Generalization of Vinen’s equation including transition to superfluid turbulence

Abstract

A phenomenological generalization of the well known Vinen equation for theevolution of vortex line density in superfluid counterflow turbulence is proposed. Thisgeneralization includes nonlinear production terms in the counterflow velocity andcorrections depending on the diameter of the tube. The equation provides a unifiedframework for the various phenomena (stationary states and transitions) presentin counterflow superfluid turbulence: in fact, it is able to describe the laminarregime, the first-order transition from laminar to turbulent TI state, the twoturbulent states, the transition from TI to TII turbulent states, and it yields a slowerdecay of the counterflow turbulence than the classical local description. Finally, acomparison with the experimental results shows that the contribution of the new termsis prevalent in the laminar and in the turbulent TI regime, while in the fullydeveloped turbulent TII regime the equation reduces to the original Vinen equation.