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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 this 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 British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of 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 British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date P i i ABSTRACT The Fraser River plume i s the brackish surface layer formed when the Fraser River discharges into the Strait of Georgia. Two approaches to understanding the dynamics of the plume are discussed. I n i t i a l l y , a series of field observations was carried out in the plume. These consisted mainly of CSTD profiles and current profiles in the upper 10-20 meters of the water column. Also, a surface current meter was installed for 34 days at the mouth of the Fraser River. The principal conclusions of the f i e l d observations are: the plume i s strongly sheared in the vertical and strongly stra t i f i e d ; this vertical structure i s most apparent in the vicinity of the river mouth, and around the time of maximum river discharge (near low water in the Strai t ) ; and that the water moving outward from the river mouth subsequently acquires velocities and s a l i n i t i e s appropriate to the water beneath i t with length and time scales for this change of order 50 km and 8 hours. The plume thickness varies between 0 and 10 meters; the salinity varies from 0 to that of the water beneath i t (approx. 25 %o) » an°l the difference between the plume velocity and that of the water beneath i t varies from up to 3.5 m/sec to 0 m/sec, and i s typically of order 0.5 m/sec over 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 in the upper layer, the thickness of the layer, and the integrated salinity in the upper layer. The bottom of the upper layer has been tentatively defined by an isopycnal surface. The mixing across this interface i s modelled by an upward flux of salt water (entrainment), and a downward flux of brackish water (termed depletion in this work). The dynamical effects included in this model are: the local time derivative; the f i e l d accelerations; the buoyant spreading pressure gradient (including the effects of salinity on the density f i e l d ) ; the entrainment of t i d a l l y moving water and the loss by the depletion mechanism of water with the plume momentum; the fric t i o n a l stress between the plume and the water beneath i t ; the forcing due to the baroclinic t i d a l slopes; and the Coriolis force. Subsets of the f u l l model equations are examined, to cla r i f y certain aspects of the plume dynamics. Preliminary results from the numerical solution of the f u l l model eguations are presented, and a comparison i s made between the paths of lagrangian trackers produced by the model and drogue tracks observed in the plume. Future improvements to the model are discussed. iv TABLE OF CONTENTS ABSTRACT ........ .... ..... ... ............ ..... ... . ........... i i LIST OF TABLES V LIST OF FIGURES ............... ...... vi ACKNOWLEDGEMENTS ................... ................. ........xiv 1. INTRODUCTION ... ... ............ 1 2. FIELD OBSERVATIONS OF THE PLUME 9 3. A MODEL OF THE FRASER RIVER PLUME . ..,. . » V . ,y,,. . . .. . * V . . , - . 26 4. AIDS TO INTUITION ABOUT THE PLUME 43 5. NUMERICAL MODELING OF THE FRASER RIVER PLUME ............. 71 6. CONCLUDING DISCUSSION , . . 93 REFERENCES CITED . . . .... . . ....... .... ............. ........... , 97 APPENDIX ................. ... .......... ...... ...... .......... 103 TABLES ...... .114 FIGURES .. ,..,117 V LIST OF TABLES - TABLE I. HARMONIC ANALYSIS OF HIVES SPEEDS .................. 114 TABLE II. 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 v i LIST OF FIGURES Figure 1. Chart showing the Straits of Georgia and Juan de Fuca 117 Figure 2. Salinity distribution in the Strait of Georgia and Juan de Fuca Strait, 1-6 July 1968 .......118 Figure 3. Salinity distribution in the Strait of Georgia and Juan de Fuca Strait, 4-8 Dec...1967 ...................... 119 Figure 4. Chart showing the Fraser River delta, and central Strait of Georgia 120 Figure 5. The daily Fraser River discharge for 1976, measured at Hope. .....................• ................. 121 Figure 6. Chart of the river 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 tidal part of the current meter record ........125 Figure 10. Tidal elevations at Point Atkinson during the time the current meter was in operation ................. 126 Figure 11. A reconstruction of the tid a l signal from constituents obtained by harmonic analysis .................. 127 Figure 12. Profiles of S, T, sigma T and current speed for 1330 PST, Hay 8, 1976, at the current meter mooring .....128 Figure 13. The salinity distribution as a function of time at the river mouth, January 21 , 1975 129 Figure 14. Station positions and times, wind and tide for April 6, 1976 ............... .< ,. ....... .. ................130 Figure 15. Salinity section along line h-r for April 6 1976 ................................ «• .....131 Figure 16. S, T, sigma T profiles at station j , 1744 PST, April 6, 1976 132 Figure 17. S, T, sigma T profiles at station k, 1750 PST, April 6, 1976 .....................................133 Figure 18. S, T, sigma T profiles at station 1, 1758 PST, April 6, 1976 .....................................134 v i i Figure 19. Station positions and times, wind and tide aoir A.pxrxX 15 ̂  1976 •».•••*•'•<'•••••'•••••?<•.•• *•.«-• «••••••«•••••«•• 135 Figure 20. Salinity distribution along line e-m and along line m-r for April 15, 1976 ................ ..... 136 Figure 21a. S, T, sigma T profiles at station k, 1710 PST, April 15, 1976 137 Figure 21b. S, T, sigma T profile at station 1, 1713 !PST̂  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 * • *,• • # • • • • • * 138 Figure 23. Salinity section for April 28 1976, along a line from a front at station g to station h .....139 Figure 24. Station positions and times, H i n d and tide for June 4, 1976 ................. ........................... 140 Figure 25a. salinity section along line o-s, *3 XL D©. II ̂  - 1 976 « *- • * *<••'• * m'm • '• • • * • • • • « • • » " • 4 • •••>•• • • * • • • • 141 Figure 25b. Salinity section along line x-t, June 4, 1 Figure 26. Station positions and times, wind and tide for July 3, 1975 ...... 142 Figure 27a. Salinity section along line a - l , July 3, 1975 y,.... ...... 143 Figure 27b. Salinity section along line p-s, 4?u.X y. 3 -f 1975 • * • * • • • • • • • • < • • • • / « • ' at * *«• * • • * « • • ••'••••» 143 Figure 28. S, T, sigma T profiles at station a, 06 42 PST, July 3, 1 975 . 144 Figure 29. S, T, sigma T profiles at station e, -0133. • I?S T f J n X y 3 § 1975 * * •/ v ^ « • **••••>•••••*•• 145 Figure 30. S, T, sigma T profiles at station j , 1106 PST, July 3, 1975 ............. . . . . , y .......... , ....146 Figure 31. Sketch of the plume observed by aerial survey, 0900 PST, July 2, 1975 147 Figure 32. S T, sigma T profiles on the s i l t y side of "the., front, 1250 PST, July 2, 1975 ............. 148 Figure 33. S, T, sigma T profiles on deep blue side of front, 1325 PST, July 2, 1975 ............................149 Figure 34. S, T, sigma T profiles on s i l t y side of front, 0849 PST, July 4, 1975 150 v i i i Figure 35. S, T, sigma T profiles on deep blue side of front, 0853 PST, July 4, 1975 151 Figure 36. The evolution of a front, January 18, 1976 .......152 Figure 37. Station positions and times, wind and tide for July 23, 1975 ....................... i . . . . . . . . . . . . . . . 153 Figure 38a. Paths of drogues released in region A, July 23, 1975 154 Figure 38b. Paths of drogues released in region B, July 23, 1965 ...... ......................................... 154 Figure 39. S, T, sigma T profiles in region A, 1025 PST, Jult 23, 1975 155 Figure 40. Speed profile in region A, 1015 PST to 1045 PST, July 23, 1975 ..................................156 Figure 41. Station positions and times, wind and tide for July 13, 1976 157 Figure 42. ,S, T, sigma T profiles at station g, 1304 PST, July 13, 1976 15 8 Figure 43. Speed profile at station g, 1317 PST, July 13, 1976 ........ ....................................... 159 Figure 44. Polar plot of velocity vectors, station 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 profiles at station a, 0646 PST, Sept. 17 1976 162 Figure 47. Speed profile at station a 0655 PST, Sept. 17, 1 976 ..... ..........................,.. ................ 163 Figure 48. Polar plot of velocity vectors, station a, 0655 PST, Sept. 17, 1976 164 Figure 49. S, T, sigma T profiles at station j , 1512 PST, Sept., 17, 1976 165 Figure 50. Speed profile at station j , 1458 PST, Sept. 17, 1976 .... ... ...........................,... .....* «. 166 Figure 51. Polar plot of velocity vectors, station j , 1458 PST, Sept. 17, 1976 167 Figure 52. An idealized Strait of Georgia, showing contours of surface sal i n i t y ..........168 ix Figure 53. ft salinity section along AA*, Fig. 52, arid salinity profiles at three stations ..................... 168 Figure 54. The control volume, indicated by dashed lines, surrounding the plume defined by the S=25°/oo contour 169 Figure 55. Definition sketch for 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 characteristics C + and C_ with respect to the streamline direction .......... 170 Figure 56b. The region of solution, f i l l e d up with intersecting characteristics ...... 170 Figure 57. Streamlines and lines of equal thickness. from Bouse et a l , 1951 ................................. 171 Figure 58a. Schematic diagram for the model of surfacing isopycnals .,....,,...........,..,....... .......... 172 Figure 58b. The thickness of the upper layer predicted by Eqn. 4.13 ......................................172 Figure 59a. The control 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 implicit characteristic solution, Egn. 4.26 ......................... 174 Figure 60b. A characteristic intersecting the front at (s,t) . . . . . . . . . . . 174 Figure 60c. Two characteristics intersecting at a hydraulic jump .....,..............,.......,............... 174 Figure 61. The characteristic diagram for the kinematic wave solution .175 Figure 62. The distribution of u at t=l4, from the kinematic wave solution ..................................... 176 Figure 63a. The vertical distribution of salinity for K=Z*/(0.22) ..... ........ 177 Figure 63b. The vertical distribution of salinity for K=1 . . . . . . . . . . . . . . 177 Figure 64a. Depth of the upper layer as a function of time ...... .178 Figure 64b. Total salt content of the upper layer as a function of time , 178 X Figure 65. An isoconcentration curve and velocity vectors for a turbulent plane jet ....179 Figure 66a. Schematic diagram of a turbulent jet ............ 180 Figure 66b. A section through the plume, showing the effect the choice of bottom salinity has on flow through an open boundary .................................... 180 Figure 67. A typical computational element of the numerical grid used in this research ..........181 Figure 68. The complete grid used for the sguare box model ..182 Figure 69. Flux out of the open ends of the linear 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 for the model of Figs. 70, 71, 72 187 Figure 73b. Influx and efflux for the model of Figs. 7 4, 75 ...... ... . . ... ....187 Figure 74. Flow f i e l d calculated in the same way as for Fig. 70, with the addition of horizontal eddy viscosity .....188 Figure 75. Flow f i e l d calculated in the same way as for Fig. 71, with the addition of horizontal eddy viscosity ..... 189 Figure 76. The propogation of a hump of water out of the system ..... .................., ........190 Figure 77. The distribution of velocity and elevation for Fig..74, along the line from the river mouth the opposite solid boundary .................................191 Figure 78a,b. Comparison of elevation fields from a 0.33 km grid size model and a 1 km grid size 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 fields from a 0.33 km mesh model and a 1 km mesh model .................... 193 Figure 81a,b. Comparison of v-velocity fields from a 0.33 km mesh model and a 1 km mesh model ........,..>........ 193 x i Figure 82. Flow f i e l d produced by a model with variable river flow, t i d a l streams and elevations, Coriolis force, and a constant Froude number boundary condition ...................... ............ 194 Figure 83. Flow fi e l d produced by a model with constant river flow, tidal elevations, Coriolis force, fric t i o n constant of 0.005, and using ( ) " " ' F l / 3 M i = o as a boundary condition ............ .....195 Figure 84. Flow f i e l d produced by a model identical to that of Fig. 83, but with a fr i c t i o n coefficient of .001 ....196 Figure 85. Velocity vectors for a model with density effects, t i d a l elevations, no Coriolis force, and constant river flow ..................................... 197 Figure 86. The model of Fig. 85, 2 hours later ..............198 Figure 87. The model of Figure 85, 4 hours later ............ 199 Figure 88. The model of Fig. 85, 6 hours later .............. 200 Figure 89. The model of Fig. 85, 8 hours later .........201 Figure 90. The model of Fig. 85, 10 hours later ............. 202 Figure 91. The model of Fig. 85, 12 hours later 203 Figure 92. Salinity distribution at hour 186, corresponding to the flow f i e l d of Fig.,88 .................. 204 Figure 93. The salinity distribution at hour 192, corresponding to Fig. 91 205 Figure 94. The distribution of upper layer thickness at hour 186, corresponding to Fig. 88 ....................... 206 Figure 95. The distribution of upper layer thickness at hour 192, corresponding to Fig. 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 Fig. 98, using an augmented flow at the boundaries during outflow ........,...v......,........209 Figure 98. & typical 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 x i i Figure 100. The discharge out of the open boundaries of a model with tidal elevations, variable density, constant river flow, and no Coriolis force .................. 212 Figure 101. Drogue tracks produced when drogues were released at zero river flow, approaching high water .........213 Figure 102. Drogue tracks produced when drogues were released at half maximum river flow, during the ebb, when river flow i s increasing ................................213 Figure 103. Drogue tracks produced when drogues were released at maximum river 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 Fig. 105 216 Figure 107. Elevation fi e l d produced by the model with depletion ..................... . ..... ...., .... 217 Figure 108. Elevation fi e l d at 72 hours, produced by the model with depletion 218 Figure 109. Velocity f i e l d produced by the f i r s t real 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 river flow ...222 Figure 113. Normalized elevations, currents, and river discharge used in the second version of the real geometry model ...223 Figure 114. Distribution of velocities and surface slopes used in the second version of the real geometry model .......224 Figure 115. Velocity f i e l d of the model with more re 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 produced by the model of Fig. 115, 4 hours later 226 x i i i Figure 117. Velocity f i e l d produced by the model of Fig. 115, 8 hours later 227 Figure 118. Velocity f i e l d produced by the model of Fig. 115, 12 hours later 228 Figure 119. Velocity f i e l d produced by the model of Fig. 115, 16 hours later ..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 travelling in the same flow fi e l d as those in Fig. 122, but with a correction for vertical shear in calculating the drogue velocity .......23 3 Figure 124. A comparison of drogue tracks from Cordes(1977) and this 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. Distribution 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 distribution of salt as calculated by a f i r s t order scheme ...........241 Figure 132. The distribution of salt as calculated by a second order scheme ....................................... 242 x i v ACKNOWLEDGEMENT Although many people help in carrying through a thesis project, I would like in particular to acknowledge the assistance of Dr. P. B. Crean, and Dr. P. H. Leblond. Dr. Crean, of Environment Canada, was particularly helpful in guiding much of the research, both the f i e l d work and in developing the numerical model. Dr. Leblond, my thesis supervisor, always offered a positive approach to the work, particularly in the writing of the thesis. I would also like to thank A. B. Ages (OAS) for the use of the CSTD probe; and Dr. M. Miyake (OAS) for the use of the sonic current meter. The financial support of the National Research Council and the University of British Columbia, through a Macmillan family scholarship i s gratefully acknowledged. 1 CHAPTER 1 INTRODUCTION One of the most striking oceanographic features of the Strait of Georgia is the Fraser River plume. Referring to Fig. 1, the Strait of Georgia i s the body of water separating Vancouver Island from the British Columbia mainland. The Fraser River, located south of Vancouver, discharges into the eastern side of the Strait of Georgia. Particularly during the late spring and early summer, in times of large river runoff, the Fraser River plume appears to be a layer of muddy brown water floating on the dark blue Strait 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 are sometimes weaker colour discontinuities 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 salinity and momentum differences between the plume and the ambient water decrease. For purposes of this thesis, the following two-part definition 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, the ambient Strait of Georgia waterj 2) in order to be associated with the plume, this mixed water must retain a significant identity as river water, for example i t must be fresher than some arbitrary maximum salinity, for example 25°/oo {parts per thousand). The value of this arbitrary salinity is subject to variation, depending on the season, and on what properties of the plume one wants to 2 describe. . Because i t i s less dense than the ambient water, the plume is a relatively thin layer, floating on and interacting with the denser Strait of Georgia water. There are two main reasons for studying the Fraser River plume.-. As mentioned above, i t i s a striking feature of the Strait of Georgia, and for that reason alone warrants attention. The other, more practical> reason i s that the plume plays a very important role in the flushing and general circulation of the Strait of Georgia, which in turn influence the biological environment of the Strait.,Biological properties of the Strait influenced by the Fraser River induced circulation include the supply of upwelled nutrients to the surface layers; the attenuation of sunlight by suspended sediment; and the horizontal advection and vertical migration of planktonic organisms. REVIEW OF THE OCEANOGRAPHY OF- THE STRAIT OF = GEORGIA At this point we will consider briefly the oceanography of the Strait of Georgia (Fig. 1) . Waldichuk (1957) carried out an extensive study of the Strait of Georgia. Most of the circulation theory discussed below appeared in this paper. Further information comes from Crean and Ages (1971), who carried out a series of hydrographic cruises over a period of one year, occupying stations in the entire Juan de Fuca Georgia system. The Strait of Georgia is a fjord-type estuary - i t i s deep, has a freshwater source, i s connected to the sea, and i s 3 strongly stratified. The average depth is about 150 meters, but considerable areas are deeper than 200 meters. The Strait differs from simpler types of fjords in that i t i s very wide ( on the average about 30 km. ), and the major source of fresh water, the Fraser River, i s near the main outlet of the system, where strong ti d a l mixing occurs. ,• Further, there are two connections with the sea. At the northern end i s found a complicated set of narrow channels, through which tidal currents reach 6 m/s. At the south, the Strait of Georgia i s connected to Juan de Fuca Strait by another system of passes and s i l l s , where ti d a l currents reach 1.5 m/s; Juan de Fuca Strait in turn has a free connection to the Pacific Ocean., Figures 2 and 3, from Crean and Ages (1971) show salinity sections along the centreline of the Strait of Georgia for the months of July and December.: The Fraser River i s seen to be a strong source of stratification in the summer, and a weaker source in the winter. One also sees the evidence for strong mixing in the region of Haro Strait in the south, and Cape Mudge in the north, with reference to Figures 2 and 3, one wonders what proportion of the Fraser River water flows north and is mixed there; what proportion flows south and i s mixed there; and what proportion, while flowing north and south, i s mixed in the middle of the Strait? The prevailing winds are in general along the axis of the Strait - northwest and southeast ( Waldichuk, 1957 ), However, there are frequent storms with variable wind direction, and the Fraser valley in particular modifies the direction of winds in the southern Strait. 4 More than 70% of the fresh water input to the Straits of Georgia and Juan de Fuca comes from the Fraser ( Herlinveaux and Tully, 1961 ). The river discharges through a complicated system of channels in i t s delta, Fig. 4, but 80% of the flow passes through the Main Arm which i s dredged to be about 400 meters wide and 10 meters deep. Flow velocities at the river mouth are tid a l l y modulated, varying between about 0 ra/s and 3.5 m/s over a tidal cycle. Further, the river discharge is very seasonal. Fig. 5, changing by an order of magnitude from about 1000 m3/sec during winter to about 10,000 m3/s during the spring-early summer freshet. If one defines the plume to be water which i s fresher than, for example, 28 % o , then from Fig. : 2 and Fig. 3, i t s thickness i s about 10 meters, and i t covers most of the central Strait of Georgia during summer, and a considerably smaller area during winter. During summer, the fresh water fraction in this plume volume i s much greater than during winter. Also, because of the lower st a b i l i t y of the plume in winter, i t is more liable to be mixed away by storms, which are stronger in the winter. There are very strong tides in the Strait of Georgia ( Crean, 1976 ), and they exert considerable influence on the plume. First, the tide modulates the river flow, so that maximum river discharge occurs near low water, and the river i s effectively shut off at high water. The river water, as i t replenishes the plume already existing, i s then acted on by the tides in the Strait. Frictional interaction with the t i d a l currents in the underlying water 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 rise to vertical mixing, partially inhibited by the vertical density gradient. The barotropic surface slope will not only tend to move the plume up and down the Strait, but also gives a considerable cross channel forcing { due to the Coriolis force in the barotropic equations ). Consequently, the plume moves , about in a complicated manner, losing fresh water by mixing at about the same rate { t i d a l l y averaged ) that i t gains i t from the river discharge. Near the river mouth, where there i s the most available kinetic energy, there i s intense mixing, but the resulting mixed water i s fresh enough, and has the necessary river-directed momentum, to form an outgoing plume. This mixing produces an upwelling of nutrient rich salt water, which i s very important from a biological point of view < de Lange Boom, 1976 ). As the river water proceeds outwards and mixes, the sta b i l i t y of the plume decreases, so that near the t i d a l mixing passes, mixing is almost complete. The seaward transport of salt water in the plume i s compensated by a return flow of salt water beneath the plume, as in a l l estuarine circulations. Thus, studying the plume w i l l aid in explaining the movements of deep water in the Strait of Georgia. , PREVIOUS STUDIES OF THE PLUME Because of i t s interaction with the general circulation of the Strait, the plume has recieved considerable study. Giovanda and Tabata ( 1970 ) presented the results of tracking drogues which were released near the Fraser River mouth, and followed for periods ranging from 2 to 33 hours. 