Mechanics of Lake Ice - Freezing of Lakes and the Evolution of their Ice Cover - page 164

Wind

Air stress

Ice velocity

T a

Ice stress

σ

T w

Coriolis
Water stress

Internal friction

Fig. 5.13 Lake ice drift and the physical problem. Ice is driven by forcing from the atmosphere and
lake water body, and the ice responds to forcing through its internal stress field. A typical force
diagram is also illustrated

X

q D u

Dt þ 2 X u

¼ rr þ F k ð 5 : 41 Þ

k

= 7.292 × 10 - 5 s - 1 ) is the angular

ʩ ʩ

where D/Dt is the material time derivative, (

velocity of the Earth, and F k ' s are the external forces. In the integration, the inertia,

advection and Coriolis acceleration are just multiplied by ice thickness. Vertical com-
ponent results in the Archimedes law, i.e. how the ice floats, while the horizontal com-

ponent gives the motion on the lake surface plane. The horizontal component of Coriolis
acceleration is further simpli

ed to hfk× u , where f =2 ʩ sin ˕ is the Coriolis parameter

is the latitude. 6 Integration of the internal ice stress gives the internal friction and
wind and water stresses at the top and bottom boundaries. If the lake water surface is
tilting, a pressure gradient force results from the hydrostatic water pressure on ice

and ˕

oes.
Finally, the general form of the equation of motion of ice on the lake surface plane is
(Fig. 5.13 ):

q h o u

o t þ u ru þ f k u ¼ rr þ s a þ s w q gh rn

ð 5 : 42 Þ

In very large lakes the Coriolis acceleration becomes signi cant in ice drift (the der-

ivation is shown, e.g., in Cushman-Roisin 1994).
Figure 5.13 also shows a schematic force diagram of ice drift. Usually, wind is the
driving force, balanced by the ice

water drag and the internal friction of the ice. Coriolis

-

6 Latitude is taken positive in the northern hemisphere and negative in the southern hemisphere.