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The circulation and energetics of the Sechelt Inlet system, British Columbia Tinis, Scott Wayne 1995

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THE CIRCULATION AND ENERGETICS OF THESECHELT INLET SYSTEM, BRITISH COLUMBIAByScott Wayne TinisB. Sc. (Physics) University of Victoria, 1990A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESOCEANOGRAPHYWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIA1995© Scott Wayne Tinis, 1995In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall riot be allowed without my writtenpermission.(Signature)____________________Department of_________________________The University of British ColumbiaVancouver, CanadaDateDE-6 (2/88)AbstractThe sill at the entrance to Sechelt Inlet, 50 km northwest of Vancouver, is tuned almostperfectly to extract the maximum power from the barotropic tide. However, despite theenormous up-inlet energy flux from the barotropic tide (—‘42 MW), the diffusive processesin the deep-water of the inlet basin are weak, and the energy flux of the internal tidegenerated at the sill is relatively small (< 200 kW). The large tidal energy flux is almostcompletely dissipated at the sill, and most of the energy that manages to escape the sillregion does so in the form of a turbulent jet that dissipates at mid-depth near the sill;very little energy is left over for mixing the deep-water. Using estimates of the changein potential energy of the water column by vertical diffusion, the mixing efficiency (fluxRichardson Number) of the breaking internal tide is determined to be between 3 and8%.Because the transfer of energy to diffusive processes is so inefficient, the vertical diffusion of salt decreases the deep-water density by only —‘0.0l kg m3 per month. Thedecrease of the deep-water density conditions the inlet for eventual deep-water replacement; however, hydrographic surveys in Sechelt Inlet suggest that while the mid-depthbasin water is replaced once a year, the deep-water (below .--150 m) is replaced onlyabout every five years.The wind has relatively little influence on the circulation. Much of the wind energyis contained in the diurnal seabreeze, and its effect on the overall baroclinic energy fluxis estimated to be no more than 5%. The coherence of the wind to the currents in thelower frequency bands (f < 0.929 cpd) is small over most of the water column, with thefrequency band between 0.2 f 0.66 cpd having the highest coherence.11Empirical orthogonal function (EOF) analysis also indicates that the wind is notprimarily responsible for the low frequency variability: the EOF analysis could not consistently identify the wind-driven mode as one that accounted for a significant percentageof the low frequency variance. The four-layer flow that was identified as the dominantmode of low frequency variability is also a persistent feature of the monthly mean currents, and is probably formed by density gradients created by the tidal mixing near thesill. The mean circulation is similar to that proposed in an earlier study of Sechelt Inletby Lazier (1963).111AbstractTable of ContentsUList of TablesList of FiguresAcknowledgements2 Instrumentation and Data2.0.1 Hydrographic surveys.2.0.2 Current Meter Moorings2.1 Data processing2.1.1 Cyclesonde2.1.2 Aanderaa RCM2.1.3 InterOcean S42.1.4 Anemometers2.1.5 Endeco oxygen meters2.2 Datavuixxiv1178121314 17181819192020201 Introduction1.1 Fjord estuaries1.2 Shallow-silled fjords1.3 The oceanography of Sechelt Inlet1.4 Motivationiv2.2.1 Currents 212.2.2 Density 272.2.3 Harmonic Analysis 302.2.4 Wind 332.2.5 Runoff 342.2.6 Oxygen 363 Tidal Choking 393.1 Inviscid tidal choking 403.2 Choking with friction 443.3 Tidal Choking in Sechelt Inlet 464 Tidal Energy Partition 534.1 Energy Sinks 534.2 Barotropic Flux Model 544.21. Energy flux of the barotropic tide 554.3 Internal Tide 604.3.1 Theory of normal modes 644.3.2 Normal mode fitting 674.3.3 Energy flux of internal modes 794.4 Friction 844.4.1 Derivation of dissipation expression 844.5 Tidal Jet 884.6 The Energy Partition 925 Vertical Diffusion 935.1 Introduction 93V5.2 Vertical Diffusion 945.3 Mixing Efficiency in the Basin 1016 Low Frequency Circulation 1086.1 Filtering and spectral analysis 1086.2 Wind 1106.3 Runoff 1276.4 Mean Circulation 1316.5 Empirical orthogonal function analysis 1366.6 Deep-Water Renewal 1467 Conclusions 152A Hourly Data Plots 157viList of Tables2.1 Instrument uncertainties 162.2 The tidal constituents used for the harmonic analysis 314.1 M2 and K1 barotropic tidal parameters from the harmonic analysis ofEgmont and Porpoise Bay, and barotropic energy fluxes. 594.2 Modal fits to the velocity and perturbation density profiles of the M2 tidefor January 744.3 Modal fits to the velocity and perturbation density profiles of the K1 tidefor January 744.4 Modal fits (amplitude and phase) for the M2 tide 804.5 Modal fits (amplitude and phase) for the K1 tide 814.6 Net up-inlet baroclinic tidal energy flux at station 3 (MW) for (a) M2 and(b) K1 constituents in January, February, April and May 835.1 Summary of diffusion calculations for K,, = 965.2 Modal energy fluxes for the first four modes of the M2 and K1 constituents. 1036.1 Mean monthly discharge from the Clowhom River dam for January to May1991 1306.2 Mean velocities and net volume fluxes at the Basin mooring 1356.3 The first four EOF eigenfunctions for the Basin mooring in January. . . . 1406.4 The first four EOF eigenfunctions for the Basin mooring in February. . . 1416.5 The first four EOF eigenfunctions for the Basin mooring in April 142vii6.6 The first four EOF eigenfunctions for the Basin mooring in May 1436.7 The first four EOF eigenfunctions for the Salmon mooring in January 1446.8 The first four EOF eigenfunctions for the Salmon mooring in February 1446.9 The first four EOF eigenfunctions for the Salmon mooring in April . 1456.10 The first four EOF eigenfunctions for the Salmon mooring in May 145vi”List of Figures1.1 Southern British Columbia and Vancouver Island 21.2 The Sechelt Inlet system 31.3 Lazier’s proposed circulation for inlets with shallow sills 101.4 Temperature, salinity and oxygen for Sechelt (100 and 200 m) and NarrowsInlet (75 m) 112.1 Moorings deployed during the Sechelt Inlet experiment 152.2 Along-channel Basin currents at selected depths for January 232.3 Along-channel Salmon currents in the surface layer for January 242.4 Power spectra of the along-channel Basin currents for January 252.5 Power spectra of the along-channel Salmon currents for January 262.6 Basin densities () at selected depths for January 282.7 Salmon densities (as) in the surface layer for January 292.8 Sea-level spectra from the pressure records of the Basin cyclesonde . . . 322.9 Wind record from Salmon Inlet for January 342.10 Estimated monthly runoff from: (a) all sources (Trites, 1955), and (b) theClowhom River dam (from daily discharge rates from 1985 to 1990). . . 352.11 Oxygen values (ml 1_i) from the moored Endeco oxygen meters 372.12 Power spectra of Egmont oxygen time-series 383.1 Choking of flow through a constriction 403.2 Relation between amplitude reduction and phase shift of the surface tideacross a sill 41ix3.3 Choking coefficient, cos , versus topography-tide factor, 443.4 Skookumchuck Narrows 473.5 Tidal amplitudes at Boom Islet (solid line) and Rapid Islet (dotted line),from January 31 to February 10, 1984 483.6 Tidal currents versus hydraulic head across Skookumchuck Narrows. . 503.7 Velocity measurements in Skookurnchuck Narrows for July 9 and 10, 1983 524.1 Model for the generalized phase analysis 564.2 Analysis of the modified barotropic tidal flux model 604.3 Profiles of Basin along-channel velocity and perturbation density from theM2 constituent for January 624.4 Profiles of Basin along-channel velocity and perturbation density from theIs constituent for January 634.5 TS diagrams for January, February, April and May 684.6 Log Brunt-Vãisãlä frequency profiles for January, February, April and May 694.7 Vertical velocity modes (w(z)) for January 704.8 Horizontal velocity modes (u(z)) for January 714.9 Perturbation density modes (p(z)) for January 724.10 In-phase (a.) and in-quadrature (b) velocity profiles of the M2 tide forJanuary 764.11 In-phase (a.) and in-quadrature (b) perturbation density profiles of the M2tide for January 774.12 Net up-inlet baroclinic tidal energy flux (MW) for (a) M2 and (b) K1constituents in January, February, April and May 854.13 Schematic of a single channel inlet showing the coordinate system used inthe derivation of frictional power loss 86x4.14 Frictional dissipation rate computed using the one dimensional sill modelversus estimated power loss from the modified barotropic tidal flux model 894.15 Along channel velocity profile of the tidal jet during flood tide 904.16 Kinetic energy flux of the turbulent jet entering Sechelt Inlet versus totalbarotropic tidal flux 915.1 Eddy diffusion coefficients for December1990 to January 1991 (125-215 m). 985.2 Eddy diffusion coefficients for January to February 1991 (120-220 m). . 985.3 Eddy diffusion coefficients for February to March 1991 (150-220 m). . . 995.4 Eddy diffusion coefficients for March to April 1991 (170-220 m) 995.5 Eddy diffusion coefficients for April to May 1991 (170-215 m) 1005.6 Eddy diffusion coefficients for May to June 1991 (135-220 m) 1005.7 Eddy diffusion coefficients for December 1990 to March 1991 (125-215 m). 1015.8 Work done against buoyancy forces, W, as a function of baroclinic tidalenergy flux, E, for January to May 1065.9 Work done against buoyancy forces up-inlet of station 3 as a function ofnet up-inlet baroclinic tidal energy flux for January to May 1076.1 Smoothed wind power spectra for (a) Salmon (January-March), (b) Salmon(April-June) and (c) Basin (April-June) 1136.2 Filtered Basin along-channel wind and currents in April 1166.3 Filtered Basin along-channel wind and currents in May 1176.4 Filtered Salmon along-channel wind and currents in January 1186.5 Filtered Salmon along-channel wind and currents in February 1186.6 Filtered Salmon along-channel wind and currents in April 1196.7 Filtered Salmon along-channel wind and currents in May 1196.8 Filtered densities (o) in Salmon Inlet, January 120xi6.9 Filtered densities (o) in Salmon Inlet, May 1206.10 Coherence and phase spectra at Basin for April 1226.11 Coherence and phase spectra at Basin for May 1236.12 Coherence and phase spectra at Salmon for January 1246.13 Coherence and phase spectra at Salmon for February 1246.14 Coherence and phase spectra at Salmon for April 1256.15 Coherence and phase spectra at Salmon for May 1256.16 Hourly discharge from the Clowhom dam 1286.17 Lag correlations between discharge from the Clowhom River dam and the2 m along-channel currents in Salmon Inlet 1296.18 Mean along-channel velocities for January, February, March, April, andMay 1336.19 Profiles of Basin along-channel velocity and perturbation density from theMSf constituent for January 1346.20 Temperature, salinity and dissolved oxygen in Sechelt Inlet from 1957 to1993 1486.21 Along-channel section contours of density and oxygen, for February andMarch 1990 1506.22 Along-channel section contours of density and oxygen, for February andMarch 1991. 151A.1 Along-channel Basin currents for February. . 158A.2 Along-channel Basin currents for April 159A.3 Along-channel Basin currents for May 160A.4 Along-channel Salmon currents in the surface layer for February 161A.5 Along-channel Salmon currents in the surface layer for April 161xliA.6 Along-channel Salmon currents in the surface layer for May 162A.7 Basin densities (o) for February 163A.8 Basin densities (crt) for April 164A.9 Basin densities (oj) for May 165A.1O Salmon densities (as) in the surface layer for February 166A.11 Salmon densities (os) in the surface layer for April 166A.12 Salmon densities (Jt) in the surface layer for May 167XIIIAcknowledgementsI would first like to thank Dr. Steve Pond for his guidance and advice throughout thisproject, and for giving me the opportunity to participate in the Sechelt Inlet program.I am also very grateful for Dr. Pond’s support during the final months of preparingthis thesis, in these times of limited funding. The other members of my supervisorycommittee deserve many thanks as well; in particular I thank Mr. Dario Stucchi for thenumerous helpful discussions during the data analysis, and Drs. Susan Allen and PaulLeBlond for their comments and suggestions during the preparation of this thesis. DavidJones, Hugh Maclean and Arjoon Ramnarine deserve much recognition for their technicalsupport during the instrument deployments and for the many water samples that wereanalysed. The assistance from the staff, especially Mike Woodward and Anne Ballantyne,at Tides and Currents (Institute of Ocean Sciences, Sidney, B.C.) in providing tide gaugedata and permitting the use of the tidal rapids model was much appreciated.Many thanks to Peter Baker for his help in data processing and for the many usefuldiscussions over coffee. Thanks to the beer garden/coffee room gang for giving me plentyto think about, to Roger Pieters for his helpful presentation suggestions, and to JenniferShore for keeping me alert and for enlightening me on the many dangers of nerf footballsin tight places. The defensive cover-ups from the guys on the hockey team were timely;thanks to the summer softball squad for showing up to most of the games.Personal support came in many ways, and from many people. My parents and sister,Michelle, always gave me good reason to get away for a while, and they never let me losemy direction. Geoff, Shirley, Dan and Susie became my second family and taught me athing or two about marathon drives and playing cards. Livvy, Jessie, Dexter and Nessiexivnever let me forget the importance of taking the dog for a walk now and again. Finally,a huge thank you to Sally for all of the late night dinners, the generous shoulder rubs,and for the weekend hikes that got me out of the house and showed me places where theair is clearer and the perspective is better.I thank the officers and crew of the research vessels C.S.S. Vector and R.B.Young fortheir cooperation during the field program. This work was supported by Natural Scienceand Engineering Research Council (NSERC) research grants to Dr. Pond. I wouldfinally like to acknowledge NSERC for its personal support through the post-graduatescholarship program.xvChapter 1IntroductionSechelt Inlet is a fjord estuary on the southern mainland coast of British Columbia (Fig.1.1). There are three inlets in the system (Sechelt, Salmon and Narrows, Fig. 1.2) ofwhich Sechelt and Narrows have shallow sills connecting their waters to the outside.Most notable for the impressive tidal rapids at Skookumchuck Narrows, Sechelt Inlet is apopular spot for recreational boaters and white-water enthusiasts. The inlet also attractsaquaculture farms, particularly shellfish, because of its relatively calm, productive watersand close proximity to the lower mainland markets. Though not heavily populated,the Sechelt Peninsula is a popular vacation destination, and the number of permanentresidents is growing. As communities along the inlet grow and rely increasingly on theinlet waters for industry, recreation and waste disposal, the water quality of Sechelt Inletand its ability to assimilate pollution will become important issues.A comprehensive field study of the Sechelt Inlet system was undertaken by the University of British Columbia Department of Oceanography beginning in January 1990,and continuing until April 1992. Hydrographic surveys were made in winter and springof each year, and moored current meters equipped with temperature and conductivitysensors were deployed over six months, beginning in December 1990.1.1 Fjord estuariesFjord estuaries have been carved from high-latitude river valleys over several glaciations.The glacially scoured U-shaped walls are steep, and the sills which often appear near1Chapter 1. Introduction 2128°W 123°W52 °N jr + + + + L52 °NBritishColumbiaSeymour Inlet+ + Knight Io4et + +Bute-—InletTobaInlet÷ Jervi +. Inlet SecheltInletN Howe SoundIndian• Vancouver44, 0 Arm÷ Alberni÷ -U.S.Pacific InletOcean SaanichInlet48 °N ÷÷8°N128°W 123°WFigure 1.1: Southern British Columbia and Vancouver Island.Chapter 1. Introduction 349.5°N—]0123.5 WTzoonieInletS echeltInlet InletLEGENDkilometreso HydrographicStation0 GeodyneI Cyclesonde. AanderaaPorpoiseBay124°WSecheltFFigure 1.2: The Sechelt Inlet system.Chapter 1. Introduction 4the seaward end of the channel are the remnants of the terminal moraine (or, perhaps,immovable bedrock) left at the glacier’s furthest extent. Most fjords are deep, have flatbottoms (from accumulations of sediment), and have a source of fresh water input at thehead. Fjords are found in the high latitudes of Scandinavia, Russia, Alaska, Chile, NewZealand and Canada.The TJBC Department of Oceanography (formerly the UBC Institute of Oceanography) has been surveying B.C. inlets since 1951 (Pickard, 1961). Hydrographic surveys arenormally made at 5 nautical mile (9 km) intervals; measurements of temperature, salinityand dissolved oxygen are taken at up to thirteen depths with variable spacing to allow forlarge scalar gradients in the surface layer. Pickard (1975) discusses the circulation andlong term characteristics of B.C. fjord basin waters and notes some general differencesbetween the northern and southern inlets (see also Pickard and Stanton, 1980). Inletssouth of Queen Charlotte Sound were found to have less dense basin water than theirnorthern counterparts. This difference was attributed to the fact that northern inletsopen directly onto the Pacific Ocean, whereas southern inlets open onto the partiallyenclosed Strait of Georgia. The basin water of Sechelt Inlet was even less dense thanmost of the southern inlets (by 1 to 2 kg m3); Pickard suggested that the deep-waterhas a long residence time because of the shallow sill, and is gradually freshened by vertical diffusion of salt. The shallow sill also limits the density of inflowing water throughvigorous mixing with the fresh surface layer.The typical fjord circulation is a two-layer estuarine flow driven mainly by the freshwater source at the head. The pressure gradient created at the head forces the buoyantsurface layer to flow out of the estuary; the surface layer entrains salt from the moresaline layer beneath as it travels seaward, and deeper water is drawn into the inlet toconserve salt. The surface layer becomes more saline as it travels down the channel andthe sharp interface between the layers becomes less defined.Chapter 1. Introduction 5Tidal influence on the two-layer circulation can range from a simple flow modulationin a deep-silled fjord, to complete blocking of inflow during ebb tide in a shallow-silledfjord (Gade and Edwards, 1980). The amount of water exchanged during a tidal cycle ina deep-silled fjord may be greater than the amount exchanged in a shallow-silled fjord ofequal up-inlet area due to choking at the entrance. During ebb tide over a deep sill, thesurface outflow will be enhanced, while the deeper inflow will be slowed, or perhaps evenreversed. However, when the sill is shallow and the outfiowing surface water dominatesthe flow over the sill, an ebb tide may block the deep inflow altogether. The slowing(or blocking) of inflow during an ebb tide and the subsequent enhancement during floodtide modulate the two-layer flow at tidal frequencies. Modulations and pulsations suchas the above may also occur with a fortnightly period, since the spring/neap cycle bringsperiods of enhanced and reduced tidal flow. Although tides exhibit the most periodicand strongest high frequency variation, short and long period winds can also affect thecirculation.Friction, internal hydraulic processes, and the generation of internal waves near thesill of an inlet remove energy from the barotropic tide and are, therefore, importantwhen calculating the energy budget of a fjord. Freeland and Farmer (1980) found that asstratification increased so did the power withdrawn from the barotropic tide in KnightInlet, suggesting that the production of internal waves is an important sink of barotropictidal energy. Stacey (1984) also studied the removal of energy from the barotropic tideby sill interactions in Observatory Inlet, using a progressive internal tide model based onthe discussion of Stigebrandt (1976, 1980). The internal tide was found to be the largestsink of power, while high frequency internal waves and internal hydraulic jumps (plus asmall amount to dissipation by friction) accounted for the rest. Whereas high frequencyinternal waves, internal hydraulic processes, and mixing by friction near the sill provideenergy for mixing near the sill, the breaking of the baroclinic tide on the bottom andChapter 1. Introduction 6along the lateral boundaries of the channel provides the energy for vertical diffusion awayfrom the sill (Stigebrandt, 1976; Stacey, 1984).Low frequency variations in the circulation in fjords are mainly driven by seasonalchanges in offshore density (Stigebrandt, 1990), wind and runoff (Gade and Edwards,1980). Because most of the water in the inlet is eventually replaced by water from outsideof the sill, density changes in offshore water can have a large effect on fjord circulation.For example, upwelling off the west coast of Vancouver Island increases the density of thewater outside the sill of Alberni Inlet: the strength of the inflow is related to the strengthof the upwelling on the shelf outside the mouth of the inlet (Stucchi, 1983). In KnightInlet, Baker (1992) found that current structures were coherent with the wind as deepas 270 m. However, the deep currents were very weak during times of high stratificationwhen most of the wind energy was trapped in the surface layer. When the stratificationwas weaker, there was deeper transfer of momentum and the currents were stronger. Theresponse of the current to changes in runoff was not as clear. As the river dischargeincreased, the increased surface density gradient was found to suppress entrainment ofsalt from the lower layer. Although there was a significant increase in the velocity shearnear the surface, no substantial increase in volume transport was measured.Over time, the density in the deep-water of fjords decreases, mainly from the upwarddiffusion of salt (Pickard, 1961). Periodically, water that is denser than the’ residentbasin water enters the inlet, and the new water sinks to replace the old basin water,and either pushes it up or towards the head of the inlet. The timing and strength ofdeep-water renewal are highly dependent on the density of the water outside the inletand the strength of mixing at the sill. For example, if there is little mixing over the sill,then spring tides often provide the extra energy required to pump dense deep-water overthe sill. If the mixing over the sill is vigorous, deep-water renewal can only take placeat times of low runoff, and often only during neap tides when mixing energy is reduced.Chapter 1. Introduction 7Depending on the residence time of the existing basin water, it may be extremely low indissolved oxygen due to decomposition of organic material. New water usually containsmore dissolved oxygen than the existing basin water; hence, during renewal the oxygenconcentration of the deep-water usually increases.1.2 Shallow-silled fjordsThe tidal rapids at Skookumchuck Narrows near the entrance to Sechelt Inlet have thefastest currents on the coast of British Columbia at 16 knots (800 cm s1) . The constriction over the sill, which is as shallow as 5 m, limits the volume of the tidal exchangewater, and the intense mixing of the inflowing water with the relatively fresh surface layerlimits the exchange water density. As a result, inflowing water is rarely dense enough topenetrate to the bottom of the inlet. The shallow depths at the sill and strong tidal flowalso pose a hazard to navigation; slack water is often the only time that the waters ofthe Narrows are passable.Shallow sills are not only a hazard to navigation, but may also be responsible forthe degradation of water quality by limiting the exchange of the basin water. Bacterialconsumption of oxygen as sinking organic material is decomposed can reduce dissolvedoxygen to trace levels in the deep-water. The ability of the basin water to assimilateand oxidize waste material is subsequently reduced. During rare exchange events wherethe basin water is replaced, the existing low-quality water may be displaced towardsthe surface (particularly near the head) where it can interfere with fish communities orcontaminate water used for industry or recreation.In Scandinavia, limited exchange of basin water due to narrow or shallow constrictionsat the entrance to heavily populated fjords has caused a significant water oxygen problem(Glenne and Simensen, 1963). The waters that are used for sewage and waste disposalChapter 1. Introduction 8tend to accumulate in the stagnant deep layer. One method of increasing the flushing rate(thus increasing the assimilation capacity) of the basin water is to tune the constriction(by dredging) to maximize the tidal energy flux. This approach is particularly effectivewhen the basin is fairly shallow.1.3 The oceanography of Sechelt InletThe Sechelt Inlet system (Fig. 1.2) is located 50 km northwest of Vancouver, B.C., andopens onto the lower part of Jervis Inlet. Sechelt Inlet is separated from Jervis by ashallow sill (5 to 20 m), and the sill separating Narrows and Sechelt Inlet is also veryshallow (14 m). There is no sill separating Sechelt and Salmon Inlets. The typical basindepths of the inlets are 275 m for Sechelt and Salmon, and 85 m for Narrows.Due to the free communication of waters between Sechelt and Salmon Inlet, they havehistorically been considered as one basin. The section of Sechelt Inlet from Nine MilePoint to Porpoise Bay has been historically disregarded with respect to the circulationbecause its runoff is much smaller than that of Salmon Inlet. However, it comprisesnearly 20 % of the total surface area and its sloping bottom makes it a likely place forinternal wave dissipation.Mean fresh water input into Sechelt is 110 m3 s1, estimated by precipitation andrunoff basin area (Pickard, 1961; Trites, 1955). There are two runoff peaks per year: onein the winter from rainfall and one in early summer caused by snow melt. Input is largelydue to the Clowhom and Tzoonie rivers and from creeks draining along the inlets. TheClowhom River at the head of Salmon Inlet is dammed for power by B.C. Hydro andflow