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Observations and linear analysis of sill-generated internal tides and estuarine flow in Haro Strait Pawlowicz, Rich 2002

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Observations and linear analysis ofsill-generated internal tides and estuarine flowin Haro StraitRich PawlowiczDepartment of Earth and Ocean Sciences, University of British Columbia, Vancouver, B.C., CanadaReceived 21 June 2000; revised 12 September 2000; accepted 7 November 2001; published 25 June 2002.[1] Current meter records from two sets of observations 20 years apart in Haro Strait,British Columbia, Canada, are analyzed. Haro Strait is a deep channel separated fromlarger bodies of water on either side by relatively shallow sills. The estuarine flow inthis region is in approximate geostrophic balance and is apparently unaffected by thetidally driven spring/neap cycle in vertical stratification. Strong baroclinic variabilitywith an amplitude 1 m sC01(close to the amplitude of the barotropic tide) is present. Alinear theory for internal tides in silled basins is developed using no-bottom-flowboundary conditions to represent sill effects. By comparison with observations it isconcluded that an internal tide is generated at the seaward sill and propagates inshore.Although local damping appears to be weak, no return signal is found from the inshoresill, suggesting that either dissipation is strong in that area or that the internal tide istransmitted across that sill. INDEX TERMS: 4235 Oceanography: General: Estuarine processes;4544 Oceanography: Physical: Internal and inertial waves; 4203 Oceanography: General: Analyticalmodeling; KEYWORDS: phase, spring/neap, ellipse1. Introduction[2] Although a meand surface outflow and deep infloware required to maintain stratification in an estuary, themechanisms by which water masses are modified and salt istransported may involve unsteady, albeit regular, tidallyforced motions. Tidal cycles themselves are usually thestrongest signal in time series derived from moored instru-ments. The relatively rapid tidal timescales can causealiasing problems when standard shipboard observationaltechniques are employed. It is often difficult to generate asuitably ‘‘instantaneous’’ view of the physical state of agiven area, especially when internal variability is strong. Onthe other hand, tidal forcing is inherently deterministic andfollows predictable patterns, and so it is possible that theproducts of such forcing can assume deterministic andpredictable temporal patterns. It may then be possible tocombine observations made during different days and evendifferent seasons in order to synthesize an understanding ofthe total physical state of a given area.[3] In this paper the summertime internal motions in aunique and complex estuarine system are described usingtwo sets of observations taken 20 years apart. This system ischaracterized by the presence of two sills some 50 km apart,separating deeper basins. Tidal velocities are on the order of1–2 m sC01. Mean flows are consistent with estuarinedynamics and consist of a surface outflow and a deepinflow. Over fortnightly scales the vertical stratification ismodulated by the spring/neap cycle of available tidalenergy. At tidal timescales the interaction of the barotropictide with the sill results in a large internal signal within thebasin. It is shown that the motions are consistent withgeneration of a propagating internal wave at the larger sill,with only weak local damping. The amplitudes of isothermdisplacements are large but not inconsistent with the forc-ing, and there is no evidence of a resonant or near-resonantbasin mode. Since the basin is short, this implies that eitherdissipation is strong at the northern end or that the northernsill does not provide a strong barrier for the internalmotions. Although the linear model can explain many ofthe observed features, it is clear that the amplitude of themotions is large enough that nonlinear effects must bestrong; analysis of such effects will appear elsewhere.2. Oceanographic Background[4] The Strait of Georgia is a large open body of water(200 km long by 30 km wide with depths of up to 400 m)that empties into the Pacific Ocean to the south of Vancou-ver Island through the relatively simple geometry of theStrait of Juan de Fuca (20 km wide, 100 km long, and 200 mdeep), which is itself joined to the Pacific Ocean by acanyon cutting through the continental shelf (Figure 1).However, the connection between the two Straits is inter-rupted by a complex region of islands and channels, thelargest of which is Haro Strait. This deep channel, with awidth of about 10 km and depths of up to 350 m, isseparated from its neighbors by a broad area of shallowwater southeast of Vancouver Island called the Victoria Silland a more abrupt sill to the north in Boundary Pass. Saddledepths are slightly greater than 100 m in both cases. Thetopography of Haro Strait is asymmetric, with a steepJOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. C6, 3056, 10.1029/2000JC000504, 2002Copyright 2002 by the American Geophysical Union.0148-0227/02/2000JC000504$09.009 - 19 - 2 PAWLOWICZ: INTERNAL TIDES IN HARO STRAITboundary on the eastern side for much of its length and moregradual, stepped, slopes on the western side (Figure 1c). Inorder to approach the Boundary Pass sill the main channelmakes a sharp eastward bend. Directly to the north of thisbend, the waters shoal into a network of relatively shelteredand isolated channels that are generally 50 m or less indepth. Even in this quieter region, however, there are twolocations at which depths are greater than 200 m.