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Internal gravity waves in a vertically sheared flow Healey, David Andrew 1968

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INTERNAL GRAVITY WAVES IN A VERTICALLY SHEARED FLOW by ' DAVID ANDREW KEALEY B.Sc.j University of B r i t i s h Columbia, 1965 •A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February, 1968 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 an 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 and 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 P h y s i c s The 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, C a n a d a D a t e F e b r u a r y 19, 1968 ABSTRACT We investigate the propagation of i n t e r n a l gravity waves i n a rotating f l u i d with horizontal and v e r t i c a l s t r a t i f i c a t i o n . The modification of these waves by the presence of a v e r t i c a l l y sheared geostrophic current i s determined, and the rate of energy exchange between waves and current i s estimated and compared to exchange rates of other i n t e r a c t i o n mechanisms. The e f f e c t of boundary conditions on the range of frequencies allowed for wave propagation i s also considered. The wave amplitude has horizontal exponential depend-ence due to the horizontal density v a r i a t i o n as well as to exchange of energy with the mean shear flow. The solution also shows a phase difference from surface to bottom. For waves propagating normally to a v e r t i c a l l y sheared geostrophic current, the energy exchange mechanism i s found to be weak when compared to other exchange mechanisms and i s l i k e l y to be of l i t t l e importance i n the ocean. The imposition of boundary conditions on the wave solution a l t e r s the frequency range over which solutions may e x i s t . i i i TABLE OF CONTENTS Page ABSTRACT . . . i i LIST OF TABLES i v LIST OF FIGURES v ACKNOWLEDGEMENTS v'i INTRODUCTION 1 THEORY 3 a) Mean Motion ... 3 b) The Wave Equation 9 PROPERTIES OF THE WAVE EQUATION .. 13 SOLUTION OF THE WAVE EQUATION 23 PHYSICAL INTERPRETATION 42 a) V e r t i c a l S t r a t i f i c a t i o n Only 42 b) Horizontal and V e r t i c a l S t r a t i f i c a t i o n ..... 43 ENERGY CONSIDERATIONS 49 a) Energy of Internal Waves 49 b) Energy Equation 49 CONCLUSIONS 54 LITERATURE CITED 55 LIST O F TABLES Page TABLE I Values for a and b for three regions .... 8 TABLE II Q f o r two frequencies and for three d i f f e r e n t exchange mechanisms .... 53 V LIST OF FIGURES Page F i g u r e 1: C r o s s - s e c t i o n of tem p e r a t u r e t h r o u g h the G u l f Stream 5 F i g u r e 2: C r o s s - s e c t i o n of temp e r a t u r e t h r o u g h t h e K u r o s h i o 5 F i g u r e 3: S u r f a c e , L e v e l S u r f a c e , and Bottom 6 F i g u r e 4: P r o p a g a t i o n diagram f o r b=V=^=0, B o u s s i n e s q case 14 F i g u r e 5: P r o p a g a t i o n diagram f o r .b=V= — =0, n o n - B o u s s i n e s q case 15 F i g u r e 6: P r o p a g a t i o n l i n e s f o r «*». =f , w i t h the d i r e c t i o n of the c h a r a c t e r i s t i c s i n d i c a t e d by arrows 20 F i g u r e 7: P r o p a g a t i o n diagram f o r b+0, non-B o u s s i n e s q c a s e , w i t h w< (ga - 16bVa* )V*. • • 22 F i g u r e 8: P r o p a g a t i o n diagram f o r b£0, non-B o u s s i n e s q case, w i t h w> (ga -16bVa*) V * . 22 F i g u r e 9: P r o p a g a t i o n curve f o r w * = g a + f 46 ACKNOWLEDGEMENTS The author wishes to express his gratitude to Dr. P.H. LeBlond who suggested the problem and gave valuable guidance throughout the course of t h i s study. Sincere thanks are given to the Ins t i t u t e of Oceanography, which provided f i n a n c i a l assistance from the National Research Council, Grant BT-100. 1 I INTRODUCTION The t h e o r y o f 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 f l u i d was f i r s t d e v e l o p e d as an e x t e n t i o n o f t h e t h e o r y o f the o s c i l l a t i o n s o f the common boundary between two superposed f l u i d s of d i f f e r i n g d e n s i t i e s . B u r n s i d e (1889) s t u d i e d the s m a l l a m p l i t u d e o s c i l l a t i o n s o f a heterogeneous f l u i d , i n which he approximated a c o n t i n -u o u s l y v a r y i n g d e n s i t y by the l i m i t o f a number o f f i n i t e s t r a t a of d i f f e r i n g d e n s i t i e s . L o v e ( l 8 9 l ) f i r s t c o n s i d e r e d wave motion 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 f l u i d , w i t h d e n s i t y v a r y i n g e x p o n e n t i a l l y w i t h d e p t h . F j e l d s t a d ( 1 9 3 3 ) extended Love's s o l u t i o n t o i n c l u d e r o t a t i o n , a l l o w i n g a p p l i c a t i o n of the t h e o r y t o the oceans o r the atmosphere. The d e n s i t y o f t h e oceans, however, v a r i e s not o n l y w i t h depth but a l s o h o r i z o n t a l l y . I n o r d e r t o unde r s t a n d i n t e r n a l wave motions i n a h o r i z o n t a l l y as w e l l as v e r t i c a l l y s t r a t i f i e d r e g i o n , a t h e o r y i s r e q u i r e d which i n c l u d e s den-s i t y v a r i a t i o n i n the h o r i z o n t a l as w e l l as the v e r t i c a l d i r e c t i o n . H o r i z o n t a l d e n s i t y v a r i a t i o n s w i l l as a r u l e be accompanied by .a v e r t i c a l l y sheared mean h o r i z o n t a l g e o s t r o p h i c c u r r e n t , f l o w i n g i n a d i r e c t i o n normal t o the d e n s i t y g r a d i e n t . P h i l l i p s ( l 9 6 6 ) found t h a t t h e r e i s an energy i n t e r -a c t i o n between a l o c a l l y u n i f o r m mean shear and i n t e r n a l waves p r o p a g a t i n g p a r a l l e l to t h a t shear f l o w . H i s model i n c l u d e d o n l y v e r t i c a l d e n s i t y v a r i a t i o n , and the a n a l y s i s 2 was c a r r i e d out f o r a n o n - r o t a t i n g f l u i d . F o r P h i l l i p s ' c a s e , t h e r e i s no i n t e r a c t i o n w i t h waves p r o p a g a t i n g perpen-d i c u l a r l y t o the