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Internal tides in Dixon entrance Carrasco, Ana 1998

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I N T E R N A L TIDES IN D I X O N E N T R A N C E By Ana Carrasco M. Sc. Oceanography CICESE Ensenada Baja California Mexico.  A T H E S I S S U B M I T T E D IN PARTIAL F U L F I L L M E N T O F THE REQUIREMENTS FOR THE DEGREE OF D O C T O R OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES EARTH AND OCEAN SCIENCES (OCEANOGRAPHY)  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA March 1998 © Ana Carrasco, 1998  In presenting this thesis  in partial  fulfilment  of  the  requirements  for  an advanced  degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of department  or  by  his  or  her  representatives.  It  is  understood  that  copying  my or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department of The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  30~  MQTCV\ —  Abstract  Semidiurnal (M2) internal tides are studied in and near Dixon Entrance. Two complementary approaches are presented: a data analysis and a numerical study. The data consist of current records as well as hydrographic profiles. The derived baroclinic velocities represent a considerable portion of the total signal. The baroclinic velocities change little in time and variations in the vertical suggest the presence of a vertical mode. A nonlinear, frictionless, two-layer, finite-difference numerical model forced by a barotropic tidal wave was applied over an idealized topography representing Dixon Entrance. Specifically, Dixon Entrance was modelled as a coastal east-west oriented, shallow channel connected to a deep flat open ocean through a very steep continental slope. Several numerical experiments are presented. The main result the model offers is a possible explanation, in terms of waves, of the generation of internal tides. The passage of long barotropic Kelvin waves travelling north on the the open ocean triggers a baroclinic diffraction process which gives rise to cylindrical Poincare waves travelling towards the deep ocean, westward, Kelvin waves propagating along the coast, northward, and Kelvin waves propagating inside the channel, eastward. This wave pattern was described by Buchwald (1971). The Kelvin wave pattern always seems to be present inside the channel; however, internal waves are very sensitive to variations in the topography, and waves of short wavelengths are also generated. It was possible from the numerical experiments to explain some of the features found in the data analysis. The velocity magnitudes and main generation regions were comparable. The results of the model were unable to explain the detail of the quasi-steady pattern of baroclinic currents.  11  Table of Contents  Abstract  ii  List of Tables  vi  List of Figures  viii  Acknowledgement 1  Introduction  1  1.1  Background  3  1.1.1  Description of the region  3  1.1.2  Low frequency circulation  6  1.1.3  Seasonal changes  9  1.1.4  Tides and predicted Tides  1.2 2  11  Objectives  20  D a t a analysis  21  2.1  Data description  21  2.1.1  Moorings  23  2.1.2  CTD casts  29  2.2  Harmonic Analysis: currents 2.2.1  3  xx  33  Baroclinic  field  Long waves in a channel  38 49  in  3.1  3.2 4  Formulation  50  3.0.3  Energy Fluxes  55  Calculation of the vertical normal modes  56  3.1.1  Baroclinic fields  64  3.1.2  Generation Regions  71  Summary  77  Numerical Model  79  4.1  Description of the model  79  4.1.1  The code  79  4.1.2  Topography  83  4.1.3  Boundary conditions  83  4.1.4  Forcing  86  4.1.5  Baroclinic Signal  87  4.1.6  Internal Wave Energetics  87  4.1.7  The two layer approximation  89  4.2  4.3 5  3.0.2  Results  90  4.2.1  Base case  90  4.2.2  Run with more realistic topography  129  4.2.3  Comparison with data  135  Summary  143  Conclusions  146  Appendices  149  A  149 A.l  Harmonic Analysis  149 iv  A.2 Baroclinic field  153  A.3 Time series  156  A.3.1  165  Variance  v  List of Tables  2.1  Information about the moorings. The name of the mooring, depth of the instrument and duration in days of the records are listed for the first (A) and second ( 0 ) periods. The nominal depth is the depth assigned to those moorings that have big vertical displacements  3.1  For the first vertical mode, i.e.n  24  = 1 the following values are listed: the  phase velocity Ci, the Rossby radius Ri = Ci/f,  the longitudinal Kelvin  wavelength 2ir/K, the decay along x of the first trapped Poincare mode Dm+i — &i,m+i -1 , the longitudinal Poincare wavelength 27r/Aj1]m and the maximum free Poincare mode m possible 3.2  57  Values of: the amplitude of Kelvin wave reflected, \/3R\; the Poincare wave reflected, |-yx |; energy due to the Poincare wave, j ^ (Eq. 3.30); and magnitude of the velocity u at the end of the channel over the averaged u everywhere else, for the representative density profiles  64  4.1  Seasonal variations of stratification  90  4.2  Velocities range and mean of the upper value from the model results and from the data analysis. Velocity are in cms'1  VI  143  A.l  Results from the harmonic analysis performed for station D04 at 52m at the location 54° 26', 132° 0' over the period of 6:00 am 3 1 / 5/84 to 6:00 am 28/ 6/84. The name and frequency of the 32 constituents are listed in the first two columns. The tidal current ellipses parameters are listed in the following four column, these are: the semi major axis, Ma; semi minor axis, M;; the inclination, ip; and Greenwich phase, G. The sense of rotation of the current is determined by the sign of M;. When Mi < 0 it rotates clockwise  154  vn  L i s t of F i g u r e s  1.1  T h e N o r t h coast of British Columbia  1.2  Geography and depth contours (m) for the west coast of Canada and the southern coast of Alaska (from Foreman, Henry, Walters and Ballantyne 1993).  4  . .  1.3  M a p of Dixon Entrance, depths in meters  1.4  F r o m B o w m a n et al. (1992) sketch of t h e m e a n surface w a t e r circulation (solid arrow) a n d deep circulation ( d o t t e d arrow) in Dixon E n t r a n c e .  1.5  5 7  . .  8  From Jacques (1997); seasonal variations averaged over 32 years of: a) wind velocity magnitude (solid line) and direction (dashed line); b ) mean daily freshwater flow for the Nass (solid line) and Skeena (dashed line) rivers; and c) mean daily sea surface salinity (solid line) and temperature (dashed line) (SST) at Triple Island station  1.6  10  at, a); t e m p e r a t u r e , 6); a n d salinity c); profiles from spring (April 1984), u p p e r frames a n d for s u m m e r ( J u n e 1985) lower frames. T h i c k lines are from s t a t i o n s n e a r t h e t h e m o u t h (close t o t h e o p e n ocean) a n d t h i n lines are from stations n e a r t h e head. N o t e t h a t t h e scales are different. T a k e n from T h o m s o n , Crawford a n d H u g g e t t (1988)  1.7  12  Modelled barotropic tides: co-amplitudes (cm) (solid curves) and co-phases (degrees)(dashed curves) for the M2 constituent. From Foreman et al. (1993).  vni  14  1.8  Contours of the rms M2 baroclinic current on a near-surface velocity sigma level. The field is shaded at the 8, 16 and 24 cms-1  levels. Here the baro-  clinic current is defined as the difference between the total and the depthaveraged current. The intensity of the near-surface baroclinic component of the tidal currents gives an indication of the strength of the internal tides. Dashed contours are for the 200 and 1000 m isobaths. From Cummins and Oey (1997) 1.9  16  Surface current ellipses (computed from drifter tracks) representing the entire semi-diurnal signal. Empty ellipses denote clockwise rotation, while shaded ellipses denote counterclockwise rotation.  From Crawford et al.  1997  17  1.10 Surface current tidal ellipses (computed from drifter tracks) representing the entire semi-diurnal signal over detailed bathymetric contours of the northeast part of Dixon Entrance. The depth contours are at 1, 10, 30, 50, 70, 90, 120, 160, 200 fathoms. From Crawford et al. 1997 2.1  18  Locations of the moorings. The symbol + correspond to the first period (Apr-Oct 84) and [] to the second (Oct 84 -July 85). The moorings joined by solid lines were used for the analysis  2.2  Part of a time series of temperature in degrees (T), east-west (u) and north-south (v) velocities, in cm/s at the mooring D05 at 100 m depth. .  2.3  22  25  The Fast Fourier Transform of the a) east-west and b) north-south velocities for the instrument at 52 m from the mooring D04. They are normalized such that the sum of the magnitudes squared equals one  2.4  26  Same as previous figure but for the instrument at 152 m from the mooring D04  27  IX  2.5  Same as previous figure but for the instrument at 279 m from the mooring D04  28  2.6  Locations and labels of CTD stations from cruise 8411 (April 1984). Spring. 29  2.7  Locations and labels of CTD stations from cruise 8414 (October 1984). Autumn  30  2.8  Locations and labels of CTD stations from cruise 8510 (June 1985). Summer. 30  2.9  Density, as at in kg /m3, sections at similar locations but different times, (a) from cruise 8411 (spring), (b) 8414 (autumn) and (c) 8510 (summer). At the top of each figure the number of each station (St) as well as the value of the reduced gravity (gr) are indicated. The curve with diamonds indicates bottom depth  31  2.10 Same as previous figure but density sections here are located farther east. (a) from cruise 8411 (spring), (b) 8414 (autumn) and (c) 8510 (summer).  32  2.11 Period A. Time series of the amplitudes and phases of the east-west and north-south velocities at M2 for the mooring D04 estimated from overlapping 28 day records. The solid lines represent the amplitudes (Ut, Vt) in cm/s,  and the dashed lines the phases ( $ t , 6t) in degrees (GMT)  35  2.12 Period 0 . Time series of the amplitudes and phases of the east-west and north-south velocities at M2 for the mooring D05 estimated from overlapping 28 day records. The solid lines represent the amplitudes (Ut, Vt) in cm/s,  and the dashed lines the phases ( $ t , 9t) in degrees (GMT)  36  2.13 Period 0 . Time series of the amplitudes and phases of the east-west and north-south velocities at M2 for the mooring QF4 estimated from overlapping 28 day records. The solid lines represent the amplitudes (Ut, Vt), in cm/s,  and the dashed lines the phases ( $ t , 6t) in degrees (GMT)  37  2.14 Baroclinic time series of the east-west and north-south velocities for the mooring D04 for the first period. The solid line represents the amplitudes (Ub, Vb), in cm/s, and the dashed line the phases ($;,, 8b) in degrees. . . .  39  2.15 Same as previous but for the mooring D05 for the second period  40  2.16 Same as previous but for the mooring QF4 for the second period  41  2.17 The two upper frames present the averaged values of the east-west velocity Ub and phase (j>b, and the two lower frames, the averaged values of the north-south velocity Vb and phase 8b in a section along Dixon Entrance for the first period [A). The lowest line is the water depth. The dots mark the location of the instruments  44  2.18 Same as previous figure but for a north-south section  45  2.19 Same as Figure 2.17 but for the second period, (O). The two upper frames present the averaged values of the east-west velocity Ub and phase 4>b, and the two lower frames, the averaged values of the north-south velocity Vb and phase 8b in a section along Dixon Entrance for the second period  (0).  The lowest line is the water depth. The dots mark the location of the instruments  46  2.20 Same as previous figure but a north-south section 3.1  Superposition of flat channel geometry (width W  47 — 55 km and length  L — 150 km) upon Dixon Entrance. Contours represent bottom bathymetry in meters 3.2  51  Density profiles from cruise 8411. The location of the CTD stations can be seen in Figure 2.6  58  XI  3.3  Averaged at profile ([kg/m3]) AT2 profiles (rad/s),  from cruise 8411, (a) with corresponding  (b). The vertical displacement (j)n (thick line) and ^f-  (thin line) for the first, n = 1 vertical mode assuming a constant depth of 320 m are shown in (c) with arbitrary units 3.4  Density profiles from cruise 8414. The location of the CTD stations can be seen in 2.7  3.5  59  60  Averaged at profile ([kg/m3]) N2 profiles (rad/s),  from cruise 8414, (a) with corresponding  (b). The vertical displacement <j)n (thick line) and ^~  (thin line) for the first, n = 1 vertical mode assuming a constant depth of 343 m are shown in (c) with arbitrary units 3.6  Density profiles from cruise 8510. The location of the CTD stations can be seen in 2.8  3.7  61  62  Averaged at profile ([kg/m3]) N2 profiles (rad/s),  from cruise 8510, (a) with corresponding  (b). The vertical displacement 4>n (thick line) and -j^p-  (thin line) for the first, n = 1 vertical mode assuming a constant depth of 343 m are shown in (c) with arbitrary units 3.8  63  Amplitude and phase of a) the vertical displacements (£), b) longitudinal velocity (u) and c) cross channel velocity (v) for the first baroclinic mode at 50 m depth obtained from the averaged density profile shown in figure 3.5, cruise 8414. Note that the contour lines are not evenly spaced. Phases in degrees and amplitudes in arbitrary units  3.9  66  Amplitude and phase of a) the longitudinal velocity (u) and b) cross channel velocity (v) obtained from cruise 8414, along y = 27.5 km. The constant depth is 321 m.  Amplitude contour lines are not evenly spaced.  Phases in degrees and amplitudes in arbitrary units  xn  67  3.10 Line of points where amplitudes and phases have been calculated from the analytical model. The circles mark the approximate location of the moorings  68  3.11 Modelled amplitude and phase of a) the longitudinal velocity u and b) cross channel velocity v for cruise 8414 along the line of moorings. Phases in degrees and amplitudes in arbitrary units. Both the real bottom depth at each mooring as well as the constant depth (321 m) assumed are indicated. Dots mark the locations of instruments  69  3.12 Duplicate of Figure 2.17 from the data analysis. The two upper frames present the averaged values of the east-west velocity Ub and phase (f>b, and the two lower frames, the averaged values of the north-south velocity Vf, and phase Of, in a east-west section along Dixon Entrance for the first period (A). The lowest line is the water depth. The dots mark the location of the instruments  70  3.13 Scatter plots of the relative observed versus  modelled values, a) of the  amplitude of u, b) of the phase of u c) of the amplitude of v, b) of the phase off. The values of the correlation coefficient r and the variance var are indicated at each frame  72  3.14 From the CTD station 58, cruise 8414, a) the density as <rt} b) the averaged density, c) N 2 (z) obtained from the density and d) N 2 (z) obtained from the averaged density  74  3.15 Regions of possible generation of internal tides estimated from the cruise 8414  75  3.16 Regions of possible generation of internal tides estimated from the cruise 8510  76  xin  4.1  Sketch showing the definitions of the variables for each layer  81  4.2  A plan view showing the grid locations where the numerical code calculates the velocities u, v and the height field h, near the edge of the domain. . .  4.3  82  Bathymetric chart of Dixon Entrance System. Depth contours are in meters with intervals of 50 and 500 m for regions shallower and deeper than 500 m respectively. Four regions are shown: depths less than 100 m (-), between 100 and 300 m (blank), between 300 and 500 m ( + ) , and greater than 500 m (>)  4.4  84  A plan view of the topography used for the base case. The points indicate the locations at which time-series were stored. Three east-west sections are indicated by arrows at the eastern end; near the northern wall of the channel, in the centre and near the southern wall. The conditions for the boundaries are indicated  4.5  85  Contours of the interface height r/2 in meters for the step channel case at t — IT. The contour interval is 2.5 metres. Dashed lines represent negative values, solid lines zero or positive values  91  4.6  Same as previous but at t = AT. The contour interval is 2.5 metres.  ...  92  4.7  Same as previous but at t — QT . The contour interval is 4 metres  93  4.8  Same as previous but at t = 8r. The contour interval is 4 metres  94  4.9  Same as previous but at t = lOr. The contour interval is 4 metres  95  4.10 Contour of the baroclinic velocity Ubc in [ms - 1 ] for the step channel case at t = IT.  The contour interval is 0.025[m.s -1 ]. Dashed lines represent  negative values, solid lines zero or positive values  96  4.11 Same as previous but at t = AT. The contour interval is 0.025 [ m s - 1 ] . . .  97  4.12 Same as previous but at t — QT. The contour interval is 0.025 [ m s - 1 ] . . .  98  4.13 Same as previous but at t = 8r. The contour interval is 0.04 [ m s - 1 ] . . . .  99  xiv  4.14 Same as previous but at t = lOr. The contour interval is 0.04 [ m s - 1 ] . . .  100  4.15 Contour of the baroclinic velocity Vbc in [ms - 1 ] for the step channel case at at t = 2r. The contour interval is 0.02 [ m s - 1 ] . Dashed lines represent negative values, solid lines zero or positive values  101  4.16 Same as previous but at t = AT. The contour interval is 0.025 [ m s - 1 ] . . .  102  4.17 Same as previous but at t = 6r. The contour interval is 0.04 [ m s - 1 ] . . . .  103  4.18 Same as previous but at t = 8r. The contour interval is 0.04 [ m s - 1 ] . . . .  104  4.19 Same as previous but at t = lOr. The contour interval is 0.05 [7ns -1 ]. . .  105  4.20 Three East-West sections of interface height r}2, in meters, across the domain at t = lOr. Their location is indicated by arrows in Figure 4.4. Very short Poincare waves created when the first interaction with the channel occurred are still in the domain on their way out. The distance in the x direction is in meters  109  (  4.21 Same as previous figure but for Ubc- The distance along the x axis is in meters and the velocity along y is in [7ns -1 ]  110  4.22 Same as before but for Vbc- The distance along the x axis is in meters and the velocity along y is in [ms - 1 ]  Ill  4.23 Contours of the surface height r/i in meters for the step channel case at t = lOr. The contour range is from -1.25 to -0.05 m every 0.05 m. Dashed lines represent negative values, solid lines zero or positive values  114  4.24 Contours of the surface height 7/1 in meters for the step channel case at t — 10.25r. The contour range is from 1.65 to 2.75 m every 0.05 m. Dashed lines represent negative values, solid lines zero or positive values 4.25 Velocity vectors in m s - 1 of the upper layer Uj = (ui,vi)  at t = lOr. The  blank spaces represent very small velocity values 4.26 Velocity vectors in m s - 1 of the upper layer Uj = (ui,vi), xv  115  116 at t = 10.25r. .  117  4.27 View from above and three dimensional view of the topography with a bank. The bank is 13.8 by 18.4 km at its base and 18.4 by 4.6 km at the top. The top is 15 m from the sea surface  118  4.28 For spring conditions, contours of the interface rj2 in metres for the topography shown in Figure 4.27 at t = lOr. The contour interval is 4 m. . . .  119  4.29 Contour plot of the east-west velocity Ubc in [ma - 1 ] at t = lOr for the topography shown in Figure 4.27. The contour interval is 0.05 [ m s - 1 ] .  .  120  4.30 Contour plot of the north-south velocity Vbc in [ma - 1 ] at t — lOr for the topography shown in Figure 4.27. The contour interval is 0.05 [ m s - 1 ] .  .  121  4.31 Three east-west sections of the velocity of Vbc across the domain at t — lOr for the bank case. The location of the sections is the same as in Figure 4.22. The distance along the x axis is in meters and the velocity along y is in  [ms-1]  123  4.32 Baroclinic energy-flux vectors at M 2 , in [ I f m " 1 ] , near the mouth of the channel for the step channel case  124  4.33 Baroclinic energy-flux vectors at M 2 , in [Wm" 1 ], near the mouth of the channel for the plus bank case  125  4.34 Values of the baroclinic energy  126  4.35 Values of the barotropic energy  127  4.36 Values of the barotropic energy for the unstratified case  128  4.37 For summer conditions, contours of the interface rj2 in metres for the topography shown in Figure 4.27 at t = lOr. The contour interval is 2 m. .  130  4.38 Contour plot of the east-west velocity Ubc in [ms - 1 ] at t — lOr for the topography shown in Figure 4.27. The contour interval is 0.04 [ m s - 1 ] .  .  131  4.39 Contour plots of the north-south velocity vbc in [ms - 1 ] at t — lOr for the topography shown in Figure 4.27. The contour interval is 0.04 [ m s - 1 ] . xvi  .  132  4.40 View from above of the entire topography domain used for the more realistic case. The dots indicate the locations at which time-series were stored 133 4.41 The topography inside the channel. View from above and three dimensional perspective. The printed depths in the upper figure are in meters.  134  4.42 Velocity Ubc at time t = r, in the upper frame and at t = 8r in the lower frame. The contour interval is 0.04 m s - 1 and 0.05 m s " 1 respectively.  . .  136  4.43 Velocity vic at time t = r , in the upper frame and at t = 8r in the lower frame. The contour interval is 0.04 m a - 1 and 0.05 m s - 1 respectively.  . .  137  4.44 Spring modelled values of baroclinic velocities along an east-west vertical section. The location of these points is indicated in Figure 4.40. The thickness of the upper layer Hi is 100 m and g' = 0.0147. The two upper frames represent the amplitude and phase of the east-west baroclinic velocity and the two lower frames represent the north-south baroclinic velocity. The values were printed at the middle depth of each layer. The lines indicate the actual and the modelled water depth  139  4.45 Duplicate of Figure 2.17 from the data analysis. The two upper frames present the averaged values of the east-west velocity Ub and phase (f>b, and the two lower frames, the averaged values of the north-south velocity Vb and phase 0b in a east-west section along Dixon Entrance for the first period (A). The lowest line is the water depth. The dots mark the location of the instruments  140  xvii  4.46 Summer modelled values of barocUnic velocities along an east-west vertical section. The location of these points is indicated in Figure 4.40. The thickness of the upper layer Hi is 50 m and g' — 0.0235. The two upper frames represent the amplitude and phase of the east-west baroclinic velocity and the two lower frames represent the north-south baroclinic velocity. The values were printed at the middle depth of each layer. The lines indicate the actual and the modelled water depth  141  4.47 Duplicate of Figure 2.19 from the data analysis. The two upper frames present the averaged values of the east-west velocity Ub and phase fa, and the two lower frames, the averaged values of the north-south velocity Vb and phase 6b in a east-west section along Dixon Entrance for the second period (0). The lowest line is the water depth. The dots mark the location of the instruments A.l  Tidal ellipse parameters: Ma,Mi  142 denote the major and minor semi-axes;  tp is the angle of inclination and g is the phase lag from t = t0 to when the ellipse gets its maximum amplitude, g is related to the phase lag of the maximum current behind the maximum tidal potential by G = V(t0) + g.  152  A.2 The procedure used to get the new time series of amplitudes and phases at M2 frequency  155  A.3 Time series of the east-west velocity Ub (cm/s) for the first period. The lowest line is the water depth. The name of the moorings is indicated in the first frame and the values are printed at the location of the instruments. 