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INTERNAL WAVES IN A RANDOMLY STRATIFIED 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 IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e Department o f M a t h e m a t i c s We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1972 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 m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f ITlcit&iswastiCsas T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada D a t e IAaaXZ /+/ 72 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 waves i n a r o t a t i n g s t r a t i f i e d unbounded ocean w i t h randomly 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 t o o b t a i n t h e d i s p e r s i o n r e l a t i o n f o r t h e mean wave f i e l d c o r r e c t t o second 2 2 o r d e r i n £ when N i s o f t h e form where - c o n s t a n t , 0 < " £ ^ < 1 and yx 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 . e i t h e r d e pth 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 r e l a t i o n 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 the change i n phase speed 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 the random f l u c t u a t i o n s yU . These f o r m u l a s a r e dependent on t h e s t a t i s t i c s of /u o n l y t h r o u g h the a u t o c o v a r i a n c e f u n c t i o n . The phase speed change and growth r a t e f o r m u l a s f o r depth dependent yu , w h i c h c o n s t i t u t e s a model o f the t e m p e r a t u r e and s a l i n i t y f i n e - s t r u c t u r e i n t h e ocean, a r e p r e s e n t e d f o r v a r i o u s s p e c i a l c a s e s i n c l u d i n g t h e l i m i t i n g c a s e s o f c o r r r 1 a t i o n l e n g t h s 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) i n d i c a t e t h a t , 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 " and 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 of the s t a t i o n P d a t a i t i s e s t i m a t e d t h a t , a l t h o u g h t h e phase speed changes a r e g e n e r a l l y s m a l l , t h ^ a m p l i t u d e o f a wave i n c r e a s e s ( d e c r e a s e s ) s i g n i f i c a n t l y i n p r o p a g a t i n g upward (downward) t h r o u g h a d e p t h 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 t h a t t h e mean e f f e c t o f t h e depth dependent f l u c t u a t i o n s /U i s t o i n c r e a s e t h e e f f e c t i v e B r u n t - V a i s a l a . f r e q u e n c y , or " s t i f f e n " t h e f l u i d . T h i s 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 f r e q u e n c y s p e c t r a o f i n t e r n a l waves do n o t e x h i b i t a sharp c u t - o f f a t A/0, t h e d e t e r -m i n i s t i c t h e o r e t i c a l upper bound f o r t h e wave f r e q u e n c y . F i n a l l y an a t t e m p t i s made t o a s s e s t h e range o f v a l i d i t y o f K e l l e r ' s method i n t h e c o n t e x t o f t h e p r e s e n t problem. TABLE OF CONTENTS i v Chapter 0: I n t r o d u c t i o n - 1 Chapter I : The Depth Dependent Case 3 1.1 F o r m a l D i s p e r s i o n R e l a t i o n 3 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 R e l a t i o n 5 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 R e l a t i o n 9 1.4 S p e c i a l Cases 12 Chapter I I : The Time Dependent Case 19 11.1 F o r m a l D i s p e r s i o n R e l a t i o n 19 11.2 S i m p l i f y i n g t h e D i s p e r s i o n R e l a t i o n .20 Chapter I I I F u r t h e r 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 B i b l i o g r a p h y 37 A p p e n d i c e s : 39 A- The 0 t h Order D i s p e r s i o n R e l a t i o n 39 B- D e r i v a t i o n o f t h e Stream F u n c t i o n E q u a t i o n 40 C- K e l l e r ' s Method 44 D- The L i m i t i n g Case L << ~XQ 49 E- The L i m i t i n g Case ^a <'< L . • 51 LIST OF TABLES v T a b l e I Sample Computed A u t o c o v a r i a n c e F u n c t i o n 13 LIST OF FIGURES F i g u r e 1 Growth and A t t e n u a t i o n R a t e s F i g u r e 2 R e l a t i v e Phase Speed Change f o r Passband I : -f2< Q~2< F i g u r e 3 R e l a t i v e Phase Speed Change f o r Passband l l : A o < < T * ' ACKNOWLEDGMENT v i i I am p l e a s e d t o thank my s u p e r v i s o r , p r o f . L.A.Mysak, p r e s e n t l y a S e n i o r V i s i t o r i n t h e Department o f A p p l i e d Mathe-m a t i c s and T h e o r e t i c a l P h y s i c s a t Cambridge U n i v e r s i t y , f o r h i s g r a c i o u s encouragement and a s s i s t a n c e i n t h e w r i t i n g o f t h i s t h e s i s . I n a d d i t i o n I s h o u l d l i k e t o thank P r o f . P.H.LeBlond f o r e x a m i n i n g t h e t h e s i s , and Dr.. R.E.Thomson f o r t h e use o f t h e d a t a . T h i s r e s e a r c h was a c c o m p l i s h e d w i t h f i n a n c i a l a s s i s t s - - e i n the form o f an H.R.MacMillan F a m i l y F e l l o w s h i p . CHAPTER 0: INTRODUCTION 1 I t has l o n g been known t h a t i n t e r n a l g r a v i t y wave m o t i o n jjf. e x i s t s i n t h e ocean (see C h a p t e r 5 o f P h i l l i p s ) . I n o r d e r t o p r o p a g a t e , t h e s e waves r e q u i r e a s t r a t i f i e d medium such as t h e ocean where l e s s dense s u r f a c e l a y e r s r e s t on more dense deeper l a y e r s . A measure o f t h e s t r e n g t h o f t h e s t r a t i f i c a t i o n i s t h e B r u n t - V a i s a l a f r e q u e n c y , N , d e f i n e d by t h e e q u a t i o n where o 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 and paii) i s t h e r e s t d e n s i t y o f t h e w a t e r as a f u n c t i o n o f depth H, w h i c h i s p o s i t i v e upwards. O b s e r v a t i o n s i n d i c a t e t h a t i n t e r n a l waves f r e q u e n t l y do n o t p r o p a g a t e w i t h c o n s t a n t a m p l i t u d e b u t grow t o t h e p o i n t o f b r e a k -i n g o r become damped o u t . 