Weak solution of the merged mathematical equations of the polluted atmosphere

TL;DR

This paper extends the mathematical analysis of atmospheric fluid dynamics by incorporating pollution effects into the hydrostatic approximation of the Navier-Stokes equations, providing a rigorous foundation for polluted atmospheric models.

Contribution

It generalizes existing convergence results from ocean models to atmospheric models with pollution, including an additional convection-diffusion equation for pollutants.

Findings

Established the existence of weak solutions for the polluted atmosphere equations.

Extended convergence theorems to include pollution effects.

Provided mathematical validation for pollutant transport modeling in the atmosphere.

Abstract

Considered as a geophysical fluid, the polluted atmosphere shares the shallow domain characteristics with other natural large-scale fluids such as seas and oceans. This means that its domain is excessively greater horizontally than in the vertical dimension, leading to the classic hydrostatic approximation of the Navier-Stokes equations. The authors of the \cite{azerad2001mathematical} article have proved a convergence theorem for this model with respect to the ocean, without considering pollution effects. The novelty of this present work is to provide a generalisation of their result translated to the atmosphere, extending the fluid velocity equations with an additional convection-diffusion equation representing pollutants in the atmosphere.