Description and evolution of anisotropy in superfluid vortex tangles with counterflow and rotation

We examine several vectorial and tensorial descriptions of the geometry of turbulent vortex tangles. We study the anisotropy in rotating counterflow experiments, in which the geometry of the tangle is especially interesting because of the opposite effects of rotation, which orients the vortices, and counterflow, which randomizes them. We propose to describe the anisotropy and the polarization of the vortex tangle through a tensor, which contains the first and second moments of the distribution of the unit vector s(') locally tangent to the vortex lines. We use an analogy with paramagnetism to estimate the anisotropy, the average polarization, the polarization fluctuations, and the geometrical contribution to the entropy of the tangle in terms of angular velocity and counterflow. We explore the influence of the geometry on the evolution of the vortex line density and propose evolution equations for the geometry of the tangle.