rates are monitored hourly. The Tzoonie River, at the head of Narrows Inlet, is notgauged, but is glacier fed and presumably has its maximum runoff period in the summer.One of the earliest descriptions of the Sechelt Inlet system was given by Carter (1934).Chapter 1. Introduction 9In his surveys of B.C. southern inlets, he noted that Narrows Inlet had very low dissolvedoxygen levels below the surface layer (although he may have been misled by infrequentsampling into thinking that this was a permanent feature). The oxygen level droppedsharply (to nearly 0 ml 1—1) and hydrogen sulfide associated with anaerobic conditionswas present in the deep-water and the black mud sampled from the bottom. Only aphysiographical description of Sechelt and Salmon Inlet was given.Sechelt Inlet was visited once by the UBC Institute of Oceanography in 1957, andthirteen times between 1961 and 1964. The area was not revisited until 1981, when,between 1981 and 1986, four cruises to Sechelt Inlet were made (one in each of 1981 and1985, and two in 1986).The first detailed look at the Sechelt Inlet system was made by Lazier (1963). Secheltwas visited in four consecutive years starting in 1961. Using hydrographic measurements,a circulation for shallow-silled inlets was proposed to explain the similarity of the observations between Sechelt, Narrows and Princess Louisa Inlet: in these shallow-silled inlets,water enters as a turbulent jet during tidal exchange. The appearance of turbulent jetsdue to tidal inflow has been directly observed by current meters in basins with shallowsills (Stucchi, 1980). Because of the constriction at the sill, the flow becomes highlyturbulent and causes mixing that breaks down the normal density stratification (Smithand Farmer, 1980).The source of the inflowing water (i.e. the surface water outside the sill) has aseasonal density cycle, where the density is largest in the winter. Lazier proposed thatas the inflowing water gets lighter in spring, it floats on the dense homogeneous deeplayer and forms a layer just below the brackish water at the surface (Fig. 1.3(a)). Astidal water density increases in the autumn and winter, the inflowing layer penetratesdeeper (Fig. 1.3(b)). Horizontal pressure gradients caused by the influx of the mixedwater induce density currents, which flow as indicated by the arrows in Fig. 1.3.Chapter 1. Introduction 10(a) (b)Figure 1.3: Lazier’s proposed circulation for inlets with shallow sills: (a) inspring/summer, when densities outside the sill are low, and (b) in winter, when densities outside the sill are high (from Lazier, 1963, Fig. 5).Lazier had hydrographic measurements during a particularly strong deep-water renewal in Sechelt Inlet. Temperature and oxygen contours show marked changes in Secheltbetween November 1961 and March 1962, and the salinity profiles indicate a deep homogenization of the water column. The deep-water oxygen levels increased from below1 ml 1_i to over 4 ml M between November and March, and sigma-t values increased by0.4 kg m3. Dissolved oxygen levels decayed to less than 2 ml J’ again by July 1962.Pickard (1975) presented a summary of all four years of data from 1961 to 1964.Figure 1.4 shows the time-series of the data taken at 100 and 200 m in Sechelt (at anaverage of 5 to 7 stations) and at 75 m in Narrows Inlet. The temperature data forSechelt 100 m and Narrows 75 m show a similar annual cycle (maximum in winter); noannual cycle is evident at 200 m. At 100 m, oxygen data show an annual cycle, peakingin late winter or early spring. The cycles suggest that while bottom water renewal isperhaps a rare event, replacement of water at mid-depths occurs annually in late winter(consistent with Lazier’s circulation).Chapter 1. Introduction 11Figure 1.4: Temperature, salinity and oxygen for Sechelt (100 and 200 m) and NarrowsInlet (75 m), from cruises between 1961 and 1964 (Pickard (1975), Fig. 3).Narrows Inlet oxygen data indicate an annual renewal cycle at 75 m. In January1962, the renewing water had a density much greater than the resident water. In May1963, the renewing water had the same salinity as the resident water, but it was nearly0.5 degrees cooler. The oxygen decay rate is greater in Narrows than in Sechelt.Chapter 1. Introduction 121.4 MotivationThe study of Sechelt Inlet by the UBC Department of Oceanography was designed togather biological, chemical and physical information about the inlet system as a whole.Aquaculture stocks, particularly salmon, have been plagued by chronically low oxygenlevels and by blooms of toxic or gill-damaging phytoplankton (Black, 1989). As well,the growing population of the Sechelt Peninsula will eventually put pressure on theassimilative capacity of the inlet for pollution and waste disposal (Arber, 1993). Thenon-physical aspects of the study have been discussed by Sutherland (1991), who lookedat the formation and movement of phytoplankton blooms in the region, and Timothy(1994), who examined the production and assimilation of organic carbon in order toestimate oxygen production and consumption in the water column. The goal of thephysical study was to answer three questions: (a) what are the main sources of energyfor mixing in the inlet, (b) what are the diffusive and deep-water exchange characteristicsof the basin waters, and (c) how is the circulation affected by wind and low frequencyforcing.The aim of this thesis is to present and analyse the data collected in Sechelt Inlet inan effort to address the goals of the study. Chapter 2 presents sample data and discussesthe data processing techniques used. Chapter 3 addresses the problem of tidal chokingat the sill. The tidal circulation is discussed in Chapter 4: first, a theoretical discussionis given of the energy flux extracted from the tide at the sill; second, the energy fluxof the baroclinic tide, the turbulent tidal jet, and energy dissipation by friction in thesill region, are quantified. Chapter 5 examines the diffusive processes in the deep-water,and analyses the efficiency of the possible sources of mixing energy. The low frequencycirculation and deep-water exchange are presented in chapter 6, and a summary of resultsis given in chapter 7.Chapter 2Instrumentation and DataThe Sechelt Inlet study began in January 1990 and continued until April 1992. Duringthis period, hydrographic surveys were made approximately monthly using a conductivity-temperature-depth (CTD) probe coupled with a water sampling rosette system at eachof the 9 oceanographic stations (Fig. 1.2) from late autumn to early summer. Threecurrent meter moorings were deployed landward of the sill between January and June1991 (see Fig. 2.1). A mooring was also deployed seaward of the sill from December 1990to June 1991 and from December 1991 to April 1992, to monitor the exchange water.The CTD surveys coincided with the servicing of the moorings so that the temperatureand conductivity sensors of the moored instruments could be inter-calibrated with theCTD probe. Historical tide gauge data from the region were provided by the Tides andCurrents section of the Institute of Ocean Sciences (lOS), Sidney, B.C.Instrument uncertainties are listed in Table 2.1. The current meter conductivity andtemperature sensors are less accurate in general than those on the CTD probes. Thecurrent meters were of two types: rotor (Aanderaa, cyclesonde) and electromagnetic sensor (InterOcean S4). The Aanderaas and cyclesondes use average rotor counts over aspecified time interval and spot direction readings to compute velocity. The InterOceanS4 current meters use voltage potentials created by the water flowing through an electromagnetic field set up around the instrument. One advantage of the S4 is that it canaverage several readings of velocity components over the sampling interval and computea true vector average of the velocity. Rotor-type current meters also have a stall speed13Chapter 2. Instrumentation and Data 14(--2 cm s’ for the Aanderaas and cyclesondes) under which the rotor will not turn. Theuncertainties for the cyclesondes are similar to those for the Aanderaa; however, whenthe cyclesonde is vertically profiling while measuring currents, the rotors are turning andthe stall speed problem is reduced. The correction of the rotor speed for the fall speedincreases the uncertainty somewhat for low current speeds.2.0.1 Hydrographic surveysA series of hydrographic surveys was made from January to June 1990, prior to the mainmooring deployment period. One station was located just seaward of the sill (station 1),five in Sechelt Inlet (stations 2-6), two in Salmon Inlet (stations 7-8), and one in NarrowsInlet (station 9). Some measurements were made at station 0, located in Jervis Inlet,but are not discussed in the analysis.A Guildline model 8705 CTD probe was used for all surveys, except February 1991,when the Guildline model 8709 CTD probe was used. The sampling rate for the 8705probe is 25 hz, and 4 hz for the 8709. The probe was attached to the General Oceanicsmodel 1015 rosette bottle cluster.During a cast, the CTD probe samples continuously. Initially, the probe and rosetteassembly is lowered to between 5 and 20 m from the bottom, and the downcast datafrom the probe provide a continuous profile of temperature, salinity and density. Duringthe upcast, bottle samples are taken at several depths and later analysed for oxygen (byWinkler titration) and salinity (using the Guildline Autosal 8400 salinometer at UBC).The temperature samples from the CTD probe at each bottle depth are combined withthe bottle salinities to produce a second, independent density profile. The CTD probedata are binned into 0.1 m bins (0.2 m bins for the 8709). Finally, the binned data aredecimated to 1 m intervals for depths shallower than 50 m, and every 5 m thereafter.tJ:‘1 I,_io0-IE r4)200-300-40200DistancefromtheHead(km)Figure2.1:MooringsdeployedduringtheSecheltInletexperiment.Depthscalecorrespondstobottomtopographyandonlyapproximatelytothedepthoftheinstruments.IIII3010Table2.1:Instrumentuncertainties.Valuesforconductivityaregiveninequivalentsalinity(indicatedby(S)).tForthecyclesonde, AanderaaandInterOceaninstrumentstheconductivityvaluesgivenaretheprecision.Theaccuracyvaluesaresomewhat largerdependingonhowwelltheycanbecorrectedtotheCTDvalues.0.02forsomeinstrumentswithexpandedtemperaturerange.C,) IInstrumentCondTempPressOxygenSpeedDirRotor(equivalentS)(°C)(dB)(ppm)(cms1)(°)TypeCuildline8705CTD.01.011.5————Cuildline8709CTD.04.025.5———Cyclesondet.01.02.25—1.03.0SavoniusAanderaaRCM4t.02.15*51.05.0Paddle/(2.0stallspeed)(7.5<5cms’)SavoniusInterOceanS4.01.022—1±2%ofreading2.0ElectromagneticEndecoO2—.2—.4———CeodyneAnemometer—2.0(%)——2.0(%)5.0Cup0)Chapter 2. Instrumentation and Data 172.0.2 Current Meter MooringsA subsurface mooring was deployed near Egmont between December 1990 and. June1991, then again between December 1991 and April 1992 (see Fig. 2.1). The depth ofthe mooring was 75 m and two Aanderaa RCM4s (recording current meters) were placedat approximately 30 m and 50 m. Between December 1990 and June 1991, an Endecooxygen meter was placed at 30 m. The current and oxygen meters were serviced everytwo months.Three mooring sites landward of the sill were maintained between January and June1991: one near station 2, one near station 3 in the main basin of Sechelt inlet (consistingof one surface and one subsurface mooring closely spaced) and one in Salmon Inlet. Thesesites will be hereafter referred to as “Sill” ,“Basin” and “Salmon” respectively.At the Sill site, one cyclesonde profiling current meter was deployed on a subsurfacemooring. The instrument used an inflatable bladder to move up and down the mooringline from -20 to 180 m depth (van Leer et al, 1974), recording velocity, depth, conductivity and temperature. The instrument normally takes less than one-half hour to completea 150 m profile.There were two moorings deployed at the Basin site. One mooring was identical tothe Sill mooring with the addition of three Aanderaa RCM4s at 208, 238 and 268 m. Thesecond mooring employed a Geodyne buoy (equipped with an anemometer) suspendingfive S4 current meters at 2, 4, 6, 9 and 12 m. The Geodyne mooring required servicingevery two months and was deployed twice (January to March and April to June).The mooring in Salmon Inlet was a Geodyne buoy suspending five S4 current meterssimilar to the one at Basin. No cyclesonde was deployed here, since it was felt that thecurrents would be too small to be accurately measured (S. Pond, pers. comm.).Chapter 2. Instrumentation and Data 182.1 Data processing2.1.1 CyclesondeThe cyclesonde collects temperature, conductivity, pressure and velocity records once perminute while profiling (1 hour every 3 hours) and once every 5 minutes while stationary.The vertical resolution is between 5 and 10 m. Profiling once every 3 hours allows thecyclesonde to operate for 30 days before requiring servicing. All data channels weredigitally recorded on cassette tape.Once removed from the cyclesonde, the tape was read using a tape reader connectedto a portable computer. There, the data were checked in binary form for missing bits inthe data stream and corrected. The corrected binary files were then transferred to theUBC MTSG mainframe.On the mainframe, the raw data channels in each file were converted into engineering units, plotted and checked for spurious data points (glitches). A glitch is definedto be any data point which varies beyond a set tolerance from a linear fit to the localtemperature-salinity curve. Glitches were examined individually then interpolated if required. CTD measurements of conductivity and temperature near the bottom of thecyclesonde mooring were used to correct any drift in the cyclesonde sensors. The cyclesondes were adjusted to agree with the deep CTD values at the beginning and end ofthe deployments. The profile data were then used to create time-series of velocity, temperature, salinity and density at regular intervals (every 5 m between 20 and 50 m, andevery 10 m below 50 m), with an approximate sampling interval of three hours. Bakerand Pond (1995) discuss the interpolation procedure for creating the profiles; essentially,scalar values are linearly interpolated, while velocities are vector averaged in 10 m binsand then linearly interpolated.Chapter 2. Instrumentation and Data 192.1.2 Aanderaa RCMThe Aanderaas recorded conductivity, temperature and velocity (some also measuredpressure) at 10 minute intervals. The temperature and conductivity records of the Aanderaas moored below 200 m at the Basin site were adjusted to the CTD values takenbefore and after deployment. The Aanderaas deployed seaward of the sill (near station1) were compared to the CTD values at station 1, but showed too much variability inconductivity and temperature to make any reliable adjustments.Each data set sampled at 10 minute intervals was filtered to create an hourly time-series. Seven data points centered on the hour were used to calculate the hourly value: thetwo highest and two lowest values were removed, and the remaining three were averaged(see Baker and Pond, 1995). This type of filter is useful when dealing with sensors thatmay be contaminated by biological material and, hence, subject to intermittent erroneoussamples. A straight average of the seven values might include one or more values whichare significantly in error.2.1.3 InterOcean S4The InterOcean S4 electromagnetic current meters were all deployed in the surface layer( 12 m) and were set to record a one minute average of velocity and spot readings ofconductivity and temperature every 10 minutes. The advantage of the electromagneticcurrent sensor is that it is not prone to rotor pumping from wave action. In addition,the vector averaging feature of the current meter allows accurate measurement of the netcurrent over the sampling interval. The data were converted to hourly values using thesame method as with the Aanderaas (see section 2.1.2).Because the surface layer has large fluctuations in temperature and conductivity,adjusting the S4 mean values to the CTD values at the beginning and end of deploymentChapter 2. Instrumentation and Data 20was impractical. Instead, the instruments were made self-consistent by ensuring that themean density increased with depth, and strong wind-mixing events were identified (whichpresumably temporarily homogenized the upper water column) and used to match thedensities between the instruments.2.1.4 AnemometersThe anemometers on the Geodyne buoys recorded air temperature, wind speed and winddirection on magnetic tape (similar to the Aanderaas) every 10 minutes. The same hourlyfilter used for the Aanderaas and the S4s was used for the anemometer data. The filtershould help to reduce the noise inherent in the wind direction observations. Velocitycomponents in all instruments were rotated to geographical north-south and east-westcoordinates.2.1.5 Endeco oxygen metersThe Endeco oxygen meters recorded oxygen and temperature every 15 minutes. Theoxygen values were corrected for in situ temperature when processed. The original processing also includes a salinity correction using a constant, typical salinity value. Theoxygen data were reprocessed to correct for in situ salinity, using the salinity record ofthe nearest current meter. When the reprocessed data were compared to those processedwith the constant salinity, oxygen values changed by no more than 2 %.2.2 DataData will be presented in SI units (MKS) with the following exceptions: currents will bein cm s1 and spectral frequencies will be in cycles per day (cpd). Salinities are derivedfrom conductivity relative to standard sea water and are given as values on the PracticalChapter 2. Instrumentation and Data 21Salinity Scale (no units). Densities are computed from salinity and temperature usingthe UNESCO (1983) equation of state for sea water, and are presented as o= p — 1000kg m3.Monthly data sets are named by the month in which they were deployed; hence,“January” contains data from approximately January 23 to February 21, and likewisefor the subsequent data sets. January data are presented and discussed in this section,and a complete presentation of all data sets is made in the appendix.2.2.1 CurrentsIn a fjord, cross-channel currents are generally small so that the analysis of currents maybe performed using one component (Webb, 1985; de Young, 1986), provided that the dataare rotated in the direction which minimizes the variance in the cross-channel component(i.e. along the principal axis). However, the principal axis changes over depth, due mainlyto topographic steering of the water. It is not possible, then, to choose one angle thatminimizes all of the cross-channel currents over the water column: a compromise mustbe made.For the 21 depths at the Basin site, the angle which minimized the cross-channelcomponent of the current was calculated. The current directions were closely alignedwith the V component, with a mean rotation angle of -6.6° with a standard deviationof 9.5° (i.e. by rotating the north-south (V) velocity component counter-clockwise by6.6°, the variance of U was minimized). At the Salmon site, the principal axis was foundto be quite close to the direction of the east-west (U) velocity component. The meanrotation for Salmon Inlet was found to be -14.3° with a standard deviation of 5.3°. Inboth cases it was decided that the angle of rotation was small enough not to requirerotating the data (cos 14° = 0.97), so the currents were left in geographic coordinates;the V component was used for the Sechelt basin currents and the -U component was usedChapter 2. Instrumentation and Data 22for Salmon Inlet; the choice of -U for Salmon Inlet was made so that positive currentsin both inlets were seaward. The along-channel component of velocity at selected depthsfor January is shown for the Basin site in Fig. 2.2, and for the Salmon site in Fig. 2.3;the cross-channel components after rotation are small compared to the along-channelcomponents and are not shown.Power spectra of the along-channel currents. are shown in Figs. 2.4 (Basin) and 2.5(Salmon). The spectra are presented in power-preserving format: the x-axis (frequency)is plotted on a logarithmic scale to expand the lower frequencies. The spectral densitiesmust, therefore, be scaled by the frequency in order that the area beneath the curvein each frequency band is proportional to the variance in that band. The spectra weresmoothed over 9 frequencies, using a boxcar filter, giving 18 degrees of freedom to eachspectral value (computation of the 95% confidence band is discussed in chapter 6). Thecurrents have a strong tidal signal but also have low frequency energy, especially near thesurface. The nature of the tidal signal also changes over the water column; some depthshave a strong diurnal signal, while others are predominantly semidiurnal.Chapter 2. Instrumentation and Data 234020E0-20-40L il. A ,ti ..R.A. A1lgJ9i( f w’ V --30E 10c’J -10-3030E 10QI.’)-10-30301o.10-303010t.2.io-303010-301/23/91 2/1/91 2/8/91 2/15/91 2/23/91Figure 2.2: Along-channel Basin currents at selected depths for January 1991. Positivecurrents indicate flow towards the sill (units are cm _1). Note the change in scale.4020-20-401/23/91 2/1/91 2/8/91 2/15/91 2/23/91r,4IVf ilj tF%..-f-.”40E 20cj0— -20JMMA-J4J“d Jf/ A/iAJ Af’—’- j-, 4 .J_? 1 LyP3010-10-30-1/23/91 2/1/91 2/8/91 2/15/91 2/23/91Chapter 2. Instrumentation and Data 244040E 20ci0— -202/1/91 2/8/91 2/15/91 2/23191V\MIy7/WVWAV’wVs--.--.WYVY\fVV1/23/914020E0-20-4020E0-20-404020E0CD-20-40AV1/23/91 2/1/91 2/8/91 2/15/91 2/23/91Figure 2.3: Along-channel Salmon currents in the surface layer for January 1991. Positivecurrents indicate flow towards the sill (units are cms1).CDi-I,CD CDCD C) CD‘CDCD C) CD CD C CD 0 CD C Cu C) CD 0 ‘-.3 CDfrequencyxspectraldensity(cm22)268m208m150mlOOm50m20m12m6m2mCD---))O)---.ooa,o-r,ccno-.røooiouoooooocor%)CD”CI)00o..,.....o‘-CDCD-r.).0a:“,.....“..IIiiCDIIIIIIIIIIIIIIIIIIIIIIIIIIIIICD,II.cC1C.r—‘o..I.144.3..4...4l—.JL•I.3.3...iC)CDoIIIIIII.‘Ig0..4....d,.,...,—.IIIIII——•1.......tIIIIIiI3%IcICDI)4.II—.03I0......,IIC3IIII—-I.-I—CDCDII_I—.._%III—I—<C)——,————-.—————14—‘ICl)‘—s:,,Cl.—III*II—‘‘......—...._i.I-P‘-t%)(WI’.)rCDC)So—.3ocI,)J)jD-’-.._3..;‘CDcupro40CJIC.)CC.)—0c-’-C.))C.)4—3.1%4pp.,— 0_.c-0,.0..CI)CD.C H --00CDfrequencyxspectraldensity(cm2s2)12m6m4m2mc)r%or%)ocDCY1OCY109000000000000000‘-CD”CoCDb..0pCDCo0001--CDIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII_CDii0::::::::::::::::::::::::::::::::::::::::::::::::::::::::::t:.::::t:::::::::::::::::;:::CDII—CDC4.1.14....1.1./_iIIICI..“1CD—..1..‘:..-Ja0,‘II—i:::____________010Co n—__________00CDCCDChapter 2. Instrumentation and Data 272.2.2 DensityThe surface waters of fjords are generally highly stratified, and the density fluctuationscaused by wind and tidally generated internal waves may be large near the surface andgrow smaller with depth. The density records for selected depths at the Basin and Salmonmoorings are shown in Figs. 2.6 and 2.7 respectively (note the change in scale betweenthe 2 to 12 m depths and the deeper depths). Density fluctuations in January haveamplitudes on the order of 2 kg m3 at 2 m, but drop quickly in magnitude; at 12 m, thefluctuations are only about 10% as large as those at 2 m. The size of the near-surfacefluctuations does not change significantly between Salmon Inlet and Sechelt Inlet.Although the density fluctuations appear mostly tidal, there is some effect from runoffand wind. For instance, the 2 m January density records show a distinct smoothing of thevariability between February 1 and February 6 which may be due to homogenization ofthe surface layer. The reduced variability coincides with the sudden, large discharge fromthe Clowhom River dam (see Fig. 6.16), and also with the onset of persistent outflowwinds in Salmon Inlet (see Fig. 2.9). Discussion of the wind and runoff responses is givenin chapter 6.Chapter 2. Instrumentation and Data 281/23/91 2/1/91 2/8/91 2/15/91Ei8‘° 141022E 18c’J‘— 14101/23/91 2/1/91 2/8/91 2/15/9122u, 2120E2120-,- ‘nfl --E 2202120E222120E 22211/23/91 2/1/91 2/8/91 2/15/91Figure 2.6: Basin densities (o) at selected depths for January 1991.Chapter 2. Instrumentation and Data 291/23/91 2/1/91 2/8/91 2/15/91 2/23/91E18141022_________E 18° 141022E 1814101/23/91 2/1/91 2/8/91 2/15/91 2/23/91Figure 2.7: Salmon densities (o) in the surface layer for January 1991.Chapter 2. Instrumentation and Data 302.2.3 Harmonic AnalysisThe current and density records from the moored instruments were analysed for tidalamplitudes and phases using harmonic analysis (see Godin, 1972). Records from tidegauge deployments of six months or longer at Boom Islet, Rapid Islet and Porpoise Baywere available to aid the tidal analysis.The constituents that are resolvable by harmonic analysis are limited both by thesampling interval of the instrument and the record length. The Nyquist frequency, definedas 1/2Lkt, where it is the sampling interval, represents the high frequency cutoff for theconstituents. The inverse of the record length represents the low frequency cutoff. Sincethe cyclesondes had the coarsest sampling rate (3 hours), the Nyquist frequency was4 cycles per day— sufficiently high enough to include the MK3 and M4 shallow waterconstituents.The choice of record length for the harmonic analysis is difficult, since the tidal current signals are often not stationary over long periods (their amplitudes and phases maychange as the stratification changes). The record length must be chosen sufficiently longto resolve the lowest frequency of interest and separate the main diurnal and semidiurnal constituents (as discussed below), but not so long that changes in stratificationsignificantly affect the stationarity of the higher frequency constituents. In the end, arecord length of one month was chosen so that constituents with frequencies as low asthe monthly tide, Mm, were resolved. A list of tidal constituents used in the analysis isincluded in Table 2.2.Power spectra of the pressure records from the Basin cyclesonde (when it was at reston the lower bumper) are shown for January, February, April and May (overlayed) in Fig.2.8. The spectra are plotted with a logarithmic y-axis so that the smaller constituentsmay be clearly seen (the plot does not preserve power). Spectral peaks are visible atChapter 2. Instrumentation and Data 31Table 2.2: The tidal constituents used for the harmonic analysis of the current velocityand density data. The amplitudes and Greenwich phases given by the harmoniè analysisof the long tide gauge records at Boom Islet (file skoo84OlOb) and Porpoise Bay (gaugestation 7852) are included. Data are courtesy of the Tides and Currents section of theInstitute of Ocean Sciences, Fisheries and Oceans Canada, Sidney, B.C.Boom Islet Porpoise BayConstituent Frequency Amp Phase Amp Phase(cyc hour1) (m) (°) (m) (°)Mm 0.0015121000 0.047 15 0.004 31MSf 0.0028220000 0.008 222 0.040 680 0.0387306544 0.495 153 0.310 193K1 0.0417807462 0.898 168 0.605 211N2 0.0789992488 0.222 137 0.094 200M2 0.0805114007 0.999 162 0.481 226S2 0.0833333333 0.252 184 0.103 252MK3 0.1222921469 0.012 124 0.036 17M4 0.1610228013 0.011 169 0.023 289fortnightly, diurnal, semidiurnal and higher tidal frequencies, but since there is only onepoint in the monthly band, it cannot be included on a smoothed spectrum. The spectralvalues were smoothed over 3 frequencies (degrees of freedom (v) = 6).There is one final restriction on the choice of tidal constituents that concerns constituents that are very close in frequency: if two constituents violate the Rayleigh criterion, that is, the frequency difference is less than the inverse of the record length,the harmonic analysis program cannot distinguish them and the fitting procedure maybecome unstable. In order to choose which of the two close constituents to use in the0cJE>U)CuC.)ci0Figure 2.8: Sea-level spectra from the pressure records of the Basin cyclesonde (notplotted in power-preserving format). Pressure records were taken when the cyclesondewas at rest on the bottom bumper (approximately 190 m). The spectral values weresmoothed over 3 frequencies (ii = 6).analysis, the harmonic analysis of the 202 day pressure gauge record at Boom Islet (outside Skookumchuck Narrows) and the 363 day record at Porpoise Bay (inside the inlet)were examined to establish which constituent had the larger amplitude. The most significant pairs that require a 182 day record to separate for the Sechelt Inlet data areMf-MSf,Pl-Kl, MKS2-M and K2-S. The MSf, K1, M2, and S2 constituents werechosen for the analysis.Apparent changes in the K1 constituent may be due to the addition of the P1 constituent; for example, the pressure records show an increase in K1 between January andJune 1991. Using the long deployment tide gauge information, inference can be used toChapter 2. Instrumentation and Data peri (days)Inf 1 0.5 0.33 0.2532000S08080000 1 2 3 4frequency(cpd)Chapter 2. Instrumentation and Data 33separate close constituents in short records (Foreman, 1979). However, since the phase ofthe baroclinic tide at a given depth often bears no resemblance to that of the barotropictide, no attempt was made to separate K1 and P1 in the current or density records.The MKS2 constituent influences the M2 similarly to the way that P1 influences K1.However, the MKS2 amplitude is much smaller than the M2 (< 1 %), so the effect isvery small.2.2.4 WindFour months of meteorological data were measured from the Salmon Geodyne buoy (January to March and April to June), but an anemometer failure during the January toMarch deployment left only the two month April to June record from the Basin Geodyne. The same convention was used as for the current meters: the V component wasanalysed for the Basin data, and the -U component was analysed for the Salmon data.The topographic steering in Salmon Inlet by the steep walls makes this a reasonable approximation; however, the irregular topography around the Basin site allows a fairly largecross-channel wind to develop, particularly when the wind is from the east. Despite theclose proximity of the Geodyne anemometers, the wind records between the two stationsare not well correlated.A sample record from the Salmon Geodyne in January is shown in Fig. 2.9. Thewind appears to be dominated by the diurnal seabreeze, but it is punctuated by periodsof constant outflow (away from the head). Winter outflow conditions occasionally occurin mainland fjords caused by a persistent high pressure arctic air mass that settles overthe British Columbia interior. The air is funneled out through the river valleys (andeventually out through the fjords), and can cause extremely high winds. A discussion ofthe spectral characteristics of the wind is presented in chapter 6.Chapter 2. Instrumentation and Data 341000750500250.Co-250-500-750-10001/23/91 2/1/91 2/8/91 2/15/91 2/23/91Figure 2.9: Wind record from Salmon Inlet for January (units are cm s’). Positivespeed indicates seaward flow.2.2.5 RunoffThe runoff in Sechelt Inlet was characterized by Pickard (1961) as having two peaks peryear, fed in the early summer by snow melt and in the early winter by rainfall. Trites(1955) used rainfall statistics and the drainage basin area to estimate the total amountof runoff into the system (Fig. 2.10 (a)). The average yearly precipitation was estimatedto be 110 m3 sW’.The Clowhom River is dammed by B.C. Hydro, which keeps hourly flow statistics.The level in the reservoir remains nearly constant, so the flow statistics from the dam area good measure of the drainage from the river (Arber, 1993). The average monthly flowfrom the reservoir between 1985 and 1990 is shown in Fig. 2.10(b). The two peaks areclearly visible, and the mean flow is about one-third of that estimated by Trites (1955)for the entire system. The hourly discharge during the time that the moorings weredeployed and its effect on the circulation and density structure are presented in chapter6.1/23/91 2/1/91 2/8/91 2/15/91 2/23/91Chapter 2. Instrumentation and Datac’JEa,ix0L1(B)35Figure 2.10: Estimated monthly runoff from: (a) all sources (Trites, 1955), and (b) theClowhom River dam (from daily discharge rates from 1985 to 1990).0)C,)EG)-Cu —ixLLU,Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Decc’JJan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecChapter 2. Instrumentation and Data 362.2.6 OxygenTwo oxygen meters were deployed during the study: one at 30 m on the Egmont mooring(outside the sill) and the other at 10 m on the Basin Geodyne mooring. The meter atEgmont was deployed three times for two months (beginning in December 1990, February1991, and April 1991) and the meter at the Basin mooring was deployed once for twomonths beginning in January 1991.The time-series from the four oxygen meter deployments are shown in Fig. 2.11. Bothrecords show a marked high frequency response along with lower frequency peaks, whichappear to correspond to the spring/neap tidal cycle. At Egmont, the mean oxygen valueremains steady at between 4 and 5 ml 1_i from December to mid March, when there is asharp increase to nearly 6 ml 1’. After mid April, the mean decreases again to 5 ml 1’.The change in the mean value is likely in response to the increase in production of oxygenby phytoplankton with the onset of spring. Although the mean oxygen values decreaseafter April, the fluctuations remain larger than in the winter, indicating that gradients ofoxygen are higher in the water column than in winter. The Basin oxygen fluctuations aresmaller than at Egmont, but also show an increase in amplitude after about mid March(the high values at the very end of the record are probably in error).A spectrum of the Egmont data is presented in Fig. 2.12. The tidal frequencies ofdiurnal and higher frequency are well resolved and the low frequency variations (periodslonger than 10 days) are also well defined. An analysis of the variance shows that 67.0%is contained in the low frequency band, 8.5% around K1, 5.9% around M2, 1.9% at MK3,and 0.4% at M4. There is very little spectral energy outside of these bands.Another feature of the power spectrum is the apparent tidal response at frequencieshigher than 4 cpd. The inset at the top right of the time-series plot (Fig. 2.11) showsan expanded section of the data over 8 days. The oxygen values do not vary strictlyChapter 2. Instrumentation and Data 371/1/91 2/1/91 3/1/91 4/1/91 5/1/91 6/1/91I I I I8C54_______________________________Sf91 5111191 511191 51151913 + + + + + ÷ + 1. ÷ + + +Jan 2 Jan15 Jan30 Feb12 Feb27 Mar20 Apr2 Apr17 Apr30 May15 May29 Jun158 + + + + + - * + + * + +E700ESC)LIJ43 I IFigure 2.11: Oxygen time-series from the moored Endeco oxygen meters. The data forthe Basin mooring (top) and the Egmont mooring (bottom) are shown. Times of springtidal currents through Skookumchuck Narrows are marked (+). The inset on the upperright of the plot is an expansion of eight days in May from the Egmont record to illustratethe shape of the oxygen signal.sinusoidally, but flatten off near the top of a fluctuation, and can be very sharp near thebottom. If one considers the spectrum of a square wave, peaks occur at the principalfrequency and at all odd harmonic frequencies; the amplitudes of the harmonics fall offas f2. Similar harmonics are created in the spectrum of a triangular wave, but theamplitudes fall off faster, as f4. Because the data appear to have square and triangularwave characteristics, the spectral peaks are probably harmonics of the K1 and M2 tides,and not real tidal responses. In fact, the surprisingly large peak at MK3 is probablyenhanced by the first odd harmonic of K1.The peculiar shape of the oxygen fluctuations may be explained by their origin. Ifthe daily oxygen fluctuations at Egmont are caused by an oxygen-rich turbulent tidaljet passing by the sensor, then the oxygen values will increase fairly quickly from their1/1/91 2/1/91 3/1/91 4/1/91 5/1/91 6/1/91Chapter 2. Instrumentation and Data 38000I(0.073)Figure 2.12: Power spectra of Egmont oxygen time-series. The 95% confidence interval(dashed lines) is based on 18 degrees of freedom (v = 18); the spectral densities are scaledby frequency so that the plot preserves power. The y-axis units are m12 1—2.ambient level at 30 m to the value of the jet. The oxygen will remain relatively steadyuntil the jet recedes and oxygen levels decrease. As the jet decreases in strength, theremay be over compensation of the returning oxygen-poor water that is displaced by thejet, and the oxygen values will drop. The actual response time of the Endeco meterappears to be fast enough that sharp fluctuations in oxygen are recorded. If the responseof the meter to changes in oxygen were slow, the data would be smoother, and the highfrequency spectral harmonics would be reduced.0.05 0.10 0.50 1.00 5.00 10.00frequency (cpChapter 3Tidal ChokingA tidal oscillation in an inlet can be represented by a pure standing wave if there is noenergy lost up-inlet of the entrance. However, some energy is lost to boundary friction,internal wave generation at the sill, and possibly internal hydraulic jumps, giving thebarotropic tide a progressive wave component. The energy loss results in a shift in tidalphase and, in some cases, a reduction of the amplitude of the surface tide landward ofthe sill. Although most British Columbia coastal inlets have phase shifts which are lessthan 15°, some have constrictions which choke the barotropic tidal wave, causing largetidal phase shifts and significant differences between the tidal range inside and outsidethe sill. In Sechelt Inlet, phase changes across the sill at Skookumchuck Narrows are 62°and 42° for the M2 and K1 tidal constituents respectively. This chapter presents a reviewof previous work on tidal choking and how the problem of tidal choking was approachedin Sechelt Inlet.The problem of tidally choked fjords has been looked at extensively in Scandinaviawhere there exist many fjords connected by constricted waterways. Glenne and Simensen(1963) used a one-dimensional viscous channel model to examine the tidal choking, whileMcClimans (1977) employed a purely inviscid model. Stigebrandt (1980) combined thetwo models by attributing some of the pressure gradient along the entrance channel tofriction and the rest to the gradient required for the inviscid acceleration of the flow (Fig.3.1). The tidal flow across the sill at Skookumchuck Narrows requires a frictional modelto account for the choking of the inflow.39Chapter 3. Tidal Choking 40Outer InnerBasin BasinH I u— L—)Figure 3.1: Choking of flow through a constriction (Stigebrandt (1980), figure 2).3.1 Inviscid tidal chokingIf one considers the situation in Fig. 3.1 without the drop due to frictional resistance,zh, then Bernoulli’s equation can be applied (assuming velocity is negligible upstreamof the constriction and uniform atmospheric pressure over the region):Ugi0 = grj +-,(3.1)where i73(t) and i7(t) are the sea-level seaward and landward of the constriction, and U2is the cross sectionally averaged velocity of the tidal stream. Thus, solving for U yieldsU = 2g?73—i. (3.2)Chapter 3. Tidal Choking 41Figure 3.2: Relation between amplitude reduction and phase shift of the surface tideacross a sill (McClimans (1977), figure 3). i0(t) is the sea-level seaward of the sill constriction; i(t) is the sea-level landward of the sill.Figure 3.2 illustrates a sinusoidal variation in external sea-level, i(t) which is responsible for the filling and draining of an inlet. The filling of the inlet will cease whenthe sea-level on either side of the constriction is the same at some time after high tideis reached outside the sill (similarly when draining). The ratio of tidal ranges and thephase shift across the sill, , are related by the expression cos = h/h0.The sea-levels outside and inside the inlet are given by=.sin(wt)= --sin(wt--).170(t)(3.3)Chapter 3. Tidal Choking 42The total amount of water entering the fjord through the constriction during floodtide must be equal to the tidal prism by continuity; therefore,S1h2 = £2 AU(t) dt. (3.4)S is the inlet surface area landward of the sill, and A is the cross stream area of thechannel. This relation was used by McClimans (1977) for a set of eight Norwegian fjordswhich ranged from inlets with little tidal choking to those with exposed sills at low tide.Substituting (3.2) and (3.3) into (3.4):Sh =AJj2[hosin(wt) — hsin(wt —Noting that cos = h/h0, the above may be written asSh1 =A/j2[sin(wt) — cos5 sin(wt— )] dt.Following McClimans (1977), let 0 = wt — (8 is the phase of the inside tide),dO = w di, and integrating from the start to the end of flood tide landward of the sill(— 8 ):Sh =A5mn2cos4 8 dO. (3.5)A topography-tide factor, , may be defined by— Si362ATVg’ (.)where T = 2i-/w. Using cosq! = h/h0, and (3.6), (3.5) can be rewritten= sin4cos4 8 dO. (3.7)4ircos _Chapter 3. Tidal Choking 43The integral on the right has a numerical value of 2.396. Finally,Sjfl2= 0.191. (3.8)cosThis relationship is plotted in Fig. 3.3, along with the tidal data from several Norwegianfjords.One explanation for the deviation of the field data from the inviscid theory in Fig.3.3 is that there is some contraction of the flow as it enters the constriction. McClimansstates that, “Due to the abrupt change of cross sectional area in most inlets, the lateralaccelerations lead to a contraction of the flow”. The contraction coefficient, K, for flowsthrough an orifice is defined as the ratio of the cross sectional area of the ensuing streamto the area of the orifice, and ranges from about 0.5 to 1 (Lamb, 1945). Stigebrandt(1977) shows experimentally that K = 0.75 is reasonable for most natural inlets; however, Stigebrandt (1980) states that for Nordasvatnet, which has L/R > 70, contractionis not important, where L is the channel length and R is the hydraulic radius. ForSkookumchuck Narrows, L/R 50, so contraction is not considered important, and Kis taken to be equal to 1. Contraction would have the effect of reducing the expectedflow rate; therefore, in (3.4), U should be replaced by KU:= 0.191K. (3.9)cosThe “contracted” curve for K = 0.75 is also plotted in Fig. 3.3 and appears to serveas an upper limit to the field data. Deviations from this curve, as in the data fromStørstraumen and Framvarden, are most likely due to friction (McClimans, 1977).Chapter 3. Tidal Choking 44- .Drammensfjordenslofjordencrstraumen8 +Kio X\ Bo nflorden.2X“Rrestva en0 +M2X ‘Nqrdaasvao—Soc------ K1X StorstraumenK= 75X Framvarden00I I I I0.0 0.2 0.4 0.6 0.8 1.0Topography-Tide Factor, = (Si/2AT)(hc’gfFigure 3.3: Choking coefficient, cos ‘, versus topography-tide factor, . Data shown arefield tidal averages (McClimans (1977), figure 4). The solid curve (K=1) corresponds tothe theoretical inviscid curve. Sechelt Inlet M2 and K1 tides are included.3.2 Choking with frictionIf one moves away from the completely inviscid theory of tidal choking and adds anadditional water level drop due to friction (as in Fig. 3.1), h, Bernoulli’s equationbecomes (Stigebrandt, 1980)U2—— z.h = i—. (3.10)The along-channel pressure force due to h is balanced by the force of friction. The forceof friction is given by F1 = WLr6,where W, is the wetted perimeter of the channel. TheChapter 3. Tidal Choking 45bed stress, Tb, has the form, Tb = pCdU2,where Cd is the non-dimensional coefficient offriction. If z.h is small compared to the depth of the channel (i.e. the flow is hydrostatic),the pressure force due to the water level drop isF ApgLih. (3.11)Defining the hydraulic radius, R = A/Wy, one can solve for zih:zh = CdU2L (3J2)RgResubstituting (3.12) into (3.10) and solving for U2,u2= (l)I71o?1iI (3.13)where \ = 2CdL/R. This expression may be used instead of (3.2) in the case whenfriction is important. Equation (3.9) may be rederived using (3.13) to yield— 0.191 K sin3 141+ cos (. )This new expression for the topography-tide factor represents a set of curves whichdepend on ), for which the inviscid curve is an upper limit. K is the coefficient ofcontraction (see section 3.1).Stigebrandt (1980) also includes a term involving fresh water runoff in the continuityequation,= Q + Q. (3.15)S is the surface area of the inlet, and Q and Qj are the volume fluxes for the tide andrunoff respectively. He showed that if a factor S = QfT/aaS: > 0.2 (T = tidal period,Chapter 3. Tidal Choking 46a0 tidal amplitude outside the constriction), then fresh water runoff could cause thelength of the ebb to be significantly longer than the length of the flood, and the meanwater level inside the fjord to be greater than outside. For Sechelt Inlet, QfT/a0S= 0.05and 0.10 for the M2 and K1 tides. For this range of S, the deviations caused by Qj aresmall, and it can be ignored in (3.15) to a good approximation.If the tidal forcing comes primarily from one constituent, as it does in the NorthAtlantic, Cd could be estimated by plotting the choking coefficient versus the topography-tide factor on a plot similar to Fig. 3.3; the curve that passes through that point willgive the value of \ (hence, Cd). For example, the M2 and K constituents for SecheltInlet are plotted on Fig. 3.3 — the large deviation from the inviscid curve would suggesta large value for Cd. However, this analysis fails when there is more than one importantconstituent as in the mixed tides on the coast of British Columbia. The M2 and K1constituents are nearly the same amplitude at the entrance to Sechelt Inlet, causing highvariability in the amplitude of the tidal forcing at the sill (see Fig. 3.5). Because frictionis a non-linear process, the harmonic tidal components cannot be analysed individuallyto ascertain the friction coefficients. Instead, a single channel friction model developedby M. Woodward (Tides and Currents Division, lOS) was adapted and used to analysethe flow through Skookumchuck Narrows. The model will be discussed in detail in thefollowing section.3.3 Tidal Choking in Sechelt InletDirect measurements of the currents in Skookumchuck Narrows were made in 1983 at thepoint of strongest flow in the channel by the Tides and Currents section of the Instituteof Ocean Sciences (105), Sidney, B.C. Together with pressure gauge measurements fromthe Boom Islet and Rapid Islet stations (Figs. 3.4 and 3.5), the data were compared toChapter 3. Tidal Choking 47Figure 3.4: Skookumchuck Narrows. Locations of the tide gauges at Boom Islet andRapid Islet are included. Depth contours are in metres.a simple model based on frictionally dominated hydraulic flow. The model results weremade available courtesy of Mike Woodward (Tides and Currents section, lOS). Afterexamining the results from the 105 model runs, it was decided that, for the purposes ofthis study, the current speeds should be scaled to reflect cross-channel averages, ratherthan currents at the point of strongest flow. Subsequently, the 105 model was reappliedto the scaled data at UBC.The model balances the pressure gradient across the sill by the frictional bed stress.The friction is expressed using linear and quadratic terms (7u and CduIuI):9 . 1JometresIsletRapidChapter 3. Tidal ChokingE-c0I.Julian Day48Figure 3.5: Tidal amplitudes at Boom Islet (solid line) and Rapid Islet (dotted line),from January 31 to February 10, 1984.1ÔF g/ / 1 ‘\H+r [6+7u+Cduluj} (3.16)The 6-term arises from the uneven leveling of the two gauges.The pressure head (z) and flow speed (u) data are fitted in a least squares sense to(3.16) to determine the values of 6,-y, and Cd (see Fig. 3.6). The 7-term affects the fitprimarily near the x-axis: a strong linear term reduces the slope of the fitted curve nearthe x-intercept.Although the dominant balance for most of the data is between the pressure gradientand friction, acceleration effects become important during periods of weak flow and smallC%J0C’jJ32 34 36 38 40Chapter 3. Tidal Choking 49pressure head. The acceleration/deceleration of the flow is the source of the hysteresis inthe raw data near the origin (Fig. 3.6(a)). To correct the data for the acceleration effects,estimates of the velocity before and after an actual velocity observation are made usingthe pressure data from the tide gauges and the Bernoulli equation (3.2). An estimateof the acceleration of the velocity field may then be made, which is less noisy than anestimate made using the direct velocity observations. The pressure gradients responsiblefor the acceleration are then calculated usinglou317Ox gOtThe pressure corrections are applied only where the hysteresis is pronounced; for the datain Fig. 3.6, the correction is applied where -150 cm s < u < 300 cm s (Fig. 3.6(b)).With the hysteresis removed, a fit of (3.16) is made to find the values of the parameters8, y, and Cd. Part of the velocity record from Skookumchuck Narrows is reconstructedusing the fitted model parameters and plotted against the actual velocity observations inFig. 3.7.The lOS model is designed to predict the maximum currents expected for a giventidally active channel, since that is the property of greatest interest to mariners. In orderto analyse the bulk friction effects on the flow in Skookumchuck Narrows, the 105 datawere modified at UBC: the tidal currents shown in Fig. 3.6 were scaled based on theaverage tidal flow expected over the sill calculated by conservation of volume,uA = (3.18)where u = Umar cos(wt) is the channel-averaged velocity, A is the cross sectional area ofthe channel, 77 = 71mar sin(wt) is the sea-level inside the inlet, and S is the surface areaChapter 3. Tidal Choking 50Figure 3.6: Tidal currents versus hydraulic head across Skookumchuck Narrows. Tidegauge data were taken at Boom Islet and Rapid Islet stations; velocities were measuredby boat mounted current meter. The solid lines are the model fits.Speed [cm1(a)Sechelt Rapids 1983Skookumchuck (Rapid Islet)Skookumchuck (Boom Islet)-1600FLOOD1600EBBHydraulic Head [mm]-500Acceleration terms not includedSpeed [cm1](b)Sechelt Rapids 1983Skookumchuck (Rapid Islet)Skooknmchuck (Boom Islet)-1600FLOOD1600EBBHydraulic Head [mm].500Acceleration terms includedChapter 3. Tidal Choking 51of the inlet landward of the sill. From (3.18) and the definitions of u and , it is clearthat Umax = 7lmaxWS/A. Using the tide gauge data to obtain 7max and w for several tidalevents, the maximum value of Umax, the channel-averaged flood current, was found tobe 185 cm s1. Because the maximum flood current in the lOS data is 770 cm s1, thecurrents were scaled before fitting by a factor of 0.24. The best fit of (3.16) to the scaledflood tide data yielded:Cd = 0.08= 0.04 ms’S = 0.03 m2sThe values of the drag coefficient, Cd, found for the flood and ebb tide may be different,depending on the geometry of the sill. For example, in Skookumchuck Narrows the waterflows over a reef south of Boom Islet during flood tide and is convergent near the middleof the stream; the ebb flow, however, is broader and is strongest just downstream ofthe tip of Boom Islet (Mike Woodward, pers. comm.). The maximum channel flowsfor the flood and ebb tides need to be scaled separately, in order to examine the bulkfriction effects. For simplicity, only the flood events are used in the subsequent dissipationanalysis (chapter 4).Equation (3.16) assumes that the drop in water level is due exclusively to the frictionalstress. Stigebrandt’s model (Fig. 3.1), however, includes an inviscid water level drop; ifthe inviscid water level drop is taken into account, the value of Cd obtained by fitting(3.16) to the data is an overestimate. Considering the arguments in section 3.2 concerningthe origin of the two separate water level drops in Stigebrandt’s model, the coefficientof the quadratic term in (3.16) is actually (Cd + R/2L), where R H is the hydraulicradius. Numerically, R/2L 0.01, meaning that in the case of flood tide the value of Cdmay be overestimated by about 12%.Chapter 3. Tidal Choking 52Figure 3.7: Velocity measurements in Skookumchuck Narrows for July 9 and 10, 1983.The solid line is the velocity predicted by the lOS model (courtesy of Mike Woodward,Tides and Currents section).With the fitted values of the friction parameters, the dissipation rate due to frictionmay be calculated. The frictional dissipation rate at Skookumchuck Narrows will bepresented in comparison to other energy sinks in chapter 4.Sechelt Rapids 1983500It+Acceleration terms includedJuly 9, 1983 July 10, 1983Chapter 4Tidal Energy Partition4.1 Energy SinksA potentially large source of energy for mixing in an inlet is the barotropic tide. Energyfrom the tide is extracted at the sill and along the sides and bottom of the inlet; Theresulting up-inlet energy flux is balanced by internal wave generation and dissipationby friction. In narrow and deep-silled inlets (e.g. Knight and Observatory), the greaterpart of the barotropic energy flux is balanced by internal hydraulic disturbances andthe generation of a progressive internal tide (Freeland and Farmer, 1980; Stacey, 1984)and only a small amount (< 5%) by frictional dissipation. In Indian Arm-Burrard Inlet(Vancouver Harbour), much of the up-inlet tidal energy flux between First Narrows andthe sill is dissipated by friction in Vancouver harbour, but internal wave generation isthe largest sink landward of the sill itself (de Young and Pond, 1987). The breaking ofthe