Mechanics of Lake Ice - Freezing of Lakes and the Evolution of their Ice Cover - page 173

the compressive strength of ice. The former tune the free drift velocity, while the com-
pressive strength tunes the length scale and the mobility in the presence of internal
friction. There are also other rheology parameters, for the dependence of the ice strength
on the ice state and for the shear strength. When the plastic flow is approximated by an

elastic-plastic (Coon et al. 1974) or a viscous-plastic (Hibler 1979) rheology, there is an
additional rheological formulation for the stresses below the yield level. The numerical
design includes the choice of the grid and, since the system is highly non-linear, the
stability of the solution may require smoothing techniques. Since the minimum continuum
particle size D is fairly large, the grid size can be taken as ʔx * D . The grid size can be

down to less than 1 km in basins with small floe size and in the melting season when oes

break into smaller pieces.
As an example, the numerical values of the Hibler (1979) sea ice model parameters are
shown in Table 5.4 based on field data and model experiments in lakes. Almost twice as

large air and water drag coef cients were used in the Lake Erie model (Wang et al. 2010)

than in Lake Peipsi model (Wang et al. 2006). The free drift wind factor has been about
2 %, as supported by observations. In the Lake Erie model, the standard plastic strength
constant was used (compressive strength of ice of unit thickness equal to 25 kPa). The ice
was quite thin, less than 10 cm, and the standard strength may have resulted in a too stiff
ice cover. The transition between viscous and plastic regimes must be very low to avoid
production of unrealistic creep of ice; here has been taken as 10 − 10 s − 1 , much lower than
strain-rates in major deformation events.

Table 5.4 The parameterisation of the viscous-plastic lake ice models
Parameter Value Comments

1.4 - 2.3 × 10 − 3 I Air - ice drag coefficient, C a Surface wind

1 - 2 × 10 − 3 Ice - water drag coefficient, C w Shirasawa et al. (2006)

Ice - water Ekman angle, ʸ w

[0, 25°] Depth dependent

Compressive strength P b h b II 25 kPa At h =1m,½ ≤ b ≤ 2

Shear strength 12.5 kPa Half of the compressive strength

Change of strength by e − 1 due to e-folding scale for opening 0.05 ʔ A

Maximum viscous creep rate, ʔ o <10 − 10 s − 1 Viscous - plastic transition
III Demarcation thickness, h o 5 - 10 cm Open water — ice cover transition
IV Spatial grid size, ʔ x = ʔ y ʔ x ≫ d 1 - 5 km, much more than floe size d
Time step, ʔ t 3h - 1 d Stability requirement
Smoothing Diffusion Harmonic and bi-harmonic
The parameter groups are I atmospheric and oceanic drag parameters, II rheology parameters
(Hibler ' s 1979 viscous - plastic rheology, III ice state redistribution parameters, and IV numerical
design parameters