MATHEMATICAL MODELLING OF THE CHLOROPHYLL DISTRIBUTION IN THE FRASER RIVER PLUME* BRITISH COLOMBIA BODO RUDOLF\de LANGE BOOM B,Sc, University of V i c t o r i a , 1970 A Thesis Submitted in P a r t i a l Fulfilment of the Requirements for the Degree of Master of Science i n the Department of Physics and In s t i t u t e of Oceanography We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA by July, 1976 (c) Bodo Rudolf de Lange Boom, 1976 In p resent ing t h i s t he s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permiss ion fo r ex tens i ve copying of t h i s t he s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r ep re sen ta t i ve s . It i s understood that copying or p u b l i c a t i o n of t h i s t he s i s f o r f i n a n c i a l gain s h a l l not be al lowed without my w r i t t e n permis s ion. Depa rtment The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date 3 i i ABSTRACT The horizontal chlorophyll a d i s t r i b u t i o n observed i n the S t r a i t of Georgia near the mouth of the Fraser River appears to r e f l e c t the influence of the r i v e r discharge. Mathematical models are developed to attempt to explain the observed d i s t r i b u t i o n i n terms of such factors as the velocity f i e l d , the available l i g h t and the grazing and sinking of the phytoplankton population. A steady state, two dimensional model i s developed f o r the upper layer. The downstream velocity i s modelled using a modified form of the downstream velocity i n a j e t ; the v e r t i c a l entrainment i s represented by an empirical expression, while the cross-stream v e l o c i t y i s calculated from the v e r t i c a l l y integrated continuity eguation- A v e r t i c a l l y integrated conservation equation i s written for the chlorophyll concentration by balancing advection against the source-sink term (net production minus grazing and sinking). Temperature effects are not modelled d i r e c t l y and nutrients are not considered as l i m i t i n g . The f i r s t model i s s i m p l i f i e d by assuming: a constant depth of the upper layer, v e r t i c a l entrainment proportional to the downstream velocity, and a uniform v e r t i c a l d i s t r i b u t i o n of chlorophyll. In model II the layer depth varies with distance from the r i v e r mouth, a more complex r e l a t i o n for the v e r t i c a l entrainment i s used and more r e a l i s t i c v e r t i c a l p r o f i l e s are employed for the horizontal velocity and the chlorophyll concentration. Although the observed downstream maximum in the horizontal chlorophyll d i s t r i b u t i o n i s not reproduced, the r e s u l t s indicate that the velocity f i e l d , the available l i g h t i n the water column and the value of the maximum production rate {a function of water temperature) are the most important parameters influencing the d i s t r i b u t i o n . Sinking i s of secondary importance while grazing appears to be r e l a t i v e l y unimportant- X V TABLE OF CONTENTS A b s t r a c t . . . i i L i s t Of T a b l e s ... v L i s t Of F i g u r e s ........................................... v i Acknowledgements x Chapter 1. G e n e r a l O u t l i n e 1 Background 1 Problem: The H o r i z o n t a l C h l o r o p h y l l D i s t r i b u t i o n ........ 7 The Approach To The Problem 13 Chapter 2. The P h y s i c a l Component: The Flow F i e l d ........ 16 Chapter 3. The C h l o r o p h y l l C o n s e r v a t i o n E g u a t i o n ......... 25 Chapter 4. G e n e r a l Method Of S o l u t i o n .................... 28 Chapter 5. Sources And S i n k s Of C h l o r o p h y l l ..............30 Chapter 6. Data 41 Chapter 7. Model I : F o r m u l a t i o n 50 Chapter 8. Model I : R e s u l t s .............................. 64 D i s c u s s i o n 82 Chapter 9. Model I I : Refinements 83 Chapter 10. Model I I : R e s u l t s 93 D i s c u s s i o n ...................................110 Chapter 11. C o n c l u s i o n s ..................................113 R e f e r e n c e s 117 Appendix: Temperature And S a l i n i t y Data 120 V LIST OF TABLES Table I- E v a l u a t i o n of entrainment from c r u i s e G ulf 1 data 54 Table I I . Model I parameters h e l d constant. ............. 68 Table I I I . Seasonal v a r i a t i o n of model I parameters. .... 74 Table IV. Model II parameters h e l d constant- ............ 96 Table V. Seasonal v a r i a t i o n of model I I parameters- ..... 99 LIST OF FIGURES Fig. 1. Map showing the general study area. 2 Fig. 2. Detailed map of the study area 4 F i g . 3. seasonal variation of the Fraser Hiver daily mean discharge measured at Hope, B.C. 5 Fig, 4. The Fraser River plume position as derived from a e r i a l photographs, after Tabata, 1972. 7 Fig. 5. Horizontal d i s t r i b u t i o n of chlorophyll a, (A) and zooplankton, (B) in the S t r a i t of Georgia. 8 F i g . 6. Horizontal d i s t r i b u t i o n of chlorophyll a-in terms of r e l a t i v e fluorescence; March, 1973 10 Fig. 7.. Horizontal d i s t r i b u t i o n of chlorophyll a showing patchiness; July, 1973. 11 F i g . 8. Temperature at 1 m as a function of distance from the r i v e r mouth. 14 Fig . 9. The co-ordinate system employed i n the model. ... 18 F i g . 10. The non-dimensionalized downstream v e l o c i t y d i s t r i b u t i o n (LTh/U0h0) for h = constant. .............. 20 Fig, 11. Chlorophyll production rate, P, as a function of l i g h t intensity, I; equation (5.3) 3 3 F i g . 12. Comparison of the curves from equations (5.4) and (5.5) 35 F i g . 13. Location of stations, cruise Gulf 1; November, 1971. 42 F i g . 14. Location of stations, cruise Gulf 2; February, 1972 44 F i g . 15. Station positions, cruise Gulf 3 and subseguent cruises. 45 F i g . 16. S a l i n i t y p r o f i l e s from cruise Gulf 1. 46 F i g . 17. Light i n t e n s i t y (% of surface value) as a function of depth. 48 F i g . 18. Non-dimensionalized extinction c o e f f i c i e n t , j x / J J L 0 , as a function of distance from the r i v e r mouth 49 Fig . 19. A segment of the upper layer, showing the quantities used to derive Table I. 53 Fig . 20. E l l i p t i c a l d i s t r i b u t i o n of contours of r = constant (from equation (7.16)). .................. 58 F i g . 21. Average zooplankton d i s t r i b u t i o n as a function of distance from the r i v e r mouth. 60 F i g . 22. The grazing r e l a t i o n of model I , based on egn. (7. 20) 61 Fig. 23. Streamline pattern of the horizontal v e l o c i t y ; model I with h = 2 m. 65 F i g . 24. Streamline pattern of the horizontal v e l o c i t y ; model I with h •= 5 m. 66 F i g . 25. Variation of M along the axial streamline; model I May conditions with h = 2 m and h = 5 m. .... 69 Fig. 26. Variation of M along y = 0; model I May conditions with f u l l and no d i l u t i o n . 71 F i g . 27. Variation of M along y = 0; model I May conditions showing the effect of an increased sinking rate. 72 Fig. 28. Variation of M along y = 0; model I showing seasonal v a r i a t i o n . ... 73 F i g . 29. Variation of M along y = 0; model I (May) showing the e f f e c t of no grazing (Z = 0). 76 Fig. 30. Variation of M along y = 0; model I May conditions with the effect of increased velocity and maximum d i l u t i o n 77 F i g . 31. Horizontal d i s t r i b u t i o n of M for model I May conditions 79 F i g . 32. Horizontal d i s t r i b u t i o n of M for model I May conditions with 0o = 2 m/sec and -1/= 0. 81 Fig. 33. The variation of the surface as a function of distance from the r i v e r mouth 84 F i g . 34. The normalized upper layer depth as a function of distance from the r i v e r mouth 87 F i g . 35. V e r t i c a l p r o f i l e s of current speed, after Tabata et a l . , 1970. 89 Fig. 36. Comparison of the e f f e c t of d i f f e r e n t values of A on egn. (9.9). 90 F i g . 37. V e r t i c a l p r o f i l e s of chlorophyll, after Fulton et a l . , 1968. .... 91 Fig. 38. Streamline pattern of the horizontal v e l o c i t y ; model II with U0 = 1 m/sec. 94 F i g . 39. Streamline pattern of the horizontal v e l o c i t y ; model II with UQ •= 2 m/sec. 95 Fig. 40. Variation of M along y = 0 (model II) showing seasonal variations. 98 F i g . 41. Variation of M along y = 0 (model I I ) ; the e f f e c t of changes i n the upper layer depth, h 101 Fig, 42. Variation of M along y = 0 (model I I ) ; the e f f e c t of changes i n the velocity f i e l d . .....................103 ix F i g . 43. Variation of M along y •= 0 (model II) ; the e f f e c t of changes i n the production rate. ....105 Fig. 44. Variation of .8 along y = 0 (model II) ; the e f f e c t of changes i n the grazing term. ....................... 106 Fig. 45. Variation of M along y = 0 (model I I ) ; the e f f e c t of increasing the sinking rate. .......................108 Fig. 46. Horizontal d i s t r i b u t i o n of M for model I I ; 0o = 1 m/s, x 0= 10 km. ................109 Fig. 47. Comparison of pJVdz/h and Jpi/dz/h as a function of layer depth. .....112 X ACKNOWLEDGEMENTS T h i s work was made p o s s i b l e by t h e a s s i s t a n c e of a number of people. F i r s t and foremost I would l i k e t o e x p r e s s my g r a t i t u d e t o my s u p e r v i s o r Dr. P a u l H. L e B l o n d , whose a s s i s t a n c e , encouragement and p a t i e n c e e n a b l e d t h i s work t o be completed. I am g r a t e f u l t o Dr. T. R. Parsons f o r s u g g e s t i n g t h i s t o p i c and f o r h i s a s s i s t a n c e and c r i t i c i s m . Dr. S. Pond d e s e r v e s t h a n k s f o r h i s comments and c r i t i c i s m . Many of t h e s t a f f and s t u d e n t s of the I n s t i t u t e of Oceanography a l s o c o n t r i b u t e d t o t h i s work, p a r t i c u l a r l y i n the dat a c o l l e c t i o n - The o f f i c e r s and crew of the C.S.S. V e c t o r deserve t h a n k s f o r t h e i r c o o p e r a t i o n . Much i n v a l u a b l e a d v i c e i n d e v e l o p i n g the computer programs was p r o v i d e d by the s t a f f of th e 0.B.C. Computing C e n t r e . F i n a l l y I wish t o thank Mary f o r her encouragement and u n d e r s t a n d i n g - F i n a n c i a l s u p p o r t f o r t h i s r e s e a r c h was p r o v i d e d by t h e N a t i o n a l Research C o u n c i l of Canada through a p o s t g r a d u a t e s c h o l a r s h i p and r e s e a r c h g r a n t s and by the Westwater Research C e n t r e . 1 CHAPTER 1. GENERAL OUTLINE Background This work deals with the interaction of physical and b i o l o g i c a l processes in the ocean on a mathematical basis. As described by Parsons and da Lange Boom (1972), a great number of interactions are possible between the physical and the b i o l o g i c a l components of a marine ecosystem. In the present discussion, the horizontal d i s t r i b u t i o n of chlorophyll a (a measure of the phytoplankton concentration) in the estuary of the Fraser River w i l l be examined. In t h i s s i t u a t i o n , the physical e f f e c t s on the b i o l o g i c a l parameters (e.g- advection of chlorophyll a ) are much more pronounced than the b i o l o g i c a l e f f e c t s on the physical parameters (e.g. l i g h t absorption by phytoplankton), and the interaction i s e s s e n t i a l l y one-sided, the physical acting on the b i o l o g i c a l component. The area of i n t e r e s t i s the S t r a i t of Georgia, located between Vancouver Island and the mainland coast of B r i t i s h Columbia (Fig. 1 ). Waldichuk (1957) and Tully S Dodimead (1957) have described the physical oceanography of t h i s body of water. The l o n g i t u d i n a l axis of the S t r a i t of Georgia l i e s i n a north-west to south-east d i r e c t i o n . Access to the P a c i f i c i s through r e s t r i c t e d passes having strong t i d a l streams, both i n the south via the Gulf Islands and Juan de Fuca S t r a i t and i n the north via the passages leading to Johnson S t r a i t . The land-locked nature of the S t r a i t of Georgia and the large amount of fresh water inflow from various r i v e r s leads to t y p i c a l estuarine conditions. The s t r a t i f i c a t i o n i s strongest in summer and weakest i n winter, coinciding with variat i o n s i n riv e r discharge- The lar g e s t r i v e r emptying into the S t r a i t of Georgia i s the Fraser River (Fig. 2 ). Its discharge varies seasonally and yearly (Fig- 3 ), minimum outflow generally occurring i n February or March and maximum outflow in June. Both the magnitude of the maxima and minima as well as the date on which they occur varies from year to year. The mean of the yearly maxima i s about 8.5 x 10 3 i 3 / s with a mean yearly discharge of 3-2 x 10 3 m3/s. It i s not uncommon for the discharge to vary by nearly an order of magnitude between extremes- Between 80% and 90% of the t o t a l outflow of the r i v e r i s via the Main (South) Arm (Giovando and Tabata, 1970)- At the mouth of the Main Arm (at Sand Heads), the surface velocity does not r e f l e c t the large seasonal changes i n discharge. Instead the variations in the vel o c i t y are mainly t i d a l l y induced, although a seasonal component i s present. A s a l t wedge i s found i n the r i v e r (Hodgins, 1974), penetrating as far as New Westminster at times of low river flow. The large , discharge of the Fraser River exerts a considerable influence on the surface waters of the S t r a i t of Georgia, p a r t i c u l a r l y i n the v i c i n i t y of the r i v e r delta. Among the more obvious e f f e c t s are the s i l t content of the r i v e r water (giving the surface waters th e i r t y p i c a l muddy brown colour near the r i v e r ) , the low s a l i n i t y values, and the surface v e l o c i t i e s due to the momentum of the r i v e r water- Nutrient l e v e l s are also low r e l a t i v e to the more saline water of the S t r a i t of Georgia- 4 4 0 0 1 , 1 1 1 r - 1 r— 1—: 1 l ~ Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec F i g . 3. Seasonal v a r i a t i o n of the Fraser River d a i l y mean discharge measured at Hope, B.C. The surface layer of water d i r e c t l y influenced by the r i v e r i s often c a l l e d the Fraser River plume- The bottom boundary of the plume i s taken to be the bottom of the h a l o c l i n e , the thickness being i n the order of 2 to 10 m. Horizontal bounds are harder to f i x since there are other r i v e r s discharging into the s t r a i t of Georgia and mixing tends to smooth out the distinguishing c h a r a c t e r i s t i c s of the Fraser River plume- Aside from r i v e r discharge, the position, c h a r a c t e r i s t i c s and extent of the plume are also determined by wind and tide as well as such modifying factors as the C o r i o l i s e f f e c t , c e n t r i f u g a l force and topography- The s i l t content of the water i s not always an indicat i o n of the dynamical extent of the plume (S. Pond, pers- com-)- In summer the plume can extend right across to the Gulf Islands (Fig. 4 ), as far north as Howe Sound and south of Active Pass, while i n winter the extent i s much smaller- Mixing due to winds acts to further decrease the extent of the plume- Problem: The Horizontal Chlorophyll D i s t r i b u t i o n Chlorophyll concentration i s a measure of the abundance of phytoplankton, the f i r s t step of the aquatic food web. Measurements taken in 1967 and reported by Parsons, Stephens and LeBrasseur (1969) and Parsons, LeBrasseur, Fulton and Kennedy (1969) indicate maxima of chlorophyll a and zooplankton concentrations associated with the Fraser River plume (Fiq- 5 ). The chlorophyll a maximum .appears to form an arc centered on the mouth of the Main Arm of the Fraser River- The hiqhest concentrations of zooplankton are further from the r i v e r mouth and there i s not the d e f i n i t e arc found i n the chlorophyll a 7 4 9 ° OO'N HOURS / 1 JUNE 1950 POINT ATKINSON 1209 Mayne^l.') 7ZA Fresh Water Bands ^""-Foam Mixed Water («M mix)| Mixed Water (saltiest) Mixed Water (fresh- 1I| Sea Water est) 123° 30 W I23°00 'W F i g . 