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On the mechanics of lake circulation Fofonoff, Nicholas Paul 1951

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L  ON THE MECHANICS OF LAKE CIRCULATIONS by NICHOLAS PAUL FOFONOFF  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department o f PHYSICS  We accept t h i s t h e s i s as conforming to the standard r e q u i r e d  from candidates f o r the  degree o f MASTER OF ARTS.  Members o f the Department o f PHYSICS  THE UNIVERSITY OF BRITISH-COLUMBIA September, 1951  c  i  By  ABSTRACT Four aspects o f the problem o f d e s c r i b i n g the phenomena governing  the flow p a t t e r n i n a lake are  gated with the a i d o f simple mathematical  models.  are p o i n t e d out between the two-dimensional  investiAnalogies  models and  the  corresponding problems i n the e l a s t i o deformation o f t h i n rigid A.  plates.  The  dependence o f the flow p a t t e r n i n a l o n g i t u d i n a l  v e r t i c a l c r o s s - s e c t i o n o f a l a k e on a g i v e n  temperature  d i s t r i b u t i o n w i t h i n the l a k e i s s t u d i e d with the a i d o f a two-dimensional section.  l a k e model o f s e m i c i r c u l a r v e r t i c a l o r o s s -  A stream f u n c t i o n s a t i s f y i n g the  two-dimensional  non-homogeneous Mharmonio e q u a t i o n i s i n t r o d u c e d . are found c o n s i s t i n g o f a throughflow souroe  Solutions  function having a  and s i n k at the ends o f the diameter  plus a o i r o u l a -  t i o n f u n c t i o n whioh i s o b t a i n e d i n terms o f a Green's f u n c t i o n and 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 .  The  total  stream f u n c t i o n s a t i s f i e s the c o n d i t i o n s o f no s l i p p i n g a t a s o l i d boundary and zero s t r e s s a t a free s u r f a o e .  Two  oases o f simple d e n s i t y s t r u c t u r e are considered, and s t r e a m l i n e s are drawn f o r s e v e r a l r a t i o s o f c i r c u l a t i o n to throughflow. B.  The  departure o f the l a k e surfaoe from the h o r i z o n t a l  r e q u i r e d to m a i n t a i n a pressure head s u f f i c i e n t to o v e r oome eddy v i s o o s i t y i s estimated by c o n s i d e r i n g the pressure I  d i s t r i b u t i o n i n the case o f the throughflow f u n c t i o n i n t r o d u c e d above* C.  The e f f e o t o f C o r i o l i s f o r e e s on the flow • i s i n v e s t i g a t e d w i t h the a i d o f a  s e m i - o y l i n d r i o a l model.  i n a canal  three-dimensional  I t i s shown t h a t the e f f e o t on the  shape o f the c a n a l s u r f a c e i s n e g l i g i b l e .  For a  suffi-  c i e n t l y low value o f v e r t i c a l d e n s i t y s t r a t i f i o a t i o n the e f f e o t o f G o r i o l i s f o r o e s may show up i n the t i l t i n g o f constant D.  density  surfaoes.  The e f f e c t o f wind s t r e s s on o i r o u l a t i o n i n a v e r t i c a l lake o r o s s - s e o t i o n i s i n v e s t i g a t e d mathematically  the a i d o f a two-dimensional model.  with  The r e s u l t s are found  to agree quite c l o s e l y w i t h experiments on the deformation o f a s e m i c i r c u l a r p l a t e r e c e n t l y r e p o r t e d i n the l i t e r a t u r e .  II  ACOOWLEDGrEMMT  I would l i k e  to express my s i n c e r e a p p r e c i a t i o n  f o r the guidance and encouragement g i v e n me by P r o f e s s o r G. M. V o l k o f f , who has devoted many hours to d i r e c t i n g my research.  I a l s o wish to thank P r o f e s s o r s G. L . P i c k a r d  and W. M. Cameron o f the I n s t i t u t e o f Oceanography, who suggested the problem and have generously  made the  f a c i l i t i e s o f the I n s t i t u t e a v a i l a b l e to me d u r i n g the summer months, and Dr. C. C. G o t l i e b o f the Computation Centre, McLennan L a b o r a t o r y , U n i v e r s i t y o f Toronto, who has g r a c i o u s l y undertaken an e x t e n s i v e  program o f computations  a r i s i n g out o f the work r e p o r t e d i n t h i s I am indebted  to the N a t i o n a l Research C o u n c i l  who have granted me a bursary c a r r y on t h i s  thesis.  and a summer s c h o l a r s h i p to  research.  Ill  TABLE OF CONTENTS Page ABSTRACT  I  ACKNOWLEDGEMENT . . 1.  INTRODUCTION  8.  FORMULATION OF GENERAL PROBLEM  I l l 1  (a) (b) (a) (d)  S i m p l i f y i n g Assumptions Eddy Y i s o o s i t y Equations o f Motion Decomposition o f Flow i n t o Throughflow plus C i r c u l a t i o n (e) Analogy to a Problem i n E l a s t i c i t y 3.  Thermal  Circulations  (a) The Model Used (b) The Throughflow F u n c t i o n (o) The C i r c u l a t i o n F u n c t i o n i n terms o f the Green's F u n o t i o n (d) The Flow P a t t e r n f o r a "Two-Layer Lake" . . (e) The Flow P a t t e r n f o r a ?Lake? w i t h a U n i f o r m H o r i z o n t a l Temperature Gradient . . . . . . . . . .  16  B.  P r e s s u r e D i s t r i b u t i o n f o r the Throughflow Funotion . ~. . . . I ii ii 7~~~. ii . . . . . . .  18  C.  C o r i o l i s Forces (a) (b) (c) (d)  D.  Model Used L o n g i t u d i n a l Flow . . . Transverse E f f e c t s Pressure D i s t r i b u t i o n . . . . .  Wind-Driven V e r t i o a l C i r o u l a t i o n  .  . . . . . . .  11 11 12 14  El 23 24 26 28  DISCUSSION OF RESULTS (a) (b) (o) (d)  5.  8 9  SPECIAL MODELS  A"  4.  3 4 5  Thermal C i r c u l a t i o n s Pressure E f f e c t s C o r i o l i s Forces Wind-Driven V e r t i c a l C i r o u l a t i o n  BIBLIOGRAPHY  . . . . .  32 34 36 38 40  APPENDIX  I  41  APPENDIX I I  47  ILLUSTRATIONS Plate I  f o l l o w i n g page  F i g . 1. S t r e a m l i n e s o f the throughflow  12  funotion  F i g , 2. V e l o c i t y p r o f i l e along v e r t i o a l r a d i u s o f F i g . 1. Plate II  .  f o l l o w i n g page  16  F i g . 3. S t r e a m l i n e s o f the thermal o i r c u l a t i o n f u n o t i o n V • 0.192 f o r the oase o f a "two-layer l a k e " with a d e n s i t y d i s continuity along - e « 135°. F i g . 4. S t r e a m l i n e s f o r (j> « F i g . 5. Streamlines f o r  s 0  f  •+ 0 . 1 9 2 ^  a  0  t  — 0.192  F i g . 6. The dynamic surfaoe o f the " l a k e " F i g . 7. S e m l - c i r o u l a r oanal model  . .  . . . . . .  f o i l O H i n g page  Plate I I I  20 21 31  F i g . 8. P r o f i l e s o f wind-driven c i r c u l a t i o n f u n c t i o n s o f F i g s . 10 and 11 along r a d i u s 0 » 90°. F i g . 9. P r o f i l e s o f wind-driven c i r o u l a t i o n f u n c t i o n s o f F i g s . 10 and 11 along r a d i i Q • 45° o r 6 * 1 3 5 ° . P l a t e IV  .  f o l l o w i n g page  F i g . 10. S t r e a m l i n e s f o r wind-driven (theoretical)  oiroulation  F i g . 11. Streamlines f o r wind-driven (experimental)  oiroulation  32  1. Little  INTRODUCTION  work has been done i n the past on  the  i n v e s t i g a t i o n o f the mechanics o f lake c i r c u l a t i o n s . Munk and E . R. Anderson^ ^ have s e t up a theory o f  H.  the  5  thermooline  W.  and have a p p l i e d i t to the oase o f Sweetwater  l a k e , but the b a s i o flow caused by a p o t e n t i a l g r a d i e n t with the p o s s i b l e thermal c u r r e n t s set up by d i f f e r e n c e s i n the l a k e has r e c e i v e d l i t t l e Sinoe the v e l o c i t y o f the flow  density attention.  i n a lake i s s m a l l ,  d i r e c t measurements o f the v e l o o i t y d i s t r i b u t i o n ourrent meters and  ourrent drags o f t e n f a i l  and  must be made to i n d i r e c t methods i n v o l v i n g the s t r u o t u r e and  with reoourse  density  c o n s i d e r a t i o n o f the f o r c e s a c t i n g on  the  f l u i d i n i t s motion. Seasonal changes i n temperature c o n d i t i o n s a f f e c t the flow,  will  f o r i t i s found that d u r i n g the summer  temperature g r a d i e n t s are c o n s i d e r a b l e .  During  winter  months and e a r l y s p r i n g the lake i s u s u a l l y very homogeneous and  the flow i s due  predominantly to the p o t e n t i a l g r a d i e n t .  