[5] The Victoria Sill is irregular and asymmetric. Itswestern slopes rise relatively smoothly to a saddle depthof 110 m. To the east and northeast of this saddle thetopography rises to a series of banks with depths less than20 m and also drops in a series of channels that eventuallydrain into Haro Strait. A shallow channel close to Vancou-ver Island drops steeply, but other channels curve farthereastward and drop more gradually. The channel axis (i.e.,the main path of flow) is not obvious, and several possiblechoices are shown in Figure 1b, with bathymetric reliefalong each illustrated in Figure 1d. Although one mightexpect the deeper channels to be important for the estuarineflow, results from a constant density tide model [Foreman etal., 1995] suggest that much of the tidal energy flux for theM2tide passes over the shallow regions of the sill. TheBoundary Pass Sill is more straightforward, but inshore ofthis sill the channel is partially blocked at the entrance toStrait of Georgia by a steep-sided ridge that extends acrosshalf the channel (illustrated in Figure 1d).[6] The overall bathymetry is thus extremely compli-cated, and it is not clear how one might physically idealizethis region. It is also not clear if any one idealization wouldsuffice for all purposes. It is easily possible to imagine thatmean flows travel along pathways different than thoseimportant for tidal motions.[7] On the other hand, the general description of theestuarine circulation over the entire region is well known[Herlinveaux and Tully, 1961; Thomson, 1981; Pawlowicz,2001]. In the summer, snow melt in the mainland interiorprovides C25104m3sC01of fresh water to the Strait ofGeorgia, forming a fresh and warm upper layer with adepth of about 10 m. This fresh water exits the strait onboth sides of Vancouver Island, to the north throughJohnstone Strait and to the south via Haro Strait. Thevigorous tidal motions in these channels result in mixingand dilution of the fresh layer, so that the surface outflowin the Strait of Juan de Fuca occupies a much greaterportion of the water column, with interface depths downto 100 m on the northern side of the strait. The southernpart of the circulation around Vancouver Island hasreceived the most attention, partly because previous workindicated that the layer transport in the Strait of Juan deFuca, of order 10–20 C2 104m3sC01is rather larger thanthe 3 C2 104m3sC01that occurs in Johnstone Strait.Recent work, however, suggests that much of this flowresults from entrainment and recirculation of deeper waterwithin the Haro Strait and that the actual outflow from theStrait of Georgia is much smaller (see discussion byPawlowicz and Farmer [1998]). Note that the deep inflowis rather directly connected to the deep Pacific Oceanthrough Juan de Fuca Canyon, bypassing the continentalshelf. The dynamics of Haro Strait are thus important incontrolling the overall oceanography of this region, notonly because of its role in controlling the estuarinecirculation but also because it provides an importantmechanism by which nutrients from the deep PacificOcean are brought to the surface [Crawford, 1991; Paw-lowicz, 2001].[8] Another, smaller channel (Rosario Strait) also linksthe Straits of Georgia and Juan de Fuca. This channel ismuch shallower (with a saddle depth of 50 m) and moreconvoluted than Haro Strait, and it is thought that thethroughflow here is minimal.[9] Oceanographic investigations in the Haro Straitregion have generally been confined to process studies ofturbulent mixing processes [e.g., Gargett and Moum, 1995;Farmer et al., 1995, 2002]. The effect of these processes onthe large-scale circulation has when necessary been para-meterized in some bulk fashion [e.g., LeBlond, 1994], and itis known through such studies that a marked spring/neapcycle of mixing occurs governed by the fortnightly varia-bility in available tidal energy but also affected by wind-driven changes in the stratification of inflowing waters[Griffin and LeBlond, 1990]. Here we attempt to understandmore directly the internal motions of water masses due toestuarine and tidal forcing.3. Mean Flows[10] Although there is a lack of detailed knowledge aboutthe dynamics of Haro Strait, observational efforts have beenmade in the past. In 1976 a large array of current meters andthermistor strings was deployed across the strait [Webster,1977]. Positions are shown in Figures 2a–2c and 1c.Observations are divided into three depth ranges: near-surface (0–50 m, nominally 25 m), midwater (50–150 m,nominally 100 m), or deep (>150 m, nominally 200 m).Plotted in Figure 2 are progressive vectors of the measuredcurrents at each instrument, normalized by record length sothat the total displacements are an indication of meancurrents according to the scale shown. In general, theprogressive vectors are fairly straight, indicating that sub-tidal flows tend to be aligned in a particular direction.Surface instruments show a southward estuarine outflow,and deep instruments show an inflow to the northwest alongthe channel axis. Midwater instruments show inflow on theeastern side of the array and outflow on the western side. Asin the Strait of Juan de Fuca, the interface between deepinflowing water and surface outflow slopes downwardtoward Vancouver Island as a consequence of the CoriolisFigure 1. (opposite) (a) An overview of the general features and bathymetry surrounding Haro Strait. (b) A detailed mapof the complex bathymetry and geometry of the Haro Strait region. Thick curves indicate various possible channel axes.The easternmost possibility is the deepest but is also the most indirect. (c) Bathymetry across Haro Strait north of theVictoria Sill, with locations of the moored instruments in 1976 shown by open circles. (d) Bathymetry along Haro Strait,with location of moored instruments in 1996 shown by open circles. At each sill the various curves illustrate the variabilityin cross-channel depths using the different channel axes shown in Figure 1b.PAWLOWICZ: INTERNAL TIDES IN HARO STRAIT 9 - 3effect. In order to balance an observed gradient in verticalvelocity of 50 cm sC01in 200 m (Figure 3b) a thermal windbalance,f@v@zC24gr0@r@x; ð1Þrequires that cross-channel density changes be of order0.2 kg m–3over 10 km. Observed density gradients appearto be of this order, but more precise comparison is precludedby offset problems in instrument calibration.