UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Water quality modelling in estuaries 1974

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
UBC_1974_A1 J69.pdf [ 8.79MB ]
Metadata
JSON: 1.0062998.json
JSON-LD: 1.0062998+ld.json
RDF/XML (Pretty): 1.0062998.xml
RDF/JSON: 1.0062998+rdf.json
Turtle: 1.0062998+rdf-turtle.txt
N-Triples: 1.0062998+rdf-ntriples.txt
Citation
1.0062998.ris

Full Text

WATER QUALITY MODELLING IN ESTUARIES by CHRISTOPHER STEWART JOY B.Eng., Mohash University, 1968 M.App.Sci., Monash University, 1972 A THESIS. SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF•PHILOSOPHY i n the Department o f C i v i l Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January, 1974 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e at t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Q 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 C o l u m b i a V a n c o u v e r 8. Canada Date A B S T R A C T A t i d a l l y varying and a t i d a l l y averaged mass transport model are applied to the Fraser River Estuary to investigate 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 the concentrations r e s u l t i n g from assumed e f f l u e n t discharges. The t i d a l l y averaged model i s due to Thomann [1963]. The t i d a l l y varying model i s developed from f i r s t p r i n c i p l e s . A hydrodynamic model was used to determine the t i d a l l y induced temporal v a r i a t i o n i n the lo n g i t u d i n a l v e l o c i t y and cr o s s - s e c t i o n a l area along the estuary. A l l models are "mathematical" and one-dimensional. F i n i t e d i f f e r e n c e techniques are used to solve the underlying p a r t i a l d i f f e r e n t i a l equations of a l l three models. The problems of s t a b i l - i t y and numerical d i s p e r s i o n are examined. Numerical di s p e r s i o n i s seen to r e s u l t from the s o l u t i o n of the mass transport equation over a f i x e d space g r i d rather than along the advective c h a r a c t e r i s t i c s . Advantages of solving the equation along the c h a r a c t e r i s t i c s are: no numerical dispersion; the advective and d i s p e r s i v e transport processes are u s e f u l l y separated; l a t e r a l d i s p e r s i o n can be p a r t i a l l y assessed with a one-dimensional model; and time dependent behaviour i n 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 can be taken into account. The t i d a l l y varying flows along the estuary are seen to 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 of a discharged e f f l u e n t . This, together with the e f f e c t s of t i d a l flow r e v e r s a l produces spikes i n the concentration p r o f i l e along the estuary. The concentration of these spikes i s then r e - duced by the dispersion process, the peak concentration during the f i r s t two i i i i i t i d a l cycles being sensitive to the form and magnitude of the coefficient of longitudinal dispersion. Time dependent variations in this coeffi- cient are considered. The effect of the lateral dispersion process on the estimated concentrations is also considered and secondary flows are tentatively explained in terms of the generation and advection of vor- t i c i t y . The predicted peak t i d a l l y varying concentration was found to be significantly greater than the ti d a l l y averaged value. TABLE OF CONTENTS Page LIST OF TABLES v i i LIST OF FIGURES v i i i LIST OF SYMBOLS x i CHAPTER INTRODUCTION. . . . 1 1. PRELIMINARY CONSIDERATIONS 5 1.1 THE ONE-DIMENSIONAL MASS TRANSPORT EQUATION 5 1.2 DETERMINATION OF PARAMETERS 7 1.3 THE EFFECTS OF THE TIDE ON MASS TRANSPORT PROCESSES 9 1.4 ACCURACY OF A ONE-DIMENSIONAL MODEL 12 2. LITERATURE REVIEW 17 2.1 ANALYTICAL SOLUTIONS 17 2.2 NUMERICAL SOLUTIONS 21 2.2.1 Finite Difference Solutions 21 2.2.2 "Box Model" Solutions 24 2.2.3 Finite Element Solutions 25 2.3 PHYSICAL AND ANALOGUE MODEL SOLUTIONS 27 2.3.1 Physical Model Solutions 27 2.3.2 Analogue Model Solutions 28 2.4 STOCHASTIC SOLUTIONS 29 2.5 SUMMARY 32 3. A DESCRIPTION OF THE HYDRODYNAMIC AND MASS TRANSPORT MODELS . . . 34 3.1 THE HYDRODYNAMIC MODEL 34 3.1.1 The Hydrodynamic Equations 34 33.1.2 Assumed Quasi-Steady Hydraulic Conditions 38 3.1.3 The Model Estuary of the Hydrodynamic Equations . . . 39 3.2 THE TIDALLY VARYING MASS TRANSPORT MODEL 39 3.2.1 Method of Solution 39 3.2.2 The Model Estuary 42 3.3 TIDALLY AVERAGED MASS TRANSPORT MODEL 44 3.3.1 Method of Solution 44 3.3.2 The Model Estuary 47 iv V CHAPTER Page 4. VERIFICATION OF THE HYDRODYNAMIC AND MASS TRANSPORT MODELS. . . . 48 4.1 THE HYDRODYNAMIC MODEL 48 4.1.1 Data Available 48 4.1.2 Water Surface Elevations f o r Low Flows . . . . . . . . 49 4.1.3 Water Surface Elevations f o r High Flows 49 4.1.4 V e r i f i c a t i o n f o r the Conditions of January 24, 1952 . 54 4.2 THE TIDALLY VARYING MASS TRANSPORT MODEL. 65 4.2.1 Advective Transport 65 4.2.2 Dispersive Transport 67 4.3 THE TIDALLY AVERAGED MASS TRANSPORT MODEL 71 5. COMPARISON AND DISCUSSION OF RESULTS 77 5.1 EFFECTS OF LATERAL DISPERSION 78 5.2 THE INITIAL DILUTION OF EFFLUENT 79 5.3 PEAK EFFLUENT CONCENTRATIONS 81 5.4 UPSTREAM EFFLUENT TRANSPORT 84 5.5 CHANNEL INTERACTIONS^ , 86 5.6 SUMMARY 89 6. SUMMARY AND CONCLUSIONS 92 REFERENCES. 98 -APPENDICES A. DERIVATION OF THE ONE-DIMENSIONAL MASS TRANSPORT EQUATIONS. . . .106 A . l GENERAL 106 A.2 LONGITUDINAL ADVECTIVE TRANSPORT 107 A.3 LONGITUDINAL DISPERSIVE TRANSPORT 109 A. 4 SOURCE-SINK EFFECTS 113 A.5 THE ONE-DIMENSIONAL MASS TRANSPORT EQUATION 114 A. 6 THE LAGRANGIAN FORM OF THE ONE-DIMENSIONAL ' MASS TRANSPORT EQUATION 115 B. A DESCRIPTION OF THE FRASER RIVER ESTUARY 117 B. l GENERAL 117 B.2 THE LOWER FRASER RIVER SYSTEM. . . 117 B.2.1 Channels of the River System .117 B.2.2 Tides i n the S t r a i t of Georgia 129 B.2.3 River Discharges 129 B.2.4 Saltwater Intrusion 134 vi APPENDIX Page C. NUMERICAL DISPERSION AND STABILITY 138 C l SURFACE GEOMETRY OF PARTIAL DIFFERENTIAL EQUATIONS . . . . . 138 C.2 NUMERICAL DISPERSION 146 C.3 STABILITY . 154 C. 4 SUMMARY 159 D. DETAILS OF THE SOLUTION SCHEMES OF THE HYDRODYNAMIC AND MASS TRANSPORT EQUATIONS 161 D. l NUMERICAL SOLUTION OF THE HYDRODYNAMIC EQUATIONS 161 D.2 NUMERICAL SOLUTION OF THE TIDALLY VARYING MASS TRANSPORT EQUATION 164 D. 3 NUMERICAL SOLUTION OF THE TIDALLY AVERAGED 'MASS TRANSPORT EQUATION 167 E. ESTIMATION OF LATERAL DISPERSION 174 E. l EXISTING ESTIMATES OF LATERAL MIXING 174 E. 2 VORTICITY ESTIMATE OF SECONDARY CURRENTS 180 P. ESTIMATION OF LONGITUDINAL DISPERSION 191 F. l GENERAL 191 F.2 TIDALLY AVERAGED COEFFICIENTS OF LONGITUDINAL DISPERSION . . 192 F.3 TIME DEPENDENT LONGITUDINAL DISPERSION COEFFICIENTS 199 F.4 SENSITIVITY OF PREDICTED CONCENTRATIONS 201 F.5 SUMMARY 204 LIST OF TABLES T a b l e Page 4.1 TIDALLY AVERAGED MASS BALANCE OF PREDICTED DISCHARGES FOR JANUARY 24, 1952 64 5.1 RATIO OF TIDALLY VARYING CONCENTRATION TO TIDALLY AVERAGED VALUE AT POINT OF DISCHARGE 85 B . l RIVER FLOW VOLUMES AND TIDAL PRISMS ON JANUARY 15 and JUNE 16, 1964 135 > E . l ESTIMATION OF COEFFICIENTS OF LATERAL DISPERSION 179 F . l ESTIMATED COEFFICIENTS OF LONGITUDINAL DISPERSION DUE TO LATERAL VELOCITY GRADIENTS 198 v i i 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 on Concentration 10 1.2 E f f e c t of M u l t i p l e Dosing ;(Due to Flow Reversal) on Concentration 11 1.3 Movement of E f f l u e n t Through A Diverging Junction 14 3.1 The Hydrodynamic Estuary 36 3.2 S i m p l i f i e d Model Estuary of the Mass Transport Equations. . . 43 4.1 Equivalent Stage Response f o r Two D i f f e r e n t Types of Cross-Section 50 4.2 Observed and Predicted Stages for June 16, 1964 52 4.3 Observed and Predicted Stages f o r June 16, 1964 53 4.4 T i d a l l y Varying Stage and Discharge — Main Arm 55 4.5 T i d a l l y Varying Stage and Discharge — Main Arm 56 4.6 T i d a l l y Varying Stage and Discharge — Main Arm 57 4.7 T i d a l l y Varying Stage and D i s c h a r g e — Main Arm 58 4.8 T i d a l l y Varying Stage and Discharge — North Arm 59 4.9 T i d a l l y Varying Stage and Discharge — Middle Arm, Canoe Pass. 60 4.10 T i d a l l y Varying Stage and Discharge — P i t t River 61 4.11 T i d a l l y Varying Stage — P i t t Lake 62 4.12 Advection Of A Slug Load Down the Main Arm - Main Stem . . . . 66 4.13 Slug Inputs f o r A n a l y t i c and Predicted Dispersion Solutions. . 69 4.14 Dispersion of A Slug Load i n the Main Stem 70 4.15 Influence of the T i d a l l y Averaged Dispersion C o e f f i c i e n t on Predicted T i d a l l y Averaged Concentrations .73 v i i i i x Figure Page 4.16 Maximum Upstream Excursion During Flow Reversal In Main Arm - Main Stem 75 4.17 Maximum Upstream Excursion During Flow Reversal In the North Arm and P i t t River 76 5.1 I n i t i a l D i l u t i o n at Station Nos. 10, 22, 50 and 102 80 5.2 Dispersion of A Concentration Spike 83 5.3 Predicted Concentrations i n One Channel Caused by E f f l u e n t Discharge i n Another Channel 87 5.4 Predicted Concentrations Due to Two E f f l u e n t Discharges i n the Main Stem 88 A . l Elemental Cross-Sectional S l i c e of A River or Estuary. . . . 108 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. ....... . 110 B. l The Drainage Basin of the Fraser River 118 B.2 The Fraser River From Hope to Vancouver 119 B.3 The Fraser River Delta 121 B.4 Network of Stations Used i n the Numerical Solution of the Hydrodynamic Model . 124 B.5 Typ i c a l Channel Cross-Sections 125 B.6 Cross-Sectional Parameters of Main Arm - Main Stem 126 B.7 Cross-Sectional Parameters of the North Arm, Middle Arm and Canoe Pass 127 B.8 Cross-Sectional Parameters of. P i t t River and P i t t Lake. . . 128 B.9 T y p i c a l Tides at Steveston . 130 B.10 Local Low and High Tide:Envelopes 131 B . l l Tide Gauging Stations i n the Fraser River Estuary 132 B.12 Mean Monthly Flows at Hope (1912-1970 Inclusive) 133 B.13 S a l i n i t y P r o f i l e s i n the Main Arm on February 13-14, 1962 . 136 X Figure Page C l Concentration Surfaces f o r the Advection of A Slug Load. . . 140 C.2 Time Rate of Concentration Along A Curve i n the (x,t) Plane) 142 C.3 Dispersion of A Slug Load 144 C.4 Forward and Backward Time Differences 148 C.5 Upstream, Central and Downstream Space Differences 150 C. 6 Stable and Unstable E x p l i c i t Advective Schemes 158 D. l E x p l i c i t F i n i t e Difference G r i d of the Hydrodynamic Equations 163 D.2 Segments of Thomann's T i d a l l y Averaged Model 168 D. 3 The Matrix A of Equation (D.8) f o r the Fraser River Estuary. 173 E. l Bends Along the 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 . v 178 E.3 Assumed Linear D i s t r i b u t i o n of Secondary V e l o c i t i e s 181 E.4 Interaction of u and E, i n A Straight River 184 z E. 5 Interaction of u and 5 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 at Stations 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 to the C o e f f i c i e n t of Longitudinal Dispersion 202 L I S T OF SYMBOLS a : H e i g h t o f e s t u a r y bed above l e v e l datum„ A : C r o s s - s e c t i o n a l a r e a , A : M a t r i x o f t h e e f f e c t s o f a d v e c t i o n , d i s p e r s i o n and d e c a y o n 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^ ^+^: A v e r a g e c r o s s - s e c t i o n a l a r e a between segments i and i+1 (Thomann's ' S o l u t i o n ) A j : C r o s s - s e c t i o n a l a r e a a t p o i n t j A x a t t i m e nAt (hydrodynamic e g u a t i o n s ) c i- C r o s s - s e c t i o n 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 b : W i d t h o f e s t u a r y C : Chezy's f r i c t i o n c o e f f i c i e n t c : C o n c e n t r a t i o n i n segment i (Thomann's S o l u t i o n ) C : M a t r i x o f segment c o n c e n t r a t i o n s (Thomann's S o l u t i o n ) C j : C r o s s - s e c t i o n 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 a t p o i n t j A x a t t i m e n A t . 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 y 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 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 v e l o c i t y 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 E f c : 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 v e l o c i t y 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 E : E f f e c t i v 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 o v e r a t i d a l c 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 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 v e r t i c a l v e l o c i t y y g r a d i e n t s E^ : C o e f f i c i e n t o f pseudo d i s p e r s i o n ( n u m e r i c a l d i s p e r s i o n ) E i i + l : " ^ ^ H Y a v e r a g e d 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 between ' segment i and i+1 (Thomann's S o l u t i o n ) x i X l 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 H e i g h t o f w a t e r s u r f a c e above l e v e l datum H e i g h t 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 nAt Von Kantian's c o n s t a n t R a te o f decay o f s u b s t a n c e i n segment i (Thomann's S o l u t i o n ) A v erage l e n g t h o f segments i and i + l (Thomann's S o l u t i o n ) Manning's "n" ( f o o t - s e c o n d u n i t s ) F r e s h w a t e r d i s c h a r g e 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 s u b s t a n c e due?to t h e i ' t h s o u r c e - s i n k p r o c e s s Time C r o s s - s e c t i o n a l l y a v e r a g e d l o n g i t u d i n a l v e l o c i t y M a g n i t u d e o f t h e s t e a d y component 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 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 v e l o c i t y E f f e c t i v e l o n g i t u d i n a l v e l o c i t y o v e r a t i d a l c y c l e C r o s s - s e c t i o n a l l y a v e r a g e d l o n g i t u d i n a l v e l o c i t y a t p o i n t j A x a t t i m e nAt Shear v e l o c i t y 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 E f f e c t i v e s h e a r v e l o c i t y o v e r a t i d a l c y c l e Volume o f segment i (Thomann's S o l u t i o n ) C r o s s - c h a n n e l 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 f l o w s R a te o f waste d i s c h a r g e i n t o segment i (Thomann's S o l u t i o n ) M a t r i x o f segment was t e d i s c h a r g e s (Thomann's S o l u t i o n ) L o n g i t u d i n a l d i s t a n c e a l o n g t h e e s t u a r y x i i i y :. V e r t i c a l d i s t a n c e i n t h e e s t u a r y z : L a t e r a l d i s t a n c e i n t h e e s t u a r y U*Y a : F a c t o r g i v e n by a = l n { — — } 11. C v a : F a c t o r r e l a t i n g d i s p e r s i o n c o e f f i c i e n t s i n d e p t h and s h e a r v e l o c i t y , as i n E = a y U # T i ^ a l exchange c o e f f i c i e n t between segments i and i+1 (Thomann's S o l u t i o n ) Y : S t a n d a r d i z e d c o n c e n t r a 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 a v e r a g e d model Y^_v : S t a n d a r d i z e d c o n c e n t r a 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 v a r y i n g model At : Time i n c r e m e n t u s e d i n 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 Ax : Space i n c r e m e n t u s e d i n 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 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 : 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 f l o w s a s s o c i a t e d w i t h U f 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 due t o t h e s e c o n d a r y f l o w s a s s o c i a t e d w i t h U £ c : E f f e c t i v 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 o v e r a t i d a l c y c l e n : R e l a t i v e d e p t h , g i v e n b y 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 v e l o c i t y g r a d i e n t s ? x : 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 v o r t i c i t y T : C i r c u l a t i o n a r o u n d 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 x "''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 by 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 and t h e f r e s h w a t e r f l o w p e r t i d a l c y c l e . A C K N O W L E D G E M E N T S The a u t h o r i s v e r y g r a t e f u l t o Dr. 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 o f t h i s t h e s i s . Mr. D o n a l d 0. H o d g i n s , who d e v e l o p e d and programmed t h e hydrodynamic model, i s a l s o t h a n k e d . S p e c i a l t h a n k s t o go Mr. R i c h a r d B run f o r d r a f t - i n g t h e m u l t i t u d e o f dia g r a m s 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 and p r o o f r e a d i n g s o i r e e s . The 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 s u p p o r t e d by t h e Westwater R e s e a r c h C e n t r e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . The a u t h o r e x p r e s s e s h i s a p p r e c i a t i o n f o r t h e i r a s s i s t a n c e , w i t h s p e c i a l t h a n k s t o P r o f e s s o r I r v i n g K. Pox. Mr. F. A. Koch and Mr. G. S. Sheehan o f t h e Westwater R e s e a r c h C e n t r e a r e a l s o t h a nked f o r t h e i r a s s i s t a n c e d u r i n g t h i s s t u d y . F i n a l l y , t h e a u t h o r g r a t e f u l l y acknowledges t h e p a t i e n c e and f o r b e a r a n c e 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 c o u r s e o f t h i s s t u d y . x i v INTRODUCTION Throughout h i s t o r y , centers of 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 located along r i v e r s and estuaries. The p r i n c i p a l uses of these surface water resources i n ea r l y times were water supply and navigation; the r i v e r provided an accessible source of 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 of surface water resources include naviga- t i o n and water supply 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 - d u s t r i a l purposes. In addition, r i v e r s and estuaries provide habitat f o r w i l d l i f e , breeding and rearing areas f o r f i s h and s h e l l f i s h and areas f o r general r e c r e a t i o n . Associated with each use of surface water resources i s a set of quantity and q u a l i t y constraints that determine whether the water i s s a t i s - f a c tory f o r that p a r t i c u l a r use. The q u a l i t y c o n s t r a i n t s consist., of the maximum allowable l e v e l s of various deleterious substances that may be pre- sent i n the water. Generally, the constraints f o r each use are d i f f e r e n t , and 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 multiple purposes. A common example i s the c o n f l i c t between the competing uses of waste disp o s a l , w i l d l i f e habitat and rec r e a t i o n . I t i s recognized that the quantity and q u a l i t y c o n f l i c t s are related; ̂  however, t h i s t hesis only considers q u a l i t y aspects, and i n p a r t i c u l a r , only those q u a l i t y aspects that are determined One obvious way of improving water q u a l i t y i s by low flow augmen- t a t i o n , as has been investigated by Worley et al.- [1965]. 1 2 by the concentration of a dissolved d e l e t e r i o u s substance i n the water. To investigate a s i t u a t i o n of e x i s t i n g or 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 "water-quality" model, or as i t s h a l l be re 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 concentration of deleterious substance to be predicted throughout the r i v e r or estuary, and i s instrumental i n assessing the effectiveness of possible c o n t r o l measures to improve water q u a l i t y . The simplest type of estuarine mass-transport model i s one-dimen- sional (1-D) and only admits a v a r i a t i o n of parameters and v a r i a b l e s i n the lo n g i t u d i n a l d i r e c t i o n . In a " t i d a l l y averaged" model, parameters and var- i a b l e s are assigned t h e i r average values over a t i d a l c y c l e , whereas i n a " t i d a l l y varying" model, they are allowed to vary throughout the t i d a l c y c l e . The temporal r e s o l u t i o n of the t i d a l l y varying model i s much f i n e r than i t s t i d a l l y averaged counterpart, but considerably more e f f o r t i s re- quired i n i t s development than f o r the l a t t e r . Thus, i t seems relevant to enquire as to (1) whether the d i f f e r e n c e s between the r e s u l t s of both models are s i g n i f i c a n t ; and (2) whether the extra e f f o r t involved i n deve- loping and applying the t i d a l l y varying model i s j u s t i f i e d by i t s f i n e r r e s o l u t i o n . In t h i s t h e s i s the s i g n i f i c a n c e of these d i f f e r e n c e s i s i n v e s t i - gated by applying both a t i d a l l y averaged and a t i d a l l y varying mass transport model to the Fraser River Estuary, a t i d a l estuary i n the Province of B r i t i s h Columbia, Canada. The t i d a l l y averaged model was developed by Thomann [1963] and the t i d a l l y varying model was developed from f i r s t p r i n c i p l e s . Both mass transport models were developed as part of a larger i n t e r d i s c i p l i n a r y study 3 b y t h e Westwater R e s e a r c h C e n t e r o f t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a t o 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 t h e w a t e r q u a l i t y 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 t h e s i s c o n s i s t s o f s i x c h a p t e r s . The t i d a l l y v a r y i n g and t i d a l l y a v e r a g e d forms o f t h e o n e - d i m e n s i o n a l mass t r a n s p o r t equa^- t i o n a r e d i s c u s s e d i n C h a p t e r 1. The e x p e c t e d d i f f e r e n c e s L b e t w e e n t h e r e s u l t s o f b o t h models and t h e a p p l i c a b i l i t y o f a o n e - d i m e n s i o n a l 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 r e a l s o c o n s i d e r e d , . t h e r e . I n C h a p t e r 2 t h e l i t e r a t u r e i s r e v i e w e d 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 o n e - 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 . I n o r d e r t o s o l v e t h e t i d a l l y v a r y i n g form o f t h e e q u a t i o n , i t was n e c e s s a r y t o d e v e l o p 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 a r e a a l o n g t h e e s t u a r y . The hydrodynamic model and 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 a v e r a g e d mass t r a n s p o r t models 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 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 C h a p t e r 4 and t h e r e s u l t s o f a p p l y i n g b o t h mass t r a n s p o r t models t o t h e E s t u a r y 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- t e r 5. F i n a l l y , 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 C h a p t e r 6, There a r e s i x a p p e n d i c e s t o t h i s t h e s i s . 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 e q u a t i o n f o r u n s t e a d y n o n - u n i f o r m f l o w i s d e r i v e d i n Ap p e n d i x A. 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 a r e d i s c u s s e d t h e r e i n , as a r e ' t h e a s s u m p t i o n s i n t h e d e r i v a t i o n o f t h e e q u a t i o n . A p p e n d i x B c o n - s i s t s o f a d e s c r i p t i o n o f t h e F r a s e r R i v e r E s t u a r y . The v a r i o u s c h a n n e l s o f t h e e s t u a r y , t h e f r e s h w a t e r 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 a l l d e s c r i b e d . 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 a r e used t o s o l v e t h e mass t r a n s p o r t e q u a t i o n , , t h e r e a r e pro b l e m s w i t h n u m e r i c a l d i s p e r s i o n and s t a b i l i t y . I n A p p e n d i x C b o t h o f t h e s e p r o b l e m s a r e seen t o o c c u r when a f i x e d g r i d i s used 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 a l o n g 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 . D e t a i l s 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 hydrodynamic e q u a t i o n s and b o t h forms 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. I n A p p e n d i x E, i t i s see 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 d i s p e r s i o n a p p a r e n t - l y u n d e r e s t i m a t e t h e l a t e r a l m i x i n g i n t h e F r a s e r R i v e r E s t u a r y . Secondary 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 , and on t h e b a s i s o f v e r y l i m i t e d f i e l d d a t a , t h e p r e d i c t e d s e c o n d a r y f l o w s a g r e e w e l l w i t h t h o s e ob- s e r v e d i n t h e e s t u a r y . F i n a l l y , 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 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 . An a p p r o x i m a t e method i s g i v e n t o a l l o w f o r t h e v a r i a b l e c o n t r i b u t i o n s o f t h e 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 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 c o m p l e t e . CHAPTER 1 PRELIMINARY CONSIDERATIONS 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 e q u a t i o n i s s t a t e d and i t s t i d a l l y v a r y i n g and t i d a l l y a v e r a g e d forms a r e b r i e f l y d i s c u s s e d . The p r o b l e m o f d e t e r m i n i n g t h e p a r a m e t e r s o f b o t h forms o f t h e e q u a t i o n i s c o n s i d e r e d , and e x p e c t e d 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 t i d a l l y v a r y i n g and a 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 model a r e d e s c r i b e d . The a b i l i t y 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 t h e mass t r a n s p o r t p r o c e s s e s i n t h e F r a s e r R i v e r E s t u a r y i s a l s o c o n s i d e r e d . 1.1 THE ONE-DIMENSIONAL MASS TRANSPORT EQUATION The o n e - d i m e n s i o n a l 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 non- 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 o v e r a n 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 o f t h e e s t u a r y . Mass i s t r a n s p o r t e d t h r o u g h t h e s l i c e by t h e mass t r a n s p o r t p r o c e s s e s o f a d v e c t i o n and d i s p e r s i o n , and t h e s e p r o c e s s e s , 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 s u b s t a n c e u n d e r g o e s , d e t e r m i n e t h e c o n c e n t r a t i o n o f t h e s u b s t a n c e w i t h i n t h e s l i c e . As i t i s a o n e - d i m e n s i o n a l 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 a r e a s s i g n e d t h e i r a v e r a g e c r o s s - s e c t i o n a l v a l u e s . The e q u a t i o n i s d e r i v e d i n A p p e n d i x A and t h e a s s u m p t i o n s 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 . The one-? 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 i s g i v e n by | £ = - u | i +• ^ { A E f e + ? S. (1.1 3t 9x A dx 9x . , 1 1=1 5 6 where c i s t h e mean 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 o f d i s s o l v e d s u b s t a n c e ; u i s t h e mean c r o s s - s e c t i o n a l v a l u 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 ; A i s t h e c r o s s - s e c t i o n a l a r e a ; 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 c o e f f i c i e n t i n t h e l o n g i t u d i n a l d i r e c t i o 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 volume o f w a t e r due t o t h e i t n 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 t h e r e a r e n s o u r c e - s i n k p r o c e s s e s ; x i s t h e l o n g i t u d i n a l d i s t a n c e ; and t i s t h e t i m e . F o r t h e sake 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 . The h i g h f r e q u e n c y t u r b u - 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 c o e f f i c i e n t E and t h e c r o s s - s e c t i o n a l a r e a A w i l l be r e f e r r e d t o as t h e p a r a m e t e r s o f E q u a t i o n (1.1). These q u a n t i t i e s a r e p a r a m e t e r s i n t h e sense t h a t t h e y a r e d e f i n e d o u t s i d e t h e e q u a t i o n by t h e c r o s s - s e c t i o n a l geometry and h y d r a u l i c s o f t h e p a r t i c u l a r e s t u a r y . These t h r e e p a r a m e t e r s can v a r y w i t h b o t h x and t Thus, E g u a t i o n (1.1) i s seen t o be a s e c o n d - o r d e r 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 v a r i a b l e c o e f f i c i e n t s . The t h r e e terms on t h e r i g h t - h a n d s i d e o f t h e e q u a t i o n w i l l 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 and t h e s o u r c e - s i n k terms r e s p e c t i v e l y . 7 E q u a t i o n (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 m o d e l s o f r i v e r s and e s t u a r i e s . I n an e s t u a r y , t h e r i s e and f a l l o f t h e t i d e c a u s e s t e m p o r a l v a r i a t i o n s i n t h e p a r a m e t e r s u, A and E. I n a t i d a l l y v a r y i n g model, t h i s t e m p o r a l v a r i a t i o n i n t h e p a r a m e t e r s i s t a k e n i n t o a c c o u n t as E q u a t i o n (1.1) i s s o l v e d t o d e t e r m i n e 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 t h e t i d a l c y c l e . I n a t i d a l l y a v e r a g e d model, t h e p a r a m e t e r s a r e a s s i g n e d 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 , and 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 i s d e t e r m i n e d o v e r p e r i o d s o f a t i d a l c y c l e . A v e r a g i n g E q u a t i o n (1.1) o v e r a t i d a l c y c l e does n o t a l t e r t h e fo r m o f t h e e q u a t i o n , b u t m e r e l y 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 p a r a m e t e r s . F o r 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 and i s d e t e r m i n e d by 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. i t s h o u l d be not e d 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 a v e r a g e d model i s n o t 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 — t h i s c a n o n l y be d e t e r m i n e d by a v e r - a g i n g t h e 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 o v e r t h e t i d a l c y c l e . The m a j o r i t y o f t i d a l l y a v e r a g e d models 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 e s t u a r y a t t i m e s o f s l a c k - w a t e r . The r e a s o n f o r t h i s i s t h e ease o f s a m p l i n g a t 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-: 1.2 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 u n s t e a d y o s c i l l a t o r y component due t o t h e t i d e s uperimposed on a s t e a d y component due t o f r e s h w a t e r i n f l o w . B e f o r e a s o l u t i o n c a n 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 f o r m o f E q u a t i o n (1.1), i t i s n e c e s s a r y t o d e t e r m i n e t h e t i d a l l y i n d u c e d t e m p o r a l 8 v a r i a t i o n i n t h e p a r a m e t e r s u, A and E. A o n e - d i m e n s i o n a l h y d r o d y n a m i c model was d e v e l o p e d t o d e t e r m i n e t h e t e m p o r a l v a r i a t i o n i n u and A. I n t h i s model, t h e e q u a t i o n s o f m o t i o n 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 and s o l v e d t h r o u g h o u t t h e t i d a l c y c l e . The h y d r o - 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. The t e m p o r a l v a r i a t i o n i n E d u r i n g t h e t i d a l c y c l e was r e l a t e d t o t h e t e m p o r a l v a r i a t i o n i n u, a s i s d i s c u s s e d i n A p p e n d i x F. In a t i d a l l y a v e r a g e d model, t h e p a r a m e t e r s u, A and E a r e a v e r - aged o v e r a t i d a l c y c l e . T h i s r e d u c e s t h e u n s t e a d y t i d a l f l o w f i e l d t o 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 f r o m t h e e q u a t i o n o f c o n t i n u i t y a l o n e (the e q u a t i o n o f c o n t i n u i t y i s a p p l i e d t o 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 ) . The t i d a l l y a v e r a g e d 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 t h e e f f e c t s o f u p s t r e a m a d v e c t i o n on t h e f l o o d t i d e , a s " i s d i s c u s s e d i n C h a p t e r 4. W i t h a t i d a l l y v a r y i n g m o del, 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 t h r o u g h o u t t h e 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 model, 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 o v e r p e r i o d s o f a t i d a l c y c l e . I n e f f e c t , t h e 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 a v e r a g e d model. Because o f t h e i r u n s t e a d y n a t u r e , t h e e s t i m a t i o n o f t h e t i d a l l y v a r y i n g p a r a m e t e r s u and A i n v o l v e s s i g n i f i c a n t l y more e f f o r t t h a n t h e e s t i m a t i o n o f t h e i r s t e a d y t i d a l l y a v e r a g e d c o u n t e r - p a r t s . T h i s i s a p p a r e n t from t h e d i s c u s s i o n o f C h a p t e r s 3 and 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 g r e a t e r e f f o r t r e q u i r e d t o e s t i m a t e t h e t i d a l l y v a r y i n g p a r a m e t e r s . 9 1.3 THE EFFECTS OF THE TIDE ON MASS TRANSPORT PROCESSES I n an e s t u a r y , t h e t i d a l r i s e and f a l l o f t h e w a t e r s u r f a c e c a u s e s t e m p o r 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 f l o w . Because o f t h i s temp- o r a l v a r i a t i o n o f f l o w , t h e 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 a l s o v a r i e s d u r i n g t h e t i d a l c y c l e . 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 1.1 f o r t h e s t e a d y d i s c h a r g e o f e f f l u e n t i n t o an e s t u a r y d u r i n g t h e f l o o d t i d e . The e f f e c t o f t h e f l o o d i n g t i d e i s t o c o n t i n u o u s l y r e d u c e t h e seaward f l o w o f w a t e r p a s t t h e e f f l u e n t o u t f a l l , and t h u s t h e s l u g o f w a t e r dosed i n any 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 . However, s i n c e t h e e f f l u e n t d i s c h a r g e i s s t e a d y , t h e same mass o f e f f l u e n t i s added t o each o f t h e d o s e d s l u g s and t h i s g i v e s r i s e t o t h e s p a t i a l d i s t r i b u t i o n o f c r o s s - s e c t i o n 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 shown i n F i g u r e 1.1. As t h e t i d e c o n t i n u e s t o f l o o d , f l o w r e v e r s a l w i l l o c c u r a t t h e e f f l u e n t o u t f a l l and p r e v i o u s l y d o s e d s l u g s w i l l move u p s t r e a m p a s t t h e o u t f a l l and be dosed a g a i n , a s i s i l l u s t r a t e d i n F i g u r e 1.2. On t h e f o l l o w i n g ebb t i d e , u p s t r e a m s l u g s w i l l move down- st r e a m p a s t t h e o u t f a l l and be d o s e d y e t a g a i n , as i s a l s o shown i n F i g u r e 1.2. 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 c a n a c c o u n t f o r 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 and m u l t i p l e d o s i n g , whereas a t i d a l l y a v e r - aged model c a n n o t . 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 a l v a r i a t i o n i n f l o w i s t o i n - t r o d u c e " s p i k e s " i n t o 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 t h e e s t u a r y (as i s a p p a r e n t f r o m F i g u r e s 1.1 and 1.2). 