Lagrangian Averaged Stochastic Advection by Lie Transport for Fluids (original) (raw)

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 (Proc R Soc A 471(2176):20140963, 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 devel...