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Observation and analysis of shear instability in the Fraser River estuary Tedford, E. W.; Carpenter, J. R.; Pawlowicz, Rich; Pieters, R.; Lawrence, Gregory A. 2009-11-30

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Observation and analysis of shear instability in theFraser River estuaryE. W. Tedford,1J. R. Carpenter,1R. Pawlowicz,2R. Pieters,2and G. A. Lawrence1Received 2 February 2009; revised 1 May 2009; accepted 14 July 2009; published 4 November 2009.[1] We investigate the occurrence of shear instability in the Fraser River estuary.Instabilities observed with an echo sounder are compared with a linear stability analysisbased on observed velocity and density profiles. We find that each set of observedinstabilities coincides with an unstable mode predicted by the Taylor-Goldstein equation.Each of these instabilities occurs in a region where the gradient Richardson number is lessthan the critical value of 1/4. Both the Taylor-Goldstein predictions and the echosoundings indicate the instabilities are concentrated either above or below the densityinterface. This ‘‘one sidedness’’ is in contrast to the archetypal Kelvin-Helmholtzinstability. Although the dominant source of mixing in the estuary appears to be caused byshear instability, when the tide produces strong near-bed velocities, small-scaleoverturning due to boundary layer turbulence is apparent throughout the depth.Citation: Tedford, E. W., 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., 114, C11006, doi:10.1029/2009JC005313.1. Introduction[2] Shear instabilities occur in highly stratified estuariesand can influence the large-scale dynamics by redistributingmass and momentum. Specifically, shear instabilities havebeen found to influence salinity intrusion in the Fraser Riverestuary [Geyer and Smith, 1987; Geyer and Farmer, 1989;MacDonald and Horner-Devine, 2008]. We describe recentobservations in this estuary and examine the shear andstratification that lead to instability. The influence of longtime scale processes such as freshwater discharge and thetidal cycle are also discussed.[3] Rather than relying on a bulk or gradient Richardsonnumber to assess stability we use numerical solutions of theTaylor-Goldstein (TG) equation based on observed profilesof velocity and density. This approach has been used withsome success in the ocean [e.g., Moum et al., 2003] but,with the exception of the simplified application by Yoshidaet al. [1998], has not been applied in estuaries. Solving theTG equation provides the growth rate, wavelength, phasespeed and mode shape of the instabilities. We compare thesepredicted wave properties with instabilities observed usingan echo sounder.[4] Geyer and Farmer [1989] found that instabilities inthe Fraser River estuary were most apparent during ebb tidewhen strong shear occurred over the length of the salinityintrusion. They outlined a progression of three phases ofincreasingly unstable flow that occurs over the course of theebb. In the first phase, strain sharpens the density interface;shear is stronger than during flood, but insufficient to causeshear instability. In the second phase, the lower layerreverses and shear between the fresh and saline layersincreases. Shear instability and turbulent mixing are con-centrated at the pycnocline rather than in the bottomboundary layer. By the third phase of the ebb, shearinstability has completely mixed the two layers leavinghomogeneous water throughout the depth. During floodthere is some mixing, however it is concentrated at thefront located at the landward tip of the salinity intrusion.Similarly, MacDonald and Horner-Devine [2008], studyingmixing at high freshwater discharge (7000 m3sC01), foundthat two to three times more mixing occurred during ebbtide than during flood. The present analysis is focused onthe ebb tide at high and low freshwater discharge, althoughsome results during flood tide are also presented.[5] The paper is organized as follows. The setting andfield methods are described in section 2. The generalstructure of the salinity intrusion is described in section 3.In section 4 we present the background theory needed toperform stability analysis in the Fraser River estuary. Insection 5 predictions from the stability analysis are com-pared with observations. In section 6 the source of relativelysmall-scale overturning is briefly discussed. In section 7 theresults of the stability analysis are discussed followed byconclusions in section 8.2. Field Program2.1. Site Description and Data Collection[6] Data were collected in the main arm of the FraserRiver estuary, British Columbia, Canada (Figure 1). Theestuary is 10 to 20 m deep with a channel width of 600 to900 m. Cruises were conducted on 12, 14, and 21 June 2006and 10 March 2008. Here we present one transect from eachof the June 2006 cruises and three transects from the MarchJOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, C11006, doi:10.1029/2009JC005313, 20091Department of Civil Engineering, University of British Columbia,Vancouver, British Columbia, Canada.2Department of Earth and Ocean Sciences, University of BritishColumbia, Vancouver, British Columbia, Canada.Copyright 2009 by the American Geophysical Union.0148-0227/09/2009JC005313$09.00C11006 1of142008 cruise (see Table 1). The freshwater discharge duringthe June 2006 transects was typical of the freshet atapproximately 6000 m3sC01. During the March 2008 trans-ects, freshwater discharge was near the annual minimum at850 m3sC01. In June 2006, transects were made during bothebb and flood tide. In March 2008, transects cover most of asingle ebb tide (Figure 2). The tides in the Strait of Georgiahave M2 and K1 components of similar amplitude (approx-imately 1 m) resulting in strong diurnal variations. The tidalrange varies from approximately 2 m during neap tides toapproximately 4.5 m during spring tides. During both the2006 and 2008 observations the tidal range was approxi-mately 3 m.