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

=
ρ i h
ρ i - ρ m
= -δ h . (2.7)
The parameter δ is of the order of 1/3. The ice thickness is (1+δ)h and we can substitute
this in eq. (2.2):
ρ g (1+δ) h d h
d x = τ 0 . (2.8)
Therefore the solution now becomes
h(x) =
2 τ 0
ρ g (1+δ) x , (2.9)
and we conclude that the profile remains parabolic with a slightly modified 'plasticity
parameter' µ isos
µ isos =
2 τ 0
ρ g (1+δ)
. (2.10)
The perfectly plastic ice-sheet model can be used to demonstrate how the HMB-feedback
leads to hysteresis when we consider the response of a bounded ice-sheet to climate
change. Suppose that the specific balance rate is a linear function of height:
b n = β (h - E) , (2.11)
where E is the equilibrium-line altitude and β the balance gradient (assumed to be
constant). Now the total mass budget of an ice cap will be positive when
h > E (2.12)
So for h < E an ice sheet cannot exist. However, in the case that h > E > 0 a situation
without an ice sheet also represenst an equilibrium state. For E < 0 there has to be an ice
sheet.
Altogether this can be visualised in a solution diagram (Fig. 2.2), showing the
equilibrium states as a function of the climatic conditions. In this simple model the ice
sheet can have only two states (for given L). Climate change is represented by a shift of
the equilibrium line. The critical points are shown by black dots.
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