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

Note that values of L for which N < 0 are spurious and should not be considered.
An example is shown in Fig. 6.1. Parameter values are: µ = 9 m, ν = 20, s = 0.06 .
Because L = 0 is a stable solution for E ' > 0 (equilibrium line above the bed
everywhere), there are two stable branches and one unstable branch (dotted). In terms of
catastrophe theory this model represents a fold, but because we add the condition that L
should be positive it appears as a distorted cusp. The branching of the steady-state
solutions that shows up here was in fact found numerically a long time ago (Oerlemans,
1981). It should also be mentioned that the dynamics of the present glacier model are
similar to those of a perfectly plastic ice sheet on a flat bed with a sloping equilibrium line
Weertman (1961).
The range of E '-values for which two stable solutions exist is
0 ≤ E ' < E ' crit , (6.7)
where the critical point E' crit is found by setting Det = 0:
E ' crit =
µ
2 s (1 + ν s)
. (6.8)
Therefore, for increasing slope of the bed, the HMB-feedback becomes weaker and the
critical point approaches the origin. The result for the linear analytical model is also
shown. In this caseτ 0 /ρg was set to 9 m, because this is consistent with eq. (6.1) for
s = 0. The squares in Fig. 6.1 shows the dependence of glacier length on E' as found by
a numerical plane-shear model for a glacier of constant width. Clearly, the nonlinear
analytical model matches the numerical result very well.
Fig. 6.1.
14
12
10
L (km)
8
6
4
linear model
2
0
nonlinear model
•
•
-200 -150 -100 -50 0 50
E' (m)
21