An analytical and numerical study of surface and bottom boundary layers with variable forcing and application to the North Sea

A study is made concerning the evolution of a tidally generated bottom layer and a wind-driven surface layer with a time-dependent surface stress in view of its application to thermal stratification problems in shelf seas. Three test cases are defined. In the first two a linear stratification is assumed initially while a stress is applied either at the bottom or at the surface. Analytical solutions can then be derived using simplifying assumptions. They show how the evolution depends on the forcing parameters, i.e. initial density gradient, rotation, forcing period, amplitude of the tidal current or magnitude of the wind stress, ellipticity and polarisation of the tidal or surface (stress) ellipse. It is shown that the analytical theory is in good agreement with results obtained using a numerical model incorporating a turbulence closure scheme. An important result is the occurrence of a resonance and an increased growth of the surface layer when the time scale of the surface stress is comparable with the rotation period. No such resonance has been found for the bottom layer forced by the semi-diurnal tide. In a third test case a simulation is made of the seasonal cycle at station CS in the North Sea. The results are compared with the temperature data from the North Sea Project. A series of numerical experiments is performed to verify parts of the theoretical analysis.