THE PROPAGATION OF T H E INTERNAL TIDE AROUND A BEND IN KNIGHT INLET, B.C. by ANTHONY JAMES WEBB B.A., University of Cambridge, England, 1980 M.Sc, University College of North Wales, 1981 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF T H E REQUIREMENTS FOR T H E DEGREE OF DOCTOR OF PHILOSOPHY i Q T H E FACULTY OF GRADUATE STUDIES Oceanography We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA SEPT 1985 © ANTHONY JAMES WEBB, 1985 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f QC&AKiO 6r£APHV The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 6 flc-feWr n2£~ D E - 6 n/Rn A b s t r a c t This thesis seeks to answer the question "how much of the internal tide propagating up Knight Inlet, B.C. is reflected by a right-angled bend ?" The internal tide in Knight Inlet is generated by the interaction of the barotropic tide with a shallow sill seaward of the bend. It then propagates in both directions as a travelling Kelvin wave. The up-inlet propagating wave then encounters the bend, where some of it may be reflected. This question is looked at with both theory (Chapter 2) and observations (Chapter 3). The theoretical study investigates the propagation of a Kelvin wave around a bend in a channel. The solution of the linearised long wave equations is expressed as a truncated series of cross-channel modes in each of three regions. The solution is matched across the two common boundaries by a least squares point matching process. The "rectangular bend" gives unsatisfactory results because of a singularity in the velocity field at the sharp inside corner. However, the "annular bend" gives good results. The bend acts as a diffraction grating, with total transmission for certain bend angles and "lobes" of high reflection for intermediate angles. Bends to the left give rise to the same reflection as bends to the right of the same angle, even if the incident Kelvin wave has a small cross-channel decay scale. For the parameters corresponding to the M<x internal tide in Knight Inlet, the energy flux reflection coefficient is very small, much less than 1%. The method of solution breaks down inexplicably for diurnal tides. The observational study is based upon 2~ months of cyclesonde current meter data from four stations in Knight Inlet taken during the summers of 1981 and 1983. The vertical profiles of amplitude and phase of the M 3 constituent of longitudinal velocity and density fluctuations are found to be the same from month to month at a particular station, but different from station to station. These complex amplitude profiles are decomposed into a truncated series of normal modes for waves propagating both up-inlet and down-inlet. The phase speeds and zero-crossing depths of the normal modes do not agree with those calculated by Farmer and Smith (1980a) using 1977 data. Two simple models of the stratification are used to explain the differences in the normal modes in terms of differences ii in the Brunt-Vaisala frequency profile used to calculate them. At the two stations up-inlet of the sill, acceptable fits can be obtained using only two up-inlet propagating waves, indicating that the data are compatible with the low reflection found in the theoretical study. When down-inlet waves are included in the fit, their amplitudes are found to be small, although care must be taken to avoid trying to fit too many modes since some of them are highly correlated in the depth range for which data are available. At the two stations seaward of the sill, the up-inlet energy flux is of the same order of magnitude as the down-inlet flux, indicating a second source of internal tide seaward of those two stations. The results indicate that only 30-50% of the power removed from the barotropic tide is being fed into the internal tide in the summer. This conclusion is at variance with that of Stacey (1985), whose model shows that most of the power lost from the barotropic tide is being fed into the internal tide. A consistent interpretation of these results is that most of the power lost from the barotropic tide initially goes into the internal tide, but that 50-70% of it is rapidly dissipated through turbulence within a couple of kilometers of the sill. The low frequency component of the velocity signals from the cyclesonde data is also presented (Chapter 4), with a brief analysis . The monthly averaged residual velocity profiles are also found to be fairly repeatable from month to month at a particular station, but different from station to station. These profiles could be useful in validating general circulation models. iii Table of Contents 1. Introduction 1 1. Historical Background 1 2. Physical Oceanography of Knight Inlet 3 1. Physical Description of the Inlet and Freshwater Input 3 2. Salinity 5 3. Temperature 5 4. Time Variations in Temperature and Salinity 7 5. Early Current Measurements . 9 6. The Interaction of the Barotropic Tide With The Sill 10 7. Tides 12 3. Aim of this Thesis 15 1. Summary : . . 17 2. The propagation of a Kelvin wave around a bend in a channel 18 1. Introduction 18 2. Governing Equations 19 3. Special Solutions in a Straight Channel 20 4. Method of Solution in a "Rectangular Bend" 22 1. First Stage 23 2. Second Stage 25 5. Results for the Rectangular Bend 26 1. Parameter Search 1: r = 0.803, 6 = 1, variable k2 27 2. Parameter Search 2: k2 = 7.02 x 10~3, 6=1, variable T 31 3. Parameter Search 8: k2 = 7.02 x I O - 3 , r = 0.803, variable 6 32 6. Discussion of the Solution Near the Inside Corner 34 7. Method of Solution in an "Annular Bend" 35 8. Discussion of the Results of the Eigenvalue Problem in the Annulus . . . . 38 9. Results for the Annular Bend 39 1. Parameter Search 1: r = 0.803, <f> = +90° , r 0 = 0.5, variable k2 . . . . 39 2. Parameter Search S: r = 0.803, r 0 = 0.5, various k2, variable <f> . . . . 43 3. Parameter Search S: k2 = 7.02 x 10 - 3 , r 0 = 0.5, <f> — 90°, variable r . . 47 4. Parameter Search ^ : ro = 0.5, various k2, various r, variable <f> 48 5. Parameter Search 5: r = 0.803, various k2, various ro, variable <f> . . . . 53 10. Conclusion 56 3. Observations of the Internal Tide in Knight Inlet 58 1. Introduction 58 2. Description of the Data 59 3. Harmonic Analysis of the Data 65 1. Harmonic Analysis of Velocity Data 69 2. Harmonic Analysis of Density Data 82 3. Harmonic Analysis of Surface Elevations 83 4. Normal Modes — Theory 85 5. Modai Decomposition 89 6. Normal Mode Fitting — Theory 92 1. Least Squares Fitting at a Single Station in Mid-Channel 93 2. Least Squares Fitting at Two Stations on Opposite Sides of the Channel . . 94 7. The Estimation of the Barotropic Component 95 8. Energy Fluxes 98 iv 9. The Calculation of the Normal Modes 100 10. Comparison with Farmer and Smith's Normal Modes 106 11. Results of the Normal Mode Fitting 117 1. Results of the Fitting at Stations Up-inlet of the Sill 124 2. Resvlts of the Fitting at Stations Down-inlet of the Sill 128 12. Comparison of Fluxes with Stacey's Model 129 13. Conclusion 131 4. Low Frequency Currents in Knight Inlet 133 1. Power Spectra of Currents 133 2. Time Series of Longitudinal Velocities with the Tide Removed 137 3 . T h e MSf Tide 138 4. Residual Velocity Profiles 145 5. S u m m a r y 148 Appendix A: Simple Models of The Stratification 149 1. Model 1 — Two Layers of Constant N2 • • • • 149 1. "a-Modes" or "Upper Layer Modes" 151 2. "P-Modes" or "Lower Layer Modes" 152 3 . Summary 153 2. Model 2 — Well mixed surface layer overlying a deep layer of constant N2 154 1. The Surface Mode 156 2. "p-modes" or "Lower Layer Modes" 156 3 . Summary 157 Bibliography 158 v List of Figures 1. Plan view and longitudinal depth profile of Knight Inlet 4 2. T-S curve near the sill : 6 3. Representative temperature, salinity and buoyancy profiles 8 4. Detailed bathymetry around the sill 10 5. Schematic representation of the phenomena occurring at the sill 13 6. Schematic diagram of the "rectangular bend* 20 7. R vs k2 for constant r = 0.803 27 8. Blowup of Figure 7 for small k2 28 9. Pressure field for k2 = 7.02 x I O - 3 , r = 0.803, 6= 1 29 10. Velocity field corresponding to Figure 9 29 11. Pressure field for k2 = 2.25, T = 0.803, 6 = 1 . 30 12. Velocity field corresponding to Figure 11 30 13. R vs r for constant k2 = 7.02 x 10 - 3 31 14. Rvsb for constant r = 0.803 and k2 = 7.02 x 10~3 32 15. Pressure field for k2 = 7.02 x I O - 3 , r = 0.803, 6 = 0.3 33 16. Velocity field corresponding to Figure 15 33 17. Schematic diagram of the "annular bend" . . 36 18. R vs k2 for constant r = 0.803 40 19. Blowup of Figure 18 for small k2 40 20. Pressure field for k2 = 7.02 x I O - 3 , r = 0.803, r 0 = 0.5, 4> = +90° 41 21. Velocity field corresponding to Figure 20 41 22. Pressure field for k2 = 2.25, r = 0.803, r 0 = 0.5, <f> = +90° 42 23. Velocity field corresponding to Figure 22 42 24. R vs (f> for various values of A;3 43 25. Pressure field for k2 = 2.467, r = 0.803, r 0 = 0.5, <p = +30° 45 26. Velocity field corresponding to Figure 25 45 27. Pressure field for k2 = 2.467, T = 0.803, r 0 = 0.5, <f> = - 3 0 ° 46 28. Velocity field corresponding to Figure 27 46 29. R vs r for constant k2 = 7.02 x I O - 3 and <f> = +90° 47 30. Pressure field for k2 = 7.02 x I O - 3 , r = 0, r 0 = 0.5, <f> = +90° 49 31. Velocity field corresponding to Figure 30 49 32. R vs <f> for various values of k2 and T 50 33. Bend in a straight coastline studied by Packham and Williams (1968) 52 34. 100% -T vs <f> for Packham and Williams' bend 52 35. Angle of first zero reflection vs A for r 0 = 0.5 53 36. R vs <f> for various values of ro and k2 .54 37. Pressure field for k2 = 2.467, T = 0.803, r 0 = 2.0, (f> = +30° 55 38. Velocity field corresponding to Figure 37 55 39. Detailed map of the first bend in Knight Inlet 59 40. Longitudinal velocity vs time at Tom-N 62 41. Transverse velocity vs time at Tom-N 63 42. Isopycnals in the time-depth plane at Tom-N 64 43. Density at the standard depths vs time at Tom-N 66 44. Depth of the bottom bumper vs time at Tom-N 67 45. Loci of M * complex amplitudes of velocity at Protection Point 70 46. Loci of Af4 complex amplitudes of velocity at Lull Bay 70 47. Loci of Af 4 complex amplitudes of velocity at Tomakstum Island 71 v i 48 . Loci of M 4 complex amplitudes of velocity at Adeane Point 71 49 . Loci of S% complex amplitudes of velocity at Protection Point 72 50. Loci of complex amplitudes of velocity at Lull Bay 72 5 1 . Loci of S% complex amplitudes of velocity at Tomakstum Island 73 52 . Loci of S 2 complex amplitudes of velocity at Adeane Point 73 53 . Loci of M2 complex amplitudes of velocity and density at Protection Point . . 74 54. Loci of M2 complex amplitudes of velocity and density at Lull Bay 74 55. Loci of M 2 complex amplitudes of velocity and density at Tomakstum Island . 75 56. Loci of M 2 complex amplitudes of velocity and density at Adeane Point . . . . 75 57 . Loci of N2 complex amplitudes of velocity at Protection Point 76 58 . Locus of JV2 complex amplitudes of velocity at Lull Bay 76 59. Loci of N2 complex amplitudes of velocity at Tomakstum Island 77 60 . Loci of N2 complex amplitudes of velocity at Adeane Point 77 6 1 . Loci of Ki complex amplitudes of velocity at Protection Point 78 62 . Loci of Ki complex amplitudes of velocity at Lull Bay 78 63 . Loci of Ki complex amplitudes of velocity at Tomakstum Island 79 64. Loci of Kx complex amplitudes of velocity at Adeane Point 79 65 . Loci of Oi complex amplitudes of velocity at Protection Point 80 66. Loci of O i complex amplitudes of velocity at Lull Bay 80 67 . Loci of Ox complex amplitudes of velocity at Tomakstum Island 81 68 . Loci of Oi complex amplitudes of velocity at Adeane Point 81 6 9 . A/2 complex amplitudes of surface elevation 84 70 . CTD density profiles at Tomakstum Island in 1983 102 7 1 . logl0N2 vs depth at Protection Point 103 72. logxoN2 vs depth at Lull Bay 103 73 . logyoN2 vs depth at Tomakstum Island 104 74. logioN2 vs depth at Adeane Point 104 75 . w-eigenfunctions at Tomakstum Island including upper 20 m 105 76. u-eigenfunctions corresponding to Figure 75 105 77. /?-eigenfunctions at Protection Point excluding upper 20 m 107 78. u-eigenfunctions corresponding to Figure 77 107 79. p-eigenfunctions at Lull Bay excluding upper 25 m 108 80. u-eigenfunctions corresponding to Figure 79 108 8 1 . p-eigenfunctions at Tomakstum Island excluding upper 20 m 109 82 . u-eigenfunctions corresponding to Figure 81 109 83 . /?-eigenfunctions at Adeane Point excluding upper 20 m 110 84. u-eigenfunctions corresponding to Figure 83 110 85. Farmer and Smith's density profile and w-eigenfunctions I l l 86 . Mode 1 phase speed and zero crossing depth vs JV 2 115 87. Mode 2 phase speed and zero crossing depths vs 115 88 . Mode 3 phase speed and zero crossing depths vs JV 2 116 89 . Results of Fit # 6 at Tomakstum Island 125 90 . Results of Fit # 6 at Adeane Point 125 9 1 . Results of Fit # 4 at Lull Bay 127 9 2 . Results of Fit # 4 at Protection Point 127 93 . Results of Fit # 6 at Protection Point 128 94 . Power spectra of longitudinal velocity at Protection Point 133 9 5 . Power spectra of longitudinal velocity at Lull Bay 134 vii 96. Power spectra of longitudinal velocity at Tomakstum Island 135 97. Power spectra of longitudinal velocity at Adeane Point 136 98. Power spectrum of velocity at shallowest depth averaged over all stations . . 137 99. Low-pass filtered longitudinal velocity at Protection Point in July 1983 . . . 139 100. Low-pass filtered longitudinal velocity at Lull Bay in July 1983 139 101. Low-pass filtered longitudinal velocity at Tomakstum Island in July 1983 . . 140 102. Low-pass filtered longitudinal velocity at Adeane Point in July 1983 . . . . 140 103. Low-pass filtered longitudinal velocity at Protection Point in September 1983 141 104. Low-pass filtered longitudinal velocity at Adeane Point in September 1983 . . 141 105. Low-pass filtered longitudinal velocity at Tom-N in September 1983 . . . . 142 106. Low-pass filtered longitudinal velocity at Tom-S in September 1983 142 107. Complex amplitudes of the MSf component of velocity at Protection Point . 143 108. Complex amplitudes of the MSf component of velocity at Lull Bay . . . . 143 109. Complex amplitudes of the MSf component of velocity at Tomakstum Island 144 110. Complex amplitudes of the MSf component of velocity at Adeane Point . . 144 111. Residual longitudinal velocity profiles at Protection Point 146 112. Residual longitudinal velocity profile at Lull Bay 146 113. Residual longitudinal velocity profiles at Tomakstum Island 147 114. Residual longitudinal velocity profiles at Adeane Point 147 115. Density and N structure used in Model 1 149 116. 0(1) eigenfunctions for a-modes 151 117. 0(1) eigenfunctions for /?-modes 152 118. Density and N structure used in Model 2 154 119. 0(1) eigenfunctions for the surface mode 156 viii List of Tables 1. Standard depths at each station 61 2. Tidal constituents used in the harmonic analysis 68 3. Estimates of the barotropic tidal currents at each station 97 4. Correlations between the modes below 20 m at each station 118 5. Results of the fitting — non-dimensional amplitudes aJJp and a^nan 121 6. Results of the fitting — phases <f>%p and «/»^ou"» in degrees 122 7. Results of the fitting — energy fluxes in MW 123 ix Acknowledgements I would like to thank my research supervisor, Professor S. Pond, for his steadfast moral support and guidance over the last four years, Professors P.H. LeBlond and L.A. Mysak for their help and teaching, Doctors D.M. Farmer and H.J. Freeland for originally suggesting the problem and for their continued interest in it, B. de Young for the use of his spectral analysis program and for hours of discussion, Doctor M. Foreman for the use of his harmonic analysis program, the officers and crew of the C.S.S. Vector for their assistance in the data collection, D. Toffoli for typing and proof-reading this thesis and my parents for their encouragement. This research was supported by the Natural Sciences and Engineering Research Council of Canada (Grant A-8301). I appreciate personal support from the Canadian Common-wealth Scholarship and Fellowship Plan. x 1. Introduction 1.1 Historical Background The first oceanographic measurements in Knight Inlet were made by G.L. Pickard in a series of surveys of B.C. inlets by the Institute of Oceanography at U.B.C. starting in 1949. The results are described in Pickard (1956, 1961). The first of these two papers is mainly a description of the geometries of B.C. inlets, whereas the second is a thorough account of their water characteristics ( salinity, temperature, oxygen and turbidity ) and changes thereof. He also carried out the first current measurements in Knight Inlet using current drag and Ekman current meters (Pickard and Rodgers 1959). Since then U.B.C. personnel returned every year except 1960-62 up to 1974 as part of their survey of the water characteristics of B.C. fjords. Pickard (1975) gives an account of the long term variations of the deep water properties in Knight Inlet as well as elsewhere, using these data. Current measurements were not, however, routinely done. Before the arrival of in-ternally recording current meters in the 1960s, any kind of direct current measurement required an enormous amount of effort, since the ship had to anchor. Since such measure-ments could be done for at most a few days, the effects of variations in wind stress and the lunar cycle could not be isolated from the "mean" circulation, so there was little point in any greater effort. For example, it was easier to measure the river discharge by inferring it from the heat budget of the inlet (based upon temperature and salinity measurements), than by direct measurement (Pickard and Trites 1957). Knight Inlet is distinguished by being a long, regular inlet with a large fresh water input at the head, and a shallow sill. The interaction of the sill with the high summer stratification produces large internal waves, both of short period (a few minutes) and of tidal period (the "internal tide"). The short period internal waves were the subject of more recent extensive investigation by Farmer and Smith (1978, 1980a, 1980b). They used a large selection of modern instruments, including a string of Aanderaa current meters fitted with conductivity cells and fast response thermistors, three orthogonally mounted 1 precision mechanical current meters, an electromagnetic current meter and an acoustic current meter. In addition they took CTD surveys of the inlet, and used acoustic sounding techniques to generate images of the hydraulic jumps that occur in the vicinity of the sill. This was the first measurement of currents in Knight Inlet since Pickard and Rodgers' comparatively primitive attempt in July 1956. In the last seven years a considerable amount of work has been done on Knight Inlet. A recent review of fjords by Farmer and Freeland (1983) is an excellent reference, as are the proceedings of the NATO conference on Fjord Oceanography, edited by Freeland, Farmer and Levings (1980). Both of these publications contain many references to Knight Inlet, both as an object of study in its own right, and as an example illustrating the mechanisms under discussion. Knight Inlet has very energetic processes occurring at the sill which generate large internal waves and tides, and for that reason is a popular site to investigate such phenomena. It has also become a testing ground for several different theories of fjord circulation; such comparisons are more easily made using data from a single fjord. The energy-transfer processes occurring between the tide, the resulting internal tides and the tidal-mean circulation play an important role in generating the mixing that is an essential element in determining the net fluxes of matter in fjords. It is the adequate parameterisation of mixing that remains the major deficiency in any modelling attempt to make quantitative predictions of the fluxes and resulting distribution of substances that are introduced into an estuary. This study therefore seeks to investigate the governing processes in a particular type of estuary, namely a deep, strongly stratified fjord. 2 1.2 Physical Oceanography of Knight Inlet J.2.1 Physical Description of the Inlet and Freshwater Input Knight Inlet is generally regarded as the most spectacular fjord in the SW sector of the province, towered over by snow-capped peaks as high as 3300 m. It is roughly 100 km long, measured from the head to the first subsidiary inlet, with an average width of 2.5 km. Knight Inlet is the third longest of the B.C. mainland inlets. Figure 1 shows a plan view and longitudinal depth profile. The sill is clearly visible 75 km from the head, where the depth falls to 60 m. The sill is in the middle of a long straight reach, while the rest of the inlet from the head to the bend between Tomakstum Island and Adeane Point is sinuous. The sides of the inlet are quite steep, with typical slopes of 30° towards the mouth and 60° towards the head. The average mid-channel depth is 400 m, and the greatest depth is 550 m. In comparison Jervis Inlet, the deepest on the B.C. coast, has an average depth of 500 m and a greatest depth of 725 m. Most of the freshwater enters at the head, with a mean annual discharge of 400 m 3 /s. There are usually two peaks in the year, one in June of about 800m3/s due to snowmelt, and a lesser one in November due to rain. Pickard (1961) calls the summer peak "stored runoff," and the winter peak "direct runoff." The typical winter runoff is 50 m 3/a. Knight has the seventh largest freshwater discharge of all B.C, inlets, with Portland Inlet having the largest with an annual mean of 990 m 3/a. For this reason, a two-layer density structure is a good description of the inlet for most of the year. In fact, two two-layer estuary models have selected Knight as an archetype (Long 1975, Pearson and Winter 1975). The mean (non-tidal) circulation is a consequence of the fresh-water discharge at the head. It forms a low salinity surface layer which flows seaward, gaining volume by the entrainment of saline water from below and gaining in speed. There is also a deepening of the brackish layer towards the mouth. An up-inlet flow at depth must compensate for the entrainment. The wind also has a large effect on the non-tidal circulation, but its speed and direction are unpredictable (varying on a time scale of a few days). When comparing circulation models with real data, the wind is often assumed to have a zero mean effect if 3 Figure 1 Plan view and longitudinal depth profile of Knight Inlet. (After Blackford 1984.) 4 averaged over a sufficiently long period. 1.2.2 Salinity According to Pickard's (1961) classification of B.C. fjords, Knight is a type A l inlet, with a low (less than 2 %o) surface salinity at the head, and a high (around 20 %o) surface salinity at the mouth. A type A l inlet means one with a large runoff, in most cases from glaciers or snowfields. In the winter, however,- the surface salinity and temperature are nearly constant along the inlet, because of the low runoff at that time. In terms of the vertical salinity profile, Knight has a type 1(a) profile at the head, and a type 1(b) profile elsewhere. This classification means that Knight Inlet has a homogeneous surface layer with a distinct halocline, which is sharper at the head of the inlet than elsewhere. The depth of the halocline increases from around 5 m at the head to around 15 m at the mouth. In the range of values of temperature and salinity found in Knight Inlet, salinity has a much greater effect on density than temperature. In fact, density profiles take on more or less the same shape as salinity profiles. Below the halocline, the salinity increases monotonically to the bottom. The salinity at 20 m is typically 29.8 %o, at 50 m it is about 30.7 % 0 , and at 100 m it is 31.0 %o. This deep water value lies between typical values of 28 to 30 %o for southern mainland inlets, and of 32 to 34 %o for northern inlets. Pickard (1961) suggests that this northwards increase in deepwater salinity is associated with the directness or otherwise of the connection with the Pacific (as the source of the most saline water). 1.2.3 Temperature In the summer, temperature profiles in the "shallow zone" (the upper 20 m or so) are a mirror image about a vertical axis of the salinity profile because the water in the shallow zone is a result of mixing of the warmer fresh surface water with the cold saline deep water. Figure 2 shows an excellent example of a straight T-S curve arising from the mixing of two water types. Typical summer figures are 14° C at the surface and 8° C in the underlying water. 5 S A L I N I 16 10 —i— T Y 20 ( % o ) 30 SFC Figure 2 T-S curve near the sill. (From Pickard 1961.) Below the thermocline associated with the halocline, the temperature profiles often show maxima or minima at intermediate depths. A temperature minimum at about 50 to 100 m in depth is present in Knight Inlet, and is a general feature of B.C. inlets (most noticable in Bute Inlet). The temperature decreases from a summer value of roughly 9° C at 20 m to a minimum of 8° C at 50 m, and then gradually rises again to a bottom value of 8.5° C. This temperature minimum has been attributed to winter cooling and wind mixing of the upper part of the water column, followed in spring and summer by a warming of the surface. The higher density stratification in spring and summer tends to inhibit wind mixing and thus serves to preserve the effects of the distributed heat loss that occurred in winter. 