ALBERT

Description: 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 streamwise 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.

Description: In this paper we discuss numerical simulations of the generation of large-amplitude solitary waves in a continuously stratified fluid by flow over isolated topography. We employ the fully nonlinear theory for internal solitary waves to classify the numerical results for mode-1 waves and compare with two classes of approximate theories, weakly nonlinear theory leading to the Korteweg-deVries and Gardner equations and conjugate flow theory which makes no approximation with respect to nonlinearity, but neglects dispersion entirely. We find that both weakly nonlinear theories have a limited range of applicability. In contrast, the conjugate flow theory predicts the nature of the limiting upstream propagating response (a dissipationless bore), successfully describes the bore's vertical structure, and gives a value of the inflow speed, cj, above which no upstream propagating response is possible. The numerical experiments demonstrate the existence of a class of large-amplitude response structures that are generated and trapped over the topography when the inflow speed exceeds cj. While similar in structure to fully nonlinear solitary waves, these trapped disturbances can induce isopycnal displacements more than 100% larger than those induced by the limiting solitary wave while remaining laminar. We develop a theory to describe the vertical structure at the crest of these trapped disturbances and describe its range of validity. Finally, we turn to the generation of mode-2 solitary-like waves. Mode-2 waves cannot be truly solitary owing to the existence of a small mode-1 tail that radiates energy downstream from the wave. We demonstrate that, for stratifications dominated by a single pycnocline, mode-2 wave dissipation is dominated by wave breaking as opposed to mode-1 wave radiation. We propose a phenomenological criterion based on weakly nonlinear theory to test whether mode-2 wave generation is to be expected for a given stratification. © 2005 Cambridge University Press.

Description: We consider the stability with respect to three-dimensional perturbations of columnar vortices on the f-plane and as a function of the strength of a stabilizing density stratification parallel to the axis of the vortex. We seek to understand the dynamics of the processes through which such a vertically oriented barotropic vortex may be destabilized. As models of the basic vorticity distribution we consider both Kelvin-Helmholtz vortices in shear and 'Kida-like vortices in strain. In the case of rotating unstratified flow, an isolated anticyclonic vortex column is shown to be strongly destabilized to three-dimensional perturbations by small values of the background rotation, while rapid rotation strongly stabilizes both anticyclonic and cyclonic columns, as expected on the basis of the Taylor-Proudman theorem. The dominant instability mechanism which drives the destruction of anticyclonic vortices in the presence of slow background rotation may be understood to constitute a three-dimensional inertial (centrifugal) instability. Through explicit analysis we show that sufficiently strong density stratification stabilizes the two-dimensional columnar structures to disruption by this and additional modes of instability that exist even in the absence of rotation. We furthermore demonstrate that there exists a second fundamental mode of instability in the presense of background rotation which affects only anticyclonic vortex columns whose cross-sections are elliptical. Only when the ellipticity of the vortex is sufficiently high does this mode dominate the centrifugal mode. The process whereby anticyclonic vortices may be selectively destroyed appears to provide a possible explanation of an asymmetry that is sometimes observed to be characteristic of the atmospheric von Kármán vortex streets that are observed in the lee of oceanic islands. The anticyclonic branch of the street often seems to be absent. More generally, the centrifugal mechanism for the selective destruction of anticyclones discussed herein very clearly explains a number of recent results obtained from both laboratory experiments and numerical simulations.

Description: We investigate the influence of the ellipticity of a columnar vortex in a rotating environment on its linear stability to three-dimensional perturbations. As a model of the basic-state vorticity distribution, we employ the Stuart steady-state solution of the Euler equations. In the presence of background rotation, an anticyclonic vortex column is shown to be strongly destabilized to three-dimensional perturbations when background rotation is weak, while rapid rotation strongly stabilizes both anticyclonic and cyclonic columns, as might be expected on the basis of the Taylor-Proudman theorem. We demonstrate that there exist three distinct forms of three-dimensional instability to which strong anticyclonic vortices are subject. One form consists of a Coriolis force modified form of the 'elliptical' instability, which is dominant for vortex columns whose cross-sections are strongly elliptical. This mode was recently discussed by Potylitsin & Peltier (1998) and Leblanc & Cambon (1998). The second form of instability may be understood to constitute a three-dimensional inertial (centrifugal) mode, which becomes the dominant mechanism of instability as the ellipticity of the vortex column decreases. Also evident in the Stuart model of the vorticity distribution is a third 'hyperbolic' mode of instability that is focused on the stagnation point that exists between adjacent vortex cores. Although this short-wavelength cross-stream mode is much less important in the spectrum of the Stuart model than it is in the case of a true homogeneous mixing layer, it nevertheless does exist even though its presence has remained undetected in most previous analyses of the stability of the Stuart solution.

