Stochastic mesoscale circulation dynamics in the thermal ocean

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.

Extended shallow-water theories with thermodynamics and geometry

“…This feature was extensively used to simulate mixed-layer (i.e., the topmost part of the ocean, above the thermocline) dynamics, particularly through the 1980s and 1990s (e.g., Beier 2 , McCreary, Kundu, and Molinari 32 , Schopf and Cane 66 ) realistically at a relatively low computational cost. Moreover, recent high-resolution numerical simulations 6,20 of a quasigeostrophic (QG) approximation of the system 60 revealed the formation of Kelvin-Helmholtz-like rollup filaments (Fig. 1) that resemble quite well submesoscale (1-10 km) features identified in satellite-derived chlorophyll distributions on the surface of the ocean.…”

“…The renewed interest in thermodynamically-active rotating shallow-water modeling 5,6,13,15,20,35,[71][72][73][74] motivated this work, which presented extended rotating shallow-water theories with thermodynamics and geometry. With a focus on the ocean mixed-layer, the topmost part of the ocean, a general model was introduced and interpreted as such, considering in addition several submodels, ranging from the shallow-water equations themself.…”

“…This paper is motivated by renewed interest 5,6,13,15,20,35,[71][72][73][74] in "thermal shallow-water modelling" of geophysical flows. Due to Pedro Ripa's investigation of its geometric structure and stability properties, [57][58][59][60]62 the core model has been known to many 10,43,52,65 as Ripa's model.…”

Carriers of \emph{Sargassum} and mechanism for coastal inundation in the Caribbean Sea

“…Setting ψ 2 = 0 and making H → ∞ while keeping rH finite, leads to the inhomogeneous-layer reduced-gravity QG model originally developed in [Ripa, 1996]. See [Beron-Vera, 2021d,a;Holm et al, 2020] for recent discussions on geometric aspects of this model as well as on sustained "thermal" instabilities [Gouzien et al, 2017], and [Crisan et al, 2021] for the construction of unique solutions. Well-posedness of ( 6) is here supported on its uniqueness of solutions and manifest generalized (noncanonical) Hamiltonian structure (cf.…”

“…This is based on the proposition of a minimal model for Caribbean Sea vortex dynamics, whose properties are discussed in the Appendix. The model builds on an old recipe, which used to be very common in ocean dynamics [Ripa, 1993] and is regaining momentum [Kurganov et al, 2020;Beron-Vera, 2021b,d,a;Holm et al, 2020], to include thermodynamics in the two-dimensional rotating shallow-water model. The paper is closed with a summary and some concluding remarks in Section 5.…”