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

Problem HMB-feedback and response time (P 1)
Conservation of ice volume V can be expressed as
d V
d t
= d A H m
d t
= H m d A
d t
+ A dH m
d t
. (P 1.1)
Here A is the glacier area and H m the mean ice thickness. Now we write
V = V 0 + V', A = A 0 +A' and H m = H m,0 + H m ', where the reference state is defined
as (V 0 , A 0 , H m,0 ). By neglecting higher-order terms we then obtain the perturbation
equation
d V'
d t
= H m,0 d A'
d t
+ A 0 dH m '
d t
. (P 1.2)
Again we assume:
A' = w f L' , (P 1.3)
Here w f is the characteristic width of the glacier tongue. Now we include the height-mass
balance feedback by assuming that the change in mean thickness is proportional to the
change in glacier length:
H m ' = η L' . (P 1.4)
So for the change of ice volume with time we find
d V'
d t
= η A 0 + w f H m,0 d L'
d t
= amount of mass added . (P 1.5)
We now have three contributions to the mass added to or removed from the glacier:
change of volume = A 0 B' + A 0 β H m ' + w f B f L' = A 0 B' + A 0 β η' + w f B f L' .
(P 1.6)
B' is the perturbation of the balance rate (constant over the glacier), β the balance gradient
and B f the characteristic balance rate at the glacier front. The first term is simply the
amount of ice added or removed by the balance perturbation on the reference glacier area.
The second term represents the feedback between balance rate and surface elevation. The
third term is the amount of ice lost or gained because the length of the glacier deviates
from that of the reference state. Combining the equations yields
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