4. The" Fraser River plume position.as derived from a e r i a l photographs, ".after Tabata, 1972. 8 Fig.' 5.: Horizontal d i s t r i b u t i o n of c h l o r o p h y l l a., CA) and zooplankton, (B) i n the S t r a i t of Georgia; a f t e r Parsons, Stephens and LeBrasseur, 1969. 9 d i s t r i b u t i o n . Further measurements taken i n 1972 also show a maximum in the chlorophyll d i s t r i b u t i o n with distance from the r i v e r mouth (Fig. 6 ) (unpublished data; Parsons, pers. com.). The d i s t r i b u t i o n of chlorophyll a i s in actual f a c t not so simple since phytoplankton d i s t r i b u t i o n s are in themselves •patchy* (Fig. 7 ) i . e . variations i n concentration occur over length scales between 10 and 10 3 m (Piatt, 1972). These variations are probably due to both physical and b i o l o g i c a l processes although no satisfactory explanation as yet e x i s t s . The question arose as to whether i t was possible to account for the observed chlorophyll d i s t r i b u t i o n in terms of the Fraser River outflow as well as such factors as the available l i g h t , grazing and sinking. B i o l o g i c a l factors must be considered since chlorophyll i s not a conservative property i n the same way as s a l i n i t y . The understanding of the relationship between the Fraser River plume and the chlorophyll d i s t r i b u t i o n i s important i f the impact of man-made changes (such as damming the Fraser River or discharging more effluent into the river) i s to be assessed. At t h i s point i t may be worth mentioning a few of the features of phytoplankton. Ec o l o g i c a l l y , the role of phytoplankton i n the aquatic environment i s eguivalent to that of green plants in the t e r r e s t r i a l environment. By photosynthesis, phytoplankton organisms transform nutrients into c e l l u l a r material using the sun's energy. Herbivorous zooplankton i n turn grazes on the phytoplankton. Phytoplankton populations are made up of sing l e c e l l organisms, although some species have complex external, F i g . 6. Horizontal d i s t r i b u t i o n of chl o r o p h y l l a. i n terms of r e l a t i v e fluorescence; March, 1973. 11 123° 30'W 123° 00* Fig. 7. Horizontal distribution of chlorophyll a showing patchiness; July, 1973. 12 structures (e.g. dinoflagellates) or form long chains. Generally speaking they are almost neutrally bouyaBt and immobile. any motion r e l a t i v e to the water i s by sinking. An exception to t h i s rule are the f l a g e l l a t e s which can move through the water using their f l a g e l l a and at t a i n speeds comparable to phytoplankton sinking rates (Parsons and Takahashi, 1973). Sinking rates vary according to species as well as environmental conditions such as nutrient l e v e l s . Thus the motion of phytoplankton i s determined mainly by the movement of the surrounding water. As one might expect, l i g h t plays an important part i n determining the growth of a phytoplankton population. The l i g h t i n t e n s i t y at any point depends on surface l i g h t i n t e n s i t y , the transparency of the water and the depth. Another important factor i s the nutrient concentration, with low concentrations decreasing the photosynthatic rate. The most important nutrients are n i t r a t e s , phosphates and s i l i c a t e s although trace elements and organic compounds i n small quantities are also important. In the Fraser River estuary nitrogen i s the l i m i t i n g nutrient i n most cases (Takahashi et a l . , 1973). Temperature i s another variable a f f e c t i n g the rate of photosynthesis. Provided other factors are not l i m i t i n g , increasing temperature increases the photosynthatic rate up to an optimum temperature (which varies with species), above which the rate decreases with temperature. / Factors tending to decrease phytoplankton biomass are r e s p i r a t i o n , sinking and grazing. Respiration i s the use by the 13 organism of stored energy to maintain the l i f e processes. The respi r a t i o n rate i s not constant but varies with environmental conditions (Parsons and Takahashi, 1973). S i m i l a r l y , sinking rates vary with environmental conditions. Grazing i s due to zooplankton feeding and i s dependent on both the concentration of the food source and the concentration of the grazers. As the food supply increases the grazing rate (fraction of zooplankton body weight ingested by an organism per unit time) increases, asymptotically approaching a maximum rate. The Approach To The Problem In order to make the problem tractable i t was necessary to quantify the factors discussed above. A model was put together, consisting of mathematical expressions for the relationships which t i e d the physical and b i o l o g i c a l components together, A conservation equation was written for chlorophyll which included advection as well as sources and sinks of chlorophyll. The source term was the net photosynthesis which included the eff e c t of re s p i r a t i o n . Nutrients were not considered to be l i m i t i n g during the time period that was modelled (mid-winter to pre-freshet spring) (Parsons et a l , , 1970; Takahashi et a l . , 1973). S i m i l a r l y the temperature was not included d i r e c t l y . Temperature was included i n d i r e c t l y by using d i f f e r e n t values for the maximum photosynthetic rate at dif f e r e n t times of the year. When one considers the amount of scatter i n the temperature r e l a t i o n (Takahashi et a l . , 1973) plus the fact that (at any given time) the temperature within the plume does not vary more than a few degrees (Fig. 8 ), then TEMPERATURE vs DISTANCE FROM RIVER MOUTH 12- 10- 9H o o 0 o o T ° C e-\ 7A 6H 5i • • D • • ~T~ 10 I 20 r(km) —r~ 30 40 F i g . 8. Temperature at 1 m as a function of distance from the r i v e r mouth ( o November, 1971; •Fe b r u a r y , 1972; m March, 1972; © May, 1972). . ' M -P>- 15 t h i s approach i s not unreasonable. The sink terms used were grazing and sinking. Other possible losses (such as natural mortality) were assumed not to be important. Since the zooplankton population (the grazers) was not i t s e l f modelled, certain assumptions, based on observational data, had to be made about the zooplankton d i s t r i b u t i o n . An a r c - l i k e horizontal d i s t r i b u t i o n was assumed with the maximum value occurring at some distance from the r i v e r mouth (determined from available data ). For the sinking speed of phytoplankton ax constant value was used. The natural s i t u a t i o n i s too complex to j u s t i f y greater precision since size and shape of the organism as well as environmental conditions a f f e c t the sinking speed (Parsons and Takahashi, 1973). The approach i n modelling was to use a s l i g h t modification of the downstream velocity in a jet as discussed by Wiegel, (1970). Continuity was then used along with an experimental expression for the v e r t i c a l v e l o c i t y to calculate the cross-stream component of the horizontal velo c i t y . The eff e c t of the barrier of the Gulf Islands was not included, i . e . a semi - i n f i n i t e sea i s assumed in the horizontal plane. 16 CHAPTER 2 . THE PHYSICAL COMPONENT: THE FLOW FIELD As the aim of t h i s study i s to examine and compare the r e l a t i v e i n f l u e n c e s of p h y s i c a l and b i o l o g i c a l f a c t o r s i n determining the d i s t r i b u t i o n s of a s c a l a r g u a n t i t y ( c h l o r o p h y l l a ) we s h a l l have to make a number of assumptions which w i l l allow us to see through the c o m p l e x i t i e s of the v a r i o u s i n t e r a c t i o n s . The most sweeping assumptions concern.the nature of the flow p a t t e r n i s s u i n g from the mouth of the F r a s e r . There i s no e x i s t i n g adequate d e s c r i p t i o n of the t i d a l l y pulsed outflow of a f r e s h water stream i n t o a broad s a l i n e body of water. Even the s t e a d y - s t a t e case i s not w e l l understood; although a number of s t u d i e s of thermal plumes have been c a r r i e d out, they cannot be d i r e c t l y a p p l i e d to the flow out of a r i v e r coming out at a n e a r l y c r i t i c a l i n t e r n a l v e l o c i t y over a s a l t wedge. Ne v e r t h e l e s s , i n order to o b t a i n some r e p r e s e n t a t i o n of the flow, we s h a l l f i r s t assume s t e a d y - s t a t e c o n d i t i o n s , i . e . , that 1) the net f r e s h water outflow i s independent of time, and t h a t 2) the i n f l u e n c e o f t i d a l v a r i a t i o n s may somehow be c o n s i d e r e d as averaging out over the time s c a l e i n v o l v e d i n s e t t i n g up a d i s t r i b u t i o n p a t t e r n corresponding to the p r e v a i l i n g steady c o n d i t i o n s . The f i r s t s t e a d y - s t a t e assumption may not be too t r a g i c , s i n c e short-^period f l u c t u a t i o n s i n r i v e r d i s c h a r g e are of r e l a t i v e l y s m a l l amplitude. N e g l e c t i n g the r a p i d and important t i d a l v a r i a t i o n s f i n d s j u s t i f i c a t i o n only i n our ignorance of how to account f o r them and i n the r a t h e r l i m i t e d aim of t h i s type of study, which i s not to work out a good d e s c r i p t i o n of the varying plume p a t t e r n but to study the 17 response of phytoplankton to the presence of a (mean) current of a reasonable form. In the absence of a correct two-dimensional description of ri v e r flow into a saline basin, we chose what we thought was the most appropriate jet flow . pattern available i n the l i t e r a t u r e . wiegel (1970) has reviewed the studies of je t s and r i v e r plumes and we have used a Gaussian j e t flow from his work. To specify t h i s flow pattern, l e t us f i r s t introduce a Cartesian coordinate system (x,y,z) as shown i n F i g . 9 , with x increasing downstream from the r i v e r mouth, y positive southwards and z positive upwards. The components of the v e l o c i t y vector u are denoted by (u,v,w) i n the three coordinate di r e c t i o n s . The r i v e r plume w i l l be assumed to extend from the surface z = 0 to some depth z = -h(x,y). The average horizontal v e l o c i t y component over that layer w i l l then be aiegel (1970) gives an empirical formula for the a x i a l v e l o c i t y of an axisymmetric j e t issuing from an o r i f i c e of diameter D0 into an unbounded body of f l u i d : C 4 i s an experimental constant, x i s the distance from the o r i f i c e in the downstream d i r e c t i o n , and r i s the r a d i a l distance from the j e t axis. Results due to Abraham (1960) indicate that a s i m i l a r expression may be used for the discharge 18 F i g . 9. The co-ordinate system employed i n the model. 19 of a r i v e r on the s u r f a c e of a body of r e c e i v i n g water, provided y i s s u b s t i t u t e d f o r r- The form which we s h a l l adopt, and which al l o w s f o r plume spreading as w e l l as i t s slowing down with d i s t a n c e from the r i v e r mouth, w i l l be U I. = kt exp (- kz f/u + x 0 f ) (z. 3) x + x 0 The upper l a y e r downstream t r a n s p o r t thus decays away from the mouth and spreads to g i v e a Gaussian t r a n s v e r s e p r o f i l e - The parameter x c . i s i n t r o d u c e d to i n s u r e that the t r a n s p o r t remains f i n i t e at x = 0; i t s value was chosen to make the width of the j e t , as measured between the p o i n t s where the Gaussian f a l l s to 0.38 of i t s peak value, equal to one k i l o m e t e r at the r i v e r mouth (x = 0). For kz, the value employed by S i e g e l (1970) was a l s o used here- Thus x c = 5 x 103 m ; = 96[1.0 + 0. 1 9 ( ^ w / ^ o - 1) 3-2 ~ 96 where ^ 0 i s the d e n s i t y of the d i s c h a r g e d water and ^ w that of the s a l t water underneath the plume- Since ( t 3 w / ^ 0 " ^ i s s m a l l , k^ i s well approximated by k j , ^ 96. The value of k, i s a d j u s t e d to the value of v e l o c i t y 0 o and depth h 0 at the c e n t r e of the r i v e r mouth (x = y = 0) : The magnitude of U c can then be v a r i e d to model v a r i o u s flow c o n d i t i o n s . The non-dimensionalized downstream v e l o c i t y d i s t r i b u t i o n 0h/U oh o i s i l l u s t r a t e d i n F i g - 10 f o r h = c o n s t a n t . P l o t s of the s t r e a m l i n e p a t t e r n cannot be c o n s t r u c t e d before f u r t h e r assumptions have allowed us to s p e c i f y the cross-stream v e l o c i t y component, V. 21 The influence of the C o r i o l i s force i s neglected e n t i r e l y . This assumption may be tenable near the r i v e r mouth, where the i n e r t i a l terms dominate the flow, but cannot r e a l l y be expected to hold f a r downstream, after the plume has slowed down. The e f f e c t of the sloping bottom on the plume i s also ignored, as the bottom slopes quite steeply o f f Sand Heads, and the presence of the s a l t water beneath e f f e c t i v e l y i s o l a t e s the upper layer from the bottom. F i n a l l y , l a t e r a l f r i c t i o n and entrainment are not considered: the plume i s so t h i n compared to i t s width and the area of i t s underside so large compared to that of i t s l a t e r a l edges that i t i s reasonable to assume that everywhere i n the plume, except very near the edges, entrainment and f r i c t i o n w i l l occur only at the bottom of the plume. Only the downstream velocity d i s t r i b u t i o n i s given by (2.3); to construct a two dimensional flow f i e l d , some assumptions have to be made concerning the v e r t i c a l entrainment v e l o c i t y found at z = -h. Letting f o r brevity, we use a r e l a t i o n s h i p obtained by Keulegan (1966) for the v e r t i c a l v e l o c i t y across the interface of a model s a l t wedge estuary: where m i s a constant; Uc , the c r i t i c a l v e l o c i t y , i s given by with c' = constant, i> = the v i s c o s i t y of the lower la y e r ,A w(x,y,-h) = w(-h) 22 t h e d e n s i t y d i f f e r e n c e between t h e l o w e r and t h e u p p e r l a y e r and ^ t h e d e n s i t y o f t h e u p p e r l a y e r . E q u a t i o n (2.4) i s o n l y v a l i d f o r s u p e r - c r i t i c a l f l o w , - I t i s t h e n p o s s i b l e t o c o m p l e t e t h e d e s c r i p t i o n o f t h e f l o w f i e l d by u s i n g t h e c o n t i n u i t y e q u a t i o n . I t w i l l be c o n v e n i e n t t o w r i t e t h e h o r i z o n t a l v e l o c i t y components as U s tfCx^i) U(x;y) / ^ where we assume t h e same v e r t i c a l v e l o c i t y p r o f i l e Y ( x , y , z ) f o r b o t h components, B e c a u s e o f t h e d e f i n i t i o n s ( 2 . 