Nonlinear evolution of linear optimal perturbations of strongly stratified shear layers

Abstract

The Miles–Howard theorem states that a necessary condition for normal-mode instability in parallel, inviscid, steady stratified shear flows is that the minimum gradient Richardson number, $Ri_{g,min}$, is less than $1/4$ somewhere in the flow. However, the non-normality of the Navier–Stokes and buoyancy equations may allow for substantial perturbation energy growth at finite times. We calculate numerically the linear optimal perturbations which maximize the perturbation energy gain for a stably stratified shear layer consisting of a hyperbolic tangent velocity distribution with characteristic velocity $U_{0}^{\ast }$ and a uniform stratification with constant buoyancy frequency $N_{0}^{\ast }$. We vary the bulk Richardson number $Ri_{b}=N_{0}^{\ast 2}h^{\ast 2}/U_{0}^{\ast 2}$ (corresponding to $Ri_{g,min}$) between 0.20 and 0.50 and the Reynolds numbers $\mathit{Re}=U_{0}^{\ast }h^{\ast }/\unicode[STIX]{x1D708}^{\ast }$ between 1000 and 8000, with the Prandtl number held fixed at $\mathit{Pr}=1$. We find the transient growth of non-normal perturbations may be sufficient to trigger strongly nonlinear effects and breakdown into small-scale structures, thereby leading to enhanced dissipation and non-trivial modification of the background flow even in flows where $Ri_{g,min}>1/4$. We show that the effects of nonlinearity are more significant for flows with higher $\mathit{Re}$, lower $Ri_{b}$ and higher initial perturbation amplitude $E_{0}$. Enhanced kinetic energy dissipation is observed for higher-$Re$ and lower-$Ri_{b}$ flows, and the mixing efficiency, quantified here by $\unicode[STIX]{x1D700}_{p}/(\unicode[STIX]{x1D700}_{p}+\unicode[STIX]{x1D700}_{k})$ where $\unicode[STIX]{x1D700}_{p}$ is the dissipation rate of density variance and $\unicode[STIX]{x1D700}_{k}$ is the dissipation rate of kinetic energy, is found to be approximately 0.35 for the most strongly nonlinear cases.