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The influence of wind on the surface waters of Alberni Inlet Farmer, David Malcolm 1972

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THE INFLUENCE OF WIND ON THE SURFACE WATERS OF ALBERNI INLET . by DAVID MALCOLM FARMER B.Com. McGill University, 1967 M.Sc. McGill University, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Physics and Institute of Oceanography We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April , 1972 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The University of B r i t i s h Columbia Vancouver 8 , Canada TABLE OF CONTENTS Page ABSTRACT iv LIST OF TABLES V LIST OF FIGURES v i ACKNOWLEDGEMENTS • v i i i Chapter 1. INTRODUCTION 1 Survey of Previous Work 3 PART I INSTRUMENTATION AND OBSERVATIONS 2. INSTRUMENTATION 7 Disposit ion of the Instruments 9 Anemometers 9 Current Meters 9 Conductivity Prof i le rs 10 Conductivity Prof i le r . 11 Principles of Operation . 12 Instrument Housing and Protection . . 16 Calibration 18 3 . OBSERVATIONS 20 Wind and Current 25 Wind 25 Currents 25 i i i Chapter Page Conductivity Records 30 Parameterisation 32 Fresh Water Thickness and Potential Energy Difference 35 PART II ANALYSIS AND DISCUSSION Fourier Analysis of Time Series Data 44 4. WIND AND CURRENT 48 Wind and Current Spectra 48 Wind effects and the Diurnal Tide 51 The Wind Driven Current 55 5. SURFACE LAYER THICKNESS . 61 A Linear Model 64 Solving the Equation 68 Estimating the Parameters 77 Comparing Theory with Observations 83 6. SUMMARY . 89 BIBLIOGRAPHY 91 ABSTRACT Observations of wind, current and surface layer thickness in Alberni Inlet have helped elucidate some of the ways in which the system responds to a surface stress. The energy of the wind and also the current at 2 meters depth i s strongly diurnal . Cross-spectral analysis has shown that the two are closely coupled at this frequency. On the basis of simple time scale considerations I have used phase angles between wind and current to estimate a bulk eddy v iscosity for the upper two or three meters of the in le t . This method has yielded values between 1 and 10 cm2/sec. On the other hand most of the energy associated with changes in the surface layer thickness is of s igni f icant ly lower frequency. Strong up- inlet winds induce a sudden thickening in the surface layer at the in let head and the disturbance appears to propagate back down the in let suffering an attenuation as i t travels. The return to equilibrium can take several days. A simple two-layer f r i c t iona l model is able to explain much of what is observed and can be used to predict the surface layer thickness on the basis of measured wind speeds. LIST OF TABLES Table Page I. Dates and Locations of Instrumentation . . . . . . . . . 10 II. T idal Amplitudes and Ratios at Port Alberni and Tofino 52 III. Coefficients of Kinematic Eddy Viscosity 60 LIST OF FIGURES Figure Page 1. Map of Alberni Inlet 8 2. Block Diagram of Instrument Electronics . 15 3. Construction Diagram of Buoy 17 4. Comparison of Wind at different points along inlet . . . 26 5. Current Measurements from Sproat Narrows 28 6. Drift Pole Measurements 29 7. Contours of Constant Conductivity 31 8. Fresh Water Thickness (33 days) 36 9. Potential Energy Difference (33 days) 37 10. Fresh Water Thickness (5 days) 39 11. Potential Energy Difference (5 days) 40 12. Comparison of Sproat North and Sproat South Fresh Water Thickness 42 13. Spectra of Sproat 2m Currents and Wind-Stress 49 14. Coherence Sq. and Phase: Sproat 2m Currents 50 15. Current Spectra at Tidal Frequencies 54 16. Coherence Sq. and Phase: Wind-Stress and Current . . . 57 17. Spectra of Wind-Stress and Surface Layer Thickness . . . 62 18. Coherence Sq. and Phase: Wind-Stress and Surface Layer Thickness 63 19. Diagram of Inlet Model 65 20. Response of Model showing effect of Friction and End Wave 74 21. Model displacement against Friction at time t = t . . . 75 v i i Figure Page 22. Decay of Free Wave for different Friction Coefficients 79 23. Model Response for different K and X 81 24. Frequency Model Response for different K and X 82 25. Model Solution for Fresh Water Thickness (5 days) . . . 84 26. Model Solution for Fresh Water Thickness (33 days) . . . 86 27. Comparison between Theory and Observations i n Frequency Space 87 LIST OF PLATES Plate Page I Conductivity Profiler showing Probes, Electronics and Chart Recorder . 14 II Vessels used i n Alberni Inlet Project 22 III Field Work in Alberni Inlet 24 ACKNOWLEDGEMENTS A project of this nature could scarcely have been undertaken without the active part ic ipation and assistance of many indiv iduals. I am part icular ly grateful to Mr. S. Wigen and the staff of the Canadian Hydrographic Service for combining their own study of Alberni Inlet with mine and for freely providing me with their tide and current data and for the loan of their anemometers. The ship and barge f a c i l i -t ies provided by the Fisheries Research Board were essential to the study; I am especially grateful to Dr. Parker for h is ready cooperation in making these f a c i l i t i e s available from his own programme. I wish to thank Mr. R. Herlinveaux for providing wind data and for his help and interest throughout the programme. It i s also a pleasure to acknowledge the invaluable assistance of Mr. R. Page, Master of the M/V "Caligus", to whose s k i l l , knowledge of the area and interest in the project I am greatly indebted. The Fisheries Research Board of Canada and the National Research Council provided f inancia l assistance. I am most grateful to Drs. Osborn, Burl ing, LeBlond and Pond for their help and advice and for c r i t i c a l l y reading the manuscript. Also to Heinz Heckl for successfully building the instrument housings and buoys in time for the f i e l d programme and to Peter Merchant for putting together the instrument electronics. F inal ly I wish to thank Brian Stewart who, with the patience of Job, careful ly d ig i t ised one third of a mi l l ion points. Without his help the time series analysis with which this thesis concludes could not have been contemplated. Chapter 1 INTRODUCTION This thesis describes an experiment to determine the influence of wind on the surface waters of Alberni Inlet. Anyone famil iar with the in lets of Canada's Pac i f i c coast is aware of the strong winds that can occur on these long, steep-sided arms of the sea. Sometimes they take the form of diurnal sea breezes, springing up in the morning and dying away in the evening (Pickard, 1961), or they may be katabatic in nature, sweeping down with great intensity especially during the winter on certain mainland in le ts . Superimposed on these shorter period effects there are frequently winds of one to a few days duration associated with the passage of weather patterns moving in from the Pac i f ic Ocean. In common with many other in lets of the Paci f ic North West, Alberni Inlet has a thin (3 meter) comparatively fresh layer overlying water of nearly oceanic sa l in i t y . This strong ver t ica l sa l in i ty gradient dominates the near surface density structure and thus exercises a major control on the system's response to wind. These wind effects are important. Observations in this study show that changes in the surface layer thickness by a factor of two or more in a few hours are not uncommon. An understanding of the way in which wind affects an in let w i l l aid oceanographers in their interpre-tation of the data they col lect for b io logical and physical models of in let processes. If the wind influence is predictable there is also some hope that by understanding i t we may learn how to diminish the 2 environmental impact of an effluent by timing and distr ibut ing i t s release so as to optimise i t s disposal . Alberni Inlet i s a part icular ly good place in which to study these processes. The Pac i f ic Oceanographic Group of the Fisheries Research Board in Nanaimo has collected much valuable background in for -mation from the area since 1939. Unlike many of the mainland in le ts , Alberni Inlet is also readily accessible and has the advantage of being comparatively free of sudden changes in direction or width. The ready co-operation of Dr. Parker in making Fisheries Research Board vessel and barge f a c i l i t i e s available from his own Alberni Inlet project has made i t possible to carry out the long time-series study which is essential in identifying the strongly time-dependent effects of wind action. This project began with a p i lo t study during August 1970 in which I used a thermistor chain and chart recorder, borrowed from the Pac i f i c Oceanographic Group, to measure temperature continuously at eight depths from the F.R.B. barge at Port Alberni . The results con-firmed the bel ief that wind has an important influence on the thickness of the surface layer and also provided the background data necessary in designing instrumentation to measure the ef fect . Towards the end of the p i lo t study i t became clear that with the seasonal decline in surface temperatures thermistors could not provide a re l iab le means of estimating the thickness of the upper layer. Further observations required the preparation of special instruments to monitor conductivity; these instruments are described in Chapter 2. The main thrust of the experiment was timed to overlap the Hydrographic Service current and tide programme carried out during February and March 1971, though the field-work of my study extended into May. Chapter 3 includes a description of these observations i n addition to some of the relevant current data. In Chapters 4 and 5 I subject the data to time series analysis and draw on simple linear theory to explain some of the more important results. The thesis concludes with a summary of the work accomplished. Survey of Previous Work Alberni Inlet i s one of the more extensively studied estuaries on Canada's Pacific coast. In 1939 and 1941 the Pacific Oceanographic Group (1957) took a series of observations which formed the basis of an important study by Tully (1949) in which the nature of the circu-lation was described with a view to predicting the effects of pollution from a pulp m i l l . Tully's paper occupies a significant place in the development of our understanding of fjord circulation and other workers have used his results as input to theoretical models (e.g. Stommel, 1951 and Rattray, 1967). Since Tully's study, intermittent observations have continued (Pickard, 1963; Waldichuk, Meikle and Hyslop, 1968). Pickard (1963) reviewed some of these data with special reference to the exchange of deep water. More recently efforts have been made to monitor water quality in Alberni Harbour (Waldichuk, Markert and Meikle, 1969) and the inlet has also become the focus of an intensive biological programme. Severe damage at Port Alberni resulting from the tsunami of March 1964 prompted Murty and Boilard (1969) to develop a barotropic model of the inlet using numerical techniques. 