shear f l o w . In a r o t a t i n g f l u i d , an i n t e r -a c t i o n i s p o s s i b l e f o r waves p r o p a g a t i n g i n a d i r e c t i o n normal t o a shear f l o w , and t h i s case w i l l be i n v e s t i g a t e d h e r e . I t i s v e r y i m p o r t a n t to know of a l l i n t e r a c t i o n mechanisms f o r i n t e r n a l waves, i n o r d e r t o understand the g e n e r a t i o n , growth and decay of t h e s e waves and the r o l e t h a t t h e y p l a y i n the o c e a n i c energy budget. A number of energy exchange mechanisms have been i n v e s t i g a t e d a l r e a d y ( P h i l l i p s ( 1 9 6 6 ) f o r a l i t e r a t u r e s u r v e y ), but not the one t o be s t u d i e d h e r e . We w i l l f i n d how i n t e r n a l waves ar e m o d i f i e d by the p r e s e n c e of a g e o s t r o p h i c c u r r e n t , e s t i m a t e the r a t e o f energy exchange between waves and c u r r e n t , and compare i t t o the exchange r a t e a p p r o p r i a t e t o o t h e r exchange mechanisms. II THEORY a) Mean Motion In a rotating system, a current normal to the horizontal component of the density gradient i s required to support a steady density s t r a t i f i c a t i o n i n the horizontal well as the v e r t i c a l . Assuming that current to be geostrophic in nature, the zeroth order geostrophic and hydrostatic equations are V + ^ * 0 , (1) ox = O (2) and /p>Q^ - . (3) A Cartesian co-ordinate system has been chosen, with x p o s i t i v e eastwards, y p o s i t i v e southwards, and z p o s i t i v e v e r t i c a l l y downwards. An equilibrium l e v e l surface has been chosen at z=0, with the bottom at z=H. The subscripts x, y, and z denote p a r t i a l d i f f e r e n t i a t i o n with respect to x, y, and z respectively. The geostrophic v e l o c i t y i s given by V, the mean density by yOe, and the C o r i o l i s parameter by f, where f = 2 f l s i n ^ . St i s the angular v e l o c i t y of the earth, ^ the geographic l a t i t u d e , and g the acceleration due to gravity. The C o r i o l i s parameter w i l l henceforth be considered l o c a l l y uniform and i t s v a r i t i o n with l a t i t u d e not incorporated i n the analysis. The density s t r a t i f i c a t i o n i s chosen as where a and b are the c o n s t a n t s ^ r i & , ' ( 5 ) t> ' ± , (6) and jO^ i s the d e n s i t y a t x=0 and z=0. I f a and b are s u f f i c i e n t l y s m a l l , the e x p o n e n t i a l dependence i s weak and c l o s e l y a p p r o x i m a t e s a l i n e a r d e n s i t y g r a d i e n t . Two w e l l known r e g i o n s o f s t r o n g h o r i z o n t a l and v e r t i c a l d e n s i t y s t r a t i f i c a t i o n are the G u l f Stream and t h e K u r o s h i o . I n t h e s e r e g i o n s the d e n s i t y of the sea-w a t e r i s d e t e r m i n e d p r i m a r i l y by t e m p e r a t u r e , as the s a l i n i t y changes l i t t l e t h r o u g h the r e g i o n s . I n the c r o s s -s e c t i o n s o f t h e G u l f Stream and the K u r o s h i o ( f i g u r e s 1 and 2 ) t h e i s o t h e r m s thus approximate i s o p y c n a l s . The d e n s i t y v a r i e s a p p r o x i m a t e l y l i n e a r l y a c r o s s the s e c t i o n s and a l s o l i n e a r l y w i t h d epth f o r the upper f i v e hundred m e t e r s . A mean v e r t i c a l l y sheared h o r i z o n t a l v e l o c i t y normal t o the h o r i z o n t a l g r a d i e n t o f d e n s i t y i s found i n bo t h c a s e s . ' A l s o , t h e r e e x i s t s i n both cases a s u r f a c e s l o p e o p p o s i t e i n s i g n t o the s l o p e o f the i s o p y c n a l s . I n r e g i o n s o f t h e ocean w i t h such a d e n s i t y f i e l d , t h e c u r r e n t a t the s u r f a c e i s non- z e r o . I n o r d e r t h a t a g e o s t r o p h i c c u r r e n t be non-zero a t the s u r f a c e , a s u r f a c e s l o p e i s r e q u i r e d t o p r o v i d e a p r e s s u r e g r a d i e n t which b a l a n c e s the C o r i o l i s f o r c e . W i t h the s u r f a c e a t z = -TiOO ( f i g u r e 3 ), the p r e s s u r e a t a depth z i s 5 I , 1 —T 20 60 100 Nautical miles Figure 1: Cross-section of temperature through the Gulf Stream (from Worthington(l954b, f i g . 4 ) ) . Figure 2: Cross-section of temperature through the Kuroshio (according to Wust(1936a)). 6 — z = H F i g u r e 3: S u r f a c e , L e v e l S u r f a c e , and Bottom. where fo^is the a t m o s p h e r i c p r e s s u r e . S u b s t i t u t i n g f o r ^© 0from (4) (8) a D i f f e r e n t i a t i n g ^ w i t h r e s p e c t t o x, and s u b s t i t u t i n g f o r f»0% i n (1) or The G u l f Stream i s chosen as a p h y s i c a l example, i n o r d e r t o det e r m i n e the maximum o r d e r s o f magnitude of a, b and ^? . The c r o s s stream b a l a n c e o f the G u l f Stream may be c o n s i d e r e d t o be e s s e n t i a l l y g e o s t r o p h i c (Stommel(1965)), w i t h the v e l o c i t y d e s c r i b e d by ( 9 ) . Stommel e s t i m a t e s the 7 d i f f e r e n c e i n s u r f a c e e l e v a t i o n a c r o s s the G u l f Stream t o be of the o r d e r o f 100 c e n t i m e t e r s . The w i d t h of the stream i s a p p r o x i m a t e l y 200 k i l o m e t e r s , and the depth 4 k i l o m e t e r s . The change of d e n s i t y both a c r o s s and from - 3 s u r f a c e to bottom i s of the o r d e r o f 10 grams p e r c u b i c c e n t i m e t e r . R e p r e s e n t i n g the w i d t h o f the stream by ^ x , one can w r i t e the change of s u r f a c e e l e v a t i o n a c r o s s the stream as dnzix = -10 cm. a* o r 1 o 2 6 " = - 1 0 c m'_. =-5<10*. (10) 2 x 10 cm. The o r d e r s o f magnitude of a and b i n t h i s case are P 10s a = J- = i £ 3 = IcT'cm/ 1 (11) and 3 b = S - .. 1 U = -5 *10 cm. . /• » * 2 . 