[11] Mean velocities change gradually rather thanabruptly with depth. Strongest flows occur near the sideand bottom boundaries of the channel, and so in spite of thelarge number of instruments in the array, it is difficult tojudge the balance between inflow and outflow, i.e., the nettransport [Pawlowicz and Farmer, 1998]. This has generallybeen true for all attempts using direct measurements ofFigure 2. Location of current meters with respect to areal features from (a) shallow, (b) midwater, and(c) deep 1976 measurements and from (d) shallow, (e) midwater, and (f) deep 1996 observations.Measured velocities are shown as progressive vector diagrams with total lengths normalized by time sothat mean currents can be read using the scale provided.9 - 4 PAWLOWICZ: INTERNAL TIDES IN HARO STRAITcurrents in this estuarine system [e.g., Thomson,1976;Godin et al., 1981].[12] A more limited mooring program took place in thesummer of 1996 (Table 1 and Figure 1). Instruments wereplaced along the channel rather than across it. The directionand magnitude of mean currents are shown in Figures 2d–2f. The instruments are again categorized as near-surface,midwater, or deep. The southernmost 1996 mooring was co-located with a 1976 mooring, and mean currents from bothyears are very similar. The line showing near-surface out-flow from this southernmost mooring appears jaggedbecause the record is relatively short (4 days) because ofa rotor loss. The daily variations are thus more prominentrelative to the cumulative drift. The general pattern ofsurface outflow and deep inflow is seen at all moorings.4. Spring/Neap Cycling[13] Figure 3a shows temperature records at selecteddepths from both current meters on subsurface mooringsand thermistor strings deployed below surface floats in the1976 program. Temperature and salinity are strongly corre-lated so the pattern of salinity (and density) changes isalmost identical. An increase of 1C176C corresponds to adensity decrease of about 0.9 kg mC03. Records have beenlow-pass-filtered with a cutoff period of 2 days to removetidal variability of about 1C176C. There is a marked fortnightlycycle in vertical stratification that is related to spring/neapcycling in tidal velocities. The timing of the spring/neapcycle can be identified from the along-channel velocityrecord shown in Figure 3c. In general, when tidal velocitiesare large, the stratification decreases. The opposite occurswhen velocities are small.[14] However, although the vertical mixing is modulated,the estuarine flow itself does not appear to be affected bythe spring/neap cycle in the manner suggested by Lindenand Simpson [1988]. Figure 3b shows low-pass-filteredalong-channel currents at selected depths. Northward veloc-ities are positive. Although there is some variability in theserecords, it is not as regular nor is it as pronounced as thatseen in the vertical stratification. No relationship was foundbetween variability in cross-channel density gradients (notshown) and either the spring/neap cycle or the along-channel currents.Figure 3. Spring/neap cycling in Haro Strait. (a) Low-pass-filtered temperature records for a variety ofinstruments. The instrument depth is labelled to the right of the curve. (b) Low-pass-filtered along-channel flow. (c) Observed tidal velocities at 35 m (northward flooding velocities are positive). Note the 2week spring/neap cycle in tidal amplitude.PAWLOWICZ: INTERNAL TIDES IN HARO STRAIT 9 - 5[15] Finally, note that the degree to which mixing duringspring tides homogenizes the water column is not the samefor each cycle. Griffin and LeBlond [1990] show thatstrong pulses of fresh water can be observed in recordsof surface salinity west of the Victoria Sill. They explainthe presence of this pulse by suggesting that a criticalcutoff mechanism applies to mixing here, such thatincreased stratification in Boundary Pass due to northerlywinds in the Strait of Georgia can suppress tidal mixingand allow fresh water to pass through the system relativelyunmixed. One such fresh pulse can be seen around 20 Julyin Figure 3a and also in Figure 9 by Griffin and LeBlond[1990].5. Internal Tides[16] At tidal timescales there is a marked variability inwater column properties with a complex spatial structure.Velocities at different locations are not in phase, and thedifference appears to change even over a day with thestrength of the tide. This is further complicated by the natureofthetide,whichusuallyconsistsofonestrongebb/floodandone weak ebb/flood in every day (see Figure 3c).[17] Figure 4a shows near-surface and deep along-chan-nel currents in Haro Strait during the summer of 1976. Thetwo (carefully selected) time series are quite different, butthe following decomposition, suggested on the basis of theanalysis described later, is instructive. First, we subtracttime means (i.e., the estuarine circulation) and form a half-sum and half-difference time series from these two records.The new time series are very similar (Figure 4b); indeed, ifTable 1. 