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'" 2 and i n t e r a c t i o n w i t h c u r r e n t s ' f In t h i s t h e s i s we s h a l l examii.. the e f f e c t on t h e waves o f random 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 s t u d y i n g t h e d i s p e r s i o n r e l a t i o n . Many s t u d i e s o f i n t e r n a l waves have assumed a s t r a t i f i c a t i o n such t h a t l\j d) — N0 , a c o n s t a n t . T h i s i s done f o r two r e a s o n s : f i r s t , 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 on depth, w h i c h may be a r e a s o n a b l e a p p r o x i m a t i o n t o t h e a c t u a l depen-dence a t l e a s t s e c t i o n a l l y ; second, i t r e d u c e s a "E-dependent c o e f f i c i e n t i n the i n t e r n a l wave e q u a t i o n t o a c o n s t a n t , i . e . , a 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 p r o p e r t i e s o f f r e e wave 2 s o l u t i o n s f o r t h i s case a r e w e l l known and a r e summarized i n Appendix A. Here we s h a l l assume N1 i s s u b j e c t t o s m a l l random f l u c t u a t i o n s about a c o n s t a n t mean Nl. i n C h a p t e r I t h e random f l u c t u a t i o n s w i l l be depth dependent and i n C h a p t e r I I t i m e dependent f l u c t u a t i o n s w i l l be c o n s i d e r e d . I n C h a p t e r I I I we d i s -c uss g r a p h s o f t h e r e l a t i v e phase speed change and growth r a t e p l o t t e d u s i n g r e a l o c e a n i c d a t a i n t h e c o r r e s p o n d i n g f o r m u l a s f o r " w h i t e n o i s e " f l u c t u a t i o n s d e r i v e d i n C h a p t e r I . CHAPTER I : THE DEPTH DEPENDENT CASE 3 1.1 Formal D i s p e r s i o n R e l a t i o n I n A ppendix B we b e g i n w i t h t h e e q u a t i o n s o f mass and 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 , i n v i s c i d f l u i d i n a r i g h t - h a n d e d 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 r o t a t i n g w i t h a n g u l a r f r e q u e n c y "£? about t h e 2 - a x i s , w h i c h i s v e r t i c a l and 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 s m a l l p e r t u r b a t i o n s i n t h e f l u i d v e l o c i t y , p r e s s u r e and d e n s i t y a r e imposed. I f t h e r e s u l t i n g e q u a t i o n s a r e l i n e a r i z e d and 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 m o t i o n then t h e stream f u n c t i o n i s shown t o s a t i s f y t h e e q u a t i o n C« + + = 0 (i-1) where x i s the h o r i z o n t a l c o o r d i n a t e and ± i s t h e t i m e . We assume depth dependent random f l u c t u a t i o n s i n N2 o f t h e form where yu(2) i s 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 " and £ i s a s i z e parameter such t h a t 0< €.*<< 1 . R e s t r i c t i n g a t t e n t i o n t o the case o f h a r m o n i c time dependence we s e t T h i s g i v e s e q u a t i o n 1-1 t h e form where a d e t e r m i n i s t i c o p e r a t o r , and a " s m a l l " random o p e r a t o r . E q u a t i o n 1-2 i s i n a form 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 method' ' ' o f w h i c h we g i v e an a c c o u n t i n Appendix C. There we show t h a t , c o r r e c t t o second 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 the mean wave < > (where < > i n d i c a t e s ensemble average) i s where JA.'1 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 by CO -oo w i t h Gr , t h e Green's f u n c t i o n , b e i n g the s o l u t i o n o f JAG- =• Scx-x'/fa-z') S i n c e ^/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 and Z, t h e Green's f u n c t i o n i s a d i s p l a c e m e n t k e r n e l i n cM.1, t h a t i s , GrCx.K', 2,Z') = Crcx-x', z - z.'J so i t i s n e c e s s a r y t o c o n s i d e r o n l y E q u a t i o n 1-4 can be s o l v e d by t a k i n g a d o u b l e F o u r i e r t r a n s f o r m o f b o t h s i d e s , i . e . m u l t i p l y b o t h s i d e s by Q~c(kx- ^^v) and i n t e g r a t e from -oo t o -hco s u c c e s s i v e l y w i t h r e s p e c t t o * and 2, . The i n v e r s i o n i n JL 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 the F o u r i e r t r a n s f o r m i n o f Q , w h i c h we denote by G(k,z) , so t h a t U(K,IL) - ... -where H i s the H e a v i s i d e f u n c t i o n and we have s e t d = »vg and f o r c o n v e n i e n c e . I n t h e d e t e r m i n i s t i c t h e o r y t h e r e a r e two passbands o f cJ 3 d e f i n e d by i n e q u a l i t i e s : r e p r e s e n t s passband I waves and r e p r e s e n t s passband I I waves. I n e i t h e r case CZ > O. 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 R e l a t i o n 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 At, J/i'1 Jf i n t o 1-3 t o f i n d t h e d i s p e r s i o n r e l a t i o n e x p l i c i t l y . S e t 7 = e - « Y * « * ' » ^ e < « x , / e ; and Then and <9U(*)L j±f*d*. -<43*lJJGrtx-x.', 2 - 2 7 — CO S i m p l i f y i n g -co T h i s can be w r i t t e n CO — OC3 Make t h e change o f v a r i a b l e s OL" - x- x' Then t h i s g i v e s Now t a k i n g i n t o a c c o u n t t h a t t h e i n t e g r a l i s a f u n c t i o n o f Z, the d i f f e r e n t i a t i o n can be pe r f o r m e d 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 t h e a u t o c o v a r i a n c e f u n c t i o n , Tc?";, by Vet") = )yjLC I-?.")> Now 8 and -$UCZj/4z*S2--z<';> ^ ~ P2» £u< zJyU ( * -Thus — Q3 - 0 0 and i n each case the i n t e g r a t i o n o v e r produces GcCk.z") i n t h e above i n t e g r a l s . So, c o r r e c t t o second 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 r e d u c e s t o -oo U s i n g i n t e g r a t i o n by p a r t s we can w r i t e -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 and CO . C o l l e c t i n g terms we o b t a i n -e\t[0.-<:**]fed% 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 f o r t h e mean f i e l d <'V'), c o r r e c t t o Oce1). We n o t e t h a t t h e Oct*) terms, w h i c h a r e t h e c o r r e c t i o n s t o the d i s p e r s i o n r e l a t i o n f o r i n t e r n a l waves i n t h e non random f l u i d ( i . e . t h e above w i t h £ - O ), a r e dependent on the passband o f &? 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 R e l a t i o n 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 i n Appendix A we employ a p o l a r r e p r e s e n t a t i o n o f t h e wavenumbers, i . e . k = Wo co<u9 and where £g0 i s r e a l . We want 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 t o t h e OcelJ d i s p e r s i o n r e l a t i o n , i . e . s e t k^^C^G (1-6) < 10 and I - Xs+me -cd/z (1-7) where 6 i s a r e a l a n g l e and B€ may be complex. S u b s t i t u t i n g 1-6 and 1-7 i n t o T - 5 and s i m p l i f y i n g we o b t a i n r ^ L ^w* -<llfrr: ^ ( ^ ^ % * ) e ****** -co CI-8) Now i f we f i x an a n g l e o f p r o p a g a t i o n , 0 , and s p e c i f y t h e q u a n t i t i e s ffjf, A/0) (/ and f c ? ; t h e n e q u a t i o n 1-8 i s an e q u a t i o n f o r ?6 w h i c h can be s o l v e d by i t e r a t i o n (see K e l l e r and Veronis") , S e t t i n g € = 0 g i v e s the 0 t h o r d e r s o l u t i o n o f Appendix A d?o - • ^ , • CT-Q) S u b s t i t u t i n g ^ 0 f o r i n t h e 0 C£l) terms g i v e s t h e f i r s t i t e r a t i o n s o l u t i o n (which i s v a l i d i f the Oc(?) terms a r e s m a l l , see d i s c u s s i o n Appendix C) S i n c e swtfe**^- and CoxL,(a z j a r e even f u n c t i o n s o f oc , t h e a b s o l u t e v a l u e s i g n s can be dropped. F o r c o n v e n i e n c e s e t o< = Mo s*m e and T a k i n g t h e square r o o t o f b o t h s i d e s o f 