6 Tabata ( 1972 ) attempted to identify the different types of water in the plume from aerial photographs. DE Lange Boom ( 1976 ) described a mathematical model of chlorophyll distribution in the plume, with the hydrodynamic flow pattern being somewhat arbitrarily prescribed. Cordes ( 1977 ) described the results of a rather sophisticated drogue tracking procedure in the plume., STUDIES OF SIMILAR SYSTEMS Recently Long (1975b), and Winter et al (1977), have discussed one layer models of fjords, similar i n many respects to the model discussed in this thesis. There also exists a considerable body of literature on thermal plumes due to cooling-water discharge from power plants (Koh and Fan, 1970; Stolzenbach and Harleman, 1971). Numerical and physical models exist for these plumes, but they are not applicable to the Fraser River plume. Usually there i s only one independent variable, the distance along the plume axis; the other horizontal dimension being taken care of by empirical spreading coefficients and profiles of properties..The models are time-independent, and not adaptable to time-dependent situations; they are not adaptable to a system with partially enclosing solid boundaries. Takano { 1954a, 1954b, 1955 ), in a series of papers, discussed the spreading of a river plume issuing into an unbounded ocean. His model was time-independent, and involved a balance of Coriolis force, hydrostatic pressure gradient, and horizontal eddy viscosity; i t predicted a bending to the right 7 of the plume, although the discharge remained symmetric about the entrant direction of the river. The choice of horizontal eddy viscosity was probably inappropriate because the most important friction is likely the interaction of the plume with the underlying water, and not with i t s e l f and water to the sides as in the case of horizontal eddy viscosity. Wright and Coleman { 1971 ) discussed the Mississippi River plume, which i s in some ways similar to the Fraser River plume, the principal difference being that tides are about 1/10 as strong in the Gulf of Mexico as in the Strait of Georgia. They f i t a model developed by Bondar (1970), similar to cooling-water plume models, to some fie l d data. One would conclude from their work that buoyant spreading and entrainment are the two most important forces governing a plume. Garvine (Garvine and Monk, 1974, Garvine, 1974, Garvine, 1977 ) has described f i e l d work done in the Connecticut River plume, and has developed a model to explain the propagation of the leading edge of a plume., The Connecticut River is much smaller than the Fraser in terms of discharge, but the plumes have many similarities - strong tidal currents act on them, and they both form distinct fronts. In tne Garvine model frontal dynamics are controlled by the salinity and density profiles behind the front. 8 ISIS RESEARCH Although Garvine lays great stress on fronts as controlling the dynamics of plumes, the approach taken in this thesis is that before one can develop a model which includes fronts, one has to have a good model to describe the flow between the river mouth and the front, i.e..a time-dependent model of a thin continuous upper layer. In this thesis, two complementary methods to increase our understanding of the plume have been used. A f a i r l y extensive, but exploratory, set of f i e l d measurements was carried out, using mainly a CSTD, and on a few occasions a profiling current meter. Based on the f i e l d work, a two dimensional, vertically integrated ( over the plume thickness )> time-dependent numerical model was developed. The model includes: the effects of barotropic t i d a l slopes and streams; the effects of vertical mixing on the distribution of salt and momentum; the Coriolis force; the stress between the upper and lower layers; and the buoyant spreading and i n e r t i a l effects in the plume. This model is exploratory, in that i t cannot predict nature very accurately, but allows us to check the effects and importance of various forces acting on the plume. Further fi e l d work wi l l allow a better adjustment of the mixing and stress in the model. 9 QM.hi.lIR 2 FIELD OBSERVATIONS OF TflE PLUil As mentioned in chapter 1, various people have already made measurements of the Fraser River plume. There are a few salinity profiles available ( Waldichuk, 1957; de Lange Boom, 1976 ) , obtained by discrete sampling. There are also extensive drogue tracks 1 available { Giovando and Tabata, 1972; Cordes, 1977 ) , but they lack information about the vertical structure of velocity and salinity in the plume. In developing the observational aspects of this research, i t was f e l t that a useful contribution to knowledge of the plume would be to investigate the vertical structure of velocity and salinity in the plume, with particular reference to location in the Strait of Georgia and stage of tide. In this chapter are described some of the f i e l d observations obtained using a continuous recording CSTD probe, ( conductivity, salini t y , temperature, and depth ), and a profiling current meter. The observations were carried out in cooperation with P.B. Crean ( Ocean and Aguatic sciences, Environment Canada ) at various times between January 1975 and September 1976. Two boats were used i n this f i e l d work. ,.In 1975 the CSV Richardson, a 20 meter vessel, was used to obtain CSTD profiles. In 1976 the Brisk, an 8 meter launch was used. As the Brisk required no operating crew other than the two scientists, ( P.B. Crean and myself ), and was always available, i t s operation allowed considerable f l e x i b i l i t y . Using this boat, which was equipped with radar for accurate positioning, we did CSTD profiles, velocity proliles, and measured positions of the colour front associated with the plume. 10 A l l of the sali n i t y and sigma-T profiles presented here were calculated from conductivity, temperature and depth data obtained with an Interocean model 513 CSTD probe and a chart recorder. The conductivity and temperature traces were digitized on the Hechanical:Engineering digitizer at U.B.C., and salinity and sigma-T profiles calculated from them. The CSTD was designed to operate from 0 to 30 meters, so was ideal for our purposes. However, for a variety of reasons, the accuracy of the probe and chart recorder system i s not reliable. The accuracy varied with time, depending on such factors as which boat was being used, the ambient temperature, and how long the chart recorder had been operating that particular day. For these reasons, the salinity profiles can only be considered to be accurate to about 0,5 % o unless one i s comparing a series of successive casts. Fortunately, the error is not random, but systematic ( e.g., the zero might be offset on the chart recorder ), and the size of the salinity error i s much smaller than the size of salinity variation in the plume, so that these low accuracy salinity profiles reveal almost a l l the pertinent salinity 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, let us f i r s t look at conditions at the river mouth. It is easier to interpret the salinity measurements in the plume when we know the temporal relationship between the river discharge and the barotropic tide in the 11 Strait. As a preliminary stage in acquiring this information in f u l l , the Canadian Hydrographic Service installed a bottom- moored surface current meter ( Neyrpic design ) at the mouth of the Fraser Biver, Figure 6. The installation position chosen i s a compromise between putting the current meter in an out-of-the- way location safe from shipping, and putting i t in the main channel, subject to the f u l l river flow. The current meter was in place for 34 days, from April 6 to Hay 11, 1976. During this time, the river discharge increased fron 1100 m3/sec to 7700 m3/sec, ( Fig.,5 ), a typical variation during the onset of the spring freshet. At the end of each 10 minute interval the meter recorded the number of revolutions of i t s propellor { which converts to a speed ), and the instantaneous magnetic heading at the end of the 10 minute interval. Because of the large mass of steel in the current meter mooring, the magnetic direction has to be used with caution. During times of significant outflow velocity, when the flow was presumably along the line of the Sand Heads jetty, { 215° magnetic ), the current meter indicated a direction of 306° magnetic, approximately perpendicular to the actual flow.,During the high water part of the t i d a l cycle, when the actual surface velocity approaches zero, and sometimes assumes a direction toward the jetty, ( a change of 90° ), the magnetically determined direction changed by only 35°, to about 270°. In order to convert the speed signal into a velocity signal, the current meter record was treated as follows. For magnetic directions greater than 300°, corresponding to outflow conditions, the velocity was assumed to be entirely parallel to the channel, and directed downstream. For directions less than 12 300°, the velocities were very small, and i t was decided to scale the magnetic variation of 30° {300° to 270°) into a variation of 180°, so that most of these low velocity periods were treated as flows up the channel.,Because the velocity signal i s sampled at 1 hour intervals in the subseguent harmonic analysis, i t i s impossible to reproduce the shape of the speed record very well, and the above treatment gives a smoother velocity record than i f the small flows at high water were taken to have zero component in the direction of the river channel., Assuming the above conversion from speed to velocity, the record was prepared for harmonic analysis, in order to determine the relation between the river flow and the tides in the Strait. The signal was f i r s t band pass fi l t e r e d by attenuating components with frequencies higher than one cycle per hour, and lower than one cycle per day. The signal resulting from removal of high frequency (periods less than one hour ) components is shown in 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, is a running mean of length 6 applied to the time series, and At i s the spacing of sampled points. The low frequency component of the signal was obtained with the A24A2*IA25/24«24«25, At=1 hour f i l t e r , ( Godin 1970), and i s shown in Figure 8. The t i d a l band of frequencies was obtained by subtracting the low frequency signal, Figure 8, from the smoothed signal. Fig. 7, and i s shown in Figure 9. This ti d a l record was then harmonically analysed. Because the record was not long enough, i t was impossible to separate some constituents, for example the group K1, P1, Sl.,It was assumed 13 that the relations between the amplitudes and phases of these constituents in the river speed record were the same as for the same constituents obtained from an analysis of one year of observations of the elevation at Point Atkinson, a nearby permanent tide gauge. The results are shown in Table 1, and the results of harmonically analysing the observed Point Atkinson elevations 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 significant mean as well as MM and MSf constituents. Finally, the velocity record was reconstructed from the harmonic analysis, and i s shown in Figure 11. If the surface current consisted only of tidal and shallow water constituents. Fig. 9 and Fig.11 should be identical. That they are not indicates a more complicated situation. The following observations about the harmonic analysis should be made. 1) . We would like to know when the maximum discharge occurs, relative to the water levels in the Strait., Comparing the records of surface current with the sea levels at Point Atkinson, maximum current occurs about 0.7 hours before low water at Pt. Atkinson. Using the phases from tables 1 and 2, for the M2 tida l constituent, low water occurs at cot=34 0° { 180° after maximum ). Maximum current occurs at <^>t=312°. The difference, 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 in the above 14 calculation of the time of maximum river 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 from the interaction 2M2-N2, a diurnal shallow water constituent identical in freguency to L2 ( Godin, 1970 ) . There i s a significant change in the L2 amplitude over the 18.6 year nodal period. The harmonic analysis program corrected the apparent L2 amplitude using this factor, whereas i t should probably have applied the correction for 2M2-N2, a much smaller correction. 3) Although there was a seven-fold increase of river discharge while the meter was in operation, the mean current. Fig.8, showed only a slight increase. One must conclude that the thickness of the outflowing layer increased over the 34 days the meter was installed. 4) . There were two events during which the measured water velocity was abnormally low for a day or two { Fig. 7 ). One occurred about day 106, the other day 129. Two possible explanations suggest themselves; one i s meteorological forcing, the other is the presence of debris caught in the current meter propellor. On day 105-106, there was a storm, with NW winds of up to 52 mph. This would certainly affect the surface flow, and probably the measurement of i t . However, day 129 ( May 8 ) was calm. By fortunate coincidence. Dr.,P. Crean happened to be doing current profiles while tied up to the current meter bouy. The measured surface current, 50 cm/sec around noon and early afternoon, agrees with that measured by the moored meter. Thus, for this period, there appears no ready explanation in terms of mteteorological forcing, or meter malfunction. 15 B. VERTICAL CURRENT AND SALINITY STRUCTURE Weather and other work never seemed to permit a very good measurement of the profiles of current at the river mouth during the time the surface current meter was in place. Figure 12 indicates the salinity and current structure present at the current meter mooring, Fig. 6, on Hay 8, 1976 (Julian day 129), at 1330 PST. Positive speeds refer to an outflow, and since only speed was measured by the current meter ( a propellor-type General Oceanics model 2031, rigidly attached to a lead " f i s h " with large orienting fins 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 sal i n i t y structure oyer 10 hours at Sand Heads on January 21, 1975. The dashed line in the inset diagram indicates, as a function of time, the fraction of fresh water in the water column, considering pure salt water to have the salinity of the deepest water in each cast. Like the surface current, the fraction of fresh water reaches a maximum shortly before ( about 2 hours before ) low water. 2A. .SALINITY AND DENSITY STRUCTURE OF THE PLUME The salinity information obtained in this research is in general agreement with that reported by laldichuk (1957) and de Lange Boom (1976). It was however obtained with a continuously recording CSTD as opposed to discrete bottle casts. As will be apparent from the profiles in this section, the density structure very closely follows the salinity structure, the temperature structure having a relatively minor effect on the 1 6 density in the upper layer. Ap_ril 6j_ 1976 This is the f i r s t day the current meter was in position. The Fraser discharge at Hope was 1100 m3/sec, and winds were light. Figure 14 indicates the positions of stations , labelled by letters. Figure 15 i s a plot of salinity contours along the line h-r. The t i d a l curve ( inset, Fig.14 ), shows that these stations were done on a rising tide after a large f a l l . It is interesting to note that we passed a very weak colour front (slight colour change in the water) in the vicinity of station k, station k being on the fresh or eastern side of the front. Figures 16 , 17 , and 18 show the S, T, sigma-T profiles for stations j , k, 1. There i s a slight indication of fresher water at the surface at station k. Fig. 17 , and a definite freshening of the surface layer at station 1, Fig. , 1 8 . This surface layer becomes thicker and fresher as one proceeds from station 1 towards the river mouth at Sand Heads. In a l l three salinity curves, Figs. 16, 17, and 18, the pycnocline i s found at about S=25 % o . He should also notice that as one proceeds outward from Sand Heads, the isopycnals gradually rise to the surface. A ^ r i l 1 5 A 1926 This series of CSTD's was taken along a rather complicated path. Fig. 19. The line g-h-i-j-k-l^m-n represents the boundary of a colour front. The lines f-g and n-o-p-g represent paths through the plume. Stations g , j , l , and m are on the plume side of the front, and stations h,i,k, and n are on the salt water side of the front. The salinity along line e-f-g-j-l-m i s plotted in Figure 2 0 , along with a continuation along m-o-p-g. 17 We note that these profiles were done on a rising tide, and that there had been a 10 hour period of strong northwest winds prior to the measurements. One notes that the profiles along the front ( g#j*l# m ) are fa i r l y similar, and that comparing them to those along line m-o-p-g-r, the plume i s considerably fresher and thicker in the inner regions than around the edges.. Figures 21a and 21b i l l u s t r a t e the salinity profiles at stations k and 1 respectively, k being outside the plume, and 1 being adjacent to k (about 50 m away), but on the plume side of the front. We note that for the profile in the plume, Fig. 21b, there appears to be a three layered structure to the profile - a surface brackish layer about 0.7 meters thick, lying on top of a layer of intermediate salinity about 2.5 meter thick, lying on top of the almost homogeneous lower water. .,- Atoril 28x 1976 Referring to Fig. 22 for position information. Figure 23 shows a salinity section taken from a front at station g back toward Sand Heads, along a line g-h. It i s interesting to note how nearly horizontal the isohalines are, and how the salinity varies continuously in the vertical through the plume. June 4 A 1976 The profile of salinity along line o-p-q-t-s { Fig 24 ) i s shown in f i g . 25a. The winds were light and the profiles were done about an hour after low water, at which time the river flow is quite strong. The daily discharge at Hope was 6,200 m3/sec, a f a i r l y high value. We see that near Sand Heads ( station t, located 0.5 nautical miles from Sand Heads into the Strait ), the water is almost entirely fresh and at station g, located 1 18 nautical mile from Sand Heads, the salinity varies from near zero at the surface to 27 °loo at 7 meters. Three possibilities suggest themselves as causes for the rise of isohalines as one proceeds downstream. One i s that there i s extensive lateral spreading of the flow near the river mouth. However, the drogue tracks of Cordes ( 1977 ) suggest that there is very l i t t l e lateral spread of flow at times of high discharge. a second reason would be that there i s extensive mixing at the river mouth. While doing the profiles at the river mouth On June 4 the water surface indicated considerable mixing activity- large diameter upwelling regions, and considerable patchiness in colour. The third possible reason i s that the salt wedge { Hodgins, 1974 ) has been flushed out, and we are seeing the toe of the wedge outside the river mouth. However,the flushing out of the wedge i s intimately connected with mixing, so the second and third reasons mentioned above are related. In order to draw more detailed isopycnals near the river mouth, we did a series of CSTD casts while drifting out from Sand Heads, as shown in Fig 25b. Note that the horizontal scale i s expanded in Fig. 25b. Station g of Fig.,25a i s at the same location as station x of Fig. 25b. Stations x and t are separated by 0.5 nautical miles, or about 1 km. The isopycnals in Fig. 25b a l l rise uniformly as one proceeds outward. ^ a l l 3,1976 Referring to Figure 26 for geographical locations, figure 27a shows a plot of salinity along line a-b-c-d-j-1, and figure 27b i s a plot of salinity along s-q-m-o-p. July 1975 was a time of high runoff,( 6,800 m3/sec on July 1 ), and one observes a 1 9 considerable amount of fresh water in these profiles. It i s interesting to observe the changes in the actual profiles as one proceeds downstream. Figure 28, 29, and 30 show three of the profiles used to draw Fig. 27a { stations a, c, and j ) . Going from station a to station c, the upper layer does not thicken very much, but becomes approximately 5 % o saltier. Going from station c 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 in the salinity sections Fig. 27a, and 27b. fts on April 6 1976, Fig.15, as one proceeds away from the river mouth, one observes the fresh water layer getting thinner. More important, perhaps, one notices that the surface salinity increases, and generally a l l isopycnals rise toward the surface. This significant feature of the plume will play an important role in subsequent chapters. Ji-ZEQETS KM THE PLUME A feature of the plume frequently commented upon is the sharp colour front, usually found on the northern boundary of the plume. Because of i t s possible dynamical significance, some effort was made to understand the nature of this front. On July 2, 1975, a strong colour front was found by aerial survey, and Figure 31 indicates the main features of the plume at about 0900 PST. Figures 32 and 33 show profiles of S, T, and sigma-T on either side of this front, close to Porlier Pass. The plume appears to be a layer of relatively fresh water about 1 meter thick advancing onto the st r a t i f i e d ambient water. Along the front there i s a very strong convergence as the rapidly 20 moving s i l t y water approaches the clear, dark blue salty water. Consequently, the front i s a very good collector of debris of varying size, ranging from air bubbles and foam to large logs. There i s also considerable lateral shear at these fronts, and large scale eddies capable of turning the 8 meter Brisk in a complete ci r c l e in less than 5 minutes. The apparently strong vertical circulation at the front suggested using dye as a tracer. We tried to obtain quantitative results using a fluorimeter, but were unsuccessful. However, on July 3, 1975 { Fig.,26 ) we were able to set up a very interesting experiment at stations g and r. A rather intense front ( in terms of vertical circulation , though not in terms of colour ) was found at stations g and r, heading south., Red rhodamine dye was released on the water surface, about 100 m upstream of the fronts This dye advanced along the surface to the front, and then disappeared by downwelling. There was no evidence of i t recirculating up into the surface waters, as i t completely vanished at the front. However, when the CSV Richardson, of 2 meter draft, was manoeuvering about 100 meters upstream of the front, dye was stirred up to the surface by the ship's screw, indicating the dye was being l e f t behind by the front. Figures 34 and 35 show two sets of profiles for July 4, 1975, taken in about the same location as those of Figs. 32 and 33. They again represent conditions on either side of a colour front, Fig. 34 representing conditions on the fresher, s i l t i e r 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 result 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, in spite of wind mixing. Figure 36 is an attempt to depict the evolution of a front in the winter, when, because of low river flow, the front does not extend over as large a region as during the summer. The front was readily identified, despite the low discharge of 1100 m3/sec, because of a good colour contrast and a large amount of debris along i t . We travelled along the front as i t spread outward, obtaining the ship's position and hence the front position by radar, and the approximate orientation of the front by compass. The curves of Fig. 36, indicating successive assumed positions of the front, were drawn using the numbered points where position and orientation of the front were known. The ti d a l currents indicated in Fig. 36 were roughly inferred from point Atkinson elevations. One sees the front, i n i t i a l l y inclined toward the south, gradually curving outward and northward as the river flow builds up and the ti d a l currents turn from ebb to flood. I t CURRENTS IN THE PLUME A predominant feature of currents in the plume is their very strong vertical shear. One method of determining the current structure i s by use of Langrangian trackers, or drogues. Drogues of three depths were used : a surface drogue (S), with drag element about 0.5 m in depth ; a medium drogue, (M), with drag element suspended from 0 to 2 meters ; and a deep drogue, (D), with 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 Fig. 38a are shown the paths of the above described drogues, released simultaneously at the front on an ebbing tide in region A, Fig. 37., The deep drogue travelled southward the fastest, and the shallow drogue stayed with the front. The drogues were picked up and re-installed in region B, considerably behind the front, on the following flood; their paths are shown in figure 33b. Now, the shallow drogue travels northward the fastest. Figure 39 is a salinity profile corresponding to the ebbing conditions, and one certainly notices that, as in velocity, there i s a great change in salinity as one descends from 0 to 4 meters. Figure 40 shows a relative speed profile obtained with an Ott propellor-type current meter, on the morning of July 23, 1975, in region a of figure 37. The profile was obtained from a very shallow draft ( approx. 