Description: We examine the stability characteristics of a two-dimensional flow which consists initially of an inflexionally unstable shear layer on an f-plane. Under the action of the primary instability, the vorticity in the shear-layer initially coalesces into two Kelvin-Helmholtz vortices which subsequently merge to form a single coherent vortex. At a sequence of times during this process, we test the stability of the two-dimensional flow to fully three-dimensional perturbations. A somewhat novel approach is developed which removes inconsistencies in the secondary stability analyses which might otherwise arise owing to the time-dependence of the two-dimensional flow. In the non-rotating case, and before the onset of pairing, we obtain a spectrum of unstable longitudinal modes which is similar to that obtained previously by Pierrehumbert & Widnall (1982) for the Stuart vortex, and by Klaassen & Peltier (1985, 1989, 1991) for more realistic flows. In addition, we demonstrate the existence of a new sequence of three-dimensional subharmonic (and therefore ‘helical’) instabilities. After pairing is complete, the secondary instability spectrum is essentially unaltered except for a doubling of length- and timescales that is consistent with the notion of spatial and temporal self-similarity. Once pairing begins, the spectrum quickly becomes dominated by the unstable modes of the emerging subharmonic Kelvin-Helmholtz vortex, and is therefore similar to that which is characteristic of the post-pairing regime. Also in the context of non-rotating flow, we demonstrate that the direct transfer of energy into the dissipative subrange via secondary instability is possible only if the background flow is stationary, since even slow time-dependence acts to decorrelate small-scale modes and thereby to impose a short-wave cutoff on the spectrum. The stability of the merged vortex state is assessed for various values of the planetary vorticity f. Slow rotation may either stabilize or destabilize the columnar vortices, depending upon the sign of f while fast rotation of either sign tends to be stabilizing. When f has opposite sign to the relative vorticity of the two-dimensional basic state, the flow becomes unstable to a new mode of instability that has not been previously identified. Modes whose energy is concentrated in the vortex cores are shown to be associated, even at non-zero f with Pierrehumbert’s (1986) elliptical instability. Through detailed consideration of the vortex interaction mechanisms which drive instability, we are able to provide physical explanations for many aspects of the three-dimensionalization process. © 1994, Cambridge University Press

Description: We study the competition between various secondary instabilities that co-exist in a preturbulent stratified parallel flow subject to Kelvin-Helmholtz instability. In particular, we investigate whether a secondary braid instability might emerge prior to the overturning of the statically unstable regions that develop in the cores of the primary Kelvin-Helmholtz billows. We identify two groups of instabilities on the braid. One group is a shear instability which extracts its energy from the background shear and is suppressed by the straining contribution of the background flow. The other group, which seems to have no precedent in the literature, includes phase-locked modes which grow at the stagnation point on the braid and are almost entirely driven by the straining contributions of the background flow. For the latter group, the braid shear has a negative influence on the growth rate. Our analysis demonstrates that the probability of finite-amplitude growth of both braid instabilities is enhanced with increasing Reynolds number and Richardson number. We also show that the possibility of emergence of braid instabilities decreases with the Prandtl number for the shear modes and increases for the stagnation point instabilities. Through detailed non-separable linear stability analysis, we show that both braid instabilities are fundamentally three dimensional with the shear modes being of small wavenumbers and the stagnation point modes dominating at large wavenumber. © 2012 Cambridge University Press.

Description: The linear stability analyses described in Mashayek & Peltier (J. Fluid Mech., vol. 708, 2012, 5-44, hereafter MP1) are extended herein in an investigation of the influence of stratification on the evolution of secondary instabilities to which an evolving Kelvin-Helmholtz (KH) wave is susceptible in an initially unstable parallel stratified shear layer. We show that over a wide range of background stratification levels, the braid shear instability has a higher probability of emerging at early stages of the flow evolution while the secondary convective instability (SCI), which occurs in the eyelids of the individual Kelvin cats eyes, will remain a relevant and dominant instability at high Reynolds numbers. The evolution of both modes is greatly influenced by the background stratification. Various other three-dimensional secondary instabilities are found to exist over a wide range of stratification levels. In particular, the stagnation point instability (SPI), which was discussed in detail in MP1, may be of great potential importance providing alternate routes for transition of an initially two-dimensional KH wave into fully developed turbulence. The energetics of the secondary instabilities revealed by our simulations are analysed in detail and the preturbulent mixing properties are studied. © 2012 Cambridge University Press.

