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UBC Theses and Dissertations

On the inertial stability of coastal flows Helbig, James Alfred 1978

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ON THE INERTIAL STABILITY OF COASTAL FLOWS by JAMES ALFRED HELBIG M. Sc., University of B r i t i s h Columbia, 1977  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Physics and Institute of Oceanography)  We accept t h i s thesis as conforming to the required  standard  THE UNIVERSITY OF BRITISH COLUMBIA November, 1978 (c) James A l f r e d Helbig, 1978  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l l m e n t o f the requirements f o r  an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree the L i b r a r y s h a l l make i t f r e e l y  a v a i l a b l e f o r reference  and  that  study.  I  f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may representatives.  be  g r a n t e d by  the Head of my  Department or by  I t i s understood t h a t copying or p u b l i c a t i o n o f  thesis for f i n a n c i a l gain  s h a l l not be  permission.  I n s t i t u t e o f Oceanography Department o f P h y s i c s  and  The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, B r i t i s h Columbia V6T 1W5 Canada  27 December  1978  a l l o w e d w i t h o u t my  written  his  this  Abstract  T h i s t h e s i s i n v e s t i g a t e s two  separate but r e l a t e d problems.  P a r t I a study i s made o f the p r o p a g a t i o n o f c o n t i n e n t a l s h e l f waves  In and  b a r o t r o p i c Rossby waves i n a steady, l a t e r a l l y sheared c u r r e n t of the V + £W,  where  W  i s a centred  c o r r e l a t i o n length of  W  random f u n c t i o n and  £ <<  1.  form  I f the  i s s m a l l compared w i t h the c h a r a c t e r i s t i c  h o r i z o n t a l l e n g t h s c a l e o f the system; f o r example, the s h e l f width o r a channel w i d t h , the waves are u n s t a b l e .  T h e i r growth r a t e i s l a r g e l y  determined by the magnitude o f the c o r r e l a t i o n l e n g t h , w h i l e the phase speed i s g i v e n by  the sum  l a t e r a l gradient Brooks and Mooers  o f weighted averages o f the mean c u r r e n t  of p o t e n t i a l v o r t i c i t y .  V  and  A p p l i c a t i o n of the theory  the to  the  (1977a) model of the F l o r i d a S t r a i t s y i e l d s wave parameters  t h a t are i n a c c o r d w i t h those measured by Duing  (1975).  In P a r t I I , an attempt i s made to understand the dynamics g o v e r n i n g observed low-frequency c u r r e n t s  i n the S t r a i t o f G e o r g i a  t w o - l a y e r model i n d i c a t e s t h a t the mean c u r r e n t s b a r o c l i n i c a l l y stable.  i n GS  (GS).  A  simple  are p r o b a b l y  A b a r o t r o p i c s t a b i l i t y model i m p l i e s t h a t a shear  i n s t a b i l i t y might be o f some importance.  However, the a n a l y s i s of  current  meter data shows t h a t the v e l o c i t y components o f the f l u c t u a t i o n s are n e a r l y i n phase o r c l o s e to 180° are not due  out o f phase; t h i s means t h a t the motions  t o the type of waves c o n s i d e r e d  r e l a t i o n s h i p between the winds and domains i m p l i e s t h a t the wind may  either  currents  here.  A n a l y s i s of  the  i n both the frequency and  p l a y an i n d i r e c t r o l e i n f o r c i n g GS  time motions.  It i s conjectured that the wind and tide interact with the Fraser River outflow to modulate the estuarine circulation i n the system and force low-frequency currents. Direct nonlinear interaction between t i d a l constituents produces a coherent fortnightly variation i n the currents, but cannot account for the observations.  TABLE OF CONTENTS  ABSTRACT  i i  LIST OF TABLES  vi  LIST OF FIGURES  v i i  ACKNOWLEDGEMENT  xi  Section 1  Introduction to Thesis . . .  1  PART I 2  Introduction to Part I  2  3  Formal Theory f o r S h e l f Waves i n a Channel  7  4  The V o r t i c i t y  5  The Channel Mode  25  6  The C o n t i n e n t a l S h e l f Model  61  7  Rossby Waves i n a Random Z o n a l Flow  67  8  Summary and C o n c l u d i n g  71  and Energy Balances  17  Remarks t o P a r t I PART I I  9  Introduction to Part I I  72  10  P h y s i c a l Oceanography  76  11  I n e r t i a l I n s t a b i l i t y Models  12  A n a l y s i s o f Data  114  13  Nonlinear  145  14  Summary o f P a r t I I  o f the S t r a i t o f Georgia  93  T i d a l Interactions  158  BIBLIOGRAPHY APPENDIX A:  160 Order o f Magnitude E s t i m a t e s Terms i n (3.25)  o f the I n t e g r a l 166  iv  APPENDIX B: :  APPENDIX C:  The F i r s t - O r d e r S o l u t i o n s Evaluation  o f the I n t e g r a l Terms f o r a Simple  Flow Model APPENDIX D:  170  Baroclinic Instability  173 i n a 2-Layer System  v  178  LIST OF TABLES  Table I  Order of magnitude estimates of terms i n the v o r t i c i t y balance equations  II  .19 T = 1/0,  The c h a r a c t e r i s t i c growth times wave  f o r a 200-km  (k/27T = .15)  59  III Calculated coherence squared and phase between v e l o c i t y components f o r the 136-day period of analysis IV  133  Results of the harmonic analysis of t i d a l elevations at Point Atkinson f o r the 38-day period beginning 6 A p r i l 1976  V  146  Relative magnitudes of the terms i n (C.9) and (C.10)  vi  176  LIST OF FIGURES  Figure 5.1 5.2 5.3 5.4  5.5  5.6  5.7  5.8  The Brooks and Mooers model of bottom topography and mean current Graphical solution of (5.14) for b = 3.0 and I = 2.5 Behaviour of the f i r s t mode nondimensional growth rate 0.^ as a function of O and k Behaviour of the nondimensional growth ratefi^as a function of k for the f i r s t three modes for 0=5 and e = 0.5, (A) channel model (B) shelf model Dispersion curves for the f i r s t three modes (A) channel model (B) shelf model Behaviour of the nondimensional phase speed as a function of k for the f i r s t three modes (A) channel model (B) shelf model The mass transport stream function for (A) channel mode 1 (B) channel mode 2 (C) channel mode 3 (D) shelf mode 1 (E) shelf mode 2 (F) shelf mode 3 Profiles of u and v for: (1) channel mode 1 taken along the line 6/2TT = 0.8 i n Fig. 7A, (2) channel mode 2 taken along 6/2TT = 0.9, and (3) channel mode 3 taken along 6/2TT = 0.37  5.9  26 32 37  38 39 40 41  42 43 45 46 47 48 49 50  51  Plan view of the Florida Straits showing lines I and I I along which the sections i n Fig. 5.10 are taken vii  53  5.10  Sections along lines I and I I of (A) a (B) alongshore velocity Plan view of the west coast of B r i t i s h Columbia and adjoining waters Plan view of the Strait of Georgia showing lines of topographic cross sections (1-10) presented i n Fig. 10.2 . . Topographic cross sections: (A) Upper panels: 1-9; (B) Lower panel:~10 Longitudinal section of a for t  9.1 10.1 10.2 10.3  (A) December 1968 10.4 10.5 10.6 10.7 10.8  11.1 11.2 11.3 11.4  .  (B) July 1969 Rotary spectrum of the winds at Sand Heads for the 600-day period beginning 3 January 1969 Cross section H showing placement of current meters Mean currents along l i n e H for the 533-day period  73 77 78 79 80 83 84  beginning 16 A p r i l 1969 Current spectra for line H for the 533-day period beginning 16 A p r i l 1969 Rotary coherence and phase between currents from  85  (A) v e r t i c a l l y seperated locations (B) horizontally seperated locations . . . . . . The baroclinic i n s t a b i l i t y model Mode 1 s t a b i l i t y boundaries for the baroclinic model as a function of the topographic parameter T  88 89 95 101  Mode 1 s t a b i l i t y boundaries for the baroclinic model as a function of the internal Froude number V = + F^  102  87  Baroclinic model mode 1 dispersion curves for S = 0.5 and S = 1.5  11.5  54 55  104  The baroclinic model mode 1 phase speed as a function of topographic parameter for k/2iT = 0.1, 0.5, and 1.0 . . . .  105  11.6  The barotropic i n s t a b i l i t y model  107  11.7  The region i n (k,S) space i n which continental shelf waves exist  11.8  I l l  Computed barotropic mean currents along l i n e H for the 18-month period beginning A p r i l 1969 viii  112  11.9 12.1 12.2 12.3  12.4  12.5 12.6 12.7 12.8  12.9 12.10  12.11 12.12 13.1 13.2 13.3  Barotropic model dispersion curves for S=0.5 Plan view of the Strait of Georgia showing current meter locations Periods of existent current meter records Current spectra for the 26-day period beginning (A) 2 May 1969 (B) 29 August 1969 Mean currents and the 6-32-day band current ellipses for the 26-day period beginning (A) 2 May 1969 (B) 29 August 1969 Spectrum of the wind stress at Sand Heads for the 500-day  113  period beginning 4 A p r i l 1969 Line H current spectra for the 136-day analysis period. . . . Coherence and phase between the wind stress and currents at H26, 50m and H16, 50m Coherence and phase between l i n e H currents and the wind stress for (A) the 13-day band (B) the 34-day band Mean currents along line H for the 136-day analysis period. . Line H current ellipses for: (A) the 13-day band, upper layer (B) the 13-day band, lower layer (C) the 34-day band, upper layer (D) the 34-day band, lower layer Computed barotropic and upper layer baroclinic mean currents for the 136-day analysis period Low-pass f i l t e r e d time series of wind stress at Sand Heads and currents along l i n e H Daily barotropic residual t i d a l flow along line H The residual barotropic t i d a l flow averaged over 10 days. . . Time series of (A) predicted t i d a l height and t i d a l range  126 128  at Pt. Atkinson (B) calculated residual current magnitude and direction along line H ix  116 117 119 120  122 123  129 130 131 135 136 137 138 139 141 143 148 151  152 153  The Fraser River discharge approximately 60 miles upstream at Agassiz, B r i t i s h Columbia Low-pass f i l t e r e d time series of river speed at the Fraser River mouth  x  155  ACKNOWLEDGEMENT  Many people c o n t r i b u t e t o a t h e s i s .  My foremost g r a t i t u d e goes  t o P r o f e s s o r L. A. Mysak f o r h i s steady encouragement, p e r s i s t e n t i n t e r e s t , p r o f i c i e n t a d v i c e , and, most o f a l l ,  h i s inordinate patience  and k i n d n e s s .  I would a l s o l i k e t o thank P r o f e s s o r s G. L. P i c k a r d , P. H. LeBlond, and G. S. Pond f o r t h e i r c r i t i c i s m s o f an e a r l i e r v e r s i o n o f t h i s t h e s i s . Each found a s p e c i f i c area t h a t needed c l a r i f i c a t i o n o r f u r t h e r development. My warmest a p p r e c i a t i o n extends t o Dr. P. B. Crean f o r a l l o w i n g me f r e e access Georgia  t o the r e s u l t s o f the n u m e r i c a l system; Mr. P. J . Richards  a c c e s s i b l e t o me.  was most h e l p f u l i n making these r e s u l t s  Dr. Crean a l s o r e q u i r e s r e c o g n i t i o n f o r encouragement o f  a more moral n a t u r e .  I would a l s o l i k e t o thank Dr. J . A. S t r o n a c h f o r  many u s e f u l c o n v e r s a t i o n s work.  model o f the Juan de F u c a - S t r a i t o f  and f o r p r o v i d i n g me w i t h  some r e s u l t s o f h i s  Mr. P. Y. K. Chang demands s p e c i a l thanks f o r a l l o w i n g me use o f h i s  e d i t e d v e r s i o n s o f wind and c u r r e n t d a t a ,  and f o r p r o v i d i n g me w i t h  enlarged  c o p i e s o f many o f the f i g u r e s i n h i s t h e s i s . Above a l l ,  I must r e g i s t e r my most h e a r t f e l t a p p r e c i a t i o n t o my  w i f e , N e l i a , f o r h e r s a c r i f i c e s d u r i n g my long tenure as a graduate and  f o r her constant  u n d e r s t a n d i n g and t h o u g h t f u l a d v i c e .  student  I want  to thank her and my daughter E r i k a f o r the c o n s i d e r a t i o n extended t o me d u r i n g the f i n a l stages my p a r e n t s  of this thesis.  f o r t h e i r constant Much a p p r e c i a t e d  F i n a l l y , I express my g r a t i t u d e t o  f a i t h i n me.  f i n a n c i a l support  has been p r o v i d e d  by the  N a t i o n a l Research C o u n c i l o f Canada and the U n i v e r s i t y o f B r i t i s h  xi  Columbia.  1  1.  I n t r o d u c t i o n to T h e s i s  Two  s e p a r a t e but  examined i n t h i s t h e s i s . which c o n t a i n a s m a l l , this i s primarily  r e l a t e d problems i n p h y s i c a l In P a r t I, the  oceanography  i n e r t i a l i n s t a b i l i t y of  currents  randomly f l u c t u a t i n g component i s examined.  a theoretical investigation,  the  summarizes an attempt to understand  low-frequency c u r r e n t s observed i n the inertial  S t r a i t of Georgia.  To  i n s t a b i l i t y , wind f o r c i n g , r e s i d u a l t i d a l f l o w , and  e s t u a r i n e c i r c u l a t i o n are are p r o v i d e d f o r each  considered.  part.  While  theory i s applied  o b s e r v a t i o n s made i n the F l o r i d a S t r a i t s w i t h encouraging r e s u l t s . l a t t e r part of t h i s thesis  are  to The  the this  end,  modified  Separate, more d e t a i l e d ,  introductions  2  2.  Introduction to Part I  Under c e r t a i n c o n d i t i o n s a p l a n e t a r y wave p r o p a g a t i n g r e g i o n o f mean c u r r e n t shear i s capable  through a  o f e x t r a c t i n g energy from the flow.  T h i s was p o i n t e d o u t i n t h e p i o n e e r i n g work o f Kuo (1949) f o r Rossby waves i n a zonal current.  I n p a r t i c u l a r , he showed t h a t an extremum i n t h e  p o t e n t i a l v o r t i c i t y d i s t r i b u t i o n was a necessary e x i s t e n c e o f u n s t a b l e waves. on a  S i n c e then,  c o n d i t i o n f o r the  numerous models o f u n s t a b l e  flows  3-plane have been c o n s i d e r e d , p a r t i c u l a r l y f o r t h e atmosphere (see  the review by Kuo, 1973).  By comparison, s c a n t a t t e n t i o n has been p a i d t o  the study o f the m o d i f i c a t i o n o f another c l a s s o f p l a n e t a r y motions, namely c o n t i n e n t a l s h e l f waves (CSWs), by sheared  mean f l o w s , although  t h e theory  o f CSWs i n t h e absence o f mean c u r r e n t s has been e x t e n s i v e l y i n v e s t i g a t e d (see LeBlond and Mysak, 1977, f o r a r e v i e w ) . always e x i s t , t h i s r e p r e s e n t s  S i n c e mean c o a s t a l flows  a s e r i o u s gap i n our knowledge o f c o n t i n e n t a l  s h e l f dynamics. N i i l e r and Mysak and  a piecewise  (1971) c o n s i d e r e d a s t e p - l i k e c o n t i n e n t a l s h e l f  l i n e a r v e l o c i t y p r o f i l e and showed the e x i s t e n c e o f two  c l a s s e s o f motions, m o d i f i e d CSWs and "shear" waves whose e x i s t e n c e depends on t h e mean c u r r e n t shear.  F o r a c e r t a i n s h o r t wavelength range, t h e two  modes c o a l e s c e i n t o a s i n g l e u n s t a b l e wave t r a v e l l i n g i n the d i r e c t i o n o f the c u r r e n t .  I n a more formal  study Grimshaw (1976) extended many o f the  r e s u l t s o f b a r o t r o p i c i n s t a b i l i t y theory t o i n c l u d e u n s t a b l e g e n e r a l depth and v e l o c i t y p r o f i l e s .  McKee (1977) has c a l c u l a t e d t h e  s t a b l e response o f t h e c o n t i n e n t a l s h e l f t o t r a v e l l i n g disturbances  CSWs f o r q u i t e  atmospheric  and showed t h a t i t i s comprised o f a s u p e r p o s i t i o n o f d i s c r e t e  3  normal modes, a continuous s e t of t r a n s i e n t s o l u t i o n s p o s s e s s i n g speeds i n the range o f the mean c u r r e n t , and Brooks and Mooers  c u r r e n t on CSWs, but solutions.  d i r e c t l y f o r c e d motions.  (1977a, h e r e i n a f t e r r e f e r r e d t o as BrM),  the F l o r i d a S t r a i t s , c o n s i d e r e d  Thus they were l i m i t e d t o m o d i f i e d  model p r e d i c t i o n s and  sheared  f o r any p o s s i b l e  unstable  CSWs w i t h phase v e l o c i t i e s  l e s s than the minimum of the mean c u r r e n t v e l o c i t y . Duing  i n a model o f  the e f f e c t o f an i n t e n s e , l a t e r a l l y  they e v i d e n t l y d i d not search  though, t h a t S c h o t t and  phase  I t should be mentioned,  (1976) found e x c e l l e n t agreement between  observations  i n the F l o r i d a S t r a i t s f o r the  BrM  10-30-day  wave p e r i o d band. In P a r t I we shore c u r r e n t and  consider  (2) a z o n a l flow w i t h r e s p e c t to CSW  perturbations, respectively. steady and  the b a r o t r o p i c s t a b i l i t y of and  (1) an  along-  Rossby wave  In each case the b a s i c c u r r e n t i s assumed  t o be composed o f a sheared mean component w i t h  a small,  s p a t i a l l y random p a r t superimposed upon i t . Although t h i s c h o i c e might seem t o f u r t h e r complicate  an a l r e a d y d i f f i c u l t mathematical problem, i t t u r n s  out t h a t the mathematics g r e a t l y s i m p l i f i e s , and solved provided  the problem may  be  easily  t h a t the c o r r e l a t i o n l e n g t h of the f l u c t u a t i n g c u r r e n t i s  s u f f i c i e n t l y small.  T h i s approach was  adopted by Manton and Mysak  (1976)  f o r the case o f p l a n e Couette flow, and P a r t I i s an outgrowth o f t h a t work. The  r a t i o n a l e f o r choosing  a random c u r r e n t i s as f o l l o w s .  The  s m a l l - s c a l e f e a t u r e s o f the b a s i c c u r r e n t are g e n e r a l l y unknown and c e r t a i n l y v a r y i n both space and  time.  b a s i c c u r r e n t p r o f i l e so c o m p l i c a t e d by a simple mathematical e x p r e s s i o n model the c u r r e n t by the sum  Moreover, these f e a t u r e s make the  as t o render a d e s c r i p t i o n o f the impossible.  Thus i t i s r e a s o n a b l e  o f a smooth, d e t e r m i n i s t i c p r o f i l e and  a  flow to small  4  irregular part.  I t i s mathematically  component as a random f u n c t i o n .  convenient  Thus we  s t r u c t u r e of the c u r r e n t p r o f i l e and  to r e p r e s e n t the  irregular  i g n o r e the a c t u a l , s m a l l - s c a l e  c o n c e n t r a t e on i t s s t a t i s t i c a l  properties, i n p a r t i c u l a r , i t s variance. I f t h i s decomposition nec e s s a r y  i s t o be p h y s i c a l l y r e a l i s t i c ,  i t is  t h a t the "random" f e a t u r e s o f the b a s i c c u r r e n t be d i s t i n c t  the motions p r e d i c t e d by the ensuing i n l a r g e p a r t , to the b a s i c flow. by t h i s t h e o r y be s e p a r a t e d  theory,  T h i s r e q u i r e s t h a t the "waves"  i n both  frequency  the random component o f the b a s i c c u r r e n t . s p e c t r a l gap  s i n c e these motions are  from due,  admitted  and wavenumber space from  That  i s , t h e r e must be a  o r r a p i d change i n s l o p e i n the v e l o c i t y s p e c t r a .  adequate d a t a , i s not p r e s e n t l y a v a i l a b l e t o t e s t v a l i d i t y o f  Unfortunately,  this  representation. As a f i r s t  s t e p i n a more e x t e n s i v e study, we  concentrate  on the l a t e r a l s p a t i a l v a r i a t i o n s and h e n c e f o r t h i g n o r e f l u c t u a t i o n s i n the b a s i c flow. which the t h e o r y developed flow i n t o d e t e r m i n i s t i c and  solely  temporal  In the case o f the F l o r i d a C u r r e n t ,  here w i l l be a p p l i e d , the decomposition  of  to the  random components i s e s p e c i a l l y a p p r o p r i a t e ,  s i n c e i n the r e g i o n o f the F l o r i d a S t r a i t s , the c u r r e n t i s s t i l l a d j u s t i n g to an almost Florida. i n t o the  90° northward t u r n i n i t s passage around the s o u t h e r n  t i p of  T h i s i s a p r o c e s s which s h o u l d i n t r o d u c e a l a r g e amount o f n o i s e flow. The  the frequency  assumption t h a t the superimposed wave f i e l d range of i n t e r e s t i s supported  e x p e r i m e n t a l l y by  i n both an i n t e n s e western boundary c u r r e n t (Diiing, 1975; Brooks, 1977)  and  Kundu and A l l e n ,  i n a weak e a s t e r n boundary c u r r e n t 1976;  i s barotropic in  Wang and Mooers, 1977).  observations  Mooers  and  (Huyer, e t a l . , 1975;  In a t h e o r e t i c a l  analysis,  5 Allen  (1976) has  shown t h a t the c o n t i n e n t a l s h e l f may  support  both  b a r o t r o p i c and b a r o c l i n i c motions, the l a t t e r trapped w i t h i n an Rossby r a d i u s of deformation  of the c o a s t .  We  expect  internal  t h i s assumption t o  be more q u e s t i o n a b l e f o r Rossby waves i n the open ocean where b a r o c l i n i c instability  i s l i k e l y t o be an important  theory r e p r e s e n t s a f i r s t instability  N e v e r t h e l e s s , the  flow.  c o n t i n e n t a l s h e l f models are c o n s i d e r e d , one  s h e l f i s bounded by a w a l l p a r a l l e l i n which i t i s not  present  step i n a study of combined b a r o t r o p i c - b a r o c l i n i c  i n a randomly p e r t u r b e d  Two  factor.  ( s h e l f model).  t o the c o a s t  (channel model), and  In both cases the BrM  topography and mean c u r r e n t i s employed.  i n which the one  model o f bottom  Attention i s primarily  focused  on the channel model s i n c e the o b s e r v a t i o n s t o which the t h e o r y i s t o be compared were made i n the F l o r i d a S t r a i t s . convenience  T h i s model a l s o has  of b e i n g l e s s complex m a t h e m a t i c a l l y  motion i s i s o l a t e d from the ocean i n t e r i o r and considered.  the added  s i n c e the c o a s t a l trapped  thus no c o u p l i n g need be  A channel model i s a l s o assumed i n the Rossby wave case.  Each of the two  c o n t i n e n t a l s h e l f models admits a c l a s s of  u n s t a b l e m o d i f i e d CSWs f o r long wavelengths p r o v i d e d t h a t the h o r i z o n t a l c o r r e l a t i o n l e n g t h o f the f l u c t u a t i n g b a s i c flow i s s u f f i c i e n t l y s h o r t compared w i t h the s h e l f width.  These waves may  propagate i n e i t h e r  direction  along the c o a s t depending on the s t r e n g t h of the mean c u r r e n t ; t h e i r phase velocity  i s g i v e n by the weighted average of the d i f f e r e n c e between the  mean v e l o c i t y and  the c r o s s - s t r e a m g r a d i e n t o f p o t e n t i a l v o r t i c i t y .  growth r a t e i s p r o p o r t i o n a l to the amplitude c u r r e n t and  o f the random component o f the  i n v e r s e l y p r o p o r t i o n a l to i t s c o r r e l a t i o n l e n g t h .  t h a t u n s t a b l e s o l u t i o n s e x i s t i s e s p e c i a l l y important velocity profile  i s almost  The  s i n c e the  The  fact  BrM  c e r t a i n l y b a r o t r o p i c a l l y s t a b l e ; t h a t i s , the  6  classical  t h e o r y p r e d i c t s the e x i s t e n c e o n l y o f s t a b l e m o d i f i e d CSWs.  A p p l i c a t i o n o f the channel model t o the F l o r i d a S t r a i t s p r e d i c t s wave parameters t h a t a r e i n good agreement w i t h o b s e r v a t i o n s The  present  Florida  (1975).  theory may thus account, i n p a r t , f o r meanders observed i n the  current. The  p l a n o f P a r t I i s as f o l l o w s .  for modified  CSWs i n a channel i s p r e s e n t e d ,  differential  equation  In S e c t i o n 3 the formal and a c o m p l i c a t e d  i s d e r i v e d f o r the m a s s - t r a n s p o r t  S c a l i n g arguments a r e employed t o reduce t h i s e q u a t i o n form.  made by Duing  A more p h y s i c a l d e r i v a t i o n o f t h i s e q u a t i o n  vorticity  i n the system i s g i v e n i n S e c t i o n 4.  derived.  A p e r t u r b a t i o n s o l u t i o n i s obtained  theory  integro-  stream f u n c t i o n . t o a more manageable  based on the b a l a n c e o f  An energy e q u a t i o n  i s also  i n S e c t i o n 5 f o r the BrM  model, and the b a s i c r e s u l t s f o r the growth r a t e s and phase speeds a r e given.  The r e s u l t s a r e a p p l i e d t o o b s e r v a t i o n s  In S e c t i o n 6, we b r i e f l y  consider  i s examined.  Straits.  the c o n t i n e n t a l s h e l f model, and i n  S e c t i o n 7 the s t a b i l i t y o f a z o n a l flow on a perturbations  made i n the F l o r i d a  6-plane  t o Rossby wave  A summary i s g i v e n i n S e c t i o n 8.  7  3.  Formal Theory f o r S h e l f Waves i n a Channel  In t h i s s e c t i o n the e q u a t i o n s governing the p r o p a g a t i o n o f small-amplitude, f r e e , modified current  are derived.  CSWs i n a l a t e r a l l y  The b a s i c c u r r e n t  c o n t i n e n t a l boundary o f i n f i n i t e  length  The  and i s composed o f a sheared mean Only the channel model i s  o f the s h e l f model i s d e f e r r e d  dynamics o f CSWs d e r i v e  barotropic  i s assumed t o flow a l o n g a  component and a s m a l l s p a t i a l l y random p a r t . t r e a t e d here; c o n s i d e r a t i o n  sheared,  to Section  from the c o n s e r v a t i o n  6.  of p o t e n t i a l  v o r t i c i t y , b u t r a t h e r than t o p r o c e e d d i r e c t l y from the c o n s e r v a t i o n  law,  i t proves convenient f o r l a t e r purposes t o b e g i n w i t h the e q u a t i o n s o f motion.  I t i s a l s o d e s i r a b l e t o work i n terms o f nondimensional  q u a n t i t i e s and the f o l l o w i n g s c a l e f a c t o r s which are r e p r e s e n t a t i v e o f continental shelf conditions be  a r e chosen:  the s h e l f width  l e s s than the channel w i d t h , see F i g u r e  coordinates  (x,y),  averaged v e l o c i t y time  L/U,  and  C o r i b l i s parameter and s t a t e which e x a c t l y  f o r the sea s u r f a c e g  v (x) B  f o r z, (u,v),  elevation.  a vertically an  Here  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 .  s a t i s f i e s the n o n l i n e a r ,  motion i s s p e c i f i e d by the b a s i c c u r r e n t b a s i c sea s u r f a c e  H  f o r the h o r i z o n t a l v e l o c i t i e s  fLU/g  (which may  5.1) f o r the h o r i z o n t a l  the maximum channel depth U  L  f  advective i s the A basic  f r i c t i o n l e s s equations of  V (x)  and i s r e l a t e d t o the  s l o p e by  = n  B x  (x).  (3.1)  8  The shallow-water  equations o f motion  linearized  about the b a s i c  state  are:  Ro(u.  Ro(v  (hu)  Here  h(x)  x  + V u ) - v = - n a y x  (3.2)  R  x.  t  + V v B  + uV ) + u = - r)  y  + h(v)  B x  y  (3.3)  y  = 0.  (3.4)  i s the nondimensional  depth and  Ro = U / f L  number f o r the b a s i c flow and i s n o t assumed s m a l l .  i s the Rossby  To o b t a i n (3.4) we 2 2  invoked the nondivergent a p p r o x i m a t i o n , which i s good t o o r d e r (10 ^  i n Florida  f L /gH  S t r a i t s ) , and t h i s a l l o w s t h e d e f i n i t i o n o f a mass-  t r a n s p o r t stream f u n c t i o n g i v e n by hu = - T  y  (3.5) hv = ¥  In terms o f  ¥  x  the l i n e a r i z e d  potential  R o ( 9 t + V B 9 y ) [h V F X  2 ,  - 1 [Roh V" _ 1  where a prime denotes  v o r t i c i t y equation i s  (h'/h )^] 2  - (1 + R o V ' ) h ' / h ] = 0  (3.6)  2  B  B  d i f f e r e n t i a t i o n with respect to  wave d i s t u r b a n c e s o f the form  x.  For t r a v e l l i n g  9  f  = *(x)e  with p o s i t i v e  k  i k ( y  -  (3.7)  C t )  and p o s s i b l y complex phase speed  (V_ - c)D$ - Q $ = hi  Here  c,  0.  -  2  (3.8)  (h'/h )d/dx - h \ 2  v o r t i c i t y o f the p e r t u r b a t i o n  Q =  (Ro  1  so t h a t  2  P$  (3.9)  B  s c a l e d by  Ro.  are o b t a i n e d by r e q u i r i n g t h a t t h e r e be no  sidewalls;  hence  . . $(x) •= 0  E(V )  such t h a t  B  = V(x)  +  = V  and  e  2  s m a l l and  E(W)  i s taken t o be  the  a s t a t i o n a r y random v a r i a b l e ,  fluctuating parts  =0 W.  where The  E  as  represents  the average over  nondimensional parameter  i s r e l a t e d t o the v a r i a n c e s  = var v / v a r W. B  f l o w through  eW(x)  an ensemble o f r e a l i z a t i o n s o f assumed t o be  boundary  (3.10)  s e p a r a t e d i n t o i t s mean and  B  The  x = 0,1.  