157 A.4 Continuation of previous figure  158  A.5 Time series of the phase east-west fa (degrees)  159  A.6 Continuation of previous figure  160  XVlll  A.7 Time series of the north-south velocity Vj, (cm/s)  161  A.8 Continuation of previous  162  figure  A.9 Time series of the north-south phase 8b (degrees)  163  A.10 Continuation of previous  164  figure  A. 11 Spatial pattern of the first two modes ei,e2.  167  A. 12 The time series of the amplitude and phase of the first principal components ai(i), d2(t) . The continuous lines are associated with lif, and $(, and the dot-lines are associated with Vb and 8b  168  A.13 Same as Figure A.11 but for the second period  169  A. 14 Same as Figure A. 12, but for the second period  170  xix  Acknowledgement  First and foremost I thank my supervisor, Paul Leblond, for the support and incentive he gave me during this endeavour. I gratefully acknowledge the facilities he offered to travel to conferences. I am also thankful to the members of my committee. To William Crawford for suggesting the topic of my thesis and to Mike Foreman for the recommendations and comments. I thank Susan Allen for being on my side in the battle in favour of using the numerical model and for making extra time to help when difficulties arose. I owe a special debt of gratitude to Steve Pond who technically, but not officially, was a member of my committee. His clear sighted approach enabled me to go further with this research than originally envisioned. His guidance and support were crucial to the success of this work. I would like to thank Denis Laplante, Chris Mewis and Carol Leven for their technical help, especially Denis who always had the right answers. I gratefully acknowledge the financial support I received from CONACYT, Paul Leblond and Steve Pond during my graduate studies. Thanks go to Steve, Ian, Diana, Adam, Greg, and Loretta for their friendship and for sharing with me this adventure of being in Vancouver.  Warmth and special thanks  go to my soul mate Gabriela, Alida and Pal for many enjoyable moments and their unmeasurable support and encouragement. I thank Barbara Bourget and Jay Hirabayashi of KOKORO DANCE for existing. Finally I thank my family for their enthusiasm and for their unconditional support and love.  xx  Chapter 1  Introduction  The tides measured in the ocean contain both a barotropic and a baroclinic component. The barotropic component, always present, originates from the gravitational forces of the sun and the moon acting on the earth and affects the water column equally from the surface to the bottom. As far as this component is concerned the density changes in the water column are negligible. The baroclinic component is a secondary effect that can become important with stratification. Dixon Entrance is an east-west oriented coastal oceanic region on the west coast of British Columbia, Canada with dominant semi-diurnal  M 2 surface tides.  Local  bathymetry and stratification are favourable to the generation and propagation of baroclinic (internal) tides. Internal tides have been recognized as internal waves excited at or near tidal periods. An explanation of their existence is that energy from barotropic tides is scattered to internal tides by bottom roughness (Hendershott, 1981). Wunsch (1975) described internal tides as a more majestic and impressive phenomenon than surface tides, since sub-surfaces of constant density can be deformed into waves 30 m high or more, marching in long slow lines through the ocean. These waves play an important role in vertical mixing of nutrients, sediments, and other water mass properties and materials. This thesis seeks to give an explanation of the generation and scattering of internal tides at the semi-diurnal frequency M 2 , (equivalent to a period of 12.42 hrs) inside Dixon Entrance. Two complementary approaches are presented; a data analysis (chapters 2 and  1  Chapter 1.  Introduction  2  3) and a numerical study (chapter 4). The data used in this study were collected as part of the North Coastal Ocean Dynamics Experiment (NCODE) during the period 1984-1985. They consist of two sets of current records, each approximately six months long, from several moorings, as well as CTD casts taken during mooring deployment and recovery. The data analysis focuses on extracting the baroclinic field at M 2 frequency and delineating the possible generation regions. Although there are seasonal variations, the main results are that the baroclinic motions, estimated at the locations of the instruments, change little in time and that variations in the vertical suggest the presence of a vertical mode. This quasi-steady pattern appears to be associated with the propagation of a weak Poincare wave and two dominant Kelvin waves travelling in opposite directions along Dixon Entrance. An analytical model, (following Taylor 1921) was used to estimate the reflection of internal waves in a closed channel (chapter 3). The potential generation regions are over the continental shelf, and near the sill located at the mouth. To better study the generation of such waves a non-linear, frictionless, layered, finite difference numerical model forced by a barotropic tidal wave was applied over an idealized topography representing Dixon Entrance. Specifically Dixon Entrance was modelled as a coastal east-west oriented, shallow channel connected to a very deep and steep continental shelf in a two-layer system. The generation and scattering of internal tides can be explained by the passage of long barotropic Kelvin waves travelling north on the continental shelf perpendicular to the mouth of the shallow-channel-like region. Internal waves are generated at the shelf break, and propagate towards both the open ocean and the channel entrance. Inside the channel, it was found that due to the difference in water depth and strong stratification, such barotropic Kelvin waves travelling north constantly generate, at the entrance of the channel, short baroclinic Kelvin waves that travel east, trapped to the  Chapter 1.  Introduction  3  southern wall. These waves get to the end of the channel, turn and continue travelling west trapped to the northern wall, as Taylor (1921) described. When they get to the end of the channel they turn the corner and travel northwards trapped to the shelf coast (Buchwald; 1968). A related phenomenon happens outside the channel. A baroclinic diffraction process takes place and a system consisting of Kelvin waves propagating along the coast (towards north) together with long cylindrical Poincare waves radiating into the deep ocean are generated, as Buchwald (1971) described. Since the system is constantly fed by long barotropic Kelvin waves, there are baroclinic waves constantly radiating from the channel, making the channel itself an internal tide generation region for the open ocean. This pattern is intensified during the summer when the stratification reaches its maximum.  1.1  Background  1.1.1  D e s c r i p t i o n of t h e region  Dixon Entrance, Hecate Strait and Queen Charlotte Sound are three connected bodies of water that separate the Queen Charlotte Islands from the mainland on the west coast of Canada (see Figure 1.1). A notable feature of the region is the highly irregular mainland coastline which contains numerous fjords and inlets. The continental shelf is very narrow in the entire region as can be seen in Figure 1.2. The 1000 m depth contour lies within 25 km of most of the coastline. To the north, along Graham Island, the shelf widens and reaches a maximum width of roughly 30 km at 54° N latitude. There, it merges with the broader continental shelf west of Dixon Entrance and the continental slope becomes less steep. Further north, the shelf widens to approximately 100 km.  Chapter 1.  Introduction  4  56°N \ Cordova f t y j  *§ S J ?)f*SC>ian&iL-  f  . , "S Dundaai.' x Cap*Chacon *• 'Y? Dixon 9  Entranca Mdnfrri  ^,  54 8  Nast  , JS^Ch&lhamSound  « -  *®  Graham I.  BRITISH COLUMBIA Moresby I Houston Stewart Channel  52° Pacific Ocean  Kimtfiitl. Cape**. Karouardlt. St James Queen Charlotte Sound  100km  50°  136°W  134°  132°  130*  128°  Figure 1.1: The North coast of British Columbia.  126°  Chapter 1.  Introduction  5  56*N  F i g u r e 1.2: Geography a n d depth contours (m) for the west coast of C a n a d a and the southern coast of Alaska (from Foreman, Henry, Walters and Ballantyne 1993).  Chapter 1.  Introduction  6  Dixon Entrance is an east-west oriented depression bounded over much of its length by two mountainous Alaskan Islands (ranging from 300-1500 m) to the north and by Graham Island to the south (Figures 1.1 and 1.3).  It is indeed a gap between two  mountain barriers. The channel is approximately 150 km long and 55-65 km wide. At the seaward end of Dixon Entrance the depression is divided into two channels (350 400 m in depth) by Learmonth Bank, which rises to within some 35 m of the surface. Learmonth Bank is about 19 km long and about 10 km wide. From the mouth, the bottom slopes gradually upward towards the east.  Clarence  Strait is a long, narrow, trench and about 400 m deep over the greater part of its length located in the north-eastern part, see Figure  1.3. A shallow bank (20 to 40 m deep)  labelled Rose Spit Sill, separates Dixon Entrance from Hecate Strait. Detailed descriptions of the physical oceanography of the region have been provided by several studies, e.g. Crean (1967), Dodimead (1980), Thomson (1981) and Crawford and Thomson (1991).  1.1.2  Low frequency circulation.  One of the most important features of the surface circulation (confined to the upper 50-100 m) is the presence in summer and winter of a cyclonic (counterclockwise) eddy labelled the Rose Spit Eddy (Figure 1.4) which fills the entire eastern end of Dixon Entrance and has a period of 5-40 days per cycle, (Bowman et al; 1992).  Ballantyne  et al. (1996) and Jacques (1997) numerically simulated the complete formation of the Rose Spit Eddy when buoyancy-driven flows where calculated. They suggested that such eddy arises from a) nonlinear interaction of tides with bathymetry and b) from nonlinear interaction of tides in combination with density gradients with bathymetry. In addition a clockwise circulation around Learmonth Bank in the western part has been observed. This second eddy is not always present, but when it is, it appears to combine  Chapter 1.  Introduction  7  130  Figure 1.3: Map of Dixon Entrance, depths in meters.  129  Chapter 1.  Introduction  8  -S3*»'  Figure 1.4: From Bowman et al. (1992) sketch of the mean surface water circulation (solid arrow) and deep circulation (dotted arrow) in Dixon Entrance.  Chapter 1.  Introduction  9  with the western side of the Rose Spit Eddy to carry surface water southward. Bowman, Visser and Crawford (1992) gave an explanation of the dynamics of these eddies based on tidal rectification over topographic features.  They also gave a sketch, Figure 1.4,  of the low-frequency upper (solid arrow) and deep currents (dotted arrow), interpreted from various measurements, model predictions and theory. Jacques (1997) refined these calculations.  1.1.3  Seasonal changes  Due to the mountainous terrain that surrounds Dixon Entrance the prevailing winds are constrained to blow parallel to the coast, i.e. east-west. The main input of fresh water is provided by two main rivers, the Skeena and the Nass (see locations in Figure 1.1). The influence of freshwater discharge is felt more in Dixon Entrance than in Hecate Strait. The main seasonal conditions to which Dixon Entrance is exposed can been extrapolated from data at one station, see Figure 1.5. Figure 1.5 a from Jacques (1997) shows the mean daily wind speed (solid line) and direction (dashed line) at Triple Island station. The wind direction is defined as the direction from which the wind blows and angles are measured clockwise from true North. The local winds are then mainly south-to-southeasterly in winter and north-to-northwesterly in summer. Figure 1.5 b is the mean daily discharge of the the Nass (solid line) and Skeena (dashed line) rivers. Finally Figure 1.5 c shows the mean daily sea surface salinity (solid line) and temperature (dashed line) at Triple Island station. The SST is affected by the seasonal cycle and runoff temperatures. In contrast to the surface waters, deep waters are saltier in summer than in winter due to upwelling on the continental shelf. The more important aspects of seasonal behaviour affecting Dixon Entrance are summarised as follows: Spring (April and May), is characterised by lower wind speed and changes in direction. The freshwater discharge into the region increases rapidly. Heat transfer at the air-sea  Chapter 1.  Introduction  10  b)  Jan. c) 32  CO CO CO  30  o_  28 Jan.  J-  Mar.  May  July  Sept  Nov.  Figure 1.5: From Jacques (1997), seasonal variations averaged over 32 years of: a) wind velocity magnitude (solid line) and direction (dashed line); b) mean daily freshwater flow for the Nass (solid line) and Skeena (dashed line) rivers; and c) mean daily sea surface salinity (SST) (solid line) and temperature (dashed line) at Triple Island station.  Chapter 1.  Introduction  11  interface is small. In summer (June through August), there is an intrusion of cold oceanic water at depth due to flushing of fresh water at the surface. The discharge of fresh water reaches a peak in June, due mainly to snow melting, then declines throughout July and August. Because surface heating, weaker summer winds and freshwater runoff stratify the water column, the upper layer is thinnest. The higher SST regions are associated with areas of low-salinity. In autumn (September and October), an abrupt increase in mean wind speeds is typical. A secondary maximum in fresh water discharge occurs associated with heavy precipitation and the heat transfer at the surface is small. In winter (November through March), the south-to-southeasterly winds reach their maximum, and advect relatively warm water north through Hecate Strait. The freshwater discharge is small and there is a net heat loss at the sea surface. Representative density, temperature and salinity profiles for spring and summer are shown in Figure 1.6. Note that throughout the year, salinity, rather than temperature plays the dominant role in determining the density in Dixon Entrance (Crean; 1967). In spring, when stratification is weak, the increase of freshwater discharge is felt down to approximately 80 m depth. Stratification plays an important role in determining the nature of the internal tide.  1.1.4  T i d e s and predicted Tides  Tides in these connected bodies of water are mixed, predominantly semi-diurnal, and co-oscillate with tides in the adjoining North Pacific Ocean. The sea level amplitudes of tides at the diurnal K^ and semidiurnal M2 frequency increase from South to North on the continental shelf. They travel north as large-scale Kelvin waves in the North Pacific (Platzman 1979). In Dixon Entrance and Hecate Strait there is a large increase in the  Chapter 1.  Introduction  25.5 26.5 sigmat(Kg/mA3)  23  24 25 26 sigmat(Kg/mA3)  12  5.0  6.0 7.0 temperature degrees  6 8 10 temperature degrees  8.0  32.5 33.5 salinity PSS  31 32 33 salinity PSS  Figure 1.6: at, a); temperature, b); and salinity c); profiles from spring (April 1984), upper frames and for summer (June 1985) lower frames. Thick lines are from stations near the the mouth (close to the open ocean) and thin lines are from stations near the head. Note that the scales are different. Taken from Thomson, Crawford and Huggett (1988).  Chapter 1.  Introduction  13  semi-diurnal M2 constituent, so that tides here are more semi-diurnal in character than on the open coast (Petro-Canada; 1983). As these long tidal waves travel up the Pacific coast, they enter Hecate Strait from the South and Dixon Entrance from the West, to meet at the northern end of Hecate Strait. Shoaling effects (Hecate Strait is very shallow) cause the mean range of sea level to increase from 4 m across the mouth of Queen Charlotte Sound to 6 m midway along Hecate Strait, and from 5 m at the mouth of Dixon Entrance to 7 m at the mouth of Chatham Sound, near the junction of Dixon Entrance and Hecate Strait (Canadian Hydrographic Service ). Surface tides in the northern waters of British Columbia have been modelled by several authors such as Foreman et al. (1993), Ballantyne et al. (1996) and Cummins and Oey (1997). Foreman et al. (1993) employed a finite element technique, over the entire north coast of British Columbia, which gave a very good fit of the irregular coastline and bathymetry. The resolution of the triangled grid goes from 35 km to 250 m using in the vertical 11 sigma-coordinate layers. Figure 1.7 shows the output tidal sea level due to the M2 constituent, from Foreman et al. (1993). It can be seen from the figure that the amplitude of the semi-diurnal constituent, M2, increases eastward in Dixon Entrance and northward in Hecate Strait. The diurnal constituent, Ki (not shown here), also increases in amplitude in the same region but to a lesser degree than M2. This model successfully predicted the sea level amplitudes and phases, in the entire domain, of the eight largest tidal constituents and a mean tidally rectified current. It was not successful in reproducing the total tidal currents in Dixon Entrance. For the semi-diurnal constituent M 2 discrepancies were found that may be attributable to the presence of baroclinic tides. However in regions where baroclinicity is not important, as in the well mixed Hecate Strait, the total tidal currents predicted agreed with observations. Tidal currents are complicated as a result of vertical density gradients, nonlinear  Chapter 1.  Introduction  14  Figure 1.7: Modelled barotropic tides: co-amplitudes (cm) (solid curves) and co-phases (degrees)(dashed curves) for the M2 constituent. From Foreman, 1993.  Chapter 1,  Introduction  15  effects, coastal geometry and interaction of the flow with bottom topography. Therefore it is more difficult to simulate them. It has been suggested that in Dixon Entrance, measured tidal currents are stronger than predicted when baroclinic dynamics have been ignored (Crawford and Thomson, 1991 and Foreman et al, 1993). An improved version of the Foreman et al. (1993) model was presented by Ballantyne et al. (1996). The tidal elevations and velocities obtained were as accurate as those from the previous model. While some baroclinic effects were modelled by Ballantyne et al. (1996), internal tides were not allowed to exist because vertical displacements could not occur. Cummins and Oey (1997) used a prognostic three-dimensional model (Princeton Ocean Model), with a horizontal grid of 5 km.  In the vertical 21 sigma levels were  used. They modelled the surface tides with and without stratification. Inclusion of vertical stratification allowed an internal tide to be generated, affecting the velocities in the water column, and improving the prediction of the semi-diurnal M2 currents in several locations. Cummins and Oey (1997) modelled internal tides in fully three dimensions. They calculated the energy flux in order to identify regions of internal tide generation and propagation. Among many other results, they showed that one of the regions influenced by internal tides is the western end of Dixon Entrance and offshore region emanating from it (see Figure 1.8). Recently Crawford, Cherniawsky, Cummins and Foreman (1997) presented a detailed description of semi-diurnal currents estimated from near surface drifter observations in waters around the Queen Charlotte Islands. Their results are valuable since the data coverage was very dense. The tidal current ellipses, which represent the path followed by the horizontal velocity at semi-diurnal frequency, are shown in Figure 1.9 and Figure 1.10. The drifters were centred at 3 m depth in Dixon Entrance and at 10 m depth everywhere else. The data where collected during the summers of several years. The data in Dixon  Chapter 1.  Introduction  600 -  •Si 400  ZOO -  0  too  300 200 X (km)  400  Figure 1.8: Contours of the rms M2 baroclinic current on a near-surface velocity sigma level. The field is shaded at the 8, 16 and 24 cms'1 levels. Here the baroclinic current is defined as the difference between the total and the depth-averaged current. The intensity of the near-surface baroclinic component of the tidal currents gives an indication of the strength of the internal tides. Dashed contours are for the 1000 and 200 m isobaths. From Cummins and Oey 1997.  Chapter 1.  Introduction  17  55N  52  \ 100 cm s" Major Axis: Min = 3.6, Max = 106.3 j Mean = 35.6, StDev = 18.3 cm s-' 51 134W  133  132  131  130  129  Figure 1.9: Surface current ellipses (estimated from drifter tracks) representing the entire semi-diurnal signal. Empty ellipses denote clockwise rotary, while shaded ellipses denote counterclockwise rotary. From Crawford et al. 1997.  Chapter 1.  Introduction  200  observed 6? \ * f » 54.7N  54.6  54.5  54.4  Major Axis: Min = 21.1, Max = 76.9, Mean = 43.5, StDev = 17.0 cm s' 1  X  Figure 1.10: Surface current tidal ellipses (computed from drifter tracks) representing the entire semi-diurnal signal over detailed bathymetric contours of the northeast part of Dixon Entrance. The depth contours are at 1, 10, 30, 50, 70, 90, 120, 160, 200 fathoms. From Crawford et al. 1997.  