0.2 meter ) boat tied to a drogue similar to the deep one described above. The zero crossing i s only inferred from the observed minimum in the speed. One has considerable confidence in the relative surface speeds measured, because the boat had such a shallow draft that i t probably did not interfere with the flow. The salinity structure at the time of the speed profile is that shown in figure 39. li,SONIC CURRENT METER ME&SUREMENTS For two weeks, in July and September of 1976, we borrowed a sonic current meter from M. Miyake (then of UBC, now of.OAS). This meter measures the two horizontal components of flow relative to the meter, and the magnetic heading of the meter, 23 with a sampling frequency of 1 hz. Current profiles were obtained from the Brisk in the following way. To eliminate the large wire angles which result when the boat dr i f t s with the surface current and one i s measuring the deep ( 5-10 meter ) currents, the boat was restrained by a sea anchor ( a conical drag element, suspended at about 8 meters ). Conseguently, the ship d r i f t was intermediate between the surface and deep velocities, and the relative speeds were such that the wire angle was always less than 20°, and usually considerably less. Ship d r i f t was monitored by radar, and added to the measured currents, giving currents relative to a fixed coordinate system. The analog output of the current meter was recorded on magnetic tape, electronically digitized, and vector averages were calculated over the 2 to 5 minute period the current meter was held at each depth. The measurements on July 13, 1976 and Sept. 17, 1976 are particularly interesting. Mil l i t 1976 Figure 41 shows the d r i f t path of the ship, from station b to station i inclusive. The Hope discharge on July 11 was 8,600 m3/sec, a high value for July. A front heading westward passed the ship at about 1225 PST, and figure 42 shows a profile of salinity at 1304 PST, station g, after the front had passed, and on the freshwater side of the front. Figure 43 shows the water speed at various depths at station g. The vertical structure of salinity and water current are remarkably similar. Figure 44 shows the result of plotting the currents at succesive 1 meter depths on a polar co-ordinate system, with the dots representing the tips of the current vectors. There i s a strong westward flow 24 at the surface, and a relatively weak flow to the southwest in the deeper water. September Y]_ 1.976 Figure 45 shows the ship drift on this day. There are two sets of observations presented , identical in format to those of July 13. Thus, figures 46, 47, and 48 refer to conditions at 0646 PST, station a, and figures 49, 50, and 51 refer to conditions at 1512 PST, station j . We note that over this 8.5 hour period, the surface salinity has increased markedly, the thickness of the upper layer decreased, and the strong velocity shear present in the morning has disappeared. Some of the decrease in thickness i s due to horizontal spreading and advection* but some is due to mixing, as indicated by the increased surface salinity. A l l of these attenuations in plume properties ( vertical shear, thickness, salinity difference ) occurred in the absence of significant winds. & i A SHORT SUMMARY OF THE OBSERVATIONS The plume is a brackish layer of thickness ranging from less than 1 to 10 meters. This layer i s freshest and thickest near Sand Heads, and becomes thinner and saltier as one proceeds away from Sand Heads. At the river mouth, the momentum of the river water i s directed in the direction of the jetty, except at high water, when the river flow i s shut off. As one proceeds outward, this river momentum is lost, and the plume acquires the velocity of the tidal streams. The vertical structure of the plume is quite complicated. 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 relatively unmixed river water. (Because of the velocity shear and vertical salinity gradient, the transport of fresh water i s much greater in the upper part of the plume than the lower. ) Because the river flow i s tidally modulated, the plume i s fed with pulses of fresh water at tidal frequencies. These bursts of fresh water advance onto, and incorporate themselves into, the existing plume. If, because of strong wind mixing, there i s no distinct plume in the Strait, then a front forms between the new discharge of river water and the ambient Strait water* the front being characterized by a significant colour change across i t , and a strong convergence along i t . ,. If s i l t y water already exists in the Strait, a somewhat weaker front forms, s t i l l being characterized by a vertical circulation. 26 CHAPTER 3 A DEL OF THE FRASES RIVER PLUME The f i e l d data described in Chapter 2 may be combined with basic hydrodynamic principles to construct a theoretical model of the plume. Such a model will 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 distribution of salt water entrainment into the plume. The results of the model may also be useful in resolving questions bearing on the nature of the mixing of fresh and salt water over large parts of the Strait and in particular the mechanism by which fresh water leaves the Strait. The above points that our model must deal with are of ultimate concern in studies of the Fraser River Plume. However, a model may also be useful i f i t only points up where there are gaps in our understanding of the physical system, when the model f a i l s to describe reality, and the failure is not due to the inadequacy of the mathematics, then we know that we must improve the model, usually by extension and generalization of the physics, inspired by f i e l d measurements. The numerical solutions of our model described below agree with f i e l d measurements to within an order of magnitude, and usually within a factor of 2; and indicate what should be measured in future f i e l d t rips. In this chapter, I will discuss the physical concepts that are to be incorporated in the model, and develop the mathematical equations that incorporate these physical principles. In Chapter 4 are presented some simple solutions of subsets of the f u l l equations; in Chapter 5 the solution of the model eguations by numerical means i s discussed. 27 A CONTROL VOLUME DESCRIPTION OF THE PLUHI In order to in i t i a t e discussion of the plume, consider an idealized Strait of Georgia, with mean contours of surface salinity schematized in the plan view of Figure 52. Figure 53 shows a salinity section along line AA* of Figure 52, and the associated salinity profiles at the indicated locations. The usual procedure in developing a 2-layer model of a strongly stratified f l u i d i s to consider the interface between the two layers to be associated with the pycnocline (Long, 1975b), or with a particular density interface (charney, 1955). In order to be specific, I chose to define the base of the upper layer in terms of an isohaline surface which (as seen in the previous chapter) i s eguivalent to using a density surface. The explicit specification of an upper layer depth is important for 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 fluxes of salt, mass, and momentum. In the case of the Fraser River plume, the isopycnal chosen corresponds approximately to the base of the pycnocline, so in a sense this model is compatible with both definitions of upper layer thickness. Choosing the S=25°/Oo contour as the plume .lower boundary, l e t us f i r s t treat the plume in the control volume approach (Figure 54). The control volume i s bounded by the S=25%o contour, and receives fresh water from the river at a tidally modulated rate. It also receives salt water from below, as indicated by the increase of salinity downstream.,Averaged over 28 a few t i d a l cycles, the volume of the control volume i s constant, so these two inflows must be matched by an outflow. This outflow presumably occurs mainly at the downstream extremity of the plume. As one follows a streamline near the outer edge of the plume, the salinity along this streamline increases, by means of small scale mixing, until i t exceeds our agreed upon limit for the boundary of the plume, and passes out of the control volume. The inflow of salt water into the plume has been called entrainment (Turner, 1974), and i t appears that a name has to be invented for flow out of the control volume. Perhaps depletion i s the most evocative term for this phenomenon. Thus, when the river flow is strong and there i s vigorous entrainment, the control volume expands somewhat, and spreads outward. The outer edges are then eroded by gentler mixing, expressed mathematically as the sum of entrainment and relatively stronger depletion. (Both terms are needed, entrainment to increase the salinity of the layer, and depletion to balance the volume added by entrainment.) Considering now the momentum balance, the water in the control volume, as well as the boundary of that volume, moves back and forth in the Strait for a variety of reasons. These reasons f a l l into 3 categories: 1) bodily fluxes of momentum - the river momentum, the momentum of the entrained water, and the momentum removed by depletion, i.e., when the water leaves the control volume, i t takes i t s momentum with i t ; 2) contact forces - the effects of wind stress are omitted in the model developed here, so the only contact force i s the stress at 29 the interface between two countermoving fluids, f r i c t i o n and form drag; 3) body forces - the barotropic tide, 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 real cases where i t is unimportant. Wind effects can be added later as a further refinement once the windless model has been sufficiently developed. , DIFFERENTIAL EQUATIONS DESCRIBING THE PLUME This section deals with the derivation of the vertically integrated equations describing the motion of a thin layer of lighter water floating on and interacting with a very deep layer of denser water. Two sets of eguations are derived - one for the entire water column, which reduce to the barotropic t i d a l equations in the 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) , salinity s=s (z), and thickness h=f>(-£, where /r[ and £ are the upper and lower interfaces respectively. The z axis i s positive upward, the mean value of i s 0, and the mean value of % i s -h, the negative of the layer thickness., The lower layer has uniform properties p o , u 0, s 0. The eguations wi l l be derived for only one horizontal dimension; they are quite simply qeneralized to include the second horizontal dimension. 1. CONTINUITY EQUATION For either layer, the continuity equation is 30 ^ 2 * 3.1. Integrating eguation 3.1 over the upper layer depth, we have 3 X L 3.2a. Expanding the integral, we get 5 * J 2 x ? X 3. 2 b. Re can apply the kinematic boundary condition at the upper surface z=/^ D t C ^ 2* 3.3.. to simplify eguation 3.2b. At z= 1= , ^ C ̂  ~ ̂  ) describes the motion of f l u i d relative to the interface z=£. As discussed earlier in this chapter, f l u i d crosses the interface , entering or leaving the plume, by means of entrainment and depletion. It i s convenient to represent the net effect of these two competing processes as the sum of the two effects expressed separately. Writing entrainment as wp, and depletion as w0, we have Dt 3.4. Thus, 3. 5. 31 and the continuity equation for the upper layer becomes: 3 * * 3T£ * 3.6.. The continuity equation for the entire water column i s A [ ll <-2 (a J ? - fk = o. * 3.7. M. Adding and subtracting J ^'^z, we have 2~t- -Q 3x 3.8. The l e f t hand side of equation 3.8 is the continuity equation in a fluid with depth independent velocity, and the right hand side is a correction due to the presence of the river flow in the upper layer. 2i SALT EQUATION Assuming molecular diffusion is negligible compared to turbulent effects the salt conservation eguation i s given by: 2tr 3x 3 * 3.9.. where <us> includes the mean and turbulent fluxes in the horizontal direction, and <ws> denotes them in the vertical. Integrating eguation 3.9 over the upper layer, we have, 32 T ^ D i 3.10. Assuming that <us^ = (us)^ , ( where for notational simplicity guantities without angle brackets denote mean flow properties ), that i s , neglecting horizontal turbulent fluxes, and assuming that <wsy= (»s)^, we can use the kinematic boundary contition 3.3 to simplify 3. 10. At the interface z=j"", the quantity <ws>^ has two components. One i s associated with motion of the interface, denoted by v s^. The other i s due to the flux of salt relative to the interface., The flux associated with entrainment we write wpsc, where s e i s the salinity of the lower layer; and the flux associated with depletion we write wns, where s i s a salinity appropiate to the negative flux of salt due to depletion. Thus ^ J ^ - - ^ i « t s g + W - S ' " w " ^ 3.11a or % • ^ J ' 3.11b. Using equation 3.11b and the kinematic boundary condition 3.3, 3. 10 simplifies to : 3.12. The salt equation i s usually not included in barotropic t i d a l calculations, as the salinity i s assumed uniform, so i t i s not written out here. However, i t i s interesting to note the salt equation for the lower layer alone takes the form 33 Z- f S Jg +2. ( SOfdz = - Sa \A>? + S VOn . -D ^ ^ 3.13. Thus, as one expects, the roles of entrainment and depletion are reversed for the lower layer, and the plume acts as both a source and a sink of salt (in different places of course) for the lower layer. 3t.THE HYDROSTATIC EgOATION It i s assumed that the balance of vertical forces i s adequately described by the hydrostatic equation, IB = - a* . 2? I d 3.14. At a depth z in the upper layer, p = - / > 3 ^ ' = / > 3 • * 3.15a. At a depth z in the lower layer, £ 3.15b. We should re c a l l , from the salinity and sigma-T profiles presented in Chapter 2, that the density structure i s controlled almost entirely by the salinity structure, and in fact, cr T » k s 3. i e . 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 i t to be only close, not exact. The agreement i s within 5% for winter conditions, but during summer, when the water i s considerably warmer and not as isothermal, a reasonable value of k would vary from 0.55 near the surface to 0.6 near the bottom of the plume. iii HORIZONTAL MOMENTUM EQUATION The horizontal momentum equation, ignoring molecular diffusion, i s : •*. I <<uu > 4- l_ <u uJ> +± -± JJL* ~-Q, 2t dz P dx C ^>z 3.17. where is the stress in the x-direction and p i s the pressure. Integrating over the upper layer, ignoring horizontal turbulent fluxes, and using the kinematic boundary condition 3.3, the f i r s t three terms become: <P f U dz + 2. f OlZ d- \aJ? U 0 t W „ U where u c and u have similar meaning to sa and s. Ignoring wind stress, the integral 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 and form drag in addition to wpu0- wnu which i s a momentum flux associated with mass transfer across the interface. The remaining term to evaluate is the pressure gradient term, 35 9 Z'*2 Because the analysis becomes very complicated otherwise, we will assume that the density in the upper layer i s vertically homogeneous, a form of the Boussinesq approximation. The pressure force in the upper layer i s then: Thus, the upper layer momentum eguation i s dt i 3 x e r 3 ? 9 x o 9 x J 3. 18. Note that for the upper layer we have 3 eguations, 3.6, 3.12, and 3.18, and 4 independent variables, u, s,/^, and £ . In order to solve the upper layer equations, we require a relation between/^ and£, which we obtain from the equations for the barotropic tide. The simplest way to obtain the relation i s by observing that in the deep water of the Strait, the horizontal pressure gradients must be those calculated in a barotropic t i d a l model, since the flows so calculated agree with observations ( Crean, 1976 ). Thus, O ̂  ' I o i/O-e r l a ye r where £>-ry i s the surface slope obtained from a barotropic model. Then using equation 3.15b for the pressure at a point in the 36 lower layer, 1 ' 3.19a. Defining$ - ̂ 0-^, the above becomes, after rearranging, 3. 20. Eguation 3.20 i s the fourth eguation reguired to complete the problem. The pressure gradient term in the integrated upper layer eguation 3.18. becomes then (ignoring variations in p when not differentiated) t f 3.21. Now to derive the momentum eguation for the entire water column. A l l terms except the pressure gradient term are quite straightforward. The pressure term i s : ± If d*- + ( 5 - L ^ ? = f ° 3.22a Substituting p z p o - ̂  # and doing a bit of rearranging, we get the above expression to egual: ^ ( D + nr\ ) t\ x + ^ C D + * 1 ) £ c5 ( / * ( - £ ) ) x I 3.22b. 37 Note that i f we consider the terms proportional to D + r*\ , and require them to equal pbojCr^, we get p - as in eguation 3.20. The momentum eguation for the entire water column i s then (Oo "I 3. 23. The l e f t hand side represents the momentum terms in the barotropic t i d a l equation. The f i r s t terra on the riqht hand side represents the correction made to g•( 0 * x because i t contains the pressure gradient required to maintain the river flow. The pressure gradient associated with the river flow i s (#/p»X <^/*""^ * * a n d this i s subtracted from r\x , in eguation 3.23, to give <^£-r^, the gradient calculated in a model with no river effects. The second term on the right represents a correction to the barotropic pressure because the upper layer i s somewhat lighter. The other terms, proportional to - j£ have l i t t l e effect on the barotropic motion, whose driving terms, the le f t hand side of 3.23, are proportional to D. Thus the presence of the river flow has negligible effects on a barotropic model, and the surface slopes and velocities obtained from that model 38 may be used as forcing for the upper layer model. To summarize, we will rewrite the eguations for the upper layer in terms of a variable z which is zero at the bottom o f the upper layer and increases upwards, and also w i l l include the second horizontal dimension, and the Coriolis force. Note that the thickness of the upper layer i s h=/^-.£. CQJTIJJOITY * A r r -T" J ° * + f- / vdx = \AJf - W „ 3.24. SALT A I jsd* + i. f usdz + A. fvsdz So % - 5 \AJ„ . 0 3.2 5. DIRECTED MOMENTUM h 4 ^ V, 2J o o pa 3. 26 X-BISECTED MOMENTUM A ~ " ~~ A A- ^ 3.27. 39 There are several things which have yet to be specified in this model. 1. We have to provide the profiles of u, v, and s, so that, for example, we can relate J ^ " " ^ ^ to J u . 