1 ) , t h e p r o f i l e f u n c t i o n must o f c o u r s e s a t i s f y C* * dx = h (z.j) I n an i n c o m p r e s s i b l e f l u i d , V- u = o (z.s) so t h a t , i n t e g r a t i n g (2.8) o v e r t h e upper l a y e r d e p t h , s u b s t i t u t i n g f r o m (2.6) and l e t t i n g U = (Ucx.y^VCx,/)) we have The s u r f a c e v e r t i c a l v e l o c i t y w(x,y,0) v a n i s h e s and (2-9) may be 23 integrated into the form V-(LIh) = w ( - K ) + * ( - ^ U ' * f l (2./0) The right hand side of t h i s r e l a t i o n i s recognized as the velocity component normal to the sloping interface h(x,y) and into the upper layer. Expanding (2.10) and writing i t as a d i f f e r e n t i a l equation for V, the transverse horizontal v e l o c i t y , we have i V + f(x,y)V = 3(x,y) (z./l) where ft^y) = / - M-h) Ak ( 2 - / 2 ) and fi h ^x Given 0(x,y) from (2.3) and w (-h) from (2.4), and an e x p l i c i t form for o'(x,y,z), (2.11) becomes a d i f f e r e n t i a l eguation for V(x,y). Since w(-h) contains V 2 , i t i s not s t r i c t l y possible to integrate (2.11) d i r e c t l y . However, i n areas where V 2 < U 2, such as near the axis of the plume, an i t e r a t i o n technique can readily be used to obtain successively better estimates for V, star t i n g from V 2 << U 2, so that w (-h) = m(U-0c). The f i r s t approximation for small V i s then found by integrating (2.11); 24 Two models w i l l be considered below: a simple one, followed by a more complex one. For each we s h a l l specify e x p l i c i t dependences for tf(x,y,z) and values of the constants m and c*. More precise estimates of the transverse flow velocity w i l l then be found for each one of the models. 2 5 CHAPTER 3. THE CHLOROPHYLL CONSERVATION EQUATION Phytoplankton, and hence the chlorophyll concentration used to quantify i t s density, i s safely assumed to be a passive scalar variable, advected by the flow but not modifying i t i n any fashion. The b i o l o g i c a l - p h y s i c a l interaction i s i n that case u n i d i r e c t i o n a l : a l l from the physics to the biology. Let us write the chlorophyll concentration n(x,y,z) as n(x,y,z) = ^(x,y,2} M(x,y) (3./) where M(x,y) = x C n J.% (3.2) K \ i s then the average concentration over the upper layer. I t follows that the p r o f i l e function l/(x,y,z) must s a t i s f y CV <Li = h (3.3) A steady-state conservation equation for chlorophyll may be written as V - ( u n ) = Q (3.4) where Q i s a source strength function, which may depend on u and n as well as space-coordinates. The function Q w i l l include the growth rate, the sinking rate, zooplankton grazing and any other process a f f e c t i n g the chlorophyll density in a non-conservative manner. 26 As we are interested in what happens in the upper layer -h < z < 0, we integrate (3.4) over that layer: Using (2.1) and (3.1), Leibnitz's rule, and the condition w(0) = 0, (3.5) becomes: .0 r w i t h 4/(-h) =1/(x,y,-h) ana ^ ° Q 6 (3.0 xi . (x,y) = £ *V C L ^ (3.7) Combining (2.10) and (3.6) so as to eliminate the v»U terms we f i n d U - v M = j _ C ° G U * + M J U-vK - U-v.ru h ) which i s further abbreviated as U - v M = H (M,U,V,x,y) (3.9) 27 where H (f3 /U rV,x,y) i s the right hand side of ( 3 . 8 ) . As jn_ always turns out to be proportional to h i n the example chosen, i t i s clear that the f i r s t two terms in the bracket prefixed by M/XL cancel out and that ( 3 . / 0 ) (w(-W + i f ( - M U - v O Any net i s then through increase or decrease in the concentration of chlorophyll due to 1) i n t e r n a l sources (the Q term) and 2) advection the bottom of the upper layer (the second term). 28 CHAPTER 4. GENERAL METHOD OF SOLUTION Let t be the time elapsed i n t r a v e l l i n g from the r i v e r mouth to some point (x fy) along a streamline (streamlines and pathlines are i d e n t i c a l i n t h i s steady state s i t u a t i o n ) . The rata of change of position along a streamline i s then given by 2ht = U (x,y) (4.l) 3>i 2l = V ( x , y ) (4.2) Dt Since D/Dt = uA/<ix •+ Vi/^y, (3.9) may be written as 3 M = H ( M , U , V , x , y ) (+.3) 2> t Given functional forms , and i n i t i a l values for 0, V and M, i t i s possible to integrate the above equations step by step along streamlines to obtain a map of the horizontal d i s t r i b u t i o n of velocity and chlorophyll. This method of solution i s broadly applicable i n the above form to any kind of scalar f i e l d M(x,y) for which a source-sink function Q(x,y,z) can be defined. It could for example be readily applied to provide a quantitative account of sediment load i n the plume, or of concentrations of chemical species, such as observed for trace elements by Thomas (1975). Alternately, the inverse problem of determining the velocity f i e l d which leads to an observed d i s t r i b u t i o n M(x,y) might be attempted using (4.1) to (4.3), although i t might not be possible, depending on the form of Q(x,y,z), to f i n d a unique solution to that problem. 30 CHAPTER 5. SOURCES AND SINKS OF CHLOROPHYLL A number of influences are covered by the source strength function Q(x,y,z), and they w i l l now be discussed and given appropriate parameterizations i n terms of environmental factors. Three sources and sinks of chlorophyll i n the upper layer are considered: primary production, zooplankton grazing and sinking. The production of particulate organic matter by phytoplankton occurs at a rate P usually called' the photosynthetic rate and expressible i n terms of grams of chlorophyll produced per unit time per gram of existing chlorophyll. The usual units i n which P i s given are i n terms of grams of carbon fixed per unit time per gram of chlorophyll: we can transform from one set of units to the other using a conversion factor (g chlorophyll/g carbon). Productivity i s l i g h t sensitive and an expression o r i g i n a l l y suggested by Steele (1962) and used by Takahashi at a l . (1973) i s employed here: T>=*fc7U expO-fcl) (s.i) P i s the chlorophyll production rate, i n units of ( t i m e ) - 1 , ©< converts from carbon units, i n which P m i s expressed, to chlorophyll units; b i s a constant with the dimensions of minutes/langley while I i s the l i g h t i n t e n s i t y in langleys/minute- It i s clear from (5.1) that P has a maximum value (<<Pm) at an optimal l i g h t i n t e n s i t y J 31 Takahashi et a l . (1973) found from a best f i t of available experimental data, a value of = 0.18 ly/min, so that from (5.2), b = 5.56 min/ly. This d i r e c t l y calculated value for b gives a better f i t to the experimental curves than that computed by Takahashi et a l . (1973) (b = 5.37 min/ly) by an improper numerical technique which does not s a t i s f y (5.2). The maximum rate of carbon f i x a t i o n P̂_ varies with, nutrient a v a i l a b i l i t y and temperature. As mentioned e a r l i e r the temperature i n the Fraser River plume does not vary by more than a few degrees at any one time but does vary with the season (Fig. 8). Given the scatter observed by Takahashi et a l . (1973) in the P^ (T) observations i t i s quite j u s t i f i a b l e to take P m •= constant everywhere in the plume for any one simulation. Observations by Parsons et a l . (1970) show that nutrient le v e l s in the S t r a i t of Georqia are high enough not to be l i m i t i n g factors i n production, so that we w i l l completely neglect the dependence of P^ on nutrient concentrations. Possible values of P^ w i l l range from 4.4 x 1Q-* to 12.4 x 10-* g carbon/g chlorophyll/sec, depending on the mean temperature of the plume (and thus on the time of the year). To take into account the ef f e c t of respiration ( i . e . that there exists a minimum energy requirement to maintain l i f e without growth) the concept of a compensation l i g h t i n t e n s i t y I c (Parsons and Takahashi, 1973, p. 64) i s introduced into (5.1), which now becomes ?=*!>?„, (I-Ic) exP6-fc(I-lj) This equation i s v a l i d only for I > I c . For I < I c , P w i l l be taken as equal to zero (Fig- 11 )- Values of I t measured by Parsons, Stephens and LeBrasseur (1969) over four months vary from 0.006 to 0.01 ly/min. A constant value consistent with those data w i l l be taken for any one simulation. As indicated by Caperon (1967), the concept of constant res p i r a t i o n implied by (5.3) i s not l i k e l y to be va l i d for a l l l i g h t i n t e n s i t i e s . However, in the abscence of a better expression, equation (5.3) accounts for the effe c t of res p i r a t i o n . The carbon to chlorophyll r a t i o varies from 25 for vigorously growing phytoplankton i n the presence of excess n i t r a t e to 60 for unhealthy organisms in nitr a t e depleted water (Antia et a l . , 1963). A f a i r l y conservative value of 40 has been used here, thus giving a conversion factor oi = .025 = 1/40. An expression for zooplankton grazing of phytoplankton has been given by Ivlev (1961). The rate of chlorophyll removal by grazing ( "$ = mg of chlorophyll/m 3/time) i s = Z G (/ - e x f f - c i . r i ) (5.4) where Z i s the zooplankton (wet weight) density i n mg/m3, G the maximum grazing rate i n units of milligrams of chlorophyll per milligram of zooplankton per unit time, d, i s a constant with units of m3/mg of chlorophyll and n(x,y,z) i s the chlorophyll concentration as before. A nearly eguivalent expression has been used here, 33 P F i g . 11. Chlorophyll production rate, P, as a function of l i g h t i n t e n s i t y , I ; equation (5.3). 34 i ? = Z G n (f.f) where i s another constant with the dimensions of milligrams of chlorophyll per unit volume. The choice of (5.5) instead of (5-4) i s primarily motivated by the fact that the second expression i s easier to integrate over the upper layer f o r the v e r t i c a l dependences of n chosen below. The expression (5-5) shows a similar behaviour to Ivlev's r e l a t i o n (Fig- 12 ), but increases more slowly. Over a limi t e d range of n, the two expressions may be made to agree closely by appropriately selecting the constant d^; th i s i s indeed the case over the region of i n t e r e s t , with n generally varying less than an order of magnitude (Parsons, Stephens and LeBrasseur, 1969; Parsons et a l - , 1970). On the basis of figures given by Parsons and Takahashi (1973) a value of d^ = 5 mg/m3 was used. This results in an ingestion rate of half the maximum rate for chlorophyll concentrations of 5 mg/m3 and about 0.83 of the maximum rate at 25 mg/m3. The voracity of zooplankton organisms varies with the species considered and with the l i f e stage of any one species. Figures quoted by Parsons and Takahashi (1973) led us to use an ingestion rate equal to 70% of the wet weight per day. Combining t h i s with an average dry to wet weight r a t i o of about 0-2 and a carbon to dry weight r a t i o of 0.5, as drawn from the data given by the same authors, and with the carbon to chlorophyll conversion factor ( o( = 1/40) used above, we 36 calculate G as G = 0>? X O.Z X 0 , 5 " X J[__ — ZX/0 rn̂ c U o r o p l y l / £4X3600 40 ^ EoopUhkton- j e e A somewhat lower value (G - 1 x 10 _ a) i s found i f the results of Stephens et a l - (1969) are used for the wet to dry weight conversion- As a l l these factors are l i k e l y to be guite variable (especially the ingestion rate), we w i l l s t i c k to the G = 2 x l O - 8 mg chlorophyll/mg zooplankton-sec value. The sinking rate i s usually written as following Riley et a l . . (1949) , with ws a sinking speed. Smayda (1970) gives values of ŵ between 0 and 30 m/day for sinking rates of l i v i n g phytoplankton, based on observations on about 25 species. Values used in the present calculations range from 1.2 x 10~ s m/sec to 5.8 x 10 _ s m/sec (1 to 5 m/day). The l o c a l source strength Q i s then the sum of the above three e f f e c t s : Q = nl + 3 + S (s.7) What i s needed i s the i n t e g r a l of Q over the upper layer. As the upper layer i s continuously agitated by wind waves and by in t e r n a l waves (Gargett, 1976), the turbulence l e v e l i n the upper few meters i s quite high, and the near-surface phytoplankton crop w i l l be carried back and forth v e r t i c a l l y over a depth range of a few meters by mechanical mixing. Besides, phytoplankton from near the surface w i l l also gradually 37 sink down with a small ve l o c i t y ws . A ty p i c a l phytoplankton organism w i l l thus experience, over a period of a few hours, l i g h t conditions which are averaged over a certa i n depth. We s h a l l assume conditions i n the upper layer to be turbulent enough to use the averaged l i g h t intensity T, given by with JUL the extinction c o e f f i c i e n t (in m-1) and Ia the l i g h t i n t e n s i t y at the surface, as representative of conditions experienced by the whole upper layer phytoplankton population. The photosynthetic available radiation (PAR) l i e s in the wave length range 400-700 nm. Following Takahashi et a l . (1973), we use Strickland's (1958) assumption that the PAR at the sea surface i s one half the t o t a l solar radiation at the surface. The radiation i n t e n s i t y varies with cloud cover and sea roughness but the best we can do here i s to use monthly mean insol a t i o n values for la, as computed by Parsons, Stephens and LeBrasseur (1969). I 0 w i l l then denote only the PAR; i t s values range from 0.03 to 0-10 ly/min. The average value T i s readily estimated from (5-8) as T = (/ - e x p ( - M « ) Values ofytt have been calculated from unpublished data provided by T.R- Parsons for the Fraser River plume i t s e l f and range from 0.3 to 0.8 m_1 for the period of in t e r e s t . These values agree clo s e l y with other measurements i n t h i s region (Parsons, 1965). 38 The integrated value of nP w i l l then be where P i s the photosynthatic rate corresponding to the average l i g h t i n t e n s i t y T- Using (3.3), The i n t e g r a l of zooplankton grazing w i l l depend on the v e r t i c a l dependence used for n, i . e . on the function i/(x,y,z). The integrated sinking rate i s simply Since the flux through the upper surface, at z = 0, must be zero (we can integrate to z = 0 + £ , where n (0 + £ ) =0, since that i s in the a i r , above the water surface, and l e t 6 0 to show that the f i r s t term must vanish), the net rate of sinking out of the upper layer i s - I i The integrated source strength i s then In a completely horizontally non-divergent uppar layer and 39 with horizontally independent - j / , the right hand side of (5-12) would be the only contribution to changes in chlorophyll- The l o c a l r e l a t i v e importance of photosynthetic growth rate, zooplankton grazing and sinking would then completely determine the d i s t r i b u t i o n of chlorophyll a i n the upper layer- One could then write ( 5 .12 ) as C Q r M ^Cx,y) and i f ^ (x,y) were a constant, integrate ( 3 - 9 ) , or rather i t s time dependent formulation (4-3) to fin d M r r M o C X p J ^ (51/3) The simple exponential growth represented by (5w13) i s readily understood as ar i s i n g from the balance of the various source terms which make up ij> (x,y). In a non-uniform flow f i e l d , (5-13) may s t i l l be regarded as determining instantaneous l o c a l chlorophyll variations- This purely l o c a l behaviour may of course be completely masked by the other terms present in H(M ,U,V,x,y) (eguation 4 . 3 ) , a r i s i n g from the non-homogeneity of the flow f i e l d - The simplest example of t h i s masking e f f e c t i s obtained by comparing the sinking term -Mws (-h) -V (-h) with the v e r t i c a l advection term Mw(-h) l/(-h) which occurs i n ( 3 . 8 ) ; i t i s obvious that the two v e r t i c a l transport terms are opposite i n their action and that sinking or ascent takes place according to the sign of (wff (-h) - w(-h)). More detailed comparisons of the r e l a t i v e influence of l o c a l sources to flow divergence on phytoplankton d i s t r i b u t i o n w i l l be given l a t e r - The obvious 40 lesson that we may expect to learn from solving (4.1) to (4.3) along pathlines is that the kinematics of the flow field may play a very significant role in establishing the observed pattern of phytoplankton distribution. It is a comforting thought however, that since the advective processes merely redistribute phytoplankton and neither create nor destroy it, a chlorophyll balance performed over the whole volume of interest will be independent of the flow pattern and will reflect the net effect of the source term Q, integrated over that volume. Our assumption of time-independence thus implies that the total quantity of chlorophyll in the volume of water considered is constant and that, over the whole volume, a balance has been reached between production, grazing and sinking: S S j Q ' x - y > ^ <** J y «k = 0 Although this is not true over a period in the order of months, over a few days this is certainly valid. 