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Unstable waves on a sheared density interface Carpenter, Jeffrey Richard 2009

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UNSTABLE WAVES ON A SHEARED DENSITY INTERFACEbyJeffrey Richard CarpenterM.A.Sc., University of British Columbia, 2005B.Sc.Eng., University of Guelph, 2003A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate Studies(Civil Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2009c Jeffrey Richard Carpenter 2009iiAbstractThe Holmboe instability is known to occur in stratified shear layers that exhibit a relatively thindensity interface compared to the shear layer thickness. At finite amplitude the instability appearsas cusp-like propagating internal waves. The evolution and identification of these unstable wavesis the subject of this thesis. The results are presented in four parts.First, the basic wave field resulting from Holmboe’s instability is studied both numericallyand experimentally. In comparing basic descriptors of the wave fields, a number of processes areidentified that are responsible for differences between the simulations and experiments. These arerelated to variations in the mean flow that arise due to the different boundary conditions.Holmboe waves are known to produce vertical ejections of interfacial fluid from the wavecrests. This ‘ejection process,’ in which stratified fluid is transported against buoyancy forces, iscaused by the formation of a vortex couple (i.e. two vorticies of opposite sign that travel as a pair).Results obtained by means of direct numerical simulations also show that the process is primarilytwo-dimensional and does not require the presence of both upper and lower Holmboe modes.Shear instability is then studied in the highly stratified Fraser River estuary. The observationsare found to be in good agreement with the predictions of linear theory. When instability occurs, itis largely as a result of asymmetry between regions of strong shear and density stratification. Thestructure of the salinity intrusion is found to depend on the strength of the freshwater discharge, inaddition to the phase of the tidal cycle. This has implications for whether estuarine mixing takesplace through shear instability or boundary layer turbulence.Finally, the asymmetric stratified shear layer, which exhibits a vertical shift between the den-sity interface and the shear layer centre, is examined by the formulation of a diagnostic that isbased on the ‘wave interaction’ mechanism of instability growth. This allows for a quantitativeassessment of Kelvin-Helmhotz and Holmboe-type growth mechanisms in stratified shear layers.The predictions of the diagnostic are compared to results of nonlinear simulations and observa-tions in the Fraser River estuary.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xCo-authorship statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction and overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Holmboe wave fields in simulation and experiment . . . . . . . . . . . . . . . . . . 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Linear stability of stratified shear layers . . . . . . . . . . . . . . . . . . . . . . 142.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.1 Description of the numerical simulations . . . . . . . . . . . . . . . . . 152.3.2 Description of the laboratory experiment . . . . . . . . . . . . . . . . . 172.4 Wave structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5 Phase speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.6 Spectral evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.6.1 Frequency shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.6.2 Wave energy spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.7 Wave growth and amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.7.1 Wave growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.7.2 Comparison of saturated amplitudes . . . . . . . . . . . . . . . . . . . . 282.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Ejection process in Holmboe waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3 Ejection mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40ivBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 Shear instability in the Fraser River estuary . . . . . . . . . . . . . . . . . . . . . 434.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Site Description and Data Collection . . . . . . . . . . . . . . . . . . . . . . . . 444.3 General Description of the Salinity Intrusion . . . . . . . . . . . . . . . . . . . . 464.3.1 High Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3.2 Low Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.4 Stability of Stratified Shear Flows . . . . . . . . . . . . . . . . . . . . . . . . . 504.4.1 Taylor-Goldstein Equation . . . . . . . . . . . . . . . . . . . . . . . . . 504.4.2 Miles-Howard criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.4.3 Mixing Layer Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.4.4 Solution of the TG equation for observed profiles . . . . . . . . . . . . . 524.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.6 Small scale overturns and bottom stress . . . . . . . . . . . . . . . . . . . . . . 594.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 Unstable modes in asymmetric stratified shear layers . . . . . . . . . . . . . . . . 695.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.2 Wave interaction interpretation of instability . . . . . . . . . . . . . . . . . . . . 735.3 Formulation of a diagnostic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.3.1 Taylor–Goldstein equation . . . . . . . . . . . . . . . . . . . . . . . . . 765.3.2 Partitioning into kinematic and baroclinic fields . . . . . . . . . . . . . . 775.3.3 Piecewise profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.3.4 Smooth profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.3.5 Classification of modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.4.1 Symmetric profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.4.2 Asymmetric profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.5 Application to field profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.6 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96vList of Tables2.1 Important parameters in simulations and experiment . . . . . . . . . . . . . . . . 174.1 Transect details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44viList of Figures1.1 Profiles of a Kelvin-Helmholtz unstable stratified shear layer . . . . . . . . . . . 41.2 Profiles of a stratified shear layer with R = 3 . . . . . . . . . . . . . . . . . . . 51.3 Profiles of an asymmetric stratified shear layer (a = 0:5, R = 3) . . . . . . . . . 62.1 Growth rate and dispersion relations for the DNS . . . . . . . . . . . . . . . . . 152.2 Evolution of the background profiles . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Spatial changes in U(z) and layer depths in the laboratory . . . . . . . . . . . . 182.4 Sample density fields from the laboratory experiments and the simulations . . . . 192.5 Wave characteristics in the laboratory experiments and the simulations . . . . . . 212.6 Traced wave characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.7 Holmboe wave pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.8 Spectral evolution in the simulation and experiment . . . . . . . . . . . . . . . . 252.9 Growth of kinetic energy in the simulations . . . . . . . . . . . . . . . . . . . . 283.1 Laboratory photograph of a Holmboe wave ejection . . . . . . . . . . . . . . . . 363.2 Illustration of Holmboe wave formation . . . . . . . . . . . . . . . . . . . . . . 373.3 Time sequence of ejection process . . . . . . . . . . . . . . . . . . . . . . . . . 383.4 Wave interactions in the ejection process . . . . . . . . . . . . . . . . . . . . . . 394.1 Map of the lower 27 km of the Fraser River . . . . . . . . . . . . . . . . . . . . 444.2 Observed tides at Point Atkinson and New Westminster . . . . . . . . . . . . . . 454.3 Echo soundings observed during high discharge . . . . . . . . . . . . . . . . . . 474.4 Echo soundings observed during low discharge . . . . . . . . . . . . . . . . . . 494.5 Transect 1 stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.6 Transect 2 stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.7 Transect 3 stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.8 Transect 4 stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.9 Transect 5 stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.10 Transect 6 stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.11 Selected density profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.12 KH and one-sided instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.1 Stratified shear layer profiles considered . . . . . . . . . . . . . . . . . . . . . . 705.2 Stability of the symmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.3 Stability of the asymmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . 725.4 KH and H stability diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.5 Partial growth rates of piecewise symmetric profiles . . . . . . . . . . . . . . . . 815.6 Partial growth rates of smooth symmetric profiles . . . . . . . . . . . . . . . . . 825.7 Partial growths along maximum growth curve . . . . . . . . . . . . . . . . . . . 835.8 Partial growth rates of piecewise asymmetric profiles . . . . . . . . . . . . . . . 845.9 Partial growth rates of smooth asymmetric profiles . . . . . . . . . . . . . . . . 855.10 Partial growths along maximum growth curve . . . . . . . . . . . . . . . . . . . 855.11 Profiles from field measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 87viiList of SymbolsRoman Characters:a . . . . . . . . . . . Dimensionless asymmetry, wave amplitudearms . . . . . . . . Root-mean-square wave amplitudec . . . . . . . . . . . Complex phase speedcd . . . . . . . . . . Speed of internal wave on density interfacecg . . . . . . . . . . Group speedci . . . . . . . . . . . Imaginary component of the complex phase speedcr . . . . . . . . . . Real component of the complex phase speedcv . . . . . . . . . . Vorticity wave speedD . . . . . . . . . . Interfacial region domaind . . . . . . . . . . . Dimensional asymmetry offsetE . . . . . . . . . . .Wave energy densityF . . . . . . . . . . .Weight functionG . . . . . . . . . . .Green’s functiong . . . . . . . . . . . Gravitational accelerationg0 . . . . . . . . . . .Reduced gravitational accelerationh . . . . . . . . . . . Shear layer thicknessi . . . . . . . . . . . . Imaginary unit,p 1J . . . . . . . . . . . Bulk Richardson numberK . . . . . . . . . . .Wave kinetic energyK0 . . . . . . . . . .Initial kinetic energyK2d . . . . . . . . . Two-dimensional kinetic energyK3d . . . . . . . . . Three-dimensional kinetic energyk . . . . . . . . . . . Dimensional wavenumberkmax . . . . . . . .Dimensional wavenumber of maximum growthLx . . . . . . . . . . Domain length in the streamwise directionLy . . . . . . . . . . Domain length in the spanwise directionLz . . . . . . . . . . Domain length in the vertical directionm . . . . . . . . . . Density interface indexN2 . . . . . . . . . Squared buoyancy frequencyN2 . . . . . . . . . Modified square buoyancy frequencyn . . . . . . . . . . . Vorticity interface indexPr . . . . . . . . . Prandtl numberp . . . . . . . . . . . Pressure, interface index~p . . . . . . . . . . . Perturbation pressure p . . . . . . . . . . . Hydrostatic background pressureq . . . . . . . . . . . Perturbation vorticityqK . . . . . . . . . .Kinematic portion of perturbation vorticityqB . . . . . . . . . . Baroclinic portion of perturbation vorticityR . . . . . . . . . . .Scale ratioviiiRe . . . . . . . . . .Reynolds numberRi . . . . . . . . . . Gradient Richardson numberr . . . . . . . . . . . Shear layer spreading rateS . . . . . . . . . . . Wave stretchingt . . . . . . . . . . . .TimeU(z) . . . . . . . .Background velocity profileU (z) . . . . . . . Modified background velocity profileU1;U2 . . . . . . Layer velocities U . . . . . . . . . Velocity difference between layers U . . . . . . . . . . .Velocity of the mean flowu . . . . . . . . . . . Velocity component in the streamwise directionu . . . . . . . . . . . Velocity field vector~u . . . . . . . . . . . Perturbation horizontal velocityu1d . . . . . . . . . One-dimensional velocityu2d . . . . . . . . . Two-dimensional velocityu3d . . . . . . . . . Three-dimensional velocityvb . . . . . . . . . . Velocity of the boatw . . . . . . . . . . . Velocity component in the vertical direction~w . . . . . . . . . . . Vertical perturbation velocityx . . . . . . . . . . . Streamwise component of Cartesian coordinate systemy . . . . . . . . . . . Spanwise component of Cartesian coordinate systemz . . . . . . . . . . . Vertical component of Cartesian coordinate systemGreek Characters: . . . . . . . . . . . Dimensionless wavenumber max . . . . . . . Dimensionless wavenumber of maximum growth . . . . . . . . . . . Density interface thickness . . . . . . . . . . . Interface elevation, displacement field^ . . . . . . . . . . . Displacement eigenfunction^  . . . . . . . . . . Modified displacement eigenfunction . . . . . . . . . . . Phase function . . . . . . . . . . . Diffusion coefficient of density . . . . . . . . . . . Wavelength  . . . . . . . . . . Apparent wavelength . . . . . . . . . . . Coefficient of kinematic viscosity . . . . . . . . . . . Density 0 . . . . . . . . . . Reference density  . . . . . . . . . Density difference between layers 1; 2 . . . . . . . Layer densities (z) . . . . . . . . Density profile . . . . . . . . . . . Intrinsic frequency, dimensionless growth rate T . . . . . . . . . . Density less 1000 kg m 3 KH . . . . . . . . KH partial growth rate H . . . . . . . . . H partial growth rateix self . . . . . . . . Self interaction partial growth rate . . . . . . . . . . . Perturbation stream function^ . . . . . . . . . . . Perturbation stream function eigenfunction K . . . . . . . . . Kinematic portion of perturbation stream function^ K . . . . . . . . . Kinematic portion of perturbation stream function eigenfunction B . . . . . . . . . Baroclinic portion of perturbation stream function^ B . . . . . . . . . Baroclinic portion of perturbation stream function eigenfunction! . . . . . . . . . . . Wave frequencyList of Abbreviations:DNS . . . . . . . . Direct numerical simulationKH . . . . . . . . . Kelvin-HelmholtzH . . . . . . . . . . . HolmboeT . . . . . . . . . . . TaylorTG . . . . . . . . . Taylor-GoldsteinxAcknowledgementsI would like to thank both of my supervisors Greg Lawrence and Neil Balmforth for giving me agreat deal of support and advice. The interactions that I’ve had with them has changed my attitudeand approach to research and teaching for the better.I also want to thank Ted Tedford for suggesting that we work together on a couple of confer-ence papers, which turned out to be a sizable portion of this thesis, and an enjoyable experience.Half of that collaboration would not have been possible if Rich Pawlowicz had not invited us totake part in the Fraser River project. Roger Pieters has also been a great source of help in all sortsof areas during my stay at UBC. The numerical model that has been used to generate a signif-icant portion of the results in this thesis was kindly provided by Kraig Winters and Bill Smyth.In addition, Bill Smyth has provided the Taylor-Goldstein code, as well as contributed valuablediscussions in the early stages of this thesis. I’ve had the opportunity to take part in a number ofinformal seminar series which have given me excellent feedback on my work. I’d therefore liketo thank the Complex Fluids group, EFM group, and the Physical Oceanography group for theirfeedback. And of course thank you to my family and friends.xiCo-authorship statementThe four chapters comprising the main body of this thesis have each been part of a collaborativeeffort. Each is intended, or has already been submitted, as a journal publication. The contributionsof each author involved are outlined below.Chapter 2 has been submitted for publication in the Journal of Fluid Mechanics with myselfas the primary author as well as E.W. Tedford, M. Rahmani and G.A. Lawrence as coauthors. Theresearch program was initiated by E.W. Tedford and myself under the guidance of G.A. Lawrence.The data analysis was performed primarily by myself and E.W. Tedford with additional help fromM. Rahmani. I am responsible for writing the initial manuscript with considerable edits and revi-sions from all of the coauthors, E.W. Tedford in particular.The authors of chapter 3 are myself and G.A. Lawrence. It is currently in preparation forpublication. I am responsible for initiating the research and implementing the simulations andanalysis. I am also responsible for writing the manuscript with editing and revisions by G.A.Lawrence.Chapter 4 has been accepted for publication, subject to revision, in the Journal of GeophysicalResearch, with E.W. Tedford, myself, R. Pawlowicz, R. Pieters and G.A. Lawrence as the authors.R. Pawlowicz, E.W. Tedford and myself are responsible for initiating the research. I helped withthe collection and early analysis of the data. The initial manuscript was prepared by E.W. Tedfordand myself with substantial revisions and refinements from all of the coauthors.The research of Chapter 5 was initiated by myself, under the guidance of G.A. Lawrence. Theanalysis was developed by myself in conjunction with N.J. Balmforth. I am responsible for thewriting of the manuscript with editing and revisions by N.J. Balmforth and G.A. Lawrence.1Chapter 1Introduction and overview1.1 IntroductionIn recent years, increasing attention has been paid to the processes shaping our weather and cli-mate. This has led to the widespread use of global models to predict oceanic and atmosphericcirculation. However, the utility of global models relies on our ability to represent small ‘subgrid-scale’ mixing processes in terms of large-scale flow properties. Therefore, an understanding of themechanisms in which small-scale mixing is generated is necessary in order to accurately predictlarge-scale transport processes.Fundamental to large-scale transport processes is the generation of instability by shear. Theinstability process is responsible for extracting energy from large-scale motions and transferringit to smaller scales where it will ultimately end up being converting into viscous dissipation (i.e.frictional heating) and mixing of the buoyancy field. Such a small-scale process is not merely aconsequence of the large-scale dynamics, but also a cause. For example, small scale mixing isknown to be responsible for the upwelling of deep water that drives the thermohaline circulationof the worlds oceans.Despite the crucial role played by small-scale mixing in density stratified environments, ourpresent understanding of the process is incomplete (Ivey et al., 2008). The complexity of theproblem is nicely illustrated by considering the evolution of the stratified shear layer. This ide-alized flow represents a simplified model of a shear-driven mixing event as it may occur in theoceans, lakes, or atmosphere. Previous studies of stratified shear layers have shown that the linearanalysis originally performed by Taylor (1931) and Goldstein (1931) is sufficient to accuratelycapture the onset of instability (Thorpe, 1973; Pouliquen et al., 1994) as well as providing a firstorder description of the developing flow structures (Thorpe, 1973; Lawrence et al., 1991; Tedfordet al., 2009). For these reasons, results of the linear theory are often used as a basis for developingconceptual models of mixing. Therefore, a background of the linear analysis of stratified shearflows is now briefly described.Description of linear stability analysisAs a first step in building the linear model of a sheared stratified current, it is necessary to describethe equilibrium (background state) of the flow. This state must be a solution of the equations ofmotionux +wz = 0 (1.1)Chapter 1. Introduction and overview 2ut +uux +wuz = px (1.2)wt +uwx +wwz = pz  g (1.3) t +u x +w z = 0 (1.4)where (u;w) are the velocities in the horizontal and vertical directions (x;z),tis time,prepresentsthe pressure,  the fluid density, g is the gravitational acceleration, and the subscripts representpartial differentiation. The first equation (1.1) expresses the law of conservation of mass in thecase of an incompressible fluid. Both (1.2) and (1.3) express the conservation of horizontal andvertical momentum in the absence of viscous forces, respectively. The final equation (1.4) isa statement of the conservation of a scalar quantity in an incompressible flow that exhibits notransport of mass by molecular diffusion. However, it has been written in terms of assuming thatthe equation of state relating them is a linear function.We now consider a parallel shear flow with u = U(z) and w = 0 that is also density stratifiedin the vertical so that  =  (z). Substituting this into (1.1 - 1.4) above demonstrates that, as longas there are no imposed pressure gradients other than hydrostatic equilibrium, any form of thehorizontal velocity U(z), and density stratification  (z), are acceptable solutions. In other words,we are free to choose the U(z) and  (z) profiles as desired. The choice of U(z) and  (z) profilesis made to represent the typical background flow of interest, and is usually done with a particularapplication in mind.Regardless of U(z) and  (z), we now move beyond a description of the background flow toconsider the evolution of perturbations from this equilibrium state. In keeping with our linearapproximation, we assume that the perturbations are small superpositions on the much largerbackground flow, so that we can writeu = U + ~u; w = ~w; p =  p+ ~p;  =  + ~ ; (1.5)where d p=dz =   g describes a hydrostatic background pressure, and the tilde quantities repre-sent the perturbation quantities. A single equation describing the evolution of the perturbationscan be obtained after performing the following steps:1. Substitute (1.5) into (1.1 - 1.4) and neglect the product of perturbation quantities (as justifiedby the linearity assumption).2. Define a perturbation streamfunction  , such that ~u =  z and ~w =  x, and the continuityequation (1.1) is satisfied.3. Reduce the set of equations to a single partial differential equation for  .4. Take the normal mode form of the solution  (x;z;t) = ^ (z)eik(x ct).Chapter 1. Introduction and overview 3These steps lead to the Taylor-Goldstein (TG) equation^ 00 +h N2(U c)2  U00U c k2i^ = 0; (1.6)where the primes indicate ordinary differentiation with respect toz. The TG equation describes aneigenvalue problem for the eigenvaluecand the eigenfunction ^ (z). While ^ describes the verticalstructure of the mode, it is c that determines the stability of the flow: when Im(c) ci6= 0 thenthe flow is said to be unstable since perturbations grow exponentially in time at the rate kci. Theperturbations may also propagate with respect to the flow with a phase speed given by Re(c) cr.The stratified shear layerThe TG equation is sufficiently general to describe the linear stability of any stratified shear flowfrom which we can define U(z) and  (z). However, we shall focus on a U(z) that has two free-stream velocities that are connected by a smooth (shear) layer in which the majority of the shearis concentrated. Similarly,  (z) is taken as a statically stable stratification that is composed ofhomogeneous upper and lower layers that are separated by a diffuse density interface. The flowis also taken to be vertically unbounded so that it is free from any influence of the boundaries.It is common to represent this simple stratified shear layer by hyperbolic tangent profiles (e.g.Holmboe, 1962; Hazel, 1972; Alexakis, 2005) of the formU(z) =  U2 tanh 2(z d)h and  (z) =  0   2 tanh 2z  (1.7)where  U represents the total difference in velocity between the upper and lower streams, hgives a measure of the thickness of the shear layer,  0 is a reference density, and   denotes thetotal change in density across the interface of thickness  . The profiles of (1.7) also allow for avertical offset d, between the shear layer and the density interface. From these scales we are ableto identify three important dimensionless parameters defined asJ g  h 0( U)2; R h ; and a 2dh : (1.8)The bulk Richardson number J is a measure of the relative strength of the density stratification tothe shear, whereas R and a represent the relative thickness of the shear layer to that of the densityinterface, and the vertical asymmetry between the profiles, respectively.These three dimensionless parameters (J;R;a) have important consequences for the stabilityof the flow, and for the subsequent development of nonlinear structures and the turbulent mixingof the density field. These consequences are now briefly described.Chapter 1. Introduction and overview 4(d) Density fieldhΔρΔU JR(a)  U(z) (b)  ρ(z) (c)  Ri(z)Figure 1.1: Profiles of an unstable stratified shear layer with R = 1 and a = 0. The U(z) and (z) profiles are shown in (a,b) as well as Ri(z) in (c). For J < 1=4 the flow develops Kelvin-Helmholtz instabilities. A representative density field of the instability is shown in (d), once it hasreached a large amplitude nonlinear form of development.R 1: Kelvin-Helmholtz instabilityIn the first instance we shall take profiles with both R = 1 and a = 0 (although only R 1 isrequired), as shown diagrammatically in figure 1.1(a,b). In this very idealized circumstance, wherethe shear layer and the density interface have the same thickness and are vertically coincident, theultimate outcome of the shear layer is determined entirely by J: when J < 1=4 the flow is subjectto the Kelvin-Helmholtz (KH) instability, which leads to the formation of vorticies that eventuallybreak down and drive turbulent mixing (figure 1.1d). When J > 1=4 the density stratification(  =h) dominates the shear ( U=h) and the flow is stable.The onset of instability in the simple profiles of figure 1.1(a,b) once J < 1=4 is a particularresult of a more general criteria that may be expressed in terms of the gradient Richardson numberRi N2(U0)2; (1.9)where N2(z) = ( g= 0) 0 is the squared buoyancy frequency. It was shown by Miles (1961)and Howard (1961) that if Ri > 1=4 for all z, then the flow must be stable to small amplitude(linear) disturbances. The stability of the profiles in figure 1.1(a,b) onceJ > 1=4 can be seen withreference to the Ri(z) plot in figure 1.1(c). Hazel (1972) has shown that as long as R<p2 anda = 0, the Ri(z) profile exhibits a single minimum at the centre of the layers, with a minimumvalue of Rimin = JR. Hence, it is necessary for J < 1=4 for instability to be possible whenR = 1. Direct solution of the TG equation has found that J < 1=4 is in fact a sufficient