internal tide on a sloping bottom has been identified as the primary source of mixingenergy in the deep basin waters (Stigebrandt, 1976).The removal of energy from the barotropic tide results in a phase shift of the surfaceelevation such that high tide inside Skookumchuck Narrows may occur two or morehours later than high tide outside the Narrows. Judging from the phase shifts in surfaceelevation across the sill, the barotropic tidal energy flux into Sechelt Inlet is very large.Three sinks for the energy flux have been identified: frictional dissipation in the sill region,the kinetic energy flux of the turbulent tidal jet, and the internal tide. Dissipation due53Chapter 4. Tidal Energy Partition 54to internal hydraulic processes (e.g. hydraulic jumps) are included in the turbulent jetflux, but estimates of the energy fluxes of high frequency internal waves are not. Staceyand Zedel (1986) and de Young (1986) estimated that the high frequency internal wavesgenerated at the sill have a flux between 5 and 15% of the energy flux in the internal tidesof Observatory Inlet and Indian Arm. This flux may provide another source of energydirectly to the deep basin water, and may be important for diffusive processes. Nearlyall of the energy flux into Sechelt Inlet is balanced by frictional dissipation near the sill;however, contrary to Stigebrandt and Aure’s (1989) claim that inlets which have strongturbulence at the sill do not transfer mixing energy via progressive internal waves, it isfound that a small but significant internal tide is produced.This chapter discusses the partition of energy between the three tidal energy sinksidentified in Sechelt Inlet. In addition, a modified analytical model for estimating theup-inlet flux of the barotropic tide using tide pressure gauge measurements is presented.4.2 Barotropic Flux ModelWhen power is extracted from the barotropic tide at a sill, the currents and tidal elevations seaward of the sill fall out of quadrature by some phase angle, , and the phasesof the surface elevations across the sill differ by an angle (see Fig 4.1). Freeland andFarmer (1980) showed how and are related under certain conditions; their analysisshowed how phase shifts determined from tide gauge data may be used to infer the energyflux from the surface tide. This section presents a more general derivation of the phaserelationship and describes how it may be used to analyse the power loss in fjords wherethe phase shifts across the sill are large.One assumption that Freeland and Farmer (1980) used in deriving their phase relationwas that the surface displacement and velocity at section 2 were in quadrature. StaceyChapter 4. Tidal Energy Partition 55(1984) generalized the model so that the quadrature assumption was not required, andde Young (1986) further generalized the derivation by assuming a linear change in channelwidth along the inlet. The above relationships, however, did not take into account thedifference between the tidal range at section 1 and section 2, which proves to be importantwhen considering the total power that may be extracted from the tide. Since the differencein tidal ranges is not negligible in Sechelt Inlet, it is necessary to re-derive the energyflux theory.4.2.1 Energy flux of the barotropic tideThe net flux of tidal energy through a section perpendicular to the inlet channel (A),averaged over a tidal cycle (denoted by an overbar), is given by (Garrett, 1975; Freelandand Farmer, 1980):P= fJ pgdA, (4.1)where p is the mean density of the fluid, g is the acceleration due to gravity, u is thealong-channel component of velocity and i is the surface elevation.The velocities (ui, u2) and tidal heights 2) discussed in the following derivationare illustrated in Fig. 4.1; the tidal heights are assumed to be independent of x, thealong-channel coordinate. If the velocity and surface elevation were in quadrature, thenthe average energy flux over a tidal cycle would be zero. Therefore, if energy is dissipatedin the inlet, u and must be out of quadrature by some phase angle, &The average over a tidal cycle can be shown to be1 p2 1U1?7i =— J U177 dwt = sin . (4.2)2ir o 2chapter 4. Tidal Energy Partition 5611=ijsth(0t)1=a1cos(t -12 2sin(ot -u2=2cos(oX -Figure 4.1: Model for the generalized phase analysis. S2 represents the entire surfacearea up-inlet of the sill.Equation (4.1) evaluated at section 1 becomesP JJ pgii11 sine dA1.A1(4.3)To get the barotropic part of F, the barotropic tidal velocity, u1, is used; u can becalculated by conservation of fluid volume between sections 1 and 2. From Fig. 4.1:u1A— u2A =(1S)+ (4.4)Expanding the trigonometric terms and equating coefficients, and noting that=2w(S — S)/A2 (from the balance of volume flux at section 2):Section 2Chapter 4. Tidal Energy Partition 5711A sin = 2wS sin i +,2w(S — S) sin 42 (4.5)?1Acos = i1S +i2wScos q5i +i2w(S — S)cos2. (4.6)Using (4.5) and (4.6), the velocity,€t1 and the phase, , may be found:= + (i2S)+2(S — S) +2iSiS cos+2i1i2S(S— S) cos 2 + 2iS(S — S) cos(ç1— 2)] (4.7)-1 I2wS sin ‘i + u2A sin 2= tan. (4.8)iiiwSi +i12wS cos +u2A cos 2Normally, the barotropic tide inside an inlet is very close to being in quadrature (implying that the energy extracted from the surface tide up-inlet of section 2 is small. Thetide gauge data in Sechelt Inlet support this approximation; the small phase differences insurface elevation between Rapid Islet and Porpoise Bay imply that there is little energylost from the barotropic tide inside the inlet. If the expression for u2 at section 2 iscombined with the approximation= = , S may be eliminated from 4.8, and theexpressions for i and e become:—1 2wSsin= tan, (4.9)i71wS1 +i72wS2cos= (4.10)Chapter 4. Tidal Energy Partition 58Substituting (4.9) and (4.10) into (4.3), the total power loss becomes= P’7i/S)2+ (i2S)2 +2i,SiScosq sin (tan_i [ 2wSsin2 ‘71Si + ‘72wS cos(4.11)Moreover, if it is assumed that ñ2S >> ‘i Si, then the power loss expression reducesfurther to:1P = pgwi2Ssin. (4.12)The tidal amplitudes and the phase shift across the sill are related by ñ2/i3i cos(see Fig. 3.2). Using this expression, (4.12) becomes1‘2 sin2çSP = pgw77S22 (4.13)In the past, the development of the energy flux expression was made without considering the change in surface elevation, 2• If (4.13) were derived with 2 equal to ,then the power loss would be pgw1S2 sin , where pgwS2 was considered to be thetotal available tidal power and sin was the percentage of this power that was extracted(Freeland and Farmer, 1980; de Young, 1986). However, as the tide becomes chokedand is no longer a small angle, the flow through the constriction is reduced such that‘72 is significantly smaller than ,. From (4.13) it is evident that the maximum tidalenergy flux will occur when the phase shift, , is 45°, and will be only half of what waspreviously considered to be the total available power. Therefore, in a dissipative inlet,the total available tidal energy flux is not pgwfS2,butThe total available power and the energy flux given by expression (4.13) from theM2 and K1 tides are shown in Table 4.1. The total power loss computed in this manner( 42 MW) may only be considered an average, since the tides interact and cannot trulyChapter 4. Tidal Energy Partition 59Table 4.1: Al2 and K1 barotropic tidal parameters from the harmonic analysis of Egmontand Porpoise Bay (courtesy of the Tides and Currents division, lOS). The tidal parameters are used to calculate the total available barotropic energy fluxes, and thç phasedifferences between the sections, , are used to compute the fractional power loss basedon the modified tidal flux model.Total AvailableConstituent w Power Power Loss %(m) (rad s_i) (°) (MW) (MW)Al2 0.9992 1.4 x iO 61.5 34.3 28.7 84K1 0.8980 0.7 x iO’ 42.1 13.9 13.7 99be considered separately. Individual ebb and flood events can occur during spring tideswhich have dissipation rates computed from (4.13) approaching 100 MW.The model behaviour in extreme choking situations should yield physically plausibleresults. For the analysis, the width of the inlet will be taken as constant (so that thelength is directly proportional to the surface area), and the length of the inlet is assumedto be 50 km from head to sill. Figure 4.2(a) shows as a function of from the expression(4.9). Three locations for section 1 were chosen for the comparison: 5, 10 and 25 km fromthe sill. reaches a maximum at a value of that is dependent on S1/S2, then decreasesto zero as approaches 90°. When the phase shift of the tide across the sill approaches90° the tide is being held back enough so that very little water is actually getting pastthe sill; when ç = 90° no water passes the sill, and the sill effectively becomes a solidwall. At section 1, the tide then becomes a standing wave where the surface oscillationand tidal currents are in quadrature ( = 0). Hence, the model returns a result which isintuitively correct for large phase shifts.Figure 4.2(b) shows how the model behaves for = 45°, as section 1 is moved seawardChapter 4. Tidal Energy Partition 60fl\0 20 40 60 80 0 50 100 150 2000 (degrees) Distance of Station 1 from SIN (km)Figure 4.2: Analysis of the modified barotropic tidal flux model using an inlet of constantwidth with the sill located 50 km from the head. (a) f as a function of q for section 1 (seeFig. 4.1) located at 5, 10 and 25 km from the sill. (b) The behavior of as a function ofthe distance of section 1 from the sill (q = 45°).of the sill. The total available energy in the barotropic wave landward of section 1 willincrease as the surface area S1 + 52 increases. Therefore, the energy lost over the sill (aconstant) will be proportionally less than the total upstream energy the further section 1is moved away, and the tidal wave will begin to look more like a standing wave. Therefore,as section 1 is moved arbitrarily far away, should (and does) asymptotically approachzero.4.3 Internal TideBased on tidal amplitudes of 1 to 2 m, barotropic tidal currents at station 3 (Fig. 1.2)in Sechelt Inlet should be on the order 2 cm s’ or less; however, tidal currents are often10 cm s or more, particularly in the upper water column. Figures 4.3 (a) and (b) showthe baroclinic (zero mean) M2 tidal velocities at station 3 (Basin) for January both “inphase” and “in-quadrature” with respect to a fixed reference phase. Figures 4.3 (c) andChapter 4. Tidal Energy Partition 61(d) show the associated M2 perturbation density profiles. To calculate a velocity profile,the tidal amplitudes and phases from the harmonic analysis at all depths at station 3were transformed into two orthogonal profiles using a reference phase; the mean wasthen removed from each profile to eliminate the barotropic mode. The density profileswere computed in a similar fashion, but the mean was not removed; the barotropic modestretches the water column, and the resulting effect seen at a fixed-depth instrument ismuch smaller than the baroclinic fluctuations. The baroclinic currents are larger thanthe barotropic currents at some depths, and the velocity profiles contain several zerocrossings. Figure 4.4 illustrates the K1 velocity and perturbation density profiles forJanuary.The internal tides of Knight Inlet (Webb, 1985) and Indian Arm (de Young, 1986)were described in terms of a dynamic modal decomposition based on the stratification ofthe water column. The modal decomposition normally requires an infinite, flat-bottomedocean for its domain. Since an inlet does not generally satisfy either requirement, theuse of the dynamic mode method must be justified.Webb (1986) outlined the requirements for such a justification: (1) the ray slopesmust everywhere be superior to the bottom slope to ensure reflection about the vertical,and (2) variations in the domain must be slow with respect to all modal solutions (WKBapproximation). The first requirement may be examined by comparing the Kelvin waveray slope, w//N2 2 (de Young, 1986), to the slopes of the main basin floor, where Nis the Brunt-Väisälä frequency and w is the tidal frequency. Using a typical near-bottommid-basin value of N2 = iO rad s2, the ray slope for the M2 constituent is 0.08and 0.O4 for the K1 constituent, compared to -0.02 on the bottom.The WKB approximation may be checked by comparing basin floor variations awayfrom the boundaries (and the sill) to the wavelengths of the lowest baroclinic modes. Thelength scale of bottom variations is about H/VH = 130 km (Webb, 1985), compared toFigure4.3:ProfilesofBasinalong-channelvelocityandperturbationdensityfromtheM2constituentforJanuary1991;(a)in-phasevelocity, (b)in-quadraturevelocity,(c)in-phasedensity, and(d)in-quadraturedensity.I.(a)(b)(c)(d)-4-2024V(cm1)V(cm)-4-20240.00.10.20.30.4-3Density(kgm)0.00.10.20.30.4-3Density(kgm)t%Z(a)(b)(c)(d)Figure4.4:ProfilesofBasinalong-channelvelocityandperturbationdensityfromtheK1constituent forJanuary1991;(a)in-phasevelocity, (b)in-quadraturevelocity,(c)in-phasedensity, and(d)in-quadraturedensity.t C, C,-.E C Qtoor-V(cm.1)V(cm1)-0.3-0.2-0.10.00.1-0.3-0.2-0.10.00.1.3-3Density(kgm)Density(kgm)0) CA3Chapter 4. Tidal Energy Partition 6434 km and 18 km for modes 1 and 2 respectively. Sechelt Inlet reasonably satisfies bothof the above criteria.4.3.1 Theory of normal modesThe equations for the dynamic (normal) modal decomposition may be derived from theshallow water equations (e.g. LeBlond and Mysak (1978), (8.16) to (8.19)). Assumingan ewt dependence for all variables, the shallow water equations reduce topo(—iwu—fv) = —ppo(—iwv+fu) =po(w2— N2)w = ZWPzux+vy+wz = 0.The field variables are horizontal velocity (u, v), vertical velocity (w), and pressure (p);the unperturbed density, po, varies with z. f is the Coriolis parameter, w is the angularfrequency of the disturbance, and N is the Brunt-Väisãlä frequency defined by N2 =29., where p is the average density of the water column and g is the acceleration dueto gravity. Now one can assume a form of the solution which will lead to a separation ofvertical and horizontal variables. The following assumptions are made:(u(x,y,z) (uh(x,y)p01 I = D(z)iv(x,y,z)) V’(x,y)p(x,y,z) = D(z)P(x,y)w(x,y,z) = -Z(z)P(x,y).The shallow water equations separate into the following horizontally and verticallyChapter 4. Tidal Energy Partition 65dependent equations, where h is the separation constant with the dimensions of depth,12 is the angular velocity of the earth, and the Boussinesq approximation has been made:iwUh— 2!2 x Oh = VP (4.14)V.U = (4.15)gh&Z N2_Wz— 4 6dz2 gh .1The boundary conditions for (4.16) areZ(—H) = 0 (4.17)ÔZ Z—= 0 at z = 0. (4.18)tiZ uCondition (4.18) is necessary only for the computation of the barotropic mode, andessentially reduces to Z(0) = 0 for the baroclinic modes.Equations (4.16) to (4.18) represent an eigensystem where the solutions to the systemare the vertical mode shapes and the eigenvalues are proportional to the phase speed ofthe mode. The solutions for the horizontal velocity and pressure terms are= _hU1L(x,y) (4.19)p(x,y,z) = —pohP(x,y). (4.20).Additional information about the modal structure of the water column can be attainedby knowing the time evolution of the perturbation density field. Since8Po 4.21at azChapter 4. Tidal Energy Partition 66we can use (4.21) to relate the density perturbations to the vertical velocity field (assuming an c” dependence):= _?N2(z)Z(z)p(x,y). (4.22)The cross-channel flow in a fjord is expected to be much smaller than the along-channel flow; therefore, the cross-channel velocity (V”(x, y)) is set to zero. The generalsolution to (4.14) and (4.15) is the set of Kelvin waves:U(x, y) = U0e”V(x,y) = 0P(x, y) = Uocnex_h’,where f = 22 sin is the Coriolis parameter, is the latitude, c, = \/7 is the phasespeed of the mode and U0 is an arbitrary scaling constant. The final solutions for thefield variables are:u(x, y, z) = —ha dZ(z)U0e” (4.23)wn(X,y,Z) iwZn(Z)U(iwxfy)/c (4.24)p(x, y, z) = jUocnN2(z)Zn(z)e_M (4.25)The vertical eigensystem is solved for each Z(z). Using the finite centered differenceform of (4.16) and varying the eigenvalue, c = \/ç, the boundary condition (4.18)was satisfied to find Z(z). The vertical eigensystem resembles a simple harmonic wavethat is modified by the vertical density profile, N2(z). Indeed, if N2(z) were constantthe solutions would be Z(z) x sin(), n = (1,2,...).Chapter 4. Tidal Energy Partition 67The N2(z) profiles were obtained using both CTD data and the salinities and temperatures from the current meters. Averages of the temperature and salinity values weremade from the moored instruments and then plotted on temperature- salinity (TS) diagrams (Fig. 4.5). The TS diagrams show that there is a lower layer of water (belowabout 150 m) that is constant in temperature and salinity throughout the year, and thatthe layers above undergo a transformation from cold to relatively warm water betweenFebruary and April. In Fig. 4.5, three distinct water types are visible correspondingto the three layers of water seen in the density profiles: (I) the surface layer with largescalar gradients and high annual variability (0-10 m); (II) the mid-depth layer drivenby mixing from the turbulent jet (10-150 m); (III) the deep basin water with little variability and small scalar gradients (the very small tail on the high salinity end of the TSdiagram, 150-275 m). There is a profound change in the middle and upper layer watertypes between February and April due mainly to temperature changes above 150 m.For each month, these moored-instrument-averaged TS diagrams were compared toTS diagrams made from the CTD profile averages at the beginning and end of thedeployment period, and the averaged temperature and salinity data were adjusted ifnecessary. The log N2(z) profiles that were produced by the adjusted temperatures andsalinities were interpolated to 1 m intervals (Fig. 4.6); they represent hybrid profiles madefrom the CTD profiles (which are not always reliable in the surface layer because of rapidtemporal changes) and the moored instrument averages. The TS diagram for March 1991is omitted, since no surface layer data were available from moored instruments.4.3.2 Normal mode fittingThe normal modes described by (4.23) to (4.25) may be normalized so that they obeythe following orthogonality conditions:Chapter 4. Tidal Energy Partition 68January February April • May......(‘.J c’J • c’J ‘I’1—— ...(I)a,.(Ill) (Ill) (N)1(II)6 18 2226 1 26 18 22Salinity Salinity Salinity SalinityFigure 4.5: TS diagrams for January, February, April and May 1991. The layers aredenoted as follows: (I) the upper layer (0-10 m), (II) the middle layer (10-150 m), and(III) the bottom layer (150-275 m).1Hun(z)um(z) dz = (4.26)1 0 nmJ Wn(Z)Wm(Z)(N2( — w2)dz = = (4.27)1 (MKS units) n = mfFp(z)p(z)(1V(7J)d= o. (4.28)The first four normalized baroclinic w(z), u(z) and p’(z) modes are plotted for January1991 in Figs. (4.7) to (4.9).Chapter 4. Tidal Energy Partition 69o 0 0 00U)Eo o 0 0o o 0 0c’j c.J cI c’Jo o o 0U) U) U) U)c’J-6-5-4-3-2-5-4-3-2 -5-4-3-2 -5-4-3-2Log B-V Ireq Log B-V Ireq Log B-V Ireq Log B-V treqFigure 4.6: Log Brunt-Väis.lä frequency profiles for January, February, April and May1991.If the assumption is made that the velocity field can be described as a superpositionof normal modes (i.e. u(z) = au(z), p’(z) = bp(z)), then the complexconstants, a and b, may be found by computing the inner products:= 1Hb= jw)dz. (4.29)Tables 4.2 and 4.3 show the coefficients for the in-phase and in-quadrature velocity profiles(ar’, ama9) and the in-phase and in-quadrature perturbation density profiles (b’1,b’’)for the January M2 and K1 data.-c 0) (30 U)0 0 0 U)0 0 c’J 0 U) c..J0I—10-505Velocity(ms1)10Figure4.7:Verticalvelocitymodes(w,(z))forJanuary1991.Onlythefirstfourmodesareshown.0-c 0. ci)0 U)0 0 0 0 C’]-40 U,0 U)C’]-0.4-0.20.00.20.4Velocity(ms1)Figure4.8:Horizontal velocitymodes(u(z))forJanuary1991.Onlythefirstfourmodesareshown.oMode18-Mode2Mode3Mode40 L()C%JIIII-0.06-0.020.00.020.040.06Density(kgn13)Figure4.9:Perturbationdensitymodes(p(z))forJanuary1991.Onlythefirstfourmodesareshown.Chapter 4. Tidal Energy Partition 73The fraction of the variance that each mode explains for the velocity and density datawas computed. For velocity, the total variance isu(z)u(z)dz = 11Hau(z) au(z) dz, (4.30)where a = a’1 + iama9, and the modes are real so that u(z) = u(z). By applyingthe orthogonality condition (4.26),1 1H u(z)u*(z)dz = a,a = (a’)2+ (a”)2. (4.31)Since the weighting function for the density function, p(z), is not unity, a differentapproach must be taken to compute the total variance of the system. One approach isto create a new set of orthogonal functions, C(z) = p(z)1 -w2 and find the set ofcomplex coefficients, b, such that= j p’(Z)(n(Z). (4.32)Because the weighting function for C(z) is unity, the variance of p’(z) is given by,The fractions of the variance that each mode contributes to the total varianceof the velocity and perturbation density profiles are included in Tables 4.2 and 4.3.The velocity and perturbation density profiles are reproduced well using only thefour lowest modes. Figures 4.10 and 4.11 present the M2 velocity and density proffles forJanuary; the superpositions of the first four modes found by the inner product methodare plotted for comparison.The inner product method may only be used when data are available over the entirewater column, since there is no orthogonality condition which exists over a partial watercolumn. When no surface layer data are available, one must use a least squares fittingChapter 4. Tidal Energy Partition 74Table 4.2: Dimensionless coefficients from the modal fits to the M2 tidal velocity andperturbation density profiles for January 1991.Mode c7 {u} {u} % Variance ?{p’} Z{p’} % Variance(cm s)1 65.3 0.024 0.059 5.4 31.8 22.5 25.92 42.7 0.109 -0.105 30.7 17.7 -10.7 32.93 27.8 0.180 -0.036 45.4 24.2 -9.2 26.84 20.1 0.061 0.047 7.9 5.8 8.2 9.05 16.2 -0.034 0.063 6.9 -0.2 1.8 2.16 13.6 -0.016 0.029 1.5 4.8 -1.5 1.57 11.8 0.019 -0.009 0.6 2.8 1.2 1.08 10.3 -0.002 0.017 0.4 0.3 1.6 0.39 9.3 -0.020 0.006 0.6 -0.3 0.7 0.310 8.3 -0.017 0.013 0.6 -1.9 -1.4 0.2Table 4.3: Dimensionless coefficients from the modal fits to the K1 tidal velocity andperturbation density profiles for January 1991.Mode c f{u} {u} % Variance {p’} {p’} % Variance(cm s)1 65.3 0.395 0.033 77.7 22.3 -20.6 24.32 42.7 0.161 0.021 13.1 3.3 3.8 37.33 27.8 -0.057 0.003 1.6 -13.5 -0.1 22.54 20.1 -0.096 -0.029 4.9 -1.9 -8.8 9.65 16.2 -0.007 0.007 0.0 2.3 0.3 2.96 13.6 0.029 0.022 0.7-4.3 4.5 2.87 11.8 -0.023 0.007 0.3 -4.4 0.8 0.18 10.3 -0.021 0.013 0.3 -2.0 -2.4 0.19 9.3 0.022 0.018 0.4 -1.7 -1.1 0.110 8.3 0.039 0.022 1.0 -2.0 -0.4 0.3Chapter 4. Tidal Energy Partition 75procedure and make some assumptions about the number of up- and down-inlet travelling modes (Webb, 1985; de Young, 1986). Since much of the difference in the modalstructures of velocity and density is in the surface layer, the absence of surface layer databrings considerable uncertainty to the modal fits using the least squares method.The data from the harmonic analysis described in chapter 2 were used to createvelocity and perturbation density profiles for January, February, April and May 1991.In March, the absence of S4 instruments in the surface layer means that the verticalprofiles are incomplete and therefore unsuitable for the inner product method. Thebarotropic effect on the density structure through the stretching of the water column wascomputed and was found to be negligible compared to the density variations measured.The velocities were vertically averaged and the mean subtracted to remove the barotropicmode. The vI2 and K1 constituents were by far the largest contributors to the varianceof the data. The January profiles are shown in Fig. 4.3 (M2) and Fig. 4.4 (K1).If one separates the waves into up- and down-inlet travelling waves, a simple modelmay be used to fit the modes to the data. Taking the origin (x= y = 0) at the mooringstation:u(z) = u(z) [ae” + (4.33)p’(z) = p(z) [a’e — ae”] . (4.34)Since u(z) = U0h and p(z) = U0cp.N2Z, the change in sign of c in the down-inlettravelling wave causes a minus sign to appear only in (4.34).Chapter 4. Tidal Energy Partition 76(a) (b)0___00 0I I-4 -2 0 2 4-4 -2 0 2 4V (cm s V (cm sFigure 4.10: In-phase (a) and in-quadrature (b) velocity profiles of the M2 tide forJanuary 1991. The dotted lines represent the approximations to the profiles given byusing the first four mode fits obtained by the inner product method.The two velocity (in-phase and in-quadrature) and two perturbation density profilesare fitted to each mode to yield four independent equations in four unknowns a,...I.up..dn, e’ ):= {a7e:’ +{u} ==—= {ae’ — ae’}.Chapter 4. Tidal Energy Partition 77(a) (b)E-C00 0Q 0cl c,1— I I I I0.0 0.1 0.2 0.3 0.4Density (kg mFigure 4.11: In-phase (a) and in-quadrature (b) perturbation density profiles of the M2tide for January 1991. The dotted lines represent the approximations to the profiles givenby using the first four mode fits obtained by the inner product method.If only the velocity data are available, one must assume something about the reflectionfrom the head before getting a fit; however, with the density data one can compute anamplitude and phase for both an up- and down-inlet wave. With the barotropic moderemoved, there is no need to include it in the fitting procedure.The normalized functions of u(z) and p(z) were not used to determine the amplitudes and a as they were previously in determining the amount of variance eachmode accounted for, since normalizing the functions removes their relative scaling. Instead, the function Z(z) was normalized for each mode, and the functions u(z) and00U)00U)8E-C000U)c..J0U)c’.J0.0 0.1 0.2 0.3 0.4Density (kg mChapter 4. Tidal Energy Partition 78p(z) were computed by (4.23) and (4.25), with U0 = 1 m s. The functions u(z) andp(z) have units of velocity and density, making the amplitudes of the fitting coefficientsdimensionless. By creating u(z) and p(z) in this fashion, the coefficients that resultfrom the velocity and density fits are scaled properly. Since each Z(z) is normalized,the fitted amplitudes for each mode may be compared to each other in terms of energy:([aPJ2— [a92)c is proportional to the up-inlet energy flux in mode n. Values ofa, qP, and for the fits of the first four modes to the M2 and K1 data are tabulatedin Tables 4.4 and 4.5.The contributions of the dynamic modes to the M2 and K1 profiles are significantlydifferent (see Tables 4.2 and 4.3). The M2 variability is dominated by the second andthird modes; the first mode appears to undergo significant reflection landward of station 3.Although the fourth mode also appears important from the tabulated