This project seeks to construct models for the study of upper ocean phenomena with two characteristics: 1) to be capable of incorporating thermodynamic processes while maintaining the 2-d structure of the rotating shallow-water equations, a paradigm of ocean dynamics on scales longer than a few hours; and 2) to preserve the geometric (generalized Hamiltonian/Euler–Poincare) structure of the exact 3-d models from which they derive. Characteristic 1) promises fundamental understanding of ocean processes difficult---if not impossible—--to be attained using general circulation models. Characteristic 2) enables application of a recent flow-topology-preserving framework for building parameterizations of unresolvable submesoscale motions. The overall goal is to derive such geometric thermal shallow-water theories to investigate the effects of submesoscale circulations on transport at resolvable mesoscales, a subject of active research.
The work will build on two existing thermal shallow-water theories. One theory enjoys the required geometric characteristic, but falls short at representing important mixed-layer phenomena such as restratification induced by baroclinic instability. The second theory represents an important improvement over the first theory in that it is capable of simulating mixed-layer restratification, yet it does not possess the needed geometric structure. To build a theory that it is capable of describing the tendency of buoyancy gradients to slump from the horizontal to the vertical in a geometrically consistent manner, we will start from the Hamilton’s principle for the Euler-Boussinesq equations for stratified fluid, and make appropriate approximations (e.g., truncations in the vertical structure of the dynamical fields) in the associated action functional, approach that is guaranteed to preserve geometry.
Once settled on appropriate deterministic geometric thermal shallow-water theories, stochastic versions thereof will be derived. This will be followed by high-resolution (deterministic) simulations and extraction of optimal stochastic approximations equivalent to known deterministic parameterizations. Applications for the various model sets will be developed to probe their adequacy for different purposes at simulations on different scales.
Nonlinear saturation of thermal instabilities, F.J. Beron-Vera, Physics of Fluids
Multilayer shallow-water model with stratification and shear, F.J. Beron-Vera, Revista Mexicana de Física, 2020