1 0 ' ( 1 2 ) C o n s i d e r now the g e o s t r o p h i c v e l o c i t y V g i v e n by ( 9 ) . S i n c e az(max) = a H « 1, e**:* 1 + az . (13) S u b s t i t u t i n g (13) i n ( 9 ) , Comparing b and a (14) TABLE I Va l u e s f o r a and b f o r t h r e e r e g i o n s ( f o r G u l f Stream and K u r o s h i o , a and b are c a l c u l a t e d from d a t a o b t a i n e d from c r o s s - s e c t i o n s , f i g u r e s 1 and 2 ) . a (cm."*') b (cm."') G u l f Stream 10**. - 5 * 1 0 " " K u r o s h i o 10"* -5 * 10" u Open Ocean 10'* 10"' 3 b = -5 * 10 cm. a £S = -5 * 10"'* cm."' a l l o w s a i t ? t o be n e g l e c t e d w i t h r e s p e c t t o b. C o n s i d e r now the r e l a t i v e s i z e s of b*7 and $3 . b ^ * . = " 5 * 1 0 ' " * - 1 0 * = 5 * 1 0 " 9 = -5 x 10" 6 a* b>i may th e n be n e g l e c t e d , w i t h r e s p e c t t o £3 . V may then a* be a p p r o x i m a t e d by V - If • (15) b and £3 o c c u r t o g e t h e r w i t h the same s i g n , so t h a t (15) r e p r e s e n t s a c u r r e n t which v a r i e s l i n e a r l y w i t h d e p t h and i n c r e a s e s w i t h d e p t h . By adding a c o n s t a n t v e l o c i t y V t o ( 1 5 ) , V may be made t o d e c r e a s e w i t h d e pth and change s i g n a t some i n t e r m e d i a t e d e p t h , o r to v a n i s h a t z=H. The v e l o c i t y may then be w r i t t e n as 9 b) The Wave E q u a t i o n C o n s i d e r now s m a l l p e r t u r b a t i o n s u, v, w, p and 'on (0 , V , 0 ) , p e and ^ o0 . The e q u a t i o n s o f motion and c o n t i n u i t y up to f i r s t o r d e r i n the p e r t u r -b a t i o n s from the g e o s t r o p h i c c u r r e n t a re U * * V U V " * C V < V ) * - 1 " 0 , (17) V< • / M + V V J ( + J L - f ^ ' L « o ' (18) 4 * + • S O (20) and ft' * V» *M/%**V |^. 3 0 > (21) where t h e s u b s c r i p t t denotes p a r t i a l d i f f e r e n t i a t i o n w i t h r e s p e c t t o t i m e . I n e q u a t i o n s ( 1 7 ) - ( 2 1 ) , the f l u i d i s assumed t o be i n c o m p r e s s i b l e and i n v i s c i d . The f i r s t o r d e r e q u a t i o n s are o b t a i n e d by s u b t r a c t i n g e q u a t i o n s ( l ) - ( 3 ) from ( 1 7 ) - ( 1 9 ) . F o r example, (17) minus (1) g i v e s ui + v M - fv * - ± - . - r ' K - -p.* - o . (22) S i n c e f , f>0% f J L JL 7 + 7a' e f^* f-.fi.'] • [/- f'7 . To f i r s t o r d e r t h i s i s ' i^lf' + 1£* . (23) From ( 1 ) , (23) may then be w r i t t e n - * V + ^ (24) Assuming that the waves are plane and propagate perpen-d i c u l a r l y to the current, that — = 0, the f i r s t order equation of motion for the x-direction may be written • Ut - -f v - * v > ' + ^ = o S i m i l a r l y , the other f i r s t order equations are (24-1) - t i n * M^- O f . (25) - j£# + f> ° > (26) + W j , = 0 (27) and ^ ' + + 2 ^ ( 3 ^ s O • (28) The f i v e equations (24-l)-(28) can be reduced to one equation i n one unknown. Eliminating v f i r s t by d i f f e r e n t i a t i n g (24-1) with respect to t, and substituting for from (25), we fi n d «<t + f\ + J v V ^ -iVfi t = o . ( 2 9 ) From (2.6 ): W * - f>o**t + 9p' , (30) ?V = - <°o**tA fr"*** 9fr (31) and ' "1°"* ^ " ^ ^ t x +3C*%-  (32) Multiplying (29) by >^0 and then d i f f e r e n t i a t i n g with respect to z, we find that - f ^ r i -f^et* v e 0 • ( 3 3 ) The B o u s s i n e s q a p p r o x i m a t i o n has not been used i n the d e r i v a t i o n of ( 3 3 ) . As L o n g ( l 9 6 5 ) has p o i n t e d o u t , i t i s no t j u s t i f i a b l e t o make t h i s a p p r o x i m a t i o n when l o o k i n g f o r e f f e c t s which may be as s m a l l as those a s s o c i a t e d w i t h t h e d e n s i t y g r a d i e n t terms t r a d i t i o n a l l y n e g l e c t e d when B o u s s i n e s q ' s a p p r o x i m a t i o n i s used. S u b s t i t u t i n g f o r , ft , and ft^ i n ( 3 3 ) , we have " W w - f> 0 ^ t * + 0^ = 0 ( 3 4 ) S u b s t i t u t i n g f o r i n ( 3 4 ) , d i v i d i n g ( 3 4 ) by >^0 , then d i f f e r e n t i a t i n g w i t h r e s p e c t to x, we now f i n d an e q u a t i o n i n v o l v i n g u and w o n l y . * 3 * ^ x * - J f t t V x = O ( 3 5 ) F i n a l l y , we can e l i m i n a t e the h o r i z o n t a l v e l o c i t y component by r e p l a c i n g u % by -w4 i n ( 3 5 ) . 12 i n ( S ) . If . + 3 b ^ * - s f l W x x - jftfo w x =o ( 3 6 ) Assuming e x p o n e n t i a l time dependence, w « e , (36) becomes 1 ^ *»* (37) I l l PROPERTIES OF THE WAVE EQUATION The wave equation (37) has wave solutions when i t i s of hyperbolic type. We w i l l f i r s t consider the case of v e r t i c a l s t r a t i f i c a t i o n ( v e r t i c a l v a r i a t i o n of density) only, i n the absence of boundaries. In t h i s case, b=V •=^ 2?=0, and (37) reduces to The equation (38) i s hyperbolic when That i s , equation (38) i s hyperbolic for the frequency range f 1 «.wa<ga when f*< ga and for the range ga< w'< f 1 when f * > ga. Waves with frequencies i n the second range w i l l not be considered here, as they do not occur i n the oceans except perhaps at great depths where the s t r a t i f i c a t i o n i s very weak. -A propagation diagram can be obtained by assuming w «* exp [ i ( l . x - * ) t ) ] , (39) where JL = (k,o,8 ). Substituting (39) i n (38), we have - < ! ! f t / w l - f l ) - J l ' r w ^ f 1 ) + k X C g a . - w » ) = O. ( 4 0 ) In the Boussinesq case, terms containing a and b are neglected i f not multiplied by g; i s real (Krauss (1964)), 14 and the r e a l part of (40) i s This i s the equation of two straight l i n e s through the o r i g i n . The propagation vector, 1, i s the radius vector from the o r i g i n of the (k,0, y )-space to a point on the. propagation diagram. The l i n e s are d i s t i n c t for the f r e -quency range f ga , which corresponds to the frequency range found by Sandstrom(l966). Figure 4: Propagation diagram for b=V= £ 2 = 0 , Boussinesq case. a* I n the n o n - B o u s s i n e s q c a s e , tf i s complex, and the r e a l and i m a g i n a r y p a r t s o f (40) are and The i m a g i n a r y p a r t o f (40) has the s o l u t i o n A 2 S u b s t i t u t i n g f o r ^ i n the r e a l p a r t of ( 4 0 ) , we have D i v i d i n g by ( - f )a /4, t h i s becomes T h i s i s t h e e q u a t i o n o f a h y p e r b o l a ( f i g u r e 5) which i s symmetric t o the axes k and )fr and has k i n t e r c e p t s F i g u r e 5: P r o p a g a t i o n diagram f o r b=V= •— =0, n o n - B o u s s i n -esq c a s e . 