1-10 and u s i n g the b i n o m i a l e x p a n s i o n on the R.H.S. (assuming the Oct1) terms a r e s m a l l r e l a t i v e t o 1 ) , we o b t a i n o —00 °^ - c a Hence i t r e a d i l y f o l l o w s t h a t <>• -Too . - e^i^Vd) %Xn l^idi (I-11) -co and 2 6 0 - ffcfrt*)(l-Co<L2*2)dti -<=Mfrti)(l + Co«.2^)di (1-12) -°° -co 1-4 S p e c i a l Cases 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.) a r e " w h i t e n o i s e " so t h a t O b s e r v a t i o n s t a k e n on F e b r u a r y 23, 1971 and J u l y 29, 1970 a t weather s h i p s t a t i o n P (50°N, 145°W) and a n a l y z e d by R i c h a r d Thomson o f Environment Canada 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 j u s t i f i e d o v e r c e r t a i n depth r a n g e s . The q u a n t i t y N2(D was computed a t 10 meter i n t e r v a l s from t h e ocean s u r f a c e t o a depth 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 /\J2d) e x h i b i t s a s t r o n g t r e n d w i t h d e pth i n t h e upper 700 m e t e r s . However, between the d e p t h s o f 700 m and 1500 m /\/ 2<Z) a p p e a r s t o have l i t t l e t r e n d w i t h mean s*e~Z and s t a t i o n a r y f l u c t u a t i o n s , Cju.it), about M> w i t h €^^0(^L) . F u r t h e r m o r e , t h e a u t ^ - j o v a r i a n c e f u n c t i o n s f o r b o t h t h e summer and w i n t e r d a t a computed u s i n g t h e mean, /V0 , over t h i s range i n d i c a t e t h e €/ua)'s a r e " w h i t e n o i s e " t o a good a p p r o x i m a t i o n . We i n c l u d e f o r i l l u s t r a t i o n i n t a b l e I the a u t o c o v a r i a n c e f u n c t i o n computed from t h e w i n t e r d a t a . 13 TABLE I, SAMPLE COMPUTED AUTOCOVARIANCE FUNCTION DEPTH LAG, Z /m €2 T< 2 ) 0 0.19580 10 .... 0.09219 20 0.01881 30 0.06659 40 0.02786 50 0.00000 60 0.02475 70 0.0659 3 80 0.02659 90 0.03095 100 0.06968 110 0.04162 120 0.01218 130 0.0597" 140 0.0750b 150 0.01380 S u b s t i t u t i n g f o r i n f o r m u l a s 1-11 and 1-12 y i e l d s and U s i n g e q u a t i o n 1-9 and 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 t h e d i m e n s i o n l e s s q u a n t i t y dV by cT where I f 5 0 i s the phase speed o f t h e Oth o r d e r wave and S i s t h e phase speed o f t h e Oct2) wave t h e n and when i s n e g a t i v e t h e wave grows, t h u s we can r e w r i t e 1-13 and 1-14 as - ^ _ ^ a - i s ) and . .Equations 1-15 and 1-16 a r e d i s c u s s e d a t l e n g t h i n C h a p t e r I I I where t h e y appear as e q u a t i o n s I I I - l and I I I - 2 . A t t h i s p o i n t we j u s t m e n t i o n t h a t from 1-16 i t i s c l e a r t h a t , i n d e p e n d e n t o f the (j2 passband, waves p r o p a g a t i n g w i t h an upward component o f d i r e c t i o n grow and waves p r o p a g a t i n g w i t h a downward component decay w i t h t h e magnitude o f t h e growth or decay dependent on t h e passband o f <T o n l y t h r o u g h 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 . Next we c o n s i d e r the s p e c i a l case o f 1-11 and 1-12 f o r w h i c h 15 w h i c h i s t h e a u t o c o v a r i a n c e f u n c t i o n o f a U h l e n b e c k - O r n s t e i n 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 as t h e y have the 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 and 1-12 become T? f t y , 1-14- j£T£JLr*£. . Co*d - ^ 7 2 Kef 4t*\ - I + ^ ^ ^ ^ L<r*-fl 4 . C c * c o * M ) - S ^ , * 6 ) Z j + J7^ + * ' U l W ' « c * a « e a ' 1 7 ) and I f we s e t 1 'ZL and then as L—>0 we have Mjym P ^ - r Su) L e t t i n g t h e d i m e n s i o n l e s s q u a n t i t i e s dLjCt.L—^0 we have t h e l i m i t i n g case o f 1-17 and 1-18 16 and w h i c h i s a r e c a p t u r i n g o f e q u a t i o n s 1-15 and 1-16 o f t h e " w h i t e n o i s e c a s e . F i n a l l y , we d e f i n e a q u a n t i t y , L_, c a l l e d t h e c o r r e l a t i o n 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 such t h a t i z i > L ==> 17*) ^ o I t I s now . p o s s i b l e t o deduce the form 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 c a ses o f s h o r t and l o n g c o r r e l a t i o n l e n g t h 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 . F i r s t we examine t h e case 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 . ^~/~^0 ^  1 where Wo = 27!/7\0- N o n - d i m e n s i o n a l i z i n g t h e depth c o o r d i n a t e Z w i t h 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 r e w r i t e 1-11 and 1-12 as -.£*<xLfrCLi.*) (2<xL**)di* ('I-19) and -°° -co -£ zc<Ljr(L?.*) con (2o<Lz*) Jz* (1-20) -CO By Ap p e n d i x D we have t h a t i n t h e l i m i t i n g case L < < & f % J = l - ^j^Hud* + 2*5Bfh*,d* +Ocp> (1-22; _oo ^ -co and = -2€*<xfr<»df: +0(p? (1-22) -co where I f the t e r m i n 1-21 i n v o l v i n g t h e i n t e g r a l -co i s s m a l l (which seems p l a u s i b l e f o r s h o r t c o r r e l a t i o n l e n g t h s ) t h e n 1-21 and 1-22 a r e q u a l i t a t i v e l y the same as t h e " w h i t e n o i s e " f o r m u l a s 1-15 and 1-16 s i n c e the i n t e g r a l f R i ] ^ > 0 as i t i s one h a l f o f t h e power spectrum of ju(±) e v a l u a t e d a t t h e o r i g i n . Now i n t h e case o f l o n g c o r r e l a t i o n l e n g t h , i . e . L / ^ > > I 18 we can 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 t h e i n t e g r a l s i n 1-11 and 1-12 i n v o l v i n g t h e s i n e and c o s i n e f u n c t i o n s by r e p e a t e d i n t e g r a t i o n by p a r t s . By Appendix E we have t h a t i n t h e l i m i t i n g case L>> , w i t h 3eo - ' d and = J.f Cayd* -oo -p sty 7 - i . £*r'°>. f-^- CaC*6 £lH°) and v a l i d f o r 6 n o t near 6= O or 9 - 77". I n v i e w o f t h e e x p e r i m e n t a l d a t a mentioned e a r l i e r i n t h i s c h a p t e r and t h e remarks o f Appendix C we cannot e x p e c t t o t r e a t t h e case L > > i n t h e oceans. However, f o r m u l a s 1-23 and 1-24 may have some r e l e v a n c e t o i n t e r n a l wave m o t i o n i n s t e l l a r a tmospheres. CHAPTER I I : THE TIME DEPENDENT CASE 19 I I . 1 Formal D i s p e r s i o n R e l a t i o n As i n Cha p t e r I t h e e q u a t i o n 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 i « + i „ « + +f - % itt - q.'^ = o (u-i) T h i s time we assume HZ=-N0(l+^/i^) where yuct) i s a g a i n 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 g i v e s e q u a t i o n I I - l t h e form ' IJL + Jf) $ = 0~ (11-2) where and I n t h i s case t h e d i s p e r s i o n r e l a t i o n t o second o r d e r i n € becomes where eAC1 i s the i n t e g r a l o p e r a t o r d e f i n e d by CO — CO w i t h Gr, t h e Green'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 As i n Ch a p t e r I , the Green's f u n c t i o n i s a d i s p l a c e m e n t k e r n e l Q(X.X',i , ^ ' , t , t ' ) - &<X.