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

7. Steady state ice-sheet profiles
The perfectly plastic ice-sheet model can be improved by calculating the profile for simple
plane shear (Vialov, 1958). In this case the deviatoric stress tensor S ij has only one
nonzero component:
S xz (z) = (H-z) ρ g sin α . (7.1)
Here H is the ice thickness, z the height above the bed, ρ ice density and α the surface
n
slope. This can be combined with Glen's law for simple shear (du/dz = 2 A S xz ) to give
du
dz
= 2 A (H-z) ρ g sin α n . (7.2)
Integrating this equation twice with respect to z yields the vertical mean 'horizontal' ice
velocity U
U = A* H n+1 sin α n + U s (7.3)
In this equation U s is the sliding velocity. A* is the effective flow parameter, which can
be expressed in A, ρ, g and n (see Problems).
Next we consider an axi-symmetric configuration (Fig. 7.1). The vertically-integrated
continuity equation then takes the form:
∂ H
∂ t
= -∇⋅(H U) + b → ∂ H
∂ t
= - ∂(r H U r )
r ∂ r
+ b (7.4)
therefore for the steady state we find:
d (r H U r ) = b r d r → U r = b 2 H-1 r (7.5)
Fig. 7.1
H
φ
R
r
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