Multi-Scale Perturbation Analysis in Hydrodynamics of the Superfluid Turbulence. Derivation of the Dresner Equation

The Hydrodynamics of Superfluid Turbulence (HST) describes the flows (or counterflows) of HeII in the presence of a chaotic set of vortex filaments, so called superfluid turbulence. The HST equations govern both a slow variation of the hydrodynamic variables due to dissipation related to the vortex tangle and fast processes of the first and second sound propagation. This circumstance prevents an effective numerical simulations of the problems of unsteady heat transfer in HeII. By virtue of a pertinent multi-scale perturbation analysis we show how one can eliminate the fast processes to derive the evolution equation for the slow processes only. We then demonstrate that the long-term evolution of a transient heat load of moderate intensity obeys the nonlinear heat conductivity equation, often referred to as the Dresner equation. We also compare our approach against the Dresner phenomenological derivation and establish a range of validity of the latter.