Laboratory, Field and NumericalInvestigations of Holmboe’s InstabilitybyEdmund W. TedfordB.A.Sc., The University of New Brunswick, 1997M.A.Sc., The University of British Columbia, 1999A 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)April 2009©Edmund W. Tedford 2009AbstractThe instabilities that occur at a sheared density interface are investigatedin the laboratory, the Fraser River estuary and with Direct Numerical Simulations (DNS).In the laboratory, symmetric Holmboe instabilities are observed duringsteady, maximal two-layer exchange flow in a long channel of rectangularcross section. Internal hydraulic controls at each end of the channel isolatethe subcritical region within the channel from disturbances in the reservoirs.Inside the channel, the instabilities form cusp-like waves that propagate inboth directions. The phase speed of the instabilities is consistent with linear theory, and increases along the length of the channel as a result of thegradual acceleration of each layer. This acceleration causes the wavelengthof any given instability to increase in the direction of flow. As the instabilities are elongated new instabilities form, and as a consequence, the averagewavelength is almost constant along the length of the channel.In the Fraser River estuary, a detailed stability analysis is conductedbased on the Taylor-Goldstein (TG) equation, and compared to direct observations in the estuary. We find that each set of instabilities observed coincides with an unstable mode predicted by the TG equation. Each of theseinstabilities occurs in a region where the gradient Richardson number is lessthan the critical value of 1/4. Both the TG predictions and echosoundingsindicate the instabilities are concentrated either above or below the density interface. These ‘one-sided’ instabilities are closer in structure to theHolmboe instability than to the Kelvin-Helmholtz instability. Although thedominant source of mixing in the estuary appears to be caused by shearinstability, there is also evidence of small-scale overturning due to boundarylayer turbulence when the tide produces strong near-bed velocities.Many features of the numerical simulations are consistent with lineartheory and the laboratory experiments. However, inherent differences between the DNS and the experiments are responsible for variations in thedominant wavenumber and amplitude of the wave field. The simulationsexhibit a nonlinear ‘wave coarsening’ effect, whereby the energy is shifted tolower wavenumber in discrete jumps. This process is, in part, related to the11Abstractoccurrence of ejections of mixed fluid away from the density interface. In thecase of the laboratory experiment, energy is transferred to lower wavenumber by the ‘stretching’ of the wave field by a gradually varying mean velocity.This stretchuig of the waves results in a reduction in amplitude comparedto the DNS. The results of the comparison show the dependence of the nonlinear evolution of a Holmboe wave field on temporal and spatial variationsof the mean flow.111Table of ContentsAbstract iiTable of Contents ivList of TablesviiList of Figures . .. viiiAcknowledgments . . xStatement of Co-Authorship .xi1 IntroductionStability Analysis: the Taylor-Goldstein EquationThe Three Basic Unstable ModesSymmetric Holmboe InstabilityAsymmetry and One-sidednessSmooth ProfilesEigenfunctionsBibliography162 Holmboe Instabilities in a Laboratory Exchange Flow2.1 Introduction2.2 Background Theory2.2.1 Hydraulics of Exchange Flow2.2.2 Dispersion Relation and Instability2.3 Experimental Setup2.4 Evolution of Mean Flow2.5 Wave Evolution2.6 Summary and ConclusionsBibliography 381.1 Linear1.1.11.1.21.1.31.1.41.1.5239101113181821212124273036ivTab1 of Contents3 Shear Instability in the Fraser River Estuary 403.1 Introduction 403.2 Site Description and Data Collection 413.3 General Description of the Salinity Intrusion 443.3.1 High Discharge 443.3.2 Low Discharge 463.4 Stability of Stratified Shear Flows 483.4.1 Taylor-Goldstein Equation 483.4.2 Miles-Howard Criterion 483.4.3 Mixing Layer Solution 493.4.4 Solution of the TG Equation for Observed Profiles . . 503.5 Results 513.6 Small Scale Overturns and Bottom Stress 593.7 Discussion 603.8 Conclusions 62Bibliography 644 Holmboe Wave Fields in Simulation and Experiment . . . 674.1 Introduction 674.2 Linear Stability of Stratified Shear Layers 694.3 Methods 704.3.1 Description of the Numerical Simulations 704.3.2 Description of the Laboratory Experiment 734.4 Wave Structure 734.5 Phase Speed 764.6 Spectral Evolution 774.6.1 Frequency Shifting 784.6.2 Wave Energy Spectra 804.7 Wave Growth and Amplitude 824.7.1 Wave Growth 834.7.2 Comparison of Saturated Amplitudes 844.8 Conclusions 86Bibliography 885 Conclusion 915.1 Summary 915.2 The Occurrence of Holmboe and Holmboe-Like Instabilities 92vTable of Contents5.3 Contributions to the Study of Shear Instabilities in StratifiedFlows 925.3.1 Laboratory Experiments 925.3.2 Field Experiments 935.3.3 Direct Numerical Simulations 945.4 Future Research 94Bibliography 96viList of Tables2.1 Experimental parameters 242.2 Summary of laboratory experiments 263.1 Transect details 424.1 Basic parameters of the simulations and laboratory experiments 72viiList of Figures1.1 Schematic of piecewise linear profiles 41.2 Dispersion relation for piecewise linear profiles 51.3 Sketches of three basic instabilities 71.4 Piecewise linear fit of laboratory profiles 91.5 Stability diagram 111.6 Dispersion relation for piecewise linear and smooth profiles 121.7 Eigenfunctions 141.8 Displacement of dye lines 152.1 Close up image of interface 202.2 Definition sketch for piecewise linear profiles used in the analysis of Holmboe instabilities 222.3 Celerity dispersion relations 232.4 Experimental setup 252.5 Phase speed 282.6 Interface statistics 292.7 Wave characteristics 312.8 Traced characteristics and statistics 332.9 Spectral evolution 352.10 Schematic of wave stretching and formation 363.1 Map of the lower 27 km of the Fraser River 413.2 Observed tides at Point Atkinson and New Westminster . 433.3 Echo soundings observed during high discharge 453.4 Echo soundings observed during low discharge 473.5 Transect 1 stability analysis523.6 Transect 2 stability analysis 533.7 Transect 3 stability analysis 553.8 Transect 4 stability analysis 563.9 Transect 5 stability analysis 573.10 Transect 6 stability analysis 583.11 Selected density profiles 60viiiList of Figures4.1 Growth rate and dispersion relations for the DNS 704.2 Evolution of the background profiles 714.3 Spatial changes in U(z) and layer depths in thelaboratory 744.4 Sample density fields from the laboratory experiments andthe simulations 754.5 Wave characteristics in the laboratory experimentsand thesimulations 774.6 Traced wave characteristics 784.7 Spectral evolution in the laboratory experimentsand the simulations 804.8 Growth of kinetic energy in the simulations84ixAcknowledgmentsI would like to thank my supervisory committee for its guidance. Specialthanks to my supervisor, Greg Lawrence, for his unyielding support and forgiving me all the freedom I could hope for. Special thanks to Roger Pietersfor his support in the laboratory and the field and to Rich Pawlowicz forinviting me on the initial cruises on the Fraser River. In addition, I wouldlike to thank my family and friends for their patience and support over thecourse of this work.xStatement of Co-AuthorshipThe authors of Chapter 2 are myself, R. Pieters and G. Lawrence. Withthe exception of extensive discussion and editing, I was responsible for allaspects of the research and the manuscript preparation. A version of thischapter has been accepted for publication subject to revision in the Journalof Fluid Mechanics.The authors of Chapter 3 are myself, J.R. Carpenter, R. Pawlowicz, R.Pieters and G.A. Lawrence. A version of this chapter has been submittedfor publication in the Journal of Geophysical Research. My contributionsto the work are as follows:• The research program was developed by R. Pawlowicz, myself and J.R.Carpenter.• I participated in every cruise and was responsible for the largest partof the data collection.• I was responsible for all of the data analysis.• I prepared the initial manuscript. J.R. Carpenter made additions particularly in describing the theory and the results of the analysis.The authors of Chapter 4 are J.R. Carpenter, myself, M. Rahmani andG.A. Lawrence. A version of this chapter is in preparation for submissionfor publication.• J.R. Carpenter and I initialized the research.• I described the relevant details of the laboratory experiments.• I developed the analyses of the laboratory data and contributed toadapting the analyses to the simulations. J.R. Carpenter and I, together, identified the basic concepts used to interpret the results.• J.R. Carpenter prepared the manuscript and I made additions.xiChapter 1IntroductionThe primary motivation for studying shear instabilities, such as the Hoimboe instability, is to better understand and predict mixing. Although suchinstabilities are not the only mechanism that causes mixing, they are, inmany cases the dominant one. The Fraser River estuary is a good example of a stratified shear flow where shear instabilities control mixing [Geyer& Farmer, 1989]. The frequent occurrence of strong vertical gradients indensity and velocity in the estuary provides ideal conditions for generatingthese instabilities.Before examining instabilities in a system as complex as the Fraser Riverestuary it is helpful to examine them in the laboratory. The first componentof this study is therefore, to conduct laboratory experiments that generateshear instabilities that are similar to those that occur in nature. The experiments were carried out in the two-layer exchange flow facility used by ZhuLawrence [2001]. The shear instabilities form at the density interface oftwo layers of water of different salinity (density). The two layers are flowingin opposite directions so that the vertical gradient of the streamwise velocity(shear) is centered on, and maximized at, the density interface. The sill inZhu & Lawrence [2001] was removed resulting in simplified flow and to allowa more thorough study of the instabilities. With the aid of dye and particles, illuminated by laser light, images of the instabilities were captured.The combined use of digital imaging and spatial-temporal filtering alloweda more thorough description of the instabilities than has been previouslyachieved. The background for, and results of, the laboratory experimentsare described in Chapter 2.The second component of this study is observation of shear instabilitiesin the Fraser River estuary and comparison of these observations with predictions from linear stability analysis. The observations were collected inthe salinity stratified region of the estuary during periods of strong shear.They include: echosoundings that show the structure of the instabilities,velocity measurements to quantify the shear, and temperature and conductivity measurements to quantify the density stratification. Results from theFraser River estuary are described in Chapter 3.1Chapter 1. IntroductionThe third component of this study is a comparison of the results from thelaboratory experiments with direct numerical simulations (DNS). Becausethe linear stability analysis used in Chapters 2 and 3 is most accurate whenthe instabilities are very small, this analysis does not provide a completedescription of the development of the instabilities. The nonlinear effectsthat become important at larger amplitudes are accurately described withDNS. There are, however, some differences between DNS and real flowsassociated with boundary and initial conditions, as well as computing powerlimitations, that limit the predictive capabilities of DNS. The results of thecomparison between the laboratory experiments and DNS are discussed inChapter 4.The background information specific to each of these three componentswill be reviewed at the beginning of each of the respective chapters. In allthree cases (Chapters 2, 3 and 4), predictions based on the Taylor-Goldsteinequation are used to understand the basic behaviour of the instabilities; i.e.phase speed, wavelength and vertical structure. Because of its importancegenerally in stratified shear flows and particularly in this study, several illustrative solutions of the equation are discussed in detail in the remainingsections of this chapter.1.1 Linear Stability Analysis: theTaylor-Goldstein EquationThe Taylor-Goldstein (TG) equation results from application of the methodof normal modes to simplified equations of motion for stratified shear flow.It is assumed that the fluid is inviscid, incompressible, non diffusive andthat the background flow is parallel (density and velocity are horizontallyuniform in the background). The Boussinesq approximation is also made.Although the TG equation was originally derived for studying atmosphericdynamics [Taylor, 1931; Goldstein, 1931] it is also applicable to the flow ofstratified water considered here. For a thorough derivation and a descriptionof the assumptions see Drazin Reid [19821. The Taylor-Goldstein equationis:d2i4’ N2 d2U/dzk2— o 1 1L(U_c)2U-c -‘ (.)where the streamfunction of the perturbation is given byb(x,z,t)= z)e(z_ct).(1.2)2Chapter 1. IntroductionThe vertical coordinate is given by z. U(z) is the background profile of thehorizontal velocity. N(z) is the profile of the Brunt Vaisala frequency, givenby N2 =— (g/po)(dp/dz), where g is the gravitational acceleration, p is thedensity andp0is a reference density. With the addition of boundary conditions at the top and bottom(1.1) defines an eigenvalue problem for thecomplex phase speed, c, given the wavenumber, k. The resulting eigenfunction ‘(z) gives the vertical mode shape, from which we can determine wherein the vertical the greatest displacements will occur. The dispersion relation, c(k), will be used in this section to illustrate the three basic instabilitymechanisms that occur in a stably stratified shear flow. In later chapterscalculated dispersion relations will be compared with observations. In thesecases the Taylor-Goldstein equation is solved analytically for piecewise linear profiles [Drazin & Reid, 1982] and numerically for continuous profiles[Mourn et al., 2003). The results presented here are for vertically boundedflow (i.e. the vertical velocity of the perturbation and therefore b are equalto zero).1.1.1 The Three Basic Unstable ModesThe three sets of piecewise linear profiles shown in figure 1.1 can be considered the basic building blocks in the study of shear instability in stratifiedflow. With piecewise linear profiles there is one eigenvalue for each step invorticity and two for each step in density. The vorticity steps are located atthe kinks in the velocity profile, i.e. where the shear, dU/dz, changes. Thetwo steps in vorticity in figure 1.la will support one mode each. The twodensity steps in figure 1.lb will support four modes in total, two for eachdensity step. The set of profiles in figure 1. ic will support three modes,one for the step in vorticity and two for the step in density. The three setsof profiles are referred to here as the Rayleigh, Taylor and Holmboe casesrespectively.In figure 1.2 the calculated eigenvalues, c, are plotted as solid linesagainst wavenumber, k, giving the dispersion relation for each set of profilesin figure 1.1. The wave number is nondimensionalized by the length scaleh and the velocity is nondimensionalized by U/2. In all three cases IUis the change in velocity that occurs over h (see figure 1.1). The dispersionrelation shown for the Rayleigh case is therefore general as all the dimensions have been accounted for. Because the nondimensionalization does notaccount for changes in density the dispersion relations of the Taylor andHolmboe cases will change when the size of the density step is changed. Thedispersion relations for these two cases are general, but only in a qualitative3Chapter 1. Introduction(a) Rayleigh (b) Taylor (c) HolmboeuFigure 1.1: Vertical profiles of horizontal velocity and density for the threebasic flow cases. In all three cases the velocity is zero at mid-depth (indicatedby the dotted line).sense, i.e. both cases will have the same unstable and stable modes no matter the size of the density step(s) but specific details of the dispersion curveswill vary. The inclusion of the density into the non dimensionalization willbe discussed further in section 1.1.2. In the cases consideredhere the lengthscale associated with the total depth is only important when consideringwaves at or near the longwave limit.For all three cases, the dispersion relation was also calculated for eachstep separately, as if the other step in the proffle did not exist and theseareshown in figure 1.2 as dashed lines. At shorter wavelengths the modesoneach step act in isolation (the solid line equals the dashed line).At greaterwavelengths the modes on the different steps interact with the eachother(the solid lines diverge from the dashed lines). In each of the threecases it/4Chapter 1. Introductiona.Rayleigh — ——1-1.5_...-—zI I I I I I1*a1Holmboe —— IGravity0 2 06 08 112 1416 18 2Wavenumber (rad)Figure 1.2: Dispersion relations for the three basic flow cases (solid lines).The dashed lines show the modes supported by individualvorticity (thin)and density (heavy) interfaces. The phase speed has been nondimensionalized by the U/2 and the wavenumber by h (see figure1.1).5Chapter 1. Introductionis the interaction of the steps (density or vorticity) at longer wavelengths,that causes instability.Looking first at the Rayleigh [1896] case, at short wavelength there aretwo stable modes shown by solid lines at high wavenumber in figure 1.2a.These stable modes are associated with the vorticity steps and are referredto as vorticity modes. The mode focussed on the upper vorticity step ispropagating to the right (positive c) and the mode focused on the bottomstep is propagating to the left. As the wavelength increases (wavenumber,k, decreases) the phase speed of the two stable modes goes to zero. At thiswavelength the two stable modes change into two unstable modes. Bothunstable modes have the same phase speed (c = 0) but one is decayingand the other is growing. The decaying mode is generally ignored. In thisexample N2(z) = 0 but this need not be the case for the Rayleigh instabilitymechanism to occur. As long as the stratification is relatively weak the twovorticity steps will interact creating an unstable mode.In the Taylor case (figure 1. lb and 1. 2b) at the shortest wavelengthsthere are four stable modes [Taylor, 1931]. These modes are simple gravitywaves propagating with equal and opposite velocity relative to the velocityat the density step. The two modes propagating to the right lie on theupper density step and the two modes propagating to the left lie on thelower density step. At longer wavelengths, two of the modes, one from eachinterface, interact and merge into a stationary unstable mode. This unstablemode is referred to as a Taylor instability.In the Holmboe case (figure l.lc and l.2c) at the shortest wavelengththere are three stable modes. The mode with the fastest phase speed ispropagating to the right and represents a vorticity wave focussed on theupper vorticity step. This mode is identical to the positive vorticity modein the Rayleigh case (figure l.2a). The other two stable modes are gravitymodes that lie on the lower density step. These two modes are identicalto the leftward propagating modes of the Taylor case. As the wavelengthincreases (k J,) the vorticity mode and one of the gravity modes interactand merge into an unstable mode. I will refer to this type of instability asa Holmboe instability. This unstable mode was first examined by Holmboe[1962] although in his analysis there was an additional vorticity step belowthe density step. The simpler case shown in figure 1. lc was first discussedin Baines & Mitsudera [1994].In the Rayleigh case N2(z) 0 and in the Taylor cased2U/dz = 0.The result, in both cases, is that equation 1.1 is greatly simplified. In thecase of the Holmboe instability both these terms are non-zero, indeed, it isthe interaction of vertical gradients in stratification, N2(z), and vorticity,6Chapter 1. Introductiona. Rayleigh or Kelvin HelmholtzEjectedmixed fluidc. HolmboeFigure 1.3: Sketches of the three basic instabilities that occur in stratifiedshear flows: a Rayleigh, b Taylor and c Holmboe. The arrows indicate theprimary vortical motion. The sketches are based on DNS and laboratoryobservations. The gray shading in c indicates fluid of intermediate density.d2U/dz2,that generates the Holmboe instability.Finite Amplitude Appearance and MixingAlthough the focus of future chapters is on comparing the predicted wavedispersion with observations it should be noted that these three types ofinstabilities have other differences of practical importance. At finite amplitude the Rayleigh instability has a spiralling billow that resembles thesketch in figure 1.3a. As the instability grows, neighboring billows interactb. TaylorLower densityinterface7Chapter 1. Introductionand combine (‘pair’) to form a new billow with twice the wavelength andincreased amplitude [Browand & Winant, 1973]. In the absence of densitystratification pairing will cause the wavelength and amplitude to increaseuntil boundaries are reached. The presence of a density interface centredwithin the shear layer restricts the pairing and reduces the amplitude of individual billows (they become more elliptical than circular). The amplitudeof the billows and the resultant mixing decrease as the strength of the density stratification increases [Thorpe, 1973]. The 2D spiral structure shownin figure 1.3a is most persistent at low Reynolds number. At high Reynoldsnumber the billow breaks down into 3D turbulence before a well definedspiral can form [Brown & Roshko, 1974).It must be noted that instabilities that occur on a density interface andthat are a result