16 V r i s r e a l by d e f i n i t i o n , - £ + JLf£-L? > o . ( 4 i ) T h i s i n e q u a l i t y h o l d s f o r a l i m i t e d r e g i o n of k. (w»- f ) * (41-1) or Waves w i l l then p r o p a g a t e f o r the f r e q u e n c y range f 2 < < go. but the h o r i z o n t a l wavenumbers k are r e s t r i c t e d t o v a l u e s such t h a t (41-1) h o l d s . That i s , t h e r e i s a lower l i m i t on wavenumbers k a l l o w e d f o r wave p r o p a g a t i o n , as i n d i c a t e d i n f i g u r e 5. . We w i l l c o n s i d e r now the case of h o r i z o n t a l as w e l l as v e r t i c a l s t r a t i f i c a t i o n , a g a i n i n the absence of bound-a r i e s . E q u a t i o n (37) i s of h y p e r b o l i c type when [ i^+fVcl* - if £ ( W*--f a-fV40(w*-S»)] >0 ( 4 2 ) The l i m i t s on the f r e q u e n c y range are o b t a i n e d by e q u a t i n g (42) to zero, and solving for w*. w * - ( g a * f a * w * + gaf fV-f f fc) [ifb + fV*]** 0. ( 43 ) (43) has roots u// 8 9 * £ 9 * - . f * - m J (44) The upper bound of the frequency range i s greater than the Vaisala frequency, whereas the lower bound i s less than the C o r i o l i s parameter, as the t h i r d term on the r i g h t hand side of (45) i s greater i n magnitude than the preceding term'. The range of possible frequencies i s extended, at both ends over the range determined for waves i n a f l u i d s t r a t i f i e d only i n the v e r t i c a l . The frequency range i s now f - <UJ < ga + 2> where "f and $ are p o s i t i v e . The wave propagation curve i n t h i s case can be ob-tained by substituting (39) i n (37): - 2 le [ ? g b • * V«, J + / ' H [ t w ^ j f t ) b + f Va% ] - K * [ w » - = 0 (46) We must allow for complex wave numbers i n both d i r e c t i o n s : k •= k. + ik- , and 8 = tfr +i . 18 T a k i n g r e a l and i m a g i n a r y p a r t s o f (46) g i v e s two e q u a t i o n s : (47) and - ( \ K > * * r ) f 2 9 b + -f VaJ + hr [ Cw**ja) * f Ve*7 - 2 <«;H P[»; 1"5AJ = O . ' ( 4 8 ) Due t o t h e c o m p l e x i t y of (47) and ( 4 8 ) , we w i l l f i r s t c o n s i d e r e q u a t i o n (46) u s i n g the B o u s s i n e s q a p p r o x i m a t i o n . T h i s w i l l g i v e us an approximate i d e a o f t h e p r o p a g a t i o n c u r v e . • - I7 ( u , 2 - -P> - 2 ab 3 k - k*(w*-ga) = O ( 4 6 - l ) The d i f f e r e n t i a l e q u a t i o n (37) under the B o u s s i n e s q approx-i m a t i o n becomes f c « * - - f * ) i ! i 2$bjL + Cw*-**) ^  ) -UT O (37-1) l a * 3 * ax* J and i s h y p e r b o l i c f o r the range u 3°- ~ * F o r = f * , (46-1) r e p r e s e n t s two s t r a i g h t l i n e s : A • o Jk 2 3 b and 22-= — . F o r U J*= ga, (46-1) a l s o r e p r e s e n t s two s t r a i g h t l i n e s : w - 0 I t s c h a r a c t e r i s t i c s have s l o p e s F o r W = f 2 C = 0 and 1* - J - f - <3*<3{ The c h a r a c t e r i s t i c s have s l o p e s 0 o r - & That i s , the c h a r a c t e r i s t i c s are p e r p e n d i c u l a r to the wave p r o p a g a t i o n c u r v e s and d i r e c t e d i n t h e d i r e c t i o n of i n c r e a s -i n g ta . T h i s i s t r u e s i n c e energy t r a v e l s i n t h a t d i r e c t i o n and t h e group v e l o c i t y i s a l o n g c h a r a c t e r i s t i c s . I n f i g u r e 6, the d o t t e d l i n e s r e p r e s e n t the wave p r o p a g a t i o n l i n e s f o r w = f . F o r UJ i n c r e a s i n g , the wave p r o p a g a t i o n l i n e s move i n the d i r e c t i o n i n d i c a t e d by the ar r o w s . The l i m i t s on w * a r e the f r e q u e n c i e s a t w h i c h t h e wave p r o p a g a t i o n l i n e s c o l l a p s e i n t o one l i n e . T h i s o c c u r s f o r v a l u e s o f u> l e s s t h a n f and f o r v a l u e s o f g r e a t e r t h a n ga. Thus, i n an unbounded medium, i n the B o u s s i n e s q a p p r o x i m a t i o n , the f r e q u e n c y range i s extended as was i n d i c a t e d by the c o n d i t i o n of h y p e r b o l i c i t y . * r f * 3 2 0 F i g u r e 6: P r o p a g a t i o n l i n e s f o r u;*= f 2 , w i t h the d i r e c t i o n o f the c h a r a c t e r i s t i c s i n d i c a t e d by ar r o w s . We c o n s i d e r now the non-B o u s s i n e s q case. The propag-a t i o n c u r v e s may be o b t a i n e d from (47) by r e g r o u p i n g t e r m s . + f * • [ a i w 1 - f 5 ) ' f V » f c ] " * / [ w l - f * - f V b ] ' \ - ^ [2^ i > f f V c T h i s i s the e q u a t i o n o f a h y p e r b o l a ( f i g u r e 7) which i s s y m m e t r i c a l t o a s e t o f axes r o t a t e d a t an an g l e © such t h a t cot 2 e - 2 Q b ~ "f ^ * from the axes tff and k f Upon r o t a t i o n the e q u a t i o n o f the h y p e r b o l a becomes 21 + Z * { c o s 1 © ( w » - - f '-fV't) " Sine Cos& ( 2 g b + -f l/a.) + Sin l0 < w * - j » ) } -+ { * / [ ft(wJ - -fVab] -2/ [ ^ ^ - - T'-fVb] - t f ^ . [ z j t t/l/cuj • [ ( W ' * , a ) k W l / f l ' J - V f ^ 5 f t J j ' O . ( 4 7 _ i } The hyperbola has X intercepts ' [ < w * + g O b * fVa*] * A/ I " } / { Cos*© [ uA § A ] X*J$m*0 [w*-f*-^b] * im&cose I 2 3 b * * V a J + eosa0 £ w1-^] J + 7* J coS*e [ w * - f * - - f V b J - * .n0c* 6 © [ a ^ b W l / c J + Jm^lV-jft]] ( 4 7_2) - 0 • In equation (47-1), the term { y ; [ » t w w *> -fv/*b] c w x - f a - m j -v^jfe;[zgb <-fv/* J i s a constant for a given frequency, but changes sign at VJ- Jqa - 16b*/a*'. For frequencies w< jga - 16b2/a* , (47-1) i s a hyperbola with X intercepts as shown in figure 7. For u.'