A S T U D Y OP L A T E R A L C I R C U L A T I O N I N A N I N L E T by Neil John Campbell A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in PHYSICS We accept this thesis as conforming to the standard required from candidates for the degree of DOCTOR OF PHILOSOPHY Members of the Department of Physics 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 December, 1954 ABSTRACT On the thesis that the lateral f r i c t i o n a l stresses should play an important part in determining the horizontal circulation in landlocked bodies of water, a mathematical model of the circulation in an inlet i s developed. The c i r -culation i s described by a fourth order differential equation A + A = o where y i s a stream function. A solution representing a commonly observed circulation i s ob-tained for a rectangular bay. The significance of Coriolian, f r i c t i o n a l , and mass f i e l d forces in maintaining such a c i r -culation i s discussed. The theoretical model i s tested by means of data avail-able for Burrard Inlet. The data indicate the existence of two net circulations, one at 50 feet which has been attributed to the tides, and the other at the surface which i s influenced strongly by the influx of brackish water from the Fraser River. Relaxation methods are introduced to test the current fields for verification of the differential equation. Despite the more complex boundaries of Burrard Inlet as compared with the rectangular bay investigation, the actual circulation in Burrard Inlet i s found to satisfy the fourth order differential equation within the limit of observational error. This agreement suggests that the lateral f r i c t i o n a l stresses do play an important part in the circulation in inlets. Farther applications of relaxation methods are urged for oceanographic studies. Suggestions are made as to where oceanographic observations should be taken for a study of lateral circulations in inlets or bays. The reasonable agreement of the mathematical model and prototype suggests that lateral effects can be described by the mathematical theory. THE UNIVERSITY OF BRITISH COLUMBIA F a c u l t y of Graduate Studies PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY NEIL JOHN CAMPBELL B. Sc: (McMaster) 1950 M. Sc. (McMaster) 1951 FRIDAY, JANUARY 7th,'1955 a t 3:00 P.M.. IN ROOM 301, PHYSICS BUILDING of COMMITTEE IN CHARGE:' H.F. Angus, Chairman A.R. C l a r k F. H. Kaempffer G. L. P i c k a r d G.M. Shrum W .11. Cameron J.R. Mackay • M. R i r s c h J.L. Robinson E x t e r n a l Examiner - M. Ra t t r a y U n i v e r s i t y of Washington A STUDY OF LATERAL' CIRCULATION IN AN INLET Abs t r a c t Oh the t h e s i s that l a t e r a l f r i c t i o n a l s tresses p l a y an important -part i n determining the h o r i z o n t a l c i r c u l a t i o n " i n landlocked bodies of water, a mathematical'model of the c i r c u l a t i o n i n an i n l e t i s developed. T h e " c i r c u l a t i o n i s described by a f o u r t h order d i f f e r e n t i a l equation. A s o l u t i o n representing a commonly o b s e r v e d ' c i r c u l a t i o n i s obtained f o r a r e c t a n g u l a r i n l e t , T h e - t h e o r e t i c a l model i s ' t e s t e d by means of data a v a i l a b l e f o r Burrard I n l e t . The data i n d i c a t e the existence of two net c i r c u l a t i o n s , one at 50 f e e t which has been a t t r i b u t e d to the t i d e s and the other at the surface which i s c o n t r o l l e d mainly by the i n f l u x of b r a c k i s h water from the Fraser R i v e r . R e l a x a t i o n methods are introduced to t e s t the current f i e l d s f o r v e r i f i c a t i o n of the d i f f e r e n t i a l equation. Despite the .complex bound-a r i e s of Burrard I n l e t as compared w i t h the r e c t -angular model, the a c t u a l c i r c u l a t i o n i s - f o u n d to s a t i s f y the equation. This agreement suggests that the ' l a t e r a l f r i c t i o n a l s t r e s s e s p l a y an important p a r t i n t h e c i r c u l a t i o n i n an i n l e t . 0 GRADUATE STUDIES F i e l d of Study: Physics Nuclear Physics Atomic and Molecular Spectroscopy Theory of Measurement Quantum Mechanics Dynamic Oceanography F l u i d Mechanics Waves and Tides Geophysics Other S t u d i e s : Chemical Oceanography Synoptic Oceanography B i o l o g i c a l Oceanography D i f f e r e n t i a l Equations M.W. Johns and F. Gulbis A.B. McLay A.M. Crooker A.J. Dekker G.L. Pickard; G.L. P i c k a r d G.L. P i c k a r d A.. Pi. Clark J.P. T u l l y W.M. Cameron W.M. Cameron T.E. H u l l ACKNOWLEDGMENTS The author would like to express his gratitude to: Dr. W.A. Clemens for extending the f a c i l i t i e s of the Institute of Oceanography and for his sympathetic interest i n this work, Dr. G.L. Pickard under whose direction and encouragement this investigation was carried out, Dr. W.A. Cameron for his much helpful advice and c r i t i -cism during the progress of this work, Mr. R.F. Hooley, Department of C i v i l Engineering, and to his fellow grad-uate students, particularly Mr. R.W. Trites, and f i n a l l y to the Defence Research Board of Canada for the financial assistance which enabled this work to be undertaken. TABLE OF CONTENTS Page I INTRODUCTION . 1 Classification of estuaries 1 Coastal plain estuaries 3 Fiord estuaries 5 II THE DIFFERENTIAL EQUATIONS OF MOTION 8 III MATHEMATICAL MODEL OF THE CIRCULATION IN A BAY 14 The transport f i e l d 14 The Coriolian and Frictional Fields . . . . 17 The Coriolian f i e l d 18 The f r i c t i o n a l f i e l d 19 The mass f i e l d 20 IV CONTINUITY OF MASS 24 Transport and mass fields with a salt balance 31 Summary from the study of the mathematical models 34 V THE OCEANOGRAPHY OF BURRARD INLET 36 Oceanographic observations 37 Factors influencing the oceanography . . . . 38 Tides 38 Wind 39 Fresh Water 39 Current measurements in Burrard Inlet . . . 41 Current distribution 42 Large ebb 42 Small ebb 43 TABLE OF CONTENTS (Continued) Page Large flood 43 Small flood 44 Summary of the Circulation . 44 The net circulation 45 The distribution of s a l i n i t y i n Burrard Inlet. 47 Survey N 28/ii/51 48 Survey K 20-27/vii/50 49 Survey G 3l/v/50 and l/vi/50 50 Survey M 27/ix/50 53 Summary of sali n i t y distribution . . . . . 54 Mean sa l i n i t y distribution 54 VI VECTOR AND SCALAR FIELDS OF THE CIRCULATION IN BURRARD INLET 57 Verification of a scalar f i e l d with a fourth order differential equation 59 Results 62 Discussion 64 VII SUMMARY AND CONCLUSIONS 68 REFERENCES 73 APPENDIX Discharge through the First Narrows Current Measurements in the F i r s t Narrows Fresh water drainage LIST OF FIGURES Figure l a . \j/ evaluated at the mouth of the bay lb. M evaluated at the mouth of the bay ax 2. Transport or Coriolian f i e l d for the bay 3. Frictional (©) f i e l d for the bay 4. Mass f i e l d with = 1 x 10 5 gm/cm/sec. 5 5. Mass f i e l d with = 2 x 10 gm/cm/sec. 6. Transport or Coriolian f i e l d which satisfies a lateral salt balance 7. Frictional f i e l d which satisfies a lateral salt balance 8. Mass f i e l d which satisfies a lateral salt balance 9. Lower mainland of British Columbia 10. Burrard Inlet 11. Cloud of Fraser River water intruding into Burrard Inlet 12. Cloud of Fraser River water entering F i r s t Narrows 13. Clouds of muddy Fraser River water in the Strait of Georgia 14. Current metering stations Burrard Inlet 15. Current distribution, Large ebb, 5 feet 16. Current distribution, Large ebb, 18 feet 17. Current distribution, Large ebb, 50 feet 18. Current distribution, Small ebb, 5 feet 19. Current distribution, Small ebb, 18 feet 20. Current distribution, Small ebb, 50 feet LIST OF FIGURES (Continued) Figure 21. Current distribution, Large flood, surface 22. Current distribution, Large flood, 18 feet 23. Current distribution, Large flood, 50 feet 24. Current distribution, Small flood, 5 feet 25. Current distribution, Small flood, 18 feet 26. Current distribution, Small flood, 50 feet 27. Net circulation 5 feet 28. Net circulation 18 feet 29. Net circulation 50 feet 30. Station Locations Survey N 31. T-S relationships Survey N 32a. Salinity distribution Survey N surface 32b. Salinity distribution 6 feet 32c. Salinity distribution 18 feet 33a. Station locations Survey K 33b. Salinity distribution Survey K surface 33c. Salinity distribution 6 feet 34. Station locations Survey G 31/V/50 35. T-S relationships Survey G 3l/v/50 36a. Salinity distribution Survey G 31/V/50 surface 36b. Salinity distribution 6 feet 36c. Salinity distribution 18 feet 37. Station locations Survey G l/vi/50 38. T-S relationships Survey G l/vi/50 LIST OF FIGURES (Continued) Figure 39a. Salinity distribution Survey G l/vi/50 surface 39b. Salinity distribution 6 feet 39c. Salinity distribution 18 feet 40. Station locations Survey M 41. T-S relationships Survey M 42a. Salinity distribution Survey M surface 42b. Salinity distribution 8 feet 42c. Salinity distribution 18 feet 43. Mean sa l i n i t y distribution Burrard Inlet 44. Scalar f i e l d of ^ calculated from the velocity f i e l d at 5 feet 45. Relaxed scalar f i e l d of $ ,5 feet, satisfying the differential equation 46. Scalar f i e l d of f calculated from the velocity f i e l d at 50 feet 47. Relaxed scalar f i e l d of ^ ,50 feet, satisfying the differential equation 48. Average yearly r a i n f a l l distribution (inches) Howe Sound, Burrard Inlet. I . INTRODUCTION One aspect of oceanographic research which has been p a r t i c u l a r l y active i n recent years has been the study of estuarine c i r c u l a t i o n . An estuary may be defined as a semi-enclosed coastal body of water having a free connection with the open sea and containing a measurable quantity of sea s a l t (1). Oceanographic investigations on the Canadian West Coast have been focussed pr i m a r i l y on the problems of coastal ocean-ography. As part of t h i s programme the University of B r i t i s h Columbia has c a r r i e d out extensive oceanographic surveys of the B r i t i s h Columbia i n l e t s , which are estuaries within the above d e f i n i t i o n . In conjunction with the studies of estuarine c i r c u -l a t i o n a number of t h e o r e t i c a l models have been proposed to explain some of the physical problems of mixing and dynamics of i n l e t c i r c u l a t i o n . C l a s s i f i c a t i o n of Estuaries Recent studies have brought f o r t h a number of new concepts i n estuarine problems. Some of these advances apply to a l l estuaries, but at the same time many differences have appeared. In order to compare phenomena i t i s necessary to c l a s s i f y estuaries i n terms of fresh water inflow, evapora-t i o n , and geomorphology. 2 Estuaries i n which the fresh water drainage exceeds the evaporation have been termed " p o s i t i v e * estuaries. For those i n which evaporation exceeds the land drainage and r a i n -f a l l the term "inverse* estuary has been applied. A t h i r d group, termed "neutral estuaries 1 1, e x i s t s i n which neither of the above mentioned processes dominate. Only the "positive type* estuary w i l l be considered further. Geomorphic differences i n structure have le d to three further c l a s s i f i c a t i o n s of i n l e t s : coastal p l a i n , f i o r d , and bar-built estuaries. Coastal p l a i n estuaries characterize most of the larger i n l e t s on the East Coast. They are the r e s u l t of subsidence of the land or r i s e of sea l e v e l . These estuaries are usually long and shallow and exhibit a d e n d r i t i c pattern. The g l a c i a l cut f i o r d s found i n B r i t i s h Columbia and Norway are t y p i f i e d by t h e i r extreme length, depth, and steep sides. Bar-built estuaries are formed by the development of an offshore bar with a small channel connecting the basin with the open sea. They are generally located o f f shore-lines of low r e l i e f and shallow water. P a r t i c u l a r attention has been directed toward a study of coastal p l a i n estuaries and the f i o r d s of t h i s contin-ent. The oceanography of Chesapeake Bay, a t y p i c a l coastal p l a i n estuary, has been described by Pritchard (2), who has stressed the mechanism of a s a l t balance i n that estuary. Alberni I n l e t i n Vancouver Island has been described by T u l l y (3), and recent investigations of several i n l e t s on the B r i t i s h Columbia mainland have been reported by Pick-3 ard (4). Cameron (5) and Stommel (6) have proposed theore-t i c a l models of the c i r c u l a t i o n of a f i o r d estuary. Coastal P l a i n Estuaries The oceanographic observations i n Chesapeake Bay and i t s t r i b u t a r y estuaries, c a r r i e d out by Pritchard, permitted a quantitative study of the physical structure and c i r c u l a -t i o n i n the i n l e t . I t i s a two layer system; the deep zone of high s a l i n i t y water has a net motion up the estuary, while the upper zone containing appreciable quantities of fresh water has a net seaward flow. For a steady state the transport of fresh water i n the upper zone i s equal to the fresh water i n -flow, and since the s a l i n i t y of t h i s layer increases to the seaward, a net upward transfer of s a l t takes place from below. Pritchard considered the factors c o n t r o l l i n g the dynamics and s a l t balance i n coastal p l a i n estuaries from a semi-empirical study. He employed as complete a set of equa-tions as possible and used f i e l d data to evaluate the s i g n i -ficance of the measurable quantities. He assumed that a steady state existed i n regard to a time mean taken over one or more t i d a l cycles. The instan-taneous v e l o c i t y and s a l i n i t y terms were replaced by the sura of a mean and random v e l o c i t y , and mean and random s a l i n i t y respectively. Terms involving the products of the instantan-eous v e l o c i t i e s represent the Reynold's or eddy stresses. In oceanography they have been generally replaced by f r i c t i o n a l terms involving an eddy c o e f f i c i e n t and the gradient of the mean v e l o c i t y . The transverse gradient i n s a l i n i t y and slope of the surface of no net motion suggested that the C o r i o l i s and trans-verse pressure forces were s u b s t a n t i a l l y balanced. Extensive measurements at three cross-sections i n the James River estu-ary indicated that more than 75% of the C o r i o l i s force, related to the mean longitudinal motion, was balanced by the mean trans-verse pressure gradient. The remaining part of t h i s pressure gradient was associated with the gradient of the l a t e r a l com-ponent of the eddy stress. The eddy stress components ap-peared to be related to the t i d a l v e l o c i t i e s . The entrainment of s a l t i n the upper layer and the processes involved i n the exchange of fresh and s a l t water were also examined by Pritchard. S u f f i c i e n t measurements were taken to evaluate the significance of the terms involved i n these processes. The mean l o c a l change of s a l t concentration i s a function of the mean advection of the mean s a l t concentration and random f l u x which i s generally termed eddy d i f f u s i o n i n oceanography. Pritchard found that the most important c o n t r i -butions to the control of fresh and s a l t water d i s t r i b u t i o n arose from the mean long i t u d i n a l advection and the v e r t i c a l random f l u x of s a l t . This l a t t e r process appeared to be re -la t e d to the instantaneous t i d a l v e l o c i t i e s rather than to the mean v e l o c i t i e s . 