of the Rayleigh mechanism described above are invariablyreferred to as Kelvin-Helmholtz (KR) instabilities rather than Rayleigh instabilities. This potential source of confusion was emphasized in Lawrenceet al. [1991]. In later chapters these instabilities will also be referred to asKR instabilities rather than Rayleigh instabilities. Strictly speaking the KRinstability is the result of coincident steps in both the density and velocityprofile [Kelvin, 1871]. The step in velocity, in this case, means the flow is always unstable, no matter the strength of the density stratification. Caulfield[1994] describes all of the unstable modes that result from the interactionof two vorticity interfaces separated by a density interface.At finite amplitude the Taylor instability (figure 1.3b) has a series of vortices located between the two density interfaces similar to rollers betweena conveyer belt (the upper and lower density interfaces). Non-linear simulations of Taylor instabilities suggest they are longer lived and cause muchslower mixing than Rayleigh instabilities [Lee & Caulfield, 2001]. UnlikeRayleigh instabilities, their development did not cause complete overturning of the density interface.The Rolmboe instability (figure 1.3c) features cusping waves somewhatresembling surface water waves At the cusp of the wave mixed fluid accumulates, eventually being ejected as a wisp into the upper layer (or the lowerlayer if there is a vorticity step below the density step). Like the Taylor instability the Holmboe instability does not cause complete overturning of thedensity interface [Carpenter et al., 2007]. In direct numerical simulations,Smyth & Winters [2003] found that although Rolmboe instabilities growmore slowly than Rayleigh (KR) instabilities the total amount of mixingmay be comparable.It is worth noting that some authors [e.g. Koop, 1976] refer to any instability that occurs in stably stratified shear flow as a KH instability.8EC)E000-ca)zChapter 1. IntroductionFigure 1.4: Smooth profiles observedmatching the symmetric Holmboe case.in the lab with a piecewise linear fit1.1.2 Symmetric Holmboe InstabilityThe piecewise linear profiles shown in figure1.4 were originally analyzed bySmyth [1986]. They represent a bounded versionof the flow configurationconsidered by Holmboe [1962]. In this chapter I will refer tothis case as thesymmetric Holmboe case. In later chapters, and in theliterature in general,it is referred to simply as the Holmboe case.The sharp density interfacepositioned within a uniform shear layer approximatesconditions observed insalt stratified shear flows at laboratory scales (thesmooth proffles are fromthe experiments discussed in chapter 2). This casewill support a singleRayleigh instability or two Holmboe instabilities dependingon the strengthof the shear compared to the strength of thestratification.In this case the strength of the stratification is characterizedwith thereduced gravity: g’=g1p/po, whereP0is the average density and p is the109872n654399 999.5 1000p (kg m)1000.5 —2 —1 0 1Velocity (cm s1)29Chapter 1. Introductiondensity difference between the layers. The bulk stability is then given bythe Richardson number, J = g’h/(U)2, IU is the total shear and h is theshear layer thickness. To obtain a representative value of h the piecewiselinear velocity profile is fit to the maximum layer velocities (see figure 1.4).In figure 1.5 the regions where Holmboe and Rayleigh instability willoccur is mapped in Richardson number - wave number space. The stability bounds shown are determined by calculating the dispersion relation for arange of J values. For weak stratification (J < 0.07) the Rayleigh instabilitywill occur and at higher Richardson number only the Holmboe instabilitywill occur. In the laboratory flow considered in chapters 2 and 4 the bulkRichardson number is 0.3 (the horizontal line in figure 1.5), well above theupper limit for a Rayleigh instability. The dispersion relation for the twoHolmboe modes at this value of J is plotted in figure 1.6. It is very similar to the dispersion relation for the Holmboe case (figure 1.2c). At shortwavelengths there is an additional (stable) vorticity mode associated withthe second, lower, vorticity step (not shown in figure 1.6). At longer wavelengths this additional vorticity mode merges with the leftward propagatinggravity mode to form a second Holmboe instability. This second Holmboemode propagates in the opposite direction (negative phase velocity) and hasa nearly identical growth rate (figure 1 .6b).1.1.3 Asymmetry and One-sidednessIf the density interface shown in figure 1.4 is not centred within the shearlayer, e.g. if the density interface is closer to one vorticity interface than theother, then the growth rates and dispersion relations of the two Holmboemodes will differ. For example, in the splitter plate experiments of Lawrenceet at. [1991], the density interface was positioned closer to the lower vorticity interface than to the upper vorticity interface. This asymmetry in theprofiles resulted in the upper Holmboe mode having a greater growth ratethan the lower Holmboe mode. The more unstable mode tended to dominate such that at finite amplitude the instability resembled the Holmboeinstability sketched in figure 1.3. These asymmetric instabilities are typically referred to as ‘one-sided’. Because the density and velocity profilesthat occur in nature often include some asymmetry one-sided instabilitiesare common (see chapter 3).10Chapter 1. IntroductionFigure 1.5: Stability diagram for the symmetric Holmboe case with the totaldepth, H 5h. The growth rate of unstable modes is contoured and shaded.The regions with the darkest shade of gray are stable. The horizontal lineindicates the value of J of interest.1.1.4 Smooth ProfilesBefore discussing solutions to the TG equation based on smooth profiles,the relationship between piecewise linear and smooth profiles, especially forthe Holmboe profiles, should be emphasized. In the piecewise linear profiles,unstable modes resulted from the interaction of a step in the density profileand a kink in the velocity profile. Similarily, in smooth profiles, unstablemodes result from the interaction of a maximum in the vertical gradient ofp (N2 in the TG equation) and a maximum in the curvature of thevelocityprofile (d2U/dz in the TG equation).Velocity and density profiles measured in the lab (see chapter 3) areplotted in figure 1.4. Using the method outlined in Moum et at. [2003]the dispersion relation for this set of profiles was calculated. The method10 15Wave number (cycles m1)11Chapter 1. IntroductionFigure 1.6: Phase speed (a) and growth rate (b) for the symmetric Holmboecase (solid line) and smooth profiles observed in the lab (dotted solid line).The small differences between the leftward and rightward propagating modesare due to slight asymmetry in both the observed and fit profiles.includes the effect of viscosity. The viscosity plays a secondary role in theflows considered here in that it tends to slightly stabilize short waves. In thecase shown in figure 1.6 the velocity and density profiles have been measuredat a fine enough vertical resolution such that on the scale of the entire depththey appear smooth. On the scale of the vertical resolution they will havesteps in the density and vorticity just as the piecewise linear profiles did.As in the piecewise linear cases each of these small steps will support twogravity modes and one vorticity mode. Most of these modes are due only tothe details of the resolution or discretization.Following Moum et al. [2003],these spurious modes are rejected using a kinetic energy criteria.For the smooth set of profiles shown in figure 1.4 there are two Holmboe0.1510 15Wave number (cycles m1)12Chapter 1. Introductionmodes (i.e. unstable modes) with similar phase speeds to those that occurfor the piecewise linear fit. The dispersion relations for the smooth andpiecewise linear profiles are plotted together in figure 1.6a. The phase speedfor the smooth profiles shows more variation over k with long waves propagating more quickly than in the piecewise linear case and shorter wavespropagating more slowly. The growth rates (the imaginary part of c) aresimilar in that the peak occurs at approximately the same wave number(k =15 cycles m1 for the smooth and k =14 cycles m1 for the piecewise linear). The magnitude of the growth rate is considerably less for thesmooth profiles. This difference is primarily due to the greater thicknessof the density interface in the case of the smooth profiles (i.e. finite ratherthan a step). For a description of the dependence of growth rate on densityinterface thickness see Smyth & Winters [2003] and Haigh[19951.1.1.5 EigenfunctionsSo far I have discussed only the eigenvalue, c, and not the associated eigenfunction b. Two quantities derived from are particularly useful, the displacement function n(z) = w(z)/(U — c) and the shear production uw. Bythe definition of the streamfunction the vertical velocity of the perturbation, w(z), is equal to b multiplied by an arbitrary constant and phaseshift. The horizontal velocity of the perturbation, u(z), is equal to s/k.The displacement function shows where in the vertical we can expect to seethe largest vertical defiections e.g. where isopycnals will show the greatestvertical movement. The shear production shows where in the vertical kinetic energy is transferred from the mean flow to the instability. Figure 1.7aand c show that the displacement and shear production have a maximumamplitude just above the elevation of the density interface (the dotted linein figure 1.7). It is in this region, between the maximum curvature in thevelocity profile, d2U/dz2,and the maximum gradient in the density, N2,that the interaction between the vorticity and stratification is strongest. Infigure 1.7b the phase of the displacement function is also plotted.To aid in the interpretation of the displacement function it is plotted inan alternative form in figure 1.8. The figure shows sinusoidal waves withrelative amplitudes and phases matching the displacement function. Thesesinusoids can be thought of as dye streaks. The reader should be remindedthe TG equation is for infinitesimal waves, so the waves in figure 1.8 havebeen given finite amplitude for illustrative purposes. Hazel [1972] used asimilar diagram for illustrating various shear instabilities.13E0E00)0)0)=Figure 1.7: Eigenfunction derived quantities for the rightward propagatingHolmboe mode at the wave number of maximum growth. The phase is inradians/Tr. The eigenfunctions were calculated using the smooth profiles infigure 1.4. The dash line indicates the height of the density interface.Chapter 1. Introduction(c)0.5 1 0 0.5 1 —1 —0.5Amplitude of displacement Phase of displacement uw014Chapter 1. Introduction4ci)2_____ _____1••10C.)C-1—20 2 4 6 8 10Along wave distance (cm)Figure 1.8: Displacement of dye lines for the rightward propagating Holmboemode at the wave number of maximum growth.15BibliographyBAINES, P.G. & MITSUDERA, H. 1994 On the mechanism of shear flowinstabilities. J. Fluid Mech. 276, 327—342.BROWAND, F.K. & WINANT, C.D. 1973 Laboratory observations of shear-layer instability in a stratified fluid. Boundary-Layer Met. 5, 67—77.BROWN, F.L. & ROSHKO, A. 1974 On the density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775—816.CARPENTER, J.R., LAWRENCE, G.A. & SMYTH, W.D. 2007 Evolutionand mixing of asymmetric Holmboe instabilities. J. Fluid Mech. 582, 103—132.CAULFIELD, C. p. 1994 Multiple linear instability of layered stratified shearflow. J. Fluid Mech. 258, 255—285.DRAZIN, P.G. & REID, W.H. 1982 Hydrodynamic Stability, first paperback edn. Cambridge University Press.GEYER, W.R. & FARMER, D.M. 1989 Tide-induced variation of the dynamics of a salt wedge estuary. J. Phys. Oceanogr. 19, 1060—1672.GOLDSTEIN, S. 1931 On the stability of superposed streams of fluids ofdifferent densities. Proc. R. Soc. Lond. A 132, 524—548.HAIGH, S .P. 1995 Non-symmetric Holmboe waves. PhD thesis, Universityof British Columbia.HAZEL, P. 1972 Numerical studies of the stability of inviscid stratifiedshear flows. J. Fluid Mech. 51, 39—61.HOLMBOE, J. 1962 On the behavior of symmetric waves in stratified shearlayers. Geofys. Publ. 24, 67—112.KELVIN, W. 1871 Hydrokinetic solutions and observations. Philos. Mag.42, 362—377.16BibliographyKoop, C.G. 1976 Instability and turbulence in a stratified shear layer.Department of Aerospace Engineering, University of Southern California.LAWRENCE, G.A., BROWAND, F.K. & REDEKOPP, L.G. 1991 The stability of a sheared density interface. Phys. Fluids 3 (10), 2360—2370.LEE, V. & CAULFIELD, C.P. 2001 Nonlinear evolution of a layered stratified shear flow. Dyn. Atmos. Oceans 24, 173—182.MOUM, J.N., FARMER, D.M., SMYTH, W.D., ARMI, L. & VAGLE, S.2003 Structure and generation of turbulence at interfaces strained by internal solitary waves propagating shoreward over the continental shelf. J.Phys. Oceanogr. 33, 2093—2112.RAYLEIGH, J.W.S. 1896 Theory of Sound, 2nd edn. Macmillan.SMYTH, W. D. 1986 M. Sc. thesis. Department of Physics, University ofToronto.SMYTH, W.D. & WINTERS, K.B. 2003 Turbulence and mixing in Holmboewaves. J. Phys. Oceanogr. 33, 694—711.TAYLOR, G .1. 1931 Effect of variation in density on the stability of superposed streams of fluid. Proc. R. Soc. Lond. A 132, 499—523.THORPE, S. A. 1973 Experiments on instability and turbulence in a stratified shear flow. J. Fluid Mech. 61, 731—751.ZHU, D. & LAWRENCE, G.A. 2001 Holmboe’s instability in exchangeflows. J. Fluid Mech. 429, 391—409.17Chapter 2Symmetric HolmboeInstabilities in a LaboratoryExchange Flow12.1 IntroductionFlows in the environment often consist of well defined layers of differentdensity. A density difference can result from salinity (e.g. in an estuary orthe ocean), temperature (e.g. in a lake), sediment (e.g. gravity current) orother factors. Studies of geophysical flows have shown that wavelike features occur at the interface between sheared layers [Wesson & Gregg, 1994;Geyer & Smith, 1987; Tedford et al., 2007]. As these interfacial features orinstabilities grow, fluid is exchanged vertically between the layers. Mixingbetween layers is important because it controls the vertical transfer of salt,heat, nutrients, pollutants and momentum.Stratified shear flows in the laboratory also exhibit a variety of wave-likefeatures. The most well known are the Kelvin-Helmholtz (KR) instabilitiesobserved in the classic experiments of Thorpe [1971]. The shear between twohomogeneous layers of differing salinity causes instabilities that are exceptionally uniform in wavelength and amplitude. These instabilities quicklygrow into stationary billows which, in turn, break down into three dimensional turbulence. Of the shear instabilities that occur in stratified flows,the KH instability has been studied most extensively, but in recent yearsincreasing attention has been paid to the Holmboe instability.Holmboe[19621analyzed the stability of a sharp density interface subjected to shear. He predicted that when stratification is strong enough tosuppress the KH instability, two wave trains develop that travel with equaland opposite phase speeds with respect to the mean flow. An example of‘This chapter has been accepted for publication subject to revision in: E. W. Tedford,R. Pieters and G.A. Lawrence (2009), Symmetric Holmboe Instabilities in a LaboratoryExchange Flow, J. Fluid Mech.18Chapter 2. Holmboe Instabilities in a Laboratory Exchange FlowHolmboe’s instability from the present experiments is shown in figure 2.1.The potential importance of Holmboe instabilities was recently highlightedby the direct numerical simulations of Smyth & Winters [2003], who foundthat, although Holmboe instabilities grow less rapidly than KR instabilities,the total amount of mixing can be greater [see also Smyth, 2006; Smythet at., 2007; Carpenter et at., 2007]. Note that while Holmboe [1962] assumed a density step, Alexakis [2005] has shown that Holmboe instabilitiescan occur providing the thickness of the velocity interface is more than double the thickness of the density interface. Holmboe instabilities are thoughtto occur in natural flows such as the exchange flow through the Strait ofGibraltar [Farmer & Armi, 1998] and the salinity intrusion in a stronglystratified estuary [Yoshida et at., 1998].Several techniques have been used to study Holmboe instabilities in thelaboratory. In the splitter plate experiments of Koop & Browand [1979]and Lawrence et at. [1991] only one of the two modes predicted by Holmboeappeared. A series of cusps from which wisps of interfacial fluid were occasionally ejected formed on only one side of the interface. This ‘one-sidedness’was a result of a vertical displacement between the sharp density interfaceand the shear, an inherent condition in splitter plate experiments. WhileCarpenter et at. [2007] have postulated that one-sided instabilities may bean important source of mixing, we will restrict our attention to symmetric(two-sided) instabilities in the present study.Using immiscible fluids and varying viscosity, Pouliquen et at. [1994]conducted tilting tube experiments to generate Holmboe instabilities. Dueto the slow growth of the instabilities and the inherently short duration oftilting tube experiments they were only able to observe the early onset ofinstabilities and, unlike Thorpe [1971], they were required to use regularlyspaced obstacles to force uniformity.The one-sidedness of splitter plate experiments, and the short durationof tilting tube experiments, can be avoided by using exchange flow. Zhu &Lawrence [2001] studied Holmboe instabilities in exchange flow through achannel of uniform width with a sill. However, symmetric Holmboe instabilities were only a transient feature of these experiments. Hogg & Ivey [2003]studied exchange flow through a contraction. However, this contraction wasrelatively short so that only a small number of instabilities were presentat any given time, and the background flow conditions changed over a single wavelength. In the present study we use a long channel of rectangularcross-section in which many waves are present at any given time.19Chapter 2. Holmboe Instabilities in a Laboratory Exchange FlowFigure 2.1: Close up image of the interface between two layers. The top,fresh layer is moving to the right and the bottom, saline layer is movingto the left. The upward pointing cusp is a positive, rightward propagatingHolmboe instability and the downward pointing cusp is a negative, leftwardpropagating Holmboe instability. Colour varies from blue to red markinghigh to low fluorescence of dye. The decrease in fluorescence below theinterface is caused by the dissipation of light. To generate particle streaksthe shutter speed of the camera was set to 0.5 seconds.The goals of this study are to carry out experiments in the laboratorythat generate Holmboe instabilities and to compare the properties of theseinstabilities with the predictions of Holmboe [1962]. In the next section,the linear model of Holmboe [1962] is described. Section 3 describes thelaboratory setup and methods. In section 4, the evolution of the meanflow and the observed wave characteristics are described. In section 5, theobservations are compared with the linear predictions.—84 —83 —82 —81 —80 —79Distance from mid—channel (cm)20Chapter 2. Holmboe Instabilities in a Laboratory Exchange Flow2.2 Background Theory2.2.1 Hydraulics of Exchange FlowThe basic features of exchange flow can be described by two-layer hydraulics[Armi, 1986]. Here we briefly review the concept of internal hydraulic controls and their relevance to the instabilities. In single-layer flows the conceptof ‘hydraulic control’ is used to determine how flow rate relates to channelgeometry. A single layer control can occur where the flow exits a restriction,such as a horizontal expansion or an increase in bed slope. At the controlthe flow speed is equal to the long wave speed and is therefore said to be critical. In subcritical flow, waves may propagate in both directions, upstreamor downstream. In supercritical flow waves can only propagate in one direction, downstream. At the control there is a transition from subcritical tosupercritical flow.Two-layer flows also exhibit hydraulic controls, but their occurrence iscomplicated by factors such as flow in both directions, channel geometryinfluencing each layer differently, shear influencing the long wave speed andmixing between the layers. Although waves can form on both the free surfaceand on the interface between the layers, here we are solely concerned withinterfacial (internal) waves and instabilities. Similar to single layer flows aninternal control occurs at a transition from subcritical to supercritical flow.In subcritical flow, internal waves, including instabilities, may propagatein both directions. In supercritical flow they can propagate in only onedirection. In the present study we focus on maximal exchange flows, whichare characterized by a control at each end of the channel [Gu & Lawrence,2005]. The flow is subcritical within the channel and supercritical outsideof it. In the supercritical regions just outside of the channel, waves canonly propagate away from the channel, i.e. waves from the reservoirs cannotenter the channel.2.2.2 Dispersion Relation and InstabilityTo investigate the dynamics of instabilities, Holmboe [1962] analyzed thepiecewise linear velocity and density profiles shown in figure 2.2. The sharpdensity interface within a uniform shear layer approximates conditions observed in salt stratified shear flows at laboratory scales; sample profiles fromthe present experiments are shown in figure 2.2. Holmboe’s analysis assumesthe density interface is centred within the shear layer and does not accountfor the influence of the boundaries. The key parameters in the stabilityanalysis are the velocity difference between the layers, U = U1 — U2, the21Chapter 2. Holmboe Instabilities in a Laboratory Exchange FlowFigure 2.2: Definition sketch for piecewise linear proffles used in the analysisof Holmboe instabilities. Also shown are sample density and velocity profilesfrom the current study.shear layer thickness, h = U/(dU/dz)max and the reduced gravitationalacceleration, g’ = gIp/po (where zp is the density difference andP0is theaverage density). The subscripts 1 and 2 indicate the upper and lower layer,respectively. To characterize the total shear across the interface, U1 and U2are defined as the maximum or free stream velocity in each layer.The shear and stratification parameters are combined to form the bulkRichardson number, J = g’h/zU2.Following Holmboe’s analysis, Lawrenceet al. [1991] used the Taylor-Goldstein (TG) equation to relate the complexphase speed (c = c.,. + ic) to wave number (cr) and J:2—al±/al—4a2c= 2(2.1)whereai =/3+/3_—ri2,a2=n2/3, i3±=[e±(1—o)1/o,n2=2J/crU222Chapter 2. Holmboe Instabilities in a Laboratory Exchange FlowC.)