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 t h e n r e d u c e d by t h e 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 . Thus, d i f f e r e n c e s between t h e r e s u l t s f r o m a t i d a l l y v a r y i n g and 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 model w i l l depend on t h e m agnitude o f t h e t i d a l l y v a r y i n g d i s p e r s i o n p r o c e s s . A n o t h e r d i f f e r e n c e between t h e 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 v a r y - i n g model c o r r e c t l y a c c o u n t s f o r t h e e f f e c t s o f u p s t r e a m a d v e c t i o n on t h e 10 Steady effluent discharge t = 0 Rising Tide E S T U A R Y SEA t = At C l f t = 2At C2 C l t = 3 At C3 C2 C l 1 C4 t = 4At C4 C3 C2 C l u Time varying dilution C3^T ime average dilution F i g u r e 1.1 E f f e c t o f V a r i a b l e I n i t i a l D i l u t i o n on C o n c e n t r a t i o n 11 S t e a d y e f f l u e n t d i s c h a r g e t = 4 A t i R i s i n g T i d e C4 C 3 C 2 C l S E A u = 0 f R i s i n g T i d e t = 5 A t C4 C5 C 3 C 2 C l F a l l i n g T i d e t = 6 A t C | C 3 C 2 C l 1 W C 4 + C 5 + C 6 E f f e c t o f m u l t i p l e d o s i n g T i m e a v e r a g e c o n c e n t r a t i o n m////////////////A c i F i g u r e 1.2 E f f e c t o f M u l t i p l e D o s i n g (Due t o F l o w R e v e r s a l ) On C o n c e n t r a t i o n 12 fl o o d t i d e , whereas a t i d a l l y averaged model can only simulate t h i s upstream transport v i a the t i d a l l y averaged d i s p e r s i o n process. 1.4 ACCURACY OF A ONE-DIMENSIONAL MODEL Having b r i e f l y discussed the t i d a l l y varying and t i d a l l y averaged forms of the mass transport equation, the problem of how well a one-dimension- a l model describes the mass transport processes i n the Fraser River Estuary i s now considered. When e f f l u e n t i s discharged i n t o a r i v e r or estuary, the time required f o r complete c r o s s - s e c t i o n a l mixing to occur depends on the c o e f f i c i e n t of l a t e r a l d i spersion, the width of the r i v e r or estuary and the 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 [Ward, 1973]. In the i n i t i a l period before 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 concentra- t i o n gradients across the r i v e r or estuary. A one-dimensional mass trans- port model p r e d i c t s the c r o s s - s e c t i o n a l l y averaged concentrations and does not "see" these l a t e r a l gradients. Thus, i n the i n i t i a l period before cross- s e c t i o n a l mixing i s complete, the predicted concentrations w i l l underestimate the peak l a t e r a l values i n the r i v e r or estuary. Also, during t h i s i n i t i a l p eriod the dis p e r s i o n of 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 of concen- t r a t i o n along the r i v e r or estuary rather than the Gaussian d i s t r i b u t i o n pre- di c t e d by the t i d a l l y varying mass transport model (see Appendix A). When the c r o s s - s e c t i o n a l mixing i s complete, l o n g i t u d i n a l gradients dominate the transport processes, and a one-dimensional model w i l l provide a good descrip- t i o n of the actual concentration p r o f i l e along the r i v e r or estuary. 13 The 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 o f t h e seven p r i n c i p a l c h a n n e l s d e s c r i b e d i n A p p e n d i x B. The j u n c t i o n s o f t h e s e v a r i o u s c h a n n e l s i n f l u e n c e how e f f l u e n t i s a d v e c t e d t h r o u g h t h e e s t u a r y . I f 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 c o m 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 , t h e b u l k o f t h e e f f l u e n t l o a d may be a d v e c t e d down one o f t h e c h a n n e l s as shown i n F i g u r e 1.3. 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 c a n n o t r e p r o d u c e t h i s b e h a v i o u r a t t h e j u n c t i o n . The s i t u a t i o n shown i n F i g u r e 1.3 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 t i d a l f l o w r e v e r s a l and t h e p r e s e n c e o f s e c o n d a r y f l o w s a t t h e j u n c t i o n . On t h e f l o o d t i d e some e f f l u e n t i s c a r r i e d back u p s t r e a m p a s t t h e j u n c t i o n , and a d d i t i o n a l c r o s s - s e c t i o n a l m i x i n g o c c u r s b e f o r e t h e e f f l u e n t i s a d v e c t e d back t h r o u g h t h e j u n c t i o n on t h e ebb t i d e . I f s e c o n d a r y f l o w s a r e p r e s e n t , t h e s i t u a t i o n may 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 bank u p s t r e a m f r o m a d i v e r g i n g j u n c t i o n and 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 , as i s shown i n F i g u r e 1.3. (On t h e ebb t i d e , marked s e c o n d a r y f l o w s o f 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 Arm-North Arm j u n c t i o n shown i n F i g u r e B.3). Thus, i n t h e 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 c o m p l e t e , 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 R i v e r E s t u a r y i s a complex t w o - d i m e n s i o n a l 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 t h e t i d a l c y c l e . To r e p r o d u c e t h e s e e f f e c t s , a o n e - d i m e n s i o n a l model wou l d have 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 and s u c h m o d i f i c a t i o n may r e q u i r e c o n - s i d e r a b l e f i e l d d a t a . I t i s n o t e d t h a t t h e 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 and s e c o n d a r y 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 j u n c t i o n s . The a b i l i t y 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 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 t h e v a r i o u s c h a n n e l s o f t h e 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 t h e t i m e r e q u i r e d f o r c r o s s - s e c t i o n a l m i x i n g t o o c c u r . 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 - s e c t i o n a l m i x i n g i s c o m p l e t e , Movement o f F i g u r e . 1 . 3 E f f l u e n t Through A D i v e r g i n g J u n c t i o n 15 the dispersion process i s skewed rather than Gaussian, the peak l a t e r a l concentration i s s i g n i f i c a n t l y greater than the predicted c r o s s - s e c t i o n a l l y averaged value and the movement of e f f l u e n t through the junctions may not be according to the simple mass balance of the one-dimensional equation, the l a t t e r two e f f e c t s probably being the most important. The time required f o r 80 per cent c r o s s - s e c t i o n a l mixing to occur has been estimated i n Appen- dix E. The techniques of Fischer [1969a] and Ward [1972] gave an estimate of 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 considerations gave an estimate of f i v e hours, the bulk of the cr o s s - s e c t i o n a l mixing being due to the influence of secondary flows. I t i s noted that t h i s l a s t value has not been confirmed by f i e l d experiments. Because of tlje highly asymmetrical nature of the t i d e s of the Fraser River Estuary, a more r e a l i s t i c estimate of the time of c r o s s - s e c t i o n a l mixing i n the lower reaches of the estuary i s probably 1-2 t i d a l c y c l e s . A l l of these aspects of l a t e r a l mixing are discussed i n Appendix E. Unfortunately, time and expense have precluded using dye studies to measure the act u a l rate of c r o s s - s e c t i o n a l mixing i n the estuary. To sum up, i n t h i s study i t i s recognized that l a t e r a l mixing and the complex two and three-dimensional flow c h a r a c t e r i s t i c s through the junctions can have a considerable influence on the concentrations along the various channels of the Estuary. These aspects are considered', and t h e i r influence i s p a r t i a l l y assessed, but the main thrust of the study has been to obtain accurate solutions to the t i d a l l y varying and t i d a l l y averaged forms of the one-dimensional mass transport equation so that the influence of the t i d e s on the predicted concentrations can be assessed. To some extent, 16 Che two- and three-dimensional effects can be estimated as further modifi- cations of the one-dimensional results, as is discussed further in Chapter 5. CHAPTER 2 LITERATURE REVIEW The l i t e r a t u r e i s reviewed to investigate the various solutions and applications of one-dimensional mass transport models. Solutions to the mass transport equation are c l a s s i f i e d into the categories of analy- t i c a l s o lutions, numerical solutions, p h y s i c a l and analogue model solutions and stochastic solutions. The advantages and disadvantages of each c l a s s of s o l u t i o n are also discussed. 2.1 ANALYTICAL SOLUTIONS The one-dimensional mass transport equation f o r unsteady non- uniform flow i n a r i v e r or estuary i s given by |£ = -up- + I ^ - f A E ^ } + E S., (2.1 3t 3x A 9x d X . . l ' 1=1 where the terms are as defined i n Section 1.2. Equation (2.1) i s a second order 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 equation with v a r i a b l e c o e f f i c i e n t s (the source-sink terms are generally linear,- and u, A and E are dependent on x and t ) . As such, no completely 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 solutions have been obtained under a number of s i m p l i f y i n g assumptions. Before reviewing various solutions, the so-called "slackwater" concentra- tions predicted by the t i d a l l y averaged models are discussed. The majority of the a n a l y t i c a l solutions discussed here are f o r the steady-state response ( d c / d t = 0) of Equation (2.1). This s i m p l i f i c a t i o n 17 18 reduces the p a r t i a l d i f f e r e n t i a l equation to an ordinary d i f f e r e n t i a l equation i n x alone. In determining t h i s steady state".response, the freshwater discharge and e f f l u e n t inputs are assumed to remain steady or constant for a period of time equal to the residence time of the estuary. Further, any steady-state s o l u t i o n of Equation (2.1) i s a t i d a l l y aver- aged s o l u t i o n . (The equation i s now independent of time and cannot r e - solve the within t i d e temporal f l u c t u a t i o n s i n c, u, A and E). However, the steady state solution to Equation (2.1) does not represent the t i d a l l y averaged concentration p r o f i l e along the r e a l estuary. In f a c t , i t repre- sents the concentration p r o f i l e along a model estuary that i s t i d e l e s s and has a high degree of l o n g i t u d i n a l d i spersion. (The t i d a l l y averaged E i s generally much higher than the t i d a l l y varying E, as i s discussed i n Sec- t i o n 3.3). The t i d a l l y averaged concentration p r o f i l e can only be obtained from the r e s u l t s of a tidally varying mass transport model. In the majority of t i d a l l y averaged mass transport models, the steady state s o l u t i o n to Equation (2.1) i s assumed to represent the concen- t r a t i o n p r o f i l e along the estuary at times of slackwater. In other words, the t i d a l l y averaged mass transport processes are used to determine the concentration p r o f i l e along the estuary at a p a r t i c u l a r phase of the t i d e . The reason f o r working with slackwater p r o f i l e s i s the r e l a t i v e ease of sampling tracers i n the estuary at these times [;0'Connor, 1960] . Any corres- pondence between the steady-state s o l u t i o n of Equation (2.1) and the slack- water concentration p r o f i l e along the r e a l estuary i s because the t i d a l e f f e c t s are small, and under these conditions the steady-state s o l u t i o n i s an adequate representation of the concentration p r o f i l e i n the r e a l estuary 19 at any phase of the t i d e , or because the steady state s o l u t i o n i s "forced" to conform to the slackwater p r o f i l e by "adjusting" the parameters of Eguation (2.1). I t i s i n t e r e s t i n g to note that Preddy and Webber [1963] developed a t i d a l l y averaged model that p r e d i c t s t i d a l l y averaged concentra- tions rather than slackwater values. This model i s described i n Section 2.2.2. In a t i d a l l y varying s i t u a t i o n , quasi-steady-state conditions are said to be achieved when the v a r i a b l e of i n t e r e s t (for example, c) undergoes a c y c l i c a l r e p e t i t i o n of the same values from t i d e c y c l e to t i d e c y c l e . Thus, i n a t i d a l l y varying model one speaks of a quasi-steady-state response, rather than a steady-state response as with 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 of Equation (2.1) e n t a i l s i m p l i f i c a t i o n s i n which various terms of the equation are ignored (for example, the d i s p e r s i o n term) and the parameters of the u, A and E are repre- sented as simple functions of x and p o s s i b l y t . E f f l u e n t inputs are treated as boundary conditions, and s o l u t i o n s have been obtained f o r both 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 inputs [O'Connor, 1965] . Steady-state s o l u t i o n s to Equation (2.1) have been obtained f o r steady e f f l u e n t discharge i n t o (1) steady uniform flows [O'Connor, 1960, 1962]; (2) steady non-uniform flows where the area i s a simple f u n c t i o n of x [O'Connor, 1965, 1967]; and (3) steady non-uniform flows where the freshwater discharge v a r i e s exponentially with x due to land run-off [O'Connor, 1967]. Transient solutions have been obtained f o r steady e f f l u e n t discharge i n t o (1) non-steady freshwater flows during the recession limb of the hydrograph (which was approxi- 20 mated as negative exponential i n time) [O'Connor, 1967]; and (2) steady flows where the d i u r n a l photosynthetic production of DO i s taken into account ( i t was approximated as a h a l f sine wave) [O'Connor and Di Toro, 1970]. To obtain these t r a n s i e n t solutions, i t was necessary to ignore dispersion. Kent [1960] used the method of separation of variables to ob- t a i n the general s o l u t i o n to Equation (2.1) f o r a slug input i n t o steady uniform flows, and the p a r t i c u l a r solutions f o r a s l u g input i n t o steady non-uniform flows (u, A and E were assumed to be l i n e a r i n x ) . Di Toro and O'Connor [1968] obtained the t r a n s i e n t s o l u t i o n f or steady e f f l u e n t input into unsteady non-uniform flows 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 could be separated into independent functions of x and t (dispersion was ignored). L i [1962] used the method of c h a r a c t e r i s t i c s to obtain a s o l u t i o n for non-steady e f f l u e n t discharge (sinusoidal v a r i a t i o n ) into steady u n i - form flow-, (dispersion was ignored) . He l a t e r used the method of perturba- tions to obtain a s o l u t i o n f or the same case with d i s p e r s i o n included [ L i , 1972] . Holley [T969b] transformed the mass transport equation f o r BOD i n t o the elementary d i f f u s i o n equation. (By t r a v e l l i n g with the water mass, the only transport process an observer "sees" i s dispersion — see Section A.6). He took the standard s o l u t i o n f o r a slug input and used the convolution method to obtain the s o l u t i o n f o r the continuous (but not neces- s a r i l y steady) e f f l u e n t discharge into unsteady uniform flow. Bennet [1971] used the convolution method to obtain the s o l u t i o n f o r the complete BOD-DO 21 system f o r the same e f f l u e n t discharge and flow conditions as Holley. 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 so l u t i o n of p a r t i a l d i f f e r e n t i a l eguations can be c l a s s i f i e d into (1) fixed mesh methods i n which the so l u t i o n i s obtained at f i x e d predetermined points i n a rectangular mesh of time and distance; (2) characteristic methods, i n which the solu t i o n i s obtained at mesh points along the c h a r a c t e r i s t i c curve(s) i n the time-distance planets), the p o s i t i o n of the mesh points being determined as the so l u t i o n progresses; and (3) combined methods, i n which the s o l u t i o n i s followed along the c h a r a c t e r i s t i c curve(s) and then extrapolated back onto a f i x e d mesh of uniformly spaced points [Amein, 1966]. In c h a r a c t e r i s t i c methods, the mesh points are generally non-uniformly spaced i n time or distance, and consequently the book-keeping of r e s u l t s i s somewhat untidy. Combined methods sim p l i f y t h i s bookkeeping by extrapolating the r e s u l t s back onto a f i x e d uniformly spaced mesh. F i n i t e differenceemethods are discussed i n some d e t a i l i n Section C.2. The problems of stability and convergence are considered, and i t i s seen that implicit f i n i t e d i f f e r e n c e schemes are generally unconditionally stable, whereas explicit schemes are at most c o n d i t i o n a l l y stable. The s t a b i l i t y requirements of e x p l i c i t schemes are discussed i n d e t a i l i n Section C.3, and are seen to impose l i m i t s on the r e l a t i v e s i z e of Ax and At, the g r i d spacing. Generally, 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 simulate the advective transport process 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 d i s - persive process being superimposed on the actual advective and d i s p e r s i v e 22 processes occurring i n the r i v e r t o r estuary. This s o - c a l l e d numerical dispersion i s discussed i n d e t a i l i n Section: c.2 and i s seen to be e l i m i - nated by using 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] developed a one-dimensional, t i d a l l y varying mass transport model of the t i d a l portion of the Potomac River. A f i x e d mesh, i m p l i c i t scheme with c e n t r a l differences was used to solve the mass transport equation. (The various types of space and time d i f f e r - ences are described i n Section C.2). T i d a l v e l o c i t i e s were ca l c u l a t e d from discharge and t i d a l records, and the Taylor d i s p e r s i o n equation [Taylor, 1954], modified for 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 gradients, was used to determine 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 s . The f i n i t e d i f f e r - ences?' s o l u t i o n simulated the r e s u l t s of dye studies i n the estuary reason- ably w e l l . However, Prych and Chidley [1969] showed that the numerical dis 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 approximately 30 times greater than the modified Taylor dispersion, and was of the order of the same magnitude as the actual dispersion occurring i n the estuary. (The modified Taylor equation neglects the e f f e c t s of transverse v e l o c i t y grad- ients and grossly underestimates the dispersion c o e f f i c i e n t for streams and estuaries, as Fischer [1966b] has demonstrated). The agreement between the dye r e s u l t s and 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 apparently only f o r t u i - tous. This example i l l u s t r a t e s the s i g n i f i c a n c e of numerical dispersion. B e l l a and Dobbins [1968] investigated f i x e d mesh f i n i t e d i f f e r - ence solutions to the one-dimensional mass transport equation describing the BOD - DO system. They applied a t i d a l l y varying f i n i t e d i f f e r e n c e model to a hypothetical estuary and compared the r e s u l t s with an a n a l y t i c a l t i d a l l y averaged s o l u t i o n . 23 Dornhelm and Woolhiser [1968] obtained a s o l u t i o n to the one- dimensional, t i d a l l y varying mass transport model f o r conservative sub- stances. They used a f i x e d mesh i m p l i c i t scheme with c e n t r a l d i f f e r e n c e s . A hydrodynamic model, solved by the same f i n i t e d i f f e r e n c e scheme, was used to determine the t i d a l l y varying parameters. The hydrodynamic model was v e r i f i e d against a steady-state a n a l y t i c a l s o l u t i o n , but i t exhibited i n s t a b i l i t i e s when applied to the Delaware estuary. (According to a l i n e a r s t a b i l i t y analysis, t h e i r i m p l i c i t scheme was unconditionally stable. How- ever, the non-linear nature of the hydrodynamic equations may require more stringent s t a b i l i t y conditions, as i s discussed i n S e c t i o n D . l ) . The mass transport model was v e r i f i e d f o r steady uniform flow and was applied i n t i d a l l y varying form to a hypothetical estuary. To overcome the problems of numerical dispersion, Gardiner et al. [1964] used the method of c h a r a c t e r i s t i c s to solve the two-dimensional mass transport equation describing the movement of a solvent through sand saturated with o i l . A combined f i n i t e d ifference method was used, with a mesh of uniformly d i s t r i b u t e d moving points superimposed on a f i x e d two- dimensional space mesh. At the beginning of a time step, the moving points are assigned the concentration of the nearest f i x e d g r i d point, and then ad- vected along the c h a r a c t e r i s t i c s according to the equation analagous to Equation (A.9). A f t e r advection, the moving points are assigned to the clo s e s t g r i d p o i n t to determine the concentration there. An e x p l i c i t form of the equation analagous to Equation (A.8) i s then used to disperse the concentration over the f i x e d g r i d points. The complete process i s then repeated for the next time step. Pinder and Cooper [1970] used the same 24 technique to ca l c u l a t e the t r a n s i e n t p o s i t i o n s of the s a l t water f r o n t i n a coastal a g u i f i e r . Di Toro [1969] recognized that-the method of c h a r a c t e r i s t i c s r e - duced the mass transport equation f o r the BOD-DO system into two ordinary d i f f e r e n t i a l equations. (He ignored dispersion, and so the equation anal- agous to Equation (A.8) was a "true" ordinary d i f f e r e n t i a l equation). He noted th a t the numerical s o l u t i o n of ordinary d i f f e r e n t i a l equations i s more exact than f o r p a r t i a l d i f f e r e n t i a l equations, and that the accuracy 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" Solutions. Another type of mass transport model i s the so-called "box model" i n which the estuary i s divided i n t o f i n i t e segments or boxes. Each segment i s assumed completely mixed and the concentration of substance i n a segment i s determined by the discharge of substance into the segment, the advective and mixing processes occuring between adjacent segments, and any source-sink e f f e c t s that the substance undergoes. Callaway [1971] noted the box models are conceptually very s i m i l a r to 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 , the l a t t e r segment the estuary into well mixed boxes centred around the g r i d points. Another s i m i l a r i t y i s that the box model representation of the estuary r e s u l t s i n a set of di f f e r e n c e equations r e l a t i n g the concentration i n any box to the concentration i n neighboring boxes (similar to i m p l i c i t d i f f e r e n c e schemes). Thomann [1963], [1965], developed a t i d a l l y averaged box model which was applied to the Delaware estuary. The estuary was segmented i n t o 30 boxes whose length varied from two to four miles. A l l hydraulic parameters 25 were assumed steady, but the concentration within each box was allowed to vary with time so that the e f f e c t of time varying inputs could be studied. The concentration i n each box i s determined from a set of simultaneous d i f f e r e n c e - d i f f e r e n t i a l equations. For' steady-state inputs, the temporal d e r i v a t i v e s are set equal to zero and the system of equations reduces to a set of d i f f e r e n c e equations s i m i l a r to those of an i m p l i c i t f i n i t e d i f f e r - ence scheme. Thomann's model i s described i n more d e t a i l i n Sections 3.3 and D.3. Pence et ai. [1968] extended the model to account f o r time varying 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 over a t i d a l c y c l e ) . Like most t i d a l l y averaged models, Thomann's model i s a "slack water" model with concentrations being determined at succeeding slack waters. Preddy and Webber [1963] developed a t i d a l l y averaged box model of the Thames estuary t o p r e d i c t the true t i d a l l y averaged value of the DO concentration. In t h e i r model, the boxes were two miles long and the t i d a l displacement was s i x miles to e i t h e r side of a box. Thus, the t i d a l l y averaged concentration within a box depends on the concentrations within the three boxes upstream and downstream of i t . This model i s one of the few that determine t i d a l l y averaged concentrations rather than slack water values. 2.2.3 F i n i t e Element Solutions. F i n i t e element methods have been used to obtain s o l u t i o n s to the one-dimensional mass transport equation. Examples include P r i c e et al. [1968], Guymon [1970], Pinder and F r i e n d [1972] and Nalluswami et al. [1972]. In f i n i t e element so l u t i o n s , the estuary i s segmented and the concentration p r o f i l e i s approximated as a s e r i e s of func- tions. 26 In. some respects the method is similar to implicit f i n i t e d i f f e r - ence methods and box models; in the latter two methods the estuary is also segmented, but the concentration profile is approximated as a point value rather than a function in each segment. Further, the solution of a l l three methods is 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 d i f f e r - 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 variation 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 characteristics of Gardiner et at. [1964] However, the accuracy of the latter technique was found to be adequate, 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 st a b i l i t y and numerical dispersion in f i n i t e element methods [Price et ai., 1968]. In closing this section, i t is mentioned that two-dimensional fin 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 in the horizontal plane, whereas the box model of Pritchard [1969] was developed to describe two-dimensional mass transport in the vertical plane. 27 2.3 PHYSICAL AND ANALOGUE MODEL SOLUTIONS 2.3.1 P h y s i c a l Model Solutions. Physical models of estuaries were o r i g i - n a l l y developed to investigate sediment erosion and deposition i n t i d a l water- ways. The use of these models to investigate mass transport processes i n the prototype estuary was a natural development, and examples include O'Connor [1962], [1965]; Diachishin [1963]; O'Connell and Walter [1963] and Lager and Tchobanoglous [1968]. In these model investigations of the mass transport process, a quantity of dye i s introduced a t the p o i n t being investigated and i t s d i s t r i b u t i o n throughout the model estuary i s recorded over succeeding t i d a l c y c l e s . P h y s i c a l modelling involves the scaled-down reproduction of the more important processes th a t a f f e c t the parameter being modelled. For mass transport i n estuaries, these processes are advection, dispersion and source- sink e f f e c t s . Physical models can s a t i s f a c t o r i l y reproduce one-dimensional advection and the two-dimensional density dependent s a l i n i t y i n t r u s i o n pro- cess [Simmons, 1960; Harleman, 1965]. However, the use of d i s t o r t e d cross- . s e c t i o n a l space scales 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 v e l o c i t y d i s t r i b u t i o n s . From the discussion of Section A.3, i t i s apparent that 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 reproduction of-the! d i s p e r s i o n process. This has been discussed by Harleman [1965], Harlemen-et al. [1968], and Fischer and Holley [1971]. Even i n p h y s i c a l models with undistorted c r o s s - s e c t i o n a l space scales, the d i s p e r s i o n process may not be reproduced c o r r e c t l y . In t h i s case, the reproduction of the d i s p e r s i o n process depends on the r a t i o of the time of c r o s s - s e c t i o n a l mixing to the t i d a l period i n the prototype estuary [Fischer and Holley, 1971]. 28 No attempt i s made to reproduce any source-sink reactions i n model studies, a "conservative" dye being used for the t e s t s . However, the dye does adsorb onto surfaces, and t h i s must be allowed f o r i n i n t e r - p r e t i n g t e s t r e s u l t s . O'Connell and Walter [1963] developed a method to account for a f i r s t - o r d e r decay r e a c t i o n . 2.3.2 Analogue Model Solutions. E l e c t r i c a l analogue models have been used to obtain s o l u t i o n s to the mass transport equation [Rennerfelt, 1963 and Leeds, 1967] and the hydrodynamic equations [Harder, 1971]. The estuary i s d i v i d e d into segments, and the space d e r i v a t i v e s of the mass transport equation are approximated as f i n i t e d i f f e r e n c e s over those seg- ments. This reduces Equation (2.1) to a set of simultaneous d i f f e r e n c e - d i f f e r e n t i a l equations (the time d e r i v a t i v e i s continuous) s i m i l a r to those of Thomann [1963]. The e l e c t r i c a l analogue of t h i s system of equations can then be constructed i n the form of a s o - c a l l e d ladder network [Leeds and Bybee, 1967]. Analogue models can determine the steady- state and t r a n s i e n t solutions f or 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 steady non-uniform flows [Leeds, 1967 and Leeds and Bybee, 1967], As the flow i s assumed steady, these models are t i d a l l y averaged, but i t may be p o s s i b l e to develop an analogue f o r s i n u s o i d a l l y varying flows. Depending on the f i n i t e d i f f e r e n c e approximations 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 also s u f f e r from numerical d i s p e r s i o n , as was recognized by B e l l a [1968]. For example, the analogue of Leeds and Bybee [1967] used c e n t r a l d i f f e r e n c e s to approximate the space d e r i v a t i v e s of Equation (2.1). From the d i s c u s s i o n of Section C.2, i t i s apparent that t h i s w i l l r e s u l t i n numerical d i s p e r s i o n . 29 2.4 STOCHASTIC SOLUTIONS Before reviewing stochastic solutions of the mass transport equa- t i o n , i t i s necessary to define several terms. A process i s any phenomena that undergoes changes with respect to time, an example being d a i l y r i v e r flow. If the chance of occurrence i s taken into account, a process i s s a i d to be stochastic or probabalistic. A p r o b a b i l i s t i c process i s time-independent and the v a r i a b l e s are considered pure-random. A stocha s t i c process i s time- dependent and the variables may be pure-random or non-pure-random. If non- pure-random, the process i s composed of a deterministic and a pure-random component. 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 of the random v a r i a b l e remains constant throughout the process, the process i s s a i d to be stationary, other- wise i t i s non-stationary. Non-stationary stochastic processes are very complicated mathematically, and i n order o f increasing s i m p l i f i c a t i o n they are treated as stationary, p r o b a b a l i s t i c , and de t e r m i n i s t i c [Chow, 1964]. Stochastic mass transport models attempt to p r e d i c t both the mean concentration p r o f i l e and the v a r i a t i o n around the mean. This v a r i a - t i o n i s due to the st o c h a s t i c nature ofC . t h V underlying processes that deter- mine the concentration, namely the v a r i a t i o n s that occur i n the freshwater and e f f l u e n t discharges, e f f l u e n t concentrations, etc. during the period of ana l y s i s . Deterministic models use the mean value of each of these stoc h a s t i c v a r i a b l e s to determine the mean concentration p r o f i l e . D iaschishin [1963] assumed t i d a l mixing to be a pure-random pro- cess that can be characterized by a mixing length. On t h i s basis he used a random walk formulation to obtain a t i d a l l y averaged s o l u t i o n f o r waste d i s - posal i n purely t i d a l waters. He allowed f o r the e f f e c t of mixing 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 well as i n the l o n g i t u d i n a l d i r e c t i o n . The s o l u t i o n of Equation (2.1) f o r a slug input into steady uniform flow i s a Gaussian d i s t r i b u t i o n . Harris [1962, 1963] assumed t h i s d i s t r i b u t i o n to r e s u l t from a pure-random dis p e r s i o n process, and on t h i s basis he obtained maximum l i k e l i h o o d estimates of u and E. He used the convolution method to obtain the s o l u t i o n f o r a continuous re- lease. However, i n the i n i t i a l non-Fickian period, the c r o s s - s e c t i o n a l l y averaged concentration p r o f i l e i s skewed and not Gaussian (see Section A.3), as Fischer [1966a] noted. Loucks and Lynn [1966] obtained the t r a n s i e n t and steady state p r o b a b i l i t y d i s t r i b u t i o n s of the DO concentrations at the point of minimum DO i n a stream (dispersion was ignored). The sequence of d a i l y streamflows was assumed to be a f i r s t - o r d e r Markov process (a s t o c h a s t i c non-pure-random process), and the sewage flows, ultimate BOD and source-sink parameters were assumed to be dependent on the d a i l y streamflows (via c o n d i t i o n a l p r o b a b i l i - t i e s ) . E s s e n t i a l l y , the technique i s as follows. Given a streamflow, the set of sewage flows, source-sink parameters, etc. t h a t r e s u l t i n the minimum DO f a l l i n g below some prescribed value i s determined (by a modified Streeter- Phelps equation). The p r o b a b i l i t y of t h i s set of events occurring i s then determined, the process being repeated for each d i s c r e t e streamflow v a r i a t e , and the p r o b a b i l i t i e s summed to give the t o t a l p r o b a b i l i t y of the minimum DO being l e s s than the prescribed value. Loucks [1967] used a s i m i l a r technique to investigate the e f f e c t of various treatment plant operating p o l i c i e s on the d i s t r i b u t i o n of minimum DO i n a stream. Thayer and Krutchkoff [1967] obtained a stochastic s o l u t i o n f o r the concentration p r o f i l e s of BOD and DO i n a stream (dispersion was ignored, and 31 the streamflow was assumed uniform and steady). The basic processes of decay, reaeration, etc. occurring i n the modified Streeter-Phelps eguation were treated as stochastic, but freshwater and e f f l u e n t discharges were assumed to be d e t e r m i n i s t i c . Concentration values were divided i n t o d i s - crete u n i t s of siz e A, and the p r o b a b i l i t y of a change of A occurring i n the concentration during a small time i n t e r v a l was assumed to follow a Pois- son d i s t r i b u t i o n . This allowed them to lump a l l of the stochastic v a r i a t i o n into the parameter A (the sum of a number of independent Poisson processes i s i t s e l f Poisson), which was determined by f i t t i n g the predicted variance i n the model estuary to the observed variance i n the r e a l estuary. E s s e n t i a l l y , they used the same set of r e s u l t s to both estimate and v e r i f y the model. Under these conditions, any model w i l l reproduce the observed r e s u l t s , i r r e s - pective of whether the underlying processes are c o r r e c t l y modelled. Whereas other stochastic solutions use the e x p l i c i t v a r i a t i o n i n the underlying stochastic processes to p r e d i c t the v a r i a t i o n i n the concentration p r o f i l e s , Thayer and Krutchkoff based t h e i r s o l u t i o n on the i m p l i c i t v a r i a t i o n i n the stocha s t i c processes (the actual v a r i a t i o n i s not even measured). Their predicted mean concentration value i s simply the:'solution to the (determinis- t i c ) modified Streeter-Phelps equation. As such, t h e i r technique i s no bet t e r than any det e r m i n i s t i c model that recognizes and incorporates s t o c h a s t i c v a r i - a t i o n through measurements of f i e l d values. Custer and Krutchkoff [1969] used a random walk formulation to extend the technique to include dispersion (the flow was assumed uniform but unsteady), and Schofield and Krutchkoff [1972] further extended the technique to account f o r v a r i a b l e stochastic parameters along the estuary. 