-X ,JT:-±',-t-t.') so i t i s n e c e s s a r y t o c o n s i d e r o n l y M G = JcxxfcvStt) (JT-+) We s o l v e by 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 o f b o t h s i d e s o f I I - 4 and d e f i n e Gr(kt*.a) •= JJJ G<.x.*tt) e dxdidt -ca U s i n g 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 <r can be p e r f o r m e d 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 i n x and ~z. o f G : „ , -H(t) Usn cut w i t h d= ^/g and // A/ok1- +W-+Uf\L f o r c o n v e n i e n c e . I I . 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 R e l a t i o n 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~J and c//" i n t c T l -t o f i n d t h e d i s p e r s i o n r e l a t i o n e x p l i c i t l y . S e t and Then and V _ -c(kxi- ?z -crt) CO ' e dx'dt'c/t' S i m p l i f y i n g , CO -A3 T h i s can be w r i t t e n CO — C o d x ' c / i ' d l t ' Make t h e change o f v a r i a b l e s x"-x-x' 2" = Z - Z' Then t h i s g i v e s T2 = - £ » Y . « r - * v . y ^ - « ^ 7 e - " " " " - " ^ , g . ^ ^ . i - C O Now t a k i n g a c c o u n t o f the i n t e g r a l b e i n g a f u n c t i o n o f t , t h e d i f f e r e n t i a t i o n can be per f o r m e d t o g i v e CO U s i n g t h e s t a t i o n a r i t y o f yU(±) we d e f i n e t he a u t o c o v a r i a n c e f u n c t i o n Fct") by Then T2 =,-e*/V*C-kz + i-jq -u^jy^a.ttyrtthe'^t* -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 second o r d e r i n £, i s t h u s CO A 0 0 S e t t i n g £ = 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 the Oth o r d e r d i s p e r s i o n r e l a t i o n as i n Appendix A, where we had Cf, k , 4% r e a l and JL - M -td/z I t i s a g a i n p h y s i c a l l y m e a n i n g f u l t o s u b s t i t u t e t h e above form f o r Jl i n t o e q u a t i o n I I - 5 , t h e g e n e r a l d i s p e r s i o n r e l a t i o n t o OCe2) , and t h e n s o l v e the e q u a t i o n f o r (T . Making 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 T h i s r e d u c e s t o g-Z(k'+ Si + <t*/4) - fli'l? - /VAV <*%.) * £2A/„*[kx- <%«r'-r>-± (r'-i'Jl*-E q u a t i o n I I - 6 i s an e q u a t i o n f o r & , and can be s o l v e d by-i t e r a t i o n . S e t t i n g € = O g i v e s t h e Oth o r d e r s o l u t i o n where we choose the p r i n c i p a l b r a n c h o f t h e square r o o t f u n c t i o n f o r d e f i n i t e n e s s . R e p l a c i n g <T by (T0 i n t h e Oct*) terms and r e a r r a n g i n g y i e l d s t h e a p p r o x i m a t e s o l u t i o n 25 ,00 D i v i d i n g b o t h s i d e s by (f0 and u s i n g the b i n o m i a l a p p r o x i m a t i o n f o r t h e square r o o t we can w r i t e o — • j / $jyyxO-0ir • -£<T0£K I (t + ir0d r ' c t ; + CM r'U, + <h r ? t i j etVr' W ( I T - 7 ) F o r a g i v e n c h o i c e o f a u t o c o v a r i a n c e f u n c t i o n f t t ) , "Rel^-l] g i v e s t h e r e l a t i v e change i n the phase speed o f t h e mean wave, and ^/m{%-B~^-} r e p r e s e n t s t h e growth or a t t e n u a t i o n f a c t o r . I f b o t h a r e p o s i t i v e t h e n t h e mean wave t r a v e l s f a s t e r and grows, i t i s a l s o p o s s i b l e from e q u a t i o n I I - 7 t o d e r i v e a s y m p t o t i c r e s u l t s f o r the l i m i t i n g c a s e s o f l o n g and s h o r t ( w i t h r e s p e c t t o t h e wave p e r i o d ) c o r r e l a t i o n t i m e s s i m i l a r t o e q u a t i o n s 1-21, 1-22, 1-2 3 and 1-24 o f Chapter I . CHAPTER I I I : FURTHER EXAMINATION OF THE CASE OF DEPTH 26 DEPENDENT "WHITE NOISE" FLUCTUATIONS I I I . l I n t r o d u c t i o n I n t h i s c h a p t e r we s h a l l be co n c e r n e d p r i m a r i l y w i t h t h e case o f " w h i t e n o i s e " y " ( 2 l ) 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.) w h i c h a r e f o r m u l a s 1-15 and 1-16 r e s p e c t i v e l y o f Ch a p t e r I . I n o r d e r 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 c o r r e s p o n d i n g t o t h e s e f o r m u l a s we employed t h e d a t a from s t a t i o n P (50°N, 145°W) 2 -8 _ y d i s c u s s e d i n s e c t i o n 1.4. We p u t T -1.22 *I0 sec t h r o u g h o u t 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 %c2 f o r t h e passband I ca s e , and i n the absence o f any d a t a we chose No = 10 $*c f o r t h e passband I I c a s e . R e p r e s e n t a t i v e v a l u e s o f cr1 were s e l e c t e d f o r b o t h passbands t o g i v e t h e v a l u e s o f asymptote a n g l e , 9# = to**1\fc^ i n d i c a t e d on the c u r v e s . We t h e n l e t 9 r u n o v e r t h e range 0, o.ie^,... 0.98# as I I I - 1 ( I I I - 2 ) i s even (odd) i n 9 , and computed by machine t h e R.H.S. o f I I I - l and 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 hand p l o t t e d on graph paper, smooth c u r v e s were drawn t h r o u g h 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 o n t o t r a c i n g paper t o g i v e f i g u r e s 1, 2 and 3. O.Q45 = 1.54-- i r- 1 " «~ O.f- J--^ quHD. JL.m Q, TRO'PR&tlTiON ANGLE CRRDtRtfS) S R ™ £ Z ' m FIG- 1 Gr'RoW/TH (6>o) flUV ftTTtMUaitON (6 <0) 7?flT£S I l l . 2 D i s c u s s i o n o f t h e "White N o i s e " Growth C u r v e s 2 I n o r d e r t o i n t e r p r e t t h e graph o f '/'e2d. v e r s u s t h e r a d i a n a n g l e o f p r o p a g a t i o n f o r b o t h passbands o f <J2 , i t s h o u l -A be u n d e r s t o o d 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 €.2 d w i l l have a d i f f e r e n t v a l u e f o r each o f t h e two passbands. We r e c a l l 0 0 -A r d zz d T:*)dz -co - dV f o r fcz) = nTi2j and f o r t h e a p p r o x i m a t e v a l u e 5 - /03 orn tit'2 we have d - 3" /0~t csm'1 f o r t h e passband I c a s e , and d - I0~n On'1 . , 2 f o r t h e passband I I c a s e , w i t h t h e above v a l u e s o f Na . E x a m i n i n g f i g u r e 1 i t i s c l e a r t h a t waves p r o p a g a t i n g w i ~ h an upward component o f d i r e c t i o n grow b u t t h a t waves w i t h a downward component o f d i r e c t i o n a r e a t t e n u a t e d , s i n c e the growth r a t e i s an odd f u n c t i o n o f t h e p r o p a g a t i o n a n g l e . T h i s b e h a v i o u r i s a l s o f o u n d i n t h e 0 t h o r d e r case where an e x p l a n a t i o n f o r i t i s g i v e n i n Appendix A. There we saw t h a t t o 0 t h o r d e r a pl-^ne wave has 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 w h i c h g i v e s the form e / z Now i n the 0 C6\> d i s p e r s i o n r e l a t i o n we have a l l o w e d d£ t o be complex, i . e . and t h e mean wave has the form ^d.