Linear inviscid damping for stably stratified Boussinesq flows
Alberto Enciso, Marc Nualart
TL;DR
The paper analyzes the linearized two-dimensional Boussinesq equations in a periodic channel around a monotone, stably stratified shear flow and proves linear inviscid damping with decay rates that depend on the local Richardson number ${\mathcal{J}}(y)$. By employing a limiting-absorption-principle framework and a detailed spectral analysis of the reduced Taylor–Goldstein operator, the authors derive resolvent bounds and oscillatory-integral estimates that yield explicit time-decay rates across four stratification regimes, including logarithmic corrections near threshold values of ${\mathcal{J}}(y)$. They also establish sublinear growth controls for vorticity and density gradients and show that the linearized operator has purely continuous spectrum under the stated hypotheses. The work marries Green’s-function techniques, Whittaker-function representations, and robust operator bounds to give a comprehensive description of linear damping in stratified shear flows, informing both stability theory and potential nonlinear extensions. The results provide a unified, regime-by-regime quantitative framework for how stable stratification modulates inviscid damping in the periodic channel geometry.
Abstract
We study the linear asymptotic stability of stably stratified monotone shear flows for the Boussinesq equations in the periodic channel. By means of the limiting absorption principle, we obtain a precise description of the inviscid damping experienced by the perturbed velocity field and density, with time-decay rates that depend on the local Richardson number $\mathcal{J}(y)$ and split into four stratification regimes (non-stratified, weak, mild, and strong) reflecting qualitative changes in the structure of the Green's function at the critical thresholds $\mathcal{J}(y)=0$ and $\mathcal{J}(y) = \frac14$. The velocity and density decay estimates are later used to prove quantitative sub-linear growth of the vorticity and gradient of density. As a byproduct of our analysis, we show that, under mild hypotheses on the underlying shear-type equilibrium, the spectrum of the linearised Boussinesq operator is purely continuous.