32 Koivo and P h i l l i p s [1971] regarded the measured BOD and DO concen- t r a t i o n p r o f i l e s along a stream to be corrupted by "noise" due to the stochas- t i c nature of the underlying processes. They developed a method based on non- l i n e a r regression analysis to determine the values of the stochastic para- meters th a t gave the best o v e r a l l f i t to the observed concentration p r o f i l e s . 2.5 SUMMARY Most a n a l y t i c a l solutions to the one-dimensional mass transport equation are 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 e f f l u e n t discharge i n t o estuaries with simple geometries. (When the number of segments with d i f f e r e n t geometries i s greater than four or f i v e , the matching of bound- ary conditions a t t h e i r ends becomes cumbersome [O'Connor et al, , 1968]) . In t i d a l l y averaged s o l u t i o n s , upstream transport 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 not adequately reproduce the i n t e r a c t i o n of two or more e f f l u e n t o u t f a l l s on the f l o o d t i d e . Numerical solutions provide a means of so l v i n g the one-dimensional t i d a l l y varying mass transport equation, but the problems of s t a b i l i t y and numerical d i s p e r s i o n must be considered. Of the various numerical methods, the method of c h a r a c t e r i s t i c s , methods with higher order approximations and f i n i t e element methods are s a t i s f a c t o r y with respect to accuracy of s o l u t i o n . The method of 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 advective process. Undistorted 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 three-dimensional aspects of the f l o w - f i e l d at important or s e n s i t i v e sections of the estuary, but because of t h e i r inaccurate and in c o n s i s t a n t reproduction 33 of the d i s p e r s i o n process ( in both d i s t o r t e d and u n d i s t o r t e d mode ls ) , a t bes t they a re on l y an ad junct t o a mathemat ica l form o f s o l u t i o n . As deve - l o p e d , analogue models are t i d a l l y averaged and s u f f e r from numer i ca l d i s - p e r s i o n , a l though i t may be p o s s i b l e t o overcome both these l i m i t a t i o n s . I t i s not c l e a r whether the v a r i ous 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 combinat ion o f s t o c h a s t i c and c r o s s - s e c t i o n a l v a r i a t i o n . I f 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 g rea t e r than the 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 mean ingfu l t o model the l a t e r a l component of the d i s p e r s i o n process w i t h a two-dimensional d e t e r m i n i s t i c model . CHAPTER 3 A DESCRIPTION OF THE HYDRODYNAMIC AND MASS TRANSPORT MODELS 3.1 THE HYDRODYNAMIC MODEL To solve the one-dimensional t i d a l l y varying mass transport equation, i t i s necessary to know both the s p a t i a l and temporal v a r i a t i o n i n the parameters u, A and E of the eguation, as i s discussed i n Section 1.3. A one-dimensional hydrodynamic model was developed to p r e d i c t the s p a t i a l and temporal v a r i a t i o n i n the parameters u and A. (The temporal and s p a t i a l v a r i a t i o n of the parameter E i s discussed i n Appendix F. In the hydrodynamic model, the equations of motion and c o n t i n u i t y were applied to the water mass of the estuary, and solved throughout the t i d a l c y c l e . River Estuary, the equations of motion and c o n t i n u i t y are given by [Dronkers, 1969] 3.1.1 The Hydrodynamic Eguations. As applied to the Fraser 9u 3t dh u u (3.1) + u- •g. (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 height of the water surface 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 depth; 34 35 A i s the c r o s s - s e c t i o n a l area; b i s the 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 acceleration; and C i s Chezy's f r i c t i o n f a c t o r . Several of these terms are i l l u s t r a t e d i n Figure 3.1. Note that i n the hydrodynamic equations, x increases i n the upstream d i r e c t i o n , whereas i n the mass transport equation, x increases i n the downstream d i r e c t i o n . The two terms on the left-hand side of Equation (3.1) are the l o c a l and convective (or Bernoulli) accelerations r e s p e c t i v e l y . The two terms on the right-hand side of the equation are the forces causing these accelerations, the net pressure force due to the slope of the water surface and the f r i c t i o n force r e s p e c t i v e l y . The f i r s t term of Equation (3.2) i s the net outflow from an-;elemental c r o s s - s e c t i o n a l s l i c e of the estuary, while the second term represents the accompanying change i n storage within the s l i c e . The assumptions made i n d e r i v i n g equations (3.1) and (3.2) are l i s t e d i n Dronkers [1964]. The most important assumption i s that the storage width i s equal to the advective width. The v a r i a t i o n of advective and stor- age widths along the major channels of the estuary i s shown i n Figures B.6 to B.8. The d i f f e r e n c e between the advective and storage widths i s exagger- ated since these values were determined from the c r o s s - s e c t i o n a l areas below l o c a l low water. (In the lower reaches of the estuary, the t i d a l l y averaged depth i s some 10 fe e t above l o c a l low water). Consequently, the assumption of equal advective and storage widths seems reasonable f o r the Fraser River Estuary. (This i s also discussed i n Section 4.1 with regard to the v e r i f i - 36 Figure 3.1 The Hydrodynamic Estuary 37 c a t i o n o f t h e h y d r o d y n a m i c m o d e l ) . The o n e - d i m e n s i o n a l n a t u r e o f t h e e q u a t i o n s i s j u s t i f i e d s i n c e t h e v a r i o u s c h a n n e l s o f t h e e s t u a r y a r e much l o n g e r t h a n t h e y a r e wide (see A p p e n d i x B ) . I n f a c t , C a l l a w a y [1971] has c l a s s i f i e d t h i s t y p e o f e s t u a r y as a " t i d a l r i v e r . " I n d e r i v i n g Equa- t i o n s (3.1) and (3.2) t h e p r e s e n c e o f t h e s a l t wedge has been i g n o r e d be- cause o f t h e c o m p l i c a t e d n a t u r e o f i t s dynamics (see A p p e n d i x B ) . A c c o r d - i n g t o Odd [1971], t h e Chezy f o r m u l a s h o u l d a d e q u a t e l y r e p r e s e n t f r i c t i o n a l e f f e c t s i n f a s t f l o w i n g , w e l l m i x e d e s t u a r i e s . The F r a s e r R i v e r e s t u a r y i s f a s t f l o w i n g , b u t t h e Chezy f o r m u l a may n o t be a c c u r a t e i n t h e h i g h l y s t r a - t i f i e d l o w e r r e a c h e s . From F i g u r e 3.1, i t i s s e e n t h a t h = a + y (3.3) where 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 above t h e same l e v e l datum t h a t h i s r e f e r r e d t o . Thus, E q u a t i o n s (3.1) and (3.2) a r e seen t o be a p a i r o f c o u p l e d 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 , t h e dependent v a r i a b l e s b e i n g u and 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 v a r i a b l e s b e i n g x and t . The w i d t h b, C h e z y ' s f r i c t i o n f a c t o r C, t h e a r e a A and t h e f a c t o r a a l l a p p e a r i n t h e e q u a t i o n s as p a r a m e t e r s . 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 been assumed t h a t t h e dependent v a r i a b l e s and p a r a m e t e r s v a r y i n a c o n t i n u o u s manner a l o n g t h e e s t u a r y . I t i s n o t e d t h a t b o t h e q u a t i o n s c o n t a i n v a r i a b l e c o e f f i - c i e n t s and t h a t E q u a t i o n (3.1) c o n t a i n s n o n - l i n e a r t e r m s . 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 D r o n k e r s [1969J was u s e d 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 t h e h y d r o d y n a m i c e q u a t i o n s . De- 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 t h e f i x e d mesh o f " s t a t i o n s " u s e d i n 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 i s shown i n F i g u r e B .4 . Because t h e 38 s o l u t i o n scheme of the hydrodynamic equations i s e x p l i c i t , s t a b i l i t y r e - quirements determine the r e l a t i v e s i z e of Ax and At. In solving the hydrodynamic eguations, Ax was set equal to 5,000 feet, except i n the deeper waters of P i t t Lake where Ax was increased to 15,000 feet f o r s t a b i l i t y reasons. I t was found that with t h i s space g r i d , a At of 90 seconds was s a t i s f a c t o r y as regards s t a b i l i t y . The s t a b i l i t y requirements are discussed i n d e t a i l i n Section C.3. 3.1.2 Assumed Quasi-Steady Hydraulic Conditions. For e f f l u e n t discharge into an estuary, the quasi-steady state response of an estuary i s u s u a l l y a condition of i n t e r e s t . .Tidally. varying mass transport models usu a l l y require many t i d a l cycles to achieve quasi-steady-state conditions, whereas hydrodynamic models only require several t i d a l c y c l e s . The reason f o r t h i s i s the r e l a t i v e speed of information propagation i n both systems. In the hydrodynamic model, the information c o n s i s t s of small changes i n sur- face elevation that propagate along the estuary as elemental surges with speeds of u±/gf-f whereas i n the mass transport model, the I information con- s i s t s of changes i n concentration ( f i n i t e or otherwise) that propagate at speeds of u ( i f dispersion i s ignored). This i s discussed further i n Sec- t i o n C.3. Thus, i n applying a mass transport model to determine the quasi- steady state response of an estuary f o r some waste loading condition, i t i s necessary to run the model for a succession of t i d a l c y c l e s . In s o l v i n g the hydrodynamic and t i d a l l y varying mass transport equations f o r the Fra'ser River Estuary, the freshwater discharge and t i d a l conditions are assumed to be quasi-steady. Thus, the model estuary "sees" a constant freshwater inflow and a succession of i d e n t i c a l t i d e s . In actual f a c t , "slow" v a r i a t i o n s occur i n the freshwater inflow and t i d a l 39 conditions, and the estuary may never achieve s t r i c t quasi-steady state conditions. However, as the residence time of the Fraser River Estuary i s only four to s i x days, i t seems reasonable to t r e a t the freshwater and t i d a l conditions as constant for t h i s period of time. 3.1.3 The Model Estuary of the Hydrodynamic Equations. The Fraser River Estuary i s described i n d e t a i l i n Appendix B. The model estuary, as described byfthe hydrodynamic equations, d i f f e r s to some ex- tent from the r e a l estuary because of the assumptions made i n d e r i v i n g the hydrodynamic equations. The model estuary consists of the same seven p r i n c i p l e s as the r e a l estuary, although the Canoe Pass Area has been s i m p l i - f i e d into a s i n g l e channel, as i s shown i n Figure B.4. The channels of the model estuary are assumed to be rectangular i n cross-section (equal storage and advective widths), and the s a l i n i t y i s assumed to be everywhere - zero. Because the s a l t wedge has been ignored, the predicted v e l o c i t i e s i n the lower reaches of the estuary are probably somewhat low. The freshwater d i s - charge i n the model estuary i s constant, and the t i d a l r i s e and f a l l of the water surface i s quasi-steady. 3.2 THE TIDALLY VARYING MASS TRANSPORT MODEL 3.2.1 Method of Solution. The t i d a l l y varying mass transport equation was solved 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 t h i s method, the equation i s solved along the c h a r a c t e r i s t i c curves of the advec- t i v e transport process, as i s discussed i n Appendix C. The accuracy and directness of s o l u t i o n were instrumental i n t h i s choice; there i s no numeri- c a l dispersion (see Appendix C) and the advection process i s simulated 40 d i r e c t l y and independently of the d i s p e r s i v e process. I t was not p o s s i b l e to use the usual technique's of dye patch studies or measurement of e x i s t i n g tracers to v e r i f y the mass transport models. The large d i l u t i o n a l capacity of the r i v e r made extensive dye patch studies too expensive, and as the seven channels of the estuary are e s s e n t i a l l y unpolluted [Benedict, et al. } 1973], there are no s u i t a b l e e x i s t i n g p o l l u t i o n a l t r a c e r s . Because of the high degree of s t r a t i f i c a - t i o n and the complicated nature dynamics of the s a l i n i t y i n t r u s i o n (see Section B.2.4), s a l t could not be used as a t r a c e r . Thus, i t was not pos s i b l e to v e r i f y the mass transport models by a d i r e c t comparison of predicted and observed concentrations. An attempt was made to " v e r i f y " the underlying mass transport processes of advection and d i s p e r s i o n inde- pendently of each other, as i s discussed i n Section 4.2. The method of c h a r a c t e r i s t i c s leads to a natural separation of the advective and di s p e r - s i v e transport processes, and t h i s was one of the reasons f o r using t h i s method to solve the t i d a l l y varying mass transport equation. The one-dimensional mass transport equation i s derived i n Appendix A and the assumptions i n i t s d e r i v a t i o n are discussed there. The equation was transformed i n t o Lagrangian or c h a r a c t e r i s t i c form (see Section A.6) to give i t . =' - n — = r"5 [EAr—} + 2 S ( 3 > 5 ) dt Adx 3x 1 i = l Equation (3.5) was then separated i n t o i t s component d i s p e r s i v e and source- sink e f f e c t s to give 41 dc dt 1 3 A 3x (3.6) and dc_ dt n Z S. (3.7) 1=1 1 I n i t i a l l y , a g r i d of moving points was assigned throughout the estuary. In the advection step, a f i n i t e d i f f e r e n c e form of Equation (3.4) was used to advect the points along the estuary f o r a time increment, the or through converging junctions. In the d i s p e r s i o n step, an e x p l i c i t f i n i t e d i f f e r e n c e form of Equation (3.6) was used to adjust the concentration of the moving points for the e f f e c t s of dispersion during the time increment. F i n a l l y , i n the source-sink step, Equation (3.7) was used to adjust the concentration of the moving points f o r the e f f e c t s of the source-sink pro- cesses during the time increment. (The source-sink step can be done analy- t i c a l l y — f o r a BOD-DO system, Equation (3.7) i s the usual Streeter-Phelps equation). The sequence of these three steps was then repeated f o r the next time increment, and so the s o l u t i o n progresses through time. At the boundaries of the estuary (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 estuary as d i c t a t e d by the advective boundary conditions. A time increment of one hour was used i n solving the t i d a l l y varying mass trans- port equation, and at the end of each hour,the concentrations were extrapolated off the g r i d of moving points onto the 5,000 foot f i x e d g r i d of Figure 3.1. The method of s o l u t i o n i s very s i m i l a r to the combined method of Gardiner et al. concentration of the points being adjusted as they passed e f f l u e n t o u t f a l l s 42 [1964], except that whereas Gardiner et al, extrapolate the moving points onto the f i x e d g r i d , the moving points i n the Fraser River Estuary s o l u t i o n always remain on t h e i r respective c h a r a c t e r i s t i c s and only t h e i r concentra- t i o n s are extrapolated onto the f i x e d g r i d . D e t a i l s of the s o l u t i o n of Equations (3.4), (3.6) and (3.7) are given i n Appendix D. An advantage of 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 that a d d i t i o n a l moving points can be added to the estuary t o more c l o s e l y de- f i n e regions of r a p i d v a r i a t i o n i n concentration, as occur at times of 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- sary moving points can be removed. Once the moving points are i n the estuary, t h e i r subsequent p o s i t i o n s are determined by the advective transport process, and they are not uniformly spaced along the estuary in'the d i s p e r s i v e step. Both an e x p l i c i t and 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 investigated f o r the dispersive step. The e x p l i c i t scheme was s l i g h t l y slower, but was chosen for reasons of s i m p l i c i t y (see Appendix D). The s t a b i l i t y requirements of the e x p l i c i t scheme are discussed i n Appendix C, and generally the time i n - crement f o r s t a b i l i t y was less than one hour; the dispersive step then con- s i s t e d of a number of " i n t e r n a l " i t e r a t i o n s within the basic time increment of one hour, as discussed i n Appendix D. 3.2.2 The Model Estuary. For the sake of convenience, the model estuary of both the t i d a l l y averaged and t i d a l l y varying mass transport equa- tions has been s i m p l i f i e d to the three major channels of the r e a l estuary, the Main Arm - Main Stem, the North Arm and the P i t t System, as shown i n Figure 3.2. The s t a t i o n s along these channels are the same as those of the Pitt L a k e ' 1 / fir _ Ma in S t e m Figure 3.2 S i m p l i f i e d Model Estuary of the Mass Transport Equations 44 hydrodynamic model. I t i s p o s s i b l e to include a l l seven channels of the r e a l estuary, but i t would have made the computer programming of the t i d a l l y varying mass transport equation considerably longer and more involved. Because of t h e i r short length and smaller flows, the concentration p r o f i l e along the minor channels i s e s s e n t i a l l y determined by the p r o f i l e along the major channels, and the approximation of ignoring the minor channels seems reasonable. In the model estuary, the dispersion process i s assumed to be 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 i s assumed to be complete at the junctions. The s a l i n i t y of the model estuary i s zero, but because of 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 estuary, the predicted v e l o c i t y f i e l d and advective transport i n t h i s region i s probably under- estimated. The estuary i s highly s t r a t i f i e d i n the region of the s a l i n i t y i n t r u s i o n and l i t t l e mixing occurs between the fresh and ^.saltwater. This w i l l tend to minimize any chemical or b i o l o g i c a l e f f e c t s the saltwater may have on the d i s s o l v e d substance i n question. As i n the model estuary of the hydrodynamic equations, the freshwater discharge i s constant and the t i d a l conditions are quasi-steady. 3.3 TIDALLY AVERAGED MASS TRANSPORT MODEL 3.3.1 Method of Solution. The one-dimensional, t i d a l l y averaged mass transport model of Thomann [1963] was used to determine the steady state t i d a l l y averaged response of the Fraser River Estuary for various waste loading conditions. (As the model i s t i d a l l y averaged, the response i s steady-state rather than quasi-steady s t a t e ) . In t h i s model, the estuary i s divided i n t o 45 a number of l o n g i t u d i n a l segments or "boxes", as shown i n Figure D.2. Each segment i s assumed to be completely mixed, and the t i d a l l y averaged advection and disp e r s i o n processes transport dissolved substance i n t o and out of each segment. If the t i d a l l y averaged transport processes and e f f l u - ent discharges remain steady, a mass balance about segment i gives a. . .-c + a. . c. + a. ...c' = W . (3.8) i , a - l i - l 1 , 1 i i , i + l l + l i where and c^ i s the concentration i n segment i ; i s the mass of e f f l u e n t discharged into segment i per t i d a l cycle; a. . ̂  i s a c o e f f i c i e n t accounting f o r the t i d a l l y averaged ' transport of substance between segments i and i - l , s i m i l a r l y f o r a^ ̂ +^; a. . accounts f o r dispersion out of segment i and any sink ' e f f e c t s the substance undergoes. D e t a i l s of t h i s eguation and i t s c o e f f i c i e n t s are given i n Appendix D. An equation s i m i l a r to (3.8) can be written f o r each of the n segments of the estuary, and i n matrix notation the system of equations can be written A • C = W (3.9) where | i s a (n x 1) column matrix of the t i d a l l y averaged waste loads into.each segment per t i d a l - c y c l e ; C i s a (n x 1) column matrix of the t i d a l l y averaged concentration i n each segment; 46 and ^ i s a (n x n) t r i - d i a g o n a l matrix containing the t i d a l l y averaged transport terms and any sink e f f e c t s . Thus, the steady state t i d a l l y averaged response of the estuary i s described by a system of n simultaneous l i n e a r equations. Equation (3.9) represents the concentrations of a conservative substance or a substance-^undergoing f i r s t order decay, such as BOD. To investigate the steady-state BOD-DO re s - ponse of an estuary, a system of equations s i m i l a r to (3.9) and coupled to i t are obtained for the DO concentrations (see Thomann [1971] f o r d e t a i l s ) . Thomann's steady-state s o l u t i o n to the one-dimensional t i d a l l y averaged mass transport equation 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 d i f f e r - ence s o l u t i o n s i m i l a r to that of an i m p l i c i t d i f f e r e n c e scheme (see Appendix D). There i s no s t a b i l i t y requirement for the steady-state s o l u t i o n (see Section C.3), but there i s a non-negativity requirement t h a t imposes r e l a - t i v e l i m i t s on the sizes of the advective and d i s p e r s i v e transport processes. This i s discussed i n Section D.3, and i f t h i s requirement i s v i o l a t e d , the concentration i n a segment becomes negative. Thomann's steady-state -solution does not s u f f e r from numerical dispersion, but f o r a rather unusual reason. Because of the f i x e d g r i d nature of his f i n i t e d i f f e r e n c e scheme, the advec- t i v e process w i l l not be c o r r e c t l y simulated and numerical d i s p e r s i o n should occur. However, from Section C.2, i t i s seen that f o r numerical d i s p e r s i o n to occur, i t i s necessary for the concentration at a f i x e d g r i d point to change with time. Thomann's steady-state s o l u t i o n admits no temporal changes, and thus no numerical dispersion occurs. Numerical di s p e r s i o n does occur i n h i s t r a n s i e n t s o l u t i o n where the concentration at the f i x e d g r i d points does change with time, as Thomann [1971] recognizes. 47 3.3.2 The Model Estuary. The model estuary of the t i d a l l y aver- aged equation consists of the same three major channels and stations as the model estuary of the t i d a l l y varying equation and c r o s s - s e c t i o n a l mixing i s assumed to be complete at the junctions. The t i d a l l y averaged model estuary has no%tides and has higher d i s p e r s i o n than i t s t i d a l l y varying counterpart. Thomann [1971] l i s t s t y p i c a l values 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 . They range from 1 - 2 0 square miles per day with a mean value of about 10 square miles per day. These values seem high compared to the t i d a l l y varying values of d i s p e r s i o n c o e f f i c i e n t s , and apparently r e f l e c t the influence of t i d a l advection i n determining the t i d a l l y average disper- sion c o e f f i c i e n t . As i n the t i d a l l y varying s o l u t i o n , the influence of the s a l i n i t y 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 of the hydrodynamic model i s to ensure that the model estuary, as represented by the hydrodynamic equations, adequate- l y reproduces the v a r i a t i o n i n water surface elevations and discharges (or advective 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 " consists of adjusting the f r i c t i o n factors and the cr o s s - s e c t i o n a l widths and depths of the model estuary u n t i l an adequate f i t i s obtained between predicted and observed r e s u l t s . (In d e r i v i n g the hydrodynamic eguations, the advective flow was assumed to be uniformly 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 , the cross-section i s not rectangular and the flow i s concentrated i n the deeper sections. To compensate for these e f f e c t s , i t i s necessary to adjust the cr o s s - s e c t i o n a l geometry of the model estuary). 4.1.1 Data A v a i l a b l e . The network of permanent tide-gauging stations throughout the estuary i s shown i n Figure B . l l , and provides a l i m i t e d but adequate record of the water surface elevations of the estuary throughout the t i d a l cycle. F i e l d measurements of v e l o c i t i e s and discharges are sporadic and inadequate f o r v e r i f i c a t i o n purposes. The only e x i s t i n g data adequate for a complete v e r i f i c a t i o n of the hydrodynamic model under high tide-low flow conditions i s due to Baines [1952]. In t h i s study, water surface elevations were recorded at half-hour i n t e r v a l s a t 43 stations 48 49 throughout the estuary f o r the high tide-low flow conditions of January 24, 1952. The freshwater discharge at C h i l l i w a c k was 36,500 cubic f e e t per second and the t i d a l range a t Steveston was 11 f e e t . Baines used the method of cubature to estimate the t i d a l l y varying discharges at the 43 s t a t i o n s . 4.1.2 Water Surface Elevations f o r Low Flows. The high t i d e - low flow conditions of January 15, 1964 were used i n an i n i t i a l attempt to reproduce the recorded water surface elevations i n the r e a l estuary. The freshwater discharge at C h i l l i w a c k was 53,500 cubic f e e t per second and the t i d a l range a t Steveston was 10 f e e t . The water surface elevations of the model estuary were found to be r e l a t i v e l y i n s e n s i t i v e to the e f f e c t s of f r i c t i o n and c r o s s - s e c t i o n a l geometry. In f a c t , the response of narrow sections with high f r i c t i o n was found to be equivalent to that of broader sections with low f r i c t i o n . This i s i l l u s t r a t e d i n Figure 4.1 which shows the predicted responses f o r both types of section a t S t a t i o n No. 18 (New Westminster) on the Main Arm - Main Stem. The average values of widths, depths and Manning's "n" along the Main Arm —. Main Stem are also shown. Note that while Chezy's formula for f r i c t i o n was used i n s o l v i n g the hydro- dynamic equations, f r i c t i o n c o e f f i c i e n t s are reported as values of Manning's "n" (the values of n are f o r feet-second u n i t s ) . 4.1.3 Water Surface Elevations for High Flows. To separate out the independent e f f e c t s of f r i c t i o n and c r o s s - s e c t i o n a l geometry, an attempt was made to reproduce the water surface elevations of the high t i d e - h i g h flow conditions of June 16, 1964. The freshwater discharge at C h i l l i w a c k was 463,000 cubic f e e t per second and the t i d a l range at Steveston was eight f e e t . I t was thought that the greater v e l o c i t i e s under high discharge 50 Observed ° o Predicted No.I (Narrow section,high friction) * x Predicted No.2 ( Broad section,low friction) CJ> O CO o o Q = 5 2 , 4 0 0 c f s (Chill iwack ) T idal Range of Steveston =10 , J J a n . 1 5 , 1 9 6 4 0 ^ 8 12 16 Hours 2 0 M Average values along Main Arm - Main Stem Resul ts Average Mannings •i II n Average Width (feet ) Average Depth (feet ) Pred ic ted No. 1 0 . 0 4 2 1350 3 0 . 6 P red ic ted N o . 2 0 . 0 3 0 1870 2 2 . 9 Figure 4.1 Equivalent Stage Response f o r Two D i f f e r e n t Types of Cross-Section 51 conditions would be more s e n s i t i v e to f r i c t i o n a l e f f e c t s . This was found to be the case, the t i d a l l y averaged or mean water l e v e l s being very sen- s i t i v e to f r i c t i o n and e s s e n t i a l l y independent of width. However, the width was found to govern the range of the water surface f l u c t u a t i o n s about the mean water l e v e l . The predicted and observed water surface elevations are shown i n Figures 4.2 and 4.3, and the reproduction of the observed r e - s u l t s i s seen to be s a t i s f a c t o r y . The gross c r o s s - s e c t i o n a l values of depths and areas shown i n Figures B.6 to B.8 were used i n obtaining these r e s u l t s . The width was determined by d i v i d i n g the area by the depth, and was a s a t i s f a c t o r y compromise between the narrower advective sections and the broader sections more representative of the storage width. Manning's "n" v a r i e d from 0.022 i n the lower reaches to 0.027 i n the upper reaches. These values are reasonable, and i n d i c a t e that the Main Arm - Main Stem i s h y d r a u l i c a l l y smooth [Chow, 1959]. Under these high discharge conditions, the flow i n the upper reaches of the Main Stem i s steady, and the pr e d i c t e d water surface slope can be used to check that the hydrodynamic model s a t i s f a c t o r i l y reproduces the f r i c t i o n a l e f f e c t s . The p r e d i c t e d value of Manning's "n", as determined by the average area, depth and predicted water surface slope between stati o n s 40 and 60 on the Main Stem, was 0.0269. This agrees c l o s e l y with the ac t u a l value of 0.027 used i n the hydrodynamic model over t h i s s e c t i o n of 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- metries were then used i n a second attempt to reproduce the water surface elevations f or the low flow conditions of January 15, 1964. To obtain a sat- i s f a c t o r y f i t between the predicted and observed r e s u l t s i t was necessary to I I I I 1 I I M 4 8 12 16 20 M H o u r s Figure 4.2 Observed and Predicted Stages for June 16, 1964 Q= 463,000 cfs (Chilliwack) Tidal range at Steveston s 8' June 16, 1964 20 8 I 6 Main Arm - Station No.22 ( Port Mann) 2 4 |~ Main Arm - Station No. 32 ( Port Hammond) 22 20 0 b s e r v e d o o P r e d i c t e d 2 2 I" Pitt - Station No. 144 ( Port Coquitlam) O A o O O ° O ° rt o r> ~ 20<K°. • - 2 — ° n n n n n n n I L I 8 2 2 r Pitt - Station No. 162 ( Al vi n ) 20 I 8 o o M 8 12 Hours 16 20 M Figure 4.3 Observed and Predicted Stages for June 16, 1964 54 increase Manning's "n" by approximately 0.005 throughout the estuary. Under these conditions the value of Manning's "n" along the Main Arm- Main Stem v a r i e d from 0.027 i n the lower reaches to 0.03 2 i n the upper reaches, and once again these values are reasonable. The increase i n f r i c - t i o n that low flows experience r e l a t i v e to higher flows also seems reason- able. During the f a l l i n g limb of the annual hydrograph sediment: i s deposited i n the lower channels of the estuary [Pretious, 1972], and thus the low winter flows experience bed forms defined by the higher summer flows. I t i s expected that the r e l a t i v e roughness of these bed forms w i l l be greater f o r the low flows — and hence the greater 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 Conditions of January 24, 1952. Having made an independent check on the f r i c t i o n c o e f f i c i e n t s and e f f e c t i v e cross- s e c t i o n a l geometries, an attempt was made to reproduce the recorded water surface elevations and cubature discharges of Baines [1952]. With minor adjustments to the f r i c t i o n c o e f f i c i e n t s , the water surface elevations and discharges shown i n Figures 4.4 to 4.11 were obtained. The p r e d i c t e d d i s - charge curves were found to be i n s e n s i t i v e to changes i n the c r o s s - s e c t i o n a l geometries, but r e l a t i v e l y s e n s i t i v e to changes i n the f r i c t i o n c o e f f i c i e n t s . In c e r t a i n sections of the estuary, the low-flow f r i c t i o n c o e f f i c i e n t s of the section have been s l i g h t l y lowered to obtain a be t t e r f i t between the pre- d i c t e d and observed peak discharges. Generally, the f i t between the p r e d i c t e d and observed water surface elevations i s s a t i s f a c t o r y . In the upper reaches of the Main Stem, the range of predicted water surface elevations i s somewhat greater than that observed, and i s probably due to "ending" the model estuary before the t i d a l r i s e and 55 Q = 36,500 c f s (Chilliwack) M 4 8 12 16 20 M H o u r s Figure 4.4 T i d a l l y Varying Stage and Discharge - Main Arm Q = 36,500 cfs (Chilliwack) m - j T r , x . r . i . x . January 24, 1952 T i d a l Range at Steveston =11' J ' 56 E o •D U> O <» • o c o CO OJ a> 'T 12 I I 10 9 8 7 6 5 4 3 2 Main Arm - Stat.No.14 ( St. Mungo Cannery) O 3 O o 240 -200 -160 -120 -80 -40 0 40 80 120 160 </) 200 ° O O 240 O a> a> 14 c n ° 13 CO I 2 h I I 101 9 8 7 6 5 4 3 \ // / o Z3 O o _ l Lu z Main Arm - Stat. No.18 (New Westminster R.R.Bridge) I I I I 240 £ in -200 a -160 -120 -80 -40 0 40 80 120 180 2 0 0 M 8 12 Hours 16 20 M Figure 4.5 T i d a l l y Varying Stage and Discharge - Main Arm Q = 36,500 cfs (Chilliwack •- q> cn I i- o M 4 8 12 16 20 M H o u r s Figure 4.6 Tidally Varying Stage and Discharge - Main Arm e O T3 W TJ O c o CO CO q> cu o> o CO 15 14 13 12 I 1 10 9 8 7 6 5 4 3 2 Q = 36,500 c f s (Chilliwack) m - j - i r , January 24, 1952 T i d a l Range at Steveston =11' J ' O b s e r v e d S t a g e P r e d i c t e d S t a g e C u b a t u r e d i s c h a r g e I B a i n e s ) P r e d i c t e d d i s c h a r g e Main Arm - Stat. No.40 ( Whonock) J_ 20 M 8 12 16 Hours Figure 4.7 T i d a l l y Varying Stage and Discharge - Main Arm * "I o -i U- Z 58 140 H-120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 u O O O I -14,0 o> t_ o -120 x: o «A -100 Q -80 - 60 -40 -20 0 20 40 60 80 1 0 0 Q = 36,500 c f s ( C h i l l i w a c k ) 59 m - j - i r , j - o x . n . } J a n u a r y 2 4 , 1 9 5 2 T i d a l Range a t S t e v e s t o n = 1 1 ' 1 1 M 4 8 12 16 20 M Hours F i g u r e 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 15 ' t 13 h 12 I I I— 10 Q = 36,500 c f s (Chilliwack) T i d a l Range at Steveston = 11' } January 24, 1952 •'/ C u b a t u r e d i s c h a r g e t B a i n e s ) \ 1/ P r e d i c t e d d i s c h a r g e \\J Middle Arm - Stat. No. 133 ( Airport Floats ) 60 •30 25 20 - 15 - 10 5 I. 10 -J 15 20 H 2 5 30 35 Canoe Pass - Stat. No. 124 ( Canoe Pass) o -I u. I- o 5 o _i • b . Z -70 -60 -50 -40 - 30 - 20 - 10 0 10 20 30 40 50 M 4 8 12 16 20 M Hours Figure 4.9 T i d a l l y Varying Stage and Discharge - Middle Arm, Canoe Pass Q = 36,500 c f s (Chilliwack) T i d a l Range at Steveston = 11' } January 24, 1952 2 12 I I 10 9 8 7 6 5 4 3 2' 15 14 h- 13 12 11 10 9 8 7 6 5 4 3 \\ Pitt - Stat.No. 144 (CPR Bridge Coquitlam) 61 -120 -100 - 8 0 - 6 0 - 40 - 20 0 20 40 60 8 0 100 120 • j Pitt - Stat. No. 151 ' • (Gilley's Quarrey) - 140 H - 1 2 0 -100 - 80 - 60 - 40 - 20 0 20 40 60 80 100 u O O O Urn a o M M 8 12 16 Hours 20 M Figure 4.10 T i d a l l y Varying Stage and Discharge - P i t t River Q = 36,500cfs (Chilliwack ) Tidal range at Steveston = l l ' Observed Stage Predicted Stage M 4 8 12 16 20 M H o u r s Figure 4.11 T i d a l l y Varying Stage — P i t t Lake j Jan. 24,1952 63 f a l l of the water water surface has d i s s i p a t e d . The f i t between the pre- d i c t e d discharges and the cubature discharges of Baines i s not as good as the f i t between predicted and observed water surface e l e v a t i o n s . There seems to be a phase d i f f e r e n c e between the discharges, the p r e d i c t e d d i s - charge tending to occur e a r l i e r than the cubature discharge. This phase di f f e r e n c e was i n s e n s i t i v e to changes i n c r o s s - s e c t i o n a l geometry and f r i c - t i o n c o e f f i c i e n t s . The cubature discharge curves behave somewhat e r r a t i c a l l y between the times of high-high-water and low-high-water. I t i s not c e r t a i n whether t h i s behaviour represents the true discharge response of the estuary, or i s due to inaccuracies i n the cubature discharge c a l c u l a t i o n s . The t i d a l l y averaged discharges were found to s a t i s f y c o n t i n u i t y at the Main Arm - North Arm junction, the North Arm - Middle Arm junction and the Main Arm - Canoe Pass j u n c t i o n . A t i d a l l y averaged mass balance was made over the e n t i r e estuary and was found s a t i s f a c t o r y . The r e s u l t s of t h i s mass balance are shown i n Table 4.1. To sum up, the behaviour of an estuary i s determined by the i n t e r - a c t i o n of the e f f e c t s of r i v e r flow, t i d e s , bed forms and c r o s s - s e c t i o n a l geo- metry. The freshwater discharge of the Fraser River Estuary undergoes a large annual f l u c t u a t i o n (see Appendix B) and consequently the bed forms of the var- ious channels, 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 geometries, are i n a c o n t i n - u a l state of dynamic equilibrium with the freshwater flows and t i d e s . Bearing t h i s i n mind, and the f a c t that Baines obtained 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 walls, e t c . have been constructed i n the i n t e r i m , i t i s impossible f o r the simple one-dimensional hydrodynamic model to exactly reproduce the behavior he observed i n the estuary. The 64 TABLE 4.1 TIDALLY AVERAGED MASS BALANCE OF PREDICTED DISCHARGES FOR JANUARY 24, 1952 CHANNEL PREDICTED VALUES DISCHARGE IN (1000*s cfs) MAIN STEM 36.5 DISCHARGE OUT (1000*s cfs) MAIN ARM NORTH ARM MIDDLE ARM CANOE PASS 27.8 2.9 0.5 1.2 TOTAL 32.4 INTERNAL STORAGE (1000's cfs) PITT SYSTEM O T H E R 2.9 M).0 TOTAL 2.9 r e s u l t s shown i n Figures 4.4 to 4.11 represent the "best o v e r - a l l f i t " between predicted and observed water surface elevations and discharges. As such, the predicted r e s u l t s are close to the l i m i t of r e s o l u t i o n of the one-dimensional hydrodynamic model, or i n other words, the predicted r e s u l t s are the "best" the model i s capable o f . 