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Theoretical studies of the generation of surface waves and the propagation of internal waves in the sea Manton, Michael John 1970

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THEORETICAL STUDIES OF THE GENERATION OF SURFACE WAVES AND THE PROPAGATION OF INTERNAL WAVES IN THE SEA by MICHAEL JOHN MANTON B. E. , U n i v e r s i t y of Sydney, 1966 M. Eng. Sc., Un i v e r s i t y of Sydney, 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 19 70 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I a g ree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree tha p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha 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 not 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 . Department o f The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada Date fO<*U 4 u A , t?70 Chairman: Professor Robert W. Stewart. i i ABSTRACT The theory of three d i s t i n c t problems a r i s i n g i n geophysical f l u i d dynamics i s considered. P a r t I concerns the generation of sea waves. The sea st a t e and the a i r flow above the sea during a c t i v e wave o generation are discussed; i n p a r t i c u l a r , i t i s shown that a s i g n i f i c a n t f r a c t i o n of the momentum f l u x from the a i r i s t r a n s f e r r e d to the sea i n the form of wave drag. D i f f e r e n t mechanisms f o r t r a n s f e r r i n g energy and momentum int o a wave f i e l d are considered. P h i l l i p s ' resonance model of the i n i t i a l generation of waves i s modified to include the d i s p e r s i v e e f f e c t s of the wave f i e l d . I t i s shown that the i r r o t a t i o n a l flow caused by surface waves i n t e r a c t s with the turbulent v e l o c i t y f l u c t u a t i o n s i n the a i r to produce an i n s i g n i f i c a n t energy f l u x to the sea. M i l e s ' theory, which involves the i n t e r a c t i o n of the wave-induced fluctuation::in the a i r with the mean v e l o c i t y f i e l d , i s discussed and the streamline c o n f i g u r a t i o n near the c r i t i c a l height i s considered. We show that a closed streamline "cat's-eye" may l i e e i t h e r over the wave c r e s t or over the wave trough, depending upon the behaviour of the mean wind p r o f i l e . The most u n s a t i s f a c t o r y s i m p l i f i c a t i o n i n M i l e s ' theory would seem to be the neglect of a i r turbulence. By considering the f u l l s e t of energy equations and c l o s i n g them with the aid of some simple p h y s i c a l assumptions, i t i s found that the e f f e c t s of turbulence are e s s e n t i a l l y r e s t r i c t e d to the neighbourhood of the c r i t i c a l height. The turbulence acts to d i f f u s e momentum across t h i s c r i t i c a l l a y e r i n a manner analogous to the a c t i o n of molecular v i s c o s i t y . When the c r i t i c a l height is small compared with the wavelength of the surface wave, the c r i t i c a l l a y e r extends to the sea surface, and the streamline cat's-eye lies over the wave trough. , The propagation of two-dimensional waves i n a r o t a t i n g Boussinesq f l u i d with constant Brunt-Vaisala frequency and v a r i a b l e depth i s considered i n Part I I . A meth.od i s developed f o r the i n v e s t i g a t i o n of the problem, which involves f i r s t reducing i t to f i n d the s o l u t i o n of a c e r t a i n f u n c t i o n a l equation. For a l i n e a r depth p r o f i l e , an exact s o l u t i o n to t h i s equation i s obtained, which agrees with the pr e v i o u s l y known s o l u t i o n . F o r a more general p r o f i l e , an a n a l y t i c s o l u t i o n ' i s derived i n a form i n v o l v i n g i n f i n i t e s e r i e s that converge provided that the depth v a r i a t i o n s are s t r i c t l y transmissive. For a slowly varying depth, the s o l u t i o n i s also obtained by means of a two-scale perturbation expansion. F i n a l l y , exact s o l u t i o n s of the f u n c t i o n a l equation which correspond to marginally transmissive depth p r o f i l e s are obtained. In P a r t I I I the d i f f r a c t i o n of i n t e r n a l waves by a semi-i n f i n i t e v e r t i c a l b a r r i e r i n a uniformly r o t a t i n g Boussinesq f l u i d with constant Brunt-VSisala frequency, N , and constant depth i s discussed. For the frequency passband f < o < N , where f and o are r e s p e c t i v e l y the i n e r t i a l and wave frequencies, the presence of r o t a t i o n gives r i s e to i n t e r n a l K e l v i n waves which propagate without attenuation away from the b a r r i e r and which have amplitudes that decay exponentially i n the d i r e c t i o n along the b a r r i e r . TABLE OF CONTENTS PAGE Abstract i i Table of Contents l v L i s t of Tables v i i L i s t of Figures v i i i Preface x PART I - THE GENERATION OF SEA WAVES 1. Introduction 1 2. The a i r - s e a i n t e r f a c e 3 2.1 The sea surface 3 2.2 The a i r flow above the sea 9 3. I n i t i a l generation of waves by turbulent pressure f l u c t u a t i o n s 17 3.1 Introduction 17 3.2 Ph y s i c a l d e s c r i p t i o n of the resonance mechanism 18 3.3 Mathematical d e s c r i p t i o n of the resonance mechanism 20 4. Generation of waves by a i r turbulence 27 4.1 Introduction 27 4.2 Equations of motion 28 4.3 F i r s t order equations 32 4.4 Second order equations 35 4.5 F i r s t order wave s o l u t i o n 40 V . 5. Wave growth caused by mean wind 47 5.1 Introduction 47 5.2 Mathematical formulation of problem 49 5.3 Streamline, configuration 61 6. Wave growth caused by turbulent wind 72 6.1 Introduction 72 6.2 Equations of motion i n the a i r 72 6.3 Mathematical formulation of the problem 85 6.4 Solution near the c r i t i c a l height 91 6.5 Matching of inner and outer solutions 105 6.6 Normal stress at the a i r - s e a i n t e r f a c e 114 6.7 Comparison with observation 122 Bibliography 127 Appendix: Formulation of laminar i n s t a b i l i t y problem i n c u r v i l i n e a r coordinates 130 PART II - PROGRESSIVE INTERNAL WAVES IN WATER OF VARIABLE DEPTH 1. Introduction 140 2. Equations of motion 142 3. Solutions f o r transmissive depth p r o f i l e s : f u n c t i o n a l equation approach 144 4. Solution for slowly varying depth p r o f i l e s : m ultiple scale approach 159 5. Discussion of s o l u t i o n 164 6. Solutions for s p e c i f i c depth p r o f i l e s 167 7. Special c l a s s of exact solutions 170 Bibliography 175 v i . PART I I I - THE DIFFRACTION OF INTERNAL WAVES BY A SEMI-INFINITE BARRIER 1. Introduction 176 2. Formulation of boundary value problem 177 3. Integral representation of s o l u t i o n f o r f < a < N 182 4. Asymptotic s o l u t i o n for f < a < N 187 5. Asymptotic s o l u t i o n f o r N < a < f 190 6. Discussion of i n t e r n a l K e l v i n waves 192 Bibliography 195 v i i . LIST OF TABLES TABLE PAGE I I n t e g r a l s of the Function Im $" 116 I I The Function G(c) f o r t, = -8(1)8 116 I I I V a r i a t i o n of 0 w i t h u,/c o * n 119 IV Mean Wind C o n d i t i o n f o r Measurements of Dobson (1969) 126 v i i i . LIST OF FIGURES PART I FIGURE 1 Relationship between wave height and phase speed at peak of wave spectrum: data points from Stewart (1961). 2 Relationship between wind duration and phase speed at peak of wave spectrum: data points from Stewart (1961). 3 Closed streamline cat's-eyes at c r i t i c a l height. 4 General behaviour of (a) F^(z) and (b) X_^(z) . 5 General behaviour of F (z) as Y •+ 0 . r 6 General behaviour of (a) F (z) and (b) X (z) as V increases when J(z) i s monotonic increasing. 7 Phasor diagrams for p, w and n when J(z) i s monotonic inc r e a s i n g : (a) V i s small, and (b) V i s large. 8 Streamline configuration when J(z) i s monotonic increasing. 9 Shape of J(z) for a logarithmic mean v e l o c i t y p r o f i l e . 10 General behaviour of (a) F (z) and (b) X (z) for small values T when J i s given by (5.3.6). 11 General behaviour of (a) F (z) and (b) X (z) for r = r r r s 12 General behaviour of (a) F (z) and (b) X (z) for V > T r r s 13 Phasor diagram for p, w and n when J(z) i s given by (5.3.6) and when T i s large. 14 Streamline configuration when J(z) i s given by (5.3.6) and when T i s large. 15 V a r i a t i o n of r e l a t i v e phase of pressure at surface with U^/c^: data points from Dobson (1969). 16 Streamline configuration f o r a l i n e a r mean p r o f i l e when (a) kz > 1 and (b) z < A (1 - kz ). c c cJ i x . PART II FIGURE 1 Streamlines (a) and wave p r o f i l e s (b) f o r mode n = 1 corres-ponding to depth p r o f i l e (6.1) 2 Streamlines (a) and wave p r o f i l e s (b) for mode n = 2 corres-ponding to depth p r o f i l e (6.1) 3 Streamlines (a) and wave p r o f i l e s (b) for mode n = 1 corres-ponding to depth p r o f i l e (6.2) 4 Streamlines (a) and wave p r o f i l e s (b) for mode n = 1 corres-ponding to depth p r o f i l e (6.3) 5 Streamlines (a) and wave p r o f i l e s (b) for mode n = 1 corres-ponding to depth p r o f i l e (7.2) i n which a = 1 = 8 6 Time evolution of streamlines f o r mode n = 1 corresponding to depth p r o f i l e (7.3) i n which a = 1 = B 7 Wave p r o f i l e s f o r mode n = 1 corresponding to depth p r o f i l e (7.3) i n which a = 1 = 6 8. Streamlines (a) and wave p r o f i l e s (b) for mode .n = 1 corres-ponding to depth p r o f i l e (7.4) i n which a = 1/2 and 3 = 1 PART III FIGURE 1 Plan view of wave approaching s e m i - i n f i n i t e b a r r i e r y = 0 , x < 0 . 2 The X-plane corresponding to (3.10) for f < a < N 3 The £-plane corresponding to (4.1) when f < a < N f o r the cases (a) cos <j> > 0 , (b) cos <j> •= 0 and (c) cos <j> < 0 The path of steepest descent i s denoted by T g . 4 Regions around' b a r r i e r i n which terms of asymptotic s o l u t i o n (equation (4.3)) become important for the case f < a < N and where (a) 0 < 6 < TT/2 and (b) TT/2 < 9 < TT 5 Maximum amplitude of i n t e r n a l K e l v i n wave as a function of Y = f/o for various values of 9 X . PREFACE This d i s s e r t a t i o n considers the theory of three d i s t i n c t problems a r i s i n g i n geophysical f l u i d dynamics. Part I concerns the generation of sea waves and. i t represents part of the continuing i n v e s t i g a t i o n of the a i r - s e a i n t e r f a c e undertaken by the I n s t i t u t e of Oceanography, U n i v e r s i t y of B r i t i s h Columbia. In §3 of Part I, . P h i l l i p s ' (1957) theory i s modified to include the e f f e c t s of wave disp e r s i o n , based on an idea of R.W. Stewart. The propagation of i n t e r n a l waves over a v a r i a b l e topography i s discussed i n Part I I . With Dr. L.A. Mysak as co-author, a paper presenting the mathematics of §3 of t h i s Part has been prepared, and accepted for p u b l i c a t i o n i n the Journal of Mathematical Analysis and App l i c a t i o n s . Part I I I i s also concerned with i n t e r n a l waves: the d i f f r a c t i o n of i n t e r n a l - i n e r t i a l waves by a s e m i - i n f i n i t e b a r r i e r i n a r o t a t i n g system i s considered. With Dr. L.A. Mysak and Mr. R.E. McGorman as co-authors, a paper describing t h i s a n alysis has been accepted for p u b l i c a t i o n i n the Journal of F l u i d Mechanics. The problems discussed i n Parts II and I I I arose from a graduate course (Mathematics 534) given by Dr. L.A. Mysak during the academic year 1968-69. The author wishes to express h i s appreciation to Prof. R.W. Stewart and Dr. L.A. Mysak for t h e i r guidance throughout the course of this work. PART I THE GENERATION OF SEA WAVES 1 1. 1. INTRODUCTION Although the problem of wave generation on the sea surface has been s e r i o u s l y studied f o r over a century (e.g. Helmholtz 1868), the l a s t f i f t e e n years have produced the most s i g n i f i c a n t contributions to our understanding of the processes involved. In a volume commemor-ating the 70th birthday of G.' I. Taylor, U r s e l l (1956) wrote a c r i t i c a l survey of 'wave generation by wind'. His conclusion was that 'the present state of our knowledge i s profoundly u n s a t i s f a c t o r y ' . I t was not c o i n c i d e n t a l that the following year produced two d i f f e r e n t , but complementary, theories of wave generation. One theory by P h i l l i p s (1957) describes the appearance of waves on an i n i t i a l l y smooth surface, while the other by Miles (1957) considers the growth of waves already present on the a i r - s e a i n t e r f a c e . While P h i l l i p s and Miles have provided the backbone of our understanding, t h e i r theories require at l e a s t some mod i f i c a t i o n before the wave generation process may be considered as known. In the present work, these theories are c r i t i c a l l y examined and modified theses are presented. There i s a general discussion i n §2 of the sea state and the a i r flow above the sea during a c t i v e wave generation. In p a r t i c u l a r , i t i s shown that a s i g n i f i c a n t f r a c t i o n of the momentum f l u x from the a i r i s transferred to the sea'in the form of wave drag. D i f f e r e n t mechanisms f or t r a n s f e r r i n g energy and momentum into a wave f i e l d are then considered i n the remaining sections. Although P h i l l i p s ' resonance model of the i n i t i a l generation of waves i s correct i n p r i n c i p l e , i t i s shown i n §3 that i n c l u s i o n of the di s p e r s i v e e f f e c t s of the wave f i e l d produces an expression d i f f e r e n t from P h i l l i p s ' f o r the rate of increase 2. of wave energy. A wave present on the sea surface produces some disturbance i n the a i r which i s capable of i n t e r a c t i n g with the ambient a i r flow. Thus, there i s the p o s s i b i l i t y of a feedback process such that the wave energy may grow exponentially with time. The e f f e c t of the turbulence f i e l d i n the a i r on the wave f i e l d i s considered i n §4. I t i s found that the i r r o t a t i o n a l flow, caused by the surface waves, i n t e r a c t s with the turbulent v e l o c i t y f l u c t u a t i o n s to produce an i n s i g n i f i c a n t energy f l u x to the sea. M i l e s ' theory involves the i n t e r a c t i o n of the wave-induced f l u c t u a t i o n i n the a i r with the mean v e l o c i t y f i e l d , and i t in d i c a t e s the importance of the c r i t i c a l height at which the mean wind speed equals the wave phase speed. In §5, we formally derive and discuss M i l e s ' expression f o r the wave-induced shear s t r e s s . The streamline c o nfiguration near the c r i t i c a l height i s also considered, and we show that the closed streamline 'cat's-eye' may l i e e i t h e r over the wave crest (as suggested by L i g h t h i l l 1962) or over the wave trough, depending upon the behaviour of the mean wind p r o f i l e . The most uns a t i s f a c t o r y s i m p l i f i c a t i o n i n Miles' theory would seem to be the neglect of a i r turbulence. However, i n §6 i t i s shown that the e f f e c t s of turbulence are e s s e n t i a l l y r e s t r i c t e d to the neighbourhood of the c r i t i c a l height. The turbulence acts to d i f f u s e momentum across t h i s c r i t i c a l layer i n a manner analogous to the action of molecular v i s c o s i t y . When the c r i t i c a l height i s small compared with the wave-length of the surface wave, the c r i t i c a l l ayer extends to the sea surface. In t h i s case, the streamline cats'-eyes l i e over the wave troughs, and the predicted wave-induced pressure at the sea surface i s not inconsistent with the measurements of Dobson (1969). ( 3. 2. THE AIR-SEA INTERFACE 2.1 The Sea Surface When there i s a c t i v e wave generation, the wave energy s p e c t r a l density i s always very sharply peaked. This means there i s some v a l i d i t y i n mathematically modelling the sea surface n by a simple sinusoid; i . e . n(x,t) = A cos k.(x - ct) , where x defines a h o r i z o n t a l plane, t i s time, A i s the wave amplitude, k i s the wavenumber and c i s the phase v e l o c i t y . When the wavelength i s much less than the depth of the ocean, the phase speed c = |c| and wavenumber k = |k| are r e l a t e d simply by (e.g. P h i l l i p s 1966, §3.2) c 2 = g/k . (2.1.1) Longuet-Higgins (1963) has shown that a progressive wave must break i f the maximum wave slope kA i s greater than about 0.5. Thus, the wave slope must ne c e s s a r i l y be much less than unity; i n f a c t , kA - 0.1 i n general. The wave energy per un i t area i s E = pT7 g <n > (2.1.2) i 4. where p i s the d e n s i t y of water, g i s the g r a v i t a t i o n a l a c c e l e r a t -w i o n , and < > denotes an average value . A dimensionless measure of the r a t e of i n c r e a s e i n wave energy i s given by Q _ _1 _dE b " oE dt » where a = k.c i s the wave frequency. C l e a r l y , S i s the f r a c t i o n a l i n c r e a s e i n energy per r a d i a n . Therefore, a dimensionless time s c a l e f o r wave growth i s T = (2TTS) 1 which i s the number of wave c y c l e s r e q u i r e d f o r the energy to i n c r e a s e by a f a c t o r of e - 2.7. Measurements of wave growth (Snyder and Cox 1966, Barnett and Wilkerson 1967, Dobson 1969) i n d i c a t e t h a t i n general S f o l l o w s the e m p i r i c a l law S = (p /p )(U-/.c - 1) , (2.1.3) a w _> where p a i s the d e n s i t y of a i r and U,. i s r e p r e s e n t a t i v e of the mean wind speed some d i s t a n c e above the a i r - s e a i n t e r f a c e . The r e l a t i o n (2.1.3) i s i n v a l i d f o r c/U,. gr e a t e r than u n i t y and f o r very h i g h frequency waves ( i . e . f o r c/U^ << 1). I t i s seen from (2.1.3) t h a t , -3 -3 s i n c e p/p = 1.3 * 10 , S i t s e l f i s of order 10 . Thus, the a w 2 time s c a l e T ~ 10 and hence the wave energy i n c r e a s e s by a f a c t o r of e over a few hundred wave c y c l e s . i 5. The wave s p e c t r a l density function F (a) i s observed to have a well-defined low-frequency cut-of f , and i t v a r i e s approx-imately as a ^  for a > a . This power-law behaviour of F i s P nn consistent with the assumption that growth i s l i m i t e d by a maximum allowable wave slope; then ( P h i l l i p s 1958) C mg a , a > a F _ ( a ) = \ . P (2.1.4) P -2 Measurements suggest that m - 1.3 >< 10 . Equation (2.1.4) implies that the t o t a l wave energy per u n i t area i s E(a ) = pT7 g / F (a)da , P w ' 0 nn 4 p w § c p 3 - 4 ( 2 - 1 - 5 > The c u t - o f f frequency a i s found to decrease as the wind duration t P increases. This could be implied d i r e c t l y from (2.1.3) which states that the rate of energy increase i s greatest f o r the highest frequencies. On the other hand, there are at l e a s t two e f f i c i e n t mechanisms for t r a n s f e r r i n g energy down the spectrum. The f i r s t , described by Benjamin and F e i r (1967), i s based on the f a c t that a progressive Stokes wave i s unstable. Consequently, the energy contained i n side-band frequencies on ei t h e r side of a given Stokes wave frequency ' increases at a maximum rate given by 1 dE 1 , 2 . 2 — — = -r k A oE dt 4 2.5 10 -3 for kA = 0.1 6. where IcA i s the maximum slope of the Stokes wave. But t h i s i s of the same order of magnitude as the observed rate of energy input from the mean wind. Hence, wave growth j u s t ahead of the front face of the spectrum should be much greater than that at lower frequencies. The second mechanism, developed by Longuet-Higgins (1969a), involves the steepening of short waves near the crests of longer waves. This i s pr i m a r i l y due to the contraction of the surface near the crests of long waves (Longuet-Higgins and Stewart 1960). When the slope of the long waves i s not small, t h i s steepening e f f e c t i s large and i t can cause the short waves.to break on the forward face of the long waves. Con-sequently, the momentum of the short waves i s transferred to the longer wave. Now Stewart (1961) has used some wave growth data ( o r i g i n a l l y compiled by Groen and Dorrestein 1958). to show that a s i g n i f i c a n t f r a c t i o n of the wind stress d i r e c t l y generates surface waves. From t h i s data, i t i s found that the wave slope slowly decreases as a P decreases; i n p a r t i c u l a r , (see f i g u r e 1) (kH) c 0 ' 1 5 - 0.45 ( m . / s e c . ) 0 , 1 5 for 0.2 < c /U_ < Q.75 , p p 5 where H i s the quoted wave height. This implies that the wave energy -3. 