5 The Fiord Estuaries The water structure of a f i o r d d i f f e r s from that of a coastal p l a i n estuary i n that three d i s t i n c t zones may be recognized rather than two. The upper and middle zones cor-respond to the upper and lower zones of the two-layered system where exchange processes take place. The deep zone has been considered as playing a r e l a t i v e l y i n a c t i v e r o l e i n the dynamics of f i o r d c i r c u l a t i o n (3). Recent observations, however, i n d i -cate the existence of substantial currents i n t h i s zone (4 I I ) . The dynamics of a t y p i c a l f i o r d i n l e t have been i n -vestigated t h e o r e t i c a l l y by Cameron who considered a two dimensional model with l a t e r a l homogeneity. He combined the horizontal and v e r t i c a l components of the equation of motion f o r a steady state. A p a r t i c u l a r feature of his study was the retention of the non-linear f i e l d acceleration terms. The i n c l u s i o n of these terms permitted a study of the influence of f i e l d accelerations i n the c i r c u l a t i o n and led to the impor-tant concept of a c r i t i c a l or upper l i m i t i n g v e l o c i t y i n an i n l e t . The existence of a fresh water source i n h i s model introduces the p r i n c i p l e of continuity of fresh water trans-port and at the same time, the p r i n c i p l e of continuity of s a l t . Both of these p r i n c i p l e s were s a t i s f i e d by Cameron's model. The f i e l d acceleration terms are required to main-t a i n a constant fresh water transport. I f there i s a de-crease of fresh water concentration down the i n l e t then an 6 acceleration of the surface water i s required to assure the constancy of fresh water transport. Since the pressure gra-dients are associated with the fresh water distribution they have to be just sufficient to accelerate or decelerate the flow, as the case may be, in order to maintain continuity. Cameron's model closely resembles the mean state in Alberni Inlet. The velocity f i e l d was in accordance with the actual mass f i e l d i n the in l e t . Stommel has also treated a simple two dimensional steady state model of a deep fiord. He considers a two layer system in which the upper layer i s brackish water with the den-sity increasing seaward. The lower layer consists of undiluted sea water of constant density throughout the in l e t . Exchange processes across the interface are permitted only in the up-ward direction. He develops several relationships involving the vertical velocity of mixing of deep water into the upper layer, the thickness of this upper layer, and the density of the two layers. Stommel's model also predicts a c r i t i c a l velocity at which the model breaks down. He points out that this phenomenon occurs when the velocity of the upper layer i s just equal to the velocity of an internal wave at the interface. The horizontal distribution of density calculated from Stommel*s model agrees very closely with that found in Alberni Inlet. His treatment, however, does not take into account the vertical distribution of density as i n Cameron's study. 7 Both models were compared with Alberni I n l e t and the agreement found supports the reasonableness of the assumptions. Although basic differences e x i s t between the hypotheses, s i m i l a r r e s u l t s were obtained. Both solutions l e d to the novel con-cept of a l i m i t i n g v e l o c i t y at the mouth. A l l these investigations have dealt with two dimen-sion a l models i n the v e r t i c a l l o n g i t u d i n a l plane. Cameron and Stommel were forced to assume l a t e r a l homogeneity i n t h e i r models to avoid mathematical complications. Pritchard's study of a coastal p l a i n estuary involved measurements of s a l i n i t y and currents i n transverse sections. I t was designed p r i n c i -p a l l y f o r an investigation of the l o n g i t u d i n a l v e l o c i t y . Harked l a t e r a l differences i n mass d i s t r i b u t i o n and currents have been noted i n various i n l e t s on the West Coast and elsewhere (2) (5). These observations suggest that the effects of side boundaries, horizontal differences i n s a l i n -i t y , and shearing stresses are frequently of such magnitude that they cannot be ignored. The present thesis i s a contribution to the descrip-t i v e and t h e o r e t i c a l knowledge of estuarine oceanography. I t introduces a mathematical model to study some of these l a t -e r a l e f f e c t s . I t includes a l a t e r a l s a l t balance condition, and employs the use of relaxation methods to test a fourth order d i f f e r e n t i a l equation derived from the equations of motion. Two solutions are presented, one representing a solu-t i o n to the net t i d a l c i r c u l a t i o n and another associated with a v a r i a t i o n of the mass f i e l d . 8 I I . THE DIFFERENTIAL EQUATIONS OF MOTION Let the i n l e t be a serai-enclosed embayment with approximately uniform horizontal dimensions and open to the sea. The o r i g i n of the left-handed coordinate system w i l l be taken at the head with the y axis directed seaward along the length and the x axis across the i n l e t . The z axis w i l l be p o s i t i v e downward and reckoned from a l e v e l surface approxi-mating the sea surface. Consider a steady state condition i n which the ver-t i c a l v e l o c i t y and the i n e r t i a l terms are neglected. The motion i s then described by the following system of d i f f e r e n -t i a l .equations: together with the steady state equation of continuity of mass In equations (la,b) u and v denote the horizontal v e l o c i t y components i n the x and y d i r e c t i o n s , p i s the pres-sure, p i s the density, and c = 2w s i n , the C o r i o l i s parameter, A x, A , and A 2 are the dynamic eddy c o e f f i c i e n t s 9 of v i s c o s i t y associated with the turbulent character of large scale flows. In accordance with the usual practice i t w i l l be assumed that the eddy c o e f f i c i e n t s associated with horizon-t a l shear (A^ and A y) are equal. I t w i l l be further assumed that they are independent of p o s i t i o n , A and A w i l l then be x y replaced by A^. The c o e f f i c i e n t associated with the v e r t i c a l shear, A , w i l l be assumed to be a function of z. z Integrating these equations v e r t i c a l l y from the sur-face of the sea to a depth h of no motion, where the stresses vanish, and assuming that the hydrostatic equation holds equations (la,b) become and "" (3) + ^S?) + T v - c p s x = q j ^ , b l\ ^x1 J yv * K >y where {3 i s the mean density of the water. The stresses [A.^Lj and (A have been J $ t c ^ S f c . equated to the components T and T of the wind stress at the / surface while [A i U | and / A are zero since h i s a depth of no motion. The quantities S and S are the components of x y t o t a l volume flow: J u d z a n d Sx = / ^ z . (4) 5f c . Sh. 10 and the quantity Q i s related to the density by the expression (5) Sf< ttc. The equation of continuity of mass transport i s approximat ely j y u v (6) Since the v a r i a t i o n of p i s small i n the x and y as compared \ f o r p r a c t i c a l purposes d i r e c t i o n with the variati o n s i n S and 5 then x y j x } y (7) Eliminating the quantities a n d - 4 % f r o m equations (3a,b) by c r o s s - d i f f e r e n t i a t i o n and subtraction, one obtains the equation Ux* byyVix ay/ iTx jy^/ A i . <8> Let ^(xy) be a transport stream function related to the components of transport by the equations (9) 11 Substituting i n equation (8) one obtains (10) where the three terms on the left-hand side comprise the b i -harmonic function; the term on the right-hand side i s the v e r t i c a l component of the c u r l of the wind stress. This equation i s formally the same as the equation governing the bending of a plate under the action of exter-nal forces. I t i s s i m i l a r to that derived by Shtokman (tf) (8). I t was treated by Hunk (9) who included the planetary v o r t i c i t y term or rate of change of the C o r i o l i s parameter with l a t i t u d e . Hunk solved t h i s equation and introduced the actual wind stress f o r the A t l a n t i c and f o r the Pacific.Oceans. His s o l u t i o n accounted f o r many of the gross features of the general ocean c i r c u l a t i o n . However, the treatment did not include a study of the forces associated with the d i s t r i b u t i o n of mass i n the ocean. These forces are r e l a t i v e l y more impor-tant i n coastal regions where there i s s i g n i f i c a n t d i l u t i o n of sea water by r i v e r discharge. Another feature of coastal c i r c u l a t i o n which should be taken into account are the t i d a l currents which can give r i s e to a net flow i n an i n l e t . I t i s known from physical observations that c i r c u l a -tions e x i s t i n i n l e t s and bays even i n the absence of the wind stress. Assuming also that the middle term of the biharmonic 12 equation i s small, equation (10) then reduces to a homogeneous equation A + ±± = o . (11) JX 4 )y 4 The v e r t i c a l l y integrated equations of motion are now of the form: or and (12) T3T ' 1 :>y3 Mx ^ and (13) These equations show the r e l a t i o n s between the gradients of the mass f i e l d and the f r i c t i o n a l and C o r i o l i a n f i e l d s . D i f f e r e n t i a t i n g equations (123) twice with respect to x and (12b) twice with respect to y and s u b s t i t u t i n g i n equations (12b) and (12a) one obtains V 3 A and (14) Cross differentiating equations (14a,b) and combining, i t will be seen that the integrated mass f i e l d Q also satisfies the reduced biharmonic equation 14 I I I . MATHEMATICAL MODEL OF THE CIRCULATION IN A BAY The Transport F i e l d A special case of an i n l e t i s a bay i n which there i s no fresh water source at the head. The mouth of the bay i s presumed to adjoin the sea. The shorelines are assumed to be symmetrically straight and to extend p a r a l l e l to each other f o r the f u l l length of the bay. The o r i g i n of the coordin-ate system w i l l be taken at the head of the bay with the y axis directed seaward and the x axis across the bay (figure i ) . V (p -O t/> = O figure i The equation JX* + i f = ° <"> i s solvable by Fourier's method i f the boundary conditions are known. The boundary conditions f o r the sides and head of 15 the bay are *u " ° ^ U " ° ' (16) The f i r s t equation states that the boundary i t s e l f i s a streamline; the second equation states that no s l i p takes place at the boundary. The boundary conditions f o r the figure are aX at x = °, at x = L, = 0 = o = O , ax * = o , at y = o, and at y = L = o M P = u> = O, a"> = yV = O . * • ay ' The s o l u t i o n to equation (11) i s expressed i n a product form ^ x(x)yfy), (17) where X(x) i s a function of x alone and Y(y) i s a function of y alone. I t can be shown that the product solutions are of the form V -z. (cos^yt-coskXxVsinXL - sinUu - (sinXx - smli A * ) / C M X L . - COJ)»XL) a and (18) Si fx fi Si' Si Si 16 where sink A w l = 2 Q m Jt (»n+i)tT f l \ * L f J rr d and x n 2L 2 L n = i , 2 , 5 , S h t o k m a n ( 8 ) p # 406. m = '-a.* B can be evaluated i f the function d/ = K(X) i s known at m ' the mouth. The Transport F i e l d Let k(x) at the mouth be (19) »k3fX)(iin»IT - S»nl»3Tlj-( Mh3TtX - sink |Ux] Jcojjn - Cotk jTt] I - sin iTJ to*k j(£ 4 2 iin 3TT iwk JH" which i s the f i r s t term of the Fourier series of the product solution This simple symmetric function of \)f at the mouth i s s u f f i c i e n t to demonstrate the a p p l i c a b i l i t y of the model. I t y i e l d s a c i r c u l a t i o n which i s of p a r t i c u l a r i n t -erest. Other solutions, and hence other c i r c u l a t i o n s , are possible, but they are r e l a t i v e l y ponderous and tedious to calculate. The form of d> and ) f as evaluated at the mouth are shown i n figures la,b. The transport f i e l d or c i r c u l a t i o n evaluated f o r the bay i s shown i n figure 2 where the length of the bay 1 has been taken as 1.41 times the width L. This r a t i o a r ises from the J 2 i n the denominator of the terms comprising the Ym part of the product solution. The simplest case i s that chosen. Other ratios are possible with corres-pondingly different values of Bm. However, the length to width ratio should be approximately unity in order to justify the assumption of equal horizontal eddy coefficients. The direction of flow in figure 2 i s determined by the relations and L - . W - (9) The direction of flow may be reversed by changing the sign of Bm. Two circulations are evident in figure 2, a deep penetration of water into the inlet on one side and a simi-l a r flow out on the other side. A large back eddy exists deep in the bay which i s powered by lateral shear in the region between the two circulations. The largest transports per unit width occur at the mouth of the bay, and these de-crease continuously toward the head. Coriolian and Frictional Fields From the equations *^y* r ^ and (20) I ^ X3 t }y b i t i s possible to determine the lateral distribution of the vertically integrated mass function Q where 18 0 JAzjfAz. (5) ifc- ih Integration of either (20a) l a t e r a l l y or (20b) l o n g i t u d i n a l l y w i l l lead to the desired r e s u l t . In each case two separate f i e l d s are obtained, one a r i s i n g from the f r i c t i o n a l term and the other from the C o r i o l i a n term. The sum of these f i e l d s y i e l d s the Q f i e l d i f the value of the eddy c o e f f i c i e n t , A^, i s known. A l t e r n a t e l y i f the Q f i e l d and the v e l o c i t y f i e l d are known i t i s possible to evaluate A,. The Co r i o l i a n F i e l d : Consider the case of an established transport f i e l d defined by <// (figure 2). Neglecting the f r i c t i o n a l forces, equation (20a) becomes d = ^ J^ d*. <2D After integration (22) and s i m i l a r l y from (20b) q ( Q - 0 > o] = c f f. D The mass f i e l d pattern w i l l be s i m i l a r to the transport f i e l d (figure 2). The above r e l a t i o n implies that i n the absence of f r i c t i o n flow must take place along the l i n e s of constant property. The uv f i e l d represents the C o r i o l i a n f i e l d whose di r e c t i o n and magnitude i s given by the ascendant of y^ . 