65 4.2 THE TIDALLY VARYING MASS TRANSPORT MODEL Because of the lack of f i e l d data (see Section 3.2.1), a thorough v e r i f i c a t i o n of the t i d a l l y varying mass transport'-, model was not pos s i b l e . However, the model i s computationally stable, free from the e f f e c t s of numerical dispersion and s u f f i c i e n t l y f l e x i b l e to be adjusted to f i e l d data when a v a i l a b l e . I t reproduces the advective transport process c o r r e c t l y and demonstrates good agreement with the standard a n a l y t i c a l r e s u l t f o r the dispe r s i o n of a slug load. To obtain preliminary notions of the t i d a l l y varying behaviour of the estuary, other peoples' r e s u l t s have been used to estimate 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 dispersion. 4.2.1 Advective Transport. In the t i d a l l y varying model, the advective transport was simulated by a g r i d of moving points on the advec- t i v e c h a r a c t e r i s t i c s , as described i n Chapter 3. This process d i r e c t l y simu- l a t e s the advective transport occurring i n the actual estuary and does not suf f e r from numerical dispersion (see Appendix C). The hydrodynamic model was used to p r e d i c t the v e l o c i t i e s throughout the estuary at half-hour i n t e r - v a l s and these v e l o c i t i e s were then used to advect the moving points along. Thus, the t i d a l l y varying mass transport:: model w i l l simulate the advective transport as accurately as the hydrodynamic model simulatesv the v e l o c i t i e s , and the v e r i f i c a t i o n of the hydrodynamic mode] can also be regarded as a p a r t i a l v e r i f i c a t i o n of the advective transport process. The predicted ad- vect i v e transport was found to be r e l a t i v e l y i n s e n s i t i v e to inaccuracies as i n the advective v e l o c i t i e s . This i s i l l u s t r a t e d i n Figure 4.12 which shows the advection of a slug load down the Main Arm - Main Stem of two d i f f e r e n t model estuaries. The freshwater discharge and quasi-steady t i d a l conditions are the same for both estuaries, but the higher f r i c t i o n of one estuary January 24, 1952 - Flow at Chilliwack « 36,500cfs 20 h Predicted No.I ( (Marrow Section with high friction) S— Predicted No. 2 ( Broad Section with low friction) AVERAGE VALUES ALONG MAIN ARM - MAIN STEM RESULTS AVERAGE MANNINGS " N " AVERAGE WIDTH ( feet ) AVERAGE DEPTH ( feet ) PREDICTED No. 1 0 . 0 4 2 1 3 50 30.6 PREDICTED No.2 0 .0 30 18 70 22. 9 20 40 60 Hours 80 100 120 Figure 4.12 Advection Of A Slug 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 to the v e l o c i t y f i e l d of the other. The only d i f f e r e n c e between both sets of r e s u l t s i s i n the t i d a l excursion i n the lower reaches of the estuaries, and t h i s i s not that s i g n i f i c a n t . When the t i d a l l y varying model was run with zero dispersion, the d i l u t i o n and advection of e f f l u e n t through the junctions was found to be co r r e c t ( i t being r e c a l l e d that c r o s s - s e c t i o n a l mixing i s assumed to be complete a t the junctions). The discrepancy i n the mass balance? over any t i d a l cycle was found to be l e s s than 6% and was due to the d i s c r e t e spacing between the moving points. 4.2.2 Dispersive Transport. A sl u g load was introduced at Station No. 50 on the Main Stem to v e r i f y the c a p a b i l i t y of the t i d a l l y varying mass transport model to simulate the di s p e r s i o n process. The f r e s h - water discharge at Ch i l l i w a c k was 36,500 cubic feet per second and the d i s - persion c o e f f i c i e n t was set equal to 500 square f e e t per second and assumed constant i n x and t . The a n a l y t i c a l s o l u t i o n f o r the dispersion of a slug load i s given by [Fischer, 1966a] c = ,m exp{- — } Vwvt  4 E t where M i s the mass per u n i t area introduced into the flow. For the given conditions, t h i s reduces to 136 r 3.5X2-, c = " T t e x P { — } where c i s the concentration i n milligrams per l i t r e ; 68 t i s the elapsed time i n hours; and x i s the distance e i t h e r side of the mean value i n s t a t i o n co-ordinates (5,000 foot segments). Figure 4.13 shows the form of the slug inputs f o r the a n a l y t i c and t i d a l l y varying solutions. (The e f f l u e n t discharge i n the t i d a l l y varying model sim- ulates continuous discharges, hence the form of the slug input i n Figure 4.13). The predicted and a n a l y t i c solutions f o r the d i s p e r s i o n of the slug load are shown i n Figure 4.14 and the agreement between both sets of r e s u l t s i s good. The higher peak concentrations of the a n a l y t i c s o l u t i o n are due to the higher i n i t i a l concentrations of the a n a l y t i c slug load. I t i s r e c a l l e d that the di s p e r s i o n process i s assumed to be Gaussian i n i t s e n t i r e t y , and t h i s i s i l l u s t r a t e d by the r e s u l t s of Figure 4.14. The estimation 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 dispersion i s discussed i n d e t a i l i n Appendix F. In obtaining the t i d a l l y varying r e s u l t s of t h i s study, 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 i s assumed to be given by E = (6 + S£)y E = ay U 0 < t < T t > T (4.1) and where = 0.06u (4.2) E/ Yr u * a n < ^ u are the "instantaneous" values of the respective parameters during the t i d a l cycle; t i s the time that has elapsed since a "parcel" of the e f f l u e n t was discharged i n t o the estuary; 2 T = b /e and i s the time scale of l a t e r a l mixing (an edge d i s - charge i s assumed); 69 Discharge at S ta t ion No. 50 Q = 3 6 , 5 0 0 c f s (Chi l l iwack) T ida l Range at Steveston = 11 feet J \ Jan.24,1952 Mass = 1650x10 lbs. 150 Slug load of Analytic Soluti on Slug load of Predicted Solution 21 22 Time ( hours ) Figure 4.13 Slug Inputs for A n a l y t i c and Predicted Dispersion Solutions Q = 36 ,500 cfs (Chilliwack) ) j a n 24 1952 Tidal Range at Steveston = II feet J ' 3 HOURS AFTER INJECTION Predicted + Analytic 136 . . f-3.47x" C = exp.{ — } t in hours x in station coordinates J L 50 Stations 55 40 r - 20 0 35 6 HOURS AFTER INJECTION 45 Stations 50 18 HOURS AFTER INJECTION 45 Stations Figure 4.14 Dispersion Of A Slug Load i n The Main Stem 71 and a i s a measure of 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 on the d i s p e r s i o n process and i s tabulated i n Tables F . l . Equations (4.1) allow f o r the increasing contribution of the l a t e r a l v e l o - c i t y gradients on the dispersion process as the e f f l u e n t spreads across the section and the v a r i a t i o n of the c o e f f i c i e n t during the t i d a l c y c l e . These aspects are discussed i n Appendix F and i t i s seen tha t that the peak t i d a l l y varying concentration i s quite s e n s i t i v e to assumptions about the form and magnitude of d i s p e r s i o n c o e f f i c i e n t . 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 of Equations (4.1) are thought to be a reasonable approximation of what a c t u a l l y occurs i n the estuary. Because the t i d a l l y varying mass transport equation hastbeen solved along the advective c h a r a c t e r i s t i c s , the p o s i t i o n of each e f f l u e n t "parcel" and the time that i t has spent i n the estuary i s known. (This i n - formation i s "masked" i n a f i x e d g r i d s o l u t i o n ) . Consequently, the t i d a l l y varying mass transport model can account f o r time dependent behaviour of i n - d i v i d u a l "parcels" of e f f l u e n t , as i s assumed i n Equations (4.1). 4.3 THE TIDALLY AVERAGED MASS TRANSPORT MODEL The t i d a l l y averaged dispersion c o e f f i c i e n t includes the e f f e c t s of advective transport due to t i d a l flow r e v e r s a l and l o n g i t u d i n a l dispersion. As such, i t has no r e a l p h y s i c a l meaning and the v e r i f i c a t i o n of a t i d a l l y averaged mass transport model e s s e n t i a l l y consists of f o r c i n g the model to f i t f i e l d data. According to Holley et al. [1970] there i s no way of making an a -priori estimate of t h i s c o e f f i c i e n t . Ward, and Espey [1971] suggest using the r e s u l t s of a t i d a l l y varying model to estimate the c o e f f i - 72 c i e n t . In the absence of f i e l d data, t h i s l a t t e r approach has been adopted to obtain estimates of the t i d a l l y averaged di s p e r s i o n c o e f f i c i e n t s f o r the Fraser River Estuary. Thomann [1971] l i s t s values 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 f o r various estuaries. They range from 1 - 2 0 square miles per day with a mean value of around 10 square miles per day. Figure 4.15 shows the influence of the t i d a l l y averaged dispersion c o e f f i c i e n t on the predicted concentrations f o r a steady, continuous discharge of a conservative e f f l u e n t at Station No. 40 on the Main Stem of the Estuary. (The concentrations are standardized by d i v i d i n g by the t i d a l l y averaged concentration based on the t o t a l mass of e f f l u e n t discharged during the t i d a l cycles and the freshwater flow at Chil l i w a c k . This i s p l o t t e d as the y , the subscript s i g n i f y i n g t a that the r e s u l t s have been obtained from the t i d a l l y averaged model). The only s i g n i f i c a n t d i f f e r e n c e between the r e s u l t s i s i n the upstream excursion. The concentration gradients are small downstream of the e f f l u e n t discharge, and the di s p e r s i o n has l i t t l e e f f e c t . I t i s r e c a l l e d that the r e s u l t s of Figure 4.15 have been obtained f o r a conservative e f f l u e n t . I f the concen- t r a t i o n of the e f f l u e n t decays with time, as with BOD, there w i l l be concentra- t i o n gradients downstream of the discharge point, and the downstream concentra- tions w i l l also depend on the values of the t i d a l l y averaged dispersion c o e f f i - c i e n t (although the downstream concentrations^are not very s e n s i t i v e to the values of the disp e r s i o n c o e f f i c i e n t ) . The decrease i n concentrations around Stations Nos. 18 and 24 i s due to the e f f e c t s of the junctions and introduces an error of about 10% i n t o the predicted concentrations. I t i s not known why t h i s e f f e c t occurs. Q = 3 6 , 5 0 0 c f s ( C h i l l i w a c k ) ) J a n . 2 4 > | 9 5 2 T i d a l R a n g e at S teveston =11 fee t J l.2r- 1.0 - 0.8 - 0.4 - 0.2 0 E = lOsquare miles per day 10 20 30 Stat ions along Main Arm 4 0 50 Main Stem Figure 4.15 Influence Of The T i d a l l y Averaged Dispersion C o e f f i c i e n t Oh Predicted T i d a l l y Averaged Concentrations 74 Figures 4.16 and 4.17 show the maximum upstream excursion on the f l o o d flow for e f f l u e n t released along the various channels. In the lower reaches of the Main Arm i t i s assumed that 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 i s equal to 20 square miles per day, and t h i s value i s reduced along the Main Arm - Main Stem according to the r e s u l t s of Figure 4.16. Along the North Arm the t i d a l l y averaged dispersion i s assumed to equal 10 square miles per day, and along P i t t River i t i s assumed to equal 30 square miles per day (the rapid decrease i n the upstream excursion along the P i t t River, as shown i n Figure 4.17, i s due to the d r a s t i c decrease i n advective v e l o c i t y i n P i t t Lake). Although the absolute values of the t i d a l l y averaged d i s p e r s i o n may not be correct, and t h i s i s discussed i n Chapter 5, the r e l a t i v e values along the estuary should be reasonable. Maximum Upstream Excursion (Stations) Q - 36 ,500c f s (Chilliwack) ) J a n . 2 4 i l 9 5 2 Tidal Range at Steveston = 11 feet J Effluent moves into Pitt Lake 01 c o o CO o 2 8 c o « 6 i _ 3 O X LU E 4 t_ to Q. Z> E 2 3 E 0 Effluent moves through N.Arm - M.Arm Junction J L Pitt Lake "00 110 118 140 150 Position of Effluent Discharge along North Arm and Pitt River ( Station Numbers ) Figure 4.17 Maximum Upstream Excursion During Flow Reversal In The North Arm And P i t t River CHAPTER 5 COMPARISON AND DISCUSSION OF RESULTS The three models discussed i n the l a s t two chapters are now used to obtain'a.preliminary i n d i c a t i o n of 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 predicted concentrations i n the Fraser River Estuary. The flow and t i d a l conditions of January 24, 1952 were used i n t h i s i n v e s t i g a t i o n . These are t y p i c a l high t i d e - low flow conditions and were used i n the v e r i f i c a t i o n of the hydrodynamic model. The freshwater discharge at Chilliwack was 36,500 cubic f e e t per second and the t i d a l range at Steveston 11 f e e t . A conservative e f f l u e n t and an edge discharge were assumed. The hydrodynamic model was used to obtain the t i d a l l y varying 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 along the various channels of the estuary. These values, together with estimated dispersion c o e f f i c i e n t s , were then used to p r e d i c t the t i d a l l y varying concentrations. The freshwater discharge at Chilliwack, the t i d a l l y averaged areas and estimated dispersion c o e f f i c i e n t s were used to pre- d i c t the t i d a l l y averaged concentrations. Before presenting the r e s u l t s of t h i s i n v e s t i g a t i o n , the assumptions of the mass transport models are b r i e f l y r e c a l l e d : both models are one- dimensional 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 concentrations; the c r o s s - s e c t i o n a l mixing i n both models i s assumed to be complete at the junc- tions; the influence of the saltwedge i s ignored i n both models; the dispersion process of the t i d a l l y varying model i s assumed to be Gaussian i n i t s e n t i r e t y ; and i n the t i d a l l y averaged model, a l l e f f e c t s of the t i d e s are lumped in t o the t i d a l l y averaged di s p e r s i o n c o e f f i c i e n t . 77 78 5.1 THE EFFECTS OF LATERAL DISPERSION In the i n i t i a l period before 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 concentration w i l l be considerably higher than the value predicted by e i t h e r mass transport model. While t h i s e f f e c t has not been q u a n t i t a t i v e l y assessed, i t i s noted that the t i d a l l y varying mass transport model can e a s i l y be adapted to give an approximate estimate of t h i s e f f e c t . The l a t e r a l concentration p r o f i l e depends on the c o e f f i c i e n t of l a t e r a l dispersion, the 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 i n the estuary cross-section and the period of time that a "parcel" of e f f l u e n t has spent i n the estuary [Ward, 1972]. Because the t i d a l l y varying mass transport model has been solved along i t s advective c h a r a c t e r i s t i c s , the p o s i t i o n of each e f f l u e n t "parcel" and the time that i t has spent i n the estuary i s known, and the predicted concentrations can be adjusted to account f o r the e f f e c t s of l a t e r a l d i s persion. (This approach was used i n allowing for 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 of l o n g i t u d i n a l d i s p e r s i o n ) . In the lower reaches of the estuary, there are three d i s t i n c t periods i n each double t i d a l cycle of 25 hours; a period of strong f l o o d flows; a per- iod of weak ebb and weak f l o o d flows and a period of strong ebb flows. This i s apparent i n Figures 4.4 to 4.10 and the duration of each of the three per- iods i s seen to be approximately eight hours. The c r o s s - s e c t i o n a l mixing i s p r i n c i p a l l y due to the e f f e c t s of secondary flows, as discussed i n Appendix E, and w i l l be greatest during the periods of strong ebb and flo o d flows. If c r o s s - s e c t i o n a l mixing i s not complete at the Main Arm - North Arm junction during the strong ebb flow, or at the Main Stem - P i t t River junction during 79 the strong flo o d flow, the e f f l u e n t w i l l not be advected through these junc- t i o n s according to the simple flow balance of the one-dimensional mass trans- port models. The i n t e r a c t i o n of the time dependent behaviour of the l a t e r a l mixing process with the advection of e f f l u e n t through the junctions i s very involved and can only be r e l i a b l y determined from f i e l d studies. I f s u f f i - c i e n t f i e l d data were a v a i l a b l e , i t may be possible to e m p i r i c a l l y allow f o r t h i s e f f e c t i n the t i d a l l y varying model. 5.2 THE INITIAL DILUTION OF EFFLUENT When e f f l u e n t i s discharged i n t o an estuary, the t i d a l l y varying flows 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 during the t i d a l c y c l e . This process i s described i n Section 1.3 and i s seen to generate concentration spikes a t times of slackwater. Figure 5.1 shows t y p i c a l v a r i a t i o n s i n the i n i t i a l d i l u t i o n during the double t i d a l c ycle at four s t a t i o n s along the estuary. During a double t i d e cycle there are four slackwaters i n the lower reaches of the estuary (see Figures 4.4 to 4.10) and the r e s u l t i n g concentration peaks at Stations Nos. 10, 22 and 102 are apparent. At Station No. 50, the t i d a l flows are s i g n i f i c a n t l y smaller i n r e l a t i o n to the freshwater 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 only two e f f e c t i v e peaks during the double t i d a l c y c l e . Because of the smaller flows along the North Arm, the t i d a l l y varying concentrations at Station No. 102 are s i g n i f i c a n t l y higher than the values along the Main Arm - Main Stem. The t i d a l l y averaged d i l u t i o n i s also shown i n Figure 5.1, the higher values at Stations Nos. 10 and 102 r e f l e c t i n g the d i v i s i o n of the freshwater discharge between the North Arm and Main Arm. Q = 3 6 , 5 0 0 c f s ( C h i l l i w a c k ) \ j a n . 2 4 , 1 9 5 2 8 ° T i d a l Range at S t e v e s t o n = l l f e e t j 4 2 0 4 r c o 0 Station No. 10 12 18 24 (Hours) 12 - Station No. 22 — • — * ~ v 1 ' i ^ r— 18 24(Hours) Station No.50 18 24(Hours) Station No.102 6 12 Figure 5.1 I n i t i a l D i l u t i o n At Station Nos. 10, 22, 50 and 102 24(Hours) 81 The peak t i d a l l y varying concentrations i n Figure 5.1 are seen to be about twice the t i d a l l y averaged values. The minimum t i d a l l y varying concentration occurs during the strong ebb and f l o o d flows, and i s r e f l e c t e d as the f l a t sections of the curves of Figure 5.1. At Stations Nos. 10 and 22, t h i s minimum concentra- t i o n i s approximately h a l f the minimum value at S t a t i o n No. 50. This i s due to the presence of P i t t Lake; discharges downstream from the Main Stem- P i t t River j u n c t i o n are d i l u t e d by the combined e f f e c t s of the t i d a l prisms of the P i t t system and the Main Stem above the j u n c t i o n , whereas discharges upstream of the j u n c t i o n are d i l u t e d only by the l a t t e r . 5.3 PEAK EFFLUENT CONCENTRATIONS As w e l l as 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 , c e r t a i n slugs of water are dosed with e f f l u e n t several times due to the e f f e c t s of t i d a l flow r e v e r s a l . This process i s described i n Section 1.3 and a l s o gen- erates concentration spikes. However, 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 multiple dosing combine to produce a compound spike whose concen- t r a t i o n i s higher than th a t of the i n d i v i d u a l component spikes. This i s w e l l i l l u s t r a t e d by an e f f l u e n t discharge at S t a t i o n No. 50. The v a r i a t i o n of v e l o c i t y at S t a t i o n No. 50 i s shown i n Figure E.2; note the slackwaters at hours 4 and 5 and that the minimal v e l o c i t y at hour 6. This i s the reason f o r the increased concentration at hours 4 and 5 and the spike at hour 6 i n Figure 5.1. Now consider the behaviour of the water mass around S t a t i o n No. 50 during t h i s time. During hour 4 a slug of water moves downstream past the o u t f a l l and i s dosed, during hour 5 i t moves back upstream and i s dosed again 82 and f i n a l l y during hour s i x i t moves downstream past the o u t f a l l i s dosed yet again. The r e s u l t of t h i s i s that the concentration i n t h i s slug of water i s 12 times greater than the t i d a l l y averaged value as i s shown i n Figure F.2 for the curve with zero l o n g i t u d i n a l d i spersion. This spike i s composed of the three subspikes generated at hours 4, 5 and 6, t h e i r i n d i v i - dual magnitudes being 3, 3 and 6 as shown i n Figure 5.1. The spikes due to 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 multiple ?dosing combine because both e f f e c t s occur around the same phase of the t i d e . 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 of the double t i d a l cycle i s the period of weak ebb and flood flows where there are three slackwaters i n the space of eight hours. The v e l o c i t i e s between these times of slackwater are r e l a t i v e l y low (see Figures 4.4 to 4.10). Once a spike i s generated 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 d i l u t i o n , multiple dosing or a combination of both, i t s peak concentration begins to be reduced by the l o n g i t u d i n a l d i s p e r s i o n process. This i s i l l u s - t r a ted i n Figure 5.2 f o r a steady e f f l u e n t discharge at Station No. 50. The compound spike i s generated a t hour 6 with a peak concentration that i s 11 times greater than the t i d a l l y averaged value. (The decrease from the pre- vious value of 12 i s due to the e f f e c t s of d i s p e r s i o n on the subspikes gen- erated at hours 4 and 5 ) . However, as the spike i s advected down the estuary, the d i s p e r s i o n process reduces the peak concentration as shown. The r e s u l t s of Figure 5.2 seem to indicate that mass i s not conserved since the area under the spike i s not constant. This i s only an apparent e f f e c t due to extrapolating the concentrations o f f the advective c h a r a c t e r i s t i c s onto the standard 5,000 foot space g r i d , as described i n Section 3.2. In actual f a c t , the i n i t i a l base width of the spike i s only 500-800 f e e t , which i s too f i n e to be resolved 12 Q = 36,500cfs at Chilliwack ^ , o y l l f t C . 0 J ^ . I Jan. 24,1952 Tidal Range at Steveston II J 10 8 Vtv 6 o 1 — Stations along Main Stem Figure 5.2 Dispersion Of A Concentration Spike 1 hour 03 CO 84 by the f i x e d g r i d . (Note that the spike i s c o r r e c t l y resolved on the char- a c t e r i s t i c s ) . The most s i g n i f i c a n t 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 multiple dosing occur i n the f i r s t 1-2 t i d a l cycles a f t e r a " p a r c e l " of e f f l u e n t has been discharged i n t o the estuary. This i s due to the high i n i t i a l concentration gradients of the spike and assumed time-dependent increase i n 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 dispersion (see Appendix F ) . These aspects are apparent from Figure 5.2 and are also i l l u s t r a t e d i n Appendix F. For the steady discharge of a conservative e f f l u e n t , the peak t i d a l l y varying concentration was found to be from 1 to 10 times greater than the value predicted by the t i d a l l y averaged model. These r e s u l t s are summarized i n Table 5.1" and were obtained f o r a s i n g l e e f f l u e n t discharge at the designated s t a t i o n . (The peak t i d a l l y varying discharge occurs a t the discharge p o i n t ) . The r e s u l t s of Table 5.1 are s e n s i t i v e to 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 of l o n g i t u d i n a l d i s - persion, as discussed and i l l u s t r a t e d i n Appendix F. In obtaining the r e s u l t s of Table F . l , 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 was assumed to be as given i n Equation (4.1). The peak c r o s s - s e c t i o n a l concentration w i l l be s i g n i f i c a n t l y higher than these values. 5.4 UPSTREAM EFFLUENT TRANSPORT In a t i d a l l y varying mass transport model, e f f l u e n t i s transported upstream by the e f f e c t s of d i s p e r s i o n and advection during t i d a l flow r e v e r s a l . Because of the large t i d a l flows, advective transport i s the more important CHANNEL Main Arm - Main Stem North Arm Pitt River STATION 8 14 22 3 0 4 0 50 6 0 106 116 142 150 ( Ttv/ *Xa) max. 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 Ratio of Peak T i d a l l y Varying Concentration to T i d a l l y Averaged Value At Point Of E f f l u e n t Discharge 86 component f o r the Fraser River Estuary, as i s apparent from Figures 4.16 and 4.17. In a t i d a l l y averaged model, upstream transport i s by the t i d a l l y averaged d i s p e r s i o n process. Figures 5.3 and 5.4 show the predicted t i d a l l y varying and t i d a l l y averaged concentrations at Stations upstream from e f f l u e n t discharges. In a l l cases, the t i d a l l y averaged value i s s i g n i f i c a n t l y l e s s than the t i d a l l y varying values. A better f i t between the r e s u l t s could be obtained by increasing the values of the t i d a l l y averagedMispersion c o e f f i - c i e n t . The r e s u l t s of Figure 5.4 are f o r simultaneous e f f l u e n t discharges at Stations Nos. 10 and 14. Once again, 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 multiple dosing cause a concentration spike at both discharge points, spike A being generated at Station No. 10 and spike B at Station No. 14. Nbte that between hours 6 and 17 the t i d a l l y varying concentration at Station No. 6 i s zero due to uncontaminated seawater moving upstream on the f l o o d t i d e . The concentration boundary condition at the sea was assumed to be zero, a more r e a l i s t i c boundary condition being p o s s i b l y as shown. The t i d a l l y averaged concentration i s not a good i n d i c a t i o n of the t i d a l l y varying response upstream of Station No. 14, between the two s t a t i o n s or downstream of Station No. 10. 5.5 CHANNEL INTERACTIONS Consider now the e f f e c t of an e f f l u e n t discharge i n one channel on the water q u a l i t y i n another channel. The s i g n i f i c a n c e of t h i s e f f e c t w i l l depend on the proximity of the e f f l u e n t discharge to the junction. I f the discharge point i s s u f f i c i e n t l y f ar upstream from a junction, the e f f l u e n t Discharge at Stat. No. 14 ( Ma in Arm) 6 12 18 24(Hours) Discharge at Stat. No.22 ( Main Arm) 6 12 18 24(Hours) Figure 5.3 Predicted Concentrations i n One Channel Caused By E f f l u e n t Discharge In Another Channel Q = 36 ,500c f s (Chilliwack) Tidal Range at Steveston = Ufeet j Jan.24,1952 Station No.17 2 r 24 (Hours) 0 8 r Station No.12 ' 1^ spi ke B - /*» 1 1 1 1 0 6 12 Station No.6 18 24( Hours) More Realistic Boundary Condition -Assumed Boundary Condition . spike A spike B 12 18 Figure 5.4 24( Hours) Predicted Concentrations Due to Two E f f l u e n t Discharges In The Main Stem 8 9 w i l l be dispersed both l a t e r a l l y and l o n g i t u d i n a l l y by the time i t reaches the junction. Under these conditions, the t i d a l l y averaged concentration i n both channels downstream of the junction w i l l be a good measure o f the t i d a l l y varying values. Of more i n t e r e s t i s the case where the e f f l u e n t discharge i s downstream from a j u n c t i o n . This i s i l l u s t r a t e d i n Figure 5.3 f o r i n t e r a c t i o n s between the Main Arm and North Arm and between the Main Stem and P i t t River. Consider the e f f l u e n t discharged at Station No. 22: at hour 8 a spike i s advected up P i t t River past Station No. 144 on the strong f l o o d flow and returns past S t a t i o n No. 144 a t hour 23 on the strong ebb t i d e , i t s concentration being reduced by the e f f e c t s of d i s p e r s i o n dur- ing i t s residence i n P i t t River. For the two cases of Figure 5.3, upstream advection causes a s l u g of contaminated water to be fed i n t o the other channel during each double t i d e cycle of 25 hours. The zero concentration i n the North Arm at hour 22 marks the end of one slug and the beginning of another. In both cases, the t i d a l l y averaged concentration 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 varying values, although as remarked i n a previous section, the f i t can be improved by 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 c o e f f i c i e n t . 5.6 SUMMARY In t h i s chapter the r e s u l t s from the t i d a l l y varying and t i d a l l y averaged mass transport models have been compared to obtain a preliminary i n d i c a t i o n of 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 the predicted concentra- t i o n s i n the Fraser River Estuary. 90 The t i d a l l y varying flows 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 of a discharged e f f l u e n t . This, and the e f f e c t s of t i d a l flow r e v e r s a l , independently generate concentration spikes. Because both e f f e c t s occur around the same phase of the t i d e , the period of weak flood and ebb flows i n each double t i d a l cycle being e s p e c i a l l y s e n s i t i v e , the i n d i v i d u a l spikes combine-to generate a spike whose concentration was estimated to be from 1 to 10 times greater than the predicted t i d a l l y averaged concentration. Because of incomplete l a t e r a l mixing, the peak c r o s s - s e c t i o n a l concentration w i l l be considerably higher than these values. The e f f e c t s of concentration spikes are most s i g n i f i c a n t i n the f i r s t 1-2 t i d a l cycles a f t e r t h e i r gener- at i o n . After t h i s period of time, the l o n g i t u d i n a l dispersion i s due to the e f f e c t s of l a t e r a l concentration gradients and the concentration of the spike i s r a p i d l y reduced. The movement of water through the Fraser River Estuary i s a com- plex process that i s influenced by the large t i d a l e f f e c t s , the s i g n i f i c a n t freshwater discharge (even at low flows), the various channels and junctions of the estuary and the presence of P i t t Lake. I f an e f f l u e n t discharge i s s u f f i c i e n t l y f a r upstream from the junctions, the spikes w i l l be f l a t t e n e d and the e f f l u e n t dispersed over the cross-section by the time i t i s advected to the junction. Under these conditions, the predicted t i d a l l y averaged be- haviour at and downstream of the junction i s a good estimate of the t i d a l l y varying behaviour, and both should be a reasonable approximation to the actual behaviour i n the estuary. E f f l u e n t discharges i n the Main Stem above Station No. 45 are expected to behave i n t h i s manner. Throughout the lower reaches of the estuary,,;including P i t t River, the predicted t i d a l l y averaged behaviour 91 i s not a good i n d i c a t i o n of the t i d a l l y varying behaviour. Because of the e f f e c t s of incomplete l a t e r a l mixing and junctions, as discussed i n Section 5.1, the actual response of the estuary may be somewhat d i f f e r e n t from the predicted t i d a l l y varying behaviour. By adjusting the t i d a l l y averaged dispersion 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 " between the predicted t i d a l l y varying and t i d a l l y averaged concentrations. However, since the t i d a l l y averaged d i s - persion 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, there i s probably no unique set of values f o r the estuary. Rather, the best f i t dispersion co- e f f i c i e n t s w i l l vary with the p o s i t i o n and number o f e f f l u e n t discharges. CHAPTER 6 SUMMARY AND CONCLUSIONS Mass transport models are commonly used to investigate s i t u a t i o n s of e x i s t i n g or 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 estuaries. Such models are used to p r e d i c t the concentration of the offending substance through- out the estuary and are e s s e n t i a l l y of two types: those that c o r r e c t l y allow f o r t i d a l e f f e c t s and those that do not. While t i d a l l y varying models c o r r e c t l y allow f o r the e f f e c t s of the t i d e s , t h e i r development and a p p l i c a t i o n involves s i g n i f i c a n t l y more work than f o r t h e i r t i d a l l y averaged counterparts. In t i d a l l y averaged models, a l l t i d a l e f f e c t s are lumped into 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 , and while such models are r e l a t i v e l y easy to develop and apply, they give no i n d i c a t i o n of 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 the predicted concentrations. In t h i s study, both a t i d a l l y averaged and a t i d a l l y varying mass transport model have been developed and applied to the Fraser River Estuary. The p r i n c i p a l object of t h i s study was to assess the a b i l i t y of the t i d a l l y averaged model to describe the t i d a l l y varying concentrations. This was investigated by comparing the predicted concentrations from both models f o r assumed e f f l u e n t discharges. The t i d a l l y averaged model used i n t h i s study i s due to Thomann [1963]. The t i d a l l y varying model was developed from f i r s t p r i n c i p l e s , and during i t s development i t was necessary t o consider the problems of numerical dispersion, s t a b i l i t y , the s i g n i f i c a n c e of l a t e r a l 92 93 dispersion and the time dependent behaviour of 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 during the i n i t i a l p eriod before c r o s s - s e c t i o n a l mixing i s complete. The p r i n c i p a l conclusions to emerge from t h i s study are: 1. Numerical d i s p e r s i o n can be eliminated from the f i n i t e d i f f e r e n c e solution of mass transport equations by solving the equations along the advective c h a r a c t e r i s - t i c s . • 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 ifference schemes have been shown to be r e l a t e d to the speed of information propagation. Advective i n s t a b i l i - t i e s are eliminated by solving the mass transport equa- t i o n along the advective c h a r a c t e r i s t i c s . 