4 It has been known for a long time that wind influences the surface structure of inlets. ' Sandstrom (1904) found that the d i s t r i -bution of low sa l in i ty water in the Gullmarfjord depended upon wind direct ion. Pettersson (1920) studied the same phenomenon using regression analysis on a long series of daily sa l in i t y measurements taken at Borno. In his analysis of Alberni Inlet Tully b r ie f l y describes some of the wind ef fects . He found that up- inlet winds tend to increase the depth of the upper zone and he includes a diagram to show that the surface layer increases in thickness from 3 to about 7 meters following three days of strong up- in let winds. He also attempted to relate the wind effect to other factors such as r iver discharge, but these results were not conclusive. In an intensive set of current measurements taken from an anchored ship in Knight Inlet, Pickard and Rodgers (1959) found some evidence of deep (40-100 m) compensating currents associated with wind-stress, in addition to surface effects . They also found that up- inlet winds could only reverse the surface flow for a l imited time suggesting the build-up of pressure gradients to oppose the wind action. Gade (1963) and Johannessen (1968) describe certain wind effects measured in the Oslofjord area of Norway. Both observers found surpr is -ingly long phase-lags of about 10 hours between the wind and the near-surface current. Gade (1970) discusses current measurements in the Vestfjord taken after a few hours of fresh up- in let winds which indicate a three-layer current system with inflow at the surface and at depth and with an outflow between. He also demonstrates that persistent winds set up strong horizontal sa l in i ty gradients along the axis of the f jord. 5 Rattray (1967) used measurements from East Sound, Orcas Island, during 10 hours of strong up-inlet winds to derive estimates of eddy viscosity. Assuming the equations of motion to be a balance between horizontal pressure gradient and the ver t i c a l stress gradient he found eddy viscosities ranging from 6 cm2/sec at the bottom to 90 cm2/sec at the surface. He also includes wind-stress in his similarity model of fjord circulation, and shows that an up-inlet stress displaces the down-inlet flow to greater depths. Practical engineering aspects have inspired a recent attempt to understand the phenomenon. Wada (1966) describes a study of wind effects from the point of view of designing cooling water intake structures. Henry and Murty (1971) are using numerical techniques i n their analysis of wind effects on Departure Bay, B r i t i s h Columbia. Most of these studies have concentrated on the steady state effects produced by the wind; yet i t appears l i k e l y that in fjords the st r a t i f i e d waters are in a state of almost continuous dynamic imbalance as the surface i s subjected to a rapidly changing wind-stress. It is a central aim of this thesis to explore the time-dependent structure of this response and to demonstrate i t s behaviour under the influence of a changing wind-field. PART I INSTRUMENTATION AND OBSERVATIONS Chapter 2 INSTRUMENTATION Any experiment designed to measure time dependent effects must necessarily take account of the more important variables involved. In this experiment I have assumed that the forcing functions dominating changes in the structure of the in let surface waters are the wind-stress, the tide and the r iver run-off . Long-term t ida l measurements are taken at Port Alberni ; however the Hydrographic Service greatly extended the col lect ion of t ida l data during the programme with the insta l la t ion of seven supplementary gauges along the Inlet, (Figure 1). MacMillan Bloedel Ltd. col lect r iver flow data from a gauge on the Somass River at the in le t head and they kindly passed on their measurements for the duration of the programme. The Hydrographic Service provided four Lambrecht anemometers (Plate III, 2), page 24. These are clockwork instruments that run for one month between chart replacements, providing a record of wind speed and direction from which hourly values may be taken. The response of the in let to wind, tide and run-off w i l l be in the form of fluctuations of the current, density structure and sea surface slope along the in le t . Recording current meters instal led as part of the Hydrographic Service programme supplemented by d r i f t pole measurements, provided a means of observing currents. Specially bu i l t conductivity prof i lers were used to estimate the density structure. In pr inciple at least , t ida l data from along the in let should provide a 8 Figure 1. Map of Alberni Inlet, showing location of instruments. 9 record of the sea-surface slope. These slopes are small , however, and could not re l iably be separated from the background noise in this study, Disposition of the Instruments Anemometers The four Lambrecht anemometers recorded wind speed and direction at prominent points along the in le t . Prior to the insta l la t ion of these instruments, data were available from an anemometer mounted on the roof of the F.R.B. barge "Ve le l la" , (Figure 1). In choosing the most representative points along an in let in which loca l topography greatly influences the winds, I re l ied on the advice of Mr. R. Page, Master of the"Caligus" who has considerable experience of wind conditions in the area. The instruments were mounted on 2 meter poles grouted into the rock in positions as exposed as possible. The intention was to obtain data that would be representative of wind-stress over the water. Obviously this was possible only in the loosest sense, since the rock i t s e l f must influence the measured wind-speed; nevertheless f i e l d experience did indicate that the locations were reasonably representative of wind conditions over the water. Current Meters Only the current records from Sproat Narrows were used in this study. These meters were located at 2m, 15m and 40m as shown in Figure 1. The necessity of avoiding the shipping route and the danger of anchoring buoys too near the submarine cables influenced the choice of mooring locations. Canadian bu i l t Neyrpic current meters were used; 10 these instruments record integrated 10 minute currents on punched paper tape. Some dr i f t -po le and drogue measurements were also taken. Conductivity Profilers These instruments were bu i l t special ly for this project and w i l l be described in a separate section. One was moored to a f loat ing dock adjoining Hohm Island and the remaining two were attached to log -booms located near the South shore just up- in let of Stamp and Sproat Narrows (Figure 1). During the last month of the experiment the Stamp Narrows instrument was moved to Spencer Creek. In addit ion, measurements were taken for a week with an instrument moored on each side of Sproat Narrows, to search for transverse isohaline slopes. The period of observation extended from February to May 1971 as shown in Table I. Table 1 Dates and Locations of Instruments Instrument Location Depth of Mooring Date In Date Out Conductivity Prof i le r 1 Hohm Island 16 m Feb 12 May 12 Conductivity Prof i le r 1 Sproat Narrows 20 m May 12 May 20 North Conductivity Prof i le r 2 Stamp Narrows 20 m Feb 12 Apr 7 Conductivity Prof i le r 2 Spencer Creek 36 m Apr 7 May 11 Conductivity Prof i le r 3 Sproat Narrows 29 m Feb 8 May 20 South Current Meters* (2m) Sproat Narrows Feb 9 May 10 Anemometers 1 Pocahontas Point Feb 24 May 11 Anemometers 2 Point 8 Feb 24 May 11 Anemometers 3 Hocking Point Feb 24 May 20 Anemometers 4 Lone Tree Point Feb 25 May 20 instruments instal led and maintained by Canadian Hydrographic Service. 11 Conductivity Prof i le r This instrument was bu i l t to monitor the conductivity prof i le at frequent intervals so as to obtain an almost continuous record of the surface layer thickness. In summer and during certain winter conditions the temperature distr ibut ion alone may be used to infer the main features of the density d ist r ibut ion. Such is not the case, however, in the autumn, late winter and spring, when the water column can be almost thermally homogeneous. In these conditions i t i s necessary to monitor the density prof i le by conductivity measurements. It i s preferable, of course, to measure both conductivity and temperature, from which the sa l in i ty and density prof i les may be determined. In typical oceanic conditions the effect of temperature on conductivity makes sa l in i ty estimates based on conductivity observations alone almost worthless. However, the surface waters in an in let with s ignif icant run-off such as Alberni Inlet, have an intense sa l in i ty s t ra t i f i ca t ion that completely dominates the conductivity p ro f i l e . In turn, i t i s the sa l in i ty distr ibut ion which dominates the density p ro f i l e . One may therefore infer the main features of the sa l in i t y and density distr ibut ion by measuring the conductivity prof i le alone, thus permitting the use of a much simpler instrument than would other-wise be required. The prof i le estimate can be improved, of course, by incorporating such background information on temperatures as may be avai lable. It should be emphasised that the instrument w i l l not provide real ly accurate data, in the sense to which we have become accustomed using c lass ica l oceanographic methods of observing sa l in i t y and 12 temperature. Rather i t was designed to measure the gross ef fects , such as a thickening of the surface layer by a factor of more than two in a few hours. The data obtained have confirmed the value of the approach for this problem. Principles of Operation The instrument measures the e lec t r i ca l conductivity of sea-water at fourteen depths by successively interrogating, for one minute at a time, each of the fourteen conductivity probes on the chain. There i s a f i f teenth probe inside the instrument housing which serves as a check on any d r i f t in the response of the electronics. Nasmyth (1970) has described the method used to measure conduc-t i v i t y . B r ie f l y , a 1.5kHz osc i l la tor supplies a signal to the primary c i rcu i t of a toroidal transformer. The magnitude of the signal induced in the secondary c i rcu i t depends upon the e lec t r i ca l conductivity of the sea-water loop which surrounds and passes through the center of the transformer. The toroidal probe casing is made entirely of plexiglass. Plate I shows how the parts are sealed together using ethylene dichloride in a hypodermic syringe, and also includes a photograph of the completed probe in posit ion. An e lec t r i ca l l y wound hobbyist's clock controls the timing. This inexpensive unit has an accuracy of better than 4 minutes per week. A photo-transistor in the clock-face and a small lamp just above i t are so mounted that once each minute the clock's second hand cuts the l ight path between the two, (Plate I, 3)- The photc-transistor f i res a Schmidt Trigger which operates the counting logic and switches into the c i rcu i t the appropriate probe (Figure 2). / PLATE I Conductivity Prof i le r (1) Sealing the probe casing with ethylene dichloride (2) Probe in position on cable (3) Layout of electronics (4) Output on chart recorder CLOCK SCHMIDT TRIGGER BINARY COUNTER AND BINARY TO DECIMAL CONVERTER OSCILLATOR AND PHASE SENSITIVE DETECTOR FOR MEASURING CONDUCTIVITY CHART RECORDER PROBE] (CALIBRATION NO.I5y PROBE) Figure 2. Block diagram of electronics for conductivity p ro f i le r . 