-oa)a)0.0a)0a)0C.)a)a)00Figure 2.3: (a) Dispersion relation for the Holmboe flow configuration atJ 0.3. The labels H, v, and g indicate line segments associated with Hoimboe (unstable), vorticity, and gravity modes respectively. (b) Exponentialgrowth rate (ccj) of the Holmboe mode. The wavenumber of maximumgrowth (c = 1.9) corresponds to a dimensional wavelength A 7 cm forthe present experiments. The phase speed is shown nondimensionalized byU/2; the wave number is nondimensionalized by the shear layer thickness,h; and the growth rate is nondimensionalized by 2h/iU.For brevity, all of the terms in (2.1) are non-dimensional and the phase speedis relative to the mean of the free stream velocities, U (U1 + U2)/2. Thedimensional phase speed c = c zU/2 + U and the dimensional wavelengthA = 2Trh/o.The dispersion relation (2.1) is plotted in figure 2.3 for J = 0.3 corresponding to conditions in our laboratory experiments (Table 2.1). At highwavenumber the flow supports two stable gravity modes and two stablevorticity modes (so called because wave propagation is governed by buoyancy in the first instance and by the vorticity gradient in the second). The0.5 1 1.5Wave number, c2 2.5 323Chapter 2. Holmboe Instabilities in a Laboratory Exchange FlowH (cm) L (cm) W (cm) g’ (cm s2) h (cm) LU (cm s’) J10.8 200 10 1.39 2.1 3.1 0.3Table 2.1: Experimental parameters.rightward propagating vorticity mode is associated with the upper vorticityinterface (upper kink in the velocity profile) and the leftward propagatingvorticity mode is associated with the lower vorticity interface. As wavenumher decreases the vorticity mode and gravity mode propagating in the samedirection merge into one unstable mode (a 2.6). At lower wavenumber(a 0.7) the dispersion relation bifurcates back to four stable modes. Unlike the Kelvin-Helmholtz instability, the unstable mode particular to theHolmboe configuration is non-stationary.When J = 0.3 the maximum growth rate of the Holmboe instability,acj 0.28, occurs at a = 1.9 (figure 2.3b) with a corresponding phasespeed Cr = ±0.53. Dimensionalizing by h = 2.1 cm and U/2 = 1.56cm s1, as observed in our experiments, yields a maximum growth rate ata wavenumber k 2ir/A 0.9 cm1 (A = 7 cm) and a phase speed ofc = U ± 0.83 cm s1. The dimensional growth rate kcj = 0.2 s1 resultsin a doubling time of 3.5 s. In the following sections we will compare theobserved wave characteristics with predictions from (2.1), particularly at thewavenumber of maximum growth.2.3 Experimental SetupA schematic of the laboratory setup is shown in figure 2.4. The overalltank was 370 cm long, 106 cm wide and 30 cm deep as in Zhu Lawrence[2001]. The tank was divided into two equally sized reservoirs and connectedby a channel 10 cm wide and 200 cm long. The water was well mixedbetween the reservoirs to ensure uniform temperature(20°C) throughoutthe tank. A removable gate was placed in the middle of the channel isolatingthe left and right side. Salt was mixed into the right reservoir to provide adensity difference of 1.41 kg m3 (g’ of 1.39 cms2). The basic experimentalparameters are provided in Table 2.1.The experiments differ from those of Zhu & Lawrence [2001] in that thewater depth was kept relatively shallow (H=10.8 cm) and there was no sillin the channel (fiat bottom). These changes resulted in more gradual horizontal variations in U and J, and therefore more uniform wave properties.24Chapter 2. Holmboe Instabilities in a Laboratory Exchange FlowI0.108 mI________________________________________________Figure 2.4: (a) Plan and (b) side view of the experimental setup and (c)wave characteristics plot. The left and right reservoirs initially contain freshand saline water respectively. The side view (b) includes a sample imageof the interface over the entire length of the channel plus a portion of eachreservoir at t = 400 s. The lower layer contains dye and is illuminated fromabove with a laser generated light sheet. The characteristics (c) represent acompilation of interface heights observed in several thousand images. Theshading is scaled such that black indicates the interface is near the bottomof the channel and white indicates the interface is near the free surface.Diagonal light and dark streaks represent interfacial waves.3.7 mI1O6roI-(a)Left reservoir Right reservoir‘1p2I L=2.OmL- —... ——1 —0.5 0 0.5x (m)25Chapter 2. Holmboe Instabilities in a Laboratory Exchange FlowExperiment Measurement Location Replicates Parameter1-7 LIF —0.9 <x <0.9 78-12 PIV x = 0 5 h, U1, U213 LIF —1.2 <x < 1.2 114 PIV and LIF x = 0 1 h, U1, U215 Particle streak x = —0.8 116 Bottle samples x = —1,0, 1 1Table 2.2: Summary of laboratory experiments.In addition, the shallower depth resulted in a prolonged period of maximal exchange. Experiments using yet shallower depths or smaller densitydifferences were attempted, but resulted in the suppression of instabilities(presumably due to viscous effects). Larger density differences were alsoused, but were subject to a number of problems including shortened experiment duration, large changes in the index of refraction at the densityinterface and diminished image quality (due to the higher shutter speedsrequired to capture faster waves). To gather the data used in this study theexperiment was repeated 16 times with depth and p held constant (seeTable 2.2).Laser induced fluorescence (LIF) was used to visualize the density interface by illuminating fluorescein dye in the lower layer with a continuous 4watt argon ion laser. The laser beam was passed through a Powell lens togenerate a downward radiating light sheet along the centre of the channel.Images were collected using a digital camera; a sample image is shown infigure 2.4 b. The interface was identified by locating the maximum verticalgradient in light intensity.A Dantec particle image velocimetry (PIV) system was used to measurethe velocity of pliolite VT-L particles (Goodyear Chemical Co.). The particles were pulverized and sieved to diameters less than 0.24 mm. Althoughsome particles did settle out there was sufficient quantity to perform PIVthroughout the duration of the experiment. Pairs of images (t = 0.04s)were collected every 3 seconds. A 3-step adaptive correlation algorithm wasused to calculate velocities. The final interrogation areas were 32 pixelswide by 16 pixels high (2.8 mm x 1.4 mm) with a 50 % overlap resulting ina 0.7 mm vertical spacing of vectors. The total image size was 11 cm wideby 9 cm high. Small scratches on the acrylic wall limited PIV to a singlelocation (x = 0). The Dantec system was also used to determine density byquantitative measurement of dye fluorescence.26Chapter 2. Holmboe Instabilities in a Laboratory Exchange FlowIn experiment 13, LIF was performed over the entire viewable regionof the tank (—1 .2m< x < 1 .2m). This experiment allowed us to includethe critical and supercritical regions of the flow in our qualitative description of the wave characteristics. In experiment 14 both PIV and LIF wereperformed simultaneously to confirm that the density interface was muchthinner than the shear layer (figure 2.2). Particle streak images (figure 2.1)were collected in experiment 15 to examine the structure of individual instabilities. Finally, fluid samples were collected with a syringe and analyzedin a densitometer to verify LIp between the layers.2.4 Evolution of Mean FlowThe experiment begins when the gate separating the fresh and salt water atthe centre of the channel is removed. Two gravity currents immediately formand propagate in opposite directions; these gravity currents exit the channelat t 60 s (figure 2.4c). The gravity currents generate mixed fluid which isgradually flushed out of the channel leaving two uniform layers separated byan interface approximately 2 mm thick (t = 200 s). In addition, a Helmholtzoscillation [Miles & Munk, 1961] is generated when the gate is removed andremains noticeable in velocity measurements until t 400 s (figure 2.5)This oscillation has a period of 28 s and is reflected in the measured phasespeed of the instabilities.Once the Helmholtz oscillations dampen out (t = 300 to 400s), the flowenters a long period of relatively steady maximal exchange, which ends whenthe control at the right end of the channel is flooded and the flow becomessubcritical in the right reservoir. After the control is lost, disturbances canenter the channel from the reservoir (not visible in figure 2.5). In the presentstudy we focus on instabilities generated within the channel during the longperiod of maximal exchange when conditions are steady (400 s < t < 800s).During this period interfacial wave properties remain relatively constant.For the steady period the mean interface height shows a gradual slope(0.02) throughout most of the channel (figure 2.6a); a steeper slope occursat the ends of the channel consistent with the presence of controls. Theobserved interface height compares well with the analytical predictions ofGu & Lawrence [2005] using their c = 0.41 and r = 1. The time averagedvelocity profile observed at x = 0 is shown in figure 2.6a. At this point theaverage flow speed in each layer is 1.1 cm s1. This is just over half the flowspeed predicted by the inviscid two-layer theory /11’/2 =1.9 cm s’).The observed velocity profile can be approximated by a piecewise linear27Chapter 2. Holmboe Instabilities in a Laboratory Exchange Flow0)8C.)>C)0800Figure 2.5: Phase velocity of rightward (positive) and leftward (negative)propagating waves at x = 0. The phase velocities were calculated usingthe cross correlation of the interface between successive images. A low passfilter (removing periods < 20 s) was applied to remove variability due toindividual waves. The horizontal lines are the phase speeds predicted usingthe linear theory.fit with free-stream (maximum) velocities in the upper and lower layers ofU1 = 1.55 cm s1 and U2 —1.57 cm s1, respectively. The subscript cdenotes the centre of the channel (x = 0). Because of the bottom boundarylayer the lower layer has a slightly greater maximum velocity than the upperlayer resulting in U -0.01 cm s1. The velocity profile has a shear layerthickness h = 2.1 cm.To understand the evolution of the waves discussed in the next section itis useful to estimate the mean, U(x)= Ui(x)--U2(x)along the entire lengthof the channel. We assume that Ui (x) and U2 (x) can be estimated from thevelocities observed at the centre of the channel using: U (x) = (x),where the layer thicknesses, y, are based on the observed interface height(figure 2.6a). The resulting velocity estimates are plotted in figure 2.6b andwill be used below to describe the evolution of the waves.0 100 200 300 400 500 600 700lime (s)28Chapter 2. Hohnboe Instabilities in a Laboratory Exchange FlowE020.0.0.0C)320-1—20.025r0.020.0150.010.00500.30.2-)0.10—1 —0.8 —0.6 —0.4 —0.2 0 0.2 0.4 0.6 0.8 1x(m)Figure 2.6: (a) Mean interface height along the channel during the period ofsteady exchange. The interface height predicted by two layer hydraulics (seetext) is shown as a dashed line. Also shown is the average velocity profileobserved at x = 0 and the piecewise linear profile used in the stabilityanalysis. The maximum speeds in the upper and lower layers at x = 0 areU1 = 1.55 cm s and U2 = —1.57 cm s respectively. (b) Estimates of thefree stream (maximum) velocities, U1 (x) and U2 (x), the total shear, U(x),and the mean velocity, U(x). (c) Horizontal gradient in the mean velocity,8U(z)/&r. (d) Bulk Richardson number. The vertical dotted lines showthelocations of the channel ends.I I I(c)•1(d)I [ — I I I — I I I I29Chapter 2. Holmboe Instabilities in a Laboratory Exchange Flow2.5 Wave EvolutionThe two waves in figure 2.1 exhibit the classic features of fuiiy developedHolmboe instabilities. The upward pointing (positive) cusp is moving tothe right with the upper layer and the downward pointing (negative) cuspis moving to the left with the lower layer. The positive cusp is ejecting awisp of interfacial fluid into the upper layer. The particle streaks indicatean elliptical vortex leading the positive cusp. The centre of the vortex hasnearly stationary particles and is well above the density interface. Suchvortices are typically present in numerical simulations of these flows [e.g.Smyth Winters, 2003] and play an important role in the generation ofthe wisps. The vortex carries partially mixed interfacial fluid back towardthe cusp where there is a horizontal convergence. The convergence at thecusp carries the fluid vertically away from the interface. These wisps ofmixed fluid can either get caught in the leading vortex or, in some cases, areejected above the vortex into a region of decreased shear and higher velocity.A similar vortex leads the lower cusp. Its presence is masked by the dye inthe lower layer.During the steady period (400-800s) there is a roughly even distributionof rightward and leftward propagating waves as can be seen in the characteristics diagram figure 2.7b. The characteristics represent a compilation ofthe interface height observed in a sequence of several thousand images (e.g.the image in figure 2.4b). The time averaged interface height (figure 2.6a)was removed and the shading is scaled such that black indicates the troughof an instability and white indicates the crest. White and black diagonallines represent the propagating cusps of positive and negative instabilities,respectively. In general the instabilities form quickly (<20 s) and maintaina nearly constant amplitude while they are within the channel. Despite irregularities in the characteristics the instabilities can be filtered into distinctrightward (figure 2.7a) and leftward (figure 2.7c) propagating componentsusing the two dimensional fast Fourier transform (FFT).The influence of the controls can be seen in the characteristics at theends of the channel (x = ±1 m, figure 2.7a and c). Within the channel,disturbances move in both directions (subcritical) and beyond the ends ofthe channel they only move outwards into the reservoirs (supercritical). Although difficult to see, both upward and downward cusping modes are propagating outwards in the supercritical regions (e.g. figure 2.7c, x = —1.05 m,t = 625 — 650 s). As expected the controls block disturbances from enteringthe channel, i.e. waves propagating within the channel have formed thererather than within the reservoirs. Because one of the two Holmboe modes30Chapter 2. Holmboe Instabilities in a Laboratory Exchange FlowFigure 2.7: Characteristics during the period of steady exchange. The shading indicates the deviation of the interface elevation from the mean. Purewhite (black) indicates a positive (negative) deviation greater than 3 mm.The characteristics in (b) were split into rightward (a) and leftward (c)propagating components using the two dimensional FFT. The ends of thechannel are at x = ±1 m.(a) (b) (c)00)0)6000I—1 0 1 —1 0(m)1 —1 0 131Chapter 2. Holmboe Instabilities in a Laboratory Exchange Flowis stationary near each control, the separation of the modes using the twodimensional FFT is less effective near the ends of the channel. In addition,the nearly stationary waves near the ends of the channel have a very lowfrequency resulting in very few waves per experiment and therefore greateruncertainty in quantifying wave properties. For these reasons our analysisbelow will focus on —0.9 m< x < 0.9 m.To further illustrate the wave evolution we have traced out the crestsof a set of the positive, rightward propagating waves (figure 2.8a). At theleft end of the channel four wave crests pass x = —0.9 m over a period ofapproximately 110 s indicating a wave period of 37 s (frequency, w = 0.027Hz). At x = +0.9 m, there is still 110 s between the first and last wavecrest, however, here there are 13 wave crests in total indicating an averagewave period of 9.2 s (w 0.11 Hz). This increase in frequency is a result ofnew waves forming throughout the channel.The frequency evolution is quantified by counting all of the zero crossingsthat occur over the period of steady exchange (400 seconds) and averagingover seven experiments (Experiments 1 to 7). The characteristics were lowpass filtered (wavelengths > 1 cm) before counting the zero crossings tominimize the upward bias associated with noise. As in the traces, the zerocrossings show an increase in the number of positive waves from left toright (figure 2.8b). The negative waves show the same accumulation in theopposite direction.The formation of new waves is related to the changes in phase velocitythat the waves undergo as they propagate along the channel. The phasevelocity of a wave is given by the inverse of the slope of its characteristic(dx/dt). A nearly vertical line indicates a slow moving wave while a nearlyhorizontal line indicates a fast moving wave. The positive (rightward propagating) waves shown in figure 2.8a therefore accelerate from left to right(the traced lines become more horizontal). The dominant phase velocity ofthe waves (i.e. the slope of the wave characteristics) is determined by crosscorrelating time series of the interface height at adjacent locations alongthe channel. This phase velocity is calculated for experiments 1 to 7 andthen averaged (figure 2.8c). The average shows that both the positive andnegative waves accelerate as they propagate along the channel.The wave acceleration is most easily understood by considering the vanation in the mean velocity, U, over the length of the channel (figure 2.6b).The mean velocity is replotted in figure 2.8c and shows a slope that issimilar to the slope of the observed phase speeds. In other words, with respect to a frame of reference moving at the mean velocity, the velocity ofboth the rightward and leftward propagating waves remains approximately32Chapter 2. Holmboe Instabilities in a Laboratory Exchange Flow0EII00a)C)Ea 200)C. 10—1 —0.5 0 0.5 1x (m)Figure 2.8: (a) Characteristics of rightward propagating waves; gray shading indicates a wave trough and white indicates a crest. The lines weretraced by hand following individual wave crests. (b) The average frequencyof the rightward(w+)and leftward (w_) propagating waves based on zerocrossings. (c) The thick solid lines represent the phase velocity of the observed rightward (positive) propagating waves, (c) and leftward (negative)propagating waves, (cr). The thin line shows U calculated using the velocity profile and interface height shown in figure 5. The dashed lines are thepredicted phase speed of Holmboe instabilities. (d) The distribution of thewavelength for the rightward propagating waves with the average plotted asa heavy line and the 10 and 90 percentiles as thin lines.33Chapter 2. Holmboe Instabilities in a Laboratory Exchange Flowconstant. By adding U to the predicted phase speed for the Holmboe instability (equation 2.1) we predict the phase speed over the whole channel.This phase speed is shown for both the rightward and leftward propagatingwaves, c(x) = U(x) ± 0.83 cm s1, in figure 2.8c and matches closely theobserved phase speed evolution.This prediction of the phase speed (figure 2.8c) does not take into accountpossible changes in J along the length of the channel. However, over thecentral region of the channel, the variation in J is too small (figure 2.6d)to have a noticeable effect on the wavenumber of maximum growth and thecorresponding phase speed.The distribution of wavelength with respect to position is shown in figure 2.8d for rightward propagating waves. The average wavelength andposition of all the waves was determined using zero crossings (similar to thefrequency in figure 2.8b). The wavelength remains nearly constant (A 10cm) throughout x. This is because the two processes, wave formation andwave acceleration, tend to cancel each other out.Acting by itself, the increase in frequency associated with wave formationwill shorten the average wavelength, AB = AA, where the subscripts A andB represent different locations in x. The effect of the convective acceleration(UU/Ox) on the wavelength is not so obvious. As is commonly observed insurface waves [e.g. Peregrine,19761,the acceleration will stretch the waves,increasing their wavelength, i.e. AA = -AB.The wave stretching is most apparent in the temporal evolution of thewavenumber spectrum (figure 2.9 a and b). Similar to the characteristics, thespectral evolution was determined by compiling the spectrum of the interfaceheight at each time and then contouring. Note that the horizontal axis infigure 2.9 is wavenumber not distance. The dark (high energy) diagonalstreaks represent energy associated with waves moving through the channelin time (the vertical axis). The slope of the streaks is a result of individualwaves stretching, i.e. waves are continuously decreasing in wavenumber(increasing in wavelength).The time averaged spectra (figure 2.9 c and d) show the peaks in the waveenergy occurring at approximately 0.5 cm1 (A = 12.5 cm). The temporalevolution and average spectrum together show the waves form near the wavenumber of maximum growth (k 0.9 cm1,A 7 cm) get stretched andstart to lose energy near the lower stability boundary (k 0.36 cm1,A17 cm).The two processes, wave stretching and wave formation, are illustratedin the simplified schematic shown in figure 2.10. The schematic shows theinterface elevation at three times. The reference frame (x = 0 in the figure)34Chapter 2. Holmboe Instabilities in a Laboratory Exchange Flow0Figure 2.9: Spectrum of waves during the steady period between x = —0.8in and x = 0.8 m. (a) and (b) Temporal evolution of the spectrum forthe rightward and leftward propagating waves respectively. The shadingis scaled logarithmically from white (low energy) to black (high energy).