Open Collections

INTERNAL WAVES I N A RANDOMLY S T R A T I F I E D OCEAN by ROBERT ERNEST McGORMAN B.Sc.  (Honours M a t h e m a t i c s ) , M c G i l l U n i v e r s i t y ,  1968  A THESIS SUBMITTED I N P A R T I A L FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  i n t h e Department of Mathematics  We a c c e p t t h i s t h e s i s a s c o n f o r m i n g required  to the  standard  THE U N I V E R S I T Y OF B R I T I S H COLUMBIA A p r i l , 1972  In p r e s e n t i n g an  advanced  the I  degree  Library  further  for  shall  agree  scholarly  by  his  of  this  written  this  thesis  in p a r t i a l  fulfilment  at  University  of  the  make  that  it  permission  purposes  may  representatives. thesis  for  be  It  financial  for  is  of  ITlcit&iswastiCsas  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  IAaaXZ /+/ 72  by  the  understood  gain  Columbia  for  extensive  granted  shall  the  requirements  B r i t i s h Columbia,  available  permission.  Department  Date  freely  of  copying Head o f  that  not  reference  be  and  of my  I agree  this  or  allowed  without  that  study. thesis  Department  copying  for  or  publication my  ABSTRACT  i i  I n t h i s t h e s i s we c o n s i d e r t h e p r o p a g a t i o n o f i n t e r n a l w a v e s in a rotating  s t r a t i f i e d unbounded ocean w i t h r a n d o m l y v a r y i n g  B r u n t - V a i s a l a frequency,N . K e l l e r ' s method i s u s e d dispersion relation  f o r t h e mean wave f i e l d  2 order  i n £ when N  0<"£^< 1  a n d yx  .either depth  to obtain the  c o r r e c t t o second  2  i s o f t h e form  where  - constant,  i s a c e n t e r e d s t a t i o n a r y random f u n c t i o n o f  o r t i m e s e p a r a t e l y . From t h e d i s p e r s i o n  relation  t h e r e a r e d e r i v e d g e n e r a l f o r m u l a s f o r t h e change i n phase and g r o w t h  o r a t t e n u a t i o n r a t e s due t o t h e r a n d o m f l u c t u a t i o n s yU .  T h e s e f o r m u l a s a r e d e p e n d e n t o n t h e s t a t i s t i c s of /u the a u t o c o v a r i a n c e The  only  through  function.  phase speed  d e p e n d e n t yu , w h i c h salinity  speed  change and growth  r a t e formulas f o r depth  c o n s t i t u t e s a model o f t h e temperature  fine-structure  i n t h e ocean,  s p e c i a l cases i n c l u d i n g the l i m i t i n g  and  are presented f o r various cases o f c o r r r a t i o n 1  lengths  o f JJ. t h a t a r e l o n g o r s h o r t w i t h r e s p e c t t o t h e w a v e l e n g t h . O b s e r v a t i o n s a t s t a t i o n P (50°N, 145° W)  indicate that,  t o a good  a p p r o x i m a t i o n , t h e y\x a r e " w h i t e n o i s e " a n d a c l o s e e x a m i n a t i o n i s made o f t h e t h e o r e t i c a l r e s u l t s f o r t h i s c a s e . W i t h t h e a i d o f the s t a t i o n P data i t i s e s t i m a t e d t h a t , changes a r e g e n e r a l l y s m a l l , (decreases) a depth  significantly  a l t h o u g h t h e phase  t h ^ a m p l i t u d e o f a wave i n c r e a s e s  i n p r o p a g a t i n g upward  (downward)  o f a few k i l o m e t e r s . I n a d d i t i o n i t i s f o u n d  e f f e c t o f t h e depth  speed  d e p e n d e n t f l u c t u a t i o n s /U  through  t h a t t h e mean  i s to increase the  e f f e c t i v e Brunt-Vaisala. frequency, or " s t i f f e n " may  e x p l a i n why  some r e c e n t l y o b s e r v e d  frequency  the f l u i d . spectra of  i n t e r n a l w a v e s do n o t e x h i b i t a s h a r p c u t - o f f a t A/ , 0  the deter-  m i n i s t i c t h e o r e t i c a l u p p e r b o u n d f o r t h e wave f r e q u e n c y . an a t t e m p t  i s made t o a s s e s t h e r a n g e  method i n the c o n t e x t o f t h e p r e s e n t  of v a l i d i t y of problem.  This  Finally  Keller's  TABLE OF CONTENTS  i v  C h a p t e r 0:  Introduction  Chapter I :  The D e p t h D e p e n d e n t C a s e  3  1.1  Formal Dispersion  3  1.2  S i m p l i f y i n g the Dispersion  1.3  Polar Representation  1.4  S p e c i a l Cases  12  The Time D e p e n d e n t C a s e  19  11.1  Formal Dispersion  19  11.2  S i m p l i f y i n g the Dispersion  Chapter I I :  Chapter I I I  Further  -  Relation Relation  of the Dispersion  1  5 Relation  Relation Relation  9  .20  E x a m i n a t i o n o f t h e Case o f Depth  Dependent "White N o i s e " F l u c t u a t i o n s  26  111.1 I n t r o d u c t i o n  26  111.2 D i s c u s s i o n  o f t h e "White N o i s e " Growth Curves  27  111.3 D i s c u s s i o n  o f t h e "White N o i s e " Phase  Speed  Curves  32  Bibliography  37  Appendices:  39 A- The 0 t h O r d e r D i s p e r s i o n  Relation  B- D e r i v a t i o n o f t h e S t r e a m F u n c t i o n C- K e l l e r ' s M e t h o d D-  39 Equation  40 44  The L i m i t i n g C a s e  L << ~X  E- The L i m i t i n g C a s e  ^ <'< L a  49  Q  .  •  51  L I S T OF TABLES T a b l e I Sample Computed A u t o c o v a r i a n c e  v  Function  13  L I S T OF  FIGURES  Figure  1 Growth and A t t e n u a t i o n  Rates  Figure  2 R e l a t i v e P h a s e S p e e d C h a n g e f o r P a s s b a n d I : -f < Q~ <  Figure  3 R e l a t i v e Phase Speed Change f o r P a s s b a n d l l : A o  2  2  < < T  *'  ACKNOWLEDGMENT I am p l e a s e d presently a Senior  t o t h a n k my s u p e r v i s o r , Visitor  thesis.  I n a d d i t i o n I should  This research  L.A.Mysak, Mathe-  a t Cambridge U n i v e r s i t y , f o r h i s  encouragement and a s s i s t a n c e  examining the t h e s i s ,  prof.  i n t h e Department o f A p p l i e d  matics and T h e o r e t i c a l Physics gracious  v i i  like  i n the w r i t i n g of t h i s  t o thank P r o f . P.H.LeBlond f o r  a n d Dr.. R.E.Thomson f o r t h e u s e o f t h e d a t a .  was a c c o m p l i s h e d w i t h  the f o r m o f an H . R . M a c M i l l a n F a m i l y  financial  Fellowship.  assists--e i n  CHAPTER 0: I t has  INTRODUCTION  1  l o n g b e e n known t h a t i n t e r n a l g r a v i t y w a v e m o t i o n  jjf. exists  i n the ocean  (see C h a p t e r 5 o f P h i l l i p s  ). In order  to  p r o p a g a t e , t h e s e w a v e s r e q u i r e a s t r a t i f i e d medium s u c h a s ocean where l e s s dense s u r f a c e l a y e r s . A measure of the Brunt-Vaisala  where  o  l a y e r s r e s t on more d e n s e  s t r e n g t h of the  f r e q u e n c y , N , d e f i n e d by  i s t h e a c c e l e r a t i o n due  stratification  the  the  deeper i s the  equation  t o g r a v i t y and  d e n s i t y o f the w a t e r as a f u n c t i o n o f depth  H,  i s the  p ii) a  rest  which i s p o s i t i v e  upwards. Observations propagate w i t h  i n d i c a t e t h a t i n t e r n a l w a v e s f r e q u e n t l y do  constant  amplitude but  i n g o r become damped o u t .  grow t o t h e p o i n t o f  e f f e c t on studying  and  2  s h a l l examii..  the  t h e w a v e s o f r a n d o m v a r i a t i o n s i n t h e q u a n t i t y N*  by  the  c u r r e n t s ' f I n t h i s t h e s i s we  dispersion  relation.  