6.6 Brine Pockets and Internal Energy of Sea Ice

Shortwave radiative heating within the sea ice and conduction warms the sea ice and opens brine pockets, melting the ice internally and storing latent heat. This storage of latent heat is accounted for explicitly by using a heat capacity and thermal conductivity that depend on temperature and salinity following the work of Maykut and Untersteiner [125] and Bitz and Lipscomb [20]. The equation for the heat capacity for sea ice $c$ was first postulated by Untersteiner [176] and then later derived from first principles by Ono [132]:

$$c(T,S) = c_o + \frac{L_i \mu S}{T^2},$$ (6.40)

where $c_o$ is the heat capacity for fresh ice, $S$ is the sea ice salinity, $T$ is the temperature, and $\mu$ is an empirical constant relating the freezing temperature of sea water linearly to its salinity ( $T_f=-\mu S$).

Equation 6.40 can be multiplied by the sea ice density and integrated to give the amount of energy $Q$ required to raise the temperature of a unit volume of sea ice from $T$ to $T'$:

$$Q(S,T,T')=\rho_i c_o(T'-T)-\rho_i L_o\mu S \left({1 \over T'} - {1 \over T}\right).$$ (6.41)

If we take $T'$ to be the melting temperature of ice with salinity $S$, then at $T'$ sea ice consists entirely of brine; that is, the brine pockets have grown to encompass the entire mass of ice. The amount of energy needed to melt a unit volume of sea ice of salinity $S$ at temperature $T$, resulting in meltwater at $T_f$, is equal to

$$q(S,T)=\rho_i c_o(-\mu S-T)+\rho_i L_o \left(1+{\mu S \over T}\right).$$ (6.42)

$q$ is referred to as the energy of melting of sea ice, and it appears in Eqs. 6.4 and 6.5.

The thermal conductivity for sea ice $k$ is

$$k(S,T) = k_o + \frac {\beta S} {T}$$ (6.43)

where $k_o$ and $\beta$ are empirical constants from Untersteiner [176].

The vertical salinity profile is prescribed based on the work of Maykut and Untersteiner [125] to be

$$S(w) = 1.6 \left[ 1 - \cos \left( \pi w^{\frac{0.407}{0.573 + w}} \right) \right]$$ (6.44)

with the normalized coordinate $w = z/h$. This results in a profile that varies from 0 ppt at ice surface increasing to $3.2$ ppt at ice base. Snow is assumed fresh.

Shortwave radiative heating within the sea ice $Q_{SW}$ is equal to the vertical gradient of the radiative transfer within the sea ice:

$$Q_{SW} = - \frac{d}{dz} \{ I_{0vs} e^{-\kappa_{vs} z} + I_{0ni} e^{-\kappa_{ni} z} \}$$ (6.45)

where $I_{0vs}$ and $I_{0ni}$, the visible and near infrared radiation fluxes that penetrate the surface, are reduced according to Beer's law with the sea ice spectral extinction coefficients $\kappa_{vs}$ and $\kappa_{ni}$, respectively. For simplicity no shortwave radiation is allowed to penetrate through snow and all of the near-infrared radiation and 30% of the visible radiation is assumed to be absorbed at the surface of sea ice (Gary Maykut, personal communication):

$$I_{0vs} = 0.70 F_{SWvsn} (1-f_{s})$$ (6.46)

$$I_{0ni} = 0.0$$ (6.47)

where $f_{s}$ is the horizontal fraction of surface covered by snow (see Eq. 6.11).