Propagation of random excitations in the polar representation of strongly nonlinear four-wave systems of turbulence.

Abstract

The present work is concerned with the uncertainty propagation of the wave turbulent system. In particular, we study the temporal development and long-term behavior of the probability with respect to the amplitude and phase of complex-valued waves constituting the generic four-wave system of turbulence. Our approach to approximating the target distribution function is via the three steps: (i) to grasp the physical process described by the true turbulence model as random process, (ii) to determine the stochastic differential equation whose solution exhibits statistically similar behavior with the underlying turbulent signal, and (iii) to solve the corresponding Kolmogorov forward equation. Our implementation of the methodology is distinguished by employing a number of simplified stochastic models and applying one of them in the adaptive fashion which varies subject to the different parameter regime of the true dynamical system model. Accordingly, we become able to demonstrate the effectiveness of this reduced-order modeling framework for the analysis of the turbulent system characterized by not only weak but strong interactions among the nonlinear waves. We numerically corroborate our theoretical predictions in the context of the generalized Majda-Mclaughlin-Tabak wave turbulence prototype.