conditionfor instability in this case.The very general and practical result, thatRi> 1=4 everywhere in the profile ensures stability,has come to be known as the Miles-Howard criteria. It is often used as the basis of various mixingChapter 1. Introduction and overview 5(d) Density fieldδhΔρΔU JR(a)  U(z) (b)  ρ(z) (c)  Ri(z)Figure 1.2: Profiles of an unstable stratified shear layer with R = 3 and a = 0. The U(z) and (z) profiles are shown in (a,b) as well as Ri(z) in (c). In this case the flow is susceptible to theHolmboe mode of instability, a schematic of which is shown in the density field of (d). Arrows onthe upper and lower wave crests in (d) indicate the direction of wave propagation.parameterizations which often involve a formulation of mixing that is dependent exclusively onRi, which must be computed as a local bulk value between grid points. It is also common to utilizea critical Richardson number (sometimes taken as 1/4), above which the turbulence is suppressedby buoyancy forces (e.g. Ivey et al., 2008; Thorpe, 2005,x7.3).R > 2: The Holmboe instabilityThe breakdown of simple mixing parameterizations such as those described above becomes ap-parent when considering the slightly more complicated, yet still highly idealized, set of symmetric(a = 0) background profiles shown in figure 1.2(a,b). In this case, the thickness of the densityinterface separating the homogeneous upper and lower layers is thinner than the shear layer, andhas been chosen such that R = 3. Profiles that possess a relatively sharp density gradient re-gion, become susceptible to the Holmboe (1962) mode of instability, in addition to the KH. (Inthe case of ‘tanh’ profiles Alexakis (2005) has found that R > 2 is necessary for the presence ofthe Holmboe instability.) The Holmboe instability is present when J is sufficiently large (i.e. forrelatively strong stratification), and consists of cusp-like propagating internal waves, as shown infigure 1.2(d). In contrast to the R = 1 profiles shown in figure 1.1(a,b), the Holmboe instabilityis not stabilized by increases in J, and may persist in flows where J is large (there being no upperlimit on J when the flow is inviscid). This is in full agreement with the Miles-Howard criteria,as can be seen from the plots of Ri(z) in figure 1.2(c). Whereas the KH profile of Ri in figure1.1(c) reaches a minimum in the centre of the shear layer where Rimin = JR, on the other hand,the Holmboe profile leads to a maximum at the shear layer centre with Rimax = JR, and Rivanishing on either side whenever R > 2. This ability of the Holmboe instability to develop atlarge J is a result of significant shear being present above and below the density interface, andmakes it a potentially relevant process in many geophysical flows.Chapter 1. Introduction and overview 6(d) Density fieldhΔρΔU(a)  U(z) (b)  ρ(z) (c)  Ri(z)dFigure 1.3: Profiles of an asymmetric stratified shear layer with R = 3 and a = 0:5. The U(z)and  (z) profiles are shown in (a,b) as well as Ri(z) in (c). In this case the flow is susceptibleto propagating asymmetric ‘one-sided’ instabilities, a schematic of which is shown in the densityfield of (d). Arrows on the upper and lower waves in (d) indicate the direction of wave propagation.Asymmetric flows: ‘One-sided’ instabilityAn additional means of generating instability at relatively large J is through a vertical displace-ment of the shear layer relative to the density interface position (figure 1.3a,b). This asymmetry,measured by the parametera, produces lowRion the strongly sheared side of the density interfaceand greater Ri on the reverse side (e.g. figure 1.3c). Instability is then found to be concentrated inthe strongly sheared region of lowRi(figure 1.3d). These ‘one-sided’ flows, where instability andmixing are confined to a region either above or below the density interface, have been observed inlaboratory experiments for some time (e.g. Thorpe, 1968; Koop and Browand, 1979). However, itwas not confirmed that ‘one-sided’ behaviour was caused by asymmetry in the profiles until thestability analysis and laboratory experiments of Lawrence et al. (1991). Since then, Haigh (1995)and Haigh and Lawrence (1999) have performed a detailed linear stability analysis of asymmetricflows for different ranges of J, R, and a. A central question that remains is what relation theseinstabilities have to the better known KH and Holmboe instabilities of the symmetric (a = 0)stratified shear layer. This question will be returned to in chapter 5 of this thesis.Implications for mixingIn the symmetric (a = 0) case, the presence of two different instability types when R > 2 – theKH at low J, and the Holmboe at high J – has important consequences on the mixing of massand momentum that are not fully understood. Smyth et al. (2007) show that the eddy viscosityand diffusivity can drop by more than an order of magnitude as the flow transitions from the KHto the Holmboe instability. On the other hand, Smyth and Winters (2003) and Carpenter et al.(2007) show that under certain conditions the mixing in KH and Holmboe instabilities can becomparable. Not only the amount of mixing, but also the vertical distribution of mixing is foundChapter 1. Introduction and overview 7to be dependent on the type of instability that results (Smyth and Winters, 2003; Carpenter et al.,2007). Mixing in KH instabilities is focused within the region of strong stratification centred onthe density interface, and tends to promote the formation of an intermediate mixed layer (Caulfieldand Peltier, 2000). In contrast, the Holmboe instability concentrates mixing on either side of thedensity interface, tending to broaden the interface (Smyth and Winters, 2003; Carpenter et al.,2007).In our analysis thus far, no mention has been made of viscous or diffusive effects. These areof course crucial to the mixing process, and are represented by the dimensionless Reynold’s andPrandtl numbersRe  Uh and Pr   ; (1.10)respectively. Here  represents the kinematic viscosity and  the diffusivity of the stratifyingagent. In geophysical mixing events the stratification is usually caused by gradients in tempera-ture or salinity resulting in Prandtl numbers of Pr 9 and Pr 700, respectively. The approxi-mate range of Re that is of interest in geophysical scale applications is large, and can roughly bebounded by 103 . Re . 108. It is therefore important to note that the mixing studies discussedabove utilize DNS, and are limited to low values of Re and Pr by computational constraints.Nonetheless these preliminary investigations of mixing suggest that significant changes in mix-ing behaviour may occur with only small changes in the flow parameters (i.e. J, R, or a), andillustrates that the characteristics of a shear-driven mixing event are linked to the complicated andsubtle instability mechanisms of the stratified shear layer. In other words, these results suggest thatthe mixing of mass and momentum can depend sensitively on details of the background densityand velocity structure. This is perhaps not surprising given that the stratified shear layer gives riseto a number of instability types, with stratification acting alternatively as a stabilizing and desta-bilizing influence on the shear flow (Howard and Maslowe, 1973). This dependence of mixingcharacteristics on the type of instability underscores the need to achieve a better understanding ofthe basic nonlinear behaviour of the instabilities. These results may then be used to develop moresophisticated parameterizations of mixing that account for the various instability modes that mayoccur.1.2 OverviewIn comparison to the large body of literature devoted to the KH instability (see Peltier and Caulfield,2003; Thorpe, 2005,x3, for recent reviews), the Holmboe instability has received relatively lessattention. Previous studies have generally focused on the applicability of linear theory to observa-tions (Lawrence et al., 1991; Pouliquen et al., 1994; Hogg and Ivey, 2003; Tedford et al., 2009),the modeling of secondary circulations (Smyth and Peltier, 1991; Smyth, 2006), or preliminaryinvestigations of the turbulence and mixing characteristics (Koop and Browand, 1979; Smyth andChapter 1. Introduction and overview 8Winters, 2003; Smyth et al., 2007; Carpenter et al., 2007). In particular, the evolution of the wavefields that result once the Holmboe instability reaches a finite amplitude have not previously beendescribed, and are therefore the subject of the following chapter. This chapter also confronts theissues of comparing laboratory results to those obtained by direct numerical simulations (DNS).Since the earliest laboratory observations of the Holmboe instability (Thorpe, 1968; Koop andBrowand, 1979) it has been noted that mixing occurs largely by entrainment of interfacial fluidinto the upper and lower layers (see also Smyth and Winters, 2003; Carpenter et al., 2007). Thishas been observed to occur through a number of different processes, however, perhaps the mostconspicuous of these is by the ejection of interfacial fluid from the wave crests. Although this‘ejection process’ is generally considered to be important for turbulence and mixing (Koop andBrowand, 1979; Smyth and Peltier, 1991; Smyth and Winters, 2003), as well as the evolution ofthe wave field (Chapter 2), no mechanism has been described to explain this process. Chapter 3,is devoted to providing such a description using the high-resolution observations from DNS.Chapter 4 is devoted to observations of shear instabilities in the highly stratified Fraser Riverestuary. Previous work of Geyer and Smith (1987) is extended by performing a direct compari-son of observations of shear instability in the estuary with the predictions of linear theory. Closeagreement of the linear predictions with the observations demonstrate that the theory is applicableeven in an evolving turbulent environment. Asymmetry between the regions of high density gra-dients and strong shear is found to be the dominant source of instability. This motivates the studyof asymmetric instabilities discussed in chapter 5, as they are of relevance to the dynamics of theFraser River estuary, and are found to be important in other highly stratified estuaries (e.g. Seimand Gregg, 1994; Yoshida et al., 1998; Sharples et al., 2003).Unlike certain viscous wall-bounded instabilities of pipe and channel flows (see Trefethenet al., 1993), the essentially inviscid instability of a stratified shear layer is well described, to firstorder, by linear stability theory (Thorpe, 1973; Lawrence et al., 1991; Pouliquen et al., 1994; Ted-ford et al., 2009). This provides a useful point of departure for studying the nonlinear aspects ofthe instabilities. This approach is adopted in chapter 5, where linear theory is used to infer charac-teristics of unstable modes that do not fit nicely into the KH/Holmboe classification. A blurring ofthe boundary between KH and Holmboe instabilities appears to be a general feature of stratifiedshear layers that possess some type of asymmetry. This asymmetry may be produced in a numberof ways such as in non-Boussinesq flows (Umurhan and Heifetz, 2007), in spatially developing in-stabilities (Pawlak and Armi, 1998; Ortiz et al., 2002), or if the density interface is vertically offsetfrom the shear layer centre (Lawrence et al., 1991; Haigh and Lawrence, 1999; Carpenter et al.,2007), to name a few. Choosing an asymmetric stratified shear layer with displaced profiles, therelationship between unstable asymmetric modes and the more familiar KH and Holmboe modesis examined. It is found that the asymmetric instabilities exhibit a gradual transition from modesof KH- to Holmboe-type as the stratification is increased (i.e. for increasing J). The analysisChapter 1. Introduction and overview 9is also applied to profiles measured in the Fraser River estuary to demonstrate the approach in ageophysically relevant scenario.10BibliographyAlexakis, A. (2005). On Holmboe’s instability for smooth shear and density profiles. Phys.Fluids, 17:084103.Carpenter, J., Lawrence, G., and Smyth, W. (2007). Evolution and mixing of asymmetric Holm-boe instabilities. J. Fluid Mech., 582:103–132.Caulfield, C. and Peltier, W. (2000). The anatomy of the mixing transition in homogenous andstratified free shear layers. J. Fluid Mech., 413:1–47.Geyer, W. and Smith, J. (1987). Shear instability in a highly stratified estuary. J. Phys. Oceanogr.,17:1668–1679.Goldstein, S. (1931). On the stability of superposed streams of fluids of different densities. Proc.R. Soc. Lond. A, 132:524–548.Haigh, S. (1995). Non-symmetric Holmboe Waves. PhD thesis, University of British Columbia.Haigh, S. and Lawrence, G. (1999). Symmetric and nonsymmetric Holmboe instabilities in aninviscid flow. Phys. Fluids, 11(6):1459–1468.Hazel, P. (1972). Numerical studies of the stability of inviscid stratified shear flows. J. FluidMech., 51:39–61.Hogg, A. M. and Ivey, G. (2003). The Kelvin-Helmholtz to Holmboe instability transition instratified exchange flows. J. Fluid Mech., 477:339–362.Holmboe, J. (1962). On the behavior of symmetric waves in stratified shear layers. Geofys.Publ., 24:67–112.Howard, L. (1961). Note on a paper of John W. Miles. J. Fluid Mech., 10:509–512.Howard, L. and Maslowe, S. (1973). Stability of stratified shear flows. Boundary-Layer Met.,4:511–523.Ivey, G., Winters, K., and Koseff, J. (2008). Density stratification, turbulence, but how muchmixing? Ann. Rev. Fluid Mech., 40:169–184.Koop, C. and Browand, F. (1979). Instability and turbulence in a stratified fluid with shear. J.Fluid Mech., 93:135–159.Bibliography 11Lawrence, G., Browand, F., and Redekopp, L. (1991). The stability of a sheared density interface.Phys. Fluids, 3(10):2360–2370.Miles, J. (1961). On the stability of heterogeneous shear flows. J. Fluid Mech., 10:496–508.Ortiz, S., Chomaz, J., and Loiseleux, T. (2002). Spatial Holmboe instability. Phys. Fluids,14:2585–2597.Pawlak, G. and Armi, L. (1998). Vortex dynamics in a spatially accelerating shear layer. J. FluidMech., 376:1–35.Peltier, W. and Caulfield, C. (2003). Mixing efficiency in stratified shear flows. Ann. Rev. FluidMech., 35:135–167.Pouliquen, O., Chomaz, J., and Huerre, P. (1994). Propagating Holmboe waves at the interfacebetween two immiscible fluids. J. Fluid Mech., 266:277–302.Seim, H. and Gregg, M. (1994). Detailed observations of naturally occurring shear instability. J.Geophys. Res., 99:10,049–10,073.Sharples, J., Coates, M., and Sherwood, J. (2003). Quantifying turbulent mixing and oxygenfluxes in a Mediterranean-type, microtidal estuary. Ocean Dyn., 53:126–136.Smyth, W. (2006). Secondary circulations in Holmboe waves. Phys. Fluids, 18:064104.Smyth, W., Carpenter, J., and Lawrence, G. (2007). Mixing in symmetric Holmboe waves. J.Phys. Oceanogr., 37:1566–1583.Smyth, W. and Peltier, W. (1991). Instability and transition in finite-amplitude Kelvin-Helmholtzand Holmboe waves. J. Fluid Mech., 228:387–415.Smyth, W. and Winters, K. (2003). Turbulence and mixing in Holmboe waves. J. Phys.Oceanogr., 33:694–711.Taylor, G. (1931). Effect of variation in density on the stability of superposed streams of fluid.Proc. R. Soc. Lond. A, 132:499–523.Tedford, E., Pieters, R., and Lawrence, G. (2009). Holmboe instabilities in a laboratory exchangeflow. J. Fluid Mech., page submitted.Thorpe, S. (1968). A method of producing a shear flow in a stratified fluid. J. Fluid Mech.,32:693–704.Thorpe, S. (1973). Experiments on instability and turbulence in a stratified shear flow. J. FluidMech., 61:731–751.Bibliography 12Thorpe, S. (2005). The Turbulent Ocean. Cambridge University Press, first edition.Trefethen, L., Trefethen, A., Reddy, S., and Driscoll, T. (1993). Hydrodynamic stability withouteigenvalues. Science, 261:578–584.Umurhan, O. and Heifetz, E. (2007). Holmboe modes revisited. Phys. Fluids, 19:064102.Yoshida, S., Ohtani, M., Nishida, S., and Linden, P. (1998). Mixing processes in a highly strat-ified river. In Physical Processes in Lakes and Oceans, volume 54 of Coastal and EstuarineStudies, pages 389–400. American Geophysical Union.13Chapter 2Holmboe wave fields in simulation and experiment 12.1 IntroductionGeophysical flows often exhibit stratified shear layers in which the region of density variation isthinner than the thickness of the shear layer (e.g. Wesson and Gregg, 1994; Yoshida et al., 1998;Sharples et al., 2003). In these circumstances, when the stratification is sufficiently strong (mea-sured by an appropriate Richardson number), Holmboe’s (1962) instability develops. At finiteamplitude the instability is characterized by cusp-like internal waves (referred to herein as Holm-boe waves) that propagate with respect to the mean flow. Accurate modelling of these instabilitiesis important for the correct parameterization of momentum and mass transfers occurring in flowsof this nature.Previous studies on the nonlinear behaviour of Holmboe waves have adopted one of two meth-ods: either an experimental approach in which the instability is studied under specified laboratorysettings (Pouliquen et al., 1994; Zhu and Lawrence, 2001; Hogg and Ivey, 2003), or a numericalapproach that allows for a detailed description of the flow in an idealized stratified mixing layer(Smyth et al., 1988; Smyth and Winters, 2003; Smyth, 2006; Smyth et al., 2007). It is difficult tomake a meaningful comparison of laboratory and numerical results for a number of reasons. In thecase of laboratory experiments, Holmboe waves often arise as a local feature of a larger-scale flow,such as an exchange flow between two basins of different density (Pawlak and Armi, 1996; Zhuand Lawrence, 2001; Hogg and Ivey, 2003), or an arrested salt wedge flow (Sargent and Jirka,1987; Yonemitsu et al., 1996). In many of these experiments the mean flow varies appreciablyover length scales that are comparable to the wavelength of the waves. For this reason, it can bedifficult to isolate the dynamics of the waves from that of the mean flow.Investigations utilizing numerical simulations comprises a great majority of the literature onthe nonlinear dynamics of Holmboe waves. The first verification of two oppositely propagatingcusp-like waves of equal amplitude, predicted by the Holmboe (1962) theory, was made throughthe numerical simulations of Smyth et al. (1988). Since then, increases in computational resourceshave led to fully three-dimensional direct numerical simulations (DNS) of Holmboe waves thatresolve the smallest scales of variability. These simulations have been used to understand turbu-lence and mixing characteristics (Smyth and Winters, 2003; Smyth et al., 2007; Carpenter et al.,2007), as well as the growth of secondary circulations and the transition to turbulence (Smyth,1A version of this chapter has been submitted for publication. J.R. Carpenter, E.W. Tedford, M. Rahmani andG.A. Lawrence (2009) Holmboe wave fields in simulation and experiment.Chapter 2. Holmboe wave fields in simulation and experiment 142006). However, partly due to computational constraints, only a single wavelength of the primaryinstability has been reported in the literature. Furthermore, no attempt has been made to comparethe results of numerical simulations with laboratory experiments.In this chapter, we undertake a combined numerical and experimental study of Holmboewaves. The experiments, originally described by Tedford et al. (2009), consist of an exchangeflow through a relatively long channel with a rectangular cross-section. The experimental designallows for a detailed study of the Holmboe wave field within a steady mean flow that exhibitsgradual spatial variation relative to the wave properties. The DNS of the present study were de-signed to correspond as closely as possible to the conditions present in the experiments to effecta meaningful comparison between the two methods. To our knowledge, this is the first studyto compare experimental and numerical results, as well as the first to perform DNS for multiplewavelengths of the instability. We focus on comparing basic descriptors of the wave fields suchas phase speed, wavenumber, and wave amplitude, in order to gain a fuller understanding of theprocesses affecting the nonlinear behaviour of the waves.The chapter is organized as follows. Section 2.2 gives a background on the stability of strat-ified shear flows. This is followed by a description of the numerical simulations, and laboratoryexperiments in x2.3. We then discuss comparisons between the simulations and experiments interms of the basic wave structure (x2.4), phase speed (x2.5), wave spectral evolution (x2.6), andwave amplitude and growth (x2.7). Conclusions are stated in the final section.2.2 Linear stability of stratified shear layersIn both experiment and simulation, the mean flow exhibits the characteristics of a classic stratifiedshear layer. The velocity profile undergoes a total change of  U, over a length scale h, that isclosely centred with respect to the density interface. Similarly, the density profile changes by   between the two layers, over a scale of  . This suggests using an idealized model of the horizontalvelocity and density profiles that is given byU(z) =  U2 tanhh2(z z0)hiand  (z) =  0   2 tanhh2(z z0) i; (2.1)respectively. The density profile  (z) is measured relative to a reference density  0, with z thevertical coordinate. A necessary condition for the growth of Holmboe’s instability is that thethickness ratio R h= > 2 (Alexakis, 2005).In addition to R, we may define three more important dimensionless parametersRe  Uh ; J g0h( U)2; and Pr   ;where g0 =   g= 0 is the reduced gravitational acceleration,  is the kinematic viscosity, andChapter 2. Holmboe wave fields in simulation and experiment 150 0.5 1 1.500.511.5c r (cm s-1 )(a)0 0.5 1 1.500.40.81.2 (b)0 0.5 1 1.500.050.10.15(c)σ (rad s-1 )kc i (rad s-1 )k (rad cm-1)k (rad cm-1)k (rad cm-1)Figure 2.1: Plots of (a) phase speed cr, (b) frequency  , and (c) growth rate kci, of the profiles in(2.1). Conditions in the experiment are shown as thick lines, and the three-dimensional simulationat Pr = 25 and R = 5 as thin lines. No noticeable difference between the simulation andexperiment can be seen in (b). The locations of the wavenumber of maximum growth are markedwith a vertical dotted line in (c). The dashed line in (b) indicates  /k. the diffusivity of the stratifying agent. These are the Reynolds, bulk Richardson, and Prandtlnumbers, respectively.Linear stability analysis of the profiles in (2.1) has been performed in numerous studies (e.g.Hazel, 1972; Smyth et al., 1988; Haigh, 1995). For the flows considered here, the effects ofviscosity and mass diffusion have been included. The resulting equation is a sixth order eigenvalueproblem originally described by Koppel (1964). Like the better known Taylor–Goldstein equation,Koppel’s equation gives predictions of the complex phase speed c = cr + ici, and vertical modeshape, as a function of wavenumber k. Results of the stability analysis are shown in figure 2.1,which includes the dispersion relation in terms of phase speed cr(k), and frequency  (k), as wellas the temporal growth rate, kci. This is done for the idealized profiles (2.1) using Re = 630,J = 0:30, Pr = 700, and R = 8, matching the dimensionless parameters in the laboratoryexchange flow. As discussed in the next section, computational constraints limited our three-dimensional DNS to a Pr = 25 and R = 5, resulting in slightly different results (figure 2.1).Although no appreciable changes are seen in the predicted phase speed cr and frequency  , thereare differences in maximum growth rate and the location of the wavenumber of maximum growthkmax.2.3 Methods2.3.1 Description of the numerical simulationsNumerical simulations were performed using the DNS code described by Winters et al. (2004),which has been modified to include greater resolution of the density scalar field by Smyth et al.Chapter 2. Holmboe wave fields in simulation and experiment 16-2 0 20246810U (cm s-1)z (cm)-1 0 1ρ−ρ0 (kg m-3)  (a) (b) Experimentt = 0 st = 100 st = 195 sSimulation:Figure 2.2: Evolution of the background profiles in both simulation and experiment. Plots (a) and(b) show temporal changes to U(z) and  (z)  0 profiles in the simulation with experimentalprofiles taken from the channel centre (x = 0) in thick lines overlain for comparison.(2005). The simulations were designed to reproduce conditions present in the laboratory exper-iment as closely as possible, while still conforming to the general methodology used in recentinvestigations of nonlinear Holmboe waves (Smyth and Winters, 2003; Smyth, 2006; Smyth et al.,2007; Carpenter et al., 2007).The boundary conditions are periodic on the streamwise (x) and transverse (y) boundaries,and free-slip on the vertical (z) boundaries. Simulations are initialized with profiles in the formof (2.1) that closely match what is observed in the experiment. Figure 2.2 shows a sequence ofrepresentativeU and    0 profiles at three different times during a simulation, as well as profilesfrom the experiment for comparison. The periodic boundary conditions of the simulations causethe flow to ‘run down’ over time, i.e. there is a continual loss of kinetic energy from the shearlayer due to viscous dissipation and mixing. This results in an increase of h and  over time, ascan be seen in figure 2.2. To indicate conditions at the initial time step (t = 0 s) of the simulationswe will use a zero subscript (e.g. h0).In order to initiate growth of the primary Holmboe instability, the flow is perturbed with arandom velocity field at the first time step. The noise is distributed evenly in the x;y directions,but given greater amplitude near the centre of the shear layer and density interface, in the samemanner as Smyth and Winters (2003). The amplitude of the random perturbation was chosen largeenough such that the instability grows to finite amplitude with minimal diffusion of the backgroundprofiles, yet is still small enough to satisfy the conditions for numerical stability given our choiceof time step.While an ideal comparison between simulation and experiment would involve matching allChapter 2. Holmboe wave fields in simulation and experiment 17Parameters Linear Theory ResultsPr R Lx Ly kmax cr arms(cm) (cm) (rad cm 1) (cm s 1) (cm)Experiment 700 8 200 10 0.91 0.79 0.31(0.51-0.84)I Simulation (3D) 25 5 128 5 0.79 0.84 0.62II Simulation (2D) 700 8 64 0 0.91 0.79 0.48III Simulation (3D) 25 5 64 10 0.79 0.84 0.62IV Simulation (2D) 25 5 64 0 0.79 0.84 0.74Table 2.1: Values of various important parameters for the simulations and experiment. The pa-rameters listed in the simulations are evaluated using the initial conditions. In all cases we haveJ0 = 0:3, Re0 = 630, and Lz = 10:8 cm. Also included are kmax and cr from the results of thelinear stability analysis, and the root mean square saturated amplitude observations. The values inparentheses under the experimental arms column indicate the predicted range of wave amplitudesonce wave stretching is accounted for (seex2.7).four of the relevant dimensionless parameters, we are constrained by the high computational de-mands of DNS. Of particular difficulty is the fine grid resolution required for high Pr flows. Forthis reason we have chosen a Pr = 25, opposed to Pr = 700 for the laboratory salt stratification.Large values of R also place a high demand on the computational resources, and we have there-fore chosen R0 = 5, opposed to the R = 8 observed in the experiments. The effects of Pr andR have been tested by performing a two-dimensional simulation at Pr = 700 and R0 = 8. Theremaining two parameters, Re0 = 630 and J0 = 0:30, have been matched to the experimentalvalues. Computational constraints also limit the size of the simulation domain. In all cases thevertical depth Lz, has been matched to the 10.8 cm of the experiments. The simulation widthLy = 5 cm, has been reduced to half of that in the experiment (Ly = 10 cm), but was not foundto effect the results presented. This reduction in the width of the computational domain enableda larger length Lx = 128 cm, allowing for approximately 16 wavelengths of the most amplifiedmode, and compares well with the Lx = 200 cm in the experiment. A summary of the parametersin the experiment and the simulations is shown in table 2.1.In addition to the three- and two-dimensional simulations already mentioned (labeled I and IIin table 2.1, respectively), two supplementary simulations (III and IV) were also performed totest the effects of Ly;R and Pr. Unless explicitly stated, we will refer to simulation I simply as‘the simulation’, hereafter.2.3.2 Description of the laboratory experimentThe laboratory experiment was performed in the exchange flow facility described by Tedford et al.