6 1.2.4 Time Variations in Temperature and Salinity The most noticeable signal in temperature and salinity time series is the internal tide. It is caused by the barotropic tide forcing stratified water to flow over the sill leading to a vertical oscillation at tidal period of the isopycnals and isohalines by as much as 50 m peak-to-peak in the straight section of the inlet. Equivalently there are temperature deviations of 0.05° C and salinity deviations of 0.4 %o at depths of 50-150 m. The signal is modulated on the spring/neaps cycle. An annual signal is also present at most depths. The variability decreases with increas-ing depth. The only available data to show this annual signal well are a set of 23 cruises over the period 1972-74 (Pickard 1975). Both temperature and salinity below 100 m show a maximum around December, with peak-to-peak values of 0.5° C and 0.6 %o respectively. Pickard describes the signal in Knight to be "sinusoidal", as opposed to "peaked" or "saw tooth" as found elsewhere. The amount of dissolved oxygen also shows an annual sinusoidal variation of lml/l peak-to peak, with a maximum around March. The dissolved oxygen in Knight Inlet ranges from 3 to 5 m/// below 100 m, with higher values above 100 m. It is nowhere anoxic. Pickard suggests that these annual variations are caused by longitu-dinal advection, rather than vertical advection, diffusion etc, and that this longitudinal advection occurs during a late winter replacement of the deep water. The annual variation of river runoff and air temperature have a direct effect on the salinity and temperature of the surface waters. In the winter the halocline and associated pycnocline are almost non-existent (see Figure 3). In the spring, they are evident at the head of the inlet, but not at the mouth. As for longer term variations (i.e. over several years), certain trends are evident. The temperature at 100 m rose by about 1° C from 1951 to 1964 and then fell by about 1.2° C to 1974, whereas the changes at 300 and 500 m were of the order of 0.8° C. Pickard suggests that this trend is associated with a rise in the average winter air temperature of about 2° C from 1952 to 1963 and a fall by the same amount to 1973. Because vertical heat exchange with the surface is very small below 100 m, he attributes the change to 7 Temperature in °C Salinity in p.p.t. Buoyancy Period (minutes) Figure 3 Representative temperature, salinity and buoyancy profiles. Su = summer, W = winter, Sp = spring. Note the change of scale at 50 m. (From Farmer and Freeland 1983.) longitudinal advection due to the mean estuarine circulation. The fact that any water flowing up-inlet at depth has to surmount the sill at 60 m means that heat exchange with the surface is possible. i 8 1.2.5 Early Current Measurements As was stated earlier, the first current measurements in the 1950s (Pickard and Rodgers 1959) were rather crude, and not much more was done until 1976 in connection with the work by Farmer et al. on internal waves at the sill. Generally, the currents can be described as an oscillating component at tidal period superimposed on a mean flow. It might seem that the oscillating component is due to tidal forcing from the ocean (and is independent of river runoff), and the mean flow is due to gravitational circulation (and is independent of tidal forcing). However, when interpreting these two components of the currents, one must bear in mind that when both forcing terms are present, the system may not respond with a sum of the responses to each individual forcing term, because the equations describing fluid flow are non-linear. Therefore the observed mean circulation would not necessarily be the same in the absence of tidal forcing. Pickard and Rodgers describe the currents at two stations, taken every half hour for three days in July 1956. Over the sill, the currents were up to 150 cm/sec at the surface, and 50 c m / s e c at 70 m. The oscillatory component was in the same phase from surface to bottom. With no wind, the mean current profile was the classic two-layer system, with outflow in the upper half and inflow below. With an up-inlet wind, the mean profile was 3-layer, with up-inlet flow in the upper 10 m and the bottom 20 m. At a deeper station (350 m) between the sill and the first bend, speeds up to 25 cm/sec were still observed at depths as great as 300 m. The oscillatory component of the current was not constant in phase with respect to depth. The mean profile was a 3- or 4-layer flow, depending on the wind direction (amongst other things). Had more data been available to them, however, they would have seen a great deal of variability in such 25 hour means. They found that the mean circulation was sufficiently strong at the deep station to overcome the oscillatory component at the surface and wipe out any up-inlet flow there, but that this was not the case at the sill. 9 1.2.6 The Interaction of the Barotropic Tide With The Sill The basic mechanism of flow interaction with the sill in Knight Inlet was first deter-mined in August 1977 by Farmer and Smith (1978, 1980a, 1980b). Figure 4 shows the detailed bathymetry around the sill. Strong currents during, say, an ebb tide, produce su-percritical conditions over the sill, with a return to subcritical flow behind the sill because of the presence of an internal hydraulic jump or a large amplitude lee wave. As the tide slackens, the disturbance moves slowly back over the sill and travels upstream, evolving as an internal surge or bore. The precise response depends upon two things: the stratification and'the strength of the tidal current. Figure 4 Detailed bathymetry around the sill. Depth contours are shown in metres. (From Farmer and Freeland 1983.) For a given energy level, conservation of mass and momentum implies that a two-layer flow has two possible modes, one as a supercritical flow and the other as a subcritical flow. In supercritical flow, the Froude number (which is defined as the ratio of the flow speed in the upper layer to the internal phase speed) exceeds unity, whereas in subcritical flow it is less than unity. The fluid velocity is greater and the upper layer thickness is less in the supercritical mode. The situation is complicated somewhat in the continuously stratified case by the existence of an infinite set of internal wave phase speeds corresponding to the internal modes, and these give rise to an infinite set of Froude numbers for a given flow. The Froude numbers increase with increasing mode number, so that it is possible, for 10 example, for a flow to be subcritical with respect to the first mode, but supercritical with respect to the second mode. A change in flow conditions from supercritical to subcritical is called a hydraulic jump. These jumps can be initiated by the presence of topography (e.g. at the sill) or a sudden widening of the channel. Farmer and Smith's observations were made with profiling current meters and CTDs, with moored current meters and thermistor chains, with recording pressure (tide) gauges, and with high frequency (100-200 kHz) echo soundings. Ship location was determined using microwave positioning equipment. The echo sounder was used to obtain acoustic images of the structure of the flow over the sill and the resulting surges and intrusions, by making traverses of the sill with the ship. The acoustic signal would be reflected not only by the bottom, but also by biological organisms and density changes (microstructure), so that details of the flow could be traced. In addition, a variety of fast response current meters were used, which sampled at a high rate and gave useful information about the vertical component of the velocity. In their first paper Farmer and Smith (1978) describe the results of the pilot study to determine the scales necessary to plan a more detailed experiment. This experiment included CTD measurements along the channel with some time series observations (using CTD and profiling current meters etc) at a station 6 km up-inlet of the sill. This distance is equivalent to a few tidal excursions, so the chaotic motions at the sill have had time to organise themselves into propagating internal waves in the upper layer. They observed amplitudes in excess of 10 m (based on the excursions of the isohalines) and periods of the order of 10 minutes. As the strongly stratified part of the water column was only 15-20 m deep, it became evident that non-linear effects were important. They interpreted the data using a model of non-linear (because of finite-amplitude effects) internal waves in a strongly sheared flow, and examined conditions leading to shear flow instability. They found that the calculated phase speeds of the waves were consistent with the observed phase speeds, and that the wave properties were strongly influenced by the shear. 11 In the other two papers Farmer and Smith (1980a,b) discuss acoustic images of the flow over the sill, and classify the wide variety of responses according to the Froude numbers for each of the internal modes. In mid-winter even the smallest tides will produce flows supercritical with respect to all internal modes, while in mid-summer most flows will be subcritical with respect to the first mode but may be supercritical with respect to the second. Figure 5 shows schematically some of the phenomena that they found, including lee waves, hydraulic jumps, blocking, boundary layer separation and the spread and collapse of turbulent mixing layers shed from the sill crest. Blocking occurs when there is insufficient kinetic energy in the flow to raise the denser deeper fluid over the crest. Flow separation occurs when the boundary layer, in which the tangential velocity drops from its value in the interior of the fluid to zero on the fjord bed, becomes detatched from the bottom. The high shear across the boundary layer leads to instabilities and turbulence. The development of a large lee wave behind the sill tends to inhibit boundary layer seperation. 1.2.7 Tides The tides in Knight Inlet are principally semi-diurnal, with the Af2 constituent of the surface elevation oscillation increasing in amplitude from 1.527 m at the mouth to 1.582 m at the head. The total range is 3-5 m. For a pure barotropic standing wave the phase would be constant along the inlet, and the currents in quadrature. In fact, there is a difference of 1 or 2° because of friction and other energy sinks. The external Rossby radius is of the order of 2000 km, which is much wider than the inlet, so rotation can be ignored for the barotropic tide. The standing wave can be thought of as a superposition of two Kelvin waves (practically plane waves) of equal amplitude propagating in opposite directions, one being the incident wave and the other being its reflection from the head. Jamart and Winter (1980) made a finite element computation of the barotropic tide in Knight Inlet. They ignored the Coriolis term and had a simple linear dissipation term, but their results fairly well approximated the observed tidal flow, with complex flows in the area of the sill. They tuned their dissipation parameter so that the model produced the 12 A B (A) . Schematic diagram of a mode 1 lee wave response to tidally forced flow over a sill. Upstream of the sill there is blocking (b) of the deep flow and separation (s) of the boundary layer beneath the trough of the lee wave downstream of the sill. A s the tidal current slackens the lee waves are released successively to form a nonlinear wavetrain that travels over the sill Crest and back along the fjord. (B) . Schematic representation of tlow that is subcritical with respect to mode 1, but critical with respect to mode 2. In the early stages of ine tidal How the boundary layer separates from the sill crest, but subsequently separation is suppressed as mode 2 lee waves evolve. The upstream flow is modified by a small mode 1 internal wave that travels up-inlet away from the sill. The growing lee wave or jump eventually collapses to spread out as an undular bore as the flow slackens. A small wave (c) travels downstream. A l l of these features have been observed in . Knight Inlet, B .C . (C) . Sketch of tidal mixing over a longer sill leading to collapse of water o f intermediate density. The mixed water spreads outwards as an undular bore between the two layers from which it was formed. A small disturbance (c) travels in the downstream direction. same phase difference between the mouth and the head as was calculated by Freeland and Farmer (1980) from tide gauge data. The tidal excursion inside the inlet is of the order of 1 or 2 km. As was previously mentioned, the internal tide is most easily seen as an oscillation in the depth of isopycnal surfaces at tidal period. A typical phase speed of 0.8 m/s for the first M<2 internal mode for summer stratification (this concept arises from a linear, flat-bottomed model with a separable solution) gives rise to a wavelength of 36 km and an internal Rossby radius of 7 km. The internal tide is generated by the passage of barotropic tidal flow over topography, such as a sill. The effect of sills in generating internal tides was first shown by Zeilon (1912, 1934), as referenced in LeBlond and Mysak (1978). Halpern (1971a,b) identified groups of higher frequency waves as well as a semi-diurnal internal tide generated by tidal flow over a bank. Baines (1982) discusses the generation of internal tides by the continental rise in the ocean, and Blackford (1978) presents a non-linear model of the generation of the internal tide by tidal flow over a sill in a channel. When there is flow over topography a vertical component of velocity is required for the fluid to surmount the obstacle. This vertical component of velocity oscillates at that frequency, and acts as a forcing function to generate internal waves at tidal frequency. With the highly energetic and turbulent processes that occur at the sill in Knight Inlet, such vertical components of velocity are bound to be chaotic, at least on the downstream side of the sill, causing both a wide spectrum of modes and an irregular waveform to be generated. However, as the internal tide propagates away from the energetic region of generation into the "far field", the higher modes will dissipate due to their large vertical shear, and the waveform will become more regular. Another way of looking at the generation mechanism is that as the deep flow is blocked, the more dense fluid will tend to rise up the sill with the flow, but then fall again as the tidal current eases. This behaviour is depicted in Figure 5a to the left of the sill. This oscillation of isopycnals at tidal period at depths below the sill crest will generate an internal tide. 14 1.3 A i m of this Thesis The main question that will be addressed in this thesis is how much of the internal tide, which is supposedly generated at the sill and propagates away from it in both directions, is reflected when it hits the first bend up-inlet of the sill, and how much is transmitted. In an analysis of four current meter records, Farmer and Freeland (1983) concluded that the amplitude of the outgoing internal tide at Tomakstum Island (see Figure l)was 0.93 times that of the incoming wave. Because they only had measurements at two depths they were forced to assume that only one internal mode was present (see later discussion in Section 3.4). Farmer and Freeland assumed that this mode was the first. Later, Freeland (1984) modified the calculation to assume that this mode was the second, and obtained an amplitude ratio of 0.70. However, their conclusion that the internal tide undergoes substantial reflection at the bend is in direct conflict with a result obtained by Blackford (1984). He modified his non-linear generation model to include a variable amount of reflection at a specified distance from the sill, and compared the resulting waveform of the internal tide with an observed waveform based upon the depth of the 30.5 %o isohaline which was obtained from a 24 hour series of CTD observations at two stations. He found much poorer agreement by using a high reflection coefficient than by using no reflection, which suggests that very little reflection occurs at the bend. Another piece of evidence in favour of a low reflection coefficient at the bend comes from a numerical simulation of internal tide generation in sill fjords by M0rk and Gjevik (1982). Their model was two-layered and depth integrated, and simulated a barotropic Kelvin wave in the open ocean propagating along the coast past the mouth of the fjord. It induced a barotropic standing wave in the fjord, which in turn generated baroclinic waves at the sill. They found that with the parameters used, a 90° bend in the fjord had negligible effect on the propagation of the internal tide, in the sense that the baroclinic interface displacements as a function of time at two points on either side of the bend were the same as if the fjord was straight. 15 This thesis seeks to resolve the conflict between Freeland and Farmer on the one hand and Blackford and M0rk and Gjevik on the other. In order to do so, a theoretical study of the propagation of a Kelvin wave around two types of bend in a channel of rectangular cross-section was carried out. The method of solution was essentially numerical. It is described in Chapter 2. In addition, cyclesonde current meter and CTD data from Knight Inlet were analysed. It was attempted to express the vertical profiles of the phase and amplitude of the Af2 constituent of longitudinal velocity and density oscillations as a superposition of the first few internal modes, both up- and down-inlet propagating. In this way an estimate of the reflection coefficient at the bend for each mode was made. This analysis is described in Chapter 3. The energetics of Knight Inlet are also discussed in Chapter 3. Freeland and Farmer (1980) estimated the power lost from the barotropic tide in the straight section from Montagu Point (near Protection Point) to the first bend (which includes the sill), and in the sinuous section from the first bend to the head. These estimates were based upon the phase of the surface elevation. They found that the power los3 in the section containing the sill was approximately 20 times that in the sinuous section. They concluded that this high loss was due to the generation of internal waves and tide and hydraulic jumps at the sill. The power loss from the barotropic tide in the sill section was of the order of 7-9 MW. Stacey (1985) developed a simple model to predict the amplitude of the internal modes generated by the sill (see Section 3.11). He found that the total power, summed over the first ten internal modes, radiated by the sill plotted monthly for 18 months followed the total power loss from the barotropic tide as calculated from the phase of the surface elevation very closely. This result implies that most of the energy being removed from the barotropic tide is being fed into the internal tide, and that therefore the internal tide, rather than high frequency internal waves or turbulence at the sill, is the most energetic response resulting from the interaction of the tides with the sill. Any energy in the internal tide that does not radiate out into the open ocean must be 16 dissipated by bottom and side friction or through turbulence to cause mixing. Freeland and Farmer (1983), using CTD data, calculated that the divergence of the total energy flux of the mean circulation (i.e. the rate of work being fed into the mean circulation) was much greater in the straight section than in the sinuous section. The total energy flux of the mean circulation consists of both potential and kinetic energy flux, although the former is an order of magnitude greater than the latter. The mean circulation is driven by the buoyancy input from river runoff, but turbulent mixing causes the potential energy of the mean state to increase. The energy flux divergence in the straight reach was about 2% of the barotropic tidal energy loss. This result is consistent with the idea of a multistep energy transfer (tidal energy loss converted to internal motions which lose energy to turbulence which mixes the water column) where efficiency could not be expected to be high. 1.3.1 Summary This thesis will describe an attempt to resolve the conflict between the results of Free-land (1984) and those of Blackford (1984). The former indicate a significant reflection of the internal tide at the first bend in Knight Inlet, whereas the latter indicate a negligible reflection. The question will be approached both theoretically in Chapter 2 and exper-imentally, using cyclesonde current meter and CTD data from Knight Inlet, in Chapter 3. The energetics of Knight Inlet will also be discussed in Chapter 3. In Chapter 4 the low-frequency component of the velocity signals from the cyclesondes will be presented, with a brief analysis. 17 2. The propagation of a Kelvin wave around a bend in a channel 2.1 Introduction The problem of the reflection of a Kelvin wave from the end of a channel was first solved by Taylor (1920). He found that it was not possible for a pair of Kelvin waves propagating in opposite directions to satisfy the condition of zero flow normal to the end wall, and that a whole spectrum of Poincare waves had to be involved in the reflection process. The problem was re-examined by Hendershott and Speranza (1971) and Brown (1973). In this thesis Taylor's method was used to investigate the behaviour of a Kelvin wave incident upon a bend in a channel. Previous models of Kelvin waves propagating around bends have not included an op-posite wall; they have been bends in straight coastlines. Buchwald (1968) considered a right-angled bend, and Packham and Williams (1968) solved the problem for a bend of general angle. Because there was no opposite wall, a reflected Kelvin wave was not pos-sible, and any energy not transmitted as a Kelvin wave had to be radiated as cylindrical Poincare waves from the corner. They found that there was complete transmission in the case of sub-inertial frequency Kelvin waves, since the Poincare' waves are evanescent. In the case of super-inertial frequency Kelvin waves, Packham and Williams found that there was complete transmission only for wedge angles 2 * + 1 , where n = 1,2,3,... (see Figure 34). Buchwald (1971) also considered the problem of a Kelvin wave propagating along a straight coastline past the mouth of a narrow channel. A Kelvin wave propagating into the channel was generated, as well as cylindrical Poincare waves in the semi-infinite ocean provided that the frequency was super-inertial. He found that the amplitude of the Kelvin wave generated in the channel was proportional to the ratio of the channel width to the wavelength in the channel. In this chapter, then, solutions of the linearised long wave equations in a flat-bottomed channel of rectangular cross-section are presented for the case of a Kelvin wave incident upon a bend. 18 2.2 Governing Equations The linearised long wave equations, for a harmonic time dependence e , w t, separated into vertical and horizontal dependences, are iuU- fV = -Px, (2.1) iuV + fU = -Py, (2.2) ghn(Ux + %) = -iuP. (2.3) (LeBlond and Mysak, 1978, equations 10.34-10.38) (U, V) and P are the horizontal dependences of velocity and specific pressure (pres-sure/density) respectively, and hn is the equivalent depth as derived from a vertical modes calculation. In the case of the barotropic tide, hn is the depth of the channel. Equations 2.1 and 2.2 can be solved for U and V in terms of P: ,~ iuj ( d . d \ = , -Ti iw / 3 . 3 \ s . where T = - . (2.6) The equation of continuity (equation 2.3) then yields the wave equation in P: (V 2 + ~k?)P =0, (2.7) where # ^ feiLz£l. (2.8) At a boundary, the requirement that there be no normal velocity is expressed by the condition where n denotes the outward normal to the boundary, and s the direction along the boundary. 19 2.3 Special Solutions in a Straight Channel The channel width is taken to be 2L, and the length scale L is used to make the space co-ordinates non-dimensional. The co-ordinate system shown in Figure 6 is adopted, where Region 1 is the channel under consideration, with a width of 2 units. The problem then becomes (V2 + k2)P = 0, ( 2 . 1 0 ) with (s- < T^)'-° l l I = 0'2' (2-n) where *2 = ^JLL\ ( 2 . 1 2 ) gK Note that k2 is a non-dimensional parameter. y A (0,0) (0,-2b) u Region 1 "g o r - l QJ erf Poincare modes '(2,0) /f/f/////// I. Region 2 A ^ 7 * M ^ Region 3 jtt f Transmitted / ' / / / / / / ' / / \,f2,-2^>y' * * * * * * f *'* Figure 6 Schematic diagram of the "rectangular bend". The co-ordinate system used in describing solutions in a straight channel is shown. The solution for a Kelvin wave propagating in the direction of increasing y is P = e x p { ( r [ x - l ] - i y ) - 7 r i = f } , ( 2 . 1 3 ) 20 tta(h.-iTh)p"°- (2-14) where U, V and P are proportional to U, V and P. The corresponding expressions for the mth Poincare mode are P =e-k™» (costmx - ^ ^ s i n * m x ^ , (2.16) / T 2 l . 2 _ *2 \ U =e-fc"" ( mf m j sinimx, (2.17) V = « " * - » ^(r2 - l)A:mcos<mz - ^ s i n < m i ) , (2.18) for m = 1,2,3,..., where and km = Wt2m ~ (2.20) In this thesis it is assumed that all the Poincare modes are evanescent, i.e. k2 < (|)2 = 2.4674, and that the frequencies are super-inertial, i.e. r = £• < 1. Equations 2.12 and 2.20 show that all sub-inertial frequency Poincare waves (e.g. diurnal tides) are evanescent, whereas super-inertial frequency Poincare waves (e.g. semi-diurnal tides) will only propagate if the channel is wide enough or the phase speed y/ghn is small enough. Kelvin waves propagate at all frequencies. The corresponding expressions for Kelvin waves propagating in the opposite direction can be obtained be replacing k by —k in 2.13-2.15, or for Poincare waves decaying in the opposite direction by replacing km by — km in 2.16-2.18. For reference, expressions for the non-dimensional wavelength of the incident Kelvin wave A, and the non-dimensional Rossby radius Rn, are A = 2ir^—r , (2.21) Rn = (2.22) 2 1 The appropriate length scale to convert these to dimensional quantities is L, the half-width of the channel. 2.4 Method of Solution in a "Rectangular Bend" The domain of interest is a right-angled bend in a channel, where the width of the channel is two units before the bend, and a different width after the bend (see Figure 6). The domain is split up into three regions: two straight channels (Regions 1 and 3) and a rectangle (Region 2). The solution in Regions 1 and 3 are expressed as a sum of a Kelvin wave propagating away from the boundary with Region 2, and a truncated infinite series of evanescent Poincare modes, which decay away from the same boundary. These waves all have unknown complex coefficients that must be determined. In addition, the solution in Region 1 has a further component, namely an incident Kelvin wave of unit amplitude and zero phase. The solution in Region 2 is expressed as a sum of Kelvin waves propagating in the positive and negative y-direction, the corresponding Poincare modes, and Kelvin waves propagating in the positive and negative z-direction, along with the corresponding Poincare modes. Hence, if the Poincare modes are truncated after M terms, then 6(M+ 1) complex coefficients are required to describe the solution everywhere. All the basis functions satisfy the wave equation, and they satisfy the boundary condi-tion of zero normal velocity everywhere along the boundaries except on the left-hand and lower boundaries of Region 2. The vanishing of normal velocity along these two boundaries remains to be imposed, as do the matching of pressure and velocity along the other two boundaries of Region 2. In fact, it is only necessary for the pressure and normal velocity to be matched across the other two boundaries, since if these two properties match then all other properties, including tangential velocity, must also match. Suppose that the pres-sure matches on either side of a boundary. Then the tangential component of the pressure gradient ^ must match. Yet if the normal velocity (§£ — 1 T § 7 ) matches, then |£ also matches. Hence V? matches, and so therefore does the tangential velocity (^£ + tV|£). 22 Also (V 2 + k 2 ) P — 0 on both sides of the boundary, which implies after differentiation that (V 2 + k2) VP = 0, (V 2 + k2) V2P = 0,..., etc on both sides. Given that P and VP match, it follows by induction that V2P, V3P, V4P,..., etc, also match. The matching process was split up into two stages, which reduced the problem from the inversion of a (complex) matrix of order 6 (M + 1) to the inversion of three matrices of order 2(M + 1). If the second channel is the same width as the first, then two of these three matrices are the same, and there are then only two different matrices to invert. 2.4.1 First Stage The first stage is the matching of normal velocity. The problem of a rectangular basin enclosed on three sides (zero normal velocity) with a prescribed normal velocity along the remaining open boundary is considered. The solution inside the rectangle can be expressed as a sum of a Kelvin wave propagating away from the open boundary as if in a channel, and its associated M Poincar£ waves decaying away from the open boundary, along with a Kelvin wave propagating towards the open boundary and its associated Poincare modes. The solution requires 2(M -I- 1) complex coefficients to be determined. The boundary conditions of zero normal velocity along the two side walls are automatically satisfied by the choice of basis functions, but the zero normal velocity along the fourth boundary, and the prescribed normal velocity along the open boundary must be forced by a suitable choice of coefficients. The (complex) coefficients were calculated using a least squares method, minimising the sum of squares of the moduli of the difference between the normal velocities corresponding to the "fitted" superposition of basis functions and the required normal velocity. The sum of squares was calculated by dividing each boundary into N — 1 intervals, and then evaluating the basis functions and required normal velocities at iV points distributed evenly along each of the two boundaries. The value of N was chosen to be sufficiently large (> 2M) that aliasing of the Poincare modes did not occur. Hence the 2(M + 1) unknowns were calculated by a least sum of squares at 2N locations. 