19 The F r i c t i o n a l F i e l d : By defining a function 6 such that and (23) A± = A, b we may plot a f i e l d (figure 3) analogous to the C o r i o l i a n f i e l d which w i l l also describe the d i r e c t i o n and magnitude of the f r i c t i o n a l forces i n the model. The C o r i o l i a n f i e l d bears a simple re l a t i o n s h i p to the v e l o c i t y , but the f r i c t i o n a l f i e l d does not bear t h i s simple r e l a t i o n s h i p because the f r i c t i o n a l forces can act i n the d i r e c t i o n of the flow as well as i n the opposite d i r e c -t i o n . The 9 l i n e s (figure 3) have the appearance of rec-tangular hyperbolae with the o r i g i n near the centre of the bay. At the boundary the l i n e s are orthogonal to the stream-l i n e </> = o. This c h a r a c t e r i s t i c a r i s e s from the second boundary condition Ai£ -O where n i s normal to the boundary. The d i r e c t i o n i n which the f r i c t i o n a l forces are acting i s along the ascendants of 0. These forces w i l l be a maximum whose magnitude i s determined by the rate of change ©. The f r i c t i o n a l forces generally act i n the opposite d i r e c t i o n to that of the flow, except i n the upper region of the gyral where they act i n the same d i r e c t i o n as the flow. Thus when the flow i s orthogonal to the Q l i n e s 20 the f r a c t i o n a l forces are large and the magnitude of these forces i s determined by the rate of change of Q. When the flow i s not orthogonal to the Q l i n e s then the f r i c t i o n a l forces are smaller. The Mass F i e l d : The mass f i e l d can be determined by adding alge-b r a i c a l l y the Co r i o l i a n and f r i c t i o n a l f i e l d s . The forces associated with a d i s t r i b u t i o n of mass must balance the Coriolian and f r i c t i o n a l forces. Since the l a t t e r forces are directed along the ascendants of ip and 6, the mass f i e l d forces must be directed along the gradients of Q. 1 1 3X 3X (24) and s i m i l a r l y q 1 £ = c f + , D then Q s y ( < f * "I"6) • (25) To effect the combination of the Cor i o l i a n and f r i c -t i o n a l f i e l d the eddy c o e f f i c i e n t must be known. The nor-mal practice i n oceanography i s to determine the mass f i e l d f i r s t , from which the eddy c o e f f i c i e n t i s estimated. Pritchard examined the rel a t i o n s h i p between the mean transverse pressure force associated with a mass f i e l d and the horizontal component of the C o r i o l i s force f o r three sections of the James River Estuary. His computations showed that the transverse pressure gradient was approximately 75% 21 of the C o r i o l i s force, the remaining 25% was balanced by the f r i c t i o n a l forces. Assuming that a s i m i l a r balance of pressure gradients and C o r i o l i a n forces could e x i s t f o r the model, the f r i c t i o n a l forces were evaluated to bring about a s i m i l a r balance of forces. This was achieved by choosing an eddy c o e f f i c i e n t 5 A^ = 1 x 10 gm./cm./sec. f o r a bay of width 6.5 km. The two f i e l d s were combined to give the integrated mass f i e l d of figure 4. The f i e l d does not change s i g n i f i c a n t l y by varying the value of the eddy c o e f f i c i e n t . The effect of using a 5 / s l i g h t l y larger eddy c o e f f i c i e n t , A 1 = 2 x 10 gm./cm./sec., i s shown i n figure (5). (See note 1, page 23.) The flow with respect to these f i e l d s i s perpendi-cular or at some angle to the l i n e s of constant property, the angle of intersection of the streamlines and Q l i n e s increasing with larger values of A^. The prominent features of each mass f i e l d are preserved i n the two figures (4 and 5) with 5 5 Aj = 1 x 10 and 2 x 10 gm./cm./sec. respectively. The central portion of each f i e l d has the basic character of the Co r i o l i a n f i e l d where t h i s term i s large. However, the whole f i e l d i s warped by the presence of the f r i c t i o n a l forces which have the effect of f o r c i n g the l i n e s of constant property to int e r s e c t the boundaries. The highest values of Q are found at the mouth on the side where the water enters the bay. From t h i s region the f i e l d slopes asymmetrically downward to a l l sides. The flow of water at the mouth i s ^ from-points of high Q to low Q or 22 down the sloping sea surface. Large values of Q correspond to regions i n which there i s a high concentration of brackish water and low values of Q to regions where there i s more s a l i n e water. The gyral centred deep i n the bay i s coupled to the c i r c u l a t i o n at the mouth by the l a t e r a l shearing stresses (per unit area) which are not zero between the two c i r c u l a t i o n s . These forces are proportional to the second derivative of the stream function, 'x M y 1 and r - A b (26) The second of these terms, f y , i s zero between the gyrals where v i s zero, but f x remains f i n i t e and becomes zero only at the centre of the gyral. The f r i c t i o n a l forces (per unit volume), on the other hand, are zero between the two c i r c u l a -tions since these forces are proportional to the t h i r d d e r i -vative of the stream function. The flow of water i n the gyral i s down the sloping surfaces of the Q f i e l d (figures 4 and 5) and approximately C o r i o l i a n , but i n order f o r the water to complete i t s path i t must move up the sloping surfaces of Q at some other point. This movement of water i s possible because the forces asso-ciated with l a t e r a l shear between the two gyrals are large enough to drive the water up the sloping sea surface. The stream function y and i t s r e l a t i o n to S x and S i n the model have been based on the assumption of steady 23 flow of an incompressible f l u i d Ai> + L b = o . ( 7 ) This relation considers only the continuity of volume, satisfaction of this condition can be seen from the transport f i e l d of figure 2. However, i t does not follow that the principle of continuity of mass has been met. This condi-tion w i l l be examined in the next section. Note 1. The values of the eddy coefficients A ^ , 1.0-2.0x10 gm./cm./sec., calculated for a rectangular inlet correspond-ing to the dimensions of Burrard Inlet compare favourably with those found in other regions. (Recent investigations of the ratio of A^ to the dimensions of the body of water, McEwen 1950, indicate that this ratio i s approximately 0.3-1.0. The ratio calculated for Burrard Inlet agrees with these results.) McEwen, G.F. 1950. Trans. A.G.U. vol. 31, no. 1, p. 35. 24 IV. CONTINUITY OP MASS For an i n l e t i t i s permissible to neglect the i n -fluence of the small v a r i a t i o n i n temperature and to consider the density as simply a function of s a l i n i t y so that con-t i n u i t y of mass and continuity of s a l t are synonymous. The s a l i n i t y i s a function of x, y, z and t. The processes which maintain or tend to a l t e r the d i s t r i b u t i o n of s a l t i n an i n l e t are advection and d i f f u s i o n . For a stationary d i s t r i b u t i o n of mass these processes are ex-pressed mathematically by the equation where p and u, v, and w are the mean density and mean velo-c i t i e s ; p' and u*, v', and w' are the variations from the mean. The products of the terms Cp'ii'y etc., are the eddy d i f f u s i o n terms. Integrating equation (27) f i r s t with respect to x across the i n l e t , the l a t e r a l d i f f u s i o n and advection terms w i l l drop out since there can be no d i f f u s i o n or advection through the side boundaries. S i m i l a r l y , the v e r t i c a l d i f -fusion and advection terms can be removed by integration from the surface (z = s) to the bottom (z = h), on the assumption that no v e r t i c a l d i f f u s i o n or advection takes place through the surface or the bottom. 25 Equation (27) becomes - / / - L ^ v ) dy - J ^ _L<fV>dxdz =0, (28) This i s an equation f o r the longitudinal advection and d i f -fusion of s a l t . Let usfLrst consider t h i s equation at the mouth of the i n l e t and examine the mass transport through a transverse v e r t i c a l section, assuming that i n t h i s region the longitudinal d i f f u s i o n of s a l t i s small. Then at the mouth h A >^v dx dz = o . (29) Sfc. * Let the density p be i n the form f = ft + *P ' (30) where ^ i s a constant and A p i s an anomaly which may be a function of x, y and z. Equation (29) becomes P . / / v J x J z * / Af V C U C ' Z (31) Sfc ° Sf. ° Equating the f i r s t term of t h i s equation to zero we have the continuity of volume equation which was considered i n the c i r c u l a t i o n of a bay (section I I I ) . The second term equated to zero i s the equation of continuity of mass ano-maly. I t remains to show that t h i s equation can be s a t i s f i e d 26 f o r the c i r c u l a t i o n i n a bay. Variations of density, along the length and width of an i n l e t , give r i s e to a system of forces capable of main-tai n i n g a l a t e r a l c i r c u l a t i o n . The magnitudes of these forces depend on the r e l a t i v e differences i n density and are p a r t i -c u l a r l y large i n the upper zone where appreciable quantities of brackish water e x i s t owing to r i v e r run-off. The l a t e r a l differences i n density can be studied by considering the non-integrated equations of motion. The equations of horizontal motion including the assumptions made i n section I I , and s t i l l assuming that the hydrostatic equation holds, are o - + cpw + A | 1" > o = -^P - c<5u + t b ( 3 2 ) and e n where the effects of v e r t i c a l stress are not being consid-ered. To these must be added the equation of continuity of mass for a steady state. J X u ' yy For an incompressible f l u i d v H + j ? ( r ) z °- ( 3 3 ) >X 3y ' <84> 27 and i t i s possible to define a stream function U - - 1 1 a n d v = ± £ , ( 3 5 a f 6 ) The assumption of incompressibility requires that i n equation (31) the term rh r L (36) p j dx dz = o, and i t remains to prove that (37) Sfc • f o r a s a l t balance. Gross-differentiating equations (32a,b) with r e s -pect to x and y, subtracting, and using equation (34) along with the r e l a t i o n s (35a,b) we obtain ± 1 r ± ± =o, , 3 8 ) which i s the same form as the equation derived i n (section I I ) . The s o l u t i o n w i l l be of the form * - *y (39) Using equations (30) and (35a,b) and d i f f e r e n t i a t i n g (32a) with respect to z and (32c) with respect to x and bearing i n mind that the product terms involving v and the derivative 28 off) are small, then (32a) becomes ^ ° 5xTz 1 iyiJz ' ( 4 0 ) ^ must also be a function of z. Let 5^ be i n the form of a product solution * = ( 4 1 ) s a t i s f y i n g equation (38). Integrating equation (40) l a t e r a l l y one obtains 6 ^ 0 - A f . ) = c ? — - AW — d x ' ( 4 2 ) and equation (37) becomes / / ^ e i i . - Ax. ( J x ) V<Jxdz =o, ( 4 3 ) or i n terms of the product solution of (38) equation (43) becomes t£Jl(lZ<li ( X X ' D X - A L Y " ' N / ( Z Z ' < J Z | X ' C J X J X C ) X = 0 . ( 4 4 ) 3 i . y» 5 V . K '° This form of the steady state continuity of mass equation which i s maintained by advective processes can be made zero by imposing r e s t r i c t i o n s on the v e r t i c a l l y i n t e -grated term common to both parts of the equation. Thus i f / h ZZ'dz = o (45) Sfc-equation (44) has been s a t i s f i e d . Ap i s a function of Z and the v e l o c i t y i s a function of Z. Then from equation (35b) v = x' N/Z. a and from (40) (46) * f = x v z". b t X Y and XT are constant at any point i n the i n l e t and f o r a l l p r a c t i c a l purposes we can write V = K l a and *p = V . b (47) or then (45) becomes 1 f VV Vdz - J_V 2 - °» (48) Sfc ™ or v l _ v l = o h $fc This r e s t r i c t i o n on the v e l o c i t i e s at the surface and the bottom appears too unreal and contributes no further l i m i t a t i o n on the mass d i s t r i b u t i o n . The solution obtained f o r the mathematical model of a bay (section I I I ) s a t i s f i e d the p r i n c i p l e of volume 30 cont i n u i t y , but i t did not necessarily follow that the s o l u -t i o n would indicate a mass d i s t r i b u t i o n maintained i n the steady state by advective processes alone. The reason why t h i s condition was not met follows from an examination of equation (44). The f i r s t term of t h i s equation i s zero since (49) where X = o on the l a t e r a l boundaries. However, the second term of equation (44) i s not 1 1 1 i d e n t i c a l l y zero. I f t h i s were the case then either Y or would have to be zero everywhere. This would reduce the problem to a simple study of C o r i o l i a n flow. A l a t e r a l s a l t balance f o r the model was achieved by incorporating the condition f o r a s a l t balance i n one of the kinematic boundary conditions at the mouth. Thus i f one « • • assumes no f r i c t i o n at the mouth Y i s zero and a l a t e r a l s a l t balance e x i s t s through t h i s cross-section. Elsewhere i n the bay continuity demands that eddy d i f f u s i o n p a r t i c i p a t e s i n the maintenance of a s a l t balance. The assumption made to obtain continuity of s a l t i s not unreal. Cameron*s i n v e s t i -gation of the dynamics of i n l e t c i r c u l a t i o n s showed that the existence of a c r i t i c a l v e l o c i t y at the mouth represented a solution to the equation of motion i n the absence of f r i c t i o n s a t i s f y i n g the condition of a constant fresh water transport. 