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

6. Including feedback between glacier length and ice thickness
In the analysis of section 3 we made thickness a function of the bed slope, but not of the
glacier length. This is an obvious shortcoming. A more advanced analysis can be made
by using the relation:
H m =
1/2
µ L
, (6.1)
1+ ν s
where µ and ν are positive constants. Actually, eq. (6.1) fits rather well results form a
numerical glacier model in which s and L are systematically varied (Oerlemans, 2001).
Note that for s = 0 eq. (6.1) reduces to the relation between ice thickness and glaciers
length for a perfectly plastic ice sheet on a flat bed (section 2). In the simple model, the
expression for L was:
L = 2 (H m + b 0 - E)
s
. (6.2)
By substituting eq. (6.1) we obtain (E ' = E -b 0 ):
L = 2 s
1/2
µ L
- E ' . (6.3)
1 + ν s
This quadratic equation is most conveniently solved by setting N = L 1/2 . We then have
N 2 -
2 µ 1/2
s (1 + ν s) 1/2
N + 2 E '
s
= 0 . (6.4)
The determinant is
Det =
4 µ
s 2 (1 + ν s)
- 8 E '
s
. (6.5)
Real solutions exist only when Det ≥ 0. The first term is always positive. Therefore a real
solution exists even for small positive values of E ', that is, when the equilibrium line is
higher than the highest part of the bed at x = 0, but below the glacier surface. This
nonlinearity, of course, reflects the HMB-feedback. The solution for L reads
L =
µ 1/2
s (1 + ν s) 1/2 ± µ
s 2 (1 + ν s)
- 2 E '
s
1/2
2
. (6.6)
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