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

L = 2 s
τ 0
ρ g s + b 0 - E . (3.5)
The solution is shown in Fig. 3.2 (parameter values b 0 -E = 500 m, τ 0 /ρ g = 10 m). For
reference the solution for zero ice thickness is also plotted. In this case the intercept of
the equilibrium line and bed is simply at x = L/2. For the full solution ice thickness
increases with decreasing slope, which implies an upward shift of the equilibrium point
(intercept of equilibrium line and glacier surface). There is no solution for a flat bed,
unless the equilibrium line is allowed to slope upwards.
Fig. 3.2
80
60
L (km)
40
20
full solution
0
H m
= 0
-20
0 0.05 0.1 0.15 0.2 0.25
Slope of bed
The simple model can be used to make an order-of-magnitude estimate of climate
sensitivity. Differentiating Eq. (3.5) with respect to E yields:
d L
dE
= - 2 s
. (3.6)
So glaciers on a bed with a smaller slope are more sensitive in an absolute sense. It is
tempting to use eq. (3.6) to make a first-order estimate of the response of glacier length
to a change in free atmospheric temperature Ta. We assume that the equilibrium-line is
linked to this temperature, which implies that d E/d T a = -γ -1 . Here γ is the temperature
lapse rate in the atmosphere, typically -0.007 K km -1 ). It follows that
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