[7] The distance salinity intrudes landward of SandHeads, i.e., the total length of the salinity intrusion, variesconsiderably with tidal conditions and freshwater discharge.Ward [1976] found the maximum length of the intrusionoccurred just after high tide and varied from 8 km athigh discharge (9000 m3sC01) to 31 km at low discharge(850 m3sC01). Geyer and Farmer [1989] found that, ataverage discharge (3000 m3sC01), the maximum length ofthe intrusion matched the horizontal excursion of the tides(10 to 20 km) and, similar to Ward [1976], occurred just afterhigh tide. Kostachuk and Atwood [1990] found that theminimum length of the salinity intrusion typically occurred1 h after low tide. The longest intrusion they observed at lowtide was approximately 20 km. They predicted that completeflushing of salt from the estuary would occur on most daysduring the freshet (freshwater discharge >5000 m3sC01).2.2. Field Methods[8] Data along the six transects were collected by driftingseaward with the surface flow while logging velocity andecho sounder data and yoyoing a conductivity-temperature-depth (CTD) profiler. The velocity measurements weremade with a 1200 kHz RDI acoustic Doppler currentprofiler (ADCP) sampling at 0.4 Hz with a vertical resolu-tion of 250 mm. The velocities were averaged over 60 s toremove high-frequency variability. The echo soundingswere made with a 200 kHz Biosonics sounder sampling at5 Hz with a vertical resolution of 18 mm. Profile data werecollected with a Seabird 19 sampling at 2 Hz. Selected echosounder, ADCP and CTD data are shown in Figure 3. Asindicated by the superimposed density profiles, stronggradients in density are generally associated with a strongecho from the sounder.[9] The CTD was profiled on a load bearing data cablethat provided constant monitoring of conductivity, temper-ature and depth. These data allowed us to quickly identifythe front of the salinity intrusion and avoid direct contact ofthe instrument with the bottom. To increase the verticalresolution of the profiles, the CTD was mounted horizon-tally with a fin to direct the sensors into the flow. In thisconfiguration, the instrument was allowed to descend rap-idly and then was raised slowly (0.2–0.4 m sC01) relying onhorizontal velocity of the water relative to the CTD to flushthe sensors. The upcast, which had higher vertical resolu-tion, was in reasonable agreement with the echo intensityfrom the sounder. On the few occasions that the higher-resolution upcast did not coincide with the appearance ofinstabilities in the echo sounder, we used the downcast. Thetotal number of CTD casts we were able to perform variedfrom transect to transect depending on field conditions(surface velocity, shear, ship traffic, woody debris).3. General Description of the Salinity Intrusion[10] We observed important differences in the structure ofthe salinity intrusion between high and low freshwaterdischarge. At high discharge, our observations were similarto those described by Geyer and Farmer [1989], where thesalinity intrusion had a two-layer structure resembling aclassic salt wedge. At low discharge, however, the salinityintrusion exhibited greater complexity.3.1. High Discharge[11] During flood tide, mixing was concentrated near thesteep front at the landward tip of the salt wedge (2.7–3.03 km in Figure 3c). During ebb tide, the steep front wasreplaced by a gently sloping pycnocline (Figure 3b, land-ward of 11.6 km) and there was no apparent concentrationof mixing at the landward tip of the salt wedge (not shown).[12] We will focus on the wave-like disturbances thatoccur on the pycnocline especially during ebb tide. TheFigure 1. Map of the lower 27 km of the Fraser River. The locations of the six transects are markedT1–T6. The mouth of the river (Sand Heads) is located at 49C176 60N and 123C176 180W.Table 1. Details of Transects Shown in Figures 1 and 2aTransectDischarge(m3sC01) Tidex(km)DU(m sC01)Dr(kg mC03) h (m) J1 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 0.3aThe location indicates the distance upstream from the mouth (SandHeads).C11006 TEDFORD ET AL.: INSTABILITY IN THE FRASER RIVER ESTUARY2of14C11006largest of these were observed during transect 1 (Figure 3a,8.7 to 8.9 km, between depths of 3 and 10 m). Thesedisturbances occurred within the upper layer as it passedover the nearly stationary water below a depth of 10 m.Smaller amplitude disturbances were observed during tran-sect 2 (Figure 3b, 11.05 km). In our application of the TGequation we will show that disturbances like these are aresult of shear instability.[13] Not all of the disturbances on the pycnocline are aresult of shear instability. For example, for most of thevelocity and density profiles collected during transect 3(Figure 3c) the TG equation does not predict instability. Thedisturbances seen from 2.5 to 3.0 km are caused by the largesand waves on the bottom (the thick black line in the echosounding). The crests of the sand waves were typically 30 mapart and 1 to 2 m high, and were found over most of theriver surveyed during high discharge (2.5 km to 15 km).During flood tide, flow over these sand waves causedparticularly regular disturbances on the pycnocline.3.2. Low Discharge[14] At low discharge, at the beginning of the ebb, thefront of the salinity intrusion was located between 28 and30 km from Sand Heads. Unlike the observations at highdischarge a well defined front was not visible in the echosounder, and CTD profiles were needed to identify itslocation. Seaward of the front (Figure 4a), the echo sounderand the CTD profiles show a multilayered structure withmore complexity than was observed at high discharge. Atthis early stage of the ebb, the CTD profiles generally showpartially mixed layers separated by several weak densityinterfaces.[15] Later in the ebb, during transect 5 (Figure 4b), nearbottom velocities turn seaward and the velocity shearbetween the top and the bottom increases. At maximumebb (transect 6, Figure 4c), the shear increases further,reaching a maximum of approximately 2.5 m sC01over adepth of 12 m. Mixed water occurs at both the surface andthe bottom resulting in an overall decrease in the verticaldensity gradient. By the time transect 6 is complete the ebbflow is decelerating. The salinity intrusion continues topropagate seaward until low tide but, given its length andvelocity it does not have sufficient time to be completelyflushed from the estuary. During the next flood the mixedwater remaining in the estuary allows a complex densitystructure to develop similar to that seen early in theobserved ebb. This differs from the behavior at highfreshwater discharge when nearly all of the seawater isflushed completely from the estuary at least once a day.4. Stability of Stratified Shear Flows4.1. Taylor-Goldstein Equation[16] Following Taylor [1931] and Goldstein [1931], weassess the stability of the flow by considering the evolutionof perturbations on the background profiles of density andhorizontal velocity, denoted here by r(z) and U(z), respec-tively. If the perturbations to the background state aresufficiently small they are well approximated by the linearequations of motion. It then suffices to