at  the b a s i c c u r r e n t  V (x)  relative  + V' )/h  conditions  be  i s the  and  i s the b a s i c s t a t e p o t e n t i a l v o r t i c i t y  i t may  reduces t o  X  V = h V/dx  As  (3.6)  of  V  0  and  e W  is by  (3.11)  10  In the present case we choose be made for the shelf model. disturbance  var W = 1  although a d i f f e r e n t choice w i l l  Since the basic state i s random and the  interacts with i t , i t necessarily follows that the perturbation  must also contain a random component; we decompose  $ as  $'(x) = ip(x) + £<M*)  E 0|>) =  with  (3.13)  and E (<})) = 0 .  random part of $  by  £, |  Although i t i s not necessary to scale the  w i l l generally be large compared with £(f>.  We are primarily concerned here with deriving a closed form equation for  . With these d e f i n i t i o n s the v o r t i c i t y equation may be cast  into the form  (L + e.M) 0> + e<J>) = 0,  where  L  and M  (3.14)  are deterministic and random d i f f e r e n t i a l operators  respectively defined by  L = (V - c)V M = WP - q  Q  and q  vorticity  x  - Q  (3.15)  x  ;  (3.16)  are the respective gradients of mean and f l u c t u a t i n g p o t e n t i a l (scaled by Ro),  Q  = h V " - (Ro _1  x  1  + V'Vh'/h  2  (3.17)  11  q  = h W"  - W'h'/h .  1  x  (3.18)  2  The boundary c o n d i t i o n s become  UJ =  0 f  at  0,1.  x =  (3.19)  <j> = 0  S t o c h a s t i c boundary v a l u e problems o f the type d e f i n e d by (3.14)-(3.19) have been i n v e s t i g a t e d by a number o f workers, and s e v e r a l t e c h n i q u e s a r e a v a i l a b l e t o d e a l w i t h them u s e f u l t o decompose ensemble a v e r a g i n g  (see Mysak, 1978, f o r a r e v i e w ) .  I t proves  (3.14) i n t o i t s mean and f l u c t u a t i n g components.  By  (3.14) and s u b t r a c t i n g t h e r e s u l t a n t e x p r e s s i o n from i t  we o b t a i n as f o l l o w s :  Lip + e2EM(j) = 0  (3.20)  L<j> + Mip + e[M(f> - E(M<(>) ] = 0.  (3.21)  A f o r m a l s o l u t i o n o f (3.20)-(3.21) was f i r s t g i v e n by T a t a r s k i i and G e r t s e n s h t e i n  (1963) and i s  00  Li|> = -  eEM  I  [-£(/-  E)L  X  M]n+V  (3.22)  n=0  Here  7  i s t h e i d e n t i t y o p e r a t o r and  L  i s the operator inverse to  L.  12  The sum i n (3.22) i s convergent  provided that  denotes an a p p r o p r i a t e o p e r a t o r norm.  e||L " * " M | | < 1,  where  T h i s c l e a r l y l i m i t s the amplitude  the f l u c t u a t i n g p a r t o f the b a s i c flow, and i t i s h e n c e f o r t h assumed £ «  1.  ||(*)|| of  that  In the p r e s e n t a n a l y s i s we r e t a i n o n l y the f i r s t term i n (3.22)  giving  b\> = £ E [ M L M ] ^ . 2  (3.23)  _1  T h i s c o n s t i t u t e s the " f i r s t - o r d e r smoothing" o r " l o c a l Born"  approximation  and i s e q u i v a l e n t t o i g n o r i n g the b r a c k e t e d terms i n (3.21).  Howe (1971)  has g i v e n a c l e a r p h y s i c a l i n t e r p r e t a t i o n o f t h i s Essentially,  approximation.  the n e g l e c t e d terms i n v o l v e the i n t e r a c t i o n o f the f l u c t u a t i n g  component o f the b a s i c f i e l d w i t h the random p a r t o f the p e r t u r b a t i o n a t d i s t a n c e s exceeding in  t h e i r mutual c o r r e l a t i o n l e n g t h , whereas the o t h e r terms  (3.20)- (3.21) are determined It  function  i s convenient  G(x,^)  for  by the l o c a l v a l u e s o f the two  fields.  t o r e w r i t e (3.13) i n terms o f the Green's  (V - c)  which  satisfies  PG(x,5) - (V - c ) Q G ( x , £ ) = 6(x - ?) - 1  x  G(0,£) = G U £ ) 7  S u b s t i t u t i o n o f (3.15) and terms o f  G(x,£)  (3.24)  = 0.  (3.16) i n t o  (3.23) and e x p r e s s i o n o f  L  r e s u l t s i n an i n t e g r o - d i f f e r e n t i a l e q u a t i o n f o r  in ,  13  [(V - c ) - e R(0)]lty - (V - c)Q i> - e h 2  2  2  x  _ 1  [ (hVh)R'(0)  I  - R"(0)]Tp = £ Q  / (V - c) G(x,£) [R(x 0  2  _1  X  - (h'/h )R'(x - ^)UJ - h R"(x - £ ) M d £ 2  _1  + £ (h'/h )(V - c) / 2  (V - c) G(x,£)[R'(x - £)tty  2  _1  0  -  (h'/h )R"(x - £)IJJ - h R 2  _ 1  (x - 04>]dE,  , M  £ - e h ( v - c) / (V - c) G(x,£) [R"(x - OVi> 0 - (h'/h )R"' (x - h~ R (x - 5 ) M o £ . 2  _ 1  -1  2  1  Here the c o r r e l a t i o n function  R(£)  (3.25)  ,v  i s defined as  R(£) = E[W(x)W(x +  O].  This equation also holds f o r the shelf model i f the upper l i m i t of integration i s extended to i n f i n i t y . W  For the channel model we assume that  i s a homogeneous random function so that  R (o) 1  vanishes; further, we  2 choose  R(o) = 1  and define  O  = - R"(o).  of the nondimensional c o r r e l a t i o n length  Then  L /L c  I/O  i s representative  of the f l u c t u a t i n g current.  2 (In f a c t ,  R"(o) = - 2/X  f o r a Gaussian process described by  R(?) =  ex (-C /X ).) 2  2  P  Certainly, (3.25) i s much too complex to be dealt with d i r e c t l y , and, indeed,  G(x,£)  w i l l generally be unknown a n a l y t i c a l l y ,  our i n a b i l i t y to solve the associated deterministic problem.  expressing Hence some  approximate analysis must be adopted, and i t c l e a r l y would be advantageous  14  to e l i m i n a t e the i n t e g r a l terms i n (3.25). t h a t the  rhs  of  We w i l l p r e s e n t l y demonstrate  (3.25) i s an o r d e r of magnitude s m a l l e r than the  and thus, t o a f i r s t approximation,  may  be i g n o r e d .  Appendix A t h a t a l l the i n t e g r a l s i n (3.25) are o f R(x)  and i t s d e r i v a t i v e s are a l s o  p o i n t s o f the e q u a t i o n where and  0(1).  lhs  I t i s shown i n 0(1)  provided that  Hence, away from the of  singular 2  V = c ± £,  the  rhs  (3.25) i s  0(£ )  i s thus n e g l i g i b l e compared w i t h the  lhs  which c o n t a i n s terms o f  0(1)  2 (and a l s o terms o f  0(£  )).  Near the s i n g u l a r p o i n t s , however, the  i s c o n s i d e r a b l y more c o m p l i c a t e d , but we ijj  and  G(x,£)  are  safely neglected. which  0(£)  claim that either  t h e r e and hence t h a t the  rhs  Q of  o r t h a t both  x  (3.25) may  To see t h i s , c o n s i d e r the l i m i t i n g case o f  (3.25) reduces  analysis  £ = 0  for  to  (V - c)tty - Q i> = 0.  (3.26)  x  Since  (3.26) must h o l d p o i n t w i s e i t f o l l o w s t h a t t h e r e are two  at p o i n t s  x  c  where  V(x ) c  = c  g r a d i e n t of p o t e n t i a l v o r t i c i t y vanishes there.  We  singular derivatives x = xc)[  be  which must be c o n s i d e r e d . Q  o r the c r o s s - s t r e a m  x  possibilities  Either  the  velocity  d i s r e g a r d the t h i r d p o s s i b i l i t y o f s o l u t i o n s w i t h (that i s , those c o r r e s p o n d i n g t o a v o r t e x sheet a t  f o r two r e a s o n s .  F i r s t , McKee (1977) has demonstrated f o r s t a b l e  mean v e l o c i t y p r o f i l e s t h a t , i n t h i s case, p a r t o f the e i g e n v a l u e spectrum  of  L  c  belongs  t o the  continuous  and the c o r r e s p o n d i n g e i g e n f u n c t i o n  r e p r e s e n t s o n l y a t r a n s i e n t component o f the complete s o l u t i o n . arguments are r e a d i l y extended  t o cases w i t h u n s t a b l e v e l o c i t y  McKee's profiles.  Second, and more i m p o r t a n t l y , L i n (1961) has e l e g a n t l y demonstrated t h a t the i n c l u s i o n o f a s m a l l amount o f  (molecular o r eddy) v i s c o s i t y p r e c l u d e s  15  the e x i s t e n c e o f a continuous friction,  solutions.  Now i f e away from  i s sufficiently  V = c,  either  Q  shows t h a t i n t h e l a t t e r case from  £ = x  since  Q  of  and  x  ,  c  the  rhs  £,  £  from  x = x  ;  c  In  G(x,£)  case, where  l h s are  2 0(£ )  if  c  G(x,£)  c  x = x  is of  G(x ,^) c  We a g a i n argue t h a t a t a  0(E).  L  and away  c  i s a continuous f u n c t i o n  c  (The above comments  a l s o imply t h a t we may  having s i n g u l a r d e r i v a t i v e s a t  case i n which 2 0(£ ) and  Q  = 0(£);  x  x = x  exclude c  ,  G  are  which may thus be i g n o r e d .  An e n t i r e l y  3 0(E ) .  g r e a t e r than  argument h o l d s  equation  2  2  x  2  - 1  ^ = 0,  which i s s u b j e c t t o t h e boundary c o n d i t i o n s (3.19). d e r i v a t i o n o f (3.27) i s g i v e n i n the next In  t h r e e terms on  0 (£""'")  analogous  3 0(£ ).  We a r e t h e r e f o r e l e d t o the  - c ) - £ ]£ty - (V - c ) Q ^ - £ 0 h 2  are  terms i n (3.25) a r e  i s complex w i t h a s m a l l imaginary p a r t .  [(V  rhs  0 ( £ ) , the f i r s t  l h s c o n t a i n s terms a t l e a s t  c o n s i d e r a t i o n o f the s i m p l i f i e d  then the terms on  w h i l e those on the  and a l l the remaining  Hence i n a l l cases t h e rhs  at  see McKee, 1977.)  the second  the  0(£) :  G(x , x ) =0.  spectrum  l h s o f (3.25) a r e a l l  the  i s also  As  , G(x, E,)  C o n s i d e r the f i r s t the  0 ( E ) . A s i m i l a r argument  ij; i s o f  G(x,£)  i t follows that  the p o s s i b i l i t y o f c  or  x  o f (3.24) v a n i s h e s and hence so does  c o n c e r n i n g the continuous  E, ? x  small, i t follows that at a distance  i s nonzero by assumption.  x  distance  That i s , i n the l i m i t o f v a n i s h i n g  the s i n g u l a r s o l u t i o n s o f the i n v i s c i d theory are n o t o b t a i n e d as  l i m i t s of v i s c i d  £  spectrum.  (3.27)  A more p h y s i c a l  section.  the a n a l y s i s p r e s e n t e d i n the f o l l o w i n g s e c t i o n s , approximate  16  s o l u t i o n s are o b t a i n e d  assumptions made i n the p r e c e d i n g R(x)  and  0".  f o r the case o f l a r g e  of  the  s c a l i n g argument i s v i o l a t e d , namely t h a t  i t s f i r s t f o u r d e r i v a t i v e s are a l l  0(1).  r e l a t i v e s i z e s of the terms i n the analogue o f flow on a  Thus one  $-plane i s made i n Appendix C.  terms are g e n e r a l l y s m a l l e r than the o t h e r  An  evaluation of  (3.25) f o r a z o n a l random  I t t u r n s out t h a t the terms, although  integral  i n some cases  they are o f s i m i l a r magnitude f o r c e r t a i n r e g i o n s o f the channel. i n c r e a s e s and Thus  O  0  decreases,  must be  t i o n s to the f i r s t equation.  if  s a f e l y i g n o r e d to a good Comparison of d e t e r m i n i s t i c case manifests at  V = c  of  V  ,  itself  O  latter  as approxima-  integro-differential  (3.27) w i t h the c o r r e s p o n d i n g  ways.  equation  F i r s t , the s i n g l e c r i t i c a l V = c ± £.  i s r e a l ) , and  In any  t h i s i s expressed,  p o i n t s each removed by the 2  f o r the  p o i n t of given  second e f f e c t appears i n the term  rms  (3.26)  realization  c = V +  i n the mean, by  v a l u e of  £W  £W  the  from  V = c.  2-1  £ a h  iL»  and depends not o n l y  the s t r e n g t h of the f l u c t u a t i n g c u r r e n t but a l s o on i t s c o r r e l a t i o n The  be  approximation.  t h e r e would no doubt e x i s t numerous p o i n t s a t which  e x i s t e n c e o f two  so  (3.26) shows t h a t the randomness o f the b a s i c c u r r e n t  i n two  c  justified.  i s not l a r g e , then the i n t e g r a l terms may  i s b i f u r c a t e d i n t o the p a i r  (provided t h a t  The  In the  i n t h i s t h e s i s are b e s t regarded  i t e r a t i v e s o l u t i o n of the f u l l  Of course,  k  a p e r t u r b a t i o n expansion but not  l a r g e t h a t the i n t e g r a l terms become o v e r l y s i g n i f i c a n t . case, the s o l u t i o n s o b t a i n e d  As  the n e g l e c t o f these terms i s b e t t e r  l a r g e enough t o p e r m i t  the  on  length.  p h y s i c a l s i g n i f i c a n c e o f t h i s term i s more f u l l y d i s c u s s e d i n the next  s e c t i o n where the v o r t i c i t y b a l a n c e  is  considered.  17  4.  The V o r t i c i t y  and Energy Balances  In t h i s  s e c t i o n the v a r i o u s v o r t i c i t y  i n the system a r e examined. a more e x p l i c i t  Consider  first  and energy b a l a n c e s  present  (3.20) and (3.21) r e w r i t t e n i n  form.  (v - c)Vi> - QJ) + e2E[wP(f)] - e 2 E [ g d>] = o  (4.1)  (v - c)V<t> - Q (J> +  (4.2)  wpTp  - q  u>  = - e(wD<j) - E[wP<j)]).  To an o b s e r v e r moving w i t h the wave speed, the terms i n (4.1) correspond respectively to: Pijj  (1) alongshore . a d v e c t i o n o f mean d i s t u r b a n c e  by the mean b a s i c f l o w ,  vorticity  (2) c r o s s - s t r e a m a d v e c t i o n o f mean b a s i c  v o r t i c i t y by t h e mean d i s t u r b a n c e , and the c o r r e l a t e d p a r t s o f (3) a d v e c t i o n by t h e f l u c t u a t i n g b a s i c flow o f the random d i s t u r b a n c e v o r t i c i t y  D<j),  and  (4) t h e c r o s s - s t r e a m a d v e c t i o n o f random b a s i c v o r t i c i t y by t h e f l u c t u a t i n g disturbance.  S i m i l a r l y , the terms i n (4.2) are i n t e r p r e t e d a s : (1)  a d v e c t i o n by the mean b a s i c flow o f f l u c t u a t i n g d i s t u r b a n c e  vorticity,  (2) a d v e c t i o n o f mean b a s i c v o r t i c i t y by the random d i s t u r b a n c e , (3) a d v e c t i o n o f mean d i s t u r b a n c e v o r t i c i t y  by the random b a s i c f l o w , (4)  c r o s s - s t r e a m a d v e c t i o n by the mean d i s t u r b a n c e o f f l u c t u a t i n g b a s i c vorticity,  and (5) t h e alongshore  by the f l u c t u a t i n g b a s i c c u r r e n t .  a d v e c t i o n o f random d i s t u r b a n c e  vorticity  The l a s t term i n (4.2) i s the o n l y one  q u a d r a t i c i n the random f i e l d s and thus might be expected t o be s m a l l . In f a c t , W  i t c o n s i s t s o f the d i f f e r e n c e o f the t o t a l  and t h a t p a r t o f  Wpcf) which i s c o r r e l a t e d .  advection o f  V$  by  Since i t p r i m a r i l y i n v o l v e s  18  the i n t e r a c t i o n o f the two f i e l d s a t d i s t a n c e s exceeding  t h e i r mutual  c o r r e l a t i o n l e n g t h , i t p l a y s an i n s i g n i f i c a n t r o l e i n t h e v o r t i c i t y b a l a n c e expressed by (4.2) and i s h e n c e f o r t h n e g l e c t e d . We now c o n s i d e r the r e l a t i v e magnitudes o f the v a r i o u s terms i n (4.1) of  and (4.2) and g i v e a h e u r i s t i c  T?<J>,  as determined  d e r i v a t i o n o f (3.27).  by (4.2), i n t o  (v - c)Vi> - Qj  Substitution  (4.1) y i e l d s t h e analogue o f (3.25),  - e ( v - c ) l t y - e a h ( v - c)~\ 2  - 1  2  2  - 1  + e Q (v - c) E[wcj>] - e h E [w"<j>] + e (h'/h )E[W'cj)] = 0. (4.3) 2  -1  2  _ 1  2  2  x  In o r d e r t o determine  the magnitudes o f  cf>(x) = - / (V - c ) 0  which i s o b t a i n e d from now 0(1)  - 1  <J>  and D<j>,  we use t h e e x p r e s s i o n  G ( x , 5 ) [Wtty - q ljJ]d£  (4.4)  x  (3.21) i n t h e l o c a l Born a p p r o x i m a t i o n .  Consider  the cases examined i n the p r e c e d i n g s e c t i o n c o r r e s p o n d i n g t o and V - c = 0(e).  (If c  (V - c) =  i s complex w i t h s m a l l imaginary p a r t ,  then t h e l a t t e r case i s e q u i v a l e n t t o V - c The r e s u l t s a r e summarized i n T a b l e I .  r  = 0(e),  and  c^ = 0(e).)  In t h e f i r s t case t h e random f i e l d s  p l a y an i n s i g n i f i c a n t r o l e i n t h e b a l a n c e o f mean v o r t i c i t y , and Vty exceeds eVty by an o r d e r o f magnitude.  On t h e o t h e r hand, t h e alongshore  of f l u c t u a t i n g d i s t u r b a n c e v o r t i c i t y by the random b a s i c flow i s important  advection  (4.1, term 3)  i n the l a s t two cases, b u t the c r o s s - s t r e a m a d v e c t i o n o f t h e  random background v o r t i c i t y by t h e f l u c t u a t i n g d i s t u r b a n c e remains unimportant.  (4.1, term 4)  Near t h e c r i t i c a l p o i n t the random component o f  19  Table I.  A.  B.  Order of magnitude e s t i m a t e s of terms i n the v o r t i c i t y balance equations. The r h s o f (4.2) i s n e g l e c t e d , and the magnitudes o f (j> and V§ are c a l c u l a t e d from (4.4).  1  l  Vi>,ty = o(i)  (4.2)  1  l  t?<|>,4> =  (4.3)  1  l  e  (4.1)  e  e  e  (4.2)  l  e  1  (4.3)  e  e  e  (4.1)  e  e  e  (4.2)  l  e  1  (4.3)  e  e  e  0(1)  o(i)  V - c =  Q  x  0(e)  =  2  2  e 1  1  2  Jl  „2  e  e  2  e  0(1)  tty = 0(i/e)  2  e 1 2  e  2  e^  2  e  e^  v - c = 0(e) = Tty  =  2  e  0(E)  =  c.  e  (4.1)  V - c =  0(e) 0(1)  P<J> = 0(l/e)  3  e e 2  e  2  e  3  e  3  e  20  disturbance  vorticity  £#<}>  i s the same o r d e r o f magnitude as t h e mean  component, a l t h o u g h  £<f>  remains s m a l l .  tends t o c o n c e n t r a t e  near p o i n t s where  Thus t h e f l u c t u a t i n g v o r t i c i t y V - c  i s small.  l a s t t h r e e terms i n (4.3) may be d i s r e g a r d e d , (3.27).  Manton and Mysak  <f>  and i t s d e r i v a t i v e s t o be s m a l l , as suggested by  (1976).  An energy e q u a t i o n i s now d e r i v e d .  expression.  f o r the mean component o f t h e p e r t u r b e d  field  I t i s e a s i e s t t o p r o c e e d d i r e c t l y from the nonaveraged  e q u a t i o n s o f motion  (3.2) - (3.4)  and then t o average the r e s u l t i n g  I n the u s u a l f a s h i o n we take the s c a l a r p r o d u c t o f the momentum  equations with  u_ t o o b t a i n  (h/2)(u  2  + v  2  ) + (h/2)V (u t  + v  2  B  = - V' huv - Ro h(u(;  2  )  _1  B  i n t e g r a t i o n over the r e g i o n  wavelength i n the and  (4.3) reduces t o  Note t h a t n e g l e c t o f these terms i n no way r e q u i r e s the  c o r r e l a t i o n s between  An  and hence  In a l l cases, the  R  y - d i r e c t i o n with  x  y  + vC ) -  (4.5)  y  d e f i n e d by t h e channel w i d t h and one a p p l i c a t i o n o f the boundary  t h e assumed p e r i o d i c n a t u r e o f the d i s t u r b a n c e  conditions  gives  I  0/9 ) t  /  (h/2) ( u + v ) = / 2  R  B  1 2  dx.  (4.6)  0  Here the Reynolds s t r e s s  =  V (x)T  2  T.. „  y+A ~ ! huvdy.  Y  i s d e f i n e d by  (4.7)  21  Thus i f a wave i s u n s t a b l e ,  the Reynolds s t r e s s must e x t r a c t k i n e t i c energy  from the shear of the b a s i c  current.  By e x p r e s s i n g  the r e a l q u a n t i t i e s  terms of the d e t e r m i n i s t i c and  u, v  and  T  h (|H - 1  2  +  in  random components of the stream f u n c t i o n  (3.7), and by ensemble a v e r a g i n g the r e s u l t a n t e x p r e s s i o n  n /  i n (4.6)  we  find,  k M )dx 2  2  0  J  + e?Q  i  h ( | c f ) ' | + k |4>| )dx _ 1  2  2  2  0  I  I  = + k J  h" V'F(i)j ,^ )dx + e k 1  2  R  I  /  0  h"  VElPt^^jJldx  0  2 + e kE  ^ /  1  h  W'ImF(i|;*,cf))dx.  (4.8)  0 Here the s u b s c r i p t s  R  and  I  (or  and  F(f,g) = f g ' - f'g  f u l l Reynolds s t r e s s assumes the  T  The  1 2  =  (k/2h) [F  i n t e r p r e t a t i o n of  01^,1^)  (4.8)  and  Q = kc^  imaginary p a r t s o f a q u a n t i t y , disturbance,  r  i)  r e f e r t o the r e a l  i s the growth r a t e o f  i s the Wronskian o f  +  F(<$> A ) + ImF(lp*,4>)  note t h a t i t i s not energy c o n s e r v i n g  R  X  equation consistent with  and  g.  The  (4.9)  1 •  i s not pursued here; however,  s i n c e no p r o v i s i o n was  m o d i f i c a t i o n of the b a s i c f i e l d which t h e r e f o r e a c t s as an  I t i s not c l e a r how  the  form  i s d i r e c t and  r e s e r v o i r of energy f o r the  f  and  we  made f o r the infinite  disturbance. to approximate (4.8)  i n order  (3.27); i . e . , i t i s d i f f i c u l t  to obtain  an  to t r a n s f e r s c a l i n g  22  arguments from the v o r t i c i t y domain t o the energy domain. problem we  form another energy e q u a t i o n from  (V - c)tty - £ ( V  M u l t i p l i c a t i o n of  (4.10) by  ty*  2  2  x  this  (3.27) r e w r i t t e n as  £ a h (V  - c ) " ^ - 0 u> -  2  To a v o i d  _1  - c ) " ^ = 0.  (4.10)  1  and i n t e g r a t i o n over the channel w i d t h  y i e l d s an e x p r e s s i o n whose i m a g i n a r y p a r t i s  £ ft /  £ h~  (|TJJ'|  £  2  + k |^| ) d x = k / 2  [(V "  + £ k /  £ 0  /  A c a r e f u l comparison o f rf) as g i v e n by  - 1  c. ] 2  ~  VF(lJJ ,^ )dx R  (V - c ) V ' | l | , ' | 7 h|v - c|  - £ ~ /f. — 0  of  -  2  ~  + e Q /  2  o)  h V ' F ( i ^ , ^ ) dx  I  hlv - c l  0  2  2  2  r  |  ;  2  + k |i|^ 2  ^  2  2  dx + £ o  Q  £ j  \ l  2  xl)  —  0  1  —  j  1  h|v -  dx  c|  2  dx.  (4.11)  hlv - c l  (4.8) and  (4.11) w i t h p a r t i c u l a r r e g a r d t o the form  (4.4) r e v e a l s t h a t terms 1 and 2 i n (4.11) c o r r e s p o n d  e x a c t l y t o terms 1 and 3 i n (4.8), and t h a t terms 3 and 4 and terms 5 and 6 in  (4.11) r e p r e s e n t terms 4 and 5 i n (4.8), r e s p e c t i v e l y .  large  0",  which w i l l be c o n s i d e r e d l a t e r , i t i s c l e a r from  In the case o f (4.11)  t h a t the  energy source f o r the u n s t a b l e p e r t u r b a t i o n i s the shear o f the random p a r t  23  o f the b a s i c  flow.  One the formal  c o u l d proceed, a t t h i s p o i n t , t o d e r i v e the e q u i v a l e n t s  r e l a t i o n s h i p s developed by Grimshaw (1976).  necessary conditions f o r i n s t a b i l i t y speeds.  These concern  and bounds t o growth r a t e s and  I t i s s u f f i c i e n t here to d e r i v e o n l y a g e n e r a l i z e d  condition for i n s t a b i l i t y .  Rather than use  Grimshaw, s i n c e they are t e d i o u s unaveraged R a y l e i g h  <i J  V  the techniques  i n the p r e s e n t  case, we  Q  x  (4.12), we  B "  Q dx = *x  C  (3.26) by  (V - c + £W)  ^  /  ip*/(V - c)  and  (c^ ^ 0 ) ,  $  i n terms o f  i n a binomial  Q>  by the  (4.12)  the i n t e g r a l must v a n i s h .  if  by  a  series.  n  d  $  a n c  Ignoring  ^  (0,£).  expand  This To  (V  B  implies  generalize  - c)  1  =  t r i p l e c o r r e l a t i o n s i n the o b t a i n to  0(£  3  )  dx  V - c  r  2Q  v - c EEW"Re(#)]  T  E  M * I  ] - -T  (V - c )  2  E[WRe(#)]  V - c  20" | ^ | (V - c ) -g [ dx = 2  +  h V - c  obtained  i n t e g r a t i n g over the channel w i d t h .  ensemble average of the r e s u l t a n t e x p r e s s i o n we  C i  employed  0.  must v a n i s h a t l e a s t once i n the i n t e r v a l express  Rayleigh  condition  I f a system i s u n s t a b l e that  phase  proceed from  T h i s r e l a t i o n s h i p i s the imaginary p a r t of the e x p r e s s i o n multiplying  of  h V - c  0.  (4.13)  24  In the l i m i t o f l a r g e  0  2 2 €0 = 0(1)),  (i.e.,  2e 0 (v - c ) r 2  Q  V - c  Equation  +  2  flow u n s t a b l e .  (4.14) shows t h a t i n s t a b i l i t y does not demand t h a t  p o s s e s s s u f f i c i e n t r e l a t i v e v o r t i c i t y t o render the  In more p h y s i c a l terms,  Q  x  = 0).  Lin  (1945) has  demonstrated  of such p o i n t s i s n e c e s s a r y f o r i n s t a b i l i t y ;  otherwise  a p a r t i c l e d i s p l a c e d from i t s e q u i l i b r i u m p o s i t i o n w i l l always be to a net r e s t o r i n g f o r c e .  As  O  i n c r e a s e , the r e l a t i v e  a s s o c i a t e d w i t h the random flow i n c r e a s e s and  therefore  l i k e l i h o o d o f f i n d i n g l o c a l maxima and minima i n  r  < 0  mean c u r r e n t . and  c  r  basic  l o c a l extrema i n t o the p o t e n t i a l v o r t i c i t y  ( i . e . , p o i n t s a t which  t h a t the e x i s t e n c e  (V - c )  the  In t h i s sense, the i n s t a b i l i t y d e s c r i b e d more f u l l y i n the  the random component i n t r o d u c e s  If  the  s i n c e the random p a r t of  next s e c t i o n i s e s s e n t i a l l y a shear i n s t a b i l i t y .  distribution  (4.14)  dx = 0  V - c  mean p o t e n t i a l v o r t i c i t y d i s t r i b u t i o n v a n i s h , b a s i c flow may  t h i s reduces t o  c^ ^ 0  and  Q  > 0  x  everywhere,  somewhere and hence t h a t However, i f  i s simply  Q  x  < 0  c  r  subject  vorticity so does the  Q.  (4.14) i m p l i e s  that  must l i e i n the range o f  everywhere then  (V - c )  bounded above by the maximum value o f  r  V;  c o u l d then, i n p r i n c i p l e , propagate a g a i n s t the mean flow.  > 0  the  somewhere  unstable  waves  25  5.  The  Channel Model  The  boundary v a l u e problem d e f i n e d by  encompasses a l l and more of the d i f f i c u l t i e s b a r o t r o p i c a l l y unstable c o e f f i c i e n t s but V(x)  = c ± £.  (3.27) and  (3.19)  inherent i n a d e t e r m i n i s t i c ,  system, f o r not o n l y does  (3.27) have v a r i a b l e  i t a l s o p o s s e s s e s a p a i r of s i n g u l a r p o i n t s a t  Since  £ «  1,  an obvious approach to  attempt a p e r t u r b a t i o n s o l u t i o n i n  £.  (3.27) would be  However, the r e s u l t i n g  to  equations  would c o n t a i n a l l the mathematical d i f f i c u l t i e s of the nonrandom problem, and  the s o l u t i o n s , as i n t e r e s t i n g as they might be, would r e p r e s e n t  d e v i a t i o n s from the d e t e r m i n i s t i c s o l u t i o n s . d i f f e r e n t c l a s s of s o l u t i o n s to case.  I f the parameter  i s small) we  then we  may  0  We  are i n t e r e s t e d i n a  (3.27) which does not e x i s t i n the nonrandom  i s large  ( i . e . , i f the c o r r e l a t i o n l e n g t h o f  t r y a p e r t u r b a t i o n expansion i n the l i m i t o f  s h a l l show t h a t the r e s u l t i n g s o l u t i o n s are u n s t a b l e .  classic barotropic i n s t a b i l i t y s t a b l e , u n s t a b l e waves may  small  theory may  still  W  0 -»- ; 00  Hence even though  i n d i c a t e a g i v e n system t o be  e x i s t i f there i s s u f f i c i e n t  " n o i s e " i n the  mean c u r r e n t . In o r d e r t o make the f o l l o w i n g r e s u l t s more s p e c i f i c , we Brooks and Mooers  V =  xe  (1977a) model of the F l o r i d a S t r a i t s  1-x  (Figure  adopt  the  5.1):  (5.1)  2b(x-l) e  0  5 x 5 1  (5.2)  h =  I  1  1 5 x 5  I,  27  b = 1.385, I  w i t h the parameters  = 2.5  and  Ro = 0.3.  T h i s model i s  chosen s i n c e i t employs n o n t r i v i a l but r e a l i s t i c v e l o c i t y and p r o f i l e s , and Although  V  s i n c e we  wish t o apply  bathymetric  our r e s u l t s t o the F l o r i d a S t r a i t s .  s a t i s f i e s a l l the n e c e s s a r y c o n d i t i o n s  for i n s t a b i l i t y , i t i s  extremely u n l i k e l y t h a t t h i s model i s u n s t a b l e  as the subsequent argument  shows.  that  The  Rayleigh  condition  (4.12) r e q u i r e s  <=i / Q H' /|V - c| d x = 0. 2  (5.3)  2  X  0  T h i s requirement i s u s u a l l y s t a t e d i n the form: must v a n i s h vanish  at  satisfied. it  a t l e a s t once i n the x = 2,  but  A p l o t of  interval  t h i s i s not Q  (0,2).  (0,£).  c^ ^ 0,  then  In f a c t ,  Q  s u f f i c i e n t to ensure t h a t  T h i s means t h a t  (2,&  \i>\  = 2.5)  must be  2)  <p(J£) = 0.  were s a t i s f i e d w i t h  c^ ^ 0,  t r a p p e d a g a i n s t the outer w a l l would be  r e s p e c t to a p p l i c a t i o n o f the p r e s e n t  theory  is  the  extremely l a r g e i n the  l i k e l y as the boundary c o n d i t i o n s  alongshore c u r r e n t s  x  shows t h a t  compared w i t h  former i n t e r v a l which i s not That i s , i f (5.3)  Q  does  x  (5.3)  (Brooks and Mooers, 1977a, F i g u r e  x  i s extremely s m a l l i n the i n t e r v a l  interval  if  require  very  large  necessary.  t o the F l o r i d a S t r a i t s , we  With note,  however, t h a t a more r e a l i s t i c b a t h y m e t r i c p r o f i l e r e s u l t s i n a p o t e n t i a l v o r t i c i t y d i s t r i b u t i o n that i s probably unstable Unfortunately,  Brooks and Mooers a p p a r e n t l y  (Brooks and Mooers, 1977a).  