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Subsections

# 6.9 Numerics

The heat content change within the sea ice over the time interval to corresponding to temperatures and , respectively, allowing for temperature dependent heat capacity, thermal conduction (see section 6.6) and internal absorption of penetrating solar radiation, is given by:

 (6.52)

The heat equation is discretized using a backwards-Euler, space-centered scheme. Using a staggered grid with representing the layer temperature and representing conductivity at the layer interfaces, for interior layers we have

 (6.53)

where , the conductivity is

 (6.54)

and the absorbed solar radiation is

 (6.55)

See Figure 6.1 for a diagram on the vertical level structure.

For a purely implicit backward scheme, should be evaluated at the time level. However, when is evaluated at time level , experiments show that the solution is stable and converges to the same solution one gets when evaluating at .

The discrete heat equation for the surface layers is modified slightly from Eq. 6.53 to maintain second-order accuracy for . The equation for the bottom layer () is

 (6.56)

where the interface in contact with the underlying ocean is assumed to be at temperature C, and where the conductivity is simply . The equations for the top surface depend on the surface conditions, of which there are four possibilities, as outlined in Table 6.2.
 snow accumulated melting case I yes no case II no no case III yes yes case IV no yes

## 6.9.1 Case I: Snow accumulated with no melting

The discrete heat equation for the uppermost layer (i.e, the snow layer) is

 (6.57)

The heat equation solver is formulated for the general case where the heat capacity of snow may be specified, although it is taken to be 0. The parameters and are defined to give second-order accurate spatial differencing for across the changing layer spacing at the snow/ice boundary;

 (6.58)

The conductivity at the snow-ice interface is found by equating conductive fluxes above and below the interface;

 (6.59)

Because is below melting, a flux boundary condition is used, and an additional equation is required in the coupled set:

 (6.60)

where is the sum of all terms on the right-hand side of Eq. 6.2 except . The net surface flux is approximated as linear in ; thus

 (6.61)

with

 (6.62)

To simplify our set of equations, we define

 (6.63)

where the hat implies that depends on as well as on , and

 (6.64)

Also, let

 (6.65) for and (6.66) (6.67)

and suppress the index for , so that for interior layers (),

 (6.68) and at the bottom layer (6.69)

where . The equation describing the snow layer is written

 (6.70)

Finally, the flux boundary condition becomes

 (6.71)

The complete set of coupled equations for case I can be written with all of the terms that explicitly depend on temperature at the time step gathered on the right-hand side:

 (6.72)

These equations are subsequently related to the following abbreviated form

 (6.73)

The first two rows can be combined to eliminate the coefficient on in the first row, allowing the set to be written in tridiagonal form:

 (6.74)

Because the matrix A depends on , which in turn depends on , the system of equations is solved iteratively. An initial guess is used for the temperature dependence of , and then is updated successively after each iteration. Under most conditions the method approaches a solution in less than four iterations with a maximum error tolerance of for with an initial guess of .

## 6.9.2 Case II: Snow free with no melting

Nearly the same method applies when the ice is snow free, except one less equation is needed to describe the evolution of the temperature profile. The equation for the uppermost ice layer is written

 (6.75)

where . After the definitions from Eqs. 6.63-6.65 are applied, Eq. 6.75 becomes

 (6.76)

The flux boundary condition follows after linearizing in :

 (6.77)

The complete set of coupled equation includes Eqs. 6.72 for layers 2 to L with the following two equations for the surface and upper ice layer:

 (6.78)

which can be written

 (6.79)

These two equations can be combined to eliminate the coefficient on , allowing the set to be written in tridiagonal form:

 (6.80)

As for case I, this system of equations must be solved iteratively.

## 6.9.3 Case III: Snow accumulated with melting

Case III describes melting conditions in the presence of a snow layer at the surface. Here a temperature boundary condition is used, which simplifies the solution because the first row in Eqs. 6.72 is not needed and C in the second row. Hence the complete set of coupled equations is identical to Eqs. 6.72 for layers 1 to L, with the addition of an equation for the snow layer,

 (6.81)

This set of equations can be written in tridiagonal form, without the need to eliminate any terms, as was required in cases I and II. However, the solution must still be iterated.

## 6.9.4 Case IV: No snow with melting

Like case III, case IV describes melting conditions, but here the sea ice is snow free. Hence, the first two rows of Eqs. 6.72 are not needed, and for . The set of coupled equations comprises those from Eqs. 6.72 for layers 2 to L and the following equation for layer 1:

 (6.82)

As in case III, this set of equations can immediately be written in the tridiagonal form and solved iteratively.

## 6.9.5 Temperature Adjustment Due to Melt/Growth

The energy of melting of the ice and snow layers needs to be adjusted when the layer spacing changes after growth/melt, evaporation/sublimation, and flooding (see Figure 6.2). This calculation is only made when CAM 3.0 is coupled to a mixed layer ocean. The adjusted energy of melting is

 (6.83)

where are weights computed from the relative overlap of layer with each layer from the old layer spacing and is the new layer spacing.

Next: 7. Initial and Boundary Up: 6. Sea Ice Thermodynamics Previous: 6.8 Snow-Ice Conversion   Contents
Jim McCaa 2004-06-22