Dynamical Stochastic Modeling of Turbulence

Abstract

Fully developed Navier-Stokes turbulence has proved resistant to methods of contemporary nonlinear dynamics that have been successful for small systems. An irreducibly large number of modes is excited in high-Reynolds-number turbulence. A set of tools that permits some progress is dynamical stochastic modeling, by which I mean the exact solution of model dynamical systems that have some relation to the true dynamics. The fact that a system is being solved exactly ensures some important consistency properties. Dynamical stochastic models whose tractability comes from intrinsic randomicity have given qualitatively and, in some cases, quantitatively good approximations to major features of fully developed turbulence: sensitivity to initial conditions, eddy viscosity, spectral energy cascade, and vorticity intensification. The present brief summary outlines three kinds of stochastic models: systems with random coupling coefficients, decimation models, and a new kind of stochastic model based on nonlinear mapping of Gaussian fields into dynamically evolving non-Gaussian fields.