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UBC Theses and Dissertations

Water quality modelling in estuaries Joy, Christopher Stewart 1974

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WATER QUALITY MODELLING IN ESTUARIES  by  CHRISTOPHER STEWART JOY B.Eng., Mohash U n i v e r s i t y , 1968 M.App.Sci., Monash U n i v e r s i t y , 1972  A THESIS. SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF•PHILOSOPHY  i n the Department of  C i v i l Engineering  We accept t h i s t h e s i s as conforming t o the required standard  THE UNIVERSITY OF BRITISH COLUMBIA January, 1974  In p r e s e n t i n g an a d v a n c e d  this  thesis  degree at  t h e L i b r a r y s h a l l make I further  agree  for scholarly by h i s of  this  written  it  for  available  financial  for  for extensive  may be g r a n t e d It  British  is understood gain  of  Q  shall  that  not  U \ ^ k S f (jrS&^oJic,y>  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8. Canada  Columbia  the  requirements  Columbia, reference  copying of  I agree and this  for that  study. thesis  by t h e Head o f my D e p a r t m e n t  permission.  Department  Date  freely  that permission  purposes  fulfilment of  the U n i v e r s i t y of  representatives. thesis  in p a r t i a l  or  copying or p u b l i c a t i o n  be a l l o w e d w i t h o u t my  A B S T R A C T  A t i d a l l y v a r y i n g and a t i d a l l y averaged mass t r a n s p o r t model are a p p l i e d t o the Fraser River Estuary t o i n v e s t i g a t e the s i g n i f i c a n c e of  t i d a l e f f e c t s on t h e c o n c e n t r a t i o n s r e s u l t i n g from assumed  discharges.  effluent  The t i d a l l y averaged model i s due t o Thomann [1963].  t i d a l l y v a r y i n g model i s developed  from f i r s t p r i n c i p l e s .  model was used t o determine the t i d a l l y induced  A hydrodynamic  temporal v a r i a t i o n i n t h e  l o n g i t u d i n a l v e l o c i t y and c r o s s - s e c t i o n a l a r e a a l o n g t h e e s t u a r y . are  "mathematical" and  The  A l l models  one-dimensional.  F i n i t e d i f f e r e n c e techniques  a r e used t o s o l v e t h e u n d e r l y i n g  p a r t i a l d i f f e r e n t i a l e q u a t i o n s o f a l l t h r e e models. i t y and n u m e r i c a l d i s p e r s i o n a r e examined.  The problems o f s t a b i l -  Numerical d i s p e r s i o n i s seen t o  r e s u l t from t h e s o l u t i o n o f t h e mass t r a n s p o r t e q u a t i o n over a f i x e d g r i d r a t h e r than a l o n g t h e a d v e c t i v e c h a r a c t e r i s t i c s . the e q u a t i o n a l o n g t h e c h a r a c t e r i s t i c s a r e :  Advantages o f s o l v i n g  no n u m e r i c a l d i s p e r s i o n ; t h e  a d v e c t i v e and d i s p e r s i v e t r a n s p o r t p r o c e s s e s a r e u s e f u l l y separated; d i s p e r s i o n c a n be p a r t i a l l y a s s e s s e d w i t h a one-dimensional time dependent behaviour taken  into  space  lateral  model; and  i n c o e f f i c i e n t o f l o n g i t u d i n a l d i s p e r s i o n c a n be  account. The  t i d a l l y v a r y i n g f l o w s a l o n g t h e e s t u a r y a r e seen t o cause a  v a r i a t i o n i n the i n i t i a l d i l u t i o n o f a discharged  effluent.  This,  together  w i t h the e f f e c t s o f t i d a l f l o w r e v e r s a l produces s p i k e s i n the c o n c e n t r a t i o n p r o f i l e along the estuary.  The c o n c e n t r a t i o n o f t h e s e s p i k e s i s then r e -  duced by t h e d i s p e r s i o n p r o c e s s , t h e peak c o n c e n t r a t i o n d u r i n g t h e f i r s t two  ii  iii  t i d a l cycles being sensitive to the form and magnitude of the c o e f f i c i e n t of longitudinal d i s p e r s i o n . Time dependent v a r i a t i o n s i n t h i s c o e f f i cient are considered.  The e f f e c t of the l a t e r a l dispersion process on  the estimated concentrations i s also considered and secondary flows are tentatively explained i n terms of the generation and advection of vorticity.  The predicted peak t i d a l l y varying concentration was found to  be s i g n i f i c a n t l y greater than the t i d a l l y averaged value.  TABLE OF CONTENTS Page LIST OF TABLES  vii  LIST OF FIGURES  viii  LIST OF SYMBOLS  xi  CHAPTER  1.  INTRODUCTION. . . .  1  PRELIMINARY CONSIDERATIONS  5  1.1 1.2 1.3 1.4 2.  5 7 9 12  LITERATURE REVIEW  17  2.1 2.2  17 21 21 24 25 27 27 28 29 32  2.3  2.4 2.5 3.  THE ONE-DIMENSIONAL MASS TRANSPORT EQUATION DETERMINATION OF PARAMETERS THE EFFECTS OF THE TIDE ON MASS TRANSPORT PROCESSES ACCURACY OF A ONE-DIMENSIONAL MODEL  ANALYTICAL SOLUTIONS NUMERICAL SOLUTIONS 2.2.1 F i n i t e Difference Solutions 2.2.2 "Box Model" Solutions 2.2.3 F i n i t e Element Solutions PHYSICAL AND ANALOGUE MODEL SOLUTIONS 2.3.1 Physical Model Solutions 2.3.2 Analogue Model Solutions STOCHASTIC SOLUTIONS SUMMARY  A DESCRIPTION OF THE HYDRODYNAMIC AND MASS TRANSPORT MODELS . . .  34  3.1  34 34 38 39 39 39 42 44 44 47  THE HYDRODYNAMIC MODEL 3.1.1 The Hydrodynamic Equations 33.1.2 Assumed Quasi-Steady Hydraulic Conditions 3.1.3 The Model Estuary of the Hydrodynamic Equations . . . 3.2 THE TIDALLY VARYING MASS TRANSPORT MODEL 3.2.1 Method of Solution 3.2.2 The Model Estuary 3.3 TIDALLY AVERAGED MASS TRANSPORT MODEL 3.3.1 Method of Solution 3.3.2 The Model Estuary  iv  V  CHAPTER 4.  Page  VERIFICATION OF THE HYDRODYNAMIC AND MASS TRANSPORT MODELS. 4.1  4.2  5.  6.  THE HYDRODYNAMIC MODEL  48  4.1.1 4.1.2 4.1.3  Data A v a i l a b l e Water S u r f a c e E l e v a t i o n s f o r Low Flows . . . . . . . . Water S u r f a c e E l e v a t i o n s f o r High Flows  48 49 49  4.1.4  V e r i f i c a t i o n f o r the C o n d i t i o n s o f January 24, 1952 . 54  THE TIDALLY VARYING MASS TRANSPORT MODEL. 4.2.1 Advective Transport 4.2.2 D i s p e r s i v e T r a n s p o r t THE TIDALLY AVERAGED MASS TRANSPORT MODEL  4.3  . . . 48  65 65 67 71  COMPARISON AND DISCUSSION OF RESULTS  77  5.1 5.2 5.3 5.4 5.5 5.6  78 79 81 84 86 89  EFFECTS OF LATERAL DISPERSION THE INITIAL DILUTION OF EFFLUENT PEAK EFFLUENT CONCENTRATIONS UPSTREAM EFFLUENT TRANSPORT CHANNEL INTERACTIONS^ , SUMMARY  SUMMARY AND CONCLUSIONS  REFERENCES.  92  98  -APPENDICES A.  DERIVATION OF THE ONE-DIMENSIONAL MASS TRANSPORT EQUATIONS. . . .106 A.l A.2 A.3 A. 4 A.5 A. 6  B.  GENERAL LONGITUDINAL ADVECTIVE TRANSPORT LONGITUDINAL DISPERSIVE TRANSPORT SOURCE-SINK EFFECTS THE ONE-DIMENSIONAL MASS TRANSPORT EQUATION THE LAGRANGIAN FORM OF THE ONE-DIMENSIONAL ' MASS TRANSPORT EQUATION  A DESCRIPTION OF THE FRASER RIVER ESTUARY B. l B.2  GENERAL THE LOWER FRASER RIVER SYSTEM. . . B.2.1 Channels o f t h e R i v e r System B.2.2 T i d e s i n t h e S t r a i t o f G e o r g i a B.2.3 River Discharges B.2.4 S a l t w a t e r I n t r u s i o n  106 107 109 113 114 115  117 117 117 .117 129 129 134  vi  APPENDIX C.  Page  NUMERICAL DISPERSION AND STABILITY C l  SURFACE GEOMETRY OF PARTIAL DIFFERENTIAL EQUATIONS  C.2 NUMERICAL DISPERSION C.3 STABILITY C. 4 SUMMARY  D.  DETAILS OF THE SOLUTION SCHEMES OF THE HYDRODYNAMIC AND MASS TRANSPORT EQUATIONS D. l D.2  E.  P.  138 . . . . . 138 146 . 154 159  161  NUMERICAL SOLUTION OF THE HYDRODYNAMIC EQUATIONS NUMERICAL SOLUTION OF THE TIDALLY VARYING MASS TRANSPORT EQUATION D. 3 NUMERICAL SOLUTION OF THE TIDALLY AVERAGED 'MASS TRANSPORT EQUATION  167  ESTIMATION OF LATERAL DISPERSION  174  E. l E. 2  174 180  EXISTING ESTIMATES OF LATERAL MIXING VORTICITY ESTIMATE OF SECONDARY CURRENTS  ESTIMATION OF LONGITUDINAL DISPERSION F. l F.2 F.3 F.4 F.5  161 164  191  GENERAL 191 TIDALLY AVERAGED COEFFICIENTS OF LONGITUDINAL DISPERSION . . 192 TIME DEPENDENT LONGITUDINAL DISPERSION COEFFICIENTS 199 SENSITIVITY OF PREDICTED CONCENTRATIONS 201 SUMMARY 204  L I S T OF T A B L E S  Table  4.1  5.1  B.l  Page  T I D A L L Y AVERAGED MASS BALANCE OF PREDICTED DISCHARGES FOR JANUARY 24, 1952  64  RATIO OF T I D A L L Y VARYING CONCENTRATION TO T I D A L L Y AVERAGED VALUE AT POINT OF DISCHARGE  85  R I V E R FLOW VOLUMES AND T I D A L PRISMS ON JANUARY 15 a n d JUNE 16, 1964  135  >E.l  ESTIMATION OF C O E F F I C I E N T S OF L A T E R A L D I S P E R S I O N  179  F.l  ESTIMATED C O E F F I C I E N T S OF LONGITUDINAL D I S P E R S I O N DUE TO LATERAL V E L O C I T Y GRADIENTS  vii  198  LIST OF FIGURES  Figure  Page  1.1  E f f e c t of Variable I n i t i a l D i l u t i o n  1.2  Effect  of M u l t i p l e Dosing  on C o n c e n t r a t i o n  ;(Due t o Flow  10  Reversal)  on C o n c e n t r a t i o n  11 Through A D i v e r g i n g J u n c t i o n  1.3  Movement of E f f l u e n t  3.1  The  3.2  Simplified  4.1  E q u i v a l e n t Stage Response f o r Two  14  Hydrodynamic E s t u a r y  36  Model E s t u a r y o f the Mass T r a n s p o r t E q u a t i o n s . Different  . .  43  Types  of Cross-Section  50  4.2  Observed and P r e d i c t e d Stages f o r June 16,  1964  52  4.3  Observed and  1964  53  4.4  T i d a l l y Varying  Stage and D i s c h a r g e  —  Main Arm  55  4.5  T i d a l l y V a r y i n g Stage and D i s c h a r g e  —  Main Arm  56  4.6  T i d a l l y V a r y i n g Stage and D i s c h a r g e  —  Main Arm  57  4.7  T i d a l l y Varying  4.8  T i d a l l y V a r y i n g Stage and D i s c h a r g e  —  N o r t h Arm  4.9  T i d a l l y Varying  Stage and D i s c h a r g e  —  M i d d l e Arm,  4.10  T i d a l l y V a r y i n g Stage and D i s c h a r g e  —  P i t t River  4.11  T i d a l l y Varying  4.12  A d v e c t i o n Of A Slug Load Down the Main Arm  4.13  S l u g Inputs  4.14  D i s p e r s i o n o f A S l u g Load i n the Main Stem  4.15  I n f l u e n c e o f the T i d a l l y Averaged D i s p e r s i o n on P r e d i c t e d T i d a l l y Averaged C o n c e n t r a t i o n s  P r e d i c t e d Stages f o r June 16,  Stage and D i s c h a r g e —  Stage —  Main Arm  58 59 Canoe Pass.  P i t t Lake  f o r A n a l y t i c and  viii  60 61 62  - Main Stem . . . .  66  P r e d i c t e d D i s p e r s i o n S o l u t i o n s . . 69 70 Coefficient .73  ix  Figure  4.16  4.17  Page  Maximum Upstream E x c u r s i o n During In Main Arm - Main Stem  Flow  Maximum Upstream E x c u r s i o n During Flow  Reversal 75 Reversal  In t h e North Arm and P i t t R i v e r  76  5.1  I n i t i a l D i l u t i o n a t S t a t i o n Nos. 10, 22, 50 and 102  80  5.2  Dispersion of A Concentration  83  5.3  P r e d i c t e d C o n c e n t r a t i o n s i n One Channel Caused by E f f l u e n t D i s c h a r g e i n Another Channel  87  5.4  Predicted Concentrations i n t h e Main Stem  88  Spike  Due t o Two E f f l u e n t  Discharges  A.l  Elemental  Cross-Sectional S l i c e of A River o r Estuary.  A. 2  Dispersive Effects of V e r t i c a l V e l o c i t y Gradients.  B. l  The D r a i n a g e B a s i n o f t h e F r a s e r R i v e r  118  B.2  The F r a s e r R i v e r From Hope t o Vancouver  119  B.3  The F r a s e r R i v e r D e l t a  121  B.4  Network o f S t a t i o n s Used i n t h e N u m e r i c a l S o l u t i o n o f the Hydrodynamic Model  . . . 108  ....... . 110  .  124  B.5  T y p i c a l Channel C r o s s - S e c t i o n s  125  B.6  C r o s s - S e c t i o n a l Parameters o f Main Arm - Main Stem  126  B.7  C r o s s - S e c t i o n a l Parameters o f the North Arm, M i d d l e Arm and Canoe Pass  127  B.8  C r o s s - S e c t i o n a l Parameters of. P i t t R i v e r and P i t t Lake. . .  128  B.9  T y p i c a l Tides a t Steveston  130  B.10  L o c a l Low and High T i d e : E n v e l o p e s  131  B.ll  T i d e Gauging S t a t i o n s i n t h e F r a s e r R i v e r E s t u a r y  132  B.12  Mean Monthly Flows a t Hope (1912-1970 I n c l u s i v e )  133  B.13  Salinity Profiles  136  .  i n t h e Main Arm on F e b r u a r y 13-14, 1962 .  X  Figure  Page  C l  Concentration  Surfaces  f o r the A d v e c t i o n  C.2  Time Rate o f C o n c e n t r a t i o n A l o n g A Curve  o f A S l u g Load. . .  140  i n t h e (x,t) Plane)  142  C.3  D i s p e r s i o n o f A S l u g Load  144  C.4  Forward and Backward Time D i f f e r e n c e s  148  C.5  Upstream, C e n t r a l and Downstream Space D i f f e r e n c e s  150  C. 6  S t a b l e and U n s t a b l e  158  D. l  E x p l i c i t F i n i t e D i f f e r e n c e G r i d o f the Hydrodynamic Equations  163  D.2  Segments o f Thomann's T i d a l l y Averaged Model  168  D. 3  The M a t r i x A o f E q u a t i o n  173  E. l  Bends A l o n g t h e Main Arm--: Main Stem  177  E.2  T i d a l l y Varying V e l o c i t i e s  178  E.3  Assumed L i n e a r D i s t r i b u t i o n o f Secondary V e l o c i t i e s  181  E.4  Interaction of u  and E, i n A S t r a i g h t R i v e r  184  E. 5  I n t e r a c t i o n of u  and 5  E x p l i c i t Advective  Schemes  (D.8) f o r the F r a s e r R i v e r E s t u a r y .  .  v  z Around A Bend  186  z F. l  L a t e r a l V e l o c i t y P r o f i l e s a t S t a t i o n s Nos. 14 and 15, Main Arm  194  F.2  S e n s i t i v i t y of T i d a l l y Varying Concentrations t o the C o e f f i c i e n t o f L o n g i t u d i n a l D i s p e r s i o n  202  L I S T OF  a  :  SYMBOLS  H e i g h t o f e s t u a r y b e d a b o v e l e v e l datum„  A :  Cross-sectional  area,  A :  M a t r i x o f t h e e f f e c t s o f advection, d i s p e r s i o n and decay on t h e t i d a l l y a v e r a g e d c o n c e n t r a t i o n s (Thomann's S o l u t i o n ) .  A^ ^ ^: '  Average c r o s s - s e c t i o n a l Solution)  Aj  Cross-sectional  +  :  c  i-  Cross-sectionally  b  :  Width o f estuary  C :  Chezy's f r i c t i o n  c  :  a r e a b e t w e e n s e g m e n t s i a n d i+1  area a t p o i n t j A x a t time nAt (hydrodynamic averaged  coefficient  C o n c e n t r a t i o n i n s e g m e n t i (Thomann's S o l u t i o n ) M a t r i x o f segment c o n c e n t r a t i o n s  Cj  :  Cross-sectionally  e  :  Coefficient of turbulent  E  eguations)  concentration  C :  y  (Thomann's  :  (Thomann's  Solution)  averaged c o n c e n t r a t i o n a t p o i n t jAx a t time n A t . diffusion  Coefficient of longitudinal  dispersion  E^  :  C o e f f i c i e n t o f l o n g i t u d i n a l d i s p e r s i o n due t o t h e l a t e r a l g r a d i e n t s o f t h e s t e a d y v e l o c i t y component  velocity  E  :  C o e f f i c i e n t o f l o n g i t u d i n a l d i s p e r s i o n due t o t h e l a t e r a l g r a d i e n t s o f t h e o s c i l l a t o r y v e l o c i t y component  velocity  :  Effective coefficient of longitudinal dispersion c y c l e due t o l a t e r a l v e l o c i t y g r a d i e n t s  :  Coefficient gradients  of longitudinal  E^  :  Coefficient  o f pseudo d i s p e r s i o n  i  i+l '  fc  E c  E y  E  :  dispersion  over a t i d a l  due t o t h e v e r t i c a l  (numerical  dispersion)  " ^ ^ H Y averaged c o e f f i c i e n t o f l o n g i t u d i n a l d i s p e r s i o n s e g m e n t i a n d i+1 (Thomann's S o l u t i o n )  xi  velocity  between  X l l  Gravitational  acceleration  Height  o f w a t e r s u r f a c e above l e v e l datum  Height  o f w a t e r s u r f a c e above l e v e l datum a t p o i n t j A x a t t i m e  Von Kantian's  constant  Rate o f decay o f substance  i n segment i  (Thomann's  A v e r a g e l e n g t h o f s e g m e n t s i a n d i + l (Thomann's Manning's "n" (foot-second Freshwater  nAt  Solution)  Solution)  units)  discharge  R a d i u s o f c u r v a t u r e o f a bend Rate o f p r o d u c t i o n o f substance  due?to t h e i ' t h s o u r c e - s i n k  process  Time Cross-sectionally  averaged l o n g i t u d i n a l  Magnitude o f t h e steady  velocity  component o f t h e l o n g i t u d i n a l  velocity  A m p l i t u d e o f o s c i l l a t o r y component o f t h e l o n g i t u d i n a l Effective  longitudinal  Cross-sectionally Shear  v e l o c i t y over a t i d a l  averaged l o n g i t u d i n a l  velocity  cycle  v e l o c i t y a t p o i n t jAx a t time  velocity  Shear v e l o c i t y due t o Shear v e l o c i t y due t o U Effective  shear v e l o c i t y over a t i d a l  Volume o f segment i Cross-channel  (Thomann's  cycle  Solution)  s u r f a c e v e l o c i t y due t o s e c o n d a r y  Rate o f waste d i s c h a r g e  into  segment i  M a t r i x o f segment waste d i s c h a r g e s  (Thomann's  (Thomann's  Longitudinal distance along the estuary  flows Solution)  Solution)  nAt  xiii  y  :.  Vertical distance  i n the  z  :  Lateral distance  a  :  F a c t o r g i v e n by a = l n { — —  a  :  Factor r e l a t i n g dispersion c o e f f i c i e n t s v e l o c i t y , as i n E = a y U  i n the  estuary estuary  U*Y  11. C v  } i n d e p t h and  shear  #  Ti^al Y Y^_  v  e x c h a n g e c o e f f i c i e n t b e t w e e n s e g m e n t s i a n d i+1  (Thomann's  :  Standardized  concentrations"'' p r e d i c t e d by t h e t i d a l l y averaged  :  Standardized  concentrations^" p r e d i c t e d by t h e t i d a l l y v a r y i n g  At  :  Time i n c r e m e n t used  in finite  Ax  :  Space i n c r e m e n t used  e  :  Coefficient of lateral  :  C o e f f i c i e n t o f l a t e r a l d i s p e r s i o n due t o t h e s e c o n d a r y associated with U  flows  :  C o e f f i c i e n t o f l a t e r a l d i s p e r s i o n due t o t h e s e c o n d a r y associated with U  flows  :  Effective coefficient of lateral  Solution)  model model  difference solutions  i n finite difference solutions dispersion  f  e  £  c  n  :  £  ?  x  T x  d i s p e r s i o n over a t i d a l  cycle  R e l a t i v e d e p t h , g i v e n by n = y/y :  V o r t i c i t y g e n e r a t e d i n t h e l a t e r a l d i r e c t i o n due t o t h e v e r t i c a l velocity gradients  :  L o n g i t u d i n a l component o f t h e l a t e r a l  :  C i r c u l a t i o n around  vorticity  a c r o s s - s e c t i o n due t o t h e l o n g i t u d i n a l v o r t i c i t y  "''The p r e d i c t e d c o n c e n t r a t i o n s a r e s t a n d a r d i z e d b y d i v i d i n g t h e c o n c e n t r a t i o n d e t e r m i n e d f r o m t h e mass o f e f f l u e n t a n d t h e f r e s h w a t e r f l o w p e r tidal cycle.  A C K N O W L E D G E M E N T S  The  author i s very g r a t e f u l  t o D r . M. C. Q u i c k f o r h i s g u i d -  ance, encouragement and c r i t i c i s m s d u r i n g t h e development Mr.  of this  thesis.  D o n a l d 0. H o d g i n s , who d e v e l o p e d a n d p r o g r a m m e d t h e h y d r o d y n a m i c  model,  i sa l s o thanked.  S p e c i a l thanks t o go Mr. R i c h a r d Brun f o r d r a f t -  i n g t h em u l t i t u d e o f diagrams o f t h i s  t h e s i s , and t o M i s s Susan  Aizenman  f o r many p l e a s a n t t y p i n g a n d p r o o f r e a d i n g s o i r e e s . The the Westwater  l a t t e r p a r t o f t h i s s t u d y was f i n a n c i a l l y  supported by  Research Centre a t the U n i v e r s i t y o f B r i t i s h Columbia.  author expresses h i s a p p r e c i a t i o n f o r t h e i r assistance, w i t h s p e c i a l  The thanks  t o P r o f e s s o r I r v i n g K. P o x . M r . F . A. K o c h a n d M r . G. S. S h e e h a n o f t h e Westwater this  Research Centre a r e a l s o  thanked f o r t h e i r a s s i s t a n c e  during  study. Finally,  the author g r a t e f u l l y  acknowledges  t h e p a t i e n c e and  forbearance o f h i s w i f e , G l o r i a , and son, N i c h o l a s , d u r i n g t h e course o f this  study.  xiv  INTRODUCTION  Throughout h i s t o r y , c e n t e r s  o f urban, a g r i c u l t u r a l and i n d u s t r i a l  development have commonly been l o c a t e d a l o n g r i v e r s and e s t u a r i e s . p r i n c i p a l uses o f these s u r f a c e water r e s o u r c e s supply  and n a v i g a t i o n ;  the r i v e r provided  The  i n e a r l y t i m e s were water  a n a c c e s s i b l e source o f water  f o r domestic and a g r i c u l t u r a l needs and a r e l a t i v e l y cheap and easy means of bulk transport.  Modern uses o f s u r f a c e water r e s o u r c e s  t i o n and water supply d u s t r i a l purposes.  include  naviga-  and waste d i s p o s a l f o r domestic, a g r i c u l t u r a l and i n -  I n a d d i t i o n , r i v e r s and e s t u a r i e s p r o v i d e  habitat f o r  w i l d l i f e , b r e e d i n g and r e a r i n g a r e a s f o r f i s h and s h e l l f i s h and a r e a s f o r general  recreation. Associated  w i t h each use o f s u r f a c e water r e s o u r c e s  i s a set of  q u a n t i t y and q u a l i t y c o n s t r a i n t s t h a t determine whether t h e water i s s a t i s f a c t o r y f o r t h a t p a r t i c u l a r use. maximum a l l o w a b l e sent and  i n t h e water.  The q u a l i t y c o n s t r a i n t s consist., o f the  l e v e l s o f v a r i o u s d e l e t e r i o u s substances t h a t may be p r e Generally,  t h e c o n s t r a i n t s f o r each use a r e d i f f e r e n t ,  c o n f l i c t s may a r i s e when water i s to.be used f o r m u l t i p l e purposes.  A  common example i s t h e c o n f l i c t between t h e competing uses o f waste d i s p o s a l , w i l d l i f e h a b i t a t and r e c r e a t i o n .  I t i s recognized  t h a t t h e q u a n t i t y and  q u a l i t y c o n f l i c t s a r e r e l a t e d ; ^ however, t h i s t h e s i s o n l y c o n s i d e r s aspects,  quality  and i n p a r t i c u l a r , o n l y those q u a l i t y a s p e c t s t h a t a r e determined  One o b v i o u s way o f improving water q u a l i t y i s by low f l o w augment a t i o n , as has been i n v e s t i g a t e d by Worley et al.- [1965].  1  2  by t h e c o n c e n t r a t i o n o f a d i s s o l v e d d e l e t e r i o u s substance i n t h e water. To i n v e s t i g a t e a s i t u a t i o n o f e x i s t i n g o r p o t e n t i a l water q u a l i t y c o n f l i c t s i t i s usual to'develop  a " w a t e r - q u a l i t y " model, o r as i t s h a l l be  r e f e r r e d t o i n t h i s t h e s i s , a mass-transport  model.  Such a model  enables  the c o n c e n t r a t i o n o f d e l e t e r i o u s substance t o be p r e d i c t e d throughout the r i v e r o r e s t u a r y , and i s i n s t r u m e n t a l i n a s s e s s i n g the e f f e c t i v e n e s s o f p o s s i b l e c o n t r o l measures t o The sional  improve  water q u a l i t y .  s i m p l e s t type o f e s t u a r i n e m a s s - t r a n s p o r t  (1-D) and o n l y admits a v a r i a t i o n o f parameters and v a r i a b l e s i n the  longitudinal direction. i a b l e s are assigned  In a " t i d a l l y averaged" model, parameters and v a r -  t h e i r average v a l u e s over a t i d a l c y c l e , whereas i n a  " t i d a l l y v a r y i n g " model, they a r e a l l o w e d cycle.  model i s one-dimen-  t o v a r y throughout t h e t i d a l  The temporal r e s o l u t i o n o f t h e t i d a l l y v a r y i n g model i s much f i n e r  than i t s t i d a l l y averaged c o u n t e r p a r t , b u t c o n s i d e r a b l y more e f f o r t i s r e q u i r e d i n i t s development than f o r t h e l a t t e r .  Thus, i t seems r e l e v a n t t o  e n q u i r e a s t o (1) whether the d i f f e r e n c e s between the r e s u l t s o f b o t h models a r e s i g n i f i c a n t ;  and (2) whether the e x t r a e f f o r t i n v o l v e d i n deve-  l o p i n g and a p p l y i n g t h e t i d a l l y v a r y i n g model i s j u s t i f i e d by i t s f i n e r resolution. In t h i s t h e s i s t h e s i g n i f i c a n c e o f these d i f f e r e n c e s i s i n v e s t i gated  by a p p l y i n g both a t i d a l l y averaged and a t i d a l l y v a r y i n g mass t r a n s p o r t  model t o t h e F r a s e r R i v e r E s t u a r y , a t i d a l e s t u a r y Columbia, Canada. and  i n the Province  of B r i t i s h  The t i d a l l y averaged model was developed by Thomann [1963]  t h e t i d a l l y v a r y i n g model was developed from f i r s t p r i n c i p l e s . Both mass  t r a n s p o r t models were developed a s p a r t o f a l a r g e r i n t e r d i s c i p l i n a r y  study  3  by  t h e Westwater Research  Center  of the University of B r i t i s h  to  i n v e s t i g a t e t h e e f f e c t s o f p o s s i b l e p a t t e r n s o f f u t u r e development  on  the water q u a l i t y o f the Fraser River  Estuary.  This t h e s i s consists o f s i xchapters. and  The t i d a l l y v a r y i n g  t i d a l l y averaged forms o f t h e one-dimensional  tion  a r e d i s c u s s e d i n C h a p t e r 1.  Columbia  The e x p e c t e d  mass t r a n s p o r t equa^-  differencesLbetween  r e s u l t s o f both models and t h e a p p l i c a b i l i t y o f a one-dimensional to  the F r a s e r R i v e r Estuary a r e a l s o considered,.there.  literature  i s reviewed  one-dimensional varying  the model  I n Chapter 2 the  t o i n v e s t i g a t e t h e v a r i o u s ways o f s o l v i n g t h e  mass t r a n s p o r t e q u a t i o n .  form o f the equation,  In order  i t was n e c e s s a r y  to solve the t i d a l l y  t o develop  a hydrodynamic  model t o p r e d i c t t h e t i d a l v a r i a t i o n s i n t h e l o n g i t u d i n a l v e l o c i t y and c r o s s - s e c t i o n a l area along the estuary.  The h y d r o d y n a m i c m o d e l a n d t h e  t i d a l l y v a r y i n g a n d t i d a l l y a v e r a g e d mass t r a n s p o r t m o d e l s a p p l i e d t o t h e Fraser River Estuary  a r e d e s c r i b e d i n C h a p t e r 3.  The v e r i f i c a t i o n o f a l l  t h r e e models i s d e s c r i b e d i n Chapter 4 and t h e r e s u l t s o f a p p l y i n g mass t r a n s p o r t m o d e l s t o t h e E s t u a r y ter  5.  Finally,  a r e d e s c r i b e d and d i s c u s s e d i n Chap-  c o n c l u s i o n s a b o u t t h e d i f f e r e n c e s and a p p l i c a b i l i t y o f  b o t h models a r e g i v e n i n Chapter  6,  There a r e s i x appendices t o t h i s mass t r a n s p o r t e q u a t i o n A.  thesis.  The  one-dimensional  f o r unsteady non-uniform flow i s d e r i v e d i n Appendix  The a d v e c t i v e a n d d i s p e r s i v e t r a n s p o r t p r o c e s s e s  as a r e ' t h e  both  are discussed  assumptions i n the d e r i v a t i o n o f the equation.  s i s t s of a description of the Fraser River Estuary. the estuary, the freshwater  therein,  Appendix B con-  The v a r i o u s c h a n n e l s  of  f l o w s , t h e t i d e s and t h e s a l i n i t y i n t r u s i o n a r e  4  all  described.  When f i n i t e d i f f e r e n c e t e c h n i q u e s  mass t r a n s p o r t e q u a t i o n , , t h e r e and  stability.  a r e used t o s o l v e t h e  a r e problems w i t h numerical d i s p e r s i o n  I n Appendix C both o f these problems a r e seen t o occur  when a f i x e d g r i d i s u s e d t o s o l v e t h e e q u a t i o n , r a t h e r t h a n a g r i d t h e more f u n d a m e n t a l c h a r a c t e r i s t i c s o f i n f o r m a t i o n p r o p a g a t i o n . o f t h e f i n i t e d i f f e r e n c e schemes u s e d t o s o l v e t h e h y d r o d y n a m i c and  both  l y underestimate  the l a t e r a l mixing  f i e l d data, t h e p r e d i c t e d secondary i n the estuary.  Finally,  apparent-  Secondary  and on t h e b a s i s o f very  flows agree w e l l w i t h those ob-  An a p p r o x i m a t e method i s g i v e n t o  allow f o r the variable contributions of the effects of v e r t i c a l velocity gradients during the i n i t i a l complete.  In  i n A p p e n d i x F, e s t i m a t e s a r e made o f t h e  coefficients of longitudinal dispersion.  is  equations  dispersion  i n the Fraser River Estuary.  v e l o c i t i e s a r e e x p l a i n e d i n terms o f v o r t i c i t y ,  served  Details  f o r m s o f t h e mass t r a n s p o r t e q u a t i o n a r e g i v e n i n A p p e n d i x D.  A p p e n d i x E, i t i s s e e n t h a t e x i s t i n g e s t i m a t e s o f l a t e r a l  limited  along  and l a t e r a l  period before c r o s s - s e c t i o n a l mixing  CHAPTER 1  P R E L I M I N A R Y CONSIDERATIONS  The tidally  one-dimensional  v a r y i n g and t i d a l l y  problem o f determining  ability  i s s t a t e d and i t s  forms a r e b r i e f l y  discussed.  averaged  tidally  mass t r a n s p o r t m o d e l a r e d e s c r i b e d .  mass t r a n s p o r t m o d e l t o d e s c r i b e t h e m a s s  i n the Fraser  River Estuary  i s also considered.  one-dimensional  e q u a t i o n f o r mass t r a n s p o r t i n u n s t e a d y  u n i f o r m f l o w i n a n e s t u a r y i s o b t a i n e d by t a k i n g a mass b a l a n c e elemental cross-sectional slice  slice  o f the estuary.  b y t h e mass t r a n s p o r t p r o c e s s e s  these processes,  over  Mass i s t r a n s p o r t e d  the c o n c e n t r a t i o n o f the substance  i ti s a one-dimensional  are assigned t h e i r  d i m e n s i o n a l mass t r a n s p o r t e q u a t i o n  =  through  substance  w i t h i n the s l i c e .  The e q u a t i o n  i s derived i n  Appendix A and t h e assumptions i n i t s d e r i v a t i o n a r e d i s c u s s e d t h e r e .  3t  an  e q u a t i o n , t h e dependent v a r i a b l e and t h e p a r a m e t e r s  average c r o s s - s e c t i o n a l v a l u e s .  |£  non-  o f a d v e c t i o n and d i s p e r s i o n , and  t o g e t h e r w i t h any s o u r c e - s i n k r e a c t i o n s t h a t t h e  undergoes, determine As  The  THE ONE-DIMENSIONAL MASS TRANSPORT EQUATION The  the  The  forms o f t h e equation i s  d i f f e r e n c e s between t h e r e s u l t s o f a  o f a one-dimensional  transport processes  1.1  averaged  t h e parameters o f both  c o n s i d e r e d , and expected v a r y i n g and a t i d a l l y  mass t r a n s p o r t e q u a t i o n  -  u  | i  9x  +•  T h e one-?  i s g i v e n by  ^ { A A dx  5  E  f e  9x  +  ., 1=1  ? S.  1  (1.1  6  where c i s t h e mean c r o s s - s e c t i o n a l substance;  concentration of dissolved  u  value of the l o n g i t u d i n a l  i s t h e mean c r o s s - s e c t i o n a l velocity;  A i sthe cross-sectional  area;  E i s t h e mean c r o s s - s e c t i o n a l d i s p e r s i o n the l o n g i t u d i n a l d i r e c t i o n ;  coefficienti n  S,. i s t h e r a t e o f p r o d u c t i o n p e r u n i t v o l u m e o f w a t e r d u e to the i s o u r c e - s i n k p r o c e s s , i t b e i n g assumed t h a t there are n source-sink processes; t  x  n  i s the longitudinal  distance;  and t  i s t h e time.  F o r t h e s a k e o f b r e v i t y , t h e "mean" v a l u e o f a p a r a m e t e r o r v a r i a b l e  i s now  t a k e n t o r e f e r t o i t s "mean c r o s s - s e c t i o n a l " v a l u e . T h e h i g h f r e q u e n c y  turbu-  l e n t f l u c t u a t i o n s a r e assumed t o have been a v e r a g e d o u t o f u and c. The  mean v e l o c i t y u , t h e mean l o n g i t u d i n a l d i s p e r s i o n  E and t h e c r o s s - s e c t i o n a l E q u a t i o n (1.1).  area A w i l l  be r e f e r r e d t o as t h e parameters o f  These q u a n t i t i e s a r e parameters i n t h e sense t h a t they a r e  defined o u t s i d e t h e equation by t h e c r o s s - s e c t i o n a l of the p a r t i c u l a r estuary. Thus, E g u a t i o n  coefficient  (1.1)  These t h r e e p a r a m e t e r s c a n v a r y w i t h  i s s e e n t o be a s e c o n d - o r d e r  equation with variable c o e f f i c i e n t s . of the equation w i l l s o u r c e - s i n k terms  geometry and  linear partial  hydraulics b o t h x and t differential  The t h r e e t e r m s o n t h e r i g h t - h a n d  be r e f e r r e d t o as t h e a d v e c t i v e , t h e d i s p e r s i v e  respectively.  side  and t h e  7  Equation  (1.1) was d e r i v e d f o r t h e g e n e r a l c a s e o f u n s t e a d y  non-  u n i f o r m f l o w and i s t h e b a s i s o f a l l o n e - d i m e n s i o n a l mass t r a n s p o r t models of  r i v e r s and e s t u a r i e s .  I n an estuary, t h e r i s e and f a l l  temporal v a r i a t i o n s i n t h e parameters t h i s temporal v a r i a t i o n  i n the parameters  (1.1) i s s o l v e d t o d e t e r m i n e the t i d a l  cycle.  u , A a n d E.  of the tide  causes  I n a t i d a l l y v a r y i n g model,  i s taken i n t o account as Equation  t h e mean c o n c e n t r a t i o n a l o n g t h e e s t u a r y d u r i n g  In a tidally  averaged model, t h e parameters  are assigned  t h e i r a v e r a g e v a l u e s o v e r a t i d a l c y c l e , a n d t h e mean c o n c e n t r a t i o n a l o n g the estuary i s determined over p e r i o d s o f a t i d a l (1.1) o v e r a t i d a l  cycle.  Averaging  Equation  c y c l e does n o t a l t e r t h e form o f t h e e q u a t i o n , b u t merely  changes t h e i n t e r p r e t a t i o n o f the^dependent  v a r i a b l e and t h e parameters.  For  example, u becomes t h e t i d a l l y a v e r a g e d v e l o c i t y a n d i s d e t e r m i n e d b y t h e f r e s h w a t e r d i s c h a r g e t h r o u g h t h e t i d a l l y a v e r a g e d a r e a A. noted t h a t t h e c o n c e n t r a t i o n p r e d i c t e d by a t i d a l l y the " t i d a l l y averaged"  concentration —  i tshould be  averaged  model i s n o t  t h i s c a n o n l y be determined by a v e r -  aging the r e s u l t s o f a t i d a l l y v a r y i n g model over the t i d a l c y c l e . of  tidally  a v e r a g e d m o d e l s s u p p o s e d l y p r e d i c t t h e mean c o n c e n t r a t i o n a l o n g t h e  estuary a t times o f slack-water. at  1.2  The m a j o r i t y  The r e a s o n f o r t h i s i s t h e e a s e o f s a m p l i n g  t i m e s o f s l a c k w a t e r , and i s d i s c u s s e d f u r t h e r i n S e c t i o n  2.-1-:  DETERMINATION OF PARAMETERS The  f l o w f i e l d o f an e s t u a r y c o n s i s t s o f an unsteady  c o m p o n e n t due t o t h e t i d e  superimposed  oscillatory  on a s t e a d y component due t o f r e s h w a t e r  inflow.  Before a s o l u t i o n can be o b t a i n e d t o t h e t i d a l l y v a r y i n g form o f  Equation  (1.1),  i t i s necessary t o determine the t i d a l l y  induced  temporal  8  variation  i n t h e parameters  u , A a n d E.  m o d e l was d e v e l o p e d t o d e t e r m i n e  A one-dimensional  the temporal v a r i a t i o n  t h i s model, t h e e q u a t i o n s o f motion  i n u a n d A.  throughout the t i d a l  dynamic model i s d e s c r i b e d i n d e t a i l  i n C h a p t e r 3.  c y c l e was r e l a t e d  i s d i s c u s s e d i n Appendix In  a tidally  cycle.  The h y d r o -  The t e m p o r a l  variation  t o the temporal v a r i a t i o n  i n u, as  F.  averaged model, t h e parameters  aged over a t i d a l c y c l e .  In  and c o n t i n u i t y were a p p l i e d t o t h e  w a t e r mass o f t h e e s t u a r y a n d s o l v e d  in E during the t i d a l  hydrodynamic  This reduces the unsteady  u, A and E a r e a v e r -  tidal  flow f i e l d to a  s t e a d y f r e s h w a t e r f l o w f i e l d , and c o n s e q u e n t l y u c a n be d e t e r m i n e d from t h e equation o f c o n t i n u i t y alone  (theequation o f c o n t i n u i t y  freshwater discharge through t h e t i d a l l y averaged  dispersion coefficient  on t h e f l o o d t i d e ,  i sapplied to the  averaged a r e a A ) .  The  tidally  i n c l u d e s t h e e f f e c t s o f upstream a d v e c t i o n  a s " i s discussed i n Chapter  4.  W i t h a t i d a l l y v a r y i n g m o d e l , mean c o n c e n t r a t i o n s a r e d e t e r m i n e d throughout the t i d a l  c y c l e , whereas w i t h a t i d a l l y  a v e r a g e d m o d e l , mean  concentrations are determined over periods o f a t i d a l c y c l e . the  In effect,  t e m p o r a l r e s o l u t i o n o f t h e t i d a l l y v a r y i n g m o d e l i s much h i g h e r t h a n  t h a t o f a t i d a l l y averaged model.  Because o f t h e i r unsteady  estimation of the t i d a l l y varying parameters more e f f o r t parts.  than the estimation o f t h e i r  This i s apparent  u and A i n v o l v e s  steady t i d a l l y  nature, the significantly  averaged  counter-  f r o m t h e d i s c u s s i o n o f C h a p t e r s 3 a n d 4.  Thus t h e  g r e a t e r t e m p o r a l a c c u r a c y o f t h e t i d a l l y v a r y i n g model i s o f f s e t by t h e greater effort required  to estimate the t i d a l l y varying  parameters.  9  1.3  THE  E F F E C T S OF I n an  THE  T I D E ON  estuary,  MASS TRANSPORT PROCESSES  the t i d a l r i s e  causes temporal v a r i a t i o n s i n the oral variation of flow,  the  varies during  cycle.  steady  the t i d a l  discharge  e f f e c t of the water past  f a l l of  initial  and  thus the  t i m e i n c r e m e n t becomes i n c r e a s i n g l y s m a l l e r . i s steady,  the  s l u g s and  this gives rise  averaged concentration flow reversal w i l l  in  F i g u r e 1.2.  On  occur  the  to the  at the the  A tidally  variation aged model  t h e n r e d u c e d by  1.1.  o u t f a l l and  seaward f l o w  of  any  However, s i n c e t h e e f f l u e n t  be  dosed  As  be  the  tide continues  and  to flood,  p r e v i o u s l y dosed  dosed again,  as  is  slugs  illustrated  t i d e , u p s t r e a m s l u g s w i l l move down-  dosed y e t again,  as  i s a l s o shown i n F i g u r e  account f o r the  m u l t i p l e dosing,  the  1.1  e f f e c t of the  e f f e c t s of  whereas a t i d a l l y  aver-  depend on  t i d a l variation i n flow i s to i n -  concentration p r o f i l e along and  1.2).  The  the magnitude o f the  Another d i f f e r e n c e between the  the  estuary  concentration of these  the d i s p e r s i v e t r a n s p o r t process.  the r e s u l t s from a t i d a l l y v a r y i n g and  ing  The  s l u g o f water dosed i n  effluent outfall  d i l u t i o n and  " s p i k e s " i n t o the  apparent from Figures  will  the  cannot.  Essentially, troduce  for  flood tide.  reduce the  v a r y i n g mass t r a n s p o r t m o d e l c a n  in initial  1.1  s p a t i a l d i s t r i b u t i o n of c r o s s - s e c t i o n a l l y  f o l l o w i n g ebb  stream p a s t t h e o u t f a l l and 1.2.  temp-  same m a s s o f e f f l u e n t i s a d d e d t o e a c h o f t h e  shown i n F i g u r e  w i l l move u p s t r e a m p a s t  the  this  effluent also  i n Figure  estuary during  i s to continuously  surface  Because o f  d i l u t i o n of a discharged  This i s i l l u s t r a t e d  the e f f l u e n t o u t f a l l ,  discharge  the water  longitudinal flow.  o f e f f l u e n t i n t o an  flooding tide  and  tidally  (as i s  spikes i s  Thus, d i f f e r e n c e s between  a v e r a g e d mass t r a n s p o r t m o d e l  t i d a l l y varying dispersion  process.  r e s u l t s o f b o t h models i s t h a t a t i d a l l y  model c o r r e c t l y accounts f o r the  e f f e c t s o f upstream a d v e c t i o n on  vary-  the  10  Steady  effluent discharge Rising Tide  t = 0  t  ESTUARY  = At  SEA  Cl  f  t = 2At  t = 3 At  t = 4At  C2  C3  C4 C3  Cl  Cl  C2  Cl  C2  u  1  C4  Time varying  dilution  C 3 ^ T i m e average  dilution  Figure Effect of Variable I n i t i a l  1.1 D i l u t i o n on C o n c e n t r a t i o n  11  Steady effluent  discharge  i t =4 A t  Rising  C4 C 3  Tide SEA  Cl  C2  u= 0 f t=5At  C4 C 3 C5  t =6At  C|C3  Rising  Tide  Falling  Tide  Cl  C2  Cl  C2 1  W C 4 + C 5 + C6  Effect of multiple dosing  Time average concentration  m////////////////A F i g u r e 1.2 Effect of Multiple (Due  t o Flow Reversal)  Dosing  On C o n c e n t r a t i o n  ci  12  f l o o d t i d e , whereas a t i d a l l y averaged  model can o n l y s i m u l a t e t h i s upstream  t r a n s p o r t v i a t h e t i d a l l y averaged  d i s p e r s i o n process.  1.4  MODEL  ACCURACY OF A ONE-DIMENSIONAL  Having b r i e f l y d i s c u s s e d t h e t i d a l l y v a r y i n g and t i d a l l y  averaged  forms o f t h e mass t r a n s p o r t e q u a t i o n , t h e problem o f how w e l l a one-dimensiona l model d e s c r i b e s t h e mass t r a n s p o r t p r o c e s s e s i s now c o n s i d e r e d .  i n the Fraser River Estuary  When e f f l u e n t i s d i s c h a r g e d i n t o a r i v e r o r e s t u a r y , t h e  time r e q u i r e d f o r complete c r o s s - s e c t i o n a l m i x i n g t o occur depends on t h e c o e f f i c i e n t o f l a t e r a l d i s p e r s i o n , t h e w i d t h o f t h e r i v e r o r e s t u a r y and t h e p o s i t i o n of the e f f l u e n t o u t f a l l c r o s s - s e c t i o n a l mixing  [Ward, 1973].  