=t/2 ^i[dt^ coo-ce-V) -<rt] -Xxfc°*-<e-Y>) But, s i n c e i n the " w h i t e n o i s e " case C'ct or then t h e mean wave form can be w r i t t e n as e 2 -e N w h i c h i m p l i e s t h a t growth o r a t t e n u a t i o n o c c u r s o n l y i f the wave has a v e r t i c a l component o f p r o p a g a t i o n ( i . e . 6 ^  o, 7T) and t h a t the growth r a t e depends on the square o f the 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 the a n g l e o f p r o p a g a t i o n approaches t h e asymptote a n g l e we f i n d t h a t t h e growth o r a t t e n u a t i o n r a t e becomes l a r g e because < t f o 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 t h a n 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 the p l a n e z = 0 t o the p l a n e z = /?. I t i s r e a s o n a b l e t h a t s h o r t e r waves s h o u l d be more a f f e c t e d by the p r e s e n c e o f t h e " w h i t e n o i s e " f l u c t u a t i o n s . The a n a l y s i s i s n o t u n i f o r m l y v a l i d i n the w a v e l e n g t h , as d i s c u s s e d i n Appendix C; v e r y s h o r t waves a r e e x c l u d e d from c o n s i d e r a t i o n . S i n c e d=-fa/^a we d e f i n e J0p , a l e n g t h s c a l e d e t e r m i n e d by the v a r i a t i o n i n >^ , by A l s o l e t us d e f i n e and by a n a l o g y w i t h t h e above we d e f i n e JL^jf. by Now dejp > d so £0<!if < ; i n words, t h e mean e f f e c t o f t h e random i n h o m o g e n e i t i e s i s t o s h o r t e n t h e l e n g t h s c a l e o f t h e p v a r i a t i o n s . C o n s e q u e n t l y by a n a l o g y w i t h t h e c o r r e s p o n d i n g r e s u l t o f Appendix A growth o r a t t e n u a t i o n o f t h e mean wave i s n e c e s s a r y t o p r e s e r v e k i n e t i c energy i n the p r e s e n c e 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 s a c l e £oetf . S i m i l a r l y d e f i n i n g /V0 efr by de* = M<8. c, we see t h i s can be i n t e r p r e t e d as an i n c r e a s e i n t h e B r u n t -V a i s a l a f r e q u e n c y o r 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 the 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 a t t e n u a t i o n r a t e s and the q u a n t i t i e s dcff. and /\£W . F o r b o t h summer and w i n t e r i n t h e 700 - 1500 m range we f i n d CO £2J Tczjdz - ^ 2*IG*c*n -oo -9-1 S i n c e d i K 10 om we have rfotf^ - i = 2e*TMoS^'e/^ ^ io'z XoZ S^yxz& Z a l k a n 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 t h a t >; 21T x IO~SC/m'2 o r A 0 £. 10*<y>*\ = 1 k>m . Thus u n l e s s B6. to'2 r a d i a n . Thus the s t o c h a s t i c e f f e c t s a r e dominant. S i n c e t h e v e r t i c a l growth r a t e i s e s s e n t i a l l y £z V z& and t h e h o r i z o n t a l growth r a t e i s ezT^o s^7^ t h e n w i t h £ l f - 2*A3?c*« we f i n d v e r t i c a l and h o r i z o n t a l growth r a t e s r e s p e c t i v e l y o f a p p r o x i m a t e l y to  SSMSGorn1 and 1 f o r a 1 km wave, and i°  Js^*i*eam'1 and ^/o'tu^iecm1 f o r a 100 m wave. 31 If the waves are e s s e n t i a l l y h o r i z o n t a l l y propagating, as Zalkan' 8 finds, i . e . Q« 1 , e.g. Q-o.l then ^ ez and i ^ ^ e - SQ and the rates for a 100 m wave are approximately /0~5 cm ~J and /0 CsrrC . This represents an e-folding length of 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^ = de# y then with our data. Thus Ke#./A/a ^ /0b<Xo ls*"-&f ^ -27T for 9 o.l and a wavelength of 1 km. Hence the mean e f f e c t of the random fluctuations i s to sub s t a n t i a l l y " s t i f f e n " the f l u i d rendering the Brunt-Vaisala frequency e f f e c t i v e l y greater. This suggests inter n a l waves i n passband I might e x i s t with frequency <r > N0 the deterministic cut-off. In fact some recently observed in t e r n a l wave frequency spectra do not ex h i b i t a sharp cut-off A/ 5"' 7'" at N0. Looking at the curves of figure 1 we can account for t h e i r o v e r a l l appearance as follows: 1) Independence of — 2 ™ { * % g j / o n the passband of cr 2 follows as i t i s only c 1 = (A/*-a-i)/((r:i-fzJ which determines the asymptotes, not the parameters i n d i v i d u a l l y . 2) Growth or attenuation requires v e r t i c a l motion of the wave; hence for 0= O,Tr we have no growth or attenuation. QR-031 (10 V* = 0.+5 . 6 5 fl.Ql) .78 (1.51) 0.9S- (l-oz) l.lo (0.63) 1.1-1 (o.os; j y i O.f 0.8 1-2. (kuRV. JL, zzr Q P R O P R G-ftTION R N & L E (7Z4D/#A/S) SRMF PS X, IX FIG. 2 RELRTIVE 7>H#SE SPEED CHRMGE &/s -l)/e*£ 0 -0.5 - 0 . 6 0.+ . 0.8 1.2 i r "i 1 6 7 . 5 5 (8.11) 0.6S (4-.53J 4-5 (2.33) l.+l (U.S) 031 (1,23) 0.7 -O.Q - 0 . 9 U G* = o.lS {10*CT = 0,37 5£c~*J FIG. 3 XELRTl\JE PH4SE 5PEED QHRNGE E°K TRSSdAHT) 7J; /\/Z < a~ * < -f* ' .' •• 32 3) On a g i v e n c u r v e as |el 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 have a h i g h e r growth o r a t t e n u a t i o n r a t e . 4) As 6fj d e c r e a s e s from c u r v e t o cu r v e t h e waves t r a v e l l i n g i n a f i x e d d i r e c t i o n 6 become s h o r t e r and t h u s have a h i g h e r growth o r a t t e n u a t i o n r a t e . 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 gr a p h s i n f i g u r e s 2 and 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 and I I r e s p e c t i v e l y . S i n c e the passband I ca s e , /"z< < N02, i s t he 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 . We n o t e t h a t when t h e o r d i n a t e (s°/s ~ V/G'£ >  0 t h i s means <'c o r t h e mean phase speed t o Oct*) i s l e s s t h a n t h e phase speed i n the non random c a s e . For our d a t a ±^ &*/o~6 o r ^ / 0 " r and so Se/s-1 :=2 /0~*(' 0(1) -0(10)) on e x a m i n i n g f i g u r e 2. Thus phase speed 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 d e f i n e F<ej = R . H . S . o f e q u a t i o n I I I - l , t h e n F^e; i s monotonic 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 j- No j R 0 ) = + ' ^ i n the l i m i t i n g case 8 —>'7P2 , we have <rz-l -f1 w i t h •£*<< so Fco) ^ - - z / ^ t i n passband I , and c r 2 ? / 1 w i t h /VcX so Fco)^-2/z i n passband I I . I n 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 t o Oct*) i s nowhere g r e a t e r t h a n t h e Oth o r d e r phase speed o c c u r s when F(o)-o i . e . cr2- = ^l/z . For fz<< No t h i s means crz - (Af*+-flJ/z and <tg-<ry«r*~fV *1 o r c 2 1 ; t h u s Qn - . On e x a m i n i n g f i g u r e 2 we see t h a t when ^ - l . ^ l c o r r e s p o n d i n g t o /06a-1 - O.o?, t h e phase speed t o Oct1) i s g r e a t e r t h a n t h e Oth o r d e r phase speed e x c e p t f o r a s m a l l range o f S n e a r . S i n c e f o r l a r g e we have cr11 -fz i t a p p e a r s t h e i n c r e a s e i n phase speed i s a r e s u l t o f t h e r o t a t i o n . However, as 3—> 6# t h e waves become s h o r t e r and a r e r e l a t i v e l y slowed down, r e f r a c t i o n e f f e c t s a p p a r e n t l y becoming dominant. 