(2009). A complete discussion of the experimental procedures and apparatus can be found in thatstudy, however, we now provide a summary of the pertinent features.Chapter 2. Holmboe wave fields in simulation and experiment 180510z (cm)(a)-100 -50 0 50 100-1-0.500.51x (cm)U (cm s-1 )(b)  ExperimentSimulationLinear fitρ1ρ2>ρ1U1U2Figure 2.3: Spatial changes in U(z) and layer depths that occur along the laboratory channel areshown in (a), along with the corresponding distribution of  U(x) in (b). A linear fit to  U(x) in thecentral portion of the channel is shown as the dashed line, and the mean velocity in the simulationdomain is given by the thin solid line.The apparatus consists of two reservoirs connected by a rectangular channel 200 cm in length,and 10 cm in width. The reservoirs are initially filled with fresh and saline water (  = 1:41 kgm 3) such that the depth in the channel is 10.8 cm. A bi-directional exchange flow is initiatedby the removal of a gate from the centre of the channel. After an initial transient period in whichgravity currents propagate to each reservoir, and mixed interfacial fluid is advected from the chan-nel, the flow enters a period of steady exchange where the density interface is found to displayan abundance of Holmboe wave activity. In contrast to the run-down conditions in the DNS, thestorage of unmixed water in the reservoirs maintains a steady exchange flow for approximately600 s. Our comparison is restricted to instabilities observed during the period of steady exchange.The exchange flow exhibits internal hydraulic controls at the entrance to each of the reser-voirs, effectively isolating the channel from disturbances in the reservoirs, and enforcing radiationboundary conditions at the channel ends. Friction between the layers leads to a gradually slopingdensity interface (figure 2.3) that produces anx-dependent mean velocity,  U = (U1 +U2)=2, withthe upper (U1) and lower (U2) layer velocities defined as the maximum and minimum free-streamvelocities (Tedford et al., 2009). The gradual variation of  U(x) along the laboratory channel isa result of the acceleration in each of the layers due to the sloping interface. This variation isshown in figure 2.3(b), and is found to be a near-linear function of x for the central portion ofthe channel. In contrast,  U is identically zero throughout the domain in the simulation, due to theperiodic boundary conditions. This difference in mean flow is found to have important effects onChapter 2. Holmboe wave fields in simulation and experiment 19z (cm)0510z (cm)0510z (cm)0510z (cm)x (cm)-20 -10 0 10 200510(a)(b)(c)(d)Figure 2.4: Representative plots of the density field for the experiment (a) along with simulation II(b), and simulation I (c,d). The plot in (d) is taken at a later time when two ejections are underway,indicated by arrows. The x-axis has been shifted by Lx=2 to the left in (d) to better display theejection process.the nonlinear development of the Holmboe wave field.2.4 Wave structureIn the first instance, it is beneficial to perform a simple visual comparison of the density struc-ture of the waves. This is shown in figure 2.4, where a representative photograph of the labora-tory waves is displayed above plots of the density field from the two-dimensional simulation II(figure 2.4b) and three-dimensional simulation I (figure 2.4c,d). The density structure in figures2.4(a,b) is very similar, as each has an identical set of dimensionless parameters, differing only inthe initial and boundary conditions. In all panels of figure 2.4 it can be seen that although many ofthe waves display the typical form of the Holmboe instability, and consist of cusps projecting intoChapter 2. Holmboe wave fields in simulation and experiment 20the upper and lower layers, others appear sinusoidal in form. The waves focused above the densityinterface (upwards pointing cusps) are moving from left to right, in the same direction as the flowin the upper layer, while the waves in the lower layer (downwards cusps) move at an equal butopposite speed with respect to the mean velocity.An important feature of nonlinear Holmboe waves is the occasional ejection of stratified fluidfrom the wave crests into the upper and lower layers. Two such ejections are shown in figure 2.4(d)where indicated, and can be characterized by thin wisps of fluid being drawn from the wave crestand advected by the mean flow. These wisps often settle back to the interface level, contributingto the accumulation of mixed fluid there. This accumulation is observed to a much greater extentin figure 2.4(c,d), and should be expected due to the larger value of R0, as well as the higherdiffusion that comes with the lower Pr used in this simulation. Although ejections are observedin both the laboratory experiment and high Pr and R simulation (II), there is a greater frequencyof occurrence in the lower Pr = 25 and R = 5 simulation (I).Holmboe’s instability has the uncommon property that, under certain conditions, the growthof the primary instability may take place as a three-dimensional wave. Such a wave wouldtravel obliquely to the orientation of the shear, and produce significant departures from a two-dimensional wave. One of the conditions for this three-dimensional growth is that Re be suffi-ciently low (Smyth and Peltier, 1990). As the laboratory experiments are carried out at low Re,and show some variation in the spanwise direction, it must be questioned whether the growthof the primary Holmboe instability is three-dimensional. This is easily tested by the simulationresults, which show a clear two-dimensional growth (see x2.7 as well), even to an initially ran-dom perturbation as described above. We can therefore confirm that the primary instability istwo-dimensional for the conditions examined in the present study.2.5 Phase speedMany of the basic features in the wave field are revealed by an x t characteristics diagram of thedensity interface elevation, shown in figure 2.5 for both the simulation and experiment. Althoughthe interface consists of contributions from both upper and lower Holmboe wave modes (eachtravelling in opposite directions), we have filtered the characteristics using a two-dimensionalFourier transform to reveal only the upper, rightward propagating wave modes.Certain differences between the simulation characteristics (figure 2.5a) and the experimentalcharacteristics (figure 2.5b) are immediately apparent. The experimental characteristics exhibit agreater degree of irregularity. Since each plot represents a two dimensional slice from a three-dimensional field, this may be a result of greater variability in the transverse direction in the caseof the experiments. Since the waves in the simulation develop from an initial random perturbationat t = 0, there is also a temporal growth of the average wave amplitude in figure 2.5(a) that is notpresent in the experimental characteristics.Chapter 2. Holmboe wave fields in simulation and experiment 21x (cm)t (s)(a) Simulation0 20 406080100120140160180x (cm)(b) Experiment-20 0 20Figure 2.5: Rightward propagating wave characteristics for the simulation (a) and experiment (b).Shading represents the elevation of the density interface with red indicating a high (crest) and blueindicating a low (trough), and has a different scale in each of (a,b). Solid black lines indicate thecharacteristic slope given by the linear prediction of phase speed cr. In the case of the laboratoryexperiment, the cr has a slight curvature since the changes in  U along the channel have beenincluded. The dark circles indicate locations and times of ejections.Despite these apparent differences in the characteristics, the phase speeds (inferred from theslope of the characteristics) are in good agreement. The observed phase speeds in both the simu-lation and experiment are found to be slightly greater than the predictions of linear theory (solidlines), which has been noted in previous studies (Haigh, 1995; Hogg and Ivey, 2003; Tedfordet al., 2009). However, the observations also suggest an increase in phase speed with wave am-plitude. This is a quintessential feature of nonlinear wave behaviour (e.g. Stokes waves). Notethat a ‘pulsing’ of the wave amplitude and phase speed is present in both sets of characteristicsin figure 2.5. This is a well known feature of Holmboe waves due to the interaction between thetwo oppositely propagating modes (Smyth et al., 1988; Zhu and Lawrence, 2001; Hogg and Ivey,2003).Sudden decreases in wave amplitude can be seen in both sets of characteristics at a number oftimes and locations. It is often the case (though not always) that these sudden amplitude changesare a result of the ejection process. Instances where ejections occur have been identified in fig-ure 2.5, and are denoted by circles. It is generally observed that the ejection process preferentiallyacts on the largest amplitude waves, similar to the breaking of surface waves.Chapter 2. Holmboe wave fields in simulation and experiment 22x (cm)t (s)(a) Simulation-50 0 50050100150x (cm) (b) Experiment-50 0 50Figure 2.6: Rightward propagating wave characteristics for the simulation (a) and experiment (b).White indicates a wave crest while grey indicates a wave trough. In each panel a number of wavecrests are indicated by solid and dashed lines. In (a), the dashed lines correspond to waves that are‘lost’ over the duration of the simulation, whereas in (b), the dashed lines correspond to waves thathave formed within the channel. Only the central portion of the laboratory channel correspondingto the simulation domain has been shown. Circles and squares indicate locations and times ofejections and pairing events, respectively.2.6 Spectral evolutionThis section concerns the distribution and evolution of wave energy with k. It will be shownthat there are two different processes acting separately in the simulation and experiment that areresponsible for a shifting of wave energy to lower k (i.e. longer waves).2.6.1 Frequency shiftingIn order to gain an understanding of the wave spectrum, it is first useful to carefully examinethe characteristic diagrams. Figure 2.6 shows rightward propagating characteristics from bothsimulation and experiment that highlight the location of wave crests (in white) and troughs (ingrey). Characteristics from the simulation (figure 2.6a) are discussed first, and are shown for theentire computational domain.Beginning with the initial random perturbation at t = 0, energy is extracted from the meanChapter 2. Holmboe wave fields in simulation and experiment 23x (cm)z (cm)-40 -35 -30 -250510x (cm)-30 -25 -20 x (cm)-20 -15 -10(a) (b) (c)-0.50 0.51 1.5Figure 2.7: Vorticity field from the simulation illustrating the vortex pairing process. Colour barindicates the magnitude of the spanwise (y) vorticity field nondimensionalized by  U=h withpositive values corresponding to clockwise rotation. Times shown are (a) t = 80 s, (b) t = 87 s,and (c) t = 97 s. Black contours denote isopycnals with a spacing of   =8, and arrows indicatethe position of vorticies in the upper layer that undergo pairing. Note that two vorticies are pairingin the lower layer in (c) as well.flow by the instability and fed into the wave field at, or very close to, the wavenumber of maximumgrowth, kmax. This results in approximately 16 waves in the computational domain (given byLxkmax=2 ) for early times. We see however, that as the simulation proceeds wave crests arecontinually being ‘lost’ over time. This feature is highlighted by the solid and dashed lines thatare used to trace the wave crests in figure 2.6(a). The dashed lines indicate wave crests that are‘lost’, while the solid lines correspond to crests that persist. This process of losing waves resultsin an observed frequency, !, that is continually shifted downwards. Because previous numericalstudies of Holmboe waves simulated only a single wavelength, this process has not been describedbefore. This ‘frequency downshifting’ or ‘wave coarsening’ has, however, been noted previouslyin other nonlinear wave systems (e.g. Huang et al., 1999; Balmforth and Mandre, 2004).In general, for all of the simulations performed, the ejection process typically results in a lossof waves and a downshift in frequency (figure 2.6a). This observation mirrors similar findingsin the frequency downshifting of nonlinear surface gravity waves, where the occurrence of wavebreaking is related to lost waves (Huang et al., 1996; Tulin and Waseda, 1999).Close examination of the vorticity field also suggests that the Holmboe waves undergo a vortexpairing process. Each interfacial wave is led by a vortex formed from the ‘rolling up’ of the initialshear layer vorticity. This leading vortex propagates with the waves in the upper and lower layers,and can be seen in the vorticity field plotted in figure 2.7(a). The pairing process is seen to act onthe two vorticies (indicated by arrows) in the upper layer of figure 2.7; they draw closer together,rotating about each other (figure 2.7b) and eventually merge into a single vortex (figure 2.7c).Although the pairing of adjacent vorticies is a well known feature of homogenous and weaklystratified shear layers (Browand and Winant, 1973), it has received little attention in Holmboewaves, the only observation being that of Lawrence et al. (1991). This is an additional means toChapter 2. Holmboe wave fields in simulation and experiment 24effect a shift of wave frequency, and is denoted by square symbols in figure 2.6(a).In contrast, figure 2.6(b) shows that the experimental characteristics display a distinctly dif-ferent behaviour. In this case, new wave crests are continually being formed as the waves traversethe channel. Again, this process is highlighted by the tracing of crests by solid and dashed lines.Now, the dashed lines represent new wave crests that have been formed within the channel. Thisprocess results in an increasing ! with x in the experiments.Tedford et al. (2009) explain the formation of new waves as follows. As waves propagatethrough the channel they are accelerated by the increasing mean velocity  U(x). This leads toa ‘stretching’ of the waves that decreases k from near kmax, where the waves initially formed,to lower values (i.e. longer wavelengths). Once a sufficiently low k is achieved, the Holmboeinstability mechanism acts between the wave crests to form additional waves. This feeds energyback into the wave field near kmax, resulting in an average k that is constant across the channel,and an increasing !.The two processes, wave coarsening in the simulation, and wave stretching in the experiments,can be described quantitatively using wave spectra.2.6.2 Wave energy spectraDifferences between the processes responsible for modifying k in both the simulation and experi-ment can be seen in figure 2.8. It demonstrates how wave energy (indicated by the dark bands) isredistributed in k over time.The spectra of the simulation (figure 2.8a), which has been normalized by the variance in orderto remove the time-dependent growth of the waves, shows a discrete transfer of wave energy tolowerk. The simulation spectra is required to evolve in discrete steps due to the periodic boundaryconditions (i.e. in wavenumber increments of  k = 2 =Lx). As a point of comparison, the kmaxprediction from linear stability theory is plotted in red. The predicted kmax has been discretizedaccording to the boundary conditions, and decreases in time due to the diffusion of the backgroundprofiles, i.e. the increase in the shear layer thickness h(t).The spectral evolution plot (figure 2.8a) compliments the characteristics diagram of figure 2.6(a),showing an initial input of energy atkmax(t = 0), and a subsequent shifting of that energy to lowerk. It is interesting to note that thekmax(t) curve shows the same general trend as the concentrationof wave energy (shown by the dark ‘blocks’ in figure 2.8a). Although the details are unclear, wespeculate that the shift in wave energy to lower k is the result of nonlinear processes such as theejections and vortex pairing.It is apparent from the wave spectra in figure 2.8(b) that the process responsible for the redis-tribution of wave energy in the experiments is a continuous one. Energy at any given time is foundto be focused in a number of ‘bands’. These bands originate near kmax, and move towards lowerk in time. In addition, they all appear to have a similar trajectory inkt-space. Tedford et al. (2009)Chapter 2. Holmboe wave fields in simulation and experiment 25  0t (s)(a) Simulation0 0.5 1 1.50100200k (rad cm-1)(b) Experiment0 0.5 1 1.50100200300400500600stretching predictionkmax predictionk (rad cm-1)10-110cm2cm2Figure 2.8: Spectral evolution of the rightward propagating waves from simulation (a) and ex-periment (b). Dark colours denote a high in energy which is proportional to the mean squareamplitude of the interface displacement. The wave energy has been normalized by the variancein (a) to remove the time dependent wave growth. Linear stability theory is used to predict kmax(red lines), which changes in time for the simulations. The predicted stretching of wave energy inthe experiment by  U(x) to lower k is shown as the yellow dashed lines in (b). The beginning ofthe steady period of exchange is referenced to t = 0 s in (b).hypothesize that these bands are a result of the stretching of wave energy to lower k by  U(x). Wenow formulate a simple model in order to quantify this hypothesis.Wave stretching predictionThe changes in k that result from wave stretching by  U(x) can be described by an application ofgradually varying wave theory. This theory assumes that the density interface elevation  (x;t),may be expressed in terms of a gradually varying amplitude a(x;t), and a rapidly varying sinu-soidal component viz. (x;t) = Refa(x;t)ei (x;t)g: (2.2)The local wavenumber and frequency are defined in terms of the phase function  (x;t) by k  @ =@x and !   @ =@t, respectively. We assume, for the moment, that  (x;t) is a smoothChapter 2. Holmboe wave fields in simulation and experiment 26function. This implies that waves are conserved, giving@k@t +@!@x = 0: (2.3)Recognizing that !, which is the frequency that a stationary observer would measure, includesboth an intrinsic portion  (k), and an advective portion k U, leads to! =  (k) +k U(x): (2.4)Substituting into (2.3) givesDkDt = Sk; (2.5)where the material derivative, defined asDDt @@t + (cg + U) @@x;denotes changes in time while moving at the speed cg +  U, and cg d =dk is the intrinsic groupspeed. This is the speed that wave energy, i.e. the dark bands in figure 2.8(b), is expected topropagate through the channel. We have also defined S  d  U=dx, which is found to be verynearly constant in the central portion of the laboratory channel (see figure 2.3b). Choosing aLagrangian frame of reference, that moves at the speed cg +  U through the channel, allows for asimple integration of (2.5) to givek(t) = k e S(t t ); (2.6)where k = k(t ) is some initial value of k that wave energy begins the stretching process at. Adirect comparison is now possible between the prediction of (2.6) and the bands of energy in theobserved spectral evolution. The prediction is shown by the yellow dashed lines in figure 2.8(b),and is found to be in agreement with the observations. This validates the hypothesis that thespectral shift towards lower k is a result of wave stretching. The excellent agreement betweenthe predictions and observations also reveals that our assumption of wave conservation is justi-fied. This is not in contradiction with the formation of new waves described inx2.6.1 since waveconservation is applied only after energy is fed into the wave field by the instability mechanism.2.7 Wave growth and amplitudeThe final basic parameter that we intend to compare is the wave amplitude, a. This feature ofthe wave field is determined when the linear growth reaches some level where it saturates. It is anonlinear property of the waves, and may involve three-dimensional effects as well as interactionChapter 2. Holmboe wave fields in simulation and experiment 27with the mean flow. We begin by discussing the various phases of wave growth.2.7.1 Wave growthIn the simulation, the instability mechanism causes the growth of waves from an initial randomperturbation into a large-amplitude nonlinear wave form. This growth process is best illustratedby considering the kinetic energy of the waves, K. Following Caulfield and Peltier (2000), wepartitionKinto a two-dimensional kinetic energyK2d associated with the primary Holmboe wave,and a three-dimensional componentK3d, that provides a measure of the departures from a strictlytwo-dimensional wave. By this partitioning we haveK=K2d +K3d; (2.7)whereK2d =hu2d u2d=2K0ixz and K3d =hu3d u3d=2K0ixyz; (2.8)and we have usedu1d(z;t) = huixy; (2.9)u2d(x;z;t) = hu u1diy; (2.10)u3d(x;y;z;t) = u u1d u2d; (2.11)withh ii representing an average in the direction i, andK0 the total kinetic energy at t = 0.TheK2d andK3d components are plotted on a log-scale in figure 2.9 for the simulation. Theplot indicates that after a start up period where the energy of the initial perturbation rapidly de-cays, a stage of exponential growth is achieved inK2d. This stage of exponential growth can becompared to the prediction of linear theory, and is found to be less than the prediction (kci = 0:10s 1 versus the observed kci = 0:06 s 1). One possible explanation of this discrepancy in kci isdue to the fairly rapid spreading rate of the shear layer, measured by r = h 1dh=dt, during thestage of linear growth up to t 65 s. This spreading is due to the viscous diffusion of vorticityin the shear layer, and may be compared with the growth rate of the instability by the ratio r=kci.Linear theory implicitly assumes that r=kci  1 indicating that there is no time dependence ofthe background profile. While r=kci  0:1 is relatively small during the linear growth period, itmay be sufficient to explain the differences in observed growth rates.The wave growth is entirely two-dimensional until the waves have reached a finite amplitude(t 65 s), at which point the growth of three-dimensional secondary structures results (see Smyth2006 for a discussion of this process in Holmboe waves). However, the waves remain primarilytwo-dimensional, withK3d at least an order of magnitude smaller thanK2d. There is not a welldefined transition to turbulence, as is found in other studies of stratified shear layers (e.g. CaulfieldChapter 2. Holmboe wave fields in simulation and experiment 280 50 100 150 20010-610-510-410-310-210-1Kinetic Energyt (s)  K2dK3dlinear growthFigure 2.9: Growth of K2d and K3d for the simulation. The thick line gives the linear growthrate prediction of the growth ofK2d, which is a function of time due to the changing backgroundprofiles.and Peltier, 2000; Smyth et al., 2001), likely due to the low Re. Once the saturated amplitude isreached, there is a slow decline ofKover the remainder of the simulation.In the laboratory experiments we have focused only on the period of steady exchange, andtherefore do not observe a time-dependent growth of the wave field on average. However, asdiscussed previously, the instability is constantly acting to produce new waves along the channel.It is difficult to measure the growth rate of these waves, but they appear to reach a saturatedamplitude rapidly, suggesting that they are strongly forced by disturbances within the channel(Tedford et al., 2009).2.7.2 Comparison of saturated amplitudesAlthough the transient growth of the instability is difficult to quantify in the experiments, it ispossible to measure the mean amplitude of the waves. This is done by using the root mean squareamplitude of the interface elevation  (x;t), given byarms(x) =s1TZT 2 dt; (2.12)where T denotes the duration of the steady period of exchange. When averaged over a numberof experiments arms is found to display little dependence on x. A similar arms can be definedfor the simulations, however, the temporal average is replaced by a spatial average in x. TheChapter 2. Holmboe wave fields in simulation and experiment 29growth of arms in time in the simulations shows a similar behaviour toK2d; an exponential initialgrowth, followed by a saturation, and subsequent decay. The saturated (maximum) amplitudereached during each of the simulations is shown in table 2.1, along with the mean amplitude in theexperiments.The first feature to note is that the waves of the two-dimensional simulation (II) at R = 8and Pr = 700 (matching the conditions in the experiment) have a lower amplitude than of allthe other simulations, especially the two-dimensional simulation (IV) at R = 5, Pr = 25. Thisindicates that there is a possible dependence of the saturated amplitude on R;Pr. Most impor-tantly, the amplitude measured in the experiments is significantly smaller than any of the saturatedamplitudes reached in the simulations. The small amplitudes observed in the experiments can beexplained by, once again, appealing to the effects of wave stretching.Wave stretching effects on amplitudeTo understand the effects