In t h e i n i t i a l p e r i o d b e f o r e  i s complete t h e r e a r e s i g n i f i c a n t  t i o n gradients across the r i v e r o r estuary.  A one-dimensional  p o r t model p r e d i c t s t h e c r o s s - s e c t i o n a l l y averaged not "see" these l a t e r a l g r a d i e n t s . s e c t i o n a l mixing  lateral  concentra-  mass t r a n s -  c o n c e n t r a t i o n s and does  Thus, i n t h e i n i t i a l p e r i o d b e f o r e c r o s s -  i s complete, t h e p r e d i c t e d c o n c e n t r a t i o n s w i l l  the peak l a t e r a l v a l u e s i n t h e r i v e r o r e s t u a r y .  underestimate  Also, during t h i s  initial  p e r i o d t h e d i s p e r s i o n o f e f f l u e n t r e s u l t s i n a skewed d i s t r i b u t i o n o f concent r a t i o n a l o n g t h e r i v e r o r e s t u a r y r a t h e r than t h e G a u s s i a n d i c t e d by t h e t i d a l l y v a r y i n g mass t r a n s p o r t model t h e c r o s s - s e c t i o n a l mixing  d i s t r i b u t i o n pre-  (see Appendix A ) .  When  i s complete, l o n g i t u d i n a l g r a d i e n t s dominate t h e  t r a n s p o r t p r o c e s s e s , and a one-dimensional  model w i l l p r o v i d e a good d e s c r i p -  t i o n o f t h e a c t u a l c o n c e n t r a t i o n p r o f i l e along t h e r i v e r o r e s t u a r y .  13  The described how  F r a s e r R i v e r e s t u a r y c o n s i s t s of the  i n Appendix  effluent  The  at a diverging junction,  down o n e  reproduce  shown i n F i g u r e 1.3  the b u l k of the e f f l u e n t A  o f secondary  b a c k t h r o u g h t h e j u n c t i o n o n t h e ebb t i d e .  flows at the j u n c t i o n .  I f secondary  j u n c t i o n and  tidal the  and  i s advected  flows are present, bank  upstream  i s t h e n a d v e c t e d down t h e o p p o s i t e c h a n n e l ,  (On t h e ebb  t i d e , marked secondary  t h i s n a t u r e a r e commonly o b s e r v e d a t t h e M a i n A r m - N o r t h Arm F i g u r e B.3).  The  On  past the junction,  a r i s e where e f f l u e n t i s r e l e a s e d f r o m one  a s i s shown i n F i g u r e 1.3.  be  one-dimensional  a d d i t i o n a l c r o s s - s e c t i o n a l mixing occurs before the e f f l u e n t  from a d i v e r g i n g  l o a d may  t h i s behaviour a t the j u n c t i o n .  some e f f l u e n t i s c a r r i e d b a c k u p s t r e a m  t h e s i t u a t i o n may  influence  i s f u r t h e r c o m p l i c a t e d by t h e e f f e c t s o f  f l o w r e v e r s a l and t h e p r e s e n c e flood tide  channels  I f cross-sectional mixing i s  o f t h e c h a n n e l s a s shown i n F i g u r e 1.3.  mass t r a n s p o r t model c a n n o t situation  j u n c t i o n s of these v a r i o u s channels  i s advected through the estuary.  not complete advected  B.  seven p r i n c i p a l  flows of  j u n c t i o n shown i n  Thus, i n the p e r i o d b e f o r e c r o s s - s e c t i o n a l m i x i n g i s complete,  t h e movement o f e f f l u e n t t h r o u g h t h e v a r i o u s j u n c t i o n s o f t h e F r a s e r  River  E s t u a r y i s a complex two-dimensional p r o c e s s t h a t a l s o v a r i e s d u r i n g the tidal  cycle.  To r e p r o d u c e  these e f f e c t s ,  h a v e t o be m o d i f i e d a t t h e j u n c t i o n s a n d siderable f i e l d data. and  secondary The  I t i s noted  a one-dimensional model would s u c h m o d i f i c a t i o n may  t h a t the e f f e c t s of t i d a l  f l o w w i l l enhance t h e c r o s s - s e c t i o n a l m i x i n g a t ability  reversal  junctions.  along the v a r i o u s channels of the F r a s e r R i v e r  E s t u a r y e s s e n t i a l l y depends on the time r e q u i r e d occur.  flow  con-  o f a o n e - d i m e n s i o n a l mass t r a n s p o r t model t o d e s c r i b e  the concentration p r o f i l e  to  require  In the i n i t i a l  for cross-sectional  mixing  p e r i o d before c r o s s - s e c t i o n a l mixing i s complete,  Figure.1.3 Movement o f E f f l u e n t T h r o u g h A D i v e r g i n g  Junction  15  t h e d i s p e r s i o n p r o c e s s i s skewed r a t h e r than G a u s s i a n , t h e peak  lateral  c o n c e n t r a t i o n i s s i g n i f i c a n t l y g r e a t e r than t h e p r e d i c t e d c r o s s - s e c t i o n a l l y averaged  v a l u e and t h e movement o f e f f l u e n t through t h e j u n c t i o n s may n o t  be a c c o r d i n g t o the simple mass balance o f t h e one-dimensional the l a t t e r two e f f e c t s p r o b a b l y b e i n g t h e most important. f o r 80 p e r c e n t c r o s s - s e c t i o n a l mixing d i x E.  equation,  The time r e q u i r e d  t o o c c u r has been e s t i m a t e d i n Appen-  The t e c h n i q u e s o f F i s c h e r [1969a] and Ward [1972] gave an e s t i m a t e  o f 55 hours, whereas c a l c u l a t i o n s based  on v o r t i c i t y c o n s i d e r a t i o n s gave an  estimate o f f i v e hours, t h e b u l k o f the c r o s s - s e c t i o n a l m i x i n g b e i n g due t o the i n f l u e n c e o f secondary been c o n f i r m e d by f i e l d  flows.  I t i s noted t h a t t h i s l a s t v a l u e has n o t  experiments.  Because o f tlje h i g h l y  asymmetrical  n a t u r e o f t h e t i d e s o f t h e F r a s e r R i v e r E s t u a r y , a more r e a l i s t i c o f t h e time o f c r o s s - s e c t i o n a l m i x i n g i s p r o b a b l y 1-2 t i d a l c y c l e s . d i s c u s s e d i n Appendix E.  estimate  i n t h e lower reaches o f t h e e s t u a r y  A l l o f these a s p e c t s o f l a t e r a l mixing a r e  U n f o r t u n a t e l y , time and expense have p r e c l u d e d  u s i n g dye s t u d i e s t o measure t h e a c t u a l r a t e o f c r o s s - s e c t i o n a l m i x i n g i n the e s t u a r y . To sum up, i n t h i s study i t i s r e c o g n i z e d t h a t l a t e r a l  mixing  and t h e complex two and t h r e e - d i m e n s i o n a l f l o w c h a r a c t e r i s t i c s through t h e j u n c t i o n s can have a c o n s i d e r a b l e i n f l u e n c e on t h e c o n c e n t r a t i o n s a l o n g the v a r i o u s channels o f t h e E s t u a r y .  These a s p e c t s a r e considered', and t h e i r  i n f l u e n c e i s p a r t i a l l y a s s e s s e d , b u t t h e main t h r u s t o f t h e study has been t o o b t a i n a c c u r a t e s o l u t i o n s t o t h e t i d a l l y v a r y i n g and t i d a l l y forms o f t h e one-dimensional  averaged  mass t r a n s p o r t e q u a t i o n so t h a t the i n f l u e n c e  of the t i d e s on t h e p r e d i c t e d c o n c e n t r a t i o n s can be a s s e s s e d .  To some e x t e n t ,  16 Che  two-  and three-dimensional  effects can be estimated as further modifi-  cations of the one-dimensional r e s u l t s , as i s discussed further i n Chapter  5.  CHAPTER 2  LITERATURE REVIEW  The and  l i t e r a t u r e i s reviewed t o i n v e s t i g a t e the v a r i o u s s o l u t i o n s  a p p l i c a t i o n s o f one-dimensional  mass t r a n s p o r t models.  the mass t r a n s p o r t e q u a t i o n a r e c l a s s i f i e d tical and  s o l u t i o n s , numerical  stochastic solutions.  of s o l u t i o n  2.1  Solutions to  i n t o the c a t e g o r i e s o f analy-  s o l u t i o n s , p h y s i c a l and analogue model s o l u t i o n s The advantages and d i s a d v a n t a g e s  o f each c l a s s  are also discussed.  ANALYTICAL SOLUTIONS The  uniform  one-dimensional  mass t r a n s p o r t e q u a t i o n  f o r unsteady non-  f l o w i n a r i v e r o r e s t u a r y i s g i v e n by  |£ 3t  =  -up3x  +  I^-fAE^} + A 9x  d X  E S.,  where the terms a r e as d e f i n e d i n S e c t i o n 1.2. order l i n e a r  (2.1  . . l '  1=1  Equation  (2.1) i s a second  p a r t i a l d i f f e r e n t i a l e q u a t i o n w i t h v a r i a b l e c o e f f i c i e n t s (the  s o u r c e - s i n k terms a r e g e n e r a l l y linear,- and u, A and E a r e dependent on x and t ) .  As such,  no c o m p l e t e l y  general a n a l y t i c a l s o l u t i o n e x i s t s , but  s o l u t i o n s have been o b t a i n e d under a number o f s i m p l i f y i n g  assumptions.  B e f o r e r e v i e w i n g v a r i o u s s o l u t i o n s , the s o - c a l l e d " s l a c k w a t e r "  concentra-  t i o n s p r e d i c t e d by t h e t i d a l l y averaged models a r e d i s c u s s e d . The m a j o r i t y o f t h e a n a l y t i c a l s o l u t i o n s d i s c u s s e d here a r e f o r the s t e a d y - s t a t e response  ( d c / d t  =  0) o f E q u a t i o n  17  (2.1).  This  simplification  18  reduces the p a r t i a l d i f f e r e n t i a l e q u a t i o n t o an o r d i n a r y equation i n x alone. freshwater constant  discharge  In d e t e r m i n i n g  differential  t h i s steady state".response,  the  and e f f l u e n t i n p u t s a r e assumed t o remain steady  f o r a p e r i o d o f time equal to the r e s i d e n c e time of the  F u r t h e r , any  steady-state s o l u t i o n of Equation  aged s o l u t i o n .  (The e q u a t i o n  i s now  (2.1)  s o l v e the w i t h i n t i d e temporal f l u c t u a t i o n s i n c, u, A and t h e steady s t a t e s o l u t i o n t o E q u a t i o n  (2.1)  estuary.  is a tidally  independent o f time and  aver-  cannot r e -  E).  However,  does n o t r e p r e s e n t the  averaged c o n c e n t r a t i o n p r o f i l e a l o n g the r e a l e s t u a r y .  or  tidally  In f a c t , i t r e p r e -  sents the c o n c e n t r a t i o n p r o f i l e a l o n g a model e s t u a r y t h a t i s t i d e l e s s has a h i g h degree o f l o n g i t u d i n a l d i s p e r s i o n .  and  (The t i d a l l y averaged E i s  g e n e r a l l y much h i g h e r than the t i d a l l y v a r y i n g E, as i s d i s c u s s e d i n Section  3.3).  The  t i d a l l y averaged c o n c e n t r a t i o n p r o f i l e can o n l y be  from the r e s u l t s o f a  tidally  varying  obtained  mass t r a n s p o r t model.  In the m a j o r i t y o f t i d a l l y averaged mass t r a n s p o r t models, the steady  state s o l u t i o n to Equation  (2.1)  i s assumed to r e p r e s e n t the concen-  t r a t i o n p r o f i l e a l o n g the e s t u a r y a t times o f s l a c k w a t e r . the t i d a l l y averaged mass t r a n s p o r t p r o c e s s e s  In o t h e r words,  a r e used t o determine  the  c o n c e n t r a t i o n p r o f i l e a l o n g the e s t u a r y a t a p a r t i c u l a r phase o f the  tide.  The r e a s o n f o r working w i t h slackwater p r o f i l e s i s the r e l a t i v e ease of sampling  t r a c e r s i n t h e e s t u a r y a t these times  [ 0'Connor, 1960] . ;  pondence between the s t e a d y - s t a t e s o l u t i o n o f E q u a t i o n  (2.1)  and  water c o n c e n t r a t i o n p r o f i l e a l o n g the r e a l e s t u a r y i s because the  Any  corres-  the s l a c k tidal  e f f e c t s a r e s m a l l , and under these c o n d i t i o n s t h e s t e a d y - s t a t e s o l u t i o n i s an adequate r e p r e s e n t a t i o n o f the c o n c e n t r a t i o n p r o f i l e i n the r e a l  estuary  19  a t any phase o f the t i d e , o r because the steady to  state solution i s "forced"  conform t o the s l a c k w a t e r p r o f i l e by " a d j u s t i n g " t h e parameters o f  Eguation  (2.1).  developed  I t i s i n t e r e s t i n g t o note t h a t Preddy and Webber [1963]  a t i d a l l y averaged  model t h a t p r e d i c t s t i d a l l y averaged  t i o n s r a t h e r than s l a c k w a t e r v a l u e s .  concentra-  T h i s model i s d e s c r i b e d i n S e c t i o n  2.2.2. In a t i d a l l y v a r y i n g s i t u a t i o n , q u a s i - s t e a d y - s t a t e c o n d i t i o n s a r e s a i d t o be a c h i e v e d when the v a r i a b l e o f i n t e r e s t undergoes a c y c l i c a l cycle.  r e p e t i t i o n o f the same v a l u e s from t i d e c y c l e t o t i d e  Thus, i n a t i d a l l y v a r y i n g model one  response,  ( f o r example, c)  r a t h e r than a s t e a d y - s t a t e response  speaks of a q u a s i - s t e a d y - s t a t e as w i t h a t i d a l l y  averaged  model. E s s e n t i a l l y , the a n a l y t i c a l s o l u t i o n s o f E q u a t i o n  (2.1)  entail  s i m p l i f i c a t i o n s i n which v a r i o u s terms o f the e q u a t i o n are i g n o r e d ( f o r example, the d i s p e r s i o n term) and  the parameters o f the u, A and E a r e r e p r e -  sented as simple f u n c t i o n s o f x and p o s s i b l y t . as boundary c o n d i t i o n s , and  E f f l u e n t inputs are treated  s o l u t i o n s have been o b t a i n e d f o r b o t h p o i n t and  d i s t r i b u t e d e f f l u e n t i n p u t s [O'Connor, 1965] . Steady-state s o l u t i o n s to Equation steady e f f l u e n t d i s c h a r g e i n t o (2) steady non-uniform 1965,  1967]; and  (2.1) have been o b t a i n e d f o r  (1) steady u n i f o r m f l o w s  [O'Connor, 1960,  f l o w s where the a r e a i s a simple f u n c t i o n o f x  (3) steady non-uniform  v a r i e s e x p o n e n t i a l l y w i t h x due  1962];  [O'Connor,  f l o w s where the f r e s h w a t e r d i s c h a r g e  t o l a n d r u n - o f f [O'Connor, 1967].  s o l u t i o n s have been o b t a i n e d f o r s t e a d y e f f l u e n t d i s c h a r g e i n t o f r e s h w a t e r f l o w s d u r i n g the r e c e s s i o n limb o f the hydrograph  Transient  (1) non-steady  (which was  approxi-  20  mated as n e g a t i v e e x p o n e n t i a l i n time) [O'Connor, 1967]; and f l o w s where the d i u r n a l p h o t o s y n t h e t i c p r o d u c t i o n o f DO account  ( i t was  approximated  1970].  To o b t a i n these t r a n s i e n t s o l u t i o n s , i t was  (2)  steady  i s taken  into  as a h a l f s i n e wave) [O'Connor and D i  Toro,  necessary t o ignore  dispersion. Kent [1960] used t h e method o f s e p a r a t i o n of v a r i a b l e s t o t a i n the g e n e r a l s o l u t i o n t o E q u a t i o n  ob-  (2.1) f o r a s l u g i n p u t i n t o  steady  u n i f o r m f l o w s , and the p a r t i c u l a r s o l u t i o n s f o r a s l u g i n p u t i n t o  steady  non-uniform  flows  (u, A and E were assumed t o be l i n e a r i n x ) .  D i Toro  and O'Connor [1968] o b t a i n e d the t r a n s i e n t s o l u t i o n f o r steady  effluent  i n p u t i n t o unsteady  non-uniform  f l o w s where the v a r i a t i o n i n c r o s s - s e c t i o n a l  area c o u l d be s e p a r a t e d i n t o independent  f u n c t i o n s o f x and t  (dispersion  was  ignored). Li  [1962] used the method o f c h a r a c t e r i s t i c s t o o b t a i n a  f o r non-steady e f f l u e n t d i s c h a r g e form flow-, ( d i s p e r s i o n was  solution  ( s i n u s o i d a l v a r i a t i o n ) i n t o steady  ignored) .  uni-  He l a t e r used the method o f p e r t u r b a -  t i o n s t o o b t a i n a s o l u t i o n f o r t h e same case w i t h d i s p e r s i o n i n c l u d e d [ L i , 1972] . H o l l e y [T969b] t r a n s f o r m e d i n t o t h e elementary  the mass t r a n s p o r t e q u a t i o n f o r  d i f f u s i o n equation.  (By t r a v e l l i n g w i t h the water  mass, the o n l y t r a n s p o r t p r o c e s s an o b s e r v e r S e c t i o n A.6).  "sees" i s d i s p e r s i o n —  see  He took the s t a n d a r d s o l u t i o n f o r a s l u g i n p u t and used  c o n v o l u t i o n method t o o b t a i n the s o l u t i o n f o r the c o n t i n u o u s s a r i l y steady) e f f l u e n t d i s c h a r g e i n t o unsteady used  BOD  uniform flow.  (but not  the neces-  Bennet [1971]  the c o n v o l u t i o n method t o o b t a i n the s o l u t i o n f o r t h e complete BOD-DO  21  system f o r the same e f f l u e n t d i s c h a r g e and f l o w c o n d i t i o n s as H o l l e y .  2.2  NUMERICAL SOLUTIONS 2.2.1  F i n i t e Difference Solutions.  F i n i t e d i f f e r e n c e methods f o r  the s o l u t i o n o f p a r t i a l d i f f e r e n t i a l e g u a t i o n s can be c l a s s i f i e d  fixed mesh methods i n which t h e s o l u t i o n i s o b t a i n e d a t f i x e d p o i n t s i n a r e c t a n g u l a r mesh o f time and d i s t a n c e ; (2)  i n t o (1)  predetermined  characteristic  methods,  i n which the s o l u t i o n i s o b t a i n e d a t mesh p o i n t s along the c h a r a c t e r i s t i c c u r v e ( s ) i n t h e t i m e - d i s t a n c e p l a n e t s ) , t h e p o s i t i o n o f t h e mesh p o i n t s b e i n g determined  a s t h e s o l u t i o n p r o g r e s s e s ; and (3)  combined methods, i n  which t h e s o l u t i o n i s f o l l o w e d a l o n g the c h a r a c t e r i s t i c c u r v e ( s ) and then e x t r a p o l a t e d back onto a f i x e d mesh o f u n i f o r m l y spaced p o i n t s  [Amein, 1966].  In c h a r a c t e r i s t i c methods, t h e mesh p o i n t s a r e g e n e r a l l y non-uniformly i n time o r d i s t a n c e , and consequently untidy.  spaced  t h e book-keeping o f r e s u l t s i s somewhat  Combined methods s i m p l i f y t h i s bookkeeping by e x t r a p o l a t i n g the  r e s u l t s back onto a f i x e d u n i f o r m l y spaced mesh. F i n i t e differenceemethods C.2.  The problems o f  seen t h a t  implicit  s t a b l e , whereas  stability  and  a r e d i s c u s s e d i n some d e t a i l i n S e c t i o n  convergence a r e c o n s i d e r e d , and i t i s  f i n i t e d i f f e r e n c e schemes a r e g e n e r a l l y u n c o n d i t i o n a l l y  explicit  s t a b i l i t y requirements  schemes a r e a t most c o n d i t i o n a l l y s t a b l e .  The  o f e x p l i c i t schemes a r e d i s c u s s e d i n d e t a i l i n  S e c t i o n C.3, and a r e seen t o impose l i m i t s on t h e r e l a t i v e s i z e o f Ax and At, the g r i d  spacing. G e n e r a l l y , f i x e d mesh f i n i t e d i f f e r e n c e schemes do not s i m u l a t e  the a d v e c t i v e t r a n s p o r t p r o c e s s c o r r e c t l y , and r e s u l t i n an a d d i t i o n a l  dis-  p e r s i v e p r o c e s s b e i n g superimposed on the a c t u a l a d v e c t i v e and d i s p e r s i v e  22  processes occurring  dispersion  i n the r i v e r t o r e s t u a r y .  i n d e t a i l i n S e c t i o n : c.2 and  i s discussed  nated by u s i n g  This s o - c a l l e d  numerical  i s seen t o be e l i m i -  c h a r a c t e r i s t i c f i n i t e d i f f e r e n c e methods.  Harleman et al.  [1968] d e v e l o p e d a o n e - d i m e n s i o n a l ,  tidally  v a r y i n g mass t r a n s p o r t model o f the t i d a l p o r t i o n o f the Potomac R i v e r . A f i x e d mesh, i m p l i c i t scheme w i t h c e n t r a l d i f f e r e n c e s was the mass t r a n s p o r t  equation.  ences are d e s c r i b e d d i s c h a r g e and  i n S e c t i o n C.2).  t i d a l records,  1954], m o d i f i e d  f o r the  t o determine the  and  t y p e s o f space and  the  Taylor d i s p e r s i o n equation  the r e s u l t s o f dye  However, Prych and C h i d l e y  The  [Taylor, was  finite  s t u d i e s i n the e s t u a r y  [1969] showed t h a t the  a p p r o x i m a t e l y 30  greater  was  T a y l o r d i s p e r s i o n , and  same magnitude as the a c t u a l d i s p e r s i o n o c c u r r i n g  i e n t s and  Taylor equation neglects  r e s u l t s and  tous.  the  [1966b] has  e f f e c t s of t r a n s v e r s e  The  f i n i t e d i f f e r e n c e s o l u t i o n was  times the (The  v e l o c i t y grad-  Dobbins  system.  a hypothetical  averaged s o l u t i o n .  and  only  and the  fortui-  s i g n i f i c a n c e of n u m e r i c a l d i s p e r s i o n .  [1968] i n v e s t i g a t e d f i x e d mesh f i n i t e  They a p p l i e d a t i d a l l y v a r y i n g  estuary  agreement between  apparently  ence s o l u t i o n s t o the o n e - d i m e n s i o n a l mass t r a n s p o r t - DO  reason-  o f the order o f  i n the e s t u a r y .  demonstrated).  T h i s example i l l u s t r a t e s the B e l l a and  BOD  differ-  g r o s s l y u n d e r e s t i m a t e s the d i s p e r s i o n c o e f f i c i e n t f o r streams  e s t u a r i e s , as F i s c h e r dye  the  used  numerical  d i s p e r s i o n i n t h e i r f i n i t e d i f f e r e n c e scheme was  modified  differ-  T i d a l v e l o c i t i e s were c a l c u l a t e d from  longitudinal dispersion coefficients.  t h a n the m o d i f i e d  solve  time  e f f e c t s of v e r t i c a l v e l o c i t y gradients,  ences?' s o l u t i o n s i m u l a t e d ably w e l l .  (The v a r i o u s  used to  differ-  equation d e s c r i b i n g  the  f i n i t e d i f f e r e n c e model t o  compared the r e s u l t s w i t h an a n a l y t i c a l t i d a l l y  23  Dornhelm and Woolhiser dimensional, stances.  [1968] o b t a i n e d  a s o l u t i o n to the  one-  t i d a l l y v a r y i n g mass t r a n s p o r t model f o r c o n s e r v a t i v e  sub-  They used a f i x e d mesh i m p l i c i t scheme w i t h c e n t r a l d i f f e r e n c e s .  A hydrodynamic model, s o l v e d by the  same f i n i t e d i f f e r e n c e scheme,  used t o determine the t i d a l l y v a r y i n g parameters. was  The  was  hydrodynamic model  v e r i f i e d a g a i n s t a s t e a d y - s t a t e a n a l y t i c a l s o l u t i o n , but i t e x h i b i t e d  i n s t a b i l i t i e s when a p p l i e d t o the Delaware e s t u a r y . s t a b i l i t y analysis, their implicit ever, the n o n - l i n e a r nature  scheme was  (According t o a  unconditionally stable.  of the hydrodynamic e q u a t i o n s  may  v e r i f i e d f o r steady u n i f o r m  flow and was  How-  r e q u i r e more  s t r i n g e n t s t a b i l i t y c o n d i t i o n s , as i s d i s c u s s e d i n S e c t i o n D . l ) . t r a n s p o r t model was  linear  The  mass  applied i n  t i d a l l y v a r y i n g form t o a h y p o t h e t i c a l e s t u a r y . To overcome the problems of n u m e r i c a l al.  d i s p e r s i o n , Gardiner  [1964] used the method o f c h a r a c t e r i s t i c s t o s o l v e the  mass t r a n s p o r t e q u a t i o n saturated with o i l .  two-dimensional  d e s c r i b i n g the movement o f a s o l v e n t through sand  A combined f i n i t e d i f f e r e n c e method was  used, w i t h  mesh of u n i f o r m l y d i s t r i b u t e d moving p o i n t s superimposed on a f i x e d dimensional are assigned  space mesh.  A t the b e g i n n i n g  the c o n c e n t r a t i o n o f the n e a r e s t  (A.9).  A f t e r advection,  f i x e d g r i d p o i n t , and  analagous t o E q u a t i o n  (A.8)  c o n c e n t r a t i o n over t h e f i x e d g r i d p o i n t s . repeated  f o r the next time s t e p .  two-  to  explicit  i s t h e n used t o d i s p e r s e The  then  analagous t o  the moving p o i n t s a r e a s s i g n e d  c l o s e s t g r i d p o i n t t o determine the c o n c e n t r a t i o n t h e r e . An o f the equation  a  o f a time s t e p , the moving p o i n t s  v e c t e d a l o n g the c h a r a c t e r i s t i c s a c c o r d i n g t o the e q u a t i o n Equation  et  complete p r o c e s s  the form the  i s then  P i n d e r and Cooper [1970] used the same  ad-  24  technique  t o c a l c u l a t e t h e t r a n s i e n t p o s i t i o n s o f t h e s a l t water f r o n t  in a coastal aguifier. D i Toro [1969] r e c o g n i z e d duced t h e mass t r a n s p o r t e q u a t i o n d i f f e r e n t i a l equations. agous t o E q u a t i o n  t h a t - t h e method o f c h a r a c t e r i s t i c s r e -  f o r the BOD-DO system i n t o two o r d i n a r y  (He i g n o r e d d i s p e r s i o n , and so the e q u a t i o n  anal-  (A.8) was a " t r u e " o r d i n a r y d i f f e r e n t i a l e q u a t i o n ) .  noted t h a t t h e n u m e r i c a l  He  s o l u t i o n o f ordinary d i f f e r e n t i a l equations i s  more e x a c t than f o r p a r t i a l d i f f e r e n t i a l e q u a t i o n s ,  and t h a t t h e a c c u r a c y  can be c o n t r o l l e d by p r e d i c t o r - c o r r e c t o r methods.  2.2.2  "Box Model" S o l u t i o n s .  Another type o f mass t r a n s p o r t  model i s t h e s o - c a l l e d "box model" i n which t h e e s t u a r y i s d i v i d e d i n t o finite  segments o r boxes.  Each segment i s assumed c o m p l e t e l y  mixed and  the c o n c e n t r a t i o n o f substance i n a segment i s determined by t h e d i s c h a r g e o f substance i n t o t h e segment, t h e a d v e c t i v e and m i x i n g p r o c e s s e s between a d j a c e n t undergoes.  segments, and any s o u r c e - s i n k  Callaway  e f f e c t s t h a t the substance  [1971] noted t h e box models a r e c o n c e p t u a l l y  s i m i l a r t o f i n i t e d i f f e r e n c e models, as i n e f f e c t , t h e l a t t e r estuary  occuring  i n t o w e l l mixed boxes c e n t r e d around the g r i d p o i n t s .  very  segment t h e Another  s i m i l a r i t y i s t h a t t h e box model r e p r e s e n t a t i o n o f the e s t u a r y r e s u l t s i n a s e t o f d i f f e r e n c e equations concentration i n neighboring Thomann [1963],  r e l a t i n g t h e c o n c e n t r a t i o n i n any box t o t h e boxes  ( s i m i l a r t o i m p l i c i t d i f f e r e n c e schemes).  [1965], developed a t i d a l l y averaged box model  which was a p p l i e d t o t h e Delaware e s t u a r y .  The e s t u a r y was segmented i n t o  30 boxes whose l e n g t h v a r i e d from two t o f o u r m i l e s .  A l l h y d r a u l i c parameters  25  were assumed steady,  but t h e c o n c e n t r a t i o n w i t h i n each box  was  allowed  v a r y w i t h time so t h a t t h e e f f e c t o f time v a r y i n g i n p u t s c o u l d be The  concentration  i n each box  For' s t e a d y - s t a t e  d e r i v a t i v e s a r e s e t e q u a l t o z e r o and  ence scheme. and D.3.  i n p u t s , the  t h e system o f e q u a t i o n s  s i m i l a r to those  temporal  reduces t o  o f an i m p l i c i t f i n i t e  Thomann's model i s d e s c r i b e d i n more d e t a i l  Pence et ai.  studied.  i s d e t e r m i n e d from a s e t o f s i m u l t a n e o u s  d i f f e r e n c e - d i f f e r e n t i a l equations.  a set of d i f f e r e n c e equations  to  differ-  i n Sections  3.3  [1968] extended the model t o a c c o u n t f o r time v a r y i n g  f r e s h water i n f l o w s ( t i d a l parameters are s t i l l  averaged o v e r a t i d a l  cycle).  L i k e most t i d a l l y averaged models, Thomann's model i s a " s l a c k water" model with  c o n c e n t r a t i o n s b e i n g determined a t s u c c e e d i n g  s l a c k waters.  Preddy and Webber [1963] d e v e l o p e d a t i d a l l y averaged box o f t h e Thames e s t u a r y t o p r e d i c t the t r u e t i d a l l y averaged v a l u e concentration. displacement  In t h e i r model, the boxes were two  was  s i x m i l e s t o e i t h e r s i d e o f a box.  averaged c o n c e n t r a t i o n w i t h i n a box t h r e e boxes upstream and  downstream o f i t . T h i s model i s one  F i n i t e Element S o l u t i o n s .  and  Nalluswami et al.  segmented and tions.  [1972].  tidal  tidally  o f the few  the that  r a t h e r than s l a c k water v a l u e s . F i n i t e element methods have been  used t o o b t a i n s o l u t i o n s t o the o n e - d i m e n s i o n a l mass t r a n s p o r t Examples i n c l u d e P r i c e et al.  the  DO  depends on the c o n c e n t r a t i o n s w i t h i n  d e t e r m i n e t i d a l l y averaged c o n c e n t r a t i o n s 2.2.3  o f the  m i l e s l o n g and Thus, the  model  equation.  [1968], Guymon [1970], P i n d e r and F r i e n d  [1972]  In f i n i t e element s o l u t i o n s , the e s t u a r y i s  the c o n c e n t r a t i o n p r o f i l e  i s approximated as a s e r i e s of  func-  26  In. some respects the method i s s i m i l a r to i m p l i c i t f i n i t e d i f f e r ence methods and box models;  i n the l a t t e r two methods the estuary i s also  segmented, but the concentration p r o f i l e i s approximated as a point value rather than a function i n each segment.  Further, the solution of a l l three  methods i s given by a set of simultaneous equations governed by a square banded matrix of bandwidth three (see Section D.3 for Thomann's version of this matrix).  Price et al. [1968] compared f i n i t e element, f i n i t e  differ-  ence and method of characteristics solutions for the case of a constant discharge into steady uniform flow.  The f i n i t e element method was  found to  be more accurate than the f i n i t e difference approximations, but this i s only to be expected as this technique can follow v a r i a t i o n within a segment, whereas f i n i t e difference solutions cannot.  The f i n i t e element method was  faster and more accurate than the method of c h a r a c t e r i s t i c s of Gardiner et at. be  [1964]  adequate,  However, the accuracy of the l a t t e r technique was  found to  and any inaccuracies probably arise from the procedure of  extrapolating the moving points back onto the fixed grid points.  Apparently,  there are no problems with s t a b i l i t y and numerical dispersion i n f i n i t e element methods [Price et ai.,  1968].  In closing this section, i t i s mentioned  that  two-dimensional  f i n i t e difference f i n i t e element and box models have been developed.  Fischer  [1970], Oster et al. [1970] and Leendertse [1971a] are among those who have obtained f i n i t e difference solutions for two-dimensional mass transport i n the horizontal plane, whereas the box model of Pritchard [1969] was to describe two-dimensional mass transport i n the v e r t i c a l plane.  developed  27  2.3  PHYSICAL AND  2.3.1  ANALOGUE MODEL SOLUTIONS  P h y s i c a l Model S o l u t i o n s .  P h y s i c a l models of e s t u a r i e s were o r i g i -  n a l l y developed t o i n v e s t i g a t e sediment e r o s i o n and ways.  The  use  i n t i d a l water-  o f t h e s e models t o i n v e s t i g a t e mass t r a n s p o r t p r o c e s s e s i n  prototype estuary  was  a n a t u r a l development, and  [1962], [1965]; D i a c h i s h i n Tchobanoglous [1968]. process, a quantity  [1963]; O ' C o n n e l l and  o f dye  i s introduced  Walter  [1963] and  Lager  and  transport  a t the p o i n t b e i n g i n v e s t i g a t e d  and  i s r e c o r d e d over s u c c e e d i n g  cycles. P h y s i c a l m o d e l l i n g i n v o l v e s t h e scaled-down r e p r o d u c t i o n  more important p r o c e s s e s t h a t a f f e c t the parameter b e i n g m o d e l l e d . transport  i n e s t u a r i e s , these p r o c e s s e s a r e a d v e c t i o n ,  sink e f f e c t s . advection cess  the  examples i n c l u d e O'Connor  In t h e s e model i n v e s t i g a t i o n s of the mass  i t s d i s t r i b u t i o n throughout the model e s t u a r y tidal  deposition  and  P h y s i c a l models can  the  F o r mass  d i s p e r s i o n and  source-  s a t i s f a c t o r i l y reproduce o n e - d i m e n s i o n a l  the two-dimensional d e n s i t y dependent s a l i n i t y  [Simmons, 1960;  of  Harleman, 1965].  However, t h e use  i n t r u s i o n pro-  of d i s t o r t e d cross- .  s e c t i o n a l space s c a l e s i n the model r e s u l t s i n d i s t o r t e d c r o s s - s e c t i o n a l velocity distributions.  From the d i s c u s s i o n of S e c t i o n A.3,  t h a t t h i s w i l l r e s u l t i n an i n c o r r e c t r e p r o d u c t i o n T h i s has and  been d i s c u s s e d  Holley  [1971].  by Harleman  Even i n p h y s i c a l models w i t h u n d i s t o r t e d  case, the r e p r o d u c t i o n  n o t be  Holley,  [1968], and  Fischer  cross-sectional  reproduced c o r r e c t l y .  In t h i s  o f the d i s p e r s i o n p r o c e s s depends on the r a t i o o f  time o f c r o s s - s e c t i o n a l m i x i n g t o the t i d a l p e r i o d [ F i s c h e r and  of-the! d i s p e r s i o n p r o c e s s .  [1965], Harlemen-et al.  space s c a l e s , the d i s p e r s i o n p r o c e s s may  i t i s apparent  1971].  i n the p r o t o t y p e  the  estuary  28  No attempt i s made t o reproduce any s o u r c e - s i n k r e a c t i o n s i n model s t u d i e s , a " c o n s e r v a t i v e " dye b e i n g used f o r the t e s t s .  However,  the dye does adsorb onto s u r f a c e s , and t h i s must be a l l o w e d f o r i n i n t e r preting test results.  O ' C o n n e l l and W a l t e r  a c c o u n t f o r a f i r s t - o r d e r decay 2.3.2  Analogue  [1963] d e v e l o p e d a method t o  reaction.  Model S o l u t i o n s .  E l e c t r i c a l analogue models  have been used t o o b t a i n s o l u t i o n s t o the mass t r a n s p o r t e q u a t i o n 1963  and Leeds, 1967]  and the hydrodynamic  equations  [Rennerfelt,  [Harder, 1971].  The  e s t u a r y i s d i v i d e d i n t o segments, and the space d e r i v a t i v e s o f the mass t r a n s p o r t e q u a t i o n a r e approximated ments.  as f i n i t e d i f f e r e n c e s over those s e g -  T h i s reduces E q u a t i o n (2.1) t o a s e t o f simultaneous  d i f f e r e n t i a l equations o f Thomann [1963].  difference-  (the time d e r i v a t i v e i s c o n t i n u o u s ) s i m i l a r t o those  The e l e c t r i c a l analogue o f t h i s system o f e q u a t i o n s  can t h e n be c o n s t r u c t e d i n the form o f a s o - c a l l e d l a d d e r network and Bybee,  [Leeds  1967]. Analogue models can determine the steady- s t a t e and  transient  solutions f o r e i t h e r constant or s i n u s o i d a l l y varying e f f l u e n t discharge i n t o s t e a d y non-uniform  flows  [Leeds, 1967  and Leeds and Bybee, 1967],  As  t h e f l o w i s assumed steady, these models a r e t i d a l l y averaged, b u t i t may be p o s s i b l e t o d e v e l o p an analogue Depending  f o r s i n u s o i d a l l y varying flows.  on the f i n i t e d i f f e r e n c e a p p r o x i m a t i o n s used f o r the  s p a t i a l d e r i v a t i v e s , these models may as was Bybee  r e c o g n i z e d by B e l l a  [1968].  a l s o s u f f e r from n u m e r i c a l d i s p e r s i o n ,  F o r example, the analogue  [1967] used c e n t r a l d i f f e r e n c e s t o approximate  o f Equation (2.1). this w i l l  the space  From the d i s c u s s i o n o f S e c t i o n C.2,  r e s u l t i n numerical d i s p e r s i o n .  o f Leeds  and  derivatives  i t i s apparent  that  29  2.4  STOCHASTIC SOLUTIONS Before reviewing s t o c h a s t i c  t i o n , i t i s necessary t o d e f i n e  s e v e r a l terms.  t h a t undergoes changes w i t h r e s p e c t flow. t o be and  s o l u t i o n s o f t h e mass t r a n s p o r t A  process i s any phenomena  t o time, an example b e i n g d a i l y r i v e r  I f t h e chance o f o c c u r r e n c e i s taken i n t o account, a p r o c e s s i s s a i d  stochastic  or  probabalistic.  A p r o b a b i l i s t i c process i s  t h e v a r i a b l e s a r e c o n s i d e r e d pure-random.  time-independent  A s t o c h a s t i c process i s  dependent and t h e v a r i a b l e s may be pure-random o r non-pure-random. pure-random, t h e p r o c e s s i s composed o f a d e t e r m i n i s t i c component.  mean c o n c e n t r a t i o n  mass t r a n s p o r t  [Chow, 1964].  p r o f i l e and t h e v a r i a t i o n around the mean.  mine t h e c o n c e n t r a t i o n ,  This  processes that  e f f l u e n t concentrations,  e t c . during  v a r i a b l e s t o determine t h e mean c o n c e n t r a t i o n  stochastic  profile.  [1963] assumed t i d a l m i x i n g t o be a pure-random p r o -  c e s s t h a t c a n be c h a r a c t e r i z e d  i n purely  deter-  the period of  D e t e r m i n i s t i c models u s e t h e mean v a l u e o f each o f t h e s e  random walk f o r m u l a t i o n  varia-  namely the v a r i a t i o n s t h a t o c c u r i n t h e f r e s h w a t e r  e f f l u e n t discharges,  Diaschishin  other-  models attempt t o p r e d i c t b o t h t h e  t i o n i s due t o the s t o c h a s t i c nature o f C . t h V u n d e r l y i n g  posal  stationary,  s i m p l i f i c a t i o n they  a r e t r e a t e d as s t a t i o n a r y , p r o b a b a l i s t i c , and d e t e r m i n i s t i c Stochastic  remains  Non-stationary stochastic processes are very  c o m p l i c a t e d m a t h e m a t i c a l l y , and i n o r d e r o f i n c r e a s i n g  analysis.  I f non-  I f the p r o b a b i l i t y d i s t r i b u t i o n o f the random v a r i a b l e  wise i t i s non-stationary.  time-  and a pure-random  c o n s t a n t throughout t h e p r o c e s s , t h e p r o c e s s i s s a i d t o be  and  equa-  by a m i x i n g l e n g t h .  On t h i s b a s i s  he used a  t o o b t a i n a t i d a l l y averaged s o l u t i o n f o r waste d i s -  t i d a l waters.  He a l l o w e d f o r t h e e f f e c t o f m i x i n g i n the \ver-  30  t i c a l and l a t e r a l d i r e c t i o n s , as w e l l as i n the l o n g i t u d i n a l The  s o l u t i o n of Equation  (2.1)  u n i f o r m flow i s a G a u s s i a n d i s t r i b u t i o n .  direction.  for a slug input into Harris  [1962, 1963]  steady assumed  t h i s d i s t r i b u t i o n t o r e s u l t from a pure-random d i s p e r s i o n p r o c e s s ,  and  on t h i s b a s i s he o b t a i n e d maximum l i k e l i h o o d e s t i m a t e s o f u and E.  He  used the c o n v o l u t i o n method t o o b t a i n the s o l u t i o n f o r a continuous  re-  lease. averaged  However, i n the i n i t i a l n o n - F i c k i a n p e r i o d , the  cross-sectionally  c o n c e n t r a t i o n p r o f i l e i s skewed and not G a u s s i a n  as F i s c h e r [1966a]  (see S e c t i o n A.3),  noted.  Loucks and Lynn [1966] o b t a i n e d t h e t r a n s i e n t and p r o b a b i l i t y d i s t r i b u t i o n s o f t h e DO DO was  i n a stream  ( d i s p e r s i o n was  steady s t a t e  c o n c e n t r a t i o n s a t the p o i n t o f minimum  ignored).  The  sequence o f d a i l y  assumed t o be a f i r s t - o r d e r Markov p r o c e s s  p r o c e s s ) , and the sewage f l o w s , u l t i m a t e BOD  streamflows  (a s t o c h a s t i c non-pure-random  and  s o u r c e - s i n k parameters were  assumed t o be dependent on the d a i l y streamflows  (via conditional probabili-  ties) .  E s s e n t i a l l y , the t e c h n i q u e  i s as f o l l o w s .  Given  a streamflow,  the  s e t o f sewage f l o w s , s o u r c e - s i n k parameters, e t c . t h a t r e s u l t i n the minimum DO  f a l l i n g below some p r e s c r i b e d v a l u e i s determined  Phelps e q u a t i o n ) . determined,  (by a m o d i f i e d S t r e e t e r -  The p r o b a b i l i t y o f t h i s s e t o f events o c c u r r i n g i s t h e n  t h e p r o c e s s b e i n g r e p e a t e d f o r each d i s c r e t e streamflow  variate,  and the p r o b a b i l i t i e s summed t o g i v e t h e t o t a l p r o b a b i l i t y o f the minimum b e i n g l e s s t h a n the p r e s c r i b e d v a l u e .  Loucks  t o i n v e s t i g a t e the e f f e c t of v a r i o u s treatment the d i s t r i b u t i o n o f minimum DO Thayer and  technique  plant operating p o l i c i e s  on  i n a stream.  Krutchkoff  c o n c e n t r a t i o n p r o f i l e s of BOD  [1967] used a s i m i l a r  DO  [1967] o b t a i n e d a s t o c h a s t i c s o l u t i o n f o r the  and DO  i n a stream  ( d i s p e r s i o n was  ignored,  and  31  t h e streamflow  was assumed u n i f o r m and s t e a d y ) .  The b a s i c p r o c e s s e s o f  decay, r e a e r a t i o n , e t c . o c c u r r i n g i n the m o d i f i e d S t r e e t e r - P h e l p s e g u a t i o n were t r e a t e d as s t o c h a s t i c , b u t f r e s h w a t e r and e f f l u e n t d i s c h a r g e s were assumed t o be d e t e r m i n i s t i c . C o n c e n t r a t i o n v a l u e s were d i v i d e d i n t o  dis-  c r e t e u n i t s o f s i z e A, and t h e p r o b a b i l i t y o f a change o f A o c c u r r i n g i n the c o n c e n t r a t i o n d u r i n g a s m a l l time son d i s t r i b u t i o n .  