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Observational and modelling studies of the Fraser River plume Stronach, J. A. 1977

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OBSERVATIONAL AND MODELLING STUDIES OF THE PHASER RIVER PLUME by JAMES ALEXANDER STRONACH M.Sc,  University of Saskatchewan, 1972  A THESIS SUBMITTED IN PABTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS INSTITUTE OF OCEANOGRAPHY  We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA December, 1977 copyright  James Alexander Stronach, 1977  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of B r i t i s h 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 my Department or by his representatives.  It  is understood that copying or publication  of this thesis for financial gain shall not be allowed without my written permission.  Department of  Physics  The University of B r i t i s h Columbia 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  Date  P  ii  ABSTRACT The Fraser River plume i s the brackish surface layer formed when the Fraser River discharges into the S t r a i t of Georgia.  Two  approaches to  are  understanding  discussed. I n i t i a l l y , out  in  the  the  dynamics  a  plume  carried  plume. These consisted mainly of CSTD p r o f i l e s and meters of the water  column.  surface current meter was i n s t a l l e d for 34 days at the  mouth of the Fraser River. field  the  a series of f i e l d observations was  current p r o f i l e s i n the upper 10-20 Also,  of  observations  are:  The  principal  the  conclusions  of  plume i s strongly sheared  v e r t i c a l and strongly s t r a t i f i e d ;  this  vertical  the  in the  structure  is  most apparent i n the v i c i n i t y of the r i v e r mouth, and around the time and  of maximum r i v e r discharge that  the  subsequently  water acquires  moving  (near low water i n the S t r a i t ) ;  outward  from  the  river  mouth  v e l o c i t i e s and s a l i n i t i e s appropriate to  the water beneath i t with length and time scales for t h i s change of order 50 km and 8 hours. The plume thickness varies between 0 and 10 meters; the s a l i n i t y varies from 0 to that of  the  water  beneath i t (approx. 25 %o) » n°l the difference between the plume a  velocity  and that of the water beneath i t varies from up to  m/sec to 0 m/sec, and i s t y p i c a l l y of order 0.5 m/sec over  3.5  much  of the plume area. Inspired by the f i e l d data, a model of the thin upper layer was  developed. The independent variables are the two components  of transport i n the upper layer, the thickness of the layer, and the integrated s a l i n i t y i n the upper layer. The upper  layer  has  been  tentatively  surface. The mixing across this  defined  interface  is  bottom by  an  of  the  isopycnal  modelled  by  an  upward  flux of s a l t water (entrainment), and a downward flux of  brackish water (termed depletion i n t h i s effects the  included  work).  f i e l d accelerations; the buoyant spreading pressure gradient  entrainment  of t i d a l l y  depletion mechanism frictional  stress  of  moving water  between  water with  and  field);  the loss  the plume  the  by the  momentum; the  the plume and the water beneath i t ;  forcing due to the b a r o c l i n i c t i d a l slopes; and the C o r i o l i s  force. Subsets of the f u l l clarify  certain  aspects  model of  equations  the plume  are examined, to  dynamics. Preliminary  results from the numerical solution of the f u l l model are  dynamical  i n t h i s model are: the l o c a l time derivative;  (including the effects of s a l i n i t y on the density  the  The  presented,  and  a  comparison i s made between the paths of  lagrangian trackers produced by observed discussed.  i n the plume.  eguations  Future  the model  and drogue  tracks  improvements to the model are  iv  TABLE OF CONTENTS  ABSTRACT ........ .... ..... ... ............ ..... ... . ........... i i LIST OF TABLES  V  LIST OF FIGURES ............... ......  vi  ACKNOWLEDGEMENTS ................... ................. ........xiv  1. INTRODUCTION  ... ... ............  2. FIELD OBSERVATIONS OF THE PLUME  1 9  3. A MODEL OF THE FRASER RIVER PLUME . ..,. . » V . ,y,,. . 4. AIDS TO INTUITION ABOUT THE PLUME  . .. . * V . . , - . 26 43  5. NUMERICAL MODELING OF THE FRASER RIVER PLUME ............. 71 6. CONCLUDING DISCUSSION  , . . 93  REFERENCES CITED . . . .... . . ....... .... ............. ........... , 97 APPENDIX ................. ... .......... ...... ...... .......... 103 TABLES FIGURES  ...... .114 ..  ,..,117  V  LIST OF TABLES TABLE I. HARMONIC ANALYSIS OF HIVES SPEEDS .................. 114 TABLE I I . HARHONIC ANALYSIS OF POINT ATKINSON ELEVATIONS. .. . .. .. .. . . . / . v . y,v«V-'VW»-. . . . . v. ..-115 TABLE III.,SCALE ANALYSIS OF TERMS IN THE EQUATION OF MOTION. . .... . .  ^ , »vv. ... ... ..... . .116  vi  LIST OF FIGURES Figure 1. Chart showing the S t r a i t s of Georgia and Juan de Fuca  117  Figure 2. S a l i n i t y d i s t r i b u t i o n i n the S t r a i t of Georgia and Juan de Fuca S t r a i t , 1-6 July 1968 .......118 Figure 3. S a l i n i t y d i s t r i b u t i o n i n the Strait of Georgia and Juan de Fuca S t r a i t , 4-8 Dec...1967 ...................... 119 Figure 4. Chart showing the Fraser River and central S t r a i t of Georgia  delta, 120  Figure 5. The daily Fraser River discharge f o r 1976, measured at Hope. .....................• ................. 121 Figure 6. Chart of the r i v e r mouth area, showing the location  of the current meter mooring  .......122  Figure 7. Smoothed current meter record ..................... 123 Figure 8.the low frequency  component of the  current meter record ....................• • . •  124  Figure 9. The t i d a l part of the current meter record ........125 Figure 10. Tidal elevations at Point Atkinson during the time the current meter was i n operation ................. 126 Figure 11. A reconstruction of the t i d a l signal from constituents obtained by harmonic analysis .................. 127 Figure 12. P r o f i l e s of S, T, sigma T and current speed for 1330 PST, Hay 8, 1976, at the current meter mooring .....128 Figure 13. The s a l i n i t y d i s t r i b u t i o n as a function of time at the r i v e r mouth, January 21 , 1975  129  Figure 14. Station positions and times, wind and tide for A p r i l 6, 1976 ............... . < ,. ....... .. ................130 Figure 15. S a l i n i t y section along l i n e h-r f o r A p r i l 6 1976 ................................ Figure 16. S, T, sigma T p r o f i l e s at station j , 1744 PST, April 6, 1976  «•  .....131 132  Figure 17. S, T, sigma T p r o f i l e s at station k, 1750 PST, A p r i l 6, 1976 .....................................133 Figure 18. S, T, sigma T p r o f i l e s at station 1, 1758 PST, A p r i l 6, 1976 .....................................134  vii  Figure 19. Station positions and times, wind and tide aoir A.pxrxX 15 ^ 1976 •».•••*•'•<'•••••'•••••?<•.•• *•.«-• «••••••«•••••«•• 135 Figure 20. S a l i n i t y d i s t r i b u t i o n along l i n e e-m and along l i n e m-r f o r A p r i l 15, 1976 ................  ..... 136  Figure 21a. S, T, sigma T p r o f i l e s at station k, 1710 PST, April 15, 1976  137  Figure 21b. S, T, sigma T p r o f i l e at station 1, 1713 !PST^ Aprx X 15 f 1976- * - * • » ^ » *'# • • #• *. • »•••.•: * # • • • # • • • * » *J 3 7 ;  ;  Figure 22. Station positions and times, wind and tide £ o IT A pxr x X 2 8 ^ 1 976 ••/•*«•••••• •: * •: v * • *,• • # • • • • • * Figure 23. S a l i n i t y section for A p r i l 28 1976, along a l i n e from a front at station g to station h  13  .....139  Figure 24. Station positions and times, H i n d and t i d e for June 4, 1976 ................. ........................... 140 Figure 25a. s a l i n i t y section along l i n e o-s, *3 XL D©. II ^ - 1 976 « *- • * *<••'• * m'm • '• • • * • • • • « • • » " • 4 •  •••>•• • • * • •  Figure 25b. S a l i n i t y section along l i n e x-t, June 4, 1 Figure 26. Station positions and times, wind and t i d e f o r July 3, 1975 Figure 27a. S a l i n i t y section along l i n e a - l , July 3, 1975  ...... 142 y,.... ...... 143  Figure 27b. S a l i n i t y section along l i n e p-s, 4?u.X y. 3 -f 1975 • * * *«• * • • * « • • ••'••••» 143 •  *  •  •  •  •  •  •  • • < • • • • / « • ' at  Figure 28. S, T, sigma T p r o f i l e s at station a, 06 42 PST, July 3, 1 975 .  144  Figure 29. S, T, sigma T p r o f i l e s at station e, -0133. • I?S T f J n X y 3 § 1975  * * •/ v  ^ « • **••••>•••••*•• 145  Figure 30. S, T, sigma T p r o f i l e s at station j , 1106 PST, July 3, 1975 ............. . . . . , y .......... , ....146 Figure 31. Sketch of the plume observed by a e r i a l survey, 0900 PST, July 2, 1975 147 Figure 32. S T, sigma T p r o f i l e s on the s i l t y side of "the., front, 1250 PST, July 2, 1975 ............. 148 Figure 33. S, T, sigma T p r o f i l e s on deep blue side of front, 1325 PST, July 2, 1975 ............................149 Figure 34. S, T, sigma T p r o f i l e s on s i l t y side of front, 0849 PST, July 4, 1975  150  viii  Figure 35. S, T, sigma T p r o f i l e s on deep blue side of front, 0853 PST, July 4, 1975  151  Figure 36. The evolution of a f r o n t , January 18, 1976 .......152 Figure 37. Station positions and times, wind and tide f o r July 23, 1975 ....................... i . . . . . . . . . . . . . . . 153 Figure 38a. Paths of drogues released i n region A, July 23, 1975 154 Figure 38b. Paths of drogues released i n region B, July 23, 1965 ...... ......................................... 154 Figure 39. S, T, sigma T p r o f i l e s i n region A, 1025 PST, J u l t 23, 1975  155  Figure 40. Speed p r o f i l e i n region A, 1015 PST to 1045 PST, July 23, 1975 ..................................156 Figure 41. Station positions and times, wind and t i d e f o r July 13, 1976 Figure 42. ,S, T, sigma T p r o f i l e s at station g, 1304 PST, July 13, 1976  157 15 8  Figure 43. Speed p r o f i l e at station g, 1317 PST, July 13, 1976 ........ ....................................... 159 Figure 44. Polar plot of v e l o c i t y vectors, s t a t i o n g, 1317 PST, July 13, 1976 .....................................160 Figure 45. Station positions and times, wind and tide for Sept. 17, 1976 ................. 161 Figure 46. S, T, sigma T p r o f i l e s at station a, 0646 PST, Sept. 17 1976  162  Figure 47. Speed p r o f i l e at s t a t i o n a 0655 PST, Sept. 17, 1 976 ..... ..........................,.. ................ 163 Figure 48. Polar plot of v e l o c i t y vectors, s t a t i o n a, 0655 PST, Sept. 17, 1976  164  Figure 49. S, T, sigma T p r o f i l e s at station j , 1512 PST, Sept., 17, 1976  165  Figure 50. Speed p r o f i l e at station j , 1458 PST, Sept. 17, 1976 .... ... ...........................,... .....* «. 166 Figure 51. Polar plot of v e l o c i t y vectors, s t a t i o n j , 1458 PST, Sept. 17, 1976  167  Figure 52. An i d e a l i z e d S t r a i t of Georgia, showing contours of surface s a l i n i t y ..........168  ix  Figure 53. ft s a l i n i t y section along AA*, Fig. 52, arid s a l i n i t y p r o f i l e s at three stations ..................... 168 Figure 54. The control volume, indicated by dashed l i n e s , surrounding the plume defined by the S=25°/oo contour  169  Figure 55. D e f i n i t i o n sketch f o r the eguations cl.6 IT x v€s (3. xu  C t i c i p ton  3  •  •••••  •*  • • • • • • • • • • • * * • • * ^16 9  Figure 56a. The orientation of the two c h a r a c t e r i s t i c s C and C_ with respect to the streamline d i r e c t i o n .......... 170 +  Figure 56b. The region of solution, f i l l e d up with intersecting c h a r a c t e r i s t i c s  ...... 170  Figure 57. Streamlines and l i n e s of equal thickness. from Bouse et a l , 1951 .................................  171  Figure 58a. Schematic diagram f o r the model of surfacing isopycnals .,....,,...........,..,....... .......... 172 Figure 58b. The thickness of the upper layer predicted by Eqn. 4.13 ......................................172 Figure 59a. The c o n t r o l volume used to obtain conditions at a strong discontinuity  ,  ...173  Figure 59b. Sketch of upper layer conditions used to obtain the integrated pressure term  ...173  Figure 60a. An element of the i m p l i c i t c h a r a c t e r i s t i c solution, Egn. 4.26 .........................  174  Figure 60b. A c h a r a c t e r i s t i c intersecting the front at (s,t) . . . . . . . . . . .  174  Figure 60c. Two c h a r a c t e r i s t i c s intersecting at a hydraulic jump .....,..............,.......,............... 174 Figure 61. The c h a r a c t e r i s t i c diagram for the kinematic wave solution  .175  Figure 62. The d i s t r i b u t i o n of u at t=l4, from the kinematic wave solution ..................................... 176 Figure 63a. The v e r t i c a l d i s t r i b u t i o n of s a l i n i t y for K=Z*/(0.22) .....  ........ 177  Figure 63b. The v e r t i c a l d i s t r i b u t i o n of s a l i n i t y f o r K=1 . . . . . . . . . . . . . . 177 Figure 64a. Depth of the upper layer as a function of time ...... Figure 64b. Total s a l t content of the upper layer as a function of time  .178 ,  178  X  Figure 65. An isoconcentration curve and velocity vectors f o r a turbulent plane j e t  ....179  Figure 66a. Schematic diagram of a turbulent jet ............ 180 Figure 66b. A section through the plume, showing the effect the choice of bottom s a l i n i t y has on flow through an open boundary .................................... 180 Figure 67. A t y p i c a l computational element of the numerical grid used i n t h i s research ..........181 Figure 68. The complete grid used f o r the sguare box model ..182 Figure 69. Flux out of the open ends of the l i n e a r model ....183 Figure 71. Velocity vectors after 100 timesteps of the square box model with non-linear terras .................. 184 Figure 71. The same as Fig. ,70, but at timestep 500  185  Figure 72. The same as Fig. 70, but at timestep 1000 ........ 186 Figure 73a. Influx and efflux f o r the model of Figs. 70, 71, 72 Figure 73b. Influx and e f f l u x f o r the model of Figs. 7 4, 75 ...... ... . . ...  187 ....187  Figure 74. Flow f i e l d calculated i n the same way as for Fig. 70, with the addition of horizontal eddy v i s c o s i t y .....188 Figure 75. Flow f i e l d calculated i n the same way as for Fig. 71, with the addition of horizontal eddy v i s c o s i t y ..... 189 Figure 76. The propogation of a hump of water out of the system ..... ..................,  ........190  Figure 77. The d i s t r i b u t i o n of velocity and elevation for Fig..74, along the l i n e from the r i v e r mouth the opposite s o l i d boundary .................................191 Figure 78a,b. Comparison o f elevation f i e l d s from a 0.33 km grid size model and a 1 km grid s i z e model  192  Figure 79a,b. Comparison of entrainment velocity from a 0.33. km mesh model and a 1 km mesh model .................. 192 Figure 80a,b. Comparison of u-velocity f i e l d s from a 0.33 km mesh model and a 1 km mesh model .................... 193 Figure 81a,b. Comparison of v - v e l o c i t y f i e l d s from a 0.33 km mesh model and a 1 km mesh model ........,..>........ 193  xi  Figure 82. Flow f i e l d produced by a model with variable river flow, t i d a l streams and elevations, C o r i o l i s force, and a constant Froude number boundary condition ...................... ............ 194 Figure 83. Flow f i e l d produced by a model with constant r i v e r flow, t i d a l elevations, C o r i o l i s force, f r i c t i o n constant of 0.005, and using ()""'F /3 i = o as a boundary condition ............ .....195 l  M  Figure 84. Flow f i e l d produced by a model i d e n t i c a l to that of Fig. 83, but with a f r i c t i o n c o e f f i c i e n t of .001 ....196 Figure 85. Velocity vectors f o r a model with density effects, t i d a l elevations, no C o r i o l i s force, and constant r i v e r flow ..................................... 197 Figure 86. The model of Fig. 85, 2 hours l a t e r ..............198 Figure 87. The model of Figure 85, 4 hours l a t e r ............ 199 Figure 88. The model of Fig. 85, 6 hours l a t e r .............. 200 Figure 89. The model of Fig. 85, 8 hours l a t e r  .........201  Figure 90. The model of F i g . 85, 10 hours l a t e r ............. 202 Figure 91. The model of F i g . 85, 12 hours l a t e r  203  Figure 92. S a l i n i t y d i s t r i b u t i o n at hour 186, corresponding to the flow f i e l d of Fig.,88 .................. 204 Figure 93. The s a l i n i t y d i s t r i b u t i o n corresponding to Fig. 91  at hour 192, 205  Figure 94. The d i s t r i b u t i o n of upper layer thickness at hour 186, corresponding to Fig. 88 ....................... 206 Figure 95. The d i s t r i b u t i o n of upper layer thickness at hour 192, corresponding to F i g . 91 ....................... 207 Figure 96. Drogue tracks produced over a 24 hour period by the flow f i e l d of Figs. 85, 91 ..............  .208  Figure 97. Drogue tracks produced over 24 hours by. the flow f i e l d of F i g . 98, using an augmented flow at the boundaries during outflow ........,...v......,........209 Figure 98. & t y p i c a l velocity f i e l d produced by a model with augmented flow at the open boundaries during outflow ............................, ..........210 Figure 99. Drogue tracks for drogues released shortly before.low water, (from Cordes, 1977) .......................211  xii  Figure 100. The discharge out of the open boundaries of a model with t i d a l elevations, variable density, constant r i v e r flow, and no C o r i o l i s force .................. 212 Figure 101. Drogue tracks produced when drogues were released at zero r i v e r flow, approaching high water .........213 Figure 102. Drogue tracks produced when drogues were released at half maximum r i v e r flow, during the ebb, when r i v e r flow i s increasing ................................213 Figure 103. Drogue tracks produced when drogues were released at maximum r i v e r flow, near the end of the ebb .....214 Figure 104. Drogue tracks produced when drogues were released at half maximum river flow during the flood stage of the tide ...................214 Figure 105. Velocity f i e l d produced by a model with depletion ................ ...... ,.......  ..,215  Figure 106. Drogue tracks produced over 12 hours by the model which produced the flow f i e l d of F i g . 105 Figure 107. Elevation f i e l d produced by the model with depletion ..................... . ..... ...., Figure 108. Elevation f i e l d at 72 hours, produced by the model with depletion  216 .... 217 218  Figure 109. Velocity f i e l d produced by the f i r s t r e a l geometry model, at time of maximum river flow  ...219  Figure 110. Distribution of upper layer thickness at the end of the ebb  ...220  Figure 111. Distribution of upper layer thickness at the end of the flood ........................................221 Figure 112. Drogue tracks produced over 12 hours by drogues released at maximum r i v e r flow  ...222  Figure 113. Normalized elevations, currents, and discharge used in the second version of the r e a l geometry model  ...223  river  Figure 114. Distribution of v e l o c i t i e s and surface slopes used in the second version of the r e a l geometry model .......224 Figure 115. Velocity f i e l d of the model with more r e a l i s t i c t i d a l forcing, at 8 hours ......................... 225 Figure 116. Velocity f i e l d Fig. 115, 4 hours l a t e r  produced by the model of  226  xiii  Figure 117. Velocity f i e l d produced by the model of Fig. 115, 8 hours l a t e r  227  Figure 118. Velocity f i e l d produced by the model of Fig. 115, 12 hours l a t e r  228  Figure 119. Velocity f i e l d produced by the model of Fig. 115, 16 hours l a t e r  ..229  Figure 120. Tracks produced by drogues released at hour 6, approximately high low water  230  Figure 121. Tracks produced by drogues released at hour 12, low high water .....................................231 Figure 122. Tracks produced by drogues released at 18 hours, at maximum river discharge, near low low water .232 Figure 123. Tracks produced by drogues t r a v e l l i n g i n the same flow f i e l d as those i n F i g . 122, but with a correction for v e r t i c a l shear i n c a l c u l a t i n g the drogue velocity .......23 3 Figure 124. A comparison of drogue tracks from Cordes(1977) and t h i s model  234  Figure 125. Distribution of upper layer thickness at hour 8 ....................  ..235  Figure 126. Distribution of upper layer thickness at hour 12 ..............................  236  Figure 127. D i s t r i b u t i o n of upper layer thickness at hour 16 ,. , ......................  237  Figure 128. Distribution of upper layer thickness at hour 20 .... ...................... ,... ................. .... 238 S figure 129. Distribution of upper layer thickness at hour 24 ..................................................  239  Figure 130. Schematic diagram of a possible extension of the upper layer model to 2 layers ........................ 240 Figure 131.,The d i s t r i b u t i o n of s a l t as calculated by a f i r s t order scheme ...........241 Figure 132. The d i s t r i b u t i o n of s a l t as calculated by a second order scheme ....................................... 242  xiv  ACKNOWLEDGEMENT Although many people help project,  I  would  like  in  in  carrying  particular  through to  a  thesis  acknowledge  the  assistance of Dr. P. B. Crean, and Dr. P. H. Leblond. Dr. Crean, of Environment Canada, was p a r t i c u l a r l y helpful i n guiding of  the  research,  numerical model. offered  a  both Dr.  positive  the  field  Leblond, approach  my  work and in developing the thesis  supervisor,  thank  A.  (OAS) f o r the use of the CSTD probe; and Dr. M. Miyake  The  always  to the work, p a r t i c u l a r l y i n the  writing of the thesis. I would also l i k e to  the use of the sonic current  much  B.  Ages  (OAS) f o r  meter.  f i n a n c i a l support of the National Research Council and  the University of B r i t i s h Columbia, through a scholarship i s g r a t e f u l l y acknowledged.  Macmillan  family  1  CHAPTER 1 INTRODUCTION One  of  the most  S t r a i t of Georgia F i g . 1,  striking  i s the  the S t r a i t  oceanographic features of the  Fraser  River  plume.  Referring  of Georgia i s the body of water separating  Vancouver Island from the B r i t i s h Columbia mainland. The River,  located  and  early  Georgia.  summer,  Particularly  during  the  late  i n times of large r i v e r runoff, the  Fraser River plume appears to be a layer of floating  Fraser  south of Vancouver, discharges into the eastern  side of the S t r a i t of spring  to  muddy  brown  water  on the dark blue S t r a i t of Georgia water; the plume i s  freguently bounded by a sharp colour discontinuity.; The boundary between the two water masses may also be-diffuse,' and there sometimes  weaker  are  colour d i s c o n t i n u i t i e s within the body of the  plume. Colour i s , however, a misleading indicator of the plume, because  sediment  does not necessarily sink at the same rate at  which the s a l i n i t y and momentum differences  between  the plume  and the ambient water decrease. For purposes of t h i s t h e s i s , the following two-part d e f i n i t i o n of the plume w i l l fee used: 1) the plume  i s the mixed  water formed when the Fraser  River discharges into, and mixes with, Georgia 2)  the  ambient  Strait  of  waterj i n order  to  be associated with the plume, t h i s mixed  water must retain a s i g n i f i c a n t identity  as  river  water, f o r  example i t must be fresher than some a r b i t r a r y maximum s a l i n i t y , for  example  25°/oo  {parts  per thousand).  arbitrary s a l i n i t y i s subject to season,  and  on  what  properties  variation, of  the  The value of t h i s depending plume  on the  one wants to  2  describe. . Because i t i s less dense than the ambient water, the plume i s a r e l a t i v e l y thin layer, f l o a t i n g on and interacting with the denser S t r a i t of Georgia water. There  are two  main reasons f o r studying the Fraser River  plume.-. As mentioned above, i t i s a  striking  feature  of the  S t r a i t of Georgia, and for that reason alone warrants attention. The other, more practical> reason i s that the plume plays a very important  role  S t r a i t of  Georgia,  environment  i n the flushing and general c i r c u l a t i o n of the which  i n turn  influence  the b i o l o g i c a l  of the S t r a i t . , B i o l o g i c a l properties of the Strait  influenced by the Fraser River induced c i r c u l a t i o n supply  of  upwelled  nutrients  attenuation  of sunlight  horizontal  advection  by  include the  to the surface suspended  and v e r t i c a l  layers; the  sediment; and  migration  the  of planktonic  organisms.  REVIEW OF THE OCEANOGRAPHY OF- THE STRAIT OF = GEORGIA At t h i s point we w i l l consider b r i e f l y the oceanography of the  S t r a i t of Georgia (Fig. 1) . Waldichuk (1957) carried out an  extensive  study  circulation  of the S t r a i t  theory  Further information carried  out a  discussed comes  below Crean  appeared  Most  of  the  i n t h i s paper.  and Ages  (1971),  who  series of hydrographic cruises over a period of  one year, occupying stations Georgia  from  of Georgia.  i n the entire  Juan  de  Fuca  system.  The S t r a i t of Georgia i s a fjord-type estuary - i t i s deep, has  a  freshwater  source,  i s connected  to the sea, and i s  3  strongly s t r a t i f i e d . The average depth i s about 150 meters, considerable  areas  are  deeper  than  200  meters.  d i f f e r s from simpler types of fjords i n that ( on  the  average about 30 km.  it  The  is  strong  tidal  mixing  connections with the  sea.  Strait  very  wide  ), and the major source of fresh  water, the Fraser River, i s near the main outlet of the where  but  occurs. ,• Further, At  the  northern  there end  system, are  is  two  found  a  complicated set of narrow channels, through which t i d a l currents reach 6 m/s.  At the south, the S t r a i t of Georgia i s connected  to  Juan de Fuca S t r a i t by another system of passes and s i l l s , where tidal  currents reach 1.5 m/s;  Juan de Fuca S t r a i t i n turn has a  free connection to the P a c i f i c Ocean., Figures Crean  and  Ages  (1971)  show  salinity  2  and  sections  3, along  the  July  and  centreline of the S t r a i t of Georgia for the months of December.: The  Fraser  River  is  from  seen to be a strong source of  s t r a t i f i c a t i o n i n the summer, and a weaker source i n the winter. One also sees the evidence f o r strong mixing i n Haro  Strait  in  the  south,  the  region  of  and Cape Mudge i n the north, with  reference to Figures 2 and 3, one wonders what proportion of the Fraser  River  proportion  water  flows  north  and  is  mixed  there;  what  flows south and i s mixed there; and what proportion,  while flowing north and south, i s mixed i n  the  middle  of  the  Strait? The  prevailing  winds are in general along the axis of the  S t r a i t - northwest and southeast there  ( Waldichuk,  1957 ),  However,  are frequent storms with variable wind d i r e c t i o n , and the  Fraser valley i n p a r t i c u l a r modifies the direction of the southern S t r a i t .  