Topological noise scalings in superfluid and classical turbulence

We calculate the topological noise characterizing the direction of line vortices in superfluid and classical turbulence by finding the intersection of line vortices with square surfaces of edge length $l_s$ positioned normal to three orthogonal axes. In the case of homogeneous superfluid turbulence in thermal counterflow, we find that the noise scales like $l_s$ along the two directions normal to the counterflow and like $l_s^{3/2}$ along the direction parallel to it. In homogeneous isotropic superfluid turbulence at $T\rightarrow 0\,$K, the noise scales like $l_s^{7/4}$. In homogeneous isotropic classical turbulence the scaling is $l_s^{2}$. We offer possible interpretations of the computed scalings, as well as, justification for their differences.