  Holm (Proc. Roy. Soc 2015) introduced a variational framework for
stochastically parametrising unresolved scales of hydrodynamic motion. This
variational framework preserves fundamental features of fluid dynamics, such as
Kelvin's circulation theorem, while also allowing for dispersive nonlinear wave
propagation, both within a stratified fluid and at its free surface. The
present paper combines asymptotic expansions and vertical averaging with the
stochastic variational framework to formulate a new approach for developing
stochastic parametrisation schemes for nonlinear wave fields. The approach is
applied to a variety of shallow water equations which descend from Euler's
three-dimensional fluid equations with rotation and stratification under
approximation by asymptotic expansions and vertical averaging. In the entire
family of nonlinear stochastic wave-current interaction equations derived here
using this approach, Kelvin's circulation theorem reveals a barotropic
mechanism for wave generation of horizontal circulation or convection
(cyclogenesis) which is activated whenever the gradients of wave elevation
and/or topography are not aligned with the gradient of the vertically averaged
buoyancy.
