The Geometry of Non-Ideal Fluids

TL;DR

This paper extends Arnold's geometric approach to ideal fluids to include dissipation and stochastic forcing, developing tractable models and analyzing symmetry breaking and most likely transition paths in fluid dynamics.

Contribution

It generalizes the geometric framework to non-ideal fluids with dissipation and noise, introducing finite-dimensional toy models and deriving equations for transition paths.

Findings

Constructed finite-dimensional toy models for non-ideal fluids.

Demonstrated spontaneous symmetry breaking in these models.

Derived PDEs for instantons in Navier-Stokes equations.

Abstract

Arnold showed that the Euler equations of an ideal fluid describe geodesics on the Lie algebra of incompressible vector fields. We generalize this to fluids with dissipation and Gaussian random forcing. The dynamics is determined by the structure constants of a Lie algebra, along with inner products defining kinetic energy, Ohmic dissipation and the covariance of the forces. This allows us to construct tractable toy models for fluid mechanics with a finite number of degrees of freedom. We solve one of them to show how symmetries can be broken spontaneously.In another direction, we derive a deterministic equation that describes the most likely path connecting two points in the phase space of a randomly forced system: this is a WKB approximation to the Fokker-Plank-Kramer equation, analogous to the instantons of quantum theory. Applied to hydrodynamics, we derive a PDE system for Navier-Stokes instantons.