2. wp,w,0, u, s, and r̂«-f must be specified in terms of flow properties such as the density and velocity differences between the upper and lower layer, and the thickness of the upper layer. ry # u0# and v G must be obtained from a barotropic tidal model (Crean, 1977), which is 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 in terms of the large scale properties of the flow. Item 3 can be considered as external driving of the flow. However, the extent of the forcing produced by u Q and v 0 depends on the type of turbulent interactions specified in items 1 and 2. Having gone through considerable algebra to get to the above eguations, we should check that they agree with the physics we want to model. First, consider the continuity and salt equations, as in the control volume approach described earlier, the guantity of water in the plume changes, ̂ V^r, because of horizontal divergences, in particular the river flow, and by fluxes relative to the boundary, Wp and wn. Similarly, the salt content can change by internal rearrangement, or by influxes of salt, w ps 0, and effluxes, ŵ s. In the momentum eguation, we see that the change in no momentum of a column of flu i d i s approximately given by: Or J buoyant spreading j t>/^^ I/^OJS/I7') * f r i c t i o n a l interaction^ * gain or loss of water and i t s associated momentum, wfUt>-\aJ„ ia , + forcing by the barotropic t i d a l slopes, aj S r ^ . It i s d i f f i c u l t to estimate the relative importance of these terms, since the plume i s spatially and temporally variable. However, Table 3 presents very coarse estimates of the order of magnitude of the terms in the momentum equation..The f i r s t part of the table l i s t s the scaling parameters, and the second part l i s t s the sizes of the various terms in the momentum eguation, for the region near the river mouth, and for the far fi e l d . For the salinity and continuity eguations, we notice that the ratio of the advective terms to the source terms i s uh/(wL), which i s also given in Table 3. Wp i s estimated from the numerical model of Chapter 5, where wp was calculated according to the formula wP = 0.0001u. Except for the action of winds and possibly horizontal eddy viscosity and diffusivity, the eguations derived in this chapter appear to have a l l the necessary terms to describe the plume. BOONDaHY CONDITIONS In solving any differential equation system, one has to specify the appropriate boundary conditions. The actual boundary conditions used will be discussed in chapter 5, but I would like to discuss here the theoretical boundary condition requirements. Consider a simplification of the above equations, in x-t space 41 only, (we define a new variable, =s„-s, the salinity defect.) £A t 2 ( uA) - o. dt 3* 3x These equations can be thought of as homogeneous eguations the behaviour of whose solutions dictate the behaviour of the eguations with forcing and dissipation, as long as the forcing and dissipation do not contain derivatives of the same or higher order. Writing g=gk, these may be put in matrix form A \ I k 1 " o 0 { 1* o o O (A or H^+fiHy=0. The eigenvalues TV # and l e f t eigenvectors, £• , of A are: Multiplying the matrix form of the differential eguations by i J , we get: I; l-i: Since ,4 tj - J: ~}\ S c ' i , there results 3 Thus, in the direction = <9 , the d t characteristic form of the differential equations. Explicitly, 42 the characteristic equations are : ^ dt J The reason for putting the equations i n characteristic form i s that the required boundary conditions at open boundaries become more obvious. The basic requirement i s that one should prescribe as many boundary conditions as there are characteristics pointing into the region under consideration. Thus, one must prescribe salinity on an open boundary i f i t s characteristic i s directed into the region. Since the characteristic speed i s u, the flow velocity, 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. If |uj i s greater than c, and u is an inflow, then both u+c and u-c point into the computational region, and two independent pieces of information about u and c, in addition to s discussed above, must be specified. If |u| i s less than than c, then one of u+c, u-c points into the region, and one points out, so either u or c, or a relation between them, must be specified. If u=c, a very complicated situation arises, in which a boundary in x-t space becomes a characteristic. This i s a situation which becomes very tricky in problems with a time-dependent boundary condition, and requires recourse to further aspects of the physical system. 43 CHAPTER 4 AIDS TO INTDITION ABOUT THE PLUME Before presenting the numerical model in the next chapter, i t i s worthwhile to look at some simple models of plume dynamics. These models do not include a l l terms in the eguations of Chapter 3, so can't be expected to describe the plume adequately, but they are useful.in clarifying various aspects of the plume's behaviour. The f i r s t 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 fri c t i o n and entrainment are important features of the plume near the river mouth. The second model, a time-independent model of surfacing isopycnals, illustrates the roles of depletion and entrainment in causing the isopycnals to rise as one proceeds downstream in the plume. The third model, conditions at a strong frontal discontinuity, discusses the motion of the strongly contrasted colour fronts frequently found in the Strait, and suggests that these fronts induce considerable vertical circulation. The fourth model, a kinematic wave approach to frontal motion, is intended to i l l u s t r a t e , in a very simple manner, the way in which fronts arise in a time- dependent situation, due to the tidally varying river flow. The f i f t h model , mixing and fluxes across an interface, shows how an upper layer model, as developed in this thesis, i s compatible with a diffusive {eddy diffusivity) model of the vertical salinity distribution. The sixth section, analogy with turbulent jets, is an attempt to motivate the use of entrainment and 44 depletion by showing how they arise in a more accessible system, a turbulent plane jet. i i . COHPfiESSIBLE FLOW ANALOGY There i s an exact analogy between the frictionless flow of a compressible f l u i d and the frictionless flow of a liquid with a free surface. The method of solution of the equations derived below was developed for compressible flow (e.g. Shapiro and Edelman, 1947), and later adopted for use in hydraulic engineering (Ippen,1951). Using this method, one i s able to predict the velocity and thickness of a f l u i d discharged from a channel into an unbounded region. The solution for water flowing oyer a solid surface (as developed in Rouse et a l , 1951) i s identical to the solution for lighter fluid flowing over heavier fl u i d (as required in a plume theory), i f q, the acceleration of qravity, i s replaced by g*=g , as shown in Chapter 3., This then i s a buoyancy spreading model, 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, frictionless plume are: u a „' * v IA Y v- g 'h x = o , 4m 2 u i A + V * ̂ ' ^ y = °- 4.3. Fundamental to this method is the requirement that the vorticity of the flow be zero, 3 ' l / * - O . 4. 4. This approximation i s assumed to be valid in a region around the river 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. where c7'=• h The continuity eguation may be put in the form ( is*--C*) ry, da. + C • *• - c * ) d is = O , where ^ = ^ ^ +. / ^ i ^ z . ^ i 1 4.6. U2- - tz du , (pi + ^ U ) <_/ d i/ = ( d-¥ + m l y ) d x Thus, in the direction /dv. - m, du and dv satisfy the ordinary differential eguation 4.6. Since there are two types of m, depending on the sign of the square root, there are two eguations, in addition to the Bernoulli equation, from which to obtain u, v, and c 2. However, a l l this depends on m being real, that i s on u 2+v 2 being greater than c 2. Thus; this model applies to supercritical flow only. There i s a fair amount of evidence (Wright and Coleman, 1971; Garvine, 1977) that the flow at a river mouth is internally c r i t i c a l or supercritical, i.e. that (u 2+v 2)>c 2. Thus, one can use eguation 4.6 to obtain a solution around the river mouth, as long as u2+v2 remains greater than or egual to c 2. Figure 56a illustrates the orientation of the two characteristics, C+ and C- (defined by / j x ̂  or m_ ) , and a streamline, with respect to an x-y coordinate system. 46 Defining u=gcos0, v=gsinO, c=gsin/w, and ~X-+= <9 ± / * , (Fig. 56a) one can eventually manipulate eguation 4.6 into the form 9 ± P(m) =constant 4.7. on ^/dx-Wi: - , and One interesting thing about the above solution i s that i t i s possible to integrate analytically the differential eguations along the characteristics. The problem i s set up as follows. An opening in a solid wall i s assumed, through which water is flowing with v=0, u=c. One then f i l l s up the computational region with the two families of characteristics, and uses tabulated values of to obtain values of u and v at the intersection of characteristics. (Fig. 56b) . Rouse et al, (1951), worked out by hand the solution to this problem, with F = u/c = 1, 2, 4, (Fig 57). The results, for the values of F examined , are a l l quite similar - as one proceeds outward from the river 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 river mouth slows down, rather than speeds up. The inadequacy of the model in this respect points out the importance of retarding forces at the river mouth. The retarding forces could come from three sources: 1) an adverse pressure gradient in the barotropic tide, caused by the large geostrophic slope during the ebb cycle of the tide in the Strait; 2) entrainment of water with zero momentum in the downstream direction of the plume; 3) f r i c t i o n a l interaction between the upper layer and lower layer. To determine how important these terms are, we f i r s t estimate ^' h x from the analytical solution. Fig. 57. For F2=1, 3'h- - x ( ^ ; J ' where L^/<>^t^j~i is measured from the diagram, h „ i s the i n i t i a l depth of flow, and b i ) 2 _ i s the halfwidth of the river mouth. The factor Ch>/bv ) converts from the non-dimensional units of the analytic solution to units appropriate to the Fraser River. Taking the width of the river to be 600 meters, and g* to be 10m/sec*x.01=0.1m/sec2, and h 0 to be 8 meters, we can evaluate c>hf$% 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 = O.l/Z-S. Thus, g*h„~7.5x10-* m/secz. in the region where h changes from 0.3 to 0.1, the pressure gradient , calculated in the same manner, i s 2.0x10-* m/sec. From the geostrophic relation, fv=g £>< , the crosschannel pressure gradient for a current of 1 m/sec (an upper limit) i s about 1x10-* m/sec*. This slope results in the water level being higher on the western side of the Strait of Georgia during an ebb, which constitutes an adverse pressure gradient to the river flow in the vicinity of the river mouth. Thus i t appears that the cross channel barotropic t i d a l slope is the same order of magnitude as the buoyant spreading pressure gradient. 48 One can estimate the relative effect of vertical entrainment as follows. If the contribution of entrainment to the continuity eguation i s written as dt 3 and one considers the vertically integrated momentum equation, then the average momentum equation is Jjt _ 1 & ~ MS u . Dt h In Thus, entrainment acts as linear f r i c t i o n , with f r i c t i o n coefficient w / k . . He wi l l assume w=Eu, and E=2x10_* (Keulegan,1966}. This order of magnitude for E was verified by both Cordes (1977) and de Lange Boom (1976) for the Fraser Biver plume. An estimate of u from the Bernoulli equation i s At h/ho=0.6, u ̂ 1.2 m/sec. This i s also a reasonable value for the measured speed near the river mouth. Hith this value for u, and h=.6x8m, the retarding force due to entrainment, wu/h, i s 0.4x10-* m/sec*. This value is one f u l l order of magnitude less than $'hx near the river mouth and somewhat closer to g*h v at points further downstream. It appears that entrainment has a significant but not dominant effect on the plume near the river 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 pressure gradient driving u is (j/zk) Cs'* • D u e t o entrainment, since g' decreases, and h has a tendency to increase (opposed by i t s L\9 buoyant spreading tendency) i t is d i f f i c u l t to predict this pressure gradient without a more detailed model, such as that discussed in Chapter 5. The third possible retarding force is f r i c t i o n . If one assumes quadratic f r i c t i o n , and equates hg*ĥ . to Ku|u|, then to be important, Ku2/h must be close to 7,5x10-* m/sec2. Assuming Ku2/h=7.5x10-* m/sec2, we get K~2x10~3. This value of K i s similar to the value of drag coefficient used in many calculations. For instance, the drag coefficient for wind over water is about 1.5x10-3, and the drag coeffiecient for bottom fr i c t i o n in a t i d a l channel i s about 2-4x10-3, and the interfacial drag coefficient in a laboratory scale flow i s of order 10 - 3 (Lofguist, 1960). Thus i t appears that f r i c t i o n plays an essential role in the plume dynamics. Is., h lI«IrINDEPENDENT MODEL OF SURFACING ISOPYCNALS As demonstrated in 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 river mouth. The model discussed here is an attempt to explain this phenomenon in terms of entrainment and depletion. This model applies to the region from around station c to station j . Fig. 27a. Here the plume i s thought to be more or less uniform across the Strait, and advected back and forth by the tide, with a small mean velocity to carry river water out of the Strait. The tidal excursion in this area i s about 10 km, so the plume i s advected back and forth by the tide a rather large distance. One could imagine performing an average 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 Strait (and hence across the plume), or else assume lateral uniformity. The plume i s fed water at i t s upstream end, and this water leaks out from the forward or leading end by means of the depletion mechanism discussed in Chapter 3. The equations for this model of the plume, a simplified form of 3.24, 3.25, and 3.26 are: + VUn - \AJp = O - 4.7. . 2 (Us) + \AJ„ 5 - \AJP So = O • 4.8. h h 4.9. , where T(j0-s) - j( 9°'f) i s an equation of state, 0 i s a transport, (vertically integrated velocity), and K' i s a coefficient of linear f r i c t i o n . A schematic drawing of the model is shown in Figure 58a. The use of linear f r i c t i o n i s not unrealistic in that 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 this model very easy, assume ŵ  and Wpare constants, and that w n>Wp, which i s valid near the outer edge of the plume., The continuity equation, 4.7, has the immediate solution UL -- Ua ~ C v u n - M J P ) * Y where D e i s the transport at x=0, and where x=0 is taken at the upstream boundary of the region of applicability of this model (Fig. 58a) . Defining <̂ =ŵ ,-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 salt ahd continuity eguation may be combined to give: 51 UL S * +\AS?(S-S*)^ O. 4. jo. Define s-s0= 2T,<0; y=L-x. Then, the above equation becomes - o with solution =Ay ; the salinity defect i s proportional to the distance from the leading edqe, y, to the power vpA<. Imposing the boundary condition that at x= 0 s=si , we get A= (s^ - s 0)L In terms of the variables y and , the momentum eguation becomes <2y k K With .2_=Ay , we get (3 Trying a solution h=Byr gives A 6 y ^ £ * ̂ ; + Z£ C w„ + /< J y Equating, powers of y: 0 = (2 - ̂ ^L)/j. &nd finding a value of B to make the l e f t hand side equal to zero: - A ( z ̂  4 w?/^J Because i t is a non-dimensional number, and perhaps the most important one for the plume, the internal Proude number is of some interest. It i s given by: 3 = 52 a constant.. To summarize the solution U -- a C L - * ) 4. 1 1. 4. 12. In = 4.13. (S.-SJ l - ^ p U C f/3 - 'h**^ And the velocity, u, i s given by 4.14., Thus, the transport, 0, decreases in the downstream direction, as plume water becomes sal t i e r and is redefined as lower layer water; the average salinity of the plume increases downstream; and the thickness of the layer decreases downstream. It i s reassuring to note that the f l u i d velocity, u, also decreases downstream, going to zero at x=L. The data for July 3, 1975 (Fig. 27a) seem a good choice for comparison with this model, in that the salinity section for that day shows the surfacing of isopycnals very clearly. Referring to Fig. 27a, consider station d as x=0. L i s then 23 km, i f the plume boundary i s defined as the 25 0/oO contour. With 53 s Q = 25 °/oo, the average salinity at station d, slr is 18.6 */oi> , from an integration of the actual salinity profile, fe need an estimate for U0. Assume that on average, half the river discharge leaves through the northern channels and half through the southern channels. Assuming that the river discharge is 8000m3/sec, and that by the time river water reaches station d i t has entrained an egual volume of salt water then the transport to the south i s 8000m3/sec, flowing along the axis of the Strait. At station d, the Strait i s about 16000m wide. Assuming uniform discharge across the Strait, 0o= (8000/16000)=0.5 m2/sec. Thus <=< =0 „ A = . 5/ (23x 105) =2 . 2x10-s m/sec. We can get an estimate of vp/*. by f i t t i n g the salinity change from station d to station j . At station j , the average salinity in the upper layer i s 23"/"° and x=11km;,We obtain, from eguation 4.12, vp /U-1.8. With =2.2x10~s m/sec, we find wp= 4x10-5 m/sec, and w„= 6x10~s m/sec. This fixes h (*•) 3 ha ( LJU)°'z/3using y - o}((J»y)/(S>>-s) =.83, {average of salinity profiles for stations d, j, 1) we calculate K*, the linear f r i c t i o n coefficient, to be 5.3x10~3 m/sec* Eguating K'u and Cu|u| at x=0, we get an eguivalent C=2.6x10—2. This i s about 10 times the usual value of a drag coefficient, 3x10~3. However, K* i s really C U r m 5 , (Groen and Groves, 1966), and the rms value of velocity is probably quite a bit higher than the t i d a l mean used here. The computed and observed distribution of h is shown in Figure 58b. The f i t uses the two h points at the upstream and downstream ends of the plume, so there is only one point left to check the f i t . To show the sensitivity of the f i t , the curve h= ti „ (^-[r)3 i s plotted as a dashed line. The model is substantially correct in that i t predicts the rising of isopycnals (modelled as a thinning of the layer and an increase in average salinity of the layer), but i t would be extremely unlikely that the entrainment and depletion processes can be modelled very well by constant values of wn and w,o. The model would be improved by obtaining data for f i t t i n g that were truly cross-channel and t i d a l l y averaged; and by obtaining, empirically, better formulae for ŵ  and Vp. 2 j l CONDITIONS AT A STRONG I1QNTAL DISCONTINUITY As discussed in Chapter 2, particularly with reference to the data of July 2, 1975, Fig. 31, there i s often a distinct colour front bounding the plume. The purpose of this section i s to c l a r i f y the role these fronts play in determining the motion of the water behind them. One would like to have a model of a front suitable for use with a larger scale model of the entire plume, and for these purposes a detailed model of the circulation at the front i s not reguired, but rather relations between frontal velocity and the fluxes of mass, momentum and salt into the frontal region. r With reference to the dye experiment described in Chapter 2, on July 3,1975, a "tank tread" model of frontal circulation suggests i t s e l f . Thus, i f one considers a military 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 is stationary.„This is similar to the velocity profiles measured in the plume, eg. Fig. HO, where the bottom part of the plume is "attached" to the lower water, and the upper part i s moving at a significantly different speed. Continuing with the tank 55 tread analogy, and changing the co-ordinate system to that of an observer sitting 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 travelling 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 travelling away from him, on the under side of the plume (like the dye which was later found at depth, behind the front). Also, to an observer on the tank, the ground on which the tank is travelling i s coming toward him, and then passing under him; similarly, to an observer at the front, the dark blue water appears to travel toward him, and then flow under the plume. A model to describe this motion can be developed from a control 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 this level, water i s flowing into the control volume, and below this level, water is flowing out. The speed of water in the control volume or frontal co-ordinate system i s u-cT. Conservation of mass requires J (u - cr) J ? + Q --. IC cr- u \ dz ̂ or 56 A c^A - j u c/z - Q = O. 0 4.15a. Q represents a rate of entrainment specifically due to frontal processes and i s entirely different from vp discussed previously. wp- and wn have been ignored because the control volume length i s assumed very small, of order tens of meters, relative to the entire plume. Similarly, conservation of salt reguires: A h' j Ct*-cr)sdz + & 5> * J $(tA-cr)dz = O 5 or A h '& ( sd* - f us dz ~ O s0 = 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 ' ' / u l 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 in Chapter 3, the net pressure force is 1/2gh 2^£ , i f P i s independent of depth in the layer. It i s possible to obtain the pressure force f a i r l y simply for a 57 variablep (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 in the water beneath the plume), we reguire h \ , k or £>k -Cp»-(p) k where p = J_ f J^ The net pressure force, divided by the average density i i 4 p P9 dz ' d ? -  Z 4. 18. For the case p = 7̂ = a constant, this becomes i ^f-c -£1 - dk- H , p 2 L { y» J 2, p as in Chapter 3. If p - ^s 2- /j^ * a linear profile, then the net pressure force i s '//̂  (p0-ps) /p0 , correct to f i r s t order in If we had used the average density in the formula 'l^g^. &fIp , we would have J/^ t p0-pj) [ p* ) significantly different from / / 2 . C fo'fs^)/p> • Now, to apply the eguations to some fie l d data. On July 23, 1975,. we measured a current profile and a salinity profile at about 200 meters from the front, in the plume. (Fig, 39 and Fig. 58 40). The current profile was actually a speed profile, so the zero crossing and direction were only inferred. Referring the velocities to 17m, the deepest measured, the resulting front speed is that relative to tid a l l y moving deep water. Assuming a plume depth of 6m, p e =1.017 , ps = 1.003, and assuming a linear density profile, p - ps ^o.pjj ' t f ee pressure term 4,18, turns out to be . 42m2s-2. The integrals of u and s are: k j. udz = 1.25m2/s 6 k J u2dz = 0.72 mVs 2 k J sdz = 100 m ppt o A f usdz = 12.9 ppt m2/s. 0 With s =23.4, we get, by solving eguations 4.15a, 4.16a, 4.17a, ^ =.41m/s Q= 1.21 m2/s K = 0.63 m3/s 2; recalling that CT i s the front speed, Q is the extra frontal entrainment, and K is the extra form 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^Judz. That i s , a front extends i t s e l f , or propagates, by mixing egual parts of plume water and salt water. Also, one could write K=1/2CD h. 59 where C p i s a drag coefficient, which turns out to be 1.2 in this case. Since values of C 0 for flow over blunt objects such as cylinders are about 1, i t appears that K may be identified with the drag exerted by the salt water as i t flows under the blunt leading edge of the plume. It is informative to re-derive eqns. 4.15a, 4.16a, and 4.17a from the differential form of the equations. Chapter 3 . Thus, in one horizontal dimension, suppose that for x<cr(t), there exists a plume satisfying the eguations: . A ?A * A- f u dz i- \AJ„ - UJP -6?<$(x-G~)=>o. dt 2* J 4.19. A ^ 4.20. / A dt 0 2x 0 ^ •<- uj„U - uop ua * K <S ( x - c r j = o . 4.21. e£(x-C) i s the Dirac delta function, used to make the quantities Q, Qs6, and K non-zero only at the front, x=tf"(t). We shall integrate equations 4.19, 4.20, and 4.21 from x= <r-& to x= ; set quantities at x= o">6 equal to zero, since there is no plume ahead of the front; and finally take ^ ^ o . There results: * 4.15b. A A ir f 5 dz - f u sd? - Q $b ~ o. J J 4.16b. O A 60 - y w * -9*% ' k " °- Notice that the quantities involving ŵ , wA, and Ru|u| disappear, since they are proportional to<£, which approached zero. The justification for this procedure is that as one proceeds from Sand Heads to the front, properties change only gradually, and are appropriately related to each other by partial differential equations.,Then, at the front, the fields of thickness, velocity and salinity change drastically in a very short distance, of the order of a small boat length, and can only be related to each other in terms of weak solutions. (A weak solution i s a set of relations between the changes of various properties across a discontinuity, tfhitham, 1 9 7 4 ) . We can compare (approximately) the speed of the front to the calculated speed, 0.41m/s. Referring to Fig.38a, a l l three drogues were inserted at the front, and the shallow drogue, S, stayed with the front. Assuming the deep drogue travelled with the bottom water (not exactly true), the relative 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, is not very small compared to the tidal period, (nor was the current profile measurement time), so great accuracy can't be expected.. Summary of frontal circulation The following description of a front emerges from the equations derived above. The front i s a mixing region of water which i s pushed outward by the momentum flux and the pressure gradient, and retarded by the mixing of ambient water and the 61 form drag i t 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 level flows into the mixing region at the front. Rater below this level, flowing away from the front, is quite salty, and could only have picked up this salt 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 in which fronts, described in Chapter 2, develop in a time-dependent plume model. 