Chapter 1.  Introduction  Entrance are from the summer of 1991. Crawford et al.  19  (1997) gave an exhaustive  comparison of this velocity field with two different models: Foreman et al. (1993) and Cummins et al. (1997). In the region of Dixon Entrance the authors suggested that the strong surface currents all through this region are a manifestation of strong semidiurnal internal tides. Although the model that includes internal tides gave better agreement with the data, this current pattern has been underestimated by the two models. They argued that this discrepancy might be due to the lack of horizontal density variation in the models as well as low spatial resolution in the horizontal. Apparently, topographic features neglected in a grid of 5 by 5 km can be relevant to the dispersion of internal tides. They mentioned two regions. A reef structure in the northeast part of Dixon Entrance (Figure 1.10) as well as Learmonth Bank at the western end (Figure 1.3).  The reef  presents a very steep topography that rises from 400 m to the surface in less than 5 km and Learmonth Bank which has a minimum depth of about 40 m in several 1 km wide domes. Abrupt changes in bottom slopes in conjunction with big density gradients are the required conditions for the generation of internal waves. Crawford et al. (1997) suggested that the surface semi-diurnal currents found near the reef structure, Figure 1.10, represent strong internal tides radiating away from these topographic features. Although a big baroclinic signal has been found by models and data analysis, it has not been clear, in all these works, how such internal waves propagate or arise inside Dixon Entrance. In this thesis the propagation behaviour of internal tides inside Dixon Entrance will be studied using a very simple numerical model as well as observational data.  Chapter 1.  1.2  Introduction  20  Objectives  Semidiurnal internal tides in and near Dixon Entrance are studied using two complementary approaches: a data analysis and a numerical study. The objective of the data analysis is to detect internal tides and study their temporal and spatial variability in Dixon Entrance using current meters observations and CTD data from the North Coast Oceanic Dynamics Experiment (NCODE) during the period 1984-85. The methodology for this includes the extraction of the baroclinic field from the total signal and the estimate of the possible tidal modes from vertical stratification. The objective of the numerical experiments is to find an explanation of how internal tides originate and propagate in Dixon Entrance. The model is nonlinear, frictionless, two-layer, finite-difference and is forced by a barotropic tidal wave over an idealized topography. The idea is to explore how much can be explained from the results of a reasonably simple model.  Chapter 2  D a t a analysis  The goal of this section is to determine the variations in time and space of the baroclinic tidal currents in Dixon Entrance using velocity records from several moorings. Various methods of isolating the internal tide have been implemented in the literature. In this thesis the extraction of the baroclinic velocity field from the raw time series was done by subtracting modelled barotropic currents from observed currents at M2 frequency, as Marsden (1986) and Drakopoulos et al. (1993) suggested. The barotropic signal used came from a finite element model developed by Foreman et al. (1993) for the north coast of British Columbia. Its triangular grid reproduced very well the irregular coastline and bathymetry. Therefore high accuracy was reached in regions where the velocity field changed rapidly.  2.1  D a t a description  The data used in this study consist of two sets of current-meter records from several moorings, the locations of which are shown in Figure 2.1. CTD data taken at the deployment and retrieval of the moorings are also used. The data were taken during the North Coast Oceanic Dynamics, NCODE, experiment from April 1984 to June 1985. (Huggett, Thomson and Woodward, 1992 ).  21  22  Chapter 2. Data analysis  (M  in  o  -134  -133  -132  -131  Degrees Longitude  Figure 2.1: Locations of the moorings. The symbol + correspond to the first period (Apr-Oct 84) and [] to the second (Oct 84 -July 85). The moorings joined by solid lines were used for the analysis.  Chapter 2. Data analysis  2.1.1  23  Moorings  There are two periods to be considered for the time series: the first one, which we called (A), is about 185 days from April to October 1984, and the second, ( 0 ) , about 195 days from October 1984 to July 1985. For the first period we have 18 moorings, with a total of 34 instruments, and for the second 16 moorings with 25 instruments. Current meters used were Aanderaa RCM4/RCM5 and Dumas Neyrpic CMDR. There were, as well, temperature and conductivity sensors in each instrument.  Unfortunately the  conductivity sensor failed in most of the cases and temperature itself was not sufficient to determine the changes in density. The positions of the moorings are marked in Figure 2.1 with ' + ' for the first (A) and [] for the second ( 0 ) period respectively. With the exception of 6 moorings, most of the time there were only two instruments per mooring: see Table 2.1. This low resolution in the vertical is not ideal for the detection of internal tides; however the length of the records was useful in following their variability in time. The time series of all the instruments are shown in the data report (Huggett, Thomson and Woodward, 1992). Typical observations are shown in Figure 2.2. The variations of the temperature can be associated with vertical semidiurnal displacements giving valuable information on the internal tides. A semidiurnal signal appears as the dominant frequency in the velocity records. To illustrate the main features of the time series in the frequency domain, the F F T of the east-west, |f/(o»)|, and north-south, |V(u;)|, velocity for the mooring D04 at 52 m, 152 m and 279 m respectively are displayed in Figures 2.3, 2.4 and 2.5. instruments revealed a prominent peak at frequency M 2  The three  (0.080511 cycles/hour). The  signal at frequency Ki (0.0417 cycles/hour), is smaller and seems more energetic in deeper waters. It was found that between 45 and 70 % of the total variance of the horizontal  24  Chapter 2. Data analysis  Mooring  D01 D02 D03 D04 D04 D05 D06 D09  Depth (m) 18, 48, 98  146 150 52, 152, 279 46, 278 50, 150 50, 100, 150 53, 103 50, 100  152  Period (A) (days) Period (O) (days) 129, 104, 125  135 155 185 250 185 198,240, 240  184 250 188  150,284  D10  50  D12 D13 D14  100, 150 150, 356 45, 145 59, 109  218, 241  184 119, 183  176 184 181 202  105 D15 D22 D23 D24 QF1  22, 52 20, 50 150, 250  250, 104  173 176 206  100  48, 148 50, 300, 357 (nominal) 65,315, 357 50, 150, 350 (nominal) 65, 165, 350 QF2 55, 155, 305,405 (nominal) 60, 160, 305, 405  QF3 QF4  188, 104  177 177 256 256 177 177  405 150  258 266  60,160,310,760  133,255,229,165  Table 2.1: Information about the moorings. The name of the mooring, depth of the instrument and duration in days of the records are listed for the first (A) and second (O) periods. The nominal depth is the depth assigned to those moorings that have big vertical displacements.  Chapter 2. Data analysis  25 '  Figure 2.2: Part of a time series of temperature in degrees (T), east-west (u) and north-south (v) velocities, in cm/s at the mooring D05 at 100 m depth.  Chapter 2. Data  26  analysis  Station D04 at 52m (A)  a)  K,  M2  CO  d  2 6  JtU—  o d  0.0  0.1  0.2  0.3  0.4  0.5  0.4  0.5  Frequency (cycles/hr)  b)  K,  Ma  CO  d  o d  0.0  0.1  0.2  0.3  Frequency (cycles/hr)  Figure 2.3: The Fast Fourier Transform of the a) east-west and b) north-south velocities for the instrument at 52 m from the mooring D04. They are normahzed such that the sum of the magnitudes squared equals one.  27  Chapter 2. Data analysis  Station D04 at 152m (A)  a)  K,  Mz  03  d  o d 0.0  0.1  0.2  0.3  0.4  0.5  0.4  0.5  Frequency (cycles/hr)  b)  K,  M  CO  d  o d 0.0  0.1  0.2  0.3  Frequency (cycles/hr)  Figure 2.4: Same as previous figure but for the instrument at 152 m from the mooring D04.  Chapter 2. Data analysis  28  Station D04 at 279m (A)  a)  K,  M  0.0  0.1  0.2  0.3  0.4  0.5  Frequency (cycles/hr)  b)  .  I *2  •  o 0.4  IV(w)l  CO  K,  q  I* 0.0  L _ J ^——~-n.  ^ . . . . . W  0.1  i.,.  rt,  0.2  ^ . . ^  ,.„•  0.3  • • „ . , •  i, •  0.4  • • , _ ,- • ,..  .  ...  -  0.5  Frequency (cycles/hr)  Figure 2.5: Same as previous figure but for the instrument at 279 m from the mooring D04.  Chapter 2. Data analysis  29  \  5656 57 56 596091 66 6634' 50  86  46  81  4«  86*  84  3867  86  40  -132 Degrees Longitude  Figure 2.6: Locations and labels of CTD stations from cruise 8411 (April 1984). Spring. velocity signal was at M2, at all moorings. Knowing that the signal at M 2 is prominent at all depths I shall try to extract from it the baroclinic part.  2.1.2  C T D casts  Data are available from three hydrographic cruises. The first one is labelled 8411 (spring) in April 84, the second 8414 (autumn) in October 84, and the third 8510 (summer) in June 85. The locations and numbers of the CTD stations from these cruises are shown in Figures (2.6), (2.7) and (2.8). Information about the CTD casts is given by Thomson, Crawford and Huggett (1988). In order to see the spatial density variation, sections of crt across Dixon Entrance for the three cruises are presented in Figures (2.9) and (2.10). Figure (2.9) shows south-north density sections at similar locations but different times. The number of each CTD station, St, is at the top of each figure. The locations are shown in Figures (2.6), (2.7) and (2.8). The next row labelled gr is the reduced gravity estimated  Chapter 2. Data analysis  30  7*17*7674 76767-7 76 7630 S6 *6  1S1 182  #  66  Se  9l1«S ^  16 -J  t  86^, 84  P  ae  g 3  6*1  -132 Degrees Longitude  Figure 2.7: Locations and labels of CTD stations from cruise 8414 (October 1984). Autumn.  -132 Degrees Longitude  Figure 2.8: Locations and labels of CTD stations from cruise 8510 (June 1985). Summer.  Chapter 2. Data analysis  31  diatanc* km  b) Or  105 O.OQ59  distance k m  c)  distance k m  Figure 2.9: Density, as <rt in kg /m3, sections at similar locations but different times, (a) from cruise 8411 (spring), (b) 8414 (autumn) and (c) 8510 (summer). At the top of each figure the number of each station (St) as well as the value of the reduced gravity (gr) are indicated. The curve with diamonds indicates bottom depth.  Chapter 2. Data analysis  a)  St pr  73 O.OQ13  32  60 0.OO82  88 O.Q123  87 O.OI 5 5  88 O.Q1 7  8S 0.0219  84 O.Q22  SS 0.0108  O.Q3  67 Q-Q31  74 O.Q37  __—**£> 8 :  §:  9: distance k m  distance k m  c)  S :  1. a  distance k m  Figure 2.10: Same as previous figure but density sections here are located farther east, (a) from cruise 8411 (spring), (b) 8414 (autumn) and (c) 8510 (summer).  Chapter 2. Data analysis  for each station according to: gr = —(p(surface)  33  — p(bottom)).  Here p0 is the density-  average and g is the acceleration due to gravity. The value of gr, in units m/s2,  gives  an estimate of the total stratification in the water column. Figure 2.10 presents density sections located farther east than those in figure 2.9. From these figures it is easy to see that the density often has its major variations in the upper 100 meters. As expected, stratification increases in the summer due to the weak winds and massive runoff event described in the previous chapter.  2.2  H a r m o n i c Analysis: currents  First it was necessary to get the total signal at M2.  To obtain it, a standard har-  monic analysis by blocks, according to Foreman (1977), was performed upon the velocity records. All the records were sub-sampled hourly, since the instruments recorded data every 30 minutes. The length of the blocks was chosen as 28 days, because this is the shortest period in which the separation of the M 2 constituent from its neighbours, S2 and N2, is ensured (see Appendix A). The amplitudes of S 2 and N2 ranged from 0.1 to 0.35 times M 2 . The constituent K% was inferred from £2 and a total of 32 constituents where included in the analysis. A list of the constituents used is presented in Appendix A. Is important to mention that the amplitudes and phases of each tidal constituent are assumed to be constant over the period used for the harmonic analysis. A half block overlapping was selected and a new time series of amplitudes and phases every 14 days at M 2 was thus obtained. The amplitude and phase (or constituents) of the M 2 east-west velocity were named Ut, $< and the north-south Vt and 9t. The subscript t stands for total velocity Moorings QF1 and QF2 had large vertical displacements.  signal.  Analysing the pressure  time series of these moorings, a new nominal depth, see Table 2.1, was assigned to some  Chapter 2. Data analysis  34  of the instruments. This new depth was the depth at which the instrument was most of the time and data that departed by more than 10 meters from such depth were removed. The least squares technique used by the harmonic analysis accepts gaps in the records, Appendix A. Three examples of time series of amplitudes and phases are presented in Figures 2.11, 2.12 and  2.13 for moorings D04, D05 and QF4 respectively.  The moorings D04 and  D05 are inside Dixon Entrance, the first in the central region and the second close to the coast. QF4 is on the continental slope. Except for the phase 9t of the instrument at 100 m from the mooring D05, the time series of all phases, inside and outside Dixon Entrance, had small variations in time around a mean value. The magnitudes were also smooth, but had more variation in time. For the moorings inside Dixon Entrance the total current along the channel (Ut) was stronger than that across it (Vt) at all depths. This result is consistent with what has been found by others. Huggett et al. (1992) reported that in Dixon Entrance tidal current ellipses at M2 are oriented with the major axis almost parallel to the isobaths, generally in an east-west direction. The external Rossby radius of deformation, RQ =  y/gH/f,  (457 km) is much bigger than the width of Dixon Entrance (55-60 km), indicating that the main flow is geometrically constrained and currents along the channel should be more dominant than across since rotation effects are barely felt. If the flow at M2 was purely barotropic the magnitudes and phases would all have the same value in the vertical. Differences in the vertical can be associated to baroclinicity and friction near the bottom. In the mooring D04 both velocities decrease with depth. In D05, Ut showed higher values at 100 m than at 50 m. On the continental slope the dynamics are different and for the mooring outside Dixon Entrance, QF4, the total east-west (Ut) and north-south (Vt) currents show similar values at each depth, although at the upper instrument Ut is slightly bigger than Vt. Although  Chapter 2. Data analysis  35  D04  (A)  n o r t h - s o u t h  360  Figure 2.11: Period A. Time series of the ampUtudes and phases of the east-west and north-south velocities at M2 for the mooring D04 estimated from overlapping 28 day records. The solid lines represent the ampHtudes (Ut, Vt) in cm/s, and the dashed lines the phases ( $ t , 6t) in degrees (GMT).  Chapter 2. Data analysis  -  5 0  \  40  ^  30 • 20  36  D05  east-west b0m  F  (0)  north—south  E-  I 10 F  -*-*~x _1  1  1  x x x I  I  I  L_  1 00n  50  360  1 5C  40  270  - ^ e ^ ^ ^ ^ ^ l  30  1 80  20 10 I i i i I  0  28  56  84 1 1 2 1 4 0 1 6 8 1 9 6 2 2 4 time  0  28  56  84112140168196224  (days )  Figure 2.12: Period 0 . Time series of the amplitudes and phases of the east-west and north-south velocities at M2 for the mooring D05 estimated from overlapping 28 day records. The solid lines represent the amplitudes (Ut, Vt) in cm/s, and the dashed lines the phases ( $ t , 6t) in degrees (GMT).  Chapter 2. Data analysis  37  QF4 ( o :  east-west 50  north-south  360  60m  ~m 4 0  270 1  ^jr^^^^^S^^l I 11 _j  L _  i i i i  50  _!  I I I  L _  100m  40  —iS360 270  30  180  20  J  10  -w^*  0 50  _l  •*-*--*' NK-*e--*  I I I I I 1  L _  _l  I I I I I I I I I I I I ' Q  360  310m  40 270  30 -  20  ---18-&-  10 0 50  _i  i i  L .  _j  i i i i  L .  760m  40  270  30 180  20  90  10 0  i^^Tfr^T^T^M^MM^i 28  i i ' ' i i ' '  56 84 112 1 4 0 1 6 8 1 9 6 2 2 4  28  56 84 112140 1 6 8 1 9 6 2 2 4  Figure 2.13: Period 0 . Time series of the ampUtudes and phases of the east-west and north-south velocities at M 2 for the mooring QF4 estimated from overlapping 28 day records. The solid lines represent the amplitudes (Ut, Vt), in cm/s, and the dashed lines the phases ( $ t , 6t) in degrees (GMT).  Chapter 2. Data analysis  38  $t and 6t are very similar in the upper three levels (upper three instruments) we can also see the presence of possible baroclinicity from the phase shift and the amplitude variation with depth. The time series of harmonic amplitudes and phases represent a superposition of barotropic and baroclinic components. The barotropic tide (being a shallow water wave) has a constant horizontal velocity as a function of depth.  Therefore a simple vector  subtraction of the barotropic coefficients from the total should yield the baroclinic component (Marsden, 1986 and Drakopoulos and Marsden, 1993) see Appendix A.  This  subtraction is most inaccurate near the generation regions since linear theory must be inadequate here. The barotropic coefficients that were subtracted were obtained from the finite element model developed by Foreman et al. (1993). This technique might be seen as poor since it does not give an error estimate. Nevertheless the barotropic modelled currents are assumed to have a small error compared to the errors associated with the observations. The major contributor to the surface elevation is the barotropic component and Foreman's model reproduced the sea surface elevations within a very reasonable range (4.0 cm for the ampUtude and 6.0° for the phase). This error estimate was obtained by comparison with observations. The barotropic model reproduced tidal currents very well in regions where baroclinicity is negligible as in the well-mixed Hecate Strait. These modelled barotropic currents are calculated satisfying volume conservation in the region taking into account variations in the bottom topography. The barotropic velocities obtained in this way are the best we can possibly get. Unfortunately there is not a way to offer an estimate of the error.  2.2.1  Baroclinic field  Figures 2.14, 2.15 and 2.16 show the result of such subtraction for the same moorings as in Figs. 2.11 - 2.13  Chapter 2. Data analysis  39  east—west _  5&  D04  (A)  north-south  b2m  IS)  ^  3b •  2i  CL  3 ic r. 3 6 0  1 b2r /  270 1 80 90  X  X  X  X ^*  360  2/9r  270 1 80 *- X _J  0  I  28  i  I  56  l  I  84  i  X  * -*-  -*-^-*90 J  L  1 1 2 140  1 68 time  1 96  56  84  112  140  i  L  168  19f  (days  Figure 2.14: Baroclinic time series of the east-west and north-south velocities for the mooring D04 for the first period. The solid line represents the amplitudes (Ub, Vb), in cm/s, and the dashed line the phases ($;,, 9b) in degrees.  Chapter 2. Data analysis  40  east-west  _ 50 \ 40  ^  F  j  D05  to:  north-south 360  ^  |  270  30 E-  1 80  2i  90  1 0 E-  -050  360  1 00m  40 E-  270  30 r 20  r  1 80  |  r  90 -  10  ^ T t V ^ ^ ^ ^ ^ ^ ^«50 p  360  150m  40 E-  270  30 180  20 10 " I  0  28 56 84112140168196224 time  0 28 56  84112140168196224  (days )  Figure 2.15: Same as previous but for the mooring D05 for the second period.  Chapter 2. Data analysis  41  QF4  east-west 5i  (0!  north-south  360  60m  270 180  *-*-*-  ' SK-JK--*-*-*""  I 1! =c  -J  50  I  I  I  I  I  1  I  I  I  I  1 L_  _J  I  I  I  I  I  I  I  I  I  I  I  L  360  100m  40  270  30  180  20 10 0 50  _)  I  I  I  I L_  31!  40 30 20 10 0 50  760n  40  270 h  30 180  20  90  10 i i i I i i~ r ~ ' 7 I  28  56  iT T T "  iiIiiiIiiiIii i  84 1 1 2 1 4 0 1 6 8 1 9 6 2 2 4  .^TTTtr^^wr^M^ i I i i i I i g i 28 56  84112140168196224  Figure 2.16: Same as previous but for the mooring QF4 for the second period.  Chapter 2. Data analysis  For the moorings inside Dixon Entrance, Figures  42  2.14 and  2.15, the baroclinic  components of the current, (Ub, Vb), had similar magnitudes at each depth. This is what we expect since the baroclinic Rossby radius of deformation, Ri = 20 km, is smaller than the width of Dixon Entrance: both Poincare and Kelvin baroclinic waves can exist. For D04, the velocity Ub reached values of 30 cm/s, at 52 m, and 20 cm/s at 279 m. These values represented 100 and 70 % of the barotropic velocity (Uo) and 76 and 77 % of the total velocity (Ut) respectively. Vb is similar in magnitude to Vt in the three instruments. Notice that in the deepest instrument Vb is bigger than Vt, which is unexpected but not uncommon. Drakopoulos and Marsden (1993) also found that, in some locations, the velocity associated with the baroclinic field was greater than the barotropic. While this situation might be due in part to viscous boundary effects near the bottom, it can also be associated with the uncertainty that comes from the assumptions used in the subtraction (the errors associated with the subtraction are smaller than the baroclinic velocity obtained). In Figure 2.14, both the east-west and north-south baroclinic velocities have a vertical pattern that seems to be associated with a first vertical mode. The changes in phases are bigger than 180° from top to bottom. Both Ub and Vb decrease from 52 m to 152 m and increase again from 152 m to 279 m. This feature was observed in those mooring that have more than two instruments in the central region of Dixon Entrance. For the mooring D05, Figure 2.15, Ub and Vb showed variations in time, over the second period, that can be associated with variations in the density field. Ub is approximately 74 %, 44 % and 56 % of Ut and Vb is approximately 48 %, 50 % and 130 % of Vt at 50, 100 and 150 m respectively. The baroclinic phase 6b also shows seasonal variations. The phase difference from the instruments at 150 m to the instrument at 50 m is bigger than 180° as it was in D04. For the mooring on the slope, Figure 2.16, (Ub, Vb), have almost the same magnitudes  Chapter 2. Data analysis  43  at each depth as (Ut, Vt) did. Although Ut is bigger than Ub at 60 m depth the general pattern of the baroclinic field is very similar to that of the total signal. In order to study the spatial variability of the baroclinic field in time, vertical sections, of simultaneous values, along lines of moorings were selected. For the east-west sections the line was formed by the moorings QF1 - D12 - D10 - D04 - D09 - D15 for the first period and for the second period by QF1 - D22 - D10 - D04 - D09 - D15 (see Figure 2.1). For the north-south sections the fine was formed by the moorings D01 — D02 — D03 — D04 — D05 — D06 for the first period and for the second period only by D04 — D24 — D05 — D06. These sections had the best distribution of instruments in the vertical. Along the central line of moorings the baroclinic field is almost seasonally invariant. The time series of M2 coefficients for Ub, <f>b and Vt,, Ob, are shown in Appendix A. The seasonally averaged variance of the data in this vertical section was very small compared to the mean. An analysis of the way the variances behave in time and space is presented in Appendix A. We will concentrate our attention on representative pictures of these baroclinic fields. Figures 2.17-2.18  show an along and across section of the averaged baroclinic velocities  for the first period (A) respectively and Figures 2.19 - 2.20 for the second period  (0).  