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 by t h e random i n h o m o g e n e i t i e s i n t h e medium and hence the f a r t h e r i t must t r a v e l t o g e t from one p l a c e t o a n o t h e r . T h i s appears as a d e c r e a s e i n the mean phase speed. As Qff d e c r e a s e s from c u r v e t o c u r v e i n f i g u r e 2 , Cz d e c r e a s e s and, from Appendix A, t h i s means d€o i n c r e a s e s f o r f i x e d 0 , i . e . t h e w a v e l e n g t h i s d e c r e a s e d . Thus t h e r e f r a c t i o n e f f e c t s become more i m p o r t a n t r e s u l t i n g i n the upper c u r v e s o f f i g u r e 2 l y i n g w h o l l y above t h e 6 a x i s . On any g i v e n c u r v e f o r O n e a r i n g &» the ends o f the c u r v e t u r n up 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 men t i o n e d i n t h e d i s c u s s i o n o f f i g u r e 1, :r,e s i z e o f the d i m e n s i o n l e s s q u a n t i t y € 2c? w i l l depend on t h e passband c f C*~. To i n d i c a t e w h i c h passband we a r e c o n s i d e r i n g we s h a l l use t h e s u b s c r i p t I o r I I . We have 34 ell - 3 * 10 * Cm_/ and t h u s djL — JO 7 3 cw ~2 I t seems u n l i k e l y t h a t the r a t i o €^ £'/Q c o u l d be s m a l l enough t o y i e l d a v a l u e o f ^jcci^ l a r g e r t h a n £ x c t x . However, i n the absence o f e x p e r i m e n t a l d a t a f o r t h e passband I I case we s h a l l 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 i n o r d e r t o o b t a i n some i d e a o f t h e i m p o r t a n c e o f t h e f l u c t u a t i o n s i n f o r passband I I r e l a t i v e t o passband I . Now 2 z 3. •_ /Vox ~ 0~x crx2 - /-or - J.22 * JO~a and o r C - ~ NCJC •O- ~7l ~— ,2 OJC — JO L e t Cx = c£ w i t h t h e i r common v a l u e . Now l e t us i n c r e a s e by 1 0 % i n each case and l e t A Cz be t h e i n c r e m e n t i n Cz . Then a-/ -1.22*10-* and Now III-3 and III-4 can be s o l v e d f o r CT^ and and t h e r e s u l t s s u b s t i t u t e d i n t o t h e above t o y i e l d Arz - -$xl0~7( 1+ C2) and Hence ^2 L0 Z Thus t h e asymptotes o f the Oth o r d e r d i s p e r s i o n r e l a t i o n a r e much more a f f e c t e d by changes i n t h e B r u n t - V a i s a l a f r e q u e n c y i n passband I t h a n i n passband I I . T h i s a n a l y s i s s u g g e s t s t h a t t h e change i n phase speed o f t h e mean wave f o r passband II w i l l be n e g l i g i b l e , so we s h a l l n o t c o n c e r n o u r s e l v e s w i t h i n t e r p r e t i n g f i g u r e 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 t h e b e h a v i o u r o f waves o f passbands I and n 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 c l a s s e s o f i n t e r n a l waves have 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 e x i s t 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 t o be more s e n s i t i v e t o f l u c t u a t i o n s i n t h a n passband I I waves, w h i c h a r e i n e r t i a l and can e x i s t i n an u n s t r a t i f i e d ocean. BIBLIOGRAPHY 3 7 1 . B a t c h e l o r , G.K., An I n t r o d u c t i o n t o F l u i d Dynamics, 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 random e q u a t i o n s , P r o c . Symp. A p p l . Math., XVI, p p . 4 0 - 6 9 , 1 9 6 4 . 3. Boyce, W.E., Random e i g e n v a l u e problems, P r o b a b i l i s t i c . Methods i n A p p l i e d M a t h e m a t i c s , I , p p . 1 - 7 3 , E d i t e d by A.T.Bharucha-Reid, Academic P r e s s , New Y o r k , 1 9 6 8 . 4. C a r r i e r , G.F., Krook, M., and P e a r s o n , C.E., F u n c t i o n s o f a Complex V a r i a b l e : Theory and Technique, M c G r a w - H i l l , 1 9 6 6 . 5„ F a f o n o f f , N.P., R o l e o f th 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 th e N o r t h A t l a n t i c , P r o c . F i r s t S c i e n c e  A d v i s o r y M e e t i n g , N a t i o n a l Data Buoy Development P r o j e c t , U.S. C o a s t Guard, 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 random media. P r o b a b i l i s t i c Methods i n A p p l i e d M a t h e m a t i c s , I , p p . 7 5 - 1 9 8 , E d i t e d by A.T.Bharucha-Reid, Academic P r e s s , New Yo r k , 1 9 6 8 . 7 . G a r r e t t , C. and Munk, W., I n t e r n a l wave s p e c t r a i n t h e p r e s e n c e 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 0 , I , pp. 1 9 6 - 2 0 2 , 1 9 7 1 . 8 o K e l l e r , J.B., Wave p r o p a g a t i o n i n random media, P r o c . Symp.  A p p l . Math., X I I I , p p o 2 2 7 - 2 4 6 , 1 9 6 2 . 9 . 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 and wave p r o p a g a t i o n i n random media, P r o c . Symp. A p p l . Math., XVI, p p . 1 4 5 - 1 7 0 , 1 9 6 4 . 1 0 . K e l l e r , J.B., 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 , p p . 8 2 3 - 8 3 4 , E d i t e d by R.L.Rowell 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 . 1 1 . K e l l e r , J.B. and V e r o n i s , G., Rossby waves i n t h e p r e s e n c e o f random c u r r e n t s , J . Geophys. Res., 7 4 , 8 , p p . 1 9 4 1 - 5 1 , A p r i l 1 5 , 1 9 6 9 . 1 2 c L e B l o n d , P „ H o , On the damping o f i n t e r n a l g r a v i t y waves 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 , Holden-Day I n c . , San F r a n c i s c o , 1964. 14. P h i l l i p s , O.M., The Dynamics o f t h e Upper Ocean, C.U.P., 1966. 15. T a y l o r , A.E., Advanced C a l c u l u s , B l a i s d e l l Pub. Co., T o r o n t o , 1955. 16. 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 waves w i t h an u n s t e a d y 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, IV, pp. 761-771, 1969. 17. Webster, T.F., L e c t u r e s , Second C o l l o q u i u m on t h e Hydrodynam-i c s o f t h e Ocean, L i e g e U n i v e r s i t y , C a h i e r s de mechanique mathematique, 26, pp. 20-53, 1970. 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 waves i n the P a c i f i c Ocean, Deep-Sea R e s e a r c h , 17, pp. 91-108, Feb. 1970. APPENDICES 39 A- The Oth Order D i s p e r s i o n R e l a t i o n F o r a p l a n e wave p r o p a g a t i n g i n a s t r a t i f i e d f l u i d w i t h c o n s t a n t B r u n t - V a i s a l a . f r e q u e n c y A 4 t h a t i s 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 ( z ) a x i s , t h e d i s p e r s i o n r e l a t i o n i s where Cz = (tf-<r\)/(o-x-fx) , <t = rtZ/3 , •£ = c o r i o l i s p a r a meter, X. = h o r i z o n t a l d i s t a n c e , t = t i m e . P u t t i n g H^fx+i^x w i t h JK, r e a l and r e q u i r i n g k r e a l (as t h e r e i s no p h y s i c a l r e a s o n f o r growth o r a t t e n u a t i o n o f the wave a l o n g t h e h o r i z o n t a l d i r e c t i o n ) g i v e s 4 r = - ^ and c * f = (A-l) A p l a n e wave now has t h e form e Q . Waves p r o p a g a t i n g 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 component o f d i r e c t i o n a r e a t t e n u a t e d . Now h a v i n g /V - N0 , a c o n s t a n t , i m p l i e s {>cc2)°<e . The k i n e t i c energy o f a wave i s p r o p o r t i o n a l t o $>A2 where A i s t h e wave a m p l i t u d e . Hence t h e f a c t o r Qdi/,z 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 t h e k i n e t i c energy o f t h e wave. The e q u a t i o n e.