Abstract
Holm (Proc R Soc A Math Phys Eng Sci 471(2176):20140963, 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 waves in fluid dynamics. The approach is applied to two sequences of shallow water models 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.
Highlights
Weather forecasting, climate change prediction and global ocean circulation all face the same fundamental challenge to create models which incorporate the effects of measurement error and uncertainty due to unresolved scales, unknown physical phenomena and incompleteness of observed data
A protocol for applying the stochastic advection by lie transport (SALT) approach in combination with data assimilation based on comparing fine-scale and coarse-scale computational simulations has recently been established in Cotter et al (2018, 2019a). These results demonstrate the capability of the SALT approach to successfully reduce forecast uncertainty in a variety of test problems for fluid dynamics in two spatial dimensions
The present paper aims to use the SALT approach for fluid dynamics described above to provide a barotropic description of wave–current interaction (WCI) in a stratified incompressible fluid flow, by incorporating stochastic fluid transport and circulation with nonlinear dispersive wave propagation internally and on the free surface
Summary
Weather forecasting, climate change prediction and global ocean circulation all face the same fundamental challenge to create models which incorporate the effects of measurement error and uncertainty due to unresolved scales, unknown physical phenomena and incompleteness of observed data.
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