In the east-west central region the baroclinic amplitudes represent almost 50 % of the total current. Both amplitudes and phases showed spatial changes associated with a vertical modal structure. That is, the magnitudes presented an extended core of lower values (around 150 - 200 m deep), and the difference in phase in the vertical (bigger than 180°) is maintained along the east-west section. In the north-south section the big velocities at the moorings D01 and D02 might be due to the fact that they are close to the entrance of Clarence Strait where the flow is channelled into the strait. Ignoring these two moorings, we can say that the instruments near the shore presented lower magnitudes than those near the central region and that  Chapter 2. Data analysis  44  east-west amplitude (cm/s) QF1  D12  D10 D04  D09  east-west phase (degrees)  D15  o o  o o E  a a>  •o  a o o co  o o co  0  20  40  60  80  100  120  0  20  40  60  80  100  distance km  distance km  north-south amplitude (cm/s)  north-south phase (degrees)  120  o o  o o  a  a T3  '  •o  o o co  o o CO  0  20  40  60  80  distance km  100  120  0  20  40  60  80  100  120  distance km  Figure 2.17: The two upper frames present the averaged values of the east-west velocity Ub and phase c/>{,, and the two lower frames, the averaged values of the north-south velocity Vb and phase 6b in a section along Dixon Entrance for the first period (A). The lowest line is the water depth. The dots mark the location of the instruments.  Chapter 2. Data analysis  45  east-west phase (degrees)  east-west amplitude (cm/s) D01 DQ2 D03  DQ4  D05 D06  o E 2  .c  a  +-»  a> o "o o CM  o o -10  -10  0  0  10  20  30  40  50  10  20  30  40  distance km  distance km  north-south amplitude (cm/s)  north-south phase (degrees)  10  20  30  distance km  40  50  10  20  30  40  50  50  distance km  Figure 2.18: Same as previous figure but for a north-south section.  Chapter 2. Data analysis  46  east-west amplitude (cm/s) QF1  D12  D10 D04  east-west phase (degrees)  D09 D15  o o  o o  E  '  I  a CD •D  o o CO  CO  0  O O  E Q. <D •o  20  40  60  80 100 120  0  20  40  60  80  100 120  distance km  distance km  north-south amplitude (cm/s)  north-south phase (degrees)  .14^-^jty//  .10 / O O  E r a  '  o o co  o o  T3  AS! 0  20  O O CO  ^ ^ 40  60  80 100 120  distance km  0  20  40  60  80 100  120  distance km  Figure 2.19: Same as Figure 2.17 but for the second period, ( 0 ) . The two upper frames present the averaged values of the east-west velocity Ub and phase </>&, and the two lower frames, the averaged values of the north-south velocity Vb and phase 6b in a section along Dixon Entrance for the second period (0). The lowest line is the water depth. The dots mark the location of the instruments.  Chapter 2. Data analysis  47  east-west amplitude (cm/s) D04  D24  east-west phase (degrees) D05 DQ6  O  288 - 2 I 3 = = = ^ » ^ g 6  E .c a ID  /T^w^  O 7-  /  X  o o  \  <M I  »  /  /~^ J  o o  CD •  10  20  30  40  -10  50  40  north-south phase (degrees)  50  O  26,,  ,5  f l^^  -  z. T3  30  north-south amplitude (cm/s)  26  Q.  20  distance km  O O T  10  distance km  O  E  0  O O  6  y  E Q. CD •o  O O 1—  o o 1  o o  o  03 i  i  -10  0  10  20  30  distance km  40  50  -10  0  10  20  30  40  50  distance km  Figure 2.20: Same as previous figure but a north-south section.  Chapter 2. Data analysis  48  a lateral modal structure is thus present. It is difficult to arrive at a definite conclusion about a vertical pattern since there is only one station with more than two instruments in the first period and none in the second. The two main features of these results are that the baroclinic velocities are large and that a prevalent pattern is present along the central region of Dixon Entrance in both periods. East-west and north-south baroclinic currents have similar magnitudes in the central region of Dixon Entrance and they are bigger than the ones near the coast. Due to the low resolution in space it is hard to arrive at solid conclusions. There is the question of how well the field has been spatially sampled and an estimation of the possible waves in the region is necessary.  Chapter 3  Long waves in a channel  In this chapter I aim to give a order of magnitude illustration of the possible internal waves at the semidiurnal frequency M2. Using density profiles, an estimate of the possible internal tides in Dixon Entrance is made by representing it as a closed channel of constant width and flat bottom as suggested by Forrester (1973) and Foreman et al. (1994). The possible modes are Poincare modes and Kelvin waves. The dispersion of Kelvin waves in a closed channel was studied by Taylor (1921). 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. In this section the problem solved by Taylor (1921) is repeated for a vertically stratified channel, of width W = 55 km located at the same latitude as Dixon Entrance 54.5°iV (therefore / = 1.1872 • 10~ 4  s~x) . An averaged  density profile is used and assumed constant in the horizontal. The problem of ocean waves in channels has been re-examined by Hendershott and Speranza (1971), Brown (1973) and Ripa (1997), among others. In this section only the first vertical mode will be described, and it will be assumed that it represents, as a first approximation, the vertical baroclinic field. Generation regions estimated from the deepest buoyancy frequency in conjunction with the bottom slope are delineated.  49  Chapter 3. Long waves in a channel  3.0.2  50  Formulation  In a flat-bottomed ocean the solutions to the linearized (i.e. infinitesimally small disturbances from the mean state) equations of motion can be separated into vertical and horizontal dependencies. For a harmonic time dependence (here elwt) and with the Boussinesq approximation, the horizontal part is defined by the shallow water equations with rotation (Leblond and Mysak 1978, equations 10.34-10.38)  ««>Un - JVn  -  iuVn + fUn = dUn dx  (  dVn dy  dPn dx  (3.1)  BPn dy  (3.2)  -iujPn ghn  (3.3)  subject to the boundary conditions Vn = 0  at  y=  W  (3.4)  and Un = Q The geometry is shown in Fig. 3.1;  at x  directions, (Un(x, y), Vn(x, y)) and Pn(x,y)  x = 0. and  y  (3.5)  are the along and cross channel  are the horizontal dependences of velocity  and specific pressure (pressure/density), respectively, / is the Coriolis parameter, g is the acceleration due to gravity, UJ is the semi-diurnal wave frequency and hn is the equivalent depth as derived from a vertical mode calculation and is the link between the horizontal and vertical dependences. Assuming that the vertical velocity w can be expressed as w = Wn(x,y)cj>(z), baroclinic problem for the vertical part is defined as:  the  Chapter 3. Long waves in a channel  51  -150  -100  -50  ( x ) E-W distance in km  •  "O CO  Kelvin Wave .  • -150  -100  -50  ( x ) E-W distance in km  Figure 3.1: Superposition of flat channel geometry (width W = 55 km and length L = 150 km) upon Dixon Entrance. Contours represent bottom bathymetry in meters.  Chapter 3. Long waves in a channel  52  **• + EL^„ dz2 ,  =  ghn  „  (3.6)  with the bottom boundary condition <j>n = 0  at z = - H  (3.7)  at z = 0,  (3.8)  and the surface boundary condition ^ - ^ = 0 dz h,_  where N 2 (z) is the Brunt-Vaisala frequency defined as 9 dp 2 Nr2 = -2-f, podz  (3.9)  g is the acceleration due to gravity, p0 is an averaged density value and p(z) is the density profile. The eigenvector (j)n(z) gives the variation with z of the vertical displacement of a fluid particle and the eigenvalue ghn is the squared phase velocity, both due to the nth vertical mode. The equivalent depth hn is the depth for an equivalent homogeneous system (Gill, 1982). The vertical displacement and horizontal velocities are separated as:  9 un =  Un(x,y)h,  vn =  Vn(x,y)hr  d(f>n{z) dz  d(f>n{z) dz  Equations 3.1 and 3.2 can be solved for Un and Vn in terms of Pn  Un =  ^^P^-%T^y-) u2 - f2  ay  ox  (3 10)  -  Chapter 3. Long waves in a channel  53  where / r = -  (3.12)  The equation of continuity 3.3 then yields the wave equation in Pn: (V 2 + ^ ^ ) P gnn  n  = 0.  (3.13)  Choosing a longitudinal dependence etKX the horizontal problem satisfying the boundary condition Eq. 3.4 becomes ^  + «»P B = 0  (3.14)  with dPn  + KrPn = 0  dy wnere  at  / ,2  y = 0,W  f2  ghn Solutions to this problem are conditioned by (u,2 - f)  sm{aW){u2  - ghnK2) = 0  (3.15)  (Pedlosky 1987). This product describes the three possible horizontal modes of propagation along the channel. The first, ui2 = f2, represents inertial waves (which are not the subject of this thesis) the second, sin(aW)  = 0, are Poincare waves, and finally  « 2 = K2 = ^ ghn  (3.16)  describes Kelvin waves. Free Kelvin waves can always be excited but for free Poincare waves, only specific along-channel wavenumbers are permitted: 9  ,9  "=*-=  a; 2 - / 2  A  m27r2  -WT  (  "7)  Chapter 3. Long waves in a channel  54  In order to have along-channel propagation the right side of this equation must be positive. It is clear from Eq. 3.17 that the Poincare modes decay exponentially up-channel of the obstruction when m 2  >  " "^ ^ - . The cross-channel wavenumber associated with  a normal horizontal mode m is am = — .  (3.18)  Although we are interested in the signal at M2 it is noticeable that at the diurnal frequency K x only evanescent Poincare waves exist since  ui  <  f and fc£m is always  negative. We will assume that there is a baroclinic Kelvin wave of vertical mode n and phase velocity cn = -\/ghn with coastal amplitude /?/, travelling into the channel with the wall at y = 0 to the right of its direction of propagation (Fig.3.1). The outgoing waves consist of one or more Poincare modes as well as another Kelvin wave of coastal amplitude (3R, having the channel wall at y = W to the right of its direction of travel. For the n t h vertical mode, the vertical displacement and horizontal velocity associated with the incoming Kelvin wave are  Ci{xty,z,t)  Ul(x,y,z,t)  =/3/e-£v<""-£"Vn(*)  (3-19)  = A_e-£v(*-£-)^!i. cnp0 dz  (3.20)  vI(x,y,z,t)  = 0.  (3.21)  The vertical displacement and horizontal velocity associated with the outgoing waves are Cn(x,y,z,t) = ( - ^ r i ^ ^ S ' + £ V m=l  ~,mFpm(y)eik-A e^M*) /  (3-22)  Chapter 3. Long waves in a channel  uR(x,y,z,t)  = (-£!Le-&-")«Z' V  vR{x,y,z,t)  55  ^Fum(y)eik^A  + ±  C  nPo  m=l  = i £ - ((f2a}^f2k\) ^ i Po \{k2 + a^waclj  e * ^ ,  Po  J  (3.23)  dz  s H a ^ y ^ - * ^ . dz  (3.24)  where Fpm{y) = cos(amy)  ^— amuj  knm  sin(amy),  f  FUm{y) = -^cosfamy)  sin(amy).  Satisfying the boundary condition at the end of the channel, Eq. 3.5, yields  f  t  °° w  pie-£y _ pRe-kb- )  + Cn J2 lmFum{y) m=l  =0  (3.25)  for a l l y e [0,W]. Setting /?i = I and truncating the summation to a finite number of terms N , the method of collocation,  suggested by Brown (1973) is used to find the coefficients f3R and  7 m for m = 1,2, ..N.  3.0.3  Energy Fluxes  The energy density E of an internal wave is defined as the mean perturbation energy per unit volume:  E = ±Po(u'+v*+  *>) + £ £ ; .  (3.26)  Chapter 3. Long waves in a channel  56  where the over-bar means average over one period. Assuming a flat-bottomed channel, for each Kelvin wave the following expression is obtained for the vertically and laterally integrated energy density per unit length;  c  + H{N2+  '' *=SI (- £&'* i  liz  ^* ) •  (327)  -  And for the m propagating Poincare mode  A = k2(l - r 2 ) 2 + r 2 ( — + a m ) 2 + ( a m + r 2 — ) 2  (3.29)  where The energy flux associated with each mode can be calculated by multiplying the energy density by the group velocity. In the case of Kelvin waves the group velocity is equal to the phase velocity cn, thus the energy flux is Jk =  EkCn, and in the case of  Poincare modes the energy flux is J p m = E p m ^ - 1 . Energy conservation requires  E  "yJpm + ] § £ = 1, Jki \Pif where M is the number of propagating Poincare modes.  3.1  (3.30)  Calculation of t h e vertical normal m o d e s  The eigenvalue problem defined by equations (3.6) to (3.8) was solved numerically from an averaged profile of the buoyancy frequency N2(z).  This second order boundary value  problem with one fixed boundary value (0 at the bottom) is solved by successive iterations of updating eigenvalues (i.e. phase speeds) to find the second boundary value.  Four  Chapter 3. Long waves in a channel  Season  St  Cl  Spring Autumn Summer  8411 8414 8510  m/s 0.902 0.962 0.878  Ri = ci/f km 7.6 8.1 7.4  57  2TV/K  km 40 43 39  Dm+1 — l/kitTn+\ km 12.8 12.0 13.2  2TT /kliTn km 103.5 117.8 98.7  m 1 1 1  Table 3.1: For the first vertical mode, i.e.n = 1 the following values are listed: the phase velocity ci, the Rossby radius Ri = ci/f, the longitudinal Kelvin wavelength 2TT/K, the decay along x of the first trapped Poincare mode Dm+i = kiiTn+i~ > the longitudinal Poincare wavelength 27r/A;1|m and the maximum free Poincare mode m possible. stations were averaged in order to get a representative density profile from each cruise and 4>n and ^f- were calculated from it, (see Figures 3.2, 3.3, 3.4, 3.5, 3.6, 3.7). The amplitude of </>„ and - g 1 is arbitrary since we did not have measurements of the density perturbation. The shape of -j21 is proportional to the horizontal velocities u and v. In Table 3.1, for the first vertical mode n = 1, values of Rn, the decay distance along the channel Dm+\,  as well as the longitudinal wavelengths of the Kelvin and Poincare  waves were calculated from the density profiles shown in Figures 3.3, 3.5 and 3.7. The phase velocity values for the three cruises are very similar but the reduced gravity values are not. The reduced gravity values (see Figures 2.9 and 2.10) are approximately 0.014, 0.02 and 0.028 m s " 2 for the cruises 8411, 8414 and 8510 respectively. In a twolayer system the thickness of the upper layer changes from approximately 100 m in spring (cruise 8411) to 35 m in summer (8510). For these averaged profiles only the first Poincare mode, m waves. For m  — 1 travels as a free  > 1 the modes are evanescent. The value of Rn which gives the cross  channel e-folding scale at which Kelvin waves are trapped to the longitudinal walls of the channel, is approximately 16 % of the width channel, so that there is plenty of room for Kelvin wave propagation in both directions. The longitudinal Kelvin wavelengths range from 43 km during the autumn to 39 km during the summer. Dm+i  — fci|7n+1-1 is the  Chapter 3. Long waves in a channel  St 8411-48  24.5  25.0  25.5  26.0  sigmaltKg/m^)  26.5  St 8411-80  27.0  24.5  25.0  25.5  26.0  sigmat(K()'mA3)  26.5  St 8411-41  27.0  24.5  25.0  25.5  26.0  sigmat(Kg/iTr*3)  26.5  St 8411-87  27.0  24.5  25.0  25.5  26.0  26.5  27.0  sigmaWg/m^)  Figure 3.2: Density profiles from cruise 8411. The location of the CTD stations can be seen in Figure 2.6.  Chapter 3. Long waves in a channel  Averaged density 8411  n=1  Figure 3.3: Averaged <rt profile ([kg/m3]) from cruise 8411, (a) with corresponding iV2 profiles [rad/s), (b). The vertical displacement (j>n (thick line) and -^p- (thin line) for the first, n — 1 vertical mode assuming a constant depth of 320 m are shown in (c) with arbitrary units.  Chapter 3. Long waves in a channel  St 8414-99  St 8414-51  60  St 8414-59  St 8414-82  25 sigmatlKp'rrc^)  26  27  sigmaltKa'm^)  Figure 3.4: Density profiles from cruise 8414. The location of the CTD stations can be seen in 2.7.  Chapter 3. Long waves in a channel  Averaged density 8414  n=1  Figure 3.5: Averaged crt profile ([kg/m3]) from cruise 8414, (a) with corresponding N2 profiles (rad/s), (6). The vertical displacement </>„ (thick line) and ^p- (thin line) for the first, n = 1 vertical mode assuming a constant depth of 343 m are shown in (c) with arbitrary units.  Chapter 3. Long waves in a channel  St 8510-49  23  24  25  sigmatfKcynTO)  26  St 8510-65  St 8510-55  27  23  24  25  sigma!(K<^nA))  23  24  25  sigmat(K^m"3}  26  St 8510-70  27  23  24  25  26  27  sigmat(K9'm*3}  Figure 3.6: Density profiles from cruise 8510. The location of the CTD stations can be seen in 2.8.  Chapter 3. Long waves in a channel  Averaged density 8510  23  24  25 sigmattKg'nrt))  26  n= 1  27  0.0  0.0002  0.0006 N2 (lad's)  0.0010  -  1  0  1  2  3  w idw/dz  Figure 3.7: Averaged crt profile ([kg/m3]) from cruise 8510, (a) with corresponding N2 profiles (rad/s), (b). The vertical displacement (j)n (thick Hne) and -£p- (thin hne) for the first, n = 1 vertical mode assuming a constant depth of 343 m are shown in (c) with arbitrary units.  Chapter 3. Long waves in a channel  Season Spring Autumn Summer  64  1/3*1  ITII  0.9994 0.9849 0.9982  7.7-lO" 0.04675 0.01327  3  J|rT  3  2.42 • 10~ 7.49 • lO - 2 7.74 • lO" 3  \ux=o\/\u\ 9.644 • lO" 8 1.401 • 10 - 7 1.406 • lO" 7  Table 3.2: Values of: the amplitude of Kelvin wave reflected, |/3R|; the Poincare wave reflected, |-yxIj energy due to the Poincare wave, j ^ (Eq. 3.30); and magnitude of the velocity u at the end of the channel over the averaged u everywhere else, for the representative density profiles. maximum distance of influence of the non-propagating Poincare modes trapped to the meridional wall [i.e at a; =  0) which in all cases is slightly smaller (by about 10 % )  than the length of the channel. Only one longitudinal Poincare wavelength fits inside the channel since the wavelengths are as long as the channel. In summary, in Dixon Entrance, due to its stratification and dimensions, two free Kelvin modes travelling in opposite directions as well as one free and an infinite number of trapped Poincare modes are excitable.  3.1.1  Baroclinic fields  The first vertical mode, n = 1, describes best the general features found in the data analysis as examined in the previous chapter. The following values are constant for all the cases W  =  55 km, u = 1.41 • 10~4 s _ 1 and / = 1.1872 • 10~ 4 a - 1 . In Table 3.2  values obtained from the representative density profiles are presented. The amplitude of the outgoing Kelvin mode, \/3R\ is much bigger than the amplitude of the outgoing Poincare mode, |7i|. The contribution of the Poincare mode, i.e. -f^-, (see Eq. 3.30) to the total energy reflected is very small. It goes from 0.02 % in the spring to 7.5 % in Autumn. An estimate of the error using the collocation technique is given by the value of the ratio |ti x = 0 |/|ii|, the closer to zero the better. The averaged values of u at the end  Chapter 3. Long waves in a channel  65  of the channel (l^a^ol) were very small compared to those away from this boundary (|tt|). The vertical displacement , ( = £/ + (R, the longitudinal velocity, u = uj + UR, and the cross channel velocity, v = vj + VR, for the first vertical mode (see Equations  3.19  to 3.24) are calculated and decomposed into amplitude and phase, i.e. assumed to be of the form \F\elu,i+8f  where | F | is the amplitude and Of is the phase. In Figure 3.8 these  fields are plotted at 50 m depth using the density profile shown in Fig. 3.5, cruise 8414. Due to the arbitrariness of the values of <f>i and -£- the ampUtudes are arbitrary. What we want to show is the variation in space of such fields. As a signature of the Kelvin modes travelling in opposite directions, both the vertical displacement and longitudinal velocity reach their maximum values at the walls parallel to x. The amphidromic regions, i.e where the ampUtudes reach a smaU but non-zero value and the phase contour join are located along the centre band of the channel. The cross channel velocity, much smaller than \u\, varies in the opposite way. \v\ reaches its maximum along the centre of the channel decreasing towards the walls. This is due to the presence of the free Poincare mode. The high values of |i>| at the end of the channel mark the region of influence of the trapped Poincare modes. The parallel phase contours of the velocity v are related to the Poincare mode travelUng along the channel. The equivalent horizontal results associated with the cruises 8411 and 8510 present the same features. A vertical section of the amplitudes and phases obtained from this analytical solution is presented in figure 3.9 along the middle of the channel, i.e along y = 27.5 km Since we have only kept the first vertical mode n  =  1, both the ampUtude and  phase give the vertical variations associated with that mode.  These are: a core, at  depth, of lower magnitude values with higher values towards the top and bottom and a vertical change in phase of 180°. In spring and in summer the wave pattern is almost exclusively due to two baroclinic Kelvin waves travelUng in opposite direction. So, out of the influence region where the Poincare modes are trapped |i>| is much smaUer than  Chapter 3. Long waves in a channel  -100  66  -150  -50  100  x (km)  -100  -50  0  -100  -150  x (km)  -100  -50  x(km)  -50  x(km)  9)  -150  -50  x (km)  0  -150  45  "5  -100  -50  x (km)  Figure 3.8: Amplitude and phase of a) the vertical displacements (£), b) longitudinal velocity (u) and c) cross channel velocity (v) for the first baroclinic mode at 50 m depth obtained from the averaged density profile shown in figure 3.5, cruise 8414. Note that the contour lines are not evenly spaced. Phases in degrees and amplitudes in arbitrary units.  67  Chapter 3. Long waves in a channel  a) 90  o o  270  o o E  a  a  CD  •D  •D  o o  o o  CO  * -150  -100  -50  -100  -150  W0  *  -50  distance km  distance km  b) O  o  6 E  o  o o V  \  O O  E r.  