Linear decay of the beta-plane equation near Couette flow on the plane

TL;DR

This paper establishes new decay estimates for the linearized beta-plane equation near Couette flow, combining inviscid damping and Rossby wave dispersion, with detailed analysis of oscillatory integrals and critical frequency effects.

Contribution

It introduces novel decay estimates that unify inviscid damping and Rossby wave dispersion for the beta-plane equation near Couette flow, including analysis of critical frequency effects.

Findings

Profiles decay polynomially on compact sets

Mixing dominates for certain frequencies

Dispersive effects are significant along critical rays

Abstract

We prove new time decay estimates for the linearized $\beta$-plane equation near the Couette flow on the plane that combine inviscid damping and the dispersion of Rossby waves. Specifically, we show that the profiles of the velocity field components (i.e. $u(t,x+ty,y)$) decay pointwise on any compact set with polynomial rates. While mixing dominates for streamwise frequencies that are $O(1)$, dispersive effects need to be extracted for low streamwise frequencies that appear along a critical ray in frequency space. Our proof entails the analysis of oscillatory integrals with homogeneous phase and multipliers that diverge in the infinite time limit. To handle this singular limit, we prove a Van der Corput type estimate, followed by two delicate asymptotic analyses of the phase and multipliers: one that is of ``boundary layer" type, featuring sharp gradients that grow in $t$ across the critical ray, and one that is of ``multi-scale" type, which extracts a governing analytic profile function for the phase.