A modified pitchfork bifurcation as a model of the TI–TII transition in liquid helium counterflow with external noise

Abstract

An amended pitchfork bifurcation is introduced to model recent experiments by Griswold and Tough on superfluid turbulence in liquid helium counterflow subject to strong external noise. We adopt the generalized white noise limit of Blankenship and Papanicolaou to take a short-correlation-time limit of the nonlinear noise which enters into the model, and we implement this limit by means of the wideband perturbation expansion. Novel boundary conditions are applied to the resultant diffusion process in order to obtain behavior in qualitative agreement with the observations at low vortex line density. We are able to account for the sharp peak in probability observed experimentally at a small positive line density. The drift and diffusion of our diffusion process may be estimated experimentally; we describe how to do this.