012435 t 72 2 ;11:265 + 0:00084934 t 82 3 ;7:3426 + 0:0015749 t 44 (T-28) ; 07 2 ;2:7114 10 t 49 (T-29) +2:0281 10;11 t3 Table 1: Cold source temperature as a function of time t in seconds, and distance from cold source to receiving thermistors. Thermistor number is indicated in parenthesis when more than one.

0.05 0 -0.05

Temperature [oC]

-0.1 -0.15 -0.2 -0.25 -0.3 -0.35 Experiment 1 Experiment 2

-0.4 -0.45

Experiment 3, T-28 Experiment 3, T-29

-0.5 -0.55 -0.6

0

2000

4000

6000

Time [s]

Figure 2: Temperature record from receiving thermistors for the three experiments.


5

1. The ice-water mixture is homogeneous. 2. The mixture is in nitely large. 3. The cold source is an in nite cylinder. The problem is therefore one-dimensional. (The problem is also solved for an the case of an in nite sheet and a sphere). 4. The contact between the cold source and the ice is perfect. 5. The metallic cold source is a perfect conductor at a uniform temperature.

3.2 Mathematical Formulation Let be the density of the ice (assume no volume change upon phase transformation) and , K and C be, respectively, the thermal diusivity, the thermal conductivity, and the heat capacity of the cold ice. These thermal properties can be functions of temperature. Let L be the latent heat of freezing of ice and ! the water content in the ice-water mixture. Let (; ) be the temperature in the cold ice, the spatial position, c the radius of the cold source, and the time. Let ( ) be the position of the moving phase-boundary. Then the initial boundary value problem is: nd (; ) and ( ), c ( ); > 0, such that: @ @ @ C ( ) = K () + ( ; 1) K () @ ; (1a) @

@

@

@

with the following boundary conditions = c( ) @ = L ! _ @ = m

= c ;

(1b)

= ( );

(1c)

and initial conditions (c ; 0) = c(0); (0) = c :

(1d) (1e) is an integer that takes on the value 1, 2, 3 for Cartesian, cylindrical and spherical geometry respectively. An analytical solution of the above problem exists for Cartesian coordinates ( = 1), constant cold source temperature, constant physical properties , K and C , m = 0 and c = 0 [Carlsaw and Jaeger, 1959]: ( ) = 2 ; ( )1=2 (; ) = c + c erf

erf ;

; 2 ( )1=2

(2a) in o C

(2b)

where ; is the root of the transcendental equation 2 ; e ; erf ; = C 1c=2 : L!

(2c)


6

3.3 Dimensionless Equations Let u; x and t be the dimensionless temperature, position and time de ned by u ( ; m )=o x =o t =o

where L! Cm o 1m=2 o 1

o

The initial boundary value problem 1 becomes

@u @ @u c(u) = k(u) + ( ; 1) k(xu) @u ; @t @x @x @x

(3a)

with boundary conditions u = uc(t) @u = s_ @x u=0

x = xc ;

(3b)

x = s(t);

(3c)

and initial conditions u(xc ; 0) = uc(0); s(0) = xc ;

(3d) (3e)

where xc = c =o uc(t) = c(t)=o um = m =o C (uo + c) c(u) = C (um o + c ) K (uo + c ) k(u) = K (umo + c ) s(t) = (to )=o

3.4 Coordinate Transformation One of the diculties in solving the above initial boundary value problem numerically is the changing spatial domain due to the moving phase-boundary. The position of the moving boundary


7

can be xed by an appropriate coordinate transformation. The new set of equations can then be solved numerically in a xed-domain. The coordinate transformation of Garofalo described in Hutter et al. [1990] changes the variable u(x; t) to u(x; s(t)). This transformation is valid when the moving boundary position changes monotonically with time. A dierent transformation is used here. The moving boundary position s(t) can be eliminated by the following transformation of coordinates [Crank, 1984]: =

x ; xc ; s ; xc

0 1:

(4)

u(x; t) is now u( (t); t). Some useful intermediate results are @u = 1 @u ; @x (s ; xc ) @ 1 @2u ; @2u = 2 @x (s ; xc )2 @ 2 @u = ; (s ; s_x ) @u + @u : @t c @ @t

With this transformation, the initial value problem becomes: nd u(; t) and s(t), 0 1; t > 0, such that @u @ @u s_ ( ; 1)k(u) @u = ( s ; xc );2 k(u) + ( s ; xc );1 + (5a) @t @ @ (s ; xc ) (s ; xc ) + xc @ with the following boundary conditions u = uc(t) @u = s_ @x u=0

= 0;

(5b)

= 1:

(5c)

This transformation is also valid only when s is a monotonically increasing function of t. This transformation, when used with the numerical scheme described below, gives better results than the method of Hutter et al. [1990] when compared to the analytical solution 2 (see Section 6.3.6).