winds  in  4  More  than  70%  of the fresh water input to the S t r a i t s of  Georgia and Juan de Fuca comes from the Fraser ( Herlinveaux and T u l l y , 1961  ). The r i v e r discharges through a complicated system  of channels i n i t s delta, F i g . 4, but through  the  Main  Arm  80% of  the  flow  passes  which i s dredged to be about 400 meters  wide and 10 meters deep. Flow v e l o c i t i e s at the r i v e r mouth tidally  modulated, varying between about 0 ra/s and 3.5 m/s  a t i d a l cycle. Further, the r i v e r discharge  is  very  during  winter  to  about  10,000  m /s  thickness  is  about  10  central Strait of Georgia smaller  area  during  f r a c t i o n in t h i s winter.  Also,  during  of  water  which  then from Fig. : 2 and F i g .  summer,  During  volume  because  3  is 3,  meters, and i t covers most of the  winter.  plume  m /sec  during the spring-early  3  summer freshet. If one defines the plume to be fresher than, for example, 28 % o ,  over  seasonal.  Fig. 5, changing by an order of magnitude from about 1000  its  are  is  and  a  summer,  much  considerably  the fresh water  greater  than  during  the lower s t a b i l i t y of the plume i n  winter, i t i s more l i a b l e to be mixed away by storms, which  are  stronger i n the winter. There  are  very  strong  tides  in  the  S t r a i t of Georgia  ( Crean, 1976 ), and they exert considerable  influence  on  the  plume. F i r s t , the tide modulates the r i v e r flow, so that maximum river  discharge  occurs  near  e f f e c t i v e l y shut o f f at high replenishes  water,  water.  The  and river  the  river  water,  as  is it  the plume already existing, i s then acted on by the  tides i n the  Strait.  currents  the  in  low  Frictional  underlying  interaction  water  with  the  tidal  w i l l drag the plume to the  south during an ebb, and to the north during a  flood.  Velocity  5  differences between the plume and the water underneath give r i s e to  v e r t i c a l mixing, p a r t i a l l y inhibited by the v e r t i c a l density  gradient. The barotropic surface slope w i l l move  the  plume  up  and  down  the  not  Strait,  only  but  considerable cross channel forcing { due to the in  the  barotropic  equations  , about i n a complicated about  the  ).  to  also gives a  Coriolis  force  Consequently, the plume moves  manner, losing fresh water by  mixing  at  same rate { t i d a l l y averaged ) that i t gains i t from  the r i v e r discharge. Near the r i v e r mouth, where most  tend  there  is  the  available k i n e t i c energy, there i s intense mixing, but the  resulting mixed water i s fresh enough, river-directed  and  has  the  necessary  momentum, to form an outgoing plume. This mixing  produces an upwelling of nutrient r i c h s a l t water, which i s very important 1976 ).  from a b i o l o g i c a l As  the  river  point  water  of  view  < de  Lange  Boom,  proceeds outwards and mixes, the  s t a b i l i t y of the plume decreases, so that near the t i d a l  mixing  passes, mixing i s almost complete. The seaward transport of s a l t water in the plume i s compensated by a return flow of s a l t water beneath  the  plume,  as  in  a l l estuarine c i r c u l a t i o n s . Thus,  studying the plume w i l l aid i n explaining the movements of  deep  water i n the S t r a i t of Georgia. ,  PREVIOUS STUDIES OF THE PLUME Because  of i t s interaction with the general c i r c u l a t i o n of  the S t r a i t , the plume has recieved considerable study. Giovanda and Tabata ( tracking  drogues  which  1970  were  )  presented  released  near  the  results  of  the Fraser River  mouth, and followed for periods ranging from 2 to 33 hours.  6  Tabata  ( 1972 ) attempted t o identify the d i f f e r e n t  types  of water i n the plume from a e r i a l photographs. DE  Lange  Boom  ( 1976 ) described a mathematical model of  chlorophyll d i s t r i b u t i o n i n the plume,  with  the hydrodynamic  flow pattern being somewhat a r b i t r a r i l y prescribed. Cordes  ( 1977  )  described  the results  sophisticated drogue tracking procedure  of a  rather  i n the plume.,  STUDIES OF SIMILAR SYSTEMS Recently discussed  Long (1975b),  one  and  Winter  et a l  (1977),  have  layer models of fjords, s i m i l a r i n many respects  to the model discussed i n t h i s thesis. There also exists a thermal  considerable  body  of  physical  models  exist  1971).  Numerical  for these plumes, but they are not  applicable to the Fraser River plume. Usually there i s only independent  variable,  the distance  other horizontal dimension spreading are  coefficients  time-independent,  situations;  they  enclosing s o l i d  being  and and  taken  along the plume axis; the care  of  not adaptable  to  by empirical  time-dependent  are not adaptable to a system with p a r t i a l l y  boundaries.  the spreading  of  a  ), i n a river  series  plume  unbounded ocean. His model was time-independent, balance  one  p r o f i l e s of properties..The models  Takano { 1954a, 1954b, 1955 discussed  on  plumes due to cooling-water discharge from power plants  (Koh and Fan, 1970; Stolzenbach and Harleman, and  literature  of C o r i o l i s  of  issuing  papers, into an  and involved  a  force, hydrostatic pressure gradient, and  horizontal eddy v i s c o s i t y ; i t predicted a bending to the  right  7  of  the plume,  although the discharge remained symmetric about  the entrant direction of the r i v e r . eddy  viscosity  was  probably  The  choice  inappropriate  of  horizontal  because  the most  important f r i c t i o n i s l i k e l y the interaction of the plume  with  the underlying water, and not with i t s e l f and water to the sides as i n the case of horizontal eddy v i s c o s i t y . Wright and Coleman { 1971 ) discussed the M i s s i s s i p p i River plume,  which i s i n some ways s i m i l a r to the Fraser River plume,  the p r i n c i p a l difference being that strong  tides  are about  1/10 as  i n the Gulf of Mexico as i n the S t r a i t of Georgia. They  f i t a model developed by Bondar (1970), s i m i l a r to cooling-water plume models, t o some f i e l d data. One would conclude from work  that  buoyant  spreading  their  and entrainment are the two most  important forces governing a plume. Garvine 1977 )  has  (Garvine and Monk, 1974, Garvine, described  1974,  Garvine,  f i e l d work done i n the Connecticut River  plume, and has developed a model to explain the propagation the  leading  edge  of  a  plume., The Connecticut River i s much  smaller than the Fraser i n terms of discharge, have  of  but  the  plumes  many s i m i l a r i t i e s - strong t i d a l currents act on them, and  they both form d i s t i n c t fronts. In tne Garvine dynamics  are controlled  behind the front.  by  model  frontal  the s a l i n i t y and density p r o f i l e s  8  ISIS RESEARCH Although Garvine  lays great stress on fronts as c o n t r o l l i n g  the dynamics of plumes, the approach taken that  before  in  this  thesis  one can develop a model which includes f r o n t s , one  has to have a good model to describe the flow between the mouth  and  the  continuous methods used.  front,  upper  to A  fairly  i.e..a  layer.  increase  measurements  In  our  time-dependent this  thesis,  understanding  extensive,  but  river  model of a thin  two  complementary  of the plume have been  exploratory,  set  of  field  was carried out, using mainly a CSTD, and on a few  occasions a p r o f i l i n g current meter. Based on the f i e l d work, two  dimensional,  vertically  integrated  ( over  thickness )> time-dependent numerical model was model  includes:  the  effects  of  barotropic  streams; the e f f e c t s of v e r t i c a l mixing on the salt  and  momentum;  in  the  effects  and  a  plume The  t i d a l slopes and distribution  of  the C o r i o l i s force; the stress between the  plume.  This  importance  and  inertial  model i s exploratory, in that i t  cannot predict nature very accurately, but allows the  the  developed.  upper and lower layers; and the buoyant spreading effects  is  of  various  us  to  forces acting on the  plume. Further f i e l d work w i l l allow a better adjustment of mixing and stress in the model.  check  the  9  QM.hi.lIR 2 FIELD OBSERVATIONS OF TflE PLUil As mentioned i n chapter 1, various people have already made measurements of the Fraser River plume. There are a few s a l i n i t y profiles  available  ( Waldichuk,  1 9 5 7 ; de Lange Boom, 1976 ) ,  obtained by discrete sampling.  There are also  tracks  and Tabata,  available  1  but they  lack  velocity  and  { Giovando  information salinity  about i n the  observational aspects of t h i s useful  contribution  to  plume.  of  In  structure of developing  i t was  felt  the plume  investigate the v e r t i c a l structure of v e l o c i t y and the  plume,  drogue  1972; Cordes, 1977 ) ,  the v e r t i c a l  research,  knowledge  extensive  the  that  a  would be to salinity  in  with p a r t i c u l a r reference to location i n the S t r a i t  of Georgia and stage of tide. In t h i s chapter are described some of the f i e l d observations obtained using a continuous CSTD  recording  probe, ( conductivity, s a l i n i t y , temperature, and depth ),  and a p r o f i l i n g current meter. The observations were carried out in cooperation with P.B. Crean ( Ocean Environment  Canada  and  Aguatic  sciences,  ) at various times between January 1975 and  September 1976. Two boats were used i n t h i s f i e l d work. ,.In 1975 the  CSV  Richardson,  a 20 meter vessel, was used to obtain CSTD  p r o f i l e s . In 1976 the Brisk, an 8 meter launch was used. As the Brisk  required no operating crew other than the two s c i e n t i s t s ,  ( P.B. Crean operation  and  allowed  myself  ),  and  considerable  was  always  flexibility.  available, i t s Using t h i s boat,  which was equipped with radar f o r accurate positioning, CSTD  we did  p r o f i l e s , velocity p r o l i l e s , and measured positions of the  colour front associated with the plume.  10  A l l of the s a l i n i t y and were  calculated  obtained  from  sigma-T  profiles  presented  here  conductivity, temperature and depth data  with an Interocean model 513 CSTD  probe  and  a  chart  recorder. The conductivity and temperature traces were d i g i t i z e d on  the Hechanical:Engineering  d i g i t i z e r at U.B.C., and s a l i n i t y  and sigma-T p r o f i l e s calculated from them. The CSTD was designed to operate from 0 to 30 meters, so was ideal f o r our  purposes.  However, for a variety of reasons, the accuracy of the probe and chart  recorder system i s not r e l i a b l e . The accuracy varied with  time, depending on such factors as which boat the  ambient  temperature,  been operating that  and  particular  unless  being  day. For these  reasons, the  to be accurate to about  one i s comparing a series of successive casts.  Fortunately, the error i s not random, but systematic zero might be o f f s e t on the chart recorder ), and the  salinity  error  reveal  ( e.g., the  the s i z e of  i s much smaller than the s i z e of s a l i n i t y  variation i n the plume, so profiles  used,  how long the chart recorder had  s a l i n i t y p r o f i l e s can only be considered 0,5 % o  was  that  these  low  accuracy  salinity  almost a l l the pertinent s a l i n i t y structure of  the plume.  l i CONDIIIQIS hi THE RIVER MOUTH A. Currents at the river mouth Before proceeding to  a  discussion  of  the  salinity  and  velocity structure of the plume, l e t us f i r s t look at conditions at  the river  mouth.  It i s easier to interpret the s a l i n i t y  measurements i n the plume when we know the temporal relationship between the r i v e r discharge  and  the barotropic  tide  i n the  11  Strait.  As a preliminary stage i n acquiring this information i n  f u l l , the Canadian moored  Hydrographic  Service  installed  a  bottom-  surface current meter ( Neyrpic design ) at the mouth of  the Fraser Biver, Figure 6. The i n s t a l l a t i o n p o s i t i o n chosen  is  a compromise between putting the current meter i n an out-of-theway  location  safe  from  shipping,  and putting i t i n the main  channel, subject to the f u l l r i v e r flow. The current in  place for 34 days, from A p r i l 6 to Hay  time, the r i v e r discharge increased fron m /sec,  11, 1976. 1100  meter  was  During t h i s  m /sec  to  3  7700  ( Fig.,5 ), a t y p i c a l variation during the onset of the  3  spring freshet. At the end of each 10 minute i n t e r v a l the recorded  the  number  of  revolutions  meter  of i t s propellor { which  converts to a speed ), and the instantaneous  magnetic heading at  the end of the 10 minute i n t e r v a l . Because of the large mass steel  in  the current meter mooring, the magnetic d i r e c t i o n has  to be used with caution. During velocity,  of  when  the  flow  times  of  significant  outflow  was presumably along the l i n e of the  Sand Heads j e t t y , { 215° magnetic ), the current meter indicated a direction of 306° magnetic, approximately actual flow.,During  the high water part of the t i d a l c y c l e , when  the actual  surface  assumes  direction toward the j e t t y ,  a  perpendicular to the  velocity  approaches  zero,  and  sometimes  ( a change of 90° ), the  magnetically determined d i r e c t i o n changed by only 35°, to 270°.  In  order  to  convert  the  signal, the current meter record was magnetic  the  speed s i g n a l into a velocity treated  as  follows.  d i r e c t i o n s greater than 300°, corresponding  conditions, the velocity was channel,  about  For  to outflow  assumed to be e n t i r e l y p a r a l l e l  to  and directed downstream. For d i r e c t i o n s less than  12  300°, the v e l o c i t i e s were very small, scale  the  magnetic  variation  of  and 30°  treated  as  flows  up  the  was  {300°  variation of 180°, so that most of these were  it  low  decided  to  to 270°) into a  velocity  periods  channel.,Because the velocity  signal i s sampled at 1 hour i n t e r v a l s i n the subseguent harmonic analysis, i t i s impossible to reproduce the shape of record  very  well,  and  the  above  the  speed  treatment gives a smoother  velocity record than i f the small flows at high water were taken to  have zero component in the d i r e c t i o n of  the  river  channel.,  Assuming the above conversion from speed to v e l o c i t y , the record was  prepared  for  harmonic analysis, in order to determine the  r e l a t i o n between the r i v e r flow and the tides in the S t r a i t . signal was f i r s t band pass f i l t e r e d with  by  attenuating  The  components  frequencies higher than one cycle per hour, and lower than  one cycle per day. The signal resulting frequency  from  removal  of  high  ( p e r i o d s less than one hour ) components i s shown i n  figure 7. The A6A6A7/6»6»7, At=10 min, f i l t e r of Godin ( 1970 was  used  to remove the high frequency components. The operator  A6, for example, i s a running mean of length 6 time  )  series,  frequency  and  applied  to  the  A t i s the spacing of sampled points. The low  component  of  A24A2*IA25/24«24«25, At=1  the  signal  hour  was  filter,  obtained (  with  the  Godin 1970), and i s  shown i n Figure 8. The t i d a l band of frequencies was obtained by subtracting smoothed  the  signal.  low  frequency  F i g . 7,  signal,  Figure  long  enough,  it  from  the  and i s shown in Figure 9. This t i d a l  record was then harmonically analysed. Because not  8,  was  impossible  the  to  constituents, for example the group K1, P1, S l . , I t  record  separate was  was some  assumed  13  that  the r e l a t i o n s  between the amplitudes and phases of these  constituents i n the river speed record were the same as for the same  constituents  observations of permanent  obtained  from  the elevation  an  at  analysis  Point  of one year of  Atkinson,  a  nearby  tide gauge. The results are shown i n Table 1, and the  results of harmonically analysing the observed elevations  Point  Atkinson  for the same period of time, (Figure 10), are shown  in Table 2. Note that the Point Atkinson elevations did not have the low frequencies f i l t e r e d out, so have a s i g n i f i c a n t mean well  as  MM  and MSf constituents. F i n a l l y , the velocity record  was reconstructed from the harmonic analysis, and Figure  11.  shallow  i s shown in  I f the surface current consisted only of t i d a l and  water  identical.  as  constituents.  That  they  Fig. 9  and  are not indicates  Fig.11 a  should  be  more complicated  situation. The following  observations  about  the harmonic  analysis  should be made. 1) .  We  would  like  to  know  when  the maximum discharge  occurs, r e l a t i v e to the water l e v e l s i n the S t r a i t . , Comparing the  records  of surface  current  Atkinson, maximum current occurs  with the sea levels a t Point about  0.7  hours  before  low  water at Pt. Atkinson. Using the phases from tables 1 and 2, f o r the after  M2  tidal maximum  difference,  constituent, ). Maximum  low water occurs at cot=34 0° { 180° current  occurs  at <^>t=312°.  The  28°, corresponds to 1 hour. For the S2 constituent,  low water occurs at <*>t=358°, and maximum current at <ot=337°, for a difference of 0.7 hour. 2) . I didn't consider  the L2  constituent  i n the above  14  calculation of the time of maximum r i v e r flow, even though i t i s the  second strongest diurnal constituent, because i t s amplitude  and phase are subject to some doubt. The L2 probably comes the  interaction  2M2-N2,  i d e n t i c a l i n freguency significant  a to  diurnal L2  from  shallow water constituent  ( Godin,  1970 ) .  There  is a  change i n the L2 amplitude over the 18.6 year nodal  period. The harmonic analysis program corrected the apparent amplitude  using  this  factor,  L2  whereas i t should probably have  applied the correction for 2M2-N2, a much smaller correction. 3) Although discharge  there  while  was  a  seven-fold  increase  of  river  the meter was i n operation, the mean current.  Fig.8, showed only a s l i g h t increase. One must conclude that the thickness of the outflowing layer increased over the 34 days the meter was i n s t a l l e d . 4) . There were two events during which the velocity  was  occurred  about  explanations  abnormally day  low  measured  water  f o r a day or two { Fig. 7 ). One  106, the other  day  129. Two  possible  suggest themselves; one i s meteorological forcing,  the other i s the presence of debris caught in the current propellor.  On  meter  day 105-106, there was a storm, with NW winds of  up to 52 mph. This would c e r t a i n l y affect the surface flow, probably  the measurement of i t . However, day 129 ( May 8 ) was  calm. By fortunate doing  and  coincidence.  Dr.,P. Crean  happened  to be  current p r o f i l e s while t i e d up to the current meter bouy.  The measured surface current, 50 cm/sec around afternoon,  noon  and  early  agrees with that measured by the moored meter. Thus,  for t h i s period, there appears no ready explanation i n terms of mteteorological f o r c i n g , or meter malfunction.  15  B. VERTICAL CURRENT AND Weather  and  SALINITY STRUCTURE  other work never seemed to permit a very good  measurement of the p r o f i l e s of current at the r i v e r mouth during the time the surface current indicates  the  salinity  meter  and  was  current  in  place.  speed  (Julian day  129),  PST. P o s i t i v e speeds refer to an outflow, and since only was  measured  by  the  General Oceanics model 2031, with  12  structure present at the  current meter mooring, Fig. 6, on Hay 8, 1976 at 1330  Figure  large  orienting  current  meter ( a propellor-type  r i g i d l y attached to a lead "  fish"  f i n s and vanes ), the zero crossing at 5  meters i s only inferred from the minimum in speed at that depth. Figure 13 indicates the evolution of the s a l i n i t y structure oyer 10 hours at Sand Heads on January 21, 1975. in the inset diagram indicates, fraction  of  fresh  as  a  function  Like  the  of  time,  the  water in the water column, considering pure  s a l t water to have the s a l i n i t y of the cast.  The dashed l i n e  surface  current,  deepest  water  in  each  the fraction of fresh water  reaches a maximum shortly before ( about 2  hours  before )  low  water.  2A. .SALINITY AND The  DENSITY STRUCTURE OF THE PLUME  salinity  information  obtained i n t h i s research i s i n  general agreement with that reported by laldichuk (1957) and Lange  Boom  (1976). I t was however obtained  with a continuously  recording CSTD as opposed to discrete bottle casts. As apparent  from  structure very temperature  the  profiles  closely  structure  follows having  in  this  the  de  section,  salinity  the  will  be  density  structure,  the  a r e l a t i v e l y minor e f f e c t on the  16 density i n the upper layer. Ap_ril 6j_ 1976 This i s the f i r s t day the current meter The  Fraser  discharge  at  was  in  position.  1100 m /sec, and winds were  Hope was  3  l i g h t . Figure 14 indicates the positions of stations , by  Figure 15 i s a plot of s a l i n i t y contours along the  letters.  l i n e h-r. The t i d a l curve stations  were  done  ( inset, Fig.14 ) ,  colour  shows  a  very  weak  16,  these  colour  front  change in the water) i n the v i c i n i t y of station  k, station k being on the fresh or eastern side Figures  that  on a r i s i n g tide after a large f a l l . It i s  interesting to note that we passed (slight  labelled  17,  18  and  show  of  the  front.  the S, T, sigma-T p r o f i l e s for  stations j , k, 1. There i s a s l i g h t i n d i c a t i o n of fresher at  water  the surface at station k. F i g . 1 7 , and a d e f i n i t e freshening  of the surface layer at station 1, F i g . , 1 8 . This becomes  thicker  and  fresher  as  one  towards the r i v e r mouth at Sand Heads. Figs. 1 6 , 1 7 ,  curves,  layer  proceeds from station 1 In  a l l three  salinity  and 1 8 , the pycnocline i s found at about  He should also notice that  S=25 % o .  surface  as  one  proceeds  outward  from Sand Heads, the isopycnals gradually r i s e to the surface. A^ril 15  A  1926  This  s e r i e s of CSTD's was taken along a rather  path. Fig. 1 9 . The l i n e g-h-i-j-k-l^m-n of  a  colour  front.  represents the  the  front,  boundary  The l i n e s f-g and n-o-p-g represent paths  through the plume. Stations g , j , l , and m are on the of  complicated  plume  side  and stations h,i,k, and n are on the s a l t water  side of the  front.  plotted  Figure 2 0 , along with a continuation along m-o-p-g.  in  The  salinity  along  line  e-f-g-j-l-m  is  17  We note that these p r o f i l e s were done on a r i s i n g t i d e , and that there had been a 10 hour period of strong northwest winds  prior  to the measurements. One notes that the p r o f i l e s along the front ( g#j*l# along  m  ) are f a i r l y similar, and that comparing them t o those  line  m-o-p-g-r,  the  plume  i s considerably fresher and  thicker i n the inner regions than around the edges.. Figures 21a and 21b i l l u s t r a t e stations  the  salinity  profiles  k and 1 respectively, k being outside the plume, and 1  being adjacent to k (about 50 m away), but on the plume side the  at  of  front. We note that f o r the p r o f i l e in the plume, F i g . 21b,  there appears to be a three layered structure to the p r o f i l e - a surface brackish layer about 0.7 meters t h i c k , lying on top of a layer of intermediate s a l i n i t y about 2.5 meter thick,  lying  on  top of the almost homogeneous lower water. .,Atoril 28  x  Referring  1976 to  F i g . 22  f o r position information. Figure 23  shows a s a l i n i t y section taken from a front at toward  Sand  station  g  back  Heads, along a l i n e g-h. I t i s interesting to note  how nearly horizontal the isohalines are, and how  the  salinity  varies continuously in the v e r t i c a l through the plume. June 4 The  A  1976  p r o f i l e of s a l i n i t y along l i n e o-p-q-t-s { Fig 24 ) i s  shown in f i g . 25a. The winds were l i g h t and  the  profiles  were  done about an hour after low water, at which time the r i v e r flow i s quite strong. The d a i l y discharge at Hope was 6,200 m /sec, a 3  fairly  high  value.  We  see  that near Sand Heads ( station t,  located 0.5 nautical miles from Sand Heads into the  water  the  Strait  ),  i s almost e n t i r e l y fresh and at station g, located 1  18  nautical mile from Sand Heads, the zero at the surface to 27 Three  °loo  possibilities  salinity  varies  from  near  at 7 meters.  suggest  themselves as causes for the  r i s e of isohalines as one proceeds downstream. One i s that there i s extensive l a t e r a l spreading of the flow near the river mouth. However, the drogue tracks of Cordes ( 1977 ) suggest is  very  little a  discharge.  lateral  second  spread  reason  of  would  flow  at  that there  times  of  high  be that there i s extensive  mixing at the river mouth. While doing the p r o f i l e s at the r i v e r mouth On June 4 the water surface indicated considerable activity-  large  diameter  upwelling  mixing  regions, and considerable  patchiness i n colour. The third possible reason i s that the s a l t wedge { Hodgins, 1974 the  toe  of  the  ) has been flushed out, and we are  wedge  outside  the  seeing  r i v e r mouth. However,the  flushing out of the wedge i s intimately connected  with  mixing,  so the second and t h i r d reasons mentioned above are related. In  order  to  draw more detailed isopycnals near the river  mouth, we did a series of CSTD casts Sand  while  drifting  out  from  Heads, as shown in Fig 25b. Note that the horizontal scale  i s expanded in Fig. 25b. Station g of Fig.,25a i s location  as  station  separated by 0.5  x  of  Fig.  25b.  at  Stations  nautical miles, or about 1 km.  The  the  x  same  and t are isopycnals  in Fig. 25b a l l r i s e uniformly as one proceeds outward. ^ a l l 3,1976 Referring  to  Figure 26 f o r geographical locations, figure  27a shows a plot of s a l i n i t y along l i n e a-b-c-d-j-1, 27b  and  figure  i s a plot of s a l i n i t y along s-q-m-o-p. July 1975 was a time  of high runoff,( 6,800 m /sec on July 1 ), and 3  one  observes  a  19  considerable  amount  of  fresh  water  i n these p r o f i l e s . It i s  interesting to observe the changes in the actual p r o f i l e s as one proceeds downstream. Figure 28, 29, and 30 profiles  used  show  much,  station  c  of  the  to draw Fig. 27a { stations a, c, and j ) . Going  from station a to station c, the upper layer very  three  does  not  thicken  but becomes approximately 5 % o s a l t i e r . Going from to  station  j , the  upper  layer  almost  entirely  disappears, and becomes almost as salty as the lower layer. This erosion  of  the  upper  layer shows up i n the s a l i n i t y sections  Fig. 27a, and 27b. fts on A p r i l 6 1976, Fig.15, as away  from  the  one  r i v e r mouth, one observes the fresh water layer  getting thinner. More important, perhaps, one notices surface  salinity  increases,  play an important role i n subsequent  A  the  plume  will  chapters.  