Coupling of Damped and Growing Modes in Unstable Shear Flow

TL;DR

This paper investigates how stable modes influence the saturation of Kelvin-Helmholtz instability, revealing their significant role in energy transfer and momentum flux reduction in shear flows.

Contribution

It demonstrates that stable modes are nonlinearly excited and significantly impact energy balance and momentum transport during shear flow instability saturation.

Findings

Stable modes are nonlinearly excited during KH instability.

Energy transfer to stable modes is comparable to small-scale transfer.

Stable modes can substantially reduce momentum flux.

Abstract

Analysis of the saturation of the Kelvin-Helmholtz (KH) instability is undertaken to determine the extent to which the conjugate linearly stable mode plays a role. For a piecewise-continuous mean flow profile with constant shear in a fixed layer, it is shown that the stable mode is nonlinearly excited, providing an injection-scale sink of the fluctuation energy similar to what has been found for gyroradius-scale drift-wave turbulence. Quantitative evaluation of the contribution of the stable mode to the energy balance at the onset of saturation shows that nonlinear energy transfer to the stable mode is as significant as energy transfer to small scales in balancing energy injected into the spectrum by the instability. The effect of the stable mode on momentum transport is quantified by expressing the Reynolds stress in terms of stable and unstable mode amplitudes at saturation, from which it is found that the stable mode can produce a sizable reduction in the momentum flux.