The discussions in chapters 2-4 are salient to many aspects of sea-i ce thermodynamics. Brine content influences the thermal properties of sea ice, relative to freshwater ice, but temperature evolution and ice thickness are still broadly governed by vertical diffusion. Heat advection from percolation of brine and meltwater can also be significant.
Similar to lake ice, heavy snow cover on a thin platform of sea ice can submerge the ice, causing flooding that creates "snow ice." Freezing of seawater in these conditions is fundamentally different from basal ice accretion, and it can lead to high levels of salt entrainment. This process is particularly strong in Antarctica, due to pack ice divergence that creates a relatively thin ice cover that is vulnerable to submergence. The crystal structure and radiative properties of snow ice differ from those of accretion ice. Snow ice also has a high thermal conductivity relative to snow, promoting sea-ice growth.
Ocean heat flux, Qw, is also more variable and potent than the geothermal heat flux into the base of terrestrial ice masses or the basal heat fluxes normally experienced by freshwater ice. Typical values are 1-2 W m-2 but can be higher where warm water masses (e.g., 0°C or even a few degrees Celsius) penetrate to high latitudes. This is known to transpire in the Northern Hemisphere through intermittent forays of North Atlantic water into the Arctic basin. Eddies from the Antarctic Circumpolar current also deliver warm water to coastal Antarctica. Warm water can upwell buoyantly or through Ekman divergence, delivering large heat fluxes to sea ice. A water mass of this type is found in the Southern Ocean and is called circumpolar deep water. This warm, salty water has its origins in the North Atlantic overturning circulation.
The thermodynamic formulations described in chapter 3 apply to sea ice or a combined sea ice and snow layer, with some modifications for the effects of salinity content. Combining the sensible and latent heat content of the sea ice, the effective heat capacity, or "thermal inertia," of sea ice with salinity S and temperature T is written cJT, S) Ci + (5.1)
where ci and Lf are the specific heat capacity and latent heat of fusion of freshwater ice, S is in parts per thousand, and T is in degrees Celsius. The coefficientm comes from the linear approximation to the effects of salinity on melting-point depression: Tm = — m S.
Couched in these terms, one can simulate the energy required to raise the temperature or melt a given volume of sea ice based on its temperature and salinity. For ice density psi, integration of (5.1) over temperature gives an equation for the energy per unit volume required to warm a parcel of sea ice from temperature Tx to T2:
Equation (5.2) neglects the temperature dependence of the specific heat capacity of pure ice, although this can be included in numerical models. If T2 = Tm, the melting temperature, this implies a complete phase change from ice to liquid brine. Equation (5.2) can then be written in terms of the total energy required to melt a volume of sea ice with an initial temperature T and salinity S:
This is equivalent to the enthalpy per unit volume of the ice. These equations are stable because T < 0°C for saline ice. For freshwater ice, the energy of melt is Em =pi [ci (Tm - T) + Lf].
Ice growth or basal melting, including the effects of brine content, can then be modeled as a function of enthalpy. This is an adaptation of the equation for the growth of freshwater ice given in Eq. (4.6). For ice thickness H,
where Qin and Qout refer to the upward-directed vertical heat flux into and away from the ocean-ice interface.
Qin is equal to the ocean heat flux, and Qout is the heat conducted upward into the sea ice. This equation is combined with the surface energy balance at the upper boundary (the snow/ice-atmosphere interface), as described in chapter 3, and the conservation of enthalpy within the ice volume. Substituting enthalpy for internal energy (temperature) in Eq. (3.5), the thermodynamic evolution of sea ice follows where J accounts for solar radiative heating and latent energy release/consumption from internal refreezing/ melting. A term can also be added to account for heat advection from brine percolation, given a model of that process.
To define fully sea-ice thermodynamics, salinity evolution S (z,t) also needs to be accounted for. Along with (5.4) and (5.5), this gives a system of coupled equations for the temporal evolution of sea ice of thickness H, with vertical temperature and salinity structure T (z) and S (z). Similar to the heat advection associated with brine migration, this is difficult to measure or model. At present, most sea-ice models prescribe salinity or hold it fixed.
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