1996 Haro Strait Current Meter ProgramaInstrument SerialNumberDepth,mDeploymentDateRecord Length,daysMooring S: 48C17630.04’N 123C17610.05’W, 271 m3684 25 16 June 1996 24 (4 days speed)5093 120 16 June 1996 242525 231 16 June 1996 24Mooring M: 48C17633.60’N, 123C17612.20’W, 270 m314 25 12 June 1996 286938 120 12 June 1996 281935 230 12 June 1996 28 (6 days speed)Mooring N: 48C17637.50’N, 123C17613.51’W, 242 m7893 25 16 June 1996 242581 120 16 June 1996 243695 202 16 June 1996 24 (9 days speed)Mooring E: 48C17639.14’N, 123C17610.25’W, 133 m736 25 12 June 1996 287897 70 12 June 1996 283503 123 12June 1996 28aAll meters are Aanderaa RCM-4. Strong currents resulted in the loss ofseveral rotors early in the deployment so that some velocity records arequite short.Figure 4. (a) Along-channel velocities near the surface and near the bottom. (b) Sum and Difference ofthe two time series. (c) Sum and difference with time mean subtracted and difference series aligned3 hours leftward.9 - 6 PAWLOWICZ: INTERNAL TIDES IN HARO STRAITwe advance the half-difference signal by 3 hours, they arevirtually identical (Figure 4c). If we identify the half-sumtime series as an estimate of the barotropic tidal signal andthe half-difference as an estimate of a first baroclinic mode,this suggests the presence of a strong internal tide.[18] The relatively short wavelengths of the lowest inter-nal mode (about 20–40 km for typical stratification) suggestthat coherent along-channel structure should be observablein velocity and density fields. In 1996, moorings wereplaced at three different locations along the channel (andone outside the channel (see Figures 1 and 2 and Table 1).Each mooring was equipped with three Aanderaa RCM-4current meters. Unfortunately, the rotors on three of theinstruments were lost within a few days, and the velocityrecords on other instruments are of low quality, especiallyduring periods of strong currents, apparently because ofstrumming problems. Sum and difference time series exhibitthe same general qualities discussed above, but the corre-spondence is less convincing. Instead, in Figure 5 the meantemperature field in a vertical along-channel section con-structed from the 3 C2 3 instrument grid is shown for hourlyblocks of time. Each box thus represents the hourly aver-aged temperature field over an along-channel distance of14.4 km and a depth of 300 m. Each row contains 25 hoursof data, and subsequent rows are offset slightly so that thepeak tides, which occur about 42 min later each day throughsprings, are aligned vertically (an abrupt shift in thisprogression occurs during neaps). Their general time periodFigure 5. Hourly averaged vertical sections of along-channel temperature for a 2 week period from27 June to 9 July 1996. (a) Observed currents for consecutive 24 hour 42 min periods for dates indicatedon the left-hand side. (b) Each row contains 25 subplots, each of which shows the mean vertical andalong-channel temperature field for the hour labelled (contour interval 0.25C176C). The vertical axis of eachsubplot represents depths from 0 to 270 m. Mooring S is at the left edge, and mooring N is at the rightedge of each subplot. Subplots showing maximum currents in subsequent days are thus vertically aligned.Rows correspond with currents displayed in Figure 5a. The dark outlines encompass periods of strongflow. Spring tides occur around the 30 June.PAWLOWICZ: INTERNAL TIDES IN HARO STRAIT 9 - 7is indicated by the large rectangular box outlines. Largestspring tides occur near 1 July. When presented in this way,it is obvious that the baroclinic field varies with surprisingregularity. During strong ebbs the temperature isothermsbegin to slope upward to the south. Strong floods areassociated with a relaxation of the southward slant andtransition to a northward slant. During the weak tides,isotherm slope changes are much less extreme. At anyparticular stage in the tide (i.e., along points on a verticalthrough Figure 5), isotherm slopes are greater near springtides and lesser near neaps. At the same time there is adecrease in the number of contour lines toward spring tides,indicating the decrease in stratification and an increaseduring neaps. This is most visible on the right-hand sideof Figure 5. The isotherm displacements strongly suggestthe presence of low mode number internal wave.[19] For further insight a harmonic analysis was carriedout for currents at different depths in the deepest part of thechannel using the 1976 observations. The resulting tidalellipses are narrow (semiminor axes are about 1 order ofmagnitude shorter than semimajor axes) and are inclined inthe along-channel direction. Flows are thus predominantlyalong the channel axis with little rotational tendency.[20] Figure 6 shows the ellipse semimajor axes amplitudeA and Greenwich Phase G for the four most significantcomponents of the analysis. The amplitudes of the diurnal O1and K1components generally decrease with depth. On theother hand, the amplitude of the semidiurnal components M2and S2have a minimum at middepths. The Greenwich phase(the phase of the local response in comparison to the phaseofthe forcing at a reference longitude of 0C176, defined such that alarger angle implies later arrival of wave crests) tends todecrease roughly linearly with depth for all constituents withbottom currents leading surface currents by about 3 hours. Inthe theoretical analysis carried out in section 6 it is shownthat plotting the depth variation of the complex numberAexp(iG) is a useful diagnostic for the internal behavior.Here we note only that such plots (Figure 6c) show the depthvariations to lie roughly along a straight line in the complexplane, not too far from the origin. Harmonic analysis of thetwo longer records at the southern mooring in 1996 showthat constituent amplitudes and Greenwich phases are gen-erally consistent with those found in 1976, although uncer-tainties are much larger because of the shortness of therecords and increased noise levels.