ON THE BOUNDARY-LAYER ARISING IN THE SPIN-UP OF A STRATIFIED FLUID INA CONTAINER WITH SLOPING WALLS

In this paper we consider the boundary layer that forms on the sloping walls of a rotating container (notably a conical container), filled w ith a stratified fluid, when flow conditions are changed abruptly from some initial (uniform) state. The structure of the solution valid awa y from the cone apex is derived, and it is shown that a similarity-typ e solution is appropriate. This system, which is inherently nonlinear in nature, is solved numerically for several flow regimes, and the res ults reveal a number of interesting and diverse features. In one case, a steady state is attained at large times inside the boundary layer. In a second case, a finite-time singularity occurs, which is fully ana lysed. A third scenario involves a double boundary-layer structure dev eloping at large times, most significantly including an outer region t hat grows in thickness as the square-root of time. We also consider di rectly the nonlinear fully steady solutions to the problem, and map ou t in parameter space the likely ultimate flow behaviour. Intriguingly, we find cases where, when the rotation rate of the container is equal to that of the main body of the fluid, an alternative nonlinear state is preferred, rather than the trivial (uniform) solution. Finally, ut ilizing Laplace transforms, we re-investigate the linear initial-value problem for small differential spin-up studied by MacCready & Rhines (1991), recovering the growing-layer solution they found. However, in contrast to earlier work, we find a critical value of the buoyancy par ameter beyond which the solution grows exponentially in time, consiste nt with our nonlinear results.