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UBC Theses and Dissertations

A laboratory and mathematical study of the 'thermal bar' 1970

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A LABORATORY AND MATHEMATICAL STUDY OF THE 'THERMAL BAR' by G.H. ELLIOTT B.Sc, University of Toronto, 1965 M.Sc., University of Bri t i s h Columbia, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Physics and Institute of Oceanography We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1970 Iti p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e tha t 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 r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f p h y s i r s Institute of Oceanography, The U n i v e r s i t y o f B r i t i s h Co lumbia V a n c o u v e r 8, Canada ABSTRACT The 'migrating thermal bar' phenomenon, known to occur i n c e r t a i n large, d i m i c t i c , freshwater lakes, has been studied i n laboratory and mathematical models. The temperature f i e l d s observed i n the laboratory agreed with those observed i n Lake Ontario and a l i n e a r p h y s i c a l model for the speed of the 'thermal bar', based on n e g l i g i b l e h o r i z o n t a l ad- vection and d i f f u s i o n of heat, gave reasonable values f o r both the laboratory model and Lake Ontario. Observations were also made of the associated v e l o c i t y f i e l d . On the basis of t h i s laboratory model, which suggests that h o r i z o n t a l advection and d i f f u s i o n of heat were not of primary importance, mathematical models were developed. F i r s t the temperature f i e l d was c a l c u l a t e d from the one-dimensional heat d i f f u s i o n equation. Then the v e l o c i t y f i e l d was c a l c u l a t e d assuming that the flow was driven by buoyancy forces and balanced by viscous forces. On the b a s i s of the s i m i l i t u d e between the temperature f i e l d s found i n my models and those observed i n the lakes, i t seems pos s i b l e that the v e l o c i t y f i e l d of the models also provides a good approximation to the c i r c u l a t i o n associated with the bar i n lakes. There are no d i r e c t measurements of the v e l o c i t i e s associated with the bar i n lakes and they w i l l be d i f f i c u l t to obtain as such v e l o c i t i e s are expected, i n Lake Ontario, to be only of the order of 1 cm sec X . i i i TABLE OF CONTENTS page ABSTRACT i i TABLE OF CONTENTS i i i L I S T OF FIGURES v ACKNOWLEDGEMENTS v i i i 1. INTRODUCTION 1 2. LABORATORY MODEL 7 2.1 A p p a r a t u s 7 2.2 E x p e r i m e n t a l T e c h n i q u e s 7 2.3 D e s c r i p t i o n o f t h e 'Bar' i n t h e Tank 12 2.4 S i m i l i t u d e t o Lake 14 2.5 L i n e a r M o d e l f o r t h e Speed o f t h e B a r 16 2.6 Compa r i s o n o f t h e L i n e a r M o d e l t o Lake O n t a r i o 24 2.7 D i s c u s s i o n 27 2. 8 Summary 29 3. MATHEMATICAL MODELS 30 3.1 The Te m p e r a t u r e F i e l d 30 i ) The deep c o n v e c t i v e s i d e 31 i i ) The s h a l l o w s t a b l e s i d e 34 i i i ) C omparison w i t h the t a n k 35 3.2 The V e l o c i t y F i e l d 39 3.3 V a l i d i t y o f t h e V e l o c i t y M o d e l 52 3.4 E x t e n s i o n t o a Lake 53 4. SUMMARY 57 • BIBLIOGRAPHY 59 i v page APPENDIX A: Cooling Experiment 61 APPENDIX B: Temperature Data: constant slopes' 63 APPENDIX C: I n v e s t i g a t i o n of the E f f e c t s of Bottom I r r e g u l a r i t i e s .... 79 APPENDIX D 85 1. S l i d i n g Door Experiment 85 v 2. Heating Water Warmer than 4°C 87 APPENDIX E: C a l c u l a t i o n of the Stream Function 89 V L I S T OF FIGURES F i g u r e page 1 The d e n s i t y o f f r e s h w a t e r as a f u n c t i o n o f t e m p e r a t u r e (Handbook) 3 2 Temperature s e c t i o n s f o r Lake O n t a r i o f o r two N-S m i d - l a k e s e c t i o n s ( S e c t i o n E, Rodgers 1966a; S e c t i o n D, R o d g e r s , u n p u b l i s h e d , 1965) 4 3 E x p e r i m e n t a l t a n k 8 4 Heat f l u x e s used i n t h e e x p e r i m e n t s p l o t t e d a g a i n s t t o t a l t i m e o f h e a t i n g 11 5 P e r c e n t a g e o f t o t a l s u r f a c e h e a t r e a c h i n g d i f f e r e n t d e p t h s . . 13 6 G e n e r a l i z e d c u r r e n t and t e m p e r a t u r e r e s u l t s f o r t h e h e a t e d s y s t e m . A v e r a g e d v e l o c i t i e s a r e shown by a r r o w s , t h e i r l e n g t h i n d i c a t i n g t h e m o t i o n o b s e r v e d i n a m i n u t e ( i . e . , cm min *) i n t h e same s c a l e s as used f o r t h e axes 15 7 P o s i t i o n o f t h e b a r , measured and p r e d i c t e d , f o r s t a n d a r d h e a t i n g and b o t t o m s l o p e s o f 2.5°, 5°, 7.5°. The SLOPE o f t h e s e c u r v e s i s BAR SPEED 18 8 P o s i t i o n o f t h e b a r , measured and p r e d i c t e d , f o r 5° b o t t o m s l o p e and t h e two h e a t i n g r a t e s ( ^ 9 x 1 0 - 3 and ^ 12 x 1 0 " 3 c a l c m - 2 s e c - 1 ) . The SLOPE o f t h e s e c u r v e s i s BAR SPEED 20 9 P o s i t i o n o f t h e b a r , measured and p r e d i c t e d , f o r 7.5° b o t t o m s l o p e and t h e two h e a t i n g r a t e s ( ^ 9 x 10 3 and ^ 12 x 1 0 " 3 c a l c m - 2 s e c - 1 ) . The SLOPE o f t h e s e c u r v e s i s BAR SPEED 21 10 P o s i t i o n o f t h e b a r , measured and p r e d i c t e d , f o r 5° b o t t o m s l o p e , s t a n d a r d h e a t i n g , and d i f f e r e n t AT, L c o m b i n a t i o n s . The SLOPE o f t h e s e c u r v e s i s BAR SPEED 22 11 T e m p e r a t u r e s e c t i o n , mean t e m p e r a t u r e s ( v e r t i c a l l y a v e r a g e d ) , and changes i n h e a t c o n t e n t f o r Run A: 5° b o t t o m s l o p e and s t a n d a r d h e a t i n g . The i n i t i a l t e m p e r a t u r e ( T i ) when the h e a t lamps were t u r n e d on was 0.2°C 23 12 Changes i n h e a t c o n t e n t f o r Lake O n t a r i o ( f o r s e c t i o n s shown i n F i g u r e 2, p.4 ) . A l s o a s i m i l a r p l o t f o r t h e t a n k f o r Run A 25 v l Figure page 13 Temperature anomaly sections f or the tank experiment ( f o r Run A) and f o r Lake Ontario ( f o r sections i n Figure 2, p. 4) 26 14 Temperature s e c t i o n (°C) c a l c u l a t e d from equations 3.1.13... 37 15 9 i n C° against distance along the tank f o r f i x e d values of z ( i n cm) (a) generalized measured values 38 (b) c a l c u l a t e d from equations 3.1.13 38 v 16 A n a l y t i c a l temperature approximation, equation 3.2.9, used f o r the v e l o c i t y c a l c u l a t i o n s : (a) c r oss-section (T i n °C) 45 (b) 9 i n C° against x i n cm f o r f i x e d values of z ( i n cm) 45 17a Calculated v e l o c i t y f i e l d . V e l o c i t i e s are shown by arrows, t h e i r length i n d i c a t i n g the motion i n a minute ( i . e . cm min - 1) i n the same scales as used for the axes. The a n a l y t i c a l temperature f i e l d used i s shown by dashed l i n e s (°C). The s o l i d curves are the h o r i z o n t a l v e l o c i t y p r o f i l e s . Compare with Figure 6 (p. 15) 46 17b Calculated v e l o c i t y f i e l d . V e l o c i t i e s are shown by arrows, t h e i r length i n d i c a t i n g the motion i n a minute ( i . e . . cm min - 1) i n the same scales as used f o r the axes. The a n a l y t i c a l temperature f i e l d used i s shown by dashed l i n e s (°C). The s o l i d curves are f o r the v e r t i c a l v e l o c i t y 47 18 . Actual 6 ( i n C° ) against distance along tank ( x, i n cm) p l o t t e d from laboratory observations 49 19 Calculated stream function <{> i n cm2 m i n - 1 . The a n a l y t i c a l temperature f i e l d used i s shown by dashed l i n e s (°C) 50 20 Temperature s e c t i o n from Lake Ontario (Rodgers, 1966a); mid-lake N-S s e c t i o n taken mid-January 1966. Temperature and temperature anomaly sections f o r the case of c o o l i ng ( f a l l simulation) with 5° bottom slope 62 21a Temperature s e c t i o n and mean temperatures ( v e r t i c a l l y averaged) f o r Run B: 5° bottom slope, standard heating, and T i , 0°C 64 21b Changes i n heat content and temperature anomaly s e c t i o n f o r Run B 65 22a Temperature s e c t i o n and mean temperatures for Run I: 5° bottom slope, standard heating, and T i , 1.4°C 66 22b Changes i n heat content and temperature anomaly f o r Run I... 67 v i i F i g u r e page 23a Tem p e r a t u r e s e c t i o n and mean t e m p e r a t u r e s f o r Run H: 5° b o t t o m s l o p e , i n c r e a s e d h e a t i n g and T i , 0°C 68 23b Changes i n h e a t c o n t e n t and t e m p e r a t u r e anomaly f o r Run H... 69 24a Temp e r a t u r e s e c t i o n and mean t e m p e r a t u r e s f o r Run C: 2.5° b o t t o m s l o p e , s t a n d a r d h e a t i n g and T i , 0°C 70 24b Changes i n h e a t c o n t e n t and t e m p e r a t u r e anomaly f o r Run C... 71 v 25a Tempera t u r e s e c t i o n and mean t e m p e r a t u r e s f o r Run D: 7.5° b o t t o m s l o p e , s t a n d a r d h e a t i n g and T i , 0°C 72 25b Changes i n h e a t c o n t e n t and t e m p e r a t u r e anomaly f o r Run D... 73 26a Temperature s e c t i o n and mean t e m p e r a t u r e s f o r Run E: 7.5° b o t t o m s l o p e , s t a n d a r d h e a t i n g and T i , 0°C 74 26b Changes i n h e a t c o n t e n t and t e m p e r a t u r e anomaly f o r Run E... 75 o 27a Temperature s e c t i o n and mean t e m p e r a t u r e s f o r Run F: 7.5° b o t t o m s l o p e , i n c r e a s e d h e a t i n g and T i , 0°C 76 27b Changes i n h e a t c o n t e n t and t e m p e r a t u r e anomaly f o r Run F... 77 28 Changes i n h e a t c o n t e n t d u r i n g t h e e a r l y s t a g e s o f t h e t h e r m a l b a r f o r Lake O n t a r i o ( f o r s e c t i o n s i n F i g u r e 2, p.4) and f o r t h e t a n k e x p e r i m e n t 78 29a Temp e r a t u r e s e c t i o n and mean t e m p e r a t u r e s f o r Run G: 5°-0° b o t t o m s l o p e , s t a n d a r d h e a t i n g and T i , 0°C 80 29b Changes i n h e a t c o n t e n t and t e m p e r a t u r e anomaly f o r Run G... 81 30a T e m p e r a t u r e s e c t i o n and mean t e m p e r a t u r e s f o r Run J: 5°-0° b o t t o m s l o p e w i t h d r o p o f f , s t a n d a r d h e a t i n g and T i , 0°C 82 30b Changes i n h e a t c o n t e n t and t e m p e r a t u r e anomaly s e c t i o n f o r Run J 83 31 P o s i t i o n o f t h e b a r , measured and p r e d i c t e d , f o r 5°-0° b o t t o m s l o p e s and s t a n d a r d h e a t i n g . The SLOPE o f t h e s e c u r v e s i s BAR SPEED 84 32 Temperature s e c t i o n and mean t e m p e r a t u r e s f o r s l i d i n g d o o r e x p e r i m e n t (5° b o t t o m s l o p e ) 86 33 Tempe r a t u r e s e c t i o n f o r h e a t i n g w a t e r warmer t h a n 4°C, w i t h a 5° b o t t o m s l o p e , s t a n d a r d h e a t i n g and T i , about 7°C 88 ACKNOWLEDGEMENTS The author would like to express her gratitude to Dr. P.H. LeBlond who suggested the problem and provided advice, encouragement and funds for the experimental work; to Dr. G.K. Rodgers of the Great Lakes Institute, University of Toronto who generously provided observational material from his work on the Great Lakes; to Dr. E.L. Lewis of the Frozen Sea Research Group, Department of Energy, Mines and Resources who kindly permitted the use of the cold chambers; to a l l who took their time to discuss and advise including Dr. R.W. Stewart, Dr. A. G i l l , Dr. R.W. Burling, Dr. Z. Rotem, Dr. L.A. Mysak and Dr. G.W. Bluman; and to Dr. J.A. E l l i o t t for his invaluable assistance i n the experimental work. The author was most fortunate to be financially supported during this work by a National Research Council of Canada Studentship and a MacMillan Family Fellowship. Finally the author wishes to thank her husband, Dr. J.A. E l l i o t t , for his patience and help in the preparation of this thesis. 