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 is similar to a Froude number, but has the dimensions of an" acceleration to simplify subsequent mathematical expressions. Although the assumption that F is constant may not be a good approximation for the plume, for the present purposes the simplicity of the resulting mathematical analysis more than compensates for the physical inadeguacy of that assumption. The approach used here i s called the kinematic wave method because a l l the dynamic interactions are summarized in a simple rule, equation 4.23, and we look at the motion prescribed by the continuity eguation 4.22 (Whitham, 1974) . 62 Substituting for u in the continuity eguation we find: 4.24. The lines (3/2) u are the characteristics. At x=0, representing the river mouth, assume the velocity versus time graph i s a series of triangles, representing t i d a l modulation of river 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. s 4.25. The implicit solution of 4.23, 4.24. And 4.25 i s h=h0( ? ) , u= u 0 { T ) , on the line given by x= (3/2) \ia { 'T.) (t- -? ) 4.26. where h^('C) and u ,>(-£:) are the values of u and h at x=0, t= f . The solution i s displayed in a characteristic diagram in the x-t plane - f i g . 60a. However, two complications arise. 1). We will assume here that i n i t i a l l y there is no fresh water in the Strait (perhaps because of a windy period before model time t=0), so that water issuing from x=0 in this model forms a front. This front i s a boundary to the region of validity of the plume equations, 4.26, and i t s motion must be found. Referring to Fig. 60b, at a point (s,t) on the front, ̂ ^/dt ~ front speed=u, the local water speed. Since the point (s,t) i s a termination point for a characteristic, s and t are related by 63 the implicit solution 4.26. For ^<3, that i s for water discharged before t=3, the front i s given by 4. 27. dt 4.28. Eguation 4.28, with i n i t i a l condition s=t=0 has solution s=2/9tz. To continue the front beyond t-3t different formulae than 4.27 and 4.28 must be used, reflecting the fact that the functional form u(f) has changed. Eguation 4.28 must then be integrated numerically, as solutions can't be found analytically. 2). The other complication i s the formation of shocks or hydraulic jumps. These occur whenever a faster moving characteristic overtakes a slower moving one. For example, the characteristics for >6 have speeds which are increasing functions of time, whereas those for t<6 decrease. Consider two characteristics, one for H>6, the other for L!<6. (Figure 60c) The curve (s,t), the path of the hydraulic jump, must satisfy both characteristic relations: S, C- tl, +L)CL—C,) 4.29. r = Lt-L - T O 4.30. . We further reguire a jump relation..Integrating the continuity eguation across this jump: 5+ or 64 5+<6 The integral approaches zero as e tends to zero, and we are l e f t with 4.31. , for u2/h = F. 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 in the plume, for instance at station q,r July 2, 1975, Figure 26. Since the water on either side of the jump took different times to arrive at the discontinuity (t- % is less than t-tz) , the amount of s i l t i n suspension, and hence the colour, will be different. These fronts have been termed internal fronts, to distinguish them from the true front, which i s the boundary of validity of the plume equations. Proceeding on in this manner, the diagram in Figure 61 may be drawn. Figure 62 is a plot of u versus x at t=14 - an instantaneous photograph of the distribution of downstream velocity in the plume. Becall that h i s proportional to u 2, so this i s also a plot of plume thickness. Using a rather mixed but convenient set of units, x could be measured in nautical miles, t in hours, u in knots, and h in meters. F would then be in (knot)2/m. In Figure 61, there is a strong colour front, bounding the region of solution. At s = 30.5, the front speeds up, since faster water, originating from'2=9.9 catches up with 65 i t . There are hydraulic jumps originating from ̂ -6, tl -12, etc. Looking at Figure 62, one could imagine the following sequence of observations, proceeding out from the river mouth. At s=1, there i s an internal front, corresponding to faster moving water catching up with slower moving water. The slow water has spent more time since i t l e f t the river mouth, so s i l t could have settled out, and this front might show up as a colour discontinuity. Again, at S=11.8, another internal 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 in reality, because of dissipation and spreading, i t would be very weak. As mentioned at the beginning of the section, the model described here i s based on grossly oversimplified dynamics. However, i t c l a r i f i e s mathematically the formation of fronts in a time dependent system., 5j_ MIXING AND FLUXES ACROSS AN INTERFACE The introduction of an isohaline as a boundary for our plume model presents complications in trying to model the vertical fluxes of salt, water, and momentum across that interface. In many fluid mechanics calculations the turbulent mixing of a scalar guantity i s assumed to be governed by a diffusion term, ̂  ^ .(/< ^ / ^ J where one attempts to choose K in such a way that <̂ >X(. (/< <^V<)*(-)-z -^u'd<), where the primes refer to •turbulence* quantities. In this section we will assume that a diffusion equation governs the distribution of salt, ..Dt' dz' ' 4.32 where 66 D t ' " ^ ' and where here primes denote dimensional quantities and u and v' are components of some appropriate advection velocity. Thus 4.32 describes the downstream evolution of a salinity profile. Equation 4.32 will be solved numerically, (in a co-ordinate system travelling at (u', v) , so D /ofc 'is replaced by t>/dt) ; then choosing 0.8 times the maximum salinity as the bottom of the upper layer we obtain, from the numerical solution, <̂ /6z?and 6>s/)t f o r this upper layer, we then try to model the evolution of the upper layer thickness and salinity using the two fluxes discussed in Chapter 3, entrainment and depletion. Two choices of K are considered in the solution of 4.32; K= a constant and K=az2, where z represents the vertical distance from the free surface, positive downwards. Scaling s with S e, the maximum salinity, and t with T^, a characteristic time, and z' with Z„, the depth where the salinity i s approximately constant, we want to solve it dZ 2* 4.33. where s*=sS0, t'=tT6, z*=zZ0, and K*=KZ2/T0, guantities without primes being dimensionless. The i n i t i a l conditions are: s=0 for z<.2, t=0 s=1 for ,2<z<1, t=0. Thus i n i t i a l l y there i s an upper layer 0.2Ze> thick, composed of fresh water, floating on a layer 0.8Z* thick, of salinity Ŝ , At z=0, the free surface, the condition of no flux K JS/^gzOr was imposed; at z=1, representing the deep, well- 6 7 mixed water, s=1. Equation 4.33 with the above boundary and i n i t i a l conditions was solved numerically. Figure 63a shows the results for K=z 2/(.2) 2, at times 0, 20At, 1 2 0 A t , 2 2 0 A t , and 320At, where At=.4(Az) 2, and Az, the vertical grid size, equals 0.02. Figure 63b shows the results for K=1, at times 0, 204t, 2 2 0 A t , 580&t, and 920 At..To compare these profiles with quantities available in an upper layer model of the plume, I chose 0.8S to be the salinity characterizing the base of 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 total salt content S, $Sdx above the level s=.8Sc, versus time. As i s evident from Figure 64a, the upper layer i n i t i a l l y increases in thickness, then decreases - remember that at t=0 i t s thickness was 0 .2z o « There is considerable difference in the s versus t curves of Fig. 63a and 6 3b, with Fig. 63a, K<* z 2 appearing to be more like the observed profiles of salinity. It i s interesting to speculate why this i s so. First, Kc/z2 i s partially the functional form prescribed by Prantdtl's mixing length hypothesis (Launder and Spalding, 1972), /Since we are not dealing with a velocity profile, we can't however define a characteristic time such as ) to complete the specification of K in terms of mean flow properties as in the complete Prandtl mixing length theory, where K ~ Second, the K of z 2 form implicitly accounts for the effects of vertical stratification - near the surface the stratification i s strongest, and hence mixing is inhibited 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 small near the surface, as i t i s when K i s proportional to z 2. We wish to model the behaviour of s and h by the two eguations, simplifications of egns. 3.24 and 3.25, 4.34. where w p i s the entrainment velocity, w „ i s the depletion velocity, °< is a salinity greater than 0.8So, and @ is a salinity less than or egual to 0.8So. S denotes the total salt content in the layer, J 5 c ^ 2 . .The quantities ^/^fc and were obtained from the K=z 2/(,2) 2 solution and plotted versus s, the average salinity in the upper layer. They were roughly fitted with wp = 1150 w =2600s i f s<.5 w^=2600s-8000 (s-.5) 2 ifs>.5 =0. 9, § =0.8. Wp and wK are converted to dimensional units by multiplying by the ratio Az/T0 , where T„ i s of order 8 hours, and A z is of order 0.5 m. The results of numerical integration of 4.34 and 4.35 using the above vp, ŵ , u , and P, are shown as the dashed curves- of Fig. 64a and 64b. There is qualitative agreement with the results from the diffusion eguation. There i s not much point adjusting wp , wn, and (3 to make the agreement better because in the real world we don't know K(z), and there are important effects due to velocity shear. However this example demonstrates 69 the use of two types of exchange across an interface (u^ and wn) to model a mixing situation, is. ANALOGY WITH TURBULENT JETS Another way to view the phenomenon of entrainment and depletion, with reference to the plume, i s to consider a turbulent plane jet (Abromovitch, 1963). Figure 65 shows a plot of an isoconcentration curve of a passive scalar discharged by the jet, (X i s downstream, Y is cross-stream), and velocity vectors at various points. We see that for X less than about 5, the net flow across the isoconcentration curve is into the jet as defined by the iso-concentration curve; whereas for x greater than 5, the net flow i s out of the jet region. Schematically, the situation i s shown in Figure 66a. The curves AD and A*D' are the usually defined boundaries of the jet, representing the interface between turbulent and non-turbulent flow. If one wanted to consider only a region where there are significantly characteristic properties of the jet, as defined for instance by the concentration of a scalar guantity discharged by the jet, that region would be bounded by ABCB'A*. As Fig. 65 indicates, along AB and A* B* there is net flow into the jet, and along BCB' there i s net flow out. How does this concept apply to the plume? Consider a section through the plume. Fig. 66b. Because we are going to solve the plume eguations in a f i n i t e region, a computational open boundary i s indicated in Figure 66b. If we choose s=25 °l*t> as our plume boundary then a l l of the Fraser River water , plus entrained water, flows out of the plume solely by means of the depletion mechanism, since the flow components (u,v) do not 70 cross the interface in the shallow water eguations developed here. If, however, we choose s=28 °/oo as the plume boundary, part of the inflow i s balanced by depletion, but there i s also horizontal outflow over the depth h ., The computational aspects of these two choices are considerably different. For the case where the iso-haline comes to the surface within the computational region, the actual computational boundary becomes the line along which the isohaline intersects the surface. sThus, one has to numerically move this 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 in particular the problem of specifying u, s, and h on an inflow, and also the problem of generation and reflection of false waves at the boundary. 71 CHAPTER 5 N2HISICAL MODELLING OF THE FRASER RIVER PLUME The numerical modelling described below f a l l s into 2 categories. First a model for a small rectangular region was developed. This model had the benefit of being inexpensive to run, and quick to show up any problems at boundaries, particularly the open outflow boundaries. Once a l l the terms described in Chapter 3 were in the small model, a few runs using a larger model simulating the real Strait of Georgia geometry were made. The aim of this chapter i s to demonstrate that a reasonably flexible model has been developed, which has the potential to become an accurate tool in understanding the circulation in the Strait of Georgia. The equations used in this model are 3.24 - 3.27, with the simplification that a l l properties were assumed homogeneous in the upper layer. Thus: 3h,9U8V = W W 3t 3x 3y 1 N 5. 1. 3S _3 US 3 VS 3t 3x h 3y h 5.2. + 3 3x - fV + + h _3 3x A _2 3x U where 0, V are vertically integrated transports, g'=g Ap/p =24-0. 8S/h a = horizontal eddy viscosity, discussed later in this chapter £ =barotropic t i d a l elevation S=vertically integrated sa l i n i t y u„, v© = t i d a l streams 0"ie-' V^=relative transports=0-u 0h, 7-v0h. K = quadratic fri c t i o n coefficient vf = entrainment velocity wn = depletion velocity s e = sali 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 in divergence form. That i s , they are of the form £_f + \7- F CP ) + Q C ? ) = O-) where a l l spatial derivatives are exact differentials. This form of the differential equations allows one to write the f i n i t e difference equations in such a way that there are no spurious 73 sources or sinks in the derivative terms. The numerical scheme used as a starting point was the semi- i o p l i c i t scheme of Heaps (Flather and Heaps, 1975). Although this scheme works very well for t i d a l calculations, there was some doubt about how well i t would work with the upper layer model, in which the f l u i d velocity i s close to the internal small amplitude gravity wave velocity. However, there i s no indication that high flow velocities (internal Froude numbers close to one) are a problem, as long as one satisfies the s t a b i l i t y requirements of the scheme. A typical element of the computational grid i s shown in Figure 67, and Figure 68 shows the entire computational grid for the rectangular model. Note in Pigure 68 that an extra row of aeshes i s provided around the outer edge of the mesh area, to f a c i l i t a t e calculations near boundaries. Also note in Figure 67 that only one subscript i s used to denote the physical location of a mesh point., The systematics of this indexing scheme i s apparent in Figure 68, and the reason for i t s use i s to increase computational efficiency. Certain spatial averages are defined below, for use in the f i n i t e difference equations which follow. A l l f i e l d s are considered to be at the same time level, and r e c a l l that changing a subscript by n, the number of columns in the grid, changes the row, 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 <vi + Vi +l> = h <vi + Vi-n> h <Zi + Zi+1> [Z = h] h <zi + W = k <Ui + V l + i+n l - = h <Vi + V l + V. + V. i-n i--n+1) = h <Zi + Z i +1 + Z, + Z. i-n i -W h <zi + 2Z1+1 + W h <zi-•a + 2 Z i + W With A i l as the spatial grid size, A t : as the timestep, we obtain the following set of f i n i t e difference eguations. The finite difference representation of the continuity equation is: [Z±(t+At) - Z ±(t)] 1/At = - [ u ^ t ) - u ^ C t ) + v ±_ n(t) - v ±(t)] I/AA + WP., (t) + WN4 (t) 75 The finite difference representation of the "salt equation is: [S±(t+At) - Si(t)] 1/At = . ru±(t) S* (t) - U1_1(t) S*^ (t) z*(t) + v 1 - n(t> s-y (t> _ v ± (t> z\ (t) + WP,(t) S - WN,(t) S i ( t ) 1 . ° z±(t) 5.6 The finite difference representation of the reduced gravity calculation is: yt+At) • 24 - .8 S i ( t + A t ) 1 Zi(t+At) 5.7 The finite difference representation of the x-directed momentum equation is: |u±(t+At) - U±(t)Jl/At = f V± (t) Gi+1(t+At) (Z ± + 1(t+At)) 2 - G±(t+At) (Z±(t+At)): 2A*. - K (U±<t) - uT(t) Z*(t) Z* (t) (0\(t) - II (t)Z*(t)) 2 + <V.(t) - V t p(t)Z^(t)) 2 1_ Afc 1_ AJt (u*(t))2 ( ^ ( t ) ) 2 zi(t) z*(t) &T(t> v*_n(t) û (t) vj(t) i+n' 'i-n 76 - g TSX ±(t) + WPj(t) U 0(t) - WN^(t) U l ( t ) Z*(t) A I + I u i + i ( t ) V f c> z ^ + 1 ( t ) z* (t) u i - •i(t) . U ±(t) . i ( t ) z*(t) U.(t) z*(t) + i+n U ±(t) Z*(t) 5.3 The f i n i t e difference representation of the y-directed momentum equation Is: V± (t+At) - V ±(t) 1/At = fU ±(t+At) 1 2M G ±(t+At) (Z ±(t+At)) 2 - G i + Q(t+At) ( Z 1 + n ( t + A t ) ) 2 - K (V±(t) - v T(t) z^c t ) ) ( Z ^ ( t ) ) 2 if (v.(t) - v T ( t ) z ^ C t ) ) 2 + (u ±(t) - u T(t) z ^ C t ) 2 l AX, ^ f n - l ^ ^ - 1 ( t ) W l ( t ) _ L AJt ( v ^ C t ) ) 2 (Vj(t))2 zj(t) 77 i+n ( t ) V0<t> ~ ™ihi ( t> V i ( t ) " § TSY^t) (t) (t) V.(t) \ + A i-n 5.9. where TSX^ and TSY C are the slopes of the water surface as obtained from a barotropic t i d a l model of the same area.. The time structure of these equations i s quite important. In each computational cyc l e , the thickness Z L a n d s a l t content from the previous timestep. Then, 0 t are calculated, using the derivatives of Z ,• and sc (or g,) from the current timestep, and the previous values of v ̂  i n the C o r i o l i s term. F i n a l l y , V 4 are calculated, using derivatives of the current Z/, and S c, and also the current 0 ; in the C o r i o l i s term. In the entrainment, depletion, and f r i c t i o n functions, and the non-linear and eddy vi s c o s i t y terms, the values of uy, V,, and s- from the previous timestep are always used. Hhen using f i n i t e difference methods to solve non-linear p a r t i a l d i f f e r e n t i a l equations, one i s always concerned with the accuracy of the solution and under what conditions the scheme chosen i s stable. Accuracy i s best assessed by comparing the S t are calculated using values of the derivatives of U / and v, 78 numerical solution to an analytic solution, or possibly to real observed data. Stability analysis, discussed in the appendix, i s usually done for the linearized equations, in the hope that the requirements in the non-linear case are 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; It 9y q'h dk +. r V * O- at 17 The river flow was specified by U 0tanh(t/T 0), where T0 was 200 time steps. -Mo was 50,000 cm2/sec, Ax was 10s cm, so that U Ax, the river discharge, was 50Q0m3/sec, approximately one half the freshet value, q' was 10 cm/sec2, and the timestep was 480 sec. Hith friction given by r=.005/h sec-r* the thickness over the entire area varied only between 503 and 515 cm, and velocities varied from 30cm/sec at the river mouth to 3-5 cm/sec at the outflow boundaries., These are not unreasonable values of velocity and thickness for a time-averaged plume - Chapter 2. Figure 69 shows a plot of the flux out of the open ends versus time, il l u s t r a t i n g the i n i t i a l l y rapid, and then very slow, approach to equilibrium. At the open ends, the boundary condition used was ^ ^Idi^-O, where n i s the direction normal to the outflow boundary (y-direction in this case). This boundary 79 condition was chosen as being the simplest one which stated that there was l i t t l e spatial change of important flow properties near the open boundary, and s t i l l allowed there to to be time- varying conditions at these open boundaries. The river mouth boundary condition was the specification of a transport. We are thus assuming subcritical flow, since only one boundary condition i s specified. The internal Froude number i s j302/(10x500) =0.45, so the flow, as determined by the frict i o n in the system, i s indeed subcritical. The next step was to add the convective acceleration terms to the equations of motion. Since the real plume i s near added, the grid adjacent to the river mouth became very unstable, with the depth rapidly decreasing and the velocity rapidly increasing as the river flow was turned on. When sguare law f r i c t i o n and entrainment were added, this problem was eliminated. The entrainment was written as wp=0.00640"2/(g'h3). This i s a deviation from the intent that Wp should be written as Eu, where E is a constant times the Froude number sguared, U 2/(g ,h 3), which would make wp proportional to U 3 (Long, 1975a). Interestingly, the real geometry model, discussed later, indicated that 2 was too large a power of 0 in the entrainment formula. Values of the fri c t i o n coefficient between 0.001 and 0.007 were used in the course of the modelling discussed below. With the addition of these two momentum dissipators, (entrainment and sguare law fri c t i o n ) , the region around the river mouth was stable. However, the outflow boundary was now c r i t i c a l near the river mouth, the gradient of u 2 is as important as the gradient of 1/2g'h2. When these terms were 80 u n s t a b l e . T h i s problem was cured by s p e c i f y i n g the Froude number at the outflow to be a constant. F i g u r e s 70, 71, and 72 show the flow f i e l d produced by the model at t h i s stage. Ax was .33 km. At was 120 s e c , g* was 10 cm/sec. The r i v e r d i s c h a r g e was given by 0=0.5x0 o(1-cos(2 t/T„), a t t a i n i n g i t s maximum val u e of 2000 i 3 / s e c at 6 hours, and remaining at that value t h e r e a f t e r . T h i s r i v e r i n f l o w was d i v i d e d among three g r i d meshes, f o r a r i v e r mouth width of 1 km. The outflow Froude number was 0.333, and the f r i c t i o n constant was 0.001. When the model i n c o r p o r a t i n g the n o n - l i n e a r terms was run f o r an extended peri o d of time, i t developed what was presumably the well-known n o n - l i n e a r i n s t a b i l i t y - the p r o d u c t i o n of short waves by the n o n - l i n e a r terms, and t h e i r r e t e n t i o n w i t h i n the system because o f t h e i r slow phase speed. Note the no o d l i n g of v e l o c i t y v e c t o r s i n F i g . 