UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

A laboratory and mathematical study of the 'thermal bar' Elliott, Gillian Hope 1970

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

Item Metadata

Download

Media
[if-you-see-this-DO-NOT-CLICK]
UBC_1970_A1 E44_2.pdf [ 3.66MB ]
Metadata
JSON: 1.0302462.json
JSON-LD: 1.0302462+ld.json
RDF/XML (Pretty): 1.0302462.xml
RDF/JSON: 1.0302462+rdf.json
Turtle: 1.0302462+rdf-turtle.txt
N-Triples: 1.0302462+rdf-ntriples.txt
Original Record: 1.0302462 +original-record.json
Full Text
1.0302462.txt
Citation
1.0302462.ris

Full Text

A LABORATORY AND MATHEMATICAL STUDY OF THE 'THERMAL BAR' by G.H. ELLIOTT B.Sc, University of Toronto, 1965 M.Sc., University of B r i t i s h Columbia, 1967  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Physics and I n s t i t u t e of Oceanography  We accept t h i s thesis as conforming to the required standard  THE  UNIVERSITY OF BRITISH COLUMBIA October, 1970  Iti  presenting  this  an a d v a n c e d d e g r e e the I  Library  further  for  agree  in  at  University  the  make  it  partial  freely  that permission for  this  representatives. thesis  for  It  financial  of  gain  physi rs  Institute of Oceanography, The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  of  of  Columbia,  British for  extensive by  the  Columbia  shall  not  the  requirements  reference copying of  Head o f  is understood that  written permission.  Department  fulfilment  available  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 h i s of  shall  thesis  I  agree  and this  that  study. thesis  my D e p a r t m e n t  copying or  for  or  publication  be a l l o w e d w i t h o u t my  ABSTRACT  The large,  ' m i g r a t i n g t h e r m a l b a r ' phenomenon, known t o o c c u r i n c e r t a i n  dimictic,  f r e s h w a t e r l a k e s , has been s t u d i e d i n l a b o r a t o r y  m a t h e m a t i c a l models.  The  temperature  fields  observed i n the  and  laboratory  agreed w i t h those observed i n Lake O n t a r i o and a l i n e a r p h y s i c a l model f o r the speed o f the 'thermal b a r ' , based on n e g l i g i b l e h o r i z o n t a l v e c t i o n and d i f f u s i o n o f h e a t , gave r e a s o n a b l e v a l u e s f o r b o t h l a b o r a t o r y model and Lake O n t a r i o . associated velocity suggests  temperature  was  the  O b s e r v a t i o n s were a l s o made o f the  On the b a s i s o f t h i s l a b o r a t o r y model, which  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 heat were n o t o f  p r i m a r y importance,  equation.  field.  ad-  mathematical  f i e l d was  models were developed.  First  c a l c u l a t e d from the o n e - d i m e n s i o n a l h e a t  Then the v e l o c i t y  f i e l d was  the diffusion  c a l c u l a t e d assuming t h a t the flow  d r i v e n by buoyancy f o r c e s and b a l a n c e d by v i s c o u s f o r c e s .  b a s i s o f the s i m i l i t u d e between the temperature  fields  On  the  found i n my  models  and those observed i n the l a k e s , i t seems p o s s i b l e t h a t the v e l o c i t y field  o f the models a l s o p r o v i d e s a good a p p r o x i m a t i o n t o the  a s s o c i a t e d w i t h the b a r i n l a k e s . the v e l o c i t i e s  circulation  There are no d i r e c t measurements o f  a s s o c i a t e d w i t h the b a r i n l a k e s and they w i l l be  t o o b t a i n as such v e l o c i t i e s a r e e x p e c t e d , i n Lake O n t a r i o , t o be the o r d e r o f 1 cm s e c  X  .  difficult only of  i i i TABLE OF  CONTENTS page  ABSTRACT T A B L E OF CONTENTS  i  L I S T OF FIGURES  i  i  i  i v  ACKNOWLEDGEMENTS  viii  1.  INTRODUCTION  1  2.  LABORATORY MODEL  7  3.  2.1  Apparatus  2.2  Experimental  2.3  D e s c r i p t i o n o f t h e ' B a r ' 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  2.6  Comparison o f t h e L i n e a r Model t o Lake O n t a r i o  24  2.7  Discussion  27  2. 8  Summary  29  Techniques  7  of the Bar  MATHEMATICAL MODELS 3.1  30  The d e e p c o n v e c t i v e  ii)  The s h a l l o w  iii)  16  30  The T e m p e r a t u r e F i e l d i)  4.  7  side  31  stable side  34  Comparison w i t h  the tank  35  3.2  The V e l o c i t y F i e l d  39  3.3  Validity  52  3.4  Extension  SUMMARY  BIBLIOGRAPHY  o f the V e l o c i t y Model t o a Lake  53 57 • 59  iv page  APPENDIX A:  Cooling Experiment  61  APPENDIX B:  Temperature D a t a : c o n s t a n t slopes'  63  APPENDIX C:  I n v e s t i g a t i o n o f t h e E f f e c t s o f Bottom I r r e g u l a r i t i e s  APPENDIX D  ....  79 85  1.  S l i d i n g Door E x p e r i m e n t  85  2.  H e a t i n g Water Warmer t h a n 4°C  87  v  APPENDIX E:  C a l c u l a t i o n o f t h e Stream F u n c t i o n  89  V  L I S T OF FIGURES Figure 1  2  page The d e n s i t y o f f r e s h w a t e r (Handbook)  as a f u n c t i o n o f t e m p e r a t u r e 3  T e m p e r a t u r e s e c t i o n s f o r L a k e O n t a r i o f o r t w o N-S sections ( S e c t i o n E , R o d g e r s 1 9 6 6 a ; S e c t i o n D, u n p u b l i s h e d , 1965)  3  Experimental tank  4  Heat f l u x e s  mid-lake Rodgers, 4 8  used i n t h e experiments p l o t t e d  against total  time of h e a t i n g  11  5  Percentage  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  7  8  of t o t a l surface heat  reaching different results  depths..  13  f o r the heated  system. A v e r a g e d v e l o c i t i e s a r e shown b y 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 the motion observed i n a minute ( i . e . , cm m i n *) i n t h e same s c a l e s as u s e d f o r t h e a x e s 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 a n d 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 P o s i t i o n o f t h e b a r , m e a s u r e d a n d p r e d i c t e d , f o r 5° s l o p e a n d t h e two h e a t i n g r a t e s ( ^ 9 x 1 0 and ^ 12 x 1 0 " c a l c m sec ). The SLOPE o f t h e s e c u r v e s i s BAR SPEED  15  18  bottom  - 3  3  9  - 2  - 1  20  P o s i t i o n o f t h e b a r , m e a s u r e d a n d 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 a n d t h e two h e a t i n g r a t e s ( ^ 9 x 10 a n d ^ 12 x 1 0 " c a l c m sec ). The SLOPE o f t h e s e c u r v e s i s BAR SPEED 3  3  - 2  - 1  21  10  P o s i t i o n o f t h e b a r , m e a s u r e d a n d 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 ) , a n d c h a n g e s 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 t h e h e a t l a m p s w e r e 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 the tank f o r Run A 25  vl Figure 13  page Temperature anomaly s e c t i o n s f o r the tank experiment ( 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 i g u r e 2, p. 4)  14  Temperature  15  9 i n C° a g a i n s t d i s t a n c e a l o n g the tank f o r f i x e d v a l u e s of z ( i n cm) (a) g e n e r a l i z e d measured v a l u e s (b) c a l c u l a t e d from e q u a t i o n s 3.1.13  38 38  A n a l y t i c a l temperature a p p r o x i m a t i o n , e q u a t i o n 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 o s s - s e c t i o n (T i n °C) (b) 9 i n C° a g a i n s t x i n cm f o r f i x e d v a l u e s o f z ( i n cm)  45 45  v 16  17a  section  (°C) c a l c u l a t e d  26  from e q u a t i o n s 3.1.13... 37  Calculated velocity f i e l d . V e l o c i t i e s are shown by arrows, t h e i r l e n g t h i n d i c a t i n g the motion i n a minute ( i . e . cm m i n ) i n the same s c a l e s 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 profiles. Compare w i t h F i g u r e 6 (p. 15) 46 - 1  17b  Calculated velocity f i e l d . V e l o c i t i e s are shown by arrows, t h e i r l e n g t h i n d i c a t i n g the motion i n a minute ( i . e . . cm m i n ) i n the same s c a l e s 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 - 1  18  19  20  21a  21b  22a  22b  . A c t u a l 6 ( i n C° ) a g a i n s t d i s t a n c e a l o n g tank ( x, i n p l o t t e d from l a b o r a t o r y o b s e r v a t i o n s  cm) 49  C a l c u l a t e d s t r e a m f u n c t i o n <{> i n cm m i n . 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  Temperature s e c t i o n from Lake O n t a r i o (Rodgers, 1966a); m i d - l a k e N-S s e c t i o n taken mid-January 1966. Temperature and temperature anomaly s e c t i o n s f o r the case of c o o l i n g ( f a l l s i m u l a t i o n ) w i t h 5° bottom s l o p e  62  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 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  64  Changes i n h e a t content and temperature anomaly f o r Run B  65  2  - 1  section  Temperature s e c t i o n and mean temperatures f o r Run bottom s l o p e , s t a n d a r d h e a t i n g , and T i , 1.4°C  I : 5°  Changes i n h e a t c o n t e n t and temperature anomaly f o r Run  66 I . . . 67  vii Figure 23a  page Temperature  section  5° b o t t o m s l o p e , 23b  Changes i n h e a t  24a  Temperature  a n d mean t e m p e r a t u r e s  f o r Run H:  i n c r e a s e d h e a t i n g a n d T i , 0°C  68  c o n t e n t a n d t e m p e r a t u r e a n o m a l y f o r Run H... 69  section  a n d 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 a n d T i , 0°C  v  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  25a  Temperature  s e c t i o n a n d mean t e m p e r a t u r e s  70  f o r R u n C... 71  f o r R u n 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 a n d T i , 0°C  o  72  25b  C h a n g e s i n h e a t c o n t e n t a n d t e m p e r a t u r e a n o m a l y f o r Run D...  26a 26b  T e m p e r a t u r e s e c t i o n a n d mean t e m p e r a t u r e s f o r R u n 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 a n d T i , 0°C 74 C h a n g e s i n h e a t c o n t e n t a n d t e m p e r a t u r e a n o m a l y f o r R u n E . . . 75  27a  Temperature  s e c t i o n a n d mean t e m p e r a t u r e s  73  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 a n d T i , 0°C  76  27b  C h a n g e s i n h e a t c o n t e n t a n d t e m p e r a t u r e a n o m a l y 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 L a k e 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) a n d f o r t h e t a n k e x p e r i m e n t T e m p e r a t u r e s e c t i o n a n d 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 a n d T i , 0°C  80  29b  C h a n g e s i n h e a t c o n t e n t a n d t e m p e r a t u r e a n o m a l y 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 a n d 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 a n d T i , 0°C  82  Changes i n h e a t f o r Run J  83  29a  30b  31  32  33  c o n t e n t and t e m p e r a t u r e anomaly  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 , 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 T e m p e r a t u r e s e c t i o n a n d mean t e m p e r a t u r e s 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 )  78  section  f o r 5°-0° 84  for sliding  T e m p e 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 w a r m e r 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 a n d T i , a b o u t  86  7°C 8 8  ACKNOWLEDGEMENTS  The author would l i k e to express her gratitude to Dr. P.H. LeBlond who suggested the problem and provided advice, encouragement and funds f o r the experimental work; to Dr. G.K. Rodgers of the Great Lakes I n s t i t u t e , University of Toronto who generously provided observational m a t e r i a l 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 t h e i r 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 f o r h i s invaluable assistance i n the experimental work.  The author was most  fortunate to be f i n a n c i a l l y supported during this work by a National Research Council of Canada Studentship and a MacMillan Family Fellowship. F i n a l l y the author wishes  to thank her husband, Dr. J.A. E l l i o t t ,  for h i s patience and help i n 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 s c i e n t i s t (Forel, 1880) t r i e d 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 i c e at the shore.  The 4°C isotherm at the  surface was s i t u a t e d j u s t on the shallow side of a drop o f f 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 b a r r i e r . However most of the l i t e r a t u r e on the thermal bar i s from the l a s t 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. to  The bar was observed to move from near shore  the centre of the lake where i t disappeared.  