C o r i o l i s f o r o e s a o t i n g upon the moving water masses may  produce observable  p a t t e r n , although o f the low  e f f e c t s i n the lake  flow  any expected e f f e c t s w i l l be s m a l l beoause  velooities involved. In a d d i t i o n to the p o t e n t i a l g r a d i e n t and  Coriolis  foroes the meohanios o f lake flow i s f u r t h e r complicated  by  - 2 interactions  with the atmosphere i n heat exchange, wind  stresses, e t c .  These e f f e c t s w i l l not be c o n s i d e r e d  e x p l i c i t l y here, w i t h the e x c e p t i o n o f wind s t r e s s , an example o f which i s c o n s i d e r e d f o r comparison with r e s u l t s o b t a i n e d from the e l a s t i c deformation o f t h i n r i g i d  plates.  2. (a)  FORMULATION OF GENERAL PROBLEM  S i m p l i f y i n g Assumptions Assuming that a steady state temperature  distri-  b u t i o n has been e x p e r i m e n t a l l y determined, the d e n s i t y f i e l d may be o b t a i n e d as a f u n o t i o n o f the p o s i t i o n c o ordinates on  of a point  the temperature.  is negligible.)  through the dependence o f the d e n s i t y (The dependence o f d e n s i t y  on pressure  Heat t r a n s f e r must be such as to m a i n t a i n  t h i s temperature d i s t r i b u t i o n , i . e . the water gains o r l o s e s heat as i t flows along the s t r e a m l i n e s a t j u s t the r i g h t r a t e t o assume a t eaoh p o i n t The  the observed temperature.  t r a n s f e r o f heat between v a r i o u s  p a r t s o f the f l u i d and  heat exchanges w i t h the atmosphere may i n p r i n c i p l e be determined once the temperature d i s t r i b u t i o n and v e l o c i t i e s i n the lake consider  are known.  T h e r e f o r e , i t i s n o t necessary to  heat t r a n s f e r i n the f o l l o w i n g d i s c u s s i o n .  observed temperature d i f f e r e n c e s are s m a l l f r a c t i o n a l density  be  so that the  changes w i l l a l s o be s m a l l .  p o i n t o f view o f the equation o f c o n t i n u i t y  The  From the  the d e n s i t y  c o n s i d e r e d constant, the v a r i a t i o n s being taken i n t o  account only  i n o a l c u l a t i n g the e f f e c t s o f g r a v i t y .  Mathematically t h i s i s equivalent  to s a y i n g  that the  a c c e l e r a t i o n o f g r a v i t y v a r i e s i n a g i v e n way throughout the  lake.  will  - 4 -  (b)  Eddy Y i s o o s i t y In d e s o r i b i n g the flow equations  i n v o l v i n g eddy  v i s c o s i t y are used, sinoe l a r g e soale flows suoh as those found i n l a k e s are g e n e r a l l y o f a t u r b u l e n t oharaoter. Turbulent flow i s c h a r a c t e r i z e d by the presence  o f numerous  eddies superimposed on the mean flow c o n t i n u a l l y t r a n s p o r t i n g f l u i d masses i n t o r e g i o n s o f d i f f e r e n t v e l o c i t y .  These  eddies are very i r r e g u l a r and complex so that the steady s t a t e i n a t u r b u l e n t flow does not e x i s t i n the same sense as i n laminar  flow.  Wherever a v e l o c i t y g r a d i e n t occurs i n the mean flow there w i l l  be a t r a n s p o r t o f momentum across the  constant momentum s u r f a c e s by the t u r b u l e n t v e l o c i t y components.  P r a n d t l ^ has introduoed a "mixing l e n g t h " ,  analogous to the mean f r e e path o f a gas moleoule, i n which he v i s u a l i z e s f l u i d elements c a r r i e d by t u r b u l e n t v e l o c i t i e s a d i s t a n c e " 1 " before l o s i n g t h e i r i d e n t i t y by mixing the surrounding f l u i d .  with  T h i s oonoept has been c r i t i c i s e d by  G. I . T a y l o r ^ ) , who c o n s i d e r s v o r t l o i t y  transfer rather  than momentum t r a n s f e r i n d e f i n i n g the eddy v i s c o s i t y . two  The  d e f i n i t i o n s l e a d t o the same r e s u l t i f the eddy  v i s c o s i t y i s a oonstant, and a d i s c u s s i o n o f the d i f f e r e n c e s r e s u l t i n g from the two concepts  i s given i n Taylor's  paper^  An e x c e l l e n t d i s c u s s i o n o f the v a r i o u s t h e o r i e s o f t u r b u lenoe i s g i v e n by R o u s e ^  from which the f o l l o w i n g s h o r t  - 5 resume o f P r a n d t l ' s d e f i n i t i o n o f eddy v i s c o s i t y i s adapted. Consider  i n the x y plane a mean motion Vx i * i t  the x d i r e c t i o n having a v e l o c i t y g r a d i e n t i n the y d i r e c t i o n . Assuming that a u n i t f l u i d mass i s c a r r i e d (2h T 1 1  P  (IP  x  +  1 <* Vx dv *  a  *  V 0  point  V  component Yy, the n e t  x  momentum t r a n s f e r to  x  (2) i s  d i s t a n c e "1" by a , , turbulent v e l o c i t y  - fYy l 4 Vx .  The time average o f t h i s  q u a n t i t y g i v e s the mean s t r e s s along the constant momentum surfaces.  The q u a n t i t y  "jpVy~i" ^  a  d-ini  s  6118  *-  0118  °f  dynamic v i s o o s i t y and i s termed the eddy v i s c o s i t y A.  In  g e n e r a l i t i s a f u n o t i o n o f p o s i t i o n , being a f f e c t e d by the s c a l e and type o f t u r b u l e n c e , s t a b i l i t y c o n d i t i o n s , magnitude o f mean flow, p r o x i m i t y o f boundaries, Sinoe turbulenoe  i s u s u a l l y h o r i z o n t a l l y i s o t r o p i c only  the h o r i z o n t a l A^ and v e r t i o a l A differentiated.  and o t h e r f a o t o r s .  v  eddy c o e f f i c i e n t s are  The r a t i o -^v/A^ i s roughly o f the o r d e r o f  magnitude o f the r a t i o o f the v e r t i c a l and h o r i z o n t a l dimensions o f the water c u r r e n t s (Rouse (o)  p. 186).  Equations o f Motion In terms o f the eddy v i s o o s i t y the hydrodynamic  equations  f o r an incompressible f l u i d  are:  - 6 -  In the l a k e models to he considered the eddy v i s c o s i t y i s assumed oonstant and i s o t r o p i c : A^ =• Ay * A The assumption that A i s i s o t r o p i c i s reasonable when the v e r t i c a l and h o r i z o n t a l dimensions are o f the same order o f magnitude.  There i s l e s s p h y s i c a l b a s i s f o r the assumption  that A i s independent o f p o s i t i o n but i t i s necessary to s i m p l i f y the mathematios. workers (e.g. M u n k ^ )  I t has been found by other  that assuming A independent o f p o s i -  t i o n l e a d s to s o l u t i o n s b e a r i n g a good d e a l o f resemblance to the observed  flow.  Taking the e x t e r n a l f o r c e s to be those o f g r a v i t y and G o r i o l i s f o r o e s , the hydrodynamio equations a r e :  *  (2)  V  V-V* - ° Consider: —9  (1) steady s t a t e flow  ^  Y  s o  (2) s m a l l mean v e l o c i t y o f flow, i . e . f 5 * V ' neglected.  can be  The equations reduoe t o : (2a)  * o  y.  — ?  Introduce a v e c t o r p o t e n t i a l p ^ r VKi?  and ohoose  sides o f equation for  V - =  0.  y  such that  Taking the c u r l o f both  (2a) a s i n g l e three dimensional  equation  i s obtained *-  13)  This e q u a t i o n w i l l be used i n s e c t i o n C below to show that C o r i o l i s f o r o e s are r e l a t i v e l y unimportant  i n lake flows.  As two dimensional models w i l l p l a y an important r o l e below, e q u a t i o n  (3) i s s p e c i a l i z e d to two  dimensions  i n the x,y plane, w i t h y p o s i t i v e downwards, x p o s i t i v e to the l e f t  (the t h i r d z a x i s i s then d i r e c t e d out o f the page). Choose  V-  m 0,  where  ^  N7-  a - 0.  J^(x,y)tf. Then  ^  This s a t i s f i e s = Vx ?  - -  ,  i s the u s u a l two dimensional stream f u n c t i o n , i . e .  9* = constant d e f i n e s a s t r e a m l i n e , and  V% - %  • rate o f  flow per u n i t width o f l a k e between s t r e a m l i n e s B and A. The C o r i o l i s f o r o e s are i n g e n e r a l n e g l i g i b l e , and i n t h e corwpowri/" m ihe vertical pktt? is p a r t i o u l a r a b s e n t i f the flow i s taken i n the north-south A  plane.  - 8  -  Under these c o n d i t i o n s , e q u a t i o n »v*v*1f The  = ^?  where  7 » l  boundary c o n d i t i o n s a r e :  ^  + ^  (3)  wind s t r e s s f ( x ) « 0.  to: (4)  x  (t = tangent,  n = normal)  r f ( x ) along l a k e surfaoe y » 0 .  (a)  reduces  I n the presence  In absence o f o f wind s t r e s s  f (x) i s a g i v e n f u n o t i o n o f x. ss 0 along l a k e bottom  (b)  at s o l i d  0,  i . e . no  slipping  boundary).  (o)  ¥  • 0 a l o n g lake s u r f a o e .  (dj  t  * ff/b along bottom  (IP » r a t e o f flow through  aotual  l a k e , b = average width o f a o t u a l l a k e F/b (d)  Decomposition The  parts Here  ^  o f Flow i n t o Throaghflow p l u s C i r c u l a t i o n  t o t a l stream =  * r a t e o f flow per u n i t width)  %{  f u n c t i o n i s s p l i t up  into  two (5)  4>,  i s assumed to be  the s o l u t i o n o f the homogeneous  biharmonio e q u a t i o n o b t a i n e d by s e t t i n g the r i g h t - h a n d side of equation  (4)  equal to z e r o .  conditions a with f(x) a 0,  ^satisfies  the boundary  b, o, and d, r e p l a c e d by  tf>  t  s 1  a l o n g bottom. Thus  ^  r e p r e s e n t s the flow p a t t e r n o f a l a k e  having no wind s t r e s s , no direction  temperature v a r i a t i o n i n the  (although there may  t i o n ) , and a u n i t throughflow  be a v a r i a t i o n i n the y per u n i t time per u n i t  x direc-  width  from r i g h t t o l e f t . Consider F / b ^ homogeneous biharmonio  to be t h a t s o l u t i o n o f the non-  equation  boundary c o n d i t i o n s (a,b,c) bottom.  T h i s corresponds  (4) whioh s a t i s f i e s the  together w i t h ^  a 0 along l a k e  to zero n e t throughflow and  d e s c r i b e s the o i r o u l a t i o n i n a " l a k e " w i t h a g i v e n temperature d i s t r i b u t i o n o r a g i v e n wind s t r e s s but w i t h no i n t a k e or o u t l e t .  I n a oase where the temperature  distribution i s  simple enough to s e t up a s i n g l e o i r o u l a t i o n o n l y ( ^ h a s a s i n g l e p o s i t i v e maximum o n l y ) , clockwise c i r c u l a t i o n ,  ^<  defines a counter-  0 d e f i n e s a olookwise  circula-  tion. *Vb \ T % _  i  s t  h  e  o i r o u l a t i o n r a t e o f flow per  u n i t width o f l a k e . i s chosen to normalize p£ such t h a t t  = 1.  Then: ^ ( ) Q  Analogy  s  oiroulation/  t h r o u g h f l o w  .  to a Problem i n E l a s t i o i t y  The e l a s t i o deformations o f a t h i n p l a t e o f uniform f l e x u r a l r i g i d i t y perimeter, hinged  damped a l o n g a p o r t i o n o f i t s  (with a g i v e n bending moment) along the  balance o f i t s perimeter, and s u b j e c t t o a v a r i a b l e normal l o a d , are governed by the same e q u a t i o n w i t h the same boundary c o n d i t i o n s as the above stream f u n c t i o n (Love, A.E.  - 10 H.^ , page 464). 2)  I n the oase o f the e l a s t i c model analogy d i s c u s s e d above constant <f r e p r e s e n t s contour l i n e s . s t r e a m l i n e s are analogous f o r m a t i o n produced  Throughflow  to the contour l i n e s o f the de-  i n the absence o f normal l o a d when the  damped p o r t i o n o f the perimeter i s d i s p l a c e d w i t h r e s p e c t to the hinged p a r t .  C i r o u l a t i o n s t r e a m l i n e s are analogous  to the oontour l i n e s o f the deformation produced  by a g i v e n  normal l o a d d i s t r i b u t i o n o r a g i v e n bending moment a t the hinge when the clamped and hinged p a r t s o f the perimeter are at the same l e v e l .  11 -  3. A.  SPECIAL MODELS  Thermal C i r c u l a t i o n s (a)  The Model Used We are i n t e r e s t e d i n o b t a i n i n g some idea o f the  i n f l u e n c e o f the temperature -inhomogeneity i n a lake on the flow p a t t e r n .  The mathematical d i f f i c u l t i e s i n t r e a t i n g  the a c t u a l three-dimensional i n order  so that  to o b t a i n a f i r s t o r i e n t a t i o n we c o n s i d e r a two-  dimensional choosing  oase are c o n s i d e r a b l e ,  model.  We f u r t h e r s i m p l i f y the mathematics by  f o r the p r o f i l e o f the "lake bottom" i n the two-  dimensional  r s a.  v e r t i c a l c r o s s - s e o t i o n the s e m i o i r o l e  We now l o o k f o r s o l u t i o n s o f equation  (4) d i s -  oussed above s u b j e c t t o the boundary c o n d i t i o n s a, b, o, d. As d i s c u s s e d above the s o l u t i o n f a l l s i n t o two parts: determination  4>  o f the throughflow f u n o t i o n  s a t i s f y i n g the homogeneous biharmonio equation, determination  separate t  and the  o f a o i r o u l a t i o n f u n o t i o n s a t i s f y i n g the non-  homogeneous biharmonio equation ( 4 ) . (b)  The Throughflow J u n c t i o n Using  "a" as the u n i t o f l e n g t h we ^ = x/a,  I t may be v e r i f i e d  )  s  y/a  P  (see Appendix I) t h a t :  a  introduoe r  /a.  - 12 satisfies  v*"*7 ^/  • o  z  and the boundary o o n d i t i o n s :  (a) y ~  a 0 along  &  (b)  \&  a 0 along  p  (o)  ^  - 0 along  o  (d)  ^  s 0 along  =  © , 7 t  «  l °J  TC  m  p a l  4>, r e p r e s e n t s the throughflow f u n o t i o n f o r a lake o f homogeneous h o r i z o n t a l d e n s i t y s t r u c t u r e .  Streamlines  c o r r e s p o n d i n g to <#> = constant together w i t h the v e l o c i t y t  profile figs.  v~  a  x  ~$Jb a l o n g x a 0  are g i v e n on P l a t e I ,  (1) and ( 2 ) . (o)  The C i r o u l a t i o n F u n c t i o n I n Terms o f the Green's Funotion To determine  <f>^ equation  (4) i s r e w r i t t e n i n  terms o f c o o r d i n a t e s i n which ''a" i s a u n i t o f l e n g t h .  ^  V  2  *  Z  Z*  (8) AF-£  *  I  w i t h the boundary o o n d i t i o n s :  2  (a)  3  e 0 along 6  4> 2  a  Q 7U t  3 h 2 (b)  ^  (o)  ^>  (d)  «^2  2  s o along p - 1 B 0 along 88 0  a  l° S n  * s 0,7cr  f " 1  For any g i v e n d e n s i t y d i s t r i b u t i o n / g , to make  7*2.max  - !•  T h i s value o f  i s to be ohosen then determines the  PLATE I  - 13 r e l a t i v e magnitude o f the e i r o u l a t i n g flow and the flow.  (8) f o r any f g can be  The s o l u t i o n o f e q u a t i o n  expressed G(p, e>  through-  as an i n t e g r a l i f the Green's f u n c t i o n 0 ) i s known, where p »  r  O  /a,  q »  r , c  /a.  The Green's f u n o t i o n s a t i s f i e s : V  V  2  G  2  =  *(*-*•)  (9)  and the above boundary o o n d i t i o n s . The d e l t a f u n o t i o n may  /"T  boundary o o n d i t i o n  (^  be r e p l a c e d by the &  >'  < * l ?  K  where  ^ ]  Oca C->o  i s an elementary  displacement  i n the counterclockwise  d i r e o t i o n along a curve surrounding  the p o i n t (q, © ) , the 0  l i m i t being taken as the curve i s shrunk around t h a t point. The Green's f u n c t i o n s a t i s f y i n g the above c o n d i t i o n s i s : (See Appendix I)  where  S+_  •tl  I t may  -  P  f  *  1 t  2  q p  2  - 2 p q ^ o (© ± e© )  2  q  2  - 2pqCos  be v e r i f i e d t h a t  {G+  ©. )  0 i n the s e m i c i r c l e .  In terms o f t h i s Green's f u n c t i o n : a 4  n  -  *  3  b  Sr  J qdq J  G (/>,*; q e ) 0  D(q, *>)  (11)  - 14  -  where D(g.,3j s  evaluated at p  To determine (d)  ^2*  C P , the form o f  The  » q,  © r ©  must be known.  Flow P a t t e r n f o r a "Two-Layer Lake" (Case  The  f o l l o w i n g simple case  f>3 i s constant i n two continuity along  e  p a r t s o f the lake w i t h a d i s -  * &,  (two-layer l a k e ) .  and i s the most elementary Choose  f g  « foj»  fg  s  T h i s means that  This c o r r e s analogy  model o f the a c t u a l s i t u a t i o n . for  o s « $ ©.  fo 3© (1-f) f o r <s>,«.*£^ * 0,  >J±  = - f %t  <T  a  ( 0 ) .  since  Ul we  |^  e  Cos6 ^  -  5  -^*  obtain  Thus from equation ^  2  a  AF •  a  g b  AF-£  (11)  ^  becomes  f S l n e,  fo ^ 1 6  Ij  i s considered:  ponds to a c o n c e n t r a t e d l i n e l o a d i n the e l a s t i o  Now,  (11a)  0  ^  g»aef Sin^> 16 T c r  /B( ,© p  ;  q  , jdq  Ufa.*;*/)  e  < > 12  15 where U(P,0; O, ) =  /Jdq  B(p,©;q,e,  ).  (13)  Here B i s the f u n c t i o n d e f i n e d by equation (10). To make <f>  2  s  max  a 3b  AF  -  f ° %  l  set:  f Sine, 16 7C  u max  •  circulation throughflow  (14)  Sinoe U> 0 i n the s e m i c i r c l e , ^ has the same s i g n as f . Thus the c i r c u l a t i o n i s clockwise clockwise  f o r f<^0  f o r f/• 0 i n agreement with  ment that dense l i q u i d s i n k s while  and counter-  the p h y s i o a l r e q u i r e -  light liquid rises.  The e v a l u a t i o n o f the i n t e g r a l  (13) i s t e d i o u s  but s t r a i g h t forward and the r e s u l t may be w r i t t e n as: U(p,d; e, )  = F ( p , £ ) - f ( p , fy) where <*a O 1L e, (15)  and  - 16  The  -  steps i n the i n t e g r a t i o n are g i v e n i n Appendix I I .  