Non-axisymmetric instabilities in basic state spherical Couette flow

We consider the flow induced in a spherical shell by fixing the outer sphere and rotating the inner one, with the aspect ratio β = (r_o-r_i)/r_i ranging from 0.1 to 10. The basic state consists of a jet in the equatorial plane, carrying fluid from the inner sphere to the outer, and involving also an azimuthal component. The azimuthal component dominates for β≲1, the radial component for β≳1. The basic state is otherwise much the same over the entire range 0.1 ≤ β ≤ 10. We next linearize the Navier–Stokes equation about this basic state, and compute the linear onset of non-axisymmetric instabilities. For 0.1 ≤ β ≤ 3.8 the instabilities have the opposite equatorial symmetry as the basic state, and consist of a series of waves on the initially flat radial-azimuthal jet. The azimuthal wavenumber decreases monotonically from m = 12 at β = 0.1 to m = 2 at β = 3.8, but with a puzzling transition between β = 0.27 and 0.28, where one m = 6 mode is replaced by another, very similar one. For 3.8 ≤ β ≤ 10 we obtain an m = 2 mode having the same equatorial symmetry as the basic state. This instability is further differentiated from the previous ones in that it consists of a modulation in the strength of the return flow after the jet has reached the outer sphere, rather than an instability of the jet itself. Finally, we solve the fully three-dimensional Navier–Stokes equation, and consider the equilibration of some of these modes in the supercritical regime. For β = 0.5 we can achieve Re = 1.15 Re_c, and obtain results in excellent agreement with experiments. For β = 0.8, 1.5 and 2.5 we can achieve up to 1.9 Re_c, and find that in all three cases secondary bifurcations occur, in which the solutions develop a time-dependence more complicated than a simple drift in φ. The precise nature of the bifurcation is different in the three cases.