Modeling wind-induced circulation in the homogeneous Lake Constance using k-ɛ closure

Barotropic models of wind-driven circulation commonly use vertical eddy-viscosities which are prescribed functions of depth and applied wind speed rather than a parameterization that is dictated by the turbulence intensity. We use a first order k-ɛ-closure to do this and demonstrate important qualitative differences with this classical turbulence closure in the pattern of wind induced barotropic currents in ideal basins, as well as in Lake Constance: penetration and Ekman layer depths respond with some delay time to the applied wind history as does the attenuation of inertial oscillations established by the wind. This also affects the topographically-induced current pattern. We demonstrate these features by subjecting the homogeneous Lake Constance to impulsive and spatially uniform external wind forcing of different strengths in the long direction. Specifically, the shallow-water equations are coupled via the vertical eddy-viscosity with the correspondingly approximated balance laws of turbulent kinetic energy and its dissipation, which are then solved numerically.