Dynamics of the vortex line density in superfluid counterflow turbulence

Describing superfluid turbulence at intermediate scales between the intervortex distance and the macroscale requires an acceptable equation of motion for the density of quantized vortex lines L. The closure of such an equation for superfluid inhomogeneous flows requires additional inputs besides L and the normal and superfluid velocity fields. In this paper, we offer a minimal closure using one additional anisotropy parameter Il0. Using the example of counterflow superfluid turbulence, we derive two coupled closure equations for the vortex line density and the anisotropy parameter Il0 with an input of the normal and superfluid velocity fields. The various closure assumptions and the predictions of the resulting theory are tested against numerical simulations.