Modeling Kelvin Wave Cascades in Superfluid Helium

Abstract

We study two different types of simplified models for Kelvin wave turbulence on quantized vortex lines in superfluids near zero temperature. Our first model is obtained from a truncated expansion of the Local Induction Approximation (Truncated-LIA) and it is shown to possess the same scalings and the essential behaviour as the full Biot-Savart model, being much simpler than the latter and, therefore, more amenable to theoretical and numerical investigations. The Truncated-LIA model supports six-wave interactions and dual cascades, which are clearly demonstrated via the direct numerical simulation of this model in the present paper. In particular, our simulations confirm presence of the weak turbulence regime and the theoretically predicted spectra for the direct energy cascade and the inverse wave action cascade. The second type of model we study, the Differential Approximation Model (DAM), takes a further drastic simplification by assuming locality of interactions in k-space via a differential closure that preserves the main scalings of the Kelvin wave dynamics. DAMs are even more amenable to study and they form a useful tool by providing simple analytical solutions in the cases when extra physical effects are present, e.g. forcing by reconnections, friction dissipation and phonon radiation. We study these models numerically and test their theoretical predictions, in particular the formation of the stationary spectra, and the closeness of the numerics for the higher-order DAM to the analytical predictions for the lower-order DAM .

Weak turbulence of Kelvin waves in superfluid He

“…This may explain why the previous numerical experiments seem to agree with the KS spectrum, obtained numerically in Ref. 15. However, by inspection one can also see that these results also agree with the LN slope.…”

“…(8), (19), and solved it to obtain a new wave spectrum (20). We proved that this new spectrum is local, and therefore it is a valid solution of the kinetic equation, which should replace the nonlocal (and therefore invalid) Kosik-Svistunov spectrum (15) in the theory of quantum turbulence. In particular, it is now necessary to revise the theory of the classicalquantum crossover scales and its predictions for the turbulence dissipation rate Refs.…”

“…-The difference between the LN-exponent -5/3 (see (20)) from the KS-exponent -7/5 (see (15)) is 4/15 which is rather small. This may explain why the previous numerical experiments seem to agree with the KS spectrum, obtained numerically in Ref.…”

“…The LNE model is similar but not identical to the Truncated LIA model of [15] (these models become asymptotically identical for weak KWs).…”

“…17-20. Further, a similar revision is needed for the analysis of laboratory experiments and numerical simulations of superfluid turbulence, which have been done over the last five years with reliance on the un-physical KS spectrum (15).…”