i n t e r v a l was assumed t o f o l l o w a P o i s -  T h i s a l l o w e d them t o lump a l l o f t h e s t o c h a s t i c v a r i a t i o n  i n t o the parameter A (the sum o f a number o f independent P o i s s o n p r o c e s s e s i s i t s e l f P o i s s o n ) , which was determined the model e s t u a r y t o the observed  by f i t t i n g  the p r e d i c t e d v a r i a n c e i n  v a r i a n c e i n the r e a l e s t u a r y .  Essentially,  they used t h e same s e t o f r e s u l t s t o b o t h e s t i m a t e and v e r i f y t h e model. Under t h e s e c o n d i t i o n s , any model w i l l reproduce p e c t i v e o f whether t h e u n d e r l y i n g p r o c e s s e s  t h e observed  results,  a r e c o r r e c t l y modelled.  irres-  Whereas  o t h e r s t o c h a s t i c s o l u t i o n s use t h e e x p l i c i t v a r i a t i o n i n the u n d e r l y i n g s t o c h a s t i c p r o c e s s e s t o p r e d i c t the v a r i a t i o n i n the c o n c e n t r a t i o n p r o f i l e s , Thayer and K r u t c h k o f f based t h e i r s o l u t i o n on t h e i m p l i c i t v a r i a t i o n i n t h e s t o c h a s t i c processes  (the a c t u a l v a r i a t i o n i s n o t even measured).  Their  p r e d i c t e d mean c o n c e n t r a t i o n v a l u e i s simply t h e : ' s o l u t i o n t o t h e ( d e t e r m i n i s t i c ) modified Streeter-Phelps equation.  As such, t h e i r technique  i s no b e t t e r  than any d e t e r m i n i s t i c model t h a t r e c o g n i z e s and i n c o r p o r a t e s s t o c h a s t i c v a r i a t i o n through measurements o f f i e l d v a l u e s . a random walk f o r m u l a t i o n t o extend  along the e s t u a r y .  [1969] used  the t e c h n i q u e t o i n c l u d e d i s p e r s i o n (the  f l o w was assumed u n i f o r m b u t u n s t e a d y ) , f u r t h e r extended t h e t e c h n i q u e  C u s t e r and K r u t c h k o f f  and S c h o f i e l d and K r u t c h k o f f  t o account  [1972]  f o r v a r i a b l e s t o c h a s t i c parameters  32  Koivo and P h i l l i p s  [1971] regarded t h e measured BOD  t r a t i o n p r o f i l e s a l o n g a stream t o be c o r r u p t e d by t i c n a t u r e o f the u n d e r l y i n g p r o c e s s e s .  " n o i s e " due  They developed  l i n e a r r e g r e s s i o n a n a l y s i s t o determine  concen-  t o the  stochas-  a method based  the v a l u e s o f the s t o c h a s t i c  meters t h a t gave the b e s t o v e r a l l f i t t o the observed  2.5  and DO  on non-  para-  concentration p r o f i l e s .  SUMMARY Most a n a l y t i c a l s o l u t i o n s t o t h e one-dimensional  e q u a t i o n a r e t i d a l l y averaged  and may  be u s e f u l f o r i n v e s t i g a t i n g  d i s c h a r g e i n t o e s t u a r i e s w i t h simple geometries. w i t h d i f f e r e n t geometries  effluent  (When the number o f segments  i s g r e a t e r t h a n f o u r o r f i v e , the matching o f bound-  a r y c o n d i t i o n s a t t h e i r ends becomes cumbersome t i d a l l y averaged  mass t r a n s p o r t  [O'Connor et al, , 1968]) . In  s o l u t i o n s , upstream t r a n s p o r t i s by d i s p e r s i o n , and thus a  t i d a l l y averaged model may  n o t a d e q u a t e l y reproduce  or more e f f l u e n t o u t f a l l s on t h e f l o o d Numerical  the i n t e r a c t i o n o f  two  tide.  s o l u t i o n s p r o v i d e a means o f s o l v i n g the  one-dimensional  t i d a l l y v a r y i n g mass t r a n s p o r t e q u a t i o n , b u t the problems o f s t a b i l i t y n u m e r i c a l d i s p e r s i o n must be c o n s i d e r e d .  and  Of t h e v a r i o u s n u m e r i c a l methods,  the method o f c h a r a c t e r i s t i c s , methods w i t h h i g h e r o r d e r a p p r o x i m a t i o n s f i n i t e element methods are s a t i s f a c t o r y w i t h r e s p e c t t o a c c u r a c y o f  and  solution.  The method o f c h a r a c t e r i s t i c s has the a d d i t i o n a l advantage of d i r e c t l y  simula-  t i n g the a d v e c t i v e p r o c e s s . U n d i s t o r t e d p h y s i c a l models may  be u s e f u l i n r e s o l v i n g complex  t h r e e - d i m e n s i o n a l a s p e c t s o f the f l o w - f i e l d a t important o r s e n s i t i v e s e c t i o n s o f the e s t u a r y , but because of t h e i r i n a c c u r a t e and  inconsistant reproduction  33  of the d i s p e r s i o n process  ( i n b o t h d i s t o r t e d and u n d i s t o r t e d m o d e l s ) ,  b e s t they a r e o n l y an a d j u n c t t o a m a t h e m a t i c a l form o f s o l u t i o n .  at  As d e v e -  l o p e d , analogue models a r e t i d a l l y averaged and s u f f e r from n u m e r i c a l d i s p e r s i o n , a l t h o u g h i t may be p o s s i b l e t o overcome b o t h t h e s e l i m i t a t i o n s . It  i s n o t c l e a r whether the v a r i o u s s t o c h a s t i c s o l u t i o n s  attempt  t o model " t r u e " s t o c h a s t i c v a r i a t i o n , o r a c o m b i n a t i o n o f s t o c h a s t i c and cross-sectional variation.  If  the c r o s s - s e c t i o n a l v a r i a t i o n i s  greater  t h a n t h e a c t u a l s t o c h a s t i c v a r i a t i o n , i t may be more m e a n i n g f u l t o model the l a t e r a l component o f the d i s p e r s i o n p r o c e s s w i t h a t w o - d i m e n s i o n a l d e t e r m i n i s t i c model.  CHAPTER 3 A DESCRIPTION OF THE  3.1  THE  HYDRODYNAMIC AND  MASS TRANSPORT MODELS  HYDRODYNAMIC MODEL To s o l v e the one-dimensional t i d a l l y v a r y i n g mass t r a n s p o r t  e q u a t i o n , i t i s n e c e s s a r y t o know b o t h the s p a t i a l and temporal v a r i a t i o n i n t h e parameters 1.3.  u, A and E o f the e g u a t i o n , as i s d i s c u s s e d i n S e c t i o n  A one-dimensional hydrodynamic  model was  developed t o p r e d i c t  s p a t i a l and temporal v a r i a t i o n i n t h e parameters u and A. and s p a t i a l v a r i a t i o n o f the parameter In  the hydrodynamic  the  (The temporal  E i s d i s c u s s e d i n Appendix  F.  model, the e q u a t i o n s of motion and c o n t i n u i t y were  a p p l i e d t o t h e water mass o f the e s t u a r y , and s o l v e d throughout the  tidal  cycle. 3.1.1  The Hydrodynamic E g u a t i o n s .  As a p p l i e d t o the F r a s e r  R i v e r E s t u a r y , t h e e q u a t i o n s o f motion and c o n t i n u i t y are g i v e n by  [Dronkers,  1969]  9u + u3t  dh •g.  u  u  (3.1)  (3.2)  where u  i s the mean l o n g i t u d i n a l v e l o c i t y ;  h i s the h e i g h t o f the water s u r f a c e above an a r b i t r a r y l e v e l datum; y i s the mean c r o s s - s e c t i o n a l water 34  depth;  35  A i s the c r o s s - s e c t i o n a l area; b i s t h e c r o s s - s e c t i o n a l width; g i s the l o c a l g r a v i t a t i o n a l a c c e l e r a t i o n ; and C i s Chezy's f r i c t i o n  factor.  S e v e r a l o f t h e s e terms are i l l u s t r a t e d  i n F i g u r e 3.1.  Note t h a t i n the  hydrodynamic e q u a t i o n s , x i n c r e a s e s i n the upstream d i r e c t i o n , whereas i n the mass t r a n s p o r t e q u a t i o n , x i n c r e a s e s i n the downstream d i r e c t i o n . The two  terms on the l e f t - h a n d s i d e of E q u a t i o n  l o c a l and c o n v e c t i v e  (3.1) are  (or B e r n o u l l i ) a c c e l e r a t i o n s r e s p e c t i v e l y .  the  The  terms on the r i g h t - h a n d s i d e o f the e q u a t i o n are t h e f o r c e s c a u s i n g  two these  a c c e l e r a t i o n s , the net p r e s s u r e f o r c e due  t o the s l o p e o f the water s u r f a c e  and t h e f r i c t i o n f o r c e r e s p e c t i v e l y .  f i r s t term o f E q u a t i o n  the net o u t f l o w from an-;elemental  The  (3.2) i s  c r o s s - s e c t i o n a l s l i c e o f the e s t u a r y ,  w h i l e the second term r e p r e s e n t s the accompanying change i n storage w i t h i n the  slice. The  listed width  assumptions made i n d e r i v i n g e q u a t i o n s  i n Dronkers  [1964].  i s e q u a l t o the a d v e c t i v e w i d t h .  B.8.  storage storB.6  exagger-  from the c r o s s - s e c t i o n a l a r e a s below  depth i s some 10 f e e t above l o c a l low w a t e r ) .  Estuary.  s t o r a g e widths i s  (In the lower reaches o f t h e e s t u a r y , the t i d a l l y  equal a d v e c t i v e and  are  of the e s t u a r y i s shown i n F i g u r e s  a t e d s i n c e these v a l u e s were determined  of  (3.2)  The v a r i a t i o n o f a d v e c t i v e and  The d i f f e r e n c e between the a d v e c t i v e and  l o c a l low water.  and  The most important assumption i s t h a t the  age widths along the major channels to  (3.1)  Consequently,  s t o r a g e widths seems r e a s o n a b l e  (This i s a l s o d i s c u s s e d i n S e c t i o n 4.1  for  the  averaged  assumption  the F r a s e r R i v e r  w i t h r e g a r d t o the  verifi-  36  Figure  3.1  The Hydrodynamic  Estuary  37  c a t i o n of the hydrodynamic model). equations i s j u s t i f i e d  tions  one-dimensional  since the v a r i o u s channels  longer than they are wide classified  The  (see A p p e n d i x B ) .  t h i s t y p e o f e s t u a r y as a " t i d a l  (3.1)  and  (3.2)  the presence  river."  [1971], t h e C h e z y f o r m u l a  [1971] h a s  In d e r i v i n g  Equa-  (see Appendix B ) .  should adequately  f a s t f l o w i n g , b u t t h e C h e z y f o r m u l a may lower  much  o f t h e s a l t wedge h a s b e e n i g n o r e d  e f f e c t s i n f a s t f l o w i n g , w e l l mixed e s t u a r i e s .  tified  the  of the e s t u a r y are  In f a c t , Callaway  cause o f the c o m p l i c a t e d nature o f i t s dynamics i n g t o Odd  nature of  n o t be  The  represent  be-  Accordfrictional  Fraser River estuary i s  accurate i n the h i g h l y  stra-  reaches.  F r o m F i g u r e 3.1,  i t i s seen t h a t  (3.3)  h = a + y  w h e r e a i s t h e h e i g h t o f t h e r i v e r b o t t o m a b o v e t h e same l e v e l d a t u m t h a t h i s referred to.  Thus, E q u a t i o n s  of coupled p a r t i a l d i f f e r e n t i a l and  f a c t o r C,  e q u a t i o n s as p a r a m e t e r s .  c i e n t s and  The  a r e s e e n t o be  v a r i a b l e s b e i n g x and  t h e a r e a A and  and  I t i s noted  that Equation  (3.2)  (3.1)  a  pair  parameters that both  finite  The  width  been assumed  vary i n a continuous  equations  contains non-linear  f i x e d mesh, e x p l i c i t  t.  manner  contain variable  coeffi-  terms.  d i f f e r e n c e method o f Dronkers  used t o o b t a i n a numerical s o l u t i o n t o the hydrodynamic equations.  t a i l s o f t h e method a r e g i v e n i n A p p e n d i x D and used i n the f i n i t e  difference  u  the f a c t o r a a l l appear i n the  I n d e r i v i n g t h e s e e q u a t i o n s , i t has  t h a t the dependent v a r i a b l e s along the estuary.  and  equations, the dependent v a r i a b l e s being  e i t h e r y o r h, and t h e i n d e p e n d e n t  b, C h e z y ' s f r i c t i o n  was  (3.1)  t h e f i x e d mesh o f  s o l u t i o n i s shown i n F i g u r e B.4.  [1969J De-  "stations"  Because  the  38  s o l u t i o n scheme o f t h e hydrodynamic e q u a t i o n s  is explicit, stability  quirements determine t h e r e l a t i v e s i z e o f Ax and hydrodynamic e g u a t i o n s ,  Ax was  reasons.  seconds was  I t was  stability.  are d i s c u s s e d i n d e t a i l i n S e c t i o n  discharge  i n c r e a s e d t o 15,000 f e e t f o r  found t h a t w i t h t h i s space g r i d , a At o f  s a t i s f a c t o r y as regards  3.1.2  In s o l v i n g the  s e t equal t o 5,000 f e e t , e x c e p t i n the  deeper waters o f P i t t Lake where Ax was stability  At.  The  stability  90  requirements  C.3.  Assumed Quasi-Steady H y d r a u l i c C o n d i t i o n s .  i n t o an e s t u a r y , the q u a s i - s t e a d y  i s u s u a l l y a c o n d i t i o n of i n t e r e s t .  For  effluent  s t a t e response o f an  estuary  . T i d a l l y . v a r y i n g mass t r a n s p o r t models  u s u a l l y r e q u i r e many t i d a l c y c l e s t o a c h i e v e  quasi-steady-state  whereas hydrodynamic models o n l y r e q u i r e s e v e r a l t i d a l c y c l e s . for  re-  t h i s i s t h e r e l a t i v e speed o f i n f o r m a t i o n p r o p a g a t i o n  conditions, The  reason  i n b o t h systems.  In the hydrodynamic model, the i n f o r m a t i o n c o n s i s t s o f s m a l l changes i n s u r f a c e e l e v a t i o n t h a t p r o p a g a t e a l o n g the e s t u a r y as e l e m e n t a l  surges w i t h  speeds o f u±/gf- whereas i n the mass t r a n s p o r t model, t h e I i n f o r m a t i o n f  s i s t s of changes i n c o n c e n t r a t i o n  ( f i n i t e or otherwise) t h a t propagate a t  speeds o f u ( i f d i s p e r s i o n i s i g n o r e d ) . t i o n C.3. steady  con-  T h i s i s d i s c u s s e d f u r t h e r i n Sec-  Thus, i n a p p l y i n g a mass t r a n s p o r t model t o determine the q u a s i -  s t a t e response o f an e s t u a r y f o r some waste l o a d i n g c o n d i t i o n , i t  i s necessary  to run the model f o r a s u c c e s s i o n o f t i d a l c y c l e s .  In s o l v i n g the hydrodynamic and equations  f o r the Fra'ser R i v e r E s t u a r y , the f r e s h w a t e r  c o n d i t i o n s are assumed t o be a constant  t i d a l l y v a r y i n g mass t r a n s p o r t  freshwater  quasi-steady.  discharge  and  Thus, the model e s t u a r y  i n f l o w and a s u c c e s s i o n of i d e n t i c a l t i d e s .  a c t u a l f a c t , "slow" v a r i a t i o n s o c c u r i n t h e f r e s h w a t e r  i n f l o w and  tidal "sees"  In tidal  39  c o n d i t i o n s , and the e s t u a r y may  never a c h i e v e s t r i c t q u a s i - s t e a d y s t a t e  c o n d i t i o n s . However, as t h e r e s i d e n c e time o f the F r a s e r R i v e r E s t u a r y i s o n l y f o u r t o s i x days, i t seems r e a s o n a b l e  t o t r e a t the f r e s h w a t e r  t i d a l c o n d i t i o n s as c o n s t a n t f o r t h i s p e r i o d o f 3.1.3  and  time.  The Model E s t u a r y o f the Hydrodynamic E q u a t i o n s .  F r a s e r R i v e r E s t u a r y i s d e s c r i b e d i n d e t a i l i n Appendix B.  The  The model  e s t u a r y , as d e s c r i b e d b y f t h e hydrodynamic e q u a t i o n s , d i f f e r s t o some ext e n t from t h e r e a l e s t u a r y because o f t h e assumptions made i n d e r i v i n g the hydrodynamic e q u a t i o n s .  The model e s t u a r y c o n s i s t s o f the same seven  p r i n c i p l e s as the r e a l e s t u a r y , a l t h o u g h the Canoe Pass Area has been f i e d i n t o a s i n g l e channel, as i s shown i n F i g u r e B.4.  The  channels  simplio f the  model e s t u a r y are assumed t o be r e c t a n g u l a r i n c r o s s - s e c t i o n (equal s t o r a g e and a d v e c t i v e w i d t h s ) , and the s a l i n i t y i s assumed t o be everywhere - z e r o . Because t h e s a l t wedge has been i g n o r e d , t h e p r e d i c t e d v e l o c i t i e s i n t h e lower reaches  o f the e s t u a r y a r e p r o b a b l y  somewhat low.  The  freshwater  charge i n the model e s t u a r y i s c o n s t a n t , and t h e t i d a l r i s e and  fall  dis-  of  t h e water s u r f a c e i s q u a s i - s t e a d y .  3.2  THE  TIDALLY VARYING MASS TRANSPORT MODEL 3.2.1  e q u a t i o n was  Method o f S o l u t i o n .  The  t i d a l l y v a r y i n g mass t r a n s p o r t  s o l v e d by a c h a r a c t e r i s t i c f i n i t e d i f f e r e n c e method.  In  this  method, the e q u a t i o n i s s o l v e d a l o n g the c h a r a c t e r i s t i c c u r v e s o f the advect i v e t r a n s p o r t p r o c e s s , as i s d i s c u s s e d i n Appendix C.  The  accuracy  and  d i r e c t n e s s o f s o l u t i o n were i n s t r u m e n t a l i n t h i s c h o i c e ; t h e r e i s no numeric a l d i s p e r s i o n (see Appendix C) and the a d v e c t i o n p r o c e s s i s s i m u l a t e d  40  d i r e c t l y and i n d e p e n d e n t l y  o f the d i s p e r s i v e process.  I t was n o t p o s s i b l e t o use the u s u a l technique's  o f dye p a t c h  s t u d i e s o r measurement o f e x i s t i n g t r a c e r s t o v e r i f y the mass t r a n s p o r t models.  The l a r g e d i l u t i o n a l c a p a c i t y o f the r i v e r made e x t e n s i v e dye  patch s t u d i e s too expensive, e s s e n t i a l l y unpolluted  and as t h e seven channels  [ B e n e d i c t , et al.  existing pollutional tracers. t i o n and the c o m p l i c a t e d  o f the e s t u a r y a r e  1973], t h e r e a r e no s u i t a b l e  }  Because o f t h e h i g h degree o f s t r a t i f i c a -  nature  dynamics o f the s a l i n i t y  S e c t i o n B.2.4), s a l t c o u l d n o t be used as a t r a c e r .  i n t r u s i o n (see  Thus, i t was n o t  p o s s i b l e t o v e r i f y the mass t r a n s p o r t models by a d i r e c t comparison o f p r e d i c t e d and observed  concentrations.  the u n d e r l y i n g mass t r a n s p o r t p r o c e s s e s pendently  An attempt was made t o " v e r i f y " o f a d v e c t i o n and d i s p e r s i o n i n d e -  o f each o t h e r , as i s d i s c u s s e d i n S e c t i o n 4.2.  The method o f  c h a r a c t e r i s t i c s l e a d s t o a n a t u r a l s e p a r a t i o n o f t h e a d v e c t i v e and d i s p e r sive transport processes,  and t h i s was one o f the reasons  f o r using  this  method t o s o l v e the t i d a l l y v a r y i n g mass t r a n s p o r t e q u a t i o n . The  one-dimensional  mass t r a n s p o r t e q u a t i o n  i s d e r i v e d i n Appendix  A and the assumptions i n i t s d e r i v a t i o n a r e d i s c u s s e d t h e r e . was t r a n s f o r m e d  i n t o Lagrangian  o r c h a r a c t e r i s t i c form  The e q u a t i o n  (see S e c t i o n A.6) t o  give  i t . =' -  —  dt  Equation  n  =  r"5 [EAr—} + Adx  (3.5) was t h e n s e p a r a t e d  sink e f f e c t s t o give  3x  2 S 1  (  3 > 5  )  i=l  i n t o i t s component d i s p e r s i v e and s o u r c e -  41  dc dt  1 3 A 3x  (3.6)  and n Z S. 1=1  dc_ dt  Initially, estuary. was  (3.7)  1  a g r i d o f moving p o i n t s was  a s s i g n e d throughout the  In the a d v e c t i o n s t e p , a f i n i t e d i f f e r e n c e form o f E q u a t i o n  used t o advect  the p o i n t s a l o n g the e s t u a r y f o r a time increment,  c o n c e n t r a t i o n o f t h e p o i n t s b e i n g a d j u s t e d as they passed e f f l u e n t o r through c o n v e r g i n g  junctions.  d i f f e r e n c e form o f E q u a t i o n  outfalls  i n the s o u r c e - s i n k s t e p , E q u a t i o n  (3.7) was  increment.  used t o a d j u s t  the  c o n c e n t r a t i o n o f the moving p o i n t s f o r the e f f e c t s o f the s o u r c e - s i n k c e s s e s d u r i n g the time increment. tically — equation).  time increment,  (3.7)  i s the usual  sequence o f t h e s e t h r e e s t e p s was and  pro-  (The s o u r c e - s i n k s t e p can be done a n a l y -  f o r a BOD-DO system, E q u a t i o n The  finite  used t o a d j u s t the c o n c e n t r a t i o n of  the moving p o i n t s f o r the e f f e c t s of d i s p e r s i o n d u r i n g the time Finally,  the  In the d i s p e r s i o n s t e p , an e x p l i c i t  (3.6) was  (3.4)  Streeter-Phelps  then repeated  so t h e s o l u t i o n p r o g r e s s e s through time.  f o r the  A t the  next  boundaries  of  the e s t u a r y  (the sea, P i t t Lake and C h i l l i w a c k ) , moving p a r t i c l e s were added  to  and removed from the e s t u a r y as d i c t a t e d by t h e a d v e c t i v e boundary c o n d i t i o n s .  A time increment of one hour was p o r t e q u a t i o n , and  used i n s o l v i n g the t i d a l l y v a r y i n g mass t r a n s -  a t the end o f each hour,the c o n c e n t r a t i o n s were e x t r a p o l a t e d  o f f the g r i d o f moving p o i n t s onto the 5,000 f o o t f i x e d g r i d o f F i g u r e 3.1. method o f s o l u t i o n i s v e r y s i m i l a r t o the combined method o f G a r d i n e r  et  The al.  42  et  [1964], except t h a t whereas G a r d i n e r  al,  e x t r a p o l a t e the moving p o i n t s  onto the f i x e d g r i d , the moving p o i n t s i n the F r a s e r R i v e r E s t u a r y s o l u t i o n always remain on t h e i r r e s p e c t i v e c h a r a c t e r i s t i c s and t i o n s a r e e x t r a p o l a t e d onto the f i x e d g r i d . Equations  (3.4), An  (3.6)  and  (3.7)  only t h e i r  concentra-  D e t a i l s of the s o l u t i o n of  a r e g i v e n i n Appendix  D.  advantage o f the c h a r a c t e r i s t i c method of s o l u t i o n i s t h a t  a d d i t i o n a l moving p o i n t s can be added t o the e s t u a r y t o more c l o s e l y  de-  f i n e r e g i o n s o f r a p i d v a r i a t i o n i n c o n c e n t r a t i o n , as occur a t times o f slackwater  a t e f f l u e n t o u t f a l l s , and where the v a r i a t i o n i s slow unneces-  s a r y moving p o i n t s can be removed.  Once the moving p o i n t s are i n the  t h e i r subsequent p o s i t i o n s are determined by the a d v e c t i v e and  they are not u n i f o r m l y  Both an e x p l i c i t and the d i s p e r s i v e step.  estuary,  transport  process,  spaced a l o n g t h e e s t u a r y i n ' t h e d i s p e r s i v e s t e p .  i m p l i c i t f i n i t e d i f f e r e n c e scheme were i n v e s t i g a t e d f o r The  f o r reasons o f s i m p l i c i t y  e x p l i c i t scheme was (see Appendix D).  s l i g h t l y slower, The  but was  chosen  s t a b i l i t y requirements o f  the e x p l i c i t scheme are d i s c u s s e d i n Appendix C, and g e n e r a l l y the time i n crement f o r s t a b i l i t y was  l e s s than one  hour; the d i s p e r s i v e s t e p then con-  s i s t e d o f a number o f " i n t e r n a l " i t e r a t i o n s w i t h i n t h e b a s i c time increment of one  hour, as d i s c u s s e d i n Appendix 3.2.2  The Model E s t u a r y .  D. F o r the sake o f convenience, the model  e s t u a r y of b o t h the t i d a l l y averaged and  t i d a l l y v a r y i n g mass t r a n s p o r t equa-  t i o n s has been s i m p l i f i e d t o the t h r e e major channels of the r e a l t h e Main Arm  - Main Stem, t h e N o r t h Arm  F i g u r e 3.2.  The  s t a t i o n s along these  and  the P i t t  estuary,  System, as shown i n  channels are t h e same as those  of  the  Pitt  Lake  '1/  F i g u r e 3.2 S i m p l i f i e d Model E s t u a r y o f the Mass T r a n s p o r t  Equations  fir _  Main  Stem  44  hydrodynamic model.  I t i s p o s s i b l e t o i n c l u d e a l l seven channels  of  r e a l e s t u a r y , but i t would have made t h e computer programming o f the  the tidally  v a r y i n g mass t r a n s p o r t e q u a t i o n c o n s i d e r a b l y l o n g e r and more i n v o l v e d . Because o f t h e i r s h o r t l e n g t h and along the minor channels major c h a n n e l s ,  and  s m a l l e r f l o w s , the c o n c e n t r a t i o n p r o f i l e  i s e s s e n t i a l l y determined by the p r o f i l e a l o n g  the a p p r o x i m a t i o n  o f i g n o r i n g the minor channels  the  seems  reasonable. In the model e s t u a r y , the d i s p e r s i o n p r o c e s s F i c k i a n i n i t s e n t i r e t y and  the c r o s s - s e c t i o n a l mixing  complete a t t h e j u n c t i o n s .  The  i s assumed t o be i s assumed t o be  s a l i n i t y of the model e s t u a r y i s zero,  but because o f the s a l i n i t y i n t r u s i o n i n the r e a l e s t u a r y , t h e p r e d i c t e d v e l o c i t y f i e l d and estimated.  The  i n t r u s i o n and  advective transport i n t h i s region i s probably  estuary i s highly s t r a t i f i e d  salinity  l i t t l e m i x i n g o c c u r s between the f r e s h and ^.saltwater.  w i l l tend t o minimize any  hydrodynamic e q u a t i o n s , c o n d i t i o n s are  This  c h e m i c a l o r b i o l o g i c a l e f f e c t s the s a l t w a t e r  have on the d i s s o l v e d substance  3.3  i n the r e g i o n of the  under-  i n question.  may  As i n t h e model e s t u a r y o f the  the freshwater discharge  i s c o n s t a n t and  the  tidal  quasi-steady.  TIDALLY AVERAGED MASS TRANSPORT MODEL 3.3.1  Method o f S o l u t i o n .  The  one-dimensional,  mass t r a n s p o r t model of Thomann [1963] was  tidally  averaged  used t o determine the steady  state  t i d a l l y averaged response o f the F r a s e r R i v e r E s t u a r y f o r v a r i o u s waste l o a d i n g conditions.  (As the model i s t i d a l l y averaged, the response i s s t e a d y - s t a t e  r a t h e r than q u a s i - s t e a d y  state).  In t h i s model, the e s t u a r y i s d i v i d e d i n t o  45  a number o f l o n g i t u d i n a l segments o r "boxes",  as shown i n F i g u r e  Each segment i s assumed t o be c o m p l e t e l y mixed, and t h e t i d a l l y  D.2. averaged  a d v e c t i o n and d i s p e r s i o n p r o c e s s e s t r a n s p o r t d i s s o l v e d substance i n t o and out  o f each segment.  I f the t i d a l l y averaged t r a n s p o r t p r o c e s s e s and e f f l u -  ent  d i s c h a r g e s remain s t e a d y , a mass b a l a n c e about segment i g i v e s  a. . .-c + a. . c. + a. ...c' = i,a-l i - l 1,1 i i,i+l l+l  (3.8)  W . i  where c^ i s t h e c o n c e n t r a t i o n i n segment i ; i s t h e mass o f e f f l u e n t d i s c h a r g e d i n t o segment i p e r t i d a l cycle; a. '  . ^ i s a c o e f f i c i e n t a c c o u n t i n g f o r t h e t i d a l l y averaged t r a n s p o r t o f substance between segments i and i - l , s i m i l a r l y f o r a^ ^ ^ ; +  and a. '  . accounts f o r d i s p e r s i o n o u t o f segment i and any s i n k e f f e c t s t h e substance undergoes.  D e t a i l s o f t h i s e g u a t i o n and i t s c o e f f i c i e n t s a r e g i v e n i n Appendix e q u a t i o n s i m i l a r t o (3.8)  D.  An  c a n be w r i t t e n f o r each o f t h e n segments o f the  e s t u a r y , and i n m a t r i x n o t a t i o n t h e system o f e q u a t i o n s c a n be w r i t t e n  A • C  =  W  (3.9)  where |  i s a (n x 1) column m a t r i x o f t h e t i d a l l y i n t o . e a c h segment p e r t i d a l - c y c l e ;  averaged waste l o a d s  C i s a (n x 1) column m a t r i x o f t h e t i d a l l y averaged c o n c e n t r a t i o n i n each segment;  46 and  ^ i s a (n x n) t r i - d i a g o n a l m a t r i x averaged t r a n s p o r t terms and any  Thus, t h e steady  c o n t a i n i n g the sink e f f e c t s .  s t a t e t i d a l l y averaged response o f the e s t u a r y i s d e s c r i b e d  by a system o f n simultaneous  l i n e a r equations.  the c o n c e n t r a t i o n s o f a c o n s e r v a t i v e substance f i r s t o r d e r decay, such as BOD.  f o r the DO  concentrations  Thomann's s t e a d y - s t a t e  Equation or a  (3.9)  represents  substance-^undergoing  To i n v e s t i g a t e t h e s t e a d y - s t a t e BOD-DO r e s -  ponse o f an e s t u a r y , a system o f e q u a t i o n s i t are obtained  tidally  similar to  (3.9)  and c o u p l e d  (see Thomann [1971] f o r d e t a i l s ) .  s o l u t i o n t o t h e one-dimensional  tidally  averaged mass t r a n s p o r t e q u a t i o n i s e s s e n t i a l l y a f i x e d g r i d f i n i t e ence s o l u t i o n s i m i l a r t o t h a t o f an i m p l i c i t d i f f e r e n c e scheme D).  There i s no s t a b i l i t y requirement  S e c t i o n C.3),  to  differ-  (see Appendix  f o r the steady-state s o l u t i o n  but t h e r e i s a n o n - n e g a t i v i t y requirement  (see  t h a t imposes  rela-  t i v e l i m i t s on the s i z e s o f the a d v e c t i v e and d i s p e r s i v e t r a n s p o r t p r o c e s s e s . T h i s i s d i s c u s s e d i n S e c t i o n D.3,  and i f t h i s requirement  c o n c e n t r a t i o n i n a segment becomes n e g a t i v e .  i s violated,  Thomann's s t e a d y - s t a t e -solution  does n o t s u f f e r from n u m e r i c a l d i s p e r s i o n , but f o r a r a t h e r unusual  reason.  Because o f the f i x e d g r i d nature o f h i s f i n i t e d i f f e r e n c e scheme, the t i v e p r o c e s s w i l l not be c o r r e c t l y s i m u l a t e d and occur.  However, from S e c t i o n C.2,  to occur,  i t i s necessary  change w i t h time.  the  advec-  numerical d i s p e r s i o n should  i t i s seen t h a t f o r n u m e r i c a l d i s p e r s i o n  f o r the c o n c e n t r a t i o n a t a f i x e d g r i d p o i n t t o  Thomann's s t e a d y - s t a t e s o l u t i o n admits no temporal changes,  and thus no n u m e r i c a l  dispersion occurs.  N u m e r i c a l d i s p e r s i o n does o c c u r i n  h i s t r a n s i e n t s o l u t i o n where t h e c o n c e n t r a t i o n a t t h e f i x e d g r i d p o i n t s does change w i t h time,  as Thomann [1971] r e c o g n i z e s .  47  3.3.2  The Model E s t u a r y .  The model e s t u a r y o f t h e t i d a l l y  aged e q u a t i o n c o n s i s t s o f the same t h r e e major channels  and  aver-  s t a t i o n s as  the  model e s t u a r y o f the t i d a l l y v a r y i n g e q u a t i o n and c r o s s - s e c t i o n a l m i x i n g i s assumed t o be complete a t the j u n c t i o n s .  The t i d a l l y  averaged  model e s t u a r y  has n o % t i d e s and has h i g h e r d i s p e r s i o n than i t s t i d a l l y v a r y i n g c o u n t e r p a r t . Thomann [1971] l i s t s t y p i c a l v a l u e s o f the t i d a l l y averaged cient.  They range from 1 - 2 0  about 10 square  m i l e s p e r day.  dispersion coeffi-  square m i l e s p e r day w i t h a mean v a l u e o f These v a l u e s seem h i g h compared t o t h e  t i d a l l y v a r y i n g v a l u e s o f d i s p e r s i o n c o e f f i c i e n t s , and a p p a r e n t l y the i n f l u e n c e of t i d a l advection i n determining sion c o e f f i c i e n t . salinity  As  reflect  the t i d a l l y average d i s p e r -  i n t h e t i d a l l y v a r y i n g s o l u t i o n , the i n f l u e n c e o f the  i n t r u s i o n i s ignored.  CHAPTER 4  VERIFICATION OF THE HYDRODYNAMIC AND MASS TRANSPORT MODELS  4.1  THE HYDRODYNAMIC MODEL The v e r i f i c a t i o n o f t h e hydrodynamic model i s t o ensure t h a t  the model e s t u a r y , as r e p r e s e n t e d by t h e hydrodynamic e q u a t i o n s , l y reproduces  adequate-  t h e v a r i a t i o n i n water s u r f a c e e l e v a t i o n s and d i s c h a r g e s (or  a d v e c t i v e v e l o c i t i e s ) observed  i n the r e a l estuary.  The " v e r i f i c a t i o n "  c o n s i s t s o f a d j u s t i n g t h e f r i c t i o n f a c t o r s and t h e c r o s s - s e c t i o n a l w i d t h s and depths o f t h e model e s t u a r y u n t i l an adequate f i t i s o b t a i n e d between p r e d i c t e d and observed  results.  (In d e r i v i n g t h e hydrodynamic  eguations,  the a d v e c t i v e flow was assumed t o be u n i f o r m l y d i s t r i b u t e d over an assumed rectangular cross-section.  In r e a l s i t u a t i o n s , t h e c r o s s - s e c t i o n i s n o t  r e c t a n g u l a r and t h e flow i s c o n c e n t r a t e d i n the deeper s e c t i o n s . compensate f o r these e f f e c t s , i t i s n e c e s s a r y  To  t o adjust the c r o s s - s e c t i o n a l  geometry o f t h e model e s t u a r y ) .  4.1.1  Data A v a i l a b l e .  s t a t i o n s throughout  The network o f permanent t i d e - g a u g i n g  t h e e s t u a r y i s shown i n F i g u r e B . l l , and p r o v i d e s a  l i m i t e d b u t adequate r e c o r d o f t h e water s u r f a c e e l e v a t i o n s o f t h e e s t u a r y throughout  the t i d a l cycle.  a r e s p o r a d i c and inadequate  F i e l d measurements o f v e l o c i t i e s and d i s c h a r g e s f o r v e r i f i c a t i o n purposes.  The o n l y  existing  d a t a adequate f o r a complete v e r i f i c a t i o n o f the hydrodynamic model under h i g h t i d e - l o w flow c o n d i t i o n s i s due t o Baines  [1952].  I n t h i s study, water  s u r f a c e e l e v a t i o n s were r e c o r d e d a t h a l f - h o u r i n t e r v a l s a t 43 s t a t i o n s  48  49  throughout  the e s t u a r y f o r the h i g h t i d e - l o w f l o w c o n d i t i o n s o f January  1952.  freshwater  The  d i s c h a r g e a t C h i l l i w a c k was  second and t h e t i d a l range a t S t e v e s t o n was method o f cubature  36,500 c u b i c f e e t  11 f e e t .  B a i n e s used  t o e s t i m a t e the t i d a l l y v a r y i n g d i s c h a r g e s  24,  per  the  a t the  43  stations. 4.1.2 low to  Water S u r f a c e E l e v a t i o n s f o r Low  flow c o n d i t i o n s o f January reproduce  15, 1964  Flows.  The  high  were used i n an i n i t i a l  the r e c o r d e d water s u r f a c e e l e v a t i o n s i n the r e a l  The  f r e s h w a t e r d i s c h a r g e a t C h i l l i w a c k was  and  the t i d a l range a t S t e v e s t o n was  attempt estuary.  53,500 c u b i c f e e t p e r  10 f e e t .  tide-  second  The water s u r f a c e e l e v a t i o n s  of  the model e s t u a r y were found t o be r e l a t i v e l y i n s e n s i t i v e t o the  of  friction  and c r o s s - s e c t i o n a l geometry.  s e c t i o n s w i t h h i g h f r i c t i o n was s e c t i o n s w i t h low  friction.  the p r e d i c t e d responses  found  In f a c t , the response  i n F i g u r e 4.1  - Main Stem.  The  broader  which shows  f o r b o t h t y p e s o f s e c t i o n a t S t a t i o n No.  Westminster) on the Main Arm  o f narrow  t o be e q u i v a l e n t t o t h a t o f  This i s i l l u s t r a t e d  effects  average v a l u e s o f  18  (New  widths,  depths and Manning's "n" a l o n g the Main Arm —. Main Stem are a l s o shown. Note t h a t w h i l e Chezy's f o r m u l a dynamic e q u a t i o n s , "n"  f o r f r i c t i o n was  f r i c t i o n c o e f f i c i e n t s a r e r e p o r t e d as v a l u e s o f Manning's  (the v a l u e s o f n a r e f o r f e e t - s e c o n d 4.1.3  units).  Water S u r f a c e E l e v a t i o n s f o r High Flows.  the independent e f f e c t s o f f r i c t i o n and was  made t o reproduce  c o n d i t i o n s o f June 16,  I t was  thought  To s e p a r a t e  c r o s s - s e c t i o n a l geometry, an  out  attempt  the water s u r f a c e e l e v a t i o n s o f the h i g h t i d e - h i g h flow 1964.  The  f r e s h w a t e r d i s c h a r g e a t C h i l l i w a c k was  463,000 c u b i c f e e t p e r second and the t i d a l feet.  u s e d i n s o l v i n g the h y d r o -  range a t S t e v e s t o n was  t h a t the g r e a t e r v e l o c i t i e s under h i g h  eight  discharge  50  Observed ° o  Predicted  No.I (Narrow section,high friction)  * x Predicted No.2 ( Broad section,low friction) o o  CJ>  Q = 5 2 , 4 0 0 c f s (Chilliwack )  O  CO  T i d a l R a n g e of S t e v e s t o n =10  8  12  , J  16  Jan.15,1964 ^ 0  20  M  Hours  Average values  Stage  Arm - M a i n  Average Width (feet )  Average Depth (feet )  No. 1  0.042  1350  30.6  Predicted No. 2  0.030  1870  22.9  Figure  Equivalent  Main  Average Mannings •i I I n  Results Predicted  along  Response f o r Two  4.1  D i f f e r e n t Types o f Cross-Section  Stem  51  c o n d i t i o n s would be more s e n s i t i v e t o f r i c t i o n a l to be the c a s e , the t i d a l l y sitive  to f r i c t i o n  w i d t h was  averaged o r mean water  and e s s e n t i a l l y  mean water  level.  are  shown i n F i g u r e s 4.2  The  depths and areas shown i n F i g u r e s B.6 The w i d t h was  However, the  surface  about  elevations  and the r e p r o d u c t i o n o f the o b s e r v e d r e -  s u l t s i s seen t o be s a t i s f a c t o r y .  results.  found  surface fluctuations  The p r e d i c t e d and o b s e r v e d water and 4.3,  T h i s was  l e v e l s being very sen-  independent o f w i d t h .  found t o govern the range o f the water  the  effects.  gross c r o s s - s e c t i o n a l values of t o B.8  were used i n o b t a i n i n g  these  determined by d i v i d i n g the a r e a by the d e p t h ,  and  was  a s a t i s f a c t o r y compromise between the narrower a d v e c t i v e s e c t i o n s  and  the  b r o a d e r s e c t i o n s more r e p r e s e n t a t i v e o f the s t o r a g e w i d t h .  Manning's  "n" v a r i e d from 0.022 i n the lower reaches t o 0.027 i n the upper These v a l u e s a r e r e a s o n a b l e , and i n d i c a t e t h a t the Main Arm hydraulically  smooth  reaches.  - Main Stem i s  [Chow, 1959].  Under these h i g h d i s c h a r g e c o n d i t i o n s , the f l o w i n the upper reaches o f the Main Stem i s s t e a d y , and the p r e d i c t e d water can be used t o check t h a t the hydrodynamic the  frictional  effects.  surface slope  model s a t i s f a c t o r i l y  reproduces  The p r e d i c t e d v a l u e o f Manning's "n", as d e t e r m i n e d  by the average a r e a , d e p t h and p r e d i c t e d water 40 and 60 on the Main Stem, was  0.0269.  v a l u e o f 0.027 used i n the hydrodynamic  s u r f a c e s l o p e between s t a t i o n s  T h i s agrees c l o s e l y w i t h the a c t u a l model over t h i s s e c t i o n o f the  estuary. The  "high flow" f r i c t i o n  c o e f f i c i e n t s and  c r o s s - s e c t i o n a l geo-  m e t r i e s were t h e n used i n a second attempt t o reproduce the water e l e v a t i o n s f o r t h e low f l o w c o n d i t i o n s of January 15,  1964.  surface  To o b t a i n a  i s f a c t o r y f i t between t h e p r e d i c t e d and observed r e s u l t s i t was  sat-  necessary to  I  M  I  I  4  8 Figure  Observed  and P r e d i c t e d  I  1  I  12  16  20  Hours 4.2  Stages f o r June 16,  1964  I  M  Q= 4 6 3 , 0 0 0 cfs (Chilliwack) Tidal range at Steveston 20  June 16, 1964  8'  s  Main Arm - Station No.22 ( Port Mann)  8 I6  2  |~  4  Main Arm - Station No. 32 ( Port Hammond)  22 20 0 bserved o 2  I"  2  O A  o  Predicted  Pitt - Station No. 144 ( Port Coquitlam) o  20<K°.  O  O  •  °  O  °  rt  o  - — 2  r>  °  ~  n  n  n  n  n n  n  I L  I8  2  r  2  Pitt - Station o  20 I 8  M  No. 162 ( Al vi n )  o  8  12 Hours  16  F i g u r e 4.3 Observed and P r e d i c t e d  Stages f o r June 16, 1964  20  M  54  i n c r e a s e Manning's "n" by a p p r o x i m a t e l y 0.005 throughout  the e s t u a r y .  Under these c o n d i t i o n s t h e v a l u e o f Manning's "n" a l o n g the Main  Arm-  Main Stem v a r i e d from 0.027 i n t h e lower r e a c h e s t o 0.03 2 i n t h e  upper  r e a c h e s , and once a g a i n t h e s e v a l u e s are r e a s o n a b l e .  The  increase i n f r i c -  t i o n t h a t low f l o w s e x p e r i e n c e r e l a t i v e t o h i g h e r f l o w s a l s o seems r e a s o n able.  D u r i n g the f a l l i n g  limb o f the annual hydrograph  i n the lower c h a n n e l s o f the e s t u a r y [ P r e t i o u s , 1972], f l o w s e x p e r i e n c e bed  and thus t h e low w i n t e r  forms d e f i n e d by the h i g h e r summer f l o w s .  t h a t t h e r e l a t i v e roughness flows —  sediment: i s d e p o s i t e d  I t i s expected  o f t h e s e bed forms w i l l be g r e a t e r f o r the  low  and hence the g r e a t e r f r i c t i o n . 4.1.4  V e r i f i c a t i o n f o r the C o n d i t i o n s o f January 24,  made an independent  check on the f r i c t i o n  s e c t i o n a l g e o m e t r i e s , an attempt was  t o 4.11  [1952].  were o b t a i n e d .  The p r e d i c t e d  s e c t i o n have been s l i g h t l y  and dis-  cross-sectional  s e n s i t i v e t o changes i n the f r i c t i o n  In c e r t a i n s e c t i o n s o f the e s t u a r y , the low-flow  d i c t e d and o b s e r v e d peak  With minor  the water s u r f a c e e l e v a t i o n s  charge c u r v e s were found t o be i n s e n s i t i v e t o changes i n the geometries, but r e l a t i v e l y  cross-  made t o r e p r o d u c e t h e r e c o r d e d water  t o the f r i c t i o n c o e f f i c i e n t s ,  d i s c h a r g e s shown i n F i g u r e s 4.4  Having  c o e f f i c i e n t s and e f f e c t i v e  s u r f a c e e l e v a t i o n s and c u b a t u r e d i s c h a r g e s o f B a i n e s adjustments  1952.  