U(p,©; 6,) f o r & =135° has been computed on a desk (  oalculator.  U  m a x  f o r t h i s case i s found to be 0.192.  s t r e a m l i n e s f o r U and f o r the t o t a l f  = <f>, -f- ?l  II, figs.  streamfunotion - ± 0.192  are g i v e n f o r  (3), (4) and  The  on P l a t e  (5).  Dr. C. C. Grotlieb, o f the Computation Centre, MoLennan L a b o r a t o r y , U n i v e r s i t y o f Toronto, has to compute XT f o r  ©,^ 0(10°) 90°.  undertaken  Unfortunately this  m a t e r i a l w i l l not be a v a i l a b l e i n time f o r i n o l u s i o n i n this  t h e s i s and w i l l be p u b l i s h e d s e p a r a t e l y . (e)  The glow P a t t e r n f o r a "Lake" with a Uniform H o r i z o n t a l Temperature Gradient (Case I I ) The next oase c o n s i d e r e d i s a uniform h o r i z o n t a l  temperature  The  g r a d i e n t a l o n g the l a k e , i . e .  vertioal variation i s arbitrary.  analogy  t h i s corresponds  This model corresponds of  In the  to a u n i f o r m l o a d per u n i t  This gives: =  area.  somewhat b e t t e r to the a o t u a l s t a t e  affairs.  D(q, ©o )  elastio  =  M  0  C  PLATE I I  Fig.3.  S t r e a m l i n e s o f the thermal c i r c u l a t i o n f u n c t i o n Ms 0 . 1 9 2 02. f o r the case o f a "two-layer l a k e " with (Contours a d e n s i t y d i s c o n t i n u i t y along 9 s 135^ correspond to i n t e r v a l s o f 0 . 0 2 forV")  F i a 4 Streamlines f o r ¥> s <P-,+0.1920^. • ( D i r e c t i o n o f flow from r i g h t to l e f t , counterclockwise c i r o u l a t i o n ) y  Fig. 5  Streamlines f o r f  e ^ - 0 . 1 9 2 $%.  ( D i r e c t i o n o f flow from r i g h t to l e f t , clockwise c i r o u l a t i o n )  - 17 From e q u a t i o n (11) $ g becomes a^bc  A  *  2  *  IFY" ^  B  Q  T , p , e )  (16)  where  V(p,e>) =  J^^J  B(  P» ? l» o) e  <  (16a)  a  As before the r a t i o o f o i r o u l a t i o n to throughflow i s given by  setting.  X  - a ^ s u y - 3 o vmax  =  AF  u  The  (l?)  16  i n t e g r a t i o n over q i s c a r r i e d out e x p l i c i t l y (See  Appendix I I ) while the i n t e g r a t i o n over  &  0  i s to be done  numerically.  v(p,e)  =  /  £"(P,*; e  (is)  0  where  £ C P , © ; © > = hip/*.) 0  *P  I  - H p / _ ) +p ( ^ ^ F c ^ _ ) - C s ^ / r ^ j j d s )  3  *p  px  r  J  - 18  and  <f±  F(p,$),  -  are d e f i n e d by e q u a t i o n  (15a) above.  V ( p , © ) i s a l s o b e i n g computed by Dr. C. G o t l i e b and w i l l B.  C.  be p u b l i s h e d s e p a r a t e l y .  Pressure D i s t r i b u t i o n f o r the Throughflow F u n o t i o n The  function  flow p a t t e r n r e p r e s e n t e d by the  throughflow  must be maintained by pressure g r a d i e n t s i n  the " l a k e " . determine  I t i s important  to oonsider the pressure to  the dynamio surfaoe o f the lake s i n c e the  energy  necessary to overcome v i s c o u s s t r e s s e s and to m a i n t a i n  the  flow i n a l a k e having a homogeneous h o r i z o n t a l d e n s i t y s t r u c t u r e must oome from a l o s s o f g r a v i t a t i o n a l of  the water, i . e . the surfaoe assumes a slope  to  supply the neoessary  energy.  c o r r e c t n e s s o f the assumption  We  potential  sufficient  oan a l s o v e r i f y  the  made a t the o u t s e t t h a t  terms i n the square o f the v e l o o i t y  (field  acceleration  terms) are n e g l i g i b l e . Consider the pressure d i s t r i b u t i o n neoessary  to  m a i n t a i n the flow g i v e n by the throughflow f u n c t i o n ^  =  B"/b ^/  where <f>t i s d e f i n e d by e q u a t i o n  For a g i v e n steady s t a t e flow the pressure i s from the hydrodynamlc e q u a t i o n :  (7) above.  determined  - 19 Here the l e f t - h a n d side r e p r e s e n t s the f i e l d a c c e l e r a t i o n c o n t r i b u t i o n , while the l a s t term on the r i g h t the v i s c o s i t y  represents  effect.  C o n s i s t e n t l y w i t h our o r i g i n a l assumption o f small mean v e l o c i t y we take the l e f t - h a n d s i d e equal to zero, and l o o k f o r the i n t e g r a l o f  with  - ££K ^^J >X  *  and <f> g i v e n by e q u a t i o n ( 7 ) . t  T h i s may be i n t e g r a t e d e x p l i c i t l y  to g i v e :  A l o n g the l i n e y » 0 the i n t e g r a l i n the above e x p r e s s i o n v a n i s h e s , and the pressure P d i f f e r s from the atmospheric  pressure by the second term.  T h i s may be  r e p r e s e n t e d by the water p i l i n g up above the h o r i z o n t a l s u r f a c e near the i n t a k e and dropping below the h o r i z o n t a l surface near the o u t l e t .  The e q u a t i o n o f t h i s s u r f a c e i s ;  A p l o t o f <j i s shown i n f i g . 6. s  the curve of  ^  The dotted p o r t i o n o f  i s to be d i s r e g a r d e d beoause the i n f i n i t e  values  a t the two ends are due to the u n p h y s i o a l f e a t u r e o f  our model h a v i n g o r i f i c e s o f zero area l e a d i n g to i n f i n i t e intake and o u t l e t  velocities.  - 20 To i n v e s t i g a t e the order o f magnitude o f the c o n t r i b u t i o n s to the pressure o f the f i e l d a c c e l e r a t i o n terms we note t h a t sinoe our u n i t o f l e n g t h i s "a", the o r d e r o f magnitude o f the v e l o c i t y i s F ^ Q . f i e l d a c c e l e r a t i o n terms are o f order  fF /a  while the v i s c o u s term i s o f order A F / a 2 ^ , . the two i s t h e r e f o r e g i v e n by  Thus the 2  2  b , 2  The r a t i o o f  *^F/AJ,. / /  i  i/  i i j i Fig.  6.  The dynamic surfaoe o f the " l a k e "  - 21 C.  Coriolis (a)  -  Foroes  Model Used We  are i n t e r e s t e d i n s t u d y i n g the e f f e c t s o f  C o r i o l i s f o r o e s on the flow p a t t e r n i n a c a n a l to decide whether o r not these f o r o e s play an important determining  role i n  the v e l o c i t y p r o f i l e i n the o a n a l .  As the  dimensional model c o n s i d e r e d i n s e c t i o n 3 A i s not f o r t h i s i n v e s t i g a t i o n we now  two-  suitable  introduce a t h r e e -  dimensional model c o n s i s t i n g o f a s e m i - o y l i n d r i o a l " o a n a l " running i n the North-South d i r e c t i o n .  We  consider  ( 3 ) i n c l u d i n g the C o r i o l i s term  equation  A «7 v Z  » 2 V  z  V-  x[-£'xr|Vx4\]]  - 0  = 0  ^  T h i s equation admits a simple approximate s o l u t i o n f o r the oase o f a "oanal" o f s e m i - o i r o u l a r transverse o r o s s - s e c t i o n . The  coordinate axes are ohosen with Z p o s i t i v e }  to the South and \r , O  forming a r i g h t - h a n d e d  system.  South «  S e m i - c i r c u l a r c a n a l model F i g . 7.  coordinate  - 22 The L a p l a o i a n o p e r a t o r i s d e f i n e d i n c u r v i l i n e a r c o o r d i n a t e s hy: V  a V {VV?  2  s - 7 z [ ? x + ) sinoe  ) • V z ( V x ^ )  >  V - * ^ r 0.  We ohoose V  ¥  suoh t h a t :  R  »  H>G  =  This s a t i s f i e s  0  ^ ( r )  V-  (  2 3  )'  s 0.  The components o f - 7 x ( 7 x f  ) are reduced so t h a t :  the r oomponent i s zero the 0 oomponent i s the Z oomponent i s  In expanding  j>_ / 1_ S (r f'o)) 1 5 _ ir S 9 ^ i  the C o r i o l i s term  v7 i J^iC  x$[yx^j]  —>  terms c o n t a i n i n g d e r i v a t i v e s o f -XL these w i l l  involve a faotor  are n e g l e c t e d sinoe  where R  i s the r a d i u s o f  the e a r t h . x?jyx<f^]J  To t h i s approximation f component  equal to r  & oomponent equal to Z component equal to -  d_ I pe* L Sin©  Sin©  has: Y  0  - CosO  Y1 T  J  Sin© 3*V  5 r  Y  e  - Cos© V" r J  - 23 the v e l o c i t i e s V , Y  Sinoe the  r  are much s m a l l e r than Y  e  z component i s c o n s i d e r e d i n the f i r s t  Equation  only  &  approximation.  (3) under these assumptions reduces to two and ^ 4  equations f o r ^  .  (25)  7 * 7 * ^  - E A e sin© i v * 3r  s  where 7*- =  (b)  1 a C  >?3  +  ~5~x 1_  r  (25a)  L o n g i t u d i n a l Flow The l o n g i t u d i n a l flow i n the t.  direction  subject  to the o o n d i t i o n s that no s l i p p i n g 00curs a t the oanal bottom and that the maximum v e l o c i t y i n the oanal i s */  0  i s o b t a i n e d by i n t e g r a t i n g the €> oomponent o f e q u a t i o n (25). The g e n e r a l i n t e g r a l i s K  r ^  (r)  r  3  ^ ^Ur  f  + (v/  x  - c/.  ) r -f  «V r  and =  1 r  2*r  ( ^») r  •  B 4  Z r  +  c  A  ,  r  -f B  E v a l u a t i n g the constants i n terms o f the boundary  oondi-  t i o n s above we o b t a i n : /  t  =  (1 -  )  (26)  - 24  -  and  V = «M Y  d  4  2  1  r - r3}  a  (26a)  a3J  In terms o f the v e c t o r p o t e n t i a l through  any  c l o s e d contour C i s by Stokes'  a  F  f  the  flux  theorem  r where <ts i s an  elementary  displacement along oontour C. Taking the contour i n the r , © plane, the t o t a l f l u x through  F  the c a n a l i s g i v e n by  =  f % r<Le f o r  r s a (27)  .2  Solving for V V  -  I F  =  R F/  2"  °  . o r o s s - s e o t i o n a l area o f oanal i  (27a)  S u b s t i t u t i n g the v a l u e s o f F and c r o s s - s e c t i o n a l area o f Kootenay Lake we o f magnitude o f 20 (o)  0  o f the order  ft/hr.  Transverse Equation  o b t a i n values of V  Effects  (25a)  i s o f the same form as  (8) s o l v e d i n s e c t i o n 3 A.  We  equation  can o b t a i n a g e n e r a l  s o l u t i o n i n terms o f the Green's f u n o t i o n d e f i n e d i n  - 25  -  equation. (10). ^  =  a  )  o  qdq J C L © G (p © ; q, © J D (q, e ) o  (28)  D  0  where D(q, © J  =  >  f  " 4**  3  V. $  sin6>e  I f a positive v e r t i o a l density gradient  exists  i n the l a k e , the C o r i o l i s f o r o e s w i l l he balanced 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 sinoe the transverse t i o n induced  by the C o r i o l i s f o r o e s w i l l  In the steady  oiroulation, i . e . ^  t  oiroula-  produce a 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 by t i l t i n g the constant surfaces.  hy a  density  s t a t e there i s no transverse  z 0.  Since G(p,e ; $,*©) £ °  i n the  s e m i c i r c l e t h i s i m p l i e s D(p,©- ) « 0, i . e .  lis  i s x ^ i  I n t e g r a t i n g t h i s equation we o b t a i n an e x p r e s s i o n  f o r the  density structure.  The slope if o f the constant  density  may be o b t a i n e d :  where  X  -  ~  °  3  surfaces  -  26  -  The constant d e n s i t y s u r f a c e s are g i v e n by f( i ) + r  ^ *  U c  = c  (29b)  ~ f ( V •) r *  I f the l a k e i s completely homogeneous i n i t s d e n s i t y s t r u c t u r e the C o r i o l i s f o r c e s give r i s e to a t r a n s verse c i r o u l a t i o n whose stream f u n o t i o n i s :  This i n t e g r a l has n o t been e v a l u a t e d sinoe C o r i o l i s  foroes  are very small (see s e c t i o n 4 ( c ) ) and do not c o n t r i b u t e s i g n i f i c a n t l y to the flow p a t t e r n . (d)  Pressure  Distribution  The pressure term i s c o n s i d e r e d to study the e f f e c t o f C o r i o l i s foroes on the transverse dynamic surfaoe o f the c a n a l i n the oase where the C o r i o l i s foroe i s balanced by a 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 . N e g l e c t i n g the f i e l d a c c e l e r a t i o n terms the pressure  21  i s g i v e n by the hydrodynamio  -  =  A oos6>  p V  -ASin0£V  o  0  (1-  equations:  + ?9  (1- £)+HCose  Sin0  (30)  - 27 -  The  z  oomponent i s simply  maintain  the p o t e n t i a l drop neoessary  the l o n g i t u d i n a l flow.  to  I t i s i n t e r e s t i n g to  oompare the slope o f the oanal surfaoe i n the l o n g i t u d i n a l d i r e c t i o n w i t h the slope o f the " l a k e " surfaoe i n the model d i s o u s s e d i n s e c t i o n 3 B. at  J  while  » 0  The e x p r e s s i o n f o r the slope  i s 16 A F / ^ t ,  (30a)  the slope f o r the oanal i s U V a  .  o  0 2  ^  16 A F  (30b)  _______  7ca  4  f 9  W r i t i n g the pressure as P  =  -  * AY,  a we now proceed The  pressure  *  -f  P(r,  ©)4-P.  2  to f i n d an e x p l i c i t  i n t e g r a l f o r P ( r , © ).  term i n v o l v i n g C o r i o l i s f o r o e s w i l l be s m a l l  so t h a t the mathematical s i m p l i f i c a t i o n o f c o n s i d e r i n g the v a r i a t i o n i n the term  ?_) to l i e i n the a c o e l e r a t i o n o f  g r a v i t y w i l l not a p p r e c i a b l y a f f e c t the f i n a l r e s u l t but w i l l greatly simplify Let  a  the e x p r e s s i o n f o r P ( r , © )•  J«,gCS) "  «f*A o\j«$ v  Making t h i s s u b s t i t u t i o n f o r $3 i n t o equation  (30) the  integration gives: s1  —»  (31)  - 28 at  y  0  a  P (r,6) 2 $ > Vo<^{\ ~ 0  This pressure y a 0  i s produced by a wedge o f water p i l e d above  on the west s i d e and a d e p r e s s i o n o f the surfaoe on  the east side o f the o a n a l .  The oanal surfaoe under the  i n f l u e n c e o f C o r i o l i s f o r o e s i s g i v e n by the e q u a t i o n  "  3  1 *  (3ib)  F i  The t o t a l d i f f e r e n c e i n h e i g h t o f the surfaoe between the west and east s i d e s o f the oanal i s fD.  Wind-driven V e r t i o a l C i r o u l a t i o n The lake models c o n s i d e r e d i n the p r e v i o u s  s e c t i o n l e n d themselves n i o e l y to the c o n s i d e r a t i o n o f v e r t i o a l o i r o u l a t i o n due to wind s t r e s s e s . i n a l a k e o f homogeneous d e n s i t y d i s t r i b u t i o n .  These have been  p r e v i o u s l y n e g l e c t e d but must be c o n s i d e r e d as a d i s turbance o f the lake  flow.  The wind s t r e s s on the lake s u r f a c e depends on the wind v e l o o i t y at the l a k e s u r f a o e .  I f the mean wind  v e l o c i t y v^, were g i v e n a t eaoh p o i n t o f the l a k e  surfaoe  the s t r e s s on the surfaoe may be o a l o u l a t e d from the equation  (Munk %  "  CD J a i r  (  \  -  -  29  where Cj, i s the drag  coefficient.  s t r e s s as a f u n o t i o n o f x  In e f f e c t the surfaoe  i s given  from atmospheric data and i n terms o f t h i s f u n o t i o n the v e r t i o a l o i r o u l a t i o n i n the lake o r c a n a l may mined.  be d e t e r -  A s e m i - o i r o u l a r v e r t i o a l c r o s s - s e o t i o n w i l l be  oonsidered.  T h i s may be e i t h e r the l o n g i t u d i n a l c r o s s -  s e c t i o n o f s e c t i o n 3 A o r the transverse  cross-section of  s e o t i o n 3 G. The wind s t r e s s must be balanoed by a v e l o c i t y shear at the surfaoe T;(x)  Ts(x)  We w r i t e surfaoe  - A  s  y s  _J_x  =  0, i . e .  - - A  T t(£")  s t r e s s , and  j*  y  where  0  for  y.O  (32)  i s a unit of  a dimensionless  function.  Then -  *1±  _L2>  -- -  K tCfc)  f o ^ - o  (  3  2  a  The v e r t i o a l c i r c u l a t i o n s are g i v e n by the streamfunction  <fs whioh s a t i s f i e s the homogeneous biharmonio  equation  V*-? 2?  Consider  the f u n c t i o n  <f  =  a 0.  - I k y Re£TC«>}  2  where *o =  V  (33)  - 30 -  I t i s shown i n Appendix I t h a t any f u n o t i o n o f the form  *j 4> where £  i s a harmonic f u n c t i o n *  satisfies  the biharmonio e q u a t i o n . Choosing T(&>) suoh t h a t :  M'S}  for  >  a  O  (33a)  and w r i t i n g (34) we r e q u i r e  that the boundary  (a)  = -<t(£) f o r i hi  (b) s  \;  « 0  *» 0 f o r p s i  (o)  ^  s 0 for  (d)  ^  =0  be  conditions  )  s  for p s  O 1  satisfied. must t h e r e f o r e U  =  )  < b>  12L  -  5 ft  (o)  /t =  (d)  /  =  satisfy  oonditions  f o r J; = 0  0  f - {i k L R e  O  for  r t v a i  i  3}for p  = i  = O  1 Jr R e ( T c - ) }  for p  Suoh a f u n c t i o n may be c o n s t r u c t e d i n Appendix I .  the boundary  = 1  by the method o u t l i n e d  - 31 The  simple oase o f uniform wind s t r e s s on the  lake surfaoe i s c o n s i d e r e d below mainly  to make a  q u a n t i t a t i v e oomparison o f t h i s model w i t h the e l a s t i o analogy o f a t h i n s e m i c i r c u l a r p l a t e o f u n i f o r m  flexural  r i g i d i t y deformed by a constant bending moment a p p l i e d a t the hinged diameter.  Deformation  contours f o r t h i s oase  were e x p e r i m e n t a l l y o b t a i n e d ( H i d a k a ^ ) u s i n g a t h i n s t e e l p l a t e clamped along the ciroumferenoe, r e s t i n g on a k n i f e edge.  the diameter  The bending moment.was a p p l i e d  by extending the s t e e l p l a t e past the k n i f e edge i n a s e r i e s o f s t r i p s on which weights  o f known s i z e were p l a o e d .  The  contour l i n e s obtained e x p e r i m e n t a l l y are reproduced i n F i g . 