[21] The nearby Victoria Sill (see Figure 2d) is likelyresponsible for the generation of this tide. Intuitively, oneexpects that stronger (spring) tides should be associated withmore intense baroclinic motions, but the fate of these internalwaves as they propagate away from the sill is not clear. Notfar to the north is a sharp bend in the channel (near TurnPoint), and past that, the Boundary Pass sill, both of whichcould act as strong reflectors. Is the amplitude of theresponse explained by the presence of a resonant internalmode, or is dissipation strong enough that the internal signaldoes not reflect and return? If dissipation is important, is itwidespread, or does it occur only in certain regions?6. Linear Analysis of the Internal Tide[22] The regularity of the baroclinic feature seen inFigure 5 and the relative simplicity of the results ofharmonic analysis suggest that a simple dynamical systemshould be able to capture the essential physics of theprocess. Consider a stratified channel dominated by thebarotropic and first baroclinic modes, in which a tidalsignal propagates past two sills separated by a distance L(Figure 7). For the situation considered here the wavelengthof the barotropic mode is much greater than that of thebaroclinic mode, and the isopycnal displacement due to thebarotropic mode is much less than that due to the baroclinicmode. We therefore assume the wavelength of the baro-tropic mode to be essentially infinite and its isopycnaldisplacement to be negligible so the net velocity uNETanddisplacement hNETcan be writtenuNET¼ uBAROTROPICþ u ð2ÞhNET¼ h ð3Þwhere uBAROTROPIC= u0exp(C0ist) is the barotropic flowthat oscillates with a tidal frequency s. The magnitude of thecomplex number u0is the amplitude of the barotropic tide,and its angle is the Greenwich phase. The baroclinicvelocity u(x, z, t) and displacement h(x, z, t) are functions ofspace as well as time and can be found for a channel ofrectangular cross section by assuming linear shallow-waterdynamics with a density field r = r0+ C22r(z)+r0between thebottom at z = 0 and the surface at z = H:ut¼C0pxr0C0 Ru;0 ¼C0pzC0 gr0;uxþ wz¼ 0; ð4Þgr0tC0 r0N2w ¼ 0;ht¼ w;where N2= C0gC22rz/r0is the buoyancy frequency, which wetake to be constant, p is the pressure, w is the verticalvelocity, and the x axis lies along the channel. Dissipationhas been parameterized using a linear damping factor R. Theeffects of rotation have been ignored; implications of thiswill be discussed later. As the equations are linear, we canassume an exp(C0ist) dependence in all variables.[23] In the vicinity of even a simple sill the baroclinicbehavior can have many forms [Baines, 1995]. The issue isfurther clouded here by the spatial complexity of theVictoria Sill and the time-varying nature of the forcing.However, we shall attempt to parameterize the overalleffects of the topographic interaction in the followingway. Since the effect of the sill will be to prevent theunimpeded flow of deep water, we assume that the tidalvelocity near the bottom at the sill edge is approximatelyzero and all of the motion is in the upper part of the watercolumn; that is, at x =0,L we haveu ¼ u0; z ¼ H;u ¼C0u0or uNET¼ 0ðÞ; z ¼ 0;ð5Þthe estuarine component of mean velocity being for themoment ignored. This is obviously a strong interaction.Similar boundary conditions have been used before for sillgeneration of internal tides in fjords [e.g., Stigebrandt,9 - 8 PAWLOWICZ: INTERNAL TIDES IN HARO STRAITFigure 6. Harmonic analysis for observations in June/July 1976 taken from the easternmost instrumentsthat were in the deepest part of the channel. Only the most significant four constituents are shown.(a) Ellipse semimajor axis amplitude. (b) Greenwich Phase. Dotted curves are 95% confidence intervals.(c) Amplitude and phase plotted as complex vectors for all constituents. The origin is marked with a plus.Thin lines are the analysis according to linear theory. The solid line extending from the origin representssill conditions according to the model and, after rotating by the angle indicated, approximately matchesthe observations. The dotted line shows the spatial evolution of an undamped surface response, and thedashed line shows the evolution of a strongly damped response (see text for details). Record lengths are39, 39, 33, and 31 days for the observations in order of shallowest to deepest.PAWLOWICZ: INTERNAL TIDES IN HARO STRAIT 9 - 91976; Stacey, 1985]. The theory can be extended in obviousways to account for an incomplete blocking of the bottomflow, but the observations considered here are notcomprehensive enough to justify such complications.[24] Assuming horizontally propagating plane wave sol-utions proportional to exp(ikx), equation (4) reduces towzzþN2k2s21 þ iRsðÞw ¼ 0 ð6Þwith boundary conditions w(z)=0atz =0,H. Lowest-modefree-wave solutions for the horizontal velocity u outside thebasin areu ¼C0u0cospzHC16C17eikx; ð7Þwhere the horizontal wave number k is complex and givenbyk ¼C6spNHffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ iRsr; ð8Þthe positive root being used for x > L and the negative rootbeing used for x < 0. The real part of k expresses thespatially oscillating signal and the imaginary part anexponential decay. In the limit of large friction the realand imaginary parts have equal magnitude.[25] In between the two sills (0 < x < L) the solution iscomposed of the sum of right and left going waves, and wehave after some manipulationu ¼C0u0sin kxðÞC0sin kxC0 LðÞ½C138sin kLðÞcospzHC16C17ð9Þkeeping in mind that k may be complex.