of wave stretching on amplitude in the experiments, we apply principlesthat have been established for waves on slowly varying currents (e.g. Peregrine, 1976). In doingso, we assume that the Holmboe waves may be represented by a simple train of linear internalwaves that satisfy the dispersion relation in figure 2.1(c). We are then able to track the changes inwave amplitude that occur as a result of the spatially varying mean velocity  U(x), i.e. the wavestretching. In this simplified model it is the conservation of wave action density that is relevant.This is given as E= , where E is the wave energy density, and recall that  (k) is the intrinsicwave frequency. Substitution into the conservation law, and following a similar procedure tox2.6.2 leads to a similar resultDDt E  = S E  ; (2.13)which describes changes in action density due to the stretching by  U. In arriving at (2.13) wehave neglected a term that is proportional to d2 =dk2, which is small in the range of k that we areinterested in (see figure 2.1b). Taking S to be constant once again, allows for simple integrationof (2.13) to give  E  = E   e S(t t ):For linear internal waves E/a2, so that we have an estimate of the reduction in wave amplitudedue to stretching ofaa =r   e S(t t )=2: (2.14)If we now take the intrinsic frequency  / k, as suggested by the linear dispersion relation infigure 2.1(b), it is possible to write the right hand side of (2.14) as e S(t t ), where we have usedthe spectral prediction in (2.6). By inspection of figure 2.8(b), we can estimate a time interval, t, that wave energy spends in the channel (i.e. the average time interval that the dark bandsChapter 2. Holmboe wave fields in simulation and experiment 30appear for) to be between 100 and 200 s. The amplitude reduction is therefore in the range 0:37 <e S t < 0:61.Given this reduction, and assuming that no other processes are taking place that may affectthe wave amplitudes, we would expect amplitudes in the range of that shown in table 2.1, givenin parentheses. This adjusted amplitude is comparable to the observations in the simulations, anddemonstrates that – in the absence of other processes – the stretching of wave action is significantin reducing the experimental wave amplitudes.2.8 ConclusionsWe have compared simulations of Holmboe wave fields with the results of laboratory experiments.A meaningful comparison was possible since both methods exhibit only gradual variations in themean flow. In the laboratory experiment, the mean flow is spatially varying, whereas the numericalsimulations display a temporal variation. Focusing on basic descriptors of the waves, such asphase speed, wavenumber, and amplitude, we have identified a number of processes affecting thenonlinear behaviour of Holmboe wave fields.Similarities between results of the two methods include the basic structure of the waves, andthe phase speeds. The observations show slightly greater phase speeds when compared with thepredictions of linear theory, in agreement with previous studies (Haigh, 1995; Hogg and Ivey,2003; Tedford et al., 2009). Further departures from the linear predictions are attributed to anonlinear dependence of the phase speed on amplitude.The greatest differences between simulation and experiment are found in the spectral evolutionand wave amplitudes. In simulations, a transfer of wave energy to lowerk was found to result fromwave coarsening, which caused waves to be ‘lost’ through discrete merging events. These eventswere typically found to result from both vortex pairing as well as the ejection process. The latterof which is suggested to be similar to wave breaking in surface waves, since it appears to actpreferentially to reduce the amplitude of the largest (steepest) waves. A detailed investigation ofboth ejections and vortex pairing in Holmboe waves is currently underway.The shift of wave energy to lower k that was observed in the experiments can be attributedto the ‘stretching’ of the wave field by the spatially accelerating mean flow. This suggestion ofTedford et al. (2009) has been confirmed by a simple application of gradually varying wave theory,which is able to accurately predict the time dependence of the spectral shift.The wave stretching process is also expected to have a significant effect in reducing the waveamplitudes observed in the experiments. This conclusion appears sufficient to explain discrepan-cies between wave amplitudes in experiment and simulation, and is based on a simple applicationof the conservation of wave action. In this application we have assumed that no other processesare actively influencing the wave amplitude, however, a dependence of wave amplitude on R andPr has been noted.Chapter 2. Holmboe wave fields in simulation and experiment 31Our comparisons of simulation and experiments reveal ways in which the nonlinear evolutionof a Holmboe wave field is dependent on the mean flow, and hence, on the boundary conditions.This must be considered when studying the nonlinear behaviour of the Holmboe instability.32BibliographyAlexakis, A. (2005). On Holmboe’s instability for smooth shear and density profiles. Phys.Fluids, 17:084103.Balmforth, N. and Mandre, S. (2004). Dynamics of roll waves. J. Fluid Mech., 514:1–33.Browand, F. and Winant, C. (1973). Laboratory observations of shear-layer instability in a strat-ified fluid. Boundary-Layer Met., 5:67–77.Carpenter, J., Lawrence, G., and Smyth, W. (2007). Evolution and mixing of asymmetric Holm-boe instabilities. J. Fluid Mech., 582:103–132.Caulfield, C. and Peltier, W. (2000). The anatomy of the mixing transition in homogenous andstratified free shear layers. J. Fluid Mech., 413:1–47.Haigh, S. (1995). Non-symmetric Holmboe Waves. PhD thesis, University of British Columbia.Hazel, P. (1972). Numerical studies of the stability of inviscid stratified shear flows. J. FluidMech., 51:39–61.Hogg, A. M. and Ivey, G. (2003). The Kelvin-Helmholtz to Holmboe instability transition instratified exchange flows. J. Fluid Mech., 477:339–362.Holmboe, J. (1962). On the behavior of symmetric waves in stratified shear layers. Geofys.Publ., 24:67–112.Huang, N., Long, S., and Shen, Z. (1996). The mechanism for frequency downshift in nonlinearwater wave evolution. Adv. Appl. Mech., 32:59–117.Huang, N., Shen, Z., and Long, S. (1999). A new view of nonlinear water waves: the Hilbertspectrum. Ann. Rev. Fluid Mech., 31:417–457.Koppel, D. (1964). On the stability of a thermally stratified fluid under the action of gravity. J.Meth. Phys., 5:963–982.Lawrence, G., Browand, F., and Redekopp, L. (1991). The stability of a sheared density interface.Phys. Fluids, 3(10):2360–2370.Pawlak, G. and Armi, L. (1996). Stability and mixing of a two-layer exchange flow. Dyn. Atmos.Oceans, 24:139–151.Peregrine, D. (1976). Interaction of water waves and currents. Adv. Appl. Mech., 16:9–117.Bibliography 33Pouliquen, O., Chomaz, J., and Huerre, P. (1994). Propagating Holmboe waves at the interfacebetween two immiscible fluids. J. Fluid Mech., 266:277–302.Sargent, F. and Jirka, G. (1987). Experiments on saline wedge. J. Hydraul. Engng ASCE,113(10):1307–1324.Sharples, J., Coates, M., and Sherwood, J. (2003). Quantifying turbulent mixing and oxygenfluxes in a Mediterranean-type, microtidal estuary. Ocean Dyn., 53:126–136.Smyth, W. (2006). Secondary circulations in Holmboe waves. Phys. Fluids, 18:064104.Smyth, W., Carpenter, J., and Lawrence, G. (2007). Mixing in symmetric Holmboe waves. J.Phys. Oceanogr., 37:1566–1583.Smyth, W., Klaassen, G., and Peltier, W. (1988). Finite amplitude Holmboe waves. Geophys.Astrophys. Fluid Dyn., 43:181–222.Smyth, W., Moum, J., and Caldwell, D. (2001). The efficiency of mixing in turbulent patches: In-ferences from direct simulations and microstructure observations. J. Phys. Oceanogr., 31:1969–1992.Smyth, W., Nash, J., and Moum, J. (2005). Differential diffusion in breaking Kelvin-Helmholtzbillows. J. Phys. Oceanogr., 35:1004–1022.Smyth, W. and Peltier, W. (1990). Three-dimensional primary instabilities of a stratified, dissi-pative, parallel flow. Geophys. Astrophys. Fluid Dyn., 52:249–261.Smyth, W. and Winters, K. (2003). Turbulence and mixing in Holmboe waves. J. Phys.Oceanogr., 33:694–711.Tedford, E., Pieters, R., and Lawrence, G. (2009). Holmboe instabilities in a laboratory exchangeflow. J. Fluid Mech., page submitted.Tulin, M. and Waseda, T. (1999). Laboratory observations of wave group evolution, includingbreaking effects. J. Fluid Mech., 92:197–232.Wesson, M. and Gregg, M. (1994). Mixing at the Camarinal Sill in the strait of Gibraltar. J.Geophys. Res., 99:9847–9878.Winters, K., MacKinnon, J., and Mills, B. (2004). A spectral model for process studies ofrotating, density-stratified flows. J. Atmos. Ocean. Tech., 21:69–94.Yonemitsu, N., Swaters, G., Rajaratnam, N., and Lawrence, G. (1996). Shear instabilities inarrested salt-wedge flows. Dyn. Atmos. Oceans, 24:173–182.Bibliography 34Yoshida, S., Ohtani, M., Nishida, S., and Linden, P. (1998). Mixing processes in a highly strat-ified river. In Physical Processes in Lakes and Oceans, volume 54 of Coastal and EstuarineStudies, pages 389–400. American Geophysical Union.Zhu, D. and Lawrence, G. (2001). Holmboe’s instability in exchange flows. J. Fluid Mech.,429:391–409.35Chapter 3Note on the ejection process in Holmboe waves 13.1 IntroductionThe Holmboe instability is one of a number of possible instability types that may evolve fromthe stratified shear layer (Holmboe, 1962; Howard and Maslowe, 1973). It results from the in-teraction between disturbances in the shear layer vorticity and the density stratification (Bainesand Mitsudera, 1994; Caulfield, 1994), and requires a relatively thin density interface comparedto the shear layer thickness. It is able to grow in strong stratifications (i.e. large bulk Richardsonnumbers), making it a potentially important process in geophysical flows. At finite amplitude,the instability develops into a series of propagating internal waves that often appear cusp-like inform. Depending on conditions, the Holmboe waves may form on one, or both sides of the densityinterface.Since the earliest laboratory studies of Holmboe waves, investigators have observed a pro-cess whereby interfacial fluid is drawn away from the density interface and entrained into thehomogeneous layers on either side (see e.g. Thorpe, 1968; Browand and Winant, 1973; Koop andBrowand, 1979). The resulting structures have been referred to by many different names such aswisps, plumes, puffs and ejections, and it is likely that they result from different physical pro-cesses. However, we shall focus on what we observe to be the most conspicuous of these events,in which interfacial fluid is ejected from the wave crests and often travels a considerable dis-tance from the density interface. A snap-shot of this ‘ejection process’ is shown in the laboratoryphotograph of figure 3.1. Although the ejection process is generally considered to be importantfor the generation of three-dimensional motions (Smyth and Peltier, 1991), mixing of the densityfield (Koop and Browand, 1979; Smyth and Winters, 2003), and the basic evolution of the Holm-boe wave field (chapter 2), relatively little is known of the mechanism by which it occurs. It istherefore, the goal of this chapter to provide a mechanistic description of the ejection process. Indoing so, we will also address a number of questions raised in the literature regarding the ejections(see Smyth and Peltier, 1991; Hogg and Ivey, 2003, in particular). These include: (i) whether ornot ejections are dependent on three-dimensional effects; (ii) if the interaction between the up-per and lower waves may trigger an ejection; and (iii) whether the ejections can be a source ofthree-dimensional motion. The present work also compliments that of Smyth (2006) on secondarycirculations in Holmboe waves.1A version of this chapter is in preparation for publication. J.R. Carpenter and G.A. Lawrence (2009), Note on theejection process in Holmboe waves.Chapter 3. Ejection process in Holmboe waves 36Figure 3.1: Laboratory photograph of a Holmboe wave ejection. The colours are representativeof the fluid density and are measured by the light intensity of fluorescing dye. See Tedford et al.(2009) for details of the experimental apparatus.3.2 MethodsOur examination is based on direct numerical simulations (DNS) of the stratified shear layer.The code has been described previously by Winters et al. (2004), but has been modified as inSmyth et al. (2005). The DNS solves the three-dimensional incompressible Boussinesq equa-tions of motion in a rectangular domain that has periodic boundary conditions in the streamwise(x), and spanwise (y) directions, and a free-slip condition on the vertical (z) boundaries. Theflow is initialized with random noise in the velocity field in the manner described by Smythand Winters (2003). This is superimposed on ‘tanh’ background profiles of horizontal velocityU(z) = ( U=2) tanh[2(z d)=h], and density  (z) =  0 (  =2) tanh(2Rz=h), where  Uis the velocity difference across the shear layer, h is the shear thickness,   is the density dif-ference across the interface,  0 is a reference density, and R is the ratio of the shear thickness tothe thickness of the density interface. For reasons discussed below, we allow the density inter-face to be vertically offset from the centre of the shear layer by a distance d, and measure thisdegree of asymmetry by a = 2h=d. Denoting the kinematic viscosity of the fluid by  , and thediffusivity of the stratifying agent by  , we identify three more important dimensionless parame-ters J =   gh= 0( U)2, Re =  Uh= , and Pr =  = , as bulk Richardson, Reynold’s, andPrandtl numbers, respectively.We have chosen conditions in the laboratory experiments of Tedford et al. (2009) as a guide,giving Re = 630 and J = 0:30. Since the large values of R = 8 and Pr 700 in Tedford et al.(2009) place a high demand on computational resources, we have taken R = 5 and Pr = 25;sufficiently large to ensure that Holmboe’s instability develops, yet low enough to adequately re-solve the density interface within our limited computational resources. Two different simulationsChapter 3. Ejection process in Holmboe waves 37x(a)(b)(c)leading vortexbaroclinic negative vorticitypositive shear layer vorticitydensity interfaceFigure 3.2: Illustration of the initial formation of asymmetric Holmboe waves. Time is increasingin each panel from (a-c), and each is in a frame of reference moving with the wave speed.are performed, one with a = 0 that results in ‘symmetric’ Holmboe waves both above and belowthe density interface, and the other an ‘asymmetric’ case with a = 0:5 where waves only form inthe upper layer. Ejections occurred most often when multiple waves were present in the domain.We therefore chose a domain length of Lx = 64h for the symmetric simulation, and Lx = 16h inthe asymmetric case, which allow for the initial growth of 16 waves and 4 waves, respectively. Toaccommodate the large number of waves in the a = 0 simulation we choose a width Ly = 2:5h,whereas Ly = 5h in the asymmetric case. In both simulations the depth is Lz = 5:4h as inTedford et al. (2009).3.3 Ejection mechanismFrom the initial random noise of the velocity perturbation, the Holmboe instability acts togetherwith the periodic streamwise boundary conditions to selectively amplify the most unstable wave-length from linear theory. The instability growth consists of a ‘rolling-up’ of the shear layervorticity on one or both sides of the density interface, depending on a. This process is illustratedin figure 3.2 for a = 0:5, where the waves are formed only in the upper layer. The positive(clockwise) shear layer vorticity indicated by the light grey shading can be seen to concentrateinto spanwise oriented leading vorticies that travel with the waves on the density interface (in thiscase from left to right). The baroclinic generation of vorticity that occurs in regions of horizontaldensity gradients results in the accumulation of negative (counter-clockwise) vorticity in the wavecrests, and positive vorticity in the troughs, similar to a linear internal wave. However, the negativevorticity becomes concentrated in a narrow region about the wave crests, as seen in figure 3.2(c).The time sequence of density and spanwise vorticity fields shown in figure 3.3 demonstratesthe ejection process in asymmetric Holmboe waves. A thin dense wisp of fluid is drawn from theChapter 3. Ejection process in Holmboe waves 3810-1-2z/h2 72  52 7z/h5 102  55 10z/h7 122  57 12z/hx/h10 152  5x/h10 15(a) (e)(b) (f)(c) (g)(d) (h)Figure 3.3: Sequence of density (a-d) and vorticity (e-h) fields taken at times of t U=h =f88;97;105;116g demonstrating the ejection process. The vorticity scale, made dimensionlessby h= U, is given below with the thin dark contours in (e-h) representing isopycnals spacedevery   =8.Chapter 3. Ejection process in Holmboe waves 39x/ht ΔU/h (a)0 20 40 6080100120140160180200x/h(b)0 20 40 60Figure 3.4: Characteristics of the right- and left-propagating waves for the symmetric simulationin (a) and (b), respectively. Colour shading represents the elevation of the density interface, buthas been reversed in (a) and (b) so that red indicates a wave crest (either upwards deflections in(a), and downwards deflections in (b)) and blue a wave trough. The approximate locations andtimes of ejections are marked by circles, which occur within the region indicated by the dashedrectangle.wave crest and transported into the homogeneous upper layer where it becomes advected with thebackground flow. Inspection of the vorticity field in figure 3.3 shows that the vertical transport ofejected fluid against buoyancy forces is accomplished by the formation of a vortex couple, i.e. twovorticies of opposite sign. The vortex couple is composed of the negative baroclinically generatedvorticity of the wave crest and the positive vorticity in the leading vortex that originated from therolling-up of the shear layer. The two oppositely signed vorticies induce an upwards velocity ineach other that results in an upwards translation of the couple. Simple models of buoyant vortexcouples have been proposed by Turner (1960) and Saffman (1972), as well as others. In thesemodels, a dense vortex couple is able to rise against the downward buoyant force by consumingthe linear impulse of the vorticity distribution. Although the application of these models in thepresent context is complicated by many factors such as complex vorticity and density fields, aswell as the presence of a mean shear flow, the basic mechanism is the same.Our observation of asymmetric ejections, that occur in the absence of waves in the lower layer,indicates that the interaction between upper and lower wave modes is not a necessary ingredientin the process. However, it is found that the presence of a second wave mode is effective intriggering ejections. This is demonstrated in figure 3.4, with an x–t characteristic diagram of thedensity interface elevation in the a = 0 simulation. A two-dimensional Fourier filter has beenapplied to separate the wave characteristics into only the right-propagating waves in figure 3.4(a)Chapter 3. Ejection process in Holmboe waves 40and the left-propagating waves in 3.4(b). Four ejections are observed in both the upper and lowerwaves at similar times and locations. They are triggered as two large-amplitude wave packets inthe upper and lower waves meet in the region identified by the dashed box. Interaction of theupper and lower wave modes is known to produce oscillations in phase speed and wave amplitude(Smyth et al., 1988; Zhu and Lawrence, 2001; Hogg and Ivey, 2003), which can also be seen in the‘wiggling’ and ‘pulsing’ of the characteristics in figure 3.4. These periodic changes in amplitudewill also periodically increase the strength of the negative vorticity in the wave crests, thus leadingto a stronger vortex couple and a greater chance of ejection.A feature of the ejections, as seen in the wave characteristics, is the sudden reduction in waveamplitude that results. This can also be seen in the reduced amplitude of the wave in the final stageof the ejection process shown in figure 3.3(d,h). Since the ejections are found to occur in only thelargest amplitude (steepest) waves, it was proposed in chapter 2 that the process be considereda type of Holmboe wave breaking. This is in general agreement with the mechanism discussedabove since a steep wave crest is able to generate stronger concentrations of negative baroclinicvorticity required in the formation of the vortex couple.Previous investigations have noted the ejection process to occur erratically, raising specula-tion that the cause may be linked to three-dimensional effects (Hogg and Ivey, 2003). Since theHolmboe instability is found to be a two-dimensional instability of the basic stratified shear layerover most of parameter space (Smyth and Peltier, 1990; Haigh, 1995), and the mechanism justdescribed is also two-dimensional, there is no need to appeal to the third dimension in order toexplain the occurrence of ejections. Indeed, the ejections have been observed in strictly two-dimensional simulations (not shown). However, three-dimensional motions are observed to formon the ejected fluid even when the wave crests are highly two-dimensional, suggesting that theymay be a source of three-dimensionality. This three-dimensionality has also been observed inlaboratory experiments (Thorpe, 1968; Maxworthy and Browand, 1975). It is possible that the er-ratic occurrence of the ejections is related to both interactions between the upper and lower modesas shown above, as well as from energy transfers occurring from within the wave field (e.g. amodulational instability).3.4 ConclusionsWe have a described a mechanism to explain the ejection process that is commonly observed inHolmboe waves. The basic mechanism is two-dimensional, relying on the formation of a vortexcouple to transport interfacial fluid against buoyancy. Although it is not necessary for two Holm-boe wave modes to be present, ejections can be triggered by an interaction between the upper andlower wave modes. The ejections also appear to be a source of three-dimensional motions, inagreement with observations made in laboratory experiments.41BibliographyBaines, P. and Mitsudera, H. (1994). On the mechanism of shear flow instabilities. J. FluidMech., 276:327–342.Browand, F. and Winant, C. (1973). Laboratory observations of shear-layer instability in a strat-ified fluid. Boundary-Layer Met., 5:67–77.Caulfield, C. (1994). Multiple linear instability of layered stratified shear flow. J. Fluid Mech.,258:255–285.Haigh, S. (1995). Non-symmetric Holmboe Waves. PhD thesis, University of British Columbia.Hogg, A. M. and Ivey, G. (2003). The Kelvin-Helmholtz to Holmboe instability transition instratified exchange flows. J. Fluid Mech., 477:339–362.Holmboe, J. (1962). On the behavior of symmetric waves in stratified shear layers. Geofys.Publ., 24:67–112.Howard, L. and Maslowe, S. (1973). Stability of stratified shear flows. Boundary-Layer Met.,4:511–523.Koop, C. and Browand, F. (1979). Instability and turbulence in a stratified fluid with shear. J.Fluid Mech., 93:135–159.Maxworthy, T. and Browand, F. (1975). Experiments in rotating and stratified flows: oceano-graphic application. Ann. Rev. Fluid Mech., 7:273–305.Saffman, P. G. (1972). The motion of a vortex pair in a stratified atomosphere. Stud. Appl. Maths,51(2):107–119.Smyth, W. (2006). Secondary circulations in Holmboe waves. Phys. Fluids, 18:064104.Smyth, W., Klaassen, G., and Peltier, W. (1988). Finite amplitude Holmboe waves. Geophys.Astrophys. Fluid Dyn., 43:181–222.Smyth, W., Nash, J., and Moum, J. (2005). Differential diffusion in breaking Kelvin-Helmholtzbillows. J. Phys. Oceanogr., 35:1004–1022.Smyth, W. and Peltier, W. (1990). Three-dimensional primary instabilities of a stratified, dissi-pative, parallel flow. Geophys. Astrophys. Fluid Dyn., 52:249–261.Bibliography 42Smyth, W. and Peltier, W. (1991). Instability and transition in finite-amplitude Kelvin-Helmholtzand Holmboe waves. J. Fluid Mech., 228:387–415.Smyth, W. and Winters, K. (2003). Turbulence and mixing in Holmboe waves. J. Phys.Oceanogr., 33:694–711.Tedford, E., Pieters, R., and Lawrence, G. (2009). Holmboe instabilities in a laboratory exchangeflow. J. Fluid Mech., page submitted.Thorpe, S. (1968). A method of producing a shear flow in a stratified fluid. J. Fluid Mech.,32:693–704.Turner, J. S. (1960). A comparison between buoyant vortex rings and vortex pairs. J. FluidMech., 7:419–432.Winters, K., MacKinnon, J., and Mills, B. (2004). A spectral model for process studies ofrotating, density-stratified flows. J. Atmos. Ocean. Tech., 21:69–94.Zhu, D. and Lawrence, G. (2001). Holmboe’s instability in exchange flows. J. Fluid Mech.,429:391–409.43Chapter 4Observation and analysis of shear instability inthe Fraser River estuary 14.1 IntroductionShear instabilities occur in highly stratified estuaries and can influence the large scale dynamics byredistributing mass and momentum. Specifically, shear instabilities have been found to influencesalinity intrusion in the Fraser River estuary (Geyer and Smith, 1987; Geyer and Farmer, 1989;MacDonald and Horner-Devine, 2008). We describe recent observations in this estuary and exam-ine the shear and stratification that lead to instability. The influence of long time scale processessuch as freshwater discharge and the tidal cycle are also discussed.Rather than relying on a bulk or gradient Richardson number to assess stability we use numer-ical solutions of the Taylor-Goldstein (TG) equation based on observed profiles of velocity anddensity. This approach has been used with some success in the ocean (e.g. Moum et al., 2003) but,with the exception of the simplified application by Yoshida et al. (1998), has not been applied inestuaries. Solving the TG equation provides the growth rate, wavelength, phase speed and modeshape of the instabilities. We compare these predicted wave properties with instabilities observedusing an echosounder.Geyer and Farmer (1989) found that instabilities in the Fraser River estuary were most ap-parent during ebb tide when strong shear occurred over the length of the salinity intrusion. Theyoutlined a progression of three phases of increasingly unstable flow that occurs over the courseof the ebb. In the first phase, strain sharpens the density interface; shear is stronger than duringflood, but insufficient to cause shear instability. In the second phase, the lower layer reversesdirection causing shear between the fresh and saline layers to increase. Shear instability and tur-bulent mixing are concentrated at the pycnocline rather than in the bottom boundary layer. Bythe third phase of the ebb, shear instability has completely mixed the two layers leaving homoge-neous water throughout the depth. During flood there is some