fits, its contributionto the energy flux is very small (higher modes even more so). Contributions to the energyflux for the K1 constituent, on the other hand, are dominated by the first mode. In thiscase, the second mode appears to be undergoing very strong reflection up-inlet of station3. The reflection coefficients in Tables 4.4 and 4.5 are the ratios a/a’ for each mode;reflection coefficients greater than 1.0 may represent a transfer of energy between modesupon reflection. The overall up-inlet flux must be greater than the down-inlet flux, butthe down-inlet flux of individual modes may in fact be greater than the up-inlet flux dueto contributions transferred from other modes at the point of reflection. The distributionof modes for the M2 and K1 constituents does not change significantly over the fourmonths of observation.Prior studies using the dynamic mode fitting employed the least squares technique,since there was no information about the surface layer (Webb, 1985). Because the modeswere difficult to distinguish in the lower layers, this procedure was not simple; someassumptions were necessary concerning the number of modes travelling up and down theChapter 4. Tidal Energy Partition 79inlet in order to help the mode fitting. In addition, the barotropic mode would have to beincluded in the fitting process. A least squares fit of the dynamic modes was performedfor comparison to the inner product method.From the variance analysis it is apparent that the first four modes account for most ofthe variability in the observations. Because the least squares fitting procedure minimizesthe residual variance without ensuring that the total variance of the solution is the sameas the original signal, using too many modes for the fit may result in unreasonably largeamplitudes. The signs of these amplitudes will be such that they cancel to give the correctfit to the data, but the sum of their individual variances will far exceed the variance ofthe observations. This behaviour was observed, particularly in the density fits, whenten modes were fit to the data. When the number of modes was reduced to four, as issuggested to be sufficient by the modal analysis, the amplitudes of the up- and down-inletwaves were very similar to those found by the inner product method (see Tables 4.4 and4.5). If the data from the surface layer were omitted from the least squares procedure,the fitted amplitudes behaved in much the same way as when too many modes were usedto fit the complete data set. The absence of data at the surface would mean much greateruncertainty in the modal amplitudes, and, hence, the energy flux estimates.4.3.3 Energy flux of internal modesFrom the modal solutions for velocity and perturbation density, the energy density andenergy flux of the internal modes may be computed. From Gill (p. 140) the energydensity per unit area of an internal wave is given by:___________________272Ea=—po(u +v --w )+ 2 ( .5)2 2poNThe overbar represents an average over wavelength. The internal waves generated at theTable4.4:Modalfits(amplitudeandphase)fortheM2tide.Fitsusingboththeinnerproductandleastsquaresmethodareshown.IInnerProductLeastSquaresMonthModec,aaqçb!’ReflectionaaqReflection(cms—’)165.33.282.4940-1500.763.142.3640-1500.75Jan242.73.990.80-371770.204.000.71-35-1750.18327.87.701.39-171360.188.121.69-171420.21420.14.310.7950-790.184.731.2545-1170.26164.23.722.1244-1510.573.742.1444-1510.57Feb235.56.590.58-181060.096.790.57-171030.08324.26.570.6130740.096.770.6028710.09417.16.391.04126-330.166.180.46123-480.07158.45.102.2944-1260.455.262.0240-1150.38Apr237.45.920.61-10590.106.261.01-9310.16325.511.040.40-11140.0410.601.0401310.10417.94.290.8451-710.203.391.50551800.44172.74.122.5646-1680.624.122.5147-1680.61May239.37.990.88721020.118.000.85-45780.11330.110.420.61104-1170.0610.420.65-38-570.06421.06.021.0664-1480.185.060.3534-1050.0700Table4.5:Modalfits(amplitudeandphase)fortheK1tide.Fitsusingboththeinnerproductandleastsquaresmethodareshown.FInnerProductLeastSquaresMonthModecaqPqReflectiona’açbReflection(cms’)165.34.752.19-16540.464.962.17-15570.44Jan242.72.261.4317-80.632.501.5113-40.60327.83.351.3418030.403.701.59173-100.43420.14.062.39-1351450.594.251.34-1451340.32164.25.021.84-26230.374.961.96-26210.40Feb235.52.662.62-5-260.992.682.90-6-251.08324.22.640.49-149500.183.080.79-153-1750.26417.12.043.86-871471.892.532.55-1291661.01158.47.004.11-19570.596.984.15-17600.59Apr237.43.673.810611.043.503.704641.07325.55.390.63172230.125.950.51175-1750.09417.93.174.42-1221681.392.864.39-1061281.53172.76.682.67-30280.406.712.77-30270.41May239.32.372.04-5310.862.762.23-5650.81330.14.820.30-1771400.065.390.20-1741010.04421.01.202.34561141.960.123.001719025.0Chapter 4. Tidal Energy Partition 82sill start off as plane waves and adjust to the rotation of the earth slowly. The Rossbyradius of the barotropic mode is approximately 500 km; because it is much larger thanthe inlet width (< 2 km), the effect on the shape of the wave is negligible. An internalwave with a phase speed of 1 m s1 has a Rossby radius of 10 km, which is still muchlarger than the inlet width. The highest significant mode appears to be the fourth (c20 cms1), with a Rossby radius of 2 km. The cross-channel dependence of the Kelvinwave is ignored for all modes and the energy density expression for plane waves is used toapproximate the modal energy density. Taking the real part of each term and integratingover the depth, H, and width, W(z), of the inlet (recognizing that i2 =c2U H= ° f W(z) {cZ2(z) +w2Z(z) + N2(z)Z(z)j dz. (4.36)The energy flux is simply Ecu, where c is the group speed of the mode.A measure of the net up-inlet energy flux independent of the modal decompositionmay be made by integrating the baroclinic velocity and perturbation density fields asfollows (Gill, p. 141):j W(z) (u(z, t) p’(z, t)) dz (4.37)p’(z, t)= i: —p’(z’, t)g dz’,where (u(z, t) p’(z, t)) represents an integral over one tidal cycle. The net up-inlet energyfluxes of the M2 and K1 tidal constituents from the inner product fits to the normalmodes are compared to (4.37) in Table (4.6). The data are also plotted in Fig. 4.12.The energy flux estimates from the modal analysis are about twice as large as thosefound from the velocity-pressure analysis; both methods show a general increase in theup-inlet energy flux from January to May. The disagreement between the two estimatesmay stem from the fact that the modal flux estimate is an approximation based on aChapter 4. Tidal Energy Partition 83Table 4.6: Net up-inlet baroclinic tidal energy flux at station 3 (MW) for (a) M2 and (b)K1 constituents in January, February, April and May 1991. Modal flux errors are basedon a 5% error in velocity; < up’ > flux errors are based on a 5% error in both velocityand p’.M2 K1Month Flux (Modal) Flux ((up’)) Flux (Modal) Flux ((up’))MW MW MW MWJan—Feb 0.026 ± 0.002 0.012 ± 0.003 0.035 ± 0.004 0.022 ± 0.003Feb—Mar 0.035 ± 0.004 0.015 ± 0.003 0.036 ± 0.004 0.015 ± 0.003Apr—May 0.055 ± 0.006 0.009 ± 0.004 0.044 ± 0.004 0.039 ± 0.006May—Jun 0.077 ± 0.008 0.030 ± 0.008 0.101 ± 0.010 0.052 ± 0.007theoretical model of the circulation, and that uncertainties in calculating quantities suchas p’(z) in (4.37) may be somewhat larger than stated in Table 4.6. The fact that thetotal energy flux never exceeded 0.2 MW in spite of the large estimates for the barotropictidal flux suggests that the energy flux of the internal tide is not the dominant energysink for the tidal flux. The energy fluxes for Sechelt Inlet are about 50% as large as thosefound by de Young (1986) in Indian Arm. Since the combined basin volume of Secheltand Salmon Inlets is nearly twice that of Indian Arm, one would expect that the mixingin Sechelt Inlet would be less vigorous. The bottom density in Sechelt Inlet decreases by0.01 kg m3 per month when no deep- or mid-water renewal takes place, in contrastto Indian Arm which decreases by 0.02 kg m3 per month. This difference supportsthe assertion that the mixing energy per unit volume of basin water is lower in SecheltInlet.The strong diurnal sea breeze (see chapter 2) could potentially create currents whichinterfere with the K1 tidal currents. To understand the extent to which the K1 tidalChapter 4. Tidal Energy Partition 84currents are contaminated, ratios of K to O tidal currents were compared to K /01ratios of surface elevation. If the response to both constituents is the same, the ratio ofthe currents should equal the ratio of the forcing.It was observed that the K1/0 ratios were quite different down to about 12 m,suggesting that some contamination by the sea breeze is present in the surface layer.To assess the impact of the wind-generated currents on the estimated total energy flux,sensitivity tests were run using current profiles with (a) currents above 12 m reducedby 50% and (b) currents above 12 m enhanced by 50 %. The change in total energyflux (including the M2 contributions) was.— ±5%. Hence, it was concluded that windcontamination does not significantly affect the energy flux estimates.In chapter 5, the energy flux of the internal tide is compared to the change in potentialenergy of the water column due to vertical diffusion, giving an estimate of the fluxRichardson number (mixing efficiency) for the Sechelt Inlet basin.4.4 FrictionWhen a strong flow passes through a constriction, friction from the sides and bottomof the channel can cause the flow to become turbulent. Energy extracted from the flowis eventually dissipated, going into heat and mixing processes. The power lost by thesurface tide to friction in the sill region has been calculated for Knight, Observatory andBurrard Inlets, and was shown to equal 3 to 15% of the total available power (Freelandand Farmer, 1980; Stacey, 1984; de Young, 1986). Using the dissipation expressiondeveloped by Freeland and Farmer (1980), the power lost due to friction in the sill regioncan be estimated, provided that a good parameterization of friction is available. Thissection is in two parts: (1) a derivation of the general frictional dissipation expression,and (2) the results from the dissipation model used for Skookumchuck Narrows.(a)(b)0.12.0.120.1<u(z)p’(z)>‘-e-i0.1<u(z)p’(z)>14—iNonnalMode1+-INo’maIMode1+-i0.080.080.060.06ItI0.040,04I0.020,02I00•1.1.1.1..020406080100120140160180020406080100120140160180JulianDay(1991)JulianDay(1991)Figure4.12:Netup-inlet baroclinictidalenergyflux(MW) for(a)M2and(b)K1constituentsinJanuary,February,AprilandMay1991.00Chapter 4. Tidal Energy Partition 86Basinx2 x1Figure 4.13: Schematic of a single channel inlet showing the coordinate system used inthe derivation of frictional power loss.4.4.1 Derivation of dissipation expressionBeginning with the shallow water equations for rectilinear flow,= —g (4.38)— Ohu439at— Ox’ (.where h(x) is the channel depth, an energy equation can be obtained by adding phu.(4.38) and pgr (4.39):+ (pghui) = 0, (4.40)where E = (phu2 + pg7)2). Integrating (4.40) from the head to the mouth of the inlet(see Fig. 4.13),X2 OEj --dx + pghur I2 —pghu,, 1o 0. (4.41)Chapter 4. Tidal Energy Partition 87The first term is the time rate of change of energy per unit width and the second andthird terms represent the energy fluxes at the boundaries to balance the first term. Thethird term is zero, since there is no energy flux through a solid boundary. Averaging(4.41) over a tidal cycle (indicated by an overbar) and assuming no dissipation in theinlet (i.e. /ät = 0, see Garrett, 1975):pghifl7 = 0. (4.42)Equation (4.42) implies that u and are in quadrature in the lossless case.In the case where there is dissipation due to boundary friction, the energy loss canbe explicitly included in the shallow water equations by including a stress term:ãu ôq D(x)+ g— + h(x) 0, (4.43)where D(x) has the dimensions of v2 (Freeland and Farmer, 1980). Equation (4.40) thenbecomesÔE 8-- + — (pghu,) + puD = 0. (4.44)Integrating over the surface area of the inlet, and averaging over a tidal cycle (assumingno cross-channel variations in u, i and D, and /öt = 0), one obtains:fX2pgWoHij+ I w(x)puDdx = 0, (4.45)Jowhere w(x) is the width, H = h(x2), and W0 = w(x2). The second term is the dissipation due to friction. Because friction is only important in the sill region, the limits ofintegration can be reduced to x1 x x2 (see Fig. 4.13). The dissipation function,D(x), was parameterized by Freeland and Farmer (1980) as D = CduIuI, where.Cd is adimensionless friction coefficient or drag coefficient.Chapter 4. Tidal Energy Partition 88With the form of the friction term given by (3.16), the dissipation expression (4.45),and the fitted values of the friction parameters (7 = 0.04 and Cd = 0.08), one can nowcalculate the dissipation rate due to friction, assuming that the width has a constantvalue in the narrows:P1033 = pWoL[’y(u2+ CdM3}. (4.46)The dissipation rates given by (4.46) were compared to the theoretical estimate givenby (4.13) for each flood event found in six months of tide gauge data from SkookumchuckNarrows (see Fig. 3.5 for sample tidal records). From each flood event, a phase shiftin surface elevation, tidal period, and tidal amplitudes inside and outside the sill werederived. To a very good approximation, the tidal elevations can be considered sinusoidalover the flood period, thus justifying the use of (4.13).From the linear regression of the data plotted in Fig. 4.14 it would appear that thedissipation rate due to friction is 1.1 ± 0.1 that of the estimated barotropic tidal energyflux. Because of the uncertainties in picking values such as average depth and width ofthe constriction, and the uncertainty in the value of Cd from the simple friction model,the error in the frictional dissipation rate is somewhat higher than the linear regressionwould suggest. The importance of the result lies in the fact that the frictional dissipationrate is comparable in magnitude to the tidal energy flux.4.5 Tidal JetNot all of the turbulent energy is dissipated directly over the sill. Some of the energywill be advected into the inlet during a flood tide in the form of a turbulent jet. Thisjet will be negatively buoyant with respect to surface water and will tend to flow underthe surface layer. The advected mixed water from the sill and mixing by the jet areChapter 4. Tidal Energy Partition 890C001d000U-Figure 4.14: Frictional dissipation rate computed using the one dimensional sill modelversus estimated power loss from the modified barotropic tidal flux model. Flood eventsfrom six months of tide gauge data taken in 1984 were used for the analysis. The solidline is the least squares fit to the data (slope = 1.1 ± 0.1).responsible for the middle layer in the characteristic three-layer density profiles of inletswith strong mixing at the sill.Lazier (1963) was the first to propose that the tidal jet in Sechelt Inlet was responsiblefor the formation of the middle density layer. Direct measurements of the tidal jet weremade at the entrance mooring with the cyclesonde profiling current meter (Fig. 4.15).The currents are quite strong at 20 m during flood tide, and a strong return flow isobserved below the inflow.800CD0c’J00 20 40 60 80 100Theoretical Barotropic Flux (MW)Chapter 4. Tidal Energy Partition 900-0.U)I8-CJ-10 0 10Velocity (cmfs)Figure 4.15: Along channel velocity profile of the tidal jet during flood tide. The profilewas measured with a cyclesonde profiling current meter approximately three hours aftermaximum flood in Skookumchuck Narrows at 9:00 pm, May 12, 1991. Negative velocityindicates flow towards the head of the inlet, positive velocity is towards the sill.Stigebrandt and Aure (1989) derived the following expression for estimating the energy flux of a turbulent tidal jet:(4.47)where p is the average density of the inflow, w is the angular frequency of the tide, ijis the tidal amplitude in the inlet, A3 is the cross sectional area of the sill and Af is thesurface area landward of the sill.Chapter 4. Tidal Energy Partition 91‘CLI.>,0Cw00Ca,-‘Barotropic Tidal Energy Flux (MW)Figure 4.16: Kinetic energy flux of the turbulent jet entering Sechelt Inlet versus totalbarotropic tidal flux. The slope of the linear regression is 0.05 ± 0.01.As with the frictional dissipation across the sill described in section 4.4, the kineticenergy flux of the turbulent tidal jet was estimated for each inflow period using the tidegauge pressure data. The jet flux at each event is compared to the theoretical estimateof barotropic tidal flux in Fig. 4.16. There is a nearly linear relation between the jet fluxand the surface tidal flux. Overall, about 5% of the tidal energy flux can be attributedto the tidal jet.LI)00 20 40 60 80 100Chapter 4. Tidal Energy Partition 924.6 The Energy PartitionThe barotropic tidal energy flux in Sechelt Inlet is very large with values approaching100 MW on the extreme flood tides. The tidal energy is dissipated mainly by frictionthrough Skookumchuck Narrows, while a comparatively small amount is transferred tothe kinetic energy flux of the tidal jet near the entrance and an even smaller amount tothe progressive internal tide.The dissipation by friction at the Narrows almost completely balances the tidal energyflux. The tidal jet and progressive internal tide are much smaller in magnitude and havevery different effects on the mixing in the inlet. The tidal jet, which accounts for 5 %of the dissipation, is responsible for the formation of the middle of three distinct layersin the water column. The three-layer density profile is seen in other fjords on the BritishColumbia coast (e.g. Seymour Inlet) where there is strong tidal mixing at the sill.The progressive internal tide, which accounts for 0.5 % of the dissipation, transfersits energy to heat and mixing in the deeper basin waters. The mechanism for this transferwas proposed by Stigebrandt (1976) to be the breaking of the internal waves on the slopingbottom; in Sechelt Inlet, the breaking probably occurs primarily in Porpoise Bay. It willbe shown in chapter 5 that the transfer of mechanical energy from the internal tide towork against buoyancy forces in the water column is quite inefficient and that most ofthe energy flux is dissipated as heat.The energy partition for Sechelt Inlet is in contrast to the partition in deep-silled fjordswhere the majority of the tidal energy flux is dissipated by the generation of progressiveinternal waves near the sill. Moreover, the fraction of the available barotropic tidal fluxthat is dissipated in Sechelt Inlet is much larger.Chapter 5Vertical Diffusion5.1 IntroductionVertical diffusion in deep-water is driven by turbulence and is much larger than moleculardiffusion. Salt is diffused upwards and heat is generally diffused downwards, so that thebasin water becomes warmer and less saline with time. The changes in temperature andsalinity both decrease the deep-water density; as the density decreases, the basin waterbecomes more susceptible to intrusions of dense water that enter over the sill. In this way,vertical diffusion conditions the basin water for deep-water renewal. Stigebrandt (1976)asserts that the energy which drives the turbulent diffusion in fjords comes primarily fromthe breaking of the baroclinic tide, which in turn gets its energy from the barotropic tide.This transfer of mechanical energy from the internal tide to do work against buoyancyis inefficient. The ratio of the work against buoyancy to the available energy in thebaroclinic tide is the flux Richardson number, Rf; Stigebrandt (1980) and Stigebrandtand Aure (1989) found be R1 0.06 in Norwegian fjords, and de Young (1986) foundRf 0.05 —. 0.10 for Indian Arm.This chapter discusses the results of the diffusion analysis in Sechelt Inlet and compares the potential energy change in the water column to the estimated energy fluxes ofthe baroclinic tide. An estimate of the flux Richardson number is made following theanalysis by Stigebrandt and Aure (1989).93Chapter 5. Vertical Diffusion 945.2 Vertical DiffusionTurbulent diffusion of scalar seawater properties has traditionally been parameterized ina fashion analogous to molecular diffusion, where the turbulent diffusivity, is muchgreater than the molecular diffusivity. The complete scalar conservation equation hasthe form (following Gargett (1984)):Kh 0 0K3= 0 K 00 0 Kac ac a ac+ U— = —(K,—) + F, (5.1)(1) (2) (3) (4)i = (1, 2, 3); repeated indices indicate a sum over that index. The terms in the conservation equation are: (1) local time rate of change of C, (2) advection, (3) turbulent diffusionand (4) source/sink term. If a time period is chosen when there is no deep-water renewal(i.e. no advection), then term (2) vanishes. Horizontal gradients across the channel arenegligible; however, along-channel gradients are not always small, especially near thesurface. In the deep-water, though, along-channel gradients may be considered small,leaving only the K,, term in (3). If, in addition to the above assumptions, the scalarquantity is assumed to be conservative, term (4) vanishes and the conservation equationbecomesac a11cat az” aIn many studies of turbulent diffusion processes, I<, is considered constant. However,recently it has become apparent that K,, is not constant, but varies systematically withChapter 5. Vertical Diffusion 95the Brunt-Väisälä frequency, N. Gargett and Holloway (1984) show that if the source ofmixing energy is from the breaking of the local internal wave field, then,= 1 R (53)where Rf is the flux Richardson number and f is the dissipation. Gargett and Hollowayassume that Rf is a constant, limiting value throughout the water column. For the broadband internal wave spectrum of the ocean interior, N’5 (K,, N°5); for internalwaves of a single frequency, e ‘‘ N’° (K,, N’°). Diffusivities from temperature andsalinity data in many fjords and lakes (where K,, =a0N-) have typical exponent valuesof 0.8 < q < 2.0, and a0 is found to be a site dependent constant determined by theamount of energy in the internal wave field. If R1 is not constant, q could be greaterthan one; however, q must be less than 2 in order for density to decrease with time (Pondet at, 1995). In general, there is more scatter in the temperature data because the errorsrelative to the gradients are larger than for salinity.The technique used for the diffusion study was the budget method described byde Young and Pond (1988): the conservation equation (5.2), integrated vertically andhorizontally (over the deep basin),UjhA(z)SdzK,,(h) = (It‘ (5.4)—A(h)was evaluated using the basin-averaged temperatures and salinities from several CTDsurveys. The area function, A(z), was determined from hydrographic charts. Deep-waterrenewal in Sechelt Inlet is rare, but, in 1991, a mid-water intrusion occurred betweenMarch and April that was confined to the upper 150 m. The period between DecemberChapter 5. Vertical Diffusion 96Table 5.1: Summary of diffusion calculations for K =a0N; values of q are given forboth the temperature and salinity data.Month Depths Levels ao(T) q(T) ao(S) q(S)(m)Dec—Jan 125—215 19 —7.46 ± 0.45 1.50 ± 0.17 —7.10 ± 0.20 1.35 ± 0.08Jan—Feb 120—220 21 —5.94 ± 0.14 0.95 ± 0.06 —6.28 ± 0.16 1.09 ± 0.06Feb—Mar 150—220 15 —7.23 ± 0.16 1.41 ± 0.06 —6.11 ± 0.23 0.95 ± 0.09Mar—Apr 170—220 11 —6.42 ± 0.23 1.10 ± 0.08 —6.27 ± 0.31 0.96 ± 0.11Apr—May 170—215 10 —7.46 ± 0.48 1.54 ± 0.18 —6.07 ± 0.43 1.01 ± 0.16May—Jun 135—220 18 —7.13 ± 0.56 1.49 ± 0.23 —5.75 ± 0.16 0.93 ± 0.07Dec—Mar 125—215 19 —6.69 ± 0.17 1.33 ± 0.13 —6.44 ± 0.13 1.05 ± 0.081990 and March 1991 was judged to have the best undisturbed deep-water data set forthe diffusion analysis.The CTD data below 50 m were binned and processed in 5 m intervals. Becausevertical gradients of scalar quantities are small in basin waters, estimates of N2 andaS/az at depth h were made using data at h— 10 m and h + 10 m. Depths where thevalues of K were extremely scattered (normally deep) or where the results were obviouslyaffected by advection (normally shallow) were omitted. The period from December 1990to March 1991 was the longest period where there were no disturbances of the deep-waterbelow 125 in.The results of the diffusion study (Figs. 5.1 to 5.7) are summarized in Table 5.1. Theerrors associated with the slopes of the least squares regression are the standard errorsgiven by the fitting routine. The temperature and salinity data were analysed monthly:Chapter 5. Vertical Diffusion 97in general, the q values for temperature (1.54 > q > 0.95) were larger and more variablethan those for salinity (1.35 > q> 0.93). Because the changes in the deep-water ( 0.01per month for salinity) are near the instrument resolution, analysis over longer periodsyields more reliable results. The best estimate for q, based on the salinity data betweenDecember 1990 and March 1991, is q = 1.05 ± 0.08.The study by Stigebrandt and Aure (1989.) showed a great variation in the valueof q = 1.50 ± 0.35 (mean ± standard deviation) based on salinity data in 29 basins,de Young and Pond (1988) reported values of q = 1.6 and 1.8 based on salinity data forIndian Arm and Saanich Inlet respectively. Since the value of q for Sechelt Inlet is nearthe narrow-band limit of q = 1 determined by Gargett and Holloway, the relationshipbetween K,, and N suggests that Sechelt Inlet mixing is driven primarily by internalwaves of a single frequency; the internal tide is the most likely candidate for driving themixing. There may be other sources of mixing energy (e.g. wind), but their effects aresmall. It is possible that the frequency band containing most of the energy in the wind isclose to a tidal frequency and the wind input is therefore masked. However, in chapter 4it was argued that even if the diurnal surface currents were reduced or increased by 50%that the change in overall energy flux was only 5%. In Sechelt Inlet, the diurnal periodwinds contain a large fraction of the wind energy (see section 6.2).Chapter 5. Vertical DiffusioncJEio>Lba• 100>1T: Slope = -1.5------ S: Slope = -1.35Brunt-Vaisala freq (rad sFigure 5.2: Eddy diffusion coefficients (temperature (Lx) and salinity (0)) for Januaryto February 1991 (120-220 m). The dashed lines are the least squares fits of the data(the solid line represents a slope of -1).16298Brunt-Vaisala freq (rad sFigure 5.1: Eddy diffusion coefficients (temperature (z) and salinity (Q)) for December1990 to January 1991 (125-215 m). The dashed lines are the least squares fits of the data(the solid line represents a slope of -1).162E>Lba1000>1T: Slope = -0.95- S: Slope = -1.09‘Si 1d2Chapter 5. Vertical Diffusion0CVE>‘>U,c0>Brunt-Vaisala freq (rad sFigure 5.4: Eddy diffusion coefficients (temperature () and salinity (Q)) for March toApril 1991 (170-220 m). The dashed lines are the least squares fits of the data (the solidline represents a slope of -1).T: Slope = -1.41S:Slope = -0.9599Brunt-Vaisala freq (rad sFigure 5.3: Eddy diffusion coefficients (temperature (La.) and salinity (Q)) for Februaryto March 1991 (150-220 m). The dashed lines are the least squares fits of the data (thesolid line represents a slope of -1).