Description: In order that it be correctly characterized, irreversible turbulent mixing in stratified fluids must distinguish between adiabatic 'stirring' and diabatic 'mixing'. Such a distinction has been formalized through the definition of a diapycnal diffusivity, Kρ (Winters & D'Asaro, J. Fluid Mech., vol. 317, 1996, pp. 179-193) and an appropriate mixing efficiency, ε (Caulfield & Peltier, J. Fluid Mech., vol. 413, 2000, pp. 1-47). Equivalent attention has not been paid to the definitions of a corresponding momentum diffusivity Km and hence an appropriately defined turbulent Prandtl number Prt = Km/Kρ. In this paper, the diascalar framework of Winters & D'Asaro (1996) is first reformulated to obtain an 'Osborn-like' formula in which the correct definition of irreversible mixing efficiency ε is shown to replace the flux Richardson number which Osborn (J. Phys. Oceanogr., vol. 10, 1980, pp. 83-89) assumed to characterize this efficiency. We advocate the use of this revised representation for diapycnal diffusivity since the proposed reformulation effectively removes the simplifying assumptions on which the original Osborn formula was based. We similarly propose correspondingly reasonable definitions for Km and Prt by eliminating the reversible component of the momentum production term. To explore implications of the reformulations for both diapycnal and momentum diffusivity we employ an extensive series of direct numerical simulations (DNS) to investigate the properties of the shear-induced density-stratified turbulence that is engendered through the breaking of a freely evolving Kelvin-Helmholtz wave. The DNS results based on the proposed reformulation of Kρ are compared with available estimations due to the mixing length model, as well as both the Osborn-Cox and the Osborn models. Estimates based upon the Osborn-Cox formulation are shown to provide the closest approximation to the diapycnal diffusivity delivered by the exact representation. Through compilation of the complete set of DNS results we explore the characteristic dependence of Kρ on the buoyancy Reynolds number Reb as originally investigated by Shih et al. (J. Fluid Mech., vol. 525, 2005, pp. 193-214) in their idealized study of homogeneous stratified and sheared turbulence, and show that the validity of their results is only further reinforced through analysis of the turbulence produced in the more geophysically relevant Kelvin-Helmholtz wave life-cycle ansatz. In contrast to the results described by Shih et al. (2005) however, we show that, besides Reb, a vertically averaged measure of the gradient Richardson number Rib may equivalently characterize the turbulent mixing at high Reb. Based on the dominant driving processes involved in irreversible mixing, we categorize the intermediate (i.e. Reb = O(101-102)) and high (i.e. Reb〉O(102)) range of Reb as 'buoyancy-dominated' and 'shear-dominated' mixing regimes, which together define a transition value of Rib ∼ 0.2. Mixing efficiency varies non-monotonically with both Reb and Rib, with its maximum (on the order of 0.2-0.3) occurring in the 'buoyancy-dominated' regime. Unlike Kρ which is very sensitive to the correct choice of ε (i.e. Kρ α ε/(1 - ε), we show that Km is almost insensitive to the choice of ε (i.e. Km α 1/(1 - ε)) so long as ε is not close to unity, which implies Km ≈RibReb for the entire range of Reb. The turbulent Prandtl number is consequently shown to decrease monotonically with Reb and may be (to first order) simply approximated by Reb itself. Assuming Prt = 1, or Prt = 10 (as is common in large-scale numerical models of the ocean general circulation), is also suggested to be a questionable assumption. © 2015 Cambridge University Press.

Description: We consider numerically the transition to turbulence and associated mixing in stratified shear flows with initial velocity distribution and initial density distribution away from a hydrostatic reference state. When the ratio of the characteristic length scales over which the velocity and density vary is equal to one, this flow is primarily susceptible to the classic well-known Kelvin-Helmholtz instability (KHI). This instability, which typically manifests at finite amplitude as an array of elliptical vortices, strongly 'overturns' the density interface of strong initial gradient, which nevertheless is the location of minimum initial gradient Richardson number, where and is a bulk Richardson number. As is well known, at sufficiently high Reynolds numbers, the primary KHI induces a vigorous but inherently transient burst of turbulence and associated irreversible mixing localised in the vicinity of the density interface, leading to a relatively well-mixed region bounded by stronger density gradients above and below. We explore the qualitatively different behaviour that arises when, and so the density interface is sharp, with now being maximum at the density interface. This flow is primarily susceptible to Holmboe wave instability (HWI) (Holmboe, Geophys. Publ., vol. 24, 1962, pp. 67-113), which manifests at finite amplitude in this symmetric flow as counter-propagating trains of elliptical vortices above and below the density interface, thus perturbing the interface so as to exhibit characteristically cusped interfacial waves which thereby 'scour' the density interface. Unlike previous lower-experimental and numerical studies, when is sufficiently high the primary HWI becomes increasingly more three-dimensional due to the emergence of shear-aligned secondary convective instabilities. As increases, (i) the growth rate of secondary instabilities appears to saturate and (ii) the perturbation kinetic energy exhibits a spectrum for sufficiently large length scales that are influenced by anisotropic buoyancy effects. Therefore, at sufficiently high, vigorous turbulence is triggered that also significantly 'scours' the primary density interface, leading to substantial irreversible mixing and vertical transport of mass above and below the (robust) primary density interface. Furthermore, HWI produces a markedly more long-lived turbulence event compared to KHI at a similarly high. Despite their vastly different mechanics (i.e. 'overturning' versus 'scouring') and localisation, the mixing induced by KHI and HWI is comparable in both absolute terms and relative efficiency. Our results establish that, provided the flow Reynolds number is sufficiently high, shear layers with sharp density interfaces and associated locally high values of the gradient Richardson number may still be sites of substantial and efficient irreversible mixing. © 2016 Cambridge University Press.