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.

Testing linear marginal stability in stratified shear layers

“…It is important to remember that we have only considered the nonlinear development of a linear normal-mode perturbation. As noted in the introduction, Kaminski et al (2017) have shown that sufficiently large amplitude perturbations with the structure of a linear optimal perturbation can still develop into a 'KH-like' billow state for flows with Ri m as high as 0.4, which may perhaps explain why the maximum energy in figure 3a does not tend to zero as Ri m → 1/4. Indeed, with a specific type of large forcing, a saturated nonlinear overturning billow structure may possibly develop at Ri m = 1/4 through a different (and inherently transient, non-normal) mechanism, distinct from perturbation by a classical normal-mode linear instability.…”

“…For inviscid flows susceptible to KHI, as Ri m approaches 1/4 from below, the exponential growth rate of the linear instabilities is predicted to drop to zero (Hazel 1972). However, this prediction does not in itself preclude the possibility that the finite amplitude 'billow' can still have significant amplitude for flows with such 'marginal' Richardson numbers, particularly in light of the results of Kaminski et al (2017). Using a direct-adjoint-looping method, they demonstrated that billow-like structures with nontrivial amplitude can still be triggered by 'linear optimal' perturbations of small initial perturbation energy (i.e.…”

Evolution of thermally stratified turbulent open channel flow after removal of the heat source

“…Howland, Taylor & Caulfield (2018) investigated marginal stability associated with the formation of K–H waves from laminar initial conditions and showed the marginal stability limit to be . In fact Kaminski, Caulfield & Taylor (2017) have shown that Kelvin–Helmholtz-like billows can form for up to 0.4 in the presence of perturbations that are sufficiently large and that have the optimal structure for amplification.…”

“…1995; Caulfield & Peltier 2000; Salehipour & Peltier 2015; Kaminski et al. 2017; Howland et al. 2018, for example).…”

Kelvin–Helmholtz billows above Richardson number

“…There is various evidence that for flows susceptible to Kelvin-Helmholtz instability (KHI), complex nonlinear behaviour exists when Ri m > 1/4. Kaminski et al (2017) showed that perturbations which grow transiently before decaying in the linearised setting can lead to turbulent-like irreversible mixing with Ri m > 1/4 when nonlinearity is included. Howland et al (2018) showed that as Ri m → 1/4 from below, the maximum amplitude of a saturated Kelvin-Helmoltz billow does not tend to zero, but to some finite value.…”