Lagrangian Averaged Stochastic Advection by Lie Transport for Fluids

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 nonlocal, in both physical and probability space. Without taking this average, the equations recover the Stochastic Advection by Lie Transport (SALT) fluid equations introduced by Holm [1]. Remarkably, the introduction of the non-locality in probability space in the form of advecting the velocity by its own mean 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 [2] in the case when the noise correlates are taken to be the constant basis vectors in 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 nonequilibrium stochastic linear response theory for fluctuations in SALT fluids with advected quantities.

A Hamiltonian Interacting Particle System for Compressible Flow

“…Recent advances in the stochastic modeling of geophysical flows include the "location uncertainty" approach of Mémin, Resseguier and collaborators [23][24][25][26], as-well as the "stochastic advection by Lie transport (SALT)" theory of Holm and collaborators [6,[27][28][29][30]. The SALT approach is based on the observation that subgrid phenomena represent unknown physical processes, and should therefore be derived from a stochastic variational principle.…”

“…Since (32)- (33) are obtained as a mean field limit of a Hamiltonian IPS, and can be viewed as mean field generalization of the stochastic fluid system in [53], there should be a Kelvin Circulation Theorem: Proposition 2. Let C be a smooth closed loop which is transported by the Lagrangian flow Φ t , defined through the mean field limit (29) and characterized by…”

“…of [27,29], where u L t is a stochastic velocity field. However, there are a few crucial differences-while (50) is the starting point for the LA SALT theory, the HIPS formulation is based on the ensemble Hamiltonian (22), and the Lie transport along (49) in the mean field Equations (32)-(33) is a consequence of the Hamiltonian structure of the IPS (and the ensuing passage to the mean field limit).…”

Symmetry actuated closed-loop Hamiltonian systems

“…cit. Examples of finite dimensional stochastic Hamiltonian (and almost-Hamiltonian) systems are in [24,17,18,3] and examples of stochastic Hamiltonian fluid dynamical systems can be found in [22,12,19,13]. The stochastic energy-Casimir method, whose deterministic version has been successfully applied to stabilize closed-loop systems in [6,7,8,9,10,20], has been developed by [4].…”

Introduction to the Special Issue on the Statistical Mechanics of Climate

“…Alonso-Oran et al [4] propose in the case of two-dimensional Euler-Boussinesq equations a closed theory of weather and climate-intended as statistics of fluctuations and expectation value of the quantities of interest, respectively. This is achieved by taking into account the corresponding Lagrangian averaged stochastic advection by Lie transport equations, which are nonlinear and non-local, in both physical and probability space [28]. This formalism is further used to explore the properties of a stochastically perturbed low-dimensional chaotic dynamical system.…”