d i d not s e a r c h  for  unstable  solutions. If a perturbation 2 2 r e s u l t s , the term k  (3.27) i s t o l e a d to  nontrivial  —1  £ O h  t h i s requires that  expansion o f  c  must be b a l a n c e d by be  0 (CT) .  another term.  For f i x e d  I t might appear t h a t t h i s c o u l d  lead  28  to a c o n t r a d i c t i o n o f the s e m i c i r c l e theorem  (Grimshaw, 1976) which s t a t e s  t h a t f o r each r e a l i z a t i o n o f the b a s i c flow  V_ ,  [c  - 1/2(V  r  where  V j ^ and  values  of  v  B  V" , V  independent o f Nevertheless,  = 0. 0.  c^  + c  2  2 ±  < [1/2(V  - V  B M  B m  ) - 1/2 c  w  ]  2  c  w  i s t h e phase speed o f the f i r s t mode  Thus i f W  i s bounded,  we assume t h a t  cr  c  i s bounded  the magnitude o f  0  CSW  in  above  i s limited.  i s l a r g e enough t o p e r m i t i t s use as an  On t h e o t h e r hand, the v a l u e s  of  c^  and  c  r  i n t h i s s e c t i o n f a l l w e l l w i t h i n t h e bounds o f t h e s e m i c i r c l e  theorem. „•  )]  Hence, once again  expansion v a r i a b l e . computed  B m  a r e , r e s p e c t i v e l y , t h e a l g e b r a i c maximum and minimum  m  and  B  the case o f  g  + V  BM  Moreover, i t t u r n s o u t t h a t s u c c e s s i v e  • •,  -2  d i m i n i s h as  0  corrections to  c  r  and  "I not 0  In t h e f o l l o w i n g development i t i s convenient t o expand b o t h t h e r e a l and imaginary p a r t s o f  c  s e p a r a t e l y ; we take 00  -m m=0  m=0  The c h o i c e o f e i t h e r to t r i v i a l  solutions.  = 0  If  at  c  rm  c^ = 0 ( 0 ) , c^ = 0(1)  I  = 0  ±  m=0  or  The boundary c o n d i t i o n s  c  r  ' i c  =  0  0  -m  (5.4)  c. lm  ( ) a  leads  only  (3.19) become  x = 0,£.  the system under c o n s i d e r a t i o n  contains  d i s c o n t i n u i t i e s i n the gradient  29  of p o t e n t i a l v o r t i c i t y ,  then the equations must be s o l v e d i n each r e g i o n  and matched a c r o s s the p o i n t o f d i s c o n t i n u i t y .  I n g e n e r a l , the matching  conditions are  [i>] = o (5.6)  [{(V - c) - e }ip'/h] - [(V - c)Qj] = 0 2  where  [ ( * ) ] r e p r e s e n t s t h e jump i n  ensure  the c o n t i n u i t y o f mass f l u x and sea s u r f a c e e l e v a t i o n a c r o s s t h e  discontinuity. and  Q  Their v a l i d i t y  (•). P h y s i c a l l y , these c o n d i t i o n s  requires that  V  be c o n t i n u o u s .  Both  h  a r e continuous i n the p r e s e n t case and (5.6) reduces t o  m  =o > a t x = 1.  W]  (5.7)  = o  The t h r e e lowest o r d e r equations are  Vty + Q  h  Vip + h i  _ : L  (e/ci0)2i|)o  1  (e/c )\ ±0  0  =  1  = - c. " {2[c 1  0  (5.8)  i l  + i(V - c  x Q  )W  0  ~ iQ ^ } x  0  30  + h~ (Z/c )\  = c  1  ±Q  2  - 2i[(V - c ) c r 0  2c  c  -1 ± 0  [  iQ 2x 2  -  { (V - c^) rO' 2  c. c 0  + i(V - c  C i l  [ ( V  i ; L  2 i 0  " r0> C  i  c  r Q  r l  ]}^  )]^  i l ^  + 0  - 2c, c, - e 'i2 I O i l 2  0  n  2  C j l  0  1  i c  (5.10)  i0 Qx^l _ 1  The zeroth order equation (5.8) defines a Sturm-Liouville problem for which an i n f i n i t e number of solutions  , (n) y Q  exist such that the nth mode has (n) 2  exactly  n  zeros, and the corresponding eigenvalues  ordered and tend to i n f i n i t y as  n -*• . 00  ( s /  The superscript  c  i o  (n)  '  a  r  e  i s henceforth  dropped. Solutions to (5.8) s a t i s f y i n g both the boundary and jump conditions are given by: b  (  x  _  1  )  e  •  w  •  1  s m Ax/sm A  0 5 x < 1 (5.11)  sin a(x - £)/sin a ( l -I)  .  2  A  2  (5.12)  i Q  2  a  / c 2 - k2 - b2 = £ /  / = e / 2  2 c i 0  , ~  2  "\ ^ 2  =  1 5 x < I  ,2  (5.13)  31  provided  that  A  and  a  satisfy  b + A/tan A = a / t a n a ( l - £) .  If  A  i s negative,  (5.14)  the s o l u t i o n s o v e r the s h e l f a r e h y p e r b o l i c and a r e  o b t a i n e d by r e p l a c i n g  A  with  iA.  e/c^  the a d m i s s i b l e v a l u e s o f  For a given choice of  k, b,  are determined i m p l i c i t l y by  t o g e t h e r w i t h e i t h e r (5.12) o r (5.13).  H,  and  (5.14)  A g r a p h i c a l s o l u t i o n o f (5.14) i s  shown i n F i g u r e 5.2 f o r a case i n which h y p e r b o l i c s o l u t i o n s are found. The growth r a t e s o f the h y p e r b o l i c modes, i f they e x i s t , exceed those o f the t r i g o n o m e t r i c s o l u t i o n s .  F o r the v a l u e s o f  b  and  £  appropriate to  the F l o r i d a S t r a i t s o n l y t r i g o n o m e t r i c modes are found. F u r t h e r i n f o r m a t i o n may be e x t r a c t e d from actually solving for  and  ^  .  2  (5.8)-(5.10)  S i n c e these equations  without  a l l have  i d e n t i c a l homogeneous p a r t s , the Fredholm a l t e r n a t i v e i m p l i e s t h a t must be o r t h o g o n a l  c  i0  t o the r e s p e c t i v e inhomogeneous  = /( e  = <V>  X  +  +  We thus o b t a i n  (5.15)  +b )  k  (c  terms.  2 i 0  /2£ )<hQ >  (5.16)  2  x  (5.17)  c. _ , c = 0 il r l  = - (3/2) <(V - c  -  i  c  i 0  <  (  V  ) >-e /2 2  r Q  "roHV c  2  " i(c  - (c  3 i Q  2 i0  /2£ )<(V - c ) h Q > 2  r Q  /2£ )<hQ |V 2  x  x  (5.18)  32  F i g u r e 5.2  G r a p h i c a l s o l u t i o n o f (5.14) f o r b = 3.0 and SL = 2.5. The ( l i g h t , heavy) s o l i d l i n e i s the l o c u s o f b + A / t a n A f o r ( r e a l , imaginary) A, w h i l e the d o t t e d l i n e r e p r e s e n t s a / t a n a ( l - Z). I n t e r s e c t i o n s w i t h the ( l i g h t , heavy) l i n e c o r r e spond t o ( t r i g o n o m e t r i c , h y p e r b o l i c ) s o l u t i o n s . F o r F l o r i d a S t r a i t parameters (b = 1.385, I = 2.5) t h e f i r s t t h r e e s o l u t i o n s a r e a = 1.650, a„ = 2.809, and a , = 3.891.  33  where  £ <f (x)| g(x)>  £ h - 1 i j ; ( x ) f (x)g(x)dx//  = /  h  0  0  -  1  ^ (x) dx  (5.19)  2  0  and  <f(x)>  E <f(x)|^ (x)>  (5.20)  0  are weighted c r o s s - c h a n n e l averages.  These r e s u l t s a r e c o m p l e t e l y  and are not l i m i t e d t o the BrM model.  F o r t h i s model, however,  c  general r  Q  may  be r e w r i t t e n as  c  r 0  = <V> + <hQ >/2(X  2  x  + k  2  + b )  (5.21)  2  which i s s t r i k i n g l y s i m i l a r t o the e x p r e s s i o n f o r a CSW i n a c o n s t a n t current  V  over an e x p o n e n t i a l  c = V - 2b/(A  where  hQ  x  = - 2b  2  + k  2  shelf,  + b ) = <V> + <hQ >/(A 2  x  + k  2  + b )  ( c f . Buchwald and Adams, 1968, f o r the case  We see t h a t the weighted average  <V>  replaces  v o r t i c i t y term i n (5.21) i s d i m i n i s h e d by o f the Doppler  2  s h i f t e d wave  c Q - <V> r  1/2.  V,  (5.22)  2  V = 0).  w h i l e the p o t e n t i a l  T h i s means t h a t the speed  i s reduced by t h e presence o f  random i r r e g u l a r i t i e s i n the .basic c u r r e n t .  The e x p l a n a t i o n i s c l e a r :  the d i s t u r b a n c e must t r a v e r s e a l o n g e r path l e n g t h i n t r a v e l l i n g from one p o i n t t o another current.  s i n c e i t i s b u f f e t e d about and s c a t t e r e d by the f l u c t u a t i n g  T h i s phenomenon i s common t o wave p r o p a g a t i o n  i n random media  34  (Howe, 1971). In order to determine  c^  2  i t i s necessary to f i r s t evaluate  i|> ^ . This i s a straightforward, although tedious task, and the complete results are summarized  i n Appendix B.  The solutions i n the onshore and  offshore regions take the form  r  A  i^o o i + iA  0  p  5 x ^ 1 (5.23)  Vo  where  and  P  +  i A  1 5 x 5  0 2 P  I  are p a r t i c u l a r solutions that s a t i s f y the boundary  2  conditions; the factor of  i  P ^ and  ensures that  solutions must be matched across  x = 1,  P  2  are r e a l .  These  and the matching conditions (5.7)  in matrix form are  A  ll"  M  _  = iA A  P  2- 1 =  Q  P '  12_  P  2  -  (5.24)  P  P ^  where  sin  s i n a ( l - l) (5.25)  M =  b sin X + A cos  and a l l quantities are evaluated at  a cos a ( l - &)  x = 1.  However,  M  i s singular,  35  s i n c e i n matching the z e r o t h o r d e r s o l u t i o n s we r e q u i r e d t h a t I f a s o l u t i o n t o (5.24) i s t o e x i s t , i t i s n e c e s s a r y  that  P  d e t M = 0. be o r t h o g o n a l  to each l i n e a r l y independent s o l u t i o n o f the a s s o c i a t e d homogeneous a d j o i n t equation,  M'  Since and  M  i s o f rank one, t h e r e i s o n l y one independent s o l u t i o n o f  - P = a~ t a n a ( l x  2  I)  (P'  1  X  .  f u l f i l l m e n t o f t h i s r e s t r i c t i o n was used as a check o f the n u m e r i c a l i n F i g u r e s 5.3-5.7.  r e p r e s e n t e d by (5.24) serves t o f i x  A  A-^  12  =  A  ll  s  i  ^/  n  and we choose ^  s i n  A-^ = 0.  a  r  e  P  u  r  e  l  v  r  e  a  l  ,  - P ) / s i n ot(l - I) ,  2  A^  (5.28)  1  o f (5.10) r e v e a l s c ^  c o u l d be chosen so t h a t  S i m i l a r c o n s i d e r a t i o n s apply t o the h i g h e r  c r  a  equation  I|JQ does n o t c o n t r i b u t e t o  Alternatively,  were o r t h o g o n a l .  0  2  A c a r e f u l examination  proportional to  o b v i o u s , one f i n d s t h a t //A  A^  Q  e i g e n f u n c t i o n s and e i g e n v a l u e s .  ^2n+l  The one independent  ct (1 - 5,) - i A ( P  remains i n d e t e r m i n a t e .  t h a t the p a r t o f  and  (5.27)  - P',) A  r e s u l t s presented  but  (5.26),  i t leads t o the a u x i l i a r y c o n d i t i o n  P  The  (5.26)  = 0.  n  2n+l d  F i n a l l y , although a n c  ^  c  imaginary  i2n+l  v  a  n  i  s  n  i t i s not an  &  that  IJJQ  order  immediately ^ n ^ O  quantities, respectively.  A N <  This  ^  36  means t h a t s u c c e s s i v e c o r r e c t i o n s to We  now  c  and  r  c^  d i m i n i s h by  dependence o f the growth r a t e  Q-^ = k(0c^g  f i r s t mode i s shown i n F i g u r e 5.3.  As  0  + 0 ^c^)  0=3  and  0  above which u n s t a b l e waves e x i s t . 0=4;  mode number.  f o r a given  F i g u r e 5.4a  f i r s t t h r e e modes.  We  1-3.  k  F o r mode  1  on  k  f o r the  i n t e r e s t i n g t h a t the r e g i o n s  be s i g n i f i c a n t .  The most s t r i k i n g  equal  documented i n the l i t e r a t u r e  The  be  higher  frequencies and  5.6a,  i n t e r s e c t , which i m p l i e s the p o s s i b l e ( t h i s p o t e n t i a l l y was  f o r which the phase  f o r a l l modes; see S e c t i o n 7). (see Mysak, 1978)  also inherent  I t has  been  f o r a v a r i e t y of p h y s i c a l  systems, t h a t modes which are uncoupled i n the d e t e r m i n i s t i c case become coupled when randomness i s i n t r o d u c e d i n t o the problem.  may  Here, however,  are d e a l i n g w i t h d i s t u r b a n c e s which do not e x i s t i n the d e t e r m i n i s t i c  case; moreover, i t i s c l e a r t h a t the h i g h e r order expansion w i l l not l e a d t o mode c o u p l i n g . of  of  f e a t u r e i n them i s the  i n the Couette flow model o f Manton and Mysak, 1976,  we  and  i t l i e s between  are p l o t t e d i n F i g u r e s 5.5a  e x i s t e n c e o f a "resonance" i n t e r a c t i o n  and  increases  note, however, t h a t f o r l a r g e wavenumbers,  e x i s t e n c e o f p o i n t s where the curves  constant  f o r the  t h a t f o r s h o r t wavelengths the h i g h e r modes may  and phase speeds as f u n c t i o n s o f  speed was  0  There i s a t h r e s h o l d  e x h i b i t s the dependence o f  o r d e r terms which a r e n e g l e c t e d here may  r e s p e c t i v e l y , f o r modes  Straits.  t h i s v a l u e decreases w i t h i n c r e a s i n g  It is particularly  i n s t a b i l i t y o v e r l a p and the more u n s t a b l e .  k  on  0,^  increases,  the wave number range over which i t i s nonzero widens. value of  ).  examine the r e s u l t s i l l u s t r a t e d i n F i g u r e s 5.3-5.7 which  were computed f o r the parameters c h a r a c t e r i s t i c o f the F l o r i d a The  2  0(1/0  terms i n the p e r t u r b a t i o n  Hence a more c a r e f u l a n a l y s i s  (3.27) i s r e q u i r e d to r e s o l v e t h i s p o i n t , perhaps u s i n g  a b a s i s s e t f o r the r e a l and  imaginary  p a r t s of  We  and  a l s o note  the  ^  as  0.5  k/27T  F i g u r e 5.3  (  Behaviour o f the f i r s t mode nondimensional growth r a t e ti^ as a function of O and k. The d i m e n s i o n a l v a l u e s g i v e n c o r r e s p o n d to F l o r i d a S t r a i t parameters (b = 1.385, I = 2.5, Ro = 0.3) . The curves are l a b e l e d by the v a l u e o f 0 " .  WAVELENGTH (KM)  k/277"  Figure  5.4A  Behaviour o f the nondimensional growth r a t e ft^ as a f u n c t i o n of k f o r the f i r s t three modes f o r 0 = 5 and £ = .5; (A) channel model (b = 1.385, I = 2.5, Ro = 0.3), (B) s h e l f model (b = 1.385, U = 1.0, Ro = 0.3).  w  oo  0.0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0  k/27T  Figure 5.4B  U)  40  W A V E L E N G T H 3 0 0  F i g u r e 5.5A  150  100  (KM) 75  6 0  D i s p e r s i o n curves f o r the f i r s t t h r e e modes, (A) channel model, (B) s h e l f model. Beyond k/2ir = 0.6 the curves are e s s e n t i a l l y l i n e a r .  3  WAVELENGTH (KM) 600 300 150 100  1  »  0.0 0.1  •  0.2  •  0.3  75  60  50  '  i  i  0.4  0.5  0.6  40  «  30  •  0.7 0.8  •  0.9  «  1.0  1  1.1  •  1.2  1  o  k/2?r F i g u r e 5.6A  Behaviour o f the nondimensional phase speed as a f u n c t i o n o f k f o r the f i r s t three modes, (A) channel model, (B) s h e l f model.  to  '  0.0 0.1 0.2 0.3  I  0.4  I  1  1  I  1  1  0.5 0.6 0.7 0.8 0.9 1.0 k / 2 7T Figure 5.6B  1  L-  1.1 1.2  44 r e l a t e d study by A l l e n (1975) of c o a s t a l trapped waves i n a s t r a t e f i e d ocean where i t was shown that CSWs may be coupled t o i n t e r n a l K e l v i n waves. Figure 5.6A i n d i c a t e s that the phase speed i s p o s i t i v e f o r a l l k  and thus that the waves propagate northward; i . e . , i n the d i r e c t i o n of  the mean flow.  However, i f the Rossby number were small enough, the  v o r t i c i t y term i n (5.16) would predominate, and the disturbances t r a v e l southward.  could  (Consider, f o r example, the case of V = 0, hQ = - 2b.) x  This i s i n marked contrast t o the s t a b l e CSW's admitted by the BrM model which propagate only southwards. From the slopes of the d i s p e r s i o n and phase speed curves, we i n f e r that the group v e l o c i t y i s p o s i t i v e and exceeds that  c Q -»• <V> r  as k -*• °°  cQ r  f o r a l l k. F i n a l l y , we note  and thus that the waves are simply advected by  the mean current i n t h i s l i m i t . Contour p l o t s of the mass transport stream function f o r the f i r s t three modes are shown i n Figure 5.7. An important feature i s the t i l t i n g of the gyre axes toward the coast since i t i s r e l a t e d to the sign of the mean disturbance Reynolds s t r e s s phase sin 0 .  0 = k(x - c t) ,  T  = (k/2h) F (ij; , 4 ^ ) .  1 2  In terms of the  the stream f u n c t i o n i s given by 4 = ty cos 9 - 4>j 1  R  The l i n e - a l o n g which i t vanishes i s determined by tan 0 = 4 / ^ I /  with slope  J  J  R  d0/dx  = - (cos  2  8/4^  2  ) F ( ^  R  /  ^ j ) .  Hence  Figure 5.7 shows that i t i s everywhere negative.  T  1 2  Since  a - d 0 / d x and V* < 0  (4.11) reveals the rather s u r p r i s i n g r e s u l t that over the s h e l f ,  f o r x < 1, T  1 2  acts t o remove energy from the nonrandom p a r t of the disturbance and t o strengthen the mean shear.  This i s i n concert with the f i n d i n g of N i i l e r  and Mysak (1971) that the c o n t i n e n t a l s h e l f acts as a s t a b i l i z i n g f a c t o r . Of course, the l a r g e s t source term i n (4.11) i s the one p r o p o r t i o n a l t o a  which shows that the disturbance energy i s extracted p r i m a r i l y from the  f l u c t u a t i n g part of the basic current. P l o t s of the mass transport stream function are of l i m i t e d value  0.6 9  F i g u r e 5.7A  The (B) (E) The  0.5 0.4 /  2  0.0  IT  mass t r a n s p o r t stream f u n c t i o n f o r (A) channel mode 1, channel mode 2, (C) channel mode 3, (D) s h e l f mode 1, s h e l f mode 2, (F) s h e l f mode 3. Here 9 = k ( y - c ) . amplitude i s a r b i t r a r y . r t  en  0/2 TT Figure 5.7B  Figure 5.7C  Figure 5.7D  0/2  TT  Figure 5.7F  UI O  51  1.0 M O D E  0.5  I  U x io  0.0  0.0  0.5  1.0 X 1 . 5 ^ ^ 2 0 ^ 2 5  -0.5h-1.0 1.0 2  M O D E  0.5 0.0 0.0  x 0 . 5 K10___^--t5 /  s —  2.0  v. 25  x io  0.5 h-1.0 1.0 \ V  3  M O D E  0.5 X 0.0  0.0 V)5  Sfe  / J  -0.5  2.0  N  * 10  -1.0 o f u and v f o r : (1) channel mode l i n e 9/2TT = 0.8 i n F i g . 7a, (2) channel mode 2 taken a l o n g 6/2TT = 0.9, and (3) channel mode 3 taken a l o n g 9/27T = 0.37. I n each case the v a l u e s a r e n o r m a l i z e d by V(x = 0 ) . 3  52  i n v i s u a l i z i n g t h e v e l o c i t y s t r u c t u r e over the s h e l f . selected p r o f i l e s of  u  and  v  F i g u r e 5.8 shows  c o r r e s p o n d i n g t o F i g u r e 5.7.  In a l l cases  the motion i s trapped a g a i n s t t h e c o a s t . These r e s u l t s a r e now compared w i t h o b s e r v a t i o n s made i n t h e Florida Straits.  The o c c u r r e n c e o f f l u c t u a t i o n s i n the F l o r i d a C u r r e n t  w i t h p e r i o d s r a n g i n g from a few days t o s e v e r a l weeks i s w e l l known, and the f o l l o w i n g review i s n o t i n t e n d e d t o be e x h a u s t i v e ; the r e a d e r i s r e f e r r e d t o the papers  r e f e r e n c e d here f o r a more e x t e n s i v e d i s c u s s i o n .  p l a n view o f the F l o r i d a S t r a i t s c h a n n e l - l i k e topography  i s shown i n F i g u r e 5.9.  I t r e v e a l s the  and i l l u s t r a t e s t h e sharp t u r n the F l o r i d a  must make on i t s northward passage.  Cross s e c t i o n s o f  v e l o c i t y a r e p r e s e n t e d i n F i g u r e 5.10.  A  O  Current  and alongshore  I n a d d i t i o n , t o showing the h i g h l y  b a r o c l i n i c n a t u r e o f the mean f l o w , i t a l s o i n d i c a t e s some o f the l o n g i t u d i n a l v a r i a t i o n s i n bathymetry and i n the d e n s i t y and v e l o c i t y The  fields.  s t r a t i f i c a t i o n i s compressed over t h e s h e l f , and a t y p i c a l v a l u e o f the  B r u n t - V a i s a l a frequency Brooks,  i n the p y c n o c l i n e i s  2  x  10  _2  rad s  -1  (Mooers and  1977). In a marked c o n t r a s t t o low-frequency  motions i n the open ocean  t h a t a r e c h a r a c t e r i z e d by a r e d spectrum, t h e r e appears  t o be  a s p e c t r a l gap between motions w i t h p e r i o d s o f about 25 days and 1 y e a r (Brooks and N i i l e r ,  1977; Diiing e t a l . ,  1977; Wunsch and Wimbush, 1977).  Duing e t a l . (1975) e s t i m a t e from mid-channel v e l o c i t y measurements t h a t approximately days.  80% o f the n o n t i d a l v a r i a n c e o c c u r s a t p e r i o d s exceeding 8  I n g e n e r a l , the low-frequency  motions may be broken i n t o t h r e e  s c a l e s , 8-25 days, 4-5 days,  and 2-3 days  i s t r e a t e d s e p a r a t e l y here.  Seasonal  have a l s o been observed  (Diiing e t a l . ,  time  1977) each o f which  f l u c t u a t i o n s i n the F l o r i d a Current  ( N i i l e r and R i c h a r d s o n ,  1973).  F i g u r e 5.9  P l a n view o f the F l o r i d a S t r a i t s .^showing l i n e s I and I I a l o n g which the s e c t i o n s i n F i g . 5.10 are taken (from Mooers and Brooks, 1977).  I.  n. DISTANCE  F i g u r e 5.10A  (km)  S e c t i o n s along l i n e s I and I I of (A) G v e l o c i t y (from Mooers and Brooks, 1977). t  and  (B)  alongshore  n. DISTANCE (kir.) 40  60  80  0  MEAN AXIAL FLOW (V)  20  IN c m s '  Figure 5.10B  40  1  60  80  56  The 8-25-Day Band  From the a n a l y s i s o f y e a r - l o n g r e c o r d s o f sea l e v e l ,  sea  temperature, and atmospheric p r e s s u r e , Brooks and Mooers (1977b) the e x i s t e n c e o f southward w i n t e r and 12—14  t r a v e l l i n g waves w i t h p e r i o d s o f 7-10  demonstrated days i n  days i n summer and speeds o f 100 cm s ^ o r g r e a t e r .  Strong  coherence between sea l e v e l and temperature and the atmospheric v a r i a b l e s showed t h a t these d i s t u r b a n c e s were wind generated; a f i t o f the BrM model t o these o b s e r v a t i o n s was  only p a r t i a l l y  p r e d i c t s wave speeds l e s s than 50 cm s  CSW  s u c c e s s f u l as the model  S c h o t t and Duing  (1976) a p p l i e d  a s i n g l e b a r o t r o p i c wave model t o the r e s u l t s o b t a i n e d from the a n a l y s i s of 65 days o f c u r r e n t measurements taken c o n c u r r e n t l y a t s t a t i o n s s e p a r a t e d i n the a l o n g s h o r e d i r e c t i o n and found a s t a t i s t i c a l l y  significant f i t for  the 10-13-day wave p e r i o d band and a m a r g i n a l f i t f o r the 7-10-day band. In the former case they c a l c u l a t e d a wavelength phase speed o f 17 cm s  -1  , and an amplitude o f 14 cm s  i n e x c e l l e n t agreement w i t h the BrM model. f o r the 7-10-day band.  o f 270 km, -1  a  southward  , v a l u e s which  are  The wave parameters were s i m i l a r  Duing e t a l . (1977) concluded t h a t 9-20-day  o s c i l l a t i o n s p o s s e s s e d amplitudes r a n g i n g from 15-25  cm s "*"; they a l s o -  showed t h a t d i s t u r b a n c e s i n the 10-14-day band o c c u r r e d i n t e r m i t t e n t l y phase-coherent wave p a c k e t s c o n s i s t i n g of 4-6 was  cycles.  S i n c e h i g h coherence  observed between the v e l o c i t i e s and the atmospheric v a r i a b l e s ,  the wind  stress c u r l ,  i t appears  l i k e l y these motions  as  especially  are, i n i t i a l l y  at  l e a s t , a t m o s p h e r i c a l l y f o r c e d and r e p r e s e n t s t a b l e , m o d i f i e d s h e l f waves.  The 4-5-Day Band  In the 4-5-day wave p e r i o d band, Diiing  (1975) d e s c r i b e d a n e a r l y  57  b a r o t r o p i c wave, 160-240 km i n l e n g t h , t h a t p r o p a g a t e s northward w i t h a mean speed o f 45 cm s  and an amplitude o f about 10 cm s \  An  interesting  m a n i f e s t a t i o n o f t h i s d i s t u r b a n c e i s t h e r e v e r s a l o f the b a r o c l i n i c mean flow a t depth on the western s i d e o f the channel t h a t accompanies i t s passage.  Based on the a n a l y s i s of s i x months o f c u r r e n t , temperature, and  bottom p r e s s u r e measurements taken i n 1974, Wunsch and Wimbush (1977) have a l s o d e s c r i b e d a northward t r a v e l l i n g 4-7-day wave about 60 km i n l e n g t h . Diiing e t a l . (1977) showed t h a t l i k e the 10-14-day motion, a 4-5-day d i s t u r b a n c e o c c u r s i n t e r m i t t e n t l y as a wave p a c k e t o f about 4 c y c l e s  and  t h a t i t i s s i g n i f i c a n t l y c o r r e l a t e d w i t h the wind s t r e s s c u r l and o t h e r atmospheric v a r i a b l e s . was  No i n d i c a t i o n o f the d i r e c t i o n o f wave p r o p a g a t i o n  given..  The 2-3-Day Band  Lee  (1975) and Lee and Mayer  (1977) have documented the e x i s t e n c e  o f w a v e - l i k e meanders o f the mean f l o w and the t r a n s i e n t o c c u r r e n c e o f c y c l o n i c " s p i n - o f f " e d d i e s i n the 2-3-day band.  These e d d i e s a r e t r a p p e d  a g a i n s t the c o n t i n e n t a l boundary, have a l a t e r a l l e n g t h s c a l e o f about 10 km and a l o n g i t u d i n a l one 2 t o 3 times g r e a t e r .  They o c c u r a t  a p p r o x i m a t e l y weekly p e r i o d s , propagate northward a t speeds r a n g i n g between 20-40 cm s  1  and p e r s i s t f o r up t o 3 weeks.  A k i n e m a t i c a l model o f a p a i r  o f v o r t i c e s superimposed on the mean flow gave a good r e p r e s e n t a t i o n o f the observed n e a r - s u r f a c e c u r r e n t .  The meanders a l s o propagate northward  but a t speeds between 65 and 100 cm s~^~. I t i s f o r the motions i n the 4-5-day band t h a t the p r e s e n t t h e o r y might o f f e r a p o s s i b l e e x p l a n a t i o n .  Indeed, b a r o t r o p i c i n s t a b i l i t y o f the  58  mean flow has been suggested by Duing these motions. determined  We  (1975) as a l i k e l y mechanism f o r  s h o u l d note, however, t h a t Brooks  and N i i l e r  t h a t , i n the mean, the F l o r i d a C u r r e n t i s i n an  (1977)  equilibrium  s t a t e and t h a t the net i n t e r c h a n g e o f energy between the mean c u r r e n t and the f l u c t u a t i o n s superimposed  upon i t i s extremely s m a l l .  Of c o u r s e , t h i s  does not r u l e out the p o s s i b i l i t y t h a t d i s t u r b a n c e s i n some frequency ranges may  e x t r a c t energy from the mean flow.  shown t h a t the p r i m a r y source o f energy paper  Furthermore,  f o r the motions  i t has been  described i n this  i s the s m a l l , sheared, f l u c t u a t i n g component o f the b a s i c c u r r e n t .  N e v e r t h e l e s s , Brooks  and N i i l e r ' s work i n d i c a t e s t h a t  along-stream  v a r i a t i o n s i n the f l o w , as w e l l as i t s b a r o c l i n i c n a t u r e , may F o r the parameters  be  significant.  a p p r o p r i a t e t o the F l o r i d a S t r a i t s , F i g u r e  5.6a  i n d i c a t e s phase speeds o f about 40 cm s"'" f o r modes 1 and 3, and 25 cm s~^~ -  f o r mode 2 f o r the wavelength independent  of  £  and  0".  range o f 160-240 km.  These r e s u l t s a r e  However, the c o r r e s p o n d i n g growth r a t e s are  s t r o n g f u n c t i o n s o f these f a c t o r s as i s i l l u s t r a t e d  i n Table I I .  In g e n e r a l ,  the h i g h e r modes are more u n s t a b l e , and i n p a r t i c u l a r , i t i s seen t h a t the t h i r d mode c o u l d grow s i g n i f i c a n t l y w i t h i n one wave p e r i o d f o r a wide of values of  £  and  0~. Diiing's  (1975) p l o t s o f the eastward  component imply the e x i s t e n c e o f a second or t h i r d  velocity  (or h i g h e r ) mode;  u n f o r t u n a t e l y , h i s measurements extend o n l y over 2/3 Wunsch and Wimbush  o f the channel w i d t h .  (1977) have c a l c u l a t e d v e l o c i t y c r o s s - s p e c t r a , and  phase d i f f e r e n c e a t 5 days between northward  i n each case.  the  This i s consistent with  a t h i r d mode u n s t a b l e wave but not a second mode d i s t u r b a n c e . o f these r e s u l t s we  the  v e l o c i t i e s measured a t the  c o n t i n e n t a l boundary and the s h e l f break, and a t the s h e l f break and e a s t e r n w a l l i s approximately 180°  range  On the b a s i s  conclude t h a t a mode 3 f l u c t u a t i o n as d e s c r i b e d by the  59  Table I I .  The c h a r a c t e r i s t i c growth times T = l/ft f o r a 200-km wave (k/27T = . 1 5 ) . The threshold values a „ are also given.  Mode 1  £ a  Mode 2  Mode 3  .1  .25  .50  .1  .25  .50  .1  .25  .'50  16.0  6.60  3.50  13.0  5.50  3.30  5.0  3.30  3.00  T a  T (days)  3.0  -  -  -  4.0  -  -  5.0  -  -  .86 •  7.5  -  -  .37  -  2.64  10.0  -  -  .25  -  .78  2.40  -  -  -  -  -  -  2.60  -  7.00  2.50  -  1.20  -  3.20  1.40  .56  5.5  1.50  .72  .38  3.1  1.00  .50  -  4  .  4  0  60  present  theory p r o v i d e s  a p o s s i b l e e x p l a n a t i o n o f Diiing's  observations.  I t i s p a r t i c u l a r l y i n t r i g u i n g t h a t the mode 1 and mode 3 d i s p e r s i o n curves c r o s s i n t h e range o f i n t e r e s t , b u t i t would be improper t o draw any c o n c l u s i o n s from t h i s  observation.  It i s interesting  t o s p e c u l a t e t h a t some r e l a t i o n s h i p  between a mode 3 wave i n t h e 2-3-day band and t h e s p i n o f f by Lee.  The p r o p a g a t i o n  little  as  t h e same s i z e as t h e observed  F i g u r e 5.7 corresponds t o k  described  speeds a r e s i m i l a r and the " i n n e r g y r e " i l l u s t r a t e d  i n F i g u r e 5.7 i s approximately (Although  eddies  might e x i s t  i s increased.)  