consider sinusoidalperturbations, represented by the normal mode form eik(xC0ct),where x is the horizontal position and t is time. Here k =2p/lis the horizontal wave number with l the wavelength, c =cr+ iciis the complex phase speed. If we further assume thatFigure 2. Observed tides at Point Atkinson (heavy line) and New Westminster (thin line) for the 4 daysof field observations. The Point Atkinson data are representative of the tides in the Strait of Georgiabeyond the influence of the Fraser River. New Westminster is located 37 km upstream of the mouth of theriver at Sand Heads (see Figure 1). The records are both referenced to mean sea level at Point Atkinson.The duration of the six transects are marked T1–T6.C11006 TEDFORD ET AL.: INSTABILITY IN THE FRASER RIVER ESTUARY3of14C11006Figure 3. Echo soundings observed during high discharge on (a) transect 1, ebb tide; (b) transect 2, ebbtide; and (c) transect 3, flood tide. The shading scales with the log of the echo intensity with blackcorresponding to the strongest echoes. Selected velocity profiles (red) from the ADCP and densityprofiles (blue dashed) from the CTD are superimposed (not all are shown). The black line indicates thelocation of the boat in the middle of the cast, as well as the zero reference for the velocity and st. Thevelocity profile was calculated as a 1 min average centered on the time of the CTD cast. The undulationsin the bed of the river (thick black line at the bottom of the echo soundings) are a result of sand waves.C11006 TEDFORD ET AL.: INSTABILITY IN THE FRASER RIVER ESTUARY4of14C11006Figure 4. Echo soundings during low discharge observed during (a) transect 4, early ebb; (b) transect 5,mid ebb; and (c) transect 6, late ebb. The shading scales with the log of the echo intensity with blackcorresponding to the strongest echoes. Note that the scale of the shading is the same in Figures 4a–4c.Velocities (red) from the ADCP and densities (blue dashed) from the CTD are superimposed. The blackline indicates the location of the boat in the middle of the cast, as well as the zero reference for thevelocityandst.Thevelocityprofilewascalculatedasa1minaveragecenteredonthetimeoftheCTDcast.C11006 TEDFORD ET AL.: INSTABILITY IN THE FRASER RIVER ESTUARY5of14C11006the flow is incompressible, Boussinesq, inviscid, and non-diffusive, we arrive at the Taylor-Goldstein (TG) equationd2^ydz2þN2U C0 cðÞ2C0d2U=dz2U C0 cC0 k2"#^y ¼ 0; ð1Þwhere the stream function is given by y(x, z, t)=^y(z)eik(xC0ct)and N2(z)=C0(g/r0)(dr/dz) represents theBoussinesq form of the squared buoyancy frequency with areference density, r0.[17] Solutions to the TG equation consist of eigenfunc-tion-eigenvalue sets {^y(z), c}, for each value of k. Each set{^y(z), c} is referred to as a mode, and the solution mayconsist of the sum of many such modes for a single k. Thebackground flow, represented by U(z) and r(z), is then saidto be unstable if any modes exist that have ci6¼ 0. In thiscase the small perturbations grow exponentially at a rategiven by kci. In general, unstable modes are found over arange of k, and it is the mode with the largest growth ratethat is likely to be observed. Although they are based onlinear analysis, TG predictions of the wave properties, k andc, typically match those of finite amplitude instabilitiesobserved in the laboratory [Thorpe, 1973; Lawrence et al.,1991; Tedford et al., 2009].4.2. Miles-Howard Criterion[18] A useful criterion to assess the stability of a givenflow without solving the TG equation was derived by Miles[1961] and Howard [1961]. They found that if the gradientRichardson number, Ri(z)=N2/(dU/dz)2, exceeds 1/4everywhere in the profile, then the TG equation has nounstable modes; that is, cimust be zero for all modes. Inother words, Ri C21 1/4 everywhere is a sufficient conditionfor stability, referred to as the Miles-Howard criterion. Notethat if Ri < 1/4 at some location, instability is possible, butnot guaranteed.[19] Despite the inconclusive nature of the Miles-Howardcriterion for determining instability, it is often employed asa sufficient condition for instability in density stratifiedflows, and has been found to have reasonable agreementwith observations [Thorpe, 2005, pp. 201–204]. Lookingspecifically at the Fraser River estuary, Geyer and Smith[1987] were able to compute statistics of Ri and show thatdecreases in Ri were accompanied by mixing in the estuary.4.3. Mixing Layer Solution[20] Since the TG equation is an eigenvalue problem withvariable coefficients, analytical solutions can only beobtained for the simplest profiles, and recourse is usuallymade to numerical methods [e.g., Hazel, 1972]. However,the available analytical solutions are often a useful point ofdeparture. We look at one such solution that closelyapproximates conditions found in the estuary during highdischarge. This solution is based on the simple mixing layermodel of Holmboe (described by Miles [1963]).[21] Inageneralformofthemodel,thevelocityanddensityprofiles are represented by the hyperbolic tangent functionsUzðÞ¼DU2tanh2 z C0dðÞhC18C19andr zðÞ¼C0Dr2tanh2zdC18C19þr0: ð2Þwhere h is the shear layer thickness, d is the thickness of thedensity interface. The parameterd allows for a vertical offsetinthe positions ofthe shearlayer and density interface. Inthesimplest case the shear layer and density stratification haveequal thickness, giving R=h/d = 1, and they coincide in theirvertical positions so that the asymmetry a =2d/h = 0. In thiscase, Ri(z) is at its minimum at the center of the mixing layer(z = 0), and is equal to the bulk Richardson number J = gDrh/r0(DU)2. When the bulk Richardson number (i.e., theminimum Ri) drops below 1/4, flows with R = 1 and a =0become unstable. The resulting instabilities are of theKelvin-Helmholtz (KH) type, in which the shear layer rollsup to form an array of billows that are stationary with respectto the mean flow, and which display large overturns indensity [Thorpe, 1973].[22] It is not generally the case that J C21 1/4 results instability. For example, if the density interface is relativelysharp (R > 2) an additional mode of instability, the Holmboemode, is excited [Alexakis, 2005]. In this case, the range ofJ over which instability occurs extends above 1/4. That is,Ri < 1/4 somewhere in z at the same time as J > 1/4. While itis generally true that flows with higher J are subject to lessmixing by shear instabilities, by itself, J does not indicatewhether or not a flow is unstable.