[26] The spatial variation of the total velocity field can beunderstood in terms of the evolution of u0+ u, which is acomplex number whose magnitude represents the amplitudeof the total response and whose angle represents the Green-wich Phase of the total [Farmer and Freeland, 1983]. Inparticular, we consider the variations with depth of this sumat a particular x. Three cases are shown schematically inFigure 8. First, consider a standing wave response whendissipation is negligible. In this case the baroclinic velocityresponse is at a phase angle of either 0C176 or 180C176 compared tothe barotropic forcing. Thus, in the complex plane thevariation with depth must be collinear with the vectorrepresenting the barotropic signal u0. The magnitude ofthe response may be greater or smaller than the magnitudeof the barotropic signal depending on the degree to which aresonance is being forced (in this case, signified by themagnitude of 1/sin kL in equation (9)). If the internalresponse is a pure propagating wave, then the surface andbottom amplitudes appear at opposite ends of the diameterof a circle, which is centered on u0and touches the origin.This diameter is rotated through an angle kx compared withthe angle of the barotropic forcing. The magnitude of theresponse thus lies somewhere between 0 and 2|u0| depend-ing on the distance from the sill. If the signal is a dampedwave, then the velocities appear on a rotating bar as before,but the length of the bar decays exponentially with a factorImag{kx} as it rotates through an angle Real{kx}. In thelimit of strong friction (large R/s) the real and imaginaryparts of k have approximately the same magnitude (seeequation (8)) so that the amplitude at a distance of, forexample, one-quarter wavelength away from the generationFigure 7. Conceptual model for baroclinic basin modes.At the sills (x =0,L) the assumption that near-bottom flowis blocked provides the coupling between barotropic andbaroclinic modes. Away from the sills the baroclinic signalpropagates in both directions as a free wave.Figure 8. Spatial evolution of total velocity in the basinmodel. (a) Standing internal wave in which the phase angleof all velocities is the same and equal to the phase angle ofthe barotropic tide (denoted by q). On the right is shown afictitious set of ‘‘data’’ from a standing wave system plottedon the complex plane. (b) Propagating internal wave. Herethe amplitude/phase of measured velocities is along a linethat does not go through the origin but can be rotated aboutits mean by angle kx to touch the origin. (c) Propagating anddecaying wave. Here the internal wave has decayed, andamplitudes approach that of the barotropic tide alone (seetext for details).9 - 10 PAWLOWICZ: INTERNAL TIDES IN HARO STRAITzone will have decayed to exp(C0p/2) C25 0.2 of its originalamplitude. Once the wave has propagated far enough awayfrom the sill, only the mean barotropic velocity structureremains.[27] Comparing Figure 8 to the observations plotted inthe same fashion in Figure 6c, we immediately see that themajor effect of the Victoria Sill is to generate an internalwave propagating northward. The observations lie along abar rotated about 90C176 or 1/4 wavelength for the semidiurnalresponse and about half that for the diurnal response (thecentral point of the rotation represents the amplitude andGreenwich Phase of the barotropic tide). This represents thedistance of the 1976 array from the ‘‘effective’’ sill. Sincewe expect the wavelength of the diurnal constituents to beabout twice that of the semidiurnal constituents (see equa-tion (8)), these two independent estimates of the sill locationare consistent. Alternatively, one might consider the internalwave sensed at the mooring to have travelled for a time ofone quarter the period of the semidiurnal tide from itssource region; that is, the sensed internal signal is delayed3 hours in relation to the barotropic tide. This matches theestimate illustrated in Figure 4.[28] Also shown in Figure 6c are arcs representing thespatial evolution of an undamped (dotted arc) and stronglydamped (dashed arc) response away from the sill. Althoughthe dominance of the propagating signal implies that littleenergy is reflected back into this region from the northernsill (or indeed generated there), it appears that local dis-sipation is quite weak. The observed amplitudes of thebaroclinic signal alone are still relatively large; that is, thediameter of the observed ‘‘bars’’ has not diminished toexp(C0p/4) C25 0.5 of its original size for the diurnal responseor exp(C0p/2) C25 0.2 for the semidiurnal as would be the caseif damping was strong, although for frequencies other thanM2, they appear to be slightly smaller than would beexpected in the undamped case.[29] In theory a similar analysis of the 1996 along-channel observations could be used to estimate the actualwavelength or phase speed of the signal by comparing theanalyzed phase shift of the internal signal kx with the knowndistance x. Unfortunately, the uncertainty in the harmonicanalysis of those velocity records is large enough that nouseful conclusion can be drawn in this way. Instead, we willattempt to use temperature records as a crude proxy forisotherm displacements hb. Success is not guaranteed sincethere will be an ambiguity between the effects of horizontaladvection of a spatially varying temperature field andvertical displacement due to internal waves.[30] Internal wave displacements are found by integratingthe vertical velocity, which is related to the horizontalvelocity via the continuity equation. The dominant responseis apparently a propagating wave, and therefore the iso-pycnal displacements arehb¼ iu0suxð10Þhb¼C0u0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ iR=spNsinpzHC16C17eikxð11ÞFor right going free waves the isopycnal displacement isnegative when the near-surface velocities are rightward. Thepresence of damping alters this situation by delaying theisopycnal displacements in relation to the horizontalvelocity signal. Whereas with no damping the wave crestsand troughs coincide with maxima and minima of horizontalbaroclinic velocity, in strongly damped (R/s > 1) systemsthey appear about one-eighth wavelength (45C176) later. Notealso that in strongly damped systems the amplitude of thedisplacement is larger and the wavelength shorter than forundamped waves of the same frequency. In weakly dampedsystems the wavelength is independent of damping.