Thermodynamics of Seasonal Lake Ice - Freezing of Lakes and the Evolution of their Ice Cover - page 126

temperature change during 200 days. Thus we need a large change in the temperature or
the winter length to get a signi

cant reduction of the equilibrium thickness.

4.4 Numerical Models

4.4.1 Structure of Models

The analytic modelling approach to the growth and decay of lake ice has limitations. First
of all, the snow cover cannot be properly considered. The insulating capacity of snow
depends on snow accumulation and metamorphosis, which may also lead to slush and
snow-ice formation. The resulting lake ice has then a multi-layer structure. Secondly,
analytic models are coarse. High time-space resolution of temperature would give a more
realistic heat conduction through ice and snow, and with more accurate surface temper-
ature, the surface heat balance comes out much better (e.g., Cheng et al. 2003; Yang et al.
2012).
Numerical lake ice models are of two basic types: (1) Quasi-steady models, where the
ice sheet is divided into layers with a quasi-steady heat ow through the system; and (2)

Time-dependent models, where the vertical equation of heat conduction is directly inte-
grated (Eqs. 4.24 , 4.25a , b , c ). In both cases the boundary conditions are properly treated.
The present lake ice models are largely based on the sea ice models of Semtner (1976) and
Maykut and Untersteiner (1971) for congelation ice representing the basic types (1) and
(2), respectively. Later a snow-ice layer was added for subarctic seas (Saloranta 2000;
Shirasawa et al. 2005). Turbulent air - ice heat transfer based on the Monin - Obukhov
similarity theory was considered in Cheng (2002) and Cheng et al. 2003). Recent
approach is a two-phase model, which includes porosity of ice for each grid cell and
produces a realistic solution to ice deterioration in the melting season (Lepp

fl

ranta 2009b).
Here we focus on the freshwater bodies. Saline lake ice models are similar apart from
the salinity effects, which can be taken from the existing sea ice models. Since the length
scale of heat diffusion in ice is less than 10 m in a year, lateral heat conduction is generally
not an important issue, and numerical lake ice models are kept vertical. For freshwater
lake ice, the thermal properties

ä

— density, speci c heat and thermal conductivity — can be

taken as constants for congelation ice and snow-ice layers. Optical properties — albedo and

diffuse attenuation coef cient — however, undergo strong evolution in the melting season.

first numerical models were based on the quasi-steady heat conduction law with
time-dependent snow accumulation and heat

The

ranta (1983)
employed a quasi-steady model to examine snow-ice formation and sensitivity of ice
thickness and stratigraphy to thermal properties of snow. Later the quasi-steady approach
was taken by Liston and Hall (1995) and Duguay et al. (2003).
Time-dependent models are a recent effort in lake ice modelling (Lepp

fl flux from the water. Lepp ä

ranta and
Uusikivi 2002; Yang et al. 2012, 2013), adapted from existing sea ice models (Maykut
and Untersteiner 1971; Flato and Brown 1996; Saloranta 2000; Cheng et al. 2002, 2003).

ä