Lagrangian averaged stochastic advection by Lie transport for fluids

TL;DR

This paper introduces LA SALT, a novel stochastic PDE framework for ideal fluids that incorporates non-locality in probability space, leading to a regularized, closed-form expectation equation related to Navier-Stokes dynamics.

Contribution

It formulates the LA SALT equations, extending SALT by including non-local probability effects and advected quantities, providing a new regularization mechanism for fluid models.

Findings

LA SALT equations recover SALT equations before averaging.

The expectation field satisfies a Navier-Stokes-like equation with Lie-Laplacian dissipation.

The framework models the statistical evolution of fluctuations in stochastic fluid flows.

Abstract

We formulate a class of stochastic partial differential equations based on Kelvin's circulation theorem for ideal fluids. In these models, the velocity field is randomly transported by white-noise vector fields, as well as by its own average over realizations of this noise. We call these systems the Lagrangian averaged stochastic advection by Lie transport (LA SALT) equations. These equations are nonlinear and non-local, in both physical and probability space. Before taking this average, the equations recover the Stochastic Advection by Lie Transport (SALT) fluid equations introduced by Holm (2015). Remarkably, the introduction of the non-locality in probability space in the form of momentum transported by its own mean velocity gives rise to a closed equation for the expectation field which comprises Navier--Stokes equations with Lie--Laplacian "dissipation". As such, this form of non-locality provides a regularization mechanism. The formalism we develop is closely connected to the stochastic Weber velocity framework of Constantin and Iyer (2008) in the case when the noise correlates are taken to be the constant basis vectors in $\mathbb{R}^3$ and, thus, the Lie--Laplacian reduces to the usual Laplacian. We extend this class of equations to allow for advected quantities to be present and affect the flow through exchange of kinetic and potential energies. The statistics of the solutions for the LA SALT fluid equations are found to be changing dynamically due to an array of intricate correlations among the physical variables. The statistical properties of the LA SALT physical variables propagate as local evolutionary equations which when spatially integrated become dynamical equations for the variances of the fluctuations. Essentially, the LA SALT theory is a non-equilibrium stochastic linear response theory for fluctuations in SALT fluids with advected quantities.