=. ^ga-16b / a * , the h y p e r b o l a c o l l a p s e s i n t o two s t r a i g h t l i n e s , g i v e n by ( 4 7 - 2 ) . For UJ > ^ g a -16b*/a"* , e q u a t i o n (47-2) i s a g a i n a h y p e r b o l a , but now has Z i n t e r -c e p t s r a t h e r than X i n t e r c e p t s . F i g u r e 8: P r o p a g a t i o n diagram f o r b£0, n o n - B o u s s i n e s q c a s e , w i t h v j > (ga-16b*/a* ) V*. IV SOLUTION OF THE WAVE EQUATION The wave equation i s given by (37). (37) The boundary conditions on (37) are that the v e l o c i t y perpendicular to z= - fl and z=H be zero. Sandstrom(l966) has shown that a free surface i t s e l f i s excited into motion by the incidence of simple waves. This e f f e c t i s more pronounced for long waves, whereas for short waves the free surface behaves very much l i k e a r i g i d plane. The surface boundary condition may be approximated by w(-">j)=0, as the surface slope i s small. Expanding the surface boundary condition i n a Taylor's series about z=0, we have Since >? i s small, w(->* ) may be approximated by w(0). The boundary conditions for (37) then become Equation (37) i s separable. We look for a solution of the form ' , . t ... (49) w(0) = w(H) = 0. (50) Substituting (50) into (37), we have 24 + A fa [ 2 3 b f \/<a. 7 j£. 4 [ cui% + <} A * [ w l * J*l J - <f<^> * O (51) E q u a t i o n (51) has s o l u t i o n s i n terms of c o n f l u e n t h y p e r g e o m e t r i c f u n c t i o n s . However, i t i s p o s s i b l e t o s i m p l i f y ( 5 1 ) . I n the f i r s t term a <w*-f*> >>-a4Vb p r o v i d e d w l - f > 5 * 10 , so t h a t afVb may be n e g l e c t e d . S i m i l a r l y , fVb may be n e g l e c t e d i n the second term. When we s u b s t i t u t e f o r V i n ( 5 1 ) , the wave e q u a t i o n i s o f the form Aj"itf + CAj + 4,3>f<}>* ( A , r o ( 5 2 ) w i t h A, : w ' - f 2 , and ^ t» = kg*" b D i v i d i n g (52) by A,, we have The f i r s t d e r i v a t i v e of (53) may be e l i m i n a t e d by the t r a n s f o r m a t i o n . r / ^ UV Z ?CPe V (54) (53) t h e n becomes A 2 - v *>3 * 2 A.) o , , 5 5 ) w h i c h may be w r i t t e n (56) w i t h . (57) £ s (58) and - f f ^ I 8 * ^ | 1 * 1 -' T v A • A r t . ' The boundary c o n d i t i o n s f o r the problem have been g i v e n i n terms of w. We have / * / - (Re J ft^) 6 J-a -FrC|> COS ( h x * « t > - f $ m (fcfc-wt) As w(0) = w(H) = 0 f o r a l l x and t , f ^ c r t * ft <o) O and ft<H> = ft < H ) = O (59) Upon i n t e g r a t i o n , (54) becomes • . / I C 2*, * %l X S u b s t i t u t i n g f o r A , , A 3 , and A y and r e p l a c i n g k by k r + i k-, we have . . #••.,*. . .«fc,- fca f q s- n * \ — ^ - T ^ J L e t e and K = ( _ ^ J * ) , ( 6 0 ) f ( z ) may now be w r i t t e n (61) The boundary c o n d i t i o n s on f ( z ) are ft to> = ft fo^ = O (62) and 4r < < 0 s ft <H> s O . ( 6 3 ) The r e a l and i m a g i n a r y p a r t s of f ( z ) a r e and ft.(*> = e @ ^ C o S K j • frttf e 0 1 f *,»X* . (62) i m p l i e s $r <o) 8 0 , <o> - O . (64) (63) i m p l i e s 0 r <H > COS 3 / * * 0 , (65) ^ $.<H> COH^ ^ r f H ) ^ V ° " ( 6 6 ) M u l t i p l y i n g (65) by c o s X H and (66) by s i n t f H and a d d i n g , we f i n d t h a t $ . C t f ) ( cos7XH + sm2 XH) - O. S i m i l a r l y , ^•iH) ( c o ^ ' H H + sin7 X H ) = 0. The second boundary c o n d i t i o n on <|> ( z ) i s th e n <pr (H) * <H) s O' . ( 6 7 ) U s i n g t h e p e r t u r b a t i o n t e c h n i q u e of B e l l m a n ( 1 9 6 4 ) , we l o o k f i r s t f o r a s o l u t i o n of t h e reduced e q u a t i o n $0 <V + *° 0° (V " 0 • (68) The g e n e r a l s o l u t i o n of (68) i s p0<r = A e * 0 e where A and B are a r b i t r a r y complex c o n s t a n t s . S e p a r a t i n g i n t o r e a l and i m a g i n a r y p a r t s : and = C I'***'* t o * / \ 2 + C/?,- S.r, y • The boundary c o n d i t i o n (52) g i v e s and <°> = O a- r - 6- . S i n c e A and B are a r b i t r a r y c o n s t a n t s , we choose A^ and 2 8 A r such t h a t <f>0{z) t a k e s a s i m p l e form. A s u i t a b l e c h o i c e r 0o(z) the n becomes i>Q <p = ^/"/^ \ . ( 6 9 ) From the boundary c o n d i t i o n ( 6 7 ) , A 0 = «!£' : - i + * . - « * A . I t now remains t o t r a n s f o r m the s o l u t i o n from^„(z) i n t o w ( z ) . From ( 5 0 ) , 'TAA = (Re j -T e j : (Re { u^e e j . ( 7 0 ) S u b s t i t u t i n g i n ( 7 0 ) , the e x p r e s s i o n f o r f ( z ) g i v e n by ( 6 1 ) , R e p l a c i n g $ e ( z ) by s i n / ^ z we f i n d t h a t o r * <? * 6 Sin rur ^  cos(k% + Xy-vt ) # (71) We now l o o k f o r a s o l u t i o n of (56) which i n c l u d e s terms of f i r s t o r d e r i n 6 . The e q u a t i o n w i t h the homogeneous boundary c o n d i t i o n s - O a t z=0 and z=H w i l l have s o l u t i o n € ? «sn 1>oS > <>(* 2> (72) and vN„ = ^ j j * + € in +• OC€Z) % (73) where i s the s o l u t i o n of the s i m p l i f i e d e q u a t i o n (68) c o r r e s p o n d i n g t o the n t h v e r t i c a l mode. The f i r s t o r d e r d e p a r t u r e from the s i m p l i f i e d s o l u t i o n i s w r i t t e n i n terms of t h e complete s e t of e i g e n f u n c t i o n s of t h e s i m p l i f i e d e q u a t i o n ( 6 8 ) . Expanding the term ( z ) $ i n an o r t h o n o r m a l s e r i e s i n terms of the f0%, a F o u r i e r s e r i e s i n t h i s c a s e , we write § A K and /?<V ^ $>s= fp I n t e g r a t i n g from top t o bottom: But fioS and $ o p are o r t h o g o n a l : f o r s £ p, '0 Then From ( 6 7 ) , ^ o 5 i s g i v e n by (74) 30 cs S i n 4? 