41 CHAPTER 6. DATA In order to obtain r e a l i s t i c values for the parameters i n the model i t was necessary to make simultaneous measurements of the most important parameters. Although quite a large amount of data had been c o l l e c t e d in Georqia S t r a i t , the nature of our problem required that the b i o l o g i c a l parameters be measured i n the Fraser River plume. Since the major variations occured i n a downstream d i r e c t i o n i t was decided to take measurements along the axis of the plume. This presented some problems since the plume i s influenced by both wind and tide and the area to be covered was quite large. The C.S.S. Vector was the vessel used for the measurements. The data were c o l l e c t e d i n conjunction with work being done by T.R. Parsons of the I n s t i t u t e of Oceanography, U.B.C- in the Fraser River plume. Temperature and s a l i n i t y p r o f i l e s were measured as well as the photosynthetic radiation- As part of the b i o l o g i c a l program chlorophyll and zooplankton samples were also collected- At a l a t e r date an attempt was made to measure the horizontal d i s t r i b u t i o n of chlorophyll a using a fluorimeter. We had no success since the scatter in the c a l i b r a t i o n curve was of the same magnitude as the observed fluctuations i n the fluorimeter output.. For the f i r s t cruise in the series (Gulf 1, November, 1971) the plume position was determined v i s u a l l y from a small aeroplane. The boundaries and general extent of the plume were relayed to the ship. A series of ten stations (Fig. 13 ) were then occupied as rapidly as possible up to 32 km from the mouth of the Main Arm • of the Fraser River. For the second cruise (February, 1972), a di f f e r e n t method of determining the plume position was attempted since i t was not possible to obtain the use of an a i r c r a f t . The method was to take s a l i n i t y and temperature p r o f i l e s in the upper 20 meters i n a coarse grid of stations and then deduce the plume position (Fig. 14 ). The drawback of t h i s approach was that one does not obtain an instantaneous picture and that, by the time the ship i s i n position to st a r t the main series of stations, the plume may have changed s i g n i f i c a n t l y . This time, a series of stations was occupied along l i n e s radiating out from the mouth of the r i v e r . No success was achieved i n following the axis of the plume. Visual observation from the ship was also unsuccessful i n determining the plume position due to the small angle between the l i n e of sight and the water surface. For s i m p l i c i t y , l a t e r cruises occupied stations whose positions were unchanged for the remainder of the program. These stations (Fig. 15 ) were chosen to extend from the r i v e r mouth to the north west- Although these stations were not always i n the same location r e l a t i v e to the plume, the positions were consistant from cruise to cruise and time was not spent attempting to locate the plume each time- S a l i n i t y and temperature p r o f i l e s were measured with an I n d u s t r i a l Instruments RS 5- The accuracy for these measurements i s taken to be ± 0. 1%»and ±0-1 C°- Fig- 16 shows the s a l i n i t y p r o f i l e s from cruise Gulf 1, while a l l the s a l i n i t y and temperature data are presented i n ' the Appendix. The v e r t i c a l extinction c o e f f i c i e n t s were determined using a 2t< l i g h t meter f i t t e d with a selenium c e l l . With this instrument the l i g h t intensity at depth i s compared with the i n t e n s i t y at F i g . 14. Location of s t a t i o n s , c r u i s e Gulf 2; February., 1972. 45 49° F i g . 15. Station p o s i t i o n s , cruise Gulf .3 and subsequent cru i s e s . F i g . 16. S a l i n i t y p r o f i l e s from cruise Gulf 1. the surface, hence extinction c o e f f i c i e n t s may be calculated. The expected accuracy of the extinction c o e f f i c i e n t s i s ± 0.05 m-1- F i g . 17 gives some sample p r o f i l e s of the l i g h t i n t e n s i t y while F i g . 18 shows the variation of the extinction c o e f f i c i e n t with position i n the plume. Aside from the extinction c o e f f i c i e n t , other parameters were derived from the data. The s a l i n i t y and temperature p r o f i l e s were used to determine the depth, h, and the density,^ , of the plume as a function of distance from the r i v e r mouth. The expressions (described later) were f i t t e d using the data from the Gulf 1 cruise since t h i s was the only cruise where the stations were known to be reasonably close to the axis of the plume. While data from one cruise can not be representative of a whole year, c e r t a i n l y during the winter and spring pre-freshet period one would expect the basic c h a r a c t e r i s t i c s to remain unchanged- Hence the same functions were used for the whole period modelled but the magnitude of the parameters was varied as appropriate. Light Intensity (% of surface value) F i g . 17. Light i n t e n s i t y (% of surface value) as a function of depth. 0 10 " X " 20 T r (km) A 2 / l / 7 l • 2 0 / 3 / 7 2 H 11/5/72 I 3 0 I 40 18. Non-dimensionalized ext i n c t i o n coefficient, J JL/ J J .^ , as a function of distance from the r i v e r mouth. 50 CHAPTER 7. MODEL I: FORMULATION The p a i r o f models f o r which r e s u l t s a r e now p r e s e n t e d may be c o n s i d e r e d a s t h e f i r s t two s t a g e s i n a s equence which w i l l h o p e f u l l y c o n v e r g e i n a s m a l l number o f s t e p s t o a r e a l i s t i c r e p r e s e n t a t i o n o f b i o l o g i c a l - p h y s i c a l i n t e r a c t i o n s i n t h e a r e a o f i n t e r e s t . The f i r s t model i s o v e r l y s i m p l i s t i c : t h e f l o w f i e l d p l a y s a p u r e l y a d v e c t i v e r o l e i n a g r e a t l y i d e a l i z e d s e t o f c o n d i t i o n s , c a r r y i n g p h y t o p l a n k t o n t h r o u g h a r e a s o f d i f f e r e n t v a l u e s o f t h e i n t e g r a t e d s o u r c e t e r m . T h i s model i s i d e a l i z e d on p u r p o s e , t o p r e s e n t us w i t h a c l e a r l y c o m p r e h e n s i b l e s i t u a t i o n , where t h e i n f l u e n c e o f t h e v a r i o u s p a r a m e t e r s i s e a s i l y i n t e r p r e t e d . T h i s f i r s t a t t e m p t may be c o n s i d e r e d as an i n t r o d u c t i o n t o t h e s e c o n d , more compl e x model- The b a s i c p r e m i s e s on which model I i s b a s e d a r e l i s t e d i n t h i s s e c t i o n , t o g e t h e r w i t h a d i s c u s s i o n o f t h e i r c o n s e q u e n c e s - N u m e r i c a l v a l u e s f o r t h e p a r a m e t e r s a r e a l s o i n t r o d u c e d and t h e i r c h o i c e j u s t i f i e d . The a c t u a l r e s u l t s and t h e i r i n t e r p r e t a t i o n a p p e a r i n t h e n e x t c h a p t e r . The p a r a m e t e r i z a t i o n s a s s o c i a t e d w i t h t h e g e o m e t r y and t h e c u r r e n t p a t t e r n a r e d i s c u s s e d f i r s t ( i - i v ) , f o l l o w e d by t h e b i o l o g i c a l components ( v - v i i ) _ i ) The d e p t h o f t h e u p p e r l a y e r i s e v e r y w h e r e t h e same: The o b s e r v e d d e p t h o f t h e u p p e r l a y e r a c t u a l l y v a r i e s down t h e plume, b u t t h i s c o m p l i c a t i o n w i l l be i n c l u d e d i n t h e s e c o n d model. I n model I, u n i f o r m v a l u e s o f h between 2 m and 30 m w i l l be u s e d . 51 i i ) From (7-1) and (2.7), i t follows that the p r o f i l e function X/(x,y,z) must also be independent of horizontal position. Experiments performed by Stefan and Schiebe (1970) on the discharge of hot water into a tank suggest a simple parameterization of the p r o f i l e i n the upper layer i n terms of the read i l y integrable function = ex p (^0 (7.2) In view of equation (2.7) , (̂ ^ must s a t i s f y <*>k + exp (-^) s / (7.3) i i i ) The entrainment ve l o c i t y i s simply written which implies that the downstream velocity U i s much larger than the cross-stream component V, and also much larger than the c r i t i c a l v e locity 0 C. Both assumptions are probably j u s t i f i a b l e near the r i v e r mouth, before there i s any appreciable spreading of the plume- Once more, the complexities of the f u l l entrainment formula (2-4) are reserved for the more r e a l i s t i c second model. A numerical value of m was estimated from the s a l t balance of the plume. Assuming that the increase i n s a l i n i t y observed along the axis of the plume i s due uniquely to v e r t i c a l entrainment from the lower layer, and not from l a t e r a l mixing, an estimate of the entrainment v e l o c i t y w(-h) may be found as follows. Consider a longitudinal segment of the upper layer, as 52 shown in Fig- 19 - The mass balance i s s a t i s f i e d by 0, h, •= 0 6h o + w(-h) L and the s a l t balance by U, h, S, = U 0h 0S 0 + w(-h)LS6 Eliminating U, h we f i n d U 0 L S6 - S 0 (7-7) Estimates of the quantities entering the right hand side of (7-7) were made from data gathered by the author on the Gulf 1 cruise already discussed in Chapter 6. Values of the s a l i n i t y differences between pairs of stations and of the appropriate depth h 0 and separation L are shown i n Table I - The r a t i o w(-h)/Uc varies over a wide range of values (from 10 - 5 to 4 x 10 - 3)- Due to the very low s t r a t i f i c a t i o n at downstream distances greater than about 25 km, i t i s probable that the thickening of the upper layer observed beyond Station 8 may be due i n part to wind mixing and not to upward entrainment. Accordingly, only the f i r s t seven values of Table I were used to form an estimate of m, finding a value of This estimate i s very close to that of Keulegan (1966), who obtained a value of m = 2,12 x 10 - 4 from experiments i n a small scale model- Under the assumptions (7.1), (7-2) and (7-4), the average horizontal velocity components i n the upper layer now obey a s i m p l i f i e d form of (2.10): m = 2.4 x 10-* a * ^y K | - L — 1 ho h. So t w( -h) t S b F i g . 19. A segment of the upper l a y e r , derive Table I. showing the q u a n t i t i e s used to 54 TABLE I. Evaluation of entrainment from cruise Gulf 1 data. T~ Stn- | Separation pair | L (m) . 1 - 2 2 - 3 3 - 4 4 - 5 5 - 6 6- 7 4-8 x 103 3-2 x 103 3.2 x 103 3.2 x 103 3.2 x 103 3-2 x 103 7 - 8 i 3.2 x 103 Depth h c (m) 2 1 5 5 5 7 7 (S, -S0 ) %m 0.4 0.8 ~ 0.01 0-2 ~ 0- 01 1.3 0.2 (S, - s 0 ) too 1. 1 0.7 1.2 1.3 1.4 0. 8 0.8 1. 5 x 10-* 3.6 x 10-* ~ 10-s 2.4 x 10-* ~ 10-5 3-6 x 10-* 5-5 x 10-* 8 - 9 | 3-2 x 103 I 9 - 10 j 3-2 x 10 3 15 30 0-7 0-2 0. 8 1,4 4.1 x 10-3 1.3 x 10-3 55 Since U > 0, the upper layer flow i s everywhere divergent, pathline separation increases downstream and, in the absence of source terms, the density of any passive scalar carried by the flow w i l l decrease downstream. This decrease i s a d i r e c t consequence of d i l u t i o n with entrained water. Only i n the case where the lower layer i s as r i c h as the upper one in that passive scalar w i l l there be no d i l u t i o n and hence no downstream decrease i n concentration. Choosing w(-h) independent of V allows d i r e c t integration of (2.11). Using (7.1) and (7.4), the c o e f f i c i e n t s f(x,y) and 9(x#y)# given by (2.12) and (2-13) take e x p l i c i t forms f <x,y} » o 3<x,y) = m U - \U (?-/0) Hence, V(x,y> =^ (^mJi "-^-) + (?") In the abscence of the C o r i o l i s force, V w i l l be antisymmetric about the downstream axis, so that we may assume V(x,0) = 0, which f i x e s the constant of integration. fiecaliing U(x,y) as written i n (2.3), (7.11) becomes V ( X , y ) = fm -h I \ VUdy I n X +X0) A (x +X< 56 Numerical values for V(x,y) are calculated from the resulting analytic expression- The three dimensional structure of the r i v e r plume i s now completely specified by equations (2.3), (7.4) and (7-12). Typical flow f i e l d s and streamline patterns are depicted i n the next section (Figs- 23,24,31,32). iv) The plankton p r o f i l e function y(x,y,z) i s also taken horiz o n t a l l y uniform. In addition, the v e r t i c a l structure i s ignored and we use -V = I -A < z < o (j./z) The only j u s t i f i c a t i o n behind t h i s choice i s i t s extreme si m p l i c i t y . More complex p r o f i l e s , based on data, w i l l be used in model I I . The i n t e g r a l of the product of the p r o f i l e functions, as defined i n (3.7), reduces to The upper-layer chlorophyll density equation (3.8) then takes the p a r t i c u l a r l y simple form ^ y fi ^ f, with i/ = 1, as per (7. 13) , and continuous across z = -h, the l a s t term on the right hand side of (7.15) vanishes. There i s then no d i l u t i o n of chlorophyll concentration due to entrainment and the only contribution to changes i n M i s from the l o c a l source terms. The role of the flow f i e l d i s then simply to carry parcels of water through areas of varying strength of the 57 source term- Such an advective role may of course be extremely important i n determining the o v e r a l l shape of the chlorophyll d i s t r i b u t i o n , since the amount of time spent i n regions of positive or negative source strength, and hence the ultimate concentrations reached due to the e f f e c t of such sources, w i l l depend d i r e c t l y on the l o c a l strength of the flow- At the opposite extreme, we might consider a v e r t i c a l chlorophyll p r o f i l e with -V = 1 for -h < z < 0, -j/ = 0 for z < -h- In that case, there would be a velocity dependent d i l u t i o n e f f e c t i n (7-15), decreasing with U away from the mouth of the r i v e r and away from the axis of the plume. An, examination of both extreme cases w i l l provide us with an estimate of the role.of d i l u t i o n by entrainment. Me now pass to a discussion of the b i o l o g i c a l parameters, v) A considerable amount of s i l t i s usually found i n suspension at the mouth of the Fraser River. The extinction c o e f f i c i e n t u i s increased by the presence of suspended particulate matter and t h i s dependence affects the mean l i g h t i n t e n s i t y I and i n turn the average photosynthetic rate "P. The s i l t load i s pictured as decreasing away from the r i v e r mouth according to an e l l i p t i c a l d i s t r i b u t i o n i l l u s t r a t e d i n F i g . 20 . Thus i f s (x,y) i s the s i l t load, i t takes constant values on the e l l i p s e s with s(r) a decreasing function of r. This d i s t r i b u t i o n i s not meant to r e f l e c t any observed conditions but- merely gives a plausible pattern i n the area of the r i v e r mouth- Direct measurements of the extinction c o e f f i c i e n t were X Z + 4y2 = F i g . 20. E l l i p t i c a l d i s t r i b u t i o n of contours of r = constant (from equation (7.16)). 59 taken in the fraser River plume (Chapter 6 ) and suggest a di s t r i b u t i o n of jx according to M = M 0 0 - _ £ _ ) o ± r < r 0 o o — as shown in t i g . la, wixn r e = z. o x tu» m- values or jx9 nave been taken i n the range 0.3 m_1 < jx^ < 0.8 m_l based on the measurements. vi) It has already been seen in Fig. 5 that there i s a semi-annular maximum i n the zooplankton d i s t r i b u t i o n o ff the mouth of the Fraser River. Data collected during the cruises c show sim i l a r maxima (Fig. 21 ). This kind of d i s t r i b u t i o n has been represented by the Gaussian form z = z . + z centered about r, csr 8 x 10 3 m, with cz = 5,0 x 10~ 8 m - 2 and with r as given in (7.16). The zooplankton concentrations Z, and Z m vary seasonally from minimum values of 15 and 35 mg/m3 i n mid-winter to 450 and 1050 mg/m3 i n May and June. v i i ) The integrated zooplankton feeding term in (5.12) reduces, for -J/ = 1, to 3 «*» = - M G Z h (7-/9) For the purpose of th i s f i r s t model, t h i s has been s i m p l i f i e d further by approximating the M dependence by a pair of straight l i n e s (see Fi g . 