Q.  /  •o  O O co  J  ]  I  ^  2  m ^  ii  ^^^  Q. 0) O O CO 1  I  1  150  -100  -50  distance km  0  93  150  -100  i 0  -50  0  ciistance km  Figure 3.9: Amplitude and phase of a) the longitudinal velocity (u) and b) cross channel velocity (v) obtained from cruise 8414, along y = 27.5 km. The constant depth is 321 m. Amplitude contour lines are not evenly spaced. Phases in degrees and amplitudes in arbitrary units.  68  Chapter 3. Long waves in a channel  -ISO  -lOO  -so  o  Figure 3.10: Line of points where amplitudes and phases have been calculated from the analytical model. The circles mark the approximate location of the moorings. \u\.  However along the middle region of the channel there are small vertical regions  where both magnitudes coincide in value. In autumn, Fig 3.9, the free Poincare mode has a higher contribution to the energy reflected (see Table 3.2) and along the middle of the channel, both magnitudes |ii| and \v\ reach similar values in the vertical.  The  phase variations in the horizontal are related to the wavelengths of the Kelvin waves. The phase changes 180° within approximately 20 km (half of a wavelength). The results from the model look rather different from the observations. The model is simplified, of course, but also the observations are very sparse in space. To obtain a more reasonable comparison of the model and the observations, values from the model at the points where observations were taken only are used. The amplitudes and phases were calculated using the density from cruise 8414, in a vertical section which viewed from above is shown in Figure 3.10, simulating the mooring line shown in Figure 2.17. The results are shown in Figure 3.11. Comparison of Figure 3.11 with Figure 3.12 (a duplicate of Figure 2.17) shows many similarities: i) the amplitudes of u and v are comparable,  69  Chapter 3. Long waves in a channel  o o £ a a)  0. 0) T3  •o  -120  -80 -60 -40 -20  o o  0  -120  distance km  -80  -60 -40  -20  0  -20  0  distance km  o o E a a) •o  Q.  o o  to  -120  -80  -60  distance km  -40 -20  0  -120  -80 -60  -40  distance km  Figure 3.11: Modelled amplitude and phase of a) the longitudinal velocity u and b) cross channel velocity v for cruise 8414 along the line of moorings. Phases in degrees and amplitudes in arbitrary units. Both the real bottom depth at each mooring as well as the constant depth (321 m) assumed are indicated. Dots mark the locations of instruments.  Chapter 3. Long waves in a channel  70  east-west amplitude (cm/s) QF1  D12  D10 D04  D09  east-west phase (degrees)  D15  o o  Q. CD •o  o o  a o  •D  o o CO  0  20  40  60  80  100 120  0  20  40  60  80  100 120  distance km  distance km  north-south amplitude (cm/s)  north-south phase (degrees)  o o  o o  a •D  o o ro  a CD  o o co  0  20  40  60  80  distance km  100 120  o o co  0  20  40  60  80  100 120  distance km  Figure 3.12: Duplicate of Figure 2.17 from the data analysis. The two upper frames present the averaged values of the east-west velocity Ub and phase (f>b, and the two lower frames, the averaged values of the north-south velocity Vb and phase 8b in a east-west section along Dixon Entrance for the first period (A). The lowest line is the water depth. The dots mark the location of the instruments.  Chapter 3. Long waves in a channel  71  ii) the amplitudes of u and v are larger at the top and bottom than those at mid depth, iii) the phase changes by about 180° from the upper layer to the bottom. We can not expect the modelled values shown in Figure 3.11 to be in direct agreement with those shown in Figure 3.12 since the model is highly simplified, and the locations chosen, for comparison with the moorings, are somewhat arbitrary. What can be shown is an agreement in pattern through a scatter plot of relative values, Figure 3.13. The relative values plotted are the difference between the value of a variable at a point and its value at its nearest neighbouring points in the vertical and the horizontal. From the values of the correlation coefficients r presented in Figure 3.13 it can be seen that the patterns described by the relative values are in reasonable agreement for the amplitude and phase of u and for the amplitude of v but less so for the phase of v. The model has a flat bottom and includes only the first baroclinic mode. The observations may be influenced by the bottom topography and may have contributions from other modes. The spatial distribution of the data is insufficient to show that the baroclinic motions are predominantly first mode Kelvin waves. Nevertheless, the observations are not inconsistent with this interpretation.  3.1.2  Generation Regions  Generation of internal tides can occur when a stratified fluid is forced to oscillate over sloping bottom topography.  One optimal condition for generation is that the bottom  slope matches the paths in which the wave energy propagates (Baines 1973). The particle path of the forcing aligns with the particle path of the free wave and a resonance condition is created (Foreman, Crawford and Marsden 1994). A suitable location for this generation is at the shelf break and sills not only because of their slope shape but also because usually they are located near the pycnochne where the density stratification is maximum  Chapter 3. Long waves in a channel  72  a) r = 0.702, var = 3.67  b) r = 0.849, var = 60.06  40  $400 CD i_  E 30 o_  0  E300 •D  O  T3 O  E  E 20 0  0  310  CO  OO  a 100  0  8  0  0°  > CO  0° 10 20 30 relative u obs.(cm/s)  40  ^  01  0 — 0 —  0  1  100 200 300 400 relative phase u obs.(degrees) d) r = 0.0709, var =105.6  c) r = 0.888, var = 3.32  30  $400 o>  25  S300 •d o 0 >200  '20 O  O  CD  0  E 15 > > 10  0  (/) . .0  QOQ.  0  0)  0  >  0  D200  CD  CD  0  D) 0)  en  ..0. °  <D 0) CO  0  o 5 D oOg 2  10 20 relative v obs. (cm/s)  a) >  30  O 0  O  0  O  0  a 100  c  O  0 D  0  0  1  &'  0  '  100 200 300 400 relative phase v obs.(degrees)  Figure 3.13: Scatter plots of the relative observed versus modelled values, a) of the amplitude of u, b) of the phase of u c) of the amplitude of v, b) of the phase of v. The values of the correlation coefficient r and the variance var are indicated at each frame.  Chapter 3. Long waves in a channel  73  (Drakopoulos and Marsden, 1993). Rays paths, or characteristics, are group velocity vectors for internal tides. Generation occurs when the slope of the characteristics C is equal to the bottom slope, i.e. when  where / is the Coriolis parameter, u> is the frequency (M 2  in this study), N(z)  is  the Brunt-Vaisala frequency defined in equation 3.9 and h is the bottom contour. This two dimensional formulation was given by Baines (1974). Mooers (1975) proposed a modified formulation that includes density changes in the horizontal, but for this study I consider the horizontal density gradients as negligible (figures 2.9 and 2.10 ). The deepest N(z),  (O(10 - 5 ) rad/s), values were obtained by averaging CTD density profiles. An  example of such average is presented in Fig. 3.14. This CTD profile comes from the station number 58 from the cruise 8414. Figure 3.14 a) is the density profile, as crt, obtained from the temperature and conductivity raw data. Figure 3.14 c) is N 2 (z) obt ained from the density in  a), which is plotted in logarithmic  scale so the variations have been enlarged. Because of the difficult task of getting a value that represents the trend of N 2 (z) near the bottom, a smoothed version was necessary. The second plot, b), is the density averaged every 21 meters with values assigned each 10 meters and the plot, d), in this figure is the N 2 (z) profile obtained from the density averaged, in b). The line at 6.0 1 0 - 7 marks the noise level. This procedure was performed upon all the deep CTD stations for the cruises 8414 and 8510. The bottom slopes, ^ , where x is the east-west direction, in Dixon Entrance were estimated from a 5 km depth grid with a bottom slope error of O(10~~4) ra. The deepest value of N 2 (closest to the bottom) and ^ were used to estimate C'. Note that the other terms in equation 3.31 are constant. The distribution of the deepest values of N 2 (i.e the CTD distribution) was very scarce  74  Chapter 3. Long waves in a channel  a)  b)  \  24  25  26  sigmat(Kg/mft3)  27  d)  C)  average (21 data) values every 10m  from data st 58*8414*  24  \  \  \  25  \  \ \ \ \  from ave. density delz = 20 values every 10 m  \  26  sigmaUKg/m^)  from data (st58)  27  10"-6  10M  log N2 (N2 in rad/s)  Figure 3.14: From the CTD station 58, cruise 8414, a) the density as crt, b) the averaged density, c) N 2 (z) obtained from the density and d) N 2 (z) obtained from the averaged density.  Chapter 3. Long waves in a channel  -134  75  -133  -132  -131  degrees Longitude  Figure 3.15: Regions of possible generation of internal tides estimated from the cruise 8414. compared with the regular distribution of ^ . It is for this reason that ranges of C where extrapolated. Regions where dh dx \  i  c was satisfied a generation region was assigned. The generation regions estimated in this way for the cruises 8414 and 8510 are shown in Figures 3.15 and 3.16 The plots show that the continental shelf and Learmonth Bank are the two potential east-west generation regions in our domain.  76  Chapter 3. Long waves in a channel  -132 degrees Longitude  Figure 3.16: Regions of possible generation of internal tides estimated from the cruise 8510.  Chapter 3. Long waves in a channel  3.2  77  Summary  Although in Dixon Entrance there is a considerable seasonal variation of density, baroclinic motions at the semidiurnal frequency M 2 presented an almost steady pattern throughout the year. The structure in the vertical seems predominantly associated with the first vertical mode. The phase velocity for the first internal mode is very similar for the three different seasons. In a two layer system one explanation could be that the depth of the upper layer and the stratification change in time in such a way that they produce the same internal phase velocity c\ ~ \ / y H i * ^ / ( H x + H 2 ) , where Hi and Hz are the upper and lower depths of the layers and g' is the reduced gravity. If Dixon Entrance can be approximated by a zonal flat channel connected to a steep continental shelf, the wave behaviour in the horizontal can be explained as follow: A baroclinic Kelvin wave, of vertical mode n  =  1, is generated at the continental shelf  (Figures 3.15 and 3.16), and travels inside Dixon Entrance along the southern wall towards the east, (Figure 3.1). At the end of the channel reflection occurs and two kinds of waves travelling west occur, another Kelvin wave together with a very weak Poincare wave and a set of evanescent Poincare modes trapped to the end of the channel. The amplitude of the reflected Poincare wave is approximately 0.04 times the amplitude of the reflected Kelvin wave (Table 3.2). The presence of this free Poincare wave allows the longitudinal velocity Ub and the cross v\, to have almost the same magnitude in a longitudinal central band along Dixon Entrance. If we have velocities of 20 c m s " 1 in the central region we should expect velocities of 70 cms-1  near the walls since Kelvin waves  reach their maximum value near the walls. This assumption breaks down when analysing the data from a north-south section (see Figure 2.18). Though in this exercise u and v coincide in magnitude it is difficult to believe that  Chapter 3. Long waves in a channel  78  simple linear Kelvin and Poincare waves travelling over a flat bottom represent the real situation shown in Figure 2.17. The internal tidal dynamics inside Dixon Entrance seems more complicated and needs to be explained by a more sophisticated model where nonlinearities and variations in the bottom are taken into account. The distribution of the moorings, along the section described is too sparse to detect the variations in phase of the Kelvin and Poincare waves. The continental slope and Learmonth Bank are the two potential east-west generation regions in our domain.  Chapter 4 Numerical Model  The generation of internal waves in the ocean can be complicated by many interrelated factors including nonlinearities in the governing equations, spatially varying tidal current strength and phase, spatially and temporally varying stratification, boundary-layer and turbulent effects, overturning waves and associated mixing, and background residual currents (Lamb, 1994). In this section I present results of idealized numerical experiments of the wave generation process, where only few of the already mentioned factors are taken into account. The goal is to develop an understanding of the generation and dispersion of nonlinear internal tides in Dixon Entrance using a simple model. The model used is frictionless, non-linear, two-layered, finite difference and forced by a barotropic wave over idealized topography. In order to study the effects of different bathymetric features in the dispersion of internal waves results for several topographies are presented.  4.1 4.1.1  D e s c r i p t i o n of t h e m o d e l The code  The numerical code used was originally described by Gill et al (1986) for a flow over a step and for two-layers over a slope by Allen (1988) and is based on an explicit leap-frog method. The code uses a second-order space finite difference scheme, which improves the simulation of nonlinear aspects of the flow, for the shallow water equations that conserves  79  Chapter 4. Numerical  Model  80  both potential enstrophy and total energy with variable bottom topography (Arakawa and Lamb 1981). The primitive shallow water equations for a two-layer, inviscid, nonlinear flow, are: Du2  ,  Dv2  ~dt  +  ^((Hz  + m  ^  -w" if  +  di]!  ,07/2  drji  5^((H2  + V2  Dux Dt  drj! ox  Dvx  drj!  -m+fui d  +  ,dr]2  =  ^  =  °  -'Si'  1 ^ + * " *>"•>+ | « H ' + " - * * > = °  where 772 is the interface elevation, 771 is the surface elevation, H 2 (x, y) and Hi are the depths of the lower and upper layers respectively, (1^2,1*2) and («i, Vi) are the horizontal velocities in the lower and upper layers, / is the Coriolis parameter, g is the gravitational acceleration and g' is the reduced gravity, g(p2 —  pi)/p0-  The domain is rectangular and the grid spacing, 8d, is chosen to be a small fraction of the Rossby radius. The horizontal velocity for each layer, and the layer thicknesses hi = 771 — 772 + Hi and h2 — r/2 + H 2 are calculated on a square grid (Figure 4.1). The two layers are coupled as the surface height variation affects both. The upper layer variables are denned at the same horizontal positions as the lower layer variables. A staggered grid, the C grid, is used as Arakawa and Lamb (1981) recommended, (see Figure 4.2). Arakawa and Lamb (1981) presented, in detail, the finite difference expansions for the shallow water equations as functions of the kinetic energy, the potential energy and the total potential vorticity.  Chapter 4. Numerical  Model  Figure 4.1: Sketch showing the definitions of the variables for each layer.  81  Chapter 4. Numerical  Model  82  v(1,2)  v(2,2)  h(1,2)  u(1,2)  •  v(1,1)  h(2,2)  u(2,2)  •  •  v(2,1) i  domain  .  N ^  boundary  h(1,1)  u(1,1)  h(2,1)  u(2,1)  •  •  •  •  Figure 4.2: A plan view showing the grid locations where the numerical code calculates the velocities u, v and the height field h, near the edge of the domain.  Chapter 4. Numerical  Model  83  The code has no explicit viscous or frictional damping; however, there is some implicit numerical damping. The advantage of this code is its accuracy and the fast execution per time step. A disadvantage is the short time step required for stability.  4.1.2  Topography.  The model topography approximates Dixon Entrance and the adjacent continental shelf in the Pacific Ocean at about 54.5°N, 131.5°W. With the purpose of analysing the impact of bathymetric features in the scattering of internal tides, different topographies will be presented. In all of them Dixon Entrance is assumed to be a zonal channel closed at its eastern end and connected to the shelf at the western end. Hecate Strait and Clarence Strait are ignored. The sharp coastline bend at the western side of Graham Island and the adjacent steep continental slope (Figure 4.3) inspired modelling Dixon Entrance in this way. Crawford et al. (1997) indicated the presence of a very narrow and shallow, north-south oriented, reef structure at the eastern end of Dixon Entrance (Figure 1.10). It is approximately 24 km long and less than 3 km wide. This reef ("Celestial Reef" plus "West Devil Rocks") could also be regarded as a wall that partially closes Dixon Entrance at its eastern end. Hecate Strait is neglected based on the argument that, in a two-layer system where the interface happens to be deeper than 50 m, as far as the internal waves dynamics are concerned Hecate Strait does not exist since it is only 40 m deep where it connects with Dixon Entrance.  4.1.3  B o u n d a r y conditions  Boundary conditions were used to calculate the edge points after each time step. The boundary condition imposed at the eastern side and the channel walls was free-slip. For the north and west open boundaries a criterion of wave radiation was applied, (Figure 4.4). A linear transformation, from a layered to a modal system (and vice-versa),  Chapter 4. Numerical  134 W  Model  133 W  84  132 W  131 W  130W  55N  55 N  54N  54 N  53N  53N 134 W  133 W  132 W  131 W  130W  Figure 4.3: Bathymetric chart of Dixon Entrance System. Depth contours are in meters with intervals of 50 and 500 m for regions shallower and deeper than 500 m respectively. Four regions are shown: depths less than 100 m (-), between 100 and 300 m (blank), between 300 and 500 m ( + ) , and greater than 500 m ( > ) .  Chapter 4. Numerical  Model  85  T  8  RADIATION  FREE SLIP  o  LAND  CO  60 km  •}  •o  : e-w sections  (300 m)  RADIATION  4 o  (2000 m) FREE SLIP  O 01  LAND  :  1«  151.8km  FORCING  -&-  •IS-  I l i l l l l I I I 111 I I l l l i i l I I 111  20  40  ii 111 I I I I I I I 111 n i l 111 I U H I ii i n 111 ii i  60  80  100  120  gridx  Figure 4.4: A plan view of the topography used for the base case. The points indicate the locations at which time-series were stored. Three east-west sections are indicated by arrows at the eastern end; near the northern wall of the channel, in the centre and near the southern wall. The conditions for the boundaries are indicated.  Chapter 4. Numerical  Model  86  is used in the open radiation conditions assuming that locally the motion is one dimensional in a non-rotating system (Leblond and Mysak, 1978, chapter 16). The criterion of wave radiation consisted in applying the Sommerfeld conditions to the barotropic and baroclinic variables obtaining the edge values and from them the edge values at each layer were found. The equation used is of the form  dx  dx  at  os  n  where Q n is an imposed local phase velocity. The derivatives are with respect to time, t, and the coordinate perpendicular to the open boundary, s respectively. The variable % was used for the correspondent surface elevation as well as for the horizontal velocities associated with the barotropic and baroclinic modes. The local barotropic phase velocity used was Q 0 = Jg(rli  4.1.4  + H2)St/Sd  and the baroclinic Q I = Jg'rli  * H2/(Hi +  rI2)5t/Sd.  Forcing  The model was continuously forced at the southern open boundary by a barotropic wave at M 2 frequency travelling north. Platzman (1979) shows that a Kelvin wave describes the barotropic tide well.  Such a wave has the following surface perturbation at the  southern boundary: C = Co sin{ujt - 0Oe ( "" / f l ) where R is the barotropic Rossby radius in the deep part of the domain, w is the M2 frequency and the values of the amplitude £0  an  d the phase #; are based on observations  (Thomson, 1981). Different values of (o and 0; are used to test the sensitivity of the results.  87  Chapter 4. Numerical Model '  4.1.5  Baroclinic Signal  The currents generated in each layer, V1 = (t£i,i>i) and U 2 = (^2,^2), contain both a barotropic and a barocUnic component. The velocity in the upper layer is: Ux = U° + H 2 * (u bc , v b c )/(H 1 + H 2 )  (4.2)  and in the lower U2 = U ° - H 1 * ( u b c , v b c ) / ( H 1 + H 2 ) where U° = (Hi * l]1 + H 2 * U 2 )/(Hi + H 2 ) is the barotropic velocity vector and Ubc{x,y) = u1-u2  vbc(x,y)  = vx - v2.  (4.3)  can be thought as the amplitude of the baroclinic mode. As an aid to interpreting the model results, Ubc and Vbc will be presented.  Time series of these baroclinic velocity  components are tidally analysed at selected locations.  4.1.6  Internal Wave Energetics  An estimate of the baroclinic energy is made following Cummins and Oey (1997). The equation for the baroclinic perturbation energy can be written as -{KE  + PE) + V-J  = S  where KE = -p2U{ubc2  + vbc2)  is the perturbation kinetic energy density term and  P E  = ^  ~ ^1)^22  (4.4)  Chapter 4. Numerical  Model  88  is the perturbation potential energy. The last term in the left hand side in Eq. 4.4 is the divergence of the vector J(x, y) = -{p2 - pi)grj2H-{ubc, vbc),  (4.5)  which represents the flux of baroclinic energy given in units of [Wm" 1 ]. S represents a conversion of energy from one form to another. These equations are correct to second order, i.e. the surface elevation rji has been neglected in comparison to the interface T/2 as well as the advective terms. When the model achieves a steady state the equation 4.4 is satisfied and, if there is no numerical dissipation, S is zero and the vector J is non divergent. The horizontal velocity (ubc,Vbc) is given by Eq. 4.3 and the depth H by  H=  HlHz  The energy flux vector averaged over a tidal period (T = — ) , is: < J > =  1  r/0  fT  Jdt  (4 6)  -  By the use of a harmonic analysis the horizontal velocity components and interface displacement can be considered as  u'bc = ucos(a>£ — G) v'bc = ysin(u;£ — G)  7/2 = rj2 cos(u;i - 0„) where u, v are the velocity magnitudes at frequency w in a coordinate system (x1, y') such that the x' axis is oriented in the direction of the positive semi-major axis of the tidal ellipse, see Appendix A. G is the phase lag of maximum current behind maximum  Chapter 4. Numerical  Model  89  tidal potential (Eq. A.5) and T]2, 6V are the amplitude and phase of the interface. The components of the energy flux vector < J' > are < J' > = -^flfH(p 2 - p 1 )7/ 3 (ucos(^ - G ) , v s i n ( ^ - G))  (4.7)  <j'> = GM) The energy fluxes in the (x,y) coordinates of the model are: cos(V')  — sin( , 0) \  Jt  sin(-0)  cos(V')  *',  )  where tp is the inclination of the tidal ellipse, see Appendix A.  4.1.7  T h e two layer approximation  In chapter 3 representative density and N 2 profiles obtained from CTD cruises in Dixon Entrance were shown, Figures 3.3, 3.5 and 3.7. The vertical displacement, (4>n), and its derivative with respect to the vertical ((4>n)z = fa) for the first, i.e. n = 1, vertical mode calculated assuming constant depth are also shown. From those figures the assumption of two layers is weak since the averaged density profiles do not show an abrupt change at one specific depth. Since the value of (<f>n)z is proportional to the horizontal velocities (see Equations 3.23 and 3.24) the depth at which this function changes signs (the velocity changes direction) can be considered as the lower limit that divides the two layer system. Based on this depth and on the reduced gravity values (see Figures 2.9, 2.10), estimated from the data, a range of seasonal conditions as indicated in Table 4.1 is defined. Spring and summer are modelled and assumed to represent the periods (A) and ( 0 ) presented in the second chapter.  