AbstractThe spherical Couette system is a spherical shell filled with a viscous fluid. Flows are driven by the differential rotation between the inner and the outer boundary that rotate with $\Omega $ and $\Omega + \mathrm{\Delta} \Omega $ about a common axis. This setup has been proposed for second-generation dynamo experiments. We numerically explore the different instabilities emerging for rotation rates up to $\Omega = (1/ 3)\times 1{0}^{7} $, venturing also into the nonlinear regime where oscillatory and chaotic solutions are found. The results provide a comprehensive overview of the possible flow regimes. For low values of $\Omega $ viscosity dominates and an equatorial jet in meridional circulation and zonal flow develops that becomes unstable as the differential rotation is increased beyond a critical value. For intermediate $\Omega $ and an inner boundary rotating slower than the outer one, new double-roll and helical instabilities are found. For large $\Omega $ values Coriolis effects enforce a nearly two-dimensional fundamental flow where a Stewartson shear layer develops at the tangent cylinder. This shear layer is the source of nearly geostrophic non-axisymmetric instabilities that resemble columnar Rossby modes. At first, the instabilities differ significantly depending on whether the inner boundary rotates faster $( \mathrm{\Delta} \Omega \gt 0)$ or slower $( \mathrm{\Delta} \Omega \lt 0)$ than the outer one. For very large outer boundary rotation rates, however, both instabilities once more become comparable. Fast inertial waves similar to those observed in recent spherical Couette experiments prevail for larger $\Omega $ values and $ \mathrm{\Delta} \Omega \lt 0$ in when $ \mathrm{\Delta} \Omega $ and $\Omega $ are of comparable magnitude. For larger differential rotations $ \mathrm{\Delta} \Omega \gg \Omega $, however, the equatorial jet instability always takes over.
The Boussinesq approximation in rapidly rotating flows