Many s t u d i e s o f i n t e r n a l w a v e s h a v e a s s u m e d a s u c h t h a t l\j d) — N  0  first,  break-  P o s s i b l e mechanisms have been advanced  t o a c c o u n t f o r t h i s b e h a v i o u r i n c l u d i n g d i f f u s i v e processes'" interaction with  not  , a constant.  This  stratification  i s done f o r two  reasons:  i t i m p l i e s a smoothly v a r y i n g dependence o f d e n s i t y  d e p t h , w h i c h may  be  a reasonable  dence a t l e a s t s e c t i o n a l l y ; c o e f f i c i e n t i n the  approximation  to the  second, i t reduces a  i n t e r n a l wave e q u a t i o n  p u r e l y m a t h e m a t i c a l c o n v e n i e n c e . The  on  actual  depen-  "E-dependent  to a constant,  properties of  i.e., a  f r e e wave  2 solutions  f o r t h i s c a s e a r e w e l l known a n d a r e s u m m a r i z e d  A p p e n d i x A. H e r e we  s h a l l assume N  i s s u b j e c t t o s m a l l random  1  f l u c t u a t i o n s a b o u t a c o n s t a n t mean Nl. fluctuations w i l l dependent  be d e p t h d e p e n d e n t  fluctuations w i l l  in  i n C h a p t e r I t h e random and i n C h a p t e r I I t i m e  be c o n s i d e r e d . I n C h a p t e r I I I we  cuss g r a p h s o f t h e r e l a t i v e phase  speed change and g r o w t h  dis-  rate  p l o t t e d u s i n g r e a l oceanic data i n the corresponding formulas f o r "white noise"  f l u c t u a t i o n s derived i n Chapter I .  CHAPTER I : THE DEPTH DEPENDENT CASE 1.1  3  Formal D i s p e r s i o n R e l a t i o n In  A p p e n d i x B we b e g i n w i t h t h e e q u a t i o n s  momentum c o n s e r v a t i o n o f an i n c o m p r e s s i b l e ,  o f mass and  inviscid  fluid  right-handed  system o f C a r t e s i a n c o o r d i n a t e s u n i f o r m l y  with angular  frequency  and  in a  rotating  "£? a b o u t t h e 2 - a x i s , w h i c h i s v e r t i c a l  p o s i t i v e upward. A b a s i c s t a t e o f h y d r o s t a t i c e q u i l i b r i u m i s  assumed and upon t h i s pressure  small perturbations i n the f l u i d  velocity,  and d e n s i t y a r e imposed. I f the r e s u l t i n g e q u a t i o n s  l i n e a r i z e d a n d a t t e n t i o n i s r e s t r i c t e d t o two d i m e n s i o n a l then  t h e s t r e a m f u n c t i o n i s shown t o s a t i s f y  C« where  x We  motion  the equation  =0  + + i s the h o r i z o n t a l c o o r d i n a t e and  ±  are  (i-1)  i s the time.  assume d e p t h d e p e n d e n t r a n d o m f l u c t u a t i o n s  in N  2  of the  form  w h e r e yu(2) i s a z e r o - m e a n , w i d e - s e n s e s t a t i o n a r y r a n d o m p r o c e s s " and  £  i s a s i z e parameter such t h a t  a t t e n t i o n t o the case o f harmonic time  This gives equation  1-1  the form  0< €.*<< 1 . R e s t r i c t i n g d e p e n d e n c e we s e t  where  a d e t e r m i n i s t i c operator,  a "small"  random  and  operator.  E q u a t i o n 1-2 i s i n a f o r m s u i t a b l e f o r t h e a p p l i c a t i o n o f K e l l e r ' s m e t h o d ' ' ' o f w h i c h we g i v e T h e r e we show t h a t ,  a n a c c o u n t i n A p p e n d i x C.  c o r r e c t t o s e c o n d o r d e r i n £" , t h e d i s p e r s i o n  r e l a t i o n o f t h e mean wave <  >  (where < > i n d i c a t e s e n s e m b l e  average) i s  w h e r e JA.' i s t h e i n t e g r a l o p e r a t o r d e f i n e d b y 1  CO  -oo  with  Gr , t h e G r e e n ' s f u n c t i o n , b e i n g t h e s o l u t i o n o f  JAG- =• Since  Scx-x'/fa-z')  ^/A, i s t r a n s l a t i o n a l l y i n v a r i a n t i n x  Green's f u n c t i o n GrCx.K',  so  i sa displacement kernel  2,Z') = Crcx-x', z - z.'J  i t i snecessary t o consider  only  i n cM. , 1  a n d Z, t h e that i s ,  E q u a t i o n 1-4 transform  can be s o l v e d by t a k i n g a d o u b l e  of both sides, i . e . m u l t i p l y both sides by  and i n t e g r a t e f r o m  the F o u r i e r so  -oo  t o -hco  2, . The i n v e r s i o n i n JL  and  Fourier  transform  in  Q~  c(kx  successively with respect  - ^^v)  to  *  can be r e a d i l y p e r f o r m e d t o y i e l d of  Q ,  w h i c h we d e n o t e b y  G(k,z) ,  that  U(K,IL)  where  H  -  ...  i s the Heaviside  d  =  -  f u n c t i o n a n d we h a v e s e t  »v  g  and  for convenience. In the d e t e r m i n i s t i c theory passbands o f cJ d e f i n e d by 3  passband  I waves and  represents  passband  I I waves. I n e i t h e r case  1-2 S i m p l i f y i n g t h e D i s p e r s i o n  C  Z  >  O.  Relation  I t i s now p o s s i b l e t o s u b s t i t u t e  Set  a r e two  inequalities:  represents  f i n d the d i s p e r s i o n r e l a t i o n  there  At, J/i'  explicitly.  1  Jf  i n t o 1-3 t o  7  = e - «  Y  * « * ' » ^  e  < « x , /  e  ;  and  Then  and  <9 * L U(  )  j±f*d*. -<43*lJJGrtx-x.', 2 - 2 7 —  CO  Simplifying  -co  T h i s can be w r i t t e n  CO  — OC3  Make t h e c h a n g e o f v a r i a b l e s OL" - x-  Then t h i s  Now t a k i n g the  x'  gives  into  account that the i n t e g r a l  i s a function of  Z,  d i f f e r e n t i a t i o n can be performed t o g i v e  — CO  CO  —CO  U s i n g t h e s t a t i o n a r i t y o f jua) we d e f i n e function,  Tc?";, b y Vet")  Now  =  )yjLC  I-?.")>  the autocovariance  8  and -$UCZj/4 *S2--z<';> z  ^ ~ P » £u< zJyU ( * 2  Thus  — Q3  -00  and i n e a c h c a s e t h e i n t e g r a t i o n o v e r  produces  GcCk.z")  above i n t e g r a l s . So,  correct  t o s e c o n d o r d e r i n •€ , t h e d i s p e r s i o n  reduces t o  -oo  U s i n g i n t e g r a t i o n b y p a r t s we c a n w r i t e  relation  i n the  -co as the i n t e g r a t e d term v a n i s h e s a t 0 we  and  CO  . Collecting  terms  obtain  -e\t[0.-<:**]fe % d  o^JT^^zjrczje^Jz^O  (1-5)  ~ 03  as the d i s p e r s i o n r e l a t i o n We n o t e t h a t t h e Oct*) the  dispersion relation  (i.e.  t h e above w i t h  f o r t h e mean f i e l d  <'V'),  c o r r e c t t o Oce  1  terms, which are the c o r r e c t i o n s t o  f o r i n t e r n a l waves i n t h e non random  £- O  ), a r e dependent on t h e p a s s b a n d  1-3 P o l a r R e p r e s e n t a t i o n o f t h e D i s p e r s i o n  of  &?  Relation  I n d i s c u s s i n g t h e Oth o r d e r d i s p e r s i o n r e l a t i o n we e m p l o y a p o l a r r e p r e s e n t a t i o n  fluid  i n Appendix  o f t h e wavenumbers, i . e .  k = Wo co<u9 and  where  £g  i s real.  0  We w a n t t o e x t e n d t h i s r e p r e s e n t a t i o n relation,  i.e.  k^^C^G  t o t h e Oce J d i s p e r s i o n l  set  (1-6)  A  <  10  and  I  - Xs+me -cd  (1-7)  /z  6  where a n d 1-7  i s a r e a l a n g l e a n d B€  may  i n t o T - 5 a n d s i m p l i f y i n g we  be complex. S u b s t i t u t i n g obtain  -<llfrr:  r ^ L ^w*  1-6  ^ ( ^ ^ % * )  e ******  -co  CI-8) Now  quantities for  ?6  f i x an a n g l e o f p r o p a g a t i o n , 0 , a n d s p e c i f y  i f we ffjf,  A/  0)  (/  w h i c h c a n be  Setting  € = 0  f c ? ; t h e n e q u a t i o n 1-8  s o l v e d by i t e r a t i o n  g i v e s the 0th order  d?o -  •  Substituting  ^  iteration  and  ^  0  solution  ,  for  see d i s c u s s i o n A p p e n d i x  C)  (see K e l l e r  equation  and Veronis") ,  s o l u t i o n of Appendix  A  CT-Q)  •  i n the  (which  i s an  the  i s valid  0 C£ ) l  i f the  terms g i v e s the Oc(?)  