72, p a r t i c u l a r l y along the s o l i d boundaries. As d i s c u s s e d i n t h e appendix,for the system used o s c i l l a t i o n s have on the i n f l u x to the system ( r i v e r flow p l u s t o t a l entrainment), and the t o t a l e f f l u x out of the open boundaries, we would l i k e the two curves to approach each other, and then remain f l a t . To ac h i e v e a steady s t a t e , h o r i z o n t a l eddy v i s c o s i t y was i n t r o d u c e d . At s o l i d boundaries the eddy v i s c o s i t y was taken t o be zero; elsewhere 10 3 cm 2/sec, so t h a t there was no net change i n the momentum of the system due t o s i d e w a l l f r i c t i o n . F i g u r e 74 and 75 show the r e s u l t i n g model v e l o c i t y f i e l d s at two times, with other c o n d i t i o n s i d e n t i c a l t o F i g u r e s 70 and 71. F i g u r e 73b shows how the i n f l u x 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 l e v e l . The q u e s t i o n a r i s e s , what i s the c o r r e c t value t o s p e c i f y as the outflow Froude number? I t t u r n s out that i t doesn't matter very much. F i g u r e 7 6 shows t h e r e s u l t of a l l o w i n g an i n i t i a l hump of water (the d i s t r i b u t i o n of e l e v a t i o n i s h=50exp ( - y 2 / 6 2 ) , with y measured i n u n i t s o f hjl ) , shown as small dashes, t o propagate outward ( e l e v a t i o n s were uniform i n the x- d i r e c t i o n , the g r e a t e s t t h i c k n e s s being o f f the r i v e r mouth). The l a r g e dashes show the hump j u s t as i t i s passing through the open boundary, 12 meshes from y=0, f o r F 2=0.33., For F 2 = 1 , the curve i s i d e n t i c a l , except t h a t the depth i n the l a s t mesh, i n d i c a t e d by a s o l i d dot, i s 11 cm l e s s than f o r F 2=.33. S i m i l a r l y , the v e l o c i t y c u r ves ( s o l i d l i n e ) were i d e n t i c a l , except f o r the l a s t mesh, where the v e l o c i t y was 2 cm/sec f a s t e r f o r the F 2 = 1 case. I t appears t h a t a flow l i k e the plume t r a v e l s l i k e a kinematic wave - the dynamics are mainly c o n t r o l l e d by f r i c t i o n , and we need p r e s c r i b e a boundary c o n d i t i o n a t the upstream end only. T h i s of course a p p l i e s o n l y i f the flow i s reasonably s t r o n g l y outward at the open boundaries. In h i s book, Whitham ( 1 9 7 4 ) d e s c r i b e s how the e f f e c t i v e order of a p a r t i a l d i f f e r e n t i a l eguation decreases, under the e f f e c t o f f r i c t i o n , f o r a l i n e a r case, and the 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 case. We are concerned with d i s p e r s i o n i n a numerical scheme. In the l i n e a r case, we want, f o r example, a l l d i s t u r b a n c e s t o propagate at j-g~̂ 1b71 I t i s d i f f i c u l t to check how w e l l a scheme i s working i n the n o n - l i n e a r case, because o f the lack of a n a l y t i c s o l u t i o n s . However, we know that i f there i s a di s t u r b a n c e 82 propagating i n t o a r e g i o n , with the f i e l d s c o n t i n o u s , but t h e i r s p a t i a l d e r i v a t i v e s d i s c o n t i n o u s , then t h a t d i s t u r b a n c e t r a v e l s at the l o c a l c h a r a c t e r i s t i c speed, b e f o r e i t breaks (Whitham, 1974). Thus, a " g e n t l e " d i s t u r b a n c e propagates at u+ Jg * h \ We can see t h i s i n Figure 77, which i s a p l o t , i n the d i r e c t i o n normal to the r i v e r opening, of u and h from F i g u r e 74. The d i s t u r b a n c e i s a bulge of water propagating out from the r i v e r mouth. I t has t r a v e l l e d about 10Ax, and (u+f^h ) 100 A t , evaluated a t the f r o n t , i s 9.95Ax. The agreement i n the c a l c u l a t i o n o f the p o s i t i o n of the f r o n t i s of course a crude v e r i f i c a t i o n , but i t i s encouraging t o note the f r o n t of the d i s t u r b a n c e moving a t a speed g r e a t e r t h a t fg*~h . U i s about 25% of iTg"̂ ? , so the f a s t e r propagation speed due to the n o n - l i n e a r e f f e c t should be apparent, which i t i s . We have i m p l i c i t l y assumed t h a t u and h a t the f r o n t were con s t a n t at a l l times. Figure 77 i s drawn f o r timestep 100. At timestep 50, u was 5 cm/sec (the same as at timestep 100), and h was 42 cm (versus 44 cm at timestep 100), so there i s l i t t l e temporal v a r i a t i o n of p r o p e r t i e s at the f r o n t . , Another g u e s t i o n , r e l a t e d t o d i s p e r s i o n , concerns the e f f e c t of g r i d s i z e on the s o l u t i o n . In F i g u r e s 78, 79, 80, and 81 we compare the r e s u l t s of two d i f f e r e n t s o l u t i o n s to the same problem. Part A of each f i g u r e shows the d i s t r i b u t i o n of h w, u, and v from the model d i s c u s s e d so f a r (Ax=0.33 km. A t =120 sec, Q K M (a ) ( =2000m 3/sac) , and part B shows the same f i e l d s at t h e same time f o r a l a r g e r - s c a l e model (6x=1km, At=240 sec, Q ^ « = 2000 ra3/sec). Each number i n p a r t A was o b t a i n e d by averaging over 9 meshes, corresponding t o 1 mesh f o r the p a r t B f i e l d s . In 83 g e n e r a l , the agreement i s b e t t e r than 10%, except i n the outer meshes of the plume. Here, one i s g e t t i n g c l o s e t o the boundary i n the 0.33 km mesh model, and the 1 km mesh model i s having t r o u b l e r e s o l v i n g the s i t u a t i o n a t the f r o n t . IIDAL EFFECTS IN THE BOX MODEL The next items t o be i n c l u d e d were the t i d a l e f f e c t s . F i r s t the e f f e c t s o f t i d a l e l e v a t i o n s , then t i d a l c u r r e n t s , and f i n a l l y v a r y i n g r i v e r flow were i n t r o d u c e d . The phases and amplitudes o f the t i d e approximated those i n the S t r a i t o f Georgia, f o r an M2 c o n s t i t u e n t only. The p e r i o d was taken to be 12 hours, f o r computational ease. The t i d a l parameters chosen were: cos C cot - z¥°) ; v ^ ZD co s ( L>t - 132 6) ; JJL ±0_ (Los C LO£- 2&i°) . Dy 2 f A x Note that low water corresponds to maximum ^/dy , maximum r i v e r flow occurs about 0.7 hours (2H degrees of phase) before low water (as determined from t h e s u r f a c e c u r r e n t meter. Chapter 2), and maximun streams l e a d the e l e v a t i o n by 72 degrees, a t y p i c a l value f o r the S t r a i t . Cross channel s l o p e s and streams were assumed to be zero, f o r s i m p l i c i t y o n l y . When these t i d a l e f f e c t s were added, i t was found t h a t the flows near boundaries were anomalous. F i g u r e 82 shows the flow 84 f i e l d from a model with the constant Froude number boundary c o n d i t i o n and C o r i o l i s f o r c e . The presence of the C o r i o l i s f o r c e causes the flow t o have a l a r g e c r o s s - c h a n n e l component which would not be present i f a more r e a l i s t i c t i d e with c r o s s - c h a n n e l s l o p e s were p r e s c r i b e d . The flow near the i n f l o w boundary bears l i t t l e r e l a t i o n to the flow i n the i n t e r i o r flow f i e l d . The o b j e c t i n s e l e c t i n g a boundary c o n d i t i o n was t h a t the flow at the open boundaries should look l i k e a smooth e x t r a p o l a t i o n of the flow i n the i n t e r i o r . I t was found t h a t s p e c i f y i n g e) at the open boundary was a s a t i s f a c t o r y boundary c o n d i t i o n i n t h i s r e s p e c t - the flow reversed d i r e c t i o n s (due to t i d a l f o r c i n g ) at about the same time everywhere i n the model, and d i d n ' t p i l e up at t h e boundaries. F i g u r e 83 shows the flow f i e l d with t h i s boundary c o n d i t i o n , with t i d a l e l e v a t i o n s , 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 flow. F i g u r e 84 shows the same flow f i e l d , but f o r much lower f r i c t i o n (K=0.005 i n the f i r s t c ase, 0.001 i n t h i s case.) During the course of subseguent experiments, i t was found t h a t a b e t t e r outflow boundary c o n d i t i o n ( i n terms o f the u n i f o r m i t y of the flow f i e l d ) was to c a l c u l a t e the v e l o c i t y based on o> ̂  fdnl~0 t and then average t h i s with the v e l o c i t y one mesh i n from the boundary. SALINITI CALCULATION At t h i s p o i n t , the c a l c u l a t i o n of t h e s a l i n i t y d i s t r i b u t i o n was s t a r t e d . I n i t i a l l y , the l e a p f r o g scheme was used, but was dropped, because o f the computational mode, d i s c u s s e d i n the Appendix. I t was intended t o use the fiichtmeyer scheme, but due t o a programming e r r o r , the second order c o r r e c t i o n was omitted, so i n e f f e c t only a f i r s t order scheme was used f o r the s a l t 85 advection equation. The i n c l u s i o n of the d e n s i t y e f f e c t of the s a l t i n t r o d u c e d no problem, as long as the v a r i a b l e g' was used i n the c a l c u l a t i o n of the Froude number f o r the outflow boundary c o n d i t i o n . The s a l i n i t y boundary c o n d i t i o n was t h a t on i n f l o w the s a l i n i t y took on the value i t had on the l a s t o u t f l o w . Figures 85 to 91 show a sequence of v e c t o r diaqrams f o r a run with t i d a l e l e v a t i o n s , constant r i v e r flow, d e n s i t y based on s a l i n i t y , and no C o r i o l i s f o r c e . The f r i c t i o n c o n s t a n t was 0. 005, A. t was 120 sec. Ax was 10 s cm, (1 km), the r i v e r flow was 2000m 3/sec, and the entrainment f l u x v a r i e d between 2000 and 7000 m 3/sec, depending on the stage of the t i d e . F i g u r e s 92 to 95 show how the v e l o c i t y f i e l d advected the s a l i n i t y and t h i c k n e s s f i e l d s back and f o r t h due t o the t i d e . There should be p e r f e c t l e f t - r i g h t symmetry between f i g u r e s 92 and 93, and between 94 and 95, i n the absence of a C o r i o l i s f o r c e , and with a 12 hour t i d e . I t was found l a t e r t h a t s p e c i f y i n g a somewhat high {20 ppt) s a l i n i t y on i n f l o w removed some of the asymmetry present, but there was s t i l l some l e f t (presumably a s t a r t - u p t r a n s i e n t that p e r s i s t s , which »e don't as yet know how t o damp out e f f e c t i v e l y ) . ROLE OF DEPLETION I t was at t h i s stage that the reguirement of the d e p l e t i o n mechanism was becoming apparent. The f i r s t c l u e was t h a t with wp p r o p o r t i o n a l to 1/g', there was too much entrainment i n the f a r f i e l d , where g' was s m a l l . The plume i n c r e a s e d i n both t h i c k n e s s and s a l i n i t y as one proceeded out from the r i v e r mouth - F i g u r e s 92 to 95. The r a t e of i n c r e a s e of volume was decreased somewhat by r e p l a c i n g 1/g* with 1/24 i n the entrainment f u n c t i o n . There 86 a r e two sources of water to the plume, r i v e r flow and entrainment, and there was no l o s s i n the model, so the plume c o n t i n u a l l y got t h i c k e r , i n the absence of d e p l e t i o n . In an e f f o r t t o improve t h i s s i t u a t i o n , I s p e c i f i e d t h a t the outflow be a s l i g h t b i t l a r g e r (outflow Froude number i n c r e a s e d by 0.03) than t h a t c a l c u l a t e d by 3 F The r e s u l t s a r e i n t e r e s t i n g . F i g u r e 96 shows drogue t r a c k s r e l e a s e d i n t o a flow i n which ^ ^ r t 1 ^ »as the boundary c o n d i t i o n ; the flow f i e l d was t h a t of the s e r i e s i n F i g s i 85 to 91. F i g u r e 97 shows the r e s u l t i n g drogue t r a c k s when the open boundary Froude number was c a l c u l a t e d to make ^ f~*/}n2'-0t and then had 0.03 added to i t during outflow c o n d i t i o n s . T h i s was done to c r e a t e an e x t r a flow on the outflow stage, t o balance the entrainment. F i g u r e 98 shows a t y p i c a l v e l o c i t y f i e l d . One cannot help but f e e l t h a t the way i n which the drogues diverge i n F i g u r e 97 i s very u n l i k e t h e i r behaviour i n the plume as shown i n F i g u r e 99 from Cordes (1977). Part of the reason f o r the divergence i s that the excess outflow d r a i n e d the system, so that the average depth was 250 cm f o r F i g 97, and about 450 cm f o r F i g . 96. T h i s would tend to favour any buoyant spreading tendency i n F i g 97. I t seems that d e p l e t i o n might do a b e t t e r job, i n t h a t i t removes mass and momentum at the same r a t e , l e a v i n g the v e l o c i t y f i e l d unchanged. I f then 87 As mentioned with r e s p e c t t o the s a l i n i t y d i s t r i b u t i o n , i t was d i f f i c u l t t o get r i d o f t h e l e f t - r i g h t asymmetry i n the d e n s i t y , e l e v a t i o n and v e l o c i t y f i e l d s . One expects, i n the absence of the C o r i o l i s f o r c e , t h at t h e r e should be o n l y a 6 hour p e r i o d i c i t y i n the t o t a l d i s c h a r g e out of the model with constant r i v e r flow. F i g u r e 100 i s a p l o t o f the d i s c h a r g e out of a model with constant r i v e r flow and no C o r i o l i s f o r c e , but with t i d a l streams and e l e v a t i o n s . The amplitude of the 12 hour p e r i o d i c i t y i s dec r e a s i n g but not as f a s t as one would l i k e . . As a f i n a l t e s t of the sguare box model without d e p l e t i o n , drogues were r e l e a s e d i n t o t he flow a t four d i f f e r e n t stages o f the t i d e - Fi g u r e s 101 to 104. The model had v a r i a b l e r i v e r flow, C o r i o l i s f o r c e , and t i d a l s l o p e s and streams. The drogue paths are not very s i m i l a r t o those of f i g u r e 99. There are two reasons f o r t h i s . F i r s t , the average r i v e r d i s c h a r g e , 1000 m3./sec, i s much lower than the f r e s h e t c o n d i t i o n s the drogue t r a c k s apply t o . Second, because t h e r e was no d e p l e t i o n , the upper l a y e r had grown very t h i c k over the time before the drogues were i n s t a l l e d (about 10 meters), so the r i v e r momentum was r a t h e r i n s i g n i f i c a n t compared t o the t i d a l momentum. I t i s ne v e r t h e l e s s i n t e r e s t i n g t o note t h a t drogues r e l e a s e d at d i f f e r e n t stages of the t i d e occupy d i f f e r e n t r e g i o n s of the model S t r a i t . 88 TESTING DEPLETION In an e f f o r t t o understand the e f f e c t s o f adding d e p l e t i o n , un, a few experiments were c a r r i e d out. The entrainment v e l o c i t y , Wp , was kept the same, and the d e p l e t i o n v e l o c i t y was s p e c i f i e d as w 0 =.0001S/200.Thus, d e p l e t i o n was p r o p o r t i o n a l t o the product of s a l i n i t y and t h i c k n e s s . As p r e d i c t e d , i t served to t h i n the plume i n the outer r e g i o n s . , A run was done with t i d a l e l e v a t i o n s , constant r i v e r flow, and no C o r i o l i s f o r c e . C o n d i t i o n s are the same as f o r f i g u r e s 85 t o 91, and s i n c e the v e l o c i t y f i e l d s a r e almost i d e n t i c a l , o n l y one i s shown. The v e l o c i t y f i e l d at 62 hours, corresponding t o the same t i d a l phase as F i g u r e 86, i s shown i n F i g u r e 105, and F i g u r e 106 shows the drogue t r a c k s c o r r e s p o n d i n g t o Figure 96. F i g u r e 107 shows the e l e v a t i o n f i e l d c o r r e s p o n d i n g t o Figure 94, and F i g u r e 108 corresponds t o 95. The d e p l e t i o n v e l o c i t y was r a t h e r l a r g e , but the o b j e c t of these runs was t o o b t a i n a very s i g n i f i c a n t d e p l e t i o n e f f e c t f o r demonstration purposes. Since F i g u r e s 107 and 108,show a t h i n n i n g plume i t i s apparent that d e p l e t i o n i s having i t s p r e d i c t e d e f f e c t . The drogue t r a c k s spread out more than we would perhaps l i k e , but t h i s i s due to the e x c e s s i v e t h i n n i n g produced by d e p l e t i o n , g i v i n g buoyant spreading g r e a t e r r e l a t i v e importance. The f a c t t h a t F i g u r e s 107 and 108 are mir r o r images of each other whereas 94 and 95 are not i n d i c a t e s t h a t d e p l e t i o n i s perhaps an e f f e c t i v e way to damp out s t a r t u p t r a n s i e n t s which a r i s e when one tu r n s on t i d a l f o r c i n g . 89 I I G E O M E T R Y MODEL Since a l l the terms e s s e n t i a l to the plume dynamics had been i n v e s t i g a t e d with the sguare box model, i t 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 the behaviour of the model i n a r e a l geometry s i t u a t i o n . U n f o r t u n a t e l y , the p h y s i c a l t r a n s f e r of t i d a l f o r c i n g from the b a r o t r o p i c model had not been worked out i n time, so an ad hoc approximation to the M2 t i d e was used, (the t i d a l f o r c i n g was s i m i l a r t o t h a t i n the sguare box model, with the t i d a l s l o p e s and streams being i n c r e a s e d i n the region of the r i v e r mouth, to model the e f f e c t of the narrowing o f the S t r a i t i n t h a t area. In r e a l i t y t h e r e i s a l a r g e r c r o s s - c h a n n e l s l o p e than down-channel, but the M2 t i d e used had only a downchannel slope.) The g r i d s i z e used was 2 km which was the same as the t i d a l model of Crean (1977). Only two passes i n the south were kept open- Haro S t r a i t and Boundary Pass, and the shallow banks near Sand Heads were r e p l a c e d with s o l i d w a l l s . The mean r i v e r flow was 4000 m 3/sec. F i g u r e 109 shows a p l o t of v e l o c i t y v e c t o r s at the time of maximum r i v e r flow. F i g u r e s 110 and 111 show the growth and advection of a bulge of water during 6 hours of f l o o d t i d e . F i g u r e 112 shows the drogue t r a c k s , f o r drogues r e l e a s e d at maximum r i v e r flow. I t was found when working with the l a r g e r r i v e r v e l o c i t i e s of t h i s model that the square-law entrainment was too l a r g e near the r i v e r mouth - s t a r t i n g with 0 °/oo s a l i n i t y everywhere, the model was e n t r a i n i n g more s a l t water i n the mesh adjacent to the r i v e r than t h e r e was f r e s h water flowing i n . I t was decided to switch to an entrainment v e l o c i t y which was l i n e a r l y p r o p o r t i o n a l to the flow v e l o c i t y , based on the f o l l o w i n g 90 argument. During the summer, the plume i s f a s t e s t and f r e s h e s t . Thus, s u ̂ , where i s probably s m a l l e r than 1. The s a l i n i t y i s approximately given by s * ̂ °?/u.', the r a t i o of entrainment v e l o c i t y t o flow r a t e . Combining these we get I - ii not w<*u2, and c l o s e r to w«u*. The entrainment v e l o c i t y was given by wP=0.0001 ̂ u 2*v 2' , where u,~ and v^ are the v e l o c i t i e s r e l a t i v e t o the t i d a l streams. The d e p l e t i o n v e l o c i t y was given by wrt =0 i f g»> 12; ŵ  =0.00025 (24-g») i f g* < 12. Thus, d e p l e t i o n i s assumed to a c t on l y when the s a l i n i t y i s g r e a t e r than 15 °/<>°, (g* = i2 at s=15 °/oo) , and i n c r e a s e s as the s a l i n i t y i n c r e a s e s . The eddy v i s c o s i t y was i n c r e a s e d to 10* cm 2/sec. The r i v e r mouth boundary c o n d i t i o n was to s p e c i f y two components of t r a n s p o r t , 0 and V, i n the r a t i o such that U/V-tan I , where I i s the i n c l i n a t i o n of the Sand Heads j e t t y t o the g r i d system. For the momentum eguations one mesh downstream ( i n both x and y d i r e c t i o n s ) , a c o r r e c t i o n , c o r r e s p o n d i n g t o r e p l a c i n g ^6^(0 2/h) with 1/b ^/2)y{D2b/h) was made, where b i s the pro j e c t e d width of the r i v e r i n each d i r e c t i o n (a very ambiguous q u a n t i t y ) . I t was f u r t h e r necessary to s e t the cross-stream t r a n s p o r t of downstream r i v e r momentum { J/zy (ov/h) ) egual t o zero f o r the two v e l o c i t y l o c a t i o n s adjacent t o , but not downstream o f , the r i v e r mouth, ( e s s e n t i a l l y because the g r i d c ould not r e s o l v e the very sharp cross-stream g r a d i e n t of downstream v e l o c i t y , t r a n s p o r t i n g t o o much momentum to the two adjacent meshes, and d r a i n i n g them). 91 The f i n a l t e s t of the model was to s p e c i f y a t i d e made up of the two most important c o n s t i t u e n t s i n the S t r a i t of Georgia, M2 and K1, and to use a mean r i v e r flow of 8000 m 3/sec, t o match the c o n d i t i o n s of Cordes' experiment (1977). F i g u r e 11.3 shows the t i d a l e l e v a t i o n s ( p r o p o r t i o n a l to the n e g a t i v e of the downchannel s l o p e s ) , t i d a l streams ( p r o p o r t i o n a l t o the c r o s s c h a n n e l s l o p e s by the g e o s t r o p h i c r e l a t i o n , s i n c e the cross-stream v e l o c i t y i s very s m a l l ) , and r i v e r flow used i n t h i s run. The e l e v a t i o n s and v e l o c i t i e s were normalized so t h a t the M2 c o n s t i t u e n t had amplitude u n i t y ; the r i v e r d i s c h a r g e was normalized so t h a t the mean mass discharge r a t e was u n i t y . The s l o p e s and v e l o c i t i e s used i n t h i s experiment were obtained by m u l t i p l y i n g the normalized value by an approximate magnitude, obtained by d i v i d i n g the model i n t o 7 areas and determining the magnitudes of s l o p e s and streams by v i s u a l i n s p e c t i o n of the output from Crean's (1977) model. Fi g u r e 114 shows the d i s t r i b u t i o n of v e l o c i t i e s (cm/sec) and s l o p e s (cm/2km) to be m u l t i p l i e d by the phase f a c t o r of F i g u r e 11.3 to o b t a i n the t i d a l f o r c i n g used. The v e l o c i t y f i e l d s and drogue t r a c k s are shown i n f i g u r e s 115 to 122. The drogue t r a c k s were s t i l l not q u i t e l i k e those of F i g u r e 99. The t r a c k s of F i g u r e 122 were repeated using a c o r r e c t i o n f o r the f a c t t h a t the drogues were t r a v e l l i n g i n a v e r t i c a l l y sheared flow. As shown by Buckley (1977), a drogue t r a v e l l i n g i n a l i n e a r l y sheared flow t r a v e l s at the average v e l o c i t y over i t s depth. I t can be shown that i f the drogue i s not as deep as the upper l a y e r , and one assumes a l i n e a r shear, with the v e l o c i t y a t the bottom of the upper l a y e r b e i n g the t i d a l v e l o c i t y , the speed of t h e drogue i s given by 92 H i t h t h i s c o r r e c t i o n , the drogue t r a c k s o f f i g u r e 123 were obtained. F i g u r e 124 i s a r e p l o t t i n g of F i g u r e 123, t o c o i n c i d e with Figure 99. There i s c o n s i d e r a b l e agreement between Figures 99 and 124. F i g u r e s 125 to 129 show the growth and break-up of a bulge of water d i s c h a r g e d by the r i v e r . Although the model has not been v e r i f i e d i n a l l r e s p e c t s , i t appears to be capable o f adjustment to f i t Nature. The top p r i o r i t y i s t o work out the l o g i s t i c s of using a c t u a l t i d a l f o r c i n g from the model of Crean (1977). The ad hoc t i d a l p a r a m e t e r i z a t i o n used to generate F i g u r e s 115-129 was unfortunate i n t h a t the t i d a l streams p r e s c r i b e d tended to p u l l upper l a y e r water away from c o a s t l i n e s p e r p e n d i c u l a r t o the main a x i s of the S t r a i t . Consequently, s e v e r a l areas had to be removed from the model, r e s u l t i n g i n the blank area north of Haro S t r a i t i n , f o r example, F i g u r e 117-119. , 93 CHAPTER 6 CONCLUDING DISCUSSJON I t i s d i f f i c u l t t o draw p r e c i s e c o n c l u s i o n s from the work d e s c r i b e d i n t h i s t h e s i s ; s i n c e the dynamics of the upper l a y e r are very complicated, the work d e s c r i b e d here can only be c o n s i d e r e d as a f i r s t stage. With the above g u a l i f i c a t i o n , I would l i k e now t o summarize what t h i s t h e s i s accomplished, and then d i s c u s s what t y p e s of work i t l e a d s to. C o n s i d e r a b l e i n s i g h t i n t o the plume was obtained by a c g u i r i n g the f a i r l y s y n o p t i c STD s e c t i o n s o f Chapter 2. The v e l o c i t y p r o f i l e s o b t a i n e d , although few i n number, were c r u c i a l i n d e c i d i n g that the plume c o u l d be modelled s u c c e s s f u l l y as a separate upper l a y e r . The simple model of a d i s c o n t i n o u s f r o n t . Chapter 4, s e c t i o n 2 , p o i n t e d out the p o s s i b i l i t y of a great d e a l of mixing at a f r o n t , and the l a r g e form drag at the f r o n t . Many models of plumes and j e t s have used the concept of entrainment a c r o s s a permeable i n t e r f a c e , but the concept of d e p l e t i o n i s , I t h i n k , r e l a t i v e l y r e c e n t (winter, Pearson, and Jamart, 1977; Stronach, Crean, and Leblond, 1977). F i n a l l y , the development of a numerical model was a major e f f o r t of t h i s work. The immediate aim i n developing the model was to have a system of e q u a t i o n s , and the c o r r e s p o n d i n g computer code, which i n c l u d e d a l l the terms which were thought of s i g n i f i c a n c e (with the e x c e p t i o n of winds); and to l e a r n how t o d e a l with open outflow boundaries i n a n o n - l i n e a r flow. These goals appear to have been s u c c e s f u l l y accomplished. Indeed, the drogue t r a c k s produced by the model compare g u i t e f a v o r a b l y t o those of Cordes (1977), when a reasonable approximation to the b a r o t r o p i c t i d e 94 i s used, and without any adjustment o f the parameters of the model other than those r e q u i r e d to o b t a i n s t a b i l i t y of the s o l u t i o n . Now, to d i s c u s s the most immediate improvements to be c o n s i d e r e d . F i r s t , one can always use more f i e l d measurements i n r e f i n i n g a model. There are two r e g i o n s where the dynamics are p a r t i c u l a r l y p u z z l i n g . One i s i n the v i c i n i t y of the outflow boundaries, where we r e a l l y know very l i t t l e of the temporal nature of the upper l a y e r f l u x e s of mass, momentum, and s a l t . The other area i s the complicated geometry around the r i v e r mouth1 (Figure 4) . We would l i k e t o i n c l u d e the other minor openings (Canoe Pass, North firm, Middle Arm), and a l s o i n c l u d e , i n some way, t h e e f f e c t s o f the flow of shallow, b r a c k i s h water over the banks, p a r t i c u l a r l y Roberts Bank. Even p u t t i n g a s i d e these g e o m e t r i c a l c o m p l i c a t i o n s , i n the model developed here, we d i d not use a c r i t i c a l or s u p e r c r i t i c a l boundary c o n d i t i o n at the r i v e r mouth. There i s probably an e n t r a i n i n g h y d r a u l i c jump at the r i v e r mouth, f o l l o w e d by s u b c r i t i c a l flow downstream of the jump, so t h a t s p e c i f i c a t i o n of o n l y one flow parameter at the r i v e r mouth, (the d i s c h a r g e ) , i s perhaps adeguate. However, s i n c e there i s such i n t e n s e mixing due t o t h i s presumed jump at the r i v e r mouth, one would l i k e to have a b e t t e r idea of what i s going on there. S i n c e we observe ( F i g . 