k / 2 7 T = .15,  eddies.  i t s form changes r e l a t i v e l y  61  6.  The C o n t i n e n t a l S h e l f Model  We now t u r n t o the c o n t i n e n t a l s h e l f model mentioned i n t h e introduction.  In o r d e r  t o examine a c o a s t a l phenomenon such as the  m o d i f i c a t i o n o f CSWs by a mean c u r r e n t , i t i s necessary t h a t any mechanisms capable o f a l t e r i n g t h e p o t e n t i a l v o r t i c i t y d i s t r i b u t i o n be l o c a l i z e d respect t o the coast.  with  I n a l l s t u d i e s o f CSWs i n the absence o f a b a s i c 2  flow, the g r a d i e n t o f the background p o t e n t i a l v o r t i c i t y ,  h'/h , has  always d i e d o u t away from the c o a s t so t h a t t h i s requirement was automatically f u l f i l l e d . I f one i n c l u d e s a sheared mean c u r r e n t , however, the p o t e n t i a l v o r t i c i t y g r a d i e n t becomes  -1  h  -1 2 V" - (Ro + V')h'/h , i t  2 i s no longer  s u f f i c i e n t that only  (1971) avoided  decay as  t h i s problem by employing a p i e c e w i s e  became c o n s t a n t Mooers  h'/h  a t a small distance offshore.  (1977a) have avoided  x -*• °°.  N i i l e r and Mysak  linear current  that  McKee (1977) and Brooks and  t h i s problem by adopting  a channel model,  a l t h o u g h Brooks and Mooers b r i e f l y d i s c u s s e d a c o n t i n e n t a l s h e l f model u t i l i z i n g an e x p o n e n t i a l l y decaying the mean c u r r e n t t o d i m i n i s h Our edge.  current.  Grimshaw (1976) a l s o r e q u i r e d  exponentially.  approach i s t o r e q u i r e  V"  B  t o be bounded away from the s h e l f  S p e c i f i c a l l y , we choose  V(x)  = s(x)v(x)  W(x)  = s(x)w(x)  (6.1)  where x -* oo,  s (x)  i s a deterministic function that s a t i s f i e s  and we assume  w(x)  s (x) ->• 0  i s a homogeneous random f u n c t i o n .  as  In terms o f  62  r (E,) = E[w(x)w(x + E,) ]  the c o r r e l a t i o n f u n c t i o n  R'(o)  2 = s r'(o)  + ss'r(o)  (6.2)  R" (o) = s s " r ( o ) + 2 s s ' r ' ( o )  With the c h o i c e o f  r(o) = 1  (3.27) t o  and  C  2  2 + s r"(o)  = - r"(o)  we  f i n d the  equation  be  - £ s ]Vty - (V - c ) Q ^  [(V - c)  2  2  x  - £ h [s a 2  The  finds that  2 = s r(o)  R(o)  e q u i v a l e n t to  one  boundary c o n d i t i o n s  1  2  - s s " + s s ' h ' / h ] ^ = 0.  2  (6.3)  are  ty(x) = 0  at  x = 0  i>(x) -»- 0  as  x -»• °°.  (6.4)  The  i n t e r p r e t a t i o n of  (6.3)  i s i d e n t i c a l to t h a t o f  (3.27), b u t we  note  2 2 t h a t the term corresponding i n v o l v e s a c o n t r i b u t i o n from The  to  e a  i n (3.27) i s more complex here  and  R'(0).  a p p r o p r i a t e matching c o n d i t i o n s  are  (6.5) [{(V - c )  2  - e s } ^ ' / h ] - [(V - c)p>] 2  2  - £ [ss'^/h] = 2  0.  63  The  g e n e r a l form o f the BrM model i s r e t a i n e d  r x e V(x)  l-x  and we choose  0  5 x 5 1  = s(x)v(x) = <  (6.6) U(l-x) x e  x > 1  r  0  5 x 5 1 (6.7)  s(x) = < y(i-x)  x > 1  2b(x-l) h(x)  0  5 x 5 1 (6.8)  = < x > 1  where  y  i s positive. As b e f o r e , we seek a p e r t u r b a t i o n s o l u t i o n o f the form (5.4)  The  t h r e e lowest order equations are  Vi> + h ( £ s / c 1  Q  V^> + h  -1  1  2  i 0  + h  1 ±  0  {2[c  i l  + i(V - c  (es/c . ) i f = c 0  2i[(V - c -2 lO  (6.9)  = 0  ( )  2 (es/c )i j ^  = - c  V4>  ) ^ 2  i 0  r  0  {Q t(v --x' Y  )  2  C  i  l  - c  -  r Q  i  0  ) ] ^  {(V - c  0  c c ]W i 0  r  r l  0  r 0  -  0  )  - 2c  i 0  [  6 10  x  - 2c  "  (- )  iQ V  C i l  i 2  c  i 0  - c  + i(V -  2 2 - e s  n  c ^ ) ] ! ^  2. -1 ) - i c . J - e h s s " + e^ss'h'/h^ifj 'il (6.11)  64  For  x < 1,  (6.9)-(6.11) a r e i d e n t i c a l w i t h ( 5 . 8 ) - ( 5 . 1 0 ) .  The z e r o t h o r d e r  s o l u t i o n s a t i s f y i n g both the boundary and matching c o n d i t i o n s i s :  b(x-l) e ^0 =  A  . y , . y s i n Ax/sin A (6.12)  0< J (pC)/J„(p) v  .2  provided  2,  A  = £  v  = k/y  /C^Q  2  -  ,2  ,2 - b  k  = £/yc  p  (6.13)  X  (6.14)  (p)/J (p)  (6.15)  that  b + X/tan X = - k + ( e / c  If  y(1-x) e  C =  iO  i  0  )J  v  +  1  v  i s n e g a t i v e , the s o l u t i o n s a r e h y p e r b o l i c over the s h e l f and may  o b t a i n e d by r e p l a c i n g  X  with  iX.  The g r a p h i c a l s o l u t i o n o f  c l o s e l y resembles t h a t o f (5.14), b u t note t h a t s e p a r a t e l y f o r each c h o i c e of for  k = 0  (i.e. , at  J  since 5  Q  £, y,  and  k.  (6.15) must be  be  (6.15) solved  T h i s s o l u t i o n i s not v a l i d  cannot s a t i s f y the boundary  condition at  x =  0 0  = 0).  The f i r s t o r d e r s o l u t i o n i s o f the form  r  n%  A  +  i A  o i p  (6.16) 12 0  0 2  65.  and the p a r t i c u l a r s o l u t i o n s il>2_  t  o  are s p e c i f i e d i n Appendix B.  s a t i s f y the matching c o n d i t i o n s ,  must a g a i n be s a t i s f i e d ;  In o r d e r f o r  the c o n s i s t e n c y c o n d i t i o n  i t was used as a check on the n u m e r i c a l  (5.27) results  d e r i v e d h e r e , and i t p r o v e d extremely s e n s i t i v e t o the a c c u r a c y of the r o o t s o f (6.15).  We  a g a i n choose  Application  A-^ = 0  and  A  i s s p e c i f i e d by  1 2  o f the Fredholm a l t e r n a t i v e t o (6.9)-(6.11)  (5.28)  implies  that  c  = <V>  r Q  c  il' rl  C  i2 iO  c  C  =  =  +  (c  2 i 0  /2e )<hQ /s > 2  (6.17)  2  x  (6.18)  0  ~  +  (3  /2)<(V - c  (c  i 0  (c  i0  2  2  - i(c  ) > 2  r Q  /2£ )<(V 2  -  (e /2)<s > 2  2  - c )hQ /s > 2  r Q  x  - (c  /2)<h's'/hs> - i c < ( V - c i 0  3 i 0  /2£ )<hQ /s 2  x  2  |  r Q  2 i 0  /2)<s"/s>  ) | T);^  \l> >  (6.19)  ±  where  <f (x) | g(x)> = /  h ^-s ^ 2  0  (x)f(x)g(x)dx//  h  - 1  s V>  2  (x) dx  (6.20)  0  and  <f(x)> = <f(x) | ^ ( x ) > Q  (6.21)  66  by comparison w i t h  (5.17)- (5.19) , we  see t h a t the terms i n v o l v i n g the  p o t e n t i a l v o r t i c i t y g r a d i e n t are emphasized s i n c e Otherwise the i n t e r p r e t a t i o n o f  s < 1  o f f the  (6.17)- (6.19) remains unchanged.  A comprehensive study o f the dependence o f the v a r i o u s on  y  was  not c a r r i e d o u t , and  the c h o i c e  comparison w i t h the channel mode. to t h a t shown i n F i g u r e 5.3, of  0"  of  k  and  are i l l u s t r a t e d  similarities.  Two  ]i = 1  was  dependence o f  3 and  The  on  facilitate o~  s i n c e as  k  decreases;  i s similar  threshold  values  and  5.6b  cQ  and  and  a d e t a i l e d comparison  r  as  are  fi^  functions  r e v e a l s n e a r l y as many d i f f e r e n c e s as  g e n e r a l c o n c l u s i o n s may  be drawn, however.  As e i t h e r  or the mode number i n c r e a s e s , the d i s p a r i t i e s between the two  increase for  results  l a r g e r f o r mode 2 than they  P l o t s of  i n F i g u r e s 5.4b  made t o 0,^  i s not shown here.  channel modes.  w i t h t h e i r channel c o u n t e r p a r t s  k  The  are s m a l l e r f o r modes 1 and  f o r the c o r r e s p o n d i n g  shelf.  and decrease f o r  c  r Q  .  models  This i s p h y s i c a l l y reasonable  or the mode number i n c r e a s e s , the e f f e c t i v e wavelength t h u s , the wave should become l e s s s e n s i t i v e t o the o u t e r boundary  and more s e n s i t i v e to the b a s i c c u r r e n t p r o f i l e . i n t i m a t e l y on t h i s p r o f i l e and  so the two  d i s p a r a t e a t s m a l l wavelengths.  Now  fi^  depends  models should be i n c r e a s i n g l y  S i m i l a r l y , the wave frequency  does not  depend on the d e t a i l s o f the b a s i c c u r r e n t but i t c e r t a i n l y depends s t r o n g l y on the p o s i t i o n o f the channel w a l l . the two  models should d i f f e r  Hence the phase speeds p r e d i c t e d by  f o r long wavelengths.  We  a l s o note t h a t  growth r a t e s f o r the d i f f e r e n t models no l o n g e r o v e r l a p . t h a t the group v e l o c i t y i s always p o s i t i v e , and curves  i n F i g u r e 5.6b  we  F i g u r e 5.5b  from the s l o p e o f  the shows  the  see t h a t , w i t h the e x c e p t i o n o f the f i r s t mode a t  s m a l l wavenumbers, i t exceeds the phase v e l o c i t y . t r a n s p o r t stream f u n c t i o n i s shown i n F i g u r e 5.7; i t s channel c o u n t e r p a r t a l t h o u g h  the a x i s t i l t  F i n a l l y , the mass i t g e n e r a l l y resembles  i s increased.  67  7.  Rossby Waves i n a Random Z o n a l Flow  In t h i s s e c t i o n we examine the i n t e r a c t i o n o f s m a l l - a m p l i t u d e , n o n d i v e r g e n t , f r e e b a r o t r o p i c Rossby waves w i t h a s t o c h a s t i c , sheared current.  F o r convenience we assume t h a t the flow  long channel w i t h s i d e w a l l s a t  y = 0,L.  to t o p o g r a p h i c Rossby waves p r o v i d e d enough so t h a t  aL/H  i s confined  The theory  t o an i n f i n i t e l y  i s also applicable a  t h a t the bottom s l o p e  i s o f the o r d e r  o f the Rossby number.  a n a l y s i s i s e n t i r e l y analogous t o t h a t o f the p r e c e d i n g  i s small The f o l l o w i n g  s e c t i o n s , and i t  i s t h e r e f o r e p r e s e n t e d i n as s u c c i n c t a f a s h i o n as p o s s i b l e . r e l a t e d study o f K e l l e r and V e r o n i s  zonal  We note the  (1969) who examined the p r o p a g a t i o n o f  Rossby waves i n a weak random c u r r e n t on an i n f i n i t e  $-plane.  For a zonal  flow o f zero mean they found t h a t the waves were damped and the wave speed reduced. In terms o f the v e l o c i t y stream f u n c t i o n d e f i n e d by  u (7.1) v  x  the nondimensional, l i n e a r i z e d v o r t i c i t y e q u a t i o n i s  0  + uB3 )? T + V B 2  T  x  where the z o n a l c u r r e n t  Ug(y)  we choose an average c u r r e n t  U  T U'V =  defines for  (7.2)  0  the b a s i c s t a t e .  (u,v),  As s c a l e f a c t o r s  the channel width  L  for  68.  (x,y),  and  L/U  f o r the time;  3  2 L /U  i s n o n d i m e n s i o n a l i z e d by  the d i m e n s i o n l e s s C o r i o l i s parameter i s  f = 1 +  Ro3y.  so t h a t  For a t r a v e l l i n g  wave s o l u t i o n o f the form  ik(x-ct) V = <Hy)e  (7.3)  (7.1) reduces t o  (U  - c ) ( $ " - k $ ) + (3 - U " ) $ = 0  (7.4)  2  B  B  which i s p r e c i s e l y the e q u a t i o n f i r s t  c o n s i d e r e d by Kuo (1949) .  The  boundary c o n d i t i o n s r e q u i r e no flow normal t o the channel w a l l , hence  <3?(y) = 0  We decompose  U  where  B  U  B  at  + £W(y)  E(W) = 0,  and  o f the t h e o r y developed of  (7.5)  i n t o i t s mean and f l u c t u a t i n g components as  = U(y)  E(U_,) = U  y = 0,1.  (7.6)  and we choose  E(W ) = 1.  i n S e c t i o n 3 l e a d s t o an e q u a t i o n f o r the mean p a r t  $,  [(u - c ) - e ]RJj + (u - c)Q 4> - e o 4> = o. 2  2  2  2  y  Here  Then a p p l i c a t i o n  F = d /dy 2  2  - k  2  , cr2 = - R" (0) ,  A p e r t u r b a t i o n expansion  and  Q  y  o f the form  (7.7)  = (3 - U") . (4.4) l e a d s t o t h e f o l l o w i n g  69  results,  c c  = e /K 2  i 0  (7-8)  2  = <U> + <Q >/2K  r Q  (7.9)  2  y  c , ,c , = 0 il r l c  i  2  c  i  (7.10)  = - (3/2XCU - c  0  ) >  - £ /2  2  r Q  2  + (1/2K )<(U - c ) Q y > - i c 2  r 0  + i(c /2K )<Q 2  i 0  y  i 0  <(U  - c  r Q  ) | tyf  | tyf .  (7.1D  Here  1  1  <f(y) | g(y)> = /  (y) f (y) g (y) dy/J  0  2  iJJ (y)dy  (7.12)  0  0  and  <f (y)> = <f (y) | 4>Q(y)>  (7.13)  2 where ty  Q  = A  Q  sin(nTry)  v a n i s h e s and t h a t  and  ^n^O  a n <  K ^  2 = k  2 2 + n TT .  ^2n+1^0  a  r  e  rea  A g a i n one f i n d s t h a t  -'-  a n c  c  2 +i n  ^ imaginary q u a n t i t i e s  respectively< These e x p r e s s i o n s a r e g e n e r a l i z a t i o n s o f those found by Manton and Mysak putting  (1976) f o r p l a n e Couette flow; t h e i r r e s u l t s may be r e c o v e r e d by U = y  and  Q  v  = 0.  I n p a r t i c u l a r , they showed t h a t a l l modes  t r a v e l l e d w i t h the same c o n s t a n t phase speed;  the i n c l u s i o n o f a nonzero  70  v o r t i c i t y g r a d i e n t i n the p r e s e n t model serves t o s e p a r a t e the d i s p e r s i o n curves as i s r e v e a l e d by (7.9) .  Somewhat more s p e c i f i c r e s u l t s a r e g i v e n  i n Appendix C f o r a p a r a b o l i c flow model,  U = 3y (y -  D/2.  F i n a l l y , we note t h a t the p r e s e n t t h e o r y i s n o t i n c o n f l i c t w i t h the study o f K e l l e r and V e r o n i s which p r e d i c t s wave damping. r e q u i r e two-dimensional,'  Their results  t r a n s l a t i o n a l i n v a r i a n c e o f the b a s i c s t a t e  ( K e l l e r , 1967), a c o n d i t i o n which cannot be s a t i s f i e d by a system w i t h a sheared mean c u r r e n t o r by one c o n f i n e d t o a c h a n n e l .  Hence t h e i r  do not apply t o the p r e s e n t system, even i n the l i m i t i n g case o f  results U = 0.  71  8.  Summary and C o n c l u d i n g  Remarks t o P a r t I  I t has been demonstrated t h a t s h e l f and Rossby waves through a r e g i o n o f b a s i c sheared W  c u r r e n t o f the form  i s a c e n t r e d random f u n c t i o n may be u n s t a b l e  length of  W  T h i s i s t r u e whether o r n o t V  necessary  conditions for barotropic i n s t a b i l i t y .  length.  Vg = V + £W  i f the l a t e r a l  where  correlation  i s s m a l l compared t o the c h a r a c t e r i s t i c l e n g t h s c a l e o f the  problem.  disturbances  propagating  s a t i s f i e s t h e well-known The growth r a t e o f these  i s p r i n c i p a l l y determined by the i n v e r s e o f the c o r r e l a t i o n  The phase speed i s t h e sum o f weighted c r o s s - s t r e a m  averages o f  the mean c u r r e n t and the mean g r a d i e n t o f p o t e n t i a l v o r t i c i t y .  Depending  on the Rossby number o f the system, the waves may t r a v e l w i t h o r a g a i n s t the mean flow. When t h i s theory  i s a p p l i e d t o a model o f the F l o r i d a  Straits,  u n s t a b l e CSWs a r e found w i t h p r o p e r t i e s t h a t a r e i n good agreement w i t h observations  made by Duing  (1975).  I t may, t h e r e f o r e , o f f e r an e x p l a n a t i o n  f o r some o f the observed meanders o f t h e F l o r i d a The p r e s e n t  theory  Current.  c o u l d o b v i o u s l y be extended i n many ways.  d e t a i l e d comparison i s needed w i t h an i n t r i n s i c a l l y u n s t a b l e to compare growth r a t e s .  A  system i n o r d e r  One c o u l d a l s o i n t r o d u c e a s m a l l , random c r o s s -  stream v e l o c i t y i n t o t h e b a s i c f l o w .  The problem o f mode-coupling has y e t  to be r e s o l v e d as does the e f f e c t o f temporal o r along-shore  variations i n  the b a s i c c u r r e n t .  only a f i r s t  Of course,  the present  theory represents  s t e p i n a more comprehensive e x a m i n a t i o n o f the e f f e c t s o f random p o t e n t i a l v o r t i c i t y d i s t r i b u t i o n s on b a r o t r o p i c - b a r o c l i n i c i n s t a b i l i t i e s .  72  9.  Introduction to Part II  The waters l y i n g between Vancouver B r i t i s h Columbia, and the S t a t e o f Washington from economic,  I s l a n d , the mainland c o a s t o f (see F i g u r e 9.1)  e n v i r o n m e n t a l , n a v i g a t i o n a l , and r e c r e a t i o n a l p o i n t s o f view.  O c e a n o g r a p h i c a l l y i t i s a complex e s t u a r i n e system. major  are i m p o r t a n t  In a d d i t i o n t o the  i n f l u e n c e s o f t i d e s , f r e s h water i n f l o w , topography, C o r i o l i s  force,  winds and o t h e r atmospheric v a r i a b l e s , one must c o n s i d e r the i n t e n s e m i x i n g t h a t o c c u r s i n the channels t h a t s e p a r a t e the S t r a i t of G e o r g i a (GS) Juan de Fuca S t r a i t  from  i n the south, and Queen C h a r l o t t e Sound i n the n o r t h .  P a r t I I o f t h i s t h e s i s r e p r e s e n t s an attempt t o understand some of the r e s u l t s p r e s e n t e d by Chang e t a l . (1976; see a l s o Chang, 1976). From the a n a l y s i s o f 18 months o f c u r r e n t r e c o r d s c o l l e c t e d a l o n g l i n e i n GS  H  (see F i g u r e 10.1), Chang showed t h a t n e a r l y o n e - h a l f o f the k i n e t i c  energy a s s o c i a t e d w i t h h o r i z o n t a l motions  i s contained i n  broad-banded,  low-frequency c u r r e n t f l u c t u a t i o n s c h a r a c t e r i z e d by p e r i o d s r a n g i n g from 4 to over 200 days.  No f o r c i n g mechanisms were e v i d e n t from Chang's a n a l y s i s  as the coherences between the c u r r e n t s and the wind, atmospheric p r e s s u r e , sea l e v e l , and water temperature were a l l c a l c u l a t e d t o be v e r y s m a l l . In an e a r l i e r attempt t o understand the low-frequency of GS, H e l b i g and Mysak both bottom  dynamics  (1976) c o n s t r u c t e d an a n a l y t i c model t h a t  topography and d e n s i t y s t r a t i f i c a t i o n .  included  T h i s model admits  n o r t h w a r d - t r a v e l l i n g t o p o g r a p h i c p l a n e t a r y waves w i t h p e r i o d s t h a t l i e i n the observed range, but i t i n c o r r e c t l y p r e d i c t s the v e r t i c a l o f h o r i z o n t a l energy.  H e l b i g and Mysak  i n s t a b i l i t y o f the mean f l o w was  distribution  (1976) suggested t h a t b a r o c l i n i c  a l i k e l y mechanism t o account f o r observed  f l u c t u a t i o n s , and i t i s from t h i s premise t h a t the p r e s e n t study commenced.  F i g u r e 9.1  P l a n view of the west c o a s t o f B r i t i s h and a d j o i n i n g w a t e r s .  Columbia  74  As  i t t u r n s out, t h i s c o n j e c t u r e i s p r o b a b l y  presented  i n S e c t i o n s 11 and  12 shows.  Two  i n c o r r e c t as the a n l y s i s simple  s t a b i l i t y models were  c o n s t r u c t e d o f a p u r e l y b a r o c l i n i c and b a r o t r o p i c system, r e s p e c t i v e l y . For the b a r o c l i n i c system the r e s u l t s i n d i c a t e t h a t the mean flow i s unstable  f o r o n l y a narrow band o f wave numbers.  An u n s t a b l e  e x i s t s a t a l l wavelengths i n the b a r o t r o p i c system and has  an  e - f o l d i n g time o f about 8 days.  shear wave  f o r a 15-day p e r i o d ,  However, as i s shown i n S e c t i o n  the observed C a r t e s i a n v e l o c i t y components are g e n e r a l l y i n phase t h a t the motions are not composed of the types r e s u l t s imply  12,  indicating  o f waves s t u d i e d here.  These  t h a t i n e r t i a l i n s t a b i l i t y p l a y s o n l y a minor r o l e , a t most,  i n the dynamics o f  GS.  A d d i t i o n a l c u r r e n t d a t a c o l l e c t e d by the Canadian H y d r o g r a p h i c S e r v i c e a t p o i n t s not a l o n g S i n c e these  line  H  (see F i g u r e 10.1)  r e c o r d s were of l i m i t e d l e n g t h  i s s u b j e c t to severe  (about  statistical limitations.  were a l s o examined.  30 days) t h e i r a n a l y s i s  I t i n d i c a t e s , however, t h a t  d u r i n g the o b s e r v a t i o n p e r i o d an a n t i c y c l o n i c gyre e x i s t e d i n the h a l f o f GS.  I n t e r e s t i n g l y , t h i s c i r c u l a t i o n was  t h a t p o s t u l a t e d by Waldichuck The  longer-term  o f the o p p o s i t e  i n v e s t i g a t e d by Chang were a l s o reexamined  w i t h the o b j e c t of g a i n i n g f r e s h i n s i g h t s .  In p a r t i c u l a r , Chang c a l c u l a t e d  r o t a r y s p e c t r a which y i e l d no d i r e c t i n f o r m a t i o n about the As mentioned, the p r e s e n t  the motion i s not comprised of simple waves. c u r r e n t s and wind s t r e s s were computed.  The  individual results indicate  C r o s s - s p e c t r a between the  While g e n e r a l l y low  found, a c o n s i s t e n t phase p a t t e r n seemed t o emerge. o f i n t e r e s t , the along-channel  sense t o  (1957).  records  C a r t e s i a n v e l o c i t y components.  southern  c u r r e n t s a r e 180°  coherence  In the frequency  was range  o u t o f pria.se with, trie wind..  c o n j e c t u r e i s made t h a t the f o r c i n g o f the low-frequency motions i s not  75  d i r e c t but r a t h e r t h a t the winds i n t e r a c t n o n l i n e a r l y w i t h the t i d e s and Fraser River outflow  t o modulate the e s t u a r i n e c i r c u l a t i o n o f the  An examination i n the time domain o f winds and  system.  c u r r e n t s suggests t h a t the  water column responds most d i r e c t l y t o the wind along the e a s t e r n s i d e of GS w i t h a l a g of about f i v e days.  The  response elsewhere i s not  clear.  R e s i d u a l t i d a l c u r r e n t s were c a l c u l a t e d from the time s e r i e s o f b a r o t r o p i c t i d a l streams generated from the n u m e r i c a l F u c a - S t r a i t of Georgia  system developed by Crean  model o f the Juan de  (1976, 1978).  A  p a t t e r n of r e s i d u a l s t h a t v a r i e d w i t h the f o r t n i g h t l y t i d a l c y c l e found.  These c u r r e n t s were i n s u f f i c i e n t l y l a r g e and  to e x p l a i n the o b s e r v a t i o n s , The  was  of the wrong d i r e c t i o n  however.  o u t l i n e of P a r t I I i s as f o l l o w s .  p h y s i c a l oceanography of GS  coherent  A b r i e f d e s c r i p t i o n of  i s g i v e n i n S e c t i o n 10 and  includes a  d i s c u s s i o n of the p o s s i b l e c h a r a c t e r of the observed low-frequency and  an enumeration o f v a r i o u s f o r c i n g mechanisms t h a t might be  Two  simple  i n e r t i a l i n s t a b i l i t y models are c o n s i d e r e d  the d a t a a n a l y s i s i s p r e s e n t e d r e s i d u a l s are c a l c u l a t e d , and  i n S e c t i o n 12.  summarized.  currents  important.  i n S e c t i o n 11,  In S e c t i o n 13,  while  tidal  a b r i e f development of the concept o f  modulated e s t u a r i n e flow i s g i v e n . I I are  the  In S e c t i o n 14,  the key p o i n t s of P a r t  76  10.  P h y s i c a l Oceanography of the S t r a i t of  Georgia  Although the p h y s i c a l oceanography o f GS has r e c e i v e d treatment elsewhere  ( c f . Waldichuck, 1957), i t i s important  here some o f i t s p r i n c i p a l f e a t u r e s t o p r o v i d e study.  t o summarize f o r the f o l l o w i n g  Some p a r t s o f t h i s d e s c r i p t i o n are a b s t r a c t e d from H e l b i g A p l a n view o f GS  average w i d t h o f GS 250  a motivation  comprehensive  km.  i s shown i n F i g u r e 10.1.  I t reveals that  i s about 30 km w h i l e i t s l e n g t h i s s l i g h t l y  Thus, the a s p e c t r a t i o of channel l e n g t h t o w i d t h i s  8:1.  Bathymetric  10.2,  and were e x t r a c t e d from a t o p o g r a p h i c  Dr. P. B. Crean  s e c t i o n s along the l i n e s  map  approximately  are p r e s e n t e d of GS  compiled  i n Figure by  Even though s m a l l - s c a l e f e a t u r e s  smoothed, the bathymetry e x h i b i t s g r e a t i r r e g u l a r i t y ,  i n the n o r t h e r n  sector.  In g e n e r a l , extremely steep  the  l e s s than  ( p e r s o n a l communication) g i v i n g average depths over  squares throughout the S t r a i t . implicitly  1-10  (1977) .  2-km  are particularly  s l o p e s c h a r a c t e r i z e GS -2  along i t s western boundary, w h i l e are common along the e a s t . one  s l o p e s n e a r l y as steep  North o f l i n e 4, two  t o the e a s t o f Texada I s l a n d and  (exceeding  channels e x i s t :  a much wider one  10  a narrow  on the western s i d e .  South of l i n e 4 the topography becomes p r o g r e s s i v e l y smoother; l i n e s 7 8 i l l u s t r a t e the marked e f f e c t of F r a s e r R i v e r s e d i m e n t a t i o n banks on the e a s t .  The  )  as  and  extensive  l o n g i t u d i n a l s e c t i o n 10 r e v e a l s t h a t although  the  a x i a l bathymetry i s somewhat smoother than the t r a n s v e r s e bathymetry, i t s t i l l p o s s e s s e s a h i g h degree o f i r r e g u l a r i t y and  e x h i b i t s slopes that often  -2 exceed  10 F i g u r e 10.3  summer.  shows l o n g i t u d i n a l s e c t i o n s of d e n s i t y f o r w i n t e r  In the upper 50 metres near l i n e 7 (see F i g u r e 10.1)  and  there e x i s t s  a s t r o n g s e a s o n a l v a r i a t i o n which i s a s s o c i a t e d w i t h the o u t f l o w o f f r e s h  F i g u r e 10.1  P l a n view o f the S t r a i t o f G e o r g i a showing l i n e s o f t o p o g r a p h i c c r o s s s e c t i o n s (1-10) p r e s e n t e d i n F i g . 10.2.  78  • — 3 0 0  4 0 0  DISTANCE  FROM  NORTHERN  BOUNDARY  (Km)  400  i  F i g u r e 10.2  Topographic c r o s s s e c t i o n s : (A) Upper panels: 1-9; (B) lower p a n e l : 10. The v e r t i c a l e x a g g e r a t i o n i s 30:1 i n (A) and 150:1 i n (B). The i n s e t s i n d i c a t e slopes of 10~ . 2  C. FLATTERY  Figure  10.3A  BOUNDARY  C.MUDGE  Longitudinal section of O f o r (A) December 1968; (B) J u l y 1969 (from Crean and Ages, 1971). t  F i g u r e 10.3B  03  o  81 1/2 water from the F r a s e r R i v e r .  The  g e n e r a l l y l i e s i n the range of  Brunt-Vaisala  3 x 10~  to  3  3 x i o ~  the water column which i s thus w e l l s t r a t i f i e d . density gradient,  p^  the a c c e l e r a t i o n due  frequency 2  Here  rad S P  z  [-gp /p^] z  throughout  - 1  i s the  vertical  i s a r e p r e s e n t a t i v e v a l u e o f the d e n s i t y , and  g  is  to g r a v i t y .  The winds i n GS are s t r o n g l y a f f e c t e d by the mountainous t e r r a i n , and  surrounding  they are predominantly up- o r d o w n - s t r a i t ;  to the northwest o r s o u t h e a s t , During  N =  r e s p e c t i v e l y (Kendrew and K e r r ,  that i s ,  1955).  the w i n t e r months o f November through March the p r e v a i l i n g wind i s  u p - s t r a i t while  i n the summer months of June t o September i t i s  down-strait.  In a l l seasons the s t r o n g e s t winds are from a s o u t h e r l y d i r e c t i o n .  There  i s of course  Although  Waldichuck  a g r e a t d e a l o f v a r i a t i o n about t h i s average p a t t e r n .  (1957) i n d i c a t e s t h a t a c y c l o n i c gyre e x i s t s over the  s t r a i t d u r i n g the w i n t e r ,  the evidence  f o r t h i s seems weak.  We  southern do  note,  however, t h a t the p r e v a i l i n g wind a t Vancouver i s u s u a l l y t o the west. The  r o t a r y spectrum o f the winds from Sand Heads computed by Chang  (1976) i s shown i n F i g u r e 10.4. u,  A r o t a r y spectrum of a v e c t o r p r o c e s s ,  i s o b t a i n e d by r e s o l v i n g each frequency  F o u r i e r transformed  vector  u  i n t o two  r o t a t e s w i t h a p o s i t i v e frequency negative  frequency  (clockwise).  clockwise  sense.  (The r e a d e r  component o f the d i s c r e t e  o t h e r components, one  ( a n t i - c l o c k w i s e ) and  i s r e f e r r e d t o Chang  p l o t o f the spectrum m u l t i p l i e d by the frequency  the curve  a  representing  t o move i n an a n t i - c l o c k w i s e o r (1976) or Mooers  f o r a comprehensive d i s c u s s i o n o f r o t a r y c u r r e n t s p e c t r a . )  l o g a r i t h m o f the frequency  o f which  the o t h e r w i t h  This gives a p a i r of spectra  the r e s p e c t i v e tendency of the p r o c e s s  say  (f o S)  (1973)  Notice that a against  the  i s v a r i a n c e p r e s e r v i n g ; t h a t i s , the area under  i s d i r e c t l y p r o p o r t i o n a l t o the  variance.  