31 In t h i s respect both t h e o r e t i c a l models are s i m i l a r although i n t h i s thesis the condition of no f r i c t i o n at the mouth must be introduced i n the form of a boundary condition. I t ap-pears then that a more complete solution would have been re a l i z e d i f the f i e l d accelerative terms had been included i n the i n i t i a l equations ( l a , b ) . The assumption of no f r i c t i o n at the mouth implying C o r i o l i a n flow i s based upon a study of Portland Canal by Cameron (10). He found that the mean transverse pressure d i s t r i b u t i o n approximated that required f o r C o r i o l i a n flow. Therefore C o r i o l i a n flow at the mouth of the model i n l e t implies that a l a t e r a l s a l t balance e x i s t s . Transport and Mass F i e l d s with a Sal t Balance Previously the boundary conditions assigned at the mouth were V = and <py -o which implies no c r o s s - i n l e t flow (figure 2). The product solutions f o r the f i r s t term of the Fourier's series are X - ( C O S \ x - c o s h X.*)(sin\L- S i n k X.l) ~ (smX.X - smKX.x) ( c « \ L - c o s t \ L J } a and (50) Si Si Si fit ^ Si S m K X . l where ct - is J I_ c Si 32 With the s a l t balance condition 3 3 £ = o at the mouth the Y term becomes -J = B U ^ h i *»KXij - s io^y cosh)i£ ) (51) Ji J I £ JT ' Evaluating these terms for the i n l e t , a s o l u t i o n represent-ing the c i r c u l a t i o n i n the bay was obtained (figure 6). The transport f i e l d (figure 6) thus s a t i s f i e s the p r i n c i p l e s of volume continuity and mass continuity. I t d i f f e r s s l i g h t l y from that f i r s t derived (figure 2). The depth of penetration into the bay i s much less than i n the previous case. At the same time the whole pattern of the c i r c u l a t i o n has contracted towards the mouth of the bay. Cross i n l e t flow can now e x i s t at the mouth since * ° • The f r i c t i o n a l f i e l d (figure 7) s a t i s f i e s the con-d i t i o n that i t must be zero at the mouth. Consequently the 6 l i n e s intersect the side boundaries rather than cross the boundary at the mouth. High and low values of © are there-fore found inside the bay and near the boundaries. The two f r i c t i o n a l f i e l d s (figure 3 and figure 7) have several differences. The pattern of the f r i c t i o n a l f i e l d associated with a s a l t balance i s s h i f t e d s l i g h t l y toward the head of the bay. The region where t h i s f i e l d i s a minimum i s not between the two c i r c u l a t i o n s as i n the pre-vious example, but rather i s near the lower t i p of the gyral. Comparing the d i r e c t i o n of the f r i c t i o n force with the d i r e c t i o n of flow into the bay the f r i c t i o n a l forces f i r s t oppose the motion and then reverse. This ef f e c t i s evident from figure 7 because the f r i c t i o n a l forces are directed along the ascendants of Q. The magnitude of the forces i s determined by the rate of change of ©. The f r i c t i o n a l forces also act i n the same dir e c -t i o n as the flow along the upper end of the gyral. Elsewhere i n the c i r c u l a t i o n of the gyral they oppose the motion. The c i r c u l a t i o n of the gyral (figure 6) i s more pronounced than i n the previous case (figure 2) i n d i c a t i n g that the l a t e r a l stresses are more e f f e c t i v e i n maintaining the c i r c u l a t i o n . The resultant mass f i e l d which i s obtained by combining the Co r i o l i a n and f r i c t i o n a l f i e l d s i s shown i n figure 8. The value of the eddy c o e f f i c i e n t calculated f o r t h i s example i s 1.5 x 1G 5 gm./cm./sec. An i n t e r e s t i n g balance of the forces involved i n the system i s seen by considering the flow with respect to the mass f i e l d . The flow at the mouth i s predominately C o r i o l i a n or along the l i n e s of constant property. Elsewhere the flow i s across the isopleths of the mass f i e l d . The d i s -t i n c t i o n i n these flows arises from the r e l a t i v e magnitudes of the Coriolian and f r i c t i o n a l forces i n balance with the pressure gradients. Cross isobaric flow occurs i n regions where the f r i c t i o n a l forces are dominant. This flow i s gen-e r a l l y down the sloping sea surfaces; the f r i c t i o n a l forces are thus directed up the slopes. The only exception to t h i s type of flow i s i n the gyral near the mouth where the flow and d i r e c t i o n of the f r i c t i o n a l forces are the same. There are regions i n the mass f i e l d (figure 8 ) and the other two f i e l d s (figures 4 , 5 ) where the Corio-l i a n and f r i c t i o n a l forces are i n balance. Consequently the f i e l d i s divided i nto regions where e i t h e r one of the two forces dominates. The f r i c t i o n a l forces are most con-spicuous near the boundaries while the C o r i o l i a n force i s dominant i n the central area. Summary from the Study of the Mathematical Models The two simple models demonstrate how a l a t e r a l c i r c u l a t i o n can be maintained by a d i s t r i b u t i o n of mass and the associated forces which balance the C o r i o l i a n and f r i c -t i o n a l forces. The mass f i e l d s require a source of fresh water at the mouth to maintain the proper pressure gradients needed for such c i r c u l a t i o n s . Commonly the inflow of fresh water occurs at the head of an i n l e t , and therefore, boundary conditions describ-ing the discharge of a r i v e r would be required. This problem was considered f o r an i n l e t , but an a d d i t i o n a l complication a r i s e s with the boundary conditions at the head. Mathemati-c a l l y there i s a wide choice of such functions describing a r i v e r flow, but these functions must represent p h y s i c a l l y acceptable flow conditions. This would require that the model also s a t i s f y the p r i n c i p l e s of volume and s a l t contin-u i t y . Consequently, some r e l a t i o n must e x i s t between the functions describing the c i r c u l a t i o n at the head and at the mouth. In practice i t would be necessary to determine these functions by physical measurements. The development of the theory i s s u f f i c i e n t to per-mit a comparison of the t h e o r e t i c a l model with f i e l d obser-vations i n an appropriate i n l e t . S u f f i c i e n t data for the f i o r d type of i n l e t are not yet a v a i l a b l e , but current ob-servations and oceanographic data are available f o r Burrard I n l e t . The i n l e t i s approximately rectangular i n shape ( f i g -ures 9 and 10) corresponding to the mathematical model. The place of the r i v e r at the head i s taken by the F i r s t Narrows through which s a l i n e water flows p e r i o d i c a l l y . This periodic flow r e s u l t s from the t i d e , the effects of which cause intense mixing of surface and deep water. The Fraser River supplies a large amount of fresh water at the mouth of the i n l e t . Before a comparison can be made a study of the f i e l d data i s necessary. These data w i l l be considered i n the next section. 36 V. THE OCEANOGRAPHY OF BURRARD INLET I f the theoreti c a l development and discussion of the mathematical model i s to represent natural events i n an i n l e t i t i s necessary to r e l a t e the unknown functions to the v e l o c i t y measurements, and to compare the theo r e t i c a l mass f i e l d with the actual d i s t r i b u t i o n of s a l i n i t y . The oceanographic observations would normally be car r i e d out with a view to s a t i s f y i n g as many of these conditions as pos-s i b l e . The measurements, however, are frequently influenced by external factors, the most important of which are wind, t i d e , and r i v e r run-off. During the F a l l of 1949 and continuing u n t i l the early Spring of 1951 an extensive oceanographic survey of the Fraser River estuary was undertaken by the P a c i f i c Oceano-graphic Group, Nanaimo, B.C. Their cruises included observa-tions i n Vancouver Harbour, Burrard I n l e t , False Creek, and the S t r a i t of Georgia from jus t south of Pt. Roberts to Bowen Island (figure 9). In 1950, i n conjunction with these observa-t i o n s , the Canadian Hydrographic Service c a r r i e d out a three months summer programme of current measurements i n Burrard I n l e t and Vancouver Harbour. A e r i a l photographic surveys of the i n l e t were made by the B r i t i s h Columbia Lands and Forests Department on June 1 and June 10, 1950. The photographs show marked differences i n colour between the Fraser River water and Georgia S t r a i t water. 37 The programme of in v e s t i g a t i o n was stimulated by the Vancouver and D i s t r i c t s Joint Sewerage and Drainage Board to guide t h e i r proposed expansion of the sewage disposal system of Greater Vancouver. The current measurement phase was intended f o r the preparation of hourly current charts as an a i d to mariners (11). The entire project was arranged so that seasonal changes i n the c i r c u l a t i o n and the influence of the Fraser River could be examined i n Burrard I n l e t . Oceanographic Observations Twelve oceanographic stations were established by the P a c i f i c Oceanographic Group i n Burrard I n l e t , where com-plete oceanographic observations were taken at d i f f e r e n t times of the year. Each s t a t i o n was occupied on opposite phases of the tide whenever possible. A complete oceanographic s t a t i o n consisted of meteor-o l o g i c a l observations, v e r t i c a l measurements of s a l i n i t y , tem-perature, and sometimes oxygen. The discussion of t h i s aspect of the oceanography of Burrard I n l e t w i l l be confined to the s a l i n i t y observations. Each of the current observation stations established by the Canadian Hydrographic Service (figure 14) was occu-pied f o r a complete t i d a l cycle of twenty-five hours on two occasions. Gurley-Price current meters were used at a l l the anchor stations f o r measurements of speed. The current d i r e c -t i o n was observed, by one of three methods depending on the 38 s i t u a t i o n namely, (1) a captive d r i f t pole buoy, twelve feet i n length, used f o r surface observations, or (2) metal vanes attached by a l i g h t l i n e to a captive buoy and set f o r speci-f i c depths, or (3) the metal vanes and l i n e were secured d i r -e c t l y to a winch on board the ship (12). Factors Influencing the Oceanography Oceanographic observations are affected by wind, t i d e , and r i v e r run-off, a l l of which are variable. Quite frequently i t i s impossible to estimate t h e i r effects on the observations. A true comparison of the physical parameters with the proposed mathematical model requires that the data be reduced to a steady state. A true steady state does not exist i n the presence of the o s c i l l a t o r y t i d a l currents, but a mean or net state can be described from the data. The i n -fluence of wind, t i d e , and fresh water on the c i r c u l a t i o n must be considered f i r s t . Tides: The diurnal i n e q u a l i t y i n time and height of the tide (figure i i ) i n t h i s region i s controlled predominantly by the moon's declination north or south of the equator. The diurnal v a r i a t i o n i n height of the tide w i l l be distinguished by the range of the tide between high water and low water, since the terms "neap1* and "spring" tides are not applicable i n the usual sense. 6 12 18 & 12. \g T i m e H r s . figure i i Time \nrs. The tid a l currents vary in a similar manner as the tides. However, at the surface the effects of wind and fresh water drainage are superimposed and sometimes prevail. Wind: Reliable current measurements can be made only dur-ing periods of light wind, consequently the observations were restricted to these occasions. Therefore, the effects of wind on the present current observations are assumed negligible. Fresh Water: It w i l l be demonstrated that the effects of fresh water are confined largely to the surface layers of the inlet where vertical and lateral mixing of saline water with the surface water produce large horizontal variations i n s a l i n i t y and hence pressure gradients. The i n t r u s i o n and mixing of muddy Fraser River water i n Burrard I n l e t i s v i s i b l e i n the surface waters (figures 11,12). Although the discharge of the r i v e r varies appreciably during the summer months there i s always s u f f i -cient discoloration to i d e n t i f y the water. The monthly mean discharge of the r i v e r during 1950, metered at Hope, B.C., varied from a low of 23,600 c f s . i n March to a high of 314,000 c f s . i n June (13). During the period of freshet the discharge rate can increase by as much as 30,000 c f s . per day. The maximum discharge recorded was 440,000 c f s . on June 20, 1950. The occurrence of the freshet i n May, June or July i s governed p r i n c i p a l l y by the amount of snow cover and meteorological conditions e x i s t i n g i n the i n t e r i o r of the Province. The heavy annual r a i n f a l l and large drainage area west of Hope add considerably to the t o t a l discharge of the Fraser, increasing i t by an average of approximately 40# at New Westminster (14). The r i v e r discharges into the S t r a i t of Georgia through three main branches: the North, Middle, and South Arms. During the freshet period the d i s t r i b u t i o n of muddy r i v e r water extends well over the S t r a i t (figure 13). The major portion of the discharge takes place through the South Arm, but approximately 6% of the t o t a l discharge occurs from the North Arm (15). 41 Current Measurements in Burrard Inlet Burrard Inlet i s considered as the region bounded by the North Shore, Stanley Park, and Vancouver City including English Bay. The west and east boundaries are shown in f i g -ure 10. Hourly current measurements were taken at a l l sta-tions (figure 14) at depths of 5, 18 and 50 feet by the Hydro-graphic Service. A few measurements were made at 25 feet on three stations, but the observations there have been included with the remainder at 18 feet. During the period of these observations the Fraser River run-off exceeded a discharge of 170,000 cfs. except when station 2 was occupied; the discharge at this time was 135,000 cfs.. The current velocities on the flood and ebb phases of the tides were averaged separately for the large and small ranges of height to give four mean pictures of flood and ebb velocities in the inlet. The net circulation was obtained by averaging a l l the data for the 50 hour period of observa-tion. The mean circulations for the various phases of the tide are presented to point out the characteristic features of the currents in the inlet. The net circulation i s related to these features and i s included to show further detail of the tidal and surface circulations. 