Remarks on the principles of statistical fluid mechanics

TL;DR

This paper provides a survey of statistical fluid mechanics focusing on the Hopf functional differential equation, illustrating approaches to turbulence theory with the Burgers equation and discussing finite-dimensional approximations.

Contribution

It offers an idiosyncratic review highlighting connections between turbulence, wave propagation in random media, and finite-dimensional approximations.

Findings

Review of functional integration approaches to turbulence

Identification of contributions from wave propagation research

Discussion of finite-dimensional approximation methods

Abstract

This is an idiosyncratic survey of statistical fluid mechanics centering on the Hopf functional differential equation. Using the Burgers equation for illustration we review several functional integration approaches to theory of turbulence. We notice in particular that some important contributions have been brought about by researchers working on wave propagation in random media, among which Uriel Frisch is not an exception. We also discuss a particular finite-dimensional approximation for the Burgers equation.