Jökull - 01.12.1969, Page 61

where
0 = the cubical dilation = \7 • *
é; = displacement vector
L p = Lamé’s constants
Substituting (14) into (13) results in the expres-
sion
R = h[(l + (t)V(V-ð + iiV2f] (15)
It is suggested that (15) could be more ap-
propriate than (12) for describing the internal
stresses in a solid sheet of ice. However, both
expressions assume continous sheets of ice and
are therefore not valid for low concentration
°f ice. For the present discussion Equation (15)
will be used, and the case of low ice concentra-
tion left for later study.
All forces acting on the ice have now been
defined, and the equation of motion (1) be-
comes
01h -r^- = Oa Ca Var val. + esCs v51. Vsl.
dt2 11 1
+ h <j Qi [f Vj X k + V (AD)0]
+ [(*• + [t) V (V • i) + lt V~€\ 1”
Some of the terms in this equation lrave
coefficients wiiich normally are taken as con-
stants, but which generally must be considered
to be functions of the ice concentration. The
tnternal stresses have been discussed briefly
above frorn this standpoint. It also seems prob-
able that the friction coefficients, Ca and Cs,
will both increase as ice concentration de-
creases. The equation is therefore coupled with
the ice concentration which again is coupled
with the ice movenrent as shown in the next
section.
EQUATIONS FOR ICE CONCENTRATION
The ice concentration is definecl as the frac-
tion of the sea surface covered by ice and will
here be denoted by c. Zero ice concentration
will therefore indicate ice free sea and 1.0
rneans a continuous, solid ice cover. In practice
the concentration is given as the number of
tenths of the surface area covered (e.g. c =
6/10).
Figure 3 shows a point on the sea surface,
P (x, y), surrounded by tlie surface element
ABCD of size Ax ■ Ay. The ice concentration
at P at tlie time t is c and the ice velocity is
Vj = u i + v j
where u and v are the velocity components in
the x and y directions respectively, and i and
j are the unit vectors in the same directions.
The area of the ice at the time t within tlie
element ABCD is therefore
A(t) = c -Ax • Ay (IV)
Somewhat later, at time t + At, the area is
A (t + At) = (c + — At) • Ax • Ay
= c•Ax• Ay
+ ice inflow
— ice outflow
+ ice formation
— ice melting
y (i8)
From Fig. 3 the following expression is ob-
tained
Inflowr — Outflow
= — V . (cvj) -Ax-AyAt (19)
where all higher order terms have been omitt-
ed. Formation of fresh ice or melting of ice
depends on weather conditions, and generallv
one or the other is taking place. The net in-
crease of ice area per unit sea area will be
denotecl by Q. Its value is positive for the case
JÖKULL. 19. AR 57