23 It is interesting here to compare this problem with that considered by Pnueli and Pekeris (1967), where they calculated the normal modes in a fully enclosed rectangular basin. The least squares analysis in both problems gives rise to a matrix equation of the form Ax = b, where A is a 2(M + 1) x 2(M + 1) complex matrix, 6 is a 2(M + 1) complex vector, and x is the 2(M + 1) vector of complex coefficients that is to be determined, b is a linear transformation of the prescribed normal velocity along the open boundary of the rectangle, and would be zero for a fully enclosed rectangle. Hence, the relation that Pnueli and Peckeris calculated between k2 and r arises from the vanishing of the determinant of A. For a given k2, r is an eigenvalue of A. However, in this thesis, k2 and T are regarded as given, and the prescribed normal velocity is non-zero. If r happens to be the eigenvalue of A for the particular value of A:2, then this method breaks down. The solution inside the rectangle would not be uniquely defined for a prescribed normal velocity; the addition of any superposition of normal modes of the corresponding closed basin would also be a solution. It should be noted that the problem of a rectangular basin with prescribed normal velocity along any number of its sides is only a small extension to this problem. Because of the linearity of the equations, the solution for prescribed normal velocity along many sides is equal to the sum of the solutions for prescribed normal velocity along each side in turn. The first stage, then, consists in applying this procedure 2Af+3 times with the following prescribed normal velocities to Region 2: (i) The normal velocity corresponding to an incident Kelvin wave of unit amplitude and zero phase in Region 1. (ii) The normal velocity corresponding to a reflected Kelvin wave of unit amplitude and zero phase in Region 1. (iii) The normal velocity corresponding to each of M evanescent Poincare modes, each of unit amplitude and zero phase, in Region 1. 24 (iv) The normal velocity corresponding to a transmitted Kelvin wave of unit amplitude and zero phase in Region 3. (v) The normal velocity corresponding to each of M Poincare modes, each of unit amplitude and zero phase, in Region 3. Note that if Region 2 is square, then (iv) and (v) are identical to (ii) and (iii). This is the reduction of work that was referred to earlier. 2.4.2 Second Stage The second stage is the matching of pressure. For each of the 2M + 3 types of wave in the straight channels, the first stage of the calculation returns the pressure field inside Region 2 such that the normal velocity matches. For each of the 2M + 3 types of wave of unit amplitude and phase, the difference between the pressure in Region 2 and that in Regions 1 and 3 is calculated along their two common boundaries. If the wave exists in Region 1, then the pressure in Region 3 is taken to be zero, and vice-versa. The aim of the second stage is to find a combination of the 2M + 3 types of wave in the straight channels that makes the sum of these "pressure anomalies" equal to zero. The amplitude and phase of the incident Kelvin wave are arbitrarily set to unit and zero, leaving 2(M + 1) complex coefficients to be calculated. This calculation is also done by a least sum of squares method, minimising the sum of squares of the modulus of the "pressure anomalies* at 2N locations. This procedure gives rise to a matrix equation of the same order as in the first stage, of the form Ay = 6. Here y is the vector of complex coefficients and 6 is a linear transformation of the incident Kelvin wave. If there were no incident wave then 6 would be zero, implying that y = 0 (the trivial solution). If the incident wave contained some decaying Poincare modes, then 6 would be changed. Loosely speaking, the first stage of the calculation describes how a rectangle responds to various forcing terms, and the second stage describes how the rectangle and the two straight channels are connected. This method is an adaptation of the collocation method. However, in the collocation method boundary conditions are applied at the same number 25 of points as there are unknown coefficients. In this method at least twice as many points are used, and the conditions are applied in a least squares sense. 2.5 Results for the Rectangular Bend The following results were obtained using 15 Poincare modes ( M =15), and doing a least squares fit at 100 points along each boundary (N = 100). The success of each run was judged by the extent to which the pressure and velocity fields were continuous across the boundaries, and the extent to which the sum of the reflected and transmitted energy fluxes equaled the incident flux. The continuity of velocity can be evaluated by looking at the plot of current ellipses in Figures 10, 12 and 16: for points on the common boundary between two regions two ellipses are plotted according to the solution in each region. The two ellipses should be indistinguishable. When the two channels were of equal width, the relative energy fluxes were calculated by simply squaring the absolute value of the coefficient of the relevant wave in the solution, be it the incident, reflected or transmitted wave. The reflected R, and transmitted T fluxes were expressed as a percentage of the incident flux. There should be no energy loss, since there is no dissipation in the model. The Poincare' modes transmit no energy, since they are evanescent. There are three free parameters in the problem, namely T, the period of the oscillation, k, which is proportional to the ratio of the channel width to the Rossby radius, and 6, the ratio of the second channel width to the first. Hence, the width of the second channel is 2b units in the model. In the approach used, two of these parameters were kept constant, and the third was varied. The standard values used to start each parameter search were r = 0.803, fc2 = 7.02 x 10 - 3 , 6= 1. These were chosen to represent the first internal mode of the M2 constituent of the internal tide in Knight Inlet. The following parameters were used to arrive at these values: u =1.40518 x 10~4 rad/'sec, f =1.12887 x 10 - 4 rad/sec (at latitude 50.7° N), L =1 km, cn = \/ghn =100 cm/sec. 2.5.1 Parameter Search 1: r = 0.803, 6 = 1, variable k2 Figure 7 shows a plot of the reflection coefficient R against k2 for constant r and 6. k is remarkably small (< 10%) until just before the cutoff wavenumber at which Poincare waves start to propagate, where there is a pronounced peak. For small k2 (< 0.2), R is roughly linear in k2, and the curve approaches the line R = 54.4 A2 % as k2 —* 0 (see the blowup in Figure 8). As a check on the computation, R and 100% — T are generally in excellent agreement, except near the peak. Figure 7 R vs k2 for constant r = 0.803. 6 = 1. The value of k2 for which the first Poincare mode starts to propagate is marked with an arrow. Circles = R, triangles = 100% — T . 27 o , , , , , , , , , , 0 0 . 0 5 0 . 1 0 . 1 5 0 . 2 0 . 2 5 k 2 Figure 8 Blowup of Figure 7 for small A:2. The straight line is R — 54.4 A;2 %, which is the least squares fit for k2 < 0.05. The default values for Knight Inlet lie well within the linear part of the relationship. Figures 9 and 10 show the pressure and velocity fields for these values of the parameters. Note that because of the assumed etwt time dependency, waves propagate in the direction of decreasing phase (Arg P). Velocity matching is not perfect, and there is a singularity in the velocity at the inside corner (see discussion in Section 2.6). The current ellipses in this and all figures showing velocity fields are scaled so that they are as large as possible without having to be plotted on top of each other. Figures 11 and 12 show the pressure and velocity fields for A;2 = 2.25. This value lies in the peak of the curve of Z vs A;2. Note the presence of amphidromes where the magnitude of the pressure oscillation is zero, and hence its phase is ambiguous. Non-zero normal velocity is visible in two of the ellipses, whereas the singularity at the inside corner has decreased. This latter phenomenon could be due to the decreased Rossby radius which causes an increased cross-channel decay of Kelvin waves, although the reflected wave is 28 Figure 9 Pressure field for k? = 7.02 x I O - 3 , T = 0.803, 6 = 1. Solid lines = \P\ contours, dashed lines = Arg P contours. A = 44.7, Rn = 8.8, Z = 0.38 %, 100% - T = 0.38 %. i I \ \ \ \ \ \ \ 1 \ \ \ \ \ \ \ \ \ \ Figure 10 Velocity field corresponding to Figure 9. 29 Figure 11 P r e s s u r e f i e l d f o r k2 = 2 . 2 5 , r = 0 . 8 0 3 , 6 = 1 . S o l i d l i n e s = \P\ c o n t o u r s , d a s h e d l i n e s = Arg P c o n t o u r s . A = 2 . 5 , Rn = 0 . 4 9 , Z = 2 4 . 4 8 % , 1 0 0 % - T = 2 4 . 2 2 % . 0 1 1 0 0 © 1 t o CO - 1 / t? CP 1 o O o • o o O 0 e 0 c & • 0 O O 0 . G G 0 G O 0 0 o o 0 6 e 0 O o / & o 5! Figure 12 V e l o c i t y field c o r r e s p o n d i n g t o Figure 11. 30 °0 0.2 0.4 0^6 ' 0.8 l7o F i g u r e 13 Z vs r for constant k2 = 7.02 x 1 0 - 3 . 6 = 1 . Circles = Z, triangles = 100% - 7 . The curve is Z = j^fr which is the least squares fit to a function of that form, also of considerable magnitude. 2.5.2 Parameter Search 2: k2 = 7.02 x 1 0 ~ 3 , 6 = 1 , variable r Figure 13 shows a plot of Z vs r for constant k2 and 6. A dependency was observed, with the constant of proportionality being 0 .135. This result, combined with the constant of proportionality from the linear portion of the relationship between Z and k2 leads to the empirical relationship 19 3 A: 2 Z = '_ t 2 % for small k2 ( < 0.2). (2.23) This is equivalent to je = 1 9 . 3 — ( 2 . 2 4 ) or je = 1 . 9 6 r 2 % , (2.25) where r is the ratio of the channel width to the wavelength of the incident wave. 2.5.3 Parameter Search 3: k2 = 7.02 x 10 - 3 , r = 0.803, variable b Figure 14 shows the relationship between R and b. As expected, the amount of energy transmitted through the second channel decreases to zero as the width of the second channel decreases to zero. There is, however, a large discrepancy between R and 100% — 7 , indicating that the solution is not reliable for 6 < 1. This breakdown is confirmed by Figures 15 and 16, where it can be seen that the velocity fields do not match. Figure 14 vs fe for constant r = 0.803 and k2 = 7.02 x 10~3. Circles = R, triangles = 100% - 7 . The energy fluxes can not be simply calculated from the square of the magnitude of the relevant coefficient in the solution expansion, because the channels are no longer of the same width. The pressure and velocity fields both have a cross-channel structure proportional to txlRn, and so the energy flux will depend on the channel width as fb (es'Rn)2 dx = Rn sinh (2.26) J-b Rn Hence, the square of the magnitude of the coefficient of the transmitted wave was multiplied by a factor of ^ ^ ^ J R " ' ) D e ^ o r e plotting to make it compatible with the fluxes obtained 32 20° 15° 10° 5° Figure 15 Pressure field for k2 = 7.02 x 10 - 3 , r = 0.803, 6 = 0.3. Solid lines = \P\ contours, dashed lines = Arg P contours. A = 44.7, Rn = 8.8, Z = 35.72%, 100% - T = 22.89%. Figure 16 Velocity field corresponding to Figure 15. 33 by squaring the magnitudes of the coefficients of the incident and reflected waves. For large values of Rn (0(1) or larger), this factor is approximately equal to b. 2.6 Discussion of the Solution Near the Inside Corner Pnueli and Pekeris (1968) showed that near a corner of angular opening (fi < 1), the solution is singular, and the pressure is approximated by the expression P—* ^(T cos fiO — ism fid) + constant as r —> 0, (2.27) where (r, 9) are polar coordinates whose origin is at the apex of the corner. Thus velocities are 0(r'*""1), and are clearly singular in the present case where fi = |. The basis functions chosen for our solution are incapable of producing such a singular-ity, since a Fourier series representation of r~* does not converge. This difficulty manifests itself in the fact that it is impossible to match velocities in the region of the inside corner. The corner itself is the only point in the whole domain that is in each of the three regions, and hence there are three expressions for the velocity there. In Region 1 a condition of zero flow in the x-direction has been imposed, and in Region 3 a condition of zero y-direction flow has been imposed. The velocity in Region 2 has been constrained to match these velocities, at least in a least squares sense. The only way in which all these conditions can be consistent is if there is zero velocity at the inside corner. However, this constraint does not admit solutions of the form given by equation 2.27. Figure 10 demonstrates the problem well. At the inside corner three current ellipses are plotted. The two calculated from the expressions for the solution in Regions 1 and 3 are simply straight lines parallel to the local boundary, whereas the third in Region 2 has its axis aligned at a 45° angle between them, as a consequence of the least squares matching. This observation might indicate that the solution shown in Figure 10 is actu-ally approaching the correct solution in a bend with a slightly smoothed inside corner. Unfortunately, there is the added deficiency of the failure of the current ellipses to match perfectly close to the inside corner. As one looks further away from the inside corner the matching improves, and becomes satisfactory after about ^ of the channel width. 34 All that can really be said is that the solution is inadequate in the region of the inside corner. Hopefully, however, the details of the flow near the inside corner do not affect the general character of the solution, so that the solutions presented here for the rectangular bend give reasonable approximations to the amount of energy reflected and transmitted. Because of this problem at the inside corner, a bend without a sharp discontinuity in the direction of the boundary was considered: the so-called "annular bend." As will be seen later, the results from the annular bend study are in general agreement with those from the rectangular bend, giving support to the idea that the results for the rectangular bend are in some sense close to the "true" solutions. 2.7 Method of Solution in an "Annular Bend" Figure 17shows the geometry of the annular bend. Again, the domain was split up into three regions, namely two straight channels and a sector of an annulus. This type of bend has two advantages over the rectangular bend . First, it is of variable angle <j>, and, second, it has no obtuse angles in the boundary. It has the slight disadvantage of requiring that the two channels be of equal width. The previous free parameter of the second channel width has been replaced by two new free parameters: the angle of the bend <f>, and the sharpness of the inside corner as measured by the inner radius of the annulus, r 0 . The solution in Regions 1 and 3 was represented in the same way as in the rectangular bend, but in Region 2 it was expressed as a superposition of "radial" Kelvin and Poincare modes. If the solution P(r, 6) is separated into P(r,9) = R(r)&(9), (2.28) then the wave equation becomes e" = -i/2e (2.29) and R" + -R' + (-^r + k 2 ) R = 0. (2.30) i/2 is the separation constant. The solution to the 9 problem is 0 = e±iv0. (2.31) 35 Figure 17 Schematic diagram of the "annular bend". The boundary conditions of zero normal flow at the inner and outer radii of the annulus become TU at r = r0, r 0 + 2. (2.32) Ur = R' + —R = 0 U r Equations 2.30 and 2.32 form an eigenvalue problem, whose eigenfunctions form a complete set. The corresponding expression for the velocity component parallel to the channel is Vg = —R + irR'. r (2.33) Equation 2.30 is Bessel's equation. For radial Kelvin waves, the eigenvalue v is real, and the solution would be R(r) = AJv{kr) + BYu(kr), (2.34) where Jv and Yu are Bessel functions of order v, and A and B are undetermined complex coefficients. The boundary conditions at r = r0, r 0 + 2 would determine these coefficients. 36 However, for evanescent radial Poincare modes, u is nearly imaginary (it is a com-plex number whose phase is close to 90° or 270° — see Section 2.8), and the solution of equation 2.30 would be in terms of Bessel functions of complex order. Because of the dif-ficulty of computing such functions it was instead decided to solve the eigenvalue problem numerically. A fourth order Runge-Kutta scheme was used with iterative refinement of the eigen-value to satisfy the boundary condition at the endpoint of the integration (a "shooting" technique). This scheme was also used for the radial Kelvin waves, since no extra computer programming was required. The scheme was tested on an annulus with a very large inner radius compared to the width, and it was indeed found that the eigenvalues approached those of Kelvin waves and Poincare modes in a straight channel (see Section 2.8). The solution in the annulus was therefore expressed as a superposition of two radial Kelvin waves propagating in opposite directions around the annulus, and the two corre-sponding truncated sets of Poincare modes, decaying in opposite directions. Again, the method of solution for the problem of an incident Kelvin wave of unit amplitude and zero phase was to solve for the 4(M + 1) complex coefficients in two stages. Each stage involved inverting a square complex matrix of order 2(M 4- 1). The first stage is the matching of normal velocities across the boundaries between Regions 1 and 2, and between Regions 2 and 3, by the method of least squares at N locations along each boundary. For each of the 2M + 3 types of wave in the straight channels, the corresponding normal velocity was prescribed at the appropriate boundary of the annulus (zero on the opposite boundary), and the response of the annulus was calculated. The second stage is the matching of pressure, again by a least squares fit of the pressure anomalies for each of the 2M + 3 straight channel waves, to make the total pressure anomaly vanish. 37 2.6 Discussion of the Results of the Eigenvalue Problem in the Annulus The eigenvalues i/, for the radial Kelvin and Poincare modes were found to be of the same order of magnitude as those in a straight channel of the same width. A value must be assigned to the radius when converting from polar coordinates to cartesian coordinates; this value was taken to be the average radius of the annulus (r0 + 1). As the radius of the annulus tended to infinity, the eigenvalues tended to the eigenvalues in a straight channel, as expected. The eigenvalues for Kelvin waves in a straight channel are real, and for evanescent Poincare modes are imaginary. In both cases, the eigenvalues corresponding to waves propagating (or decaying) in the opposite direction are the negative of the original eigen-values. However, in a bend, when there is rotation, a "splitting" occurs. As / is increased from zero, or as r 0 is decreased from infinity, the eigenvalue for a Kelvin wave propagating around a bend to the left increases slowly above its original value. The eigenvalue for a Kelvin wave propagating around a bend to the right decreases slowly below its original value. For sharp bends (ro = 0.1 say), this discrepancy between the eigenvalues for bends of opposite senses is large. This effect can be intuitively understood in terms of the differ-ence in path lengths for a Kelvin wave bound tightly to the right-hand side of the channel. Such a wave propagating around a bend to the right will have a larger angular velocity than a wave propagating around a bend to the left for a given linear phase speed because of the shorter path length. Hence its angular wavenumber will be smaller. The eigenvalues for Kelvin waves do, however, remain real. The cross-channel pressure distribution also stays real, and the along-channel velocity component is still imaginary (see equation 2 .15) . However, a small, real cross-channel velocity component is introduced. This quantity is small at first, but increases as ro decreases. The eigenvalues for the evanescent Poincare modes are no longer imaginary, but a small real part is introduced as the sharpness of the bend is increased. It can be shown that the eigenvalue for a Poincare mode decaying around a bend to the right is the negative of the complex conjugate of the eigenvalue for one decaying around a bend to the left. This small real part to the eigenvalues 38 is equivalent to a long wavelength oscillation which is quickly damped out by the presence of the imaginary part. The real part always has a sign such that the oscillation propagates around the bend in an anticyclonic direction (clockwise in the northern hemisphere). 2.9 Results for the Annular Bend The following results were obtained using 10 Poincare modes ( M = 10), and doing a least squares fit at 20 points along each boundary (N = 20). It was found that it was not necessary to use such a high value for M as was used in the rectangular bend analysis (where M = 15) since energy conservation could be achieved with a lower number of Poincare modes. Also N was reduced to the lowest number such that aliasing of the cross-channel structure of the highest Poincare mode would not occur. These steps were taken to reduce computing costs. Again the approach was to vary one of the parameters while keeping the other three constant. The standard values were the same as in the rectangular bend, with the addition of ro = 0.5 and <f> = +90° (a right-angled bend to the left). 2.9.1 Parameter Search 1: r = 0.803, <f> = +90° , r 0 = 0.5, variabie k2 Figure 18 shows a plot of the reflection coefficient R against k2 for a right-angled bend to the left with constant r and r 0 . The shape of the curve is very similar to that in Figure 7, with a pronounced rise just before the cutoff. Again, the relationship is roughly linear for small k2 (k2 < 0.025), and the curve approaches the line R = 5.34 k2 % as k2 —• 0. There is an order of magnitude difference in R between Figure 18 and Figure 7; this difference can be attributed to the difference in sharpnesses of the two types of bend (see Subsection 2.9.5). In all of the cases comprising Figure 18 the current ellipses were observed to match very well, and no normal velocity component was observed at the boundaries. For example, in Figures 21 and 23, which correspond to Figures 10 and 12, the current ellipses match very well along the common boundaries between regions. The pattern of amphidromes in Figure 11 is different from that in Figure 22 because of the difference in reflection coefficients. Agreement between R and 100% — T was very good. 39 CO esi-Figure 18 Z vs k2 for constant r = 0.803. r 0 = 0.5, $ = +90°. The value of k2 for which the first Poincare mode starts to propagate is shown by the right-hand border. Solid line = Z, dashed line = 100% — T . Figure 19 Blowup of Figure 18 for small &2. The straight line is Z = 5.34 A;2 %, which is the least squares fit for A:2 < 0.025. 40 A = 44 7 R l-l To™' d a s h e d = l / Z c ^ f ' r ° = ° - 5 ' * = + 9 0 c ' n ~ 8-8> * = °-°38%, 100% - T-omy"'' Figure 21 Velocity field corresponding to ^ r e m 41 Figure 22 Pressure field for k2 = 2.25, T = 0.803, r0 = 0.5, <f> = +90° . Solid lines = \P\ contours, dashed lines = Arg P contours. A = 2.5, Rn = 0.49, je = 0.394%, 100% - 7 = 0.408%. Figure 23 Velocity field corresponding to Figure 22. 42 2.9.2 Parameter Search 2: r = 0.803, r0 = 0.5, various k2, variable <p Figure 24 shows the effect of varying the bend angle <f>. Two statements can be made. The first is that there are certain angles for which there is zero reflection. The second is that there is no significant difference in reflection coefficient for a bend of a given angle to the left and a bend of the same angle to the right; i.e. Figure 24 would be symmetrical about the £-axis <f> = 0. Results for negative <f> are not actually plotted because they are the mirror image of the values plotted. However, the results for negative <f> would follow the R curve for positive <j> as closely as the 100% — T curve does. Figure 24 R vs <p for various values of k2. r = 0.803, r 0 = 0.5. Solid lines = R, dashed lines = 100% - T . The effect of increasing k2 is to compress the peaks laterally (i.e. make the zeros closer together), and to make them higher. The case k2 = 2.467 was also calculated, but was not included in Figure 24 because it would have gone off scale. It can be seen in Figure 32 on different axes. Examination of Figure 24 reveals why there is a dip in Figure 18 at around k2 = 0.75. Figure 18 is a plot of R against k2 for constant <f> = 90°, and Figure 24 shows that the dip at k2 » 0.75-1.00 is due to the passing of the first zero reflection over the value 4> = 90°. Figures 25-28 illustrate the fact that a bend of 30° to the left and a bend 43 of 30° to the right have similar but not identical solutions, yet the captions show that they have the same reflection coefficient (36%). Energy is preserved, and the pressure and velocities match. The value used for ft2 is 2.467, which is just before the cutoff value at which the first mode Poincare wave propagates. This value was chosen to give a large J2. In fact, the e-folding length for the decay of the Poincar6 mode is 50L (i.e. 25 times the channel width), and the solution is dominated by this mode (its amplitude is an order of magnitude higher than the Kelvin waves). This is the reason for the characteristic change in phase of the pressure in mid-channel by 180°, and the maximum in velocities in mid-channel in Figures 25-28 (see equations 2.16 and 2.17 with very small A:2). There is no along-channel phase propagation because the Poincare mode is evanescent. The Poincare mode does not contribute any energy fiux, but it does dominate the solution near the bend (within, say, 1001 of the bend). The generation of a large amplitude Poincar£ mode seems to be associated with the occurrence of high reflection coefficients for the Kelvin wave and occurs when the Poincare mode has a large decay scale (almost propagating). It is interesting to note that Brown (1973) found a similar result in his study of the reflection of a Kelvin wave from the end of a channel (the Taylor problem). He allowed the first Poincare mode to propagate, so that any energy not reflected as a Kelvin wave was reflected as a Poincare wave. He found that just above the critical frequency at which the first Poincare mode starts to propagate (i.e. k2 just above (f )2), this mode becomes the principal energy-reflection mechanism. The Poincare mode rapidly asserts itself as the dominant reflection mechanism. It therefore seems likely that soon after the first PoincarS mode starts to propagate it will rapidly become a dominant energy radiation mechanism in the bend too, although in which direction is open to speculation. The amplitude of the Poincare mode has already been seen to become very large for k2 just less than (f )2, although as long as the mode is evanescent it cannot contribute to the energy flux budget. However, as A:2 increases above i 44 Figure 25 Pressure field for k2 = 2.467, r = 0.803, r 0 = 0.5, <j> = +30°. Solid lines = \P\ contours, dashed lines = Arg P contours. A = 2.4, Rn = 0.47, £ = 35.93%, 100% - T = 36.00%. o o o o o Incident wave Figure 26 Velocity field corresponding to Figure 25. The curved line at the inside corner is a result of several short ellipses running into each other. 45 Figure 27 Pressure field for k2 = 2.467, r = 0.803, r 0 = 0.5, <f> = - 3 0 ° . Solid lines = \P\ contours, dashed lines = Arg P contours. A = 2.4, Rn = 0.47, R = 35.88 %, 100% - T = 36.42 %. Figure 28 Velocity field corresponding to Figure 27. 