The spontaneous breaking of axisymmetry in shallow rotating flows

We show that the axial symmetry of a shallow rotating flow is spontaneously broken in the absence of an externally forced velocity gradient. It is caused by an instability excited by the gradients that arise from the axisymmetric counter-rotating vortices. The experimental setup consists of an electrolyte poured into a cylindrical container with radius R and height h and subject to electromagnetic forcing caused by an axial magnetic field and a radial current (J) leading to an azimuthal rotation V-theta. The flow motion is considered to be two-dimensional at large aspect ratio (R/h) and low Reynolds number, Re = V(theta)h/nu, where nu is the kinematic viscosity. At a moderate aspect ratio, we record the existence of an axisymmetric vortex at the edge caused by the no-slip boundary condition at the walls. When Re is increased by changing h or J, the flow becomes unstable at the radial position where gradients exist due to the edge vortices at a critical Reynolds number of about 220. The most unstable mode of this nonaxisymmetric instability is found to be m = 1 followed by m = 2 and other higher mode numbers. Using perturbation theory, we found that two counter-rotating vortices that are in azimuthal motion are unstable when subject to nonaxisymmetric perturbations with the onset of low azimuthal mode numbers in agreement with the experiment. We conclude that the axial symmetry breaking in shallow rotating flows occurs at relatively low Reynolds numbers caused by the gradients generated by the vortices in the height-radial plane.