Formation of a condensate in two-dimensional turbulence on a sphere

We simulate the formation of a condensate on a sphere, generated by an inverse energy cascade originating from a stochastic forcing at spherical harmonic wavenumber $ l_{\!f} \gg 1$ . The condensate forms as two pairs of oppositely signed vortices lying on a great circle that is randomly rotating in three dimensions. The vortices are separated by $ 90^\circ$ and like signed vortices are located at opposite poles. We show that the configuration is the maximum energy solution to a Hamiltonian dynamical system with a single degree of freedom. An analysis in wavenumber space reveals that interactions between widely separated scales of motions dominate the formation process. For comparison, we also perform freely decaying simulations with random initial conditions and prescribed spectra. The late time solutions consist of four coherent vortices, similar to the solutions of the forced simulations. However, in the freely decaying simulations the vortex configuration is not stationary but exhibits periodic motions.