3. E x i s t i n g theories have been shown to apparent- l y underestimate the s i g n i f i c a n c e of secondary currents on the l a t e r a l mixing process. Secondary currents have been explained i n terms of the generation and advection of vor- t i c i t y , and estimated values show good agreement with l i m i t e d f i e l d data. Revised estimates of l a t e r a l d isper- sion indicate 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. If the mass transport equation i s solved along the advective c h a r a c t e r i s t i c s , the time dependent behaviour of 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 during the i n i t i a l period before c r o s s - s e c t i o n a l mixing i s complete can be taken into account. Also, a one-dimensional equa- t i o n can be used to estimate l a t e r a l concentration p r o f i l e s . 5. The e f f e c t of the t i d e has been shown to i n t r o - duce "spikes" into the concentration p r o f i l e along the estu- ary. These spikes 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 of a discharged e f f l u e n t and multiple dosing 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 resources are i n - volved i n the development and a p p l i c a t i o n of a t i d a l l y varying mass transport model than f o r a t i d a l l y averaged model. The time required f o r development, the amount of f i e l d data required f o r v e r i f i c a t i o n and the amount of computer time required to analyze an i d e n t i c a l s i t u a t i o n i s estimated to be an order of magnitude greater f o r the t i d a l l y varying model as compared to the 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 no t 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 - i n g r e s p o n s e o f t h e F r a s e r R i v e r E s t u a r y . The peak c r o s s - s e c t i o n a l l y a v eraged c o n c e n t r a t i o n 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 model was found t o be fr o m one 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 . I n t h i s s t u d y two mass t r a n s p o r t models and a hydrodynamic model were 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 u s e d t o p r e d i c t t h e t e m p o r a l v a r i a t i o n i n t h e p a r a m e t e r s 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. F i n i t e d i f f e r e n c e methods have been u s e d 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. Impor- t a n t a s p e c t s o f t h i s t y p e o f s o l u t i o n a r e t h e p r o b l e m s o f n u m e r i c a l d i s - p e r s i o n and s t a b i l i t y . There has been 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 and 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 . I n t h i s s t u d y , n u m e r i c a l d i s p e r s i o n has been shown t o r e s u l t f r o m 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 o v e r a f i x e d space g r i d r a t h e r t h a n a l o n g t h e more f u n d - amental a d v e c t i v e c h a r a c t e r i s t i c s ; i f t h e e q u a t i o n i s s o l v e d a l o n g t h e ad- v e c t i v e c h a r a c t e r i s t i c s , n u m e r i c a l d i s p e r s i o n i s e l i m i n a t e d . 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 and has no n u m e r i c a l d i s p e r s i o n because i t i s i n d e p e n d e n t o f t i m e . I n a d d i t i o n t o t h e p r o b l e m o f n u m e r i c a l d i s p e r - s i o n , t h e r e i s a l s o t h e p r o b l e m o f s t a b i l i t y i 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 schemes. I n Appendix C t h i s i s seen t o be r e l a t e d t o t h 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 a l o n g t h e c h a r a c t e r i s t i c s o f t h e r e s p e c t i v e 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 . S t a b i l i t y r e q u i r e m e n t s govern* t h e r e l a t i v e s i z e o f t h e space and t i m e i n c r e m e n t s i n t h e hydrodynamic e q u a t i o n s 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 . T h e r e i s no s t a b i l i t y r e q u i r e m e n t f o r Thomann's s o l u t i o n as i t i s i n d e p e n d e n t o f t i m e . However, 95 there i s a l i m i t on the r e l a t i v e magnitudes of the t i d a l l y averaged advec- t i v e and dispersive transport processes, and i f v i o l a t e d , the concentration i n the offending segment w i l l become negative. There are a number of advantages to solving the t i d a l l y varying mass transport equation along the advective c h a r a c t e r i s t i c s ; i t r e s u l t s i n a d i r e c t simulation of the advective transport occurring i n the estuary, a u s e f u l separation of the advective and di s p e r s i v e transport processes i s achieved, and the p o s i t i o n of each separate " e f f l u e n t p a r c e l " and the time that i t has spent i n the estuary i s known. This l a s t piece of i n f o r - mation has been used to account f o r an assumed > time-dependent increase i n 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 dispersion i n the i n i t i a l period before c r o s s - s e c t i o n a l mixing i s complete, and also allows the t i d a l l y varying mass transport model to be adapted to p r e d i c t l a t e r a l concentration p r o f i l e s . A most u s e f u l feature of solving the equation along thej-advective charac- t e r i s t i c s i s that a d d i t i o n a l moving points can be placed on the character- i s t i c s to more accurately define regions of rapid v a r i a t i o n i n concentra- t i o n , such as occur at e f f l u e n t o u t f a l l s at times of slackwater, and un- necessary moving points can be removed from regions of slow v a r i a t i o n . The concentration spikes generated 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 d i l u - t i o n and multiple dosing are i n i t i a l l y very sharp and only some 500 - 800 feet wide at the base. To achieve adequate r e s o l u t i o n of such a spike with a fi x e d g r i d s o l u t i o n would require a space g r i d approximately 10 times f i n e r than the standard 5,000 foot space g r i d used i n t h i s study. The advantages of sol v i n g the mass transport equation along the advective c h a r a c t e r i s t i c s must be balanced against the somewhat untidy "book- keeping"of s o l u t i o n .results inherent to c h a r a c t e r i s t i c s methods of s o l u t i o n . 95a In the t i d a l l y varying model of the Fraser River Estuary, t h i s book-keeping i s complicated by the muIti-channeled nature of the estuary and the i n t e r - actions of these channels at t h e i r junctions. But while awkward, the book- keeping of r e s u l t s was not excessively d i f f i c u l t . Of the three models used i n t h i s study, only the hydrodynamic model has been v e r i f i e d with any degree of thoroughness. Because 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 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 forced to reproduce measured f i e l d r e s u l t s rather than being r i g o r o u s l y v e r i f i e d . Lack of f i e l d data prevented t h i s , and a l s o pre- vented the complete v e r i f i c a t i o n of the t i d a l l y varying mass transport model. However, a l l three models are s u f f i c i e n t l y f l e x i b l e to simulate the range of flow and t i d a l conditions of the Fraser River Estuary and can be adjusted to f i t 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 , the e f f e c t of the t i d e s i s t o cause spikes i n the concentration p r o f i l e along the estuary. In a one-dimensional model, the t i d a l l y varying flows 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 of a d i s - charged e f f l u e n t , the concentration being greatest at times of slackwater. This generates spikes i n the concentration p r o f i l e along the estuary. The e f f e c t s of t i d a l flow r e v e r s a l r e s u l t i n c e r t a i n slugs of water being dosed with e f f l u e n t several times, and t h i s a lso generates concentration spikes. Because of the asymmetric nature of the t i d e s there i s a p e r i o d of weak flood and ebb flows once i n each double t i d a l c y c l e . During t h i s time there are three slackwaters and the v e l o c i t i e s are low, and 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 dosing i n t e r a c t to form a compound spike 96 whose concentration i s s i g n i f i c a n t l y greater than that of the component spikes. A f t e r a spike has been generated, i t s concentration i s reduced by the e f f e c t s of 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 "parcel of e f f l u e n t " has been discharged into the estuary, i s due p r i n c i p a l l y to 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 gradients. However, when the e f f l u e n t i s mixed over the cr 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 gradients 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 . To .account f o r t h i s e f f e c t , i t was assumed that 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 - persion increased between these two extremes as described i n Appendix F. Also, 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 was assumed to vary d i r e c t l y as the absolute v e l o c i t y . While s i m p l i s t i c , these v a r i a t i o n s are thought to be a reasonable approximation of what occurs i n the estuary. The peak t i d a l l y varying concentration at the point of e f f l u e n t discharge was estimated to be from one to 10 times: greater than the value predicted from the t i d a l l y averaged model. Because the e f f l u e n t i s not uniformly d i s t r i b u t e d over the cro s s - s e c t i o n , the peak l a t e r a l concentration w i l l be much greater than these values. A f t e r a spike has been i n the estuary f o r several t i d a l c y c l es, i t s concentration has been g r e a t l y reduced by the l o n g i t u d i n a l d i s p e r s i o n process. In the estuary with r e l a t i v e l y short residence times such as the Fraser, l a t e r a l mixing w i l l be of considerable importance. For example, i n the lower reaches of the estuary, the residence time i s only 2-4 t i d e cycles, and e f f l u e n t may not be completely mixed over the cros s - s e c t i o n even when i t leaves the estuary. In the i n i t i a l p eriod before c r o s s - s e c t i o n a l mixing i s complete,the peak l a t e r a l concentration 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 value predicted by the t i d a l l y varying model. However, the model can be adapted to p r e d i c t l a t e r a l concentration p r o f i l e s as was discussed p r e v i o u s l y . I f c r o s s - s e c t i o n a l mixing i s not com- ple t e at a diverging junction, the e f f l u e n t may not be advected through the junction according to the simple flow balance of the mass transport models. To account f o r t h i s e f f e c t i n a model would require considerable f i e l d data. Because of the influence o f secondary currents, c r o s s - s e c t i o n a l mixing i s thought t o be r e l a t i v e l y r a p i d i n the Fraser River Estuary. Secondary flows have been t e n t a t i v e l y explained i n terms of the generation and advection o f v o r t i c i t y , and on the basis of l i m i t e d f l o a t studies, show good agreement with values measured i n the estuary. To sum up, the movement of water through the Fraser River Estuary i s a complex phenomenon that i s a f f e c t e d by the t i d e s , the freshwater d i s - charge, the various channels and junctions of the estuary and the presence of P i t t Lake. The advective and d i s p e r s i v e transport processes t h a t d i s t r i b u t e an e f f l u e n t throughout the- water mass of the estuary are s i m i l a r l y complex, and i n p a r t i c u l a r are s i g n i f i c a n t l y a f f e c t e d by the t i d e s . A t i d a l l y aver- aged model, while simpler to develop and apply to the estuary, does not give a good i n d i c a t i o n o f the t i d a l l y varying concentrations. This study does not pretend to solve the t o t a l problem of c a l c u l a t i n g e f f l u e n t concentrations i n an estuary as complex as the Fraser. Rather, i t has concentrated on deve- loping stable mathematical models free from numerical d i s p e r s i o n to allow a preliminary i n v e s t i g a t i o n of 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 . In addi- t i o n an assessment i s made of the importance of l a t e r a l d i s p e r s i o n , and i t i s seen that with simple modifications, the t i d a l l y varying model can at l e a s t p a r t i a l l y account for the e f f e c t s of l a t e r a l d i s p e r s i o n . R E F E R E N C E S Baines, W. D. [1952]. Water Surface Elevations and Tidal Discharges in the Fraser River Estuary, January 23 and 24, 1952, Report No. MH-32, National Research Council of Canada. Baines, w. D. [1953]. Survey of Tidal Effects on Flow in the Fraser River Estuary, June 10 and 11, 1952, Report No. MH-40, National Research Council of Canada. B e l l a , D. A. [1968]. "Solution of Estuary Problems and Network Programs," Discussion, Proc. A.S.C.E., J. Sanit. Eng. Div., 94 (SA1):180-181. B e l l a , D. A. and Greriney [1970]. " F i n i t e - D i f f e r e n c e Convection E r r o r s , " Proc. A.S.C.E., J. Sanit. Eng. Div., 96(SA6): 1361-1375. B e l l a , D. A. and Dobbins, W. E. [1968]. "Difference Modelling o f Stream P o l l u t i o n , " Proc. A.S.C.E., J. Sanit. Eng. Div., 94'(SA5): 995-1016. Benedict, A. H., H a l l , K. J . and Koch, F. A. [1973]. A Preliminary Water Quality Survey of the Lower Fraser River System, Technical Report No. 2, Westwater Research Center, U n i v e r s i t y of B r i t i s h Columbia. Bennet, J . P. [1971]. "Convolution Approach to the Solution f or the Dissolved Oxygen Balance i n a Stream," Water Resources Res., 7(3): 580-590. Bird, R. B., Stewart, w. E., and Lightfoot, E. N. [I960]. Transport Pheno- mena, (New York: John Wiley & Sons, 1960). Callaway, R. J . [1971]. "Application of Some Numerical Models to P a c i f i c Northwest Estuaries," Proceedings, 1971 Technical Conference on Estuaries of the Pacific Northwest, C i r c u l a r No. 42, Engineering Experiment Station, Oregon State Un i v e r s i t y , C o r v a l l i s . Chow, V. T. [1959]. Open Channel Hydraulics, (New York: McGraw-Hill, 1959). Chow, v. T. [1964]. Handbook of Applied Hydrology, (Ed.) V. T. chow. (New York: McGraw-Hill, 1964), Section 8. 98 99 Courant, R. and H i l b e r t , D. [1962].', Methods of Mathematical Physics (2 vols; New York: Interscience, 1962), V o l . I I . Custer, S. W. and Krutchkoff, R. G. [1969] . "Stochastic Model for BOD and DO i n Estuaries," Proc. A.S.C.E., J. Sanit. Eng. Div., 95(SA5): 865-885. Diachishin, A. N. [1963], "Waste Disposal i n T i d a l Waters," Proc. A.S.C.E., J. Sanit. Eng. Div.3 89(SA4): 23-43. Di.Torb, D. M. and O'Connor, D. J . [1968]. "The D i s t r i b u t i o n of Dissolved Oxygen i n a Stream with Time Varying V e l o c i t y , " Water Resources Res., 4(3): 639-646. Di Toro, D. M. [1969]. "Stream Equations and The Method of C h a r a c t e r i s t i c s , " Proc. A.S.C.E., J. Sanit. Eng. Div., 95(SA4): 699-703. Dornhelm, R. B. and Wodlhiser, D. A. [1968]. " D i g i t a l Simulation of Estuarine Water Quality," Water Resources Res., 4(6): 1317-1328. Dronkers, J . J . [1964]. Tidal Computations in Rivers and Coastal Waters, (Amsterdam: North-Holland Publishing Co., 1964). Dronkers, J. J . [1969]. " T i d a l Computations f o r Rivers, Coastal Areas and Seas," Proc. A.S.C.E., J. Hyd. Div., 95(HY1): 29-77. E i n s t e i n , H. A. [1972]. Sedimentation, edited and published by Hsieh Wen Shen, Department of C i v i l Engineering, Colorado State University, Colorado. Elder, J . W. [1959]. "The Dispersion of Marked F l u i d i n Turbulent Shear Flow," J. Fluid Mech., 5: 544-560. Fischer, H. B. [1966a]. "A Note on the One-Dimensional Dispersion Model," Air Wat. Poll. Int. J., 10(6/7): 443-452. Fischer, H. B. [1966b]. Longitudinal Dispersion in Laboratory and Natural Streams, Report No. KH-R-12, D i v i s i o n of Engineering and Applied Science, C a l i f o r n i a I n s t i t u t e of Technology, Pasadena, C a l i f o r n i a . 100 Fischer, H. B. [1967]. "The Mechanics of Dispersion i n Natural Streams," Proceedings: ASCE Environmental Engineering Conference, Dallas, Texas, February 6-9, 1967. Fischer, H. B. [1969a], "The E f f e c t s of Bends on Dispersion i n Streams," Water Resources Res.t 5(2): 496-506. Fischer, H. B. [1969b]. "Cross-Sectional Time Scales and Dispersion i n Estua r i e s , " Paper C. 19, Proceedings Volume 3 (Subject C), International Association for Hydraulic Research,, Thirteenth Congress, Kyoto, Japan, 31 August - 5 September, 1969. Fischer, H. B. [1970]. AlMethod for Predicting Pollutant Transport in Tidal Waters, Water Resources Center Contribution No. 132, Hydraulic Engineering Laboratory, College of Engineering, U n i v e r s i t y of C a l i f o r n i a , Berkeley. Fischer, H. B. and Holley, E. R. [1971], "Analysis of the Use of Distorted Hydraulic Models for Dispersion Studies," Water Resources Res., 7(1): 46-51. Fox, D. G. [1971]. " F i n i t e Difference Convection E r r o r s , " Discussion, Proc. A.S.C.E., J. Sanit. Eng. Div., 97(SA5): 766-771. Gardiner, A. 0., Peaceman, D. W. and Pozzi, J r . , A. L. [1964]. "Numerical C a l c u l a t i o n of Multi-Dimensional M i s c i b l e Displacement by the Method of C h a r a c t e r i s t i c s , " J. Soc. Pet. Eng., 4(1): 26-36. Goldie, C. A. [1967]. Pollution and the Fraser: Report I Dealing with Pre- liminary Investigations of Waste Disposal to the Lower Fraser River," prepared f o r the P o l l u t i o n Control Board, V i c t o r i a , B r i t i s h Columbia. Guymon, G. L. [1970], "A F i n i t e Element Solution of the One-Dimensional Diffusion-Advection Equation," Water Resources Res., 6(1): 204-210. Harder, J . A. [1971]. "Solution Techniques: 1; Analogue and Hybrid Tech- niques," Chapter 6 of Estuarine Modelling: An Assessment, Ed. G. H. Ward and W. H. Espey, J r . ( l i s t e d under e d i t o r s ) . 1G1 Harleman, D. R. F. [1965]. "The S i g n i f i c a n c e o f Longitudinal Dispersion i n the Analysis of P o l l u t i o n i n Estuaries," Advances in Water Pollution Research, Ed. 0. Jaag (3 vols; Oxford: Pergamon Press, 1965), I, pp. 279- 290. Harleman, D. R. F., L e e , C. H. and H a l l , L. C. [1968], " N u m e r i c a l S t u d i e s o f U n s t e a d y D i s p e r s i o n i n E s t u a r i e s , " P r o c . A.S.C.E., J. Sanit. Eng. Div., 94(SA5): 897-911. Harris, E. K. [1963]. "A New S t a t i s t i c a l Approach to the One-Dimensional D i f f u s i o n Model," Air Wat. Poll. Int. J.,7(8): 799-812. Harris, E. K. [1964]. "Note on the One-Dimensional D i f f u s i o n Model and the Log-Normal D i s t r i b u t i o n , " Air Wat. Poll. Int. J., 8(5): 421-423. Henderson, F. M. [1966]. Open Channel Flow (New York: The MacMillan Co., 1966). Hinze, J . O. [1959]. Turbulence: An Introduction to its Mechanism and Theory (New York: McGraw-Hill, 1959). Holley, E. R. [1969a]. "Unified View of D i f f u s i o n and Dispersion," Proc. A.S.C.E., J. Eyd. Div., 95(HY2): 621-631. Holley, E. R. [1969b]. "Difference Modelling and Stream P o l l u t i o n , " Proc. A.S.C.E., J. Sanit. Eng. Div., 95(SA5): 968-972. Holley, E. R., Harleman, D. R. F. and Fischer, H. B. [1970]. "Dispersion i n Homogenous Estuary Flow," Proc. A.S.C.E., J. Hydv Div., 96(HY8).: 1691-1709. Kent, R. [1960]. " D i f f u s i o n i n a S e c t i o n a l l y Homogenous Estuary," Proc. A.S.C.E., J. Sanit. Eng. Div., 86(SA2): 15-47. Koivo, A. J . and P h i l l i p s , G. R. [1971]. " I d e n t i f i c a t i o n of Mathematical Models f o r DO and BOD Concentrations i n Polluted Streams From Noise Corrupted Measurements," Water Resources Res., 7(4): 853-862. 102 Lager, J . A. and Tchobanoglous, G. [1968]. " E f f l u e n t Disposal i n South San Francisco Bay," Proc. A.S.C.E., J. Sanit. Eng. Div., 94(SA2): 213-236. Leeds, J . V. [1967]. "Accuracy of Discrete Models Used to Predict Estuary P o l l u t i o n , " Water Resources Res., 3(2): 481-490. Leads, J . V. and Bybee, H. H... [1967]. "Solution of Estuary Problems and Network Programs," Proc. A.S.C.E., J. Sanit. Eng. Div., 93*'(SA3) : 29-36. Leendertse, J . J . [1971a]. "Environmental Simulation as a Tool i n a Marine Waste Disposal Study of Jamaica Bay," Advances in Water Pollution Research, Ed. S. H. Jenkins (2 v o l s ; Great B r i t a i n : Pergamon Press, 1971), U , Sec- t i o n I I I , pp. 4/1-4/7. Leendertse, J . J . [1971b], "Solution Techniques: 2; F i n i t e Differences," Chapter 6 of Estuarine Modelling: An Assessment, Ed. G. H. Ward and W. H. Espey, J r . ( l i s t e d under e d i t o r s ) . L i , W. H. [1962]. "Unsteady Oxygen Sag i n a Stream," Proc. A.S.C.E., J. Sanit. Eng. Div., 88(SA3): 75-85. L i , W. H. [1972]. " E f f e c t s of Dispersion on DO Sag i n Uniform Flow," Proc. A.s.C.E., J. Sanit. Eng. Div., 98(SA1): 169-182. Loucks, D. P. [1967]. "Risk Evaluation i n Sewage Treatment Plant Design," Proc. A.S.C.E., J. Sanit. Eng. Div., 93(SA1): 25-39. Loucks, D. P. and Lynn, W. R. [1966]. " P r o b a b l i s t i c Models f o r P r e d i c t i n g Quality," Water Resources Res., 2(3): 593-605. NalTuswami,; M.̂ , .Longenbaugh,; R. A. and Sunada, D." K. [1972]. " F i n i t e Element Method f o r the Hydrodynamic Dispersion Equation with Mixed P a r t i a l Deriva- t i v e s , " Water Resources Res., 8(5): 1247-1250. O'Connell, R. L. and Walter, C. M. [1963]. "Hydraulic Model Tests of E s t u a r i a l Waste Dispersion," Proc. A.S.C.E., J. Sanit. Eng. Div., 89(SA1): 51-65. O'Connor, D. J . [I960]. "Oxygen Balance of an Estuary," P r o c , A.S.C.E., J. Sanit. Eng. Div., 86(SA3): 35-55. 103 O'Connor, D. J . [1962]. "Organic P o l l u t i o n of New York Harbour - Theoreti- c a l Considerations," J. Water Poll. Control Fed.3 34(9): 905-919. O'Connor, D. J . [1965]. "Estuarine D i s t r i b u t i o n of Non-Conservative Sub- stances," Proc. A.S.C.E., J. Sanit. Eng. Div., 91(SA1): 23-42. O'Connor, D. J . [1966]. "An Analysis of the Dissolved Oxygen D i s t r i b u t i o n i n the East River," J. Water Poll. Control Fed., 38(11): 1813-1830. O'Connor, D. J . [1967]. "The Temporal and S p a t i a l D i s t r i b u t i o n s of Dissolved Oxygen i n Streams," Water Resources Res.3 3(1): 65-79. O'Connor, D. J . , St. John, J.P. and Di Toro, D. M. [1968]. "Water Quality Analysis of the Delaware Estuary," Proc. A.S.C.E., J. Sanit. Eng. Div.3 94(SA6): 1225-1252. O'Connor, D. J . and Di Toro, D. M. [1970]. "Photosynthesis and Oxygen Balance i n Streams," Proc. A.S.C.E., J. Sanit. Eng. Div.3 96(SA2): 547-571. Odd, N. V. M. [1971]. " P r a c t i c a l A p p l i c a t i o n of Mathematical Model Techniques to Predict E f f e c t s of Engineering Works on T i d a l Flow i n Estuaries With A Large T i d a l Range," Proc. Inst. Civ. Engrs.3 50(December): 507-517. Oster, C. A., Sannichsen, J . C. and Jaske, R. T. [1970]. "Numerical Solution to the Convective-Diffusion Equation," Water Resources Res.3 6(6): 1746-1752. Pence, G. D. J r . , J e g l i c , G. D. and Thomann, R. V. [1968], "Time Varying Dissolved Oxygen Model," Proc. A.S.C.E., J. Sanit. Eng. Div.3 94(SA2): 381-402. Pinder, G. F. and Cooper, J r . H. H. [1970]. "A Numerical Technique f o r Cal c u l a t i n g the Transient P o s i t i o n of the Saltwater Front," Water Resources Res., 6(3): 875-882. Pinder, G. F. and Friend, G. 0. [1972]. "Application of Galerkin's Procedure to A q u i f i e r A n a l y s i s , " Water Resources Res., 8(1): 108-120. Preddy, W. S. and Webber, B. [1963]. "The C a l c u l a t i o n of P o l l u t i o n of the Thames Estuary by a Theory of Quantized Mixing," Air Wat. Poll. Int. J., 7(8): 829-843. 104 Pretious, E. S. [1972]. Downstream Sedimentation Effects of Dams on the Fraser River, B.C., Water Resource Series Report No. 6, Department of C i v i l Engineering, U n i v e r s i t y of B r i t i s h Columbia. Pr i c e , H. S., Cavendish, J. C. and Varga, R. S. [1968], "Numerical Methods of Higher-Order Accuracy f o r Diffusion-Convection Equations," J. Soc. Pet. Eng., 8(3): 293-303. Pritchard, D. W. [1969]. "Dispersion and Flushing of Pollutants i n Estuaries," Proc. A.S.C.E., J. Hyd. Div., 95(HY1): 115-124. Prych, E. A. and Chidley, T. R. E. [1969]. "Numerical Studies of Unsteady Dispersion i n Est u a r i e s , " Discussion, Proc. A.S.C.E., J. Sanit. Eng. Div., 95(SA5): 959-964. Quick, M. C. [1973], "The Mechanics of Streamflow Meandering," paper submitted f o r p u b l i c a t i o n . Rennerfelt, J. G. W. [1963]. "An Analogue Computer f o r Oxygen Sag Calcu- l a t i o n s , " Air Wat. Poll. Int. J., 7(4/5): 367-372. Richtmyer, R. D. and Morton, K. w. [1967]. Difference Methods for Initial Value Problems. 2nd Ed. (New York: Interscience, 1967). Sawyer, W. W. [1966]. A Path to Modem Mathematics (London: Penguin Books Ltd., 1966). Schofield, W. R. and Krutchkoff, R. G. [1972]. "Stochastic Model for BOD and OD i n Segmented Estuary," Proc. A.S.C.E., J. Sanit. Eng. Div., 98 (SA5): 745-762. Simmons, H. B. [1960]. "Application and L i m i t a t i o n of Estuary Models i n P o l l u t i o n Analyses," Proceedings: First International Conference on Waste Disposal in the Marine Environment, Ed. E. A. Pearson (Oxford: Pergamon Press, 1960), pp. 540-546. Streeter, H. w. and Phelps, E. B. [1925]. A Study of the Pollution and Natural Purification of the Ohio River, Public Health B u l l e t i n 146, U.S. Pu b l i c Health Service, Washington, D.C. 1 0 5 Taylor, G. I. [1954] . "The Dispersion-, of Matter i n Turbulent Flow Through a Pipe," Proa. Roy. Soc. London, 223 (Series A): 446-468. Thayer, R. P. and Krutchkoff, R. G. [1967]. "Stochastic Model f o r BOD and DO i n Streams," Proc. A.S.C.E., J. Sanit. Eng. Div., 93(SA3): 59-72. Thomann, R. V. [1963]. "Mathematical Model f o r Dissolved Oxygen," Proc. A.S.C.E., J. Sanit. Eng. Div., 89(SA5): 1-30. Thomann, R. V. [1965]. "Recent Results From A Mathematical Model of Water P o l l u t i o n Control i n the Delaware Estuary," Water Resources Res., 1(3);: 349-359. Thomann, R. V. [1971]. Systems Analysis and Water Quality Management (New York: Environmental Science Services D i v i s i o n , Environmental Research and Applications, 1971). Waldichuk, M. Market, J . R., and Meikle, J . H. [1968]. Fraser River Estuary, Burrard Inlet, HoweSound and Malaspina Strait Physical and Chemical Ocean- ographic Data, 1957-1966: Volume 1, September 1957 to February 1962, Manuscript Report Series No. 939, F i s h e r i e s Research Board of Canada. Ward, G . H. and Espey, w. H. [ 1 9 7 1 ] . Estuarine Modelling: An Assessment, Ed. G . H. Ward and W. H. Espey, Report No. 1 6 0 7 0 DZV 0 2 / 7 1 , Water P o l l u t i o n Research Series, Environmental Protection Agency, U.S.A. Ward, P. R. B. [1972]. Transverse Dispersion of Pollutants in Oscillatory Open-Channel Flow, Report No. WHM-2, Department of C i v i l Engineering, U n i v e r s i t y of C a l i f o r n i a , Berkeley. Ward, P. R. B. [1973]. "P r e d i c t i o n of Mixing Lengths f o r River Flow Gauging," Proc. A.S.C.E., J. Hyd. Div., 99(HY7): 1069-1081. Worley, J . L., Burgess, F. J . and Towne, W. W. [1965]. " I d e n t i f i c a t i o n of Low-Flow Requirements f o r Water Quality Control by Computer Techniques," J. Water Poll. Control Fed., 37(5): 659-673. A P P E N D I C E S APPENDIX A DERIVATION OF THE ONE-DIMENSIONAL MASS TRANSPORT EQUATION A. 1 GENERAL The mass transport processes of advection, turbulent d i f f u s i o n and molecular d i f f u s i o n transport and d i s t r i b u t e any dissolved substance throughout the water mass of a r i v e r or estuary. (Advection i s also re- ferred to as convection or forced convection i n the l i t e r a t u r e ) . At the time and space scales of i n t e r e s t , the contribution to mass transport 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 to the other processes, and i s not considered f u r t h e r . The i n a b i l i t y of the advective or bulk v e l o c i t y to describe the d i s t r i b u t i o n of v e l o c i t y over a cross-section of the r i v e r or estuary makes, i t necessary to postulate another transport mechanism, namely di s p e r s i o n . The mechanisms of advective and di s p e r s i v e transport are described i n Sections A.2 and A.3. The one-dimensional mass transport equation i s derived by taking a mass balance of dissolved substance over an element c r o s s - s e c t i o n a l s l i c e of the estuary. Dissolved substance i s transported into and out of t h i s volume by the processes of advection and dispersion, and these processes, together with any source-sink e f f e c t s that the substance undergoes within the water mass, determine i t s concentration within the elemental volume. The only s p a t i a l v a r i a t i o n that i s admitted i n the one-dimensional mass transport equation 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 concentration, l o n g i t u d i n a l v e l o c i t y , depth and dis p e r s i o n c o e f f i c i e n t are 106 107 a l l assigned t h e i r average cro s s - s e c t i o n a l values (c, u, y and E respec- t i v e l y ) . For the sake of b r e v i t y , the average c r o s s - s e c t i o n a l value of any parameter w i l l be r e f e r r e d to as i t s "mean" value. The high f r e - quency turbulent f l u c t u a t i o n s are assumed to have been averaged out of the mean values u and c. Figure A . l shows an elemental c r o s s - s e c t i o n a l s l i c e of an estuary. The d i r e c t i o n of the co-ordinate axes x, y and z are as shown, the p o s i t i v e x axis p o i n t i n g seawards and the p o s i t i v e y axis v e r t i c a l l y down. Note that the co-ordinate axes are Eulerian, or f i x e d i n space. The flow i s assumed to be both unsteady and non-uniform so that c, u, y, E and the c r o s s - s e c t i o n a l area A vary both with time (t) and l o n g i t u d i n a l distance (x). The width of estuary b i s assumed to vary with x only. I t i s assumed that the 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 con- tinuous. A.2 LONGITUDINAL ADVECTIVE TRANSPORT The d i s t r i b u t i o n of l o n g i t u d i n a l v e l o c i t y i n an estuary i s three-dimensional. V a r i a t i o n s i n the l o n g i t u d i n a l d i r e c t i o n occur due to changes i n the c r o s s - s e c t i o n a l area of the estuary, and over any cross- section, v a r i a t i o n s 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 occur due to the f r i c t i o n a l influence of neighbouring f l u i d layers with each other and the s o l i d bed. In a r i v e r or estuary, the l a t e r a l d i s t r i b u t i o n of l o n g i - tudinal v e l o c i t y i s modified by the presence of bends, and i n an estuary, the v e r t i c a l d i s t r i b u t i o n of l o n g i t u d i n a l v e l o c i t y i s modified by the a d d i t i o n a l influence of s a l i n i t y . 