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

$$\rho c { \partial T \over \partial t} = \left( { \partial \over \partial z} k { \partial T \over \partial z} + Q_{SW} \right)$$ (6.1)

where $\rho$ is the density, $c$ is the heat capacity, $T$ is the temperature, $k$ is the thermal conductivity, $Q_{SW}$ is shortwave radiative heating, $z$ is the vertical coordinate, and $t$ is time. Note that $\rho$, $c$, and $k$ 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

$$F_{TOP}(T_s) = F_{SW}-I_{SW} + F_{LW} + F_{SH} + F_{LH} + k {dT \over dz}$$ (6.2)

where $T_s$ is the surface temperature, $F_{SW}$ is the absorbed shortwave flux, $I_{SW}$ is the shortwave flux that penetrates into the ice interior, $F_{LW}$ is the net longwave flux, $F_{SH}$ is the sensible heat flux, and $F_{LH}$ is the latent heat flux. All fluxes are taken as positive down. If $F_{TOP}(T_s=0) \ge 0$, then the surface is assumed to be melting and a temperature boundary conditions (i.e., $T_s=0$) is used for the upper boundary with Eq. 6.1. However if $F_{TOP}(T_s=0) < 0$ 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 $F_{TOP}(T_s) = 0$ in the latter.

Snow melt and accumulation is computed from

$$\rho_s {dh_s \over dt} = {-F_{TOP} \over L_i} + {F_{LH} \over L_i+L_v} + F_{SNW}$$ (6.3)

where $h_s$ is the snow depth, $\rho_s$ is the snow density, $L_i$ and $L_v$ are the latent heats of fusion and vaporization, and $F_{SNW}$ 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

$${dh_i \over dt} = {F_{TOP} \over q} + {F_{LH} \over - q+\rho_i L_v}$$ (6.4)

where $h_i$ is the ice thickness, $\rho_i$ is the ice density, and $q$ is the energy of melting of sea ice ($q<0$ by definition, see section 6.6 on brine pockets). Basal growth or melt is computed from

$${dh_i \over dt} = {F_{BOT} \over q} - {k \over q}{dT \over dz}$$ (6.5)

where $F_{BOT}$ 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 $A$:

$${dA \over dt} = {\cal A}$$ (6.6)

where $\cal A$ 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.

Table 6.1: List of Physical Constants

Symbol	Description	Value
$\rho_s$	Density of snow	330 kg m$^{-3}$
$\rho_i$	Density of ice	917 kg m$^{-3}$
$\rho_o$	Density of surface ocean water	1026 kg m$^{-3}$
$C_p$	Specific heat of atmosphere dry	1005 J kg$^{-1}$ K$^{-1}$
$C_{pwv}$	Specific heat of atmosphere water	1810 J kg$^{-1}$ K$^{-1}$
$C_o$	Specific heat of ocean water	3996 J kg$^{-1}$ K$^{-1}$
$c_s$	Specific heat of snow	0 J kg$^{-1}$ K$^{-1}$
$c_o$	Specific heat of fresh ice	2054 J kg$^{-1}$ K$^{-1}$
$z_i$	Aerodynamic roughness of ice	5.0x10$^{-4}$ m
$z_{ref}$	Reference height for bulk fluxes	10 m
$q_1(ice)$	saturation specific humidity constant	11637800
$q_2(ice)$	saturation specific humidity constant	5897.8
$k_{s}$	Thermal conductivity of snow	0.31 W m$^{-1}$ K$^{-1}$
$k_{o}$	Thermal conductivity of fresh ice	2.0340 W m$^{-1}$ K$^{-1}$
$\beta$	Thermal conductivity ice constant	0.1172 W m$^{-1}$ ppt$^{-1}$
$L_{i}$	Latent heat of fusion of ice	3.340x10$^5$ J kg$^{-1}$
$L_{v}$	Latent heat of vaporization	2.501x10$^6$ J kg$^{-1}$
$T_{melt}$	Melting temperature of top surface	0 $^\circ$C
$\mu$	Ocean freezing temperature constant	0.054 $^\circ$C ppt$^{-1}$
$\sigma_{sb}$	Stefan-Boltzmann constant	5.67x10$^{-8}$ W m$^{-2}$ K$^{-4}$
$\varepsilon$	Ice emissivity	0.95
$\kappa_{vs}$	Ice SW visible extinction coefficient	1.4 m$^{-1}$
$\kappa_{ni}$	Ice SW near-ir extinction coefficient	17.6 m$^{-1}$

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