26 results
Parametrization of irreversible diapycnal diffusivity in salt-fingering turbulence using DNS
- Yuchen Ma, W. R. Peltier
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- Journal:
- Journal of Fluid Mechanics / Volume 911 / 25 March 2021
- Published online by Cambridge University Press:
- 25 January 2021, A9
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We employ direct numerical simulations of salt fingering engendered turbulent mixing to derive a parameterization scheme for the representation of this physical process in low-resolution ocean models and compare the results with those previously suggested on empirical grounds. In this analysis we differentiate between the reversible and irreversible contributions to diapycnal diffusivity associated with the turbulence generated by this mechanism. The necessity of such a distinction has been clearly recognized in connection with shear-driven density stratified turbulence processes: only irreversible processes can contribute to the effective turbulent diapycnal diffusivity. We expand the formalism herein to the more complicated salt-fingering case as a first step towards analysis of the general case. The irreversible fluxes are determined in the case of salt fingering related turbulence by examining high-resolution direct numerical simulation (DNS)-derived turbulence data sets based upon two different models: namely the ‘unbounded gradient model’ and the ‘interface model’ with depth-dependent gradients of temperature and salinity. By fitting the irreversible diapycnal fluxes in the unbounded gradient model (for equilibrium states) as a function of density ratio (the governing non-dimensional parameter), we derive a functional form that can be used as a basis for a next generation salt-fingering parametrization scheme. By applying this scheme to the interface model, we demonstrate that the local fluxes predicted agree well with those obtained from the numerical simulations based upon this more complex model. We compare this new DNS-derived turbulence parameterization with those that have been derived empirically.
An investigation of the possibility of non-Laurentide ice stream contributions to Heinrich event 3
- Jesse Velay-Vitow, W. Richard Peltier, Gordan R. Stuhne
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- Journal:
- Quaternary Research / Volume 101 / May 2021
- Published online by Cambridge University Press:
- 24 November 2020, pp. 13-25
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The ocean floor sedimentological signature of Heinrich event 3 (H3) is markedly different from that of other Heinrich events that are known to have originated in Hudson Strait. It has therefore been suggested that the H3 contribution to iceberg flux may have been delivered by ice streams located in the eastern sector of the North Atlantic, from the Fennoscandian or British Isles ice sheets. To investigate this possibility and whether the instability involved may have been tidally induced, as seems to have been the case for H1, we consider several eastern Atlantic sector possibilities: a hypothetical Barents Sea ice stream, the Norwegian ice stream, and the Irish Sea ice stream. We find that the extremely high amplitude of the M2 tidal constituent in the western North Atlantic that appears to have forced H1 did not exist in the northeastern Atlantic. This suggests that, with one possible exception, if destabilized ice streams in this region did contribute to H3, tidal forcing was most probably not the cause. The single exception to this general conclusion may be the Irish Sea ice stream, and we comment on the probability of a contribution to H3 from this source.
Deep learning of mixing by two ‘atoms’ of stratified turbulence
- Hesam Salehipour, W. R. Peltier
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- Journal:
- Journal of Fluid Mechanics / Volume 861 / 25 February 2019
- Published online by Cambridge University Press:
- 04 January 2019, R4
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Current global ocean models rely on ad hoc parameterizations of diapycnal mixing, in which the efficiency of mixing is globally assumed to be fixed at 20 %, despite increasing evidence that this assumption is questionable. As an ansatz for small-scale ocean turbulence, we may focus on stratified shear flows susceptible to either Kelvin–Helmholtz (KHI) or Holmboe wave (HWI) instability. Recently, an unprecedented volume of data has been generated through direct numerical simulation (DNS) of these flows. In this paper, we describe the application of deep learning methods to the discovery of a generic parameterization of diapycnal mixing using the available DNS dataset. We furthermore demonstrate that the proposed model is far more universal compared to recently published parameterizations. We show that a neural network appropriately trained on KHI- and HWI-induced turbulence is capable of predicting mixing efficiency associated with unseen regions of the parameter space well beyond the range of the training data. Strikingly, the high-level patterns learned based on the KHI and weakly stratified HWI are ‘transferable’ to predict HWI-induced mixing efficiency under much more strongly stratified conditions, suggesting that through the application of appropriate networks, significant universal abstractions of density-stratified turbulent mixing have been recognized.
