Thermodynamics of Seasonal Lake Ice - Freezing of Lakes and the Evolution of their Ice Cover - page 109

d q s
dt ¼ C q s ; T a ; U a

Þ; q s ¼ q s0

ð ð 4 : 28a Þ

refers to new snow. The density of new snow is normally around 100 kg m − 3 ,
while strong packing may lead to the level of 400 kg m − 3 for old seasonal snow.
The temperature structure of the water column beneath the ice develops in the cooling
season. The mixing conditions determine the deep-water temperature and the depth of the
winter thermocline. In fresh and brackish waters, inverse strati

where q s

cation forms after the
surface temperature has passed the temperature of maximum density. But long-lasting,
strong mixing cools the deep water, in freshwater lakes down to 1

C, and deepens the
thermocline. After the ice cover has formed, the circulation slows down, heat exchange
through the surface weakens remarkably, and the heat

2 °

-

fl flux from the lake bottom becomes

signi flux from water body to the ice bottom has been estimated as less

than 5 W m − 2 (e.g., Shirasawa et al. 2006; Kirillin et al. 2012; Yang et al. 2012).
Beneath a stable ice cover, heat

cant. The heat fl

fl flux from water to ice retards the growth rate or melts

ice. This heat fl flux is expressed as

z¼h

Q w ¼ k w @ T
@ z

ð 4 : 28b Þ

where k w is the heat conductivity in the water beneath the ice. In the case of laminar fl ow,

the molecular heat conductivity k w = 0.6 W m − 1 C − 1

° is employed. Then for ∂ T/

Wm − 2 . In turbulent conditions, the conductivity (due to
eddy diffusion) is much larger, k w *

Cm − 1 , we have Q w * ½ ∂ z 1 °

*

10 4 Wm − 1 C − 1 (e.g., McPhee 2008). However, the °

water flow beneath a stable lake ice cover is often neither laminar nor turbulent but rather in

the laminar

fl

turbulent transition regime, where the value of conductivity is not well
established. Petrov et al. (2006) employed an effective conductivity o f k w =20Wm − 1

-

C − 1

°

in a medium-size, shallow lake in Russian Karelia.
In the presence of turbulent boundary layer under the ice, the exchange of momentum,
heat and salts can be examined with the boundary layer theory similar to the atmospheric
case (see Sect. 4.1 ). The heat transfer is then approximated by the bulk formula:

U w z ðÞ

Q w ¼ q w c w C Hw z ðÞ T w z ðÞ T f

ð : Þ

4 29

where z = z − h is the distance from ice bottom, C Hw is the ice water heat exchange

′ -

coef cient, T w is the water temperature, and U w is the current speed (relative to the ice).

This approach approximates the pro

le formula with effective heat conductivity repre-

10 − 3 ,

sented by ˁ w c w C Hw z U w . Lake P ää j ä rvi station data gave the estimate of C Hw = 0.4 ×

′

corresponding to k w =16.8Wm − 1 C − 1 for z = 1 m and U w = 1 cm s − 1 , and the heat ° fl ux

5Wm − 2 (Shirasawa et al. 2006).
The salinity balance in the boundary layer beneath the ice is important under a stable
ice cover even in freshwater lakes. Growing ice rejects most of the impurities, but a small

magnitude of 2

-