Stochastic Wave–Current Interaction in Thermal Shallow Water Dynamics

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.

Computational modeling for high-fidelity coarsening of shallow water equations based on subgrid data

“…In the SALT approach, one investigates differences only in the velocity variables, since one introduces stochasticity in the vector fields that carry the flow properties. Results of [13] imply that for this situation, obtaining the ξ i in one dimension and extending their domain to two dimensions corresponds to ξ i obtained from the two-dimensional translation-invariant setting.…”

“…The SW equations, also called the Saint-Venant equations, describe the behaviour of a fluid in a shallow channel with a free surface and bottom topography. This model can be derived by vertically integrating the incompressible free surface Euler-Boussinesq equations over the shallow domain in the small aspect ratio limit, as is demonstrated in [13]. The SW model is nonlinear and consists of two coupled equations.…”

A Local-in-Time Theory for Singular SDEs with Applications to Fluid Models with Transport Noise

“…Transforming the equation into a partial differential equation with random coefficients, the well-posedness of the stochastic CH equation with some special transport noise has been studied in Albeverio et al (2021). We can extend this result to a far more complex system: the stochastic two-component CH system (see Holm and Luesink (2021) for the related models), i.e.,…”

“…To set the stage, in Sect. 3, we consider two models governing ideal flows with particularly interesting stochastic perturbation, namely -the two-component Camassa-Holm (CH) system with transport noise (Holm and Luesink 2021), see (3.4), -a nonlinear transport equation with non-local velocity, referred as the Córdoba-Córdoba-Fontelos (CCF) model (Córdoba et al 2005), with transport noise, see (3.10).…”

Existence and Uniqueness of Maximal Solutions to SPDEs with Applications to Viscous Fluid Equations

“…In these papers Holm and Memin establish a new class of stochastic equations driven by transport noise which serve as fluid dynamics models by adding uncertainty in the transport of the fluid parcels to reflect the unresolved scales. The significance of such equations in modelling, numerical schemes and data assimilation continues to be well documented, see ( [6], [7], [32], [31] [49], [37], [9], [15], [36], [5], [20], [21], [2]). In contrast there has been limited progress in proving well-posedness for this class of equations: Crisan, Flandoli and Holm [8] have shown the existence and uniqueness of maximal solutions for the 3D Euler Equation on the torus, whilst Crisan and Lang ([10], [11], [12]) extended the well-posedness theory for the Euler, Rotating Shallow Water and Great Lake Equations on the torus once more.…”