Numerical Solutions of the Isotropic 3-Wave Kinetic Equation

TL;DR

This paper introduces a new numerical method for solving the isotropic 3-wave kinetic equation, revealing insights into energy dissipation, bottleneck effects, and thermalization phenomena in wave turbulence.

Contribution

A novel numerical algorithm for the 3-wave kinetic equation is developed and validated, enabling detailed study of wave turbulence phenomena including dissipation and thermalization.

Findings

Dissipation becomes independent of cut-off in finite capacity cascades.

Bottleneck structure varies with interaction coefficient.

Evidence of mixed solutions combining turbulence and equilibrium spectra.

Abstract

We show that the isotropic 3-wave kinetic equation is equivalent to the mean field rate equations for an aggregation-fragmentation problem with an unusual fragmentation mechanism. This analogy is used to write the theory of 3-wave turbulence almost entirely in terms of a single scaling parameter. A new numerical method for solving the kinetic equation over a large range of frequencies is developed by extending Lee's method for solving aggregation equations. The new algorithm is validated against some analytic calculations of the Kolmogorov-Zakharov constant for some families of model interaction coefficients. The algorithm is then applied to study some wave turbulence problems in which the finiteness of the dissipation scale is an essential feature. Firstly, it is shown that for finite capacity cascades, the dissipation of energy becomes independent of the cut-off frequency as this cut-off is taken to infinity. This is an explicit indication of the presence of a dissipative anomaly. Secondly, a preliminary numerical study is presented of the so-called bottleneck effect in a wave turbulence context. It is found that the structure of the bottleneck depends non-trivially on the interaction coefficient. Finally some results are presented on the complementary phenomenon of thermalisation in closed wave systems which demonstrates explicitly for the first time the existence of so-called mixed solutions of the kinetic equation which exhibit aspects of both Kolmogorov-Zakharov and equilibrium equipartition spectra.