Inverse cascade anomalies in fourth-order Leith models

TL;DR

This paper investigates fourth-order Leith models related to wave turbulence, revealing anomalous inverse cascade scaling and computing associated exponents, with implications for nonlinear Schrödinger and gravitational wave models.

Contribution

It introduces a detailed analysis of inverse cascade anomalies in fourth-order Leith models, highlighting their non-scaling behavior and calculating anomalous exponents.

Findings

Inverse transfer exhibits anomalous scaling, not constant-flux scaling.

Anomalous exponents are explicitly computed.

Analysis links exponents to dynamical systems theory.

Abstract

We analyze a family of fourth-order non-linear diffusion models corresponding to local approximations of 4-wave kinetic equations of weak wave turbulence. We focus on a class of parameters for which a dual cascade behaviour is expected with an infrared finite-time singularity associated to inverse transfer of waveaction. This case is relevant for wave turbulence arising in the Nonlinear Schrodinger model and for the gravitational waves in the Einstein's vacuum field model. We show that inverse transfer is not described by a scaling of the constant-flux solution but has an anomalous scaling. We compute the anomalous exponents and analyze their origin using the theory of dynamical systems.