THE PLUME  feature  of  the  plume frequently commented upon i s the  sharp colour front, usually found on the the  that  and generally a l l isopycnals r i s e  toward the surface. This s i g n i f i c a n t feature of the  Ji-ZEQETS KM  proceeds  northern  boundary  of  plume. Because of i t s possible dynamical s i g n i f i c a n c e , some  e f f o r t was made to understand the nature of t h i s front. On July 2, 1975, a strong colour front was found by survey,  and  aerial  Figure 31 indicates the main features of the plume  at about 0900 PST. Figures 32 and 33 show p r o f i l e s of S, T,  and  sigma-T on either side of t h i s front, close t o P o r l i e r Pass. The plume  appears  to  be a layer of r e l a t i v e l y fresh water about 1  meter thick advancing onto the s t r a t i f i e d ambient the  front  there  is  a  water.  Along  very strong convergence as the rapidly  20  moving s i l t y water approaches the c l e a r , dark blue salty Consequently,  the  front  i s a very good c o l l e c t o r of debris of  varying size, ranging from a i r bubbles and There  is  also  water.  foam to  large  logs.  considerable l a t e r a l shear at these f r o n t s , and  large scale eddies capable of turning the 8  meter  Brisk  in  a  complete c i r c l e in l e s s than 5 minutes. The  apparently  strong  vertical  c i r c u l a t i o n at the front  suggested using dye as a tracer. We t r i e d to obtain quantitative results using a fluorimeter, but were unsuccessful. However, July  3,  1975  {  Fig.,26  )  we  were  able  interesting experiment at stations g and front  ( in  r.  A  rather  dye  was  released  heading  front,  and  then  south.,  disappeared  the  surface  by downwelling. There was  evidence of i t r e c i r c u l a t i n g up into the surface waters, completely  vanished  at  the  front.  However,  when  Richardson, of 2 meter draft, was manoeuvering about 100 upstream  intense  Red  on the water surface, about 100 m  upstream of the fronts This dye advanced along the  set up a very  terms of v e r t i c a l c i r c u l a t i o n , though not i n terms  of colour ) was found at stations g and r, rhodamine  to  on  as the  to no it CSV  meters  of the front, dye was s t i r r e d up to the surface by the  ship's screw, i n d i c a t i n g the dye was  being l e f t  behind  by  the  front. Figures 1975, 33.  34  and  35  show two sets of p r o f i l e s for July 4,  taken i n about the same location as those of Figs. 32 They  and  again represent conditions on either side of a colour  front, Fig. 34 representing conditions on the  fresher,  siltier  side of the front. There was a f a i r l y strong Northwest wind ( 20 mph  )  blowing  July  4,  and  we  see  that  one  r e s u l t i s to  21  homogenise the upper 2 to 3 meters on either side of the front. Interestingly,  there  was  s t i l l a guite strong colour contrast  across the front, i n spite of wind mixing. Figure 36 i s an attempt to depict the evolution of a in  the  winter, when, because of low river flow, the front does  not extend over as large a region front  front  as  during  the  summer.  The  was readily i d e n t i f i e d , despite the low discharge of 1100  m /sec, because of a good colour contrast and a large amount 3  debris  along  i t . We  travelled  outward, obtaining the position  ship's  along  position  of  the front as i t spread and  hence  the  front  by radar, and the approximate orientation of the front  by compass. The curves of F i g . 36, indicating successive assumed positions of the front, were drawn where  position  and  orientation  using of  the  numbered  the front were known. The  t i d a l currents indicated in Fig. 36 were roughly point  Atkinson  inclined northward  toward as  elevations.  One  the  gradually  the  south,  river  points  sees  the  inferred  front,  curving  from  initially  outward  and  flow builds up and the t i d a l currents  turn from ebb to flood.  It  CURRENTS IN THE PLUME A predominant feature of currents i n  very  strong  vertical  shear.  One  method  the  plume  of  is  their  determining the  current structure i s by use of Langrangian trackers, or drogues. Drogues of three depths were used : a surface drogue drag  element  about 0.5 m i n depth ; a medium drogue, (M), with  drag element suspended from 0 to 2 meters ; and a (D),  with  (S), with  deep  drogue,  drag element suspended from 2 to 4 meters. Figure 37  22  shows the wind and tide for July 23, 1975, and the location of two drogue tracking experiments. In F i g . 38a are shown the paths of  the above described drogues, released simultaneously  front on an ebbing t i d e i n region A, Fig. 37., The travelled  southward  deep  at the drogue  the fastest, and the shallow drogue stayed  with the front. The drogues were picked up and r e - i n s t a l l e d i n region B, considerably behind the front, on the following flood; their  paths  are shown  i n figure 33b. Now, the shallow drogue  travels northward the fastest. Figure 39 i s a corresponding  to the ebbing  notices that, as  i n velocity,  conditions, there  salinity  and one  is a  great  profile  certainly change i n  s a l i n i t y as one descends from 0 t o 4 meters. Figure  40  shows a r e l a t i v e speed p r o f i l e obtained with an  Ott propellor-type current meter, on the morning 1975,  of July  23,  i n region a of figure 37. The p r o f i l e was obtained from a  very shallow draft similar  ( approx. 0.2 meter ) boat t i e d t o a  drogue  to the deep one described above. The zero crossing i s  only inferred from the observed minimum i n the speed.  One has  considerable confidence in the r e l a t i v e surface speeds measured, because  the boat had such a shallow draft that i t probably did  not interfere with the flow. The s a l i n i t y structure at the time of the speed p r o f i l e i s that shown in figure 39.  li,SONIC CURRENT METER ME&SUREMENTS For two weeks, i n July and September of 1976, we borrowed a sonic  current  This meter relative  meter  measures  from M. Miyake (then of UBC, now of.OAS). the two  horizontal  components  of  flow  to the meter, and the magnetic heading of the meter,  23  with a  sampling  obtained  from  frequency  of  1  hz.  the Brisk i n the following way.  large wire angles which result when the surface  Current  current  and  currents, the boat was  one  boat  ship  drift  was  intermediate  velocities,  and  the  with  (  the  meter )  a  conical  suspended at about 8 meters ) . Conseguently, the  relative  between speeds  the  was  monitored  surface  and  deep  were such that the wire  angle was always less than 20°, and usually drift  drifts  restrained by a sea anchor  element,  were  To eliminate the  i s measuring the deep ( 5-10  drag  Ship  profiles  considerably  less.  by radar, and added to the measured  currents, giving currents r e l a t i v e to a fixed coordinate system. The analog output of the current meter was recorded on tape,  electronically  digitized,  and  vector  averages  calculated over the 2 to 5 minute period the current held  at each depth. The measurements on July 13, 1976  17, 1976  Mil  magnetic were  meter  was  and Sept.  are p a r t i c u l a r l y i n t e r e s t i n g .  l i t 1976 Figure 41 shows the d r i f t path of the ship, from station  to  b  station i i n c l u s i v e . The Hope discharge on July 11 was 8,600  m3/sec, a high value f o r July. A front heading the  ship  salinity  at  about  at 1304  on the freshwater speed  PST,  passed  and figure 42 shows a p r o f i l e of  station g, after the front had passed, and  side of the front. Figure 43 shows  the  water  at various depths at s t a t i o n g. The v e r t i c a l structure of  salinity shows  PST,  1225  westward  and water current are  the  remarkably  similar.  Figure  44  result of p l o t t i n g the currents at succesive 1 meter  depths on a polar co-ordinate system, with the dots representing the t i p s of the current vectors. There i s a strong westward flow  24  at the surface, and a r e l a t i v e l y weak flow to the  southwest  in  shows the ship d r i f t on this day. There are  two  the deeper water. September Y]_ 1.976 Figure  45  sets of observations presented  , i d e n t i c a l i n format to those of  July 13. Thus, figures 46, 47, and 48  refer  0646  49,  PST,  station  conditions at 1512 hour  period,  the  a,  and  figures  to 50,  conditions  and 51 refer to  PST, station j . We note that  over  this  present  decrease  in  advection*  in  the  thickness  but  some  morning  has  is  to  is  due  due  to  disappeared. horizontal  mixing,  velocity  Some of the  spreading  ( vertical  shear,  and  as indicated by the  increased surface s a l i n i t y . A l l of these attenuations properties  8.5  surface s a l i n i t y has increased markedly, the  thickness of the upper layer decreased, and the strong shear  at  in  plume  thickness, s a l i n i t y difference )  occurred in the absence of s i g n i f i c a n t winds.  & i A SHORT SUMMARY OF THE OBSERVATIONS The plume i s a brackish layer less  than  1  of  thickness  ranging  from  to 10 meters. This layer i s freshest and thickest  near Sand Heads, and becomes thinner and s a l t i e r as one proceeds away from Sand Heads. At the r i v e r mouth, the  momentum  of  the  r i v e r water i s directed in the direction of the j e t t y , except at high  water,  when  the  r i v e r flow i s shut o f f . As one proceeds  outward, t h i s river momentum i s l o s t , and the plume acquires the velocity of the t i d a l streams. The plume  i s quite complicated.  vertical  structure  of  the  The bottom part undergoes extensive  mixing with the deeper water, while, near  the  river  mouth  at  25  least,  the top part i s fed with r e l a t i v e l y unmixed r i v e r water.  (Because of the velocity shear and v e r t i c a l s a l i n i t y the  transport  gradient,  of fresh water i s much greater i n the upper part  of the plume than the lower. ) Because the r i v e r flow i s t i d a l l y modulated, the plume i s fed with pulses of fresh water at frequencies.  These  bursts  of  fresh  water  tidal  advance onto, and  incorporate themselves into, the existing plume. I f , because strong  wind  mixing,  there i s no d i s t i n c t plume i n the S t r a i t ,  then a front forms between the new discharge of r i v e r water the  ambient  Strait  water*  i t . ,. I f  silty  water  somewhat weaker front forms, vertical circulation.  and  the front being characterized by a  s i g n i f i c a n t colour change across i t , and along  of  already still  a  strong  exists  being  in  convergence the S t r a i t , a  characterized  by  a  26  CHAPTER 3 A The  DEL OF THE FRASES RIVER PLUME  f i e l d data described i n Chapter 2 may be combined with  basic hydrodynamic p r i n c i p l e s t o construct a of  t h e o r e t i c a l model  the plume. Such a model w i l l be successful i f i t can t e l l us  about the  horizontal  circulation  patterns  around  the  river  mouth, and the amount and d i s t r i b u t i o n of s a l t water entrainment into  the  plume. The r e s u l t s of the model may also be useful i n  resolving questions bearing on the nature of the mixing of fresh and s a l t water over large parts of the S t r a i t and i n p a r t i c u l a r the mechanism by which fresh water leaves the S t r a i t . The  above  points  that  our  model  must deal with are of  ultimate concern i n studies of the Fraser River Plume. a  However,  model may also be useful i f i t only points up where there are  gaps i n our understanding of the physical system, when the model f a i l s to describe r e a l i t y , and the f a i l u r e i s  not  due  to the  inadequacy of the mathematics, then we know that we must improve the  model,  usually  by  extension  and  generalization  physics, inspired by f i e l d measurements. The numerical of our model described below agree with within  field  of the  solutions  measurements  an order of magnitude, and usually within a f a c t o r of 2;  and indicate what should be measured in future f i e l d  trips.  In t h i s chapter, I w i l l discuss the physical concepts are  to  to  be  mathematical principles.  incorporated equations  in  that  the  model eguations by numerical  and  incorporate  In Chapter 4 are presented  subsets of the f u l l equations;  model,  develop  these  that the  physical  some simple solutions of  in Chapter 5 the solution of the  means i s discussed.  27  A CONTROL VOLUME DESCRIPTION OF THE PLUHI In  order  to i n i t i a t e discussion of the plume, consider an  idealized S t r a i t of salinity  Georgia,  schematized  with  mean  contours  salinity  profiles  fluid  2-layer  model  (Long,  with a particular density i n t e r f a c e (charney, specific,  I  strongly  1975b), or  1955). In order to  chose to define the base of the upper layer i n  terms of an isohaline surface which (as seen chapter)  of a  i s to consider the interface between the two  layers to be associated with the pycnocline  be  52, and the  at the indicated locations. The  usual procedure i n developing a stratified  surface  i n the plan view of Figure 52. Figure 53  shows a s a l i n i t y section along l i n e AA* of Figure associated  of  i n the  previous  i s eguivalent to using a density surface. The e x p l i c i t  s p e c i f i c a t i o n of an upper  layer  depth  i s important  f o r two  reasons. One i s that the degree of turbulent interaction between the  two  layers  depends  on where the interface i s chosen. The  other reason i s that one has to know how to compare computed and observed quantities, such as layer depth, and mass,  and  of  salt,  momentum. In the case of the Fraser River plume, the  isopycnal chosen corresponds approximately pycnocline,  fluxes  so i n a  to the base  sense t h i s model is compatible  of the with both  d e f i n i t i o n s of upper layer thickness. Choosing the S=25°/ o contour as the plume .lower O  let  us  first  treat  the plume i n the control volume approach  (Figure 54). The c o n t r o l volume contour,  and receives  modulated rate. It also indicated  boundary,  i s bounded  by  the  S=25%o  fresh water from the r i v e r at a t i d a l l y receives  salt  water  from  below, as  by the increase of s a l i n i t y downstream.,Averaged over  28  a few  tidal  constant, This  cycles,  so  outflow  extremity  these  the volume  occurs  the plume.  outer edge of the plume, increases,  volume i s  two inflows must be matched by an outflow.  presumably  of  of the control  mainly  at  the  downstream  As one follows a streamline near the the s a l i n i t y  along  this  streamline  by means of small scale mixing, u n t i l i t exceeds our  agreed upon l i m i t f o r the boundary of the plume, and passes out of  the control volume. The inflow of s a l t water into the plume  has been called entrainment a  name  Perhaps  (Turner, 1974), and i t appears  has to be invented for flow out of the control volume. depletion  phenomenon.  Thus,  i s the most when  evocative  term  outward.  The  for  outer  edges  somewhat, and  are then eroded by gentler  mixing, expressed mathematically as the sum of entrainment relatively  stronger  this  the river flow i s strong and there i s  vigorous entrainment, the control volume expands spreads  that  depletion.  (Both  terms  and  are needed,  entrainment to increase the s a l i n i t y of the layer, and depletion to balance the volume added by entrainment.) Considering now the momentum control  volume,  balance,  the water  i n the  as well as the boundary of that volume, moves  back and forth i n the S t r a i t f o r a  variety  of reasons.  These  reasons f a l l into 3 categories: 1)  bodily fluxes of momentum - the r i v e r momentum, the momentum  of the entrained water, and the momentum removed i.e.,  when  the water  by  depletion,  leaves the control volume, i t takes i t s  momentum with i t ; 2) contact forces - the effects of wind stress are omitted i n the model developed here, so the only contact force i s the stress at  29  the interface between two  countermoving  fluids,  friction  and  form drag; 3)  body  forces - the barotropic t i d e , the bouyancy tendency of  the plume to spread into  a  thinner  layer,  and  the  Coriolis  force. Omitting  the  wind  reduces  the  complexity somewhat, and  there are many r e a l cases where i t i s unimportant. Wind can  be  added  later  effects  as a further refinement once the windless  model has been s u f f i c i e n t l y developed. , DIFFERENTIAL EQUATIONS DESCRIBING THE PLUME This section deals with the derivation integrated  equations  describing  l i g h t e r water f l o a t i n g on and of denser water. Two entire water equations  the  the  vertically  the motion of a thin layer of  interacting with a very deep layer  sets of eguations are derived - one for the  column,  in  of  which  reduce  to  the  barotropic  tidal  absence of an upper layer, and a second set,  which describe the motion of the upper layer. Consider an upper layer. Figure 55,  with  density p ^ p c z ) ,  velocity u=u (z) , s a l i n i t y s=s (z), and thickness and  £  are  lower  i s -h, the negative  layer  has  w i l l be derived quite  where  /r[  the upper and lower interfaces respectively. The z  axis i s p o s i t i v e upward, the mean value of value of %  h=f>(-£,  simply  uniform  for only qeneralized  of  the  i s 0, and the mean layer  thickness.,  properties p , u , s . The o  one to  horizontal include  0  0  dimension; the  second  dimension. 1. CONTINUITY EQUATION For either layer, the continuity equation i s  The  eguations they  are  horizontal  30  2*  ^  3.1.  Integrating eguation 3.1 over the upper layer depth, we have  3 X  L  3.2a. Expanding the i n t e g r a l , we  5  *  2x  J  Re can apply the surface  ?X  kinematic  boundary  3. 2 b. condition  at  the  upper  z=/^  Dt  to  get  simplify  2*  ^  C  eguation 3.2b.  motion of f l u i d r e l a t i v e to earlier  i n t h i s chapter,  3.3..  At z= 1= , ^ the  C^~^ )  interface  describes  z=£.  As  the  discussed  f l u i d crosses the interface , entering  or leaving the plume, by means of entrainment and depletion. i s convenient to represent processes  as  the  sum  p  Dt  the net e f f e c t of these two competing  of the two e f f e c t s expressed  Writing entrainment as w ,  It  and depletion as w , 0  separately.  we have  3.4.  Thus,  3. 5.  31  and the continuity equation for the upper layer becomes:  3*  *  3T£  *  3.6..  The continuity equation f o r the entire water column i s  A  [ ll  <-2  (a  J  - fk  ?  = o.  *  3.7. M.  J ^'^z,  Adding and subtracting  2~t-  -Q  we have  3.8.  3x  The l e f t hand side of equation 3.8 i s the continuity equation i n a f l u i d with depth independent  v e l o c i t y , and the r i g h t hand side  is a correction due to the presence of the r i v e r  flow  i n the  upper layer.  2 i SALT EQUATION Assuming  molecular  diffusion  i s negligible  compared to  turbulent e f f e c t s the s a l t conservation eguation i s given by:  2tr  3x  where <us>  3.9..  3*  includes  the  mean  and  turbulent  fluxes  i n the  horizontal d i r e c t i o n , and <ws> denotes them i n the v e r t i c a l . Integrating eguation 3.9 over the upper layer, we have,  32  ^  T  Assuming  D  that  3.10.  i  <us^ = (us)^ ,  ( where  for notational s i m p l i c i t y  guantities without angle brackets denote mean flow properties ), that i s , neglecting horizontal turbulent fluxes, that  we  <wsy= (»s)^,  and  assuming  can use the kinematic boundary contition  3.3 to simplify 3. 10. At the interface z=j"", the quantity has  two  components.  One  interface, denoted by v salt  relative  entrainment  i s associated  to the interface., The f l u x  we write w s , where s p  s  motion  of the  s^. The other i s due to the flux  c  layer; and the flux associated where  with  <ws>^  associated  of with  i s the s a l i n i t y of the lower  e  with  depletion  we  write  w s, n  i s a s a l i n i t y appropiate to the negative flux of salt  due to depletion. Thus ^  J  ^ - -  ^ i « t  s  g  +  W  -  ' "  S  w  "  ^  3.11a  or  %  •  ^  J  Using equation 3.11b and the kinematic boundary  '  3.11b.  condition 3.3,  3. 10 s i m p l i f i e s to :  3.12. The  salt  equation  i s usually not included i n barotropic t i d a l  calculations, as the s a l i n i t y i s assumed uniform, so i t written  out here.  i s not  However, i t i s i n t e r e s t i n g t o note the s a l t  equation for the lower layer alone takes the form  33  Z-  f S Jg  +2.  -D  ( SOfdz  ^  = - S  a  \A>  ?  + S VO  n  .  3.13.  ^  Thus, as one expects, the roles of entrainment and depletion are reversed f o r the lower layer, and source  and  a  sink of s a l t  the  plume  acts  as  both  a  (in d i f f e r e n t places of course) for  the lower layer.  3t.THE HYDROSTATIC EgOATION I t i s assumed  that  the  balance  of  vertical  forces  is  adequately described by the hydrostatic equation,  = - a*  IB 2?  I  .  3.14.  d  At a depth z i n the upper layer, p  =  -  /  > 3 ^ '  =  />3  •  *  3.15a.  At a depth z i n the lower layer,  £ We  should  3.15b. recall,  from  the  salinity  and  sigma-T  profiles  presented i n Chapter 2, that the density structure i s controlled almost e n t i r e l y by the s a l i n i t y structure, and in f a c t ,  cr  T  » k s  3.ie.  i s a very good approximation, with k=0.8. This eguation of state was  chosen  for  computational  simplicity;  comparison  with  34  salinity  and  sigma-T  traces  shows  exact. The agreement i s within 5% during  it  for  to be only close, not  winter  conditions,  but  summer, when the water i s considerably warmer and not as  isothermal, a reasonable value of k would vary the surface to 0.6  from  0.55  near  near the bottom of the plume.  iii HORIZONTAL MOMENTUM EQUATION The  horizontal  momentum  equation,  ignoring  molecular  diffusion, i s : •*. I  2t where  <<uu >  turbulent  <P  uJ>  +±  -±  dx  P  C  ^>  z  JJL*  i s the stress i n the x-direction and p i s the Integrating  3.3,  4- l_ <u  dz  over  fluxes,  and  the  upper  using  layer,  ignoring  ~-Q,  3.17.  pressure. horizontal  the kinematic boundary condition  the f i r s t three terms become:  f U dz  where u  c  +  2.  f Ol dZ  \aJ  ?  t  U0  U  W „  and u have s i m i l a r meaning to s  a  and s.  Ignoring  wind  stress, the i n t e g r a l of the stress term becomes  This i s the stress at the interface due to f r i c t i o n a l forces form  drag  in  addition  to  w u - wu p  0  n  and  which i s a momentum f l u x  associated with mass transfer across the interface. The remaining term to evaluate term,  is  the  pressure  gradient  35  9  Z'*2  Because the analysis becomes very complicated assume  that  the  homogeneous,  a  density form  of  in  the  the  upper  otherwise, we  layer  Boussinesq  will  i s vertically  approximation.  The  pressure force in the upper layer i s then:  Thus, the upper layer momentum eguation i s dt i  3x  e  r  3  99 xx  ?  o  J  3. 18. Note  that  and  3.18,  to  solve  for  the upper layer we have 3 eguations,  and 4 independent variables, u, s,/^, and £ . In the  upper  layer  equations,  between/^ a n d £ , which we obtain barotropic  tide.  The simplest  from way  we the  equations  pressure  gradients  model,  observations  O ^  since  ( Crean, 1976  ' I o i/O-e r  where  £>-r  Then  using  y  the  be  those  flows  order  for  the  to obtain the r e l a t i o n i s by  that i n the deep water of the S t r a i t , must  3.12,  require a r e l a t i o n  observing  tidal  3.6,  the  horizontal  calculated i n a barotropic  so  calculated  agree  with  ). Thus,  layer  i s the surface slope obtained equation  3.15b  from a barotropic model.  for the pressure at a point in the  36  lower layer,  '  1  3.19a.  Defining$ - ^ -^, the above becomes, after rearranging, 0  3. 20. Eguation 3.20 problem.  i s the fourth eguation reguired  The  pressure  layer eguation 3.18. not differentiated)  gradient  term  to  complete  the  in the integrated upper  becomes then (ignoring variations i n p when  t  f  3.21.  Now column.  to derive the momentum eguation for A l l terms  the  entire  water  except the pressure gradient term are quite  straightforward. The pressure term i s :  ±  If d*-  (  +  5  -L  ^  ?  =  f °  3.22a Substituting p  z  p  - ^ # nd doing a b i t of a  o  rearranging,  we  get  the above expression to egual: ^  (  D  +  I  nr\  )  t\  x  +  ^  C  D  + * 1  )  £  c5  (  /*(  -  £  ) )  x  3.22b.  37 Note that i f we consider the terms proportional to D r*\  , and  +  require them to equal  p ojC ^, b  we get  r  p-  as i n eguation 3.20. The  momentum eguation f o r the entire water column i s then  "I  (Oo  3. 23. The  left  hand  side  represents  barotropic t i d a l equation.  the momentum  terms  The f i r s t terra on the riqht hand side  represents the correction made to g•( 0 *  because i t contains  x  the pressure gradient required to maintain the r i v e r pressure  gradient  (#/p»X ^ *""^ * <  3.23,  /  to give  river  *  a  n  d  associated  effects.  with  the  t h i s i s subtracted from  <^£-r^,  The  i n the  flow.  river r\  x  The  flow  , i n eguation  the gradient calculated i n a model with second  term  on  is  the right  no  represents a  correction to the barotropic pressure because the upper layer i s somewhat l i g h t e r . The other terms, proportional to  - j£  have  l i t t l e e f f e c t on the barotropic motion, whose driving terms, the l e f t hand side of 3.23, are proportional to D. Thus the presence of  the r i v e r flow has n e g l i g i b l e effects on a barotropic model,  and the surface slopes and v e l o c i t i e s obtained from  that  model  38  may be used as forcing for the upper layer model. To  summarize,  we w i l l rewrite  the eguations f o r the upper  layer i n terms of a variable z which i s zero at the bottom o f the upper layer and increases upwards, and also w i l l include the second  horizontal  the thickness  dimension, and the C o r i o l i s force. Note that  of the upper layer i s h=/^-.£.  CQJTIJJOITY *  -T"  rr  J  A °  +  *  f-  = \AJ - W „  / vdx  f  3.24. SALT A  I  jsd*  + i. f usdz  + A.  fvsdz 0  So  %  -  5  \AJ„ .  3.2 5. DIRECTED MOMENTUM  h  ^  4 J  2  V,  o  o  p  a  3. 26  X-BISECTED MOMENTUM A  ~  "  ~~ A  A-  ^  3.27.  39  There are several things which have yet to be specified  in  t h i s model. 1. We have to provide the p r o f i l e s of u, v, and s, so that, f o r example, we can relate J ^ " " ^ ^ 2. w,w,, u, p  s,  0  to J u  and ^r«-f  must be specified i n terms of  flow properties such as the density between  the upper  and  lower  .  and  velocity  differences  layer, and the thickness of the  upper layer. r # u # and v y  0  G  must be obtained from a barotropic  t i d a l model (Crean, 1977), which i s the solution of eguations of the type 3.8, 3.23, with the right hand sides set to zero. Items 1 and 2 above are related to properties of the flow, being  related  to the turbulent structure present, and can only  be parameterized i n terms of the large scale properties flow.  of the  Item 3 can be considered as external driving of the flow.  