Optimal excitation of two dimensional Holmboe instabilities

TL;DR

This paper investigates the excitation mechanisms of Holmboe instabilities in stratified shear layers, revealing that adjoint modes can induce large transient growth even in stable conditions, with implications for mixing in stratified environments.

Contribution

It introduces a generalized stability analysis showing Holmboe instabilities can be excited by adjoint modes at high Richardson numbers, and identifies potential for large transient growth.

Findings

Holmboe instabilities can be excited by adjoint modes at high Richardson numbers.

Large transient growth is possible even in stable flows.

Holmboe quasi-modes persist as stable structures in certain parameter regimes.

Abstract

Highly stratified shear layers are rendered unstable even at high stratifications by Holmboe instabilities when the density stratification is concentrated in a small region of the shear layer. These instabilities may cause mixing in highly stratified environments. However these instabilities occur in limited bands in the parameter space. We perform Generalized Stability analysis of the two dimensional perturbation dynamics of an inviscid Boussinesq stratified shear layer and show that Holmboe instabilities at high Richardson numbers can be excited by their adjoints at amplitudes that are orders of magnitude larger than by introducing initially the unstable mode itself. We also determine the optimal growth that is obtained for parameters for which there is no instability. We find that there is potential for large transient growth regardless of whether the background flow is exponentially stable or not and that the characteristic structure of the Holmboe instability asymptotically emerges as a persistent quasi-mode for parameter values for which the flow is stable.