(0CME>U,0(UC.,0>162111T:Slope=-1.1- S: Slope = -0.961 1 161Chapter 5. Vertical DiffusionU,c.JE>>U,D0(UC.,a’>Brunt-Vaisala treq (rad sFigure 5.6: Eddy diffusion coefficients (temperature (Lx) and salinity (Q)) for May toJune 1991 (135-220 m). The dashed lines are the least squares fits of the data (the solidline represents a slope of -1).100162111T: Slope = -1.54S:Slope = -1.011o3 1o2 161Brunt-Vaisala freq (rad sFigure 5.5: Eddy diffusion coefficients (temperature (Lx) and salinity (0)) for April toMay 1991 (170-215 m). The dashed lines are the least squares fits of the data (the solidline represents a slope of -1).162U,E.>U,01001T:Slope=-1.49S:Slope = -0.931 161Chapter 5. ‘vrtica1 Diffusion 101C’,c’1E>‘>00a,>i2Brunt-Vaisala freq (rad sFigure 5.7: Eddy diffusion coefficients (temperature () and salinity (0)) for December1990 to March 1991 (125-215 m). The dashed lines are the least squares fits of the data(the solid line represents a slope of -1).5.3 Mixing Efficiency in the BasinThe net up-inlet energy fluxes of the K1 and M2 internal tides were computed using twoseparate methods in chapter 3. The fluxes were found (a) using a theoretical normalmode fit to the measured baroclinic velocities and density fluctuations, and (b) fromthe baroclinic velocity and perturbation pressure expression (4.37). Although the resultsfrom the two methods differ somewhat (see Figure 4.12), they are both of the same orderof magnitude, and both methods show a general trend towards higher energy fluxes inearly summer. Robinson (1969) showed theoretically that the baroclinic modes of anexponentially stratified fluid are almost completely reflected from a barrier whose heightis > 90% of the channel depth (as in Sechelt Inlet). Since the down-inlet energy has littlechance to escape over the sill, it will eventually dissipate inside the inlet and contributeChapter 5. Vertical Diffusion 102to the mixing. Therefore, the up-inlet energy fluxes given in Table 5.2 will be used toexamine the mixing efficiency. The errors associated with the work against buoyancywere derived using the statistical uncertainties of ao and q; the error in the modal fluxis estimated to be 10% based on a rather optimistic estimate of an average 5% errorin field measurements. It should be noted that the estimates of up-inlet energy flux atstation 3 will underestimate the total baroclinic energy flux generated at the sill becauseof dissipation between the generation point and the mooring.Stigebrandt and Aure (1989) define the total rate of work done against buoyancyforces, W, by (5.5). It was found that W = W0 +R1E, where E is the barodinic energyflux and W0 is a background rate of work due to other mixing processes. Using (5.5), therate of work for Sechelt Inlet is calculated and compared to the total estimated energyflux of the internal tide, E, to obtain a value for the flux Richardson number.W= j pK(z)N2(z)A(z) dz, (5.5)where p is the average density of the water column, K(z) is the diffusivity given by(5.4), and A(z) is the inlet area as a function of depth. The lower bound, b, was chosento be the bottom (275 m), and the choice for the upper bound, u, is discussed later inthis section.Stigebrandt and Aure concluded that R1 = 0.056±0.011 for inlets with “well-behavedvertical profiles of N” where most of the tidal energy flux went into the internal tide(these inlets were termed “wave basins”). The breaking of the internal tide along theboundaries, particularly along a sloping bottom, was assumed to be the principal sourceof energy for vertical diffusion in the deep basin water.There were a few Norwegian fjord basins where the tidal flow entered the mouth ofthe inlet as a turbulent jet (“jet basins”). These were treated separately, since it was notTable5.2:Modalenergyfluxes(up- anddown-inlet)forthefirstfourmodesoftheM2andK1constituents.Thetotalup-inlet fluxiscomparedtotheworkdoneagainstbuoyancy.M2K1TotalWorkPhaseUp-inletAgainstMonthModeSpeedUp-inletDown-inletUp-inletDown-inletFluxBuoyancyFluxFluxFluxFlux(cms’)(MW)(MW)(MW)(MW)(MW)(MW)165.3.0189.0109.0397.0084Jan—Feb242.7.0093.0004.0030.0012.0834.0065327.8.0089.0003.0017.0003+.0083±.0001420.1.0010.0000.0009.0003164.2.0228.0074.0415.0055Feb—Mar235.5.0141.0001.0023.0022.0872.0041324.2.0043.0000.0007.0009±.0087±.0001417.1.0014.0000.0001.0005158.4.0338.0068.0637.0220Apr—May237.4.0133.0001.0057.0055.1344.0078325.5.0136.0000.0032.0000±.0134±.0004417.9.0007.0000.0004.0005172.7.0431.0167.1136.0181May—Jun239.3.0270.0003.0024.0018.2135.0109330.1.0206.0001.0044.0000±.0214±.0005.420.1.0023.0001.0001.0003I IChapter 5. Vertical Diffusion 104believed that a significant internal tide could be generated in such a basin. The energyfluxes were calculated using (4.45) and were much larger than the fluxes estimated forthe wave basins. Hence, the estimated value of the flux Richardson number in the jetbasins, Rf 0.01, was much smaller than for the wave basins. Since R1 is a measureof the efficiency of the mixing process, tidal jet basins were said to be less efficient attransferring energy from the tidal flow to vertical mixing.Sechelt Inlet is both a wave and a jet basin: there is a baroclinic tide produced despitethe tidal jet, the effects of the latter being confined to the upper 125 m. In chapter 4,it is shown that an average estimate for the total power extracted from the M2 and K1barotropic tidal constituents is approximately 42 MW. The maximum total energy fluxfrom both of the dominant internal tidal constituents is estimated to be not more than0.21 MW. Although the internal energy flux is much smaller than the energy extractedfrom the surface tide, it is still comparable to the internal fluxes in Indian Arm (0.13 to0.30 MW) (de Young, 1986). Since Sechelt Inlet has a larger volume than Indian Arm,vertical mixing will be weaker given the same baroclinic energy flux.The rate of work done against buoyancy forces was estimated for the four periodswhen the energy flux of the internal tide was calculated from the moored instruments(Table 5.2). The K,, = a0N relationships attained from the diffusion analysis weresubstituted into (5.5), and the upper bound, u, was chosen to be the depth in the watercolumn unaffected by the jet water 150 m). Stigebrandt (1979) asserts that the effectsof the breaking internal tide will be felt mostly in the deep-water since laboratory experiments suggest that breaking occurs mainly on regions of sloping bottom topography. Alinear regression of T’V as a function of E (Fig. 5.8) yields the relation:W = (0.043 ± 0.009)E + (0.002 ± 0.001).Chapter 5. Vertical Diffusion 105Hence, the flux Richardson number has a value of 0.043 and the “background” rate ofwork, Wo, is 0.002 MW.Another measure of Rf may be made by comparing the net up-inlet tidal flux to thework done against buoyancy up-inlet of station 3 (Fig. 5.9). This estimate of R1 hasthe advantage of being independent of the energy lost between the generation pointand station 3. Two measures of the net up-inlet flux from station 3 were given insection 4.3.3: (a) an estimate from (4.37) and (b) the net modal flux. These were bothcompared to the work done against buoyancy up-inlet of station 3 (Fig. 5.9), which wasdetermined by (5.5) using an adjusted area function, A(z). Rf is found to be 0.082±0.016and 0.034±0.010 respectively, and the background rate of work is 0.002±0.001 MW.Combining all of the estimates for R and W0 in Sechelt Inlet, R1 = 0.034- 0.082, andthe background rate of work, 14 = 0.002±0.001 MW.Two additional sources of uncertainty should be noted. Although the M2 and K1 tidalconstituents are the key contributors to the tidal energy flux, they are not the only ones.From the harmonic analysis of the tide gauge data, M2 and K1 contribute 80% to the totalvariance. Estimates of J?f should be multiplied by 0.80 to allow for the contributions ofthe other tidal constituents. The second, and more important uncertainty, comes from thechoice of the upper bound for the integration in (5.5). Because 150 m was the shallowestdepth that the K versus N relationship held, and the assertion by Stigebrandt (1979)that the mixing energy is confined to the lower layers, the integration stopped at 150 m.However, this covers a little less than half of the water column, and it could be arguedif the mixing effect extends shallower than 150 m that the estimate of the work done inthe water column is low. Hence, the range of 0.03 to 0.08 for Rf could be considered anunderestimate.There is, of course, considerable uncertainty the estimates for W0 and R1, but theyshow a relationship similar to what Stigebrandt and Aure (1989) found for inlets whoseChapter 5. Vertical Diffusion 1060.0060.0040.0020Figure 5.8: Work done against buoyancy forces, W, as a function of baroclinic tidalenergy flux, E, for January to May 1991. The dashed line is the least squares regressionW = 0.002 + 0.043E.diffusion was driven by the internal tide. Their results suggestedW = (0.056 ± 0.011)E + 0.0018(the units of iV m2 used by Stigebrandt and Aure (1989) for W0 were converted to MWfor comparison to the Sechelt data). Stigebrandt and Aure argue that the backgroundrate of work is due primarily to wind; however, another possible source of energy is thehigh frequency internal wave field generated at the sill. de Young and Pond (1989) foundthat these internal waves had an energy flux ‘— 10% as large as the baroclinic tidal flux. Ifthe energy flux of the high frequency internal wave flux remains relatively steady duringthe year. it may contribute to the constant background rate of work.0.0140.0120.010.0080 0.05 0.1 0.15 0.2 0.25Barodnic TdaI Fhix (MW)Chapter 5. Vertical Diffusion 107:: zzz00e (b)i.f1/0.004 JNet Modal Flux i-—*0.0022 .. 1.00 0.05 0.1 0.15 0.2 0.258aroctkdc That Flux (MW)Figure 5.9: Work done against buoyancy forces up-inlet of station 3 as a functionof net up-inlet baroclinic tidal energy flux for January to May 1991. Two estimates are shown: (a) the up-inlet flux derived from the < u(z)p’(z) > (regression isW— (0.002 ± 0.001) + (0.034 ± O.010)E) and (b) the net modal flux (regression isW = (0.002 ± 0.001) + (0.082 ± 0.016)E).Chapter 6Low Frequency CirculationThe low frequency circulation is taken to be that part of the circulation which has variations with periods greater than 25 hours. The 25-hour cutoff period eliminates all diurnaland shorter period tides and fluctuations caused by diurnal period winds. The along-channel currents, densities, wind and runoff records were low-pass filtered using a spectralfilter with a cutoff frequency of 0.929 cyc day’.The main goal of the low frequency analysis is to determine the effect that wind andrunoff have on the circulation. Baker (1992) used statistical coherences to identify thewind-driven circulation in Knight Inlet, using data similar to that presented in this thesis.The determination of the effect of runoff in Knight Inlet, however, proved more difficult.The approach taken here is to examine the coherence and phase spectra between thecurrent and wind records to determine the strength and extent of the wind influence overthe water column. Results from an empirical orthogonal function (EOF) analysis of thevariability are also presented in an effort to identify and explain the dominant modes ofvariability. The mean circulation and the deep-water renewal cycle are also discussed.6.1 Filtering and spectral analysisThe method of low pass filtering chosen to smooth the data used a spectral filter witha cutoff frequency of 0.929 cyc day1. The data were Fourier transformed, then thefrequencies higher than the cutoff, f, were set to zero and a cosine taper was applied forfrequencies 0.8f < f < f. The cutoff frequency was chosen not only to be low enough108Chapter 6. Low Frequency Circulation 109to exclude the diurnal and shorter period variability, but also to fall in a relatively lowenergy frequency band to reduce the ringing caused by cutting the spectrum off sharplyin a band with significant amounts of energy.Raw and smoothed power spectra were generated for each time-series. A discreteFourier decomposition of the function F(t) (t = nLt, where Lt is the sampling interval)may be defined (see Press et al (1986) for a complete discussion):N/2F(t) = a cos(2irf2t)+ b sin(2irft), (6.1)where f, is the th discrete frequency and N is the number of samples in F(t). The valueof the power spectrum at a frequency, f, is then defined asP(f)f = (a + b), (6.2)where if = 1/(Nzt), and f = izS.f. The spectra were smoothed over 9 frequenciesusing a boxcar filter, giving each spectrum 18 degrees of freedom. The 95% confidenceinterval for the spectral value, S(f), with v degrees of freedom is2 S(f) S(f) 2 S(f). (6.3)Xzi,O.025 Xv,o.975In this case, i’ = 18,x80.25 = 31.53 and Xs,o.97s = 8.23. The power spectra are plottedon a logarithmic frequency axis so that the lower frequencies are expanded. In order thatthe area under each band remain proportional to the total energy, the spectral values arescaled by their frequency.Another useful method of examining the relationship between two signals is throughtheir coherence spectra, C(f), and phase spectra, (f). First, the cross-spectra of twotime-series were computed using the series’ Fourier coefficients, a2 and b. The co- andquadrature spectra of time-series F(t) and F’(t) are defined byChapter 6. Low Frequency Circulation 110C3(f)zf + bb)Q3(fjzXf =—ab:).The coherence squared and phase spectra are then defined asC2 C(f)+Q(f)(fJ- P(f)P’(f)•——‘ (—3sf,) an C,(f)The coherence squared must be smoothed over a band of frequencies, since thecoherence of two processes which occur at the same frequency is identically 1. Thesmoothing allows the estimate of what is termed the 95% noise level (Chang et a!, 1976),= 1 — 0•5M, where M = 2/(v — 2). The 95% noise level is the level above whichthe coherence of two hypothetical time-series with no true coherence would lie 5% of thetime. The number of degrees of freedom, ii, is 2N3, where N3 is the number frequenciessmoothed over. N3 was chosen to be 9 (similar to the power spectra), and the frequencieswere smoothed using a boxcar filter. The 95% confidence interval is computed followingthe method outlined by Jenkins and Watts (1968), chapter 9.3. When coherence (or coherence squared) is plotted with an arctanh y-axis, the confidence interval has a constantwidth over all coherences.6.2 WindStudies of many British Columbia mainland fjords have shown that the wind has a stronginfluence on the circulation, particularly near the surface (Baker and Pond, 1995; Buckleyand Pond, 1976). The wind stress transfers momentum through the water column, andChapter 6. Low Frequency Circulation 111also causes mixing in the upper layer through shear-generated turbulence and breakingsurface waves. While the direct effect of wind stress is felt near the surface, barocliniccompensation can result from the surface pressure gradients caused by the accumulationof water at one end of the inlet. The compensating flow can reach right to the bottom,and it is not unusual for currents deep in the water column to be coherent with the wind.When the wind is steady, the surface currents and compensating flow will only last untilpressure gradients develop completely to balance the wind stress. Synoptic scale weatherdisturbances in the 2 to 10 day band are the source of much of the low frequency contentin the wind, and the energy of these bands is normally comparable to or greater than theenergy at diurnal and higher frequencies. The diurnal seabreeze, however, can becomeimportant in the warmer seasons, when the daily heating and cooling of the interior landmasses becomes more pronounced than during cooler seasons.The anemometers aboard the Geodyne buoys measured the wind at approximately4 m from the air/water interface. Smoothed wind power spectra (v = 18) for the twomonth records from January to March (Salmon), and April to June (Salmon and Basin)are shown in Fig. 6.1. For convenience, the January to March deployment will betermed “winter” and the April to June record will be termed “spring”. Despite the closeproximity of the Salmon and Basin anemometers, the along-channel spring winds arepoorly correlated: at zero lag, the linear correlation coefficient is only —0.21. It couldbe argued that the depression in the topography around the Basin mooring allows someof the along-channel Salmon wind to leak into the cross-channel Basin winds; in fact,the cross-channel wind variance at Basin is 64% as large as that of the along-channelvariance. The correlation between the along-channel Salmon and cross-channel Basinwinds is 0.43. The correlation is stronger, but the sign indicates that the two componentsare anti-correlated at zero lag. Because the +U component is used at Basin and the—U component is used at Salmon, one would expect a negative correlation. A possibleChapter 6. Low Frequency Circulation 112explanation for the non-negative correlation would be a topographically generated eddynear the Basin anemometer.A variance analysis shows that the spring Salmon winds are 1.1 times as energetic asthe winter winds, and 3.6 times as energetic as the spring Basin winds. The energy in thediurnal band increases at the Salmon site by 2.1 times between winter and spring, andthe spring value is 7.3 times as large as the Basin spring diurnal band energy. In termsof the total variance of the individual spectra, the low frequency wind energy constitutes51.4% of the variance of the Salmon wind in winter, 37.7% of the Salmon wind variancein spring, and 47.5% of the Basin wind variance in spring. By contrast, the diurnal bandenergy constitutes 21.8%, 40.9% and 20.6% of the variance in the wind in those threerecords. The diurnal band energy actually exceeds that of the low frequency winds inthe spring winds in Salmon Inlet.Unfortunately, the wind-driven currents associated with the diurnal seabreeze and thecurrents associated with the diurnal tides occur very closely together in frequency, andare, therefore, almost inseparable. During the outflow wind conditions in early February,the diurnal seabreeze almost disappears (see section 2.2.4). However, the diurnal periodcurrents remain as strong during the outflow as before or after. It appears then that thedominant forcing for the diurnal period currents in the surface layer is from the tides,and not from the wind. The sensitivity tests to the barodinic tidal fluxes mentioned inchapter 4 assumed that the winds contributed half of the energy to the surface currents.Since the actual effect of the wind appears to be smaller than 50%, the 5% estimatederror in energy flux due to wind contamination is probably high.The filtered along-channel velocities are shown in Figs. 6.2 to 6.7. The filteredalong-channel winds (divided into individual monthly records) are included at the topfor comparison (the Basin January and February data are not included due to the lackof wind records). After the strong outflow winds in early February, the surface layerCD—‘qI-1:j&CI)n—.CD’—-(Th0‘CD CD-aHCD •00 C))CD—.C-I‘-I÷OCD ,Q_C.,.f*S(f)01000020000300004000050000III>0)00).,..—.Co01..0 CD CDS(f)0100020003000400050006000III0 0 a,C)-D LQ.0 0Ia 00 a (If*S(f)05000100001500020000I0•0 0 01.1’) 0pC 0 C)01 CII..-,o01000Ia8 0 80 ‘acn 8•0 0 0011%),p0.pp.. •2aet-.•-‘.lr0 p 10 0Chapter 6. Low Frequency Circulation 114appears to have thickened. There is an increased response in the 6 m and 12 m densityrecords to high frequency fluctuations (see Figs. 2.6 and 2.7), and the filtered densityrecord (Fig. 6.8) from Salmon Inlet shows a short period of increasing density at 2 mduring the period of strong winds, followed by a significant decrease (-..‘3 kg m3). Theinitial increase in density indicates a homogenization of the surface layer and the periodof decreasing density coincides with the large discharge of water from the Clowhom Riverdam (330 m3 s’) on February 4. The decrease in high frequency fluctuations in the 2 mrecord (see Figs. 2.6 and 2.7) is due to the decrease in the stratification due to mixing;the increase in response to high frequency fluctuations at 6 and 12 m is likely caused bythe deepening of the pycnocline from mixing and the subsequent increase in stratificationbelow 4 m. The effect of the strong winds did not substantially affect the densities below6 m.In the Salmon along-channel velocities, an oscillating flow with a period of about 2days appears twice, once beginning on January 30 (Fig. 6.4) and once on May 30 (Fig.6.7). These oscillations last for approximately a week, and do not appear to be correlatedwith the wind. In January, the oscillating currents are seen down to 12 m, but in Maythey are limited to the 2 m record. One possible explanation for these oscillations isthat Salmon Inlet may be prone to internal seiching, where, under the right conditions, afreely oscillating basin scale internal wave may be produced. For example, the seiche inJanuary could have been triggered by the sudden discharge of fresh water from the dam,or, perhaps, by the strong outflow wind event. The period of the seiche would be that ofa freely oscillating baroclinic wave whose wavelength is twice that of Salmon Inlet. Theinlet is approximately 20 km long, and the period of the oscillation is roughly 2 days,giving a phase speed for the seiche of about 23 cm s1.The phase speed of the seiche is the same order of magnitude as the theoreticalbaroclinic modes discussed in chapter 4 (approximately the same as mode 3 or 4). TheChapter 6. Low Frequency Circulation 115seiche can be examined using a simple model: simplifying Salmon Inlet into a two-layersystem, an internal phase speed,c — (6.4)may be computed from hydrographic data; p is the average water column density, 1p isthe change in density across the interface, g is the acceleration due to gravity, and h0 isthe upper layer thickness. Depending on the thickness of the upper layer that is chosenfrom the CTD data, the two-layer internal phase speed lies between 16 and 54 cm s1.A seiche would also have some signature in the isopycnal displacement near the surface: the isopycnals would be displaced most at the ends of the inlet, and there would bea node near the center. The location of the mooring in Salmon Inlet is nearly halfwayalong the inlet, which would be a node for the isopycnal displacement. The filtered density records for January and May, shown in Figs. 6.8 and 6.9, do not show significantoscillations having a 2 day period during the times in question. The lack of a strongsignature in the density records suggests that an internal seiche is a possible explanationfor part of the low frequency variability in Salmon Inlet.The coherence and phase spectra between wind and currents for the Basin and Salmonmoorings are shown in Figs. 6.10 to 6.15. Because the anemometer failed on the BasinGeodyne buoy in January, only the April and May Basin data are shown. Coherence andphase spectra were created for all four months of the Salmon mooring deployment. Theunfiltered current velocities and the winds were gridded to a 3 hour sampling interval,corresponding to the coarsest field sampling (from the cyclesondes). The number ofFourier frequencies smoothed over is 9, giving a 95% noise level of 0.31. A positive phaseindicates that the wind is leading the currents.2010-10-202010E0CD-10-20—‘.,--—‘-- -— —‘- \-\/ \__-‘ %-10EoO14)-1010E80-1010S00U)-1010S0-1 C10S-10. /ThCrf—_____/_-- —--——‘-- -—- —— —-----Figure 6.2: Filtered Basin along-channel wind and currents inof the filter is 0.929 cpd.April. The cutoff frequencyChapter 6. Low Frequency Circulation4/23/91 5/1/91 5/8/91750-250-250-7505/15/91 5/23/91116\Z2010cj0-10-20—-Th -------4/23/91 5/1/91 5/8/91 5/15/91 5/23/91- —--------.-‘ —‘.‘/ ‘-F W—-—---- w-----__ --4/23/91 5/1/91 5/8/91 5/15/91 5/23/91Chapter 6. Low Frequency Circulation2010E0-10-202010°-10-201172010‘ -10t6/1/91 6/8/91 6/15/91 6/23/91Figure 6.3: Filtered Basin along-channel wind and currents in May.of the filter is 0.929 cpd.The cutoff frequency750250-250-7506/1/91 6/8/91 6/15/91 6/23/91F— z--—\__—_ - \., — —E00E0U)E0c’.J:100-—-1010-----10100—.-——-—--10100—-—-— ——————--———--106/1/91 6/8/91 6/15/91 6/23/91Chapter 6. Low Frequency Circulation 118If°I2016 a1(.4-,.—---- F\ -40200-20 \_/ \J ‘__ \_‘ -..--- .— — -.._, ‘—.---.40I—401/23191 2/1/91 2/8/91Figure 6.4: Filtered Salmon along-channel windfrequency of the filter is 0.929 cpd.2123191 3/1191 3/8/91 3/15/91°I2016 at(4 I4020Eo.e-20.4040200.20.40.\__- -.-- —- ---A- A“,--—--- --- ‘-, ‘—, \I \J-I4020Eo——-—----————-20.4040620(40-204C2123191 3/1/91 3/8191 3/15191 3i23191Figure 6.5: Filtered Salmon along-channel wind and currents in February. The cutofffrequency of the filter is 0.929 cpd.1123/91. 5000500-10CC2/1/91 2/8191 2/15/91 2/23/910a-500-100C2/15191 2/23/91and currents in January. The cutoff3123/91Chapter 6. Low Frequency Circulation500•500.400C4020-20-404020‘C-20-40119Figure 6.7: Filtered Salmon along-channel wind and currents in May. The cutoff frequency of the filter is 0.929 cpd.21/91 5/1/91 5/8/91 5/15/91 5/23/91‘— J-50000-100C4020c-204020-20.4014020-204020E0-2040140E20-20\_/-—.----- “—----- -.-------- ‘—.----- ‘—--——-Figurequency4123/91 5/1)91 5/6/91 5/15/91 5/23/916.6: Filtered Salmon along-channel windof the filter is 0.929 cpd.6/1/91 6/8/91and currents in April. The cutoff fre6/15/91 6/23/91F\--,_/-, \_/ — \_/ - ‘-, -,S.—--4020-20-404020E00,-20-4040E20C4O-20——r---—---- —-— ------------6/1/91 6/8/91 6/15/91 6/23)91Chapter 6. Low Frequency Circulation 1201/23/91 2/1/91 2/8/91 2/15/91 2/23/911!