The processes which lead to vertical mixing in stratified fluids operate in a multiparameter space, much of which has not been explored theoretically or in laboratory experiments. This review focuses on the transitional phenomena which precede, and may eventually lead to, the broadband spectra associated with turbulent flow. Conditions of static instability are known to be produced in wave-wave, wave-current, and wave-boundary interactions, but little is known of the subsequent stages of transition. Whilst there has been considerable progress in describing transition in Rayleigh-Benard and Taylor-Couette flows, that in Kelvin-Helmholtz instability is relatively unexplored, and in consequence, our knowledge of energy exchange and of density and momentum fluxes is extremely sparse. Attention is drawn here to the flow disturbance caused by sidewalls in laboratory experiments, and some recently discovered examples of transitional phenomena in Kelvin-Helmholtz instability are described. More information is available about fluxes in double-diffusive convection, but here too the transitional processes are as yet poorly known. Applying this information to the natural environment (and emphasis is given to the oceans) poses the fundamental difficulty of estimating the probability of conditions favoring the development of any particular kind of instability. Knowledge of transition in conditions in which the background field varies in space and time is deficient.

Transitional phenomena and the development of turbulence in stratified fluids: A review

This paper reviews work done over the last twelve years on wave overreflection in shear flows, and on the development of a mechanical picture of how overreflection and instability work It is argued that the wave geometry of a flow configuration is the primary determinant of its stability It is also argued that the primary mechanism for wave amplification is the Orr mechanism

Instability of plane parallel shear flow (toward a mechanistic picture of how it works)

Eddy‐zonalflowinteractionsintheannularmodesareinvestigatedinthisstudyusingamodifiedbeta-plane multilayer quasigeostrophic (QG) channel model. This study shows the different response of high- and lowphase-speed (frequency) eddies to the zonal wind anomalies and suggests a baroclinic mechanism through which the two eddies work symbiotically maintaining the positive eddy feedback in the annular modes. Analysis also indicates that the different roles played by these two eddies in the annular modes are related to the differences in their critical line distributions. Eddies with higher phase speeds experience a low-level critical layer at the center of the jet. They drive the zonal wind anomalies associated with the annular mode but weaken the baroclinicity of the jet in the process. Lower-phase-speed eddies encounter low-level critical lines on the jet flanks. While their momentum fluxes are not as important for the jet shift, they play an important role by restoring the lower-level baroclinicity at the jet center, creating a positive feedback loop with the fast eddies that extends the persistence of the jet shift. The importance of the lower-level baroclinicity restoration by the low-phase-speed eddies in the annular modesis furtherdemonstrated in sensitivity runs,in whichsurface frictionon eddiesisincreasedto selectively damp the low-phase-speed eddies. For simulations in which the low-phase-speed eddies become inactive, the leading mode of the zonal wind variability shifts from the position fluctuation to a pulsing of the jet intensity. Further studies indicate that the response of the lower-level baroclinicity to the zonal wind anomalies caused by the low-phase-speed eddies can be crucial in maintaining the annular mode‐like variations.

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Annular Mode-Like Variation in a Multilayer Quasigeostrophic Model

This paper investigates the stability of a columnar vortex in an inviscid stratified fluid. By means of a WKBJ analysis for large axial wave number, we demonstrate that the normal modes can be stable or unstable owing to the emission of internal waves from the vortex. This phenomenon is shown to be analogous to the radioactive decay of nuclei in quantum mechanics. The destabilized character of the wave emission is shown to be associated with the presence of a critical point in the radial structure of the normal mode. The theoretical predictions for the frequency and growth rate of the normal modes are shown to be in good agreement with numerical results for two examples.

Radiative instability in stratified vortices

Context. With the progress of observational constraints on dynamical processes in stars, it becomes necessary to understand the angular momentum and the rotation profile history. In this context, internal waves constitute an efficient transport mechanism over long distances in stellar radiation zones. Indeed, they could be one of the mechanisms responsible for the quasi-flat rotation profile of the solar radiative region up to 0.2 R ⊙ .Aims. Angular momentum transport induced by internal waves depends on the properties of their excitation regions and of their dissipation during propagation. Then, the bottom of convective envelopes (the top of convective cores, respectively) are differentially rotating magnetic layers while radiation zones may host fossil magnetic fields. It is therefore necessary to understand the modification of internal wave mechanisms by both rotation and magnetic fields.Methods. We continue our previous work by proceeding step by step. We analytically built a complete formalism that treats the angular momentum transport by internal waves while taking into account both the Coriolis acceleration and the Lorentz force in a non-perturbative way for an axisymmetric toroidal field. We assumed a uniform Alfven frequency and a weak differential rotation to isolate the transport properties as a function of the Rossby and Elsasser numbers.Results. We examined the different possible approximations to describe low-frequency internal waves modified by the Coriolis acceleration and the Lorentz force in a deep spherical shell. The complete structure of these waves, which become magneto-gravito-inertial waves, is given assuming the quasi-linear approximation first in the adiabatic case and then in the dissipative one. Vertical and equatorial trapping phenomena appear that favor retrograde waves. The efficiency of the induced transport as a function of the Rossby and Elsasser numbers is then obtained. Conclusions. A complete study of the transport of angular momentum induced by magneto-gravito-inertial waves in stellar radiative regions is achieved for an axisymmetric toroidal magnetic field for a uniform Alfven frequency and a weak differential rotation. General differential rotation, complex azimuthal magnetic fields, and poloidal and mixed fields will be examined in follow-up studies.

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Low-frequency internal waves in magnetized rotating stellar radiation zones - II. Angular momentum transport with a toroidal field

We reexamine the related problems of instability of parallel shear flows and over-reflection of internal waves at a critical level, concentrating on the stratified case. Our primary aim is to delineate the specific aspects of a flow that permit overreflection and instability. A related and partly realized aim is to develop a mechanistic ‘picture’ of how over-reflection and instability work. In the course of this study we have also uncovered some new results concerning the instability of stratified shear flows – showing how regions of enhanced static stability and enhanced damping can destabilize otherwise stable flows.For the scattering of steady plane waves, we show that, of the conditions found by Lindzen & Tung (1978), in the unstratified case, only the existence of wave-propagation regions above and below the critical level is always necessary for over-reflection (at least in the absence of damping), although a trapping region around the critical level and a reflecting surface bounding the upper wave region may play crucial roles in some cases. Our results suggest that the role of the upper wave region may be to allow a wave flux through the critical level. Moreover, we show numerically that the effect of an upper wave region can be mimicked by a region of localized damping which leads to over-reflection as well.We also consider an initial-value problem, using numerical methods. When a wave is incident on the incident level, the reflection and transmission coefficients grow smoothly to their final values. The rate of growth depends on the flow parameters, but there is some evidence to suggest there is a characteristic timescale involved that depends only on the shear (and not on wave travel time). This fits a mechanistic picture of over-reflection and instability that we describe, in which the essential part is a kinematic interaction between wave and mean flow at the critical level, depending only on shear.

John W. Barker

Papers

Instability and wave over-reflection in stably stratified shear flow