[23] For simplicity, the analytical solution of Holmboe’smixing layer model assumes the flow is unbounded in thevertical. In our analysis we include boundaries at the topand bottom where^y must satisfy the boundary condition^y = 0. The presence of these boundaries tends to extendthe range of unstable wave number to longer wavelengths[Hazel,1972].However,inthecasesconsideredhere,atthewave number of maximum growth, the boundaries havelittle or no impact on k and c.4.4. Solution of the TG Equation for Observed Profiles[24] We use the numerical method described by Moum etal. [2003] to generate solutions to the TG equation based onmeasured velocity and density profiles. Whenever possiblewe use velocity and density profiles collected at the up-stream edge of apparent instabilities in the echo soundings.The velocity profile, a 60 s average, is an average over oneor more instabilities (the instabilities have periods <60 s).This averaging reduces the influence of individual instabil-ities on the velocity profile, which in the TG equation, istaken to represent the background velocity profile. Thevelocity profile is then smoothed in the vertical using alow-pass filter (removing vertical wavelengths <2 m). Thedensity profile is smoothed by fitting a linear function, andone or more tanh functions (one for each density interface).By using smooth profiles we are effectively ignoringinstability associated with small-scale variations in theprofiles.[25] Because the point of observation moves in time, i.e.,the boat is drifting seaward, predicted wavelengths from theTG equation cannot be compared directly to the wavelengthof instabilities as they appear in the echo soundings. Thewavelength predicted with the TG solution must be shiftedto account for the speed of the instabilities with respect tothe speed of the boat:l* ¼vbcrC0 vbC12C12C12C12C12C12C12C12l: ð3ÞC11006 TEDFORD ET AL.: INSTABILITY IN THE FRASER RIVER ESTUARY6of14C11006Here vbis the velocity of the boat and crand l are the phasespeed and wavelength predicted with the TG equation. Thepredicted apparent wavelength, l*, is directly comparableto observations made from the moving boat. Seim andGregg [1994] used a similar approach for estimating thewavelength of observed features.[26] As well as giving a wavelength, phase speed, andgrowth rate for each unstable mode, the TG solutions alsogive an eigenfunction that describes the vertical structure ofthe growing mode. The vertical displacement eigenfunction^h(z)=C0^y/(U C0 c) is particularly useful. At the location in zwhere j^hj is a maximum we expect to see evidence ofinstabilities in the echo soundings.5. Results[27] In this section we use J, Ri(z) and solutions of the TGequation to assess the stability of six sets of velocity anddensity profiles (one from each of the six transects). Eachset of profiles was chosen to coincide with evidence ofinstability in the echo soundings.5.1. Ebb During High Discharge: Transect 1[28] The selected velocity and density profiles fromtransect 1 are shown in Figure 5. The corresponding valueof J for these profiles is 0.29 (see Table 1). The stabilityanalysis yields two modes of instability. The fastest growingmode is unstable for wavelengths greater than 11 m and hasa peak growth rate of 0.025 sC01(doubling time of 28 s)occurring at a wavelength of 21 m. The phase speed of theinstability at this wavelength is C01.02 m sC01, where thenegative indicates a seaward direction. Given this phasespeed and the seaward drift of the boat (C02.2 m sC01), anapparent wavelength of 39 m is calculated.[29] Echo soundings collected at the same time, Figure5c, show clear evidence of instabilities. The prediction isfound to be similar to, although shorter than, the approxi-mately 50 m wavelength of the observed instabilities. Themaximum displacement of the predicted instabilities islocated at a depth of 7.6 m (indicated by the horizontalline), closely matching the depth of the observed instabil-ities. Both the observed and predicted instability occurwithin the region of shear above the maximum gradient inr (at a depth of 9 m). As indicated by the gray shading, thisregion of high shear and low gradient in r corresponds to Ri< 1/4.[30] For the set of profiles shown in Figure 5 the TGequation predicts a second, weaker, unstable mode locatedat a depth of 2.5 m. This mode is associated with theinflection point (d2U/dz2= 0) in the velocity profile at thisdepth. Because there is very little density stratification andFigure 5. (a) Velocity and (b) density profiles observed during transect 1 (12 June 2006, 0805 LT,8.9 km upstream of Sand Heads). The smooth profiles used in the stability analysis are shown asthick black lines, and the observed data are plotted as points. The gray shading indicates regions inwhich Ri < 1/4. The black horizontal line indicates the location of maximum displacement (j^hj) forthe most unstable mode predicted with the TG equation. The thin lines in Figure 5b show thedisplacement functions for each of the unstable modes. The functions are scaled in proportion to thegrowth rate. (c) A close up of the echo sounding logged near the location of the profiles is shownand includes a scale indicating the apparent wavelength predicted by the TG equation. The arrow atthe top of image indicates the approximate location of the density and velocity measurements. In thiscase, the velocity is averaged over a distance of approximately 130 m.C11006 TEDFORD ET AL.: INSTABILITY IN THE FRASER RIVER ESTUARY7of14C11006Figure 6. (a) Velocity and (b) density profiles observed during transect 2 (14 June 2006, 0821 LT,11.1 km upstream of Sand Heads). See Figure 5 for details. In this case, the velocity is averaged overapproximately 110 m.Figure 7. (a) Velocity and (b) density profiles observed during transect 3 (21 June 2006, 1238 LT,2.66 km upstream of Sand Heads). See Figure 5 for details. In this case, the velocity is averaged overapproximately 30 m.C11006 TEDFORD ET AL.: INSTABILITY IN THE FRASER RIVER ESTUARY8of14C11006hence weak echo intensity at this depth we are unable toconfirm or deny the presence of this mode in the echosoundings.5.2. Ebb During High Discharge: Transect 2[31] In transect 2 a single hyperbolic tangent gives a goodfit to the measured density profile (Figure 6b). Because ofdifficulties in profiling, the density profile at this locationwas missing data below 12 m. Data from the previous cast,taken 60 m upstream, were used below 12 m. This cast isexpected to be sampling water of similar density below thisdepth.