[Paper Review] Linear inviscid damping for shear flows near Couette in the 2D stably stratified regime

Roberta Bianchini, Michele Coti Zelati+1 more|arXiv (Cornell University)|May 18, 2020

Fluid Dynamics and Turbulent Flows9 citations

TL;DR

This paper establishes nearly optimal linear inviscid damping rates for 2D incompressible stably stratified shear flows near Couette in the Boussinesq and exponentially stratified regimes. Using a frequency-space pointwise approach and weighted energy estimates, it proves decay rates matching Hartman's 1975 predictions for Couette flow and extends them to general shear flows close to Couette, under the Miles-Howard stability criterion.

ABSTRACT

We investigate the linear stability of shears near the Couette flow for a class of 2D incompressible stably stratified fluids. Our main result consists of nearly optimal decay rates for perturbations of stationary states whose velocities are monotone shear flows $(U(y),0)$ and have an exponential density profile. In the case of the Couette flow $U(y)=y$, we recover the rates predicted by Hartman in 1975, by adopting an explicit point-wise approach in frequency space. As a by-product, this implies optimal decay rates as well as Lyapunov instability in $L^2$ for the vorticity. For the previously unexplored case of more general shear flows close to Couette, the inviscid damping results follow by a weighted energy estimate. Each outcome concerning the stably stratified regime applies to the Boussinesq equations as well. Remarkably, our results hold under the celebrated Miles-Howard criterion for stratified fluids.

Motivation & Objective

• To establish rigorous linear stability and decay rates for perturbations of monotone shear flows near Couette in 2D stably stratified incompressible fluids.
• To extend known inviscid damping results—previously limited to homogeneous fluids—to the case of variable-density, stably stratified flows governed by the Boussinesq equations.
• To prove nearly optimal decay rates for vorticity and velocity components in the $L^2$ norm under the Miles-Howard criterion for stratified fluids.
• To develop a unified framework using weighted energy estimates and frequency-space analysis applicable to both Couette and nearby shear flows.
• To demonstrate Lyapunov instability in $L^2$ for vorticity as a byproduct of the optimal decay rate results.

Proposed method

• Adopt a pointwise approach in frequency space to analyze the linearized dynamics of perturbations in the Couette flow case.
• Introduce symmetric variables and a tailored energy functional to control the evolution of perturbations in the frequency domain.
• Construct a weight function $w$ and a modified weight $m = w^{- ho} m_1^{-1}$ to control growth and decay in the energy estimate.
• Apply weighted energy estimates with careful commutator control to handle nonlinear terms and perturbations of the shear profile.
• Use the Biot-Savart law to recover velocity from vorticity and relate streamfunction, vorticity, and density perturbations via the linearized Boussinesq system.
• Establish bounds on the residual terms (e.g., $\mathcal{R}_8$) via commutator estimates and smallness of $\varepsilon = \|U' - 1\|_{H^6} + \|U''\|_{H^5}$ to close the energy inequality.

Experimental results

Research questions

• RQ1 What are the optimal decay rates for vorticity and velocity perturbations in 2D stably stratified shear flows near Couette?
• RQ2 Can linear inviscid damping be rigorously established for non-homogeneous, stratified fluids under the Miles-Howard criterion?
• RQ3 How do perturbations of the Couette flow (i.e., $U(y) \approx y$) behave asymptotically in the presence of stable stratification?
• RQ4 What is the role of the density profile and stratification parameter $\beta$ in determining the long-time behavior of the system?
• RQ5 Can the $L^2$-stability and decay rates be quantified for general shear flows close to Couette using energy methods with carefully chosen weights?

Key findings

• For the Couette flow ($U(y) = y$), the paper recovers the decay rates predicted by Hartman (1975) via a direct pointwise analysis in frequency space.
• The $L^2$-norm of the vorticity decays like $\langle t \rangle^{-1/2}$, and the velocity and density perturbations decay at the same rate, confirming optimal decay.
• For general shear flows satisfying $\|U' - 1\|_{H^6} + \|U''\|_{H^5} \leq \varepsilon$ with small $\varepsilon$, the same decay rates are obtained via weighted energy estimates.
• The results hold under the Miles-Howard criterion, which ensures spectral stability for stratified flows.
• The analysis implies Lyapunov instability in $L^2$ for the vorticity, as the decay rate is sharp and cannot be improved.
• The framework applies to both the Boussinesq approximation ($\beta = 0, R > 1/4$) and the exponentially stratified regime ($\beta > 0, R = \beta \mathfrak{g} > 1/4$).