1 1. INTRODUCTION The term 'thermal bar' (or rather 'barre thermique') was f i r s t coined 90 years ago when a Swiss scientist (Forel, 1880) tried to explain a curious winter temperature pattern i n Lake Geneva. The deeper part of the lake remained above 4°C, the temperature of maximum density of fresh water, while v during severe winters there was ice at the shore. The 4°C isotherm at the surface was situated just on the shallow side of a drop off to deep water and was quite stationary. The water was not observed to mix across this isotherm, which thus seemed to mark some sort of limnological barrier. However most of the literature on the thermal bar is from the last decade. In 1963 Tikhomirov published a paper describing a migrating thermal bar observed on Lake Ladoga during both the spring heating-up period and the f a l l cooling-off period. The bar was observed to move from near shore to the centre of the lake where i t disappeared. The most detailed observa- tional studies have been done by G.K. Rodgers of the Great Lakes Institute, University of Toronto, who for many years has been studying the migrating thermal bar i n Lake Ontario (Rodgers, 1963, 1965a, 1966a, 1966b, 1967, 1968). A migrating thermal bar thus occurs in some large freshwater lakes in the spring and late f a l l (e.g., Lake Michigan: Church, 1942; Great Lakes: Richards et al, 1969; Lake Ontario: Rodgers, 1966a; Lake Ladoga: Tikhomirov, 1963). It is a thermal structure i n which a surface 4°C isotherm appears f i r s t near the shores and then progresses towards the centre of the lake, where i t eventually disappears. Strong surface temperature gradients, a marked change i n turbidity, and indications of a convergence are usually associated with the 'bar 1, defined by the 4° surface isotherm, which s e p a r a t e s s t a b l e , s t r a t i f i e d , s h o r e w a r d w a t e r from a l m o s t homogeneous deeper w a t e r ( R o d g e r s , 1966a). The t h e r m a l b a r e x i s t s b e c a u s e f r e s h w a t e r a t t a i n s i t s maximum d e n s i t y above i t s f r e e z i n g p o i n t ( F i g u r e 1 ) . D u r i n g h e a t i n g o r c o o l i n g t h r o u g h t h e t e m p e r a t u r e o f maximum d e n s i t y i n c e r t a i n l a r g e l a k e s , i n w h i c h the ' b a r ' e x i s t s , t h e v e r t i c a l m i x i n g does n o t o c c u r u n i f o r m l y o v e r t h e w h o l e l a k e . I n s t e a d t h e m i x i n g i s f i r s t c o m p l e t e d n e a r t h e s h o r e s w h i c h t h e n become s t a b l e . The m i g r a t i n g t h e r m a l b a r e x i s t s a t the t r a n s i t i o n between the s t a b l e and u n s t a b l e r e g i o n s . i The s i t u t a t i o n i s w e l l i l l u s t r a t e d by what happens i n Lake O n t a r i o . I n e a r l y s p r i n g Lake O n t a r i o i s everywhere c o l d e r t h a n 4°C, t y p i c a l l y b e l o w 2°C. As the l a k e h e a t s up, t h e mean w a t e r t e m p e r a t u r e a t t h e s h o r e i n c r e a s e s more r a p i d l y t h a n i n t h e dee p e r p a r t s o f t h e l a k e . The s u r f a c e 4°C i s o t h e r m appears f i r s t n e a r t h e s h o r e a r o u n d t h e l a k e and g r a d u a l l y moves i n towards t h e c e n t r e o f t h e l a k e , r e m a i n i n g r o u g h l y p a r a l l e l t o the d e p t h c o n t o u r s . I t t a k e s one and a h a l f t o two months from i t s a p p e a r a n c e a t t h e s h o r e u n t i l i t d i s a p p e a r s b e l o w t h e s u r f a c e a t t h e c e n t r e o f t h e l a k e . By t h e b e g i n n i n g o f June a t e m p e r a t u r e s e c t i o n a c r o s s Lake O n t a r i o i s s i m i l a r t o t h o s e shown i n F i g u r e 2 ( R o d g e r s , 1966a and Rod g e r s , u n p u b l i s h e d , 1965). F l o t s a m a l o n g t h e s u r f a c e 4°C i s o t h e r m s u g g e s t s some a s s o c i a t e d c o n v e r g e n c e . The h i g h e r t u r b i d i t y on the s h o r e w a r d s i d e o f t h e 4°C i s o t h e r m i n d i c a t e s t h a t t h i s w a t e r does n o t mix q u i c k l y w i t h t h e r e m a i n d e r o f t h e l a k e . T h i s i s t h e r e a s o n f o r th e name ' t h e r m a l b a r ' meaning ' b a r r i e r ' . I n t h e warmer w a t e r on t h e s h o r e w a r d s i d e o f t h e 4°C i s o t h e r m a t h e r m o c l i n e and a s t r o n g s u r f a c e t e m p e r a t u r e g r a d i e n t e x i s t ( F i g u r e 2) (see R o d g e r s , 1966a, 1966b). T h i s phenomenon can a l s o o c c u r i n t h e f a l l as the t e m p e r a t u r e 3 Figure 1. The density of fresh water as a function of temperature (Handbook) Lake Ontario Temperature Sections (°C) Figure 2. Temperature s e c t i o n s f o r Lake Ontario f o r two N-S mid-lake s e c t i o n s ( S e c t i o n E, Rodgers 1966a; S e c t i o n D, Rodgers, unpublished, 1965) n e a r t h e s h o r e d e c r e a s e s more r a p i d l y t h a n n e a r t h e m i d d l e o f t h e l a k e r e s u l t i n g i n a 4°C i s o t h e r m t h a t t h e n p r o g r e s s e s towards t h e c e n t r e o f t h e l a k e i n a manner s i m i l a r t o t h e s i t u a t i o n i n t h e s p r i n g ( C h u r c h , 1942; R i c h a r d s et al, 1969; R o d g e r s , 1966a). T h i s i s n o t as p r o m i n e n t a c o n d i t i o n s i n c e the s t a b l e s h o r e w a r d s i d e cannot become as s t r o n g l y s t r a t i f i e d , t h e maximum t e m p e r a t u r e d i f f e r e n c e b e i n g 4C°. The o n l y p r e v i o u s m a t h e m a t i c a l c o n s i d e r a t i o n o f t h e t h e r m a l b a r phenomenon was t h e r e c e n t work p e r t a i n i n g t o Lake M i c h i g a n by Huang ( 1 9 6 9 ) . H i s a p p r o a c h assumed t h a t the p r e s s u r e g r a d i e n t s i n d u c e d by t h e r m a l e f f e c t s were b a l a n c e d by C o r i o l i s f o r c e s ( t h a t i s , he u s e d t h e t h e r m a l w i n d e q u a t i o n ) e x c e p t i n t h i n v i s c o u s b o u n d a r y l a y e r s . The e q u a t i o n s were l i n e a r i z e d by o r d e r i n g i n terms o f t h e Rossby number; th e z e r o t h and f i r s t o r d e r s o l u t i o n s were c o n s i d e r e d . These s o l u t i o n s a r e n o t a t a l l t i m e dependent, as the m i g r a t i n g t h e r m a l b a r o b v i o u s l y i s . The c o n f i n i n g o f v i s c o u s e f f e c t s t o t h i n b o undary l a y e r s c o n f i n e s t h e h o r i z o n t a l f l o w p r e p e n d i c u l a r t o t h e b a r t o t h e s e b o u n d a r y l a y e r s , w h i c h does n o t seem r e a s o n a b l e i n t h e l i g h t o f o u r model s t u d i e s . A l s o h i s c a l c u l a t e d f l o w i s i n t h e form o f c e l l s , one on e i t h e r s i d e o f t h e b a r , w i t h m i x i n g and s i n k i n g a t t h e b a r . T h i s was n o t what I f o u n d i n my e x p e r i m e n t a l o b s e r v a t i o n s . A s i m p l e p r e d i c t i o n scheme f o r t h e t i m e o f d i s a p p e a r a n c e o f t h e t h e r m a l b a r i n Lake O n t a r i o was d e v e l o p e d by S a t o (1969) u s i n g o b s e r v a - t i o n a l d a t a . 'He f o u n d t h e time t o depend p r i m a r i l y on t h e h e a t c o n t e n t i n t h e c e n t r a l p o r t i o n s o f 'the l a k e i n the e a r l y s p r i n g ( a r b i t r a r i l y u s i n g A p r i l 1 ) ; t h a t i s , t h e t i m e i n t e r v a l was t h a t n e c e s s a r y t o h e a t t h e c e n t r a l w a t e r s t o t h e p o i n t where t h e s u r f a c e t e m p e r a t u r e was I n my o p i n i o n a t h e o r e t i c a l a p p r o a c h t o t h e p r o b l e m o f t h e t h e r m a l b a r was hampered by t h e l a c k o f knowledge o f t h e v e l o c i t y f i e l d i n v o l v e d i n t h e phenomenon; f o r example, no measured c u r r e n t s have been d i r e c t l y r e l a t e d t o t h i s phenomenon. F o r t h i s r e a s o n I have s t u d i e d t h e t h e r m a l b a r i n a l a b o r a t o r y model ( E l l i o t t and E l l i o t t , 1969, 1970) ( s e e s e c t i o n 2 ) . T h i s s t u d y compared f a v o u r a b l y w i t h l a k e o b s e r v a t i o n s . From t h e s e l a b o r a t o r y s t u d i e s and t h e o b s e r v a t i o n s t a k e n by o t h e r s o f t h e t h e r m a l b a r i n l a k e s , a l i n e a r p h y s i c a l model f o r t h e s p e e d o f t h e m i g r a t i n g b a r was d e v e l o p e d . A t e m p e r a t u r e f i e l d was d e v e l o p e d m a t h e m a t i - c a l l y w h i c h compared r e a s o n a b l y w e l l w i t h t h e l a b o r a t o r y model s t u d i e s . U s i n g an a n a l y t i c a l a p p r o x i m a t i o n o f t h e o b s e r v e d t e m p e r a t u r e f i e l d , a v e l o c i t y f i e l d was then f o u n d , u s i n g an a p p r o x i m a t e d v o r t i c i t y e q u a t i o n , w h i c h a g a i n a g r e e d r e a s o n a b l y w e l l w i t h the o b s e r v a t i o n s f r o m t h e work i n t h e l a b o r a t o r y ( s e e s e c t i o n 3 ) . These r e s u l t s were t h e n s c a l e d , on the b a s i s o f t h e l a b o r a t o r y s t u d i e s and compared t o t h e c o n d i t i o n s i n Lake O n t a r i o d u r i n g t h e p r e s e n c e o f t h e t h e r m a l b a r . The l a b o r a t o r y and t h e o r e t i c a l s t u d i e s p r o d u c e r e a s o n a b l e a p p r o x - i m a t i o n s t o the t e m p e r a t u r e f i e l d s o b s e r v e d i n Lake O n t a r i o and t h e a s s o c i a t e d v e l o c i t y f i e l d i s n o t u n r e a s o n a b l e , b u t has n o t b e e n d i r e c t l y measured i n t h e l a k e s . 7 2. LABORATORY MODEL 2.1 A p p a r a t u s The e x p e r i m e n t s were c o n d u c t e d i n a r e c t a n g u l a r t a n k 1.5 meters l o n g and 30 c e n t i m e t e r s w i d e t h a t was i n s u l a t e d on t h e s i d e s and b o t t o m ( F i g u r e 3 ) . The end w a l l s and b o t t o m were p l y w o o d w i t h 4 c e n t i m e t e r s t y r o f o a m i n s u l a t i o n . The l o n g w a l l s were a d o u b l e t h i c k n e s s o f 0.6 c e n t i m e t e r p l e x i g l a s s e p a r a t e d by a 1.2 c e n t i m e t e r a i r s p a c e . A d e s i c c a t i n g a g e n t was i n s e r t e d i n the a i r s p a c e t o p r e v e n t c o n d e n s a t i o n . To p r o v i d e a s l o p i n g b o t t o m , wedges o f s t y r o f o a m c o v e r e d w i t h s h e e t s o f p l a s t i c were s e a l e d i n t o t h e t a n k w i t h m a s k i n g t a p e and a s i l i c o n e s e a l a n t . T h i s t a n k was u s e d t o s i m u l a t e b o t h t h e s p r i n g and f a l l c o n d i t i o n s . The s p r i n g c o n d i t i o n o r h e a t i n g p e r i o d was s i m u l a t e d b y f i r s t c o o l i n g w a t e r i n t h e t a n k down t o about 0°C and t h e n h e a t i n g t h e s u r f a c e f r o m above w i t h h e a t lamps ( W e s t i n g h o u s e , 250 w a t t , r e f l e c t o r , i n f r a r e d h e a t ) . The f a l l o r c o o l i n g p e r i o d was s i m u l a t e d by c o o l i n g c o l d (6° t o 8°C) tap w a t e r i n a c o l d chamber ( f a c i l i t i e s o f t h e F r o z e n Sea R e s e a r c h Group, DEMR, V i c t o r i a ) . S i n c e t h e s p r i n g c o n d i t i o n s have been more f r e q u e n t l y r e p o r t e d f o r t h e G r e a t L a k e s and t h e h e a t e d s y s t e m i s e a s i e r t o s i m u l a t e i n t h e l a b o r a t o r y t h i s s y s t e m was s t u d i e d i n more d e t a i l . 2.2 E x p e r i m e n t a l T e c h n i q u e s F o r t h e h e a t i n g e x p e r i m e n t s t h e tan k was f i r s t f i l l e d w i t h f r e s h t a p w a t e r . About 1 c c . o f l i q u i d d e t e r g e n t was added; t h i s was n e c e s s a r y t o d e c r e a s e s u r f a c e t e n s i o n e f f e c t s d u r i n g t h e a d d i t i o n o f dye. I f t h e d e t e r - g e n t was n o t p r e s e n t when t h e f l e c k s o f dye came i n c o n t a c t w i t h t h e w a t e r , s u r f a c e v e l o c i t i e s r e s u l t e d w h i c h d i s t o r t e d any ' t h e r m a l b a r e f f e c t s ' Figure 3. Experimental tank oo present. Crushed i c e was s t i r r e d i n t o the water to lower the temperature of the water to near 0°C; then excess i c e was removed, the mean temperature measured, and the heat lamps turned on. In approximately one h a l f hour the shallow end had reached a temperature s l i g h t l y above 4°C. A temperature and current pattern developed (see subsection 2.3). which maintained i t s e l f throughout the experiment, progressing from the shallow end to the deep end i n a further h a l f hour. The current s t r u c t u r e was followed by dropping fl e c k s of Rhodamine B dye on the surface; these sank to the bottom leaving v e r t i c a l red traces which were subsequently d i s t o r t e d by the currents. The dye trace was positioned with respect to a centimeter g r i d on the back w a l l of the tank. The motion was timed with a stop-watch. Rela t i v e motions were obtained from time s e r i e s photographs and from 16 mm movies of the dye streaks. To obtain temperature p r o f i l e s a thermistor bead (VECO 32A5), attached to the end of a 40 cm, t h i n glass rod with centimeter markings, was lowered v e r t i c a l l y through the water. The resistance of the thermistor was measured d i r e c t l y with a F a i r c h i l d Multimeter (Model 7050). This instrument uses s u f f i c i e n t l y small current (1 ua) that s e l f heating of the thermistor i s n e g l i g i b l e . To prevent thermal contamination from water dragged by the glass rod, measurements were made only while lowering the thermistor. Resistances were usually read at depths of 0.5 cm, 1 cm, 2 cm, 3 cm, et c . , to the bottom. The thermistor was c a l i b r a t e d against a mercury and glass thermometer before, during, and a f t e r the experiments. I t s c a l i b r a t i o n d i d not change. A c a l i b r a t e d , battery-operated thermistor thermometer was s i t u a t e d at the