Self-organized criticality of turbulence in strongly stratified mixing layers
- Hesam Salehipour, W. R. Peltier, C. P. Caulfield
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- Journal:
- Journal of Fluid Mechanics / Volume 856 / 10 December 2018
- Published online by Cambridge University Press:
- 02 October 2018, pp. 228-256
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Motivated by the importance of stratified shear flows in geophysical and environmental circumstances, we characterize their energetics, mixing and spectral behaviour through a series of direct numerical simulations of turbulence generated by Holmboe wave instability (HWI) under various initial conditions. We focus on circumstances where the stratification is sufficiently ‘strong’ so that HWI is the dominant primary instability of the flow. Our numerical findings demonstrate the emergence of self-organized criticality (SOC) that is manifest as an adjustment of an appropriately defined gradient Richardson number, $Ri_{g}$, associated with the horizontally averaged mean flow, in such a way that it is continuously attracted towards a critical value of $Ri_{g}\sim 1/4$. This self-organization occurs through a continuously reinforced localization of the ‘scouring’ motions (i.e. ‘avalanches’) that are characteristic of the turbulence induced by the breakdown of Holmboe wave instabilities and are developed on the upper and lower flanks of the sharply localized density interface, embedded within a much more diffuse shear layer. These localized ‘avalanches’ are also found to exhibit the expected scale-invariant characteristics. From an energetics perspective, the emergence of SOC is expressed in the form of a long-lived turbulent flow that remains in a ‘quasi-equilibrium’ state for an extended period of time. Most importantly, the irreversible mixing that results from such self-organized behaviour appears to be characterized generically by a universal cumulative turbulent flux coefficient of $\unicode[STIX]{x1D6E4}_{c}\sim 0.2$ only for turbulent flows engendered by Holmboe wave instability. The existence of this self-organized critical state corroborates the original physical arguments associated with self-regulation of stratified turbulent flows as involving a ‘kind of equilibrium’ as described by Turner (1973, Buoyancy Effects in Fluids, Cambridge University Press).
Role of overturns in optimal mixing in stratified mixing layers
- A. Mashayek, C. P. Caulfield, W. R. Peltier
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- Journal:
- Journal of Fluid Mechanics / Volume 826 / 10 September 2017
- Published online by Cambridge University Press:
- 08 August 2017, pp. 522-552
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Turbulent mixing plays a major role in enabling the large-scale ocean circulation. The accuracy of mixing rates estimated from observations depends on our understanding of basic fluid mechanical processes underlying the nature of turbulence in a stratified fluid. Several of the key assumptions made in conventional mixing parameterizations have been increasingly scrutinized in recent years, primarily on the basis of adequately high resolution numerical simulations. We add to this evidence by compiling results from a suite of numerical simulations of the turbulence generated through stratified shear instability processes. We study the inherently intermittent and time-dependent nature of wave-induced turbulent life cycles and more specifically the tight coupling between inherently anisotropic scales upon which small-scale isotropic turbulence grows. The anisotropic scales stir and stretch fluid filaments enhancing irreversible diffusive mixing at smaller scales. We show that the characteristics of turbulent mixing depend on the relative time evolution of the Ozmidov length scale $L_{O}$ compared to the so-called Thorpe overturning scale $L_{T}$ which represents the scale containing available potential energy upon which turbulence feeds and grows. We find that when $L_{T}\sim L_{O}$, the mixing is most active and efficient since stirring by the largest overturns becomes ‘optimal’ in the sense that it is not suppressed by ambient stratification. We argue that the high mixing efficiency associated with this phase, along with observations of $L_{O}/L_{T}\sim 1$ in oceanic turbulent patches, together point to the potential for systematically underestimating mixing in the ocean if the role of overturns is neglected. This neglect, arising through the assumption of a clear separation of scales between the background mean flow and small-scale quasi-isotropic turbulence, leads to the exclusion of an highly efficient mixing phase from conventional parameterizations of the vertical transport of density. Such an exclusion may well be significant if the mechanism of shear-induced turbulence is assumed to be representative of at least some turbulent events in the ocean. While our results are based upon simulations of shear instability, we show that they are potentially more generic by making direct comparisons with $L_{T}-L_{O}$ data from ocean and lake observations which represent a much wider range of turbulence-inducing physical processes.