Chapter 4. Numerical Model  Season Spring Autumn Summer  Cruise 8411 8414 8510  90  g' [ms~2] 0.0147 0.0220 0.0235  Thickness upper layer [m] 100 100 50  Table 4.1: Seasonal variations of stratification. 4.2  Results  4.2.1  B a s e case  S t e p channel In this run a very basic topography with a (120,108) grid is used, see Figure 4.4. The grid spacing 5d is 4.6 km. In this case Dixon Entrance is modelled as a flat channel 300 m deep connected by a very steep continental slope to a flat (2000 m) open ocean. The width of the channel is 60 km. The amplitude and phase of the barotropic Kelvin wave are (o — 2.5 m and B{ — 331.9° at point number 1 in Figure 4.4. Its wavelength is ^Ko — 6242 km. The spring conditions are selected, that is g' — 0.0147 and Hi = 100m. The computational domain goes from 2.4 to 562.1 km in the x direction and from 2.4 to 506.1 km in the y direction. The model was run to a steady state. Contour plots of the interface height rj2 and amplitude of the baroclinic velocity, Ubc,Vbc, every two periods (i.e  r = 0.517 day) are shown in Figure  4.5 - 4.9,  4.10  - 4.14 and 4.15 - 4.19, respectively. In these plots the broken lines represent negative values, the continuous lines zero or positive values. This convention is valid for all the contour plots presented in this chapter. As the long barotropic Kelvin wave travels north, a complex baroclinic pattern that can be associated with a wave field emerges in the region near the mouth of the channel. Inside t h e channel  Chapter 4. Numerical Model  Interface  91  Height  at  1.035d  506.1  2.4 2.4  562.1  Figure 4.5: Contours of the interface height 772 in meters for the step channel case at t = 2r. The contour interval is 2.5 metres. Dashed lines represent negative values, solid lines zero or positive values.  Chapter 4. Numerical Model  Interface  92  Height  at  2.070d  506.1  562.1  Figure 4.6: Same as previous but at t = AT. The contour interval is 2.5 metres.  Chapter 4. Numerical Model  Interface  93  Height  at  3 . 106d  506.1  2.4 2.4  562.1  Figure 4.7: Same as previous but at t = 6r . The contour interval is 4 metres.  Chapter 4. Numerical Model  Interface  94  Height  at  4 . 141d  506.1  562.1  Figure 4.8: Same as previous but at t = 8r. The contour interval is 4 metres.  Chapter 4. Numerical  95  Model  Interface Height at 5.176 506.1  2.4 2.4  l^  v  V*  562.1  Figure 4.9: Same as previous but at t = lOr. The contour interval is 4 metres.  Chapter 4. Numerical  Model  u1 - u2 a t  96  1.035d  Figure 4.10: Contour of the baroclinic velocity Ubc in [mi" 1 ] for the step channel case at t — 2r. The contour interval is 0.025[ms - 1 ]. Dashed lines represent negative values, solid lines zero or positive values.  Chapter 4. Numerical Model  u1 - u2 a t  97  1 .553d  506.1  2.4 2.4  562.1  Figure 4.11: Same as previous but at t = AT. The contour interval is 0.025 [ms 1 ].  Chapter 4. Numerical  Model  u1  98  -  u2  at  3.106d  for  2  506.1  2.4 2.4  •J62.1  Figure 4.12: Same as previous but at t = 6r. The contour interval is 0.025 [ms -n  Chapter 4. Numerical Model  99  u1 - u2 at 4 . 141d 506.1  2.4 2.4  562.1  Figure 4.13: Same as previous but at t — 8r. The contour interval is 0.04 [ms : ] .  Chapter 4. Numerical Model  u1 - u2 a t  100  5.176d  506.1  2 .2i A  562.1  Figure 4.14: Same as previous but at t = lOr. The contour interval is 0.04 [ms 1}.  Chapter 4. Numerical  Model  101  v2 a t  1.035d  506.1  ^62.1  Figure 4.15: Contour of the baroclinic velocity Vbc in [ms - 1 ] for the step channel case at at t = 2 T . The contour interval is 0.02 [ m s - 1 ] . Dashed lines represent negative values, solid lines zero or positive values.  Chapter 4. Numerical Model  -1 - v2 a t 2.070d  102  for 2  506.1  2.4 2.4  562.1  Figure 4.16: Same as previous but at t = 4r. The contour interval is 0.025 [ms 1}.  Chapter 4. Numerical Model  •1 -  103  v2  at  3.106d  506.1  2.32 . 4  562.1  Figure 4.17: Same as previous but at t = 6r. The contour interval is 0.04 [ms - 1 ].  Chapter 4. Numerical Model  104  -1 - v2 at 4 .1 41 d 506.1  2.4 2.4  562.1  Figure 4.18: Same as previous but at t = 8r. The contour interval is 0.04 [ms 1}.  Chapter 4. Numerical Model  v1 - v2 a t  105  5.176d  506.1  2.4 2.4  562.1  Figure 4.19: Same as previous but at t = lOr. The contour interval is 0.05 [ms *]  Chapter 4. Numerical  Model  106  At the entrance of the narrow channel the long barotropic Kelvin wave generates, perpendicular to its direction of propagation, short baroclinic waves. These waves travel east, trapped to the southern wall inside the channel. The velocity pattern formed by the internal waves inside the channel (see Figure 4.12) can be associated with baroclinic Kelvin waves of wavelengths of approximately Xkch = 42 km, averaged ampUtude of 26 m and a phase velocity of approximately 0.9 m a - 1 . These values are consistent with those found in the previous chapter (see Table 3.1). By t = 4r these baroclinic Kelvin waves have already hit the end of the channel and turned around, see Figure 4.18. In this reflection process a long and weak Poincare mode is created as well. The sign of its presence is the weak asymmetry of the two Kelvin wave patterns. That is, there exists an amphidromic region along the central channel instead of a straight line of amphidromic points (Brown, 1973). As Taylor (1921) suggested, when he studied the reflection of Kelvin waves in a flat channel: the Kelvin wave progressing down one side of the channel, takes some time to cross the end of the channel before returning along the other side changing phase. In turning at right angles in order to cross the end of the channel, the wave produces a bigger rise and fall at the corners (see Figure 4.7 for t — AT). The presence of a wave structure associated with a Poincare mode can be appreciated in Figure 4.17 for t = 6r since the velocity Vbc is not zero along the channel. As it has been shown in the previous chapter, at this latitude and with the stratification observed in Dixon Entrance, Poincare modes can exist. The wavelength of this mode is as long as the length of the channel and its amplitude is approximately 8 m. This magnitude is much bigger than predicted by linear theory, see Table 3.2. O u t s i d e t h e channel At the same time that the baroclinic Kelvin waves are generated at the mouth of the channel, fronts moving westward, away from the discontinuity in the deep ocean are  Chapter 4. Numerical  Model  107  also generated (see figures at t = 2r). As time goes on, these transient waves carry energy with them, so for any finite region energy is lost through the sides by radiation of short Poincare waves until the energy left is that associated with a solution that tends to oscillate about an equilibrium state. Such equilibrium state consists of Kelvin waves propagating along the coast (towards the north) together with long cylindrical Poincare waves radiating into the deep ocean (see figures at t = lOr). The cylindrical Poincare waves can been seen in almost all the figures, since they have an east-west and north-south component, while their connection to the northward Kelvin waves is most clearly seen in Figures 4.15 to 4.19. The cyHndrical Poincare wavelength is approximately twice that of the Kelvin waves. The Kelvin waves travelling north have a wavelength of \k0 = 51 km with a maximum amplitude at the coast of 9 m. When the Kelvin waves inside the channel have gotten to the northern end, turn the corner and travel northwards trapped to the shelf coast (as Buchwald; 1968 described) they constantly feed the wave pattern already formed in that region. Since the system is constantly fed by long barotropic Kelvin waves, there are baroclinic waves constantly radiating from the channel, making the channel itself an internal tide generation region for the open ocean. The frequency of the baroclinic waves generated is mainly M 2 as can be seen from the figures since a new wave is generated each period. Harmonic analysis was performed at the locations marked in Figure 4.4. Outside the channel around 80 percent of the total baroclinic signal is associated with the frequency M2 and inside the channel from 60 to 70 percent. The leak of energy to some other frequencies, mainly M4 (period of 6.21 hrs), M 6 (4.14 hrs), represents the effect of the non-linear advection terms in the momentum equations (Holloway 1996).  At the entrance of the channel, where the generation of  internal waves happens, the M4 signal becomes significant and because of the absence of dissipation in the momentum equations its presence is felt wherever the internal waves  Chapter 4. Numerical  Model  108  propagate. The ratio of the signal at M 4 with M 2 goes from 0.12 to 0.2. In the actual data, the M4 did not have such a large contribution. Some reflection at the domain boundaries occurred, especially at the southern boundary. It can mainly be noticed in the interface elevation 772 contours which presented a north-south asymmetry. Such reflection is small compared to the Kelvin and Poincare wave heights and did not seem to affect the wave pattern formed in and near the channel. Three East-West sections in the entire domain of the height 772, the velocities Ubc and Vbc are shown in Figures 4.20, 4.22 and 4.21 respectively. Their location is indicated by arrows in Figure 4.4, near the northern wall of the channel, in the centre and near the southern wall. As expected, the waves in the oceanic region are weaker and faster than the waves inside the channel. The phase velocity for the first mode in the open ocean is C\ = 1.2 m s - 1 and inside the channel is Ci = 0.9 m s - 1 The wavelength of the Poincare waves in the open ocean measured along a line in the middle of the channel ( Figure 4.20), is approximately 100 km and the amplitudes are 5.4 m near the continental slope and around 0.9 m towards the open ocean. In the channel the Kelvin waves have wavelengths of 42 km and amplitudes of 20 m near the mouth to 48 m at the eastern end.  The  sharp peaks that appear in Figures 4.22 and 4.21 are associated with the corners at the entrance of the channel showing the inability of the model to produce realistic values at these points. From the middle section shown in Figure 4.22 it is noticeable that the Poincare wave inside the channel co-oscillates with the cylindrical Poincare wave outside the channel. The cross channel velocity Vbc reaches values of 0.18 m s - 1 at the eastern end and 0.11 m s - 1 at the western end of the channel. The along channel velocity Ubc near the walls of the channel reaches values of 0.33 m s - 1 whereas in the open ocean they have values of 0.06 m s - 1 near the mouth to 0.03 m s - 1 far from it. Steep continental slopes are well known (e.g. Baines 1982) for being likely places for  Chapter 4. Numerical  E-W  Model  profile  of Interface  109  Height  at  5.176d  Figure 4.20: Three East-West sections of interface height r)2, in meters, across the domain at t — lOr. Their location is indicated by arrows in Figure 4.4. Very short Poincare waves created when the first interaction with the channel occurred are still in the domain on their way out. The distance in the x direction is in meters.  Chapter 4. Numerical  E-W  2371.7  Model  profile  110  oful  -  u2  at  5 . 176d  562091.4  Figure 4.21: Same as previous figure but for Uj,c. The distance along the x axis is in meters and the velocity along y is in [ m s - 1 ] .  Chapter 4. Numerical  Model  E-W p r o f i l e  111  ofvl  -  v2  at  5 . 176d  562091.4  Figure 4.22: Same as before but for vbc. The distance along the x axis is in meters and the velocity along y is in [ms" 1 ].  Chapter 4. Numerical  Model  112  the generation of high amplitude internal waves. Maze (1987) simulated the generation and propagation of internal waves induced by a tidal wave which, in the absence of stratification, would propagate linearly over a shelf break. In the absence of stratification the barotropic tide produces velocities parallel to the edge of the shelf. Maze (1987) used a two layer model over a shelf edge and found that the amplitude of the internal waves reached its maximum at the shelf edge and decreased away from it on both sides. The internal wave amplitudes increase linearly with the thickness of the upper layer and decrease when stability increases. The generation process in the channel-shelf system is triggered by the interaction of the barotropic wave with the slope at the mouth of the channel and of course by the presence of stratification. Once the internal waves are generated they are diffracted by the presence of the channel. The diffraction of long waves by a narrow channel was studied analytically by Buchwald (1971). Buchwald (1971) used a Green function to study the barotropic linear case of waves originated at the mouth of a infinitely long channel connected to a semi-infinite ocean. In his case, the forcing at the mouth simulated a uniform barotropic tide height that went up and down at a certain frequency. He found that the diffracted waves consisted of Kelvin waves, inside the channel and along the coast, as well as cylindrical Poincare waves radiating into the open ocean at the same frequency. In our case the long barotropic Kelvin wave travelling north produces at the entrance of the channel a velocity gradient which initiates the baroclinic channel-diffraction process. M a n i f e s t a t i o n s on sea surface The diffraction process described by Buchwald does not occur barotropically, i.e when stratification is not taken into account, in Dixon-Entrance since the barotropic Rossby radius inside the channel ( around 456 km) is much bigger than the width (60 km) of the channel. In the two layer system the internal tides influence is felt at the sea surface. Figures 4.23  Chapter 4. Numerical  Model  113  to 4.26 show the surface elevation rji as well as the velocity vectors of the upper layer Uj at two moments, when t = lOr and t = 10.25r. The effect of the internal tide is minimal in the sea surface elevation whereas it is quite noticeable in the surface velocity (Sandstrom, 1991).  The barotropic tide (not shown) produces velocities parallel to  the edge of the continental slope outside the channel and longitudinal velocities inside the channel. The presence of internal waves modify the surface velocity pattern adding gradients associated with the internal wave system explained above. At some locations the velocity \J1 reaches values of 50% grater than the barotropic velocity. The magnitude of the surface height r)i differs from the barotropic surface heigh by only 0.05 to 0.25 cm. In Figures 4.23 and 4.24 the inclination of the r}\ contour lines outside the channel is associated with the pass of the long barotropic Kelvin wave 1/4 r apart.  A d d i t i o n of a bank In order to study the influence of a bank in the internal wave pattern the topography presented in Figure 4.27 is now used. The idea is to simulate Learmonth Bank, the bank located at the eastern end of Dixon Entrance (Figure 4.3). The same forcing and stratification as for the previous case are used: £o = 2.54 m, 8{ = 331.9°, g' = 0.0147 and Hi = 100m. Contour plots of the interface 7/2 and the velocities, Ubc,V(,c at t — 10r are shown in Figures 4.28, 4.29 and 4.30 respectively. The inclusion of the narrow bank at the entrance of the channel has modified the pattern of Vbc in the central part of the channel producing larger magnitudes than in the previous case (Figure 4.19 and 4.30). Near the bank values 50 % larger than in the previous case are reached (Figure 4.31 and 4.22). The Kelvin wave pattern associated with Ubc inside the channel has remained essentially unchanged ( see Figure 4.29). An explanation could be that the bank is located in the region where the magnitude of Kelvin waves is negligible, i.e beyond an internal Rossby radius (around 10 km) from the walls.  Chapter 4. Numerical  Model  Interface  114  Height  at  5.176d  506.1I-<T  562.1  Figure 4.23: Contours of the surface height rji in meters for the step channel case at t = lOr. The contour range is from -1.25 to -0.05 m every 0.05 m. Dashed lines represent negative values, solid lines zero or positive values.  Chapter 4. Numerical  Model  Interface  115  Height  at  5.305d  506.1  2.4 2.&  562. 1  Figure 4.24: Contours of the surface height r]i in meters for the step channel case at t = 1 0 . 2 5 T . The contour range is from 1.65 to 2.75 m every 0.05 m. Dashed lines represent negative values, solid lines zero or positive values.  Chapter 4. Numerical  Ve I  116  Model  vectors  at  5.176d  506.1  iiiii  2.4 2.4  562.1 0.334E>I MAXIMUM VECTOR  Figure 4.25: Velocity vectors in ms 1 of the upper layer U 1 blank spaces represent very small velocity values.  (ui.vi) at t = lOr. The  Chapter 4. Numerical  Model  Vel 506.1  iiii mm* lijMiftfi n? • • •  +t+  m  + * + * + + + *,* +  m  117  vectors  at  5.305d  Hm iii  •{I  ::!  •  VtiVU  il  titti" Z.'**4 miiiiti'ti-.  ": Jifil!  us i?  Hi!  m  i ti  tin  1  2.4 2  562.1  0.374E + I  MAXIMUM VECTOR  Figure 4.26: Velocity vectors in ms  x  of the upper layer \J1 = (ui,t>i), at t = 10.25r.  Chapter 4. Numerical Model  118  tiiii.iiiiiitini.iiii.niiiiiniiiiiiii-iinim.iliniiuiiiim-mniiiiiii.litfciii.iiiuiui.iiinii.iiii.iiw  >*"" " "  I I I M I I M . I H . M . I I M tlM-lllt  MI.IMM!-  IMII1-1  Figure 4.27: View from above and three dimensional view of the topography with a bank. The bank is 13.8 by 18.4 km at its base and 18.4 by 4.6 km at the top. The top is 15 m from the sea surface.  Chapter 4. Numerical Model  Interface  119  Height  at  5 . 176d  506.1  2.4 2.4  562.1  Figure 4.28: For spring conditions, contours of the interface rj2 in metres for the topography shown in Figure 4.27 at t = lOr. The contour interval is 4 m.  Chapter 4. Numerical Model  u1 - u2 a t  120  5.176d  506.1  2.4 2.4  -62. 1  Figure 4.29: Contour plot of the east-west velocity Ubc in [ms x] at t = lOr for the topography shown in Figure 4.27. The contour interval is 0.05 [ms - 1 ].  Chapter 4. Numerical  Model  0  121  - v2 a t  5.176d  506.1  2._4  2.4  562.1  Figure 4.30: Contour plot of the north-south velocity Vbc in [ms J] at t = lOr for the topography shown in Figure 4.27. The contour interval is 0.05 [ m s - 1 ] .  Chapter 4. Numerical  Model  122  Outside the channel the presence of the bank has increased the magnitude of Ubc and Vbc by about 10 %. The interface rj2 does not have noticeable changes (Figures 4.28 and 4.9). From the harmonic analysis performed at the same locations as in the previous case we found that at the locations very close to the the mouth of the channel, which now includes the bank, the currents at M 4 reach values of about 26 % of those at M 2 . Everywhere else the ratio of the signal at M 4 with M 2 goes from 0.12 to 0.2 as in the previous case. In this case, at some locations, the baroclinic velocities have larger magnitudes than the barotropic. The barotropic run was done for the model without stratification. Energy  fluxes  The baroclinic energy-flux vectors at the entrance of the channel, at M2, are calculated for the case without and with the bank. For the spring conditions the Rossby number is around 0.09 indicating that the non-linear terms can be neglected and Equation 4.5 can be used. The baroclinic energy fluxes provide insight into propagation and magnitude of the internal tide. The vertically averaged energy-flux vectors, calculated at the locations shown in Figure 4.4, for the step channel case, i.e. with no bank are shown in Figure 4.32 and for the case with the bank in Figure 4.33. For the case without the bank the largest values of energy flux are near the southern wall of the channel. Both the sharp bend and the difference in depth contribute to the generation of internal waves in that region. In the deep water the arrows are associated with the radiation of the cylindrical Poincare waves. The energy flux vectors presented in Figure 4.33 indicate that the presence of the bank produces a change at both sides of the channel entrance. The energy flux has a considerable increase in the magnitude, and change in direction, inside the channel. In both Figures, 4.32 and 4.33, the energy flux vectors are divergent, indicating that the term S in Equation 4.4 is not zero. In this case the values of S can be a sign of a)  Chapter 4. Numerical Model  E-W  p r o f i l e  123  ofvl  562091.4  Figure 4.31: Three east-west sections of the velocity of Vbc across the domain at t — lOr for the bank case. The location of the sections is the same as in Figure 4.22. The distance along the x axis is in meters and the velocity along y is in [ms - 1 ].  Chapter 4. Numerical  Model  124  4* 4«  *  *~~~.. . — i >  «r*  1000 W/m  70  80  90  110  100  gridx  Figure 4.32: Baroclinic energy-flux vectors at M2, in [Wm channel for the s t e p channel case.  1  ] , near the mouth of the  Chapter 4. Numerical  Model  125  gridx  Figure 4.33: Baroclinic energy-flux vectors at M 2 , in [Wm channel for the plus bank case.  1  ] , near the mouth of the  Chapter 4. Numerical  Model  126  Figure 4.34: Values of the baroclinic energy. energy exchange to M 4 , b) how important the numerical diffusion is for the baroclinic field or/and c) how unsteady the final state is. For the locations very close to the change in depth and near the bank the divergence can be explained because the nonlinear terms are bigger there than everywhere else. Many of the time series did not reach a steady state as time went by. An estimate of the divergence per unit area was made and in the worst case it reaches approximately 0.14 of the baroclinic energy flux. In the next three figures the sum of kinetic plus potential energy, at M 2 , for the case with the bank are presented. Figures 4.34 and 4.35 show values of the baroclinic and barotropic energy respectively evaluated from the same run and Figure 4.36 is the barotropic energy obtained when the model was run without stratification.  