Shell Models of Superfluid Turbulence

Superfluid helium consists of two inter-penetrating fluids, a viscous normal fluid and an inviscid superfluid, coupled by a mutual friction. We develop a two-fluid shell model to study superfluid turbulence and investigate the energy spectra and the balance of fluxes between the two fluids in a steady state. At sufficiently low temperatures a ’bottle-neck’ develops at high wavenumbers suggesting the need for a further dissipative effect, such as the Kelvin wave cascade. 1. Two-fluid model of liquid helium Superfluidity is a property of Bose-Einstein condensed systems, such as He, He and ultracold atomic gases. Superfluids are notable for their singular vorticity, which is confined to thin filaments. In this work we are concerned about He. According to Landau’s two-fluid theory, below the critical temperature Tc = 2.17K, liquid helium consists of two inter-penetrating fluid components, a viscous normal fluid and an inviscid superfluid; coupled by a mutual friction. The normal fluid is governed by a Navier-Stokes-like equation and the superfluid by an Eulerlike equation. In the presence of a tangle of superfluid vortices the two-fluid equations are given by