Aggregation-Fragmentation Processes and Wave Kinetics

TL;DR

This paper establishes a link between wave turbulence and fragmentation processes, analyzing the spectral evolution in 3-wave systems with constant interaction kernels, revealing new divergence behaviors and decay laws validated by simulations.

Contribution

It introduces a novel analytical approach connecting wave kinetics with fragmentation theory, providing new insights into spectral divergence and decay in 3-wave turbulence.

Findings

In forced turbulence, the spectrum diverges as x^{-3/2} consistent with Kolmogorov-Zakharov theory.

In decaying turbulence, the spectrum exhibits unexpected algebraic and logarithmic divergences.

Theoretical predictions are confirmed through high-quality numerical simulations.

Abstract

There is a formal correspondence between the isotropic 3-wave kinetic equation and the rate equations for a non-linear fragmentation--aggregation process. We exploit this correspondence to study analytically the time evolution of the wave frequency power spectrum. Specifically, we analyzed a 3-wave turbulence in which the wave interaction kernel is a constant. We consider both forced and decaying turbulence. In the forced case, the scaling function diverges as $x^{-3/2}$ as expected from Kolmogorov-Zakharov theory. In the decaying case, the scaling function exhibits non-trivial, and hitherto unexpected, divergence with both algebraic and logarithmic spectral exponents which we calculate. This divergence leads to non-trivial decay laws for the total wave action and the number of primary waves. All theoretical predictions are verified with high quality numerical simulations of the 3-wave kinetic equation.