42 Current Distribution Large Ebb: At a l l depths, 5, 18, and 50 feet, (figures 15, 16, 17) the ebb pictures show the existence of a strong nar-row jet which extends from the Fir s t Narrows to Pt. Atkinson. The central core of the stream l i e s offshore except in the region between Reardon Pt. and the Fi r s t Narrows. The high-est velocities of the stream occur close to shore between these two points. The jet stream broadens out along i t s path and narrows again at Pt. Atkinson. South of the jet, and west of Stanley Park, a large counter-clockwise gyral exists. It i s clearly defined at 50 feet (figure 17) by stations 17, 12, 13, and 15, but at 18 feet (figure 16) the gyral i s noticeably more elongated. This effect appears to be associated with the higher average ebb velocities of the jet recorded at this depth. The eddy does not sweep the shore of Kitsilano beach, because a return circulation l i e s between the gyral and the shore. Some of the water from this gyral i s returned to the jet stream while part, when sweeping southward, i s deflected outward just north of Jericho Beach and Spanish Banks adding to the return circulation from Kitsilano and False Creek. This broad flow parallels Spanish Banks and moves out in a westerly direction. Ebb waters south of Pt. Grey and in the Strait of Georgia flow northward while the ebb from Howe Sound i s south-ward. The three broad streams from Pt. Grey, Howe Sound, and 43 Burrard Inlet meet west of the outer boundary and flow out into the Strait as one stream. Small Ebbs The current direction during the small ebb (figures 18, 19, 20) exhibits most of the gross features found for the large ebb. The jet persists along the North shore, but with reduced average velocities. The back eddy off Stanley Park i s less well defined than that of the large ebb. The westward movement of water north of Spanish Banks follows the same path as the ebb from the larger tide. Off Pt. Grey the water fans out from the mouth of the Fraser. The direction of flow off Pt. Atkinson, however, i s reversed on this ebb as compared with that of the large ebb. Large Flood: The flood waters move into the inlet from the north, south, and west (figures 21, 22, 23). The three streams join at the mouth and flow into the i n l e t . The main body of the water sets towards the Narrows, while the water at the lateral extremities of the streams fans shoreward. At the surface a large back eddy forms along the North Shore, but at 18 feet and 50 feet this eddy i s not maintained, and a resultant ret-urn circulation takes i t s place. A similar situation occurs in English Bay on a smaller scale. A return circulation dev-elops west of Kitsilano Beach and extends along Spanish Banks. This movement generally takes place during the last stages of flood and i s not very pronounced. The large flood t i d a l range can temporarily seal off the river, preventing any further discharge at the mouth. However, for discharges of the order of 200,000 cfs. or more, there i s some discharge of the river into the Strait. This phenomenon is quite small compared with the effect on smaller flood tides. Small flood: The situation for the flood on the small tide i s similar to that of the large flood. The main part of this c i r -culation i s again directed towards the Narrows with the weak circulation along the south shore s t i l l persisting. A return circulation along the North shore from Sandy and Pilot Coves and out by Pt. Atkinson i s evident at a l l depths. The circu-lation around Pt. Atkinson i s directed up Howe Sound on this flood, while for the large flood the direction of flow i s from Howe Sound into the i n l e t . A strong southerly current i s observed at station 1 similar to that on the small ebb. Both the flood and ebb currents off Pt. Grey (stations 1, 3 and 16), show a divergent flow of water. The smaller range of the flood i s insufficient to contain the river, and hence the brackish and muddy water spreads out into the Strait. Summary of the Circulation: The aerial photographs (figures 11, 12, 13) show that the surface movement of brackish water can be traced visually. However, the current measurements are necessary to 45 yield quantitative information. The direction of ebb remains essentially the same as at other depths, but the ebb flows are more nearly uniform over the whole inlet as opposed to the streaming nature in the deeper waters. Ebb velocities in the jet are greater at 18 feet than at the surface. Off Pt. Grey in the vic i n i t y of the Fraser River, the surface velocities are affected by the fresh water discharge and are two and three times as large as the deeper velocities. At station 14 west of Pt. Atkinson, where there i s l i t t l e brackish water the average velocity i s nearly the same at a l l depths. The combined effects of flood tide and Fraser dis-charge give rise to much higher speeds along Pt. Grey, Span-ish Banks, and the northern reaches of English Bay. The sta-tions which are swept by fresh water exhibit continually higher hourly surface speeds than those at 18 feet. The Net Circulation: The flood and ebb circulations differ at each depth of measurement and reveal the existence of two net circulations in the inlet. Characteristic circulations appear for both the upper and lower depths of Burrard Inlet (figures 27, 28, 29). The net velocities at each station were obtained from the vector sum of a l l the hourly flood and ebb velocities taken during the periods of observation. The magnitudes of the velocities of the net circulation are approximately 10$ of the maximum velocities recorded at the current metering 46 stations. The jet stream which i s a marked characteristic on the ebb tide i s s t i l l evident in the net circulation. The counter-clockwise gyral off Stanley Park remains as part of the deep circulation. The gyral at 18 feet i s more extensive and relatively more dominant than that at 50 feet. This i s probably associated with the higher average velocities and larger lateral shearing forces at 18 feet. The flow of the deeper water (figure 29) from Pt. Grey into the inlet divides into two streams, one branch f o l -lowing around Spanish Banks and flowing towards the Narrows between the gyral and the shore-line, and the other sweeping across the mouth and joining the jet stream south of Pt. Atkinson. The region in the immediate vi c i n i t y of station 10 exhibits l i t t l e character because of the opposing flows. The surface circulation shows an entirely different appearance. The back eddy off Stanley Park has completely disappeared because the main flow from Pt. Grey sweeps directly into the Narrows and the jet stream (figure 27). The circula-tion centred in English Bay i s either in the form of an eddy or a return circulation close along the shore. The observa-tions are too few in this area to establish definitely the nature of flow. Visual observations and photographs from Mt. Hollyburn and West Point Grey indicate that the basic circula-tion in English Bay i s i n the form of a gyral, but that i t can extend westwards as far as the Inner Beacon on Spanish Banks. The difference between the circulation at the surface 47 (figure 27) and at 18 and 50 feet (figures 28, 29) appears to be associated with the large quantities of brackish water flow-ing into the inlet i n the upper 10 feet. The circulation at 50 feet i s attributed to the mean pressure f i e l d associated with the tides, since the fresh water pressure gradients act-ing at this depth are small (see Mean Salinity Distribution). Bearing in mind that the side boundaries are essentially the same for both circulations, the differences in circulation must be due to different forcing functions present at the seaward boundary and the F i r s t Narrows. The current d i s t r i -bution at 50 feet can be considered as representing the net tidal circulation of the inlet while the surface circulation represents the combined circulation related to the fresh water f i e l d and the tides. The relative differences in the distribution of mass at the surface and 50 feet become apparent from the oceano-graphic observations. The inferred movement of water from these observations adds more detail to the progressive changes of the direction of flow taking place during a flood or ebb tide and wi l l be considered next. Distribution of Salinity i n Burrard Inlet The oceanographic observations that were carried out by the Pacific Oceanographic Group from 1949-1951 included locations other than those in Burrard Inlet (16). Unfortun-ately, some of these surveys in the adjacent areas did not overlap with a l l the stations in the inlet (figures 30, 34). 48 A synoptic picture could not be determined from these surveys primarily because of the different relative t i d a l times of ob-servations and the daily changes in the Fraser River discharge. However, data from four oceanographic surveys taken during 1950 and 1951 i n Burrard Inlet yield representative s a l i n i t y d i s t r i -butions for river discharges ranging from approximately 24,000 cfs. in February to 200,000 cfs. in June. Survey N 28/ii/51 (Figures 30, 31 and 32 a,b,c) This survey was carried out over a three hour period between the times of high water and maximum ebb on a large ebb tide (inset figure 31). The average discharge of the Fraser River was 24,500 cfs. The lack of fresh water i s partly res-ponsible for the simple s a l i n i t y distributions (figures 32a,b,c). The surface isohalines (figure 32a) along the North Shore extend parallel to the shore from the Narrows to Pt. Atkinson in the corresponding region of the jet (figure 15). The average ebb velocity of 1.2 knots in the jet i s sufficient for a particle of water to be transported from Reardon Pt. to Pt. Atkinson in three to four hours. The flow becomes para-l l e l to the isohalines after this time. In the region off Stanley Park the distribution of sal i n i t y indicates that the water pushes out into the in l e t as a continually expanding cloud or eddy. The current measurements (figure 15) confirm this deduction, since the movement of water i s largely circula-tory. The eddy gradually expands and moves westwards with the circulatory motion of i t s boundaries decaying. North of Spanish Banks the isohalines become tangent to the Banks i n 49 the direction of flow. The flow eventually crosses the iso-halines near the west boundary of the inlet. A similar picture i s evident at 6 feet (figure 32b) except that the isbhalines appear as tongues along the North Shore. At 18 feet (figure 32c) the difference i s more pron-ounced. These changes are caused partly by the i n i t i a l time lag between the ebb at the surface and the deeper water. The ebb velocities at 18 feet are less than those at the surface for the f i r s t hour of ebb and they do not become larger until the second hour. The sa l i n i t y picture at 18 feet shows part of the structure from the preceding flood north of Spanish Banks, while the distribution of sali n i t y from the Narrows i s just beginning to follow a similar pattern to that at 6 feet. The character of the water masses below 15 feet i s plotted in a T-S diagram (figure 31). The remnants of the previous flood from Ft. Grey are distinguished by the warm water mass at stations 11, 15, 18 and 17, while the colder ebb water from the Narrows i s shown at stations 8, 12, 13, and 16. The high velocities and turbulent conditions existing in the Narrows cause vertical mixing of the warm surface water with the deeper colder water. The resulting water i s well mixed and colder. The surface T-S relations for stations 8, 10, 13, and 16 are related to those of the mixed cold water from the Narrows. Survey K 20-27/vii/50 (Figures 33a,b,c) This survey i s represented in terms of the average 50 distribution of sa l i n i t y for a period of one week. The obser-vations were taken over a three hour interval between the times of maximum ebb and low water on a large ebb tide. The average discharge of the Fraser was 110,000 cfs. for the period of observations. The surface isohalines which extended from Jericho Beach to Pt. Ferguson as indicated in survey N (figures 32a, b,c) do not move appreciably until the later stages of ebb when they begin to bulge out into the central section of the inlet as shown i n figures 33b,c of survey K. The water from English Bay i s the last large mass of water to move in the inlet during the ebb tide. It flows northward into the middle section of the inlet and turns southward off Spanish Banks. However, i t i s not always completely flushed out of the inlet during a single ebb, and i s frequently returned to the inlet on the following flood appearing as a large mass of water on the t i p of the intruding tongue which floods in around Spanish Banks. Survey G 3l/v/50 (Figures 34,35, and 36,a,b,c) l/vi/50 (Fig-ures 37, 38, and 39a,b,c) These two sets of observations were taken on succes-sive days: the f i r s t on 3l/v/50 during the period between low water and maximum flood (inset figure 35), and the second on l/vl/50 between high water and maximum ebb (inset figure 38). The average Fraser River discharge for these two days was 200,000 cfs. 51 The surface isohalines (figure 36a) along the North Shore have a similar distribution to those of Survey N ( f i g -ure 32a) indicating that during a long ebb this pattern becomes characteristic and possibly approximates a steady state dis-tribution. This distribution is not immediately destroyed by the f i r s t stages of flood. The direction of flow as shown by the current measurements (figure 15) would have been along these lines of constant property. The flood i s f i r s t apparent near Spanish Banks as shown in figure 36a. This movement of water takes place early in the flood before any appreciable changes occur in the northern sections of the inlet. (The hourly current measurements at stations 5, 9, and 16, and 11, 17, and 21, show that ebb currents per-s i s t along the North Shore until at least one hour after the time of low water, while flood currents are observed i n the south almost immediately after the change of tide). The low sal i n i t y water at the t i p of the tongue ( f i g -ure 36a) is probably a cloud of brackish water discharged from the North Arm of the Fraser, while the remainder of the tongue i s mixed water from Sturgeon Banks and the South Arm of the river (17) (18). The north side of this large tongue presses over to the North Shore while the south side wedges into English Bay (figure 36a) pushing water westwards along Kitsilano Beach and Spanish Banks and also northwards past Stanley Park. Once the brackish flood water has moved over Spanish Banks and has reached the Narrows the small secondary tongues along the shore-lines begin to recede. This effect i s represented partly by the isohalines extending from Jericho Beach to Stanley Park which are so characteristic of English Bay. The same overall pattern of the distribution of sal i n i t y i s observed at 6 feet and 18 feet. The isohalines at 6 feet along the North Shore have shifted southward from Pt. Atkinson and shrunk to Reardon Pt. (figure 36b), and those at 18 feet to Sandy Cove (figure 36c). The ebb characteristics on survey G l/vi/50 ( f i g -ures 39 a,b) reveal several interesting features. The eddy of brackish water which formerly appeared close to the Narrows (figure 36a).appears to have been transported along the North Shore during the time of the half tides. It i s indicated by the circular salinity pattern at the surface, 6 feet, and 18 feet. The T-S relations (figure 35) for 3l/v/50 show the distinction between the high sal i n i t y cold water of the jet at stations 30, 35, and 36 and the warm less saline water i n the central and southern regions. Almost the same T-S charac-te r i s t i c s exist on l/vi/50 (figure 38) as on the previous day except that there is no indication of the presence of cold saline water along the North Shore. Nearly a l l the water in the inlet appears to be flood water from the previous day. The T-S diagrams for station 37 on 3l/v/50 identify what i s probably the North Arm water. The same T-S relation-ship occurs at station 35 on l/vi/50, indicating perhaps that this mass of water did actually move along the North Shore. 