Jökull - 01.12.1955, Síða 5

therefore flow at an averaged shear stress lower
than the above value of S0, and this will great-
ly increase the rate of strain in the boundary
layer.
Measurements on valley glaciers (Orvig, 1953)
have actually shown that the maximun shear
within the ice inferred from the surface slope
is of the order of 0.5 to 1.5 bar, that is, a great
deal lower than the above critical S0 = 1.6 bar.
It is on the other hand to be expected that
pressure increases the compactness and strength
of the ice in the boundary layer and thus in-
creases the critical shear stress there.
Summing up the above considerations we
can conclude that the rate of strain in the flow-
ing ice-sheet will be by far greatest in the
boundary layer, and that the ice above it will
move more like a solid. The conditions will actu-
ally be somewhat similar to the gliding of a
body on a slightly lubricated plane, and we
may therefore infer that the shear stress at the
top of the boundary layer can be expressed by
the above relation
= kvw (h — f),
where k is the coefficient of friction and (h — f)
the thickness. The dependence on the height
is underlined by the effect of pressure on the
strength of the boundary layer.
It is on the other hand to be stated that this
relation does not hold at very low or zero
velocity, but this is not harmful when we are
dealing with glaciers which actually glide on
the bed.
THE NEWTONIAN ICE
Although of theoretical interest only the
first step will be the deduction of a differential
equation for the form of an ice-sheet moving
on a horizontal plane and behaving as a New-
tonian fluid with constant viscosity.
If the vector of flow at (x, y) is denoted by F
the following equation of continuity is obtained:
(2)
where C is the volume of accumulation and B the
volume of ablation per unit area and unit
time. The figures C and B are generally func-
tions of the coordinates and of the time.
In accordance with the above approximations
(a), (b) and (c) the differential equation for the
velocity vector V can be written::
ju - grodp = wgrac/h
(3)
where p is the coefficient of viscosity for the
ice. This equation has to be integrated with
the boundary conditions dV/dz = 0 at z = h,
and kVowh = — wh (gracl h) at the bed. The
solution is therefore:

if w is assumed constant, ancl this gives the flow:
r ’ jn*=-(j£J(5)
Inserting F from (5) into (2) we obtain the
final approximate differential equation for the
thin Newtonian ice-sheet with constant viscosity
moving on a horizontal plane:
■*'((£'* tír"")* c-‘-$ <6)
which in the one-dimensional case becomes:
It is to be underlined that the equations (6)
and (7) are derived by the approximations (a)
to (c), constant specific weight, and furthermore
that the specific accumulation and ablation C
and B are measured in volumes. Nevertheless,
these are non-linear partial differential equa-
tions which are quite difficult to solve
numerically.
THE REAL ICE
In the case of real ice the equation (1) has
to be applied instead of the constant viscosity
relation above. The treatment here will for
simplicity at first be restricted to the one-
dimensional ice-sheet moving on a horizontal
plane as this furnishes all the essentials needed
for the final result.
By the approximation (c) the equation (1)
gives
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