25b) c o n s i d e r a b l e change over a short d i s t a n c e near the r i v e r mouth, and are using a r a t h e r coarse g r i d , a d e s c r i p t i o n of the r i v e r mouth dynamics, s u i t a b l e f o r a coarse g r i d model, needs t o be developed. As f a r as the numerical model i s concerned, the s p e c i f i c a t i o n o f f r i c t i o n , entrainment, and d e p l e t i o n i s always 95 open t o question. To a c e r t a i n extent, we don't even know the f u n c t i o n a l forms to g i v e t o the s e e f f e c t s , l e t alone the co n s t a n t i n f r o n t . The c l o s e n e s s of the model p r e d i c t i o n s and f i e l d r e s u l t s i n d i c a t e t h a t we have the s i z e s of these e f f e c t s approximately c o r r e c t . He would l i k e t o i n c o r p o r a t e the e f f e c t s of winds i n the model. The s i m p l e s t t h i n g i s to put a l l the wind s t r e s s , 1/2C DD2, i n t o the upper l a y e r . C o m p l i c a t i o n s a r i s e , f o r example, i f the wind blows f o r too l o n g , and mixes away the plume/ a problem which cannot be handled at t h i s stage of model development. I t has r e c e n t l y been observed, (Chang, 1976) , that t h e r e i s c o n s i d e r a b l e energy i n low frequency (periods g r e a t e r than 4 days) o s c i l l a t i o n s . The e f f e c t o f t h i s temporal v a r i a b i l i t y on plume motion ineeds t o be as s e s s e d , as does the amount and e f f e c t of b a r o c l i n i c i t y i n the c u r r e n t s and pressure g r a d i e n t s beneath the plume. As mentioned i n Chapter 4, s e c t i o n 6, t h e r e i s a p o s s i b i l i t y t h a t we should l e a r n how to deal with s u r f a c i n g i s o p y c n a l s i n the model. T h i s b r i n g s up the i n t r i g u i n g p o s s i b i l i t y of a 2 - l a y e r plume, a s e c t i o n through which i s shown i n F i g u r e 130. For arguments sake, we w i l l assume t h a t at the northern end of the plume, t h e r e i s an outflow boundary, a s i t u a t i o n we have d e a l t with i n the model so f a r . At the southern end, there i s a s u r f a c i n g i s o p y c n a l , and some development of the model must be done to accomodate t h i s s i t u a t i o n . There may be an a d d i t i o n a l boundary, i n d i c a t e d NP i n Figure 130. T h i s i s the boundary between the e x i s t i n g plume ( r e p r e s e n t i n g s e v e r a l days d i s c h a r g e ) , and the much f r e s h e r 96 water discharged d u r i n g the c u r r e n t t i d a l c y c l e . The boundary of t h i s c u r r e n t plume would be the i n t e n s e c o l o u r f r o n t found i n the S t r a i t . N umerically, we would have to l e a r n how to move t h i s f r o n t , and how to e v e n t u a l l y i n c o r p o r a t e "new plume" i n t o " e x i s t i n g plume". When the upper l a y e r model i s thought to be adeguately r e f i n e d , i t cou l d be used as the upper l a y e r of a co n v e n t i o n a l 3-dimensional model (Leendertse, Alexander, and L i u , 1973). The topmost l a y e r of the of the 3-dimensional model would have as i t s f r e e s u r f a c e the i n t e r f a c e between the upper and lower l a y e r s as d e f i n e d i n t h i s r e s e a r c h . The i n t e r f a c i a l fluxes; of mass, s a l t and momentum found i n the plume model would then be a p p l i e d to the 3-dimensional model, with of course t h e i r s i g n s r e v e r s e d . I n t h i s way one i s l e t t i n g the ph y s i c s of the s i t u a t i o n (a str o n g p y c n o c l i n e at a v a r i a b l e depth) d i c t a t e the type of numerical schemes employed to model i t . 9 7 SIIIMICES CITED Abramovich, G. N. 1963. The Theory of T u r b u l e n t J e t s . The H.I.T. Press, Cambridge, Mass. 671p. Bondar, C. 1970. C o n s i d e r a t i o n s t h e o r i g u e sur l a d i s p e r s i o n d•un courante l i g u i d e de d e n s i t e r e d u i t e e t a niveau l i b r e , dans un b a s s i n contenant un l i g u i d d'un plus grand d e n s i t e . 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Paper presented at t h 9th L i e g e Colloguim on Ocean Hydrodynamics. Wright, L. D., and J . M. Coleman. 1971. E f f l u e n t expansion and i n t e r f a c i a l mixing i n the presence of a s a l t wedge, M i s s i s s i p p i River d e l t a . J . Geophys. Res. 76: 8649-8661. 103 APPENDIX LINEAR STABILITY ANALYSIS C e r t a i n p a r t s o f the f o l l o w i n g d i s c u s s i o n are based on unpublished notes by R. F l a t h e r . Consider the s e t o f eguations: dk + ^ dy _ Q dt 2* j-cfho^jk --pu i-r 1/ =• C D . I f 6. i s h a l f the g r i d s p a c i n g , and "P the ti 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 = h t/-?. U - U -h \J - 1/ . J . UL -- u - tjjho ( k _^ ) - f v £. - ru ?_ . 2 A V * i/ - HqA* ( k - h ) * i U t - r ^ Z a. A. Assume that at any time,the f i e l d s of h, u, and v are w r i t t e n as F o u r i e r s e r i e s , i . e . , Then c o n s i d e r only one F o u r i e r component, and a l s o d e f i n e a new h^x egual to - j T ^ A o ^-kJi • D e f i n i n g ia r ^JJA7 ' sin l<A y do ?JJh7 C 5;„ J A 5 A A and dropping the k.,1 s u b s c r i p t s , we have: 104 l ft |- rt-tx -a t , -a ft ' -^-"loft J t V I f we c o n s i d e r the case r=f=0 , we a r e l e f t with the simpler system: k \/ V clo -a. )o T h i s i s of the form S - C* s , where s U I 6 0 o 1 and G i s the above matrix. I f we f i n d a s e t o f e i g e n v a l u e s 7- , and a s e t of e i g e n v e c t o r s E L f o r G, then the time s t r u c t u r e becomes and " K i t t -O - (71.-)' E . C l e a r l y , f o r the s o l u t i o n to be bounded TW must be bounded, which i s t r u e i f ) ^ J i 1 + OCC), which i s the von Neumann s t a b i l i t y c o n d i t i o n (Bichtmeyer and Morton, 1967). The term of 0 ( t ) allows f o r t h e p o s s i b i l i t y of e x p o n e n t i a l growth, which 105 one would not i n ge n e r a l want to exclude. Note that E t i s a gu a n t i t y of the form so t h a t knowing °<t", fi> i t 7 t, (which i s r e a d i l y done), one c o u l d o b t a i n h, u, and v from the t h r e e E ^ s . C a l c u l a t i n g the ^i a c c o r d i n g to det (G-"A 1) =0, we o b t a i n : I f a-^- i s l e s s than 2, the modulus of ^*-,3 t u r n s out to be e x a c t l y one. T h i s i s what we want i n t h a t the time f a c t o r f o r l i t * ? the a n a l y t i c case, going from t t o t t t must be e f o r a system without f r i c t i o n , and t h e modulus of e i s one./A,=1 i s a s s o c i a t e d with the e i g e n v e c t o r -bu+av, which can be i d e n t i f i e d with t JJkl ( S'H^-L \A. - sinMV;^j , which f o r s m a l l kA, I A , (l a r g e wavelengths), can be i d e n t i f i e d with the v o r t i c i t y , which i s an i n v a r i a n t s i n c e the C o r i o l i s f o r c e i s absent. He can compare A 1,3 with e . w i s g i v e n by iJ q h0 Lx +~** J . T h u s , e 3 4 ligk.?*- t 3 t r * ^ ^ / 3 [ . . . . & A 2 A 106 For J?4<<1, izt <<1, we have Thus the numerical and a n a l y t i c f a c t o r s d i f f e r i n the t h i r d order, so one cou l d say the scheme i s of second order accuracy i n time. T h i s i s only true i f we can r e p l a c e sin(-ir'l) with Jt A , and s i n ( ^ A - ) with A ; t h a t i s i f t h e r e are many meshes per wavelength. Thus, the accuracy o f the space d i f f e r e n c i n g (which gave r i s e t o the term £ sfa r a t h e r than c -*c. , f o r the a n a l y t i c e x p r e s s i o n f o r the l o g a r i t h m i c d e r i v a t i v e of a s i n u s o i d e ) a f f e c t s our time f a c t o r . One could a l s o c o n s i d e r A * - . 3 as Lick the expansion of <£ ( i n one dimension), where c i s the computational phase speed, which t u r n s out to be J aA? Sj_nj0 . The requirement t h a t # 2+# ̂  i s s a t i s f i e d i f Z or ?J~giu> ^ t T z ^ A a R e a l i z i n g t hat A i s one h a l f the g r i d spacing A / , we get as a s t a b i l i t y c r i t e r i o n fPCJ^l <"A^. T h i s i s the s t a b i l i t y requirement f o r a l i n e a r model, and we would l i k e to expand 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 i s to r e c o g n i z e [y~k7 as the wavespeed f o r the l i n e a r case, and r e p l a c e i t with (fu^+v? +/gh , the maximum expected wave speed f o r the n o n - l i n e a r case. Thus, our f i n a l c o n d i t i o n i s I f r=0, f * 0 , the s t a b i l i t y reguirement can be shown t o be: 107 As long as f2.«1 (a very easy condition to satisfy), the Coriolis force places no restriction on the scheme. If f=0, r£0, i t can be shown that the requirements imposed by fric t i o n are rf<2, and which i s a more severe form of the t!-4-^ relation. One can also see the requirements imposed by friction as follows. Consider the simple eguation with the f i n i t e difference approximation (/ - a - r c u . Writing 'u ^7\u , we can obtain 7i=1-rC. Bote that these are the f i r s t two terms in the expansion of e , the analytic value for.7\. If r't-is 2, the error in > is e~2-(1-2)= 1. 14, about 7 times the true value of ~h . If r t =1, the error is .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 less than one. Physically, f r i c t i o n 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 . When applied to 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^+v2"1 M<.% 1 < 1. 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 f i n i t e d i f f e r e n c e r e p r e s e n t a t i o n c t \ u ' ^ - 3*° C ti 2A Proceeding as before we get ik u where a =• I CO. h u t l j j k 7 Sin kA , lo , * cos k A A The eigenvalues are The q u e s t i o n i s , do these e i g e n v a l u e s s a t i s f y the von Neumann s t a b i l i t y c r i t e r i o n , I7sj<1*0(-e)? We w i l l assume b i s s m a l l ( f o r accuracy) and expand about b=0. Thus Z Z V ii 3 ; _ ̂  t if f^r^p f / - it fSfjLz •2- 4- 2 £<Z. fJZ^f lo / " i f ± 1 109 Now |cj=1, and |D| = 0 ( t ) , s i n c e b i s O(^) and a i s 0 ( f A ) , which i s 0(1). Thus s i n c e |>|<|C|+|D|, the i n c l u s i o n of entrainment r e s u l t s i n a s t a b l e scheme i f the entrainment i s s m a l l . He f u r t h e r should be aware of the s t a b i l i t y l i m i t a t i o n s imposed by the eddy v i s c o s i t y equation. Consider ^ _ A ^ _ 0 with f i n i t e d i f f e r e n c e r e p r e s e n t a t i o n Again, u=ue g i v e s ^ / * ZA^_ ( cosCk6) - / J . Cos(kA)-1 v a r i e s from 0 t o -2, thus 7\ v a r i e s from 1 to ]-4 A A*- and f o r s t a b i l i t y , we want <1. Because i t i n v o l v e s o n l y one eguation (assuming u i s known p e r f e c t l y ) the s a l t a d v e c t i o n eguation i s u s e f u l to demonstrate the phase e r r o r i n t r o d u c e d by numerical schemes. In order to see what kind of accuracy t o expect, and as an a i d i n s e l e c t i n g a f i n i t e d i f f e r e n c e scheme, s e v e r a l schemes were t r i e d i n a one- dimensional prototype. The eguation s o l v e d was ds + dj : o, u was dt o t taken t o be 40cm/s, A T =120sec. , AX=1km=10 5 cm. At x=0, the s a l i n i t y grew l i n e a r l y from 0 t o 30 i n 180 timesteps (6 hours), and then decreased l i n e a r l y back t o zero i n the next 180 timeste p s , a f t e r which i t remained zero. T h i s t r i a n g u l a r shaped p u l s e then propogated along i n the d i r e c t i o n of i n c r e a s i n g x, s i n c e D was p o s i t i v e . A l l the schemes t e s t e d conserved s a l t , but some appeared more s t a b l e than others. I n a l l of them, i t was p o s s i b l e t o see tha t waves o f d i f f e r e n t wavelength propagated a t d i f f e r e n t v e l o c i t i e s , so that the o r i g i n a l t r i a n g u l a r shape was d i s p e r s e d 110 i n t o a rounded bump, with e i t h e r l e a d i n g or t r a i l i n g r i p p l e s , depending on the s i g n of the phase e r r o r . T h i s d i s p e r s i o n i s produced by the phase e r r o r o f the f i n i t e d i f f e r e n c e scheme. 1*. Forward time d i f f e r e n c i n g The f i n i t e d i f f e r e n c e form i s Thus i f , . ^ . n d -j^ - j - Sir* K ^ A 3 3 i ' ,. 1* iy - L' LA. ^ K A . . * For s m a l l kA, > ~ ' ' 1 U L 1 A~ 3T The a n a l y t i c i s € I - c u tk - k w -g . . . Zl Thus t h i s scheme i s only f i r s t order a c c u r a t e . I d e n t i f y i n g the computational ^ with an expansion of e * 4 , we see t h a t the computational phase speed i s CA l ^ J i 6 f and that t h e r e f o r e s m a l l e r waves, with l a r g e r k, t r a v e l slower. Thus, we p r e d i c t r i p p l e s (short waves) emerging from the t r a i l i n g edge of the s a l t pulse. As we see i n F i g u r e 131, t h i s i s indeed the case. Checking f o r s t a b i l i t y , we see t h a t | I = 1*0 (-?). Thus, although the system conserves s a l t , the amplitude of the v a r i o u s F o u r i e r components grows with time. S i n c e there i s no source term i n the o r i g i n a l eguation, t h i s growth of the v a r i o u s modes i s an e r r o r . 111 v A J ' A The l e a d i n g terra i s ± I ~ l ^ sm kA . T h e r e are two 7)'s with 2i. Leapfrog time The scheme used i s t+-c i - r t t . 5 * ' 5 * - * L ^ ~ S J - J • 7.1 D e f i n i n g S=Se'*X , a= zis,« kA) crt f w e g e t and i f a 2 <4, |?>{ = 1. Thus none of the waves grows i n amplitude, which i s what we want. Expanding A , we get is = t Cl - *JL1La ( < ± l f + . . . ) - 'ii* l e a d i n g terms TN , = I ' 'JL? Sin k A , " - - ^~ we a s s o c i a t e with the r e a l p h y s i c a l wave, with true I S - E = <2 . He see t h a t f o r t h i s T\, again the short waves t r a v e l slowest, as F i g , 132 shows. "/Wis a s s o c i a t e d with the computational mode, and i s approximately -e , thus i t i s a s s o c i a t e d with a high freguency (period =2?) s i g n a l t e r n a t i o n . From the d i s p e r s i o n r e l a t i o n t o =uk, the wavelength a s s o c i a t e d most s t r o n g l y with t h i s mode i s 7\ -- 2 W^o -- 2.7TU / Z 7 7 / t , - u •£ -- DC A ) *, t h a t i s the s h o r t e s t wave. The computational scheme does not i n i t i a t e the computational mode, but i f th e r e are short wavelengths present, they w i l l e x c i t e t h i s mode. One u s u a l l y plans on doing a b i t of f i l t e r i n g t o e l i m i n a t e t h i s problem. 112 3. Rightmeygr scheme (Richtmyer and Morton, 1967) T h i s i s i n some sense an i t e r a t i v e scheme. The b a s i c i d e a i s to c a l c u l a t e 5 ~ s as a f u n c t i o n of s p a t i a l d e r i v a t i v e s of s at t+1/2 t . Thus t . // / ^ * ^ * ^ r ^ Sj* ' / 2 . xs approximated as / i l i j We o b t a i n f o r Sy : *j '- S* ~£Z t 5̂ ' -6r' ^ CS;., <•*,-,)). W i t h ^ - ^ . , we get f o r 7s 7 \ = I _ cosCkA) -i ) , Again we see t h a t s m a l l waves t r a v e l slower.. so t h a t numerical d i s s i p a t i o n i s present. The shape of the S d i s t r i b u t i o n f o r t h i s scheme i s i d e n t i c a l t o the Leapfrog scheme and i s not shown., APPLYING A FINITE DIFFERENCE SCHEME TO THE PLUME MODEL A problem develops when one t r i e s t o apply a f i n i t e d i f f e r e n c e scheme t o the plume model. U i s not const a n t , so the formulae look messier. The l e a p f r o g scheme i s the most d i r e c t , s i n c e one doesn't have to i n t e r p o l a t e s p a t i a l d e r i v a t i v e s as i n the Richtmeyer scheme^ However, i t was not used i n the model because of i t s e x t r a time l e v e l requirement - i t i s a three l e v e l scheme. Also, i n developing a model , one o f t e n runs f o r a few timesteps with a c e r t a i n s e t of c o n d i t i o n s , then saves the output as i n i t i a l c o n d i t i o n s f o r a f u r t h e r run. I f the succeeding run i n v o l v e s c o n s i d e r a b l e changes (which one hopes are improvements) there i s a very good o p p o r t u n i t y to generate a 113 computational mode. For t h i s reason i t was intended t o use the Eichtmeyer scheme. However, because of a programming e r r o r , d i s c o v e r e d a f t e r the modeling disc u s s e d here was completed, i n f a c t only a f i r s t o rder a c c u r a t e scheme f o r the s a l t a d v e c tion eguation was used. T h i s probably accounts f o r the c o n s i d e r a b l e s m a l l s c a l e f l u c t u a t i o n s present a f t e r a long run, showing up i n p a r t i c u l a r as small n e g a t i v e s a l i n i t i e s . Thus the s o l u t i o n of the c o n v e c t i v e equation i n a 2-dimensional, s p a t i a l l y and temporally v a r y i n q flow has not been c a r r i e d out as w e l l as one would l i k e . The problems are t w o - f o l d : .1) g e t t i n g a high enough order a c c u r a t e scheme; 2) a p p l y i n g t h i s scheme t o the already e x i s t i n g g r i d of staggered t r a n s p o r t s and t h i c k n e s s e s . , There i s a f u r t h e r , more s u b t l e , p o i n t about the s a l i n i t y e guation. The s a l i n i t y a d v e c t i o n equation i s a l s o a d e n s i t y advection equation, and thus a f f e c t s the momentum eguation. I t was assumed, and confirmed only by the l i m i t e d experience d i s c u s s e d i n t h i s t h e s i s , t h a t a s t a b l e scheme would r e s u l t i f t h e s a l i n i t y f i e l d were t r e a t e d the same as the e l e v a t i o n f i e l d , as f a r as time l e v e l s i n the numerical scheme were concerned. Thus, the o l d v e l o c i t y f i e l d i s used to update the s a l i n i t y f i e l d , and the new d e n s i t y f i e l d i s used i n t o update the v e l o c i t y f i e l d s . 114 TABLE I HARMONIC ANALYSIS OF BIVER SPEEDS ^ , ^ , , , — 1 , NAME I FREQUENCY | j c y c l e / d a y J — i - i AMPLITUDE | cm/sec | i PHASE degree ! T t 1 ZO ! 0.0 | 0.0860 | 0.00 I MM 1 0.03629164 | 0.2318 J 2 74.16 j MSF I 0.06772637 ) 0.4556 J 176.66 I 2Q1 1 0.85695237 { 2.8886 | 28. 19 I ' Q l | 0.89324397 f 3.3717 | 167.73 I 01 | 0.92953563 | 24.0625 | 280.06 I N01 I 0.96644622 | 8.2796 | 59.74 I P1 I 0.99726212 | 10.7181 | 304.67 INFERRED (K1) S1 | 1.00000000 | 1. 7835 | 2 57.24 INFERRED (K1) K1 | 1.00273705 | 33.1047 | 302.74 | J1 I 1.03902912 | 2.7960 | 169.53 | 001 | 1.07594013 | 4.7931 | 197.50 | MNS2 I 1.82825470 J 2.5522 | 7.42 | MU2 | 1.86454678 \ 6. 4664 | 108.74 N2 | 1.89598083 | 6.7790 | 2 34.64 | NU2 | 1.90083885 | 1.3024 | 239.74 INFERRED (N2) M2 J 1.93227291 | 48.3670 | 311.66 ] L2 | 1.96856499 | 25.7985 | 320.57 j T2 | 1.99726295 | 0.8353 | 313.44 INFERRED (S2) S2 | 1.99999905 | 13.2946 | 3 36.78 } K2 I 2.00547504 | 3.5766 | 3 35.88 INFERRED (S2) 2SM2 I 2.06772518 | 5.8606 | 183.44 | M03 | 2.86180973 | 5.7899 | 154.61 ] M3 | 2.89841080 | 1.5712 | 3 25.21 | MK3 | 2.93500996 | 10.6364 | 173.79 j SK3 t 3.00273800 | 3.2154 | 1 71.84 MN4 I 3.82825470 \ 1.0384 | 178.85 M4 | 3.86454678 | 4.3592 J 346.81 SN4 | 3.89598179 | 2.3171 J 166.22 | MS 4 I 3.93227291 J 6.3899 | 189.51 | S4 I 4.0O000000 | 2.8767 | 172.03 2BN6 j 5.76052761 | 1.4951 | 44.20 | M6 | 5.79681969 | 1.9540 | 185.45 | MSN6 I 5.82825565 | 1.1174 | 19.73 j 2MS6 | 5.86454582 | 2.2803 { 129.23 | 2SM6 | 5.93227386 j 1.2324 | 329.69 3HN8 | 7.69280148 | 0.2242 | 167.19 M8 J 7.72909451 | 1.0761 | 180.33 | 3MS8 | 7.79681969 | 0.2408 | 248.67 ] M12 | 1 1.59364128 J 0.0446 1 263.10 I I L , 1 X i J 115 TABLE I I ANALYSIS OF POINT -MMIISJ! ELEVATIONS NAME FREQUENCY cycle/day ZO | 0.0 MM J 0.0 3629164 MSF | 0.06772637 2Q1 | 0.85695237 Q1 | 0.89324397 01 f 0.92953563 'HOI \ 0.96644622 P1 | 0.99726212 51 | 1.00000000 K1 | 1.00273705 J1 | 1.03902912 001 J 1.07594013 MNS2 | 1.82 825470 MU2 | 1.86454678 N2 | 1.89598083 NU2 | 1.90083885 M2 | 1.93227291 L2 | 1.96856499 T2 | 1.99726295 52 | 1.99999905 K2 | 2.00547504 2SM2 f 2.06772518 M03 | 2.86180973 M3 | 2.89841080 MK3 | 2.93500996 SK3 | 3.00273800 MN4 | 3.82825470 M4 | 3.8 6454678 SN4 | 3.89598179 MS 4 j 3.93227291 S4 | 4.00000000 2MN6 | 5.76052761 M6 | 5.79681969 MSN6 f 5.82825565 2MS6 | 5.86454582 2SM6 | 5.93227386 3MN8 | 7.69280148 M8 | 7.72909 451 3MS8 | 7.79681969 M12 | 11.59364128 AMPLITUDE cm 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 PHASE degree 0.0 349.62 3 23.83 141.43 137 . 9 1 153.47 220.03 166.24 1 18.81 164.31 200.06 2 38.04 336.40 94.61 131.42 136.52 159.49 209.59 155.02 178.25 177.35 110.66 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 INFERRED (K1) INFERRED <K1) INFERRED (N2) I I INFERRED (S2) I INFERRED (S2) 116 TABLE I I I SCALE ANALYSIS OF TERHS IK -THE EQUATION OF MOTION parameter u, flow v e l o c i t y u 0 , t i d a l stream h, plume t h i c k n e s s L, h o r i z o n t a l l e n g t h s c a l e T, time s c a l e f , C o r i o l i s parameter g / r e d u c e d g r a v i t y ^ ,* . Wp, entrainment v e l o c i t y ?uh/0 t , time d e r i v a t i v e 2 {u 2h) / D X , a d v e c t i v e term 2(g' h 2/2) /?xf pressure g r a d i e n t fuh, C o r i o l i s term Ku2> {K=0. 