2r ( 7 5 ) ri S u b s t i t u t i n g (75) i n ( 7 4 ) , we have . H A We now r e p l a c e $ i n (76) by $ n from ( 7 2 ) . H '0 S u b s t i t u t i n g ^ n and A n f r o m (72) and (73) i n t o e q u a t i o n (76) t/3in = 2. £ rMyrf<1> ^ 0>, ^ s + 0 ( £ a ) ( 7 7 ) H v 0 ( 5 6 ) , we have . •> T A W Si y% C O R e p l a c i n g ^ ( z ) by 2 - / ^ J ros , and e q u a t i n g c o e f f i c i e n t s o f £ , or t * / V 0 o n + € E = ' (78) From ( 7 8 ) , we see t h a t rH ^ 31 5. (V- s 1) t o terms of 0(€ ). Thus the r e q u i r e d e x p a n s i o n s are and where the c o e f f i c i e n t s j3sn are g i v e n by (80) S u b s t i t u t i n g f o r ^ e n a n d 6 i n (79) and (80) g i v e s e s,n ~f if - ^ r b i ~ i — ~ V " + ( 8 D W1 and A n " 2 3 * - A , » + 0 (C2) . (82) / /* T r a n s f o r m i n g t h e s o l u t i o n i n terms of w ( z ) , we have from (70-1) 1*S = (^e j fc,c^ev*e e e j • (70-1 S u b s t i t u t i n g f o r 0>j( z) from ( 8 1 ) , (70-1) becomes ® * - fa**- r / •_ £ /*«» *<*tf*\ v ^ + K v " ' " i ( 8 3 , rk / * " e^*-**** ' } 32 k and ^ S n are complex. W r i t i n g k and f?Sri as a sum of r e a l and i m a g i n a r y p a r t s : (83) t h e n becomes u i 5 # r i — - — — — ~ + - ' J rtx Sm f-X^ " M t > J . ( 8 4 ) fii-n i s g i v e n by - f J rt I n t e g r a t i n g , ( 8 5 ) b e c o m e s * -<+*9*>>>_!L% f -11— - "-£JL- ] ^ 2 * < u>*-_ f *) * f'H ( ± I - I) f _J _ _ J L _ 7 Z + ( * * - o r — — - - i ] , w h e r e ± 1 = +1 i f n + s i s e v e n , = -1 i f n + s i s o d d . k _ a n d k - a r e d e t e r m i n e d f r o m (82). x- = - ^ r V ^ ( 8 2 ) F r o m ( 7 7 ) , + . 2 £ft<ui»--f*7 *2Xkyh +Aka.($ p 5 * * l j > ] % l *m 8 2f f ^ ^ . ( 8 6 ) I n t e g r a t i n g ( 8 6 ) , we have fin - -q%<**1+»- f J. - -L- 1 a. H - I 2 <u^-f*) S u b s t i t u t i n g f o r ^ ? w w and ^ i n ( 8 2 ) : _ ± c i ft<u,». f') * .2 & g b ^ / < ? a C9 ^ f l / D ) J 2 "V f f ^ J (87) I n o r d e r t o s o l v e f o r , we c o n s i d e r the i m a g i n a r y p a r t of ( 8 7 ) . . i f x k r J * b <w*- • 2 ( 3 | S +f -8^ k-kr j ' b * - SUk- \ j a b ( g W O 2 v«v*J — ~ — (88) D i v i d i n g (88) by i k ^ and r e g r o u p i n g terms g i v e s + | c j^/jf" ' + q H (89) S i m i l a r l y , k may be d e t e r m i n e d from the r e a l p a r t of ( 8 7 ) . 36 • - " 0 ) ] - V (90) u , v, ^>' and p ' may now be de t e r m i n e d i n terms of the v e r t i c a l v e l o c i t y w. From the c o n t i n u i t y e q u a t i o n ( 2 7 ) , D i f f e r e n t i a t i n g w w i t h r e s p e c t t o z and i n t e g r a t i n g w i t h r e s p e c t t o x, ^ S I N T ? » \ ) + w r + h r £ ^ 5 / H 7 T > \1 + 2*j  cos njT 37 - sm ( hr*. + - <W) - k r c o s ffc,. * *• xj, - *>*/j j (91) where @ = L (®\) and K' = i ( X ^ ) . The e q u a t i o n f o r the y-component of momentum, ( 2 5 ) , r e l a t e s u, v, and w t h r o u g h I n t e g r a t i n g w i t h r e s p e c t t o t g i v e s l A s - f ( -fu -f w ) d-t from w h i c h v = • e ®* e - M Ir®7 ,„ + 3 a h ( k, Z fi~"'ft '+ K- £  i m 1?* ) ) + yfff c o s n j r H* 38 j i i i sin (K* + cos (/^ ^ - ^ t ) 1 - K ' * 0 « - , -»Wn-£ ^ * m y » r /•«-«-iftx r " -j fZ I L - \ »•» " T V ? , £ 2 (w*-^ ; ^ J L J [ • C O 5 C *% X - " ^ J J . ( 9 2 ) E q u a t i o n (27) relates^©', u, and w. fik * " f t x * ^ f l i = 0 (28) D i v i d i n g (28) by f0 and i n t e g r a t i n g w i t h r e s p e c t t o t : 3 9 j s^>o ( u b + "ura, ) di P e r f o r m i n g the i n t e g r a t i o n , we f i n d t h a t + o c u b so- / h.ZL ' « * + /? r 21 ?J. ) £ ' £ • u*t) + hr cos f r. ®'^b( hr z. /*«»- "w */* ^ £ sr* A -3<,b 1* ( h, £. /9s«r COS S*^ £ C > . « f t ^ ] - c o s ( k r * - u j t ) ( h r x • - w t ) y j l 40 . H* rtv | cos ( k r % + - ) J j ! . (93 p ' may be d e r i v e d from e q u a t i o n s ( 2 4 ) , (25) and (28), D i f f e r e n t i a t i n g w i t h r e s p e c t t o time and s u b s t i t u t i n g f o r v t a n c* ft from (25) and ( 2 8 ) , we have from w h i c h I n t e g r a t i n g w i t h r e s p e c t t o x and t : H \ £-'(«*-$*> SLlln^S*)/ H" + K ^ ( b-K.) s/n (kr~K + - f c p t o j ( f ? r * • tf^. - a > * ))J ^ -»* rt* (94) 42 V PHYSICAL INTERPRETATION a) V e r t i c a l S t r a t i f i c a t i o n Only I n view of the com-p l e x i t y of the s o l u t i o n , we w i l l f i r s t c o n s i d e r waves i n a f l u i d s t r a t i f i e d o n l y i n the v e r t i c a l . The s o l u t i o n f o r the v e r t i c a l v e l o c i t y w of i n t e r n a l waves i n a f l u i d w i t h e x p o n e n t i a l v e r t i c a l s t r a t i f i c a t i o n i s o b t a i n e d from (84) w i t h b=V= =0. TAJ- - e~* Sm n/r c o s ( /?rx - ufh) (g^) H where (96) H 2 f < u j * - - f a ) (95) may a l s o be w r i t t e n * I > r w-~ 1 €. 2 J Sin (J? r\ + w ^ - k r f ) - Sm(hr* - ^ " ^ ~ u/t)J . ( 9 7) Sandstrom(1966) has shown t h a t (97) may be i n t e r p r e t e d as t h e the sum of two p l a n e waves p r o p a g a t i n g i n a wave g u i d e , a wave of the f i r s t type becoming a wave of the second t y p e upon r e f l e c t i o n a t the r i g i d b o u n d a r i e s z=0 and z=H. However, (38) has wave s o l u t i o n s i n a bounded f l u i d o n l y when the r i g h t hand s i d e of (96) i s g r e a t e r t h a n z e r o . nr v3 > 0 /no\ C*/*-f> * ( 9 8 ) T h i s i s the same c o n d i t i o n as ( 4 1 ) , and waves w i l l p r o -pagate i n a wave gu i d e f o r f r e q u e n c i e s such t h a t 4 3 r 2 2 The two s i m p l e waves which make up the s o l u t i o n i n a wave g u i d e must have the same x and t dependence, and v e r t i c a l wave numbers of o p p o s i t e s i g n s . From f i g u r e 5, we see t h a t t h i s can o n l y o c c u r f o r a g a i n p l a c i n g a lower l i m i t on the v a l u e s of the h o r i z o n t a l wavenumbers k. b) H o r i z o n t a l and V e r t i c a l S t r a t i f i c a t i o n We now c o n s i d e r the wave s o l u t i o n f o r h o r i z o n t a l as w e l l as v e r t i c a l s t r a t i f i c a t i o n . The v e r t i c a l v e l o c i t y component w may be w r i t t e n 2 I ' " . - Sin ( fcrx + (X - ™r>> -ui) - stn ( hr x + c y- S£)i -out) f H  J + H r l - ( sm ( hr x + (K-t sjr)^ ~*t ) - g a b / k r £ l j f f ^ I ( c o s ( f e r * + ( x - sjr; 3 - a , * ) - c o s ( k r x + ( X + ) i - tut J -KT.A To z e r o t h o r d e r i n ^  (£=kgab), the n t h mode of the s o l u t i o n c o n s i s t s of two p l a n e waves w i t h the same x and t dependence, but w i t h v e r t i c a l wavenumbers w h i c h d i f f e r i n magnitude as w e l l as i n s i g n . The terms of o r d e r £ are g i v e n by an e x p a n s i o n of the e i g e n f u n c t i o n s of the s o l u t i o n t o the z e r o t h o r d e r i n £ . They are thus made up of a s e r i e s of terms w i t h the same p e r i o d and h o r i z o n t a l wavenumber, but h i g h e r v e r t i c a