16 Output from the conductivity measuring c i rcu i t i s fed to a Rustrak chart recorder. This type of record has to be laboriously d ig i t ised but was chosen in place of a magnetic tape for reasons of cost, s impl ic i ty , r e l i a b i l i t y and ava i lab i l i t y . The deepest probe i s interrogated f i r s t , then the next deepest and so on. In this way the depth corresponding to a part icular reading can be found from i t s posit ion relat ive to the rest of the group. A second pen tracing a l ine in one of two positions along the right hand margin of the chart paper, i s switched on with the eighth probe and thus serves as a check on the probe sequence. Plate I shows photographs of the electronics and also the chart recorder with typical output on i t . Instrument Housing and Protection Water-tight containers of P.V.C. tubing separately house the batteries and instrument. Four low speci f ic gravity 6 volt lead-acid batteries provide power. A converted f ibre-g lass marker buoy with a l i d at one end and an inner sheath of p last ic foam affords shock protection and f loatat ion for the battery and instrument cases (Figure 3). This arrangement permits f a i r l y easy servicing from a small vessel. I recharged the batteries every two weeks and replaced the chart-paper every week. When replacing the chart-paper alone, i t was possible to l i f t the instrument box out of the buoy and on to the ship's deck, since there was a suff ic ient length of cable to the battery case. Battery replacement required the use of the ship's winch to l i f t the battery container on board. At a l l times the buoy remained moored to the log-boom or f loat ing dock to which i t was attached. Figure 3. Construction diagram of conductivity p rof i le r buoy. 18 Calibration Each probe i s calibrated by immersing i t in a large container of salt water of known temperature, for which the conductivity may be accurately determined by other means. After noting the output, the c e l l i s removed from the salt solution and a conducting loop in series with a potentiometer i s passed through the probe. In this way i t i s possible to f ind a c e l l constant Kc in terms of the solution conduc-t i v i t y Sc and the 'equivalent resistance' Req of the potentiometer that just yields the same output as that obtained with the c e l l in solution: i . e . : Kc = Req • Sc The c e l l constant is independent of the solution conductivity or the electronics response, being determined solely by the geometry of the c e l l casing. In practice i t was determined from several such cal ibrat ion tests using solutions of different sa l in i t i es and an average value was taken. Having found the c e l l constant the actual output reading corresponding to different conductivities is found by simply adjusting the potentiometer through a range of values and recording the output. The reading on the chart recorder depends almost l inear ly on the con-ductivity being measured. A convenient functional approximation between conductivity and measured output may be obtained by a least squares f i t to the equation Conductivity = Kc {a + a x + a x2} o i 2 where x defines the recorded output and the coeff icients a. are 19 determined from the range of values produced by different potentiometer settings. Nasmyth quotes a short term relative sensitivity better than or equivalent to 0.005 0"^, and a shift in absolute accuracy over 3 weeks of 0.07 0" t . No such high level of accuracy was sought with the present instrument, although i t should presumably be attainable, since the conductivity measuring part i s basically similar, a calculations require a knowledge of temperature, and this was not obtained. More-over the Rustrak chart recorder limits the repeatability of a single measurement to about one part i n a hundred. It is estimated that the recorded output had a relative accuracy between probes of better than ± 1 millimho/cm and an absolute accuracy better than 2 millimhos/cm. The range measured was from a l i t t l e over 0 millimho/cm to about 35 millimoho/cm. Chapter 3 OBSERVATIONS A project of this type could scarcely have been undertaken without the active support of the Fisheries Research Board whose ship and barge f a c i l i t i e s in Alberni Inlet provided the essential log is t i c support. The observation period of February 12th to May 20th, 1971, involved eighteen f i e l d - t r i p s to the in le t . I used the F.R.B. barge " V e l e l l a " , which was docked in Port Alberni as a base; most of the field-work was done from the Fisheries Research vessel "Caligus" (Plate II, 2 and 4). Captained by Mr. R. Page this maneuverable vessel was ideal ly suited to in let work of this type. When the Caligus was unavailable I used the F.R.B. "Melibe" (Plate II, 1). The f i e l d - t r i p s each involved one f u l l day, and were mainly concerned with maintaining the conductivity p ro f i le rs . Batteries on these instruments had to be recharged every two weeks. To reduce the work load, the change-over was staggered so that no more than two sets of batteries were replaced on any one day. The rest of the day was taken up with changing the conductivity recorder charts, cleaning the conductivity probes, taking in s i tu sa l in i ty and temperature measure-ments and also replacing the anemometer charts as required. The conductivity prof i l ing instruments were completed just in time for the experiment and were therefore used without prior f i e l d testing. Their maintenance involved considerable effort and the PLATE II Vessels used in Alberni Inlet project (1) F.R.B. M/V "Melibe" (2) F.R.B. barge "Ve le l la" with M/V "Melibe" and M/V "Caligus" (3) C.S.S. "Parizeau" ( 4 ) M/V "Caligus" PLATE III F ie ld Work in Alberni Inlet Raising conductivity p rof i le r buoy Lambrecht Anemometer Lowering conductivity probes into water at Stamp Narrows 24 2 5 conductivity records include several gaps. Failure of the chart re-corders, failure of the switching lamps and leakage of water into the instrument housings were the main sources of trouble. Nevertheless, the instruments did yield useful data for about 60% of the time that they were in the water. The Canadian Hydrographic service undertook the current measure-ment and t i d a l programme. This work was principally carried out from the C.S.S. "Parizeau" (Plate II, 3). Wind and Current Wind Figure 4 shows data taken from the four anemometers. The wind-speed is shown as either up-inlet or down-inlet, since at the four locations chosen the transverse component was negligible. This section of data is typical and demonstrates two features that were consistent throughout the observation period. The winds have a strong diurnal component and they are also similar at each of the four stations, with the exception of the Pocahontas record, which is similar in shape but is biassed i n the 'Down-Inlet' direction. Calm waters are often seen off Uchucklesit Inlet 4 miles west of Pocahontas Point, while winds of 5 meters/sec occur up-inlet of this area. Currents I have only considered the current measurements taken at Sproat Narrows in this analysis. The data are in the form of components resolved along the major axis of the measured distribution of directions and were supplied by the Canadian Hydrographic Service. Figure 4. Sample record of wind measurements at four locations. & 2 7 Histograms showing the distribution of currents with respect to direction from each instrument indicate that there i s usually only a small trans-verse component. These and other details largely irrelevant to the present programme are described by the Canadian Hydrographic Service (1972). Figure 5 shows a typical section of Sproat Narrows current data at depths of 2, 15 and 40 meters. Only one each of the two 2m and 15m records are included, both from the North side of the channel, but they are representative of measurements on the South side, which they follow closely. Superimposed on the 15m current record i s a plot based on hourly values of t i d a l height measured at Port Alberni. The figure also includes wind observations and average daily river discharge figures for the Somass River. The 2m current data show a net down-inlet flow on which is superimposed a t i d a l fluctuation. The most striking feature of the record, however, is the strong up-inlet flow which appears to be strongly correlated with the wind. There is also a f a i r amount of high frequency energy in the current. At 15m the current is closely related to the t i d a l height which . i t precedes by about 2 hours, but there does not appear to be any obvious or consistent relationship with the wind. The current at 40m is remarkable in that i t flows down-inlet on the ebb, yet has l i t t l e or no component on the flood. There is no obvious explanation for this result and i t may be due to instrumental error. The Canadian Hydrographic Service also undertook a number of 28 10 Lone Tree Point Wind 0 (m/sec) 40 m UP INLET . . . . . M l . A , V r \/ v l LI 1 DOWN INLET 3275 4010 5160 RIVER FLOW (cu.ft/sec) 5160 Figure 5. Wind, current and mean dai ly r iver discharge. Measured t i d a l elevation at Port Alberni is superimposed on 15m current. The ver t i ca l grey l ine indicates time of d r i f t pole measurements shown in Figure 6. 29 SPROAT NARROWS Conductivity Profiler speed in knots DRIFT POLE MEASUREMENTS March 11,1971 Figure 6. Dr i f t pole and drogue measurements at Sproat Narrows. 1 knot = 51.48 cm/sec. (Courtesy Canadian Hydrographic Service) 30 drift-pole measurements in the Sproat Narrows area. The purpose of these was to gain some insight as to the extent to which the 2m current records were representative of the general surface flow. In addition some 11 and 20m drogue measurements were taken. Figure 6 shows a typical set of drift-pole data collected on March 11th. A v e r t i c a l grey line i n Figure 5 indicates the timing of the measurements. The records show that the current is approximately 10 to 20% greater in the middle of the inlet than at the sides. The upstream flow indicated by the 11m drogue seems consistent with the 15m current measurements shown in Figure 5. Conductivity Records Each chart record was digitised with a chart dig i t i s e r and the output punched onto paper-tape. The paper tape was transferred to magnetic tape on the University of British Columbia IBM 360 computer and converted back into conductivity using the relationship found during the calibration procedure. This procedure results in a three-dimensional array of points, the respective axes being depth, time and conductivity. Using an interpolation between the points one may search for contours of constant conductivity on a chart of the conductivity at each depth against time. The result i s a graphic description of the detailed changes occurring in the measured water column. Figure 7 shows one such plot taken near Hohm Island in February 1971. The contours clearly demonstrate the sudden thickening of the surface layer associated with strong up-inlet winds on February 14th and also the internal t i d a l o s c i l l a t i o n . The v e r t i c a l spread of contours Figure 7. Contours of constant conductivity taken at Hohm Island. Wind and current measurements for this data are shown in Figure 10. 