7 E i s approximately proportional to a * , and hence i t i s quite w e l l described by (2.1.5). D i f f e r e n t i a t i n g (2.1.5), the rate of increase i n t o t a l wave energy i s seen to be dt a dt P (2.1.6) ( 7. where da /dt i s the r a t e at which the f r o n t face of the spectrum moves. P We now assume that energy from the mean wind i s absorbed at the peak of the spectrum only and at a r a t e given by (2.1.3). We f u r t h e r assume that only a f r a c t i o n 3 of t h i s energy i s t r a n s f e r r e d down the spectrum; the remainder i s r e q u i r e d to make up l o s s e s due to wave bre a k i n g , e t c . Hence, dt" = 3 ( P a / p w ) ( U 5 a p / g - l ) a p E . (2.1.7) C l e a r l y , t h i s e x p r e s s i o n i s v a l i d only f o r a p somewhat l e s s than g/U 5 where c /U 5 = 1 . Equating (2.1.6) and (2.1.7), the r a t e of movement of the f r o n t face of the spectrum i s given by da „ ^ - ( 3 / 4 ) ( p a / p w ) ( V p / g - l ) a 2 . This f i r s t order o r d i n a r y d i f f e r e n t i a l equation i s r e a d i l y solved to y i e l d c p / U 5 + An CI - C p / U 5 ) = - ( 3 / 4 ) ( p a / p w ) ( g t / U 5 ) . (2.1.8) The r a t i o - (gt/U,.)/ [c p/U,. + £n(l - Cp/U,_) ] f o r the data given by Stewart i s p l o t t e d i n f i g u r e 2, and i t i s found to have an average 4 -2 value of about 3.5 x 10 . This i m p l i e s that 3 - 9 x 10 ; that i s , only about 10% of the energy gained from the wind i s r e t a i n e d by the wave f i e l d . This i s c o n s i s t e n t w i t h the assumption made by 8. Longuet-Higgins (1969b) i n p r e d i c t i n g the value of m i n (2.1.4) that most of the energy l o s t by wave breaking goes d i r e c t l y into turbulence. The growth equation (2.1.8) also implies that f o r small wind durations t , C c / l O 2 - (B/2)(p /p )(gt/U,) ; (2.1.9) p j a w J i n p a r t i c u l a r , we require that gt/U^ << 17,500 and t h i s corresponds with a duration of about 5 hours f o r a 10 m./sec. wind. From (2.1.5) and (2.1.9), i t i s seen that the wave energy E grows approximately 2 as t f or small durations. For an i r r o t a t i o n a l wave, the energy density E and the momentum density M .are simply r e l a t e d by (e.g. P h i l l i p s 1966, §3.2) E = Mc . (2.1.10) Assuming that (2.1.10) may be applied to the peak of the wave spectrum, we f i n d that J ^ C T + ( E / C ) , dt p w p d t where T = dM/dt i s the momentum f l u x to the waves (or the wave-w induced shear s t r e s s ) . Thus, from (2.1.5), i t i s seen that x = (3/4c ) ~ . (2.1.11) w p dt Using (2.1.3), equation (2.1.11) becomes Tw - I <<VPw> C V C p " X ^ E / C p (2.1.12) Now the t o t a l shear stress T i n the a i r above the sea surface may be represented by (see §2.2) T = C D p a U 2 , (2.1.13) -3 where i s a drag c o e f f i c i e n t approximately equal to 1.2 x 10 Thus, from (2.1.5), (2.1.12) and (2.1.13), the r a t i o of the wave-induced shear stress to the t o t a l s t r e s s i s given by T /T = (3m/16C Wc / l U C l - c /UP) . (2.1.14) w D p i p J The r a t i o (2.1.14) a t t a i n s a maximum value at c /U r =0.5 , such that P 5 (T /T) = 3m/64 C = 0.52 . w max D This suggests that over 50% of the momentum f l u x from the a i r i s tran s f e r r e d to the sea i n the form of wave drag. In f a c t , measure-ments by Dobson (1969) imply that up to 80% of the momentum f l u x i n the a i r appears as form drag at the sea surface. 2.2 The A i r Flow Above the Sea The a i r flow over the sea i s i n v a r i a b l y turbulent with a mean time scale of the order of a day (10^ s e c ) . The atmospheric pressure f i e l d has a h o r i z o n t a l scale of the order of hundreds of kilometres i 10. (10^ cm.) and, for a d i a b a t i c conditions, the v e r t i c a l length scale i s of order 10^ cm. Thus, when considering the motions of surface waves 3 with wavelength of order 10 metres (10 cm.) and period of order 3 seconds, the a i r flow may be treated as a f u l l y developed, zero pressure gradient boundary la y e r . But such a flow supports a constant shear stress- x, say. Far above the a i r - s e a i n t e r f a c e , viscous e f f e c t s are n e g l i g i b l e and so x i s maintained by the turbulence, which has a •velocity s c a l e u_<_ . I f the a i r density i s , then we have that 2 T = p u, , and t h i s stress must be transmitted through the i n t e r f a c e , a *'>" If i t i s assumed that the momentum transfe r from the a i r to the water i s purely turbulent, the f l u c t u a t i o n s of order u^ are required i n the water, such that 2 2 T = p U.,. = p U , a " w "W where p i s the density of sea water. That i s , w J ' = (p„/p ) u. ~w a w A _3 Now because p / p - 1 0 , i t i s seen that u, ~ 30 u, . In the a w * *w 2 water, there i s a k i n e t i c energy per unit mass of order u, . Thus ~w neglecting viscous e f f e c t s and because p / p i s small, t h i s energy a w could give r i s e to o s c i l l a t i o n s at the i n t e r f a c e of amplitude , where 11. H T ~ U*w / g = ^ a V (u*/g) ' C l e a r l y , H^ i s a measure of the maximum v e r t i c a l turbulent length scale near the i n t e r f a c e . On the other hand, the viscous transfer length scale i n the a i r i s of order v./u, where v i s the v i s c o s i t y ° a K a of a i r , so that the Reynolds number of the v e r t i c a l motion j u s t above the i n t e r f a c e i s R a " t p a / p w ) ( u 2 / v a g ) . S i m i l a r l y , the Reynolds number for the v e r t i c a l turbulent motion j u s t below the i n t e r f a c e i s R - ul /v g = (p /p ) 1 / 2 (v /v )R , w *w w a w a w a where v i s the v i s c o s i t y of water. Now u, i s t y p i c a l l y of order w J x 2 2 20 cm./sec; thus, taking v -0.15 cm. / s e c , v -0.01 cm. /sec. a w and g - 10 3 cm./sec. 2 , we f i n d that R - 0.05 and R - 0.03 . b a w Hence, near the i n t e r f a c e , viscous e f f e c t s cannot be neglected and the v e r t i c a l motion of the turbulence i s expected to be attenuated. I t would therefore appear that the i n t e r f a c e may be treated as r i g i d with respect to the v e r t i c a l turbulent motions i n the a i r . Although the a i r turbulence i s unable to generate turbulence i n the water by mechanically d r i v i n g the a i r - s e a i n t e r f a c e , momentum might be transferred to the water v i a the viscous shear s t r e s s . The viscous stresses i n the water and i n the a i r are i n equilibrium at the i n t e r f a c e when 12. , 9 u N where 9 u / 9 z i s the o v e r a l l r a t e of shear s t r a i n and u i s the dynamic v i s c o s i t y . But ( 9 u / 9 z ) ^ i s a measure of the response time of the flow and , 9 u x - l , , N , 9 u x - l , 9 u . - l ( 9 l } w = W ( 9 i } a ~ 7 0 ( 9 ¥ } a ' Therefore, the response time of the motion i n the water i s much greater than that i n the a i r . This i m p l i e s that the i n e r t i a of the water i s so great that the a i r - s e a / i n t e r f a c e may be t r e a t e d as a r i g i d w a l l w i t h respect to high frequency h o r i z o n t a l t u r b u l e n t f l u c t u a t i o n s i n the a i r . However, the water surface i s able to respond to s l o w l y v a r y i n g f l u c t u a t i o n s , and so the mean speed at the i n t e r f a c e i s not n e c e s s a r i l y zero. , Consider the response of a s e m i - i n f i n i t e body of water, i n i t i a l l y at r e s t , to a constant shear s t r e s s suddenly a p p l i e d at the s u r f a c e . Mathematically, we t r e a t the i n i t i a l - b o u n d a r y value problem: u = v u ( z , t ) f o r t > 0 and -°° < z < 0 ; t w zz u(z,0) = 0 = u(-°°,t) 2 and u (0,t) = (p /p )(u./v ) . This i s r e a d i l y s o l v e d and the v e l o c i t y at the sur f a c e i s found to be u(0,t) = ( 2 / T T 1 / 2 ) ( P /p ) ( u 5 t / v ) 1 / 2 u A • a w " w * , 9 u . W d Z W 13. The 'momentum thickness of the shear l a y e r at the surface may be defined as Z J = (TTV t ) 1 / 2 d w 2 such that the net s t r e s s p z ,u (0,t) ( = p u.) remains constant. yw d t a * Let k , a and c be the wavenumber, frequency and phase speed of a t y p i c a l deep water wave on the i n t e r f a c e . Then kz = 1 when t = t , ^ d d and - 1 3 3 a t d = TT ( u A / v w g ) (c/u A) ; i . e . f o r c/u.,. ~ 10 , about 10^ radians are r e q u i r e d before the shear l a y e r t h i c k n e s s approaches the wavelength of the s u r f a c e wave. I f the wave energy per u n i t area i s E , then measurements (e.g. Snyder dE 3 and Cox 1966) i n d i c a t e that a E / — i s of order 10 . Thus, the time dt s c a l e a s s o c i a t e d w i t h the v i s c o u s surface l a y e r t ^ i s much grea t e r than that a s s o c i a t e d w i t h wave growth E / " ^ j r • Consequently, the e f f e c t of t h i s s u r f a c e l a y e r i s probably not important to the wave generation process. We a l s o note that v i s c o s i t y i s capable of producing but a l i n e a r growth i n the t o t a l momentum per u n i t area of the water (p uz , ~ t) , whereas the momentum per u n i t area of surface waves i s Kw d observed to grow e x p o n e n t i a l l y . Since the a i r - s e a i n t e r f a c e appears as a s o l i d b a r r i e r to the a i r turbulence, the usual arguments l e a d i n g to the 'log law of the w a l l ' should apply f a r above the i n t e r f a c e . Thus, the turbulence i s c h a r a c t e r i s e d by a s i n g l e l e n g t h s c a l e z , the v e r t i c a l d i s t a n c e above 14. the i n t e r f a c e , and a s i n g l e v e l o c i t y scale u A . Assuming that there i s a balance between the l o c a l rate of turbulence production, which 2 varies as u A dU^/dz (where U^(z) i s the mean wind speed), and the 3 l o c a l rate of turbulence d i s s i p a t i o n , which varies as u^/z , i t i s seen that du\ u. 1 _ _* dz KZ (2.2.1) where K i s the von Karman constant. Thus, the mean wind v e l o c i t y i s given by U 2(z) = (U A/K) £n ( Z / Z Q ) , (2.2.2) where Z q i s the v i r t u a l o r i g i n of the mean wind. For a r i g i d smooth w a l l , the s t r e s s near the surface i s supported by viscous e f f e c t s and then z ~ v /u. . However, we have shown that f l u i d v i s c o s i t y i s o a * incapable of producing the observed rate of increase of momentum i n the water, and so some other mechanism must support the stress close to the a i r - s e a i n t e r f a c e . There i s some evidence that the sea surface appears 'rough' to the a i r motion above (e.g. see P h i l l i p s 1966, §4.8); i . e . although the surface slope i s generally f a r too small f o r flow separation to occur, v i s c o s i t y does seem to play a n e g l i g i b l e part i n the actual momentum transfe r process. Stewart (1961) recognised that 'roughness' of the i n t e r f a c e implies that the momentum transfe r i s v i a pressure forces, which produce i r r o t a t i o n a l motion. Since the only i r r o t a t i o n a l motion that decays away from the surface i s wave-like, i t i s c l e a r that the apparent roughness of the i n t e r f a c e , i t s e l f , 15. i m p l i e s that a l a r g e f r a c t i o n of the t o t a l momentum t r a n s f e r to the water i s i n the form of wave generation. Recent measurements of the pressure at the i n t e r f a c e (Dobson 1969) show t h a t , indeed, up to 80% 2 of the t o t a l s t r e s s (p u A) t r a n s f e r s momentum d i r e c t l y i n t o surface waves by means of pressure f l u c t u a t i o n s . Because the t o t a l s t r e s s must be independent of h e i g h t , Stewart f u r t h e r deduced that the t u r b u l e n t c o n t r i b u t i o n to the s t r e s s must decrease as the surface i s approached from above, i . e . as the wave-generating shear s t r e s s i n c r e a s e s . Thus, the slope of the mean v e l o c i t y , which i s p r o p o r t -i o n a l to the t u r b u l e n t s t r e s s , should correspondingly decrease, and hence the mean v e l o c i t y i s probably not t r u l y l o g a r i t h m i c near the s u r f a c e . Now the wave-generating shear s t r e s s i s present only below the c r i t i c a l h e i g h t where U^Cz^) = c , the wave phase speed (see §5). But z i s o f t e n l e s s than the wave amplitude; consequently, c u n t i l v e l o c i t y measurements are made very c l o s e to the water s u r f a c e , the a c t u a l shape of the mean v e l o c i t y i n the neighbourhood of the c r i t i c a l h e i g h t remains open to con j e c t u r e . On the other hand, the l o g a r i t h m i c behaviour of the mean v e l o c i t y above the c r i t i c a l height i s q u i t e f i r m l y e s t a b l i s h e d , and, f o r wind speeds l e s s than about 2 2 10 m./sec, the drag c o e f f i c i e n t = u^/Uj. (U,. i s the mean v e l o c i t y -3 5 metres above the surface) i s approximately equal to 1.2 x 10 (Miyake et a l 1970). A constant value of i m p l i e s that the length s c a l e Z q i s constant. However, equation (2.2.2) shows that the mean v e l o c i t y depends upon £n z^ ; hence, the drag c o e f f i c i e n t i s r a t h e r i n s e n s i t i v e to the p r e c i s e value of z . P h i l l i p s (1966) quotes o 2 unpublished r e s u l t s of Sheppard which suggest that z ^ u /g . Such o behaviour of z seems c o n s i s t e n t w i t h the concept that the s u r f a c e 16. i s aerodynamically rough, and i t had previously been suggested by Chamock (1955) on dimensional grounds. On the other hand, the con-stancy of the maximum surface slope at the peak of the spectrum suggests that may be constant. Since the surface roughness ( i . e . the sea surface) i s e s s e n t i a l l y s i n u s o i d a l , the surface slope, rather than the wave height, determines the degree of roughness. Hence, the drag c o e f f i c i e n t may appear to be constant because the maximum wave slope i s approximately constant f o r a f u l l y developed sea. 17. 3. INITIAL GENERATION OF WAVES BY TURBULENT PRESSURE FLUCTUATIONS 3.1 Introduction It i s c l e a r that because turbulent v e l o c i t y f l u c t u a t i o n s are greatly attentuated near the a i r - s e a i n t e r f a c e (see §2.2), they are unable to generate waves on an i n i t i a l l y smooth surface. However, turbulent pressure f l u c t u a t i o n s are transmitted (almost unattentuated by v i s c o s i t y ) down to the i n t e r f a c e . P h i l l i p s (1957, 1966 §4.2) has demonstrated that turbulent pressure f l u c t u a t i o n s passing over a deep ocean can generate surface g r a v i t y waves. The generating process involves a resonant coupling between waves of wavenumber k and frequency a and turbulent pressure f l u c t u a t i o n s of the same wave-number that are advected over the surface with an average v e l o c i t y Up such that a = k.Up . Because the turbulent pressure f i e l d i s not r i g i d l y advected, P h i l l i p s deduced that resonance occurs only as long as a group of pressure f l u c t u a t i o n s remains coherent. Thus, the wave s p e c t r a l density function was found to grow l i n e a r l y with time and to be proportional to the i n t e g r a l time scale T^ of the turbulent pressure component with wavenumber k . However, the d i s p e r s i v e nature of the wave f i e l d i t s e l f has been neglected i n t h i s d e s c r i p t i o n . A wave group of bandwidth Ak , group speed c and phase speed c w i l l remain coherent f o r a time TT, of order (|c - c I Ak) . W 1 g 1 Assuming that Ak ^ k , the group wavenumber, i t follows that T o i s of order unity, where a i s the group frequency. Experiments of Willmarth and Wooldridge (1962) suggest that, for turbulent pressure f l u c t u a t i o n s , kU T i s of order 20, so that at the resonant wavenumbers, 18. T i s expected to be much greater than T . This implies that, i n r W general, a wave group, w i l l become incoherent before the f o r c i n g turbulent pressure f i e l d becomes incoherent; therefore, the important time scale associated with wave growth i s T rather than T . W r 3.2 P h y s i c a l Description of the Resonance Mechanism We consider a wave group of wavenumber k and frequency 1/2 a = (kg) propagating on deep water. At some insta n t i n time, a group of pressure f l u c t u a t i o n s , also of wavenumber k , i s t r a v e l l i n g over the surface such that the resonance condition, a = k.Up , i s s a t i s f i e d , where Up i s the advection v e l o c i t y of the turbulence. If the wave group were non-dispersive or contained a single, wavenumber and i f the pressure f l u c t u a t i o n s were stationary i n a frame moving with v e l o c i t y Up , then t h i s state of resonance would continue i n d e f i n i t e l y such that the wave amplitude would grow l i n e a r l y with time and the wave energy would increase as the square of time. However, none of these conditions i s s a t i s f i e d i n r e a l i t y ; the phase speed of deep water waves i s twice t h e i r group speed, l i n e spectra do not e x i s t i n p r a c t i c e and there i s not a unique r e l a t i o n s h i p between the frequency and x^avenumber of turbulent f l u c t u a t i o n s . The resonance condition implies that the component of the turbulence advection v e l o c i t y i n the k - d i r e c t i o n i s equal to the wave phase v e l o c i t y c . On the other hand, the wave group i t s e l f t r a v e l s at v e l o c i t y c = y c , ~g z ~ so that a f t e r a short time the group of f o r c i n g pressure f l u c t u a t i o n s moves ahead of the wave group and energy ceases to be fed in t o the wave 19. group. Some time l a t e r , another group of turbulent pressure f l u c t -uations comes in t o resonance with the wave group so that the l a t t e r gains energy u n t i l these pressure f l u c t u a t i o n s move ahead of the wave group. Since the r e l a t i v e phase of each group of f o r c i n g pressure f l u c t u a t i o n s i s random, the input of energy to the waves proceeds i n a manner reminiscent of a c l a s s i c a l random walk and hence the wave energy i s expected to increase l i n e a r l y with time. Now P h i l l i p s (1957) has shown that while the turbulent pressure f l u c t u a t i o n s remain coherent, the amplitude of the forced wave component also grows l i n e a r l y with time. Hence, we are presented with an apparent paradox i n which both the amplitude and energy of a wave group increase l i n e a r l y with time. The paradox i s resolved by considering the d e t a i l e d behaviour of the i n t e r a c t i o n between the turbulence and wave f i e l d s . I t has been shown i n §3.1 that the di s p e r s i v e nature of the wave f i e l d i s more important than that of the turbulence. Thus, we consider a group of pressure f l u c t u a t i o n s that are r i g i d l y advected with v e l o c i t y Up . At time t = 0 , the pressure f l u c t u a t i o n s appear to be i n resonance with a group of waves of bandwidth Ak , say. However, as time proceeds, the wave group gradually disperses such that the f r a c t i o n of the group a c t u a l l y i n resonance with the pressure f l u c t u a t i o n s decreases. This phenomenon i s a consequence of the well-known uncertainty p r i n c i p l e of f o u r i e r transform theory that the bandwidth-duration product of a s i g n a l must not be less than a c e r t a i n constant (e.g. Bracewell 1965, chpt. 8). That i s , as time increases, the band-width of the wave group that i s gaining energy from the turbulence i s 20. decreasing. I f t h i s decrease i s l i n e a r i n time, then i t i s c l e a r that both the amplitude and energy of the wave group w i l l grow l i n e a r l y with time. 3.3 Mathematical Description of the Resonance Mechanism To the f i r s t approximation, nonlinear and r o t a t i o n a l e f f e c t s may be neglected when describing the motion of surface gravity waves on a deep ocean ( P h i l l i p s 1966, §3.1). Thus, there e x i s t s a v e l o c i t y p o t e n t i a l <j> such that u = V<j> , where u i s the E u l e r i a n v e l o c i t y vector. From the i n c o m p r e s s i b i l i t y condition V.u = 0 , the equation of motion becomes Equation (3.3.1) applies w i t h i n the domain R^  : - r o < x , y < 0 0 , -oo < z < 0 and 0 < t < 0 0 ; where (x,y) are the h o r i z o n t a l c a r t e s i a n coordinates; z i s the v e r t i c a l coordinate such that z = 0 i s the undisturbed free surface of the ocean; t i s time. On the boundaries of R , the following conditions are imposed: V d) = 0 . (3.3.1) <J> 0 as z -> -°° , (3.3.2) iv = d> on z = 0 , (3.3.3) p/p + gn + 4> = 0 on z = 0 . (3.3.4) w and n = 0 = n at t = 0 (3.3.5) 21. In these equations, n i s the surface e l e v a t i o n , p the pressure f l u c t u a t i o n at the surface due to a i r turbulence, p the water w density, and g the 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 . We define f o u r i e r transforms of the dependent va r i a b l e s such that /co j_k . X dx e - ~ d)(x,z,t) , (k,t) = /"dx e 1 - ' 5 n(x,t) , (3.3.6) and p(k,t) = / dx e^~'~ p(x,t) , where x = (x,y) and k = (k^jk^) . Because the turbulent pressure p i s a st o c h a s t i c v a r i a b l e , i t follows from equation (3.3.4) that <j) and n are also s t o c h a s t i c . Hence, the transform v a r i a b l e s p , $ and f\ e x i s t as generalised functions only. Transforming the equations of motion, i t i s f i n a l l y found from equations (3.3.2) -(3.3.6) that $ = k 1 n t ( k , t ) e k z , (k = |k|) , and f, + kg fKk,t) = -kp(k,t)/pv7 , (3.3.7) t t ~ w where n(k,0) = 0 = fj t(k,0) . The complete s o l u t i o n of (3.3.7) i s given by fi(k,t) = -(p c) 1 d? p(k,0 s i n o(t-c) . (3.3.8) ~ w o 22. 