The issue of the physical mechanism(s) that control the efficiency with which the density field in stably stratified fluid is mixed by turbulent processes has remained enigmatic. Similarly enigmatic has been an explanation of the numerical value of ∼0.2, which is observed to characterize this efficiency experimentally. We review recent work on the turbulence transition in stratified parallel flows that demonstrates that this value is not only numerically predictable but also that it is expected to be a nonmonotonic function of the Richardson number that characterizes preturbulent stratification strength. This value of the mixing efficiency appears to be characteristic of the late-time behavior of the turbulent flow that develops after an initially laminar shear flow has undergone the transition to turbulence through an intermediate instability of Kelvin-Helmholtz type.

We investigate the detailed nature of the ‘mixing transition’ through which turbulence may develop in both homogeneous and stratified free shear layers. Our focus is upon the fundamental role in transition, and in particular the associated ‘mixing’ (i.e. small-scale motions which lead to an irreversible increase in the total potential energy of the flow) that is played by streamwise vortex streaks, which develop once the primary and typically two-dimensional Kelvin–Helmholtz (KH) billow saturates at finite amplitude.Saturated KH billows are susceptible to a family of three-dimensional secondary instabilities. In homogeneous fluid, secondary stability analyses predict that the stream-wise vortex streaks originate through a ‘hyperbolic’ instability that is localized in the vorticity braids that develop between billow cores. In sufficiently strongly stratified fluid, the secondary instability mechanism is fundamentally different, and is associated with convective destabilization of the statically unstable sublayers that are created as the KH billows roll up.We test the validity of these theoretical predictions by performing a sequence of three-dimensional direct numerical simulations of shear layer evolution, with the flow Reynolds number (defined on the basis of shear layer half-depth and half the velocity difference) Re = 750, the Prandtl number of the fluid Pr = 1, and the minimum gradient Richardson number Ri(0) varying between 0 and 0.1. These simulations quantitatively verify the predictions of our stability analysis, both as to the spanwise wavelength and the spatial localization of the streamwise vortex streaks. We track the nonlinear amplification of these secondary coherent structures, and investigate the nature of the process which actually triggers mixing. Both in stratified and unstratified shear layers, the subsequent nonlinear amplification of the initially localized streamwise vortex streaks is driven by the vertical shear in the evolving mean flow. The two-dimensional flow associated with the primary KH billow plays an essentially catalytic role. Vortex stretching causes the streamwise vortices to extend beyond their initially localized regions, and leads eventually to a streamwise-aligned collision between the streamwise vortices that are initially associated with adjacent cores.It is through this collision of neighbouring streamwise vortex streaks that a final and violent finite-amplitude subcritical transition occurs in both stratified and unstratified shear layers, which drives the mixing process. In a stratified flow with appropriate initial characteristics, the irreversible small-scale mixing of the density which is triggered by this transition leads to the development of a third layer within the flow of relatively well-mixed fluid that is of an intermediate density, bounded by narrow regions of strong density gradient.

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Two-dimensional finite-amplitude Kelvin–Helmholtz waves are tested for stability against three-dimensional infinitesimal perturbations. Since the nonlinear waves are time-dependent, the stability analysis is based upon the assumption that they evolve on a timescale which is long compared with that of any instability which they might support. The stability problem is thereby reduced to standard eigenvalue form, and solutions that do not satisfy the timescale constraint are rejected. If the Reynolds number of the initial parallel flow is sufficiently high the two-dimensional wave is found to be unstable and the fastest-growing modes are three-dimensional disturbances that possess longitudinal symmetry. These modes are convective in nature and focused in the statically unstable regions that form during the overturning of the stratified fluid in the core of the nonlinear vortex. The nature of the instability in the high-Reynolds-number regime suggests that it is intimately related to the observed onset of turbulence in these waves. The transition Reynolds number above which the secondary instability exists depends strongly on the initial conditions from which the primary wave evolves.

We analyse the stability of horizontally periodic, two-dimensional, finite-amplitude Kelvin-Helmholtz billows with respect to infinitesimal three-dimensional perturbations having the same streamwise wavelength for several different levels of the initial density stratification. A complete analysis of the energy budget for this class of secondary instabilities establishes that the contribution to their growth from shear conversion of the basic-state kinetic energy is relatively insensitive to the strength of the stratification over the range of values considered, suggesting that dynamical shear instability constitutes the basic underlying mechanism. Indeed, during the initial stages of their growth, secondary instabilities derive their energy predominantly from shear conversion. However, for initial Richardson numbers between 0.065 and 0.13, the primary source of kinetic energy for secondary instabilities at the time the parent wave climaxes is in fact the conversion of potential energy by convective overturning in the cores of the individual billows. A comparison between the secondary instability properties of unstratified Kelvin-Helmholtz billows and Stuart vortices is made, as the latter have often been assumed to provide an adequate approximation to the former. Our analyses suggest that the Stuart vortex model has several shortcomings in this regard.