terms are  first small,  Since  swtfe**^-  the absolute o< = Mo  s*m  e  value  and  CoxL,(a  zj  oc ,  a r e even f u n c t i o n s o f  s i g n s c a n be d r o p p e d . F o r c o n v e n i e n c e s e t  and  T a k i n g t h e s q u a r e r o o t o f b o t h s i d e s o f 1-10 a n d u s i n g t h e binomial  e x p a n s i o n o n t h e R.H.S. ( a s s u m i n g t h e Oct ) t e r m s a r e 1  s m a l l r e l a t i v e t o 1 ) , we  obtain o —00  °^  -ca  Hence i t r e a d i l y  follows  that  <>• -Too  . - e^i^Vd) -co  %Xn l^idi  (I-11)  and 2  6  0  - ffcfrt*)(l-Co<L2*2)dti  -<=Mfrti)(l + Co«.2^)di  -°°  -co  (1-12)  1-4 S p e c i a l C a s e s F i r s t o f a l l we s h a l l assume t h a t t h e f l u c t u a t i o n s i n jud.) are  "white noise"  so t h a t  O b s e r v a t i o n s t a k e n o n F e b r u a r y 23, 1 9 7 1 a n d J u l y 29, 1 9 7 0 a t w e a t h e r s h i p s t a t i o n P (50°N, 145°W) a n d a n a l y z e d  by  Richard  Thomson o f E n v i r o n m e n t C a n a d a i n d i c a t e t h a t t h e a s s u m p t i o n i s w e l l justified  o v e r c e r t a i n d e p t h r a n g e s . The q u a n t i t y  N (D  was  2  c o m p u t e d a t 10 m e t e r i n t e r v a l s f r o m t h e o c e a n s u r f a c e o f 1500 m e t e r s . The r e s u l t s i n d i c a t e t h a t  /\J d) 2  t o a depth  exhibits a  s t r o n g t r e n d w i t h d e p t h i n t h e u p p e r 700 m e t e r s . H o w e v e r , b e t w e e n  /\/ <Z)  t h e d e p t h s o f 700 m a n d 1500 m s*e~  t r e n d w i t h mean a b o u t M>  with  2  Z  €^^0(^L)  appears t o have  a n d s t a t i o n a r y f l u c t u a t i o n s , Cju.it),  . Furthermore, the aut^-jovariance  f u n c t i o n s f o r b o t h t h e summer a n d w i n t e r mean, /V  0  little  data  computed u s i n g t h e  , o v e r t h i s r a n g e i n d i c a t e t h e €/ua)'s a r e " w h i t e  t o a good a p p r o x i m a t i o n . the autocovariance  We i n c l u d e f o r i l l u s t r a t i o n  f u n c t i o n computed from t h e w i n t e r  noise"  i n table I data.  13 TABLE I, SAMPLE COMPUTED AUTOCOVARIANCE FUNCTION DEPTH LAG, Z /m  €  2  0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150  ....  Substituting  for  T< 2 )  0.19580 0.09219 0.01881 0.06659 0.02786 0.00000 0.02475 0.0659 3 0.02659 0.03095 0.06968 0.04162 0.01218 0.0597" 0.0750b 0.01380  i n f o r m u l a s 1-11 a n d 1-12  yields  and  U s i n g e q u a t i o n 1-9 a n d r e c a l l i n g t h e v a l u e s o f  Fl  and  oi. we  can. w r i t e  and  //m i 4?ol = — £  /  1 , i. , 2 \l C*C<rz*Q-U»\*Q  (1-1+)  14 We  denote the d i m e n s i o n l e s s q u a n t i t y  5  If  phase  i s the phase  0  speed o f the  a n d when 1-13  dV  where  s p e e d o f t h e O t h o r d e r wave a n d Oct )  S  i s the  wave t h e n  2  i s n e g a t i v e t h e wave g r o w s ,  a n d 1-14  cT  by  t h u s we  can  rewrite  as  -  ^  _  ^  a  -is)  and  . . E q u a t i o n s 1-15  a n d 1-16  are discussed a t length i n Chapter I I I  where t h e y appear as e q u a t i o n s I I I - l j u s t m e n t i o n t h a t f r o m 1-16 (j  2  and I I I - 2 . A t t h i s p o i n t  i t i s clear  that,  we  independent of the  p a s s b a n d , w a v e s p r o p a g a t i n g w i t h an u p w a r d  component o f  d i r e c t i o n g r o w a n d w a v e s p r o p a g a t i n g w i t h a downward c o m p o n e n t decay w i t h t h e magnitude o f the growth or decay dependent passband of N e x t we  <T  only through the dimensionless q u a n t i t y  c o n s i d e r t h e s p e c i a l c a s e o f 1-11  a n d 1-12  on t h e £ d  .  f o r which  15 which i s the autocovariance  function  ofa  Uhlenbeck-Ornstein  p r o c e s s . The r e s u l t i n g i n t e g r a l s a r e e a s i l y e v a l u a t e d a s t h e y have t h e form o f L a p l a c e t r a n s f o r m s . E q u a t i o n s  1-11 a n d 1-12  become  T? f t y , 1-14Kef  4t*\  -I  +  j£T£JLr*£. +  ^  ^  J7^  .  ^  ^  L<r*-f  +  *'  l  U l  Co  *  -^ 7  d  Z  4 . C c * c o * M ) - S ^ , * 6 )  j  W'« * « c  a  2  a e  '  1 7 )  and  I f we s e t 'ZL  1  and  then as  L—>0  Mjym P ^ Letting limiting  we h a v e  -  r Su)  the dimensionless quantities c a s e o f 1-17 a n d 1-18  dLjCt.L—^0  we h a v e t h e  16  and  which i s a recapturing o f equations noise  1-15 a n d 1-16 o f t h e " w h i t e  case.  F i n a l l y , we d e f i n e a q u a n t i t y ,  L_,  called the correlation  l e n g t h o f yuciL} ; i t i s t h e s m a l l e s t l e n g t h  izi > L  ==> 17*) ^  such  that  o  I t I s now . p o s s i b l e t o d e d u c e t h e f o r m o f t h e d i s p e r s i o n r e l a t i o n i n t h e two l i m i t i n g with respect  cases o f s h o r t and l o n g  correlation  length  t o the wavelength.  F i r s t we e x a m i n e t h e c a s e o f s h o r t c o r r e l a t i o n l e n g t h , i . e .  ^~/~^ ^ 1  where  0  coordinate rewrite  Z  with  Wo = /7\ - N o n - d i m e n s i o n a l i z i n g 27!  0  L_, i . e . s e t t i n g  Z = L ^ * i t i s now p o s s i b l e t o  1-11 a n d 1-12 a s  -.£*<xLfrCLi.*)  and  the depth  (2<xL**)di*  ('I-19)  -°°  -co  -£ zc<Ljr(L?.*) -CO By A p p e n d i x  con (2o<Lz*)  Jz*  (1-20)  D we h a v e t h a t i n t h e l i m i t i n g c a s e  ^j^Hud*  & f % J = l -  + 2*5Bfh*,d*  _oo  ^  L <<  +Ocp>  (1-22;  -co  and  = -2€*<xfr<»df: +0(p?  (1-22)  -co  where  I f t h e t e r m i n 1-21 i n v o l v i n g  the i n t e g r a l  -co  is  small  ( w h i c h seems p l a u s i b l e  f o r short correlation  lengths)  t h e n 1-21 a n d 1-22 a r e q u a l i t a t i v e l y t h e same a s t h e " w h i t e  noise"  f o r m u l a s 1-15 a n d 1-16 s i n c e t h e i n t e g r a l  f  R  i  ]  ^  >  a s i t i s one h a l f  0  o f t h e p o w e r s p e c t r u m of ju(±) e v a l u a t e d  at the  origin. Now i n t h e c a s e o f l o n g  correlation  length,  i . e . L / ^>> I  18 we c a n o b t a i n t h e a s y m p t o t i c e x p a n s i o n s  f o r the integrals i n  1-11 a n d 1-12 i n v o l v i n g t h e s i n e a n d c o s i n e f u n c t i o n s b y r e p e a t e d i n t e g r a t i o n b y p a r t s . By A p p e n d i x E we h a v e t h a t i n t h e l i m i t i n g case  L>>  , with  3e o  'd  and  = J.f Cayd* -oo  -p sty  7 -  i .  £* '°>. f-^r  *  CaC  6  £lH°)  and  valid  for 6  n o t near  6= O  o r 9 - 77".  In view of the experimental data mentioned e a r l i e r i n t h i s c h a p t e r a n d t h e r e m a r k s o f A p p e n d i x C we c a n n o t e x p e c t t o t r e a t the  case  L  > >  i n t h e o c e a n s . H o w e v e r , f o r m u l a s 1-23 a n d 1-24  may h a v e some r e l e v a n c e t o i n t e r n a l w a v e m o t i o n i n s t e l l a r atmospheres.  CHAPTER I I : THE TIME DEPENDENT CASE I I . 1 Formal  19  Dispersion Relation  As i n Chapter  I the equation  s a t i s f i e d by t h e stream f u n c t i o n  $(X,*,t) i s  +f - % i - q.'^ o (u-i)  i« i„« + +  tt  T h i s t i m e we assume H =-N (l+^/ ^) Z  i  0  where  =  yuct)  i sagain a  zero-mean, w i d e - s e n s e s t a t i o n a r y random p r o c e s s . T h i s equation I I - l  gives  t h e form  ' IJL + Jf) $  =  0~  (11-2)  where  and  In t h i s  case  the dispersion relation  t o second order  becomes  w h e r e eAC i s t h e i n t e g r a l o p e r a t o r d e f i n e d b y 1  CO  —  with  CO  Gr, t h e G r e e n ' s f u n c t i o n , b e i n g t h e s o l u t i o n o f  i n €  As  i n Chapter I , the Green's f u n c t i o n i s a displacement  kernel Q(X.X',i  ,^',t  ,t')  so i t i s n e c e s s a r y  M G  -  &<X.