1/2 The phase speed and frequency of f r e e s u r f a c e waves are c = (g/k) 1/2 and a = (kg) , r e s p e c t i v e l y . This s o l u t i o n f o r the wave amplitude was f i r s t derived by P h i l l i p s (1957). Assuming that the turbulence f i e l d i s s t o c h a s t i c a l l y s t a t i o n a r y ( f o r a l l t ) t and h o r i z o n t a l l y homogeneous, we may define the time l a g covariance f u n c t i o n ft(k,x) of the pressure f l u c t u a t i o n s at the wavenumber k by < p ( k , t 1 ) p * ( k ' , t 2 ) > = 4 T r 2 f t ( k , t 1 - t 2 ) 6(k' - k) . (3.3.9) 6(k) i s the two-dimensional D i r a c d e l t a f u n c t i o n , * denotes the complex conjugate of a q u a n t i t y , and < > denotes the mathematical e x p e c t a t i o n of a f u n c t i o n . The homogeneity of the turbulence i m p l i e s that the wave f i e l d i s a l s o s t o c h a s t i c a l l y homogeneous. The s p e c t r a l d e n s i t y f u n c t i o n of the wave amplitude F ^ ( k , t ) may t h e r e f o r e be def i n e d such that <fi(k,t)n*(k',t)> = 4TT 2 F (k,t) <S(k' - k) . (3.3.10) From equations (3.3.8) - (3.3.10), i t i s found that the wave s p e c t r a l d e n s i t y i s r e l a t e d to the t u r b u l e n t pressure covariance f u n c t i o n by the r e l a t i o n t This extension of the time domain to t < 0 f o r t u r b u l e n t q u a n t i t i e s may be j u s t i f i e d by a s s e r t i n g t h a t the water surface i s " r i g i d " f o r t < 0 . 1 ~*2 t t F (k,t) = 2 ( pw c ) f d h f d ? 2 " ? 2 ) [ c O S " ^ cos a ( 2 t - ? - C 2 ) ] . Transforming the i n t e g r a t i o n v a r i a b l e s to T = t,^. ~ and 1 s = ^(C-^ + ' a n d n o t i n g that fi(k, -T) = fi(k,x) , the expression for F becomes nn ~" 2 t F (k,t) = (p c) / dx (t - x) ft(k,x) cos ax nn - w o 1 2 2 - " l t + —(p c a) / dx ft(k,x)[sin ax - s i n a(2t - x)] Z w ' ~ C l e a r l y , f o r at >> 1 , the behaviour of F w i l l be dominated by nn the f i r s t i n t e g r a l , so that F (k,t) = (p c) 2 / f c dx(t - x) fi(k,x)cos ax . (3.3.11) nn ~ o ~ The form of the turbulent pressure covariance function fi(k,x) may be estimated by invoking Taylor's 'frozen f i e l d ' hypothesis that on the average, the pressure f l u c t u a t i o n s are advected by the mean wind at a speed Up . Thus, i f the mean wind p r e v a i l s i n the x ^ - d i r e c t i o n , then n(k,x) = F (k) cos k nU x , (3.3.12) pp ~ I P where F (k) i s the s p e c t r a l density function at any i n s t a n t i n time, pp -Equation (3.3.12) i s v a l i d only for wavenumbers at which k^U pT p(k) >> 1 where i s the i n t e g r a l time scale of the pressure f l u c t u a t i o n s . As i n d i c a t e d i n §3.1 however, Willmarth and Wooldridge (1962) have found that this condition i s w e l l s a t i s f i e d by pressure f l u c t u a t i o n s w i t h i n a turbulent boundary layer. Assuming that ft(k,t) i s ade-quately described by (3.3.12), equation (3.3.11) becomes F n n ( k . t ) = F (k)/(2pV)' {[1 - cos (a + k ^ t ] /(a + k l U p ) 2 + [1 - cos(a - k 1 U p ) t ] / ( a - k ^ ) 2 } . (3.3.13) Thus, the wave s p e c t r a l density reaches a maximum along the curves defined by a(k) + k 2U = 0 , (3.3.14) i . e . at the resonant wavenumbers k^ = k^ + where k1 ± = ± (g/U 2) [1 + J<V ki ) 2 + ° ( k 2 / k l ) 4 ] At the resonant wavenumbers, i t i s seen that the s p e c t r a l 2 density grows as t ; i n p a r t i c u l a r , 2 2 2 F (k.. ,k„,t) = t F (k)/(4p c ) + o s c i l l a t o r y terms, nn 1± 2 pp ~ w J • However, away from these wavenumbers ( i . e . for k^ = k^ + + 6, 2 6 > 0) , F (k,t) i s purely o s c i l l a t o r y . Therefore, the energy i n a f i n i t e wavenumber band i n the neighbourhood of k^ + must grow at a rate less than t . To determine t h i s rate of growth, we consider the t o t a l energy at a given k-wavenumber ! 2 ( k 2 , t ) = p w g / dk x F n (k,t) Since F (k) i s sharply peaked at k^ = k^ + , E 2 ( k 2 , t ) consists e s s e n t i a l l y of the energy i n these peaks. I t i s convenient to assume that F (k,,k_) = F (-k l 5k 0) , so that from (3.3.13) , E 0 ( k 0 , t ) pp 1 I pp 1 I 1 1 may be approximated by /co 9 dk. (1 - cos o)t)/a> , (3.3.15) c <- iv v v - . 1 where k, i s evaluated at k., = k. . and u = a - k1U_1 . We now —r 1 IT- 1 if transform the i n t e g r a t i o n v a r i a b l e i n (3.3.15) to a) , noting that 3Ur> _3w _ TT _ v -8k 1 C g i P 1 3k ' where c = 8a/3k i s the component of the wave group v e l o c i t y i n the x ^ - d i r e c t i o n . Neglecting the 'dispersive' nature of the turbulence i . e . assuming that Up 3k x < < we f i n d J 2(k 2,t) - ( k + / p w ) ( U p - c g i + ) _ 1 F p (k +) f dw (1 - cos u t ) / u 2 26. Therefore, the t o t a l wave energy at a given k^-wavenumber i s E 2 ( k 2 , t ) = ( k + / P w ) ^ p " c ^ r 1 *t F p p ( k + ) . (3.3.16) The wave amplitude s p e c t r a l density function corresponding to (3.3.16) is given by V k ' t } * i ( p w c ) ~ 2 ( u p - W 1 ^ F P P ( k ) [ 6 s ( k i - k i + } + 6 g ( k 1 - k±J] , (3.3.17) 1/2 -1 2 where (k) = (TT Ak) exp[-(k/Ak) ] , say, i s a 'smudged' Dirac d e l t a function and Ak i s the bandwidth of the wave spectrum. Equation (3.3.17) implies that the wave energy grows l i n e a r l y with time and that i t has a time scale of order T 7 = [(U_ - c,, ) A k ] ^ . W P g 1+ For waves t r a v e l l i n g with the mean wind ( k 2 =0) , i t i s seen from equation (3.3.14), that T = [(c - c )Ak] 1 ; i . e . the time scale w & for wave growth i s of the order of the time f o r which a wave group of bandwidth Ak i s expected to remain coherent. Although the time scale of the turbulence T p has been neglected e x p l i c i t l y i n t h i s a n a l y s i s , i t s e f f e c t upon the bandwidth of the wave spectrum may be included; i n p a r t i c u l a r , Ak should be a decreasing function of 4. GENERATION OF WAVES BY A I R TURBULENCE 4.1 Introduction Since turbulent pressure f l u c t u a t i o n s i n the a i r are able to generate waves on an i n i t i a l l y smooth a i r - s e a i n t e r f a c e (see §3), i \ i t would seem to be of at l e a s t academic i n t e r e s t to i n v e s t i g a t e the e f f e c t of turbulent v e l o c i t y f l u c t u a t i o n s on surface waves. To t h i s end, we assume that while a non-uniform mean wind i s required to main-t a i n the a i r turbulence, the former does not i n t e r a c t with the f l u c t -uations produced by the surface wave. Hence, there i s a primary flow i n the a i r c o n s i s t i n g of a uniform mean wind U q and a turbulence f i e l d , which i s characterised by a v e l o c i t y scale u,. . Because there are two v e l o c i t y scales associated with the problem, a two scale perturbation analysis i s -carried out with a ' f a s t ' time scale propor-t i o n a l to U and a 'slow' time scale proportional to u. (e.g. o see Cole 1968, §3). The r a t i o U.--/UD serves as the ordering parameter for the a n a l y s i s . We have in d i c a t e d i n §2.2 that the i n t e r f a c e appears as a s o l i d b a r r i e r to the turbulence and that e x p l i c i t viscous e f f e c t s are confined to a t h i n layer (of order v /u, thick) at the i n t e r f a c e . a Therefore, the v e r t i c a l component of the turbulent v e l o c i t y i s taken to be zero at the surface and the a i r i s treated as i n v i s c i d with respect to the secondary f l u c t u a t i o n s set up by the wave, which has a length scale much greater than v /u, . There i s no primary motion a x i n the water, which i s also taken to be i n v i s c i d . From the perturb-a t i o n a n a l y s i s , i t i s found that turbulent v e l o c i t y f l u c t u a t i o n s con-t r i b u t e i n s i g n i f i c a n t l y to the wave generation process. 28. 4.2 Equations of Motion We consider the f l u i d motion within the two s e m i - i n f i n i t e domains and R^ , where R^ : -°° < x^, x 2 < » , n < z < '» , t > 0 , R : -°° < x , x„ < » , -a> < z < n , t > 0 . Here, (x^,x 2,z) are c a r t e s i a n coordinates such that z increases v e r t i c a l l y upwards; z = n(x^,x 2,t) i s the a i r - s e a i n t e r f a c e ; t i s time. For an i n v i s c i d , incompressible f l u i d the equations of motion are Y t + v.Vv + V II/p = -g 3 Y.v = 0 (4.2.1) where v = ( v ^ v ^ v ^ ) i s the E u l e r i a n v e l o c i t y vector, n i s the pressure, p i s the f l u i d density, g i s the g r a v i t a t i o n a l acceler-a t i o n , and 3 = (0,0,1) i s a v e r t i c a l u nit vector. Equations (4.2.1) are s a t i s f i e d w i t h i n the domains R^ ( a i r ) and R 2 (water). On the common boundary of R^ and R 2 , v i z . z = n , we apply the conditions that the i n t e r f a c e must remain coherent and that the normal stress must be continuous across the i n t e r f a c e , i . e . v„ = dn/dt and [Jl]Z n + = 0 at z = n , (4.2.2) J z=n~ where n+ = l i m n + 6 , 6 > 0 . We also require that the disturbance 6+o dies out away from the i n t e r f a c e , i . e . II -»- p - pzg + P Q as z -> + °° , (4.2.3) where p^ represents the f l u c t u a t i n g pressure associated with the primary turbulence f i e l d and P^ i s the mean pressure at z = 0 . I n i t i a l l y , the i n t e r f a c e i s f l a t and at r e s t , i . e . n = 0 = n t at t = 0 . (4.2.4) Although there i s no c h a r a c t e r i s t i c length s c a l e , there are two v e l o c i t y scales c h a r a c t e r i s i n g the problem, v i z . U o and u.,{ and t h e i r r a t i o U*/U 0 =6 i s generally small. The v e l o c i t y vector and the pressure may be normalised such that v = U A [ 1 + 6 u' + u'] 0 - ~o (4.2.5) and n = P + pU 2[-zg/U 2 + 6 2p' + p'] , o o b o 'o r where 1 = (1,0,0) and (u^,p^) represent the primary turbulence f i e l d . Hence, the disturbance caused by the wave f i e l d i s given by (y',p') . The appearance of two v e l o c i t y scales suggests that two independent time scales are required i n the problem. Thus, we i n t r o -duce two new independent time v a r i a b l e s s = U t (4.2.6) and s* = 6s , 30. which have the dimensions of length. Now the temporal d i f f e r e n t i a l operator transforms as 3 -> U (3 + 6 3 , ) . (4.2.7) t os s* We seek an asymptotic s o l u t i o n to the system (4.2.1) - (4.2.4) v a l i d as 6 approaches zero, and so we expand u',p' and n i n asymptotic power se r i e s i n 6 ; i . e . , n (n) , u = £ 0 v ( X , Z , S , S " ) , n=l °° r \ P ' = I <5n p W ( x , z , s , s * ) , (4.2.8) n=l 00 and T) = 1 &n n^ n^(x,s,S") , n=l where x = ( x ^ j X ^ ) • Substituting (4.2.5) - (4.2.8) into (4.2.1) -(4.2.4), assuming that (u^,p^) s a t i s f y the equations of motion independently of (u',p f) , and equating the c o e f f i c i e n t s of l i k e powers of 6 to zero, we obtain a sequence of systems of equations for ( u ^ n \ p ^ ) . The f i r s t order ( i n 6 ) system of equations i s v ( 1> + V p C l ) = \ -v ( 1> i n P ; ~ x l 1 { 0 i n R 2 ; V . v ( 1 ) = 0 i n ,R2 ; ( D I I p 0 as | z | 31. n ( 1 ) at z x l 0+ ; (4.2.9) 0 at z = 0-r (1) (D/T^, r CD (D/TT2! P J P - gn / u 0 ] z = 0 + = pw[p - gn /u 0] z = 0_ n ( 1 ) = 0 = rig1"* at s = 0 = s* ; where p and p are the a i r and water d e n s i t i e s , r e s p e c t i v e l y , a w In boundary condition (4.2.9d), i t has been assumed that the v e r t i c a l turbulent v e l o c i t y vanishes at the a i r - s e a i n t e r f a c e , i . e . u' (x,0,s) = 0 . The second order system of equations i s v(2) + V p ( 2 ) + v ( l ) + v ( l ) . v v ( 1 ) = - v ( 2 ) - u'.Vv ( 1 ) - v ( 1 ) . v u ' i n R ; ~o ~~ ~ ~~o 1 0 i n R 2 ; V.v (2) = 0 i n \ > ^ 2 ; (2) ->- 0 as I z I -> °° ; .(2) _ n ( 2 ) _ n a ) + n a > v a ) . vd). 7 na) J 3,z n - n ( 1 ) u ' x 1 o 1 3, z (4.2.10) (1) -u'.Vn o ~ at z = 0+ ; 0 at z = 0- ; P a [ p P > ( 2 ) " Bn(2>/u2 + n P z ]z=0- > n ( 2 ) = Q = (2) + (1) a t s = Q „ s i S S' c 32. Because the systems (4.2.9) and (4.2.10) are l i n e a r , they are r e a d i l y solved by transform methods. Thus, we define f o u r i e r transforms of the dependent v a r i a b l e s such that CO T - ( N ) - f J ik.x (n) , „ (n) ,, ~ -,, N F v s = J dx e - ~ v '(x.z.s.s*) = v /(k,z,s,s- {) , (4.2.11) —oo ' where k = (k^jk^) . S i m i l a r l y , we l e t r (n) ~(n) E , j- , *• , r (n) - (n) ,. 1 0 . Fp = p , F i r = i r ,• rjv = and Fn = n . (4.2.12) Because the turbulence f i e l d (u ,p ) i s s t o c h a s t i c and because the wave f i e l d i s driven by the turbulence, the transform v a r i a b l e s e x i s t only as generalised functions. 4.3 F i r s t Order Equations Taking the divergence of (4.2.9a) and using (4.2.9b), i t i s seen than the f i r s t order pressure s a t i s f i e s Laplace's equation, 2 (1) V p v = 0 i n \ , ^ 2 • Thus, from (4.2.12), - k 2 p ( 1 ) ( k , z , s , s * ) = 0 , where k = |k| . Boundary condition (4.2.9c) implies that 33. p ( 1 ) = (k,s,s*) exp{-sgn(z)kz] . (4.3.1) Using equations (4.2.9a) and (4.3.1) and boundary c o n d i t i o n (4.2.9d), i t i s found that (3 g - i k 1 ) v ^ 1 ) = k p ( 1 ) = (3 - i k ^ 2 f , ( 1 ) at z = 0+ , (4.3.2) and that ~(D , -(1) -(1) n ( l , ,v v„ = -kp = n at z = 0- . (4.3.3) 3,s ss Putting (4.3.2) and (4.3.3) i n t o the transform of boundary condition (4.2.9c), the surface displacement i s found to be given by {p [ 3 2 + kg/U 2] + p [(3 - i k j 2 - k g / U 2 ] } n ( 1 > = 0 W S O cl o JL O Therefore, nv i o ' s f , ( 1 ) = N (k,s*)e m m where m = 1,2 and the E i n s t e i n summation convention holds. The nat u r a l 'frequencies' a' are the roots of n m a'2 - kg/U2 + (p /p )[a' 2 - 2k.a' + kg/U2 + k 2 ] = 0 . (4.3.4) O cl W X o x _3 Since p /p ^ 1 0 , these roots are approximately a w 34. 1,2 = + ( k g )1 / 2 / U (4.3.5) Equation (4.3.5) breaks doxvTi only at very high wavenumbers when Kelvin-Helmholtz i n s t a b i l i t y occurs (e.g. see Lamb 1932, §232). The i n i t i a l conditions (4.2.9f) imply that N (k,0) = 0 , m = 1,2 m ~ . (4.3.6) Thus, without loss of ge n e r a l i t y , we assume that = 0 , and so we l e t ^(1) W 1 i a f s n = N(k,s*)e (4.3.7) where N(k,0) = 0 and a' i s the p o s i t i v e root of (4.3.5). From equations (4.2.9) and (4.3.1) - (4.3.7), the trans-forms of the f i r s t order pressure and v e l o c i t y components are found to be (1) ' -k 1 ( a ' - k 1 ) 2 N ( k , s * ) e i a ' S k z i n R^ J ,-1 ,2 J N ia fs+kz . _ k ar N(k,s*)e i n R 0 ; f-(k /k)(a'-k.) N ( k , s * ) e i a ' s k z i n R. ; m 1 ' 1 v = 4, m ,(k /k)a' N ( k , s * ) e i a ' S + k z i n R 0 m 2 (4.3.8) v ( 1 ) 3 1 ( 0 ' - ^ ) N(k,s- :)e i n R^ ; . , „,, , s la 's+kz . _ i a ' N(k,s*)e i n R 2 ; 35. where m = 1,2 . The behaviour of N(k,s*) must be determined from the second order ( i n 6) equations. 4.4 Second Order Equations P h i l l i p s (1960) has demonstrated that resonant i n t e r a c t i o n s cannot occur between two i n f i n i t e s i m a l surface waves which s a t i s f y the d i s p e r s i o n r e l a t i o n (4.3.5). This means that any i n t e r a c t i o n between two such waves produces only forced waves whose amplitudes do not increase with time. Since we are presently i n t e r e s t e d i n growing waves, the terms i n system (4.2.10) i n v o l v i n g two f i r s t order wave qu a n t i t i e s are therefore non-resonant terms (N.R.T.) and they may be neglected i n the a n a l y s i s . On the other hand, the terms in v o l v i n g a f i r s t order wave quantity and a turbulent quantity act as sources f o r the second order wave terms and so they cannot be neglected. Taking the divergence of equation (4.2.10a) and using (4.2.10b), the second order pressure i s given by v 2 P ( 2>== - 2 u ' v ( 1 ) - 2 u ' v ( 1 ) - 2 u ' v ^ o n,x 0„ m,z o 3,x m,x m 3,x m, z m ii m -2u' v ^ + N.R.T. i n R, , 0 o 3, z 1 3,z N.R.T. i n R. , where n = 1,2 and m = 1,2 . Thus, applying the f o u r i e r convolution 36. theorem and using (4.3.8), the transform of the pressure satisfies the equation p ( 2> - k 2 p ( 2 ) = zz -TT 1 J dK[a'(K) - K ]N(K,s*) ia '(K)s -*D ,(K,k-K, z,s) e + N.R.T. in R N.R.T. in R 2 ' 1 ' (4.4.1) where D, = [(K /K)K (k -K )u' (k-K,z,s) - iK (k -K )u r (k-K,z,s) + 1 m n n n o ~ ~ n n n o~ ~ ~ m 3 K u' (k-K,z,s) - iKu' (k-K,z,s)]e mo ~ ~ o~ - ~ m,z 3,z -Kz Using boundary condition (4.2.10c), equation (4.4.1) is readily solved and the transform of the pressure gradient at the air-sea interface is found to be (2) -kp ( 2 ) + TT 1 j a d K . e ± a , ( i V S [ a ' (K)-K ]N(K,s*) x / dze D1(K,k-K,z,s) + N.R.T. at z = 0+. (4.4.2) (2) kp + N.R.T. at z = 0-Transforming the momentum equation (4.2.10a) and using (4.3.8), the transform of the ver t i c a l velocity at the interface i s given by OO Og - i k 1 ) v ^ 2 ) = - P ^ 2 ) - v ^ ; V - ( 2 T T ) " 1 J dK[a'(K) - K^NCK.s*) Y — O O i n ' C K " ) S x [K u' (k-K,0,s) + i u' (k-K,0,s)]e ~ + N.R.T. at z = 0+ , m o ~ - o 0 ~ ~ m 3, z (4.4.3) and by 37. v i 2 ) = - p ^ 2 ) - viX\ + N.R.T. at z = 0- . 3, s z 3, s * ( 4 . 4 . 4 ) The kinematic boundary condition (4.2.10d) r e l a t e s the v e r t i c a l v e l o c i t y to the surface displacement such that :(2) 'O - i k 1 ) n ( 2 ) + n ( P - (2TT) X / dK N(K,s*)[iK u' (k-K,0,s) s 1 S " m o ~ ~ -o> m \ + u' ( k - K , 0 , s ) ] e i a ' ( ~ ) S + N.R.T. at z = 0+ . (4.4.5) o„ ~ ~ 3,z + + N.R.T. at z = 0- . S S " -1 ~(2) ~(2) Elim i n a t i n g v.: and p from (4.4.2) - (4.4.5), the transform of the pressure at the i n t e r f a c e i s found to be given by kp<2> f(3 8 - i k l ) 2 n ( 2 ) + O s - i k , ) ^ + ^  + ^  / dK : ( 1 ) ^ ~(1) ^ -1 r ^  io'(K)s x [a'(K) - K 1]N(K,s - ) [ J dze k ZD 1(K,k-K,z,s) o CO + K u ' (k-K,0,s)] -(2 T T) _ 1 / dKe i C T' (~ ) sN(K,s^) m o x O - i k . ) [ i K u' (k-K,0,s) + u' (k-K,0,s)] s l m o ~ ~ o. m + N.R.T. at z = 0+ . 3,z ( 4 . 4 . 6 ) (2) .(1) n ; - v: + N . R . T . at z = 0- . S S S S " 3 , S Putting (4.4.6) in t o the transform of boundary condition (4.2.10e), we f i n d 38. -{p [3 + kg/U ] + p tO.. " i k . ) - kg/U ] } TT + N.R.T. W S O 3. S X O oo = P + v^ 1 } . , J + p [(3 - i k . ) n ( P + v j ^ . + r r " 1 / dKe Kw s s " 3 , S " z=o- K a s 1 s-' 3 , s * -1 i a ' (K) s {a'(K) - K }N(K,s*) {/ dze k zD 1(K,k-K,z,s) + (k-K,0,s)} o m oo - ( 2 ^ ) _ 1 / d K e i a ' ( ~ ) s N(K,s*)(3 - i l ^ ) {IK ^  (k-K,0,s) -oo m + u' (k-K,0,s) +kp']' , • (4.4.7) o„ ~ ~ r o z=o+ 3,z To develop the a n a l y s i s f u r t h e r , i t i s necessary to make some assumption about the temporal dependence of the t u r b u l e n t q u a n t i t i e s u^ and p^ . Thus, we adopt Ta y l o r ' s 'frozen f i e l d ' hypothesis which s t a t e s that the turbulence i s s t a t i o n a r y i n a coordinate frame moving w i t h the l o c a l mean v e l o c i t y . Although the hypothesis cannot be tru e e x a c t l y , there i s consi d e r a b l e t h e o r e t i c a l (e.g. L i n 1953) and experimental (e.g. Favre et a l 1957) evidence that i t i s a very good approximation provided that the mean s t r a i n r a t e i s s m a l l . In keeping w i t h the i n i t i a l approximation of the mean v e l o c i t y , we assume that the turbulence i s advected w i t h the constant v e l o c i t y U : i . e . we assume that o ik^s"\ u^(k,z,s) = W(k,z)e ik.. s and p'(k,0,s) = Y(k)e o ~ (4.4.8) P u t t i n g (4.3.8) and (4.4.8) i n t o (4.4.7), the transform of the second "(2) order s u r f a c e displacement n must s a t i s f y the equation 39. - { p i T [ 3 ? + kg/U2] + p J O - i k . ) 2 - k g / U 2 ] } n ( 2 ) + N.R.T. W S O cl s X o • t • t -I i k 1 s = i2p a'N . (k,s*)e 1 C T S + p [i2(a'-k.)N ,(k,s*)e l a S + Tr e *w s* ~ K a 1 s* ~ i{o'(K)-K }s ik s , q ^ x / dK D (k,K)N(K,s*)e ~ + k Y(k)e ] ' W-1*^) — OO where D„(k,K) = [a' (K) - K.].