# Comparing the effects of Temperature, Fabrics and Stress exponent

Throstur Thorsteinsson
University of Washington

December 10, 1996

## 1 Introduction

Many factors can affect the effective viscosity of ice, . Important ones in ice streams include temperature, fabric and the non-linearity of the flow law. Other factors such as impurities (ionic), dust, effect of mass balance on temperature distribution are not considered here.

## 2 The Physics

### 2.1 Glens flow law

Glen's Flow Law [Glen (1958)] has for a long time been widely accepted as the constitutive relation for ice. It is given by the relation where is the strain rate, A = , is the effective shear stress, and is the stress deviator, , where .

We will throughout this paper assume that Glen's flow law is the correct law for isotropic ice.

We define as ### 2.2 Temperature effects

The temperature effects are described through the Arrhenius factor where Q is the activation energy for creep (0.68e V), k is the Boltzmann constant and T is the temperature in Kelvin. varies a little with pressure according to where V is the activation volume for creep ( ). This effect is very small, and in ice sheets it is negligible.

The relation (eq. 3) holds for temperatures lower than -10 degrees Celsius. Above that one must use a different relation.

Good data on the value of A are given by Paterson [Paterson (1994)], here tabulated in Table 1.

Table 1. ### 2.3 Fabric

We choose to look at the fabric dependence by using Azuma's [Azuma (1994)] flow law for anisotropic polycrystalline ice.

It is assumed that all the crystals deform only by glide in the basal plane and that the stress is homogeneous. Using these definitions one can calculate the enhancement that happens when the ice is anisotropic.

### 2.4 Power law for creep or a Newtonian fluid ?

In ice streams the deviatoric stresses vary from about 1 bar at the edges to 0.1 bar at the center. This is just at the suggested threshold for n=1 instead of n=3 in Glen's flow law, equation(1).

## 3 Ice Streams

In ice streams the temperature gradient near the surface is small and most of the temperature variation happens in the bottom of the thickness. Assuming that the temperature changes from -20 C to 0 C, we get that the viscosity is Similarly for the whole temperature range we get For the fabric dependence we get, according to Azuma's [Azuma (1994)] flow law, that the ratio of for isotropic versus anisotropic ice is at maximum For the stress dependence we have that the stresses are predominantly shear stresses. The magnitude is changing from the center (0.1 bar) to the edge (1 bar) of the ice stream. For a power fluid (n=3) this results in changes in  If ice were a Newtonian fluid (n=1) there would be no change in as the stress changes.

## 4 Conclusion

Temperature, fabric and stress dependence are all important factors for flow calculations of ice.

If we were to assume that the ice in the ice streams were isothermal at (-25 C) and isotropic, we could be off, close to the bed, by a factor of 650. Ignoring the nonlinearity of the ice could cost us a factor of 100. So all these factors are important and should be accounted for in flow calculations.

## References

Azuma (1994)
Azuma, N. (1994). A flow law for anisotropic ice and its application to ice sheets. Earth and Planetary Science Letters, (128), 601-614.
Glen (1958)
Glen (1958). The flow law of ice. In IAHS, pages 171-183. IAHS Publ. no. 47.
Paterson (1994)
Paterson, W. S. B. (1994). The Physics of Glaciers. Pergamon, 3rd edition.

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