However, the extent of the forcing produced by u  Q  and v  0  on the type of turbulent interactions specified i n items  depends 1 and  2. Having  gone  through  above eguations, we physics  we  want  should  considerable check  that  they  to get to the  agree  with the  to model. F i r s t , consider the continuity and  s a l t equations, as i n the control earlier,  algebra  the guantity  of  water  volume  approach  described  i n the plume changes, ^V^r,  because of horizontal divergences, i n particular the r i v e r flow, and by fluxes r e l a t i v e t o the boundary, Wp the  salt  content  and  w. n  Similarly,  can change by internal rearrangement, or by  influxes of s a l t , w s , and effluxes, w^s. p  In the momentum  0  eguation,  we  see that  the change i n  no  momentum of a column of f l u i d i s approximately given by: Or  J  buoyant spreading  j  t>/^^  I/^OJS/I ') 7  * f r i c t i o n a l interaction^ * gain or loss of water and i t s associated momentum, w Ut>-\aJ„ ia , f  + forcing by the barotropic t i d a l slopes, It these  i s difficult terms,  since  to estimate the plume  aj S r ^  .  the r e l a t i v e importance of  i s spatially  and  temporally  variable. However, Table 3 presents very coarse estimates of the order  of  magnitude  of the terms i n the momentum  f i r s t part of the table l i s t s the scaling  equation..The  parameters,  and the  second part l i s t s the sizes of the various terms i n the momentum eguation,  for the region near the r i v e r mouth, and f o r the f a r  f i e l d . For the s a l i n i t y and continuity eguations, we notice that the r a t i o of the advective terms to the source terms i s uh/(wL), which i s also given i n Table numerical  3.  Wp  model of Chapter 5, where w  p  i s estimated  from  the  was calculated according  to the formula w = 0.0001u. P  Except for the action of winds and possibly horizontal eddy viscosity and d i f f u s i v i t y , the eguations derived i n t h i s chapter appear to have a l l the necessary terms to describe the plume.  BOONDaHY CONDITIONS In solving any d i f f e r e n t i a l equation  system,  one  has to  specify the appropriate boundary conditions. The actual boundary conditions used w i l l be discussed i n chapter 5, but I would l i k e to discuss here the theoretical boundary condition requirements. Consider  a  s i m p l i f i c a t i o n of the above equations, i n x-t space  41  only, (we define a new variable,  =s„-s, the s a l i n i t y defect.)  £A t 2 ( uA) - o.  dt  3*  3x  These equations can be thought of as homogeneous behaviour  of  whose  solutions  dictate  the  eguations  behaviour  eguations with forcing and dissipation, as long as  the  the  of the forcing  and dissipation do not contain derivatives of the same or higher order. Writing g=gk, these may be put i n matrix form  1  o  "  0  {  A \ I k O  o o  1*  (A  or H^+fiH =0. The eigenvalues TV # and l e f t eigenvectors, £• , of A y  are:  Multiplying i  J  the  matrix  form  of the d i f f e r e n t i a l eguations by  , we get:  I; l-i: Since  , 4 j - J: ~}\ S ' i , there results 3  Thus,  in  c  t  the  = <9 ,  direction  the  d t  characteristic  form  of the d i f f e r e n t i a l equations. E x p l i c i t l y ,  42  the c h a r a c t e r i s t i c equations are :  ^ dt  J  The reason for putting the equations i n c h a r a c t e r i s t i c form i s that the required become  more  prescribe  boundary  obvious.  as  boundary  pointing  into  conditions  is  open  boundaries  directed  as  should  there  are  the region under consideration.  Thus, one must prescribe s a l i n i t y on an characteristic  at  The basic requirement i s that one  many  characteristics  conditions  into  open the  boundary region.  if its  Since  the  c h a r a c t e r i s t i c speed i s u, the flow v e l o c i t y , one must prescribe s on an inflow, and must devise a way  for i t to be determined at  an outflow boundary by flow conditions in the  interior  of  the  computational region. I f |uj i s greater than c, and u i s an inflow, then both and u-c point into the computational region, and two pieces  independent  of information about u and c, i n addition to s discussed  above, must be s p e c i f i e d . I f |u| i s l e s s than than c, of  u+c,  u+c  u-c  points  into  the  then  one  region, and one points out, so  either u or c, or a r e l a t i o n between them, must be s p e c i f i e d . If u=c,  a very complicated  s i t u a t i o n a r i s e s , i n which a boundary i n  x-t space becomes a c h a r a c t e r i s t i c . This i s becomes  very  a  situation  which  tricky in problems with a time-dependent boundary  condition, and requires physical system.  recourse  to  further  aspects  of  the  43  CHAPTER 4 AIDS TO INTDITION ABOUT THE PLUME Before it is  presenting the numerical model i n the next chapter,  worthwhile  to  look  at  some  simple  models  of  plume  dynamics. These models do not include a l l terms i n the eguations of  Chapter  3,  so  can't  be  expected  to  describe the plume  adequately, but they are useful.in c l a r i f y i n g various aspects  of  the plume's behaviour. The  first  sub-model, the compressible flow analogy, shows  that near the river mouth, the  predictions  of  a  frictionless  model are at variance with observations, leading one to conclude that  friction  and  entrainment  are  important features of the  plume near the r i v e r mouth. The second model, a time-independent model  of  surfacing  depletion one  and  proceeds  conditions  isopycnals,  illustrates  the  roles  of  entrainment i n causing the isopycnals to r i s e as downstream  at  a  strong  in  the  plume.  frontal  The  third  model,  discontinuity, discusses the  motion of the strongly contrasted colour fronts frequently found in  the  Strait,  and  suggests  that  these  fronts  induce  considerable v e r t i c a l c i r c u l a t i o n . The fourth model, a kinematic wave approach to f r o n t a l motion, i s intended to i l l u s t r a t e , in a very  simple  manner,  the  way i n which fronts arise i n a time-  dependent s i t u a t i o n , due to the t i d a l l y varying r i v e r flow. fifth  model  The  , mixing and fluxes across an interface, shows how  an upper layer model, as developed i n t h i s thesis, i s compatible with a  diffusive  {eddy  diffusivity)  model  of  the  vertical  s a l i n i t y d i s t r i b u t i o n . The s i x t h section, analogy with turbulent jets,  is  an  attempt  to  motivate  the use of entrainment and  44  depletion by showing how they arise i n a more accessible system, a turbulent plane j e t . i i . COHPfiESSIBLE FLOW ANALOGY There i s an exact analogy between the f r i c t i o n l e s s flow a  compressible  f l u i d and the f r i c t i o n l e s s flow of a l i q u i d with  a free surface. The method of s o l u t i o n of the equations below  was  Edelman,  developed  1947),  engineering  of  f o r compressible  and  later  (Ippen,1951).  adopted Using  flow  (e.g. Shapiro and  f o r use  this  derived  method,  in  hydraulic  one i s able to  predict the v e l o c i t y and thickness of a f l u i d discharged from  a  channel into an unbounded region. The solution f o r water flowing oyer  a  solid  surface  (as developed i n Rouse et a l , 1951) i s  i d e n t i c a l to the solution for l i g h t e r f l u i d flowing over heavier fluid  (as required i n a plume theory), i f q, the acceleration of  qravity, i s replaced by g*=g then  is a  buoyancy  spreading  , as shown i n Chapter model,  3., This  representing a balance  between the convective accelerations and the spreading  tendency  of the pressure gradient. The equations  of continuity and momentum conservation for a  steady-state, f r i c t i o n l e s s plume are:  u a „'  u  iA  v  *  IA  +V  Y  v-  g 'h x  *^  ' ^ y  =  o, =  °-  4m 2  4.3.  Fundamental to this method i s the requirement that the v o r t i c i t y of the flow be zero, 3  ' l/*  - O .  4. 4.  This approximation i s assumed to be valid i n a region around the r i v e r mouth.  45  The  method  of  solution i s as follows. From 4.2, 4.3, and  4.4 one obtains the Bernoulli eguation,  Z  Z  4.5.  c '=•  h  7  where  The continuity eguation may be put in the form  ( is*--C*) ry, da. + C • *• - c * ) d is = O , where  ^  +.  ^ ^  =  /  ^  i  ^  z  .  ^  i  4.6.  1  U - - tz 2  du d  i/  , (pi =  ^  +  U ) <_/  ( -¥ m l y ) d d  +  Thus, i n the direction  x  /dv. - m, du and dv s a t i s f y the ordinary  d i f f e r e n t i a l eguation 4.6. Since depending  on  the sign  of  there  are two  the square  root,  types there  of  are two  eguations, i n addition to the Bernoulli equation, from which obtain  2  2  2  being greater than c . Thus; this model applies 2  to s u p e r c r i t i c a l flow only. There i s a f a i r amount and  Coleman,  (u +v )>c . 2  2  of  evidence  1971; Garvine, 1977) that the flow at a  r i v e r mouth i s i n t e r n a l l y c r i t i c a l or 2  to  u, v, and c . However, a l l t h i s depends on m being r e a l ,  that i s on u +v  (Wright  m,  supercritical,  i . e . that  Thus, one can use eguation 4.6 to obtain a solution  around the r i v e r mouth, as long as u +v 2  2  remains greater than or  egual to c . 2  Figure  56a  characteristics,  illustrates  the orientation  C+ and C- (defined by  /j ^ x  of  the  two  or m_ ) , and a  streamline, with respect to an x-y coordinate system.  46  Defining u=gcos0, v=gsinO, 56a)  and ~X-+= <9 ± / * , (Fig.  c=gsin w, /  one can eventually manipulate eguation 4.6 i n t o the form  9 ± P(m)  =constant  on ^/dx-Wi:  One  4.7.  -  , and  interesting  thing  about  the above solution i s that i t i s  possible to integrate a n a l y t i c a l l y  the  differential  eguations  along the c h a r a c t e r i s t i c s . The  problem  i s set up as follows. An opening i n a s o l i d  wall i s assumed, through which water i s flowing with  v=0, u=c.  One then f i l l s up the computational region with the two families of  c h a r a c t e r i s t i c s , and uses tabulated values of  to obtain  values of u and v at the intersection of c h a r a c t e r i s t i c s . (Fig. 56b) . Rouse  et  a l , (1951),  worked out by hand the solution to  t h i s problem, with F = u/c = 1, 2, 4, (Fig 57). The r e s u l t s , for the values of F examined ,  are a l l quite  similar  -  as one  proceeds outward from the r i v e r mouth, the upper layer thins and spreads.  Since the downstream thinninq of the layer constitutes  a pressure gradient, the flow accelerates. In contrast  to  this  model we observe, (Cordes, 1977), that the flow outward from the r i v e r mouth slows down, rather than speeds up. The inadequacy of the model i n t h i s respect points out the importance of retarding forces  at the r i v e r mouth. The retarding forces could come from  three sources: 1) an adverse pressure gradient by  the large geostrophic  i n the barotropic  tide,  caused  slope during the ebb cycle of the tide  in the S t r a i t ; 2) entrainment of water with zero  momentum  i n the downstream  direction of the plume; 3)  frictional  interaction  between  the upper layer and lower  layer. To determine estimate ^' h x  3' -  how  important  depth  are, we  ( ^ ; J '  i s measured from the diagram, h „ i s the i n i t i a l  of flow, and b i _ i s the halfwidth of the r i v e r mouth. The )2  factor Ch>/b ) converts from v  analytic  solution  the non-dimensional  to units  appropriate  10m/sec*x.01=0.1m/sec , 2  c>hf$%  and h  0  units  of the  to the Fraser River.  Taking the width of the r i v e r to be 600 meters,  and  g*  to be  to be 8 meters, we can evaluate  for the region along the centreline where h changes from 1  to 0.3, a distance of 2.5 halfwidths:  9  V3-x") ^  f 7  Thus, g*h„~7.5x10-* m/sec . i n the region where h z  0.3  first 2  x  where L^/<>^t^j~i  terms  from the a n a l y t i c a l solution. F i g . 57. For F =1,  -  h  these  to  0.1, the pressure  gradient  =  O.l/Z-S.  changes  from  , calculated i n the same  manner, i s 2.0x10-* m/sec. From the geostrophic r e l a t i o n , pressure  gradient  fv=g £>< ,  the  crosschannel  f o r a current of 1 m/sec (an upper limit) i s  about 1x10-* m/sec*. This slope results i n the water l e v e l being higher on the western side of the S t r a i t of  Georgia  during  an  ebb, which constitutes an adverse pressure gradient to the r i v e r flow  i n the v i c i n i t y of the river mouth. Thus i t appears that  the cross channel barotropic t i d a l slope i s the same magnitude as the buoyant spreading pressure gradient.  order  of  48  One  can  estimate  the  relative  effect  entrainment as follows. I f the contribution  of  vertical  of entrainment  to  the continuity eguation i s written as  dt  3  and one considers the v e r t i c a l l y integrated momentum equation,  then the average momentum equation i s  Jjt Dt  1 h  _  Thus,  &  ~ MS u . In  entrainment  coefficient w / k . . (Keulegan,1966}. both Cordes  acts He  This  as  linear  will  friction,  assume  w=Eu,  with  friction  and  E=2x10 * _  order of magnitude f o r E was v e r i f i e d by  (1977) and de Lange Boom (1976) for the Fraser Biver  plume. An estimate of u from the Bernoulli equation i s  At h/h =0.6, u ^1.2 m/sec. This i s also a reasonable  value for  o  the  measured speed near the r i v e r mouth. Hith this value for u,  and h=.6x8m, the retarding force due to entrainment, 0.4x10-*  m/sec*. This value i s one f u l l order of magnitude less  than $'h near the r i v e r mouth and somewhat closer x  points  wu/h, i s  further  downstream.  to g*h  v  at  I t appears that entrainment has a  s i g n i f i c a n t but not dominant effect on the plume near the r i v e r mouth.  There  i s an  indirect  effect  also.  As  entrainment  proceeds, g', proportional to the density difference, decreases, and h has a tendency to increase. The actual driving  u  i s (j/zk)  Cs'*  •  decreases, and h has a tendency  D  u  e  t  o  pressure  entrainment,  to increase  gradient  since  (opposed  g'  by i t s  L\9  buoyant  spreading  tendency)  it  is  d i f f i c u l t to predict t h i s  pressure  gradient without a more detailed model,  such  as  that  discussed i n Chapter 5. The  third  possible  retarding  force  i s f r i c t i o n . I f one  assumes quadratic f r i c t i o n , and equates hg*h^. to Ku|u|, then be  important,  Ku /h must be close to 7,5x10-* m/sec . Assuming 2  2  Ku2/h=7.5x10-* m/sec , we get  K~2x10~ .  similar  drag  2  to  the  value  3  of  This  value  coefficient  of  used  calculations. For instance, the drag c o e f f i c i e n t for water  is  about  friction  in  interfacial order 10  K  is  in  many  wind  over  1.5x10 , and the drag c o e f f i e c i e n t for bottom -3  a  tidal  drag  channel  is  coefficient  about  2-4x10 , -3  and  the  in a laboratory scale flow i s of  (Lofguist, 1960). Thus i t appears that f r i c t i o n  -3  to  plays  an e s s e n t i a l role i n the plume dynamics. Is., h lI«IrINDEPENDENT MODEL OF SURFACING ISOPYCNALS As demonstrated i n Chapter 2, figures 15, 27a, and  27b  in  particular, the surfacing of isopycnals i s a dominant feature of the  plume  in  regions  away  from  the  r i v e r mouth. The model  discussed here i s an attempt to explain t h i s phenomenon i n terms of entrainment and depletion. This model applies to from  region  around station c to s t a t i o n j . Fig. 27a. Here the plume i s  thought to be more advected  or  less  uniform  across  the  area  is  about  10  km,  and  an  average  excursion  in  so the plume i s advected back and  forth by the tide a rather large performing  Strait,  back and forth by the tide, with a small mean v e l o c i t y  to carry r i v e r water out of the S t r a i t . The t i d a l this  the  distance.  One  could  imagine  over a few t i d a l cycles and obtaining a  set of data describing a stationary plume.  One  also  needs  to  50  perform averages across the S t r a i t (and hence across the plume), or else assume l a t e r a l uniformity. The plume i s fed water at i t s upstream  end, and t h i s  water  leaks  out from the forward or  leading end by means of the depletion  mechanism  discussed i n  Chapter  model  the plume, a  3.  The  equations  for this  of  simplified form of 3.24, 3.25, and 3.26 are:  + VU - \AJ = O n  p  4.7. .  2  (Us) + \AJ„ 5 - \AJ So = O • P  4.8.  h  h where T(j -s) - j( 9°'f)  i s an equation of state, 0 i s a transport,  0  ( v e r t i c a l l y integrated v e l o c i t y ) , and K' linear  friction.  A  is a  coefficient  of  schematic drawing of the model i s shown i n  Figure 58a. The use of linear f r i c t i o n that  4.9. ,  i s not u n r e a l i s t i c i n  the t i d a l average of square law f r i c t i o n i s linear in the  residual flow (Gruen and Groves, 1966). To make the solution of t h i s model very easy, assume w^ and constants, and that w >Wp, which i s v a l i d near  Wpare  n  the outer  edge of the plume., The continuity equation, 4.7, has the immediate  UL -- U  ~ C  a  where  D  e  vu  n  -  M J  P  )  *  solution  Y  i s the transport at x=0, and where x=0 i s taken at the  upstream boundary of the region of a p p l i c a b i l i t y of t h i s  model  (Fig. 58a) . Defining <^=w^,-Wp, and I^Uo/^, we have 0=«<{L-x). Note that 0=0 at x=L, so the length of the plume i s L., The s a l t ahd continuity eguation may be combined to give:  51  UL S *  +\AS (S-S*)^  4. jo.  O.  ?  Define s-s = 2T,<0; y=L-x. Then, the above equation becomes 0  with  solution  o  =Ay  ; the s a l i n i t y defect i s proportional to  the distance from the  leading  edqe,  y,  to  the  power  v A<. p  the boundary condition that at x= 0 s=s , we get A= (s^ -  Imposing  i  s )L 0  In terms of the variables y and  , the  momentum  eguation  becomes  <2y  k  With .2_=Ay  K  , we get  (3 r  Trying a solution h=By  A 6 y  gives  ^ £*^;  + Z£  C w„ +  /< J y  Equating, powers of y: 0 = (2 - ^^L)/j. & d finding a value of B to n  make the l e f t hand side equal to zero: 3 =  - A (z ^  4  w?/^J  Because i t i s a non-dimensional number, important  one  and  perhaps  the  most  f o r the plume, the i n t e r n a l Proude number i s of  some interest. It i s given by:  52  a  constant.. To summarize the solution  U -- a C  L -  * )  4. 1 1.  4. 12.  In =  (S.-SJ l - ^  p  U  C f/3 -  'h**^  4.13.  And the v e l o c i t y , u, i s given by  4.14., Thus, the transport, 0, decreases i n the as  plume  the  thickness  of  the  layer  increases  downstream, going to zero at  that  day  Referring km,  shows to  this the  u,  also  decreases  x=L.  The data for July 3, 1975 with  downstream;  decreases downstream. It i s  reassuring to note that the f l u i d v e l o c i t y ,  comparison  direction,  water becomes s a l t i e r and i s redefined as lower layer  water; the average s a l i n i t y of the plume and  downstream  (Fig. 27a)  model, surfacing  seem a good choice for  i n that the s a l i n i t y section for of  isopycnals  very  Fig. 27a, consider station d as x=0.  i f the plume boundary i s defined as the 25 /o 0  O  clearly.  L i s then 23 contour. With  53  s = 25 °/oo, the average s a l i n i t y at station  d, s  Q  from  an  for  discharge  U.  Assume  0  that  on  8000m /sec,  and  3  transport  an  egual  S t r a i t . At station  We  the  volume  of  can  salinity  in  river  through  discharge i s  salt  water  then  the  m2/sec.  about  across  16000m the  wide. Strait,  Thus <=< =0 „ A = . 5/ (23x 105) =2 . 2x10-s  d to station upper  j . At station  p  and  h ( JU)°' /3using for  the  =2.2x10~  m/sec,  s  w„= 6x10~s  m/sec.  average  linear f r i c t i o n c o e f f i c i e n t ,  d, j, 1)  we  find fixes  {average  s  stations  we  This  y - o}((J»y)/(S>>- ) =.83,  z  a  profiles  j,  layer i s 23"/"° and x=11km;,We obtain,  v /U-1.8. With  m/sec,  salinity  is  p  from eguation 4.12,  L  Strait  get an estimate of v /*. by f i t t i n g the s a l i n i t y  the  4x10-5  the  discharge  change from station  3  d,  uniform  o  h (*•)  that  river  3  0 = (8000/16000)=0.5  p  the  to the south i s 8000m /sec, flowing along the axis of  Assuming  w=  half  that by the time r i v e r water reaches station d  entrained  m/sec.  average,  leaves through the northern channels and half  the southern channels. Assuming  the  18.6 */oi> ,  integration of the actual s a l i n i t y p r o f i l e , fe need an  estimate  i t has  is  lr  calculate K*, the  to be 5.3x10~ m/sec* Eguating 3  K'u  and Cu|u| at x=0, we get an eguivalent C=2.6x10— . This i s about 2  10 times the usual value of a drag c o e f f i c i e n t , 3x10~ . However, 3  K* i s r e a l l y C U of  velocity  r m 5  , (Groen and Groves, 1966), and the rms value  i s probably quite a b i t higher than the t i d a l mean  used here. The computed and observed distribution  of h i s  shown  in Figure 58b. The f i t uses the two h points at the upstream and downstream ends of the plume, so there i s only one point l e f t to check  the  f i t . To  h= ti „ (^-[r)  3  show the s e n s i t i v i t y of the f i t , the curve  i s plotted as a dashed l i n e .  of  The model i s substantially correct i n that i t predicts  the  r i s i n g of isopycnals (modelled as a thinning of the layer and increase  in  average  salinity  of  the layer), but i t would be  extremely unlikely that the entrainment and depletion can  be  modelled very well by constant  processes  values of w  and w,o.  n  model would be improved by obtaining data for f i t t i n g that truly  cross-channel  and  tidally  averaged;  empirically, better formulae for w^ and 2jl  and by  The were  obtaining,  Vp.  CONDITIONS AT A STRONG I1QNTAL DISCONTINUITY As discussed i n Chapter 2, p a r t i c u l a r l y with  the  data  of  July  2, 1975,  reference  to  F i g . 31, there i s often a d i s t i n c t  colour front bounding the plume. The purpose of t h i s section to  an  is  c l a r i f y the role these fronts play in determining the motion  of the water behind them. One would l i k e to have a front  suitable  plume,  and  for  circulation  at  these  purposes  a  a  detailed  model  of  the  the front i s not reguired, but rather r e l a t i o n s  s a l t into the f r o n t a l region. reference  itself.  mass,  momentum  and  r  to the dye experiment described i n Chapter  2, on July 3,1975, a "tank tread" model of suggests  of  f o r use with a larger scale model of the entire  between f r o n t a l v e l o c i t y and the fluxes of  With  model  frontal  circulation  Thus, i f one considers a m i l i t a r y tank moving  over the ground at speed V, to an observer  on  the  ground  the  upper tread i s moving at 27, the tank at V, and the bottom tread i s stationary.„This i s s i m i l a r to the v e l o c i t y p r o f i l e s measured in  the  plume, eg.  i s "attached" at  a  Fig. HO, where the bottom part of the plume  to the lower water, and the upper part  significantly  different  speed. Continuing  is  moving  with the tank  55  tread analogy, and changing the co-ordinate system to that of an observer s i t t i n g on the tank, the top tread i s moving at V,  the  bottom at -V, and the tank i t s e l f appears stationary. Similarly, an  observer t r a v e l l i n g with the front sees water at the surface  of the plume coming toward him (as the dye on  the  surface  ran  toward the front, July 2), and sees deeper water t r a v e l l i n g away from him, on the under side of the plume ( l i k e the dye which  was  l a t e r found at depth, behind the f r o n t ) . Also, to an observer on the  tank,  the ground on which the tank i s t r a v e l l i n g i s coming  toward him, observer  at  and the  then  passing  front,  under  him;  similarly,  to  an  the dark blue water appears to t r a v e l  toward him, and then flow under the plume. A model to describe t h i s motion can control  be  developed  from  a  volume approach. Consider a cross-section of the front,  and draw a control volume around  i t . Fig. ., 59a. , Assuming  the  front moves ater, the control volume also moves atG". Taking z=0 at  the  base  of  the plume, at some z=h'  (Fig. 59a), the water  speed u (relative to a stationary co-ordinate system) equals the front speed CT . Above t h i s l e v e l , control  water  is  flowing  control  volume  or  frontal  system i s u-cT. Conservation of mass requires  or  the  volume, and below t h i s l e v e l , water i s flowing out. The  speed of water i n the  J  into  (u - cr) J ? + Q --. IC cr- \ dz ^ u  co-ordinate  56  A - j u c/z  c^A  - Q  = O. 4.15a.  0  Q  represents  processes  a rate of entrainment  and i s e n t i r e l y  previously.  wp  and  w  n  s p e c i f i c a l l y due to frontal  different  have  from  v  discussed  p  been ignored because the control  volume length i s assumed very small, of order  tens  of meters,  r e l a t i v e to the entire plume. Similarly, conservation of salt reguires:  A  h'  j Ct*-cr)sdz  + & 5>  *J  $(tA-cr)dz  = O  5  or  A '& ( sd*  h - f us dz  ~ O s  =  0  o .  4.16a.  Conservation of momentum reguires:  [  (u -cr)*dz-  + f (jj-crfj  z  - O'er + A p - K = O ,  A' or, using the continuity eguation,  A  ' ' l  u  ds.  /  ~& J u d z  + A p  - l< ~- O . 4.17a.  Here, K i s the excess momentum reaction at the front, due mainly to form drag. as  derived  1/2gh ^£ , i f P 2  possible  i n Chapter  3,  i s independent  to obtain  the net pressure  force i s  of depth i n the layer.  the pressure  force  fairly  It i s  simply f o r a  57  v a r i a b l e p (z). Consider a plume of thickness h, with i t s base at z=- (h-Ah)  (Fig. 59b). For zero pressure gradient at depth,  (the  assumed condition i n the water beneath the plume), we reguire \  h  , or £>k -Cp»-(p)  p  where  k  = J_  f  k  J^  The net pressure force, divided by the average density  4  ii  p  dz ' d ? P9  Z  4. 18. For the case p  ^f-  i  -£1  c  p  = ^7 = a constant, this becomes  2 y» as i n Chapter 3. L  {  If p - ^  - dJ  s  we  H  2,  p  2- /j^  *  (p -p )  pressure force i s '//^  If  k  0  s  ,  l i n e a r p r o f i l e , then the net  a  /p  , correct to f i r s t  0  order  had used the average density i n the formula 'l^g^.  we would have J/^  t p -pj)  s i g n i f i c a n t l y different Now,  0  [ p*  in  &fIp ,  )  from / / 2 . C fo'fs^)/p> •  to apply the eguations to some f i e l d data. On July 23,  1975,. we measured a current p r o f i l e and a  salinity  profile  at  about 200 meters from the front, i n the plume. (Fig, 39 and Fig.  58  40).  The  current  p r o f i l e was actually a speed p r o f i l e , so the  zero crossing and d i r e c t i o n were only velocities speed  to  17m,  inferred.  Referring the  the deepest measured, the r e s u l t i n g front  i s that r e l a t i v e to t i d a l l y moving deep water.  a plume depth of 6m, p =1.017 , p = 1.003, and assuming density  p - p ^ .p j  profile,  a  s  e  s  o  j  '  tfe  e  pressure  Assuming linear  term 4,18,  turns out to be . 42m s- . 