82  In t h e p r e s e n t of the wind. has  case, F i g u r e 10.4 i n d i c a t e s the c y c l o n i c tendency  The s p e c t r a a r e b r o a d l y peaked about 3-5 days b u t the wind  s i g n i f i c a n t energy t o p e r i o d s as l a r g e as 25 days.  percent  o f the v a r i a n c e i s c o n t a i n e d  Approximately 10  i n the 10-20-day band and o n e - t h i r d  of t h e v a r i a n c e i s i n p e r i o d s exceeding 7 days. Some r e s u l t s o f Chang's a n a l y s i s o f GS c u r r e n t s a r e p r e s e n t e d i n F i g u r e s 10.5-10.8. H06, at  The c u r r e n t r e c o r d s examined were c o l l e c t e d a t s t a t i o n s  H16 and H26 as shown i n F i g u r e s 10.1 and 10.5. Meters were p o s i t i o n e d  3, 50 and 200m a t the western  3, 50 and 140 m i n the e a s t near s u r f a c e i n s t r u m e n t s .  (H06) and c e n t r a l (H16) l o c a t i o n s and a t  (H26).  Chang d i d n o t analyze  r e c o r d s from the  Most o f the c u r r e n t r e c o r d s were o b t a i n e d  with  Aanderaa Model 4 c u r r e n t meters, b u t s e v e r a l Geodyne Model 850 meters were employed.  The c u r r e n t s were sampled e i t h e r every  (Geodyne) minutes.  A subsurface  10 (Aanderaa) o r 15  buoy mooring was used f o r the i n i t i a l  o f the experiment, b u t was r e p l a c e d t h e r e a f t e r by a s u r f a c e buoy, mooring.  taut-rope  A l t h o u g h t h e t h r e s h o l d l e v e l o f these meters i s 1.5 cm s  t h i s p r e s e n t s minimal d i f f i c u l t i e s  year  \  i n the d e t e c t i o n o f s m a l l , low-frequency  c u r r e n t s s i n c e s t r o n g e r t i d a l c u r r e n t s were superposed on these f l u c t u a t i o n s . The mean c u r r e n t s computed over the 18-month p e r i o d a r e shown i n F i g u r e 10.6.  There a r e two s i g n i f i c a n t f e a t u r e s .  cross-channel  f l o w a t the 50-m c e n t r a l l o c a t i o n , and t h e second i s t h e  v e r y s t r o n g c u r r e n t found a t 140 m i n t h e e a s t .  The f i r s t  i s the s t r o n g ,  The mean speed t h e r e i s  f i v e times g r e a t e r than t h a t found a t the o t h e r deep l o c a t i o n s , w h i l e the r o o t mean square v e l o c i t y i s twice as l a r g e . both shallow  In the e a s t and the west,  and deep c u r r e n t s a r e c l o s e l y a l i g n e d w i t h  the l o c a l  topography.  F i g u r e 10.7 shows the c u r r e n t s p e c t r a o b t a i n e d by summing the respective  p o s i t i v e and n e g a t i v e p a r t s o f the r o t a r y s p e c t r a computed by  PERIOD (DAYS) —i  510 2550100200500  r- 1  \  \  \ \*  1'  H i l l  F—•—i  1  i  i  CM  0) CM  ifJ  £ 4.0-  j  1.0 0.0 -1.0 "2.0 -3.0 -4.0 -3.0 -2.0 -1.0 0.0 ~ log (freq /1 cpd) •  •  f  F i g u r e 10.4  i  1.0  + f  Rotary spectrum of the winds a t Sand Heads f o r the 600-day p e r i o d b e g i n n i n g 3 January 1969 (from Chang, 1976).  oo  I  Figure 10.5  Cross section H showing placement of current meters. The moorings are spaced 10 km apart. The deep meters ' are (from west to east) 50, 80, and 25 m from the bottom (from Tabata et a l . , 1971).  ^  Figure  10.6  Mean c u r r e n t s along l i n e H f o r the 533-day p e r i o d b e g i n n i n g 16 A p r i l 1969.  86 Chang.  The  a r e a under the curve  o f the s i g n a l .  i s thus p r o p o r t i o n a l t o the t o t a l  Examination of t h i s f i g u r e r e v e a l s the complex nature  the low-frequency c u r r e n t s i n GS,  f e a t u r e s i n F i g u r e 10.7 peak about 15-25  days, and  the 200-m r e c o r d s  separated  are:  s i g n i f i c a n t t o 95%.  (2) i n c o n t r a s t to the 140-m  o n l y about 0.3.  separated  The  H26, records.  There the upper- and be  low  h i g h e s t v a l u e o f the squared  c u r r e n t s was  observed i n the  lower-layer  rotary v e l o c i t i e s  l o c a t i o n s the v e r t i c a l coherence was  very s m a l l and  o r no  the h o r i z o n t a l coherences were below the 95% Chang a l s o analyzed  water temperature r e c o r d s  c o u p l i n g between the upper  c o l l e c t e d by  In a l l cases  u n c o r r e l a t e d w i t h the c u r r e n t s .  The  found between the c u r r e n t s and  .3 f o r b o t h 50 and  140 m),  level. and  temperatures were  f o r c i n g mechanisms are  and  these q u a n t i t i e s were e s s e n t i a l l y  h i g h e s t v a l u e o f the squared coherence  the wind a t the e a s t e r n l o c a t i o n  (about  which suggests t h a t the s u r f a c e wind s t r e s s  be a p o s s i b l e f o r c i n g mechanism.  It  The  In a l l  the Aanderaa meters which were equipped t o sample c u r r e n t s  temperatures c o n c u r r e n t l y .  what o t h e r  noise  sea l e v e l , atmospheric p r e s s u r e , wind,  f o r the 18-month p e r i o d .  At  the phases  lower l a y e r s and hence i m p l i e s mainly b a r o c l i n i c motions t h e r e .  cases  east  i n d i c a t i v e o f a b a r o t r o p i c motion.  were s c a t t e r e d ; t h i s r e s u l t suggests l i t t l e  was  s i g n a l from  both h o r i z o n t a l l y and v e r t i c a l l y were g e n e r a l l y s m a l l a t  were n e a r l y i n phase which may  and  appear t o  (1976) found t h a t coherences between c u r r e n t s a t p o s i t i o n s  coherence between v e r t i c a l l y  the other  significant  energy i n comparison w i t h the 50-m  f r e q u e n c i e s as i s shown i n F i g u r e 10.8.  and was  The  (1) the s p e c t r a are broadbanded and  contain l i t t l e  of  but i t must be emphasized t h a t most o f  the f i n e s t r u c t u r e i s not s t a t i s t i c a l l y  Chang  variance  I t i s not c l e a r from Chang's a n a l y s i s important.  i s apparent, however, t h a t the low-frequency c u r r e n t s must  may  87 PERIOD (DAYS) -. 2001-0050251510 5  50.0 200IQQ 50 251510 5  500 20010050251510 5 I - i 1—i i i t i  11* r H 06,50 m  0.0  H 26,50 m  -3.0  -2.0  -1.0  log(f/lcpd)  F i g u r e 10.7  0.0  •3.0  -2.0  C u r r e n t s p e c t r a f o r l i n e H f o r the 533-day p e r i o d b e g i n n i n g 16 A p r i l 1969 (adapted from Chang, 1976) .  -1.0  0.0  88  (A)  0.07 0.95  EASTERN  -1  H  0.9  .  -2.5  -1.5  -1  0.8 0.7 O.S 0.3  A  i  95V.  • o.i -i 1 o J -0.5  0.5  FREQUENCY ( C P D )  1  1.5  FREQUENCY ( C P D )  180  T5  90  4>  O ' - 9 0 -180  (B)  0.97. 0.95 0.9 • 0.8 • 0.7 0.5 0.3 0.1 -  CENTRAL  :  :  -2.5  •1.5  -1  o-  — 1 —  •0.5  —i— 0.5  1.5  0.5  1.5  2.5  o -90  (C ) WESTERN  0 . 9 7 -1 0.95 • 0.9 H 0.8 0.7 0.5 0.3 - 0.1  3  -2.5  •1.5  -1  •0.5  s  o 18090 • O •  •  - 9 0 • -180  F i g u r e 10.8A  ;  Rotary coherence and phase s p e c t r a between c u r r e n t s , from (A) v e r t i c a l l y s e p a r a t e d l o c a t i o n s and (B) h o r i z o n t a l l y s e p a r a t e d locations. N o t i c e t h a t the frequency here extends t o much h i g h e r v a l u e s than a r e d i s cussed i n the t e x t . The s o l i d l i n e i n d i c a t e s a 95% n o i s e l e v e l (from Chang, 1976).  T5  89  (A)  •2.5  CENTRAL- EASTERN 50m  -1.5 FREQUENCY  -1  0.97 0.95 0.9 0.8 0.7 0.5 0.3 -. 0.1 T 0  -95V.  -0.5  0.5  (CPD)  1 FREQUENCY  .  1.5 (CPD)  2.5  90  O -90  -180  0.97 0.95 0.9 0.8 0.7 0.5 0.3 0.1  ( B ) CENTRAL-WESTERN 50 m .  2.5  •1.5  -1  • ' • • • :  • -  o-  -0.5  T —r-~ 0.5  —i— 1.5  2.5  —i— 0.5  1.5  2.5  180 •  90 •  O -90  (C)CENTRAL 200m  0.97 0.95 0.9  WESTERN  . -  2.5  "2  •1.5  1  -0.5  -i n u  0.8 0.7 0.5 0.3 0.1  •  -  o 180 90 •  O -90  -180  F i g u r e 10.8B  •  90  r e s u l t from more or l e s s continuous f o r c i n g o f some k i n d ; f r i c t i o n would q u i c k l y damp out the motions.  otherwise,  P r i o r to the enumeration o f  v a r i o u s p o s s i b l e f o r c i n g mechanisms, i t i s u s e f u l to c o n s i d e r what the low-frequency motions might p o s s e s s .  F i r s t they c o u l d be  character  wavelike.  T h i s c l a s s i f i c a t i o n i n c l u d e s b o t h a s u p e r p o s i t i o n o f p l a n e waves (as i n P a r t I) i n which the dependence on the h o r i z o n t a l c o o r d i n a t e s and more complex wavetypes  (eddies)  is  i n which i t i s i n s e p a r a b l e .  separable Waves  c o u l d be d i r e c t l y f o r c e d , f o r example, by the wind, and move w i t h speed o f the atmospheric d i s t u r b a n c e , c h a r a c t e r i s t i c frequency.  or they c o u l d be  Moreover, waves c o u l d occur  wave p a c k e t s o r e x i s t almost c o n t i n u o u s l y . i n c l u d e i n t e r n a l K e l v i n and  topographic  the phase  f r e e and have a intermittently in  P o s s i b l e s u b i n e r t i a l waves  p l a n e t a r y waves.  Second, the  low-frequency c u r r e n t s might be m a n i f e s t a t i o n s  of t r a n s i e n t s t h a t c o u l d  i n i t i a t e d by a v a r i e t y o f d r i v i n g mechanisms.  T h i r d , they c o u l d c o n s i s t o f  a s u p e r p o s i t i o n o f any  of these t y p e s .  describable only i n s t a t i s t i c a l Any  be  F i n a l l y , the motions might be  terms.  mechanism c a p a b l e o f a l t e r i n g the d i s t r i b u t i o n o f momentum,  v o r t i c i t y , or mass i n the system might f o r c e the low-frequency  currents.  Such mechanisms i n c l u d e the wind s t r e s s and wind s t r e s s c u r l which impart momentum and v o r t i c i t y , r e s p e c t i v e l y , to the system through the sea In a d d i t i o n , the wind s t r e s s may  i n t r o d u c e anomalies i n t o the  d i s t r i b u t i o n by f o r c i n g water columns a c r o s s b a t h y m e t r i c s t r e t c h i n g o r compressing v o r t e x  lines.  a c t i n a s i m i l a r manner a t the sea s u r f a c e . mass d i s t r i b u t i o n s may i s i n e r t i a l l y unstable. c o n s t i t u e n t s may  be  The  vorticity  contours,  Atmospheric p r e s s u r e  surface.  thus  differences  momentum, v o r t i c i t y ,  a l t e r e d i n t e r n a l l y i f the mean flow o f the In a d d i t i o n , n o n l i n e a r  i n t e r a c t i o n s between  r e s u l t i n r e s i d u a l flows and produce t i d a l  stresses  and system tidal  91 analogous t o the u s u a l Reynolds s t r e s s e s t i d e s may  (Heaps, 1978).  S i m i l a r l y , the  i n t e r a c t w i t h the topography t o g e n e r a t e i n t e r n a l motions.  F i n a l l y , freshwater i n f l u x e s or i n t r u s i o n s o f s a l i n e o c e a n i c water i n d e n s i t y d i f f e r e n c e s which i n t u r n d r i v e  result  currents.  I f the magnitude o f any o f these mechanisms v a r i e s i n time, the r e s u l t a n t motions should v a r y i n a s i m i l a r f a s h i o n .  Thus one might expect  the spectrum o f c u r r e n t s d r i v e n d i r e c t l y by the winds t o be peaked about 3 t o 5 days as i s the wind spectrum.  T h i s n o t i o n i g n o r e s the p o s s i b l e  importance o f the s p a t i a l c h a r a c t e r i s t i c s o f the wind f i e l d , however, and it  i s c o n c e i v a b l e t h a t t h e s e motions might peak a t some o t h e r frequency f o r  which the l e n g t h s c a l e s o f the winds and c u r r e n t s were comparable.  It  seems u n l i k e l y , however, t h a t the s p a t i a l s c a l e o f the wind d e c r e a s e s w i t h decreasing frequency.  On the o t h e r hand, a r e l a t i v e l y modest s p e c t r a l  component o f the wind might be capable o f e x c i t i n g a f r e e wave  at i t s  c h a r a c t e r i s t i c frequency. In the case o f t i d a l f o r c i n g , any p r o c e s s dependent upon the s t r e n g t h o f the t i d a l streams s h o u l d v a r y w i t h a f o r t n i g h t l y p e r i o d .  This  i n c l u d e s the t u r b u l e n t m i x i n g t h a t o c c u r s i n the c o n s t r i c t e d c h a n n e l s s e p a r a t i n g Juan de Fuca S t r a i t from GS  ( F i g u r e 10.1).  Thus i n t r u s i o n s o f  i n t e r m e d i a t e d e n s i t y water i n t o GS r e s u l t i n g from the m i x i n g o f more dense, r e l a t i v e l y deep Juan de Fuca water w i t h o u t f l o w i n g , c o m p a r a t i v e l y l i g h t water c o u l d g e n e r a t e c u r r e n t s o f f o r t n i g h t l y  period.  In f a c t ,  GS  Herlinveaux  (1957, 1969) has noted t h a t semimonthly v a r i a t i o n s o c c u r i n the s u r f a c e s a l i n i t y and temperature a t v a r i o u s l o c a t i o n s i n the Juan de F u c a - S t r a i t o f G e o r g i a system, and t h a t these v a r i a t i o n s are most e v i d e n t near the c o n n e c t i n g passages.  Webster and Farmer  o b s e r v a t i o n from the a n a l y s i s o f a l o n g  (1976) have s u b s t a n t i a t e d timeseries  of lighthouse  this station  92 data.  These f i n d i n g s suggest t h a t the degree o f mixing depends on  t i d a l range and  hence v a r i e s w i t h a f o r t n i g h t l y p e r i o d .  I t i s a l s o p o s s i b l e t h a t Juan de Fuca S t r a i t and dynamically  the  coupled and  t h a t i n f l u e n c e s i n one  f o r c e motions i n the o t h e r .  may  GS  are  d i r e c t l y or  I n t e r e s t i n g l y , F i s s e l and  Huggett  indirectly (1976) have  shown t h a t low-frequency c u r r e n t f l u c t u a t i o n s o f about a 15-day p e r i o d a l s o o c c u r i n Juan de Fuca  Strait.  Motions c o u l d a l s o be mechanisms.  d r i v e n by one  For example, the F r a s e r R i v e r o u t f l o w  i n t e r a c t n o n l i n e a r l y w i t h the t i d a l c u r r e n t s modulation o f the b a s i c e s t u a r i n e Finally,  That i s , n o n l i n e a r  above-mentioned  (Figure 9.1)  might  resulting in a fortnightly  flow.  i t i s p o s s i b l e t h a t a s i g n i f i c a n t f r a c t i o n o f the  low-frequency c u r r e n t s  i n GS  can o n l y be c l a s s i f i e d as g e o s t r o p h i c  i n t e r a c t i o n s between both l a r g e - and  i r r e s p e c t i v e o f t h e i r s o u r c e , may has  or more o f the  turbulence.  s m a l l - s c a l e motions,  be a predominate i n f l u e n c e .  demonstrated t h a t i n a g e o s t r o p h i c a l l y t u r b u l e n t system,  f l u c t u a t i o n s tend to e v o l v e  observed  Rhines  (1975)  small-scale  i n t o l a r g e r - s c a l e , more w e l l - d e f i n e d ,  planetary  w a v e l i k e motions. C l e a r l y , t h i s d i s c u s s i o n of the c h a r a c t e r low-frequency c u r r e n t s  o f the  observed  and p o s s i b l e f o r c i n g mechanisms i s not  exhaustive.  Perhaps many o r a l l o f the mentioned mechanisms p l a y a s i g n i f i c a n t r o l e i n GS dynamics. flow and few  In P a r t I I o f t h i s t h e s i s , the i n e r t i a l s t a b i l i t y o f the mean  the r e s i d u a l t i d a l c i r c u l a t i o n are examined i n g r e a t e r  comments are a l s o made concerning  F r a s e r R i v e r o u t f l o w , and  the winds.  detail.  A  the p o s s i b l e i n t e r a c t i o n o f the t i d e s ,  93  11.  I n e r t i a l I n s t a b i l i t y Models  As mentioned i n S e c t i o n 9, a t the c o n c l u s i o n of a p r e v i o u s ( H e l b i g and Mysak, 1976), i t was  s t r o n g l y suspected  i n s t a b i l i t y o f the mean flow w i t h i n GS was  facts.  inertial  an agent r e s p o n s i b l e f o r a major  p r o p o r t i o n o f the observed low-frequency energy. two  that  study  T h i s b e l i e f was  based  F i r s t , the phase speed o f a low-frequency wave of moderate  wavelength would be argument shows.  comparable t o mean c u r r e n t speeds as a p u r e l y  Consider  a 14-day wave of l e n g t h  X.  A  is  for  f o r a 100-km wave, a speed w i t h i n  range of the c u r r e n t s .  8 cm s **"  .08A;  expressed  cm s ^  example, a v a l u e o f  i s g i v e n by  If  kinematic  i n k i l o m e t r e s , the phase speed i n  t h e r e was  on  this gives,  Second, based on the f i n d i n g s o f Chang  the  (1976),  no apparent f o r c i n g mechanism f o r the f l u c t u a t i o n s . In p a r t i c u l a r , i t was  f e l t t h a t the i n s t a b i l i t y would be p r i m a r i l y  b a r o c l i n i c , the l a t e r a l shear of the c u r r e n t s p l a y i n g a r e l a t i v e l y minor role.  This hypothesis  was  based on two  premises.  First, vertical  were g e n e r a l l y observed t o be l a r g e r than h o r i z o n t a l shears e x c e p t i o n o f the deep e a s t e r n s t a t i o n ) . showed t h a t f o r an i d e a l i z e d model o f GS, with  (with  Second, H e l b i g and Mysak topographic  The  the (1976)  p l a n e t a r y waves e x i s t  f r e q u e n c i e s t h a t l i e i n the observed range f o r r e a s o n a b l e  the wave l e n g t h .  shears  choices  v e r t i c a l d i s t r i b u t i o n o f h o r i z o n t a l k i n e t i c energy  ( i . e . , t h a t a s s o c i a t e d w i t h the h o r i z o n t a l motion) f o r these waves opposite felt  of  to t h a t o b s e r v e d .  was  That i s , the waves were bottom t r a p p e d .  It  was  t h a t p e r t u r b a t i o n s of t h i s form, perhaps i n i t i a t e d by the winds, might  grow i n time by e x t r a c t i n g p o t e n t i a l energy from the mean flow w i t h r e s u l t a n t enhancement of u p p e r - l a y e r  kinetic  energy.  a  94  Baroclinic  Instability T h e r e f o r e , the f i r s t  s t e p i n the a n a l y s i s o f the i n e r t i a l  s t a b i l i t y o f GS was t o extend the model o f H e l b i g and Mysak t w o - l a y e r system  A  c o n f i n e d t o a channel w i t h a s l o p i n g bottom and w i t h a  c o n s t a n t mean v e l o c i t y i n each l a y e r was adopted parameters  (1976).  ( F i g u r e 11.1).  c h a r a c t e r i s t i c o f GS, the r e s u l t s below i n d i c a t e t h a t  can o c c u r o n l y f o r a narrow band o f wavelengths.  For instability  The p r i m a r y reason f o r  t h i s i s t h e s t r o n g s t a b i l i z i n g e f f e c t t h a t the narrow channel has on t h e system,  as i t l i m i t s t h e e f f e c t i v e wavelength  o f any p e r t u r b a t i o n s .  It is  i n t e r e s t i n g t h a t t h i s e f f e c t was a l s o l a r g e l y r e s p o n s i b l e f o r the h i g h degree  o f bottom t r a p p i n g found by H e l b i g and Mysak. T h i s model has been a p p l i e d by Mysak and S c h o t t  (1977) and Mysak  (1977) t o t h e Norwegian c u r r e n t and the C a l i f o r n i a u n d e r c u r r e n t , r e s p e c t i v e l y , with considerable success. of  A l t h o u g h the p r e s e n t development  t h i s model was c a r r i e d o u t i n d e p e n d e n t l y , i t s d e t a i l s are r e s t r i c t e d t o  Appendix D s i n c e the model has appeared  i n the l i t e r a t u r e .  The dynamics o f b a r o c l i n i c i n s t a b i l i t y d e r i v e from the c o n s e r v a tion of p o t e n t i a l v o r t i c i t y . first  d e r i v e d by Pedlosky  The g o v e r n i n g e q u a t i o n s e x p r e s s i n g t h i s were  (1964), and the d e r i v a t i o n p r e s e n t e d i n Appendix  D i s s i m i l a r a l t h o u g h i t d i f f e r s i n some r e s p e c t s . s p e c i f i e d by the c o n s t a n t c u r r e n t s  V  and  The b a s i c s t a t e i s  (see F i g u r e 11.1) which  are i n g e o s t r o p h i c b a l a n c e w i t h the mean s u r f a c e and i n t e r f a c i a l ments.  displace-  A p e r t u r b a t i o n w i t h i n i t i a l v e l o c i t i e s s m a l l compared w i t h mean  c u r r e n t s i s a p p l i e d t o the system.  I f i t grows i n time t h e system  to  equations g o v e r n i n g the p e r t u r b e d  are  be u n s t a b l e .  The nondimensional  i s said state  95  Z  / /  upper layer p  / / / / / / /  /  lower layer p  /  z  h  2  (x)  /  2 0  /  /  /  /  /  / / / / /  *  /  A  F i g u r e 11.1  The b a r o c l i n i c i n s t a b i l i t y model. The s l o p i n g s u r f a c e e l e v a t i o n and i n t e r f a c i a l displacement are i n geostropic balance w i t h the mean c u r r e n t s .  96  9  + V 3 ][V $  + F ($  2  t  1  y  1  1  2  -  - $ [V  - F (V  n  ly  1  1  - V )] = 0  1  2  (11.1)  t  [d  +  V  2 y 3  ]  [ V  ^2  " 2 F  ( $  2 "  " 2y $  [ V  "2  +  F  2  ( V  1  _  V  2  + T] = 0.  }  (11.2)  Here  and  i  U  $  = "  a  r  stream f u n c t i o n s  e  2  f o r the p e r t u r b a t i o n  velocities,  (11-3)  $ i  y  (11.3) i  v  where  = ix  (ll- )  $  i = 1,2  4  s p e c i f i e s the l a y e r , and  o f the p e r t u r b a t i o n  $  and  ^  are d e f i n e d  s u r f a c e and i n t e r f a c i a l d i s p l a c e m e n t s  i n terms  and  ?  ,  2  respectively,  (11.4) }  2 = h  h  +  The f o l l o w i n g s c a l e f a c t o r s were used i n the n o n d i m e n s i o n a l i z a t i o n : f o r the h o r i z o n t a l c o o r d i n a t e s velocities £^  and  £  (u,v), 2  ,  an a d v e c t i v e  respectively.  l a y e r i n t e r n a l Froude numbers 9"'  =  9"(P2 ~ Pl^/P2  (x,y),  "' '  :  s t  i e  re(  a t y p i c a l current  time  L/U,  and  fUL/g  U  F^ = f L / g ' h - ^ 2  2  and  F  fUL/g'  for  (11.2) are the  = f L /g'h Q 2  2  f o r the  and  A l s o a p p e a r i n g i n (11.1) and  L  2  3 u c e d a c c e l e r a t i o n due t o g r a v i t y ;  2  T  where  is a  97 t o p o g r a p h i c parameter d e f i n e d by Ro = U / f L that  T  T = - (L/RQI^Q)(dh /dx ^ ), 2  i s the Rossby number f o r the f l o w .  d  m  and  The n e g a t i v e s i g n ensures  i s o f the same s i g n as the bottom s l o p e , and the s u b s c r i p t "dim"  denotes a d i m e n s i o n a l v a r i a b l e . For  w a v e l i k e p e r t u r b a t i o n s o f the form  (V  - c) [<(>»! - k ^ ] - (^[V'-L - F ( V  ik(y-ct) <J)^e ,  (11.1) - (11.2)  reduce t o  1  +  (V  2  - c) [ c f ) " 2  - k^(f)2]  -  and  (j)-^ = A  V  2  ,  2  ) ]  n  cj) [V" 2  2  constant  1  - c) (o>, 2 - <j>) = 0  - F (V  For  -V  2  X  2  2  + F (V 2  - c)  (<J>  2 2  - V )  1  2  -  +  T]  f ^ ) = 0.  B  (11.6)  the s o l u t i o n i s  s i n nTTx  n  n = 1,2, . . . > (J> ~ n  (11.5)  s:  2  "-  (11.7)  n n 7 r x  where  K  2  + F  x  V  x  c - V,  and  K  = k  unstable,  A  + n TT n  and  i s the " t o t a l " B  n  A„  wave number.  (11.8)  I f the s o l u t i o n s a r e  w i l l be complex and the v e l o c i t i e s i n t h e upper and  98  lower l a y e r s w i l l be o u t o f phase.  c = V, +  „-  2K Z(K^  1  ±  T { (K  2  + F  x  !  where T  S = V-^ - V  2  +  FT  speed i s g i v e n by  ) (T - SK ) 2  -  1  SK  F,  2  1  2  Z  4F F sr  2  + F )  ±  ([(K + F )(T  1  The phase  [T -  SK ) + Fi SK 2]2 2  n  SK ]) 2  1 / 2  }  (11.9)  2  i s the v e r t i c a l  "shear."  i s r e s t r i c t e d t o be p o s i t i v e w h i l e  S  With no l o s s o f g e n e r a l i t y ,  may  have e i t h e r  The r a t i o o f the h o r i z o n t a l k i n e t i c energy  sign.  (HKE) p e r u n i t depth  i n the upper l a y e r to t h a t i n the lower l a y e r averaged over the a r e a d e f i n e d by the channel width and one wavelength  i n the  y-direction i s  (V-, - c j ^ + cV  A R = (K  2  + F)  2  ±  [(V  ±  - c ) r  - F S/(K  By HKE we mean the k i n e t i c energy- a s s o c i a t e d of motion.  An i n t e g r a t i o n o f  R =  so t h a t  R  X  Two  + F )] x  2  + c  w i t h the h o r i z o n t a l  2 ±  (11.10)  components  (h /h )R 1  .  (11.10) over the l a y e r depths g i v e s  (11.11)  2 0  r e p r e s e n t s the r a t i o o f the t o t a l HKE  i n the lower  2  i n the upper l a y e r t o t h a t  layer. l i m i t i n g cases are o f i n t e r e s t .  In the f i r s t we s e t  99 V  x  = V  2  = 0  to obtain  T(K  2  + F ) (11.12)  K (K 2  + F1  2  +  F ) 2  2 F R  l  72  = 7~2 (IT +  < !•  (11.13)  These r e s u l t s are e s s e n t i a l l y those o b t a i n e d by H e l b i g and Mysak  (1976) f o r  bottom i n t e n s i f i e d , t o p o g r a p h i c p l a n e t a r y waves i n a 2 - l a y e r c h a n n e l . the second  case, we p u t  c l a s s i c a l Eady  c = V  -  n  R = h  of HKE  (11.11) we  t o o b t a i n the 2 - l a y e r analogue o f the  (1949) s o l u t i o n ,  X  From  T = 0  In  2 0  /h  =  2(K Z  1  {(K +  F±  2  + 2F-,) ±  (K  + F )  4  - 4F,F ) 9  1 / 2  }.  2  .  see t h a t  (11.14)  X  (11.15)  R = 1  and  the two  l a y e r s c o n t a i n e q u a l amounts  i r r e s p e c t i v e of t h e i r thicknesses. In the g e n e r a l case, the s t a b l e s o l u t i o n s l i e between these  extremes.  For s u f f i c i e n t l y  large  T,  one  root of  (11.8) corresponds  two to a  shear m o d i f i e d t o p o g r a p h i c wave w h i l e the o t h e r r e p r e s e n t s a t o p o g r a p h i c a l l y m o d i f i e d shear wave.  Whether o r not these waves are more i n t e n s e i n the  upper o r lower l a y e r depends upon the c h o i c e o f parameters.  In the p r e s e n t  case a t wavelengths f o r which the system i s s t a b l e , one wave i s c o n c e n t r a t e d i n each l a y e r .  U n s t a b l e waves are found t o be more e n e r g e t i c i n the upper  100 layer. Schott  On  the o t h e r hand, f o r Norwegian c u r r e n t parameters, Mysak  (1977) found u n s t a b l e waves t o be bottom i n t e n s i f i e d .  study, Wright  (1978) has  From  In a r e c e n t  t r e a t e d t h i s q u e s t i o n i n much g r e a t e r  (11.9) we  see t h a t i f  S  and  detail.  i s p o s i t i v e a necessary  but  not  s u f f i c i e n t condition for i n s t a b i l i t y i s  S > T/K .  (11.16)  2  The bathymetry thus a c t s t o s t a b i l i z e the flow i f the bottom s l o p e s upward to the e a s t and  i s a d e s t a b i l i z i n g i n f l u e n c e i n the o p p o s i t e  agrees w i t h the f i n d i n g s o f Blumsack and G i e r a s c h s t r a t i f i e d system.  c o n s i d e r a system w i t h  F  Eady wave (see 11.14).  Then  = F  ±  and  if  16F S 1  l -l6Fi; +  > 9T  continuously  and  T  K  = 4F F  4  1  1/2 X / Z  ±  (i.e.,  a flow u n s t a b l e . 2  To see  corresponding  this to a n e u t r a l  (11.9) becomes  1  V  render  2  3T - 8F S =  (1972) f o r a  This  However, t h i s n o t i o n must be q u a l i f i e d , f o r the  presence of weak topography may  C  case.  [9T -  lSF^S]  1^  V2  '  S > 9T/8K ) ,  (  the system i s u n s t a b l e w i t h  1  1  -  1  7  )  a  1/2 growth r a t e p r o p o r t i o n a l t o a c t i o n o f Eady waves w i t h  T  .  DeSzoeke  presented.  Figures  (11.2) and  b o u n d a r i e s f o r the system.  The  p o s i t i v e bottom s l o p e .  T  i n s i z e and  As  s t u d i e d the  inter-  topography.  Numerical r e s u l t s c o r r e s p o n d i n g are now  (1975) has  to the g e n e r a l r e l a t i o n  (11.3) i l l u s t r a t e t y p i c a l  (11.9) stability  former shows the s t a b i l i z i n g e f f e c t of a i n c r e a s e s , the r e g i o n o f i n s t a b i l i t y  s h i f t s t o s m a l l e r wavelengths.  shrinks  There are no u n s t a b l e waves f o r  F i g u r e 11.2  Mode 1 s t a b i l i t y boundaries f o r the b a r o c l i n i c model as a f u n c t i o n o f the t o p o g r a p h i c parameter T.  103 negative parameter of  S  ( c f . 11.9).  V = F^ + F  the channel w i d t h  F i g u r e 11.3 which may  2  L  2  =••  the dependence on the  be r e w r i t t e n as the square o f the  ratio  t o the i n t e r n a l Rossby r a d i u s o f d e f o r m a t i o n  f L (h _ 2  V = F, + F  illustrates  + h  2  )  r^ ,  2 =  (J±)  .  (11.18)  *' l 20 h  For  small  V,  h  i . e . , f o r narrow channels o r s t r o n g s t r a t i f i c a t i o n ,  region of i n s t a b i l i t y  is relatively  narrow.  As  V  the  increases, corresponding  to an i n c r e a s e i n the channel w i d t h , a decrease i n the s t r a t i f i c a t i o n , o r a t h i n n i n g upper l a y e r , t h i s r e g i o n spreads out and s h i f t s  t o h i g h e r wave-  numbers . Parameters c h a r a c t e r i s t i c o f GS are .14, F  and  =  2  T = 7.4  (1 - A)V.)  2  GS,  = - 0.5V-^).  2  ( H e l b i g and Mysak, 1976).  (Note t h a t  F  = Av  f o r the cases  In the f i r s t  S = 0.5  (V" = O.SV^) 2  and  S =  and  and i s i n t e n s i f i e d  1.5  case, which i s g e n e r a l l y r e p r e s e n t a t i v e o f  the system i s u n s t a b l e o n l y i n the wavelength band o f 40-46 km.  most u n s t a b l e wave has an  The  e - f o l d i n g time o f 78 days, a p e r i o d o f 11 days,  i n the upper l a y e r  (R = 1.3).  