The most d e t a i l e d observa-  t i o n a l studies have been done by G.K. Rodgers of the Great Lakes  Institute,  University of Toronto, who f o r 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 i n some large freshwater lakes in  the spring and l a t e f a l l (e.g., Lake Michigan: Church, 1942; Great Lakes:  Richards et al, 1969; Lake Ontario: Rodgers, 1966a; Lake Ladoga: Tikhomirov, 1963).  I t i s 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 t u r b i d i t y , and indications of a convergence  are usually  associated with the 'bar , defined by the 4° surface isotherm, which 1  separates stable, deeper w a t e r  stratified,  shoreward water  (Rodgers, 1966a).  water attains  f r o m a l m o s t homogeneous  The t h e r m a l b a r e x i s t s b e c a u s e  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  (Figure 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 in  certain  large  lakes, i nwhich  the 'bar' e x i s t s ,  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 first  completed near t h e shores which  thermal b a r e x i s t s  In  2°C.  O n t a r i o i s everywhere  i s o t h e r m appears  depth  than i n t h e deeper p a r t s  u n p u b l i s h e d , 1965).  section  convergence.  'barrier'.  4°C  The h i g h e r t u r b i d i t y that this water  does  I n t h e warmer w a t e r on t h e  ( F i g u r e 2) ( s e e R o d g e r s ,  T h i s phenomenon c a n a l s o o c c u r i n t h e f a l l  on  This i s the reason f o r  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 a n d a s t r o n g  temperature gradient e x i s t  at the  Flotsam along the surface  the remainder of the lake.  ' t h e r m a l b a r ' meaning  the surface  a temperature  o f t h e 4°C i s o t h e r m i n d i c a t e s  not mix q u i c k l y w i t h  shoreward  t h e l a k e and g r a d u a l l y  t o t h o s e shown i n F i g u r e 2 ( R o d g e r s ,  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 shoreward s i d e  The s u r f a c e  t o two months f r o m i t s  By t h e b e g i n n i n g o f June  Ontario i s similar  1966a and Rodgers,  name  of the lake.  a t t h e shore u n t i l i t disappears below  a c r o s s Lake  Ontario.  remaining roughly p a r a l l e l t o  I t t a k e s one and a h a l f  centre of the lake.  the  and u n s t a b l e  temperature a t the shore  f i r s t near the shore around  the centre of the lake,  contours.  appearance  the  The m i g r a t i n g  c o l d e r t h a n 4°C, t y p i c a l l y  As t h e l a k e h e a t s u p , t h e mean w a t e r  moves i n t o w a r d s the  t h e n become s t a b l e .  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  i n c r e a s e s more r a p i d l y 4°C  Instead the mixing i s  i  e a r l y s p r i n g Lake  below  density  the v e r t i c a l mixing  a t the t r a n s i t i o n between the s t a b l e  regions. The  lake.  fresh  1966a,  surface  1966b).  as t h e 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 F i g u r e 2.  (°C)  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 sections ( S e c t i o n E, Rodgers 1966a; S e c t i o n D, Rodgers, u n p u b l i s h e d , 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 r e s u l t i n g i n a 4°C  isotherm that  the l a k e i n a manner s i m i l a r 1942;  R i c h a r d s et  al,  a condition since stratified, The  1969;  then p r o g r e s s e s towards  to the s i t u a t i o n Rodgers,  as  prominent  c a n n o t b e c o m e as  difference being  strongly  4C°.  the 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 assumed t h a t  t h e r m a l e f f e c t s were b a l a n c e d by thermal wind  the zeroth  and  first  Coriolis  forces  o r d e r i n g i n terms  ( t h a t i s , he  as  t o t h i n boundary  The  solutions obviously  layers confines  the 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 the b a r to these boundary which his  f l o w i s i n the form of c e l l s ,  b a r , w i t h m i x i n g and s i n k i n g my  layers,  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 .  calculated  at the b a r .  one  on e i t h e r s i d e  T h i s was  Also  of  the  n o t what I found i n  experimental observations. 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  thermal b a r i n Lake t i o n a l data. in  the  number;  These  the m i g r a t i n g t h e r m a l b a r  confining of viscous effects  used  layers.  o f the Rossby  order s o l u t i o n s were c o n s i d e r e d .  at a l l time dependent,  Huang  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  equation) except i n t h i n v i s c o u s boundary  e q u a t i o n s were l i n e a r i z e d by  The  (Church,  only previous mathematical c o n s i d e r a t i o n of the thermal b a r  (1969). H i s approach  is.  side  This i s not  lake  the c e n t r e of  i n the s p r i n g  1966a).  the s t a b l e shoreward  t h e maximum t e m p e r a t u r e  p h e n o m e n o n was  are not  than near the middle of the  'He  O n t a r i o was  Sato  (1969)  found t h e time t o depend p r i m a r i l y  the c e n t r a l p o r t i o n s  using A p r i l  developed by  1); that i s ,  the c e n t r a l waters  t h e t i m e i n t e r v a l was  using observa-  on t h e h e a t  o f 'the l a k e i n the e a r l y s p r i n g  the  content  (arbitrarily  that necessary to heat  to the p o i n t where the s u r f a c e temperature  was  In  my  o p i n i o n a t h e o r e t i c a l approach  b a r was h a m p e r e d b y in  the l a c k of knowledge o f the v e l o c i t y  t o t h i s phenomenon.  b a r i n a l a b o r a t o r y model 2).  For this  (Elliott  and E l l i o t t ,  1969, 1970)  involved directly  the thermal (see s e c t i o n  lake observations.  and t h e o b s e r v a t i o n s t a k e n by  others  the thermal 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 the speed  m i g r a t i n g b a r was c a l l y which  developed.  A temperature  compared r e a s o n a b l y w e l l w i t h  f i e l d was  velocity which  f i e l d was  the laboratory basis  reasonably w e l l with (see s e c t i o n  3).  of the l a b o r a t o r y s t u d i e s  Lake O n t a r i o d u r i n g the presence The imations  temperature  t h e n f o u n d , u s i n g an a p p r o x i m a t e d  again agreed  laboratory  associated velocity  field  measured i n the l a k e s .  of the  mathematistudies.  field,  vorticity  a  equation,  the o b s e r v a t i o n s from the work  These r e s u l t s were  then s c a l e d ,  on  and compared t o t h e c o n d i t i o n s i n  of the thermal b a r .  and t h e o r e t i c a l s t u d i e s produce  t o the temperature  developed  the l a b o r a t o r y model  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  the  field  reason I have s t u d i e d  T h i s s t u d y compared f a v o u r a b l y w i t h From t h e s e l a b o r a t o r y s t u d i e s  in  of the thermal  t h e phenomenon; f o r e x a m p l e , no measured c u r r e n t s h a v e b e e n  related  of  to the problem  fields  reasonable  approx-  o b s e r v e d i n L a k e O n t a r i o and t h e  i s n o t unreasonable, but has not been  directly  7 2.  LABORATORY MODEL  2.1  Apparatus The  and  e x p e r i m e n t s w e r e 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 m e t e r s  30 c e n t i m e t e r s w i d e  3).  The e n d w a l l s  insulation. plexiglas was  t h a t was i n s u l a t e d  and bottom were plywood w i t h  The l o n g w a l l s w e r e a d o u b l e  s l o p i n g bottom, sealed into used  w i t h h e a t lamps  in  for  2.2  (Westinghouse,  Since thespring  t h e Great Lakes  laboratory  this  agent  To p r o v i d e a  sealant.  were  This  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  chamber ( f a c i l i t i e s  Victoria).  A desiccating  tank  conditions. cooling  0°C a n d 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 250 w a t t , r e f l e c t o r ,  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  a cold  o f 0.6 c e n t i m e t e r  tape and a s i l i c o n e  the s p r i n g and f a l l  w a t e r i n t h e t a n k down t o a b o u t  fall  thickness  t o prevent condensation.  t h e tank w i t h masking  spring  (Figure  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  to simulate both  The  and bottom  4 centimeter styrofoam  s e p a r a t e d b y a 1.2 c e n t i m e t e r a i r s p a c e .  i n s e r t e d i n the a i r space  was  on t h e s i d e s  long  infrared heat).  above The  (6° t o 8°C) t a p w a t e r  o f the Frozen Sea Research  G r o u p , DEMR,  c o n d i t i o n s h a v e b e e n more f r e q u e n t l y r e p o r t e d  and t h eheated system i s e a s i e r t o s i m u l a t e i n t h e  s y s t e m was s t u d i e d i n more  detail.  Experimental Techniques For  water.  theh e a t i n g experiments About 1 c c .  of liquid  t h e t a n k was f i r s t  fresh tap  d e t e r g e n t w a s a d d e d ; t h i s was n e c e s s a r y t o  decrease surface tension effects  d u r i n g t h e a d d i t i o n o f dye.  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 surface velocities  f i l l e d with  r e s u l t e d which  I f the deter-  o f d y e came i n c o n t a c t w i t h  distorted  the water,  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 w a t e r t o lower the temperature  o f the water  t o n e a r 0°C; then excess i c e was removed, the mean  measured, and the h e a t lamps t u r n e d on.  temperature  I n a p p r o x i m a t e l y one h a l f  the s h a l l o w end h a d reached a temperature s l i g h t l y  above 4°C.  hour  A temperature  and c u r r e n t p a t t e r n developed (see s u b s e c t i o n 2.3). which m a i n t a i n e d  itself  throughout the experiment, p r o g r e s s i n g from the s h a l l o w end t o the deep end i n a f u r t h e r h a l f hour. The  c u r r e n t s t r u c t u r e was f o l l o w e d by d r o p p i n g f l e c k s o f Rhodamine B  dye on the s u r f a c e ;  these sank  t o the bottom  leaving v e r t i c a l red traces  which were s u b s e q u e n t l y d i s t o r t e d by the c u r r e n t s .  The dye t r a c e was  p o s i t i o n e d w i t h r e s p e c t t o a c e n t i m e t e r g r i d on the back w a l l o f the tank. The motion was timed w i t h a stop-watch. from time s e r i e s photographs  R e l a t i v e motions were o b t a i n e d  and from 16 mm movies  To o b t a i n temperature p r o f i l e s  o f the dye s t r e a k s .  a t h e r m i s t o r bead  (VECO  32A5),  a t t a c h e d t o the end o f a 40 cm, t h i n g l a s s r o d w i t h c e n t i m e t e r markings, was lowered v e r t i c a l l y  through the w a t e r .  The r e s i s t a n c e o f the t h e r m i s t o r  was measured d i r e c t l y w i t h a F a i r c h i l d M u l t i m e t e r (Model 7050). i n s t r u m e n t uses s u f f i c i e n t l y thermistor i s n e g l i g i b l e .  small current  This  (1 ua) t h a t s e l f h e a t i n g o f the  To p r e v e n t t h e r m a l c o n t a m i n a t i o n from w a t e r  dragged by the g l a s s r o d , measurements were made only w h i l e l o w e r i n g the thermistor.  R e s i s t a n c e s were u s u a l l y r e a d a t depths o f 0.5 cm, 1 cm, 2 cm,  3 cm, e t c . , t o t h e bottom. and g l a s s  The t h e r m i s t o r was c a l i b r a t e d  a g a i n s t a mercury  thermometer b e f o r e , d u r i n g , and a f t e r the e x p e r i m e n t s . I t s  c a l i b r a t i o n d i d n o t change.  A calibrated, battery-operated thermistor  thermometer was s i t u a t e d a t the deep end o f the tank a t a l l times i n o r d e r t o measured the temperature h e r e p e r i o d i c a l l y  d u r i n g the e x p e r i m e n t s .  The h e a t f l u x was o b t a i n e d from the d i f f e r e n c e i n h e a t c o n t e n t o f 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 t h e h e a t l a m p s w e r e t u r n e d on a n d j u s t the  heat  lamps w e r e t u r n e d o f f .  N o r m a l l y t h e t a n k was  heated with  after  three  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  meters  (Figure  above t h e w a t e r s u r f a c e  3) .  The w a t e r was  also heated  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 . of  this  radiation  and  c o n d u c t i o n w e r e e v a l u a t e d as  l a m p s o f f , a t a n k o f 0°C gave a h e a t g a i n e d by  w a t e r was  an i n s u l a t e d  t o p , was  h e a t i n g r a t e was -3  c a l cm  -2 cal  cm  -2  through the w a l l s .  a b o u t 9 x 10 sec  -1  was  -3  c a l cm  through the w a l l s the f i r s t  r a t e o f 9 x 10  sec  -2  sec  c o u l d be  fifth  -1  (Figure 4).  Of  and  this  2 x  10  -3  obtained using five heat  and n o t t h r o u g h t h e s u r f a c e  series  lamps.  