46 (f )2, it is only necessary for the magnitude of that mode to remain large for the energy flux to increase rapidly, since the group velocity cg will increase rapidly anyway. Brown showed that if k2 = ( § ) 2 (1 + 5), then cg = 0[6$). 2.9.3 Parameter Search 3: k2 = 7.02 x I O - 3 , r 0 = 0.5, <f> = 90°, variable T Figure 29 shows the effect of varying r with a constant, small k2 in a right-angled bend. Just as in the rectangular bend (Figure 13 ), there is good agreement with a curve of the form at least for r < 1. The constant of proportionality is 1.33 x 10 - 3 for the case ro = 0.5. This is equivalent to Z = 0.191 r2%, (2.35) where again r is the ratio of the channel width to the wavelength of the incident Kelvin wave. This result can be compared to 1.96 r 2 % (equation 2.25) for the rectangular bend. 0.2 0.8 1.0 0 .4 0.6 T Figure 29 Z vs r for constant k2 = 7.02 x I O - 3 and (p = +90° . r 0 = 0.5. Circles = Z, triangles = 100% - T . The curve is Z = 1 - 3 3 ^ 3 — %, which is the least squares fit to a function of that form. The method of solution breaks down for r > 1 (sub-inertial frequencies). For these 47 values of r, k2 was taken to be —7.02 x 10 - 3 . This choice preserves the wavelength of the Kelvin waves and the Rossby radius. Poincare waves can never propagate at sub-inertial frequencies. There was found to be a large discrepancy between Z and 100% — 7 . Also, inspection of the current ellipses revealed that velocity matching was bad. Figures 30 and 31 show the solution in the case T = 0, which corresponds to no rotation. The Rossby radius is infinite, and there is therefore no cross-channel decay of the Kelvin wave. The Kelvin wave becomes a regular gravity wave. There is very little variation in the amplitude of P in the whole domain; the solution is more or less a plane wave propagating directly around the bend with no reflection. The current ellipses are almost linear, which is associated with the fact that the Poincare modes are of very small amplitude and short decay scale (an e-folding length of 0.6£). Kelvin waves have no cross-channel velocity in straight channels, and very little cross-channel velocity in annular bends. On the other hand, odd Poincare modes have a maximum in the cross-channel velocity at the centre of the channel, which gives rise to round current ellipses there. The presence of large, slowly decaying Poincare modes seems to be associated with large reflection coefficients for the Kelvin wave, which therefore correlates with round current ellipses. 2.9.4 Parameter Search 4: r0 = 0.5, various k2, various r, variabie <j> Figure 32 shows plots of Z against bend angle <f> for various values of k2 and r. For small k2 (Figure 32a), larger values of T lead to larger values of Z for all angles. The system is in the j—^ regime. However, for large k2 (Figure 32c), larger values of r lead to smaller values of Z. For intermediate values of k2 (Figure 32b), the curves seem to start off in the regime for small angles (say <p < 40°), but move into the large A;2 regime for larger angles. Figure 32c is in keeping with the results of Packham and Williams (1968). Their Figure 1 is presented in Figure 34 on reversed axes to conform with Figure 32c. They investigated the propagation of a Kelvin wave around a wedge of variable angle. In their 48 28° Figure 30 P r e s s u r e f i e l d f o r k2 = 7 . 0 2 x I O - 3 , r = 0 , r 0 = 0 . 5 , <j> = + 9 0 ° . S o l i d l i n e s = \P\ c o n t o u r s , d a s h e d l i n e s = Arg P c o n t o u r s . A = 7 5 . 0 , Rn = o o , Z = 0 . 1 4 1 %, 100% - T — 0 . 1 4 4 % . Figure 31 V e l o c i t y field c o r r e s p o n d i n g t o Figure 30. 49 Figure 32 Jl vs <f> for various values of k2 and r. ro 50 = 0.5. case no reflected Kelvin wave was possible. Energy that was not transmitted as an ongoing Kelvin wave was radiated as cylindrical Poincar£ waves (see Figure 33). These Poincare waves could propagate because the frequency was super-inertial, and there was no channel. They also found angles at which there was complete transmission, and that larger values of r gave lower values for Jl. Their problem gives no meaning to A:2, since there is no natural length scale with which to non-dimensionalise the incident wavelength. However, large k2 ( as in Figure 32c) in the annular bend would be expected to correspond most closely with their problem, since then the Poincare modes are almost propagating, and they do have propagating Poincar6 modes. The observation that the peaks in Figures 32 and 24 were compressed as A;2 increased led to the idea of plotting the first angle of complete transmission <f>0, against the incident wavelength A. The result is shown in Figure 35, which includes results from various values of A;2 and r. A remarkable linear relationship was found. This result means that the width of the "lobes" in the Jl-<f> plots is directly proportional to the incident wavelength. This result is also found in diffraction theory in other fields such as optics and electromagnet ism (e.g. Hecht and Zajac 1974 Chapter 10). It suggests that perhaps a better way to represent the data would be to scale the c^ -axes with A, since then the angles of complete transmission would lie at the same position. Then only the amplitudes and shapes of the diffraction patterns would vary; some secondary lobes are bigger than the primary lobe. This relationship between the width of the lobe and A is also consistent with the linear relationship between Jl and A - 2 for large A and fixed <f>. The relationship between Jl and <t> is almost quadratic for small <f> (i.e. Jl oc <f>2, see, for example Figure 32a), so scaling <f> with A is almost equivalent to scaling Jl with A 3 . This means that plots of Jl against X4> for various values of A:2 and r will be approximately the same for small \<j>. After the initial portion of the curve, the plots will diverge according to their different peak values. 51 0 Cylindrical Figure 33 Bend in a straight coastline studied by Packham and Williams (1968). o o Figure 34 100% - 7 vs <f> for Packham and Williams' bend. (After Packham and Williams 1968.) 52 w ~ l I I I I I I 0 1 2 3 4 5 6 7 X Figure 35 Angle of first zero reflection vs A for r 0 = 0.5. The straight line is the least squares fit. 2.9.5 Parameter Search 5: r = 0.803, various k2, various r0, variabie <f> So far all the results for the annular bend have been for a fixed sharpness, ro = 0.5. Figure 36 shows the effect of varying this inside radius for medium and large values of k2. In both cases increasing the sharpness (decreasing ro) gives rise to higher reflection coefficients. Not only are the peak values of the lobes increased but the constant of proportionality between the width of the lobe <f>Q and the incident wavelength A is increased. For very sharp bends (ro = 0.1) there is a significant discrepancy between R and 100% — T and the velocities fail to match precisely. Also there are large velocities in the region of the inside bend, especially for small A;3. In effect, all the problems encountered with the rectangular bend start to show. Figures 37 and 38 show the solution in a gradual bend. The parameters are the same as in Figures 25 and 26 except for the inside radius, which has been increased from 0.5 to 2.0. This change has caused the reflection coefficient to drop from 36% to 0.7%. The 53 Figure 38 Z vs <f> for various values of r 0 and k2. r = 0.803. Solid lines = Z, dashed lines = 100% - T. 54 Figure 37 Pressure field for Jfc2 = 2.467, T = 0.803, r 0 = 2.0, <f> = +30°. Solid lines = \P\ contours, dashed lines = Arg P contours. A = 2.4, R„, = 0.47, Z = 0.733%, 100% - 7 = 0.738%. 0 0 o 0 o 0 o o o o o 0 0 0 ° n Q u O O 0 0 0 o o o o O u Q o ^ 0 7 0 0 Incident wave Figure 38 Velocity field corresponding to Figure 37. 55 solution is no longer dominated by Poincare modes. It is remarkable that bends to the left and bends to the right of the same angle (e.g. Figures 25-28) have similar solutions and identical reflection coefficients, whereas changing the inside radius and leaving the angle alone (e.g. Figures 37 and 38) has a major impact upon the solution and reflection coefficient. 2 . 1 0 Conclusion It was found that the method of solution did not work in a bend with an obtuse discontinuity in the direction of the boundary (i.e. the rectangular bend) because there is a singularity in the velocity field there. The method worked well in a bend with smooth sides (i.e. the annular bend). The bend acts as a "diffraction grating." For certain bend angles there is total transmission, and between these angles there are "lobes" of high reflection. These lobes form a diffraction pattern which spreads out as the incident wavelength increases. The width of the lobes is proportional to the wavelength, as in other types of waves in optics, electromagnetism etc. The constant of proportionality depends on the sharpness of the bend. It was found that bends to the left give rise to the same reflection coefficient as bends to the right of the same angle, irrespective of the degree to which the incident Kelvin wave is trapped against one wall (as measured by the Rossby radius). For bends of fixed angle the reflection coefficient was found to be proportional to r 2 for small r, where r is the ratio of the channel width to the incident wavelength. (This result is similar to that of Buchwald (1971) for a Kelvin wave propagating past the mouth of a narrow channel (see Section 2.1).) The constant of proportionality increases as the bend becomes sharper. Just below the critical frequency at which PoincarS mode propagation becomes possible, this mode starts to dominate the solution, although while this mode is still evanescent it cannot contribute to the energy flux budget. This is consistent with Brown's (1973) result just above the critical frequency in a closed channel. The method of solution breaks down inexplicably for sub-inertial frequency waves. The 56 method also breaks down as the inside radius of the annular bend tends to zero, because of the singularity in the velocity field near corners whose angular opening is greater than 180°, as shown by Pnueli and Pekeris (1967). To solve the problem correctly in a geometry that has such a corner, the form of the solution must be capable of expressing a singularity. The reflection coefficient of the internal tide in Knight Inlet at the 90° bend was found to be very small. It would be much less than 1% even for modes of order 4 or 5. However, the results of this chapter are equally applicable to a bend in any channel, and to barotropic tides. 57 3 . Observations of the Internal Tide in Knight Inlet 3 . 1 Intro dnction In this Chapter data taken from Knight Inlet in the summers of 1981 and 1983 will be used to try to determine how much of the internal tide propagating up-inlet from the sill is reflected by the first bend, and how much is transmitted. The first few theoretical normal modes for a flat-bottomed ocean (calculated using measured IV^-profiles) are fitted to the vertical profiles of amplitude and phase of the Af2 constituent of longitudinal velocity and density oscillations. In this way the amplitude and phase of the first few normal modes can be estimated at each station, for waves propagating both up-inlet and down-inlet. The results of this analysis at the station just before the bend should, in theory, give a reflection coefficient for each mode. Sections 3.2-3.3 describe the data and give the results of the harmonic analysis. The amplitude and phase of each constituent are included for completeness, although only those of the M 2 constituent will be used in the subsequent analysis. Sections 3.4-3.7 give the theory of normal modes and normal mode fitting, including the treatment of the barotropic mode. Section 3.8 describes how the amplitude af the various modes can be converted into energy flux estimates for each mode. The actual normal modes used in the fitting are presented in Section 3.9, and a discussion of how they differ from Farmer and Smith's (1980) modes given in Section 3.10. Finally, the results of the fitting are presented in Section 3.11. In Section 3.12 the energetics of Knight Inlet are discussed, with reference to the work of Stacey (1985) and Freeland and Farmer (1980). The Chapter is concluded in Section 3.13. 58 3.2 Description of the Data In an attempt to study the internal tide in Knight Inlet, four cyclesonde current meters were deployed in the summers of 1981 and 1983. In total they produced 11 time series: three 12-day records from 30th July - 11th Aug 1981 (Julian days 211 - 223), four 32-day records from 20th July - 21st Aug 1983 (Julian days 201 - 233) and four 26-day records from 1st Sept - 27th Sept 1983 (Julian days 244-270). The positions of the stations are shown in Figure 1. In 1981 the instruments were located at Protection Point (towards the mouth of Knight Inlet), Lull Bay (2 km seaward of the sill), and at Tomakstum Island (8 km seaward of the first big bend). In July 1983 an additional instrument was deployed off Adeane Point (3 Arm landward of the bend). All these stations were in mid-channel. However, in September 1983 the mooring at Lull Bay was replaced by a second mooring at Tomakstum Island. The two moorings at Tomaks-tum Island were placed on either side of the channel, each approximately 500 m from the centreline of the channel. The locations of these two stations are shown in Figure 39. Figure 39 Detailed map of the first bend in Knight Inlet. The cyclesondes are automatic-profiling current meters (Van Leer et al., 1974). Built into the body of the instrument is an inflatable bladder and a helium tank. The valve to inflate and deflate the bladder with helium is under electronic control. When the bladder is inflated the instrument has a positive buoyancy of about 500 g, whereas when it is deflated 5 9 it has a similar negative buoyancy. Ballast can be added to or removed from the instrument before each deployment to compensate for an operating environment of different density from its previous deployment. The cyclesonde moves up and down a plastic coated steel wire, which passes through a block in the body of the instrument, under its own buoyancy. The wire is attached to a railway wheel at the bottom, and is kept tight by a subsurface buoy at a depth of approximately 10 m. The vertical travel of the instrument is constrained by two bumpers, the upper one at around 12— 17 m, depending on the state of the tide, and the lower one at no more than 200 m for these deployments. Sometimes the instrument did not reach the upper bumper if the pycnocline was relatively deep, because of insufficient buoyancy. The current is measured by two Savonius rotors which provide independent information on the speed of the water relative to the instrument. In addition there is an internal compass, and temperature, conductivity, pressure and pitch sensors. Salinity and density can be calculated from the conductivity and temperature data. In 1981 the profiles were set to commence every l£ hours at Tomakstum Island and Protection Point, and every 1 hour at Lull Bay. The sampling interval, that is the interval over which revolutions of the rotors are counted and data recorded on cassette tape, was set to be 1 minute. However, in 1983 the profiles were set to occur every 3 hours at all stations, and a variable sampling rate was used. In the variable rate mode, the data were recorded every minute for one hour during the profile, and every 5 minutes for the next 2 hours while the instrument was sitting against the top or bottom bumper. This mode is more efficient on tape usage than the fixed rate mode by not wasting tape when not profiling. Each profile took approximately 30-45 minutes, depending on the depth of the bottom bumper and on the weight of helium remaining in the tank. The data were transferred back in the laboratory from cassette tape to computer tape, so that the processing could be done on the mainframe computer. After the data had been converted into physical units they were edited on the terminal. Suspect features such as spikes in the conductivity record (which could be caused by organisms swimming 60 in the conductivity cell) or in the heading (which could be caused by a faulty reading from the compass) were listed and looked at, with the option of replacing the values with interpolated values based on previous and succeeding points. The average water speed past the instrument over the one-minute period was calculated from the number of revolutions of the rotors over that period. If the two rotors did not differ by more than 3 cm/s the average of the two was used, otherwise the higher value was used on the assumption that the other rotor had become stuck for some time. The vertical component of the velocity due to rising or falling of the instrument was calculated from the rate of change of depth (given by the pressure sensor), and the square of this vertical component was subtracted from the square of the speed of water past the intrument. The absolute vertical component of velocity of the water was assumed to be small enough to be ignored, since according to theory the vertical component of velocity for the internal tide is at most ^ times the horizontal component (see Figures 75-76). Average values for the northwards and eastwards components of the horizontal velocity for that minute were then calculated by assuming that the heading varied linearly with time from the beginning of the interval to the end. These velocities were vector-averaged over intervals of a little less than the standard depth interval (10 or 15 m) and then interpolated to standard depths. TaWe 1 lists the standard depths for each station. Table 1 Standard depths at each station station bottom depth (m) shallowest standard depth (m) standard depth interval (m) deepest standard depth (m) Protection Point 180 20 15 155 Lull Bay 164 25 10 125 Tomakstum Island 340 20 15 170 Adeane Point 530 20 15 170 These standard depths were chosen so that the profiles usually reached the upper and lower depths. The principal axes of the velocities at each standard depth were then cal-culated, and their average orientation over the water column used to resolve the velocities 61 into along-channel (longitudinal) and cross-channel (transverse) components. In practice, the principal axes at Protection Point, Lull Bay and Tomakstum Island were all within 5° of true east, and those at Adeane Point were all within 5° of true north. Figures 40 and 41 show the longitudinal and transverse components of the velocity at Tom-N in September 1983 at the standard depths. The dominant signal is the Af2 tide , with a period of 12.42 hours. The signal is modulated at a fortnightly frequency. This springs to neaps cycle can be thought of as a beating of the M 2 signal with the 5 2 signal which has a period of 12.00 hours. 20m 35m 50m 65m 80m S* 95m 110m 125m CO 140m "B u 155m 5 170m o I A A A* A. Aj A A n * A. A I A « . . . . A A r t / i J , 1 - A A A « / 1 1 WVyyv w l \TV V / y V \ / U \ 7 V < V V » r V V W ^ V V V V V v y y y y y v v yy 1 1 r „ r A _ ^ L » . » A /, pi A A A A A A/V>A A A ~ A r A J T - A , A A A A A A I * A A A A - , A 4 * - A A A A 1 1 I ^ v ^ v ' ^ ^ r v Y V ^ Y V v T VVv V V V v r v w v V v T V "'v v y v y Y y v v v\A/y v v v l/V 1 1 A/v*iA A A A ^ A AA A A A A A / I ' V A V I - / i f - n A A 1 A A A A A A A A A A A A A A A A AI 1 1 /yv»-' wv\yw\iy v y y ^ y*>y v y y v v v ' v y v f v v *"*/ Vvuui/VyVvl/ \ju i „ / V ^ A / < / M A A n A A A A A A A A A AA *A .„ „ A A ^ A A A _ A - A A A A A A A A A A f t A fl A AA „ / i 1 ^ A A A / A I A A / \ A A A AA A A/1 A A A A I A A A AA A A J A ^ ^ A A A / \ A A A A A A A A A A A A A A A I 1 I A A A A A ^ I A A A A A A A A A / 1 A yy yvvivy vw\/ v\r u • wvyuvvvvvuuvuijvui/y 1 A A A A A A A A * A A A / \ ^ AAAAAAAAAAAAAAAAAAAA I i / w u v w vvyyvvyvyyy i A A , A A A A | A A A A A AAA A A A l/uyvv'uvvwv vwv vvuuvvvvvuuuvyyuuVV AAAAHAAAAAAATAAAAAAAAAAAAAAAAAAAAA I 1 A A A A AAiArt A A AAAAA A/ y V V v W v v V u V " v V u , v v v v v v y y y y y M y y y v v v 1 IAAAAA/IA^MM^MMAAAAAAAAAAMAAAAA I MAMMAAAAAA/I/U/ yyvui/ul/vv,/v ^^^vyvvvyyvyyyiyyyyvvv ' IAAAAAAAA^AMI**AAAAAAAAAAAAAAAAAAAA I \aAAAAAji\AAAAAA/l/lA/ yyvWVVW '^vvv/vvvuvvVVl/vi/VVvVVVvvv i lAAnA AA AA A A A . A I A 1 A A AAAAAAA AAA A IAAAA A A 1 wvvvvyvyvVvvWy yvVu\/VVVVW wwvuvvvvvvVvvyvVvvvvyi 1 243 247 251 255 259 263 Time ( J u l i a n Days) Figure 40 Longitudinal velocity vs time at Tom-N. September 1983. Positive is up-inlet (eastwards). 267 271 The density was calculated from the conductivity and temperature data, assuming i 62 20m 35m 50m 65m 80m o 110m 125m 05 140m o 155m ° + 170m o 1 f -1 • - - 1 _ . . - L A A /^ A.-A ^ 1 -A.—.. - i 1 , - . - - 1 1 - -• 1 ^ ' / " V - v * V v - ta ' | • " V • - T 1 v ••"> i 1 - 1 -1- • - - y _ | 1 •- - ~ - l J - /\ * .1 . I , . . . | -|.-_ 1 - V | . . . . ^ 1 v — V - • V | ^ . U u - p j ^ . \ i J - ' 1 - - -" " J • *»P^ _ - I -y ^ v v v v ^ p > • » 1 A — V • r-. - 1 A 243 247 251 255 259 263 Time ( J u l i a n Days) 267 271 Figure 41 Transverse velocity vs time at Tom-N. September 1983. Positive is northwards. that both sensors provided intantaneous readings. This assumption is satisfactory for the temperature, where the thermistor was well ventilated, but not so good for the conduc-tivity, because the conductivity cell did not flush well at low water speeds. However, the velocities in Knight Inlet are large enough that the cell should be sufficiently well flushed. In coastal waters such as these, salinity rather than temperature dominates density varia-tions. Figure 42 shows the isopycnals in the time-depth plane at Tom-N in September 1983. Again the signal is dominated by an oscillation at period with a very large amplitude (sometimes a 50 m peak-to-peak isopycnal displacement). Large amplitude oscillations in isopycnal depth at tidal period like these are charateristic of an internal tide. Figure 43 shows the same data in a different form — it shows the density as a function of time at 63 each standard depth. The average density at each depth is given on the right-hand side of the plot. There is a small offset in the conductivity values due to calibration drift, so these average density values can not be compared between records. The cyclesondes also give information on surface elevation variations due to the tide in 1983, because for over two hours out of every six they were resting against the bottom bumper. The bottom bumper is a fixed distance from the bottom, so that any variations in the depth of the instrument as calculated from pressure sensor data are due to variations in surface elevation. The pressure sensor is 2 m from the block, so that the depth data must be corrected for the pitch of the instrument, which varies by as much as ±10°. Figure 44 shows the depth of the bottom bumper as a function of time at Tom-N in September 1983. The dots are data points from the pressure sensor, and the solid line is the reconstructed signal after a harmonic analysis was done on the data (see Section 3.3.3). In addition to the cyclesonde density information from 20 m or below, CTD data were also available for this study. CTD casts were done at each station at each deployment and retrieval of the cyclesondes. Unfortunately the timing of these casts was somewhat sporadic due to the limitations of ship time. What the CTDs gain in terms of depth coverage the cyclesondes make up for in temporal coverage. 3.3 Harmonic Analysis of the Data Harmonic analysis was done on the longitudinal velocity and density data at the stan-dard depths, and on the surface elevation data. Nine constituents were included in the analysis, as listed in TaWe 2. These components were chosen to be the smallest set that could be reasonably be ex-pected to describe the data. With short records only a few constituents should be included in the analysis. Foreman's (1977, Tables 1-3) guidelines on which constituents should be included for a given record length were followed. In practice the number of constituents included was not found to affect the phase and amplitude of the M2 constituent, and this component is the one of primary interest in this study because it contains the most energy. 65 m <r CN r o o v o o -d- oo — i m O L O P ^ OO O> cr\ O o o —< Fignre 43 Density at the standard depths vs time at Tom-N. September 1983. 66 ' "S861 Jaquia:)cfog 'Jtf-inoj, $u suit!) EA ladranq uio^oq aq; jo q^dsQ ff a J u S i j DEPTH OF BOTTOM BUMPER 180.0 182.0 184.0 186.0 188.0 Table 2 Tidal constituents used in the harmonic analysis tidal Darwin period harmonic name (hours) Shallow water constituents M4 MK3 6.21 8.18 Semi-diurnal Principal solar s2 12.00 Principal lunar M2 12.42 Longer lunar elliptic N2 12.66 (Not in 1981) Diurnal Soli-lunar declinational Kx 23.93 Main lunar Ox 25.82 Long period Luni-solar fortnightly MSf 354.4 (Not in 1981) Mean Z0 CO The basic criterion was whether the M2 signal could be separated from the 52 signal. The Rayleigh criterion requires the record length to be greater than the reciprocal of the difference of their frequencies. In other words, the record must be at least as long as the period of the beat frequency (14.76 days — the MSf tide). This was certainly true in 1983, and almost true in 1981. The N2 constituent could not be included in the analysis of the 1981 data because the frequency is too close to that of the M2 constituent. It was found that between 60% and 80% of the total variance of the longitudinal velocity signal was at the M2 frequency, and that the amplitude of the M2 constituent was always at least twice that of the S2 constituent, which was the next largest. Two shallow water constituents were included in the analysis. These constituents arise from the non-linear terms in the hydrodynamic equations, and are products of the interac-tions between the main constituents. Only two shallow water constituents were included because the sampling interval was 3 hr3 in 1983, and constituents whose periods are shorter than twice that value cannot be resolved without aliasing (the Nyquist criterion). M 4 arises from the interaction of M2 with itself, and MK$ from the interaction of M2 with K\. 68 3.3.1 Harmonic Analysis of Velocity Data Figures 45-68 show the results of the harmonic analysis of the longitudinal velocity at the various standard depths at each station. The Greenwich phase and nodally corrected amplitudes of the various constituents at each depth are plotted in the complex plane. If <f> is the Greenwich phase and r is the nodally corrected amplitude at a particular depth, then these values are plotted at the point (r, <f>) in polar coordinates. The Greenwich phase is the phase relative to the passage of the sun or the moon (depending on the constituent under consideration) over the Greenwich meridian, and is therefore a standard reference for the three different time periods. The nodally corrected amplitude is corrected for long period modulations in the tide-generating force due to other astronomical cycles. This nodal correction is an attempt to standardise the amplitude of the response of the inlet in a situation where the amplitude of the tide-generating force at a particular frequency is not the same for the two years 1981 and 1983. However, this small correction makes the implicit assumption that the amplitude of the response of the inlet at that frequency is linear in the amplitude of the tide-generating force. Figures 45-68 show the loci of these (r, <f>) points at the standard depths (10 m in-tervals for Lull Bay, 15 m for the others). The point corresponding to the shallowest standard depth (25 m for Lull Bay, 20 m for the others) is marked by a larger symbol. The constituents are arranged in decreasing order as in Tabie 2, except that the complex amplitudes (i.e. amplitudes and phases) of the MK3 constituent are not given because they were always small (< 2cm/s). The corresponding plots for the MSf constituent appear later in Chapter 4 (Figures 107-110). The scale factors used in the figures were allowed to vary so that the loci could be distinguished from each other, although each of the four figures for a particular constituent were plotted on the same scale for comparison. These loci have strikingly similar shapes for the three time periods at a given station, whereas different stations have quite different loci. The repeatability of the loci at any given station provides motivation to understand why they have their characteristic shapes, and shows that they are not simply random patterns. Even the shapes of the loci for the 69 Figure 45 L o c i of M j c o m p l e x a m p l i t u d e s of veloci ty at P r o t e c t i o n P o i n t . o J u l y 1983, A S e p t e m b e r 1983, * J u l y 1981. T h e shallowest s t a n d a r d d e p t h is m a r k e d b y a larger s y m b o l . 