22 ), so that the zooplankton feeding term over 61 F i g . 22. The grazing r e l a t i o n of model I, based on eqn. (7.20). 62 the upper layer i s written as C°S4» = -Ma'Zn = - / S a'Zk (7 - *° ) The constant a« = 1.35 x 1Q-»; 15a» = G = 2 x 10~« mg of chlorophyll per mg of zooplankton per sec, the value introduced for the maximum feeding rate i n Chapter 5. The average source strength 1/h ^ Q dz then has the form M ( ( P - a ' Z ) - W.I-MA] M (? - wsU)A) - / s a ' Z M > is ^ P i s defined as i n (5.3), with the average l i g h t intensity obtained from (5.9) and the extinction c o e f f i c i e n t ytt given by (7.17). For the lower range of M, the whole right hand side of (7.15) i s proportional to M. In i t s time dependent form (i. e . along a pathline), that eguation then reads DM = M Ft U) Dt where c k u 3 i s a function of time only along a pathline through the dependence of the coordinates x and y on the time elapsed while moving along a pathline- Thus F, (t) i s the l o c a l exponential 63 growth rate and M w i l l decay or increase l o c a l l y according to whether F, (t) i s negative or positive. The influence of each one of the factors at work i s c l e a r l y i d e n t i f i a b l e i n F; (t) and can be estimated at every point of the f i e l d . For higher concentrations, H > 15 mg/m3, (7.15) may be written DM = MFzii) - / s V Z with The chlorophyll concentration i s then subject to an exponential growth rate F9 (t) and a l i n e a r decay at a rata 15a'Z. 64 CHAPTER 8. MODEL I : RESULTS Streamline patterns r e s u l t i n g from the assumed downstream velocity (2.3) and the si m p l i f i e d entrainment law (7.4) are shown in Fig . 23 and Fig. 24 for two depths of the upper layer, h = 2 m and 5 m respectively. I t i s obvious that the rate of spreading of thi s type of plume i s strongly dependent on the value of h. The or i g i n of t h i s dependence i s readily found. On the axis of the plume (y = 0) we have, from (2.3) and (7.9) (8-') The second term, due to entrainment •, i s a constant and i t s e f f e c t on the spreading of streamlines away from the axis does not decrease downstream. With x 0 = 5 x 10 3 m and m = 2.4 x 10~ 4, the divergence term due to entrainment exceeds the f i r s t term for x > 3.3 km when h = 2 m, but only for x > 15 km when h = 5 m. The premature appearance of an appreciable transverse velocity f or h = 2 m pushes water pa r t i c l e s off the top of the Gaussian downstream velocity p r o f i l e , U rapidl y decreases and the streamlines begin to diverge very early (Fig. 23). For larger values of h, t h i s divergence i s retarded- The variation of M along a streamline i s determined by the sign of the right-hand-side of (7.22) (or (7.24) for M > 15 mg/m3). Looking at the growth rate as written i n (7.23) we note that ws (-h) , h and ̂ / (-h) do not change along a streamline. The other parameters: U, Z and P* vary along streamlines according to functional forms given above. The 0.05 0 5 10 x ( k m ) 15 2 0 F i g . 23. Streamline pattern of the h o r i z o n t a l v e l o c i t y ; model I with h = 2 m. f i e l d of H has been computed following the scheme outlined in Chapter 4, for a range of values of a l l these parameters. These ranges correspond to various conditions, such as to be expected in d i f f e r e n t months of the year and under maximum and minimum growth rates, sinking (ws (-h) ) or d i l u t i o n (Vt-h)) rates and zooplankton grazing. The influences of the parameters on the phytoplankton d i s t r i b u t i o n have been isolated and w i l l be presented below. In Table II , we l i s t the values of those parameters which are not varied i n the examples discussed below; while the varied parameters w i l l be given for each example. a) Variation i n upper layer depth. The influence of the upper layer depth on the flow f i e l d has already been noted above. Changes i n h also a f f e c t the photosynthetic rate P through t h e i r influence on the average l i g h t i n t e n s i t y I, as given by (5-9); they also influence the sinking and d i l u t i o n terms (the l a s t two terms) i n (7.23). Fig. 25 shows the variation of M along the a x i a l (y = 0) streamline in summer conditions ( P^ = 2.2 x 10 _ s ) and with a low sinking rate for hc - 2 m and h e = 5 m, in the absence of any chlorophyll d i l u t i o n due to entrainment (-)/(-h) = 1).. DM/Dt > 0 everywhere, but i s larger, due to increased average l i g h t i n t e n s i t y , for the thinner layer. b) Chlorophyll d i l u t i o n by the entrained flow. The parameter <f(-h) can take values from 0 to 1, depending on the chlorophyll concentration just below the upper layer. When -V(-h) = 1, there i s no d i l u t i o n of the upper layer TABLE II- Model I parameters held constant- Parameter (m) b (min/ly) Value 96 2-4 x 10-* 5.0 x 10-a 2.5 x 10* 5.0 x 103 5-56 1.35 x 10-9 70 chlorophyll content M; at the extreme end of the range, •j/(-h) =0, one finds a maximum degree of d i l u t i o n . That such d i l u t i o n i s s u f f i c i e n t to reverse the growth trend of M i s seen from Fig. 26 where M i s plotted on the downstream axis f o r May conditions for 1/(-h) = 0 and l/(-h) = 1. c) Sinking rates. The obvious e f f e c t of an increased sinking rate, given otherwise i d e n t i c a l conditions i s shown i n Fig. 27 along the a x i a l streamline. Under May conditions, no d i l u t i o n by entrainment and a 5 m upper layer depth, a f i v e - f o l d increase i n sinking rate i s s u f f i c i e n t to transform a net growth to a net decay of chlorophyll concentration. d) Seasonal v a r i a t i o n . The variation of phytoplankton concentration M along the a x i a l streamline i s shown in Fig. 28 under three sets of conditions, t y p i c a l of the months of January, March and May respectively. The values of the parameters which change from curve to curve are shown i n Table I I I . The main fa c t o r s which d i f f e r e n t i a t e the three sit u a t i o n s are seen from Table III to be: 1) The mean upper layer depth, which i s greater i n l a t e spring, due to increased runoff (Fig. 3). An increased depth would tend to decrease the rate of growth of M, as seen i n F i g . 25; the influence of the upper layer depth variation i s obviously more than overcompensated by other factors! 2) The zooplankton biomass increases from January to May, corresponding to an increasing chlorophyll 27. V a r i a t i o n of M along y = sinking rate. 0; model I, May conditions showing the e f f e c t of an increased 74 TABLE III, Seasonal variation of model I parameters. 1 Parameter ] January j. + _ 1-0 2. 0 0 o (m/sec) | I (m) (m) | 8.0 x 10 3 i Z, (mg/m3) J 15 1 I 35 (mg/m3) (S8C-1) | 1.1 X 10-5 J I e (ly/min) | 0.6 x 10-2 I I e (ly/min) 1 3.0 x 10"2 0.3 March 1.0 2.0 8-0 x TO3 150 350 1-3 x l O " 5 0,7 x 10-2 4.0 x 10-2 0.4 ~ i — i - j . - May 2-0 5.0 1.5 x 10* 450 1050 2.2 x 10-5 1.0 x 10-2 1.0 x 10-i 0.8 withdrawal term. Again, t h i s f a c t o r cannot be of fundamental importance to the r e l a t i v e shape of the th r e e curves, s i n c e the tre n d from winter t o l a t e s p r i n g i s i n a d i r e c t i o n o p p o s i t e to t h a t which would r e s u l t from the v a r i a t i o n s of zooplankton alone. 3) The net p r o d u c t i v i t y i n c r e a s e s markedly from January to Hay, through i n c r e a s e s i n P m, a s s o c i a t e d with the heating of the s u r f a c e waters, and i n I f t , the i n p u t of s o l a r r a d i a t i o n . I t i s t h i s i n c r e a s e i n p r o d u c t i v i t y which determines the seasonal change i n c h a r a c t e r of the curves of F i g . 28. . e) Zooplankton g r a z i n g . As observed above, i n c r e a s e s i n the zooplankton s i n k term i n (7.23) or (7.25) are overcompensated on a seasonal b a s i s by i n c r e a s e s i n p r o d u c t i v i t y . In order t o estimate the i n f l u e n c e of zooplankton g r a z i n g by i t s e l f , the May curve of F i g . 28 i s compared to the a x i a l d i s t r i b u t i o n o f M under the . same c o n d i t i o n s but i n the absence of any zooplankton (Z, = Z m = 0) ( F i g . 29 ) . This f i g u r e has been p l o t t e d on the same s c a l e as many of the other f i g u r e s to show the r a t h e r n e g l i g i b l e i n f l u e n c e that zooplankton g r a z i n g has i n t h i s model on c h l o r o p h y l l c o n c e n t r a t i o n during high p r o d u c t i v i t y c o n d i t i o n s . f) Strength of the mean flow. In order t o i s o l a t e the i n f l u e n c e of the magnitude of the flow v e l o c i t y , the a x i a l c h l o r o p h y l l c o n c e n t r a t i o n was c a l c u l a t e d f o r two d i f f e r e n t r i v e r outflow v e l o c i t i e s (0*o = 1 m/sec and 2 m/sec) f o r May c o n d i t i o n s , as shown i n F i g . 30 . At any g i v e n d i s t a n c e from the mouth, the value of M v =0 U 0 " l m / sec 10 x(km) 20 30 40 Fig. 30. Variation of M along y •and maximum dilution. = 0; model I, May conditions with the effect of increased velocity •4 78 i s increased for a decreased flow f i e l d . Looking back at the e f f e c t of the velocity in (7.23), i t i s clear that i n the absence of d i l u t i o n (1/ - 1), the flow f i e l d plays a purely advective role and that i f the net source-sink term i s positive the rate of growth at any point i s unchanged by decreasing the flow v e l o c i t y . The value of M should then increase since i t takes longer to reach any given point when 0o i s reduced. In_ the case of maximum d i l u t i o n , (V= 0), a decrease i n 0o also decreases the sink term with the effect shown i n F i g . 30 (compare with F i g . 26}. The chlorophyll concentration decreases i n i t i a l l y because of the higher d i l u t i o n rate, but recovers after f a l l i n g to a minimum value. g) Lateral d i s t r i b u t i o n of chlorophyll. Looking back once more at the source terms (7.23) or (7.24), one notices that the variables U, Z and ~P which vary along any one streamline because of their s p a t i a l dependence w i l l also change i n passing from a streamline to another. The variation of M along the a x i a l streamline may thus not be representative of what happens over the rest of the (x,y) plane. Although M was calculated along a number of streamlines i n each case above for which only i t s variation along the axis y = 0 has been displayed, only two types of l a t e r a l d i s t r i b u t i o n emerged from the integrations. In a l l cases but one, the monotonicity exhibited by the M variation along the axis was mimicked on the other streamlines. The M contours shown on Fig. 31 correspond to the high productivity May conditions holding for the h e = 2 m curve of x (km) F i g . 31. Horizontal d i s t r i b u t i o n of M for model I, May conditions. 80 Fig. 25 and to the streamline pattern of F i g . 23. In these circumstances, the chlorophyll concentration increases uniformly along.each streamline and, in the (x,y) plane, thus increases i n a l l d i r e c t i o n s away from the mouth of the r i v e r . The chlorophyll d i s t r i b u t i o n has the form of an elongated r i s i n g trough oriented along the axis of the flow. The corresponding d i s t r i b u t i o n for those cases where a uniform decrease i n fl i s found (the V = 0 curve of f i g . 26; the larger sinking rate curve of Fig. 27; the March and January curves of Fig. 28) i s not i l l u s t r a t e d . The s p a t i a l d i s t r i b u t i o n i s very similar to that shown for uniformly increasing M, except that there i s now a descending ridge. The only case where a non-monotonic behavior was found along any streamlines was for the f u l l d i l u t i o n (•!/ = 0) May conditions curve shown i n Fig . 26 and F i g . 30. For the high flow rate (0o = 2 m/sec) a uniform decrease i n M i s found there only along the axis; on the other streamlines (Fig. 32 ) an i n i t i a l diminution of chlorophyll concentration i s always followed by an eventual recovery and an increase i n M. In order to see whether the mimimum in M on the non-axial streamlines i s associated with zooplankton grazing, the e l l i p s e on which the zooplankton density i s a maximum, according to (7.18), has been traced as a thi n dotted l i n e on F i g . 32. I f the zooplankton were responsible for the chlorophyll depletion, one would expect the minima of M, as indicated by crosses on the various streamlines, to f a l l on or near the e l l i p s e - This i s c l e a r l y not the case. I t seems most l i k e l y that the diminution of M along the streamline segments lying near the axis i s associated I 0 I 10 I 15 i 20 x (km) I 25 i 30 F i g . 32. Horizontal d i s t r i b u t i o n of M for model I, May conditions with U G = 2 m/sec and v = 0, 82 w i t h t h e d i l u t i n g e f f e c t o f t h e e n t r a i n m e n t o f c h l o r o p h y l l - f r e e water f r o m below. The d i l u t i o n i s most p r o n o u n c e d n e a r t h e a x i s s i n c e i t i s p r o p o r t i o n a l t o 0, and f a l l s r a p i d l y o f f t h e a x i s a c c o r d i n g t o t h e G a u s s i a n f o r m c h o s e n f o r U i n ( 2 . 3 ) . The p o s i t i o n o f t h e minima o f M a l o n g c u r v e s which n e a r l y p a r a l l e l t h e a x i s s t r o n g l y s u p p o r t s t h i s i n t e r p r e t a t i o n . D i s c u s s i o n The s i m p l e model j u s t e x p l o r e d h as shown t h e r e l a t i v e e f f e c t s o f many o f t h e p a r a m e t e r s a f f e c t i n g c h l o r o p h y l l c o n c e n t r a t i o n . I t a p p e a r s i n p a r t i c u l a r t h a t t h e s e a s o n a l v a r i a t i o n i s p r i m a r i l y d e t e r m i n e d by c h a n g e s i n p r o d u c t i v i t y t h r o u g h i n c r e a s e d i n s o l a t i o n and warming o f t h e upper l a y e r s . T h i s f a c t i s o f c o u r s e w e l l known and i t i s c e r t a i n l y n o t w o r t h c o n s t r u c t i n g a n u m e r i c a l model t o c o n f i r m i t . More s u r p r i s i n g i s t h e v e r y weak i n f l u e n c e o f z o o p l a n k t o n g r a z i n g on t h e c h l o r o p h y l l d e n s i t y ; t h e M c u r v e s a r e a l m o s t u n i q u e l y d e t e r m i n e d by p r o d u c t i v i t y f a c t o r s and dynamic f a c t o r s s u c h as d i l u t i o n e n t r a i n m e n t . . F u r t h e r m o r e , i n none o f t h e above r e s u l t s i s t h e r e any i n d i c a t i o n o f t h e f o r m a t i o n o f a downstream maximum i n M, as a p p e a r s 'in F i g . 5 , a f e a t u r e which we s e t f o r t h t o e x p l a i n i n c o n s t r u c t i n g t h e model. I n view o f t h i s t h e model has been r e f i n e d , a s p r e s e n t e d below, m a i n l y t o y i e l d b e t t e r e s t i m a t e s o f dynamic e f f e c t s . 0 83 CHAPTER 9- MODEL II : .REFINEMENTS I n o r d e r t o b r i n g t h e p r e m i s e s o f t h e model i n t o c l o s e r a g r e e m e n t w i t h t h e o b s e r v a t i o n s t a k e n i n t h e G u l f 1 t o G u l f 3 c r u i s e s , a number o f a p p r o x i m a t i o n s and s i m p l i f i c a t i o n s used above have been abandoned. What were deemed more a p p r o p r i a t e f o r m s f o r t h e e n t r a i n m e n t f u n c t i o n , t h e d e p t h o f t h e u p p e r l a y e r , and t h e v e r t i c a l p r o f i l e s o f v e l o c i t y and c h l o r o p h y l l d e n s i t y , have been u s e d and a r e p r e s e n t e d below. i ) I n s t e a d o f t h e s i m p l i f i e d form ( 7 . 