(c) and (d) Time averaged spectrum. The heavy vertical line indicates thewavenumber of maximum growth (k 0.9 cm1,A 7 cm) and the thinvertical lines indicate the stability boundaries (k 0.36 cm1,A 17 cmandk1.3cm’,A5cm).is moving at the speed of the trailing wave crest. As shown in figure 2.8cwaves undergo the same convective acceleration as U(x). In the centralregion of the channel this acceleration is approximately 0.005 s_i (see 8U/Dxin figure 2.6 c). As the pair of crests propagate through the channel thehorizontal variation in U gives the leading crest a slightly greater phasespeed than the trailing crest (A 8U/Ox 0.035 cm srn’). This difference inphase speed allows the leading crest to pull away from the trailing crest. As(c) (d)101EE100Co0.5 1 1.5 0 0.5 1 1.5Wave number (cm1) Wave number (cm1)35Chapter 2. Holmboe Instabilities in a Laboratory Exchange Flow140a)700Distance (cm)Figure 2.10: Schematic of stretching and formation of rightward propagating waves. The interface elevation is shown at three times tracking the samewave. The horizontal distance at each time is relative to the trailing crestof the wave. The wave is shown initially with a wavelength equal to thewavelength of maximum growth. Eventually the wave is stretched to twicethis length. At the same time new waves form, also at the wavelength ofmaximum growth. The resulting interface has waves of mixed amplitudeand wavelength. The time scale shown for doubling of the wavelength (approximately 140 s) is based on the observations (see figure 2.9a and b).the spacing between the two crests.)increases, their growth rate decreases(i.e. they are no longer at the wavelength of maximum growth). On theother hand, as the spacing increases the interface between the crests becomesunstable to shorter waves (i.e. waves that are closer to the wavenumber ofmaximum growth) and a new wave forms. The new waves grow and stretchand eventually the process repeats itself (see figure 2.8a).2.6 Summary and ConclusionsInstabilities were investigated using an exchange flow through a long rectangular channel with a rectangular cross section. The channel connectedtwo large fresh water and salt water reservoirs. A long period of steadymaximal exchange occurred after the cessation of Helmholtz resonance andended when one of the controls was flooded. During this time symmetricHolmboe instabilities were observed. These instabilities evolved into cuspswith a leading elliptical vortex. The vortices drew mixed fluid from the cuspinto the free stream. The observed density interface was sharper than, andcentred within, the shear layer.The gradual slope of the interface along the length of the channel re0 7 1436Chapter 2. Holmboe Instabilities in a Laboratory Exchange Flowsuited in the convective acceleration of each layer. The Hoimboe instabilitiesalso accelerated as they propagated through the channel. This accelerationcaused the distance between successive cusps to increase and new wavesformed. The new waves formed uniformly along the channel such that theaverage wavelength remained nearly constant.By focusing on the central section of the channel we selected the regionwhere the Bulk Richardson number is relatively constant. This, togetherwith the prolonged period of steady exchange and simple channel geometry,resulted in instabilities that had average wave properties that were in goodagreement with the linear predictions of Holmboe.37BibliographyALEXAKIS, A. 2005 On Holmboe’s instability for smooth shear and densityprofiles. Phys. Fluids 17, 084103.ARMI, L. 1986 The hydraulics of two flowing layers with different densities.J. Fluid Mech. 163, 27—58.CARPENTER, J.R., LAWRENCE, G.A. & SMYTH, W.D. 2007 Evolutionand mixing of asymmetric Holmboe instabilities. J. Fluid Mech. 582, 103—132.FARMER, D.M. & ARMI, L. 1998 The flow of Atlantic water through theStrait of Gibraltar. Frog. Oceanogr. 21, 1—98.GEYER, W.R. & SMITH, J.D. 1987 Shear instability in a highly stratifiedestuary. J. Phys. Oceanogr. 17, 1668—1679.Gu, L. & LAWRENCE, G. 2005 Analytical solution for maximal frictionaltwo-layer exchange flow. J. Fluid Mech. 543, 1—17.HOGG, A. MCC. & IVEY, G.N. 2003 The Kelvin-Helmholtz to Holmboeinstability transition in stratified exchange flows. J. Fluid Mech. 477, 339—362.HOLMBOE, J. 1962 On the behavior of symmetric waves in stratified shearlayers. Geofys. Pubi. 24, 67—112.Koop, C. G. & BROWAND, F.K. 1979 Instability and turbulence in astratified fluid with shear. J. Fluid Mech. 93, 135—159.LAWRENCE, G.A., BROWAND, F.K. & REDEKOPP, L.G. 1991 The stability of a sheared density interface. Phys. Fluids 3 (10), 2360—2370.MILES, J. & MUNK, W. 1961 Harbor paradox. J. Wat Ways Harb. Am.Soc. Civ. Engrs WW3 87, 111—130.PEREGRINE, D. H. 1976 Interaction of water waves and currents. Adv.Appl. Mech. 16, 9—117.38BibliographyPOULIQUEN, 0., CHOMAZ, J. M. & HUERRE, P. 1994 Propagating hoimboe waves at the interface between two immiscible fluids. J. Fluid Mech.266, 277—302.S MYTH, W. D. 2006 Secondary circulations in Holmboe waves. Phys. Fluids18 (064104), 1—13.SMYTH, W.D., CARPENTER, J.R. & LAWRENCE, G.A. 2007 Mixing insymmetric Holmboe waves. J. Phys. Oceanogr. 37, 1566—1583.SMYTH, W. D. & WINTERS, K. B. 2003 Turbulence and mixing in Hoimboe waves. J. Phys. Oceanogr. 33, 694—711.TEDFORD, E., CARPENTER, J., PAWLOWICZ, R. & LAWRENCE, G. 2007Linear stability analysis in a salt wedge. In Proceedings of the Fifth International Symposium on Environmental Hydraulics. Tempe, Arizona, USA.THORPE, 5. 1971 Experiments on instability of stratified shear flows: miscible fluids. J. Fluid Mech. 46, 299—319.WESSON, J. C. & GREGG, M. C. 1994 Mixing at Camarinal Sill in theStrait of Gibraltar. J. Geophys. Res. 99, 9847—9878.YOSHIDA, S., OHTANI, M., NISHIDA, S. & LINDEN, P.F. 1998 Mixingprocesses in a highly stratified river. In Physical Processes in Lakes andOceans, Coastal and Estuarine Studies, vol. 54, pp. 389—400. AmericanGeophysical Union.ZHU, D. & LAWRENCE, G.A. 2001 Holmboe’s instability in exchangeflows. J. Fluid Mech. 429, 391—409.39Chapter 3Observation and Analysis ofShear Instability inthe Fraser River Estuary23.1 IntroductionShear instabilities occur in highly stratified estuaries and can influence thelarge scale dynamics by redistributing mass and momentum. Specifically,shear instabilities have been found to influence salinity intrusion in theFraser River estuary [Geyer & Smith, 1987; Geyer & Farmer, 1989; MacDonald & Horner-Devine, 2008]. We describe recent observations in thisestuary and examine the shear and stratification that lead to instability.The influence of long time scale processes such as freshwater discharge andthe tidal cycle are also discussed.Rather than relying on a bulk or gradient Richardson number to assessstability we use numerical solutions of the Taylor-Goldstein (TG) equationbased on observed profiles of velocity and density. This approach has beenused with some success in the ocean [e.g. Moum et al., 2003] but, withthe exception of the simplified application by Yoshida et al. [1998], has notbeen applied in estuaries. Solving the TG equation provides the growthrate, wavelength, phase speed and mode shape of the instabilities. We compare these predicted wave properties with instabilities observed using anechosounder.Geyer & Farmer [1989] found that instabilities in the Fraser River estuarywere most apparent during ebb tide when strong shear occurred over thelength of the salinity intrusion. They outlined a progression of three phasesof increasingly unstable flow that occurs over the course of the ebb. Inthe first phase, strain sharpens the density interface; shear is stronger than2This chapter has been submitted for publication in: E.W. Tedford, J.R. Carpenter,R. Pawlowicz, R. Pieters and G.A. Lawrence (2009),Observation and Analysis of ShearInstability in the Fraser River Estuary, J. Geophys. Res.40Chapter 3. Shear Instability in the Fraser River EstuaryFigure 3.1: Map of the lower 27 km of the Fraser River. The locations ofthe six transects are marked T1-T6. The mouth of the river (Sand Heads)is located at 49° 6’ N and 123° 18’ W.during flood but insufficient to cause shear instability. In the second phase,the lower layer reverses direction causing shear between the fresh and salinelayers to increase. Shear instability and turbulent mixing are concentratedat the pycnocline rather than in the bottom boundary layer. By the thirdphase of the ebb, shear instability has completely mixed the two layersleaving homogeneous water throughout the depth. During flood there issome mixing, however it is concentrated at the front located at the landwardtip of the salinity intrusion. Similarly, MacDonald & Horner-Devine[20081,studying mixing at high fresh water discharge (7000m3s1),found that twoto three times more mixing occurred during ebb tide than during flood.The present analysis is focused on the ebb tide at high and low freshwaterdischarge, although some results during flood tide are also presented.The paper is organized as follows. The setting and field methods aredescribed in section 3.2. The general structure of the salinity intrusion isdescribed in section 3.3. In section 3.4 we present the background theoryneeded to perform stability analysis in the Fraser River estuary. In section3.5 predictions from the stability analysis are compared with observations.In section 3.6 the source of relatively small scale overturning, is briefly discussed. In section 3.7 the results of the stability analysis are discussedfollowed by conclusions in section 3.8.3.2 Site Description and Data CollectionData were collected in the main arm of the Fraser River estuary, BritishColumbia, Canada (figure 3.1). The estuary is 10 to 20 m deep with achannel width of 600 to 900 m. Cruises were conducted on June 12, 14 and41Chapter 3. Shear Instability in the Fraser River EstuaryDischarge Tide x LUiph J(m3 s’) (km) (m s1) (kg m3) (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 3.1: Details of transects shown in figures 3.1 and 3.2. The locationindicates the distance upstream from the mouth (Sand Heads).21, 2006 and March 10, 2008. Here we present one transect from each ofthe June 2006 cruises and three transects from the March 2008 cruise (seeTable 3.1). The freshwater discharge during the June 2006 transects wastypical of the freshet at approximately 6000 m3s1.During the March 2008transects, freshwater discharge was near the annual minimum at 850 m3s1.In June 2006, transects were made during both ebb and flood tide. InMarch 2008, transects cover most of a single ebb tide (figure 3.2). The tidesin the Strait of Georgia have M2 and Ki components of similar amplitude(approximately 1 m) resulting in strong diurnal variations. The tidal rangevaries from approximately 2 m during neap tides to approximately 4.5 mduring spring tides. During both the 2006 and 2008 observations the tidalrange was approximately 3 m.The distance salinity intrudes landward of Sand Heads, i.e. the totallength of the salinity intrusion, varies considerably with tidal conditionsand freshwater discharge. Ward [1976], found the maximum length of theintrusion occurred just after high tide and varied from 8 km at high discharge(9000 m3s’) to 31 km at low discharge (850 m3s1). Geyer & Farmer [1989]found that, at average discharge (3000 m3s1), the maximum length of theintrusion matched the horizontal excursion of the tides (10 to 20 km) and,similar to Ward [1976], occurred just after high tide. Kostachuk Atwood[1990] found that the minimum length of the salinity intrusion typically occurred approximately one hour after low tide. The longest intrusion theyobserved at low tide was approximately 20 km. They predicted that complete flushing of salt from the estuary would occur on most days during thefreshet (freshwater discharge> 5000 m3s’).42EC)G)z20)zE0)a)z20)a)zChapter 3. Shear Instability in the Fraser River EstuaryFigure 3.2: Observed tides at Point Atkinson (heavy line) and New Westminster (thin line) for the four days of field observations. The Point Atkinsondata is representative of the tides in the Strait of 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 3.1). The records areboth referenced to mean sea level at Point Atkinson. The duration of thesix transects are marked T1-T6.Field MethodsData along the six transects were collected by drifting seaward with thesurface flow while logging velocity and echosounder data and yoyoing a CTD(conductivity, temperature and depth) profiler. The velocity measurementswere made with a 1200 kHz RDI Acoustic Doppler Current Profiler (ADCP)sampling at 0.4 Hz with a vertical resolution of 250 mm. The velocitieswere averaged over 60 seconds to remove high frequency variability. Theecho soundings were made with a 200 kHz Biosonics sounder sampling at 56 8 10 12 14 16 18Local Time (hours, PDT)43Chapter 3. Shear Instability in the .&aser River EstuaryHz with a vertical resolution of 18 mm. Profile data was collected with aSeabird 19 sampling at 2 Hz. Selected echosounder, ADCP and CTD dataare shown in figure 3.3. As indicated by the superimposed density profiles,strong gradients in density are generally associated with a strong echo fromthe sounder.The CTD was profiled on a load bearing data cable that provided constant monitoring of conductivity, temperature and depth. This data allowedus to quickly identify the front of the salinity intrusion and avoid direct contact of the instrument with the bottom. To increase the vertical resolutionof the profiles, the CTD was mounted horizontally with a fin to direct thesensors into the flow. In this configuration, the instrument was allowed todescend rapidly and then was raised slowly (0.2 - 0.4 m s’) relying on horizontal velocity of the water relative to the CTD to flush the sensors. Theupcast, which had higher vertical resolution, was in reasonable agreementwith the echo intensity from the sounder. On the few occasions that thehigher resolution upcast did not coincide with the appearance of instabilities in the echosounder, we used the downcast.3.3 General Description of the Salinity IntrusionWe observed important differences in the structure of the salinity intrusionbetween high and low freshwater discharge. At high discharge, our observations were similar to those described by Geyer Farmer [1989] at averagedischarge (3000 m3 s’), where the salinity intrusion had a two-layer structure resembling a classic salt-wedge. At low discharge, however, the salinityintrusion exhibited greater complexity.3.3.1 High DischargeDuring flood tide, mixing was concentrated near the steep front at the land-ward tip of the salt-wedge (2.7 to 3.03 km in figure 3.3c). During ebb tide,the steep front was replaced by a gently sloping pycnocline (figure 3.3b land-ward of 11.6 km) and there was no apparent concentration of mixing at thelandward tip of the salt-wedge (not shown).We will focus on the wave-like disturbances that occur on the pycnodine especially during ebb tide. The largest of these were observed duringtransect 1 (figure 3.3a 8.7 to 8.9 km, between depths of 3 and 9 m). Thesedisturbances occurred within the upper layer as it passed over the nearlystationary water below a depth of 10 m. Smaller amplitude disturbances44Chapter 3. Shear Instability in the Fraser River EstuaryEa,Figure 3.3: Echo soundings observed during high discharge on: (a) transect1, ebb tide; (b) transect 2, ebb tide; (c) transect 3, flood tide. The shadingscales with the log of the echo intensity with black corresponding to thestrongest echos. Selected velocity profiles (red) from the ADCP and densityprofiles (blue dash) from the CTD are superimposed (not all are shown).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 o. The velocity profile wascalculated as a 1 minute average centred on the time of the CTD cast. Theundulations in the bed of the river (thick black line at the bottom of theechosoundings) are a result of sandwaves.2.3 2.4 2.5 2.6 2.7 2.8 2.9Distance (km)45Chapter 3. Shear Instability in the Fraser River Estuarywere observed during transect 2 (figure 3.3b 11.05 km). In our applicationof the TG equation we will show that disturbances like these are a result ofshear instability.Not all of the disturbances on the pycnocline are a result of shear instability. For example, for most of the velocity and density profiles collectedduring transect 3 (figure 3.3c) the TG equation does not predict instability.The disturbances seen from 2.5 to 2.8 km are caused by the large sand waveson the bottom (the thick black line in the echo sounding). The crests of thesand waves were typically 30 m apart and 1 to 2 m high, and were foundover most of the river surveyed during high discharge (2.5 km to 15 km).During flood tide, flow over these sand waves caused particularly regulardisturbances on the pycnocline.3.3.2 Low DischargeAt low discharge, at the beginning of the ebb, the front of the salinityintrusion was located between 28 and 30 km from Sand Heads. Unlikethe observations at high discharge a well defined front was not visible inthe echosounder, and CTD profiles were needed to identify its location.Seaward of the front (figure 3.4a), the echosounder and the CTD profilesshow a multilayered structure with more complexity than was observed athigh discharge. At this early stage of the ebb, the CTD profiles generallyshow partially mixed layers separated by several small density interfaces.Later in the ebb, during transect 5 (figure 3.4b), near bottom velocities turn seaward and the velocity shear between the top and the bottomincreases. At maximum ebb (transect 6, figure 3.4c), the shear increasesfurther, reaching a maximum of approximately 2.5 m s1 over a depth of12 m. Mixed water occurs at both the surface and the bottom resulting inan overall decrease in the vertical density gradient. By the time transect 6is complete the ebb flow is decelerating. The salinity intrusion continues topropagate seaward until low tide but, given its length and velocity it doesnot have sufficient time to be completely flushed from the estuary. Duringthe next flood the mixed water remaining in the estuary allows a complexdensity structure to develop similar to that seen early in the observed ebb.This differs from the behaviour at high freshwater discharge when nearly allof the seawater is flushed completely from the estuary at least once a day.46Chapter 3. Shear Instability in the Fraser River Estuary11Figure 3.4: Echo soundings during low discharge observed during: (a) transect 4, early ebb; (b) transect 5, mid ebb; and (c) transect 6, late ebb. Theshading scales with the log of the echo intensity with black correspondingto the strongest echos. Note that the scale of the shading is the same inall three panels. Velocities(red) from the ADCP and densities(blue dashed)from the CTD are superimposed. The black line indicates the location of theboat in the middle of the cast, as well as the zero reference for the velocityando.The velocity profile was calculated as a 1 minute average centredon the time of the CTD cast.17.5 18 18.5 19 19.5 20 20.5(c) Transect 67.5 8 8.5 9 9.5 10 10.5Distance (km)47Chapter 3. Shear Instability in the Fraser River Estuary3.4 Stability of Stratified Shear Flows3.4.1 Taylor-Goldstein EquationFollowing Taylor [1931] and Goldstein [1931] we assess the stability of theflow by considering the evolution of perturbations on the background profilesof density and horizontal velocity, denoted here by p(z) and U(z), respectively. If the perturbations to the background state are sufficiently smallthey are well approximated by the linear equations of motion. It then sufficesto consider sinusoidal perturbations, represented by the normal mode formec(a),where x is the horizontal position and t is time. Here k = 27r/Ais the horizontal wave number with A the wavelength, c= Cr + Cj is thecomplex phase speed. If we further assume that the flow is incompressible,Boussinesq, inviscid, and non-diffusive, we arrive at the Taylor-Goldstein(TG) equation+N —d2U/dz— k21— 0 3 1dz2L(U_c)2U-c, (.)where the stream function is given by &(x, z, t)= (z)e(x_ct)and N2(z) =(g/po)(dp/dz) represents the Boussinesq form of the squared buoyancy frequency with a reference density,P0.Solutions to the TG equation consist of eigenfunction-eigenvalue sets{‘(z), c}, for each value of k. Each set {(z), c} is referred to as a mode, andthe solution may consist of the sum of many such modes for a single k. Thebackground flow, represented by U(z) and p(z), is then said to be unstableif any modes exist that have Cj 0. In this case the small perturbationsgrow exponentially at a rate given by kcj. In general, unstable modes arefound over a range of k, and it is the mode with the largest growth rate thatis likely to be observed. Although they are based on linear analysis, TGpredictions of the wave properties, k and c, typically match those of finiteamplitude instabilities observed in the laboratory [Thorpe, 1973; Lawrenceet al., 1991, and Chapter 2].3.4.2 Miles-Howard CriterionA useful criterion to assess the stability of a given flow without solving theTG equation was derived by Miles [1961] and Howard [1961]. They foundthat if the gradient Richardson number, Ri(z) =N2/(dU/dz),exceeds 1/4everywhere in the profile, then the TG equation has no unstable modes, i.e.c must be zero for all modes. In other words, Ri > 1/4 everywhere is a48Chapter 3. Shear Instability in the Fraser River Estuarysufficient condition for stability, referred to as the Miles-Howard criterion.Note that if Ri < 1/4 at some location, instability is possible, but notguaranteed.Despite the inconclusive nature of the Miles-Howard criterion for determining instability, it is often employed as a sufficient condition for instabilityin density stratified flows, and has been found to have reasonable agreementwith observations [Thorpe, 2005,p. 201-2041.Looking specifically at theFraser River estuary, Geyer c Smith [1987] were able to compute statisticsof Ri and show that decreases in Ri were accompanied by mixing in theestuary.3.4.3 Mixing Layer SolutionSince the TG equation is an eigenvalue problem with variable coefficients,analytical solutions can only 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 theestuary during high discharge. This solution is based on the simple mixinglayer model of Holmboe [described in Miles, 1963].In this model, the velocity and density profiles are represented by hyperbolic tangent functions,2z 2zU(z) = —i-- tanh(--)and p(z) = —-i— tanh(--)+ po. (3.2)In the simplest case the shear layer thickness, h, and the density interfacethickness, S are equal, giving R = h/S = 1. In this case, Ri(z) is at itsminimum at the center of the mixing layer (z = 0), and is equal to thebulk Richardson number J = gph/po(U)2.When the bulk Richardsonnumber (i.e. the minimum Ri) drops below 1/4, flows with R = 1 becomeunstable. The resulting instabilities are of the Kelvin-Helmholtz (KH) type,in which the shear layer rolls up to form an array of billows that are stationary with respect to the mean flow, and which display large overturns indensity [Thorpe, 1973].It is not generally the case that J> 1/4 results in stability. For example,if S is reduced such that R > 2, an additional mode of instability, theHolmboe mode, is excited [Alexakis, 2005]. In this case, the range of J overwhich instability occurs extends above 1/4. That is, Ri < 1/4 somewherein z at the same time as J> 1/4. While it is generally true that flows with49Chapter 3. Shear Instability in the Fraser River Estuaryhigher J are subject to less mixing by shear instabilities, by itself, J doesnot indicate whether or not a flow is unstable.For simplicity, the analytical solution of Holrnboe’s mixing layer modelassumes the flow is unbounded in the vertical. In our analysis we includeboundaries at the top and bottom where b must satisfy the boundary condition & = 0. The presence of these boundaries tends to extend the range ofunstable wavenumber to longer wavelengths [Hazel,19721.However, in thecases considered here, at the wavenumber of maximum growth, the boundaries have little or no impact on k and c.3.4.4 Solution of the TG Equation for Observed ProfilesWe use the numerical method described in Mourn et al.