mixing, however it is concentratedat the front located at the landward tip of the salinity intrusion. Similarly, MacDonald and Horner-Devine (2008), studying mixing at high fresh water discharge (7000 m3s 1), found that two tothree times more mixing occurred during ebb tide than during flood. The present analysis is fo-cused on the ebb tide at high and low freshwater discharge, although some results during floodtide are also presented.1A version of this chapter has been accepted for publication. E.W. Tedford, J.R. Carpenter, R. Pawlowicz, R. Pietersand G.A. Lawrence (2009) Observation and analysis of shear instability in the Fraser River estuary, J. Geophys. Res.Chapter 4. Shear instability in the Fraser River estuary 44Sand Heads (0 km) Massey           Tunnel (18 km)T1 T2T3 T4T5T6N6 km0 2 4Figure 4.1: Map of the lower 27 km of the Fraser River. The locations of the six transects aremarked T1-T6. The mouth of the river (Sand Heads) is located at 49 60 N and 123 180 W.Discharge Tide x  U   h J(m3 s 1) (km) (m s 1) (kg m 3) (m)1 6400 Ebb 8.6 1.6 14.3 5.2 0.292 6500 Ebb 11 1.65 20 3.5 0.253 5700 Flood 2.2 1.5 23.1 3.5 0.354 850 Ebb 24.5 1.5 12.9 12 1.35 850 Ebb 19 1.5 12.9 12 1.36 850 Ebb 10.5 2.5 7.3 12 0.3Table 4.1: Details of transects shown in figures 4.1 and 4.2. The location indicates the distanceupstream from the mouth (Sand Heads).This chapter is organized as follows. The setting and field methods are described in section4.2. The general structure of the salinity intrusion is described in section 4.3. In section 4.4 wepresent the background theory needed to perform stability analysis in the Fraser River estuary. Insection 4.5 predictions from the stability analysis are compared with observations. In section 4.6the source of relatively small scale overturning is briefly discussed. In section 4.7 the results ofthe stability analysis are discussed followed by conclusions in section 4.8.4.2 Site Description and Data CollectionData were collected in the main arm of the Fraser River estuary, British Columbia, Canada (fig-ure 4.1). The estuary is 10 to 20 m deep with a channel width of 600 to 900 m. Cruises wereconducted on June 12, 14 and 21, 2006 and March 10, 2008. Here we present one transect fromeach of the June 2006 cruises and three transects from the March 2008 cruise (see Table 4.1).The freshwater discharge during the June 2006 transects was typical of the freshet at approxi-mately 6000 m3s 1. During the March 2008 transects, freshwater discharge was near the annualminimum at 850 m3s 1. In June 2006, transects were made during both ebb and flood tide. InMarch 2008, transects cover most of a single ebb tide (figure 4.2). The tides in the Strait of Geor-Chapter 4. Shear instability in the Fraser River estuary 450 2 4 6 8 10 12-202Height (m) T1June 12, 2006. Qfresh = 6400 m3s-12 4 6 8 10 12 14-202Height (m) T28 10 12 14 16 18-202Height (m)T36 8 10 12 14 16 18-202Height (m)Local Time (hours, PDT)T4 T5 T6June 14, 2006. Qfresh = 6500 m3s-1June 21, 2006. Qfresh = 5700 m3s-1March 10, 2008. Qfresh = 850 m3s-1Figure 4.2: Observed tides at Point Atkinson (heavy line) and New Westminster (thin line) for thefour days of field observations. The Point Atkinson data is representative of the tides in the Straitof Georgia beyond the influence of the Fraser River. New Westminster is located 37 km upstreamof the mouth of the river at Sand Heads (see figure 4.1). The records are both referenced to meansea level at Point Atkinson. The duration of the six transects are marked T1-T6.gia have M2 and K1 components of similar amplitude (approximately 1 m) resulting in strongdiurnal variations. The tidal range varies from approximately 2 m during neap tides to approxi-mately 4.5 m during spring tides. During both the 2006 and 2008 observations the tidal range wasapproximately 3 m.The distance salinity intrudes landward of Sand Heads, i.e. the total length of the salinityintrusion, varies considerably with tidal conditions and freshwater discharge. Ward (1976), foundthe maximum length of the intrusion occurred just after high tide and varied from 8 km at highdischarge (9000 m3s 1) to 31 km at low discharge (850 m3s 1). Geyer and Farmer (1989) foundthat, at average discharge (3000 m3s 1), the maximum length of the intrusion matched the hori-zontal excursion of the tides (10 to 20 km) and, similar to Ward (1976), occurred just after hightide. Kostachuk and Atwood (1990) found that the minimum length of the salinity intrusion typi-Chapter 4. Shear instability in the Fraser River estuary 46cally occurred approximately one hour after low tide. The longest intrusion they observed at lowtide was approximately 20 km. They predicted that complete flushing of salt from the estuarywould occur on most days during the freshet (freshwater discharge > 5000 m3s 1).Field methodsData along the six transects were collected by drifting seaward with the surface flow while loggingvelocity and echosounder data and yoyoing a CTD (conductivity, temperature and depth) profiler.The velocity measurements were made with a 1200 kHz RDI Acoustic Doppler Current Profiler(ADCP) sampling at 0.4 Hz with a vertical resolution of 250 mm. The velocities were averagedover 60 seconds to remove high frequency variability. The echo soundings were made with a 200kHz Biosonics sounder sampling at 5 Hz with a vertical resolution of 18 mm. Profile data wascollected with a Seabird 19 sampling at 2 Hz. Selected echosounder, ADCP and CTD data areshown in figure 4.3. As indicated by the superimposed density profiles, strong gradients in densityare generally associated with a strong echo from the sounder.The CTD was profiled on a load bearing data cable that provided constant monitoring ofconductivity, temperature and depth. These data allowed us to quickly identify the front of thesalinity intrusion and avoid direct contact of the instrument with the bottom. To increase thevertical resolution of the profiles, the CTD was mounted horizontally with a fin to direct thesensors into the flow. In this configuration, the instrument was allowed to descend rapidly and thenwas raised slowly (0.2 - 0.4 m s 1) relying on horizontal velocity of the water relative to the CTDto flush the sensors. The upcast, which had higher vertical resolution, was in reasonable agreementwith the echo intensity from the sounder. On the few occasions that the higher resolution upcastdid not coincide with the appearance of instabilities in the echosounder, we used the downcast.The total number of CTD casts we were able to perform varied from transect to transect dependingon field conditions (surface velocity, shear, ship traffic, woody debris).4.3 General Description of the Salinity IntrusionWe observed important differences in the structure of the salinity intrusion between high andlow freshwater discharge. At high discharge, our observations were similar to those describedby Geyer and Farmer (1989), where the salinity intrusion had a two-layer structure resembling aclassic salt-wedge. At low discharge, however, the salinity intrusion exhibited greater complexity.4.3.1 High DischargeDuring flood tide, mixing was concentrated near the steep front at the landward tip of the salt-wedge (2.7 to 3.03 km in figure 4.3c). During ebb tide, the steep front was replaced by a gentlyChapter 4. Shear instability in the Fraser River estuary 47(a) Transect 18.6 8.7 8.8 8.9 9 9.1 9.2 9.3 9.4 9.5 9.624681012Depth (m)(b) Transect 21 m s-1 25 kg m10.8 10.9 11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.86810121416Distance (km)(c) Transect 32.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.124681012 -3Figure 4.3: Echo soundings observed during high discharge on: (a) transect 1, ebb tide; (b) tran-sect 2, ebb tide; (c) transect 3, flood tide. The shading scales with the log of the echo intensitywith black corresponding to the strongest echos. Selected velocity profiles (red) from the ADCPand density profiles (blue dash) from the CTD are superimposed (not all are shown). The blackline indicates the location of the boat in the middle of the cast, as well as the zero reference forthe velocity and  T. The velocity profile was calculated as a 1 minute average centred on the timeof the CTD cast. The undulations in the bed of the river (thick black line at the bottom of theechosoundings) are a result of sandwaves.Chapter 4. Shear instability in the Fraser River estuary 48sloping pycnocline (figure 4.3b, landward of 11.6 km) and there was no apparent concentration ofmixing at the landward tip of the salt-wedge (not shown).We will focus on the wave-like disturbances that occur on the pycnocline especially duringebb tide. The largest of these were observed during transect 1 (figure 4.3a, 8.7 to 8.9 km betweendepths of 3 and 10 m). These disturbances occurred within the upper layer as it passed over thenearly stationary water below a depth of 10 m. Smaller amplitude disturbances were observedduring transect 2 (figure 4.3b, 11.05 km). In our application of the TG equation we will show thatdisturbances like these are a result of shear instability.Not all of the disturbances on the pycnocline are a result of shear instability. For example, formost of the velocity and density profiles collected during transect 3 (figure 4.3c) the TG equationdoes not predict instability. The disturbances seen from 2.5 to 3.0 km are caused by the large sandwaves on the bottom (the thick black line in the echo sounding). The crests of the sand waves weretypically 30 m apart and 1 to 2 m high, and were found over most of the river surveyed during highdischarge (2.5 km to 15 km). During flood tide, flow over these sand waves caused particularlyregular disturbances on the pycnocline.4.3.2 Low DischargeAt low discharge, at the beginning of the ebb, the front of the salinity intrusion was located be-tween 28 and 30 km from Sand Heads. Unlike the observations at high discharge a well definedfront was not visible in the echosounder, and CTD profiles were needed to identify its location.Seaward of the front (figure 4.4a), the echosounder and the CTD profiles show a multilayeredstructure with more complexity than was observed at high discharge. At this early stage of theebb, the CTD profiles generally show partially mixed layers separated by several weak densityinterfaces.Later in the ebb, during transect 5 (figure 4.4b), near-bottom velocities turn seaward and thevelocity shear between the top and the bottom increases. At maximum ebb (transect 6, figure 4.4c),the shear increases further, reaching a maximum of approximately 2.5 m s 1 over a depth of 12m. Mixed water occurs at both the surface and the bottom resulting in an overall decrease in thevertical density gradient. By the time transect 6 is complete the ebb flow is decelerating. Thesalinity intrusion continues to propagate seaward until low tide but, given its length and velocityit does not have sufficient time to be completely flushed from the estuary. During the next floodthe mixed water remaining in the estuary allows a complex density structure to develop similar tothat seen early in the observed ebb. This differs from the behaviour at high freshwater dischargewhen nearly all of the seawater is flushed completely from the estuary at least once a day.Chapter 4. Shear instability in the Fraser River estuary 49(a) Transect 4  21.5 22 22.5 23 23.5 24 24.5 25051015Depth (m)(b) Transect 517.5 18 18.5 19 19.5 20 20.5051015Distance (km)(c) Transect 67.5 8 8.5 9 9.5 10 10.5 110510152 m s-1 20 kg m -3Figure 4.4: Echo soundings during low discharge observed during: (a) transect 4, early ebb; (b)transect 5, mid ebb; and (c) transect 6, late ebb. The shading scales with the log of the echointensity with black corresponding to the strongest echos. Note that the scale of the shading isthe same in all three panels. Velocities (red) from the ADCP and densities (blue dashed) from theCTD are superimposed. The black line indicates the location of the boat in the middle of the cast,as well as the zero reference for the velocity and  T. The velocity profile was calculated as a 1minute average centred on the time of the CTD cast.Chapter 4. Shear instability in the Fraser River estuary 504.4 Stability of Stratified Shear Flows4.4.1 Taylor-Goldstein EquationFollowing Taylor (1931) and Goldstein (1931) we assess the stability of the flow by consider-ing the evolution of perturbations on the background profiles of density and horizontal velocity,denoted here by  (z) and U(z), respectively. If the perturbations to the background state aresufficiently small they are well approximated by the linear equations of motion. It then sufficesto consider sinusoidal perturbations, represented by the normal mode form eik(x ct), where x isthe horizontal position and t is time. Here k = 2 = is the horizontal wave number with  the wavelength, c = cr + ici is the complex phase speed. If we further assume that the flow isincompressible, Boussinesq, inviscid, and non-diffusive, we arrive at the Taylor-Goldstein (TG)equation^ 00 +h N2(U c)2  U00U c k2i^ = 0; (4.1)where primes represent ordinary differentiation with respect to height (z), the stream function isgiven by  (x;z;t) = ^ (z)eik(x ct) and N2(z) = g 0= 0 represents the Boussinesq form of thesquared buoyancy frequency with a reference density,  0.We will find solutions to the TG equation that consist of eigenfunction-eigenvalue setsf^ (z);cg,for each value of k. Each setf^ (z);cgis referred to as a mode, and the solution may consist ofthe sum of many such modes for a single k. The background flow, represented by U(z) and  (z),is then said to be unstable if any modes exist that have ci6= 0. In this case the small perturbationsgrow exponentially at a rate given by kci. In general, unstable modes are found over a range ofk, and it is the mode with the largest growth rate that is likely to be observed. Although they arebased on linear analysis, TG predictions of the wave properties, k and c, typically match thoseof finite amplitude instabilities observed in the laboratory (Thorpe, 1973; Lawrence et al., 1991;Tedford et al., 2009).4.4.2 Miles-Howard criterionA useful criterion to assess the stability of a given flow without solving the TG equation wasderived by Miles (1961) and Howard (1961). They found that if the gradient Richardson number,Ri(z) = N2=(U0)2, exceeds 1=4 everywhere in the profile, then the TG equation has no unstablemodes, i.e. ci must be zero for all modes. In other words, Ri 1=4 everywhere is a sufficientcondition for stability, referred to as the Miles-Howard criterion. Note that if Ri < 1=4 at somelocation, instability is possible, but not guaranteed.Despite the inconclusive nature of the Miles-Howard criterion for determining instability, itis often employed as a sufficient condition for instability in density stratified flows, and has beenfound to have reasonable agreement with observations (Thorpe, 2005, p. 201-204). LookingChapter 4. Shear instability in the Fraser River estuary 51specifically at the Fraser River estuary, Geyer and Smith (1987) were able to compute statistics ofRi and show that decreases in Ri were accompanied by mixing in the estuary.4.4.3 Mixing Layer SolutionSince the TG equation is an eigenvalue problem with variable coefficients, analytical solutions canonly be obtained for the simplest profiles, and recourse is usually made to numerical methods (e.g.Hazel, 1972). However, the available analytical solutions are often a useful point of departure. Welook at one such solution that closely approximates conditions found in the estuary during highdischarge. This solution is based on the simple mixing layer model of Holmboe (described inMiles, 1963).In a general form of the model, the velocity and density profiles are represented by the hyper-bolic tangent functionsU(z) =  U2 tanhh2(z d)hiand  (z) =   2 tanh 2z  + 0: (4.2)where h is the shear layer thickness,  is the thickness of the density interface, and  U,   corre-spond to the total change in velocity and density across the layers, respectively. The parameter dallows for a vertical offset in the positions of the shear layer and density interface. In the simplestcase the shear layer and density stratification have equal thickness, giving R = h= = 1, andthey coincide in their vertical positions so that the asymmetry a = 2d=h = 0. In this case, Ri(z)is at its minimum at the centre of the mixing layer (z = 0), and is equal to the bulk Richardsonnumber J = g  h= 0( U)2. When the bulk Richardson number (i.e. the minimum Ri) dropsbelow 1=4, flows with R = 1 and a = 0 become unstable. The resulting instabilities are of theKelvin-Helmholtz (KH) type, in which the shear layer rolls up to form an array of billows that arestationary with respect to the mean flow, and which display large overturns in density (Thorpe,1973).It is not generally the case that J  1=4 results in stability. For example, if the densityinterface is relatively sharp (R > 2) an additional mode of instability, the Holmboe mode, isexcited (Alexakis, 2005). In this case, the range of J over which instability occurs extends above1=4. That is, Ri < 1=4 somewhere in z at the same time as J > 1=4. While it is generallytrue that flows with higher J are subject to less mixing by shear instabilities, by itself, J does notindicate whether or not a flow is unstable.For simplicity, the analytical solution of Holmboe’s mixing layer model assumes the flow isunbounded in the vertical. In our analysis we include boundaries at the top and bottom where ^ must satisfy the boundary condition ^ = 0. The presence of these boundaries tends to extendthe range of unstable wavenumber to longer wavelengths (Hazel, 1972). However, in the casesconsidered here, at the wavenumber of maximum growth, the boundaries have little or no impactChapter 4. Shear instability in the Fraser River estuary 52on k and c.4.4.4 Solution of the TG equation for observed profilesWe use the numerical method described in Moum et al. (2003) to generate solutions to the TGequation based on measured velocity and density profiles. Whenever possible we use velocity anddensity profiles collected at the upstream edge of apparent instabilities in the echosoundings. Thevelocity profile, a 60 second average, is an average over one or more instabilities (the instabilitieshave periods< 60 seconds). This averaging reduces the influence of individual instabilities on thevelocity profile, which in the TG equation, is taken to represent the background velocity profile.The velocity profile is then smoothed in the vertical using a low pass filter (removing verticalwavelengths < 2 m). The density profile is smoothed by fitting a linear function, and one ormore tanh functions (one for each density interface). By using smooth profiles we are effectivelyignoring instability associated with small scale variations in the profiles.Because the point of observation moves in time, i.e. the boat is drifting seaward, predictedwavelengths from the TG equation cannot be compared directly to the wavelength of instabilitiesas they appear in the echosoundings. The wavelength predicted with the TG solution must beshifted to account for the speed of the instabilities with respect to the speed of the boat:  =   vbcr vb    : (4.3)Here vb is the velocity of the boat and cr and  are the phase speed and wavelength predicted withthe TG equation. The predicted apparent wavelength,   , is directly comparable to observationsmade from the moving boat. Seim and Gregg (1994) used a similar approach for estimating thewavelength of observed features.As well as giving a wavelength, phase speed, and growth rate for each unstable mode, the TGsolutions also give an eigenfunction that describes the vertical structure of the growing mode. Thevertical displacement eigenfunction ^ (z) = ^ =(U c) is particularly useful. At the location inz wherej^ jis a maximum we expect to see evidence of instabilities in the echosoundings.4.5 ResultsIn this section we use J, Ri(z) and solutions of the TG equation to assess the stability of sixsets of velocity and density profiles (one from each of the six transects). Each set of profiles waschosen to coincide with evidence of instability in the echosoundings.Ebb during high discharge: Transect 1The selected velocity and density profiles from transect 1 are shown in figure 4.5. The corre-Chapter 4. Shear instability in the Fraser River estuary 53-2 -1 002468101214Depth (m)U (m s   )(a)0 10 20 ρ - ρ  (kg m   )-3(b)Distance (m)(c) ↓0 50 100 150 200 2500-1Figure 4.5: Velocity (a) and density (b) profiles observed during transect 1 (June 12, 2006, 8h05PDT, 8.9 km upstream of Sand Heads). The smooth profiles used in the stability analysis are shownas thick black lines and the observed data are plotted as points. The gray shading indicates regionsin which Ri < 1=4. The black horizontal line indicates the location of maximum displacement(j^ j) for the most unstable mode predicted with the TG equation. The thin lines in (b) show thedisplacement functions for each of the unstable modes. The functions are scaled in proportion tothe growth rate. A close up of the echosounding logged near the location of the profiles is shownin (c), and includes a scale indicating the apparent wavelength predicted by the TG equation.The arrow at the top of image indicates the approximate location of the density and velocitymeasurements. In this case, the velocity is averaged over a distance of approximately 130 m.sponding value of J for these profiles is 0.29 (see table 4.1). The stability analysis yields twomodes of instability. The fastest growing mode is unstable for wavelengths greater than 11 m andhas a peak growth rate of 0.025 s 1 (doubling time of 28 s) occurring at a wavelength of 21 m.The phase speed of the instability at this wavelength is -1.02 m s 1, where the negative indicatesa seaward direction. Given this phase speed and the seaward drift of the boat (-2.2 m s 1), anapparent wavelength of 39 m is calculated.Chapter 4. Shear instability in the Fraser River estuary 54Echosoundings collected at the same time, figure 4.5(c), show clear evidence of instabilities.Our interpretation of the echosoundings follows that described by Browning (1971) and Browninget al. (1973). The prediction is found to be similar to, although shorter than, the approximately 50m wavelength of the observed instabilities. The maximum displacement of the predicted instabil-ities is located at a depth of 7.6 m (indicated by the horizontal line), closely matching the depth ofthe observed instabilities. Both the observed and predicted instability occur within the region ofshear above the maximum gradient in  (at a depth of 9 m). As indicated by the gray shading, thisregion of high shear and low gradient in  corresponds to Ri< 1=4.For the set of profiles shown in figure 4.5 the TG equation predicts a second, weaker, unstablemode located at a depth of 2.5 m. This mode is associated with the inflection point (U00 = 0) inthe velocity profile at this depth. Because there is very little density stratification and hence weakecho intensity at this depth we are unable to confirm or deny the presence of this mode in theechosoundings.Ebb during high discharge: Transect 2In transect 2 a single hyperbolic tangent gives a good fit to the measured density profile (figure4.6b). Due to difficulties in profiling, the density profile at this location was missing data below 12m. Data from the previous cast, taken 60 m upstream, was used below 12 m. This cast is expectedto be sampling water of similar density below this depth.In this case the stability analysis of the profiles results in a single mode of instability. The modeis unstable for wavelengths from 10 m to 35 m with a peak growth rate of 0.02 s 1 (doubling timeof 35 s) occurring at a wavelength of 17 m. The phase speed of the instability at this wavelength is-0.51 m s 1. Given the drift velocity of -1.9 m s 1, an apparent wavelength of 24 m is calculated.This prediction is found to be similar to, although longer than, the approximately 18 m wavelengthof the small instabilities appearing in the echosounding (figure 4.6c). The maximum displacementof the predicted instabilities is located at a depth of 10.6 m, closely matching the depth of theobserved instabilities.Flood during high discharge: Transect 3Despite the occurrence of Ri < 1=4 the stability analysis of the profiles in figure 4.7(a) and4.7(b) does not find any unstable modes. Echosoundings collected during the flood generallyshow features on the pycnocline that were well correlated with sand waves (figure 4.7c). Thesecorrelated features are likely controlled by the hydraulics of the flow over the sand waves.There was very little evidence of instabilities independent of these sand waves. There appearto be some wave-like features on the pycnocline that are shorter ( 10 m) than the sandwaves,however, these are not well resolved by the echosounder (e.g. depth of 9 m atx = 60 m). ProperlyChapter 4. Shear instability in the Fraser River estuary 55-1.5 -1 -0.5 002468101214160 10 20↓0 20 40 60 80 100Depth (m)U (m s   )(a) ρ - ρ  (kg m   )-3(b)Distance (m)(c)0-1Figure 4.6: Velocity (a) and density (b) profiles observed during transect 2 (June 14, 2006, 8h21PDT, 11.1 km upstream of Sand Heads). See figure 4.5 for details. In this case, the velocity isaveraged over approximately 110 m.assessing the flow over these sandwaves would require at least two or three sets of density andvelocity profiles per sandwave, more than we were able to obtain.Low freshwater dischargeEarly ebb during low discharge: Transect 4At low discharge, during the ebb tide, shear and density stratification are spread over the entiredepth (see figure 4.4). The bulk shear layer thickness,h, is therefore greater than at high discharge,where shear and stratification were concentrated at a single, relatively thin interface. The increasein the vertical extent of the shear results in a greater bulk Richardson number despite a decreasein the overall strength of the density stratification,   (see Table 4.1).The density and velocity profiles collected early in the ebb (transect 4, figure 4.8) consist of anumber of layers. The stability analysis yields two modes of instability. The most unstable modeChapter 4. Shear instability in the Fraser River estuary 56-0.5 0 0.502468101214 0 10 20↓0 50 100 150 200Depth (m)U (m s   )(a) ρ - ρ  (kg m   )-3(b)Distance (m)(c)0-1Figure 4.7: Velocity (a) and density (b) profiles observed during transect 3 (June 21, 2006, 12h38PDT, 2.66 km upstream of Sand Heads). See figure 4.5 for details. In this case, the velocity isaveraged over approximately 30 m.has a peak growth rate of 0.023 s 1 occurring at a wavelength of 10.3 m with a phase speed of-0.86 m s 1. Given this phase speed and the seaward drift of the boat (1.6 m s 1), an apparentwavelength of 22 m is calculated. This is very similar to the wavelength of the largest instabilityin figure 4.8(c). This