[32] In this case the stability analysis of the profilesresults in a single mode of instability. The mode is unstablefor wavelengths from 10 m to 35 m with a peak growth rateof 0.02 sC01(doubling time of 35 s) occurring at a wave-length of 17 m. The phase speed of the instability at thiswavelength is C00.51 m sC01. Given the drift velocity ofC01.9 m sC01, an apparent wavelength of 24 m is calculated.This prediction is found to be similar to, although longerthan, the approximately 18 m wavelength of the smallinstabilities appearing in the echo sounding (Figure 6c).The maximum displacement of the predicted instabilities islocated at a depth of 10.6 m, closely matching the depth ofthe observed instabilities.5.3. Flood During High Discharge: Transect 3[33] Despite the occurrence of Ri < 1/4 the stabilityanalysis of the profiles in Figures 7a and 7b does not findany unstable modes. Echo soundings collected during theflood generally show features on the pycnocline that werewellcorrelatedwithsandwaves(Figure7c).Thesecorrelatedfeatures are likely controlled by the hydraulics of the flowover the sand waves.[34] There was very little evidence of instabilities inde-pendent of these sand waves. There appear to be some wave-like features on the pycnocline that are shorter (C2510 m) thanthe sand waves, however, these are not well resolved by theecho sounder (e.g., depth of 9 m at x = 60 m). Properlyassessing the flow over these sand waves would require atleast two or three sets of density and velocity profiles persand wave, more than we were able to obtain.5.4. Low Freshwater Discharge5.4.1. Early Ebb During Low Discharge: Transect 4[35] At low discharge, during the ebb tide, shear anddensity stratification are spread over the entire depth (seeFigure 4). The bulk shear layer thickness, h, is thereforegreater than at high discharge, where shear and stratificationwere concentrated at a single, relatively thin interface. Theincrease in the vertical extent of the shear results in a greaterbulk Richardson number despite a decrease in the overallstrength of the density stratification, Dr (see Table 1).[36] The density and velocity profiles collected early inthe ebb (transect 4, Figure 8) consist of a number of layers.The stability analysis yields two modes of instability. Themost unstable mode has a peak growth rate of 0.023 sC01occurring at a wavelength of 10.3 m with a phase speed ofC00.86 m sC01. Given this phase speed and the seaward driftof the boat (1.6 m sC01), an apparent wavelength of 22 m iscalculated. This is very similar to the wavelength of thelargest instability in Figure 8c. This mode has a maximumdisplacement at a depth of 2.5 m, closely matching thelocation of the observed instabilities.5.4.2. Mid Ebb During Low Discharge: Transect 5[37] The instabilities in Figure 9c were observed 1 h laterand approximately 3 km downstream from Transect 4. Ther profile (Figure 9b) again displays a number of layersFigure 8. (a) Velocity and (b) density profiles observed during transect 4 (10 March 2008, 1120 LT,22.4 km upstream of Sand Heads). See Figure 5 for details. In this case, the velocity is averaged overapproximately 90 m.C11006 TEDFORD ET AL.: INSTABILITY IN THE FRASER RIVER ESTUARY9of14C11006Figure 10. (a) Velocity and (b) density profiles observed during transect 6 (10 March 2008, 1434 LT,7.6 km upstream of Sand Heads). See Figure 5 for details. In this case, the velocity is averaged overapproximately 130 m.Figure 9. (a) Velocity and (b) density profiles observed during transect 5 (10 March 2008, 1221 LT,19.6 km upstream of Sand Heads). See Figure 5 for details. In this case, the velocity is averaged overapproximately 140 m.C11006 TEDFORD ET AL.: INSTABILITY IN THE FRASER RIVER ESTUARY10 of 14C11006consisting of high-gradient steps. However, the layers arenot evident in the measured velocity profile (Figure 9a), aswas the case in Figure 8, and the overall shape of thevelocity profile is more linear.[38] The CTD cast is one of the few collected during thestudy where the instrument passed through an overturn inthe pycnocline (depth of approximately 3.8 m). Consistentwith the small amplitude of the instabilities in the echosounder, the overturn in the density profile has only water ofintermediate density; that is, no surface or bottom water isobserved in the overturn.[39] The TG equation predicts an unstable mode with apeak growth rate (0.03 sC01) at a wavelength of 14 m with aphase speed of C01.2 m sC01. The apparent wavelength ispredicted to be 32 m, whereas the features in the echosounder range in horizontal length from approximately 10 to40 m, with the largest being near the TG prediction (C2530 m).The predicted maximum in the displacement eigenfunctionoccurs at a depth of 4.2 m closely matching the depth of theinstabilities.5.4.3. Late Ebb During Low Discharge: Transect 6[40] In the later stages of the ebb, during transect 6(Figure 10), the shear has increased such that J is reducedto approximately 0.3. Unlike most of the other profilescollected during low or high discharge the density profilehas no homogeneous layers, and shows small-scale (i.e., onthe scale of the instrument resolution) overturning through-out the depth. In these profiles Ri is below critical through-out most of the depth aside from at the density interface.[41] The most unstable mode predicted with the TGequation is located at a depth of 5.6 m and has a maximumgrowth rate of 0.019 sC01at an apparent wavelength of 65m.This is close to, but longer than, the largest features in theecho sounder (approximately 50 m).6. Small-Scale Overturns and Bottom Stress[42] In Figure 10 there are no features in the echosoundings that are associated with the small-scale overturnsin r below a depth of 7 m, and although our solutions to theTG equation suggest unstable modes, these are both locatedwell above a depth of 7 m. To further examine the source ofthese overturns we compare selected density profiles fromeach of the low discharge transects (Figure 11). In thedensity profile from transect 4, small-scale overturns arerare or completely absent (Figure 11, T4). Approximately2 h later, during transect 5, just one profile exhibits thesesmall-scale overturns (Figure 11, T5). This cast was per-formed at the shallow constriction in the river associatedwith the Massey Tunnel (Figure 4b, 18 km). In this case thesmall-scale overturns in the profile occur only below thepycnocline suggesting that the stratification within thepycnocline is confining the overturns to the lower layer.By maximum ebb, small-scale overturns occur throughoutthe depth (Figure 11, T6).