11. The biharmonio  e q u a t i o n s u b j e c t to the above  boundary o o n d i t i o n s w i t h t ( £ ) s  C  admits  an exact  s o l u t i o n which i s g i v e n below. 7  r  L  (35)  P  Contour l i n e s o f y> n o r m a l i z e d to an a r b i t r a r y  ^  m  a  x  chosen  to agree with the experimental contours are p l o t t e d i n F i g . 10, and  y (p,$ max  0  i  s  p l o t t e d as a f u n o t i o n o f p f o r  = 90°» and 45°, together w i t h the e x p e r i m e n t a l l y  o b t a i n e d deformations unity  f o r the same angles n o r m a l i z e d to  (broken c u r v e s ) , i n F i g s . 8 and 9.  PLATE I I I  FiQ  .  8  P r o f i l e s o f wind ^-driven c i r c u l a t i o n f u n c t i o n s o f F i g s . 10 and 11 a l o n g r a d i u s 6 - 90° (normalized to u n i t y at maximum)  P r o f i l e s o f wind-driven c i r c u l a t i o n Fig. 9. f u n c t i o n s of F i g s . 10 and 11 along r a d i i 6 • 45° or & s 135°. (Same n o r m a l i z a t i o n as i n F i g . 8)  32 -  4. (a)  DISCUSSION OF RESULTS  Thermal C i r c u l a t i o n s In F i g u r e s 1, 5 and 4 o f s e c t i o n 3 A we have  shown the s t r e a m l i n e s f o r the flow i n a  two-dimensional  model lake w i t h zero thermal o i r o u l a t i o n , and w i t h a olookwise and a counterclockwise thermal o i r o u l a t i o n r e s p e c t ively.  In the clockwise o i r o u l a t i o n oase the flow i s  reduoed along the top and i n c r e a s e d a l o n g the bottom, the s t r e a m l i n e s o f F i g . 4 a l l b e i n g d i s p l a o e d downwards compared to those o f F i g . 1 on the i n l e t side o f the l a k e , and upwards on the o u t l e t s i d e .  I n the oounterolookwise  oir-  o u l a t i o n oase the flow a l o n g the top i s i n c r e a s e d , and a c l o s e d o i r o u l a t i o n i s s e t up near  the bottom.  In both  oases the thermal c i r o u l a t i o n has been taken to be 19.2fo o f the throughflow. We now d i s c u s s the above model i n the l i g h t o f o b s e r v a t i o n a l data taken from a t y p i c a l l a k e .  (In t h i s  oase the data i s taken f o r the n o r t h e r n arm o f Kootenay Lake.)  The importance  o f the conveotive o i r o u l a t i o n as  oompared to the throughflow  i s r e f l e c t e d i n the c o e f f i c i e n t  ^ whioh measures the r a t i o o f the two f l o w s .  We oonsider  the "two-layer l a k e " d e s o r i b e d i n s e c t i o n 3 A (d) (Case I ) . Here  i s g i v e n by:  1 L  a  a  b A F 2  fc 9 o * sla.0, IT max TS-TC  PLATE  1 9  l 0  S t r e a m l i n e s f o r wind-^driven - c i r c u l a t i o n ( t h e o r e t i c a l ; normalized i n same u n i t s as P i g . 11.)  IV  S t r e a m l i n e s f o r wind-driven c i r ig. 11, c u l a t i o n (experimental contours from e l a s t i c model o f r e f . (1); arbitrary normalization)  - 33 T y p i o a l observed values o f the q u a n t i t i e s are i n s e r t e d i n t o t h i s equation  t o g e t an i d e a o f the order o f magnitude o f  the temperature d i f f e r e n c e necessary c i r c u l a t i o n s important I f we take  \  to make thermal  i n determining  * 0.19S • V _  for  ma  the flow e  * 135°,  pattern* the value  used i n F i g s . 4 and 5, and solve f o r f we o b t a i n : f  a  16 7E" A j* M.a  3  b sin©,  The mean disoharge  o f the Lardeau R i v e r i n t o  Kootenay Lake d a r i n g the summer months i s 2 x 1 0 The mean width o f the l a k e i s about 2.5 Em. to be the depth o f the lake sinoe  8  cm / 3  S Q 0  .  " a " i s taken  the thermal c u r r e n t s  will  be muoh more s e n s i t i v e to depth than to the l e n g t h o f the lake. The  The average depth i s about 400 f e e t o r 1.22 x 10 cm. 4  g r e a t e s t u n c e r t a i n t y l i e s i n the value o f the eddy  v i s o o s i t y A.  The range o f v a l u e s f o r the h o r i z o n t a l eddy g  v i s o o s i t y observed i n ooean c u r r e n t s l i e s from 2 x 10 ** 4 x 1 0 gm/om seo. 8  ( S v e r d r u p l * , p. 485). 1 0  The value i s  found to deorease as the s i z e and v e l o c i t y o f the c u r r e n t s decrease.  The value o f 2 x 1 0 gm/  the C a l i f o r n i a o u r r e n t .  6  om  S Q  o . was observed f o r  Kootenay Lake, being muoh narrower  with s m a l l e r c u r r e n t s than the ocean c u r r e n t s f o r whioh the above eddy v i s c o s i t i e s were c a l c u l a t e d , would be expected t o be c h a r a c t e r i z e d by a s m a l l e r eddy v i s o o s i t y .  - 34 We take A ~ value o f  10  gm/  6  om  S  Q  0  .  The  corresponding  f is f  =  2.2  x  10~  5  whioh corresponds to a temperature drop from the intake to the o u t l e t o f the l a k e o f of  0.27°C.  a t a mean temperature  10°C Since h o r i z o n t a l temperature v a r i a t i o n s o f t h i s  o r d e r o f magnitude, o r even h i g h e r , can e a s i l y e x i s t i n the upper l a y e r s o f the l a k e d u r i n g the summer months, our model g i v e s  *i 7, 0.2,  i . e . that thermal o i r o u l a t i o n i s  s u f f i c i e n t l y l a r g e to s e t up flow p a t t e r n s with o i r o u l a t i o n s s i m i l a r to F i g . 4.  olosed  This i s c o n s i s t e n t w i t h  the observed f a o t t h a t r e l a t i v e l y h i g h v e r t i c a l  stratifi-  c a t i o n e x i s t s i n the upper l a y e r s o f the lake d u r i n g the summer while  the lower l a y e r s are quite homogeneous, the  average temperature not v a r y i n g a p p r e c i a b l y over the seasons.  Suoh a s t r a t i f i c a t i o n suggests t h a t d u r i n g the  summer the. warmer r i v e r water flows i n the upper p a r t o f the lake while a c l o s e d o i r o u l a t i o n takes plaoe  i n the  bottom homogeneous l a y e r i n a d i r e o t i o n suoh that bottom v e l o c i t i e s are i n the d i r e o t i o n o f the i n l e t , s i m i l a r to the flow p a t t e r n o f F i g . 4. (b)  Pressure E f f e c t s There i s some question as to the ohoioe o f the  - 35 value o f A i n the above d i s c u s s i o n . U n f o r t u n a t e l y there i s a l a c k o f a c c u r a t e experimental data about a o t u a l c o n d i t i o n s i n the l a k e as w e l l as o f t h e o r e t i c a l knowledge of  the v a r i a t i o n o f the eddy v i s o o s i t y w i t h  flows. observed  different  However, an estimate o f A may be made from the i n d i c a t i o n o f a s l o p i n g lake surfaoe.  In seotion  3 C we o b t a i n e d two equations f o r the slope o f the dynamic lake s u r f a o e .  Equation  (30a) r e f e r r e d to the two-dimen-  s i o n a l model o f s e c t i o n 3 A, while e q u a t i o n to  (30b) r e f e r r e d  the t h r e e - d i m e n s i o n a l c a n a l model o f s e c t i o n 3 C. The  observed drop o f the Kootenay l a k e surfaoe  as found by t a k i n g water l e v e l s r e f e r r e d to e l e v a t i o n s e s t a b l i s h e d by a Geodetic survey from a p o i n t near the i n l e t t o Queen's Bay, a d i s t a n c e o f about 60 Km., v a r i e s between 0 - 5 cms. d u r i n g the summer months. average  U s i n g the  value o f 2;? cm. f o r the drop i n the l a k e surfaoe  we o b t a i n a slope o f 4.5 x 1 0 ~ . U s i n g t h i s s l o p e , and 7  the other oonstants f o r Kootenay Lake as g i v e n i n 4 ( a ) , we f i n d A » 2.0 x 1 0 gm/om e o . f o r e q u a t i o n 5  S  (30a).  U s i n g a oanal w i t h the same c r o s s - s e c t i o n a l area as the lake we o b t a i n from e q u a t i o n  (30b), A = 4.1 x 1 0 gm/ 5  cm  S  T h i s method o f e s t i m a t i o n i s subjeot to many o r i t i o i s m s but does i n d i c a t e  that A l i e s  i n the range  ohosen f o r the d i s o u s s i o n i n 4 ( a ) . The r a t i o between the c o n t r i b u t i o n o f the f i e l d  Q  0  #  - 36  -  a c c e l e r a t i o n terms and the eddy v i s c o s i t y term was  found  i n s e o t i o n 3 B to be o f the order o f magnitude o f F / A D » U s i n g the v a l u e s o f F for  r  b and A g i v e n i n 4  the r a t i o a value o f 8 x 10~ , 4  a c c e l e r a t i o n terms are about 0.08^ and may  (a) we  so that the  obtain field  o f the v i s o o s i t y  terms  be v a l i d l y n e g l e c t e d i n d i s c u s s i n g the mean flow  pattern. (o)  Coriolis  Foroes  I n the three-dimensional model d i s c u s s e d i n s e o t i o n C we  found t h a t C o r i o l i s f o r o e s have two  main  e f f e c t s on oanal flow: the t i l t i n g o f the oanal surfaoe by p i l i n g up o f water on the west s i d e and dropping o f the surfaoe on the east s i d e , and the t i l t i n g o f the  constant  d e n s i t y surfaoes to c o u n t e r a c t the unequal a c t i o n on f l u i d o f the C o r i o l i s f o r o e s beoause o f the  the  vertioal  v e l o o i t y g r a d i e n t i n the l a k e . We  oonsider f i r s t l y  the t i l t i n g o f the  oanal  s u r f a c e u s i n g v e l o c i t i e s and d i s t a n c e s r e p r e s e n t a t i v e o f those  found i n l a k e s .  