2  2  The integrals of u and s are:  k j. udz = 1.25m /s 2  6 k  J u dz = 0.72 m V s 2  J  2  k  sdz = 100 m ppt  o A  f usdz = 12.9 ppt m /s. 2  0  With s =23.4, we get, by solving eguations 4.15a, 4.16a, 4.17a, ^  =.41m/s Q= 1.21 m /s 2  K = 0.63  m3/s ; 2  r e c a l l i n g that CT i s the front speed,  Q  entrainment,  form  and  K  i s the extra  i s the extra  frontal  drag associated with  propogation of the front. Note that the front speed i s considerably larger  than the  average speed, 1.25/6 =• .21m/s. Also note that Q ^ J u d z . That i s , a front extends i t s e l f , or propagates, by mixing egual parts of plume  water  and s a l t water. Also, one could write K=1/2C  D  h.  59  where C this  p  i s a drag c o e f f i c i e n t , which turns out  case.  Since values of C  the  drag  be  1.2 i n  for flow over blunt objects such  0  as cylinders are about 1, i t appears that K with  to  may  be  identified  exerted by the s a l t water as i t flows under the  blunt leading edge of the plume. I t i s informative to 4.17a  from  the  re-derive  eqns.  4.15a,  4.16a, and  d i f f e r e n t i a l form of the equations. Chapter 3 .  Thus, i n one horizontal dimension,  suppose  that  f o r x<cr(t),  there exists a plume s a t i s f y i n g the eguations: . A ?A * A- f u dz i- \AJ„ - UJ -6?<$(x-G~)=>o. P  2* J  dt  A  4.19.  ^  4.20. /  dt  A 2x  0  •<- uj„U  -  uo  u  p  ^  0  *  a  K  <S ( x - c r j  =  o  .  4.21. e£(x-C) i s the Dirac delta function, used to make the quantities Q,  Qs ,  and  6  K  non-zero  only  at the front, x=tf"(t). We s h a l l  integrate equations 4.19, 4.20, and 4.21 from x= <r-& to set  x=  ;  quantities at x= o">6 equal to zero, since there i s no plume  ahead of the front; and f i n a l l y take ^ ^ o . There r e s u l t s :  *  4.15b.  A  A  f 5 dz  ir J  O  - f u sd? J  A  - Q  $  b  ~  o.  4.16b.  60  Notice  that  disappear,  the since  quantities they  zero. The j u s t i f i c a t i o n proceeds  from  gradually, and partial  '  -9*%  - y w *  Sand  " °-  k  involving  w^,  w,  and  A  Ru|u|  are proportional to<£, which approached for t h i s  procedure  i s that  as one  Heads to the front, properties change only  are appropriately  differential  related  to  each  other  by  equations.,Then, at the f r o n t , the f i e l d s  of thickness, v e l o c i t y and s a l i n i t y change d r a s t i c a l l y i n a very short distance, of the order of a small only  be  related  to  boat  length,  each other i n terms of weak solutions. (A  weak solution i s a set of  relations  between  the changes  various properties across a discontinuity, tfhitham, We  can compare  and can  of  1974).  (approximately) the speed of the front to  the calculated speed, 0.41m/s. Referring to Fig.38a,  a l l three  drogues  drogue, S,  were  inserted at the front, and the shallow  stayed with the front. Assuming the deep drogue  travelled  with  the bottom water (not exactly true), the r e l a t i v e speed i s about .33 m/s. This i s 8055 of the calculated speed, but the d r i f t time of  the drogues, 1.5 and 2 hours, i s not very small compared to  the t i d a l period,  (nor was  the current  profile  measurement  time), so great accuracy can't be expected.. Summary of frontal c i r c u l a t i o n The  following  description  of  a  front  emerges from the  equations derived above. The front i s a mixing region which  i s pushed  of  water  outward by the momentum flux and the pressure  gradient, and retarded by the mixing of ambient  water  and the  61  form  drag  it  experiences  from the ambient water. The intense  mixing at the front can be visualized as  follows.  Relative  to  the depth at which u=0"i guite fresh water above this l e v e l flows into  the  mixing  region  at the front. Rater below t h i s  level,  flowing away from the front, i s quite s a l t y , and could only have picked up this s a l t by an  intense  ,  churninq  mixing  at  the  front. ta. ISINEMATIC WAVE APPROACH TO FBONTAL MOTION The  following  model i s intended to demonstrate, as simply  as possible, the way i n which fronts, described develop  in  a  time-dependent  plume  model.  in  Chapter  2,  We consider a one  dimensional model..The continuity eguation, without entrainment,  dt  1.22...  We further suppose that a l l the dynamics governing the plume can be characterized by  h  '  4.23.  where F. i s a constant.. ,F i s s i m i l a r to a Froude number, but the  dimensions  of  an"  acceleration  mathematical expressions. Although constant  the  simplify  subsequent  assumption  that  more  than  compensates  the  resulting  is  mathematical  for the physical inadeguacy of  that assumption. The approach used here i s c a l l e d the wave  F  may not be a good approximation for the plume, for the  present purposes the s i m p l i c i t y of analysis  to  has  kinematic  method because a l l the dynamic interactions are summarized  in a simple rule, equation 4.23,  and  we  prescribed by the continuity eguation 4.22  look  at  the  (Whitham, 1974) .  motion  62  Substituting f o r u i n the continuity eguation we f i n d :  4.24.  The  lines  (3/2) u  representing  are  the c h a r a c t e r i s t i c s .  the r i v e r mouth, assume the velocity  graph i s a series o f t r i a n g l e s , representing  At  x=0,  versus  time  t i d a l modulation of  r i v e r flow (plotted along the time axis, Fig. 61). Thus, on x=G: u= (2/3)t; 0<t<3 u=4-(2/3)t; 3<t<6 u=(2/3)t-4; 6<t9. The  4.25.  s  i m p l i c i t solution of 4.23, 4.24. And 4.25 i s  h=h ( ? ) , 0  u= u { T ) , 0  on the l i n e given by x= (3/2) \i {'T.)(t- -? )  4.26.  a  where The  h^('C) and u ,>(-£:) are the values of u and h at x=0, t= f .  solution i s displayed  i n a c h a r a c t e r i s t i c diagram i n the x-t  plane - f i g . 60a. However, two complications arise. 1). We w i l l assume here that i n i t i a l l y there i s no in  the S t r a i t  time t=0),  fresh  water  (perhaps because of a windy period before model  so that water issuing from x=0 i n this model forms  a  front. This front i s a boundary to the region of v a l i d i t y of the plume  equations,  to Fig. 60b, at speed=u,  the  4.26, and i t s motion must be found. a  point  local  water  (s,t) on speed.  Referring  the front, ^^/dt ~ Since  the point  front  (s,t) i s a  termination point f o r a c h a r a c t e r i s t i c , s and t are related  by  63  the  implicit  solution  4.26.  For ^<3,  that  is  for  water  discharged before t=3, the front i s given by 4. 27.  dt  4.28.  Eguation  4.28,  s=2/9t .  To  than 4.27  and 4.28  z  functional  with  initial  continue  form  integrated  condition  s=t=0  the front beyond t-3  has  different  t  must be used, r e f l e c t i n g the u(f)  has  fact  changed. Eguation 4.28  numerically,  as  solutions  solution formulae that  the  must then be  can't  be  found  analytically. 2).  The  other  hydraulic  jumps.  characteristic characteristics functions  complication These  overtakes for  >6  occur  curve  the  formation  whenever  of  a  shocks or  faster  moving  a slower moving one. For example, the have  speeds  which  are  increasing  of time, whereas those for t<6 decrease. Consider  c h a r a c t e r i s t i c s , one f o r H>6, The  is  (s,t),  the other for  L!<6.  (Figure  two 60c)  the path of the hydraulic jump, must s a t i s f y  both c h a r a c t e r i s t i c r e l a t i o n s : S,  C- tl,  r = Lt-L  +L)CL—C,) -TO  We further reguire a jump relation..Integrating eguation across t h i s jump: 5+  or  4.29. 4.30. .  the  continuity  64  5+<6  The  integral approaches zero as e tends to zero, and we are l e f t  with  4.31. , for u /h = F. 2  The  solution to egn. 4.29, 4. 30, and 4.31 i s s=. 229.. , (t-6) . 2  These  internal  jumps  may  represent  some  of the weaker  discontinuities one sees i n the plume, f o r instance q,r  July  at  station  2, 1975, Figure 26. Since the water on either side of  the jump took d i f f e r e n t times t o arrive at the discontinuity (t%  i s less than t-tz) , the amount of  hence  the colour,  will  be  silt  i n suspension, and  d i f f e r e n t . These fronts have been  termed i n t e r n a l fronts, to distinguish them from the true front, which i s the boundary of v a l i d i t y of the plume equations. Proceeding on i n t h i s manner, the diagram in Figure 61 be  drawn.  Figure  instantaneous velocity  62  is a  photograph  of  plot  of  may  u versus x at t=14 - an  the d i s t r i b u t i o n  of  downstream  i n the plume. B e c a l l that h i s proportional  to u , so  t h i s i s also a plot of plume thickness. Using a rather  mixed but  2  convenient set of units, x could be measured i n nautical t  i n hours,  (knot) /m. In 2  bounding  u  miles,  i n knots, and h i n meters. F would then be in  Figure  61,  there  is a  strong  colour  front,  the region of solution. At s = 30.5, the front speeds  up, since faster water, originating from'2=9.9 catches  up  with  65  it.  are hydraulic jumps originating from ^-6, tl -12, etc.  There  Looking at Figure 62, one could imagine the of  observations,  proceeding  following  sequence  out from the r i v e r mouth. At s=1,  there i s an i n t e r n a l front, corresponding to faster moving water catching up with slower moving water. The slow water more  time  settled  since  out,  it  and  left  this  has  spent  the r i v e r mouth, so s i l t could have  front  might  show  up  as  a  colour  discontinuity. Again, at S=11.8, another i n t e r n a l front appears, and  at  s=19.6  the true front i s found. The front at s=19.6 i s  s t i l l quite stronq, although i n r e a l i t y , because of  dissipation  and spreading, i t would be very weak. As  mentioned  described here i s However,  at  the  based  on  beginning of the section, the model grossly  oversimplified  dynamics.  i t c l a r i f i e s mathematically the formation of fronts i n  a time dependent system., 5j_ MIXING AND FLUXES ACROSS AN INTERFACE The introduction of an isohaline plume  model  vertical  presents  fluxes  interface.  In  of  complications  salt,  many  water,  such a way that <^>. (/< •turbulence*  in  trying  and  assumed  term, ^ ^ .(/< ^ / ^ J where X (  a  boundary to  momentum  for  our  model the  across  that  f l u i d mechanics calculations the turbulent  mixing of a scalar guantity i s diffusion  as  one  <^V<)*-)-z -^ 'd<),  quantities.  u  (  to  be  governed  by  attempts to choose K in  where the primes  refer  where  dz'  to  In t h i s section we w i l l assume that a  diffusion equation governs the d i s t r i b u t i o n of s a l t ,  ..Dt'  a  '  4.32  66  D t '  ^  "  '  and where here primes denote dimensional  quantities and u and v'  are components of some appropriate advection v e l o c i t y . Thus 4.32 describes Equation system  the downstream 4.32 w i l l be  travelling  evolution  solved  at  of  a  numerically,  salinity (in a  profile.  co-ordinate  so /ofc'is replaced by t>/dt) ; then  (u', v) ,  D  choosing 0.8 times the maximum s a l i n i t y as the bottom upper  layer  6>s/)t f of  o  r  we  obtain,  from  of the  the numerical solution, <^/6z?and  this upper layer, we then try t o model  the evolution  the upper layer thickness and s a l i n i t y using the two fluxes  discussed i n Chapter 3, entrainment and depletion. Two choices of K are considered  in the solution  of 4.32;  K= a constant and K=az , where z represents the v e r t i c a l 2  from  the free  distance  surface, p o s i t i v e downwards. Scaling s with S , e  the maximum s a l i n i t y , and t with T^, a c h a r a c t e r i s t i c time, z' with  Z„, the depth  where  the s a l i n i t y  and  i s approximately  constant, we want to solve  it  dZ  2*  4.33.  where s*=sS , t'=tT , z*=zZ , and K*=KZ /T , guantities 2  0  6  0  primes being dimensionless. s=0  for z<.2, t=0  s=1  for ,2<z<1, t=0.  Thus  initially  K  JS/^gzOr  The i n i t i a l conditions are:  on a layer 0.8Z* thick, of s a l i n i t y S^,  z=0, the free was  without  there i s an upper layer 0.2Ze> thick, composed of  fresh water, f l o a t i n g At  0  imposed;  surface, at  the condition  of  no  flux  z=1, representing the deep, well-  67  mixed water,  s=1.  Equation conditions for  4.33  with  the  above  and  initial  was solved numerically. Figure 63a shows the results  K=z /(.2) , at times 0, 20At, 2  2  At=.4(Az) , and Az,  120At,  and 320At, where  220At,  the v e r t i c a l grid s i z e , equals 0.02.  2  63b shows the r e s u l t s for K=1, and  boundary  at times 0, 204t,  Figure 580&t,  220At,  920 At..To compare these p r o f i l e s with quantities available  in an upper layer model of the plume, I chose 0.8S salinity  characterizing  the  base  of  the  to  be  the  upper layer. Thus,  Figure 64a shows the depth of the upper layer  versus  time  for  the variable K case, and Figure 64b shows the t o t a l s a l t content S, $Sdx Figure  above the level s=.8S , versus time. As i s evident from c  64a,  the  upper layer i n i t i a l l y  increases i n thickness,  then decreases - remember that at t=0 i t s thickness was 0 . 2 z « o  There i s considerable difference i n the s versus of  Fig.  63a and 6 3b,  with F i g . 63a, K<* z  l i k e the observed p r o f i l e s of s a l i n i t y . speculate  why  this  functional  form  hypothesis  (Launder  is  prescribed and  dealing with a velocity characteristic of  so.  First, by  time such as  )  It  we  is  Kc/z  2  is  interesting  to  partially  the  mixing  1972), /Since can't  curves  appearing to be more  Prantdtl's  Spalding, profile,  2  t  however  length  we  are not define  a  to complete the s p e c i f i c a t i o n  K i n terms of mean flow properties as in the complete Prandtl  mixing length theory, where K ~  Second, the K of z  2  form  i m p l i c i t l y accounts f o r the effects of v e r t i c a l s t r a t i f i c a t i o n near the surface the mixing  is  stratification  is  strongest,  and  hence  i n h i b i t e d the most and the supply of fresh water due  to the velocity shear i s greatest. For both these  reasons,  the  68  effective  value  of  K  should  be  relatively  surface, as i t i s when K i s proportional We wish to model the behaviour  near the  h  the  to z . 2  of  eguations, s i m p l i f i c a t i o n s of egns. 3.24  small  s  and  and  by  two  3.25,  4.34.  where  w  is  p  the  v e l o c i t y , °< i s a salinity  less  entrainment salinity  greater  than  w „ i s the depletion 0.8S ,  and  o  @  is  a  than or egual to 0.8S . S denotes the t o t a l s a l t o  content in the layer, obtained  velocity,  J ^ 5 c  2  . .The quantities ^/^fc and  from the K=z /(,2) 2  2  were  solution and plotted versus s, the  average s a l i n i t y i n the upper layer. They  were  roughly  fitted  with w = 1150 p  w =2600s i f s<.5 w^=2600s-8000 (s-.5)  2  ifs>.5  =0. 9, § =0.8. Wp  and  w  are converted to dimensional units by multiplying  K  the r a t i o Az/T , where T„ i s of order 8 0  order  0.5  m.  hours,  and  The results of numerical integration  4.35 using the above v , p  w^, , u  and P, are shown as  Az  is  of 4.34 the  by of and  dashed  curves- of Fig. 64a and 64b. There i s qualitative agreement with the results from the d i f f u s i o n eguation. There i s not much point adjusting w , w , p  in  the  real  n  and (3 to make the agreement better  because  world we don't know K(z), and there are important  effects due to velocity shear. However t h i s example demonstrates  69  the use of two types of exchange across an interface (u^ and  w) n  to model a mixing situation, is. ANALOGY WITH TURBULENT JETS Another way to depletion,  with  view  the  reference  phenomenon  to  the  of  plume,  entrainment is  to  consider a  turbulent plane j e t (Abromovitch, 1963). Figure 65 shows a of  an  and  plot  isoconcentration curve of a passive scalar discharged  the j e t , (X i s downstream, vectors  Y  is  cross-stream),  and  by  velocity  at various points. We see that for X l e s s than about 5,  the net flow across the isoconcentration curve i s into  the  jet  as defined by the iso-concentration curve; whereas f o r x greater than  5,  the  net flow i s out of the jet region. Schematically,  the situation i s shown i n Figure 66a. The curves AD and A*D'  are  the usually defined boundaries  the  interface  between  turbulent  of  the  and  jet,  representing  non-turbulent  wanted to consider only a region where there  are  flow.  If one  significantly  c h a r a c t e r i s t i c properties of the j e t , as defined for instance by the  concentration  of  a s c a l a r guantity discharged by the j e t ,  that region would be bounded by ABCB'A*. As F i g .  65  indicates,  along AB and A* B* there i s net flow into the jet, and along  BCB'  there i s net flow out. How  does  this  concept  apply  to  the  section through the plume. F i g . 66b. Because solve  the  plume  plume? Consider a we  eguations i n a f i n i t e region, a  are  going  computational  open boundary i s indicated in Figure 66b. I f we choose s=25 as  to  °l*t>  our plume boundary then a l l of the Fraser River water , plus  entrained water, flows out of the plume s o l e l y by means depletion  mechanism,  since  the  flow  of  the  components (u,v) do not  70  cross the interface i n the here.  If,  however,  we  shallow  water  eguations  developed  choose s=28 °/oo as the plume boundary,  part of the inflow i s balanced by depletion, but there  is  also  horizontal outflow over the depth h ., The  computational  aspects  of  these  two  choices  considerably d i f f e r e n t . For the case where the iso-haline to  the  surface  computational isohaline  within  boundary  intersects  the  computational  becomes  the  line  are  comes  region, the actual along  which  the  the surface. Thus, one has to numerically s  move t h i s bondary across the grid system - a not impossible feat (Kasahara,Isaacson and Stoker, 1965), but a complicated one.,For the case where the iso-haline does not surface, one then has the problem of an open boundary, and i n particular specifying  u,  s,  and  the  problem  of  h on an inflow, and also the problem of  generation and r e f l e c t i o n of f a l s e waves at the boundary.  71  CHAPTER 5 N2HISICAL MODELLING OF THE FRASER RIVER PLUME The  numerical  categories.  modelling  First  a  model  described  and  quick  to  show  falls  into  2  f o r a small rectangular region was  developed. This model had the benefit of run,  below  up  any  being  problems  p a r t i c u l a r l y the open outflow boundaries.  Once  inexpensive at  to  boundaries,  a l l the  terms  described i n Chapter 3 were i n the small model, a few runs using a  larger  model  simulating the r e a l S t r a i t of Georgia geometry  were made. The aim of t h i s chapter reasonably potential  flexible to  model  become  an  has  is been  accurate  tool  to  demonstrate  that  a  developed, which has the in  understanding  the  c i r c u l a t i o n i n the S t r a i t of Georgia. The  equations used i n t h i s model are 3.24 - 3.27,  simplification  that a l l properties were assumed  with the  homogeneous  the upper layer. Thus: 3h,9U8V 3t 3x 3y  =  WW 1  N 5. 1.  3S 3t  _3 US 3x h  3 VS 3y h 5.2. +  - fV  +  +  h _3 3x  3 3x A _2 3x  U  in  where 0,  V are v e r t i c a l l y integrated transports,  g'=g Ap/p =24-0. 8S/h a = horizontal eddy v i s c o s i t y , discussed l a t e r i n t h i s chapter £  =barotropic t i d a l elevation  S=vertically integrated s a l i n i t y u„, v© = t i d a l streams 0"ie-' V^=relative transports=0-u h, 7-v h. 0  0  K = quadratic f r i c t i o n c o e f f i c i e n t v  = entrainment  f  w  n  s  e  = depletion  velocity  velocity  = s a l i n i t y of water underneath the plume  h = plume thickness f = Coriolis  parameter  It should be noted that the equations to be modelled are i n divergence form. That i s , they are of the form  £_f  +  \7-  F CP )  + Q C ? )  = O-)  where a l l s p a t i a l derivatives are exact d i f f e r e n t i a l s . This form of  the  differential  equations  allows one to write the f i n i t e  difference equations i n such a way that there  are  no  spurious  73  sources or sinks i n the d e r i v a t i v e terms. The numerical scheme used as a s t a r t i n g point was the semiioplicit  scheme  of  Heaps  (Flather and Heaps, 1975). Although  t h i s scheme works very well for t i d a l some  doubt  about  how  amplitude  gravity  wave  is  to  one)  are  a  close  velocity.  indication that high flow v e l o c i t i e s close  was  well i t would work with the upper layer  model, i n which the f l u i d v e l o c i t y small  c a l c u l a t i o n s , there  problem,  as  to  the  internal  However, there i s no  (internal  Froude  numbers  long as one s a t i s f i e s the  s t a b i l i t y requirements of the scheme. A t y p i c a l element of the computational  grid  is  Figure 67, and Figure 68 shows the entire computational the  rectangular  model.  in  grid for  Note i n Pigure 68 that an extra row of  aeshes i s provided around the outer edge of the facilitate  shown  c a l c u l a t i o n s near boundaries.  mesh  area,  to  Also note i n Figure 67  that only one subscript i s used to denote the p h y s i c a l location of  a  mesh  point., The  systematics of t h i s indexing scheme i s  apparent i n Figure 68, and the reason for i t s use i s to increase computational  efficiency.  Certain s p a t i a l averages are defined below, for use i n finite  difference  considered to be changing  a  equations  at  the  subscript  changes the row,  same  which time  follow. level,  All and  fields recall  the are that  by n, the number of columns i n the g r i d ,  a change of 1 unit i n the y-direction.  The averaging operations for the f i n i t e difference formulations are defined as follows: ,  °i  =  *< ± U  +  1  W  74  h =  <i v  +  V  +  h  <i  +  V  h  <i  +  Z  v  Z  i l>  i-n> [Z = h]  i 1> +  h  <i  =  k  <i + V l  +  =  h  <i  +  +  =  h  <i  +  z  U  V  Z  h <zi h  W  +  V l Z  +  +  + 2Z  11  +  +  < i-•a z  i1  +  2 Z  i  +  i+n  l-  V.  i-n + V. i--n+1)  Z,  i-n + Z. i -W  W W  With A i l as the s p a t i a l g r i d s i z e , A t : as the timestep, obtain the following set of f i n i t e difference  eguations.  The finite difference representation of the continuity equation i s : [Z (t+At) - Z (t)] 1/At = ±  ±  - [ u ^ t ) - u ^ C t ) + v _ ( t ) - v ( t ) ] I/AA ±  + WP., (t) + WN (t) 4  n  ±  we  75  The finite difference representation of the "salt equation is: [S (t+At) - S (t)] 1/At = ±  .  i  ru (t)  S* (t) - U _ (t) S*^ (t)  ±  1  1  z*(t) v  +  1 - n  ( t > s-y (t> _ v  (t>  ±  z\ (t)  + WP,(t) S - WN,(t) i . ° z (t) S  ( t )  1  ±  5.6  The finite difference representation of the reduced gravity calculation is: yt+At) • 24 - .8 1  i Z (t+At) S  (  t  +  A  t  )  i  5.7  The finite difference representation of the x-directed momentum equation is:  |u (t+At) - U (t)Jl/At = f V (t) ±  ±  ±  G (t+At) (Z (t+At)) - G (t+At) (Z (t+At)) 2  i+1  ±+1  ±  ±  2A*.  -K  (U<t) - u(t)  Z*(t)  T  ±  Z* (t)  (0\(t) - II (t)Z*(t)) + <V.(t) - V (t)Z^(t)) 2  tp  (u*(t)) (^(t)) z*(t) z(t) &T(t> v*_(t) u^(t) vj(t) 2  1_ Afc  2  i  1_ AJt  n  i+n'i-n '  2  :  76  - g TSX (t) + WPj(t) U ( t ) - WN^(t) ±  U  0  l  (  t  )  Z*(t) u A  I  I  +  i+i  V >  (t)  fc  z^ (t)  z* ( t )  + 1  u  i - •i(t) .i  (  t  U (t)  .  U (t)  ±  ±  +  i+n  z*(t)  )  Z*(t)  U.(t) z*(t)  5.3 The f i n i t e d i f f e r e n c e representation of the y-directed momentum equation Is:  V  (t+At) - V ( t )  ±  1/At = fU (t+At)  ±  1  ±  G (t+At) ( Z ( t + A t ) ) ±  2  ±  - G  i+Q  (t+At)  (Z  1 + n  (t+At))  2M  (V (t) - v (t) z ^ c t ) ) ±  -K  T  (Z^(t))  if(v.(t)  2  - v (t) z ^ C t ) ) T  l  2  + (u (t) - u (t) z ^ C t ) ±  T  ^-1  ^ f n - l ^  AX,  W l  _L AJt  (v^Ct))  2  (Vj(t))2 zj(t)  (  t  )  ( t )  2  2  77  i+n  (t) V  < > ~ ™ihi > t  0  (t  V  i  (  t  " § TSY^t)  )  (t)  \  V.(t)  (t)  +  A  i-n 5.9.  where  TSX^  and  TSY  are  C  the  s l o p e s of the water s u r f a c e as  o b t a i n e d from a b a r o t r o p i c t i d a l model of the same a r e a . . The time s t r u c t u r e of t h e s e e q u a t i o n s i s In S  t  each  computational  quite  important.  c y c l e , the t h i c k n e s s Z a n d s a l t  content  L  are c a l c u l a t e d u s i n g v a l u e s of the d e r i v a t i v e s of U /  from  the  p r e v i o u s timestep. Then, 0  d e r i v a t i v e s of Z ,• and s (or g,) c  the  t  from the c u r r e n t  p r e v i o u s v a l u e s of v ^ i n the C o r i o l i s  the  current  d e p l e t i o n , and viscosity  0;  i n the C o r i o l i s  timestep,  term. F i n a l l y , Z/,  and  term. In the  the values o f uy,  V,,  and  V  and 4  S , c  are and  entrainment,  f r i c t i o n f u n c t i o n s , and the n o n - l i n e a r  terms,  v,  are c a l c u l a t e d , u s i n g the  c a l c u l a t e d , u s i n g d e r i v a t i v e s o f the c u r r e n t also  and  and  s- from the  eddy  previous  timestep are always used. Hhen u s i n g f i n i t e d i f f e r e n c e methods partial accuracy  d i f f e r e n t i a l equations, of  the  s o l u t i o n and  chosen i s s t a b l e . Accuracy  one  to  solve  non-linear  i s always concerned  with  the  under what c o n d i t i o n s the scheme  i s best  assessed  by  comparing  the  78  numerical  solution to an a n a l y t i c solution, or possibly to r e a l  observed data. S t a b i l i t y analysis, discussed i n the appendix, i s usually done for the l i n e a r i z e d equations, i n the hope that requirements  in  the  non-linear  case  are  the  not much different  (local s t a b i l i t y , Richtmyer and Morton,1967).  THE SQUARE BOX MODEL Modelling was  started  using  a  linearized  form  of  the  eguations without t i d a l forcing, and with constant density; 9y  It  q  at The  'h  dk  +. r V * O-  17  river  flow was s p e c i f i e d by U t a n h ( t / T ) , where T 0  0  time steps. -Mo was 50,000 cm /sec, Ax was 10 2  the  s  cm, so  U Ax,  3  Hith  friction  10 cm/sec , and the timestep was 480  varied  from  outflow  given  by  r=.005/h sec-r* the thickness over the  30cm/sec  at  and  thickness  and  are  not  unreasonable  approach  to  the  initially  equilibrium.  condition used was ^ ^Idi^-O, outflow  values  of  for a time-averaged plume - Chapter 2.  Figure 69 shows a plot of the f l u x out of the open illustrating  velocities  the r i v e r mouth to 3-5 cm/sec at the  boundaries., These  velocity  sec.  2  entire area varied only between 503 and 515 cm,  the  that  200  r i v e r discharge, was 50Q0m /sec, approximately one half the  freshet value, q' was  time,  was  0  At  the  rapid, open  ends  versus  and then very slow, ends,  the  boundary  where n i s the direction normal to  boundary (y-direction i n t h i s case). This boundary  79  condition was chosen as being the simplest one which stated that there was l i t t l e s p a t i a l change near  of  important  flow  properties  the open boundary, and s t i l l allowed there to to be time-  varying conditions at these open boundary thus  boundaries.  The  river  mouth  condition was the s p e c i f i c a t i o n of a transport. We are  assuming  condition  subcritical  flow,  specified.  The  is  since  only  internal  one  Froude  j30 /(10x500) =0.45, so the flow, as determined by  boundary number i s  the f r i c t i o n  2  in the system, i s indeed s u b c r i t i c a l . The  next step was t o add the convective acceleration terms  to the equations critical  near  of motion. the r i v e r  Since  mouth,  important as the gradient of added,  the grid  adjacent  plume  the gradient  1/2g'h . 