deep end of the tank at a l l times i n order to measured the temperature here p e r i o d i c a l l y during the experiments. The heat f l u x was obtained from the di f f e r e n c e i n heat content of the w a t e r i n t h e t a n k j u s t b e f o r e the h e a t lamps were t u r n e d on and j u s t a f t e r t h e h e a t lamps were t u r n e d o f f . N o r m a l l y the t a n k was h e a t e d w i t h t h r e e h e a t lamps e q u a l l y s p a c e d w i t h r e s p e c t t o t h e t a n k and a t a h e i g h t o f 1.5 meters above t h e w a t e r s u r f a c e ( F i g u r e 3) . The w a t e r was a l s o h e a t e d by l o n g wave r a d i a t i o n and h e a t c o n d u c t i o n t h r o u g h the w a l l s . The e f f e c t s o f t h i s r a d i a t i o n and c o n d u c t i o n were e v a l u a t e d as f o l l o w s . W i t h t h e h e a t lamps o f f , a t a n k o f 0°C w a t e r was a l l o w e d t o s i t f o r about an h o u r . T h i s gave a h e a t f l u x w h i c h i n c l u d e d b o t h l o n g wave r a d i a t i o n b a l a n c e and h e a t g a i n e d by c o n d u c t i o n t h r o u g h t h e w a l l s . Then a t a n k o f w a t e r a t 0°C, w i t h an i n s u l a t e d t o p , was a l l o w e d t o s i t f o r a p p r o x i m a t e l y an h o u r . T h i s gave an e s t i m a t e o f t h e h e a t f l u x t h r o u g h t h e w a l l s . U s i n g t h r e e lamps t h e -3 -2 -1 h e a t i n g r a t e was about 9 x 10 c a l cm s e c ( F i g u r e 4 ) . Of t h i s -3 -2 -1 -3 2 x 10 c a l cm s e c was f r o m t h e l o n g wave r a d i a t i o n and 2 x 10 -2 -1 c a l cm s e c f r o m h e a t c o n d u c t i o n t h r o u g h the w a l l s . A h i g h e r h e a t i n g -3 -2 -1 r a t e o f 12 x 10 c a l cm s e c c o u l d be o b t a i n e d u s i n g f i v e h e a t lamps. Thus"a maximum o f about one f i f t h o f t h e h e a t i n g o f t h e w a t e r i n t h e t a n k i s t h r o u g h the w a l l s and n o t t h r o u g h t h e s u r f a c e o f t h e w a t e r . I n the f i r s t s e r i e s o f e x p e r i m e n t s a b o t t o m s l o p e o f 5° and a h e a t i n g -3 -2 -1 r a t e o f 9 x 10 c a l cm s e c were used. T h i s s l o p e i s f i v e t o twenty t i m e s t h o s e f o u n d i n Lake O n t a r i o , the p r o t o t y p e f o r t h i s m o d e l , however i t gave a r e a s o n a b l e change i n w a t e r d e p t h o v e r t h e l e n g t h o f t h e t a n k ( a b o u t 1 cm o f w a t e r a t s h a l l o w end, 14 cm a t deep e n d ) . The h e a t i n g r a t e -3 -2 -1 i s o n l y s l i g h t l y h i g h e r t h a n t h e 7 x 10 c a l cm s e c t y p i c a l l y e x p e r i e n c e d i n the s p r i n g on Lake O n t a r i o . F i g u r e 5 shows t h e p e n e t r a t i o n of the s u r f a c e h e a t i n g f r o m t h e h e a t lamps, c a l c u l a t e d f r o m t h e s p e c t r a l e n e r g y d i s t r i b u t i o n o f t h e lamps and known a b s o r p t i o n c o e f f i c i e n t s ( S v e r d r u p et al, 1942; D o r s e y , 1940). A l s o shown i s t h e p e n e t r a t i o n o f < UJ X r X I O " 3 u <D CO * E 1 2 o o u X R 8 X t INCREASED HEATING x* " x * x x * * STANDARD 5 / * * * x * HEATING 2 0 4 0 6 0 8 0 1 0 0 T O T A L T I M E ( m i n u t e s ) Figure 4. Heat fluxes used in the experiments plotted against total time of heating 12 s o l a r r a d i a t i o n i n t o pure water (Sverdrup et at, 1942). In the tank most of the heat was absorbed i n the upper centimeter of the water; this simulates s o l a r heat absorption i n the upper few meters i n the lake. Thus Figure 5 suggests a v e r t i c a l s c a l i n g of approximately 1/1000. The 5° bottom slope, compared to the north shore of Lake Ontario, i n turn suggests a h o r i z o n t a l s c a l i n g of approximately 1/20,000. In making a comparison with the lakes, the tank could then be viewed as representing a s e c t i o n from the shore to the centre of the lake, however these tank experiments are not meant to be an example of s t r i c t dynamic modelling. The experimental work was done i n two stages. The f i r s t s e r i e s of experiments were done p r i m a r i l y to i n v e s t i g a t e the p o s s i b i l i t y of studying the thermal bar i n a laboratory model and to i n v e s t i g a t e i t s behaviour during heating and co o l i n g . The v e l o c i t y f i e l d was also studied at t h i s time. Later, a f u r t h e r s e r i e s of experiments were done to study my 'thermal bar' i n a more q u a n t i t a t i v e manner. 2.3 Description of the 'Bar' i n the Tank I n i t i a l l y the surface heating produces v e r t i c a l convection everywhere i n the tank. The temperature i n the shallow end r i s e s f a s t e r than that i n the deep end making the water i n the shallow end denser. However downslope flow due to this h o r i z o n t a l density gradient i s i n h i b i t e d by the convection. Eventually the temperature i n the shallow end reaches 4°C while i n the deep end i t i s s t i l l less than 2°C. By t h i s time some flow of dense water along the bottom slope can be observed i n s p i t e of the v e r t i c a l convection. This weak mean c i r c u l a t i o n i s superimposed on the convection; towards the deep end at the bottom and towards the shallow end at the top. Further heating produces, at the shallow end, a s t a b l e thermocline of water warmer than 4°C. 0 % of T O T A L E N E R G Y 0 2 0 4 0 6 0 8 0 100 E o CL UJ Q 4 H /T n 7 I 8h- ? I2h 16k / ? • * H E A T L A M P S 1° i o S O L A R RADIATION ' x (mult iply depths by 1000) I X I X Figure 5. Percentage of total surface, heat reaching different depths This thermocline progresses towards the deep end and i t s forward edge marks a boundary between the now stable shallow region of the tank and the deeper convecting region (Figure 6). A mean flow towards the deep end i s observed i n the thermocline with a counter-flow underneath. In the deep end of the tank a flow near the surface t r a v e l s towards the s t a b l e thermocline. This current sinks at the front edge of the thermocline and d i v i d e s , a part providing the upslope current i n the shallow end and the other part a down- slope current i n the deep end. The s i n k i n g zone (Figure 6) i n f r o n t of the s t a b l e thermocline i s made up of water between 3.5 and 4.5°C (maximum density region). This dense water i s the r e s u l t of heating colder, l i g h t e r water from the deep end of the tank and does not include water from the st a b l e thermocline on the shallow s i d e . In these experiments, t h i s s i n k i n g zone w i l l be r e f e r r e d to as the 'thermal bar' or simply the 'bar'. The 4°C isotherm does not always extend to the surface since eventually a shallow thermocline also develops on the deep sid e due to the intense heating at the surface. This thermocline i s s l i g h t l y unstable and more a c t i v e convec- t i v e mixing would d i s s i p a t e i t . The surface v e l o c i t i e s go to zero because of surface tension e f f e c t s , often r e f e r r e d to as 'surface pressure' (see Davies and Rideal, 1961, p.218); t h i s i s due to the presence of nearly unavoidable surface contamination. The temperature and current pattern maintains i t s e l f u n t i l the bar has reached the deep end of the tank. The r e s u l t s f o r the cooling system are s i m i l a r (see Appendix A). 2.4 S i m i l i t u d e to Lake Comparing these r e s u l t s from the tank to f i e l d observations, there are s e v e r a l obvious s i m i l a r i t i e s . A 'bar' moves from the shallow end or shore to the deep end or centre of the lake. A thermocline develops on the DISTANCE ALONG T A N K ( c m ) Figure 6. Generalized current and temperature results for the heated system.. Averagedvelocities are shown by arrows, their length indicating the motion observed i n a minute (I.e., .cm.min"1.) in the same scales as used for the axes. s h a l l o w s i d e and t h e f r o n t edge i s marked by w a t e r n e a r 4°C. On the deep s i d e a more u n i f o r m t e m p e r a t u r e . r e g i o n e x i s t s i n b o t h , i n d i c a t i n g c o n v e c t i o n . The m o t i o n o b s e r v e d i n t h e t h e r m o c l i n e i n t h e e x p e r i m e n t has a l s o b e e n deduced f r o m h e a t b u d g e t c a l c u l a t i o n s f o r Lake O n t a r i o ( R o d g e r s , 1 9 6 8 ) . The s i n k i n g plume o b s e r v e d i n t h e e x p e r i m e n t i n f r o n t of t h e t h e r m o c l i n e w o u l d p r o d u c e t h e convergence t h a t has b e e n s u g g e s t e d f r o m t h e f i e l d o b s e r v a t i o n s . I n t h e e x p e r i m e n t t h e r e was no h o r i z o n t a l m o t i o n a c r o s s the b a r f r o m t h e t h e r m o c l i n e r e g i o n t o t h e d e e p e r r e g i o n . T h i s a g r e e s w i t h the f a c t t h a t t h e b a r i s o b s e r v e d t o be a l i m n o l o g i c a l b a r r i e r t o o f f s h o r e m o t i o n . The s h a l l o w t h e r m o c l i n e w h i c h d e v e l o p e d on t h e deep s i d e i n t h e e x p e r i m e n t w o u l d n o t g e n e r a l l y e x i s t i n l a k e s , due t o w i n d m i x i n g . N e v e r t h e - l e s s under v e r y s t a b l e , l i g h t w i n d c o n d i t i o n s s u c h a s h a l l o w t h e r m o c l i n e has b e e n o b s e r v e d on Lake O n t a r i o ( E l d e r and Lane, 1970). A l s o t h e s u r f a c e v e l o c i t i e s w o u l d n o t , o f c o u r s e , be e x p e c t e d t o go t o z e r o . Thus the t a n k e x p e r i m e n t appears t o r e p r e s e n t a h e a t d r i v e n c i r c u l a t i o n s i m i l a r t o t h e t h e r m a l b a r phenomenon o b s e r v e d i n t h e G r e a t L a k e s . 2.5 L i n e a r M o d e l f o r t h e Speed o f t h e B a r A s i m p l e l i n e a r model b a s e d on t h e above r e s u l t s was used t o e v a l u a t e t h e s p e e d o f t h e b a r . The model assumes t h a t h o r i z o n t a l a d v e c t i o n and d i f f u s i o n o f h e a t a r e n o t o f p r i m a r y i m p o r t a n c e . T h i s means t h a t most o f t h e h e a t e n t e r i n g t h e s u r f a c e o f a u n i t column remains w i t h i n t h a t column. The p o s i t i o n o f the b a r i s t h e n a t t h e t r a n s i t i o n f r o m an u n s t a b l e t o a s t a b l e w a t e r column. The t i m e t a k e n f o r t h e b a r t o t r a v e l f r o m a p o i n t A t o a p o i n t B i s t h e t i m e t a k e n t o h e a t t h e column o f u n i t a r e a a t p o i n t B up t o 4°C. The model g i v e s the s p e e d o f t h e b a r , S, s i n c e 17 time = — = change in heat content Q as Q L S = AT pc D P (2.5.1) where Q i s heat flux through the surface in cal cm sec -1 L is the horizontal distance (in cm) between the bar and some deeper position where the mean temperature is known; AT i s the temperature difference (in C°) between 4°C and the mean temperature at the deeper position; D is the depth of the water (in cm) at the deeper position where the mean temperature i s known; and pc is the density of the water times i t s specific heat. To the Thus i f the mean temperature at some position i n the convective region is known, together with the heat input, the arr i v a l time of the bar can be calculated. The above formula was compared to actual speeds observed in the experiments and i t s dependence on the three variables D/L, Q, and AT was examined. Each of these was varied in turn. The data used in the formula were the depth at the deep end of the tank, the temperature at the deep end, usually when the bar was about 30 cm from the shallow end, and the distance from this position of the bar to the deep end. Figure 7 shows a comparison _3 of the actual and predicted speeds for a heat flux of approximately 9 x 10 - 2 - 1 1 1 cal cm sec and three different slopes: 2"2°, 5°, 7—°. The poorer observational data from the 2^-° slope resulted from the longer time required before the bar properly developed; there tended to be a lack of two- P accuracy of the experimental data this has a value of 1 cal C / • A 100 o < O § 8 0 o CO O Q. 6 0 4 0 o oo / * * A A X o o/ 2 . 5 ° S L O P E * * * ° V ~ ' 5 ° S L O P E X ^ A A O o/o * ft yx j 7 . 