Thermodynamics of Seasonal Lake Ice - Freezing of Lakes and the Evolution of their Ice Cover - page 121

thickness evolution. Since the flux from the water is fairly small in lakes, the equilibrium

thickness is not reached but the growth rate of ice is reduced. However, in geothermal
lakes, equilibrium thickness can be reached fast.

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4.3.3 Snow-Ice

Snow-ice grows down from the top of the slush, and the growth model is similar to the
case of snow-covered congelation ice (Eq. 4.42 ) (Lepp

ranta 1993). The only change in
the heat conduction equation is that, since snow-ice grows into slush, in the moving lower
boundary condition the latent heat of freezing is multiplied by the porosity of slush. The
analytic model is thus

ä

q L f m dh si

dt ¼ k T f T s

¼ k T s T 0
h s

¼ k 0 þ k 1 T a T 0 ½ ð Þ ð 4 : 48a Þ

0
si

h

t ¼ 0 : h si ¼ h si ð 0 Þ ð 4 : 48b Þ

0
si

where is the thickness of snow-ice above the slush, and the time t refers to the

beginning of the freezing of the present slush layer. With the initial condition ( 4.48b ), the
new growth from Eq. ( 4.48a ) is added to the existing older snow-ice deeper down. Snow-
ice layer is not necessarily continuous but there is a slush layer between old snow-ice and
new snow-ice. After a slush formation event, h s = constant, and the thickness of snow-ice
is

h

q

m 1 a 2 S a ðÞ b 0 t Þ 2 h si ðÞ ¼h si ðÞþ ½ þ b h s þ b 1 ð b h s þ b 1 ð Þ ð 4 : 49 Þ

ν − 1/2 ʲ

h s ≫

Here normally b 1 . Snow-ice layer grows faster by the factor of * 3/2 as

compared with congelation ice.
The upper limit of snow-ice thickness is obtained assuming that new slush forms and
freezes continuously. Then, if the slush is created by fl flooding, we have h s = ʳ h si , where

ʳ =( ˁ w − ˁ si )/ ˁ s , and, consequently:

dh si
dt ¼

k
q w L f k cm T f T s

k s

¼ k 0 þ k 1 T a T s ½ ð Þ ð 4 : 50 Þ

h s

This is similar to the growth law of congelation ice, except that the Stefan ' is coef cient

p

k s =ð k mcÞ

ν * ½ ʳ *

is multiplied by the factor of ; since k s /k * 0.1, , and 1/3 (Lep-

p ä ranta and Kosloff 2000), this factor is 0.8. Thus, forming snow-ice with continuous

*

snowfall and consequent flooding produces less ice than the growing snow-free conge-

lation ice. This is because to produce slush with growing snow-ice, we need more snow
freeboard that adds to insulation.

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