  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.