A high-resolution model of the 100 ka ice-age cycle
- L. Tarasov, W. R. Peltier
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- Journal:
- Annals of Glaciology / Volume 25 / 1997
- Published online by Cambridge University Press:
- 20 January 2017, pp. 58-65
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Significant improvements to the representation of climate forcing and mass-balance response in a coupled two-dimensional global energy balance climate model (EBM) and vertically integrated ice-sheet model (ISM) have led to the prediction of an ice-volume chronology for the most recent ice-age cycle of the Northern Hemisphere that is close to that inferred from the geological record. Most significant is that full glacial termination is delivered by the model without the need for new physical ingredients. In addition, a relatively close match is achieved between the Last Glacial Maximum (LGM) model ice topography and that of the recently-described ICE-4G reconstruction. These results suggest that large-scale climate system reorganization is not required to explain the main variations of the North American (NA) ice sheets over the last glacial cycle. Lack of sea-ice and marine-ice dynamics in the model leaves the situation over the Eurasian (EA) sector much more uncertain.
The incorporation of a gravitationally self-consistent description of the glacial isostatic adjustment process demonstrates that the NA and EA bedrock responses can be adequately represented by simpler damped-relaxation models with characteristic time-scales of 3–5ka and 5 ka, respectively. These relaxation times agree with those independently inferred on the basis of postglacial relative sea-level histories.
A Spectral Model of the Ice-Age Cycle with Glacial Isostatic Adjustment (Abstract)
- W. R. Peltier
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- Journal:
- Annals of Glaciology / Volume 5 / 1984
- Published online by Cambridge University Press:
- 20 January 2017, p. 223
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Lithospheric Thickness, Antarctic Deglaciation History, and Ocean Basin Discretization Effects in a Global Model Of Postglacial Sea Level Change: a Summary of Some Sources of Nonuniqueness
- W. R. Peltier
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- Quaternary Research / Volume 29 / Issue 2 / March 1988
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- 20 January 2017, pp. 93-112
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The global model of postglacial relative sea level variations that has been developed over the past decade is employed to investigate the constraints that it may be invoked to place on the timing of the deglaciation of West Antarctica. The analyses presented here confirm the suggestion of P. Wu and W. R. Peltier (1983, Geophysical Journal of the Royal Astronomical Society 74 , 377–450) that the model of this event employed in J. A. Clark and C. S. Lingle (1979Quaternary Research 11 , 279–298) may be simply modified to rectify the misfits between theory and observation that are otherwise obtained at Southern Hemisphere sites. A large number of Southern Hemisphere relative sea level data are shown to require that the retreat of Antarctic ice substantially lagged the retreat of Northern Hemisphere ice if the deglaciation of Antarctica was abrupt. The time of onset of Antarctic deglaciation is thereby shown to coincide with the time of most rapid Northern Hemisphere deglaciation. Sensitivity tests are performed which demonstrate that this result is relatively insensitive to the discretization employed to represent the ocean basins; the only exception to this general rule obtains at some coastal sites at which a trade-off is revealed between the delay of West Antarctic melting and the thickness of the lithosphere required to reconcile the observed local variations of relative sea level. At such sites, which are all in the far field of the ice sheets, some attention must be paid to the accuracy of the local finite element representation of the oceans and to the details of the Antarctic deglaciation history.