*k*-J* = c o r r e s p o n d s t o a d i s p e r s i o n d i a g r a i r f c o n s i s t i n g o f a r e c t a n g u l a r h y p e r b o l a o p e n i n g t o w a r d l a r g e I M v a l u e s , f o r f i x e d 0~ . We i n t r o d u c e a p o l a r c o o r d i n a t e 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 numbers. S e t (*j = fa (tote, SlsriQ) S u b s t i t u t i n g i n t o A - l , we f i n d f o r 9 6 (-Qf,,Qfl) K}(7r-e^jlTi-Qf,) where 6^ , t h e a n g l e a t w h i c h t h e asymptotes t o t h e h y p e r b o l a a r e i n c l i n e d t o t h e k - a x i s , i s g i v e n 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 C c o r r e s p o n d s t o an i n c r e a s e i n 0# or a s p r e a d i n g o f t h e a s y m p t o t e s . I n a d d i t i o n e q u a t i o n A-2 i m p l i e s t h a t f o r f i x e d 6 an i n c r e a s e i n Cz r e s u l t s i n a d e c r e a s e i n d&0 • B- D e r i v a t i o n o f t h e Stream F u n c t i o n E q u a t i o n L e t Z) be a p o i n t i n a r i g h t - h a n d e d system o f C a r t e s i a n c o o r d i n a t e s r o t a t i n g u n i f o r m l y about t h e 2 - a x i s , w h i c h i s v e r t i c a l and p o s i t i v e upward. L e t f = zQ. ^ be the C o r i o l i s p arameter, where X~L i s t h e magnitude o f t h e e a r t h ' s r o t r .'on v e c t o r and f i s the l a t i t u d e . Then t h e system r o t a t e s w i t h a n g u l a r f r e q u e n c y -f/z . If $ i s t h e magnitude o f 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 , a n t i - p a r a l l e l t o t h e B - a x i s , (u,i/,ur) a r e the f l u i d v e l o c i t y components, ^ i s the f l u i d d e n s i t y and o^. i s t h e p r e s s u r e , then t h e momentum c o n s e r v a t i o n 41 e q u a t i o n s a r e W-t + UVX + . + usu± +fu + ^ = O art + uusK + yur^ + usur^ + g + ly ^  = 0 f o r an i n v i s c i d f l u i d . The e q u a t i o n o f mass c o n s e r v a t i o n i s + ( f i L ) x -h (£(Sjv + (^ur)% - o and t h e i n c o m p r e s s i b i l i t y c o n d i t i o n i s f t + w p x + i- "Sfe - 0 I n i t a l l y s e t t i n g ( U, l/, UJ) - ( 0 , 0, 0) g i v e s w i t h ^ = ^ 0 ( H j / f ^ - - f 9 ( i ) t h e 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 n t r o d u c i n g p e r t u r b a t i o n s such t h a t ^cx, y, 2,-e; = ^>0<rz) +Q (x,\/.-t) (U.W, us) = <ut ,u},urt) f- - f-J?) + fcC^V.-t) and l i n e a r i z i n g t h e e q u a t i o n s i n t h e q u a n t i t i e s s u b s c r i p t e d w i t h 1 we o b t a i n f„u^ +$9 1- ft? = 0 (S-3) UIX f i/ly + ur^ = o (B-4-) Pit f = 0 _ • • (B-5) D r o p p i n g t h e s u b s c r i p t 1, B - l , B-2 y i e l d fotU - -fxx± -ff.? ^oLW - --p-^ +ff-x. where Thus and A p p l y i n g p0L 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 - (%°Lwi)i - 0 (B-8) E q u a t i o n B-3 g i v e s f o ^ t t -f-^69 + fit -  0 U s i n g e q u a t i o n B - 5 or fo u^. + -fit -9%* LLT - O 43 Thus and S u b s t i t u t i n g t h i s i n t o B-8 produces A l t e r n a t i v e l y , + e<*/fo <"htt +fX?°Vpur* = 0 (13-9) Put A/2= - y f 0 * ^ and dy = 0 , t h e n B-9 r e d u c e s t o Now UK -h usi - O i m p l i e s where 3 i s a stream f u n c t i o n ! Thus th e e q u a t i o n f o r § i s seen t o be , t h e same as e q u a t i o n 1-1 o f the t e x t . C- K e l l e r ' s Method 44 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 method f o r d e r i v i n g t h e d i s p e r s i o n r e l a t i o n f o r t h e mean wave i n a random medium. A more g e n e r a l t r e a t m e n t w i l l be f o u n d i n K e l l e r ' s paper i n 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 1 " I f c<f 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 X i s 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 i m p l i e s $ i s a random p r o c e s s . A p p l y i n g X t h e i n v e r s e o f oC, t o b o t h s i d e s o f e q u a t i o n C - l we o b t a i n T a k i n g ensemble a v e r a g e s , denoted by < > , t h i s becomes I n v e r t i n g <£~ 2> g i v e s <^"1>"1<$> = X (C-2) Now i f «*f i s s t a t i s t i c a l l y homogeneous and ~X.-Q e q u a t i o n C-2 has as e i g e n d i f f e r e n t i a l s t h e p l a n e wave s o l u t i o n s g i v e n by w h i c h obey t h e d i s p e r s i o n r e l a t i o n ^«tx+J?z-crt)<^l>-i^(kxi-J>z-(r-L) = Q ( c _ 3 ) S i n c e Z£-cM. •+<//' and J\T : i s " s m a l l " compared t o JA. t h e b i n o m i a l e x p a n s i o n i s used t o g e t 45 X'1 -<M~2-M-'JfM'1* M-'JiTM'1^^'1 w h i c h i s v a l i d i f . II <M i ^ H < 1 . A v e r a g i n g t h i s e q u a t i o n w i t h K.tA/'y — 0 , one o b t a i n s <X~2> =M'~L+<M-1<Jf<M-W><M-t- . . . . T h i s i s i n v e r t e d t o y i e l d w h i c h i s c o r r e c t t o second o r d e r i n JV* , and hence £ . S u b s t i t u t i n g C-4 i n t o C-3 g i v e s e " " " - " ^ - ^ ^ * " ' ' ^ o fc-s; as the d i s p e r s i o n r e l a t i o n o f t h e i n f i n i t e s i m a l a m p l i t u d e mean wave c o r r e c t t o 0((3). E q u a t i o n C-5 i s i d e n t i c a l w i t h I I - 3 . I n Chapter I we have §cx.t.t) = e" t i r t Thus i n t h i s case C-5 r e d u c e s t o e-i("'*"'{j(-<w IJf>}e''"0"'*' = o w h i c h c o i n c i d e s w i t h e q u a t i o n 1-3. The f o r e g o i n g a n a l y s i s i s dependent f o r i t s v a l i d i t y on t h e n o t e a s i l y a p p l i c a b l e c o n d i t i o n t h a t \\Ji~2<J/'ll be s m a l l . I n o r d e r t o a c h i e v e a n o t h e r p e r s p e c t i v e on t h e v a l i d i t y o f e q u a t i o n C-5 we p r e s e n t an a l t e r n a t i v e d e r i v a t i o n o f i t . P r o c e e d i n g d i r e c t l y from (M+Jf)$ - o ( C - 6 ) we have <M$ - - (c-7) A p p l y i n g uU'1 t o b o t h 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 an i n t e g r a l e q u a t i o n and the f i r s t s u b s t i t u t i o n y i e l d s T a k i n g ensemble a v e r a g e s < § > = M ~ l<JfM 'lu4r§ > Thus M<$> = <Jf<M-W$> (C-?) We now make t h e s o - c a l l e d c l o s u r e a s s u m p t i o n 3 , 6 KJfM-'Jfgy = <JTM'1J/,X§> (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? > has the p l a n e wave . s o l u t i o n s i so the approximate 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 p o i n t i n t h e a n a l y s i s i s t h e a p p r o x i m a t i o n C-10, On p. 45 W.E.Boyce 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 j u s t i f y t h i s a p p r o x i m a t i o n by an argument somewhat s i m i l a r t o t h e f o l l o w i n g o I n t h e term <"J/Ui'tV§>it i s q u i t e c l e a r t h a t t h e s t a t i s t i c s o f $ cannot be i n d e p e n d e n t o f t h o s e o f uV i n v i e w o f e q u a t i o n C-6. 