mode has a maximum displacement at a depth of 2.5 m, closely matchingthe location of the observed instabilities.Mid ebb during low discharge: Transect 5The instabilities in figure 4.9(c) were observed one hour later and approximately 3 km downstreamfrom Transect 4. The  profile (figure 4.9b) again displays a number of layers consisting of high-gradient steps. However, the layers are not evident in the measured velocity profile (figure 4.9a),as was the case in figure 4.8, and the overall shape of the velocity profile is more linear.The CTD cast is one of the few collected during the study where the instrument passed throughChapter 4. Shear instability in the Fraser River estuary 57-1 -0.5 002468101214 0 5 10 0 10 20 30 40 50 60 70Depth (m)U (m s   )(a) ρ - ρ  (kg m   )-3(b)Distance (m)(c)0-1Figure 4.8: Velocity (a) and density (b) profiles observed during transect 4 (March 10, 2008,11h20 PDT, 22.4 km upstream of Sand Heads). See figure 4.5 for details. In this case, the velocityis averaged over approximately 90 m.an overturn in the pycnocline (depth of approximately 3.8 m). Consistent with the small ampli-tude of the instabilities in the echosounder, the overturn in the density profile has only water ofintermediate density, i.e. no surface or bottom water is observed in the overturn.The TG equation predicts an unstable mode with a peak growth rate (0.03 s 1) at a wave-length of 14 m with a phase speed of -1.2 m s 1. The apparent wavelength is predicted to be32 m, whereas the features in the echosounder range in horizontal length from approximately10 to 40 m, with the largest being near the TG prediction ( 30 m). The predicted maximumin the displacement eigenfunction occurs at a depth of 4.2 m closely matching the depth of theinstabilities.Chapter 4. Shear instability in the Fraser River estuary 58-1.5 -1 -0.50246810120 5 10 15↓0 50 100 150 200Depth (m)U (m s   )(a) ρ - ρ  (kg m   )-3(b)Distance (m)(c)0-1Figure 4.9: Velocity (a) and density (b) profiles observed during transect 5 (March 10, 2008,12h21 PDT, 19.6 km upstream of Sand Heads). See figure 4.5 for details. In this case, the velocityis averaged over approximately 140 m.Late ebb during low discharge: Transect 6In the later stages of the ebb, during transect 6 (figure 4.10), the shear has increased such thatJ is reduced to approximately 0.3. Unlike most of the other profiles collected during low orhigh discharge the density profile has no homogeneous layers, and shows small scale (i.e. on thescale of the instrument resolution) overturning throughout the depth. In these profiles Ri is belowcritical throughout most of the depth aside from at the density interface.The most unstable mode predicted with the TG equation is located at a depth of 5.6 m andhas a maximum growth rate of 0.019 s 1 at an apparent wavelength of 65m. This is close to, butlonger than, the largest features in the echosounder (approximately 50 m).Chapter 4. Shear instability in the Fraser River estuary 59-2 -1 0024681012 5 10 15↓0 50 100 150 200 250Depth (m)U (m s   )(a) ρ - ρ  (kg m   )-3(b)Distance (m)(c)0-1Figure 4.10: Velocity (a) and density (b) profiles observed during transect 6 (March 10, 2008,14h34 PDT, 7.6 km upstream of Sand Heads). See figure 4.5 for details. In this case, the velocityis averaged over approximately 130 m.4.6 Small scale overturns and bottom stressIn figure 4.10 there are no features in the echosoundings that are associated with the small scaleoverturns in  below a depth of 7 m, and although our solutions to the TG equation suggestunstable modes, these are both located well above a depth of 7 m. To further examine the sourceof these overturns we compare selected density profiles from each of the low discharge transects(figure 4.11). In the density profile from transect 4, small scale overturns are rare or completelyabsent (figure 4.11, T4). Approximately two hours later, during transect 5, just one profile exhibitsthese small scale overturns (figure 4.11, T5). This cast was performed at the shallow constrictionin the river associated with the Massey Tunnel (figure 4.4b, 18 km). In this case the small scaleoverturns in the profile occur only below the pycnocline suggesting that the stratification within thepycnocline is confining the overturns to the lower layer. By maximum ebb, small scale overturnsChapter 4. Shear instability in the Fraser River estuary 600 2 4 6 8 10 12 14024681012Depth (m)ρ − ρ (kg m   )T4T5T6−3− 0Figure 4.11: Selected density profiles from transects performed at low freshwater discharge. Theprofiles were collected at t=10h53, 12h36 and 14h25, at x=26.2, 17.9 and 8.8 km (transects 4, 5and 6 respectively).occur throughout the depth (figure 4.11, T6).The presence of these small scale overturns is apparent, although not immediately obvious, inthe echosoundings in figure 4.4. Note that the scale of the shading is the same in all three panelsof figure 4.4 and that there is a gradual increase (darkening) in background echo intensity fromearly to late ebb (transects 4 to 6). This increase in echo intensity is attributed to the small scaleoverturning observed in the density profiles. Early in the ebb the dark shading associated with highecho intensity is concentrated at the density interfaces (transect 4). Otherwise, at this time, echointensity is low (light shading) corresponding to an absence of small scale overturns in the densityprofiles (e.g. figure 4.11, T4). At this stage of the ebb, near-bottom velocities are close to zero andbottom stress is expected to be negligible. In transect 5 (figure 4.4b) there is an increase in echointensity as the flow passes over the Massey Tunnel (18 km). At this location and during this stageof the ebb, near bottom velocity increases to approximately 0.2 m s 1 at 1 m above the bed. Inthis case the small scale overturns in the profile occur only below the pycnocline (figure 4.11, T5)suggesting that the stratification within the pycnocline is confining bottom generated turbulence tothe lower layer. Near maximum ebb, during transect 6, near bottom velocities reach 0.5 m s 1 at1 m above the bed. By this stage, high echo intensity and small scale overturns occur throughoutthe depth (figure 4.11, T6) suggesting that bottom generated turbulence has reached the surfaceChapter 4. Shear instability in the Fraser River estuary 61despite the presence of stratification.4.7 DiscussionAlthough combining echosoundings, velocity, and density measurements to study shear flows isnot in itself novel, even for studies in the Fraser estuary (e.g. Geyer and Smith, 1987), efforts inthe present study were focussed on simultaneously measuring the details of the flow and the shearinstabilities. Our strategy of drifting slowly with the upper-level flow allowed acoustic imaging tocapture shear instabilities similar to those observed in laboratory and numerical simulations (e.g.Tedford et al., 2009). Density and velocity measurements also allowed us to analyze these featuresusing a method more typically applied to laboratory experiments, namely direct application of theTG equation. This analysis has refined our understanding of instability and mixing in the FraserRiver estuary.One-sided instabilityIn all five of the cases that the TG equation predicted the occurrence of unstable modes, thebulk Richardson number, J, was greater than 1/4. This result suggests the mixing layer modeland associated J (see section 4.4.3) are not adequate for describing the stability of the measuredprofiles. In all of these unstable cases, both the region of Ri(z) < 1=4 and the depth of themaximum in the displacement eigenfunction (j^ (z)j) were vertically offset from the maximumgradient in density ( 0). This offset between the depth of the predicted region of instability andthe density interface is due to asymmetry between the density and velocity profiles. This suggeststhat a minimum of three bulk parameters (J, R, a) are required if the stability is to be representedby the simplified profiles of equation 4.2.Laboratory models and direct numerical simulations (DNS) of asymmetry result in one-sidedinstabilities that resemble the features in the echosoundings in figures 4.5(c), 4.6(c) and 4.8(c)(e.g. Lawrence et al., 1991; Yonemitsu, 1991; Carpenter et al., 2007). Similar observations weremade in the Strait of Gibraltar by Farmer and Armi (1998) and in a strongly stratified estuary byYoshida et al. (1998). In these cases the instabilities were attributed to one-sided modes. One-sided modes are part of a general class of instability that includes the Holmboe mode. In contrastto the classic KH mode, the Holmboe mode is a result of the unstable interaction of gradients indensity and gradients in shear (N2 and U00 in equation 4.1) and can occur at relatively high valuesof J (Holmboe, 1962).There are a number of potential sources of asymmetry in the Fraser estuary. The most obviousis the difference in the bottom and surface boundary condition. If the stress acting on these bound-aries is not equal and opposite, i.e. if it is unbalanced, then there is the potential for asymmetry.During low freshwater discharge the presence of multiple layers of varying thickness adds furtherChapter 4. Shear instability in the Fraser River estuary 62Kelvin−Helmholtzvin-HelmholtzOne-sided InstabilityDensity interfaceFigure 4.12: Schematic of a Kelvin-Helmholtz and a one-sided instability. The gray shadingindicates mixed fluid and the solid black line indicates the position of the central isopycnal (i.e.density interface).irregularity and potential asymmetry to the profiles. Although some laboratory models of strat-ified flows successfully generate symmetric conditions (e.g. Thorpe, 1973; Tedford et al., 2009)many others result in asymmetry (e.g. Lawrence et al., 1991; Yonemitsu et al., 1996; Pawlak andArmi, 1998; Zhu and Lawrence, 2001). In most of these cases asymmetry in the flow results fromthe geometry of the channel, such as a sill causing localized acceleration of the lower layer. Inthe arrested salt wedge experiments of Yonemitsu et al. (1996) asymmetry was associated withsecondary circulation. Given the common occurrence of asymmetry in the laboratory it is notsurprising to find asymmetry in nature.MixingLinear stability analysis does not provide quantitative predictions of mixing. When one-sidedinstabilities are modeled using DNS at the values of J observed here the complete overturningof the density interface normally associated with KH billows does not occur. Figure 4.12 showsa schematic of a one-sided instability and a Kelvin-Helholtz instability. Although one-sided in-stabilities are offset from the region of maximum density gradient they have been found to beresponsible for considerable mixing (Smyth and Winters, 2003; Carpenter et al., 2007). Unlikethe mixed fluid that results from the KH instability, the mixed fluid that results from one-sided in-stabilities is not concentrated at the density interface, but, is instead drawn away from the densityChapter 4. Shear instability in the Fraser River estuary 63interface (Carpenter et al., 2007).MacDonald and Horner-Devine (2008) quantified mixing in the Fraser estuary at high fresh-water discharge over approximately two tidal cycles. Using a control volume approach and over-turning analysis they estimated mean buoyancy flux, B, during the ebb to be 2.2 10 5 m2s 3.The associated mean turbulent eddy diffusivity, K = B=N2, was estimated to be 9 10 4 m2 s 1(MacDonald, 2003). Smyth et al. (2007) proposed parametrizing the mixing caused by Holmboeinstabilities as K = 0:8 10 4 h U. For transect 1 of the present study, which most closelymatches the conditions of MacDonald and Horner-Devine (2008) (see Table 4.1), this results inK = 6:7 10 4 m2 s 1 (0:8 10 4 5:2 m 1.6 m s 1). The parametrization of Smyth et al.(2007) represents the effect of a uniform distribution of instabilities and has not been validated athigh Reynolds number. Mindful of these inherent limitations of DNS and the complexity of thefield conditions, the similarity between the observed (K = 9 10 4 m2 s 1) and predicted mixing(K = 6:7 10 4 m2 s 1) is promising. A more rigorous analysis would include a description ofthe spatial and temporal distribution of instabilities. Unfortunately, our sampling was inadequateto comprehensively describe this distribution particularly at high fresh water discharge.During our survey of the estuary the mixing was apparently caused by shear instabilities actingwithin the interior of the flow and, to a lesser extent, by turbulence associated with the bottomboundary. Although we have addressed these two types of mixing separately they both originateas a shear instability. Unlike instability predicted with the TG equation the instability associatedwith the bottom boundary layer relies on viscous effects and the presence of the solid boundary.In some cases, for example during late ebb at low freshwater discharge (transect 6, figure 4.10),the two mechanisms (TG-type instabilities and viscous shear instabilities) are acting together togenerate mixing.At high freshwater discharge, during the ebb, MacDonald and Horner-Devine (2008) foundthat mixing at the pycnocline causes a collapse of the salt wedge which leads to complete flushingof seawater from the estuary. The well defined salt wedge is then regenerated during the subse-quent flood. Although we also see a well defined salt wedge at high discharge our observationsat low freshwater discharge suggest that during the ebb mixing caused by both shear instabilitiesat the pycnocline and bottom generated turbulence is not able to homogenize the water column.We therefore expect the presence of mixed water in the estuary at the beginning of the subsequentflood. The presence of this mixed water will prevent the formation of a well defined salt wedgeand the estuary will remain in a partially mixed state.Wave HeightUnlike KH instabilities, the deflection of the density interface (wave amplitude) caused by one-sided instabilities is usually smaller than the amplitude of the billows (see figure 4.12). It istherefore difficult to assess the amplitude of the instabilities using echosoundings (e.g. figure 4.5).Chapter 4. Shear instability in the Fraser River estuary 64Nevertheless, taking the vertical distance between the trough and the cusp, the observed instabili-ties vary in height from approximately 0.5 m to 2 m. The maximum height to wavelength aspectratio of the observed instabilities varies between approximately 0.025 (0.5/20, figure 4.6c) and 0.1(2/20, figure 4.5c). In the tilting tube experiments of Thorpe (1973) the maximum aspect ratioof KH instabilities varied between 0.05 and 0.6. Given the low values of J (< 1=4) in Thorpe’sexperiments this difference in aspect ratio is not surprising. Unfortunately, other than the caseof the KH instability (symmetric density and velocity profiles and J < 1=4) the height of shearinstabilities in stratified flows is not well documented.Use of echosoundings to identify instabilityOur analysis focused on periods when instabilities were evident in the echosoundings. Therewere instances where predictions from the TG equation suggested instabilities would occur, butnone were visible in the echosounder. In some cases (e.g. the secondary mode in figure 4.5), thelack of apparent instabilities in the echosoundings can be explained by the absence of the strongvariations in salinity and temperature (i.e. density stratification) that are responsible for most ofthe back scatter of sound to the instrument (see Seim, 1999; Lavery et al., 2003, for a thoroughdescription of acoustic scattering in similar environments).The quality of the visualization of the instabilities also depends on the speed of the boat relativeto the speed of the instabilities. For profiles collected at 2.2 km, during transect 3 (figure 4.3c),the TG equation predicted instability close to the depth of the pycnocline (results not shown).In this region the boat speed and predicted instability speed were almost the same (-0.28 m s 1versus -0.24 m s 1). Considering equation 4.3, the resulting apparent wavelength would be 250m. The corresponding apparent period of approximately 15 minutes (250 m / -0.28 m s 1) wouldlikely distort the appearance of an instability beyond recognition. This highlights an importantchallenge in identifying instabilities in echosoundings: if the point of observation is moving at aspeed similar to the instabilities, the appearance of the instabilities becomes greatly distorted. Onthe other hand, if the observer is moving at a much different velocity than the instabilities, i.e. theapparent wavelength and period are relatively short, the sampling rate of the echosounder may notbe sufficient to resolve the instabilities.In addition, our ability to detect shear instabilities depends on the timing of the echosoundingrelative to the stage of development of the instability. In DNS of symmetric and asymmetricinstabilities there are several stages of development beginning with rapid growth and finishingwith a breakdown into three-dimensional turbulence. Only during the stage where the instabilitieshave large two dimensional structures, e.g. billows, will they be easily recognizable with theechosounder. For example, during transect 2 (figure 4.3b) instabilities were only recognizableover a distance of approximately 100 m (11 to 11.1 km). However it is possible that instabilitiesare at a less recognizable stage of development throughout most of this transect.Chapter 4. Shear instability in the Fraser River estuary 65Because of these challenges, the echosounder is able to confirm only the presence and notthe absence of shear instability. We therefore limited our application of the TG equation to caseswhere instabilities were apparent.4.8 ConclusionsAfter performing a detailed stability analysis on six sets of velocity and density profiles using theTaylor-Goldstein equation and comparing with the echosoundings we conclude the following.1. All of the instabilities observed in the echosoundings coincided with the most unstablemode in the TG analysis. This confirms the applicability of the TG equation in predictinginstability, even in cases as complex as the Fraser River estuary.2. The location of each of the observed instabilities occurs in a region of depth where Ri <1=4. However, there are also cases that haveRi< 1=4 in which no unstable modes were ob-served. This result is in full agreement with the Miles-Howard criterion, but also highlightsthe inconclusive nature of this criterion.3. Although the observed instabilities all act on a well defined density interface, they appearto be concentrated on only one side of the interface. The maximum vertical displacementoccurs either above or below the density interface in a region of z where Ri < 1=4. Noneof the observations show Ri < 1=4 across the thickness of a density interface. This is incontrast to the archetypal KH instability described by the simple mixing layer model, inwhich Ri < 1=4 where  0 (N2) is greatest. The observed instabilities might therefore bebetter described by the so-called ‘one-sided’ modes of Lawrence et al. (1991); Carpenteret al. (2007).4. During the majority of the survey the observed mixing was due to shear instabilities at thepycnocline. In other stratified estuaries with moderate to strong tidal forcing, such as theColumbia and Hudson rivers, turbulence generated at the bottom is considered the dominantsource of mixing (Nash et al., 2008; Peters and Bokhorst, 2000). In the present study weonly observe mixing due to bottom generated turbulence during late ebb at low freshwaterdischarge.66BibliographyAlexakis, A. (2005). On Holmboe’s instability for smooth shear and density profiles. Phys.Fluids, 17:084103.Browning, K. (1971). Structure of the atmosphere in the vicinity of large-amplitude Kelvin-Helmholtz billows. Quart. J. Roy. Met. Soc., 97:283–298.Browning, K., Bryant, G., Starr, J., and Axford, D. (1973). Air motion within Kelvin-Helmholtzbillows determined from simultaneous Dopler radart and aircraft measurements. Quart. J. Roy.Met. Soc., 99:608–618.Carpenter, J., Lawrence, G., and Smyth, W. (2007). Evolution and mixing of asymmetric Holm-boe instabilities. J. Fluid Mech., 582:103–132.Farmer, D. and Armi, L. (1998). The flow of Atlantic water through the Strait of Gibraltar. Prog.Oceanogr., 21:1–98.Geyer, W. and Farmer, D. (1989). Tide-induced variation of the dynamics of a salt wedge estuary.J. Phys. 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A, 132:499–523.Tedford, E., Pieters, R., and Lawrence, G. (2009). Holmboe instabilities in a laboratory exchangeflow. J. Fluid Mech., page submitted.Bibliography 68Thorpe, S. (1973). Experiments on instability and turbulence in a stratified shear flow. J. FluidMech., 61:731–751.Thorpe, S. (2005). The Turbulent Ocean. Cambridge University Press, first edition.Ward, P. (1976). Seasonal salinity changes in the Fraser River Estuary. Can. J. Civil Eng.,3:342–348.Yonemitsu, N. (1991). The stability and interfacial wave phenomena of a salt wedge flow. PhDthesis, University of Alberta.Yonemitsu, N., Swaters, G., Rajaratnam, N., and Lawrence, G. (1996). Shear instabilities inarrested salt-wedge flows. Dyn. Atmos. Oceans, 24:173–182.Yoshida, S., Ohtani, M., Nishida, S., and Linden, P. (1998). Mixing processes in a highly strat-ified river. In Physical Processes in Lakes and Oceans, volume 54 of Coastal and EstuarineStudies, pages 389–400. American Geophysical Union.Zhu, D. and Lawrence, G. (2001). Holmboe’s instability in exchange flows. J. Fluid Mech.,429:391–409.69Chapter 5Unstable modes in asymmetric stratified shear layers 15.1 IntroductionWhen considering the possibility of turbulence production and mixing in a sheared density strati-fied environment, it is important to determine whether or not a particular flow configuration repre-sents a stable solution of the equations of motion. This problem has a long history that started withthe work of Helmholtz (1868) and Kelvin (1871) on the stability of homogeneous and stratifiedvortex sheets in the 19th century. Since then, numerous authors have examined cases of increasingcomplexity in an attempt to understand the basic instability mechanisms that are present in flowswith greater physical relevance.Significant progress was made by Rayleigh (1880) who examined a piecewise-linear repre-sentation of the homogeneous shear layer, represented by U(z), that consists of a finite shearthickness h (see figure 5.1). A stability analysis on this idealized shear layer produced resultsthat are in qualitative agreement with subsequent studies of smooth profiles, like the hyperbolictangent shear layer sketched in figure 5.1 (Michalke, 1964). This suggests that the piecewise shearlayer is sufficient to capture the basic instability mechanism that is also present in more realisticsmooth profiles. Motivated by geophysical flows, Holmboe (1962) extended Rayleigh’s analysisto include a stable density stratification. The piecewise-linear shear layer was retained and a lay-ered, piecewise-constant density profile  (z) was added with a step change in density of   at theshear layer centre (figure 5.1, with d = 0). As in the homogeneous (Rayleigh) case, these ideal-ized profiles give qualitatively similar results to the smooth profiles shown in figure 5.1 (Hazel,1972). In representing Holmboe’s (1962) piecewise profiles with smooth functions, it is necessaryto introduce another length scale, given by the density interface thickness  . Agreement betweenthe smooth and piecewise profiles is achieved when  is sufficiently small (i.e. R h= & 3 ingeneral) with the piecewise profiles representing the limit of vanishing  .Holmboe’s (1962) stability analysis shows the presence of two distinct types of unstablemodes. Which mode of instability develops is found to be dependent on the wavenumber k,made dimensionless by   kh=2, and the relative strength of the stratification, measured bythe dimensionless bulk Richardson number J g0h=( U)2, where g0 =   g= 0 is the reducedgravity, and  0 a reference density. The resulting stability diagram is shown in figure 5.2(a) forsmooth ‘tanh’ profiles with R = 5. At low J, the stratification is relatively unimportant, and1A version of this chapter has been submitted for publication. J.R. Carpenter, N.J. Balmforth and G.A. Lawrence(2009) Identifying unstable modes in stratified shear layers.Chapter 5. Unstable modes in asymmetric stratified shear layers 70ΔU/2ΔU/2h/2h/2zU(z)zρ0+Δρ/2ρ0d δρ(z)Figure 5.1: Profiles of the stratified shear layers to be considered. The thick lines indicate thesmooth profiles and the thin lines represent the piecewise profiles.the resulting mode of instability is essentially a stratified analogue of the Rayleigh instability. Itis common in the literature to refer to this mode as Kelvin-Helmholtz (KH), despite the closerassociation with Rayleigh’s shear layer, and we keep with this convention throughout. When Jis sufficiently large, the qualitative behaviour of the instability changes. Unstable Holmboe (H)modes develop due to a destabilizing influence of the stratification (Holmboe, 1962; Howard andMaslowe, 1973), reaching a peak growth rate at finite J.This distinct change in the stability properties that occurs across the KH-H transition can alsobe seen in the nonlinear development of the instabilities (Smyth et al., 1988; Smyth and Peltier,1991; Hogg and Ivey, 2003). Figure 5.2(b-d) shows plots of the density field for KH and H in-stabilities, taken from a series of direct numerical simulations2 (DNS), once they have reacheda large amplitude nonlinear stage of development. The KH instability (figure 5.2b) exhibits thewell-known billows of overturning fluid caused by a rolling up of the shear layer vorticity (as inthe homogeneous shear layer). These billows become susceptible to secondary instabilities andsubsequently breakdown to drive turbulent mixing of the density field (Thorpe, 1973; Caulfieldand Peltier, 2000). The nonlinear form of H instabilities (figure 5.2c,d) consist of cusp-like prop-agating waves that protrude into the upper and lower layers. These generally do not involve acomplete overturning of the density interface. It is perhaps not surprising that this different finiteamplitude behaviour between the KH and H modes has recently been found to have a pronouncedeffect on the mixing of mass and momentum in the shear layer; in some cases changing the ef-2The DNS results in this chapter were performed using the code of Winters et al. (2004). A two-dimensional gridwas utilized as well as periodic boundary conditions inxand free-slip on the top and bottom. Since the model is viscousand diffusive, moderate values of Re =  Uh= = 1200 and Pr =  = = 25, with  the kinematic viscosity and the diffusivity, were used. These do not result in significant changes in stability properties compared to the inviscidnondiffusive theory.Chapter 5. Unstable modes in asymmetric stratified shear layers 71αJ 0.060.040.020.160.120.060.040.25 0.5 0.75 100.10.20.3 (a)bcd(b) (c) (d)0.20.3++0.4+0.5+Figure 5.2: Stability diagram (a) of the Holmboe model of the stratified shear layer for smooth(tanh) profiles with R = 5. Dark contours are of growth rate and grey contours of phase speedwith the dark grey shading representing regions of stability. The thick lines correspond to thestability boundaries and the transition between stationary (below) and propagating (above) modes.Representative density fields from DNS are shown in (b-d). The approximate location of theinstabilities on the stability diagram is indicated with letters. The different widths of (b-d) aredue to changes in the wavenumber of maximum growth rate. The vertical domain height is takento be 10h for the stability diagram and in the DNS, which is sufficiently large to approximateunbounded domains.fective diffusivity by an order of magnitude across the KH-H transition (Smyth et al., 2007). Notonly the amount of mixing, but also the character of the mixing have been found to depend on theresulting mode type (Carpenter et al., 2007). It is therefore important to predict which mode ofinstability is to occur when quantifying mixing and momentum transfers.An important feature of Holmboe’s profiles is that they exhibit a symmetry about the shearlayer centre (once the Boussinesq approximation has been made). However, observations of strat-ified shear instabilities in the field often display some asymmetry between the shear layer centreand the vertical location of the density interface (Armi and Farmer, 1988; Wesson and Gregg,1994; Yoshida et al., 1998, chapter 4). It is also common to observe this asymmetry in laboratoryexperiments (Koop and Browand, 1979; Lawrence et al., 1991; Pawlak and Armi, 1998). The im-plications that this asymmetry has on the stability of the flow was first studied by Lawrence et al.