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

Eduard Feireisl, Piotr Gwiazda, Agnieszka Świerczewska-Gwiazda

TL;DR

The paper analyzes data assimilation for a rotating Oberbeck–Boussinesq model that arises as a singular limit ($\varepsilon\to 0$) of the 3D compressible Navier–Stokes–Fourier system with rotation and a temperature gradient. It proves global well-posedness and dissipativity for the reduced rOB system, develops a continuous nudging data-assimilation framework using an interpolant, and establishes exponential convergence to a reference trajectory; it then connects these results to the full 3D model via a relative entropy approach. By combining the reduced model analysis with relative entropy, the authors justify assimilating data at the lower-dimensional level while controlling assimilation error in the full 3D setting, offering potential computational savings for weather prediction. The framework highlights when and how to apply nudging in a physically relevant, rotation-influenced convection context, providing a robust bridge between simplified and full atmospheric models with rigorous guarantees. These results support efficient, accurate data assimilation workflows in meteorology that exploit natural dimensional reduction without sacrificing fidelity to the underlying 3D dynamics.

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.

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

TL;DR

The paper analyzes data assimilation for a rotating Oberbeck–Boussinesq model that arises as a singular limit () of the 3D compressible Navier–Stokes–Fourier system with rotation and a temperature gradient. It proves global well-posedness and dissipativity for the reduced rOB system, develops a continuous nudging data-assimilation framework using an interpolant, and establishes exponential convergence to a reference trajectory; it then connects these results to the full 3D model via a relative entropy approach. By combining the reduced model analysis with relative entropy, the authors justify assimilating data at the lower-dimensional level while controlling assimilation error in the full 3D setting, offering potential computational savings for weather prediction. The framework highlights when and how to apply nudging in a physically relevant, rotation-influenced convection context, providing a robust bridge between simplified and full atmospheric models with rigorous guarantees. These results support efficient, accurate data assimilation workflows in meteorology that exploit natural dimensional reduction without sacrificing fidelity to the underlying 3D dynamics.

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.
Paper Structure (24 sections, 6 theorems, 104 equations)

This paper contains 24 sections, 6 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Primitive system
  3. Singular limit and the target system
  4. Mathematics of rotating Oberbeck--Boussinesq system
  5. Well-posedness
  6. Transformation
  7. Global existence
  8. A priori bounds
  9. Harmonic extension of boundary data
  10. Energy bounds
  11. Maximum principle for the temperature
  12. Regularity of the velocity field
  13. Regularity of the time derivative of the temperature
  14. $L^p$-regularity of the temperature
  15. Long time behaviour
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let $\Omega_h \subset R^2$ be bounded domain of class $C^2$. Suppose the boundary temperature is a restriction of a smooth function $\vartheta_B \in C^2(R^3)$. Let the initial data belong to the class and satisfy the compatibility conditions Then there exists a global-in-time solution ${\bf u}_h$, $\Theta$ of problem t2, t3, supplemented with the boundary conditions t4, and the initial condition

Theorems & Definitions (7)

  • Theorem 2.1: Well posedness
  • Proposition 3.1: Absorbing set in $L^2 \times L^2$
  • Proposition 3.2: Levinson dissipativity
  • Remark 3.3
  • Theorem 3.4: Absorbing set in $W^{1,2} \times L^\infty$
  • Theorem 4.1: Convergence of data assimilation method
  • Theorem 5.1: Approximate data assimilation