[20031to generatesolutions to the TG equation based on measured velocity and density profiles. Whenever possible we use velocity and density profiles collected at theupstream edge of apparent instabilities in the echosoundings. The velocityprofile, a 60 second average, is an average over one or more instabilities(the instabilities have periods < 60 seconds). This averaging reduces theinfluence of individual instabilities on the velocity profile, which in the TGequation, is taken to represent the background velocity profile. The velocityprofile is then smoothed in the vertical using a low pass filter (removing vertical wavelengths < 2 m). The density profile is smoothed by fitting a linearfunction, and one or more tanh functions (one for each density interface).By using smooth profiles we are effectively ignoring instability associatedwith small scale variations in the profiles.Because the point of observation moves in time, i.e. the boat is driftingseaward, predicted wavelengths from the TG equation cannot be compareddirectly to the wavelength of instabilities as they appear in the echosoundings. The wavelength predicted with the TG solution must be shifted toaccount for the speed of the instabilities with respect to the speed of theboat:= Cr _Vj(3.3)Here v& is the velocity of the boat and C,. and ). are the phase speed andwavelength predicted with the TG equation. The predicted apparent wavelength, ), is directly comparable to observations made from the movingboat. Seim & Gregg [1994] used a similar approach for estimating the wavelength of observed features.As well as giving a wavelength, phase speed, and growth rate for eachunstable mode, the TG solutions also give an eigenfunction that describes50Chapter 3. Shear Instability in the Fraser River Estuarythe vertical structure of the growing mode. The vertical displacement eigenfunction i(z) = —‘/(U — c) is particularly useful. At the location in z whereI ñIis a maximum we expect to see evidence of instabilities in the echosoundings.3.5 ResultsIn this section we use J, Ri(z) and solutions of the TG equation to assessthe stability of six sets of velocity and density profiles (one from each of thesix transects). Each set of profiles was chosen to coincide with evidence ofinstability in the echosoundings.Ebb During High Discharge: Transect 1The selected velocity and density profiles from transect 1 are shown in figure 3.5. The corresponding value of J for these profiles is 0.29 (see Table3.1). The stability analysis yields two modes of instability. The fastestgrowing mode is unstable for wavelengths greater than 11 m and has a peakgrowth rate of 0.025 s1 (doubling time of 28 s) occurring at a wavelength of21 m. The phase speed of the instability at this wavelength is -1.02 m s1,where the negative indicates a seaward direction. Given this phase speedand the seaward drift of the boat (-2.2 m s1), an apparent wavelength of39 m is calculated.Echosoundings collected at the same time, figure 3.5c, show clear evidence of instabilities. The prediction is found to be similar to, althoughshorter than, the approximately 50 m wavelength of the observed instabilities. The maximum displacement of the predicted instabilities is locatedat a depth of 7.6 m (indicated by the horizontal line), closely matching thedepth of the observed instabilities. Both the observed and predicted instability occur within the region of shear above the maximum gradient in p (ata depth of 9 m). As indicated by the gray shading, this region of high shearand low gradient in p corresponds to Ri < 1/4.For the set of profiles shown in figure 3.5 the TG equation predicts asecond, weaker, unstable mode located at a depth of 2.5 m. This mode isassociated with the inflection point (& U/dz2 = 0) in the velocity profile atthis depth. Because there is very little density stratification and hence weakecho intensity at this depth we are unable to confirm or deny the presenceof this mode in the echosoundings.51EQ.a)Chapter 3. Shear Instability in the Fraser River EstuaryFigure 3.5: Velocity (a) and density (b) profiles observed during transect 1(June 12, 2006, 8h05 PDT, 8.9 km upstream of Sand Heads). The smoothprofiles used in the stability analysis are shown as thick black lines and theobserved data are plotted as points. The gray shading indicates regionsin which Ri < 1/4. The black horizontal line indicates the location ofmaximum displacement()for the most unstable mode predicted with theTG equation. The thin lines in (b) show the displacement functions for eachof the unstable modes. The functions are scaled in proportion to the growthrate. A close up of the echosounding logged near the location of the profilesis shown in (c), and includes a scale indicating the apparent wavelengthpredicted by the TG equation. The arrow at the top of image indicatesthe approximate location of the density and velocity measurements. In thiscase, the velocity is averaged over a distance of approximately 130 m.I.14—2 —1 0 0 10 20u (m s1)(kg m3)100 150 200Distance (m)52Chapter 3. Shear Instability in the Fraser River EstuaryE5)Figure 3.6: Velocity (a) and density (b) profiles observed during transect 2(June 14, 2006, 8h21 PDT, 11.1 km upstream of Sand Heads). See figure 3.5for details. In this case, the velocity is averaged over approximately 110 m.Ebb During High Discharge: Transect 2In transect 2 a single hyperbolic tangent gives a good fit to the measureddensity profile (figure 3.6b). Due to difficulties in profiling, the densityprofile at this location was missing data below 12 m. Data from the previouscast, taken 60 m upstream, was used below 12 m. This cast is expected tobe sampling water of similar density below this depth.In this case the stability analysis of the profiles results in a single modeof instability. The mode is unstable for wavelengths from 10 m to 35 mwith a peak growth rate of 0.02 s1 (doubling time of 35 s) occurring at awavelength of 17 m. The phase speed of the instability at this wavelength is-0.51 m s1. Given the drift velocity of -1.9 m s, an apparent wavelengthof 24 m is calculated. This prediction is found to be similar to, although—1.5 —1 —0.5 0 0 10 20 0 20 40 60 80 100u (m s1)(kg m)Distance (m)53Chapter 3. Shear Instability in the Fraser River Estuarylonger than, the approximately 18 m wavelength of the small instabilitiesappearing in the echosounding (figure 3.6c). The maximum displacement ofthe predicted instabilities is located at a depth of 10.6 m, closely matchingthe depth of the observed instabilities.Flood During High Discharge: Transect 3Despite the occurrence of Ri < 1/4 the stability analysis of the profilesin figure 3.7a and 3.Th does not find any unstable modes. Echosoundingscollected during the flood generally show features on the pycnocline thatwere well correlated with sand waves (figure 3.7c). These correlated featuresare likely controlled by the hydraulics of the flow over the sand waves.There was very little evidence of instabilities independent of these sandwaves. There appear to be some wave-like features on the pycnocline that areshorter(10 m) than the sandwaves, however, these are not well resolvedby the echosounder (e.g. depth of 9 m at x = 60 m). Properly assessing thestability of the flow over these sandwaves would require at least two or threesets of density and velocity profiles per sandwave, many more than we wereable to obtain.Low Freshwater DischargeEarly Ebb During Low Discharge: Transect 4At low discharge, during the ebb tide, shear and density stratification arespread over the entire depth (see figure 3.4). The vertical scales, h and 6,are therefore greater than at high discharge, where shear and stratificationwere concentrated at a single interface. The increase in h results in greaterJ despite a decrease in the density stratification,p(see Table 3.1).The profiles collected early in the ebb (transect 4, figure 3.8) exhibita number of homogeneous and weakly stratified layers connected by high-gradient steps, in both U andp. At some locations the steps appear tocoincide in both velocity and density, however, this is not always the case.Note that smoothing of the velocity reduces much of the step structure inthe measured profile, which occurs on the scale of the instrument resolution.Despite this smoothing the Ri profile shows four regions in which it dropsbelow critical.The stability ana’ysis yields two modes of instability. The most unstablemode has a peak growth rate of 0.023 s_i occurring at a wavelength of 10.3m with a phase speed of -0.86 m s1. Given this phase speed and theseaward drift of the boat (1.6 m s1), an apparent wavelength of 22 m is54Chapter 3. Shear Instability in the Fraser River EstuaryFigure 3.7: Velocity (a) and density (b) profiles observed during transect3(June 21, 2006, 12h38 PDT, 2.66 km upstream of Sand Heads). See figure3.5 for details. In this case, the velocity is averaged over approximately30 m.calculated. This is very similar to the wavelength of the largest instabilityin figure 3.8c. This mode has a maximum displacement at a depth of 2.5 m,closely matching the location of the observed instabilities.For these profiles there is a second, weaker, unstable mode locatedat a depth of 10.8 in. This mode is associated with the inflection point(d2U/dz = 0) in the velocity profile at this depth. Similar to the casein transect 1 (figure 3.5), the absence of strong vertical density gradientsprevents us from confirming or denying the presence of this mode in theechosounding.—0.5 0 0.5 0 10 20 0 50 100 150 200u (m s1)a(kg m)Distance (m)55Chapter 3. Shear Instability in the Fraser River EstuaryE70Figure 3.8: Velocity (a) and density (b) profiles observed during transect 4(March 10, 2008, 11h20 PDT, 22.4 km upstream of Sand Heads). See figure3.5 for details. In this case, the velocity is averaged over approximately90 m.Mid Ebb During Low Discharge: Transect 5The instabilities in figure 3.9c were observed one hour later and approximately 3 km downstream from Transect 4. The p profile (figure 3.9b) againdisplays a number of layers consisting of high-gradient steps. However, thelayers are not evident in the measured velocity profile (figure 3.9a), as wasthe case in figure 3.8, and the overall shape of the velocity profile is morelinear.The CTD cast is one of the few collected during the study where theinstrument passed through an overturn in the pycnocline (depth of approximately 3.8 m). Consistent with the small amplitude of the instabilities inthe echosounder, the overturn in the density profile has only water of intermediate density, i.e. no surface or bottom water is observed in the overturn.(a)14—1 —0.6 0 0 5 0 10 20 30 40 50 60u (m s1)(kg m)Distance (m)56Chapter 3. Shear Instability in the 1aser River EstuaryE(c) L50 100Distance (m)Figure 3.9: Velocity (a) and density (b) profiles observed during transect 5(March 10, 2008, 12h21 PDT, 19.6 km upstream of Sand Heads). See figure3.5 for details. In this case, the velocity is averaged over approximately140 m.The TG equation predicts an unstable mode with a peak growth rate(0.03 s’) at a wavelength of 14 m with a phase speed of -1.2 m s1. Theapparent wavelength is predicted to be 32 m, whereas the features in theechosounder range in horizontal length from approximately 10 to 50 m, withthe largest being near the TG prediction(30 m). The predicted maximumin the displacement eigenfunction occurs at a depth of 4.2 m closely matchingthe depth of the instabilities.As in the cases in figures 3.5 and 3.8, a second, weaker mode occurs nearthe bottom of the profile at a depth of 9.4 m. Again, this mode is associatedwith an inflection point in the velocity profile.i-t,——1.5 —1 —0.5 0 5 1 15 0u(ms1)G(kgm)150 20057Chapter 3. Shear Instability in the Fraser River EstuaryEa)Figure 3.10: Velocity (a) and density (b) profiles observed during transect 6(March 10, 2008, 14h34 PDT, 7.6 km upstream of Sand Heads). See figure3.5 for details. In this case, the velocity is averaged over approximately130 m.Late Ebb During Low Discharge: Transect 6In the later stages of the ebb, during transect 6 (figure 3.10), the shearhas increased such that J is reduced to approximately 0.3. Unlike most ofthe other profiles collected during low or high discharge the density profilehas no homogeneous layers, and shows small scale (i.e. on the scale of theinstrument resolution) overturning throughout the depth. In these profilesRi is below critical throughout most of the depth aside from at the densityinterface.The most unstable mode predicted with the TG equation is located at adepth of 5.6 m and has a maximum growth rate of 0.019 s_i at an apparentwavelength of 65m. This is close to, but longer than, the largest features in05 150—2 —1 10 100 150u (m s1)(kg m4) Distance (m)50 200 25058Chapter 3. Shear Instability in the 1aser River Estuarythe echosounder (approximately 50 m).3.6 Small Scale Overturns and Bottom StressIn figure 3.10 there are no features in the echosoundings that are associatedwith the small scale overturns in p below a depth of 7 m, and although oursolutions to the TG equation suggest unstable modes, these are both locatedwell above a depth of 7 m. To further examine the source of these overturnswe compare selected density profiles from each of the low discharge transects(figure 3.11). In the density profile from transect 4, small scale overturnsare rare or completely absent (figure 3.11, T4). Approximately two hourslater, during transect 5, just one profile exhibits these small scale overturns(figure 3.11, T5). This cast was performed at the shallow constriction in theriver associated with the Massey Tunnel (figure 3.4b 18 km). In this case thesmall scale overturns in the profile occur only below the pycnocline suggesting that the stratification within the pycnocline is confining the overturns tothe lower layer. By maximum ebb, small scale overturns occur throughoutthe depth (figure 3.11, T6).The presence of these small scale overturns is apparent, although notimmediately obvious, in the echosoundings in figure 3.4. Note that the scaleof the shading is the same in all three panels of figure 3.4 and that there isa gradual increase (darkening) in background echo intensity from early tolate ebb (transects 4 to 6). This increase in echo intensity is attributed tothe small scale overturning observed in the density profiles. Early in the ebbthe dark shading associated with high echo intensity is concentrated at thedensity interfaces (transect 4). Otherwise, at this time, echo intensity is low(light shading) corresponding to an absence of small scale overturns in thedensity profiles (e.g. figure 3.11, T4). At this stage of the ebb, near-bottomvelocities are close to zero and bottom stress is expected to be negligible. Intransect 5 (figure 3.4b) there is an increase in echo intensity as the flow passesover the Massey Tunnel (18 km). At this location and during this stage ofthe ebb, near bottom velocity increases to approximately 0.2 m s at 1 mabove the bed. In this case the small scale overturns in the profile occuronly below the pycnocline (figure 3.11 T5) suggesting that the stratificationwithin the pycnocline is confining bottom generated turbulence to the lowerlayer. Near maximum ebb, during transect 6, near bottom velocities reach0.5 m s1 at 1 m above the bed. By this stage, high echo intensity and smallscale overturns occur throughout the depth (figure 3.11, T6) suggesting thatbottom generated turbulence has reached the surface despite the presence59Chapter 3. Shear Instability in the Faser River EstuaryE0.Figure 3.11: Selected density profiles from transects performed at low freshwater discharge. The profiles were collected at t=10h53, 12h36 and 14h25,at x=26.2, 17.9 and 8.8 km (transects 4, 5 and 6 respectively).of stratification.3.7 DiscussionOne-Sided InstabilityIn all five of the cases that the TG equation predicted the occurrence ofunstable modes, the bulk Richardson number, J, was greater than 1/4. Thisresult suggests the mixing layer model and associated J (see section 3.4.3)are not adequate for describing the stability of the measured profiles. Inall of these unstable cases, both the region of Ri(z) < 1/4 and the depthof the maximum in the displacement eigenfunction(I(z)I)were verticallyGT(kg m)60Chapter 3. Shear Instability in the Fraser River Estuaryoffset from the maximum gradient in density (dp/dz). This offset betweenthe depth of the predicted region of instability and the density interfaceis due to asymmetry between the p and U profiles, i.e. deviations fromthe idealized profiles of the simple mixing layer model (equation 3.2 withR=h/6= 1).Laboratory models and direct numerical simulations of asymmetry result in one-sided instabilities that resemble the features in the echosoundingsin figures 3.5c, 3.6c and 3.8c [e.g. Lawrence et aL, 1991; Yonemitsu et al.,1996; Carpenter et aL, 2007]. Similar observations were made in the Straitof Gibraltar by Farmer Armi [1998] and in a strongly stratified estuary byYoshida et al. [1998]. In both of these cases the instabilities were attributedto one-sided modes. One-sided modes are part of a general class of instability that includes the Holmboe mode. In contrast to the classic KH mode,the Holmboe mode is a result of the destabilizing influence of the densityinterface and can occur at relatively high values of J [Holmboe, 1962].When these one-sided instabilities are modelled using DNS, at the valües of J observed here, they lack the complete overturning of the densityinterface normally associated with KH billows. Unlike the mixed fluid thatresults from the KH instability, the mixed fluid that results from one-sidedinstabilities is not concentrated at the density interface, but, is instead drawnaway from the density interface [Carpenter et al., 2007].Amplitude of the InstabilitiesUnlike KH instabilities, the deflection of the density interface caused byone-sided instabilities does not necessarily equal the amplitude of the billows. It is therefore difficult to assess the amplitude of these instabilitiesusing echosoundings (e.g. figure 3.5). Nevertheless, taking the approximatedistance between the trough and the crest, the observed instabilities vary inheight (twice the amplitude) from approximately 0.5 m to 2 m. The maximum height to wavelength aspect ratio of the observed instabilities variesbetween approximately 0.025 (0.5/20, figure 3.5c) and 0.1 (2/20, figure 3.6c).In the tilting tube experiments of Thorpe [1973] the maximum aspect ratioof KR instabilities varied between 0.05 and 0.6. Given the low values of J(< 1/4) in Thorpe’s experiment this difference in aspect ratio is not surprising. Unfortunately, other than the case of the KR instability (symmetricdensity and velocity profiles and J < 1/4) the height of shear instabilitiesin stratified flows is not well documented.61Chapter 3. Shear Instability in the Fraser River EstuaryUse of Echosoundings to Identify InstabilityIn section 3.5 our analysis focused on periods when instabilities were evident in the echosoundings. There were instances where the predictions fromthe TG equation suggested instabilities would occur when there were nonevisible in the echosounder. For example, in figures 3.5, 3.8 and 3.9 there areno apparent instabilities in the echosoundings associated with the weakerunstable modes. In