5 ° S L O P E predicted X 0 0 0 / x x ^ - m e a s u r e d O O o » X X ' A A A ^ AAA *" 6 A A STANDARD HEATING ^ X A^A A * tt A A A 10 15 2 0 2 5 3 0 3 5 4 0 T I M E (minutes) Figure 7. P o s i t i o n of the bar, measured and predicted, f o r standard heating and bottom slopes of 2.5°,5°, 7.5°.g The SLOPE of these curves i s BAR SPEED. dimensionality and current structures often were not uniform across the tank. This prevented using slopes smaller than 2^°. Figures 8 and 9 show a s i m i l a r comparison for two d i f f e r e n t surface heat fluxes at slopes of 5° and 7̂ -° • Figure 10 i s a comparison of speeds for d i f f e r e n t AT's. The predicted speeds compare w e l l with the measured speeds i n a l l cases. The increase i n the speeds as the bar approaches the end of the tank seems to be due to an i n t e r a c t i o n with the shallow thermocline which has developed on the deep side by t h i s time. The main thermocline i s then able to develop f a s t e r by incorporating the p a r t i a l l y warmed surface l a y e r from the deep side and the region r e f e r r e d to as the bar becomes more d i f f u s e . The dashed curve, Figure 8, shows the speed of the bar that r e s u l t e d when an i n s u l a t e d l i d was put on the tank at the time i n d i c a t e d by the arrow; t h i s shows the strong dependence of the speed on the surface heat input rather than on the h o r i z o n t a l density gradient or on other heat t r a n f e r s . Temperature sections were used to c a l c u l a t e mean temperatures along the tank i n order to compare with the temperatures predicted by the approximations leading to equation 2.5.1. Figure 11 i s a temperature s e c t i o n through the bar f o r a 5° bottom slope and heat input of approximately -3 -2 -1 9 x 10 c a l cm sec . The p o s i t i o n s at which observations were made are designated by the v e r t i c a l marks along the top l i n e i n the diagram. From t h i s f i g u r e the mean temperature and change i n heat content from the time the heat lamps were turned on are c a l c u l a t e d and i n the lower diagrams of Figure 11 are compared with values predicted by the model. As can be seen the model gives a good f i r s t approximation to the temperature or heat content along the tank. Differences from the predicted heat content may be accounted f o r by the presence of advective e f f e c t s (see subsection 2.7). XX XX / / / AA * ^^A 4 predic ted A& LID ON A / XX A A ' V / AAA A AA measured 5 ° S L O P E A S T A N D A R D HEATING x I N C R E A S E D HEATING • H E A T I N G S T O P P E D A A _L 10 15 2 0 25 T IME ( minutes) 3 0 3 5 4 0 Figure 8. P o s i t i o n of the bar, measured and predicted, f o r 5° bottom slope and the two heating rates ( ^ 9 x 10 3 and ^ 12 x 10" 3 c a l cm"2 s e c - 1 ) . The SLOPE of these curves i s BAR SPEED. ro o lOOh E o <80 O O I < O h- co60 O CL 40 predicted 7.5° SLOPE a STANDARD HEATING x INCREASED HEATING 5 10 15 20 25 30 35 40 TIME (minutes ) Figure 9. P o s i t i o n of the bar, measured and predicted, f o r 7.5° bottom slope and the two heating rates ( 'v 9 x IO" 3 and ̂  12 x 10" 3 c a l cm"2 s e c " 1 ) . The SLOPE of these curves i s BAR SPEED. 100 £ o 2 < H O 8 0 2 O _ l < if) £ 6 0 4 0 * p r e d i c t e d ^ . © ' measured 5° S L O P E S T A N D A R D H E A T I N G * A T = IC° , L = 1 0 0 c m * A T = I C° t L = I I 0 cm o A T = l . 5 C ° , L =105 c m 0 Figure 10. 5 10 15 2 0 2 5 3 0 3 5 4 0 T I M E ( m i n u t e s ) Position of the bar, measured and predicted, for 5° bottom slope, standard heating and different AT, L combinations. The SLOPE of these curves is BAR SPEED. to Observations RUN A 4 0 8 0 D I S T A N C E A L O N G TANK ( c m ) _I2 O o UJ cc < CC UJ < UJ 2 _ 0 u measured predicted BAR o u 40 UJ O U 20 »- ui I meosured BAR predicted Figure 11. 40 80 D I S T A N C E A L O N G T A N K ( c m ) 120 Temperature section, mean temperatures (vertically averaged), and changes in heat content for Run A: 5° bottom slope and standard heating. The i n i t i a l temperature (Ti) when the heat lamps were turned on was 0.2°C. S i m i l a r d a t a f r o m t h e o t h e r e x p e r i m e n t s a r e c o n t a i n e d i n A p p e n d i x B. The r e s u l t s a t t h e d i f f e r e n t s l o p e s and h e a t i n g r a t e s a r e s i m i l a r t o t h e above. 2.6 Comparison o f t h e L i n e a r M o d e l t o Lake O n t a r i o The model f o r t h e speed o f t h e b a r was a l s o compared w i t h d a t a t a k e n i n Lake O n t a r i o . The d a t a used t o make t h e c o m p a r i s o n were t a k e n f r o m Rodgers (1965b, 1966a, 1968) and Rodgers and A n d e r s o n ( 1 9 6 1 ) . The n e t s u r f a c e h e a t t r a n s f e r i s a p p r o x i m a t e l y u n i f o r m o v e r t h e s u r f a c e o f t h e L a k e , as assumed i n t h e c a l c u l a t i o n s . U s i n g t h e t e m p e r a t u r e d a t a f r o m Rodgers ( 1 9 6 6 ) , the s p e e d o f t h e b a r was c a l c u l a t e d and compared t o t h e v a l u e p r e d i c t e d by t h e model. C a l c u l a t e d v a l u e s were 0.95 cm s e c ^ ( n o r t h s h o r e ) and 0.3 cm s e c ̂  ( s o u t h s h o r e ) . The p r e d i c t e d v a l u e s were 0.7 and 0.4 r e s p e c t i v e l y . The model a l s o gave a r e a s o n a b l e a p p r o x i m a t i o n t o t h e s p e e d o f t h e b a r i n Lake M i c h i g a n from t h e work o f Church ( 1 9 4 2 ) . To make a c l o s e r check on t h e a c c u r a c y o f t h e model, b e t t e r d a t a f r o m t h e l a k e s a r e needed. V a l u e s o f t h e change i n h e a t c o n t e n t i n a u n i t column f o r a c r o s s s e c t i o n t h r o u g h t h e t h e r m a l b a r f r o m Rodgers (1968) a r e p l o t t e d i n F i g u r e 12. A l s o shown, f o r c o m p a r i s o n , i s a p l o t o f s i m i l a r v a l u e s f o r a t a n k e x p e r i m e n t ( f r o m F i g u r e 1 1 ) . As see n f r o m t h e f i g u r e t h e a p p r o x i m a t i o n o f n e g l i g i b l e a d v e c t i o n o f h e a t ( u n i f o r m change i n h e a t c o n t e n t ) i s n o t as good f o r t h e Lake as i t i s f o r t h e t a n k . However t h e d i s t r i b u t i o n o f h e a t i s s t r i k i n g l y s i m i l a r i n b o t h c a s e s . The e f f e c t o f a d v e c t i o n i s i l l u s t r a t e d i n more d e t a i l i n F i g u r e 13 w h i c h shows the c r o s s s e c t i o n t e m p e r a t u r e a n o m a l i e s f o r t h e t a n k and t h e Lake. The ' t e m p e r a t u r e anomaly' i s t h e d i f f e r e n c e between t h e o b s e r v e d t e m p e r a t u r e and t h e mean t e m p e r a t u r e p r e d i c t e d f r o m t h e h e a t f l o w t h r o u g h t h e s u r f a c e , a s s u m i n g no 25 o 20 < LU oi cCT 30 • C ° 'C* BAB o SECTION D — x ^ ^ - ^ . 0 ^ J T ^ _ ° I '* ' 'D' BAR T5 UJ o 'Or / MEAN for LAKE 0 / / S E C T I O N E L A K E ONTARIO ui 5 o" " 1 1 10 20 30 40 D I S T A N C E from N O R T H S H O R E (km) £ h E X P E R I M E N T 0 . R U N A £ 4 0 . measured 2 1 it! r O 2 0 h - ^ X " "BAT MEAN for TA o N K - ^ <J 0 40 60 120 D I S T A N C E along T A N K ( c m ) Figure 12. Changes in heat content for Lake Ontario (for sections shown in Figure 2, p.4). Also a similar plot for the tank for Run A. 26 DISTANCE along TANK Figure 13. Temperature anomaly s e c t i o n s f o r the tank exper iment ( f o r Run A) and f o r Lake O n t a r i o ( f o r s e c t i o n s i n F igu re 2 , p . 4 ) . a d v e c t i o n . As can be s e e n f r o m the f i g u r e , t h e s e c t i o n s a r e s i m i l a r . T h e r e f o r e a d v e c t i v e e f f e c t s s i m i l a r t o t h o s e o b s e r v e d i n t h e t a n k c o u l d o c c u r i n t h e L a k e . Anomaly s e c t i o n s f o r t h e o t h e r e x p e r i m e n t s a r e shown i n A p p e n d i x B. 2.6 D i s c u s s i o n As has been shown a r e a s o n a b l e f i r s t a p p r o x i m a t i o n t o t h e h e a t c o n t e n t o f a w a t e r column and the s p e e d o f t h e b a r f o r b o t h the t a n k and Lake O n t a r i o can be o b t a i n e d f r o m t h e a s s u m p t i o n o f n e g l i g i b l e h o r i z o n t a l a d v e c t i o n and d i f f u s i o n . I n b o t h c a s e s , i n the d e e p e r r e g i o n s , most o f t h e h e a t i s c a r r i e d by c o n v e c t i v e p r o c e s s e s . These s t r o n g v e r t i c a l c o n v e c t i o n s i n h i b i t h o r i z o n t a l a d v e c t i o n and p r o d u c e t h e b a s i c f e a t u r e s o f t h e t h e r m a l b a r . A f u r t h e r r e f i n e m e n t t o t h e f i r s t a p p r o x i m a t i o n w o u l d r e q u i r e i n c l u d i n g t h e e f f e c t s o f h o r i z o n t a l a d v e c t i o n . The i m p o r t a n c e o f t h e s e e f f e c t s i s i l l u s t r a t e d by t h e d i f f e r e n c e s between t h e a c t u a l and p r e d i c t e d v a l u e s s e e n i n t h e Lake and i n t h e t a n k . T h r e e i m p o r - t a n t a d v e c t i v e r e g i o n s e x i s t ( F i g u r e 6, p . 1 5 ) . One i s the a d v e c t i o n o f warm w a t e r towards t h e b a r i n t h e s h a l l o w end. T h i s i s a f l o w o f t h e l i g h t , u pper t h e r m o c l i n e w a t e r r e p l a c i n g p a r t o f t h e s i n k i n g w a t e r a t t h e b a r . T h i s r e s u l t s i n a d v e c t i o n o f h e a t towards t h e b a r f o r b o t h t h e t a n k and t h e L a k e . T h i s f l o w i s a i d e d by the h o r i z o n t a l d e n s i t y g r a d i e n t t o w a r d t h e b a r , b u t f o r t h e t a n k t h i s d e n s i t y g r a d i e n t i s n o t s u f f i c i e n t f o r t h e f l o w t o c r o s s t h e b a r . T h e r e i s a c o u n t e r - f l o w b e n e a t h t h e above a d v e c t i o n . T h i s b r i n g s c o l d w a t e r t o t h e s h a l l o w end f r o m t h e v i c i n i t y o f t h e b a r ; t h a t i s , h e a t f l u x o f t h e same si g n . The other advective region i s the downslope flow of 4° water from the v i c i n i t y of the bar toward the deep end. This advects heat to the deep regions. For the tank this flow was found to be weaker for smaller bottom slopes. Its e f f e c t was also reduced by turbulent mixing. Figure 12 (p.25) suggests that there was some heat advected towards the centre of Lake Ontario. However the impor- tance of this flow i n lakes i s d i f f i c u l t to judge because of the smaller slope, the unknown nature of the convective mixing, and i r r e g u l a r i t i e s i n the bottom. In s p r i n g , the net e f f e c t of advection i s that the heat content i s higher than would otherwise be expected i n the c e n t r a l portions of the Lake and i n the regions next to the bar, while regions near the shore have been depleted of heat. The advective v e l o c i t i e s observed i n the tank are a l l of the same order of magnitude as the speed of the bar. The in f l u e n c e of bottom i r r e g u l a r i t i e s was tested i n two experiments. In both a slope of 5° was used for h a l f the tank, followed by a f l a t s e c t i o n i n one and by a v e r t i c a l drop of 12 cm to a f l a t s e c t i o n i n the other. The main influence of bottom i r r e g u l a r i t y was on the speed of the bar; advection produced an averaging e f f e c t over the i r r e g u l a r i t i e s . The speed observed was roughly that which, from equation 2.5.1, would have been associated with a bottom slope that was the weighted average of the a c t u a l slopes. Otherwise the temperature and current s t r u c t u r e were s i m i l a r to the others. The r e s u l t s are given i n Appendix C. Thus bottom i r r e g u l a r i t i e s do not seem to be of primary importance i n the development of the thermal bar i n lakes. An important e f f e c t not studied i n t h i s experiment i s wind mixing. In smaller lakes the r e s u l t i n g wind driven h o r i z o n t a l advections would likely keep the temperature uniform and not permit a bar to develop. A trip was made to the Merritt region of British Columbia to see i f the bar could be found in smaller lakes, however they were found to be too thoroughly wind mixed. In the larger lakes where the thermal bar develops, this wind induced advection might result in mixing across the bar. If this brings water below 4°C adjacent to water above 4°C, more active convection would be expected which would include water from the warmer side. An experiment that brought together 8°C water and 2°C water, by removing a barrier separating the two, pro- duced convection at a stationary mixing zone (see Appendix D). This indicated that mixing across the bar would immediately produce convec- tion but would not dissipate the bar. Further experimental studies on the thermal bar should await more detailed observations of this phenomenon in lakes. Information on the current structure i s particularly necessary. 2.8 Summary In this laboratory study, a temperature structure has been pro- duced which is similar to that associated with the thermal bar i n lakes. The existence of my 'bar' in the tank depends entirely on the temperature dependence of the density and on the presence of heat flux and a sloping bottom. A simple model for the speed of the bar, which also applies to the bar in the lakes, is based on horizontal heat advection and diffusion not being of primary importance. The success of this model indicates that effects of wind mixing, Coriolis force, etc., which exist for lakes but not i n the model, are not important factors in the formation of the 1 thermal bar'. 