(1991) using the piecewise model of Holmboe (1962) with a density interface located a distanceChapter 5. Unstable modes in asymmetric stratified shear layers 72J0.160.120.08 0.040.120.080.0400.10.20.3α0.5 1 1.5(a)bcd(b) (c) (d)0.30.20.1Figure 5.3: Stability diagram of the asymmetric stratified shear layer for smooth profiles withR = 5 and a = 0:5. See caption of figure 5.2 for further details.d below the shear layer centre, as shown in figure 5.1. They found that two distinct branches ofinstability were present, each consisting of propagating modes. No distinct transition from KH toH modes is apparent. One of the two modes always consists of larger growth rates than the other(referred to as the dominant mode), and has often been found to be the only mode observed at largeamplitudes (Lawrence et al., 1991; Haigh, 1995). Stability results for the dominant mode of theasymmetric stratified shear layer using smooth profiles with R = 5, and the asymmetry parametera  2d=h = 0:5, are shown in figure 5.3. No distinct transition between KH and H modes isapparent from the results of linear stability theory when an asymmetry is present (a6= 0). Theasymmetric laboratory observations of Lawrence et al. (1991, 1998) show a continuous change inbehaviour from overturning billows to cusp-like waves. This can also be seen in the simulationresults shown in figure 5.3(b-d), where instabilities resemble KH at low J, and become more likeH instabilities at larger J.The lack of any distinct transition in the stability properties when the flow is asymmetricdemonstrates the difficulty in distinguishing between KH- and H-like instabilities in this case, andraises the question of how to appropriately define what is meant by KH and H modes. Whenthe flow is perfectly symmetric the marginal curve separating stationary (cr = 0) modes frompropagating (cr 6= 0) modes also coincides well with distinct changes in growth rate. TheseChapter 5. Unstable modes in asymmetric stratified shear layers 73changes in stability characteristics have been found to match reasonably well with changes inthe nonlinear behaviour of the instabilities, with stationary billows occurring in the KH (cr = 0)region and cusp-like propagating waves in the H (cr 6= 0) region. It should be noted however,that nonlinear behaviour characteristic of both types of instabilities has been observed near thetransition between the two mode types (Smyth and Peltier, 1991; Hogg and Ivey, 2003). Thisled Smyth and Peltier (1991) to hypothesize that nonlinear effects can modify the location ofthe transition. The fact that a transition between KH and H behaviour, based on the nonlineardynamics, is observed in asymmetric instabilities, suggests that we must reexamine the distinctionbased on phase speed between the two modes. We therefore develop a diagnostic to interpret theunstable modes of asymmetric stratified shear layers.The purpose of this chapter is to utilize linear theory to predict the occurrence of KH- andH-type modes in asymmetric flows where the distinction between these two modes is blurred. Themotivation for this is based on three previous findings:1. KH and H instabilities result from two different linear growth mechanisms (to be discussedfully in the next section) (Holmboe, 1962; Baines and Mitsudera, 1994; Caulfield, 1994);2. it is common to find asymmetry in geophysically relevant flows (Armi and Farmer, 1988;Wesson and Gregg, 1994; Yoshida et al., 1998, chapter 4); and3. the type of instability that develops can have a significant influence on turbulent mixing andvertical transports (Smyth and Winters, 2003; Smyth et al., 2007; Carpenter et al., 2007).We use results from a linear stability analysis of both piecewise and smooth profiles of the asym-metric stratified shear layer to predict the occurrence of KH and H modes. These results are basedon the ‘wave interaction’ interpretation of shear instability that is reviewed in the following sec-tion. This is followed by the formulation of a diagnostic inx5.3 that is used to distinguish betweenthe contributions that KH and H modes make to the stability of the flow. Results of applying thediagnostic to the symmetric (a = 0) and asymmetric (a6= 0) cases shown above are outlined anddiscussed inx5.4. We then examine a set of profiles measured from a highly stratified estuary inorder to illustrate the applicability of the formulation in a geophysical context. Conclusions arestated in the final section.5.2 Wave interaction interpretation of instabilityThe wave interaction interpretation attributes instability in stratified and homogeneous shear flowsto a mutual interaction between otherwise freely propagating stable waves in the profiles. Themajority of work that has utilized the wave interaction interpretation has been carried out on theidealized piecewise -linear representation of the stratified shear layer (Holmboe, 1962; Cairns,1979; Caulfield, 1994; Redekopp, 2001), since this is the easiest possible geometry to understandChapter 5. Unstable modes in asymmetric stratified shear layers 74and apply the theory. Piecewise profiles of U and  are particularly simple because they havedelta function behaviour of vorticity gradients U00, and density gradients, represented by N2 = g 0= 0, where primes denote differentiation with respect to z. At these locations, referred to asinterfaces, wave motion may occur. The phase speed of waves on vorticity and density interfacesin a stationary frame of reference is given bycv =  q2k and cd =  g02k 1=2; (5.1)respectively, where  q = U0(z+v ) U0(z v ) denotes the jump in vorticity across the vorticityinterface. Note that the vorticity interface supports a single unidirectional mode of propagation,whereas the density interface supports two oppositely propagating modes.The wave interaction interpretation requires that two interfaces must be present, each sup-porting an oppositely propagating wave mode, in order for instability to be possible. The twointerfacial waves are then able to interact such that (i) they are stationary relative to one another,and (ii) in a ‘phase-locked’ position such that they may cause mutual growth. It is only possi-ble for (i) to occur between two oppositely propagating wave modes when there is shear in thebackground profile. Condition (i) suggests that the region of instability in the  J-plane should beclose to the locus of points where the two freely propagating interfacial wave modes have equalphase speeds. Although this is not strictly true, as each wave will interact and adjust the othersphase speed, in the limit of large  this interaction vanishes and the approximation becomes ac-curate. This large- approximation has proven useful in identifying different instability modes inprevious studies (Caulfield, 1994; Baines and Mitsudera, 1994; Redekopp, 2001).This method of identifying the wave interactions that lead to instability provides an interpre-tation of the KH and H modes that are observed in the piecewise symmetric stratified shear layerprofiles of Holmboe (1962). Figure 5.4(a) shows the resulting stability diagram with the large- approximation shown as a dashed line. The curve is obtained by equating the upper (lower) vor-ticity wave speed with the rightward (leftward) propagating internal wave speed on the densityinterface. The close agreement between the large- approximation and the region of instabil-ity indicates that the H modes are caused by an interaction between vorticity and internal wavemodes. As J vanishes, we approach Rayleigh’s homogeneous shear layer, where instability mustresult from the interaction of the two vorticity modes. This was clearly illustrated by Baines andMitsudera (1994) in considering the same profiles as Holmboe (1962) except with the lower vor-ticity interface removed, shown in figure 5.4(b). By comparing the stability diagrams in figure5.4, we see that the H region of instability remains largely unchanged, however, the KH region iscompletely eliminated. Since only two interfacial waves are able to interact – the upper vorticitywave and the rightward propagating internal wave – only one type of unstable mode (the H mode)is present. The KH instability is not present since it is caused by the interaction of the upper andlower vorticity modes.Chapter 5. Unstable modes in asymmetric stratified shear layers 75α0 1 2 3 4z U(z)ρ1ρ2 > ρ1αJ0 100.3J012α0 1 2 3 4αJ0 100.3z U(z)ρ1ρ2 > ρ1(a) (b)Figure 5.4: Stability diagram after Baines and Mitsudera (1994) showing (a) the piecewise sym-metric stratified shear layer of Holmboe (1962), and (b) the profiles of Baines and Mitsudera(1994). Contours are of growth rate with the grey shading representing regions of stability. Thickdashed line represents the large- approximation obtained by equating the speeds of an internalwave and a vorticity wave. Profiles are shown as insets in the upper left. The dotted lines denotethe boundary of the close-up regions shown as insets in the lower right.In the above description, we have only concentrated on piecewise profiles. However, the samemechanisms are still believed to apply to smooth profiles. In this case, rather than having deltafunction behaviour of U00 and N2 at the interfaces, those functions take on smooth distributionsthat attain extrema in an ‘interfacial region’. The KH instability is now a result of the inflectionpoint in the U-profile that separates two regions of oppositely signed vorticity gradients. Like-wise, the H instability is the interaction of the region of strong vorticity gradients (U00) with astrong density gradient region (N2). It does not require the presence of an inflection point. Notethat this is not a violation of Rayleigh’s (1880) inflection point theorem since it applies only tohomogeneous flows. Similarities between smooth profiles and piecewise profiles in terms of waveinteractions has been discussed previously by Baines and Mitsudera (1994), and will also be seenin the results to follow.Stratified shear layers consisting of two density interfaces may also be susceptible to a thirdinstability type that was first described by Taylor (1931). These unstable Taylor (T) modes re-sult from the interaction of two oppositely propagating waves on the density interfaces that maybecome phase-locked due to a background shear (Caulfield, 1994). Similar to the H modes, theT modes do not require the presence of an inflection point (in fact U00 may be identically zerothroughout the domain). In general, the instability of a stratified shear layer may be described interms of these three interaction types (i.e. KH, H, and T). In the following section we formulate adiagnostic in order to quantify the strength of the three types of wave interactions.Chapter 5. Unstable modes in asymmetric stratified shear layers 765.3 Formulation of a diagnosticIn this section we utilize condition (ii) from x5.2, that the interacting waves must cause mutualgrowth in each other, to formulate a diagnostic used to interpret unstable modes of the stratifiedshear layer. The formulation is general, and may be applied to any profiles in which distinctinterfaces can be identified. This allows for a classification of the unstable modes in terms of KH-,H- and T-types, following Caulfield (1994).5.3.1 Taylor–Goldstein equationWe will be concerned with the small amplitude motions of an incompressible inviscid Boussinesqfluid, with perturbations taken about the basic profiles that are small enough to be well approxi-mated by the linearized equations of motion. Following the framework of Holmboe (1962), wepartition the total perturbation vorticity of the flow,q, into a kinematic portionqK, and a baroclinicportion, qB. The kinematic vorticity is created by the vertical displacement of vorticity gradientsin the U-profile, and is given in the linear approximation byqK = U00 ; (5.2)where  denotes the vertical displacement field. In Boussinesq fluids, baroclinic vorticity is pro-duced by the horizontal tilting of the constant density surfaces of the  -profile. This baroclinicproduction of vorticity may be written, within the linear approximation, asDqBDt = N2@ @x; (5.3)where the material derivative here and throughout the remainder of the paper has been linearized,and is given byDDt @@t +U@@x:The total perturbation vorticity can now be written as the sum of the kinematic and baroclinicportions viz.q =r2 = qK +qB: (5.4)Here we have used a stream function representation of the perturbation velocity field u = (u;w),such thatu = @ =@z andw = @ =@x. This ensures that the incompressible continuity equationfor the perturbation velocity field is satisfied.Changes in the vertical displacement field may be related to the vertical velocity through thekinematic condition D Dt = @ @x; (5.5)Chapter 5. Unstable modes in asymmetric stratified shear layers 77and allows the problem, given by (5.2) through (5.4), to be expressed in terms of a single equationfor the stream function. Perturbations can now be taken to be of the normal mode form, i.e. (x;z;t) = ^ (z)eik(x ct); (5.6)for the stream function, where k is the horizontal wave number and c = cr + ici is the complexphase speed. Substituting this form results in the well known Taylor–Goldstein (TG) equation^ 00 + N2(U c)2  U00U c k2 ^ = 0: (5.7)This equation, together with the condition that ^ vanishes on the boundaries (which may be takenat z =  1), describes an eigen-problem for the eigenvalue c, and the eigenfunction ^ (z). Theflow given by the basic profiles is said to be unstable if the eigenvalue has ci > 0, which indicatesthat the perturbations grow at an exponential rate  = kci.5.3.2 Partitioning into kinematic and baroclinic fieldsOnce the normal modes are determined from solving the TG equation, it is possible to use thisinformation to examine the roles that the kinematic and baroclinic fields play in the growth of theresulting instabilities. This may be accomplished by first noting that the  due to the kinematicvorticity alone, may be determined directly from  . By defining this stream function field as K(x;z;t), (5.4) implies that =  K + B; and similarly ^ = ^ K + ^ B; (5.8)so that (5.2) may be written as^ 00K k2 ^ K = U00^ : (5.9)A similar form may be found for the baroclinic field from (5.3), and expressed as^ 00B k2 ^ B = N2U c^ : (5.10)Since ^ and c are both known from the solution to the TG equation, (5.9) and (5.10) can beexpressed in the general formL[ ^ ] = f(z); (5.11)where the linear operator L = d2=dz2  k2, and f(z) may be regarded as some known forcingfunction given by the right hand sides of (5.9, 5.10). We may now solve for ^ (z) by inverting theChapter 5. Unstable modes in asymmetric stratified shear layers 78operator L by the relation^ =ZDG(s;z)f(s)ds; (5.12)where D is the domain, and G(s;z) is the appropriate Green’s function for L, which depends onthe type of domain and boundary conditions. For an unbounded domainG(s;z) = e kjz sj=2k,otherwiseG(s;z) =  1ksinh[k(zu zl)] (sinh[k(zu z)] sinh[k(s zl)] , z>ssinh[k(zu s)] sinh[k(z zl)] , z<sfor domains with upper and lower boundaries at zu and zl, respectively.From (5.12) it is possible to partition the ^ into kinematic and baroclinic effects. The contri-bution of each field to the growth rate and phase speed of the normal mode disturbance can nowbe explicitly solved for by rearranging the kinematic condition (5.5) and substituting the normalmode form (5.6), to givecr = U + Re ^ K + ^ B^  and  = kIm ^ K + ^ B^  : (5.13)The relation for  in (5.13) will be used throughout the remainder of the chapter to assess thecontributions of the kinematic and baroclinic fields in the growth of unstable modes.5.3.3 Piecewise profilesWe now consider the idealized piecewise profiles in which the vorticity and density gradients (U00and N2) exhibit delta function behaviour at a number of discrete interface locations. This allowsus to writeU00(z) =nXj=1 qj  (z zj) and N2(z) =mX‘=1g0‘ (z z‘) (5.14)where we are considering general piecewise profiles consisting ofnvorticity interfaces with jumps qj, and m density interfaces with jumps g0‘, at the vertical locations zj and z‘, respectively.When the delta-function forms in (5.14) are substituted into (5.12) the integrals for ^ K and^ B reduce to sums, where each term represents the contribution of a particular interface. If wenow choose an interface of interest, the pth interface say, and apply (5.13), we are able to breakthe total growth rate of the normal mode  , into the individual contributions of each interfacialwave. This allows us to write =nXj=1 pKj +mX‘=1 pB‘; (5.15)where each term of the sums will be referred to as a partial growth rate, and are given explicitlyChapter 5. Unstable modes in asymmetric stratified shear layers 79by pKj = k Im( qj G(zj;zp) ^ (zj)^ (zp))and  pB‘ = k Im(g0‘G(z‘;zp)U(z‘) c^ (z‘)^ (zp)): (5.16)Note that for piecewise profiles a vorticity interface cannot cause growth in itself, i.e.  pKp = 0,since  qp is a real number.5.3.4 Smooth profilesThe same partial growth rate diagnostic is now developed for continuous distributions of U00 andN2. In doing so we presume that the domain can be split into a number of ‘interfacial regions,’where either U00 or N2 reaches some extrema. A ^ K;B can then be defined for each interface,using (5.12), as^ Kj(z) = ZDjG(s;z)U00(s)^ (s) ds (5.17)and^ B‘(z) =ZD‘G(s;z) N2(s)U(s) c ^ (s) ds; (5.18)where the Dj;‘ denotes the domain of the jth kinematic vorticity region and the ‘th baroclinicvorticity region. Using (5.13) we can write =nXj=1 Kj(z) +mX‘=1 B‘(z); (5.19)with  Kj(z) = k Im( ^ Kj=^ ) and  B‘(z) = k Im( ^ B‘=^ ). Finally, to apply this condition tothe pth interfacial region, we multiply both sides of (5.19) by a suitable weight function F(z),integrate over Dp, and rearrange to give a direct analogy to (5.15) for smooth profiles, =nXj=1hF KjiphFip +mX‘=1hF B‘iphFip (5.20)=nXj=1 pKj +mX‘=1 pB‘ (5.21)wherehip indicates integration overDp. A natural choice for the weight function is eitherF = U00if p corresponds to a vorticity interface, or F = N2 if p is a density interface, and we will keepwith this convention throughout.Chapter 5. Unstable modes in asymmetric stratified shear layers 805.3.5 Classification of modesWe have described above, a method whereby the growth rate of each interface  , can be expressedas a sum of the contributions from all of the other interfaces present. This allows us to quantifywhether or not a particular growth mechanism is present if two interfaces are causing mutualgrowth in one another. If the interfaces are both vorticity interfaces interacting across an inflectionpoint in theU profile, then the growth mechanism is classified as KH-type, for example. The othertwo possible mechanisms (following Caulfield, 1994) can be due to the interaction of a vorticity-internal wave (i.e. of the H-type), or an internal wave-internal wave (i.e. of the T-type). However,for reasons of brevity we shall limit our application of the partial growth rates in quantifying thestrength of only KH and H mechanisms in the following sections.5.4 ResultsWe now apply the partial growth rate formulation for piecewise and smooth profiles to the stratifiedshear layers in figure 5.1. The symmetric case is examined first, followed by the asymmetric casewhere we let a = 0:5. This particular value for a was chosen to match the asymmetry thatLawrence et al. (1991) observe in their laboratory experiments. In the case of smooth profiles, wewill takeU(z) =  U2 tanh 2zh and  (z) =  0   2 tanh 2Rzh +a ; (5.22)and examine a single value of the interfacial thickness ratio R = 5, which is large enough topermit a significant region of unstable propagating modes when a = 0.The profiles consist of two vorticity interfaces and a single density interface. Therefore, the Tmode that results from the interaction of two density interfaces can immediately be disregarded.The partial growth rate diagnostics in (5.15) or (5.20) reduces to a sum of three terms for eachinterface. However, we will limit our attention to only the rightward propagating H mode, the KHmode, as well as the dominant asymmetric mode. In doing so, it will suffice to apply the partialgrowth rate diagnostic only to the upper vorticity interface, since each unstable mode will consistof mutual growth between the upper vorticity interface, and one of the lower vorticity or densityinterfaces. The diagnostic equation then becomes =  KH + H + self; (5.23)where the terms on the right hand side represent the partial growth rate due to the interactionbetween the upper vorticity interface and (from left to right) the lower vorticity, density, andupper vorticity interfaces. These terms are identified with KH, H, and self-interaction components,respectively.Chapter 5. Unstable modes in asymmetric stratified shear layers 81J(a) σ0.120.080.040.160.120.0800.10.20.30.12 0.20.080.16-0.04-0.080 0.08-0.08α0.25 0.5 0.75 α0.25 0.5 0.75 α0.25 0.5 0.750 00(b) σKH (c) σHFigure 5.5: Partial growth rates for the symmetric case using piecewise profiles. Contours of  are shown in (a), which is given as the sum of  KH in (b), and  H in (c). The thick solid linedenotes the stability boundary and transition from stationary (cr = 0) to propagating (cr 6= 0)modes. Dark grey shading indicates a region of stable modes, whereas light grey shading denotesregions where the partial growth takes negative values.5.4.1 Symmetric profilesBeginning with the stability properties of the piecewise symmetric stratified shear layer, we plotthe partial growth rates from (5.23) in figure 5.5. Recall that a vorticity interface cannot interactwith itself, therefore, for all the piecewise results  self = 0 and only  KH and  H are required inthe sum, i.e. the normal mode growth rate can be expressed as a KH and H component accordingto (5.23).In the region of stationary (cr = 0) instability, growth is due entirely to  KH (figure 5.5b),while the density interface acts as a stabilizing influence, indicated by the negative values of  H(figure 5.5c). In addition, the asymptotic result of an unstable H mode consisting predominantlyof a  H component is recovered for large  (and J), as expected. However, as J increases to-wards the transition to propagating modes,  KH increases to a maximum value on the transition.Once we cross over to the propagating region of the diagram  KH gradually asymptotes to zero.This behaviour indicates that the interaction of vorticity interfaces, normally associated with thestationary KH mode, does contribute to the growth of the propagating modes, particularly near thetransition between the two. This KH contribution is greatest in the low- portion of the propagat-ing modes where  H < 0 (figure 5.5c).Note that a similar result was observed inx5.2 when the stability diagram of Holmboe (1962)was compared to the profiles of Baines and Mitsudera (1994), consisting of only the upper vorticityinterface and density interface (figure 5.4): removal of the lower vorticity interface eliminatesthe stationary modes but also alters the normal mode growth  into the propagating region. Inother words, despite the propagating modes being primarily an H-type vorticity-internal waveinteraction, the presence of additional interfaces will affect changes in  . This also reflects theChapter 5. Unstable modes in asymmetric stratified shear layers 82αJ 0.060.040.020.160.120.060.040.250.50.75100.10.20.3α0.150.10.050-0.05-0.10.250.50.751α0.050.10.150.20.250.50.751(a) σ (b) σKH (c) σHσselfFigure 5.6: Partial growth rates for the symmetric case using smooth profiles. All notation as infigure 5.5 except  self (not shown) must be included to complete the sum.fact, which has been observed in nonlinear studies of the transition (Smyth and Peltier, 1991;Hogg and Ivey, 2003), that the KH mechanism of vorticity wave interaction exerts an influenceinto the region of propagating modes.The same general behaviour is also seen in the partial growth rates of the smooth profilesshown in figure 5.6. Note however, the differences between the normal mode growth rate  , in thepiecewise and smooth profiles in figures 5.5(a) and 5.6(a). An important difference in obtainingsolutions to the TG equation for smooth profiles is that it must be done numerically on a finitedomain. A domain height of 10h was chosen since it is large enough to closely approximate solu-tions on an unbounded domain, however, it has the effect of destabilizing the low- disturbances(Hazel, 1972). In addition, when examining smooth profiles,  self must be included in order tocomplete the sum in (5.23). Although  self can be a large term, it is always found to be a negative(stabilizing) influence, and does not significantly affect the interpretation of smooth profiles. Forthis reason we do not show the details of  self in the results.Just as in the piecewise results in figure 5.5 the continuous profiles show that the stationarymodes are dominated by the KH component of the partial growth rates. This persists into thepropagating region, particularly at small  , where  H continues to be negative. As J is increasedthe H interaction between shear layer