these cases, this is explained by the absence of strongvariations in salinity and temperature (i.e. density stratification) that areresponsible for the back scatter of sound to the instrument [see Seim, 1999;Lavery et al., 2003, for a thorough description of acoustic scattering in similar environments].There was one notable case where the TG equation predicted an unstable mode in the presence of stratification while there was no clear evidenceof instabilities in the echosoundings. For profiles collected at 2.2 km, duringtransect 3 (figure 3.3 c), the TG equation predicted instability close to thedepth of the pycnocline (results not shown). In this region the boat speedand predicted instability speed were almost the same (-0.28 m s versus-0.24 m s1). Considering equation 3.3, the resulting apparent wavelengthwould be 250 m. The corresponding apparent period of approximately 15minutes (250 m/-0.28 m s’) would likely distort the appearance of aninstability beyond recognition. This highlights an important challenge inidentifying instabilities in echosoundings: if the point of observation is moving at a speed similar to the instability, the appearance of the instabilitybecomes greatly distorted. On the other hand, if the observer is moving ata much different velocity than the instability, i.e. the apparent wavelengthand period are relatively short, the sampling rate of the echosounder maynot be sufficient to resolve the instabilities.3.8 ConclusionsWe successfully conducted a field program in the Fraser River estuary aimedat studying the details of shear instabilities. A bulk stability analysis showedthe flow was least stable during mid and late ebb, consistent with the findingsof previous investigators. Performing a detailed stability analysis on sixsets of velocity and density profiles using the Taylor-Goldstein equation andcomparing with the echosoundings we conclude the following.1. All of the instabilities observed in the echosoundings coincided withthe most unstable mode in the TG analysis. This confirms the appli62Chapter 3. Shear Instability in the Fraser River Estuarycability of the TG equation in predicting instability, even in cases ascomplex as the Fraser River estuary.2. The location of each of the observed instabilities occurs in a regionof depth where Ri < 1/4. However, there are also cases that haveRi < 1/4 in which no unstable modes were observed. This result isin full agreement with the Miles-Howard criterion, but also highlightsthe inconclusive nature of this criterion3. Although the observed instabilities all involve the mixing of a welldefined density interface, they appear to be concentrated on only oneside of the interface. The maximum of,occurs either above or belowthe density interface in a region of z where Ri < 1/4. None of theobservations show Ri < 1/4 across the width of a density interface.This is in contrast to the archetypal KR instability described by thesimple mixing layer model, in which Ri < 1/4 where dp/dz (N2) isgreatest. The observed instabilities might therefore be better describedby the so-called ‘one-sided’ modes of Lawrence et al. [1991]; Carpenteret al. [2007], or the layered model of Caulfield [1994]; Lee & Caulfield[2001].4. When there is active bottom generated turbulence in the water column, as in figure 3.10, we observe regions of z with near linear gradients in U and p and Ri 1/4. In other stratified estuaries withmoderate to strong tidal forcing, such as the Columbia and Hudsonrivers, turbulence generated at the bottom is considered the dominantsource of mixing [Nash et al., 2008; Peters & Bokhorst, 2000]. Thecommon occurrence of overturning caused by bottom generated turbulence in the late ebb of the present study suggests that this mixingprocess may be important in the Fraser River estuary.63BibliographyALEXAKIS, A. 2005 On Holmboe’s instability for smooth shear and densityprofiles. Phys. Fluids 17, 084103.CARPENTER, J.R., LAWRENCE, G.A. & SMYTH, W.D. 2007 Evolutionand mixing of asymmetric Holmboe instabilities. J. Fluid Mech. 582, 103—132.CAULFIELD, C.P. 1994 Multiple linear instability of layered stratified shearflow. J. Fluid Mech. 258, 255—285.FARMER, D.M. & ARMI, L. 1998 The flow of Atlantic water through theStrait of Gibraltar. Frog. Oceanogr. 21, 1—98.GEYER, W.R. & FARMER, D.M. 1989 Tide-induced variation of the dynamics of a salt wedge estuary. J. Phys. Oceanogr. 19, 1060—1672.GEYER, W.R. & SMITH, J.D. 1987 Shear instability in a highly stratifiedestuary. J. Phys. Oceanogr. 17, 1668—1679.GOLDSTEIN, S. 1931 On the stability of superposed streams of fluids ofdifferent densities. Proc. R. Soc. Lond. A 132, 524—548.HAZEL, P. 1972 Numerical studies of the stability of inviscid stratifiedshear flows. J. Fluid Mech. 51, 39—61.HOLMBOE, J. 1962 On the behavior of symmetric waves in stratified shearlayers. Geofys. Pubi. 24, 67—112.HOWARD, L.N. 1961 Note on a paper of John W. Miles. J. Fluid Mech.10, 509—512.KOSTACHUK, R.A. & ATWOOD, L.A. 1990 River discharge and tidal controls on salt-wedge position and implications for channel shoaling: FraserRiver British Columbia. Can. J. Civil Eng. 17, 452—459.64BibliographyLAVERY, A.C., SCHMITT, R.W. & STANTON, T.K. 2003 High-frequencyacoustic scattering from turbulent oceanic microstructure: The importanceof density fluctuations. J. Acoust. Soc. Am. 114 (5), 2685—2697.LAWRENCE, G.A., BROWAND, F.K. & REDEKOPP, L.G. 1991 The stability of a sheared density interface. Phys. Fluids 3 (10), 2360—2370.LEE, V. & CAULFIELD, C.P. 2001 Nonlinear evolution of a layered stratified shear flow. Dyn. Atmos. Oceans 24, 173—182.MACDONALD, D.G. & HORNER-DEVINE, A.R. 2008 Temporal and spatial variability of vertical salt flux in a highly stratified estuary. J. Geophys.Res. 113.MILES, J.W. 1961 On the stability of heterogeneous shear flows. J. FluidMech. 10, 496—508.MILES, J.W. 1963 On the stability of heterogeneous shear flows, part 2. J.Fluid Mech. 16, 209—227.MOUM, J.N., FARMER, D.M., SMYTH, W.D., ARMI, L. & VAGLE, S.2003 Structure and generation of turbulence at interfaces strained by internal solitary waves propagating shoreward over the continental shelf. J.Phys. Oceanogr. 33, 2093—2112.NASH, J.D., KILCHER, L. & MOUM, J.N. 2008 Turbulent mixing in theColumbia River Estuary: structure and consequences for plume composition. J. Geophys. Res. p. submitted.PETERS, H. & BOKHORST, R. 2000 Microstructure observations of turbulent mixing in a partially mixed estuary. part 1: Dissipation. J. Phys.Oceanogr. 30 (6), 1232—1244.SEIM, H.E. 1999 Acoustic backscatter from salinity microstructure. J. Atmos. Ocean. Technol. 16, 1491—1498.SEIM, H.E. & GREGG, M.C. 1994 Detailed observations of naturally occurring shear instability. J. Geophys. Res. 99 (C5), 10049—10073.TAYLOR, G.I. 1931 Effect of variation in density on the stability of superposed streams of fluid. Proc. R. Soc. Lond. A 132, 499—523.THORPE, S.A. 1973 Experiments on instability and turbulence in a stratified shear flow. J. Fluid Mech. 61, 731—751.65BibliographyTHORPE, S.A. 2005 The Turbulent Ocean, 1st edn. Cambridge UniversityPress.WARD, P.R.B. 1976 Seasonal salinity changes in the Fraser River Estuary.Can. J. Civil Eng. 3, 342—348.YONEMITSU, N., SWATERS, G.E., RAJARATNAM, N. & LAWRENCE, G.A.1996 Shear instabilities in arrested salt-wedge flows. Dyn. Atmos. Oceans24, 173—182.YOSHIDA, S., OHTANI, M., NI5HIDA, S. & LINDEN, P.F. 1998 Mixingprocesses in a highly stratified river. In Physical Processes in Lakes andOceans, Coastal and Estuarine Studies, vol. 54, pp. 389—400. AmericanGeophysical Union.66Chapter 4Holmboe Wave Fields inSimulation and Experiment4.1 IntroductionGeophysical flows often exhibit stratified shear layers in which the region ofdensity variation is thinner than the thickness of the shear layer [e.g. ArmiFarmer, 1988; Wesson & Gregg, 1994; Yoshida et at., 1998, Chapter 3]. Inthese circumstances, when the stratification is sufficiently strong (measuredby an appropriate Richardson number), Holmboe’s instability develops. Atfinite amplitude the instability is characterized by cusp-like internal waves(referred to herein as Holmboe waves) that propagate at equal speed andin opposite directions with respect to the mean flow. Accurate modelling ofthese instabilities is important for the correct parameterization of momentum and mass transfers occurring in flows of this nature.Previous studies on the nonlinear behaviour of Holmboe waves haveadopted one of two methods: either an experimental approach in which theinstability is studied under specified laboratory settings [Zhu & Lawrence,2001; Hogg & Ivey, 2003], or a numerical approach that allows for a detaileddescription of the flow in an idealized stratified mixing layer [Smyth et at.,1988; Smyth Winters, 2003; Smyth, 2006; Smyth et at., 2007]. It is difficult to make a meaningful comparison of laboratory and numerical resultsfor a number of reasons. In the case of laboratory experiments, Holmboewaves often arise as a local feature of a larger-scale flow, such as an exchangeflow between two basins of different density [Pawlak & Armi, 1996; Zhu &Lawrence, 2001; Hogg & Ivey, 2003], or an arrested salt wedge flow [Sargent& Jirka, 1987; Yonemitsu et at., 1996]. In many of these experiments themean flow varies appreciably over length scales that are comparable to the.wavelength of the waves. For this reason, it can be difficult to isolate thedynamics of the waves from that of the mean flow.3This chapter is in preparation for publication in: J.R. Carpenter, E. W. Tedford, M.Rahmani and GA. Lawrence (2009), Holmboe Wave Fields in Simulation and Experiment.67Chapter 4. Holmboe Wave Fields in Simulation and ExperimentThe use of numerical simulations has been advantageous in this regard,and comprises a great majority of the literature on the nonlinear dynamics of Holmboe waves. The first verification of two oppositely propagatingcusp-like waves of equal amplitude, predicted by the Holmboe[19621theory, was made through the numerical simulations of Smyth et al. [1988].Since then, increases in computational resources have led to fully three-dimensional direct numerical simulations (DNS) of Holmboe waves that resolve the smallest scales of variability. These simulations have been used tounderstand turbulence and mixing characteristics (Smyth & Winters 2003;Smyth, Carpenter & Lawrence 2007; Carpenter, Lawrence & Smyth 2007),as well as the growth of secondary circulations and the transition to turbulence [Smyth, 2006]. However, partly due to computational constraints,only a single wavelength of the primary instability has been reported in theliterature. Furthermore, no attempt has been made to compare the resultsof numerical simulations with laboratory experiments.In this paper, we undertake a combined numerical and experimentalstudy of Holmboe waves. The experiments, originally described by Tedford,Pieters & Lawrence (2009) (Chapter 2), consist of an exchange flow througha relatively long channel with a rectangular cross-section. The experimental design allows for a detailed study of the Holmboe wave field within asteady mean flow that exhibits gradual spatial variation relative to the waveproperties. The DNS of the present study were designed to correspond asclosely as possible to the conditions present in the experiments to effect ameaningful comparison between the two methods. To our knowledge, thisis the first study to compare experimental and numerical results, as well asthe first to perform DNS for multiple wavelengths of the instability. Wefocus on comparing basic descriptors of the wave fields such as phase speed,wavenumber, and wave amplitude, in order to gain a fuller understandingof the processes affecting the nonlinear behaviour of the waves.The paper is organized as follows. Section 4.2 gives a background onthe stability of stratified shear flows. This is followed by a description ofthe numerical simulations, and laboratory experiments in section 4.3. Wethen discuss comparisons between the simulations and experiments in termsof the basic wave structure (section 4.4), phase speed (section 4.5), wavespectral evolution (section 4.6), and wave amplitude and growth (section4.7). Conclusions are stated in the final section.68Chapter 4. Holmboe Wave Fields in Simulation and Experiment4.2 Linear Stability of Stratified Shear LayersIn both experiment and simulation, the mean flow exhibits the characteristics of a classic stratified shear layer. The velocity profile undergoes a totalchange of U, over a length scale h, that is closely centred with respect tothe density interface. Similarly, the density profile changes by zp betweenthe two layers, over a scale of S. This suggests using an idealized model ofthe horizontal velocity and density profiles that is given byf2(z—zo)i — i2(z—z)iU(z)=—.--tanh[hjand p(z)=po_—--tanh[(4.1)respectively. The density profile (z) is measured relative to a referencedensityP0,with z the vertical coordinate. A necessary condition for thegrowth of Holmboe’s instability is that the thickness ratio R h/S 2[Alexakis, 2005].In addition to R, we may define three more important dimensionlessparametersUh g’h iiRem—,2’and Pr—,(AU) kwhere g’= pg/pois the reduced gravitational acceleration, “is the kinematic viscosity, and ic the diffusivity of the stratifying agent. These are theReynolds, bulk Richardson, and Prandtl numbers, respectively.Linear stability analysis of the profiles in (4.1) has been performed innumerous studies [e.g. Hazel, 1972; Smyth et al., 1988; Haigh, 1995]. For theflows considered here, the effects of viscosity and mass diffusion have beenincluded. The resulting equation is a sixth order eigenvalue problem originally described by Koppel [1964]. Like the better known Taylor—Goldsteinequation, Koppel’s equation gives predictions of the complex phase speedc Cr+Cj, and vertical mode shape, as a function of wavenumber k. Resultsof the stability analysis are shown in figure 4.1, which includes the temporalgrowth rate, kcj, as well as the dispersion relation in terms of phase speedc7.(k), and frequency o-(k). This is done for the idealized profiles (4.1) usingRe = 630, J = 0.30, Pr = 700, and R = 8, matching the conditions inthe laboratory exchange flow (thick lines). As discussed in the next section,computational constraints limited our three-dimensional DNS to a Pr = 25and R = 5, resulting in slightly different results (figure 4.1, thin lines). Although no appreciable changes are seen in the predicted phase speed c andfrequency a, there are differences in maximum growth rate and the location69Chapter 4. Holmboe Wave Fields in Simulation and Experimento.eI0Cu00.4Cu1.50.5170.5 1k (rad cm)“0 0.5 1 1.5 “0 0.5 I 1.5k (rad cm1) k (rad cm”)Figure 4.1: Plots of (a) growth rate kc, (b) phase speed Cr, and (c) frequencya of the profiles in (4.1). Conditions in the experiment are shown as thicklines, and the three-dimensional simulation at Pr = 25 and R = 5 as thinlines. No noticeable difference between the simulation and experiment canbe seen in (c). The location of the wavenumber of maximum growth in eachcase is marked with a vertical dotted line. The dashed line in (c) indicatesaock.of the wavenumber of maximum growth kmaz.4.3 Methods4.3.1 Description of the Numerical SimulationsNumerical simulations were performed using the DNS code described byWinters, MacKinnon & Mills (2004), which has been modified to includegreater resolution of the density scalar field by Smyth, Nash & Mourn (2005).The simulations were designed to reproduce conditions present in the laboratory experiment as closely as possible, while still conforming to the general methodology used in recent investigations of nonlinear Holmboe waves[Smyth & Winters, 2003; Smyth, 2006; Smyth et aL, 2007; Carpenter et al.,2007].The boundary conditions are periodic on the strearnwise (x) and transverse (y) boundaries, and free-slip on the vertical (z) boundaries. Simulations are initialized with profiles in the form of (4.1) that closely match whatis observed in the experiment. Figure 4.2 shows a sequence of representative U and— P0profiles at three different times during a simulation, aswell as profiles from the experiment for comparison. The periodic boundaryconditions of the simulations cause the flow to ‘run down’ over time, i.e.70Chapter 4. Holmboe Wave Fields in Simulation and ExperimentE0NFigure 4.2: Evolution of the background profiles in both simulation andexperiment. Plots (a) and (b) show temporal changes to U(z) and (z)— Poprofiles in the simulation with experimental profiles taken from the channelcentre (x = 0) in thick lines overlain for comparison.there is a continual loss of kinetic energy from the shear layer due to viscousdissipation and mixing. This results in an increase of h and S over time,as can be seen in figure 4.2. To indicate conditions at the initial time step(t = 0 s) of the simulations we will use a zero subscript (e.g. ho).In order to initiate growth of the primary Holmboe instability, the flowis perturbed with a random velocity field at the first time step. The noiseis distributed evenly in the x, y directions, but given greater amplitude nearthe centre of the shear layer and density interface, in the same manner asSmyth & Winters [2003]. The amplitude of the random perturbation waschosen large enough such that the instability grows to finite amplitude withminimal diffusion of the background profiles, yet is still small enough tosatisfy the conditions for numerical stability.While an ideal comparison between simulation and experiment wouldinvolve matching all four of the relevant dimensionless parameters, we areconstrained by the high computational demands of DNS. Of particular difficulty is the fine grid resolution required for high Pr flows. For this reasonwe have chosen a Pr 25, opposed to Pr = 700 for the laboratory saltstratification. Large values of R also place a high demand on the computa0U (cm s1)0— p0 (kg m3)71Chapter 4. Holmboe Wave Fields in Simulation and ExperimentParameters Linear Theory ResultsPr R L L Crarms(cm) (cm) (rad cm’) (cm s1) (cm)Laboratory 700 8 200 10 0.91 0.79 0.31(0.51-0.84)SimulationsI (3D) 25 5 128 5 0.79 0.84 0.62II (2D) 700 8 64 0 0.91 0.79 0.48III (3D) 25 5 64 10 0.79 0.84 0.62IV (2D) 25 5 64 0 0.79 0.84 0.74Table 4.1: Values of the various important dimensionless parameters for boththe simulation and experiment. The parameters listed in the simulationare evaluated using the initial conditions. In all cases we have Jo = 0.3,Re0 = 630, and L = 10.8 cm. Also included are kmax and Cr from theresults of the linear stability analysis, and the root mean square saturatedamplitude observations. The bracketed value ofarmsis for the laboratoryexperiments with the effect of wave stretching taken into account.tional resources, and we have therefore chosen1?o = 5, opposed tothe R = 8observed in the experiments. The effects of Pr and R have been tested byperforming a two-dimensional simulation (II) at Pr = 700 and R0 = 8. Theremaining two parameters, Re0 630 and Jo = 0.30, have been matchedto the experimental values. Computational constraints also limit the size ofthe simulation domain. In all cases the vertical depth L, has been matchedto the 10.8 cm of the experiments. The simulation width L = 5 cm, hasbeen reduced to half of that in the experiment (L = 10 cm), but was notfound to effect the results presented. This reduction in the width of thecomputational domain enabled a larger length L = 128 cm, allowing forapproximately 16 wavelengths of the most amplified mode, and compareswell with the L = 200 cm in the experiment. A summary of the parametersin the experiment and the simulations is shown in table 4.1.In addition to the three- and two-dimensional simulations already mentioned (labeled I and II in table 4.1, respectively), two supplementary simulations (III and IV) were also performed to test the effects of L and R, Pr.Unless explicitly stated, we will refer to simulation I simply as ‘the simulation’, hereafter.72Chapter 4. Holmboe Wave Fields in Simulation and Experiment4.3.2 Description of the Laboratory ExperimentThe laboratory experiment was performed in the exchange flow facility described by Tedford et al. [2009] (Chapter 2). A complete discussion of theexperimental procedures and apparatus can be found in that study, however,we now provide a summary of the pertinent features.The apparatus consists of two reservoirs connected by a rectangularchannel 200 cm in length, and 10 cm in width. The reservoirs are initiallyfilled with fresh and saline water (p = 1.41 kg m3) such that the depthin the channel is 10.8 cm. A bi-directional exchange flow is initiated by theremoval of a gate from the centre of the channel. After an initial transientperiod in which gravity currents propagate to each reservoir, and mixed interfacial fluid is advected from the channel, the flow enters a period of steadyexchange where the density interface is found to display an abundance ofHolmboe wave activity. In contrast to the run-down conditions in the DNS,the storage of unmixed water in the reservoirs maintains a steady exchangeflow for approximately 600 s. Our comparison is restricted to instabilitiesobserved during the period of steady exchange.The exchange flow exhibits internal hydraulic controls at the entranceto each of the reservoirs, effectively isolating the channel from disturbancesin the reservoirs, and enforcing radiation boundary conditions at the channel ends. Friction between the layers leads to a gradually sloping densityinterface that produces an x-dependent mean velocity, U = (U1+ (12)72(figure 4.3). The upper (U1) and lower((72)layer velocities are the maximumand minimum free-stream velocities (see Chapter 2). The gradual variationof U(x) along the laboratory channel is a result of the acceleration in eachof the layers due to the sloping interface. This variation is shown in figure4.3(b), and is found to be a near-linear function of x for the central portionof the channel. In contrast, U is identically zero throughout the domain inthe simulation, due to the periodic boundary