108 Figure A . l Elemental Cross-Sectional S l i c e of A River or Estuary 109 In one-dimensional mass transport models, i t i s assumed that a c r o s s - s e c t i o n a l s l i c e or slug of water moves along the r i v e r or estuary at the mean cr o s s - s e c t i o n a l v e l o c i t y u. The transport of dissol v e d sub- stance i n t h i s slug of water i s termed longitudinal advection. The advective transport i n a r i v e r i s always i n the downstream d i r e c t i o n . In an estuary, the net advective transport over a t i d a l cycle i s i n the downstream d i r e c t i o n because of the e f f e c t of freshwater inflow. However, because of flow r e v e r s a l due to t i d a l e f f e c t s , the advective transport w i l l be i n the upstream d i r e c t i o n over p o r t i o n of the t i d a l c y c l e . The mass of dissolved substance (M) advected through any cross- section of a r i v e r or estuary i n time 6 t i s given by M = uAc 6 t , and thus the net mass (M \ remaining i n the elemental volume of Figure net A . l due to advective transport during time 6 t i s g - -M = - r—[uAc } 6x6 t (A.l) net 9x A.3 LONGITUDINAL DISPERSIVE TRANSPORT Advection accounts f o r the transport of dissolved 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 to the assumed uniform d i s t r i b u t i o n of the v e l o c i t y over the cross-section, 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 that e x i s t over any cross-section r e s u l t i n small "parcels" of water preceding and lagging the l i n e of advective advance, as defined by u. This i s i l l u s t r a t e d f o r a v e r t i c a l v e l o c i t y p r o f i l e t y p i c a l of a r i v e r 110 Vertical Distribution of Velocity U A t V7 A o — ~ o o -^Line source Y position after o Time At 0 >v 0 > a 0 o i 0 0 ° o ° 1 ° o o o o o o o o o o A Effect of Diffusion f ^ ^ ^ - - A d v e c t i o n Due to Velocity Gradient Figure 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 I l l i n Figure A.2. I t i s seen that a slug input from a l i n e source i s not advected downstream as a slug load, but i s spread around the l i n e of mean advance. The e f f e c t of t h i s i s to reduce the peak concentration and f l a t t e n the concentration gradients, as predicted on the basis of advection alone. Superimposed on the e f f e c t s of v e l o c i t y gradients i s the process of turbulent d i f f u s i o n . The mass transport associated with the turbulent 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 concentration gradients. The e f f e c t df turbulent d i f f u s i o n over the v e r t i c a l section "AA" i s also i l l u s t r a t e d i n Figure A.2. The combined e f f e c t s of v e l o c i t y gradients and turbulent d i f f u s i o n i n spreading the dissolved substance around the l i n e of advective advance (u) i s termed longitudinal dispersion. Unless . otherwise q u a l i f i e d , the terms "advection" and "dispersion" w i l l be taken to mean ad- vection and 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 necessary to represent the dispersion process mathematically i f i t i s to be included i n the mass transport equation. Holley [19 69a]dis- cussed the underlying s i m i l a r i t y between the mechanisms of molecular d i f f u - sion, turbulent d i f f u s i o n and d i s p e r s i o n . Both molecular d i f f u s i o n [Bird, et al.s 1960] and turbulent d i f f u s i o n i n homogeneous i s o t r o p i c free turbu- lence [Hinze, 1959] can be represented mathematically by a F i c k i a n d i f f u s i o n equation. Taylor [1954] showed that a f t e r a s u i t a b l e time had elapsed, the 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 pipe can also be represented by a one-dimensional F i c k i a n d i f f u s i o n equation. Elder [1959] found a s i m i l a r r e s u l t to hold i n model experiments of open channel flow. Fischer [1966a] i l l u s t r a t e d that i n the i n i t i a l non-Fickian period following the release of trac e r , the e f f e c t s of v e l o c i t y gradients outweighed the e f f e c t s of turbu- l e n t d i f f u s i o n , and l e d to a skewed d i s t r i b u t i o n of mean concentration i n 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 Figure A.2. According to Fick's Law, the d i s t r i b u t i o n of concentration should be Gaussian, but Fischer [1966a] has shown that t h i s only occurs a f t e r c r o s s - s e c t i o n a l mixing has reduced the s p a t i a l v a r i a t i o n of concentration over the cr o s s - s e c t i o n to a value much smaller than the mean c r o s s - s e c t i o n a l value. In d e r i v i n g and applying the one-dimensional mass transport equation, the disp e r s i o n process i n i t s e n t i r e t y i s assumed to be F i c k i a n . In a c t u a l f a c t , the l o n g i t u d i n a l d i s p e r s i o n of e f f l u e n t i n a r i v e r or estuary i s a complex phenomenom that i s i n i t i a l l y c o n t r o l l e d by 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 gradients. As turbulent d i f f u s i o n and secondary flows d i s t r i b u t e the e f f l u e n t mass over the cro 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 exert an inc r e a s i n g influence on the l o n g i t u d i - n a l d i s p e r s i o n process, The l a t e r a l mixing due to turbulent d i f f u s i o n and secondary flows i s discussed i n Appendix E and an estimate 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 of the estuary i s given i n Appendix F. (The e f f e c t s of 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 are also discussed i n Appendix F). 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 (and the d i s p e r s i o n process i s Fickian) 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 are of the order of 20 - 100 times greater than 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 gradients [Fischer, 1966a]. In an estuary, the disp e r s i o n process i s modified by the a d d i t i o n a l influence of s a l i n i t y and t i d a l e f f e c t s . The influence of t i d a l o s c i l l a t i o n s on the dis p e r s i o n c o e f f i c i e n t has been in v e s t i g a t e d by Holley et al. [1970] , and t h e i r method has been used i n estimating the disp e r s i o n c o e f f i c i e n t s f o r the Fraser River Estuary i n Appendix F. According to Fick's law of d i f f u s i o n , the net mass of d i s s o l v e d substance transported through u n i t c r o s s - s e c t i o n a l area i n time 6t i s given by [Bird et al., 1960] M = - E — f i t (A. 2) 3x 113 the minus sign a r i s i n g because the net transport i s i n the d i r e c t i o n of decreasing concentration. E i s the disp e r s i o n c o e f f i c i e n t , and as i s expected from the previous discussions of the dispersion process, depends on the d i s t r i b u t i o n of v e l o c i t y over the cross-section and the c o e f f i c i e n t of turbulent d i f f u s i o n (see, f o r example, Taylor [1954]; Fi s c h e r [1966a], [1967] and Holley et al. [1970]. According to equation (A . 2 ) , the net mass remaining i n the elemental volume due to di s p e r s i v e transport during time 6t i s M . = ^ - t A E ^ K x S t . (A.3) net 3x 3x A.4 SOURCE-SINK EFFECTS The source-sink terms account f o r any processes that r e s u l t i n the production or removal of the dissolved substance from the water body. In the BOD-DO system, source processes include reoxygenation by atmospheric oxygen d i s s o l v i n g i n the water and the oxygen produced photosynthetically by aquatic plants; the sink processes include oxygen uptake by b a c t e r i a breaking; down the substances exerting the BOD and oxygen uptake f o r aquatic plant r e s p i r a t i o n . The source-sink processes that a substance undergoes i n the aqua- •tic, environment are s p e c i f i c t o the p a r t i c u l a r substance, and depend on the ph y s i c a l , chemical and b i o l o g i c a l processes that the substance undergoes i n the p a r t i c u l a r aquatic environment. Consequently, source-sink e f f e c t s are included i n the mass transport equation dh a general manner. Suppose that the p a r t i c u l a r substance i s undergoing n processes that r e s u l t i n i t s produc- t i o n or removal from the water body. If the rate of production per u n i t 114 volume of water due to the i process i s given by S^, removal processes being regarded as negative production, then the net, mass of substance pro- duced i n the elemental volume i n time 6t i s n M = A&x&tl S. (A.4) net . , 1 i = l A.5 THE ONE-DIMENSIONAL MASS TRANSPORT EQUATION The mass of di s s o l v e d substance contained i n the elemental volume i s M = A<5xc As a r e s u l t of the advection, d i s p e r s i o n and source-sink e f f e c t s that occur i n time 6t, the change i n mass of di s s o l v e d substance i n the e l e - mental volume i s 3 — 6M = T--{Ac}6x6t (A.S) dt Summing equations ( A . l ) , (A.3) and (A.4) and s u b s t i t u t i n g i n t o equation (A.5) gives |-CAc}> 4-iAuc"} + |-[AE|£-} + A E S. (A.6) 9t 3x 3x 3x . , l i= l Expanding the f i r s t two d e r i v a t i v e s and using the equation of conservation of f l u i d mass i n the form ' 3 <• - i 3 A N (Au) + — = 0, 3x dt equation (A.6) reduces to 3c 3c 1 3 f 3c •, . s r = -ur— + 7 - 1 <AE;r— } + I S., (A.7) 3t 3x A 3x 3x . , l 1=1 115 the one-dimensional mass transport equation f o r unsteady non-uniform flow. The terms on the right-hand side of Equation (A.7) s h a l l be r e - ferred to r e s p e c t i v e l y as the advective, the d i s p e r s i v e and the source- sink terms. A.6 THE LAGRANGIAN FORM OF THE ONE-DIMENSIONAL MASS TRANSPORT EQUATION Equation (A.7) was derived r e l a t i v e to an E u l e r i a n co-ordinate system (xyz) that i s f i x e d i n space. To transform the equations r e l a t i v e to a co-ordinate system that i s advected along at the mean v e l o c i t y u (a Langrangian co-ordinate system), i t i s necessary to transform the E u l e r i a n co-ordinates according to § - = u (A.S) dt Relative to the curve (A.8), the time rate of change of concentration i s given by (see Section C l ) dc _ 3c_ -3c dt " 3t + U3x and thus, r e l a t i v e to Langrangian co-ordinates, Equation (A.7) becomes £ = | T-CAE& + Z S. (A.9) dt A 3x 3x , i x=l Equation (A.8) i s the c h a r a c t e r i s t i c curve of advective information propagation, and i s discussed f u r t h e r with regard to numerical d i s p e r s i o n and s t a b i l i t y i n Appendix C. Equation (A.7) and Equations (A.8) and (A.9) repre- sent the same s i t u a t i o n but from d i f f e r e n t viewpoints; i n Equation (A.7) the "observer" i s stationed on the bank of the r i v e r or estuary and observes changes 116 i n the concentration as the water flows past him, whereas i n the l a t t e r two equations, the observer i s advected along with the water mass [Equation (A.8)] and observes changes i n the water mass around him [Equation (A.9)]. I f d i s p e r s i v e transport i s ignored, Equation (A.8) reduces to d t i-1 1 which i s the o r i g i n a l one-dimensional mass transport equation developed by Streeter and Phelps [1925] to p r e d i c t the e f f e c t of substances exerting a b i o l o g i c a l oxygen demand (BOD) on the concentration of d i s s o l v e d oxygen (DO) i n a r i v e r . In t h e i r model, n was equal to two; the only source of DO was reaeration through the water surface and the only sink was the 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 Fraser River i s the 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 drainage area of 90,000 square miles i s shown i n Figure B . l and occupies almost one-quarter of the Province. The r i v e r r i s e s in. the Rocky Mountains near Jasper and flows north-west f o r some 330 miles. • North of Prince George the r i v e r changes d i r e c t i o n and flows south some 400 miles to Hope. At Hope the r i v e r emerges from the Coast Mountains and flows westward through the Fraser V a l l e y f o r almost 100 miles to enter the S t r a i t of Georgia at Vancouver. I t i s t h i s l a t t e r section of the r i v e r , from Hope to Vancouver, that i s of d i r e c t concern to t h i s t h e s i s , and d i s - cussion i s now r e s t r i c t e d to i t , B.2 THE LOWER FRASER RIVER SYSTEM The Lower Fraser River from Hope to Vancouver i s shown i n Figure B.2. A convenient way of describing t h i s section of the r i v e r i s i n terms of the factors that a f f e c t the flow. These factors include: (1) the var- ious channels of the r i v e r system; (2) the tid e s i n the S t r a i t of Georgia; (3) the r i v e r discharge; and (4) saltwater i n t r u s i o n . B.2.1 Channels of the River System. The Fraser River deposits 20 - 25 m i l l i o n tons of sediment i n t o the S t r a i t of Georgia annually [Pretious, 1972], and has b u i l t a large d e l t a with a seaward perimeter of some 30 miles. 117 00 The Fraser River From Hope to Vancouver 120 The d e l t a i s shown i n Figure B.3 and i t i s seen that the r i v e r has cut a net- work of channels through i t . At New Westminster the r i v e r t r i f u r c a t e s i n t o the Main Arm, which enters the S t r a i t o f f Steveston; Annacis Channel, which flows around the back of Annacis Island to reconnect with the Main Arm; and the North Arm, which enters the S t r a i t near Point Grey. Annacis Channel i s not treated as a separate channel, but as part of the Main Arm. Around Sea Island, the North Arm divides to form the Middle Arm. North-west of Ladner, a complex of shallow channels and sloughs (backwater areas) interconnect with each other, the three most important channels being Ladner Reach, Sea Reach and Canoe Pass. This area?; drains through Canoe Pass to provide the Main Arm with another connection to the S t r a i t of Georgia. The percentage of r i v e r discharge that enters the S t r a i t through the Main Arm, the North Arm, the Middle Arm and Canoe Pass i s approximately 85 per cent, 5 per cent, f i v e per cent and f i v e per cent r e s p e c t i v e l y [Goldie, 1967-]. This i s only a general i n d i c a t i o n of the percentage d i s t r i b u t i o n of flows, the summer d i s - t r i b u t i o n of flows probably d i f f e r i n g from the winter d i s t r i b u t i o n . The section of r i v e r between New Westminster and Hope i s r e f e r r e d to as the Main Stem. Of the various islands present i n the Main Stem (see Figure B.2), only Douglas and Barns ton Islands form s i g n i f i c a n t a d d i t i o n a l channels. The channels at the "back" of the other i s l a n d s are treated as part of the Main Stem. Of the various major lakes that drain i n t o the Main Stem (see F i g - ure B.2), P i t t Lake i s of 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 lake i s a f f e c t e d both by the t i d e s i n the S t r a i t of Georgia and the fr e s h - water discharge of the Fraser (the discharge of P i t t River i n t o the north end 121 Point Ladner Reach L A D N E R V ; ' v ' : : " ' ; - - ' : ' " • - " • ^ • ' . f ' Canoe 'Pass' S C A L E : 1 2 3 4 5 M I L E S LEGEND : ^Y::i:;; - Delta C A N A D A U , S . A Figure B.3 The Fraser River Delta 122 of the lake i s n e g l i g i b l e compared to the Fraser discharge. At low f r e s h - water flows i n the Fraser, the maximum t i d a l range of the lake i s several feet, and as the lake has a surface area of some 25 square miles, the volume of water stored i n the lake on the flood t i d e and released to the Fraser on the ebb t i d e i s very large (see Table B . l ) . The large reverse d e l t a at the entrance to P i t t Lake (see Figure B.2) i s evidence of the large flows that enter the lake. The P i t t River - P i t t Lake system obviously c o n s t i - tute an important channel of the lower Fraser,[/system and w i l l be r e f e r r e d to as the Pitt System. The other lakes draining i n t o the Main Stem are of s u f f i c i e n t elevation to be independent of t i d a l influence and freshwater discharge i n the Fraser. Thus, the lower Fraser River system consists of the following seven p r i n c i p a l channels: 1. Main Arm - Main Stem 2. North Arm 3. Middle Arm 4. Canoe Pass 5. P i t t System 6. Douglas Channel, and 7. Barnston Channel. To some extent the channel geometries are determined by the many .-•.. dykes, j e t t i e s and t r a i n i n g walls that have been constructed i n the lower Fraser system. Each Spring the Fraser V a l l e y i s subject to pos s i b l e flooding during the freshet (time of high runoff due to snowmelt), and an extensive system of dykes has been constructed f o r flood p r o t e c t i o n . Ocean-going ships 123 use the Main Arm to enter and leave New Westminster, an overseas shipping terminal. To maintain the shipping channel, a number of dykes and t r a i n i n g walls have been constructed to increase l o c a l scour. Cross-sectional areas, depths and widths were evaluated at the net- work of stations shown i n Figure B.4. The stations are 5,000 fee t apart, except i n the deeper water of P i t t Lake where the spacing has been increased to 15,000 feet. (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 solutions of the hydrodynamic equation and the t i d a l l y varying and t i d a l l y averaged mass transport equations, as i s discussed i n Chapter 3). Soundings charts supplied by the Department of Public Works of Canada were used to deter- mine the channel geometries. A t y p i c a l cross-section of each of the seven channels i s shown i n Figure B.5. Widths and depths are recorded i n feet, and depths are r e l a t i v e to l o c a l low water. Figures B.6 to B.8 show the var- i a t i o n i n depths, widths and c r o s s - s e c t i o n a l areas along the f i v e major channels. In an attempt to estimate the advective area ( e f f e c t i v e area of discharge), i t was a r b i t r a r i l y assumed that any depth of section less than 10 feet deep di d not contribute to the advective area. This i s i l l u s t r a t e d for Station No. 35 i n Figure B.5, where areas less than 10 feet deep are shown hatched. The storage width of section i s AB, but the advective width i s only CD. The advective area was used to determine the advective hydraulic radius (which i s greater than the hydraulic radius determined from the gross area of s e c t i o n ) . From Figures B.6 to B.8, i t i s seen that the advective area i s not much smaller than the gross area, but the advective width i s often considerably smaller than the storage width. The r i v e r sections are e s s e n t i a l l y wide and shallow and the terms hydraulic radius and mean depth are now used synonymously. FIRST AND LAST STATIONS • Main Arm - Main Stem I North Arm 101 Canoe Pass 11 9 Middle Arm 129 Pitt System 140 Douglas Channel 180 Barnston Channel 184 , North Arm 101 S J06 HQ, •Middle Arm ®jS "Main Arm *I28 Pitt Lake Junction Stations 5 0 0 0 * spacing 15,000' spacing • C a n o e P a s s Figure B.4 Network of Stations Used i n the Numerical Solution of the Hydrodynamic and Mass Transport Equations 1000 2000 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 Channel) Figure B.5 Ty p i c a l Channel Cross-Sections (Widths and Depths i n Feet, Depths Relative to Local Low Water) 60,000 h 50,000 o 40,000 30,000 20,000 50 CD < CL OJ Q 10 3,000 sz 2,000 • D 1,000 0 Gross Values Advective Values D E P T H S A N D A R E A S R e l a t i v e to l o c a l l o w w a t e r . 10 20 30 Stations 40 50 Figure B.6 Cross-Sectional Parameters of Main Arm - Main Stem 60 Figure B„7 Cross-Sectional Parameters of North Arm, Middle Arm and Canoe Pass D E P T H S A N D A R E A S R e l a t i v e to l o c a l l ow w a t e r . Pitt River 4 0 , 0 0 0 i ; 3 0 , 0 0 0 o 2 0 , 0 0 0 a> k. < 1 0 , 0 0 0 0 4 0 £ 3 0 2 0 I 0 0 3 , 0 0 0 - 2 , 0 0 0 CL a> Q •o , 0 0 0 o z o i n o z Pitt Lake i n m i n o z o (0 o z co in ro o Z /A y. r^yjr§e_ Delta 1 \\ H - \ » \ - V — ' » / — / /^\\ / V » \ ~ — v 7 \ \ § W ' S N — \ % \ t \ % \ V* — i I / — H 1 1 — Gross Values Advective Values 2 , 0 0 0 , 0 0 0 ~ 1 , 5 0 0 , 0 0 0 ~ 1 , 0 0 0 , 0 0 0 < 5 0 0 , 0 0 0 0 2 1 0 ^ 2 0 0 ~ 1 9 0 1 8 0 CL Q 1 0 , 0 0 0 8 , 0 0 0 _ 6 , 0 0 0 - JZ 4 , 0 0 0 5 2 , 0 0 0 03 Figure B.8 Cross-Sectional Parameters of P i t t River and P i t t Lake 129 B.2.2 Tides i n the S t r a i t of Georgia. The S t r a i t of Georgia i s t i d a l , being connected to the P a c i f i c Ocean by the Juan de Fuca S t r a i t (see Figure B . l ) . The t i d e s are of the mixed type c h a r a c t e r i s t i c of much of the coast of Northwest America. The t i d a l range at Steveston for mean and large tides i s r e s p e c t i v e l y 10 f e e t and 15 feet and t y p i c a l tides at Steveston are shown i n Figure B.9. The e f f e c t s of the t i d e s on the flows i n the Fraser system depends both on the t i d a l range a t Steveston and the r i v e r discharge. This i s i l l u s - trated i n Figure B.10 [after Baines, 1953] which shows the l o c a l low and high t i d e envelopes of the Main Arm and Main Stem f o r discharges of 27,000 and 250,000 cubic feet per second at Hope. The greater influence of the tide during low flow conditions i s r e a d i l y apparent, and i t i s seen that the upstream l i m i t of t i d a l influence i s around Chilliwack, some 60 miles from Steveston. The section of the Fraser system from the S t r a i t of Georgia to Chilliwack w i l l be referred to as the Fraser River Estuary, although i t i s noted that the Fraser i s more properly designated as a t i d a l r i v e r [Callaway, 1971]. During low flow-high t i d e conditions, flow r e v e r s a l occurs i n the Fraser Estuary. The cubature study of discharges by Baines [1952] predicted flow r e v e r s a l at Mission, some 50 miles upstream of Steveston. Downstream of Chilliwack there i s a network of t i d e gauging s t a t i o n s , . some.equipped with continuous recorders and others only recording the maximum .and minimum l e v e l s during each 24 hour period. The p o s i t i o n and type of gauge at each s t a t i o n i s shown i n Figure B . l l . • : B.2.3 River Discharges. The v a r i a t i o n of t^g'meap^n^n.thly" "dis^* charge'~?at Hope, as c a l c u l a t e d f o r the period 1912-1970;, is':sriowh;.-in Figure-''. B. 12. .'The .mean monthly flows vary 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 ' 1 1 1 1 1 1 1 I 0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0 Hours Figure B.9 Ty p i c a l Tides at Steveston Steveston New Mission Chilliwack Hope Westminster Figure B.J.0 Local Low and High Tide Envelopes [After Baines, 1953] Figure B . l l Tide Gauging Stations i n the Fraser River Estuary to 133 280i 240 (/> "2001 w O O — I 601 o L L . 120 80 40 0' JAN. FEB. MAR. APR. MAY JUNE JULY AUG. SER OCT. NOV. DEC. Figure B.12 Mean Monthly Flows at Hope (1912-1970 Inclusive) 134 fr e s h e t 7:to a winter minimum of around 30,000 c f s . Recorded extremes at Hope are 536,000 c f s on May 31, 1948 and 12,000 cfs on January 8, 1916. The drainage area of the Fraser River below Hope i s some 6,000 square miles, and between Hope and the S t r a i t of Georgia various other r i v e r s flow into the Fraser system, the most important being the Harrison River (see Figure B.2). The e f f e c t of t h i s a d d i t i o n a l inflow i s to increase the flow at New Westminster by some 15 per cent over the flows at Hope during the freshet, and up to 50 per cent during winter. I t should be apparent that the Fraser River Estuary i s somewhat unusual, being characterized by both high r i v e r discharges and large t i d a l e f f e c t s . This 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 of the t i d a l prism and t o t a l r i v e r discharge between the times of low-low-water and high-high-water for the t i d a l cycles of January 11 and June 16, 1964. These t i d a l cycles and the freshwater discharges are shown i n Figure B.9. The volumes of the t i d a l prisms and r i v e r flows are shown i n Table B".l. B.2.4 Saltwater Intrusion. The s a l i n i t y of the S t r a i t of Georgia i s e s s e n t i a l l y some 30 parts per thousand (ppt), and under low r i v e r discharge conditions saltwater intrudes a considerable distance into the four channels that emerge from the d e l t a . The saltwater 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 Figure 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 four stations along the Main Arm. (Note: the s t a t i o n designation i n Figure B.13 i s that of Waldichuk, et al. 3 and should not be confused with the s t a t i o n designation i n Figure B.4). The p o s i t i o n of the stati o n s , the t i d e at Steveston and the time of t i d e the observations were taken are also shown. The four observations c l o s e l y occur around the same phase of t i d e and can be regarded as "simultaneous." The freshwater discharge TABLE B . l RIVER FLOW VOLUMES AND TIDAL PRISMS ON JANUARY 15 AND JUNE 16, 1964 (Volumes Calculated for Time Between Low-Low-Water and High-High-Water) DATE RIVER FLOW (cfs) TIDAL RANGE (feet) RIVER FLOW v o l . ( f t ) TIDAL PRISM ( f t 3 ) RIVER FLOW TIDAL PRISM JANUARY 15 53,500 10.4 12 x 10 8 66 x 10 8 0.18 JUNE 16 463,000 7.6 104 x 10 8 25 x 10 8 4.2 10 20 30 • ° • - J L . 1 10 20 30 10 20 30 100 200 3 0 0 L All depths in feet River Discharge - 57,600 cfs at Hope 10 20 30 —I 1 1 Stat. No. 3 Figure B.13 S a l i n i t y P r o f i l e s i n the Main Arm on February 13-14, 1962 01 137 at Hope i s 57,000 cfs and i t i s apparent that under these freshwater d i s - charge and t i d a l conditions, the Main Arm i s highly s t r a t i f i e d with the toe of the s a l t wedge somewhere between stat i o n s 2 and 3. Recent studies have involved monitoring the s a l i n i t y p r o f i l e con- tinuously at three s t a t i o n s along the Main Arm. This was done for the two periods February 1 - 1 6 and March 16 - 30, 1973, during which the average discharge at 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 that the e f f e c t of the tides was to move the wedge bo d i l y up and down the Main Arm [D. O. Hodgins, p r i v a t e communication], the maximum excursion of the toe of the wedge probably being to somewhere around Annacis Island. Thus, f or low r i v e r flows, the Main Arm of the Fraser Estuary i s highly s t r a t i f i e d with the s a l t wedge moving b o d i l y up and down the estuary under the influence of the t i d e s . During times of flow r e v e r s a l , the wedge may move as far upstream as Annacis Island but during seaward discharge i t i s washed downstream past Steveston. For high r i v e r flows, the wedge prob- ably does not penetrate'-past Steveston, i f that f a r . Saltwater intrudes i n t o the other channels of the d e l t a , and s i m i l a r s i t u a t i o n s probably occur there, although the saltwater movement may be modified by the slower v e l o c i t i e s through these channels. S a l i n i t y p r o f i l e s i n these other channels show them to be t y p i c a l l y s t r a t i f i e d , but perhaps not quite as s t r a t i f i e d as the Main Arm (probably due to the lower discharge v e l o c i t i e s through these channels). APPENDIX C NUMERICAL DISPERSION AND STABILITY The dependent v a r i a b l e of a p a r t i a l d i f f e r e n t i a l equation de- f i n e s a surface over the plane of the independent v a r i a b l e s . In the plane of the independent v a r i a b l e s there are various curves, or " c h a r a c t e r i s t i c s " that describe the propagation of information through the system that the p a r t i a l d i f f e r e n t i a l equation represents. ' The one-dimensional forms of the advective transport equation, the dispersive transport equation and the hydrodynamic equations are of i n t e r e s t to t h i s t h e s i s . The forms of t h e i r respective surfaces are b r i e f l y described and the equations of t h e i r respec- 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 of numerical d i s p e r s i o n a r i s e s from solving the one-dimensional advective transport equation (or the one-dimensional mass transport equation) r e l a t i v e to a f i x e d space g r i d , rather than along the advective c h a r a c t e r i s - t i c s . 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 also a r i s e s from solving the above three equations r e l a t i v e to a f i x e d g r i d rather than along t h e i r respective c h a r a c t e r i s t i c s . C l SURFACE GEOMETRY OF PARTIAL DIFFERENTIAL EQUATIONS The following p a r t i a l d i f f e r e n t i a l equations are of i n t e r e s t 3c 3t ( C l ) (C.2) 138 139 9u 9t * 3 7 - ? f = 9a u u c y (C.3) 9y_ 9t -9u Equation ( C l ) describes one-dimensional advective transport and Equation (C.2) nent transport processes of the one-dimensional mass transport equation, and the sum of t h e i r e f f e c t s i s the one-dimensional mass transport equation with- out the source-sink terms. The two coupled equations (C.3) are the hydrodynamic equations and are used to p r e d i c t the temporal v a r i a t i o n s i n the parameters u and A of the t i d a l l y varying mass transport equation. [Equations (C.3) are ob- tained by s u b s t i t u t i n g Equation (3.3) and A = by into Equations (3.1) and (3.2)]. Equations ( C l ) and (C.2) define surfaces c(x,t) over the (x,t) plane and Equations (C.3) define a p a i r of coupled surfaces u(x,t) and y(x,t) over the (x,t) plane. Consider now the form of the surface defined by the advective transport Equation ( C l ) . Figure C l shows the shape of t h i s surface for a slug input i n t o (a) steady uniform flow; (b) steady non-uniform flow along a r i v e r of contracting cross-section; and (c) the mixture of steady and o s c i l l a t o r y flow c h a r a c t e r i s t i c of an estuary (estuary flow). These sur- faces are solutions to Eguation ( C l ) for a slug input into the various types of flow. I t i s apparent from Figure C l that there are various curves i n the (x,t) plane that r e s u l t i n a considerable s i m p l i f i c a t i o n of Equation (C.l) — the concentration i s constant along each of the curves AB of Figure C . l while everywhere else i n the plane i t i s zero. For any continuous, smoothly varying surface c ( x , t ) , the time rate of change of c(x,t) along the v e r t i c a l plane defined by the curve one-dimensional dispersive transport.'"These two equations represent the compo- 140 c(x,t) Figure C l Concentration Surfaces for the Advection of a Slug Load 141 'Sr = f ( x , t ) (G.4) at i n the (x,t) plane i s given by dc _ 3c 3c_ dx dt 3t 3x dt l C ' ' The graphical i n t e r p r e t a t i o n of t h i s elementary formula of calculus i s i l l u s - trated i n Figure [C.2. Substituting Equation ( C l ) i n t o (C.5) gives dt " { " u + d t } 3 7 ( c ' 6 ) and along the curve defined by dx dt = u ( C ' 7 ) the time rate of change of concentration i s given by §£• = 0 (C.S) Equations (C.7) and (C.8) are the well-known Langrangian form of the advective transport equation and curve (C.7) obviously defines the three curves AB of Figure C l . In deriving Equation ( C l ) i t i s assumed that the v a r i a t i o n of c and u with x and t i s continuous. For the cases shown i n Figure C l , the v a r i - a t i o n of c i s discontinuous 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 are not defined everywhere i n the (x,t) plane. Thus, Equation ( C l ) does not c o r r e c t l y represent the advection of a slug load (this i s the fundamental reason for numerical d i s p e r s i o n ) . I t i s only when the s o l u t i o n to Equation ( C l ) i s ex- amined along the curve (C.7) that the problem of the non- d e f i n i t i o n of the par- t i a l d e r i v a t i v e s i s avoided. [See Equation (C.6)]. 142 i f 3t = | f 8t Sx dt at ax dc _ ac + ac ^x dt at ax dt Figure C.2 Time Rate of Change of Concentration Along A Curve i n the (x,t) Plane 143 Consider disturbing the system represented by Equation ( C l ) by introducing an elemental slug load of Ac. This disturbance, or "information," w i l l propagate through the system according to Equation (C.7). Thus, Equation (C.7) i s of 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 along t h i s curve the so l u t i o n to the advective transport equation reduces to the simple form of Equation (C.8). The f i x e d (Eulerian) space g r i d used to derive Equation ( C l ) "masks" the under- l y i n g physics of the transport process and the s i m p l i c i t y of the s o l u t i o n . Now consider the equation of dispersive transport (C.2). I f both E and A are constant i n x and t, Equation (C.2) reduces to | | - E 2 i IC.9> ax The surface defined by Equation (C.9) for the dispersion of a slug load i s shown i n Figure C.3. The equation of t h i s surface i s [Fischer, 1966a] S(x ft) ='-^~ exp{-'-—-} (CIO) 4IlE t where M i s the mass per u n i t area;6f tracer released. At any t i m e t 1 the shape of the surface i s a Gaussian d i s t r i b u t i o n and c(x,t.') i s uniquely defined by the l o c a t i o n of the point of standard deviation. This i s i l l u s t r a t e d i n Figure C.3. Thus, the locus of the standard deviation i n the (x,t) plane d i r e c t l y des- c r i b e s the underlying mass transport process. From Equation (CIO) the locus of the standard deviation i s the parabola x 2 = 2E ( C l l ) or i n the form of Equation (C.4) 144 Figure C.3 Dispersion of A Slug Load 145 If t h i s system i s disturbed by introducing an elemental slug load of Ac, the e f f e c t of t h i s slug load, or i t s "information", w i l l propagate through the system according to Equation (C . l l ) or (C.12). Once again, the curve (C.12) i s of fundamental s i g n i f i c a n c e and along t h i s curve the concentration i s given by _ M r l c(x,t) = - 7 = — - ' (C.13) ^SHe X F i n a l l y , consider the hydrodynamic Equations (C.3). The well-known c h a r a c t e r i s t i c s of t h i s system are given by [Courant and H i l b e r t , 1962] §T = u ± (C.14) dt If t h i s system i s disturbed by introducing a small change i n the water sur- face elevation Ay, t h i s "information" w i l l propagate through the system accord- ing to Equation (C.14) and along t h i s curve Equations (C.3) reduce to [Henderson, 1966] d ; dh . |u|u — - i u ± 2/gy} dt ---'w> = -9 3 7 + J - 1 r <c-15> c y In summary, there are c e r t a i n curves i n the (x,t) plane that d i r e c t l y describe the propagation of information through the systems that Equations ( C . l ) , (C.2) and (C.3) represent. These curves can be c a l l e d charaeteristies of infor- mation propagation, and along these curves the solutions to the respective equa- ti o n s are g r e a t l y s i m p l i f i e d . This underlying s i m p l i c i t y i s masked by the Eu l e r i a n nature of the d e r i v a t i o n of the equations. The problems of numerical 146 dispersion and s t a b i l i t y a r i s e d i r e c t l y from viewing the numerical solu- tions to Equations ( C l ) to (C.3) from a f i x e d Eulerian space g r i d , rather than from a space g r i d along the c h a r a c t e r i s t i c s . C.2 NUMERICAL DISPERSION Before discussing numericalddispersion i n d e t a i l , i t i s necessary to b r i e f l y consider a few fundamentals of f i n i t e d i f f e r e n c e solutions to p a r t i a l d i f f e r e n t i a l equations. In using f i n i t e d i f f e r e n c e methods to solve d i f f e r e n t i a l equations, d e r i v a t i v e s are approximated by f i n i t e d i f f e r - ence expressions and the d i f f e r e n t i a l equation i s reduced to a differe n c e equation. The j u s t i f i c a t i o n of repla c i n g the d e r i v a t i v e s by differe n c e ex^r- pressions can be i l l u s t r a t e d by the Taylor's Series expansion of the v a r i a b l e U(x,t) with respect to t i n the neighborhood of the point (x = jAx, t = nAt). This can be written as where the p a r t i a l d e r i v a t i v e s are assumed continuous and 0 < 0 < 1 ;[Richtmyer and Morton, 1967, p. 19]. By making At small enough, the di f f e r e n c e expression of Equation (C.16) can be made to approximate the d e r i v a t i v e to any desired degree of accuracy. The term on the right-hand side of Equation (C.16) i s the truncation error t h a t r e s u l t s when the d e r i v a t i v e 8U/3t i s replaced by the di f f e r e n c e expression of Equation (C.16). When a p a r t i a l d i f f e r e n t i a l equation i n x and t i s approximated by a p a r t i a l d i f f e r e n c e equation, the truncation error consists of terms containing .n+1 (C.16) 147 Ax and At r a i s e d to various powers. The conditions under which the truncation error tends to zero i s the problem of convergence. That i s , as the mesh i s r e f i n e d (Ax, At -»• 0) , does the numerical s o l u t i o n converge to the true solu- t i o n of the p a r t i a l d i f f e r e n t i a l equation? The behaviour of thes truncation error as the s o l u t i o n progresses through time i s the problem of stability. If the d i f f e r e n c e scheme i s unstable, errors generated during the s o l u t i o n (for example, round-off errors) may become so magnified during the c a l c u - l a t i o n s as to make the f i n a l r e s u l t s meaningless. There are many f i x e d mesh di f f e r e n c e schemes f o r approximating the d e r i v a t i v e s of p a r t i a l d i f f e r e n t i a l equations, but e s s e n t i a l l y they can be c l a s s i f i e d into e x p l i c i t and i m p l i c i t schemes. An explicit scheme uses forward time dif f e r e n c e s (see Figure C.4), and each diffe r e n c e equation con- t a i n s only one unknown variable which can be solved f o r e x p l i c i t l y . Implicit schemes use backward time d i f f e r e n c e s (see Figure C.4) and each diffe r e n c e equation contains several unknown v a r i a b l e s . To obtain a s o l u t i o n at any time step, the r e s u l t i n g system of simultaneous-iequations must be solved over the e n t i r e space g r i d . Generally, i m p l i c i t schemes are unconditionally stable, whereas e x p l i c i t schemes are a t most c o n d i t i o n a l l y s t a b l e . This i s discussed further i n Section C.3. Consider now the numerical s o l u t i o n of the one-dimensional mass transport equation ( C l ) . This equation i s a component of the one-dimensional mass transport equation. Fixed mesh f i n i t e d i f f e r e n c e schemes generally do not simulate the advective transport process 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 d i s p e r s i v e process being superimposed bn the actual advective and dispersive processes occurring i n the r i v e r or estuary. This s o - c a l l e d numerical disper- sion can be i l l u s t r a t e d by considering the truncation error associated with 148 Forward Time Differences s U i n + I = function { u"-,, u" U" } -o- time = ( n + I )At time = n^t ( j-l)Ax jAx (j+ I )AX Backward Time Differences : u n + I _ _ r n n*l n+l - funct ion { U j , U j + | , U|_, } ( J - I ) A X — o — j AX ( J + Q I ) A X time = (n + l )A t time = nAt Figure C.4 Forward and Backward Time Differences 149 repl a c i n g the p a r t i a l d i f f e r e n t i a l equation of advective transport (C.l) with i t s corresponding e x p l i c i t d i f f e r e n c e equation ~n+l „n ,, »-,_n „n . n „n. C. - C. . (1-a (C . - C ) q (C - C .) where the flow i s assumed to be uniform and steady, x = jAx, t = nAt and a i s a weighting f a c t o r such that a = 0 f o r upstream d i f f e r e n c e s , a = 1/2 f o r cen- t r a l d i fferences and a = 1 f o r downstream d i f f e r e n c e s . These various d i f f e r - ences are i l l u s t r a t e d i n Figure C.5. Eguation (C.17) i s an example of an ex- p l i c i t d i f f e r e n c e scheme that uses forward time differences (the concentrations at time step n are used t o determine the concentrations at time step n+1). A Taylor's Series expansion of the d e r i v a t i v e s of Equation (C.l) shows that the d i f f e r e n c e equation (C.17) approximates the d i f f e r e n t i a l equation ar n ar n a 2f n a 2r n 2 2 O . + U(TT-) • = %UAx(l-2a) ( ± 4 ) • " + O(Ax) + O(At)^ the terms on the right-hand side being the truncation e r r o r . As C i s a solu- t i o n of Equation (C.l) i t also s a t i s f i e s = 0 (C.19) 9 t 9x 2 Substituting Equation (C.19) in t o (C.18) and neglecting terms of order (Ax) . 