3. MATHEMATICAL MODELS These t h e o r e t i c a l studies are an attempt to explain more q u a n t i t a t i v e l y the d e t a i l s of the temperature and v e l o c i t y f i e l d s observed i n the laboratory studies and to understand the dynamic balance of the flow. The r e s u l t s may then be t e n t a t i v e l y extended to describe the s i t u a t i o n i n lakes. 3.1 The Temperature F i e l d From the two-dimensional laboratory model reasonable f i r s t appro- ximations to the heat content and to the speed of the bar were obtained by assuming that h o r i z o n t a l advection and d i f f u s i o n of heat were not of primary importance. This suggests that the temperature d i s t r i b u t i o n may be derived from the one-dimensional heat d i f f u s i o n equation: where z i s taken v e r t i c a l l y upwards, t i s the time, T i s the temperature i n °C, and K i s the thermal d i f f u s i v i t y with boundary conditions 9T 3t K (3.1.1) 9T at z = 0 (top) (3.1.2) K 3-dz = q = 0 at z = -D (bottom) (3.1.3) where q i s t h e h e a t f l u x t h r o u g h t h e s u r f a c e i n C° cm s e c 1 ( = Q/pc^, e q u a t i o n 2.5.1, p . 1 7 ) . D = ax where a i s the s l o p e o f the b o t t o m and x i s the d i s t a n c e a l o n g t h e tan k from t h e s h a l l o w end. ( T h i s assumes t h a t D = 0 a t the s h a l l o w end, e q u i v a l e n t t o a s h o r e ) . F i n a l l y , T = 4°C a t z = 0 a t the 'bar' . S i n c e when one i s d i s c u s s i n g t h e t h e r m a l b a r a l l t e m p e r a t u r e s a r e r e l a t i v e t o t h e t e m p e r a t u r e o f maximum d e n s i t y o f f r e s h w a t e r , t h e e q u a t i o n and boundary c o n d i t i o n s can be r e w r i t t e n : 2 3 9 9 8 at = K 7T ( 3 - 1 ' 4 ) 38 a t z = 0 K ~ = q (3.1.5) dz a t z = -D | i = 0 (3.1.6) dz a t z = 0, a t t h e ' b a r 1 , 8 = 0 (3.1.7) where 8 = T - 4°C. From t h e l a b o r a t o r y work t h e t e m p e r a t u r e f i e l d may be c o n s i d e r e d i n two r e g i o n s , on e i t h e r s i d e o f t h e b a r : a d e e p e r , h i g h l y c o n v e c t i v e r e g i o n and a s h a l l o w e r , s t a b l e r e g i o n . F o r t h e p u r p o s e s o f t h e t h e o r e - t i c a l work, the b a r w i l l be d e f i n e d by t h e 4° s u r f a c e i s o t h e r m as i s done i n l a k e s t u d i e s . i ) The deep, c o n v e c t i v e s i d e The aim h e r e i s t o p r o v i d e a r o u g h , b u t r e a s o n a b l e , a p p r o x i m a t i o n t o t h e t e m p e r a t u r e f i e l d on t h e u n s t a b l e s i d e o f t h e b a r . No a t t e m p t w i l l be made to consider convective e f f e c t s i n d e t a i l , nor to model the shallow thermocline that develops over t h i s end i n the l a t t e r stages of the b a r i n the tank and o c c a s i o n a l l y forms on Lake O n t a r i o . Instead my reasoning w i l l be based on the observation that the temperature i n the deeper region of both tank and l a k e tends to be n e a r l y uniform v e r t i c a l l y , i m p l y i n g a thorough mixing process and hence a h i g h v e r t i c a l eddy (or convective) d i f f u s i v i t y . I n t e g r a t i n g equation 3.1.4 v e r t i c a l l y from z = -D to z = 0 one obtains ,,36 89 dt dz 0 (3.1.8) -D and u s i n g the z boundary c o n d i t i o n s : _96 = £ 3t D or 9 = - 6 + S£ o D where -0 i s the temperature i n C° at t = 0. Using the simple model f o r the speed of the bar (equation 2.5.1, p.17) S - (2.5.1) o where 0 q = AT, the temperature d i f f e r e n c e , i n t h i s case between the shallow end and a p o s i t i o n x along the tank, at time t = 0 and choosing the o r i g i n of time to be when the 'bar' i s ' a t ' x = 0, the temperature 33 f o r t h e dee p e r r e g i o n can be w r i t t e n : 6 = 0 ( — - 1 ) (3.1.9) o x T h i s c o u l d have b e e n w r i t t e n e m p i r i c a l l y f r o m the o b s e r v a t i o n a l d a t a . I f t = 0 i s d e f i n e d , as above, as t h e ti m e when t h e ' b a r ' i s ' a t ' x = 0, t h e n 8 i s c l e a r l y a f u n c t i o n o f x, t h e d i s t a n c e down t h e t a n k , o The a s s u m p t i o n i s made t h a t a t some t i m e -t^> b e f o r e h e a t i n g b e g i n s , the e n t i r e f l u i d i s a t a u n i f o r m t e m p e r a t u r e ; t h i s i s o b v i o u s l y t r u e f o r t h e t a n k e x p e r i m e n t s , and r o u g h l y v a l i d f o r t h e l a k e s . I f h e a t i n g i s u n i f o r m o v e r t h e s u r f a c e o f the w a t e r , i f t h e b o t t o m has a c o n s t a n t s l o p e , and i f t h e a s s u m p t i o n i s made t h a t h o r i z o n t a l a d v e c t i o n and d i f f u s i o n o f h e a t a r e n o t o f p r i m a r y i m p o r t a n c e , t h e n t h e t e m p e r a t u r e a t any t i m e u n t i l t h e b a r appears w i l l be a l i n e a r f u n c t i o n o f x. Thus a t t = 0 one can w r i t e n e x " e o - " X where 0 i s t h e t e m p e r a t u r e a t p o s i t i o n X and X i s t h e l e n g t h o f t h e t a n k o r t h e d i s t a n c e f r o m t h e s h o r e t o t h e dee p e r p o r t i o n s o f a l a k e . Thus e q u a t i o n 3.1.9 may be r e w r i t t e n as 0 = f ( S t - x) f o r x > S t (3.1.10) 34 i i ) The s h a l l o w , s t a b l e s i d e / The t e m p e r a t u r e f i e l d on t h i s s i d e o f t h e b a r i s c l e a r l y more c o m p l i c a t e d t h a n on t h e d e e p e r s i d e . S i n c e i t i s d e s i r a b l e t o match the two r e g i o n s , t h e b o u n d a r y c o n d i t i o n a t t h e 'bar' ( e q u a t i o n 3.1.7) w i l l be r e w r i t t e n as where t h e o r i g i n o f x i s a t t h e s h a l l o w end and t h e o r i g i n o f t i m e i s d e f i n e d as when t h e 'bar' i s ' a t ' x = 0. The s o l u t i o n t o 6 = 0 a t t - f (3.1.11) 99 3 t K (3.1.4) f o r a s e m i - i n f i n i t e body w i t h b o u n d a r y c o n d i t i o n s a t z = 0 K 89 8z = q (3.1.5) and 3.1.11 i s 9 S x < S t (3.1.12) I f an e r r o r i n t h e b o t t o m b o u n d a r y c o n d i t i o n (3.1.6) i s ' t o l e r a t e d ' up to a heat flux that is 20% of the surface heat flux, then the above represents the temperature to within 20 cm of the shallow end of the tank for a 5° bottom slope. This 'error' in bottom heat flux is li k e l y not more than the error involved in neglecting horizontal advection of An attempt to consider the diffusion equation for a wedge (see Arlinger, 1965) would not improve the temperature solution as the bottom effects tend to make the shallower portions more uniform in temperature vert i c a l l y . This is not the case in the observations. Neglecting the bottom heat flux i n this manner is equivalent to removing some of the heat input in the shallower portions and thus approximates the advective effects observed in this region. In order to calculate actual values from this mathematical model i t is necessary to choose a value for K. Since the motion appears to be laminar in the tank, K ought to have the value of molecular thermal -3 2 - 1 di f f u s i v i t y , 1.4 x 10 cm sec . However, in assuming that a l l the heating occurred at the surface, the penetrative effects of the surface heating (see Figure 5, p.13) have been neglected. These are included roughly in the above model by increasing K. • - \ i i i ) Comparison with the tank The temperature model developed above is heat in the shallow end. 9 x < St (3.1. To compare this to the measured temperature f i e l d the position of the bar had to be set. In the generalized temperature f i e l d shown in Figure 6 (p.15), extending the 4° isotherm to the surface shows the 'bar' at 90 cm. This defines the value of t as 90 times S, the speed of the bar. The value for the speed of the bar is taken to be -2 -1 the average value observed in the tank; that i s , 3.9 x 10 cm sec -3 -1 q is taken to be the standard heating rate of 9 x 10 C° cm sec -1 -3 (assuming pc^ - 1 cal C° cm ). The value of 0 is obtained using the formula for the speed of the bar (2.5.1), the average observed value of S, the standard value for q, the length of the tank, and the depth at the end of the tank; this gives a value of 2.6 C°. It remains to determine a value of K; a reasonable f i t for the temperature cross-section was found using a value for K of -3 -2 -1 5.8 x 10 cm sec , see Figures 14 and 6(p.l5). This reasonable f i t with the tank results together with general similarity with the lake suggests the temperature model is good for the lake too. This w i l l be discussed in more detail at the end of the theoretical section (see subsection 3.4). In order to further check on the degree to which the model matched the general picture in the tank, 0 was plottted against distance down the tank at fixed values of depth for both the tank and the mathematical model (see Figure 15). In order to make the plot for the tank the 4.5 and 3.5°C isotherms were extended to intersect with the surface rather than form a shallow thermocline over the deeper region. The reason for the interest i n the 9 versus x plot w i l l become apparent when the vorticity equation is considered in the next subsection. As Figure 14. Temperature s e c t i o n (°C) calculated from equations 3.1.13. 38 Figure 15. 6 i n C° against distance along the tank f o r f i x e d values of z ( i n cm). (a) generalized measured values (b) c a l c u l a t e d from equations 3.1.13 can be seen the curves are similar. 3.2 The Velocity Field The velocity f i e l d i s most readily investigated through the vorticity equation: where w is the vorticity vector, v is the velocity vector, p i s the density, p i s the pressure, and V is the viscosity. Assuming two-dimensional flow, neglecting non-linear advective effects and making the hydrostatic approximation for pressure, equation 3 . 2 . 1 reduces to ( 3 . 2 . 1 ) 9w. p 3x 9t + v V 2 w. ( 3 . 2 . 2 ) where g is the value of the acceleration due to gravity, and co„ = Tr— where u and w are the x and z velocity 2 dz dx components respectively. Introducing a stream function, (}>, where u = 3£ 9z w 3£ 3x ' 40 e q u a t i o n 3.2.2 may be r e w r i t t e n |_ v2<f> = * | £ + v V > (3.2.3) d t p dx where V1* = V 2 . V 2 . The f i r s t t erm i n e q u a t i o n 3.2.3 i s t h e t i m e r a t e o f change o f the v o r t i c i t y and t h e l a s t t e r m r e p r e s e n t s e f f e c t s o f v i s c o u s d i f f u s i o n . However t h e f i r s t t erm on t h e r i g h t hand s i d e , t h e buoyancy term i s n o t s o f a m i l i a r and per h a p s needs f u r t h e r d i s c u s s i o n . R e c a l l i n g t h a t t h e buoyancy t e r m i n v e c t o r n o t a t i o n i s -~r Vp x Vp, i t i s e v i d e n t t h a t p r e s s u r e f o r c e s a r e b e i n g c o n s i d e r e d . I n t h e p r e s e n c e o f g r a v i t y , a h o r i z o n t a l change i n d e n s i t y r e s u l t s i n a h o r i z o n t a l p r e s s u r e g r a d i e n t w h i c h tends t o p r o d u c e a f l o w down t h e d e n s i t y g r a d i e n t . f l o w s t e n d t o x l o w e r p . M . h i g h e r p Vp +ve v o r t i c i t y The e f f e c t o f t h e buoyancy t e r m I f an h o r i z o n t a l d e n s i t y g r a d i e n t o f t h e same s i g n o c c u r s a t a l l d e p t h s i n a f l u i d , t h e p r e s s u r e g r a d i e n t s a t t h e b o t t o m w i l l be l a r g e r t h a n t h o s e n e a r e r t h e s u r f a c e due t o t h e s t r o n g v e r t i c a l d e n s i t y g r a d i e n t ( -pg under the hydrostatic approximation, valid for the tank and lake). In the tank gradients in the directions shown above are set up on the shallow side due to the effects of surface heating. Flow towards the shallow end results i n a slight surface slope downwards towards the bar which i s sufficient to produce the flow i n the positive x-direction i n the upper regions of the tank. In the deeper end the density gradients are reversed, resulting i n flows in the opposite directions (and vorticity of the opposite sign). Because of viscosity a shear develops which opposes the flow, producing vorticity of the opposite sign to that produced by the buoyancy term. This occurs mainly on the boundaries of the tank although the inter i o r contribution, for the low Reynolds number flow considered, is also important. Consider the operator V1* = 31* + 23 23 2 + 31* in equation 3.2.3. z x z X Since the v e r t i c a l shears greatly exceed horizontal shears almost everywhere i n the tank (with the main exception of side walls and the region of the bar i t s e l f ) , I neglect the last two terms of V and approximate equation 3.2.3 by vortici t y shallow end i _ 3 f i = IIP + v i ^ i (3.2.4) 3t 3z 2 p 3x dzh Because the flow changes only slowly, i t appears that the buoyancy term and the viscous term roughly balance each other. Thus equation 3.2.4 may be approximated, at least heuristically, by i ! i = _ _ 8 _ i £ (3.2.5) dz1* V p 3x Given a density f i e l d , p(t,x,z) and appropriate boundary conditions, equation 3.2.5 may be solved for the stream function and hence for the flow pattern of this model. The density of fresh water in the region of i t s maximum density may be expressed as p = p ( 1 -• A 6 2 ) (3.2.6) —3 —6 —2 where p^ is the density i n gm cm at 4°C and A - 8 x 10 C° Substituting 3.2.6 into equation 3.2.5 gives 1 ^ - 2 g A e ^ (3.2.7) dzk v 3x where I have used p - p when not differentiated. 