and  a heating  -1 sec  were used.  those found i n Lake O n t a r i o ,  This slope i s five  to  twenty  the p r o t o t y p e f o r t h i s model, however of the  tank  cm a t d e e p e n d ) . The h e a t i n g -3 -2 -1 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 sec typically  e x p e r i e n c e d i n the s p r i n g  14  on L a k e O n t a r i o .  the s u r f a c e h e a t i n g from the h e a t lamps, distribution  ( S v e r d r u p et  al,  F i g u r e 5 shows t h e calculated  o f t h e l a m p s a n d known a b s o r p t i o n  1942;  tank  of the water.  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°  -2 c a l cm  A higher heating  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 the l e n g t h  energy  with  the  of the h e a t i n g o f the water i n the  ( a b o u t 1 cm o f w a t e r a t s h a l l o w e n d ,  of  heat  -1  c a l cm  -3  is  This  T h i s gave  U s i n g t h r e e lamps  from the l o n g wave r a d i a t i o n  -2  T h u s " a maximum o f a b o u t one  it  an h o u r .  T h e n a t a n k o f w a t e r a t 0°C,  from heat c o n d u c t i o n through the w a l l s .  r a t e o f 12 x 10  times  the heat  -1 sec  In  With  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 .  -3  is  follows.  by  effects  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  an e s t i m a t e o f t h e h e a t f l u x  2 x 10  The  a l l o w e d t o s i t f o r about  conduction through the w a l l s .  1.5  Dorsey,  1940).  from the  rate  penetration spectral  coefficients  A l s o shown i s t h e p e n e t r a t i o n  of  rXIO"  3  u  <D CO  *  E  t  12  INCREASED HEATING  o o u X  R  X  8  x* " 5/*  x  *  x **  x  x  *  * *  STANDARD HEATING  < UJ X  20  4 0  TOTAL  Figure 4.  6 0  TIME  8 0  100  (minutes)  Heat fluxes used i n the experiments p l o t t e d against t o t a l time of heating  12 s o l a r r a d i a t i o n i n t o pure w a t e r (Sverdrup et at, of  the h e a t was  absorbed i n the upper  s o l a r h e a t a b s o r p t i o n i n the upper  1942).  In the tank most  c e n t i m e t e r o f the water;  few meters i n the l a k e .  suggests a v e r t i c a l s c a l i n g o f a p p r o x i m a t e l y 1/1000.  this simulates  Thus F i g u r e 5  The 5° bottom s l o p e ,  compared to the n o r t h s h o r e o f Lake O n t a r i o , i n t u r 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 p p r o x i m a t e l y 1/20,000. t h e tank c o u l d then be viewed  In making a comparison w i t h the  as r e p r e s e n t i n g a s e c t i o n from the shore  the c e n t r e o f the l a k e , however t h e s e tank experiments an example o f s t r i c t  to  are n o t meant t o be  dynamic m o d e l l i n g .  The e x p e r i m e n t a l work was experiments  lakes,  done i n two s t a g e s .  were done p r i m a r i l y  The  f i r s t series  t o i n v e s t i g a t e the p o s s i b i l i t y  of  of studying  the t h e r m a l b a r i n a l a b o r a t o r y model and t o i n v e s t i g a t e i t s b e h a v i o u r d u r i n g h e a t i n g and c o o l i n g . time.  The v e l o c i t y  f i e l d was  also studied at this  L a t e r , a f u r t h e r s e r i e s o f experiments were done t o study my  'thermal  b a r ' i n a more q u a n t i t a t i v e manner.  2.3  D e s c r i p t i o n o f the 'Bar' i n the Tank Initially  in  the tank.  the s u r f a c e h e a t i n g produces The  temperature  v e r t i c a l c o n v e c t i o n everywhere  i n the s h a l l o w end r i s e s  f a s t e r than t h a t i n  the deep end making the w a t e r i n the s h a l l o w end denser. f l o w due  t o t h i s 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 i s i n h i b i t e d by  Eventually  the temperature  end i t i s s t i l l  less  the c o n v e c t i o n .  i n the s h a l l o w end reaches 4°C w h i l e i n the deep  than 2°C.  the bottom s l o p e can be  By t h i s  time some flow o f dense w a t e r a l o n g  observed i n s p i t e o f the v e r t i c a l c o n v e c t i o n .  weak mean c i r c u l a t i o n i s superimposed  a t the s h a l l o w end,  This  on the c o n v e c t i o n ; towards the deep  end a t the bottom and towards the s h a l l o w end a t the top. produces,  However downslope  Further heating  a s t a b l e t h e r m o c l i n e o f w a t e r warmer than  4°C.  % 0  0  4H  20  of T O T A L 40  ENERGY 60  80  100  /T n 7 I  E o  8h- ? /  CL UJ Q  1° ' I2h  ? i x  •  * HEAT  LAMPS  o S O L A R RADIATION ( m u l t i p l y depths by 1 0 0 0 )  I  X  I  X  16k  Figure 5.  Percentage of t o t a l surface, heat reaching d i f f e r e n t depths  This  thermocline progresses  towards the deep end and i t s forward edge marks  a boundary between the now s t a b l e s h a l l o w r e g i o n o f t h e tank and t h e deeper convecting region (Figure 6). in  A mean f l o w towards t h e deep end i s observed  the t h e r m o c l i n e w i t h a c o u n t e r - f l o w underneath.  tank a flow n e a r  the s u r f a c e t r a v e l s  I n the deep end o f the  towards the s t a b l e t h e r m o c l i n e .  This  c u r r e n t s i n k s a t the f r o n t edge o f t h e t h e r m o c l i n e and d i v i d e s , a p a r t p r o v i d i n g the upslope  c u r r e n t i n t h e s h a l l o w end and the o t h e r p a r t a down-  s l o p e c u r r e n t i n the deep end.  The s i n k i n g zone ( F i g u r e 6) i n f r o n t o f  the s t a b l e t h e r m o c l i n e i s made up o f w a t e r between 3.5 and 4.5°C (maximum density region).  T h i s dense w a t e r i s the r e s u l t o f h e a t i n g c o l d e r , l i g h t e r  w a t e r from t h e deep end o f the tank and does n o t i n c l u d e w a t e r from t h e s t a b l e t h e r m o c l i n e on the s h a l l o w s i d e . zone w i l l be r e f e r r e d  t o as the 'thermal b a r ' o r s i m p l y  i s o t h e r m does n o t always extend t h e r m o c l i n e a l s o develops the s u r f a c e .  2.4  u n s t a b l e and more a c t i v e  The s u r f a c e v e l o c i t i e s  The 4°C  convec-  go t o z e r o because  o f t e n r e f e r r e d t o as ' s u r f a c e p r e s s u r e ' ( s e e  and R i d e a l , 1961, p.218); t h i s i s due t o the p r e s e n c e  unavoidable s u r f a c e contamination.  results  the 'bar'.  sinking  to the s u r f a c e s i n c e e v e n t u a l l y a shallow  This thermocline i s s l i g h t l y  of s u r f a c e tension e f f e c t s ,  maintains  this  on the deep s i d e due t o the i n t e n s e h e a t i n g a t  t i v e m i x i n g would d i s s i p a t e i t .  Davies  I n these e x p e r i m e n t s ,  The temperature  i t s e l f u n t i l the b a r has reached  f o r the c o o l i n g system a r e s i m i l a r  of nearly  and c u r r e n t p a t t e r n  t h e deep end o f the tank.  The  ( s e e Appendix A ) .  S i m i l i t u d e to Lake Comparing these r e s u l t s  s e v e r a l obvious  similarities.  from the tank t o f i e l d  observations, there are  A 'bar' moves from the s h a l l o w end o r shore  to t h e deep end o r c e n t r e o f the l a k e .  A t h e r m o c l i n e develops  on t h e  DISTANCE  Figure 6.  ALONG  TANK  (cm)  Generalized current and temperature results f o r the heated system.. A v e r a g e 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 observed i n a minute (I.e., .cm.min".) i n the same scales as used f o r the axes. 1  s h a l l o w s i d e and deep s i d e  t h e f r o n t e d g e i s m a r k e d b y w a t e r n e a r 4°C.  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  convection.  The  from heat budget  calculations  observations.  the convergence  In the experiment  the b a r from the t h e r m o c l i n e r e g i o n the f a c t  that  motion.  The  f o r Lake O n t a r i o (Rodgers,  t h e r e was  shallow thermocline which  under very s t a b l e , l i g h t wind  h a s b e e n o b s e r v e d on L a k e  region.  appears  a limnological barrier  L i n e a r Model  due  1970).  Also  position  stable water  t o 4°C.  The  on t h e a b o v e r e s u l t s was  m o d e l assumes  Thus t h e  tank  t o the  the s u r f a c e of a u n i t  column.  The  the speed  to evaluate  column  T h i s means t h a t m o s t o f  remains w i t h i n  the column  that  column.  f r o m an u n s t a b l e t o a  time taken f o r the b a r to t r a v e l  model gives  used  t h a t h o r i z o n t a l a d v e c t i o n and  o f the b a r i s then at the t r a n s i t i o n  The  surface  f o r the Speed o f t h e B a r  to a p o i n t B i s the time taken to heat up  the  Lakes.  d i f f u s i o n of heat are not of primary importance.  The  Neverthe-  conditions such a s h a l l o w thermocline  to represent a heat driven c i r c u l a t i o n s i m i l a r  of the b a r .  the heat e n t e r i n g  to offshore  to wind mixing.  O n t a r i o ( E l d e r and L a n e ,  A s i m p l e l i n e a r model based the speed  with  d e v e l o p e d on t h e d e e p s i d e i n 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  2.5  the  This agrees  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 , b e e x p e c t e d t o go t o z e r o . experiment  the  no h o r i z o n t a l m o t i o n a c r o s s  to the deeper  the b a r i s o b s e r v e d t o be  has  t h a t has been s u g g e s t e d from  experiment would not generally e x i s t i n lakes, less  indicating  s i n k i n g plume o b s e r v e d i n the e x p e r i m e n t i n f r o n t of  thermocline would produce field  the  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  a l s o been deduced 1968).  i n both,  On  from a p o i n t  A  of u n i t area a t p o i n t  B  o f t h e b a r , S,  since  17 time =  as  S  =  —  =  change i n heat content Q  Q L AT pc D P  (2.5.1)  where Q i s heat flux through the surface i n c a l cm  sec  -1  L i s the h o r i z o n t a l distance ( i n cm) between the bar and some deeper p o s i t i o n where the mean temperature i s known; AT i s the temperature difference ( i n C°) between 4°C and the mean temperature at the deeper p o s i t i o n ; D i s the depth of the water ( i n cm) at the deeper p o s i t i o n where the mean temperature i s known; and  pc i s the density of the water times i t s s p e c i f i c heat. P  To the  accuracy of the experimental data this has a value of 1 c a l C Thus i f the mean temperature at some p o s i t i o n i n the convective region i s known, together with the heat input, the a r r i v a l time of the bar can be calculated. The above formula was compared to actual speeds observed i n the experiments and i t s dependence on the three variables D/L, Q, and AT was examined.  Each of these was varied i n turn.  The data used i n 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 p o s i t i o n of the bar to the deep end. Figure 7 shows a comparison  _3 of the actual and predicted speeds f o r a heat flux of approximately 9 x 10 - 2 - 1 1 1 c a l cm sec and three d i f f e r e n t 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-  /• A  * *  o oo  / 2.5°  o o/ 100  SLOPE  *  ° V  ~  '  A  X  *  A  *  5 ° SLOPE  o  *X ^  o/o  A A  < O  ft  §80  yx j  7.5°  SLOPE  predicted O  X  o CO O Q.  /  0 0 0  x x  ^-measured  O O  »XX  o  60  ^ ' A AA AAA  *"  ^ X  6  A  STANDARD HEATING  A  A^A A *  40 tt  F i g u r e 7.  AA  A  10  15  20 TIME  25 (minutes)  30  35  40  P o s i t i o n o f the 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 bottom s l o p e s o f 2.5°,5°, 7.5°.g The SLOPE of these curves i s BAR SPEED.  d i m e n s i o n a l i t y and c u r r e n t s t r u c t u r e s o f t e n were not u n i f o r m a c r o s s the tank.  T h i s p r e v e n t e d u s i n g s l o p e s s m a l l e r than 2^°.  show a s i m i l a r comparison of 5° and  7^-° •  o f speeds  fluxes at slopes  for different  compare w e l l w i t h the measured speeds  i n c r e a s e i n the speeds  to be due  d i f f e r e n t s u r f a c e heat  F i g u r e 10 i s a comparison  The p r e d i c t e d speeds The  f o r two  F i g u r e s 8 and 9  as the b a r approaches  AT's.  i n a l l cases.  