Bifurcations in narrow-gap spherical Couette flow

Abstract

The incompressible Navier–Stokes equations in spherical coordinates are solved using a pseudo-spectral method to simulate the problem of spherical Couette flow. The flow is investigated for a narrow-gap ratio with only the inner sphere rotating. We find that the flow is sensitive to the initial conditions and have used various initial conditions to obtain different branches of the bifurcation curve of the flow. We have identified three different branches dominated respectively by axisymmetric flow, travelling wave instability and equatorial instability. The axisymmetric branch shows unsteadiness at large Reynolds numbers. The travelling wave instability branch shows spiral instability and is prominent the near poles. The travelling wave instability branch further exhibits a reversal in the propagation direction of the spiral instability as the Reynolds number is increased. This branch also exhibits a multi-mode equatorial instability at larger Reynolds numbers. The equatorial instability branch exhibits twin jet streams on either side of the equator, which become unstable at larger Reynolds numbers. The flow topology on the three branches is also investigated in their phase space and found to exhibit chaotic behaviour at large Reynolds numbers on the travelling wave instability branch.