Jökull - 01.12.1955, Page 7

which gives the time-dependent solution. The
steady state solutions are consequently to be
found by the equation:
£íí£) - °IH-» ■ <17>
The flow of ice through the section at the
point x is F = — — dh/dx.
Equation (17) is a non-linear differential
equation of the second order. The general
solution of such equations generally includes
two arbitrary constants which are fixed by two
boundary conditions.
The first boundary condition is obtained at
the top of the glacier, which is here put at
x= 0, wliere h = H, that is, the lieight of the
top of the glacier has to be equal to the height
of the firn line above the base.
The second boundary condition is obtained
at the end of the glacier where the flow of ice
is zero, that is, if the length of the glacier is
L then we have the condition F = 0 at x = L.
In the present case L is, however, not
fixed but has to be determined from the
condition h = 0. The second boundary
condition is tlierefore F = 0 for h = 0, which
actually means that the amount of ice flowing
into the glacier at x = 0 has to be ablated
from its surface.
In accordance with the character of equa-
tion (17) the above boundary conditions define
one and only one solution which is admittable
from the physical point of view. In other words,
for each set of values of the constants a, k and
H there exists in fact only one stable form of
the thin linear glacier moving on a horizontal
plane.
Tliis implies that the flow of ice into the
glacier at x = 0, is uniquely fixed by the
constants a, k and H, and the stable glacier
can consequently only dissipate a fixed amount
of ice per unit time. This important fact not
only holds for the special type of equation (17)
but will be valid for a great group of similar
equations, and it can thus be expected to be
valid for glaciers in general although the physi-
cal behaviour differs in some degree from that
described above.
The solution of (17), which is the only stable
solution of (16) is very simple:
h = h)/- 0.73-1 - 087)fJ ) • (18)
where:
and the flow of ice into the glacier at x = 0 is:
r= 0.46 aHL = Q587/jff- = 0.58H, (20)
where A = aH is the ablation at the end of
the glacier.
As the boundary condition at the bed is
sb = kvwli = — whdh/dx, the velocity of the
glacier becomes:
-£f073 + a54rl ■ (2i)
that is, proportional to the slope of the surface.
Furthermore the flow through the section at
x is:
fí(° 75 + a54z)(l - a7Jf ~ a?7(fi) •(22)
The relations for the profile (18), velocity
(21) and the flow (22) are in Figure 2 applied
to a central section through the glacier Brúar-
jökull in the northern Vatnajökull in Iceland,
where H is about 600 meters and L = 20000 m.
This glacier appears to flow on a level surface.
The total flow of ice into the glacier has been
derived from recent measurements of the accu-
mulation which give an average value of C —
B = 1.5 meters of water in the accumulation
region. The average density of the ice in the
glacier is assumed to be 0.8. The fit of the
theoretical profile is quite good although the
Brúarjökull is comparatively thick.
The finding of the stable solutions is quite
more tedious in the case of uneven bed or vari-
able k, and the stabilty conditions also become
rnore obscure and require a more detailed ana-
lysis, which will not be discussed in the present
paper .The same applies in a greater degree
to tlie two-dimensional cases where equation
(15) has to be applied.
In the case of the thin one-dimensional glacier
moving on a bed witli a slope down the glacier,
the height of the firn line H to. be applied in the
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