In the second case, i n  which the c u r r e n t s are d i r e c t e d i n o p p o s i t e d i r e c t i o n s , the f i r s t mode i s u n s t a b l e f o r a l l wavelengths a p e r i o d o f 70 days and an the  upper l a y e r  =  F o r these parameters, the f i r s t - m o d e d i s p e r s i o n c u r v e s are  shown i n F i g u r e 11.4 (V  V = 7.5, A = h-^/(h-^ + h Q>  exceeding 93 km.  The most u n s t a b l e wave has  e - f o l d i n g time o f 39 days, and i s s t r o n g e s t i n  (R = 2.8).  The h i g h e r modes are s t a b l e i n each c a s e .  F i g u r e 11.5 d r a m a t i c a l l y i l l u s t r a t e s two r o o t s o f (11.9) f o r GS parameters. wave i s almost independent of  T,  the d i s p a r a t e n a t u r e o f the  While the phase  speed o f the shear  t h a t o f the t o p o g r a p h i c wave v a r i e s  250 100  50  F i g u r e 11.4  WAVELENGTH  (km)  25  B a r o c l i n i c model, for S = 0.5 and i n d i c a t e the most l e n g t h and p e r i o d scale factors of  mode 1, d i s p e r s i o n curves S = 1.5. The s o l i d c i r c l e s u n s t a b l e waves. The waveare c a l c u l a t e d u s i n g the U = .5 cm s and L = 25 km. -  x  105  F i g u r e 11.5  The b a r o c l i n i c model, mode 1, phase speed as a f u n c t i o n o f t o p o g r a p h i c parameter f o r k/2TT = 0.1, 0.5, and 1.0.  106 l i n e a r l y with  Barotropic  T.  Instability A simple b a r o t r o p i c model i s now  c o n s i d e r e d i n o r d e r t o g a i n some  i d e a o f the importance o f h o r i z o n t a l shear. F i g u r e 11.6. simplicity,  Although this  an e x p o n e n t i a l bottom p r o f i l e  £ = 0,  (3.8)  governing reduces  (V - c) [  We  i s chosen f o r  c h o i c e does not s e v e r e l y l i m i t the c o n c l u s i o n s drawn h e r e .  Indeed, the s l o p i n g topography has The  The model i s i l l u s t r a t e d i n  (^1)  equations  l i t t l e e f f e c t on the u n s t a b l e waves. are a b s t r a c t e d d i r e c t l y  from P a r t I ; w i t h  to  V  •  1  +  V  '  (j) = 0.  (11.19)  specify  fv. V(x)  0 < x < d  =<  (11.20) d < x 5 1  and  h(x)  and  = e  2b(x-l)  thus o b t a i n a c o n s t a n t c o e f f i c i e n t  As b e f o r e , the boundary c o n d i t i o n s are  cf> = 0  at  x = 0,1.  (11.21)  d i f f e r e n t i a l e q u a t i o n i n each r e g i o n . (cf.  3.19)  (11.22)  107  z  1> x y  v,  h [x)  ^  x=d  x =0  F i g u r e 11.6  The  barotropic  instability  model.  ^  ^  x  =1  108 The  s o l u t i o n i s g i v e n by  r  A^ s i n XjK  0 < x < d  b(x-lW  <f> (x)  (11.23) A  2  sinA ( x - 1 ) 2  d < x 5 1  where  2b i  R ( i v  0  (k  " <=)  + b )  2  i  2  = 1,2.  (11.24)  In o r d e r t h a t the normal f l u x e s of mass and momentum be continuous a t the material interface centred at  V - c  x = d,  <f> (x)  must s a t i s f y  = 0  (11.25a) at  [ (V - c)<j>' - V<1>]  x = d.  = 0  (11.25b)  (These r e l a t i o n s h i p s a r e d e r i v e d i n LeBlond and Mysak, 1978; p. 429. However,  (11.25b) d i f f e r s from t h e i r  that  i s continuous.  $  (45.9) s i n c e they e f f e c t i v e l y assumed  In the p r e s e n t case i t i s n o t , and one must p r o c e e d  from the i n t e g r a l r e l a t i o n s h i p p r e c e d i n g t h e i r c o n d i t i o n s l e a d s t o the i m p l i c i t d i s p e r s i o n  2-,  (V, - c) A,  ± k tan A-^d  1 tan  A  2  2 _ (d - 1)  b  [  (  v  A p p l i c a t i o n o f these  relation  2  (v- - c) A,  (45.9).)  _ Z c)  j  _  Z  ( v  c)  ^  ]  ~ =  Q  _  (H.26)  109 S e v e r a l l i m i t i n g cases a r e o f i n t e r e s t . = V A  2  tan A d = t a n A ( d - 1)  to o b t a i n  2  = nir  2  independent o f  c = V„ RQ  1  2  d.  (k  p  In the f i r s t we  put  which has the s o l u t i o n  This gives  + b  2b p  z  (11.27)  + n  )  TT  which i s the d i s p e r s i o n r e l a t i o n f o r a f r e e c o n t i n e n t a l s h e l f wave i n a mean current  V  ( c f . 5.21) .  2  V  l  +  Q V  2  i 0 1 + Q 1  (1 + Q)  where  In the second case we p u t  /  2  i  1  21  (11.28)  and i s p o s i t i v e .  These waves t r a v e l a t  average o f the mean c u r r e n t s and grow i n time  a t a r a t e p r o p o r t i o n a l to the shear. k -> «>,  to obtain  I 1  Q = - tanh kd/tanh k ( d - 1)  a speed g i v e n by a weighted  b = 0  Finally,  i n the s h o r t wave l i m i t o f  (11.26) reduces t o  V, + V(V, c = -^-r— ' ± i 2 2  V ) 9  .  £  n  Equations  (11.28) and  (11.2-  (11.29) r e p r e s e n t a p a i r o f shear waves, one o f which  i s u n s t a b l e and another which decays  i n time.  As these s p e c i a l cases suggest, t h e r e are a t most t h r e e s o l u t i o n s to  (11.26).  For nonzero  b  and  p a i r o f complex r o o t s f o r a l l k, m o d i f i e d shear waves.  S,  where  S = V  2  ,  there e x i s t a  c o r r e s p o n d i n g t o a m p l i f i e d and damped  Provided that both  A^  and  r e a l r o o t e x i s t s and r e p r e s e n t s a shear m o d i f i e d CSW. be expressed as  -  A  2  are r e a l , a t h i r d , This r e s t r i c t i o n  may  110  2b  c > V 1  /  R (k  2  + b )  2  The r e g i o n i n  k, S  Along the l i n e  .  2  Q  H  = V,,  f lHl [h  o n l y a CSW i s found.  +  h  1  x  u  (11.31)  2  was chosen as 50 m and  = 0.5  a p p r o p r i a t e t o GS, the mean c u r r e n t s  2( ) 2^  h  2  was determined f o r each mooring.  r e s u l t s a r e shown i n F i g u r e 11.8. V  S  + h (x)]  ±  h-^  3 0 )  were depth averaged as  h  BT  .  space i n which CSWs e x i s t i s shown i n F i g u r e 11.7.  To determine a v a l u e o f along l i n e  ( 1 1  A reasonable choice i s V  (with a s c a l e v e l o c i t y o f 5 cm s  - 1  ),  d = 0.66,  The  = 1.0,  and  b = - 0.3.  The d i s p e r s i o n r e l a t i o n f o r t h e s e v a l u e s i s shown i n F i g u r e 11.9. seen t h a t a CSW e x i s t s f o r wavelengths shear wave o f 15-day p e r i o d has an  g r e a t e r than 55 km.  An u n s t a b l e  e - f o l d i n g time o f about 8 days, a phase  speed o f about 4 cm s "*" and a wavelength o f 44 km. -  It is  I t i s possible  t h a t a shear i n s t a b i l i t y o f t h i s type might p l a y some r o l e i n GS  therefore  dynamics.  However, i n the next s e c t i o n i t i s shown t h a t the motions i n t h e 15-day band are  p r e d o m i n a n t l y nonwavelike i n t h e sense o f the waves s t u d i e d h e r e .  i m p l i e s shear i n s t a b i l i t y i s o f l i m i t e d importance i n GS.  This  F i g u r e 11.7  The r e g i o n i n (k,S) space i n which c o n t i n e n t a l s h e l f waves e x i s t .  112  Figure  11.8  Computed b a r o t r o p i c mean c u r r e n t s a l o n g l i n e H f o r the.18-month p e r i o d b e g i n n i n g A p r i l 1969.  WAVELENGTH (km) 250 IU i — i  F i g u r e 11.9  100 1  50  25  1  1  1 1 3  15 —  —  1  B a r o t r o p i c model d i s p e r s i o n curves f o r S = 0.5. The growth time i s d e f i n e d as the e - f o l d i n g time, u> i s the f r e q u e n c y , and 9. i s d e f i n e d as k • Im c. The wavelength and p e r i o d are computed u s i n g t h e s c a l e f a c t o r s o f u = 5 cm s and L = 25 km. _ x  116  F i g u r e 12.1  P l a n view of the S t r a i t o f G e o r g i a showing c u r r e n t meter l o c a t i o n s . These l i n e s s h o u l d not be confused w i t h those of F i g . 10.1. Winds were r e c o r d e d a t Sand Heads.  1970  1969  .A. M . J . J . A . S . 0 . N . D  STATION DEPTH l _ J (m)  H26  I  I  I  I  50  99  J] 35 £  140  99  -] 3 5  [  I  I  I  J F . M A M . J . J . A . S I  I  I  I  I  E— -] 2 8 £  1 A 7  10A  ] 29 [  50  62  t  H5  200  0 Q  "  £  108  79  £  — 3  I  —3  ]  39-Jf-29]  3  196  •258  I  ]  525  3 50  1 0 0  [-36-311 [-  200  |H06  I  158  3 |H16  I  [-42-3  33  E  382  Figure 12.2A Periods of existent current meter records.  E-59-3  49  150  3  3  1969 STATION DEPTH(m)| H 06 H 16 H 26 31 33 36  42 43 44 45 46 47 51 52 53 54 55 56 61 62 63 64 65  50 200 50 200 50 140  | MAY  | J U N | JUL | AUG | S E P I OCT 2 5 8 3 8 2 5 2 5 145  I  100 100 100 50 84  9 9  -147-  9 9  104  - 3 7 -37-  E—29 """"1-15-3  100 300 100 300  E-I5--3  •36  ...  100 200 100 200  -41  E-27—3  100 50 100 50 130 100 200  F  3 6 b —  250 100 200 100 180  l=8=l_ E-I7H  2 9 — - j  50 100 50 100 50 100 50 100 100  APR  3 5  J  3  3 5  U H  r  , t - 1 5 - d  E-ie-3 - 3 8 - 3 0 -  I  20-3 Figure 12.2B  ,  E-15-3 3  PERIOD (DAYS) 16 854 2  10^  16 854 2  119  —i—r—n—|— 31  A - northward  100 m  O - eastward  10 CP  10°lr  ^Q* I 2  i  10 |  i i mill  i  I I inn  1—r-n—r  2  41 84 m  ~  10  1  CvJ CvJ  o  10% o  o  o  10"  1  ,-2  10  10"  2  10"  1  10° 10"  2  10"  1  10°  FREQ (cpd) F i g u r e 12.3A  C u r r e n t s p e c t r a f o r the 26-day p e r i o d b e g i n n i n g and (B) 29 August 1969.  (A) 2 May 1969  10'  16 8 54 2 —i  i—n—|—  °-  1 0 1 — i i i mill 2  10  PERIOD (DAYS) 16 854 2  H06 50 m  4  •  2  FREQ (cpd) Figure  12.3B  16 854 2  120  121 course, these  none o f the peaks a r e s t a t i s t i c a l l y  s p e c t r a do i n d i c a t e the e x i s t e n c e o f low-frequency energy a t l o c a t i o n s  n o r t h o f l i n e H. due  s i g n i f i c a n t t o 95 p e r c e n t b u t  The c o m p a r a t i v e l y  q u i e t spectrum from S t a t i o n 41 may be  t o the f a c t t h a t the meter l i e s i n t h e "shadow" o f a t o p o g r a p h i c  high  j u s t t o i t s south. S p e c t r a computed from l i n e s 5, 6 and H r e c o r d s period beginning  f o r the 26-day  29 August 1969 a r e shown i n F i g u r e 12.3b.  These time s e r i e s  were t r e a t e d as above, and the s p e c t r a a l l i n d i c a t e low-frequency energy. The  r e c o r d from S t a t i o n 64, 50 m i s o f dubious q u a l i t y and thus i s o f l i m i t e d  value  f o r comparison w i t h H26, 50 m. Mean c u r r e n t s were c a l c u l a t e d d i r e c t l y  current e l l i p s e s  from the time s e r i e s , and  (see, f o r example, Stone, 1963) were c o n s t r u c t e d from t h e  average o f the lowest  two s p e c t r a l bands.  The e l l i p s e s a r e i l l u s t r a t e d i n  F i g u r e 12.4 by t h e i r major and minor axes, although l a t t e r i s t o o s h o r t t o be v i s i b l e .  i n s e v e r a l c a s e s , the  Due t o the s t a t i s t i c a l l i m i t a t i o n s o f  the d a t a , no i n d i c a t i o n i s g i v e n o f e i t h e r the d i r e c t i o n o f r o t a t i o n o f the o s c i l l a t i n g c u r r e n t v e c t o r around t h e e l l i p s e o r o f t h e r e l a t i v e phases between e l l i p s e s . contamination  Of course,  by trends d u r i n g the p e r i o d o f a n a l y s i s s i n c e these a f f e c t t h e  lowest-frequency  s p e c t r a l estimates,  must be viewed w i t h cross-channel  the e l l i p s e parameters a r e s u b j e c t t o  caution.  oscillating  and thus the e l l i p s e s c a l c u l a t e d here  Of p a r t i c u l a r i n t e r e s t i n F i g u r e  12.4A i s t h e  flow suggested a t H16, 50 m and S t a t i o n 43, 100 m.  The p a t t e r n o f mean c u r r e n t s i l l u s t r a t e d i n F i g u r e 12.4B i s extremely i n t e r e s t i n g s i n c e i t i n d i c a t e s a c l o s e d , c l o c k w i s e , mean c i r c u l a t i o n i n the lower s t r a i t . support  While t h i s may n o t be t r u e f o r longer p e r i o d s , i t lends  t o Waldichuck's  however, o f the o p p o s i t e  (1957) c o n j e c t u r e  t h a t a gyre e x i s t s .  sense t o t h a t i n d i c a t e d by Waldichuck.  It is, During  this  122  Figure  12.4A  Mean c u r r e n t s and the 6-32-day band c u r r e n t e l l i p s e s f o r the 26-day p e r i o d b e g i n n i n g (A) 2 May 1969 and (B) 29 August 1969. The e l l i p s e s are i n d i c a t e d by t h e i r major and minor axes.  F i g u r e 12.4B  124 p e r i o d , the f l o w a t H26, 50 m i s southward whereas the 18-month mean i s northward  (Figure 10.6).  permanent f e a t u r e .  T h i s i m p l i e s t h a t the gyre may n o t be a  A c l o s e d c i r c u l a t i o n i s n o t i n d i c a t e d f o r the c e n t r a l  s t r a i t d u r i n g May 1969 (Figure 12.4A). H26,  flow  The s t r o n g  a x i a l current present at  50 m i s n o t observed a t S t a t i o n 47, 100 m.  Wind-Driven Motions The  dynamics o f low-frequency, l a r g e - s c a l e motions are due i n  l a r g e p a r t t o the c o n s e r v a t i o n s t r e s s c u r l that enters  o f p o t e n t i a l v o r t i c i t y , and i t i s the wind  the v o r t i c i t y e q u a t i o n as a f o r c i n g f u n c t i o n .  In  a d d i t i o n , i f the system under c o n s i d e r a t i o n p o s s e s s e s s i g n i f i c a n t bottom topography, the wind s t r e s s i t s e l f  may induce v o r t i c i t y by f o r c i n g water  columns a c r o s s b a t h y m e t r i c c o n t o u r s thus squeezing o r s t r e t c h i n g lines.  Indeed, i n a b a r o t r o p i c system the v o r t i c i t y i n p u t by t h i s mechanism  may f a r exceed t h a t due t o the wind s t r e s s c u r l Chang currents  along  frequencies.  days  ( G i l l and Schumann, 1974).  (1976) c a l c u l a t e d c r o s s - s p e c t r a between the winds and l i n e H and found the coherence t o be g e n e r a l l y s m a l l a t low  However, the use o f r o t a r y s p e c t r a does n o t r e v e a l  s h i p s between the v a r i o u s winds.  vortex  rectangular  components o f the c u r r e n t s  relationand the  Moreover, the f a c t t h a t the c u r r e n t s p e c t r a a r e peaked a t about 14  (at l e a s t f o r the e a s t e r n  and western s t a t i o n s ) , t h a t the motion may  be b a r o t r o p i c a t H26, and t h a t the p e r i o d o f a f r e e CSW f o r GS parameters is  about 14 days f o r a v a r i e t y o f bottom p r o f i l e s  Mysak, 1977; o r Csanady, 1976) suggests v e r y might f o r c e m o d i f i e d  CSW's.  (see, e.g., LeBlond and  s t r o n g l y t h a t the wind s t r e s s  I t i s t h e r e f o r e s e n s i b l e t o examine t h e  r e l a t i o n s h i p between the wind s t r e s s and the c u r r e n t s by computing components spectra.  S u f f i c i e n t d a t a do  n o t e x i s t t o adequately determine the wind  125 s t r e s s c u r l , which a t any  rate i s a d i f f i c u l t  task due  t o the e f f e c t  the  complex orography of the B r i t i s h Columbia c o a s t has upon the winds.  That  i s , measurements taken a t land-based s t a t i o n s a r e not n e c e s s a r i l y r e p r e s e n t a t i v e o f c o n d i t i o n s a t sea.  One  c o u l d attempt t o e v a l u a t e  the  wind s t r e s s c u r l from atmospheric s u r f a c e - p r e s s u r e maps, but i t i s a t e d i o u s e x e r c i s e and ensuing wavelike  i s not pursued here.  We  note as b e f o r e , however, t h a t  a n a l y s i s i m p l i e s t h a t the low-frequency c u r r e n t s are not motions o f the type  studied i n this thesis.  In t h i s  simple  respect,  t h e r e f o r e , i t i s u n l i k e l y the wind s t r e s s c u r l p l a y s a s i g n i f i c a n t F i g u r e 12.5 for  the 500-day p e r i o d b e g i n n i n g  a r e l a t i v e l y f l a t region.  4 A p r i l 1969.  Sand Heads i s l o c a t e d i n  influences.  u s i n g a q u a d r a t i c law w i t h a v a l u e of exact v a l u e  (see F i g u r e 12.1)  adjacent  Thus winds measured t h e r e s h o u l d be  t i v e l y f r e e of l o c a l t o p o g r a p h i c  Its  role.  shows the spectrum of the wind s t r e s s a t Sand Heads  shallow water a t the mouth o f the F r a s e r R i v e r to  the  1.5  compara-  The wind s t r e s s e s were computed  * 10  f o r the drag  coefficient.  i s unimportant i n t h i s d i s c u s s i o n s i n c e i t e n t e r s o n l y as a  scale factor. F i g u r e 12.5A the wind s t r e s s . i s present  shows the t r u e northward and  eastward components of  Both a r e peaked a t about 3 days, but s i g n i f i c a n t  to p e r i o d s up t o a t l e a s t 20 days.  d i r e c t e d approximately  50° west o f n o r t h , and  are shown i n F i g u r e 12.5B.  The  mean a x i s o f GS  variance is  the s p e c t r a r o t a t e d by  50°  S i n c e s i g n i f i c a n t l y h i g h e r coherences were  found i n t e s t runs u s i n g the r o t a t e d wind s t r e s s time s e r i e s , they were employed i n the f o l l o w i n g a n a l y s i s . l i n e H,  the o t h e r hand, i n the v i c i n i t y  the topography runs n e a r l y n o r t h - s o u t h .  were not r o t a t e d . and  On  nonrotated  Therefore  the  of  currents  A l l subsequent f i g u r e s r e f e r to the r o t a t e d wind s t r e s s  currents.  126  PERIOD (days) 50 25 10 5 2 1  10  (A)  IE-  .—i-r  10V  if  10  O - eastward A - northward  i i i mill  10»-5 3.  1 0  2  1  p — i  1—i  i i i mill r  (B)  :  A  10"'r  ,  :  Y  ia !  Figure  12.5  O-cross-strait A-along-strait  • i«""il  i 11 nml  10"2 10"' 10' FREQ (cpd)  Spectrum o f the wind s t r e s s a t Sand Heads f o r the 500-day p e r i o d b e g i n n i n g 4 A p r i l 1969: (A) the n o r t h and e a s t wind s t r e s s components; (B) the wind s t r e s s components r o t a t e d a n t i - c l o c k w i s e by 50°.  127  The  209-day p e r i o d b e g i n n i n g  29 August 1969  was  a n a l y s i s , as r e c o r d s e x i s t e d a t a l l s t a t i o n s along l i n e H  selected for ( F i g u r e 12.2).  Large gaps i n the time s e r i e s were d e l e t e d from good as w e l l as bad  records  and the time s e r i e s were l i n e a r l y i n t e r p o l a t e d a c r o s s s h o r t gaps o f the o r d e r o f a day o r two.  The  r e s u l t i n g r e c o r d was  136 days i n l e n g t h .  C u r r e n t s p e c t r a computed from these time s e r i e s are shown i n F i g u r e 12.6; F i g u r e 12.5. c u r r e n t s and  the spectrum Two  o f the wind s t r e s s i s e s s e n t i a l l y t h a t shown i n  exemplary p l o t s o f coherence  and phase between the  the a l o n g - s t r a i t wind s t r e s s are shown i n F i g u r e 12.7.  i l l u s t r a t e the g e n e r a l l y low coherence i n c r e a s i n g frequency and the tendency  observed which decreases  They  with  f o r the c u r r e n t s t o be e i t h e r i n  phase o r 180° out of phase w i t h the a l o n g - s t r a i t component o f the wind A more d e t a i l e d p r e s e n t a t i o n o f the coherence s h i p s i s shown i n F i g u r e 12.8 s i g n i f i c a n t coherences  f o r the 34- and  and phase  13-day bands.  stress.  relation-  Statistically  are found i n many c a s e s , although i n some, due  r e s p e c t must be p a i d t o the amount o f energy  i n the g i v e n s i g n a l .  Thus,  2 f o r example, the meaning o f h i g h v a l u e of c a l c u l a t e d f o r H06, the spectrum  there  200 m,  i s u n c l e a r due  ( F i g u r e 12.6).  y  = .6  i n the 34-day band  t o the c o r r e s p o n d i n g  low v a l u e o f  The most s t r i k i n g f e a t u r e , however, i s  the c o n s i s t e n c y w i t h which the phase e s t i m a t e s c l u s t e r about e i t h e r 0° o r 180°.  T h i s tendency  t o g e t h e r w i t h the f a c t t h a t phase d e t e r m i n a t i o n may  good even though the c o r r e s p o n d i n g coherences Diiing, 1976)  instills  are i n s i g n i f i c a n t  be  (Schott and  some degree o f c o n f i d e n c e i n the c a l c u l a t e d phases.  More s p e c i f i c a l l y , phases between c u r r e n t s and the a l o n g - s t r a i t wind s t r e s s tend t o be c l o s e t o 180°, w h i l e those between the c u r r e n t s and the c r o s s channel wind s t r e s s l i e near 0 ° .  The a l o n g - s t r a i t wind s t r e s s i s , o f c o u r s e ,  c o n s i d e r a b l y more e n e r g e t i c than the c r o s s - c h a n n e l component.  128  PERIOD ( D A Y S ) 34 13 7 3  1  34 13 7  3  1  34 13 7 3  H16  50 m  A-northward O - eastward  I I I I itlll  w 10  I I I I Mil  E—r H06 200 m  4  10 l  10"  10  h  10  4>  -2  • • • ""I  101-1  » i • u m  10° 1 0"2  10T1 FREQ  F i g u r e 12.6  10° 1  (cpd)  L i n e H c u r r e n t s p e c t r a f o r the 136-day analysis period. The v e r t i c a l bars i n d i c a t e 95% c o n f i d e n c e l i m i t s .  1  PERIOD (DAYS) 13. 6.5 2.8  1.4  PERIOD (DAYS) 2.8 13. 6.5 34.  0.6  i  i  0.5  i  HI6 50m  i  •-u •V  0.4 0.3  1.4  _  •  •  0.2 0.1 0 -2.0  -1.0 log (freq / I cpd)  i t -* -1.0 log ( f r e q / I cpd) A  270  T  180  F i g u r e 12.7  Coherence and phase between the wind s t r e s s and c u r r e n t s a t H26, 50 m and H16, 50 m. A p o s i t i v e phase i n d i c a t e s the c u r r e n t l e a d s the wind. Here u and v r e f e r t o the eastward and northward v e l o c i t y components. The s o l i d  line indicates  the 95% noise  level.  o.c  130  0.6  O  U-TX  • u-ry 0.5 L A v - r y  , A v-rx  UPPER LAYER  13 DAYS  0.6 1  0.5  -  LOWER ' 13 DAYS LAYER  O UJ  cr <  0.4  U)  0.3  z> o  UJ o LxJ  A  0.3  A  0.2  o °  1  -  O  A  A  1  •  •  1  n  0.1 r  0.1  • 1  0.0  H06  H 16  H26  0.0  270  270  180  180  -  H16  H06  I  #  A  H26  •  A  I  o  A  A  a  UJ Q UJ  90  90 h  to  < I 0_  0  •90  F i g u r e 12.8A  O  A  O  IO  .  A  •90  Coherence and phase between l i n e H c u r r e n t s and the wind s t r e s s for: (A) the 13-day band, and (B) the 34-day band. The s o l i d l i n e s i n the coherence p l o t s i n d i c a t e the 95% n o i s e l e v e l s . A p o s i t i v e phase i n d i c a t e s the c u r r e n t l e a d s the wind.  131  0.6 O  0.5  U - T X  •  U - T V  A  V  A  -  T  X  v-ry  UPPER LAYER  34 DAYS  0.6  LOWER LAYER  0.5  34 DAYS  Q L U  < ZD  cr  0.4  -  •  a  in L U  0.3  A  -  I-  o L U  cr  0.2  0.2  ,  A  L U  •  X  o °  0.1  0.1 h A_  0.0 H06  H16  H26  -©•A.  180  o L U  O  I  H06  270  270  180  0.0  •  90  90 h  L U  < X  0_  •90  •90  Figure  12.8B  -e—A-  A  O  A  O  H16  H26  .  132  These phase r e s u l t s should not be  i n t e r p r e t e d t o mean, f o r example,  t h a t the c u r r e n t s flow down-channel when the winds blow up-channel. imply  simply  They  t h a t the c u r r e n t s are i n o p p o s i t i o n to the g i v e n s p e c t r a l  component o f the a l o n g - s t r a i t wind.  The  r e l a t i o n s h i p i n the time domain  between the c u r r e n t s and winds w i l l be examined s h o r t l y . Table  III l i s t s  the coherences and phases c a l c u l a t e d between  v e l o c i t y components f o r the 13- and  34-day bands.  o f the types of waves s t u d i e d here,  then the phase d i f f e r e n c e between  and  v  should be somewhere near 90°.  caused by 4).  friction  (see, e.g.,  Deviations  Csanady, 1978)  With the e x c e p t i o n o f the v a l u e o f  days, examination o f T a b l e found, the c o r r e s p o n d i n g  be b a r o t r o p i c .  We  On  34  t h a t i s , the exception,  f o r which the p r e v i o u s  evidence  the o t h e r hand, the r e s u l t s f o r  (1976).  c h a r a c t e r and  t h i s s p e c u l a t i o n , and  type of  the motion i s j u s t as l i k e l y  note, however, t h a t CSWs of 34-day p e r i o d s h o u l d  mean c u r r e n t s and  34  i t i s enticing  There i s no d i r e c t evidence,  wavelengths i n excess of the l e n g t h o f GS The  (see S e c t i o n  140 m a t  t h a t t h i s might be a bottom-enhanced wave of the  t o support  be  T h i s i s t r u e , i n p a r t i c u l a r , f o r the  do suggest a wavelike  d e s c r i b e d by H e l b i g and Mysak course,  f o r H26,  T h i s i m p l i e s , w i t h the noted  s t r o n g l y suggested the c o n t r a r y .  to s p e c u l a t e  or by i n s t a b i l i t y  <J> = 117°.  observed 13-day c u r r e n t o s c i l l a t i o n s a t H26  m,  from t h i s v a l u e may  phase i s e i t h e r near 0° or 180°,  t h a t the motion i s not w a v e l i k e .  140  u  I I I shows t h a t i f a s i g n i f i c a n t coherence i s  motion i s l i n e a r l y p o l a r i z e d .  days f o r H26,  I f a motion i s composed  (see, e.g.,  current e l l i p s e s 12.10.  Figure  f o r the  13-  to have  11.5). and  34-day  bands are shown i n F i g u r e s 12.9  and  While the mean flow i s s i m i l a r  to t h a t c a l c u l a t e d f o r the f u l l  18 months ( F i g u r e 10.6), the deep c u r r e n t s  i n both the e a s t and west are c o n s i d e r a b l y s t r o n g e r .  Indeed, a t H06  the  133  T a b l e I I I . C a l c u l a t e d coherence squared and phase between v e l o c i t y components f o r the 136-day p e r i o d o f a n a l y s i s . A p o s i t i v e phase i n d i c a t e s t h a t v leads u.  13 days  Station  H06  H16  H26  Depth  Y  2  34 days  <j) (deg)  Y  2  <f> (deg)  50  .36  - 4  .08  52  200  .24  - 11  .10  50  .01  3  .05  39  200  .29  179  .69  179  50  .37  5  .18  177  140  .51  2  .48  117  -  178  134  upper- and l o w e r - l a y e r mean flows are n e a r l y the same. bear a s t r i k i n g resemblance  The c u r r e n t  to the r e s p e c t i v e mean v e l o c i t i e s .  t r e n d s d u r i n g the p e r i o d o f a n a l y s i s may  contaminate  ellipses  Although  the 34-day band  i s averaged over the second t o s i x t h f r e q u e n c i e s ) , they should e x e r t i n f l u e n c e on the 1 3 - d a y band These r e s u l t s thus may fluctuating  (which i s averaged over f r e q u e n c i e s  minor  7-14).  imply a dynamical r e l a t i o n s h i p between the mean and  f l o w s , a p o i n t which  i s d i s c u s s e d f u r t h e r i n the next  I t i s e v i d e n t from the f i g u r e s  section.  t h a t the channel b o u n d a r i e s e x e r t a s t r o n g  t o p o g r a p h i c i n f l u e n c e on the near-shore Approximate  (which  currents.  b a r o t r o p i c and b a r o c l i n i c time s e r i e s were formed  depth a v e r a g i n g the 1 3 6 - d a y r e c o r d s .  Indeed, the p e r i o d of a n a l y s i s  the treatment o f the data r e c o r d s were s e l e c t e d f o r t h i s purpose.  by  and  The  v e l o c i t y time s e r i e s were combined as  Hfi  =  (  T  h  l^l  +  h  2^2  ) / ( h  l  +  V  and  HBC =  h  2 ^l" U2 (  to g i v e b a r o t r o p i c respectively.  (HBT^  ) / ( h  a n <  l  +  V  3 upper-layer b a r o c l i n i c  The u p p e r - l a y e r depth  h-^  was  ^Hgc)  records,  chosen as 50 m s i n c e  this  corresponds t o a r e a s o n a b l e f i t o f a two-layer model t o the observed density d i s t r i b u t i o n was  ( H e l b i g and Mysak,  1976).  The  l o w e r - l a y e r depth  then simply o b t a i n e d from the t o t a l depth a t each mooring.  vertical velocity profile  i s , i n r e a l i t y , much more complex.  h  The I f a greater  number o f meters had been used a t each s t a t i o n , the method o f e m p i r i c a l o r t h o g o n a l f u n c t i o n s c o u l d have been employed t o r e s o l v e the v e r t i c a l structure  (see, e.g., Mooers and Brooks, 1 9 7 8 ) .  2  No o t h e r c h o i c e s o f  h^  F i g u r e 12.9  Mean c u r r e n t s a l o n g analysis period.  l i n e H f o r the  136-day  F i g u r e 12.10A  L i n e H c u r r e n t e l l i p s e s f o r the 13-day band: (A) upper l a y e r , (B) lower l a y e r ; and the 34-day band: (C) upper l a y e r , (D) lower l a y e r .  F i g u r e 12.10B  138  F i g u r e 12.IOC  F i g u r e 12.10D  140  were t r i e d . Each time s e r i e s was  s p e c t r a l l y analysed,  but the b a r o t r o p i c  b a r o c l i n i c s p e c t r a were almost i n d i s t i n g u i s h a b l e from the lower- and layer spectra, respectively.  The  c o r r e l a t i o n between the wind and  and  upper-  r e s u l t s were i n c o n c l u s i v e w i t h r e g a r d currents.  to  In some cases h i g h e r v a l u e s  of  2  Y  were found but i n o t h e r s the c o r r e l a t i o n was  to determine, t h e r e f o r e , i f the s e p a r a t i o n was  diminished.  It i s d i f f i c u l t  successful.  The b a r o t r o p i c and b a r o c l i n i c mean c u r r e n t s f o r the 136-day p e r i o d are shown i n F i g u r e 12.11.  