4 ) , t h e e n t r a i n m e n t v e l o c i t y w(-h) was e x p r e s s e d i n t e r m s o f t h e c o m p l e t e e x p r e s s i o n ( 2 . 4 ) , w i t h U c as g i v e n i n ( 2 . 5 ) . R e p e a t i n g t h e s e e x p r e s s i o n s f o r c o n v e n i e n c e , where m = 2.4 x 10~* a s b e f o r e ; g = 9.8 m /sec 2 and Vz - 1 0 - 6 m 2 / s e c . K e u l e g a n (1966) g i v e s two v a l u e s f o r c»: one (c* = 7.3) f o r a r r e s t e d s a l t wedges, t h e o t h e r ( c ' = 5.6) f o r s t a g n a n t s a l t p o o l s . The l a t t e r v a l u e was c h o s e n h e r e as more a p p r o p r i a t e t o t h e plume. The d e n s i t y c o n t r a s t AO^ between t h e l o w e r and t h e upper l a y e r d i m i n i s h e s downstream, and t h i s v a r i a t i o n has been t a k e n i n t o a c c o u n t . The v a r i a t i o n o f ^ (c£ = (̂ - 1) x 10 3) a t t h e s u r f a c e as a f u n c t i o n o f d i s t a n c e from t h e r i v e r mouth i s p l o t t e d i n F i g . 33 f r o m d a t a t a k e n i n G u l f 1. The f i t t e d c u r v e 22 H 21- 20 H 19- °t 18" 1 7 H 16 H 15 V / A / / / V A o / o o ® Q—• -A- —I —A- ' G 0 © Observed A Fitted Curve 1 r 1 ~ I 0 10 , 20 30 40 x (km) 33. The v a r i a t i o n of the surface «^ as a function of distance from the r i v e r mouth. 85 with k = 0.935 x 10 - 4 i s also shown i n Fig. 33. This curve was chosen for i t s simplicity;' the o v e r a l l f i t of (9.3) to the data points i s tolerable, although (9.3) i s well above the observational values for 12 < x < 25 km. In the lower layer a constant density of ,̂ = 1.0235 was used. Now that w(-h) includes V, (2.10) becomes non-linear i n V and i s no longer simply integrated to y i e l d (7.12) for V i n terms of 0. The velocity f i e l d was now computed using the following procedure. Given U(x,y) in (2.3) and Uc in (9.2), the continuity equation (2.10) was integrated to find V, with the help of (9.1), through the following i t e r a t i v e process. 1- for a given value of x and st a r t i n g on y = 0 (where V = 0} , w(x,0,-h) was evaluated from (9.1). 2- at a point off the axis, y = £, i t was assumed that w(x,£,-h) = w(x,0,-h) which allows the ca l c u l a t i o n of V(x,£) from (2.10). 3- using the computed V, an updated value of w(x,£,-h) was calculated from (9-1). 4- at y = 2&, w(x,2&,-h) was found by extrapolation from the values of v e r t i c a l v e l o c i t i e s at y = 0 and y = &- V(x,2£) i s then calculated from (2.10). 5- an updated w(x,2&,-h) i s estimated from (9.1) using V(x,2c0- 6- at y = 3&, w(x,3&,-h) i s obtained by extrapolation and the process continues. The velocity f i e l d was mapped i n this fashion for various values of £. A value of & = 10 m was found, by comparison with f i n e r grid computations, to give s u f f i c i e n t accuracy. 86 In routine integration of the b i o l o g i c a l - p h y s i c a l model, an even simpler method of integration was used. at any point (x,y), w(x,y,-h) was estimated from (9.1) with U = U(x,y) and V = 0. V(x,y) was then calculated from (2.10) for that value of w(-h). The results of this simpler method agreed with the i t e r a t i o n process outlined above within IS for |v| < U. In the b i o l o g i c a l c a l c u l a t i o n s (3.8), w(-h) was updated with the value of V substituted back into (9.1). i i ) The depth of the upper layer, i d e n t i f i e d with the depth of the bottom of the halocline, frequently increases rapidly around x = 25 km. From the s a l i n i t y p r o f i l e s f o r Gulf 1 shown i n Fig. 16, the thickness of the upper layer (normalized with respect to h 6 = 15 m) have been plotted i n F i g . 34 . The rapid deepening of the upper layer has been modelled with the curve with r as given by (7.17). The o r i g i n of the hyperbolic tangent was always chosen at r d = 25 km and the steepness factor B = 3.5. For r >> r 0 , h 0 ( f ( + D while for r << r, K K (f, " From which f , In most runs, h^ was kept constant at 2 m and only h + was 88 varied. The rapid change of depth embodied in (9.4) should be expected to have some important consequences on the flow f i e l d and on the chlorophyll concentration. I f h increases rapidly i n of entrainment and d i l u t i o n . An increased mixed layer depth also leads, from (5.9), to a decreased mean l i g h t i n t e n s i t y and thus to decreased productivity. i i i ) In an attempt to include more r e a l i s t i c v e r t i c a l p r o f i l e s of u and v, current meter data from Tabata et a l . (1970) were examined. These are shown i n Fi g . 35 together with a f i t t e d curve of the form The value of A was adjusted to provide the best v i s u a l f i t to the current p r o f i l e s . Curves of X for various values of A are shown i n Fig. 36 A = 1 gave the best f i t and i s the curve shown i n Fig- 35. The requirement (2.7) that the i n t e g r a l of tf(z) equal the depth of the upper layer imposes the r e l a t i o n Thus, f o r A = 1, K= 1-434. Examples of v e r t i c a l chlorophyll variation -j/(x,y,z) i n the region of in t e r e s t were drawn from Fulton et a l . (1968) and are shown i n Fig. 37 . Once more a curve of the form (9.9) with A = 1 provides a good f i t . Using these forms for X and V , the function xi.(x,y) as (2.3), U w i l l decrease accordingly, thereby decreasing the rate R 89 Tf(z,h) 0 .5 1.0 15 z/h F i g . 35. V e r t i c a l p r o f i l e s of current speed; the curve represents eqn. (9.9) with A = 1, (after Tabata et a l . , 1970). 90 0 0.5 * ( z ' h ] |.0 1.5 z/h F i g . 36. Comparison of the e f f e c t of d i f f e r e n t values of A on eqn. (9.9). -Mz.h) 4 ^ 5^ F i g . 37. V e r t i c a l p r o f i l e s of c h l o r o p h y l l , the curve represents eqn. (9.9) with A = 1, (after Fulton et a l . , 1968). 92 defined i n (3-7) becomes -a- = J_ [ (/+ f*nJ>(j.+ /)) eii - /.oi4n foil) and the conservation equation (3-8) takes the form K K ( 3 iv) Since we now have an analytic expression for 4/(x,y,z) , the i n t e g r a l i n the grazing term of equation (5-12) can be evaluated. Using (9.9) we obtain C 3 = K ( fl + U cosh (A + A^) - U cosh Plt j («5./3) where R^ = (trctctnr) Hence equation (5.12) becomes \ V ftU2X + aM)( - t» cosn fyj - W i(-K)^(-n)^ (?./5") Equations (9,12) and (9.15) may then be used to solve for values of M alonq pathlines. The concentration of M w i l l increase or decrease depending on wether the integrated source term (the Q term) i s large enough to overcome the entrainment d i l u t i o n term- 93 CHAPTER 10. MODEL I I : RESULTS Direct comparison of the streamlines (pathlines) calculated in model II with those of model I i s d i f f i c u l t . The problem ari s e s from the fact that calculated v e l o c i t i e s are dependent on h(x,y); i n model I, h i s constant while i n model II , h increases away from the r i v e r mouth. Fig . 38 and F i g . 39 show the streamlines for two di f f e r e n t i n i t i a l v e l o c i t i e s , U 0 = 1 m/sec and U6 = 2 m/sec. On the axis of the plume (y = 0) we can write, using (2.11), (2.3) and (2.4) and r e c a l l i n g that Uc < U, V(x,0) = 0 From (9-9) we know that X(-h) - 1/>̂ > 0. Thus we have divergent flow since the right-hand-side of (10.1) i s always >-0. Also we see that the rate of spreading depends not only on the layer depth, h, but also on the gradient of h. Hence we would expect the rate of spreading to increase when U0 (and thus U) i s increased, as i s demonstrated by F i g . 38 and Fig. 39. Similarly a larger value of h would decrease the rate of spreading. The variation of M along a streamline depends upon the sign of the right-hand-side of (9.12). I t can be seen that the entrainment d i l u t i o n term i s always a l o s s term. Since most of the parameters i n (9.12) and (9.15) vary along a streamline, i t i s not easy to determine th e i r net e f f e c t on M. As with model I, the f i e l d of M was calculated for d i f f e r e n t values of the various model II parameters. For model I I , the parameters held constant are given i n Table IV . We w i l l now discuss the influences of the varied parameters on the chlorophyll 96 TABLE IV. Model II parameters held constant- i 3 -1 Parameter Value d l (mg/m3) 5.0 B 3.5 c' 5.6 * t (m2/sec) 1.002 x 10~ 6 k (in-1) 9.35 x 10-s 96 m 2.4 x 10-* cz (ffl-2) 5.0 x 10-a ro (m) 2. 5 x 10* x e (m) 5.0 x 10 3 b (min/ly) 5.56 a 1 1.35 x 10-9 97 d i s t r i b u t i o n . a) Seasonal variation The variation of M along the a x i a l streamline i s i l l u s t r a t e d i n F i g . 40 for conditions representative of the months of January, March and May respectively. For each curve, the values of parameters which varied are given i n Table V. In a l l three cases a sinking speed of v$ = 1.2 x 10 - 5 m/sec was used. Refering to Table V , i t can be seen that the basic differences i n the three cases are: 1) increased r i v e r discharge in late spring which increases the vel o c i t y , Ud and increases the upper layer depth near the mouth (due to increased s t a b i l i t y the layer deepens l e s s rapidly downstream). 2) The increase of the maximum production rate, P̂ ,, and the incident solar radiation, I 0 , towards summer. The resultant increase i n productivity i s counterbalanced by an increase i n the compensation l i g h t i n t e n s i t y , I e , and the extinction c o e f i c i e n t , J J L 0 . 3) The increased zooplankton grazing towards summer. Of the above e f f e c t s , the increase of P_ and I„ when coupled with a more gradual increase i n the layer depth tends to increase the chlorophyll concentration while the increased values of U0 , I e , and the layer depth near the mouth tend to increase the chlorophyll sink term. The curves shown i n Fig- 40 r e f l e c t the balance attained by the source and sink terms in the chlorophyll equation. The results indicate that except f o r May, a l l the curves show a steady decrease of chlorophyll away from the r i v e r mouth- In May there i s an i n i t i a l decrease with a TABLE V- Seasonal variation of model II paramaters. i ] 3 1 T j Parameter | January | March j May I I j. ^ (. ^ j h c (m) I 15.0 | 15-0 J 1.0 | i I i I I J f, I 1.13 | 1.13 J 5.00 j i i l l I | U 0 (m/sec) I 1.0 | 1-0 j 2.0 J I I I I I | P m (sec-*) | 1. 1 x 10-s j 1.3 x lo-s j 2.2 x 10~s j I I I I I I I c (ly/min) j 0.6 x 10~* I 0.7 x 10-* I 1-0 x 1Q-* J I i I i I j I 0 (ly/min) | 3.0 x 10~* j 4.0 x 10-2 j 1.0 x 1Q-» 1 1 I I i i i u0 (m-M I 0.3 | 0.4 J 0.8 J I I I i I I Z, (mg/m3) j 15 I 150 I 450 j I i I I I J Z m (mg/m3) j 35 I 350 j 1050 I I I I I I I r, (m) J 8.0 x 10 3 j 8.0 x 1Q3 j 1.5 x 10* J 1 : J J I J 100 minimum at about 25 km, then there i s a gradual increase. The discussion which follows shows the effect of varying some of the parameters i n d i v i d u a l l y . The reference curve i n the discussion below i s that obtained by choosing parameter values to maximize the source terms and minimize the sink terms. This produces a curve where M increases with increasing distance from the r i v e r mouth, i . e . s i m i l a r to the comparison curve of model I. The effect of changes i n the parameter values i s then demonstrated by changing one of the parameters i n the reference curve and comparing the res u l t i n g curve with the reference curve. The parameter values fo r the reference curve are those of Table IV and h0 = 5 m, f, = 2.00, Ue = 1 m/sec, P m = 3.1 x 10~5 sec--1, JJL0 - 0.3 m-i, I 0 = 1.0 x 1 0 - i ly/min, I c = 0.6 x 10-* ly/min, Z, = 15 mg/m3, Z^ = 35 mg/m3 and w$ = 1.2 x 10 - 5 m/sec. b) Changes i n upper layer depth In Fig. 41 the effect of changes i n the depth of the upper layer are compared. With a l l other factors being kept constant, the chlorophyll d i s t r i b u t i o n s for three upper layer depth p r o f i l e s are compared: (&) h 0 = 5 m, f, - 2.00 (reference curve) which gives 5m < h < 15m; (B) h 0 - 15 m, f, = 1.13 which gives 2m < h < 30m; and (C) h Q = 5 m, f, - 1.40 which gives 2m < h < 12m. It i s clear from equations (9.12) and (9.15) that variations in the depth of the upper layer are i n s i g n i f i c a n t i n the l o c a l production and grazing terms. The main eff e c t of variations in h occurs in the hydrodynamic d i l u t i o n terms 3 H 2H O) E A ho=5m , ^=2.00; 5<h<15 B ho^lSm, f f l . 1 3 ; 2<h<30 C h0=5m, fpl .40; 2<h<12 I 5 10 x I km) 15 20 25 F i g . 41. V a r i a t i o n of M along y = 0 (model I I ) ; the e f f e c t of changes i n the upper l a y e r depth 102 (proportional to 1/h) in (9.12). Comparing curves A and C for example, i t i s clear that for x < 15 km, where h a* constant, the chlorophyll growth rate of curve A should be more rapid than that of curve C since 1/hfl < 1/hc . On the other hand, once the steep gradient of the upper layer depth i s reached (x as 15 - 25 km) , curve C catches up and passes curve A because (with f, = 2.00 f o r A, as compared to 1.40 for curve C) the gradient sink-term O-yh i s larger i n A than in C. The r e l a t i v e behavior of curves B and C i s s i m i l a r at small x since the o r i g i n a l upper layer depths are equal; curve B, with a smaller d i l u t i o n by divergence term (f, = 1.13 for B compared to f, = 1.40 for curve C) , outdistances C in the region of ,the upper layer depth gradient. c) Variations i n the velocity f i e l d In r i g . 42 the re s u l t s of changing the strength of the vel o c i t y f i e l d are i l l u s t r a t e d . The curves compared have values of parameters Ud = 1 m/sec and x 0 = 5 x 103 m (the reference (upper) curve) , U0 = 1 m/sec and x e = 1 x 10 4 m (middle curve) and U„ = 2 m/sec and x e = 5 x 10 3 m (lower curve) . The lower curve i l l u s t r a t e s the effect of increasing the downstream velo c i t y at the r i v e r mouth; such as happens when the r i v e r discharge increases,. The s i t u a t i o n for a less rapid decrease i n U downstream i s i l l u s t r a t e d by the middle curve. The less rapid increase of a with distance can be explained by the fact that; 1) the d i l u t i o n by entrained water from below i s increased, 2) with the increased velocity a phytoplankton organism spends l e s s time in t r a n s i t and for s i m i l a r l o c a l i n : i i r 0 5 10 15 20 25 x(kmi) . 42. V a r i a t i o n of M along y = 0 (model I I ) ; the e f f e c t of changes i n the v e l o c i t y f i e l d . 104 growth rates, would not attain equally high concentrations at a given distance downstream. d) Variations in the production term The production term has been varied in two ways; by changing the value of the maximum production rate, P m and by changing the value of the extinction c o e f f i c i e n t , j^g. The result i n g curves are i l l u s t r a t e d i n F i g . 43 , The reference curve (top) has values of P m = 3. 1 x 10~s s e c - 1 , jAa- 0.3 m_1 while the middle curve has P m = 3.1 x 10 - 5 s e c - 1 , JA0 - 0.8 m_1 and the bottom curve has P m = 1.1 x 10 _ s s e c - 1 , j i 0 - 0.3 m-1. Although both and ju.0 were changed by about the same amount (just l e s s than a factor of 3), the d i s t r i b u t i o n of M appeared less sensitive to changes in jx0 than to changes in P m. Increasing JJ.0 decreased M as did decreasing P m, as one would expect. e) Variations in the grazing term Fig. 44 i l l u s t r a t e s the effe c t of increasing the grazing rate by increasing the zooplankton biomass by a factor of 30. The top curve i s the reference curve (Z, = 15 mg/m3; Z m = 35 mg/m3) while the bottom curve (Z, = 450 mg/m3; Z m - 1050 mg/m3) has the increased grazing term. Although there i s a large increase in the grazer population, the chlorophyll concentration i s not decreased very much. when the i n i t i a l concentration of M, Me = M(1,y) (ie. at x = 1 km) i s increased to 3 mg/m3 from 1 mg/m3 then the curve of M/M0 l i e s between those for the above two cases. Thus i t 1 0 x(km) 15 20 25 F i g . 