the end o f the tank seems  t o an i n t e r a c t i o n w i t h the s h a l l o w t h e r m o c l i n e which has  on the deep s i d e by  this  time.  developed  The main t h e r m o c l i n e i s then a b l e t o  develop f a s t e r by i n c o r p o r a t i n g the p a r t i a l l y warmed s u r f a c e l a y e r from  the  deep s i d e and the r e g i o n r e f e r r e d t o as the b a r becomes more d i f f u s e . The  dashed  curve,  F i g u r e 8, shows the speed o f the b a r t h a t  when an i n s u l a t e d l i d was  resulted  put on the tank a t the time i n d i c a t e d by  the  arrow; t h i s shows the s t r o n g dependence o f the speed on the s u r f a c e h e a t i n p u t r a t h e r t h a n on 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 or on o t h e r h e a t tranfers. Temperature s e c t i o n s were used t o c a l c u l a t e mean temperatures the tank i n o r d e r t o compare w i t h the temperatures approximations s e c t i o n through -3 9 x 10  l e a d i n g t o e q u a t i o n 2.5.1.  F i g u r e 11 i s a  the  temperature  the b a r f o r a 5° bottom s l o p e and h e a t i n p u t o f a p p r o x i m a t e l y -2  c a l cm  are d e s i g n a t e d by From t h i s  p r e d i c t e d by  -1 sec  .  The p o s i t i o n s a t which o b s e r v a t i o n s were made  the v e r t i c a l marks a l o n g the top l i n e i n the  f i g u r e the mean temperature  diagram.  and change i n heat content from  time the h e a t lamps were t u r n e d on are c a l c u l a t e d and i n the lower of F i g u r e 11 a r e compared w i t h v a l u e s p r e d i c t e d by seen the model g i v e s a good f i r s t c o n t e n t a l o n g the tank. be accounted  along  f o r by  As  a p p r o x i m a t i o n t o the temperature  Differences  the presence  the model.  diagrams can be or heat  from the p r e d i c t e d heat c o n t e n t  of a d v e c t i v e e f f e c t s  the  may  (see s u b s e c t i o n 2.7).  XX  XX  /  /  / AA *  ^^A  4  predicted  LID ON  A&  AAA  A  AA  measured  A  V A  F i g u r e 8.  XX  5 ° SLOPE A S T A N D A R D HEATING x I N C R E A S E D HEATING • HEATING STOPPED  / A  A  /  '  _L  A  10  20 25 T I M E ( minutes)  15  30  35  P o s i t i o n o f the b a r , measured and p r e d i c t e d , f o r 5° bottom s l o p e and the two h e a t i n g ( ^ 9 x 10 and ^ 12 x 10" c a l cm" s e c ) . The SLOPE o f these curves i s BAR SPEED. 3  3  2  - 1  40  rates  ro o  lOOh  E o  <80 O O I  <  predicted  O  h-  co60 O  CL  7.5° SLOPE  a STANDARD HEATING x INCREASED HEATING  40  5  10  15  20  25  TIME F i g u r e 9.  30  35  40  (minutes )  P o s i t i o n o f the b a r , measured and p r e d i c t e d , f o r 7.5° bottom s l o p e and the two h e a t i n g r a t e s ( 'v 9 x I O " and ^ 12 x 10" c a l cm" s e c " ) . The SLOPE o f these curves i s BAR SPEED. 3  3  2  1  100  £ o  2 < H  O 80 2  O _l <  *  predicted^.©'  if) £  measured  60 5° S L O P E STANDARD HEATING * A T = IC° , L= 100 cm * A T = I C°  t  L = I I 0 cm  o A T = l . 5 C ° , L =105 c m 40  0  5  10  15 TIME  Figure 10.  20 25 (minutes)  30  35  40  P o s i t i o n of the bar, measured and predicted, f o r 5° bottom slope, standard heating and d i f f e r e n t AT, L combinations. The SLOPE of these curves i s BAR SPEED.  to  RUN A  Observations  40 DISTANCE  ALONG  80 TANK  ( cm )  _I2 O  o  UJ  cc  <  measured  CC  UJ  predicted  <  BAR  UJ 2  _ u o u  0  meosured  40 UJ O  U 20 »ui I  BAR  40 DISTANCE Figure 11.  80 ALONG TANK  predicted  120 (cm)  Temperature section, mean temperatures ( v e r t i c a l l y averaged), and changes i n 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 data from the other experiments results  2.6  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  Comparison The  model f o r t h e speed Ontario.  Rodgers  (1965b,  surface heat  The d a t a u s e d  and Anderson  (1961).  taken  from  The n e t  t r a n s f e r i s approximately uniform over the surface of the  (1966), the speed  Using the temperature  0.7 a n d 0.4 r e s p e c t i v e l y . to the speed  data  C a l c u l a t e d v a l u e s w e r e 0.95 cm s e c ^ The p r e d i c t e d v a l u e s w e r e  The m o d e l a l s o g a v e a r e a s o n a b l e a p p r o x i m a t i o n  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  To make a c l o s e r c h e c k  from  o f t h e b a r was c a l c u l a t e d a n d compared t o t h e  ( n o r t h s h o r e ) a n d 0.3 cm s e c ^ ( s o u t h s h o r e ) .  on t h e a c c u r a c y o f t h e m o d e l ,  (1942).  b e t t e r data from t h e  a r e needed. V a l u e s o f t h e change i n h e a t  section 12.  data  t o make t h e c o m p a r i s o n w e r e t a k e n  1966a, 1968) and Rodgers  v a l u e p r e d i c t e d by t h e model.  lakes  The  t o t h e above.  o f t h e b a r was a l s o compared w i t h  as a s s u m e d i n t h e c a l c u l a t i o n s .  Rodgers  are s i m i l a r  B.  o f t h e L i n e a r Model t o Lake O n t a r i o  i n Lake  Lake,  are contained i n Appendix  content i n a unit  t h r o u g h t h e t h e r m a l b a r from Rodgers  A l s o shown, f o r c o m p a r i s o n ,  experiment  (from Figure 11).  of n e g l i g i b l e  As s e e n  advection o f heat  as good f o r t h e Lake heat i s s t r i k i n g l y  i s a plot  from the f i g u r e  i n both cases.  content) i s not  However t h e d i s t r i b u t i o n o f The e f f e c t  i n more d e t a i l i n F i g u r e 1 3 w h i c h  temperature  anomalies  of advection i s  shows t h e c r o s s s e c t i o n  f o rt h e tank and the Lake.  anomaly' i s t h e d i f f e r e n c e between t h e observed from t h e heat  i n Figure  the approximation  ( u n i f o r m change i n h e a t  illustrated  temperature p r e d i c t e d  are p l o t t e d  o f s i m i l a r values f o ra tank  as i t i s f o r t h e t a n k .  similar  (1968)  column f o ra c r o s s  flow through  The  'temperature  temperature  a n d t h e mean  the s u r f a c e , assuming  no  25  cCT 30 • C  °  o o °  —x^^-^. ^JT^_  SECTION D 20  T5  I  '*  'C* B A B 0  '  'D' BAR  /  UJ  0  /  /  MEAN for LAKE  SECTION E  o 'Or LAKE  ONTARIO  ui  5  o"  "  10  DISTANCE  £  1  30  from N O R T H S H O R E  h .  0  1  20  (km)  EXPERIMENT RUN A  £40.  measured  2 it!  1 r  O20h o  - ^  X  "  "BAT  MEAN for TA  N K - ^  < LU  <J  oi0  40  DISTANCE  Figure 12.  60  along T A N K  120  (cm)  Changes i n heat content f o r Lake Ontario ( f o r sections shown i n Figure 2, p.4). Also a s i m i l a r plot f o r the tank f o r Run A.  40  26 DISTANCE  Figure 13.  along TANK  Temperature anomaly s e c t i o n s f o r the tank e x p e r i m e n t ( 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 i g u r e 2 , p . 4 ) .  advection. Therefore occur  As  c a n be  advective effects  i n the Lake.  shown i n A p p e n d i x  2.6  seen from the  figure,  s i m i l a r to those  Anomaly s e c t i o n s f o r the  content and  i n the  tank  other experiments  could  are  B.  b e e n shown a r e a s o n a b l e  o f a w a t e r column and  Lake O n t a r i o  horizontal  can be  strong v e r t i c a l  the b a s i c f e a t u r e s of the A  f u r t h e r refinement  including these and  the e f f e c t s  effects  tant advective  thermal to the  In both  light,  the  density not  This  tank  and  gradient sufficient  tank  negligible  i n the  deeper  and  These produce  bar. first  approximation The  would  require  importance  the d i f f e r e n c e s between the  ( F i g u r e 6,  i n the  p.15).  tank. One  i n the s h a l l o w end.  results  actual  Three  i s the This  of  impor-  advection  i s a flow  i n a d v e c t i o n o f heat towards the b a r  the Lake.  This  f l o w i s a i d e d by  toward the bar, but f o r the  flow  from the v i c i n i t y  f o r the  to cross  f l o w beneath the above a d v e c t i o n . s h a l l o w end  the  convective processes.  s e e n i n t h e L a k e and  regions e x i s t  heat  of  upper thermocline water r e p l a c i n g p a r t of the s i n k i n g w a t e r  the bar.  both  cases,  of h o r i z o n t a l advection.  o f warm w a t e r t o w a r d s t h e b a r the  f o r both  i n h i b i t h o r i z o n t a l advection  i s i l l u s t r a t e d by  predicted values  t o the  from the assumption of  diffusion.  convections  approximation  the speed o f the b a r  obtained  a d v e c t i o n and  first  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  is  observed  similar.  Discussion As h a s  at  the s e c t i o n s are  tank  the b a r .  This brings  of the b a r ;  for  the h o r i z o n t a l this  density  There i s a  cold water to  gradient  counterthe  that i s , heat f l u x of  the  same s i g n .  The  o t h e r a d v e c t i v e r e g i o n i s the downslope flow o f 4°  w a t e r from the v i c i n i t y  o f the b a r toward  heat t o the deep r e g i o n s .  F o r the tank t h i s  weaker f o r s m a l l e r bottom s l o p e s . turbulent mixing.  the deep end. flow was  I t s e f f e c t was  slope, in  found t o be  a l s o reduced  F i g u r e 12 (p.25) s u g g e s t s t h a t t h e r e was  h e a t a d v e c t e d towards the c e n t r e of Lake O n t a r i o . tance o f t h i s  flow i n l a k e s i s d i f f i c u l t  by  some  However the  t o judge because  the unknown n a t u r e o f the c o n v e c t i v e m i x i n g , and  the bottom.  This advects  impor-  o f the s m a l l e r  irregularities  In s p r i n g , the n e t e f f e c t o f a d v e c t i o n i s t h a t the h e a t  c o n t e n t i s h i g h e r than would o t h e r w i s e be e x p e c t e d i n the c e n t r a l p o r t i o n s of  the Lake and i n the r e g i o n s next t o the b a r , w h i l e r e g i o n s n e a r  s h o r e have been d e p l e t e d o f h e a t .  The  advective v e l o c i t i e s  the  observed i n  the tank are a l l of the same o r d e r o f magnitude as the speed o f the b a r . The i n f l u e n c e o f bottom i r r e g u l a r i t i e s was In b o t h a s l o p e o f 5° was s e c t i o n i n one other.  used  t e s t e d i n two  f o r h a l f the tank, f o l l o w e d by a  flat  and by a v e r t i c a l drop o f 12 cm t o a f l a t s e c t i o n i n the  The main i n f l u e n c e of bottom i r r e g u l a r i t y was  the b a r ; a d v e c t i o n produced The speed observed was  an a v e r a g i n g e f f e c t  roughly that which,  the a c t u a l s l o p e s .  Otherwise  were s i m i l a r t o the o t h e r s . Thus bottom i r r e g u l a r i t i e s  The  of  irregularities.  from e q u a t i o n 2.5.1, would  the temperature results  on the speed  over the  have been a s s o c i a t e d w i t h a bottom s l o p e t h a t was of  experiments.  the w e i g h t e d  average  and c u r r e n t s t r u c t u r e  are g i v e n i n Appendix  C.  do n o t seem t o be of p r i m a r y importance i n  the development o f the t h e r m a l b a r i n l a k e s . An i m p o r t a n t e f f e c t n o t s t u d i e d i n t h i s experiment In  i s wind m i x i n g .  s m a l l e r l a k e s the r e s u l t i n g wind d r i v e n h o r i z o n t a l a d v e c t i o n s would  l i k e l y keep the temperature uniform and not permit a bar to develop. A t r i p was made to the M e r r i t t region of B r i t i s h Columbia to see i f the bar could be found i n 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 i n mixing across the bar.  I f 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 b a r r i e r separating the two, produced 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 d e t a i l e d observations of this phenomenon i n lakes.  Information on the  current structure i s p a r t i c u l a r l y necessary.  2.8  Summary In this laboratory study, a temperature structure has been pro-  duced which i s s i m i l a r to that associated with the thermal bar i n lakes. The existence of my  'bar' i n the tank depends e n t i r e l y 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 i n the lakes, i s based on h o r i z o n t a l heat advection and d i f f u s i o n not being of primary importance.  The success of this model indicates  that effects of wind mixing, C o r i o l i s force, etc., which e x i s t for lakes but not i n the model, are not important factors i n the formation of the 1  thermal bar'.  3.  MATHEMATICAL MODELS  These t h e o r e t i c a l s t u d i e s are an attempt quantitatively  the d e t a i l s  o f the temperature  t o e x p l a i n more and v e l o c i t y  fields  o b s e r v e d i n the l a b o r a t o r y s t u d i e s and to u n d e r s t a n d the dynamic b a l a n c e o f the flow.  The  results  may  then be  tentatively  extended  t o d e s c r i b e the s i t u a t i o n i n l a k e s .  3.1  The Temperature F i e l d From the two-dimensional  l a b o r a t o r y model r e a s o n a b l e f i r s t  appro-  x i m a t i o n s to the h e a t c o n t e n t and t o the speed o f the b a r were o b t a i n e d by assuming 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 were n o t o f p r i m a r y importance.  