The interaction of Kelvin waves and the non-locality of the energy transfer in superfluids

TL;DR

This paper develops a detailed theory of Kelvin Wave turbulence in superfluids, correcting previous misconceptions, deriving a new simplified nonlinear equation, and clarifying the non-local nature of energy transfer in superfluid turbulence.

Contribution

It provides a comprehensive calculation of Kelvin Wave interactions, introduces a new local nonlinear equation, and refutes the applicability of the Kozik-Svistunov spectrum due to non-local energy transfer.

Findings

Derived a new local nonlinear equation for Kelvin waves.

Showed the Kozik-Svistunov spectrum is irrelevant for superfluid turbulence.

Demonstrated weak non-locality and logarithmic corrections in the inverse cascade spectrum.

Abstract

We argue that the physics of interacting Kelvin Waves (KWs) is highly non-trivial and cannot be understood on the basis of pure dimensional reasoning. A consistent theory of KW turbulence in superfluids should be based upon explicit knowledge of their interactions. To achieve this, we present a detailed calculation and comprehensive analysis of the interaction coefficients for KW turbulence, thereby, resolving previous mistakes stemming from unaccounted contributions. As a first application of this analysis, we derive a new Local Nonlinear (partial differential) Equation. This equation is much simpler for analysis and numerical simulations of KWs than the Biot-Savart equation, and in contrast to the completely integrable Local Induction Approximation (in which the energy exchange between KWs is absent), describes the nonlinear dynamics of KWs. Secondly, we show that the previously suggested Kozik-Svistunov energy spectrum for KWs, which has often been used in the analysis of experimental and numerical data in superfluid turbulence, is irrelevant, because it is based upon an erroneous assumption of the locality of the energy transfer through scales. Moreover, we demonstrate the weak non-locality of the inverse cascade spectrum with a constant particle-number flux and find resulting logarithmic corrections to this spectrum.