[k-K+k K /K] / dze z ~ ~ ~ l m m ' -(k+K)z ox [K W (k-K,z)-iKW 0(k-K,z)] - -| K.k W (k-K,0) . (4.4.10) n n ~ ~ 3 ~ ~ 2 1 m m ~ ~ The f i r s t term i n the expression (4.4.10) for T>2 i s due to the i n t e r a c t i o n of the f i r s t order wave with the turbulent v e l o c i t y f l u c t u a t i o n s i n the a i r . The second term i s caused by the turbulent f l u c t u a t i o n s advecting the i n t e r f a c e . Because the wavenumber k^ spans the r e a l a x i s , i t i s cl e a r that the inhomogeneous f o r c i n g terms on the r i g h t hand side of. (4.4.9) -(2) can produce secular contributions to n , which has a nat u r a l frequency of a' ( c . f . equation (4.3.4)). Such secular terms a r i s e when 0 ' (k) = k^ , i . e . when the wave phase v e l o c i t y equals the turbulence advection v e l o c i t y . Thus, we define w' = (o'-k 1)/6 , (4.4.11) and assume that N(k,s*) i s such that the r i g h t hand side of (4.4.9) i s zero; i . e . we assume that CO i 2 [ o ' + Y(o'-k 1)jN A ( k , s * ) + yn'1 j dK D (k,K)N(K,s*) ~ —OO i{w'(K)-<o'(k)}s* -iw's* _ , 1 0 . xe ~ ~ + yk Y(k)e = 0 , (4.4.12) 40. where y = o 10 • From (4.3.7), N must s a t i s f y the i n i t i a l ' a *w J condition N(k,0) = 0 . (4.4.13) Therefore, the f i r s t order surface displacement i s given by CO n ( 1 ) ( x , s , s * ) = U T T ) " 1 / dk N ( k , s * ) e 1 { c r f ( k ) s " k ' ? } , (4.4.14) — CO where N(k,s*) i s determined from the system (4.4.12) - (4.4.13) and where a' i s the p o s i t i v e root of (4.3.4). 4.5 F i r s t Order Wave Solution The complete f i r s t order wave solution i s given by equations (4.4.12) - (4.4.14). Since the air-sea interface i s i n i t i a l l y at rest, i t i s clear that for small times the i n t e g r a l term i n (4.4.12) w i l l ! be small compared with, the other terms, i . e . i u ' s * i2Io' + yCa'-tyl N^ Ot.s*) - Y(k)e Thus, using condition C4.4.13), N(k,s*). = | Y ( k / a i ' ) [ a ' + Y C o - ' - k ^ ] "'(1 - e x w *") Y(k) (4.5.1) for small s* . We define the wave and turbulent pressure spectral 41. density functions by <n(1) (k,s,s*)n ( 1 )*(K,s,s*)> = 4TT 2 F^ (k,s*)6(K-k) , (4.5.2) and <Y(k) Y*(K)> = 4 T r 2 F p , p , ( k ) S ( K - k ) , (4.5.3) r e s p e c t i v e l y , where 6(K) i s the two-dimensional Dirac d e l t a function; <f> denotes the ensemble average of f ; f * denotes the complex conjugate of the function f . Thus, from (4.5.1) - (4.5.3) and (4.3.7), i t i s found that, for small s* , F (k,s*) = | ( Y / a J ' ) 2 [ c ' + Y ( c ' - k / k ] " 2 ( l - cos w's*)F , , (k) , (4.5.4) T)T) ~ Z -L P p where c r = a'/k i s the normalised wave phase speed. Equation (4.5.4) describes the i n i t i a l wave growth due to the P h i l l i p s (1957) resonance mechanism (see §3). Comparing t h i s equation with (3.3.13), 2 i t i s seen that c i n the denominator of the l a t t e r i s replaced by 2 [ c ' + y(c'-k^/k)] i n the former. This occurs because the present an a l y s i s accounts for the wave-induced pressure f l u c t u a t i o n s i n the a i r while that of §3 does not. Thus, the wave amplitude grows l i n e a r l y with time due to the energy input from the turbulent pressure f l u c t u a t i o n s u n t i l i t i s f i n a l l y large enough to i n t e r a c t with the turbulent v e l o c i t y f l u c t u a t i o n s . At t h i s stage, the i n t e g r a l term i n (4.4.12) becomes important and, being a homogeneous term i n the equation, i t describes a feedback type 42. of growth mechanism,. A p a r t i c u l a r s o l u t i o n of (4.4.12) i s N = N (k)e X U J , (4.4.5) P ~ where N p s a t i s f i e s the inhomogeneous. i n t e g r a l equation of the second k i n d 2w'[a'+y(a'-l< 1)]N(k) + Y ^ " 1 / dK D (k,K)N (K) + yk Y(k) =• 0 . ^ — CO ^ (4.5.6) The homogeneous p a r t of (4.4.12) has e i g e n f u n c t i o n s o l u t i o n s of the form N = N c(k; ^ e 1 * * ' - " ' * 8 * , (4.5.7) where the eigenvalue 9' i s independent of k and where s a t i s f i e s the homogeneous equation of the second k i n d 2 (9'-a> 0 [a'+y(a'-k ) ]N (k;<|>') = yv 1 J dK (k,K)N c(K; 9 ') (4.5.8) Thus the complete s o l u t i o n of (4.4.12) - (4.4.13) may be w r i t t e n f o r m a l l y as N(k,s*) = [N (k) + / d(J>' B (9')N ( k ; 9 ' ) e 1 < f > ' S * ] e " l w r S > < , (4.5.9) p ~ Q c 43. where 9' spans some space Q and where / d* ' B ( 9 ' ) N ( k ; c f , ' ) = -N (k) . (4.5.10) n C ~ P ~ A s o l u t i o n of (4.5.6), v a l i d as y approaches zero, may be constructed by expanding N p i n an asymptotic power s e r i e s i n y , s u b s t i t u t i n g into the equation and equating the c o e f f i c i e n t s of l i k e powers of y to zero; i . e . we l e t N = I Y n N ( n ) , (4.5.11) P n-1 P where N ( 1 ) = -Y(k)/ (2c 'to') , P N ( n> P CO -(.1-K/o ' ) N ( n _ 1 ) ( k ) - (2Tra'w') _ 1 / dK D 0 ( k , K ) N ( n - 1 ) (K) , 1 p ~ LM - 2 ~ - P n > 1 , where the i n t e g r a l s are assumed to e x i s t (as singular i n t e g r a l s , at l e a s t ) . S i m i l a r l y , an asymptotic power s e r i e s s o l u t i o n may be con-structed for the eigenfunctions of (4.5.8). However, i t i s c l e a r that the r e s u l t i n g zeroth order eigenf unction i s simply S(9'-u)'(k)) ; i . e . a Dirac d e l t a function which i s non-zero only at wavenumbers where to'(k) = 9 ' , the eigenvalue. This implies that the eigenvalue 9 ' spans the r e a l axis and i t i s therefore pure r e a l . Thus, the subsequent power s e r i e s s o l u t i o n w i l l y i e l d an algebraic growth (at most) of the wave amplitude. On the other hand, a feedback type of mechanism described by (4.4.12) suggests that an exponential growth 44. may be possible; i . e . the eigenvalue <j>' of (4.5.9) may have an imaginary part. In f a c t , from equation (4.5.9), the imaginary part of <j> ' i s given by CO 2(<j>'-4)'*)[o'+Y(a'-k1)]N (k)N*(k) = yn'1 / dK x [D 0(k,K)N (K)N*(k) - D*(k,K)N*(K)N (k)] z ~ ~ c ~ c ~ I ~ ~ c ~ c ^ Neglecting the c r o s s - c o r r e l a t i o n between <j>' and N^ , an estimate of the mean imaginary part of <j>' may be found from <*•'-<(.'*> [ a ' + Y ( r f ' k 1 ) ] F n n ( k ) = Y / dK[F d n n(k,K) - F * n n ( k , K ) J , — CO (4.5.12) 2 where <N (k)N*U)> = 4TT F (k)5(£-k) , c ~ c - nn ~ ~ ~ <D„ (k,K)N (K)N*U)> = 8T T 3 F j (k,K)6(£-k) 2 ~ ~ c ~ c ~ dnn ~ ~ ~ ~ From (4.5.10) and (4.5.11), i t i s seen that the. major c o n t r i b u t i o n to the complete s o l u t i o n of (4.4.12) comes from those eigenfunctions that vary somewhat l i k e N and hence l i k e Y . Thus, an estimate of the P mean growth rate of the wave amplitude ((i/2) <<+> '*-<}> '> ) may be found by repl a c i n g N c by Y i n (4.5.12); i . e . «j>'-<!> , * > k [ a ' + Y ( a ' - k 1 ) ] F p , p , ( k ) = y j d K [ F d p , ? , (k,K) - F * p f p , ( k , K ) ] (4.5.13) Neglecting the quad-spectra contributions to F ^ p , , (such contributions 45. are i n v a r i a b l y small f o r turbulent q u a n t i t i e s ) , we f i n d from (4.4.10) that F , , ,(k,K) - F * , ,(k,K) = -12Kla'(K)-Kj[k-K + k K /K] dp'p ' ~ ~ dp'p' ~'~ L ~ 1 J m m 00 x / d z e ~ C k + K ) z F , f(k,K,z) , (4.5.14) where <W (k-K, z)Y (K)Y* (_%)> = 8TT 3F , , (k,K, z) S (£-k) . J ~ ~ ~ ~ u^P P ~ ~ ~ ~ Thus, from (4.5.13) and (4.5.14) , 1 -1 2 i<cj>'*-c|>'>k = -Ylcr'+YCa'-kp] / dK Kla'CK)-^] ~ —CO x Ik-K + k K /KJ [ d z e " ( k + K ) z F , ,(k,K,z ) / F , , (k) . m m J u^p'p ' ~ ~ P P ~ The main c o n t r i b u t i o n to the wave growth therefore appears to a r i s e from a turbulent v e r t i c a l energy transport caused by the pressure f l u c t u a t i o n s , V7hich i n i t i a l l y induce the wave f i e l d , i n t e r a c t i n g with the v e r t i c a l v e l o c i t y f l u c t u a t i o n s i n the a i r above the i n t e r f a c e . While i t i s not cl e a r that even the sign of \r i<9 '> i s such as to cause wave growth ( i . e . negative), the maximum magnitude of the d 2 2 f r a c t i o n a l rate of increase i n wave energy ("j^ l^ l /1HI ) 1 S seen to be yku^ ,. from equations (4.2.6), (4.3.7), (4.5.9) and (4.5.13). This rate of growth i s much smaller than the observed rate of approx-imately Yk(U Q-c) , where c i s the wave phase speed (e.g. Snyder and Cox 1966). That the a i r turbulence alone i s unable to generate waves at the observed rate i s explained by Stewart (1967), who showed •that the wave energy i s " t y p i c a l l y comparable with, or more than the 46. energy of turbulence i n the e n t i r e troposphere". He suggests that the a i r turbulence i s more l i k e l y to act as a c a t a l y s t i n the tr a n s f e r of the mean wind energy into the waves. 47. 5. WAVE GROWTH CAUSED BY MEAN WIND 5.1 Introduction That small amplitude f l u c t u a t i o n s of a f l u i d can extract energy from a mean flow has long been recognised (e.g. Rayleigh 1880). Indeed, t h i s concept i s the basis of the theory of f l u i d i n s t a b i l i t y , which attempts to pre d i c t when s i g n i f i c a n t secondary motions can e x i s t through t h e i r i n t e r a c t i o n with a laminar mean (or primary) flow (e.g. see L i n 1955). A ' s i g n i f i c a n t ' motion i s one which does not decay i n time; thus, the boundary of s i g n i f i c a n t motions consists of n e u t r a l f l u c t u a t i o n s , i . e . f l u c t u a t i o n s whose amplitudes remain constant i n time. I f the laminar mean motion i s steady and two-dimensional, and i f the amplitude of the secondary motion i s so small that the n o n - l i n e a r i t i e s of the equations of motion may be neglected, then Squire (1933) has shown that the secondary motion may be treated as two-dimensional. In p a r t i c u l a r , only the component of the mean flow i n the d i r e c t i o n of propagation of the secondary f l u c t u a t i o n a f f e c t s the secondary motion. Thus, i t i s u s e f u l to assume that the secondary motion i s p e r i o d i c and of the form i|>(z) exp [ i k ( x - c t ) J , where k and c are the wavenumber and phase speed of the motion, (x,z) are ca r t e s i a n coordinates, and t i s time. The f l u i d i s unbounded i n the x - d i r e c t i o n and the mean v e l o c i t y (U^,0) i s a function of z alone. The amplitude function Tp i s determined from the l i n e a r i s e d equations of motion and associated boundary conditions. For a ne u t r a l f l u c t u a t i o n , both k and c are r e a l q u a n t i t i e s , so that there may e x i s t a c r i t i c a l height z = z^ , where the mean v e l o c i t y equals the phase v e l o c i t y of the secondary o s c i l l a t i o n , 48. i . e . where ^ ( z ) = c . Inspection of the l i n e a r i s e d v o r t i c i t y equation (Orr-Sommerfeld equation) shows that at t h i s height the advection of v o r t i c i t y i n the z - d i r e c t i o n , which i s p r o p o r t i o n a l to 2 2 d U^/dz , i s balanced by viscous d i f f u s i o n . Consequently, there i s a shear s t r e s s produced i n the neighbourhood of the c r i t i c a l height. As the v i s c o s i t y of the f l u i d v decreases, v e l o c i t y gradients i n the z - d i r e c t i o n must correspondingly increase. In the l i m i t v •> 0 , the surface z = z^ degenerates to a vortex sheet and there i s a d i s c o n t i n u i t y i n the shear s t r e s s at t h i s height. Miles (1957) has applied the concepts of c l a s s i c a l laminar i n s t a b i l i t y theory to the problem of surface wave growth. A wave of amplitude A , present on the a i r - s e a i n t e r f a c e , produces a f l u c t u a t i o n i n the a i r (and i n the water) which has a magnitude of the order of the wave slope, kA , compared with, the magnitude of the mean motion. Since kA i s i n v a r i a b l y small, t h i s suggests the v a l i d i t y of a l i n e a r theory. The use of a l i n e a r theory allows an a r b i t r a r y f u nction (e.g. the surface displacement) to be decomposed into i t s f o u r i e r components and the analysis to be c a r r i e d out with respect to each component independently. Although the a i r turbulence maintains the mean shear flow, Miles assumes that i t does not i n t e r a c t with the perturbation. Away from the c r i t i c a l height, the Reynolds number of the perturbation (|ll^-c|/kv) i s very large, and hence viscous e f f e c t s are neglected i n both the a i r and the water. (In subsequent papers, Miles (1959) and Benjamin (1959) have shown that v i s c o s i t y does not s i g n i f i c a n t l y a f f e c t the r e s u l t s of the a n a l y s i s . ) Because the f r a c t i o n a l rate of growth of a wave i s very small compared with the 49. wave frequency, the wave motion i s treated as a neutral o s c i l l a t i o n . Thus, Miles considers the l i n e a r i s e d i n v i s c i d v o r t i c i t y equation (Rayleigh equation) i n the s e m i - i n f i n i t e domain 0 < z < OT , where z i s the v e r t i c a l distance above the a i r - s e a i n t e r f a c e , and he finds that the shear stress increases discontinuously from zero for z > z J c to be a constant value x for 0 < z < z . But the shear stress w c at z = 0 i s equal to the mean f l u x of momentum to the wave ( P h i l l i p s 1966, §4.3), and for an i r r o t a t i o n a l wave, the r a t i o of the wave energy density to the wave momentum density i s equal to the wave phase speed ( P h i l l i p s 1966, §3.2). Hence, the rate of increase i n wave energy per unit area i s cx . We now formally derive Miles' expression w for the wave-induced shear s t r e s s , and we discuss the streamline con-f i g u r a t i o n i n the neighbourhood of the c r i t i c a l height. 5.2 Mathematical Formulation of Problem It has been shown i n §2.2 that the c r i t i c a l height z^ i s often l e s s than the wave amplitude A . Using c a r t e s i a n coordinates and an analysis correct only to 0(kA) , the boundary conditions are applied at the mean water surface z = 0 , rather than at the disturbed surface. Therefore, d e t a i l s such as the streamlines i n the wave troughs are not described w e l l . Benjamin (1959) formulates the problem i n terms of c u r v i l i n e a r coordinates such that the disturbed surface i s approximately a coordinate l i n e and, f a r above the surface, these coordinates reduce to the usual c a r t e s i a n coordinates. Since Benjamin's analysis i s also correct to 0(kA) , the change i n coordinate 50. system does not a f f e c t the o v e r a l l features of the s o l u t i o n . In the present a n a l y s i s , the more straightforward cartesian coordinate system i s therefore used. However, i n the appendix, the problem i s defined i n Benjamin's coordinates and some of the differences between the two approaches are discussed. We consider two-dimensional f l u i d motion wi t h i n two semi-i n f i n i t e domains and R^ , where R^ : -co < x, t < °° and n < z < °° , R^ : -°° < x,t < oo and -°° < z < n • Here, (x,z) are c a r t e s i a n coordinates such that x i s the h o r i -zontal coordinate and z increases v e r t i c a l l y upward; t i s time; z = 0 i s the undisturbed i n t e r f a c e between the a i r (R^) and the water (R ) ; the displaced water surface i s given by z = n • In the domain R^ , there i s no primary mean motion. Thus, because the water i s assumed to be i n v i s c i d , the motion i s i r r o t a t i o n a l . A v e l o c i t y p o t e n t i a l <J> therefore e x i s t s such that u = V9 , (5.2.1) where u = (u,w) i s the E u l e r i a n v e l o c i t y vector. The incompress-i b i l i t y of the water implies that V2<j> = 0 (5.2.2) 51. The kinematic boundary conditions on the motion are that the disturbance dies out far below the surface and that the surface z = r\ remains coherent; i . e . (j) ->• 0 as z •> -oo and w = — L on z = n dt (5.2.3) We seek a wave-like s o l u t i o n of the system (5.2.1) - (5.2.3) of the form'* = A exp [ik(x-ct) ] + V (kA) nri , n=l = I (kA)n<| n=l (5.2.4) where k and A are r e a l , and the maximum wave slope kA << 1 . Putting (5.2.4) i n t o (5.2.1) - (5.2.3) and equating the c o e f f i c i e n t s of l i k e powers of kA to zero, the s o l u t i o n i s found to be n = A exp[ik(x-ct)J + O(kA) , = -iAc exp[ik(x-ct) + kz] + O(kA)' (5.2.5) The disturbance <f> produces a pressure f l u c t u a t i o n p , given by B e r n o u l l i ' s equation, v i z . t In a l l equations d e f i n i n g r e a l q u a n t i t i e s , the equality i s assumed to be between the r e a l parts only. 52. P / P W + 9 t + -|(y?) 2 = 0 , where p i s the density of the water. Thus, 2 2 p = p kAc exp[ik(x-ct) + kz] + O(lcA) w (5.2.6) In the a i r (domain R^), there i s a primary mean v e l o c i t y 2 (U^,0) where = U^(z) and e C . The wave disturbance (5.2.5a) produces a two-dimensional motion which i s described by a streamfunction \> , where the secondary v e l o c i t y components are u = ii and w = -\l> z x (5.2.7) Since the f l u i d i s i n v i s c i d , the momentum equations are u + (TJ\ 4- u ) u + w(tL + u) + p /p = 0 , t 1 x 1 z r x a w + (TT + u)w + ww + p /p = 0 , t 1 x z *z a (5.2.8) where p i s the pressure caused by the disturbance and p a i s the density of a i r . Expanding ijj and p i n asymptotic power se r i e s i n kA , s u b s t i t u t i n g into (5.2.7) and (5.2.8), and equating the c o e f f i c i e n t s of l i k e powers of kA to zero, the f i r s t order equations of motion are found to be 53. = U' + P l , x / p a = ° > (5.2.9) ( 3 t + V x ^ l . x " P l , z / p a = ° » J CO 00 where U| = dU /dz , ^ = I ( k A ) * 1 ^ and p = "> ( k A ) n p n . n=l n=l E l i m i n a t i n g p from (5.2.9), the v o r t i c i t y equation i s (5.2.10) Equation ( 5 . 2 . 1 0 ) i s a statement (to the f i r s t order i n kA) of 2 2 Helmholtz's theorem that the t o t a l v o r t i c i t y (U| + kA^7 ^ + 0(kA) ) . must be conserved along a p a r t i c l e path l i n e . The kinematic boundary conditions are that the disturbance dies out f a r above the water and that the water surface z = n remains coherent, i . e . ip ->- 0 as z -> and w = dt on z = n Thus, from (5.2.5a), the boundary conditions on \p are \p 1 ->• 0 as z -> 0 0 , - (5.2.11) and )p = i [ c - 11.