In this paper we consider the boundary layer that forms on the
walls of a
rotating container (notably a conical container), filled with a stratified
when flow conditions are changed abruptly from some initial (uniform) state.
structure of the solution valid away from the cone apex is derived, and
shown that a
similarity-type solution is appropriate. This system, which is inherently
in nature, is solved numerically for several flow regimes, and the results
a number of interesting and diverse features.
In one case, a steady state is attained at large times inside the boundary
In a second case, a finite-time singularity occurs, which is fully analysed.
scenario involves a double boundary-layer structure developing at large
significantly including an outer region that grows in thickness as the
We also consider directly the nonlinear fully steady solutions to the
map out in parameter space the likely ultimate flow behaviour. Intriguingly,
cases where, when the rotation rate of the container is equal to that of
body of the fluid, an alternative nonlinear state is preferred, rather
trivial (uniform) solution.
Finally, utilizing 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
find a critical value of the buoyancy parameter beyond which the solution
exponentially in time, consistent with our nonlinear results.