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# 6.2 Fundamental Equations

The method for computing the surface turbulent heat and radiative exchange, evaporative flux, and surface drag is integrally coupled with the formulation of heat transfer through the sea ice and snow. The equation governing vertical heat transfer in the ice and snow, which allows for internal absorption of penetrating solar radiation, is

 (6.1)

where is the density, is the heat capacity, is the temperature, is the thermal conductivity, is shortwave radiative heating, is the vertical coordinate, and is time. Note that , , and differ for snow and sea ice, and also the latter two depend on temperature and salinity within the sea ice to account for the behavior of brine pockets.

The boundary condition for the heat equation at the surface is

 (6.2)

where is the surface temperature, is the absorbed shortwave flux, is the shortwave flux that penetrates into the ice interior, is the net longwave flux, is the sensible heat flux, and is the latent heat flux. All fluxes are taken as positive down. If , then the surface is assumed to be melting and a temperature boundary conditions (i.e., ) is used for the upper boundary with Eq. 6.1. However if in Eq. 6.2, then the surface is assumed to be freezing and a flux boundary condition is used for Eq. 6.1, and Eqs. 6.1 and 6.2 are solved simultaneously with in the latter.

Snow melt and accumulation is computed from

 (6.3)

where is the snow depth, is the snow density, and are the latent heats of fusion and vaporization, and is the snowfall rate (see Table 6.1 for values of constants).

When CAM 3.0 is coupled to the mixed layer ocean and the sea ice is snow-free, sea ice surface melt is computed from

 (6.4)

where is the ice thickness, is the ice density, and is the energy of melting of sea ice ( by definition, see section 6.6 on brine pockets). Basal growth or melt is computed from

 (6.5)

where is the heat flux from the ocean to the ice (see section 6.5). Finally an equation is needed to describe the evolution of the ice concentration :

 (6.6)

where accounts for new ice formation over open water and lateral melt (see section 6.7)

Parameterizations of albedo, surface fluxes, brine pockets, and shortwave radiative transfer within the sea ice are given next. Finally, the numerical solution to Eq. 6.1 is described. Numerical methods for Eqs. 6.2 -6.6 are straight-forward and hence are not described here.

 Symbol Description Value Density of snow 330 kg m Density of ice 917 kg m Density of surface ocean water 1026 kg m Specific heat of atmosphere dry 1005 J kg K Specific heat of atmosphere water 1810 J kg K Specific heat of ocean water 3996 J kg K Specific heat of snow 0 J kg K Specific heat of fresh ice 2054 J kg K Aerodynamic roughness of ice 5.0x10 m Reference height for bulk fluxes 10 m saturation specific humidity constant 11637800 saturation specific humidity constant 5897.8 Thermal conductivity of snow 0.31 W m K Thermal conductivity of fresh ice 2.0340 W m K Thermal conductivity ice constant 0.1172 W m ppt Latent heat of fusion of ice 3.340x10 J kg Latent heat of vaporization 2.501x10 J kg Melting temperature of top surface 0 C Ocean freezing temperature constant 0.054 C ppt Stefan-Boltzmann constant 5.67x10 W m K Ice emissivity 0.95 Ice SW visible extinction coefficient 1.4 m Ice SW near-ir extinction coefficient 17.6 m

NOTE: CSIM in CAM 3.0 uses the shared constants defined in Appendix A.

Next: 6.3 Snow and Ice Up: 6. Sea Ice Thermodynamics Previous: 6.1 Basic assumptions   Contents
Jim McCaa 2004-06-22