coefficients.  f r i c t i o n c o e f f i c i e n t s o f the  lowered t o o b t a i n a b e t t e r f i t between the p r e discharges.  G e n e r a l l y , the f i t between the p r e d i c t e d and o b s e r v e d water s u r f a c e elevations i s satisfactory.  In the upper r e a c h e s o f the Main Stem, the  o f p r e d i c t e d water s u r f a c e e l e v a t i o n s i s somewhat g r e a t e r t h a n t h a t and i s p r o b a b l y due  t o "ending" the model e s t u a r y b e f o r e the t i d a l  range  observed, rise  and  55 Q = 36,500 c f s  M  (Chilliwack)  4  8  12  16  20  M  Hours Figure  4.4  T i d a l l y Varying  Stage and  D i s c h a r g e - Main  Arm  Q = 36,500 cfs m - j T r ,  (Chilliwack)  x . r . i .  January  x.  T i d a l Range a t S t e v e s t o n =11'  J  24,'  56  1952  'T  240 -200 O  12  3 O  I I  E o  •D U> O <»  •o c o CO  -160 -120  10  -80  9  -40  8  0  7  40  6  80 o  5  120  4  160  3  Main Arm - Stat.No.14 ( St. Mungo Cannery)  200  2  240  OJ  a>  14  °  13  CO  ° O O O  a>  a>  cn  </)  240  -200  I2h I I  o  -160  Z3  -120  O  101  -80  9  -40 0  8  40  7 o _l  6  Lu  5 4  \  3  Figure  Main Arm - Stat. No.18 (New Westminster R.R.Bridge)  //  M 4.5  / 8  Tidally  I 12  Hours  I 16  I 20  I  M  V a r y i n g Stage and D i s c h a r g e - Main Arm  z  80 120 180 200  £  in  a  Q = 36,500 cfs  (Chilliwack  •-  q>  cn  I  i-  o  M  4  8  12  16  20  M  Hours  Figure 4.6  T i d a l l y Varying Stage and Discharge - Main  Arm  Q = 36,500 c f s (Chilliwack) m  15  e O T3 W TJ O  c o CO  -  j  -  i  r  ,  T i d a l Range a t S t e v e s t o n =11'  January J  24, '  58  1952  140  14  H-120  13  -100  12  -80  I 1  -60  10  -40  9  -20  8  0  7 6  Observed  Stage  Predicted  Stage  Cubature Predicted  5  discharge  20 IBaines)  UZ  4 3  * "I 40 o -i  discharge  60 80  Main Arm - Stat. No.40 ( Whonock) J_  2  100 120  u O O O I  CO q>  -14,0 -120  cu o> o  o> t_  o x: o  «A  -100  CO  Q  -80 - 60 -40 -20 0 20 40 60 80  8  12 Hours  F i g u r e 4.7  16  20  M  T i d a l l y V a r y i n g Stage and D i s c h a r g e - Main Arm  100  Q = 36,500 Tidal  cfs  59  (Chilliwack)  j-ox.  m - j - i r ,  January  n . }  Range a t S t e v e s t o n = 1 1 '  M  4  8  12  2 4 ,1  1  16  1952  20  M  Hours Figure 4.8  T i d a l l y V a r y i n g Stage  and D i s c h a r g e  - North  Arm  Q = 36,500 c f s ( C h i l l i w a c k )  15  } January  60  24, 1952  T i d a l Range a t S t e v e s t o n = 11'  •30  25  't  20  13 h  - 15  12  - 10 5  I I I—  I.  10  10 -J •'/  Cubature  \\J  discharge  P r e d i c t e d  1/  t Baines ) \  15 20  discharge  Middle Arm - Stat. No. 133 ( Airport Floats )  H25  30 35 -70 -60 -50 o -I u.  -40  o  - 30  I-  - 20 - 10 0 5  10  o  _i • 20  b. Z  Canoe Pass - Stat. No. 124 ( Canoe Pass) M F i g u r e 4.9  4  8  12  Hours  16  20  30 40  M  T i d a l l y V a r y i n g Stage and D i s c h a r g e - M i d d l e Arm,  50  Canoe Pass  Q = 36,500 c f s ( C h i l l i w a c k )  2  T i d a l Range a t S t e v e s t o n  61  } January 24, 1952  = 11' -120 -100 -80  12  -60  I I  - 40  10  - 20  9  0  8  20  7 6  40  5  60  4  \\ Pitt - Stat.No. 144 (CPR Bridge Coquitlam)  3 2'  80 100 120  15  - 140  H-120  14 h13  -100  12  - 80  11  - 60  10  - 40  9  - 20  8  0  7  20  6  40  • j Pitt - Stat. No. 151 ' • (Gilley's Quarrey)  5 4 3  M  8  12  16  20  T i d a l l y Varying  Stage and D i s c h a r g e  - Pitt  80 100  M  Hours F i g u r e 4.10  60  River  u O O O  Urn  a  o M  Q =  3 6 , 5 0 0 c f s (Chilliwack )  j Jan. 24,1952  Tidal range at Steveston = l l '  M  4  Observed  Stage  Predicted  Stage  8  12  16  Hours  Figure T i d a l l y Varying  4.11  Stage —  P i t t Lake  20  M  63  f a l l o f the water water  s u r f a c e has d i s s i p a t e d .  The  f i t between the p r e -  d i c t e d d i s c h a r g e s and the c u b a t u r e d i s c h a r g e s o f B a i n e s i s n o t as good as the f i t between p r e d i c t e d and o b s e r v e d water s u r f a c e e l e v a t i o n s .  There  seems t o be a phase d i f f e r e n c e between the d i s c h a r g e s , the p r e d i c t e d charge t e n d i n g t o o c c u r e a r l i e r than the c u b a t u r e d i s c h a r g e . d i f f e r e n c e was  dis-  T h i s phase  i n s e n s i t i v e t o changes i n c r o s s - s e c t i o n a l geometry and  tion coefficients.  The  fric-  c u b a t u r e d i s c h a r g e c u r v e s behave somewhat e r r a t i c a l l y  between the times o f h i g h - h i g h - w a t e r and low-high-water.  It is  not c e r t a i n  whether t h i s b e h a v i o u r r e p r e s e n t s the t r u e d i s c h a r g e response o f the e s t u a r y , o r i s due t o i n a c c u r a c i e s i n the c u b a t u r e d i s c h a r g e c a l c u l a t i o n s . The t i d a l l y averaged d i s c h a r g e s were found t o s a t i s f y a t the Main Arm and the Main Arm was  - N o r t h Arm  j u n c t i o n , the N o r t h Arm  - Canoe Pass j u n c t i o n .  made over the e n t i r e e s t u a r y and was  o f t h i s mass b a l a n c e a r e shown i n T a b l e  A tidally  continuity  - M i d d l e Arm  junction  averaged mass b a l a n c e  found s a t i s f a c t o r y .  The  results  4.1.  To sum up, t h e b e h a v i o u r o f an e s t u a r y i s determined by t h e  inter-  a c t i o n o f the e f f e c t s o f r i v e r f l o w , t i d e s , bed forms and c r o s s - s e c t i o n a l geometry.  The f r e s h w a t e r d i s c h a r g e o f the F r a s e r R i v e r E s t u a r y undergoes  annual f l u c t u a t i o n  (see Appendix  B) and c o n s e q u e n t l y the bed  a large  forms o f t h e v a r -  i o u s c h a n n e l s , and p o s s i b l y t h e i r c r o s s - s e c t i o n a l g e o m e t r i e s , a r e i n a  contin-  u a l s t a t e o f dynamic e q u i l i b r i u m w i t h t h e f r e s h w a t e r f l o w s and t i d e s .  Bearing  t h i s i n mind, and t h e f a c t t h a t B a i n e s o b t a i n e d h i s r e s u l t s some 20  years  ago, and a d d i t i o n a l dykes, t r a i n i n g w a l l s , e t c . have been c o n s t r u c t e d i n the i n t e r i m , i t i s i m p o s s i b l e f o r the s i m p l e o n e - d i m e n s i o n a l  hydrodynamic  model t o e x a c t l y reproduce the b e h a v i o r he o b s e r v e d i n the e s t u a r y .  The  64  TABLE 4.1 TIDALLY AVERAGED MASS BALANCE OF PREDICTED DISCHARGES FOR JANUARY 24, 1952  CHANNEL  PREDICTED VALUES  DISCHARGE IN (1000*s c f s )  MAIN STEM  36.5  DISCHARGE OUT (1000*s c f s )  MAIN ARM NORTH ARM MIDDLE ARM CANOE PASS  27.8 2.9 0.5 1.2  TOTAL  INTERNAL STORAGE (1000's c f s )  PITT SYSTEM O T H E R  TOTAL  32.4  2.9 M).0  2.9  r e s u l t s shown i n F i g u r e s 4.4 t o 4.11 r e p r e s e n t t h e "best o v e r - a l l between p r e d i c t e d and o b s e r v e d As such,  fit"  water s u r f a c e e l e v a t i o n s and d i s c h a r g e s .  the p r e d i c t e d r e s u l t s are close t o the l i m i t o f r e s o l u t i o n o f  the one-dimensional  hydrodynamic model, o r i n o t h e r words, t h e p r e d i c t e d  r e s u l t s a r e t h e " b e s t " t h e model i s capable o f .  65  4.2  THE TIDALLY VARYING MASS TRANSPORT MODEL Because o f t h e l a c k o f f i e l d d a t a  (see S e c t i o n 3.2.1), a thorough  v e r i f i c a t i o n o f t h e t i d a l l y v a r y i n g mass transport'-, model was not p o s s i b l e . However, t h e model i s c o m p u t a t i o n a l l y numerical  s t a b l e , f r e e from t h e e f f e c t s o f  d i s p e r s i o n and s u f f i c i e n t l y f l e x i b l e t o be a d j u s t e d  when a v a i l a b l e .  I t reproduces the advective t r a n s p o r t process  demonstrates good agreement w i t h the standard dispersion of a slug load.  a n a l y t i c a l result f o r the  other peoples'  Advective  Transport.  t r a n s p o r t was s i m u l a t e d  r e s u l t s have been used t o  l a t e s the advective  In t h e t i d a l l y v a r y i n g model, t h e  by a g r i d o f moving p o i n t s on t h e advec-  t i v e c h a r a c t e r i s t i c s , as d e s c r i b e d  i n Chapter 3.  This process  directly  simu-  t r a n s p o r t o c c u r r i n g i n t h e a c t u a l e s t u a r y and does n o t  s u f f e r from n u m e r i c a l  dispersion  (see Appendix C ) . The hydrodynamic model  was used t o p r e d i c t t h e v e l o c i t i e s throughout t h e e s t u a r y v a l s and t h e s e v e l o c i t i e s were  a t half-hour  t h e n used t o advect t h e moving p o i n t s  Thus, t h e t i d a l l y v a r y i n g mass transport:: model w i l l s i m u l a t e t r a n s p o r t a s a c c u r a t e l y as t h e hydrodynamic model s i m u l a t e s v and  c o r r e c t l y and  the c o e f f i c i e n t s o f l o n g i t u d i n a l d i s p e r s i o n . 4.2.1  advective  data  To o b t a i n p r e l i m i n a r y n o t i o n s o f t h e t i d a l l y  varying behaviour o f the estuary, estimate  to field  t h e v e r i f i c a t i o n o f t h e hydrodynamic mode]  p a r t i a l v e r i f i c a t i o n o f the advective  interalong.  the advective the v e l o c i t i e s ,  c a n a l s o be r e g a r d e d as a  transport process.  The p r e d i c t e d a d -  v e c t i v e t r a n s p o r t was found t o be r e l a t i v e l y i n s e n s i t i v e t o i n a c c u r a c i e s as i n the advective v e l o c i t i e s . the a d v e c t i o n  T h i s i s i l l u s t r a t e d i n F i g u r e 4.12 which shows  o f a s l u g l o a d down t h e Main Arm - Main Stem o f two d i f f e r e n t  model e s t u a r i e s .  The f r e s h w a t e r  discharge  and q u a s i - s t e a d y  a r e t h e same f o r b o t h e s t u a r i e s , b u t t h e h i g h e r  tidal  conditions  f r i c t i o n o f one e s t u a r y  January 24, 1952 - Flow at Chilliwack « 36,500cfs  S—  20 h  Predicted  No.I ( (Marrow Section with high friction)  Predicted  No. 2 ( Broad Section with low friction)  AVERAGE VALUES ALONG MAIN ARM - MAIN STEM AVERAGE MANNINGS " N "  RESULTS  AVERAGE WIDTH ( feet )  AVERAGE DEPTH ( feet)  PREDICTED No. 1  0.042  1 3 50  30.6  PREDICTED No.2  0.0 30  18 7 0  22. 9  20  40  60 Hours Figure  Advection  80  100  120  4.12  Of A S l u g Load Down The Main Arm - Main Stem  CTl CT*  67  d i s t o r t s i t s v e l o c i t y f i e l d r e l a t i v e t o the v e l o c i t y f i e l d o f the other. The o n l y d i f f e r e n c e between b o t h s e t s o f r e s u l t s i s i n t h e t i d a l  excursion  i n t h e lower reaches o f t h e e s t u a r i e s , and t h i s i s n o t t h a t s i g n i f i c a n t . When the t i d a l l y v a r y i n g  model was run w i t h zero  dispersion,  the d i l u t i o n and a d v e c t i o n o f e f f l u e n t through t h e j u n c t i o n s was found t o be c o r r e c t  ( i t b e i n g r e c a l l e d t h a t c r o s s - s e c t i o n a l m i x i n g i s assumed t o be  complete a t t h e j u n c t i o n s ) .  The d i s c r e p a n c y i n the mass b a l a n c e ? o v e r any  t i d a l c y c l e was found t o be l e s s than 6% and was due t o t h e d i s c r e t e  spacing  between the moving p o i n t s . 4.2.2  Dispersive  Transport.  A s l u g l o a d was i n t r o d u c e d  at  S t a t i o n No. 50 on t h e Main Stem t o v e r i f y t h e c a p a b i l i t y o f t h e t i d a l l y varying  mass t r a n s p o r t model t o s i m u l a t e t h e d i s p e r s i o n p r o c e s s .  water d i s c h a r g e a t C h i l l i w a c k was 36,500 c u b i c persion  f e e t p e r second and the d i s -  c o e f f i c i e n t was s e t e q u a l t o 500 square f e e t p e r second and assumed  c o n s t a n t i n x and t . load  The f r e s h -  The a n a l y t i c a l s o l u t i o n f o r t h e d i s p e r s i o n o f a s l u g  i s g i v e n by [ F i s c h e r ,  c  1966a]  = ,  exp{-  m  Vwvt  — } 4  E  t  where M i s t h e mass p e r u n i t a r e a i n t r o d u c e d conditions,  i n t o the flow.  t h i s reduces t o  r  136 c = "Tt  e x  P  {  3.5X -, 2  —  }  where c i s the concentration  i n milligrams  per l i t r e ;  F o r the g i v e n  68  t  i s t h e e l a p s e d time i n hours;  and x i s the d i s t a n c e e i t h e r s i d e o f the mean v a l u e c o - o r d i n a t e s (5,000 f o o t segments). F i g u r e 4.13  shows the form o f the s l u g i n p u t s f o r the a n a l y t i c and  varying solutions. u l a t e s continuous The  (The e f f l u e n t d i s c h a r g e  tidally  i n t h e t i d a l l y v a r y i n g model sim-  d i s c h a r g e s , hence the form o f the s l u g i n p u t i n F i g u r e 4.13).  p r e d i c t e d and a n a l y t i c s o l u t i o n s f o r the d i s p e r s i o n o f the s l u g l o a d  shown i n F i g u r e 4.14 The  i n station  and  the agreement between b o t h s e t s o f r e s u l t s i s good.  h i g h e r peak c o n c e n t r a t i o n s  o f the a n a l y t i c s o l u t i o n are due  i n i t i a l c o n c e n t r a t i o n s o f the a n a l y t i c s l u g l o a d . d i s p e r s i o n process  It i s recalled that  higher the  this i s  4.14.  e s t i m a t i o n of the c o e f f i c i e n t s of l o n g i t u d i n a l d i s p e r s i o n i s  d i s c u s s e d i n d e t a i l i n Appendix F. o f t h i s study, given  t o the  i s assumed t o be G a u s s i a n i n i t s e n t i r e t y , and  i l l u s t r a t e d by t h e r e s u l t s of F i g u r e The  are  In o b t a i n i n g the t i d a l l y v a r y i n g  results  the c o e f f i c i e n t o f l o n g i t u d i n a l d i s p e r s i o n i s assumed t o  be  by  E =  (6 + S£)y  0 < t < T (4.1)  E = ay U  t > T  and (4.2)  = 0.06u where  /  E  Yr * ^ are the " i n s t a n t a n e o u s " v a l u e s o f t h e r e s p e c t i v e parameters d u r i n g t h e t i d a l c y c l e ; u  a n <  u  t i s the time t h a t has e l a p s e d s i n c e a " p a r c e l " o f the was d i s c h a r g e d i n t o the e s t u a r y ; 2 T = b /e  and i s the time s c a l e o f l a t e r a l m i x i n g charge i s assumed);  effluent  (an edge d i s -  69  Discharge  at  Station  No. 50  Q = 3 6 , 5 0 0 c f s (Chilliwack) \ Jan.24,1952 T i d a l Range at Steveston = 11 feet J  150  Mass = 1 6 5 0 x 1 0  lbs.  Slug load Soluti on  of Analytic  Slug load of Predicted Solution  21 22 Time ( hours )  Figure Slug  4.13  Inputs f o r A n a l y t i c and P r e d i c t e d D i s p e r s i o n  Solutions  Q = 3 6 , 5 0 0 cfs (Chilliwack) ) j Tidal Range at Steveston = II feet J  a  n  24 1952 '  3 HOURS AFTER INJECTION Predicted + Analytic C =  136 exp.{ . . f-3.47x" —} t in hours  x in station coordinates  50 Stations  J  L  55  6 HOURS AFTER INJECTION  45 Stations 40  50  r-  18 HOURS AFTER INJECTION 20  0  35  Stations Figure  4.14  D i s p e r s i o n Of A S l u g Load i n The Main Stem  45  71  and a i s a measure o f the e f f e c t s o f l a t e r a l v e l o c i t y g r a d i e n t s on the d i s p e r s i o n p r o c e s s and i s t a b u l a t e d i n Tables F . l . Equations  (4.1)  a l l o w f o r the i n c r e a s i n g c o n t r i b u t i o n o f the l a t e r a l  c i t y g r a d i e n t s on the d i s p e r s i o n p r o c e s s s e c t i o n and aspects  as the e f f l u e n t spreads across  the v a r i a t i o n o f the c o e f f i c i e n t d u r i n g the t i d a l c y c l e .  are d i s c u s s e d  varying concentration  i n Appendix F and  (4.1)  a c t u a l l y occurs  i t i s seen t h a t t h a t the peak t i d a l l y and  Although s i m p l i s t i c , t h e r e l a t i o n s h i p s  a r e thought t o be a r e a s o n a b l e  i n the  the  These  i s q u i t e s e n s i t i v e t o assumptions about the form  magnitude o f d i s p e r s i o n c o e f f i c i e n t . of Equations  velo-  a p p r o x i m a t i o n o f what  estuary.  Because the t i d a l l y v a r y i n g mass t r a n s p o r t e q u a t i o n  hastbeen  s o l v e d a l o n g the a d v e c t i v e c h a r a c t e r i s t i c s , the p o s i t i o n o f each e f f l u e n t " p a r c e l " and formation  the time t h a t i t has  i s "masked" i n a  spent i n the e s t u a r y  i s known.  (This i n -  f i x e d g r i d s o l u t i o n ) . Consequently, the  tidally  v a r y i n g mass t r a n s p o r t model can account f o r time dependent b e h a v i o u r o f i n d i v i d u a l " p a r c e l s " o f e f f l u e n t , as i s assumed i n E q u a t i o n s  4.3  THE  TIDALLY The  of advective As  AVERAGED MASS TRANSPORT MODEL t i d a l l y averaged d i s p e r s i o n c o e f f i c i e n t i n c l u d e s the e f f e c t s  t r a n s p o r t due  such, i t has  (4.1).  t o t i d a l f l o w r e v e r s a l and  no r e a l p h y s i c a l meaning and  longitudinal dispersion.  the v e r i f i c a t i o n o f a  tidally  averaged mass t r a n s p o r t model e s s e n t i a l l y c o n s i s t s o f f o r c i n g the model t o f i t f i e l d data. an  According  t o H o l l e y et al.  a -priori e s t i m a t e of t h i s c o e f f i c i e n t .  [1970] t h e r e Ward, and  i s no way Espey  suggest u s i n g the r e s u l t s o f a t i d a l l y v a r y i n g model t o e s t i m a t e  o f making  [1971] the  coeffi-  72  cient.  In t h e absence o f f i e l d d a t a , t h i s l a t t e r approach has been adopted  t o o b t a i n e s t i m a t e s o f the t i d a l l y averaged d i s p e r s i o n c o e f f i c i e n t s f o r the F r a s e r R i v e r  Estuary.  Thomann  [1971] l i s t s v a l u e s o f the t i d a l l y averaged d i s p e r s i o n  c o e f f i c i e n t f o r various estuaries.  They range from 1 - 2 0  day w i t h a mean v a l u e o f around 10 square m i l e s p e r day. t h e i n f l u e n c e o f the t i d a l l y  discharge o f a conservative  S t a t i o n No. 40 on t h e Main Stem o f the E s t u a r y .  s t a n d a r d i z e d by d i v i d i n g by t h e t i d a l l y  effluent  (The c o n c e n t r a t i o n s a r e  averaged c o n c e n t r a t i o n based on t h e  t o t a l mass o f e f f l u e n t d i s c h a r g e d d u r i n g t h e t i d a l flow a t C h i l l i w a c k .  F i g u r e 4.15 shows  averaged d i s p e r s i o n c o e f f i c i e n t on t h e p r e d i c t e d  c o n c e n t r a t i o n s f o r a steady, c o n t i n u o u s at  square m i l e s p e r  T h i s i s p l o t t e d as t h e y  cycles  and t h e f r e s h w a t e r  , the subscript s i g n i f y i n g ta  t h a t t h e r e s u l t s have been o b t a i n e d  from t h e t i d a l l y averaged model).  The  o n l y s i g n i f i c a n t d i f f e r e n c e between t h e r e s u l t s i s i n t h e upstream e x c u r s i o n . The  c o n c e n t r a t i o n g r a d i e n t s a r e s m a l l downstream o f t h e e f f l u e n t  and t h e d i s p e r s i o n has l i t t l e  effect.  discharge,  I t i s r e c a l l e d t h a t the r e s u l t s o f  F i g u r e 4.15 have been o b t a i n e d f o r a c o n s e r v a t i v e e f f l u e n t .  I f t h e concen-  t r a t i o n o f t h e e f f l u e n t decays w i t h time, as w i t h BOD, t h e r e w i l l be  concentra-  t i o n g r a d i e n t s downstream o f the d i s c h a r g e p o i n t , and t h e downstream  concentra-  t i o n s w i l l a l s o depend on the v a l u e s o f t h e t i d a l l y averaged d i s p e r s i o n c o e f f i cient  (although t h e downstream c o n c e n t r a t i o n s ^ a r e n o t v e r y s e n s i t i v e t o t h e  values of the d i s p e r s i o n c o e f f i c i e n t ) .  The decrease  i n concentrations  around  S t a t i o n s Nos. 18 and 24 i s due t o t h e e f f e c t s o f t h e j u n c t i o n s and i n t r o d u c e s an e r r o r o f about 10% i n t o t h e p r e d i c t e d c o n c e n t r a t i o n s . this effect  occurs.  I t i s n o t known why  Q = 3 6 , 5 0 0 c f s (Chilliwack) ) T i d a l R a n g e at S t e v e s t o n =11 f e e t J  J  a  n  .2  4  >  |952  E = lOsquare miles per day  l.2r1.0 0.8 -  0.4 0.2 0  10  20 Stations  30 along Main Arm  Figure  40  50  Main Stem  4.15  I n f l u e n c e Of The T i d a l l y Averaged D i s p e r s i o n C o e f f i c i e n t  Oh P r e d i c t e d T i d a l l y Averaged  Concentrations  74  F i g u r e s 4.16  and 4.17  show the maximum upstream e x c u r s i o n on  the f l o o d f l o w f o r e f f l u e n t r e l e a s e d a l o n g the v a r i o u s c h a n n e l s . lower reaches o f the Main Arm  i t i s assumed t h a t the t i d a l l y  d i s p e r s i o n c o e f f i c i e n t i s e q u a l t o 20 square m i l e s per day, i s reduced a l o n g t h e Main Arm 4.16.  A l o n g the N o r t h Arm  30 square m i l e s p e r day  (the r a p i d decrease  advective v e l o c i t y i n P i t t Lake). d i s p e r s i o n may  and t h i s v a l u e  d i s p e r s i o n i s assumed t o  and a l o n g P i t t R i v e r i t i s assumed t o e q u a l  t h e P i t t R i v e r , as shown i n F i g u r e 4.17,  averaged  averaged  - Main Stem a c c o r d i n g t o t h e r e s u l t s o f F i g u r e  the t i d a l l y averaged  equal 10 square m i l e s p e r day,  In the  i n the upstream e x c u r s i o n a l o n g  i s due  t o the d r a s t i c decrease i n  A l t h o u g h the a b s o l u t e v a l u e s o f the  tidally  not be c o r r e c t , and t h i s i s d i s c u s s e d i n Chapter  t h e r e l a t i v e v a l u e s a l o n g the e s t u a r y s h o u l d be  reasonable.  5,  Maximum Upstream Excursion  (Stations)  Q - 3 6 , 5 0 0 c f s (Chilliwack) ) Tidal Range at Steveston = 11 feet J  J  a  n  .  2  4  i  l  9  5  2  Effluent moves into Pitt Lake 01  Effluent moves through N.Arm - M.Arm Junction  c o 8 o CO  c o  «i _ 6 3  O X  LU E  4  Pitt Lake  t_  to Q. Z>  E 2 3  E  o 2  0 "00  110  J  L  118  Position of Effluent  140 Discharge  ( Station  150  along North Arm and Pitt River  Numbers )  Figure  4.17  Maximum Upstream E x c u r s i o n D u r i n g Flow R e v e r s a l  In The North Arm  And P i t t  River  CHAPTER 5  COMPARISON AND  The to  DISCUSSION OF RESULTS  t h r e e models d i s c u s s e d i n the l a s t two  c h a p t e r s are now  o b t a i n ' a . p r e l i m i n a r y i n d i c a t i o n o f the s i g n i f i c a n c e o f t i d a l e f f e c t s  p r e d i c t e d c o n c e n t r a t i o n s i n the F r a s e r R i v e r E s t u a r y . c o n d i t i o n s o f January 24,  1952  The  f l o w and  were used i n t h i s i n v e s t i g a t i o n .  t y p i c a l h i g h t i d e - low flow c o n d i t i o n s and were used i n the of  used  the hydrodynamic model.  The  36,500 c u b i c f e e t per second and  freshwater  tidal  These are  verification  d i s c h a r g e a t C h i l l i w a c k was  the t i d a l range a t S t e v e s t o n  c o n s e r v a t i v e e f f l u e n t and an edge d i s c h a r g e were assumed. model was  on  11 f e e t .  The  A  hydrodynamic  used t o o b t a i n t h e t i d a l l y v a r y i n g v e l o c i t i e s and c r o s s - s e c t i o n a l  areas a l o n g the v a r i o u s channels with estimated  of the estuary.  These v a l u e s ,  together  d i s p e r s i o n c o e f f i c i e n t s , were then used t o p r e d i c t the  t i d a l l y varying concentrations.  The  freshwater  d i s c h a r g e a t C h i l l i w a c k , the  t i d a l l y averaged a r e a s and e s t i m a t e d d i s p e r s i o n c o e f f i c i e n t s were used t o p r e d i c t the t i d a l l y averaged c o n c e n t r a t i o n s . Before p r e s e n t i n g the r e s u l t s o f t h i s i n v e s t i g a t i o n , the of  the mass t r a n s p o r t models are b r i e f l y r e c a l l e d :  dimensional  b o t h models are  assumptions one-  and p r e d i c t the c r o s s - s e c t i o n a l l y averaged c o n c e n t r a t i o n s ;  c r o s s - s e c t i o n a l mixing  i n both models i s assumed t o be complete a t the  the junc-  t i o n s ; the i n f l u e n c e o f t h e saltwedge i s i g n o r e d i n both models; the d i s p e r s i o n p r o c e s s o f the t i d a l l y v a r y i n g model i s assumed t o be G a u s s i a n and  i n i t s entirety;  i n the t i d a l l y averaged model, a l l e f f e c t s o f the t i d e s a r e lumped i n t o  tidally  averaged d i s p e r s i o n c o e f f i c i e n t .  77  the  78  5.1  THE  EFFECTS OF LATERAL DISPERSION In the i n i t i a l p e r i o d b e f o r e c r o s s - s e c t i o n a l mixing  i s complete,  there are s i g n i f i c a n t l a t e r a l concentration gradients across the  estuary,  and the peak c r o s s - s e c t i o n a l c o n c e n t r a t i o n w i l l be c o n s i d e r a b l y h i g h e r  than  the v a l u e p r e d i c t e d by e i t h e r mass t r a n s p o r t model.  has  not been q u a n t i t a t i v e l y a s s e s s e d ,  i t is  While t h i s e f f e c t  noted t h a t the t i d a l l y v a r y i n g  mass t r a n s p o r t model can e a s i l y be adapted t o g i v e an approximate  estimate  of t h i s e f f e c t .  coefficient  The  l a t e r a l c o n c e n t r a t i o n p r o f i l e depends on the  o f l a t e r a l d i s p e r s i o n , t h e p o s i t i o n o f the e f f l u e n t o u t f a l l i n the c r o s s - s e c t i o n and i n the e s t u a r y  estuary  the p e r i o d of time t h a t a " p a r c e l " o f e f f l u e n t has  [Ward, 1972].  spent  Because the t i d a l l y v a r y i n g mass t r a n s p o r t  model has been s o l v e d a l o n g i t s a d v e c t i v e c h a r a c t e r i s t i c s , the p o s i t i o n o f each e f f l u e n t " p a r c e l " and the time t h a t i t has known, and  spent i n the e s t u a r y i s  the p r e d i c t e d c o n c e n t r a t i o n s can be a d j u s t e d t o account f o r the  e f f e c t s of l a t e r a l dispersion.  (This approach was  used i n a l l o w i n g f o r the  assumed time dependent v a r i a t i o n i n the c o e f f i c i e n t o f l o n g i t u d i n a l d i s p e r s i o n ) . In t h e lower reaches  o f the e s t u a r y , t h e r e are t h r e e d i s t i n c t  i n each double t i d a l c y c l e o f 25 hours; i o d o f weak ebb  t o 4.10  i o d s i s seen t o be a p p r o x i m a t e l y p r i n c i p a l l y due  a p e r i o d of strong f l o o d flows; a per-  and weak f l o o d flows and a p e r i o d o f s t r o n g ebb  i s apparent i n F i g u r e s 4.4  periods  flows.  This  and t h e d u r a t i o n o f each of the t h r e e  e i g h t hours.  The  c r o s s - s e c t i o n a l mixing  peris  to the e f f e c t s o f secondary f l o w s , as d i s c u s s e d i n Appendix  E, and w i l l be g r e a t e s t d u r i n g the p e r i o d s o f s t r o n g ebb c r o s s - s e c t i o n a l mixing d u r i n g the s t r o n g ebb  i s not complete a t the Main Arm  and  flood flows.  - North Arm  If  junction  flow, or a t the Main Stem - P i t t R i v e r j u n c t i o n d u r i n g  79  the s t r o n g f l o o d flow, t h e e f f l u e n t w i l l not be advected t i o n s a c c o r d i n g t o t h e simple p o r t models.  flow b a l a n c e  through these  o f t h e one-dimensional  The i n t e r a c t i o n o f t h e time dependent b e h a v i o u r  junc-  mass t r a n s -  o f the l a t e r a l  m i x i n g p r o c e s s w i t h t h e a d v e c t i o n o f e f f l u e n t through t h e j u n c t i o n s i s v e r y i n v o l v e d and can o n l y be r e l i a b l y determined from f i e l d s t u d i e s .  If suffi-  c i e n t f i e l d d a t a were a v a i l a b l e , i t may be p o s s i b l e t o e m p i r i c a l l y a l l o w f o r t h i s e f f e c t i n t h e t i d a l l y v a r y i n g model.  5.2  THE INITIAL DILUTION OF EFFLUENT When e f f l u e n t i s d i s c h a r g e d  i n t o an e s t u a r y , t h e t i d a l l y v a r y i n g  f l o w s cause a v a r i a t i o n i n i t s i n i t i a l d i l u t i o n d u r i n g t h e t i d a l c y c l e . process at  i s d e s c r i b e d i n S e c t i o n 1.3 and i s seen t o generate  times o f s l a c k w a t e r .  This  concentration spikes  F i g u r e 5.1 shows t y p i c a l v a r i a t i o n s i n t h e i n i t i a l  d i l u t i o n d u r i n g t h e double t i d a l c y c l e a t f o u r s t a t i o n s a l o n g t h e e s t u a r y . D u r i n g a double t i d e c y c l e t h e r e a r e f o u r s l a c k w a t e r s of  the estuary  at  S t a t i o n s Nos. 10, 22 and 102 a r e apparent.  i n the lower  reaches  (see F i g u r e s 4.4 t o 4.10) and t h e r e s u l t i n g c o n c e n t r a t i o n peaks At S t a t i o n No. 50, t h e t i d a l  flows a r e s i g n i f i c a n t l y s m a l l e r i n r e l a t i o n t o t h e f r e s h w a t e r  flow, and t h e i r  combined i n t e r a c t i o n r e s u l t s i n o n l y two e f f e c t i v e peaks d u r i n g the double tidal  cycle. Because o f t h e s m a l l e r flows along t h e North Arm, t h e t i d a l l y  v a r y i n g c o n c e n t r a t i o n s a t S t a t i o n No. 102 a r e s i g n i f i c a n t l y h i g h e r than the v a l u e s a l o n g the Main Arm - Main Stem.  The t i d a l l y averaged d i l u t i o n i s a l s o  shown i n F i g u r e 5.1, t h e h i g h e r v a l u e s a t S t a t i o n s Nos. 10 and 102 r e f l e c t i n g the d i v i s i o n o f t h e f r e s h w a t e r  d i s c h a r g e between the North Arm and Main Arm.  Q = 3 6 , 5 0 0 c f s ( Chilliwack ) \ j T i d a l Range at S t e v e s t o n = l l f e e t j  4  0  12  18  4 r  . 24,1952  — • — * ~  24 (Hours)  Station No. 22  0  n  Station No. 10  2  c o  a  v  1  '  i  12  ^  r—  18  24(Hours)  Station No.50 18  24(Hours)  Station No.102  6  12  F i g u r e 5.1 I n i t i a l D i l u t i o n At S t a t i o n Nos. 10, 22, 50 and 102  24(Hours)  8  °  81  The  peak t i d a l l y  about twice  the  varying concentrations tidally  averaged  i n F i g u r e 5.1  are seen to  values.  The minimum t i d a l l y v a r y i n g c o n c e n t r a t i o n o c c u r s s t r o n g ebb curves  and  f l o o d f l o w s , and  o f F i g u r e 5.1.  due  i s r e f l e c t e d as the  At S t a t i o n s Nos.  t i o n i s approximately  Lake;  P i t t R i v e r j u n c t i o n are d i l u t e d by of the P i t t  10 and  h a l f the minimum v a l u e  to the p r e s e n c e of P i t t  system and  22,  flat  during  the  sections of  t h i s minimum  at S t a t i o n No.  discharges  the  concentra-  50.  This i s  downstream from the Main Stem-  the combined e f f e c t s o f the t i d a l  the Main Stem above the j u n c t i o n , whereas  upstream of the j u n c t i o n a r e d i l u t e d o n l y by  5.3  be  the  prisms  discharges  latter.  PEAK EFFLUENT CONCENTRATIONS As w e l l as t h e e f f e c t s o f v a r i a t i o n i n i n i t i a l d i l u t i o n , c e r t a i n  s l u g s o f water are dosed w i t h t i d a l flow r e v e r s a l . erates concentration  T h i s process spikes.  d i l u t i o n and m u l t i p l e d o s i n g t r a t i o n i s higher illustrated  for  the  Now  initial  combine t o produce a compound s p i k e whose concen-  a t S t a t i o n No.  50 i s shown i n F i g u r e E.2;  50.  The  This i s well  variation  of  note t h e s l a c k w a t e r s  t h a t the m i n i m a l v e l o c i t y a t hour 6. 5 and  T h i s i s the  at  reason  t h e s p i k e a t hour 6 i n  c o n s i d e r t h e b e h a v i o u r o f the water mass around S t a t i o n  50 d u r i n g t h i s time. and  of  and a l s o gen-  However, t h e e f f e c t s of v a r i a t i o n i n  i n c r e a s e d c o n c e n t r a t i o n a t hours 4 and  F i g u r e 5.1.  outfall  i s d e s c r i b e d i n S e c t i o n 1.3  by an e f f l u e n t d i s c h a r g e  5 and  t o the e f f e c t s  t h a n t h a t o f t h e i n d i v i d u a l component s p i k e s .  v e l o c i t y a t S t a t i o n No. hours 4 and  e f f l u e n t s e v e r a l t i m e s due  During  No.  hour 4 a s l u g o f water moves downstream p a s t  i s dosed, d u r i n g hour 5 i t moves back upstream and  i s dosed  the  again  82  and f i n a l l y d u r i n g hour s i x i t moves downstream p a s t the o u t f a l l i s dosed yet  again.  The r e s u l t o f t h i s i s t h a t the c o n c e n t r a t i o n i n t h i s s l u g o f  water i s 12 times g r e a t e r t h a n the t i d a l l y averaged v a l u e as i s shown i n F i g u r e F.2  f o r the curve w i t h z e r o l o n g i t u d i n a l d i s p e r s i o n .  composed o f t h e t h r e e subspikes generated d u a l magnitudes b e i n g 3, 3 and The multiple the t i d e .  s p i k e s due  This spike i s  a t hours 4, 5 and 6, t h e i r  6 as shown i n F i g u r e  5.1.  to the e f f e c t s o f v a r i a t i o n i n i n i t i a l d i l u t i o n  A p a r t i c u l a r l y s e n s i t i v e p o r t i o n o f the double t i d a l c y c l e i s the f l o o d flows where t h e r e are t h r e e s l a c k w a t e r s  space o f e i g h t hours.  The v e l o c i t i e s between these times  r e l a t i v e l y low  (see F i g u r e s 4.4  to  Once a s p i k e i s generated  t o be reduced  t r a t e d i n F i g u r e 5.2  o f both,  i t s peak c o n c e n t r a t i o n This i s i l l u s -  f o r a steady e f f l u e n t d i s c h a r g e a t S t a t i o n No.  v i o u s v a l u e o f 12 i s due  (The decrease  However, as the s p i k e i s advected  seem t o i n d i c a t e t h a t mass i s not conserved  the s p i k e i s not c o n s t a n t .  The  a t hour 6 w i t h a peak c o n c e n t r a t i o n t h a t i s 11 from t h e p r e -  down the  the d i s p e r s i o n p r o c e s s reduces the peak c o n c e n t r a t i o n as shown. F i g u r e 5.2  50.  t o the e f f e c t s o f d i s p e r s i o n on t h e s u b s p i k e s  5).  are  by the e f f e c t s of v a r i a t i o n i n i n i t i a l  times g r e a t e r t h a n the t i d a l l y averaged v a l u e .  e r a t e d a t hours 4 and  of slackwater  by t h e l o n g i t u d i n a l d i s p e r s i o n p r o c e s s .  compound s p i k e i s generated  i n the  4.10).  d i l u t i o n , m u l t i p l e d o s i n g o r a combination  of  and  ?dosing combine because b o t h e f f e c t s occur around t h e same phase of  p e r i o d o f weak ebb and  begins  indivi-  The  estuary, results  s i n c e the area under  T h i s i s o n l y an apparent e f f e c t due  to e x t r a p o l a t i n g  the c o n c e n t r a t i o n s o f f the a d v e c t i v e c h a r a c t e r i s t i c s onto t h e s t a n d a r d f o o t space g r i d , as d e s c r i b e d i n S e c t i o n 3.2.  gen-  In a c t u a l f a c t , the  5,000  initial  base w i d t h of the s p i k e i s o n l y 500-800 f e e t , which i s too f i n e t o be  resolved  Q = 3 6 , 5 0 0 c f s at Chilliwack ^ , . J ^ . I Jan. 24,1952 Tidal Range at Steveston II J o y l  l f t C  0  12  1 hour  10  8  Vtv  6  o  1  —  Stations  along  Figure  Main Stem  5.2  D i s p e r s i o n Of A C o n c e n t r a t i o n  Spike  03 CO  84  by t h e f i x e d g r i d .  (Note t h a t the s p i k e i s c o r r e c t l y r e s o l v e d on t h e  char-  acteristics) . The most s i g n i f i c a n t e f f e c t s o f v a r i a t i o n i n i n i t i a l and m u l t i p l e d o s i n g o c c u r i n the f i r s t of  e f f l u e n t has been d i s c h a r g e d  initial  1-2  dilution  t i d a l cycles after a "parcel"  i n t o the e s t u a r y .  c o n c e n t r a t i o n g r a d i e n t s o f t h e s p i k e and  T h i s i s due  t o the  high  assumed time-dependent  i n c r e a s e i n the c o e f f i c i e n t o f l o n g i t u d i n a l d i s p e r s i o n (see Appendix F ) . These a s p e c t s are apparent from F i g u r e 5.2  and a r e a l s o i l l u s t r a t e d i n Appendix  F. F o r the steady d i s c h a r g e o f a c o n s e r v a t i v e e f f l u e n t , the peak t i d a l l y v a r y i n g c o n c e n t r a t i o n was  found  t o be from 1 t o 10 times  t h a n t h e v a l u e p r e d i c t e d by the t i d a l l y averaged model. summarized i n T a b l e at  the d e s i g n a t e d  These r e s u l t s  5.1" and were o b t a i n e d f o r a s i n g l e e f f l u e n t  station.  the d i s c h a r g e p o i n t ) .  greater are  discharge  (The peak t i d a l l y v a r y i n g d i s c h a r g e o c c u r s a t  The  r e s u l t s of Table  5.1 a r e s e n s i t i v e t o the magni-  tude arid assumed temporal v a r i a t i o n of the c o e f f i c i e n t o f l o n g i t u d i n a l  dis-  p e r s i o n , as d i s c u s s e d and  results  of  i l l u s t r a t e d i n Appendix F.  T a b l e F . l , the c o e f f i c i e n t o f l o n g i t u d i n a l d i s p e r s i o n was  as g i v e n i n E q u a t i o n  (4.1).  assumed t o  be  The peak c r o s s - s e c t i o n a l c o n c e n t r a t i o n w i l l  s i g n i f i c a n t l y h i g h e r than t h e s e  5.4  In o b t a i n i n g the  be  values.  UPSTREAM EFFLUENT TRANSPORT In a t i d a l l y v a r y i n g mass t r a n s p o r t model, e f f l u e n t i s t r a n s p o r t e d  upstream by the e f f e c t s o f d i s p e r s i o n and a d v e c t i o n d u r i n g t i d a l Because o f the l a r g e t i d a l f l o w s , a d v e c t i v e t r a n s p o r t i s the more  flow  reversal.  important  STATION ( Ttv/ *Xa) max.  North Arm  Main Arm - Main Stem  CHANNEL  Pitt River  8  14  22  30  40  50  60  106  116  142  150  5.4  3.8  3.8  10.7  5.0  10.9  1.0  7.2  8.2  1.2  1.2  Table  5.1  R a t i o o f Peak T i d a l l y V a r y i n g C o n c e n t r a t i o n t o T i d a l l y Averaged A t P o i n t Of E f f l u e n t Discharge  Value  86  component f o r the F r a s e r R i v e r E s t u a r y , and 4.17.  as i s apparent from F i g u r e s  4.16  In a t i d a l l y averaged model, upstream t r a n s p o r t i s by the  averaged d i s p e r s i o n p r o c e s s .  F i g u r e s 5.3  and  v a r y i n g and t i d a l l y averaged c o n c e n t r a t i o n s discharges.  In a l l c a s e s ,  5.4  show t h e p r e d i c t e d  tidally tidally  a t S t a t i o n s upstream from  effluent  the t i d a l l y averaged v a l u e i s s i g n i f i c a n t l y  t h a n the t i d a l l y v a r y i n g v a l u e s .  less  A b e t t e r f i t between t h e r e s u l t s c o u l d  o b t a i n e d by i n c r e a s i n g the v a l u e s o f the t i d a l l y a v e r a g e d M i s p e r s i o n  be  coeffi-  cient. The at  r e s u l t s o f F i g u r e 5.4  S t a t i o n s Nos.  10 and  14.  are f o r simultaneous e f f l u e n t  Once a g a i n , the e f f e c t s o f v a r i a t i o n i n  d i l u t i o n and m u l t i p l e d o s i n g cause a c o n c e n t r a t i o n s p i k e a t b o t h p o i n t s , s p i k e A b e i n g generated a t S t a t i o n No. 14.  Nbte t h a t between hours 6 and  S t a t i o n No.  6 i s z e r o due  flood tide.  The  discharges initial  discharge  10 and s p i k e B at S t a t i o n  17 the t i d a l l y v a r y i n g c o n c e n t r a t i o n  t i d a l l y averaged c o n c e n t r a t i o n i s not a good i n d i c a t i o n o f the v a r y i n g response upstream of S t a t i o n No.  5.5  14,  between the two  the  assumed t o  be z e r o , a more r e a l i s t i c boundary c o n d i t i o n b e i n g p o s s i b l y as shown.  downstream of S t a t i o n No.  at  t o uncontaminated seawater moving upstream on  c o n c e n t r a t i o n boundary c o n d i t i o n a t the sea was  No.  The  tidally  s t a t i o n s or  10.  CHANNEL INTERACTIONS Consider  now  the e f f e c t o f an e f f l u e n t d i s c h a r g e  on t h e water q u a l i t y i n another c h a n n e l .  The  i n one  s i g n i f i c a n c e of t h i s  channel effect  w i l l depend on the p r o x i m i t y o f the e f f l u e n t d i s c h a r g e t o the j u n c t i o n . the d i s c h a r g e  p o i n t i s s u f f i c i e n t l y f a r upstream from a j u n c t i o n , t h e  If  effluent  Discharge at Stat. No. 14 ( Ma in Arm)  6  12  18  24(Hours)  Discharge at Stat. No.22 ( Main Arm)  6  12  Figure  18  5.3  Predicted C o n c e n t r a t i o n s i n One Channel Caused By E f f l u e n t D i s c h a r g e In Another Channel  24(Hours)  Q = 36,500cfs  (Chilliwack)  j Jan.24,1952  Tidal Range at Steveston = Ufeet  Station No.17 2 r  24 (Hours) r  Station No.12 ' 1^  -  0  8  /*»  spi ke B  1 1  1  6  12  Station  1  24( Hours)  18  No.6 spike A  spike B  More Realistic Boundary Condition  0  -Assumed Boundary Condition .  12 Figure  18  24( Hours)  5.4  P r e d i c t e d C o n c e n t r a t i o n s Due t o Two E f f l u e n t In The Main Stem  Discharges  89  w i l l be d i s p e r s e d both l a t e r a l l y the j u n c t i o n .  