-X T:-±',-t-t.') , J  to consider  only  = JcxxfcvStt)  (JT-+)  We s o l v e b y t a k i n g a t r i p l e F o u r i e r t r a n s f o r m  of both  sides  of II-4 and define  Gr(k *.a) t  •= JJJ  G<.x.* t) t  e  dxdidt  -ca  Using <r  t h e c a u s a l i t y p r o p e r t y , Cr = 0  f o r ~t < 0 , t h e i n v e r s i o n i n  can be performed t o y i e l d t h e F o u r i e r t r a n s f o r m  of  G  a n d ~z.  :  „  with  in x  d= ^/g  -H(t)  ,  Usn  cut  and  // A/ok - +W-+Uf\L 1  for II.2  convenience. Simplifying the Dispersion  Relation  I t i s now p o s s i b l e t o s u b s t i t u t e ,jA> , dC~ to find the dispersion r e l a t i o n  explicitly.  J  a n d c//" i n t c l T  Set  and  Then  and V  _  -c(kxi-  ?z  -crt)  CO  '  dx'dt'c/t'  e  Simplifying,  CO -A3  T h i s can  be w r i t t e n  CO —Co  dx'c/i'dlt'  Make t h e c h a n g e o f v a r i a b l e s x"-x-x'  2" = Z - Z'  Then t h i s  gives  = -£»Y.«r-*v.y^-«^7 -""""-"^,g.^^.i  T  e  2  - C O  Now  taking account o f the i n t e g r a l  differentiation  being a function  of t , the  can be performed t o g i v e CO  Using the s t a t i o n a r i t y function  o f yU(±)  we d e f i n e  the autocovariance  Fct") b y  Then T  =,-e*/V*C-k + i-jq  -u^jy^a.ttyrtthe'^t*  z  2  -oo  4-  * -co The d i s p e r s i o n r e l a t i o n , c o r r e c t t o s e c o n d  order i n £, i s thus  CO  A  Setting  0  £=0  0  and r e a r r a n g i n g t h e terms o f e q u a t i o n I I - 5  g i v e s t h e O t h o r d e r d i s p e r s i o n r e l a t i o n a s i n A p p e n d i x A, w h e r e we h a d  Cf, k , 4% r e a l a n d  JL - M It form to  -td/  z  i s again p h y s i c a l l y meaningful  f o r Jl  into equation  OCe ) , a n d t h e n 2  t o s u b s t i t u t e t h e above  I I - 5 , the general d i s p e r s i o n r e l a t i o n  solve the equation f o r  (T . M a k i n g t h e  s u b s t i t u t i o n we o b t a i n  -era*  - td/Jd  +  ^^[f-  -r ffj?* -  c^j]\  24  This reduces t o g-Z( '+ k  Si  +  <t* ) - fli'l? - /VAV  <*%.) * £ A/„*[k 2  /4  E q u a t i o n I I - 6 i s an e q u a t i o n iteration.  Setting  € = O  x  <%«r'-r>-±  (r'-i'Jl*-  f o r & , a n d c a n b e s o l v e d by-  g i v e s t h e Oth order  solution  w h e r e we c h o o s e t h e p r i n c i p a l b r a n c h o f t h e s q u a r e r o o t f u n c t i o n for definiteness. Replacing  <T  by  (T  0  i n t h e Oct*) t e r m s a n d  rearranging y i e l d s the approximate s o l u t i o n  25 ,00  Dividing both  sides by  (f  and u s i n g the b i n o m i a l  0  approximation  f o r t h e s q u a r e r o o t we c a n w r i t e  o  —  •  +  / $jyyxO- ir  j  ir d r ' c t ; + CM r'U, 0  0  • -£<T0 £K  <h r ? t i j e ' W  (IT- 7)  tVr  +  For a given choice o f autocovariance  function  I (t  ftt),  "Rel^-l]  g i v e s t h e r e l a t i v e c h a n g e i n t h e p h a s e s p e e d o f t h e mean w a v e , a n d ^/m{%- ~^-} B  represents  are p o s i t i v e then  t h e growth or a t t e n u a t i o n f a c t o r . I f both  t h e mean wave t r a v e l s f a s t e r a n d g r o w s , i t i s  a l s o p o s s i b l e from equation  II-7 to derive asymptotic  the l i m i t i n g cases o f l o n g and s h o r t period) and  c o r r e l a t i o n times  1-24 o f C h a p t e r I .  results for  ( w i t h r e s p e c t t o t h e wave  s i m i l a r t o equations  1-21, 1-22, 1-2 3  CHAPTER I I I : DEPENDENT  FURTHER EXAMINATION OF THE CASE OF DEPTH  26  "WHITE NOISE" FLUCTUATIONS  I I I . l Introduction I n t h i s c h a p t e r we s h a l l b e c o n c e r n e d case o f "white n o i s e " y " ( 2 l )  primarily with the  v a r i a t i o n s and t h e e q u a t i o n s  and  (nr.' 2.) which  a r e f o r m u l a s 1-15 a n d 1-16 r e s p e c t i v e l y o f C h a p t e r I . In order t o o b t a i n a g r a p h i c a l r e p r e s e n t a t i o n corresponding  t o t h e s e f o r m u l a s we e m p l o y e d t h e d a t a f r o m  -8  2  d i s c u s s e d i n s e c t i o n 1.4. We p u t  s t a t i o n P (50°N, 145°W) _y  T -1.22 *I0 sec  c o r r e s p o n d i n g t o 50°N l a t i t u d e , No - 3* io %c  2  throughout  f o r the passband I  c a s e , a n d i n t h e a b s e n c e o f a n y d a t a we c h o s e No = 10 $*c passband I I case. Representative values o f  f o r the  cr w e r e s e l e c t e d f o r 1  b o t h p a s s b a n d s t o g i v e t h e v a l u e s o f a s y m p t o t e a n g l e , 9# = to** \fc^ 1  i n d i c a t e d on t h e c u r v e s . We t h e n l e t ( I I I - 2 ) i s even of I I I - l  9  r u n over the range  (odd) i n  as I I I - 1  9 , a n d c o m p u t e d b y m a c h i n e t h e R.H.S.  a n d I I I - 2 f o r t h e s e r a d i a n a n g l e s o f p r o p a g a t i o n . The  p r i n t - o u t was h a n d p l o t t e d on g r a p h drawn t h r o u g h paper  0, o.ie^,... 0.98#  paper,  smooth c u r v e s were  t h e p o i n t s and t h e c u r v e s were i n k e d onto  t o g i v e f i g u r e s 1, 2 a n d 3.  tracing  O.Q45  = 1.54-  -i  r-  1  "  «~  O.f-  FIG-  Q,  TRO'PR&tlTiON  1  Gr'RoW/TH (6>o)  J--^ ANGLE flUV  CRRDtRtfS) ftTTtMUaitON  quHD. JL.m S  R  ™  £  (6 <0)  Z  '  m  7?flT£S  Ill.2  D i s c u s s i o n of the  "White N o i s e "  Growth Curves  In order to i n t e r p r e t the graph of r a d i a n angle of propagation  for both  2  '/'ed.  versus  2  passbands of  <J  the  , i t shoul  2  -A  be  understood  t h a t t h e d i m e n s i o n l e s s q u a n t i t y €. d  d i f f e r e n t v a l u e f o r e a c h o f t h e two -A  d  we  a  recall  r T:*)dz  zz d  -co  dV  fcz) = nTi2j and 5 -  p a s s b a n d s . We  have  00  for  will  2  /0  orn  3  f o r the approximate  value  tit'  2  have d  for  -  3"  /0~t  csm'  1  the passband I case,  d - I0~  On'  n  and  1  .,2  for  t h e passband I I case, w i t h t h e above v a l u e s o f Examining  figure 1 i t i s clear  N  .  a  t h a t waves p r o p a g a t i n g  wi~h  an u p w a r d c o m p o n e n t o f d i r e c t i o n g r o w b u t t h a t w a v e s w i t h  a  downward c o m p o n e n t o f d i r e c t i o n a r e a t t e n u a t e d ,  growth  rate  i s an o d d  i s a l s o found  f u n c t i o n of the propagation i n the 0th order  i s g i v e n i n Appendix T h e r e we  saw  s i n c e the  angle. This  behaviour  c a s e w h e r e an e x p l a n a t i o n f o r i t  A.  t h a t t o 0 t h o r d e r a pl-^ne w a v e h a s  the  form  28 For  a p o l a r r e p r e s e n t a t i o n we s e t  and  which gives  the form  Now i n t h e 0 C6\> be  complex,  and  e  / z  d i s p e r s i o n r e l a t i o n we h a v e a l l o w e d  d£  to  i.e.  t h e mean wave h a s t h e f o r m  ^d.=t/ ^i[dt^ 2  But,  coo-ce-V) -<rt] -Xxfc°*-<e-Y>)  since i n the "white noise"  case  C'ct or  t h e n t h e mean wave f o r m c a n b e w r i t t e n a s  2  e  -e N  which i m p l i e s t h a t growth or a t t e n u a t i o n occurs has  a vertical  component o f p r o p a g a t i o n  only  ( i . e . 