43 The b o u n d a r y c o n d i t i o n s a r e s i m p l y t h a t u and w v a n i s h a t t h e top and t h e b o t t o m ( r e c a l l i n g t h e e f f e c t o f s u r f a c e c o n t a m i n a t i o n ) : a t z = 0, u = 0 and w = 0 (3.2.8) and a t z = -D, u = 0 and w = 0 (bottom) where D = ax where a i s t h e s l o p e o f t h e b o t t o m . S p e c i f y i n g a s u i t a b l e t e m p e r a t u r e f i e l d i s n o t s u c h a s i m p l e m a t t e r . O b v i o u s l y e q u a t i o n 3.2.7 cannot be used w i t h a t e m p e r a t u r e f i e l d g i v e n s e p a r a t e l y f o r t h e two s i d e s o f the b a r as i t w o u l d n o t be p o s s i b l e t o match t h e d i s p l a c e m e n t s and s t r e s s e s a c r o s s t h e b a r . The t e m p e r a t u r e f i e l d 3.1.13 s u g g e s t e d p r e v i o u s l y (p.35) i s t h u s n o t o n l y awkward, due t o t h e p r e s e n c e o f t h e e r r o r f u n c t i o n , b u t unmanageable f o r i n t e g r a t i o n o f e q u a t i o n 3.2.7 t h r o u g h o u t t h e model. The s o l u t i o n w i l l b e q u i t e s e n s i t i v e t o t h e chosen t e m p e r a t u r e f i e l d , as may be s e e n f r o m the p r e s e n c e o f t h e p r o d u c t o f 6 and 8^ i n what i s e s s e n t i a l l y t h e f o r c i n g t e r m ( s e e e q u a t i o n 3.2.7). Thus much c a r e and t r i a l was n e c e s s a r y t o f i n d f o r t h e t e m p e r a t u r e f i e l d a r e a s o n a b l e compromise w h i c h w o u l d be b o t h r e a d i l y i n t e g r a b l e and a r e a s o n a b l e r e p r e s e n t a t i o n o f t h e e x p e r i m e n t a l r e s u l t s . L o o k i n g a t t h e p l o t s ( F i g u r e 15, p.38) o f 9 v e r s u s x f o r t h e t a n k and f o r t h e t e m p e r a t u r e f i e l d as g i v e n : b y ^ e q u a t i o n s 3.1.13 s u g g e s t s t h e use o f h y p e r b o l i c t a n g e n t f u n c t i o n s . The compromise w h i c h f i n a l l y p r o d u c e d a r e a s o n a b l e f i t t o t h e v e l o c i t y f i e l d f o r t h e b a r a t t h e p o s i t i o n shown i n t h e g e n e r a l i z e d o b s e r v a t i o n s ( F i g u r e 6, p.15) ( t h a t i s , a t 90 cm and a 5° b o t t o m s l o p e ) i s 9 * { [ 1 + t a n h ( 0 . 3 9 2 [ - 2 ] ) ] ( 2 . 9 ) ( 1 + 0.2z + O . O l z 2 ) - 1} ( 3 . 2 . i n C°. T h i s i s p l o t t e d i n c r o s s - s e c t i o n i n °C and as 9 v e r s u s x a t c o n s t a n t z's i n F i g u r e 16. As can be s e e n t h i s p r o d u c e s a r e a s o n a b l e a p p r o x i m a t i o n t o b o t h t h e o b s e r v e d ( F i g u r e 6, p.15 and F i g u r e 15, p.38) and t h e c a l c u l a t e d ( F i g u r e 14, p.37, and F i g u r e 15, p.38) t e m p e r a t u r e f i e l d s . The a n a l y t i c a l t e m p e r a t u r e f i e l d , e q u a t i o n 3.2.9, was t h e n used i n e q u a t i o n 3.2.7 and t h e s t r e a m f u n c t i o n and t h e v e l o c i t y f i e l d were c a l c u l a t e d ( s e e A p p e n d i x E ) . The r e s u l t i n g v e l o c i t y f i e l d i s shown i n F i g u r e s 17 a and b. As i n F i g u r e 6 (p.15) t h e l e n g t h s o f t h e a r r o w s i n d i c a t e m o t i o n e x p e c t e d i n a m i n u t e ( i . e . cm min ^) i n t h e same s c a l e s as used f o r t h e ax e s . Comparison o f F i g u r e s 17a and 6 shows c l o s e agreement between the two. A g a i n i n F i g u r e s 17 t h e t e m p e r a t u r e f i e l d has been i n c l u d e d (dashed l i n e s ) . The v e l o c i t i e s a r e a l l i n t h e c o r r e c t d i r e c t i o n s and t h e change i n d i r e c t i o n o f t h e f l o w s a l o n g t h e tank, o c c u r s s l i g h t l y b e h i n d t h e s u r f a c e 4°C i s o t h e r m as was s u g g e s t e d by t h e l a b o r a t o r y model s t u d i e s . The l a r g e s t downward v e l o c i t i e s o c c u r i n t h e r e g i o n o f maximum d e n s i t y as was o b s e r v e d i n t h e t a n k . I n g e n e r a l t h e v e l o c i t i e s a r e o f t h e same magnitude as t h o s e o b s e r v e d . The d i f f e r e n c e s between the c a l c u l a t e d and o b s e r v e d v e l o c i t y f i e l d s can be e x p l a i n e d f i r s t i n terms o f t h e i n a d e q u a c i e s o f t h e h y p e r b o l i c t a n g e n t t e m p e r a t u r e a p p r o x i m a t i o n . The magnitude o f t h e c a l c u l a t e d X (cm) Figure 16. Analytical temperature approximation, equation 3.2.9, used for the velocity calculations: (a) cross-section (T in °C) (b) 9 in C° against x in cm for fixed values of z (in cm) X ( c m ) Figure 17a. Calculated v e l o c i t y f i e l d . V e l o c i t i e s are shown by arrows, t h e i r length i n d i c a t i n g the motion i n a minute ( i . e . cm min" 1) i n the same scales as used f o r the axes. The a n a l y t i c a l temperature f i e l d used i s shown by dashed l i n e s (°C). The s o l i d curves are the h o r i z o n t a l v e l o c i t y p r o f i l e s . Compare with Figure 6 (p.15). ON X ( c m ) 0 -2 ~ - 4 E u NI •8 10- ISOTHERMS (°C) VERTICAL VELOCITY Figure 17b. Calculated velocity f i e l d . Velocities are shown by arrows, their length indicating the motion in a minute (i.e. cm min" ) in the same scales as used for the axes. The analytical temperature f i e l d used i s shown by dashed lines (°C). The s o l i d curves are for the vertical velocity. v e l o c i t i e s drops o f f a t t h e e x t r e m a o f t h e p l o t s i n c e t h e h y p e r b o l i c t a n g e n t a p p r o x i m a t i o n a t t a i n s maximum and minimum v a l u e s b e f o r e t h e ends o f t h e t a n k a r e r e a c h e d , whereas t h e o b s e r v e d t e m p e r a t u r e s d i d n o t . The v e l o c i t i e s between 50 and 70 c e n t i m e t e r s a l o n g t h e t a n k a r e up t o t w i c e as l a r g e as t h o s e shown i n the g e n e r a l i z e d v e l o c i t y f i e l d ( F i g u r e 6, p . 1 5 ) . T h i s can be a t t r i b u t e d t o t h e f a c t t h a t 90/9x i s l a r g e r t h a n t h a t f o r t h e h y p e r b o l i c t a n g e n t t e m p e r a t u r e model i n t h i s r e g i o n . An even c l o s e r match t o t h e o b s e r v e d g e n e r a l i z e d v e l o c i t y f i e l d c o u l d p r o b a b l y be made by u s i n g a more i n v o l v e d t e m p e r a t u r e f i e l d , however t h e r e i s l i t t l e p o i n t i n s u c h a t e d i o u s e n d e a v o u r i n t h a t t h e t e m p e r a t u r e f i e l d s o b s e r v e d i n t h e tan k do v a r y between runs ( s e e F i g u r e 18) and t h e a s s o c i a t e d v e l o c i t y f i e l d s w o u l d hence a l s o v a r y somewhat f r o m t h e a v e r a g e v a l u e s shown i n F i g u r e 6 ( p . 1 5 ) . A l s o t h e o b s e r v e d t e m p e r a t u r e s e c t i o n s a r e n o t an i n s t a n t a n e o u s measurement b u t r a t h e r t h e t e m p e r a t u r e s were t a k e n i n sequence f r o m t h e deep end t o t h e s h a l l o w end d u r i n g w h i c h t i m e t h e b a r p r o g r e s s e d about 10 cm. V a r i a t i o n s i n t h e t e m p e r a t u r e f i e l d s i m i l a r t o t h o s e f o u n d i n the t a n k h ave b e e n o b s e r v e d f o r t h e b a r i n Lake O n t a r i o ( R o d g e r s , 1968). I t ought t o be n o t e d t h a t a d i f f e r e n t h y p e r b o l i c t a n g e n t a p p r o x i m a t i o n w i l l need t o be made f o r any o t h e r p o s i t i o n o f t h e b a r t h a t i s n o t w i t h i n a b o ut 10 cm o f t h e p o s i t i o n chosen f o r t h e c a l c u l a t i o n . T h i s i s due t o the f a c t t h a t t h e t e m p e r a t u r e f i e l d changes w i t h t i m e , p a r t i c u l a r l y i n ma g n i t u d e . Such a p p r o x i m a t i o n s c o u l d be b a s e d on the t e m p e r a t u r e f i e l d p r e d i c t e d by t h e m a t h e m a t i c a l model ( e q u a t i o n s 3.1.13). The s t r e a m f u n c t i o n cf) i s p l o t t e d i n F i g u r e 19. There i s no n e t t r a n s p o r t a c r o s s any v e r t i c a l s e c t i o n o f t h e t a n k , s o t h a t <j> can be s e t -2 20 40 60 80 100 120 X ( c m ) Figure 18. Actual 6 ( in C° ) against distance along tank ( x, in cm ) plotted from laboratory observations. o 51 e q u a l t o z e r o a t the top and b o t t o m . F i g u r e 19 must be i n t e r p r e t e d w i t h c a r e . One must keep i n mind t h a t t h i s i s t h e p i c t u r e o f cJ) a t t = 90/S i n t h e ta n k . As t i m e i n c r e a s e s h e a t i n g c o n t i n u e s , t h e t e m p e r a t u r e f i e l d changes and t h e ' b a r ' d e f i n e d as t h e 4°C s u r f a c e i s o t h e r m p r o g r e s s e s down t h e t a n k . Thus t h e d e n s i t y d r i v e n v e l o c i t y f i e l d i s a l s o moving down t h e t a n k , c h a n g i n g s l o w l y . I n terms o f t h e s t r e a m f u n c t i o n cj) ( F i g u r e 19) t h i s means t h a t t h e two s e t s o f c l o s e d s t r e a m l i n e s a r e moving a l o n g t h e ta n k a t t h e spee d o f t h e ' b a r ' and hence must be s l o w l y e x p a n d i n g v e r t i c a l l y . The a s s u m p t i o n , used t o d e r i v e t h e c i r c u l a t i o n , t h a t the 3/9t t e r m i n e q u a t i o n 3.2.4 can be n e g l e c t e d i s thus o n l y v a l i d when changes a r e s l o w enough. The n e g a t i v e s e t o f s t r e a m l i n e s on t h e s h a l l o w s i d e must a l s o s l o w l y expand h o r i z o n t a l l y w h i l e t h e p o s i t i v e s e t on t h e deep s i d e must be s l o w l y s h r i n k i n g . W i t h t h i s i n mind one can s e e t h a t t h i s p i c t u r e does n o t c o n t r a d i c t t h e l a b o r a t o r y o b s e r v a t i o n s w h i c h c l e a r l y showed t h a t w a t e r f r o m t h e warmer, s t a b l e t h e r m o c l i n e was n o t i n v o l v e d i n any m i x i n g o r downward f l o w a t t h e b a r . T h i s s t a b l e t h e r m o c l i n e i n v o l v e s the w a t e r a t t e m p e r a t u r e above 4.5°C. L o o k i n g a t $ ( F i g u r e 19, p.49) and t h e v e l o c i t y f i e l d ( F i g u r e 17b, p.47) fr o m t h e m a t h e m a t i c a l model, and r e c a l l i n g t h a t t h e a v e r a g e b a r s p e e d u s e d i n t h i s model i s about 2.5 cm min \ i t can be s e e n t h a t t h e downward v e l o c i t i e s i n t h e r e g i o n warmer th a n 4.5°C s e r v e o n l y t o t h i c k e n t h e t h e r m o c l i n e r e g i o n . I n f a c t , b y the ti m e t h e b a r has t r a v e l l e d r o u g h l y a f u r t h e r 10 cm down t h e t a n k , t h e w v e l o c i t i e s o b s e r v e d a t , s a y , 70 cm, w i l l p r o b a b l y a l l be p o s i t i v e . The f l o w u n d e r n e a t h t h e warmer w a t e r on t h e s h a l l o w s i d e i s f e d by t h e f l o w towards t h e b a r f r o m t h e dee p e r s i d e . T h i s model i s n o t s t e a d y s t a t e , thus streamlines are not pathlines. The streamlines shown in Figure 19 (p.49) close more rapidly at the two ends of the tank in the mathematical model than they probably do i n the laboratory experiments since the hyperbolic tangent temperature approximation has reached i t s maximum and minimum values before the ends of the tank, whereas the observed temperatures has not. Such more gradual closings of the streamlines would tend to decrease the magnitudes of w away from the region of the bar. 3.3 Validity of the Velocity Model Before extending the mathematical model to compare i t with the lake i t i s necessary to consider the size of the neglected terms i n the vorticity equation. Now including advection terms, 3.2.2 becomes ^ + v . Vu, = -S^- + v ( - ^ - + -^-W (3.3.1) 3t ~ p 3x 9x 2 9 z 2 1 3u. 3w where u)„ = - - r — . The size of these terms was evaluated at several 2 3z dx positions along the tank from the velocity f i e l d calculated from equation 3.2.7. This thus provides an internal consistency check on the solution of equation 3.2.7 and therefore on the assumptions used to derive i t . 3w 3u 1) - j ^ with respect to is always less than 1:10 and usually less than 3w 3 xx 1:100. Thus the assumption that TT- is small with respect to -r— dx dz is reasonable everywhere. 9 2u 9 2 2) In the viscous term was neglected with respect to -^z . The ratio of these two terms is also 1:10 or less everywhere. 3) The 9/9t t e r m can be compared t o t h e buoyancy term and i s f o u n d t o be g e n e r a l l y l e s s t h a n 1:10 e x c e p t , o f c o u r s e , where t h e buoyancy term i s z e r o ; t h a t i s , r i g h t a t t h e b a r (8 = 0) and a t the e x t r e m a 99 o f t h e h y p e r b o l i c t a n g e n t t e m p e r a t u r e a p p r o x i m a t i o n where = 0. The l a t t e r l i e s w e l l t o t h e e x t r e m a o f t h e p l o t s . I t i s p o s s i b l e t o c a l c u l a t e t h e s i z e o f the 9/9t t e r m s i n c e t h e buoyancy t e r m depends on t i m e . 