Stratified Shear Flow Instabilities Research Articles

The structure and origin of confined Holmboe waves

Finite-amplitude manifestations of stratified shear flow instabilities and their spatio-temporal coherent structures are believed to play an important role in turbulent geophysical flows. Such shear flows commonly have layers separated by sharp density interfaces, and are therefore susceptible to the so-called Holmboe instability, and its finite-amplitude manifestation, the Holmboe wave. In this paper, we describe and elucidate the origin of an apparently previously unreported long-lived coherent structure in a sustained stratified shear flow generated in the laboratory by exchange flow through an inclined square duct connecting two reservoirs filled with fluids of different densities. Using a novel measurement technique allowing for time-resolved, near-instantaneous measurements of the three-component velocity and density fields simultaneously over a three-dimensional volume, we describe the three-dimensional geometry and spatio-temporal dynamics of this structure. We identify it as a finite-amplitude, nonlinear, asymmetric confined Holmboe wave (CHW), and highlight the importance of its spanwise (lateral) confinement by the duct boundaries. We pay particular attention to the spanwise vorticity, which exhibits a travelling, near-periodic structure of sheared, distorted, prolate spheroids with a wide ‘body’ and a narrower ‘head’. Using temporal linear stability analysis on the two-dimensional streamwise-averaged experimental flow, we solve for three-dimensional perturbations having two-dimensional, cross-sectionally confined eigenfunctions and a streamwise normal mode. We show that the dispersion relation and the three-dimensional spatial structure of the fastest-growing confined Holmboe instability are in good agreement with those of the observed confined Holmboe wave. We also compare those results with a classical linear analysis of two-dimensional perturbations (i.e. with no spanwise dependence) on a one-dimensional base flow. We conclude that the lateral confinement is an important ingredient of the confined Holmboe instability, which gives rise to the CHW, with implications for many inherently confined geophysical flows such as in valleys, estuaries, straits or deep ocean trenches. Our results suggest that the CHW is an example of an experimentally observed, inherently nonlinear, robust, long-lived coherent structure which has developed from a linear instability. We conjecture that the CHW is a promising candidate for a class of exact coherent states underpinning the dynamics of more disordered, yet continually forced stratified shear flows.

Stratified shear flow instabilities in the non-Boussinesq regime

Effects of the baroclinic torque on wave propagation normally neglected under the Boussinesq approximation is investigated here, with a special focus on the associated consequences for the mechanistic interpretation of shear instability arising from the interaction between a pair of vorticity-propagating waves. To illustrate and elucidate the physical effects that modify wave propagation, we consider three examples of increasing complexity: wave propagation supported by a uniform background flow; wave propagation supported on a piecewise-linear basic state possessing one jump; and an instability problem of a piecewise-linear basic state possessing two jumps, which supports the possibility of shear instability. We find that the non-Boussinesq effects introduce a preference for the direction of wave propagation that depends on the sign of the shear in the region where waves are supported. This in turn affects phase-locking of waves that is crucial for the mechanistic interpretation for shear instability, and is seen here to have an inherent tendency for stabilisation.

Instability in Stratified Shear Flow: Review of a Physical Interpretation Based on Interacting Waves

Instability in homogeneous and density stratified shear flows may be interpreted in terms of the interaction of two (or more) otherwise free waves in the velocity and density profiles. These waves exist on gradients of vorticity and density, and instability results when two fundamental conditions are satisfied: (I) the phase speeds of the waves are stationary with respect to each other (“phase-locking“), and (II) the relative phase of the waves is such that a mutual growth occurs. The advantage of the wave interaction approach is that it provides a physical interpretation to shear flow instability. This paper is largely intended to purvey the basics of this physical interpretation to the reader, while both reviewing and consolidating previous work on the topic. The interpretation is shown to provide a framework for understanding many classical and nonintuitive results from the stability of stratified shear flows, such as the Rayleigh and Fjørtoft theorems, and the destabilizing effect of an otherwise stable density stratification. Finally, we describe an application of the theory to a geophysical-scale flow in the Fraser River estuary.

Stratified shear flow instabilities at large Richardson numbers

Numerical simulations of stratified shear flow instabilities are performed in two dimensions in the Boussinesq limit. The density variation length scale is chosen to be four times smaller than the velocity variation length scale so that Holmboe or Kelvin–Helmholtz unstable modes are present depending on the choice of the global Richardson number Ri. Three different values of Ri were examined Ri=0.2, 2, and 20. The flows for the three examined values are all unstable due to different modes, namely: the Kelvin–Helmholtz mode for Ri=0.2, the first Holmboe mode for Ri=2, and the second Holmboe mode for Ri=20 that has been discovered recently and this is the first time that it is examined in the nonlinear stage. It is found that the amplitude of the velocity perturbation of the second Holmboe mode at the nonlinear stage is smaller but comparable to first Holmboe mode. The increase in the potential energy, however, due to the second Holmboe modes is greater than that of the first mode. The Kelvin–Helmholtz mode is larger by two orders of magnitude in kinetic energy than the Holmboe modes and about ten times larger in potential energy than the Holmboe modes. The effect of increasing Prandtl number is also investigated, and a weak dependence on the Prandtl number is observed. The results in this paper suggest that although mixing is suppressed at large Richardson numbers it is not negligible, and turbulent mixing processes in strongly stratified environments cannot be excluded.