6 ^ o, 7T)  i f t h e wave and t h a t  the g r o w t h r a t e depends on t h e s q u a r e o f t h e c o r r e s p o n d i n g  Oth  o r d e r wave number. I n f i g u r e 1 wave number i s a " h i d d e n v a r i a b l e " ; as t h e a n g l e o f p r o p a g a t i o n  a p p r o a c h e s t h e a s y m p t o t e a n g l e we  f i n d t h a t t h e g r o w t h o r a t t e n u a t i o n r a t e becomes l a r g e b e c a u s e <tfo  becomes l a r g e . So f o r a g i v e n d i s t a n c e h  s h o r t e r waves w i l J  row f a s t e r  than l o n g e r waves, p o s s i b l y t o t h e p o i n t o f b r e a k i n g , i n t r a v e l l i n g upward from t h e p l a n e It i s reasonable  not uniformly v a l i d  noise"  d=-fa/^  the v a r i a t i o n  z = /?.  a  f l u c t u a t i o n s . The a n a l y s i s i s  i n the wavelength, as discussed  s h o r t waves a r e e x c l u d e d  Since  t o the plane  t h a t s h o r t e r w a v e s s h o u l d b e more a f f e c t e d b y  the presence o f t h e "white  C; v e r y  z = 0  we d e f i n e  i n Appendix  from c o n s i d e r a t i o n .  J0p , a l e n g t h s c a l e d e t e r m i n e d b y  i n ^> , b y  Also l e t us define  and b y a n a l o g y w i t h t h e a b o v e we d e f i n e JL^jf. b y  Now dejp > d  so £ f < 0<!i  inhomogeneities variations.  ; i n w o r d s , t h e mean e f f e c t o f t h e r a n d o m  i s t o shorten  the length scale of the p  Consequently by analogy w i t h the  corresponding  r e s u l t o f A p p e n d i x A g r o w t h o r a t t e n u a t i o n o f t h e mean wave i s necessary  t o preserve  k i n e t i c energy i n the presence o f t h e  d e c r e a s e d mean d e n s i t y v a r i a t i o n  sacle  £ tf oe  . Similarly  /V fr  defining  by  0e  de*  = M<8.  c, we s e e t h i s c a n b e i n t e r p r e t e d  a s an i n c r e a s e  i n the Brunt-  Vaisala frequency or a " s t i f f e n i n g " o f t h e f l u i d . U s i n g t h e s t a t i o n P d a t a d i s c u s s e d i n 1.4 i t i s p o s s i b l e t o e s t i m a t e t h e magnitude o f t h e change i n growth o r rates  and t h e q u a n t i t i e s  d ff. a n d /\£W  attenuation  . F o r b o t h summer a n d  c  w i n t e r i n t h e 700 - 1500 m r a n g e we f i n d CO  £J  Tczjdz  2  -  -oo Since  -9-1  i K 10 om  d rfotf^  Zalkan  - i  =  we h a v e ^  2e*TMoS^'e/^  S  2  C/  B6. to'  2  l  Z  S^yx & z  Thus t h e s t o c h a s t i c  growth r a t e  e f f e c t s a r e dominant.  i sessentially  i s ezT^o ^ ^ s  that  . Thus  0  radian.  thehorizontal  £ f-  Xo  z  o r A £. 10*<y>*\ = 1 k>m  Since t h e v e r t i c a l growth r a t e and  io'  o b s e r v e d f o r i n t e r n a l waves i n t h e P a c i f i c  >; 21T x IO~ m'  unless  2*IG*c*n  ^  then  7  £ V  respectively  of approximately  f o r a 1 km wave, a n d  i°  s^*i*eam'  J  1  to SMSGorn S  and  1  and  ^/o'tu^iecm  1  z  with  2*A3 c*« we f i n d v e r t i c a l a n d h o r i z o n t a l g r o w t h ?  &  z  rates 1  f o r a 100 m w a v e .  31 I f the waves a r e e s s e n t i a l l y h o r i z o n t a l l y p r o p a g a t i n g , as Zalkan' i . e . Q«  finds, and  , e.g. Q-o.l  1  ^ e  then  and i ^ ^ e - SQ  z  the r a t e s f o r a 100 m wave a r e approximately  and  /0  /0~ cm ~ 5  J  CsrrC . T h i s r e p r e s e n t s an e - f o l d i n g l e n g t h  o f 10  wavelengths i n the v e r t i c a l and 1 wavelength i n the h o r i z o n t a l . Since  A^ ^ 2  = d# y e  w i t h our d a t a . Thus and  then  Ke#./A/  ^ /0 <Xo ls*"-&f ^ -27T f o r 9  o.l  b  a  a wavelength o f 1 km. Hence the mean e f f e c t o f the random  f l u c t u a t i o n s i s to s u b s t a n t i a l l y " s t i f f e n " the B r u n t - V a i s a l a  the f l u i d  frequency e f f e c t i v e l y g r e a t e r .  rendering  This  suggests  i n t e r n a l waves i n passband I might e x i s t w i t h frequency the d e t e r m i n i s t i c  c u t - o f f . In f a c t some r e c e n t l y  i n t e r n a l wave frequency s p e c t r a  <r > N  0  observed  do n o t e x h i b i t a sharp  cut-off  A/ "' '" 5  at  7  N. 0  Looking a t the curves o f f i g u r e 1 we can account f o r t h e i r o v e r a l l appearance as f o l l o w s : 1) follows  Independence o f — 2 ™ { * % g j / o n as i t i s o n l y  c = (A/*-a-)/((r -f J 1  i  :i  z  the passband o f cr  2  which determines the  asymptotes, n o t the parameters i n d i v i d u a l l y . 2)  Growth or a t t e n u a t i o n  wave; hence f o r 0= O,Tr  requires  v e r t i c a l motion o f the  we have no growth or  attenuation.  8  Q -031 R  (10 V * =  0.+5  fl.Ql)  .65  .78 (1.51)  0.9S-  (l-oz)  l.lo  (0.63) j  O.f  0.8  1.1-1 (o.os; y i  1-2. (kuRV. JL, zzr  Q  FIG.  P R O P R  2  G-ftTION RELRTIVE  R N & L E  (7Z4D/#A/S)  7>H#SE SPEED  SRMF  CHRMGE  PS  X,  IX  &/s  0  -l)/e*£  0.+  0.8  .  r  i  1.2 "i  -0.5 67.55  0.6S  1 l.+l  (8.11)  (4-.53J  4-5 (2.33) - 0 . 6  031  (1,23)  0.7  -O.Q  G* = o.lS  {10*CT  = 0,37  5£c~*J  -0.9U  FIG.  3  XELRTl\JE E°K  PH4SE  TRSSdAHT)  5PEED 7J;  /\/  Z  <  QHRNGE  a~ *  <  -f*  (U.S)  ' .' 3)  ••  32  On a g i v e n c u r v e a s | e l  i n c r e a s e s t h e waves a r e s h o r t e r  and t h u s h a v e a h i g h e r g r o w t h o r a t t e n u a t i o n 4)  6fj  As  travelling  rate.  d e c r e a s e s from curve t o curve t h e waves 6  i n a fixed direction  a higher growth o r a t t e n u a t i o n  become s h o r t e r a n d t h u s h a v e  rate.  I I I . 3 D i s c u s s i o n o f t h e "White N o i s e " Phase Speed C u r v e s I n t h i s s e c t i o n we s h a l l a t t e m p t t o i n t e r p r e t t h e g r a p h s i n f i g u r e s 2 a n d 3, w h i c h c o r r e s p o n d t o f o r m u l a I I I - l  f o r passbr-.-.ds  I a n d I I r e s p e c t i v e l y . S i n c e t h e p a s s b a n d I c a s e , /" <  <  z  N , 2  0  i s t h e u s u a l c i r c u m s t a n c e we s h a l l g i v e more a t t e n t i o n t o i t . ( °/s ~ V/G'£ >  We n o t e t h a t when t h e o r d i n a t e  s  Oct*)  o r t h e mean p h a s e s p e e d t o  ±^ &*/o~  /0~*(' 0(1) -0(10)) o n e x a m i n i n g f i g u r e  :=2  c  i s l e s s than t h e phase speed i n  the non random c a s e . F o r o u r d a t a  Se/s-1  t h i s means <'  0  6  or  ^ /0"  r  and so  2. T h u s p h a s e s p e e d  changes a r e s m a l l , g e n e r a l l y s p e a k i n g . L e t us define monotonic  F<ej =  R.H.S.  of equation I I I - l ,  then  F^e; i s  i n c r e a s i n g i n passband I and monotonic d e c r e a s i n g i n  passband I I . A l s o  jR  0  +  =  i n the l i m i t i n g Fco) ^ - - / ^ t z  j  No )  '  ^  c a s e 8 —>'P , we h a v e 7  2  <r -l -f  i n passband I , and c r ? / 2  z  1  1  with  with  •£*<< /Vc  X  so  so Fco)^- / 2  z  i n passband I I . In passband I t h e t r a n s i t i o n case f o r w h i c h t h e phase  speed  33 to  Oct*) i s n o w h e r e g r e a t e r t h a n t h e O t h o r d e r p h a s e s p e e d  when and  F(o)-o  i . e . cr - = ^l/  <tg-<ry«r*~fV *1  or  c  1  2  1  o r d e r phase speed for large speed  ; thus  Q  t o Oct )  1  i t appears  z  z  l  .  ^-l.