32 is indicative of the different degree of mixing present. For example the s t ra t i f i ca t ion is more intense during that part of the internal t ida l osc i l la t ion (peaks on the diagram) during which the surface layer is thinnest than during the part at which i t i s thickest (troughs on the diagram). Parameterisation Contour plots of this type provide a very detailed description of the conductivity d ist r ibut ion, but for analysing large quantities of data i t i s more convenient to summarise the results at any given time with appropriate parameters. Two parameters seem relevant: a length scale associated with the thickness of the surface layer and a parameter indicating the extent of the mixing. I define the thickness length scale as that thickness of fresh water which would occur i f the measured water column were separated into two layers: a surface layer of fresh water and a lower one whose sa l in i t y was equal to that found at the greatest depth of measurement H. The lower layer of thickness H - H_ = H , i s understood to include a l l f s the sal t in the or ig inal water column. Thus Fresh Water Thickness = = H - 1 y o S(z) dz where H is the greatest depth of measurement and S(z) i s the sa l in i t y at depth z. The second parameter APE represents the difference between the potential energy per unit cross-section of the two layer system just defined, and the potential energy per unit cross-section of the observed d ist r ibut ion. The potential energy is calculated with respect to the 33 lowest probe. Let the or igin be at the depth of the lowest probe and the posit ive z axis point ve r t i ca l l y upwards; then the Potential Energy Difference is rH zP„ (z)dz - %g V APE = g 0 m where p i s the measured density distr ibut ion and p and p.. are the m^ J s f densities of the salt and fresh-water layers respectively. Determination of sa l in i t y and density requires a knowledge of temperature. As mentioned ear l ie r , only conductivity was monitored, since the sa l in i t y was so great that i t dominated both conductivity and density p ro f i les . It i s possible to estimate both and APE by using sa l in i ty and density prof i les calculated on the basis of a constant temperature d ist r ibut ion. Some temperature data are available however, and I have incorporated them in the following way. The data are in the form of weekly observations of temperature, conductivity and sa l in i t y I took with an in s i tu salinometer alongside each instrument and also r iver water temperatures taken every few days by McMillan Bloedel who operate the pulp m i l l at Port Alberni . Interpolating between the measurements two basic temperatures were estimated: that of unmixed r iver water entering the in let and that of sea-water at 9m. The temperature of the water at 9m rarely changed more than 0.1 °C in one week. River water temperatures varied 3° or 4°C in one week in extreme conditions. I calculated temperatures at depths less than 9 meters on the assumption that the measured conductivity indicated the extent of mixing between r iver water and the water at 9 meters, and that temperature differences were l inear ly related to the conductivity differences. This is a very rough approximation, but 34 probably as good as any other under the circumstances. The worst possible error in a estimates due to lack of temperature data would be less than 10%; typical ly the error would be much less than 5%. It i s worth noting that the effect of temperature on the conduc-t i v i t y of sea-water increases with increasing sa l in i t y , yet in the case considered here, the depth of highest sa l in i ty and hence the depth most sensitive to temperature errors, was also the most thermally stable. This condition tends to favour the type of approximation made above. On the other hand, heat exchange through the surface of the in let is completely neglected. The choice of parameterisation is necessarily rather arbitrary. In addition, the two estimates of and APE are unlikely to be inde-pendent of each other. For example when the surface layer becomes very thick the conductivity gradient associated with the halocl ine may influence the conductivity at depth H. This change in turn w i l l al ter the sa l in i t y estimate for the salt-water layer and thus the APE estimate as wel l . I used the weekly average of sa l in i t y estimates in computing the density of the lower layer, in an attempt to reduce this ef fect . The problem stems from the fact that 9 meters was not always an adequate depth for monitoring the type of effects with which we are concerned. Nevertheless, the two parameters do seem to reproduce the main features indicated by the contour plots . An advantageous consequence of the ver t i ca l integration involved in the parameterisation is that i t tends to average out the effect of re -la t ive inaccuracies between the probes in any one probe chain. The para-meter estimates were based on successive observations taken 15 minutes apart. Since the data were to be compared with hourly wind values, hourly averages of each parameter estimate were taken. Straight l ines have been 35 drawn between these hourly values in the plots presented in later sections. Fresh Water Thickness and Potential Energy Difference Figures 8 and 9 show Fresh Water Thickness and Potential Energy Difference data taken from the three main stations over a period of 33 days. The Hohm Island record represents the longest unbroken stretch of data obtained. The Fresh Water Thickness appears to undergo a series of sudden increases associated with strong up- in let winds, followed by a gradual return to equilibrium. The rapid f luctuations, especially at Hohm Island, are dominantly semi-diurnal. There i s l i t t l e obvious relationship between the diurnal wind component and the Fresh Water Thickness. The Potential Energy Difference variations also seem to be associated with the wind; the effect is usually greater at Stamp and Sproat Narrows than at Hohm Island. It i s not surprising that increases in the r iver discharge associated with heavy precipitat ion frequently occur at about the same time as extra strong winds. S igni f icant ly , however, even in the absence of these f luctuations, changes in H^ and APE occur with strong winds. For example Figure 8 shows that on March 2nd an increase in Fresh Water Thickness of 2%m occurred; there was a strong up- in let wind on that occasion but as shown in Figure 9 there was no change in r iver discharge. Similar ly , Figure 9 shows a noticeable increase in Mixing Energy on that date, especially at Stamp and Sproat Narrows. It seems l i ke l y that most of the observed changes are a consequence of wind-stress. To observe these processes in greater d e t a i l , consider the situation described by Figures 10 and 11. The following sequence of 36 Figure 8. Thickness of Fresh Water Layer (33 days). 37 Figure 9. Potential Energy Difference, Lone Tree Point wind Somass River discharge (33 days). 38 events occurred. After a period of comparative calm a strong up- in let wind arose on February 14th. Within 3 hours the current records at 2m in Sproat Narrows responded with an up- inlet current of almost 1 knot. The Fresh Water Thickness increased suddenly, f i r s t at Hohm Island, then at Stamp Narrows and somewhat later Sproat Narrows. Fluctuations in thickness at Hohm Island on February 15th, which may have been associated with wind-stress changes at that time, do not markedly appear at Stamp Narrows and not at a l l at Sproat Narrows. Moving down-inlet from the Hohm Island to the Sproat Narrows stat ion, we see these fluctuations become progressively gentler. More-over the greatest change of thickness between the 14th and 15th of February occurs at Hohm Island; but a l l three stations show similar changes during the slow return to equilibrium. Figure 10 demonstrates another important feature common to most of the conductivity data. The internal semi-diurnal osc i l la t ion shows up most strongly at Hohm Island. It i s s t i l l evident at Stamp Narrows, but only with about one half the Hohm Island amplitude. At Sproat Narrows i t occasionally shows up, but at this station these fluctuations are mainly lost in the background noise. During the p i lo t study in August 1970, temperature measurements taken at the f loat plane dock on the Eastern edge of Port Alberni harbour, across from Hohm Island, indicated essential ly the same resul ts : an internal semi-diurnal osc i l la t ion of surface thickness in phase with the surface t ide. Tully (1949) also observed an internal osc i l la t ion in the harbour. It seems l ike l y that this is an internal tide generated over the gently sloping bottom contours at the northern end of the harbour. Rattray (1959) has shown how similar topographies can cause 39 Figure 10. Fresh Water Thickness, 2m Current and Wind (5 days). 40 r UJ o a: at S- in CD at •4 HOHM ISLAND 2 STAMP o: i " NARROWS UJ *o 2 SPROAT t ; NARROWS4 UJ WINO (Km/hr) UP INLET 40 20-0 20L 13 f] A 1 4 15 J V V \ 4 A r DOWN INLET i2.ooor RIVER F L Q W 8.000H (cu.ft7s) 17 FEBRUARY 1971 Figure 11. Potential Energy Difference, Wind Speed and Somass River discharge (5 days). The discharge curve is taken di rect ly from the trace produced by the recording r iver flow guage. 41 internal tides on an open coast. Outside of the generating zone the disturbance can be expected to have the properties of a progressive internal gravity wave. The lower amplitude observed at Stamp Narrows presumably indicates the presence of f r i c t i o n a l damping. Turning now to Figure 11, we see that there is a strong increase in APE when the wind picks up. Figure 11 also shows the r iver flow which i s taken direct ly from the curve traced on the or ig inal discharge record. It i s interesting to note that the flow increases and reaches i t s peak before the wind star ts , but no large change in or APE occurs at this time. These results tend to support the hypothesis that most of the changes in these two parameters depend upon wind rather than discharge. On the other hand there does appear to be a sl ight decrease in APE at Stamp and Sproat during the f i r s t twelve hours of February 14th, before the wind begins. The Potential Energy Difference also shows a marked var iat ion of semi-diurnal frequency. Tully (1949) observed a similar ef fect . On May 12th I moved the conductivity p rof i le r at Hohm Island down to the North side of Sproat Narrows (Figure 1). The purpose of this change was to search for transverse gradients in the density p ro f i l e , such as might result from i n e r t i a l or rotation ef fects . Figure 12 shows the corresponding Fresh Water Thickness records; apart from a curious discrepancy on May 17th the data i s very s imi lar . 