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Dynamic Behavior of a Multi-Layer Quasi-Geostrophic Model: Weak and Time-Periodic Solutions

Zineb Hassainia, Haroune Houamed

TL;DR

The paper analyzes the QS2L two-layer quasi-geostrophic model, proving global weak (Yudovich) existence and uniqueness via flow-map stability despite lacking a velocity formulation. It introduces a contour-dynamics framework to construct time-periodic, $m$-fold V-states bifurcating from two rotating discs, using Crandall–Rabinowitz bifurcation in a setting where Euler and shallow-water kernels interact. A detailed kernel analysis based on Bessel functions and nonlocal operators underpins the existence, regularity, and spectral results; the bifurcation diagrams exhibit a two-dimensional pattern reflecting the multi-layer coupling. The study highlights parameter regimes where Crandall–Rabinowitz applies and identifies scenarios with spectral collisions, offering insights into the extension of V-state theory to multi-layer geophysical fluids and linking to both Euler and shallow-water dynamics.

Abstract

The quasi-geostrophic two-layer (QS2L) system models the dynamic evolution of two interconnected potential vorticities, each is governed by an active scalar equation. These vorticities are linked through a distinctive combination of their respective stream functions, which can be loosely characterized as a parameterized blend of both Euler and shallow-water stream functions. In this article, we study (QS2L) in two directions: First, we prove the existence and uniqueness of global weak solutions in the class of Yudovich, that is when the initial vorticities are only bounded and Lebesgue-integrable. The uniqueness is obtained as a consequence of a stability analysis of the flow-maps associated with the two vorticities. This approach replaces the relative energy method and allows us to surmount the absence of a velocity formulation for (QS2L). Second, we show how to construct $m$-fold time-periodic solutions bifurcating from two arbitrary distinct initial discs rotating with the same angular velocity. This is achieved provided that the number of symmetry $m$ is large enough, or for any symmetry $m\in \mathbb{N}^*$ as long as one of the initial radii of the discs does not belong to some set that contains, at most, a finite number of elements. Due to its multi-layer structure, it is essential to emphasize that the bifurcation diagram exhibits a two-dimensional pattern. Upon analysis, it reveals some similarities with the scheme accomplished for the doubly connected V-states of the Euler and shallow-water equations. However, the coupling between the equations gives rise to several difficulties in various stages of the proof when applying Crandall-Rabinowitz's Theorem. To address this challenge, we conduct a careful analysis of the coupling between the kernels associated with the Euler and shallow-water equations.

Dynamic Behavior of a Multi-Layer Quasi-Geostrophic Model: Weak and Time-Periodic Solutions

TL;DR

The paper analyzes the QS2L two-layer quasi-geostrophic model, proving global weak (Yudovich) existence and uniqueness via flow-map stability despite lacking a velocity formulation. It introduces a contour-dynamics framework to construct time-periodic, -fold V-states bifurcating from two rotating discs, using Crandall–Rabinowitz bifurcation in a setting where Euler and shallow-water kernels interact. A detailed kernel analysis based on Bessel functions and nonlocal operators underpins the existence, regularity, and spectral results; the bifurcation diagrams exhibit a two-dimensional pattern reflecting the multi-layer coupling. The study highlights parameter regimes where Crandall–Rabinowitz applies and identifies scenarios with spectral collisions, offering insights into the extension of V-state theory to multi-layer geophysical fluids and linking to both Euler and shallow-water dynamics.

Abstract

The quasi-geostrophic two-layer (QS2L) system models the dynamic evolution of two interconnected potential vorticities, each is governed by an active scalar equation. These vorticities are linked through a distinctive combination of their respective stream functions, which can be loosely characterized as a parameterized blend of both Euler and shallow-water stream functions. In this article, we study (QS2L) in two directions: First, we prove the existence and uniqueness of global weak solutions in the class of Yudovich, that is when the initial vorticities are only bounded and Lebesgue-integrable. The uniqueness is obtained as a consequence of a stability analysis of the flow-maps associated with the two vorticities. This approach replaces the relative energy method and allows us to surmount the absence of a velocity formulation for (QS2L). Second, we show how to construct -fold time-periodic solutions bifurcating from two arbitrary distinct initial discs rotating with the same angular velocity. This is achieved provided that the number of symmetry is large enough, or for any symmetry as long as one of the initial radii of the discs does not belong to some set that contains, at most, a finite number of elements. Due to its multi-layer structure, it is essential to emphasize that the bifurcation diagram exhibits a two-dimensional pattern. Upon analysis, it reveals some similarities with the scheme accomplished for the doubly connected V-states of the Euler and shallow-water equations. However, the coupling between the equations gives rise to several difficulties in various stages of the proof when applying Crandall-Rabinowitz's Theorem. To address this challenge, we conduct a careful analysis of the coupling between the kernels associated with the Euler and shallow-water equations.
Paper Structure (18 sections, 18 theorems, 320 equations)

This paper contains 18 sections, 18 theorems, 320 equations.

Table of Contents

  1. Introduction and main results
  2. Functional tools and building blocks
  3. Bessel functions and asymptotic expansions
  4. Kernel estimates
  5. Hölder-continuity of some singular operators
  6. Crandall-Rabinowitz's theorem
  7. Weak and Lagrangian solutions
  8. Existence of global solutions
  9. Existence of weak-distributional solutions
  10. Existence of Lagrangian solutions
  11. Stability of the flow-maps and uniqueness
  12. Uniformly rotating solutions
  13. Contour dynamics equation
  14. Linearization around discs
  15. Regularity properties
  16. ...and 3 more sections

Key Result

Theorem 1.1

For any initial data satisfying there is a unique global weak solution $(\omega_1, \omega_2)$ to EQ enjoying the bounds as well as the conservation of norms for any $i\in \{1,2\}$, $q\in [1,\infty]$ and $t\geq 0$. Moreover, the solution is continuous in time in the sense that for all $p\in [1,\infty)$.

Theorems & Definitions (36)

  • Definition : Weak--distributional solutions
  • Definition : Lagrangian solutions
  • Theorem 1.1: Existence and uniqueness of weak solutions
  • Corollary 1.2
  • proof
  • Theorem 1.3: Periodic solutions bifurcating from simple eigenvalues
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • ...and 26 more