^l  of  S  corresponding  near  . Since  t h e i n c r e a s e i n phase  i s a r e s u l t o f t h e r o t a t i o n . H o w e v e r , a s 3—>  become s h o r t e r a n d a r e r e l a t i v e l y s l o w e d  cr - (Af*+-f J/z  i s g r e a t e r t h a n t h e Oth  1  except f o r a s m a l l range  we h a v e cr 1 -f  -  n  2 we s e e t h a t when  figure  /0 a- - O.o?, t h e p h a s e s p e e d 6  t h i s means  z  z  On e x a m i n i n g to  . F o r f << No  2  occurs  6#  t h e waves  down, r e f r a c t i o n  effects  a p p a r e n t l y b e c o m i n g d o m i n a n t . The s h o r t e r t h e wave t h e more i t i s s c a t t e r e d b y t h e r a n d o m i n h o m o g e n e i t i e s i n t h e medium a n d h e n c e the f a r t h e r  i t m u s t t r a v e l t o g e t f r o m one p l a c e t o a n o t h e r .  appears  as a decrease  i n t h e mean p h a s e  As  Qff d e c r e a s e s  from curve t o curve i n f i g u r e  decreases and, from Appendix 0  , i . e . the wavelength  become more i m p o r t a n t r e s u l t i n g l y i n g w h o l l y above t h e &»  6  speed.  A, t h i s means  i s decreased.  This  2,  C  z  d€o i n c r e a s e s f o r f i x e d  Thus t h e r e f r a c t i o n  effects  i n the upper c u r v e s o f f i g u r e  a x i s . On a n y g i v e n c u r v e f o r O  2  nearing  t h e e n d s o f t h e c u r v e t u r n u p s h a r p l y due t o r e f r a c t i o n o f  the r a p i d l y  s h o r t e n i n g waves.  As was m e n t i o n e d  i n the discussion of figure  t h e d i m e n s i o n l e s s q u a n t i t y € c? w i l l 2  To i n d i c a t e w h i c h  passband  s u b s c r i p t I o r I I . We  have  1,  :r,e s i z e o f  d e p e n d on t h e p a s s b a n d  we a r e c o n s i d e r i n g we  cf  s h a l l use t h e  C*~.  34 ell  - 3 * 10  * Cm  _/  and  djL — JO  73  cw ~2  thus  I t seems u n l i k e l y  t h a t the r a t i o  enough t o y i e l d a v a l u e o f  ^jcci^  €^ £'/Q  larger  than  c o u l d be £  x  small  c t . However, i n x  t h e a b s e n c e o f e x p e r i m e n t a l d a t a f o r t h e p a s s b a n d I I c a s e we c o n s i d e r t h e Oth o r d e r d i s p e r s i o n r e l a t i o n idea of the importance relative  i n o r d e r t o o b t a i n some  of the f l u c t u a t i o n s i n  f o r passband I I  to passband I .  Now 2  3.  /Vox ~  •_  0~  z x  cr - /2  x  or  - J.22 * JO~  a  and C  -  •O-  ~ N  CJC  ~—  ~7l  or  ,2  Let  OJC  C  x  —  = c£  JO  with  their  shall  common v a l u e . Now  l e t us  increase in  C  by 1 0 % i n each case and l e t A C  z  be t h e i n c r e m e n t  . Then  z  a-/  -1.22*10-*  and  Now  III-3 and III-4  can be  solved for  r e s u l t s s u b s t i t u t e d i n t o the above t o Ar  z  - -$xl0~ ( 1+ 7  CT^  and  and  the  yield  C) 2  and  Hence  L0  ^2  Z  Thus t h e a s y m p t o t e s o f t h e Oth o r d e r d i s p e r s i o n r e l a t i o n much more a f f e c t e d b y c h a n g e s  i n the B r u n t - V a i s a l a  are  frequency i n  passband I than i n passband II. This a n a l y s i s suggests t h a t the change i n phase negligible, figure  s p e e d o f t h e mean wave f o r p a s s b a n d I I w i l l  so we  be  s h a l l not concern ourselves w i t h i n t e r p r e t i n g  3, w h i c h i s i n c l u d e d f o r c o m p l e t e n e s s .  A d i f f e r e n c e i n the b e h a v i o u r o f waves o f passbands i s n o t s u r p r i s i n g when i t i s r e c a l l e d t h a t t h e two  I and  n  classes of  i n t e r n a l w a v e s h a v e e s s e n t i a l l y d i f f e r e n t d e p e n d e n c i e s on t h e media  t h r o u g h w h i c h t h e y p r o p a g a t e . Passband I waves cannot  exist  36 i n an u n s t r a t i f i e d ocean and so a r e l i k e l y to f l u c t u a t i o n s and c a n e x i s t  in  than passband  i n an u n s t r a t i f i e d  t o b e more  sensitive  I I waves, w h i c h a r e  ocean.  inertial  BIBLIOGRAPHY 1.  37  B a t c h e l o r , G.K., A n I n t r o d u c t i o n t o F l u i d  D y n a m i c s , C.U.P.,  1 9 6 8 .  2.  B h a r u c h a - R e i d , A.TV, On t h e t h e o r y o f r a n d o m e q u a t i o n s , Symp. A p p l . M a t h . , X V I , p p . 4 0 - 6 9 , 1 9 6 4 .  Proc.  3.  B o y c e , W.E., Random e i g e n v a l u e p r o b l e m s , Probabilistic. Methods i n A p p l i e d Mathematics, I , p p . 1 - 7 3 , E d i t e d b y A . T . B h a r u c h a - R e i d , A c a d e m i c P r e s s , New Y o r k , 1 9 6 8 .  4.  C a r r i e r , G.F., K r o o k , M., a n d P e a r s o n , C.E., Complex V a r i a b l e : T h e o r y and T e c h n i q u e ,  Functions o f a McGraw-Hill,  1 9 6 6 .  5„  F a f o n o f f , N.P., R o l e o f t h e NDBS i n f u t u r e n a t u r a l v a r i a b i l i t y s t u d i e s o f the North A t l a n t i c , Proc. F i r s t Science A d v i s o r y M e e t i n g , N a t i o n a l D a t a Buoy D e v e l o p m e n t P r o j e c t , U.S. C o a s t G u a r d , p p . 5 0 - 6 1 , 1 9 6 9 .  6.  F r i s c h , U., Wave p r o p a g a t i o n i n r a n d o m m e d i a . P r o b a b i l i s t i c Methods i n A p p l i e d Mathematics, I , p p . 7 5 - 1 9 8 , E d i t e d b y A . T . B h a r u c h a - R e i d , A c a d e m i c P r e s s , New Y o r k , 1 9 6 8 .  7.  G a r r e t t , C. a n d Munk, W., I n t e r n a l wave s p e c t r a i n t h e presence o f f i n e - s t r u c t u r e , J . Phys. 0 c e a n o q r , I , 0  pp.  8o 9.  1 9 6 - 2 0 2 ,  1 9 7 1 .  K e l l e r , J.B., Wave p r o p a g a t i o n A p p l . Math., X I I I , p p o 2 2  i n r a n d o m m e d i a , P r o c . Symp. 7 - 2 4 6 ,  1 9 6 2 .  K e l l e r , J . B . , S t o c h a s t i c e q u a t i o n s a n d wave p r o p a g a t i o n i n r a n d o m m e d i a , P r o c . Symp. A p p l . M a t h . , X V I , p p . 1 4 5 - 1  7 0 ,  1 9 6 4 .  10.  11.  1 2 c  K e l l e r , J.B., T h e v e l o c i t y a n d a t t e n u a t i o n o f w a v e s i n a r a n d o m 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 , p p . 8 2 3 - 8 3 E d i t e d b y R . L . R o w e l l and R . S . S t e i n , Gordon and B r e a c h S c i e n c e P u b l i s h e r s , New Y o r k , 1 9 6 7 . K e l l e r , J.B. a n d V e r o n i s , G., R o s s b y w a v e s i n t h e o f random c u r r e n t s , J . Geophys. Res., 7 4 , 8 , April 1 5 , 1 9 6 9 .  4 ,  presence p p .  1 9 4 1 - 5 1 ,  L e B l o n d , P „ H o , On t h e d a m p i n g o f i n t e r n a l g r a v i t y w a v e s i n a c o n t i n u o u s l y s t r a t i f i e d ocean, J . F l u i d Mech., 2 5 , I , pp.  1 2 1 - 1 4 2 ,  1 9 6 6 .  38 13.  P a r z e n , E., S t o c h a s t i c P r o c e s s e s , H o l d e n - D a y I n c . , S a n F r a n c i s c o , 1964.  14.  Phillips,  15.  T a y l o r , A.E., A d v a n c e d C a l c u l u s , B l a i s d e l l P u b . Co., T o r o n t o , 1955. Wang, Y.C., The i n t e r a c t i o n o f i n t e r n a l w a v e s w i t h a n unsteady n o n - u n i f o r m c u r r e n t , J . F l u i d Mech., 37, I V , pp. 761-771, 1969.  16.  O.M.,  The D y n a m i c s o f t h e U p p e r O c e a n , C.U.P., 1 9 6 6 .  17.  W e b s t e r , T.F., L e c t u r e s , S e c o n d C o l l o q u i u m o n t h e H y d r o d y n a m i c s o f t h e O c e a n , L i e g e U n i v e r s i t y , C a h i e r s de m e c h a n i q u e m a t h e m a t i q u e , 26, p p . 20-53, 1 9 7 0 .  18.  Z a l k a n , R.L., H i g h f r e q u e n c y i n t e r n a l w a v e s i n t h e P a c i f i c O c e a n , Deep-Sea R e s e a r c h , 1 7 , p p . 