53 Survey M 27/ix/50 (Figures 40, 41, and 42a,b,c) This survey (figures 42a,b,c) was conducted during the small flood of a half tide and shows the situation for a Fraser River discharge of 60,000 cfs. as compared with the other observations taken during higher river discharges of 110,000 and 200,000 cfs. In place of the single intruding tongue of fresh water, two tongues are present, one relatively fresh and the other more saline. This difference in the character of the flood as compared with that of survey G i s related to the much smaller discharge of the Fraser. For run-offs of the order of 200,000 cfs. the tida l rise i s insufficient to prevent some discharge of the river even at high water, and brackish water flows around Pt. Grey (figure 36a). For lower discharges of the river i t i s sealed off by a flood tide and saline water moves around Pt. Grey close to the shore (figure 42b). The brackish water which originally extended far out from the mouth of the river on the previous ebb i s cut-off by the incoming tide and moves into the inlet as a cloud or tongue next to the saline tongue at Pt. Grey. Remnants of the previous flood and ebb are s t i l l noticeable near the Narrows. The large eddy of brackish water (figures 42a,b), situated near English Bay, began i t s north-ward movement on the previous ebb but did not complete i t s westward movement, because of the short duration of this ebb. This eddy has the same T-S characteristics (figure 41) as the brackish flood water. Probably i t was brought i n on the pre-54 vious large flood tide and settled in English Bay. The water lying just offshore from Pt. Grey and Spanish Banks i s much colder and more saline, and i s definitely not associated with Fraser River water. Summary of the Salinity Distributions: The circulations inferred from the salinity d i s t r i -bution are in agreement with the current observations. The isohalines are approximately parallel to the North Shore in the region of the jet, their southward displacement (survey K) follows the movement of water in this direction late in the ebb (11). The same general features are also shown at 6 feet and 18 feet. The large tongues of saline water which extend from the Narrows are associated with the back eddy developed on the ebb tide, while the tongue-like distributions of saline water across English Bay are features of both flood and ebb tides. The change of position of these isohalines during the tides indicates some of the stages of the water movements which are obscured by considering only the average velocities over a f u l l ebb or flood. The flood movement of brackish water through the inlet i s also indicated by the large tongues of low sa l -i n i t y water, particularly in the region off Spanish Banks. Mean Salinity Distribution: An approximate mean surface s a l i n i t y distribution (figure 43) related to a high river discharge was obtained by considering the gross features of the salinity patterns for 55 flood and ebb stages of the tide on surveys G and K. Survey G (figure 36a) shows the distribution of s a l -i n i t y off Pt. Grey during flood tide, while survey K (figure 33a) shows the distribution of sal i n i t y from the Narrows dur-ing ebb tide. The tongues of brackish water near Pt. Grey and Spanish Banks indicate flow i n the directions of the North Shore, the Narrows, and English Bay. In English Bay the iso-halines generally extend across from Jericho Beach to Fer-guson Pt. for both the average flood and ebb distribution of salinity. The sal i n i t y distribution of the jet stream along the North Shore i s slightly distorted by the tongues of intrud-ing brackish water off Spanish Banks (figure 43). The tidal analyses of the oceanographic observations have revealed the general features of flow i n the in l e t . If one assumes that the tongues of saline water along the south-ern boundary indicate flow towards the Narrows and that the parallel isohalines along the North Shore indicate flow along the lines of constant property from the Narrows to Pt. Atkin-son, then this interpreted current f i e l d agrees basically with the net surface circulation (figure 27) obtained from the cur-rent measurements. The flow in and out of the inlet across the mouth is generally along the lines of constant sal i n i t y . This phenom-enon agrees with the theoretical assumption (section iv) that flow must be along the lines of constant property at the mouth in order that a salt balance be maintained. 56 At 50 feet the average lateral difference of salin-i t y for surveys G and K is only l°/oo while the surface d i f -ference across the mouth i s 14°/oo. The net circulation at 50 feet also differs from the net circulation at the surface. The pressure gradients associated with the distribution of mass at the surface decrease vertically downwards and disap-pear at some depth. At 50 feet they are small compared with the pressure gradients developed by the oscillatory sea sur-face. The circulation at 50 feet i s more l i k e l y to be asso-ciated with the mean ti d a l pressure f i e l d , while that at the surface i s influenced strongly by the large local pressure forces which arise from the larger differences in sali n i t y and higher concentration of fresh water. 57 VI. VECTOR AND SCALAR FIELDS OF CIRCULATION IN BURRARD INLET A formal solution by Fourier series to the mathe-matical model of a rectangular inlet i s possible i f J'(xy) and one of i t s derivatives are known at the river and seaward boundaries. This would require current measurements at the head and mouth of a suitably selected inlet which has symmet-r i c a l boundaries. However, most inlets are not rectangular, and not even regular i n shape. Solutions for these conditions are not practical by formal methods. Formal rules for approximating irregular boundaries by regular functions have not been worked out but, knowing the current distribution, i t will be shown that solutions can be obtained by numerical means. The current observations in Burrard Inlet i l l u s t r a t e the essential features of flow which take place during flood and ebb tides. The analysis of the data indicate different mean vector current fields at the surface and at 50 feet. On the assumption of no net discharge or flow through the F i r s t Nar-rows (see appendix) a f i e l d of ^(xy) representing the circula-tion in terms of streamlines has been obtained from the vector f i e l d s . Two fields have been derived i n this manner: one at 50 feet, which i t will be assumed shows the net tida l circula-tion, and the other at the surface which represents the com-bined net t i d a l and fresh water circulation. The scalar fields have been examined by numerical methods to determine how accurately they satisfy the fourth 58 order differential equation ±£ + ^ = O (38) which was derived earlier. In order to preserve the natural shape of the inlet and s t i l l employ the data available (section v), a numerical method of solution i s offered which is s t i l l applicable to i r -regular boundaries and has a l l the advantages of the Fourier method. The use of relaxation methods (19) (20) has provided an analytic method for testing the scalar fie l d s for v e r i f i -cation of equation (38) within the limits of accuracy of the observations. By expressing the differential equation in the form of finite-difference equations a numerical operator can be derived. This operator i s used for testing the f i e l d . The network of anchor stations (figure 14) est-ablishes a grid of observation points i n the i n l e t . The net vector velocities at each of these stations i s sufficient to indicate the general features of flow, but a quantitative vector f i e l d i s required for an analytical study. By reducing the grid size to a finer mesh and interpolating vector velo-c i t i e s at each intersection or node of the network a more de-tailed f i e l d satisfying the principle of volume continuity was developed. Since the flow i s assumed to be incompressible, a stream function may be related to the velocity f i e l d by the relations. 5 9 U = - ^ , „ H v = L I , Y y a n d i % 1 (52) S° that i*fa)-.*M * / ( v J x - u ^ ) . ( 5 3 ) The vectors are given at the points of intersection or nodes of the grid. The x and y components of the velocity are u(x,y) and v(x,y). At the s o l i d boundaries ^ =o and yfr -o where n i s the normal to the boundary, i n Equation (53) was solved for the stream function ^(xy) in the inlet by carrying out a numerical integration over the grid. Scalar fields at both 5 and 50 feet were evolved from the vector interpolations and integrations. These fields were then contoured for (figures 44, 46) and checked for satisfaction of the differential equation by re-laxation methods. The Verification of a Scalar Field with a Fourth Order Differ-ential Equation The two terms of the differential equation expressed in the form of a finite-difference approximation for square nets are (19) = f - 4 * + ~ 4* + % , (a) and (54) h ' ( ^ i ) = t - « ? (b) 60 where the coordinates of these points are the numbered inter-sections of the net (figure i i i ) . The point o may be any point remote from a boundary, but for convenience let i t be the origin of the system. 10 2 II 3 o 1 4 u -4 -4 »2 -4 1 -4 figure i i i figure i v The differential equation in the form of a unit operation operator for points remote from a boundary i s rep-resented in figure iv. For points near a boundary standard methods are used to check the boundary conditions \b - jff =Q (19). The simplest method i s to approximate each section of the boundary by a straight line of the network. 61 By applying the operator to each grid point of the f i e l d , the f i e l d may be tested for verification of the d i f -ferential equation. Since the equation i s homogeneous, the sums and differences of the products taken around the central node should be zero (figure i i i ) . Generally a residual w i l l remain which can be liquidated by slight adjustments of the values of the scalar quantity at one of the nine points. For the two scalar fields of $ examined, positive and negative residuals were found, indicating that the fields did not satisfy the differential equation identically. The residuals were liquidated or reduced to a minimum by relaxing the f i e l d at some of the grid points. Bach change of \p at a grid point affects the residuals at eight other points, con-sequently there i s a continual wash-back of residuals while the f i e l d i s being altered. The goal i s to reduce the resid-uals to a minimum without destroying the original f i e l d . The effort involved in these operations i s considerable i f the unit relaxation operator (figure iv) i s used exclusively. It is a matter of personal choice, somewhat dependent on the dis-tribution of the residuals, how to ease this laborious process. Point and line relaxation operators were the easiest to employ for relaxation near the boundaries, while simple block opera-tors were more successful in the central area. In this con-nection a certain amount of discretion i s required because the f i e l d i t s e l f has already been established independently from other sources. Thus the relaxed f i e l d satisfying the equation must s t i l l retain the basic character of the scalar f i e l d from 62 which i t i s derived. A c r i t i c a l feature of this method i s the proper choice of grid size. For ease of operation i t i s best to choose as coarse a grid as possible. It w i l l contain fewer nodes and the liquidation processes w i l l be correspondingly easier. Re-laxation of the f i e l d can always be advanced to a finer net by simply halving the network. A finer net was found to be neces-sary in regions where the most rapid changes of ^ occurred, in particular, areas adjacent to the North Shore, Stanley Park, and Spanish Banks. In these areas the behavior of ^ was studied in closer detail. It i s d i f f i c u l t to assess the possible error of the f i n a l fields because there i s one error which arises through incomplete liquidation of the residuals and a second which arises from neglect of higher order terms in the f i n i t e -difference expressions. An arbitrary residual limit was set at - IQffo of the value of \£ for any grid point with approxi-mately equal distribution of positive and negative values throughout the f i e l d . Results: The scalar fields (figures 44, 46) derived by inte-grating the current fields were successfully relaxed to satisfy the differential