While the b a r o t r o p i c means are s i m i l a r t o  l o w e r - l a y e r means of F i g u r e 12.9,  the b a r o c l i n i c means are l e s s  of a g y r e l i k e c i r c u l a t i o n than are the u p p e r - l a y e r One  c u r r e n t meter, t h a t a t H16,  over the 18-month p e r i o d . and winds b e g i n n i n g  17 A p r i l  1969  to t h a t shown i n F i g u r e 12.6.  was  means.  operated  almost  analysed  13-day bands.  The  f o r comparison w i t h  from those  spectra  shown i n F i g u r e 12.8  similar  for  I n t e r e s t i n g l y , i n the former band, the  computed phase d i f f e r e n c e s were s i m i l a r , and v  currents  c u r r e n t spectrum i s v e r y  coherence between the v e l o c i t y components i n c r e a s e d from 0.05  and  continuously  In g e n e r a l , the c a l c u l a t e d coherences between  the c u r r e n t s and winds are decreased  The  suggestive  Consequently, the 500-day time s e r i e s of  computed from s h o r t e r r e c o r d l e n g t h s .  both the 34-~ and  50 m,  the  components were n e a r l y 180°  to  0.23.  i n p a r t i c u l a r b o t h the  out of phase w i t h the wind s t r e s s .  In o r d e r t o o b t a i n an a p p r e c i a t i o n i n the time domain of how water column responds t o the wind, the two-month p e r i o d o f M a r c h - A p r i l was  s e l e c t e d f o r more i n t e n s i v e study.  reasons. period.  chosen f o r  two  F i r s t , r e c o r d s were a v a i l a b l e f o r a l l meters f o r most o f  the  T h i s p e r i o d was  Second, d u r i n g t h i s time s e v e r a l s i g n i f i c a n t storms o c c u r r e d ,  w i t h northwest and  u  some w i t h s o u t h e a s t  winds.  The  time s e r i e s o f winds  the 1970  some and  141  Figure  12.11  Computed b a r o t r o p i c and upper l a y e r b a r o c l i n i c mean c u r r e n t s f o r the 136-day a n a l y s i s p e r i o d .  142  2 c u r r e n t s were f i l t e r e d w i t h an  2  A ^ A ,j/(.24 2  • 25)  2  filter  i n o r d e r t o remove d i u r n a l and s e m i d i u r n a l o s c i l l a t i o n s . filter  and produces a r e c o r d w i t h a twelve-hour time s t e p .  shown i n F i g u r e 12.12.  (Godin, 1972)  T h i s i s a low-pass The r e s u l t s a r e  N o t i c e t h a t the wind has been advanced f i v e days  with respect t o the currents. b r i n g i t i n t o alignment w i t h  As b e f o r e , the wind s t r e s s was r o t a t e d t o channel geometry.  Seven wind events a r e i d e n t i f i e d i n F i g u r e 12.12; peak v a l u e s o f the wind s t r e s s occur f o r  (down-strait)  and  E^  (up-strait).  F o r the  50-metre r e c o r d a t H26, the s i g n a t u r e o f the wind on t h e c u r r e n t s i s c l e a r , and at  t h e c u r r e n t s l a g t h e wind by about f i v e days.  S i m i l a r l y , the response  the 140-m s t a t i o n i s apparent f o r the f i r s t month.  however, some ambiguity e x i s t s i n the assignment o f  F o r the second month, E^ - E^ .  I f the  c h o i c e i n d i c a t e d i s c o r r e c t , then a d o w n - s t r a i t  wind does n o t n e c e s s a r i l y  produce a down-channel c u r r e n t  E^ , E^ ,  i s opposite  (compare events  t o the response observed a t 50 m.  and  E ^ ) . This  I t i s not p o s s i b l e to  d e f i n i t i v e l y c o r r e l a t e c u r r e n t s and winds a t the c e n t r a l 50-m s t a t i o n , b u t the c o r r e l a t i o n a t 200 m i s c l e a r - c u t , a g a i n w i t h a 5-day l a g .  I t i s also  not p o s s i b l e t o make the assignment a t H06, 50 m f o r the one-month r e c o r d that exists.  As was the case f o r the deep e a s t e r n meter, c o r r e l a t i o n o f  winds and c u r r e n t s a t H06, 200 m, i s ambiguous i n the second month.  Unlike  the e a s t e r n s t a t i o n , however, the chosen assignment i n d i c a t e s d i r e c t  response  to  the wind w i t h a 9-day l a g .  In the f i r s t month the l a g i s about 7 days.  In an attempt t o determine i f the response t o t h e winds observed at  50 m i s r e p r e s e n t a t i v e o f t h e e n t i r e upper water column, p r o g r e s s i v e  v e c t o r diagrams o f the c u r r e n t s a t 3 m and 50 m [Tabata and S t i c k l a n d , 1972a; 1972b; 1972c; Tabata e t a l . , 1971] were compared w i t h one another and  the wind f o r the p e r i o d b e g i n n i n g  A p r i l 1970 ( F i g u r e 12.2).  143  Figure 12.12  Low-pass f i l t e r e d time series of wind stress at Sand Heads and currents along l i n e H. A s o l i d l i n e represents either the along-strait (northwestward) component of wind stress or the northward component of current. A dashed l i n e represents either the cross-channel wind stress (northwestward) or eastward current component. The wind stress time series i s advanced by 5 days.  144  At a l l s t a t i o n s t h e r e were times d u r i n g were o b v i o u s l y  o t h e r o r the wind.  which the c u r r e n t s  But  were o u t o f phase w i t h each  S i m i l a r l y , the h o r i z o n t a l r e l a t i o n s h i p s between the 3-m  were u n c l e a r .  However, t h e c u r r e n t s  h i g h l y c o r r e l a t e d and i n phase. currents,  a t both l e v e l s  c o r r e l a t e d and i n phase w i t h each o t h e r and the wind.  t h e r e were a l s o times d u r i n g  currents  which the c u r r e n t s  a t H06 and H16 were, a t times,  T h e r e f o r e , by comparison w i t h the 3-m  i t i s d i f f i c u l t t o s t a t e i f t h e measurements a t 50 m a r e  representative  o f the e n t i r e upper l a y e r .  In summary, examination o f the d a t a has i n d i c a t e d s e v e r a l i n t e r e s t ing features.  First,  the low-frequency f l u c t u a t i o n s a r e n o t i s o l a t e d t o  the v i c i n i t y o f l i n e H o r the southern s t r a i t . spectra  Second, as suggested by the  and response t o t h e wind, S t a t i o n H26 may l i e i n an oceanographic  domain d i s t i n c t from the o t h e r two s t a t i o n s .  T h i r d , the o s c i l l a t i n g  currents  bear a resemblance t o the mean flow which may i n d i c a t e t h a t the two a r e dynamically l i n k e d . are p o s s i b l e :  I f this supposition  i s v a l i d , then t h r e e  alternatives  (1) the f l u c t u a t i o n s a r e due t o the mean c u r r e n t s  (inertial  i n s t a b i l i t y ) , (2) t h e mean f l o w i s a byproduct o f the o s c i l l a t i o n s ( t r a n s i e n t s , a r r e s t e d waves; s e e , e.g., Csanady, 1978), o r (3) they a r e both caused by some o t h e r agency o f the type o u t l i n e d i n S e c t i o n  10.  p o s s i b i l i t y may be r u l e d o u t on the b a s i s o f r e s u l t s o f S e c t i o n f a c t t h a t the components o f the observed o s c i l l a t i n g c u r r e n t s phase.  The l a s t a l t e r n a t i v e i s e x p l o r e d b r i e f l y  F i n a l l y , the wind o b v i o u s l y not  clear.  plays  i n the next  The f i r s t 11 and the  tend t o be i n section.  some r o l e i n GS dynamics b u t i t s r o l e i s  145  13.  Nonlinear  T i d a l Interactions  In an e a r l i e r study, H e l b i g and Mysak  (1976) d i s c o u n t e d  the  p o s s i b i l i t y t h a t the t i d e s were r e s p o n s i b l e  f o r the low-frequency motions  i n GS.  fortnightly  However, they were r e f e r r i n g t o the  not c o n s i d e r  the p o s s i b i l i t y of n o n l i n e a r  M^  t i d e and  i n t e r a c t i o n s between  did  tidal  constituents.  In a system l i k e GS w i t h l a r g e v a r i a t i o n s i n bottom  topography and  channel geometry, i t i s l i k e l y t h a t such i n t e r a c t i o n s produce  s i g n i f i c a n t f o r t n i g h t l y v a r i a t i o n s i n the t i d e . between t i d a l c o n s t i t u e n t s  These i n t e r a c t i o n s o c c u r  through f r i c t i o n a l or a d v e c t i v e  terms, and  r e s u l t a n t o s c i l l a t i o n s are known as shallow-water c o n s t i t u e n t s . harmonic c o n s t a n t s constituents  for d i u r n a l , semidiurnal,  are l i s t e d  a 38-day r e c o r d o f t i d a l h e i g h t s Figure  12.1).  i n t e r a c t i o n s between the 0-^  and  P-^  M  2  and  given K  2  , M  i n T a b l e IV, one 2  and  S  2  , 0-^  (see, e.g. , may  and  show t h a t K-^  c o n s t i t u e n t s a l l produce shallow-water c o n s t i t u e n t s  fortnightly period.  F o r example, the  M  2  - S  2  c o n s t i t u e n t w i t h a 14.76-day p e r i o d , w h i l e the  tidal  from the a n a l y s i s o f  observed a t P o i n t A t k i n s o n  From the f r e q u e n c i e s  The  and h i g h e r - f r e q u e n c y  i n T a b l e IV and were o b t a i n e d  the  ,  and  of  i n t e r a c t i o n g i v e s the O-^ - K-^  MS^  interaction results  i n a 13.66-day o s c i l l a t i o n . In t h i s s e c t i o n two  types o f t i d a l i n t e r a c t i o n s are  The  first  I term d i r e c t , n o n l i n e a r  The  second o r i n d i r e c t , n o n l i n e a r  of the t i d e w i t h another agency.  i n t e r a c t i o n and  considered.  i s that just outlined.  i n t e r a c t i o n c o n s i s t s of the i n t e r a c t i o n In p a r t i c u l a r , I s p e c u l a t e  a c t i o n of the t i d e w i t h the F r a s e r R i v e r  upon the  inter-  outflow.  To determine the s i g n i f i c a n c e o f the f i r s t mechanism, r e s u l t s generated from the Department o f the Environment n u m e r i c a l t i d a l model of  146  T a b l e IV. R e s u l t s o f t h e harmonic a n a l y s i s o f t i d a l e l e v a t i o n s a t P o i n t A t k i n s o n f o r t h e 38-day p e r i o d b e g i n n i n g 6 A p r i l 1976. The P^ and S - L c o n s t i t u e n t s a r e i n f e r r e d from , Nu^, i s i n f e r r e d from N , T and K are inferred from S" . (Dr. J . A. S t r o n a c h , p r i v a t e communication.) 2  2  2  2  Constituent No.  Name  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38  ZO  2Q1 Ql 01 N01 PI SI Kl Jl 001 MNS2 MU2 N2 NU2 M2 L2 T2 S2 K2 2SM2 M0 3 M3 MK3 SK3 MN4 M4 SN4 MS 4 S4 2MN6 M6 MSN6 2MS6 2SM6 3MN8 M8 3MS8 M12  Frequency  (cpd)  0.0 0.85695237 0.89324397 0.92953563 0.96644622 0.99726212 1.00000000 1.00273705 1.03902912 1.07594013 1.82825470 1.86454678 1.89598083 1.90083885 1.93227291 1.96856499 1.99726295 1.99999905 2.00547504 2.06772518 2.86180973 2.89841080 2.93500996 3.00273800 3.82825470 3.86454678 3.89598179 3.93227291 4.00000000 5.76052761 5.79681969 5.82825565 5.86454582 5.93227386 7.69280148 7.72909451 7.79681969 11.59364128  Amplitude  (cm) Greenwich phase (deg) 30.2087 0.1155 0.6998 3.9706 0.4666 2.8014 0.3242 7.9255 0.4867 0.1736 0.1048 0.3987 2.0722 0.3795 9.3629 0.2994 0.1426 2.2687 0.4967 0.0464 0.0132 0.0187 0.0152 0.0095 0.0106 0.0353 0.0083 0.0283 0.0093 0.0569 0.0709 0.0220 0.0826 0.0209 0.0086 0.0177 0.0077 0.0087  0.0 96.73 326.92 215.33 276.69 347.70 154.08 95.47 359.38 221.21 322.62 317.05 249.94 228.27 148.55 262.76 24.32 298.19 220.37 1.47 137.41 247.17 87.29 203.48 276.20 139.86 281.85 341.27 291.57 146.10 33.69 266.26 198.62 334.93 51.41 97.38 327.90 99.16  147  the Juan de F u c a - S t r a i t of G e o r g i a system  (Crean, 1976;  1978)  were examined.  T h i s i s a two-dimensional, v e r t i c a l l y i n t e g r a t e d model u t i l i z i n g an f o r w a r d - s t e p p i n g , f i n i t e d i f f e r e n c e scheme; a d j o i n i n g i n l e t s and passages t o the open ocean are s i m u l a t e d the l a t e s t v e r s i o n , a 2-km  obtained  2 4  first  • 5)  5  A  filter 2  ^25^  2 4  (Godin, 1972) 2 *  and  To determine  the  smoothed t o one  then low-passed  hour  By r e s i d u a l , we  exceeding 0.8  r e f e r to the remaining low-frequency components.  o f d a t a were l o s t i n the a p p l i c a t i o n o f the two time s e r i e s .  The  r e s i d u a l flow along  filtered  f i l t e r previously described  e f f e c t i v e l y eliminates o s c i l l a t i o n s with frequencies  line  H  filters  which  cpd. Three days  l e a v i n g a 10-day  i s indicated i n Figure  13.1  3 s e p a r a t e days; the average flow over the 10-day p e r i o d i s shown i n  Figure H26,  are  2 A /(4  to 12 hours w i t h the  for  by  e l e v a t i o n s w i t h a time  been generated from t h i s model.  r e s i d u a l c u r r e n t s , the v e l o c i t y time s e r i e s was A  system i s d r i v e n  tide.  A 13-day time s e r i e s o f v e l o c i t i e s and  w i t h an  The  In  the open b o u n d a r i e s ; these e l e v a t i o n s  from a 61 harmonic c o n s t i t u e n t  step o f 15 minutes has  northern  as one-dimensional c h a n n e l s .  mesh s i z e i s employed.  s p e c i f y i n g t i d a l e l e v a t i o n s along  explicit,  13.2.  H16  The  and H06  currents  c a l c u l a t e d f o r the g r i d s encompassing s t a t i o n s  are i l l u s t r a t e d  i n Figure  13.3.  The  range based on the p r e d i c t e d t i d e s f o r P o i n t A t k i n s o n Figure  13.3 It  e x i s t s , and  t i d a l elevation  and  are a l s o shown i n  f o r the p e r i o d o f the a n a l y s i s . i s evident  t h a t a coherent p a t t e r n of r e s i d u a l c i r c u l a t i o n  t h a t i t i s dependent upon the t i d a l range.  t h a t the s t r o n g e s t Unfortunately,  r e s i d u a l flows o c c u r near the e a s t e r n  It i s also clear boundary.  the time s e r i e s i s i n s u f f i c i e n t l y long to f u l l y r e s o l v e  f o r t n i g h t l y v a r i a t i o n , and  i t i s p o s s i b l e t h a t the very  r i n g a t the b e g i n n i n g o f the a n a l y s i s p e r i o d may  be  strong  transients  a  flows o c c u r associated  J  i  l • — _ i — i —  0 H06  2 cms" MARCH 13  d  i  i  F i g u r e 13.IA  D a i l y , b a r o t r o p i c , r e s i d u a l t i d a l flow along l i n e  H. co  ?  i  i  i  i  0  .4  i  2 cms" MARCH 17  H 06  H 16  in  G A| L I A N 0  _TL VANCOUVER  I S L A| N D  FRASER RIVER  Figure  13.IB  i  0  i  i  i  i  2 cms MARCH 21  H 06 '  /  HI6  G A L I A N 0  H 26  VANCOUVER  I S L A N D  FRASER  Figure  13.1C  RIVER  7 ?  _ i — i — i  2cms AVERAGE  H06  FLOW  d  HI6  VANCOUVER  F R A S E R RIVER  F i g u r e 13.2  The r e s i d u a l b a r o t r o p i c , t i d a l flow averaged over 10 days.  152  5.0  MARCH  F i g u r e 13.3A  (1973)  Time s e r i e s o f p r e d i c t e d (A) t i d a l h e i g h t and ' t i d a l range a t P t . A t k i n s o n arid c a l c u l a t e d (B) r e s i d u a l c u r r e n t magnitude and d i r e c t i o n along l i n e H.  153  F i g u r e 13.3B  154  w i t h s t a r t i n g the model from an i n i t i a l  state of rest.  However, the model  was r u n f o r two ( t i d a l ) days p r i o r t o the 13-day p e r i o d i n o r d e r t o a v o i d t h i s problem.  A t any r a t e , i t i s c l e a r t h a t the r e s i d u a l motions a r e o f  i n s u f f i c i e n t s t r e n g t h and improper d i r e c t i o n t o serve as an e x p l a n a t i o n o f the observed low-frequency motion along  l i n e H.  On the o t h e r hand, n o n l i n e a r t i d a l i n t e r a c t i o n s may be  important  i n o t h e r ways, f o r example, i n the g e n e r a t i o n o f i n t e r n a l t i d e s o r i n i n t e r a c t i o n with the Fraser River.  While the r e s i d u a l flow does n o t r e s o l v e  the p r e s e n t problem, i t c l e a r l y m e r i t s f u r t h e r i n v e s t i g a t i o n . it  i s l i k e l y t o be s i g n i f i c a n t i n the southern  t i d a l passes  In p a r t i c u l a r ,  (Crean, 1978;  F i g u r e s 12 and 13). F i n a l l y , I s p e c u l a t e on the p o s s i b i l i t y t h a t the t i d e i n t e r a c t s n o n l i n e a r l y w i t h the F r a s e r R i v e r o u t f l o w low-frequency c u r r e n t s .  t o produce, i n p a r t , the observed  An examination o f the F r a s e r R i v e r d i s c h a r g e a t  some d i s t a n c e upstream from the mouth i n d i c a t e s no c o n s i s t e n t f o r t n i g h t l y o r monthly v a r i a t i o n s (Figure 13.4); t h e d i s c h a r g e annual peak t h a t occurs snowpack. indeed  i s dominated by the l a r g e  i n l a t e s p r i n g and i s due t o the m e l t i n g o f the  However, near t h e r i v e r mouth the t i d e modulates t h e r i v e r  the r e g i o n comprises a salt-wedge-type e s t u a r y .  flow,  I f the magnitude  o f t h i s i n t e r a c t i o n v a r i e s w i t h t i d a l range, then i t i s p o s s i b l e t h a t motions t h a t a r e d r i v e n by the p r e s s u r e  g r a d i e n t due t o F r a s e r R i v e r water  above GS water may v a r y w i t h a f o r t n i g h t l y p e r i o d . expressed  The h y p o t h e s i s  lying as  i s o b v i o u s l y crude and i g n o r e s e f f e c t s due t o d e n s i t y d i f f e r e n c e s ,  f o r example, b u t i t i s o f f e r e d as a s p e c u l a t i v e p o s s i b i l i t y t h a t c o u l d be examined i n the f u t u r e . 13.5  However, t h e r e i s some evidence  fori t .  Figure  shows the low-frequency r i v e r speed a t the mouth o b t a i n e d by low-pass  f i l t e r i n g current records.  Unfortunately,  i t i s superimposed on an i n c r e a s i n g  FRASER RIVER DISCHARGE AT AGASSIZ  F i g u r e 13.4  The F r a s e r R i v e r d i s c h a r g e approximately 60 m i l e s upstream a t A g a s s i z , B r i t i s h Columbia (from Chang, 1976).  F i g u r e 13.5  Low-pass f i l t e r e d time s e r i e s o f r i v e r speed a t the F r a s e r R i v e r mouth ( Stronach, 1977)  t— 1  157  d i s c h a r g e due  to the o n s e t o f f r e s h e t .  Nevertheless, a s i g n i f i c a n t  v a r i a t i o n i n the speed i s e v i d e n t i n the f i r s t a d d i t i o n , the a n a l y s i s o f Chang mean c u r r e n t d i r e c t i o n a t H26  12 days o f the r e c o r d .  (1976; F i g u r e s 40 and 42)  suggests  In  t h a t the  t u r n s to the south i n p e r i o d s o f h i g h - r i v e r  runoff. T h i s s p e c u l a t i o n has  the advantage of p r e d i c t i n g t h a t the most  s i g n i f i c a n t flow should occur a l o n g the e a s t e r n boundary s i n c e the p r e s s u r e head due  to the r i v e r s h o u l d be l o c a l i z e d t h e r e .  t h i s mechanism might be o f f s e t .  In a p e r i o d o f h i g h r u n o f f  Moreover, the t h e o r y a l l o w s f o r i n t e r a c t i o n  w i t h the winds which would serve t o modify the o u t f l o w .  It i s clear,  however, t h a t t h i s h y p o t h e s i s must be p a r t of a more encompassing t h e o r y o f the modulation system. mixing  o f the e s t u a r i n e flow i n the Juan de F u c a - S t r a i t of  Other e f f e c t s such as the i n f l u e n c e o f the s t r e n g t h o f i n the southern  t i d a l passes must be examined.  has y e t to be c o n s i d e r e d .  Moreover,  Georgia  tidal friction  I t i s hoped, however, t h a t the p r e s e n t work w i l l  s t i m u l a t e n u m e r i c a l modelers to work on t h i s system and t o examine not o n l y s h o r t - p e r i o d e f f e c t s b u t a l s o l o n g e r p e r i o d ones.  158  14.  Summary of P a r t I I  From the a n a l y s i s o f c u r r e n t and wind d a t a taken i n the S t r a i t o f G e o r g i a and from the c o n s i d e r a t i o n o f simple i n e r t i a l i n s t a b i l i t y  models,  the f o l l o w i n g c o n c l u s i o n s may be drawn from P a r t I I . 1. motions.  The observed f l u c t u a t i o n s a r e n o t due t o simple w a v e l i k e  T h a t i s , they a r e n o t composed o f f r e e , f o r c e d , o r u n s t a b l e  p l a n e waves o f the type c o n s i d e r e d i n t h i s t h e s i s .  This conclusion i s  based on the f i n d i n g t h a t the components o f the f l u c t u a t i n g c u r r e n t s tend to be i n phase. 2.  The o s c i l l a t i o n s may bear some dynamical r e l a t i o n s h i p t o  the mean c u r r e n t s .  T h i s n o t i o n i s based on the g e n e r a l resemblance o f  the mean and f l u c t u a t i n g 3.  currents.  As e v i d e n c e d from the s p e c t r a and t h e response t o t h e wind,  the e a s t e r n s t a t i o n may respond t o f o r c i n g d i f f e r e n t l y than the c e n t r a l and western  stations.  4. currents.  The wind p l a y s some r o l e i n d e t e r m i n i n g the low-frequency  T h i s i s suggested by the f a c t s t h a t  (a) s t a t i s t i c a l l y  signif-  i c a n t a l t h o u g h s m a l l coherences a r e c a l c u l a t e d between the c u r r e n t s and winds,  (b) t h a t the c o r r e s p o n d i n g phases c o n s i s t e n t l y l i e near 0° o r  180°, and (c) i n the time domain, the response o f the water column t o a wind event i s o f t e n e v i d e n t . 5.  B a r o c l i n i c i n s t a b i l i t y of the mean flow i s an u n l i k e l y  mechanism i n GS due t o t h e narrow r e g i o n o f i n s t a b i l i t y 6.  i n parameter  space.  A b a r o t r o p i c i n s t a b i l i t y model i n d i c a t e s t h a t shear  i n s t a b i l i t y might be o f some s i g n i f i c a n c e b u t (1) m i l i t a t e s a g a i n s t  this  159  possibility. 7. magnitude and  The  barotropic, residual  the  wrong d i r e c t i o n  tidal circulation  to account f o r the  i s of  insufficient  observations.  160  BIBLIOGRAPHY  A l l e n , J . S., 1975: C o a s t a l t r a p p e d waves i n a s t r a t e f i e d ocean. Oceanogr., 3_, 300-325.  J . Phys.  Blumsack, S. L., and P. J . G i e r a s c h , 1972: Mars: The e f f e c t s o f topography on b a r o c l i n i c i n s t a b i l i t y . J . Atmos. S c i . , 29, 1081-1089. Brooks, D. A., and C.N.K. Mooers, 1977a: F r e e , s t a b l e c o n t i n e n t a l s h e l f waves i n a sheared, b a r o t r o p i c boundary c u r r e n t . J . Phys. Oceanogr., 7_, 380-388. , 1977b: Wind-forced c o n t i n e n t a l s h e l f waves i n the F l o r i d a C u r r e n t . J . Geophys. Res., 82, 2569-2576. Brooks, I . H., and P. P. N i i l e r , 1977: E n e r g e t i c s o f the F l o r i d a C u r r e n t . J . Mar. Res., 35, 163-191. Buchwald, V. T., and J . K. Adams, 1968: The p r o p a g a t i o n o f c o n t i n e n t a l s h e l f waves. P r o c . Roy. Soc. London, A 305, 235-250. B u c k l e y , J . R., 1977: The c u r r e n t s , winds and t i d e s o f n o r t h e r n Howe Sound. Ph.D. t h e s i s , U n i v e r s i t y o f B r i t i s h Columbia, 228 pp. Chang, P.Y.K., 1976: S u b s u r f a c e c u r r e n t s i n t h e S t r a i t o f G e o r g i a , west o f Sturgeon Bank. M.Sc. t h e s i s , U n i v e r s i t y o f B r i t i s h Columbia, 183 pp. Chang, P., S. Pond, and S. Tabata, 1976: S u b s u r f a c e c u r r e n t s i n t h e S t r a i t of G e o r g i a , west o f Sturgeon Bank. J o u r n a l o f the F i s h e r i e s Research Board o f Canada, 3_3, 2218-2241. Crean, P. B.> and A. B. Ages, 1971: Oceanographic r e c o r d s from twelve c r u i s e s i n t h e S t r a i t o f G e o r g i a and Juan de Fuca S t r a i t , 1968. Department o f Energy, Mines and Resources, Marine S c i e n c e s Branch, V o l . 1-5. Crean, P. B., 1976: Numerical model s t u d i e s o f the t i d e s between Vancouver I s l a n d and the mainland c o a s t . J . F i s h . Res. Board Can., 33, 2340-2344. , 1978: A n u m e r i c a l model o f b a r o t r o p i c mixed t i d e s between Vancouver I s l a n d and the mainland and i t s r e l a t i o n t o s t u d i e s o f the e s t u a r i n e c i r c u l a t i o n . Hydrodynamics o f E s t u a r i e s and F j o r d s , e d i t e d by J . C. J . N i h o u l , pp. 283-313. E l s e v i e r S c i e n t i f i c P u b l i s h i n g Co., Amsterdam. Csanady, G. T., 1976: Topographic waves i n Lake O n t a r i o . 6, 93-103.  J . Phys.  Oceanogr.,  161  , 1978: The a r r e s t e d t o p o g r a p h i c IB, 47-62.  wave.  J . Phys. Oceanogr.  Department o f the Environment, 1972a: Data r e c o r d o f c u r r e n t o b s e r v a t i o n s , S t r a i t o f Georgia, S e c t i o n 4, G a b r i o l a I s l a n d t o Gower P o i n t , 1969-1972. Water Management S e r v i c e , Marine S c i e n c e s D i r e c t o r a t e , P a c i f i c Region, M a n u s c r i p t Report S e r i e s , V o l . 10, 153 pp. , 1972b: Data r e c o r d o f c u r r e n t o b s e r v a t i o n s , S t r a i t o f G e o r g i a , S e c t i o n 5, P o r l i e r Pass t o Sand Heads, 1969-1972. Water Management S e r v i c e , Marine S c i e n c e s D i r e c t o r a t e , P a c i f i c Region, M a n u s c r i p t Report S e r i e s , ' V o l . 11, 124 pp. , 1973a: Data r e c o r d o f c u r r e n t o b s e r v a t i o n s , S t r a i t o f G e o r g i a , S e c t i o n 6, Samuel I s l a n d t o P o i n t R o b e r t s , 1969-1970. Water Management S e r v i c e , Marine S c i e n c e s D i r e c t o r a t e , P a c i f i c Region, M a n u s c r i p t Report S e r i e s , V o l . 12^, 96 pp. , 1973b: Data r e c o r d o f c u r r e n t o b s e r v a t i o n s , S t r a i t o f G e o r g i a , S e c t i o n 3, Northwest Bay t o McNaughton P o i n t , 1968-1969. Water Management S e r v i c e , Marine S c i e n c e s D i r e c t o r a t e , P a c i f i c Region, M a n u s c r i p t Report S e r i e s , V o l . 13_R 106 pp. de Szoeke, R. A., 1975: Some e f f e c t s o f bottom topography on b a r o c l i n i c stability. J . Mar. Res., 33_, 93-122. Diiing, W.,  1973: Some evidence f o r l o n g - p e r i o d b a r o t r o p i c waves i n t h e F l o r i d a Current. J . Phys. Oceanogr. 3, 343-346.  , 1975: S y n o p t i c s t u d i e s o f t r a n s i e n t s i n t h e F l o r i d a C u r r e n t . J . Mar. Res., 33_, 53-73. Duing, W., C. N. K. Mooers, and T. N. Lee, 1977: Low-frequency v a r i a b i l i t y i n t h e F l o r i d a C u r r e n t and r e l a t i o n s t o atmospheric f o r c i n g from 1972 t o 1974. J . Mar. Res., 35, 129-161. Eady, E . T., 1949:  Long waves and c y c l o n e waves.  T e l l u s , 1, 33-52.  F i s s e l , D. E . , and W. S. Huggett, 1976: O b s e r v a t i o n s o f c u r r e n t s , bottom p r e s s u r e s , and d e n s i t i e s through a c r o s s s e c t i o n o f Juan de Fuca Strait. P a c i f i c Marine S c i e n c e Report 76-6, I n s t i t u t e o f Ocean S c i e n c e s , P a t r i c i a Bay, V i c t o r i a , B. C , U n p u b l i s h e d m a n u s c r i p t , 68 pp. G i l l , A. E., and E . H. Schumann, 1973: The g e n e r a t i o n the wind. J . Phys. Oceanogr., 4, 83-90. Godin, G., 1972: The A n a l y s i s o f T i d e s . Toronto. 264 pp.  o f long waves by  U n i v e r s i t y o f Toronto  Press,  Grimshaw, R., 1976: The s t a b i l i t y o f c o n t i n e n t a l s h e l f waves i n the p r e s e n c e o f a boundary c u r r e n t shear. Res. Rep. No. 43, S c h o o l o f Mathematical S c i e n c e s , U n i v e r s i t y o f Melbourne, 19 pp.  162  Heaps, N. S., 1978: L i n e a r i z e d v e r t i c a l l y i n t e g r a t e d e q u a t i o n s f o r r e s i d u a l c i r c u l a t i o n i n t i d a l seas. Unpublished manuscript, 39 pp. H e l b i g , J . A., and L. A. Mysak, 1976: S t r a i t o f G e o r g i a o s c i l l a t i o n s : low-frequency waves and t o p o g r a p h i c p l a n e t a r y waves. J . F i s h . Res. Board Can., 33, 2329-2339 H e l b i g , J . A., 1977: Low-frequency c u r r e n t o s c i l l a t i o n s and t o p o g r a p h i c waves i n t h e S t r a i t o f G e o r g i a . M.Sc. t h e s i s , U n i v e r s i t y o f B r i t i s h Columbia, 81 pp. Herlinveaux,R. H., 1957: On t i d a l c u r r e n t s and p r o p e r t i e s o f the s e a water a l o n g the B r i t i s h Columbia c o a s t . F i s h . Res. Board Can., P a c i f i c P r o g r e s s Report No. 108. Herlinveaux, R. H., and L. F. Giovando, 1969: Some oceanographic f e a t u r e s o f the i n s i d e passage between Vancouver I s l a n d and the mainland of B r i t i s h Columbia. F i s h . Res. Board Can., T e c h n i c a l Report No. 142, 48 pp. Howe, M. S., 1971: Wave p r o p a g a t i o n i n random media. 769-783.  