Nonlinear inviscid damping and shear-buoyancy instability in the two-dimensional Boussinesq equations

TL;DR

This paper studies the long-term behavior of the 2D inviscid Boussinesq equations near a stratified Couette flow, revealing shear-buoyancy instability with specific damping and growth rates under classical stability conditions.

Contribution

It establishes the first rigorous analysis of shear-buoyancy instability and inviscid damping in the 2D Boussinesq equations near stratified Couette flow, incorporating novel energy methods.

Findings

Density exhibits inviscid damping unlike passive scalars.

Velocity damps at a rate of O(t^{-1/2}).

Vorticity and density gradient grow as O(t^{1/2}).

Abstract

We investigate the long-time properties of the two-dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for an initial Gevrey perturbation of size $\varepsilon$. Under the classical Miles-Howard stability condition on the Richardson number, we prove that the system experiences a shear-buoyancy instability: the density variation and velocity undergo an $O(t^{-1/2})$ inviscid damping while the vorticity and density gradient grow as $O(t^{1/2})$. The result holds at least until the natural, nonlinear timescale $t \approx \varepsilon^{-2}$. Notice that the density behaves very differently from a passive scalar, as can be seen from the inviscid damping and slower gradient growth. The proof relies on several ingredients: (A) a suitable symmetrization that makes the linear terms amenable to energy methods and takes into account the classical Miles-Howard spectral stability condition; (B) a variation of the Fourier time-dependent energy method introduced for the inviscid, homogeneous Couette flow problem developed on a toy model adapted to the Boussinesq equations, i.e. tracking the potential nonlinear echo chains in the symmetrized variables despite the vorticity growth.