Stability of flow on a rotating polar cap

Two-dimensional non-divergent Navier-Stokes flow is studied on a rotating polar cap, i.e. a polar region of a rotating sphere that lies above (or below) a given plane normal to the rotation axis. The inflow and outflow on the boundary of the polar cap drive the fluid motion. The nonlinear steady solution is obtained numerically and its linear stability is studied. As the inflow is increased, the flow pattern of the nonlinear solution differs largely from that of the linear analytical solution. When the inflow is further increased, the nonlinear steady solution becomes unstable with a Hopf bifurcation to an oscillating solution.