Thermodynamics of Seasonal Lake Ice - Freezing of Lakes and the Evolution of their Ice Cover - page 134

Fig. 4.14 A two-phase model
grid for the freshwater lake ice
model. In each grid cell the
temperature (T) and porosity ( ν )
are determined; if temperature
is below the freezing point, the
porosity is kept zero, and at the
freezing point temperature is
fixed but porosity changes

TOP

T=0°C

>

ICE

T<0°C
V=0

B OTT OM

The model is based on the heat conduction law with solar radiation as the source term
and ice growth and melting resulting from phase changes. In general, a volume element in
ice sheet contains ice and liquid water. The differential heat content dE across a differ-
ential depth dz is

dE ¼ q L f vdz þ q c ð 1 v Þ Tdz ð 4 : 65 Þ

where is the liquid water content (essentially the porosity of ice). We may assume the

following. When the temperature is below the freezing point, the liquid water content is
zero and heat gains and losses are re

ν

ected in the temperature; and when the temperature
is at the freezing point, the liquid water content is positive and heat gains and losses are
re

fl

fl ected in the porosity. Consequently, the model physics can be presented by:

T \ T f : q c @ T

@ t ¼ q I ; m ¼ 0

ð : Þ

4 66a

T ¼ T f and m ¼ 0 : q L f d m Þ; q c @ T

dt ¼ max 0 ; q I

ð

@ t ¼ min 0 ; q I

ð Þ ð 4 : 66b Þ

T ¼ T f and m [ 0 : q L f d m

dt ¼ q I

ð 4 : 66c Þ