4) N o n - l i n e a r terms a r e g e n e r a l l y l e s s t h a n one t e n t h o f the buoyancy t e r m e x c e p t , o f c o u r s e , where t h e buoyancy t e r m i s z e r o , as m e n t i o n e d above. I n t h e i m m e d i a t e v i c i n i t y o f t h e 4°C i s o t h e r m (0 = 0) the v o r t i c i t y b a l a n c e w o u l d s h i f t t o i n c l u d e 9/9t, a d v e c t i v e and v i s c o u s t erms. Thus t h e a s s u m p t i o n s made i n d e r i v i n g t h e m a t h e m a t i c a l model a p p e a r t o be r e a s o n a b l e , as t h e c o m p a r i s o n between t h e c a l c u l a t e d and o b s e r v e d v e l o c i t y f i e l d s w o u l d s u g g e s t and t h e model p r o v i d e s a u s e f u l a p p r o x i m a t i o n t o t h e c i r c u l a t i o n . 3.4 E x t e n s i o n t o a Lake The l a b o r a t o r y s t u d i e s s u g g e s t e d t h a t t h e t a n k c o u l d be v i e w e d as a s e c t i o n o f a l a k e f r o m s h o r e t o d e e p e r r e g i o n s u s i n g a v e r t i c a l s c a l i n g o f a bout 1/1000 ( s e e p . 1 2 ) . The 5° b o t t o m s l o p e , compared t o t h e n o r t h s h o r e b o t t o m s l o p e s o f Lake O n t a r i o , t h e n s u g g e s t s a h o r i z o n t a l s c a l i n g o f a b o u t 1/20,000. The a v e r a g e s p e e d o f the b a r i n Lake O n t a r i o i s about 20 t i m e s t h a t i n the t a n k w i t h t h e 5° b o t t o m s l o p e . U s i n g t h e s e s c a l e s f o r z, x , and t and a s s u m i n g t h a t t h e e q u a t i o n s (3.1.13 and 3.2.7) u s e d can be a p p l i e d t o t h e l a k e s s i m p l y by making t h e a p p r o p r i a t e s c a l e changes, 2 -1 t h e v a l u e o f v e r t i c a l eddy d i f f u s i v i t y must be about 6 cm s e c and 2 -1 v e r t i c a l eddy v i s c o s i t y about 38 cm s e c . Now i t i s n o t u n r e a s o n a b l e t h a t t h e s e v a l u e s be d i f f e r e n t , e s p e c i a l l y i n a s t a b l e s i t u a t i o n as on t h e s h a l l o w s i d e o f t h e ' b a r ' . I n f a c t , e x p e r i m e n t a l v a l u e s g i v e n f o r 2 - 1 Lake M i c h i g a n (Huang, 1969) a r e 1-10 cm s e c f o r v e r t i c a l eddy d i f f u s i v i t y 2 - 1 and 1-100 cm s e c f o r v e r t i c a l eddy v i s c o s i t y . A r a t i o o f 1:2 ( v e r t i c a l eddy d i f f u s i v i t y t o v e r t i c a l eddy v i s c o s i t y ) m i g h t be more r e a l i s t i c . I f 2 -1 t h e v e r t i c a l eddy d i f f u s i v i t y i s a r b i t r a r i l y s e t a t 11 cm s e c , w h i c h , when a p p l i e d t o e q u a t i o n 3.1.13 does n o t g i v e an u n r e a l i s t i c t e m p e r a t u r e f i e l d , t h e c o r r e s p o n d i n g v e r t i c a l eddy v i s c o s i t y , f r o m e q u a t i o n 3.2.7, i s 0 1 2 -1 21 cm s e c . Such an e x t e n s i o n as t h i s s h o u l d , o f c o u r s e , o n l y be a p p l i e d t o l a r g e d i m i c t i c f r e s h w a t e r l a k e s , s u c h as t h e G r e a t L a k e s , i n w h i c h t h e b a r c o u l d be e x p e c t e d t o e x i s t . However, when e q u a t i o n 3.2.7 i s a p p l i e d t o a l a k e t h e r e a r e a d d i t i o n a l terms t h a t h ave been n e g l e c t e d , t h o s e a s s o c i a t e d w i t h t h e C o r i o l i s f o r c e . I n t h e v o r t i c i t y e q u a t i o n 3.2.1 t h e s e a r e i n c l u d e d by r e p l a c i n g O J, t h e r e l a t i v e v o r t i c i t y , by (u) + 2fi) where 20, i s the ' p l a n e t a r y v o r t i c i t y ' . From V e r o n i s ' (1963) e s t i m a t e s , b a s e d on E r t e l ' s c i r c u l a t i o n t heorem as a p p l i e d t o a s t r a t i f i e d f l u i d , t h e r a t i o o f h o r i z o n t a l t o v e r t i c a l s c a l e s i n t h e t h e r m a l b a r phenomenon i s s u c h t h a t t h e component o f p l a n e t a r y v o r t i c i t y p a r a l l e l t o t h e e a r t h ' s s u r f a c e may be s a f e l y n e g l e c t e d . B e c a u s e o f t h e p r e s e n c e o f t h e v e r t i c a l component o f p l a n e t a r y v o r t i c i t y ( f ) , t h e d e n s i t y i n d u c e d p r e s s u r e g r a d i e n t s i n a l a k e w i l l t e n d t o be 55 b a l a n c e d by a g e o s t r o p h i c c u r r e n t . However a t i m e o f 0 ( l / f ) i s n e c e s s a r y t o a c h i e v e s u c h an e q u i l i b r i u m , d u r i n g w h i c h t h e p o s i t i o n o f the b a r changes a p p r e c i a b l y . T h i s l e a d s me t o b e l i e v e t h a t g e o s t r o p h i c e q u i l i b r i u m may n o t be a t t a i n e d w h i l e t h e b a r i s moving and t h a t on a s h o r t t i m e s c a l e t h e c i r c u l a t i o n i s r o u g h l y due t o t h e same dynamic b a l a n c e as i n t h e model. A f o s s i l g e o s t r o p h i c c i r c u l a t i o n due t o t h e d e n s i t y g r a d i e n t s a s s o c i a t e d w i t h t h e b a r may c o n t i n u e t o e x i s t a f t e r t h e b a r has d i s a p p e a r e d . The v a l i d i t y o f e q u a t i o n 3.2.7 was c h e c k e d f o r Lake O n t a r i o by e v a l u a t i n g , f r o m t h e s c a l e d r e s u l t s o f 3.2.7, th e s i z e o f t h e n e g l e c t e d terms i n t h e s h a l l o w and d e e p e r r e g i o n s and i n t h e v i c i n i t y o f t h e ' b a r ' . I t i s p o s s i b l e t h a t t h e r e l a t i v e s i z e o f t h e s e terms may be d i f f e r e n t f r o m t h o s e i n t h e t a n k s i n c e t h e r e a r e d i f f e r e n t s c a l e s used f o r h o r i z o n t a l and v e r t i c a l d i s t a n c e s as w e l l as t i m e . I t was f o u n d t h a t t h e b a s i c b a l a n c e between t h e buoyancy and t h e a p p r o x i m a t e d v i s c o u s e f f e c t s as used i n e q u a t i o n 3.2.7 i s v a l i d f o r t h e l a k e , e x c e p t a g a i n r i g h t i n t h e v i c i n i t y o f t h e b a r . The n o n - l i n e a r terms a r e more i m p o r t a n t n e a r t h e b a r i n Lake O n t a r i o t h a n t h e y were i n t h e t a n k . N e v e r t h e l e s s t h e m a t h e m a t i c a l model p r o b a b l y g i v e s a r e a s o n a b l e , i f r o u g h , a p p r o x i m a t i o n o f what o c c u r s i n the l a k e s s i n c e t h e a s s u m p t i o n s a r e n e a r l y s a t i s f i e d , e x c e p t f o r t h e z e r o s u r f a c e v e l o c i t i e s ( p . 1 6 ) . C o n s i d e r i n g t h e p l o t o f t h e c a l c u l a t e d s t r e a m l i n e s , F i g u r e 19(p.49) i n t h e l i g h t o f a p o s s i b l e e x t e n s i o n t o l a k e s i t i s w o r t h r e m a r k i n g on t h e s i t u a t i o n o f t h e ' s t a t i o n a r y t h e r m a l b a r ' ; f o r e x a m p l e , t h e c a s e o f Lake Geneva ( F o r e l , 1880) ( m e n t i o n e d e a r l i e r ) o r t h e c a s e o f e f f l u e n t s warmer t h a n 4°C b e i n g dumped i n t o a l a k e t h a t i s c o o l e r t h a n 4°C ( s e e s e c t i o n 2, p.29 and A p p e n d i x D). As o b s e r v e d i n t h e l a b o r a t o r y , F i g u r e 19(p.49) s u g g e s t s two c e l l s , i n t h i s c a s e f i x e d i n p o s i t i o n , w i t h c l o c k w i s e m o t i o n on the shallow side and counter-clockwise motion on the deep side and mixing and sinking between the two cell s . It must be emphasized that this is NOT what has been observed for the 'migrating thermal bar' in the laboratory model. Streamlines are pathlines only when flow is stationary.' It would appear from the way in which the mathematical model as well as the tank can approximate the temperature fields observed in Lake Ontario, that the velocity f i e l d observed in the laboratory and calculated from a simple vorticity balance which also holds for the lake might also be that associated with the 'bar' in lakes. However i t w i l l probably be very d i f f i c u l t to determine this by direct measurements as the velocities expected, for example i n Lake Ontario, are only of the order of 1 cm sec and there are many other flows present i n lakes that are not directly associated with the thermal bar. 57 4. SUMMARY The 'migrating thermal bar' phenomenon which is known to occur in certain large dimictic freshwater lakes has been studied i n a two- dimensional laboratory model. The temperature fields agree with those observed i n the Great Lakes. A linear model is used to describe the speed of the 'thermal bar'. The effects of variation of the parameters of heat input, bottom slope, and i n i t i a l temperature on the speed of the bar were measured and found to agree with this simple model. The linear model gives a reasonable f i r s t approximation to the speed of the 'thermal bar' in both the experimental model and the Great Lakes. Since the observed temperature f i e l d and speed of the bar appear to model conditions i n the lakes i t seems possible that the associated observed velocity fields may do this as well. On the shallow side of the 'migrating thermal bar' temperature observations showed that a stable thermocline developed which progressed towards the deep end, i t s front end marking the boundary between the stable shallower region and the convecting deeper region. In the laboratory model a mean flow towards the deep end was observed i n the thermocline with a counter-flow underneath. In the deep end of the model a flow near the surface travelled toward the stable thermocline. This current sank at the front edge of the thermocline and divided, a part feeding the upslope current in the shallow end, the other part providing the downslope current i n the deep end. The water in this sinking zone or 'bar zone' was from the deep end of the model and did not include water from the thermocline 58 on the shallow side. On the basis of this laboratory model which indicated that horizontal advection and diffusion were not of primary importance a mathematical model was developed. F i r s t the temperature f i e l d was calculated from the one-dimensional heat diffusion equation. Then the velocity f i e l d was calculated assuming that the flow was driven by buoyancy forces and balanced by viscous forces. Since there is a great similarity between the calculated and observed temperature and velocity f i e l d s , the assump- tions on which the vorticity balance i s based are obviously nearly sa t i s f i e d in the laboratory model. Because of the similitude between the experimental and calculated temperature fields and those observed in lakes, the observed and calculated velocity f i e l d may model the flows associated with the thermal bar i n the lakes. In this case the balance would be between buoyancy forces and eddy viscosity effects except possibly i n the immediate v i c i n i t y of the 'bar' where non-linear terms would be important. The velocities expected i n Lake Ontario would be of the order of 1 cm sec \ From the laboratory and mathematical studies i t is also possible to describe the behaviour associated with the 'stationary thermal bar' which would be expected i n the case of waters warmer than 4°C being dumped into waters cooler than 4°C. In this case the 'bar' would be expected not to move and the circulation would consist of two cells with mixing and sinking at the 'bar'. This is quite different from the behaviour observed for the 'migrating thermal bar' i n the laboratory. I 59 BIBLIOGRAPHY A r l i n g e r , B., 1965. C a l c u l a t i o n o f t e m p e r a t u r e i n an i n f i n i t e wedge w i t h g i v e n h e a t f l u x t h r o u g h i t s b o u n d i n g s u r f a c e s . 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Handbook o f C h e m i s t r y and P h y s i c s , 1969-70. The C h e m i c a l Rubber Co., C l e v e l a n d , O h i o , page F-4. Huang, C.K., 1969. The t h e r m a l c u r r e n t s t r u c t u r e i n Lake M i c h i g a n , A t h e o r e t i c a l and o b s e r v a t i o n a l model s t u d y . U n i v e r s i t y o f M i c h i g a n , G r e a t Lakes Res. D i v . , S p e c i a l R e p o r t No. 43, 169pp. R i c h a r d s , T.L., J.G. I r b e and D.G. Massey, 1969. A e r i a l s u r v e y s o f G r e a t Lakes w a t e r t e m p e r a t u r e s A p r i l , 1966 t o March, 1968. Department o f T r a n s p o r t , M e t e o r o l o g i c a l B r a n c h , C l i m a t o l o g i c a l S t u d i e s No. 14, 55p. 60 , and , 1969-1970. U n p u b l i s h e d d a t a . Rodgers, G.K., 1965a. The t h e r m a l b a r i n t h e L a u r e n t i a n G r e a t L a k e s . P r o c . 8 t h Conf. G r e a t Lakes Res.; U n i v . M i c h i g a n , G r e a t Lakes Res. D i v . Pub. 13, 358-363. , 1965b. U n p u b l i s h e d d a t a . __, 1966a. The t h e r m a l b a r i n Lake O n t a r i o , S p r i n g 1965 and W i n t e r 1965-66. P r o c . 