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Well-posedness and long time dynamics for a quasi-geostrophic ocean-atmosphere model with radiation balance

Federico Fornasaro, Tobias Kuna, Giulia Carigi

TL;DR

This work analyzes a coupled two-layer quasi-geostrophic atmosphere and one-layer ocean model with radiation balance (MAOOAM), establishing rigorous well-posedness and long-time dynamics. Using Galerkin methods and energy estimates, it proves existence and uniqueness of weak solutions, then strengthens to quasi-strong and strong solutions, and demonstrates the existence of a finite-dimensional global attractor. It also proves the injectivity of the semigroup on the attractor and provides Foias–Prodi-type bounds leading to a finite number of determining modes, enabling ocean temperature reconstruction from velocity data. The results offer a solid mathematical foundation for reduced-order modeling and data assimilation in mid-latitude climate dynamics, showing that long-term behavior is governed by finite degrees of freedom despite the infinite-dimensional governing equations.

Abstract

We investigate a coupled atmosphere-ocean model including the mechanical and thermodynamical interaction between the two fluids for the mid-latitudes. The formulation combines a multilayer quasi-geostrophic dynamical framework with temperature equations incorporating long- and short-wave radiative forcing, as in energy balance models. Within a suitable functional framework, we establish the existence and uniqueness of solutions, and their continuous dependence on the radiation parameters. We also prove that the long-time dynamics are described by a finite-dimensional global attractor and, moreover, that the system possesses a finite set of determining modes that governs its asymptotic behaviour. In particular, we show that the long-term evolution of the ocean's temperature can be reconstructed solely from observations of the velocity fields across the model's layers.

Well-posedness and long time dynamics for a quasi-geostrophic ocean-atmosphere model with radiation balance

TL;DR

This work analyzes a coupled two-layer quasi-geostrophic atmosphere and one-layer ocean model with radiation balance (MAOOAM), establishing rigorous well-posedness and long-time dynamics. Using Galerkin methods and energy estimates, it proves existence and uniqueness of weak solutions, then strengthens to quasi-strong and strong solutions, and demonstrates the existence of a finite-dimensional global attractor. It also proves the injectivity of the semigroup on the attractor and provides Foias–Prodi-type bounds leading to a finite number of determining modes, enabling ocean temperature reconstruction from velocity data. The results offer a solid mathematical foundation for reduced-order modeling and data assimilation in mid-latitude climate dynamics, showing that long-term behavior is governed by finite degrees of freedom despite the infinite-dimensional governing equations.

Abstract

We investigate a coupled atmosphere-ocean model including the mechanical and thermodynamical interaction between the two fluids for the mid-latitudes. The formulation combines a multilayer quasi-geostrophic dynamical framework with temperature equations incorporating long- and short-wave radiative forcing, as in energy balance models. Within a suitable functional framework, we establish the existence and uniqueness of solutions, and their continuous dependence on the radiation parameters. We also prove that the long-time dynamics are described by a finite-dimensional global attractor and, moreover, that the system possesses a finite set of determining modes that governs its asymptotic behaviour. In particular, we show that the long-term evolution of the ocean's temperature can be reconstructed solely from observations of the velocity fields across the model's layers.
Paper Structure (21 sections, 12 theorems, 127 equations, 1 figure, 1 table)

This paper contains 21 sections, 12 theorems, 127 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Previous results on related models
  3. Physical description of the model, geometry of the domain and boundary conditions
  4. The equations of the model
  5. Geometry of the Domain and Boundary Conditions
  6. Energy Equations
  7. Existence and Uniqueness of the Weak Solution
  8. Existence of Weak Solutions
  9. Uniqueness of Weak Solutions
  10. Continuity with respect to radiation parameters
  11. Solutions with Higher Regularity
  12. Quasi-Strong Solution
  13. Existence of Quasi-Strong solutions
  14. Continuous dependence on initial conditions
  15. Strong Solution
  16. ...and 6 more sections

Key Result

Theorem 4.1

Assume that the short-wave radiation functions $R_a$ and $R_o$ are bounded and continuous in $L^{2}_tL^{2}_x$ and $L^{5/4}_tL^{5/4}_x$, respectively, and they satisfy xlpqsxa. Then for any initial condition $(\bm{\psi}(0),T_o(0))\in (\bm{H}^1_0\times L^2)(\Omega)$, there exists a Weak Solution for the system bb, Temp, eq:Tb. Moreover if assum:R:weaksol holds then the solution is unique and depend

Figures (1)

  • Figure 1: Schematic representation of the domains and boundary conditions for the variables $\bm{\psi}$.

Theorems & Definitions (29)

  • Remark 1
  • Definition 4.1: Weak Solution for MAOOAM
  • Theorem 4.1: Existence and Uniqueness of the Weak Solution for MAOOAM
  • Remark 2
  • Theorem 4.2
  • proof
  • Remark 3
  • Remark 4
  • Definition 5.1: Quasi-Strong Solution for MAOOAM
  • Theorem 5.1
  • ...and 19 more