70 <J.O 5 . 0 Figure 47 Loci of M 4 complex amplitudes of velocity at Tomakstum Island, o July 1983, A Tom-S September 1983, * Tom-N September 1983, o July 1981. The shallowest standard depth is marked by a larger symbol. C P r o Real a x i s i — T - 1 r 1 . 0 2 0 3 . 0 U /(CM/SECI 4 . 0 5 . 0 Figure 48 Loci of M 4 complex amplitudes of velocity at Adeane Point, o July 1983, A September 1983. 71 I 1 1 1 1 1 0.0 2 0 4.0 6 0 8.0 10 0 U /(CM/SEC) cn ro R e a l a x i s Figure 49 Loci of S 2 complex amplitudes of velocity at Protection Point. o July 1983, A September 1983, * July 1981. The shallowest standard depth is marked by a larger symbol. 4,0 6 0 U /(CM/SEC) cn ro R e a l a x i s Figure 50 Loci of 5 2 complex amplitudes of velocity at Lull Bay. o July 1983, A July 1981. 72 Figure 51 Loci of 5 2 complex amplitudes of velocity at Tomakstum Island, o July 1983, A Tom-S September 1983, * Tom-N September 1983, o July 1981. The shallowest standard depth is marked by a larger symbol. i 1 1 1 1 1 0.0 2.0 4.0 6 0 8.0 10.0 U /(CM/SEC) Real axis cn ro Figure 52 Loci of S 2 complex amplitudes of velocity at Adeane Point, o July 1983, A September 1983. 73 Figure 53 Loci of M 2 complex amplitudes of velocity and density at Protection Point. o July 1983, A September 1983, * July 1981. The shallowest standard depth is marked by a larger symbol. 5IGMfl-T 0.1 0.2 5.0 10.0 15.0 U /(CM/SEC) 0.3 0.4 20.0 Veloci ty Figure 54 Loci of M 2 complex amplitudes of velocity and density at Lull Bay. o July 1983, A July 1981. 74 i cn x ro (_ ro (Z cn Figure 55 Loci of Mi complex amplitudes of velocity and density at Tomakstum Island, o July 1983, A Tom-S September 1983, * Tom-N September 1983, o July 1981. The shallowest standard depth is marked by a larger symbol. D e n s i t y P f Real axis 0.0 0.0 SIGMA-T . 0.1 0.2 0.3 0.J 5.0 10.0 15.0 U /(CM/SEC) V e l o c i t y 20.0 Figure 56 Loci of M<z complex amplitudes of velocity and density at Adeane Point, o July 1983, A September 1983. 75 CD —I X ro R e a l a x i s Figure 57 Loci of N2 complex amplitudes of velocity at Protection Point, o July 1983, A September 1983. The shallowest standard depth is marked by a larger symbol. R e a l a x i s ° " i r 1.0 2.0 3.0 U /(CM/SEC) -1 1 4.0 5.0 cn ra Figure 58 Locus of N2 complex amplitudes of velocity at Lull Bay. o July 1983. 76 I Figure 59 Loci of complex amplitudes of velocity at Tomakstum Island, o July 1983, A Tom-S September 1983, * Tom-N September 1983. The shallowest standard depth is marked by a larger symbol. Real axis ° i 1 1 1 1.0 2.0 3.0 U /(CM/SEC) 4.0 5.0 Figure 60 Loci of _/V2 complex amplitudes of velocity at Adeane Point, o July 1983, A September 1983. 77 Real axis 2.0 d.O 6.0 U / (CM/SEC) CP ro Figure 61 Loci of K\ complex amplitudes of velocity at Protection Point. o July 1983, A September 1983, * July 1981. The shallowest standard depth is marked by a larger symbol. Figure 62 Loci of Kx complex amplitudes of velocity at Lull Bay. o July 1983, A July 1981. 78 ro C Real axis 4.0 6.0 U / (CM/SEC) Figure 63 Loci of Kt complex amplitudes of velocity at Tomakstum Island, o July 1983, A Tom-S September 1983, * Tom-N September 1983, o July 1981. The shallowest standard depth is marked by a larger symbol. 4,0 6 0 U / (CM/SEC) Real axis Figure 64 Loci of K\ complex amplitudes of velocity at Adeane Point, o July 1983, A September 1983. 79 i X I ro Imaginary 0 / R e a l a x i s 0 . 0 1 2 . 0 i i i 4 0 6 0 8 0 U /(CM/SEC) i 10 . 0 Figure 65 Loci of Ot complex amplitudes of velocity at Protection Point. o July 1983, A September 1983, * July 1981. The shallowest standard depth is marked by a larger symbol. R e a l a x i s 4 . 0 6 . 0 U /(CM/SEC) X ro ro •—* cn ro e Figure 66 Loci of Oi complex amplitudes of velocity at Lull Bay. o July 1983, A July 1981. 80 Real axis in —i x ro ro c ro —-i 1— <J0 6.0 U /(CM/SECI —i ICO Figure 6 7 Loci of Oi complex amplitudes of velocity at Tomakstum Island, o July 1983, A Tom-S September 1983, * Tom-N September 1983, o July 1981. The shallowest standard depth is marked by a larger symbol. cn ro —I 1— 4.0 6.0 U /(CM/SEC) Real axis Figure 6 8 Loci of 0\ complex amplitudes of velocity at Adeane Point, o July 1983, A September 1983. 81 M 4 constituent are repeatable from month to month, especially at Protection Point and Lull Bay. The amplitude of this shallow water constituent is as large as that of the diurnal constituents at Tomakstum Island and Adeane Point, or that of the 7V2 constituent at all stations. The next larger group consists of the diurnal constituents at Protection Point and Lull Bay, and the 52 constituent. However, the M 2 constituent is by far the largest of all. The loci at a particular station for a particular constituent are also quite similar in shape to the loci at the same station for a different constituent, as long as the two con-stituents are in the same band (i.e. diurnal or semi-diurnal). For example, the loci in Figures 51, 55 and 59 all have similar shapes, as do those in Figures 62 and 66, and in Figures 61 and 65. This property indicates that the inlet is far from resonance at these frequencies, because the transfer function is almost the same at two very close frequencies. 3.3.2 Harmonic Analysis of Density Data The oscillations of density values also give valuable information on the internal tide, so Figures 53-56 also show the loci of the M 2 constituent of density oscillations as the depth varies. An example of the original data that were used in this analysis was shown in Figure 43. The small offset in density values will not affect this analysis because only the Af2 constituent of the density signal is of interest. Later, the vertical gradient of the average density values will be used, but this is also unaffected by the offset. The foregoing comments on the repeatability of the loci of the Af2 constituent of the longitudinal velocities apply equally well to those of density. The density oscillations are caused by oscillations in the vertical component of velocity W j SO 3- harmonic analysis of density variations gives information on the phase and magnitude of the w-oscillation, as will be discussed quantitatively in Section 3.5. 82 3.3.3 Harmonic Analysis of Surface Elevations The third variable for which continuous data exist is surface elevation. It gives in-formation on the barotropic component of the tide, since the baroclinic modes have very little influence on the surface elevation. Figure 69 shows the phase and magnitude, in the complex plane, of the M 2 constituent of the surface elevation oscillation for each record. Again, Greenwich phases and nodally corrected amplitudes are used. Only a small sector of the complex plane is plotted because the surface elevation phases and amplitudes are almost constant throughout the inlet. The surface of the inlet rises and falls almost in unison because the wavelength of the Kelvin wave associated with the barotropic tide is so large (A = TsfqR » 2500 Arm when H = 300 m and T = 12.42 hours). In fact, the time series of surface elevations (as in Figure 44) from the different stations are nearly identi-cal. The barotropic tide is a standing wave, because it is a superposition of an incoming wave and a reflected wave, with complete reflection occurring at the head. In a straight, flat-bottomed channel with rectangular cross-section, the shape of the surface would be a cosine curve whose peak would occur at the head of the inlet and whose first zero-crossing (or node) would occur at a distance of A/4 from the head. Knight Inlet is much shorter than A/4, so no node occurrs. However, the amplitude of the oscillation at 50 Arm should be 99.2% of the amplitude at the head, and the amplitude at 100 A;m should be 96.7% of that at the head, assuming that H = 300 m. The difference between them, 2.5%, is consistent with the drop in amplitude from » 1.54 m at Adeane Point and Tomakstum Island to « 1.51 m at Lull Bay and Protection Point (a drop of 1.9%). However, the change in phase by 2-3° from Protection Point to Tomakstum Island cannot be explained by this simple frictionless model, because the model predicts a constant phase along the channel with 180° jumps at nodes. This change in phase is caused by energy dissipation due to transfer of energy to the internal tide and due to friction. Freeland and Farmer (1980) concluded that about 95% of the energy dissipation rate of the barotropic tide in Knight Inlet could be accounted for by internal wave and hydraulic jump generation, and that the remaining 5% is lost to bottom and side friction. They obtained phase 83 1.56 -3 1 - 5 5 -^ 1.54 a 1.53 -1.52 -1.51 1.50 1 1.49 TOMAKSTUM July 83-Tom-N 36 Sept 83 ABEANE A f July 81 frj—Wuly 83 -^Ssept 83 LULL Tom-S Sept 83 O July 81 July 83 <t> PROTECTION July 83 July 81* *Sept 83 35 32 34 33 Greenwich Phase (°) Figure 69 M<z complex amplitudes of surface elevation. o Adeane Point, A Tomakstum Island, o Lull Bay, * Protection Point. 31 30 differences of around 1.5° between Tomakstum Island and Protection Point (see Freeland and Farmer 1980 Figure 7a) using 29-day blocks of data, compared to the cyclesonde value of 2-3°. The bottom bumpers at Tomakstum Island and Adeane Point were much further from the bottom than at the other two stations, so that the drag assosciated with oscillating baroclinic currents at M-i frequency could pull them down. This effect could change the calculated amplitude and phase of the surface elevation. However, the currents at depth and therefore the associated drag are not likely to be large. 84 3.4 Normal Modes — Theory In a flat-bottomed ocean with vertical side-walls the solutions to the linearised (i.e. in-finitesimally small disturbances from the mean state) equations of motion can be separated into a vertical dependency and a horizontal dependency. A harmonic time dependency of e~tu>t is assumed, so that waves propagate in the direction of increasing phase. This sign convention was chosen so as to be consistent with the harmonic analysis program of Fore-man (1979), and with LeBlond and Mysak (1978). The opposite sign convention was used in Chapter 2 so as to be consistent with Taylor (1920), Packham and Williams (1968), Buchwald (1968), Brown (1972) and Pnueli and Pekeris (1968). With the Boussinesq approximation, the horizontal equations are: twU fc - 2t1 x VH = V P , (3.1) V . U / l = - ^ - P , (3.2) and the vertical equation is cPz ghn with the bottom boundary condition cPZ N * - U J 2 „ N + 7 Z = 0, (3.3) Z{0) = 0, (3.4) and the surface boundary condition dZ Z — - — = 0 at z = H. (3.5) dz hn (LeBlond and Mysak equations 10.39-10.42) The horizontal velocity is given by ufc(x, y, *) = -hn ^(*) U f c(z, y), (3.6) the vertical velocity by and the pressure by w(x,y,*) = yZ(z)P(z,y), (3.7) dZ p(x, y, z) = -p0hn —(z) P(x, y), (3.8) 85 where 20 = (0,0,/), (3.9) and 9 dp0 Po dz (3.10) Equations 3.3-3.5 form an eigenvalue problem which determines the vertical structure of the solution. The eigenvalue problem has an infinite set of solutions (eigenfunctions or verticcd modes) Zn(z) and corresponding eigenvalues hn (or equivalent depths). The gravest mode is the barotropic mode, and is called mode 0 because the vertical structure of the horizontal velocity does not have a zero crossing. In this mode the second term of equation 3.3 becomes negligible because hn is so large, so equation 3.3 becomes Z" = 0. The solution to this problem is ZQ(Z) = z with = H. The vertical strucure of w is linear with w = 0 at the bottom, and the vertical structure of tih is a constant function. The equivalent depth ho is equal to the total depth H. The reason for the name equivalent depth is that it is the depth of water hn that would give the same phase speed for the barotropic mode as for a higher mode n in water of depth H. It will be seen later that the phase speed of Kelvin waves for mode n is \/ghn. For the barotropic mode, this is the familiar expression for gravity waves in shallow water. If the eigenvalues of the higher modes are arranged in order of decreasing size , then the nth mode has n zero crossings in the u-eigenfunction. The first term of the boundary condition 3.5 can be discarded for the higher modes, although it is of paramount importance for mode 0. Density perturbations p'(x, y, z, t) are related to w perturbations by (3.11) (LeBlond and Mysak equation 8.2) Therefore p'(x, y,z) = (3.12) 86 The procedure for calculating the vertical structure of the normal modes is therefore as follows: (i) Calculate a "mean" density profile po(z) from either several CTD casts, or a cy-clesonde record. (ii) Calculate the iV2-profile (Brunt-Vaisala frequency squared) using definition 3.10. (iii) Solve the eigenvalue problem given by equations 3.3-3.5 to give a series of eigen-values hn and eigenfunctions Zn(z). (iv) Calculate the u-eigenfunctions using equation 3.6 and the p-eigenfunctions using equation 3.12 for each mode. The former is proportional to the vertical gradient of Z(z), and the latter is proportional to N2(z) times Z(z). Any small oscillation of a fluid in a flat-bottomed ocean can then be expressed as a superposition of these modes. The coefficients of each mode will be complex, because an e~twt time dependancy has been assumed. The horizontal problem consists of equations 3.1-3.2 along with boundary conditions of zero normal flow at the coast and matching conditions at open boundaries. A special solution in a straight channel is the Kelvin wave. In fact, this is the way the barotropic and internal tides in Knight Inlet propagate, because this solution is the only propagating wave that can exist in such a narrow channel. The effect of the evanescent Poincare modes generated at the bend is negligible, because the e-folding length scale of the first cross-channel mode is approximately 600 m, and shorter for the higher modes. The expressions for P and U f c = (U, V) for the Kelvin wave solution are: U(x, y) = U0 e(*"*-'3)/ C" (3.13) V(x,y) = 0 (3.14) ?(x,y) = U0cne(iux-f»Ve" (3.15) where cn = >/ghni and x and y are the longitudinal and transverse co-ordinates respec-tively. UQ is an arbitrary scaling constant. The Kelvin wave decays exponentially away 87 from the wall to the right of the direction of propagation with a decay scale given by = cn/f (the Rossby radius). Upon substitution into equations 3.6 and 3.12 one obtains: u(x, y, z) = -U0 hn ^-(z) <.(•«*-/»>/«» (3.16) dz v(z,y,z)=0 (3.17) p'(z,y,z) = - V ^ ^ N ^ Z W e ^ * - ™ ^ , (3.18) 9 where u h = (u,v). The use of theoretical modes for a flat-bottomed ocean in an inlet where the bottom is not flat must be justified. In a flat-bottomed ocean the normal modes approach is equivalent to the ray-tracing approach, with rays being reflected from the surface and the bottom. The upward propagating rays interfere with the downward propagating rays. However, when the bottom is not flat, it is necessary for the rays to be steeper than the bottom slope if the normal mode approach is to be retained, otherwise the rays will be reflected about the horizontal rather than the vertical (see LeBlond and Mysak 1978 p55). The ray slope is given by the ratio y/N2 — u2 : y/w2 — f2. Taking a typical value of logl0N2 = —5.5 at the bottom (see Figures 71-74 in Section 3.9), the slope of the rays near the bottom at Af2 frequency can be calculated to be approximately 1 in 21. Hence the bottom slope must be less than this value for the normal mode approach to be used. Figure 1 shows that on average, apart from irregularities in the topography, the longitudinal bottom slope does not exceed 100 m in lOfcm (1 in 100) except on the flanks of the sill. Therefore the rays will usually be reflected about the vertical, as required by normal mode theory. In addition, as the waves propagate along the inlet, it is necessary for the length scale over which the depth varies » 40 km if H = 400 m) to be greater than the wavelength of those waves if a WKB approximation is to be valid. Typical wavelengths for the internal tide are around 36 km for the first mode and 20 km for the second mode, so this criterion is in fact barely met. 88 3.5 Modal Decomposition The observed profiles of the M 2 amplitude and phase of longitudinal current and density will be decomposed into a superposition of normal modes. Moreover,a distinction between waves propagating up-inlet and waves propagating down-inlet is also desirable, so that the reflection coefficient at the bend for each mode can be estimated. To make this distinction, the phase relationship between density oscillations and longitudinal velocity oscillations will be exploited. It can be seen from equations 3.16 and 3.18 that these two quantities are in phase. However, for a wave propagating in the negative x-direction, the corresponding expressions can be obtained by replacing c„ by —cn. Thus the expression for density oscillations changes sign, which is equivalent to a 180° phase change. This phase relationship can be used to resolve the waves going up-inlet from those going down-inlet. Another way of resolving the direction of the waves is to make use of the cross channel decay of Kelvin waves, since higher amplitude oscillations will occur to the right of the direction of propagation of a single Kelvin wave in thenorthern hemisphere. This method can be used without a density record — velocity data from two stations across the channel and a few CTD casts are sufficient. In fact, this was the method used by Freeland (1984) in his analysis of current meter records from Knight Inlet. He had a total of four current meters at Tom-N and Tom-S at depths of 75 m and 300 m. Unfortunately, with current meters at only two depths, he was forced to assume that there was only one significant internal mode besides the barotropic mode. Farmer and Freeland (1983) assumed that this mode was the first mode, whereas Freeland (1984) modified the analysis by assuming that the dominant mode was the second. When only the barotropic mode is present, the locus of the complex amplitude (i.e. amplitude and phase) of the longitudinal velocity as the depth varies is a single point in the complex plane. However, when a single internal mode (the nth mode) is also present, ,this locus becomes a straight line. This line passes through the point corresponding to i 89 the barotropic tide, because the additional velocity component due to the internal mode is zero at depths where the u-eigenfunction has a zero crossing. The point corresponding to the barotropic tide should be the same for both stations, because the cross-channel decay of the barotropic tide is very small (since the external Rossby radius Ro is very large, about 2000 Am), and it is assumed that the barotropic current is laterally homogeneous (i.e. negligible topographic effects, and negligible inertial and centrifugal forces associated with the bend). Therefore, the loci at the two stations should intersect at this point in the complex plane. If a£ p and <^ JJP are the non-dimensional magnitude and phase of the up-inlet internal wave, a £ o u m and ^ £ o u m are the corresponding values for the down-inlet internal wave, and Ubt and 4>bt are the velocity and phase of the barotropic component, then the complex amplitude of the longitudinal velocity at the northern station is given by: uu + « p « - ' / « " + • ' « ' + aJ°-» e + ' ' / c « + ' " * - ° " B ) un(z) (3.19) whereas that at the southern station is: uue*" + (a^e +* // c" + ,'^P + a f w n e - ^ / c « + « ^ o w n ) U n ( 2 ) > (3.20) where 5 is the distance of the two stations from the centre of the channel (assumed to be equidistant from the centre), and tt„(2) = UQ hn —(z) (3.21) is the dimensional u-eigenvalue. U0 is fixed by some normalisation criterion, such as max un(z) = 1 m/sec. The two expressions in brackets in equations 3.19-3.20 will not in general be the same complex number. The fact that they have a different phase means that the slopes of the two loci in the complex plane are different. The quotient of the two numbers in brackets can be determined by plotting the four complex amplitudes in the complex plane; the angle between the two lines joining the 75 m-points to the 300 m-points gives the phase 90 of the quotient, and the ratio of the length of these two lines gives the magnitude of the quotient. The reflection coefficient <p = anov"l/anv can be determined from this complex quotient (see Freeland 1984 equation 3). Freeland (1984) obtained a value of <p = 0.70 ± 0.07 using the second mode, whereas Freeland and Farmer (1983) had obtained a value of tp = 0.93 using the first mode. These vales of <p correspond to energy reflection coefficients of JZ = 49% and Z = 86% respectively after squaring. However, the fact that the loci o'f the complex amplitudes of longitudinal velocity are not straight lines (e.g. see Figure 55) conflicts with their assumption of a single dominant mode, and puts their values for the reflection coefficient into doubt. One argument in support of their assumption is that the calculated phase and magnitude of the barotropic current agrees very well with the theoretical values based on the phase and magnitude of the surface elevation. However, this agreement could be coincidental. Looking at the loci of the longitudinal velocity at Tom-S and Tom-N in Figure 55, it might be tempting to deduce that the internal tide must be propagating predominantly up-inlet because the magnitude of the oscillation is always greater on the south side than on the north side, just as it would be for a single Kelvin wave propagating up a straight inlet. However, this argument is not valid, because it is possible to construct a counterex-ample where a single second mode internal tide propagating down-inlet combines with the barotropic tide to produce stronger currents on the south side than on the north side. Consider a barotropic tidal current of phase 0° and magnitude 10 cm/a at Tomakstum Island. On top of this current superimpose a second mode velocity profile that decreases monotonically from a magnitude of 6 cm/s (for example) at 20 m to Zcm/s at 120 m (see Figure 76 for an illustration of a u-eigenfunction of this form), with a phase of 180°. This internal wave is propagating down-inlet, so that its magnitude might be 6.6 cm/s (at 20 m) and 3.3 cm/s (at 120 m) at Tom-N, and 5.4 cm/a (at 20 m) and 2.7 cm/a (at 120 m) at Tom-S. However, the internal mode currents are 180° out of phase with the barotropic 91 currents, so the net currents are ZAcmJ3 (at 20m) and 6.7cm/s (at 120m) at Tom-N, and 4.6 cm/s (at 20 m) and 7.3 cm/s (at 120 m) at Tom-S. The currents are now stronger at Tom-S than at Tom-N, even though the internal wave is propagating down-inlet! Moreover, the larger currents at Tom-S might be solely due to the inertia of the water as it flows from the bend on an ebb tide, and have nothing to do with the cross-channel decay of Kelvin waves at all. However, the tidal excursion is only of the order of 1 or 2 km, and Tomakstum Island is about 6 km from the bend, so hopefully it is justifiable to ignore this effect. 3.6 Normal Mode Fitting — Theory It is assumed that the longitudinal velocity and density oscillations at a particular station can be represented by a superposition of normal modes as follows: n(y,z) = un(z) ( a - ' e - ^ - W + o f " e+*"/c"+'*'0,°n) (3.22) n=0 />'(y>z) = £ « P e - y I I C n + i K * - 4°wn e+^fe"+i*d"n) (3.23) n=0 where y is the cross-channel co-ordinate (y — 0 at mid-channel, y = +500 m at Tom-N and y = —500 m at Tom-S), and pn(z) is the dimensional p-eigenfunction for the nth mode defined by Pn(z) = U 0 ^ N2(z) Zn{z). (3.24) u n(z) is the dimensional u-eigenfunction, previously defined in equation 3.21. The dimen-sions of pn(z) are those of density, and the dimensions of u„(z) are those of velocity. The minus sign in equation 3.23 arises because equation 3.18 must be modified for down-inlet propagating waves (i.e. waves propagating in the negative x-direction) by replacing cn by — c n , as was stated earlier. For stations in the centre of the channel (that is, all stations except Tom-N and Tom-S), y must be set to 0, and 3.22 and 3.23 reduce to * ( * ) = £ u « ( * ) ( A » " + Ad™n) ( 3 - 2 5 ) n=0 92 p'{z) = £ pn(z) (A? - A f - » ) , (3.26) n-0 where An" = < p e^"' (3.27) and A n o w n = of"" 1 e1'*-""" (3.28) are the complex amplitudes of the up- and down-inlet propagating waves. 3.6.1 Least Squares Fitting at a Single Station in Mid-Channel The statistical model can be formally defined by uobl(z) = J2un(z)(KP+<OV,n) + «"(*) (3.29) n=0 Po6.(*) = X > ( * ) ( A : ' - A f " ) + «"(*). (3.30) fi=0 The e's are error terms, and uoba(z) and (2) are the complex amplitudes of longitudinal velocity and density oscillations from a harmonic analysis of the data. In practice the infinite series must be truncated after a finite number of terms N; this was usually two or three (see Section 3.10). In the case of a station in mid-channel, the least squares fit can be done independently for each of the following four quantities: (i) dt(uobl) to yield B(AS P + A d ™ n ) (ii) 3(uo6.) to yield 3(AJ* + A n ° w n ) (iii) &(pobt) to yield X(An* - A ^ w n ) (iv) 9f(/>o6.) to yield 3(A« P - A d ™ n ) . This decomposition is possible because the un's and p n ' s are all real. It can be seen that a fit of the u-eigenfunctions to the observed u-profiles yields (AJ|P + AjJ0""1), whereas a fit of the p-eigenfunctions to the observed p-profiles yields (AJJP — A f f w n ) . The sum of these two complex numbers then gives 2A£ P , and the difference gives 2Anown. 9 3 3.6.2 Least Squares Fitting at Two Stations on Opposite Sides of the Channel However, when simultaneous data are available from both sides of the channel, as at Tomakstum Island in September 1983, the four fits described above cannot be done independently if the information on the cross-channel decay of Kelvin waves is to be used. The statistical model is now + s"(y,z) (3.31) + ep{ytz). (3.32) + eu{+s,z) (3.33) + ep(+s,z). (3.34) + eu(-s,z) (3.35) + e"(-s,z). (3.36) These equations can then be solved simultaneously by least squares, using information from both stations, to give estimates for A " p and A £ o u m . The advantage of using both stations simultaneously is that there is twice the amount of data to fit to, and information on the cross-channel decay of Kelvin waves is used as well as information on the phase of p-oscillations relative to u-oscillations. i i 94 n = 0 N Pot.