Dependence of three-dimensional damping and stability on stratification strength

Determine whether the three-dimensional inviscid damping rates for nonzero modes of the stratified Boussinesq equations around Couette flow improve as the Brunt–Väisälä frequency \(\beta\) increases, and determine how the nonlinear stability threshold depends on \(\beta\).

Background

The paper notes that, for every $\beta>0$, the corresponding two-dimensional linear problem has stronger polynomial decay rates than the three-dimensional result. Its three-dimensional analysis obtains $\langle t\rangle^{-1}$ decay for the wall-normal velocity component $U^2_{\neq}$, but does not establish whether stronger stratification produces improved damping.

The authors also leave unresolved whether the nonlinear stability threshold varies with the Brunt–Väisälä frequency. This question concerns the quantitative effect of stratification on both linear inviscid damping and nonlinear stability for the three-dimensional Boussinesq system.