Jökull - 01.12.1955, Side 6

ancl by an integration:
V
( w d/,\n (h -z) c
ÍS,. ðx! ~(n+0
(9)
wliere C is a constant which has to be fixed by
the boundary condition at the bed s^ = kvwh,
where v is the velocity at the becl. A simple
calculation gives:
(h-z)n ) / ðh
n + ! / k öx
(10)
and the flow:
r = fv</z
-'o
h_ dh_
k áx
(ii)
which can also be expressecl in terms of the
shear stress on the bed s,
b:
F:
— Í—Í
n+P I SJ
kw
(12)
By inserting F frorn (11) into (2) the follow-
ing differential equation is obtained:
ðxl tn+2!50
S„ í áx / k dx/
+ C-B
áh
át
(13)
This equation is quite tedious and very
difficult to handle especially because of tlie
first term in the parenthesis. It is therefore
fortunate that this term appears in most
practical cases to be small compared to the
second term.
The writer has investigated the numerical
conditions at the Vatnajökull in Iceland and
found that if the observed values of surface
slope, which gives the shear stress at the bed sb
thickness, accumulation and ablation, are used
then the second term in equation (12) accounts
for over 90% of the flow at least in the gla-
ciers. In view of the general quality of the pre-
sent treatment it therefore appears reasonable
to cancel this term and write the equation (13)
simply as follows:
±(hɱ.) + C - B = — . (14)
dx\k dx/ dt K '
which is our final equation for the thin linear
ice-sheet moving on a horizontal plane.
The last approximation actually means that
the differential flow between top and bed ac-
counts for only a small part of the total flow
and that the real ice moves practically in solid
blocks which slide on the bounclary layer at
the bed. The last approximation is, however,
only valid for ice-sheets with a thickness less
than 500 to 700 meters, somewhat depending
on other conditions. This can be inferred from
equation (12).
Nevertheless, in spite of the number of
approximations made, the equation (14) reveals
the general mathematical character of the pro-
blem and it can be used as an approximation
under a number af circumstances.
Equation (14) which applies to the linear ice-
sheet moving on a horizontal plane can easily
be extended to the case of the thin ice-sheet
moving in two dimensions on a slightly uneven
surface defined by z = f (x,y):
divgrad(h+f)j + c - B = j( , (15)
where it is assumed that the projections of the
vectors gracl h and grad f are parallel in the
(x,y) plane. 'Ihis induces certain restrictions,
although very natural, on the distribution of the
accumulation and the ablation.
Equations (14) and (15) are non-linear par-
tial parabolic differential equations but con-
siderably easier to handle than (13). The boun-
dary conditions will be discussed below.
It is of interest to note that the first term o£
equation (14) is of the same form as encountered
in the differential equation for the height of
the surface of ground water moving in porous
rock.
CHARACTER OF SOLUTIONS AND
STABILITY
In order to study the solutions of the above
equations we will turn to (14), that is, the differ-
ential equation for the linear thin ice-sheet
moving on a horizontal plane, and restrict our-
selves at first to the part below the firn line,
that is, to the ablation region.
The volume of ablation per unit time and
unit area will in practical cases for the most
be a function of the height o£ the glacier, and
we may write B — C = a(H — h), where a
is the ablation gradient and H the form of
the firn line above the base of the glacier.
Equation (14) is under these conditions:
£0B) - * § . (16)
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