I f the e f f e c t s are n e g l i g i b l e i n a  canal o f the dimensions o f the lake then the e f f e c t s would be n e g l i g i b l e i n the lake i t s e l f . 11  The  total difference  Jl ^ between the water l e v e l s on the west and east banks  of  the oanal was 8 3"  a  V a y 3 0  found i n s e o t i o n 3 C (d) to be g i v e n by  37  f? =  _  * VQ  where If • 2a e width o f c a n a l .  W  ? 6  3  ,A s  J X s i n f6  s earth's  where  angular  velocity,  <j^s l a t i t u d e f o r Kootenay Lake. <}> * 50°. *  We  2 7T 86400  *  s i n 50°  = 5.57 x 10~ seo~]- £9 5  s 980 dynes/ o  V«  •  .13 om/seo. (Equation  W  *  2.5 x 10 om.  m  27a)  5  obtain h e  2.5 x 10"* cm. 3  T h i s d i f f e r e n c e i s completely no t i l t i n g o f the lake surfaoe  n e g l i g i b l e so that  occurs beoause o f C o r i o l i s  foroes. The  second e f f e c t a r i s i n g because o f the v e l o c i t y  g r a d i e n t i n the l a k e i s c o n s i d e r e d equation  We  recall  (29a).  if  r  = 3  _r  next.  *  _f  4  l ,  where Y-  4*  « 3 x 10""  8  where a s depth o f lake o r oanal  To get an order o f magnitude o f -if  we a r b i t r a r i l y  ohoose  the f o l l o w i n g data, whioh are more r e p r e s e n t a t i v e o f summer r a t h e r than winter  o o n d i t i o n s i n the l a k e .  Suppose that a t a depth o f 100 f e e t the lake  - 38  -  temperature were 7°C and d e c r e a s i n g at the r a t e o f 1° 100  per  feet, i . e . £ |i  The slope  «  4 x  10- / 7  f o o t  a t the oentre o f the lake  £  s 0  is  This slope corresponds  to a drop o f 3 inohes per mile o f  the constant d e n s i t y s u r f a c e s from the dynamic surfaoe o f the l a k e .  This oannot be measured as the l a k e would have  to be completely suoh a low  free from o t h e r d i s t u r b i n g e f f e o t s before  slope oould be observed.  Sinoe Ve i s  e s s e n t i a l l y i n v e r s e l y p r o p o r t i o n a l to the s t a b i l i t y faotor  h \l  vertioal  the slope might p o s s i b l y become  measurable i n the winter months when the s t r a t i f i o a t i o n i s low. (d)  However, no suoh o b s e r v a t i o n s are a v a i l a b l e a t present. Wind-driven V e r t i o a l C i r o u l a t i o n Sinoe no wind data has been s t u d i e d , no  i s made here circulations.  to o a l o u l a t e the magnitude o f The  wind-driven  oase o f u n i f o r m wind s t r e s s i s considered  f o r comparison w i t h a s e t o f oontours obtained  attempt  experimentally  ( H i d a k a ^ ) f o r the deformation  o f a damped semi-  o i r o u l a r s t e e l p l a t e to whioh a constant bending moment was  a p p l i e d at the hinged diameter.  The  agreement between  - 39 experiment  and theory i s quite good.  F i g u r e s 10 and 11  show the t h e o r e t i o a l and experimental contours w i t h the same a r b i t r a r y n o r m a l i z a t i o n .  I n F i g u r e s 8 and 9  p r o f i l e s along & » 4 5 ° and  = 90° are g i v e n w i t h the  6  f u n c t i o n s n o r m a l i z e d to u n i t y a t the maximum. I t i s found t h a t the maximum experimental def o r m a t i o n occurs a t 0.25 o f the r a d i u s along 6- » 90° while the t h e o r e t i o a l maximum l i e s a t 0.3 approximately. Along 9 a 45° the agreement i s n o t as good, the deformations being l a r g e r than t h e o r e t i o a l l y expected.  No e x p l a n a t i o n s  o f the d i f f e r e n c e between the two f u n c t i o n s i s attempted since we have no acourate knowledge o f the o o n d i t i o n s under whioh t h i s experiment  was  performed.  - 40 5.  -  BIBLIOGRAPHY  1.  Hidaka, K., and M. Koizumi, 1950: V e r t i o a l o i r o u l a t i o n as i n f e r r e d from the b u c k l i n g o f e l a s t i c plates. Geophysical Notes, V o l . 3, No. 3, Geop h y s i c a l I n s t i t u t e , Tokyo U n i v e r s i t y , Tokyo, Japan. (Included i n C o l l e c t e d Ooeanographioal Papers, V o l . 1, 1949-1950.)  2.  Love, A.E.H., 1944: A t r e a t i s e on the mathematical theory o f e l a s t i c i t y . Fourth Ed., Dover, New York.  3.  Love, A.E.H., 1929: Biharmonio a n a l y s i s and i t s applications. Proo. London Math. S o c , S e r i e s 2, V o l . 29, pp. 189-242.  «4.  Munk, W.H.,  1950: On the w i n d - d r i v e n ooean o i r o u l a t i o n . J o u r n a l o f Meteorology, V o l . 7, No. 2, pp. 79-93.  5.  Munk, W.H.,  and E.R. Anderson, 1948: Notes on the theory o f the thermooline. J o u r n a l o f Marine Research, V o l . V I I , No. pp. 276-295.  3,  6.  M u s k h e l i s h v i l i , N.I., 1949: Some fundamental problems i n the mathematical theory o f e l a s t i c i t y . (In Russian) Academy o f S c i e n o e s , U.S.S.R., Moscow.  7.  P r a n d t l , L., 1925: B e r i c h t Uber Untersuohungen zur a u s g e b i l d e t e n Turbulenz. Z. Angew. Math. Mech., V o l . 5, No. 2, p. 136.  B,  Rouse, H.,  9.  T a y l o r , G.I., 1932: The t r a n s p o r t o f v o r t i c i t y and heat through f l u i d s i n t u r b u l e n t motion. P r o c . Roy. Soo. (London), A, V o l . 135, pp. 685-702.  10.  1938: F l u i d meohanios f o r h y d r a u l i c engineers, F i r s t Ed., McGraw-Hill, New York.  Sverdrup, H.U., M.W. Johnson, and R.H. Fleming, 1942: The oceans, t h e i r p h y s i o s , chemistry and general biology. P r e n t i c e - H a l l Inc., New York.  - 41 APPENDIX I Biharmonio A n a l y s i s Biharmonio f u n c t i o n s w i t h p r e s c r i b e d boundary o o n d i t i o n s on the circumference  o f a c i r c l e may be con-  s t r u c t e d by the f o l l o w i n g g e n e r a l method. If 0 ( x , y ) , ^ i , ^ 2 V tf> Z  * 0)  then  r x^,  a  r  y^  e  harmonic f u n c t i o n s ( i . e .  , x j ^  y  e t c . are  biharmonio f u n c t i o n s s a t i s f y i n g the biharmonio V  2  V r^  = 0.  2  equation  T h i s may be v e r i f i e d by d i r e c t s u b s t i t u -  t i o n , e.g.:  ^  v y ^  ax*  T y  ^ y Sinoe ^  S <t>  and therefore  are harmonic f u n c t i o n s  *y 2v  2  i i  >y  Sinoe the equation  = v >7 yj£ *0 s  2  i s l i n e a r the sum o r d i f f e r e n c e o f two  solutions i s also a solution. Introducing we oonsider  £ s x/ , a  )  a y/ , a  i n p a r t i c u l a r the two biharmonio  p •  *7a  functions  - 42 „(2)  .^U)  n-l  , .(1)  • n  >  '  n  r  where  » P  n  sinii0 «^ | « 8  t  n p  O  oa  we o b t a i n :  n-i  =  p  n V l  f cos e s i n n e  -sinecosn*} = p  y,,(2) . a»-lJ n-i p  o  o  s  ©  © t i n » s i n n e j s p "*" 11  c  o  s  n  S  -  = ^{ n^ A  Sinoe  n  ) +  G  n * a}"  £  the s e t o f f u n c t i o n s p  n  oos(n-l)©  1  J  A s e r i e s o f suoh biharmonio f u n c t i o n s may  f  s i n ( n - l )6  n + 1  J  [V^  111  n  s i n nfi , p  be w r i t t e n  a  n  +  G  n  P  °  n + 2  cos n *  o s  n  d  ]  are a l s o  s o l u t i o n s o f the biharmonio equation we g e t the g e n e r a l s o l u t i o n o f the biharmonio e q u a t i o n t  B  n[{ n P " + A  n+  2  B  n P*]  8 1  * -9+[c  p * + D n  a  2  p ] oos n  n  Any biharmonio f u n o t i o n which i s r e g u l a r f o r p < 1  may  n*] be  expanded i n a s e r i e s o f the above f u n c t i o n s and i s d e t e r mined u n i q u e l y by the v a l u e s o f p  and  3 & g i v e n on the  boundary p s i . For the purposes o f t h i s t h e s i s we oonsider the function:  - 43  -  i s a g i v e n f u n o t i o n which may  be e i t h e r a harmonic or  a biharmonio f u n c t i o n s a t i s f y i n g the c o n d i t i o n jf ^ = Q __)  %  s 0,  s f(j) for £  where f££)  i s a given  of i .  