conditions. This difference inmean flow is found to have important effects on the nonlinear developmentof the Holmboe wave field.4.4 Wave StructureIn the first instance, it is beneficial to perform a simple visual comparisonof the density structure of the waves. This is shown in figure 4.4, where arepresentative photograph of the laboratory waves is displayed above plotsof the density field from the two-dimensional simulation II (figure 4.4b)and three-dimensional simulation I (figure 4.4c, d). The density structure in73Chapter 4. Hohnboe Wave Fields in Simulation and Experiment10ENI)EC)ID100Figure 4.3: Spatial changes in U(z) and layer depths that occur along thelaboratory channel are shown in (a), along with the corresponding distribution of U(x) in (b). A linear fit to U(x) in the central portion of the channelis shown as the dashed line, and the mean velocity in the simulation domainis given by the thin solid line.figures 4.4( a, b) is very similar, as each has an identical set of dimensionlessparameters, differing only in the initial and boundary conditions. In allpanels of figure 4.4 it can be seen that many of the waves display the typicalform of the Holmboe instability, and consist of cusps projecting into theupper and lower layers. The upward pointing cusps are moving from leftto right, in the same direction as the flow in the upper layer, while thedownward cusps move at an equal but opposite speed with respect to themean velocity. The waves do not always appear cusp-like, and many take amore sinusoidal form.An important feature of nonlinear Holmboe waves is the occasional ejection of stratified fluid from the wave crests into the upper and lower layers.Two such ejections are shown in figure 4.4(d) where indicated, and can becharacterized by thin wisps of fluid being drawn from the wave crest andadvected by the mean flow. These wisps often settle back to the interfacelevel, contributing to the accumulation of mixed fluid there. This accumulation is observed to a much greater extent in figure 4.4(c,d), and shouldC—100 —50 0 50x (cm)74Chapter 4. Holmboe Wave Fields in Simulation and Experiment2(a) .N(b).2(c)N(d)iFigure 4.4: Representative plots of the density field for the experiment (a)along with simulation 11(b), and simulation I (c,d). The plot in (d) istaken at a later time when two ejections are underway, indicated by arrows.The x-axis has been shifted by L/2 to the left in (d) to better display theejection process.be expected due to the larger value of R0, as well as the higher diffusionthat comes with the lower Pr used in this simulation. Although ejectionsare observed in both the laboratory experiment and high Pr,R simulation(II), there is a greater frequency of occurrence in the lower Pr = 25, R = 5simulation (I).Holmboe’s instability has the uncommon property that, under certainconditions, the growth of the primary instability may take place as a threedimensional wave. Such a wave would travel obliquely to the orientationof the shear, and produce significant departures from a two-dimensionalwave. One of the conditions for this three-dimensional growth is that Re be575Chapter 4. Holmboe Wave Fields in Simulation and Experimentsufficiently low [Smyth & Peltier, 1990]. As the laboratory experiments arecarried out at low Re, and show some variation in the spanwise direction, itmust be questioned whether the growth of the primary Holmboe instabilityis three-dimensional. This is easily tested by the simulation results, whichshow a clear two-dimensional growth (see section 4.7 as well), even to aninitially random perturbation as described above. We can therefore confirmthat the primary instability is two-dimensional for the conditions examinedin the present study.4.5 Phase SpeedMany of the basic features in the wave field are revealed by an x — t characteristics diagram of the density interface elevation, shown in figure 4.5for both the simulation and experiment. Although the interface consistsof contributions from both upper and lower Holmboe wave modes (eachtravelling in opposite directions), we have filtered the characteristics usinga two-dimensional Fourier transform to reveal only the upper, rightwardpropagating wave modes.Certain differences between the simulation characteristics (figure 4.5a)and the experimental characteristics (figure 4.5b) are immediately apparent. The experimental characteristics exhibit a greater degree of irregularity.Since each plot represents a two dimensional slice from a three-dimensionalfield, this may be a result of greater variability in the transverse direction inthe case of the experiments. Since the waves in the simulation develop froman initial random perturbation at t = 0, there is also a temporal growthof the average wave amplitude in figure 4.5(a) that is not present in theexperimental characteristics.Despite these apparent differences in the characteristics, the phase speeds(inferred from the slope of the characteristics) are in good agreement. Theobserved phase speeds in both the simulation and experiment are foundto be slightly greater than the predictions of linear theory (solid lines),which has been noted in previous studies [Haigh, 1995; Hogg Ivey, 2003].However, the observations also suggest an increase in phase speed with waveamplitude. This is a quintessential feature of nonlinear wave behaviour (e.g.Stokes waves). Note that a ‘pulsing’ of the wave amplitude and phase speedis present in both sets of characteristics in figure 4.5. This is a well knownfeature of Holmboe waves due to the interaction between the two oppositelypropagating modes [Smyth et al., 1988; Zhu Lawrence, 2001; Hogg & Ivey,2003].76Chapter 4. Holmboe Wave Fields in Simulation and ExperimentFigure 4.5: 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 blue indicating a low (trough),and has been optimized in each of (a, b). Solid black lines indicate the characteristic slope given by the linear prediction of phase speed c,.. In the case ofthe laboratory experiment, the c,. has a slight curvature since the changes inU across the channel have been included. The dark circles indicate locationsand times of ejections.Sudden decreases in wave amplitude can be seen in both sets of characteristics at a number of times and locations. It is often the case (thoughnot always) that these sudden amplitude changes are a result of the ejectionprocess. Instances where ejections occur have been identified in figure 4.5,and are denoted by circles. It is generally observed that the ejection processpreferentially acts on the largest amplitude waves, and in this way resemblesa wave breaking mechanism.4.6 Spectral EvolutionThis section concerns the distribution and evolution of wave energy with k.It will be shown that there are two different processes acting separately inS12077Chapter 4. Holmboe Wave Fields in Simulation and Experiment(a) Simulation—50(b) Experiment0x (cm)50Figure 4.6: Rightward propagating wave characteristics for the simulation(a) and experiment (b). White indicates a wave crest while grey indicatesa wave trough. In each panel a number of wave crests are indicated bysolid and dashed lines. In (a), the dashed lines correspond to waves thatare ‘lost’ over the duration of the simulation, whereas in (b), the dashedlines correspond to waves that have formed within the channel. Only thecentral portion of the laboratory channel corresponding to the simulationdomain has been shown. Circles and squares indicate locations and times ofejections and pairing events, respectively.the simulation and experiment that are responsible for a shifting of waveenergy to lower k (i.e. longer waves).4.6.1 Frequency ShiftingIn order to gain an understanding of the wave spectrum, it is first useful tocarefully examine the characteristic diagrams. Figure 4.6 shows rightwardpropagating characteristics from both simulation and experiment that highlight the location of wave crests (in white) and troughs (in grey). Characteristics from the simulation (figure 4.6 a) are discussed first, and are shownfor the entire computational domain.Beginning with the initial random perturbation at t = 0, energy is ex100—50/0x (cm)5078Chapter 4. Holmboe Wave Fields in Simulation and Experimenttracted from the mean flow by the instability and fed into the wave field at,or very close to, the wavenumber of maximum growth,kmax. This results inapproximately 16 waves in the computational domain (given byLxkmax/2rr)for early times. We see however, that as the simulation proceeds wave crestsare continually being ‘lost’ over time. This feature is highlighted by thesolid and dashed lines that are used to trace the wave crests in figure 4.6(a).The dashed lines indicate wave crests that are ‘lost’, while the solid linescorrespond to crests that persist. This process of losing waves results in anobserved frequency, w, that is continually shifted downwards. Because previous numerical studies of Holmboe waves simulated only a single wavelength,this process has not been described before. This ‘frequency downshifting’ or‘wave coarsening’ has, however, been noted previously in many other nonlinear wave systems [e.g. Huang et al., 1999; Balmforth & Mandre, 2004].It can be seen in figure 4.6(a) that three of the five lost waves indicated bydashed lines correspond to waves that have undergone ejections (indicatedby circles). In general, for all of the simulations performed, the ejectionprocess typically results in a loss of waves and a downshift in frequency.This observation mirrors similar findings in the frequency downshifting ofnonlinear surface gravity waves, where the occurrence of wave breaking isrelated to lost waves [Huang et at., 1996; Tulin & Waseda, 1999]. Closeexamination of the vorticity field also suggests that the Holmboe wavesundergo a vortex pairing process. Although the pairing of adjacent vorticiesis a well known feature of homogenous and weakly stratified shear layers[Browand & Winant, 1973], it has not previously been identified in Holmboeinstabilities. This is an additional means to effect a shift of wave frequency,and is denoted by square symbols in figure 4.6(a).In contrast, figure 4.6(b) shows that the experimental characteristicsdisplay a distinctly different behaviour. In this case, new wave crests arecontinually being formed as the waves traverse the channel. Again, thisprocess is highlighted by the tracing of crests by solid and dashed lines.Now, the dashed lines represent new wave crests that have been formedwithin the channel. This process results in an increasing w with x in theexperiments.Tedford et at. [2009] (Chapter 2) explain the formation of new waves asfollows. As waves propagate through the channel they are accelerated by theincreasing mean velocity (7(x). This leads to a ‘stretching’ of the waves thatdecreases k from nearkmax,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 energy back into the wave field near kma, resulting in an average79Chapter 4. Holmboe Wave Fields in Simulation and ExperimentFigure 4.7: Spectral evolution of the rightward propagating waves from simulation (a) and experiment (b). Dark colours denote a high in energy whichis proportional to the mean square amplitude of the interface displacement.The wave energy has been normalized by the variance in (a) to removethe time dependent wave growth. Linear stability theory is used to predictkmax(red lines), which changes in time for the simulations. The predictedstretching of wave energy in the experiment by U(x) to lower k is shown asthe yellow dashed line in (b).k that is constant across the channel, and an increasing w.The two processes, wave coarsening in the simulation, and wave stretching in the experiments, are best described quantitatively using wave spectra.4.6.2 Wave Energy SpectraDifferences between the processes responsible for modifying k in the simulation and experiment can be seen in figure 4.7. It demonstrates how waveenergy (indicated by the dark bands) is redistributed in k over time.The spectra of the simulation (figure 4.7a), which has been normalizedby the variance in order to remove the time-dependent growth of the waves,shows a discrete transfer of wave energy to lower k. The simulation spectra isrequired to evolve in discrete steps due to the periodic boundary conditions0.5 1k (rad cm1)0.5 1k (rad cm1)80Chapter 4. Holmboe Wave Fields in Simulation and Experiment(i.e. in wavenumber increments of /.k = 2ir/L). As a point of comparison,the kmarc prediction from linear stability theory is plotted in red. The predicted has been discretized according to the boundary conditions, anddecreases in time due to the diffusion of the background profiles, i.e. theincrease in the shear layer thickness h(t).The spectral evolution plot (figure 4.7a) compliments the characteristicsdiagram of figure 4.6(a), showing an initial input of energy at kmax (t = 0),and a subsequent shifting of that energy to lower k. It is interesting to notethat the kmaz(t) curve shows the same general trend as the concentrationof wave energy (shown by the dark ‘blocks’ in figure 4.7a). Although thedetails are unclear, we speculate that the shift in wave energy to lower k isthe result of nonlinear processes such as the ejections and vortex pairing.It is apparent from the wave spectra in figure 4.7(b) that the processresponsible for the redistribution of wave energy in the experiments is acontinuous one. Energy at any given time is found to be focused in a numberof ‘bands’. These bands originate near kmax, and move towards lower k intime. In addition, they all appear to have a similar trajectory in kt-space.Tedford et al. [2009] (Chapter 2) hypothesize that these bands are a resultof the stretching of wave energy to lower k by U(x). We now formulate asimple model in order to quantify this hypothesis.Wave Stretching PredictionThe changes in k that result from wave stretching by U(x) can be describedby an application of gradually varying wave theory. This theory assumesthat the density interface elevation (x, t), may be expressed in terms of agradually varying amplitude a(x, t), and a rapidly varying sinusoidal component viz.(x, t) = Re{a(x,t)et)}.(4.2)The local wavenumber and frequency are defined in terms of the phase function 8(x, t) by k 08/Ox and w —08/Of, respectively. We assume, for themoment, that O(x, t) is continuous. This implies that waves are conserved,givingOk Ow(4.3)Recognizing that w, which is the frequency that a stationary observer wouldmeasure, includes both an intrinsic portion u(k), and an advective portionkU, leads to -w = o(k) + kU(x). (4.4)81Chapter 4. Holmboe Wave Fields in Simulation and ExperimentSubstituting into (4.3) gives= —Sk, (4.5)where the material derivative, defined asD 0 -ôdenotes changes in time while moving at the speed Cg + U, and Cg du/dkis the intrinsic group speed. This is the speed that wave energy, i.e. thedark bands in figure 4.7(b), is expected to propagate through the channel.We have also defined S dU/dx, which is found to be very nearly constantin the central portion of the laboratory channel (see figure 4.3 b). Choosinga Lagrangian frame of reference, that moves at the speed c9 + U throughthe channel, allows for a simple integration of (4.5) to givek(t)= k*e_t_t*),(4.6)where k = k(t) is some initial value of k that wave energy begins thestretching process at. A direct comparison is now possible between the prediction of (4.6) and the bands of energy in the observed spectral evolution.The prediction is shown by the yellow dashed line in figure 4.7(b), and isfound to be in excellent agreement with the observations. This validates thehypothesis that the spectral shift towards lower k is a result of wave stretching. The excellent agreement between the predictions and observations alsoreveals that our assumption of wave conservation is justified. This is notin contradiction with the formation of new waves described in section 4.6.1since wave conservation is applied only after energy is fed into the wave fieldby the instability mechanism.4.7 Wave Growth and AmplitudeThe final basic parameter that we intend to compare is the wave amplitude,a. This feature of the wave field is determined when the linear growthreaches some level where it must saturate. It is a nonlinear property of thewaves, and may involve three-dimensional effects as well as interaction withthe mean flow. We begin by discussing the various phases of wave growth.82Chapter 4. Holmboe Wave Fields in Simulation and Experiment4.7.1 Wave GrowthIn the simulation, the instability mechanism causes the growth of waves froman initial random perturbation into a large-amplitude nonlinear wave form.This growth process is best illustrated by considering the kinetic energy ofthe waves, IC. Following Caulfield & Peltier [2000], we partition IC into atwo-dimensional kinetic energy IC2d associated with the primary Holmboewave, and a three-dimensional component IC3d, that provides a measure ofthe departures from a strictly two-dimensional wave. By this partitioningwe haveIC—IC2d+IC3d, (4.7)whereIC2d = (U2d U2d/2K0)XZandAC3d = (u3d U3d/2ICO)yz, (4.8)and we have usedUld(Z,t) = (u),(4.9)U2d(X, z, t) = (u — Uld)y, (4.10)U3d(X,y,Z,t) = UU1dU2d,(4.11)with(.)representing an average in the direction i, and K0 the total kineticenergy at t = 0.TheIC2dandK3dcomponents are plotted on a log-scale in figure 4.8for the simulation. The plot indicates that after a start up period wherethe energy of the initial perturbation rapidly decays, a stage of exponentialgrowth is achieved inK2d.This stage of exponential growth can be comparedto the prediction of linear theory (shown as a thick line), and is found tobe slightly less than the prediction. The growth is entirely two-dimensionaluntil the waves have reached a finite amplitude (t 65 s), at which point thegrowth of three-dimensional secondary structures results (see Smyth 2006for a discussion of this process in Holmboe waves). However, the wavesremain primarily two-dimensional, with lCSd at least an order of magnitudesmaller thanK2d.There is not a well defined transition to turbulence, asis found in other types of stratified shear layers (e.g. Caulfield & Peltier2000; Smyth, Mourn & Caldwell 2001), likely due to the low Re. Oncethe saturated amplitude is reached, there is a slow decline of K over theremainder of the simulation.In the laboratory experiments we have focused only on the period ofsteady exchange, and therefore do not observe a time-dependent growth of83Chapter 4. Holmboe Wave Fields in Simulation and Experiment>,a)U]C)a)C200Figure 4.8: Growth ofC2d and lC3d for the simulation. The thick line givesthe linear growth rate prediction of the growth of1C2d,which is a weakfunction of time due to the changing background profiles.the wave field on average. However, as discussed previously, the instabilityis constantly acting to produce new waves along the channel. It is difficult tomeasure the growth rate of these waves, but they appear to reach a saturatedamplitude rapidly, suggesting that they are strongly forced by disturbanceswithin the channel.4.7.2 Comparison of Saturated AmplitudesAlthough the transient growth of the instability is difficult to quantify in theexperiments, it is possible to measure the mean amplitude of the waves. Thisis done by using the root mean square amplitude of the interface elevationj(x,t), given by_______/1 rarms(x)=//i72dt, (4.12)v£where T denotes the duration of the steady period of exchange. When averaged over a number of experiments arms is found to display little dependence10-I100t (s)84Chapter 4. Holmboe Wave Fields in Simulation and Experimenton x. A similar arms can be defined for the simulations, however, the temporal average is replaced by a spatial average in x. The growth ofarms intime in the simulations shows a similar behaviour to1C2d;an exponential initial growth, followed by a saturation, and subsequent decay. The saturated(maximum) amplitude reached during each of the simulations is shown intable 4.1, along with the mean amplitude in the experiments.The first feature to note is that the waves of the two-dimensional simulation (II) at R = 8 and Pr 700 (matching the conditions in the experiment)have a lower amplitude than of all the other simulations, especially the two-dimensional simulation (IV) at R = 5, Pr 25. This indicates that thereis a possible dependence of the saturated amplitude on R, Pr. Most importantly, the amplitude measured in the experiments is significantly smallerthan any of the saturated amplitudes reached in the simulations. The smallamplitudes observed in the experiments can be explained 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 principles that have been established for waves on slowlyvarying currents [e.g. Peregrine, 1976]. In doing so, we assume that theHolmboe waves may be represented by a simple train of linear internal wavesthat satisfy the dispersion relation in figure 4.1(c). We are then able to trackthe changes in wave amplitude that occur as a result of the spatially varyingmean velocity U(x), i.e. the wave stretching. In this simplified model it isthe conservation of wave action density that is relevant. This is given asE/a, where E is the wave energy density, and recall that o(k) is the intrinsic wave frequency. Substitution into the conservation law, and following asimilar procedure to section 4.6.2 leads to a similar result= _s(), (4.13)which describes changes in action density due to the stretching by U. Inarriving at (4.13) we have neglected a term that is proportional tod2u/dk,which is small in the range of k that we are interested in (see figure 4.1 c).Taking S to be constant once again, allows for simple integration of (4.13)to give()= ()*e_t_t*85Chapter 4. Holmboe Wave Fields in Simulation and ExperimentFor linear internal waves E cx a2, so that we have an estimate of the reduction in wave amplitude due to stretching of= /ie_S(t_t*)/2.