Stochastic wave-current interaction in thermal shallow water dynamics

TL;DR

This paper develops a new stochastic parametrisation approach for nonlinear wave-current interactions in thermal shallow water dynamics, combining variational principles, asymptotic expansions, and vertical averaging to preserve key fluid features.

Contribution

It introduces a novel stochastic framework for shallow water equations derived from Euler's equations, capturing wave-current interactions while maintaining fundamental fluid dynamics properties.

Findings

Kelvin's circulation theorem reveals a barotropic mechanism for wave-induced circulation.

The approach models stochastic wave-current interactions in stratified shallow water.

New equations describe wave generation of horizontal circulation under non-aligned gradients.

Abstract

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.