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 J/1 and § a r e 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 f o r C-10 can be g i v e n . I f t h e random p r o c e s s e s yU. and ^ a r e assumed t o have t h e e r g o d i c p r o p e r t y , 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 t o ensemble a v e r a g e s , t h e n i t seems c l e a r t h a t i f t h e s c a l e s o f v a r i a t i o n o f Jf and $ a r e g r e a t l y d i f f e r e n t i t w o u l d be a good 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 J/'M.'1^ and $ 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 space o r time average i s made o f t h e o t h e r . T h i s average c o u l d be r e g a r d e d as an ensemble aver a g e . Then a l a r g e s c a l e a v e r a g e c o u l d be t a k e n o f the r e m a i n i n g p r o c e s s w i t h t h e p r e v i o u s l y a v e r a g e d q u a n t i t y b e i n g r e g a r d e d as a c o n s t a n t . T h i s second average c o u l d a l s o be i d e n t i f i e d as an ensemble average i n view o f the a s s u m p t i o n o f e r g o d i c i t y . When we used t h e Oth o r d e r s o l u t i o n , e q u a t i o n 1-9, i n the Oct1) terms and e x p e c t e d t h e f i r s t i t e r a t i o n o f e q u a t i o n 1-8 t o g i v e a good a p p r o x i m a t i o n f o r d€ i t was n e c e s s a r y t o assume t h a t the Oct2) terms be s m a l l , o r e q u i v a l e n t l y t h a t be n o t g r e a t l y d i f f e r e n t f r om 0 . T h i s a s s u m p t i o n e n a b l e d us t o use t h e 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 1-12. s i m i l a r 48 remarks a p p l y t o t h e d e r i v a t i o n o f e q u a t i o n I I - 7 from I I - 6 . Now i f we r e s u b s t i t u t e 2]]Clccr<C'6 -Sun'6 i n t o e q u a t i o n s I-11 and 1-12, i t w i l l be c l e a r t h a t t h e a s s u m p t i o n t h a t t h e Oct') terms must be 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 c l o s e l y , i . e . 6 cannot approach , d e f i n e d i n Appendix A, a r b i t r a r i l y c l o s e l y , and i n o t h e r words t h e a n a l y s i s c annot be e x p e c t e d t o h o l d f o r v e r y s h o r t waves. That t h e Oct3-) p e r t u r b a t i o n s cease t o r e m a i n s m a l l as the waves become s h o r t e r and 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 and 3 f o r " w h i t e n o i s e " yu(2). E x c l u d i n g v e r y Shockwaves' 6m the a n a l y s i s i s . c o n s i s t e n t ' w i t h k e e p i n g o n l y second o r d e r terms i n € i n e q u a t i o n C - 4 . T h i s i s so because h i g h e r powers o f J/ w o u l d add h i g h e r powers o f the wave number t o the d i s p e r s i o n r e l a t i o n and t h e s e c o u l d n o t be ex p e c t e d t o be n e g l i g i b l e f o r v e r y s h o r t waves, as has been assumed i n u s i n g C-5. As 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 e x p e r i m e n t a l d a t a c o n s i d e r e d i n Chapter I i n d i c a t e d jjid) was "whit e n o i s e " t o a good a p p r o x i m a t i o n ; and the a n a l y s i s cannot be e x p e c t e d t o be v a l i d 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 approaches the s c a l e o f v a r i a t i o n o f ydd) . T h i s 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 the 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 Case L << Xa 49 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 and 1-20 we d e f i n e -co I 3 = Lf°r<L2*) c<**(2*L-z*)dz* and Now o<= de0i*nd and we d e f i n e t h e n Ii (i = 1,2.3.+) and we want t o t a k e JjLvrx II (&) (C - 1, Z. 3. 4-J We have J-2<f) - LJ r(Lz*) sun dz* — CO r , f f ) = Lf ra.?:*) c<xL($i*)di* and L e t and Hi*) = I 2*r(Li*)\ Now t h e i n t e g r a l s and - CO converge as P i s an a u t o c o v a r i a n c e f u n c t i o n , i t i s easy t o show t h a t t h e f u n c t i o n s svn(piV/qi* , s^cp?*) , aacpzV and [l-caufpiVj/pz* -are a b s o l u t e l y bounded by 1 ind e p e n d e n t o f £J . Hence ± he z*; and Then by Theorem V I I I p. 667 o f T a y l o r 1 5 t h e improper i n t e g r a l s o f the f i r s t k i n d U ' = i , 2 , 3 . - f ) above a r e u n i f o r m l y converge! f o r a i n t-co,coJ . Hence we may t a k e JU*n under t h e (3-->0 2nt i n t e g r a l s i g n s t o o b t a i n J^n It(B) = 2L2 ft*r(Li*) (3-f>0 1 _Joo 51 a n d I2(&) - O Changing v a r i a b l e s 2 if/**("(Live!= 2 f l r ( i ) d z and CO Hence e q u a t i o n s 1-21 and 1-22 f o l l o w . E- The L i m i t i n g Case X 0 << L Here we d e r i v e f o r m u l a s 1-23 and 1-24 f o r t h e l i m i t i n g case and prove we have o b t a i n e d a t r u e a s y m p t o t i c r e s u l t . We have o C = d f o ^ n e = d u / n 6 — = 2frsunQ D e f i n e J d,i J d* Then T - T°<ir<±) L2«* . I - J ~dT'  e — CO W i t h 2 = L?* .a x J d^  e cdt* 52 L e t t i n g ^ = ^irsutiej^o/^ we can 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 4 " o f I as (3 —>oo a l o n g t h e 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 p a r t s . I t i s n e c e s s a r y t o assume P 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 t o r e s t r i c t & so i t i s n o t n e a r & - o Q r 0 = 77~ . L e t t i n g 1 cU*" -co 2 * and -oo as r i s an a u t o c o v a r i a n c e f u n c t i o n . We o b t a i n a f t e r N i n t e g r a t i o n s by p a r t s where To prove t h a t t h i s i s t h e v a l i d a s y m p t o t i c e x p a n s i o n o f I _ t i l . C oC^rcL^j as ^ _ > o o a l o n g t h e r e a l a x i s i t i s n e c e s s a r y t o prove t h a t f o r f i x e d M. = O ( E - i ) r -co 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 two 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 ) t h 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 f u n c t i o n and hence must d i e o u t a t i n f i n i t y i n any p h y s i c a l system. Thus e q u a t i o n E - l i s pr o v e d and we have o b t a i n e d the t r u e a s y m p t o t i c e x p a n s i o n o f I as ^ - > « . Ke e p i n g o n l y t h e Oth o r d e r i n ^/$ g i v e s fi-yOO J Of 2 I - C O and Hence i n t h i s l i m i t i n g case t h e f o r m u l a s 1-11 and 1-12 r e d u c e t o -f and 2* l&o where £e0-&°/d and d = d J VcDdi. v a l i d f o r Q n o t near 6 - o -"CO o r Q - IT . These a r e f o r m u l a s 1-23 and 1-24 r e s p e c t i v e l y o f t h e t e x t .