2.2. Ice thickness distribution¶
The Arctic and Antarctic sea ice packs are mixtures of open water, thin first-year ice, thicker multiyear ice, and thick pressure ridges. The thermodynamic and dynamic properties of the ice pack depend on how much ice lies in each thickness range. Thus the basic problem in sea ice modeling is to describe the evolution of the ice thickness distribution (ITD) in time and space.
The fundamental equation solved by CICE is [51]:
where \({\bf u}\) is the horizontal ice velocity, \(\nabla = (\frac{\partial}{\partial x}, \frac{\partial}{\partial y})\), \(f\) is the rate of thermodynamic ice growth, \(\psi\) is a ridging redistribution function, \(L\) is the lateral melt rate and \(g\) is the ice thickness distribution function. We define \(g({\bf x},h,t)\,dh\) as the fractional area covered by ice in the thickness range \((h,h+dh)\) at a given time and location. Icepack represents all of the terms in this equation except for the divergence (the first term on the right).
Equation (1) is solved by partitioning the ice pack in each grid cell into discrete thickness categories. The number of categories can be set by the user, with a default value \(N_C = 5\). (Five categories, plus open water, are generally sufficient to simulate the annual cycles of ice thickness, ice strength, and surface fluxes [5][32].) Each category \(n\) has lower thickness bound \(H_{n-1}\) and upper bound \(H_n\). The lower bound of the thinnest ice category, \(H_0\), is set to zero. The other boundaries are chosen with greater resolution for small \(h\), since the properties of the ice pack are especially sensitive to the amount of thin ice [36]. The continuous function \(g(h)\) is replaced by the discrete variable \(a_{in}\), defined as the fractional area covered by ice in the open water by \(a_{i0}\), giving \(\sum_{n=0}^{N_C} a_{in} = 1\) by definition.
Category boundaries are computed in init_itd using one of several
formulas, summarized in Table Lower boundary values.
Setting the namelist variable kcatbound equal to 0 or 1 gives lower
thickness boundaries for any number of thickness categories \(N_C\).
Table Lower boundary values shows the boundary values for \(N_C\) = 5 and linear remapping
of the ice thickness distribution. A third option specifies the boundaries
based on the World Meteorological Organization classification; the full WMO
thickness distribution is used if \(N_C\) = 7; if \(N_C\) = 5 or
6, some of the thinner categories are combined. The original formula
(kcatbound = 0) is the default. Category boundaries differ from those
shown in Table Lower boundary values for the delta-function ITD. Users may
substitute their own preferred boundaries in init_itd.
Table Lower boundary values shows lower boundary values for thickness categories, in meters, for the three distribution options (``kcatbound``) and linear remapping (``kitd`` = 1). In the WMO case, the distribution used depends on the number of categories used.
| distribution | original | round | WMO | ||
|---|---|---|---|---|---|
kcatbound |
0 | 1 | 2 | ||
| \(N_C\) | 5 | 5 | 5 | 6 | 7 |
| categories | lower bound (m) | ||||
| 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 2 | 0.64 | 0.60 | 0.30 | 0.15 | 0.10 |
| 3 | 1.39 | 1.40 | 0.70 | 0.30 | 0.15 |
| 4 | 2.47 | 2.40 | 1.20 | 0.70 | 0.30 |
| 5 | 4.57 | 3.60 | 2.00 | 1.20 | 0.70 |
| 6 | 2.00 | 1.20 | |||
| 7 | 2.00 | ||||