3.5 Numerical Algorithm: Variable Time-Step Method The initial boundary value problem 5 is solved in a xed domain using a numerical scheme by Asaithambi [1988] based on the variable time-step method (VTS). Given an increment in the position of the moving boundary, the time to reach that position is calculated iteratively. The xed time-step (FTS) method [Mastanaiah, 1976; Furzeland, 1980], in which one increments time and solves iteratively for the new position of the moving boundary, was tried but did not seem to converge. A nite dierence fully implicit time discretization is used, with central dierence for space, and three point forward/backward dierences at boundaries. The spatial domain 0 1 is divided into N equal subintervals (N + 1 grid points) of length , such that the coordinate of


8

grid point is i = Ni;;11 ; i = 1; ; N + 1. At each step j , given an increment in the position of the moving boundary (sj+1 ; sj ), the time interval t(p) = (t(jp+1) ; tj ) is calculated iteratively, where the superscript (p) denotes the pth iteration. Let ui;j be the temperature at the point i at step j . Then u(i;jp)+1 is the value of u at point i during the pth iteration in step j + 1. This discretization of Equations 5 yields h i u(i;p)1;j+1 (ip) ; i(p) ki(;k)1=2;j+1=2 + h i p) u(i;jp)+1 1 + i(p) ki(;p)1=2;j+1=2 + ki(+1 =2;j +1=2 + p) u(i+1 ;j +1

h

;(ip) ;

i

2iN

(6)

(p) (p) i ki+1=2;j +1=2 = ui;j

where (p) ( ; 1)t(p) ki;j i (sj+1 ; sj ) +1=2 = + i 2(sj+1 ; xc ) 2(i (sj+1 ; xc ) + xc )(sj+1 ; xc ) t(p) (ip) = 2 sj+1 2 (p)

(7a) (7b)

and ki(p)1=2;j+1=2 is the thermal conductivity, a function of the temperature u, evaluated at ki(p)1=2;j+1=2

p) k(ui;j ) + k(u(i;jp)+1 ) k(ui+1;j ) + k(u(i+1 ;j +1 ) = 21 + 2 2

!

(8)

The boundary condition at = 0 is used as the equation for i = 1 and yields u(1p;j)+1 = uc(tj )

(9)

This system of equations forms a tridiagonal system and is solved using the LAPACK subroutine DGTSV for tridiagonal matrices. The boundary condition at = 1 is used to iterate on the time step t(jp) . t(p+1) =

2(sj+1 ; xc )(sj+1 ; sj ) 3uN;j+1 ; 4u(Np;) 1;j+1 + u(Np;) 2;j+1 (p)

At the pth iteration in step j + 1, the algorithm can be described as 1. Compute ki(p)1=2;j+1=2 2. Assemble and solve tridiagonal system for u(i;jp+1) +1 +1) 3. Compute t(jp+1 +1) 4. If jt(jp+1 ; t(jp+1+) j , go back to step 1 and iterate until convergence.

(10)


9

0.02

0.018

Error [%]

Step size = 10-5 Step size = 10

0.016

-6

0.014

0.012

0.01

0

0.02

0.04

0.06

0.08

0.1

Phase-Boundary [m]

Figure 3: Error between analytical and numerical solution with w = 0:01, L = 0:3335 106 J/kg, = 917 kg/m3, c = ;30 oC, m = 0 oC, constant physical properties K = 2:2156 J/m/s/K, C = 2096:07 J/kg/K [Yen et al., 1991], and N = 101. Step size is a dimensionless length.

10


3.6 Testing of the Algorithm The algorithm was tested against the analytical solution of Carlsaw and Jaeger [1959] for Cartesian geometry (Equation 2). Figure 3 shows the percent error in the numerical solution for two dierent choices of dimensionless step size. In both cases, the error is smaller than that shown in Hutter et al. [1990] for the same problem with the algorithm of Garofalo. Notice that the error is large for small time; this is typical for this kind of Stefan problem. With increasing time, however, the error decreases rapidly and remains constant at 0.012%. Figure 4 shows the position of the moving boundary as a function of time for a speci c test case with constant physical properties.

