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

For L < L ub the solution is given by eq. (3.3):
L = 2 (H + b 0 - E)
s
. (4.1)
When the mass budget of the upper basin is positive L will be larger than Lub. The length
of the glacier is then determined by
L ub L
(H + b 0 - s x - E) dx + ξ (H + b 0 - s x - E) dx = 0
0
L ub
. (4.2)
Here ξ is the width of the glacier tongue divided by the width of the upper basin.
Evaluating the integrals yields:
- 1 2 ξ s L2 + ξ (H + b 0 - E) L + (1 - ξ) (H + b 0 - E) L ub - 1 2 s (1 - ξ) L ub
2 = 0 (4.3)
This is a quadratic equation for L which is easily solved:
L = - 1 s
E' + E' 2 + 2 s 1-ξ
ξ
1/2
E' L ub + 1 2 s L ub
2
(4.4)
In this expression E' = E - b 0 - H (note that E' < 0).
The solution is illustrated in Fig. 4.3. In this example the upper basin has a length of
5 km. Other parameter values are b 0 =2000 m, s=0.1, H=100 m. L is plotted as a
function of E for three different values of ξ. For ξ=0.25, the width of the upper basin is
four times that of the glacier tongue.
Fig. 4.3
20
L (km)
15
10
ξ =4
ξ=1
ξ =0.25
largest sensitivity
5
0
1400 1500 1600 1700 1800 1900 2000
E (m)
15