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Linear Inviscid Damping for Couette Flow in Stratified Fluid

Jincheng Yang, Zhiwu Lin

TL;DR

This work derives and sharpens the linear decay rates (inviscid damping) for Couette flow in an exponentially stratified fluid, comparing the Boussinesq approximation with the full Euler equations. By transforming to sheared coordinates and solving via explicit hypergeometric functions, the authors obtain precise decay rates in L^2 and L^∞ across regimes determined by the Richardson number B^2, including critical cases B^2 = 0, 1/4, and ∞, with minimal regularity requirements on initial data. A key contribution is showing that full Euler linear damping matches the Boussinesq predictions in weighted settings, and that dispersive decay dominates when shear is absent, highlighting distinct mechanisms: vorticity mixing under shear versus wave dispersion in stratified, non-sheared flows. The results provide a rigorous, granular baseline for potential nonlinear damping analyses and offer explicit, sharp tools (hypergeometric representations) for further spectral and time-decay studies in stratified fluid stability.

Abstract

We study the inviscid damping of Couette flow with an exponentially stratified density. The optimal decay rates of the velocity field and the density are obtained for general perturbations with minimal regularity. For Boussinesq approximation model, the decay rates we get are consistent with the previous results in the literature. We also study the decay rates for the full Euler equations of stratified fluids, which were not studied before. For both models, the decay rates depend on the Richardson number in a very similar way. Besides, we also study the dispersive decay due to the exponential stratification when there is no shear.

Linear Inviscid Damping for Couette Flow in Stratified Fluid

TL;DR

This work derives and sharpens the linear decay rates (inviscid damping) for Couette flow in an exponentially stratified fluid, comparing the Boussinesq approximation with the full Euler equations. By transforming to sheared coordinates and solving via explicit hypergeometric functions, the authors obtain precise decay rates in L^2 and L^∞ across regimes determined by the Richardson number B^2, including critical cases B^2 = 0, 1/4, and ∞, with minimal regularity requirements on initial data. A key contribution is showing that full Euler linear damping matches the Boussinesq predictions in weighted settings, and that dispersive decay dominates when shear is absent, highlighting distinct mechanisms: vorticity mixing under shear versus wave dispersion in stratified, non-sheared flows. The results provide a rigorous, granular baseline for potential nonlinear damping analyses and offer explicit, sharp tools (hypergeometric representations) for further spectral and time-decay studies in stratified fluid stability.

Abstract

We study the inviscid damping of Couette flow with an exponentially stratified density. The optimal decay rates of the velocity field and the density are obtained for general perturbations with minimal regularity. For Boussinesq approximation model, the decay rates we get are consistent with the previous results in the literature. We also study the decay rates for the full Euler equations of stratified fluids, which were not studied before. For both models, the decay rates depend on the Richardson number in a very similar way. Besides, we also study the dispersive decay due to the exponential stratification when there is no shear.

Paper Structure

This paper contains 22 sections, 8 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Linearized Euler Equation and Boussinesq Approximation
  4. Sobolev spaces
  5. Hypergeometric Functions
  6. Solutions by Hypergeometric functions
  7. Boussinesq approximation
  8. Full Euler Equations
  9. Decay estimates in the case of Boussinesq approximation
  10. The case $B^2 > 0$ and $B^2 \neq \frac{1}{4}$
  11. The case $0<B^{2}<\frac{1}{4}$
  12. The case $B^{2}>\frac{1}{4}$
  13. The case $B^{2}=\frac{1}{4}$
  14. The case $B^{2}=0$
  15. Decay estimates for the full Euler equation
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\left( \psi (t;x,y),\frac{\rho }{A}(t;x,y)\right)$ be the solution of (eqn-stream-Bou)-(eqn-density-bou) with the initial data where $y\in \mathbb{R}$ and $x$ is periodic with period $L$. Denote the velocity $\boldsymbol{v}=\nabla ^{\perp }\psi =\left( v^{x},v^{y}\right)$. Below, $f\lesssim g$ stands for $f\leq Cg$ for a constant C depending only on $R,\beta ,g.\ $We denote $\left\langle f\r

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 4.1
  • Remark 4.2
  • Lemma 6.1
  • Lemma 6.2
  • Remark 6.3