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 $U_{0}^{*}$ and a uniform stratification with constant buoyancy frequency $N_{0}^{*}$. We vary the bulk Richardson number $Ri_b$=$N_{0}^{*}$$^2$$h^{*2}$/$U_{0}^{*}$$^2$ (corresponding to $Ri_{g,min}$) between 0.20 and 0.50 and the Reynolds numbers $Re$=$U_{0}^{*}$$h^{*}$/$v^{*}$ between 1000 and 8000, with the Prandtl number held fixed at $Pr$=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 $Re$, 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 $\epsilon_p$/($\epsilon_p$+$\epsilon_k$) where $\epsilon_p$ is the dissipation rate of density variance and $\epsilon_k$is the dissipation rate of kinetic energy, is found to be approximately 0.35 for the most strongly nonlinear cases.