4 Results and Discussion Figure 5 and 6 show the temperature measured by the receiving thermistor as a function of time, together with the predicted temperature from the numerical model for two values of water content, for the two experiments in the clean cloudy ice layer. Only one thermistor was working properly during these two experiments. There are two major discrepancies between the temperature measured by the receiving thermistors and the one predicted by the model: rst, the passage of the cold wave results in a gradual decrease in temperature rather than an abrupt decrease. Second, after the passage of the cold wave, the measured temperature does not drop as fast as that predicted by the theory. These discrepancies are probably due to the assumption that temperate ice is a homogeneous mixture of ice and water. In reality, temperate ice is made up of ice crystals with water in veins at threegrain intersections [Nye and Mae, 1972; Harrison, 1972; Nye and Frank, 1973; Raymond and Harrison, 1975; Mader, 1992a]. The veins are triangular in cross section with the sides of the triangle concave inward. Because of curvature eects, the equilibrium pressure is lower in the water than in the ice. Furthermore, the equilibrium temperature, although identical in both phases, is lower than if the interface was at. Also, impurities in the water phase may decrease the equilibrium temperature. As a result, water can be present in ice whose temperature is well below the normal melting-point temperature for ice under the prevailing hydrostatic pressure. This is a major dierence between the real ice and the homogeneous model. In the homogeneous model, temperate ice is at the melting temperature and water content is constant, while cold ice is below the melting temperature and is dry. At the transition between cold and temperate ice, jump conditions are imposed: temperature is continuous but temperature gradient and water content are not. In contrast, when taking into account the inhomogeneous nature of real ice with veins, and the character of water in the veins, there is no sharp distinction between cold and temperate ice when the temperature is near the melting point. Nye [1991b] (see also earlier work by Harrison [1972] and Raymond and Harrison [1975]) has shown that ice with veins can be modelled as a continuum, heat conducting, material with smoothly varying thermal properties. As a result, water content and temperature vary smoothly across the transition zone. This explains why, in the experiments, the passage of the cold front measured by the receiving thermistors appears gradual rather than abrupt. After passage of the cold front, water is still present, although in smaller quantities, in the vein network. This water, as it freezes, acts as a heat source due to the latent heat of freezing. The decrease in temperature in the \cold" ice is therefore slowed in comparison with the predictions of

4 RESULTS AND DISCUSSION

11

1.5 1.4

Moving boundary position [m]

1.3 1.2

Cartesian Cylindrical

1.1

Spherical

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

100000

200000

300000

Time [s]

Figure 4: Position of moving boundary as a function of time for Cartesian, Cylindrical and Spherical symmetry. Same parameters as used in the test case, with c = 6:35 mm.


12

0 -0.1 -0.2

Temperature [oC]

-0.3 -0.4 -0.5 -0.6 -0.7 -0.8

w = 0.01 w = 0.015

-0.9

Experiment 1

-1 -1.1

0

1000

2000

3000

Time [s]

Figure 5: Measured and computed temperature for Experiment 1.

4000


13

0

Temperature [oC]

-0.1 -0.2

w = 0.007

-0.3

Experiment w = 0.015

-0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1 -1.1

0

2000

4000

6000

8000

Time [s]

Figure 6: Measured and computed temperature for Experiment 2. The increase in temperature in the model at time greater than about 7000 seconds is due to the linear increase of temperature of the cold source.

14


the homogeneous model. This explains the second discrepancy. A more mathematical explanation can be found in the model by Nye. Neglecting solute diusion, Nye showed that, close to the melting temperature, temperate ice has an anomalous heat capacity, up to twice its normal value, due to the presence of water in veins. This higher heat capacity slows the cooling of the \cold" ice. Nye's model has been used to study the evolution of temperature, vein size (water content), and solute concentration of a temperate ice sample brought into a warm environment [Nye, 1991a], or into contact with another ice sample at a dierent temperature [Nye, 1991b]. The results of the water content experiment in basal ice are shown in Figure 7, together with computed temperatures from the numerical model. In this experiment, two receiving thermistors were working. One of them, T-29, gave a temperature versus time curve similar to the ones observed in clean ice, with, however, a linear decrease in temperature before the passage of the cold front. The temperature recorded by the other receiving thermistor, T-28, does not show the passage of a cold front. Furthermore, even though the two thermistors are almost at the same distance from the cold source, at t = 4000 seconds, the temperature in T-28 is ;0:1 oC while the temperature in T-29 is ;0:3 oC. Clearly, the t with the homogeneous ice model is worse for basal ice. This is not surprising since basal ice is particularly non-homogeneous, containing water lenses and rock particles of various sizes. Water pockets would tend to slow the progress of the cold wave while small rock fragment are likely to be surrounded by a water layer and large rock fragments could remove heat from the ice at a higher rate. Based on the curves in the gures, comparison of the eld experiments with the homogeneous ice model indicates that the water content in clean ice is between 0.7 and 1.5%. Although there are serious problems with the model, these values are reasonable and within the range of other measurements [Vallon et al., 1976; Duval, 1977]. These values can be taken as a rst approximation to the water content of clean ice. The measurements demonstrate that ice is more complex than a homogeneous mixture, and that the theory brought forward by Nye could be used to re-evaluate the measurements. For basal ice, the homogeneous model performs poorly. The best t to the data using the homogeneous model is for water content between 2 and 4%, values much higher than in clean ice. Because of the complexity of the material, Nye's model may not be appropriate for such ice. Despite these shortcomings, these experiments show that both clean ice and basal ice contain a substantial amount of water. In basal ice, smaller grain size, water pockets, and high sediment concentration may contribute to a higher than normal water content.