-181412102220-E 18co12102220c—E 18c’i 18— 1412101/23/91 2/1/91 2.18/91 2/15/91 2/23/91Figure 6.8: Filtered densities (o) in Salmon Inlet, January.6/1/91 6/6/91 6/15/91 6/23/91222018c’J1210Elo;1412102220_________________18c16141210222018121022208 18c’i 16— 1412106/1/91 6/8/91 6/15/91 6/23/91Figure 6.9: Filtered densities (o) in Salmon Inlet, May.Chapter 6. Low Frequency Circulation 121The coherence squared between the wind and the currents at both sites is relativelystrong (0.5 to 0.7) at all depths around the diurnal and semidiurnal frequency bands.The relatively high coherence at the diurnal and semidiurnal frequencies is the result ofthe diurnal sea breeze and its harmonics having strong spectral lines near the diurnal andsemidiurnal tides. Both wind-generated and tidal currents at diurnal and semidiurnalfrequencies will be coherent with the wind; however, because they occur at the samefrequency, it is difficult (if not impossible) to separate the Fourier components associatedwith the two processes.At the Basin mooring in both April and May, the highest coherence with the wind forf 0.8 cpd is at 2 m. In April, there is a coherence squared of 0.7 just above 0.5 cpd,with a phase very near zero. In May, the strong coherence at 2 m is stretched over agreater frequency band, 0.16 f 0.5, and ranges from 0.44 to 0.54. The coherencesquared at 4 m is slightly different in nature, with peaks of 0.57 (April) and 0.51 (May)lying in the 0.2 to 0.5 cpd (2 to 5 day) band. Below 4 m, the coherence squared forf 0.8 drops off sharply until 208 m, with the exception of a peak of 0.49 near the 2day band at 100 m in May. At 208, 238 and 268 m, there appears to be an increase inthe coherence for periods between 1.5 and 2 days. The deep coherence squared valuesare typically 0.4, although at 208 m in April there is a particularly strong coherencesquared (0.64) at 0.65 cpd. The phase spectra show that the deep currents lag the windforcing slightly more than the surface currents do.The wind-generated surface currents appear to be compensated for by a deep baroclinic flow. However, the coherences near the bottom are barely above the 95% noiselevel, and the connection between the deep currents and those at the surface is tenuousat best. One possible explanation for the low coherences at 9 and 12 m for f 0.8 cpdcould be the vertical migration of the velocity zero crossing over the survey period ontime scales greater than one day. For example, if the wind response at 9 m were to shiftChapter 6. Low Frequency Circulation 122•110.‘4..)I41II.I)c.)L4)C%J .1.... .J I I I I I I I I. 0)o : I.I :t’ :: 1 4.: 1 :1 $ :. 1 . 1II II —) 1!: : 1 : :: :: : r—, )ai I)’ , ¶C%JJ I .I I .1 1 IjI I I .1 1 t 1 3 1 cf) 0.)‘4’ :t:::::i:. :i::: :::: :i::: :1:’.:i::. ::: :i::: :I:: :: ::::: ::::: IL: ::::: :: :t:: :1:, I 0 0)—0 1o oW • LU W • WL • • wg • W9 • W99 5-,(seei6ep) 9Sd -oQIt) I I I I -c H0 I I I II I I I c6 -‘I I I I II I -I II I II I I 0I I I ‘. .°I II I I I I S-II 1 1I I I I I I 0I e-I I — 0)0.)I I.I I . H,Q)I I I I —I 0.) -‘—I I — CL) CtI—I I-.-I I I—I I—I I II I—S1 1I I I I 14, 0c’J I H I IkQI Ito 3 0I I I I I I.— 0)I.......j .. .. .. .. ..•or.too o,-too o,r..u,o gor-oo ol—too or—u-to 0) Ct)W W9 WOOL W9O W9o c,— —UItoa,01.o) r—tf) 0woI-It) 0peienb eouaieiioFigure6.11:CoherenceandphasespectraforwindandcurrentsatBasinforMay.Thedashedlineinthecoherenceplotisthe95%noiselimit.Theerrorbarshownforthecoherenceateachdepthisthe95%confidenceinterval.I.9E7 :5 0 .95 .9E7 :5period(days)period(days)nt5210.50.330.25nt5210.50.330.25I..95 .9IUE.----.—.——HII--a> I Cl, a> 0 C a> 0 C).95 : C.95IL-L---k----4---V-0 I180E0--180-H-H180-:---:--180-------&c:::_____e0——.-.---.180a> a> I:---180--H----_b:.--4-:zZt: E180 V....V-180_____________________________________________________________.95E.907 :5 U..95VE.9V01.5.95E °.7I--_-00.20.52340020.51frequency(cpd)frequency(cpd)234I.Chapter 6. Low Frequency Circulation 124period (days)0.5.9C:5.3C...95:1WE7.3W0°.95.9a7°‘.30.20.5 1 2frequency (cpd).9C7.3C....95.75_________W0°.95.987°‘:5.3CInS 2 0.33H ------------- -period (days)0.25 Iiif, 5 ? 1 0.5 0.33 0.25LzzxfN. :_* IY--J11i--v. z.?c..95.9acsj .7.3EEEiiS3EI. c- 184-4 ._%_._.1 --6S... . -18 --------- ‘-r-_%.180’!00.20.5 2 3 4frequency (cpd)aFigure 6.12: Coherence and phase spectra for wind and currents at Salmon for January.The dashed line in the coherence plot is the 95% noise limit. The error bar shown forthe coherence at each depth is the 95% confidence interval.period (days) period (days)InS, 1 O5 0.33 0.25 Iiif, ? 1 05 0.33 0.25.------L : -----;;w.—I:4 .:180jc01_ —,j;--4—s’. -84 •%.6180f . E-z----Of T80f__ _..18------ I----I I sTI .: .-18 ;._f —v-..95j.9CcJ .7—.5.3.1O.2 0.5 1 2frequency (cpd)3 4 ÔO.2 0.5Figure 6.13: Coherence and phase spectra for wind and currents at Salmon for February.The dashed line in the coherence plot is the 95% noise limit. The error bar shown forthe coherence at each depth is the 95% confidence interval.2 3 4frequency (cpd)Chapter 6. Low Frequency Circulation= .950(I)‘ .753a -.0C).957°‘ :.3.957C%J.3.95.957‘ :5.3z .950Cog 57o:5.3a) t.I- :C00.2 0.5125.95 7period (days) period (days)Inf,5 ? 1 0.5 0.33 0.25 nf 5 2 1 0.5 0.33 0.25.95.957:5-1—----... z7tAdA----- ---------I_.95.9Sc..J .7.3-184- V V -18- V V-. —-T v_’ --1 -.-‘----- __%_._ -‘0)ol -is4--180f—-.. I ;.—oj : - .‘......-‘- .‘. %- -_____________________________-00.20.5 1 2 3 4frequency (cpd)ÔO.20.5 1 2frequency (cpd)3 4h’if 5 ?period (days)Figure 6.14: Coherence and phase spectra for wind and currents at Salmon for April.The dashed line in the coherence plot is the 95% noise limit, The error bar shown forthe coherence at each depth is the 95% confidence interval.0,5 0.33 0.25--_----“--V-period (days)Inf 5 2 1 0.5 0.33 0.25180jV°-—iaoF V -—‘4 -‘ — -%0)a)18.__V:.ot • V-184 _.,180Ho I V 4 •.%‘o- ----.S •. VVV-----Jz-2frequency (cpd)3 4,- V—00.20.5 1 2 3 4Figure 6.15: Coherence and phase spectra for wind and currents at Salmon for May. Thedashed line in the coherence plot is the 95% noise limit. The error bar shown for thecoherence at each depth is the 95% confidence interval.frequency (cpd)Chapter 6. Low Frequency Circulation 126by 1800 halfway through the deployment because of a change in zero crossing depth dueto changes in stratification, the wind response at 9 m would become incoherent withthe wind forcing. Migration of the zero crossing does not explain, however, the lack ofcoherence with the wind below the main pycnocline.The Salmon wind response for f 0.8 cpd is stronger than the Basin response. Thecoherence squared peaks at 2 m in January (0.78) and February (0.87) are near 0.5 cpd,with phases near zero for both. In April, the coherence is spread evenly over most of thelow frequency band, with coherence squared values of 0.50 to 0.70; however, the phase isstrongly negative for frequencies greater than 0.5 cpd. The coherence squared at 2 m inMay shows a sharp peak of 0.68 near 0.5 cpd, and values near the noise level elsewhere.The phases at 2 m in May are nearly identical to those in April. With the exceptionof February and April, where the peak coherence squared values at 4 m were 0.80 and0.78 respectively, and at 6 m in April (0.73 at 0.5 cpd), the coherence squared valuesat depths below 2 m are small (< 0.5). The phases in each of the four months for alldepths below 4 m are negative for almost all f 0.8, except in February, where thephases at the peak coherence frequencies are positive down to 6 m (they are positive at9 m also, but the peak coherence at 9 m is very small). Since there is an indication fromthe filtered current and density records that the surface layer thickened in February, thepycnocline may have deepened, allowing the wind to transfer momentum deeper in thewater column.In general, the coherence between the currents and the wind is weak for all f 0.8 cpdbelow 2 m. The surface layer thickening in February, mentioned earlier in this section, isreflected in the coherence and phase spectra of the Salmon data, with high coherence andpositive phases down to 6 m. Relatively strong coherences near the bottom (below 200 m)at Basin in April and May suggest the possibility of a deep baroclinic compensation flow,in response to the surface flow.Chapter 6. Low Frequency Circulation 1276.3 RunoffThe mean monthly runoff was described in chapter 2 as having two peaks per year (springand autumn). The hourly discharge values from the B.C. Hydro dam on the ClowhomRiver were obtained for the period spanning the main current meter deployment (Fig.6.16). It is apparent that 1991 was not a typical year for discharge, since the January andApril averages (Table 6.1) were much higher than normal, probably the result of moreprecipitation falling as rain in the winter. The discharge in May was less than average,perhaps indicating that the snow pack was smaller than normal. The values of runofffrom the Tzoonie River (Narrows Inlet) and the numerous creeks along the sides of theinlets are not known.The correlation between surface currents and discharge is small (Fig. 6.17). Thecorrelation coefficient between the runoff and the 2 m currents is typically less than 0.5,and the peak correlation often does not occur at zero or positive lags: one intuitivelyexpects that if the runoff has any influence over the surface currents, the peak correlationwould occur at zero or positive lag (i.e. the down-inlet currents would increase some timeshortly after discharge increased). This expectation may not be true, of course, if thefresh water travels in a very thin layer at the surface, which the 2 m instrument isunable to measure. Baker (1992) found that the surface currents in Knight Inlet werepoorly correlated with the runoff, and that increased stratification in the surface layerprevented large fresh water discharges from increasing seaward volume transport. As inKnight Inlet, the river discharges in Sechelt Inlet depress the surface isopycnals as thefresh water mixes downwards and thickens the surface layer, but no direct correlationbetween the surface currents and the discharge is seen.Changes in discharge alter the density structure, particularly in the surface layer, andcan, therefore, affect the modal structure of the baroclinic tide. For example, using a2000;:50c1100a: 0 U-50 0211/912/15/91f1/913/15/914/1/914/15/915/1/915/15/916/1/916/15/91Figure6.16:HourlydischargefromtheClowhomdamfortheperiodofJanuary21toJune18,1991.Dataweresmoothedusingthreepassesof a25-hourmovingaveragefilter.IIChapter 6. Low Frequency Circulation12 24 36 48129Figure 6.17: Lag correlations between discharge from the Clowhom River dam and the2 m along-channel currents in Salmon Inlet. A positive lag means that the dischargeleads the current response.-8 -36 4 -12 12 24 36 48C2>..I .•0.I .C;°•.•Z:.0..1.-48 -36 -24 -12Lag (hours)Chapter 6. Low Frequency Circulation 130Table 6.1: Mean monthly discharge from the Clowhom River dam for January to May1991 (data are courtesy of B.C. Hydro).Month Mean Monthly Runoff(m3 s’)Jan 62.6Feb 14.6Apr 50.8May 49.3linear numerical model for the internal tide (a generalization of the model discussed byStigebrandt (1980)), Stacey (1984) showed that the increase of fresh water into Observatory Inlet, B.C., in early summer was the cause of the increase in the power lost from thebarotropic tide. The change in density structure increased the power transferred to themode 2 baroclinic tide relative to the other modes, thereby increasing the total baroclinicenergy flux. Changes in the transfer of energy to the mode 1 baroclinic tide were insensitive to the increase in summer stratification, but increased during deep-water renewal. InSechelt Inlet, the total baroclinic energy flux does increase from winter to early summer.However, the partition of the energy flux between the modes in Sechelt Inlet does notchange significantly: all of the modes appear to increase in amplitude evenly (see section4.3.3). Moreover, the density structure does not change very much, since the fresh waterdischarge is relatively small. In chapter 5, the influence on the K1 tidal constituent by theP1 constituent was mentioned as a possible reason for the increase in the energy flux ofthe K1 baroclinic tide. The M2 baroclinic energy flux, on the other hand, also increasedin early summer, without significant influence from another constituent. It is unclearif the change in the stratification of the water column is responsible for the increase inbaroclinic energy flux.Chapter 6. Low Frequency Circulation 1316.4 Mean CirculationThe mean along-channel current profile is quite similar over the four months of observation. Mean profiles from January, February, March, April and May 1991 are shown inFig. 6.18. With the exception of February, there is a strong surface outflow with a returnflow beneath (the zero crossing is about 12 m). Beneath the two-layer exchange at thesurface is another two-layer exchange with a seaward flowing layer between about 30 and75 m, and an up-channel flow down to approximately 150 m. The water beneath 150 mis relatively quiescent. In February, the up-inlet return flow near the surface is shifted toseaward flow, and the seaward mid-depth flow of the other months, centred at 50 m, isreversed. The reversal of the mid-depth flow could be in response to a niid-depth renewalwhich is seen to take place between the February and March hydrographic surveys (seesection 6.6); as the renewing water enters at mid-depth, the two-layer mean surface flowis shifted to compensate.The mean along-channel velocities from the Sill cyclesonde are plotted along withthe Basin and Salmon values in Fig. 6.18. Two of the five cyclesonde deploymentsfailed (January and March). Each month shows a substantial seaward flow from 20 toabout 90 m, with a weaker up-channel flow underneath (to 180 m). Without the surfacelayer measurements it is difficult to say whether or not volume is being conserved at themooring, but it appears that the seaward volume flux above 90 m is substantially higherthan the up-channel flux below. At the Sill mooring, there may be enough cross-channelvariability in the flow that a single current meter is insufficient to properly measure themean volume flux. The cross-channel flow variability is likely caused by the complextopography near the sill (see Fig. 3.4). Exchange water is probably steered to one sideof the channel around Skookum Island.The two-layer flow near the surface is likely caused by the buoyancy-driven fresh waterChapter 6. Low Frequency Circulation 132outflow. The compensating layer beneath would be a combination of the compensationdue to entrainment and a residual flow caused by the currents associated with the tidaljet. The broader two-layer exchange flow underneath is probably caused by pressuregradients arising from the mixing of the water column near the sill by the tidal jet. Thetidal jet mixes buoyant water from near the surface with the water outside the sill toform a water mass which is denser than the surface layer, yet almost always lighter thanthe deep resident basin water. The new water mass enters with significant kinetic energywhich is dissipated as the jet water mass is mixed with the basin water. As a result,the deep isopycnals are tilted down towards the sill, and a mid-depth flow is driventowards the sill (see Fig. 6.21, March density). To conserve mass, a return flow is setup underneath. Hence, the kinetic energy of the tidal jet is ultimately responsible formaintaining the tilt in the deep isopycnals, which, in turn, drives the deeper two-layerflow.Harmonic analysis of the along-channel velocities shows that the MSf tidal velocitiesmay be as large as the M2 tidal velocities in the upper 150 m of the water column, andabout half as large as the K1 velocities (see Figs. 4.3 and 4.4). Since MSf currentsdriven directly by astronomical forcing are negligible, the presence of measurable lowfrequency tidal currents indicates that non-linear tidal interactions are important. Thevertical structure of the MS1 velocities in January, shown in Fig. 6.19, represents amultilayer flow down to approximately 120 m, with much smaller velocities between 120m and the bottom. In general, the MS velocity structure is principally a two-layerflow down to depths of 75 to 120 m, with small velocities near the surface; the zerocrossing may be anywhere from 20 to 50 m, depending on the month. The perturbationdensity amplitudes for the MS1 constituent are comparable to those of M2 and K1. Thelargest MS1 density amplitudes occur in the surface layer where the density gradientsare relatively large.—BasinSalmonFigure6.18:Meanalong-channelvelocitiesforJanuary,February,March,April,andMay1991.ThesolidlinesarefromtheBasinmooring,theheavydashedlinesarefromtheSillmooring,andthelightdashedlinesarefromtheSalmonmooring.Seawardflowispositive.---Sill0r8E0Febc..J-6-4-20246-6-4-20246-6-4-20246-6-4-20246-6-4-20246Along-channel V(cmi1)Along-channel V(cm‘)Along-channelV(cms1)Along-channel V(cm1)Along-channel V(cm1)(a)(b)(c)(d)Figure6.19:Profilesof Basinalong-channel velocityandperturbationdensityfromtheMSfconstituent forJanuary1991;(a)in-phasevelocity, (b)in-quadraturevelocity,(c)in-phasedensity, and(d)in-quadraturedensity.0 IE clQQ ‘4,to 0 0 C’,-4-2024V(cm1)-4-2Ô24U,C’,V(cms)-0.1Ô0.10.20.30.40.5-3Density(kgm)-0.160.10:20.30:40:5-3Density(kgm)Chapter 6. Low Frequency Circulation 135Table 6.2: Mean velocities and net volume fluxes at the Basin mooring.Month Mean Velocity Net Volume Flux(cm s1) (m3 s’)Jan 0.05 +163Feb -0.11-362Apr -0.02-58May -0.04 -143From the profiles of mean velocity (Fig. 6.18) and the hypsographic function forSechelt Basin, estimates of the net volume flux were made for January, February, Apriland May 1991. The net volume flux at the Basin station should equal the total amount offresh water flux from the Clowhom River, and the flux from the creeks along Salmon Inletand the portion of Sechelt Inlet from Porpoise Bay to the Mooring. Flow estimates fromthe B.C. Hydro dam on the Clowhom River put the discharge from a low of 14.6 m3 s’in February (Table 6.1) to a high of 62.6 m3 s in January. The net volume fluxes fromthe Basin mooring data are listed in Table 6.2.The volume flux estimates are much larger than the discharge rates from the ClowhomRiver, and for three out of the four months, they are the wrong sign. There are severalpossible reasons for the discrepancy between the net volume fluxes and the dischargerates. Topographic steering or deflection of the fresh surface layer may cause crosschannel variations in the flow that are not measured by a single mooring in the centre ofthe channel. A second possibility is that the uncertainties in the velocities are too large toget an accurate estimate of the net flux. If an uncertainty of 5% is assigned to the velocity,the error in the surface velocity alone (typically 5 cm s’) results in a volume flux errorwhich is considerably larger than the mean fluxes in Table 6.2. Anecdotal evidence fromChapter 6. Low Frequency Circulation 136the divers who cleaned the S4 instruments halfway through their two month deploymentssuggests that there is a large current shear between the surface and the 2 m instrument(P. Baker, pers. comm.). There could be a significant volume flux from this surface flowthat is not taken into account when computing the net volume flux.6.5 Empirical orthogonal function analysisThere are many ways to examine the variability of the low frequency circulation where theforcing (i.e. wind and runoff) and the currents are correlated. Stucchi (1990) discussedthe variability of oxygen levels arising from pulp mill effluent discharge in Neroutsos Inlet,B.C., using empirical orthogonal function (EOF) analysis. The variability of the windand the surface currents in Neroutsos Inlet was also examined using EOF analysis, anda wind-driven mode was successfully identified in several of the data sets. This sectionpresents the results of the EOF analysis of the anemometer and current meter data fromSechelt and Salmon Inlet.Empirical orthogonal function analysis provides another means to examine the lowfrequency circulation (see Kundu et al, 1975). By examining the covariance betweenthe physical variables of a system, EOF analysis can separate the dominant modes ofvariability from the data. The physical interpretation of these modes may be difficult,however, since they are only dependent on the empirical variability of the data, and notnecessarily on the dynamics of the system.After their means are removed, the low-pass filtered along-channel winds and currentsare scaled so that the variance of each record is unity. A real, symmetric covariancematrix of the scaled data is then created, with the off-diagonal elements equal to thecovariances of velocities at different depths and the diagonal elements equal to 1. Eacheigenfunction, cj, of this matrix represents an independent mode of variability in theChapter 6. Low Frequency Circulation 137data and its associated eigenvalue, ), is proportional to the contribution of cz5j to theoverall variance of the system. The relative contribution, v1, for each mode to the totalvariance (energy) is given by:(6.5)The EOF analysis not only gives the amount of variance accounted for by each eigenfunction, but the amount of variance accounted for at each depth may be determined forthe elements of the eigenfunction as well. The contribution to the variance of the signalat depth n of eigenvalue i is—________2 (.)It should be noted that the physical system could be modelled by a series expansionover any set of orthogonal functions (e.g. the set of dynamic modes); however, the seriesexpansion using the EOFs is the one which converges fastest to the system being modelled(Kundu et al, 1975).The intention of the EOF analysis is to separate the independent modes of variabilityand to try to explain their origin. The inclusion of the wind velocities in the analysiswill hopefully reveal the contribution of the wind to the variability in the low frequencycurrents. However, it was felt that an analysis of the low frequency variability of theBasin January and February currents would be useful, even though wind data were notavailable. In general, it was found that only four modes are needed to account for 95%or more of the variance.Because the instruments are not evenly spaced over the water column, the eigenfunctions are not truly orthogonal. Instead, the value of the function at a given depth shouldnot be viewed as a point measure, but rather an integral of the function over the intervalbetween adjacent instruments. For the purposes of the discussion below, the exact natureChapter 6. Low Frequency Circulation 138of the function is not important. The vertical structure of the function and its correlationwith the wind are the two real properties of interest.The Basin eigenfunctions are tabulated in Tables 6.3 to 6.6. The variance in the Basindata during January and February was split fairly evenly between the first and secondmodes. The first mode (52%) in January is essentially a three-layer flow with outflowat the surface and the zero crossings near 35 arid 125 m. The second mode (27%) is afive-layer flow, also with outflow at the surface and the first zero crossing is between 9and 20 m. In February, the first mode (39%) is, again, a three-layer flow (outflow atthe surface) with zero crossings near 30 and 115 m. The second mode (35%) is nearlyidentical to the second mode in January, with zero crossings near 9 and 60 m; however,the deep flow breaks down into a flow with several zero crossings, which contributes littleof the variance in the deep currents (0 to 3 %). The third mode in both months representsa five-layer flow which contributes 13 to 17 % of the variance, with most of the variancecoming from currents below 100 m. The anomalous distribution of variance across themodes in February could be due to a mid-depth intrusion of water, which disrupts theusual partition of variance among the modes.In April and May, the first mode of variability accounts for the bulk (70 to 80 %) ofthe variance. The second mode is much less important and accounts for 10 to 15 % ofthe variance; the third mode accounts for 5 to 10 % of the variance. The dominant modeis the same in both cases and represents a four-layer flow, with the uppermost layer (0- 30 m) flowing seaward, and zero crossings at approximately 30, 70 and 180 m. Thisfour-layer flow is very similar to the mean flow shown earlier in this chapter. The secondmode of variability represents a five-layer flow, but is slightly different in each month. InApril, the second mode has a weak surface outflow with a crossing at about 3 m, and asignificant return flow beneath that extends to approximately 15 m. In May, there is nosurface outflow, but the rest of the profile is virtually the same as in April. Based on theChapter 6. Low Frequency Circulation 139April eigenfunction, it would be tempting to say that this mode represents the estuarinecirculation. However, the difference in the surface layer in May makes this unlikely. Thecontribution to the variance in the wind is strongest in the fourth mode in April, butbecomes strongest in the first mode in May. Since the largest contribution to the windvariance is not consistently in the same mode, it appears that the definitive separation ofthe wind-driven low frequency mode is not possible in Sechelt Inlet using EOF analysis.The four months of surface layer data from the Salmon site were also analysed (Tables6.7 to 6.10). The first mode of variability accounted for 57 to 79 % of the variance in thedata, the second mode accounted for 18 to 35 %, and the third mode contributed mostof the remaining variance (2 to 6 %). With the exception of January, where there was azero crossing near 2 m, the dominant mode represents a uniformly moving surface layer.The layer is moving towards the head in January, April and May, but flows seaward inFebruary. The second mode, however, has a seaward flowing surface layer (with a zerocrossing at about 5 m), and a counter flow beneath in all months. The first mode mayrepresent the bulk motion of the upper layer in response to movement lower in the watercolumn (which is not measured); the change in sign of the first mode in February could bein response to the inflow of new water at mid-depth. The second mode is representativeof the buoyancy-driven estuarine circulation. One explanation for the estuarine modeappearing in the analysis of the Salmon data and not in the Basin data is that theestuarine circulation may be better established in Salmon Inlet, but is masked in SecheltInlet by the intense mixing near the sill. As in the Basin data, the apparent wind-drivenmode changes in each data set and is always negatively correlated with the surface flow.The wind dominated mode in January and February is mode 3, and in April and May itis mode 1. It is unlikely that the EOF analysis is able to accurately identify the winddriven mode given the inconsistency in its placement and the poor correlations seen withthe surface currents.Chapter 6. Low Frequency Circulation 140Table 6.3: The first four EOF eigenfunctions for the Basin mooring in January. Winddata were not available.Mode (Percent Variance)Depth (m) 1 (52.1) 2 (26.6) 3 (13.0) 4 (4.8)2 -0.11 (19) 0.27 (57) 0.22 (18) -0.08 ( 0)4 0.07 ( 7) 0.25 (55) 0.22 (20) -0.15 ( 3)6 0.13 (25) 0.26 (51) 0.16 (10) -0.15 ( 3)9 0.26 (73) 0.20 (21) 0.04 ( 0) -0.16 ( 2)20 0.30 (87) -0.14 ( 8) -0.07 (1) -0.05 ( 0)30 0.20 (40) -0.32 (56) -0.08 (1) -0.07 ( 0)35 0.07 ( 6) -0.38 (90) -0.06 (1) -0.06 ( 0)40 -0.08 ( 7) -0.37 (87) -0.01 ( 0) -0.01 ( 0)45 -0.20 (43) -0.30 (52) 0.01 ( 0) 0.03 ( 0)50 -0.27 (71) -0.23 (25) 0.03 (0) 0.05 ( 0)60 -0.32 (95) -0.08 ( 3) 0.02 (0) 0.09 ( 0)70 -0.33 (97) 0.04 ( 0) -0.03 ( 0) 0.13 (1)80 -0.31 (89) 0.12 ( 7) -0.09 (1) 0.14 (1)90 -0.26 (68) 0.21 (23) -0.12 ( 3) 0.20 ( 3)100 -0.21 (52) 0.24 (35) -0.14 (5) 0.20 ( 4)120 -0.04 ( 3) 0.19 (36) -0.32 (49) 0.12 ( 2)130 0.21 (56) 0.12 ( 9) -0.30 (30) 0.02 ( 0)140 0.24 (72) 0.14 (11) -0.13 ( 5) 0.22 ( 5)150 0.24 (77) 0.06 ( 2) -0.02 ( 0) 0.37 (17)160 0.19 (60) -0.07 (4) -0.01 ( 0) 0.45 (31)170 0.11 (29) -0.07 ( 6) 0.13 ( 9) 0.50 (51)180 0.09 (16) -0.04 (1) 0.30 (50) 0.36 (26)208 -0.10 (16) -0.06 ( 2) 0.44 (80) -0.00 ( 0)238 -0.01 ( 0) 0.05 ( 2) 0.44 (91) 0.04 ( 0)268 0.08 (15) -0.02 ( 0) 0.33 (64) 0.08 (1)Chapter 6. Low Frequency Circulation 141Table 6.4: The first four EOF eigenfunctions for the Basin mooring in February. Winddata were not available.Mode (Percent Variance)Depth (m) 1 (38.8) 2 (35.3) 3 (16.5) 4 (3.3)2 0.13 (18) 0.26 (66) 0.06 (1) 0.07 ( 0)4 0.14 (21) 0.24 (57) 0.06 (1) 0.17 ( 2)6 0.16 (28) 0.23 (52) 0.05 (1) 0.35 (11)9 0.17 (41) 0.12 (18) -0.06 ( 2) 0.39 (19)12 0.03 ( 2) -0.10 (19) -0.12 (12) 0.24 (10)20 0.14 (22) -0.24 (55) -0.11 ( 5) 0.35 (11)30 0.04 (1) -0.36 (94) -0.06 (1) 0.15 (1)35 -0.02 ( 0) -0.38 (97) -0.01 ( 0) 0.01 ( 0)40 -0.06 (2) -0.37 (93) 0.03 ( 0) -0.05 ( 0)45 -0.14 (16) -0.33 (77) 0.09 ( 2) -0.10 ( 0)50 -0.22 (40) -0.25 (46) 0.15 ( 7) -0.10 ( 0)60 -0.33 (84) -0.02 ( 0) 0.19 (12) -0.03 ( 0)70 -0.32 (79) 0.12 ( 9) 0.16 ( 8) -0.02 ( 0)80 -0.29 (63) 0.22 (33) 0.08 (1) -0.04 ( 0)90 -0.26 (54) 0.24 (41) -0.04 ( 0) -0.14 (1)100 -0.22 (48) 0.18 (30) -0.18 (14) -0.19 ( 3)120 0.05 ( 3) 0.05 ( 3) -0.37 (78) -0.21 ( 5)130 0.15 (25) 0.04 (1) -0.37 (65) -0.24 ( 5)140 0.24 (60) -0.03 ( 0) -0.27 (30) -0.24 ( 4)150 0.28 (83) -0.02 ( 0) -0.08 ( 3) -0.31 ( 8)160 0.25 (73) -0.01 ( 0) 0.12 ( 7) -0.28 ( 7)170 0.17 (38) 0.04 (1) 0.29 (46) -0.25 ( 6)180 0.17 (34) 0.06 ( 3) 0.34 (55) -0.09 ( 0)208 0.21 (44) -0.06 ( 3) 0.34 (48) -0.03 ( 0)238 0.20 (42) -0.02 ( 0) 0.33 (51) -0.09 ( 0)268 0.18 (52) 0.02 ( 0) 0.18 (22) -0.09 (1)Chapter 6. Low Frequency Circulation 142Table 6.5: The first four EOF eigenfunctions for the Basin mooring in April.Mode (Percent Variance)Depth (m) 1 (70.3) 2 (14.1) 3 (8.2) 4 (3.2)Wind 0.06 (18) 0.05 ( 2) -0.18 (18) -0.43 (42)2 0.10 (41) 0.01 ( 0) -0.18 (15) -0.41 (30)4 0.17 (70) -0.07 ( 2) -0.20 (11) -0.26 ( 6)6 0.21 (83) -0.14 ( 7) -0.16 (5) -0.04 ( 0)9 0.23 (84) -0.19 (11) 0.03 ( 0) 0.21 ( 2)12 0.24 (84) -0.21 (13) 0.02 ( 0) 0.10 ( 0)1.5 0.27 (95) -0.05 ( 0) 0.14 ( 2) 0.08 ( 0)20 0.26 (91) 0.03 ( 0) 0.21 ( 7) 0.04 ( 0)25 0.16 (54) 0.17 (12) 0.34 (29) -0.05 ( 0)30 -0.06 ( 9) 0.26 (43) 0.33 (40) -0.13 ( 2)35 -0.21 (73) 0.20 (14) 0.23 (10) -0.10 ( 0)40 -0.27 (93) 0.13 (4) 0.11 (1) -0.04 ( 0)45 -0.29 (98) 0.08 (1) 0.00 ( 0) 0.01 ( 0)50 -0.29 (98) 0.04 ( 0) -0.07 ( 0) 0.02 ( 0)60 -0.27 (94) -0.01 ( 0) -0.17 (4) 0.01 ( 0)70 -0.23 (88) -0.06 (1) -0.16 ( 5) -0.16 (1)80 -0.06 (16) -0.18 (27) -0.05 (1) -0.43 (35)90 0.18 (72) -0.14 ( 8) 0.10 ( 2) -0.25 ( 6)100 0.24 (90) -0.07 (1) 0.11 ( 2) -0.03 ( 0)120 0.13 (52) 0.18 (19) -0.05 ( 0) 0.21 ( 5)130 0.03 (1) 0.35 (77) -0.19 (13) 0.19 ( 5)140 0.08 (18) 0.31 (56) -0.25 (20) 0.09 (1)150 0.12 (35) 0.35 (59) -0.08 (1) 0.04 ( 0)160 0.14 (46) 0.27 (35) 0.16 ( 7) -0.12 (1)170 0.01 ( 0) 0.14 (17) 0.31 (52) -0.14 ( 3)180 -0.08 (23) -0.00 ( 0) 0.36 (56) -0.23 ( 8)208 -0.18 (62) -0.26 (26) 0.17 ( 6) 0.15 (1)238 -0.17 (62) -0.25 (27) 0.14 ( 4) 0.10 (1)268 -0.11 (37) -0.26 (41) 0.15 ( 8) 0.10 (1)Chapter 6. Low Frequency Circulation 143Table 6.6: The first four EOF eigenfunctions for the Basin mooring in May.Mode (Percent Variance)Depth (m) 1 (79.2) 2 (9.8) 3 (4.1) 4 (2.6)Wind -0.11 (51) -0.09 ( 4) -0.32 (21) 0.18 ( 4)2 0.03 ( 5) -0.23 (35) -0.39 (44) -0.06 ( 0)4 0.22 (89) -0.09 (1) -0.18 ( 3) -0.21 ( 2)9 0.27 (97) -0.08 (1) -0.14 (1) -0.02 ( 0)12 0.26 (96) -0.04 ( 0) -0.14 (1) 0.11 ( 0)15 0.25 (95) 0.01 ( 0) 0.17 ( 2) 0.10 ( 0)20 0.15 (67) 0.09 ( 2) 0.38 (22) 0.15 ( 2)25 -0.19 (76) 0.17 ( 7) 0.34 (13) 0.08 ( 0)30 -0.26 (93) 0.16 ( 4) 0.13 (1) -0.08 ( 0)35 -0.27 (96) 0.11 (1) 0.04 ( 0) -0.17 (1)40 -0.27 (97) 0.09 (1) -0.04 ( 0) -0.16 (1)45 -0.26 (95) 0.10 ( 1) -0.15 (1) -0.12 ( 0)50 -0.24 (91) 0.13 ( 3) -0.21 (3) 0.00 ( 0)60 -0.14 (64) -0.04 (0) -0.18 ( 5) 0.33 (12)70 0.05 ( 9) -0.35 (65) 0.03 ( 0) 0.31 (13)80 0.07 (16) -0.42 (75) 0.13 ( 3) 0.03 ( 0)90 0.04 ( 5) -0.35 (66) 0.22 (10) -0.31 (14)100 -0.01 ( 0) -0.20 (30) 0.20 (13) -0.48 (48)120 0.20 (85) 0.10 ( 2) -0.09 ( 0) -0.25 ( 4)130 0.22 (87) 0.18 ( 7) -0.05 ( 0) -0.21 ( 2)140 0.22 (87) 0.19 ( 8) -0.10 (1) -0.13 ( 0)150 0.23 (89) 0.16 ( 5) -0.10 ( 0) -0.11 ( 0)160 0.16 (61) 0.34 (36) -0.01 ( 0) -0.04 ( 0)170 0.12 (52) 0.26 (28) -0.00 ( 0) -0.01 ( 0)180 -0.05 (13) 0.15 (19) -0.20 (13) 0.09 (1)208 -0.22 (90) -0.10 ( 2) -0.08 ( 0) -0.28 ( 4)238 -0.19 (88) -0.11 ( 3) 0.07 ( 0) 0.03 ( 0)268 0.08 (32) 0.14 (15) 0.29 (26) 0.17 ( 5)Chapter 6. Low Frequency Circulation 144Table 6.7: The first four EOF eigenfunctions for the Salmon mooring in January.Mode (Percent Variance)Depth (m) 1 (72.8) 2 (19.2) 3 (6.4) 4 (1.5)Wind -0.02 ( 0) 0.41 (37) 0.91 (62) 0.06 ( 0)2 0.05 ( 1) 0.70 (88) -0.27 ( 4) -0.65 ( 6)4 -0.42 (73) 0.45 (22) -0.24 ( 2) 0.40 ( 1)6 -0.54 (97) 0.14 ( 1) -0.09 ( 0) 0.33 ( 0)9 -0.55 (96) -0.18 ( 2) 0.07 ( 0) -0.25 ( 0)12 -0.48 (87) -0.31 ( 9) 0.17 ( 0) -0.50 ( 1)Table 6.8: The first four EOF eigenfunctions for the Salmon mooring in February.Mode (Percent Variance)Depth (m) 1 (57.7) 2 (35.4) 3 (6.4) 4 (0.4)Wind 0.13 ( 9) -0.29 (31) 0.94 (58) -0.15 ( 0)2 0.45 (66) 0.41 (32) -0.02 ( 0) -0.63 ( 0)4 0.56 (88) 0.25 (10) -0.02 ( 0) -0.04 ( 0)6 0.58 (97) -0.09 ( 1) -0.02 ( 0) 0.60 ( 0)9 0.36 (41) -0.53 (56) -0.16 ( 0) 0.04 ( 0)12 0.08 ( 2) -0.63 (92) -0.31 ( 4) -0.47 ( 0)Chapter 6. Low Frequency Circulation 145Table 6.9: The first four EOF eigenfunctions for the Salmon mooring in April.Mode (Percent Variance)Depth (m) 1 (78.7) 2 (18.1) 3 (2.0) 4 (1.1)Wind 0.38 (76) -0.40 (18) -0.23 ( 0) -0.76 ( 4)2 -0.23 (36) 0.56 (52) -0.78 (10) -0.17 ( 0)4 -0.48 (93) 0.21 ( 4) 0.37 ( 1) -0.24 ( 0)6 -0.51 (98) -0.02 ( 0) 0.22 ( 0) -0.44 ( 0)9 -0.45 (83) -0.41 (15) -0.17 ( 0) -0.08 ( 0)12 -0.33 (59) -0.55 (37) -0.37 ( 1) 0.36 ( 0)Table 6.10: The first four EOF eigenfunctions for the Salmon mooring in May.Mode (Percent Variance)Depth (m) 1 (67.3) 2 (26.5) 3 (4.4) 4 (1.6)Wind 0.38 (69) -0.25 (11) -0.70 (15) -0.53 ( 3)2 -0.30 (43) 0.52 (50) 0.23 ( 1) -0.66 ( 4)4 -0.48 (83) 0.29 (11) -0.41 ( 3) -0.10 ( 0)6 -0.51 (93) -0.03 ( 0) -0.48 ( 5) 0.34 ( 0)9 -0.44 (67) -0.48 (32) 0.10 ( 0) -0.02 ( 0)12 -0.28 (35) -0.59 (60) 0.24 ( 1) -0.41 ( 1)Chapter 6. Low Frequency Circulation 1466.6 Deep-Water RenewalBecause of the strong mixing at the sill, replacement of the deep basin water in SecheltInlet is very infrequent. The renewing water must enter the inlet first over a 30 to 40 mdeep sill at the connecting channel between Jervis Inlet and the entrance to Sechelt Inlet.The new water then passes through a shallow basin (75 m) between the first sill and themain sill at Skookumchuck Narrows before entering the main basin. Because the mixingat the main sill is so vigorous, the density of the inflow must be significantly denser thanthe basin water it is to replace.From January 1990 to April 1992 there was no bottom water renewal in Sechelt Inlet,but mid-depth intrusions of water were seen in the late winter of 1990 and 1991, withthe new water reaching depths of 100 to 150 m. The lack of bottom replacement wasdisappointing from the point of view of measuring the hydrographic parameters duringturnover of the bottom water; however, the relative quiescence of the deep-water allowedfor good measurements of the vertical diffusion of heat and salt.Previous studies (e.g. Lazier, 1963) were fortunate enough to have measurementsduring years where the bottom water was replaced. Temperature, salinity and oxygenmeasurements at 100 and 200 m in Sechelt Inlet from UBC hydrographic surveys from1957 to 1993 are shown in Fig. 6.20. Mid-depth intrusions of water are seen to occurevery year from the 100 m data in winter, when the renewing water is densest and therunoff is low (as proposed by Lazier (1963)). The renewal cycle of the bottom wateris much less frequent: relatively high oxygen levels (> 2.5 ml 1_i) were seen in August1957, March 1962, January 1986, and August 1993. During several surveys between theserenewals (1962 to 1964 and 1990 to 1992) there is no indication of renewal at or below200 m. There is an increase in oxygen levels at 200 m in 1990, but because there is noindication in the temperature or the salinity records at 200 m that it is due to an intrusionChapter 6. Low Frequency Circulation 147of new water, the gradual increase in oxygen (see Fig. 6.20) is probably due to downwarddiffusion of oxygen from the mid-depth replacement, rather than from replacement of thedeep-water itself. It is tempting to say that, based on the hydrographic surveys, bottomwater renewal occurs with a period of approximately 5 years. However, the one surveyin 1981 suggests that there was no renewal in the previous winter, and that the existingbasin water had been resident a long time. At best, the bottom water renewal in Secheltcan be said to be infrequent, with the average residence time of the basin water below150 m being probably five years or more.Since oxygen is a good tracer of new water, changes in oxygen at depth’ usuallyindicate the presence of new water. Normally, an increase in oxygen is accompanied byan increase in density, because the replacement water is likely to have a density greaterthan the resident basin water. However, the background rate of density change in thedeep-water is about 0.01 kg m3 per month, and will tend to deepen the isopycnals overtime, even if the water remains undisturbed. Contours of oxygen and density are shownin Figs. 6.21 and 6.22, from hydrographic surveys before and after mid-depth renewal in1990 and 1991.In 1990, mid-depth renewal took place between February and March. The oxygencontours suggest that the new water penetrated to just below 150 m, displacing the resident basin water towards the head. An oxygen minimum occurs at 50 m as the relativelyoxygen-rich plume appears to push under the surface layer. The oxygen isopleths atstation 8, near the head of Salmon Inlet, are noticeably displaced upwards. The 24.4 atcontour deepens between February and March 1990, due to vertical diffusion, but the22.2 and 22.0 isopycnals are displaced upwards, indicating an increase in the mid-depthdensities associated with the renewal. There is also an increase in the tilt of the 22.2 and22.0 isopycnals between the sill and station 3, and, to a lesser extent, in the 22.4 contourline.Figure6.20:Temperature,salinityanddissolvedoxygenat100and200minSecheltInletfrom1957to1993.Significantgapsintimehavebeenremovedfromtheplot,anddatavalueswhichexceedtheboundsoftheplotareindicatedby().Inyearswithonlyonesurvey,individualdatapointsareindicatedby(Q)at100mand(D)at200m.195719611962196319641981198519861990199119921993c-) I I9.4 9.2•9.0 8.8.8.68.4.8.2 8.0 7.8 29.028.828.628.428.228.0200mo(7.61r•(7.44)• 0 • o(9.77)j (Io.o80 I::(9.98)•..0•0..A1‘::Ij::i°°1‘/::.‘ lOOm;::1oomIs\\III I I / I I I I9.4 9.2 9.0 8.8 8.6 8.4 8.2 8.0 7.8 29.028.828.628.428.228.027.827.64 3 2 0I o27.8,.27.6-4 21957200m’%,19611962196319641981198519861990199119921993Chapter 6. Low Frequency Circulation 149In 1991, mid-depth renewal also took place between February and March, but toa lesser degree than in 1990. An oxygen-rich plume can be seen penetrating to about100 m, causing an oxygen minimum at 50 m at station 3. The response of the isopycnalsin 1991 is shallower than in 1990; The effect of the inflow is seen only at depths shallowerthan 100 m. The 22.0, 22.2 and 22.3 o isopycnals all deepen slightly; the 21.5 isopycnal,however, becomes shallower with the inflowing dense water. In February, the isopycnalsare actually tilted towards the head. This behaviour could be in response to the largedischarge of water from the Clowhom River Dam in February: as the surface pressuregradient increases (tilts seaward) from the discharge, the isopycnals lower in the watercolumn will tilt towards the head to compensate. The isopycnals at station 8 becomeelevated in March, as in 1990.One of the potential problems associated with renewal of the deep-water is the upwarddisplacement of the resident low oxygen basin water. In 1962, the oxygen at station 8dropped to below 1 ml 1_i at 10 m, and remained very low until 150 m, where it increasedto 4 ml 1_i. Even in Porpoise Bay (station 6), where oxygen values are normally wellabove 2 ml 1’ down to 75 m, oxygen values fell below 2 ml M at 20 m. Although theupwelling of deep-water is visible during mid-depth replacement, it is not as severe asduring a replacement of the bottom water. The 1962 event demonstrates that it is possiblefor displaced bottom water to reach Porpoise Bay, and that may have implications ondisposal of waste in the inlet. The upwelling of basin water will need to be consideredwhen determining the assimilative capacity of Sechelt Inlet for pollution. It could alsohave a deleterious effect on salmon farming.Chapter 6. Low Frequency Circulation 1501 21.5 2 3 4 7 822—224______________________________100 )222 jI! -200 --I--—.300- (a).... Dt (as) (kg ni3)I I I I40 30 20 10 0Distance from the head (km)1 2 3 4 7 80-•.. ‘S - *—- -— —,. . —r- :• •‘“-•—--- : ,—:- 2•4. 3 - / / :100 : ..; : : > ‘ : • —- 7 “ ;___._._- .——r’—’ 1E . : -• -— - .—.-— •• •—.—- . _•.,—200’ 1.5_._....;-...— —--— : -/300- (b):::Oxygen (ml 1)I I I I40 30 20 10 0Distance from the head (km)Figure 6.21: Along-channel section contours of (a) density (kg m3) and (b) oxygen(ml 1_i), for February (solid) and March (dashed) 1990.Chapter 6. Low Frequency Circulation 1511 21 2 3 4 7 8::300- (a) Density (as) (kg n13)I I I I40 30 20 10 0Distance from the head (km)1 2 3 4 7 S0—5 :/ : 4ioo r20O I :jI_____300- (b)_4:—Oxygen (ml 1)I I I I I40 30 20 10 0Distance from the head (km)Figure 6.22: Along-channel section contours of (a) density (kg m3) and (b) oxygen(ml 1_i), for February (solid) and March (dashed) 1991.Chapter 7ConclusionsThe circulation in Sechelt Inlet is weak, despite the highly energetic tidal interactionat the sill. Because the narrow constriction at Skookumchuck Narrows chokes the tidalinflow enough to significantly reduce the tidal amplitudes landward of the sill, a morerobust analytical model was derived in order to better estimate the loss of energy fromthe barotropic tide. The barotropic tidal flux into Sechelt Inlet is very large: the tidesprovide an average up-inlet energy flux of about 42 MW, based on tide gauge records andthe estimates from the theory presented in chapter 4. The fluxes in Sechelt Inlet are anorder of magnitude larger than those of 4 MW in Knight Inlet (Farmer and Freeland,1980) and 5 MW through Burrard Inlet and Indian Arm (de Young and Pond, 1987).The three main sinks of the tidal energy flux were identified to be (a) direct frictionaldissipation over the sill, (b) the kinetic energy flux of the turbulent tidal jet, and (c) theprogressive internal tide. The apparent inefficiency of the transfer of energy from thetides to the vertical diffusive processes is explained by the frictional dissipation at thesill computed in chapter 4, which accounts for nearly all of the power extracted fromthe surface tide. Some energy (about 5% that of direct frictional dissipation) is alsodissipated close to, but not directly over the sill, through the kinetic energy flux of theturbulent tidal jet. The up-inlet flux of the internal tide based on the analysis of the M2and K1 constituents was between 0.1 and 0.2 MW, and is by far the smallest of the threesinks. The energy flux of high frequency internal waves was not measured, but may bea small fraction of that of the internal tide (de Young and Pond, 1989). The dissipation152Chapter 7. Conclusions 153of the tidal jet is responsible for forming the middle density layer (20 to 150 m), whilethe breaking of the internal tide has been suggested as the primary source of energy forvertical diffusion (Stigebrandt, 1976; Stacey, 1984).Vertical diffusion in Sechelt Inlet is weak compared to neighbouring inlets (e.g. IndianArm). When not disturbed by deep-water intrusions, the basin density decreases byonly about 0.01 kg m3 per month. A power law relationship was established betweenthe vertical diffusivity K, and the Brunt-Väisälä frequency, N, following the theoreticalwork of Gargett and Holloway (1984). Gargett and Holloway suggested that K,, =a0N,where 0.5 q 1.0, and a0 is a site-specific constant. The range of q is determinedby the source of the internal waves which provide the energy for mixing: q = 0.5 fora broad-band internal wave spectrum, and q = 1.0 when the internal waves are of asingle frequency. For Sechelt Inlet, q 1.05 ± 0.08 based on changes in salinity andq 1.33 ± 0.13 based on changes in temperature. These estimates of q are closer to thenarrow-band limit set by Gargett and Holloway (1984), than those found in other fjords,which tend to have q 1.5 (e.g. Gade and Edwards, 1980; de Young and Pond, 1988;Stigebrandt and Aure, 1989).Comparing the estimated gain in potential energy of the basin waters through verticaldiffusion of salt and heat (see chapter 5) with the energy flux of the internal tide, onefinds that they are positively correlated. Following the analysis of Stigebrandt and Aure(1989), an estimate for the flux Richardson number (mixing efficiency) was found forthe transfer of mechanical energy in the internal tide to vertical diffusive processes.For Sechelt Inlet, 0.03 Rf 0.08, with an estimated background rate of work of0.002 ± 0.001 MW. The flux Richardson number for Sechelt Inlet agrees with the rangefound by Stigebrandt and Aure for “well-behaved wave basins”. The background rate ofwork was attributed by Stigebrandt and Aure to wind-generated mixing, but in SecheltInlet, and indeed other inlets, it may also be due to the dissipation of high frequencyChapter 7. Conclusions 154internal waves which propagate away from the sill and break when they approach thedepth where their frequency approaches the Brunt-Väisälä frequency (Pond et al, 1995).The wind appears to have relatively little influence on the circulation. A large part ofthe wind energy is in the diurnal seabreeze, and its effect on the circulation was difficultto measure, since the wind-generated flow and the diurnal tides could not be separated.Sensitivity studies of the baroclinic energy flux showed that the diurnal seabreeze wouldlikely change the baroclinic tidal energy flux by no more than 5%.The coherence of the wind to the currents in the lower frequency bands (lower than0.929 cpd) was generally low, but it was higher in Salmon Inlet than in Sechelt. Coherences well above the 95% noise level (0.31) were found only shallower than 9 m; below200 m, peak coherences of 0.4 to 0.7 were found. The frequency bands that were coherent were typically 0.2 f 0.66 cpd (or 1.5 to 5 days period). The depth of thecoherent response suggests that the low frequency wind variations have some influence onthe surface layer, and may also cause some deep baroclinic response. The low coherencebetween the wind and the currents is in contrast to Knight Inlet, where the along-channelwinds are spatially correlated, and the coherence between the wind and the surface currents is generally high (Baker and Pond, 1995). The more complex geometry in SecheltInlet causes high spatial variability in the wind, which may be responsible for the lowercoherence between the wind and the currents.In Salmon Inlet, there also appeared to be a low frequency response reminiscent ofseiching. A strong outflow wind at the end of January and a sudden large dischargeof water at the beginning of February from the Clowhom River dam was followed byseveral days of oscillation with a period of approximately two days. An internal seichein Salmon Inlet would have a phase speed of about 23 cm s’, which falls in the range of16 cm s1 54 cm s estimated from a simple two-layer model. The existence of aseiche could explain the phase response to the wind in Salmon Inlet, where the currentsChapter 7. Conclusions 155frequently appear to lead the wind.Empirical orthogonal function (EOF) analysis supports the assertion that the windis not primarily responsible for the low frequency variability. The EOF analysis couldnot consistently identify the wind-driven mode as one that accounted for a significantpercentage of the low frequency variance. Instead, a four-layer flow (with outflow at thesurface), was identified as the dominant mode, accounting for 35 to 85% of the varianceat the Basin station. The four-layer flow is also a persistent feature of the monthly meancurrents, and is probably formed by density gradients created by the tidal mixing nearthe sill. The water mass formed by the tidal jet pushes mid-depth water towards the headwhile water in the surface and lower layers moves seaward to compensate, similar to thecirculation proposed by Lazier (1963). The lower layer is blocked by the sill, and a weakup-inlet counterfiow is produced underneath. The empirical modes both in Sechelt andSalmon Inlet seemed to be sensitive to the modulation of the low frequency variabilityby the mid-depth intrusion of water in February.The diffusion of heat and salt decreases the deep-water density over time, conditioningthe deep-water for eventual renewal. Hydrographic surveys in the Sechelt Inlet systemtaken since 1957 suggest that the mid-depth water is replaced once a year, but the deep-water (below 150 m) is replaced only once approximately every five years. Since there wasno bottom water replacement during the 1991 study, the exact conditions for deep-waterrenewal in Sechelt Inlet are not known.The vertical displacement of existing deep-water by intrusions of new, denser waterhas been demonstrated by the consistent tilting of the oxygen isopleths at the head ofSalmon Inlet during mid-depth replacements. During one particularly strong renewalevent in 1963, upwelling of low oxygen water was seen as far as station 6 in Porpoise Bay.The low oxygen levels in this displaced water can threaten fish stocks and cause aestheticand health problems if contaminated by bacteria. It is conceivable that pollutants whichChapter 7. Conclusions 156accumulate in the deep-water could be pumped back up near the surface, even as faras Porpoise Bay during times of bottom water flushing. If pollution is not carefullymonitored, wastes that appear to be assimilated in the deep-water of the inlet may laydormant and unoxidized for several years, only to be forced to the surface again by astrong deep-water intrusion.Appendix AHourly Data PlotsThis appendix contains the along-channel velocity and density plots of hourly data forthe months of February, April and May.Basin velocities are shown in Figs. A.1 to A.3 and Salmon velocities are shown inFigs. A.4 to A.6; positive flow is seaward. Densities are given as o = p — 1000 kg m3.Basin densities are shown in Figs. A.7 to A.9; Salmon densities are shown in Figs. A.10to A.12. Note the change in scale between the 2 to 12 m data and the deeper depths.157Appendix A. Hourly Data Plots 158A-IVW V I • r y, rt . P 1’ ‘rvsJ •‘2/23/91 3/1/91 3/8/91 3/15/91 3/23/91I’ ( 1’ 11.1 1I.J’..ftA hIAA..r A A A A ! AEcU)30102-io-3030410-1o-3030t10-10Figure A.1: Along-channel Basin currents in the surface layer for February 1991. Positive2/23/91 3/1/91 3/8/91 3/15/91i’kA.. ..3/23/9130E 10CN-3030E 10CD-10-3030E 10-i0r.- v r - rvr’.’- “vvv y ‘i.v-i4jr2c..J3010-10 v’V V V V V •‘V U V ‘ V V “ W ‘-V’’ V y y u V-302Qy44vjAJV1VM&.4rA‘rvWy4Wvr2/23/91 3/1/91 3/8/91 3/15/91 3/23/91currents indicate flow towards the sill (units are cmAppendix A. Hourly Data Plots4/23/91 5/1/91 5/8/91 5/15/91 5/23/913030-3030.qriWhJ VVAr’ v yl.(V r’11594/23/91 5/1/91 5/8/91 5/15/91 5/23/9130E 10c.J -10-3030E 10U)-10-3030‘10-10-3030102-io-30I”if’ A”V-w,\f’N-v\fJ,,IJ3010-10-30VAvr\r v--‘r4/23/91 5/1/91 5/8/91 5/15/91 5/23/91Figure A.2: Along-channel Basin currents in the surface layer for April 1991. Positivecurrents indicate flow towards the sill (units are cm s’).E 10C%J-10-302 10cD10y- ‘r2 10-10430‘10-10 “ U V V ‘1 V V • • J W U ‘ NAppendix A. Hourly Data Plots30E 10-10-3030E 10co10-3030E 10-10302 10c’4-10-30302 10LI)-10-3030102-io-3030160Figure A.3: Along-channel Basin currents in the surface layer for May 1991. Positivecurrents indicate flow towards the sill (units are cm s’).6/1/91 6/8/91 6/15/91 6/23/91M ..A i . %. 1, .. Ih(%. — . A & n r ri..••r’ viJv1rvlfl\1vPAe&fy1v/Al6/1/91 6/8/91 6/15/91 6/23/913010-10ArifiJA[fAfWt jANA-30,A, AA.rAv A,i,—- V ‘‘-‘30‘10-10-30&J’rr\tvVV”j6/1/91 6/8/91 6/15/91 6/23/91Appendix A. Hourly Data Plots 1612/23/91 3/1/91 3/8/91 3/15/91 3/23/91I /‘wvW’N400402/23/91 3/1/91 3/8/91 3/15/91 3/23/91Figure A.4: Along-channel Salmon currents in the surface layer for February 1991. Positive currents indicate flow towards the sill (units are cm s’).4/23/91 5/1/91 5/8/91 5/15/91 5/23/91qg•rwvvwfJ\-4%64/23/91 5/1/91 5/8/91 5/15/91 5/23/91Figure A.5: Along-channel Salmon currents in the surface layer for April 1991. Positivecurrents indicate flow towards the sill (units are cm s’).Appendix A. Hourly Data Plots4020-20-40162.. .Afl/i iA flVV6/1/91 6/8/91 6/15/91 6/23/91Figure A.6: Along-channel Salmon currents in the surface layer for May 1991. Positivecurrents indicate flow towards the sill (units are cm6/1/91 6/8/91 6/15/91 6/23/91\f\A/JV71V4VA1VfV’J4020-20-40402020Co-20-40402 20cj0— -20A i,A - - .4-Jvwvvv VVVVVVVWVJ VVYflfWW________________________I- A.--I . A\MiVVV vvqAppendix A. Hourly Data Plots 1632123/91 3/1/91 3/8/91 3/15/91 3/23/91E18° 141022E 1814102/23/91 3/1/91 3/8/9 1 3/15/91 3/23/9 1E 2222E0o 21202200 2120—220° 21202120E 2221202/23/91 3/1/91 3/8/91 3/15/91 3/23/91Figure A.7: Basin densities (o) for February 1991.Appendix A. Hourly Data Plots 1644/23/91 5/1/91 5/8/91 5/15/91 5/23/91CD 1410222 1814104/23/91 5/1/91 5/8/91 5/15/91 5/23/91222to 2120222° 2120- ---- - --2 22212022221202222120.4/23/91 5/1/91 5/8/91 5/15/91 5/23/91Figure A.8: Basin densities (Jt) for April 1991.Appendix A. Hourly Data Plots 1656/1/91 6/8/91 6/15/91 6123/91E 18141022E 1814106/1/91 6/8/91 6/15/91 6/23/91E22E 222120= 22QLO 2120212021206/1/91 6/8/91 6/15/91 6/23/91Figure A.9: Basin densities (a) for May 1991.Appendix A. 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