This suggests  t h a t the temperature  distribution  may  be d e r i v e d from the o n e - d i m e n s i o n a l h e a t d i f f u s i o n e q u a t i o n :  9T  K  3t  (3.1.1)  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 and  i n °C,  K i s the t h e r m a l d i f f u s i v i t y  w i t h boundary c o n d i t i o n s  9T  K 3dz  z = 0  =  q  at  =  0  at z = -D  (top)  (bottom)  (3.1.2)  (3.1.3)  where q i s t h e heat equation  flux  2.5.1, p . 1 7 ) .  through  t h e s u r f a c e i n C° cm s e c  ( = Q/pc^,  1  D = ax where a i s the s l o p e o f the bottom and  x i s the d i s t a n c e along  t h e tank  from t h e s h a l l o w end.  that D = 0 a t the shallow end, e q u i v a l e n t t o a shore).  (This  assumes  Finally,  T = 4°C a t z = 0 a t t h e ' b a r ' . 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 relative  to the temperature  equation  and boundary c o n d i t i o n s can be  3  9  at  =  9 K  8  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  2 ( 3  1  (3.1.5)  | i = dz  0  (3.1.6)  at  z = -D  at  z = 0, a t t h e ' b a r ,  8  1  =  0  (3.1.7)  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  field  may b e  considered  two r e g i o n s , o n 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  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 .  convective  F o r the purposes of the theore-  t i c a l w o r k , t h e b a r w i l l b e d e f i n e d b y t h e 4° s u r f a c e i s o t h e r m a s i s done i n l a k e s t u d i e s .  i)  The d e e p , c o n v e c t i v e  side  The a i m h e r e i s t o p r o v i d e to  4 )  q  z = 0  in  - '  =  at  where  rewritten:  7T  38 K ~ dz  are  the temperature  field  a rough, b u t reasonable,  on t h e u n s t a b l e s i d e o f t h e b a r .  approximation No  attempt  w i l l be made t o c o n s i d e r c o n v e c t i v e e f f e c t s i n d e t a i l , n o r t o model the s h a l l o w t h e r m o c l i n e t h a t d e v e l o p s o v e r t h i s end i n the l a t t e r s t a g e s o f 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 .  I n s t e a d my  r e a s o n i n g w i l l be based on the o b s e r v a t i o n t h a t the t e m p e r a t u r e i n the deeper r e g i o n o f b o t h tank and l a k e tends t o be n e a r l y u n i f o r m v e r t i c a l l y , i m p l y i n g a thorough m i x i n g p r o c e s s and hence a h i g h v e r t i c a l eddy ( o r convective) d i f f u s i v i t y .  I n t e g r a t i n g e q u a t i o n 3.1.4  vertically  from  z = -D t o z = 0 one o b t a i n s  ,,36  89  dt  dz  and u s i n g the z boundary  _96 3t  or  =  +  - 6 o  where -0  (3.1.8) -D  conditions:  £ D  =  9  0  S£ D  i s the temperature i n C° a t t = 0.  f o r t h e speed o f the b a r ( e q u a t i o n 2.5.1,  S  U s i n g the s i m p l e model  p.17)  -  (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  s h a l l o w end and a p o s i t i o n x a l o n g the t a n k , a t time t = 0 and c h o o s i n g the  origin  of time t o be when the 'bar' i s ' a t ' x = 0, t h e t e m p e r a t u r e  33 f o r the deeper  6  This  r e g i o n can be w r i t t e n :  =  0 o  ( —  x  - 1 )  (3.1.9)  c o u l d have been w r i t t e n  empirically  from the o b s e r v a t i o n a l  data. If  t = 0 i s d e f i n e d , as a b o v e , as t h e t i m e w h e n t h e ' b a r ' i s  x = 0, t h e n 8 o The  assumption  the e n t i r e  i s clearly i s made  fluid  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 ,  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 ,  i s a t a uniform temperature;  f o r the tank experiments, is  'at'  and r o u g h l y v a l i d  this  i s obviously true  f o r the lakes.  I fheating  uniform over the s u r f a c e o f the water, i f the bottom has a constant  s l o p e , and i f the assumption diffusion  of heat  a t any t i m e u n t i l  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  are not of primary importance,  then the temperature  t h e b a r appears w i l l be a l i n e a r  function  o f x.  Thus a t t = 0 o n e c a n w r i t e  e  n  "  where  0  e  o  -  "  x  X  i s the temperature  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  tank o r t h e d i s t a n c e from t h e shore  t o t h e deeper  portions  of a  lake.  T h u s e q u a t i o n 3.1.9 may b e r e w r i t t e n a s  0  =  f  ( S t - x)  f o rx > St  (3.1.10)  34  ii)  The s h a l l o w , The  stable side  temperature f i e l d  /  on t h i s s i d e  c o m p l i c a t e d t h a n on t h e deeper s i d e . the  two r e g i o n s ,  the boundary  o f t h e b a r i s c l e a r l y more  Since i ti s desirable  condition  t o match  at the 'bar' (equation  3.1.7)  w i l l b e r e w r i t t e n as  6  =  0  at  t - f  (3.1.11)  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 defined  end and t h e o r i g i n o f t i m e i s  a s w h e n t h e ' b a r ' i s ' a t ' x = 0.  The s o l u t i o n t o  99 3t  for  (3.1.4)  K  a s e m i - i n f i n i t e body w i t h boundary  at  z = 0  K  conditions  89 8z = q  (3.1.5)  a n d 3.1.11 i s  9  x < St  (3.1.12)  S  If  an e r r o r i n t h e b o t t o m boundary  condition  (3.1.6) i s ' t o l e r a t e d '  up to a heat flux that i s 20% of the surface heat flux, then the above represents the temperature to within 20 cm of the shallow end of the tank f o r a 5° bottom slope.  This 'error' i n bottom heat flux i s l i k e l y  not more than the error involved i n neglecting h o r i z o n t a l advection of heat i n the shallow end. An attempt to consider the d i f f u s i o n equation f o r a wedge (see A r l i n g e r , 1965) would not improve the temperature s o l u t i o n as the bottom e f f e c t s tend to make the shallower portions more uniform i n temperature v e r t i c a l l y .  This i s not the case i n the observations.  Neglecting the bottom heat flux i n this manner i s equivalent to removing some of the heat input i n the shallower portions and thus approximates  the advective e f f e c t s observed i n this region.  In order to calculate actual values from this mathematical model i t i s necessary to choose a value f o r K.  Since the motion appears to  be laminar i n the tank, K ought to have the value of molecular thermal -3 d i f f u s i v i t y , 1.4 x 10  2 - 1 cm  sec  .  However, i n 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 i n the above model by increasing K. • -  iii)  \  Comparison with the tank The temperature model developed above i s  9  x < St (3.1.  To compare this to the measured temperature the bar had to be set.  f i e l d the p o s i t i o n of  In the generalized temperature  f i e l d shown  i n Figure 6 (p.15), extending the 4° isotherm to the surface shows the 'bar' at 90 cm. speed of the bar.  This defines the value of t as 90 times S, the The value f o r the speed of the bar i s taken to be  -2 -1 the average value observed i n the tank; that i s , 3.9 x 10 cm sec -3 -1 q i s taken to be the standard heating rate of 9 x 10 C° cm sec -1 -3 (assuming pc^ - 1 c a l C°  cm  ).  The value of 0 i s obtained using  the formula f o r the speed of the bar (2.5.1), the average  observed  value of S, the standard value f o r q, the length of the tank, and the depth at the end of the tank; this gives a value of 2.6 C°. I t remains to determine a value of K; a reasonable f i t f o r the temperature  cross-section was  -3 5.8 x 10  -2 cm  found using a value f o r K  of  -1 sec  , see Figures 14 and 6(p.l5).  This reasonable  f i t with the tank results together with general s i m i l a r i t y with the lake suggests the temperature  model i s good f o r the lake too.  This  w i l l be discussed i n more d e t a i l at the end of the t h e o r e t i c a l section (see subsection 3.4). In  order to further check on the degree to which the model matched  the general p i c t u r e i n the tank, 0 was p l o t t t e d against distance down the tank at f i x e d values of depth f o r both the tank and the model (see Figure 15).  mathematical  In order to make the p l o t f o r the tank the  4.5 and 3.5°C isotherms were extended  to i n t e r s e c t with the surface  rather than form a shallow thermocline over the deeper region.  The  reason f o r the i n t e r e s t i n the 9 versus x p l o t w i l l become apparent when the v o r t i c i t y equation i s considered i n the next subsection.  As  F i g u r e 14.  Temperature s e c t i o n  (°C) c a l c u l a t e d from e q u a t i o n s 3.1.13.  38  Figure  15.  6 i n C° a g a i n s t d i s t a n c e a l o n g the tank f o r f i x e d v a l u e s ( i n cm). (a) g e n e r a l i z e d measured v a l u e s (b) c a l c u l a t e d from e q u a t i o n s 3.1.13  of z  can be seen the curves are s i m i l a r .  3.2  The V e l o c i t y F i e l d The v e l o c i t y f i e l d i s most readily investigated through the  v o r t i c i t y equation:  (3.2.1)  where w  i s the v o r t i c i t y vector,  the density, p  i s the pressure,  v  i s the v e l o c i t y vector,  and V  p is  i s the v i s c o s i t y .  Assuming two-dimensional flow, neglecting non-linear  advective  effects and making the hydrostatic approximation f o r pressure, equation 3 . 2 . 1 reduces to  9w.  + v V w. 2  9t  p 3x  (3.2.2)  where g i s the value of the acceleration due to gravity,  and  co„ = 2  dz  Tr— dx  where  u  and w  are the x  components respectively. Introducing  3£ u = 9z  a stream function, (}>, where  w  3£ 3x '  and z v e l o c i t y  40 equation  3.2.2 may b e r e w r i t t e n  |_ v<f>  =  * |£ p dx  V* = V . V  .  The f i r s t  2  dt  where  1  2  2  +  o f change o f t h e v o r t i c i t y viscous buoyancy  diffusion.  i s evident  of g r a v i t y , pressure  (3.2.3)  term i n e q u a t i o n  and t h e l a s t  However t h e f i r s t  3.2.3 i s t h e t i m e  term represents  forces are being  tends  further discussion.  considered.  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  gradient which  effects of  t e r m i n v e c t o r n o t a t i o n i s -~r Vp  t h a t the buoyancy that pressure  rate  t e r m on t h e r i g h t h a n d s i d e , t h e  term i s n o t s o f a m i l i a r and perhaps needs  Recalling it  v V>  Vp,  I n the presence  i n a horizontal  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  flows  x  gradient.  tend t o  x lower  p  .  M  .  higher p  +ve  vorticity  Vp  The  effect  If  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  in  a fluid,  those  nearer  the pressure the surface  o f t h e buoyancy  gradients  term  o f t h e same s i g n o c c u r s a t the bottom w i l l  due t o t h e s t r o n g v e r t i c a l  a t a l l depths  be l a r g e r  density  than  gradient  ( -pg under the h y d r o s t a t i c approximation, v a l i d f o r the tank and lake). In the tank gradients i n the directions shown above are s e t up on the shallow side due to the e f f e c t s of surface heating.  Flow  towards the shallow end results i n a s l i g h t surface slope downwards towards the bar which i s s u f f i c i e n t to produce the flow i n the p o s i t i v e x - d i r e c t i o n i n the upper regions of the tank.  vorticity  shallow end  In the deeper end the density gradients are reversed, r e s u l t i n g i n flows i n the opposite directions (and v o r t i c i t y of the opposite s i g n ) . Because of v i s c o s i t y a shear develops which opposes the flow, producing v o r t i c i t y of the opposite sign to that produced by the buoyancy term.  This occurs mainly on the boundaries of the tank  although the i n t e r i o r contribution, f o r the low Reynolds number flow considered, i s also important. Consider the operator  V* = 3 * + 2 3 3 1  1  z  2  2  x z  + 3* 1  i n equation 3.2.3.  X  Since the v e r t i c a l shears greatly exceed h o r i z o n t a l 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 l a s t two terms of V and approximate equation 3.2.3 by  i _ 3 f i  3t 3z  =  IIP  +  v  p 3x  2  i ^ i dz  (3.2.4)  h  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 l e a s t h e u r i s t i c a l l y , by  i ! i  =  __8_i£  (3.2.5)  V p 3x  dz * 1  Given a density f i e l d , p(t,x,z) and appropriate boundary conditions, equation 3.2.5 may be solved f o r the stream function and hence f o r the flow pattern of this model. The density of fresh water i n the region of i t s maximum density may be expressed as  p  =  p  ( 1 -• A 6  2  )  (3.2.6)  —3  where p^  i s the density i n gm cm  —6  at 4°C and  A - 8 x 10  —2  C°  Substituting 3.2.6 i n t o equation 3.2.5 gives  1 ^ dz  -  k  where I have used  2 g A v  ^ 3x  (3.2.7)  e  p - p  when not d i f f e r e n t i a t e d .  