9 1 - 1 0 8 , F e b . 1 9 7 0 .  39  APPENDICES A- The O t h O r d e r D i s p e r s i o n For  Relation  a p l a n e wave  fluid with  constant  propagating  Brunt-Vaisala. frequency  u n i f o r m l y r o t a t i n g about t h e v e r t i c a l  A4  (z) axis,  in a  stratified  that i s the dispersion  relation i s  C = (tf-<r\)/(o- -f )  where  z  x  , <t = rtZ/3 , •£ =  x  X. = h o r i z o n t a l d i s t a n c e , P u t t i n g H^fx+i^x  t =  parameter,  time.  J,  with  coriolis  and r e q u i r i n g k  real  K  r e a l (as  there  i s no p h y s i c a l r e a s o n f o r g r o w t h o r a t t e n u a t i o n o f t h e w a v e  along  the h o r i z o n t a l direction) gives  4r  = -  ^  and  (A-l)  c * f = A p l a n e wave now h a s t h e f o r m propagating  e  Q  . Waves  w i t h an upward component o f d i r e c t i o n grow, and waves  w i t h a downward c o m p o n e n t o f d i r e c t i o n a r e a t t e n u a t e d . /V - N  0  , a constant,  i m p l i e s {> c2)°<e c  wave i s p r o p o r t i o n a l t o $>A Hence t h e f a c t o r  Q  di/,z  2  where  A  Now  having  . The k i n e t i c e n e r g y o f a i s t h e wave  amplitude.  i s seen t o be n e c e s s a r y t o p r e s e r v e  the  k i n e t i c energy o f t h e wave. The  equation  e.*k*-J* =  corresponds t o a d i s p e r s i o n diagrairf  consisting of a rectangular hyperbola values,  for fixed  0~ . We  introduce  opening toward large a polar  coordinate  I M  40 r e p r e s e n t a t i o n o f t h e r e a l wave n u m b e r s .  (*j  = fa (tote, SlsriQ)  S u b s t i t u t i n g i n t o A - l , we  9  for  6  find  (-Qf,,Q ) K}(7r-e^jlTi-Qf,) fl  the asymptotes t o the hyperbola given  6^  where  , the angle  are i n c l i n e d to the  k  at which  -axis, i s  by  Thus i t i s c l e a r  t h a t an i n c r e a s e i n  increase i n  0#  e q u a t i o n A-2  implies that for fixed  in  Set  or a spreading  a decrease i n  d&  0  6  an i n c r e a s e i n C  z  Equation  Z) be a p o i n t i n a r i g h t - h a n d e d  v e r t i c a l and p o s i t i v e upward.  =  zQ.  rotates w i t h angular  f  frequency  X~L  i s the magnitude of the  i s t h e l a t i t u d e . Then t h e s y s t e m -f/z  . If  the e f f e c t i v e g r a v i t a t i o n a l a c c e l e r a t i o n ,  ^o.  -axis, which i s  ^  e a r t h ' s r o t r .'on v e c t o r a n d  d e n s i t y and  2  system of C a r t e s i a n  Let  be t h e C o r i o l i s p a r a m e t e r , where  B - a x i s , (u,i/,ur)  results  •  coordinates r o t a t i n g u n i f o r m l y about the  f  c o r r e s p o n d s t o an  of the asymptotes. In a d d i t i o n  B- D e r i v a t i o n o f t h e S t r e a m F u n c t i o n Let  C  are the f l u i d  $  i s the magnitude  a n t i - p a r a l l e l to the  v e l o c i t y components,  i s the pressure,  of  t h e n t h e momentum  ^  i s the  fluid  conservation  41 equations are  W-t + UV  +  X  ar  for  t  + uus  + yur^ + usur^ + g  K  + ^  u  = O  + ly ^  =  0  an i n v i s c i d f l u i d . The  equation + (fiL)  and  . + usu± +f  o f mass c o n s e r v a t i o n i s  -h (£(Sj  x  v  + (^ur)  %  -  o  the incompressibility condition i s ft  +  Initally  w  px +  i- "Sfe  -  0  setting  ( U, l/,  UJ)  - ( 0 , 0, 0)  gives w i t h ^=^ (Hj / f ^ - - f ( i ) the b a s i c s t a t e o f h y d r o s t a t i c 0  9  equilibrium  Introducing ^cx,  perturbations  such  that  y, 2,-e; = ^> <rz) +Q (x,\/.-t)  (U.W, us)  0  = <u  t  ,u},ur ) t  f- - f-J?) + fcC^V.-t) and we  linearizing obtain  the equations  i n the quantities subscripted with  1  f„u^ U  IX  +$9  ft?  1-  f i/ + ur^ ly  Pit f  (S-3)  = 0  = o  (B-4-)  = 0  _ •  D r o p p i n g t h e s u b s c r i p t 1, fotU  - -fx  ^oLW  - --p-^ +ff-x.  x±  B - l , B-2  •  (B-5)  yield  -ff.?  where  Thus  and  Applying  pL 0  t o B-4 y i e l d s  S u b s t i t u t i n g B-6, B-7 g i v e s  or"  f « z t +f-wit - (%°Lw ) i  E q u a t i o n B-3 g i v e s fo^tt  -f-^69 + fit  -  0  Using equation B-5 fo  or  u^. + -fit -9%*  LLT  - O  i  - 0  (B-8)  43 Thus  and  Substituting  this  i n t o B-8  produces  Alternatively,  + e<*/f <"htt +f ?°Vp * X  ur  o  Put  A/ =  -yf *^  U  -h us  2  0  and  dy = 0 , t h e n B-9  = 0 reduces  (13-9) to  Now i  K  -  O  implies  where  3  i s a stream f u n c t i o n !  Thus t h e e q u a t i o n f o r  seen t o be  the  same a s e q u a t i o n 1-1  §  is ,  of the  text.  44  C- K e l l e r ' s M e t h o d We now g i v e a b r i e f a c c o u n t o f K e l l e r ' s m e t h o d deriving the dispersion r e l a t i o n  for  f o r t h e mean wave i n a r a n d o m  medium. A more g e n e r a l t r e a t m e n t w i l l b e f o u n d  i n Keller's  paper  in Electromagnetic Scattering " 1  If  c<f  X  i s an i n v e r t i b l e random l i n e a r o p e r a t o r and  is  a known f u n c t i o n , t h e n t h e e q u a t i o n  implies  $  to both  Taking  sides of equation  ensemble averages,  Inverting  <£~ > 2  <^" >" <$> 1  Now  X  i s a random p r o c e s s . A p p l y i n g C - l we  the inverse of  obtain  denoted by <  > , this  becomes  gives = X  1  i f «*f  oC,  (C-2)  i s s t a t i s t i c a l l y homogeneous  a n d ~X.-Q  equation  C-2 h a s a s e i g e n d i f f e r e n t i a l s t h e p l a n e wave s o l u t i o n s g i v e n b y  w h i c h obey t h e d i s p e r s i o n r e l a t i o n ^«tx+J?z-crt) ^ -i^(kxi-J>z-(r-L) <  l>  S i n c e Z£-cM. •+<//' binomial expansion  and  =  Q  (  c  _  3  J\T i s " s m a l l " c o m p a r e d t o JA.  i s used t o g e t  :  )  the  45  X'  -<M~ -M-'JfM' *  1  2  0  1  i f . II <M i ^ H < 1  which i s v a l i d  K.tA/'y —  M-'JiTM' ^^'  1  ,  . Averaging  1  this equation with  one o b t a i n s  <X~ > =M'~ +<M- <Jf<M-W><M- 2  L  1  . ...  t  This i s inverted t o y i e l d  w h i c h i s c o r r e c t t o s e c o n d o r d e r i n JV*  , and hence  £ .  S u b s t i t u t i n g C-4 i n t o C-3 g i v e s  e " " " - " ^ - ^ ^ * " ' ' ^ as t h e d i s p e r s i o n r e l a t i o n wave  correct to In Chapter  §cx.t.t) Thus i n t h i s  0(( ). 3  fc-s;  amplitude  mean  E q u a t i o n C-5 i s i d e n t i c a l w i t h I I - 3 .  I we h a v e  = e"  tirt  c a s e C-5 r e d u c e s t o  - "'*"'{j(-<w Jf>}e''" "'*' i(  e  o f the i n f i n i t e s i m a l  o  I  = o  0  w h i c h c o i n c i d e s w i t h e q u a t i o n 1-3. The  f o r e g o i n g a n a l y s i s i s dependent f o r i t s v a l i d i t y on t h e  not e a s i l y to  applicable condition that  achieve another  \\Ji~ <J/'ll  be s m a l l .  In order  p e r s p e c t i v e o n t h e v a l i d i t y o f e q u a t i o n C-5 we  p r e s e n t an a l t e r n a t i v e d e r i v a t i o n o f i t . Proceeding  2  directly  from  (M+Jf)$  - o  ( - ) C  6  we h a v e  <M$ Applying  §  (c-7)  -  uU' t o b o t h 1  s i d e s o f C-7 we o b t a i n  = -M^JV^  (C-8)  Now C-8 i s a n i n t e g r a l e q u a t i o n a n d t h e f i r s t  substitution  yields  Taking ensemble  averages  < § > = M ~<JfM l  ' u4r§ > l  Thus  M<$>  = <Jf<M-W$>  (C-?)  We now make t h e s o - c a l l e d  KJfM-'Jfgy  closure  assumption  = <JTM' J/ X§> 1  3 , 6  (C-10)  ,  U s i n g C-10 i n C-9 g i v e s  and a g a i n  < 3? >  the approximate  h a s t h e p l a n e wave . s o l u t i o n s  i  so  d i s p e r s i o n r e l a t i o n becomes  i d e n t i c a l w i t h e q u a t i o n C-5„ The  c r u c i a l point i n the analysis  i s the approximation  C-10,  On p . 