43. V a r i a t i o n s of M along y v a r i a t i o n s i n P m and JJL0 = 0 (model I I ) ; the e f f e c t of changes i n the production rate by 107 appears that the grazing term i s not one of the more important terras. f) variations i n the sinking rate The phytoplankton sinking rate was increased from w$ = 1 m/day of the reference curve to w5 = 5 m/day. These curves are shown in F i g . 45 with the top curve being the reference curve. The increased sinking rate results i n a much reduced chlorophyll concentration. g) Lateral d i s t r i b u t i o n of chlorophyll To i l l u s t r a t e the l a t e r a l d i s t r i b u t i o n of chlorophyll we have chosen the case i l l u s t r a t e d i n Fig. 46 (U,, - 1 m/sec, x 0 = 10 km). The parameters are the same as the middle curve of Fig. 42 which shows the d i s t r i b u t i o n of M along y = 0 (the axis of the velocity f i e l d ) . In contrast to model I (Figures 31 and 32) two completely d i s t i n c t d i s t r i b u t i o n s are not found for model II. The most common pattern f o r model II (Fig. 46) resembles F i g . 31 of model I. Provided M shows either a monotonic increase or decrease, the l i n e s of constant M are convex towards positive x, i e . the l o c i i of points (x,y) of M = constant are located such that as x increases the magnitude of y decreases. The few cases that d i f f e r from Fig. 46 are those where there i s f i r s t a decrease and then an increase i n M with distance from the r i v e r mouth. Near the r i v e r mouth (where a i s decreasing) the contours of constant M are closed, while i n the region where M i s increasing the contours of M = constant resemble those of Fig. 46. If one 46. Horizontal d i s t r i b u t i o n of M for model I I ; s o l i d l i n e s are streamlines, dashed l i n e s are contours of M = constant, U e = 1 m/sec, x e = 10 km. 110 looks just at the region where M i s decreasing, then the M contours look s i m i l a r to those of F i g . 32 of model I. Discussion Model I I , which has been discussed above, has produced e s s e n t i a l l y the same results as model I, even though greater realism was introduced into model II, Probably the single most important difference between the two models i s the variation of the upper layer thickness with x and y in model II, since i t affects both the v e l o c i t y f i e l d and the production term. Using the same parameter values in both models led to lower values of M in the second model when looking at seasonal differences. Again i t became apparent that the available l i g h t , the magnitude of P m and the advection by the velocity f i e l d were the most important parameters while zooplankton grazing, had r e l a t i v e l y l i t t l e influence on M. In none of the model runs was i t possible to produce a downstream maximum such as we set out to study (Fig, 5). The reduced values of M in the second model (as compared to model I) can be explained in part by the increase in the layer depth which decreases the average l i g h t i ntensity, thus reducing the s i z e of the . production term. Another factor i s the formulation used for the depth integrated production term. It w i l l be r e c a l l e d that one of the assumptions used in the model i s that the phytoplankton population i s v e r t i c a l l y mixed over time periods that are short r e l a t i v e to the growing time, so that l i g h t of varying i n t e n s i t i e s i s experienced at dif f e r e n t depths. Thus we used a depth-averaged l i g h t i n t e n s i t y i n 111 equation (5.3). To check the effect of t h i s assumption we compared pj-vdz/h to ^Pj/ dz/h for various values of extinction c o e f f i c i e n t , ^ , and various values of layer thickness, h. Some of the res u l t i n g curves are shown i n Fig. 4 7 . It can be seen that only when the layer thickness or the extinction c o e f f i c i e n t become s u f f i c i e n t l y large, so that the average l i g h t i n t e n s i t y decreases enough, does the assumption lead to an under-estimate of the production term. The curves diverge noticeably for I ( z ) / I 0 < 0.027. Thus the lower values of M in modal II can be attributed, at least partly, to the layer depth variation and the assumption that the plankton experience a depth-averaged l i g h t i n t e n s i t y . 113 CHAPTER 11. CONCLUSIONS The two models discussed above have given an ind i c a t i o n of the r e l a t i v e importance of the various parameters that determine the chlorophyll d i s t r i b u t i o n . The two most important terms in the chlorophyll conservation equation appear to be the production term and the advection term, with the sinking term being of somewhat lesser importance and the grazing term the least important. The production term i s affected by the i n s o l a t i o n , the t u r b i d i t y of the water, the depth of the upper layer and the maximum production rate (through water temperature). The increase in the incident radiation, the decrease i n the upper layer thickness (through increased s t a b i l i t y due to greater fresh water input) and the increase in the maximum production rate a l l tend to increase production as winter changes to spring and summer. On the other hand the increased t u r b i d i t y tends to decrease the available l i g h t i n the water column, decreasing the production term. The advection term also varies with the season; r i v e r discharge increasing from winter to summer. The increased discharge tends to increase the velocity components, (u,v,w), giving r i s e to a greater flushing rate (shorter residence time) and increased mixing and entrainment. However, the increased mixing i s inhi b i t e d somewhat by the greater s t a b i l i t y of the water column as runoff increases. I t appears that the natural s t a b i l i t y of the phytoplankton population in the S t r a i t of Georgia may be attributed to the fa c t that although i n s o l a t i o n , the upper layer thickness and the 114 production rate serve to increase the chlorophyll concentration as sinter changes to summer, the increased t u r b i d i t y and advection work i n the opposite d i r e c t i o n , l i m i t i n g the size of the blooms. Only when an imbalance occurs i s there a large increase in the population. One mechanism for this imbalance (or perhaps a r e s u l t of i t ) may be patchiness. The results of these studies point to further work that could be done to improve the realism of the model. It i s f e l t that the single most important step i s to develop a better model of the velocity f i e l d for river estuaries such as the Fraser River. It has been shown that advection i s very important i n determining the chlorophyll d i s t r i b u t i o n , hence to attempt further modelling without a better velocity f i e l d model would not prove very useful. Recent measurements of flow i n the Fraser River plume have shown how the r i v e r discharge i s pulsed by t i d a l modulation. Also the downstream velocity does not appear to decay as 1/x (as the analogy with jets suggests) but rather more slowly (S. Pond, pers. com-). Further work on t h i s problem i s presently underway at t h i s I n s t i t u t e . A second deficiency of the present models i s the fact that time dependent changes are not included in the formulation. This i s not very important for long time scales (eg- seasonal variations) since the time required for the phytoplankton population to achieve 'equilibrium i s much shorter than that required for the long period variations to be f e l t . However, when such things as the diurnal variation of the i n s o l a t i o n , the diurnal v e r t i c a l migration of zooplankton and the t i d a l l y induced variations i n the velocity f i e l d are considered i n 115 conjunction with the non-linearity of some of the terms in the chlorophyll conservation eguation, the l i m i t a t i o n s of the present models can be appreciated, p a r t i c u l a r l y since the grazing i s about 180° ( i . e . 1/2 day) out of phase with the photosynthetic production. Spatial inhomogeneity must also be considered. Me have shown i n Chapter 10 that, in general, averaging the e f f e c t of the v e r t i c a l structure of the chlorophyll d i s t r i b u t i o n and the available l i g h t did not introduce large errors.. However, the combined effect of the v e r t i c a l chlorophyll d i s t r i b u t i o n and the v e r t i c a l migration of the zooplankton population must be investigated in conjunction with a time-dependent formulation. Last but not l e a s t i s the problem of choosing values for the b i o l o g i c a l parameters. Most of the b i o l o g i c a l parameters can take on a large range of values. Part of that i s due to natural variations between species, geographical areas and in time. Another i s that laboratory measurements may give d i f f e r e n t r e s u l t s than f i e l d studies. The problem i s not a simple one to resolve. However, i t indicates that r e a l i s t i c models must have input from f i e l d studies in the p a r t i c u l a r area of interest i n order to choose the correct parameter values. In our study the problems of shelf-shading and nutrient l i m i t a t i o n were not considered; they would become more important at the higher chlorophyll concentrations. In summary, although i t was not possible to produce the downstream maximum i n the d i s t r i b u t i o n of chlorophyll that we set out to explain, i t was shown that the l i g h t a v a i l a b l e in the water column, the value of P m and the velocity f i e l d are 116 important in determining the chlorophyll d i s t r i b u t i o n . The ef f e c t of changes i n these parameters must be considered when evaluating the results of natural or man-made changes to the system, such as damming the Fraser River, constructing a nuclear power plant or discharging possible pollutants. REFERENCES Abraham, G. , 1960. Jet d i f f u s i o n i n l i q u i d of greater density. J_. Hyd. Div. , Proc. ASCE 86, HY6, 1-13. Antia, N.J., CD. M c A l l i s t e r , T.R. Parsons, K. Stephens and J.D.H. Strickland 1963. Further measurements of primary production using a large- volume p l a s t i c sphere. Limnol. Oceanogr. 8_, 166-183. Caperon, J . , 1967. Population growth i n micro-organisms l i m i t e d by food supply. Ecology 48(5), 715-722. Fulton, J.D., O.D. Kennedy, K. Stephens and J . Skelding, MS 1968. Data record; p h y s i c a l , chemical and b i o l o g i c a l data, S t r a i t of Georgia, .1967. F i s h . Res. Bd. Canada, Man. Rep. No. 968. Gargett, A.E., 1976. Generation of i n t e r n a l waves i n the S t r a i t of Georgia, B r i t i s h Columbia. Deep-Sea Research 23(1), 17-32. Giovando, L.F. and S. Tabata, MS 1970. Measurements of surface flow i n the S t r a i t of Georgia by means of f r e e - f l o a t i n g current followers. F i s h . Res. Bd. Canada, Tech. Rep. No. 163, 69 pp. Hodgins, D.O., 1974. S a l i n i t y i n t r u s i o n i n the Fraser River, B r i t i s h Columbia. Ph.D. d i s s e r t a t i o n , Univ. of B r i t i s h Columbia. I v l ev, V.S., 1961. Experimental ecology of the feeding of f i s h e s . Translated by D. Scott. Yale Univ. Press, New Haven. 302 pp. Keulegan, G.H., 1966. The mechanism of an arrested s a l i n e wedge, pp. 546-574 i n Estuary and Coastline Hydrodynamics, A.T. Ippen, ed. McGraw-Hill, N.Y.• 744 pp. Parsons, T.R., MS 1965. A general d e s c r i p t i o n of some factors governing primary production i n the S t r a i t of Georgia, Hecate S t r a i t and Queen Charlotte Sound and the P a c i f i c Ocean. F i s h . Res. Bd. Canada, Man. Rep. No.;193. i Parsons, T.R. and B.R. de Lange Boom, 1972. The control of ecosystem processes i n the sea. pp. 29-58 i n The Biology of the Oceanic P a c i f i c , C.B. M i l l e r , ed. Oregon State Univ. Press, C o r v a l l i s . 155 pp. Parsons, T.R., R.J. LeBrasseur and W.E. Barraclough, 1970. Levels of production i n the pelagic environment of the S t r a i t of Georgia, B r i t i s h Columbia: a review. J . F i s h . Res. Bd. Canada 27, 1251- 1264. Parsons, T.R., R.J. LeBrasseur, J.D. Fulton and O.D. Kennedy, 1969. Production studies i n the S t r a i t of Georgia. Part I I . Secondary production under the Fraser River plume, February to May, 1967. J_. exp. mar. B i o l . E c o l . _3, 39-50. Parsons, T.R., K. Stephens and R.J. LeBrasseur, 1969. Production studies i n the S t r a i t of Georgia. Part I. Primary production under the Fraser River plume, February to May, 1967. J_. exp ./.mar. B i o l . Ecol. 3, 27-38. Parsons, T.R. and M. Takahashi, 1973. B i o l o g i c a l Oceanographic Proces- ses . Pergamon, Oxford. 186 pp. P i a t t , T., 1972. Local phytoplankton abundance and turbulence. Deep- Sea Res. 19, 183-188. R i l e y , G.A., H. Stommel and O.A. Bumpus, 1949. Quantitative ecology of the western North A t l a n t i c . B u l l . Bingham Oceanbgr. C o l l . 12, 1-169. Smayda, T.J. , 1970. The suspension and sinking of phytoplankton i n the sea. Oceanogr. Mar. B i o l . Ann. Rev. 8_, 353-414. Steele, J.H., 1962. Environmental co n t r o l of photosynthesis i n the sea. Limnol. Oceanbgr. ]_, 137-150. Stefan, H. and F.R. Schiebe, 1970. Heated discharge from flume into tank. J_. Sanit. Eng. Div., Proc. ASCE 96, SA6, 1415-1433. Stephens, K., J.D. Fulton and O.D. Kennedy, MS 1969. Summary of b i o - l o g i c a l oceanographic observations i n the S t r a i t of Georgia, 1965-1968. F i s h . Res. Bd. Canada, Tech. Rep. No. 110. Str i c k l a n d , J.D.H. , 1958. Solar r a d i a t i o n penetrating the ocean. A--- review of requirements, data and methods of measurements, with p a r t i c u l a r reference to photo synthetic p r o d u c t i v i t y . J_. F i s h . Res. Bd. Canada 15, 453-493. Tabata, S., MS 1972. The movement of Fraser River-influenced surface water i n the S t r a i t of Georgia as deduced from a seri e s of a e r i a l photographs. Marine Sciences Branch, Pac. Mar. Sc_. Rep. No, 72-6, 69 pp. Tabata, S., L.F. Giovando, J.A. Stickland 1 and J . Wong, MS 1970. Current v e l o c i t y measurements i n the S t r a i t of Georgia - 1967. F i s h . Res. Bd. Canada, Tech. Rep. No. 169, 245 pp. Takahashi, M., K. F u j i i and T.R. Parsons, 1973. Simulation study of phytoplankton photosynthesis and growth i n the Fraser River estuary. Mar. B i o l . 19, 102-116. Thomas, D.J., 1975. The d i s t r i b u t i o n of zinc and copper i n Georgia S t r a i t , B r i t i s h Columbia: E f f e c t s of the Fraser River and sediment- exchange reactions. M.Sc. d i s s e r t a t i o n , Univ. of B r i t i s h Columbia. 119 T u l l y , J.P. and A.J. Dodlmead, 1957. Properties of the water i n the S t r a i t of Georgia, B r i t i s h Columbia and i n f l u e n c i n g f a c t o r s . J_. F i s h . Res. Bd. Canada 14, 241-319. Waldichuck, M., 1957. Ph y s i c a l oceanography of the S t r a i t of Georgia, B r i t i s h Columbia. J . F i s h . Res. Bd. Canada 14, 321-486. Wiegel, R.L., 1970. Ocean dynamics, pp. 123-228 i n Hydronautics, H. E. Sheets and V.T. Boatwright J r . , ed. Academic Press, N.Y. 454 pp. 120 APPENDIX: TEMPERATURE AND SALINITY DATA Abbreviations used: CRN — cruise number, G for Gulf HR -- time observation started (GMT) STN station number DY — date, (day/month/year) Note: The f i r s t 5 stations of cruise 2 (p r o f i l e s to 20 m) correspond to the single number stations i n Fig- 14. The stations preceded by a 2 in F i g . 