Acknowledgement Funding for this project was supported by NSF grants OPP{9423422 and OPP{9713383, and by the Norwegian Water Resources and Administration. The author gratefully ackowledge the Minnesota Supercomputing Institute for providing computer resources and research environment.


15

0.04 0.02 0

Temperature [oC]

-0.02 -0.04 -0.06 -0.08 -0.1 -0.12

T-28

-0.14

w = 0.05 w = 0.07

-0.16 -0.18 -0.2

0

1000

2000

3000

4000

5000

Time [s]

0 -0.05 T-29

Temperature [oC]

-0.1

w = 0.02 w = 0.04

-0.15 -0.2 -0.25 -0.3 -0.35 -0.4

0

1000

2000

3000

4000

Time [s]

Figure 7: Measured and computed temperature for Experiment 3.

5000

16


References Asaithambi, N. S. On a variable time-step method for the one-dimensional Stefan problem.

Comp. Meth. Appl. Mech. Eng., 71:1{13, 1988. Carlsaw, H. S. and Jaeger, J. C. Conduction of heat in solids. Oxford, Clarendon Press, 1959. Crank, J. Free and moving boundary problems. Oxford, Clarendon Press, 1984. Duval, P. The role of water content on the creep rate of polycrystalline ice. In Isotopes and impurities in snow and ice. Proceedings of the Grenoble symposium, number 118, pages 29{33. IASH, 1977. Furzeland, R. M. A comparative study of numerical methods for moving boundary problems. J. Inst. Maths. Applics., 26:411{429, 1980. Harrison, W. D. Temperature of a temperate glacier. J. Glaciol., 11(61):15{29, 1972. Hutter, K., Zryd, A., and Ro thlisberger, H. On the numerical solution of Stefan problems in temperate ice. J. Glaciol., 36(122):41{48, 1990. Mader, H. M. Observations of the water-vein system in polycrystalline ice. J. Glaciol., 38(130): 333{347, 1992a. Mastanaiah, K. On the numerical solution of phase change problems in transient non-linear heat conduction. Int. J. Num. Meth. Eng., 10:833{844, 1976. Nye, J. F. The rotting of temperate ice. Journal of Crystal Growth, 113:465{476, 1991a. Nye, J. F. Thermal behaviour of glacier and laboratory ice. J. Glaciol., 37(127):401{413, 1991b. Nye, J. F. and Frank, F. C. Hydrology of the intergranular veins in a temperate glacier. International Association of Scienti c Hydrology Publication 95 (Symposium at Cambridge 1969 { Hydrology of Glaciers), pages 157{161, 1973. Nye, J. F. and Mae, S. The eect of non-hydrostatic stress on intergranular water veins and lenses in ice. J. Glaciol., 11(61):81{101, 1972. Raymond, C. F. and Harrison, W. D. Some observations on the behavior of the liquid and gas phase in temperate glacier ice. J. Glaciol., 14(71):213{233, 1975. Vallon, M., Petit, J.-R., and Fabre, B. Study of an ice core to the bedrock in the accumulation of an alpine glacier. J. Glaciol., 17(75):13{28, 1976. Yen, Y.-C., Cheng, K. C., and Fukusako, S. Review of thermophysical properties of snow, ice , sea ice and frost. In Zarling, J. P. and Fausset, S. L., editors, Proceedings of the Third International Symposium on Cold Region Heat Transfer, University of Alaska, Fairbanks, June 11-14 1991. Zryd, A. Conditions dans la couche basale temperes: contraintes, teneur en eau et frottmenet interieur. PhD thesis, Versuchsanstalt fur Wasserbrau, Hydrology und Glaziologie der Eidgenossischen Technischen Hochschule Zurich, 1991.