43 The b o u n d a r y top  conditions  and t h e b o t t o m  at  (recalling  z = 0,  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 the e f f e c t of surface contamination):  u = 0  and  w = 0  u = 0  and  w = 0  (3.2.8) and  a t z = -D, (bottom)  where D = ax  where  a i s the s l o p e of the bottom.  Specifying a suitable matter. field  O b v i o u s l y e q u a t i o n 3.2.7  given separately  be p o s s i b l e The  temperature  field  i s not such  cannot be used w i t h  f o r t h e two s i d e s  field  n o t o n l y awkward,  3.1.13 s u g g e s t e d p r e v i o u s l y  due t o t h e p r e s e n c e  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 a s may b e s e e n  i n what i s e s s e n t i a l l y  across the b a r .  (p.35)  i s thus  throughout the  t o the chosen  from 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 the f o r c i n g  term  ( s e e e q u a t i o n 3.2.7).  r e a s o n a b l e compromise w h i c h would be b o t h  readily  reasonable representation  8^ Thus  f o r the temperature  plots  model.  temperature  much c a r e a n d t r i a l was n e c e s s a r y t o f i n d  the  not  of the e r r o r function, but  u n m a n a g e a b l e 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  field,  temperature  o f t h e b a r as i t w o u l d  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  temperature  a  a simple  field  i n t e g r a b l e and a  of the experimental r e s u l t s .  Looking at  ( 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  temperature  field  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  h y p e r b o l i c tangent functions.  The  a reasonable f i tto the v e l o c i t y shown i n t h e g e n e r a l i z e d  compromise w h i c h  field  the use o f  finally  produced  f o r the bar at the p o s i t i o n  o b s e r v a t i o n s ( F i g u r e 6, p . 1 5 )  (that i s ,  a  at  90 cm a n d a 5° b o t t o m  9  *  in  C°.  slope) i s  {[ 1 + tanh(0.392[  This i s plotted  - 2 ] ) ] ( 2 . 9 ) ( 1 + 0 . 2 z + O . O l z ) - 1}  i n c r o s s - s e c t i o n i n °C a n d 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. approximation to both and t h e c a l c u l a t e d  (3.2.  2  As c a n b e s e e n  the observed  (Figure  t h i s produces  a reasonable  ( F i g u r e 6, p . 1 5 a n d F i g u r e 1 5 , p . 3 8 )  14, p.37, and F i g u r e 15, p.38)  temperature  fields. The in  analytical  temperature  field,  e q u a t i o n 3.2.9, was t h e n  e q u a t i o n 3.2.7 a n d t h e s t r e a m f u n c t i o n a n d t h e v e l o c i t y  calculated(see Appendix F i g u r e s 17 a a n d b .  E).  The r e s u l t i n g v e l o c i t y  As i n F i g u r e 6 ( p . 1 5 )  i n d i c a t e motion expected i n a minute as u s e d  f o r the axes.  Comparison  agreement between the two. has been i n c l u d e d correct  (dashed  directions  (i.e.  field  field  the l e n g t h s o f the arrows cm m i n ^) i n t h e same s c a l e s  o f F i g u r e s 1 7 a and 6 shows  The v e l o c i t i e s  and t h e change i n d i r e c t i o n  by  t h e l a b o r a t o r y model s t u d i e s .  in  t h e r e g i o n o f maximum d e n s i t y velocities The  are a l li n the  of the flows along the suggested  as w a s o b s e r v e d i n t h e t a n k . as t h o s e  d i f f e r e n c e s between the c a l c u l a t e d  can be e x p l a i n e d f i r s t i n terms  field  The l a r g e s t downward v e l o c i t i e s  a r e o f t h e same m a g n i t u d e  tangent temperature  close  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  lines).  were  i s shown i n  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  the  used  occur  In general  observed.  and observed v e l o c i t y  fields  of the inadequacies of the hyperbolic  approximation.  The m a g n i t u d e  of the calculated  X  Figure 16.  (cm)  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 (a) cross-section (T i n °C) (b) 9 i n C° against x i n cm for fixed values of z (in cm)  calculations:  X  F i g u r e 17a.  (cm)  Calculated velocity f i e l d . V e l o c i t i e s a r e shown by arrows, t h e i r l e n g t h i n d i c a t i n g the motion i n a minute ( i . e . cm m i n " ) i n the same s c a l e s 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 a r e 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 w i t h F i g u r e 6 (p.15). 1  ON  X (cm) 0  -2  ~-4 E u NI  ISOTHERMS (°C) VERTICAL VELOCITY •8  10-  Figure 17b.  Calculated velocity 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" ) 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 velocity.  velocities  drops  o f fa t the extrema  of the plot since  the hyperbolic  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 a n d m i n i m u m 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 not.  the observed  temperatures d i d  The v e l o c i t i e s b e t w e e n 50 a n d 70 c e n t i m e t e r s a l o n g t h e t a n k  are  up t o t w i c e a s l a r g e as t h o s e s h o w n i n t h e g e n e r a l i z e d  field  ( F i g u r e 6, p . 1 5 ) .  90/9x  to the fact  i s l a r g e r than that f o rthe h y p e r b o l i c tangent  model i n t h i s velocity field, the  T h i s can be a t t r i b u t e d  region.  field  An e v e n  temperature  temperature  c l o s e r match t o t h e o b s e r v e d  fields  p o i n t i n such  somewhat f r o m t h e a v e r a g e  f r o m t h e deep e n d t o t h e  time t h e b a r p r o g r e s s e d about  in  similar  field  observed f o r t h e b a r i n Lake  will  Ontario (Rodgers,  fact  chosen  that the temperature  magnitude.  field  tangent approximation  of the b a r that i s not w i t h i n  f o r the calculation. changes w i t h  Such a p p r o x i m a t i o n s c o u l d be based  p r e d i c t e d by t h e mathematical model (equations The  Variations  1968).  that a different hyperbolic  10 cm o f t h e p o s i t i o n  10 cm.  t o those found i n the tank have been  n e e d t o b e made f o r a n y o t h e r p o s i t i o n  about the  t o be noted  Also the  a r e n o t an i n s t a n t a n e o u s measurement b u t  s h a l l o w end d u r i n g which  ought  i n that  f i e l d s would hence a l s o v a r y  v a l u e s shown i n F i g u r e 6 ( p . 1 5 ) .  temperature sections  the temperature  temperature  a t e d i o u s endeavour  t h e temperatures were taken i n sequence  It  generalized  o b s e r v e d i n t h e tank do v a r y b e t w e e n runs ( s e e  F i g u r e 18) a n d t h e a s s o c i a t e d v e l o c i t y  rather  that  c o u l d p r o b a b l y b e made b y u s i n g a more i n v o l v e d  however t h e r e i s l i t t l e  observed  velocity  s t r e a m f u n c t i o n cf) i s p l o t t e d  T h i s i s due t o  time, p a r t i c u l a r l y i n  on t h e t e m p e r a t u r e  field  3.1.13).  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 a n y 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> c a n b e s e t  -2  20  40  60  80  X  Figure 18.  (cm)  100  120  Actual 6 ( i n C° ) against distance along tank ( x, i n cm ) p l o t t e d from laboratory observations.  o  51 equal  t o zero Figure  that this heating as  a t t h e top and bottom.  19 m u s t b e i n t e r p r e t e d w i t h  care.  One m u s t k e e p i n m i n d  i s t h e p i c t u r e o f cJ) a t t = 9 0 / S i n t h e t a n k .  continues,  t h e 4°C s u r f a c e  driven velocity  the temperature f i e l d isotherm  field  progresses  As t i m e  increases  changes and t h e ' b a r ' d e f i n e d  down t h e t a n k .  i s a l s o m o v i n g down t h e t a n k ,  Thus t h e d e n s i t y  changing slowly.  In  t e r m s 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 of c l o s e d s t r e a m l i n e s and  a r e moving along  hence must b e 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 .  derive the circulation, neglected  while  The a s s u m p t i o n , u s e d t o  t h a t t h e 3/9t t e r m i n e q u a t i o n  3.2.4 c a n b e  i s thus o n l y v a l i d when changes a r e s l o w enough.  set of streamlines  on t h e s h a l l o w  The  s i d e must a l s o s l o w l y expand  this  laboratory observations  flow at the bar. a b o v e 4.5°C.  This  Looking  which c l e a r l y  showed t h a t w a t e r from t h e  was n o t i n v o l v e d i n a n y m i x i n g  stable thermocline  involves  o r downward  the water a t temperature  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 g u r e 17b, p.47) from 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 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 m o d e l i s a b o u t 2.5 cm m i n \ s e e n t h a t t h e downward v e l o c i t i e s  has  horizontally  i n m i n d one c a n 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  warmer, s t a b l e t h e r m o c l i n e  only  negative  t h e p o s i t i v e s e t on t h e deep s i d e must b e s l o w l y s h r i n k i n g . With  the  the tank a t the speed o f the 'bar'  to thicken the thermocline travelled  roughly  underneath  region.  i t can be  probably  This  serve  I n f a c t , by the time the b a r the w  a l l be p o s i t i v e .  t h e warmer w a t e r on t h e s h a l l o w  the b a r from t h e deeper s i d e .  that the  i n t h e r e g i o n w a r m e r t h a n 4.5°C  a f u r t h e r 10 cm down t h e t a n k ,  o b s e r v e d a t , s a y , 70 cm, w i l l  field  velocities  The  flow  s i d e i s f e d by the flow  model i s n o t steady  s t a t e , thus  towards  streamlines are not pathlines. The streamlines shown i n Figure 19 (p.49) close more rapidly at the two ends of the tank i n 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  V a l i d i t y of the Velocity Model Before extending the mathematical model to compare i t with the lake  i t i s necessary to consider the s i z e of the neglected terms i n the v o r t i c i t y equation.  ^ + v . 3t ~  where  3u. u)„ = 2 3z  Now  Vu,  3w -r— . dx  including advection terms, 3.2.2 becomes  =  -S^p 3x  +  v ( - ^ - + -^-W 9x 9z 2  2  (3.3.1) 1  The s i z e of these terms was evaluated at several  positions along the tank from the v e l o c i t y f i e l d calculated from equation 3.2.7.  This thus provides an i n t e r n a l consistency check on the s o l u t i o n  of equation 3.2.7  1)  and therefore on the assumptions  used to derive i t .  3w 3u - j ^ with respect to i s always less than 1:10 and usually less than 3w 3 xx 1:100. Thus the assumption that TT- i s small with respect to -r— dx dz i s reasonable  everywhere. 9 u 2  2)  In the viscous term  was neglected with respect to  r a t i o of these two terms i s also 1:10  or less  9 -^z  everywhere.  2  .  The  3)  The be  9/9t  term can be  generally  less  term i s zero;  compared t o t h e buoyancy  t h a n 1:10  that i s ,  t e r m and i s f o u n d  e x c e p t , o f c o u r s e , where the  r i g h t a t the b a r  (8 = 0 )  and  to  buoyancy  at the  extrema 99  of the h y p e r b o l i c The  latter  lies well  to c a l c u l a t e depends on 4)  tangent temperature a p p r o x i m a t i o n where t o the extrema  the s i z e  o f t h e 9/9t  of the p l o t s .  term s i n c e  0.  I t i s possible  the buoyancy  term  tenth  buoyancy  time.  N o n - l i n e a r terms  are generally  less  t h a n one  term e x c e p t , of c o u r s e , where the buoyancy above.  =  I n the immediate  vicinity  v o r t i c i t y balance would s h i f t  t e r m i s z e r o , as  o f t h e 4°C  to include  of the  mentioned  i s o t h e r m (0 = 0 )  9/9t,  a d v e c t i v e and  the viscous  terms. 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 to 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 b e t w e e n t h e c a l c u l a t e d  velocity t o the  3.4  f i e l d s w o u l d s u g g e s t and  and  appear  observed  the 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  circulation.  Extension to a The  Lake  laboratory studies  suggested that  a s e c t i o n o f a lake from shore t o deeper of about  1/1000 ( s e e p . 1 2 ) .  The  of about  1/20,000.  The  average speed  times t h a t i n the tank w i t h  f o r z, x , a n d  t and  assuming  the tank c o u l d be  viewed  regions using a v e r t i c a l  5° b o t t o m s l o p e ,  s h o r e bottom s l o p e s o f Lake O n t a r i o ,  20  the mathematical model  as  scaling  compared t o t h e n o r t h  then suggests a h o r i z o n t a l  scaling  o f the b a r i n Lake O n t a r i o i s about  t h e 5° b o t t o m s l o p e .  Using these scales  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 the l a k e s s i m p l y by  the value  of v e r t i c a l  making the a p p r o p r i a t e s c a l e  eddy d i f f u s i v i t y  must be  2 vertical that  eddy v i s c o s i t y  t h e s e v a l u e s be  a b o u t 38  different,  the s h a l l o w s i d e of the Lake M i c h i g a n  2  sec  sec  .  