4-wave dynamics in kinetic wave turbulence

TL;DR

This paper develops a formalism for 4-wave kinetic turbulence, deriving evolution equations for multimode characteristic functions using diagrammatic techniques, applicable to numerical simulations and intermittency studies.

Contribution

It introduces a hierarchy of equations for 4-wave turbulence that generalizes the Boltzmann hierarchy, incorporating frequency renormalization and phase randomness.

Findings

Derived a first-order differential equation for the characteristic function Z.

Established a hierarchy of equations analogous to the Boltzmann hierarchy.

Provided a formalism suitable for numerical simulations and intermittency analysis.

Abstract

A general Hamiltonian wave system with quartic resonances is considered, in the standard kinetic limit of a continuum of weakly interacting dispersive waves with random phases. The evolution equation for the multimode characteristic function $Z$ is obtained within an "interaction representation" and a perturbation expansion in the small nonlinearity parameter. A frequency renormalization is performed to remove linear terms that do not appear in the 3-wave case. Feynman-Wyld diagrams are used to average over phases, leading to a first order differential evolution equation for $Z$. A hierarchy of equations, analogous to the Boltzmann hierarchy for low density gases is derived, which preserves in time the property of random phases and amplitudes. This amounts to a general formalism for both the $N$-mode and the 1-mode PDF equations for 4-wave turbulent systems, suitable for numerical simulations and for investigating intermittency.