[43] The presence of these small-scale overturns is appar-ent, although not immediately obvious, in the echo sound-ings in Figure 4. Note that the scale of the shading is thesame in Figures 4a–4c and that there is a gradual increase(darkening) in background echo intensity from early to lateebb (transects 4 to 6). This increase in echo intensity isattributed to the small-scale overturning observed in thedensity profiles. Early in the ebb the dark shading associ-ated with high echo intensity is concentrated at the densityinterfaces (transect 4). Otherwise, at this time, echo intensityis low (light shading) corresponding to an absence of small-scale overturns in the density profiles (e.g., Figure 11, T4).At this stage of the ebb, near-bottom velocities are close tozero and bottom stress is expected to be negligible. Intransect 5 (Figure 4b) there is an increase in echo intensityas the flow passes over the Massey Tunnel (18 km). At thislocation and during this stage of the ebb, near bottomvelocity increases to approximately 0.2 m sC01at 1 m abovethe bed. In this case the small-scale overturns in the profileoccur only below the pycnocline (Figure 11, T5) suggestingthat the stratification within the pycnocline is confiningbottom generated turbulence to the lower layer. Nearmaximum ebb, during transect 6, near bottom velocitiesreach 0.5 m sC01at 1 m above the bed. By this stage, highecho intensity and small-scale overturns occur throughoutthe depth (Figure 11, T6) suggesting that bottom generatedturbulence has reached the surface despite the presence ofstratification.7. Discussion[44] Although combining echo soundings, velocity, anddensity measurements to study shear flows is not in itselfnovel, even for studies in the Fraser estuary [e.g., Geyer andSmith, 1987], efforts in the present study were focussed onsimultaneously measuring the details of the flow and theshear instabilities. Our strategy of drifting slowly with theupper level flow allowed acoustic imaging to capture shearinstabilities similar to those observed in laboratory andnumerical simulations [e.g., Tedford et al., 2009]. Densityand velocity measurements also allowed us to analyze thesefeatures using a method more typically applied to laboratoryexperiments, namely direct application of the TG equation.This analysis has refined our understanding of instabilityand mixing in the Fraser River estuary.Figure 11. Selected density profiles from transectsperformed at low freshwater discharge. The profiles werecollected at t = 10h53, 12h36, and 14h25 at x = 26.2, 17.9,and 8.8 km (transects 4, 5, and 6, respectively).C11006 TEDFORD ET AL.: INSTABILITY IN THE FRASER RIVER ESTUARY11 of 14C110067.1. One-Sided Instability[45] In all five of the cases that the TG equation predictedthe occurrence of unstable modes, the bulk Richardsonnumber, J, was greater than 1/4. This result suggests themixing layer model and associated J (see section 4.3) arenot adequate for describing the stability of the measuredprofiles. In all of these unstable cases, both the region ofRi(z) < 1/4 and the depth of the maximum in the displace-ment eigenfunction (j^h(z)j) were vertically offset from themaximum gradient in density (dr/dz). This offset betweenthe depth of the predicted region of instability and thedensity interface is due to asymmetry between the densityand velocity profiles. This suggests that a minimum of threebulk parameters (J, R, a) are required if the stability is to berepresented by the simplified profiles of equation (2).[46] Laboratory models and direct numerical simulations(DNS) of asymmetry result in one-sided instabilities thatresemble the features in the echo soundings in Figures 5c,6c, and 8c [e.g., Lawrence et al., 1991; Yonemitsu, 1991;Carpenter et al., 2007]. Similar observations were made inthe Strait of Gibraltar by Farmer and Armi [1998] and in astrongly stratified estuary by Yoshida et al. [1998]. In thesecases the instabilities were attributed to one-sided modes.One-sided modes are part of a general class of instabilitythat includes the Holmboe mode. In contrast to the classicKH mode, the Holmboe mode is a result of the unstableinteraction of gradients in density and gradients in shear (N2and d2U/dz2in equation (1)) and can occur at relativelyhigh values of J [Holmboe, 1962].[47] There are a number of potential sources of asymme-try in the Fraser estuary. The most obvious is the differencein the bottom and surface boundary condition. If the stressacting on these boundaries is not equal and opposite, i.e., ifit is unbalanced, then there is the potential for asymmetry.During low freshwater discharge the presence of multiplelayers of varying thickness adds further irregularity andpotential asymmetry to the profiles. Although some labora-tory models of stratified flows successfully generate sym-metric conditions [e.g., Thorpe, 1973; Tedford et al., 2009],many others result in asymmetry [e.g., Lawrence et al.,1991; Yonemitsu et al., 1996; Pawlak and Armi, 1998; Zhuand Lawrence, 2001]. In most of these cases asymmetry inthe flow results from the geometry of the channel, such as asill causing localized acceleration of the lower layer. In thearrested salt wedge experiments of Yonemitsu et al. [1996],asymmetry was associated withsecondarycirculation. Giventhe common occurrence of asymmetry in the laboratory it isnot surprising to find asymmetry in nature.7.2. Mixing[48] Linear stability analysis does not provide quantitativepredictions of mixing. When one-sided instabilities aremodeled using DNS at the values of J observed here thecomplete overturning of the density interface normallyassociated with KH billows does not occur. Figure 12 showsa schematic of a one-sided instability and a Kelvin-Helholtzinstability. Although one-sided instabilities are offset fromthe region of maximum density gradient they have beenfound to be responsible for considerable mixing [Smyth andWinters, 2003; Carpenter et al., 2007]. Unlike the mixedfluid that results from the KH instability, the mixed fluidthat results from one-sided instabilities is not concentratedat the density interface, but, is instead drawn away from thedensity interface [Carpenter et al., 2007].