2 (At) and higher, i t i s found that the difference equation approximates the d i f f e r e n t i a l equation 90 9C „ 9 C 9t * + E p TI  (C'20) 9x 150 Figure C.5 Upstream, Central and Downstream Space Differences 151 where E can be considered to be a c o e f f i c i e n t of pseudo or numerical disper-P sion, and i s given by E = %UAx{ (l-2a) - 7 ^ } (C.21) p Ax Thus, the s o l u t i o n t o the d i f f e r e n c e equation (C.17) approximates the s o l u t i o n t o an advective-dispersive equation rather than the c o r r e c t ad- vective equation. As Ax ->• 0, E^ •> 0 and the mass transport associated with the numerical d i s p e r s i o n becomes i n c r e a s i n g l y smaller.- However, i n dif f e r e n c e schemes Ax i s always f i n i t e , and while the s o l u t i o n to d i f f e r e n c e equation converges to the s o l u t i o n of the correct advective equation i n the l i m i t (Ax = 0), i t converges to the s o l u t i o n of an advective-dispersive equation at the l e v e l of a p p l i c a t i o n (Ax ^ 0). It i s apparent from Equations (C.20) and (C.21) that numerical d i s - persion w i l l always occur unless e i t h e r U • 7 ^ = 1 - 2a (C.22) Ax or d X Ignoring the t r i v i a l l a t t e r case (the mass transport associated with any r e a l d i s p e r s i o n i s a l s o zero), i t i s seen that numerical d i s p e r s i o n does not occur when upstream d i f f e r e n c e s are used (a = 0) and Ax and At are defined by Ax T T = U (C.23) At 152 as was recognized by B e l l a and Dobbins [1968]. Under these conditions Equa- t i o n (C.17) reduces to C j + 1 = C j i = constant (C.24) Equations (C.23) and (C.24) are simply the f i n i t e d i f f e r e n c e versions of Equa- t i o n s (C.7) and (C.8). Thus, the use of upstream differences with the g r i d spacing defined by Equation (C.23) i s equivalent to so l v i n g Equation (C.17) along the c h a r a c t e r i s t i c curve i n the (x,t) plane. Under these conditions the underlying advective mass transport process i s c o r r e c t l y simulated., and there i s no numerical dispersion. Consider now using c e n t r a l differences (a = 1/2) to solve Equation (C.17). From-Equation (C.22) i t i s seen that numerical d i s p e r s i o n w i l l always occur except i n the t r i v i a l case of u = 0. According to Leendertse [1971b] there i s no numerical dispersion when c e n t r a l d i f f e r e n c e s are used. He apparently assumes that 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 describe the advection process, and then uses t h i s i n c o r r e c t assumption to demonstrate that numerical d i s p e r s i o n w i l l occur when upstream or downstream differences are used. The use of downstream dif 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 f f = -U (C.25) The use of downstream differences with Equation (C.25) d e f i n i n g the g r i d spacing can be interpreted as an attempt to determine upstream conditions from those downstream by working backwards through time (hence the negative sign) along the c h a r a c t e r i s t i c curve i n the (x,t) plane. Although t h i s procedure i s im- p r a c t i c a l , i t would c o r r e c t l y determine the preceeding concentration d i s t r i b u t i o n s 153 with no numerical di s p e r s i o n i f the f i n a l concentration d i s t r i b u t i o n - was known. ': B e l l a and Grenney [1970] investigated the numerical di s p e r s i o n r e s u l t - ing from the advection of a slug load by various f i x e d mesh f i n i t e difference;: schemes. They found that the c o e f f i c i e n t of numerical di s p e r s i o n was as given by Equation (C.21) and t h a t no numerical dispersion occurred when up- stream dif f e r e n c e s were used with Equation (C.23) d e f i n i n g the g r i d spacing. Numerical dis p e r s i o n always occurred with c e n t r a l or downstream d i f f e r e n c e s . Fox [1971] made a Fou r i e r s e r i e s analysis of the d i f f e r e n c e schemes of B e l l a and Grenney and showed that the e f f e c t of d i s c r e t i z a t i o n was to introduce ampli- tude and phase errors into the s o l u t i o n of the d i f f e r e n c e equations. His r e s u l t s demonstrate that neither amplitude nor phase errors occur when upstream d i f f e r - ences are used with a g r i d spacing according to Equation (C.23). In conclusion, numerical dispersion occurs 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 simulate the advective transport process. (Although numerical dispersion has conly been demonstrated f o r e x p l i c i t d ifference schemes, i t also occurs i n i m p l i c i t d i f f e r e n c e schemes). The truncation error of the simple diffe r e n c e schemes discussed here corresponds t o an actual p h y s i c a l mode of mass transport and w i l l modify the advective transport and any d i s p e r s i v e transport that i s occurring i n a r i v e r or estuary. To c o r r e c t l y simulate the advective transport process, i t i s necessary to solve the advective mass transport equation along the c h a r a c t e r i s t i c curve i n the (x,t) plane. The magnitude of the numerical dispersion depends d i r e c t l y 2 2 on the magnitude of 3 c/3x , and so i s greatest for a slug load and l e s s f o r a continuous release. A continuous release can be treated as a succession of 154 slug loads, and apparently the numerical dispersion due to any p a r t i c u l a r slug i s compensated by the numerical dispersion of neighbouring slugs, as has been noted by B e l l a and Grenney [1970]. In d e r i v i n g Equation (C.l) and w r i t i n g the d i f f e r e n c e equation (C.17), i t i s i m p l i c i t l y assumed that c(x,t) v a r i e s i n a continuous manner with x and t . This i s c e r t a i n l y not the case for a slug load, as i s seen i n Figure C . l . I t i s only along the character- i s t i c curve that c(x,t) v a r i e s continuously, as i s also apparent from Figure C . l . Thus, i n using Equation (C.17) to advect a slug load, there i s the added problem that the p a r t i a l d e r i v a t i v e s 8c/8t and 9c/9x are not defined, and the numerical di s p e r s i o n can be interpreted as an attempt to define them. Numerical dis p e r s i o n can be eliminated by solving the equation. advective-dispersive mass transport along the c h a r a c t e r i s t i c s of advective information..propagatation. .Such a.^numerical s o l u t i o n i s r e f e r r e d to as a c h a r a c t e r i s t i c method i n Section 2.2, where the problem of the bookkeeping of s o l u t i o n r e s u l t s i s discussed. Numerical dis p e r s i o n can be c o n t r o l l e d by the use of more sophisticated d i f f e r e n c i n g schemes [see Fox, 1970], or by reducing the actual d i s p e r s i o n c o e f f i c i e n t s by E according to Equation (C.21). Numerical dispersion i s apparently reduced i n f i n i t e element solutions [Price et al.3 1968; Fox, 1970], but because of the f i x e d g r i d nature of f i n i t e element solutions, the numerical di s p e r s i o n i s probably not t o t a l l y eliminated. C.3 STABILITY I f a f i x e d g r i d d i f f e r e n c e equation i s used to 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 equation with constant c o e f f i c i e n t s , the von Neumann s t a b i l i t y condition requires that the eigenvalues of the a m p l i f i c a t i o n matrix 155 should not exceed unity i n absolute value for p h y s i c a l l y s t a b l e systems [Richtmyer and Morton, 1967]. To i l l u s t r a t e these ideas, consider the ex- p l i c i t f i n i t e d i f f e r e n c e scheme of Equation (C.17) that was used to approxi- mate the advective transport equation. This d i f f e r e n c e scheme can be wri t t e n c n + 1 . c „ . _ « t , ( 1 . a l ( c » . ^ + „ ( c n + i . c n ) } ( c 2 6 ) where a = 0 f o r upstream d i f f e r e n c e s , a = 1/2 f o r c e n 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 . The von Neumann s t a b i l i t y c o n d i t i o n is based on a Fourier Series a n a l y s i s of the d i f f e r e n c e equation. At any time, the concentration p r o f i l e along the r i v e r or estuary can be considered to be composed of a number of Fo u r i e r components. The co n t r i b u t i o n of the k'th component to the concentration at point jAx at time nAt is given by „n+l „n r . , . . i C . = C . exp i lki Ax i where i = / - l . S u b s t i t u t i n g i n t o Equation (C.26) f o r the k'th F o u r i e r component and s i m p l i f y i n g gives n+1 „ n C = G C where G = l-0(l-2a)(1-cosO) + 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 of the di f f e r e n c e equation (C.26) . As C only depends on C n , the eigenvalue of the a m p l i f i c a t i o n matrix is given by the value of G from Equation (C.27) . Thus, the von Neumann s t a b i l i t y c o n d i t i o n 156 requires {l- 8(l-2a)(1-cosO)} 2 +{gsin 0 2 } < 1 Evaluating t h i s f o r the three cases of 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 given by < Ax — or At > U ' ( C I f the 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 unacceptably amplified, and the value of the concentration, as deter- mined by Equation (C.26) w i l l o s c i l l a t e with ever-increasing amplitude. Richtmyer and Morton [1967] give a very c l e a r example of t h i s o s c i l l a t i o n f o r the simple d i s p e r s i o n equation (C.9). In the hydrodynamic equations, the i n s t a b i l i t y manifests i t s e l f as an o s c i l l a t i o n of the water depths and v e l o c i t i e s (the water depths eventually becoming negative). Note that the von Neumann s t a b i l i t y c o n d i t i o n has been derived f o r l i n e a r equations with constant c o e f f i c i e n t s . This implies t h a t the U of Equation (C.26) i s con- stant, or that the flow i s uniform and steady. In a s i t u a t i o n where U var- i e s with x and t , the s t a b i l i t y c ondition i s assumed to be determined by the worst case of Equation (C.28) along the estuary. Consider the simple d i s p e r s i o n equation (C.9). This can be approx mated by the 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 c n + l _ c n = ^ { c " - 2C n + C n > 3 3 ( A x ) 2 3+1 3 3-1 157 and i f E i s constant, the s t a b i l i t y condition i s given by [Richtmyer and Morton, 1967] 2EAt . < (Ax) 2 T or -ll̂ — >_ 2E (C.29) The hydrodynamic equations (C.3) are non-linear and have non- constant c o e f f i c i e n t s . Richtmyer and Morton [1967] l i n e a r i z e the equations and obtain the following s t a b i l i t y condition ^ > U ± /qT A t _ - < c - 3 ° ) I t i s apparent that Equations (C.28), (C.29) and (C.30) are the f i n i t e d i f f e r e n c e equivalents of Equations (C.7), ( t c.ll) and (C.14), which describe the c h a r a c t e r i s t i c curve of information propagation i n the (x,t) plane (see Section C . l ) . The p h y s i c a l reason f o r t h i s can be seen from a consideration of Equation (C.26), the ^ e x p l i c i t f i n i t e d i f f e r e n c e approxima- t i o n to the advective transport equation. This equation states that the value of C at the p o i n t jAx a t time (n+l)At depends on the value of C at points (j- l ) A x , jAx and (j+l)Ax at time nAt. If t h i s advective system i s disturbed at point jAx at time nAt, as shown i n Figure C.6, the disturbance propagates through the system according to Equation (C.7). If the s t a b i l i t y condition of Equation (C.28) i s v i o l a t e d , the disturbance w i l l reach the point (j+l)Ax, 1 5 8 Stable Explicit Scheme uAt < Ax • Ac -o- Initial Position of Disturbance Position after At time = (n + I )At uAt U time = nAt Unstable Explicit Scheme uAt > Ax : Ac Position after At time = (n+ I )At Initial Position of Disturbance j - l uAt U Ax Ac time - nAt Figure C .6 Stable and Unstable E x p l i c i t Advective Schemes 159 and a l t e r the concentration there, before the concentration at point jAx has been updated.; according to Equation (C.26) . In other words, the concen- t r a t i o n at point jAx at time (n+l)At now depends on what i s occurring at a distance greater than Ax from jAx (see Figure C.6). Obviously, under these conditions the dif f e r e n c e equation (C.26) i s no longer p h y s i c a l l y meaningful, and i t i s only to be expected that i t behaves i n some strange manner. Simi- l a r reasoning holds f o r the r e l a t i o n s h i p of Equation (C.29) to (C. l l ) f o r the dis p e r s i v e equation, and Equations (C.30) to (C.14) f o r the hydrodynamic equation. The problem of s t a b i l i t y only a r i s e s with e x p l i c i t d i f f e r e n c i n g schemes, where the time d i f f e r e n c i n g i s forward. In i m p l i c i t d i f f e r e n c i n g schemes, where the time d i f f e r e n c i n g i s backward, the values are simultaneous- l y determined a t each g r i d point along the estuary. This 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 greater than Ax i n the time increment. A differ e n c e scheme i s simply a?-means of determining the value CN+^~ from the value C n. In e f f e c t , the differ e n c e equation maps C into C , and the eigenvalues of t h i s mapping (in other words, the eigenvalues of the a m p l i f i c a t i o n matrix) define the characterise 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]. I t i s only to be expected t h a t these eigenvalues should r e f l e c t the underlying physics of the problem. C. 4 SUMMARY For any one-dimensional p h y s i c a l system, there are characteristic-" curves i n the (x,t).plane that d i r e c t l y r e f l e c t s the ph y s i c a l behaviour of the 160 system. These curves are often hidden or masked when a d i f f e r e n t i a l equa- t i o n i s derived r e l a t i v e to a f i x e d g r i d i n the (x,'t) plane. The problem of numerical d i s p e r s i o n occurs because the equation of advective transport i s solved r e l a t i v e to a f i x e d g r i d , rather than along i t s appropriate charac- t e r i s t i c . In an e x p l i c i t d i f f e r e n c e scheme, the problem o f s t a b i l i t y a r i s e s for exactly the same reasons. If the equations are solved along t h e i r charac- t e r i s t i c curves (or by an i m p l i c i t method) there are no s t a b i l i t y r e q u ire- ments, and the r e l a t i v e s i z e of Ax and At are determined by the v a r i a t i o n i n the dependent v a r i a b l e . APPENDIX D DETAILS OF THE SOLUTION SCHEMES OF THE HYDRODYNAMIC AND MASS TRANSPORT EQUATIONS D.l 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 ifference method of Dronkers [1969] was used to obtain a numerical so l u t i o n t o the hydrodynamic eguations. The f i n i t e d i f f e r e n c e forms of Equations (3.1) and (3.2) are r e s p e c t i v e l y 2n+l 2n-l .At , . 2n+l,..2n-l 2n-l, At,,2n .2n -» U2m = U2m " (2Ax"} U 2m ( U2m +2 " U2m-2) ~ gAx- ( h2m +l ~ h2m-i ) c V " (D.l) Y2m and 2n+2 _ 2n At 2n+l 2n 2n+l 2n n2m+l " h2m+l " . _2n (U2m+2 A2m+2 " U2m' A 2 m ) ( D , 2 ) ^xb^ 2m+l where and h^ are re s p e c t i v e l y the mean v e l o c i t y and surface elevation a t the g r i d point given by x = mAx at time t = nAt. Note that the dif f e r e n c e scheme employs c e n t r a l differences and that the "bar" sign has been dropped from the 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 to avoid confusion with the subscripts and superscripts'. (This convention w i l l be followed when presenting f i n i t e d i f f e r e n c e q u a n t i t i e s ) . Because an 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 to solve the hydrodynamic equations, the r e l a t i v e s i z e of Ax and At i s governed by the 161 162 s t a b i l i t y c r i t e r i o n Ax • -• • r-=y A t - ' U ± / g Y This s t a b i l i t 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 of the f r i c t i o n term, the hydrodynamic equations are 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 helps maintain s t a b i l i t y . In f a c t , s t a b i l i t y was found to be de- pendent on a minimum value of f r i c t i o n . For the c r o s s - s e c t i o n a l gemoetries and flow and t i d a l conditions of Section 4.1.4, the hydrodynamic equations became unstable when Manning's "n" was les s than 0.012. (Manning's "n" was constant throughout the estuary f o r t h i s i n v e s t i g a t i o n ) . The s t a b i l i t y c r i - t e r i o n of Appendix C i s independent of the e f f e c t s of f r i c t i o n . However, i t was derived from a l i n e a r s t a b i l i t y analysis on the non-linear hydrodyna- mic equations. The true non-linear s t a b i l i t y c r i t e r i o n may be more s t r i c t than the derived 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 the necessity of a minimum l e v e l of f r i c t i o n to preserve s t a b i l i t y . The s o l u t i o n points f o r the f i n i t e d i f f e r e n c e scheme are shown i n Figure D . l . I t i s seen that the v e l o c i t y and elevation points are stag- gered i n both time and space. In e f f e c t , the values of v e l o c i t y and water surface elevations are marched forward through time i n a "leap-frog" manner. From Equation D . l , the v e l o c i t i e s at any time step 2n+l are determined by the surface elevations at time step 2n and the v e l o c i t i e s a t time step 2n-l. The surface elevations a t time step 2n+2 are then determined from the v e l o - c i t i e s at time step 2n+l by'using Equation•(D.2). The f i x e d mesh of space points or ";stations" used i n sol v i n g the hydrodynamic equations i s shown i n Figure B.4. V e l o c i t i e s are evaluated at odd -numbered stati o n s and water surface elevations are evaluated a t even- 2n+2 2n+l 2n 2n-l u 1 u s A u 1 % t ' > 1 h X ,h ? u A V A / U \ i u % f % h ? h f \ ? U (> < > u c u > 2m-3 2m-2 2m-| 2m 2m+| 2m + 2 x longitudinal distance Figure D.l E x p l i c i t F i n i t e Difference Grid of the Hydrodynamic Equations [After Dronkers, 1969] 164 numbered stations. At channel junctions, i t i s necessary that the junction s t a t i o n be a water surface elevation s t a t i o n , and t h i s accounts for the non- coincidence of the junctions of the model estuary and r e a l estuary i n several cases (see Figure B.4). In applying the hydrodynamic model, a design t i d a l cycle of selected flow and t i d a l conditions i s chosen. The freshwater inflow forms a boundary condition at Chilliwack and the t i d a l r i s e and f a l l of the water surface forms boundary conditions at the seaward ends of the four channels that emerge from the d e l t a . The v e l o c i t y and water surface elevations through- out the estuary are.-rset to i n i t i a l values, and Equations >(D.l) and (D.2) are then used to march the v e l o c i t i e s and water surface elevations through time. The i n i t i a l values of v e l o c i t y and water surface elevations do not have to be exact as the model w i l l converge to the true i n i t i a l values a f t e r several t i d a l cycles.; The hydrodynamic model was programmed i n high 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 consisted of 670 active s t a t e - ments and required approximately 40 seconds t o analyse two complete t i d a l c y c l e s . (The f i r s t t i d a l c y c l e was required f o r the estuary to converge to the true i n i t i a l values of v e l o c i t y and water surface e l e v a t i o n ) . The values of v e l o c i t y and c r o s s - s e c t i o n a l area were recorded a t half-hourly i n t e r v a l s and l a t e r used i n solving the t i d a l l y varying mass transport equation. D.2 NUMERICAL SOLUTION OF THE TIDALLY VARYING MASS TRANSPORT EQUATION From Section 3.2 i t i s seen that the s o l u t i o n of the t i d a l l y vary- ing mass transport equation over any time increment involves an advection step, a dispersion step and a source-sink step. The f i n i t e d i f f e r e n c e form 165 of Equation (3.4) used to advect the moving points along the c h a r a c t e r i s t i c s during the advection step was n+1 n n.. x_. = x.. + u At (D.3) where n+1 . x. i s the p o s i t i o n of moving point j at the end of time -1 increment n; x" i s the p o s i t i o n of moving poi n t j at the s t a r t of 2 time increment n; 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 x1? and x1?"*"̂ " during time increment n. ^ ^ The hydrodynamic model was used to obtain the l o n g i t u d i n a l v e l o c i t i e s and cro s s - s e c t i o n a l areas throughout the estuary a t half-hourly i n t e r v a l s during the t i d a l c y c l e . The value of u n can be obtained, from these v e l o c i t i e s . In the di s p e r s i o n step, the concentration of the moving points i s adjusted f o r the e f f e c t s of dispersion during the time increment. Both an 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 investigated f o r the dispersive step. The i m p l i c i t scheme was the Crank-Nicholson scheme described i n Richtmyer and Morton [1967]. Irrespective of whether an i m p l i - c i t or e x p l i c i t scheme i s used, the "information" propagates through a d i s - persive system according to x 2 = 2Et (D.4) as i s discussed i n Appendix C. The f i n i t e d i f f e r e n c e form of Equation (D.4), namely * < <§>2 defines the s t a b i l i t y c r i t e r i o n of an e x p l i c i t scheme/ as i s also discussed i n Appendix C. Equation (D.5) also determines the response of an i m p l i c i t system, and although the i m p l i c i t system i s unconditionally stable, the con- vergence c r i t e r i o n i s r e l a t e d to Equation (D.5). In other words, i f At i s s i g n i f i c a n t l y larger than the At of Equation (D.5) an i m p l i c i t scheme may converge to the wrong s o l u t i o n . Of the two schemes, the i m p l i c i t scheme was s l i g h t l y f a s t e r , but because of the large somewhat i l l - c o n d i t i o n e d matrices involved (of order 150) there 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 of round-off e r r o r s . Consequently, the simpler e x p l i c i t scheme was used. (The i l l - c o n d i t i o n e d nature of these matrices was due to the v a r i a b l e spacing of the moving points. This v a r i a b l e spacing i s discussed i n Chapter The following 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 equation was used to Equation (3.6) 3). n+1 n c. + 2At 3 A.(Ax) . 3 3 n+1 j + l , j - l A x ' j , j - l ^ j (D where c. i s the concentration of moving point j at the s t a r t of ^ time increment n; j , j - l n = x. - x 3 n j - l x. i s the p o s i t i o n of moving point j at the s t a r t of time 3 increment n; (EA) n j , j - l i s the average value of EA between moving points j and j - l during time increment n; xj,'[ and A. i s the average cr o s s - s e c t i o n a l area at moving point j during time increment n. 167 Equation (D.6) i s the usual 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 approximation except that i t i s applied over a g r i d where Ax i s not constant. The s t a b i l i t y r e - quirement of Equation (D.5) generally r e s u l t s i n a At smaller than the basic time increment of one hour. When t h i s occurred, Equation (D.6) was solved repeatedly within the hour f o r as many i t e r a t i o n s required by Equa- t i o n (D.5). (This assumes that the r e l a t i v e spacing of the p a r t i c l e s and the values of E and A remain constant during the hour — which i s a reason- able assumption). F i n a l l y , i n the source-sink step, the concentrations of the moving points are adjusted f o r the e f f e c t s of any source-sink processes occurring. The following f i n i t e d i f f e r e n c e equation can be used to approximate the source- sink equation (3.7) +1 n c. = cP + At .Z.S. (D.7) 3 3 1=1 i although i t i s noted th a t Equation (3.7) can be solved a n a l y t i c a l l y during the time increment. The t i d a l l y varying mass transport model was programmed i n high 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 of some 1,300 a c t i v e statements and required approximately 100 seconds to analyse s i x t i d a l c y c l e s . D.3 NUMERICAL SOLUTION OF THE TIDALLY AVERAGED MASS TRANSPORT EQUATION The one-dimensional, t i d a l l y averaged mass transport was solved by the method of Thomann [1963]. In t h i s s o l u t i o n , the estuary i s divided into a number of segments or boxes, as shown i n Figure D.2, and each segment i s assumed to be completely mixed. If the t i d a l l y averaged transport processes  169 and waste discharges are steady i n time, a inass balance over segment i for a substance undergoing firsfcrorder decay gives (see Thomann [1971] for de- t a i l s ) Q{a. , .c, + (1-a. . .)c.} - Q{a. _ . c . + (1-a. . J c .} * i - l , i i - 1 i - l , i i i , i + l i i , i + l i + l + E. .(c. -c.) + E. . ,(c. -c.) - K.c.V. + W. = 0 (D.8) i - l , i i - 1 l i , i + l i + l l i l l l where c i s the concentration i n segment i ; V^ i s the volume of segment i ; W. i s the mass of waste substance discharged i n t o segment i per t i d a l cycle; K^ i s the 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 discharge through the estuary (the freshwater discharge); a. i s the t i d a l exchange c o e f f i c i e n t between segments i 1 , 1 + 1 and ( i + l ) ; and • E^ i s the " e f f e c t i v e d i s p e r s i v e " transport between segments l f i and ( i + l ) . The subscript notation of the various terms i s i l l u s t r a t e d i n Figure D.2. The f i r s t two terms on the right-hand side of Equation (D.8) are the t i d a l l y averaged advective transport into and out of segment i . The factor a i s a weighting factor used to determine the concentration at the in t e r f a c e of two segments from the concentration at within each segment. In purely t i d a l flows a i s set equal to 0.5 to allow f o r the e f f e c t s of flow r e v e r s a l , whereas i n a r i v e r flow s i t u a t i o n a i s set equal to 1.0 as the flow i s always downstream. The next two terms on the right-hand side of the equation 170 represent the net disp e r s i v e transport of mass in t o segment i from the neighbouring segments. E' i s given by • " i , i + l E- _ E j , i + 1 A j , i + 1 where and L i , i + 1 E^ i s the e f f e c t i v e c o e f f i c i e n t of dispersion over a ' t i d a l period at the in t e r f a c e of segments i and (i+1); A. i s the c r o s s - s e c t i o n a l area ( t i d a l l y averaged) of the ' i n t e r f a c e between segments i and (i+1); i s the average of the l engths of segments i and X i + D • The f i n a l two terms on the right-handclside of Equation (D.8) represent the e f f e c t s of decay and waste discharge. An equation s i m i l a r to (D.8) can be written f o r each of the n segments of the estuary to give a system of n simultaneous, l i n e a r , d i f f e r e n c e equations. Equation (D.8) can be written where a. . .c. , + a..c. + a. . .c.^. = W. (D.9) i , i - l i - l i i I i , i + l i+I l a. . .. = -a. . . Q - E. , . ; i , i - l i - l , i i - l , i a-.... = Q{a. . - (1- c. , .} + E. + E. + V.K.; i i i , i + l i - l , i i - l , i i , i + l i l a. . . = (1-a. )Q - E. i , i + l i , i + l i , i + l . In matrix notation, the system of Equations (D.9) can be written k £ =  ( D - 1 0 ) where A i s a (nxn) t r i - d i a g o n a l matrix and C and W are (nxl) column matrices. ri, •  J O, r\j ' 171 Thomann [1971] gives d e t a i l s of the complete BOD-DO system of equations. Thomann's model i s e s s e n t i a l l y a f i n i t e f i x e d g r i d f i n i t e d i f - ference model that uses c e n t r a l differences for the dispersive transport, and c e n t r a l differences f o r the advective transport when a = 0.5 and up- stream differences for the advective transport when a = 1.0. (Upstream and ce n t r a l d i f f e r e n c e s are described i n Appendix C). Thomann's model i s simi~- ; l a r to 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. In both schemes the concentra- t i o n at a gridpoint or i n a segment depends on the concentrations at neigh- bouring gridpoints or segments, and consequently the response of the estuary i s determined by a square t r i - d i a g o n a l matrix i n both equations. F i n i t e element solutions are also governed by a square t r i - d i a g o n a l matrix, as i s discussed i n Section 2.2.3. Thomann's differ e n c e equations are unconditionally stable and do not s u f f e r from numerical dispersion, as i s discussed i n Chapter 3. How- ever, there i s a non-negativity requirement f o r each segment given by E. a.,i+l > 1 - X ' (D.ll) i Q 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 discharge of a waste substance into seg- ment i r e s u l t s i n a negative concentration i n the segment. The p h y s i c a l reason for t h i s i s that more substance i s being transported out of the segment per t i d e cycle than i s being added. The substance i s transported upstream by dispersion and downstream by advection and disp e r s i o n . Rearrang- ing Equation (D.ll) gives E. . n > (1-a. . ,,)Q i , i + l i , i + l x which imposes l i m i t s on the r e l a t i v e s i z e of the dispe r s i v e and advective transport processes. 172 F i n a l l y , the ease with which Thomann's approach handles the separate channels of the estuary should be mentioned. Figure D.3 shows the matrix A of Equation (D.10). Note that the three channels of the estuary are contained i n the one matrix. In e f f e c t , the matrix i s p a r t i t i o n e d i n t o three separate blocks, each of which represents a s i n g l e channel. Note that each block or channel i s uncoupled from the others except at the junction s t a t i o n s , where an a d d i t i o n a l term, which i s not t r i - d i a g o n a l , appears i n the rows and columns of the matrix. These a d d i t i o n a l terms r e f l e c t the extra boundary through which mass trans port occurs at the junctions. The t i d a l l y averaged mass transport model was programmed i n FORTRAN f o r s o l u t i o n by d i g i t a l computer. The program consisted of some 250 a c t i v e statements and required approximately 10 seconds to determine the steady state response of the estuary. 173 Figure D.3 The Matrix A of Eguation (D.8) For The Fraser River Estuary APPENDIX E ESTIMATION OF LATERAL DISPERSION E x i s t i n g theories of l a t e r a l d i s p e r s i o n were used to estimate the time of c r o s s - s e c t i o n a l mixing i n the Main Arm - Main Stem of the Fraser River Estuary. The predicted time of c r o s s - s e c t i o n a l mixing appears high f o r the conditions of t h i s study. More recent work on secondary currents i n r i v e r s was used to develop revised estimates of the c r o s s - s e c t i o n a l mixing time. These revised estimates ind i c a t e much f a s t e r mixing over the cross-section. E . l EXISTING ESTIMATES OF LATERAL MIXING For steady uniform flow i n s t r a i g h t channels, the c o e f f i c i e n t of l a t e r a l d i s p e r s i o n i s given by [Fischer, 1969a]. e z = 0.23yU^ (E.l) where y i s the mean depth of cross-section; and i s the shear v e l o c i t y . The presence of bends i n the channel of a r i v e r or estuary induces s p i r a l secondary currents that increase the rate of l a t e r a l mixing. For steady flow around a bend, the increase i n the c o e f f i c i e n t of l a t e r a l d ispersion due to these secondary currents has been estimated as [Fischer, 1969a] -2-3 Se = - U / • I (E.2) z '5 2 k R U * 174 I 175 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 ; k i s Von Kantian's constant; R i s the radius of curvature of the bend; and I i s a fac t o r (negative) that depends on channel f r i c t i o n and k, and i s evaluated by Fischer . (A t y p i c a l value of I i s -0.3). Ward [1972] measured the c o e f f i c i e n t of l a t e r a l d i s p e r s i o n from laboratory experiments of o s c i l l a t o r y flow (purely t i d a l ) i n a channel con- s i s t i n g of a series-of bends. His r e s u l t s are of the form e ' = ayU* (E.3) z * where and e i s the t i d a l l y averaged c o e f f i c i e n t of l a t e r a l d i s p e r s i o n due to o s c i l l a t o r y flow; i s the t i d a l l y averaged shear v e l o c i t y due to o s c i l l a t o r y flow; a i s a fac t o r depending on the r a t i o s of the depth and width of the channels to the radius of curvature of the bends. (For s t r a i g h t channels a = 0.5, and for wide shallow channels with a small radius of curvature a^2.0). In a t i d a l l y varying s i t u a t i o n where there i s both a steady v e l o c i t y compo- nent and a s i n u s o i d a l l y varying component, the t i d a l l y averaged shear v e l o c i t y can be estimated from where h i s Manning's "n " f i s the steady v e l o c i t y component; 176 and U i s the amplitude of the o s c i l l a t o r y v e l o c i t y component. Equation (E.4) i s obtained from the Manning formula r e l a t i n g steady v e l o c i t y to f r i c t i o n a l e f f e c t s . According to Odd [1971], the Manning or Chezy formu- l a t i o n should 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- ing, well-mixed est u a r i e s . The following estimation of dispersion c o e f f i c i e n t s i s l i m i t e d to the Main Arm - Main Stem of the Fraser River Estuary. This i s the widest channel and c a r r i e s the bulk of the flow through the d e l t a . Along t h i s chan- nel, there are 12 s i g n i f i c a n t bends whose r a d i i of curvature range from 7,000 feet to 35,000 fe e t , as i l l u s t r a t e d i n Figure E . l . The v a r i a t i o n of l o n g i - t u d i n a l v e l o c i t y throughout the t i d a l cycle at three stations along the Main Arm - Main Stem i s shown i n Figure E.2. The predominance of the t i d a l component i n the lower reaches and the steady component i n the upper reaches i s apparent. The following procedure was used to estimate the c o e f f i c i e n t s of l a t e r a l dispersion. The v e l o c i t y at each bend was divided into a steady com- ponent U f, which i s given by the freshwater discharge through the t i d a l l y averaged area, and an approximating sinusoidal component of amplitude U t. Equations ;(E-;.i) and (E.2) were used to estimate the c o e f f i c i e n t of l a t e r a l d i s p e r s i o n due to the steady component of v e l o c i t y ('e ) , and Equation ;[E.3] was used to estimate the c o e f f i c i e n t of l a t e r a l d i s p e r s i o n due to the sinu- s o i d a l or o s c i l l a t o r y component ( e ^ ) . Equation (E.4) was used to estimate f t the shear v e l o c i t i e s due to the steady (U^) and o s c i l l a t o r y (U^) v e l o c i t y components. The t o t a l l a t e r a l dispersion was assumed to be the sum of these two separate e f f e c t s . The r e s u l t s of the c a l c u l a t i o n s are shown i n Figure E . l Bends Along The Main Arm - Main Stem Q= 36,500cfs at Chilliwack Tidal Range at Steveston Jan.24,1952 178 24( Hours) 24( Hours) Stat.No.33 24(Hours) 24(Hours) - 2 l Figure E.2 T i d a l l y Varying V e l o c i t i e s TABLE E . l ESTIMATION OF COEFFICIENTS OF LATERAL DISPERSION BE ND  N O.  