42 2 _ 3 CO cc L U oo 6 cc u_ uu 4 cc U J SPROAT NORTH SPROAT SOUTH 7 12 13 14 15 16 17 18 19 20 MAY I97I Figure 12. Comparison of Fresh Water Thickness on North and South sides of Sproat Narrows. Between Apr i l 7 and May 11 the Stamp Narrows conductivity pro-f i l e r was stationed at Spencer Creek (Figure 1). This move was made to observe wind effects in the lower part of the i n l e t . The records from this station are not good; the sa l in i ty gradient is rather less than in the upper reaches of the in let and the noise leve l is much higher. During this period of observation winds of up to 14m/sec were recorded, but i t i s clear from the data, which are not presented here, that major variations in the Fresh Water Thickness are not present. On the other hand the data show quite large changes in the Potential Energy Difference associated with strong winds. PART II ANALYSIS AND DISCUSSION 44 The combined effects of wind, tide and run-off produce a dynamic condition of extraordinary complexity. Nevertheless certain important features do stand out. A str ik ing relationship exists between surface current and wind. And from the surface layer thickness data we see that strong up- in let winds produce a thickening which i s largest at the in let head, a distort ion that takes at least two or three days to disappear. These are phenomena that might profitably be considered separable from the other processes of in let c i rcu lat ion . By this I do not mean that they are unconnected. The mechanisms of turbulent entrainment and mixing associated with estuarine c i rculat ion are highly non-l inear; they undoubtedly influence and are influenced by the wind ef fects . But i f we consider only the dominant features of the process, we may yet learn something of the physics involved while retaining a description that is analyt ical ly tractable. Fourier Analysis of Time Series Data When analysing long time series i t i s often useful to consider their frequency representation. In practice this i s achieved by using such numerical techniques as the 'Fast Fourier Transform' to obtain c o e f f i -cients of the F in i te Fourier Series of the data. For the process X(t) sampled at 2N points spaced At apart where X(t) represents the deviation from the mean, the following expression defines the coeff ic ients : N X(t) = E k=l a^os (k2TrAf-t) + bkSin(k2-rrAf-t) where Af is the frequency band-width and i s equal to 2NAt ^ = ^" 45 The second order s t a t i s t i c X(t ) :X(t ) , where the overbar denotes an ensemble average, is cal led the variance. P(f)Af, the power spectral density, i s i t s frequency space representation; i t defines the contribu-tion to the variance of the observed fluctuations within the band Af. When taking the ensemble average of the product ser ies , cross terms of the form Sin(mt)•Cos(nt) m n^ disappear, as also do terms of the form Cos(mt)•Cos(nt) and Sin(mt)•Sin(nt). We are le f t with the expression P(f)Af = + b2] . For two different processes X(t) and X ' ( t ) , we define the co-variance as X(t)*X'(t) and in the same way find the Co-spectrum: Co(f)Af = J5[aka£ + b ^ ] , Normalisation of the co-spectrum with respect to the spectra of each variable leads to the quantity Co'CP'P') 2 which represents that f ract ion of the observed fluctuations in each signal in phase with the other signal over the given frequency band. Shift ing one signal forward by 90° at each frequency and taking the averaged product, we can define the Quadrature Spectrum: Q(f)Af = ^ [ a ^ - a £ b j . Co-spectra and Quadrature spectra are extensions of the concept of correlation to a frequency representation. For instance, two signals may be perfectly related in a band but have zero co-spectrum i f the fluctuations at this frequency are 90° out of phase. The normalised quadrature spectrum in this case w i l l be equal to one. An alternative representation for two spectra is provided by 46 their phase and their coherence squared. The phase spectrum i s <f>(f) = Arc tan and the Coherence Squared -Q(f) lCo(f) C 2 m _ (Co(f) 2 + (Q(f)2)  C C U P ( f ) .p ' ( f ) " In any given band the Coherence-Squared represents that f ract ion of the variance in one signal which is related to the other signal regardless of phase. F inal ly we define the Amplitude Transfer Function a This function corresponds to the gain of a l inear system which relates the two processes a and b. The spectra defined thus far are exact frequency representations of the data from which they are derived. This does not mean that they sat is factor i l y represent the process from which the data points were sampled. Averaging enables us to obtain smoother spectral estimates. In practice we may average spectral values over adjacent frequency bands, or average values from each band with the corresponding values derived from other data gathered under the same conditions, or we may use both procedures. An ar t i fact of the sampling is that the use of a f i n i t e length of data causes leakage into adjacent frequency bands from any given spectral band. A problem also occurs i f there is any signif icant energy in the sampled signal at frequencies greater than > t n e highest for which spectral estimates can be made. In this case the high frequency 47 energy may appear within the frequency range of analysis, a result known as a l ias ing . In this thesis I have plotted the product of frequency and spectrum against the logarithm of frequency. This has the advantage of allowing several decades of frequency to be shown while preserving area under the curve (Since Pdf = f -Pd(Znf)). Thus a comparison of the areas under the curve for a given band width at different frequencies provides a comparison of the contribution to the variance in each case. When plott ing spectra in this way i t i s common to average over more coeff ic ients at higher frequencies than at the lower end of the scale. In this case we obtain smoother results at higher frequencies: caution should be exercised in interpreting the low frequency estimates. Chapter 4 WIND AND CURRENT Wind and Current Spectra Three months of almost unbroken records exist for current-meters at 2m depths on both sides of Sproat Narrows. Hourly wind data also exist throughout this period. Figure 13 shows spectra computed for hourly averaged current records taken on both sides of the inlet at Sproat Narrows, and also for the quantity u|u| derived from the records of the wind-speed U at Lone Tree Point. If a square law holds for the relationship between wind-speed and stress, then U|u| should be proportional to the stress over the water. A prominent peak centered on the diurnal frequency dominates the wind-stress spectrum. There is also some energy in the bands of periodicity 4 and 8 days. The two current spectra are very similar to each other with the instrument on the South side of Sproat Narrows recording slightly more energy in nearly a l l bands. Again there i s a strong diurnal signal with a smaller peak at four days; the spectrum f a l l s off rapidly at lower frequencies. With current records derived from instruments on each side of the i n l e t , a simple test provides a means of determining the way in which current fluctuations occurring on one side are related to those on the other side. Figure 14 shows coherence and phase found.from spectral calculations of each record. Apart from a curious exception at a periodicity of 346 hours at which the North Sproat record leads the South by 24°, both data sets are essentially in phase and are highly 49 Figure 13. Spectra of 2m current at Sproat Narrows and u|u|, assumed proportional to Wind-Stress, at Lone Tree Point. 50 10 a s 0-6 0 4 0-2 o o " o COHERENCE SQ. i i i i i i_ 000 111 I I I I I I 11 I I I I I I 100 10 HOURS CO CD < t- 2 z> o CO CO o o z 180 CURRENTS (2 M)= SPROAT NORTH a SOUTH 90 - -o o ° ° O O U v \j \j O O O o -90 PHASE --180 1 i i i 000 100 10 HOURS Figure 14. Coherence Squared and Phase between 2m current on North and South side of Sproat Narrows. 51 coherent for a l l fluctuations greater than about 6 hours. Loss of coherence at higher frequencies is presumably a consequence of eddies in the stream flow being s igni f icant ly narrower than the distance between the two current-meters. Wind Effects and the Diurnal Tide The strong energy peaks in the wind and current spectra at the diurnal frequency suggest close coupling between the two. Yet 24 hours is also the period of the diurnal t ide. Before relat ing the wind and the current we must f i r s t estimate the re lat ive importance of the t i d a l contribution. The Canadian Hydrographic Service has derived t ida l con-stituents from a one year record at Port Alberni . Values from Tofino on the coast are also available as an average of 17 years of annual estimates. Table II shows constituents from both locations for the more important diurnal and semi-diurnal t ides, together with certain rat ios . In both places tides are of the mixed, semidiurnal dominant type. The amplitudes are very s imi lar ; the constituent is one tenth of a foot greater at Tofino than at Port Alberni . Clearly , the in let has no resonant surface mode response at these frequencies. This feature is consistent with Murty and Boi lard's (1969) numerical pre-dict ion that seiche action only occurs at periods less than 2% hours. If we can assume that the rat io between two different t ida l height constituents multiplied by the rat io of their respective f r e -quencies is similar to that between the corresponding t ida l currents, then i t i s possible to make some estimate of the re lat ive contributions of wind and tide to energy in different bands of the current spectrum. 52 Table II Tidal Amplitudes and Ratios at Port Alberni and Tofino Tide Period Tofino Amp Pt. Alberni Amp Pt. Alberni (Amp)2 «. 26.87 hrs .144 f t .140 f t .02 f t 2 0 l 25.82 .803 .809 .66 P l K l 24.07 23.93 .398 1.275 1.673 .400' 1.300 1.700 2.89 N 2 12.91 .658 .665 .44 M 2 12.42 3.248 3.128 9.79 S 2 K 12.00 11.97 .927' .253 1.180 .876 .247 1.123 1.26 Ratio of Tidal Energy at Port Alberni Tidal Ratio (Amp)2 (P + K )/0 4.41 l i i (P + K )/M .29 1 1 2 53 This assumption is l i ke l y to be good provided that the frequencies are close to each other, and yet distant from any natural frequencies in the in le t . We have seen that no surface mode resonance occurs at these frequencies; there remains the poss ib i l i t y of natural internal o s c i l l a -t ions. If these do occur, however, they w i l l be at much lower f r e -quencies than the diurnal t ide; they w i l l also have broad energy bands ref lect ing the changing nature of the density structure. Separation of the closely spaced t ida l constituents requires a narrow bandwidth analysis. With the data available i t i s not possible to separate the Pi and Ki or the S2 and K 2 t ides. But we can separate the Oi from the (Pi + Ki) band and the M2 from the (S2 + K 2) band. Figure 15 shows the relevant parts of the current spectrum, together with a schematic representation of the relat ive energy of the t ida l constituents shown in Table 2. The t i d a l height energy associated with each tide has been assumed to be proportional to the square of the respective constituent. In the case of two constituents f a l l i n g within the same frequency band I have taken the square of the sum. This value represents the energy that would appear i f both components were in phase and should therefore be regarded as an upper l im i t . Since the t ida l current is proportional to the derivative of the t ida l height with respect to time, we must multiply the t ida l const i tu-ents by the frequency f (and thus multiply the squared constituents by f 2 ) . The current spectrum in Figure 15 is a plot of f<j>(f) against log f using a constant band-width Af. We therefore multiply the squared constituents again by the frequency in order to make the ratios compar-able. The comparison w i l l thus be between ratios of ( t idal amplitude) 2•f 3 54 Figure 15. Spectrum of 2m current at Sproat Narrows (South), and Tidal Constituents squared. The t ida l constituents represent the amplitude of the t ida l height derived from one year's data at Port Alberni . 