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, tex2html_wrap_inline230 . 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

  equation25

where tex2html_wrap_inline232 is the strain rate, A = tex2html_wrap_inline234 , tex2html_wrap_inline236 is the effective shear stress,

displaymath238

and tex2html_wrap_inline240 is the stress deviator, tex2html_wrap_inline242 , where tex2html_wrap_inline244 .

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

We define tex2html_wrap_inline230 as

equation112

2.2 Temperature effects

  The temperature effects are described through the Arrhenius factor

  equation115

where Q is the activation energy for creep (0.68e V), k is the Boltzmann constant and T is the temperature in Kelvin.

tex2html_wrap_inline256 varies a little with pressure according to

equation117

where V is the activation volume for creep ( tex2html_wrap_inline258 ). 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.

tabular65

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 tex2html_wrap_inline280 of the thickness. Assuming that the temperature changes from -20 C to 0 C, we get that the viscosity is

equation119

Similarly for the whole temperature range we get

equation121

For the fabric dependence we get, according to Azuma's [Azuma (1994)] flow law, that the ratio of tex2html_wrap_inline230 for isotropic versus anisotropic ice is at maximum

equation124

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 tex2html_wrap_inline230

equation127

If ice were a Newtonian fluid (n=1) there would be no change in tex2html_wrap_inline230 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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