[31] A harmonic analysis for temperature records atmiddepth shows that the Greenwich phase for the M2constituent increases by 144C176 ±18C176 from south to north(14.4 km). At K1this increase is 100C176 ±51C176. Both values areconsistent with a phase velocity of about 0.8 m sC01, not toodifferent from a theoretical estimate c = NH/p C25 0.7 m sC01using equation (8) and taking as typical a density gradient ofabout 2 kg mC03over 270 m. However, many other featuresof the phase do not match the picture we have derivedabove, and it is presumed that this occurs because of effectsof horizontal advection. In particular, the phase of the S2and O1components, which in amplitude are not too muchsmaller that those of the M2and K1components, eitherdecrease northward or exhibit unreasonably large changes.Also, the phase of the M2and K1displacements, lags that ofthe associated velocities by about 50C176 and 20C176, respectively.Such a phase lag implies through equation (11) a largerdegree of damping than was found by analyzing velocitydata alone.[32] Figure 9 shows observed and modeled isopycnaldisplacements and velocities for two 25 hour periods. Thefirst, 17 July 1996, is typical for a period of spring tides.The second, 23 July 1996, is a typical neap tide period.The modeled results were found by adding the fields fornorthward propagating waves generated by the four majorcomponents of the harmonic analysis (taking the ampli-tudes and Greenwich Phases from the central points of therotations shown in Figure 6c, assuming damping to benegligible and taking the effective sill distance to be one-quarter (one-eighth) wavelength away for the semidiurnal(diurnal) components. The remaining free parameter isthe wave number k or, more conveniently, the wave speedc = s/k, which is typically about 0.7 m sC01for a densitygradient of about 2 kg m–3over 270 m. This implies awavelength of about 32 km for the semidiurnal responseand 64 km for the diurnal waves. However, a qualitativelybetter picture results from taking c =1msC01(i.e., awavelength of about 50 km for the semidiurnal response).In either case the effective sill location is centered some-where slightly north of the shallowest part of the sill in theregion of km 30–35 in Figure 1d. The longer wavelengthreduces the magnitude of isotherm displacements, makingthem more comparable with the observations, and alsoimproves the along-channel correspondence of spatialfeatures. In spite of this, modeled displacements are stillquite large; isotherms sometimes rise above the surface anddrop below the bottom! It appears that the large amplitudeof the observed motions can be explained solely by thestrength of the forcing and is not due to any additionaleffects of resonance. The comparison of along-channelvelocities is not as good, although the major features, suchas the rising of the zero-velocity region from the bottomPAWLOWICZ: INTERNAL TIDES IN HARO STRAIT 9 - 11during the transition to the strong flood tide, are wellrepresented.[33] Although the data are in general agreement with thelinear wave hypothesis, several potentially important effectshave been ignored. In particular, the problem was treated ina two-dimensional fashion, and the effects of the Earth’srotation were assumed negligible or irrelevant. Althoughthere are no free-gravity waves at diurnal frequencies forthis latitude, there is a Kelvin wave response, which willhave a similar dispersion relation and along-channel struc-ture, at least for the propagating solutions (basin modes willbe more complicated). The deformation radius c/f C25 7kmisof the same order as the width of Haro Strait. Is it possiblethat a basin mode is being excited but that its form is moresimilar to an edge wave propagating in a counterclockwisedirection around the basin? The apparent unidirectionalnature of the wave observed in the deepest part of HaroStrait would then reflect the relative proximity of the easternside and distance from the western side of Haro Strait. It isnot possible to refute this supposition unequivocally, but atpresent it seems unlikely. The western side of Haro Strait isfar more complex than the eastern side, with many shallowbanks, islands, and troughs, and although the S mooring isquite far from the western side, the M and N moorings in1996 are in much narrower parts of the channel in whichsuch effects might be more visible. Harmonic analysis of theentire array of current meters deployed in 1976 (not shown)suggests that the amplitude of the internal oscillation doesdecrease westward away from the channel edge withapproximately the deformation scale. However, it is possi-ble that this cross-channel variation arises from purelyinertial effects as the tidal stream moves northeastwardfrom the Victoria Sill and is deflected northward by thecoastline, i.e., that it arises from the geometry of the channelrather than from rotational effects.7. Conclusions[34] Although the baroclinic motions of the Haro Straitregion are complex, some aspects of this behavior appear tobe highly regular, remaining recognizably similar not onlyfrom tide to tide but also between spring/neap cycles andeven from year to year (at least within the same season).The subtidal Eulerian estuarine flow appears to be fairlysteady and, unlike the vertical stratification, does notundergo spring/neap cycling, at least during the limitedperiod of the summer observations. At tidal frequencies alarge internal wave is observed. This feature is generated atthe Victoria Sill and propagates northward with only weaklocal damping. In spite of the lack of local damping theFigure 9. Comparison between observed and modeled parameters for the along-channel section inhourly blocks as in Figure 5: (a)–(e) the daily cycle for a typical spring tide (17 July 1996) and (f)–(j)the daily cycle for a typical neap tide (23 July 1996). By this time two of the current meters