14 correspond to the 14 cruise 2 stations with p r o f i l e s to 50 m- 1 -J I CRN: G- 1 HR: 1814 | STN: 01 DY: 02/11/71 .. + + Depth (m) Temp. ( 0 C ) | Sal. i (%•) 0 8.7 | 20.4 1 9.2 | 25.6 2 9.3 j 27.0 3 9.0 | 26.4 5 9.2 I 26.6 7 9.3 ! 28.4 10 9.2 I 28.7 15 9.4 j 29.1 20 9.2 j 29.4 30 9.4 | 30.1 50 9.4 i 30.2 .1 1 1 I I CRN: G- 1 Hfi: 1856 I | STN: 02 DY: 02/11/71 r -i 1 Depth (m) I Te mp. I (°C) | Sal. 1 (%o) 0 .j | 8.8 1 24.3 1 I 8.9 I 25-5 2 | 8.8 i 26.0 5 | 9.0 | 26.3 10 | 10.0 | 28.5 15 J 9.8 J 28.8 20 | 9.7 | 28.6 30 | 9.7 I 30.1 50 | 9.7 I 30.3 75 J 9,7 1 30.8 1 1 1 1 122 CRN: G- 1 HR: 1922 STN: 03 DY: 0 2/11/71 Depth (m) 0 1 2 5 7 10 15 20 30 50 75 Temp. <°C) 8.8 8. 9 8. 8 9.0 9. 2 9. 5 9,5 9. 6 9.6 9.8 9.7 Sal. 24. 6 27.0 26.9 27.5 27.9 28.7 29.0 29.0 29.5 30.5 30.9 CRN: G- 1 HR: 1948 STN: 04 DY; 02/11/71 Depth (m) Temp. | (°C) i Sal. <%. ) 0 8.7 ] 24.3 1 8.6 | 24.2 2 8.8 | 25.7 5 9.3 i 27.3 7 ! 9-2 i 27.3 10 1 9-3 | 27.5 15 9.5 i 27.9 20 1 9.6 ] 28.4 30 1 9.6 j 29.3 50 1 9.5 | 30.2 75 1 9.8 j 30.9 CRN: G- 1 HR: 2014 STN: 05 DY: 02/11/71 Depth | Temp, (ffl) I (°C) 0 1 2 5 7 10 15 20 30 50 I 8.7 J 8.6 | 8.5 | 9.0 | 9. 1 j 9.2 j 9.4 I 9.4 I I i ] 9.4 9.1 24.2 24.4 24.7 27.4 27.6 28.4 29.3 29.5 29.8 30.2 CRN: G- 1 HR: 2047 STN: 06 DY: 02/11/71 Depth (m) Temp. (°C) 0 1 2 5 7 10 15 20 30 50 75 9.0 8.8 8.8 8.9 9.2 9.2 9.8 9.6 9.6 9.4 9.7 Sal- 25.7 25.8 26.0 26. 1 27.5 27.9 28.9 29.4 29.6 30.3 30-7 r - •• | CRM: G- 1 HR: 2 1 1 6 » 4 - 1 | CRN; G- 1 —3 HR: 2 1 4 1 J j STN: 07 DI: 0 2 / 1 1 / 7 1 i 1 STN; 0 8 DY: 0 2 / 1 1 / 7 1 J j Depth | i (m) | Temp. (OC) I Sal. I (%«) i i i Depth ! I (m) i i i Temp. (°C) j Sal. | I {%• ) I i i o i 9 . 0 | 2 7 . 0 J i I o 1 8 . 9 J 2 7 . 3 J i 1 ! 9 . 0 | 2 7 - 6 ! 1 1 8 . 8 I 2 7 . 4 | i 2 ! 8 . 9 J 2 7 . 3 i j 2 8 . 8 I 2 7 . 6 } J 5 ! 8 - 9 | 2 7 . 6 ! i 5 8 . 8 I 2 7 . 7 J I 7 ! 8 . 8 | 2 8 - 0 i J 7 8 . 8 I 2 7 . 7 1 | 10 I 9 . 0 | 2 8 . 2 1 J 10 8 . 9 j 2 7 . 7 | I 15 ! 9 . 2 I 2 8 . 5 ! j 15 9 - 2 1 2 8 - 5 | | 20 I 9 . 7 | 2 8 . 8 i | 20 9 - 2 i 2 8 - 6 J i 30 I 9 . 4 j 2 9 - 9 ! J 30 9 . 6 I 29.4 ) I 50 ! 9 . 2 | 3 0 . 2 i | 50 9 . 4 i 3 0 , 3 | I 75 ] 9 . 8 | 3 0 . 8 j I 75 9 . 8 I 3 0 . 8 J 125 CRN: G- 1 HR: 2212 STN: 09 DY: 02/11/71 + Depth (m) Temp, j (OC) | Sal. 0 9.0 | 28. 1 1 9.0 | 28. 1 2 9.0 | 28.2 5 9.0 | 28.2 7 8.9 | 28.2 10 9.1 | 28.4 15 9.2 j 28.6 20 9- 1 | 28.9 30 9.4 I 29.4 50 9.3 | 30-4 75 9.6 | 31.0 CRN; G— 1 HR; 2236 STN: 10 DY; 02/11/71 Depth (m) 0 1 2 5 7 10 15 20 30 50 75 +- -h Temp« (°C) 9.0 8.9 8.8 8.8 8.8 8.8 9.0 9.0 9.3 9.3 9.5 Sal, (%o) 28. 1 28.2 28.3 28.4 28,4 28.6 28.9 28.9 29.5 30.5 31.0 CRN: G- 1 HR: 0041 STN: 1 DY: 03/11/71 Depth (m) Temp (°C) 0 8. 1 1 8,0 2 8.0 5 8.4 7 8.7 10 8.8 Sal. (%o) 23.3 23. 2 23.5 25.7 28-4 28-6 CRN: G- 2 HR; 2120 STN: 01 DY: 09/02/72 Depth (m) 0 1 2 3 5 7 10 15 20 Temp. (°C) 4 5.2 ! 5.5 | 5.6 J 5.6 | 5.8 | 6.0 | Sal. (X.) 5.0 | 27.2 i 5.0 i 27.3 I 28.9 29.9 30.5 30.5 30.8 30.9 6.2 J 30.9 i CRN: G- 2 HR: 2220 STN: 02 DY: 09/02/72 Depth (a) Temp. (°C) Sa 1. 0 1 2 3 5 7 10 15 20 5.4 5.4 5.5- 5.5 5. 5 5.5 5.6 5.8 6.0 30. 1 30. 1 30.2 30-3 30.3 30.3 30.4 30.6 30.9 -A CRN: G- 2 HS: 2322 STN; 03 DY: 09/02/72 Temp. | Sal. (°C) | (%•) CRN: G- 2 HR: 0025 STN: 04 DY: 10/02/72 127 | CRN: G- 2 HE: 0220 i | CRN: G- 2 HR: 1205 | | ST H : 05 DY; 10/02/72 i I STN: 01 DY: 10/0 2/72 | j Depth | I (m) | Temp. (OC) i i L . Sal. J (*•) 1 j Depth I (m) Temp. ( ° C ) j Sal. J {%•) I I o .... T 3. 1 r I i 12.9 | I o 4.3 1 24.0 | I 1 ! 4.0 J I i 22.0 | I 1 5. 1 I 27.4 | I 2 I 4.1 1 i i 22.6 | I 2 5.4 I 30.3 | I 3 I 4.3 I 1 I 25.9 | I 3 5.6 I 30.6 | I 5 ! 5.1 1 1 i 30.2 | I 5 5.8 ! 30.7 | I 7 ! 5.8 1 1 1 30.5 I I 7 6. 3 31-0 | | 10 I 5.6 1 J i 30.5 | | 10 6.4 I 31-0 | | 14 I 5.7 1 1 i 30.6 j I 15 5,9 ! 31.4 | j 19 6.1 1 1 i 31.0 | • | 20 5.7 ! 31.4 | | 29 5.7 i 31.4 j J 38 I 5.7 ! 31.4 j J 48 5.5 j 31.4 J 128 CRN: G- 2 HE: 1250~1 I I j STN: 0 2 DY; 10/02/72 i | f r ^ Depth (in) Temp. | (°C) I Sal. (*•) 0 3.9 j 19.8 1 3.9 | 20. 1 2 4. 5 | 24. 1 3 4.6 | 25. 5 5 5.2 | 29.7 7 5.4 j 30. 1 10 5.9 j 31. 1 15 6. 1 j 31. 2 20 6.2 | 31. 2 30 6. 4 | 31.3 40 6. 4 | 31. 3 50 | 5.8 | 31.3 L 1 I J 1 | CBN; G- 2 HR; 1333 | J STN: 03 DY: 10/02/72 J i 1 j Depth 1 (m) 1 Temp- 1 (°C) I Sal.. | (%o) I 1 o | 3.3 1 15.8 , | 1 j 4.0 J 20.6 J I 2 1 4.2 ! 23.3 1 I 3 I 5.0 ! 28.5 | I 5 j 5.2 1 30.1 J I 7 i 5.3 ! 30.5 J j 10 J 5.6 ! 31.0 J | 15 | 5.8 ! 31.2 | J 20 i 5,8 ! 31.3 | I 30 I 6.1 ! 31-3 J j 40 | 6.1 ! 31.4 I 1 50 i | 6.1 j 31.4 | _ . i 129 CRN; G- 2 HR: 1409 STN: 04 DI: 10/02/72 i — • • I CRN: G- 2 — 9 Hfi: 1440 j j STN: 05 JL DY; 10/02/72 | 1 1 j Depth | 1 (m) 1 Temp. (°C) 1 S a l . | 1 (-oo ) 1 •-+- . 1 25.8 J 1 o 1 4.8 I 1 ! 5.0 | 28.1 j J 2 I 5.0 | 28.1 j I 3 ! .5.3 J 30.3 | i 5 I 5.3 I 30.3 J j 7 ! 5.5 I 30.5 J I 10 ! 5.6 i 30.5 I i 15 ! 5.8 i 30.7 j 1 20 1 5.9 i 30.7 | 1 30 ! 6.1 I 30.9 | | 40 I 6.3 J 31.0 i k 50 L _ J 6.6 I 31.2 j . .!„,_ , _. J CfiN: G- 2 HE: 1511 STN: 06 DY: 10/02/72 Depth } Temp. J (m) (°C) Sal. 0 1 2 3 5 7 10 15 20 30 40 50 5.1 I 5.3 | 5.0 | 5. 1 | 5. 3 j 5.4 j 5.4 J 5.5 ! 5.8 I 6.3 | 6.7 J 6.9 I 29.2 29. 2 29.0 29.0 29.9 30.3 30.4 30.5 30. 9 31.0 31.3 31.3 CfiN: G- 2 HH: 1553 STN: 07 DY; 10/02/72 CRN: G- 2 HR: 1622 STN: 08 DY: 10/02/72 Depth (m) 0 1 2 3 5 7 10 15 19 29 39 48 Temp, (OC) 4.8 5.0 5.0 5. 1 5. 1 5.2 5.4 5.5 5.6 5.9 6.3 6. 8 S a l . (%.) 27.2 29.5 29. 1 29. 5 29.9 30. 2 30.3 30.4 30.5 30.9 31. 0 31.3 CRN: G- 2 HR; 1648 STN: 09 DY: 10/02/72 Depth (m) Temp. (°C) S a l . (%• ) 4.7 28.9 5.0 29.2 5.0 29. 1 5.1 29.3 5.2 30. 1 5.2 30. 1 5.2 30.1 5.2 30.2 5.4 30.5 6.1 30.9 6.5 31.2 6.7 31.3 0 1 2 3. 5 7 10 15 20 30 40 50 CRN: G- 2 HR: 1720 STN: 10 DY: 10/02/72 Depth (m) 0 1 2 3 5 7 10 15 20 30 40 50 Temp, ( 0 C ) 3.3 4.8 5. 1 5. 1 5. 2 5.3 5. 5 5.6 5.8 6.0 6.0 6.0 S a l . (%.) 17.8 26.7 28.7 28-8 29.5 29.9 30. 2 30.5 31.0 31.4 31,4 31.4 -i 1 " | CRN: G- 2 HH: 1747 | 1 STN: 11 1 +- J Depth j j (m) | DY; 10/02/72 l Temp- (°C) J S a l . | 1 (%o) 1 1 0 1 4.2 J 23.5 j 1 1 I 4.8 1 26.8 | i 2 | 4.8 J 27.3 J 1 3 | 5.0 J 28.1 I ! 5 | 5.5 | 30.0 { 1 7 1 5.5 I 30.2 J 1 10 J 5.5 J 30.5 J 1 15 i 5.5 j 30.7 \ I 20 | 6.1 j 31.3 | I 30 1 6.6 j 31.4 | I 40 i 6.5 i 31.6 | 1 50 I i i 6-2 J 31.6 | , 4, - , - J CRN: G- 2 HR: 1816 STN: 12 DI: 10/02/72 Depth (m) Temp. <°C) Sal. 0 1 2 3 5 7 10 15 20 30 40 50 3.5 5.0 5.5 5.4 5. 8 6. 1 6. 2 15.6 27.4 30.2 30.3 31.0 31. 0 31.4 6.2 | 31.5 6 .3 I 31-5 5.9 | 31 .5 5.7 | 31.6 5.7 | 31.5 | CRN: G- 2 HS: 1846 J | STN: 13 DI: 10/02/72 j J Depth Temp. I Sal. | I (m) <°C) i (%o) I I o | 4.5 I 25-5 | j 1 4.7 I 25.2 | I 2 5. 1 J 30.1 | J 3 5.2 i 30.0 j I 5 5-4 i 30.2 | I 7 5.4 i 30.2 I I 10 I 5.4 i 30.2 | 1 15 5.8 i 30.6 | | 20 I 6.0 i 30 .7 1 | 30 6.0 i 30.9 1 } 40 6-0 1 31.1 1 j 50 6.3 1 31.1 | j CRN: G- 2 HR: 1921 I | STN: 14 DY: 10/02/72 j. j r Depth j (m) Temp. | ( 0 C ) j Sal. i%o) 0 j 4.7 | 26.6 1 5.2 | 27.6 2 5.4 | 30.0 3 5.5 | 30.1 5 5.5 | 30.2 7 5.5 j 30.2 10 5.5 I 30.2 15 5.5 | 30.2 20 5.5 I 30.4 30 5.9 | 30.6 40 6.4 | 30.9 50 6.8 | 31,1 135 1 I CRN: G- 3 HR: 1910 | t • CRN: G- 3 HR: 1958 | I SIN: 01 DT: 2 0/03/72 | i STN: 02 DY: 20/03/72 | | Depth I (m) 1 1 1 +- J l Temp. ( ° C ) L • Sal. | (%*) ! I L 1 Depth 1 On) 1 Temp. ( ° C ) * j Sal. I (%•) I r I o 7.2 1 23.0 | r 0 r 6.7 1 25,6 | 1 1 1 i i 7.0 1 23.0 | 1 1 l 6,7 ! 25.8 | I 2 1 1 1 1 f 6.8 I 23.6 | 1 2 I 6.6 ! 26. 1 | I 3 6.8 1 24.1 j ! 3 I 6.5 ! 27.0 | i 5 1 I i 6.7 1 26.8 | ! 5 i 6.5 ! 27.4 | ! 7 I 1 l 6.7 1 28.1 I I 7 I 6.5 I 27.4 | J 10 1 1 1 6.6 1 28,6 ] ! 10 ! 6.4 ! 27.9 | I 15 1 1 6.4 1 29.9 j ! 15 ! 6.2 I 30.0 | | 20 « „.. , „ 1 1 i. 6.3 30.4 | i | 20 j 6.5 _[_ 30.2 | CRN: G- 3 HH; 2025 STN: 03 DY: 20/03/7 2 Depth | Temp. (m) j (°C) -r- Sal. 0 1 2 3 5 7 10 15 20 I 6.9 6. 8 6.7 6. 6 6. 5 6.4 6. 3 6. 4 6.5 24.8 25.0 27. 4 27.8 27.7 28.6 28. 8 29.9 30.3 j CRN: G- 3 It HR; 2055 1 J STN: 04 DY: 20/03/72 { | Depth i (m) 1 Temp, 1 (°C) j Sal. | i (%o) 1 1 o I 7.5 1 18.5 I I 1 i 7.0 1 22.1 J I 2 1 7.1 i 22.7 j | 3 | 6.8 I 23.4 | | 5 I 6,6 j 26.0 I 1 7 I 6.8 I 28.4 I | 10 | 6.7 J 29. 1 i ] 15 | 6.3 | 29.9 J I 20 | 6.3 A J 30.4 | . . . i . I 136 J ~\ i 1 | CRN: S- 3 HR: 2115 I { CRN: G- 3 HS: -2149 J I STN: 05 DY: 20/03/72 | I STN: 06 DY: 20/03/72 | I b ; f -J I J- {- i | Depth J Temp- J Sal. | | Depth I Temp- | Sal- | I (m) | (°C) | (So) i | (m) | (OC) j (%.) I I + 1 : i I -I r -4 I 0 | 7.1 | 15.7 ] | 0 j 6.9 | 17.8 | I 1 | 6.7 j 20.8 J | 1 | 6.9 | 19.9 | I 2 j 6.5 | 21.6 J I 2 j 7.1 J 20.0 | | 3 | 6.6 1 22.1 | I 3 | 7.1 j 21.4 | I 5 | 6.6 | 26.8 | | 5 | 6.7 J 26.3 | I 7 | 6.8 | 28.8 j I 7 J 6.7 J 29.7 | I 10 | 6.6 J 29.9 | | 10 | 6.8 | 29.7 | i 15 | 6.4 J 30.4 | 1 15 | 6.7 j 30.5 | I 20 J 6.5 | 30.5 I | 20 | 6.7 J 30.9 | i 1 1 i I I i J J CRN: G- 3 HR: 2215 I J STN: 07 DY: 20/03/72 J j Depth | Temp. | Sal- J J (m) I <°C) I i%o) J 1 o | 7.0 J 16.4 J i 1 | 7.2 | 18-6 | | 2 I 7. 1 J 20.0 | I 3 I 7. 1 1 20.3 J I 5 | 6.6 j 29.0 I I 7 | 6- 6 | 29-2 | | 10 | 6.6 j 30-0 j I 15 | 6-8 | 30. 3 j | 20 I 6.7 \ 30.4 I i \ CRN: G- 3 HR: 2247 | STN: 0 8 DY: 20/03/72 1 j j__ | Depth | 1 (a) 1 Temp. (°C) 1 , .. .. i 1 Sal. j J {*.) 1 I 0 i 5 . 5 i 6.8 J I 1 I 5,4 I 6.9 | I 2 i 5-6 i 8.5 1 I 3 i 5.7 1 14.6 | I ^ j 6. 1 j 23.7 I I 6 | 6.5 | 27.5 J i 8 J 6.6 j 30.7 ! I 12 | 6.5 j 30.9 i I 15 1 t i_ 6.4 | 30.7 | i . J CRN: G- 4 HR: 1840 STN; 01 DY; 17/04/72 + h Depth | Temp. | S a l . (m) I (°C) 1 i (%o) 0 | 6 . 9 i 1 i 2 7 - 3 1 | I 6 . 9 i i i i 2 7 . 6 2 I 1 6 . 9 1 \ i 1 2 7 . 8 3 1 6 . 9 i 1 • 2 7 . 5 5 i j 6 . 8 1 1 i 1 2 7 . 6 7 | 6 . 8 i 1 i 2 7 . 8 10 i | 6 . 8 i l i i 2 8 . 2 15 1 j 6 . 7 i 1 1 i 2 9 .1 20 i 1 6 . 6 l i 2 9 . 6 CRN: G- 4 HR: 1915 STN: 02 DY: 17/04/72 Depth Temp. <°C) S a l . (%o) 0 1 2 3 5 7 10 15 20 6 .9 6 .9 6 .9 6 .9 6 . 9 6 .9 6 .9 6 .6 6.6 2 7 . 7 2 7 . 6 2 7 . 6 2 7 . 6 2 7 . 7 2 7 . 8 2 8 . 2 2 9 . 5 3 0 . 0 CRN: G- 4 HR: 2000 STN: 03 DY: 1 7/04/72 CRN: G- 4 HR: 2025 STN: 04 DY: 17/04/72 -+-Depth (m) 0 1 2 3 5 7 10 15 20 Temp, 7.1 7 . 0 6 .9 6 .9 6 . 7 6 .9 6 .8 7 .0 6.6 S a l . 2 8 . 2 2 8 . 3 2 8 . 2 2 8 . 1 2 8 . 1 2 8 . 1 2 8 . 3 2 8 . 5 3 0 . 3 CRN: G- 4 HR: 2059 STN: 05 DY: 17/04/72 Depth | Temp. j S a l . (n) I (°C) | (%») 0 1 2 3 5 7 10 15 20 6.8 6.7 i 27. 1 27. 1 27.0 7. 3 } 7.2 J 7.2 | 7.2 } 27.1 6.9 J 27.4 6.8 I 27.9 6.7 I 27.9 28.1 29.3 CRN: G- 4 Hfi: 2124 STN: 06 DY: 17/04/72 Depth (m) Temp. {°C) S a l . 0 1 2 3 5 7 10 15 20 7.5 7.4 7.3 7.0 6.9 6.9 6.8 6.6 6.7 26.9 27.0 27.3 27.6 27.8 27.8 28.3 30.0 30.4 1 | CRN: G- 4 HE: 2150 | j STN: 07 DY: 17/04/72 | | Depth 1 (m) 1 Temp. 1 (°C) i 1 L . S a l . | {%•> ! 1 o | 7.6 I 1 i 25.0 | | 1 | 7. 5 I 1 l 25.8 J I 2 ] 7 .5 1 i i 25.2 J i 3 | 7.5 1 1 I 25.5 | I 5 | 7. 2 1 1 1 26.0 J I 7 1 6.9 1 I I I i 27. 1 | | 10 | 6.7 29.3 | i 15 1 6.6 1 i 30.0 | | 20 i. _ _ j 6.6 i 1 1 1 30.1 J , , | CRN: G- 4 Hfi: 2241 J | STN: 08 J _ DY: 17/0 4/72 | i . i | Depth 1 1 (m) 1 Temp. (°C) r S a l . | (35.) i 1 o I 5.8 I 2.0 J 1 1 ! 5.8 ! 6.7 J I 2 ! 6.3 ! 17.3 j I 3 ! 6.5 I 21.3 1 i 5 I 6.7 ! 23.9 j I 7 ! 6.8 i 28.9 J j 10 ! 6.7 ! 29.3 J | 15 ! 6.7 ! 29.7 1 I 20 I 6.7 ! 30.1 J i J I 1 139 j CRN: G- 5 HR: 1815 j STN: 01 DY: 11/05/72 , + + Depth (a) Temp. (OC) | Sal. I (%o ) 0 9.5 „j I 10.6 1 9.6 | 13.8 2 10.5 | 24.7 3 10.6 j 26.0 5 1 0.3 \ 27.0 7 8.8 | 27.8 10 7.7 j 28-9 15 7.5 I 29.5 20 7.4 | 29.6 CRN: G- 5 HR; 1847 STN: 02 DY: 1 Depth (IQ) 0 1 2 3 5 4 10 15 20 Temp. (°C) 9.6 10. 1 10. 1 10.3 10.0 9.5 8.5 7.5 7-2 4- /05/72 Sal. (».) 11. 1 18.6 18-7 22.3 27.5 27.8 28.4 29.3 29.9 CRN: G- 5 HR: 1925 STN: 03 DY: 11/05/7 2 Depth (m) 0 1 2 3 5 7 10 15 20 Temp, (°C) 10.7 10. 8 10.6 10.4 10. 1 9. 9 8. 8 7. 9 7.3 Sal. i%o) 7.5 10.4 18.2 22. 1 27.0 27. 1 27.7 28.7 29. 3 r I CRN: G- 5 • •• — i HR: 2000 I | STN: 04 DY: 11/0 5/72 | j Depth | I (m) I i ., , , i Temp. ( O Q I Sal. | 1 <%o) | ? i 1 o 1 10.7 i 7.0 | ! 1 I 10.6 1 12.3 J i 2 I 10.8 i 16.4 J I 3 | 10.5 I 21.9 | I 5 I 10.3 1 26.9 i I 7 | 10.2 i 27.3 | I 10 I 9.0 j 27.8 I | 15 i 7-5 i 29.1 | I 20 | t J 7.1 1 29.9 | 140 I T J ? | CRN: G- 5 HE: 2020 J j CRN: G- 5 HE: 2042 j } STN: 05 DY: 1 1/05/72 | I STN: 06 DY: 1 1/05/72 | , |_ L j }. + j- j | Depth j Temp- 1 S a l . 1 I Depth | Temp- J S a l - J I (m) | (°C) | {%*) 1 I (m) | (°C) i (%o) | , 1- j. 4 . -j j- j j 0 I 11 .1 | 14 .4 | I 0 | 11 ,0 | 1 2 . 9 | I 1 J 1 1 . 1 I 15 .2 | | 1 | 11 ,0 | 1 3 . 0 | | 2 I 11.1 | 15 .9 | 1 2 | 11.1 | 1 2 . 7 | | 3 | 1 1 . 0 i 17 .1 I | 3 I 11 .3 | 1 5 . 0 J 1 5 i 1 0 . 5 J 2 6 - 8 | | 5 | 10 . 9 J 2 4 . 8 J j 7 | 9 . 7 | 2 7 . 3 | I 7 | 10 .4 | 2 7 . 2 | | 10 1 8 .2 J 2 8 . 2 | | 10 J 9.7 | 2 7 . 5 | | 15 j 7 . 7 j 2 8 . 9 | | 15 | 7.6 J 2 9 . 2 | j 20 j 7 .1 | 2 9 . 7 J | 20 | 7 .2 | 2 9 . 7 | i i i j i 1 i t I "1 I 3 3 CRN: G- 5 HR: 2100 J I CRN; G- 5 HE: 2130 j 3 STN: 07 DY: 1 1/05/72 I I STN: 08 DY; 1 1/05/72 | \ 1- 1 i I 1 i- i | Depth 1 Temp, | S a l . 1 | Depth | Temp. I S a l , | I (m) i (°C) j J | (m) J {°C) J (%e) J r + + i I 1 f i | 0 I 1 0 . 4 | 1 3 , 7 I I 0 J 8 .2 | 0 . 0 J | 1 | 1 0 . 7 3 1 6 . 5 j i 1 | 7.8 | 0 .0 J | 2 I 1 0 . 7 J 1 9 . 4 | | 2 I 9.4 1 1 5 . 8 j | 3 j 1 0 . 7 | 2 1 . 5 | 1 3 | 10.1 J 2 1 . 2 1 J 5 1 9 .8 | 2 6 . 7 | 1 5 1 7 .5 J 2 9 . 6 1 1 7 J 8 . 7 | 2 8 . 1 J \ 1 J 7.3 \ 2 9 . 7 | | 10 I 7 . 4 | 2 8 . 7 j | 10 | 7 .3 J 2 9 . 8 | J 15 J 7 . 0 | 2 9 . 7 j | 15 I 7 .2 | 2 9 . 8 3 | 20 \ 6 . 8 | 3 0 . 2 j i 16 1 7 .2 1 2 9 . 9 | i j i i . i J i i
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Mathematical modelling of the chlorophyll distribution in the Fraser River Plume, British Columbia De Lange Boom, Bodo Rudolf 1976
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Title | Mathematical modelling of the chlorophyll distribution in the Fraser River Plume, British Columbia |
Creator |
De Lange Boom, Bodo Rudolf |
Date | 1976 |
Date Created | 2010-02-08 |
Date Issued | 2010-02-08 |
Description | The horizontal chlorophyll a distribution observed in the Strait of Georgia near the mouth of the Fraser River appears to reflect the influence of the river discharge. Mathematical models are developed to attempt to explain the observed distribution in terms of such factors as the velocity field, the available light and the grazing and sinking of the phytoplankton population. A steady state, two dimensional model is developed for the upper layer. The downstream velocity is modelled using a modified form of the downstream velocity in a jet; the vertical entrainment is represented by an empirical expression, while the cross-stream velocity is calculated from the vertically integrated continuity eguation. A vertically integrated conservation equation is written for the chlorophyll concentration by balancing advection against the source-sink term (net production minus grazing and sinking). Temperature effects are not modelled directly and nutrients are not considered as limiting. The first model is simplified by assuming: a constant depth of the upper layer, vertical entrainment proportional to the downstream velocity, and a uniform vertical distribution of chlorophyll. In model II the layer depth varies with distance from the river mouth, a more complex relation for the vertical entrainment is used and more realistic vertical profiles are employed for the horizontal velocity and the chlorophyll concentration. Although the observed downstream maximum in the horizontal chlorophyll distribution is not reproduced, the results indicate that the velocity field, the available light in the water column and the value of the maximum production rate (a function of water temperature) are the most important parameters influencing the distribution. Sinking is of secondary importance while grazing appears to be relatively unimportant. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2010-02-08 |
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. |
DOI | 10.14288/1.0053126 |
Degree |
Master of Science - MSc |
Program |
Oceanography |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/19779 |
Aggregated Source Repository | DSpace |
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