2  When  to the river  unstable, with the depth rapidly rapidly  the r e a l  of u  these mouth  decreasing  i s near i s as  2  terms  were  became  very  and the velocity  increasing as the r i v e r flow was turned on. When sguare  law f r i c t i o n eliminated.  and entrainment  were  added,  this  problem  was  The entrainment was written as w =0.00640" /(g'h ). 2  3  p  This i s a deviation from the intent that Wp should be written as Eu, where E i s a U /(g h ), 2  ,  3  constant  times  which would make w  p  Interestingly,  the  real  the Froude  number  proportional t o U  geometry  model,  3  (Long, 1975a).  discussed  indicated that 2 was too large a power of 0 i n the formula.  Values  sguared,  later,  entrainment  of the f r i c t i o n c o e f f i c i e n t between 0.001 and  0.007 were used i n the course of the modelling discussed below. With  the addition  (entrainment  and sguare  of these law  two momentum  dissipators,  f r i c t i o n ) , the region around the  r i v e r mouth was stable. However, the outflow  boundary  was now  80  unstable. at  by  problem  the outflow  flow At  This  field  produced  o  river  inflow  mouth w i d t h  and r e m a i n i n g  was  d i v i d e d among t h r e e  constant  the  an e x t e n d e d  period  system  because o f t h e i r  total  then  no  the  fields  74  a t two t i m e s ,  and  71.  and  t e r m s was r u n  efflux  elsewhere  and with  the  3  the  flow  plus  the  open  t h e eddy  other,  viscosity  c m / s e c , so t h a t t h e r e 2  system  due  75 show t h e r e s u l t i n g  other  of  used  s t a t e , h o r i z o n t a l eddy  boundaries 10  system  (river  out  of  the solid  t o approach each  a steady  short  noodling  along  t o t h e system  At s o l i d  of  r e t e n t i o n w i t h i n the  particularly  total  To a c h i e v e  t o be z e r o ;  Figure  This  what was p r e s u m a b l y  appendix,for  n e t c h a n g e i n t h e momentum o f  friction.  70  F i g . 72,  and  flat.  o f 2000  0.333,  slow phase speed. Note t h e  was i n t r o d u c e d .  taken  was  - the production  we would l i k e t h e two c u r v e s  remain  viscosity  instability  on t h e i n f l u x  entrainment),  boundaries,  was  in  have  given  meshes, f o r a r i v e r  i t developed  As d i s c u s s e d i n t h e  oscillations  grid  n o n - l i n e a r t e r m s , and t h e i r  vectors  was  thereafter.  F r o u d e number  of time,  by  boundaries.  value  model i n c o r p o r a t i n g t h e n o n - l i n e a r  waves  velocity  discharge  was 0.001.  t h e well-known n o n - l i n e a r the  was .33 km.  i t s maximum v a l u e  at that  o f 1 km. T h e o u t f l o w  friction When  attaining  number  70, 71, and 72 show t h e  b y t h e model a t t h i s s t a g e . A x  0=0.5x0 (1-cos(2 t/T„), a t 6 hours,  and  Figures  was 120 s e c , g* was 10 cm/sec. The r i v e r  3  for  by s p e c i f y i n g t h e F r o u d e  t o be a c o n s t a n t .  i /sec  the  was c u r e d  model  conditions identical  F i g u r e 73b shows how t h e i n f l u x  to  to  was  sidewall velocity Figures  and e f f l u x g u i c k l y  81  approach  an e q u i l i b r i u m  The as  the  matter  question arises,  what i s t h e c o r r e c t  outflow  number?  Froude  hump  of  water  h=50exp ( - y / 6 ) , w i t h 2  direction,  the  curve  is  indicated  except  like  greatest  by a s o l i d  the F = 1 case. 2  a kinematic  friction,  and  reasonably  differential for  dot,  that  t h e depth  11  i s  cm  curves  I t appears  need  less  (solid  t h a t a flow  prescribe  This of course  describes  eguation  a linear  case,  elevation  i s  i n t h e xmouth).  through the F  2  except  s t r o n g l y outward  (1974)  Whitham  an  y=0, f o r F = 0 . 3 3 . , F o r  wave - t h e d y n a m i c s a r e  u p s t r e a m end o n l y .  of  allowing  as i t i s passing  =1,  2  i n the l a s t than  line)  mesh, where t h e v e l o c i t y  we  specify  thickness being o f f the r i v e r  velocity  f o r the l a s t  of  ( e l e v a t i o n s were u n i f o r m  1 2 meshes from  the  result  (the d i s t r i b u t i o n  outward  identical,  Similarly,  to  t u r n s out t h a t i t doesn't  the  l a r g e d a s h e s show t h e hump j u s t  open b o u n d a r y ,  value  y measured i n u n i t s o f hjl ) , shown a s s m a l l  2  dashes, t o propagate  for  It  much. F i g u r e 7 6 shows  very  initial  The  level.  for  were  2  identical,  t h e plume  mainly  mesh,  F =.33.  was 2 cm/sec  like  the  faster travels  controlled  by  a boundary c o n d i t i o n a t the  applies only i f  the  flow  i s  a t t h e open b o u n d a r i e s .  I n h i s book,  how t h e e f f e c t i v e  of a  decreases,  order  under t h e e f f e c t  of  partial  friction,  and t h e plume seems t o be an example  of t h i s  phenomenon f o r a n o n - l i n e a r c a s e . We a r e c o n c e r n e d the  linear  propagate working  case,  a t jg~^1b71 -  with  we  dispersion  want,  solutions.  However,  we  know  scheme.  In  f o r example, a l l d i s t u r b a n c e s t o  It i sdifficult  i n the non-linear case,  i n a numerical  t o c h e c k how w e l l a scheme i s  because o f t h e l a c k of that  analytic  i f there i s a disturbance  82  propagating spatial at  the  into a region, with the  d e r i v a t i v e s d i s c o n t i n o u s , then local Thus,  can  t h i s i n F i g u r e 77,  normal  to  a  the  It  evaluated  river  has at  calculation  of  which i s a  opening,  front,  the  effect  is  position  , so t h e f a s t e r  that  F i g u r e 77 cm/sec cm  effect we  and  timestep  of the  the  Q  KM(  a  which  for timestep  guestion,  of g r i d  speed  it  =  s i z e on  agreement  the  due  is.  100),  and  front  a  the crude  of  the 25%  to the  non-linear  have  implicitly  We  h was  there i s l i t t l e  to  50,  was  5  (versus  44  variation  of  42 cm  temporal  dispersion,  the s o l u t i o n . o f two  3  and  u  concerns  I n F i g u r e s 78,  different  Each  (6x=1km,  number i n p a r t A was  meshes, c o r r e s p o n d i n g  to 1  mesh  same f i e l d s At=240  obtained  for  the  79,  the  80,  and  s o l u t i o n s t o t h e same  (Ax=0.33 km. A t  p a r t B shows t h e  t i m e f o r a l a r g e r - s c a l e model 3  in  i s of c o u r s e  At t i m e s t e p  t h e model d i s c u s s e d s o f a r  2000m /sac) ,  ra /sec).  river  were c o n s t a n t a t a l l t i m e s .  100.  related  compare t h e r e s u l t s  v from )(  The  (u+f^h ) 100 A t ,  p r o b l e m . P a r t A o f e a c h f i g u r e shows the d i s t r i b u t i o n and  direction  g r e a t e r t h a t fg*~h . U i s a b o u t  propagation  so  We  F i g u r e 74.  and  t o note  h a t the f r o n t  100),  (Whitham,  the  o u t from  front  travels  a t u+ Jg * h \  in  The  their  at the f r o n t . ,  Another  81  u  i s drawn  properties  breaks  h from  10Ax,  9.95Ax.  ( t h e same a s a t t i m e s t e p  at  o f u and  but i t i s e n c o u r a g i n g  s h o u l d be a p p a r e n t ,  assumed  plot,  about  d i s t u r b a n c e moving a t a speed  iTg"^?  it  of water propagating  travelled  the  verification,  of  that disturbance  " g e n t l e " d i s t u r b a n c e propagates  d i s t u r b a n c e i s a bulge mouth.  c o n t i n o u s , but  c h a r a c t e r i s t i c speed, before  1974). see  fields  sec,  of =120  B  u,  sec,  a t t h e same Q ^ « = 2000  by a v e r a g i n g part  h w,  over  fields.  9 In  83  general,  the  agreement i s b e t t e r t h a n  meshes o f t h e plume. H e r e , in  the  0.33  km  10%, e x c e p t i n t h e o u t e r  one i s g e t t i n g  close t o the  boundary  mesh model, and t h e 1 km mesh model i s h a v i n g  trouble resolving  the s i t u a t i o n  a t the front.  IIDAL EFFECTS IN THE BOX MODEL The  n e x t i t e m s t o be i n c l u d e d  the e f f e c t s finally  of  varying  tidal  elevations,  river  amplitudes o f t h e t i d e Georgia,  were t h e t i d a l  flow  then  were  tidal  those  in  o n l y . The p e r i o d  12 h o u r s , f o r c o m p u t a t i o n a l e a s e . The  tidal  First  currents,  introduced.  approximated  f o r an M2 c o n s t i t u e n t  effects.  and  The p h a s e s and the  Strait  of  was t a k e n t o be  parameters  chosen  were:  C cot  cos  v ^  ZD  JJL  2  y  and value  (  that  z¥°)  L>t -  ;  132 ) ; 6  (Los C LO£- 2&i°)  low water  c o r r e s p o n d s t o maximum ^/dy  0.7 h o u r s  (as determined maximun s t r e a m s  (2H d e g r e e s  lead  the elevation  Cross  channel  assumed t o be z e r o , f o r s i m p l i c i t y these t i d a l  f l o w s near boundaries  of  from t h e s u r f a c e c u r r e n t  f o rthe Strait.  When  .  f Ax  f l o w o c c u r s about water  s  ±0_  D  Note  co  -  effects  , maximum  phase)  before  low  meter. C h a p t e r 2 ) ,  by 72 d e g r e e s , a  slopes  river  and  typical  streams  were  only.  were added, i t was f o u n d  were anomalous. F i g u r e 82 shows  that the the  flow  84  field  from  condition causes  and  the  would n o t slopes little  a  model  with  Coriolis  force.  be p r e s e n t  relation in  the flow  i n the i n t e r i o r .  at  open  forcing)  boundary  at  pile  up  w i t h t h i s boundary force, field, 0.001  and but  i t  river  was  velocity  this  the v e l o c i t y  t h a t the  The  flow  extrapolation  directions  (due  to  tidal  flow.  found  elevations,  F i g u r e 84  that  the a  based one  on  o>  course  better  ^ fdn ~0 l  mesh i n from  of  t  the  and  field  Coriolis  shows t h e same f l o w  (K=0.005 i n t h e  During  at  tidal  shows t h e f l o w  ( i n terms o f the u n i f o r m i t y of the the  field.  boundary c o n d i t i o n i n  F i g u r e 83  friction  case.)  calculate with  flow  bears  t h a t s p e c i f y i n g e)  a satisfactory reversed  cross-channel  t h e i n f l o w boundary  a smooth  found  which  same t i m e e v e r y w h e r e i n t h e model,  f o r much l o w e r  experiments,  I t was  c o n d i t i o n , with  this  t i d e with  interior  like  boundaries.  constant  in  condition  the  at t h e  look  was  flow  about  component  a b o u n d a r y c o n d i t i o n was  should  the  the  boundary  of the C o r i o l i s f o r c e  cross-channel  flow near  flow i n  selecting  t h i s respect -  presence  i f a more r e a l i s t i c  to the  t h e open b o u n d a r i e s  the  The  f l o w t o have a l a r g e  were p r e s c r i b e d . The  object  didn't  t h e c o n s t a n t F r o u d e number  first of  subseguent  outflow flow and  case,  boundary  field) then  was  to  average  boundary.  S A L I N I T I CALCULATION At t h i s was  started.  p o i n t , the  calculation  Initially,  of the s a l i n i t y  t h e l e a p f r o g scheme was  dropped, because o f the computational  mode,  Appendix.  fiichtmeyer  I t was  i n t e n d e d t o use  to  a programming e r r o r ,  so  in effect  the second  only a f i r s t  the  used,  discussed  used  but  was  in  the  scheme, but  order c o r r e c t i o n  o r d e r scheme was  distribution  was  for  due  omitted, the  salt  85  advection salt in  equation.  introduced  the  The  salinity Figures  run  salinity,  85  to  m /sec, how  thickness perfect  fields  12  and  hour  high  constant  velocity back and  river  stage of  t o the  between  i n the  absence of  tide.  I t was  found  but  there  transient  that  p e r s i s t s , which  was  still  figures  a  based  on  that  don't as  was  2000  and 92  salinity  and  to and  There should 92  93,  be and  with  s p e c i f y i n g a somewhat  removed some o f some l e f t  was  flow  Figures  the  tide.  for  a C o r i o l i s f o r c e , and  later  »e  inflow  constant  river  tide.  advected  f o r t h due  inflow  friction  the  on  density  between  field  present,  flow,  varied  flux  boundary  diaqrams  the  cm,  s  95,  on  vector  used  outflow.  (1 km),  symmetry  salinity  last  10  the  ppt)  the  asymmetry  (presumably a yet  start-up  know how  to  damp  effectively).  ROLE OF It  DEPLETION was  at t h i s  mechanism was proportional field,  salinity  92  t o 95.  to  1/g',  there  was  small.  one  proceeded  r a t e of  1/g*  with  the  reguirement of the  a p p a r e n t . The  was  as  The  replacing  stage that  becoming  where g'  and  by  the  that  The  was  {20  out  on  was  the  was  outflow  force.  entrainment  left-right  between 94 a  the  i t had  e f f e c t of  v a r i a b l e g'  condition  show a s e q u e n c e o f  d e p e n d i n g on  3  show  value  sec. Ax  the  as the  boundary  Coriolis  120  3  95  91  no  2 0 0 0 m / s e c , and 7000  the  long  density  F r o u d e number f o r t h e  elevations,  and  0. 005, A. t was  the  as  salinity  t o o k on  with t i d a l  i n c l u s i o n of the  problem,  c a l c u l a t i o n of  condition. the  no  The  The  in  clue  out of the  was  much e n t r a i n m e n t  plume i n c r e a s e d  increase 1/24  too  first  from t h e  volume was entrainment  that  with  i n the  i n both  river  depletion p  far  thickness  mouth -  decreased  w  Figures somewhat  function.  There  86  are  two  sources  e n t r a i n m e n t , and continually effort be  than  that  ^ ^ r t  96 1  bit larger calculated  »as  the  tracks  calculated  on  the  way  their  for  favour  91.  the  open t  This to  One  system,  the  Froude 0.03  a better  in Figure  the  outflow  the  by  0.03)  was  which that  number  an  was  to  it  extra  but 97  of  resulting  added  flow  feel  98  that unlike  99  from  Cordes  i s that  the  excess  average depth  was  250  This  would  i n F i g 97.  i n that  same r a t e , l e a v i n g  an  i s very  divergence  tendency  job,  In  entrainment. Figure  cannot help  f o r F i g . 96.  buoyant spreading  might do  that  plume  in  the  plume as shown i n F i g u r e  the  any  field  done t o c r e a t e  drogues diverge  cm  the  flow  shows  had  balance the  so  the  depletion.  a  flow  97  then  was  velocity field.  a b o u t 450  the  boundary  and  2  i n the  into  Figure  drained  momentum a t  then  to  reason f o r the  and  so  and  r e s u l t s are i n t e r e s t i n g .  released  the  depletion  If  The  of  F i g 97,  model,  flow  F r o u d e number i n c r e a s e d  condition;  ^ f~*/}n '-0  i n which t h e  (1977). P a r t outflow  85  stage,  behaviour  river  absence of  by 3 F  conditions.  shows a t y p i c a l  the  (outflow  when  outflow  plume,  l o s s i n the  in  boundary  t o make  outflow  the  no  the  this situation, I specified that  series in Figsi  during  was  to  shows d r o g u e t r a c k s  ^  drogue  water  thicker,  t o improve  Figure  the  there  got  a slight  of  i t  tend  cm to  I t seems t h a t  removes  velocity field  mass  and  unchanged.  87  As was  mentioned  difficult  density,  with  respect t o the s a l i n i t y  to get r i d of  elevation  and  absence o f t h e C o r i o l i s hour  periodicity  constant r i v e r of  a  with  in  flow.  model w i t h  tidal  left-right  velocity  force,  constant  discharge  100 i s a p l o t river  flow  i s decreasing  only  a  o u t o f t h e model of the  discharge  and no C o r i o l i s  6  with out  f o r c e , but  o f t h e 12  b u t n o t a s f a s t a s one would  hour  like..  box model w i t h o u t  depletion,  d r o g u e s were r e l e a s e d i n t o t h e f l o w  a t four d i f f e r e n t  stages  tide  -  Figures  flow, C o r i o l i s paths  test  be  i n the  o f the sguare  the  a final  i n the  One e x p e c t s ,  that there should  the t o t a l  Figure  fields.  asymmetry  s t r e a m s and e l e v a t i o n s . The a m p l i t u d e  periodicity As  the  distribution, i t  f o r this.  m ./sec,  is  3  tracks  apply  upper  layer  much  rather  t o those  First,  the  lower  different model  than  had  grown  very  (about  interesting  stages  Strait.  average  of  the  drogue  river  discharge,  1000  t h e f r e s h e t c o n d i t i o n s t h e drogue  thick  was over  10 m e t e r s ) ,  compared to  The  o f f i g u r e 99. T h e r e a r e two  t o . Second, because t h e r e  insignificant  nevertheless  s l o p e s and streams.  similar  d r o g u e s were i n s t a l l e d was  t o 104. The model had v a r i a b l e r i v e r  f o r c e , and t i d a l  are not very  reasons  101  of  note  no  the time  so the r i v e r  t o the t i d a l that  t i d e occupy  depletion,  before the momentum  momentum. I t i s  drogues  different  the  released regions  at  of the  88  TESTING  DEPLETION  I n an e f f o r t u,  a  n  few  experiments  velocity,  Wp  specified  as w  the to  t o understand were  of s a l i n i t y  elevations,  velocity  and t h i c k n e s s .  field  the  drogue elevation  corresponds the  object  depletion and  mirror that  entrainment v e l o c i t y was  was p r o p o r t i o n a l t o  A  flow,  run  was  almost  and no C o r i o l i s  identical,  only  corresponding  to  the  same  105, and F i g u r e  field  corresponding t o Figure  94, and  effect  f o r demonstration  a thinning  effect.  The  images o f e a c h o t h e r  that  but t h i s  fact  is  due  to  106 shows 107 shows  significant Figures  107  depletion  i s  spread  o u t more  the  excessive  g i v i n g buoyant spreading that  Figures  t r a n s i e n t s which a r i s e when one t u r n s  greater  107 and  w h e r e a s 94 a n d 95 a r e n o t  i s p e r h a p s an e f f e c t i v e way  108  l a r g e , but  very  The d r o g u e t r a c k s  p r o d u c e d by d e p l e t i o n , importance.  a  purposes. Since  plume i t i s a p p a r e n t  perhaps l i k e ,  depletion  obtain  the  tidal  Figure  v e l o c i t y was r a t h e r  to  force.  one i s shown. The  96. F i g u r e  was  with  since  corresponding t o Figure  runs  i t served done  tracks  of these  t h a n we would  relative  river  86, i s shown i n F i g u r e  i t s predicted  thinning  The  As p r e d i c t e d ,  regions.,  t o 95. The d e p l e t i o n  108,show  having  are  a t 62 h o u r s ,  phase as F i g u r e the  constant  depletion,  a r e t h e same a s f o r f i g u r e s 85 t o 91, and fields  velocity  out.  =.0001S/200.Thus, d e p l e t i o n  0  t h i n t h e plume i n t h e o u t e r  Conditions  carried  , was k e p t t h e same, and t h e d e p l e t i o n  product  tidal  the e f f e c t s o f adding  108 a r e  indicates  t o damp o u t s t a r t u p  on t i d a l  forcing.  89  IIGEOMETRY  MODEL  S i n c e a l l the terms e s s e n t i a l been  investigated  with  the  to  the  plume  sguare  box  model,  dynamics i t  had  appeared  a p p r o p r i a t e t o i n v e s t i g a t e t h e b e h a v i o u r o f t h e model i n a geometry tidal in  situation.  forcing  time,  with of  an  was  similar to that  i n that  s l o p e s and  area.  slope.)  same a s t h e t i d a l were  shallow The  hours  this  the  of f l o o d  was  model was  switch  M2  tide  s i z e u s e d was (1977). Only  Strait  and  tide.  f o u n d when w o r k i n g  which  two  fresh  the  passes i n  the  P a s s , and solid  entrainment the  flow  the  walls.  flow. Figures  o f a bulge of water  shows t h e d r o g u e  of 110  during  tracks, for  flow.  with the l a r g e r  water  water  a  was  river  velocities  everywhere,  t h a n t h e r e was  proportional to  2 km  w i t h 0 °/oo  more s a l t  the  only  too l a r g e  entraining  an  112  had  t h e s q u a r e - l a w e n t r a i n m e n t was  mouth - s t a r t i n g  to  region  109 shows a p l o t  3  and a d v e c t i o n  used  with  4000 m / s e c . F i g u r e  Figure  used, model,  i n the  Boundary  Heads were r e p l a c e d  a t maximum r i v e r  model t h a t  river  river  Haro  out  of the narrowing o f  v e c t o r s a t t h e t i m e o f maximum r i v e r  drogues r e l e a s e d  of  f l o w was  111 show t h e g r o w t h  It  the  grid  open-  banks n e a r Sand  velocity  6  The  of  there i s a larger cross-channel  model o f C r e a n  kept  mean r i v e r  but  worked  t i d e was  streams being i n c r e a s e d  In r e a l i t y  s l o p e than down-channel,  south  transfer  i n t h e s g u a r e box  t h e r i v e r mouth, t o model t h e e f f e c t  downchannel  physical  ad hoc a p p r o x i m a t i o n t o t h e M2  forcing  the t i d a l  Strait  and  the  f r o m t h e b a r o t r o p i c model had n o t been  so  (the t i d a l  Unfortunately,  real  salinity  the  i n t h e mesh a d j a c e n t t o t h e  flowing  i n . I t was  velocity  velocity,  near  based  which on  decided was  the  to  linearly following  90  argument. Thus, s is  During  t h e summer, t h e plume i s f a s t e s t a n d f r e s h e s t .  u ^ , where  approximately  velocity  i s probably smaller than given  t o flow r a t e .  1. T h e  by s * ^°?/u.', t h e r a t i o  salinity  o f entrainment  C o m b i n i n g t h e s e we g e t  I - ii  not  w<*u , and c l o s e r The  where  rt  entrainment  u,~ and  streams. w  t o w«u*.  2  v^  velocity are  was  the  given  by  velocities  The d e p l e t i o n v e l o c i t y  w =0.0001 ^ u * v ' , 2  2  P  relative  t o the t i d a l  was g i v e n by  =0 i f g»> 12;  w^ =0.00025 (24-g») only  when  the  i f g* < 12. Thus, d e p l e t i o n i s assumed t o a c t  salinity  is  greater  than  s=15 °/oo) , and i n c r e a s e s as t h e s a l i n i t y The  eddy v i s c o s i t y The  river  mouth  boundary 0  V,  in  ^6^(0 /h) w i t h 2  projected  width  quantity).  It  was  a  two  not  resolve  i n each  further  ratio  such  that  downstream v e l o c i t y  (a v e r y  { J/zy (ov/h) ) adjacent  (essentially sharp  to  ambiguous  to s e t t h e cross-stream  locations  mouth, very  direction  momentum  corresponding  was made, where b i s t h e  necessary  velocity  the  correction, 2  o f downstream r i v e r  f o r the  adjacent  the  t o s p e c i f y two  o f t h e Sand Heads j e t t y t o  1/b ^/2)y{D b/h)  of the r i v e r  downstream o f , t h e r i v e r could  was  s y s t e m . F o r t h e momentum e g u a t i o n s one mesh downstream  replacing  zero  increases.  condition  and  ( i n b o t h x and y d i r e c t i o n s ) ,  transport  at  2  U/V-tan I , where I i s t h e i n c l i n a t i o n grid  (g* = i 2  was i n c r e a s e d t o 10* c m / s e c .  components o f t r a n s p o r t ,  the  15 °/<>°,  to,  because  cross-stream  egual  to  but not  the  grid  gradient of  , t r a n s p o r t i n g t o o much momentum t o t h e two  meshes, and d r a i n i n g  them).  91  The  final  test  of the  of  t h e two  most i m p o r t a n t  M2  and  and  the  K1,  t o use  tidal  a mean r i v e r  elevations  downchannel  slopes),  crosschannel  slopes  cross-stream  velocity  this  run.  t h e M2  The  normalized  by  s l o p e s and  i s very  velocities  multiplying  the  dividing  magnitudes  of  output  from  the  velocities  s t r e a m s by  (1977)  figures  115  those a  to  and  slopes  11.3  drogue  The  t r a c k s of Figure  travelling velocity  sheared in  over  flow.  a  as deep a s t h e  with the v e l o c i t y velocity,  shown by  linearly  sheared  I t can  upper l a y e r ,  speed  122  be  Buckley flow  was The  obtained  by  magnitude, the  and  one  the  (cm/2km) t o  be  tidal  quite  like  were r e p e a t e d  using  not  (1977),  travels  drogue  at the  average  layer  o f t h e drogue i s g i v e n  in a  a  drogue  assumes a l i n e a r upper  the  shows  shown t h a t i f t h e  a t t h e bottom o f t h e  the  so t h a t  t h a t t h e d r o g u e s were t r a v e l l i n g  As  i t s depth.  in  t r a c k s a r e shown i n  o f F i g u r e 99.  fact  used  to o b t a i n the  d r o g u e t r a c k s were s t i l l  f o r the  the  unity.  114  The  vertically  tidal  (cm/sec) a n d  fields  since  determining  122.  correction  not  velocity  the  discharge  were  and  Figure  the phase f a c t o r o f F i g u r e The  river  the  v i s u a l i n s p e c t i o n of  multiplied  used.  velocities  model.  of  were n o r m a l i z e d  model i n t o 7 a r e a s  shows  to  flow  v a l u e by an a p p r o x i m a t e  of  forcing  11.3  negative  river  experiment  distribution by  Figure  relation,  u n i t y ; the  in this  and  Crean's  3  the  and  up  Georgia,  (proportional  small),  made  of  mean mass d i s c h a r g e r a t e was  used  slopes  a tide  8000 m / s e c , t o match  geostrophic  amplitude  normalized  o b t a i n e d by  to  streams  the  that the  of  (1977).  (proportional  e l e v a t i o n s and  so  flow  experiment  tidal  c o n s t i t u e n t had  to s p e c i f y  c o n s t i t u e n t s i n the S t r a i t  c o n d i t i o n s of Cordes'  the  model was  by  is  shear,  being  the  92  Hith  this  obtained. with  correction, Figure  F i g u r e 99.  99 and  124.  Although appears  125  to  129  forcing  from  parameterization  of the  Haro S t r a i t  away  Strait.  from  verified  logistics of  to  the  from  Crean  coincide  between  Figures  and b r e a k - u p  in  a l l  of a  respects,  of  using  actual  The  Figures  resulting  i n , f o r example, F i g u r e  areas  115-129  was  to  pull  t o t h e main had  i n the blank area  117-119. ,  tidal tidal  coastlines perpendicular several  top  ad h o c  streams p r e s c r i b e d tended  Consequently, model,  to  were  t o f i t N a t u r e . The  (1977).  generate  i n t h a t the t i d a l  l a y e r water  removed  model used  123,  123  river.  be c a p a b l e o f a d j u s t m e n t  i s t o work o u t t h e  axis  of F i g u r e  t h e model has n o t been  the  of figure  show t h e g r o w t h  by t h e  priority  upper  tracks  i s c o n s i d e r a b l e agreement  discharged  to  unfortunate  drogue  is a replotting  There  Figures  b u l g e o f water  it  124  the  to north  be of  93  CHAPTER 6 CONCLUDING DISCUSSJON It  is difficult  described are  in this  t o draw p r e c i s e  thesis;  very complicated,  considered would  as  like  a  now  since the  the  first  c o n c l u s i o n s from  work  dynamics o f  described  s t a g e . With  t o summarize  the  work  the u p p e r  layer  here  can  only  be  the above g u a l i f i c a t i o n ,  what t h i s  thesis  accomplished,  I  and  t h e n d i s c u s s what t y p e s o f work i t l e a d s t o . Considerable  insight  a c g u i r i n g the f a i r l y velocity in  profiles  deciding  Chapter  4,  layer.  of  entrainment  across  Jamart,  The  few  2 , p o i n t e d out  plumes  and  and  jets  permeable  i s , I think,  relatively  Stronach, Crean,  work.  The  immediate  system  o f e q u a t i o n s , and  aim  form used  recent  and  Leblond, a  have been s u c c e s f u l l y  the  crucial  front.  of a great front.  concept  1977).  major  effort  to  deal  the  of  this  t o have a which (with  with  open  i n a n o n - l i n e a r flow. These g o a l s appear accomplished.  Indeed,  produced  by t h e model compare g u i t e f a v o r a b l y  (1977),  when  a reasonable approximation  the  of and  Finally,  of s i g n i f i c a n c e  how  of  Pearson,  t h e c o r r e s p o n d i n g computer code,  to l e a r n  The  but t h e c o n c e p t  (winter,  a l l t h e t e r m s which were t h o u g h t  boundaries  2.  