ST AT . NU MB ER S +J <D £ 0) fa -p fa o 0) w \ i w • ID -P fa o <D W \ D fa ~G • V <D Ul \ <4-l * • D -P fa « o <u w \ +1 * • D -P fa o <u w fa \ A \ o •P * D 1 > i \ -P (J M-l * P 1 > i \ u u CO W \ fa 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 2.7 180 Table E . l , the combined c o e f f i c i e n t of l a t e r a l d ispersion being designated e . c From the r e s u l t s of Table E . l , the average value of the c o e f f i - c i e n t of l a t e r a l d i s p e r s i o n along the Main 'Arm - Main Stem i s approximately 3.3 square fe e t per second. Assuming an average width and depth of channel of 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 discharge at the channel edge, the time required f o r 80 per cent mixing i n the l a t e r a l d i r e c - t i o n i s 55 hours [Ward, 1973]. (The percentage l a t e r a l mixing i s defined as the r a t i o of the l a t e r a l root-mean-square concentration deviation to the aver- age l a t e r a l concentration). This estimate of 55 hours f o r the time of 80 per cent cross- s e c t i o n a l mixing seems very high. Strong secondary currents around bends 9, 10 and 11 are observable i n the estuary during ebb flow conditions. In fa c t , i f the fishermen lose a net i n the Main Stem, i t i s often washed ashore on the north bank of bend 11 by these secondary currents. In a surface f l o a t study during ebb t i d e conditions, the f l o a t s were also washed ashore on the north bank of bend 11. I t seems that the r e s u l t s of Table E . l underestimate the e f f e c t s of the secondary currents on the l a t e r a l mixing process. In view of t h i s , an attempt was made to estimate the v e l o c i t i e s of the secondary currents and the influence of the secondary currents on l a t e r a l mixing. E.2 VORTICITY ESTIMATE OF SECONDARY CURRENTS F i r s t consider the l a t e r a l mixing due to secondary currents. For the sake of s i m p l i c i t y , the v e r t i c a l d i s t r i b u t i o n of the secondary v e l o c i t i e s i s assumed to be l i n e a r , as shown i n Figure E.3. This v e l o c i t y d i s t r i b u t i o n should be a s a t i s f a c t o r y approximation for wide, shallow cross-sections. 181 w(?7) = Ws( !- 2 77)1 V = y / y _ Ws ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ B Figure E.3 Assumed Linear D i s t r i b u t i o n of Secondary V e l o c i t i e s 1 8 2 The v a r i a t i o n of v e l o c i t y with depth i s given by w(n) = W s(l - 2n ) (E.5) n = y/y where Wg i s t h e surface v e l o c i t y of the secondary current. In the manner of Elder [1959] and Fi s c h e r [1969a]the c o e f f i c i e n t of l a t e r a l d i s p e r s i o n can be estimated from e = -y r 2J 1w(n){f T ,w(Ti> [ J n — d n ] d n > d n (E.6) z . ' e o o o y where e ^ i s the c o e f f i c i e n t of v e r t i c a l mixing. If we assume the flow to be steady and. uniform, the v e r t i c a l p r o f i l e of the l o n g i t u d i n a l v e l o c i t y u i s logarithmic, and i s given by [Einstein, 1972] -, , U * • r9.05yU*,, u(y) = r - ln{ £ }} (E.7) is. V where y i s measured v e r t i c a l l y upwards from the bend; and v i s the kinematic v i s c o s i t y . For t h i s logarithmic p r o f i l e , the value of i s given by [Fischer, 1969a] e y =: k (l-n)ny<U* (E . 8 ) where k i s von Kantian's constant. Using the r e l a t i o n s of Equations (E.5) and ( E . 8 ) , the in t e g r a t i o n of Equation (E.6) gives e w 2 -2— = 0.42{-S-} (E.9) y U* 183 This r e s u l t c l o s e l y agrees with the value of e W - 2 — = 0.5{-^}2 y U* U * tha t Ward [1972] gives f o r the same d i s t r i b u t i o n of secondary v e l o c i t i e s . Thus, i t only remains to estimate or measure the secondary v e l o c i t y Wg to determine the c o e f f i c i e n t of l a t e r a l mixing. Quick [1973] has developed a theory of r i v e r meanders tha t n i c e l y p r e d i c t s the types of meanders that are observed i n nature. The existance of secondary flows when bends or meanders are present i n a r i v e r i s well known, and Quick's theory has been used to obtain an estimate of the secondary v e l o c i t y Wg. The basis of t h i s theory i s the generation o f v o r t i c i t y by the gradient of the v e r t i c a l d i s t r i b u t i o n of l o n g i t u d i n a l v e l o c i t y u, and the sub- sequent advection of 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 i n Figure E.4, the sense of the v o r t i c i t y being given by the right-hand-screw r u l e . Because of the large v e l o c i t y gradients of u close to the bed, the v o r t i c i t y i s gener- ated i n t h i s region as intense, small-scale, highly anisotropic v o r t i c e s . The 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 into the flow and i s advected along by u. During t h i s d i f f u s i o n and advection, the v o r t i c i t y degenerates into increasing- l y i s o t r o p i c turbulence 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 e f f e c t s . However, i n any steady flow s i t u a t i o n there must be a balance between the generation 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 logarithmic p r o f i l e of u r e f l e c t s the energy balance of a steady flow s i t u a t i o n , and presumably the v o r t i c i t y balance a l s o . Because of the v e l o c i t y gradients of t h i s pro- f i l e , the flow at any section has a "bulk" v o r t i c i t y given by P = 9u(y) 3y (E.10) average 184 Figure E.4 Interaction of u and £ In A Straight River 185 where the v e l o c i t y gradient i s averaged over the depth of flow. If t h i s argument i s correct, there i s an intermediate stage between the generation and d i s s i p a t i o n of the 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 an organized, anisotropic state. This state i s r e f l e c t e d by the v e l o c i t y gradients of 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 vector U and the v o r t i c i t y vec- to r are perpendicular to each other and do not i n t e r a c t . This i s i l l u s - t rated i n Figure E.4. However, the s i t u a t i o n i s d i f f e r e n t when the flow moves around a bend. I n i t i a l l y , the water on the in s i d e of the bend flows f a s t e r than the water on the outside of the bend, but i n the bend proper, the f a s t e r filaments of flow move to the outside of the bend, and the outside water moves f a s t e r than- the water on the in s i d e of the bend. This i s i l l u s - t r a ted i n Figure E.5. At A the fa s t e r filament i s on the in s i d e of the bend while at B i t i s on the outside of the bend. Because of the l a t e r a l v e l o c i t y gradient of u as the flow moves around the bend, the v o r t i c i t y i s being gener- ated and advected at an angle 0 to uy as i l l u s t r a t e d i n Figure E.5. Note that 0 varies as the flow moves around the bend and t h i s v a r i a t i o n of 0 w i l l depend on the radius of the curvature of the bend. Quick's theory p r e d i c t s t h i s angle?© between u and £ . Thus, there i s a streamwise component of v o r t i c i t y given by E, = k |cos0 (E.U) X z and i t i s t h i s streamwise component that gives r i s e to the secondary current as the flow moves around the':>berid. Substituting the logarithmic p r o f i l e (E.7) i n t o (E.10) and aver- aging over,.the.;depth.between the l i m i t s y = 11.0v/U* and y =,y, the lower l i m i t being an estimate of the depth of the laminar sublayer, gives 186 Figure E.5 I n t e r a c t i o n of u and 5 Around A Bend 187 a U * K = (E.12) where a = l n { r - T - — r } (E.13) 11. Ov and thus from Equation ( E . l l ) = aUJ*cos0 ;(E;14) k y Now, consider the c i r c u l a t i o n around the section ABCD of Figure E.3 due to t h i s streamwise v o r t i c i t y . This c i r c u l a t i o n i s given by «r = <j> ? • dA X ABCD ^ X = i- d U*b cosG (E.15) where b i s the width of the r i v e r . Now, the c i r c u l a t i o n around the cross- section ABCD i s also given by r = <j> U • d l (E.16) X ABCD ^ ^ Assuming there i s s u f f i c i e n t time f o r the streamwise v o r t i c i t y to "arrange" i t s e l f into the l i n e a r d i s t r i b u t i o n of secondary v e l o c i t i e s of Figure E.3, Equation (E.16) can be evaluated as T ~ 2W b (E.17) x s where the contributions over BC and DA have been ignored (the sectidn i s assumed to be wide and shallow). F i n a l l y , putting k = 0.4 and s u b s t i t u t i n g Equation (E.15) i n t o (E.17), the following expression f o r Wg i s obtained W = 1.25aU. cosG (E.18) s * t 188 The r e s u l t s o f a s u r f a c e f l o a t s t u d y i n d i c a t e a c r o s s - c h a n n e l 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 f e e t p e r second a r o u n d bend No. 11 o f F i g u r e E . l . The s t u d y was made on M a r c h 20, 1973 when t h e f r e s h w a t e r d i s c h a r g e a t Hope was 31,000 c u b i c f e e t p e r second. The f l o a t s had d r o g u e s 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 p e r i o d o f t h e s t r o n g ebb phase o f t h e t i d e . The 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- t a i n e d : 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 p e r s e c o n d (Manning's e q u a t i o n ) I n h i s s t u d y o f s t a b l e meanders, Q u i c k f o u n d t h a t f o r g e o m e t r i c a l l y s i m i l a r meanders t o bend No. 11, t h e v a l u e o f 9 v a r i e d from a b o u t 50 t o 70 d e g r e e s . T a k i n g a s above and y = 34 f e e t ( T a b l e E . l ) -5 w = 1.6 x 10 s q u a r e f e e t p e r second; and 0 = 60 d e g r e e s we o b t a i n from E q u a t i o n s (E.13) and (E.18) Wg ? 0.7 f e e t p e r s e c o n d w h i c h a g r e e s w e l l w i t h t h e v a l u e o b t a i n e d f r o m t h e f l o a t s t u d i e s . Assuming a l i n e a r v a r i a t i o n o f s e c o n d a r y f l o w w i t h d e p t h (as i n F i g u r e E.3) and u s i n g a v a l u e o f Wg o f 0.6 f e e t p e r second, t h e c o e f f i c i e n t 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) i s g i v e n by 189 e — = 6 (E.19) ? U* or e z = 30 square f e e t per second. The value of Equation (E.19) i s considerably larger than other : values that have been reported (for example, the values i n Table 1 of Ward [1972]), and i t should be emphasized that the estimated value has not been confirmed experimentally. A pos s i b l e reason f o r t h i s high value i s that the Fraser i s e s s e n t i a l l y a " t i d a l r i v e r " with both High -freshwater and t i d a l flows. The Fraser i s r e l a t i v e l y wide when compared to other r i v e r s where values of e z/y U^ have been measured, but r e l a t i v e l y narrow when com- pared to estuaries (see Table 1 of Ward). I t i s noted that the di s p e r s i o n equation (E.9) for determining the l a t e r a l d ispersion c o e f f i c i e n t i n second- ary flows i s very s e n s i t i v e to the value of (Ws/U^). Using a value of = 30 square f e e t per second, the previous estimate of 55 hours f o r 80 per cent c r o s s - s e c t i o n a l mixing i s reduced to f i v e hours. This w i l l underestimate the time of cr o s s - s e c t i o n a l mixing as the flow i s not steady during the t i d a l c y c l e . Because of the highly assymetrical nature of the t i d e s , there i s only one strong ebb and fl o o d t i d e i n each double t i d a l cycle of 25 hours to generate secondary currents (see Appendix B f o r t y p i c a l t i d e s ) . Thus, within any t i d a l c y c l e , the l a t e r - a l mixing w i l l vary from a maximum during the strong f l o o d or ebb to a minimum at times of slackwater. Consequently, an estimate of the e f f e c t of cross- se c t i o n a l mixing i s probably one t i d a l cycle (12.5 hours) f o r the lower reaches of the estuary and P i t t River and 1-2 t i d a l cycles f o r the upper 190 reaches where the t i d a l e f f e c t s are smaller. On t h i s basis, the e f f e c t i v e or t i d a l l y averaged c o e f f i c i e n t of l a t e r a l d i s p e r s i o n would be about 15 square fe e t per second i n the lower reaches and seven square f e e t per second i n the upper reaches of the Main Arm - Main Stem. On the basis of the r e l a t i v e magnitudes of the t i d a l l y averaged values of y and U^, the l a t e r a l d i s p e r s i o n i n the North Arm i s estimated to be f i v e square fee t per second and 10 square fe e t per second i n P i t t River. At t h i s stage, the theory r e l a t i n g v o r t i c i t y and secondary currents i s neither f u l l y developed t h e o r e t i c a l l y nor confirmed experimen- t a l l y , but future work i s planned i n both d i r e c t i o n s . In concluding, i t i s noted that some values quoted f o r l a t e r a l d ispersion c o e f f i c i e n t s are based onithe r e s u l t s of laboratory experiments. Cross-rsectional mixing i n the presence of secondary currents i s a complex three-dimensional pheno- mena, and i t may be that some components of t h i s process are not being scaled properly i n model experiments. APPENDIX P ESTIMATION OF LONGITUDINAL DISPERSION 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 due to the e f f e c t s of 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 are estimated for the Fraser River Estuary. Simple approximations are given for 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 during the i n i t i a l period before c r o s s - s e c t i o n a l mixing i s complete and during the t i d a l c y c l e . The predicted t i d a l l y varying concentrations during the f i r s t double t i d a l c y c l e were found to be very s e n s i t i v e to assumptions about the 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 . Neither the magnitude nor time-dependent behaviour of the di s p e r s i o n c o e f f i c i e n t s has been v e r i f i e d by f i e l d measurements, and i t i s recognized that they may be i n e r r o r . F.1 GENERAL When e f f l u e n t i s discharged i n t o a r i v e r or estuary i t i s dispersed 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 of both 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, as described i n Appendix A. The e f f l u e n t must disperse over a reasonable depth and width of the estuary before these v e l o c i t y grad- ients can exert 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 process. Most r i v e r s and estuaries are much wider than they are deep and consequently mixing i n the v e r t i c a l d i r e c t i o n i s much more ra p i d than i n mixing i n the l a t e r a l d i r e c t i o n . Thus, when a "parcel" of e f f l u e n t i s i n i t i a l l y discharged, 191 192 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 e s s e n t i a l l y due to 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 gradients. However, as the e f f l u e n t spreads across the cross- section, the l a t e r a l v e l o c i t y gradients exert an increasing influence on the l o n g i t u d i n a l d i s p e r s i o n process. In an estuary, 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 complicated by the e f f e c t s of t i d a l flow reversal;,. If the c r o s s - s e c t i o n a l mixing i s not e s s e n t i a l l y complete within a t i d a l cycle, some of the l o n g i t u d i n a l d i s p e r s i o n due to the o s c i l l a t o r y flow w i l l be "undone" by the e f f e c t s of flow r e v e r s a l [Holley et al.3 1970]. The Fraser River Estuary f a l l s into a c l a s s that Holley et al. [1970] describe as "sinuous, multi-channeled or island-studded estuaries." Because of the complicating e f f e c t s of the bends, islands and junctions of the estuary, 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 can only be r e l i a b - l y determined by f i e l d dye studies. Time and expense precluded such studies, and i n the absence of adequate f i e l d data, the work of Fischer [1966b, 1969b] and Holley et al. [1970] has been used to obtain preliminary estimates of the l o n g i t u d i n a l d i s p e r s i o n . F.2 TIDALLY AVERAGED COEFFICIENTS OF LONGITUDINAL DISPERSION For steady flow the l o n g i t u d i n a l d i s p e r s i o n due to the e f f e c t s of 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 can be r e s p e c t i v e l y estimated as E -2— = 6 (F.l) y u * 193 where u' i s the square of the v e l o c i t y d e v i a t i o n and i s given by 2 - 2 u' = (u(y,z) - u) the over-bar i n Equation (F.2) s i g n i f y i n g the average c r o s s - s e c t i o n a l value. Eguation (F.l) was obtained by Elder [1959] f o r the logarithmic v e l o c i t y p r o f i l e . In obtaining Equation (F.2), Fischer : [1966b] assumed the l a t e r a l mixing to be due to turbulent d i f f u s i o n alone. According to the t r i p l e i n t e g r a l of Equation (E.6), the l o n g i t u d i n a l dispersion due to l a t e r a l v e l o c i t y gradients w i l l vary i n v e r s e l y as the c o e f f i c i e n t of l a t e r - a l d i s p e r s i o n . This i s because mixing over the cross-section tends to "undo" the e f f e c t s of 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 . The l a t e r a l d i s - persion i n the Fraser River estuary i s thought to be quite high because of the e f f e c t s of secondary flows (see Appendix E), and Equation (F.2) i s sub- sequently adjusted f o r t h i s e f f e c t . To apply Equation (F.2) i t i s necessary to have some estimate 2 of u' 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 a t Stations Nos. 14 and 15 on A p r i l 4, 1973. The freshwater flow at Hope was 34,700 cubic fe e t per second, the t i d a l range at Steveston was 11 f e e t and the measurements were taken during the strong ebb phase of the t i d e . The depth-averaged v e l o c i t y p r o f i l e s across the sections are shown i n Figure F . l , and for these conditions, the various v e l o c i t i e s are estimated to be u = 3.6 f e e t per second (measured); = 0.18 f e e t per second (Manning's equation); = 0.9 f e e t per second; and U. = 3.3 feet per second Q = 31,000 cfs (Hope) April 4,1973 194 500 1000 Distance from North Bank ( feet ) 500 1000 1500 Distance 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 At Stations No. 14 and 15, Main Arm 195 where i t i s assumed that the t o t a l v e l o c i t y u consists of a steady compo- nent (freshwater) of magnitude U f and a s i n u s o i d a l l y varying o s c i l l a t o r y component.(tidal) of amplitude Ufc. Stations Nos. 14 and 15 are located i n bend No. 10 of Figure E . l , a region of strong secondary flows. Using the average of the measured -2 values of u' at both st a t i o n s , Equation (F.2) can be evaluated as E — = 3600 (F.3) which i s much higher than recorded values i n other r i v e r s [Fischer, 1966b, Table 1]. However, the value of Equation (F.3) has to be reduced to account f o r the greater c r o s s - s e c t i o n a l mixing i n the Fraser. The r e l a t i v e magnitude of l a t e r a l mixing due to turbulent d i f f u s i o n i s 0.23, as i n Equation ( E . l ) , whereas the r e l a t i v e magnitude f o r the e f f e c t s of secondary currents i n the region of the estuary has been estimated to be 6.0 (see Appendix E). Thus the adjusted value of Equation (F.3) i s E — — = 140 (F.4) y which i s much more reasonable when compared to the values that Fischer l i s t s . Equation (F.4) i s an estimate of the "instantaneous" l o n g i t u d i n a l dispersion due to the l a t e r a l v e l o c i t y gradients e x i s t i n g a t one p a r t i c u l a r time during the t i d a l c y c l e . According to Fischer [1969a], the e f f e c t s of the steady and o s c i l l a t o r y components on the l o n g i t u d i n a l d i s p e r s i o n are 196 separate and a d d i t i v e , and thus Equation (F.4) can be separated into the following steady and o s c i l l a t o r y components 5"f = 140 (F.5) and = 140 (F.6) where For the sake of s i m p l i c i t y , the subscript z has been dropped. Note that Equations (F.5) and (F.6) sum to give the correct combined di s p e r s i o n of Eguation (F.4). To apply the analysis of Holley et at. [1970], the o s c i l l a - t ory component has to be corrected back to a t i d a l l y averaged value. In the manner of Equation (E.4), t h i s i s estimated to be E t ^ = 120 (F.7) y & * In the absence of other f i e l d data, Equations (F.5) and (F.7) have been used to estimate the l o n g i t u d i n a l dispersion due to l a t e r a l v e l o c i t y gradients throughout the estuary. The equations are applied as i s f o r the Main Arm - Main Stem of the estuary, but have been corrected f o r the d i f f e r e n t depths and widths of the other channels to give E_ - f y u * = 75 E } North Arm (F.8) = ,65 y ^ 197 = 120 P i t t River (F.9) Y< Note that there i s no steady dispersion component i n P i t t River, the flow i s purely o s c i l l a t o r y . The estimates of the various 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 ispersion components are given i n Table F . l . If the cr o s s - s e c t i o n a l mixing i s not e s s e n t i a l l y complete within a t i d a l ; cycle, some of the 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 the e f f e c t s of t i d a l flow r e v e r s a l . To account for t h i s e f f e c t , the values of E^ have been reduced t o t h e i r e f f e c t i v e values E£ according to the procedure i n Holley et at. These r e s u l t s are also shown i n Table F . l , the e f f e c t i v e per- iod of flow o s c i l l a t i o n i s designated T and the r a t i o of t h i s value to the time scale of l a t e r a l mixing i s designated T'. F i n a l l y , the steady and e f f e c t i v e o s c i l l a t o r y d ispersion components are summed to give a combined t i d a l l y averaged 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 due to 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 ( E c ) . It i s recognized that the values of the disp e r s i o n c o e f f i c i e n t s given i n Table F . l have been obtained from very l i m i t e d f i e l d data and de- pend heavily on the u n v e r i f i e d estimate of l a t e r a l d i s p e r s i o n from Appendix E. Consequently, the estimates may be considerably i n error. However, the r a t i o E f/y and the absolute value E f are reasonable when compared to values measured i n other r i v e r s [Fischer, 1966b, Table 1] and the maximum value of the e f f e c t i v e o s c i l l a t o r y component E f c of 450 square f e e t per second agrees well with the upper l i m i t of approximately 500 square f e e t per second that Fischer [1969b] suggests. The combined dispersion 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 Discharge at Chilliwack 36,500 c f s ; T i d a l Range at Steveston 11 feet) . . u u • . - . . o o CJ CU CU u * o o o o cj CU cu — CO co D •H cu cu 'CO cu CO co co CO \ \ w • co co 10 CO co \ \ \ U CM CN 1 >i z to \ \ \ \ \ CN CN CN - 4J • u • \ z • 4-> • M-l * • 4-> * • o * • IW • 4J • N • o - W IP W 4J O < D -P D 4J D -P , D 4-> P 4J W 4J fa V Ul, 4-> En a EH fa fa w u EH CO fa ' fa fai fa fa_ fa fa 1-10 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 MAIN ARM 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 MAIN STEM 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 70 PITT RIVER 140-154 2.8 0.11 0.11 — 400 10 25 1.8 400 400 120 199 increase down the Main Stem - Main Arm due to increasing e f f e c t s of t i d a l flows i n the lower reaches, arid t h i s also seems reasonable. The s e n s i t i v i t y of the predicted concentrations to errors i n 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 i s investigated i n Section F.4. F.3 TIME DEPENDENT LONGITUDINAL DISPERSION COEFFICIENTS The t i d a l l y averaged values of the combined dis p e r s i o n c o e f f i c i e n t E c l i s t e d i n Table F . l represent the e f f e c t s of the l a t e r a l v e l o c i t y 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 process 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 following the discharge of a "parcel" of e f f l u e n t , the l o n g i t u d i n a l dispersion i s p r i n c i p a l l y due to 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 gradients alone, as discussed i n Section F . l . (The time scale of v e r t i c a l mixing i s a half-hour). According to Fischer [1969a], the e f f e c t s of 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 on the l o n g i t u d i n a l d i s p e r s i o n are separate and a d d i t i v e . To allow f o r the v a r i a b l e contribution of both components, i t i s assumed that E = E + . E t * T 0 < t < T and } (F.10) E = E t > T c where t i s the time that has elapsed since a "parce was discharged into the estuary; 1" of e f f l u e n t T i s the time scale of l a t e r a l mixing based on the width of section b (an edge discharge i s assumed); and E , E are 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 due ^ c to the e f f e c t s of 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 r e s p e c t i v e l y . 200 Equations (F.10) are a s i m p l i s t i c representation of a complex three-dimen- s i o n a l phenomenom, but they should adequately reproduce both the short- and long-term d i s p e r s i o n e f f e c t s . I t i s only possible to account f o r the e f f e c t s of such temporal v a r i a t i o n s i n E because the t i d a l l y varying mass transport equation has been solved along i t s advective c h a r a c t e r i s t i c s . Con- sequently, the p o s i t i o n of each e f f l u e n t "parcel" and the time that i t has spent i n the estuary i s known, information that i s masked by a f i x e d g r i d s o l u t i o n to the mass transport equation. As well as increasing with time due to the influence of l a t e r a l v e l o c i t y gradients as the e f f l u e n t spreads over the cross-section, 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 w i l l a l s o vary during the t i d a l c y c l e . I t w i l l be minimal during times of slackwater and greatest during the times of strong ebb and fl o o d flow. To inve s t i g a t e t h i s e f f e c t , i t was assumed that 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 given by Equations (F.9) a l s o varied d i r e c t l y as the absolute v e l o c i t y E = (t + S£) y U # 0 <_ t < T ( F . l l ) E = ay t >_ T and__ U * = ° ' 0 6 5 (F.12) where E, y, and u are the instantaneous values of the respective parameters — c during the t i d a l cycle and a i s assumed equal to the tabulated values of E c/y i n Table F . l . The r e l a t i o n (F.12) i s obtained from Manning's equation 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. The average of Equation ( F . l l ) over a t i d a l cycle wasrfound to be a s a t i s f a c t o r y estimate of Equation (F.10). 201 F.4 SENSITIVITY OF PREDICTED CONCENTRATIONS The s e n s i t i v i t y of the predicted concentrations to assumptions about 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 dispersion was investigated f o r a steady discharge of a conservative e f f l u e n t at Station No. 50 on the Main Stem. Figure F.2 shows the v a r i a t i o n i n concentration a t Station No. 50 during a double t i d a l cycle and the concentration p r o f i l e s along the Main Stem 50 hours a f t e r the i n i t i a l discharge. The predicted concentrations have been standardized by d i v i d i n g by the t i d a l l y averaged concentration ob- tained from the mass of e f f l u e n t discharged per t i d a l c ycle and the f r e s h - water discharge at Chilliwack. This i s p l o t t e d as the parameter V^ v/ s i g n i - f y i n g that the r e s u l t s have been obtained from the t i d a l l y varying model. 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 multiple dosing due to flow r e v e r s a l , as discussed i n Section 1.3, r e s u l t i n spikes i n the concen- t r a t i o n p r o f i l e along the channel. I t should be noted th a t the base of these spikes i s i n i t i a l l y only some 500 - 800 f e e t wide, and i s much "thinner" than i t appears i n Figure F.2. The reason the base appears wide i s that the predicted concentrations are extrapolated of the advective c h a r a c t e r i s t i c s onto the standard 5,000 foot space g r i d (see Section 3.2). This g r i d i s too coarse to accurately resolve the i n i t i a l forms of the spikes. I t i s noted that the spikes are c o r r e c t l y resolved on the advective c h a r a c t e r i s t i c s (where necessary, a d d i t i o n a l moving points were added to define regions of rapid v a r i a t i o n , as discussed i n Section 3.3). Note that the concentration of the most seaward spike (E = 0) has been halved by d i l u t i o n from the P i t t River i n passing through the Main Stem - P i t t River junction. Q = 36 ,500c f s (Chilliwack) ) j a n . 2 4 i | 9 5 2 Tidal Range at Steveston = 11 feet J V a r i a t i o n in C o n c e n t r a t i o n at S t a t i o n No. 5 0 6 12 18 T i m e ( H o u r s ) X tv 10 8 6 4 2 0 10 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 M a i n A r m - M a i n S t e m ( 5 0 h o u r s a f t e r i n i t i a l d i s c h a r g e ) (i) E = 0 (ii) E = E y (20 ft.2/scc.) (350 ft?/sec.) t — (iii) E =E ( (iv) E = E y • E c i (v) E = (6 • oc^)y U r 2 0 3 0 4 0 S t a t i o n s a l o n g M a i n A r m - M a i n S t e m 5 0 Figure F.2 S e n s i t i v i t y of T i d a l l y Varying Concentrations to the C o e f f i c i e n t of Longitudinal Dispersion 203 The r e s u l t s of assuming E to be constant and independent of time are shown as curves ( i ) , ( i i ) and ( i i i ) i n Figure F.2. The corresponding values of E are: . Case (i) E = 0 Case ( i i ) E = E (20 square f e e t per second); y and Case ( i i i ) E = E c ( 3 5 0 square f e e t per second). The peak concentration at Station No. 50 i s seen to be quite s e n s i t i v e to the magnitude of E. Eighteen hours a f t e r i t s generation, the spike has been advected downstream to Station No. 35 and i t s concentration i s seen to be s i g n i f i c a n t l y reduced i r r e s p e c t i v e of whether E equals E^ or E^. When E i s allowed t o vary with time according to Equations (F.10) and ( F . l l ) , the predicted concentrations are given by curves (iv) and (v) r e s p e c t i v e l y . Once again, the greatest e f f e c t i s on the peak concentrations at Station No. 50, the differences between both curves being n e g l i g i b l e a f t e r 1.8 hours when the spike i s at Station No. 35. Note the s i g n i f i c a n t increase i n the peak predicted concentration a t Station No. 50 when E i s allowed to vary with u. The reason f o r t h i s i s apparent from the v e l o c i t y v a r i a t i o n a t Station No. 50 during the t i d a l c y c l e . This i s shown i n Figure E.2, the spike being due to the low v e l o c i t i e s and flow reversals around hours 4, 5 and 6. Because of the low values of u, the value of 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 very small compared to other times dur- ing the t i d e l c y c l e . Because of the assymetric nature of the t i d e s , the v e l o c i t i e s i n the lower reaches of the estuary are quite low during the period of time 204 between high-high-water and low-high-water. This i s i l l u s t r a t e d by the v e l o c i t y v a r i a t i o n at Station No. 5i;in Figure E.2, and 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 during t h i s phase of the t i d e than during the strong ebb and flo o d flow that are seen to occur once each double t i d e c y c l e . 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 multiple dosing are most s i g - n i f i c a n t during the slackwaters"around the time between high-high-water and 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 period of the t i d e c y c l e . In conclusion, the peak concentrations at the point of e f f l u e n t discharge are very s e n s i t i v e to assumptions about the form and magnitude of 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 spersion. However, a f t e r a spike has been i n the estuary several t i d a l cycles, i t s peak concentration i s reason- ably i n s e n s i t i v e to the form and magnitude of the c o e f f i c i e n t . U n t i l adequate f i e l d data i s a v a i l a b l e , the assumed temporal v a r i a t i o n s of E i n Equation ( F . l l ) are thought to be a reasonable approximation of what occurs i n the estuary. F. 5 SUMMARY The dispersion of e f f l u e n t i n an estuary i s a complex, time-dependent three-dimensional phenomenon due to an intimate combination of the e f f e c t s of turbulent d i f f u s i o n , 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 and secondary flows. Limited f i e l d data and the r e s u l t s of other peoples' work have been used to obtain estimates 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 spersion. Due to the lack'of f i e l d data, these estimated values may be s u b s t a n t i a l l y i n e r r o r . However, they provide a basis for obtaining preliminary notions of the t i d a l l y varying response of the estuary to waste discharges. For the assumed time-dependent behaviour of 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 the predicted concentrations were found to be r e l a t i v e l y i n s e n s i t i v e to the value of E^. The predicted peak concentrations at t h i s point of e f f l u - ent discharge were found to be very s e n s i t i v e to the assumed time-dependent behaviour of 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 dispersion. It i s thought that the assumed time dependent behaviour i s a reasonable approximation of what happens i n the estuary,-

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
Japan 6 0
France 3 0
China 2 1
United States 1 0
City Views Downloads
Tokyo 6 0
Unknown 3 4
Beijing 2 0
Ashburn 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}

Share

Share to:

Comment

Related Items