Jökull - 01.12.1969, Side 60

Fig. 2. Internal stresses in an element of the
ice sheet.
in tlie ocean, for which the velocity is given
by the expression
vg= -~V (AD)0xk (9)
where (AD)0 is the geopotential anomaly at
the sea surface and V denotes the two-dimen-
sional gradient operator
V =
+
3
5y
j
(10)
For gradient flow the pressure gradient force
is equal ancl opposite to the Coriolis force.
Since the velocity (9) is independent of densitv
it is seen that the equation applies equally
well to the slightly lighter layer of ice floating
on the surface. Using (8) with the velocity
given by (9) gives then
G = Ojh V (AD)0 (11)
In order to find G the geopotential anomaly
must be known under the ice and tliis is not
easily found. This will be discussed briefly
later.
The force R resulting from internal stresses
in the ice, can have considerable influence on
the ice movement. In their study of the drift
of the ice island “Alpha”, Reed and Campbell
(1963) concluded that currents and internal
stresses were the main source of discrepancy
between the actual path of the island and the
56 JÖKULL 19. ÁR
computed path based on wind stresses only.
Nansen (1902) noted that these forces might
be of importance and Sverdrup (1928) attempt-
ed to take them into account in liis calcula-
tions. He assumed that internal stresses resnlt-
ed in a force acting opposite to the direction
of the ice movement ancl proportional to the
velocity. Reed and Campbell (1962) questioned
the validity of this assumption, since the re-
sulting force R acted only to retard the motion,
whereas both accelerations and decelerations
are to be expected from the momentum ex-
changes in the moving ice layer. They suggest-
ed viewing the ice as a film of highly viscous
fluid suspended between two less viscous fluids,
air and water. Using the viscous terms in the
Navier-Stokes equation they assumed the in-
ternal stresses on the form
R = O-.hEi V' 2Vi
(12)
where V2 is the twodimensional Laplace opera-
tor ancl Ej is the horizontal kinematic eddv
viscosity coefficient for the ice. This equation
states that regions of deficient momentum will
be accelerated and areas of excessive moment-
um are decelerated. On the other hand the
form of the equation assumes a constant value
of £j in all directions which is open to ques-
tion since momentum transfer by shear stresses
and normal stresses may not be the same.
However, this concept is probably an improve-
ment over Sverdrup’s model with friction a
linear mcdel of velocity.
Using the notation in Fig. 2 the components
of the internal stress force are given by the
general expressjons
Rx
hAxAy
3gxx
3x
3°yx
3y
R.v
hAxAy
3+y
3x
+
(13)
If the ice is assumed to be an isotropic elastic
solid the stresses are given by
3?.
a^kD + ^ —
Pi + 3l\
L 3xj T 3xj J
(14)