HYDRODYNAMICAL MODELS OF SUPERFLUID TURBULENCE

This review paper puts together some of our results concerning the application of non equilibrium Thermodynamics to superfluid liquid helium. Two of the most important situations of this quantum fluid are rotating superfluid and superfluid turbulence, both characterized by the presence of quantized vortices (vortex lines whose core is about 1 Angstrom and the quantum of circulation is $h/m$, $h$ being the Plank's constant and $m$ the mass of helium atom). In the first part of the work a non-standard model of superfluid helium, which considers heat flux as independent variable, is briefly recalled, and compared with the well known two-fluid model, in absence of vortices, proposed by Tisza and Landau more than half a century ago. The model is generalized taking into account the presence of vortices in different cases of physical interest: rotating superfluids, counterflow superfluid turbulence (a particular situation in which no mass flux but only heat flux is present) and combined situations of counterflow and rotation. Since vortices are not fixed when all the hydrodynamical fields change, an additional scalar quantity, the averaged vortex line density per unit volume $L$, {\it line density} for short, is introduced in the model as a new field variable and an evolution equation is written for it, both in linear and in nonlinear regimes, via Extended Thermodynamics. Finally, to encompass more general situations, the model is further extended considering the flux of vortex line density as an independent new variable. In all these models the propagation of harmonic waves is studied, motivated by the fact that vortex lines density is experimentally detected via the attenuation of second sound. A new kind of waves, vortex density waves, is also dealt with.