The implications of linearly varying eddy viscosity for wind-driven current profiles

Abstract

A major problem in numerical circulation modelling is the specification of vertical turbulence closure. After the simplicity of a constant eddy viscosity, the next step towards reality appears to be an eddy viscosity described by υ = κ u * z where κ is von Karman's constant, u * is a friction velocity, and z is the distance away from the sea surface or bottom. Applied to the “Ekman problem”, this eddy viscosity results in current profiles that have logarithmic boundary layers. The Ekman depth is replaced by a new depth of influence, κu */ f (where f is the Coriolis frequency). If this depth is less than the water depth, the wind-forced current profile does not reach the bottom, and is unaffected by the bottom conditions. For water depths less than the depth of surface influence, currents through the whole water column are sensitive to the value of the bottom roughness. By contrast, changes to the surface roughness only affect currents in a thin layer near the surface. Numerically, resolution of the log-layers at the sea surface and bottom requires the use of special techniques. The log-layers may be described analytically, or the vertical coordinate may be transformed to expand the near-boundary regions. For modelling the mid-water column, away from the log-layers, a constant eddy viscosity with a linear bottom slip condition represents the currents with reasonable accuracy. Appropriate values of the constant eddy viscosity and bottom friction parameter are determined empirically from the shear stress and bottom roughness by formulae that are consistent over a broad parameter range.