A variational reasoning leading to the decay of ring-like vortices

In superconducting nanoshells the order parameter is confined to a spherical film with thickness smaller than the penetration depth. When nanoshells are placed in a magnetic field, various vortex states can occur, among which ring-like vortices. These states exhibit non-uniform vorticity, and a suppression of the order parameter in a ring-like region at constant latitude on the sphere. Here, we develop a variational description suitable to compare these ring-like vortices and arrays of singly quantized vortices. This allows us to derive a variational argument leading to the breaking up of the ring-like vortex, valid for thin nanoshells of any radius and in any uniform applied magnetic field. The present variational results confirm and complement earlier numerical investigations of ring-like vortices using the time-dependent nonlinear Ginzburg-Landau equations.