Buoyant inhibition of Ekman transport on a slope and its effect on stratified spin-up

The unsteady boundary layer of a rotating, stratified, viscous, and diffusive flow along an insulating slope is investigated using theory, numerical simulation, and laboratory experiment. Previous work in this field has focused either on steady flow, or flow over a conducting boundary, both of which yield Ekman-type solutions. After the onset of a circulation directed along constant-depth contours, Ekman-type flux up or down the slope is opposed by buoyancy forces. In the unsteady, insulating case, it is found that the cross-slope transport decreases in time as ( t /τ) −½ where \[ τ = 1/S^2fcosα(1/σ + S/1+S), \] may be called the ‘shut-down’ time. Here S = ( N sin α/ f cos α) 2 , f is the Coriolis frequency, α is the slope angle, N is the buoyancy frequency, and σ is the Prandtl number. Subsequently the along-slope flow, v̂ , approximately obeys a simple diffusion equation \[ {̂}{∂ t} = ν(1/σ + S/1+S){∂^2v̂}{̂^2}, \] where t is time, ν is the kinematic viscosity, and ẑ is the coordinate normal to the slope. By this process the boundary layer diffuses into the interior, unlike an Ekman layer, but at a rate that may be much slower than would occur with simple non-rotating momentum diffusion. The along-slope flow, v̂ , is nevertheless close to thermal wind balance, and the much-reduced cross-slope transport is balanced by stress on the boundary. For a slope of infinite extent the steady asymptotic state is the diffusively driven ‘boundary-mixing’ circulation of Thorpe (1987). By inhibiting the cross-slope transport, buoyancy virtually eliminated the boundary stress and hence the ‘ fast’ spin-up of classical theory in laboratory experiments with a bowl-shaped container of stratified, rotating fluid.