Now  i t i s not  1-10  and  unreasonable  i n a stable situation  In fact, experimental values 2 - 1  are  -1  -1  especially  'bar'.  ( H u a n g , 1969)  cm  a b o u t 6 cm  changes,  cm  sec  as  given  for vertical  eddy  on for  diffusivity  2 - 1 a n d 1-100 cm sec 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 (vertical e d d y d i f f u s i v i t y t o v e r t i c a l e d d y v i s c o s i t y ) m i g h t b e more r e a l i s t i c . If 2 -1 the v e r t i c a l  eddy d i f f u s i v i t y  is arbitrarily  when a p p l i e d t o e q u a t i o n  3.1.13 d o e s n o t  f i e l d , the corresponding 2 -1 2 1 cm sec .  vertical  s e t a t 11 cm  sec  g i v e an u n r e a l i s t i c  eddy v i s c o s i t y ,  ,  which,  temperature  from equation  3.2.7, i s  0 1  S u c h a n e x t e n s i o n as dimictic be  freshwater  expected  to  this  l a k e s , such  In  the v o r t i c i t y  applied the  equation  vorticity,  From V e r o n i s '  in  by  3.2.1  (u) +  the Great  o n l y be  Lakes,  applied to large  i n which  the b a r  i s a p p l i e d to a lake there are  could  associated with  the C o r i o l i s  these  are i n c l u d e d by  r e p l a c i n g OJ, t h e  2fi) w h e r e 20, i s t h e  fluid,  the  ratio  t h e r m a l b a r phenomenon i s s u c h parallel  the d e n s i t y induced  circulation  force.  vorticity'. theorem  as  scales  t h a t t h e component o f p l a n e t a r y  of the v e r t i c a l pressure  'planetary  of h o r i z o n t a l to v e r t i c a l  t o t h e e a r t h ' s s u r f a c e may  Because of the presence  additional  those  ( 1 9 6 3 ) e s t i m a t e s , b a s e d on E r t e l ' s  to a s t r a t i f i e d  vorticity  (f),  3.2.7  t h a t have been n e g l e c t e d ,  relative  as  of course,  exist.  However, when e q u a t i o n terms  should,  be  safely  neglected.  component o f p l a n e t a r y  gradients i n a lake w i l l  tend  vorticity to  be  55 balanced  by  to achieve  a geostrophic  may  n o t be  the  circulation  fossil  This  attained while  t h e b a r may  continue  validity  and  t h a t the tank  of equation  relative size are  the buoyancy  The  and  due  to the  was  t e r m s may  tank.  Nevertheless  the p l o t  of the  of the  different  as  near the bar  the mathematical of what occurs f o r the  than  4°C  being  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  p.29  and  A p p e n d i x D). cells,  As  observed  i n this  i n the  case f i x e d  or the  in  the  vertical between  used i n e q u a t i o n of the  3.2.7  bar.  i n Lake O n t a r i o  than  model probably  gives  i n the  f o r example, the  1880)  two  (mentioned e a r l i e r )  the  lakes since  the  zero surface v e l o c i t i e s  Geneva ( F o r e l ,  suggests  from those  to lakes i t i s worth remarking  'stationary thermal bar';  evaluating,  I t i s possible  calculated streamlines, Figure  l i g h t of a possible extension  situation  associated with  terms i n  'bar'.  r i g h t i n the v i c i n i t y  assumptions are n e a r l y s a t i s f i e d , except  the  of the  the approximated viscous e f f e c t s again  model.  disappeared.  of the neglected  be  scale  i n the  found t h a t the b a s i c balance  i f rough, approximation  Considering  as  density gradients  i n the v i c i n i t y  I t was  bar  a s h o r t time  d i f f e r e n t s c a l e s u s e d f o r h o r i z o n t a l and time.  necessary  checked f o r Lake O n t a r i o by  n o n - l i n e a r t e r m s a r e more i m p o r t a n t  a reasonable,  t h a t on  the b a r has  3.2.7  of these  is  o f the  t o t h e same d y n a m i c b a l a n c e  and  f o r the l a k e , except  they were i n the  the p o s i t i o n  i s m o v i n g and  o f 3.2.7, t h e s i z e  deeper regions  since there  valid  due  of 0 ( l / f )  to b e l i e v e that geostrophic e q u i l i b r i u m  to e x i s t a f t e r  d i s t a n c e s as w e l l as  is  the b a r  circulation  from the s c a l e d r e s u l t s shallow  l e a d s me  i s roughly  geostrophic  The  However a time  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  changes a p p r e c i a b l y .  A  current.  (p.16).  19(p.49) i n on  case of  the Lake  case of e f f l u e n t s warmer 4°C  (see s e c t i o n  laboratory, Figure  i n position, with  2,  19(p.49)  clockwise  motion  on the shallow side and counter-clockwise motion on the deep side and mixing and sinking between the two c e l l s .  I t must be emphasized that  this i s NOT what has been observed f o r the 'migrating thermal bar' i n the laboratory model.  Streamlines are pathlines only when flow i s  stationary.' I t would appear from the way i n which the mathematical model as w e l l as the tank can approximate the temperature Lake Ontario, that the v e l o c i t y  f i e l d s observed i n  f i e l d observed i n the laboratory and  calculated from a simple v o r t i c i t y balance which also holds f o r the lake  might also be that associated with the 'bar' i n lakes.  i t w i l l probably be very d i f f i c u l t to determine  However  this by d i r e c t  measurements as the v e l o c i t i e s 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 d i r e c t l y associated with the thermal bar.  57  4.  SUMMARY  The  'migrating thermal bar' phenomenon which i s 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 a twodimensional laboratory model.  The temperature  f i e l d s agree with  those  observed i n the Great Lakes. A l i n e a r model i s used to describe the speed of the 'thermal bar'. The effects of v a r i a t i o n 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  to agree with this simple model.  found  The l i n e a r model gives a reasonable  f i r s t approximation to the speed of the 'thermal bar' i n 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 p o s s i b l e that the associated observed v e l o c i t y f i e l d s may  do this as w e l l .  the 'migrating thermal bar' temperature  On the shallow side of  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. the deep end was In  In the laboratory model a mean flow towards  observed i n the thermocline with a counter-flow  underneath.  the deep end of the model a flow near the surface t r a v e l l e d toward  the stable thermocline.  This current sank at the front edge of the  thermocline and divided, a part feeding the upslope current i n the shallow end, the other part providing the downslope current i n the deep end.  The water i n 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 h o r i z o n t a l advection and d i f f u s i o n were not of primary importance a mathematical model was  developed.  F i r s t the temperature f i e l d was  one-dimensional heat d i f f u s i o n equation.  calculated from the  Then the v e l o c i t y f i e l d  was  calculated assuming that the flow was driven by buoyancy forces and balanced by viscous forces.  Since there i s a great s i m i l a r i t y between  the calculated and observed temperature and v e l o c i t y f i e l d s , the assumptions on which the v o r t i c i t y balance i s based are obviously nearly s a t i s f i e d i n the laboratory model. Because of the s i m i l i t u d e between the experimental and calculated temperature f i e l d s and those observed i n lakes, the observed and calculated v e l o c i t y 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 v i s c o s i t y e f f e c t s 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 v e l o c i t i e s  expected i n Lake Ontario would be of the order of 1 cm sec  \  From the laboratory and mathematical studies i t i s 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 i n t o waters cooler than 4°C.  In this case the 'bar' would be  expected not to move and the c i r c u l a t i o n would consist of two with mixing and sinking at the 'bar'.  cells  This i s quite d i f f e r e n t from  the behaviour observed f o r the 'migrating thermal bar' i n the laboratory.  I  59 BIBLIOGRAPHY  Arlinger,  B., 1 9 6 5 . 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 w e d g e w i t h given heat f l u x through i t s bounding s u r f a c e s . S a a b TN 5 9 , S a a b Co., L i n k o p i n g , S w e d e n , 3 0 p p .  C h e r m a c k , E.E., 1970. Study of t h e r m a l e f f l u e n t s i n s o u t h e a s t e r n Lake O n t a r i o as m o n i t o r e d b y a n a i r b o r n e r a d i a t i o n thermometer. P a p e r p r e s e n t e d a t the 1 3 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 L a k e s R e s . ) , B u f f a l o , N.Y., A p r i l , 1970. C h u r c h , P.E., 1 9 4 2 . The a n n u a l t e m p e r a t u r e c y c l e o f L a k e M i c h i g a n . U n i v e r s i t y of Chicago, Department of Meteorology, M i s c . Reports Nos. 4 and 18, 148pp. D a v i e s , J.T. a n d E.K. R i d e a l , N.Y., 474pp.  1961.  I n t e r f a c i a l Phenomena.  D o r s e y , N.E., 1 9 4 0 . P r o p e r t i e s of Ordinary Water-Substance. P u b l i s h i n g C o r p . , N.Y., 327. Elder,  Academic P r e s s ,  Reinhold  F.C. and R.K. L a n e , 1970. Some e v i d e n c e o f m e t e o r o l o g i c a l r e l a t e d c h a r a c t e r i s t i c s of lake s u r f a c e temperature s t r u c t u r e . Paper p r e s e n t e d a t the 1 3 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 L a k e s R e s . ) , B u f f a l o , N.Y., A p r i l , 1 9 7 0 .  Elliott,  G.H. a n d J.A. E l l i o t t , 1 9 6 9 . S m a l l - s c a l e model of the ' t h e r m a l bar'. P r o c . 12th 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., 553-557. and , 19 70. L a b o r a t o r y s t u d i e s on t h e ' t h e r m a l bar'. P r o c . 13th Conf. Great Lakes Res.; I n t e r n a t i o n a l Assoc. Great Lakes Res., ( i n the p r e s s ) .  Forel,  F.A., 1 8 8 0 . L a c o n g e l a t i o n des l a c s s u i s s e s e t S a v o y a r d s pendant l ' h i v e r 1 8 7 9 - 1 8 8 0 , L a c Leman. L ' E c h o des A l p e s , G e n e v e , 3>, 149-161.  Handbook o f C h e m i s t r y and P h y s i c s , 1969-70. C l e v e l a n d , O h i o , p a g e F-4.  The  Chemical Rubber  Co.,  H u a n g , 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 L a k e 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 m o d e l 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 L a k e s R e s . D i v . , S p e c i a l R e p o r t No. 4 3 , 1 6 9 p p . R i c h a r d s , T.L., J.G. I r b e a n d D.G. M a s s e y , 1 9 6 9 . A e r i a l surveys of G r e a t L a k e s 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 M a r c h , 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 Branch, C l i m a t o l o g i c a l S t u d i e s No. 1 4 , 5 5 p .  60 , Rodgers,  and  , 1969-1970.  Unpublished  data.  G.K., 1 9 6 5 a . 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 . 8th Conf. Great Lakes Res.; Univ. M i c h i g a n , G r e a t Lakes Res. D i v . Pub. 13, 358-363. , 1965b.  Unpublished  __, 1966a. The W i n t e r 1965-66. G r e a t Lakes Res.  data.  t h e r m a l b a r i n L a k e O n t a r i o , S p r i n g 1965 and P r o c . 9 t h Conf. Great Lakes Res.; U n i v . M i c h i g a n , D i v . Pub. 1 5 , 3 6 9 - 3 7 4 .  , 1966b. A n o t e on t h e r m o c l i n e d e v e l o p m e n t a n d 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 , S y m p o s i u m o f G a r d a , I . A . S . H . P u b . No. 70, 401-405. , 1967. T h e r m a l r e g i m e 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 Resources o f Canada Symposia, 87-95. , 1968. H e a t 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 . 11th Conf. Great Lakes Res.; I n t e r n a t i o n a l Assoc. Great Lakes Res., 480-486. a n d D.V. A n d e r s o n , 1 9 6 1 . A p r e l i m i n a r y study of the energy b u d g e t o f L a k e O n t a r i o . J . F i s h . R e s . B d . C a n a d a , 18, 6 1 7 - 6 3 6 . and , 1963. The t h e r m a l s t r u c t u r e o f L a k e Ontario. P r o c . 6th Conf. Great Lakes Res.; Univ. M i c h i g a n , G r e a t L a k e s R e s . D i v . P u b . 10, 5 9 - 6 9 . S a t o , G.K.,  1969. P r e d i c t i o n of the time of disappearance of the thermal 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 , 1 1 4 p p .  S v e r d r u p , H.U., M.W. J o h n s o n a n d R.H. F l e m i n g , H a l l , I n c . , N.Y., 80, 81, 105.  1942.  The  Oceans.  Prentice-  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 L a k e L a d o g a . Bull. (Izvestiya) A l l - U n i o n Geogr. S o c , 95, 134-142. Am. G e o p h y s . 