Hydrodynamics of quantum turbulence in He II: Vinen's equation derived from energy and impulse of vortex tangle

Abstract

A macroscopic theory of superfluid turbulence is developed in which the vortex tangle is characterised by both its line-length density and its drift velocity. Three velocity fields are accordingly distinguished, viz. the mass velocity υ, the normal-fluid velocity υ n and the drift velocity υ e of the vortex tangle. The introduction of the drift velocity of the tangle as an additional variable seems to be new. The non-dissipative equations of motion are derived from a generalised form of Lin's variational principle. Both the energy and the impulse of the vortex tangle into account. It is shown that the effective mass density of the vortex tangle vanishes. In equilibrium the relative velocity of the vortex tangle is given by the derivative of the energy of the tangle with respect to the tangle impulse. A similar relation holds for a ring vortex in a fluid of infinite extent. When dissipative terms are added to the equations according to the thermodynamics of irreversible processes the Vinen equation follows immediately. The derivation suggests a new interpretation of the right-hand member of the Vinen equation in terms of the derivative of a potential energy. A similar potential energy has been used in investigations on vortex nucleation and critical velocities. The theory is extended by including the effects of large gradients of the line-length density. The corresponding generalisation of the Vinen equation is presented.