55 with rat ios of fc(>(f). A str ik ing feature of the spectrum in Figure 15 is the d i s -proportionate amount of current energy at the (Pi + Ki) band. If the spectral peak were representative of the t i d a l energy alone we would expect i t to be at most 4.4 times the signal at the Oi frequency. Yet the spectrum gives a value 45 times the Oi s ignal . Indeed, neither the Oi nor the Qi bands show up above the background noise. Evidently the (Pi + Ki) band has at least ten times as much energy as the tides alone can contribute. Now consider the semi-diurnal frequencies. Background noise obscures the N2 t ide, but the ( S 2 + K 2 ) band shows a small peak and the M2 band dominates this part of the spectrum. The spectral values of the current show about eighty times as much energy at the diurnal peak as we would expect on the basis of the M2 and (Pi + Ki) rat io of squared t ida l constituents. These comparisons clearly show that tides alone cannot account for the strong 24 hour component in the currents. We must attribute the difference to another diurnal factor; this is almost certainly the wind-stress. The Wind Driven Current The mountainous terrain bordering Alberni Inlet confines the wind to the i n l e t ' s longitudinal axis. The topography in turn constrains the surface waters to move in the same direction as the wind-stress. Since the in let head i s a so l id boundary a wind 'set -up ' or 'set-down' w i l l soon occur. Thus a pressure gradient develops in the water along the in let in opposition to the wind-stress. Fluctuations in the wind-stress w i l l cause fluctuations in the wind set-up which produces changes 56 in the pressure gradient; these changes propagate back down the channel as surface gravity waves. Barocl inic effects are also present, of course; their discussion follows in a later section. But to a depth of 2 meters at least , we expect current fluctuations to be governed mainly by changes in the wind-stress and barotropic pressure gradients. In this surface layer turbulent processes w i l l diffuse momentum imparted v ia the surface down through the water column. The momentum transfer may not be instantaneous; in this event a f i n i t e , measurable time w i l l elapse between a change in wind-stress at the surface and the corresponding change in velocity at 2 meters. There i s also a time interval associated with the propagation of pressure gradient changes from the in let head to the point of measurement which depends upon the surface gravity wave speed. For Sproat Narrows this w i l l be about 10 minutes. When the wind-stress fluctuations have a period long compared to this transit time, yet not too close to any natural frequencies of the basin, we expect wind-stress and pressure gradient to be nearly 180° out of phase. Figure 16 shows phase and coherence plots for the wind-stress and 2 meter current at Sproat South. The coherence drops off rather quickly for periods shorter than about 20 hours. Apart from the highest frequency band analysed, the phase angle changes almost logarithmically throughout the range. At high frequencies the wind precedes the current, but for osc i l lat ions of about 60 hours both wind and current appear to be in phase. At s t i l l lower frequencies the wind actually lags the current. This last f inding is at variance with the arguments made above 57 10 08 C-€ 0 4 0-2 COHERENCE SQ. i i_ I I I I I I I I I 1 I I I | I I 1 L_ 1000 I80r CO o £ 90 z ^ CO o < z £ o -90 -180 100 10 HOURS WIND STRESS AND CURRENT (2M) o o o o PHASE o O o o • • i > I I I • I • • I I I I I I I I I I I I L 1000 100 10 HOURS Figure 16. Coherence Squared and Phase between 2m current at Sproat Narrows and Wind-Stress derived from Lone Tree Point wind records. 58 concerning the relat ion between changes in wind-stress and current. One possible mechanism that could explain this result i s the genera-tion of long surface gravity waves by meteorological disturbances off the coast. Such waves could travel faster than the disturbance i t s e l f , thus inducing currents in advance of the wind-stress. Variations in r iver discharge w i l l also influence the spectra and may be partly responsible for the positive phase at low frequencies. For higher frequencies we may translate the discussion of ve r t i ca l momentum transfer into a simple model for deriving dif fusion coeff ic ients from observed phase angles. Eddy v iscosity i s a poor representation of turbulent processes, but in the absence of more appropriate data i t i s also the only one avai lable. If the pressure gradient changes are independent of depth and 180° out of phase with the wind-stress, they w i l l change the amplitude of the wind-driven current, but not i t s phase angle. We may therefore solve the diffusion equation for U , that part of the velocity induced direct ly by wind-stress, but excluding the pressure gradient ef fect , in order to find the phase angle between wind and current. For an average kinematic eddy v iscosity V £ , the l inearised equation of motion is 32U 3U v o e 3z 2 3t A solution decreasing with depth for the periodic surface condition U = A e i W t , i s w ' U = Ae w -z{w/2ve}^ . i(wt-z{co/2ve} ls) 59 This yields a phase angle between wind and current of <}> = z io/2v£J^ In part icular , V £ = 0)z2/2cJ>2 (4.1) Table III shows values of V £ found from measured phase angles using equation (4.1) for per iodic i t ies of 43 hours and less , together with the amplitude of the wind component. At lower frequencies the phase relat ion cannot be used in this way, for as we have seen above, we may no longer consider the current fluctuations to be purely wind driven. A further complication occurs in the highest frequency band analysed. Here also the current appears to precede the wind and I have excluded i t from the v iscosity calculat ions. This spurious result may be due to the presence of a surface seiche; Murty and Boilard (1969) predict a f i r s t mode natural frequency for the Alberni In let— Trevor Channel system just 2% lower than the center of this band, at 2.096 hours. Apart from this exception, values for per iod ic i t ies less than 2 — 1 24 hours f a l l remarkably close to 2 cm sec . The apparent eddy v iscosity is s l ight ly greater for 24 hours, a result that may be due to higher wind speed, and thus more turbulence, i n this band. Such arguments cannot hold however for the 32 and 43 hour bands; perhaps these values represent a balance between r e a l i s t i c calculations and contamination of the data with current f luctuations that are not wind driven, of the type discussed ear l ie r . In any event, the results indicate that a kinematic eddy coeff ic ient in the range 1 to 10 cm sec i s certainly consistent with the observations. 60 Table III Coefficient of Kinematic Eddy Viscosity, V £ Period Wind Speed Amplitude V e 43.0 Hrs 1.60 m/sec 9.3 32.4 2.65 13.9 24.3 3.79 5.0 18.3 2.06 2.3 13.7 1.91 1.9 10.23 1.61 2.4 7.69 1.45 1.3 5.78 1.25 1.9 4.33 1.11 1.9 3.24 .95 2.5 2.43 .84 1.1 Chapter 5 SURFACE LAYER THICKNESS Certain features of the surface thickness are of special im-portance in trying to understand the response of the s t ra t i f i ca t ion to wind-stress. The f i r s t result i s that strong up- in let winds cause a sudden thickening of the surface layer at the in le t head. Secondly, this distort ion takes at least two or three days to disappear. The effect also occurs away from the head, but here the disturbance is attenuated and appears to arrive s l ight ly later . Figure 17 shows spectra of the wind-stress and the Hohm Island surface layer thickness data; both data sets were for the total period shown in Figure 8. Coherence and phase plots in Figure 18 show a f a i r l y high coherence for lower than semi-diurnal frequencies and a phase angle that tends to increase with frequency. The spectra in part icular demonstrate another remarkable feature of the surface thickness data: despite the strong diurnal wind s ignal , most of the surface thickness energy is at lower than diurnal frequencies. There is a strong peak at 12 hours, but this is the internal ^ tide and unassociated with the wind. If we are to seek a cause and effect relationship between wind and surface layer thickness, we must assume either that the coupling is more ef f ic ient at lower frequencies, or alternatively that there is a strongly non-linear process that transfers energy down the frequency scale. The observations suggest a wind effect very similar to the response of a heavily damped mechanical system. If strong f r i c t iona l 40i r HOURS Figure 17. Spectra of wind-stress derived from Lone Tree Point wind records and surface layer thickness at Hohm Island for the 33 day period shown in Figure 8. 63 x < t- —i Is 2 I 10 o o o ° o ° o 08 o o o o o 0-6 o o 0 4 COHERENCE SQ. o o 0-2 o _ o o 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i i 1 i i i t 000 100 10 HOURS 180 -WIND STRESS AND SURFACE LAYER THICKNESS 90 -o ° ° ° ° o o o ° o o PHASE -90 — -180 • i • i i i i i 11 i i i i i i i 11 1 1 1 1 1 1 1 _ 1000 100 10 HOURS Figure 18. Coherence Squared and Phase for wind-stress (Lone Tree Point) and Fresh Water Thickness (Hohm Island). r • 64 damping i s present, internal effects originating near the mouth of the inlet w i l l have almost disappeared by the time they approach the head. In the upper reaches of the inlet at least, we expect the response to be an end effect, dominated by the character of the wind and by the consequences of a solid boundary. A simple two-layer model offers an opportunity for testing this hypothesis. It i s a remarkable fact that the steady state fjord circulation tends to maintain a f a i r l y uniform surface layer thickness from the head to the mouth. This condition favours the two-layer representation. On the other hand the density difference between the layers diminishes progressively from the head to the mouth of the i n l e t ; for this reason we expect the model to be less r e a l i s t i c further down the inl e t . The model w i l l exclude rotation effects; as indicated earlier the instrumentation could not detect them. A more drastic s i m p l i f i -cation is the neglect of the steady state circulation. Yet i t i s the time-dependent response to changing wind-stress that i s at issue; the success of the model w i l l indicate the extent to which we may consider this effect separable from the other processes occurring i n the inl e t . A Linear Model Let the inlet be approximated by a semi-infinite canal of uniform depth and width consisting of two homogeneous layers of slightly different density (Figure 19). We shall formulate the hydrodynamic equations in terms of the integrated currents in each layer and solve the 'normal mode' version (Rattray, 1964) of these equations to obtain an expression for the motion of the interface. The following notation w i l l apply: 65 WiNO-STRESS t h' 1 T U- V t h" x = d x=0 Figure 19. Two layer model of i n l e t . h' surface layer thickness h" lower iayer thickness h = h* + h" u' upper layer velocity u" lower layer velocity U' upper layer transport U" lower layer transport H1 surface displacement n" interface displacement T ' horizontal component of kinematic stress in surface layer T " horizontal component of kinematic stress in lower layer Ap/p" relative density difference The linearised equations of motion and continuity are then (Rattray, 1964) 3u' 3t = "g 3n • + 3r' 3x 3z 66 3u 3 H!l - /i Ap.8n' Ap 3n" , 9T" t " " 8 U ~ o " ; 9 x V 5 " 3x 3z 3(h'u') ^ 3n" 3n' 3x 3t ~ 3t 3(h"u") = 3rT , ' 3x "3t Integrating v e r t i c a l l y , the transports in each layer are: rh' U' = U" = u'dz o CO u"dz -4i" and the integrated equations become = -gh'37-+ ( T 1 - T 2 ) 3t 3U" , A p . S n ' . „Ap 3n" j_ , N 3t" * h ^ 3 x " - § h -fi 3x~ + ( T 2 " T 3 ) 3uT 3n" 3n' 3x 3t 3t 3U" = 3jT 3x ~ 3t (5.1) where T i , Tz and T3 represent the kinematic stress at the free surface, interface and bottom respectively. Rattray (1964) has devised a convenient transformation of these equations which allows the separation of surface and internal modes into an equation applicable for either type. This i s accomplished by the following change of variable, to the lowest order in -j^- : 67 lh = U1 + U" u 2 = £ - u» _ £_ u» h h Ti = T 1 - T 3 T 2 = ( T 1 - T 2 ) - (T 2 -Tj) • Substitution of these variables into the integrated equations y ie lds : 3 2U. 3 2U. 3T. ^ = c 2 — ^ + (5.2) 3t 2 1 3x 2 3t The surface mode, indicated by subscript " 1 " , has the gravity wave velocity cj = /gh ; subscript "2" denotes the internal mode with / Ap h'h" The model w i l l include internal f r i c t i o n under the assumption that the stress at the interface depends upon shear; in part icular , on the difference between the mean velocity in each layer. Allowance for f r i c t i o n effects along the bottom and sides of the in let can be made by including in the surface and bottom stress expressions, terms proportional to the transport in each layer. Let p and q be the external and internal f r i c t ion coeff ic ients . The kinematic stress terms then become Ti = W-pU' T 2 = q (pU ' -^ ,U" ) T3 = pU" 68 where W i s the kinematic wind-stress and p and q are of dimensions [ T _ 1 J and [ L T - 1 ] respectively. Substitution of these into the Normal Mode stress terms gives: Ti = W - pUi T z = IT W " ^ h ^ h 7 7 ^ 2 = -2- W - KU2 where K i s a f r i c t i o n a l coefficient for the Internal Mode of motion and has dimension [ T " " 1 ] • h" Dropping the subscript "2" and letting T W = — W, the internal mode equation (5.2) f i n a l l y becomes: i f E = c 2 l ! u + 3_ [ T , _ K U ] . ( 5 . 