Jökull - 01.12.1955, Page 9

Tlie height of the top is h0 = 3H/2 and the
total length:
'■ ■ 'fí • <24)

Fig. 3. Theoretical form of a thin one-dimens-
ional ice-sheet, moving on a horizontal plane,
according to equation (23).
Comparing' this with equation (17) we see
that the ratio of the length of the accumulation
area to that of the ablation area is 1.73/1.27
= 1.36.
As equation (23) represents the only stable
form of the ice-sheet we can infer that pertur-
bations of this form will lead to instability,
either growth or decrease, depending on the
character of the perturbation. If the three basic
figures a, k and H are constants in space and
time, then the removal of a small amount of
material from the theoretical ice-sheet will
bring it to a decrease or retreat ending in
complete disappearance. On the other liand
the adding of material will bring it to grow
without limit. An increase of the height of the
firn line will thus bring the ice-sheet to retreat,
and a decrease of the height will bring it to
grow.
The fact of main importance is that cause
and effect are out of proportions, that is, a
small cause results in a very great effect, and
a srnall variation in the clima can, therefore,
result in great changes of glaciers. The amount
of change depends in actual cases on the various
stabilizing factors mentioned above and also
on the mass of the ice-sheet which is a further
stabilizing factor.
Tlie numerical problem to be treated here
is the initial growth, or decrease, of the ice-sheet
in Figure (3) by a change in the height of the
firn line, that is, we will ask for the pertur-
bation u of the stable form hs in equation
(23) trhen the height of the firn line changes
from H to H + g(t), where g(t) is the time-
dependent perturbation of the height of the
firn line, which will be assumed small com-
pared to H.
Inserting (hs + u) for h, and (H + g(t))
for H in equation (16) we get:
<L/AH Mjia!) = a/Hrg/rl-^-u) * £ , (25)
dx\ k dx /
but as hs is the stable form corresponding to
H and u is to be small compared to h, we can
in the first approximation write by a rearrange-
ment:
which is a linear partial differential equation
for u. By the substitution:
u = p/r/ , ^7)
where q(x) is a function of x only, and p(t) a
particular solution of (26), that is, solution of
the equation:
ap - ~- = ag/rj , (28)
we get the general solution of (26):
u * pM ■ (29)
where the functions q (x) are the orthonormal
i
set of solutions of the Sturm-Liouville equation:
£frSJ + 'a+ m>9 = 0 • (30)
and by the boundary conditions:
Pa
— = 0, /or x = 0 and x = /. ,£,,,
dx (31)
The values of the constants mj are deter-
mined by the eigenvalues of (30) and (31), and
the constants C ; are determined by the initial
condition of (26):
u ” Zfi<//M * p/O), /or r =0, (32)
where the values of u at t = 0 are to be
inserted and the constants determined by the
usual procedure in expanding given functions
in a series of orthonormal functions.
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