equation and boundary conditions within the 10$ limit of accuracy. Further relaxation carried out to reduce the residuals another 5% resulted in an overall change of less than 1% of the values of \^ . Since there exists an 63 inherent error of approximately 10$ in the current observa-tions and one equally as large for the interpolation of the vector f i e l d , further liquidation of the residuals was not con-sidered to be significant. The f i n a l relaxed fields satisfying the theoretical assumptions and physical measurements are shown in figures 45 and 47. The extent of distortion introduced by relaxation has been confined to a smoothing effect of the function. The character of the vector f i e l d i s s t i l l preserved, and the general flow pattern has become more apparent; a feature which i s frequently lost when only the directional qualities of the vectors are used. The above tests were carried out using the reduced biharmonic equation \l + y* = o ) x 4 b V 4 In addition both the scalar fields were tested with the f u l l biharmonic operation operator derived from the equation +lJL± + It 5 0 . (55) 3*« i X v i V l >y* The values of the residuals found in using this operator were within the 10$ limit of accuracy set for the reduced biharmonic equation. However, relaxation employing the f u l l biharmonic operator i s much more tedious to perform, since i t i s more slowly convergent than the reduced form. The contri-bution of the middle term to the f i e l d does not appear to be 64 significant within the limits of accuracy of the current mea-surements and the interpolated vector f i e l d . Discussion: The net current distributions at the surface and 50 feet (figures 27, 29) were considered in section (V). The scalar fie l d s (figures 44, 46) representing these circulations provide further details which were not originally apparent in the vector f i e l d . The streamlines are crowded towards the shores while in the middle regions they are generally widely separated. This crowding of the streamlines and the associated high velocities near a boundary are common features of coastal circulations. The jet stream i s clearly defined by the streamlines at the surface and 50 feet. The deep circulation i s characterized by the back eddy west of the Narrows and the shallow penetration of the water at the mouth. The streamlines around Pt. Grey and Spanish Banks are deflected southward into English Bay before turning towards the Fi r s t Narrows. The central region now appears as a stagnation point i n the i n l e t . The surface pattern of streamlines exhibits crowd-ing along the North Shore while the streamlines from Pt. Grey extend directly into the Narrows with some turning northwards and adding to the jet stream. The large but weak back eddy shown in English Bay is merely indicated since the magnitude of the }p terms are small compared with those in the 65 remainder of the f i e l d . English Bay may be considered as another bay off the main inlet delineated by a line from Spanish Banks to Prospect Pt. The circulation in this area closely approximates that previously discussed in the mathe-matical model with a salt balance. A similar type of circus lation might be expected in Pilot and Sandy Coves. The difference between the circulations at 50 feet and the surface implies that different boundary conditions exist at the Narrows and the mouth for the two depths. Any change of the forces acting in the inlet w i l l reflect on these boundary conditions at the open ends. The end boundary conditions at the Narrows are not well defined owing to the lack of observational data in this region. The form of $ and i t s f i r s t derivative here, wi l l affect the values and distribution of ^ elsewhere in the i n l e t . However, i t i s f e l t that reasonable estimates of the velocities and hence of the functions have been made which f i t the overall current distribution of the inlet. The neces-sity for doing this serves to emphasize the importance of de-tailed measurements across the open boundaries of an i n l e t . The brackish water is generally limited to the upper ten feet and i t i s not surprising that marked gradients of sa l i n i t y exist. The variation in this mass f i e l d produces relatively large forces; forces which, averaged over a period of time, are large enough to override the mean pressure grad-ient forces associated with the tides. A characteristic sur-66 face circulation i s maintained by the influx of fresh water and the fresh water f i e l d . The extent of penetration of water into the inlet from the seaward boundary depends on the size of the eddy off the F i r s t Narrows (figures 46, 47). The shallow sweep of water at the mouth i s confined to this region by the eddy, the ef-fects of which are reflected on the boundary conditions at the mouth. If the eddy shrinks towards the Narrows, more water wi l l flow into the in l e t . The eddy may extend to the surface, but i t s size appears to be controlled by the boundary condi-tions at the mouth which change as the amount of brackish water flowing into the inlet varies. The scalar fields of represent two possible solutions to the linear homogeneous differential equation. The difference of these two solutions must also be a solution. If the tid a l circulation remains essentially the same at a l l depths and i s the fundamental circulation of the in l e t , then i t should be possible to study changes of the surface circula-tion when other forces become effective. The surface circula-tion studied in the present thesis represents a temporary mean state of water movement existing during high Fraser River discharge. Assuming that the fresh water effect for lower discharges of the Fraser also satisfies the same differential equation then i t should be possible to correlate the surface circulation with other mean discharges of the Fraser. In this way seasonal differences in circulation, controlled by the discharge of the Fraser, can be predicated from the observa-67 tions gathered on a single concentrated oceanographic survey. Reference has been made to the f u l l biharmonic equa-tion, equation (55), which governs the bending of a f l a t plate. The similar equation describing the bending of a plate and the circulation in the f l u i d model suggests that an inexpen-sive and practical plate model of the inlet could be built to assist in the f i e l d observations and interpretation of the data (21). Boundary conditions can be simulated by clamping and loading the plate. The advantage of this technique would be that the solution i s directly revealed by the deflection of the plate. The contours of the deflection could then be related to the current pattern i n the in l e t . Another advan-tage i s that the middle term of the biharmonic equation i s taken into account in the bent-plate model. However, the magnitude of this term was evaluated and found to be insignificant except in regions where marked curvature of the streamlines exists particularly at the so l i d boundaries. This was done by evaluating the residuals R^ and R2 from both equations 38 and 55. y $ . w ~ a. (38) ;> x< + T T (55) The magnitude of the middle term i s of the order of U R, - RZJ and was found to be less than 5% of either or Y9 Thus the contribution of the longitudinal f r i c t i o n a l forces to the circulation in Burrard Inlet i s small compared with that of the lateral f r i c t i o n a l forces. 68 VII. SUMMARY AND CONCLUSIONS Theoretical and practical examples of lateral c i r -culations have been presented and related to f r i c t i o n a l forces, and forces associated with pressure gradients in an inhomo-geneous medium. The presentation of these examples has re-quired theoretical and numerical solutions to the boundary-value problem of the circulation in an inl e t . Both methods of solution were required to study c r i t i c a l phases of the c i r -culation. The Fourier type solution affords a practical means of studying some of the characteristics of the Coriolian and fr i c t i o n a l fields as well as the integrated mass f i e l d s . It is only useful for symmetrically shaped models with uniform side boundaries. On the other hand, the numerical method i s applicable to natural boundaries, and i s more practical from the oceanographic point of view. Without the introduction of a salt balance the model would lack physical reality. The inclusion of this principle i n the boundary conditions has made a study of the circulation possible. Although the mass f i e l d i s undefined by virtue of the vertical integration, i t i s possible to study the relative significance of either the Coriolian or f r i c t i o n a l forces. The mass f i e l d may be described by a function Q which i s related to the density by the equation 69 Q = Jdz f p J l . ( 5 ) Sfc ff< The horizontal variation of Q is expressed by the equation L£ +- ±0. - O (15) Calculating the values of Q from oceanographic data the f i e l d could be developed and tested by relaxation methods in a simi-lar manner to that of the current f i e l d . Knowing this f i e l d and the velocity f i e l d i t would be possible to evaluate the eddy coefficient, A^. This method has the disadvantage that i t would require a long series of oceanographic observations to establish the mean mass f i e l d . In this respect direct cur-rent observations have the advantage of yielding the velocity f i e l d directly, from which scalar fields of the stream func-tions can be developed and tested. The Coriolian and f r i c t i o n a l f i e l d s could then be determined, and hence the mass f i e l d . This f i e l d could be compared with that of the inlet for further verification of the model. The introduction of relaxation methods has provided an excellent method of bridging the gap between observational data and theory. The question of dealing with asymmetric boundaries in nature and symmetric theoretical boundaries has been answered by this numerical approach. The two solutions to the differential equation des-cribing the circulations at 50 feet and at the surface have been attributed to the forces associated with a mass f i e l d . 70 The surface circulation i s primarily controlled by the influx of fresh water from the Fraser establishing a large pressure f i e l d which in turn influences the mean tida l pressure f i e l d at the surface. At 50 feet this phenomenon i s small and the circulation appears to be associated only with the mean t i d a l pressure f i e l d . The two solutions can be combined to give the net circulation associated with the inhomogeneity of the f l u i d . The surface circulation i s controlled largely by the discharge of the Fraser. The discharge varies seasonally, and i t would be expected that some change would occur i n the surface circulation of Burrard Inlet. This effect may be stu-died by considering the two solutions. A proportionate change of the solution at the surface combined with that of the tide should correlate the seasonal changes of the circulation with the discharge of the Fraser. Both the theoretical and numerical solutions provide an indication of the observational data which would be required for inlet studies. The f i e l d observations should include cur-rent measurements across the mouth and river ( i f there i s one) and for purposes of checking the model also along the longi-tudinal axis of the inlet. If a detailed study of the circula-tion i s desired the effects of boundary configuration on the flow must be included in the planning of an oceanographic survey. Another problem in physical oceanography i s that the ships and manpower available are never adequate for an exhaustive survey. It is always necessary to decide whether i t i s better to distribute the effort over a l l the inlet and 71 be satisfied with the gross features of the circulation, or to concentrate in limited regions in order to obtain a de-tailed picture of the circulation. On the basis of this study a number of suggestions are offered for f i e l d investiga-tions of lateral circulations. A preliminary survey of the area i s desirable to establish an approximate f i e l d of flow so that a coarse grid of the stream function can be obtained. Relaxation of such a f i e l d would involve a minimum of effort and s t i l l prove valuable for indicating the most significant regions for further f i e l d observations. These data may be included in the approximate f i e l d by reducing the grid size. By repeating these theoretical and observational studies, the circulation could be well established with a minimum of effort. It would seem feasible to investigate the practi-ca b i l i t y of elastic plate models for the study of lateral c i r -culations. The biharmonic equation developed for the f l u i d model (equation 55) also governs the bending of an elastic plate. By stressing the plate the vertical deflection can be contoured and related to the flow pattern i n the f l u i d proto-type. The differential equation i s satisfied without relaxa-tion, and the plate would show the complete scalar f i e l d . The information obtainable from such a model i s directly applicable to the problem of locating stations for oceano-graphic observations. In the present study wind effects were neglected for the purpose of studying the influence of fresh water and 72 tides. However, the possibility s t i l l exists that the wind stress could be included in a study of this type. By calcul-ating the value of the wind stress at each oceanographic sta-tion and interpolating over the grid pattern of the model, the value of the curl of the wind stress would be known. The relaxation residuals of the current f i e l d would have to be related to these values of the curl of the wind stress. In this manner wind effects may be considered in the circulation of an i n l e t . The agreement between the mathematical model and the oceanographic observations in Burrard Inlet indicates that lateral circulations can be described by the differen-t i a l equation The success which has been achieved by describing the circulation in Burrard Inlet demonstrates that the theore-t i c a l considerations are physically sound and reasonable enough to consider further the problems of lateral circulation in other estuaries. 73 REFERENCES 1. Pritchard, D.W. (1952). Estuarine hydrography. Advances in Geophysics, vol. 1, 243-280. Academic Press Inc. New York, N.Y.. 2. Pritchard, D.W. (1950). A review of our present know-ledge of the dynamics and flushing of estuaries. Tech. Rept. 4. Ref. 52-7. Chesapeake Bay Institute, John Hopkins University. 