Simple analytical models of glacier-climate interactions - by Prof. J ...

2. Ice deformation: perfect plasticity
Perfect plasticity provides a description of ice deformation that is very tractable in simple
analytical models of ice sheets, ice shelves and glaciers.
In the following we use a Cartesian coordinate system with the z-axis pointing upwards.
The ice velocity is in the x-direction and is denoted by u. In terms of simple shearing
flow, perfect plasticity can be considered as an asymptotic case of Glen's law when the
exponent n goes to infinity. It implies the existence of a yield stress τ 0 , such that
τ xz > τ 0 : d u
dz → ∞ , (2.1)
τ xz ≤ τ 0 : d u
dz = 0 .
Eq. (2.1) means that, as long as stress is being built up by gravity, the ice will deform in
such a way that the yield stress is approached but never exceeded. Therefore the ice
velocity is constant and all the shear is concentrated at the base where
τ xz = ρ g H
d h
dx
= τ 0 . (2.2)
Here ρ is ice density, g is the gravitational acceleration, H is the ie thickness and h the
surface elevation. Values for the yield stress used in the literature are typically in the
0.5 . 10 5 to 3 . 10 5 Pa range, where the smaller values apply to ice caps with low mass
turnover and the high values to active valley glaciers. For a given value of τ 0 , the product
of surface slope and ice thickness is constant. Eq. (2.2) can thus be used to estimate ice
thickness from the surface slope.
The simplest possible model for a symmetric ice cap resting on a flat bed is based on the
theory of perfect plasticity. In this case H=h, so
d h 2
d x
= 2 τ 0
ρ g . (2.2)
This can be integrated to give:
h 2 (x) - h 2 (0) = 2 τ 0
ρ g (x-x 0 ) . (2.3)
To construct a solution for an ice cap we have to prescribe its size (L) and the ice
thickness at the boundaries (we take it to be zero). The result is (Weertman, 1961):
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