drag at the  interface,  by  s u c c e s s f u l l y as a  i n d e v e l o p i n g t h e model was  t h e e x c e p t i o n o f w i n d s ) ; and outflow  Chapter  i n number, were  have  d e v e l o p m e n t o f a n u m e r i c a l model was  included  of  obtained  the p o s s i b i l i t y  the l a r g e  a  1977;  sections  was  s i m p l e model o f a d i s c o n t i n o u s  at a f r o n t ,  Many models  plume  t h e plume c o u l d be m o d e l l e d  section  d e a l o f mixing  depletion  s y n o p t i c STD  the  obtained, although  that  s e p a r a t e upper  into  drogue  t o those of  to  tracks Cordes  to the b a r o t r o p i c  tide  94  is  u s e d , and  model  without  any  than  those  other  adjustment of required  the  to  parameters  of  the  o b t a i n s t a b i l i t y of  the  solution. Now,  to discuss  considered. refining  First,  a  the  one  puzzling.  boundaries,  where  of the  The  other  mouth  1  in  upper  area  the the  use  would  of the  to  flow  of  little  of  followed  by  that s p e c i f i c a t i o n  mouth,  (the d i s c h a r g e ) ,  mouth, one  would  there.  Since  over  a  short  distance  coarse  grid,  specification  as  grid the  salt.  the  river minor  also include,  brackish  water  flow one  due  here,  river the  of  parameter  at  However, jump a t  idea of  what i s  considerable  change  river  o f f r i c t i o n , e n t r a i n m e n t , and  jump  downstream  mouth, and  model  at  presumed  model, n e e d s t o be numerical  we  condition  flow  to this  ( F i g . 25b) the  aside  entraining hydraulic  l i k e t o have a b e t t e r  near  and  i s perhaps adeguate.  mixing  observe  temporal  other  boundary  of only  a d e s c r i p t i o n of  s u i t a b l e f o r a coarse far  we  the  model d e v e l o p e d  subcritical  so  i s such i n t e n s e  outflow  R o b e r t s Bank. Even p u t t i n g  an  are  the  momentum,  of s h a l l o w ,  i n the  be  dynamics  geometry a r o u n d  or s u p e r c r i t i c a l  on  As  like  complications,  going  rather  complicated  particularly  mouth,  there  river  vicinity  know v e r y  mouth. T h e r e i s p r o b a b l y  jump,  since the  the  a critical  river  river  really  the e f f e c t s  geometrical  the  i s i n the  where t h e  and  these  at  regions  to  measurements i n  (Canoe P a s s , N o r t h firm, M i d d l e Arm),  banks,  river  two  more f i e l d  the  the  the  a l w a y s use  improvements  l a y e r f l u x e s o f mass,  is  over  not  immediate  include  some way,  did  One  we  ( F i g u r e 4) . We  openings  can  model. T h e r e a r e  particularly  nature  most  mouth  are  using  a  dynamics,  developed. is  concerned,  depletion i s  the  always  95  open  to  question.  functional constant field  forms in  to  give  front.  results  c o r r e c t . He  1/2C D2,  example, plume/  i f  these  we  would  The  the  wind  a p r o b l e m which  we  d o n ' t e v e n know t h e  effects,  let  o f t h e model  like  simplest  these  to incorporate  f o r too long,  c a n n o t be  the  effects  the  effects  t h i n g i s to put a l l the  layer. Complications  blows  alone  p r e d i c t i o n s and  have t h e s i z e s o f  i n t o the upper  D  extent,  The c l o s e n e s s  winds i n t h e model.  stress,  to  i n d i c a t e that  approximately of  To a c e r t a i n  handled  and  at t h i s  wind  arise,  for  m i x e s away t h e stage  of  model  development. It  has r e c e n t l y b e e n  considerable days)  energy  oscillations.  plume m o t i o n ineeds of the  baroclinicity  observed,  in  low  frequency  The e f f e c t  1976)  (Chang,  , that  (periods greater  o f t h i s temporal  t o be a s s e s s e d ,  there i s than  variability  a s d o e s t h e amount and  i n t h e c u r r e n t s and  pressure  gradients  4 on  effect beneath  plume. As  mentioned  possibility  in  that  we  should  isopycnals  in  possibility  of a 2 - l a y e r  in  130.  Figure  northern  model.  southern  we  plume,  F o r arguments  have  end,  development situation.  of  there  is  the  model  T h e r e may  130.  (representing  This  up  the  s e v e r a l days  with  the  is in  an  is  a  surfacing intriguing  be  assume t h a t  outflow  the  surfacing must  will  model  done  boundary  between  discharge),  and  at the  boundary,  a  s o f a r . At t h e  isopycnal, to  be an a d d i t i o n a l b o u n d a r y , is  there  to deal  brings  s a k e , we  with a  how  6,  a s e c t i o n t h r o u g h which i s shown  there  dealt  section  learn This  end o f t h e plume,  situation  Figure  the  4,  Chapter  and  some  accomodate  this  i n d i c a t e d NP  the e x i s t i n g the  much  in  plume  fresher  96  water d i s c h a r g e d this the  current Strait.  front,  during  plume  would be t h e i n t e n s e  Numerically,  and  how  the current t i d a l  to  eventually  plume".  adeguately  r e f i n e d , i tcould  Liu, would and  When  have  incorporate  "new  be used a s t h e model  upper  (Leendertse,  i n this  research.  layer  of  a  A l e x a n d e r , and  and momentum f o u n d  The  model  reversed.  In  (a s t r o n g  of numerical  this  way one i s l e t t i n g  pycnocline  interfacial  i n t h e plume model would  t h e n be a p p l i e d t o t h e 3 - d i m e n s i o n a l model, w i t h  type  into  a s i t s f r e e s u r f a c e t h e i n t e r f a c e between t h e upper  f l u x e s ; o f mass, s a l t  situation  plume"  l a y e r of the of the 3-dimensional  lower l a y e r s as d e f i n e d  signs  f r o n t found i n  t h e u p p e r l a y e r model i s t h o u g h t t o be  3-dimensional  1973). The t o p m o s t  colour  we w o u l d have t o l e a r n how t o move t h i s  "existing  conventional  c y c l e . The b o u n d a r y o f  of course  the physics  their of the  a t a v a r i a b l e depth) d i c t a t e  schemes employed  t o model i t .  the  97  SIIIMICES CITED  A b r a m o v i c h , G. Press,  1963.  The  Cambridge, Mass.  Bondar, C. courante bassin the  N.  1970.  de  c o n t e n a n t un  densite  liguid  Hydrology of D e l t a s ,  Ph.D.  R.  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Civil  Design  Channel  Engineers.  116:  34 7-363. ,  Shapiro,  A.  characteristics and  numerical.  H.,  and  f o r two  G.  M.  dimensional  Edelman.  1947.  supersonic flow  J o u r n a l of A p p l i e d Mechanics.  Pp  Method -  of  graphical  A154-A162.  {part  101  of  T r a n s . Am.  Stolzenbach,  Soc.  K.  Mech. Eng.  D.,  and  experimental investigation Ralph  M.  Parsons  Hydrdynamics,  Stronach, layer 9th  J . A.,  of surface  Laboratory R e p o r t no.  P.  Liege Colloguim  S.  as deduced Branch,  1972. from  Pacific  discharge  for  135.  212  The  and  water.  Resources  and  p.  Leblond.,1977.  plume. P a p e r  An  presented  upper  at  the  hydrodynamics.  movement o f F r a s e r  a series  An a n a l y t i c of heated  Hater  B. C r e a n , and P.,H.  on Ocean  1971.  of aerial  Region, p a c i f i c  River -influenced  photographs. Marine  Marine  water  Sciences  S c i e n c e R e p o r t no.  72-6.  p.  Takano, a  D. F. H a r l e m a n .  model o f t h e F r a s e r R i v e r  Tabata,  69  MIT,  V o l . 69.)  K.  river.  Takano, the  the velocity Soc. J a p a n .  K.  t h e s a l i n i t y and  K.  seaward  distribution  J . Oceanogr.  1954b. On  mouth o f a r i v e r .  Takano,  11:  1954a. On  1955.  flow  J . Oceanogr.  10:  o f f t h e mouth  of  60-64.  velocity  Soc. J a p .  A c o m p l i m e n t a r y n o t e on  o f f t h e mouth o f a r i v e r .  the  10:  distribution  off  92-98.  diffusion  J . Oceanogr.  of  Soc.  the  Japan.  1-3.  T u r n e r , J . S. (England)  1973.  Buoyancy  Univ. Press.  367  p.  Effects  in Fluids.  1973.  Cambridge  102  Waldichuk, Georgia,  M.  1957.  Physical  B r i t i s h Columbia.  oceanography  J. Pish.  Res. Bd.  of  the S t r a i t of  Canada.  14:  321-  486.  Whitham,  G.  B. , 1974.  Interscience,  New  York.  Linear  and  Non-linear  Waves.,, W i l e y -  636 p .  W i n t e r , D. F., P e a r s o n , C. E. , and B..... H. J a m a r t .  1977. Two  analysis of  fjords.  presented  Wright,  steady  circulation  in  stratified  a t t h 9th L i e g e C o l l o g u i m on Ocean  L.  interfacial  D., and J . M. Coleman.  R i v e r d e l t a . J . Geophys. Res. 76: 8649-8661.  Paper  Hydrodynamics.  1971. E f f l u e n t  mixing i n the presence of a s a l t  layer  wedge,  e x p a n s i o n and Mississippi  103  APPENDIX LINEAR STABILITY Certain unpublished  parts  the  following  discussion  a r e b a s e d on  n o t e s by R. F l a t h e r .  Consider  dk dt  of  ANALYSIS  the set o f  +  ^ dy  _  eguations:  Q  2*  j-cfho^jk  --pu i-r 1/ =• C D .  I f 6. i s h a l f t h e g r i d  spacing,  and "P t h e t i m e s t e p ,  the  scheme  under c o n s i d e r a t i o n i s : h  =  UL  h  t/-?.  -- u  U  - tjjho  ( k  - U  -h \J  _^  )  - f v  -  1/  . J.  £. - ru  ?_ .  2 A  V  * i/  - HqA* ( a. A.  Assume t h a t  k  )  - h  a t any t i m e , t h e f i e l d s  * i U  t  - ^ Z r  o f h, u, and v a r e w r i t t e n a s  Fourier series, i.e.,  Then h^x  consider egual  only  one F o u r i e r component, and a l s o d e f i n e a new  to • Defining  - j T ^ A o ^-kJi  ia  r ^JJA7 A  and  ' sin  l<A  y  do ?JJh7  C  ;„ J A  5  A  d r o p p i n g t h e k.,1 s u b s c r i p t s , we  have:  5  104 t  ft  l  |-  -at,  rt-tx  -a  ft  '  -^-"loft  V  J  I f we  c o n s i d e r the  case  r=f=0,  we  are  left  with  the  simpler  system:  k  \/  T h i s i s of t h e  U  s  I  -a. )o  clo  V  form  S  - C* s  , where  o  6 0  and and  G  i s t h e above m a t r i x .  a set of eigenvectors E  L  I f we for  find G,  1  a set of eigenvalues  then  the  time  7- ,  structure  becomes  and K i t  "  Clearly, which  is  stability 0(t)  -  for true  (71.-)' the  E .  t  solution  -O  t o be bounded  i f ) ^ J i 1 + OCC),  condition  (Bichtmeyer  allows f o r the p o s s i b i l i t y  which and of  TW is  Morton,  must the  be  bounded,  von  Neumann  1967). The  exponential  growth,  term  of  which  105  one  would  guantity  so  not  in  general  t  h,  according  I f -^a  exactly  Note t h a t E  t  i s a  o f the form  t h a t knowing °<t",fi>i 7 ,  obtain  want t o e x c l u d e .  u,  and  (which  t  v  from  i s readily  the three  done),  E^s.  one  could  C a l c u l a t i n g the  ^i  t o d e t (G-"A 1) =0, we o b t a i n :  i s less one.  than  2, t h e modulus  This  o f ^*-,3 t u r n s  i s what we want i n t h a t  out  to  be  the time f a c t o r f o r lit*?  the  a n a l y t i c case,  system  without  associated with  t JJkl  going  friction,  t to tt t  from  wavelengths),  S  L  \A.  can  si  -  nM ;^j V  A 1,3 w i t h  e  e  for a  i s one./A,=1 i s  -bu+av, w h i c h c a n be i d e n t i f i e d , which f o r s m a l l  be i d e n t i f i e d  with  f o r c e i s absent.  by iJ3q h  . w i s given  kA, I A , ( l a r g e the v o r t i c i t y ,  which i s a n i n v a r i a n t s i n c e t h e C o r i o l i s compare  be  and t h e modulus o f e  with t h e e i g e n v e c t o r  ( 'H^-  must  0  Lx  +  ~**  J  .Thus,  e 4  ligk.?*-  t  3  t  r  *  ^  ^  &  2 A  /  3  [  He c a n  . . . .  A  106  F o r J?4<<1, izt <<1, we  have  Thus t h e n u m e r i c a l and a n a l y t i c order, in  so  one c o u l d  factors  differ  s a y t h e scheme i s o f s e c o n d  t i m e . T h i s i s o n l y t r u e i f we c a n r e p l a c e sin(^A-)  and  A  with  ;  that  Thus, t h e a c c u r a c y o f t h e space  gave  to  analytic e  term  £ sfa  our time f a c t o r . Lick  expansion  computational The  o f <£  One c o u l d  ( i n one  phase speed,  requirement that  third  order accuracy  sin(-ir'l)  w i t h Jt A ,  differencing  rather  expression f o r the logarithmic  ) affects  the  the  the  i s i f t h e r e a r e many meshes p e r  wavelength. rise  in  c -*c. , f o r t h e  than  derivative also  of a sinusoid  consider  A * - . 3as  d i m e n s i o n ) , where c i s t h e  which  t u r n s o u t t o be J aA?  # +# ^  i ssatisfied i f  2  (which  Sj_nj0 .  Z o r ?J~giu> ^ A / ,  we  get  stability expand is  t T z ^ A  a  Realizing  as  a  t h a t A i s one h a l f  stability  requirement  the  c r i t e r i o n fPCJ^l  fora linear  grid  <" ^. A  spacing  This  model, a n d we would  i s the  like  to  i t now t o a n o n - l i n e a r one. The most s t r a i g h t f o r w a r d  way  t o r e c o g n i z e [y~k7 a s t h e wavespeed f o r t h e l i n e a r c a s e ,  replace speed  i t with  ( f u ^ + v ? +/gh  , t h e maximum e x p e c t e d wave  f o r t h e n o n - l i n e a r c a s e . Thus, our f i n a l  If  r=0, f * 0 , t h e s t a b i l i t y  and  reguirement  condition i s  c a n be shown t o be:  107  As long as f2.«1 (a very  easy  condition  to s a t i s f y ) , the  C o r i o l i s force places no r e s t r i c t i o n on the scheme. If  f=0, r£0, i t can be shown that the requirements imposed  by f r i c t i o n are rf<2, and  which i s a more severe form of the t!- -^ 4  see  the requirements  r e l a t i o n . One can also  imposed by f r i c t i o n as follows. Consider  the simple eguation  with the f i n i t e difference approximation  (/  -a  Writing 'u the  -r  c  u .  ^7\u , we can obtain 7i=1-rC. Bote  f i r s t two terms i n the expansion of e  for.7\. I f r't-is 2, the error i n >  that  these are  , the analytic value  i s e~ -(1-2)= 1. 14, 2  about  7  times the true value of ~h . If r t =1, the error i s .37, about the same  size as ~k . i f r t=. 5, the error i s . J , about 16% of 7*. Thus  for accuracy, one wants rt!:to be considerably Physically,  friction  less  than one.  cannot remove more momentum from a system  than was i n i t i a l l y present. Thus, ~A should never be be negative, and we should minimally replace rt! <2 with r Z o . to  When  applied  the case of non-linear f r i c t i o n , ru i s replaced with Cufuj.  Th us we r eg uir e C Ju^+v " . 1 < 1. 2  1  M<  %  We can also look at the effect of entrainment. Consider the set of equations  108  dL  +. }u O . •  In  the  scheme  adopted  here  these  have  finite  representation  ' ^  u  c  t  \  C ti  - 3*°  2A Proceeding  difference  a s b e f o r e we g e t  h  I  ik  u  u  CO.  where  a =•  t l j j k 7  Sin  kA  ,  lo ,  *  cos  k A  A The  The  eigenvalues are  question  i s , do t h e s e e i g e n v a l u e s  stability  criterion,  accuracy)  and expand a b o u t b=0. Thus  Z  I7sj<1*0(-e)? We w i l l  Z _^  ;  •2-  V  t  satisfy  t h e von Neumann  assume b i s s m a l l ( f o r  i  i  3  if f^r^p  f / - it  4-  lo / " i f 2  f  SfjL  z  ± 1  £<Z.  fJZ^f  109  and |D| = 0 ( t ) , s i n c e b i s O(^) and a i s 0 ( f A ) , w h i c h  Now  |cj=1,  is  0 ( 1 ) . Thus  results  since  i n a stable  He f u r t h e r imposed  ^  should  _  with f i n i t e  ^  scheme i f t h e e n t r a i n m e n t be aware  by t h e eddy v i s c o s i t y  _ A ^  Again,  |>|<|C|+|D|, t h e i n c l u s i o n  i s small.  the s t a b i l i t y  equation.  limitations  Consider  0  difference representation  u=ue  gives  / * ZA^_ (  Cos(kA)-1  of  o f entrainment  cosCk6)  varies  - /J.  from  0 t o -2, thus  7\ v a r i e s from  1 t o ]-4 A A*-  and  for stability,  we want  <1.  Because i t i n v o l v e s o n l y one e g u a t i o n perfectly) the  thes a l t  advection eguation  phase e r r o r i n t r o d u c e d by n u m e r i c a l  what k i n d o f a c c u r a c y finite  difference  dimensional taken  then  after  p u l s e then  i n selecting  s o l v e d was ds + dt  from  linearly  along  dj ot  back  to  zero  :  u was  o,  x=0, t h e (6 h o u r s ) ,  i n t h e next  zero. This t r i a n g u l a r  i n thedirection  of  a  i n a one-  At  0 t o 30 i n 180 t i m e s t e p s  which i t r e m a i n e d  propogated  cm.  5  t o see  180  shaped  increasing  x,  D was p o s i t i v e . All  t h e schemes  more s t a b l e that  t o demonstrate  schemes. I n o r d e r  and as an a i d  The e g u a t i o n  linearly  decreased  u i s known  scheme, s e v e r a l schemes were t r i e d  prototype.  grew  timesteps,  since  i s useful  t o be 40cm/s, A T =120sec. , AX=1km=10  salinity and  t o expect,  (assuming  waves  velocities,  than of  t e s t e d conserved  salt,  b u t some  appeared  o t h e r s . I n a l l o f them, i t was p o s s i b l e t o s e e different  wavelength  so that the o r i g i n a l  propagated  triangular  at different  s h a p e was  dispersed  110  into  a  rounded  depending produced 1*.  bump, w i t h e i t h e r  on t h e s i g n by t h e  finite  phase  phase e r r o r  Forward time The  of the  l e a d i n g or t r a i l i n g error.  of the  finite  This  ripples,  dispersion  difference  is  scheme.  differencing difference  form  is  Thus i f  ,  d  . ^ . n  -j^  -  j  -  K  Sir*  ^  A  3  ' ,. 1*  i  F o r s m a l l kA, The  >  analytic  ~  ''  1 U  L  L' LA. ^  -  iy  A~  1  3  K  A  . . *  3T  is  I - c u tk  €  - k w  -g  . . .  Zl Thus  this  scheme i s o n l y f i r s t  computational the  computational  smaller  waves, w i t h  ripples salt  ^ w i t h an  (short  p u l s e . As  Checking  for  the system  phase  speed  larger  k,  see  conserves  eguation,  travel  i n Figure we  salt, time.  this  of e  *  i s CA l ^ J i  emerging  stability,  components grows w i t h original  expansion  waves) we  order accurate. I d e n t i f y i n g  see t h a t  f  slower.  from  131,  6  4  and  we  is  Thus,  indeed  | I = 1*0 (-?).  see  that  that therefore  the t r a i l i n g  this  the amplitude  ,  the  we  predict  edge o f the  the  case.  Thus,  although  of t h e v a r i o u s  Fourier  S i n c e t h e r e i s no  source  term  i n the  growth o f t h e v a r i o u s modes i s an  error.  111  2i. L e a p f r o g The  scheme u s e d i s  t+-c *  5  time  i - r '  t  -  *  5  *  ^  L  S  ~  t  .  J -  J  •  7.1 D e f i n i n g S=Se'*  and  a <4,  i f  , a= zis,« kA) crt  X  |?>{ = 1.  2  f  w  e  g  e  t  Thus none o f t h e waves grows i n a m p l i t u d e ,  w h i c h i s what we want. E x p a n d i n g A , we g e t  is = t Cl - *JL1L ( < ± l f + . . .  ) - 'ii*  a  A  v  l e a d i n g terra i s ± I ~  The  leading  we S  kA  Sin  associate  travel  slowest,  computational associated  . here  are  T  initiate  the  two 7)'s  with  and  132  is  relation  with t h i s  physical  wave,  with  --  2.7TU  /  Z 7 7 / t  ,  shortest  the  computational  present,  p l a n s on d o i n g  true waves  shows. "/Wis a s s o c i a t e d w i t h t h e  approximately  -e  ( p e r i o d =2?)  t o =uk,  the  , sign  wavelength  thus  i t is  alternation. associated  mode i s  the  wavelengths  ^~  real  with a high freguency  W^o  is  "- -  Fig,  mode,  most s t r o n g l y  that  ,  with  as  From t h e d i s p e r s i o n  7\ -- 2  ^ sm kA  . He s e e t h a t f o r t h i s T\, a g a i n t h e s h o r t  = <2  E  l  A  terms  TN , = I ' 'JL?  I -  '  J  -  u  •£  -- DC  A  )  wave. The c o m p u t a t i o n a l  they  mode, will  but  i f  excite this  *,  scheme d o e s n o t  there  are  short  mode. One  usually  a b i t of f i l t e r i n g to eliminate this  problem.  112  3.  Rightmeygr  scheme  (Richtmyer  T h i s i s i n some s e n s e is  to calculate  at  t+1/2 t . T h u s  //  xs a p p r o x i m a t e d  Sj*'/2.  *j  '- *  ~£Z  S  With^-^.  that  dissipation  f o rthis  ^  <•*,-,)).  -i ) ,  slower..  i s p r e s e n t . The s h a p e o f t h e S  scheme i s i d e n t i c a l  t o t h e L e a p f r o g scheme  i s n o t shown.,  A problem difference formulae since  develops  when  of  scheme^  to  apply  scheme i s t h e  a  finite  time  initial  level  used  in  requirement  set of conditions,  conditions  direct,  for a  the  model  - i t i s a three  i n d e v e l o p i n g a model , one o f t e n then  further  s u c c e e d i n g run i n v o l v e s c o n s i d e r a b l e changes improvements)  most  s p a t i a l d e r i v a t i v e s as i n  However, i t was n o t  i t s extra  scheme. A l s o ,  as  tries  have t o i n t e r p o l a t e  t i m e s t e p s with a c e r t a i n  output  one  l o o k m e s s i e r . The l e a p f r o g  one d o e s n ' t  because level  DIFFERENCE SCHEME TO THE PLUME MODEL  scheme t o t h e plume model. U i s n o t c o n s t a n t , so t h e  the Richtmeyer  are  idea  f o r Sy :  CS;.,  s m a l l waves t r a v e l  numerical  APPLYING A F I N I T E  few  r  We o b t a i n  6  basic  d e r i v a t i v e s of s  ^ * ^  - r' ^  5  scheme. The  of s p a t i a l  / i l i j  cosCkA)  distribution and  t ^'  / ^ *  1967)  , we g e t f o r 7s  A g a i n we s e e t h a t  so  as  I _  =  an i t e r a t i v e  as a f u n c t i o n  s  .  t  7\  ~  5  and Morton,  runs f o r a saves  run.  (which  the  I f the  one  hopes  t h e r e i s a v e r y good o p p o r t u n i t y t o g e n e r a t e a  113  computational Eichtmeyer  after  the  only a f i r s t  eguation  For t h i s  scheme. However,  discovered fact  mode.  was  reason  because  modeling  order  accurate  the  convective  temporally would  equation  v a r y i n q flow  like.  The  a  not  problems are  applying  staggered  this  scheme  t r a n s p o r t s and  advection was  The  assumed,  discussed  and  in this  the  salinity  field  as  far  time  as  Thus, the field, velocity  and  the  fields.  to  thus  thesis,  a long  error,  advection  considerable  r u n , s h o w i n g up  Thus  been c a r r i e d  accurate the  the  out  solution  in of  spatially  and  as w e l l a s  one  scheme;  already  existing  only  that  l e v e l s i n the field  equation  affects  is  density  the  grid  of  the  momentum  by  the  same as  numerical  salinity  i s also a density eguation.  limited  a s t a b l e scheme would  were t r e a t e d t h e  new  f o r the  2-dimensional,  advection  confirmed  old velocity  and  after  the  completed, i n  salt  more s u b t l e , p o i n t a b o u t  salinity  equation,  accounts  use  thicknesses.,  There i s a f u r t h e r , eguation.  was  to  two-fold:  .1) g e t t i n g a h i g h enough o r d e r 2)  programming  a  salinities.  in  has  of  scheme f o r t h e  used. T h i s p r o b a b l y  as s m a l l n e g a t i v e  intended  d i s c u s s e d here  small scale f l u c t u a t i o n s present particular  i t was  experience result  the e l e v a t i o n  scheme were  used  to  update  field  is  used  the  It  i f  field,  concerned. salinity  i n t o update  the  114  TABLE I HARMONIC ANALYSIS OF BIVER ^  ,  NAME  ^  I FREQUENCY j cycle/day  —i  ZO MM MSF 2Q1 'Ql  01 N01 P1 S1 K1 J1 001 MNS2 MU2 N2 NU2 M2 L2 T2 S2 K2 2SM2 M03 M3 MK3 SK3 MN4 M4 SN4 MS 4 S4 2BN6 M6 MSN6 2MS6 2SM6 3HN8 M8 3MS8 M12 I  T  -  ,  | J  i  t  ! 0.0 | 1 0.03629164 | I 0.06772637 ) 1 0.85695237 { | 0.89324397 f | 0.92953563 | I 0.96644622 | I 0.99726212 | | 1.00000000 | | 1.00273705 | I 1.03902912 | | 1.07594013 | I 1.82825470 J | 1.86454678 \ | 1.89598083 | | 1.90083885 | J 1.93227291 | | 1.96856499 | | 1.99726295 | | 1.99999905 | I 2.00547504 | I 2.06772518 | | 2.86180973 | | 2.89841080 | | 2.93500996 | t 3.00273800 | I 3.82825470 \ | 3.86454678 | | 3.89598179 | I 3.93227291 J I 4.0O000000 | j 5.76052761 | | 5.79681969 | I 5.82825565 | | 5.86454582 | | 5.93227386 j | 7.69280148 | J 7.72909451 | | 7.79681969 | | 1 1.59364128 J  L  ,  1  ,  ,  —  AMPLITUDE | cm/sec | 1  i  0.0860 0.2318 0.4556 2.8886 3.3717 24.0625 8.2796 10.7181 1. 7835 33.1047 2.7960 4.7931 2.5522 6. 4664 6.7790 1.3024 48.3670 25.7985 0.8353 13.2946 3.5766 5.8606 5.7899 1.5712 10.6364 3.2154 1.0384 4.3592 2.3171 6.3899 2.8767 1.4951 1.9540 1.1174 2.2803 1.2324 0.2242 1.0761 0.2408 0.0446  1 | J J | | | | | | | | | | | | | | | | | | | | | | | | J J | | | | | { | | | | 1 X  SPEEDS ,  PHASE degree 0.00 2 74.16 176.66 28. 19 167.73 280.06 59.74 304.67 2 57.24 302.74 169.53 197.50 7.42 108.74 2 34.64 239.74 311.66 320.57 313.44 3 36.78 3 35.88 183.44 154.61 3 25.21 173.79 1 71.84 178.85 346.81 166.22 189.51 172.03 44.20 185.45 19.73 129.23 329.69 167.19 180.33 248.67 263.10 i  ! I j I I I I I INFERRED INFERRED  | | | |  (K1) (K1)  | INFERRED  (N2)  j INFERRED  (S2)  INFERRED  (S2)  ]  } | ] | j  | | | |  j  | | ] I  J  115  TABLE I I ANALYSIS OF POINT -MMIISJ! ELEVATIONS NAME  FREQUENCY cycle/day  AMPLITUDE cm  PHASE degree  ZO MM MSF 2Q1 Q1 01 'HOI P1 51 K1 J1 001 MNS2 MU2 N2 NU2 M2 L2 T2 52 K2 2SM2 M03 M3 MK3 SK3 MN4 M4 SN4 MS 4 S4 2MN6 M6 MSN6 2MS6 2SM6 3MN8 M8 3MS8 M12  | 0.0 J 0.0 3629164 | 0.06772637 | 0.85695237 | 0.89324397 f 0.92953563 \ 0.96644622 | 0.99726212 | 1.00000000 | 1.00273705 | 1.03902912 J 1.07594013 | 1.82 825470 | 1.86454678 | 1.89598083 | 1.90083885 | 1.93227291 | 1.96856499 | 1.99726295 | 1.99999905 | 2.00547504 f 2.06772518 | 2.86180973 | 2.89841080 | 2.93500996 | 3.00273800 | 3.82825470 | 3.8 6454678 | 3.89598179 j 3.93227291 | 4.00000000 | 5.76052761 | 5.79681969 f 5.82825565 | 5.86454582 | 5.93227386 | 7.69280148 | 7.72909 451 | 7.79681969 | 11.59364128  302.087 5.304 2.762 1.381 8.231 45.787 4.997 27.801 4.626 85.8747 5.079 2.411 0.883 4. 137 19.242 3.697 91.281 5.429 1.426 22.720 6. 112 0.454 0. 136 0. 180 0. 160 0. 103 0.096 0.336 0.077 0.276 0.093 0.502 0.657 0.200 0.787 0.204 0.074 0. 160 0.072 0.075  0.0 349.62 3 23.83 141.43 137 . 9 1 153.47 220.03 166.24 INFERRED 1 18.81 INFERRED 164.31 200.06 2 38.04 336.40 94.61 131.42 136.52 INFERRED 159.49 I 209.59 I 155.02 INFERRED 178.25 177.35 I INFERRED 110.66 90.45 2 63.58 167.06 1 52.38 168.62 161.73 43.39 2 32.26 51.69 49.46 66.51 38.74 100.55 105.99 3 25.71 141.13 240.77 164.79  (K1) <K1)  (N2)  (S2) (S2)  116  TABLE I I I SCALE ANALYSIS OF TERHS IK -THE EQUATION OF parameter u, f l o w u ,  r i v e r area (cgs)  velocity  tidal  0  20  I  400  | 100  10s  I 10«  4 *10*  |  10-*  I 10-*  20  I  400 >10~*  f 10»10-«  derivative  1  ! .05  a d v e c t i v e term  40  I  pressure gradient  16  1 0.1  term  4  stream  length  scale  scale  f, C o r i o l i s  parameter  g /reduced  gravity^  Wp, e n t r a i n m e n t ?uh/0 t , t i m e 2  2(g' h / 2 ) /?x 2  f  fuh, C o r i o l i s  ,* .  velocity  Ku > {K=0. 001) , i n t e r f a c i a l 2  Wp u , e n t r a i n e d 0  qhC^,  field] <cgs)  I 20  L, h o r i z o n t a l  2{u h)/DX,  far  100  h, plume t h i c k n e s s  T, t i m e  MOTION  momentum  barotropic  tidal  friction  10  20  4»10*  10  0.04  0.2  I 0  flux  4  0.02  forcing  4  4  10  2  uh/Lw, a d v e c t i o n / e n t r a i n m e n t  117  FIGURE 1. Chart showing the Straits of Georgia and Juan de Fuca, and the Fraser River.  SALINITY DISTRIBUTION IN THE STRAIT OF GEORGIA X JUAN DE FUCA STRAIT, 1-6 JUL. 1968 FIGURE 2.  Taken f r o m C r e a n and A g e s ( 1 9 7 l ) .  C. F L A T T E R Y JUAN  D E F U C A STr  BOUNDARY f-HARCH -PASS.ST.  C. -ST. O F GEORGIAFRASER RIVER  2  3  SALINITY DISTRIBUTION IN THE STRAIT OF GEORGIA & JUAN DE FUCA STRAIT, 4-8 DEC. 1967  FIGURE 3.  Taken f r o m C r e a n and A g e s ( l 9 7 l ) .  MUDGE  6  9  12  14  16  FIGURE 5. The d a i l y F r a s e r R i v e r d i s c h a r g e f o r 1976, measured a t Hope. The two arrows i n d i c a t e t h e p e r i o d d u r i n g which a c u r r e n t meter, d e s c r i b e d  i n C h a p t e r 2, was  installed.  to  FIGURE 6.  Chart of the r i v e r mouth area, showing the location of the current meter mooring  250-j  200-  J U L I A N  DAY  FIGURE 7. Smoothed current meter record obtained by using the angular c o r r e c t i o n and the smoothing f i l t e r A A A / ( 6 - 6 - 7 ) „ The break i n the time axis occurs at day 114. 