Suppression of lift-up effect in the 3D Boussinesq equations around a stably stratified Couette flow

TL;DR

This paper proves that in the 3D Boussinesq equations around a stably stratified Couette flow, the lift-up effect is suppressed, leading to faster dissipation rates due to buoyancy and stratification effects.

Contribution

The paper introduces a Fourier-based symmetrization approach to analyze the energy structure, demonstrating suppression of the lift-up effect in 3D stratified flows.

Findings

Enhanced dissipation rates surpass those in homogeneous Navier-Stokes.

Explicit solutions for streaks modes show oscillatory behavior.

Lift-up effect is eliminated in the stratified setting.

Abstract

In this paper, we establish linear enhanced dissipation results for the three-dimensional Boussinesq equations around a stably stratified Couette flow, in the viscous and thermally diffusive setting. The dissipation rates are faster compared to those observed in the homogeneous Navier-Stokes equations, in light of the interplay between velocity and temperature, driven by buoyant forces. Our approach involves introducing a change of variables grounded in a Fourier space symmetrization framework. This change elucidates the energy structure inherent in the system. Specifically, we handle non-streaks modes through an augmented energy functional, while streaks modes are amenable to explicit solutions. This explicit treatment reveals the oscillatory nature of shear modes, providing the elimination of the well-known three-dimensional instability mechanism known as the ``lift-up effect''.