We examine the transitions among the steady-state axisymmetric spherical Couette flows with zero, one, and two vortices per hemisphere. The steady flows are reflection symmetric with respect to the equator, but some of the transitions that we find break this symmetry. This is the first study to reproduce numerically the transitions to the one-vortex flow from the zero- and two-vortex flows. In our study, we use a numerical initial-value code to: (i) compute the bifurcation diagrams of the steady (stable and unstable) states, (ii) solve for the most unstable or least stable linear eigenmode and eigenvalue of a steady state, (iii) calculate the velocity field as a function of time during both the linear and nonlinear stages of the transitions, and (iv) determine the energy transfer mechanism into and out of the antireflection-symmetric components of the flow during the transitions.
Our study provides the explanation of the laboratory observation that some transitions occur only when the inner sphere of the Couette-flow apparatus is accelerated or decelerated quickly, whereas other transitions occur only when the acceleration is slow.