45 W.E.Boyce justify  g i v e s an a c c o u n t o f R . C . B o u r r e t ' s a t t e m p t t o  t h i s a p p r o x i m a t i o n b y a n a r g u m e n t somewhat s i m i l a r  to the  followingo I n t h e t e r m <"J/Ui'tV§>it i s q u i t e $ C-6.  c a n n o t be  clear uV  independent of those of  However, i f t h e s c a l e s o f v a r i a t i o n o f  g r e a t l y d i f f e r e n t t h e n some j u s t i f i c a t i o n I f t h e r a n d o m p r o c e s s e s yU. ergodic property,  and  ^  Jf  and  $  J/  and  1  f o r C-10  §  are  c a n be  $  given.  a r e assumed t o h a v e t h e  that i f the scales  are g r e a t l y d i f f e r e n t  as c o n s t a n t w h i l e a s m a l l s c a l e  made o f t h e o t h e r . T h i s a v e r a g e a v e r a g e . Then a l a r g e  to  of  i t w o u l d be  a  J/'M.' ^  g o o d a p p r o x i m a t i o n t o r e g a r d t h e more s l o w l y v a r y i n g o f and  of  i n view of equation  i . e . space and time a v e r a g e s a r e e q u i v a l e n t  e n s e m b l e a v e r a g e s , t h e n i t seems c l e a r variation of  that the s t a t i s t i c s  1  space or time average i s  c o u l d b e r e g a r d e d a s an  s c a l e a v e r a g e c o u l d be  ensemble  taken of the remaining  process w i t h the p r e v i o u s l y averaged q u a n t i t y b e i n g r e g a r d e d as a constant. T h i s second average ensemble average When we Oct ) 1  u s e d t h e Oth o r d e r s o l u t i o n ,  terms and e x p e c t e d t h e f i r s t  Oct ) t e r m s 2  d i f f e r e n t from  identified  i n view of the assumption of  g i v e a good a p p r o x i m a t i o n f o r the  c o u l d a l s o be  be 0  d€  as  an  ergodicity.  e q u a t i o n 1-9,  i n the  i t e r a t i o n o f e q u a t i o n 1-8 i t was  n e c e s s a r y t o assume  small, or e q u i v a l e n t l y t h a t  be n o t  . T h i s assumption e n a b l e d us t o use  b i n o m i a l a p p r o x i m a t i o n t o o b t a i n e q u a t i o n s 1-11  and  to that  greatly  the  1-12.  similar  48 remarks apply t o the d e r i v a t i o n o f equation Now  I I - 7 from I I - 6 .  i f we r e s u b s t i t u t e  2]]C ccr<C'6 -Sun'6 l  into equations that the  I - 1 1 a n d 1-12,  Oct') t e r m s m u s t b e s m a l l i m p l i e s t h a t t h e q u a n t i t y  C*c<xi.'G-svn*e  cannot approach 0 a r b i t r a r i l y  cannot approach and  i t w i l l be clear t h a t the assumption  closely,  i.e.  , d e f i n e d i n A p p e n d i x A, a r b i t r a r i l y  i n o t h e r words the a n a l y s i s cannot be expected  very s h o r t waves. That t h e  Oct -) 3  6 closely,  t o hold for  p e r t u r b a t i o n s cease t o remain  s m a l l a s t h e w a v e s become s h o r t e r a n d s h o r t e r i s i n d i c a t e d i n f i g u r e s 1, 2 a n d 3 f o r " w h i t e n o i s e " E x c l u d i n g very Shockwaves' w i t h keeping is  o n l y second order  yu(2).  6m t h e a n a l y s i s i s . c o n s i s t e n t '  terms i n €  s o b e c a u s e h i g h e r p o w e r s o f J/  i n equation C - 4 .  w o u l d add h i g h e r powers o f  t h e wave number t o t h e d i s p e r s i o n r e l a t i o n a n d t h e s e expected  t o be n e g l i g i b l e  f o r very  This  could n o t be  s h o r t waves, a s has been  assumed i n u s i n g C-5. A s a f i n a l n o t e we p o i n t o u t t h e f a c t t h a t t h e a n a l y z e d experimental  data  considered i n Chapter  "white n o i s e " t o a good a p p r o x i m a t i o n ; expected  t o be v a l i d  I i n d i c a t e d jjid)  was  and the a n a l y s i s cannot be  f o r v e r y s h o r t waves, i . e . waves whose s c a l e  o f v a r i a t i o n a p p r o a c h e s t h e s c a l e o f v a r i a t i o n o f ydd)  . This i s  i n agreement w i t h t h e d i s c u s s i o n o f the a l t e r n a t i v e d e r i v a t i o n o f e q u a t i o n C-5 i n v o l v i n g t h e c l o s u r e a s s u m p t i o n  C-10.  D- The L i m i t i n g C a s e  L <<  49  Xa  W i t h r e f e r e n c e t o f o r m u l a s 1-19 a n d 1-20 we d e f i n e  -co  I  3  = Lf°r<L2*)  c<**(2*L-z*)dz*  and  Now  o<= de i*nd 0  a n d we d e f i n e  then Ii  (i = 1,2.3.+)  a n d we w a n t t o t a k e  (C - 1, Z. 3. 4-J  JjLvrx II (&) We h a v e  J- <f) 2  -  LJ  r(Lz*)  sun  dz*  — CO  r , f f ) = Lf ra.?:*) c<xL($i*)di* and  Let = I  Hi*)  2*r(Li*)\  and  Now t h e i n t e g r a l s  and  -  CO  converge as  P  i s an autocovariance  that the functions  svn(piV/qi*  f u n c t i o n , i t i s e a s y t o show  , s^cp?*)  , aacpzV  -are a b s o l u t e l y b o u n d e d b y 1 i n d e p e n d e n t o f  ±  £J  [l-caufpiVj/pz*  and . Hence  he z*;  and  Then b y T h e o r e m the for  first a  V I I I p . 667 o f T a y l o r  kind  U'  i n t-co,coJ  = i,2,3.-f)  1 5  t h e improper i n t e g r a l s o f  above a r e u n i f o r m l y  . Hence we may t a k e  JU*n (3-->0  integral  signs t o obtain  J^n (3-f>0  I (B) = 2L  ft*r(Li*)  2  t  1  _oo J  c o n v e r g e2nt !  under t h e  51 I (&)  -  2  O  and  Changing  variables  2 if/**("(Live!=  2 f l r i ) d z (  and  CO  Hence e q u a t i o n s  1-21 a n d 1-22 f o l l o w .  E- The L i m i t i n g C a s e  X  0  << L  H e r e we d e r i v e f o r m u l a s  1-23 a n d 1-24 f o r t h e l i m i t i n g  a n d p r o v e we h a v e o b t a i n e d a t r u e a s y m p t o t i c We h a v e oC=dfo^ne=  d  u  /  n  —  6  =  Define  J  d,i  J  Then T - T°<ir<±)  I - J  With  — CO  ~dT'  L2«*  .  e  2 = L?* .a x  J  d^  e  cdt*  d*  2frsunQ  result.  case  52 ^ ^irsutiej^o/^  Letting as  =  (3 —>oo  we  can o b t a i n  the asymptotic expansion " of 4  a l o n g the r e a l a x i s by r e p e a t e d i n t e g r a t i o n by  I t i s n e c e s s a r y t o assume to r e s t r i c t  &  P  I  parts.  i s s u f f i c i e n t l y d i f f e r e n t i a b l e and  so i t i s n o t n e a r  & - o  Q  r  0 = 77~  .  Letting  cU*"  1  2*  -co and  -oo as  r  i s a n a u t o c o v a r i a n c e f u n c t i o n . We  i n t e g r a t i o n s by  obtain  after  N  parts  where _  To p r o v e as  ^_>oo  fixed  til.  C  oC^rcL^j  that t h i s i s the v a l i d asymptotic expansion  along the r e a l a x i s i t i s necessary  t o prove  of  I  that  for  M.  =  O  (E-i)  -co  r  Now t h e i n t e g r a l ' i n E-2 i s j u s t t h e sum o f t w o e x p r e s s i o n s p r o p o r t i o n a l t o t h e F o u r i e r s i n e and c o s i n e t r a n s f o r m s o f t h e (N+l)th d e r i v a t i v e o f an a u t o c o v a r i a n c e die  out a t i n f i n i t y  proved as  i n any p h y s i c a l  f u n c t i o n and hence must  system.  Thus e q u a t i o n E - l i s  a n d we h a v e o b t a i n e d t h e t r u e a s y m p t o t i c e x p a n s i o n  of  I  ^ - > « . Keeping  fi-yOO  I  o n l y t h e Oth o r d e r  J -CO  i n ^/$  gives  Of 2  and  Hence i n t h i s l i m i t i n g c a s e  the formulas  1-11 a n d 1-12 r e d u c e t o  -f  and 2* l&o  where  £e -&°/d 0  a n d d = d J VcDdi.  valid for Q  n o t near  6 -o  -"CO  or  Q - IT . T h e s e a r e f o r m u l a s  text.  1-23 a n d 1-24 r e s p e c t i v e l y o f t h e