6  6  7  to  CO  Ul  2  150-1  U  JULIAN FICJURE A 4A 2  2 4  A  8. 2 5  DAY  The low frequency component o f the current meteor record, obtained by using the f i l t e r . The break i n the time axis occurs a t day 1 1 4 .  /(24'24-25)  126  120  • JULIAN DAY  FIGURE 10. T i d a l e l e v a t i o n s a t P o i n t A t k i n s o n d u r i n g t h e time t h e c u r r e n t meter was i n o p e r a t i o n . The break i n t h e time a x i s o c c u r s a t day 114.  130  ty  ro  i—i o  <T>  ?r H* 3  rt  H  >  SPEED  IT H fD ro o rt H-  3  0)  O  (CM/S)  SPEED  i  3  cn  rt  o  4  X  o  cn O  o O  cn O  o  o o  o  m  r t  c  cu rt  :r ro  rt *<  Eu  H H H CO  >£• H • TO  W Hi H  C  O  g o  o  >  co rt Hrt  ro 3  C  r~  rt  CO __  &> rt  Z.  H  3  o  H* O  fl!  3  fl)  M *<  CO  AST  > -<  (CM/S)  1330 PST  MAY 8 1976 0.0  4.0  '  1  0.0  3.0  <  1  FIGURE 12.  8.0  1 .6.0  1  12.0  16.0  '  1  20.0  1  24.0  1  9.0  12.0  15.0  1S.0  1  1  1  1  28.0  i 21.0  i  32 0  I  SIGMA T  24 0  I  TEMP.  P r o f i l e s o f S,T,sigma t'. and c u r r e n t speed ( i n d i c a t e d by dots ) f o r 1330 PST. May  8, 19.76, a t t h e c u r r e n t meter mooring.  K  130  123"00-  1200 1500 1800  SE6 E6 calm  6 12 18 TIME (PST) STATION TIMES  h i  .1  k  1730 1732 1744 1750  1 rn n o  1758 1805 1814 1822 •  p q r  1832 1842 1849  FIGURE 14. S t a t i o n p o s i t i o n s and t i m e s , wind and t i d e Tor A p r i l 6.-. 1976.  STATION  FIGURE 15. ' S a l i n i t y s e c t i o n a l o n g l i n e h - r f o r A p r i l 6.  1976.  132  FIGURE 16.  S, T. sigma t prof.iles at s t a t i o n j , 1744 PST, A p r i l 6, 1976.  133  FIGURE  17.  S, T, sigma t p r o f i l e s  at station  k, 1750 PST, A p r i l  6,  1976.  134  WINDS AT SAND HEADS 0000 NW52 0400 NW44 0 700 NW42 0800 NW36 1200 N8 1800 SE6  0  6 12 18 TfME (PST)  24  STATION TIMES e 1513 f 1535 g 1553 h 1558  FIGURE  :i j k 1  1620 1625 1710 1713  m n o p  1730 1732 1755 1818  q 1831 r 1844  19. S t a t i o n p o s i t i o n s and t i m e s , wind and t i d e f o r A p r i l 15, 1976.  S T A T I O N  FIGURE 20.  Salinity distribution  a l o n g l i n e e-m  and a l o n g l i n e m-r  f o r A p r i l 15,  1976.  1 — ! 00  137  STATION  APRIL 15 197S 0.0  •s.o i  a.o  0.0  1.0  D.D  5.0  i  i  K  1710  PST  i  1G.J i  20.0  2 0  23.0 1  ?  6.0  9.0  12.3  1S.0  13.0  21.J  r  10.0 i  15.0 i  ?o.:  2S.:  J3.0  K.J  00.Q  i  i  i  i  l  i  i  l  •  SIGMA T  - 3  T W .  i  SflllNITT  —I X  FIGURE 21a. S, T, sigma t p r o f i l e s  STATION  F i P R I l 15 197G  3  O.C  4.0  8.0  12.0  IS.O  O.C  3.C  CO  9.0  12.0  0.0 - l —  a tstation  20.0  '?-  L 20.0  J  k, 1710 PST, A p r i l 15, 1976,  1713 2B.0  3  ''  3  PST n.1  SIGrtfi T  ^  TEMP.  C  SPLINITT  7  o rn •a  FIGURE 21b.  S, T, sigma t p r o f i l e a t s t a t i o n  1, 1713 PST, A p r i l 15, 1976.  WINDS AT SAM'TIE AD S" 0000 E10 0400 E4 0 700 calm 0900 W4 1200 NWS 1800 N10  6 12 18 TIME (PST)  24  STATION TIMES S 1623 h 1714 FIGURE 22. S t a t i o n p o s i t i o n s and t i m e s , wind and t i d e f o r A p r i l 28, 1976  FIGURE 23.  S a l i n i t y s e c t i o n f o r A p r i l 28, 1976, a l o n g a l i n e from a f r o n t a t s t a t i o n g t o s t a t i o n h. The s a l i n i t y i s o p y c n a l s a r e 15, 17.5, 20, 22.5, 25, and 26 p a r t s per thousand.  WINDS AT SAND HEADS 0000 0400 0700 1200 1500 1800  6 12 TIME (PST) 1  8  E2 E12 E12 SE8 SE6 S4  24  STATION TIMES o 1517 p 1530 q 1540 s 1625  t v w X  FIGURE 24.  1631 1635 1638 1640 1643  S t a t i o n p o s i t i o n s and t i m e s , vj.ind and t i d e f o r June 4, 1976.  STATION  x i-  Q_ LxJ Q  FIGURE 25a. S a l i n i t y s e c t i o n a l o n g l i n e o-s, June 4, 1976, S.H. i n d i c a t e s t h e l o c a t i o n o f Sand Heads.  STATION  FIGURE 25b. S a l i n i t y s e c t i o n a l o n g l i n e x - t , June 4, 1976. I s o h a l i n e s shown a r e 5, 10, 15, 20, 25, 26, 2 7 % . S.H. i n d i c a t e s t h e l o c a t i o n o f Sand Heads. 0  123  0  1  1  r  6  12  18  STATION TIMES Tl M E  a b c d  0642 0709 0733 0916  FIGURE 26.  3 1 rn o  ( PST  1106 1230 1302 1319  )  p q q r  132 9 1447 1457 1540  s  1625  S t a t i o n p o s i t i o n s and t i m e s , wind and t i d e f o r J u l y 3, 1975,  143  STATION  FIGURE 27a.  S a l i n i t y s e c t i o n a l o n g l i n e a-l,  J u l y 3, 1975.  STATION  9H FIGURE 2 7b.  S a l i n i t y s e c t i o n a l o n g l i n e p-s, J u l y 3, 1975.  FIGURE 28. S, 1; sigma t profiles at station a, 0642 PSTy July 3, 1975.  FIGURE 29.  S. T, sigma t p r o f i l e s a t s t a t i o n e, J u l y 3, 1975;  8733 PST,  146  JULY 3 1975 o.o 1 ., 0.0 1  0.0 o  STATION  4.0  B.O  i  l  I  3.0  CO  9.0  i  12.0  1  5.0  i  12.0  i  10.0  I  16.0  i  15.0  1  20.0  l  J  1106.  20.0  24.0  I  15.0 l  25.0 l  o m  F I G U R E 30.  S,  T,  1106  sigma t p r o f i l e s a t PST,  July  3,  1975.  station  i  19.0 I  30.0 l  PST 28.0 i  21.0 l  35.0  32.0 I  24.0 43.0 _ 1  TIME ( P S T ) FIGURE 31. Sketch of the plume observed by aerial survey, 0900 PST, July 2, 1975.  FIGURE  32.  S, T,  sigma t p r o f i l e  1250 PST,  July  2,  on t h e s i l t y  1975..  side  of the  front,  149  FIGURE 33.  S, T, sigma t p r o f i l e s on deep b l u e ' s i d e 1325 PST, J u l y 2,  1975.  of front,  150  FIGURE 34.  S, T, sigma t p r o f i l e s on s i l t y 0849 PST, J u l y 4, 19.75.  side  of front,  151  FIGURE 35.  S. T,  s i g m a t p r o f i l e s on d e e p b l u e s i d e  0 8 5 3 PST", J u l y  4, 1 9 7 5 .  of front,  152  FIGURE 36. The e v o l u t i o n of a f r o n t , J a n u a r y 18, 1976. The e s t i m a t e d t i d a l c u r r e n t s i n the i n s e t t a b l e are i n cm/sec, p o s i t i v e c u r r e n t s are f l o o d i n g .  153 30'  O H  2&  0  ,  TIME  STATION TIMES A B  1  .  12  10'  (PST)  12T0CT  1  24  '  0752 - 1121 1245 - 1425  FIGURE 37.  S t a t i o n p o s i t i o n s and t i m e s , wind and t i d e f o r J u l y 23. 1975.  •4SP05' FIGURE 38a. P a t h s o f drogues r e l e a s e d i n . r e g i o n A, J u l y 23, 1975. Speeds i n m/sec a r e i n d i c a t e d .  123° 2 5 ' FIGURE 38b. Paths o f drogues r e l e a s e d i n r e g i o n BJ u l y . 23, 1975. Speeds i n m/sec are i n d i c a t e d . cn  FIGURE 39. S, T, sigma t profiles in region A, 1025 PST, July 23, 1975.  156  SPEED  FIGURE 40.  (M/S)  Speed profile in region A, 1015 PST to 1045 PST, July 23, 1975.  157  SO"  10'  12X00-  WINDS AT SAND HEADS 0000 0400 0700 0900 1200 1500 0  6 12 18 T I M E (PST)  E10 SE12 SE12 E12 SE10 S8  24  STATION TIMES b c d e  0907 1020 1110 1222  f g h i  1226 1304 1404 1500  FIGURE 41. Station positions and times, wind and tide for July 13, 1976.  FIGURE 42.  S, T, sigma t profiles at station g, 1304 PST,  July 13,  1976.  SPEED 0 j  :  30  i  (CM/S) 60 1  124  FIGURE  43  Speed p r o f i l e a t s t a t i o n g, 1317  PST,  J u l y 13,  1976.  90  I—  FIGURE 44.'  P o l a r p l o t of v e l o c i t y v e c t o r s , s t a t i o n g, 1317  PST,  J u l y 13,  1976.  161 12T0O-  NE6 NE4 NE6  NE2 SE12 SE6  6 S T AT I ON TIME S a 0646 b 0716 c 0807 d 0857 FIGURE 45.  12  18  —^-0^-^_LE5J.) e  f g  h  0957 1113 1226 1318  i ,1  1422 1512  S t a t i o n p o s i t i o n s and t i m e s , wind and t i d e f o r Sept. 17, 1976.  FIGURE 4 6 .  S, T, s i g m a t p r o f i l e s a t s t a t i o n  a , 0646 P S T , S e p t . 1 7 , 1 9 7 6 .  FIGURE 47.  Speed p r o f i l e  a t s t a t i o n a, 0655 P S T , S e p t . 1 7 , 1 9 7 6 .  ON CO  0° "50 cm/s 0 6 5 5 PST  SEPT 17  0 m. 270°  &  _  50 cm/s  »  i  -• • i  Q  shi p drift  •  \  >  180°  ^ —  10 m.  FIGURE 48. Polar plot of velocity vectors, station a, 0655 PST, Sept. 17, 1976.  >  165  SEPT  1976  STATION  0.0  4.0  8.0 i  12.0  0.0  3.0  6.0 i  9.0 l  12.0  0.0  5.0 l  10.0  15.0  20.0  1  1  o  17  i  1  1  !  16.0  ! i  1  J  1512 20.0 i  15.0 i  25.0 i  PST 24.0 t  23.0 I  32.0  1B.0  21.0  24.0  35.0 1  4a.o  I  30.0 I  i  1 1 i  SIGMfl T TEMP. SALINITY  "o  cn. o  •  —lb •J:  CD'  b  FIGURE 49. S, T, Sigma t p r o f i l e s a t s t a t i o n j, 1512 PST, Sept. 17, 1976  0° 50  cm/s  1458 PST SEPT 17 50 cm/s  270°-^  1 3 m. ship  drift  0 m.  FIGURE 51.  P o l a r p l o t o f v e l o c i t y v e c t o r s , s t a t i o n j , 1458 PST, Sept. 17, 1976.  168  F R A S E R  FIGURE 52. salinity  An i d e a l i z e d S t r a i t o f G e o r g i a , showing c o n t o u r s o f s u r f a c e  ( 1 5 , 20, 25 p p t ) .  FIGURE 53.  R I V E R  AA' i s t h e p a t h o f a s a l i n i t y s e c t i o n , F i g . 53.  A s a l i n i t y s e c t i o n a l o n g AA', F i g . 52, and s a l i n i t y p r o f i l e s  a t t h r e e s t a t i o n s a l o n g AA'.  169  RfVER  FLUX DEPLETION  ENTRAINMENT FIGURE 54.  The c o n t r o l volume, i n d i c a t e d by dashed l i n e s ,  the plume d e f i n e d by t h e S=25  contour.  surrounding  The t h r e e c o n t r i b u t i o n s t o t h e  mass b a l a n c e , r i v e r f l o w , e n t r a i n m e n t , and d e p l e t i o n a r e i n d i c a t e d .  — —  Z  FIGURE 55. D e f i n i t i o n s k e t c h f o r t h e e q u a t i o n s d e r i v e d i n C h a n t e r 3.  :-D  FIGURE 56b. The r e g i o n o f s o l u t i o n , f i l l e d characteristics.  up w i t h  intersecting  1 7 1  \  FIGURE 57. R o . ' i s e  et  al,  Streamlines 1951.  and l i n e s o f e q u a l t h i c k n e s s , from  172  FIGURE 58a.  0 0  1  Schematic diagram f o r t h e model o f s u r f a c i n g  5  X ( KM) 10  _L_  15  isopycnals.  20  T /  -A  3 FIGURE 58b. The t h i c k n e s s o f t h e upper l a y e r , s o l i d l i n e , p r e d i c t e d Eqn. 4.13. The d o t s i n d i c a t e observed t h i c k n e s s e s , J u l y 3, 1975. The dashed l i n e i s a p l o t o f h ( L - x ) . 1 / 3  by  FIGURE 59b. S k e t c h of upper l a y e r c o n d i t i o n s used t o o b t a i n the i n t e g r a t e d p r e s s u r e term.  174  I FIGURE 60b.  1 FIGURE 60c.  — ^ ~  X  A c h a r a c t e r i s t i c intersecting the front at ( s , t ) .  3^  X  Two c h a r a c t e r i s t i c s intersecting at a hydraulic jump.  on  FIGURE 61.  The  characteristic  diagram f o r the  k i n e m a t i c wave  solution.  SALINITY  SALINITY  10-j  FIGURE 63a. The vertical distribution of salinity  FIGURE 63b. The vertical distribution of  at t=0, t=20At, 120At, 220At, and 320At, for  salinity at t=0, 20At, 220At, 500At, and 920At.  K=Z /(.2 ).  for K=l.  2  2  178  \  \  \  \  !D_  \  \  LU Q  . H  0 20 FIGURE 64a.  120  220 TIMESTEP  320  420  Depth o f t h e upper l a y e r . The s o l i d l i n e i s from t h e d i f f u s i o n e q u a t i o n s o l u t i o n , t h e dashed l i n e i s f r o m the  discrete layer  i  120  solution.  r  220 TIMESTEP  320  420  FIGURE 64b. T o t a l s a l t c o n t e n t o f t h e upper l a y e r . The s o l i d l i n e i s f r o m the d i f f u s i o n e q u a t i o n , t h e dashed from t h e l a y e r s o l u t i o n .  180  D'  FIGURE 66a. Schematic diagram of a turbulent j e t .  FIGURE 66b. A s e c t i o n through the plume, showing the e f f e c t the choice of bottom s a l i n i t y has on flow through an open boundary.  181  Y  A  X FIGURE 67. A t y p i c a l computat"onal element o f t h e n u m e r i c a l g r i d used i n t h i s r e s e a r c h . Z i s t h e v a r i a b l e name a s s i g n e d t o t h e l a y e r t h i c k n e s s , r e f e r r e d t o as h i n most e q u a t i o n s o f t h i s t h e s i s .  182  1  2  3  4  5  6  7  15  16  17  18  19  20  21  29  12  13  14  25 26  27  28  30  41  42  43  44  55  56  57  58  69  70  71  72  83  84  85  86  97  98  8  9  10  22 23  24  11  99 100  111 112  113 114  125 126  127 128  139 140  141 142  153 154  155 156  167 168  169 170  -  181 182  183 184  195 196  197 198  209 210  211 212  223 224  225 226  237 238  239 240  251 252  253 254  265 266  267 268  279 280  281 282  293 294  295 296  307 308  309 310  321 322  323 324 325 326 327 328 329 330 331 332 333 334 335 336 -  337 338 339|340 341 342 343 344 345 346 347 348 349 350  FIGURE 68. The complete g r i d used f o r t h e square box model. The r i v e r d i s c h a r g e s t h r o u g h mesh 169. P o s i t i v e v a l u e s o f y, ( s m a l l i n d i c e s ) , a r e a t t h e s o u t h e r n h a l f o f t h e model. The s o l i d b o u n d a r i e s a r e i n d i c a t e d by a t h i c k l i n e , the open b o u n d a r i e s by a heavy dashed l i n e .  FIGURE 69. Flux out of the open ends of the l i n e a r model. are equivalent to 133.33 hours.  1000  5 0 CM/SEC TIME  HRS.  3  TIME  STEP  100  MINS  0  1  W «  •« ^  -o  ~o  -e  X  \  \  6  ^Ni ^Na  i  (t  11i  \ \  20  \  \a  \  O  D  ti  6  o  i  / i  I / / / / »  J  —  *-*  l-l  «  /  </  s  <s a- o  y  ^  ^ O-  O-  B-  O-  O-  B-  O-  B-  o-  a-  B-  -€J —o —-o  FIGURE 70. V e l o c i t y vectors after 100 timesteps of the square box model with non-linear terms. The small square represents the t a i l of the vector, and also is the location f o r which the vector represents the v e l o c i t y .  -a  TIME HRS.  = 50 16  TIME STEP  500  a——o——a  CM/SEC MINS  40  o'  T V A  FIGURE 71. The same as F i g .  70, b u t a t timestep 500.  TIME HRS.  = 5 0 CM/SEC 33  TIME STEP  1000  MINS  — a  —o  20  "-n or'  FIGURE 72. The same as F i g . 70, b u t a t t i m e s t e p  1000.  o—  a —  74  0  500 TIMESTEP  1000  FIGURE 73a. Influx (solid line) and efflux (dashed line) for the model of Figs. 70, 71, and 72.  0  500 TIMESTEP  FIGURE 73b. Influx (solid line) and efflux (dashed line) for the model of Figs. 74, 75.  00  <1  TINE  HRS.  TIME  STEP  = 50 CM/SEC 3  a  e  o  o  B  O  Q  u  B  a  a  B  o  e  a  B  B  B  6  a  6  a  a  B  o  o  o  0  4  i  1 1 I  \  -a  eddy  \  \  —Q  -B  —O  —a  —e  —o  ~—-a  —o  —o  — a  ——ta  FIGURE  74.  111 111 1 1  \  \  \  \  \ \  \  0  B-  B—  •—  B—  i 0 JJ / J J / J J / a  A \  o  Flow f i e l d  viscosity.  \  \ \ \ \  -o  -e  \  \ \ \  \ \ \ \ \ \ \ \ \ V  ^B  \ \ \ \  a h  e  -o  20  100  o  -a  M1NS  calculated  i n t h e same way a s f o r F i g .  70, w i t h t h e a d d i t i o n o f h o r i z o n t a l  00 00  I TIME HRS.  =50 16  TIME STEP  • 500  CM/SEC MINS  40  cr"* cr-  i 1 / 1 / i  !  l  //  FIGURE 75. F l o w f i e l d c a l c u l a t e d eddy v i s c o s i t y .  /  /  y  /  ^  s  i n the same way as f o r F i g . 71, w i t h t h e a d d i t i o n o f h o r i z o n t a l  0 H -0  > 2  4 MESH  6 NO.  . 8  > 10  i  12  FIGURE 76. The propagation of a hump of water out of the system, the l i n e of small dashes shows the i n i t i a l d i s t r i b u t i o n of thickness. The shape as the bulge passed out of the open boundary i s shown by the large dashes, and the corresponding d i s t r i b u t i o n of v e l o c i t y by the s o l i d l i n e .  MESH  NO.  FIGURE 77. The d i s t r i b u t i o n of v e l o c i t y and elevation f o r F i g . 74, along the l i n e from the r i v e r mouth to the opposite s o l i d boundary.  192  78a.  78b.  49  44  45  125  109  68  51  177  155  115  64  230 168  128  80  89  63  48  141  121  69  49  49  196  161  112  63  48  51  234  158  120  76  51  "ks.  V  V.  v.  v  vj'  FIGURE 78a,b. Comparison o f e l e v a t i o n f i e l d s f r o m a .33 km g r i d model, 78a, and from a 1 km g r i d s i z e model, 78b.  79a.  size  79b.  140  79  3  171  162  72  218  170  427  184  123  76  10  24  199  128  90  19  144  70  3 1 1  142  125  75  131  105  4 8 5 157  117  90  FIGURE 79a,b. Comparison o f e n t r a i n m e n t v e l o c i t y from a .33 km mesh s i z e model, 79a, and from a 1 km mesh s i z e model, 79b.  193 80a.  80b.  2 17  14  4  16  10  23  16  10  24  15  35  10  37  11  8 3 V  v v x ^ ^  80a,b. Comparison o f u - v e l o c i t y and a 1 km mesh s i z e model, 80b.  FIGUPvE  80a,  f i e l d s f r o m a 0.33 km mesh s i z e model,  81a.  81b.  A  .3  7  4  8  7  10  7  14  16  10  4  19  16  12  6  14 16  17  "< v.  11 14  2  3  4  FIGURE 81a, b. Comparison o f v - v e l o c i t y f i e l d s from a 0.33 km mesh s i z e model, 81a, and a 1 km mesh s i z e model, 81b.  = 5 0 CM/SEC TIME HRS.  12  TIME STEP  360  MINS  0  B  — t t  \  \ \  \  \ \ \ \ \ \ \ \ \ l  o-  BT  h  C  1 1  i i  \ \  l  cr"  \  \  \  cr"  \  \  Q  h — o  — B  er-  d ea  ct—  FIGURE 82. F l o w f i e l d produced by a model w i t h v a r i a b l e r i v e r f l o w , t i d a l streams and e l e v a t i o n s , C o r i o l i s f o r c e , and a c o n s t a n t F r o u d e number boundary c o n d i t i o n .  = T I M E  H R S .  T I M E  S T E P  °\  * \  * \  \  \  V  ^  \  \  \  \  \  5  0  2 0  C M / S E C M I N S  6 0 0  \ \  \  \ \ \ \.\ \  \ \ \\ \ \ \ \ \ \ \ / \\\ \ \ \ \ \ \ \ \ / N  *s  ^  FIGURE 83. Flow f i e l d produced by a model with constant r i v e r flow, t i d a l f r i c t i o n constant of 0.005, and using ^2p3 =0 as a boundary condition.  N  TIME HRS.  = 50 CM/SEC 20  TIME STEP  600  V  MINS  ^  \ \ \ v % \  \\\ \ \\ \ \\ \ \ \ \ \ \ \ \\\\\W ^ \ \\ W W WW/ \\\ \\\\\\\v \  \  \  ° N  FIGURE 84. F l o w f i e l d produced by a model i d e n t i c a l t o t h a t o f F i 83 c o e f f i c i e n t o f 0.001. ' > 2  g  FIGURE 85 V e c t o r v e l o c i t y d i a g r a m f o r a model w i t h d e n s i t y e f f e c t s , no C o r i o l i s f o r c e , and c o n s t a n t r i v e r f l o w .  tidal  elevat erevar  198  r r i r r  r r r r r  rj r rr rr  J T  J  j r  j f  j  r  r  t  f  f  t r  p  / s "  IT) Z.  *—<  XL  I J  +J CO  '  CO  3  O  U UJ CO  CN  s. _  0  Q_ Ct  x  I—  in  II II I! IJ II If  oo  tc  •H  CL,  o cu  o E a  EH  a CL,  =  5 0 CM/SEC  TIME HRS.  184  TIME STEP  5520  MNS  0  A L = 1 KM ^  •—a —s —e "~-o —a  '—a ^-o  —o  ^"-B  ^B  —-a  ~*~-n \  \ i  \  •—B  B  B  \  o  ~-o —a -« —o ~-B  -a  ~-e  —o  "*B  ~a  -B  —a  "^B  ^B  "a -o -a  ^5  "B  "a  a  B  A or'  a  B  B B  -O  -a a  -e  -a -a -a  -a  —a  —a  —a  FIGURE 87. The model o f f i g u r e 85, 4 hours l a t e r . o  =  S O CM/SEC  TIME HRS.  186  TIME STEP  5580  MINS  FIGURE 88. The model o f F i g u r e 85, 6 hours l a t e r .  A L = 1 KM  to  o o  f  I  = 5 0  TIME  ,«  TIME STEP  5640  CM/SEC „  m  o  FIGURE 89. The model o f F i g u r e 85, 8 h o u r s l a t e r .  AL  = 1  KM  TIME HRS.  = 50 CM/SEC 190  TIME STEP  5700  ' MINS  0  ..  . ....  AL = 1 KM  FIGURE 90. The model of Figure 85, 10 hours later.  to o to  = 50 CM/SEC  —o  TIME HRS.  192  TIME STEP  5760  MINS  —e  —o  --a  — o  —«  ~~a  AL=1 KM -a  -a  -a  -e  -a  -a  -e  -a  -o  -o  *~a  -a  . *a  ""a  -B  -a  -a  -a  -a  -a  -a  -a  -a  -a  -a  -a  -a  -a  **B  ^B  ^B  ^B  ^a  >B  ^B  "TI ' - B  ^B  -B  -s  -a  -a  -a  "B  -e  -a  -o  -a  -a  "B  ~e  -a  -a  -e  -a  ~^B  -  e  -a  The m o d e l o f F i g u r e  -a  —a  B  F I G U R E 91.  -a  8 5 , 12 h o u r s  later.  -  e  -a  -  e  -a  -  e  -a  -  e  -a  -  a  -  e  186  HOURS  SALINITY  FIGURE 92. S a l i n i t y distribution at hour 186, corresponding to the flow f i e l d of Fig. 88. The 5 ppt and 10 ppt contours are shown.  to o  192  HOURS  SALINITY  10  /  \ 5  / /  1  / u  \  —  \  \  \  A.  FIGURE 93. The salinity distribution at hour 192, corresponding to Fig. 91. The 5 ppt and 10 ppt contours are shown. to o C/l  186  HOURS  ELEVATIONS  I  '  FIGURE 94. The d i s t r i b u t i o n o f upper l a y e r t h i c k n e s s a t hour 186, c o r r e s p o n d i n g t o F i g . 88. The 400 cm and 500 cm c o n t o u r s a r e shown.  to o  ^  192  HOURS  ELEVATIONS  FIGURE 95. The distribution of upper layer thickness at hour 192, corresponding to Fig. 91. The 400 cm and 500 cm contours are shown.  to o  to o  FTCURE 96 Drogue t r a c k s produced over a 24 hour period by the flow f i e l d of Figs. 85 - 91. Dots on thf t r a c k s are serrated by one hour, and the upper curve indicates the t i d a l elevatxon during the 24 hour period.  00  TIMESTEP 9360 TO TIMESTEP 10081  FIGURE 97. Drogue tracks produced over 24 hours by the flow f i e l d augmented flow at the boundaries during' outflow.  = 5 0 CM/SEC TIME HRS.  330  TIME STEP  9900  MINS  —o  -o  -e  ~-s  -a  -a  -e  —o  -a  -a  -a  a  e  a  e  o  a  a  FIGURE 98. A t y p i c a l v e l o c i t y f i e l d produced by a model w i t h augmented f l o w a t t h e open b o u n d a r i e s during outflow.  to O  1 nautical  mile  a)  float no.  time in  time out  15 16 17a 1Tb 18 19  58.3 58.3 58.3  58. h 58.5 58.8  20  21 22  59-0  58.7 58.8  59-0 60.3 60.  It  b) float no.  60.8 60.7 60.8 60.6 60.8 60.8  1 2  3 k 5 6  7a 7b  s  N  FIGURE 99.  Drogue t r a c k s f o r d r o g u e s { r e l e a s e d s h o r t l y  time in  time out  59-9 60.0 60.1 60.1* 6o. k 60.9  60. 5 60.5  6i.O 61.8  61.7 61.8 61.8 62.1 61.8 62.1  P a t h l i n e s o f f l o a t s i n a, S e t 16, from 58.2 h o u r s t o 60.8 h o u r s , b , S e t 17, f r o m 59-9 hours t o 62.1 h o u r s , t a k e n near l o w e r low t i d e . Winds were s o u t h h to southwest 8. The p a t h l i n e s numbered 7a and 7b, and 17a and 17b, i n d i c a t e t h a t t h o s e f l o a t s were b r i e f l y removed f o r repairs. b e f o r e low w a t e r , ( f r o m C o r d e s ,  1977)  u  TIME  (HOURS)  FIGURE 100. The discharge out o f the open boundaries o f a model w i t h t i d a l e l e v a t i o n s , v a r i a b l e density, constant r i v e r flow, and no C o r i o l i s f o r c e . The r i v e r inflow was 2000 m /sec. 3  213  FIGURE 101. Drogue tracks produced when drogues were released at zero r i v e r flow, approaching high water.  TIMESTEP 1530 TO TIMESTEP 1981  T I D f i l ELEVATION AT TSAUASSEN  FIGURE 102. Drogue tracks produced when drogues were released at half maximum r i v e r flow, during the ebb, when r i v e r flow i s increasing.  OUT  IN  TIMESTEP 1980 TO TIMESTEP 2431 TIDAL ELEVATION AT TSAUAS5EN  FIGURE 103. Drogue tracks produced when drogues were released at maximum r i v e r flow, near the end of the ebb.  IN  OUT  TIMESTEP 2430 TO TIMESTEP 2881  TIDAL ELEVATION AT TSAUASSEN  FIGURE 104. Drogue tracks produced when drogues were released at h a l f maximum r i v e r flow during the flood stage of the t i d e .  = 50 TIME  HRS.  TIME STEP  62  CM/SEC  MINS  I860  1=  Na N=  N,  S.  \  N,  N  \  \  \,  \  b  b  i  A  t4  A  \  • 1A s  \a  N.  tl  N N  b  •a  >l  V  "b  "to  "b  h  b  *b  •b  "b  b  b  fa  to  FIGURE 105. V e l o c i t y f i e l d produced by a model w i t h d e p l e t i o n . O t h e r terms a r e t h e same as t h o s e w h i c h produced F i g . 86, e x c e p t s a l i n i t y i s 20 % on i n f l o w . Q  TIMESTEP  1800 TO TIMESTEP  2161  TIDAL ELEVATION RT TSRWflSSEN  to M ON  FIGURE 106.  Drogue t r a c k s produced o v e r 12 h o u r s by t h e model w h i c h produced t h e f l o w f i e l d o f F i g . 105.  66  HOURS  ELEVATIONS  - 1 7 5 -  \ \ /  r \  I  \  200  \  \  \ \ \  FIGURE 107. E l e v a t i o n f i e l d produced by t h e model w i t h d e p l e t i o n a t 66 h o u r s , a t t h e same t i d a l phase as F i g . 94. The 175 cm and 200 cm c o n t o u r s a r e shown.  to -J  72  HOURS  ELEVATIONS  FIGURE 108. E l e v a t i o n f i e l d a t 72 h o u r s , produced b y t h e model w i t h d e p l e t i o n . i s t h e same as i n F i g . 95. The 175 cm and 200 cm c o n t o u r s a r e shown.  The t i d a l phase  to OO  219  TIME HRS. TIME STEP  FIGURE 109. V e l o c i t y f i e l d prodded' by the f i r s t r e a l geometry model, at time of maximim r i v e r flow.  220  END  OF  EBB  8 200  HOURS CM,  300  CM  C O N T O U R S  FIGURE 110.  D i s t r i b u t i o n of upper l a y e r thickness at the end of the  ebb.  END  OF  FLOOD  U  FIGURE 111.  HOURS  D i s t r i b u t i o n o f upper l a y e r t h i c k n e s s a t the  end  o f the  flood.  222  FIGURE 112. Drogue t r a c k s produced over 12 maximum r i v e r f l o w .  hours by drogues r e l e a s e d a t  223  F I G U R E 1 1 3 . N o r m a l i z e d e l e v a t i o n s , c u r r e n t s , and r i v e r the second v e r s i o n o f t h e r e a l geometry model.  d i s c h a r g e used i n  X= -.3  V=20 X=-.4  V = 25 X=-.5  V = 30 X=-.6  V= 2 5 U=10 X=-.6  V-20 X = -.56  V=15 X=-.36  FIGURE 114. D i s t r i b u t i o n of v e l o c i t i e s and surface slopes (cm/2km) used in the second version of the r e a l geometry model.  225  •  •  «  *  •  •  •  *  •  •  * f t* t * rt  8'  HOURS  FRASER  -*  /  1 \  s / 1 *  FIGURE 115. at 8 hours.  t \  •  •  t  t  4  4  4  4  •  •  •  *  •  *  •  *  •  •  •  •  •  •  •  *  4  4  4  4  4  •  4  t  4  4  4.  4  4  4.  4  4  4  4 •  4  4  4  4  4  4  4  4  4  4  4  4  4  4  4  4  4  V e l o c i t y f i e l d of the model with more r e a l i s t i c t i d a l forcing,  226  J"  •  *  *  *  * * * t *  12  t  H O U R S  *.  t 7T t \ \ \ T t rt \ \ \ \ \ 1 ** 1 \ \ \ 1 1 f *> \ *s  \  T  \ t  r  r \ ] \\ t t t>r \ i t t  FIGURE 116. V e l o c i t y f i e l d  produce  •  *  t * t1  by the model o f F i g u r e 115, 4 hours  later.  227  / t t /  T  / / / / / / / /  / / / / / /  / / / / / / / /  / / / / / / /  / / / / / / / / /  /  /  /  /  FIGURE 117.  / / /  /  / /  Velocity field produced by the model of Fig. 115, 8 hours later.  228  20  HOURS  »  *  /  /  /  I  i  /  /  /  I  I  /  /  J I  I J  /  4  *  /  <  4  /  / /  f  4  /  /  r~  r  s  I /  -r~  — f==r  *  .  • • r  \  \ v t *  ^  \  \  \  N  r  k  *  •  *  FIGURE 118. Velocity f i e l d produced by the model of Fig. 115, 12 hours later.  229  FIGURE 119.  Velocity field produced by the model of Fig. 115, 16 hours later.  230  TIDAL ELEVATION RT  HOUR  6  TO  TSAVASSEN  HOUR  12  FRASER N  FIGURE 120. T r a c k s p r o d u c e d igh low water.  by drogues  released  a t h o u r 6, a p p r o x i m a t e l y  231  T1DRL ELEVRTION RT TSRWRSSEN  HOUR  FIGURE 121.  12  TO  HOUR  18  Tracks produced by drogues released at hour 12, low high water.  FIGURE 12^-." Tracks produced by drogues r e l e a s e d a t 18 h o u r s , a t maximum r i v e r d i s c h a r g e , near low low w a t e r . __  233  TlORl ELE.VRT10N AT TSRURSSEN  HOUR  18  TO HOUR  24  FIGURE 123. Tracks produced by drogues t r a v e l l i n g i n the same flow f i e l d as those i n F i g . 122, but with a correction f o r v e r t i c a l shear in calculating'the drogue v e l o c i t y .  1 nautical  float no.  time in  time out  15 16 17a 1Tb 18 19 - 20 21 22  58.3 58.3 58.3 59-0 58.7 58.8 59.0 60.3 60.U  58.U 58.5 58.8 60.8 60.7 60.8 60.6 60.8 60.8  FIGURE 124. A comparison of drogue tracks from Cordes (1977), and this model. Both tracks correspond to drogues being released shortly before low water, and the markers on the drogue tracks are in both cases separated by one half hour.  to  8  HOURS  ELEVATIONS  FIGURE 125.  D i s t r i b u t i o n o f upper l a y e r t h i c k n e s s a t hour 8.  . 12 HOURS ELEVATIONS  FIGURE 126. Distribution of upper layer thickness at hour 12.  FIG1JKE 12 7.  D i s t r i b u t i o n of upper layer thickness at hour 16.  20  HOURS  ELEVATIONS  ^150 /  \  \ \ \  FRASER N  V  \  \ 300  \  /  150  FIGURE 128. Distribution of upper layrr thickness at hour 20.  FIGURE 129.  D i s t r i b u t i o n of upper layer thickness at hour 24.  FRONT  FIGURE 130.  FRONT  Schematic diagram of. a p o s s i b l e e x t e n s i o n o f t h e upper l a y e r model t o 2 l a y e r s .  FIGURE 131. The distribution exact solution.  of salt as calculated by a f i r s t order scheme.  The dashed line is the  FIGURE 132. The d i s t r i b u t i o n o f s a l t as c a l c u l a t e d by a second o r d e r scheme. s o l u t i o n i s shown by t h e dashed l i n e .  The e x a c t  

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