Bimodal vortex solutions on a sphere

Abstract

Steady and time-periodic solutions of the two-dimensional Navier–Stokes equations are studied on a sphere under steady forcings represented by a single spherical harmonic function. This setting is similar to the Kolmogorov problem of the two-dimensional flows under a sinusoidal forcing on a doubly periodic domain (a flat torus), where either simple steady or time-periodic unimodal solutions have been found even at high Reynolds numbers depending on the forcing profiles. In our spherical case, we investigate the bifurcation structure and find either simple steady or time-periodic bimodal solutions, consisting of only two pairs of positive and negative vortices, depending on the forcing profiles at very high Reynolds numbers.