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On the stability of viscous three-dimensional rotating Couette flow

Michele Coti Zelati, Augusto Del Zotto, Klaus Widmayer

TL;DR

This work analyzes the stability of Couette flow in the 3D Navier–Stokes equations with rotation, revealing how background shear and Coriolis forces interact through the Bradshaw–Richardson number $B_β$. By transitioning to a moving frame and introducing the principal unknowns $Q=\Delta_L U^2$ and $W=\sqrt{\frac{β}{β-1}}|\nabla_L|Ω^2$, the authors separate dynamics into double zero, simple zero, and nonzero modes and develop a nonlinear bootstrap built on a tailored energy functional ${\mathcal A}$. They prove a nonlinear transition threshold for stability in the linearly stable regime $B_β>0$ that scales as $ε\lesssim ν^{8/9}$ (or with larger $B_β$ as $ε\lesssim ν^{-1}ν^{5/6}$ in a second regime), showing enhanced dissipation for nonzero modes and dispersive decay for simple zero modes, with the stabilization increasing as the rotation speed grows (via $α=\sqrt{B_β}$). The nonlinear analysis confirms global stability and a refined threshold compared to the non-rotating case, quantifying how fast rotation enhances mixing and damping to suppress perturbation growth. The results provide a quantitative bridge between linear dispersive/mixing mechanisms and nonlinear stability in rotating shear flows, with implications for geophysical and industrial flows under strong rotation.

Abstract

We study the stability of Couette flow in the 3d Navier-Stokes equations with rotation, as given by the Coriolis force. Hereby, the nature of linearized dynamics near Couette flow depends crucially on the force balance between background shearing and rotation, and includes lift-up or exponential instabilities, as well as a stable regime. In the latter, shearing resp. rotational inertial waves give rise to mixing and dispersive effects, which are relevant for distinct dynamical realms. Our main result quantifies these effects through enhanced dissipation and dispersive amplitude decay in both linear and nonlinear settings: in particular, we establish a nonlinear transition threshold which quantitatively improves over the setting without rotation (and increases further with rotation speed), showcasing its stabilizing effect.

On the stability of viscous three-dimensional rotating Couette flow

TL;DR

This work analyzes the stability of Couette flow in the 3D Navier–Stokes equations with rotation, revealing how background shear and Coriolis forces interact through the Bradshaw–Richardson number . By transitioning to a moving frame and introducing the principal unknowns and , the authors separate dynamics into double zero, simple zero, and nonzero modes and develop a nonlinear bootstrap built on a tailored energy functional . They prove a nonlinear transition threshold for stability in the linearly stable regime that scales as (or with larger as in a second regime), showing enhanced dissipation for nonzero modes and dispersive decay for simple zero modes, with the stabilization increasing as the rotation speed grows (via ). The nonlinear analysis confirms global stability and a refined threshold compared to the non-rotating case, quantifying how fast rotation enhances mixing and damping to suppress perturbation growth. The results provide a quantitative bridge between linear dispersive/mixing mechanisms and nonlinear stability in rotating shear flows, with implications for geophysical and industrial flows under strong rotation.

Abstract

We study the stability of Couette flow in the 3d Navier-Stokes equations with rotation, as given by the Coriolis force. Hereby, the nature of linearized dynamics near Couette flow depends crucially on the force balance between background shearing and rotation, and includes lift-up or exponential instabilities, as well as a stable regime. In the latter, shearing resp. rotational inertial waves give rise to mixing and dispersive effects, which are relevant for distinct dynamical realms. Our main result quantifies these effects through enhanced dissipation and dispersive amplitude decay in both linear and nonlinear settings: in particular, we establish a nonlinear transition threshold which quantitatively improves over the setting without rotation (and increases further with rotation speed), showcasing its stabilizing effect.

Paper Structure

This paper contains 15 sections, 18 theorems, 138 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Structure of the equations and choice of unknowns
  3. Bootstrap for Theorem \ref{['thm:transition threshold']}
  4. Additional notation
  5. Linear analysis
  6. Dynamics of the simple zero modes
  7. Instability regime for $0\leq\beta\leq 1$
  8. Stable regime $B_\beta>0$: $\beta<0$ or $\beta>1$
  9. Dynamics of the nonzero modes
  10. Nonlinear analysis
  11. Preliminaries
  12. Dispersive decay of simple zero modes
  13. Double zero modes
  14. Simple zero modes
  15. Nonzero modes

Key Result

Theorem 1

Let $u\in C_t([0,\infty);L^2({\mathbb T}\times{\mathbb R}\times{\mathbb T}))$ solve eq:linearized. Then and we have the following behavior depending on $B_\beta$:

Figures (2)

  • Figure 1: The (unstable) eigenmode $\lambda^+_{\nu,\beta}(\eta,l)=-\nu\left\lvert (\eta,l) \right\rvert^2+\sqrt{\beta(1-\beta)}\frac{\left\lvert l \right\rvert}{\left\lvert (\eta,l) \right\rvert}$ as a function of $\eta\in[0,30]$ for fixed $l=2$, $\beta=0.5$ and $\nu=0.001$ (left) resp. $\nu=0.0001$ (right).
  • Figure 2: $S'\cap\{l>0\}$ (dark red) inside $S\cap\{l>0\}$ (red) for $\nu=0.001$ and $\beta=0.5$ (left) resp. $\beta=0.1$ (right).

Theorems & Definitions (36)

  • Theorem 1: Linear dynamics
  • Remark 1.1
  • Theorem 2: Nonlinear stability
  • Remark 1.2
  • Theorem 3: Bootstrap
  • proof : Overview of the proof of Theorem \ref{['thm:bootstrap']}
  • Lemma 2.1
  • proof : Proof of part (\ref{['it:linear_stable']}) Theorem \ref{['thm:linear']} -- simple zero modes
  • Lemma 2.2: Properties of ${\mathfrak{m}}$
  • proof : Proof of part (\ref{['it:linear_stable']}) Theorem \ref{['thm:linear']} -- non-zero modes
  • ...and 26 more