3 ) 3t 2 3x 2 3t w Having solved for U, the term (n"-n') = H representing departure of the surface layer thickness from equilibrium, can be recovered from equations (5.1): 3U2 = cS£ (n , , -n') + T 2 3 3t 3x or (n,,-n*) = n = — c 2 fx 3U2 , |-x g — dx - — T 2dx (5.4) 0 C 2 ''o Solving the Equation Equation (5.3) has the form of a mixed Heat and Wave equation. Fi r s t consider free wave solutions of the form U = e ^ ^ x + a ) t ^ where (0 is complex and k i s real corresponding to waves whose wave number is fixed. The dispersion relation i s 69 co2 + iKto - c 2 k 2 = 0 with the roots co = ~ i | ± J s / - K 2 + 4 c 2 k 2 • Wave solutions are possible provided 4 c 2 k 2 > K 2 , that i s , for wavelengths X such that X < 4TTC K (5.5) K2 The phase velocity i s c J l - ^ 2 ^ 2 • I t s value increases to a maximum of c with increasing wave number: the waves are anomalously dispersive. We may also solve the dispersion relat ion for complex k. The coeff ic ient defining exponential decay with distance is then the imaginary part of the root k = --/co2+iKco c This complex root can be rewritten as k = ^Kr^CosHje+ir^Sin^e} where the modulus r = /co't+a)2K2 and the argument 0 = Tan 1 decay of waves with distance is therefore of the form . The exp { - ^ w ' W K ^ S i n t e T a n 1 K )> (5.6) For forced wave solutions consider a uniform wind-stress extending from the in let head, x = 0, to a point x = d further down the in le t . In this way the stress w i l l approximate winds in Alberni Inlet that typical ly occur between Spencer Creek and the in let head. At d we 70 match the transports in each layer and the surface layer thickness. Finally U i s set equal to zero at the inlet head and for x = °°. The boundary and i n i t i a l conditions are therefore as follows: U = 0 at x = 0 U = 0 at x = 0 0 U i s piecewise continuous through x = d n i s piecewise continuous through x = d U = 0 at t = 0 T = T (t) x < d W W T = 0 x > d « w For the inner region over which the wind acts, the Laplace Transform of equation (5.3) is ^ - Y 2U = ¥ (5.7) j 2 2 W dx c 1 .—5 where y = — /s + Ks and the Laplace Transform of P(x,t), indicated by an overbar, is defined by P(x,s) = e S tP(x,t)dt 0 A solution to (5.7) is U - C l e ^ + C 2 e - ^ + s - ^ • T w Applying the boundary condition at x = 0 U r-Cge^- T e Y X+C 2e" Y X+-^ T . s+K w s+K w 71 Similar ly , the solution in the outer region, after applying the condition at x + » , i s U = D2e At d we match the transports: and the layer thickness, using (5.4): _ C 2 e Y d _ _ L T J D - C 2 e ^ d = - D 2 e ^ d s+K w Solving for the constants C 2 and D2 gives y ielding the solutions x > d . Restrict ing attention now to the inner region, equation (5.4) enables us to write the solution in terms of the surface layer thickness ri" = — T" { V ^ V ^ * * ) - Jte-YW-x) } x < d t c 2 w y 2 Y 2y F ina l l y , inverting the Laplace Transform (Abramowitz and Stegun, 1965, p. 1027) and applying the convolution theorem: n(x,t) = ± T ( t -T )e w KT 2 - hi. - hi. K I 2 (x+d) : K , (d-x)' 2 J T " c* H H "T_r*+±] c H T — dx 72 x < d (5.8) where I i s the zero order modified Bessel Function and H is the 0 Heaviside unit step function: H(t) =0 t < 0 = 1 t > 0 . The case of zero f r i c t ion (K=0) is instruct ive. Let the wind-stress be of the form T (t) = T 0 < t < t i , t i < -w c T (t) = 0 t > t i and t < 0 (5.9) w where T i s a constant. Integrating (5.8) for x = 0 gives: Tl(0,t) = iT{t - (t - t 1 )H(t -t i ) - (t4 )H(t^^-)+(t -f - t 1 )H(t -4- tj)}. The f i r s t term ref lects a constant rate of decrease in surface layer thickness for a down-inlet wind, or increase for an up- in let wind. This effect persists unt i l the wind stops at t = t i when the second term appears and the thickness remains constant. The return to equilibrium starts when the wave emanating from point d, referred to as the End Wave, reaches the in let head, and finishes with the fourth term representing the termination of the End Wave. 73 Figure 20 shows solutions numerically evaluated for different K = %Kti, for the case = 2 t i , and indicates the way in which f r i c t i o n influences the solution. The maximum displacement is reduced, the return to equilibrium begins as soon as the wind ceases, but is not complete with the termination of the End Wave and thereafter approaches zero only gradually with increasing time. As the time taken for the End Wave to reach the in let head increases i t s contribution diminishes and for suf f ic ient ly large values of ~ i t can be neglected altogether. In this case we may drop the second and third terms of (5.8); integration for a wind-stress of form (5.9) then yields the solution: n (0,t) = | r " 2 * t -e K K i I o ( f t )+ l ! ( f t ) - t 'e 2 H(t') (5.10) where t ' = t - t i Using equation (5.10) and tables of Bessel Functions we may now evaluate the displacement at time t = t i . Figure 21 shows how f r i c t i o n influences the solution. For spectral comparisons i t i s also useful to have a solution in frequency space. Consider the same system as before described by equation (5.4) and the previously used boundary conditions, but with a periodic wind-stress: T = A e i W t x < d T = 0 x > d , Separation of the x-dependence leads to the equation X" - y 2 X = - i toAc - 2 74 a of ™odel to uniform wind, showing effect Figure 20. Response of model to ui of f r i c t i o n and end wave. 75 Figure 21. Normalised f r i c t ion plotted against normalised displacement of the interface depth with respect to the surface following a step impulse wind of duration t i . The quantity c is the internal mode phase speed; T is the kinematic wind-stress and K is the internal mode f r i c t i o n . 76 where y = -^/itoK-u)2. The general solutions for the inner and outer regions respectively, are „ i w t r „ yx, _ -yx.iwk •> U = e {Cie' +C2e +r?v7r} c y < d iwtr yx, „ -yx-, U = e {C3e' +Ci»e ' } x > d . Following the same procedure as before we apply the boundary conditions at x = 0 and x = 0 0 and match U and Cn"—n *) at d. Restr ict ing attention to the region x ^ d we obtain: U(x,t) = e iut ( % e - Y d - l ) e - ^ - % e - Y ( d - x ) + l iwA c 2 y 2 x < d , Applying equation (5.5) iwt , Ae f -yx ! -y(d+x) t -y(d-x) n(x,t) = Y c ? l e -he -he- x < d of which the real part can be written n(x,t) = G Sin(wt+(J)) x < d . (5.11) In the solution (5.11) the gain G = — y^- /a2+b2 and the phase cr angle (J> = Sin -b [ / a 2 + b 2 J > where a and b are defined as follows: a = Sin f * A X : - r * ^ . ^ < a s > . r i ] . -(d+x) -%Sin •r2 - %Sin (d-x) (d-x) \ — —— * L * • r i f - — . r 2 b = cos |e-^. r i e c ^cos j e- ( ~ ) - r 1 (d+x ) r 77 and n = (uN-c^K 2 )^ Sin8 r 2 = (uWK 2) 1* Cos6 9 = HsTan-1 -K to Estimating the Parameters Since wind measurements are in the form of hourly average values, i t seems reasonable to superpose model solutions for a succession of s tep - l ike impulses such as (5.9), each one representing the hourly value of wind-stress. Given an appropriate drag coeff ic ient C we may estimate the d kinematic wind-stress W from the measured wind-speed U, thus: P a W = C. ulul — d 1 1 p w P a where —— is the density rat io of a i r to water which I have taken as w — 3 1.2 x 10 . In evaluating wind-stress from the anemometer at Lone Tree - 3 Point I used a drag coeff ic ient of 1.4 x 10 . In the case of data taken on the roof of the F.R.B. barge at Port Alberni a value of _ 3 1.3 x 10 was used, ref lect ing the s l ight ly greater height of the measurement. The speed of the undamped internal wave depends on the depths and densities of the two layers: , Ap h'h" c = ' h" With l i t t l e loss in accuracy we may set ^— = 1: even over the s i l l the lower layer is more than 10 times the upper layer thickness. The term h' -^ jy- must at best be only a rough approximation. We are using a two-layer representation for a continuous density d ist r ibut ion that varies 78 both ver t ica l l y and horizontal ly . Typically -^ jr, varies from about 1% off Uchucklesit Inlet to s l ight ly over 2% near the in let head. Choice of an equilibrium thickness h' i s also somewhat arbitrary. I have chosen a value c = 85 cm/sec which seems appropriate for the upper reaches of the in le t . This value corresponds to a density difference of 2% for an equilibrium thickness of 368 cm, or about 1.8% for a 4 meter thickness. The distance d over which the wind usually acts is about 35 km. I have taken the quantity as 15 hours; this i s rather longer than indicated by the value of c just chosen, but is intended to ref lect the smaller density difference in the lower part of the in le t . As w i l l be shown subsequently, for solutions near the head, the model is quite insensit ive to changes in the value of d. As noted in Chapter 2, the internal tide generated in the harbour appears to decay to about one half of i t s amplitude by the time i t reaches Stamp Narrows. Despite the d i f f i cu l t y of determining the decay more precisely , this value does offer an opportunity for estimating the f r i c t iona l coeff ic ient . Using the exponential decay formula (5.6) and solving i t for different values of K at the 1^ frequency we obtain the curve in Figure 22 applicable for Stamp Narrows (— = 2 hours). A decay to one half corresponds to a coeff ic ient of K = 0.8 Hrs" 1 . We can apply these values to equation (5.10) to see how the surface layer thickness responds to a 1 dyne/cm2 wind-stress last ing for 1 hour. The maximum change of 35.4 cm occurs at the moment the wind ceases. It has decayed to 9 cm in 10 hours, 4 cm in 50 hours and is down to 2.7% of i t s maximum value after 200 hours. Asymptotic expansion 79 T AMPLITUDE Figure 22. Decay of internal wave of M2 frequency between Hohm Island and Stamp Narrows (X/C = 2 hours) as a function of f r i c t i o n . 80 of the solution (5.10) shows that for large t , the displacement is . • , -h. proportional to t The effect of a f i n i t e distr ibut ion of wind-stress acting over the in let i s to s l ight ly reduce the displacement for large values of t . Figure 23(a) shows the response for different values of K, at the in let head. Figure 23(b) indicates the variat ion of the response at different points down the in le t . Over the range shown, as the distance from the head increases the shape of the curve changes and the amplitude de-creases, but for large values of time the displacements at different locations converge. Now consider the frequency space representation. Solving equation (5.11) for a periodic wind-stress at different frequencies, we can evaluate the Amplitude Transfer Function, T ( f ) . Figures 24(a) and (b) show how this function varies for different K and x. The effect of changing the f r i c t iona l coeff ic ient is more pronounced at lower frequencies, but increasing the distance from the in let head has a much greater effect on higher frequencies. The dashed curve in Figure 24(b) shows the solution for a wind-stress of i n f in i te extent at x = 0. In the frequency range shown, the effect is to s l ight ly decrease the gain; at s t i l l lower frequencies the gain increases. The Transfer Function curves demonstrate an important feature of the model: surface layer thickness responds much more readily at lower than at higher frequencies. For example, using the values derived ear l ie r , i . e . K = 0.8 Hrs a periodic wind of amplitude 1 dyne/ cm at the diurnal frequency produces a response of amplitude 89 cm; for a 300 hour per iodici ty the response i s 3.4 Meters. 