Saturation of nonaxisymmetric instabilities of magnetized spherical Couette flow

We numerically investigate the saturation of the hydromagnetic instabilities of a magnetized spherical Couette flow. Previous simulations demonstrated a region where the axisymmetric flow, calculated from a 2D simulation, was linearly unstable to nonaxisymmetric perturbations. Full, nonlinear, 3d simulations showed that the saturated state would consist only of harmonics of one azimuthal wave number, though there were bifurcations and transitions as nondimensional parameters (Re, Ha) were varied. Here, the energy transfer between different azimuthal modes is formulated as a network. This demonstrates a mechanism for the saturation of one mode and for the suppression of other unstable modes. A given mode grows by extracting energy from the axisymmetric flow, and then saturates as the energy transfer to its second harmonic equals this inflow. At the same time, this mode suppresses other unstable modes by facilitating an energy transfer to linearly stable modes.