Scaling laws for vortical nucleation solutions in a model of superflow

The bifurcation diagram corresponding to stationary solutions of the nonlinear Schrödinger equation describing a superflow around a disc is numerically computed using continuation techniques. When the Mach number is varied, it is found that the stable and unstable (nucleation) branches are connected through a primary saddle-node and a secondary pitchfork bifurcation. Computations are carried out for values of the ratio ξ/d of the coherence length to the diameter of the disc in the range 1/5-1/80. It is found that the critical velocity converges for ξ/d → 0 to an Eulerian value, with a scaling compatible with previous investigations. The energy barrier for nucleation solutions is found to scale as ξ². Dynamical solutions are studied and the frequency of supercritical vortex shedding is found to scale as the square root of the bifurcation parameter.