Inviscid vortex motions in weakly three-dimensional boundary layers and their relation with instabilities in stratified shear flows

In this paper we consider the inviscid instability of three-dimensional boundary-layer flows with a small crossflow over locally concave or convex walls, along with the inviscid instability of stratified shear flows. We show how these two problems are closely related through the forms of their governing equations. A proposed definition of a generalized Richardson number for the neutrally stable inviscid vortex motions is given. Implications of the similarity between the two problems are discussed.

Radiative instability of stratified shear flows in the Drazin model

The results of analytical stability calculations of an ideal liquid stratified shear flow described by the Drazin model /1/ are presented. It is shown that the previously found solution /1/, which gives the boundaries of the region of stability in a plane with parameters αJ, where α is a dimensionless wave number and J is the minimum value of the Richardson number are generally incomplete. The presence of constant stratification in the whole flow region enables free internal waves to exist which, emanating from the shear layer, give rise to additional radiative instability, previously examined in /2/ for a flow with tangential velocity discontinuity. The influence of the outer boundaries on the stability of the shear flow is clarified and the analytical results obtained are compared with numerical calculations carried out in /3, 4/ for certain models with smooth velocity and density profiles.

Instability and wave over-reflection in stably stratified shear flow

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.

Baroclinic instability of stratified shear flows in the ocean and atmosphere

As an extension of Eady's model, two new baroclinic models are studied in this paper: Model 1 consists of a layer with constant shear surmounting a quiescent layer, and model 2 consists of a more statically stable uniform layer of constant zonal flow surmounting a layer with constant shear. In model 1 two special cases are considered, one in which the upper shear layer is more stably stratified than the lower quiescent layer (1a) and another in which the upper layer is less stably stratified than the lower layer (1b). The stability properties of model 16 and model 2 turn out to be very similar, since for the same shear, depth ratio, and static stability ratio the growth rates are the same and the wave structure of model 1b is the symmetric image of that of model 2. The wavelengths of the stability cutoff wave and the most unstable wave and the growth rate of the most unstable wave are considerably influenced by the presence of a quiescent layer in model 1a but are only moderately influenced by this quiescent layer in model 1b or by the uniform layer in model 2. The unstable wave moves eastward with a velocity slower than that of the vertically averaged mean current in the shear layer in model 1 but faster than the mean current in the shear layer in model 2. The deviation of the phase velocity from this mean current becomes larger the longer the wave. Typical parameters for an oceanic current system, such as the Gulf Stream, are applied to model 1a, and typical parameters for an atmospheric mid-latitude zonal flow are applied to model 2. The implications regarding long waves are that just as the long atmospheric waves penetrate the tropopause into the stratosphere, so the long oceanic waves penetrate the base of the thermocline to great depths, although the horizontal scales for the two systems are considerably different.

Experiments on the instability of stratified shear flows: miscible fluids

Earlier papers have described a technique whereby a stratified shear flow may be produced under controlled conditions in the laboratory. A comparison between the experiments and theory is made here for small-amplitude unstable disturbances in an accelerating stratified free shear layer at the diffuse interface between layers of brine and water. In the early stages of the observed growth of the instability, which takes the form of growing waves, three measurable quantities can be compared with predicted values: the wavelength of the small amplitude waves, the time, which determines the flow conditions, and the growth rates of the waves. Some observations of the development of the disturbances to finite amplitude, the transition to turbulence and the resulting turbulence are reported.

Experiments on the instability of stratified shear flows: immiscible fluids

When a long rectangular tube containing two immiscible fluids is slightly tilted away from the horizontal, a uniformly accelerating flow is produced with shear at the interface. The presence of shear leads to instability, which is characterized by the spontaneous and rapid growth of almost stationary waves if the fluid depths are equal and the density difference small. The conditions for the onset of Kelvin-Helmholtz instability, taking account of the accelerating flow and the presence of a velocity transition region at the interface, are investigated theoretically and comparison made with observations. The time at which instability occurs is quite well predicted by this theory, but the wavelength of the unstable waves is rather greater than predicted in the accelerating flow. The difference between the predictions and observations may be the result of finite amplitude effects or of the development of Tollmien-Schlichting instability before Kelvin-Helmholtz.