  This paper introduces an energy-preserving 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. We also present an Eulerian-averaged SFLT
model (EA SFLT), based on decomposing the Eulerian solutions of the
energy-conserving SFLT model into sums of their expectations and fluctuations.