(0)] exp [ik(x-ct) ] on z = 0 . > Equations (5.2.10) and (5.2.11) imply that 54. i|> = F(z) exp[ik(x-ct)] , (5,2.12) where F s a t i s f i e s the Rayleigh equation (U - c ) ( F " - k^F) - UjF(z) = 0 ; F -> 0 as z -> F = k 1 [ c - 1^(0)] on z - 0 (5.2.13) When c i s r e a l ( i . e . f o r a n e u t r a l disturbance), equation (5.2.13) has a regular singular point at the c r i t i c a l height z = z^ , where U^(z ) = c . From equations (5.2.10) and (5.2.12), i t i s seen that the r a t i o of the v e r t i c a l advection time scale to the h o r i z o n t a l t advection time scale i s (U^ - c)U^/kAFTj|f , which approaches zero as z approaches z , p r o v i ded that U"(z ) ± 0 and F(z ) ± 0 . c 1 c c Thus, f o r the t o t a l v o r t i c i t y of a p a r t i c l e near the c r i t i c a l height to be conserved, the r a t i o of the v e r t i c a l l y advected v o r t i c i t y to the h o r i z o n t a l l y advected v o r t i c i t y , i . e . U^/kA(F" - k F) , must also approach zero. Because the v e r t i c a l v e l o c i t y (^F) must remain f i n i t e , t h i s condition implies that F ' ^ u ) behaves l i k e (z-z^) ^ near z . Hence, the h o r i z o n t a l v e l o c i t y i s discontinuous across c J the c r i t i c a l height. C l e a r l y , a r e a l f l u i d could not sustain such behaviour. In f a c t , the presence of a f i n i t e v i s c o s i t y v adds the 4 term W to the r i g h t hand side of equation (5.2.10) and hence the t The 'horizontal' time scale denotes the h o r i z o n t a l time scale i n a coordinate system moving at the wave phase speed c . term - i ( v / k ) ( d /dz - k ) F to the r i g h t hand side of (5.2.13). Now the v e r t i c a l advection of v o r t i c i t y i s balanced by the viscous d i f f u s i o n term near the c r i t i c a l height, and so the point z = z^ i s a regular point. But the magnitude of v i s so small that, away from the c r i t i c a l height, the viscous term i s n e g l i g i b l e , so that the equation of motion reduces to (5.2.13). This suggests that viscous e f f e c t s are important only w i t h i n a layer of order kv/U^z^) (at most) thick around the c r i t i c a l height. Thus, provided that t h i s layer does not include the lower boundary, the f l u i d v i s c o s i t y simply 'smoothes out the s i n g u l a r i t i e s ' at the c r i t i c a l height and i t should not greatly a f f e c t the o v e r a l l r e s u l t s of an ' i n v i s c i d ' a n a l y s i s . L e t t i n g p =p X(z) exp[ik(x-ct)] , (5.2.14) X(z) i s given from (5.2.9) and (5.2.12) to be X(z) = UVF - (U - c)F'(z) . (5.2.15) In a d d i t i o n to the pressure f l u c t u a t i o n caused by the wave, there i s a h y d r o s t a t i c pressure equal to P ^ - pgz , where P q i s the mean pressure at the undisturbed a i r - s e a i n t e r f a c e and g i s the 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 . At the i n t e r f a c e , the normal stress ( i . e . the pressure, i n t h i s analysis) must be continuous and hence, from (5.2.6) and (5.2.14), p ( k c 2 - g) = p (kX(0) - g) , to the f i r s t order i n kA . That i s , (1 - y)g/k + YX(0) , (5.2.16) where y = p /p << 1 . Expanding c. and X i n asymptotic power a w ser i e s i n y , s u b s t i t u t i n g into (5.2.16) and equating the c o e f f i c i e n t s of l i k e powers of y to zero, we f i n d that and 2c, = g / k > -c + X (0)/c n n n (5.2.17) where c = c + Y y c and X = X + Y y X n ^ m n u . m m=l m=l C l e a r l y , Miles' (1957) assumption that c i s r e a l i s j u s t i f i e d because c i s r e a l , and hence the f i r s t order ( i n y) c o r r e c t i o n n to c i s found by solving (5.2.13) for the case of a n e u t r a l d i s t u r b -ance (denoted by the subscript n ). For a surface wave given by equation (5.2.5a), the wave energy per un i t area i s (e.g. see P h i l l i p s 1966, §3.2) E = 2 P w 8 l n | (5.2.18) Thus, the f r a c t i o n a l increase i n energy per radian i s n (5.2.19) where a = kc i s the frequency of a free wave. From (5.2.17) and n n (5.2.19), i t i s seen that S = (p /p ) Im(X ( 0 ) ) / c 2 . (5.2.20) a w n n Thus, S i s d i r e c t l y p r o p o r t i o n a l to that part of the pressure f l u c t u a t i o n which i s i n quadrature with the water surface displacement, i . e . i n phase with the slope of the water surface. This f a c t l e d Miles (1957) to suggest that the water surface acts l i k e a mechanical spring-mass-damper system with negative damping. Since (5.2.20) i s derived simply from the condition that the normal st r e s s i s con-tinuous at the air-water surface, i t i s v a l i d for any weak i n s t a b i l i t y type of wave growth, provided that X i s inte r p r e t e d as the complete normal s t r e s s . I t i s not v a l i d for strong i n s t a b i l i t y mechanisms such as the Kelvin-Helmholtz i n s t a b i l i t y which i s due to the in-phase pressure force overcoming the g r a v i t a t i o n a l r e s t o r i n g force. The expression (5.2.20) for S may be derived more d i r e c t l y by recognizing that the rate of increase of wave energy i s equal to the rate of working by the pressure on the water surface, i . e . dE/dt = - Re(pw-) at z = n , where f * denotes the complex conjugate of f . From (5.2.1) and (5.2.5), the v e r t i c a l v e l o c i t y at the water surface i s w = -iknc + 0(kA) n 58. and, using (5.2.14), the pressure at the surface i s p = P - p gn + P.kn X (0) + 0(kA)2 o ci ci n Therefore, dE/dt = -J p k 2|n| 2c Im(X (0)) . (5.2.21) 2 a 1 1 n n Hence, from (5.2.18) and (5.2.21), i p k 2 | n | 2 c Im(X (0)) g _ 2 a 1 1 n n = (p /p )Im(X ( 0 ) ) / c 2 , i . e . equation (5.2.20) 3. w n n Equation (5.2.13) may be integrated to y i e l d CO F'(z) = -kF - / dy e _ k ( y " z ) F ( y ) U ^ 7 ( U 1 - c ) . (5.2.22) Thus, from (5.2.13) and (5.2.15), r -It 7 Im(X (0)) = [U n(0) - c ]Im[/ dze K Z F (z)U"/(U -c )] , (5.2.23) n 1 n n l l n x^here F xs found from (5.2.13) wxth c = lim c + xe , e > 0 : n n E+O i . e . the neutral disturbance F i s taken as the l i m i t i n g case of a n growing disturbance (Lin 1955, §8.5). Thus", the path of i n t e g r a t i o n i n (5.2.23) must pass below the s i n g u l a r i t y at the c r i t i c a l height z 59. where U,(z ) = c 1 c n The parameter S i s also r e l a t e d to the mean shear stress (or momentum flux) produced at the surface by the disturbance i n the a i r . The shear stress i s defined as T (z) = \ p Re(uw*) , W Z Q. | p k 3 | n | 2 Im(F*F') , (5.2.24) Z 3. from (5.2.5a), (5.2.7) and (5.2.12). Therefore, from (5.2.13) and (5.2.22), the shear stress at the surface due to a ne u t r a l disturbance i s , to the second order i n kA , oo Tw ( 0 ) = y P a k 2 | n | 2 [ U 1 ( 0 ) - C n ] I m [ / dze" k zF n(z)U^'/(U 1-c n)] , ^ p k 2 | n | 2 Im(X (0)) , (5.2.25) 2 a ' n from (5.2.23). Equations (5.2.21) and (5.2.25) imply that dE/dt = c T (0) ; (5.2.26) n w n i . e . the rate of increase i n wave energy i s equal to the product of the wave phase speed and the mean shear stress at the surface. D i f f e r e n t i a t i n g (5.2.24), the shear stress gradient i s given by dx /dz = j i p k 3 U 2 ( F F * " - F*F") w 4 a 1 1 60. Stewart 0-961) has suggested that i n the act u a l turbulent flow above the sea, there i s probably a gradient i n the turbulent shear s t r e s s to compensate for this wave-induced s t r e s s . Using equation (5.2.13), i t i s r e a d i l y found that (Lin 1955, §4.3) dx /dz = -L k 3!n I 2U"|F I 2c./[(U -c ) 2 + c 2 ] , (5.2.27) w 2 a 1 1 1' 1 I 1 r x where c = c + i c . : c ,c. are r e a l . The stress gradient associated r x r x with a ne u t r a l disturbance i s given from (5.2.27) i n the l i m i t c -> c and c. -> 0+ . Then dx /dz i s obviously zero, except near r n x w the c r i t i c a l height z^ where lim c./[(U -c ) 2 + c 2 ] = lim y/U' (z ) [ ( z - z ) 2 + y 2 ] ; c -+0+ 1 y->0+ where y = c./U' (z ) J l i e = T T 6 (Z - Z )/U'(z ) ; e.g. see L i n 1955, §8.2. c 1 c Using the condition that x + 0 as z -> 0 3 , equation (5.2.27) may be w integrated so that 0 , z > z c X w n (z) = (5.2.28) i . p / l n l ' l F j V u y u ^ ) , 0 < z < z c The subscript c c r i t i c a l height. denotes that the function i s evaluated at the The expression (5.2.28) for the shear s t r e s s can 61. be derived h e u r i s t i c a l l y (e.g. see L i g h t h i l l 1962, or P h i l l i p s 1966, §4.3) by considering the mean vortex force acting on the f l u i d . From (5.2.18), (5.2.19), (5.2^26) and (5.2.28), the f r a c t i o n a l increase i n energy per radian i s given by S = TT( P /p ) C-kIF | V / c U' ) . (5.2.29) a w 1 c 1 l c n l c The value of S may be found e i t h e r from (5.2.29) or from (5.2.20) and (5.2.23). However, both formulae require that the Rayleigh equation (5.2.13) be solved for. F . Conte and Miles (1959) have used a numerical procedure to obtain values of S when the mean wind U v a r i e s l o g a r i t h m i c a l l y with z . 5.3 Streamline Configuration We have shown i n §5.2 that u i s discontinuous at z z c such that the shear s t r e s s jumps from zero for z > z^ to a constant value x for z < z . I t would therefore seem that the stream-w c n l i n e configuration near z^ i s e s s e n t i a l to the wave generation mechanism. C l e a r l y , the complete streamline pattern can be determined only by solving the Rayleigh equation, i . e . system (5.2.13). However, some idea of the o v e r a l l pattern may be found from insp e c t i o n of the equations of motion. Any d i s c u s s i o n of the streamline configuration i s best understood by taking a coordinate system moving h o r i z o n t a l l y with the wave phase speed c . That i s , we make a G a l i l e a n trans-r n formation, x - c t = x' , say, so that the motion becomes independent 62. of time, to the f i r s t order i n y > a n& the streamlines coincide with the p a r t i c l e path l i n e s . I f the v e r t i c a l v e l o c i t y w does not vanish at the c r i t i c a l height z^ , then a sequence of closed-streamline patterns (or 'cat's-eyes') must form i n the neighbourhood of (see Figure 3). In Figure 3, the cat's-eyes are shown to be completely symmetric and t h e i r l o c a t i o n r e l a t i v e to the water surface i s not given. It i s shown below that i f there i s a mean v o r t i c i t y gradient ( i . e . 4 0) , then the cat's-eyes cannot be symmetric and that t h e i r l o c a t i o n depends strongly upon . Consider the l i n e a r i s e d v o r t i c i t y and momentum equations (5.2.13) and (5.2.9); v i z . F" = [k - U'7(c - IL ) ] F I n 1 X = IL'F - (U - c )F' , 1 I n X' = -lc (u^ - c n ) F , (5.3.1) where p/p = kAX(z)e^" k x , u = kAF' (z)e"^ 0 5 1 a n d w = -ikAF(z)e 3. i l e x 1 i l e x i k x 1 At the water surface n = Ae , we have that F(0) = (c - U )/k , n o (5.3.2) where E U^(0) . We assume that the mean v e l o c i t y p r o f i l e i s twice d i f f e r e n t i a b l e and that i t i s a monotonic increasing function of z . For a neutral disturbance, c i s r e a l and hence the r e a l n and imaginary parts of F ( v i z . F^ and F^ , resp e c t i v e l y ) s a t i s f y equations (5.3.1) and (5.3.2) independently. Assuming that i s 63. negative, i t follows that the r a t i o F" ,/F . i s p o s i t i v e d e f i n i t e r,x r,x r i n the open i n t e r v a l (0,2;^) . From equations (5.2.24) and (5.2.28), Im(F*F') = F F.' - F.F' = r x x r W, 0 < z < z c (5.3.3) 0, z > z c 2 where W = TT [ j ^ • ~ ^ ' - [ c ^ i c ) » a p o s i t i v e constant. Now, Im(F*F /) i s the Wronskian of the solutions ^ r , F ^ of equation (5.3.1a). Therefore, f or 0 < z < z^ , F^ and F_^  are l i n e a r l y independent, while above the c r i t i c a l height, they are not. Equations (5.3.1) - (5.3.3) imply that F.(0) = 0 = F'/(0) and F.'(0) = kW/(c -U ) > 0 . (5.3.4) x x x n o C l e a r l y , equation (5.3.1a) now requires that F_^(z) be p o s i t i v e d e f i n i t e and, i n f a c t , monotonic increasing i n (OjZ^) . If F_^  = 0 at some point above the c r i t i c a l height, then F^ = 0 al s o , i . e . F = 0 . The appearance of a s t r a i g h t streamline above the c r i t i c a l height i s p h y s i c a l l y improbable. . Hence, F_^  i s p o s i t i v e d e f i n i t e over the whole i n t e r v a l (0,°°) , i n general (figure 4a). From equations (5.3.1b), (5.3.1c) and (5.3.4), i t follows that the imaginary part of X s a t i s f i e s the conditions X (0) = kW > 0 and X£(0) = 0 = X^(z ) 64. Also, since i s p o s i t i v e d e f i n i t e , i s p o s i t i v e d e f i n i t e i n (0,z ) and negative d e f i n i t e i n (z ,°°) • Thus the v a r i a t i o n of X. c c i must be as shown i n f i g u r e 4b. The general behaviour of the imaginary components of F and X i s r e l a t i v e l y simple to deduce because i t i s governed by the condition (5.3.3) that the perturbation must be such that energy i s t r a n s f e r r e d towards.the surface. The behaviour of F and X , r r the r e a l components of F and X , i s l e s s r e s t r i c t e d however and i t i s determined p r i m a r i l y from the behaviour of the function J = U ' 7 k 2 ( L L - C ) . ± 1 n In general, i t i s p o s s i b l e to make J an a n a l y t i c function of a parameter T = r(kz ) over the i n t e r v a l 0 < z < z , such that J c c i s a monotonic increasing function of r and J = 0 when T = 0 . Thus, T i n d i c a t e s the extent of the influence of the c r i t i c a l height s i n g u l a r i t y on F^ over the whole i n t e r v a l 0 < z < z^ . For example, r may vary as (kz^) ^ , so that r •> 0 corresponds with kz^ -»• » ; i . e . kz^ i s so large that the disturbance produced by the wavy boundary has almost completely died out at the c r i t i c a l height. When T = Q , the equation for F (_z;r) , v i z . (5.3.1a), reduces to F" = k 2 F (z;0) . r r Using (5.3.2) and the condition that F (.°°) = 0 , the s o l u t i o n i s s imply 65. F (z;0) = k _ 1 ( c - U )e k z . (5.3.5) r n o Since J(z), and hence F(z) , are a n a l y t i c i n the open i n t e r v a l (OjZ^) and since J i s an a n a l y t i c function of T for 0 < z < z^ then (5.3.1) implies that F i s also an a n a l y t i c function of r . Therefore, the function F ( r ) should vary smoothly as T increases from zero. Assuming that the v e l o c i t y curvature i s f i n i t e , i t i s c l e a r that for every r > 0 and A > 0 , there e x i s t s a x ( r , A ) where 0 <_ X _S z c > such that J(z;T) <_ A i n the i n t e r v a l 0 _< z <_ z^ - x • Since J i s a monotonic increasing function of r , x must also be monotonic increasing for a given value of A . Thus, for very small values of T , F^ _ i s given approximately by (5.3.5), except i n the i n t e r v a l z^ - x < z < z c where J approaches i n f i n i t y and hence, from (5.3.1a), the curvature of F^ correspondingly increases (see fi g u r e 5). Now i f J. i s also a monotonic increasing f u n c t i o n of z , then A = JCz c - x>F) a n d X 1 S a monotonic increas-ing function of r . That i s , as r increases, the height at-which F^ i s no longer given by (5.3.5) correspondingly decreases and F^ _ should behave as shown i n f i g u r e 6a. Thus, i f J i s a monotonic incr e a s i n g function of z , then F^ i s expected to be p o s i t i v e d e f i n i t e for 0 < z < z and therefore, for z < z < «> also. Equation c c (5.3.1c) shows that X' i s p o s i t i v e for 0 < z < z and negative for r c z > z ; at the c r i t i c a l height, X i s zero and, since c ° r 2 X"(z ) = -k X (z ) < 0, X a t t a i n s a maximum value here. Thus, X r c r c' r r i s monotonic increasing for 0 < z < and monotonic decreasing for z > z . The value of X (0) must be found from (5.3.1b). If the c r c r i t i c a l height i s f a r above the surface (r small), then i s 66. small and F (0) i s given approximately by (5.3.5). Hence, 2 X^(0) = ~ ( c n " ~-"0) < ^ » a n c " consequently X^ passes through zero as z increases to . When z i s small (T l a r g e ) , X (0) - U'(c - U ) / k > 0 and X i s p o s i t i v e d e f i n i t e , as shown i n r o n o r fi g u r e 6b. Figure 7 depicts phasor diagrams of the general behaviour of the pressure p and v e r t i c a l v e l o c i t y w r e l a t i v e to the surface displacement n ; here, the angular displacement i s kx' which increases as x increases (t • fixed) or as t decreases (x f i x e d ) . Above the c r i t i c a l height where there i s no shear s t r e s s , the pressure simply leads the v e r t i c a l v e l o c i t y by TT/2 , and so there i s no v e r t i c a l energy transport. C l e a r l y , the pressure above the c r i t i c a l height also leads the surface displacement by an angle 0^ l e s s than TT/2 ; th i s angle probably increases as r increases. Thus, the centre of the cat's-eye, which corresponds with the pressure minimum, lags the wave crest by TT~Qc • Hence, the cat's-eye forms i n the trough of the wave such that the streamlines above the cat's-eye are v i r t u a l l y i n anti-phase with those below, as i l l u s t r a t e d i n fig u r e 8. This i s i n contrast with the s i t u a t i o n proposed by L i g h t h i l l (1962) i n which the cat's-eye l i e s above the wave cr e s t . At the surface, the v e r t i c a l v e l o c i t y n e c e s s a r i l y lags the displacement by TT/2 . As z increases however, t h i s angle decreases and so the streamline trough i n c r e a s i n g l y leads the wave trough. Now, a streamline crest above the c r i t i c a l height corresponds with a trough below. Since the cat's-eye i s of f i n i t e thickness, i t s crest leads i t s trough, making the shape asymmetric. The degree of this asymmetry c l e a r l y depends upon the magnitude of the shear stress generated at the c r i t i c a l height. When J i s not a monotonic increasing function of z , the behaviour of F and P i s not so r e a d i l y deduced. However, as r r J the c r i t i c a l height approaches the surface, J must eventually become monotonic increasing for any (well-hehaved) U^(z) and then the above discu s s i o n holds. I t i s generally assumed that the mean wind above the sea surface may be represented by a logarithmic p r o f i l e ; i . e . U1Cz) " c n = (U,JK) £nl(z + z £ ) / ( z c + zA)] , where u, , K , Z . and z are constants. Here, the usual z (see x, c §2.2) has been replaced by z + z , where z i s the height at which the logarithmic representation becomes v a l i d , so that U - c i s o n f i n i t e . Thus, the function J i s given by J = I k 2 ( z + z%)2 £n{(z c + z £ ) / ( z + z £ ) } ] _ 1 . (5.3.6) L e t t i n g T = (kz^) ^ , but keeping z^/z^ f i x e d , i t i s seen that J(z,T) i s a monotonic increasing function of T and that the boundary value of F (= (c - U )/k) i s independent of r . Thus, r -> 0 r n o 1 corresponds with kz^ -> 0 0 such that the r a t i o of the sub-layer t h i c k -ness to the true c r i t i c a l height (z /(z + z„)) remains constant. I c £ However, J i s not generally a monotonic function of z ; i n f a c t , J has a minimum at a height z , where m z + z „ = ( z + z „ ) e m I c V 1/2 Therefore, J(z) i s monotonic increasing only when z^ + z^ <_ z^e 68. At t h i s distance from the surface, the c r i t i c a l height would surely be a f f e c t e d by v i s c o s i t y and so the present analysis would be i n v a l i d . We must therefore consider the behaviour of F i n the open i n t e r v a l r 0 < z < z^ when J(z) v a r i e s as shown i n f i g u r e 9, i . e . when 1/2 zjz > e - 1 . For very small values of V , -equation (5.3.5) s t i l l describes F s a t i s f a c t o r i l y , except i n some i n t e r v a l (z - x> z c ) near the c r i t i c a l height where the curvature increases (see f i g u r e 5). However, as r increases, the value of J near the surface increases, and hence equation (5.3.5) becomes i n v a l i d i n the neighbourhood of both the surface and the c r i t i c a l height.. From the s o l u t i o n (5.3.5) for T = 0 , i t seems that near z = 0 , the behaviour of F i s given approximately by Thus, as r increases, the slope of F (0) becomes more negative. But as z increases ( T f i x e d ) , the curvature decreases more r a p i d l y than that given by (5.3.7) so that F f a l l s below t h i s curve. For z > z however, J i s monotonic increasing and hence the curvature m of F^ _ correspondingly increases, as shown i n f i g u r e 10a. From equations (5.3.1b) and (5.3.1c), i t i s seen that X i s monotonic increasing with a maximum at . We may also deduce a lower bound for X (.0) from (5.3.1b). Equation (5.3.7) implies that r r (5.3.7) X (0) - F (0){U f - k(c -r r o n U )[1 + J ( 0 ) ]1 / 2 } 69. Now since U^' < 0 , i t follows that c - U < z U' . n o c o Therefore, X (0) > U'F ( 0 ) f l - kz [1 + J ( 0 ) ] 1 / 2 } r o r c U j F r ( 0 ) U - I ( k z c ) 2 + (z c/z £) 2{£n(l + zjz^)} 1] } 1/2 But z /z„ i s greater than [£n(l + z /z„)l for z /z„ > 1 at c £ c £ c £ l e a s t ; hence, the lower bound on X^(0) i s generally negative and i t decreases as either zjz^ or r increases. This suggests that X (0) i s , i n f a c t , a monotonic decreasing function of T (and z /z„), r c £ as shown i n f i g u r e 10b. Thus, for small values of r , the streamline pattern i s as i l l u s t r a t e d i n f i g u r e 8 with the cat's-eye ' f i l l i n g ' the wave trough. Also, the phasors of p, w and r\ are as given i n fi g u r e 7a; i . e . the o v e r a l l motion i s s i m i l a r to that when J(z) i s monotonic increasing. As r increases to some value r , the curvature of F s r may decrease so quickly away from the surface that at z = the slope of F i s s t i l l large and the e f f e c t of J increasing over the i n t e r v a l z < z < z i s unable to stop F from going to zero at m c r z c ( f i g u r e 11a). Thus, equation (5.3.3) implies that F^(z; r ) E 0 for z > z and that F'(z -) = T T U " F . ( Z ) / U ' . From equations c r c c x c l c (5.3.1b) and (5.3.1c) (or from f i g u r e 10b), i t i s clear that as r -> T > the maximum value of X , located at z , approaches zero, s r c 70. so that X^(z; r ) = 0 f o r z > z c ( f i g u r e l i b ) . Thus, when r equals T g , the centre of the cat's-eye i n the streamline pattern lags the wave crest by p r e c i s e l y TT/2 . For r greater than , the height at which F goes to zero, z say, decreases, and hence ir a. both F and F" are monotonic decreasing f o r z < z < z (f i g u r e r r a c 12a). Equations (5.3.1b), (5.3.1c) and (5.3.3) show that now X^ i s negative d e f i n i t e with a maximum at z and a minimum at z , a c as shown i n fi g u r e 12b. Also, the pressure above the c r i t i c a l height leads the surface displacement by an angle greater than TT/2 (figur e 13). Therefore, the centre of the cat's-eye lags the wave cre s t by an angle le s s than TT/2 . As shown i n f i g u r e 14, th i s implies that the cat's-eye appears above the wave cr e s t and that the streamlines above the c r i t i c a l height are almost i n phase with the surface displacement; i . e . between the water surface and the c r i t i c a l height, the streamlines are s h i f t e d by more than TT/2 . It has been pointed out by Stewart (1967) that the c r i t i c a l height i s often much less than the wave amplitude. In such cases, the cat's-eyes of f i g u r e 8, and e s p e c i a l l y f i g u r e 14, must l i e very close to the surface and t h e i r h o r i z o n t a l axes must curve so as to follow the wave shape. In f i g u r e 8, the cat's-eye l i e s p r i m a r i l y over the wave trough and so i t i s r e a d i l y accommodated as the c r i t i c a l height decreases. On the other hand, the cat's-eye of f i g u r e 14 i s s h i f t e d more than h a l f -way up the face of the wave. Associated with t h i s i s a maximum i n the in-phase ( r e l a t i v e to n) part of the pressure between the surface and z^_ ( f i g u r e 13). Thus, as z^ decreases, the v e r t i c a l gradient of the streamfunction ( i . e . the f l u c t u a t i n g h o r i z o n t a l v e l o c i t y ) increases. 