[49] MacDonald and Horner-Devine [2008] quantifiedmixing in the Fraser estuary at high freshwater dischargeover approximately two tidal cycles. Using a control vol-ume approach and overturning analysis they estimated meanbuoyancy flux, B, during the ebb to be 2.2 C2 10C05m2sC03.The associated mean turbulent eddy diffusivity, K = B/N2,was estimated to be 9 C2 10C04m2sC01[MacDonald, 2003].Smyth et al. [2007] proposed parameterizing the mixingcaused by Holmboe instabilities as K = 0.8 C2 10C04hDU.For transect 1 of the present study, which most closelymatches the conditions of MacDonald and Horner-Devine[2008] (see Table 1), this results in K = 6.7 C2 10C04m2sC01(0.8 C2 10C04C2 5.2 m C2 1.6 m sC01). The parametrization ofSmyth et al. [2007] represents the effect of a uniformdistribution of instabilities and has not been validated athigh Reynolds number. Mindful of these inherent limita-tions of DNS and the complexity of the field conditions, thesimilarity between the observed (K =9C2 10C04m2sC01) andpredicted mixing (K = 6.7 C2 10C04m2sC01) is promising. Amore rigorous analysis would include a description of thespatial and temporal distribution of instabilities. Unfortu-nately, our sampling was inadequate to comprehensivelydescribe this distribution particularly at high freshwaterdischarge.[50] During our survey of the estuary the mixing wasapparently caused by shear instabilities acting within theinterior of the flow and, to a lesser extent, by turbulenceassociated with the bottom boundary. Although we haveaddressed these two types of mixing separately they bothoriginate as a shear instability. Unlike instability predictedwith the TG equation the instability associated with thebottom boundary layer relies on viscous effects and thepresence of the solid boundary. In some cases, for exampleduring late ebb at low freshwater discharge (transect 6,Figure 10), the two mechanisms (TG-type instabilities andFigure 12. Schematic of a Kelvin-Helmholtz and a one-sided instability. The gray shading indicates mixed fluid,and the solid black line indicates the position of the centralisopycnal (i.e., density interface).C11006 TEDFORD ET AL.: INSTABILITY IN THE FRASER RIVER ESTUARY12 of 14C11006viscous shear instabilities) are acting together to generatemixing.[51] At high freshwater discharge, during the ebb,MacDonald and Horner-Devine [2008] found that mixingat the pycnocline causes a collapse of the salt wedge whichleads to complete flushing of seawater from the estuary. Thewell defined salt wedge is then regenerated during thesubsequent flood. Although we also see a well defined saltwedge at high discharge our observations at low freshwaterdischarge suggest that during the ebb mixing caused by bothshear instabilities at the pycnocline and bottom generatedturbulence is not able to homogenize the water column. Wetherefore expect the presence of mixed water in the estuaryat the beginning of the subsequent flood. The presence ofthis mixed water will prevent the formation of a welldefined salt wedge and the estuary will remain in a partiallymixed state.7.3. Wave Height[52] Unlike KH instabilities, the deflection of the densityinterface (wave amplitude) caused by one-sided instabilitiesis usually smaller than the amplitude of the billows (seeFigure 12). It is therefore difficult to assess the amplitude ofthe instabilities using echo soundings (e.g., Figure 5).Nevertheless, taking the vertical distance between thetrough and the cusp, the observed instabilities vary in heightfrom approximately 0.5 m to 2 m. The maximum height towavelength aspect ratio of the observed instabilities variesbetween approximately 0.025 (0.5/20, Figure 6c) and 0.1(2/20, Figure 5c). In the tilting tube experiments of Thorpe[1973] the maximum aspect ratio of KH instabilities variedbetween 0.05 and 0.6. Given the low values of J (<1/4) inThorpe’s experiments this difference in aspect ratio is notsurprising. Unfortunately, other than the case of the KHinstability (symmetric density and velocity profiles and J <1/4) the height of shear instabilities in stratified flows is notwell documented.7.4. Use of Echo Soundings to Identify Instability[53] Our analysis focused on periods when instabilitieswere evident in the echo soundings. There were instanceswhere predictions from the TG equation suggested insta-bilities would occur, but none were visible in the echosounder. In some cases (e.g., the secondary mode inFigure 5), the lack of apparent instabilities in the echosoundings can be explained by the absence of the strongvariations in salinity and temperature (i.e., density stratifi-cation) that are responsible for most of the back scatter ofsound to the instrument (for a thorough description ofacoustic scattering in similar environments, see Seim[1999] and Lavery et al. [2003]).[54] The quality of the visualization of the instabilitiesalso depends on the speed of the boat relative to the speed ofthe instabilities. For profiles collected at 2.2 km, duringtransect 3 (Figure 3c), the TG equation predicted instabilityclose to the depth of the pycnocline (results not shown). Inthis region the boat speed and predicted instability speedwere almost the same (C00.28 m sC01versus C00.24 m sC01).Considering equation (3), the resulting apparent wavelengthwould be 250 m. The corresponding apparent period ofapproximately 15 min (250 m/C00.28 m sC01) would likelydistort the appearance of an instability beyond recognition.This highlights an important challenge in identifying insta-bilities in echo soundings: if the point of observation ismoving at a speed similar to the instabilities, the appearanceof the instabilities becomes greatly distorted. On the otherhand, if the observer is moving at a much different velocitythan the instabilities; that is, the apparent wavelength andperiod are relatively short, the sampling rate of the echosounder may not be sufficient to resolve the instabilities.[55] In addition, our ability to detect shear instabilitiesdepends on the timing of the echo sounding relative to thestage of development of the instability. In DNS of symmet-ric and asymmetric instabilities, there are several stages ofdevelopment beginning with rapid growth and finishingwith a breakdown into three-dimensional turbulence. Onlyduring the stage where the instabilities have large two-dimensional structures, for example, billows, will they beeasily recognizable with the echo sounder. For example,during transect 2 (Figure 3b) instabilities were only recog-nizable over a distance of approximately 100 m (11 to11.1 km). However it is possible that instabilities are at aless recognizable stage of development throughout most ofthis transect.[56] Because of these challenges, the echo sounder is ableto confirm only the presence and not the absence of shearinstability. We therefore limited our application of the TGequation to cases where instabilities were apparent.8. Conclusions[57] After performing a detailed stability analysis on sixsetsofvelocityanddensityprofilesusingtheTaylor-Goldsteinequation and comparing with the echo soundings we con-clude the following.