J . F l u i d Mech., 45,  Huyer, A., B. M. Hickey, J . D. Smith, R. L. Smith, and R. D. P i l l s b u r y , 1975: Alongshore coherence a t low f r e q u e n c i e s i n c u r r e n t s observed over t h e c o n t i n e n t a l s h e l f o f f Oregon and Washington. J . Geophys. Res., 80, 3495-3505. J e n k i n s , G. M. and D. G. Watts, 1968: S p e c t r a l A n a l y s i s and I t s A p p l i c a t i o n s . Holden-Day, San F r a n c i s c o , 525 pp. K e l l e r , J . B., 1967: The v e l o c i t y and a t t e n u a t i o n o f waves i n a random medium. E l e c t r o m a g n e t i c S c a t t e r i n g , R. L. Bowell and R. S. S t e i n , Eds., Gordon and Breach, 823-834. K e l l e r , J . B., and G. V e r o n i s , 1969: Rossby waves i n the p r e s e n c e o f random c u r r e n t s . J . Geophys. Res., 74, 1941-1951. Kendrew, W. G., and D. K e r r , 1955: The c l i m a t e o f B r i t i s h Columbia Yukon t e r r i t o r y , Queen's P r i n t e r , Ottawa.  and t h e  Kundu, P. K., and J . S. A l l e n , 1976: Some t h r e e - d i m e n s i o n a l c h a r a c t e r i s t i c s o f low-frequency c u r r e n t f l u c t u a t i o n s near the Oregon c o a s t . J . Phys. Oceanogr., 6_, 181-199. Kuo,  H.-L., 1949: Dynamic i n s t a b i l i t y o f two-dimensional n o n - d i v e r g e n t flow i n a b a r o t r o p i c atmosphere. J . Meteor. , £j, 105-122. , 1973: Dynamics o f q u a s i ^ g e o s t r o p h i c f l o w s and i n s t a b i l i t y Adv. A p p l . Mech., 13, 248-330.  theory.  163  LeBlond, P. H., and L. A. Mysak, 1977: Trapped c o a s t a l waves and t h e i r r o l e i n s h e l f dynamics. The Sea, v o l . 6, e d i t e d by E . D. G o l d b e r g , I . N. McCave, J . J . O'Brien, and J . H. S t e e l e , pp. 459-495, John W i l e y and Sons, New York. , 1978: Waves i n the Ocean, E l s e v i e r S c i e n t i f i c P u b l i s h i n g Co., Amsterdam. Lee, T. N., 1975: F l o r i d a C u r r e n t s p i n - o f f e d d i e s . 753-765.  Deep-Sea Res., 22,  Lee, T. N., and D. Mayer, 1977: Low-frequency c u r r e n t v a r i a b i l i t y and s p i n - o f f eddies a l o n g the s h e l f o f f Southeast F l o r i d a . J . Mar. Res., 35, 193-220. L i n , C. C , 1945: On the s t a b i l i t y of two-dimensional p a r a l l e l f l o w s . P a r t I I — S t a b i l i t y i n an i n v i s c i d f l u i d . Quart. A p p l . Math., _3, 218-234. , 1961: Some mathematical problems i n the t h e o r y o f the s t a b i l i t y of p a r a l l e l f l o w s . J . F l u i d Mech., 10, 430-438. Manton, M. J . , and L. A. Mysak, 1976: The s t a b i l i t y o f i n v i s c i d p l a n e Couette flow i n the presence o f random f l u c t u a t i o n s . J . E n g i n e e r i n g Math., 10, 231-241. McKee, W. D., 1977: C o n t i n e n t a l s h e l f waves i n the p r e s e n c e o f a sheared geostrophic current. L e c t u r e Notes i n P h y s i c s , v o l . 64, e d i t e d by D. G. P r o v i s and R. Radok, pp. 173-183, A u s t r a l i a n Academy o f Sciences, Canberra, S p r i n g e r - V e r l a g , B e r l i n . Mooers, C. N. K., 1973: A t e c h n i q u e f o r the c r o s s spectrum a n a l y s i s o f p a i r s o f complex-valued time s e r i e s , w i t h emphasis on p r o p e r t i e s o f p o l a r i z e d components and r o t a t i o n a l i n v a r i a n t s . Deep-Sea Res., 20, 1129-1141. Mooers, C. N. K., and D. A. Brooks, 1977: F l u c t u a t i o n s i n the F l o r i d a C u r r e n t , summer 1970. Deep-Sea Res., 24, 399-425. Mysak, L. A., and F. S c h o t t , 1977: E v i d e n c e f o r the b a r o c l i n i c i n s t a b i l i t y o f the Norwegian c u r r e n t . J . Geophy. Res., 82, 2087-2095. Mysak, L. A., 1977: On the s t a b i l i t y o f the C a l i f o r n i a u n d e r c u r r e n t o f f Vancouver I s l a n d . J . Phys. Oceanogr., 7_, 904-917. , 1978: Wave p r o p a g a t i o n i n random media, w i t h o c e a n i c applications. Rev. Geophys. Space Phys., 16, 233-261. N i i l e r , P. P., and L. A. Mysak, 1971: B a r o t r o p i c waves a l o n g an e a s t e r n continental shelf. Geophys. F l u i d Dyn. , 2_, 273-288. N i i l e r , P. P., and W. S. R i c h a r d s o n , 1973: Seasonal v a r i a b i l i t y o f the F l o r i d a C u r r e n t . J . Mar. Res., 31, 144-167.  164  P e d l o s k y , J . , 1964: The s t a b i l i t y o f c u r r e n t s i n the atmosphere ocean: P a r t I . J . Atmos. S c i . , 21, 201-219. Rhines, P., 1975: Waves and t u r b u l e n c e on a 69_, 417-443.  3 plane. -  and the  J . F l u i d Mech.,  S c h o t t , F., and W. Diiing, 1976: C o n t i n e n t a l s h e l f waves i n the F l o r i d a Straits. J . Phys. Oceanogr., 6_, 451-460. S i n g l e t o n , R. C , 1969: An a l g o r i t h m f o r computing the mixed r a d i x F a s t F o u r i e r T r a n s f o r m . IEEE T r a n s a c t i o n s on Audio and E l e c t r o a c o u s t i c s , AU-17, 93-103. Stone, J . M.,  1963:  Stronach, J . A.,  R a d i a t i o n and O p t i c s , M c G r a w - H i l l , New  1977:  York.  O b s e r v a t i o n a l and m o d e l l i n g s t u d i e s o f the F r a s e r  R i v e r Plume. Ph.D.  t h e s i s , U n i v e r s i t y of B r i t i s h Columbia, 260  pp.  Tabata, S., and J . A. S t i c k l a n d , 1972a: Summary o f oceanographic r e c o r d s o b t a i n e d from moored i n s t r u m e n t s i n the S t r a i t o f G e o r g i a , 1969-1970: C u r r e n t v e l o c i t y and seawater temperature from S t a t i o n H-06. Environment Canada, Water Management S e r v i c e , Marine S c i e n c e s Branch, P a c i f i c Region, P a c i f i c Marine S c i e n c e Report No. 72-7, 132 pp. , 1972b: Summary o f o c e a n o g r a p h i c r e c o r d s o b t a i n e d from moored i n s t r u m e n t s i n the S t r a i t o f G e o r g i a , 1969-1970: C u r r e n t v e l o c i t y and seawater temperature from S t a t i o n H-16. Environment Canada, Water Management S e r v i c e , Marine S c i e n c e s Branch, P a c i f i c R e g i o n . P a c i f i c Marine S c i e n c e Report No. 72-8, 144 pp. , 1972c: Summary o f oceanographic r e c o r d s o b t a i n e d from moored i n s t r u m e n t s i n the S t r a i t o f G e o r g i a 1969-1970: C u r r e n t v e l o c i t y and seawater temperature from S t a t i o n H-26. Environment Canada, Water Management S e r v i c e , Marine S c i e n c e s Branch, P a c i f i c Region, P a c i f i c Marine S c i e n c e Report No. 72-9, 141 pp. Tabata, S., J . A. S t i c k l a n d , and B. R. de Lange Boom, 1971: The program o f c u r r e n t v e l o c i t y and water temperature o b s e r v a t i o n s from moored i n s t r u m e n t s i n the S t r a i t o f G e o r g i a , 1969-1970, and examples o f r e c o r d s o b t a i n e d . F i s h e r i e s Research Board o f Canada, T e c h n i c a l Report No. 253, 222 pp. Tatarskii.  V. I . , and M. E. G e r t s e n s h t e i n , 1963: P r o p a g a t i o n of waves i n a medium w i t h s t r o n g f l u c t u a t i o n s o f the r e f r a c t i v e i n d e x . S o v i e t P h y s i c s JETP, 17, 458-463.  V e r o n i s , G., and H. Stommel, 1956: The a c t i o n o f a v a r i a b l e wind s t r e s s on a s t r a t i f i e d ocean, J . Mar. Res., 15, 43-75. Waldichuk, M., 1957: P h y s i c a l oceanography o f the S t r a i t o f G e o r g i a , B r i t i s h Columbia. J . F i s h . Res. Board Can., 14, 321-486.  165  Wang, D.-P., and C. N. K. Mooers, 1977: Long c o a s t a l - t r a p p e d waves o f f the west c o a s t o f t h e U n i t e d S t a t e s , summer 1973. J . Phys. Oceanogr., 7, 856-864. Webster,  I . , and D. M. Farmer, 1977: A n a l y s i s o f l i g h t h o u s e s t a t i o n temperature and s a l i n i t y d a t a — P h a s e I I . P a c i f i c Marine S c i e n c e Report 77-21, I n s t i t u t e o f Ocean S c i e n c e s , P a t r i c i a Bay, Sidney, B. C , U n p u b l i s h e d m a n u s c r i p t .  Wright, D. G., 1978: Mixed b a r o c l i n i c - b a r o t r o p i c applications. Ph.D. t h e s i s , U n i v e r s i t y 201 pp.  i n s t a b i l i t y w i t h ocean o f B r i t i s h Columbia,  Wunsch, C , and M. Wimbush, 1977: Simultaneous p r e s s u r e , v e l o c i t y and temperature measurements i n t h e F l o r i d a S t r a i t s , J . Mar. Res., 35, 75-104.  166  Appendix A:  Order o f Magnitude E s t i m a t e s  o f the I n t e g r a l Terms i n (3.25)  In t h i s appendix i t i s shown t h a t t h e i n t e g r a l terms i n (3.25) are a l l o f o r d e r u n i t y .  fty = £ { Q I  I t i s convenient  + (h'/h )(V  2  X  - c)I  2  1  to rewrite  - h  2  (3.25) as  ( V - c)I }  _ 1  (A.l)  3  where  H = [ ( v - c ) - e ]V 2  and  Ij , I  2  r  and  I^  -  2  (v - c)Q - e a h  = / (V - c) G ( x , ^ ) [ R ( x 0 1  x  -  (h'/h )R'(x 2  It s u f f i c e s to consider only form.  0$ -  1^  bJVty (A.3)  h R"(x _ 1  £)Md£.  since the other i n t e g r a l s are s i m i l a r i n  There a r e two types o f p o i n t s i n t h e range o f i n t e g r a t i o n t h a t must  be d e a l t w i t h , namely p o i n t s is  (A.2)  _ 1  a r e i n t e g r a l s d e f i n e d , f o r example, by  I  I  2  2  x  s i n g u l a r and p o i n t s  integrands  x+  x  a t which  c  where  c = V(x )  and the i n t e g r a n d  c  c = V(x+)  ± £.  At a l l other p o i n t s the  a r e continuous and a r e assumed t o be o f u n i t o r d e r .  To i s o l a t e  the s i n g u l a r i t i e s we p a r t i t i o n t h e i n t e g r a l as  x_-6 J  l  ~ /  x_+6 +  /  0  x -6  + / X —  x +<5  c  -0  /  +  X —  x -6  c  +0  +  X C  x +6  +  /  +  x^+o  -0  C  +  /  , / x,-0 +  T  £  , x.+o -r  (A.4) 1  2  3  4  5  6  7  167  where  6  i s some s m a l l p o s i t i v e number which we w i l l  zero.  We need determine Consider  examine o n l y  J  J  •  2  only  J  and  2  Since  ,  2  ,  first;  s i n c e they a r e o f the same form we  ip"  can have, a t worst, a  velocity 6-function-  We r e q u i r e a somewhat s h a r p e r r e s u l t , however.  O r d i n a r i l y , one c o u l d f i n d a P r o b e n i u s - t y p e determine  Jg .  i s p r o p o r t i o n a l t o the c r o s s - s t r e a m  i t must be c o n t i n u o u s , and, t h e r e f o r e , like singularity.  and  l a t e r l e t tend t o  the b e h a v i o u r o f  terms p r e v e n t s t h i s .  at  x_ ,  s o l u t i o n t o (A.l) i n o r d e r t o  b u t the presence o f the i n t e g r a l  We t h e r e f o r e look f o r an i t e r a t i v e s o l u t i o n t o (A.l)  o f the form  Hxb = 0  (A.5)  0  Hty = £ { Q I ( ^ ) + ( h ' / h ) ( V - c ) I ( ^ ) 2  X  - h  etc.  _ 1  1  0  independent  2  0  (V - c)I (^ )}, 3  A s e r i e s expansion  linearly  (A.6)  2  ±  0  o f (A.5) about  x = x_  i n d i c a t e s t h a t the two  s o l u t i o n s a r e o f the form  00  ^  = (x - x j [1 +  1 0  I  a (x - x_) ]  (A.7)  n  n  n=l CO  ^  That i s , determined ^  n  .  ( 0  2  )  =  I b ( x - x _ ) + al(; n=0 n  n  behaves no worse than by the i n t e g r a t e d v a l u e  (1) 0  i l n | x - x_| .  (x - x_)£n|x - x_| ; I^Q >  The same argument h o l d s f o r h i g h e r  (A.8)  since  ip-^ i s  i t can be no more s i n g u l a r 4*  n  ,  than  and we conclude t h a t ijj  168  shares t h i s q u a l i t y .  We a r e now ready t o e s t i m a t e  J„ ,  x_+6 / tty x_-6  J - < (l/e)max[.G(x,p ] R(e)  (A.9)  x +6 - h [ R ' (e)h'/h + R" (e) 1 / 4> 1  Since the '  ijj i s c o n t i n u o u s ,  first  term g i v e s the jump i n  In] x - x_|  i s o f the form  hence  the l a s t term v a n i s h e s  Jg ,  near  6 -»• 0 ,  and t h e r e f o r e a l s o v a n i s h e s x = x_ .  We c o n c l u d e t h a t  and since ,  and  6 -*• 0 .  b o t h v a n i s h as  Evaluation of  J  ^'/h  i n the l i m i t  i s more s t r a i g h t f o r w a r d .  We have  < max[G(x,£) ] [R(<S)tty - (h'/h )R'(6 )^ 2  4  (A.10)  -1  - h  x  R"(6)^] /  c  +  .1  6  (V - c) d £ . x  xc-6  Consider  o n l y the i n t e g r a l and p u t  c 1 = / xc~6 x  where  F(£)  6  - i (V - c) d£ = / c  +  6  '  c-<5 1  which tends t o  i s r e a d i l y evaluated  In the l i m i t o f  c ^ -> 0,  to obtain  -1 [F'(C)(?-c)] d?  i s the f u n c t i o n i n v e r s e t o  defined constant and  +  £ = V(^)  0  by a l l o w i n g we  find  with c  V ( £ ) , and 6.  Now  6'  i s an a p p r o p r i a t e l y  ( A . 7 ) i s a Cauchy  integral  t o have a s m a l l imaginary p a r t  ^  .  169  c+6'  [F' (£) (? - c) ]  I = s g n ( c ) / [ 2 F ' ( c ) ] + PV / i  (A.11)  c-S'  Here  PV  6 ->- 0,  denotes the Cauchy p r i n c i p l e v a l u e , I  reduces t o  f o l l o w s then t h a t are a l s o  0(1).  ± (l/2F'(c)) is  0(1)  and t h e r e f o r e  which i s an  0(1)  i n the l i m i t o f  quantity.  It  and hence t h a t the i n t e g r a l terms i n ( A . l )  170  Appendix B:  The F i r s t - O r d e r  Solutions  In t h i s s e c t i o n we s p e c i f y the f i r s t - o r d e r channel and s h e l f models.  F o r the channel case we f i n d :  r A  i ^ o  +  i  x<  0 5  o i  A  s o l u t i o n s f o r the  p  1 (B.l)  A k  12^0  +  i A  x5  1 5  0 2 P  I  where  c  s i n XP±  i 0  = e  b  (  + e  c  { (G^x - e D ^ c o s Ax  x  1 X  [ ( A x + B - ^ s i n Ax + (CjX + D±) cos Ax]},  s i n a ( l - £ ) P = [ G ( x - £) - e  i Q  2  2  1-x  + e  l  B  =  1  a  2a /Y )/Y  = 2Aa /Y  1  x  1  1  D  G  a  = 2A[  l  =  "  q  l  = 2£ /c  2 i 0  + 1 + 2b  + B ] s i n a ( x - I) 2  2  1  X  2  x  2  1  2  2  2  + a ^ l + 2A )]/Y  P;L  /  1  D ] c o s a ( x - I)  { [ A ( x - I)  l ^ l  = (p +  £  [ c ( x - £) + D ] c o s a.(x - I)  +  A  1  1  (B.2)  (B.3)  171  p  = - 2(1 + b)  1  %  = ~  y  = l + 4A'  A  (  e  2  = a /Y  2  3  2  2  2  r(/ i0  c  c  2  +  b  )  2  = - 2(1  a2/y )/Y  -  2  2  C  9  = 2aa2/Y " 2  D  2  = - 2a[2 - a ( l + 2 / Y ) ] / Y  G  2  = - q /2a  9  2  2  2  2  =  a  Y  2  0  2  e  2  /  c  2  e  i 0  c  2  +  1  r(/ iO c  = 1 + 4a  F o r the s h e l f case we  A  11^0  +  i A  find  0 l  ° "  P  x  "  1  (B.4)  A  where  12^0  +  1 A  x > 1  0 2 P  i s g i v e n as above and  P  2  = u (x)J (p?) + r  v  u (x)Y (p?) 2  y  (B.5)  172  (1-x) u  (x) = - ^ [ 2 y c .  J,.(p) ]  2  J  j  1  (p5)Y (p£)F(?)d£ v  (B.6)  1-x e  u (x)  = Tr[2y c J (p)] 2  2  F(C)  i 0  =  (2p ? 2  2  v  /  + 1) (1 - y hn£)  J (p?)F(Od? 2  v  - 2p c 2  r 0  ^ - 2.  (B.7)  (B.8)  173  Appendix C:  Evaluation  The  o f the  I n t e g r a l Terms f o r a Simple Flow Model  purpose o f t h i s appendix i s t o e s t i m a t e the  i n t e g r a l terms i n the mean v o r t i c i t y e q u a t i o n f o r the To do the  so we  choose the  3-plane.  (u  The  -  c)  simplest  - e ]RJj 2  +  (u  to  the  case o f l a r g e  p o s s i b l e model, a p a r a b o l i c  equation equivalent  2  size of  zonal  O.  flow  on  (3.25) i s  - c)g i(> -  e a ^ 2  y  2  1 - eQ  /  2  (u - c ) G ( y , ? ) [R(y - 5)F^ _ 1  - R"(y  - ZWiaZ  (c.D  0  Y  1  (U - c)~ G(y,5) [R"(y - K)H  - (U - c) /  - R (y - O ^ d ? ,  1  1 V  0  2 where  F  = d /dx  (3.25) , the terms two and  2  2 - k  .  Although  i n t e g r a l s are o f the  obstacles  must be  second, a form f o r  R(y  analytical solution for  (C.l) i s somewhat l e s s c o m p l i c a t e d than same form.  surmounted. - K)  G(y,£)  First  must be we  In o r d e r t o e v a l u a t e these G(y,£)  specified.  choose  U  must be To  such t h a t  obtain  determined, an  0 = 3 -  U"  v a n i s h e s , namely  U(y)  then  G(y,£)  = 3y(y  i s given  - l)/2;  by  f s i n h ky G(y,?> =  (C.2)  s i n h k ( l - £)/k  sinh k  0 < y < § S 1 (C.3)  i s i n h k^  sinh k ( l - y)/k  sinh k  0 5 £ 5 y <  1.  174  We s e l e c t  R(y)  t o be Gaussian,  R(y) = e x p ( - y a / 2 ) . 2  (C.4)  2  + a ^ty  IJJ =  We now take the s o l u t i o n  as determined  p e r t u r b a t i o n expansion o u t l i n e d i n S e c t i o n 7, s u b s t i t u t e i t i n t o  from the ( C . l ) , and  determine the r e l a t i v e v a l u e s o f the v a r i o u s terms i n the r e a l and imaginary p a r t s o f the r e s u l t i n g e x p r e s s i o n .  A  S p e c i f i c a l l y , we  find  s i n nlTy  (C.5)  IJJ0  =  ijj  = i A ( g K /2nTTc ) { ( y - y) s i n nTTy/mr  Q  2  Q  i0  (C.6) -  where  K  2  [ y / 3 - y /2 - (2c /3 + l/2n TT )y]cos nlTy] } 3  2  2  c  i 0  = £/K  c  r Q  = - (3/4)(1/nV + 1/3),  = k  2  (C.7)  + n iT . 2  [(u - c  2  rQ  The r e a l and imaginary p a r t s o f (C.l) a r e  2  ) - c 2  r  (C.8)  - e ]Bf  2  2  ±  0  + 2(u - c )c o r  1  ±  F^  1  -  e a ip 2  2  0  (C9) = - (U - ^  a~ [(u - c ) 1  r  2  (  -.  C  2 i  = - (u - c ) {i r  ^ - I ) - c.(I + I ) 4  2  3  - e ] F ^ - 2(u  c )  + i ) + c (i  - i )  2  2  3  ±  1  r  4  C i  F^  - e a^ 2  0  1  (CIO)  175  where  i  = /  1  1  , |u - c| (U  - c ) G ( y C ) [R"(y -  Z  r  r  0  l v  1 i  9  = i/a /  2  (c.ii)  - O^AdK  R (y  |u - c | " ^ ( u - c )G-(y,5) [R"(y r  o  - R (y i v  1 i  3  = c  (C12)  , |u - c | - G ( y , S ) tR"(y - ? ) F ^  /  ±  - O^ld?  z  0  0  l v R  (y  - C)^ ]d5  i  i  4  = c /a /  _  |u - c|  ±  <- > c  Q  9  13  0H±  G(y,C) [R"(y -  0  - R (y i v  - ?)^]dC-  (C14)  The r e l a t i v e magnitudes o f the terms i n (C.9) and presented i n Table V f o r selected values of T4/T3 One and a.  £,0,  and  determines whether o r not n e g l e c t o f the i n t e g r a l  sees t h a t the v a l i d i t y o f t h i s approximation improves 0  decreases.  and  k = ir/5  cases o f  ratio  terms i s j u s t i f i e d . as  k  increases  £.  Only i n the case o f  i s the rhs of - ( C l ) o f g r e a t e r magnitude  than the l h s and then, o n l y by 16 p e r c e n t . f o r the two  The  T h i s e f f e c t i v e l y puts an upper bound on the c h o i c e o f  There i s r e l a t i v e l y l i t t l e dependence on  £ = .5, a = 10,  k.  (C.10) are  The approximation i s v e r y good  £ = .5, O = 10, k = 2TT  and  £ = .5, a = 5  V  k = TT.  176  T a b l e V.  R e l a t i v e magnitudes o f the terms i n (C.9) and (C.10). Here T12, T3, and T4 r e f e r , r e s p e c t i v e l y , t o the a b s o l u t e v a l u e o f the sum o f terms 1 and 2, and the a b s o l u t e v a l u e o f terms 3 and 4 i n (C.9) and (C.10). The v a l u e s g i v e n here are symmetrical about y = 0.5.  a = 10, k =  £ = .5,  T12/T3  T4/T3  0.91 0.91 0.91 0.91 0.91  0.26 0.89 1.13 1.16 1.16  0.1 0.2 0.3 0.4 0.5  1.52* 1.22 0.70 0.53 0.63 Z  T12/T3  T4/T3  0.76 0.75 0.74 0.73 0.72  0.30 0.60 0.61 0.61 0.62  0.1 0.2 0.3 0.4 0.5  T4/T3 0.21 0.85 0.94 0.93 0.92  0.95 0.92 0.91 0.90 0.90  T4/T3 0.02 0.41 0.50 0.53 0.55  T4/T3  T12/T3  ,2,3,4  I  T4/T3 0.11 0.27 0.51 0.67 0.75  T12/T3 0 .83 0 .83 0 .83 0 .83 0 .82  TT  T12/T3  T4/T3  0.92 0.86 0.84 0.83 0.83  0.21 0.61 0.64 0.67 0.72  0.85 0.84 0.84 0.84 0.83  =  TT  (C. 10)  T12/T3  T4/T3  1.00 0.86 0.76 0.69 0.64  5, a = 5, k =  (C.9)  0.1 0.2 0.3 0.4 0.5  T12/T3  0.08 0.74 0.89 0.92 1.00  0.97 0.80 0.74 0.72 0.72  =  (C. 10)  (C .9)  £ =  y  .5, a == 10, k  £ = .1, a = 10, k  27T  (C. 10)  (C .9)  =  (c..9)  T12/T3  T4/T3  £ = .5, a = 10, k =  y  E  ( c :10)  (C .9) y  TT/5  0.17 0.66 0.69 0.66 0.69  T12/T3 0.48 0.73 0.75 0.75 0.74  TT  (C.10) T4/T3  T12/T3  3 2. 17 0. 7 0 A 0. 28 0. 13 0. 27 4  2. 44 1. 23 0. 87 0. 75 0. 75  j_ T4/T1, where TI r e f e r s t o the f i r s t term i n e i t h e r (C.9) o r (C.10), i s formed the r e s u l t i n g v a l u e s a r e 0.28, 0.60, 0.73, and 0.51, respectively. f  t  h  e  r  a  t  0  177  We conclude, approximation,  then, t h a t the n e g l e c t o f the i n t e g r a l s i s a good  b u t t h a t i n some cases, namely v e r y l a r g e  the p e r t u r b a t i o n s o l u t i o n s a r e b e s t r e g a r d e d  T12/T3  or small  equation.  i n d i c a t e s how w e l l the two-term  p e r t u r b a t i o n s o l u t i o n s a t i s f i e s the s i m p l i f i e d v o r t i c i t y e q u a t i o n S i n c e the terms r e p r e s e n t e d by  k,  as r e p r e s e n t i n g a f i r s t - o r d e r  i t e r a t i v e s o l u t i o n t o the complete i n t e g r o - d i f f e r e n t i a l The v a l u e o f  O  T12  and  T3  (7.7).  are g e n e r a l l y of opposite  s i g n , t h i s s o l u t i o n r e p r e s e n t s a good approximation t o the s o l u t i o n o f (7.7) i n most  cases.  178  Appendix  D:  B a r o c l i n i c I n s t a b i l i t y i n a 2-Layer  In t h i s appendix, S e c t i o n 11 a r e d e r i v e d . motion  System  the e q u a t i o n s g o v e r n i n g the model d e s c r i b e d i n  P r o c e e d i n g from the f u l l n o n l i n e a r equations o f  (see, e.g., V e r o n i s and Stpmmel, 1964, o r H e l b i g , 1977) , we  e s s e n t i a l l y f o l l o w the procedure developed by Pedlosky two  approaches  d i f f e r i n some r e s p e c t s .  (1964) a l t h o u g h the  The p r i n c i p a l assumption made i n  the d e r i v a t i o n o f the 2 - l a y e r equations i s t h a t the h o r i z o n t a l components o f v e l o c i t y a r e z-independent w i t h i n each  layer.  C o n s i d e r then t h i s s e t o f e q u a t i o n s :  upper  »  %  +  lt  v  (D.l)  ±  + u • V v, + f u , = - g r u -1 HI 1 ly  \  - V  lower  V  - fv = - g n l x  '  It  (  u  layer  2t  +  2t  +  n  2 t  Hi • v  +  \  - V  +  (  i  h  + n  i - V  V  H  * \  =  0  -  ( D  3 )  layer  H  2  *V u H  ^2  + u  Here the s u b s c r i p t respectively;  t  (D.2)  •  2  V  - f v = - gn  2  2  +  2  •v n H  f U  n  =  "  g  n  i y  + (h + n )V  2  2  i = 1,2  r)^ and  2  2  2  - g6 0 i  l x  "  H  g 6 ( T 1  2  -n )  2  x  "  n  (D.4)  x  i > y  (  • u + u^ • V h = o. 2  2  r e f e r s t o the upper o r lower  D  '  5  )  (D.6)  layer,  a r e the s e a s u r f a c e and i n t e r f a c i a l  displacements  179  (see F i g u r e 6 =  (p  and  V  2  H  11.1);  - p^)/p  and  h  a r e the mean l a y e r  2  e x p r e s s e s the d e n s i t y  2  thicknesses;  d i f f e r e n c e between the two  layers,  r e f e r s t o the two-dimensional L a p l a c i a n o p e r a t o r . As b e f o r e , i t i s c o n v e n i e n t t o n o n - d i m e n s i o n a l i z e these e q u a t i o n s .  The  following  s c a l e f a c t o r s are chosen:  h o r i z o n t a l coordinates  (x,y),  an a d v e c t i v e time  for  depth as  h  2  l a y e r , and  L/U  = h Qb(x,y) b(x,y)  geostrophically  i s an  by  a t y p i c a l speed  t.  hQ  quantity.  and  L  f o r the  f o r the v e l o c i t i e s ,  and  the l o w e r - l a y e r  i s the maximum depth o f the lower  2  0(1)  (fUL/g)  U  In a d d i t i o n , we w r i t e  where  2  the s h e l f w i d t h  The e l e v a t i o n s  (fUL/g6),  are  respectively.  scaled  In non-dimensional  form the e q u a t i o n s o f motion a r e :  upper  R  °  ( u  layer  it  Ro(v  l t  Ro[(n2  lower  '  -1  +  + u  l  » -  • VgV^  ±  -  V  6n ) 1  t  + u  ?  v  l  + fu  ±  = - ^lx  x  = -  • vH(n  -  2  n  ( D  -  7 )  (D.8)  l y  Sn1) ]  (D.9)  =  [B1  = -  n  2 x  -  = -  n  2 y  - (i - 6)n2y  - Ro(n2  -  &r\±) ] V  •  u  ±  layer  Ro(u  2 t  + u  Ro(v  2 t  +  Ro(n2t  + u  2  • v u ) - fv H  2  • v v ) + fu H  2  2  • Vn2)  2  2  = - [B b + Ron lV 2  2  (i - 6 ) n  • u  2  -  (D.IO)  l x  (D-ii)  • Vb.  (D.12)  180  /N  Here  Ro = U / f L  i s t h e Rossby number,  f = f/f =1  i s retained  to a i d i n t h e i d e n t i f i c a t i o n o f t h e C o r i o l i s term, and 2 2 B^ = g'h-^/f L  Burger numbers d e f i n e d by g' = go  0(6)  on t h e r h s o f (D.10)  a l s o be assumed t h a t  b  i s a function of  and  2 2 B., = g'h2/f L  i s t h e reduced a c c e l e r a t i o n due t o g r a v i t y .  i g n o r e t h e term o f will  and  B-^  B  2  are  where  We w i l l h e n c e f o r t h  and ( D . l l ) x  temporarily  since  6 «  1.  It  only.  The presence o f the Rossby number i n these e q u a t i o n s , which f o r GS i s a p p r o x i m a t e l y  4  10  x  I  (u.,^) =  ,  suggests a p e r t u r b a t i o n expansion o f t h e form  R o ^ ^ ' V ^  n=0  (D.13)  The b a s i c s t a t e must s a t i s f y t h e z e r o t h - o r d e r e q u a t i o n s ,  fv  f  U l  V  H  ( 0 )  = n  ( 0 )  = -  x  • u  fv  o  fu  2  (  )  0  (D.14),  ( 0 )  l  y  (D.14)  (  0  0  )  (D-15)  = 0  (D.16)  ( 0 )  - U  T,  +  (T,2  =  ( 0 )  V„ • U H —I  each l a y e r .  n  = (n,  ( 0 )  n  Equations  0  x  l x  ( 0 )  £ 9  (D.15),  ( 0 ) 1  >n  >„  ( 0 , 1  )  (D.17)  (D.18)  y  (0), -1 db b — - = 0. 3 n  , (D.19) n 1  X  (D.17) and (D.18) d e f i n e stream f u n c t i o n s f o r  We note t h a t w h i l e (D.16) i s i d e n t i c a l l y s a t i s f i e d ,  (D.19)  0  )  181  requires that either  2 ^  u  vanish or that  db/dx  be  0(Ro).  Although  (0) we w i l l ,  i n f a c t , choose  l e v e l , and so we p u t sign of  T  u.,  =0,  t h i s problem again  db/dx = - RoT(x).  a r i s e s a t the next  A minus s i g n i s chosen so t h a t the  c o i n c i d e s w i t h t h a t o f the bottom s l o p e ; i . e . , i f the bottom  s l o p e s upward t o the e a s t , w i t h Mysak and S c h o t t =  u± i0)  T > 0.  (1977).  This choice  also f a c i l i t a t e s  As a b a s i c s t a t e we  comparison  choose  (0,V.(x)) (D.20)  n.(°>  where  V  = H.(x)  and  ±  H  are r e l a t e d by  i  (D.14)-(D.19).  I t i s easy t o show t h a t  t h i s s t a t e a c t u a l l y s a t i s f i e s the complete n o n l i n e a r  s e t of equations  (D.7)-(D.12). The p e r t u r b a t i o n s t a t e i s governed by the f i r s t - o r d e r e q u a t i o n s  fv  f  (1)  (1) U l  (1)  fv  0  fu  2  (1) = n  x  (1)  = - n  =  (D.21)  y  x  - (ni  (1)  , = -  (n-L  (D.22)  x  (1)  . _ (1).  + n  9  + n  2  )  (D  (1)  (D.23)  x  (D.24)  x  )  y  which a g a i n d e f i n e stream f u n c t i o n s f o r each l a y e r . are i n d e t e r m i n a t e , equation  for  H]/"^  As such (D.21)-(D.24)  and i t i s n e c e s s a r y t o go t o second order a  n  d  ^ ^  '  T  °  s  e  c  o  n  d  o r <  3 e r , one f i n d s :  t o o b t a i n an  182  upper  layer  O  t  + V^yJu^  O  t  + V 3 )v  O  t  + V^y) (Tl  X  lower  (3  O  t  t  1  2  = - T1  )  ( 1 )  1  V'  - <5lli  ( 1 ) 2  (1)  + f  1  (D-25)  ( 2 ) l x  ( U  2  = - n  )  l  ) + u (H x  - 6H )  2  ( 2 )  l y  1  (D.26)  = B^V  X  • u  (  2  (D.27)  )  1  layer  2  ( 1 )  y  2  - fv  V Z )v ™  +  (3t  ( x  + u .  ( 1 )  y  V 8 )u  +  - fv  1 5  2  y  (  2  = - (H  )  2  - u ^>V  2  2  + V29y)n  2  -  ( 1 )  U  ( 1 ) 2  2  H'  fu  +  2  =  V  ( 2 ) 2  (  +  2  )  2  = -  B2bV  -  2  • u  )  )  (n  2  (  2  2  Here a prime denotes d i f f e r e n t i a t i o n w i t h r e s p e c t t o  (D.28)  x  ( 2 )  )  r)l (2))Y  +  + B U 2  x.  ( 1 2  (D.29)  ]T.  (D.30)  By c r o s s (2)  d i f f e r e n t i a t i n g the momentum e q u a t i o n s and s u b s t i t u t i n g f o r from the c o n t i n u i t y e q u a t i o n we o b t a i n the v o r t i c i t y perturbation  [3  V • u^  e q u a t i o n s f o r the  state,  + V^y] [ v  t  l x  - u  B^ 1  +  &t  +  V  2 y 3  ]  U  t 2x " 2 v  + Bj(n2  l y  2  ~ B^'  U  Y  B  [ H  f  1 1 2 1  6  H  +  6m)]  l x= ]  U  +  n  2 "2  9  V  U0  (D.31)  0  [u,H', 2 2 + B ° 2u 2T ] == 0.  L U 2 b  "  -  (  °-  3  2  )  183  Here we b(x)  have dropped the s u p e r s c r i p t  about the p o i n t  b(x)  It by  x  where  Q  b(x )  gives  (D.33)  + 0(R 0 )  = 1 + RoT(x ) (X - x ) 0  A T a y l o r s e r i e s expansion o f  = 1  Q  n  f o l l o w s then t h a t to the p r e s e n t 1.  (1).  n  o r d e r of a n a l y s i s ,  1/b  may  be  replaced  In terms o f the stream f u n c t i o n s ,  (D.34) ?  and  2  =  n  +  2  n  1  the b a s i c s t a t e v e l o c i t i e s ,  [ 9  t  V y  +  ]  [ V 2 $  1  +  F  l  (  $  (D.31)-(D.32) may  2 " 1 $  ~  ) ]  $  ly  [  V  be  "l  rewritten  " 1 F  ( V  1  " 2 V  =  [ 8  t  +  V  2 y 9  ] [ y 2 $  2  -  F  2  ( $  2  -  $  1  ) ]  -  $  2y  [ V  "2  +  F  2  ( V  1  " 2 V  and  F  2  ( D  }  +  T ]  0(6)  T h i s i s the d e s i r e d s e t o f e q u a t i o n s g o v e r n i n g  dropped.  perturbation state.  Since  respectively.  To o b t a i n these,  the  and  was  ,  (D.36)  reciprocals of  2  they express the c o n s e r v a t i o n  a term o f the  of p o t e n t i a l  vorticity  i n the system, they c o u l d a l s o have been developed d i r e c t l y  the f u l l ,  nonlinear, v o r t i c i t y  equations.  3 5 )  0,  are i n t e r n a l Froude numbers g i v e n simply by B  -  0  = where  ) ]  from  

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