Jökull - 01.12.1955, Side 4

cussion of an approximate treatment o£ the
general character of the flow of ice-sheets by the
means of equations derived by the assumption
of special boundary conditions at the bed. A
short discussion of the stability and response of
ice-sheets to climatic variations is included.
The following terminology will be used. The
height of an ice-sheet moving on a horizontal
plane (x, y) will be clenoted by h, which is a
function of the coordinates in the plane and
of the time t. If the plane is not horizontal
but characterizecl by the equation z = f (x, y)
the actual thickness of the ice-sheet will be
(h - f (x, y)).
In accordance with actual conditions in nat-
ure only thin ice-sheets will be treated, that is,
ice-sheets where the thickness is very small com-
pared to the other dimensions. The discussion is
also restricted to almost level beds.
This admits the following basic approxim-
ations, (a) that the direction of flow within the
ice can be assumed independent of z, which
means that the direction of flow above each
(x, y) is constant from z = 0 or z = f (x, y) to
z = h, and (b) that the flow lines are parallel
to the bed, and finally (c) because of the very
slow movement of the quasi-viscous ice that a
static distribution of pressure prevails, that is,
the pressure at the point z is w (h — z), where
w is the specific weight of the ice. The pressure
on the bed is consequently w (h — f (x, y)), or wli
in the case of the horizontal bed.
CONDITIONS AT THE BED
If the friction on the bed is an ordinary dry
sliding friction the Coulomb friction law gives
the shear stress on the bed s = cw (h — f), where
c is the coefficient of friction which would main-
ly depend on the character of the bed, but be
approximately independent of the sliding ve-
locity.
It is, however, known that ice melts under
pressure when the temperature is near to zero
C° and the friction on the bed can therefore not
be expected to behave according to the Cou-
lomb law. It is rather to be expected that the
coefficient of friction will also depend on the
sliding velocity as in the case of viscous friction.
Consequently we may expect that the friction
on the bed can be approximated by
s^ = kvw (h — f),
where v is the sliding velocity, h the thickness
and k a new factor, probably more or less con-
stant. If v is independent of z the above rela-
tion can be written
s, = kwF = kG,
1)
where F is the volume of flow per unit length
and G tlie weight of flow per unit length.
There are further reasons for assuming this
relation. According to Glen (1952) the rheo-
logical character of ice is expressed by a curve o£
the form A in Figure (1) showing the relation
of shear stress to the rate of strain dy/dt. The
curve B represents the Newtonian fluid with
linear behaviour, and C is the plastic body.
Fig.l. Stress-strain relation for fluids and plastic
bodies.
Glen (1952) gives for ice the relation:
where the rate o£ strain is expressed in units per
year and s and S0 in bar. For the temperature
— 1.5 °C the constants are S0 = 1.62 and n = 4.1.
According to this the behaviour of ice resembles
that o£ plastic bodies, that is, when the shear
stress is below a critical value, approximately
So, then there is very little flow which, however,
increases very much when the shear stress sur-
passes this value.
In flowing ice-sheets the greatest shear stress
is at the bed and the rate of strain is, therefore,
greatest there.
The ground moraine which the ice-sheet
transports along the bed consists of pieces of
rock which penetrate into the ice just above
the bed. This forms a shallow boundary layer
where the ice is broken by the inclusion of
rock fragments ancl its strength consequently
reduced. The ice in the boundary layer will