Under t h e s e  and  c o n d i t i o n s , the t i d a l l y  i n b o t h c h a n n e l s downstream o f the t i d a l l y varying values. discharge for  l o n g i t u d i n a l l y by t h e time i t r e a c h e s  Of more i n t e r e s t i s t h e case where t h e  i s downstream from a j u n c t i o n .  P i t t River.  concentration  j u n c t i o n w i l l be a good measure o f  i n t e r a c t i o n s between the Main Arm  Stem and  averaged  Consider  and  This i s i l l u s t r a t e d North Arm  and  the e f f l u e n t d i s c h a r g e d  r e t u r n s p a s t S t a t i o n No.  144  effluent  i n Figure  5.3  between the Main a t S t a t i o n No.  a t hour 8 a s p i k e i s a d v e c t e d up P i t t R i v e r p a s t S t a t i o n No. s t r o n g f l o o d f l o w and  the  144  on  22: the  a t h o u r 23 on the  strong  ebb  t i d e , i t s c o n c e n t r a t i o n b e i n g reduced by the e f f e c t s o f d i s p e r s i o n d u r -  ing  i t s residence i n P i t t River.  advection  F o r t h e two  causes a s l u g o f c o n t a m i n a t e d water t o be  channel d u r i n g each double t i d e c y c l e i n the N o r t h Arm another.  cases  o f 25 h o u r s .  a t hour 22 marks t h e end  o f one  o f F i g u r e 5.3, fed i n t o the The  zero  upstream other  concentration  s l u g and the b e g i n n i n g  of  In b o t h c a s e s , t h e t i d a l l y averaged c o n c e n t r a t i o n i s not a good  i n d i c a t i o n o f the t i d a l l y v a r y i n g v a l u e s , a l t h o u g h s e c t i o n , t h e f i t can be  improved by  as remarked i n a  previous  i n c r e a s i n g the t i d a l l y averaged d i s p e r s i o n  coefficient.  5.6  SUMMARY In t h i s c h a p t e r t h e r e s u l t s from the t i d a l l y v a r y i n g and  tidally  averaged mass t r a n s p o r t models have been compared t o o b t a i n a p r e l i m i n a r y i n d i c a t i o n o f the s i g n i f i c a n c e o f t i d a l t i o n s i n the F r a s e r R i v e r  Estuary.  e f f e c t s on the p r e d i c t e d  concentra-  90  The  t i d a l l y v a r y i n g flows cause a v a r i a t i o n i n t h e i n i t i a l  d i l u t i o n o f a discharged r e v e r s a l , independently  effluent.  T h i s , and t h e e f f e c t s o f t i d a l  generate concentration spikes.  flow  Because b o t h e f f e c t s  occur around the same phase o f the t i d e , t h e p e r i o d o f weak f l o o d and ebb flows i n each double t i d a l c y c l e b e i n g e s p e c i a l l y s e n s i t i v e , t h e i n d i v i d u a l s p i k e s combine-to g e n e r a t e a s p i k e whose c o n c e n t r a t i o n was e s t i m a t e d from 1 t o 10 times g r e a t e r t h a n t h e p r e d i c t e d t i d a l l y Because o f incomplete  concentration.  l a t e r a l m i x i n g , t h e peak c r o s s - s e c t i o n a l c o n c e n t r a t i o n  w i l l be c o n s i d e r a b l y h i g h e r than these v a l u e s . s p i k e s a r e most s i g n i f i c a n t i n the f i r s t ation.  averaged  t o be  The e f f e c t s o f c o n c e n t r a t i o n  1-2 t i d a l c y c l e s a f t e r t h e i r gener-  A f t e r t h i s p e r i o d o f time, t h e l o n g i t u d i n a l d i s p e r s i o n i s due t o  the e f f e c t s o f l a t e r a l c o n c e n t r a t i o n g r a d i e n t s and t h e c o n c e n t r a t i o n o f t h e s p i k e i s r a p i d l y reduced. The movement o f water t h r o u g h t h e F r a s e r R i v e r E s t u a r y i s a complex process freshwater  t h a t i s i n f l u e n c e d by the l a r g e t i d a l e f f e c t s , t h e s i g n i f i c a n t  discharge  (even a t low f l o w s ) , t h e v a r i o u s c h a n n e l s and j u n c t i o n s  of t h e e s t u a r y and the p r e s e n c e o f P i t t Lake.  I f an e f f l u e n t d i s c h a r g e i s  s u f f i c i e n t l y f a r upstream from t h e j u n c t i o n s , the s p i k e s w i l l be f l a t t e n e d and t h e e f f l u e n t d i s p e r s e d over t h e c r o s s - s e c t i o n by t h e time i t i s advected to the j u n c t i o n .  Under these c o n d i t i o n s , t h e p r e d i c t e d t i d a l l y averaged be-  h a v i o u r a t and downstream o f t h e j u n c t i o n i s a good e s t i m a t e varying behaviour,  and b o t h should be a r e a s o n a b l e  behaviour i n the estuary.  E f f l u e n t discharges  No. 45 a r e expected t o behave i n t h i s manner.  o f the t i d a l l y  approximation t o the a c t u a l  i n t h e Main Stem above S t a t i o n Throughout t h e lower reaches  o f t h e estuary,,;including P i t t R i v e r , t h e p r e d i c t e d t i d a l l y averaged b e h a v i o u r  91  i s n o t a good i n d i c a t i o n o f t h e t i d a l l y v a r y i n g b e h a v i o u r . e f f e c t s o f incomplete l a t e r a l 5.1, t h e a c t u a l r e s p o n s e  Because o f t h e  m i x i n g and j u n c t i o n s , as d i s c u s s e d  i n Section  o f t h e e s t u a r y may be somewhat d i f f e r e n t from the  p r e d i c t e d t i d a l l y varying behaviour. By a d j u s t i n g t h e t i d a l l y averaged d i s p e r s i o n c o e f f i c i e n t s , i t i s possible to obtain a " b e s t - f i t " t i d a l l y averaged c o n c e n t r a t i o n s .  between t h e p r e d i c t e d t i d a l l y v a r y i n g and However, s i n c e the t i d a l l y averaged  p e r s i o n c o e f f i c i e n t has no r e a l p h y s i c a l meaning, t h e r e unique s e t o f v a l u e s efficients  f o r the estuary.  i s probably  dis-  no  Rather, t h e b e s t f i t d i s p e r s i o n c o -  w i l l v a r y w i t h t h e p o s i t i o n and number o f e f f l u e n t d i s c h a r g e s .  CHAPTER 6  SUMMARY AND  CONCLUSIONS  Mass t r a n s p o r t models a r e commonly used t o i n v e s t i g a t e s i t u a t i o n s o f e x i s t i n g o r p o t e n t i a l water q u a l i t y c o n f l i c t s i n e s t u a r i e s .  Such models  are used t o p r e d i c t the c o n c e n t r a t i o n o f t h e o f f e n d i n g substance througho u t t h e e s t u a r y and a r e e s s e n t i a l l y o f two t y p e s : a l l o w f o r t i d a l e f f e c t s and those  those t h a t  t h a t do n o t . While t i d a l l y  correctly varying  models c o r r e c t l y a l l o w f o r t h e e f f e c t s o f the t i d e s , t h e i r development and a p p l i c a t i o n i n v o l v e s s i g n i f i c a n t l y more work than f o r t h e i r averaged c o u n t e r p a r t s .  In t i d a l l y averaged models, a l l t i d a l  lumped i n t o t h e t i d a l l y  averaged d i s p e r s i o n c o e f f i c i e n t ,  models a r e r e l a t i v e l y easy t o d e v e l o p and a p p l y , of the s i g n i f i c a n c e of t i d a l t h i s study, b o t h a t i d a l l y model have been  e f f e c t s are  and w h i l e  such  they g i v e no i n d i c a t i o n  e f f e c t s on the p r e d i c t e d c o n c e n t r a t i o n s .  developed and a p p l i e d t o the F r a s e r R i v e r E s t u a r y .  i n v e s t i g a t e d by comparing t h e p r e d i c t e d c o n c e n t r a t i o n s assumed e f f l u e n t d i s c h a r g e s .  T h i s was  from both models f o r  The t i d a l l y averaged model used i n t h i s  study  The t i d a l l y v a r y i n g model was d e v e l o p e d from  f i r s t p r i n c i p l e s , and d u r i n g i t s development the problems o f n u m e r i c a l  The  the a b i l i t y o f the t i d a l l y  averaged model t o d e s c r i b e t h e t i d a l l y v a r y i n g c o n c e n t r a t i o n s .  [1963].  In  averaged and a t i d a l l y v a r y i n g mass t r a n s p o r t  p r i n c i p a l o b j e c t o f t h i s study was t o a s s e s s  i s due t o Thomann  tidally  i t was n e c e s s a r y  dispersion, stability,  92  t o consider  the s i g n i f i c a n c e of l a t e r a l  93  d i s p e r s i o n and the time dependent b e h a v i o u r o f the c o e f f i c i e n t o f  longi-  t u d i n a l d i s p e r s i o n d u r i n g the i n i t i a l p e r i o d b e f o r e c r o s s - s e c t i o n a l is  mixing  complete. The p r i n c i p a l c o n c l u s i o n s t o emerge from t h i s study a r e : 1. Numerical d i s p e r s i o n can be e l i m i n a t e d from t h e f i n i t e d i f f e r e n c e s o l u t i o n o f mass t r a n s p o r t e q u a t i o n s by s o l v i n g the e q u a t i o n s a l o n g t h e a d v e c t i v e c h a r a c t e r i s tics.  • 2. The s t a b i l i t y requirements of e x p l i c i t f i n i t e d i f f e r e n c e schemes have been shown t o be r e l a t e d t o the speed o f i n f o r m a t i o n p r o p a g a t i o n . Advective i n s t a b i l i t i e s a r e e l i m i n a t e d by s o l v i n g the mass t r a n s p o r t equat i o n a l o n g the a d v e c t i v e c h a r a c t e r i s t i c s .  3. E x i s t i n g t h e o r i e s have been shown t o apparentl y u n d e r e s t i m a t e the s i g n i f i c a n c e o f secondary c u r r e n t s on the l a t e r a l mixing p r o c e s s . Secondary c u r r e n t s have been e x p l a i n e d i n terms o f t h e g e n e r a t i o n and a d v e c t i o n of v o r t i c i t y , and e s t i m a t e d v a l u e s show good agreement w i t h l i m i t e d f i e l d data. Revised estimates of l a t e r a l d i s p e r sion i n d i c a t e s i g n i f i c a n t l y f a s t e r l a t e r a l mixing.  4. I f the mass t r a n s p o r t e q u a t i o n i s s o l v e d a l o n g the a d v e c t i v e c h a r a c t e r i s t i c s , t h e time dependent b e h a v i o u r of the c o e f f i c i e n t o f l o n g i t u d i n a l d i s p e r s i o n d u r i n g t h e i n i t i a l p e r i o d b e f o r e c r o s s - s e c t i o n a l m i x i n g i s complete can be t a k e n i n t o account. A l s o , a one-dimensional equat i o n can be used t o e s t i m a t e l a t e r a l c o n c e n t r a t i o n p r o f i l e s .  5. The e f f e c t o f the t i d e has been shown t o i n t r o duce " s p i k e s " i n t o the c o n c e n t r a t i o n p r o f i l e a l o n g t h e e s t u ary. These s p i k e s are caused by v a r i a t i o n i n the i n i t i a l d i l u t i o n o f a d i s c h a r g e d e f f l u e n t and m u l t i p l e d o s i n g due to t i d a l flow r e v e r s a l .  6. S i g n i f i c a n t l y more work and r e s o u r c e s a r e i n v o l v e d i n the development and a p p l i c a t i o n o f a t i d a l l y v a r y i n g mass t r a n s p o r t model than f o r a t i d a l l y averaged model. The time r e q u i r e d f o r development, the amount o f f i e l d d a t a r e q u i r e d f o r v e r i f i c a t i o n and the amount o f computer time r e q u i r e d t o a n a l y z e an i d e n t i c a l s i t u a t i o n i s e s t i m a t e d t o be an o r d e r o f magnitude g r e a t e r f o r the t i d a l l y v a r y i n g model as compared t o t h e t i d a l l y averaged model.  94  7. The t i d a l l y a v e r a g e d c o n c e n t r a t i o n s were not found t o be a good i n d i c a t i o n o f t h e t i d a l l y v a r y ing response o f the Fraser R i v e r Estuary. The p e a k c r o s s - s e c t i o n a l l y averaged c o n c e n t r a t i o n p r e d i c t e d b y t h e t i d a l l y v a r y i n g m o d e l was f o u n d t o b e f r o m o n e t o 10 t i m e s g r e a t e r t h a n t h e t i d a l l y a v e r a g e d v a l u e s . The  c o n c l u s i o n s a r e now d i s c u s s e d i n d e t a i l .  In this  study  two m a s s t r a n s p o r t m o d e l s a n d a h y d r o d y n a m i c m o d e l w e r e d e v e l o p e d  and  a p p l i e d t o t h e F r a s e r R i v e r E s t u a r y , t h e hydrodynamic model b e i n g used t o p r e d i c t t h e temporal  variation  mass t r a n s p o r t m o d e l .  i n the parameters  of the t i d a l l y varying  F i n i t e d i f f e r e n c e methods have been used t o s o l v e  the -underlying p a r t i a l d i f f e r e n t i a l  equations o f a l l t h r e e models.  Impor-  tant aspects of t h i s type o f s o l u t i o n a r e the problems of numerical p e r s i o n and s t a b i l i t y .  T h e r e h a s b e e n some c o n f u s i o n i n t h e l i t e r a t u r e  o v e r t h e o r i g i n a n d means o f c o n t r o l l i n g n u m e r i c a l d i s p e r s i o n . s t u d y , n u m e r i c a l d i s p e r s i o n h a s b e e n shown t o r e s u l t f r o m t r a n s p o r t equation over a f i x e d amental  dis-  s o l v i n g t h e mass  space g r i d r a t h e r t h a n a l o n g t h e more  advective characteristics;  vective characteristics,  In this  fund-  i ft h e equation i s s o l v e d along the ad-  numerical d i s p e r s i o n i s eliminated.  As a p p l i e d ,  Thomann's s o l u t i o n i s s t e a d y s t a t e a n d h a s no n u m e r i c a l d i s p e r s i o n b e c a u s e it  i s independent  of time.  In a d d i t i o n t o t h e problem  s i o n , there i s a l s o t h e problem schemes. mation  I n Appendix C t h i s  of stability  of numerical disper-  i n explicit  i s s e e n t o be r e l a t e d  finite  difference  t o t h e speed o f i n f o r -  propagation along the c h a r a c t e r i s t i c s of the respective p a r t i a l  f e r e n t i a l equations.  S t a b i l i t y requirements  space and time increments  govern* t h e r e l a t i v e  size of the  i n t h e hydrodynamic equations and i n t h e d i s p e r s i o n  s t e p o f t h e t i d a l l y v a r y i n g mass t r a n s p o r t e q u a t i o n . requirement  dif-  T h e r e i s no  f o r Thomann's s o l u t i o n a s i t i s i n d e p e n d e n t  o f time.  stability However,  95  t h e r e i s a l i m i t on t h e r e l a t i v e magnitudes o f the t i d a l l y averaged advect i v e and d i s p e r s i v e t r a n s p o r t p r o c e s s e s , in  the o f f e n d i n g segment w i l l become  and i f v i o l a t e d , the c o n c e n t r a t i o n  negative.  There a r e a number o f advantages t o s o l v i n g t h e t i d a l l y mass t r a n s p o r t e q u a t i o n  a l o n g the a d v e c t i v e c h a r a c t e r i s t i c s ;  varying  i tresults i n  a d i r e c t s i m u l a t i o n o f t h e a d v e c t i v e t r a n s p o r t o c c u r r i n g i n the e s t u a r y , a u s e f u l separation o f the advective i s achieved,  and the  and d i s p e r s i v e t r a n s p o r t  p o s i t i o n o f each s e p a r a t e  time t h a t i t has spent  i n t h e e s t u a r y i s known.  processes  " e f f l u e n t p a r c e l " and t h e This l a s t piece o f i n f o r -  mation has been used t o account f o r an assumed > time-dependent i n c r e a s e i n t h e c o e f f i c i e n t o f l o n g i t u d i n a l d i s p e r s i o n i n the i n i t i a l p e r i o d c r o s s - s e c t i o n a l m i x i n g i s complete, and a l s o a l l o w s t h e t i d a l l y  before varying  mass t r a n s p o r t model t o be adapted t o p r e d i c t l a t e r a l c o n c e n t r a t i o n A most u s e f u l f e a t u r e o f s o l v i n g t h e e q u a t i o n  a l o n g thej-advective  profiles.  charac-  t e r i s t i c s i s t h a t a d d i t i o n a l moving p o i n t s can be p l a c e d on t h e c h a r a c t e r i s t i c s t o more a c c u r a t e l y d e f i n e r e g i o n s o f r a p i d v a r i a t i o n i n c o n c e n t r a t i o n , such as o c c u r a t e f f l u e n t o u t f a l l s a t times o f s l a c k w a t e r , necessary  and un-  moving p o i n t s can be removed from r e g i o n s o f slow v a r i a t i o n .  c o n c e n t r a t i o n s p i k e s g e n e r a t e d by the e f f e c t s o f v a r i a t i o n i n i n i t i a l t i o n and m u l t i p l e d o s i n g a r e i n i t i a l l y f e e t wide a t t h e base.  To a c h i e v e  The  dilu-  v e r y sharp and o n l y some 500 - 800  adequate r e s o l u t i o n o f such a s p i k e  w i t h a f i x e d g r i d s o l u t i o n would r e q u i r e a space g r i d approximately f i n e r t h a n the standard  The  5,000 f o o t space g r i d used i n t h i s  study.  advantages o f s o l v i n g the mass t r a n s p o r t e q u a t i o n  a d v e c t i v e c h a r a c t e r i s t i c s must be b a l a n c e d  10 times  along the  a g a i n s t t h e somewhat u n t i d y  "book-  keeping"of s o l u t i o n . r e s u l t s i n h e r e n t t o c h a r a c t e r i s t i c s methods o f s o l u t i o n .  95a  In t h e t i d a l l y v a r y i n g model o f t h e F r a s e r R i v e r E s t u a r y , t h i s book-keeping i s complicated  by t h e m u I t i - c h a n n e l e d  a c t i o n s o f these channels  n a t u r e o f t h e e s t u a r y and t h e i n t e r -  at their junctions.  But w h i l e awkward, t h e book-  k e e p i n g o f r e s u l t s was n o t e x c e s s i v e l y d i f f i c u l t . Of t h e t h r e e models used i n t h i s study, o n l y the hydrodynamic model has been v e r i f i e d w i t h any degree o f t h o r o u g h n e s s .  Because t h e t i d a l l y  averaged  d i s p e r s i o n c o e f f i c i e n t s have no r e a l p h y s i c a l meaning, a t i d a l l y  averaged  model i s f o r c e d t o reproduce  being r i g o r o u s l y v e r i f i e d . vented  measured f i e l d r e s u l t s r a t h e r than  Lack o f f i e l d  data prevented  this,  and a l s o p r e -  t h e complete v e r i f i c a t i o n o f t h e t i d a l l y v a r y i n g mass t r a n s p o r t model.  However, a l l t h r e e models a r e s u f f i c i e n t l y  f l e x i b l e t o s i m u l a t e t h e range o f  f l o w and t i d a l c o n d i t i o n s o f t h e F r a s e r R i v e r E s t u a r y and c a n be a d j u s t e d t o fit  f i e l d r e s u l t s when a v a i l a b l e .  E s s e n t i a l l y , t h e e f f e c t o f t h e t i d e s i s t o cause s p i k e s i n t h e concentration p r o f i l e along the estuary.  In a one-dimensional  t i d a l l y v a r y i n g f l o w s cause a v a r i a t i o n i n t h e i n i t i a l charged  model, t h e  dilution of a d i s -  e f f l u e n t , t h e c o n c e n t r a t i o n b e i n g g r e a t e s t a t times o f s l a c k w a t e r .  T h i s generates  s p i k e s i n t h e c o n c e n t r a t i o n p r o f i l e a l o n g the e s t u a r y .  The  e f f e c t s o f t i d a l f l o w r e v e r s a l r e s u l t i n c e r t a i n s l u g s o f water b e i n g dosed w i t h e f f l u e n t s e v e r a l t i m e s , and t h i s a l s o g e n e r a t e s  concentration spikes.  Because o f t h e asymmetric nature o f t h e t i d e s t h e r e i s a p e r i o d o f weak f l o o d and ebb f l o w s once i n each double are three slackwaters in  tidal cycle.  During  t h i s time  there  and t h e v e l o c i t i e s a r e low, and t h e e f f e c t s o f v a r i a t i o n  i n i t i a l d i l u t i o n and m u l t i p l e d o s i n g i n t e r a c t t o form a compound s p i k e  96  whose c o n c e n t r a t i o n i s s i g n i f i c a n t l y g r e a t e r than t h a t o f t h e component spikes. A f t e r a s p i k e has been g e n e r a t e d ,  i t s c o n c e n t r a t i o n i s reduced  by the e f f e c t s o f l a t e r a l and l o n g i t u d i n a l d i s p e r s i o n .  The i n i t i a l  l o n g i t u d i n a l d i s p e r s i o n a f t e r a " p a r c e l o f e f f l u e n t " has been  discharged  i n t o t h e e s t u a r y , i s due p r i n c i p a l l y t o t h e e f f e c t s o f v e r t i c a l gradients.  velocity  However, when t h e e f f l u e n t i s mixed o v e r t h e c r o s s - s e c t i o n ,  l a t e r a l v e l o c i t y g r a d i e n t s dominate i t s l o n g i t u d i n a l d i s p e r s i o n . for  To .account  t h i s e f f e c t , i t was assumed t h a t t h e c o e f f i c i e n t o f l o n g i t u d i n a l  dis-  p e r s i o n i n c r e a s e d between t h e s e two extremes as d e s c r i b e d i n Appendix F. A l s o , t h e c o e f f i c i e n t o f l o n g i t u d i n a l d i s p e r s i o n was assumed t o v a r y d i r e c t l y as t h e a b s o l u t e v e l o c i t y . a r e thought t o be a r e a s o n a b l e  While s i m p l i s t i c ,  approximation  these v a r i a t i o n s  o f what o c c u r s  i n the estuary.  The peak t i d a l l y v a r y i n g c o n c e n t r a t i o n a t t h e p o i n t o f e f f l u e n t was e s t i m a t e d  discharge  t o be from one t o 10 times: g r e a t e r than t h e v a l u e p r e d i c t e d  from t h e t i d a l l y averaged model.  Because t h e e f f l u e n t i s n o t u n i f o r m l y  d i s t r i b u t e d over t h e c r o s s - s e c t i o n , t h e peak l a t e r a l c o n c e n t r a t i o n w i l l be much g r e a t e r than t h e s e v a l u e s .  A f t e r a s p i k e has been i n t h e e s t u a r y f o r  s e v e r a l t i d a l c y c l e s , i t s c o n c e n t r a t i o n has been g r e a t l y reduced  by t h e  longitudinal dispersion process. In t h e e s t u a r y w i t h r e l a t i v e l y the F r a s e r , l a t e r a l mixing in  the lower r e a c h e s  s h o r t residence times  w i l l be o f c o n s i d e r a b l e importance. F o r example,  o f t h e e s t u a r y , t h e r e s i d e n c e time i s o n l y 2-4 t i d e  c y c l e s , and e f f l u e n t may n o t be c o m p l e t e l y mixed o v e r even when i t l e a v e s the e s t u a r y . mixing  such as  the c r o s s - s e c t i o n  In t h e i n i t i a l p e r i o d b e f o r e c r o s s - s e c t i o n a l  i s complete,the peak l a t e r a l c o n c e n t r a t i o n i s s i g n i f i c a n t l y  greater  97  than the c r o s s - s e c t i o n a l averaged v a l u e p r e d i c t e d by model.  the t i d a l l y v a r y i n g  However, the model can be adapted t o p r e d i c t l a t e r a l  p r o f i l e s as was  discussed previously.  I f c r o s s - s e c t i o n a l mixing  p l e t e a t a d i v e r g i n g j u n c t i o n , the e f f l u e n t may j u n c t i o n a c c o r d i n g t o the simple  concentration  flow balance  n o t be advected  i s n o t comthrough  the  o f the mass t r a n s p o r t models.  To a c c o u n t f o r t h i s e f f e c t i n a model would r e q u i r e c o n s i d e r a b l e f i e l d  data.  Because o f t h e i n f l u e n c e o f s e c o n d a r y c u r r e n t s , c r o s s - s e c t i o n a l m i x i n g  is  thought t o be r e l a t i v e l y r a p i d i n t h e F r a s e r R i v e r E s t u a r y .  flows  Secondary  have been t e n t a t i v e l y e x p l a i n e d i n terms o f the g e n e r a t i o n and a d v e c t i o n vorticity,  and  on t h e b a s i s of l i m i t e d f l o a t s t u d i e s , show good agreement  w i t h v a l u e s measured i n the To sum i s a complex charge,  up,  estuary.  t h e movement o f water t h r o u g h the F r a s e r R i v e r  the v a r i o u s channels The  and  a d v e c t i v e and  j u n c t i o n s o f the e s t u a r y and t h e d i s p e r s i v e t r a n s p o r t processes  that distribute  e f f l u e n t t h r o u g h o u t the- water mass o f the e s t u a r y a r e s i m i l a r l y  and  i n p a r t i c u l a r a r e s i g n i f i c a n t l y a f f e c t e d by t h e t i d e s .  aged model, w h i l e  s i m p l e r to d e v e l o p  and  A tidally  T h i s study  t o s o l v e the t o t a l problem o f c a l c u l a t i n g e f f l u e n t  aver-  does  concentrations  Rather, i t has c o n c e n t r a t e d  l o p i n g s t a b l e m a t h e m a t i c a l models f r e e from n u m e r i c a l a p r e l i m i n a r y i n v e s t i g a t i o n of the  complex,  a p p l y t o the e s t u a r y , does n o t g i v e  a good i n d i c a t i o n o f t h e t i d a l l y v a r y i n g c o n c e n t r a t i o n s .  i n an e s t u a r y as complex as t h e F r a s e r .  dis-  presence  an  not pretend  Estuary  phenomenon t h a t i s a f f e c t e d by the t i d e s , the f r e s h w a t e r  of P i t t Lake.  of  dispersion to  s i g n i f i c a n c e of t i d a l e f f e c t s .  on deveallow  In a d d i -  t i o n an assessment i s made of the importance o f l a t e r a l d i s p e r s i o n , and i t is  seen t h a t w i t h  simple m o d i f i c a t i o n s , the t i d a l l y v a r y i n g model can  l e a s t p a r t i a l l y a c c o u n t f o r the e f f e c t s o f l a t e r a l d i s p e r s i o n .  at  R E F E R E N C E S  Water Surface Elevations and Tidal Discharges in the Fraser River Estuary, January 23 and 24, 1952, Report No. MH-32,  Baines, W. 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Control Fed., 3 7 ( 5 ) : 659-673.  A P P E N D I C E S  APPENDIX A  DERIVATION OF THE ONE-DIMENSIONAL  A. 1  MASS TRANSPORT EQUATION  GENERAL The mass t r a n s p o r t p r o c e s s e s  and m o l e c u l a r  of advection, turbulent d i f f u s i o n  d i f f u s i o n t r a n s p o r t and d i s t r i b u t e any d i s s o l v e d  throughout t h e water mass o f a r i v e r o r e s t u a r y .  substance  (Advection i s a l s o r e -  f e r r e d t o as c o n v e c t i o n o r f o r c e d c o n v e c t i o n i n t h e l i t e r a t u r e ) .  A t the  time and space s c a l e s o f i n t e r e s t , t h e c o n t r i b u t i o n t o mass t r a n s p o r t by molecular  d i f f u s i o n i s i n s i g n i f i c a n t compared t o t h e o t h e r p r o c e s s e s , and  i s not considered f u r t h e r .  The i n a b i l i t y o f t h e a d v e c t i v e o r b u l k  velocity  t o d e s c r i b e t h e d i s t r i b u t i o n o f v e l o c i t y over a c r o s s - s e c t i o n o f t h e r i v e r or e s t u a r y makes, i t necessary namely d i s p e r s i o n .  t o p o s t u l a t e another  t r a n s p o r t mechanism,  The mechanisms o f a d v e c t i v e and d i s p e r s i v e t r a n s p o r t  are d e s c r i b e d i n S e c t i o n s A.2 and A.3. The a mass balance  one-dimensional  mass t r a n s p o r t e q u a t i o n  o f d i s s o l v e d substance  s l i c e o f the estuary.  over an element c r o s s - s e c t i o n a l  D i s s o l v e d substance  t h i s volume by t h e p r o c e s s e s  i s d e r i v e d by t a k i n g  i s t r a n s p o r t e d i n t o and o u t o f  o f a d v e c t i o n and d i s p e r s i o n , and these  t o g e t h e r w i t h any s o u r c e - s i n k e f f e c t s t h a t t h e substance  undergoes w i t h i n t h e  water mass, determine i t s c o n c e n t r a t i o n w i t h i n the e l e m e n t a l The  o n l y s p a t i a l v a r i a t i o n t h a t i s admitted  mass t r a n s p o r t e q u a t i o n  processes,  i n the  volume. one-dimensional  i s a l o n g i t u d i n a l v a r i a t i o n along the estuary.  The  c o n c e n t r a t i o n , l o n g i t u d i n a l v e l o c i t y , depth and d i s p e r s i o n c o e f f i c i e n t a r e  106  107  a l l a s s i g n e d t h e i r average c r o s s - s e c t i o n a l tively) .  F o r the sake of b r e v i t y ,  values  (c, u, y and  the average c r o s s - s e c t i o n a l  any parameter w i l l be r e f e r r e d t o as i t s "mean" v a l u e . quency t u r b u l e n t  The d i r e c t i o n o f t h e c o - o r d i n a t e axes x, y and  the p o s i t i v e x a x i s p o i n t i n g down. The  freout o f  c.  F i g u r e A . l shows an e l e m e n t a l c r o s s - s e c t i o n a l estuary.  value of  The h i g h  f l u c t u a t i o n s are assumed t o have been averaged  the mean v a l u e s u and  E respec-  seawards and  s l i c e o f an z are as shown,  the p o s i t i v e y a x i s  Note t h a t the c o - o r d i n a t e axes a r e E u l e r i a n ,  vertically  o r f i x e d i n space.  f l o w i s assumed t o be b o t h unsteady and non-uniform so t h a t c, u, y, E  and t h e c r o s s - s e c t i o n a l distance  (x).  a r e a A v a r y both w i t h time  (t) and  longitudinal  The w i d t h o f e s t u a r y b i s assumed to v a r y w i t h x o n l y .  It  i s assumed t h a t t h e l o n g i t u d i n a l v a r i a t i o n i n a l l these parameters i s continuous.  A.2  LONGITUDINAL ADVECTIVE TRANSPORT The d i s t r i b u t i o n o f l o n g i t u d i n a l v e l o c i t y i n an e s t u a r y i s  three-dimensional.  Variations  i n the l o n g i t u d i n a l d i r e c t i o n o c c u r  due  t o changes i n t h e c r o s s - s e c t i o n a l a r e a o f the e s t u a r y , and over any section, variations  i n the l a t e r a l and v e r t i c a l d i r e c t i o n s o c c u r due  the f r i c t i o n a l i n f l u e n c e the s o l i d bed.  cross-  of n e i g h b o u r i n g  fluid  l a y e r s w i t h each o t h e r  to and  In a r i v e r o r e s t u a r y , the l a t e r a l d i s t r i b u t i o n o f l o n g i -  t u d i n a l v e l o c i t y i s m o d i f i e d by the presence  of bends, and  i n an  t h e v e r t i c a l d i s t r i b u t i o n o f l o n g i t u d i n a l v e l o c i t y i s m o d i f i e d by additional influence  of  salinity.  estuary, the  108  Figure A . l E l e m e n t a l C r o s s - S e c t i o n a l S l i c e of A R i v e r or  Estuary  109  In one-dimensional mass t r a n s p o r t models, i t i s assumed t h a t a c r o s s - s e c t i o n a l s l i c e o r s l u g o f water moves a l o n g t h e r i v e r o r e s t u a r y a t the mean c r o s s - s e c t i o n a l v e l o c i t y u.  The t r a n s p o r t o f d i s s o l v e d sub-  stance i n t h i s s l u g o f water i s termed longitudinal The direction.  advective  advection.  t r a n s p o r t i n a r i v e r i s always i n the downstream  In an e s t u a r y ,  the net advective  t r a n s p o r t over a t i d a l  i s i n t h e downstream d i r e c t i o n because o f t h e e f f e c t o f f r e s h w a t e r  cycle inflow.  However, because o f f l o w r e v e r s a l due t o t i d a l e f f e c t s , t h e a d v e c t i v e t r a n s p o r t w i l l be i n t h e upstream d i r e c t i o n over p o r t i o n o f the t i d a l cycle. The mass o f d i s s o l v e d substance section o f a r i v e r or estuary  (M) advected through any c r o s s -  i n time 6 t i s g i v e n by  M = uAc 6 t ,  and  thus t h e n e t mass (M \ r e m a i n i n g i n t h e e l e m e n t a l volume o f F i g u r e net  A . l due t o a d v e c t i v e  transport during  M  A.3  net  =  time 6 t i s  g  - -  - r—[uAc}6x6t 9x  (A.l)  LONGITUDINAL DISPERSIVE TRANSPORT A d v e c t i o n accounts f o r the t r a n s p o r t o f d i s s o l v e d substance i n  the l o n g i t u d i n a l d i r e c t i o n due t o t h e  assumed u n i f o r m d i s t r i b u t i o n o f t h e  v e l o c i t y over the c r o s s - s e c t i o n , u.,,The v e r t i c a l and l a t e r a l v e l o c i t y gradients  t h a t e x i s t over any c r o s s - s e c t i o n r e s u l t i n s m a l l  water p r e c e d i n g u.  and l a g g i n g t h e l i n e o f a d v e c t i v e  This i s i l l u s t r a t e d  "parcels" of  advance, a s d e f i n e d by  for a vertical velocity profile typical of a river  110  Vertical Distribution of Velocity  UAt  A  V7 o o  —~  o -^Line source Y position after o0 Time At 0 a 0  >v  >  o  o  i  0  °  1  °  o  °  0  o  o  o o o o o o o  A  Figure  Effect of Diffusion  f ^ ^ ^ - - A d v e c t i o n Due to Velocity Gradient  A.2  Dispersive E f f e c t s of V e r t i c a l V e l o c i t y  Gradients  Ill  i n F i g u r e A.2.  I t i s seen t h a t a s l u g i n p u t from a l i n e source i s not  advected downstream as a s l u g l o a d , but advance.  The  i s spread  around the l i n e o f mean  e f f e c t of t h i s i s t o reduce the peak c o n c e n t r a t i o n and  the c o n c e n t r a t i o n  g r a d i e n t s , as p r e d i c t e d on the b a s i s o f a d v e c t i o n  Superimposed on the e f f e c t s o f v e l o c i t y g r a d i e n t s of t u r b u l e n t d i f f u s i o n .  i s the  The mass t r a n s p o r t a s s o c i a t e d w i t h t h e  v e l o c i t y f l u c t u a t i o n s r e s u l t s i n a f u r t h e r f l a t t e n i n g of the gradients.  The  i n F i g u r e A.2.  The  and t u r b u l e n t d i f f u s i o n i n s p r e a d i n g advance  (u) i s termed  q u a l i f i e d , the terms " a d v e c t i o n " v e c t i o n and  longitudinal  and  1960] [Hinze,  equation.  dispersion.  "AA"  line  Unless . otherwise  " d i s p e r s i o n " w i l l be t a k e n t o mean ad-  the d i s p e r s i o n p r o c e s s  and  Holley  [19 69a]dis-  dispersion.  [Bird,  t u r b u l e n t d i f f u s i o n i n homogeneous i s o t r o p i c f r e e  turbu-  can be r e p r e s e n t e d  Both m o l e c u l a r  diffu-  diffusion  1959]  Taylor  mathematically  s i m i l a r i t y between the mechanisms o f m o l e c u l a r  s i o n , t u r b u l e n t d i f f u s i o n and  lence  concentration  combined e f f e c t s o f v e l o c i t y g r a d i e n t s  i n c l u d e d i n the mass t r a n s p o r t e q u a t i o n .  c u s s e d the u n d e r l y i n g  s  turbulent  d i s p e r s i o n i n the l o n g i t u d i n a l d i r e c t i o n .  i t i s t o be  et al.  process  the d i s s o l v e d substance around the  I t i s necessary to represent if  alone.  e f f e c t d f t u r b u l e n t d i f f u s i o n o v e r the v e r t i c a l s e c t i o n  i s also i l l u s t r a t e d  of advective  flatten  m a t h e m a t i c a l l y by a F i c k i a n d i f f u s i o n  [1954] showed t h a t a f t e r a s u i t a b l e time had e l a p s e d ,  l o n g i t u d i n a l d i s p e r s i o n i n flow  through a p i p e can a l s o be r e p r e s e n t e d  one-dimensional F i c k i a n d i f f u s i o n equation.  Elder  Fischer  a  [1966a]  p e r i o d f o l l o w i n g the r e l e a s e of  t r a c e r , the e f f e c t s of v e l o c i t y g r a d i e n t s outweighed the e f f e c t s o f l e n t d i f f u s i o n , and  by  [1959] found a s i m i l a r  r e s u l t t o h o l d i n model experiments o f open channel f l o w . i l l u s t r a t e d t h a t i n the i n i t i a l n o n - F i c k i a n  the  turbu-  l e d t o a skewed d i s t r i b u t i o n of mean c o n c e n t r a t i o n  in  the  l o n g i t u d i n a l d i r e c t i o n , as i s i l l u s t r a t e d i n F i g u r e  t o F i c k ' s Law, t h e d i s t r i b u t i o n o f c o n c e n t r a t i o n Fischer has  A.2.  According  s h o u l d be G a u s s i a n , b u t  [1966a] has shown t h a t t h i s o n l y o c c u r s a f t e r c r o s s - s e c t i o n a l m i x i n g  reduced the s p a t i a l v a r i a t i o n o f c o n c e n t r a t i o n  a v a l u e much s m a l l e r applying  o v e r the c r o s s - s e c t i o n t o  than the mean c r o s s - s e c t i o n a l v a l u e .  the o n e - d i m e n s i o n a l mass t r a n s p o r t  I n d e r i v i n g and  e q u a t i o n , the d i s p e r s i o n  process  i n i t s e n t i r e t y i s assumed t o be F i c k i a n . In a c t u a l f a c t , t h e l o n g i t u d i n a l d i s p e r s i o n o f e f f l u e n t i n a r i v e r or estuary  i s a complex phenomenom t h a t i s i n i t i a l l y  the e f f e c t s o f v e r t i c a l v e l o c i t y g r a d i e n t s .  As t u r b u l e n t  c o n t r o l l e d by d i f f u s i o n and  secondary flows d i s t r i b u t e t h e e f f l u e n t mass over t h e c r o s s - s e c t i o n , the l a t e r a l v e l o c i t y gradients nal dispersion process,  e x e r t an i n c r e a s i n g i n f l u e n c e on the l o n g i t u d i -  The l a t e r a l m i x i n g due t o t u r b u l e n t  secondary flows i s d i s c u s s e d  i n Appendix E and an e s t i m a t e o f the c o e f f i -  c i e n t s o f l o n g i t u d i n a l d i s p e r s i o n o f the e s t u a r y  i s given  e f f e c t s o f v e r t i c a l and l a t e r a l v e l o c i t y g r a d i e n t s Appendix F ) .  d i f f u s i o n and  i n Appendix F.  are a l s o discussed i n  When t h e c r o s s - s e c t i o n a l m i x i n g i s e s s e n t i a l l y complete (and  the d i s p e r s i o n p r o c e s s i s F i c k i a n )  the e f f e c t s of l a t e r a l v e l o c i t y  gradients  dominate the l o n g i t u d i n a l d i s p e r s i o n and a r e o f t h e o r d e r o f 20 - 100 greater  than t h e e f f e c t s o f v e r t i c a l v e l o c i t y g r a d i e n t s  In an e s t u a r y ,  the d i s p e r s i o n p r o c e s s i s m o d i f i e d  o f s a l i n i t y and t i d a l e f f e c t s .  by t h e a d d i t i o n a l  [1970] , and  the d i s p e r s i o n c o e f f i c i e n t s f o r the  i n Appendix F.  According to Fick's substance t r a n s p o r t e d by  influence  The i n f l u e n c e o f t i d a l o s c i l l a t i o n s on t h e  t h e i r method has been used i n e s t i m a t i n g River Estuary  times  [ F i s c h e r , 1966a].  d i s p e r s i o n c o e f f i c i e n t has been i n v e s t i g a t e d by H o l l e y et al.  Fraser  (The  law o f d i f f u s i o n , the n e t mass o f d i s s o l v e d  through u n i t c r o s s - s e c t i o n a l a r e a i n time 6t i s g i v e n  [ B i r d et al., 1960] M = -E—fit 3x  (A. 2)  113  the minus s i g n a r i s i n g because t h e n e t t r a n s p o r t decreasing  concentration.  expected from t h e p r e v i o u s  i s i n the d i r e c t i o n o f  E i s t h e d i s p e r s i o n c o e f f i c i e n t , and as i s discussions o f the dispersion process,  depends  on t h e d i s t r i b u t i o n o f v e l o c i t y over t h e c r o s s - s e c t i o n and t h e c o e f f i c i e n t o f t u r b u l e n t d i f f u s i o n (see, f o r example, T a y l o r  [1954]; F i s c h e r  [1966a],  [1967] and H o l l e y et al. [1970]. According  t o equation  ( A . 2 ) , t h e n e t mass r e m a i n i n g i n t h e  e l e m e n t a l volume due t o d i s p e r s i v e t r a n s p o r t d u r i n g time 6 t i s  M  A.4  . = ^-tAE^KxSt. net 3x 3x  (A.3)  SOURCE-SINK EFFECTS The  the p r o d u c t i o n  source-sink  terms account f o r any p r o c e s s e s t h a t r e s u l t i n  o r removal o f t h e d i s s o l v e d substance from t h e water body.  In the BOD-DO system, source p r o c e s s e s i n c l u d e r e o x y g e n a t i o n by atmospheric oxygen d i s s o l v i n g i n t h e water and t h e oxygen produced p h o t o s y n t h e t i c a l l y by a q u a t i c p l a n t s ; the s i n k p r o c e s s e s i n c l u d e oxygen uptake by b a c t e r i a breaking; down t h e substances e x e r t i n g t h e BOD and oxygen uptake f o r a q u a t i c plant respiration. The  source-sink  p r o c e s s e s t h a t a substance undergoes i n t h e aqua-  • t i c , environment a r e s p e c i f i c t o t h e p a r t i c u l a r s u b s t a n c e , and depend on t h e p h y s i c a l , c h e m i c a l and b i o l o g i c a l p r o c e s s e s t h a t t h e substance undergoes i n the p a r t i c u l a r a q u a t i c  environment.  Consequently, s o u r c e - s i n k  i n c l u d e d i n t h e mass t r a n s p o r t e q u a t i o n dh a g e n e r a l manner.  e f f e c t s are Suppose t h a t  the p a r t i c u l a r substance i s undergoing n p r o c e s s e s t h a t r e s u l t i n i t s product i o n o r removal from t h e water body.  I f the r a t e o f p r o d u c t i o n  per u n i t  114  volume o f water due t o t h e i  p r o c e s s i s g i v e n by S^, removal p r o c e s s e s  b e i n g regarded a s n e g a t i v e p r o d u c t i o n , then t h e net, mass o f s u b s t a n c e duced i n t h e e l e m e n t a l volume i n time  M  A.5  THE ONE-DIMENSIONAL  pro-  6t i s  n = A&x&tl S. net . , 1 i=l  MASS TRANSPORT  The mass o f d i s s o l v e d  (A.4)  EQUATION  substance  contained i n the elemental  volume i s  M = A<5xc As a r e s u l t o f t h e a d v e c t i o n , d i s p e r s i o n and s o u r c e - s i n k e f f e c t s t h a t o c c u r i n time 6t,  t h e change i n mass o f d i s s o l v e d substance  i n the e l e -  mental volume i s  3 —  6M = T--{Ac}6x6t dt Summing e q u a t i o n s ( A . l ) , (A.5)  (A.S)  (A.3) and (A.4) and s u b s t i t u t i n g i n t o  equation  gives | - C A c } > 4-iAuc"} + |-[AE|£-} + A E S. 9t 3x 3x 3x . , l i=l  (A.6)  Expanding t h e f i r s t two d e r i v a t i v e s and u s i n g t h e e q u a t i o n o f c o n s e r v a t i o n o f f l u i d mass i n t h e form '3  <•  -  3 A  i  (Au) + — 3x dt equation  N  = 0,  (A.6) reduces t o  3c sr 3t  =  3c 1 3 3c •, - u r — + 7 - 1 <AE;r— } + I S., 3x A 3x 3x . , l f  1=1  . (A.7)  115  the one-dimensional flow.  mass t r a n s p o r t e q u a t i o n  f o r unsteady non-uniform  The terms on t h e r i g h t - h a n d s i d e o f E q u a t i o n  (A.7)  s h a l l be r e -  f e r r e d t o r e s p e c t i v e l y as the a d v e c t i v e , the d i s p e r s i v e and the s o u r c e s i n k terms.  A.6  THE LAGRANGIAN FORM OF THE ONE-DIMENSIONAL MASS TRANSPORT EQUATION Equation  (A.7) was d e r i v e d r e l a t i v e t o an E u l e r i a n c o - o r d i n a t e  system (xyz) t h a t i s f i x e d i n space.  