Energy Spectrum of the Random Velocity Field Induced by a Gaussian Vortex Tangle in HeII

Abstract

Using the Gaussian model of the vortex tangle (VT) arising in the turbulent superfluid Hell, we calculate the energy spectrum E(k) of the 3D random velocity field induced by that VT. If the VT is assumed to be a purely fractal object with Haussdorf dimension H D , the E(k) is a power-like function E(k) k -2+HD . A more realistic VT in Hell is a semi-fractal object, behaving as smooth line for small separations Δξ << R (ξ is the label coordinate, R is mean curvature )and having a random walk structure for large Δξ with H D = 2. For that case calculations give a spectrum E(k) that is k-independent for k smaller than 1/R (but larger than the inverse size of the system) and that scales as k -1 for larger k. The latter reflects the fact that for small scales a vortex filament behaves as a smooth line. Our results agree with recent numerical simulations.