Mechanics of Lake Ice - Freezing of Lakes and the Evolution of their Ice Cover - page 163

surface. The forcing is resisted by the strength of the ice as speci ed by the rheology

model. The external forces are in general given by

s a ¼ q a C a U a U a ð 5 : 40a Þ

s w ¼ q w C w U w u j j ½ cos h w ðU w uÞþ sin h w k ðU w uÞ ð 5 : 40b Þ

G ¼ q gh rn ð 5 : 40c Þ

where C a and C w are air and water drag coef ʸ w is the boundary layer angle in

water, U a and U w are the surface wind and current velocities, and

cients,

is the water level
elevation. The reference depth of the current velocity can be taken at the vertical mid-level
or bottom (zero current) in shallow lakes, while in deep lakes a convenient depth is the
bottom of the layer of frictional

ʾ

in uence of the ice. Representative values of the

C a = 1.4 × 10 − 3 for the wind speed at 10-m altitude (Andreas 1998),
C w =1×10 − 3 for shallow reference depth (Shirasawa et al. 2006), and

parameters are

ʸ w → 0 in shallow

lakes, ʸ w → - 25° (northern hemisphere) or ʸ w → 25° (southern hemisphere) in deep

lakes. In a strati ed fluid the drag parameters depend also on the stability of the strati-

fication (see Sect. 4.1 ).
When the forcing is strong enough, the ice fails and the

flow goes into the plastic

0.5 N m − 2 for a wind speed of 15 m s − 1 over a lake ice

regime. The wind stress is ˄ a *

5kNm − 1 at
the windward side of the lake, enough to break 10-cm thick ice by compression. This is
the correct magnitude according to observations in ice-covered lakes. Since water cir-
culation is weak and has changing direction under the ice, the integrated forcing by the
ice

cover. If the fetch L becomes 10 km, the stress integrates to a force of ˄ a L *

water stress never gets close to the wind forcing. On a sloping lake surface, gravity
produces a force of

-

ˁ i ghʔ ξ ʔ ξ

is the water level difference across the lake, at the
shallow end. Thus to obtain lake surface slope more than 10 − 4 would be needed to break
the ice in a 10-km size lake but such slope is most unlikely to occur.

, where

5.5.2 Equations of Drift Ice Mechanics

Even very large lakes are small enough to allow a local Cartesian co-ordinate system. The
liquid water surface serves as the zero reference level. A drift ice

field is described with

the fields of ice thickness h = h(x, y; t) and velocity u = u (x, y; t), u = u i + v j .A ' point ' in

the drift ice finite cell, where the velocity is represented by the average value

while the drift ice state is represented by the thickness distribution of ice (Thorndike et al.
1975).
Displacements in

field is a

floating lake ice are two-dimensional, horizontal. The equation of
motion can be derived by integrating,

the thickness of the ice, the three-dimen-
sional continuum law the (see Leppäranta 2011):

across