vorticity and the density interface becomes the dominantmechanism of instability growth. We can again conclude that a significant portion of the propa-gating modes experience a KH-type mechanism of inflection point growth near the phase speedtransition.Since finite amplitude instabilities in stratified shear layers have generally been found to occurat the wavenumber where the growth rate is a maximum, denoted by max, it is these disturbanceswhich we should expect to be the most relevant. It is therefore of interest to compare the relativecontributions of  KH and  H to the growth of the instability along the wavenumber of maximumgrowth curve. This is shown in figure 5.7 where the partial growth rates are plotted as a function ofChapter 5. Unstable modes in asymmetric stratified shear layers 83(a) Piecewise profiles (b) Smooth profiles  cr transitionσKHσHσ0 0.050.1 0.150.2 0.250.300.050.10.150.20.25Jgrowth rate0 0.050.1 0.150.2 0.250.3J(c)(d)(e)cddeeσH = σKHFigure 5.7: Partial growth rates for the symmetric case along  max. Piecewise results are shownin (a), while (b) gives the results for smooth profiles along with the numerical results illustratingthe nonlinear form of the instabilities. The transition between stationary and propagating modesis given by the vertical grey line (with cr = 0 for small J and cr 6= 0 at larger J). The dashedgrey line denotes where  KH =  H. Negative partial growth rates are not shown.J along the  max-curve. Note that since  max changes discontinuously across the transition fromstationary to propagating modes, figure 5.7 shows that  KH and  H are also discontinuous there.There are two locations of particular interest in these plots: the transition from stationary to prop-agating modes (vertical grey line), and the point at which  KH =  H (vertical dashed grey line).The first location denotes the commonly accepted transition between KH and H modes based onphase speed considerations, and the second shows our diagnostic describing the predominance ofeither the KH- or H-type growth mechanisms. In the case of the piecewise profiles of figure 5.7(a)these two locations are coincident. However, the smooth profiles show that H-type growth doesnot predominate until J has surpassed the transition to propagating modes. The partial growthrate diagnostic appears to provide a good description of the nonlinear dynamics for the few simu-lations performed, as shown in figure 5.7(c-e). It is also interesting to note that Smyth and Peltier(1991) have observed nonlinear characteristics of KH instabilities within the propagating modespredicted by linear stability theory in their numerical simulations of the KH-H transition.5.4.2 Asymmetric profilesWhereas the symmetric profiles show a distinct transition from stationary to propagating instabil-ities, there is no such transition in the case of asymmetric profiles. Instead, two unstable propa-gating (cr 6= 0) modes, each travelling in opposite directions with respect to the mean flow, areChapter 5. Unstable modes in asymmetric stratified shear layers 840.04(a) σ (b) σKH (c) σHJα0.160.120.080.120 0.25 0.5 0.7500.10.20.3α0.160.120.080 0.25 0.5 0.75 α00.040.080 0.25 0.5 0.75Figure 5.8: Partial growth rates for the dominant mode of the asymmetric (a = 0:5) case usingpiecewise profiles. Details as in figure 5.5.present for all J > 0. However, we note that as J!0 we recover Rayleigh’s homogeneous shearlayer instability, and so KH behaviour is expected at small J. Also, for large J (and large  ) itcan be seen that both instability regions are in alignment with the resonance condition betweenthe vorticity interfaces and the interfacial waves on the density interface (obtained by equating thephase speeds of each pair of waves, see x5.2). We therefore expect that the unstable modes areof H-type at large J, and some sort of transition between the two linear growth mechanisms musttake place.This behaviour is reflected in the partial growth rates of the piecewise profiles shown in figure5.8. Only the dominant mode consisting of larger growth rates is plotted since it has most oftenbeen found to control the development of the instabilities into the nonlinear regime (Lawrenceet al., 1991; Haigh, 1995). Figure 5.8 shows that both  KH and  H are positive over the majorityof the  J-plane where the unstable mode is present. Since  KH is a decreasing function of J,and  H is an increasing function of J, we expect a gradual transition from KH- to H-type modes.In other words both KH and H growth mechanisms are present throughout the majority of theunstable region, and they replace one another in a gradual fashion as J is increased.The resulting plots for smooth profiles (figure 5.9) show qualitatively similar results. Somedifficulty was encountered in the numerical solution of the TG equation for smooth profiles nearthe stability boundary due to the near-singular behaviour of the solutions at the critical level (seeAlexakis, 2005, for further discussion). Evidence of this can be seen in figure 5.9(c), however, wedo not expect this to have significantly affected the results.Focusing on the most amplified wave number, figure 5.10 shows a gradual transition from agrowth dominated by  KH to one where  H becomes the stronger influence, as J is increased.This is in accord with previous observations of the nonlinear behaviour of the instabilities (Lawrenceet al., 1991, 1998), and can also be seen in the results from the numerical simulations (figure 5.10c-e). A KH-like behaviour can be seen in the instability at J = 0:10, where significant overturningChapter 5. Unstable modes in asymmetric stratified shear layers 85J0.16 0.120.08 0.040.120.080.0400.10.20.3α0.10.050.5 1 1.5α0.150.20.10.050.10.050.5 1 1.5α0.5 1 1.5(a) σ (b) σKH (c) σHσselfFigure 5.9: Partial growth rates for the asymmetric (a = 0:5) case using smooth profiles. Detailsas in figure 5.6.0 0.050.1 0.150.2 0.250.300.050.10.150.20.25Jgrowth rate(a) Piecewise profiles0 0.050.1 0.150.2 0.250.3J(b) Smooth profiles  σKHσHσ (c)(d)(e)ccdd eeσH = σKHFigure 5.10: Partial growth rates for the asymmetric case along the wave number of maximumgrowth. Details as in figure 5.7.is present in the density field, and this is predicted in the larger value of  KH. By J = 0:30we have a nonlinear form of the instability that is more akin to the H instability, consisting of acusp-like wave. A result that can be seen directly from figure 5.10 is that the asymmetry extendsthe KH mechanism to larger values of J than in the symmetric case.5.5 Application to field profilesWe now give an example of how to apply the diagnostic toU and  profiles collected from the field.The profiles shown in figure 5.11 were measured in the Fraser River estuary using the methodsChapter 5. Unstable modes in asymmetric stratified shear layers 86described in chapter 4. Both of the U and  profiles can be seen to consist of a number of ‘layers’which lead to many possible interactions between interfaces. These interfaces can be identifiedby extrema in the profiles of U00 and N2 shown in figure 5.11(a) and 5.11(b), respectively. Thepresence of multiple inflection points in U, and a number of maxima in N2 indicate that any ofthe three instability types (KH, H, and T) mentioned in x5.2 may be present. A linear stabilityanalysis of these profiles predicts an unstable mode with a maximum growth rate of  = 0:024s 1 at a wavenumber of k = 0:61 m 1 and a displacement eigenfunction amplitudej^ j, shown infigure 5.11(c). These predictions of linear theory were found in chapter 4 to be in good agreementwith observations recorded by an echosounder (figure 5.11d), both in the wavenumber k, and thevertical location given by the maximum ofj^ j.We now determine which interfacial wave interactions are responsible for generating the in-stabilities seen in the echosounding of figure 5.11(d). Due to the many interfaces that are presentin the profiles, we shall simplify the analysis by noting that the dominant vertical location of theperturbations is concentrated near the peak ofj^ jat z = 10:9 m. This suggests that the instabilityis due only to the influence of the interfaces nearest this level. These consist of the two uppermostvorticity interfaces, with extrema at z  11:5 m and 10 m, as well as the density interface atz  10:5 m. Assuming this to be the case, we modify the observed profiles so that U00 and N2vanish below, with U0 remaining constant. The modified profiles will be denoted by an asterisksubscript (e.g. U ), and are shown in figure 5.11. Performing a linear stability analysis on themodified profiles yields only a 1% change in the peak growth rate, and negligible changes inkmaxandj^ j(see figure 5.11c). We conclude that our choice of modified profiles is justified by thesenegligible changes in the properties of the unstable mode.The U and N2 profiles are of similar form to the asymmetric stratified shear layer examinedin the previous section with the exception that the two vorticity interfaces are of different strength,and in the location of the vertical boundaries. Once again we identify a  KH,  H and  selfassociated with the partial growth rates of the upper vorticity interface and the lower vorticity,density, and upper vorticity interfaces, respectively. Applying the diagnostic formulated in x5.3leads tof KH; H; selfg = f0:011;0:023; 0:010gs 1. We therefore conclude that the insta-bilities observed in figure 5.11 are primarily of the H-type. This appears to be in agreement withthe wave-like features observed in the echosounding image (figure 5.11d) and the conclusions inchapter 4.5.6 Discussion and conclusionsBased on the wave interaction view of stratified shear instability (Holmboe, 1962; Baines and Mit-sudera, 1994; Caulfield, 1994), a diagnostic has been developed to interpret the wave interactionsin stratified shear layers that lead to the growth of unstable modes. It can be viewed as an ex-tension of the large- approximation used by Caulfield (1994) and Baines and Mitsudera (1994),Chapter 5. Unstable modes in asymmetric stratified shear layers 870510024681012 z (m)-1-0.500.500.51x (m)0204060(a)(b)(c)(d)Nρ − ρ02 N*2U U*U’’U*’’|η|, |η∗|^^100100Figure5.11:ProfilestakenfromfieldmeasurementsintheFraserRiverestuary(seechapter4fordetails).In(a-c)theoriginalandmodifiedprofilesaredenotedbysolidanddashedcurves,respectively.Thedensitystructureisplottedin(a)andthevelocitystructurein(b).Thenormalizedamplitudeofthedisplacementeigenfunction,isshownin(c)withthesolidhorizontallineindicatingtheverticallocationofthepeakdisplacement.Anechosoundingimagetakenatatimeclosetotheprofiles(a,b)isshownin(d),whereinstabilitycanbeseennearthepeakinj^ j.Chapter 5. Unstable modes in asymmetric stratified shear layers 88to allow for a general classification of stratified shear instabilities in terms of the KH, H and Ttypes. These types refer to vorticity-vorticity, vorticity-internal wave, and internal wave-internalwave interactions, respectively. The diagnostic has the advantage of quantifying wave growth inthe entire  J-plane, as well as being applicable to smooth profiles.The diagnostic is applied first to the symmetric and asymmetric stratified shear layers. An in-teresting result of the analysis was the presence of the KH growth mechanism in the propagatingmodes that are normally classified as H-type modes. The asymmetric stratified shear layer, whoseregion of instability consists entirely of propagating modes, was generally found to be composedof a mix of both KH- and H-type growth mechanisms. The diagnostic suggests that KH-typeinstabilities are found at larger values of J in the asymmetric case, and appears to be in quali-tative agreement with results from the numerical simulations presented here, as well as those ofCarpenter et al. (2007), which cover a range of asymmetries.A drawback of the method is the large number of wave interactions that must be accounted forin profiles with multiple interfaces. In the field profiles examined from the Fraser River estuary,the number of observed interfaces would produce an unwieldy number of interactions that must beaccounted for. This issue was dealt with by using a knowledge of the unstable mode to simplifythe profiles without significantly affecting the stability properties. This reduced the analysis tothat of the stratified shear layers examined earlier. However, this simplification may not always bepossible, and there may arise cases in which the instability involves the interaction of numerousinterfaces.Extending the use of the diagnostic to smooth profiles allows for the direct interpretation ofwave interactions in geophysically relevant profiles measured in the field. These profiles invariablydisplay some type of asymmetry, and the resulting modes of instability are likely to be of a ‘mixed’or ‘hybrid’ type, i.e. they may involve two or more interaction types as is found in this study. 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On the behavior of symmetric waves in stratified shear layers. Geofys.Publ., 24:67–112.Howard, L. and Maslowe, S. (1973). Stability of stratified shear flows. Boundary-Layer Met.,4:511–523.Kelvin, W. (1871). Hydrokinetic solutions and observations. Philos. Mag., 42:362–377.Koop, C. and Browand, F. (1979). Instability and turbulence in a stratified fluid with shear. J.Fluid Mech., 93:135–159.Bibliography 90Lawrence, G., Browand, F., and Redekopp, L. (1991). The stability of a sheared density interface.Phys. Fluids, 3(10):2360–2370.Lawrence, G., Haigh, S., and Zhu, Z. (1998). In search of Holmboe’s instability. In PhysicalProcesses in Lakes and Oceans, volume 54 of Coastal and Estuarine Studies, pages 295–304.American Geophysical Union.Michalke, A. (1964). On the inviscid instability of hyperbolic-tangent velocity profile. J. FluidMech., 19:543–556.Ortiz, S., Chomaz, J., and Loiseleux, T. (2002). Spatial Holmboe instability. Phys. Fluids,14:2585–2597.Pawlak, G. and Armi, L. (1998). Vortex dynamics in a spatially accelerating shear layer. J. FluidMech., 376:1–35.Rayleigh, J. (1880). On the stability, or instability, of certain fluid motions. Proc. Lond. Math.Soc., 12:57–70.Redekopp, L. (2001). Elements of instability theory for environmental flows. In EnvironmentalStratified Flows. Kluwer.Smyth, W., Carpenter, J., and Lawrence, G. (2007). Mixing in symmetric Holmboe waves. J.Phys. Oceanogr., 37:1566–1583.Smyth, W., Klaassen, G., and Peltier, W. (1988). Finite amplitude Holmboe waves. Geophys.Astrophys. Fluid Dyn., 43:181–222.Smyth, W. and Peltier, W. (1991). Instability and transition in finite-amplitude Kelvin-Helmholtzand Holmboe waves. J. Fluid Mech., 228:387–415.Smyth, W. and Winters, K. (2003). Turbulence and mixing in Holmboe waves. J. Phys.Oceanogr., 33:694–711.Taylor, G. (1931). Effect of variation in density on the stability of superposed streams of fluid.Proc. R. Soc. Lond. A, 132:499–523.Thorpe, S. (1973). Experiments on instability and turbulence in a stratified shear flow. J. FluidMech., 61:731–751.Umurhan, O. and Heifetz, E. (2007). Holmboe modes revisited. Phys. Fluids, 19:064102.Wesson, M. and Gregg, M. (1994). Mixing at the Camarinal Sill in the strait of Gibraltar. J.Geophys. Res., 99:9847–9878.Bibliography 91Winters, K., MacKinnon, J., and Mills, B. (2004). A spectral model for process studies ofrotating, density-stratified flows. J. Atmos. Ocean. Tech., 21:69–94.Yoshida, S., Ohtani, M., Nishida, S., and Linden, P. (1998). Mixing processes in a highly strat-ified river. In Physical Processes in Lakes and Oceans, volume 54 of Coastal and EstuarineStudies, pages 389–400. American Geophysical Union.92Chapter 6ConclusionsStratified shear instabilities are an important process in the understanding of turbulent mixingin geophysical flows. This thesis departs from the great majority of previous research on thistopic by focusing on the case of a relatively thin density interface embedded in a broader shearlayer. Although the case in which the shear and stratification have the same length scale has beenstudied more extensively, thin sheared interfaces have been observed in a number of previousstudies (e.g. Seim and Gregg, 1994; Yoshida et al., 1998; Sharples et al., 2003), all of which alsoexhibit relatively strong stratifications (J  0:5). These conditions have a richer dynamics asthey are favourable to the generation of instability by the Holmboe mechanism, as well as theKelvin-Helmholtz.In these stratified shear flows unstable waves are generated on the density interface – and itis these waves that are the topic of this thesis. Each of the four main chapters of the thesis aresummarized below, and in each case the original contributions to the study of unstable waveson a sheared density interface are described. In closing, some directions for future research areoutlined.6.1 SummaryIn the first manuscript chaper, a study of the wave fields generated by the Holmboe instability wasundertaken. Two different flows were considered: one in which the mean flow exhibited gradualtemporal variations, corresponding to the direct numerical simulations (DNS), and laboratory ex-periments where the flow displayed a gradual spatial variation. This chapter extends the currentknowledge of shear instabilities by describing the changes to the wave field that are caused bythese gradual variations in the mean flow. In the case of the laboratory experiments, the spatiallyaccelerating mean flow is responsible for ‘stretching’ the waves to longer wavelengths. The appli-cation of gradually varying wave theory has been found to provide an excellent description of thisstretching phenomena – despite the constant formation of new waves by the instability process.The effect of the stretching on wave amplitude can also be estimated using a simple application ofthe conservation of wave action, showing that significant reductions in amplitude can be expected.These experimental results are compared and contrasted to results from multiple wavelength DNS,which have not previously been performed for the Holmboe instability. This led to the identifica-tion of two important processes for the ‘coarsening’ of the wave field, namely, vortex pairing andejections. Although the vortex pairing process has been described extensively for KH instabili-Chapter 6. Conclusions 93ties, it has not been observed in symmetric Holmboe instabilities. On the other hand, the ejectionprocess has frequently been observed by numerous investigators (e.g. Thorpe, 1968; Koop andBrowand, 1979). However, the importance of ejections as a wave breaking mechanism, and itsrole in the coarsening of the wave field has been described for the first time.Given the importance of the ejection process, the next chapter focuses on describing the basicmechanism governing its occurrence. This is due to the formation of a stratified vortex couple thatconsumes the impulse of the shear layer in transporting the ejected fluid against buoyancy forces.The proposed mechanism is able to qualitatively explain observations of ejections in the DNSsuch as the dependence on interactions between the upper and lower wave modes, and the view ofejections as wave breaking events. This qualitative description of the wave breaking mechanismis a first step in quantifying important wave properties such as maximum wave amplitudes.In chapter 4, shear instabilities are identified in the important geophysical scale flow of thehighly stratified Fraser River estuary. Observations of instabilities, made through the use of anechosounder, were verified directly by a stability analysis of the measured profiles. The closeagreement between the observations and the linear predictions demonstrates that even in a highlyenergetic, evolving environment, linear theory can be used to predict features of the instabilitieswith reasonable accuracy. Previous studies in the Fraser River estuary suggest that it is the gener-ation of shear instabilities that is responsible for the majority of the mixing in the estuary (Geyerand Smith, 1987; MacDonald and Horner-Devine, 2008). These studies have generally focusedon the high freshwater discharge conditions, where the salinity intrusion more closely resemblesa classic salt wedge. However, the observations presented in chapter 4 show that the structureof the intrusion becomes much more complex, displaying multiple layers, with bottom boundarygenerated mixing playing a more important role. In addition, the observations show that instabil-ity is generally associated with an asymmetry in the locations of strong shear and strong densitygradients. This asymmetry has important consequences for the type of instability that forms, andis the subject of chapter 5.When an asymmetry is present in the profiles of sheared density interfaces it introduces am-biguity in the type of unstable modes that may arise. Focusing on the case of a single densityinterface allows both Kelvin-Helmholtz and Holmboe-type modes. Since the linear growth mech-anism of these unstable modes can be interpreted as the interaction of two different wave pairs,a diagnostic is developed that quantifies these wave interactions. This diagnostic extends previ-ous methods of mode identification so that it can now be applied to smooth profiles, and for allwavenumbers. The transition from Kelvin-Helmholtz to Holmboe modes in asymmetric stratifiedshear layers is examined, as well as an instability observed in the Fraser River estuary. Asym-metry is found to lead to ‘mixed’ or ‘hybrid’ modes that are composed of both Kelvin-Helmholtzand Holmboe growth mechanisms, thus illustrating that instability may involve the interactionbetween numerous waves.Chapter 6. Conclusions 946.2 Future workThis thesis has identified a number of possible avenues for future research. Since linear theoryhas been found successful in describing the basic features of the instabilities, it is an appropriatestarting point in the study of unstable waves on a sheared density interface. It was found inchapter 2 that predicting the nonlinear evolution of the wave field is intimately linked to boththe instability process and the mean flow. Depending on whether the mean flow is varying intime or space, different processes are expected to act on the wave field. However, it remainsunclear what overarching principles are responsible for the changes in some of the basic waveparameters such as wavenumber and amplitude. For example, the temporally varying conditionsof the simulations show a ‘wave coarsening’ effect which was found to be related to both a vortexpairing mechanism and the ejection process. But what controls the overall rate of coarsening, andhow are the instabilities responding to changes in the mean flow? This relationship between themean flow and the instabilities represents a challenging problem that is perhaps only present instratified shear layers that exhibit the Holmboe mode of instability. In this case the instability leadsto persistent finite amplitude waves that do not immediately break down into turbulent patches,as found in Kelvin-Helmholtz instabilities. However, turbulence is expected to also play a rolein the dissipation of the Holmboe wave field, particularly at larger Reynold’s numbers than thatemployed in this thesis.The greatest area requiring further study is in the application of our current understandingof shear instability to assess its role in geophysically relevant flows. While chapter 4 showedthat linear theory could be applied to geophysical observations of shear instabilities with someconfidence, it also raised a number of questions regarding the interpretation of the observationsand their relevance to the mixing of mass and momentum. Observing instabilities from a movingframe of reference will produce a distorted view due to the Doppler shifting of the wave features.In addition, it is not possible to identify instabilities (particularly with echosounders) after theyhave broken down and become sufficiently turbulent. Direct numerical simulations of stratifiedshear flow instabilities have shown that this turbulent phase of the instability life-cycle can persistfor a considerable duration (Caulfield and Peltier, 2000; Smyth and Winters, 2003). The turbulentphase is more likely to be the natural state of the high-Reynold’s number instabilities observedin geophysical flows, and makes for difficulties in quantifying the importance of the instabilityprocess in the overall dynamics of the flow.Of particular importance is the quantification of mixing due to shear instabilities, with theultimate goal being a parameterization of shear driven mixing processes. Since mixing character-istics have been found to depend on the type of instability that occurs (Smyth and Winters, 2003;Smyth et al., 2007; Carpenter et al., 2007), this task must account for the instability mechanismsin each particular flow. A method of classifying instabilities by their linear growth mechanismswas described in chapter 5 that aimed at predicting the nonlinear characteristics. This is perhapsChapter 6. Conclusions 95a first step towards assessing the type of mixing that may occur at a sheared density interface, andwhether the instabilities consist of Kelvin-Helmholtz-like billows or Holmboe-like waves. Futureresearch in this area would link the instability mechanisms of the stratified shear layer with thenonlinear mixing characteristics and wave properties of the resulting turbulent flow.96BibliographyCarpenter, J., Lawrence, G., and Smyth, W. (2007). Evolution and mixing of asymmetric Holm-boe instabilities. J. Fluid Mech., 582:103–132.Caulfield, C. and Peltier, W. (2000). The anatomy of the mixing transition in homogenous andstratified free shear layers. J. Fluid Mech., 413:1–47.Geyer, W. and Smith, J. (1987). Shear instability in a highly stratified estuary. J. Phys. Oceanogr.,17:1668–1679.Koop, C. and Browand, F. (1979). Instability and turbulence in a stratified fluid with shear. J.Fluid Mech., 93:135–159.MacDonald, D. and Horner-Devine, A. (2008). Temporal and spatial variability of vertical saltflux in a highly stratified estuary. J. Geophys. Res., 113.Seim, H. and Gregg, M. (1994). Detailed observations of naturally occurring shear instability. J.Geophys. Res., 99:10,049–10,073.Sharples, J., Coates, M., and Sherwood, J. (2003). Quantifying turbulent mixing and oxygenfluxes in a Mediterranean-type, microtidal estuary. Ocean Dyn., 53:126–136.Smyth, W., Carpenter, J., and Lawrence, G. (2007). Mixing in symmetric Holmboe waves. J.Phys. Oceanogr., 37:1566–1583.Smyth, W. and Winters, K. (2003). Turbulence and mixing in Holmboe waves. J. Phys.Oceanogr., 33:694–711.Thorpe, S. (1968). A method of producing a shear flow in a stratified fluid. J. Fluid Mech.,32:693–704.Yoshida, S., Ohtani, M., Nishida, S., and Linden, P. (1998). Mixing processes in a highly strat-ified river. In Physical Processes in Lakes and Oceans, volume 54 of Coastal and EstuarineStudies, pages 389–400. American Geophysical Union.

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