function  £  J^must s a t i s f y the 2,  0,  f =  oonditions « f£)  V' = oonstant,  for = 0  _____  J  .0  for p = 1  *P The a n a l y s i s and below may where «  constants A , n  B  n  are o b t a i n e d  by  Fourier  the r e s u l t i n g s e r i e s i n a l l oases oonsidered  be summed by s u b s t i t u t i n g w s pe'*  » ^  for p  sin n ©  imaginary p a r t o f the r e s u l t -  +  i n g complex f u n c t i o n o f  <-a  g i v e s us  the r e q u i r e d  function.  For a more d e t a i l e d d i s o u s s i o n o f biharmonio a n a l y s i s  see  Love, A . E . H . ^ or M u s k h e l i s h v i l i , N . I . ^ . (a)  The  Throughflow F u n o t i o n Choose  a 2/^ a r c t a n [" 2 p s i n 6  L  i -  P*  1.  J  This  harmonic f u n c t i o n i s w e l l known from p o t e n t i a l flow and g i v e s the s t r e a m l i n e s souroe and oirole.  o f non-viscous flow  to a  s i n k at the ends o f the diameter o f a u n i t  Choosing the c i r c u l a r streamline  boundary we  due  theory  require  that  p • 1 as a s o l i d  - 44  = 2/^ arc tan j  2 p s i n 6 1 +.  L  i - P  s a t i s f y the o o n d i t i o n s no  J  2  -  Z  ( A^ p* * *. B 1  pnJ  2  n  sin  n6  n  ^ » 1,  « 0  at p s 1  i . e . that  s l i p p i n g occur at the s o l i d boundary.  ZL.  The  resulting series i s :  H  ( p * 2  - p * * ] sin(2n4-l)©  + 1  2  3  Although t h i s s e r i e s i s not convergent f o r  P/•  at p s 1  P  (1-p ) p s i n 6 \ to whioh t h i s s e r i e s l - 2 p oosZ** p i s a t i s f i e s a l l the o o n d i t i o n s a t p = 1. 2  converges f o r p < l (b)  The  Green's  4  Funotion  This f u n c t i o n i s w e l l known from e l a s t i c  theory  and d e s c r i b e s the d e f l e c t i o n o f a clamped c i r c u l a r p l a t e due  to equal and opposite  (Love, A . E . H . ^ ) .  I t may  method by choosing s  _»  s  «-  | , {&-  a r e  point loads at conjugate p o i n t s be c o n s t r u c t e d by the above  « C £ s £ In s_  - s  2  In s } 2  where  the d i s t a n c e s from the conjugate p o i n t s 0. ); ^ ,0+e»)respectively.  the biharmonio equation t i o n s a l o n g the  and  This  satisfies  the neoessary boundary  oondi-  diameter.  The Green's f u n c t i o n i s then:  G - C [ B £ i n s£ - s  2  in s } 2  ->I|A_ p *K n  B_ p n j  sin n 0  - 45 The c o e f f i c i e n t s A ,  -  B, are determined by F o u r i e r a n a l y s i s  n  and the r e s u l t i n g s e r i e s i s ^m s i n m 0 s i n o  •h[2 p  q  3  - 4 p q ]sin © c s i n * j  - 4 p q  3  m©  3  3  which converges t o : C [ In t . - i n t _ \ 2  where  t  s l + p  2  q  2  8  - 2 p t oos  The n o r m a l i z a t i o n  (9 ±  ©•)  i s accomplished by the method  outlined i n section 3 A (o). (o)  Wind S t r e s s  Funotion  Choosing the biharmonio P  p  «  - 1  p  K  2 sin 9  funotion s a t i s f y i n g the  oonditions  we r e q u i r e t h a t the t o t a l f u n c t i o n f  s  - lKp 2  satisfy  ^  s i n © -t £ $ A *  2  2  r 0,  j>J_ > P  s 0  p " «- B n<  n  2  along  p j  sin n©  n  n  p s i .  The r e s u l t i n g s e r i e s i s : - __  X  21 n  f  fj;  * °l Lin^T  1  1 _2n+l  2n-TTj » < ^sin  (2n-»-l)EJ  r  1 2n+3l  jg-J*  J  - 46 -  which converges t o : sin© f 2 p  2  sin © t a n " ^ P 2  1  2  SJ  ^«f ]  - ( 1 - P 2 ) 2 r s j g _ 2 © l n [ l » 2 p oos»-l-p ~| 2p2 - L Z. L l - 2 p oose + p * J 2  - oos 2 ^ t a n  _ 1  f l  2p sin© i  z  p  —  1 |[ J  J  j  - 47  -  APPEMIZ I I E v a l u a t i o n o f Thermal Cirou.lati.on F u n c t i o n s Case I .  The e v a l u a t i o n o f  V"(p,0; 9 ) as d e f i n e d by t  e q u a t i o n (13). XT «  / d  q B(p,*;  q,©,)  where B ( p d ;  q^J  t  i s d e f i n e d by  equation (10),  We  l T ( p , 0 ; 6,)  oonsider  L e t ^£.a © ±fl> ,  s  0  B  ( p , 0 ; q^^©)  m a  y  t  where O  i s r e p l a c e d by £ .  f  0  r p +q -2 pq o o s ^ , t « 1+p  2  2  2  2  2  q -2pq 2  oos^  written:  e  Btp^jq^cV,)  =  l (P» 4 i J t ) "  B  l P»  B  (  ^.^)  where: B  (p, q,* )  x  =  s  2  In s / 2 2  t  Then:  where: S(Pt0 )  / ^ ( p , q, f* ) d q  =  o  Making the t r a n s f o r m a t i o n p z - (l^p ) oos^ 2  we o b t a i n :  - {z ^p sin ^/ln|_z +i> sin ^f 2  B_(p,q.*)  *(p.0  s w,  z . s ^ -^poos^, and w r i t i n g  2  2  2  2  2  -/ b fP sin ^{lnLz *.p sin f] -poos^> 1  2  2 i  2  2  2  2  -lnCwVeinVj) - l n [ w * s l n * j } dz 2  2  - 48 I n t e g r a t i n g by p a r t s and decomposing into p a r t i a l fractions  3  +  -f.  p  S  integral:  sinV*]fln£z 4-p 2  oosf  (1-p ) 3p 2  integrand  whioh may be i n t e g r a t e d d i r e o t l y , we  o b t a i n the i n d e f i n i t e |z ,  the r e s u l t i n g  z + 2  2  sinV]  2 fo-P > 3p^" 2  - ln|> «-sin <£]j 2  2  2  o o s * - U-p )sin <}>} z 2  4  2  c  ^ ( l j _ p l ) { ( l - p 2 ) o o s V ~ ad$p2) s i n 2 ^ l n L A s i n 9 t J 2  3p  4-  _  4 p  2 3p  2  3  3 4 __ ssiinn^< 3 / s i n 0/  arotanf  2  2  oosV  3  Evaluating t h i s expression and z a 1-p oos ^  1  L jsin*|J  sin <fr(3(l-p ) 2  z  L /sin<*>/.  - ( l " 3 p ) sin») 1 arotanf" w "1 |sin£| Osin^J 4  between the l i m i t s z =-p  cos#  we o b t a i n :  F(P»*) = p _ / o o s f > -f- 3 3 3  sinVoos*}  In p  2  1  , 1-p i - 2 ( l * p ) 4 P e o s * + 2 ( 2 - p ) 3-p 2  2  2  2  cos ^] 2  1  .2. „.(l-p ) QQS<?>f(l-p ) o o s ^ - 3 ( l ^ p ) s i n f l i } l n [ l 4 - p - 2 p 3 p2  2  4- 4 p sin 4 ft 3 )sin £>| 3  - 2 3p  2  3  2  2  2  cosj_)  arotanf fslngj "j Lp-oos^J  sin ^3(l-p 2 )^oos 2 2  2  - ( l - 3 p 4 ) s i n 2 / j _ l arotanf (sing/1 (sinftj t'l-oos^J P  - 49 -  Making use of multiple angle relations the above equation may be simplified, to the following: Etp,^) = P f 3 cos^ - oos 3ft} In p  2  6"  f (1-P > { -3p -tP oos / + 3p< 2  (2-p ) oos 2 ft}  2  2  2,2 f (1-P ) j (2<-p) cos 3/ - 3p 2  2  oosft} In [l#-p -2p  2  2  o o s  + J- j |3 sinft - s i n 3ft/j a r o t a n f j s i n ^ t l 3 Lp-oos^J 3  _1 f(2-3p ) s i n 3 0 * 3 p ( 2 p - l ) s i n f t j s l n ft arotan ~3p2~* |sin?| 2  2  2  f Is i nftI 1 I _L -cosft I  Case I I . The evaluation of $ (p,©; ©  0  )  a s  cLsf—iscL by  equation (18). iMp.e;  - / o  X  qaq B(p,©;q,<9 ) 0  As before, we write this as a sum of two integrals $"(?,©;©.)  *f o  B-jJp.q,^ )qdq - / Bitp,q,ftv)qdq -o . • . /  Making the transformation defined i n Case I above we obtain  - 50 -  l  •* l-pCOS«5  /  B  o  l(p»<l»^ )<iag. = J  /L-pOOSft  B;_(z,^)zdz +-p oosft/ B;_(z,ft)dz -poos <^ -^poos^  h l ( p . ^ ) f P oosft F(p,ft)  *  where F(p,ft) i s defined i n Case I above. Integrating the expression for h;_ by parts and decomposing the resultant integrand into p a r t i a l fraotions whioh may be integrated d i r e c t l y , we obtain the following i n d e f i n i t e integral:  / x ( ,ft)zdz « £ z V s l n f z j [ I n [ z + p s i n f t ] -ln[w -*-sin «ftj} B  2  z  2  2  2  2  2  2  2  p  (1-P ) oosft z ^ f ( l - p 2 ) 2 o o s V - ( l - p 4 ) s i n V }  2  3  8  (  6 p  4p'  z  J  [  ^ (l^p )oos<* ( ( l - p ) c o s f t + ( 2 p - 3 ) s i n 2 f t j z " 2p^ <• 2  +  f  1  2  2  2  4  4__. 4^ , r 2 2_, 2 P s i n <P In [_ z -M? s i n ftJ 4  |"[(l-p 8 ) 8 oos^f (l-p 4 Jslay  2  -4((l-p ) (2-p ) 2  2  4  oosV sin ftj -p sin *} ln(w -<-sin *) 2  8  4  2  ^ 2 J ( l - p ) s i n ^ cos^[(14p )sin g-(l-p )oos <l>J p4"( 2  2  2  2  2  2  2  *j 1 arotan j w ji 1 |sinft| L IsinftlJJ Evaluating t h i s expression between the l i m i t s z • -p cosft  2  * 51 and z = 1-p cos 56  we o b t a i n : s  ^(P.ft )  - p_, 4  i np  2  .(!_£_)[(lj.p ) (9-.4p -9p ) cos^ 2p ( 2p • 3p 2  2  -(l-p 2 )oos ft-4(l-p 2 )oos^ p p  4  2  f  a  2  f ( 1 - P ) | [(l-p 2 )cos Wl+p 2 )sin «J - 4(2-p 4 )oos 2 ftsinVj* 2  4  2  2  2  2  K{mfi p -2 2  p  +  4_ 2(l-p 2 ) 2 sln 2 *oosft  P  costf}  5 (H-p 2 )sin 2 ft - (l-p 2 )oos 2 ^} * x  1 arotanf fsin ft< "] |sinft| I 1 -oosft I  ( I  p Introducing  4 j L h r h i *- p'p I npp2 4 +m - ((i zl r~2P^ ) M(1+P "2') 4 4*p< 4  2  A  2  A  .  2  ^  and e x p r e s s i n g the other terms i n m u l t i p l e angle  form, we  o b t a i n a much simpler e x p r e s s i o n f o r h ( p , * ): h(P.*) = - ( l r _ l ) ( ( 5 - 9 p ) oosft - ( l - p ) c o s 2^- (1-p )oos 3f] 2p t 3 2p pk 2  2  2  (1-P ) ( f p - 2 p o o s 2ft-*-oos 4ft]I n f l + p - 2 p 2 2pr ~ 2  4  1  2  2  ~  oosftj  - [ T sk i n 4ft 4tf> - 2 2D p s i n 2^1 2^] sinft>arctan J* lsin</>| . 1 2  \aiai\  [l-oos<|,J  Finally $(p,*;*<>) = h ( p , £ ) - h(p,<^)i-p{aos^F(p, ftj-oosj4 F(p, ftj j f  

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