(4.14)aIf we now take the intrinsic frequency a k, as suggested by the lineardispersion relation in figure 4.1(c), it is possible to write the right hand sideof (4.14) ase_S(t_t*),where we have used the spectral prediction in (4.6).By inspection of figure 4.7(b), we can estimate a time interval, t, that waveenergy spends in the channel (i.e. the average time interval that the darkbands appear for) to be between 100 and 200 s. The amplitude reduction istherefore in the range 0.37 <e9t<0.61.Given this reduction, and assuming that no other processes are takingplace that may affect the wave amplitudes, we would expect amplitudes inthe range of that shown in table 4.1, given in parentheses. This adjustedamplitude is comparable to results of the simulations, and demonstratesthat — in the absence of other processes — the stretching of wave action issignificant in reducing the experimental wave amplitudes.4.8 ConclusionsWe have compared, for the first time, simulations of Holmboe wave fieldswith the results of laboratory experiments. A meaningful comparison waspossible since both methods exhibit only gradual variations in the mean flow.In the laboratory experiment, the mean flow is spatially varying, whereasthe numerical simulations display a temporal variation. Focusing on basicdescriptors of the waves, such as phase speed, wavenumber, and amplitude,we have identified a number of processes affecting the nonlinear behaviourof Holmboe wave fields.Similarities between results of the two methods include the basic structure of the waves, and the phase speeds. The observations show slightlygreater phase speeds when compared with the predictions of linear theory,in agreement with previous studies [Haigh, 1995; Hogg & Ivey, 2003]. Further departures from the linear predictions are attributed to a nonlineardependence of the phase speed on amplitude.The greatest differences between simulation and experiment are foundin the spectral evolution and wave amplitudes. In simulations, a transferof wave energy to lower k was found to result from wave coarsening, whichcaused waves to be ‘lost’ through discrete merging events. These events86Chapter 4. Holmboe Wave Fields in Simulation and Experimentwere found, at least in part, to result from the vortex pairing and ejectionprocesses. The latter of which is suggested to be similar to wave breaking insurface waves, since it appears to act preferentially to reduce the amplitudeof the largest waves. A detailed investigation of both ejections and vortexpairing in Holmboe waves is currently underway.The shift of wave energy to lower k that was observed in the experimentscan be attributed to the ‘stretching’ of the wave field by the spatially accelerating mean flow. This suggestion of Tedford et al. [2009] (Chapter 2) hasbeen 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 wave amplitudes observed in the experiments. Thisconclusion appears sufficient to explain discrepancies between wave amplitudes in experiment and simulation, and is based on a simple application ofthe conservation of wave action. In this application we have assumed thatno other processes are actively influencing the wave amplitude, however, adependence of wave amplitude on R, Pr has been noted.A general result of this comparison is that the nonlinear evolution ofa Holmboe wave field is dependent on the mean flow, and hence, on theboundary conditions. This must be considered when studying the nonlinearbehaviour of the Holmboe instability.87BibliographyALEXAKIS, A. 2005 On Holmboe’s instability for smooth shear and densityprofiles. Phys. Fluids 17, 084103.ARMI, L. & FARMER, D. 1988 The flow of Mediterranean water throughthe strait of Gibraltar. On the mechanism of shear flow instabilities. Frog.Oceanogr. 21, 1—98.BALMFORTH, N. J. & MANDRE, S. 2004 Dynamics of roll waves. J. FluidMech. 514, 1—33.BROWAND, F.K. & WINANT, C.D. 1973 Laboratory observations of shear-layer instability in a stratified fluid. Boundary-Layer Met. 5, 67—77.CARPENTER, J.R., LAWRENCE, G.A. & SMYTH, W.D. 2006 Instabilitymechanisms in asymmetric stratified shear layers. Proceedings6thInternational Symposium on Stratified Flowspp.14—19.CARPENTER, J.R., LAWRENCE, G.A. & SMYTH, W.D. 2007 Evolutionand mixing of asymmetric Holmboe instabilities. J. Fluid Mech. 582, 103—132.CAULFIELD, C.P. & PELTIER, W.R. 2000 The anatomy of the mixingtransition in homogenous and stratified free shear layers. J. Fluid Mech.413, 1—47.HAIGH, S.P. 1995 Non-symmetric Holmboe waves. PhD thesis, Universityof British Columbia.HAZEL, P. 1972 Numerical studies of the stability of inviscid stratifiedshear flows. J. Fluid Mech. 51, 39—61.HOGG, A. MCC. & IVEY, G.N. 2003 The Kelvin-Helmholtz to Holmboeinstability transition in stratified exchange flows. J. Fluid Mech. 477, 339—362.88BibliographyHOLMBOE, J. 1962 On the behavior of symmetric waves in stratified shearlayers. Geofys. Pubi. 24, 67—112.HUANG, N. E., LONG, S. R. & SHEN, Z. 1996 The mechanism for frequency downshift in nonlinear water wave evolution. Adv. Appl. Mech. 32,59—117.HUANG, N. E., SHEN, Z. & LONG, S. R. 1999 A new view of nonlinearwater waves: the Hilbert spectrum. Annu. Rev. Fluid Mech. 31, 417—457.KOPPEL, D. 1964 On the stability of a thermally stratified fluid under theaction of gravity. J. Meth. Phys. 5, 963—982.PAWLAK, G. & ARMI, L. 1996 Stability and mixing of a two-layer exchangeflow. Dyn. Atmos. Oceans 24, 139—151.PEREGRINE, D. H. 1976 Interaction of water waves and currents. Adv.Appi. Mech. 16, 9—117.SARGENT, F.E. & JIRKA, G.H. 1987 Experiments on saline wedge. J.Hydraul. Engng ASCE 113 (10), 1307—1324.SMYTH, W.D. 2006 Secondary circulations in Holmboe waves. Phys. Fluids18 (064104), 1—13.SMYTH, W.D., CARPENTER, J.R. & LAWRENCE, G.A. 2007 Mixing insymmetric Holmboe waves. J. Phys. Oceanogr. 37, 1566—1583.SMYTH, W.D., KLAASSEN, G.P. & PELTIER, W.R. 1988 Finite amplitude Holmboe waves. Geophys. Astrophys. Fluid Dyn. 43, 181—222.SMYTH, W.D., MOUM, J.N. & CALDWELL, D.R. 2001 The efficiencyof mixing in turbulent patches: Inferences from direct simulations andmicrostructure observations. J. Phys. Oceanogr. 31, 1969—1992.SMYTH, W.D., NASH, J.D. & MOUM, J.N. 2005 Differential diffusion inbreaking Kelvin-Helmholtz billows. J. Phys. Oceanogr. 35, 1004—1022.SMYTH, W.D. & PELTIER, W.R. 1990 Three-dimensional primary instabilities of a stratified, dissipative, parallel flow. Geophys. Astrophys. FluidDyn. 52, 249—261.SMYTH, W.D. & WINTERS, K.B. 2003 Turbulence and mixing in Holmboewaves. J. Phys. Oceanogr. 33, 694—711.89BibliographyTEDFORD, E.W., PIETERS, R. & LAWRENCE, G.A. 2009 SymmetricHolmboe instabilities in a laboratory exchange flow. J. Fluid Mech. p. toappear.TULIN, M. P. & WASEDA, T. 1999 Laboratory observations of wave groupevolution, including breaking effects. J. Fluid Mech. 378, 197—232.WESSON, J. C. & GREGG, M. C. 1994 Mixing at Camarinal Sill in theStrait of Gibraltar. J. Geophys. Res. 99, 9847—9878.WINTERS, K.B., MACKINNON, J.A. & MILLS, B. 2004 A spectral modelfor process studies of rotating, density-stratified flows. J. Atmos. Ocean.Tech. 21, 69—94.YONEMITSU, N., SWATERS, G.E., RAJARATNAM, N. & LAWRENCE, G.A.1996 Shear instabilities in arrested salt-wedge flows. Dyn. Atmos. Oceans24, 173—182.YOSHIDA, S., OHTANI, M., NISHIDA, S. & LINDEN, P.F. 1998 Mixingprocesses in a highly stratified river. In Physical Processes in Lakes andOceans, Coastal and Estuarine Studies, vol. 54,pp.389—400. AmericanGeophysical Union.ZHu, D. & LAWRENCE, G.A. 2001 Holmboe’s instability in exchangeflows. J. Fluid Mech. 429, 391—409.90Chapter 5Conclusion5.1 SummaryHolmboe instabilities have been studied in the laboratory, the field andwith direct numerical simulations (DNS). The instabilities were a result ofinteraction between shear and a density interface.In the laboratory the instabilities were observed on the density interfaceof a two-layer exchange flow. The analysis was focussed on the middle portion of the channel where velocity and density profiles most closely matchedthe original model of Holmboe [1962]. The simplicity of the flow resulted ina relatively uniform wave field, which, combined with the prolonged .periodof steady exchange, provided instabilities with average wave properties ingood agreement with the linear predictions of Holmboe.The gradual slope of the interface along the length of the channel resultedin convective acceleration within each layer. The Holmboe instabilities experienced an equivalent acceleration as they propagated along the channel.This acceleration caused the waves to stretch until they were approximatelytwice the most amplified wavelength allowing new waves to form. The newwaves formed uniformly along the channel such that the average wavelengthwas nearly constant and slightly greater than the most amplified wavelength.The conditions in the Fraser River salinity intrusion provided a variety of shear instabilities for investigation. Although none were identical tothe symmetric Holmboe instabilities of the laboratory many were similar inthat they had crests or troughs that cusped away from the density interface.Some of these had billows or wisps that were displaced vertically from thedensity interface. As in the case of the laboratory experiments, linear predictions based on the Taylor-Goldstein (TG) equation compared well withthe observed instabilities.In chapter 4 the results from the laboratory experiments were comparedwith Direct Numerical Simulations (DNS). The DNS provided predictions ofnon-linear aspects of the instabilties, such as the shape of the density interface and the maximum amplitude. The initial and boundary conditions ofthe DNS did not exactly match those of the laboratory experiments resulting91Chapter 5. Conclusionin significant differences. However, the density interface had a similar cusplike appearance and once the differences in the mean flow were accountedfor the amplitudes of the instabilities were well matched.5.2 The Occurrence of Holmboe andHolmboe-Like InstabilitiesIn Chapter 1, the basic instabilities that occur in shear flows with stabledensity stratification were described in terms of the interaction of two interfaces: the Kelvin-Helmholtz (KR) instability resulted from the interactionof two vorticity interfaces (Rayleigh mechanism); the Taylor instability fromthe interaction of two density interfaces; and the Holmboe instability fromthe interaction of one vorticity interface and one density interface. The instabilities in both the laboratory and field study resembled the Rolmboemode most often.The regular occurrence of Holmboe instabilities discussed in Chapters2 and 4 has not generally been observed in laboratory experiments of saltstratified exchange flows, especially those that have used relatively shortchannels. The high degree of non-uniformity that occurs in short channelsdiminishes the regularity of propagating waves (e.g. the Holmboe instabilities observed by Hogg & Ivey, 2003]. On the other hand, stationary waves(i.e. KR instabilities) are not as strongly influenced by non-uniformity.In the Fraser River estuary Holmboe-like instabilities were also present.However, the presence of multiple vorticity and density interfaces makesclassification more difficult. In some cases, the instabilities are potentiallythe result of the interaction of more than two interfaces.5.3 Contributions to the Study of ShearInstabilities in Stratified Flows5.3.1 Laboratory ExperimentsThe laboratory work described in Chapter 2 was, in its very early stages,meant to form a baseline for proposed experiments focussed on the impactof barotropic oscillations on a two-layer exchange flow. The proposed experiments were specifically intended to model the two-layer exchange flowin the Burlington Ship Canal [Lawrence et al., 2004]. Modifying the channelused by Zhu & Lawrence [2001] to more closely match the Burlington Shipcanal (i.e. removing the sill and reducing the depth) inadvertently created92Chapter 5. Conclusionoptimal conditions for generating symmetric, regularly occurring, Holmboeinstabilities. This setup, two reservoirs connected by a long straight channel,is the simplest possible for studying stratified shear instabilities in a steady,nearly uniform, background flow. The long channel also provides a lengthto depth aspect ratio that is more representative of geophysical flows. Inaddition to clearly demonstrating the occurrence of Holmboe instabilities,Chapter 2 provides a foundation for future research in similar facilities.The wave properties (i.e. phase speed, wavelength and frequency) ofthe instabilities were described quantitatively. This description relied onseveral techniques, the most important of which was filtering with the twodimensional Fourier transform (2DFFT). Because the experiments consistedof a long section of subcritical flow that was steady, the 2DFFT was effective in separating the instabilities into positive (rightward propagating)and negative (leftward propagating) modes. In comparison, in the experiments of Zhu & Lawrence [2001], the phase velocity of the negative modeschanged sign (from rightward to leftward propagating), which would renderthe 2DFFT less effective in separating the two modes. Once the instabilitieswere separated into positive and negative modes, the quantification of thewave properties was straightforward and allowed direct comparison with thelinear theory.5.3.2 Field ExperimentsShear instabilities were observed in the Fraser River estuary using simultaneous measurements of velocity, density and echo intensity. Drifting withthe upper layer allowed profiling of the CTD in a highly sheared flow whilesimultaneously logging high quality sounder and ADCP data. The use of aload bearing data cable for the CTD allowed us to quickly locate the frontof the salinity intrusion. To my knowledge, previous studies of the FraserRiver salinity intrusion did not use these techniques.In the stability analysis, a vertical low-pass filter was used on the velocityprofiles to ensure smooth profiles ofd2U/dz.Multiple hyperbolic tangentswere used to remove statically unstable features from the density profiles,while retaining step-like features. This careful processing of the velocityand density profiles allowed application of the TG equation in a relativelycomplex flow. Knowledge of the basic stable and unstable modes outlinedin Chapter 1 aided in interpreting the results of the stability analysis.93Chapter 5. Conclusion5.3.3 Direct Numerical SimulationsWe have compared, for the first time, simulations of Holmboe wave fieldswith the results of laboratory experiments. The use of multi-wavelength domains combined with random perturbations allowed a number of importantmechanisms to be modelled with DNS. These include, wave-group behaviour,vortex pairing and ejections. The use of the 2DFFT allowed us to comparethe spectral evolution of the waves. In the DNS, vortex pairing and ejections resulted in wave energy getting shifted in discrete steps to longer andlonger wavelengths. In contrast, in the laboratory exchange flow, wave energy moved continuously to longer wavelengths due to spatial acceleration.The wave stretching in the laboratory reduced the height of waves relativeto the DNS.5.4 Future ResearchThe most common criticism of laboratory experiments and DNS is their lowReynolds number. The use of larger laboratory facilities and more powerfulcomputers could address this issue in future research. During the course ofthis study we have built a larger exchange flow facility that will allow anincrease in the depth of flow to 0.6 m. This will increase the Reynolds number by a factor of approximately 5. The use of a stronger density gradientbetween the two reservoirs, i.e. more salt, will generate higher velocities andalso higher Reynolds numbers. Higher velocities will require a camera withgreater frame rate and sensitivity.Once Holmboe instabilities can be modelled at higher Reynolds numbereither physically or numerically, the associated mixing should be quantified.While it is straightforward to quantify mixing in DNS, it is challenging inlaboratory flows. Integrative methods are typically the most accurate, however, these generally sum the effects of all of the mechanisms that causemixing [see Prastowo et aL, 2008]. Specifically, mixing caused by turbulence generated at no slip boundaries (the bottom and side walls) must beseparated from mixing caused by shear instabilities. Micro conductivity-temperature (CT) probes and laser induced fluorescence could also be usedto measure mixing. CT probes require a large channel so that the action ofthe probes does not significantly alter the flow or mixing.The long term goal in the Fraser River estuary is to conduct higher resolution sampling over the full range of freshwater discharge. A CTD witha higher sampling rate would provide greater horizontal and vertical resolution of the density field. In addition, the simultaneous use of multiple94Chapter 5. Conclusionechosounders could provide a more direct measure of the phase speed andwavelength of internal waves. The work described in Chapter 3 focussed onobserving and identifying shear instabilities. Future work should include acomponent focussed on quantifying the mixing associated with these instabilities. This would require either multiple transects in rapid succession ormultiple moorings along the length of the salinity intrusion. Future studies of mixing should also examine three dimensional aspects, particularilyin the rivers bends. These features may have a strong infuence on shearinstabilities and associated mixing.95BibliographyHOGG, A. McC. & IvEY, G.N. 2003 The Kelvin-Helmholtz to Holmboeinstability transition in stratified exchange flows. J. Fluid Mech. 477, 339—362.HOLMBOE, J. 1962 On the behavior of symmetric waves in stratified shearlayers. Geofys. Publ. 24, 67—112.LAWRENCE, G.A., PIETERS, R., ZAREMBA, L., TEDFORD, T., Cu, L.,GRECO, S. & HAMBLIN, P. 2004 Summer exchange between HamiltonHarbour and Lake Ontario. Deep-Sea Res. II 51, 475—487.PRASTOWO, T., GRIFFITHS, R.W., HUGHES, G.O. & HOGG, A.MCC.2008 Mixing efficiency in controlled exchange flows. J. Fluid Mech. 600,235—244.ZHu, D. & LAWRENCE, G.A. 2001 Holmboe’s instability in exchangeflows. J. Fluid Mech. 429, 391—409.96
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Laboratory, field and numerical investigations of Holmboe's instability Tedford, Edmund W. 2009
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Title | Laboratory, field and numerical investigations of Holmboe's instability |
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Tedford, Edmund W. |
Publisher | University of British Columbia |
Date | 2009 |
Date Issued | 2009-11-09 |
Description | The instabilities that occur at a sheared density interface are investigated in the laboratory, the Fraser River estuary and with Direct Numerical Simulations (DNS). In the laboratory, symmetric Holmboe instabilities are observed during steady, maximal two-layer exchange flow in a long channel of rectangular cross section. Internal hydraulic controls at each end of the channel isolate the subcritical region within the channel from disturbances in the reservoirs. Inside the channel, the instabilities form cusp-like waves that propagate in both directions. The phase speed of the instabilities is consistent with linear theory, and increases along the length of the channel as a result of the gradual acceleration of each layer. This acceleration causes the wavelength of any given instability to increase in the direction of flow. As the instabilities are elongated new instabilities form, and as a consequence, the average wavelength is almost constant along the length of the channel. In the Fraser River estuary, a detailed stability analysis is conducted based on the Taylor-Goldstein (TG) equation, and compared to direct observations in the estuary. We find that each set of instabilities observed coincides with an unstable mode predicted by the TG equation. Each of these instabilities occurs in a region where the gradient Richardson number is less than the critical value of 1/4. Both the TG predictions and echosoundings indicate the instabilities are concentrated either above or below the density interface. These ‘one-sided’ instabilities are closer in structure to the Holmboe instability than to the Kelvin-Helmholtz instability. Although the dominant source of mixing in the estuary appears to be caused by shear instability, there is also evidence of small-scale overturning due to boundary layer turbulence when the tide produces strong near-bed velocities. Many features of the numerical simulations are consistent with linear theory and the laboratory experiments. However, inherent differences be tween the DNS and the experiments are responsible for variations in the dominant wavenumber and amplitude of the wave field. The simulations exhibit a nonlinear ‘wave coarsening’ effect, whereby the energy is shifted to lower wavenumber in discrete jumps. This process is, in part, related to the occurrence of ejections of mixed fluid away from the density interface. In the case of the laboratory experiment, energy is transferred to lower wavenumber by the ‘stretching’ of the wave field by a gradually varying mean velocity. This stretching of the waves results in a reduction in amplitude compared to the DNS. The results of the comparison show the dependence of the nonlinear evolution of a Holmboe wave field on temporal and spatial variations of the mean flow. |
Extent | 4060913 bytes |
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File Format | application/pdf |
Language | Eng |
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Electronic Theses and Dissertations (ETDs) 2008+ |
Date Available | 2009-11-09 |
Provider | Vancouver : University of British Columbia Library |
DOI | 10.14288/1.0063138 |
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Doctor of Philosophy - PhD |
Program |
Civil Engineering |
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Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2009-05 |
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UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/14718 |
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