l mode number. The o t h e r wave v a r i a b l e s a l s o i n c l u d e such e x p a n s i o n s , o n l y more c o m p l i c a t e d . I n S e c t i o n I I I we have shown t h a t i n the B o u s s i n e s q c a s e , the c h a r a c t e r i s t i c s are p e r p e n d i c u l a r t o the p r o -p a g a t i o n c u r v e s , and t h a t the p r o p a g a t i o n c u r v e s r o t a t e w i t h i n c r e a s i n g f r e q u e n c y . S i m i l a r l y , i n the n o n - B o u s s i n -esq case the asymptotes of the p r o p a g a t i o n c u r v e s r o t a t e w i t h i n c r e a s i n g f r e q u e n c y . A sum of two s i m p l e waves i s n e c e s s a r y t o s a t i s f y t h e boundary c o n d i t i o n s i n a wave g u i d e , and t h e s e waves w i l l add up i n such a way as t o remain i n phase a l o n g the wave gu i d e o n l y when they have i d e n t i c a l h o r i z o n t a l wave-45 numbers and v e r t i c a l wavenumbers of o p p o s i t e s i g n s . C o h e r e n t p r o p a g a t i o n a l o n g the g l i d e w i l l then o c c u r i f the p r o p a g a t i o n c u r v e s are such t h a t has two v a l u e s of o p p o s i t e s i g n s f o r a g i v e n k p . When the & r - a x i s of f i g u r e 7 i s an asymptote of the p r o p a g a t i o n c u r v e , has o n l y one v a l u e f o r a g i v e n k r. The f r e q u e n c y a t whi c h the ^ r - a x i s becomes an asymptote d e t e r m i n e s the lower bound of the f r e q u e n c y range f o r wave guide p r o p a g a t i o n . When the k r - a x i s i s an asymptote ( f i g u r e 8 ) , a g i v e n k r h a s two v a l u e s of tfr of o p p o s i t e s i g n s . However, f o r f r e q u e n c i e s g r e a t e r t h a n the f r e q u e n c y f o r which the k r - a x i s i s an asymptote, the p r o p a g a t i o n curve c r o s s e s the k r - a x i s p l a c i n g an upper bound on wavenumbers k^ t o which c o r r e s -pond two v a l u e s of &r d i f f e r e n t i n s i g n ( f i g u r e 9 ) . The e q u a t i o n of the asymptotes i s ' • • * o (ioo) The & r - a x i s . i s an asymptote when 2_2 - cotz e 46 F i g u r e 9 : P r o p a g a t i o n curve f o r w 1 = ga + f , f « ga. From (100), we have - 5 m 6 c o s 0 ( Z3h> + f V o . ) + s i n ^ C w ^ - j a ) } . (101) cot'© may be f a c t o r e d from each s i d e of (101) . Cw x-ga) - cot 0 (Zyb -J- fVc) + cot 7 '© (w*- f 2 - f V b ) = - { (vx- qc.) 4 t a n © C2§b* f V e ) + i a n * 0 f f Vfe)} or * vi v 47 2(u*-3^ - ( 2 g b + f Vo.) < c o t © - t o . « e ) S u b s t i t u t i n g cot 6 - t a,n 6 = 2 cat 2 6 and c o t 2® = ga. - -f *- fyfc i n ( 1 0 2 ) , we have 2 - y e O + 2 C - f Vb) + (w"*- f2-f\/t>)(cot ze «• = O or (103) w h i c h may be w r i t t e n \ sin The S r - a x i s i s then an asymptote when u>* = f + f V b . S i m i l a r l y , i t can be shown t h a t the k r - a x i s i s an asymptote when w 3 = ga. At f i r s t s i g h t , t h e r e w i l l be wave guide p r o p a g a t i o n f o r f r e q u e n c i e s <*> such t h a t i 2* f V b < 0Jr < $ Cs, The e f f e c t of the h o r i z o n t a l d e n s i t y g r a d i e n t i s t o i n -c r e a s e the lo w e r f r e q u e n c y l i m i t by - \^fVb'. Wave gui d e p r o p a g a t i o n i s a l s o p e r m i t t e d f o r «*>> ^ ga" , but f o r such f r e q u e n c i e s t h e r e i s an upper bound on the h o r i z o n t a l wavenumbers, k f ( f i g u r e 9 ) . From e q u a t i o n s (89) and (90) one f i n d s on the o t h e r hand t h a t kj and kr v a n i s h a t = f *, i n d i c a t i n g a l o w e r f r e q u e n c y l i m i t . The d i f f e r e n c e w i t h the l o w e r l i m i t d e r i v e d above i s not s u r p r i s i n g s i n c e the c o e f f i c i e n t s A, and A3 of the wave e q u a t i o n (52) have been s i m p l i f i e d t o e x c l u d e t h e term f V b , assuming i t t o be s m a l l w i t h r e s p e c t t o (<*»*-fx). T h i s a p p r o x i m a t i o n a l l o w s the s o l u t i o n t o be e x p r e s s e d i n terms of t r i g o n o m e t r i c r a t h e r t h a n con-f l u e n t h y p e r g e o m e t r i c f u n c t i o n s . Were fVb assumed i n d e -pendent of z however and kept i n the above s o l u t i o n , i t would appear as p a r t of the e x p r e s s i o n ( n / * - f * - f Vb), a f a c t o r i n and k p , thus i n d i c a t i n g the same lower f r e -quency l i m i t as found j u s t above. E q u a t i o n s (89) and (90) a l s o i n d i c a t e the upper l i m i t of f r e q u e n c y i s g r e a t e r t h a n yfga". T h i s agrees w i t h t h e above r e s u l t s f o r waves w i t h an Upper bound on the h o r i z o n t a l wa'venumber. VI ENERGY CONSIDERATIONS a) Energy of I n t e r n a l Waves. The t o t a l energy of i n t e r n a l waves ( t o second o r d e r ) i n a r e g i o n bounded by r i g i d s u r f a c e s z=0 and z=H i s where "f r e p r e s e n t s the a m p l i t u d e of the waves. b* C o n s i d e r the k i n e t i c energy: ^ v a r i e s as e , and u, v, and w v a r y as e^"*, so t h a t the k i n e t i c energy v a r i e s as e *• _ The d e n s i t y p e r t u r b a t i o n v a r i e s as e , and the wave a m p l i t u d e v a r i e s as e *' , as and w v a r i e s as e The p o t e n t i a l energy of the waves a l s o v a r i e s as e . The t o t a l energy of the waves then v a r i e s as e b) Energy E q u a t i o n F or s i m p l i c i t y of a n a l y s i s , the energy e q u a t i o n w i l l be d e r i v e d i n t e n s o r n o t a t i o n . The e q u a t i o n of motion i s where = u; + Vt- » (105). 50 and = f>o * 1*' . (107) S u b s t i t u t i n g f o r u.', ^> , and p i r : ( 1 0 4 ) , we have M u l t i p l y i n g (108) by u* and t a k i n g a time average: a* ** d*k t uivh + * c i j h si- ujuu n- uj + 5. e A * 2 - / ^ i - - ^ ^ *«« = O . (109) u « , ^ ' , and p' are p e r i o d i c i n t i m e , so t h a t E q u a t i o n (109) th e n becomes I n t e r c h a n g i n g i and j i n e q u a t i o n (110) t o get an o t h e r e q u a t i o n , then a d d i n g : at ^ a*h a * k a« f e 51 dxk " "J4 a_i?