(y,z) = £ M * ) ( K p r - s f / e n - A2~* e + " / e « ) n = 0 At Tom-S, y = +s, and these two equations become N n = 0 N Pob,(+S,z) = Pn{z) ( A y « - ' / 8 - - A f » « + " / « " ) n = 0 At Tom-N, y = —a, giving N nobt(-s,z) = X>„(z) (A«Pe+^/«- + A f " " e - ' / e » ) n = 0 N PoU-s,z) = J2 P M IA~ - Airn'-",en) 3.7 The Estimation of the Barotropic Component The component of the u- and /7-oscillations due to the barotropic tide can be estimated from a harmonic analysis of the surface elevation data and a knowledge of the geometry of the inlet. As previously described, the surface tide in an inlet is classically described as a standing wave, where the whole surface of the inlet rises and falls together. A standing wave can be thought of as a superposition of an incident wave and a reflected wave of equal magnitude. At the head, the longitudinal velocity due to the incident wave is precisely 180° out of phase with the longitudinal velocity due to the reflected wave. In this way the longitudinal velocity cancels out there as required. As Figure 69 shows, the surface elevation phases are indeed roughly constant along Knight Inlet. This observation implies a standing wave pair, in which the amplitude of the up-inlet wave equals that of the down-inlet wave at each station i.e.: alp = a do«7n- (3.37) It will be seen later in Section 3.10 that only about 1% (= 1 MW - r 700 MW) of the up-inlet or down-inlet barotropic energy flux is lost through friction and non-linear processes in the straight section, so that this assumption (3.37) can be justified a posteriori. The phase of the barotropic component of the longitudinal velocity will lead the phase of the surface elevation by 90°. Therefore, if the phase of the surface elevation is taken to be 33° (see Figure 69), then that of velocity will be 303° (= <f>bt in equations 3.19-3.20). In addition, the magnitude of the barotropic component of the velocity uu can be estimated from the magnitude r]o of the surface elevation oscillation, the cross-sectional area Sof the inlet at a particular section, and the surface area A of the inlet landward of that section. The magnitude of the barotropic velocity will be given by »H = (3.38) where u is the frequency of the Af 2 tide. Then 0"O(A*P + K " " 1 ) = «««••*', (3.39) 95 where it is assumed that the normalisation constant U0 is such that UQ(Z) = lm/sec for all z. There is one more piece of information that enables the unique determination of A" p and A o o t o n at any particular section in the inlet, namely the barotropic phase speed c0. Temporarily assuming this to be a constant, the phase of the u-oscillation due to the up-inlet wave will be given by ($> P+ ^ f) at the head, where d is the distance from the section of interest to the head, and <f>QP is the phase of the up-inlet wave at that section. Similarly, the phase of the u-oscillation due to the down-inlet wave will be (<f>*own - f-) at the head. These two quantities must be 180° out of phase there, so Equation 3.39 now becomes -up , d ( J _ Adoxon ^ <Po + — = <Po rir. Co CQ = 2tfU0 sin eW+-»/«o-/a]. (3.40) Therefore, « P _ „**.n = ^ c o s e c (M a 0 = a 0 4>QP = Ht - — + ~, du> and c0 .down _ i ,duj ir 9o -<Pbt + -• (3.41) (3.42) (3.43) (3.44) In the case of Co varying slowly along the length of the inlet, as occurs in reality, — must be replaced by a WKB approximation to the phase difference between the section of interest and the head of the inlet *head A0 - J C dx. (3.45) Section co{x) Since c2(x) = gH(x), this amounts to calculating the average value of H~% between the section of interest and the head of the inlet. This calculation was done by taking the depths 96 from the chart, which are given every kilometer or so, and calculating average values of f f ~ 5 between the four stations and the head. Tabie 3 gives the calculated values of A ^ between each station and the head. They range from 7.3° at Adeane Point to 13.2° at Protection Point. Tabie 3 also gives values of A and 5, as estimated from the chart. Assuming that rj 0 = 1.52 m, equations 3.42-3.44 were used to calculate a"p (which = aQu>u,n), ^ " p and <f>^own. Table 3 Estimates of the barotropic tidal currents at each station station A<t> A S Uo*7 <t>7 (°) (km2) (km2) (cm/s) (cm/s) (°) (°) Protection Point 13.2 234 0.41 11.9 26.0 20 226 Lull Bay 11.0 195 0.36 11.4 29.9 22 224 Tomakstum Island 9.0 156 0.50 6.7 21.4 24 222 Adeane Point 7.3 130 0.76 3.6 14.2 26 220 In the case of the barotropic mode, the Rossby radius is so large (w 2000 km) that the Kelvin wave has insignificant cross-channel decay, so it is not necessary to consider this at Tom-N and Tom-S. There is considerable error involved in these estimates, especially in because it depends on the reciprocal of 5, and in a"p ( and a*0*"1) because it depends on cosec A<f>. The second of these is not too important, because a knowledge of O Q P is only necessary for the calculation of the contribution of the barotropic mode to p-oscillations (proportional to ( A Q P - A^own) = 2 <ZQ P COS A ^ E ,'I^ , + , R/2I), which is small compared to the contribution from other modes. An error in the estimation of the magnitude of this small contribution will not affect the results of fitting the internal modes to the residual. Note that the phase of this barotropic contribution to p-oscillations can be accurately determined — it is [<t>bt + ff/2]. However, the first of the two errors, namely that of estimating Ubt, is important because the contribution of the barotropic mode to u-oscillations is considerable. All that can be done is to make the most careful estimate possible of U M given the existing data on the 97 geometry of Knight Inlet. It is still worth incorporating this information into the fitting process, even though it is of poor quality. Note that the phase fot oi the barotropic component of u-oscillations can be accurately estimated — it is the phase of the surface elevation oscillation minus Knowledge of A Q P and A Q 0 " " 1 was incorporated into the fitting procedure by sub-tracting the n = 0 terms on the right-hand sides of equations 3.29-3.30 (or equations 3.33-3.36) from the observed complex amplitude profiles on the left-hand sides, and fitting the internal modes to the residual. 3.8 Energy Fluxes The energy density E of an internal wave is defined as the mean perturbation energy per unit volume: E=$p0(u2 + v2 + v,2) + JL_!__ (3. 4 6 ) (Gill 6.7.6) Assuming a flat-bottomed channel, and using the expressions 3.16-3.18 for a Kelvin wave, the following expression is obtained for the vertically and laterally integrated energy density E' per unit length of channel for the nth mode: E _ P ^ M W F f» ^ + j r + ( 3 4 7 ) where W is the channel width and F = £ s i n h ( £ - ) (3.48) 1 (W\2 „ The first term inside the integral is the kinetic energy due to longitudinal velocity, the second term is the kinetic energy due to vertical velocity, and the third term is the potential energy associated with the displacement of isopycnals. The second term is negligible compared to the third because u>2 <S N2. It will be seen in the next Section that logioN2 varies from -1.5 near the surface to -5.5 at depth when N is measured in units of rad/s, whereas logiou2 = —7.7 for the Af2 constituent. 98 For internal waves there is an equipartition between the kinetic energy and the po-tential energy associated with the displacement of isopycnals. This was checked by direct calculation of the three terms in equation 3.47 using the numerical solutions of the eigen-value problem (see Section 3.8). It was found that the first term comprised 50-52% of the total, the second comprised around 0.001%, and the third comprised 48-50%. A fourth component becomes significant for the barotropic mode, namely the potential energy associated with the displacement of the free surface. The energy density of this component is given by: Ef =l-pQgrjilFW, (3.50) where »7o is the amplitude of the surface oscillation. This component is insignificant for the internal modes, but for the barotropic mode it becomes equal in magnitude to the kinetic energy density. Again there is equipartition of energy between kinetic and potential energy, but for the surface wave the potential energy is associated with the free surface displacement, whereas for internal modes the potential energy is associated with isopycnal displacements. The third term in equation 3.47 drops to around 1-2% of the total for the barotropic mode. The factor F takes account of the cross-channel decay of Kelvin waves. For large Rossby radii compared to the width (e.g. for the surface wave), this factor is practically equal to unity. For the first internal mode, which has a Rossby radius of roughly 7 km, F « 1.01. Even for the fourth internal mode, which has a Rossby radius of the order of 1.6 km, this factor is still only 1.28. Even though this factor was incorporated into the calculation of the energy flux, it is a minor factor compared to the inaccuracy involved in obtaining a value for W. W is the width of a channel of rectangular cross-section that would have the same energy flux as Knight Inlet, which has a trapezoidal or U-shaped cross-section (see Pickard 1961). The energy flux associated with each mode can easily be calculated from the energy density E' by multiplying by the group velocity, which in the case of Kelvin waves is equal 99 to the phase velocity cn. The units of E* are J/m, and those of energy flux are J/sec. In the case of up-inlet waves a^pU0 was used for U0 in equation 3.47, and in the case of down-inlet waves o^otoni7o was used for U0. In this way the non-dimensional amplitudes of the various modes were used to calculate the corresponding energy fluxe3. 3.0 The Calculation of the Normal Modes The eigenvalue problem for the normal modes (equations 3.3-3.5) was solved numeri-cally for observed iV^-profiles. As in Section 2.7, the numerical scheme used was a 4t,l-order Runge-Kutta scheme with iterative refinement of the eigenvalue to satisfy the boundary condition at the surface (a "shooting" technique). The biggest problem in calculating the normal modes was in knowing what to use for the iV^-profile. The linear theory requires the use of the vertical gradient of the average density. Here, average means over many tidal cycles, but not too many since the average density itself will change over the course of a year . A one-month mean would be desirable. Unfortunately, the cyclesondes only have a limited profile range, usually 20 m to around 170 m. They can provide good estimates of the monthly averaged AT2-profile over that depth range, but lack vital information on the upper 20 m. This information is contained in the CTD casts, but they were only taken be-fore deployment and after retrieval of each cyclesonde, so some aliasing will be introduced because of the different phases of the tide. A combination JV2-profile was eventually used, taking cyclesonde data wherever avail-able, and CTD data for the upper 20 m or so. There are two ways of taking averages of the various CTD casts: (i) calculate the normal modes using the average of all CTD profiles at a particular station, or (ii) calculate the normal modes using single CTD profiles, and then take the average of these normal modes. Both methods were used, and compared. Figure 70 shows all the density profiles from CTD casts at the Tomakstum Island stations (including Tom-N and Tom-S). The dashed profiles were taken at the end of September, and a weakening of the fresh surface layer is evident due to a reduction in river runoff. The density structure of the inlet is in a transition from summer stratification 100 (with a strong pycnocline) to winter stratification (with no fresh surface layer). Because this winter stratification is not typical of the period July-September, all CTD data from the end of September will henceforth be ignored. Figures 71-74 show log-plots of iV^-profiles at the various stations. In each plot, the dashed line is the mean ./V2-profile of all CTD profiles (except the end of September)at that station. The solid lines are A/^-profiles calculated from monthly averages of cyclesonde density data from 1983. The triangles on these lines indicate the deepest standard depth. The cyclesonde 7V2-profiles were linearly extrapolated in log space to the bottom using a best straight line fit to the last five standard depths, since none of the CTD casts went further than 200 m either. This extrapolation was chosen because the log^N2 profiles seemed to be reasonably linear below 60 m. The cyclesonde profiles and the CTD profiles agree well. Figures 75 and 76 show a set of five normal modes at Tomakstum Island, calculated using the average of all CTD data (except the end of September) in the upper 20 m, and using the average of all three appropriate records of cyclesonde density data. The modes in these figures are normalised so that the maximum value attained by the horizontal velocity eigenfunction is 1 m/sec. Most of the structure of these functions is in the upper 20 m, which is a pity because that is precisely where there are no cyclesonde data. Some of these eigenfunctions look very similar if their shape above 20 m is ignored. If there were data from the upper 20 m, then the least squares fitting procedure could be replaced by a method which takes the dot product of the observed velocity profiles with each of the normal modes to obtain the coefficients of each mode; i.e. exploiting the orthogonality property of the normal modes: H mn (3.51) and (3.52) 101 SIGMfl-T (KG/MXX3) 30.0 o o CD O o oo _ l I Fignre 70 CTD density profiles at Tomakstum Island in 1983. Solid lines = July and August, dashed lines = end of September. 102 Figure 71 log^N2 vs depth at Protection Point. Dashed line = CTD data, solid lines = cyclesonde data. -6.0 -5.0 LOG NXX2 -4.0 -3.0 -2.0 Figure 7 2 logioN2 vs depth at Lull Bay. Dashed line = CTD data, solid line = cyclesonde data. 1 .0 103 LOG NXK2 Figure 73 log10N2 vs depth at Tomakstum Island. Dashed line = CTD data, solid lines = cyclesonde data. H = 340 m. LOG NXX2 -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 Figure 74 log^N2 vs depth at Adeane Point. Dashed line = CTD data, solid lines = cyclesonde data. H = 540 m. 104 w-eigenfunction (cm/s) 0 Figure 75 w-eigenfunctions at Tomakstum Island including upper 20 m. H = 340 m. u-eigenfunction (m/s) o CM O o a 0 0 •u 2 °- 2 oj ^ o CN O o o oo •1.2 - 0 . 6 C i i . +0.6 + i 7 - — 2 \ 1 \ \ A' V 1 /\ > ; v / ; v / ' * / ; < a i * \ i > i \ i * t ; ' i I i j 1 i; i i i : Figure 76 u-eigenf unctions corresponding to Figure 75. 105 Unfortunately there is no orthogonality condition which uses only the portion of the normal mode profiles below 20 m, so a least squares fitting approach must be taken. The solid lines in Figures 81-82 show the same set of normal modes as in Figures 75-76 (using an average iV2-profile from CTD casts in the upper 20 m, and from cyclesonde density profiles below 20 m). However, in Figures 77-84 the profiles were plotted below 20 m (25 m at Lull Bay) only, and the modes were normalised so that each mode had the same energy density E' as given by equation 3.47. The /?-eigenfunctions were calculated using equation 3.24, and are plotted at the standard depths with straight line segments joining them. The two dashed lines for each mode are ± 1 standard deviation from the mean of profiles calculated using a single CTD cast above 20 m and a single cyclesonde profile below 20 m. Each CTD cast (MCTD of them) was combined with a cyclesonde profile ( M c y of them) to produce MCTD x MCY different JV2-profiles. These profiles were then used in turn to calculate MCTD* MCY sets of normal modes, which were then averaged to give the dashed lines shown in Figures 77-84. The mean phase speeds in m/s are also given, along with the ± 1 standard deviation values. In all cases there is good agreement between the eigenfunctions calculated using the mean composite JV -^profiles and the mean eigenfunctions calculated using single composite iV2-profiles. In the fitting stage discussed later in Section 3.10 it was the former (eigenfunctions calculated using mean composite N2-profiles) that were actually used. 3.10 Comparison with Farmer and Smith's Normal Modes Figure 85 is a reproduction of the density profiles and corresponding normal modes used by Farmer and Smith (1980a). This CTD cast was taken on August 13, 1977 "a few kilometers up-inlet of the sill." Given a bottom depth of 320 m, their location would correspond most closely with the Tomakstum Island station. Farmer and Smith argue that because their w-eigenfunction has a maximum at around 24 m for mode 1, and at around 136 m (and 4 m) for mode 2, a large disturbance at depth, such as flow over a sill, might be expected to excite mode 2 oscillations in the summer. In the winter the surface fresh layer is much weaker, and the mode 1 eigenfunction in the winter looks very similar to the 106 Density eigenfunction (kg/m3) - 0 . 0 2 - 0 . 0 1 0 0 . 0 1 0 . 0 2 Figure 77 p-eigenfunctions at Protection Point excluding upper 20 m. u-eigenfunction (cm/s) - 2 Figure 78 u-eigenfunctions corresponding to Figure 77. MCTD = 4, MCY = 2. 107 Density eigenfunction (kg/m3) -0.02 -0.01 0 0.01 0.02 ,_, H 1 1 1 1 1 1 1 1 O CM o--3" O O oo a. OJ o. o o o CN* o a o co-C o = 4 0 . 5 3 4 + / - 0 . 0 0 0 C , = 0 . 7 8 8 C j = 0 . 4 4 9 C , = 0 . 2 9 6 C . = 0 . 2 3 1 + / - 0 . 0 9 1 + / - 0 . 0 0 5 + / - 0 . 0 0 5 + / - 0 . 0 0 4 Bottom Figure 79 p-eigenfunctions at Lull Bay excluding upper 25 m. -2 o u-eigenfunction (cm/s) •1 0 1 Figure 80 u-eigenfunctions corresponding to Figure 79. MCTD = 3, MCY = 1 108 D e n s i t y e i g e n f u n c t i o n (kg/m 3) -0.01 0 0.01 0.02 Figure 81 p-eigenfunctions at Tomakstum Island excluding upper 20 m. H = 340 m. -2 u - e i g e n f u n c t i o n (cm/s) -1 0 1 o CM o o o JZ 0 0 4-1 CJ o a o o CM O o o oo 2 1 3 ' v ' ' / s s ^ — / I / ' 0 / < M l / \\/ ' i 1 1 \ w A\ ' ' 1 1 \ ^ / V \ \V / A Y \ A'M ' 70 \ Av \ A K A* \ W V KK * A \ \ 1 V\ v v \\ • v\ * > V\ \ \\ \ V \ V \ t \ \ * \ V ' i l \ \ *\' n i l . \ \ \ , W Figure 82 u-eigenfunctions corresponding to Figure 81. MCTD = 5, MCY =3 109 -0.02 o Density eigenfunction (kg/m3) -0,01 Q.,01 • 02 Figure 83 /-"-eigenfunctions at Adeane Point excluding upper 20 m. H = 540 m. u-eigenfunction (cm/s) Figure 84 u-eigenfunctions corresponding to Figure 83. MCTD = 3> MCY = 2. mode 2 eigenfuntion in the summer, with a maximum at 127 m. Mode 1 should therefore be excited by the sill in winter. Figure 85 Farmer and Smith's density profile and w-eigenfunctions. Phase speeds: c\ = 98 cm/s, c2 = 41 cm/s, c$ = 33 cm/s. (From Farmer and Smith 1980.) Blackford (1984) also proposes the same argument in favour of a predominantly mode 2 response, based upon the same normal modes. He also cites the zero crossing in the mode 2 u-eigenfunction at 136 m as an explanation for the 180° phase shift in the baroclinic component of the longitudinal velocity between 75 m and 300 m found by Farmer and Freeland (1983). A predominantly mode 1 response would not produce this 180° phase shift, because the mode 1 u-eigenfunction has a zero crossing at 24 m. (Note that extrema in the w-eigenfunctions correspond to zero crossings in the u-eigenfunctions.) However, inspection of Figure 76 reveals zero crossings in the u-eigenfunctions at 75 m for mode 1, and at 126 m and 4 m for mode 2. This change of the shape would partially invalidate the preceding arguments, because the sill crest would now be at the same depth as the mode 1 maximum in tu-eigenfunction, and because a mode 1 with a u-eigenfunction zero crossing below 75m is capable of producing the 180° phase shift between 75m and 300m on its own. How can the deepening of the mode 1 zero crossing from Farmer and 111 Smith's 24 m to this value of 75 m be explained? The obvious answer is that the JV2-profile taken in 1977 used by Farmer and Smith is different from the one taken in 1983 by the cyclesonde. In an attempt to understand how the JV2-profile affects the shape of the normal modes, two simple models were investigated. The first model consisted of two layers of constant JV2 so that the density profile was made up of two linear segments which matched at the interface. The upper layer was of thickness h\ with N = Ni, and the lower layer was of thickness h2 with N = N2. The second model consisted of a well mixed surface layer of constant density (so that N = 0),"overlying the remainder of the water column which had a constant N value. There was a density jump of Ap at the interface. In both models an asymptotic expansion was used, assuming that the ratio e of the lower layer N value to the mean upper layer N value (including any density jump at the interface) was small. The details of the theory are presented in Appendices A l and A2. In the first model the eigenfunctions could be categorized into so-called a-modes and /?-modes. a can be any non-negative integer, and (3 can be any positive integer. /?-modes are due to the lower layer stratification, since their O(l) phase speed c° = only depends on JV2/i2. a-modes are associated with the upper layer stratification, since their O(l) phase speed c° = ^+i)ir only depends on Nihi. The nth mode can be either an a-mode or a /?-mode, since its phase speed is the nth element of the ranked series of the two sets of phase speeds. In the second model the a-modes are replaced by a single surface mode. The surface mode is associated with the density jump at the interface, and to 0(1) has a phase speed of y/g'hi, where g' = gAp0/pQ is the reduced gravity. As before, the nth mode could either be one of the /?-modes or the surface mode. However, in the summer the first mode is most likely to be the surface mode or the a = 0 mode. The phase speeds and zero crossing depths of Farmer and Smith's normal modes and those shown in Figure 76 can be interpreted in terms of the model profiles described in Appendices A l and A2. Farmer and Smith obtained phase speeds of 0.98, 0.41 and 0.33 m/sec for the first three modes. The fact that 0.98 « 3 x 0.33 is entirely consistent with the first and third modes being or = 0 and a = 1 modes respectively, for which the 0(1) phase speeds are 2*^- and 2 ^ l i . This gives a value of Nxhi = 1.54m/sec. The second mode must therefore be the P = 1 mode, which has a 0(1) phase speed of , giving N2h2 = 1.29 m/sec. Given a total density drop of Ap = 22 kg/m3 across the upper layer (from o~t = 24 below the surface layer to o~t » 2 at the surface), values of hi = 11 m, h<2 = 309 m, Ni = 0.14 s - 1 , and N2 = 0.0042 s - 1 can then be deduced for the individual parameters. The same procedure can then be applied to the normal modes in Figure 76, where the phase speeds are 0.80, 0.51 and 0.32 m/sec. It is less obvious this time which modes are a-modes, because neither 0.80 = 3 x 0.32 (if the first and third modes were a-modes) nor 0.51 = 2 x 0.32 ( if the second and third modes were p-modes). However, assuming the first mode to be the a = 0 mode, Nihi = 1.26 m/sec is obtained, and assuming the second mode to be the P — 1 mode, N2h2 — 1.60 m/sec is obtained. Again A c t = 22, gives hx = 7.5m, h2 = 312.5m, Ni = 0.16a - 1, and N2 = 0.0052s-1. However, if the third mode is used to calculate Nihi assuming it to be the a = 1 mode, then Nxhi = 1.51m/sec, hi = 11m and Ni = 0.14 s - 1 are obtained, which are the same as calculated for Farmer and Smith. It was therefore hypothesised that the most important difference between Farmer and Smith's profile and the Figure 76 profile was in the lower layer stratification. Figures 86-88 show plots of the phase speeds and zero crossing depths as a function of N2 for the first three modes, keeping hx = 11m, h2 = 309 m, and Ni = 0.14 s - 1 constant. The vertical dashed lines indicate transitions from a-modes to /3-modes; for example, mode 2 could either be the a = 1 mode or the P = 1 mode depending on the value of N2. The horizontal dashed line indicates the depth of the interface between the two layers. The solid curves are solutions to the full non-linear dispersion relation (equation A4), whereas the short dashed curves are the 0(1) terms in the asymptotic expansion. (Note that the classification of modes into a-modes and /3-modes is not possible without the asymptotic 113 expansion.) The phase speeds and zero crossing depths are marked by circles for Farmer and Smith's profile, and by triangles for the Figure 76 profile. This simple model explains the main differences between the Farmer and Smith modes and the Figure 76 modes well: (i) The mode 1 zero crossing is deeper in Figure 76 (75 m) than in the Farmer and Smith mode 1 (24 m) because the former has a stronger lower layer stratification (i.e. larger JV"2). The zero crossing is pulled down below the interface to a value of z = N £ l when "rfc > 1 (see'Figure 116). This also occurs in model 2 when > 1 (see Figure 119). Therefore in both models, the criterion for the mode 1 zero crossing to be pulled down below the interface is > 1. For Farmer and Smith this ratio is 0.84, whereas in Figure 76 it is 1.27. (ii) The mode 2 phase speed is smaller for Farmer and Smith because of their weaker lower layer stratification. Mode 2 is the f3 = 1 mode, for which the 0(1) phase speed-is given by NlJlJ. This argument also applies for model 2. (iii) The higher mode 1 phase speed for Farmer and Smith can only be explained by a stronger surface layer stratification. This phase speed is given by 2 N i h i for model 1, or \/gTh~l for model 2. The weaker lower layer statification used by Farmer and Smith can be seen by com-paring the density gradients at depth from the density profiles. In Farmer and Smith's profile, the density changes by 0.55 kg/m3 between 20 m and 170 m (an average N value of 0.0059 s - 1 ) , whereas in the profiles in Figure 70 the corresponding density change is 1.07 kg/m3 (an average N value of 0.0083 a~l). These simple models of the stratification help to conceptualise statements like a a mode 2 eigenfunction for the summer is very similar to a mode 1 eigenfunction for November be-low the top few meters* (Farmer and Smith 1980a, p240), because with the disappearance of the fresh surface layer in the winter, the phase speed of the "surface mode" (model 2), 114 o "I n> 5' a Figure 8 8 Mode 3 phase speed and zero crossing depths vs N2. or the a = 0 mode (model 1), drops off to a value that is lower than the phase speed of the first "lower layer mode" /? = 1, causing this mode to become mode 1. Hence the j3 — 1 mode is mode 1 in the winter, yet it is mode 2 in the summer. It is obvious from this discussion that small values of N at depth cannot be neglected when calculating normal modes, for it is the depth-integrated value N2h2 that is important, not N2 itself. N2 may be very small compared to the surface value of N, but it maintains this small value over a great depth range, and N2h2 is not small. Therefore when taking CTD casts it is preferable to lower the instrument to the bottom rather than assuming that the density changes at great depth are so small that it is not worth the time to go that far. This error was made in our CTD casts in Knight Inlet, where the instrument was never lowered any further than 200 m. To make up for this missing data, a linear extrapolation of logioN2 had to be made to the bottom. To summarise this section, it has been shown that the differences between the normal modes of Farmer and Smith (1980a) and those in Figure 76 can be explained by the 116 differences in the iV2-profile used; specifically Farmer and Smith used a stronger upper layer stratification and a weaker deep-water stratification. In the mode fitting that follows, the normal modes shown in Figures 77-84 will be used, since they were calculated using 1983 data. 3.11 Results of the Normal Mode Fitting Having gained some confidence in the correctness of the calculated normal modes, the results of fitting them to the observed profiles of complex amplitudes of u- and p-oscillations can be discussed. The statistical model described by equations 3.33-3.36 was used at the Tomakstum Island stations, and that described by equations 3.29-3.30 at the rem'aining stations. In both cases, a pre-programmed least-squares fitting routine was used to yield estimates of AJ£P and A ^ o t o n for each of the modes. The pre-programmed subroutine fitted real functions to real functions, so the real and imaginary parts of AJJP and Aj[O U 7 n were solved for simultaneously as real numbers. The complex amplitude profiles from the month of September 1983 were used at each station, because there were data from both sides of the channel at Tomakstum Island, and that month was representative of the other months at the other stations. However, the July 1983 data was used at Lull Bay because there were no data from that station in September 1983. Mode 1 strongly resembles mode 2 below 20 m. Except for the fact that mode 1 has a zero crossing in the u-eigenfunction at around 75 m and that mode 2 has a zero crossing -at around 125m their shapes are very similar. It is therefore very difficult for the fitting routine to separate mode 1 from mode 2. Likewise, modes 0 and 1 have a similar p-eigenfunction shape, although this is not such a problem because the amplitude and phase of the barotropic component are fixed by other considerations (see Section 3.6). Tabie 4 gives the correlations r m n = (3.53) yV"* ««.)