001) , i n t e r f a c i a l f r i c t i o n Wp u 0 , e n t r a i n e d momentum f l u x qhC^, b a r o t r o p i c t i d a l f o r c i n g uh/Lw, advection/entrainment r i v e r area (cgs) 100 I 20 20 I 20 400 | 100 10s I 10« 4 *10* | 4»10* 10-* I 10-* 20 I 10 400 >10~* f 10»10-« 1 ! .05 40 I 0.04 16 1 0.1 4 0.2 10 I 0 4 0.02 4 4 10 2 f a r f i e l d ] <cgs) 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 from Crean and Ages(197l). C. FLATTERY J U A N DE F U C A STr BOUNDARY C . MUDGE f-HARCH -PASS.- ST. -ST. OF GEORGIA- F R A S E R R I V E R 2 3 6 9 12 14 16 SALINITY DISTRIBUTION IN THE STRAIT OF GEORGIA & JUAN DE FUCA STRAIT, 4-8 DEC. 1967 FIGURE 3. Taken from Crean and Ages(l97l).  FIGURE 5. The d a i l y F r a s e r R i v e r discharge f o r 1976, measured at Hope. The two arrows i n d i c a t e the period during which a current meter, described i n Chapter 2, was i n s t a l l e d . FIGURE 6. Chart of the river mouth area, showing the location of the current meter mooring to 250-j 2 0 0 - J U L I A N DAY FIGURE 7. Smoothed current meter record obtained by using the angular correction and the smoothing f i l t e r A 6 A 6 A 7 / ( 6 - 6-7)„ The break i n the time axis occurs at day 114. to CO Ul 2 150-1 U J U L I A N DAY FICJURE 8 . The low frequency component of the current meteor record, obtained by using the A 2 4 A 2 4 A 2 5 / ( 2 4 ' 2 4 - 2 5 ) f i l t e r . The break i n the time axis occurs at day 1 1 4 .  126 120 • 130 JULIAN DAY FIGURE 10. T i d a l e l e v a t i o n s at P o i n t Atkinson during the time the c u r r e n t meter was i n operation. The break i n the time a x i s occurs at day 114. ty i — i ro o <T> ?r H H* 3 > r t I T H fD ro o r t O H- 3 3 cn 0) r t 4 X o o o o c m r t :r cu ro r t r t *< Eu H H H CO >£• H - • TO W H i H O g o o co r t H - r t ro 3 C r t r~ CO __ &> r t Z. H 3 o H* O fl! 3 fl) M *< CO SPEED (CM/S ) SPEED ( C M / S ) i o o cn o cn O O O C > > -< AST MAY 8 1976 1330 PST 0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32 0 ' 1 1 ' 1 1 1 i I SIGMA T 0.0 3.0 . 6 . 0 9.0 12.0 15.0 1S.0 21.0 24 0 < 1 1 1 1 1 1 i I TEMP. FIGURE 12. P r o f i l e s of S,T,sigma t'. and current speed ( in d i c a t e d by dots ) f o r 1330 PST. K May 8, 19.76, a t the c u r r e n t meter mooring.  130 123"00- 1200 SE6 1500 E6 1800 calm 6 12 18 TIME (PST) STATION TIMES h 1730 i 1732 .1 1744 k 1750 1 1758 rn 1805 n 1814 o 1822 • p 1832 q 1842 r 1849 FIGURE 14. S t a t i o n p o s i t i o n s and times, wind and t i d e Tor A p r i l 6.-. 1976. STAT ION FIGURE 15. ' S a l i n i t y s e c t i o n along 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, Apr i l 6, 1976. 133 F I G U R E 17. S, T, sigma t p r o f i l e s at s t a t i o n 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 STATION TIMES 6 12 18 T f M E ( P S T ) 24 e 1513 f 1535 g 1553 h 1558 :i 1620 j 1625 k 1710 1 1713 m 1730 n 1732 o 1755 p 1818 q 1831 r 1844 FIGURE 19. S t a t i o n p o s i t i o n s and times, 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. 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. 1 — ! 0 0 137 APRIL 15 197S S T A T I O N K 1 7 1 0 P S T 0.0 •s.o a.o i i i 1G.J 20.0 i l 2 0 i 23.0 1 ? - 3 SIGMA T 0.0 1.0 6.0 i i 9.0 i 12.3 1S.0 i • 13.0 l 21.J i r T W . D.D 5.0 10.0 i i 15.0 i ? o . : 2S.: J3.0 K . J 00.Q i S f l l lNITT —I X FIGURE 21a. S, T, sigma t p r o f i l e s a t s t a t i o n k, 1710 PST, A p r i l 15, 1976, O.C O.C 0.0 3 - l — o rn •a FiPRI l 15 197G 4.0 8.0 12.0 IS.O 3.C C O 9.0 7 S T A T I O N L 1 713 P S T 20.0 20.0 2B.0 n.1 12.0 SIGrtfi T ' ? - J 3 ' ' 3 ^ C TEMP. SPLINITT 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 T I M E ( P S T ) STATION TIMES 24 S 1623 h 1714 FIGURE 22. S t a t i o n p o s i t i o n s and times, 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, along a l i n e from a f r o n t at s t a t i o n g to s t a t i o n h. The s a l i n i t y isopycnals are 15, 17.5, 20, 22.5, 25, and 26 parts per thousand. WINDS AT SAND HEADS 0000 E2 0400 E12 0700 E12 1200 SE8 1500 SE6 1800 S4 6 12 1 8 T I M E (PST) STATION TIMES 24 o 1517 p 1530 q 1540 s 1625 t v w X 1631 1635 1638 1640 1643 FIGURE 24. S t a t i o n p o s i t i o n s and times, 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 along l i n e o-s, June 4, 1976, S.H. i n d i c a t e s the l o c a t i o n of Sand Heads. STATION FIGURE 25b. S a l i n i t y s e c t i o n along l i n e x - t , June 4, 1976. I s o h a l i n e s shown are 5, 10, 15, 20, 25, 26, 2 7 % 0 . S.H. i n d i c a t e s the l o c a t i o n of Sand Heads. 123 1 1 r 0 6 12 18 STATION TIMES Tl M E ( PST ) a 0642 b 0709 c 0733 d 0916 3 1106 1 1230 rn 1302 o 1319 p 132 9 q 1447 q 1457 r 1540 s 1625 FIGURE 26. S t a t i o n p o s i t i o n s and times, 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 along l i n e a-l, J u l y 3, 1975. STAT ION 9H FIGURE 2 7b. S a l i n i t y s e c t i o n along 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 at s t a t i o n e, 8733 PST, July 3, 1975; 146 o.o 1 ., JULY 3 4.0 i 1975 B.O l 12.0 I S T A T I O N 16.0 i J 20.0 I 1106. 24.0 i P S T 28.0 i 32.0 I 0.0 1 3.0 i CO 1 9.0 i 12.0 i 15.0 l 19.0 I 21.0 l 24.0 0.0 o 5.0 I 10.0 1 15.0 l 20.0 25.0 l 30.0 l 35.0 43.0 _ 1 o m FIGURE 30. S, T, sigma t p r o f i l e s a t s t a t i o n 1106 PST, J u l y 3, 1975. 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 on t h e s i l t y s i d e o f t h e f r o n t , 1250 PST, J u l y 2, 1975.. 149 FIGURE 33. S, T, sigma t p r o f i l e s on deep blue'side of f r o n t , 1325 PST, July 2, 1975. 150 FIGURE 34. S, T, sigma t p r o f i l e s on s i l t y side of f r o n t , 0849 PST, July 4, 19.75. 151 FIGURE 35. S. T, sigma t p r o f i l e s on deep b l u e s i d e o f f r o n t , 0853 PST", J u l y 4, 1975. 152 FIGURE 36. The e v o l u t i o n of a f r o n t , January 18, 1976. The estimated t i d a l currents 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 currents are f l o o d i n g . 153 30' 2& 10' 12T0CT O H , . 1 1 0 12 2 4 T I M E ( P S T ) STATION TIMES ' A 0752 - 1121 B 1245 - 1425 FIGURE 37. S t a t i o n p o s i t i o n s and times, wind and t i d e f o r J u l y 23. 1975. FIGURE 38a. Paths of drogues released i n . r e g i o n A, J u l y 23, 1975. Speeds i n m/sec are i n d i c a t e d . •4SP05' 123° 2 5 ' FIGURE 38b. Paths of drogues released i n re g i o n B- July. 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 S P E E D ( M / S ) FIGURE 40. Speed profile in region A, 1015 PST to 1045 PST, July 23, 1975. 157 SO" 10' 12X00- WINDS AT SAND HEADS 0000 E10 0400 SE12 0700 SE12 0900 E12 1200 SE10 1500 S8 0 6 12 18 24 T I M E ( P S T ) STATION TIMES b 0907 f 1226 c 1020 g 1304 d 1110 h 1404 e 1222 i 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 (CM/S) 0 30 60 90 j : i 1 I— 124 FIGURE 43 Speed p r o f i l e at s t a t i o n g, 1317 PST, J u l y 13, 1976. 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 1 2 T 0 O - NE6 NE4 NE6 NE2 SE12 SE6 6 12 18 S T AT I ON TIME S —̂ -0̂ -̂ _LE5J.) a 0646 b 0716 c 0807 d 0857 e 0957 f 1113 g 1226 h 1318 i 1422 ,1 1512 FIGURE 45. S t a t i o n p o s i t i o n s and times, wind and t i d e f o r Sept. 17, 1976. FIGURE 46. S, T, sigma t p r o f i l e s a t s t a t i o n a, 0646 PST, S e p t . 17, 1976. FIGURE 47. Speed p r o f i l e a t s t a t i o n a, 0655 PST, Sept. 17, 1976. ON C O 0° 270° 0655 PST SEPT 17 0 m. & _ » "50 cm/s 50 cm/s i -• • i > Q shi p d r i f t • 10 m. > \ ^ — 180° FIGURE 48. Polar plot of velocity vectors, station a, 0655 PST, Sept. 17, 1976. 165 SEPT 17 1976 S T A T I O N J 1512 P S T 0.0 1 4.0 8 . 0 i 12.0 16.0 1 ! 20.0 i 24.0 t 23.0 I 32.0 1 SIGMfl T 0.0 1 3.0 i 6.0 i 9 . 0 12.0 l i 15.0 i 1B.0 I 21.0 i 24.0 1 TEMP. 0 . 0 o 5.0 l 10.0 1 15.0 20.0 ! 1 25.0 i 30.0 I 35.0 1 4a.o i 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  1458 PST SEPT 17 2 7 0 ° - ^ 0° 50 cm/s 1 3 m. 50 cm/s sh ip dr i f t 0 m. FIGURE 51. 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 j , 1458 PST, Sept. 17, 1976. 168 F R A S E R R I V E R 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 (15, 20, 25 p p t ) . AA' i s the path of a s a l i n i t y s e c t i o n , F i g . 53. FIGURE 53. A s a l i n i t y s e c t i o n along AA', F i g . 52, and s a l i n i t y p r o f i l e s at three s t a t i o n s along AA'. 169 RfVER FLUX E N T R A I N M E N T D E P L E T I O N 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 , surrounding the plume defined by the S=25 contour. The three c o n t r i b u t i o n s t o the mass balance, r i v e r f l o w , entrainment, and d e p l e t i o n are i n d i c a t e d . — — Z : -D FIGURE 55. D e f i n i t i o n sketch f o r the equations derived i n Chanter 3. FIGURE 56b. The r e g i o n of s o l u t i o n , f i l l e d up w i t h i n t e r s e c t i n g c h a r a c t e r i s t i c s . 1 7 1 \ FIGURE 57. Streamlines and l i n e s of equal thickness, from R o . ' i s e et a l , 1951. 172 FIGURE 58a. Schematic diagram f o r the model of s u r f a c i n g isopycnals. 0 0 1 -A 5 _L_ X ( K M ) 10 15 20 T / 3 FIGURE 58b. The thickness of the upper l a y e r , s o l i d l i n e , p r e d i c t e d by Eqn. 4.13. The dots 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 of h ( L - x ) 1 / 3 . FIGURE 59b. Sketch of upper l a y e r c o n d i t i o n s used to o b t a i n the i n t e g r a t e d pressure term. 174 I — ^ ~ X FIGURE 60b. A characteristic intersecting the front at (s,t). 1 3 ^ X FIGURE 60c. Two characteristics intersecting at a hydraulic jump. on FIGURE 61. The c h a r a c t e r i s t i c diagram f o r the kinematic wave solut i o n .  SALINITY 1 0 - j FIGURE 63a. The vertical distribution of salinity at t=0, t=20At, 120At, 220At, and 320At, for K=Z2/(.22). S A L I N I T Y FIGURE 63b. The vertical distribution of salinity at t=0, 20At, 220At, 500At, and 920At. for K=l. 178 ! - D_ LU Q . H \ \ \ \ \ \ 0 20 120 220 320 420 TIMESTEP FIGURE 64a. Depth of the upper l a y e r . The s o l i d l i n e i s from the d i f f u s i o n equation s o l u t i o n , the dashed l i n e i s from the d i s c r e t e l a y e r s o l u t i o n . i r 120 220 TIMESTEP 320 420 FIGURE 64b. T o t a l s a l t content of the upper l a y e r . The s o l i d l i n e i s from the d i f f u s i o n equation, the dashed from the 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 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. 181 Y A X FIGURE 67. A t y p i c a l computat"onal element of the numerical g r i d used i n t h i s research. Z i s the v a r i a b l e name assigned to the l a y e r t h i c k n e s s , r e f e r r e d to as h i n most equations of t h i s t h e s i s . 182 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 41 42 43 44 55 56 57 58 69 70 71 72 83 84 85 86 97 98 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 the square box model. The r i v e r discharges through mesh 169. P o s i t i v e values of y, ( s m a l l i n d i c e s ) , are at the southern h a l f of the model. The s o l i d boundaries are i n d i c a t e d by a t h i c k l i n e , the open boundaries by a heavy dashed l i n e . FIGURE 69. Flux out of the open ends of the linear model. 1000 are equivalent to 133.33 hours. 5 0 CM/SEC T I M E H R S . 3 T I M E S T E P 1 0 0 MINS 2 0 O D « •« ^ X \ \ \ -o \ ^Ni ^Na \ a 0 6 i (t t i 6 o 1 1 1 i i / i W \ \ I / / / / ~o -e -€J —o —-o » — *-* l-l « J / </ s <s y ^ ^ a- o O - O- B- O - O - B- O- B - o- a- B - FIGURE 70. Velocity 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 for which the vector represents the velocity. = 50 CM/SEC TIME HRS. 16 MINS 40 TIME STEP 500 -a a——o——a o ' T V A FIGURE 71. The same as F i g . 70, but at timestep 500. = 5 0 CM/SEC TIME HRS. 33 MINS 20 TIME STEP 1000 — a —o "-n or' o— a — FIGURE 72. The same as F i g . 70, but at timestep 1000. 74 0 5 0 0 1000 0 5 0 0 T I M E S T E P T I M E S T E P FIGURE 73a. Influx (solid line) and efflux (dashed FIGURE 73b. Influx (solid line) and efflux line) for the model of Figs. 70, 71, and 72. (dashed line) for the model of Figs. 74, 75. 0 0 < 1 T I N E HRS. T I M E S T E P = 50 CM/SEC 3 100 M1NS 20 a e o o B O Q u B a a B o e a o o o B 0 B 4 B i 6 1 a 1 6 I a i a 0 B a o a h \ \ \ 1 1 1 J J / e \ \ \ \ \ 1 1 1 J J / V \ \ \ \ \ 1 0 1 J J / B- B— -a ^B \ \ \ \ \ \ \ -o -a -o \ \ \ \ \ \ A -e \ \ \ \ \ \ \ —Q -B —O o — a — e — o ~—-a —o — o — a ——ta • — B— FIGURE 74. F l o w f i e l d c a l c u l a t e d i n t h e same way as 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 eddy v i s c o s i t y . 00 00 I =50 CM/SEC TIME HRS. 16 MINS 40 TIME STEP • 500 cr"* cr- i 1 / y ^ 1 i / !l / / s // / FIGURE 75. Flow f i e l d c a l c u l a t e d i n the same way as f o r F i g . 71, with the a d d i t i o n of h o r i z o n t a l eddy v i s c o s i t y . 0 H > . > i -0 2 4 6 8 10 12 M E S H NO. FIGURE 76. The propagation of a hump of water out of the system, the line of small dashes shows the i n i t i a l distribution of thickness. The shape as the bulge passed out of the open boundary is shown by the large dashes, and the corresponding distribution of velocity by the solid line. M E S H NO. FIGURE 77. The distribution of velocity and elevation for Fig. 74, along the line from the river mouth to the opposite solid boundary. 192 78a. 78b. 49 44 45 89 63 48 125 109 68 51 141 121 69 4 9 177 155 115 64 49 196 161 112 63 48 230 168 128 80 51 234 158 120 76 51 " k s . V V . v. vj ' v FIGURE 78a,b. Comparison of e l e v a t i o n f i e l d s from a .33 km g r i d s i z e model, 78a, and from a 1 km g r i d s i z e model, 78b. 79a. 79b. 140 171 218 427 79 162 170 184 3 72 144 131 24 70 105 123 199 3 1 1 485 76 128 142 157 10 90 125 117 19 75 90 FIGURE 79a,b. Comparison of entrainment 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. 1 7 23 35 14 16 10 4 10 v v x ^ ^ 16 24 37 2 10 1 5 11 8 3 V "< v. FIGUPvE 80a,b. Comparison of u - 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, 80a, and a 1 km mesh s i z e model, 80b. 81a. A 4 16 .3 8 14 17 11 14 7 7 7 81b. 10 14 19 16 16 2 4 10 12 3 4 6 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 s i z e model, 81a, and a 1 km mesh s i z e model, 81b. TIME HRS. TIME STEP = 5 0 CM/SEC 12 360 MINS 0 — t t B o - \ BT \ \ \ h C \ \ 1 \ \ 1 \ \ \ i \ \ l i \ \ l c r " \ \ \ c r " \ \ Q e r - h d e- — o — B a ct— FIGURE 82. Flow 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 constant Froude number boundary c o n d i t i o n . = 5 0 C M / S E C T I M E H R S . 2 0 M I N S T I M E S T E P 6 0 0 ° \ * \ * \ \ \ V ^ \ \ \ \ \ \ \ \ \ \ \ \.\ \ \ \ \ \\\ N *s ^ \ \ \ \ \ \ \ \ / \ \ \ \ \ \ \ \ / FIGURE 83. Flow f i e l d produced by a model with constant river 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. = 50 CM/SEC TIME HRS. 20 MINS TIME STEP 600 N V ^ \ \ \ v % \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\\\W ̂  W W W W / \ \ \ \ \ \ \ v \ \ \ ° N FIGURE 84. Flow f i e l d produced by a model i d e n t i c a l t o t h a t of F i 2 83 c o e f f i c i e n t of 0.001. g ' > FIGURE 85 V e c t o r v e l o c i t y diagram f o r a model w i t h d e n s i t y e f f e c t s , t i d a l e l e v a t no C o r i o l i s f o r c e , and constant r i v e r f l o w . erevar 198 IT) Z . *—< XL U U J C O s. _ 0 Q_ Ct I — x in J r r r r r i r r r r r J T r r j r r r j t t f r j r f r p j f r / s " I J ' I I I I I ! I J I I I f +J CO CO 3 O CN oo tc •H CL, o c u o E a EH a CL, = 5 0 CM/SEC TIME HRS. 184 M N S 0 A L = 1 KM TIME STEP 5520 ^ •—a —s —e "~-o ~-o —a -« —o —a '—a -̂o ~-B -a ~-e —o —o "̂-B ^B "*B ~a -B —a —-a ~*~-n \ \ i "̂B B̂ "a -o -a •—B B B \ 5̂ "B "a -O -e \ o a B A B B or ' a B a -a -a -a -a -a —a —a —a FIGURE 87. The model of f i g u r e 85, 4 hours l a t e r . o TIME HRS. TIME STEP = SO CM/SEC 186 5580 MINS A L = 1 K M FIGURE 88. The model of Figure 85, 6 hours l a t e r . to o o f I = 5 0 CM/SEC TIME , « „ m o A L = 1 K M TIME STEP 5640 FIGURE 89. The model of Figure 85, 8 hours l a t e r . = 50 CM/SEC TIME HRS. 190 ' MINS 0 . . . .... A L = 1 K M TIME STEP 5700 FIGURE 90. The model of Figure 85, 10 hours later. to o to = 50 CM/SEC TIME HRS. 192 TIME STEP 5760 MINS AL=1 KM -a -a -a -a -a -e -a -a -e -a —o —e — o ~~a — a - a . *a -o -o - - a *~a — o — « ~̂ B **B ^ a ""a -B ^B ^B B̂ - a >B ^B "TI ' ^B -B - a -a -a -a -a - a -a -a -a -a -B -a -a - s - a -a -a "B -e - a -o -a -a "B ~e -a -a -e -a B - e - e - e - e - e - a - e -a -a -a -a -a FIGURE 91. The model o f F i g u r e 85, 12 h o u r s l a t e r . 1 8 6 H O U R S S A L I N I T Y FIGURE 92. Salinity 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 H O U R S S A L I N I T Y / / / / 1 u 1 0 \ 5 — \ \ \ \ 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 1 8 6 H O U R S E L E V A T I O N S I ' to o FIGURE 94. The d i s t r i b u t i o n of upper l a y e r t h i c k n e s s a t hour 186, corresponding t o F i g . 88. ^ The 400 cm and 500 cm contours are shown. 192 H O U R S E L E V A T I O N S 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 FTCURE 96 Drogue t racks produced over a 24 hour period by the flow field of Figs. 85 - 91. Dots on thf t racks are serrated by one hour, and the upper curve indicates the tidal elevatxon during the 24 hour period. to o 00 TIMESTEP 9360 TO TIMESTEP 10081 FIGURE 97. Drogue tracks produced over 24 hours by the flow field augmented flow at the boundaries during' outflow. TIME HRS. TIME STEP = 5 0 CM/SEC 330 9900 MINS —o -o - e ~-s - a - a -e — o - a - a - a a e a a e o 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 flow a t the open boundaries d u r i n g outflow. to O 1 n a u t i c a l m i l e a) f l o a t time time no. i n out 15 58.3 58. h 16 58.3 58.5 17a 58.3 58.8 1Tb 59-0 60.8 18 58.7 60.7 19 58.8 60.8 20 59-0 60.6 21 60.3 60.8 22 60. It 60.8 b) f l o a t time time no. i n out 1 59-9 60. 5 2 60.0 60.5 3 60.1 61.7 k 60.1* 61.8 5 6o. k 61.8 6 60.9 62.1 7a 6i.O 61.8 7b 61.8 62.1 s N Path l i n e s of f l o a t s i n a, Set 16, from 58.2 hours t o 60.8 hours, b, Set 17, from 59-9 hours t o 62.1 hours, taken near lower low t i d e . Winds were south h to southwest 8. The path 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 those f l o a t s were b r i e f l y removed f o r r e p a i r s . FIGURE 99. Drogue t r a c k s f o r drogues{released s h o r t l y before low water, (from Cordes, 1977) u T IME (HOURS ) 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. The r i v e r inflow was 2000 m3/sec. 213 FIGURE 101. Drogue tracks produced when drogues were released at zero river flow, approaching high water. TIMESTEP 1530 TO TIMESTEP 1981 TI D f i l ELEVATION AT TSAUASSEN FIGURE 102. Drogue tracks produced when drogues were released at half maximum river flow, during the ebb, when river flow is increasing. IN OUT TIMESTEP 1980 TO TIMESTEP 2431 TIDAL ELEVATION AT TSAUAS5EN FIGURE 103. Drogue tracks produced when drogues were released at maximum river 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 half maximum river flow during the flood stage of the tide. = 5 0 CM/SEC TIME HRS. TIME STEP 62 I860 MINS Na N= N, N S. \ N, N N \ \ \, \ \ b b b • 1 A i A fa s \a tl t4 A N. >l V "b "to "b h b *b "b 1= •a •b b b 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 . Other terms are the same as those which produced F i g . 86, except s a l i n i t y i s 20 % Q on i n f l o w . to TIMESTEP 1800 TO TIMESTEP 2161 TIDAL ELEVATION RT TSRWflSSEN FIGURE 106. Drogue t r a c k s produced over 12 hours by the model which produced the flo w f i e l d of F i g . 105. to M ON 6 6 HOURS E L E V A T I O N S / - 1 7 5 - \ \ r \ I \ 2 0 0 \ \ \ \ \ FIGURE 107. E l e v a t i o n f i e l d produced by the model w i t h d e p l e t i o n a t 66 hours, a t the same t i d a l phase as F i g . 94. The 175 cm and 200 cm contours are shown. t o - J 72 HOURS E L E V A T I O N S FIGURE 108. E l e v a t i o n f i e l d at 72 hours, produced by the model w i t h d e p l e t i o n . The t i d a l phase i s the same as i n F i g . 95. The 175 cm and 200 cm contours are shown. to OO 219 TIME HRS. TIME STEP FIGURE 109. Velocity f i e l d prodded' by the f i r s t real geometry model, at time of maximim river flow. 220 END O F E B B 8 H O U R S 20 0 C M , 300 C M C O N T O U R S FIGURE 110. D i s t r i b u t i o n of upper layer thickness at the end of the ebb. E N D OF F L O O D U HOURS FIGURE 111. D i s t r i b u t i o n of upper l a y e r t h i c k n e s s a t the end of the f l o o d . 222 FIGURE 112. Drogue t r a c k s produced over 12 hours by drogues re l e a s e d at maximum r i v e r f l o w . 223 FIGURE 113. 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 d i s c h a r g e used i n t h e second v e r s i o n o f t h e r e a l geometry model. X= -.3 V=20 X=-.4 V = 25 X=-.5 V = 30 X=-.6 V= 25 U=10 X=-.6 V-20 X = -.56 V=15 X=-.36 FIGURE 114. Distribution of velocities and surface slopes (cm/2km) used in the second version of the real geometry model. 225 * f t * t * r t 8' H O U R S F R A S E R • • « * • • • * • • • • • * • * • * • • • • • • • * 4 4 -* / 1 \ s / 1 * t \ • • t t 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 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. J " • * * * * * * t * t 12 H O U R S *. *> \ T t 7 T * s \ \ t t t \ \ \ T t r \ \ \ \ \ 1 * * 1 \ \ \ 1 1 f r r \ \ \ t r * i ] \ t t > • t t t * t 1 FIGURE 116. V e l o c i t y f i e l d produce 226 by the model of Figure 115, 4 hours l a t e r . 227 / t T t / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / FIGURE 117. Velocity field produced by the model of Fig. 115, 8 hours later. 228 20 H O U R S » * / / / I i / / / / 4 * I I / / / < 4 J I / / / f 4 I J / / s r~ r I / r~ • — - • f==r * . - r \ \ v r \ \ t ^ \ N * k * • * FIGURE 118. Velocity field 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 TSAVASSEN H O U R 6 T O H O U R 12 F R A S E R N FIGURE 120. T r a c k s produced by drogues r e l e a s e d a t hour 6, a p p r o x i m a t e l y i g h l o w w a t e r . 231 T1DRL ELEVRTION RT TSRWRSSEN H O U R 12 T O H O U R 18 FIGURE 121. Tracks produced by drogues released at hour 12, low high water. FIGURE 12^-." _ _Tracks produced by drogues released at 18 hours, at maximum r i v e r discharge, near low low water. 233 TlORl ELE.VRT10N AT TSRURSSEN H O U R 1 8 T O H O U R 2 4 FIGURE 123. Tracks produced by drogues travelling in the same flow f i e l d as those in Fig. 122, but with a correction for vertical shear in calculating'the drogue velocity. 1 nautical float time time no. in out 15 58.3 58.U 16 58.3 58.5 17a 58.3 58.8 1Tb 59-0 60.8 18 58.7 60.7 19 58.8 60.8 - 20 59.0 60.6 21 60.3 60.8 22 60.U 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 ELEVAT IONS FIGURE 125. D i s t r i b u t i o n of upper la y e r thickness at hour 8. . 12 HOURS ELEVAT IONS FIGURE 126. Distribution of upper layer thickness at hour 12. FIG1JKE 12 7. Distribution of upper layer thickness at hour 16. 2 0 HOURS E L E V A T I O N S ^150 / \ \ \ \ FRASER V \ N \ \ 300 / 150 FIGURE 128. Distribution of upper layrr thickness at hour 20. FIGURE 129. Distribution of upper layer thickness at hour 24. F R O N T F R O N T FIGURE 130. Schematic diagram of. a p o s s i b l e e x t e n s i o n o f the upper l a y e r model t o 2 l a y e r s . FIGURE 131. The distribution of salt as calculated by a f i r s t order scheme. The dashed line is the exact solution. FIGURE 132. The d i s t r i b u t i o n of s a l t as c a l c u l a t e d by a second order scheme. The exact s o l u t i o n i s shown by the dashed l i n e .

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