[58] 1. All of the instabilities observed in the echosoundings coincided with the most unstable mode in theTG analysis. This confirms the applicability of the TGequation in predicting instability, even in cases as complexas the Fraser River estuary.[59] 2. The location of each of the observed instabilitiesoccurs in a region of depth where Ri < 1/4. However, thereare also cases that have Ri < 1/4 in which no unstable modeswere observed. This result is in full agreement with theMiles-Howard criterion, but also highlights the inconclusivenature of this criterion.[60] 3. Although the observed instabilities all act on awell defined density interface, they appear to be concen-trated on only one side of the interface. The maximumvertical displacement occurs either above or below thedensity interface in a region of z where Ri < 1/4. None ofthe observations show Ri < 1/4 across the thickness of adensity interface. This is in contrast to the archetypal KHinstability described by the simple mixing layer model, inwhich Ri < 1/4 where dr/dz (N2) is greatest. The observedinstabilities might therefore be better described by the so-called ‘‘one-sided’’ modes of Lawrence et al. [1991] andCarpenter et al. [2007].[61] 4. During the majority of the survey the observedmixing was due to shear instabilities at the pycnocline. Inother stratified estuaries with moderate to strong tidalforcing, such as the Columbia and Hudson rivers, turbu-lence generated at the bottom is considered the dominantsource of mixing [Peters and Bokhorst, 2000; Nash et al.,C11006 TEDFORD ET AL.: INSTABILITY IN THE FRASER RIVER ESTUARY13 of 14C110062009]. In the present study we only observe mixing due tobottom generated turbulence during late ebb at low fresh-water discharge.[62] Acknowledgments. The authors thank NSERC for grants36366-06 and 194270-06. Gregory Lawrence is also grateful for the supportof a Canada Research Chair. We would also like to thank Bill Smyth forproviding us with the TG code and for helpful discussions and MelissaRohde for assistance with the field program.ReferencesAlexakis, A. (2005), On Holmboe’s instability for smooth shear and densityprofiles, Phys. Fluids, 17, 084103, doi:10.1063/1.2001567.Carpenter, J., G. Lawrence, and W. Smyth (2007), Evolution and mixing ofasymmetric Holmboe instabilities, J. Fluid Mech., 582, 103–132.Farmer, D., and L. Armi (1998), The flow of Atlantic water through theStrait of Gibraltar, Prog. Oceanogr., 21, 1–98.Geyer, W., and D. Farmer (1989), Tide-induced variation of the dynamicsof a salt wedge estuary, J. Phys. Oceanogr., 19, 1060–1672.Geyer, W., and J. Smith (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. London 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. Publ., 24, 67–112.Howard, L. (1961), Note on a paper of John W. Miles, J. Fluid Mech., 10,509–512.Kostachuk, R., and L. Atwood (1990), River discharge and tidal controls onsalt-wedge position and implications for channel shoaling: Fraser River,British Columbia, Can. J. Civ. Eng., 17, 452–459.Lavery, A., R. Schmitt, and T. Stanton (2003), High-frequency acousticscattering from turbulent oceanic microstructure: The importance of den-sity fluctuations, J. Acoust. Soc. Am., 114, 2685–2697.Lawrence, G., F. Browand, and L. Redekopp (1991), The stability of asheared density interface, Phys. Fluids, 3, 2360–2370.MacDonald, D. (2003), Mixing processes and hydraulic control in a highlystratified estuary, Ph.D. thesis, Mass. Inst. of Technol., Cambridge.MacDonald, D., and A. Horner-Devine (2008), Temporal and spatial varia-bility of vertical salt flux in a highly stratified estuary, J. Geophys. Res.,113, C09022, doi:10.1029/2007JC004620.Miles, J. (1961), On the stability of heterogeneous shear flows, J. FluidMech., 10, 496–508.Miles, J. (1963), On the stability of heterogeneous shear flows. Part 2,J. Fluid Mech., 16, 209–227.Moum, J., D. Farmer, W. Smyth, L. Armi, and S. Vagle (2003), Structureand generation of turbulence at interfaces strained by internal solitarywaves propagating shoreward over the continental shelf, J. Phys. Ocea-nogr., 33, 2093–2112.Nash, J. D., L. F. Kilcher, and J. N. Moum (2009), Structure and composi-tion of a strongly stratified, tidally pulsed river plume, J. Geophys. Res.,114, C00B12, doi:10.1029/2008JC005036.Pawlak, G., and L. Armi (1998), Vortex dynamics in a spatially acceleratingshear layer, J. Fluid Mech., 376, 1–35.Peters, H., and R. Bokhorst (2000), Microstructure observations of turbu-lent mixing in a partially mixed estuary. Part 1: Dissipation, J. Phys.Oceanogr., 30, 1232–1244.Seim, H. (1999), Acoustic backscatter from salinity microstructure, J. At-mos. Oceanic Technol., 16, 1491–1498.Seim, H., and M. Gregg (1994), Detailed observations of naturally occur-ring shear instability, J. Geophys. Res., 99, 10,049–10,073.Smyth, W. D., and K. B. Winters (2003), Turbulence and mixing inHolmboe waves, J. Phys. Oceanogr., 33, 694–711.Smyth, W., J. Carpenter, and G. Lawrence (2007), Mixing in symmetricHolmboe waves, J. Phys. Oceanogr., 37, 1566–1583.Taylor, G. (1931), Effect of variation in density on the stability of super-posed streams of fluid, Proc. R. Soc. London A, 132, 499–523.Tedford, E., R. Pieters, and G. Lawrence (2009), Symmetric Holmboeinstabilities in a laboratory exchange flow, J. Fluid Mech., 636, 137–153.Thorpe, S. (1973), Experiments on instability and turbulence in a stratifiedshear flow, J. Fluid Mech., 61, 731–751.Thorpe, S. (2005), The Turbulent Ocean, 1st ed., Cambridge Univ. Press,Cambridge, U. K.Ward, P. (1976), Seasonal salinity changes in the Fraser River Estuary, Can.J. Civ. Eng., 3, 342–348.Yonemitsu, N. (1991), Stability and waves of a salt-wedge flow, Ph.D.thesis, Univ. of Alberta, Edmonton, Alberta, Canada.Yonemitsu, N., G. Swaters, N. Rajaratnam, and G. Lawrence (1996),Shear instabilities in arrested salt-wedge flows, Dyn. Atmos. Oceans,24, 173–182.Yoshida, S., M. Ohtani, S. Nishida, and P. Linden (1998), Mixing processesin a highly stratified river, in Physical Processes in Lakes and Oceans,Coastal Estuarine Stud., vol. 54, edited by J. Imberger, pp. 389–400,AGU, Washington, D. C.Zhu, D., and G. Lawrence (2001), Holmboe’s instability in exchange flows,J. Fluid Mech., 429, 391–409.C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0J. R. Carpenter, G. A. Lawrence, and E. W. Tedford, Department of CivilEngineering, University of British Columbia, Vancouver, BC V6T 1Z4,Canada. (ttedford@eos.ubc.ca)R. Pawlowicz and R. Pieters, Department of Earth and Ocean Sciences,University of British Columbia, Vancouver, BC V6T 1Z4, Canada.C11006 TEDFORD ET AL.: INSTABILITY IN THE FRASER RIVER ESTUARY14 of 14C11006


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