Turbulent Energy Decay Along the Path of Least Action

A fluctuation law of the energy in freely-decaying, homogeneous and isotropic turbulence is derived within a Gaussian closure ansatz for 3D incompressible flow. In particular, a fluctuation-dissipation relation is derived which relates the strength of a stochastic backscatter term in the energy decay equation to the mean of the energy dissipation rate. The theory is based on the so-called “effective action” of the energy history and illustrates a Rayleigh-Ritz method recently developed to evaluate the effective action approximately within probability density-function (PDF) closures. These effective actions generalize the Onsager-Machlup action of nonequilibrium statistical mechanics to turbulent flow and yield detailed, concrete predictions for fluctuations, such as multi-time correlation functions of arbitrary order. They also characterize the mean histories by a variational principle.