Stochastic effects of waves on currents in the ocean mixed layer

Abstract

How does one guarantee that a stochastic approach for modelling fluid dynamics will preserve its fundamental deterministic properties, such as (i) energy conservation, (ii) Kelvin circulation theorem and (iii) conserved quantities arising from the Lagrangian particle relabelling symmetry? In fact, a choice must be made. For example, the approach of stochastic advection by Lie transport (SALT) preserves the latter two properties, but SALT does not conserve the deterministic energy. This paper introduces an energypreserving stochastic model for studying wave effects on currents in the ocean mixing layer. The model is called stochastic forcing by Lie transport (SFLT). The SFLT model is derived here from a stochastic constrained variational principle, so it has a Kelvin circulation theorem. The examples of SFLT given here treat 3D Euler fluid flow, rotating shallow water dynamics and the Euler-Boussinesq equations. In each example, one sees the effect of stochastic Stokes drift and material entrainment in the generation of fluid circulation. Moreover, we present an Eulerian-averaged SFLT model (EA SFLT), based on the Eulerian decomposition of solutions of the energy-conserving SFLT model into the sums of their expectations and fluctuations.