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Time-periodic solutions to an energy balance model coupled with an active fluid under arbitrary large forces

Gianmarco Del Sarto, Matthias Hieber, Filippo Palma, Tarek Zöchling

TL;DR

The paper proves the existence of strong time-periodic solutions for a coupled Sellers-type energy balance model and three-dimensional primitive equations under time-periodic forcing and a dynamic boundary condition, without smallness restrictions on the forcing. It employs a two-step framework: first constructing a weak $T^*$-periodic solution via Galerkin methods and Brouwer's fixed point, then leveraging global strong well-posedness and a weak-strong uniqueness argument to obtain a strong $T^*$-periodic solution. The results extend to strong steady states under time-independent forcing and demonstrate global-in-time, large-force robustness for this climate-fluid system. This advances understanding of time-periodic and steady-state behavior in coupled ocean-atmosphere models with dynamic boundary interactions, with potential implications for climate dynamics under cyclic forcing.

Abstract

This article concerns time-periodic solutions to a two-dimensional Sellers-type energy balance model coupled to the three-dimensional primitive equations via a dynamic boundary condition. It is shown that the underlying equations admit at least one strong time-periodic solution, provided the forcing term is time-periodic. The forcing term does not need to satisfy a smallness condition and is allowed to be arbitrarily large.

Time-periodic solutions to an energy balance model coupled with an active fluid under arbitrary large forces

TL;DR

The paper proves the existence of strong time-periodic solutions for a coupled Sellers-type energy balance model and three-dimensional primitive equations under time-periodic forcing and a dynamic boundary condition, without smallness restrictions on the forcing. It employs a two-step framework: first constructing a weak -periodic solution via Galerkin methods and Brouwer's fixed point, then leveraging global strong well-posedness and a weak-strong uniqueness argument to obtain a strong -periodic solution. The results extend to strong steady states under time-independent forcing and demonstrate global-in-time, large-force robustness for this climate-fluid system. This advances understanding of time-periodic and steady-state behavior in coupled ocean-atmosphere models with dynamic boundary interactions, with potential implications for climate dynamics under cyclic forcing.

Abstract

This article concerns time-periodic solutions to a two-dimensional Sellers-type energy balance model coupled to the three-dimensional primitive equations via a dynamic boundary condition. It is shown that the underlying equations admit at least one strong time-periodic solution, provided the forcing term is time-periodic. The forcing term does not need to satisfy a smallness condition and is allowed to be arbitrarily large.

Paper Structure

This paper contains 5 sections, 5 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Main Results
  3. Global strong solutions
  4. Weak time periodic solutions
  5. Weak-strong uniqueness

Key Result

Theorem 2

Let $T^*>0$ and $(f_1,f_2,f_3) \in \mathbb{E}_{0,\tau}$ be, for all $\tau>0$, $T^*$-periodic given external forces. Then the system eq: primitive + EBM simplified subject to the boundary conditions eq:bc has at least one strong $T^*$-periodic solution.

Theorems & Definitions (10)

  • Definition 1
  • Theorem 2
  • Corollary 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • proof : Proof of Theorem \ref{['thm: main']}
  • Lemma 6
  • proof