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.