71. The increasing f l u c t u a t i n g h o r i z o n t a l v e l o c i t y may become large enough to i n t e r a c t d i r e c t l y with an external perturbation such as turbulence; therefore, t h i s streamline configuration may be unstable i n the presence of a i r turbulence. 72, 6. WAVE GROWTH CAUSED BY TURBULENT WIND 6.1 Introduction Measurements of the rate of wave growth imply that i t i s so large that the energy must come from the mean wind. But the only p o s i t i o n where there i s strong i n t e r a c t i o n between the wave-induced f l u c t u a t i o n i n the a i r and the mean wind i s at the c r i t i c a l height. This suggests that b a s i c a l l y M i l e s ' formulation of the problem must be c o r r e c t . However, i t i s noted i n §5.3 that the streamline con-f i g u r a t i o n , e s p e c i a l l y i n the v i c i n i t y of the c r i t i c a l height, i s probably affected by the ubiquitous a i r turbulence. Thus, we now show that a layer e x i s t s around the c r i t i c a l height i n which, the turbulence transfers momentum v e r t i c a l l y and so smoothes the v o r t i c i t y . s i n g u l a r i t y present i n the i n v i s c i d model of §5. Because turbulence i s a f a r more e f f i c i e n t d i f f u s e r of momentum than i s molecular v i s c o s i t y , t h i s c r i t i c a l l ayer i s correspondingly l a r g e r than the viscous layer u s u a l l y introduced to eliminate the s i n g u l a r i t y at the c r i t i c a l height (e.g. see L i n 1955, §8). If the c r i t i c a l height i s larger than the thickness of the c r i t i c a l l ayer d , then the turbulence has l i t t l e e f f e c t upon the flow near the sea surface. On the other hand, when z^ < d , the a i r - s e a i n t e r f a c e l i e s w i t h i n the c r i t i c a l l ayer and hence the flow near the i n t e r f a c e i s g r e a t l y influenced by the turbulence. 6.2 Equations of Motion i n the A i r The t o t a l v e l o c i t y f i e l d v. i n the a i r above the sea surface can be decomposed into a mean part U_^  and a f l u c t u a t i n g part u^ ; i . e . 73. v. = U. + u. l x i where, by d e f i n i t i o n , the mean value of u^ i s zero. Here, the mean value of a quantity f i s understood to mean, the ensemble average <f> : i . e . • 1 N < f > = lim ± y f vr N L , m where f denotes the mt*1 r e a l i s a t i o n of the flow. S i m i l a r l y , the m pressure It may be decomposed such that II = P + p where <p> = 0 . For an incompressible viscous f l u i d , the equations for the conservation of momentum and volume are 3v. 3v. - , n _ i . i . 1 3It n2 - r — + v. - — + — - — = v V v. 3t i 3x, p 3x. a x J 3 a x - eS 13 3v. x 3x. = 0 (6.2.1) where v i s the kinematic v i s c o s i t y , p i s the f l u i d density, a < a ( X ^ J X ^ J X ^ ) are ca r t e s i a n coordinates such that increases v e r t i c a l l y upward, and 6 i s the Kronecker d e l t a . Taking the mean value of (6.2.1) y i e l d s 74. DU 1 3 P <u. u. > + — ——-— = v V U. - g<5 . „ , xi p ax. a i • i 3 Dt 9x. l 1 p 3 J a I 3U. i 3x. l = 0 , (6.2.2) where TT— = •5-7 + U. T— Dt 3t j 3x, We note the presence i n (6.2.2) of the (kinematic) Reynolds st r e s s gradient against which the mean flow must work. Subtracting (6.2.2) from (6.2.1), the equation f o r the momentum f l u c t u a t i o n i s found to be Du. 3U. 3q.. . . 1 + u i + _ i l + _1_ J L E . Dt i 3x. 3x. p 3x. 3 3 a 1 v V u . , a 1 3u. 3x. = 0 (6.2.3) where q.. = u.u. - <u.u.> i s the Reynolds st r e s s f l u c t u a t i o n . 1 3 1 3 1 3 Thus, the f l u c t u a t i n g f i e l d must work against the mean s t r a i n rate f i e l d and the f l u c t u a t i n g Reynolds stress f i e l d . We now multiply (6.2.3a) by u^ , interchange the i and k i n d i c e s , and add the two r e s u l t i n g equations to y i e l d -{.u. u ) + u u. + u.u Dt i k k j 3x. 8 U k 8 q i i 8 q k i + u, — J- + u. "L i "j 3x. k. 3x. 3 3 i 3x. 3 • 1 / 3P . 3Px f „2 , _2 N + — ( u . — — + u - — ) = v (u. V u. + u. V u, ) p 1 3x, k 3x. a k 1 - 1 k a k 1 (6.2.4) Thus, the mean Reynolds st r e s s s a t i s f i e s the equation 75. D , x k , 1 9 , , N —-<u.u, > + <u, u.> + <u.u > V — (<u.q, .> + <u, q. . >) Dt i k ic j 9x. i x, 2 9x. i H k i k H i j 3 3 3 , 1 , 3 P a p . + — ( < u . ~-> + <u. ~ > ) p x 9x. k 3x. a k x v (< U lV 2u.> + <u.72xx> ) , (6.2.5) a k x l K because u.u.u, = ^ -(u.<u, u.> + u <u.u.> + u.q, . + q ) • x j k 2 x k j k x j x^kj K XJ Subtracting (6.2.5) from (6.2.4), the equation for the Reynolds stress f l u c t u a t i o n i s found to be, a f t e r some rearrangement, D q M . .311. 3IL 3u d u . xk , x . k . ^  k , x — = — + q. . T 1- q. . + <u.u.> 1- <u, u.> Dt H x i 3x. H x j 3x. x j 3x. \ i 3x. 3 3 3 J 3 , . 3u. . 3u 3u. 3u - — fp (—- + — ^ ) - < p ( — i + — £ ) > I p . 1 P dx, 9x/ P ^ 3 X l 3x. J a k x k x 8u. 3u au. 3u , ? r i k _ x_ k , a ^ x . 9x. 3x. 9x. J 3 3 3 3 2 1 3 v V q., - -r (u.q, . + u. q. . - <u.q. .> - <u. q. .>) a ^xk 2 3Xj x^kj k i j x n k j k^xj 1 9 3 3 + ~ (u, <u. u. > + u. <u u.>) - — I T — (pu, - <pu >) 2 3x, k x i x k j p 9x. k k 3 a j + Cpu^ . - <pu.>)J . (6.2.6) k Thus, the stress f l u c t u a t i o n i s affe c t e d by the mean stress working against the f l u c t u a t i n g s t r a i n rate f i e l d , i n addi t i o n to i t s e l f work-ing against the mean s t r a i n rate f i e l d . The second l a s t term on the l e f t hand side of (6.2.6) describes the working of the i s o t r o p i c 76. pressure against the f l u c t u a t i n g rate of deformation tensor; c l e a r l y , the work done by thi s i n t e r a c t i o n tends to make the stress tensor i s o t r o p i c . The l a s t term on the l e f t hand side represents the viscous decay of st r e s s . The r i g h t hand side of (6.2.6) consists of the stres s transport terms. S i m i l a r l y , i t i s po s s i b l e to determine equations f o r higher order v e l o c i t y products and covariances (e.g. u.q, . and <u.q, .>) . However, since we are looking f o r a non-1 k] I k] turbulent wave-like s o l u t i o n , we s h a l l consider equations (6.2.1) -(6.2.6) only. We also s i m p l i f y some of the terms i n these equations to y i e l d a tractable system. The mean motion above the a i r - s e a i n t e r f a c e may generally be treated as a f u l l y developed, zero pressure gradient turbulent boundary layer flow (see §2.2); that i s , a l l mean qu a n t i t i e s <f> vary i n the v e r t i c a l d i r e c t i o n only, such that <f> = <f>(x^) where increases v e r t i c a l l y upward. Thus, the mean v e l o c i t y has com-ponents u\ = {U^(x.j) jl^Cxg) ,0} i n the (x^,x^,x^)-coordinate system. Let (y'^,y2,y• ) be a new coordinate system obtained by r o t a t i n g the ( xi> x2' X3^ system through an angle 0 about the x^-axis such that the v e l o c i t y vector i n the new system acts i n the y ^ - d i r e c t i o n . We assume that the mean stress matrix <u^ u_.>^  i n the ( y ^ , y 2 > y 3 ) system i s in v a r i a n t under r e f l e c t i o n through the (y^,y^)-plane, i . e . f I <w-L> , 0 , <w^ w^ > ( < u i u j V = • 0 2 <w2> , 0 • , say 0 , 2 <w1}w3>, <w3> j In the ( X ^ J X ^ J X ^ ) system, the mean stress therefore becomes « u . u j > x ) = 2 2 2 2 2 2 <w^ > cos 9 + <w^ > s i n 8 , <vi^-'vi^> sin8cos6 , <w^ v?^ > cos6 2 2 2 2 2 ? <x<!2_~'w2> s^-n®cos® » < w - j _ > s i n ^ + < W 2 > c o s ^ » < w i w 3 > s^ n^ 2 < W 1 W 3 > c o s 8 ' < w l w 3 > s^n® » <w^> j (6.2.7) that i s , a l l components of stress are present when the two coordinate systems are not aligned. The mean v e l o c i t y components i n the new system are simply U\ = {|u|, 0, 0} where = |u| cos8 and = |u| sin0 . Therefore, from equation (6.2.2), 2 < w l w 3 > " VadIVl/dy3 = ~ u* ' the constant kinematic shear s t r e s s . But far above the i n t e r f a c e , the mean wind follows the log law of the w a l l (see §2.2), i . e . d|u|/dy = u,./Ky , where K i s the von Karman constant. Therefore, < W T W 3 > 1 S approx-imately constant f o r y„ >> v / K U , - 0.03 cm. when u, = 15 cm./sec. 3 a x * Thus, we s h a l l assume that the mean str e s s <u.u.> i s constant for i 3 a l l x^ . 78. Now the v e l o c i t y and stress f l u c t u a t i o n s , u. and q., , consist of both turbulent and wave-induced pa r t s . The l a t t e r part may be defined i n terms of coherence with the wave f i e l d ; e.g. the wave-induced v e l o c i t y i s <n(k)u.(k)>/<n2Ck)>1/2 , where n(k) and u_^(k) are the f o u r i e r transforms of the wave and v e l o c i t y f i e l d s r e s p e c t i v e l y , and k i s the x^avenumber. In p r a c t i c e , hox^ever, the wave s p e c t r a l density i s quite sharply peaked. Thus, i t i s convenient to assume that the phase average of x^ave-dependent qu a n t i t i e s exists and hence we define the wave-induced part of a quantity f to be 1 N f = lim 2 N + 1 1 f ( x 1 + 2mir/k , x 2»x ,t) , N-*» m=- N xtfhere the peak of the x^ave spectrum i s at wavenumber k = (k,0) ; i . e . x^ e take coordinates such that the x^ave i s propagating i n the x^-d i r e c t i o n . The turbulent part of a quantity f i s therefore given by f T = f - f . The quantity f i s c l e a r l y random; i t va r i e s f o r each r e a l i s a t i o n of the flow because the wave amplitude and phase vary between flox^ r e a l i s a t i o n s . We note that 79. q.. 4 u.u. - <u.u.> : that i s , the wave-induced stress does not consist of the wave-induced v e l o c i t y product only. There i s an a d d i t i o n a l c o n t r i b u t i o n due to the i n t e r a c t i o n of the turbulence and wave f i e l d s . The mean stress <u.u.> i s the t o t a l s t r e s s : i t consists of contributions from the 1 3 turbulence and the wave f i e l d s . Equations f o r u_^  and q are found by taking the phase average of (6.2.3) and (6.2.6). The phase average commutes with a l l the d i f f e r e n t i a l operators i n these equations; i n p a r t i c u l a r , I f , = l ± m z k l l t e - f ^ , x 2 , x r t ) xvhere 5 - x + 2mTr/k , 1 N-*» m 1 " i 1 , n 2 N + r Z-3T £ ( ?» x2' x3' t ) S ± n C e " f e = 1 ' N-*» m 1 8 X 1 Since we have shown that viscous stresses have n e g l i g i b l e e f f e c t upon the mean v e l o c i t y , we also neglect the viscous term i n (6.2.3) and hence u_^  i s given by Du. 9U. 9q.. n -_ i + _ J l + 1 9 i L . = o Dt j 9x. 9x. p 9x. ' 3 3 a 1 (6.2.8) 9u. 9x. 1 The log law for the mean wind i s derived on the assumption of l o c a l 80. e q u i l i b r i u m of the turbulence (see §2.2), i . e . turbulent and viscous transport processes are assumed to be zero so that l o c a l l y the production and d i s s i p a t i o n of turbulent energy are balanced. We now assume that the wave-induced stress i s also i n " l o c a l " equilibrium i n the sense that only the mean v e l o c i t y i s e f f e c t i v e i n transporting the stress f l u c t u a t i o n . Thus, q ^ i s given from (6.2.6) by the equation Dq.. tfU. 3U, 3u, 3u. — r — + q., 1- q., - — + <u.u.>-— + <u. u.>-Dt ^ i k 3x. ^ i i 3x. l i x, k i x, 3 3 3 3 ., 3u. 3u, 3u. 3u. 3u. 3u, 3u. 3u, i r /• 1 i_ k \ / 1 i k s , , „ r 1 k i k i - [ p ( i x 7 + 3 T } - < p ( i x 7 + 1 x T ) > ] + 2 v a [ 3 x T ^ 7 - 3 7 7 > ] a k x k x 3 3 3 3 = 0 . (6.2.9) To make the system (6.2.8) - (6.2.9) t r a c t a b l e , i t i s necessary to s i m p l i f y the l a s t two terms of (6.2.9). The simplest, n o n - t r i v i a l assumption i s that they are l i n e a r functions of the stress f l u c t u a t i o n . The second l a s t term i n (6.2.9) represents the working of the i s o t r o p i c pressure which tends to force the stress towards a state of isotropy. I t i s also seen that the i n c o m p r e s s i b i l i t y con-d i t i o n (6.2.3b) implies that t h i s force does not influence the o v e r a l l energy f l u c t u a t i o n q.. . Thus, we model t h i s term such that i i 1 3u. 3u, 3u. 3u, a k i k i = a (q. . 6.. - 3q.. ) (6.2.10) p mJJ l k ^ i k ' 81. where a i s a rate constant. The expression (6.2.10) has previously been suggested by Rotta (.1951) and i t i s discussed by Hinze (1959, p. 253). Since the l a s t term i n (6.2.9) represents a viscous decay of s t r e s s , we model i t byN the product of the stress f l u c t u a t i o n and a rate constant a , i . e . 9u. 3u 9u. 9u 2v [ — - — - - < — i — £ > ] = a_q.. . (6.2.11) a 3x. 3x. 9x. 9x. D i k 3 3 3 3 Putting (6.2.10) and (6.2.,11) into (6.2.9), the model equation for the wave-induced st r e s s f l u c t u a t i o n becomes Dq„ 9U. 3U, 3u, 9u. ik , - l , - k , k , l P.. + q, . T — + q. . - 7 — + <u.u.> - — + <u, u.> - — Dt ki 9x. i i 3x. i i 3x. k i 3x. 3 J J 3 a (q. .6., - 3q., ) + a„q.. = 0 . (6.2.12) p JJ i k n i k D i k Equation (6.2.12) implies that q ^ behaves v i s c o e l a s t i c a l l y . To show t h i s , we s i m p l i f y the equation by contracting the indices and we obtain an equation for the t o t a l energy f l u c t u a t i o n " ^ j ^ > v i z . 9u. 3U. ( -(<u.u.> ~ + q. . ) 1 1 dx. 11 dx. J 3 Thus, the energy produced by s t r e s s - s t r a i n f i e l d i n t e r a c t i o n s i s balanced by mean advection and by viscous d i s s i p a t i o n . The r e l a t i v e importance of the l a t t e r terms i s given by the r a t i o 82. We expect ex to be of the same order of magnitude as OL , and so p D X i s also a measure of the r e l a t i v e importance of mean advection com-pared with the tendency of q_„ towards isotropy. When X >> 1 , the mean advection dominates. Hence, the stress behaves " e l a s t i c a l l y " i n that i t responds e s s e n t i a l l y to the s t r a i n f i e l d . This case i s equivalent to Batchelor's (1953, chpt. 4) "rapid d i s t o r t i o n " problem. When X << 1 , the stress i s i n true l o c a l e q uilibrium: the stress production i s simply balanced by viscous d i s s i p a t i o n . Now the stress behaves i n a viscous manner, responding to the rate of s t r a i n f i e l d . In the present wave generation problem, there e x i s t s a c r i t i c a l height at which X goes to zero. At th i s height, the viscous d i s s i p a t i o n d e f i n i t e l y dominates, producing a "viscous" s t r e s s . C l e a r l y , the stress behaves i n a v i s c o e l a s t i c manner when X i s of order unity. Equations (6.2,8) and (6.2.12) form a closed set of ten, i n general, equations f o r the ten unknowns u. , p and q : a l l com-X XiC ponents are present when the i n t e r f a c i a l wave i s not t r a v e l l i n g i n the same d i r e c t i o n as the mean wind. For a wave t r a v e l l i n g i n the x^-d i r e c t i o n with wavenumber k , we assume that a l l q u a n t i t i e s are inde-pendent of x^ . Thus, (6.2.8b) implies t h a t there e x i s t s a stream-function IJJ such that u. - f t - and u - . (6.2.14) 1 8x„ 3 3x, The pressure f l u c t u a t i o n p i s r e a d i l y eliminated from (6.2.8a) to y i e l d 83. (6.2.15) ,„|, c o s e | i_; +_Z_ ( 5 i i.5 3 3 ) 2 2 + (7T " 7T)qi3 =0 ' D U 2 >TT|, . fl 3* , 8 q l 2 . 3 q 2 3 n — - | u|'sine ^ + — + _ - 0 , where | u ] ( x 3 ) is. the magnitude of the mean wind and 0 i s the angle between the wind and wave d i r e c t i o n s . Using (6.2.7), the components of q are given from (6.2.12) such that ( ~ + 3 c t p + a n ) C ( l 1j_ ~ q . 3 3 ) + 2 | u | ' c o s 0 q 1 3 + 2<w^ 3> c p s 0 x r l J i ^ or 2 2 Q , 2 . 2 Q , 2 . 3 2^ n x C—I + — j , - * ' 2(<w1>cos 0 + <w2> sxn 0 + <w3>) ^ ^ x = 0 ' (6.2.16a) 8x 3 3x 1 "1 3 2-( DT + % + V<13 + ' C O s 6 q33 + ^ T ~ 2 r 3x 3 2-- (<w > cos e + <w„> s i n 0 ) —I = 0 , (6.2.16b) 3 x l 2-( ^ + 2 a + a D ) i 3 3 - a ( i n + q^) - 2<^z> c o S 0 A-f 3x^ - 2 < V ^ = 0 ' ( 6- 2- 1 6 c ) 84. ( £ + % + V ^ l l + ^22 } " % ^33 + ' C O s 9 «13 + 2 ^ > ' S i n 9 ^23 2- 2-2 2 2 2 3 iij 9 ( + 2(<w. > cos 6 + <w„> s i n 8) "-""^— + 2<w 1w„> cos8 —¥r 1 I d X , d X , 1 j . Z 1 3 3x^ 2 2 3^2 3 u 2 + 2(<wl' - w>) sin8cos8 — — + 2<w.w > sin6 -r—=• = 0 , (6.2.16d) 1 . 2 dx^ 1 3 3x^ D - - 2 8 u 2 du2 (— + 3a + a_,)q0„ + lul 'sin0 q„_ + <w„> T — + <w w > c o s 8 - — Dt p D n23 ^33 3 3x^ 1 3 3x^ 2- 2-<w2 - w2> sin8 Cos0 - <w,w > sin8 — = 0 , (6.2.16e) 1 2 n. 2 1 3 dx n3x„ 3x^ 1 3 (j^T + 3 a + a D ) q 1 2 + |U| 'cos8 q^ + | u| 'sine q±3 + (<w2> cos 26 a. 2 • 2fl, 3 U 2 ^ a 3 U 2 ^ 2 2 . Q A 3 ^ + <w > s i n 8) h <w 1w„> cos8 h <w, - w > sin9cos9- f 2 1 3 1 3 + <w1w > sin8 — ^ = 0• .' (6.2.16f) 3x 3 From (6.2.15) - (6.2.16), i t i s seen that the streamfunction i s coupled to the transverse v e l o c i t y v i a the s t r e s s - s t r a i n rate i n t e r a c t i o n terms f o r q^ ; i n turn, q^ i s coupled to \p v i a the 2 isotropy rate constant . This coupling i s of order s i n 0 at most. Therefore, i t may be neglected provided that the wave tra v e l s at a small angle to the mean wind. On the other hand, the v e r t i c a l advection of the mean vortex l i n e s ( i . e . the u.j|u|'sin0 term i n (6.2.15b)) produces a transverse v e l o c i t y of order sin8 . Although t h i s v e l o c i t y component and i t s associated s t r e s s components, q.. 9 and q „ , do not 85. a f f e c t the wave generation process, we s h a l l consider them f o r the sake of completeness. 6.3 Mathematical Formulation of Problem The behaviour of the wave-induced v e l o c i t y and stress compon-ents, u_^  and , i n the a i r (0 < < K>) i s described by the system (6.2.13) - (6.2.16). From (6.2.8a), the t o t a l v e r t i c a l stress p/p a + q^^ s a t i s f i e s the equation (p/p + q - J + - ~ + = 0 . (6.3.1) 8x 3 ^ " a H33' Dt dx± The wave-induced disturbance must be attenuated far above the i n t e r -face; thus, we have the boundary condition that ijj and -* 0 as x^ -> °° . (6.3.2) At the a i r - s e a i n t e r f a c e , there i s a kinematic boundary condition that the i n t e r f a c e must remain coherent, i . e . ("3T + vj -dr*11 = u 3 a t x 3 Q n ' 3 where n(x^,X2,t) i s the instantaneous displacement of the i n t e r f a c e about i t s mean p o s i t i o n x^ = 0 . If'the. wave slope i s small, then we may l i n e a r i s e t h i s condition to obtain 86. | | = at x 3 = 0 . (6.3.3) Thus, the motion i n the a i r i s described by the l i n e a r equations (6.2.14) - (6.2.16) and (6.3.1) - (6.3.3). We assumed i n §6.2 that the wave n i s t r a v e l l i n g i n the x ^ - d i r e c t i o n with wavenumber k . Therefore, we now l e t i k ( x 1 ~ c t ) n = Ae , (6.3.4) where A i s the (random, complex) wave amplitude and c i s the (complex) wave phase speed. Thus, boundary condition (6.3.3) becomes _ i k ( x ^ - c t ) u„ = -ikA(c - U )e at x = 0 , 3 o 3 where U = IL (0) = U (0) cos0 . This implies that ( i ) since the o i ~ equations of motion are l i n e a r and U \ ^ U_^(x^,t) , a l l wave-induced functions w i l l behave as exp{ik(x^ - c t ) ] , ( i i ) the r e l a t i v e mean v e l o c i t y at the i n t e r f a c e (c - U q ) i s a c h a r a c t e r i s t i c v e l o c i t y scale f o r the problem, and ( i i i ) the wave slope kA i s an ordering parameter f o r the problem. Taking k ^ as a c h a r a c t e r i s t i c length s c a l e , we introduce new normalised v a r i a b l e s : r = k x 3 , U(r) = [ U x ( x 3 ) - U q]/(C - U Q ) , V ( r ) = [ U 2 ( x 3 ) - U q]/(C - U Q ) , 87. 