9 t h Conf. G r e a t Lakes Res.; U n i v . M i c h i g a n , G r e a t Lakes Res. D i v . Pub. 15, 369-374. , 1966b. A n o t e on t h e r m o c l i n e development and t h e t h e r m a l b a r i n Lake O n t a r i o , Symposium o f Garda, I.A.S.H. Pub. No. 70, 401-405. , 1967. T h e r m a l regime and c i r c u l a t i o n i n t h e G r e a t L a k e s . R o y a l S o c i e t y o f Canada, Water R e s o u r c e s o f Canada S y m p o s i a , 87-95. , 1968. Heat a d v e c t i o n w i t h i n Lake O n t a r i o i n s p r i n g and s u r f a c e w a t e r t r a n s p a r e n c y a s s o c i a t e d w i t h the t h e r m a l b a r . P r o c . 1 1 t h Conf. G r e a t Lakes Res.; I n t e r n a t i o n a l A s s o c . G r e a t Lakes Res., 480-486. and D.V. A n d e r s o n , 1961. A p r e l i m i n a r y s t u d y o f the e n e r g y b u d g e t o f Lake O n t a r i o . J . F i s h . Res. Bd. Canada, 18, 617-636. and , 1963. The t h e r m a l s t r u c t u r e o f Lake O n t a r i o . P r o c . 6 t h Conf. G r e a t Lakes Res.; U n i v . M i c h i g a n , G r e a t Lakes Res. D i v . Pub. 10, 59-69. S a t o , G.K., 1969. P r e d i c t i o n o f t h e t i m e o f d i s a p p e a r a n c e o f t h e t h e r m a l b a r i n L a k e O n t a r i o , U n i v . T o r o n t o , M.A.Sc. d i s s e r t a t i o n , 114pp. S v e r d r u p , H.U., M.W. J o h n s o n and R.H. F l e m i n g , 1942. The Oceans. P r e n t i c e - H a l l , I n c . , N.Y., 80, 81, 105. T i k h o m i r o v , A . I . , 1963. The t h e r m a l b a r o f Lake Ladoga. B u l l . ( I z v e s t i y a ) A l l - U n i o n Geogr. S o c , 9 5 , 134-142. Am. Geophys. U n i o n t r a n s l a t i o n , S o v i e t H y d r o l o g y : S e l e c t e d P a p e r s , No. 2. V e r o n i s , G., 1963. On t h e a p p r o x i m a t i o n s i n v o l v e d i n t r a n f o r m i n g the e q u a t i o n s o f m o t i o n f r o m a s p h e r i c a l s u r f a c e t o t h e 8-plane. I I . B a r o c l i n i c s y s t e m s . J . M a r i n e Res., 2_1, 199-204. APPENDIX A: C o o l i n g E x p e r i m e n t F i g u r e 20 shows a t e m p e r a t u r e s e c t i o n and t e m p e r a t u r e anomaly s e c t i o n f o r s u r f a c e c o o l i n g o f w a t e r i n i t i a l l y above 4°C. F o r t h e t e m p e r a t u r e anomaly ( s e e p.24) s e c t i o n i t was n e c e s s a r y t o e s t i m a t e t h e h e a t l o s s from t h e t e m p e r a t u r e s e c t i o n . The s e c t i o n i s s i m i l a r t o measurements made i n Lake O n t a r i o ( F i g u r e 2 0 ) . The f l o w p a t t e r n o b s e r v e d i n t h e l a b o r a t o r y e x p e r i m e n t was the same as t h a t e n c o u n t e r e d i n t h e h e a t i n g e x p e r i m e n t s ( s e e F i g u r e 6, p . 1 5 ) . DISTANCE from NORTH S H O R E (km) DISTANCE along T A N K ( c m ) Figure 20. Temperature se c t i o n from Lake Ontario (Rodgers, 1966a); mid-lake N-S s e c t i o n taken mid-January 1966. Temperature and temperature anomaly sections f o r the case of cooling ( f a l l simulation) with 5° bottom slope. APPENDIX B: Temperature Data: constant slopes Figures 21 to 27 are temperature data f o r runs simulating s p r i n g warming. They include a temperature s e c t i o n , mean temperatures, changes i n heat content, and a temperature anomaly s e c t i o n (p.24) f o r d i f f e r e n t slopes, d i f f e r e n t heating rates, and d i f f e r e n t p o s i t i o n s of the 'thermal bar'. Also shown are the curves f o r mean temperature and change i n heat content predicted by the assumptions used i n the l i n e a r model f o r the speed of the bar. There i s a p o s s i b l e f u r t h e r p a r a l l e l between the tank and the observations i n Lake Ontario (Rodgers 1968); the advection of heat towards the deep end of the tank or centre of the lake appears to occur mainly during the ea r l y stages when the thermal bar i s near the shore (Figure 28 p.77, and Figure 12, p.25). However th i s i s purely speculation as some of th i s could be due to the nonuniform d i s t r i b u t i o n of heat f l u x over the surface of the Lake. R U N B Observations 0 ' 40 L _ 8 0 ' 120 "cm F i g u r e 21a. Tempera t u r e s e c t i o n and mean t e m p e r a t u r e s ( v e r t i c a l l y a veraged) f o r Run B: 5° b o t t o m s l o p e , s t a n d a r d h e a t i n g , and T i , 0°C. 4 65 Figure 21b. Changes in heat content and temperature anomaly section for Run B Figure 22a. Temperature s e c t i o n and mean temperatures f o r Run I: 5° bottom slope, standard heating, and T i , 1.4°C. 67 Figure 22b. Changes in heat content and temperature anomaly for Run I. F i g u r e 23a. T e m p e r a t u r e s e c t i o n and mean t e m p e r a t u r e s f o r Run H: 5° b o t t o m s l o p e , i n c r e a s e d h e a t i n g and T i , 0°C. 69 RUN H • E u o w ^measured ~ 40 - ^ %-tC* — Z ^ - _ - * - - x UJ / z predicted^ o 2 0 | - BAR < Ul , X <3 ° 0 40 80 120 Cm Observations 0 40 80 120 cm D I S T A N C E along T A N K Figure 23b. Changes in heat content and temperature anomaly for Run H. Figure 24a. Temperature s e c t i o n and mean temperatures f o r Run C: 2.5° bottom slope, standard heating and T i , 0°C. o RUN C i i i i i i i i 0 40 80 120 cm 0 I S T A N C E along T A N K Figure 24b. Changes in heat content and temperature anomaly for Run C Observat ions RUN D 0« 1 1 1 1 i i i 0 40 80 120 cm D I S T A N C E along T A N K F i g u r e 25a. Temperature s e c t i o n and mean t e m p e r a t u r e s f o r Run D: 7.5° b o t t o m s l o p e , s t a n d a r d h e a t i n g and T i , 0°C. RUN D Figure 25b. Changes in heat content and temperature anomaly for Run D F i g u r e 26a. T e m p e r a t u r e s e c t i o n and mean t e m p e r a t u r e s f o r Run E: 7.5° b o t t o m s l o p e , s t a n d a r d h e a t i n g and T i , 0°C. 75 RUN E i i i i i 1 1 1— 0 40 80 120 cm DISTANCE along T A N K Figure 26b. Changes in heat content and temperature anomaly for Run E 76 RUN F Observations i i i 1 1 1 1 1 — 0 40 80 120 Cm Figure 27a. Temperature section and mean temperatures for Run F: 7.5° bottom slope, increased heating and T i , 0°C. 77 R U N F i 1 i i i i i i 0 40 80 120 Cm D I S T A N C E a l o n g T A N K Figure 27b. Changes i n heat content and temperature anomaly for Run F 1 o S20h Ul i 1 0 o < Ui X <» 0. LAKE ONTARIO S E C T I O N E - N ^ _ _ — - * "̂ ^ v̂. A ^ ^ - M E A N for L A K E ^ - S E C T I O N D BAR J — * J BAR 10 20 30 D I S T A N C E from N O R T H S H O R E ( k m ) 40 E u o 40h u O o < Ul 20 EXPERIMENT MEAN for TANK R U N D R U N B ^ 7 " BAR 40 80 120 D I S T A N C E along T A N K ( c m ) Figure 28. Changes in heat content during the early stages of the thermal bar for Lake Ontario (for sections in Figure 2, p.4) and for the tank experiment. 00 79 APPENDIX C: Investigation of the E f f e c t s of Bottom I r r e g u l a r i t i e s Figures 29 and 30 contain temperature data s i m i l a r to that i n Appendix B f o r the cases of a 5° slope followed by a 0° slope and a 5° slope followed by a v e r t i c a l dropoff to a 0° slope. Figure 31 shows the bar speeds from these experiments. The r e s u l t s are a l l s i m i l a r to the previous cases. However, the l i n e a r model f o r the speed of the bar i s not as good an approximation f o r these extreme changes i n bottom topography. The speed of the bar was roughly that which would have been expected from equation 2.5.1 f o r a bottom slope that was the weighted average of the bottom slopes i n each s e c t i o n . There are no abrupt changes i n bar speed, temperature f i e l d or current s t r u c t u r e associated with these bottom i r r e g u l a r i t i e s . R U N G Observations i i i i i i | i 0 40 80 120 cm Figure 29b. Changes in heat content and temperature anomaly for Run G 82 E o o. Observations O r ' 1 8 R U N J 16 BAR L T E M P E R A T U R E SECTION ( ° C ) 40 80 120 cm 16 O o Ul § 1 2 or ui a S s ui h- Z < Ul p r e d i c t e d m e a s u r e d BAR 40 80 D I S T A N C E along T A N K 120 cm F i g u r e 30a. Temperature s e c t i o n and mean t e m p e r a t u r e s f o r Run J : 5°-0° b o t t o m s l o p e w i t h d r o p o f f , s t a n d a r d h e a t i n g , and T i , 0°C. 83 R U N J _ measured CM „ _ 5 I x x J i - * - o I P-4 0 - ~ ~ i z I UJ I h- - ^ x * * ^ ° 2 0 - S J < 'BAR Ul I O 0" 1 ' 1 1 I i i 0 40 80 120 Cm predicted 7 Figure 30b. Changes in heat content and temperature anomaly section for Run J 100 0 ° S L O P E ^ * 0 ° S L O P E —J— o predicted X X oo X X XX XX » X X o o o e 5° SLOPE - 0 ° SLOPE STANDARD HEATING x o° SLOPE at a DEPTH of 8cm o o° S L O P E at a DEPTH of 20cm X X o e 10 15 20 25 T l ME ( minutes ) 30 35 4 0 Figure 31. P o s i t i o n of the bar, measured and predicted, f o r 5°-0° bottom slopes and standard heating. The SLOPE of these curves i s BAR SPEED. oo APPENDIX D To show that the thermal bar phenomenon is driven, basically, by the heating of water through i t s maximum density, as has been proposed i n the linear model for the speed of the bar, the following two experiments were performed. 1. Sliding Door Experiment Figure 32 shows the temperature section and mean temperatures 20 minutes after 2°C and 8°C water were brought together. There was no heating (heat lamps were turned o f f ) . This represents the behaviour of flow which could be expected when 'fresh' water colder than 4°C flows into water warmer than 4°C (for example, at the foot of a glacier) or when water warmer than 4°C flows into water colder than 4°C (for example, effluents). The behaviour is not similar to the migrating thermal bar i n that the resulting current pattern does not move and the active sinking zone between the two water masses involves water from both sides. This circulation results from the production of dense, 4°C water at the contact between the two water masses. The horizontal velocity profiles are S-shaped, similar to those shown in Figure 6 (p.15) for the migrating thermal bar experiments. This current structure, superimposed on the temperature section, i s a clockwise circulation i n the shallower water and a counter-clockwise circulation in the deeper water, with most of the sinking between the 3.5 and 4.5°C isotherms (as indicated by arrows). In a manner similar to the migrating thermal bar phenomenon, the sinking prevents flow due 86 F i g u r e 32. Temperature s e c t i o n and mean t e m p e r a t u r e s f o r s l i d i n g door e x p e r i m e n t (5° b o t t o m s l o p e ) . 87 to the h o r i z o n t a l density gradients. Actual examples of water warmer than 4°C being dumped i n t o water colder than 4°C were presented by Chermack (1970) at the 13th Conference on Great Lakes Research, A p r i l , 1970. His work involved airborne r a d i a t i o n thermometer studies of e f f l u e n t s from nuclear power plants on southeastern Lake Ontario. In each case these e f f l u e n t s were confined to an arc of radius of the order of a kilometer from the point of discharge. This confinement agrees with the r e s u l t s of t h i s ' s l i d i n g door' experiment. 2. Heating Water Warmer than 4°C Figure 33 shows the temperature s e c t i o n associated with heating water that i s a l l at a temperature greater than 4°C. The flow i s towards the deep end i n the upper layers and towards the shallow end i n the deeper layers. The temperature structure bears no resemblance to the thermal bar case. The v e l o c i t i e s involved are considerably l e s s than the speed of the migrating thermal bar (about 1/10). This flow i s driven by the h o r i z o n t a l density gradients set up by uniform surface heating and a sloping bottom since the temperature r i s e s f a s t e r i n the shallow end of the tank. I 1 1 1 1 1 I L 0 4 0 8 0 120 D I S T A N C E along T A N K ( c m ) Figure 33. Temperature section for heating water warmer than 4°C, with a 5° bottom slope, standard heating and T i , about 7°C. co oo APPENDIX E: C a l c u l a t i o n of the Stream Function The stream function § i s c a l c u l a t e d from equation 3.2.7, using the a n a l y t i c a l temperature function given i n equation 3.2.9 and s a t i s f y i n g the boundary conditions given i n equations 3.2.8. The streamline along z = 0 and z = -D i s s e t equal to zero. <j> » 2£& (2.9) (0.0392) sech 2(0.392[ - 2 ]) r 0.001 6 , 0.01 5 , 1 4 { - 3 6 — Z + — Z + 2 4 Z + z 3 ( 0,001 ,3 _ 0^01 D2 + 1_ D } . 2 0.001 n4 0.01 n 3 . 1 n2 . + z ( - 5 J - D - — D + 24 D ) -2.9 [ 1 + tanh(0.392[ ĵĝ  - 2 ])] , 0.00001 8 , 0.0001 7 , 0.001 6 0.01 5 [ 168 z + -Ti— z + — 6 — z + — 2 L 1 4 , 3 , 0.00001 ^5 0.001 „4 , 0.01 „3 + 24 z + z ( 28 " " 4 2 — D + -15" D - 0 . 0 1 D 2 + ^ D ) + z 2 ( ^ D 6 - ^ D 5 + 0^001 D4 _ 0^1 D 3 + 1 _ D 2 } ] } where A = 8 x 10" C° 90 3 -2 g i s the a c c e l e r a t i o n due to gravity (- 10 cm sec ), -2 2 - 1 V i s v i s c o s i t y ( = 1.5 x 10 cm sec f o r the laboratory model), S i s the speed of the bar, D = ax i s the depth of the bottom and a i s bottom slope. From t h i s cf>, u and w can be found since u _3£ 8z and w 3£ 9x

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