Energy spectra of quantum turbulence in He II and $\mathrm{^3He-B}$ – a unified view

Quantum turbulence in superfluid He II and in $\mathrm{^3He--B}$ that can be regarded as nearly isothermal, isotropic and homogeneous is discussed within the two–fluid model. A general form of the 3D energy spectrum is proposed: at large length scales, where normal and superfluid eddies are locked together by the mutual friction force, the energy spectrum is essentially classical and includes an inertial range of a Kolmogorov K62-form. With increasing wavenumber, $k$, the normal fluid part of the spectrum terminates due to finite viscosity while the superfluid part of the spectral energy density changes towards $k^{-3}$ and then back into Kolmogorov-like $k^{-5/3}$ again. Agreement with computer simulations and experiments is claimed if account is taken of the turbulent box size and of the energy decay rate.