« - 1±£' - S ^ ' i ^ ' = O (in) The mean shear V£ i s a funct ion of depth only: In the two dimensional equation, = 0. We consider the It \ z / equation; that i s , equation ( i l l ) with i = j = 2 . . « xu, dw^ + 141w3 du» + <4j u3 hVi _ Q (112) Returning to the notat ion u = (u,v ,w) , we have u. This i s the energy balance equation for the v component of v e l o c i t y . The term v " f e / represents the energy exchange between the mean v e l o c i t y shear and the pe r iod ic wave motion through the Reynold's s t r e s s , and may represent product ion or ext ract ion of wave enerqy. U / and ^ ( ¥ \ are r e d i s t r i b u t i o n terms. It i s a lso e a s i l y v e r i f i e d t h a t vw i s the o n l y energy i n t e r a c t i o n term, s i n c e V i s a f u n c t i o n of z a l o n e . 'We have shown t h a t the t o t a l energy of i n t e r n a l waves v a r i e s as e . For waves p r o p a g a t i n g i n the d i r e c t i o n o f d e c r e a s i n g d e n s i t y and normal t o the mean shear f l o w , (b-2k;) i s n e g a t i v e and energy i s e x t r a c t e d from the waves, F o r waves p r o p a g a t i n g i n the o p p o s i t e d i r e c t i o n , wave energy i s p r o d u c e d . The r a t e of energy exchange can be d e t e r m i n e d by c a l c u l a t i n g the Q of the waves, th e r a t i o of t o t a l a v e r -age energy c o n t e n t of the waves t o the amount of e n e r g y , p r o d u c e d or e x t r a c t e d per c y c l e . I n t h i s case V a l u e s of Q are p r e s e n t e d i n t a b l e I I f o r two f r e q u e n c i e s and f o r t h r e e exchange mechanisms. The l a r g e Q f o r waves p r o p a g a t i n g normal t o a shear f l o w i n d i c a t e s t h a t the energy i n t e r a c t i o n i s s m a l l . The exchange f o r waves p r o p a g a t i n g p a r a l l e l t o a shear f l o w i s much g r e a t e r , but d e c r e a s e s f o r d e c r e a s i n g < T . For l o n g waves, the i n t e r a c t i o n f o r t u r b u l e n t m i x i n g i s an o r d e r df magnitude g r e a t e r t h a n t h a t f o r p r o p a g a t i o n normal t o a shear f l o w . T r u l y , f o r the l a t t e r mechanism, Q w i l l become s m a l l when approaches f * - f V b , s i n c e k r v a n i s h e s a t t h a t f r e q u e n c y . For e x t r e m e l y l o n g waves, however, i t i s no l o n g e r j u s t i f i a b l e t o n e g l e c t the v a r i a t i o n of the C o r i o l i s parameter w i t h l a t i t u d e , as has been done h e r e , and the a n a l y s i s i s no l o n g e r a p p l i c a b l e t o the r e a l ocean. TABLE I I Q f o r two f r e q u e n c i e s and f o r t h r e e d i f f e r e n t exchange mechanisms: a) p r o p a g a t i o n normal t o a shear f l o w w i t h . r o t a t i o n , b) p r o p a g a t i o n a t a n g l e <r t o normal t o shear f l o w , no r o t a t i o n , P h i l l i p s ( 1 9 6 6 ) , and c) a t t e n u a t i o n of waves by t u r b u l e n t m i x i n g i n i n t e r i o r of f l u i d ( w i t h and w i t h o u t r o t a t i o n ) , L e B l o n d ( 1 9 6 6 ) . Q= - ( b - 2 k i ) Q ( P h i l l i p s ) Q ( L e B l o n d ) £1=0 SI =.7 ^ 10 2.12 v 1 0 6.43 * 10 f .117 4.23 47 1.6 * 10 36 3 1.35 * 10 6 7.70 * 10 .935 33.9 35 54 VII CONCLUSIONS A s o l u t i o n has been found d e s c r i b i n g i n t e r n a l waves i n a r e g i o n of h o r i z o n t a l and v e r v i c a l d e n s i t y s t r a t i f i c -a t i o n . T h i s s o l u t i o n c o n s i s t s of a s u p e r p o s i t i o n of waves w h i c h have: a) e x p o n e n t i a l dependence i n the x d i r e c t i o n i n a m p l i t u d e due t o the change of d e n s i t y of the f l u i d i n t h e h o r i z o n t a l as w e l l as t o the energy exchange mechanism, b) phase d i f f e r e n c e from s u r f a c e t o bottom. The energy exchange mechanism f o r waves p r o p a g a t i n g n o r m a l l y t o a v e r t i c a l l y sheared g e o s t r o p h i c c u r r e n t i s weak, and i s l i k e l y t o be of l i t t l e i m p o r t a n c e i n the ocean. 55 V I I I LITERATURE CITED B e l l m a n , R i c h a r d , 1964. P e r t u r b a t i o n Techniques i n Math-e m a t i c s , P h y s i c s , and E n g i n e e r i n g . H o l t , R i n e h a r t , and W i n s t o n , I n c . , New Y o r k . 118 pages. B u r n s i d e , W. , 1889. On the S m a l l Wave-Motions of a H e t e r o -geneous F l u i d under G r a v i t y . P r o c . Lond. Math. S o c , 20, p. 392. F j e l d s t a d , J . , 1933. I n t e r n e W e l l e n . G e o f y s . P u b l . 10, No. 6. . O s l o . K r a u s s , W., 1964. I n t e r n e W e l l e n i n einem e x p o n e n t i e l l g e s c h i c h t e t e n Meer. K i e l e r M e e r e s f o r s c h 20, p. 109. L e B l o n d , P.H., 1966. On the damping of 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., v o l . 25, pp. 121-142. Long, R.R., 1965. On the B o u s s i n e s q a p p r o x i m a t i o n and i t s r o l e i n the t h e o r y of I n t e r n a l 'Waves. T e l l u s 17, pp. 46-52. Love, A.E.H., 1891. Wave M o t i o n i n a Heterogeneous Heavy L i q u i d . P r o c . Lond. Math. S o c , 22, p. 307. P h i l l i p s , O.M., 1966. The Dynamics of the Upper Ocean. Cambridge U n i v e r s i t y P r e s s , London. 261 pages. Sandstrom, Helmuth, 1966. The Importance of Topography i n G e n e r a t i o n and P r o p a g a t i o n of I n t e r n a l Waves. Ph.D. D i s s e r t a t i o n , U n i v e r s i t y of C a l i f o r n i a , San Diego. Stommel, Henry, 1965. The G u l f Stream. U n i v e r s i t y of C a l i f o r n i a P r e s s , B e r k e l e y and Los A n g e l e s . 248 pages. W o r t h i n g t o n , L.V., 1954b. Three d e t a i l e d c r o s s - s e c t i o n s of the G u l f Stream. T e l l u s , 6, pp. 116-123. Wust, G., 1936a. K u r o s h i o und G o l f s t r o m . E i n e ^ v e r g l e i c h -ende hydrodyn. U n t e r s u c h u n g . V e r o f f . I n s t . Meeresk. U n i v . B e r l . A, H f t , 29. B e r l i n . 

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