To t r a n s f o r m  t o a c o - o r d i n a t e system t h a t i s advected L a n g r a n g i a n c o - o r d i n a t e system),  the e q u a t i o n s  relative  a l o n g a t the mean v e l o c i t y u (a  i t i s necessary  t o transform the E u l e r i a n  co-ordinates according to  §dt R e l a t i v e t o t h e curve  =  u  (A.S)  (A.8), t h e time r a t e o f change o f c o n c e n t r a t i o n i s  g i v e n by (see S e c t i o n C l )  dc dt and  _ "  3c_ 3t  +  U  -3c 3x  thus, r e l a t i v e t o L a n g r a n g i a n c o - o r d i n a t e s , E q u a t i o n  £ dt Equation propagation,  =  | T-CAE& A 3x 3x  (A.8)  +  becomes  Z S. , i x=l  (A.9)  i s t h e c h a r a c t e r i s t i c curve o f a d v e c t i v e  and i s d i s c u s s e d f u r t h e r w i t h r e g a r d t o n u m e r i c a l  s t a b i l i t y i n Appendix C.  Equation  (A.7) and E q u a t i o n s  sent t h e same s i t u a t i o n b u t from d i f f e r e n t v i e w p o i n t s ; "observer"  (A.7)  (A.8)  information  d i s p e r s i o n and  and (A.9)  i n Equation  repre-  (A.7) t h e  i s s t a t i o n e d on the bank o f t h e r i v e r o r e s t u a r y and observes  changes  116  in  t h e c o n c e n t r a t i o n as t h e water flows p a s t him, whereas i n t h e l a t t e r  two  equations,  the observer  (A.8)] and o b s e r v e s  i s advected  a l o n g w i t h the water mass  changes i n t h e water mass around him [ E q u a t i o n  I f d i s p e r s i v e transport i s ignored, Equation  d t  i-1  which i s the o r i g i n a l o n e - d i m e n s i o n a l S t r e e t e r and P h e l p s biological in a river.  (A.9)].  (A.8) reduces t o  1  mass t r a n s p o r t e q u a t i o n developed  [1925] t o p r e d i c t the e f f e c t o f substances  oxygen demand  [Equation  by  exerting a  (BOD) on the c o n c e n t r a t i o n o f d i s s o l v e d oxygen (DO)  In t h e i r model, n was e q u a l t o two; the o n l y source o f DO was  r e a e r a t i o n through the water s u r f a c e and the o n l y s i n k was t h e s a t i s f a c t i o n of  the BOD.  APPENDIX B  A DESCRIPTION OF THE FRASER RIVER ESTUARY  B.l  GENERAL The F r a s e r R i v e r i s t h e l a r g e s t and most important r i v e r i n  B r i t i s h Columbia.  I t s d r a i n a g e a r e a o f 90,000 square m i l e s i s shown i n  F i g u r e B . l and o c c u p i e s almost o n e - q u a r t e r o f t h e P r o v i n c e . r i s e s i n . the Rocky Mountains near J a s p e r and f l o w s north-west  The r i v e r f o r some 330  m i l e s . • North o f P r i n c e George t h e r i v e r changes d i r e c t i o n and flows south some 400 m i l e s t o Hope. and  A t Hope t h e r i v e r emerges from t h e Coast Mountains  flows westward through the F r a s e r V a l l e y f o r almost 100 m i l e s t o e n t e r  the S t r a i t o f G e o r g i a a t Vancouver.  I t i s t h i s l a t t e r section o f the r i v e r ,  from Hope t o Vancouver, t h a t i s o f d i r e c t concern t o t h i s t h e s i s , and d i s c u s s i o n i s now r e s t r i c t e d  B.2  to i t ,  THE LOWER FRASER RIVER SYSTEM The Lower F r a s e r R i v e r from Hope t o Vancouver i s shown i n F i g u r e  B.2.  A c o n v e n i e n t way o f d e s c r i b i n g t h i s s e c t i o n o f t h e r i v e r i s i n terms  o f t h e f a c t o r s t h a t a f f e c t t h e flow. i o u s channels o f the r i v e r system;  These f a c t o r s i n c l u d e :  (1) t h e v a r -  (2) t h e t i d e s i n t h e S t r a i t o f G e o r g i a ;  (3) t h e r i v e r d i s c h a r g e ; and (4) s a l t w a t e r i n t r u s i o n . B.2.1  Channels  o f t h e R i v e r System.  20 - 25 m i l l i o n tons o f sediment  The F r a s e r R i v e r d e p o s i t s  i n t o the S t r a i t o f Georgia annually  [Pretious,  1972], and has b u i l t a l a r g e d e l t a w i t h a seaward p e r i m e t e r o f some 30 m i l e s .  117  00  The  F r a s e r R i v e r From Hope t o Vancouver  120  The  d e l t a i s shown i n F i g u r e B.3  work of channels  through i t .  and i t i s seen t h a t the r i v e r has  A t New  cut a net-  Westminster the r i v e r t r i f u r c a t e s  Main Arm, which e n t e r s the S t r a i t o f f S t e v e s t o n ; Annacis Channel, which  the  flows around the back o f Annacis  I s l a n d t o r e c o n n e c t w i t h the Main Arm;  North Arm, which e n t e r s the S t r a i t near P o i n t Grey.  the  I s l a n d , the North Arm  d i v i d e s t o form the Middle  a complex o f s h a l l o w channels  and sloughs  Arm.  Around  North-west of  (backwater  Main Arm w i t h another  T h i s area?; d r a i n s through  areas) i n t e r c o n n e c t  the M i d d l e Arm  c o n n e c t i o n t o the S t r a i t o f G e o r g i a .  and Canoe Pass i s a p p r o x i m a t e l y  f i v e p e r c e n t and f i v e p e r c e n t r e s p e c t i v e l y g e n e r a l i n d i c a t i o n o f the percentage  to  The  as the  the  North  [ G o l d i e , 1967-].  T h i s i s only a  d i s t r i b u t i o n o f f l o w s , the summer d i s -  s e c t i o n of r i v e r between New  distribution.  Westminster and Hope i s r e f e r r e d  Main Stem. Of the v a r i o u s i s l a n d s p r e s e n t i n the Main Stem (see  F i g u r e B.2), o n l y channels.  percentage  85 p e r c e n t , 5 p e r c e n t ,  t r i b u t i o n o f flows p r o b a b l y d i f f e r i n g from the w i n t e r The  Sea  Canoe Pass t o p r o v i d e the  r i v e r d i s c h a r g e t h a t e n t e r s the S t r a i t through the Main Arm,  Arm,  Sea  Ladner,  w i t h each o t h e r , the t h r e e most important channels b e i n g Ladner Reach, Reach and Canoe Pass.  and  A n n a c i s Channel i s  n o t t r e a t e d as a s e p a r a t e channel, b u t as p a r t o f the Main Arm.  of  into  The  Douglas and Barns ton Islands  form s i g n i f i c a n t  additional  channels a t the "back" o f the o t h e r i s l a n d s are t r e a t e d as  p a r t o f the Main Stem. Of the v a r i o u s major l a k e s t h a t d r a i n i n t o the Main Stem (see F i g ure B.2), P i t t Lake i s o f p a r t i c u l a r s i g n i f i c a n c e .  The  surface elevation of  the l a k e i s a f f e c t e d b o t h by the t i d e s i n the S t r a i t of G e o r g i a and the  fresh-  water d i s c h a r g e o f the F r a s e r (the d i s c h a r g e o f P i t t R i v e r i n t o the n o r t h  end  121  Point  Ladner  Reach  LADNER V;'v'  : :  "' --' '"•-"•^•'.f' ;  :  Canoe ' P a s s '  S C A L E :  1  2  3  4  LEGEND :  ^Y::i: ; - Delta ;  C A N A D A U , S .  Figure  B.3  The F r a s e r R i v e r  Delta  A  5  MILES  122  o f t h e l a k e i s n e g l i g i b l e compared t o t h e F r a s e r d i s c h a r g e . water flows  A t low f r e s h -  i n t h e F r a s e r , t h e maximum t i d a l range o f t h e l a k e i s s e v e r a l  f e e t , and as t h e l a k e has a s u r f a c e a r e a o f some 25 square m i l e s , t h e volume o f water s t o r e d i n t h e l a k e on t h e f l o o d t i d e and r e l e a s e d t o the F r a s e r on t h e ebb t i d e i s v e r y l a r g e the entrance  The l a r g e r e v e r s e d e l t a a t  t o P i t t Lake (see F i g u r e B.2) i s evidence  that enter the lake. t u t e an important to  (see T a b l e B . l ) .  as t h e Pitt  The P i t t R i v e r - P i t t Lake system o b v i o u s l y  channel o f t h e lower Fraser,[/system  System.  o f the l a r g e  flows  consti-  and w i l l be r e f e r r e d  The o t h e r l a k e s d r a i n i n g i n t o the Main Stem a r e o f  s u f f i c i e n t e l e v a t i o n t o be independent o f t i d a l i n f l u e n c e and f r e s h w a t e r discharge  i n the F r a s e r . Thus, the lower F r a s e r R i v e r system c o n s i s t s o f t h e f o l l o w i n g  seven p r i n c i p a l  channels:  1.  Main Arm - Main Stem  2.  N o r t h Arm  3.  M i d d l e Arm  4.  Canoe Pass  5.  Pitt  6.  Douglas Channel, and  7.  B a r n s t o n Channel.  System  To some e x t e n t t h e channel geometries a r e determined by t h e many .-•.. dykes, j e t t i e s and t r a i n i n g w a l l s t h a t have been c o n s t r u c t e d i n the lower F r a s e r system.  Each S p r i n g t h e F r a s e r V a l l e y i s s u b j e c t t o p o s s i b l e f l o o d i n g  d u r i n g the f r e s h e t (time o f h i g h r u n o f f due t o snowmelt), and an e x t e n s i v e system o f dykes has been c o n s t r u c t e d f o r f l o o d p r o t e c t i o n .  Ocean-going s h i p s  123  use  the Main Arm  terminal.  to e n t e r and  To m a i n t a i n  l e a v e New  Westminster, an overseas  the s h i p p i n g c h a n n e l ,  shipping  a number o f dykes and  w a l l s have been c o n s t r u c t e d t o i n c r e a s e l o c a l  training  scour.  C r o s s - s e c t i o n a l a r e a s , depths and widths were e v a l u a t e d a t the work o f s t a t i o n s shown i n F i g u r e B.4. except  The  s t a t i o n s are 5,000 f e e t  net-  apart,  i n the deeper water o f P i t t Lake where the s p a c i n g has been i n c r e a s e d  t o 15,000 f e e t .  (This network of s t a t i o n s i s used i n the f i n i t e d i f f e r e n c e  s o l u t i o n s o f the hydrodynamic e q u a t i o n and averaged mass t r a n s p o r t e q u a t i o n s ,  the t i d a l l y v a r y i n g and  as i s d i s c u s s e d i n Chapter 3 ) .  tidally Soundings  c h a r t s s u p p l i e d by the Department o f P u b l i c Works o f Canada were used t o d e t e r mine the channel channels  geometries.  A t y p i c a l c r o s s - s e c t i o n o f each o f the  i s shown i n F i g u r e B.5.  Widths and depths are r e c o r d e d  and depths are r e l a t i v e t o l o c a l low water. i a t i o n i n depths, widths and channels.  the a d v e c t i v e a r e a  f o r S t a t i o n No.  35 i n F i g u r e B.5,  shown hatched.  The  radius  The  where areas  ( e f f e c t i v e area of  This i s i l l u s t r a t e d  b u t the a d v e c t i v e width  (which i s g r e a t e r than the h y d r a u l i c r a d i u s determined from the From F i g u r e s B.6  than  used t o determine the a d v e c t i v e h y d r a u l i c  t o B.8,  i t i s seen t h a t the  gross  advective  not much s m a l l e r than the gross a r e a , but the a d v e c t i v e w i d t h i s  o f t e n c o n s i d e r a b l y s m a l l e r than the storage w i d t h . e s s e n t i a l l y wide and are now  var-  l e s s than 10 f e e t deep are  s t o r a g e width o f s e c t i o n i s AB,  a d v e c t i v e a r e a was  area of s e c t i o n ) . area i s  show the  a r b i t r a r i l y assumed t h a t any depth of s e c t i o n l e s s  10 f e e t deep d i d not c o n t r i b u t e t o the a d v e c t i v e a r e a .  i s o n l y CD.  t o B.8  i n feet,  c r o s s - s e c t i o n a l areas a l o n g the f i v e major  In an attempt t o e s t i m a t e  d i s c h a r g e ) , i t was  F i g u r e s B.6  seven  shallow and  used synonymously.  The  r i v e r s e c t i o n s are  the terms h y d r a u l i c r a d i u s and mean depth  FIRST AND LAST STATIONS •• M a i n Arm - Main Stem North A r m  Pitt Lake  I 101  Canoe P a s s  11 9  Middle Arm  129  Pitt S y s t e m  140  Douglas Channel  180  Junction  Barnston  184  5 0 0 0 * spacing  101  Channel  Stations  1 5 , 0 0 0 ' spacing  , North Arm  S  J06  HQ,  •Middle Arm "Main Arm  ®jS *I28  •Canoe  Pass  Network o f S t a t i o n s Used i n t h e Numerical  F i g u r e B.4 S o l u t i o n o f t h e Hydrodynamic and Mass T r a n s p o r t  Equations  2000  1000  125  Stat. No. 35 ( Main Stem )  Stat. No. 109 ( North Arm) 1000  Stat. No. 134 ( Middle Arm ) 1000  Stat. No. 126 ( Canoe  Pass)  1000 1 —  ,  Stat. No. 145 ( Pitt)  Stat.No. l8l(Douglas Channel)  Stat. No. I87( Barnston  Figure  B.5  T y p i c a l Channel C r o s s - S e c t i o n s (Widths and Depths i n F e e t , Depths R e l a t i v e t o L o c a l Low Water)  Channel)  Gross  Values  Advective Values  60,000  h 50,000 o 40,000 CD <  30,000 20,000 50  Q  DEPTHS  AND  AREAS  Relative  to  local  low  water.  CL OJ  10 3,000 sz •D  2,000 1,000 0  10  30 Stations  20 Figure  Cross-Sectional  40  B.6  Parameters o f Main Arm - Main Stem  50  60  Figure  B„7  C r o s s - S e c t i o n a l Parameters  o f North Arm,  M i d d l e Arm and Canoe Pass  DEPTHS  AND AREAS  Relative  Pitt River 40,000 i; o a> k.  to  local  low water. Gross Values Advective Values  Pitt Lake r^yjr§e_ Delta  y.  /A  30,000  H-  \\  20,000  1  -  \  V  '  »  /  \  ' N—\ S  •o  500,000  ~—v7\\  §  W  190  % \ t \ %  \ V*  —  I 0  -  <  200  V »  / /^\\ /  CL  Q  1,000,000  — 2 10  30 20  ~  0  40  a>  1,500,000  —  0  £  ~  »\  10,000  <  2,000,000  10,000  3,000  8,000  /  6,000  —H  ~ CL Q  180  0  2,000  ^  _ JZ  —  ,000  4,000 i o  z  F i g u r e B.8  I  o in  o  z  in m  1  1  in  o (0  ro  o  o  o Z  co  in  z  z  5  2,000  C r o s s - S e c t i o n a l Parameters o f P i t t R i v e r and P i t t Lake  03  129  B.2.2  T i d e s i n the S t r a i t o f G e o r g i a .  t i d a l , b e i n g connected Figure B . l ) .  The  The  t i d a l range a t S t e v e s t o n f o r mean and 15 f e e t  are  B.9.  e f f e c t s of the t i d e s on the flows i n the F r a s e r system depends  t r a t e d i n F i g u r e B.10  [ a f t e r Baines, 1953]  of the Main Arm  which shows the l o c a l low and  The g r e a t e r i n f l u e n c e o f the  d u r i n g low f l o w c o n d i t i o n s i s r e a d i l y apparent, l i m i t of t i d a l  This i s i l l u s high  and Main Stem f o r d i s c h a r g e s o f 27,000 and  250,000 c u b i c f e e t per second a t Hope.  The  large  and t y p i c a l t i d e s a t Steveston  both on the t i d a l range a t S t e v e s t o n and the r i v e r d i s c h a r g e .  t i d e envelopes  (see  t i d e s a r e of the mixed type c h a r a c t e r i s t i c o f much o f the  t i d e s i s r e s p e c t i v e l y 10 f e e t and  The  S t r a i t of Georgia i s  t o the P a c i f i c Ocean by the Juan de Fuca S t r a i t  c o a s t of Northwest America.  shown i n F i g u r e  The  and  tide  i t i s seen t h a t the upstream  i n f l u e n c e i s around C h i l l i w a c k , some 60 m i l e s from  s e c t i o n o f the F r a s e r system from the S t r a i t o f G e o r g i a  Steveston.  to C h i l l i w a c k w i l l  be r e f e r r e d t o as the F r a s e r R i v e r E s t u a r y , a l t h o u g h i t i s noted t h a t the F r a s e r i s more p r o p e r l y d e s i g n a t e d as a t i d a l r i v e r D u r i n g low Fraser Estuary.  The  [Callaway,  1971].  f l o w - h i g h t i d e c o n d i t i o n s , flow r e v e r s a l o c c u r s i n the cubature  study o f d i s c h a r g e s by Baines  f l o w r e v e r s a l a t M i s s i o n , some 50 m i l e s upstream o f  [1952] p r e d i c t e d  Steveston.  Downstream o f C h i l l i w a c k t h e r e i s a network o f t i d e gauging . some.equipped w i t h continuous  r e c o r d e r s and o t h e r s o n l y r e c o r d i n g the maximum  .and minimum l e v e l s d u r i n g each 24 hour p e r i o d . gauge a t each s t a t i o n i s shown i n F i g u r e •  : B.2.3  stations,  River Discharges.  The p o s i t i o n and type  of  B.ll.  The v a r i a t i o n of t^g'meap^n^n.thly" " d i s ^ *  charge'~?at Hope, as c a l c u l a t e d f o r the p e r i o d 1912-1970;, is':sriowh;.-in Figure-''. B. 12. .'The .mean monthly flows v a r y from a summer maximum of-250,000""cfs  during  0.0  3.0  6.0  9.0  12.0  15.0  18.0  21.0 24.0  Hours  6.0' 0.0  1  3.0  1  6.0  1  9.0  1  1  12.0  15.0  Hours Figure  B.9  T y p i c a l Tides at  Steveston  1  18.0  1  I  21.0 24.0  Steveston  New Westminster  Mission  Chilliwack  F i g u r e B.J.0 L o c a l Low and High T i d e Envelopes  [ A f t e r Baines, 1953]  Hope  to  Figure B . l l T i d e Gauging S t a t i o n s i n the F r a s e r R i v e r  Estuary  133  280i  240 (/> "2001 w  O  O  — I 601 o  120  LL.  80 40  0'  JAN.  F E B . MAR.  APR. MAY J U N E JULY  AUG. SER  F i g u r e B.12 Mean Monthly Flows a t Hope (1912-1970 I n c l u s i v e )  OCT. NOV. DEC.  134  f r e s h e t 7:to a w i n t e r minimum o f around 30,000 c f s .  Recorded extremes a t Hope  are 536,000 c f s on May 31, 1948 and 12,000 c f s on January  8, 1916.  The d r a i n a g e area o f t h e F r a s e r R i v e r below Hope i s some 6,000 square m i l e s , and between Hope and t h e S t r a i t o f Georgia v a r i o u s o t h e r  rivers  f l o w i n t o the F r a s e r system, t h e most important b e i n g t h e H a r r i s o n R i v e r (see F i g u r e B.2).  The e f f e c t o f t h i s a d d i t i o n a l i n f l o w i s t o i n c r e a s e t h e  f l o w a t New Westminster by some 15 p e r c e n t over the flows a t Hope d u r i n g the f r e s h e t , and up t o 50 p e r c e n t d u r i n g w i n t e r . I t s h o u l d be apparent  t h a t t h e F r a s e r R i v e r E s t u a r y i s somewhat  u n u s u a l , b e i n g c h a r a c t e r i z e d by b o t h h i g h r i v e r d i s c h a r g e s and l a r g e t i d a l effects.  T h i s i s i l l u s t r a t e d by a c a l c u l a t i o n o f t h e t i d a l p r i s m and t o t a l  r i v e r d i s c h a r g e between t h e times o f low-low-water and high-high-water t i d a l c y c l e s o f January  11 and June 16, 1964.  freshwater d i s c h a r g e s a r e shown i n F i g u r e B.9. prisms  f o r the  These t i d a l c y c l e s and t h e The volumes o f t h e t i d a l  and r i v e r flows a r e shown i n T a b l e B".l. B.2.4  Saltwater I n t r u s i o n .  The s a l i n i t y o f t h e S t r a i t o f G e o r g i a  i s e s s e n t i a l l y some 30 p a r t s p e r thousand  ( p p t ) , and under low r i v e r  c o n d i t i o n s s a l t w a t e r i n t r u d e s a c o n s i d e r a b l e d i s t a n c e i n t o the f o u r  discharge channels  t h a t emerge from t h e d e l t a .  The s a l t w a t e r i n t r u s i o n i n t o the Main Arm i s  i l l u s t r a t e d i n F i g u r e B.13  [data from Waldichuk, et al. , 1968], which shows  the s a l i n i t y p r o f i l e a t f o u r s t a t i o n s a l o n g the Main Arm. d e s i g n a t i o n i n F i g u r e B.13 i s t h a t o f Waldichuk, et al. confused w i t h the s t a t i o n d e s i g n a t i o n i n F i g u r e B . 4 ) .  3  (Note:  the s t a t i o n  and s h o u l d n o t be The p o s i t i o n o f t h e  s t a t i o n s , the t i d e a t S t e v e s t o n and the time o f t i d e the o b s e r v a t i o n s were taken a r e a l s o shown.  The f o u r o b s e r v a t i o n s c l o s e l y occur around the same  phase o f t i d e and can be regarded  as "simultaneous."  The f r e s h w a t e r  discharge  TABLE B . l  RIVER FLOW VOLUMES AND TIDAL PRISMS ON JANUARY 15 AND JUNE 16, 1964 (Volumes C a l c u l a t e d f o r Time Between Low-Low-Water and High-High-Water)  DATE  JANUARY 15  JUNE 16  TIDAL PRISM (ft )  TIDAL RANGE (feet)  RIVER FLOW vol. (ft )  53,500  10.4  12 x 1 0  8  66 x 1 0  8  463,000  7.6  104 x 1 0  8  25 x 1 0  8  RIVER FLOW (cfs)  3  RIVER FLOW TIDAL PRISM  0.18  4.2  10 •°  •  20 -JL.  30  10  20  10  30  20  30  —I  1  10  20 1  30 1  Stat. No. 3  100  200  300  All depths in feet River Discharge - 57,600 cfs at  L  Figure  Hope  B.13 01  Salinity Profiles  i n the Main Arm on February 13-14, 1962  137  a t Hope i s 57,000 c f s and i t i s apparent  t h a t under these  freshwater  dis-  charge and t i d a l c o n d i t i o n s , t h e Main Arm i s h i g h l y s t r a t i f i e d w i t h the t o e o f the s a l t wedge somewhere between s t a t i o n s 2 and 3. Recent s t u d i e s have i n v o l v e d m o n i t o r i n g the s a l i n i t y p r o f i l e t i n u o u s l y a t t h r e e s t a t i o n s a l o n g the Main Arm. periods February  1-16  con-  T h i s was done f o r the two  and March 16 - 30, 1973, d u r i n g which t h e average  d i s c h a r g e a t Hope was 30,000 c f s ^and 34,000 c f s r e s p e c t i v e l y .  I t was  found  t h a t the e f f e c t o f t h e t i d e s was t o move the wedge b o d i l y up and down the Main Arm [D. O. Hodgins, p r i v a t e communication], the maximum e x c u r s i o n o f the t o e o f t h e wedge p r o b a b l y b e i n g t o somewhere around Annacis  Island.  Thus, f o r low r i v e r f l o w s , the Main Arm o f the F r a s e r E s t u a r y i s h i g h l y s t r a t i f i e d w i t h t h e s a l t wedge moving b o d i l y up and down the e s t u a r y under the i n f l u e n c e o f the t i d e s .  D u r i n g times o f flow r e v e r s a l , the  wedge may move as f a r upstream as Annacis i t i s washed downstream p a s t S t e v e s t o n .  I s l a n d b u t d u r i n g seaward d i s c h a r g e F o r h i g h r i v e r f l o w s , the wedge p r o b -  a b l y does n o t penetrate'-past S t e v e s t o n , i f t h a t f a r . the o t h e r channels  Saltwater intrudes i n t o  o f t h e d e l t a , and s i m i l a r s i t u a t i o n s p r o b a b l y occur  there,  although the s a l t w a t e r movement may be m o d i f i e d by t h e slower v e l o c i t i e s these channels.  S a l i n i t y p r o f i l e s i n these o t h e r channels  show them t o be  t y p i c a l l y s t r a t i f i e d , b u t perhaps n o t q u i t e as s t r a t i f i e d as t h e Main Arm (probably due t o the lower d i s c h a r g e v e l o c i t i e s through  through  these  channels).  APPENDIX C  NUMERICAL DISPERSION AND  STABILITY  The dependent v a r i a b l e o f a p a r t i a l d i f f e r e n t i a l e q u a t i o n def i n e s a s u r f a c e over the p l a n e o f the independent v a r i a b l e s . o f the independent v a r i a b l e s t h e r e a r e v a r i o u s c u r v e s , o r  In the p l a n e  "characteristics"  t h a t d e s c r i b e the p r o p a g a t i o n o f i n f o r m a t i o n through the system t h a t the p a r t i a l d i f f e r e n t i a l e q u a t i o n r e p r e s e n t s . ' The one-dimensional  forms o f the  a d v e c t i v e t r a n s p o r t e q u a t i o n , the d i s p e r s i v e t r a n s p o r t e q u a t i o n and hydrodynamic e q u a t i o n s a r e o f i n t e r e s t t o t h i s t h e s i s .  the  The forms o f t h e i r  r e s p e c t i v e s u r f a c e s a r e b r i e f l y d e s c r i b e d and the e q u a t i o n s o f t h e i r r e s p e c t i v e c h a r a c t e r i s t i c s are given.  In f i x e d g r i d f i n i t e d i f f e r e n c e schemes,  the problem o f n u m e r i c a l d i s p e r s i o n a r i s e s from s o l v i n g the advective transport equation  one-dimensional  (or the one-dimensional mass t r a n s p o r t  equation)  r e l a t i v e t o a f i x e d space g r i d , r a t h e r than along the a d v e c t i v e c h a r a c t e r i s tics.  In e x p l i c i t f i n i t e d i f f e r e n c e schemes, the problem  of s t a b i l i t y  a r i s e s from s o l v i n g the above t h r e e e q u a t i o n s r e l a t i v e t o a f i x e d g r i d than a l o n g t h e i r r e s p e c t i v e  C l  also rather  characteristics.  SURFACE GEOMETRY OF PARTIAL DIFFERENTIAL EQUATIONS The f o l l o w i n g p a r t i a l d i f f e r e n t i a l e q u a t i o n s are o f  3c 3t  interest (Cl)  (C.2)  138  139  9u 9t  9a  *37  u u - ? f =  c  y  (C.3) 9y_ 9t  Equation  -9u  ( C l ) d e s c r i b e s one-dimensional  one-dimensional  a d v e c t i v e t r a n s p o r t and E q u a t i o n (C.2)  d i s p e r s i v e t r a n s p o r t . ' " T h e s e two e q u a t i o n s r e p r e s e n t t h e compo-  nent t r a n s p o r t p r o c e s s e s o f t h e one-dimensional  mass t r a n s p o r t e q u a t i o n , and  the sum o f t h e i r e f f e c t s i s t h e one-dimensional  mass t r a n s p o r t e q u a t i o n w i t h -  o u t t h e s o u r c e - s i n k terms.  The two c o u p l e d e q u a t i o n s  (C.3) a r e t h e hydrodynamic  equations and a r e used t o p r e d i c t the temporal v a r i a t i o n s i n t h e parameters u and A o f t h e t i d a l l y v a r y i n g mass t r a n s p o r t e q u a t i o n . t a i n e d by s u b s t i t u t i n g E q u a t i o n Equations (x,t) p l a n e and E q u a t i o n s  [Equations  (3.3) and A = by i n t o E q u a t i o n s  (C.3) a r e ob-  (3.1) and  ( C l ) and (C.2) d e f i n e s u r f a c e s c ( x , t ) over t h e  (C.3) d e f i n e a p a i r o f c o u p l e d s u r f a c e s u ( x , t ) and  y ( x , t ) over t h e (x,t) p l a n e .  C o n s i d e r now t h e form o f t h e s u r f a c e d e f i n e d by  the a d v e c t i v e t r a n s p o r t E q u a t i o n surface f o r a slug input into  ( C l ) . F i g u r e C l shows t h e shape o f t h i s  (a) steady u n i f o r m flow;  (b) steady  non-uniform  f l o w a l o n g a r i v e r o f c o n t r a c t i n g c r o s s - s e c t i o n ; and (c) t h e mixture and o s c i l l a t o r y f l o w c h a r a c t e r i s t i c o f an e s t u a r y faces are s o l u t i o n s t o Eguation of f l o w .  I t i s apparent  (3.2)].  (estuary f l o w ) .  o f steady  These s u r -  ( C l ) f o r a slug i n p u t into the various types  from F i g u r e C l t h a t t h e r e a r e v a r i o u s c u r v e s i n t h e  (x,t) p l a n e t h a t r e s u l t i n a c o n s i d e r a b l e s i m p l i f i c a t i o n o f E q u a t i o n the c o n c e n t r a t i o n i s c o n s t a n t  along  (C.l)  —  each o f the c u r v e s AB o f F i g u r e C . l  w h i l e everywhere e l s e i n t h e p l a n e i t i s z e r o . F o r any c o n t i n u o u s , smoothly v a r y i n g s u r f a c e c ( x , t ) , t h e time r a t e o f change o f c ( x , t ) a l o n g t h e v e r t i c a l plane d e f i n e d by t h e curve  140  c(x,t)  Figure  C l  C o n c e n t r a t i o n S u r f a c e s f o r the A d v e c t i o n o f a S l u g Load  141  'Sr at i n the  =  f(x,t)  (G.4)  (x,t) p l a n e i s g i v e n by dc dt  _ 3c 3t  3c_ 3x  dx dt  The g r a p h i c a l i n t e r p r e t a t i o n o f t h i s elementary trated i n Figure  [C.2.  dt  S u b s t i t u t i n g Equation  " " {  u  dt  +  }  '  l C  '  formula o f c a l c u l u s i s i l l u s -  ( C l ) into  (C.5) g i v e s  37  (  c  '  6  )  and a l o n g the curve d e f i n e d by dx dt  =  u  ( C  '  7 )  t h e time r a t e o f change o f c o n c e n t r a t i o n i s g i v e n by §£•  Equations  (C.7) and  = 0  (C.8) are the well-known L a n g r a n g i a n  t r a n s p o r t e q u a t i o n and c u r v e Figure C l .  (C.S)  form o f the a d v e c t i v e  (C.7) o b v i o u s l y d e f i n e s the t h r e e c u r v e s AB  In d e r i v i n g E q u a t i o n  and u w i t h x and t i s c o n t i n u o u s .  of  ( C l ) i t i s assumed t h a t the v a r i a t i o n o f c F o r the cases shown i n F i g u r e C l , the  vari-  a t i o n o f c i s d i s c o n t i n u o u s and the p a r t i a l d e r i v a t i v e s 3c/3t and 3c/3x a r e not d e f i n e d everywhere i n the  (x,t) p l a n e .  r e p r e s e n t the a d v e c t i o n o f a s l u g l o a d numerical d i s p e r s i o n ) . amined a l o n g the curve  Thus, E q u a t i o n  ( C l ) does n o t  correctly  ( t h i s i s the fundamental r e a s o n f o r  I t i s o n l y when the s o l u t i o n t o E q u a t i o n  ( C l ) i s ex-  (C.7) t h a t the problem o f the n o n - d e f i n i t i o n o f the p a r -  t i a l d e r i v a t i v e s i s avoided.  [See E q u a t i o n  (C.6)].  142  i f 3t = | f 8t dt at dc _ ac dt at Figure  +  ax  Sx  ac ^x ax dt C.2  Time Rate o f Change o f C o n c e n t r a t i o n A l o n g A Curve i n the ( x , t ) Plane  143  C o n s i d e r d i s t u r b i n g t h e system r e p r e s e n t e d by E q u a t i o n i n t r o d u c i n g an e l e m e n t a l s l u g l o a d o f Ac. w i l l propagate (C.7)  ( C l ) by  This disturbance, or "information,"  through t h e system a c c o r d i n g t o E q u a t i o n  (C.7).  Thus, E q u a t i o n  i s o f d i r e c t p h y s i c a l s i g n i f i c a n c e , and a l o n g t h i s curve t h e s o l u t i o n t o  the a d v e c t i v e t r a n s p o r t e q u a t i o n reduces t o the simple form o f E q u a t i o n The f i x e d  ( E u l e r i a n ) space g r i d used t o d e r i v e E q u a t i o n  (C.8).  ( C l ) "masks" t h e under-  l y i n g p h y s i c s of the t r a n s p o r t p r o c e s s and the s i m p l i c i t y o f t h e s o l u t i o n . Now c o n s i d e r the e q u a t i o n o f d i s p e r s i v e t r a n s p o r t (C.2). and A a r e c o n s t a n t i n x and t , E q u a t i o n  | |  -  I f both E  (C.2) reduces t o  IC.9>  E 2i ax  The  s u r f a c e d e f i n e d by E q u a t i o n  shown i n F i g u r e C.3.  (C.9) f o r the d i s p e r s i o n o f a s l u g l o a d i s  The e q u a t i o n o f t h i s s u r f a c e i s [ F i s c h e r , 1966a]  S ( x t ) ='-^~ 4IlE  exp{-'-—-}  f  (CIO)  t  where M i s t h e mass p e r u n i t a r e a ; 6 f t r a c e r r e l e a s e d .  A t any t i m e t  1  the  shape o f the s u r f a c e i s a Gaussian d i s t r i b u t i o n and c(x,t.') i s u n i q u e l y d e f i n e d by t h e l o c a t i o n o f t h e p o i n t o f standard d e v i a t i o n . C.3.  This i s i l l u s t r a t e d i n Figure  Thus, t h e l o c u s o f the s t a n d a r d d e v i a t i o n i n t h e (x,t) plane d i r e c t l y  c r i b e s t h e u n d e r l y i n g mass t r a n s p o r t p r o c e s s .  From E q u a t i o n  des-  (CIO) the locus  o f the s t a n d a r d d e v i a t i o n i s t h e p a r a b o l a  x  2  =  o r i n t h e form o f E q u a t i o n  2E  (C.4)  (Cll)  144  Figure  C.3  D i s p e r s i o n o f A Slug Load  145  I f t h i s system  i s d i s t u r b e d by i n t r o d u c i n g an e l e m e n t a l s l u g l o a d o f Ac,  e f f e c t o f t h i s s l u g l o a d , o r i t s " i n f o r m a t i o n " , w i l l propagate system a c c o r d i n g t o E q u a t i o n i s o f fundamental  (C.ll) or  significance  (C.12).  through  Once a g a i n , the curve  the  the (C.12)  and a l o n g t h i s curve the c o n c e n t r a t i o n i s  g i v e n by _  M  rl  -7=—-'  c(x,t) =  ^SHe  (C.13)  X  F i n a l l y , c o n s i d e r the hydrodynamic E q u a t i o n s characteristics  o f t h i s system  §T dt I f t h i s system  =  a r e g i v e n by  The well-known  [Courant and H i l b e r t ,  1962]  u ±  (C.14)  i s d i s t u r b e d by i n t r o d u c i n g a s m a l l change i n the water s u r -  f a c e e l e v a t i o n Ay, t h i s " i n f o r m a t i o n " w i l l propagate ing to Equation  (C.3).  through the system  accord-  (C.3) reduce t o  [Henderson,  (C.14) and a l o n g t h i s curve E q u a t i o n s  1966]  d —- iu ±  dh  ;  2/gy} dt ---'w> = -9 3 7  In summary, t h e r e a r e c e r t a i n  .  + J  |u|u  -1r  <- > c  c y  c u r v e s i n the  (x,t) p l a n e t h a t d i r e c t l y  d e s c r i b e the p r o p a g a t i o n o f i n f o r m a t i o n through the systems t h a t E q u a t i o n s (C.2) and  (C.3) r e p r e s e n t .  mation propagation,  15  These curves can be c a l l e d  charaeteristies  of  (C.l),  infor-  and a l o n g these c u r v e s the s o l u t i o n s t o the r e s p e c t i v e equa-  t i o n s are g r e a t l y s i m p l i f i e d .  This underlying simplicity  E u l e r i a n nature o f the d e r i v a t i o n o f the e q u a t i o n s .  i s masked by  the  The problems o f n u m e r i c a l  146  d i s p e r s i o n and s t a b i l i t y a r i s e d i r e c t l y t i o n s t o Equations  (Cl)  than from a space g r i d  C.2  from v i e w i n g t h e n u m e r i c a l  solu-  t o (C.3) from a f i x e d E u l e r i a n space g r i d ,  rather  a l o n g the c h a r a c t e r i s t i c s .  NUMERICAL DISPERSION Before d i s c u s s i n g  to b r i e f l y consider partial  equations.  In u s i n g  pressions  solutions to  f i n i t e d i f f e r e n c e methods t o  e q u a t i o n s , d e r i v a t i v e s a r e approximated by f i n i t e  ence e x p r e s s i o n s and t h e d i f f e r e n t i a l equation.  i t i s necessary  a few fundamentals o f f i n i t e d i f f e r e n c e  differential  solve d i f f e r e n t i a l  numericalddispersion i n d e t a i l ,  differ-  e q u a t i o n i s reduced t o a d i f f e r e n c e  The j u s t i f i c a t i o n o f r e p l a c i n g the d e r i v a t i v e s by d i f f e r e n c e can be i l l u s t r a t e d  U(x,t) w i t h respect  by the T a y l o r ' s  Series  ex^r-  expansion o f the v a r i a b l e  t o t i n t h e neighborhood o f t h e p o i n t  (x = jAx,  t = nAt).  T h i s can be w r i t t e n as .n+1 (C.16)  where t h e p a r t i a l and  d e r i v a t i v e s a r e assumed c o n t i n u o u s and 0 < 0 < 1 ;[Richtmyer  Morton, 1967, p. 1 9 ] . By making At s m a l l  of Equation  expression  (C.16) can be made t o approximate t h e d e r i v a t i v e t o any d e s i r e d  degree o f a c c u r a c y .  truncation  enough, t h e d i f f e r e n c e  error  The term on the r i g h t - h a n d s i d e o f E q u a t i o n  (C.16) i s t h e  t h a t r e s u l t s when t h e d e r i v a t i v e 8U/3t i s r e p l a c e d  d i f f e r e n c e expression o f Equation When a p a r t i a l  by t h e  (C.16).  differential  e q u a t i o n i n x and t i s approximated b y  a p a r t i a l d i f f e r e n c e e q u a t i o n , the t r u n c a t i o n  e r r o r c o n s i s t s o f terms  containing  147  Ax and  At r a i s e d t o v a r i o u s powers.  The  e r r o r tends t o z e r o i s the problem o f refined  (Ax,  At -»• 0) , does the  conditions  convergence.  true  behaviour of thes  scheme i s u n s t a b l e , e r r o r s generated d u r i n g  example, r o u n d - o f f e r r o r s ) may  l a t i o n s as t o make the  the  become so m a g n i f i e d d u r i n g  solu-  truncation  stability.  s o l u t i o n p r o g r e s s e s through time i s the problem o f  I f the d i f f e r e n c e (for  The  truncation  That i s , as the mesh i s  n u m e r i c a l s o l u t i o n converge t o the  t i o n o f the p a r t i a l d i f f e r e n t i a l e q u a t i o n ? e r r o r as the  under which the  solution  the  calcu-  f i n a l r e s u l t s meaningless.  There are many f i x e d mesh d i f f e r e n c e schemes f o r a p p r o x i m a t i n g the d e r i v a t i v e s o f p a r t i a l d i f f e r e n t i a l be  classified  i n t o e x p l i c i t and  forward time d i f f e r e n c e s t a i n s o n l y one schemes use  implicit  (see F i g u r e  e q u a t i o n s , but schemes.  C.4),  and  unknown v a r i a b l e which can be  backward time d i f f e r e n c e s  time step, over the  explicit  An  solved  To  e q u a t i o n con-  and  obtain  each  difference  a solution at  the r e s u l t i n g system o f simultaneous-iequations must be  e n t i r e space g r i d .  Generally,  implicit  Implicit  for explicitly.  C.4)  schemes are  further i n Section C o n s i d e r now  transport  equation  mass t r a n s p o r t  the  (Cl).  equation.  unconditionally  n u m e r i c a l s o l u t i o n o f the This Fixed  one-dimensional mass  e q u a t i o n i s a component o f the mesh f i n i t e d i f f e r e n c e p r o c e s s c o r r e c t l y , and  d i s p e r s i v e p r o c e s s b e i n g superimposed bn i n the  This i s  C.3.  s i m u l a t e the a d v e c t i v e t r a n s p o r t  processes occurring  any  solved  s t a b l e , whereas e x p l i c i t schemes a r e a t most c o n d i t i o n a l l y s t a b l e . discussed  can  scheme uses  each d i f f e r e n c e  (see F i g u r e  e q u a t i o n c o n t a i n s s e v e r a l unknown v a r i a b l e s .  e s s e n t i a l l y they  one-dimensional  schemes g e n e r a l l y r e s u l t i n an  the a c t u a l a d v e c t i v e and  r i v e r or estuary.  This so-called  do  not  additional  dispersive  numerical  disper-  sion can be i l l u s t r a t e d by c o n s i d e r i n g the t r u n c a t i o n e r r o r a s s o c i a t e d w i t h  148  Forward Time Differences  s  U i  = function { u"-,, u"  n + I  U"  }  time = ( n + I )At  -o-  time = n^t (j-l)Ax  jAx  (j+ I  )AX  Backward Time Differences  :  un + I  _ _  -  r  o  n*l  n+l  f u n c t i o n { U j , U j | , U|_, } +  (J-I)AX —  n  j AX  —  (  J  +  Q  I ) A X  time = (n + l ) A t  time = nAt  Figure  C.4  Forward and Backward Time D i f f e r e n c e s  149  r e p l a c i n g the p a r t i a l  differential  e q u a t i o n o f a d v e c t i v e t r a n s p o r t (C.l) w i t h  i t s corresponding e x p l i c i t d i f f e r e n c e ~n+l „n C. - C.  equation  ,, »-,_n „n . (1-a (C . - C  )  where the f l o w i s assumed t o be u n i f o r m and  .n  „n. - C .)  q (C  steady, x = jAx, t = nAt and a i s  a w e i g h t i n g f a c t o r such t h a t a = 0 f o r upstream d i f f e r e n c e s , a = 1/2 t r a l d i f f e r e n c e s and a = 1 f o r downstream d i f f e r e n c e s . ences a r e i l l u s t r a t e d plicit  i n F i g u r e C.5.  Eguation  These v a r i o u s d i f f e r -  (C.17) i s an example o f an  d i f f e r e n c e scheme t h a t uses forward time d i f f e r e n c e s  a t time s t e p n are used t o determine  f o r cen-  ex-  (the c o n c e n t r a t i o n s  the c o n c e n t r a t i o n s a t time s t e p n+1).  A  T a y l o r ' s S e r i e s expansion o f the d e r i v a t i v e s of E q u a t i o n  (C.l) shows t h a t the  d i f f e r e n c e equation  equation  ar  O  n  (C.17) approximates  ar  the d i f f e r e n t i a l  a f  n  2  . + U(TT-) •  =  n  %UAx(l-2a) ( ± 4 )  a r 2  • "  n  2 + O(Ax)  the terms on t h e r i g h t - h a n d s i d e b e i n g the t r u n c a t i o n e r r o r . t i o n of E q u a t i o n  (C.l) i t a l s o  As  2  + O(At)^  C i s a solu-  satisfies  =0 9t  (C.19)  9x 2  S u b s t i t u t i n g Equation  (C.19) i n t o  (C.18) and n e g l e c t i n g terms o f o r d e r  2 (At) and h i g h e r , i t i s found t h a t the d i f f e r e n c e e q u a t i o n approximates d i f f e r e n t i a l equation 90  9t  9C  *  „  +  E  9 C  p TI  9x  (Ax) . the  '  (C 20)  150  Figure  C.5  Upstream, C e n t r a l and Downstream Space  Differences  151  where E  P  can be  s i o n , and  considered  i s given  t o be a c o e f f i c i e n t o f pseudo or n u m e r i c a l d i s p e r -  by  E  Thus, the  p  = %UAx{  (l-2a) - 7 ^ } Ax  s o l u t i o n t o the d i f f e r e n c e e q u a t i o n  t h e s o l u t i o n t o an a d v e c t i v e - d i s p e r s i v e vective equation.  (C.21)  As  Ax ->• 0, E^ •> 0 and  e q u a t i o n r a t h e r t h a n the c o r r e c t the mass t r a n s p o r t  the n u m e r i c a l d i s p e r s i o n becomes i n c r e a s i n g l y smaller.schemes Ax  i s always f i n i t e ,  converges t o the  and w h i l e the  s o l u t i o n o f the  (Ax ^  associated  ad-  with  However, i n d i f f e r e n c e  s o l u t i o n t o d i f f e r e n c e equation  correct advective  e q u a t i o n i n the  i t converges t o the s o l u t i o n o f an a d v e c t i v e - d i s p e r s i v e of application  (C.17) approximates  limit  e q u a t i o n a t the  (Ax =  0),  level  0).  I t i s apparent from E q u a t i o n s p e r s i o n w i l l always o c c u r u n l e s s  U • 7^ Ax  =  (C.20) and  (C.21) t h a t n u m e r i c a l  dis-  either  1 - 2a  (C.22)  or  dX  Ignoring  the t r i v i a l l a t t e r case  (the mass t r a n s p o r t a s s o c i a t e d  w i t h any  d i s p e r s i o n i s a l s o z e r o ) , i t i s seen t h a t n u m e r i c a l d i s p e r s i o n does not when upstream d i f f e r e n c e s are used  Ax TT At  =  U  (a = 0) and  Ax and  At are d e f i n e d  real occur  by  (C.23)  152  as was r e c o g n i z e d by B e l l a and Dobbins tion  Under t h e s e c o n d i t i o n s Equa-  (C.17) reduces t o  Cj  Equations tions  [1968].  + 1  =  j i  C  constant  =  (C.24)  (C.23) and (C.24) a r e s i m p l y t h e f i n i t e d i f f e r e n c e v e r s i o n s o f Equa-  (C.7) and (C.8).  Thus, t h e use o f upstream d i f f e r e n c e s w i t h t h e g r i d  s p a c i n g d e f i n e d by E q u a t i o n  (C.23) i s e q u i v a l e n t t o s o l v i n g E q u a t i o n  a l o n g t h e c h a r a c t e r i s t i c curve i n t h e (x,t) p l a n e .  (C.17)  Under these c o n d i t i o n s  the u n d e r l y i n g a d v e c t i v e mass t r a n s p o r t p r o c e s s i s c o r r e c t l y simulated., and t h e r e i s no n u m e r i c a l  dispersion.  C o n s i d e r now u s i n g c e n t r a l d i f f e r e n c e s (C.17).  From-Equation  (a = 1/2) t o s o l v e E q u a t i o n  (C.22) i t i s seen t h a t n u m e r i c a l d i s p e r s i o n w i l l  o c c u r except i n the t r i v i a l case o f u = 0. t h e r e i s no n u m e r i c a l d i s p e r s i o n when  According t o Leendertse  always  [1971b]  c e n t r a l d i f f e r e n c e s a r e used.  