Turbulent mixing due to the Holmboe wave instability at high Reynolds number
- Hesam Salehipour, C. P. Caulfield, W. R. Peltier
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- Journal:
- Journal of Fluid Mechanics / Volume 803 / 25 September 2016
- Published online by Cambridge University Press:
- 30 August 2016, pp. 591-621
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We consider numerically the transition to turbulence and associated mixing in stratified shear flows with initial velocity distribution $\overline{U}(z,0)\,\boldsymbol{e}_{x}=U_{0}\,\boldsymbol{e}_{x}\tanh (z/d)$ and initial density distribution $\overline{\unicode[STIX]{x1D70C}}(z,0)=\unicode[STIX]{x1D70C}_{0}[1-\tanh (z/\unicode[STIX]{x1D6FF})]$ away from a hydrostatic reference state $\unicode[STIX]{x1D70C}_{r}\gg \unicode[STIX]{x1D70C}_{0}$. When the ratio $R=d/\unicode[STIX]{x1D6FF}$ 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 $Ri_{g}(0)=Ri_{b}=g\unicode[STIX]{x1D70C}_{0}d/\unicode[STIX]{x1D70C}_{r}U_{0}^{2}$, where $Ri_{g}(z)=-([g/\unicode[STIX]{x1D70C}_{r}]\,\text{d}\overline{\unicode[STIX]{x1D70C}}/\text{d}z)/(\text{d}\overline{U}/\text{d}z)^{2}$ and $Ri_{b}$ is a bulk Richardson number. As is well known, at sufficiently high Reynolds numbers ($Re$), 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 $R\gg 1$, and so the density interface is sharp, with $Ri_{g}(z)$ now being maximum at the density interface $Ri_{g}(0)=RRi_{b}$. 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-$Re$ experimental and numerical studies, when $Re$ is sufficiently high the primary HWI becomes increasingly more three-dimensional due to the emergence of shear-aligned secondary convective instabilities. As $Re$ increases, (i) the growth rate of secondary instabilities appears to saturate and (ii) the perturbation kinetic energy exhibits a $k^{-5/3}$ spectrum for sufficiently large length scales that are influenced by anisotropic buoyancy effects. Therefore, at sufficiently high $Re$, 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 $Re$. 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.
Diapycnal diffusivity, turbulent Prandtl number and mixing efficiency in Boussinesq stratified turbulence
- Hesam Salehipour, W. R. Peltier
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- Journal:
- Journal of Fluid Mechanics / Volume 775 / 25 July 2015
- Published online by Cambridge University Press:
- 26 June 2015, pp. 464-500
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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, $K_{{\it\rho}}$ (Winters & D’Asaro, J. Fluid Mech., vol. 317, 1996, pp. 179–193) and an appropriate mixing efficiency, $\mathscr{E}$ (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 $K_{m}$ and hence an appropriately defined turbulent Prandtl number $\mathit{Pr}_{t}=K_{m}/K_{{\it\rho}}$. 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 $\mathscr{E}$ 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 $K_{m}$ and $\mathit{Pr}_{t}$ 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 $K_{{\it\rho}}$ 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 $K_{{\it\rho}}$ on the buoyancy Reynolds number $\mathit{Re}_{b}$ 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 $\mathit{Re}_{b}$, a vertically averaged measure of the gradient Richardson number $\mathit{Ri}_{b}$ may equivalently characterize the turbulent mixing at high $\mathit{Re}_{b}$. Based on the dominant driving processes involved in irreversible mixing, we categorize the intermediate (i.e. $\mathit{Re}_{b}=O(10^{1}{-}10^{2})$) and high (i.e. $\mathit{Re}_{b}>O(10^{2})$) range of $\mathit{Re}_{b}$ as ‘buoyancy-dominated’ and ‘shear-dominated’ mixing regimes, which together define a transition value of $\mathit{Ri}_{b}\sim 0.2$. Mixing efficiency varies non-monotonically with both $\mathit{Re}_{b}$ and $\mathit{Ri}_{b}$, with its maximum (on the order of 0.2–0.3) occurring in the ‘buoyancy-dominated’ regime. Unlike $K_{{\it\rho}}$ which is very sensitive to the correct choice of $\mathscr{E}$ (i.e. $K_{{\it\rho}}\propto \mathscr{E}/(1-\mathscr{E})$), we show that $K_{m}$ is almost insensitive to the choice of $\mathscr{E}$ (i.e. $K_{m}\propto 1/(1-\mathscr{E})$) so long as $\mathscr{E}$ is not close to unity, which implies $K_{m}\approx \mathit{Ri}_{b}\mathit{Re}_{b}$ for the entire range of $\mathit{Re}_{b}$. The turbulent Prandtl number is consequently shown to decrease monotonically with $\mathit{Re}_{b}$ and may be (to first order) simply approximated by $\mathit{Re}_{b}$ itself. Assuming $\mathit{Pr}_{t}=1$, or $\mathit{Pr}_{t}=10$ (as is common in large-scale numerical models of the ocean general circulation), is also suggested to be a questionable assumption.
