Turbulence of generalised flows in two dimensions

Abstract

This paper discusses the generalised least-action principle introduced by Brenier (J. Am. Math. Soc, 1989), from the perspective of turbulence modelling. In essence, Brenier's least-action principle extends to a probabilistic setting Arnold's geometric interpretation of ideal fluid mechanics, whereby strong solutions to the Euler equations are deduced from minimising an action over Lagrangian maps. While Arnold's framework relies on the deterministic concept of Lagrangian flow, Brenier's least-action principle describes solutions to the Euler equations in terms of non-deterministic "generalised flows", namely probability measures over sets of Lagrangian trajectories. Generalised flows seem naturally fit to describe turbulent Lagrangian trajectories in terms of stochastic processes, an approach that originates from Richardson's seminal work on turbulent dispersion. In particular, they seem suited to address cases when the concept of Lagrangian flow breaks down, due to Lagrangian trajectories becoming spontaneously stochastic. The purpose of the present paper is therefore to give a physical perspective on Brenier's principle, and provide a qualitative description of the hydrodynamical features of generalised flows. Using Monte-Carlo techniques, we analyse the statistical features of three classes of two-dimensional generalised flows, ranging from solid rotation and cellular flows to to freely decaying two-dimensional turbulence. Our results suggest that generalised variational formulations, if carefully used, may provide new tools to coarse-grain multi-scale hydrodynamics.