Theoretical and computational analysis of the thermal quasi-geostrophic model

TL;DR

This paper provides a rigorous mathematical framework for the Thermal Quasi-Geostrophic (TQG) model, establishing existence, uniqueness, and convergence of solutions, alongside numerical verification of the model's behavior in geophysical fluid dynamics.

Contribution

It introduces a novel analytical approach to construct strong solutions for TQG and demonstrates convergence from regularized models to the original, supported by computational simulations.

Findings

Existence and uniqueness of local-in-time strong solutions for TQG.

Convergence of regularized $ extalpha$-TQG solutions to TQG as $ extalpha ightarrow 0$.

Numerical verification of convergence rates in geophysical fluid dynamics regimes.

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 $\alpha$-TQG converge to solutions of TQG as its smoothing parameter $\alpha \rightarrow 0$ and we obtain blowup criteria for the $\alpha$-TQG model. The main contribution of the computational analysis is to verify the rate of convergence of $\alpha$-TQG solutions to TQG solutions as $\alpha \rightarrow 0$ for example simulations in appropriate GFD regimes.