Generalization of Vinen's equation including transition to superfluid turbulence

A phenomenological generalization of the well known Vinen equation for the evolution of vortex line density in superfluid counterflow turbulence is proposed. This generalization includes nonlinear production terms in the counterflow velocity and corrections depending on the diameter of the tube. The equation provides a unified framework for the various phenomena (stationary states and transitions) present in counterflow superfluid turbulence: in fact, it is able to describe the laminar regime, the first-order transition from laminar to turbulent TI state, the two turbulent states, the transition from TI to TII turbulent states, and it yields a slower decay of the counterflow turbulence than the classical local description. Finally, a comparison with the experimental results shows that the contribution of the new terms is prevalent in the laminar and in the turbulent TI regime, while in the fully developed turbulent TII regime the equation reduces to the original Vinen equation. © 2005 IOP Publishing Ltd.