Theoretical and computational analysis of the thermal quasi-geostrophic model

Abstract

This work involves theoretical and numerical analysis of the Thermal Quasi-Geostrophic (TQG) model of submesoscale geophysical fluid dynamics (GFD). Physically, the TQG model involves thermal geostrophic balance, in which the Rossby number, the Froude number and the stratification parameter are all of the same asymptotic order. The main analytical contribution of this paper is to construct local-in-time unique strong solutions for the TQG model. For this, we show that solutions of its regularized version α-TQG converge to solutions of TQG as its smoothing parameter α → 0 and we obtain blowup criteria for the α-TQG model. The main contribution of the computational analysis is to verify the rate of convergence of α-TQG solutions to TQG solutions as α → 0 for example simulations in appropriate GFD regimes.

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.…”

Existence and Uniqueness of Maximal Solutions to a 3D Navier-Stokes Equation with Stochastic Lie Transport

“…Whilst this represents a strong start in the theoretical analysis (alongside works for SPDEs with general transport noise e.g. [12], [13]), the modelling literature continues to expand in both the deterministic fluid models (see for example Figure 2 of [7] and the analysis therein) and method of stochastic perturbation (for example we may soon look to introduce nonlinearity and time dependence in the (ξ i )). The significance of an abstract approach to the well-posedness question is clear, and whilst we discuss here only an application to SALT Navier-Stokes ( [4], [9]) the hope is that other stochastic viscous fluid models can be similarly solved by simply checking the required assumptions.…”

Breakdown of strong solutions of the Thermal Quasi-Geostrophic equation

“…The existence of a unique strong solution of (1.1)-(1.4) has recently been shown in [3,Theorem 2.10] on the torus. A unique maximal solutions also exist [3, Theorem 2.14] and the result also applies to the whole space [3, Remark 2.1].…”

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The thermal Quasi-Geostrophic equation is a coupled system of equations that governs the evolution of the buoyancy and the potential vorticity of a fluid. It has a local in time solution as proved in [3]. In this paper, we give criterion for the breakdown of solutions to the Thermal Quasi-Geostrophic (TQG) equations, in the spirit of the classical Beale-Kato-Majda blowup criterion (cf.

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