Parameterization depends very much on the complexity of the ocean model:

1.1 No explicit ocean boundary layer treatment (PIPS 2.0)

Recommend Rossby similarlity law:

  (1)

where is vector ice velocity relative to undisturbed geostrophic ocean currents, is vector friction velocity (square root of the kinematic stress, in the same direction); and is the surface friction Rossby number; is von Karman's constant (0.4) and A and B are Rossby similarity constants, each equal to about 2.

1.2 Extensive OBL treatment with near ice (within meters) grid point

Law of the wall:    where and refer to the velocity and elevation of the first model gridpoint. Note that refers to a reference depth in the fluid-- not ice thickness.

1.3 Explicit, but coarse resolution, OBL treatment

Submodel based on local turbulence closure to determine eddy viscosity based on surface stress, surface buoyancy flux, and mean density structure. Described in:

McPhee, M. G., 1998. Parameterization of mixing in the ocean boundary layer, in press: J. Fluid Systems.

McPhee, M. G., C. Kottmeier, J. H. Morison, 1998. Ocean heat flux in the central Weddell Sea during winter, in press: J. Phys. Oceanogr.

Both available in Adobe pdf format by anonymous FTP. Login as anonymous to nansen.apl.washington.edu; cd anzflux/mcphee; binary; get JFS98.pdf, JPO98.pdf.

Principles (sometimes overlooked):

1. Ice can grow when the mixed layer is above freezing-- if it didn't the Weddell Sea would have no sea ice.
2. Heat transfer between the ocean and the ice depends on both the elevation of mixed layer temperature above freezing and the friction velocity (
) at the ice/ocean interface.
3. An unexpected result of turbulent heat flux measurement programs in recent years is that the turbulent Stanton number (heat transfer coefficient) shows little dependence on the undersurface roughness (
). This runs counter to expectations from laboratory studies of heat and mass transfer across hydraulically rough surfaces, but much simplifies parameterization:

2.1 Ocean Surface Heat Flux

  (2)

for where is the elevation of mixed layer temperature above freezing; is the observed range for the exchange coefficient (turbulent Stanton number. This provides the upper boundary condition for the heat conservation equation:

+ horizontal diffusion + advection (3)

2.2 Ocean Surface Salinity Flux

Salinity flux at the ice/ocean interface depends on basal melting or freezing and freshwater input from surface or internal melting which percolates through the ice or runs off at floe edges. The latter may be expressed in terms of a vertical velocity of the interface as it adjust isostatically to surface or internal melting: . The basal melt or freeze rate may be similarly expressed as a vertical interface velocity, , which depends on the ocean heat flux ( ) and heat flux conducted away from the interface in the ice cover: where is ice thermal conductivity. To a fair approximation the salinity flux may be expressed as where is the difference between mixed layer and ice salinities. Substituting, the salinity flux boundary condition is:

  (4)

where is latent heat of fusion (adjusted for brine volume) divided by specific heat of seawater ( in Chang and Preller).

McPhee, M. G., 1992. Turbulent heat flux in the upper ocean under sea ice, J. Geophys. Res., 97, 5365-5379.