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., 1 9 6 3 . 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 t h e equations of motion from a s p h e r i c a l s u r f a c e t o the 8-plane. I I . B a r o c l i n i c systems. J . M a r i n e R e s . , 2_1, 1 9 9 - 2 0 4 .  A P P E N D I X A:  Figure section  Experiment  20 s h o w s a t e m p e r a t u r e s e c t i o n a n d  for surface  temperature the h e a t  Cooling  anomaly  c o o l i n g of water i n i t i a l l y ( s e e p.24)  s e c t i o n i t was  l o s s from the temperature s e c t i o n .  t o m e a s u r e m e n t s made i n L a k e O n t a r i o  temperature  anomaly  a b o v e 4°C.  For  necessary to The  (Figure 20).  The  t h e same as  in  p.15).  experiments  ( s e e F i g u r e 6,  estimate  section i s similar  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 h e a t i n g  the  flow that  pattern encountered  DISTANCE  DISTANCE F i g u r e 20.  from N O R T H S H O R E  along T A N K  (km)  (cm)  Temperature s e c t i o n from Lake O n t a r i o (Rodgers, 1966a); mid-lake N-S s e c t i o n taken mid-January 1966. Temperature and temperature anomaly s e c t i o n s f o r the case o f c o o l i n g ( f a l l s i m u l a t i o n ) w i t h 5° bottom s l o p e .  APPENDIX B:  Temperature  F i g u r e s 21 warming.  Data: c o n s t a n t s l o p e s  to 27 are temperature d a t a f o r runs s i m u l a t i n g  They i n c l u d e a temperature s e c t i o n , mean temperatures,  i n h e a t c o n t e n t , and a temperature anomaly s e c t i o n  A l s o shown are the curves f o r mean temperature  c o n t e n t p r e d i c t e d by the assumptions speed o f the b a r .  o f the 'thermal  and change i n h e a t  used i n the l i n e a r model f o r the  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 t h e tank  and t h e o b s e r v a t i o n s i n Lake O n t a r i o (Rodgers h e a t towards  changes  (p.24) f o r d i f f e r e n t  s l o p e s , d i f f e r e n t h e a t i n g r a t e s , and d i f f e r e n t p o s i t i o n s bar'.  spring  1968);  the a d v e c t i o n of  the deep end o f the tank o r c e n t r e o f t h e l a k e appears t o  o c c u r m a i n l y d u r i n g the e a r l y s t a g e s when the t h e r m a l b a r i s n e a r the s h o r e ( F i g u r e 28 p.77, and F i g u r e 12, p . 2 5 ) . s p e c u l a t i o n as some o f t h i s  However t h i s  c o u l d be due t o t h e n o n u n i f o r m  o f h e a t f l u x o v e r the s u r f a c e o f the Lake.  i s purely distribution  RUN  B  Observations  0  Figure  '  21a.  40  L  _  80  '  120  "cm  T e m p e r a t u r e s e c t i o n a n d 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 ) f o r R u n 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 , a n d T i , 0°C.  4  Figure 21b.  65  Changes i n heat content and temperature anomaly section f o r Run B  F i g u r e 22a.  Temperature s e c t i o n and mean temperatures f o r Run I : 5° bottom s l o p e , s t a n d a r d h e a t i n g , and T i , 1.4°C.  67  Figure 22b.  Changes i n heat content and temperature anomaly f o r Run I.  Figure  23a.  T e m p e r a t u r e s e c t i o n a n d 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 a n d T i , 0°C.  RUN  69  H  •  E u o ^measured  w  ~ 40 -  ^ ^  Z  UJ  z  -  %  -tC*  _ /  —  -*--x  predicted^  BAR  o20|-  <  Ul , X <3 °0  40  80  40  80  120  Cm  Observations  0  DISTANCE  Figure 23b.  along  120  cm  TANK  Changes i n heat content and temperature anomaly f o r Run H.  F i g u r e 24a.  Temperature s e c t i o n and mean temperatures f o r Run C: 2.5° bottom 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.  o  RUN  i 0  i  i  i  40 0ISTANCE  Figure 24b.  C  i 80 along T A N K  i  i 120  Changes i n heat content and temperature anomaly f o r Run C  i cm  Observations  0« 0  1  RUN  1  1  40  1  i  80  DISTANCE Figure 25a.  D  along  i  120  TANK  T e m p e r a t u r e s e c t i o n a n d 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 a n d T i , 0°C.  i  cm  RUN  Figure 25b.  D  Changes i n heat content and temperature anomaly f o r 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 a n d 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 a n d T i , 0°C.  75  RUN  i 0  i  i 40  i DISTANCE  Figure 26b.  E  i 80 along  1  1  1—  120  cm  TANK  Changes i n heat content and temperature anomaly f o r Run E  76  RUN F  Observations  i  0  Figure 27a.  i  i  40  1  1  80  1  1  120  Temperature section and mean temperatures f o r Run F: 7.5° bottom slope, increased heating and T i , 0°C.  1—  Cm  77 R  i 0  1  i 40  U  N  i DISTANCE  Figure 27b.  F  i 80 along  i  i 120  TANK  Changes i n heat content and temperature anomaly f o r Run F  i Cm  1  o  S20h  LAKE ONTARIO SECTION  Ul  i o  E-N^__  BAR 1  J—*  0  J  —  - * "^^^v. ^ ^-SECTION  <  ^-MEAN  A  BAR  for L A K E  D  Ui X  <»  0.  10  20 DISTANCE  40  30 NORTH  SHORE  (km)  EXPERIMENT  E u o 40h u MEAN for TANK  O o  from  RUN  D  20  RUN  B  ^  7  "  BAR  < Ul  40  80 DISTANCE  Figure 28.  along T A N K  120  (cm)  Changes i n heat content during the early stages of the thermal bar f o r Lake Ontario ( f o r sections i n Figure 2, p.4) and f o r the tank experiment.  00  79 APPENDIX C:  I n v e s t i g a t i o n of the E f f e c t s  o f Bottom  Irregularities  F i g u r e s 29 and 30 c o n t a i n temperature d a t a s i m i l a r t o t h a t i n Appendix B f o r the cases o f a 5° s l o p e f o l l o w e d by a 0° s l o p e and a 5° s l o p e f o l l o w e d by a v e r t i c a l d r o p o f f t o a 0° s l o p e . shows the b a r speeds similar  from t h e s e e x p e r i m e n t s .  to the p r e v i o u s c a s e s .  Figure  The r e s u l t s  31  are a l l  However, the l i n e a r model f o r the  speed o f the b a r i s n o t as good an a p p r o x i m a t i o n f o r t h e s e extreme changes  i n bottom  topography.  The speed o f the b a r was  which would have been e x p e c t e d from e q u a t i o n 2.5.1 t h a t was  roughly that  f o r a bottom  slope  the w e i g h t e d average o f the bottom s l o p e s i n each s e c t i o n .  a r e no a b r u p t changes  i n b a r speed, temperature f i e l d  a s s o c i a t e d w i t h t h e s e bottom  irregularities.  or current  There  structure  R  U  N  G  Observations  i  0  i  i 40  i  i  80  i  |  120  i  cm  Figure 29b.  Changes i n heat content and temperature anomaly f o r Run G  82  Observations Or '  RUN  J  1  E o 8 BAR  o. 16 L  TEMPERATURE  SECTION  (°C)  40  80  120  cm  16 O o Ul  §12  predicted  or ui a  S s hui Z  < Ul  measured  BAR  40  80  DISTANCE  Figure  30a.  along  TANK  120  T e m p e r a t u r e s e c t i o n a n d 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 , a n d T i , 0°C.  cm  83  RUN _  CM  J measured  „_  5  I  o  I  ~~  P-40-  z  UJ h-  ^  -  xJi-  x  i  I  x  *  I  *^  ° 2 0 -  S  <  predicted  * -  7  J  'BAR  Ul  I O  0" 0  Figure 30b.  1  ' 40  1  1  80  I  i  120  i  Cm  Changes i n heat content and temperature anomaly section f o r Run J  100 0° SLOPE 0° SLOPE^  —J—  *  o  predicted oo  XX  XX  XX  XX  5 ° SLOPE - 0 ° SLOPE o  » o  XX  XX  o  STANDARD  e  HEATING  x o ° S L O P E at a D E P T H of 8 c m o o ° S L O P E at a D E P T H of 20cm  o  e  10  15  20  25  30  35  40  T l ME ( minutes ) F i g u r e 31.  P o s i t i o n o f the b a r , measured and p r e d i c t e d , f o r 5°-0° bottom 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 these curves i s BAR SPEED.  oo  APPENDIX D  To show that the thermal bar phenomenon i s driven, b a s i c a l l y , by the heating of water through i t s maximum density, as has been proposed i n the l i n e a r model for the speed of the bar, the following two experiments were performed.  1.  S l i d i n g Door Experiment Figure 32 shows the temperature section and mean temperatures  minutes a f t e r 2°C and 8°C water were brought together. heating (heat lamps were turned o f f ) .  There was  20 no  This represents the behaviour  of flow which could be expected when 'fresh' water colder than 4°C flows i n t o water warmer than 4°C ( f o r example, at the foot of a glacier) or when water warmer than 4°C flows i n t o water colder than 4°C ( f o r example, e f f l u e n t s ) .  The behaviour i s not s i m i l a r to the migrating  thermal bar i n that the r e s u l t i n g current pattern does not move and the active sinking zone between the two water masses involves water from both sides.  This c i r c u l a t i o n results from the production of  dense, 4°C water at the contact between the two water masses.  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 are S-shaped, s i m i l a r to those shown i n Figure 6 (p.15) f o r the migrating thermal bar experiments. current structure, superimposed  This  on the temperature section, i s a  clockwise c i r c u l a t i o n i n the shallower water and a counter-clockwise c i r c u l a t i o n i n the deeper water, with most of the sinking between the 3.5 and 4.5°C isotherms (as indicated by arrows).  In a manner s i m i l a r  to the migrating thermal bar phenomenon, the sinking prevents flow due  86  F i g u r e 32.  T e m p e r a t u r e s e c t i o n a n d mean t e m p e r a t u r e s f o r s l i d i n g e x p e r i m e n t (5° b o t t o m s l o p e ) .  door  87 to the h o r i z o n t a l d e n s i t y A c t u a l examples  gradients.  o f water warmer than 4°C b e i n g dumped i n t o w a t e r  c o l d e r than 4°C were p r e s e n t e d by Chermack (1970) at the 13th Conference on Great Lakes R e s e a r c h , A p r i l , r a d i a t i o n thermometer  1970.  H i s work i n v o l v e d  studies of e f f l u e n t s  on s o u t h e a s t e r n Lake O n t a r i o .  airborne  from n u c l e a r power p l a n t s  I n each case t h e s e e f f l u e n t s were  confined  t o an a r c o f r a d i u s o f the o r d e r o f a k i l o m e t e r from the p o i n t o f d i s c h a r g e . This  2.  confinement agrees w i t h the r e s u l t s o f t h i s  ' s l i d i n g door' experiment.  H e a t i n g Water Warmer than 4°C F i g u r e 33 shows the temperature s e c t i o n a s s o c i a t e d w i t h h e a t i n g  w a t e r t h a t i s a l l a t a temperature g r e a t e r than 4°C. the deep end i n the upper l a y e r s and towards deeper l a y e r s .  flow i s towards  the s h a l l o w end i n the  The temperature s t r u c t u r e b e a r s no resemblance t o the  thermal b a r case.  The v e l o c i t i e s  i n v o l v e d are c o n s i d e r a b l y l e s s  the speed o f the m i g r a t i n g t h e r m a l b a r (about 1/10). d r i v e n by  The  than  This flow i s  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 s s e t up by u n i f o r m s u r f a c e  h e a t i n g and a s l o p i n g bottom s i n c e the temperature r i s e s s h a l l o w end o f the tank.  f a s t e r i n the  I  0  1  1  40  1  DISTANCE  1  80 along T A N K  1  I  L  120 (cm)  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:  The  C a l c u l a t i o n o f the Stream F u n c t i o n  s t r e a m f u n c t i o n § i s c a l c u l a t e d from e q u a t i o n  a n a l y t i c a l temperature f u n c t i o n g i v e n i n e q u a t i o n the boundary c o n d i t i o n s g i v e n i n e q u a t i o n s  3.2.7, u s i n g the  3.2.9 and s a t i s f y i n g  3.2.8.  The s t r e a m l i n e  along  z = 0 and z = -D i s s e t e q u a l t o z e r o .  <j>  »  2£& (2.9) (0.0392) s e c h ( 0 . 3 9 2 [  r {  +  0.001 -36—  z  .  3  + z  2  6  1 24  0.001 4 (-5J-D  -  n  A  =  8  4 z  , +  z  2 +  D  C°  +  z  0^001 4  8 x 10"  , 1 24  +  0^01 2 D  n  1_  +  0.01 3  —  (  ,  .  D  0.0001 z  _  0^1 3 D  +  2  }  2 .  24 D n  , 0.001 — 6 —  +  z  D  )  - 2 ])]  z  3 , 0.00001 ^5 28 "  +  1  +  7  -Ti—  ^ D )  4  Z  [ 1 + tanh(0.392[ ^jg^  -0.01D  +  5  Z  _  [  L  0.01 —  +  , 0.00001 168  +  ,  Z  0,001 ,3  (  -2.9  where  - 2 ])  2  0.01 —  5  2  0.001 „4 , 0.01 „3 "42— -15" D  (  1 _  6 +  ^  D  2  D  } ]  6  }  +  D  -  ^  D  5  90 g i s the a c c e l e r a t i o n due t o g r a v i t y (- 10 -2 V i s viscosity  ( = 1.5  x 10  3  cm s e c  -2  ),  2 - 1 cm  sec  f o r the l a b o r a t o r y  S i s t h e speed o f the b a r ,  D = ax i s the depth o f the bottom and a i s bottom  From t h i s cf>, u and w can be u  _3£ 8z  and  w  found s i n c e  3£ 9x  slope.  model),  

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
China 2 6
Russia 1 0
United States 1 0
City Views Downloads
Shenzhen 2 3
Moscow 1 0
Ashburn 1 0

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

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0302462/manifest

Comment

Related Items