*} {/„"-"- «8M*} 117 where un(z) = un(z) - H _l2Qm J un(0 dc;, (3.54) and r£ , n (defined similarly) between the eigenfunctions at each of the stations. The corre-lation between the mode 1 and mode 2 u-eigenfunctions is always at least 0.96! Table 4 Correlations between the modes below 20 m at each station Mode 1 u-eigenfunctions 2 3 4 Mode 1 p-eigenfunctions 2 3 4 Protection Point 0 0.881 0.772 -0.080 -0.734 0 0.979 0.112 -0.937 -0.501 1 1.000 0.975 0.338 -0.451 1 1.000 0.312 -0.873 -0.596 2 1.000 0.537 -0.268 2 1.000 0.141 -0.476 3 1.000 0.625 3 1.000 0.549 Lull Bay 0 0.859 0.810 0.456 -0.346 0 0.996 0.816 -0.918 -0.945 1 1.000 0.986 0.715 -0.124 1 1.000 0.862 -0.886 -0.943 2 1.000 0.822 0.039 2 1.000 -0.542 -0.760 3 1.000 0.596 3 1.000 0.930 Tomakstum Island 0 0.816 0.690 -0.801 -0.742 0 0.995 0.766 -0.993 0.082 1 1.000 0.978 -0.508 -0.934 1 1.000 0.833 -0.998 -0.020 2 1.000 -0.318 -0.892 2 1.000 -0.828 -0.574 3 1.000 0.620 3 1.000 0.022 Adeane Point 0 0.598 0.452 -0.682 -0.478 0 0.993 0.230 -0.970 0.313 1 1.000 0.964 -0.881 -0.968 1 1.000 0.343 -0.992 0.200 2 1.000 -0.729 -0.966 2 1.000 -0.460 -0.846 3 1.000 0.832 3 1.000 -0.074 As was shown in Section 3.5.1, the fitting of u-eigenfunctions to observed profiles for mid-channel stations yields the sum (Anp + Anown), whereas the fitting of p-eigenfunctions yields the difference (AJJP—A^0""1). If both mode 1 and mode 2 were included in the fit, the coefficients (A* p + Anov"1) of the u-eigenfunctions would be expected to be very large and differing by 180° in phase, so that the mode 1 and mode 2 u-eigenfunctions would almost cancel each other out (analogous to including two constituents in a harmonic analysis that are too close together in frequency). If the p-eigenfunction fit yields reasonably small values for (Anp - Anown), then this leads to very large magnitudes for A ^ p and AnOXBn, 118 with A £ P « A * This was, in fact, observed to happen whenever both mode 1 and mode 2 were included in the fit, even at Tomakstum Island where information on the cross-channel structure could be incorporated. Tables 5-7 show the results of a series of fits at each station. The different fits arise by excluding various combinations of mode3 from the analysis by deleting those modes from the set of basis functions to be fitted to the observations. A blank entry in any table means that the corresponding mode was excluded from the set of basis functions, so that its amplitude can be taken to be zero. Tabie 5 gives the amplitudes a£ p and a^0""1, and TaWe 6 gives the phases <f>lP and <f>A^Dn of the fitted waves. The normalisation constant U0 for each eigenfunction was such that the a n values are amplitudes of the dimensional eigenfunctions given in Figures 77-84. The energy density E' of equation 3.47 is the same for each mode in those figures. Tabie 7 gives the energy fluxes in Megawatts as calculated using equation 3.47, integrating over the whole water column (note: Flux = E'cn). It should be noted that the fluxes are only the Af2 component of the total flux of the internal tide, and that there will be a small but significant 5 2 component. What are considered to be the most accurate and consistent estimates from the various fits are italicised in Tables 5-7. The values in italics were then averaged to give best estimates of the values in each mode at the various stations. These best estimates are given in the last column of the tables. The rows in Tabie 71abelled "Residual sum sqs" contain estimates of the residual errors. They are the sum of squares of the residuals divided by the number of equations minus the number of unknowns, i.e. they are the mean squared distance of the observed points in the complex plane from the fitted points. It was necessary to assign an equivalency between "velocity distances" and "density distances", which is the same as assigning a weight to u-observations relative to p-observations. This equivalency was chosen so that 20 cm/s of "velocity distance" is the same as 0.4kgm~3 of "density distance". This equivalency was used in Figures 53-56. 119 The inclusion of more basis functions in the fit leads to a smaller residual sum of squares, but also increases the possibility of meaningless results because of degenerate basis functions. Thus there is a trade-off between goodness-of-fit and robustness in the derived coefficients. The approach used was to find the smallest set of modes that was sufficient to give an acceptable fit to the data. 120 Table 5 Results of the Biting — non-dimensional amplitudes a„p and af^"1 Fit # 1 2 3 4 5 6 7* Best estimate Protection Point Up-inlet mode 0 26.04 26.04 26.04 26.04 26.04 26.04 26.0 1 5.05 - - — - - ? 2 5.42 - 7.95 8.04 - - 8.6 3 1.47 - 1.64 4.56 - - 4.6 4 - - 0.94 - — — ? Down-inlet mode 0 26.04 26.04 26.04 26.04 26.04 26.04 20.6 1 19.17 20.01 - - — — ? 2 6.40 7.25 13.43 9.59 8.81 13.57 9.6 3 6.59 6.09 9.13 3.84 3.91 10.65 3.8 4 - - 5.98 - - 7.82 ? Lull Bay Up-inlet mode 0 29.90 29.90 29.90 29.90 29.90 29.90 29.9 1 27.11 - - — — — ? 2 10.93 - 10.10 6.91 - - 6.9 3 7.66 - 13.56 S.9S — — 3.9 4 - — 6.22 - — — ? Down-inlet mode 0 29.90 29.90 29.90 29.90 29.90 29.90 29.9 1 21.64 9.05 - - — — ? 2 6.41 12.02 9.71 9.04 10.39 16.09 9.6 3 14.54 2.59 5.12 6.26 4.58 15.69 6.3 4 - - 3.68 - - 9.82 ? Tomakstum Island Up-inlet mode 0 21.41 21.41 21.41 21.41 21.41 21.41 21.41 21.4 1 27.82 7.94 - - - - - 7.9 2 0.46 10.54 6.43 9.33 9.08 6.96 6.98 8.8 3 7.21 5.91 7.16 7.33 7.90 8.46 8.37 7.7 4 - - 4.61 - - 4.73 4.52 4.6 Down-inlet mode 0 21.41 21.41 21.41 21.41 21.41 21.41 21.41 21.4 1 29.30 - - - — — — 0 2 7.80 - 3.67 2.09 - - - 0 3 4.32 - 1.37 1.84 - - — 0 4 - - 4.82 -• - - - 0 Adeane Point Up-inlet mode 0 14.16 14.16 14.16 14.16 14.16 14.16 14.16 14.2 1 8.30 1.41 - - - - - 1.4 2 6.26 S.99 3.59 3.55 S.41 8.21 1.04 2.9 3 2.90 2.25 2.23 2.10 1.98 2.05 1.95 2.0 4 - - 1.61 - - 0.87 0.00 0.4 Down-inlet mode 0 14.16 14.16 14.16 14.16 14.16 14.16 14.16 14.2 1 11.50 - - - - - — 0 2 5.07 — 0.18 0.77 — - — 0 3 2.94 - 0.50 0.38 - - - 0 4 — - 0.58 - - — - 0 * Phase differences fixed. 1 2 1 Table 6 Results of the Siting — phases <p%p and (pf?""1 in degrees Fit# 1 2 3 4 5 6 7* Best estimate Protection Point Up-inlet mode 0 20 20 20 20 20 20 20 1 169 - - - - — ? 2 254 - 211 198 - - 198 3 216 - 342 315 — — 315 4 - - 238 - - — ? Down-inlet mode 0 226 226 226 226 226 226 226 1 320 332 - - - - ? 2 122 178 328 334 311 317 334 3 344 349 120 102 124 139 102 4 - - 317 - - 327 ? Lull Bay Up-inlet mode 0 22 22 22 22 22 22 22 1 350 - - — — - ? 2 186 - 337 320 - - 320 3 351 - 179 171 - - 171 4 - - 359 - - -Down-inlet mode 0 224 224 224 224 224 224 224 1 156 83 - - - - ? 2 303 203 170 180 179 163 180 3 166 155 234 183 230 302 183 4 - - 141 - - 139 ? Tomakstnm Island Up-inlet mode 0 24 24 24 24 24 24 24 24 1 306 192 - - - - - 192 2 299 320 287 297 302 294 298 303 3 0 55 17 43 42 31 32 40 4 - - 140 - 147 144 145 Down-inlet mode 0 222 222 222 222 222 222 222 222 1 332 2 151 - 71 7 - - — 3 357 - 292 133 - - — 4 - - 102 - - - -Adeane Point Up-inlet mode 0 26 26 26 26 26 26 26 26 1 332 . 20 - - - - - 20 2 175 186 176 193 183 178 Ill 164 3 331 328 307 311 316 307 306 314 4 - - 96 - - 53 53 Down-inlet mode 0 220 220 220 220 220 220 220 220 1 316 2 129 - 70 95 - - -3 313 - 318 25 — - -4 - - 286 - - - -* Phase differences fixed. 122 Table 7 Results of the Sitting — energy fluxes in MW F i t # 1 2 3 4 5 6 7* Best estimate Protection Point Up-inlet mode 0 610.3 610.3 610.3 610.3 610.3 610.3 610 1 0.181 - - - - - ? 2 0.143 - 0.310 0.317 - - 0.32 3 0.007 - 0.009 0.070 - - 0.07 4 - - 0.002 - — — ? Down-inlet mode 0 610.3 610.3 610.3 610.3 610.3 610.3 610 1 2.655 2.893 - - - - ? 2 0.200 0.258 0.884 0.451 0.380 0.902 0.45 3 0.146 0.125 0.281 0.050 0.051 0.381 0.05 4 - - 0.096 - - 0.167 ? Residual sum sqs. 0.112 4.134 1.384 2.435 8.305 5.366 Lull Bay Up-inlet mode 0 667.9 667.9 667.9 667.9 667.9 667.9 668 1 5.412 - - - - - ? 2 0.489 - 0.440 0.206 - - 0.21 3 0.175 - 0.551 0.046 - - 0.05 4 - - 0.096 - — - ? Down-inlet mode 0 667.9 667.9 667.9 667.9 667.9 667.9 668 1 3.449 0.603 — — — — ? 2 0.178 0.623 0.396 0.352 0.466 1.117 0.35 3 0.633 0.020 0.078 0.117 0.063 0.738 0.12 4 - - 0.034 - - 0.239 7 Residual sum sqs. 0.512 5.101 0.700 1.125 5.272 3.748 Tomakstum Island Up-inlet mode 0 682.7 682.7 677.6 677.6 677.6 677.6 677.6 680 1 8.062 0.656 - - - - - 0.66 2 0.001 0.748 0.278 0.355 0.554 0.325 0.327 0.49 3 0.231 0.155 0.227 0.238 0.276 0.317 0.310 0.26 4 - - 0.075 - • - 0.079 0.072 0.08 Down-inlet mode 0 682.7 682.7 677.6 677.6 677.6 677.6 677.6 680 1 8.938 - - - - - — 0 2 0.410 - 0.090 0.029 - - — 0 3 0.083 - 0.008 0.015 - — — 0 4 - - 0.082 - - - - 0 Residual sum sqs. 1.091 4.665 2.137 3.681 4.867 3.255 3.256 Adeane Point Up-inlet mode 0 568.0 568.0 565.1 565.1 565.1 565.1 565.1 566 1 1.182 0.034 - - - - - 0.03 2 0.459 0.187 0.151 0.147 0.136 0.121 0.013 0.11 3 0.064 0.039 0.038 0.033 0.030 0.032 0.029 0.03 4 - - 0.016 - - 0.004 0.000 0.00 Down-inlet mode 0 568.0 568.0 565.1 565.1 565.1 565.1 565.1 566 1 2.268 — - - - - — 0 2 0.301 - 0.001 0.007 - - - 0 3 0.066 - 0.002 0.001 - - - 0 4 - - 0.002 - - - - 0 Residual sum sqs. 0.065 0.193 0.087 0.147 0.214 0.176 0.183 123 3.11.1 Results of the Fitting at Stations Up-inlet of the Sill The first column in each table refers to Fit #1, where the first three internal modes in both directions were included. As expected, mode 1 has a very large amplitude both up-inlet and down-inlet, and the up-inlet wave is often in phase with the down-inlet wave. This is a symptom of degenerate basis functions. It was therefore necessary to remove either mode 1 or mode 2 from the fit, or to introduce more information in the form of phase speeds in order to resolve them. In an attempt to exploit the difference in phase speeds between the two modes, the least squares fit was done simultaneously at Tomakstum Island and Adeane Point, and the phase difference of the up-inlet wave between the two stations was constrained to be duj/5n, where d = 11.2 km is the distance between the two stations and l/Sn is the average inverse phase speed of the nth mode over the two stations (a crude WKB approximation). For 5i = 83cm/s this phase difference is 108°, whereas for c"3 = 50cm/s it is 188°. The same procedure could not be applied to the down-inlet waves, because it is not known a priori how much of the up-inlet wave is reflected by the bend which is half-way between the two stations, so some of the down-inlet wave at Tomakstum Island could be a reflection from the bend and some could have propagated around the bend from Adeane Point. It is not even clear how the phases of the up-inlet waves are affected by passage around the bend unless the results of Chapter 2 are accepted. Despite this objection, the procedure was attempted on the up-inlet waves using a sophisticated least-squares fitting routine with non-linear constraints. Unfortunately, this additional information was insufficient to resolve the two modes, and similar results to Fit #1 were obtained. Following these two failures, it was decided to exclude mode 1 from the fit in order to try to remove the degeneracy between modes 1 and 2. This step can be justified by assuming that mode 1 is mostly a surface phenomenon due to the fresh upper layer, and it does not manifest itself a great deal below 20 m. To illustrate this point, Figure 76 shows that if the surface peak in the mode 1 u-eigenfunction takes a value of 1 m/sec, then the function has dropped to 10 cm/s at 20 m. However, if the surface peak in the mode 2 124 0.4 20.0 Figure 89 Results of Fit #6 at Tomakstum Island. Dashed lines = observed, solid lines = fitted, (ci. Figure 55.) —I X ro >* i _ ro a —i c n ro Real axis Figure 90 Results of Fit #6 at Adeane Point, (c.f. Figure 56.) 125 u-eigenfunction has the same amplitude, then the function has only dropped to 60 cm/s at 20 m. In other words, currents at the surface are 10 times the currents at 20 m for mode 1, whereas they are only twice as big for mode 2. If the total surface current is to be kept to a reasonable value, say < 1 m/sec (Pickard and Rogers 1959), then the mode 1 currents at 20 m and below must be small. Farmer and Smith also argue in favour of mode 2 dominating mode 1, as does Blackford. However, their arguments are based on the shapes of the eigenfunctions in 1977, which have been shown to be different in 1983. This assumption of mode 1 having a negligible manifestation below 20 m is compatible with the existence of a significant mode 1 energy flux, which would be mostly in the upper layer. The results obtained by excluding mode 1 from a bi-directional fit are much improved over the full bi-directional fit. The amplitudes of the various waves are more reasonable, and the energy fluxes are in the correct direction — for example, there are no longer greater down-inlet fluxes than up-inlet fluxes as there were in Fit #1. These results are given under columns #3 and #4. Fit #3 includes modes 2, 3 and 4, whereas Fit #4 only includes modes 2 and 3. They show that there is indeed a much greater up-inlet flux at Tomakstum Island than down-inlet flux, which is consistent with the very low reflection coefficients found in Chapter 2. To obtain more accurate estimates of the up-inlet energy fluxes at Tomakstum Island and Adeane Point, the down-inlet waves were specifically excluded from the fit (as in Fits #2, #5, #6 and #7), now that the reflection coefficient at the bend can be taken to be small. In Fit #2 modes 1, 2 and 3 were included, in Fit #5 modes 2 and 3 were included, and in Fit #6 modes 2, 3 and 4 were included. The fits were acceptable in all three cases. In Fit #2 modes 1 and 2 were successfully resolved by their different /-"-eigenfunctions. This was not possible in the bi-directional fit because the density information was being used to resolve the direction of propagation. Fit #2 gives supporting evidence for a dominantly mode 2 response, since the amplitude of the mode 2 up-inlet wave is greater than the mode 1 up-inlet wave. Figures 89 and 90 show that it is possible to obtain adequate fits using only up-inlet propagating modes 2, 3 and 4. The fit is only marginally worse when mode 126 X ro i_ ro (Z —i cn ro 0.0 0.0 51GMR-T 0.1 0.2 5.0 10.0 U / ICM/SEC) 0.3 15.0 0.4 20.0 Real axis Fig-are 91 Results of Fit #4 at Lull Bay. Dashed lines = observed, solid lines = fitted, (cf. Figure 54.) <_ ro ro Real axis Figure 92 Results of Fit #4 at Protection Point, (cf. Figure 53.) 127 4 is dropped. In Fit #7 the phase difference constraint described earlier was also imposed without much loss of goodness-of-fit. These four fits show that the data are compatible with the idea of zero reflection from the bend. 3.11.2 Results of the Fitting at Stations Down-inlet of the Sill Figure 93 Results of Fit # 6 at Protection Point. Dashed lines = observed, solid lines = fitted, (cf. Figure 53.) Again, Fit #1 gives unacceptable results, with unreasonably large coefficients (although the fits are good). The exclusion of mode 1, as in Fits #3 and #4, removes the degener-acy while still maintaining an acceptable goodness-of-fit, as shown in Figures 91 and 92. However, unlike at Tomakstum Island and Adeane Point, the results of Fits #3 and #4 do not justify the exclusion of waves propagating towards the sill, since the amplitudes of 128 the up-inlet waves are comparable to those of the down-inlet waves. The results of the uni-directional fits at Protection Point and Lull Bay are given under columns #2, #5 and #6, but are not considered to be useful. The residual sums of squares in TaWe 7 show that the fits are worse for the uni-directional fits than for the bi-directional fits. Figure 93 is an example of an unacceptable fit arising from the exclusion of up-inlet waves at Protection Point. These results indicate a second source of internal tide seaward of Protection Point. Indeed, when the longitudinal depth profile of the inlet is studied, a secondary sill can be seen. Its slightly gentler bottom slope can be compensated by its stronger barotropic velocities in terms of its internal tide generating potential. 3.12 Comparison of Fluxes with Stacey's Model Stacey (1984, 1985) developed a simple linear model of internal tide generation by the sill of an inlet to estimate the energy flux in each mode. The model was proposed by Stigebrandt (1976, 1980) with a linear density profile, but Stacey modified it for the case of arbitrary stratification. The sill is modeled by a vertical wall, and the bottom is assumed to be flat on either side of it, although not necessarily at the same depth on both sides. The velocity field in the vicinity of the sill is expressed as a sum of normal modes, and the crux of the model is to make the horizontal velocity at the sill vanish below sill depth, and equal to a constant barotropic value above sill depth. He obtained the formula where en is the energy flux away from the sill in the nth mode, d is the depth of the sill crest, Uu is the barotropic tidal velocity amplitude over the sill, and H is the depth of water next to the sill. The calculation of the normal modes Zn is done for the deep water on the side of the sill towards which the energy flux is required. A similar model was applied by Buckley (1980) to Alberni Inlet. 129 Applying Stacey's formula to the normal modes at Lull Bay (Figure 80) for the Aij constituent, the following flux estimates were obtained for the first three internal modes: 1.89, 1.48, and 0.02 MW. The data from Lull Bay were used because that station is the closest to the sill. These values are therefore estimates of the down-inlet flux from the sill. The flux estimates from this simple model are very sensitive to the shapes of the modes, because they are proportional to the square of the value of the w-eigenfunction at sill depth. However, the mode fitting procedure is also very sensitive to the modal shapes, so it seems that any technique for estimating energy fluxes will be sensitive to the modal shapes. The only way to escape the decomposition of the internal tidal flux into normal modes is by direct calculation of the total flux from velocity and density data. An expression for the local energy flux can be extracted from the 3-D energy conservation equation. Unfortunately, this direct flux calculation was not possible with the cyclesonde data because of the problem of the missing upper 20 m. The above values for the down-inlet fluxes associated with the first three internal modes obtained from Stacey's model can be compared with the values 0.35 and 0.12 MW for the second and third modes at Lull Bay obtained from the modal fitting. The up-inlet fluxes obtained at Tomakstum Island for the first three internal modes were 0.66, 0.49 and 0.26 MW. It seems reasonable to assume that the sill will radiate similar amounts of energy in each direction in each mode. With this assumption, the modal fitting indicates that the sill radiates about 1 A M W in each direction in the first three internal modes at M-2, frequency. Only about 0.2 MW makes it around the bend to Adeane Point. This result at the sill compares to 3.4 MW in each direction from Stacey's model. As mentioned in Section 1.3, Stacey plotted his estimates of the total flux being radiated away from the sill at Mi frequency against time from November 1976 to May 1978. The fluxes from the first 10 modes were summed, as well as the contributions from up- and down-inlet propagating waves. He found that it agreed very well with estimates of the power withdrawn from the barotropic tide between Protection Point and Tomakstum Island based on the difference in the phase of the surface elevation oscillation at those two stations. This estimate was also 130 plotted against time for the same period. The two curves followed the same seasonal cycle, with a maximum around July. The values from July to September were 7-9 MW. Because the two curves agreed so well, he concluded that most of the power being withdrawn from the M-i barotropic tide at the sill was being fed into a progressive internal tide, and that the other processes occurring at the sill, such as hydraulic jumps, turbulence and high frequency internal wave generation, were relatively unenergetic. If this were the case, however, and the results of this thesis referring to the reflection coefficient at bends are correct, then one would expect that half of the energy lost from the barotropic tide (i.e. the half that propagates up-inlet) would be dissipated in the sinuous section of the inlet, and the other half would be dissipated in the straight section or propagate out to sea. This is not consistent with the results of Freeland and Farmer (1980), who found that much more energy went into the mean circulation (i.e. much more mixing of the water column occurred) in the straight section containing the sill than in the sinuous section. One alternative possibility is that Stacey's model does not contain enough dynamics of internal tide generation mechanisms, and that his predicted amplitudes are too high. If about 40% (= 2.8 MW -f 7 MW) of the power being withdrawn from the barotropic tide was being fed into a progressive internal tide, then only 1.4 MW would have to be dissipated via the internal tide in the sinuous section, 1.4 MW via the internal tide down-inlet of the sill, and 4.2 MW directly at the sill. These values would then be in better agreement with Freeland and Farmer. Another possibility is that most of the power lost from the barotropic tide initially goes into the internal tide, but about 60% of it is rapidly dissipated through turbulence within a couple of kilometers of the sill. 3.13 Conclusion With the exclusion of mode 1 from the fitting procedure on the grounds that this mode has a negligible manifestation in currents and density oscillations below 20 m, it was found that the up-inlet energy flux at both stations landward of the sill was much larger than the down-inlet flux. When down-inlet waves were excluded from the fit, and mode 1 retained, it was still possible to obtain a good fit to the data, and the phase differences between the 131 stations for the various modes were in agreement with the phase speeds of those modes. This is consistent with the very low reflection coefficient (< 1%) at the bend predicted in Chapter 2. The reduced amplitude of the internal tide up-inlet of the bend (i.e. at Adeane Point) must therefore be attributed to dissipation, perhaps through turbulence in the presence of the large horizontal velocity shears near the inside corner of the bend predicted by Chapter 2, and to scattering of the energy by the very irregular features in the vicinity of the bend. Subsequent bends will reduce the amplitude still further, until a negligible energy flux reaches the head. It was found that the up-inlet flux at the two stations seaward of the sill was the same order of magnitude as the down-inlet flux, suggesting the existence of a subsidiary sill seaward of these stations. This sill can be seen on longitudinal depth profiles of the inlet. Its slightly gentler bottom slope can be compensated by stronger barotropic velocities in terms of its internal tide generating potential. There could be some reflection of the up-inlet waves at Lull Bay by the main sill, and the resulting down-inlet waves would interfere (either constructively or destructively) with the down-inlet waves generated by that sill. The results of the fitting indicate that only 30-50% of the power being removed from the barotropic tide (7-9 MW in 1977) is being fed into the internal tide in the summer, so that the remaining energy must be dissipated directly at the main sill. This conclusion is at variance with the results of Stacey (1980), whose model shows that most of the power is being fed into the internal tide. However, an interpretation that is consistent with both results is that most of the power lost from the barotropic tide initially goes into the internal tide, but 50-70% of it is rapidly dissipated through turbulence within a couple of kilometers of the sill. 132 4. Low Frequency Currents in Knight Inlet 4.1 Power Spectra of Currents Figures 94-97 show the power spectra of longitudinal currents at the four stations in July 1983. The spectra for September 1983 are similar and are therefore not included. The 1981 time-series were considered to be too short for use in this chapter. Figure 94 Power spectra of longitudinal velocity at Protection Point. Solid line = 20 m, long dashed line = 80 m, short dashed line = 155 m. The energy density is plotted on a log scale because the Afg peak is an order of magni-tude higher than any other peak. The use of a log scale means that the plot is not energy preserving (i.e. the area under the curve between two frequencies does not represent the 133 i n -ro -zn<=> (_> C \ J X O " > X .—, o 1 1 1 I™ c o s L 0 m r 1 m 1 •—• ZD™ I—r- • a-Q_ cc LU I D -0.0 0.02 0.04 0.06 0.08 0.1 F R E Q U E N C Y ( C P H : 0. 12 0 .14 0.16 Figure 9 5 Power spectra of longitudinal velocity at Lull Bay. Solid line = 25 m, long dashed line = 65 m, short dashed line = 125 m. energy in that band). The mean and trend of the signals were removed before the Fourier transform was calculated. The Nyquist frequency is = 0.166 cph, where At is the sampling interval (3 hrs). This frequency is just greater than the frequency of the Af4 constituent. The frequency resolution is y « 1.3 x 10 - 3 cph, where T is the record length ( « 750 hrs, depending on the record). The power spectra were smoothed with a 3-point moving average, so that the values plotted are actually the mean energy in three consec-utive energy bands. The power spectra were calculated at three depths — the shallowest standard depth (solid line), the deepest standard depth (short dashed line), and a middle depth (long dashed line). The frequencies of the constituents used in the harmonic analysis 134 r— — in "I 1 1 7 "i r 0 . 12 —i——\ r 0 . 1 4 0 . 1 6 0 . 0 I 1 I 0 . 0 2 0 . 0 4 i r 0 . 0 6 0 . 0 8 0 . 1 F R E Q U E N C Y ( C P H ; Fignre 06 Power spectra of longitudinal velocity at Tomakstum Island. Solid line = 20 m, long dashed line = 80 m, short dashed line = 170 m. are plotted as ticks on the frequency axis. The dominant feature in each plot is the peak at semi-diurnal frequency, with lesser peaks at diurnal frequency, quarter-diurnal frequency (M 4), and at low frequency (less than 0.01 cph). There is also a significant peak at M K $ frequency in most plots. The power spectra at low frequencies show peaks at different frequencies at the various stations. In an attempt to identify a peak associated with the passage of weather systems and corresponding wind shifts, which is generally taken to be at a period of 3-5 days, the mean power spectrum at the shallowest depth of all eight records from 1983 was calculated. This 135 in • m — n \ i n -X L J c o 2 o — UJ o — m -i r - l 1 1 i i i i i i i r 0 . 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 F R E Q U E N C Y ( C P H ) 0 . 12 0 . 1 4 0 . 