Abstract
In analogy with similar effects in adiabatic compressible fluid dynamics, the effects of buoyancy gradients on incompressible stratified flows are said to be “thermal.” The thermal rotating shallow water (TRSW) model equations contain three small nondimensional parameters. These are the Rossby number, the Froude number, and the buoyancy parameter. Asymptotic expansion of the TRSW model equations in these three small parameters leads to the deterministic thermal versions of the Salmon's L1 (TL1) model and the thermal quasi-geostrophic (TQG) model, upon expanding in the neighborhood of thermal quasi-geostrophic balance among the flow velocity and the gradients of free surface elevation and buoyancy. The linear instability of TQG at high wavenumber tends to create circulation at small scales. Such a high-wavenumber instability could be unresolvable in many computational simulations, but its presence at small scales may contribute significantly to fluid transport at resolvable scales. Sometimes, such effects are modeled via “stochastic backscatter of kinetic energy.” Here, we try another approach. Namely, we model “stochastic transport” in the hierarchy of models TRSW/TL1/TQG. The models are derived via the approach of stochastic advection by Lie transport (SALT) as obtained from a recently introduced stochastic version of the Euler–Poincaré variational principle. We also indicate the potential next steps for applying these models in uncertainty quantification and data assimilation of the rapid, high-wavenumber effects of buoyancy fronts at these three levels of description by using the data-driven stochastic parametrization algorithms derived previously using the SALT approach.
Highlights
In this paper we are dealing with the thermal rotating shallow water (TRSW) equations, which can be regarded as the vertically averaged version of the primitive equations with a buoyancy variable [Zei18].In the balanced 2D model hierarchy of TRSW, thermal L1 (TL1) and thermal quasi-geostrophic (TQG), we are investigating a certain stochastic model of potential vorticity dynamics as a basis for stochastic parametrisation of the dynamical creation of unresolved degrees of freedom in computational simulations of upper ocean dynamics
This mathematical foundation is needed in applying the stochastic advection by Lie transport (SALT) (Stochastic Advection by Lie Transport) approach to the derivation of stochastic fluid equations which preserve the geometric structure of fluid dynamics [Hol15]
Our motivation in this paper has been to prepare the mathematical framework for our ongoing investigations of Stochastic Transport in Upper Ocean Dynamics (STUOD) by using the stochastic data assimilation algorithms developed and applied previously to determine the eigenvectors ξi(x) in the cases of the stochastic Euler fluid equation and the 2-layer stochastic QG model in [CCH+18, CCH+19]
Summary
In this paper we are dealing with the thermal rotating shallow water (TRSW) equations, which can be regarded as the vertically averaged version of the primitive equations with a buoyancy variable [Zei18]. In preparation for meeting this challenge, the present paper develops the stochastic variational principles for the TRSW, TL1 and TQG models This mathematical foundation is needed in applying the SALT (Stochastic Advection by Lie Transport) approach to the derivation of stochastic fluid equations which preserve the geometric structure of fluid dynamics [Hol15]. In the Euler–Poincare framework, we prove the Kelvin–Noether circulation theorem and discuss steady solution properties of the deterministic TRSW equations This derivation of the TRSW equations with stochastic advection by Lie transport (SALT) is intended to be the mathematical foundation for a systematic means of introducing data-driven parametrisations of stochastic transport for uncertainty quantification and data assimilation for upper ocean dynamics. For a related deterministic discussion of a fully multilayer variational model with nonhydrostatic pressure, see [CHP10]
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