Turbulent diapycnal mixing in stratified shear flows: the influence of Prandtl number on mixing efficiency and transition at high Reynolds number
- H. Salehipour, W. R. Peltier, A. Mashayek
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- Journal:
- Journal of Fluid Mechanics / Volume 773 / 25 June 2015
- Published online by Cambridge University Press:
- 20 May 2015, pp. 178-223
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Motivated by the importance of small-scale turbulent diapycnal mixing to the closure of the large-scale meridional overturning circulation (MOC) of the oceans, we focus on a model problem which allows us to address the fundamental fluid mechanics that is expected to be characteristic of the oceanographic regime. Our model problem is one in which the initial conditions consist of a stably stratified parallel shear flow which evolves into the turbulent regime through the growth of a Kelvin–Helmholtz wave to finite amplitude followed by transition to turbulence. Through both linear stability analysis and direct numerical simulations (DNS), we investigate the secondary instabilities and the turbulent mixing at a fixed high Reynolds number and for a range of Prandtl numbers. We demonstrate that the oceanographically expected high value of the Prandtl number has a profound influence on the nature of the secondary instabilities that govern the transition process. Specifically through non-separable linear stability analysis, we discover new characteristics for the shear-aligned convective instability such that it is modified into a mixed mode that is driven both by static instability and by shear. The growth rate and ultimate strength of this mode are both strongly enhanced at higher $\mathit{Pr}$ while the growth rate and ultimate strength of the stagnation point instability (SPI), which may compete for control of the transition process, are simultaneously impeded. Of equal importance is the fact that, for higher $\mathit{Pr}$, the characteristic length scales associated with the dominant mixed mode of instability decrease and therefore there ceases to be a strong scale selectivity. In the limit of much higher $\mathit{Pr}$, we conjecture that a wide range of spatial scales become equally unstable so as to support an ‘ultraviolet catastrophe’, in which a direct injection of energy occurs into a broad range of scales simultaneously. We further establish the validity of these analytical results through a series of computationally challenging DNS analyses, and provide a detailed analysis of the efficiency of the turbulent mixing of the density field that occurs subsequent to transition and of the entrainment of fluid into the mixing layer from the high-speed flanks of the shear flow. We show that the mixing efficiency decreases monotonically with increase of the molecular value of the Prandtl number and the expansion of the shear layer is reduced as such entrainment diminishes.
Time-dependent, non-monotonic mixing in stratified turbulent shear flows: implications for oceanographic estimates of buoyancy flux
- A. Mashayek, C. P. Caulfield, W. R. Peltier
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- Journal:
- Journal of Fluid Mechanics / Volume 736 / 10 December 2013
- Published online by Cambridge University Press:
- 11 November 2013, pp. 570-593
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We employ direct numerical simulation to investigate the efficiency of diapycnal mixing by shear-induced turbulence in stably stratified free shear layers for flows with bulk Richardson numbers in the range $0. 12\leq R{i}_{0} \leq 0. 2$ and Reynolds number $Re= 6000$. We show that mixing efficiency depends non-monotonically upon $R{i}_{0} $, peaking in the range 0.14–0.16, which coincides closely with the range in which both the buoyancy flux and the dissipation rate are maximum. By detailed analyses of the energetics of flow evolution and the underlying dynamics, we show that the existence of high mixing efficiency in the range $0. 14\lt R{i}_{0} \lt 0. 16$ is due to the emergence of a large number of small-scale instabilities which do not exist at lower Richardson numbers and are stabilized at high Richardson numbers. As discussed in Mashayek & Peltier (J. Fluid Mech., vol. 725, 2013, pp. 216–261), the existence of such a well-populated ‘zoo’ of secondary instabilities at intermediate Richardson numbers and the subsequent high mixing efficiency is realized only if the Reynolds number is higher than a critical value which is generally higher than that achievable in laboratory settings, as well as that which was achieved in the majority of previous numerical studies of shear-induced stratified turbulence. We furthermore show that the primary assumptions upon which the widely employed Osborn (J. Phys. Oceanogr. vol. 10, 1980, pp. 83–89) formula is based, as well as its counterparts and