3. Tully, J.P. (1949). Oceanography and prediction of pulp mill pollution in Albemi Inlet. Fish. Res. Board, Canada. Bull. 83. 169 pp. 4. Pickard, G.L. (1953, 1954). Oceanography of British Columbia mainland inlets I, II, III, IV. Fish. Res. Board, Canada. Progress Reports of Pacific Coast Stations, 96, 97, 98, 99. 5. Cameron, W.M. (1950). On the dynamics of inlet circula-tion. Doctoral dissertation. Scripps Institution of Oceanography. University of California, La J o l l a , Calif., 1951. 6. Stommel, H. (1951) Recent studies in the study of tid a l estuaries. Tech. Rept. Ref. 51-33. Woods Hole Oceanographic Institution. 7. Shtokman, V.B. (1948). Equations for a f i e l d of total flow induced by the wind in a non-homogeneous sea. Comptes Rendus (Doklady) de l'Academcie des Sciences de l'URSS. vol. LIV, no. 5, 403-406. 8. Shtokman, V.B. (1948). Relationships between the wind-f i e l d the transport-field and the mean mass-field in a non-homogeneous ocean. Dok. Akad. Nauk SSSR vol. 59, no. 4, 675-678. Manuscript translation T 56R. Defence Scientific Information Service, DRB. Canada, 1952. 9. Munk, W.H. (1950). On the wind-driven ocean circulation. Journal of Meteorology, vol. 7, no. 2, 79-93. 10. Cameron, W.M. (1951). On the transverse forces in a British Columbia in l e t . Trans. Roy. Soc. Canada., vol. XLV, series i i i , 1-8. 11. Canadian Hydrographic Service. Tidal Current Charts, Vancouver Harbour, British Columbia. Department of Mines and Technical Surveys, Ottawa. Tidal Publication no. 22. 1950. 74 12. La Croix, G.W. (1950). MS. Report on current investigation, Burrard Inlet. Hydrographic Survey, Victoria, B.C. 13. Unpublished data (subject to revision) Dominion Water and Power Bureau, Vancouver Office. 14. Sewerage and Drainage of the Greater Vancouver Area, British Columbia, 1953. Sewerage Board, Vancouver, B.C. p. 126. 15. Ibid, p. 125. 16. Data Record. MS. Fraser River Estuary Project, 1950. Pacific Oceanographic Group, Nanaimo, B.C.. 17. F j a r l i e , R.L.D. (1950). MS. Fraser River estuary project Pacific Oceanographic Group, Nanaimo, B.C.. 18. F j a r l i e , R.L.D. (1950). MS. The oceanographic phase of the Vancouver sewage problem. Pacific Oceanographic Group, Nanaimo, B.C. 19. Shaw, F.S. (1953). An Introduction to Relaxation Methods. Dover Publications Inc. New York, N.Y.. 395 pp. 20. Arthur, D.W. de G. (1954). Relaxation Methods. McGraw-H i l l Book Company Inc. New York, N.Y.. 256 pp. 21. Hidaka, K. and Koizumi, M. (1950). Vertical circulation due to winds as inferred from the buckling experiments of elastic plates. Geophysical Notes, vol. 3, no. 3, 1-9. Geophysical Institute, Tokyo, Japan. 22. Surface Water Supply of Canada (1941-1942). Pacific Drain-age, 94. Department of Mines and Resources. Ottawa. 1946. A P P E N D I X APPENDIX The influence of the Narrows on the circulation i n Burrard Inlet has been seen from the oceanographic observa-tions. The current fields were developed on the assumption of no net discharge through the Narrows and that the flow was asymmetric. Estimates of the net discharge have been made from the total discharge on flood and ebb tides. These data were supplied by the office of the Vancouver Harbour Model Pro-ject. Discharge figures were calculated by the method of cubature, and supplemented by current measurements taken by the Water Resources Board. This information on the circulation and discharge through the Narrows i s presented to supplement the current fields of Burrard Inlet. Discharge through the Narrows: The volume discharge through the Narrows depends on the tid a l range of flood and ebb. The average volume transport on May 4, 5, 1954, (when measurements were carried out) was 9 Q approximately 7.7 x 10 cu. f t . on flood and 7.2 x 10 cu. f t . on ebb for an average range of 14 feet. The difference in volume discharge i s related to the differences of tidal range which was approximately 1 foot for the large tides of May 4 and 5, 1954. Any net transport of water through the Narrows i s temporary and dependent on the tides. It is too small to consider over a long time average of the tides. This assump-tion presumes that the volume of fresh water draining into the inlet i s not significant. This effect w i l l be considered later. Current Measurements: Figures 44 and 46 indicate an inward movement of water through the Fi r s t Narrows along Stanley Park and an outward movement along the North Shore. Current measurements along transverse sections, situated east and west of the Lions Gate Bridge, indicate that the flow has this asymmetric pattern. The flood movement of water flows past the Stanley Park side of the Narrows and crosses the Narrows striking the shore east of the bridge. The magnitudes of the flood velocities are large, and give rise to net inward components of velocity west of the bridge on the south side, and east of the bridge on the north side. The ebb movement of water i s partly from Stanley Park east of the bridge across to the North Shore on the west side of the bridge. This flow together with the ebb along the North Shore produces a net outward flow along the North Shore west of the bridge. East of the bridge the direction of the net flows i s reversed with respect to those west of the bridge. Part of this effect inside the Bridge may be the result of large back eddies formed along both shores on flood and ebb tides. Fresh Water Drainage: A study of Burrard Inlet and the Fir s t Narrows i s not complete without some mention of the fresh water drainage into the area. There are a great many small streams draining into the in l e t , particularly along the north side and Indian Arm. The discharge of these streams i s insufficient to war-rant continuous stream measurements. Four major rivers are located along the North Shore namely, the Capilano, Seymour, Indian, and Lynn. Daily discharge measurements are taken only on the Capilano and Seymour Rivers. The Indian River system was metered for a period of 8 years from 1912-1920. There are no discharge figures for the Lynn Creek. The variation of daily discharge of these rivers from season to season i s large. Some idea of the extremes of discharge for the Capilano, Seymour and the Fraser may be gained from Table I (22). River Discharge cfs. Maximum Minimum Capilano Seymour Fraser 20,800 17,200 536,000 6.5 9.3 12,000 Drainage Area Sq. Miles 76 68 85,600 Table I. One stream which i s of a size comparable with the others is the Lynn. The only satisfactory means of estimating i t s discharge i s to consider the mean annual distribution of r a i n f a l l over the drainage basin of the river (figure 48). This can be done by considering the whole Howe Sound-Burrard Inlet and Greater Vancouver area where there are a number of r a i n f a l l stations (figure 48). From the distribution of r a i n f a l l an estimate of 160 cfs. for the mean discharge of the Lynn was deduced. This figure was calculated for a yearly cycle. Similar cal-culations for the Capilano, Seymour, and Indian Rivers were made and compared with the metered values of discharge for these rivers (Table II). River Metered Discharge Estimated Discharge (yearly mean) cfs. (yearly mean) cfs. Capilano 702 704 Seymour 552 607 Indian 654 638 Lynn — 160 TABLE II The estimated and actual values of discharge agree very closely for the Capilano, Seymour, and Indian Rivers. On this basis the discharge of the Lynn was incorporated into the figures for fresh water drainage into Burrard Inlet. The total mean yearly discharge for the four rivers i s ap-proximately 2080 cfs. and for the in l e t slopes outside these drainage areas 575 cfs. The total contribution of the fresh water draining into the inlet for an averaged tidal time of 7.6 hours for 7 flood and ebb i s approximately 7..4 x 10 cu. f t . If a l l this water passed through the Narrows on ebb i t would ac-count for less than l°/o of the discharge. This small con-tribution can be neglected compared with the total discharge through the Narrows. The assumption of no net transport through the Narrows is s t i l l valid. (a) \p evaluated at the mouth of the bay (b) ° f evaluated at the mouth of the bay b X Figure 1. Transport or Coriolian f i e l d for the bay-Figure 2. Frictional (e) f i e l d for the bay Figure 3. Mass f i e l d with A, = 1.0x10 gm./cm./sec.. Figure 4. Mass f i e l d with A- = 2.0x10 gm./cm./sec. Figure 5. Transport or Coriolian f i e l d which satisfies a lateral salt balance Figure 6. Mass f i e l d which satisfies a lateral salt balance Figure 8. Lower Mainland of British Columbia Figure 9. Figure 10. Burrard Inlet Cloud of Fraser River water intruding into Burrard Inlet B.C. Government Photograph Figure 11. Cloud of Fraser River water entering First Narrows B.C. Government Photograph Figure 1 2 . Sc so 58 Clouds of muddy Fraser River water in the Strait of Georgia B.C. Government Photograph Figure 13. 13 (6 Current metering stations Burrard Inlet Figure 14. Current distribution Large ebb 5 feet Figure 15. Current distribution Large ebb 18 feet Figure 16. Current distribution Large ebb 50 feet Figure 17. Current distribution Small ebb 5 feet Figure 18. Current distribution Small ebb 18 feet Figure 19. Current distribution Small ebb 50 feet Figure 20. Current distribution Large flood 5 feet Figure 21. Current Distribution Large flood 18 feet Figure 22. Current distribution Small flood 5 feet Figure 24. Current distribution Small flood 18 feet Figure 25. Current distribution Small flood 50 feet Figure 26. Net circulation 5 feet Figure 27. Net circulation 18 feet Figure 28. Net circulation 50 feet Figure 29. 9 11 ^ \ ft • 8 12 15 14 13 hi i e 19 17 16 v ' / s-': •'*«"*.* •*•.•• # **••*# J \ * ' • , f i j t I1 m I Figure 30. Oceanographic stations Survey N. 28/ii/51 i 1 1 1 1 " 1 1 ' 2 5 2 6 2 7 2 8 2 9 S A L I N I T Y % o SURVEY N 28/11/51 Figure 31. T-S relationships SURVEY N 28/ii/51 Salinity distribution Surface Figure 32a. Salinity distribution 6 feet Figure 32b. Salinity distribution 18 feet Figure 32c. Salinity distribution' 6 feet Figure 33c. 3 5 36 3 0 41 3 4 31 3 7 4 0 3 2 3 9 3 3 3 8 Figure 34. Oceanographic stations Survey G. 3l/v/50 i 1 r — 12 14 16 n 1 1 » I i 18 2 0 22 2 4 2 6 2 8 r 6 0 SAL IN ITY % , S U R V E Y G 3I/V/50 r 45 UJ •15 cr 3 u. AT •10 U i cr in UJ CL -5 UJ < y- Q -0 r-F R A S E R R. D I S C H A R G E 2 0 0 , 0 0 0 C . F . S . Figure 35. T-S relationships SURVEY G 3l/v/50 Salinity distribution Surface Figure 36a. Salinity distribution 6 feet Figure 36b. Salinity distribution 18 feet Figure 36c. 3 5 \ £ • 3 6 ^ " ^ ^ s ; . 3 0 41 N 4 0 31 3 2 3 9 ^ « " • • . 3 3 W ' • . 3 8 \£V ' •••'; ^^rrr^s, /^T^C1' * A Figure 37. t i 1 —i Oceanographic stations Survey G . l / v i / 5 0 — i 1 1 1 — i i 1 12 14 16 18 20 22 24 26 28 SALINITY % 0 - 60 u. S U R V E Y G l/VI/50 0 • - 6 12 18 I I 1— Figure 38. T-S relationships SURVEY G l/vi/50 Salinity distribution Surface Figure 39a. Salinity distribution 6 feet Figure 39b. Salinity distribution 18 feet Figure 39c. 1* * • • A** J ** *1 • *..* . • . 10 9 11 8 15 14 12 *. • "*°.\. 13 1> * f e \ * . IV* 19 17 16 \4* •' /\ •> •* # * * * * • • •. * Figure 40. Oceanographic stations Survey M. 27/ix/50 14 16 18 I 1 1 1 1 1 2 0 2 2 2 4 2 6 2 8 3 0 S A L I N I T Y % o - 6 0 S U R V E Y M 2 7 / I X / 5 0 0 12 l - 7 _ ^ - ^ " ^ 16 13 11 - 5 5 UJ cr 1— < cr - 5 0 UJ n. 2 UJ \— - 15 -10 T I D E " 18 u. f \ I _M5 1 1 5 I 3 ^ v . 17 16 >0 1 8 " ' — • 18 F R A S E R R. D I S C H A R G E - 4 5 •^5 - 0 CC < T I M E Q K 6 12 18 1 6 0 , 0 0 0 C . F . S . Figure 41. T-S relationships SURVEY M 27/ix/50 Salinity distribution Surface Figure 42a. Salinity distribution 6 feet Figure 42b, Salinity distribution 18 feet Figure 42c. Mean salinity Burrard Figur distribution Inlet 43. Figure 44. Scalar f i e l d of ^ c a l c u l a t e d from the velocity f i e l d at 5 feet Figure 45. Relaxed scalar f i e l d of ¥ , 5 feet, satisfying the differential equation Figure 47. Relaxed scalar f i e l d of y satisfying the differential equation at 50 feet Average yearly r a i n f a l l distribution (inches) Howe Sound, Burrard Inlet Figure 48.
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A study of lateral circulation in an inlet Campbell, Neil John 1954
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Title | A study of lateral circulation in an inlet |
Creator |
Campbell, Neil John |
Publisher | University of British Columbia |
Date Issued | 1954 |
Description | On the thesis that the lateral frictional stresses should play an important part in determining the horizontal circulation in landlocked bodies of water, a mathematical model of the circulation in an inlet is developed. The circulation is described by a fourth order differential equation [formula omitted] where Ψ is a stream function. A solution representing a commonly observed circulation is obtained for a rectangular bay. The significance of Coriolian, frictional, and mass field forces in maintaining such a circulation is discussed. The theoretical model is tested by means of data available for Burrard Inlet. The data indicate the existence of two net circulations, one at 50 feet which has been attributed to the tides, and the other at the surface which is influenced strongly by the influx of brackish water from the Fraser River. Relaxation methods are introduced to test the current fields for verification of the differential equation. Despite the more complex boundaries of Burrard Inlet as compared with the rectangular bay investigation, the actual circulation in Burrard Inlet is found to satisfy the fourth order differential equation within the limit of observational error. This agreement suggests that the lateral frictional stresses do play an important part in the circulation in inlets. Further applications of relaxation methods are urged for oceanographic studies. Suggestions are made as to where oceanographic observations should be taken for a study of lateral circulations in inlets or bays. The reasonable agreement of the mathematical model and prototype suggests that lateral effects can be described by the mathematical theory. |
Subject |
Hydrodynamics Fluid mechanics |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2012-01-31 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0106240 |
URI | http://hdl.handle.net/2429/40406 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physiology |
Affiliation |
Medicine, Faculty of Cellular and Physiological Sciences, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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