16 Figure 97 Power spectra of longitudinal velocity at Adeane Point. Solid line = 20 m, long dashed line = 80 m, short dashed line = 170 m. spectrum is shown in Figure 98. The shallowest depth was chosen because it is the one most likely to show currents associated with wind stress. A broad peak at a frequency of Q.Qllcpk, corresponding to a period of 3.8 days, can indeed be observed, although its significance is questionable. 136 in O — ' i n — ro — 0 . 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 F R E Q U E N C Y ( C P H : 0 . 12 —I 0 . 1 4 I 0 . 1 6 Figure 98 Power spectrum of velocity at shallowest depth averaged over all stations. 4.2 Time Series of Longitudinal Velocities with the Tide Removed In order to extract the low-frequency signal from the longitudinal velocity records, the tidal component was reconstructed using the amplitudes and phases from the harmonic analysis and subtracted from the original signal. The seven constituents at and above diurnal frequency were used to recontruct the tidal signal. As a check on the process a harmonic analysis was done on the residual. In all cases it was found that there was indeed a negligible component (< 1 cm/s) at the frequencies that had been removed. When the power spectra of these residuals were calculated, it was found that the tidal peaks had been removed and in some cases they appeared as troughs in the power spectra. However, 137 the energy at frequencies between the tidal frequencies was unaffected. This energy was also evident in the residual time series, where there was a considerable amount of high-frequency (periods less than 1 day) noise. It was therefore necessary to smooth the residual with a 24 hr (8-point) moving average. The resultant low-pass filtered signals are shown in Figures 99-106. The power spectra of these filtered signals were found to be identical to the power spectra of the original signals at low frequency. The fortnightly oscillation of the MSf tide is plainly evident in the filtered signals, especially in September. This oscillation seems to have the same phase at all stations, with maximum near surface up-inlet flows being attained on or near Julian days 208, 223, 252 and 268. A sudden inflow occurs near the surface at Protection Point around day 253. The filtered signals are very similar at Tom-N and Tom-S in September, indicating that there is little cross-channel variability there. 4.3 The MSf Tide The complex amplitudes of the MSf constituent from a harmonic analysis of the longitudinal velocity are shown in Figures 107-110. The same results can be obtained by doing a harmonic analysis of the low-pass filtered signals shown in Figures 99-106. The loci all show the same general features, with the shallowest depth usually located in the first quadrant of the complex plane. The loci at Tomakstum Island and Adeane Point are particularly repeatable, with a doubling back of the loci at 110-140 m at Tomakstum Island, and a spiral shape at Adeane Point. The irregularity of the low-pass filtered signals makes the validity of using harmonic analysis for the MS f constituent questionable. However, the repeatability of the loci implies that the results are meaningful and worthy of future interpretation. Freeland and Fanner (1980) used Fourier analysis to determine the amplitude of the MSf velocities at Montagu Point (near Protection Point). They found similar results to those presented in Figure 107, with amplitudes of 4.8 cm/s at 15 m, 1.1 cm/s at 100 m and 3.1 cm/s at 165 m. These values compare to amplitudes ranging from 0 to 5 cm/s 138 I Figure 99 Low-pass filtered longitudinal velocity at Protection Point in July 1983. Posi-tive is up-inlet. Figure 100 Low-pass filtered longitudinal velocity at Lull Bay in July 1983. 139 Time (J u l i a n Days) Figure 101 Low-pass filtered longitudinal velocity at Tomakstum Island in July 1983. Positive is up-inlet. 206 211 216 221 226 231 236 7* Time (J u l i a n Days) Figure 102 Low-pass filtered longitudinal velocity at Adeane Point in July 1983. 140 Time (Julian Days) Figure 103 Low-pass filtered longitudinal velocity at Protection Point in September 1983. Positive is up-inlet. 20m 35m 50m 65m 80m OJ 95m Q 155m 2 170m 271 Time ( J u l i a n Days) Figure 105 Low-pass filtered longitudinal velocity at Tom-N in September 1983. Positive is up-inlet. < > . 0 5.0 Figure 107 Complex amplitudes of the MSf component of velocity at Protection Point o July 1983, A September 1983. The shallowest standard depth is marked by a larger symbol. ro Figure 108 Complex amplitudes of the MSf component of velocity at Lull Bay. o July 1983. 143 I i 1 1 1 i—1 n 0 . 0 1 . 0 2 . 0 3 . 0 4 . 0 5 o I U /(CM/SEC) Figure 109 Complex amplitudes of the MSf component of velocity at Tomakstum Island, o July 1983, A Tom-S September 1983, * Tom-N September 1983. 144 in Figure 107. The surface elevation data were also analysed for the MSf constituent. Values ranging from 6 cm to 21cm were found from the eight cyclesonde records from 1983. Freeland and Farmer obtained a value of 4.1 cm using tide gauge data. If the entire surface of the inlet is moving up and down with period 14.7 days, then a barotropic current of 0.02-0.06 cm/s (from the cyclesondes) or 0.01 cm/s (from Freeland and Farmers' tide gauge) could be expected at Protection Point. Clearly the currents observed are much larger than the calculated barotropic current. Freeland and Farmer argue that the large baroclinic MSf frequency currents are due to variations in the strength of the estuarine circulation. If more energy is available for mixing at spring rather than neap tides, then the circulation would be modulated at MSf frequency. 4.4 Residual Velocity Profiles Having observed so much variability in the non-tidal longitudinal velocities, the defi-nition and relevance of the "mean circulation" become questionable. It is certainly not a steady circulation. However, in an attempt to calculate mean velocity profiles, the eight cyclesonde records from 1983 were averaged over two MSf cycles (= 29.5 days). In July the first 29.5 days of data were used, but in September, where the records are only about 26 days long, a small amount of aliasing must be accepted. The results are shown in Figures 111-114. The profiles are quite repeatable from month to month, especially at Protection Point and Tomakstum Island. At Protection Point the profile has the classical structure of outflow at the surface and inflow at depth, with a zero crossing at 75 m (much deeper than the pycnocline)! This behaviour is also true at Adeane Point, with a zero crossing at 100 m. However at Tomakstum Island the profile takes on a 3-layered structure, with outflow at the surface and at depth, and inflow at intermediate depths. The zero crossings are at about 50 m and 150 m. Curiously, the profile at Lull Bay shows outflow at all depths. This result is not consistent with the idea of lateral homogeneity, since mass must be conserved. 145 RESIDUAL VELOCITY (CM/SEC; 5.0 Figure 111 Residual longitudinal velocity profiles at Protection Point. Solid line = July 1983, dashed line = September 1983. Positive is up-inlet. -5.0 RESIDUAL VELOCITY (CM/SEC) -2.5 o . o 2.5 5.0 in. a. LU o o in o o oo. Bottom Figure 112 Residual longitudinal velocity profile at Lull Bay. July 1983 146 Figure 113 Residual longitudinal velocity profiles at Tomakstum Island. Solid line = July 1983, long dashed line = Tom-S September 1983, short dashed line = Tom-N September 1983. Positive is up-inlet. RESIDUAL VELOCITY (CM/SEC) o-5.0 -2.5 0.0 2.5 5.0 J o LU 1 1 \ \ \ - \ \ \ \ -\ \ \ \ I \ Figure 114 Residual longitudinal velocity profiles at Adeane Point. Solid line = July 1983, dashed line = September 1983 147 A possible explanation is that tidal flows over the sill produce eddies in the vicinity of Lull Bay. These non-linear features may have non-zero means, causing lateral inhomogeneity in the mean velocity field. 4.5 Summary In this Chapter some interesting data relating to the low frequency component of the longitudinal velocity signal, which could be of use in further studies, are presented. Firstly the velocity power spectra are calculated at the various stations. In addition to validating the choice of constituents used in the harmonic analysis, they show a broad peak at a period of 3.8 days at the shallowest depth when averaged over all stations. This peak is probably associated with wind driven currents. Secondly the low-pass filtered velocity time series are presented, with the fortnightly oscillation of the MSf tide in evidence. There is scope for future work in analysing the convergences and divergences at various depths, and in interpreting sudden inflows and outflows. Finally, both the complex amplitudes of the MSf constituent and the residual velocity profiles (averaged over two MS f cycles) show repeatable features from month to month, indicating that they are worthy of further investigation. The latter are particularly repeatable, and could be of use in validating general circulation models. i 148 Appendix A : Simple Models of The Stratification A . l Model 1 — Two Layers of Constant N2 4 Figure 115 Density and N structure used in Model 1. H = h\ + h2. In this model N2 is taken to be constant in both layers, so that the density profile is piecewise linear and continuous. Figure 115 shows the density and N structure. Assuming that u2 <3C N2 and making the rigid lid approximation, the eigenvalue problem 3.3-3.5 becomes d?Z N2 with Z = 0 at z = O and H. The solution is „ . N2h2 . Nx{H-z) Z\ = sin sin — 1 - , c c . Nihi . N2z Z-2 = sin sin , c c with the following dispersion relation: . N ^ N2h2 „ . N2h2 Nihi N2 P in cos + A i sin cos = 0. (A.1) (A.2) (A.3) (AA) Suppose that (A5) i.e. that the density stratification in the upper layer is much stronger than in the lower layer. Note, however, that N2h2 is not necessarily smaller than Nihi. Expanding all 149 variables as an asymptotic expansion in powers of e, c = c° + ecx + e V + ... [A.6) Zx = Z\ + eZ\ + e2Zl + ... (A.7) Z2 = Z2° + eZ\ + £ 2 Z | + . . . . (A.S) The dispersion relation becomes . Nthi N2h2 . N2h2 Nihi e sm cos 4- sin cos = 0. (A.9) c c c c Now, sin cos is bounded, so to O(l), . N2h2 Nihi which has solutions or c° = ^ , 0 = 1,2,3... ("0-modes"), c° = N l h \ , a = 0,1,2... ("a-modes"). ( a + * ) * - ' The /?-modes are due to the lower layer stratification, since to 0(1) their phase speed depends only on iV 2/i 2. The a-modes are due to the upper layer stratification, since their phase speed depends only on Nihi. These a- and /?-modes comprise the infinite set of eigenfunctions to the problem, with the nth mode corresponding to the nth largest phase speed in the union of the sets { ^ ; ^ X , 2 , 3 . . . } U { ^ - 0 , 1 , 2 . . . } . So either the p = 1 mode (if > 2) or the a = 0 mode (if < 2) could be the first mode, depending on which has the higher phase speed. The order in which these phase speeds arrange themselves into descending order is completely determined by the ratio Eahx rauo N l h l -150 A.J.I "a-Modes" or "Upper Layer Modes" The first two terms in the asymptotic expansion of c are Nyhx o = 0,1,2.., c = ~ i v ^ c o t t e ( a + ^ V ' c 2 = ( A l l ) (-4.12) and the 0(1) term of the eigenfunction is Z 2 0 = ( - l ) « s i n ( ^ ( a + x ) ^ . ( i 4 . 1 4 ) Figure 116 illustrates the structure of these eigenfunctions for various values of jfi^. Z°(z) a = 0 mode Z°(z) a = 1 mode Figure 116 0(1) eigenfunctions for ot-modes. Case 1: ^ ( o r + i) < i, case 2: %±(a + *) = i, case 3: + *) > The asymptotic expansion will not converge if the argument of cot in equation A12 is close to I-JT; i = 1,2,3..., since then c1 —» oo; i.e. it will not converge if ^ = JV^_ f o r . = tff ( «+A )JT V ' This is true precisely when the phase speed of an a-mode is close to the phase speed of an /3-mode. Then the ranking of the phase speeds is ambiguous, because it is not clear which mode has the higher phase speed. 151 Z°(z) Z°(z) 3 = 1 mode = 2 mode Figure 117 0(1) eigenfunctions for /?-modes. The vertical gradient of Z° does not match across the interface to 0(1); the matching becomes however better and better as higher order terms of the asymptotic expansion are incorporated. There are or-fl extrema in the upper layer in an or-mode, with the remaining (n — a — 1) of the nth mode situated in the lower layer. The nth mode can be any one of the first n a-modes, or any of the first n /?-modes. For example, the 3 r d mode could be the a = 0, a = 1, a = 2, /? = 1, f3 = 2 or P = 3 mode, depending on the value of This parameter controls which mode is the nth mode, and hence also how many extrema are required in the lower layer to bring the total number up to n. A. 1.2 "(3-Modes" or "Lower Layer Modes" The first two terms in the asymptotic expansion of c are P = 1,2,3... (A.16) (A.17) and the 0(1) term of the eigenfunction is (ii.18) Figure 117 shows these eigenfunctions. 152 The asymptotic expansion will not converge if the argument of tan in equation A17 is close to (j + %)ir\ j = 0,1,2..., since then c1 —• oo; i.e. it will not converge if Hp. = " i k _ f o r i = 0,1,2... . (A.20) Again this is true precisely when the phase speed of an or-mode is close to the phase speed of a /?-mode. There are (3 extrema in the lower layer in an /?-mode, with the remaining (n — /?) of the nth mode situated in the upper layer. To find the position of the remaining extrema it is necessary to go to 0(e) in Z\, since the positions of the upper layer extrema are ambiguous to 0(1). After some algebra, one obtains Hence the extrema in the upper layer occur at Ni{H-z)0 1 3 5 N2h7 _ 2 ' 2 ' 2 ' (A.23) A.1.3 Summary /?-modes are due to the lower layer stratification, since their 0(1) phase speed c° = only depends on N2h2. a-modes are associated with the upper layer stratification, since their 0(1) phase speed c° = ^ ^ i j ^ only depends on N\h\. The nth mode can be either an a-mode or a /3-mode, since its phase speed is the nth element of the ranked series of the two sets of phase speeds. /3-modes have /3 zero crossings in their u-eigenfunctions in the lower layer at the fol-lowing fixed positions: l2 - - -h2 2'2'2' ' 2 The remaining (n — /?) zero crossings occur in the upper layer at Ni{H-z)p 1 3 5 2(n-/?)-! N2h2 2'2'2 153 » n » n > > (A.25) a-modes have (a +1) zero crossings in their u-eigenfunctions in the upper layer at the following fixed positions: hi (2a + 1) = 1,3,5,..., (2a + 1). (A.26) The remaining (n — a — 1) zero crossings occur in the lower layer at N2z (2a + 1) = 1,3,5,..., (2 (n - a - 1) - 1). (A 27) The order of ranking of the two sets of phase speeds depends solely on the ratio A .2 Model 2 — Well mixed surface layer overlying a deep layer of constant N2 z A h 2 D e n s i t y N 6 - f u n c t i o n Figure 118 Density and N structure used in Model 2. H = hi + h2. This profile was used by Baines (1982) in an analysis of internal tide generation by the continental slope, and by Stacey (1985) in his study of the internal tide in Knight Inlet. See Figure 118 for the density and N strucure. Assuming that w2 <§: AT2 and making the rigid lid approximation, as in Model 1, one obtains the solution Z 2 = s i n — , (4.29) c with the following dispersion relation: (c2 - ^ O t a n ^ 2 . + N2hlC = o. (A30) c 154 where g1 = This is the same dispersion relation as obtained by Stacey (1985, equation 13). Again, suppose that e = J V a i / - 7 « l . (A.31) V 9' By analogy with the e defined in equation A5 for Model 1, this e can be thought of as the ratio of the lower layer JV-value to the mean upper layer AT-value \J*p^f if the 5-function in N2 at the interface were smeared out over the upper layer. Expanding all variables as an asymptotic expansion in powers of e, as given by equations A6-A8, the dispersion relation becomes Noh ( c a - ^ A i ) t a n - i - i + ecy/g7^ = 0. (A.32) c It is not necessary to replace the JV2 by s when it occurs with an A 2 , because JV2/i2 is not a small quantity. To 0(1) this is (c°2 -g'ht)^^- = 0, (A33) which has solutions c° = yjg'hx (the "surface mode"), or c° = ^ 2 - , 0 = 1,2,3... ("/3-modes"). pic The surface mode is associated with the density jump at the interface, and has an 0(1) phase speed of \Jg'h\ as expected. The /3-modes are associated with the lower layer stratification as in Model 1. The collapse of the infinite series of a-modes associated with the surface layer in Model 1 to a single surface mode in this model is analogous to the way in which an infinite series of modes in a continuously stratified model collapses to a single internal mode in a model with two layers of constant density. 155 A.2.1 The Surface Mode The first two terms in the asymptotic expansion of c are c° c1 = —— cot Noh 2 "2 (it.34) (A.35) and the O(l) term of the eigenfunction is 7° - N * z *2 = S m ^ 0 ~ -(A36) (A.37) This eigenfunction is illustrated in Figure 119 for various values of N ^ a , Z°(z) Figure 119 0(1) eigenfunctions for the surface mode. C a s e l : ^ i < | , case 2: ^ = f , case 3: ^ > f. A.2.2 "(3-modes" or "Lower Layer Modes" The first two terms in the asymptotic expansion of c are c2 = 156 and the 0 ( 1 ) term of the eigenfunction is Z\ = 0, (A.40) Z% = sin Z-P- (A.41) n 2 See Figure 117. A.2.3 Summary The surface mode is associated with the density jump at the interface, and to 0 ( 1 ) has a phase speed of v V * i - The 0-modes are associated with the lower layer stratification, and to 0 ( 1 ) have phase speeds of N £ 2 . As before, the nth mode could either be one of the 0-modes or the surface mode; it will be the nth largest element of the set \y/¥hu 0 = 1 , 2 , 3 . . . } . However, in the summer the first mode is most likely to be the surface mode. The zero crossings in the u-eigenfunction will be at 2 = < 2 ' V f l ^ T 2 ' *VV f ti jr N,h, ^ * for the surface mode, and at (3z 1 3 5 2 0 - 1 , . . . M«> 2'2'2' ' 2 for the 0-modes. There will be an additional zero crossing at the interface z = h2 for 0-modes when n > 0 (e.g. the 0 = 2 mode acting as mode 3 ) . This additional zero crossing comes from the 0(e) expansion of Z\. 157 Bibliography BAINES, P.G. (1982) On internal tide generation models. Deep Sea Research, 29(3A), 307-338. BLACKFORD, B.L. (1978) On the generation of internal waves by tidal flow over a sill -a possible nonlinear mechanism. J. Mar. Res., 30(3), 529-549. (1984) Effect of a Tidal Stream on Internal Wave Observations and Predictions. Atmosphere-Ocean, 22(2), 125-143. BROWN, P.J. (1973) Kelvin wave reflection in a semi-infinite canal. J. Mar. Res., 31, 1-10. BUCHWALD, V.T. (1968) The diffraction of Kelvin waves at a corner. J. Fluid Mech., 31, 193-205. (1971) The diffraction of tides by a narrow channel. J. Fluid Mech., 46, 501-511. BUCKLEY, J.R. (1980) A linear model of internal tides in sill fjords. In: Fjord Oceanog-raphy. H.J. Freeland, D.M. Farmer, and C D . Levings, eds., Plenum Press, New York, 165-171. FARMER, D.M. and FREELAND, H.J. (1983) The physical oceanography of fjords. Progress Oceanog., 12, 147-220. FARMER, D.M. and SMITH, J.D. (1978) Nonlinear internal waves in a fjord. In: Hydro-dynamics of Estuaries and Fjords, J.NIHOUL,ed., Elsevier Oceanography Series, No. . 23, 465-494. (1980a) Tidal interaction of stratified flow with a sill in Knight Inlet. Deep Sea Research, 27A, 239-254. (1980b) Generation of lee waves over the sill in Knight Inlet. In: Fjord Oceanogra-phy. H.J. Freeland, D.M. Farmer, and C D . Levings, eds., Plenum Press, New York, 259-270. FOREMAN, M.G.G. (1977) Manual for Tidal Heights Analysis and Prediction. Pacific Mar. Sci. Report 77-10, Institute of Ocean Sciences, Patricia Bay, B.C., 101pp. (1979) Tidal Analysis Based on High and Low Water Observations. Pacific Mar. Sci. Report 79-15, Institute of Ocean Sciences, Patricia Bay, B.C., 39pp. FREELAND, H.J. (1984) The Partition of Internal Tidal Motions in Knight Inlet, B.C. Atmosphere-Ocean, 22(2), 144-150. FREELAND, H.J. and FARMER, D.M. (1980) Circulation and Energetics of a Deep, Strongly Stratified Inlet. Can. J. Fish, and Aquatic Sci., 37(9), 1398-1410. FREELAND, H.J., FARMER, D.M. and LEVINGS, C D . (1980) Fjord Oceanography, Plenum Press, New York, 715pp. 158 GILL, A.E. (1982) Atmospkere-Ocean Dynamics. Academic Press, New York, 662pp. HALPERN, D. (1971a) Observations of short-period internal waves in Massachusetts Bay. J. Mar. Res., 29, 116-132. (1971b) Semi-diurnal internal tides in Massachusetts Bay. J. Geophys. Res, 76(27), 6573-6584. HECHT, E. and ZAJAC, A. (1974) Optics. Addison-Wesley Publishing Co., Don Mills, • Ontario, 565pp. HENDERSHOTT, M.C. and SPERANZA, A. (1971) Co-oscillating tides in long narrow bays; the Taylor problem revisited. Deep Sea Research, 18, 959-980. JAMART, B.M. and WINTER, D.F. (1980) Finite element computation of the barotropic tides in Knight Inlet, B.C. In: Fjord Oceanography. H.J. Freeland, D.M. Farmer, and C D . Levings, eds., Plenum Press, New York, 283-290. LEBLOND, P.H. and MYSAK, L.A. (1978) Waves in the Ocean. Elsevier Scientific Pub-lishing Co., New York, 602pp. LONG, R.R. (1975) Circulations and density distributions in a deep, strongly stratified, two-layer estuary. J. Fluid Mech., 71, 529-40. M0RK, G. and GJEVIK, B. (1982) Numerical Simulations of Internal Wave Generation in Sill Fjords. In: Coastal Oceanography. H.G. Gade, A. Edwards and H. Svendsen, eds., Plenum Press, New York, 397-410. PACKHAM, B.A. and WILLIAMS, W.E. (1968) Diffraction of Kelvin waves at a sharp bend. J. Fluid Mech., 34, 517-529. PEARSON, C E . and WINTER, D.F. (1975) Analysis of stratified inlet flow by the method of weighted residuals. Symposium on modeling of transport mechanisms in oceans and lakes, Canada center for Inland Waters, Burlington, Ontario, Oct.6-8. PICKARD, G.L. (1956) Physical features of B.C. inlets. Trans. Roy. Soc. Can., Series m, 50, 47-58. (1961) Oceanographic features of inlets in the B.C. mainland coast. J. Fish. Res. Bd. Canada, 18(6), 907-999. (1975) Annual and longer term variations of deepwater properties of the coastal waters of southern B.C. J. Fish. Res. Bd. Canada, 32, 1561-1587. PICKARD, G.L. and RODGERS, K. (1959) Current measurements in Knight Inlet, B . C . J. Fish. Res. Bd. Canada, 16(5), 635-678. PICKARD, G.L. and TRITES, R.W. (1957) Fresh water transport determination from the heat budget with applications to B.C. inlets. J. Fish. Res. Bd. Canada, 14(4), 605-616. 159 PNUELI, A. and PEKERIS, C L . (1968) Free tidal oscillations in rotating flat basins of the form of rectangles and of sectors of circles. Philos. Trans. Roy. Soc. London, A , 263, 149-171. STACEY, M.W. (1984) The Interaction of Tides with the Sill of a Tidally Energetic Inlet. J. Phys. Oceanogr., 14(6), 1105-1117. (1985) Some aspects of the Internal Tide in Knight Inlet, B.C. (in press, J. Phys. Oceanogr., 1985. STIGEBRANDT, A. (1976) Vertical diffusion driven by internal waves in a sill fjord. J. Phys. Oceanogr., 6, 486-495. , A. (1980) Some aspects of tidal interaction with fjord constricions. Est. Coast. Mar. Set., 11, 151-166. TAYLOR, G.I. (1920) Tidal Oscillations in Gulfs and Rectangular Basins. Proc. London Math. Soc, 20, 148-181. VAN LEER, J . C ; W. DUING, R. ENRATH, E. KENNELLY and A. SPEIDEL. (1974) The Cyclesonde: An unattended vertical profiler for scalar and vector quantities in the upper ocean. Deep Sea Research, 21, 385-400. 160
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The propagation of the internal tide around a bend in Knight Inlet, B.C. Webb, Anthony James 1985
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Title | The propagation of the internal tide around a bend in Knight Inlet, B.C. |
Creator |
Webb, Anthony James |
Publisher | University of British Columbia |
Date Issued | 1985 |
Description | This thesis seeks to answer the question "how much of the internal tide propagating up Knight Inlet, B.C. is reflected by a right-angled bend ?" The internal tide in Knight Inlet is generated by the interaction of the barotropic tide with a shallow sill seaward of the bend. It then propagates in both directions as a travelling Kelvin wave. The up-inlet propagating wave then encounters the bend, where some of it may be reflected. This question is looked at with both theory (Chapter 2) and observations (Chapter 3). The theoretical study investigates the propagation of a Kelvin wave around a bend in a channel. The solution of the linearised long wave equations is expressed as a truncated series of cross-channel modes in each of three regions. The solution is matched across the two common boundaries by a least squares point matching process. The "rectangular bend" gives unsatisfactory results because of a singularity in the velocity field at the sharp inside corner. However, the "annular bend" gives good results. The bend acts as a diffraction grating, with total transmission for certain bend angles and "lobes" of high reflection for intermediate angles. Bends to the left give rise to the same reflection as bends to the right of the same angle, even if the incident Kelvin wave has a small cross-channel decay scale. For the parameters corresponding to the M₂ internal tide in Knight Inlet, the energy flux reflection coefficient is very small, much less than 1%. The method of solution breaks down inexplicably for diurnal tides. The observational study is based upon 2⅓ months of cyclesonde current meter data from four stations in Knight Inlet taken during the summers of 1981 and 1983. The vertical profiles of amplitude and phase of the M₂ constituent of longitudinal velocity and density fluctuations are found to be the same from month to month at a particular station, but different from station to station. These complex amplitude profiles are decomposed into a truncated series of normal modes for waves propagating both up-inlet and down-inlet. The phase speeds and zero-crossing depths of the normal modes do not agree with those calculated by Farmer and Smith (1980a) using 1977 data. Two simple models of the stratification are used to explain the differences in the normal modes in terms of differences in the Brunt-Väisälä frequency profile used to calculate them. At the two stations up-inlet of the sill, acceptable fits can be obtained using only two up-inlet propagating waves, indicating that the data are compatible with the low reflection found in the theoretical study. When down-inlet waves are included in the fit, their amplitudes are found to be small, although care must be taken to avoid trying to fit too many modes since some of them are highly correlated in the depth range for which data are available. At the two stations seaward of the sill, the up-inlet energy flux is of the same order of magnitude as the down-inlet flux, indicating a second source of internal tide seaward of those two stations. The results indicate that only 30-50% of the power removed from the barotropic tide is being fed into the internal tide in the summer. This conclusion is at variance with that of Stacey (1985), whose model shows that most of the power lost from the barotropic tide is being fed into the internal tide. A consistent interpretation of these results is that most of the power lost from the barotropic tide initially goes into the internal tide, but that 50-70% of it is rapidly dissipated through turbulence within a couple of kilometers of the sill. The low frequency component of the velocity signals from the cyclesonde data is also presented (Chapter 4), with a brief analysis . The monthly averaged residual velocity profiles are also found to be fairly repeatable from month to month at a particular station, but different from station to station. These profiles could be useful in validating general circulation models. |
Subject |
Tides -- British Columbia -- Knight Inlet |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-06-24 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0053114 |
URI | http://hdl.handle.net/2429/25997 |
Degree |
Doctor of Philosophy - PhD |
Program |
Oceanography |
Affiliation |
Science, Faculty of Earth, Ocean and Atmospheric Sciences, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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