derivatives, which relate buoyancy flux to dissipation rate through a (constant) flux coefficient ($\Gamma $), fail at higher Richardson numbers provided that the Reynolds number is sufficiently high. Specifically, we show that the assumptions of fully developed, stationary, and isotropic turbulence all break down at high Richardson numbers. We show that the breakdown of these assumptions occurs most prominently at Richardson numbers above that corresponding to the maximum mixing efficiency, a fact that highlights the importance of the non-monotonicity of the dependence of mixing efficiency upon Richardson number, which we establish to be characteristic of stratified shear-induced turbulence. At high $R{i}_{0} $, the lifecycle of the turbulence is composed of a rapidly growing phase followed by a phase of rapid decay. Throughout the lifecycle, there is considerable exchange of energy between the small-scale turbulence and larger coherent structures which survive the various stages of flow evolution. Since shear instability is one of the most prominent mechanisms for turbulent dissipation of energy at scales below hundreds of metres and at various depths of the ocean, our results have important implications for the inference of turbulent diffusivities on the basis of microstructure measurements in the oceanic environment.
Shear-induced mixing in geophysical flows: does the route to turbulence matter to its efficiency?
- A. Mashayek, W. R. Peltier
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- Journal of Fluid Mechanics / Volume 725 / 25 June 2013
- Published online by Cambridge University Press:
- 14 May 2013, pp. 216-261
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Motivated by the importance of diapycnal mixing parameterizations in large-scale ocean general circulation models, we provide a detailed analysis of high-Reynolds-number mixing in density stratified shear flows which constitute an archetypical example of the small-scale physical processes occurring in the oceanic interior that control turbulent diffusion. Our focus is upon the issue as to whether the route to fully developed turbulence in the stratified mixing layer is in any significant way determinant of diapycnal mixing efficiency as represented by an effective turbulent diffusivity. We characterize different routes to fully developed turbulence by the nature of the secondary instabilities through which a primary Kelvin–Helmholtz billow executes the transition to this state. We then demonstrate that different mechanisms of turbulence transition characterized in these different transition mechanisms lead to considerably different values for the efficiency of diapycnal mixing and also for the effective vertical flux of buoyancy. We show that the widely employed value of 0.15–0.2 for the efficiency of mixing in shear-induced stratified turbulence based upon both laboratory measurements and similarly low-Reynolds-number numerical simulations may be too low for the high-Reynolds-number regime characteristic of geophysical flows. Our results show that the mixing efficiency tends to a value of approximately $1/ 3$ for sufficiently large Reynolds number at an intermediate value of 0.12 for the Richardson number. This is in agreement with a theoretical predictions of Caulfield, Tang and Plasting (J. Fluid Mech., vol. 498, 2004, pp. 315–332) for the asymptotic value of mixing efficiency in stratified Couette flows. In the high-Reynolds-number regime, mixing efficiency is shown to vary over a considerable range during the course of a particular shear-induced mixing event. We explain this variation on the basis of a detailed examination of the underlying dynamics. Since values in the range 0.15–0.2 for mixing efficiency have been extensively employed to infer an effective diffusivity from ocean microstructure measurements and also in energy balance analyses of the requirements of the global ocean circulation, our findings have potentially important implications for large-scale ocean modelling. We also quantify the errors introduced by employing the Osborn (J. Phys. Oceanogr., vol. 10, 1980, pp. 83–89) formula along with an efficiency of 0.15 to infer values for effective diffusivity, and explain the logical underpinnings of this conclusion. One of the more important aspects of this work from the perspective of our theoretical understanding of stratified turbulence is the demonstration that the inverse cascade of energy, which is facilitated by the vortex-merging process that is typical of laboratory experiments and of the low-Reynolds-number simulations of shear flow evolution, is strongly suppressed by increase of the Reynolds number to values typical of geophysical flows. Based on this finding, the application of results based on low-Reynolds-number (numerical or laboratory) experiments to high-Reynolds-number geophysical shear flows needs to be reconsidered.
