325
H = Ho(x) + ¢(~ - t),
(3.7)
where
(3.8)
is a characteristic spatial coordinate (note ~ is finite at the snout). (3.7) clearly reveals
the travelling wave characteristic of the solution.
If H is increased locally (e.g. due to the surge of a tributary glacier) then a shock
travels forward. The role of the term in f is then to diffuse such shocks. A shock at x = Xs
will propagate at a rate
.
[H n +11:!:
Xs= (n+2)[HJ:!:'
(3.9)
where [ J:!: denotes the jump across Xs. When the shock reaches the snout, it then
propagates at a speed H~+l/(n + 2), which is slower than the surface speed.
In the neighbourhood of a shock (with 1J,b = 0), we put
X=Xs+IIX,
(3.10)
so that
aH _ Xs aH + ~ [{I _ :'Hx}n Hn+2] = s'(x s + IIX);
at
II ax II
II
n + 2 x
(3.11 )
if II is small, the profile rapidly relaxes to the steady travelling wave described by
xoHx = {I - HX}"_- ,
[
Hn+2]
n+2 x
(3.12)
providing we choose II = c, which thus gives the width of the shock structure. (3.12) can
be solved by quadrature.
Seasonal waves
There is no explanation of seasonal waves available. On the face of it, we might seek waves
of amplitude of velocity of 0(1) propagating at a speed 0(11 J-t), where J-t is the ratio of
one year to the convective time scale, so J-t ;S 0.05. Apparently we should associate the
variations in 11. with variations in water supply, so that a natural model would involve
only sliding, so
(3.13)
and if 1J,b = ¢(tlJ-t)Hm/(m + 1), where ¢(tlJ-t) represents the seasonal variation of water
supply and hence of N, then