On a system of equations arising in meteorology: Well-posedness and data assimilation

TL;DR

This paper analyzes a simplified meteorological model derived from the compressible Navier-Stokes-Fourier system, establishing well-posedness, stability, and convergence of data assimilation techniques, and linking the reduced model to the full system.

Contribution

It provides the first rigorous analytical framework for data assimilation in a reduced meteorological model derived from complex 3D fluid dynamics.

Findings

Proved global well-posedness of the reduced model.

Established stability and convergence of nudging data assimilation.

Extended results to the full 3D compressible system using relative entropy.

Abstract

Data assimilation plays a crucial role in modern weather prediction, providing a systematic way to incorporate observational data into complex dynamical models. The paper addresses continuous data assimilation for a model arising as a singular limit of the three-dimensional compressible Navier-Stokes-Fourier system with rotation driven by temperature gradient. The limit system preserves the essential physical mechanisms of the original model, while exhibiting a reduced, effectively two-and-a-half-dimensional structure. This simplified framework allows for a rigorous analytical study of the data assimilation process while maintaining a direct physical connection to the full compressible model. We establish well posedness of global-in-time solutions and a compact trajectory attractor, followed by the stability and convergence results for the nudging scheme applied to the limiting system. Finally, we demonstrate how these results can be combined with a relative entropy argument to extend the assimilation framework to the full three-dimensional compressible setting, thereby establishing a rigorous connection between the reduced and physically complete models.