Quasi-Markovian property of strong wave turbulence

This paper is concerned with the reduced-order modeling of the strongly nonlinear wave turbulence system. The motivation for such an attempt comes from the utility of the probabilistic coarse-grained model in facilitating the theoretical and numerical analysis of the true dynamical system model. One typical practice of simplifying the complex physical model is, in the spirit of Brownian motion, to replace the nonlinear interactions by white noise forcing and linear dissipation. For the case of slowly varying longwave, the resulting Markov process is an accurate approximate model. However, this conventional scheme is highly inappropriate for the description of shortwaves because the rapidly varying turbulent signal acquires a significantly non-Markovian character resulting from the poor timescale separation between the relevant mode and the environmental wave field. To resolve the issue, we discuss a simplification technique for which the central concept is the quasi-Markovian property; a non-Markov stochastic process is referred to as quasi-Markovian if it can be represented as a component of Markovian system made by adding a finite number of auxiliary variables. Our contribution in this work is to single out the nontrivial and near resonances from the nonlinear interactions in search of the auxiliary variable. We perform a comparison analysis of the autocorrelation matrices of the true and approximate models, and numerically demonstrate the effectiveness of our Markovian formulation of the inherently non-Markov turbulent signal.