Thermodynamics of Seasonal Lake Ice - Freezing of Lakes and the Evolution of their Ice Cover - page 115

the ice breaks into small blocks into the surface water. Then there is a rapid increase in the
rate of melting.

flux of 35 W m − 2 , in 24 h we can decrease ice thickness by
1.0 cm or increase the mean porosity of 50 cm thick ice by 2.0 %.

Example 4.3. With the heat fl

4.3 Analytic Models

4.3.1 Basic Principles

The classical problem of modelling lake ice thermodynamics is the thickness cycle of
congelation ice. Stefan (1891) derived the first ice growth model, which was based on a

quasi-steady conduction of the latent heat released in freezing through the ice. He
examined the ice in the Arctic Ocean, but in fact his model applies better for lake ice,
since the thermal properties of sea ice vary largely depending on the brine volume.
Stefan s model was further developed by Barnes (1928) for different kind of growth

conditions in Canadian freshwater bodies. Since then these models have been applied for
the

'

first-order approximations for ice growth (e.g., Anderson 1961; Ashton 1986; Lep-

p ranta 1993).

Analytic ice growth models are mostly quasi-steady. They are based on the solution of
the steady-state heat conduction law with moving lower boundary of the ice sheet. Quasi-
steady situation means that thermal inertia is ignored, or in other words there is no heat
capacity. Also in the growth season the net solar radiation is low and either ignored or
included in the surface boundary condition. The time scale of heat

ä

fl flux in the ice is

10 L 2 m − 2 day. Thus when ice thickness is less than 20 cm, the daily
temperature cycle can be resolved with quasi-steady models. In the polar winter, forcing
changes slowly, and rather the synoptic time scale is the shortest one of interest. This can
be resolved up to the ice thickness of 50 cm.
Modelling the decay of lake ice is in principle straightforward, since in the melting
season ice loss is due to the net gain of energy from external

L 2 D − 1 , i.e. T T

* *

fl fluxes and heat conduction is

less signi cant. However, the space - time variability of the albedo and light attenuation

coef cient of melting ice arise problems. Analytic melting models are simpli ed energy

balance models, where changes in these ice properties can be parameterized.
Analytic models serve well lake ice research. They can be used to obtain the scale of
ice thickness for different lakes and environmental conditions, and as well they are very
good tools for sensitivity analysis and first-order evaluation of the climate change impact.

The models also show how shallow lakes may freeze to the bottom providing information
of the phenological event of total freezing.