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

Using the Gaussian model of the vortex tangle (VT) arising in the turbulent superfluid HeII, 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 $E(k)\propto k^{-2+H_D}$. A more realistic VT in HeII is a semi-fractal object, behaving as smooth line for small separations $\Delta \xi \ll R$ ($\xi $ is the label coordinate, $R$ is mean curvature)and having a random walk structure for large $\Delta \xi $ 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. PACS numbers: 47.32.Cc, 47.37.+q, 67.40.Vs., 05.10.Gg.