Statistical properties of superfluid turbulence in He4 from the Hall-Vinen-Bekharevich-Khalatnikov model

We obtain the von K\'arm\'an--Howarth relation for the stochastically forced three-dimensional (3D) Hall-Vinen-Bekharevich-Khalatnikov (HVBK) model of superfluid turbulence in helium ($^{4}\mathrm{He}$) by using the generating-functional approach. We combine direct numerical simulations (DNSs) and analytical studies to show that, in the statistically steady state of homogeneous and isotropic superfluid turbulence, in the 3D HVBK model, the probability distribution function (PDF) $P(\ensuremath{\gamma})$, of the ratio $\ensuremath{\gamma}$ of the magnitude of the normal fluid velocity and superfluid velocity, has power-law tails that scale as $P(\ensuremath{\gamma})\ensuremath{\sim}{\ensuremath{\gamma}}^{3}$, for $\ensuremath{\gamma}\ensuremath{\ll}1$, and $P(\ensuremath{\gamma})\ensuremath{\sim}{\ensuremath{\gamma}}^{\ensuremath{-}3}$, for $\ensuremath{\gamma}\ensuremath{\gg}1$. Furthermore, we show that the PDF $P(\ensuremath{\theta})$ of the angle $\ensuremath{\theta}$ between the normal-fluid velocity and superfluid velocity exhibits the following power-law behaviors: $P(\ensuremath{\theta})\ensuremath{\sim}\ensuremath{\theta}$ for $\ensuremath{\theta}\ensuremath{\ll}{\ensuremath{\theta}}_{*}$ and $P(\ensuremath{\theta})\ensuremath{\sim}{\ensuremath{\theta}}^{\ensuremath{-}4}$ for ${\ensuremath{\theta}}_{*}\ensuremath{\ll}\ensuremath{\theta}\ensuremath{\ll}1$, where ${\ensuremath{\theta}}_{*}$ is a crossover angle that we estimate. From our DNSs we obtain energy, energy-flux, and mutual-friction-transfer spectra, as well as the longitudinal-structure-function exponents for the normal fluid and the superfluid, as a function of the temperature $T$, by using the experimentally determined mutual-friction coefficients for superfluid helium $^{4}\mathrm{He}$, so our results are of direct relevance to superfluid turbulence in this system.