81 23(b) Figure 23(a). Depth of interface with respect to surface for s tep- l ike wind impulse for different f r i c t ion coeff ic ients . K has units H r s - 1 . Figure 23(b). Depth of interface at different points down the in let for s tep- l ike wind impulse. X is distance from in let head, C is internal mode phase speed; X/C has units of hours. 82 Figure 24. Gain of the. model for different f r i c t i o n coeff ic ients and different distances from the in let head (X/C is in Hours). The dashed curve in the lower figure shows the gain at the in let head for the case of a wind-stress of in f in i te extent. 83 Comparing Theory with Observations The occurrence of strong up- in let winds following a long calm, which i s represented by the observations shown in Figure 10, page 39, offers an opportunity for comparing the model with real data. In pract ice, the convolution integral (5.8) need only be evaluated once, a succession of solutions then being superposed, each one being weighted by the successive hourly wind-stress values. The solutions are plotted in Figure 25. When comparing the theoretical solution with the observations i t i s important to keep in mind that the concept of Fresh Water Thickness is not ident ical to the model representation. In the case of the model the dynamics of the system are found for a two layer i n l e t ; the observed Fresh Water Thickness is simply a convenient parameterisation of a continuous density d ist r ibut ion. Now compare the model solutions in Figure 25 with the observa-t ions. Both show the same general features: a sudden increase and a slow return to equilibrium. However the model predicts too large a change and the return to equilibrium is too slow, especially at Sproat Narrows. This section of data was chosen for comparison because the long calm which preceded the storm gives an opportunity to observe the response from essentially zero in i t i a l conditions. However the Lambrecht anemometers had not been instal led at the time and i t was necessary to use wiiyi data collected from the "Ve le l la" . The anemometer at this location is not considered very accurate (R. Herlinveaux, personal communication) errors of 10 to 15% having been found on a previous cal ibrat ion. Moreover the location of the instrument i t s e l f , on the 84 2 r FEBRUARY 1971 Figure 25. Comparison of model response and observations corresponding to data shown in Figure 10. 85 roof of the barge, makes i t rather susceptible to acceleration effects. These conditions suggest that the drag coefficient chosen was too large and the stress overestimated. A smaller stress would have yielded results closer to the observations. On the other hand the shape of the Sproat Narrows prediction suggests that the f r i c t i o n coefficient chosen was a l i t t l e too large; a smaller f r i c t i o n coef-fic i e n t would have produced a more rapid return to equilibrium. Longer comparisons with the model predictions for Stamp and Sproat Narrows using data from the Lambrecht anemometers gave quite good results. The Hohm Island predictions were best of a l l as might be expected, for i t is the least sensitive to errors in the f r i c t i o n coefficient. For prediction purposes i t would seem useful to carefully vary the parameters and K in order to obtain the best possible f i t to observations. Using data for Hohm Island shown in Figure 8, with the solution (5.8) and wind measurements from Lone Tree Point, we can again compare the model response. Figure 26 shows the solution together with the observations. Unlike the previous case the observations do not indicate zero i n i t i a l conditions and there is an early period of two to three days before the solution corresponds to observed conditions. There-after i t follows the data rather well and, except for the high fre-quencies, reproduces almost a l l the more important features of the response. We may make frequency space comparisons by deriving an observed Transfer Function from the spectra and coherence plots for Hohm Island; these are shown in Figures 17 and 18. Figure 27 shows the model and observed Gain and Phase Functions together. Apart from some significant 28 5 10 15 20 25 30 MARCH 1971 oo Figure 26. Comparison of model and data for Fresh Water Thickness at °* Hohm Island and measured wind-speed at Lone Tree Point. Since the model starts with zero i n i t i a l conditions i t takes a few days to catch up with the data. 87 350 300 „ 250 E o \ </> CO c >. O 200 < 150 100 50 1000 HOURS GAIN • THEORETICAL • OBSERVED T(f) I , • • - I • 1 1 l—i • L I 01 PHASE MODEL o OBSERVED j i_ _1_L_ 0 1 001 001 HOURS r n • anA Phase for model and 33 days of 88 peaks in the data, the model reproduces the general trend shown by the observations. Progressively better coupling occurs as we move to lower frequencies. In addition the observed phase angles f a l l f a i r l y close to the theoretical curve. The peak in the observed Amplitude Transfer Function at the semi-diurnal frequency is almost certainly a spurious estimate due to the large internal t ide. At the diurnal frequency the observations indicate a rather poorer coupling than the model predicts. Fluctuations above and below the theoretical curve also occur at 129 and 173 hours. These points have only 4 degrees of freedom, however, and departures of this magnitude from a smooth curve cannot be considered s igni f icant . It i s of interest to consider the possible existence of internal seiches in the in le t . It i s not clear what boundary condition is appropriate at the mouth since the s t ra t i f i ca t ion becomes progressively less intense near this area due to the entrainment. If the boundary condition were similar to that used for calculating surface mode seiches, then the osc i l la t ion would represent a standing wave of four times the length of the channel. Equation (5.5) shows that for the parameters chosen a maximum wavelength is 48 kilometers; under such conditions an internal seiche could not exist . Chapter 6 SUMMARY This study represents a concentrated effort to throw light on the main features of a st r a t i f i e d inlet's response to wind. The project has included development of simple and relatively inexpensive instrumentation for monitoring the structure of surface waters in Alberni Inlet over a period of many weeks. Together with wind measurements the observations have shown how the thickness of the surface layer responds to a changing wind-stress. Strong up-inlet winds produce a rapid thickening at the inlet h ead.This distortion appears to propagate back down the inlet suffering an attenuation as i t travels. The return to equilibrium can take several days; the inlet's response i s similar to that of a heavily damped mechanical system. Using a simple parameterisation of the data i t is possible to see how wind affects the intensity of the st r a t i f i c a t i o n . As might be expected, mixing increases with strong winds, but the effect i s more noticeable away from the inlet head. Current measurements taken by the Canadian Hydrographic Service at Sproat Narrows have shown that the movement of water at a depth of 2 meters i s closely coupled to the wind. At 15 and 40 meters the current i s mainly t i d a l , but with an unexplained down-inlet bias at 40 meters. Time series analysis has put these observations into sharper 90 r e l i e f . Three months of wind and current data have shown that there is about eighty times more energy of diurnal frequency at 2 meters than can be attributed to tides alone. On the basis of simple time scale considerations I have used phase angles between wind and current to estimate a bulk eddy v iscosity for the upper two or three meters of the in le t . This method has yielded values between 1 and 10 cm2/sec. On the other hand spectral analysis of surface layer thickness data from Hohm Island shows most of the internal f luctuations to be of much lower frequency. I have used a simple two-layer f r i c t i o n a l model of an in let to explain these resul ts . Neglecting the normal processes of estuarine c i rculat ion the model accounts for the heavy damping evident in the observations by means of l inear f r i c t i o n . Despite i t s s implicity the model succeeds in describing the coupling between wind and surface layer thickness and provides a means of predicting the internal response on the basis of measured wind-stress. BIBLIOGRAPHY Abramowitz, Milton and Irene A. Stegun (eds.) (1968) Handbook of Mathematical Functions. Dover Publications, Inc. , New York. Canadian Hydrographic Service (1972) Data record of current observa-t ions, Alberni Inlet and Approaches, 1971. Canadian Hydrographic Service, Department of the Environment, Manuscript Report Series VIII. (In Press) Gade, H.G. (1963) Some hydrographic observations of the inner Oslofjord during 1959. Hvalradets skr i f te r , Nr. 46. Gade, H.G. (1970) Hydrographic Investigations in the Oslofjord, a study of Water Circulation and Exchange Processes. Report 24, Geophysical Inst itute, University of Bergen, Norway. Henry, R.F. and T.S. Murty (1971) Three-dimensional c i rculat ion in a s t ra t i f i ed bay under variable wind-stress. Third Liege Colloquium on Ocean Hydrodynamics, 3-8 May, 1971, University of Liege. Johannessen, O.M. (1968) Some current measurements in the Dr^bak Sound, the narrow entrance to the Oslofjord. Hvalradets skr i f te r , Nr. 50. Murty, T .S. and Lise Boilard (1969) The Tsunami i n Alberni Inlet caused by the Alaska Earthquake of March 1964. Proceedings of the International Symposium on Tsunamis and Tsunami research, held at Honolulu, 7-10 Oct. , 1969. East West Center Press, Honolulu, pp. 165-187. Nasmyth, P. (1970) Ocean Turbulence. Ph.D. Thesis, Institute of Oceanography, University of Br i t i sh Columbia. Pac i f i c Oceanographic Group (1957) Physical and Chemical data record Alb erni Inlet and Harbour 1939 and 1941. Joint Committee on Oceanography. Fisheries Research Board of Canada, Manuscript Report. Pettersson, H. (1920) Internal Movements in Coastal Waters and Meteor-ological Phenomena, Geografiska Annaler, Stockholm, Vol . I, pp. 32-66. Pickard, G.L. and G.K.Rodgers (1959) Current measurements in Knight Inlet , Br i t i sh Columbia. Journal of the Fisheries Research Board of Canada, Vol . 16 (5), pp. 635-678. Pickard, G.L. (1961) Oceanographic features of in lets in the Br i t i sh Columbia mainland coast. Journal of the Fisheries Research Board of Canada, Vol . 18 (6), pp. 907-999. 92 Pickard, G.L. (1962) Oceanographic Characteristics of Inlets of Vancouver Island, Br i t i sh Columbia. Journal of the Fisheries Research Board of Canada, Vol . 20 (5), pp. 1109-1144. Rattray, M., Jr . 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