Motion of three geostrophic vortices on a sphere

Accurately approximating and applying geophysical phenomena requires considering geostrophic vortices on spherical or sphere-like curved surfaces. Accordingly, this study explores the dynamics of geostrophic fluid vortices on the surface of a sphere. The geostrophic vortex model incorporates the effects of stratification and planetary rotation and is relevant to large-scale atmospheric phenomena. The governing equations of the model are expressed in terms of the Legendre functions and systematically formulated along with the corresponding Hamiltonian. We focus on the three-vortex case and investigate its dynamics, identifying the fixed and relative equilibria of the vortices. The general dynamics is analyzed using the phase diagram based on trilinear coordinates. Notably, we demonstrate the occurrence of the non-self-similar collapse of the three geostrophic vortices. This result significantly differs from that of the point vortex on a sphere, where the collapse is always self-similar. Numerical computations for the trajectories of vortices with various initial configurations reveal regular and irregular patterns of vortex motion depending on the strength and initial location of the vortices. Remarkably, we find a recurring motion of vortices converging toward and diverging from a relative equilibrium.