The ‘zoo’ of secondary instabilities precursory to stratified shear flow transition. Part 2 The influence of stratification
- A. Mashayek, W. R. Peltier
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- Journal:
- Journal of Fluid Mechanics / Volume 708 / 10 October 2012
- Published online by Cambridge University Press:
- 03 September 2012, pp. 45-70
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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.
The ‘zoo’ of secondary instabilities precursory to stratified shear flow transition. Part 1 Shear aligned convection, pairing, and braid instabilities
- A. Mashayek, W. R. Peltier
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- Journal:
- Journal of Fluid Mechanics / Volume 708 / 10 October 2012
- Published online by Cambridge University Press:
- 29 August 2012, pp. 5-44
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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.
Three-dimensionalization of barotropic vortices on the f-plane
- W. D. Smyth, W. R. Peltier
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- Journal:
- Journal of Fluid Mechanics / Volume 265 / 25 April 1994
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- 26 April 2006, pp. 25-64
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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 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.
Instability and transition in finite-amplitude Kelvin–Helmholtz and Holmboe waves
- W. D. Smyth, W. R. Peltier
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- Journal:
- Journal of Fluid Mechanics / Volume 228 / July 1991
- Published online by Cambridge University Press:
- 26 April 2006, pp. 387-415
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We investigate the transition to turbulence in a free shear layer which contains a thin central region of stable density stratification. The fluid is assumed to possess Prandtl number significantly larger than unity, and the flow may exhibit either Holmboe or Kelvin–Helmholtz (KH) instability, depending upon the intensity of the stratification. A sequence of two-dimensional nonlinear numerical simulations of flows near the KH–Holmboe transition (i.e. having bulk Richardson numbers near 1/4) clearly illustrates the structural relationship between Holmboe and Kelvin–Helmholtz waves. The time-dependent nonlinear wave states delivered by the simulations are subjected to a three-dimensional normal-mode stability analysis in order to discover the physical processes that might drive the flow towards a turbulent state. Strong secondary instability is found to persist up to large spanwise wavenumbers, with no indication of a preferred lengthscale. These results indicate that secondary instability may lead the flow directly into the turbulent state.
The influence of stratification on secondary instability in free shear layers
- G. P. Klaassen, W. R. Peltier
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- Journal:
- Journal of Fluid Mechanics / Volume 227 / June 1991
- Published online by Cambridge University Press:
- 26 April 2006, pp. 71-106
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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.
The role of transverse secondary instabilities in the evolution of free shear layers
- G. P. Klaassen, W. R. Peltier
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- Journal:
- Journal of Fluid Mechanics / Volume 202 / May 1989
- Published online by Cambridge University Press:
- 26 April 2006, pp. 367-402
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Linear stability analyses and nonlinear flow simulations reveal several important features of transverse secondary instabilities of two-dimensional Kelvin–Helmholtz billows and Stuart vortices. Vortex pairing is found to be the most rapidly amplified mode in a continuous spectrum of vortex merging instabilities. In certain not uncommon circumstances it is possible for more than two vortices to amalgamate in a single interaction, demonstrating that the phenomenon that has become known as the pairing resonance in fact has a rather low quality factor. Another form of merging instability in which a vortex is deformed and drained by its neighbours has been revealed by our linear stability analyses of nonlinear shear-layer disturbances. It appears, however, that this vortex draining instability may be important only in unstratified or very weakly stratified flows, since in moderately stratified Kelvin–Helmholtz flow, it is replaced by a highly localized instability which leads to a temporary distortion of the braids. Nonlinear simulations of vortex merging events in moderately stratified, high-Reynolds-number shear layers are compared to the theoretical predictions of our stability analyses. We investigate and quantify the sensitivity of merging events to variations in the initial conditions. The character of the flow after merging instability saturates and the nonlinear aspects of multiple merging events are also considered.
The onset of turbulence in finite-amplitude Kelvin–Helmholtz billows
- G. P. Klaassen, W. R. Peltier
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- Journal:
- Journal of Fluid Mechanics / Volume 155 / June 1985
- Published online by Cambridge University Press:
- 20 April 2006, pp. 1-35
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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.