2 R., = <u.u, >/u, , i k i k * ' ik(x..-ct) ¥(r) = ib/[(c - U D)Ae 1 ] v(r) = u 2/[kA(c - U Q)e " ] , „ ik(x..-ct) Q(r) = (p/p )/[kA(c - U ) e ] ci O 2 ikCx^-ct) S i k ( r ) = " l i k / [ l c A u ^ e i k ^ - c t ) (6.3.5) X = a /[k(c - U )] , P P ° \ = V [ k ( c " U o ) ] • Putting (6.3.5) into (6.2.14) - (6.2.16) and (6.3.2) - (6.3.3), the normalised equations of motion are found to be (U - 1)CF" -10 - U"y + e 2 ( S 1 ; L - S 3 3 ) ' - i e 2 ( S ^ 3 + S.^) = 0 , (6.3.6a) Ii ( U - 1) + 3\ + ^ D ] ( . S n i " S 3 3 ) + 2U'S 1 3 + 2 R 1 3 ( ^ " - V) + i 2 ( R N + ^ 3 3 ) ^ ' = 0 , (6.3.6b) [i(U - 1) + 3A + X _ ] S . , + IT'S,, + R „ Y " + R-.-V = 0 , (6.3.6c) [i( U - 1 ) + 2X + X n]S„„ - X ( S n 1 + S 0 9 ) + 2 R . , * - i2R„„H" = 0 , p D 33 p 11 22 13 33 (6.3.6d) l i C U - 1) + X p + X D J ( S l l + S 2 2 ) - 2 X p S 3 3 + 2U'S 1 3 + 1 2 R u r + 2R 1 3^" = 0 , (6.3.6e) (U - l)v - V'¥ + e 2 S 1 0 - i e 2 S ' = Q , C6.3.6f) :±(U - 1) + 3A + X D ] S 2 3 + V ' S 3 3 + R 3 3 v ' + i R 1 3 v + R ¥ - i R 2 3 v i " = 0 , (6.3.6g) [i(U - 1) + 3X + A j S 1 0 + U'S 0„ + V S . , + i R 7 1 v + R-.v' p • D 12 23 13 11 13 + i R 1 2 ¥ ' + R23^"' = 0 , (6.3.6h) where E = u,/(c - U ) and where terms of order s i n 9 have been w o neglected. Thus, the f i r s t f i v e equations are independent of the l a s t three. ¥(r) s a t i s f i e s the boundary conditions *F -> 0 as r -> 0 0 , no) = 1 . (6.3.7) S i m i l a r l y , the v e r t i c a l stress i s found from (6.3.1) to be Q + e 2 S 3 3 = / [(U - 1)V + i e 2 S 1 C ( ] d r 13- (6.3.8) There i s also a dynamic boundary condition to be s a t i s f i e d , namely, that the normal stress must be continuous across the a i r - s e a i n t e r f a c e . If we assume that the motion i n the water (-<*> < r < 0) i s i r r o t a t i o n a l . then the f r a c t i o n a l increase i n energy per radian i s f i n a l l y found to be (see d e r i v a t i o n of equation (5.2.20)) 89. S S Tilt - ( p a V ( 1 " U c / C n ) 2 + *Sl\-o > n where E i s the wave energy per unit area, a = kc i s the free n n 2 wave frequency, c = g/k and p /p i s the r a t i o of the de n s i t i e s of n a w a i r and water. Thus, from (6.3.8), S = (p /p )(1 - U /c ) 2Im / [(U - 1M' + i e V j d r . (6.3.9) r.7 o n 1 a w 13-I t i s also found that the dependent v a r i a b l e s Y. , etc. are determined by s o l v i n g the equations of motion f o r c = c^ , i . e . f o r a neutral disturbance. From equations (6.3.6a) and (6.3.6c), i t i s seen that the system (6.3.6) i s equivalent to a fourth order d i f f e r e n t i a l equation i n ¥ , and hence four boundary conditions are required to obtain a unique s o l u t i o n . On the. other hand, the c o e f f i c i e n t s of the highest 2 order terms are prop o r t i o n a l to e , and e i s i n v a r i a b l y small. 2 Thus, we may regard e as an ordering parameter and expand V , v , 2 etc. i n asymptotic power s e r i e s i n e equations f o r V and v reduce to In the l i m i t e -> 0 , the (U - 1) cv" - ¥) " U"T = 0 , (U - l ) v - V'V = 0 , (6.3.10) independent of the stress f l u c t u a t i o n . This implies that, throughout most of the flow, the wave-induced f l u c t u a t i o n i s e s s e n t i a l l y unaffected 90. by a i r turbulence. Thus, turbulent mechanisms such as that suggested by P h i l l i p s (1966, §4.3) are probably not important wave-generation processes. The system (6.3.7) and (6.3.10) forms the f a m i l i a r second order Rayleigh system ( c f . equation (5.2.13)) and i t has a unique s o l u t i o n . From the discussion i n §5.2, i t i s clear that both v and the gradient of ¥ are singular at a c r i t i c a l height r = r ^ where U(r ) = 1 . The s i n g u l a r i t y a r i s e s because the mean h o r i -c zontal advection of v o r t i c i t y i s unable to balance the v e r t i c a l advection of v o r t i c i t y . This suggests that near the c r i t i c a l height, where the gradients of ¥ must become large, the neglected stress f l u c t u a t i o n terms may become comparable with the mean advection terms. Thus, we introduce the stress f l u c t u a t i o n s near the c r i t i c a l height only to eliminate the singular behaviour of the "outer" s o l u t i o n ( i . e . the s o l u t i o n of (6.3.10)). This i s completely analogous to the usual procedure i n s t a b i l i t y theory where l i n e a r viscous stresses are used to smooth the s o l u t i o n near the c r i t i c a l height. Using the method of Frobenius, the s o l u t i o n of (6.3.10) may be expanded i n a power se r i e s about the c r i t i c a l height (Tollmien 1929), such that Y = B . ^ - C r ) + ¥ ¥ . ( r ) (6.3.11) 1 1 c I 2 (r - r ) + 0(r - r £ ) , l + 0 ( r - r ) 2 + (U"/U')[(r - r ) + 0(r - r ) 2]£n(r - r c c c c c where 91. and v = ¥ (V'/U')(r - r ) 1 + 0(1) + (V'U"/U f 2)[^ + 0(r - r )j£n(r-r ). c c c c c c c c c c (6.3.12) Here, the subscript c denotes that a function i s evaluated at r = r The constant r a t i o ^ i ^ c m a y ^ e determined by applying the upper boundary condition (6.3.7a). We now seek a s o l u t i o n of the f u l l system (6.3.6) which i s regular near the c r i t i c a l height and which can be matched i n some manner to the outer s o l u t i o n (6.3.11) - (6.3.12). "inner" s o l u t i o n ) , we adopt the usual approach of singular perturb-a t i o n theory (e.g. see Cole 1968, chpt. 1) of introducing a new stretched independent v a r i a b l e t,, such that 'where i s a constant to be determined. I f i s greater than zero, then the new v a r i a b l e increases the importance of the gradients of functions because 6.4 Solution Near the C r i t i c a l Height To f i n d the required s o l u t i o n of (6.3.6) near r • r (the X, = e (r - r £ ) (6.4.1) _d_ dr - v 1 J_ dC = e Expanding the mean v e l o c i t y components i n Taylor s e r i e s , i t i s seen that U - 1 = ") (d^J/Dr 1 1 1) (r - r )m/m! i c c m=l 92. mv = I (d^J/dr1") e \m/ml m=l ° (6.4.2) and 0 3 mv. V = J (d^/dr 1") e V / m ! m=o We also introduce new dependent variables: v = E h(.c) , S. . = e f . .CO 13 13 (6.4.3) where and are constants to be determined; they allow the r e l a t i v e importance of each function to vary. Putting C6.4.2) - C6.4.3) into (6.3.6), the equations of motion become v-, - 2 v i 2-v ,+Vo (e \v +...)Ce - *) " (U" +...)$ + 1 ^ ( f ^ - f „ ) ' c c 11 33 2-2v.,+v 3 2 + v 3 - i e f » 3 - ie f 1 3 = 0 , (6.4.4a) (ie 5U'+...+ 3A + X j C f n - f-,) + 2(U' +...)f c p D 11 33 c 1J -v - 2 v -v -v -v + 2 e R i 3 $ " " 2 e R!3 4 ) + i 2 e ( R n + R 3 3 ) $ ' = 0 , (4.4.4b) 93. v _ 2 v r v 3 (ie CU' +.,.+ 3A + A_)f.„ + (U' +...)f„. + e R 0 o $ " c p D 13 c 33 33 " V3 + e R 1 ; L * = 0 , (6.4.4c) (ie XCU' +...+ 2A + L ) f - A (f + f ) + 2e \ $ c p • D 33 p 11 22 13 -v -v - i2e R 3 3 $ ' = 0 » (6.4.4d) V l ( i e CU' +...+ A + A ) ( f + f ) - 2A f + 2(U' +...)f,, c p D i i 11 p 33 c 13 " V V 3 " 2 V V 3 + i2e R n $ ' + 2 e R13 $" = 0 » (6.4.4e) v 1 v„ 2+v„ CU' +...)e h - ( V +...)*+ e f._ c c 12 2-v +v - i e °f 2 f 3 = 0 , (6.4.4f) v - V 1 + V 2 " V 3 (ie ?U' +...+ 3A + A _ ) f 0 , + (V' +...)£„„ + e R 0.h' c p D 23 c 33 33 v -v -v " V v 3 + i e R 1 3 h + e J R 1 2 * - i e R 2 3 $ ' =0 ' (6.4.4g) C l e ^ V +...+ 3A p + A D ) f 1 2 + (Uj + . . . ) f 2 3 + O r + . . . ) f 1 3 + ie 11^ + e R 1 3 h ' + ± E R 1 2 $ ' -2v -v + e R 2 3 $ " = 0 ' C6.4.4h) The exponents v^, v 2 and v 3 are found from (6.4.4) by matching the terms of maximum order of magnitude i n each equation. Thus, we f i n d 94. -v., = 2 - 2v^ + _< 0 , 0 = -2v1 - v 3 < -v, + v 2 - v 3 v^+v 2 = 2 + v 3 - <_ 0 , where the i n e q u a l i t i e s ensure that the terms chosen are indeed those of maximum order magnitude. These conditions are s a t i s f i e d when v = 2/3 , v 2 = -2/3 and v 3 = -4/3 (6.4.5) Equations (6.4.1) and (6.4.5) imply that there i s a " c r i t i c a l l a y e r " 2/3 of order e i n thickness around the c r i t i c a l height r ; therefore, 2/3 when r < e th i s layer extends to the a i r - s e a i n t e r f a c e . I t i s c ~ 2/3 2 also seen that the stress f l u c t u a t i o n i s of order e kA(c - U ) n o 2 2 i n the c r i t i c a l layer (cf. e k A ( c n - U q ) away from r ) , and -2/3 that the h o r i z o n t a l v e l o c i t y f l u c t u a t i o n s are of order e kA(c - U ) n o (cf. kA(c - u ) away from r ) . n o c We now expand the dependent v a r i a b l e s i n asymptotic power 2/3 ser i e s i n e , i . e . _ r- 2m/3 (m) $ = I e $ , m=o f.. = y £ 2 m / 3 f C m ) , 1 J m=o 1 J h = I E h » ; m=o (6.4.6) su b s t i t u t e (6.4.6) into (6.4.4) and equate the c o e f f i c i e n t s of l i k e 95. powers of e to zero. Thus, the zeroth order equations are J C U c 4 » ( o ) " - i f ^ " = 0 , (6.4.7a) (A + 3A ) ( f 1 1 ( ° ) - f[°h + 2 U f f 1 ( ° ) + 2 R $ ( 0 ) " = 0, (6.4.7b) D p 11 55 c I J 13 (A + 3A )f 1 (° ) + U ' f , J o ) + R „ $ ( o ) " = 0 , (6.4.7c) D p I J C 55 55 ( X + 2\)&J - A ( f ^ + f = 0 , (6.4.7d) D p 55 p 11 22 .<*Dtv«iio) + ^ 0 > > - . 2 V 3 5 0 ) + ^£S> + 2 R 1 3 $ ( o ) " = 0 , (6.4.7e) and OT'h ( o ) - V ' $ ( o ) - i f ^ ' = 0 , (6.4.8a) c c 25 (6.4.8b) ttn + 3A )f<°> + u'f<°> + V ' f ^ + R 1,h ( o )'+ R „ $ ( 0 ) " - 0 D p 12 c 23 c 13 13 23 (6.4.8c) From the system (6.4.7), i t i s r e a d i l y found that s a t i s f i e s the equation > ( o ) l v - i R " 1 5 * C o ) " = 0 , (6.4.9) c where R c = V/^'c 1 S t n e Reynolds number,'and 96. V " I AD ( XD + 3 V R 3 3 - 2 X p U c R 1 3 ] / I X D ( X D +  3 X / " ^ c ^ i s the normalised "turbulent d i f f u s i v i t y " . Assuming that A and X are of order U' and that the mean v e l o c i t y i s logarithmic, p c 2 then the Reynolds number i s seen to be of order (r /e) . If the determinant of the algebraic equations (6.4.7b) - (6.4.7e) for the 2 2 stresses i s negative, then X (X„ + 3X ) i s less than 2X U' • D D p p c and the turbulent d i f f u s i v i t y V i s also negative. However, we have shown (see discussion of (6.2.13)) that the d i s s i p a t i o n rate constant X^ i s v i t a l to the present problem. On the other hand, the isotropy rate constant X could be neglected on mathematical p ( i f not physical) grounds. Therefore, although X and X are ^ P probably of the same order of magnitude, we s h a l l assume that X D dominates, ensuring that V i s p o s i t i v e . Two independent solutions of (6.4.9) for < J > ( O ) " are F. = C 1 / 2 H [ / 3 ( ( 2 / 3 ) ( - i / R c ) 1 / 2 ? 3 / 2 ) , i = 1,2; (6.4.10) where a r e t ^ e H a n k e x functions of order 1/3 (see Abramowitz and Stegun 1965, chpt. 9). We note that the asymptotic forms of H, as | z I ->• M are 1/ J H $ ( z ) - ( 2 / T T Z ) 1 / 2 [ 1 + 0 ( z - 1 ) ] e i ( z - 5 i r / 1 2 ) ( v a l i d f o r —rr < arg z < 2TT ) , H<2>(z) , ( 2 / K ) 1 / 2 [ l + 0 ( Z - 1 ) ] e - i ( z - 5 i r / 1 2 ) ( v a l i d f o r -2TT < arg z < TT ) (6.4.11) I t has been shown h e u r i s t i c a l l y by L i n (1945), and s t r i c t l y by Wasow (1948), that arg and arg .1, must be chosen such that the same asymptotic expansion i s v a l i d f o r both . £ greater and £ les s than zero. Thus, from (6.4.11), we require for a l l r e a l £ that - T T < arg(.(-i/R c) 1 / 2£ 3 / 2) < TT . This condition i s s a t i s f i e d i f arg(-i/R ) = 3TT/2 and arg £ = 0 , f o r £ > 0 . —TT , for £ < 0 . (6.4.12) In a l l that follows, R denotes R c 1 c 1 C l e a r l y , neither F^ nor F^ i s bounded away from the c r i t i c a l height ( i . e . F^ i s unbounded as £->-«>, while F^ i s unbounded as £ -> . On the other hand, we expect the "inner" s o l u t i o n , which i s important only i n the c r i t i c a l l a y e r , to be bounded as t, -> +«> . Also, the leading term of the outer s o l u t i o n (6.3.11) as r r i s a constant f . Therefore, the c c required s o l u t i o n of the system (6.4.7) i s Co) : (0) _ AO) "II f^~/ = Ao) = f ( o ) 22 33 13 = 0 (6.4.13) 98. Using equations (6.4.13), we now consider the f i r s t order equations associated with the f i r s t f i v e equations of the system (6.4.4); v i z . SU'$ ( 1 ) - If^ = U"1< , c 13 c c (6.4.14a) (X n + 3X ) ( f - I 1 ( 1 ) - f + 2U'f 1 ( 1: ) + 2 R 1 0 ® ( 1 ) " = 0 , (6.4.14b) D p 11 j j c 13 13' ( AD + 3 V f 1 3 1 ) + U c f 3 3 1 ) + R 3 3 $ ( 1 ) " = ° ' (6.4.14c) • ^ D + V ^ " A p ( f l i } + f 2 2 } ) - ° > (6.4.14d) 2X f ^ } + 2 U ' f ^ ) + 2 R 1 0 $ ( 1 ) " = 0 p 33 c 13 13 • (6.4.14e) From (6.4.14), the f i r s t order s t r e s s f l u c t u a t i o n s are given by '13 -(1) "11 = (.1) "22 2(A D + X p ) L 2 $ ( 1 ) " , CD _ ? , T2,(l)„ f33 ~ 2 X p L * >J (6.4.15) where L 2 = IU'R,_ - (X n + 3X )R ,J/IX (X + 3X ) 2 - 2X U' 2] c 33 D p 13 D D p p c and the turbulent d i f f u s i v i t y V i s defined i n (6.4.9). The parameter L may be considered as a dimensionless turbulent mixing length. As suggested i n §6.2, the turbulence i n the c r i t i c a l layer indeed behaves 99. i n a "viscous" manner; there i s a shear stress which i s d i r e c t l y p r o portional to the f l u c t u a t i n g shear s t r a i n rate. The f l u c t u a t -ing k i n e t i c energy -Tr f f'P also v a r i e s as $ " . I t has con-6 ; 2 u t r i b u t i o n s from the mean, shear stress working against <I>^" and from the f l u c t u a t i n g shear stress working against the mean s t r a i n rate . The normal st r e s s components ^22^ a n < ^ f ^ ^ e x i s t only because of the "isotropy coupling" and they are therefore proportional to A p . Putting (6.4.15a) into (6.4.14a), the f i r s t order streamfunction i s found to s a t i s f y the equation i t ? $ ( 1 ) l v + U = U" ( o ) , (6.4.16) c c where = V . This equation governs the v o r t i c i t y balance i n the c r i t i c a l l a y e r : the v e r t i c a l advection of v o r t i c i t y ( r i g h t hand side of (6.4.16)) must be balanced by turbulent d i f f u s i o n and h o r i -zontal advection of v o r t i c i t y ( l e f t hand side of (6.4.16)). Near the c r i t i c a l height ( i . e . as £ -> 0)', the turbulent d i f f u s i o n alone supports the v e r t i c a l v o r t i c i t y advection. But for large £ , the l a t t e r i s balanced by the mean h o r i z o n t a l advection of v o r t i c i t y , i n agreement with equation (6.3.10a). From (6.4.9) and (6.4.10), and applying the method of v a r i a t i o n of parameters (e.g. Ince 1956, §5.23), the s o l u t i o n of (6.4.16) may be wr i t t e n as > ( 1 )"U) = - ( ^ c U ^ ' / 6 P ) [ F 1 ( O l 2 ( ? ) - F 2 ( C ) I 1 ( ? ) ] , (6.4.17) where F_^  i s defined by (6.4.10) and where 100. % r+a> f o r i = 1 . I = / F (z)dz , L =1 . L L » f o r i = 2 . The l i m i t s of in t e g r a t i o n have been chosen such that i s bounded as c, -> + °°. The f i r s t two terms of the inner s o l u t i o n f o r M' ( v i z . and $ ) , together with the outer s o l u t i o n , enable us to obtain a s o l u t i o n for 'f which i s uniformly v a l i d over the whole s o l u t i o n domain 0 < r < °° . Eliminating ^23^ f r o m (6.4.8) and using (6.4.3), h^°^ i s found to s a t i s f y the equation IV h ( o ) " + U ' c h ( o ) = V'V , (6.4.18) V c c c where V = R„~/(X„ + 3A ) i s the normalised transverse turbulent d i f -v 33 D p f u s i v i t y . This equation governs the balance of the X2~component of momentum i n the c r i t i c a l l a y e r : the v e r t i c a l advection of momentum (rig h t hand side of (6.4.18)) must be balanced by turbulent d i f f u s i o n and h o r i z o n t a l advection ( l e f t hand side of (6.4.18)). Comparing (6.4.18) with (6.4.16) and using (6.4.17), we see that h ( O ) ( 0 = - ( T T ¥ C V 7 6 P v ) [ F v 1 ( C ) I v 2 ( C ) - F V 2 ( ? ) I v l U ) ] , (6.4.19) i where F . and I . are equal to F. and. I . , r e s p e c t i v e l y , but v i vx 1 x with R = V / T J ' replacing R . From (6.4.8) we also f i n d that vc v c c 101. (6.4.20) [ U ' / a + 3A ) - R-./R-JP h c D p 13 33 v (o) Thus, at the c r i t i c a l height, zeroth order stresses are. required only when the mean wind and the wave do not t r a v e l i n the same d i r e c t i o n . Equations (6.4.19) - (6.4.20) describe the leading terms i n the expansions for the v e l o c i t y and stresses produced by the non-alignment of wind and wave d i r e c t i o n s . Since these terms can be used to eliminate the major s i n g u l a r i t y i n the outer s o l u t i o n for the trans-verse v e l o c i t y ( v i z . the ( r - r ) behaviour given by (6.3.12)) and, consequently, to describe the general behaviour of th i s v e l o c i t y com-ponent near the c r i t i c a l height, we s h a l l not c a l c u l a t e the higher order terms of the expansion. In order to match the inner solutions f o r ¥ and v to t h e i r respective outer s o l u t i o n s , i t i s necessary to know the asymp-t o t i c behaviour of the inner solutions as From (6.4.10) -(6.4.12), i t may be shown that as £->+«>, F 1 ( 0 * ( 3 / 7 r ) 1 / 2 a / R C ) 1 / 4 [ 1 + 0(C 3 / 2 ) J e x p I ( 2 / 3 ) R ; -1/2 3/2 15TT/4 £ e c - 1/97T/24] (6.4.21a) F„(C) ^ ( 3 / T T ) 1 / 2 U / R ) 1 / 4 U + 0 U 3 / 2 ) ] e x p [ ( 2 / 3 ) R " I c I -1/2 3/2 ITT/4 c C 6 + iTr/24] (6.4.21b) and that as £ •> -°° , 102. F]_CO * (3/Tr ) 1 / 2 ( | c|/R c ) " ± / 4 U + 0(? J / Z ) . ] e x p [ ( 2 / 3 ) R C ± / Z U x e - i i r / 4 _ i l 3 i r / 2 4 ] } ,-1/4, -3/2, -l/2, ? i3/2 (6.4.22a) F 2 ( 0 - ^ ( 3 / T T ) ± / Z ( U | / R c ) ± / 4 H + 0(? 3 / 2 ) ] e x p [ ( 2 / 3 ) R / / Z | ? -i5u/4 -1/2|^|3/2 c x e + 17TT/24] . (6.4.22b) To f i n d the asymptotic behaviour of the i n t e g r a l s 1^ and , we rewrite them using (6.4.10), (6.4.12) and (6.4.17), as ( 2 / 3 ) R _ 1 ^ 2 r 3 / / 2 f c ^ „1/2„(1), 13TT/4), . . R H- ,1(re dr for ? > 0 , ^ c 1/ 3 M c ) -v J c 1/3 J c oo o x H ^ ( r e ± 3 7 T / 4 ) d r f o r ? < 0 , (6.4.23a) I 2 ( 0 = 4 1/2 -3TT/2„(2), - i 3 T r / 4 - H 1 / 3 ( r e ( 2 / 3 ) R / / 2 c 3 / 2 1 / 9 )dr + / C R c 1 / 2 o x H J 2 ^ ( r e l 3 T r / 4 ) d r f o r ? > 0 (?/SIR--'-/2 J 13/2 I C R 1 / 2 e - i 3 T r / 2 H { 2 > ( r e - i 3 ^ 4 ) d r for C < 0 (6.4.23b) Thus, from (6.4.11) and (6.4.23), we f i n d that as £->•+«>, 103. 1 , ( 0 - ( 3 / , ) 1 / 2 a / R c ) - 3 / 4 [ l + 0 ( r 3 / 2)]exp[(2/3)R-: 1 / 2C 3 / 2e i 5 7 T / 4 -i49Tr/24] , (6.4.24a) X2(0 - ( 3 / T r ) 1 / 2 a / R c ) - 3 / 4 I l + 0 ( r r 3 / 2 ) J e x p [ ( 2 / 3 ) R ; 1 / 2 -15TT/24J , C6.4.24b) and that as £ -°° , I, CO * ( 3 / T r ) 1 / 2 ( | ? | / R ) - 3 / 4 n + 0 ( r 3 / 2 ) ] e x p ( 2 / 3 ) R - 1 / 2 | c | 3 / 2 e - i 7 T / 4 •131TT/24] , (6.4.25a) 1 , ( 0 * (3/T0 1 / 2 (U|/R ) _ 3 / 4 U + 0 ( r 3 / 2 ) ] e x p[(2/3)R" 1 / 2| ?! 3 / 2 x e - i 5 i r / 4 + i l 3 7 T / 2 4 - j _ (6.4.25b) Therefore, equations (6.4.21) - (6.4.25) imply that as £ -> +«> F 1 ( 0 I 2 ( 0 - F 2 ( 0 I 1 ( 0 ^ -(6/TT)(C/RJ 1 + 0(5 4 ) + T.S.T. (6.4.26) where T.S.T. denotes transcendentally small terms. H o l s t e i n (1950) -5/2 shows that the apparent leading term of order £ i n the error of (6.4.26) cancels out, leaving an actu a l leading term of order £ 4 . Using the fact that H i / 3 ( z ) = H l / 3 * ( z * ) ' (6.4.27) 104. where ( )* denotes the complex conjugate of a quantity, we can derive another u s e f u l property of and I . Putting (6.4.27) into (6.4.10) and (6.4.23), i t i s r e a d i l y seen that IF. I- - F 0 I J = -IF, I - F „ I J * ; (6.4.28) 1 1 2 2 lJ<:>o 1 2 2 1 c<o ' that i s , from (6.4.17) and (6.4.28), Re i s an'odd function of £ , (1) and Im $ " i s an even function of t, . Putting (6.4.26) into (6.4.17), we f i n d that <^iy(.0 CUJJ/U/) QS /O + 0(£ 4 ) as £ + +» . (6.4.29) We note that t h i s r e s u l t was an t i c i p a t e d from inspection of the equation of motion (6.4.16); i . e . away from the c r i t i c a l height the mean h o r i z o n t a l advection of v o r t i c i t y should be j u s t balanced by the v e r t i c a l advection of mean v o r t i c i t y . Now the f i r s t order stream-function may be wri t t e n as * c i )fe) = / a - z ) $ c l ) " ( Z ) d z + c?