Although  the values of the baroclinic velocity at the frequency M2 are at some locations close to  V Chapter 4. Numerical Model  Figure 4.35: Values of the barotropic energy.  127  Chapter 4. Numerical  Model  128  :  ; 315 i  339  302  /  325 315  -  341  303  j - ^ _ _ -  324 313  303  '\  / 2jrT5l2  337  207  322  313 323  3371  313 302  ^re-213 208 V \  323 341 J  313 324  303 313  3 0  204 204  188o  332  302  :  205 206  *)  —see ^—eee See  -  337 Barotropic (KE + PE)*0.01 { N/m}  I  (without stratification)  70  80  90  100  110  gridx  Figure 4.36: Values of the barotropic energy for the unstratified case.  Chapter 4. Numerical  Model  129  or bigger than the barotropic velocity magnitudes, the baroclinic energy is about two orders of magnitude smaller than the barotropic energy. It ranges from 1 to 5 percent from the barotropic energy. The presence of the baroclinic scattering produces gradients in the barotropic energy that are absent in the unstratified case. S u m m e r case Using the topography with the bank, summer conditions were simulated.  They are  g' = 0.0235 and Hi = 50m. The forcing is the same as in the previous case. Plots of the interface height 772, the velocities Ubc and v\,c are shown in Figures 4.37, 4.38 and 4.39 respectively. Although there are some differences, the wave pattern described by u^ and Vbc for the spring conditions (Figures 4.29 and 4.30), still prevails (Figures 4.38 and 4.39). Due to the fact that the internal phase velocity c\ ~ yg'rli  * H 2 / ( H ! + H 2 ) has similar values in  both cases. The surface elevation 772, Figure 4.37, has a more complicated pattern inside the channel. The harmonic analysis performed at the same locations as in the previous case shows that inside the channel the ratio of the signal at M4 with M 2 is comparable to the spring case although there are places where the amplitude of the signal at M4 reaches values of around 30 % of the signal at M 2 . Therefore for these specific summer conditions the problem has become more non-linear than for the spring case.  4.2.2  R u n w i t h m o r e realistic t o p o g r a p h y  Assuming that the generation of internal tides happens in the south-western corner of Dixon Entrance their dispersion inside a non flat channel is simulated.  Bathymetric  features are modelled based on the map shown in Figure 4.3. On the western side of Graham Island and around the bank shallow regions, marked with (-), are adjacent to deep regions, marked with (+) and ( > ) . These abrupt changes are modelled as steep slopes. The topography used in this case is presented in Figures 4.40 and 4.41.  The  Chapter 4. Numerical Model  Interface  130  Height  at  5.176d  506.1  2.4 2.4  562.1  Figure 4.37: For summer conditions, contours of the interface r/2 in metres for the topography shown in Figure 4.27 at t = lOr. The contour interval is 2 m.  Chapter 4. Numerical Model  u1 - u2 a t  131  5.176d  506.1  2.4  i.e.  .562.1  Figure 4.38: Contour plot of the east-west velocity u^ in [ms *] at t = lOr for the topography shown in Figure 4.27. The contour interval is 0.04 [ms - 1 ].  Chapter 4. Numerical Model  v1 -  132  v2 a t  5.176d  506.1  2.4L 2.4  562.1  Figure 4.39: Contour plots of the north-south velocity v^ in [ms *] at t = lOr for- the topography shown in Figure 4.27. The contour interval is 0.04 [ms - 1 ].  Chapter 4. Numerical Model  133  ±lllllllllllll!]MlillllllllllllllMllilliII!llllllllMllllll!llli:!:!llll|[|llllllllllllllll|i|llll  (1)« miiiiiiiiiiiiiiiiiiiiiiniiHiii  inn  I  mil  II  IMIIIIIIIIIIII  Figure 4.40: View from above of the entire topography domain used for the more realistic case. The dots indicate the locations at which time-series were stored  Chapter 4. Numerical Model  134  Figure 4.37: The topography inside the channel. View from above and three dimensional perspective. The printed depths in the upper figure are in meters.  Chapter 4. Numerical  Model  135  grid in this case is 150 by 150 with a grid spacing of Sd = 2.6 km. The width of the channel is 57.2 km and the length is 145.6 km. The bank is 18.2 km wide at its base and 13 km at the top. The barotropic amplitude and phase are £o = 1-95 m and 8i = 252 at point number 1 in Figure 4.40. Note that this point is closer to the channel than in the previous cases. The summer conditions g' = 0.0235 and Hi = 50 m are used. The computational domain goes from 1.35 to 401.2 km in x as well as in y. Figures 4.42 and 4.43 show the structure of the velocity Ubc and Vbc at the beginning of the run and when the system has approached equilibrium. From Ubc at t = T, there is a Kelvin-like wave getting inside the channel through the narrow passage and from Vbc an intense cross channel flow develops near the bank. In this case the Kelvin waves have an approximate wavelength of 57 km, a phase velocity Ci of 1.2 m a - 1 , and they oscillate at M 2 frequency. When t = 8r, the clear two-Kelvin wave pattern found in a flat channel does not exist. The regions of positive and negative values of Ubc near the walls can still be associated with Kelvin waves; however short waves related to topographic features occur. Note that in this case the thickness of the upper layer, 50 m, is close to the isobaths of 40 m. Although these waves have both Ubc and Vbc components, they can be better appreciated in the contours of Vbc. Their wavelengths are from 5 to 20 km. From the harmonic analysis performed at the locations marked in Figure 4.40 it was found that there is considerable energy transfered to M 4 and M 6 therefore these topographic waves are mainly a product of non linear interactions.  4.2.3  C o m p a r i s o n w i t h data  A comparison with the baroclinic field obtained from the data analysis is made at the locations marked with dots in Figure 4.40. The idea is to represent the line described by the moorings QF1 — D15 indicated in Figure 2.1. The M2 baroclinic velocity component at each layer is obtained by subtracting the harmonically analysed barotropic component  Chapter 4. Numerical Model  136  §«©  Figure 4.42: Velocity U{,c at time t = r, in the upper frame and at t = 8r in the lower frame. The contour interval is 0.04 m s - 1 and 0.05 m a - 1 respectively.  Chapter 4. Numerical Model  137  Figure 4.39: Velocity v\,c at time t = r, in the upper frame and at t = 8r in the lower frame. The contour interval is 0.04 ms~l and 0.05 m s - 1 respectively.  Chapter 4. Numerical  Model  138  from the total. The barotropic component was obtained when the model was run without stratification. Figure 4.44 and 4.46 can be compared with Figures 4.45 (period A) and 4.47 (period 0) respectively. From these figures we can see that the baroclinic modelled velocity magnitudes are bigger in the summer than in spring. In the summer case the depth of change in density (Hi = 50 m) occurs, at some locations, very close to the depth of changes in topography (i, e where H = 40 m) producing high velocity magnitudes. The low values in the spring model results maybe associated with a deepening of the upper layer. At each season the model magnitudes of the cross channel velocity are similar to the longitudinal velocity. The phases at the locations chosen indicate that the wave pattern is complicated and undetectable from this data distribution. In the summer the baroclinic velocity magnitudes in the upper layer are similar to those found from the data analysis. At the locations D12, D10, D14 and D09 the upper baroclinic velocities represented 60% of the total velocity and were larger than the barotropic by 4%. In the spring case the baroclinic velocity magnitudes represented 20% of the total velocity and were everywhere smaller than the barotropic by about 50%. From the data analysis, Figures 4.45 and 4.47, we found that a prevalent pattern is present along the centre region of Dixon Entrance in both periods. The current magnitudes are similar in both periods. At each season the east-west and north-south baroclinic currents have similar magnitudes in the central region of Dixon Entrance. Because of the limitations of the model a direct evaluation of agreement with the data is not possible. Table 4.2 presents velocity magnitudes of observed and modelled results. The summer conditions are better modelled than the spring conditions.  Chapter 4. Numerical  QF1  D22  7  2  Model  139  east-west amplitude (cm/s) D10 D04 D09 D15 7  7  6  4  o o  / o o  /6 1  a  CD •o  east-west phase (degrees)  4  4  4  E  7  3  o o  '  / / !  o o  /  TO  TO  0  20  40  60  80  100  120  0  20  60  80  100  120  distance km  distance km  north-south amplitude (cm/s)  north-south velocity phase (degrees)  312 279  276 269  258  o o  O O  E  '  Q. CD  n  O  o  TO  o o  162  /  334  T-  E sz •Q  40  107  88  90  m  70 / / / /  TO  0  20  40  60  80  distance km  100  120  0  20  40  60  80  100  120  distance km  Figure 4.44: Spring modelled values of baroclinic velocities along an east-west vertical section. The location of these points is indicated in Figure 4.40. The thickness of the upper layer Hi is 100 m and g' = 0.0147. The two upper frames represent the amplitude and phase of the east-west baroclinic velocity and the two lower frames represent the north-south baroclinic velocity. The values were printed at the middle depth of each layer. The lines indicate the actual and the modelled water depth.  Chapter 4. Numerical  Model  140  east-west amplitude (cm/s) QF1  D12  D10 D04  D09  east-west phase (degrees)  D15  o o  I  o o  a 0) •o  o o  0  20  40  60  80  100 120  0  20  40  60  80  100 120  distance km  distance km  north-south amplitude (cm/s)  north-south phase (degrees)  o o  o o  a  a CD  o o co  <D  •o  o o co  0  20  40  60  80  distance km  100 120  o o co  0  20  40  60  80  100 120  distance km  Figure 4.45: DupHcate of Figure 2.17 from the data analysis. The two upper frames present the averaged values of the east-west velocity Ub and phase <pb, and the two lower frames, the averaged values of the north-south velocity Vb and phase 0b in a east-west section along Dixon Entrance for the first period [A). The lowest line is the water depth. The dots mark the location of the instruments.  Chapter 4. Numerical  O  QF1  D22  17  21  Model  141  east-west amplitude (cm/s) D10 D04 D09 D15  20 22  20  7  east-west phase (degrees)  /  91  T  -  E £  5  Q. CD  •D  12  18  57  26  o o  O O  5  5  E .c a  5  2  0) •o  O O  337  /  124 190  185 250  198  20  40  80  >56 o o  CO  1  0  20  40  60  80  100 120  0  100 120  distance km  distance km  north-south amplitude (cm/s)  north-south velocity phase (degrees)  12 288 o o E ' .c  o o E Q.  a CD  o o co  •o  0  60  20  40  60  80  distance km  100 120  293 316  302  265  /  %j/ 110  122 125  125  20  40  80  350 o o co  0  60  100 120  distance km  Figure 4.46: Summer modelled values of barocUnic velocities along an east-west vertical section. The location of these points is indicated in Figure 4.40. The thickness of the upper layer Hi is 50 m and g' — 0.0235. The two upper frames represent the amplitude and phase of the east-west barocUnic velocity and the two lower frames represent the north-south barocUnic velocity. The values were printed at the middle depth of each layer. The Unes indicate the actual and the modeUed water depth.  Chapter 4. Numerical  Model  142  east-west amplitude (cm/s) O  QF1 1  D12  D10 D04  D09  east-west phase (degrees)  D15 7 T  o o  O  o  a !  CD  T3  o o  O CO  0  20  40  60  80  100  120  0  20  40  60  80  100  distance km  distance km  north-south amplitude (cm/s)  north-south phase (degrees)  o o  120  o o  E  E  r. a  '  Q.  ©  •o  o  •D  o o  O O CO  CO  0  20  40  60  80  distance km  100  120  0  20  40  60  80  100  120  distance km  Figure 4.47: Duplicate of Figure 2.19 from the data analysis. The two upper frames present the averaged values of the east-west velocity Ub and phase </>{,, and the two lower frames, the averaged values of the north-south velocity Vb and phase &b in a east-west section along Dixon Entrance for the second period (O). The lowest line is the water depth. The dots mark the location of the instruments.  Chapter 4. Numerical Model  Obs. Obs. Mod. Mod.  u v u v  143  spring total range mean upper layer 6 - 22 18 4 - 22 18.25 0.2 - 10 5.5 0.2- 9 5.5  summer total range mean upper layer 5 - 22 13 3 - 24 16 0.2 - 21 17.8 17.6 0.3 - 23  Table 4.2: Velocities range and mean of the upper value from the model results and from the data analysis. Velocity are in cms-1 4.3  Summary  The model has many limitations.  Nevertheless it gives a number of features in the  internal tides that are consistent with the observations as well as a possible explanation, in terms of waves, of their generation and propagation.  It produces an internal tide  generation process, described by Buchwald (1971), which agrees with what Cummins and Oey (1997) presented for the open ocean region (Figure 1.8).  In a topography  where an abrupt continental slope is connected to a narrow channel, the propagation of a long barotropic Kelvin wave triggers a baroclinic diffraction process which gives rise to cylindrical Poincare waves travelling towards the deep ocean, Kelvin waves propagating along the coast and Kelvin waves inside the channel. The bank at the entrance of the channel seems to be responsible for an increase in the cross-channel velocity as well as an increase in the magnitude of the cylindrical Poincare waves in the deep ocean. The baroclinic energy i.e the sum of the kinetic plus potential, represents approximately 1 to 5 percent of the barotropic energy. Bathymetric features, when the stratification is strong, seem to play an important role in the increase of baroclinic velocity magnitudes. Using the parameters of chapter 3, for the base case topography, the model simulates the summer and spring conditions producing the same internal phase velocity. As expected, both give very similar wave patterns although it is important to mention that  Chapter 4. Numerical  Model  144  in the summer case the signal at M 4 becomes more important, which implies that the nonlinearities are significant. At some locations the baroclinic currents were found to be bigger in magnitude than the barotropic currents and represented more than 50% of the total, which agrees with some data results. Although inside the channel the Kelvin wave pattern always seems to be present, these internal waves are very sensitive to variations in the topography and new waves of short wavelengths are generated. The expected range of wavelengths for the internal tides ranges from 60 to 5 km inside Dixon Entrance, therefore the spatial resolution needed to properly follow their behaviour must be at least 1.25 km. Crawford et al. (1997) suggested that bathymetric features of order 1 km can drastically modified the semidiurnal internal tide. Many physical effects are missing in these idealized calculations: horizontal and vertical varying density variations, bottom friction, mixing, and many topographic features such as the presence of Clarence Strait and Hecate Strait. But the wave pattern described would have been difficult to detect in a more complicated model. The purpose of this study was to explain as much as possible with a reasonably simple model. I think it would be very interesting to continuing exploring the generation and scattering of these waves over the topographies presented in this chapter. The immediate next experiments could incorporate a more complex vertical density structure (e.g. adding another layer or a continuously stratified fluid) as well as to take into account frictional effects. Friction plays an important role in the dynamics of very shallow regions. The model was run without friction for simplicity. But is important to mention that dissipation of internal wave energy has been ascribed mainly to internal wave breaking and critical layer abortion. It would also be very interesting to localise other oceanic regions in the world similar to Dixon Entrance. Regions that because of the bathymetry, stratification and latitude  Chapter 4. Numerical Model  145  behave as a emanating source of internal tides towards the open ocean, as suggested by Dr. Steyn.  Chapter 5 Conclusions  Internal tides in, and near, Dixon Entrance at the semidiurnal frequency M 2 were studied in this thesis. Two complementary approaches were presented: a data analysis and a numerical study. The data used in this study consist of two periods of current-meter records, each approximately six months long, from several moorings as well as CTD data. The data were taken from April 1984 to June 1985. The extraction of the baroclinic velocity field from the harmonically analysed time series was done by subtracting from them modelled barotropic harmonic constituents. The barotropic signal used came from a finite element model developed by Foreman et al (1993) for the north coast of British Columbia. The results indicate that semidiurnal baroclinic tidal currents are very important in Dixon Entrance in the following sense. I.- the baroclinic velocity magnitudes were more than 50% of the total signal and at some locations, as large as the barotropic; I I . - the range of the east-west baroclinic velocity, u, in both periods, goes from 7 to 22 cms'1  and for the north-south baroclinic velocity, v, from 5 to 25  cms"1.  Along an east-west central region of Dixon Entrance it was found that: I I I . - a quasi-steady pattern is present in both periods; in the sense that the variability in time around the mean is very small,  146  Chapter 5.  Conclusions  147  I V . - both velocity amplitudes and phases showed a structure associated with a vertical modal structure; V . - both, u and v, have similar magnitudes. In Dixon Entrance there is a considerable seasonal density variation through the year. An explanation of the baroclinic quasi-steady pattern found can be that in a two layer system the thickness of the upper layer and the stratification change in such a way that they produced the same internal phase velocity c\. This can be seen from the formula cx ~ y/g'Hi *H2/(H1  + H2) where Hi and H2 are the upper and lower depths of the  layers and g' is the reduced gravity (see Table 3.1). The fact that u and v are similar in the central region of Dixon Entrance is consistent with the scattering of two baroclinic Kevin waves travelling in opposite direction plus a long Poincare mode in a channel. Unfortunately the distribution of the moorings, along the east-west section is too sparse to detect the variations in phase of the Kelvin and Poincare waves. Definite conclusions about the magnitude and variability of the baroclinic signal across Dixon Entrance, i.e. along north-south sections, were difficult to make since some of the instruments available were located in regions where bottom friction was very important. From the CTD distribution available it was estimated that the continental slope and Learmonth Bank are the two potential east-west generation regions of internal tides. In order to have a better understanding of how these internal tides originate and propagate in Dixon Entrance experiments of a numerical model were presented. The model used is frictionless, non-linear, two-layered, finite difference and forced by a barotropic wave over an idealized topography. The only modal structure possible is the first vertical mode. The main result that the model offered is a possible explanation, in terms of waves, of  Chapter 5.  Conclusions  148  the generation of internal tides. The generation process produced a wave pattern that was described by Buchwald (1971). In a topography where an abrupt continental slope is connected to a narrow channel, the propagation of a long barotropic Kelvin wave triggers a baroclinic diffraction process which gives rise to cylindrical Poincare waves travelling towards the deep ocean, Kelvin waves propagating along the coast and Kelvin waves inside the channel. The cylindrical Poincare waves travelling towards the deep ocean can be an explanation of what Cummins and Oey (1997) presented in the Figure 1.8. Inclusion of the sill (Learmonth Bank) at the entrance of the channel seems to be responsible for an increase in the cross-channel velocity as well as an increase in the magnitude of the cylindrical Poincare waves in the deep ocean. The generation region estimated from the CTD cast are the same as the ones obtained by the model. When Dixon Entrance is taken as a flat channel, the summer and spring conditions (which simulate the first and the second period of the time series respectively) give the same wave pattern, and similar baroclinic velocities are obtained.  The wave pattern  inside the channel consists of two Kelvin waves travelling in opposite direction as well as a long Poincare wave. When Dixon Entrance is taken as non flat and bathymetric changes are taken into account the summer conditions produced velocities much higher than the spring conditions. Then the wave pattern formed still has Kelvin waves together with new shorter waves related to topographic features. In this case, for the summer conditions, the features I, I I , V are reproduced but not I I I . The expected range of wavelengths for the internal tides ranges from 60 to 5 km inside Dixon Entrance, therefore the spatial resolution needed to properly follow their behaviour must be at least 1.25 km, as Crawford et al. (1997) noted.  