He  a p p a r e n t l y assumes t h a t c e n t r a l d i f f e r e n c e s c o r r e c t l y d e s c r i b e t h e a d v e c t i o n p r o c e s s , and then uses t h i s i n c o r r e c t assumption  t o demonstrate t h a t n u m e r i c a l  d i s p e r s i o n w i l l o c c u r when upstream o r downstream d i f f e r e n c e s a r e used. The use o f downstream d i f f e r e n c e s  ( d = 1) w i l l always r e s u l t i n numeri-  c a l d i s p e r s i o n except when  ff  =  -U  (C.25)  The use o f downstream d i f f e r e n c e s w i t h E q u a t i o n can be i n t e r p r e t e d as an attempt  t o determine  downstream by working backwards through time t h e c h a r a c t e r i s t i c curve i n the (x,t) p l a n e . p r a c t i c a l , i t would c o r r e c t l y determine  (C.25) d e f i n i n g t h e g r i d  upstream c o n d i t i o n s from  spacing  those  (hence t h e n e g a t i v e s i g n ) a l o n g Although  t h i s procedure  i s im-  the preceeding concentration d i s t r i b u t i o n s  153  w i t h no n u m e r i c a l d i s p e r s i o n i f the f i n a l c o n c e n t r a t i o n  d i s t r i b u t i o n - was  known. ': B e l l a and  Grenney [1970] i n v e s t i g a t e d the n u m e r i c a l d i s p e r s i o n r e s u l t -  i n g from the a d v e c t i o n schemes.  o f a s l u g l o a d by v a r i o u s  f i x e d mesh f i n i t e d i f f e r e n c e ; :  They found t h a t the c o e f f i c i e n t of n u m e r i c a l d i s p e r s i o n was  g i v e n by E q u a t i o n  (C.21) and  t h a t no n u m e r i c a l d i s p e r s i o n o c c u r r e d  stream d i f f e r e n c e s were used w i t h E q u a t i o n N u m e r i c a l d i s p e r s i o n always o c c u r r e d  as  when  (C.23) d e f i n i n g the g r i d  up-  spacing.  w i t h c e n t r a l or downstream d i f f e r e n c e s .  Fox  [1971] made a F o u r i e r s e r i e s a n a l y s i s o f the d i f f e r e n c e schemes o f B e l l a  and  Grenney and  tude and  showed t h a t the  phase e r r o r s i n t o the  e f f e c t o f d i s c r e t i z a t i o n was  to introduce  s o l u t i o n o f the d i f f e r e n c e e q u a t i o n s .  His r e s u l t s  demonstrate t h a t n e i t h e r amplitude nor phase e r r o r s o c c u r when upstream ences are used w i t h a g r i d spacing In c o n c l u s i o n ,  according  to Equation  (Although n u m e r i c a l d i s p e r s i o n has  (C.23).  the a d v e c t i v e  transport  c o n l y been demonstrated f o r e x p l i c i t  d i f f e r e n c e schemes, i t a l s o o c c u r s i n i m p l i c i t d i f f e r e n c e schemes). t r u n c a t i o n e r r o r of the  simple d i f f e r e n c e schemes d i s c u s s e d  t o an a c t u a l p h y s i c a l mode o f mass t r a n s p o r t and transport  and  any  dispersive transport  To c o r r e c t l y s i m u l a t e s o l v e the a d v e c t i v e the  (x,t) p l a n e .  the a d v e c t i v e  mass t r a n s p o r t  The  on the magnitude o f 3 c/3x  here corresponds  w i l l modify the  that i s occurring  The  advective  i n a r i v e r or  estuary.  transport process, i t i s necessary t o e q u a t i o n a l o n g the c h a r a c t e r i s t i c curve i n  magnitude o f the 2  continuous r e l e a s e .  differ-  n u m e r i c a l d i s p e r s i o n o c c u r s because f i x e d mesh  f i n i t e d i f f e r e n c e schemes do not c o r r e c t l y s i m u l a t e process.  ampli-  n u m e r i c a l d i s p e r s i o n depends d i r e c t l y  2 , and  so i s g r e a t e s t  A continuous r e l e a s e can be  f o r a s l u g l o a d and  less for a  t r e a t e d as a s u c c e s s i o n  of  154  s l u g l o a d s , and  apparently  the n u m e r i c a l d i s p e r s i o n due  t o any p a r t i c u l a r  s l u g i s compensated by the n u m e r i c a l d i s p e r s i o n o f n e i g h b o u r i n g s l u g s , has been noted by  B e l l a and  Grenney  w r i t i n g the d i f f e r e n c e e q u a t i o n  [1970].  In d e r i v i n g E q u a t i o n  f o r a s l u g l o a d , as i s seen i n F i g u r e  t.  C.l.  istic  curve t h a t c ( x , t ) v a r i e s c o n t i n u o u s l y ,  C.l.  Thus, i n u s i n g E q u a t i o n  This  i s c e r t a i n l y not  I t i s o n l y a l o n g the  mass t r a n s p o r t  information..propagatation.  of s o l u t i o n r e s u l t s i s discussed. by the use  (C.21).  9c/9x are not  eliminated  a l o n g the  by  s o l v i n g the  2.2,  where the problem o f the  N u m e r i c a l d i s p e r s i o n can be  the a c t u a l d i s p e r s i o n c o e f f i c i e n t s by  Numerical d i s p e r s i o n i s a p p a r e n t l y  [ P r i c e et al.  3  1968;  Fox,  defined,  E  according  advective a  bookkeeping controlled 1970], or to Equation  reduced i n f i n i t e element s o l u t i o n s  1970], but because o f the  f i x e d g r i d n a t u r e of  element s o l u t i o n s , the n u m e r i c a l d i s p e r s i o n i s p r o b a b l y not t o t a l l y  C.3  them.  equation.  c h a r a c t e r i s t i c s of  o f more s o p h i s t i c a t e d d i f f e r e n c i n g schemes [see Fox,  by r e d u c i n g  i s the  .Such a.^numerical s o l u t i o n i s r e f e r r e d t o as  c h a r a c t e r i s t i c method i n S e c t i o n  Figure  i n t e r p r e t e d as an attempt t o d e f i n e  N u m e r i c a l d i s p e r s i o n can be advective-dispersive  case  character-  (C.17) t o advect a s l u g l o a d , t h e r e  the n u m e r i c a l d i s p e r s i o n can be  and  c(x,t)  the  as i s a l s o apparent from  added problem t h a t the p a r t i a l d e r i v a t i v e s 8c/8t and and  (C.l)  (C.17), i t i s i m p l i c i t l y assumed t h a t  v a r i e s i n a c o n t i n u o u s manner w i t h x and  as  finite  eliminated.  STABILITY I f a f i x e d g r i d d i f f e r e n c e e q u a t i o n i s used t o approximate a l i n e a r  p a r t i a l d i f f e r e n t i a l e q u a t i o n w i t h c o n s t a n t c o e f f i c i e n t s , the von  Neumann  s t a b i l i t y c o n d i t i o n r e q u i r e s t h a t the e i g e n v a l u e s o f t h e a m p l i f i c a t i o n m a t r i x  155  s h o u l d n o t exceed u n i t y  i n absolute value for p h y s i c a l l y s t a b l e  [Richtmyer and Morton, 1967]. plicit  To i l l u s t r a t e  f i n i t e d i f f e r e n c e scheme o f E q u a t i o n  mate t h e a d v e c t i v e t r a n s p o r t  c  n  .  + 1  „  c  .  equation.  _ «t,  (  1  .  a  l  (  c  these ideas, (C.17)  +  where a = 0 f o r upstream d i f f e r e n c e s , a = 1/2 a = 1 f o r downstream d i f f e r e n c e s . based on a F o u r i e r the  concentration  composed  of  component to  ( c  n  along the r i v e r  at p o i n t  the concentration „n+l C.  Substituting  component and s i m p l i f y i n g  C  n+1  .  c  n  ) }  (  r  c  2  6  )  is  equation.  At any time,  o r e s t u a r y can be c o n s i d e r e d t o be  jAx at  „n . , .. C . exp i lki  =  + i  scheme can be w r i t t e n  The von Neumann s t a b i l i t y c o n d i t i o n  Series analysis of the difference profile  t h e ex-  f o r c e n t r a l d i f f e r e n c e s and  a number o f F o u r i e r components.  where i = / - l .  „  consider  t h a t was used t o a p p r o x i -  This difference  » . ^  systems  The c o n t r i b u t i o n time nAt is  o f t h e k'th  given by  i  Ax i  i n t o Equation  (C.26)  f o r t h e k'th  Fourier  gives „ n = GC  where  G = l - 0 ( l - 2 a ) ( 1 - c o s O ) + iBsinG  (C.27)  Ax and  0 =  ikAx  G i s the a m p l i f i c a t i o n matrix o f the d i f f e r e n c e  equation  (C.26).  o n l y depends on C , t h e e i g e n v a l u e o f t h e a m p l i f i c a t i o n m a t r i x is n  the v a l u e of G from E q u a t i o n  (C.27).  As C g i v e n by  Thus, t h e von Neumann s t a b i l i t y  condition  156  requires {l-8(l-2a)(1-cosO)}  Evaluating t h i s  2  +{gsin 0 }  < 1  2  f o r the t h r e e c a s e s o f upstream, c e n t r a l and downstream  d i f f e r e n c e s , the s t a b i l i t y c r i t e r i o n  i n each case i s g i v e n by  <  Ax  —  At  >  or U  '  ( C  I f t h e s t a b i l i t y c o n d i t i o n i s v i o l a t e d , c e r t a i n F o u r i e r components w i l l be u n a c c e p t a b l y a m p l i f i e d , and the v a l u e o f the c o n c e n t r a t i o n , as d e t e r mined by E q u a t i o n Richtmyer for  (C.26) w i l l o s c i l l a t e w i t h e v e r - i n c r e a s i n g a m p l i t u d e .  and Morton  [1967] g i v e a v e r y c l e a r example o f t h i s  the s i m p l e d i s p e r s i o n e q u a t i o n  the i n s t a b i l i t y m a n i f e s t s velocities  itself  (C.9).  oscillation  In the hydrodynamic  as an o s c i l l a t i o n  equations,  o f the w a t e r d e p t h s  (the water depths e v e n t u a l l y becoming n e g a t i v e ) .  and  Note t h a t the  von Neumann s t a b i l i t y c o n d i t i o n has been d e r i v e d f o r l i n e a r  equations  constant c o e f f i c i e n t s .  (C.26) i s con-  T h i s i m p l i e s t h a t the  s t a n t , o r t h a t t h e f l o w i s u n i f o r m and  U o f Equation  steady.  In a s i t u a t i o n where  with  U  i e s w i t h x and t , the s t a b i l i t y c o n d i t i o n i s assumed t o be determined worst case o f E q u a t i o n  explicit  c  simple d i s p e r s i o n equation  (C.9).  f i n i t e d i f f e r e n c e scheme  n l 3 +  by  the  (C.28) a l o n g t h e e s t u a r y .  Consider the mated by t h e  var-  _ n c  =  3  ^  {  c 2  (  A  x  )  "  - 2C 3+1  n +  3  C  n  > 3-1  T h i s can be  approx  157  and i f E i s c o n s t a n t , the s t a b i l i t y Morton,  c o n d i t i o n i s g i v e n by  [Richtmyer  and  1967]  2EAt (Ax)  .  2  <  T  or -ll^—  >_ 2E  The hydrodynamic e q u a t i o n s constant c o e f f i c i e n t s .  (C.29)  (C.3)  are n o n - l i n e a r and have  Richtmyer and Morton  [1967] l i n e a r i z e the  nonequations  and o b t a i n t h e f o l l o w i n g s t a b i l i t y c o n d i t i o n  ^  A t  I t i s apparent  _>  U ±-  /qT  t h a t Equations  < - °) c  (C.28),  f i n i t e d i f f e r e n c e equivalents of Equations  (C.29) and  (C.30) are  (C.7), ( t c . l l ) and  d e s c r i b e the c h a r a c t e r i s t i c curve o f i n f o r m a t i o n p r o p a g a t i o n plane  (see S e c t i o n C . l ) .  c o n s i d e r a t i o n o f Equation  The p h y s i c a l reason  o f C a t the p o i n t jAx a t time ( j - l ) A x , jAx and  of  (x,t)  T h i s e q u a t i o n s t a t e s t h a t the v a l u e  (n+l)At depends on t h e v a l u e o f C a t p o i n t s  (j+l)Ax a t time nAt.  I f t h i s a d v e c t i v e system i s d i s t u r b e d  the system a c c o r d i n g t o E q u a t i o n  Equation  i n the  f i n i t e d i f f e r e n c e approxima-  at p o i n t jAx a t time nAt, as shown i n F i g u r e C.6, through  (C.14), which  f o r t h i s can be seen from a  (C.26), t h e ^ e x p l i c i t  t i o n t o the a d v e c t i v e t r a n s p o r t e q u a t i o n .  the  (C.7).  the d i s t u r b a n c e  propagates  I f the s t a b i l i t y c o n d i t i o n  (C.28) i s v i o l a t e d , the d i s t u r b a n c e w i l l r e a c h the p o i n t ( j + l ) A x ,  3  158  Stable Explicit Scheme  uAt < Ax •  Position after At Ac  -o-  time = (n + I )At  uAt  U  Initial Position of Disturbance  time = nAt  Unstable Explicit Scheme uAt > Ax  :  Position after At Ac  time = (n+ I )At  uAt Initial Position of Disturbance  U  Ax  time - nAt  Ac j-l  Figure C.6 S t a b l e and U n s t a b l e  E x p l i c i t Advective  Schemes  159  and a l t e r t h e c o n c e n t r a t i o n t h e r e , b e f o r e t h e c o n c e n t r a t i o n a t p o i n t jAx has been updated.; a c c o r d i n g t o E q u a t i o n (C.26) . t r a t i o n a t p o i n t jAx a t time  In o t h e r words, t h e concen-  (n+l)At now depends on what i s o c c u r r i n g a t a  d i s t a n c e g r e a t e r than Ax from jAx (see F i g u r e C.6). conditions the d i f f e r e n c e equation and  O b v i o u s l y , under  (C.26) i s no l o n g e r p h y s i c a l l y  these  meaningful,  i t i s o n l y t o be expected t h a t i t behaves i n some s t r a n g e manner.  l a r reasoning holds f o r the r e l a t i o n s h i p o f Equation t h e d i s p e r s i v e e q u a t i o n , and E q u a t i o n s  Simi-  (C.29) t o ( C . l l ) f o r  (C.30) t o (C.14) f o r t h e hydrodynamic  equation. The problem o f s t a b i l i t y o n l y a r i s e s w i t h e x p l i c i t schemes, where t h e time d i f f e r e n c i n g i s f o r w a r d .  In i m p l i c i t  differencing differencing  schemes, where the time d i f f e r e n c i n g i s backward, t h e v a l u e s a r e s i m u l t a n e o u s l y determined  a t each g r i d p o i n t a l o n g t h e e s t u a r y .  T h i s technique  automa-  t i c a l l y adjusts f o r the e f f e c t of the information propagating distances g r e a t e r than Ax i n t h e time  increment.  A d i f f e r e n c e scheme i s s i m p l y a?-means  o f d e t e r m i n i n g t h e v a l u e C ^~ from t h e v a l u e C . N+  e q u a t i o n maps C  into C  n  In e f f e c t , t h e d i f f e r e n c e  , and t h e e i g e n v a l u e s o f t h i s mapping  (in other  words, t h e e i g e n v a l u e s o f the a m p l i f i c a t i o n matrix) d e f i n e t h e c h a r a c t e r i s e t i c d i r e c t i o n s along which c ^ n+  i s determined  from C  n  [see Sawyer, 1966].  It  i s o n l y t o be expected t h a t these e i g e n v a l u e s s h o u l d r e f l e c t t h e u n d e r l y i n g p h y s i c s o f t h e problem.  C. 4  SUMMARY F o r any one-dimensional  p h y s i c a l system, t h e r e a r e c h a r a c t e r i s t i c - "  curves i n the (x,t).plane t h a t d i r e c t l y r e f l e c t s the p h y s i c a l behaviour o f the  160  system.  These curves a r e o f t e n h i d d e n o r masked when a d i f f e r e n t i a l equa-  t i o n i s d e r i v e d r e l a t i v e t o a f i x e d g r i d i n t h e (x,'t) p l a n e .  The problem  o f n u m e r i c a l d i s p e r s i o n o c c u r s because t h e e q u a t i o n o f a d v e c t i v e t r a n s p o r t i s s o l v e d r e l a t i v e t o a f i x e d g r i d , r a t h e r than a l o n g i t s a p p r o p r i a t e c h a r a c teristic.  I n an e x p l i c i t d i f f e r e n c e scheme, t h e problem o f s t a b i l i t y a r i s e s  f o r e x a c t l y t h e same r e a s o n s . teristic  curves  I f the equations are solved along t h e i r  charac-  (or by an i m p l i c i t method) t h e r e a r e no s t a b i l i t y r e q u i r e -  ments, and t h e r e l a t i v e s i z e o f Ax and At a r e determined the dependent v a r i a b l e .  by t h e v a r i a t i o n i n  APPENDIX D  DETAILS OF THE SOLUTION SCHEMES OF THE HYDRODYNAMIC AND MASS TRANSPORT  D.l  EQUATIONS  NUMERICAL SOLUTION OF THE HYDRODYNAMIC EQUATIONS The f i x e d mesh, e x p l i c i t f i n i t e  d i f f e r e n c e method o f Dronkers  was used t o o b t a i n a n u m e r i c a l s o l u t i o n t o the hydrodynamic e g u a t i o n s . finite  [1969] The  d i f f e r e n c e forms o f E q u a t i o n s (3.1) and (3.2) a r e r e s p e c t i v e l y  U  2n+l 2m  =  U  2 n - l .At , . 2n+l,..2n-l 2 n - l , At,,2n .2n 2m " 2Ax" m 2m 2 " 2m-2 ~ A x - 2 m l ~ 2 m - i (  }U  (U  2  U  )  g  (h  h  +  -»  )  +  (D.l)  cV" Y  2m  and n  where  2n+2 2m+l  _ "  h  2n At 2n+l 2n 2n+l 2n 2m+l " . _2n 2m+2 2m+2 " 2m' 2 m ^xb^ 2m+l (U  A  U  A  )  and h ^ are r e s p e c t i v e l y t h e mean v e l o c i t y and s u r f a c e  the g r i d p o i n t g i v e n  by x = mAx a t t i m e t = nAt.  (  D  ,  2  )  elevation a t  Note t h a t t h e d i f f e r e n c e  scheme employs c e n t r a l d i f f e r e n c e s and t h a t t h e "bar" s i g n has been dropped from t h e c r o s s - s e c t i o n a l l y averaged v a r i a b l e s and parameters t o a v o i d w i t h t h e s u b s c r i p t s and s u p e r s c r i p t s ' . presenting  ( T h i s c o n v e n t i o n w i l l be f o l l o w e d  confusion when  f i n i t e difference quantities). Because a n e x p l i c i t f i n i t e d i f f e r e n c e scheme was used t o s o l v e the  hydrodynamic e q u a t i o n s , t h e r e l a t i v e s i z e o f Ax and At i s governed by t h e  161  162  stability  criterion Ax • -• • At-' U  This s t a b i l i t y  ±  r-=y /  g  Y  c r i t e r i o n i s discussed  i n d e t a i l i n Appendix C.  Because  o f t h e f r i c t i o n term, t h e hydrodynamic e q u a t i o n s a r e n a t u r a l l y d i s s i p a t i v e , and t h i s h e l p s m a i n t a i n  stability.  In f a c t , s t a b i l i t y was found t o be de-  pendent on a minimum v a l u e o f f r i c t i o n . and  For t h e c r o s s - s e c t i o n a l gemoetries  f l o w and t i d a l c o n d i t i o n s o f S e c t i o n 4.1.4, t h e hydrodynamic e q u a t i o n s  became u n s t a b l e constant terion  when Manning's "n" was l e s s t h a n 0.012.  throughout the estuary f o r t h i s i n v e s t i g a t i o n ) .  (Manning's "n" was The s t a b i l i t y  o f Appendix C i s independent o f the e f f e c t s o f f r i c t i o n .  i t was d e r i v e d from a l i n e a r s t a b i l i t y a n a l y s i s on t h e n o n - l i n e a r mic  equations.  The t r u e n o n - l i n e a r  stability  cri-  However, hydrodyna-  c r i t e r i o n may be more s t r i c t  t h a n t h e d e r i v e d l i n e a r c r i t e r i o n , and t h i s i s r e f l e c t e d by t h e n e c e s s i t y o f a minimum l e v e l o f f r i c t i o n t o p r e s e r v e The in Figure D . l .  stability.  s o l u t i o n p o i n t s f o r t h e f i n i t e d i f f e r e n c e scheme a r e shown I t i s seen t h a t t h e v e l o c i t y and e l e v a t i o n p o i n t s a r e s t a g -  g e r e d i n both time and space.  In e f f e c t , t h e v a l u e s  o f v e l o c i t y and water  s u r f a c e e l e v a t i o n s a r e marched forward through time i n a " l e a p - f r o g " manner. From E q u a t i o n  D . l , t h e v e l o c i t i e s a t any time s t e p 2n+l a r e determined by the  s u r f a c e e l e v a t i o n s a t time step The  2n  s u r f a c e e l e v a t i o n s a t time step  c i t i e s a t time step 2n+l b y ' u s i n g The  and t h e v e l o c i t i e s a t time s t e p 2 n - l . 2n+2 a r e then determined from the v e l o -  Equation•(D.2).  f i x e d mesh o f space p o i n t s o r " ; s t a t i o n s " used i n s o l v i n g the  hydrodynamic e q u a t i o n s i s shown i n F i g u r e B.4. odd  V e l o c i t i e s are evaluated a t  -numbered s t a t i o n s and water s u r f a c e e l e v a t i o n s are e v a l u a t e d  at  even-  u  1s 1  % t '  2n+2  2n+l  >  ?  u  2n  2n-l  /  A\  V  2m-2  <  u  i  >f  U  2m-3  U  h  u 1  ,h  %?  ?  (>  A X  h  % f A  u  h  u  2m-| 2m longitudinal  \  u c 2m+| distance >  2m + 2 x  Figure D.l E x p l i c i t F i n i t e Difference Grid o f the Hydrodynamic E q u a t i o n s [ A f t e r Dronkers,  1969]  164  numbered s t a t i o n s .  At c h a n n e l j u n c t i o n s , i t i s n e c e s s a r y t h a t the  junction  s t a t i o n be a water s u r f a c e e l e v a t i o n s t a t i o n , and  t h i s accounts f o r the  coincidence  and  cases  o f the  (see F i g u r e  j u n c t i o n s o f the model e s t u a r y  r e a l estuary  non-  i n several  B.4).  In a p p l y i n g the hydrodynamic model, a d e s i g n t i d a l c y c l e of s e l e c t e d flow and t i d a l c o n d i t i o n s i s chosen.  The  freshwater  a boundary c o n d i t i o n a t C h i l l i w a c k and the t i d a l r i s e and  i n f l o w forms  f a l l o f the water  s u r f a c e forms boundary c o n d i t i o n s a t the seaward ends o f the f o u r c h a n n e l s t h a t emerge from the d e l t a . out  The  v e l o c i t y and water s u r f a c e e l e v a t i o n s t h r o u g h -  the e s t u a r y are.-rset t o i n i t i a l v a l u e s , and  E q u a t i o n s >(D.l) and  (D.2)  are  then used t o march the v e l o c i t i e s and water s u r f a c e e l e v a t i o n s through t i m e . The  i n i t i a l values  of v e l o c i t y and water s u r f a c e e l e v a t i o n s do n o t have t o  exact as the model w i l l converge to the t r u e i n i t i a l v a l u e s  be  after several t i d a l  cycles.; The for  hydrodynamic model was  s o l u t i o n by d i g i t a l computer.  ments and cycles.  programmed i n h i g h speed FORTRAN  The  program c o n s i s t e d o f 670  r e q u i r e d a p p r o x i m a t e l y 40 seconds t o a n a l y s e (The  first  t i d a l c y c l e was  two  active state-  complete  o f v e l o c i t y and c r o s s - s e c t i o n a l a r e a were r e c o r d e d  D.2  From S e c t i o n 3.2  THE  The  values  at half-hourly intervals  l a t e r u s e d i n s o l v i n g the t i d a l l y v a r y i n g mass t r a n s p o r t  NUMERICAL SOLUTION OF  tidal  r e q u i r e d f o r the e s t u a r y t o converge t o  the t r u e i n i t i a l v a l u e s of v e l o c i t y and water s u r f a c e e l e v a t i o n ) .  and  (FORTRANH)  equation.  TIDALLY VARYING MASS TRANSPORT EQUATION  i t i s seen t h a t the  i n g mass t r a n s p o r t e q u a t i o n  o v e r any  s t e p , a d i s p e r s i o n s t e p and  a source-sink  s o l u t i o n o f the t i d a l l y  time increment i n v o l v e s an step.  The  vary-  advection  f i n i t e d i f f e r e n c e form  165  o f E q u a t i o n (3.4) used t o advect the moving p o i n t s a l o n g the c h a r a c t e r i s t i c s d u r i n g the a d v e c t i o n s t e p was n+1 x_.  n x.. +  =  n.. u At  ( .3) D  where n+1 x. -  . i s t h e p o s i t i o n o f moving p o i n t j a t the end o f time increment n;  x"  i s the p o s i t i o n o f moving p o i n t j a t the s t a r t o f time increment n;  1  2  and u  n  ' i s the average v e l o c i t y between p o s i t i o n s x ? and x?"*"^" d u r i n g time increment n. ^ ^  The hydrodynamic  1  model was  1  used t o o b t a i n the l o n g i t u d i n a l v e l o c i t i e s  and  c r o s s - s e c t i o n a l a r e a s throughout the e s t u a r y a t h a l f - h o u r l y i n t e r v a l s d u r i n g the t i d a l c y c l e .  The v a l u e o f u  can be obtained, from t h e s e v e l o c i t i e s .  n  In the d i s p e r s i o n s t e p , the c o n c e n t r a t i o n o f t h e moving p o i n t s i s a d j u s t e d f o r the e f f e c t s o f d i s p e r s i o n d u r i n g the time increment. e x p l i c i t and an i m p l i c i t f i n i t e d i f f e r e n c e scheme were the d i s p e r s i v e step.  The i m p l i c i t scheme was  d e s c r i b e d i n Richtmyer and Morton  [1967].  Both an  investigated for  the C r a n k - N i c h o l s o n scheme  I r r e s p e c t i v e o f whether an  impli-  c i t o r e x p l i c i t scheme i s used, the " i n f o r m a t i o n " p r o p a g a t e s through a d i s p e r s i v e system a c c o r d i n g t o x  2  = 2Et  as i s d i s c u s s e d i n Appendix  (D.4) C.  The  namely  *<  <§>  2  f i n i t e d i f f e r e n c e form o f E q u a t i o n  (D.4),  d e f i n e s t h e s t a b i l i t y c r i t e r i o n o f an e x p l i c i t scheme/ i n Appendix C.  as i s a l s o d i s c u s s e d  E q u a t i o n (D.5) a l s o determines t h e response o f an i m p l i c i t  system, and a l t h o u g h t h e i m p l i c i t system i s u n c o n d i t i o n a l l y s t a b l e , t h e convergence c r i t e r i o n i s r e l a t e d t o E q u a t i o n (D.5). significantly  In o t h e r words, i f A t i s  l a r g e r than t h e At o f E q u a t i o n (D.5) an i m p l i c i t scheme  converge t o t h e wrong s o l u t i o n .  Of the two schemes, t h e i m p l i c i t  was s l i g h t l y f a s t e r , but because o f t h e l a r g e somewhat matrices involved  scheme  ill-conditioned  (of o r d e r 150) t h e r e were u n c e r t a i n t i e s i n the s i g n i f i -  cance o f r o u n d - o f f e r r o r s . used.  may  (The i l l - c o n d i t i o n e d  Consequently, t h e s i m p l e r e x p l i c i t scheme  was  n a t u r e o f t h e s e m a t r i c e s was due t o t h e v a r i a b l e  s p a c i n g o f t h e moving p o i n t s .  T h i s v a r i a b l e s p a c i n g i s d i s c u s s e d i n Chapter  3). The f o l l o w i n g e x p l i c i t c e n t r a l d i f f e r e n c e e q u a t i o n was used t o E q u a t i o n (3.6) n+1  n c. +  3  2At n+1 A.(Ax) . 3 3j + l , j - l  Ax'j,j-l^j (D  where c. i s the c o n c e n t r a t i o n o f moving p o i n t j a t t h e s t a r t o f ^ time increment n; j,j-l  n n = x. - x j - l  3  x. i s t h e p o s i t i o n o f moving p o i n t j increment n;  a t t h e s t a r t o f time  3  (EA)  n  j,j-l  i s the average v a l u e o f EA between moving p o i n t s j and j - l d u r i n g time increment n; xj,'[  and A. i s t h e average c r o s s - s e c t i o n a l a r e a a t moving p o i n t j d u r i n g time increment n.  167  Equation  (D.6)  i s the u s u a l e x p l i c i t  c e n t r a l d i f f e r e n c e a p p r o x i m a t i o n except  t h a t i t i s a p p l i e d over a g r i d where Ax i s not c o n s t a n t .  The s t a b i l i t y r e -  quirement o f E q u a t i o n (D.5) g e n e r a l l y r e s u l t s i n a At s m a l l e r than the b a s i c time increment o f one hour.  When t h i s o c c u r r e d , E q u a t i o n (D.6)  was  s o l v e d r e p e a t e d l y w i t h i n t h e hour f o r as many i t e r a t i o n s r e q u i r e d by Equation the  (D.5).  (This assumes t h a t the r e l a t i v e  s p a c i n g o f the p a r t i c l e s  v a l u e s o f E and A remain c o n s t a n t d u r i n g the hour —  and  which i s a r e a s o n -  a b l e assumption). Finally,  i n t h e s o u r c e - s i n k s t e p , the c o n c e n t r a t i o n s o f the moving  p o i n t s are a d j u s t e d f o r the e f f e c t s o f any s o u r c e - s i n k p r o c e s s e s o c c u r r i n g . The f o l l o w i n g f i n i t e d i f f e r e n c e e q u a t i o n can be used t o approximate sink equation  the s o u r c e -  (3.7)  c. 3  +1  n  =  cP + At .Z.S. 3 1=1 i  (D.7)  a l t h o u g h i t i s noted t h a t E q u a t i o n (3.7) can be s o l v e d a n a l y t i c a l l y d u r i n g the  time increment. The t i d a l l y v a r y i n g mass t r a n s p o r t model was programmed i n h i g h  speed FORTRAN  (FORTRANH) f o r s o l u t i o n by d i g i t a l  computer.  The program con-  s i s t e d o f some 1,300  a c t i v e statements and r e q u i r e d a p p r o x i m a t e l y 100  to analyse s i x t i d a l  cycles.  D.3  seconds  NUMERICAL SOLUTION OF THE TIDALLY AVERAGED MASS TRANSPORT EQUATION The one-dimensional, t i d a l l y  by the method o f Thomann [1963].  averaged mass t r a n s p o r t was  In t h i s s o l u t i o n , the e s t u a r y i s d i v i d e d  i n t o a number o f segments o r boxes, as shown i n F i g u r e D.2, i s assumed t o be c o m p l e t e l y mixed.  solved  and each segment  I f the t i d a l l y averaged t r a n s p o r t p r o c e s s e s  169  and waste d i s c h a r g e s a r e steady a substance  i n time,  a inass b a l a n c e over  undergoing f i r s f c r o r d e r decay g i v e s  segment i f o r  (see Thomann [1971] f o r de-  tails)  Q{a. , .c, *  i-l,i  i-1  + (1-a. . .)c.} - Q{a. _ . c . + (1-a. . J c .} i - l , i i i,i+l i i,i+l i+l  + E. . ( c . -c.) + E. . , ( c . -c.) - K.c.V. + W. = 0 i-l,i i-1 l i,i+l i+l l i l l l  (D.8)  where c  i s the c o n c e n t r a t i o n i n segment i ;  V^ i s t h e volume o f segment i ; W.  i s the mass o f waste substance per t i d a l cycle;  d i s c h a r g e d i n t o segment i  K^ i s t h e decay c o e f f i c i e n t f o r segment i ; Q  i s the t i d a l l y averaged d i s c h a r g e through the e s t u a r y (the f r e s h w a t e r d i s c h a r g e ) ;  a. 1  ,  1  +  1  i s the t i d a l exchange c o e f f i c i e n t between segments i and ( i + l ) ;  and •  E^ l f  i s the " e f f e c t i v e d i s p e r s i v e " t r a n s p o r t between segments i and ( i + l ) .  The s u b s c r i p t n o t a t i o n o f the v a r i o u s terms i s The f i r s t  illustrated  i n Figure  two terms on the r i g h t - h a n d s i d e o f E q u a t i o n  (D.8) are  the t i d a l l y averaged a d v e c t i v e t r a n s p o r t i n t o and o u t o f segment i . factor a i s a weighting  D.2.  The  f a c t o r used t o determine the c o n c e n t r a t i o n a t the  i n t e r f a c e o f two segments from the c o n c e n t r a t i o n a t w i t h i n each segment. In p u r e l y t i d a l flows a i s s e t e q u a l t o 0.5 t o a l l o w f o r the e f f e c t s o f flow r e v e r s a l , whereas i n a r i v e r f l o w s i t u a t i o n a i s s e t e q u a l t o 1.0 as the f l o w i s always downstream.  The n e x t two terms on the r i g h t - h a n d s i d e o f the e q u a t i o n  170  r e p r e s e n t the n e t d i s p e r s i v e t r a n s p o r t o f mass i n t o segment i from the neighbouring  segments. E  E' i s g i v e n by  -• "i,i+l  _  E  j,i+1 L  A  j,i+1  i,i+1  where E^ '  i s t h e e f f e c t i v e c o e f f i c i e n t o f d i s p e r s i o n over a t i d a l p e r i o d a t t h e i n t e r f a c e o f segments i and (i+1);  '  i s t h e c r o s s - s e c t i o n a l a r e a ( t i d a l l y averaged) o f the i n t e r f a c e between segments i and (i+1);  A.  and  The  i s t h e average o f t h e l e n g t h s o f segments i and X i + D • f i n a l two terms on the r i g h t - h a n d c l s i d e o f E q u a t i o n  (D.8) r e p r e s e n t the  e f f e c t s o f decay and waste d i s c h a r g e . An e q u a t i o n s i m i l a r t o (D.8) can be w r i t t e n f o r each o f the n segments o f the e s t u a r y t o g i v e a system o f n simultaneous, d i f f e r e n c e equations.  Equation  linear,  (D.8) can be w r i t t e n  a. . .c. , + a..c. + a. . . c . ^ . = W. i,i-l i - l i iI i , i + l i+I l  (D.9)  where a. . .. = -a. . . Q - E. , . ; i,i-l i - l , i i - l , i a-.... = Q{a. . - (1- c. , .} + E. + E. + ii i,i+l i - l , i i - l , i i,i+l a. . . = (1-a. i,i+l i,i+l  )Q - E.i , i + l .  In m a t r i x n o t a t i o n , the system o f Equations  k  £  V.K.; i l  (D.9) can be w r i t t e n  =  ( D  -  where A i s a (nxn) t r i - d i a g o n a l m a t r i x and C and W a r e (nxl) column m a t r i c e s . ri,  •  J  O,  r\j  '  1 0 )  171  Thomann [1971] g i v e s d e t a i l s o f t h e complete BOD-DO system o f e q u a t i o n s . Thomann's model i s e s s e n t i a l l y a f i n i t e ference and  model t h a t uses c e n t r a l d i f f e r e n c e s f o r t h e d i s p e r s i v e  c e n t r a l d i f f e r e n c e s f o r the a d v e c t i v e  stream d i f f e r e n c e s f o r the a d v e c t i v e c e n t r a l d i f f e r e n c e s are d e s c r i b e d lar  fixed grid finite  dif-  transport,  t r a n s p o r t when a = 0.5 and up-  t r a n s p o r t when a = 1.0.  i n Appendix C ) .  t o an i m p l i c i t f i n i t e d i f f e r e n c e scheme.  (Upstream and  Thomann's model i s s i m i ~ -  ;  In both schemes the c o n c e n t r a -  t i o n a t a g r i d p o i n t o r i n a segment depends on the c o n c e n t r a t i o n s  a t neigh-  b o u r i n g g r i d p o i n t s o r segments, and consequently t h e response o f the estuary  i s determined by a square t r i - d i a g o n a l m a t r i x i n b o t h e q u a t i o n s .  F i n i t e element s o l u t i o n s are a l s o governed by a square t r i - d i a g o n a l m a t r i x , as i s d i s c u s s e d  i n Section  2.2.3.  Thomann's d i f f e r e n c e e q u a t i o n s are u n c o n d i t i o n a l l y not  s u f f e r from n u m e r i c a l d i s p e r s i o n , as i s d i s c u s s e d  ever, there  i s a non-negativity  s t a b l e and do  i n Chapter 3.  How-  requirement f o r each segment g i v e n by  a.,i+l > 1 i  E. ' Q  (D.ll)  X  I f t h i s c r i t e r i o n i s v i o l a t e d the d i s c h a r g e o f a waste substance i n t o s e g ment i r e s u l t s i n a n e g a t i v e c o n c e n t r a t i o n  i n t h e segment.  r e a s o n f o r t h i s i s t h a t more substance i s b e i n g t r a n s p o r t e d segment p e r  t i d e c y c l e than i s b e i n g added.  Equation  (D.ll)  out o f t h e  The substance i s  upstream by d i s p e r s i o n and downstream by a d v e c t i o n ing  The p h y s i c a l  transported  and d i s p e r s i o n .  Rearrang-  gives E. . > (1-a. . ,,)Q i,i+l i,i+l n  x  which imposes l i m i t s on the r e l a t i v e s i z e o f t h e d i s p e r s i v e and a d v e c t i v e transport  processes.  172  F i n a l l y , the ease w i t h which Thomann's approach  handles  the s e p a r a t e channels o f the e s t u a r y s h o u l d be mentioned.  Figure  shows the m a t r i x A o f E q u a t i o n  (D.10).  Note t h a t the t h r e e  o f the e s t u a r y are c o n t a i n e d i n the one m a t r i x .  D.3  channels  In e f f e c t , the m a t r i x  i s p a r t i t i o n e d i n t o t h r e e s e p a r a t e b l o c k s , each o f which r e p r e s e n t s a single  channel.  Note t h a t each b l o c k o r channel i s uncoupled  t h e o t h e r s except a t the j u n c t i o n s t a t i o n s , where an a d d i t i o n a l which i s n o t t r i - d i a g o n a l , appears  from term,  i n the rows and columns o f the m a t r i x .  These a d d i t i o n a l terms r e f l e c t the e x t r a boundary through which mass t r a n s p o r t o c c u r s a t the  junctions.  The t i d a l l y averaged mass t r a n s p o r t model was FORTRAN f o r s o l u t i o n by d i g i t a l computer. 250  a c t i v e statements  programmed i n  The program c o n s i s t e d o f some  and r e q u i r e d a p p r o x i m a t e l y 10 seconds  the steady s t a t e response o f the e s t u a r y .  to  determine  173  Figure The  M a t r i x A of E g u a t i o n  (D.8)  D.3 F o r The  Fraser River  Estuary  APPENDIX E  ESTIMATION OF LATERAL DISPERSION  E x i s t i n g t h e o r i e s o f l a t e r a l d i s p e r s i o n were used t o e s t i m a t e the time o f c r o s s - s e c t i o n a l m i x i n g i n the Main Arm River Estuary.  - Main Stem o f the F r a s e r  The p r e d i c t e d time o f c r o s s - s e c t i o n a l m i x i n g appears h i g h  f o r the c o n d i t i o n s o f t h i s s t u d y .  More r e c e n t work on secondary c u r r e n t s  i n r i v e r s was  used t o develop r e v i s e d e s t i m a t e s o f the c r o s s - s e c t i o n a l  m i x i n g time.  These r e v i s e d e s t i m a t e s i n d i c a t e much f a s t e r m i x i n g over the  cross-section.  E.l  EXISTING ESTIMATES OF LATERAL MIXING F o r s t e a d y u n i f o r m f l o w i n s t r a i g h t c h a n n e l s , the c o e f f i c i e n t o f  l a t e r a l d i s p e r s i o n i s g i v e n by  e  z  [ F i s c h e r , 1969a].  = 0.23yU^  (E.l)  where y i s the mean depth o f c r o s s - s e c t i o n ; and i s the shear v e l o c i t y . The presence o f bends i n the c h a n n e l o f a r i v e r o r e s t u a r y induces secondary c u r r e n t s t h a t i n c r e a s e the r a t e o f l a t e r a l m i x i n g . f l o w around due  a bend, the i n c r e a s e i n the c o e f f i c i e n t o f l a t e r a l  t o these secondary c u r r e n t s has been e s t i m a t e d as  Se  z  =  -  U  -2-3 / '5 2 k  174  R  U  *  • I  spiral  F o r steady dispersion  [ F i s c h e r , 1969a]  (E.2)  I  175  where u i s t h e mean l o n g i t u d i n a l  velocity;  k i s Von Kantian's c o n s t a n t ; R i s t h e r a d i u s o f c u r v a t u r e o f the bend; and I i s a factor  (negative) t h a t depends on c h a n n e l f r i c t i o n and  k, and i s e v a l u a t e d by F i s c h e r .  (A t y p i c a l v a l u e o f I i s -0.3).  Ward [1972] measured t h e c o e f f i c i e n t o f l a t e r a l d i s p e r s i o n  from  l a b o r a t o r y experiments o f o s c i l l a t o r y f l o w (purely t i d a l ) i n a channel s i s t i n g of a series-of e' z  =  bends.  con-  H i s r e s u l t s a r e o f the form  ayU* *  (E.3)  where e  i s t h e t i d a l l y averaged c o e f f i c i e n t o f l a t e r a l d i s p e r s i o n due t o o s c i l l a t o r y flow; i s t h e t i d a l l y averaged o s c i l l a t o r y flow;  shear v e l o c i t y due t o  and a i s a f a c t o r depending on the r a t i o s o f t h e depth and w i d t h o f the c h a n n e l s t o the r a d i u s o f c u r v a t u r e of the bends. (For s t r a i g h t c h a n n e l s a = 0.5, and f o r wide s h a l l o w channels w i t h a s m a l l r a d i u s o f c u r v a t u r e a^2.0).  In a t i d a l l y v a r y i n g s i t u a t i o n where t h e r e i s b o t h a steady v e l o c i t y component and a s i n u s o i d a l l y v a r y i n g component, t h e t i d a l l y averaged can be e s t i m a t e d from  where h i s Manning's " n " f i s t h e s t e a d y v e l o c i t y component;  shear  velocity  176  and U Equation  i s the amplitude o f t h e o s c i l l a t o r y v e l o c i t y component.  (E.4) i s o b t a i n e d from the Manning f o r m u l a r e l a t i n g steady  to f r i c t i o n a l e f f e c t s .  velocity  A c c o r d i n g t o Odd  [1971], the Manning or Chezy formu-  l a t i o n s h o u l d be an adequate d e s c r i p t i o n  of f r i c t i o n a l e f f e c t s i n f a s t flow-  i n g , well-mixed The  estuaries.  following  t o the Main Arm  e s t i m a t i o n of d i s p e r s i o n  coefficients i s limited  - Main Stem o f the F r a s e r R i v e r E s t u a r y .  T h i s i s the w i d e s t  channel and c a r r i e s the b u l k o f the f l o w through the d e l t a .  A l o n g t h i s chan-  n e l , t h e r e are 12 s i g n i f i c a n t bends whose r a d i i o f c u r v a t u r e range from 7,000 f e e t t o 35,000 f e e t , as i l l u s t r a t e d i n F i g u r e E . l . t u d i n a l v e l o c i t y throughout Main Arm  the t i d a l c y c l e  The v a r i a t i o n o f  at three s t a t i o n s  - Main Stem i s shown i n F i g u r e E.2.  longi-  a l o n g the  The predominance o f the  tidal  component i n the lower reaches and the steady component i n the upper reaches is  apparent. The f o l l o w i n g  l a t e r a l dispersion. ponent U , f  procedure was  The v e l o c i t y a t each bend was  divided  i n t o a s t e a d y com-  which i s g i v e n by the f r e s h w a t e r d i s c h a r g e through the  averaged a r e a , and an approximating E q u a t i o n s ;(E-;.i) and dispersion was  used t o e s t i m a t e the c o e f f i c i e n t s o f  due  sinusoidal  component o f amplitude  t o the steady component o f v e l o c i t y  s o i d a l o r o s c i l l a t o r y component  (e^).  the shear v e l o c i t i e s due t o the steady  these two  U . t  (E.2) were used t o e s t i m a t e the c o e f f i c i e n t o f l a t e r a l ('e ) , and E q u a t i o n ;[E.3]  used t o e s t i m a t e the c o e f f i c i e n t o f l a t e r a l d i s p e r s i o n  components.  tidally  Equation  was  t o the  sinu-  used t o e s t i m a t e  f (U^) and o s c i l l a t o r y  The t o t a l l a t e r a l d i s p e r s i o n  separate e f f e c t s .  (E.4) was  due  t (U^)  velocity  assumed t o be the sum  The r e s u l t s o f the c a l c u l a t i o n s  of  a r e shown i n  Figure E . l Bends A l o n g The Main Arm - Main Stem  Q= 3 6 , 5 0 0 c f s at Chilliwack Tidal Range at Steveston  Jan.24,1952 178  24( Hours)  24( Hours)  Stat.No.33 24(Hours)  24(Hours) -2  l  F i g u r e E.2 T i d a l l y Varying  Velocities  TABLE E . l ESTIMATION OF COEFFICIENTS OF LATERAL DISPERSION  •  o  V  <D  <D  «  o  o  •P  <u w  D  *  M-l  P  *  u  CO  STAT. NUMBERS  <u w \  BEND NO.  o  1  57-60  17,000  24  1.2  0.1  0.032  0.088  0.005  0.088  0.12  0.14  1.0  0.24  0.6  2  55r57  7,000  24  1.2  0.8  0.032  0.088  0.037  0.097  0.29  0.34  1.9  0.29  2.3  3  52-55  7,000  35  0.9  0.7  0.032  0.062  0.031  0.070  0.29  0.40  1.8  0.37  2.8  4  46-52  17,000  31  0.9  1.2  0.032  0.063  0.054  0.087  0.13  0.18  1.0  0.24  2.2  5  41-46  16,000  37  0.8  1.3  0.032  0.055  0.057  0.083  0.13  0.23  1.0  0.26  2.7  6  36-41  35,000  32  0.7  1.3  0.032  0.049  0.058  0.081  0.06  0.09  0.6  0.23  1.5  7  32-35  7,000  43  0.7  1.5  0.025  0.036  0.050  0.066  0.29  0.62  1.8  0.62  4.8  8  29-31  7,000  38  0.7  2.2  0.025  0.037  0.074  0.091  0.29  0.54  1.8  0.52  5.8  9  19-23  16,000  38  0.6  2.5  0.025  0.033  0.088  0.10  0.13  0.24  1.0  0.28  4.7  10  14-17  8,000  38  0.7  3.3  0.025  0.037  0.11  0.12  0.25  0.48  1.6  0.45  7.3  11  8-14  33,000  34  0.6  3.0  0.025  0.032  0.10  0.11  0.06  0.10  0.6  0.24  2.3  12  2-8  27,000  34  0.6  3.5  0.025  0.032  0.12  0.13  0.07  0.13  0.6  0.24  2.7  0)  +J  -p  <D  £  0) fa  w \  iw  fa  ID  •  -P fa  D  W  Ul  \  \  fa  ~G  <4-l * D  •  -P fa  +1 * • D  -P  fa  \  A  \  o  1  >i  \  -P  fa  (J  1  W > i  \  u  \  fa  180  Table E . l ,  t h e combined c o e f f i c i e n t o f l a t e r a l d i s p e r s i o n b e i n g d e s i g n a t e d  e . c From t h e r e s u l t s o f Table E . l , t h e average v a l u e o f t h e c o e f f i c i e n t o f l a t e r a l d i s p e r s i o n a l o n g t h e Main 'Arm - Main Stem i s a p p r o x i m a t e l y 3.3  square f e e t p e r second.  