In this paper we consider the boundary layer that forms on the
sloping
walls of a
rotating container (notably a conical container), filled with a stratified
fluid,
when flow conditions are changed abruptly from some initial (uniform) state.
The
structure of the solution valid away from the cone apex is derived, and
it is
shown that a
similarity-type solution is appropriate. This system, which is inherently
nonlinear
in nature, is solved numerically for several flow regimes, and the results
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 analysed.
A third
scenario involves a double boundary-layer structure developing at large
times, most
significantly including an outer region that grows in thickness as the
square-root of
time.We also consider directly the nonlinear fully steady solutions to the
problem, and
map out 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, 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
work, we
find a critical value of the buoyancy parameter beyond which the solution
grows
exponentially in time, consistent with our nonlinear results.
On the boundary layer arising in the spin-up of a
stratified fluid in a container with sloping walls
