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Upper bounds on the dimension of the global attractor of the 2D Navier-Stokes equations on the $β-$plane

Aseel Farhat, Anuj Kumar, Vincent R. Martinez

TL;DR

This paper analyzes the 2D Navier–Stokes equations on the β-plane with rotation, focusing on how fast rotation (small Rossby number $\varepsilon$) and forcing (Grashof number $\mathscr{G}$) constrain the Hausdorff dimension of the global attractor. It develops a trace-based, time-averaged framework for the linearized dynamics, incorporating a zonal/non-zonal decomposition and a rotation-absorbing transform, to derive explicit $\varepsilon$-dependent bounds in two regimes: $\dim_H\mathcal{A}^\varepsilon \lesssim \varepsilon^{1/3} \mathscr{G}^{3/2} (1+\log\mathscr{G})^{1/2}$ when $\varepsilon$ is very small, and $\dim_H\mathcal{A}^\varepsilon \lesssim \varepsilon^4 \mathscr{G}^{11} (1+\log\mathscr{G})^{2}$ in an intermediate regime. The analysis yields a decomposition of solutions into a dominant 1D zonal heat-diffusion component plus a rapidly damped non-zonal part, and proves convergence of attractors as $\varepsilon \to 0^+$ to the 1D heat attractor in the Hausdorff sense, highlighting a dynamical degeneracy to effectively 1D behavior under fast rotation.

Abstract

This article establishes estimates on the dimension of the global attractor of the two-dimensional rotating Navier-Stokes equation for viscous, incompressible fluids on the $β$-plane. Previous results in this setting by M.A.H. Al-Jaboori and D. Wirosoetisno (2011) had proved that the global attractor collapses to a single point that depends only the longitudinal coordinate, i.e., zonal flow, when the rotation is sufficiently fast. However, an explicit quantification of the complexity of the global attractor in terms of $β$ had remained open. In this paper, such estimates are established which are valid across a wide regime of rotation rates and are consistent with the dynamically degenerate regime previously identified. Additionally, a decomposition of solutions is established detailing the asymptotic behavior of the solutions in the limit of large rotation.

Upper bounds on the dimension of the global attractor of the 2D Navier-Stokes equations on the $β-$plane

TL;DR

This paper analyzes the 2D Navier–Stokes equations on the β-plane with rotation, focusing on how fast rotation (small Rossby number ) and forcing (Grashof number ) constrain the Hausdorff dimension of the global attractor. It develops a trace-based, time-averaged framework for the linearized dynamics, incorporating a zonal/non-zonal decomposition and a rotation-absorbing transform, to derive explicit -dependent bounds in two regimes: when is very small, and in an intermediate regime. The analysis yields a decomposition of solutions into a dominant 1D zonal heat-diffusion component plus a rapidly damped non-zonal part, and proves convergence of attractors as to the 1D heat attractor in the Hausdorff sense, highlighting a dynamical degeneracy to effectively 1D behavior under fast rotation.

Abstract

This article establishes estimates on the dimension of the global attractor of the two-dimensional rotating Navier-Stokes equation for viscous, incompressible fluids on the -plane. Previous results in this setting by M.A.H. Al-Jaboori and D. Wirosoetisno (2011) had proved that the global attractor collapses to a single point that depends only the longitudinal coordinate, i.e., zonal flow, when the rotation is sufficiently fast. However, an explicit quantification of the complexity of the global attractor in terms of had remained open. In this paper, such estimates are established which are valid across a wide regime of rotation rates and are consistent with the dynamically degenerate regime previously identified. Additionally, a decomposition of solutions is established detailing the asymptotic behavior of the solutions in the limit of large rotation.
Paper Structure (12 sections, 17 theorems, 177 equations)

This paper contains 12 sections, 17 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Vorticity formulation
  4. Estimation of the Dimension of the Global Attractor
  5. Treating $\langle\mathrm{Tr}[\widetilde{\mathbb{B}}\mathbb{P}_\mathrm{N}]\rangle$
  6. Treating $\langle\mathrm{Tr}[\overline{\mathbb{B}}\mathbb{P}_\mathrm{N}]\rangle$
  7. Treating $\langle \mathrm{Tr}[\mathbb{P}_\mathrm{N}\overline{\mathbb{B}}_1\partial_s\mathbb{P}_\mathrm{N}]\rangle$
  8. Treating $\left\langle \mathrm{Tr}[\mathbb{P}_\mathrm{N}(s)\overline{\mathbb{B}}_{2}(s)\mathbb{P}_\mathrm{N}(s)]\right\rangle$
  9. Culmination of the Estimates
  10. A description of the asymptotic behavior of $\omega^\varepsilon$
  11. Proof of \ref{['lem:om:timeavg']}
  12. Proof of $\langle \partial_t\mathrm{Tr}[\mathbb{P}_\mathrm{N}(t)\mathbb{T}(t)\mathbb{P}_\mathrm{N}(t)]\rangle=0$

Key Result

Proposition 1.1

Suppose $\mathscr{G}\gtrsim1$. If then and the global attractor consists of a single point.

Theorems & Definitions (31)

  • Proposition 1.1: MustafaDjoko2011
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1: MustafaDjoko2011
  • Remark 2.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 21 more