Assuming an average w i d t h and depth o f channel  o f 1800 f e e t and 30fcfeet r e s p e c t i v e l y , and an e f f l u e n t d i s c h a r g e a t t h e channel edge, t h e time r e q u i r e d f o r 80 p e r c e n t m i x i n g i n t h e l a t e r a l t i o n i s 55 hours  [Ward, 1973].  (The percentage l a t e r a l m i x i n g i s d e f i n e d as  the r a t i o o f t h e l a t e r a l root-mean-square age l a t e r a l  direc-  c o n c e n t r a t i o n d e v i a t i o n t o the a v e r -  concentration).  T h i s e s t i m a t e o f 55 hours f o r the time o f 80 p e r c e n t c r o s s s e c t i o n a l m i x i n g seems v e r y h i g h .  S t r o n g secondary c u r r e n t s around bends  9, 10 and 11 a r e o b s e r v a b l e i n the e s t u a r y d u r i n g ebb f l o w c o n d i t i o n s .  In  f a c t , i f the f i s h e r m e n l o s e a n e t i n t h e Main Stem, i t i s o f t e n washed ashore on the n o r t h bank o f bend 11 by these secondary c u r r e n t s .  In a s u r f a c e  float  study d u r i n g ebb t i d e c o n d i t i o n s , t h e f l o a t s were a l s o washed ashore on t h e n o r t h bank o f bend 11.  I t seems t h a t t h e r e s u l t s o f T a b l e E . l underestimate  the e f f e c t s o f t h e secondary c u r r e n t s on the l a t e r a l m i x i n g p r o c e s s .  In view  o f t h i s , an attempt was made t o e s t i m a t e t h e v e l o c i t i e s o f the secondary c u r r e n t s and the i n f l u e n c e o f t h e secondary c u r r e n t s on l a t e r a l m i x i n g .  E.2  VORTICITY ESTIMATE OF SECONDARY CURRENTS First  c o n s i d e r t h e l a t e r a l m i x i n g due t o secondary c u r r e n t s .  the sake o f s i m p l i c i t y , t h e v e r t i c a l d i s t r i b u t i o n o f t h e secondary i s assumed t o be l i n e a r , a s shown i n F i g u r e E.3.  This v e l o c i t y  For  velocities  distribution  s h o u l d be a s a t i s f a c t o r y a p p r o x i m a t i o n f o r wide, s h a l l o w c r o s s - s e c t i o n s .  181  w(?7) = W ( !- 2 77)1 s  V = y/y  _ W ^  ^  ^  ^  Figure  ^  ^  s  ^  ^  ^  E.3  Assumed L i n e a r D i s t r i b u t i o n o f Secondary V e l o c i t i e s  ^  ^  ^  B  182  The v a r i a t i o n o f v e l o c i t y w i t h depth i s g i v e n by  w(n)  = W ( l - 2n)  n  where W  g  (E.5)  s  = y/y  i s t h e s u r f a c e v e l o c i t y o f t h e secondary  o f E l d e r [1959] and F i s c h e r be e s t i m a t e d e  c o e f f i c i e n t o f l a t e r a l d i s p e r s i o n can  = -y J w(n){f w(Ti> [ J — d n ] d n > d n . ' e o o o y 1  (E.6)  n  T,  where e ^ i s t h e c o e f f i c i e n t o f v e r t i c a l m i x i n g . be  In the manner  from r2  z  [1969a]the  current.  I f we assume the f l o w t o  steady and. u n i f o r m , t h e v e r t i c a l p r o f i l e o f t h e l o n g i t u d i n a l v e l o c i t y u  i s l o g a r i t h m i c , and i s g i v e n by [ E i n s t e i n ,  1972]  -, , * • r9.05yU*,, u(y) = r - ln{ £ }} U  (E.7)  V  is.  where y i s measured v e r t i c a l l y upwards from t h e bend; and v i s the kinematic For t h i s logarithmic p r o f i l e ,  e  viscosity. the value o f  (E.8)  = k (l-n)ny<U* :  y  where k i s von Kantian's c o n s t a n t . and  i s g i v e n by [ F i s c h e r , 1969a]  Using the r e l a t i o n s of Equations  ( E . 8 ) , the i n t e g r a t i o n o f E q u a t i o n  e -2— y U*  w =  (E.6) g i v e s  2  0.42{- -} S  (E.5)  (E.9)  183  This result  c l o s e l y agrees w i t h the v a l u e o f  e -2— y U*  W 0.5{-^} *  =  2  U  t h a t Ward [1972] g i v e s f o r the same d i s t r i b u t i o n o f secondary  velocities.  Thus, i t o n l y remains t o e s t i m a t e o r measure the secondary v e l o c i t y W  g  determine  to  the c o e f f i c i e n t o f l a t e r a l  mixing.  Quick  a t h e o r y o f r i v e r meanders t h a t n i c e l y  [1973] has developed  p r e d i c t s the types o f meanders t h a t a r e observed o f secondary  i n nature.  The e x i s t a n c e  f l o w s when bends o r meanders are p r e s e n t i n a r i v e r i s w e l l  known, and Quick's t h e o r y has been used t o o b t a i n an e s t i m a t e o f the v e l o c i t y W. g  secondary  The b a s i s o f t h i s t h e o r y i s the g e n e r a t i o n o f v o r t i c i t y by  the  g r a d i e n t o f the v e r t i c a l d i s t r i b u t i o n o f l o n g i t u d i n a l v e l o c i t y u, and the subsequent  a d v e c t i o n o f t h i s v o r t i c i t y by u.  This i s i l l u s t r a t e d  the sense o f the v o r t i c i t y b e i n g g i v e n by the r i g h t - h a n d - s c r e w o f t h e l a r g e v e l o c i t y g r a d i e n t s of u c l o s e t o the bed,  i n Figure rule.  E.4,  Because  t h e v o r t i c i t y i s gener-  ated i n t h i s r e g i o n as i n t e n s e , s m a l l - s c a l e , h i g h l y a n i s o t r o p i c v o r t i c e s . v o r t i c i t y then d i f f u s e s v e r t i c a l l y i n t o the f l o w and  The  i s advected a l o n g by u.  D u r i n g t h i s d i f f u s i o n and a d v e c t i o n , the v o r t i c i t y degenerates  into  l y i s o t r o p i c t u r b u l e n c e and i s u l t i m a t e l y d i s s i p a t e d by f r i c t i o n a l  increasingeffects.  However, i n any steady f l o w s i t u a t i o n t h e r e must be a balance between the g e n e r a t i o n and d i s s i p a t i o n of v o r t i c i t y .  The v e r t i c a l l o g a r i t h m i c p r o f i l e  o f u r e f l e c t s t h e energy b a l a n c e o f a steady f l o w s i t u a t i o n , and the v o r t i c i t y b a l a n c e a l s o . file,  presumably  Because o f the v e l o c i t y g r a d i e n t s of t h i s p r o -  the f l o w a t any s e c t i o n has a "bulk" v o r t i c i t y g i v e n by  P  =  9u(y)  3y  (E.10) average  184  Figure Interaction  of u and  £  E.4 In A S t r a i g h t  River  185  where the v e l o c i t y g r a d i e n t i s averaged over t h e depth o f f l o w . argument i s c o r r e c t ,  If this  t h e r e i s an i n t e r m e d i a t e s t a g e between the  generation  and d i s s i p a t i o n o f t h e v o r t i c i t y , i n which the v o r t i c i t y r e t a i n s anisotropic  state.  This state  an  organized,  i s r e f l e c t e d by the v e l o c i t y g r a d i e n t s o f the  logarithmic v e l o c i t y p r o f i l e . In a s t r a i g h t r i v e r , the v e l o c i t y v e c t o r U and  the v o r t i c i t y v e c -  tor  are p e r p e n d i c u l a r t o each o t h e r and do not i n t e r a c t .  trated  i n F i g u r e E.4.  This i s  However, the s i t u a t i o n i s d i f f e r e n t when the  moves around a bend.  Initially,  illusflow  the water on the i n s i d e o f t h e bend  flows  f a s t e r than the water on the o u t s i d e o f the bend, b u t i n the bend p r o p e r , f a s t e r f i l a m e n t s o f f l o w move t o t h e o u t s i d e o f the bend, and water moves f a s t e r thantrated  i n F i g u r e E.5.  t h e water on the i n s i d e of t h e bend.  A t A the f a s t e r  Because o f the l a t e r a l v e l o c i t y  flow moves around the bend, t h e v o r t i c i t y i s b e i n g  as t h e flow moves around the bend and  depend on the r a d i u s of the c u r v a t u r e o f the bend. t h i s angle?© between u and v o r t i c i t y given  £ .  gener-  Note  t h i s v a r i a t i o n of 0 Quick's t h e o r y  will  predicts  Thus, t h e r e i s a streamwise component o f  by  E, = k X  and  This i s i l l u s -  a t an a n g l e 0 t o uy as i l l u s t r a t e d i n F i g u r e E.5.  a t e d and advected that 0 varies  the o u t s i d e  f i l a m e n t i s on the i n s i d e o f the bend  w h i l e a t B i t i s on the o u t s i d e o f the bend. g r a d i e n t o f u as the  the  z  |cos0  (E.U)  i t i s t h i s streamwise component t h a t g i v e s r i s e t o t h e secondary c u r r e n t  as the f l o w moves around the':>berid. Substituting  the l o g a r i t h m i c p r o f i l e  (E.7)  into  (E.10) and  a g i n g over,.the.;depth.between the l i m i t s y = 11.0v/U* and y =,y, l i m i t b e i n g an e s t i m a t e o f the depth o f t h e laminar  the  sublayer, gives  aver-  lower  186  Figure I n t e r a c t i o n o f u and  E.5 5  Around A Bend  187  a U  where  K  =  a  =  * (E.12)  (E.13)  ln{r-T-—r}  11. Ov and thus from E q u a t i o n  =  (E.ll) aU *cos0  ;( ;14)  J  E  k y  Now, c o n s i d e r t h e c i r c u l a t i o n around t h e s e c t i o n ABCD o f F i g u r e E.3 due t o t h i s streamwise v o r t i c i t y .  «r = X  =  <j> ? ABCD ^  T h i s c i r c u l a t i o n i s g i v e n by  • dA X  i - d U*b cosG  where b i s the w i d t h o f the r i v e r .  (E.15)  Now, t h e c i r c u l a t i o n around t h e c r o s s -  s e c t i o n ABCD i s a l s o g i v e n by  r  = X  <j> U • d l ABCD ^ ^  (E.16)  Assuming t h e r e i s s u f f i c i e n t time f o r t h e streamwise v o r t i c i t y t o "arrange" itself  i n t o t h e l i n e a r d i s t r i b u t i o n o f secondary v e l o c i t i e s o f F i g u r e E.3,  Equation  (E.16) can be e v a l u a t e d a s  T ~ x  2W b s  (E.17)  where t h e c o n t r i b u t i o n s over BC and DA have been i g n o r e d to be wide and s h a l l o w ) . (E.15) i n t o  F i n a l l y , p u t t i n g k = 0.4 and s u b s t i t u t i n g E q u a t i o n  (E.17), t h e f o l l o w i n g e x p r e s s i o n f o r W  g  W  s  (the s e c t i d n i s assumed  = 1.25aU. cosG *t  i s obtained (E.18)  188  The r e s u l t s o f a s u r f a c e f l o a t  study indicate a cross-channel e  s u r f a c e v e l o c i t y o f a p p r o x i m a t e l y 0.6 of Figure E . l .  The  s t u d y was  d i s c h a r g e a t Hope was  f e e t p e r s e c o n d a r o u n d b e n d No.  made o n M a r c h 20,  31,000 c u b i c  1973  f e e t per second.  when t h e f r e s h w a t e r The  f l o a t s had  drogues  t o m i n i m i z e t h e e f f e c t s o f w i n d and were r e l e a s e d d u r i n g t h e i n i t i a l o f t h e s t r o n g ebb p h a s e o f t h e t i d e .  The  11  period  f o l l o w i n g v e l o c i t y v a l u e s were  ob-  tained : u =3.1  f e e t p e r second  (from f l o a t s ) ;  and = 0.15  f e e t per second  (Manning's  equation)  In h i s study o f s t a b l e meanders, Quick found t h a t f o r g e o m e t r i c a l l y m e a n d e r s t o b e n d No. Taking  as above  t h e v a l u e o f 9 v a r i e d f r o m a b o u t 50  11,  t o 70  similar degrees.  and y = 34  feet  (Table E . l ) -5  w = 1.6  x 10  square f e e t per  second;  and 0 = 60 we  o b t a i n from Equations W  g  degrees  (E.13) a n d ? 0.7  (E.18)  f e e t per  second  which agrees w e l l w i t h the value obtained from the f l o a t  studies.  Assuming a l i n e a r v a r i a t i o n o f secondary flow w i t h depth F i g u r e E.3)  and u s i n g a v a l u e o f W  g  o f 0.6  l a t e r a l d i s p e r s i o n f r o m E q u a t i o n (E.9)  f e e t per second, the  i s g i v e n by  (as i n  coefficient  189  e  —  ? U*  or  e  z  = 30 square  = 6  (E.19)  f e e t p e r second.  The v a l u e o f E q u a t i o n v a l u e s t h a t have been r e p o r t e d  (E.19) i s c o n s i d e r a b l y l a r g e r t h a n o t h e r : ( f o r example, t h e v a l u e s i n T a b l e 1 o f Ward  [1972]), and i t should be emphasized t h a t t h e e s t i m a t e d v a l u e has n o t been confirmed e x p e r i m e n t a l l y .  A p o s s i b l e reason f o r t h i s high value i s t h a t  the F r a s e r i s e s s e n t i a l l y a " t i d a l r i v e r " w i t h b o t h High tidal  flows.  The F r a s e r i s r e l a t i v e l y wide  -freshwater and  when compared t o o t h e r r i v e r s  where v a l u e s o f e / y U^ have been measured, b u t r e l a t i v e l y narrow when comz  pared t o e s t u a r i e s (see Table 1 o f Ward). equation ary  I t i s noted t h a t the d i s p e r s i o n  (E.9) f o r d e t e r m i n i n g t h e l a t e r a l d i s p e r s i o n c o e f f i c i e n t i n second-  flows i s v e r y s e n s i t i v e t o t h e v a l u e o f (W /U^). s  Using a value o f  = 30 square  f e e t p e r second,  the previous  e s t i m a t e o f 55 hours f o r 80 p e r c e n t c r o s s - s e c t i o n a l m i x i n g i s reduced t o f i v e hours.  T h i s w i l l underestimate  the time o f c r o s s - s e c t i o n a l m i x i n g as  the f l o w i s n o t steady d u r i n g t h e t i d a l c y c l e .  Because o f t h e h i g h l y  a s s y m e t r i c a l n a t u r e o f t h e t i d e s , t h e r e i s o n l y one s t r o n g ebb and f l o o d t i d e i n each double  t i d a l c y c l e o f 25 hours t o generate  (see Appendix B f o r t y p i c a l t i d e s ) .  secondary  currents  Thus, w i t h i n any t i d a l c y c l e , t h e l a t e r -  a l m i x i n g w i l l v a r y from a maximum d u r i n g t h e s t r o n g f l o o d o r ebb t o a minimum at  times o f s l a c k w a t e r .  s e c t i o n a l mixing  Consequently,  an e s t i m a t e o f t h e e f f e c t o f c r o s s -  i s p r o b a b l y one t i d a l c y c l e  (12.5 hours) f o r t h e lower  r e a c h e s o f t h e e s t u a r y and P i t t R i v e r and 1-2 t i d a l c y c l e s f o r t h e upper  190  reaches where t h e t i d a l e f f e c t s a r e s m a l l e r .  On t h i s b a s i s , t h e e f f e c t i v e  o r t i d a l l y averaged c o e f f i c i e n t o f l a t e r a l d i s p e r s i o n would be about 15 square f e e t p e r second i n t h e lower reaches and seven square f e e t p e r second i n t h e upper reaches o f t h e Main Arm - Main Stem.  On the b a s i s  o f t h e r e l a t i v e magnitudes o f t h e t i d a l l y averaged v a l u e s o f y and U^, the l a t e r a l d i s p e r s i o n i n t h e North Arm i s e s t i m a t e d t o be f i v e f e e t p e r second and 10 square f e e t p e r second i n P i t t  square  River.  A t t h i s s t a g e , t h e t h e o r y r e l a t i n g v o r t i c i t y and secondary c u r r e n t s i s n e i t h e r f u l l y developed t h e o r e t i c a l l y n o r c o n f i r m e d t a l l y , b u t f u t u r e work i s planned i n b o t h d i r e c t i o n s .  experimen-  In concluding, i t  i s noted t h a t some v a l u e s quoted f o r l a t e r a l d i s p e r s i o n c o e f f i c i e n t s a r e based o n i t h e r e s u l t s o f l a b o r a t o r y experiments. in  C r o s s - r s e c t i o n a l mixing  t h e presence o f secondary c u r r e n t s i s a complex t h r e e - d i m e n s i o n a l pheno-  mena, and i t may be t h a t some components o f t h i s p r o c e s s a r e n o t b e i n g s c a l e d p r o p e r l y i n model experiments.  APPENDIX P  ESTIMATION OF LONGITUDINAL DISPERSION  The c o e f f i c i e n t s o f l o n g i t u d i n a l d i s p e r s i o n due  t o the e f f e c t s  o f v e r t i c a l and l a t e r a l v e l o c i t y g r a d i e n t s a r e e s t i m a t e d f o r the F r a s e r River Estuary.  Simple a p p r o x i m a t i o n s a r e g i v e n f o r the  time-dependent  behaviour of the l o n g i t u d i n a l d i s p e r s i o n c o e f f i c i e n t d u r i n g the  initial  p e r i o d b e f o r e c r o s s - s e c t i o n a l m i x i n g i s complete  and d u r i n g the  tidal  The p r e d i c t e d t i d a l l y v a r y i n g c o n c e n t r a t i o n s d u r i n g the  first  cycle.  double t i d a l c y c l e were found t o be v e r y s e n s i t i v e to assumptions t h e time-dependent  behaviour of the d i s p e r s i o n c o e f f i c i e n t .  magnitude nor time-dependent  F.1  Neither the  behaviour o f the d i s p e r s i o n c o e f f i c i e n t s  been v e r i f i e d by f i e l d measurements, and in  about  has  i t i s r e c o g n i z e d t h a t they may  be  error.  GENERAL When e f f l u e n t i s d i s c h a r g e d i n t o a r i v e r or e s t u a r y i t i s d i s p e r s e d  i n the l o n g i t u d i n a l d i r e c t i o n by the e f f e c t s o f b o t h v e r t i c a l and v e l o c i t y g r a d i e n t s , as d e s c r i b e d i n Appendix A.  lateral  The e f f l u e n t must d i s p e r s e  over a r e a s o n a b l e depth and width o f the e s t u a r y b e f o r e these v e l o c i t y g r a d i e n t s can e x e r t a s i g n i f i c a n t e f f e c t on the l o n g i t u d i n a l d i s p e r s i o n p r o c e s s . Most r i v e r s and e s t u a r i e s a r e much wider than they are deep and  consequently  m i x i n g i n the v e r t i c a l d i r e c t i o n i s much more r a p i d t h a n i n m i x i n g i n the lateral direction.  Thus, when a " p a r c e l " o f e f f l u e n t i s i n i t i a l l y d i s c h a r g e d ,  191  192  the  longitudinal dispersion  v e l o c i t y gradients. section, the  the  i s e s s e n t i a l l y due  However, as the  i s c o m p l i c a t e d by  the  c r o s s - s e c t i o n a l m i x i n g i s not some o f the  In an  [1970] d e s c r i b e  as  e s t u a r y , the  e s s e n t i a l l y complete w i t h i n  e f f e c t s of flow r e v e r s a l  The F r a s e r  increasing  influence  a tidal  3  "sinuous, m u l t i - c h a n n e l e d or i s l a n d - s t u d d e d  f i e l d dye  studies.  Time and  Holley  the  longitudinal  been used t o o b t a i n  v e r t i c a l and  preliminary  al.  of  reliabstudies,  [1966b, 1969b]  e s t i m a t e s of  dispersion.  TIDALLY AVERAGED COEFFICIENTS OF For  be  estuaries."  expense p r e c l u d e d such  and  F.2  et  can o n l y be  i n the absence o f adequate f i e l d d a t a , the work o f F i s c h e r [1970] has  cycle,  junctions  and  et al.  the  1970].  River Estuary f a l l s into a c l a s s that Holley  e s t u a r y , the c o e f f i c i e n t s of l o n g i t u d i n a l d i s p e r s i o n  l y determined by  If  to the o s c i l l a t o r y f l o w w i l l [ H o l l e y et al.  on  longitudinal  Because o f the c o m p l i c a t i n g e f f e c t s of the bends, i s l a n d s and the  vertical cross-  e f f e c t s of t i d a l flow reversal;,.  l o n g i t u d i n a l d i s p e r s i o n due  "undone" by the  e f f e c t s of  e f f l u e n t spreads a c r o s s the  l a t e r a l v e l o c i t y g r a d i e n t s e x e r t an  longitudinal dispersion process.  dispersion  to the  steady f l o w the  LONGITUDINAL DISPERSION  l o n g i t u d i n a l d i s p e r s i o n due  l a t e r a l v e l o c i t y g r a d i e n t s can  t o the  be r e s p e c t i v e l y  e f f e c t s of  estimated  as  E  -2— y  u  = *  6  (F.l)  193  where u'  i s the square o f the v e l o c i t y d e v i a t i o n and u'  the over-bar value.  2  - 2 (u(y,z) - u)  =  i n Equation  Eguation  (F.2) s i g n i f y i n g the average  ( F . l ) was  cross-sectional  o b t a i n e d by E l d e r [1959] f o r the l o g a r i t h m i c  velocity profile.  In o b t a i n i n g E q u a t i o n  the l a t e r a l m i x i n g  t o be due  the t r i p l e  i s g i v e n by  (F.2), F i s c h e r : [1966b] assumed  to t u r b u l e n t d i f f u s i o n alone.  i n t e g r a l of E q u a t i o n  According  (E.6), the l o n g i t u d i n a l d i s p e r s i o n due  l a t e r a l v e l o c i t y g r a d i e n t s w i l l v a r y i n v e r s e l y as t h e c o e f f i c i e n t o f al dispersion.  to to  later-  T h i s i s because m i x i n g over the c r o s s - s e c t i o n tends t o "undo"  the e f f e c t s o f d i s p e r s i o n i n the l o n g i t u d i n a l d i r e c t i o n . p e r s i o n i n the F r a s e r R i v e r e s t u a r y i s thought the e f f e c t s o f secondary  flows  sequently adjusted f o r t h i s  The  lateral  dis-  t o be q u i t e h i g h because o f  (see Appendix E ) , and E q u a t i o n  (F.2) i s sub-  effect.  To a p p l y E q u a t i o n  (F.2) i t i s n e c e s s a r y t o have some e s t i m a t e  2 o f u' Nos.  and 14 and  .  Two  v e l o c i t y p r o f i l e s were made i n the Main Arm  15 on A p r i l 4, 1973.  f e e t per second,  at Stations  The f r e s h w a t e r f l o w a t Hope was  the t i d a l range a t S t e v e s t o n was  11 f e e t and  were t a k e n d u r i n g the s t r o n g ebb phase o f the t i d e .  the measurements  The depth-averaged  p r o f i l e s a c r o s s the s e c t i o n s a r e shown i n F i g u r e F . l , and  f e e t p e r second  = 0.18 = 0.9  f e e t per  second;  = 3.3  f e e t per  second  and U.  (measured);  f e e t per second  velocity  f o r these c o n d i t i o n s ,  the v a r i o u s v e l o c i t i e s a r e e s t i m a t e d t o be u = 3.6  34,700 c u b i c  (Manning's e q u a t i o n ) ;  Q = 3 1 , 0 0 0 cfs (Hope) April 4,1973  500 1000 Distance from North Bank ( feet )  500 Distance  1000  1500  from North Bank (feet)  Figure F . l L a t e r a l V e l o c i t y P r o f i l e s A t S t a t i o n s No. 14 and 15, Main Arm  194  195  where i t i s assumed t h a t the t o t a l v e l o c i t y u c o n s i s t s o f a steady component  (freshwater) o f magnitude U  component.(tidal)  o f amplitude  f  and a s i n u s o i d a l l y v a r y i n g o s c i l l a t o r y  U . fc  S t a t i o n s Nos. 14 and 15 a r e l o c a t e d i n bend No. 10 o f F i g u r e E.l,  a r e g i o n o f s t r o n g secondary  flows.  -2 v a l u e s o f u' a t b o t h s t a t i o n s , E q u a t i o n E —  =  U s i n g the average o f t h e measured (F.2) can be e v a l u a t e d as  3600  (F.3)  which i s much h i g h e r than r e c o r d e d v a l u e s i n o t h e r r i v e r s Table 1 ] .  However,  the v a l u e o f E q u a t i o n  f o r the greater c r o s s - s e c t i o n a l mixing  [Fischer,  (F.3) has t o be reduced  i n the F r a s e r .  1966b,  t o account  The r e l a t i v e magnitude  o f l a t e r a l m i x i n g due t o t u r b u l e n t d i f f u s i o n i s 0.23, as i n E q u a t i o n whereas t h e r e l a t i v e magnitude f o r the e f f e c t s o f secondary r e g i o n o f t h e e s t u a r y has been e s t i m a t e d t o be 6.0 the a d j u s t e d v a l u e o f E q u a t i o n  E — — y  =  (E.l),  c u r r e n t s i n the  (see Appendix E ) .  Thus  (F.3) i s  140  (F.4)  which i s much more r e a s o n a b l e when compared t o t h e v a l u e s t h a t F i s c h e r lists. Equation  (F.4) i s an e s t i m a t e o f t h e " i n s t a n t a n e o u s "  longitudinal  d i s p e r s i o n due t o the l a t e r a l v e l o c i t y g r a d i e n t s e x i s t i n g a t one p a r t i c u l a r time d u r i n g the t i d a l c y c l e .  A c c o r d i n g t o F i s c h e r [1969a],  the e f f e c t s o f  the steady and o s c i l l a t o r y components on t h e l o n g i t u d i n a l d i s p e r s i o n a r e  196  s e p a r a t e and a d d i t i v e , and thus E q u a t i o n f o l l o w i n g steady and o s c i l l a t o r y  5"f  (F.4) can be s e p a r a t e d i n t o the  components  =  140  (F.5)  =  140  (F.6)  and  where  For the sake o f s i m p l i c i t y , Equations Eguation  (F.5) and (F.4).  the s u b s c r i p t z has been dropped.  (F.6) sum  t o g i v e the c o r r e c t combined d i s p e r s i o n of  To a p p l y the a n a l y s i s o f H o l l e y et at.  [1970], the  t o r y component has t o be c o r r e c t e d back t o a t i d a l l y averaged manner o f E q u a t i o n  value.  t ^ & *  y  In the  (F.7)  = 120  In the absence of o t h e r f i e l d d a t a , E q u a t i o n s been used t o e s t i m a t e the l o n g i t u d i n a l d i s p e r s i o n due g r a d i e n t s throughout  the e s t u a r y .  (F.5) and  to l a t e r a l  (F.7) have velocity  The e q u a t i o n s are a p p l i e d as i s f o r the  - Main Stem o f the e s t u a r y , but have been c o r r e c t e d f o r the  depths and widths  oscilla-  (E.4), t h i s i s e s t i m a t e d t o be  E  Main Arm  Note t h a t  different  o f the o t h e r channels t o g i v e E_ y  u  f *  =  75  =  ,65  E y ^  } North Arm  (F.8)  197  =  120  P i t t River  (F.9)  Y< Note t h a t t h e r e i s no steady d i s p e r s i o n component i n P i t t R i v e r , t h e f l o w is purely o s c i l l a t o r y .  The e s t i m a t e s o f t h e v a r i o u s v e l o c i t y components and  the steady and o s c i l l a t o r y d i s p e r s i o n components a r e g i v e n i n T a b l e F . l . I f the c r o s s - s e c t i o n a l m i x i n g  i s n o t e s s e n t i a l l y complete w i t h i n  a t i d a l ; c y c l e , some o f t h e o s c i l l a t o r y d i s p e r s i o n i s undone by t h e e f f e c t s of t i d a l flow r e v e r s a l .  To account  f o r t h i s e f f e c t , t h e v a l u e s o f E ^ have  been reduced t o t h e i r e f f e c t i v e v a l u e s E£ a c c o r d i n g t o t h e procedure i n H o l l e y et at.  These r e s u l t s a r e a l s o shown i n T a b l e F . l ,  the e f f e c t i v e per-  i o d o f f l o w o s c i l l a t i o n i s d e s i g n a t e d T and the r a t i o o f t h i s v a l u e t o t h e time s c a l e o f l a t e r a l m i x i n g  i s d e s i g n a t e d T'.  F i n a l l y , t h e steady and  e f f e c t i v e o s c i l l a t o r y d i s p e r s i o n components a r e summed t o g i v e a combined t i d a l l y averaged  c o e f f i c i e n t o f l o n g i t u d i n a l d i s p e r s i o n due t o t h e e f f e c t s  of l a t e r a l v e l o c i t y gradients  (E ). c  It i s recognized t h a t the values o f the d i s p e r s i o n c o e f f i c i e n t s g i v e n i n T a b l e F . l have been o b t a i n e d from v e r y l i m i t e d f i e l d d a t a and depend h e a v i l y on t h e u n v e r i f i e d e s t i m a t e o f l a t e r a l d i s p e r s i o n from Appendix E.  Consequently,  ratio E /y f  t h e e s t i m a t e s may be c o n s i d e r a b l y i n e r r o r .  and t h e a b s o l u t e v a l u e E  measured i n o t h e r r i v e r s  f  a r e r e a s o n a b l e when compared t o v a l u e s  [ F i s c h e r , 1966b, T a b l e 1] and t h e maximum v a l u e o f  the e f f e c t i v e o s c i l l a t o r y component E  fc  o f 450 square f e e t p e r second  w e l l w i t h t h e upper l i m i t o f a p p r o x i m a t e l y Fischer  [1969b] suggests.  However, t h e  agrees  500 square f e e t p e r second t h a t  The combined d i s p e r s i o n c o e f f i c i e n t E  c  shows an  TABLE F . l ESTIMATED COEFFICIENTS OF LONGITUDINAL DISPERSION DUE TO LATERAL VELOCITY GRADIENTS (January 24, 1952: Freshwater D i s c h a r g e a t C h i l l i w a c k 36,500 c f s ; T i d a l Range a t S t e v e s t o n 11 f e e t )  •H  w z  z  u  o cu co  •  to  < EH  \ D  •  -P  CO  .  .  •  o cu co  D  \ 4-> • 4J  fa '  .  -. o  'CO  10 \  M-l * • D  -P  fa  .  o cu  cj  CO  CO  co  \ 4-> * • , D 4-> fai  o * •  \  P  4J  fa  o  o  CJ  co  co  CO  \  IW  CN  W  •  4J  fa_  CU  \  CN  4J • V  fa fa  u  cu  CN N • Ul, 4->  fa  o  En  a  CU  co  \  U  \  u  CU CO  —  -EH  -  CM  4J  W  •  IP  fa  \  W  u  CN  • 4J  fa  u * D  1 >i \  w  O  0.6  3.5  0.025  0.11  0.12  120  450  15  25  1.7  450  570  140  11-18  0.6  3.3  0.025  0.10  0.11  130  430  15  25  1.7  430  560  140  19-24  0.6  2.5  0.033  0.088  0.10  180  400  15  25  1.7  400  580  140  25-30  0.7  2.2  0.037  0.074  0.091  190  340  15  12.5  0.8  200  390  110  31-40  0.7  1.4  0.042-  0.055  0.074  220  240  7  12.5  0.4  70  350  130  41-50  0.9  1.2  0.059  0.055  0.085  280  220  7  12.5  0.4  60  320  110  51-60  1.1  0.7  0.079  0.035  0.082  310  120  7  12.5  0.4  40  350  120  NORTH ARM  101-118  0.2  1.6  0.011  0.063  0.064  20  100  5  25  2.8  100  120  PITT RIVER  140-154  0.11  0.11  —  400  10  25  1.8  400  400  1-10  MAIN ARM  MAIN STEM  2.8  70 120  199  increase  down the Main Stem - Main Arm  due  to i n c r e a s i n g e f f e c t s o f  f l o w s i n the lower r e a c h e s , arid t h i s a l s o seems r e a s o n a b l e . o f the p r e d i c t e d  concentrations  t o e r r o r s i n the  dispersion i s investigated i n Section  F.3  c  sensitivity  c o e f f i c i e n t of l o n g i t u d i n a l  F.4.  TIME DEPENDENT LONGITUDINAL DISPERSION COEFFICIENTS The  E  The  tidal  listed  t i d a l l y averaged v a l u e s o f the combined d i s p e r s i o n c o e f f i c i e n t  i n Table F . l represent  the e f f e c t s o f the  lateral velocity  gradients  on the l o n g i t u d i n a l d i s p e r s i o n p r o c e s s when the c r o s s - s e c t i o n a l mixing i s e s s e n t i a l l y complete.  In the  i n i t i a l period  f o l l o w i n g the d i s c h a r g e of a  " p a r c e l " o f e f f l u e n t , the l o n g i t u d i n a l d i s p e r s i o n i s p r i n c i p a l l y due the e f f e c t s of v e r t i c a l v e l o c i t y g r a d i e n t s F.l.  a l o n e , as d i s c u s s e d  (The time s c a l e o f v e r t i c a l m i x i n g i s a h a l f - h o u r ) .  Fischer  [1969a], the  e f f e c t s of v e r t i c a l and  on the l o n g i t u d i n a l d i s p e r s i o n a r e  additive.  Section  According  lateral velocity  s e p a r a t e and  in  to  to  gradients  To a l l o w  for  the  v a r i a b l e c o n t r i b u t i o n o f b o t h components, i t i s assumed t h a t  E = E  + . E *  t T  0  < t < T  and  }  E = E  c  t  (F.10)  > T  where t i s the time t h a t has e l a p s e d s i n c e a was d i s c h a r g e d i n t o the e s t u a r y ;  "parce 1" o f e f f l u e n t  T i s the time s c a l e o f l a t e r a l mixing based on the w i d t h o f s e c t i o n b (an edge d i s c h a r g e i s assumed); and E , E ^  c  are the c o e f f i c i e n t s o f l o n g i t u d i n a l d i s p e r s i o n due t o the e f f e c t s o f v e r t i c a l and l a t e r a l v e l o c i t y g r a d i e n t s respectively.  200  Equations  (F.10) a r e a s i m p l i s t i c r e p r e s e n t a t i o n o f a complex  three-dimen-  s i o n a l phenomenom, b u t they s h o u l d a d e q u a t e l y reproduce b o t h t h e s h o r t and  long-term d i s p e r s i o n e f f e c t s .  e f f e c t s o f such temporal  I t i s o n l y p o s s i b l e t o account  f o r the  v a r i a t i o n s i n E because t h e t i d a l l y v a r y i n g mass  t r a n s p o r t e q u a t i o n has been s o l v e d a l o n g i t s a d v e c t i v e c h a r a c t e r i s t i c s .  Con-  s e q u e n t l y , t h e p o s i t i o n o f each e f f l u e n t " p a r c e l " and t h e time t h a t i t has spent i n t h e e s t u a r y i s known, i n f o r m a t i o n t h a t i s masked by a f i x e d  grid  s o l u t i o n t o t h e mass t r a n s p o r t e q u a t i o n . As w e l l a s i n c r e a s i n g w i t h time due t o t h e i n f l u e n c e o f l a t e r a l v e l o c i t y g r a d i e n t s as t h e e f f l u e n t spreads  over t h e c r o s s - s e c t i o n , t h e  c o e f f i c i e n t o f l o n g i t u d i n a l d i s p e r s i o n w i l l a l s o vary during the t i d a l cycle.  I t w i l l be minimal  d u r i n g times o f s l a c k w a t e r and g r e a t e s t d u r i n g  the times o f s t r o n g ebb and f l o o d flow.  To i n v e s t i g a t e t h i s e f f e c t , i t  was assumed t h a t t h e c o e f f i c i e n t o f l o n g i t u d i n a l d i s p e r s i o n g i v e n by E q u a t i o n s (F.9) a l s o v a r i e d d i r e c t l y as t h e a b s o l u t e v e l o c i t y  E = (t + S£) y U  #  0 <_ t < T (F.ll)  E = ay  t >_ T  and__ U  where E, y,  *  =  °'  (F.12)  0 6 5  and u a r e t h e i n s t a n t a n e o u s v a l u e s o f t h e r e s p e c t i v e parameters —  d u r i n g t h e t i d a l c y c l e and a i s assumed e q u a l t o t h e t a b u l a t e d v a l u e s o f E / y c  i n Table F . l .  The r e l a t i o n  (F.12) i s o b t a i n e d from Manning's e q u a t i o n and  represents a "best o v e r - a l l f i t " f o r the e n t i r e estuary. Equation of  The average o f  ( F . l l ) over a t i d a l c y c l e wasrfound t o be a s a t i s f a c t o r y  Equation  (F.10).  estimate  c  201  F.4  SENSITIVITY OF PREDICTED CONCENTRATIONS The  s e n s i t i v i t y o f the p r e d i c t e d c o n c e n t r a t i o n s  about the c o e f f i c i e n t o f l o n g i t u d i n a l d i s p e r s i o n was  investigated for a  steady d i s c h a r g e o f a c o n s e r v a t i v e e f f l u e n t a t S t a t i o n No. Stem.  F i g u r e F.2  t o assumptions  50 on the Main  shows the v a r i a t i o n i n c o n c e n t r a t i o n a t S t a t i o n No.  50  d u r i n g a double t i d a l c y c l e and  the c o n c e n t r a t i o n p r o f i l e s a l o n g t h e Main  Stem 50 hours a f t e r the i n i t i a l  discharge.  The p r e d i c t e d  concentrations  have been s t a n d a r d i z e d by d i v i d i n g by the t i d a l l y averaged c o n c e n t r a t i o n  ob-  t a i n e d from the mass o f e f f l u e n t d i s c h a r g e d per t i d a l c y c l e and the f r e s h water d i s c h a r g e a t C h i l l i w a c k .  T h i s i s p l o t t e d as the parameter V^ /  f y i n g t h a t the r e s u l t s have been o b t a i n e d The  effects of v a r i a t i o n i n i n i t i a l  from the t i d a l l y v a r y i n g model.  d i l u t i o n and m u l t i p l e d o s i n g due  f l o w r e v e r s a l , as d i s c u s s e d i n S e c t i o n 1.3, t r a t i o n p r o f i l e along the c h a n n e l . these spikes i s i n i t i a l l y  predicted concentrations onto the standard  The  r e s u l t i n s p i k e s i n the  to concen-  I t should be noted t h a t the base o f  o n l y some 500  t h a n i t appears i n F i g u r e F.2.  signi-  v  - 800  f e e t wide, and  i s much " t h i n n e r "  r e a s o n the base appears wide i s t h a t t h e  are e x t r a p o l a t e d of the a d v e c t i v e  5,000 f o o t space g r i d  c o a r s e t o a c c u r a t e l y r e s o l v e the i n i t i a l  (see S e c t i o n 3.2).  characteristics T h i s g r i d i s too  forms of t h e s p i k e s .  t h a t the s p i k e s a r e c o r r e c t l y r e s o l v e d on the a d v e c t i v e  I t i s noted  characteristics  (where n e c e s s a r y , a d d i t i o n a l moving p o i n t s were added t o d e f i n e r e g i o n s r a p i d v a r i a t i o n , as d i s c u s s e d o f the most seaward s p i k e  i n S e c t i o n 3.3).  Note t h a t the  concentration  (E = 0) has been h a l v e d by d i l u t i o n from the  R i v e r i n p a s s i n g through the Main Stem - P i t t R i v e r j u n c t i o n .  of  Pitt  Q = 3 6 , 5 0 0 c f s (Chilliwack) )j Tidal Range at Steveston = 11 feet J  a n  .  2 4 i  |9 2 5  V a r i a t i o n in C o n c e n t r a t i o n at  6  10  Concentration Main  8  Xtv  6  Profile  Arm - Main  (i) —  18  E = 0  (ii) E = E  discharge)  No. 5 0  12 Time ( Hours)  along  Stem  ( 5 0 hours after initial  Station  (iii) E = E (iv) E = E  y  (  y  (20 ft. /scc.) (350 ft?/sec.) 2  •  t E i c  (v) E = (6 • oc^)y U  4  r  2 0  10  F i g u r e F.2  20  Stations  30  along  40  Main Arm - Main  S e n s i t i v i t y of T i d a l l y Varying Concentrations  Stem  t o the C o e f f i c i e n t  50  of Longitudinal Dispersion  203  The  r e s u l t s o f assuming E t o be c o n s t a n t and independent o f  time a r e shown as c u r v e s  ( i ) , ( i i ) and  (iii)  i n F i g u r e F.2.  The  corresponding values of E are: . Case Case  (i)  E  =  (ii) E  =  0 E  (20 square  f e e t per  second);  y and Case  (iii) E  =  E c  The peak c o n c e n t r a t i o n a t S t a t i o n No. the magnitude o f E.  (  3 5 0  square  f e e t per  second).  50 i s seen t o be q u i t e s e n s i t i v e t o  E i g h t e e n hours a f t e r i t s g e n e r a t i o n , the s p i k e  been advected downstream t o S t a t i o n No. to be s i g n i f i c a n t l y reduced  has  35 and i t s c o n c e n t r a t i o n i s seen  i r r e s p e c t i v e o f whether E e q u a l s E^ o r  E^.  When E i s a l l o w e d t o v a r y w i t h time a c c o r d i n g t o E q u a t i o n s and  ( F . l l ) , t h e p r e d i c t e d c o n c e n t r a t i o n s a r e g i v e n by c u r v e s  (F.10)  ( i v ) and  (v)  respectively.  Once a g a i n , the g r e a t e s t e f f e c t i s on the peak c o n c e n t r a t i o n s  a t S t a t i o n No.  50, the d i f f e r e n c e s  between b o t h c u r v e s b e i n g  a f t e r 1.8 hours when the s p i k e i s a t S t a t i o n No.  35.  Note the  i n c r e a s e i n the peak p r e d i c t e d c o n c e n t r a t i o n a t S t a t i o n No.  significant  50 when E i s  a l l o w e d t o v a r y w i t h u.  The  v a r i a t i o n a t S t a t i o n No.  50 d u r i n g the t i d a l c y c l e .  E.2,  t o the low v e l o c i t i e s and f l o w r e v e r s a l s around  the s p i k e b e i n g due  hours 4,  5 and 6.  r e a s o n f o r t h i s i s apparent  negligible  from the  velocity  T h i s i s shown i n F i g u r e  Because o f the low v a l u e s o f u, the v a l u e o f U  #  i s very  low and  the e f f e c t i v e d i s p e r s i o n i s v e r y s m a l l compared t o o t h e r times  ing the  tidelcycle.  dur-  Because o f the a s s y m e t r i c nature o f t h e t i d e s , t h e v e l o c i t i e s i n the lower  reaches o f the e s t u a r y a r e q u i t e low d u r i n g the p e r i o d of  time  204  between high-high-water and  low-high-water.  v e l o c i t y v a r i a t i o n a t S t a t i o n No.  T h i s i s i l l u s t r a t e d by  5i;in F i g u r e E.2,  and  the  the d i s p e r s i o n w i l l  c e r t a i n l y be l e s s d u r i n g t h i s phase o f the t i d e than d u r i n g the s t r o n g and f l o o d f l o w t h a t a r e seen t o o c c u r once each d o u b l e t i d e c y c l e .  ebb  The  e f f e c t s of v a r i a t i o n i n i n i t i a l d i l u t i o n and m u l t i p l e d o s i n g are most s i g n i f i c a n t d u r i n g the slackwaters"around  the time between high-high-water  low-high-water, and t h i s can be i d e n t i f i e d as a s e n s i t i v e p e r i o d o f tide  and  the  cycle. In c o n c l u s i o n , t h e peak c o n c e n t r a t i o n s a t the p o i n t o f  effluent  d i s c h a r g e are v e r y s e n s i t i v e t o assumptions about the form and magnitude o f the c o e f f i c i e n t o f l o n g i t u d i n a l d i s p e r s i o n .  However, a f t e r a s p i k e  has  been i n t h e e s t u a r y s e v e r a l t i d a l c y c l e s , i t s peak c o n c e n t r a t i o n i s r e a s o n a b l y i n s e n s i t i v e t o the form and magnitude of the c o e f f i c i e n t .  Until  adequate  f i e l d d a t a i s a v a i l a b l e , the assumed temporal v a r i a t i o n s o f E i n E q u a t i o n are thought t o be a r e a s o n a b l e  F. 5  approximation  of what o c c u r s  i n the  (F.ll)  estuary.  SUMMARY The d i s p e r s i o n o f e f f l u e n t i n an e s t u a r y i s a complex, time-dependent  three-dimensional  phenomenon due  t u r b u l e n t d i f f u s i o n , v e r t i c a l and flows.  t o an i n t i m a t e combination  o f the e f f e c t s o f  l a t e r a l v e l o c i t y g r a d i e n t s and  secondary  L i m i t e d f i e l d d a t a and t h e r e s u l t s o f o t h e r p e o p l e s ' work have been  used t o o b t a i n e s t i m a t e s o f the c o e f f i c i e n t s o f l o n g i t u d i n a l d i s p e r s i o n . t o t h e l a c k ' o f f i e l d d a t a , these e s t i m a t e d v a l u e s may error.  be  substantially i n  However, they p r o v i d e a b a s i s f o r o b t a i n i n g p r e l i m i n a r y n o t i o n s  the t i d a l l y v a r y i n g response o f the e s t u a r y t o waste d i s c h a r g e s . assumed time-dependent behaviour  Due  For  of  the  o f the c o e f f i c i e n t o f l o n g i t u d i n a l d i s p e r s i o n  the p r e d i c t e d c o n c e n t r a t i o n s were found t o be r e l a t i v e l y i n s e n s i t i v e t o t h e v a l u e o f E^.  The p r e d i c t e d peak c o n c e n t r a t i o n s a t t h i s p o i n t o f e f f l u -  ent d i s c h a r g e were found t o be v e r y s e n s i t i v e t o the assumed time-dependent behaviour o f the c o e f f i c i e n t of l o n g i t u d i n a l d i s p e r s i o n . t h a t t h e assumed time dependent behaviour o f what happens i n t h e estuary,-  i s a reasonable  I t i s thought approximation  

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