no longerreport speeds. Contour intervals for velocity comparisons are 0.25 m sC01, and the zero-velocity line isthick. Observed isotherm displacements are shown in comparison with two modeled results, oneassuming a wave speed of 0.7 m sC01(predicted from linear theory) and one assuming a wave speed of 1.0msC01(tuned to provide a qualitatively better result). At extreme left is a comparison between theobserved mean tidal velocities (solid) and the four-component fit (dashed) for the relevant time periods.9 - 12 PAWLOWICZ: INTERNAL TIDES IN HARO STRAITwave does not appear to be reflected at the northern end ofHaro Strait, and although the Haro Strait basin is only abouta wavelength long, little or no energy returns southward.Either damping is very strong in the northern end of thisregion (because of the right-angle bend in the channel andthe shallow passages between islands to the north) or theBoundary Pass Sill is not large enough to form an effectivebarrier for wave propagation, in which case, the internaloscillation should propagate into the Strait of Georgia. Thiswould occur in addition to the well-known nonlinear high-frequency internal wave packets generated at the BoundaryPass entrance [Shand, 1953; Gargett, 1976].[35] These findings resulted from the interpretation ofgraphical diagnostics describing the essential behavior of asimple two-dimensional linear dynamical model in whichthe effect of sills was parameterized by an enforced cou-pling between the barotropic and first baroclinic mode in thevicinity of the sill such that near-bottom oscillations at thatpoint were minimal (i.e., a blocking of tidal flow near thebottom). Although the linear model captures the grossdetails of the internal tide, the amplitude is large enoughthat nonlinear effects must play a role in the detailedevolution.[36] Acknowledgments. Although the majority of this analysis wascarried out at UBC under the support of Natural Sciences EngineeringResearch Council of Canada through grant OGP0194270, personnel fromthe Institute of Ocean Sciences, Sidney, B.C., were instrumental in makingthe original observations. I wish to thank in particular Les ‘‘Woody’’Spearing for his sterling work in preparing, deploying, and recovering the1996 moorings and Kevin Bartlett for processing this data. The 1976 dataset was partly discovered by rummaging through IOS archives (with thehelp of Robin Brown) and partly resurrected from decaying magnetic tapeswith the help of Grace Kamitakahara-King.ReferencesBaines, P. G., Topographic Effects in Stratified Flows, Cambridge Univ.Press, New York, 1995.Crawford, W. R., Tidal mixing and nutrient flux in the waters of southwestBritish Columbia, in Tidal Hydrodynamics,editedbyB.B.Parker,pp. 855–869, John Wiley, New York, 1991.Farmer, D. M., and H. J. Freeland, The physical oceanography of fjords,Prog. Oceanogr., 12, 147–220, 1983.Farmer, D. M., E. A. D’Asaro, M. V. Trevorrow, and G. T. Dairiki, Three-dimensional structure in a tidal convergence front, Cont. Shelf. Res., 15,1649–1673, 1995.Farmer, D. M., R. Pawlowicz, and R. Jiang, Tilting separation flows: Amechanism for intense vertical mixing in the coastal ocean, Dyn. Atmos.Oceans, in press, 2002.Foreman, M. G. G., R. A. Walters, R. F. Henry, C. P. Keller, and A. G.Dolling, A tide model for eastern Juan de Fuca Strait and the southernStrait of Georgia, J. Geophys. Res., 100, 721–740, 1995.Gargett, A. E., Generation of internal waves in the Strait of Georgia, DeepSea Res., Part A, 23, 17–42, 1976.Gargett, A. E., and J. N. Moum, Mixing efficiencies in turbulent tidalfronts: Results from direct and indirect measurements of density flux,J. Phys. Oceanogr., 25, 2583–2608, 1995.Godin, G., J. Candela, and R. de la Paz-Vela, On the feasibility of detectingnet tranports in and out of Georgia Strait with an array of current meters,Atmos. Ocean, 19, 148–157, 1981.Griffin, D. A., and P. H. LeBlond, Estuary/ocean exchange controlled byspring/neap tidal mixing, Estuarine Coastal Shelf Sci., 30, 275–297,1990.Herlinveaux, R. H., and J. P. Tully, Some oceanographic features of Juan deFuca Strait, J. Fish. Res. Board Can., 18, 1027–1071, 1961.LeBlond, P. H., D. A. Griffin, and R. E. Thomson, Surface salinity varia-tions in the Juan de Fuca Strait: Test of a predictive model, Cont. Shelf.Res., 14, 37–56, 1994.Linden, P. F., and J. E. Simpson, Modulated mixing and frontogenesis inshallow seas and estuaries, Cont. Shelf. Res., 8, 1107–1127, 1988.Pawlowicz, R., A tracer method for determining transport in two-layersystems, applied to the Strait of Georgia/Haro Strait/Juan de Fuca Straitestuarine system, Estuarine Coastal Shelf Sci., 52, 491–503, 2001.Pawlowicz, R., and D. M. Farmer, Diagnosing vertical mixing in two-layerexchange flows, J. Geophys. Res., 103, 30,695–30,711, 1998.Shand, J. A., Internal waves in Georgia Strait, Trans. AGU, 34, 849–856,1953.Stacey, M. W., Some aspects of the internal tide in Knight Inlet, BritishColumbia, J. Phys. Oceanogr., 15, 1652–1661, 1985.Stigebrandt, A., Vertical diffusion driven by internal waves in a sill fjord,J. Phys. Oceanogr., 6, 1105–1117, 1976.Thomson, R. E., Tidal current and estuarine-type circulation in JohnstoneStrait, British Columbia, J. Fish. Res. Board Can., 33, 2242–2264, 1976.Thomson, R. E., Oceanography of the British Columbia Coast, Can. Spec.Publ. Fish. Aquat. Sci., vol. 56, Can. Dept. of Fish. and Oceans, Ottawa,Ont., 1981.Webster, I., A physical oceanographic study of Haro Strait: A data summaryand preliminary analysis, Contract Rep. Ser. 77-3, Inst. of Ocean Sci.,Sidney, B.C., Canada, 1977.C0C0C0C0C0C0C0C0C0C0C0R. Pawlowicz, Department of Earth and Ocean Sciences, University ofBritish Columbia, 6270 University Blvd., Vancouver, B.C., V6T 1Z4Canada. (rich@ocgy.ubc.ca)PAWLOWICZ: INTERNAL TIDES IN HARO STRAIT 9 - 13


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