Appendix A  A.l  H a r m o n i c Analysis  Foreman (1977) developed a complete and useful package of programs to analyse tides from time series data. The harmonic analysis applied to the current records in this thesis was done using one of his programs. The following explanation is based on the manual provided with the package, Foreman (1977), as well as from Foreman et al (1994). Barotropic tides originate from the gravitational forces of the sun and the moon acting on a rotating and orbiting earth. The oceanic response to these forces can be expressed as a linear combination of sinusoidal terms, each having a distinct amplitude, phase and frequency. Each sinusoid is referred as a constituent. Constituents with periods near 24 hours are called diurnal and those with periods near 12 hours are called semidiurnal. Harmonic analysis involves calculating the amplitudes and phases of a finite number of constituents that best fit a given time series in a least squares sense. From the current meter records the horizontal velocity at a specific depth was decomposed into north-south (v) and east-west (u) components. The northern and eastern directions are positive. The horizontal velocity VH is then taken as a complex signal by setting V H (<0 = Ui + ivi.  (A.l)  with M  Ui ~ [u0 + ^2 UjCOs(ujjti -  149  2-K^j)}  Appendix A.  150  M  Vi = [v0 + ^2 VjCos(wjti - 2-KOJ)} 3=1  where u0 and v0 are averaged values, Uj and Vj are the amplitudes, ujj the frequency and $ j and 6j are phases of each constituent j and U{ and i>; are the observations at time £». For simplicity the system is rearranged and new unknowns are defined. The harmonic analysis used, minimized the residuals eu{ and evi via least squares in the equations, M  eui = Ui — uQ — ^2[CXjcos{ojti)  +  SXjsin(ujti)]  i=i M  ev{ = Vi — v0 — ^2[CYjCos(u;ti)  + SYjsin(u>U)]  i=i  for u0, v0, CXj,  SXj,  CYj and SYj where CXj = UjCos(2n$j),  SXj = Ujsin(2-K$j)  and  CYj = VCOS(2TV6J)  (A.2)  SYj = Vsin(2Tr6j).  (A.3)  Although linear algebra only requires that the number of unknowns (2M + 1, i.e. amplitudes and phases) be no larger than the number of equations (one for each observation), short records and close frequencies in combination with noise and measurements errors result in the normal matrix equations being ill-conditioned. Foreman (1977) gives guidelines on which constituents should be included for a given record length. A nodal and an astronomical  argument  correction are applied modifying the least squares so-  lution. The former adjusts for long period variations in the forcing. The latter shifts the time origin from the centre of the analysis period to a universal Greenwich time. The horizontal velocity V H for a constituent can also be written as V H («) = Z+(t) + Z~{t) = - a + e i e + + 2 ™ + a~ei€~-2™  (A.4)  Appendix  A.  151  where  ,  rfCX  a ={  + SY\2  +  C—^—; 2  a  rfCX-SY\ ={ +  fCY-SX\\i }i  (—2—} /CY +  ~ {-^)  SX\\i  C — 2 — ; >" {  CX + SYl  {  CX-SYl  Over a time period, i.e  (27ru;)_1, the path of the vector VH traces out an ellipse, whose  respective major and minor semi axis lengths are Ma = a+ + a~  Mi = a+ — a~  and  and whose angle of inclination from the positive x axis is ip = (e + + e~)/2 radians. The maximum current occurs when the vectors Z+(t)  and Z~(t)  coincide along the major  axis. V H rotates counterclockwise if a+ < a~. The program used gives the tidal currents in terms of four ellipse parameters: the amplitudes of the major and minor semi-axes (Ma,Mi);  the angle of inclination of the  northern major semi-axis, ip; and the phase lag of the maximum current behind the maximum tidal potential, G, Figure A.l.  The Greenwich phase lag G is the phase  lag between the maximum local tidal current and the phase of the component of the corresponding tidal force at a conventional time origin, t0, on the meridian of Greenwich. G is independent of the time origin but depends on the time zone. The time zone selected was GMT. The expression that calculates G is  G = V(t0) - ^ A  (A.5)  where V(t0) is the astronomical argument. The physical meaning of G can be viewed as the phase lag between the applied astronomical force and its response. The components  Appendix  A.  152  Figure A.l: Tidal ellipse parameters: Ma, M; denote the major and minor semi-axes; tp is the angle of inclination and g is the phase lag from t = t0 to when the ellipse gets its maximum ampKtude. g is related to the phase lag of the maximum current behind the maximum tidal potential by G = V(t0) + g.  Appendix  A.  153  of the vector V R in terms of the ellipse parameters are: u \  I cos(ijj)  —sin{tj}) \ I Macos(ujt — G)  v J  y sin(ip)  cos(xjj)  (A.6)  J y Misin(uit — G)  The harmonic analysis was done by blocks. The block size was chosen to be 28 days, which was large enough to resolve M 2 from the S2 and N 2 constituents. The amplitudes of S2 and N2 ranged from 0.1 to 0.3 times M2. The constituent K 2 was inferred from S2 and a total of 32 constituents were included in the analysis. An example is shown in Table A.l. From each block the amplitudes and phases at the frequency M2 were retrieved and assigned to the middle date of the block. An overlap of 14 days was selected in such a way that a new time series of VH(£) at M 2 every 14 days was obtained, see Figure A.2. From the least squared fit standard deviations for the coefficients defined in Eq. A.2 and A.3 are estimated. At the frequency M2 such errors ranged from 1 to 10 percent of each coefficient.  A.2  Baroclinic field  Let's assume that the horizontal baroclinic velocity at M 2 can be expressed as Vb = Vt - V0,  (A.7)  where V 0 = U0cos(ut - 2TT$ 0 ) + iV0cos(ut  - 2TT0O)  (A.8)  - 2;r0t)  (A.9)  is the barotropic velocity and V t = Utcos(ujt - 27r$ t ) + iVtcos(ut  is the total horizontal velocity. Amphtudes UQ, VQ and phases $ 0 , #o are obtained from a barotropic model developed by Foreman in 1993 and Ut, Vt, $t and 9t come from the  Appendix  A.  NAME ZO MSF 2Q1 Ql 01 N01 PI Kl Jl 001 UPS1 N2 M2 S2 K2 ETA2 M03 M3 MK3 SK3 MN4 M4 MS4 S4 2MK5 2SK5 2MN6 M6 2MS6 2SM6 3MK7 M8 M10  154  FREQ (cycle/hour) 0.00000000 0.00282193 0.03570635 0.03721850 0.03873065 0.04026859 0.04155287 0.04178075 0.04329290 0.04483084 0.04634299 0.07899925 0.08051140 0.08333334 0.08356149 0.08507364 0.11924206 0.12076710 0.12229215 0.12511408 0.15951064 0.16102280 0.16384473 0.16666667 0.20280355 0.20844743 0.24002205 0.24153420 0.24435613 0.24717808 0.28331494 0.32204559 0.40255699  Ma cms-1 6.935 1.807 0.357 0.369 1.853 0.773 1.225 4.139 0.640 0.771 0.535 11.834 34.767 7.634 2.104 1.840 1.186 1.518 1.743 1.346 2.899 2.024 1.228 1.688 0.900 0.471 2.217 1.230 0.349 0.914 0.244 0.283 0.179  Mi cms~l 0.000 0.655 -0.119 0.273 0.807 -0.451 0.134 0.453 -0.347 -0.175 -0.342 -6.918 -12.104 -0.373 -0.103 0.488 -0.280 0.011 -0.834 0.049 -1.056 0.208 -0.129 -0.309 -0.402 -0.174 -0.749 -0.584 0.262 -0.469 -0.087 -0.105 -0.054  i>  '  degrees 17.6 128.6 40.5 121.7 169.2 160.2 177.4 178.9 136.8 117.3 140.4 7.9 29.9 24.4 20.6 12.1 2.1 146.9 24.5 175.2 148.8 140.8 150.8 174.4 1.3 164.0 144.1 2.9 170.8 12.9 161.8 94.7 16.1  G  Inference  180.0 316.0 323.1 95.1 183.3 115.9 224.3 INF FR Kl 222.8 4.9 5.6 308.0 312.9 329.7 351.1 355.0 INF FR S-2 96.3 192.6 357.4 234.3 199.8 127.2 192.4 175.8 252.7 21.8 250.1 88.2 340.0 138.9 323.5 17.7 332.8 232.0  Table A.l: Results from the harmonic analysis performed for station D04 at 52m at the location 54° 26', 132° 0' over the period of 6:00 am 3 1 / 5/84 to 6:00 am 28/ 6/84. The name and frequency of the 32 constituents are listed in the first two columns. The tidal current ellipses parameters are listed in the following four column, these are: the semi major axis, Ma] semi minor axis, Mi] the inclination, tp; and Greenwich phase, G. The sense of rotation of the current is determined by the sign of M;. When Mi < 0 it rotates clockwise.  Appendix  A.  original time series  155  1.0 hr.  A  111 new time series total signal at M2 frequency  I l  I l  A,  Aj  ai i ( A,  At  Aj  Ag  A7  A,  Harmonic analysis by blocks (28d) for M 2 frequency  >^=VT  0\ 01 0J 04 05 06 &1 0» \ / 14 days  Figure A.2: The procedure used to get the new time series of amplitudes and phases at M2 frequency.  Appendix  A.  156  harmonic analysis. If we set CXb = Utcos(2-K$t) - U0cos{2ir<f>0)  (A.10)  SXb = Utsin(2ir$t)  (A.11)  - U0sin{2it§0)  CYb = Vtcos(2ir6t) - V0cos{2irdo)  (A.12)  SYb = Vtsin(2irOt) - V0sin(2Tr60)  (A.13)  the baroclinic velocity can be rewritten as V b (i) = cos{u>t)[{CXb) + i{CYb)] + sin{u>t)[(SXb) + i(SYb)].  (A.14)  For each block the barocHnic east-west uu and the north-south v\> velocities are ub = UbCos(ujt — ${,)  (A.15)  vb = Vbcos(ujt - 6b)  (A.16)  where Ub = {CXb2 +  SXb2)1/2,  $ b = tan'1 {SXb I CXb), Vb = {CYb2 +  SYb2)1/2  and 6b = tan'1  A.3  {SYb/CYb).  T i m e series  The time series of the baroclinic constituents along a east-west line of moorings inside Dixon Entrance, for the first period, are presented in Figures A.3 to A. 10.  The  line is formed by the moorings Q F l - D12 - D10 - D04 - D09 - D15 and its location  Appendix  A.  157  block 1 QF1  D12  D10 D04  22  21 14  block 2 D09 D15  7  /  11  24  0  I  20  24  3  8  6  /  11  21 8  ? 12  7 10_ 0  § 7 10__  20  40  17  24 4  60  80  100  120  0  20  40  60  distance km  distance km  block 3  block 4  2  /  11  18  17  24 7  10  80  120  /  13  22 6  100  10  17 6  5  9___  7 20  40  19  26 6  60  80  100  0  120  20  40  distance km  block 5  block 6  /  14  24 6  18  22 4  10  7  7___ 60  100  15  120  /  10  1.4 5  distance km  80  24 8  18 5 40  60  distance km  80  100  120  40  60 distance km  Figure A.3: Time series of the east-west velocity Ub (cm/s) for the first period. The lowest line is the water depth. The name of the moorings is indicated in the first frame and the values are printed at the location of the instruments.  Appendix A.  158  block 7 QF1  D12  D10  D04  block f D09  18  19  3  16  30  7  9  15  5  10__ 60  8  distance km  block 9  20  16  4  block 10  7  /  14  33 10  14  6  10__ 40  60  80  100  120  distance km  Figure A.4: Continuation of previous figure.  /  Appendix  A.  159  block 1 QF1  D12  D10 D04  block 2 D09  267  241 229  244  D15  /  256  256 242  234  /  o  140  86  120  64  84  77  g o o m  58  325  306 299__ 0  323_ 20  40  60  80  58  20  120  40  block 3  block 4  236  172  / 104  90  160  100  120  /  222 74  7]  72 2 34_7_  354__ 20  8  B0  268 262  235  2 40  20  60  85  40  60  distance km  distance km  block 5  block 6 233  272 257  240  i  60 distance km  265 253  223  100  distance km  149  /  39  145  100  236  279 261  246 55  80  120  /  49  7fi  10  28 17__  20__ 20  40  60 distance km  Figure A.5: Time series of the phase east-west <f>b (degrees).  Appendix A.  160  block 1 QF1  D12  D10  D04  block 2 D09  /  267  241 229  244  D15  256  256 242  234  /  ? 140  E  86  120  64  84  77  M 58  1 306  325  299__ 0  323_ 20  40  60  80  120  20  40  60 distance km  block 3  block 4 236  265 253  223  100  distance km  /  80  120  /  222  268 262  235  100  §  58  E  i  160  7?  20  71  40  60  60  100  120  20  40  60  distance km  distance km  block 5  block 6 233  272 257  240  E  74  2 34_7_  /  8 85  149  39  145  100  120  236  279 261  246  55  80  /  49  8  H i  104  90  2 354__ 0  8  172  75 28 17__  80  •?  n  10 20__ 0  20  40  60  80  distance km  Figure A.6: Continuation of previous figure.  100  120  Appendix A.  QF1  D12  block 1 D10 D04 D09 D15  26  21  block 2  8  25  /  I 14  5  4  8  E  0  12___ J i ^ ^  20  40  24  20  3  60  1  7  ^~  _ ^  80  100  120  0  20  40  60  80  distance km  distance km  block 3  block 4  3  24  20  9  3  E  22  6  §  13  14__ 20  40  27  23  2  60  80  100  20  17  /  ^ " 40  60 distance km  block 5  block 6  6  /  -J  12 12___J£-^  120  /  14  14  /'  distance km  24  120  /  10  M 12  100  o  /  13  18  / /'  ?9  11__  /  11  22  H  12__  9  26  20  25  21  4  11  21  8  12  11  11.  10  9  12__  12__ 20  40  60 distance km  80  100  120  40  60  80  100  120  distance km  Figure A.7: Time series of the north-south velocity Vb (cm/s).  Appendix A.  162  block 7  QF1  1  D12  D10 D04 23  16  4  block?  D09  /  20  23 7  D15 24  16  5  9  6  y  19  28 10  1 1  U  UL_  8  7 11__  14__ 60  1  25  14  1  E  0  20  40  60  distance km  distance km  block 9  block 10  5  /  19  28  27  19  3  11  80  100  120  /  20  28  8  5  I' 10  1]  1 10  8 12__  14___ 40  60 distance km  80  100  120  0  20  40  60  80  distance km  Figure A.8: Continuation of previous figure.  100  120  Appendix  A.  163  block 1 o  QF1  D12  D10 D04  block 2 D09 D15  145  156 140  161 68  3  / 78  358  141  165 154  146  40  /  0  M °  *  30fi 273__ 20  §  325__ 286  271 257_ 40  60  80  169  40  60 distance km  block 3  block 4 127  170 165  137  20  distance km  125  /  82  92  100  120  114  175 173  142 8  80  /  357  t* g  330  322_  * 299  303  288_  290__ 20  40  60  80  120  0  40  60 distance km  block 5  block 6  128  /  88  80  o  8  86  20  distance km  178 172  147  100  235  340  78  120  132  186 182  154  100  /  329  8 336  9 313  30_6_ 0  341  8  319  309__ 20  40  60 distance km  80  100  120  0  20  40  60  80  100  120  distance km  Figure A.9: Time series of the north-south phase 8b (degrees).  Appendix  A.  164  block 7 QF1  D12  D10  D04  block i D09  122  183 192  164 223  69  D15  y 202  338  55  347 307  299_  300_ I  146  40  60  distance km  distance km  block 9  block 10  124  177 171  163  35  350  343  296 60  /  119  177 183  173  / 116  351  33.4  100  /  134  172 160  161 339  80  329  332_  311  306  30J__  304_ 20  40  60  80  distance km  Figure A.10: Continuation of previous figure.  100  120  Appendix  A.  165  is indicated in Figure 2.1. These time series have 10 simultaneous values in all locations, one every 14 days. The first frame with the title b l o c k l corresponds to the date 17 of May 1984 and the last frame blocklO corresponds to the 23 of August 1984.  A . 3.1  Variance  There is an average pattern in all the figures presented in the previous section. In this section this average pattern is removed and an analysis of the variances is done. In order to have an estimate of how the variations of the baroclinic field at M2 change in time and space a complex principal component analysis is used. The analysis was performed for the east-west section marked from the moorings QF1 to D15, see figure 2.1. This technique enables representation of the variations about the time averaged of the M2 amplitudes as a linear combination of empirical orthogonal functions (EOF) weighted by amplitudes named principal components (PC). Each element of this sum is called a "mode", (Preisendorfer 1988). A complex matrix using both the east-west and north-south baroclinic velocities zu = Ube~lib and zv = Vbe~l6b (see Equations A.15 and  A.16) is built. The time average is  removed and the scatter matrix is S = Z T Z* where Z is the matrix formed from the zu and zv of each instrument for each 28 days block. * denotes complex conjugate and T transpose operation. S is a Hermitian symmetric matrix. The EOF's are the set of orthonormal complex eigenvectors, E = {ej(x),j  = 1, ..p},  coming from the scatter matrix S, where p is 22, twice the number of instruments. The E O F are empirical because they arise from data and orthogonal because they are uncorrected over space. The vectors ej(x) can be thought as the unit-length directions in a  Appendix  A.  166  p-dimensional complex space along which the scatter of the complex data set has extreme values. The eigenvalues of S characterize the contribution to the variance contained in each mode. The amplitudes or P C , complex time series A = {a,j(t) = 1, ...n)} have the important property of temporal uncorrelatedness, and they carry information about the variance of the data set along the directions ej. n = 10 is the number of 28 day blocks used. The P C are obtained from: A = ZE*. And finally the original data set can be exactly represented in the form: Z=AE  T  = ^ai(i)ej(x).  (A.17)  For the first period we found that the first two modes, which are associated with zu and zv respectively, can explain together 74 % of the variance of the data set Z. e\ and e 2 are presented in Figure A.11 and a,\ and a 2 in Figure A.12. The structure of those modes can be seen by multiplying the corresponding amplitudes from the Figure A. 12 with the amplitudes of Figure A. 11 and the phases should be added. Figure A.12 represents the variations in time of the pattern presented in Figure A.11. Note that the variability in time is very small compared to the mean. The maximum velocity is 0.89 cms-1.  The phases oscillate from almost —180° to 180°. The phases  associated to both, Vb and Ub, change sign in July. Perhaps some variability is associated with the runoff during the summer. For the second period the first modes, associated with zu and Zb, can explain 68 % of the total variability of Z. Figures A.13 and maximum possible velocity is 2.22 cms-1  A.14 are for this second period.  The  and it happens in February where both phases  change sign in opposite direction. The phases oscillate, as well, from almost —180° to 180°.  Appendix  A.  QF1 o o  167  first eo > east-west amplitude (cm/s) D12 D10 D04 D09 D15 0.16 0.14  0.2 0.06  first eof > east-west phase (degrees)  /  0.26  0.23  0j3  /  /  a a)  ).18 (.24._0M^ 0  o o  20  274 o o CO  340  321 209  138  £ a •o  o o CO  o o  178  129  1ZD  67 >94_J29-  40  60  80  100 120  0  20  40  60  80  100 120  distance km  distance km  first eof > north-south amplitude (cm/s)  first eof > north-south phase (degrees) (J  0.17  0.15 0.37  <p.34  o o  217  321 266  0  1—  E r a  0.33  E  0.15 0.1  Q. CO V  o m 0.14  0J  65  o o  CO  0.21 (L 0  20  /  40  60  80  distance km  100 120  221  334  13__ 182 ?0_2_JU50  20  40  60  80  100 120  distance km  Figure A. 11: Spatial pattern of the first two modes ei, e2-  /  Appendix  A.  168  17-M en CD T3  31-M  14-J  •  28-J  12-JL  26-JL  9-A  23-A  6-S  20-S  •  o  * -^^  •  t  Q.  ^^^. • • "~"*^  E  ^^__~ • ^"^ • '—"  CO  •  o d  20  40  60  80  100  120  140  80  100  120  140  days  o o ©  w CO  o.  o o  20  40  60 days  Figure A.12: The time series of the amplitude and phase of the first principal components ai(t),a2(t) . The continuous lines are associated with Ub and $& and the dot-lines are associated with Vb and 6b  Appendix  A.  first eof > east-west amplitude (cm/s) D22 D10 D04 D09 D15  QF1  o o E  '  0.08  ).55  first eof > east-west phase (degrees)  / O O  0.07  ).24  0.18  0.19  E  /  66 234  JZ  344  226  €  /  171  95  Q.  a o o  0.Q7  0  20  40  60  80  100 120  0  40  60  80  100 120  distance km  first eof > north-south amplitude (cm/s)  first eof > north-south phase (degrees)  0.€6  0.€4  H6 °-65  0 0  0.H2  0.€3  0  /  /  E  a  Q.  •D  •o  ®  o o co  20  distance km  ).€5  0<41  0£2_ _Qi5  \AY 0  1£i-  i$y  1.3/  o o  344  O O CO  0Z-JUH  CO  344  126  249  >86 2S2  183  0 0 CO  /  383 161_  35JK 20  40  60  80  distance km  100  120  0  20  40  60  80  100  distance km  Figure A. 13: Same as Figure A. 11 but for the second period  120  Appendix A.  170  23-Nov  7-D  21-D  4-Ja  18-Ja  1-F •  15-F  1-M  15-M 29-MZ-85  t  CD  •  CO  •o  •  tw E (0  •^\  •  ^ * ~ - — •  •  __^ •  .  •  j -  20  40  60  80  100  120  140  80  100  120  140  days  O O (0  cc Q.  O O  20  40  60 days  F i g u r e A.14: S a m e as F i g u r e A.12, b u t for t h e second p e r i o d  Bibliography  ALLEN, S. 1988: Rossby adjustment England, pp. 206.  over a slope. PhD thesis, Cambridge University,  ARAKAWA, A. and V. R. LAMB, 1981: A potential enstrophy and energy conseving scheme for the shallow water equation. Mon. Wea. Rev. 109:18-36. BAINES P . G 1973: The generation of internal tides by flat-bump topography. Res. 20:179-205.  Deep-Sea  BAINES P . G 1974: The generation of internal tides over steep continental slopes. Philosophical Transactions of the Royal Society of London A277:27-58. BAINES P . G 1982: On internal tide generation models. Deep-Sea Res. 2 9 , 3A:307-338. BALLANTYNE V. A., M . G . G . FOREMAN, W . C. and R. JACQUES, 1996: Threedimensional model simulations for the north coast of British Columbia. Contin. Shelf Res. 1:1655-1682. BOWMAN, M . J . , A. and CRAWFORD, 1992: Spit Eddy in Dixon Entrance: Evidence for its existence and underlying dynamics. / . Mar. Res. 30:70-93. BROWN P . J . 1973: Kelvin-wave reflection in a semi-infinite canal. 1:1-10.  J. Mar. Res. 3 1 ,  BUCHWALD, V . T . 1968: The diffraction of Kelvin waves at a corner. J. Fluid. Mech. 31:193-205. BUCHWALD V . T . 1971: The diffraction of tides by a narrow channel. J. Fluid. Mech. 4 6 , 3:501-511. CANADIAN-HYDROGRAPHIC-SERVICE 1997: Canadian Tide and Current Tables. Barkley Sound and Discovery Passage to Dixon Entrance. Ottawa, Ontario. CRAWFORD  W.R., J.Y.  CHERNIAWSKY,  P. C. and M.  FOREMAN,  1997: Variability  of tidal currents in a wide strait as observed in drifter observations and numerical simulations. J. Geophys. Res. submitted. CRAWFORD W . R . , AND R.E.THOMSON 1991: Physical oceanography of the western Canidian continental shelf. Contin. Shelf Res. 11:669-683.  171  BIBLIOGRAPHY  172  CREAN, P . B . 1967: Physical Oceanography of Dixon Entrance, British Columbia. Bull. Fish. Res. Board Can. 156:66. CUMMINS P . F . AND LlE-YAUW O E Y 1997: Simulation of barotropic and baroclinic tides off Northern British Columbia. J. Phys. Oceanogr. 27:762-781. DODIMEAD, A . J . 1980: A general review of oceanogrphy of the Queen Charlotte Sound - Hecate Strait - Dixon Entrance. Canadian Special Publication of Fisheries and Aquatic Sciences 1574, Ottawa, pp. 248. DRAKOPOULOS, P . AND R . F . MARSDEN 1993: The internal tide off the west coast of Vancouver Island. J. Phys. Oceanogr. 23:758-775. FOREMAN, M . G . G 1977: Manual for tidal heights analysis and prediction. Pacific Marine Science Report 77-10. Institute of Ocean sciences, Sindney, B.C. Canada. F O R E M A N , M . G . G . , R . F . H E N R Y , R . A . W A L T E R S , AND V . A . B A L L A N T Y N E 1 9 9 3 : A fi-  nite element model for tides and resonance along the north coast of British Columbia. J. Geophys. Res. 98,C2:2509-2531.  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T H O M S O N AND  M.J.  WOODWARD  1992: Data record of current  observations volume XX. Dixon Entrnace, Hecate Strait and the west coast of the Queen Charlotte Islands 1983, 1984 and 1985. Institute of Ocean sciences, Sindney, B.C. Canada.  BIBLIOGRAPHY  173  JACQUES R. 1997: Modelling of the circulation of Northern PhD thesis, University of British Columbia, pp. 240.  British  Columbia  Waters.  LAMB K . G . 1994: Numerical experiments of internal wave generation by strong tidal flow across a finite amplitude bank edge. J. Geophys. Res. 99:843-864. LEBLOND, P . H . AND L. A. MYSAK 1982: Waves in the Ocean. Elsevier Science Publishers B. V., Amsterdam, pp. 662. MARSDEN, R. F . 1986: The internal tide on Georges Bank. J. Mar. Res. 44:35-50. MAZE R . 1987: Generation and propagation of non-linear internal waves induced by the tide over a continental slope. Contin. Shelf Res. 7:1079-1104. 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