-^)* 0 0 'c^) + *(1).c?1) , (6.4.30) where £^ i s an a r b i t r a r y point on the r e a l l i n e . I f t,^ < 0 i s such that the asymptotic representation f o r , v i z . (6.4.29), i s v a l i d f o r £ < r; , then the asymptotic form of i s given by 105. $ \ 0 ^ I*^ J 'Cc,) + V (U"./U')(£n £ - £n ? n - l)Jc 1 C c c 1 + - C ^ ^ ' C ^ ) + £ ^ c ( U W ) j + 0 ( f 2 ) + 0(e~ 2) + T.S.T. (6.4.31) 6.5 Matching of Inner and Outer Solutions We now have two solutions to the system (6.3.6): an outer s o l u t i o n v a l i d away from the c r i t i c a l height r = and an inner s o l u t i o n v a l i d i n the neighbourhood of r ^ . To r e l a t e , or match, these i n d i v i d u a l s o l u t i o n s , we assume that there i s a region of over-lap i n which both solutions are v a l i d (see Cole 1968, chpt. 1). In p a r t i c u l a r , we equate the solutions at a f i x e d point r = (r - r )/A(e) 2/3 where A -> 0 as e ->- 0 , but where 0(A) > 0(e ) . Thus, the outer and inner solutions are evaluated at the points, respectivel}', r - r = AF and £ = ( A / e 2 / 3 ) r . (6.5.1) c C l e a r l y , i n the l i m i t e -> 0 with r f i x e d , the outer independent v a r i a b l e approaches zero while the inner v a r i a b l e approaches i n f i n i t y . Let us f i r s t consider the transverse v e l o c i t y v . We denote the outer and inner solutions of a function by ( ) and o ( ). , r e s p e c t i v e l y . The outer s o l u t i o n i s , as A 0 by (6.3.12) 106. and (6.5.1), (v) = A 1 ( V 7 U f ) 0 ' IT) + 0(1) , O C V c (6.5.2) where the term of order £n A i s included i n the terms of order unity. From (6.4.3), (6.4.5), (6.4.6), (6.4.19), (6.4.26) and (6.5.1), the asymptotic form of the inner s o l u t i o n as A -> 0 i s We now equate the c o e f f i c i e n t s of l i k e powers of e (and hence A ) i n equations (6.5.2) and (6.5.3). This procedure must be v a l i d f o r the two representations to be true asymptotic expansions of the same function at the same point. Inspection of the terms of order A ^  shows that these are already equal; t h i s occurs because we deduced i n i t i a l l y that the inner expansion for Y has a leading term $^0^ equal to f . Thus, the inner and outer expansions have been formally matched to terms of order A ^ . An asymptotic s o l u t i o n v a l i d over the whole s o l u t i o n domain i s given by where (cp) denotes those terms common to both representations, i . e . the "common part". Putting (6.3.10b) and (6.4.19) into (6.5.4), the composite s o l u t i o n f o r the transverse v e l o c i t y component i s equal to (v) . = A 1(V7U') IT) + 0(1) . X c c c (6.5.3) v = ( y ) i + ( y ) o - (cp) , (6.5.4) 107. v ( r ) = V ' ( r ) H r ) / [ U ( r ) - IJ - 1' (V'/U f)/(r - r ) c ' c c c - £ - 2 / 3 ( ^ C V 7 6 P v ) [ F v i a ) I v 2 C O ~ F v 2 C O l v l < C > ] + 0(1) (6.5.5) From (6.5.5) and (6.4.28), we note that at the c r i t i c a l height i t s e l f , the leading term of the expansion f or v ( r ) i s pure imaginary. 2/3 Therefore, i f tan.6 = V f / l J ' i s not too small compared with e , c c then there w i l l be a s i g n i f i c a n t transverse shear stress <u2u^> produced at the c r i t i c a l height and caused by the non-alignment of the wind and wave d i r e c t i o n s . Putting (6.5.1) into (6.3.11), the outer expansion f o r the streamfunction ¥ i s as A 0 , CO •= V + Art* (U'7U r)£n(Ar) + B j + 0(A 2) , (6.5.6) o c c c c I where terms of order A™ £n A are included with terms of order A™ , (This arrangement i s v a l i d since 0(A m +^) < 0(A m £n A) < 0(A m ''").) S i m i l a r l y , the asymptotic expansion of the inner s o l u t i o n as A -> 0 i s , from equations (6.4.3), (6.4.6), (6.4.13) and (6.4.31), (*F) . = Y + Ar[¥ (U'7U')£n(Ar) + '(?.,) - V (U'7U') x c c c c I c c c x {1 + £ n ( £ 2 / V ) } ] + e 2 / 3 [ $ ( 1 ) ( £ 1 ) - '(C-,) + C, Y (U»/U')] 1 1 1 l l c c c + 0(e /A ) + 0(e ' \ ± ) , (6.5.7) 108. where £ = -a for r > 0 ; b for r < 0 ; a and b are p o s i t i v e constants. As before, we now match the c o e f f i c i e n t s of l i k e powers of A and e i n equations (6.5.6) and (6.5.7). Because we deduced previously that 0^°^ = y , the f i r s t two terms are already matched. Matching 2/3 terms of order A and of order e , r e s p e c t i v e l y , i t i s seen that $ w '(C) - V (U"/U ' ) U + a n(e 2 / 35,)] , c c c 1 1 v " l o = f ^ a j - s ^ ^ ' U J + ? i \ C u 7 u p . J (6.5.8) Equations (6.5.8) together with (6.4.30) provide s i x equations f or the s i x unknowns £^ , <3>^  (t,^) and $^^'(<; ) where £ = a , -b ; v i z . B. = $ ( 1 ) ' ( a ) - V (U"/U')I 1 + £ n ( e 2 / 3 a ) ] , (6.5.9a) 1 c c c B. = $ ( 1 ) ' ( - b ) (U"/U')[l + to(-e2/3b)J , (6.5.9b) 1 c c c $ ( 1 ) ( a ) = a l $ ( 1 ) ' ( a ) - \ ( U W ) ] , 0 ( 1 ) ( - b ) = -bI<J>(1) '(-b) - y j u W ) ] , (6.5.9c) (6.5.9d) CD (-b) = -/ ( b + z ) $ ( 1 ) " ( z ) d z - (a+b)$ ( 1 )'(a) + $ ( 1 ) ( a ) , '(-b) = / $ u ; r r ( z ) d z + -b ( I ) * , (a) , (6.5.9e) (6.5.9f) 109. where i s given e x p l i c i t l y by (6.4.17). We note that these 2/3 equations need be solved only to 0(e ) , since any residue can be accounted f or i n the next highest order terms. Adding b times (6.5.9f) to (6'.5.9e), we f i n d > ( 1 )(a) - $ ( 1 ) ( - b ) = a $ ( 1 ) , ( a ) + b $ ( 1 ) ' ( - b ) + / dz z $ ( 1 ) " ( z ) . a (6.5.10) Subtracting (6.5.9d) from (6.5.9c) also gives » ( 1 )(a) - $ ( 1 ) ( - b ) = a $ ( 1 ) , ( a ) + b $ ( 1 ) ' ( - b ) - (a+b) (Ur,/U') ¥ . c c c (6.5.11) Subtracting (6.5.10) from (6.5.11), i t i s seen that -1 ~ b m a+b = (II' /U") f dz z3>v ; " ( z ) . (6.5.12) c c c J a Now since a and -b l i e on the r e a l l i n e , the l e f t hand side of (6.5.12) i s r e a l ; consequently, we must have that Im / dz z $ ( 1 ) " ( z ) = 0 . (6.5.13) But t h i s implies that a = b from (6.4.28), because Im <j>^ "^ " i s an even function. Thus (6.5.12) becomes a = b = ¥ 1(U'/U") J dz z Re 5> ( 1 )"(z) . (6.5.14) c c c o 110. 2/3 From (6.4.29), we see that t h i s condition i s s a t i s f i e d to 0(c ) at -2/3 l e a s t , provided that a = b = 0(e ) . Elimi n a t i n g and I * ( 1 ) r ( a ) - $ ( 1 ) ' ( - b ) ] from (6.5.9a), (6.5.9b) and (6.5.9f), we f i n d that £n a - £n(-b) = -(U'/U"H' ) / $ ( 1 ) " ( z ) d z . c c c J From (6.4.12), (6.4.28) and (6.5\14), t h i s condition reduces to TT/2 = (ir/U£¥ ) / Im $ ( 1 ) " ( z ) d z (6.5.15) 2/3 As before, (6.5.15) i s s a t i s f i e d to 0(e ) provided that -2/3 a = 0(e ) . Putting (6.5.14) into (6.5.9), i t i s seen that > ( 1 )'(-a) = $ ( 1 ) ' ( a ) -iirY (U"/U') c c c > ( 1 ) (-a) = - f ( 1 ) ( a ) + i T r a ^ (U"/U') . c c c 1 (6.5.16) -2/3 Since a = 0(e ) , equations (6.5.16) e s s e n t i a l l y r e l a t e the asymp-t o t i c values of $ ^ and <3>^ ' as £ -> +<» . Hence, equations (1) C6.5.16) allow $ " , as given by (6.4.17), to be integrated uniquely. From equations (6.5.9a) and (6.5.9c), i t i s f i n a l l y found that $ a ) ( a ) = ajB, + ¥ (U"/U')to(e 2 / 3a) , 1 c c c > C 1 ) '(a) = B 1 + Y CGJ£/U|)I1 + £ n ( £ 2 / 3 a ) J , (6.5.17) 111. -2/3 where a i s a p o s i t i v e number of order e at l e a s t . Thus, the inner expansion i s given by + E 2 / 3 Q [ B ^ + ¥ (U"/U f)£n( E 2 / 3a)] + e 2 / \ (l-a)(U"/U') 1 , 0 l c c c C C C + e 2 / 3 J ? ( c-z)$ ( 1 )"(z)dz + 0 ( e 4 / 3 ) . (6.5.13) As £; -> + °° , the inner expansion c l e a r l y behaves as (V). ^ V + e 2 / 3 ? [ B . + ¥ (U"/U')£n(e 2 / 3<;)] + 0 ( e 4 / 3 ) . (6.5.19) 1 c l c c c A composite asymptotic s o l u t i o n f o r ¥ uniformly v a l i d over the whole s o l u t i o n domain i s V = O K + 00 - (cp) , (6.5.20) where (cp) i s the part common to both expansions; i . e . (cp) = V + e 2 / 3B..? + e 2 / 3 Y c(U"/U'Hn(E 2 / 3 s ) + 0 ( e 4 / 3 ) , c 1 c c c = T + ( r _ r ) B + ( r - r )Y (U"/U rHn(r - r ) + 0(r - r ) 2 c c l c c c c c c The f i r s t expansion for (cp) i s u s e f u l when considering H' at a point f a r from the c r i t i c a l height, while the second expansion i s use-f u l near the c r i t i c a l height. C l e a r l y , the expansions are equivalent: the f i r s t i s expressed in; the inner v a r i a b l e £^ while the second i s expressed i n the outer v a r i a b l e r - r . 112. We stated i n §6.3 that the r a t i o B./T may be determined 1 c by applying the upper boundary condition (6.3.7a) to the outer s o l u t i o n (6.3.11); that i s ¥_ + (B./Y )V. -»• 0 as r -> » . (6.5.21) 2 1 c' 1 Since solutions ^ and ¥ a r e r e a l f o r r > , t h i s c o n d i t i o n implies that B^/V must also be r e a l . To obtain a unique s o l u t i o n for ¥ , i t remains to f i n d the value of either or B 1 by using the lower boundary condition (6.3.7b), v i z . 0 0 + (v) - (cp) = 1 at r = 0- . (6.5.22) 2/3 Two cases may be considered when s a t i s f y i n g (6.5.22): ( i ) » E 2/3 ' 2/3 and ( i i ) r < e . The r a t i o r-/e i s that of the r e l a t i v e c ~ c error i n the two expansions f o r (cp) at the boundary r = 0 . Therefore, i n case ( i ) , the error i n the second expression i s much greater than that i n the f i r s t ; while i n case ( i i ) , (cp) i s better represented by the second expression. Thus, from (6.5.19) and (6.5.20), the boundary condition (6.5.22) i n case ( i ) becomes (¥) + 0 ( e 4 / 3 ) = 1 at r = 0 That i s , when the c r i t i c a l height i s f a r above the a i r - s e a i n t e r f a c e , the value of f/ at the i n t e r f a c e i s simply that given by the outer 2/3 s o l u t i o n ( i . e . the s o l u t i o n of (6.3.10)). Hence the case >> e 113. corresponds with the Miles (1957) s o l u t i o n which i s discussed i n §5. On the other hand, the boundary condition (6.5.22) i n case ( i i ) becomes ¥ - e 2 / 3 ? [B, + ¥ (U"/U').ta(e 2 / 3a)] - e 2 / 3 V + a)(U"/U') c o l c c c C O c c - e 2 / 3 / C ° (£ + z)§ ( 1 )"(z)dz + 0 ( e 4 / 3 ) + 0 ( r 2 ) = 1 , J o c a -2/3 where £ = e r < 1 . That i s , o c ~ ¥ = 1 + e 2 / 3 ? [(B./¥ ) + ( U " / U f ) £ n ( e 2 / 3 a ) ] c o I c c c _ r + e 2 / 3 U + a ) ( U " / U ' ) + E 2 / 3 Y _ 1 / *° (c + z ) $ ( 1 ) " ( z ) d 2 o c c c J o a + 0 ( e 4 / 3 ) + 0 ( r 2 ) . (6.5.23) c 2/3 Thus, when r ^ i s of order e the c r i t i c a l layer extends to the a i r - s e a i n t e r f a c e , and hence the turbulent flow around the c r i t i c a l height must influence tire lower boundary condition. From equations (6.3.6), (6.4.3) - (6.4.6), (6.4.13), (6.4.15) and (6.4.20), the inner and outer expansions f o r the stress f l u c t u a t i o n s may be matched i n a s i m i l a r manner. Then i t i s f i n a l l y found that S n = 2 z 2 / 3 C AD + y L 2 * ( 1 ) " c o + O(D , S „ = 2e~' 2 / 3A L 2$ C 1 )"a) + 0(1) , 11 P S_, = 2e" 2 / 3A L 2 * A ) " ( 0 + 0(1) , (6.5.24) 33 p 114. -e~ 2 / 3 p$ ( 1 ) "C C ) +0(1) , - e - 4 / 3 P h ( o ) ' ( C ) + o( £'- 2 / 3) , e " 4 / 3 D r / ( X D + 3X ) - R 1 3/R 33]P V h ( o ) ' ( C ) + 0 ( e " 2 / 3 ) . Here, the common part has been expanded in, the outer v a r i a b l e r - r c and hence i t i s i n the terms 0(1) . The asymptotic behaviour of and h ( o 5 (see (6.4.19), (6.4.26) and (6.4.29)) i s such that the leading terms i n equations (6.5.24) approach zero as £-*-+«>, i . e . the turbulent stresses are important near the c r i t i c a l height only. S13 = S23 =  S12 = 6.6 Normal'Stress at the Air-Sea Interface We now consider the normal stress at the i n t e r f a c e caused by the wave-induced motion when the c r i t i c a l layer extends to the i n t e r -2/3 face, i . e . when r c < 8 • Putting the appropriate form of (6.5.20) into (6.3.8), i t i s found that IQ + £ 2 S 3 3 J r = 0 = J (U - l ) C V ) 0 d z + 0 ( r 4 ) r c + e 4 / 3 / ds[£U' + 0 ( e 2 / 3 ) ] l V + 0 ( £ 2 / 3 ) ] + 0(e 2) _ 5 o That i s , IQ + e 2 S „ ] = / (U - 1 ) W dz - \ e 4 / \ V' r2 + 0 ( £ 2 ) + 0 ( r 2 ) 33 r=o o I c c o c r c (6.6.1) 115. 4/3 Hence, the normal stress at the i n t e r f a c e i s to 0(e ) equal to 4/3 that at the c r i t i c a l height. The c o r r e c t i o n of order e i s required to provide a c e n t r i p e d a l force because of the curvature of the streamlines near the i n t e r f a c e . Now, equation (6.3.10a) may be w r i t t e n as (U - 1) (?) = (U - 1) CO" - U"CF) > o o o = [(U - ! ) ( ! ) ; - U'(¥) o] ' Therefore, 00 j (U - DC*) dz = U ^ c ; (6.6.2) r ' c i . e . the pressure at the c r i t i c a l height i s proportional to the amplitude of the vortex force at that height. Putting (6.6.2) and (6.5.23) into (6.6.1), the normal stress at the i n t e r f a c e i s given by N + £ 2 s 3 3 ] r = o = K + e 2 / 3I<VV Uc5o + + a ) U c + £ U" £ n ( £ 2 / 3 a ) + U' f ? ° U + z ) $ ( 1 ) " ( z ) d z + 0 ( e 4 / 3 ) + 0 ( r 2 ) o c c 1 o c a (6.6.3) Thus, the r e a l and imaginary parts of (6.6.3) are Re[Q + e 2 S 3 3 ] r = o = IT + 0 ( £ 2 / 3 ) , (6.6.4) Im[Q + E 2 S „ J = E 2 /V / J°(£ +z)Im "(z)dz' + 0 ( £ 4 / 3 ) 33 r=o c ' o TABLE I Integrals of the function Im &" — + — Im *" / Im <J>"(s)ds o / slm <±>"(s)ds 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 0.8594 0.1777 -0.0704 -0.0348 -0.002.1 -0.0005 -0.0008 -0.0005 0.0000 1.1362 1.6344 1.6482 1.5883 1.5 740 1.5736 1.5727 1.5721 1.5708 0.5325 1.2227 1.2366 1.0300 0.9685 0.9662 0.9600 0.9554 0.9422 ft t From Holstein (1950) tt Obtained by Aitken's 6 -process of extrapolation (see N.P.L. 1961, chpt. 13) from the preceding three elements. TABLE II The function G(£) f o r £ = -8(1)8 00 / Im i " ( s ) d s + CO / slm i " ( s ) d s + C 8.0 -0.0013 -0.0132 -0.0028 7.0 -0.0019 -0.0178 -0.0045 6.0 -0.0028 -0.0240 -0.0075 5.0 -0.0032 -0.0263 -0.0103 4.0 -0.0175 -0.0878 -0.0178 3.0 -0.0774 -0.2944 -0.0622 2.0 -0.0636 -0.2805 -0.1533 1.0 0.4345 0.409 7 -0.0248 0.0 1.5708 0.9422 0.9422 -1.0 2.7071 0.409 7 3.1168 -2.0 3.2052 -0.2805 6.1299 -3.0 3.2190 -0.2944 9.3626 -4.0 3.1591 -0.0878 12.5486 -5.0 3.1448 -0.0263 15.6977 -6.0 3.1444 -0.0240 18.8424 -7.0 3.1435 -0.0178 21.9807 -8.0 3.1429 -0.0132 25.1300 t Calculated from Table I. 116. The l i m i t of i n t e g r a t i o n a i n (6,6.3) has now been taken equal to i n f i n i t y . H o l s t e i n (1950) has tabulated the function '"CO = (TT/6)[F 2CO / F 1 ( z ) d z - F 1 ( 0 / F 1 ( z ) d z ] • (6.6.5) where F.(0 = C 1 / 2 ( (2/3) ? 3 / 2 E I 3 L T / 4 ) , for ? = 0(0.5)8 . C l e a r l y , F and i " are simply r e l a t e d to the functions F^ and ,(1)// . ; i n f a c t , i t may be shown from (6.6.4) - (6.6.5) that Im[Q + A j = - e 2 / 3 R 1 / 3 U " G(-£ /R 1 / 3) , (6.6.6) 33 r=o c c o c where G(0 = -/ (?-S)lm $"(S)dS . Table II shows the function G ( 0 for £ = -8(1)8 ; these values are found by numerical i n t e g r a t i o n of $" using Simpson's r u l e (see Table I ) . F i t t i n g G(0 to a second order polynomial f o r r, i n the i n t e r v a l [-2,0] , we f i n d that G O O = 0.9422 - 1.7553 £ + 0.4193 2 . (6.6.7) Thus, when |£| i s of order unity, the function G(0 i s also of order unity. For large negative values of i t i s seen from (6.5.16) and Table II that G ( 0 ^ - u C (6.6.8) 117. From (6.6.4a) and (6.6.6), the normal stress at the a i r - s e a i n t e r f a c e may be w r i t t e n as [Q + e S 3 3 ] r = ; 0 = W'C + 0 ( e Z / - i ) ] e 0 , (6.6.9) where tan 0 = e 2 / 3 ( - U " / U ' ) R 1 / 3 G(-? /R 1 / 3) + 0 ( e 4 / 3 ) . o c c c o c Thus, the normal stress leads ( i n space) the surface displacement by the angle 0 q . This causes energy to be transferred to the surface at a dimensionless rate given from (6.3.9) to be S = (p /p ) e 2 / 3 ( l - U /c ) 2(-U")R 1 / 3G(-C /R 1 / 3) . (6.6.10) a w o n c c o c Now tan 0 q and S depend upon the Reynolds number which, i n turn, depends upon the turbulent d i f f u s i v i t y V. However, since t h i s dependence i s only upon the 1/3-power of R^ , discrepancies i n the value of R w i l l not a f f e c t the o v e r a l l r e s u l t g r e a t l y . Assuming that X.„ arid A are of the same order of magnitude as U' , the D p c 'value of R i s found from (6.4.4) to be c R c = ( j V c 2 ) 1 ' (6.6.11) where i s a constant (of order u n i t y ) . Thus, equations (6.6.9) -(6.6.11) r e l a t e the energy transfer process to the mean wind. The c r i t i c a l height i s the only p o s i t i o n at which there i s s i g n i f i c a n t i n t e r a c t i o n between the wave-induced and the mean flows. We have also shown that the wave-induced and turbulent motions i n t e r a c t 118. w i t h i n the c r i t i c a l l a y e r . Thus, i f th i s i n t e r a c t i o n i s very strong, then i t may tend to modify the mean v e l o c i t y f i e l d . When the c r i t i c a l 2/3 layer l i e s above the i n t e r f a c e (r > E ) , the o v e r a l l motion i s w e l l described by Miles ' i n v i s c i d laminar theory (see §6.5), and • numerical c a l c u l a t i o n s by Conte and Miles (1959) imply that the shear stress generated at the c r i t i c a l height i s small compared with the mean turbulent shear s t r e s s . In th i s case the wind p r o f i l e close to the i n t e r f a c e most probably i s unaffected by the c r i t i c a l layer i n t e r -a ctions, and i t i s therefore adequately represented by a logarithmic 2/3 v a r i a t i o n (see §2.2). In the present case (r < e ) however, the c r i t i c a l layer includes the i n t e r f a c e and the problem i s no longer represented by the laminar model. The i n t e r a c t i o n between the turbulent and wave-induced motions may therefore a f f e c t the mean flow. The time scale for the wave-induced motion wi t h i n the cat's-eye i s a ^ ; but as the f l u i d moves, i t s rate of s t r a i n i s determined n p r i m a r i l y by the mean s t r a i n r a t e f i e l d ( i . e . dU^/dx^) . Thus, the i n t e r a c t i o n should be most vigorous when the mean s t r a i n rate matches the c h a r a c t e r i s t i c frequency of the wave-induced motion; that i s , when (dU./dxJ _ 7 ^ a , (6.6.12) I 3 x„—z n 3 c where r = kz . Assuming that the mean s t r a i n rate i s s t i l l c c determined p r i m a r i l y by the turbulence f i e l d , then as i n §2.2, dlL/dx 0 = UJK.X0 , (6.6.13) 1 J ± J where = K/COS 6 i s the r a t i o of von Kantian" s constant to the TABLE I I I Va r i a t i o n of 0 with 0 U * / C n ' U*/Cn 0 o tan 0 o s i n 0 o cos 0 o 0.001 84°54' 11.2192 0.9960 0.0889 0.008 81°18' 6.5502 0.9885 0.1513 0.027 .78°36' 5.0218 0.9803 0.1977 0.064 76°48' 4.2785 0.9736 0.2284 0.125 75°24 ! 3.8494 0.9677 0.2521 0.216 74°23* 3.5772 0.9631 0.2692 0.343 73°37* 3. 3948 0.9594 0.2820 0.512 72°59' 3.2685 0.9562 0.2926 0. 729 72°32' 3.1796 0.9539 0.3001 1.0 72°13' 3.1168 0 .9522 0.3054 8.0 71°56' 3.0649 0.9507 0.3101 27.0 72°14' 3.1209 0.9523 0.3051 119. cosine of the. angle between the wind and wave d i r e c t i o n s . Thus, from (6.3.5), (6.6.12) and (6.6.13), we f i n d that U' = HA1 - U /c ) 1 and r = H0)_1(U,/C ) c 2 o n c 1 2 ^ n (6.6.14) where i s a constant (of order u n i t y ) . S i m i l a r l y , the normalised mean curvature at the c r i t i c a l height i s U" = -KJ?(c/u,)(l - U /c ) 1 c 1 2 n I n (6.6.15) Using equations (6.6.9) - (6.6.15), i t i s found that and tan 0^ = x^iuJcJ 1 / 3 G ( - X l ( u v / c n ) 1 / 3 ) , ' S = (p /p )(1 - U /c )H 0 tan 0 , a w o n 2 o (6.6.16) 1/3 -1/3 -1 where x-j_ = ^ . Using equations (6.6.7), (6.6.16a) and Tab l e I I , tan 0 and 0 may be ca l c u l a t e d as functions of v.(u,/c ) ' o o 1 - n (see Table I I I ) . I t i s seen that for X j _ ( u A / c n ) greater than about 0.03 , the pressure leads ( i n space) the surface displacement by approximately 75° ; i . e . 0 q <V 75° , so that the cat's-eye e s s e n t i a l l y l i e s i n the wave trough. We note that, although (6.6.9) states that 2/3 tan 0 i s formally of 0(e ) , equation (6.6.16) y i e l d s large values of tan 0 . This a r i s e s from assumptions (6.6.12) - (6.6.15) which o imply that, while U' ^ 1 , the mean curvature U" % (c /u,) >> 1 ; r • ' c c n ; c i n p a r t i c u l a r , we assume that 120. 2/3 (-U"/U') = e ' J / r = ? c c c -1 e ^ 1 o Now U o i s the mean wind speed "observed" by the i n t e r f a c i a l wave. For an aerodynamically rough boundary, U q i s usually taken to be the wind speed at a c e r t a i n height, proportional to the surface roughness height. In the present case, the surface "roughness" i s the i n t e r f a c i a l wave i t s e l f and hence we assume that U occurs at o a height which v a r i e s d i r e c t l y with the wavelength; i . e . where r ^ i s a dimensionless constant (probably much less than u n i t y ) . We introduce a further assumption that the average value of the s t r a i n rate over the c r i t i c a l layer v a r i e s i n the same manner as the s t r a i n rate of the c r i t i c a l height; i . e . from (6.6.12) and (6.6.17), where i s a constant. I f the mean p r o f i l e were logarithmic, then the value of tire average s t r a i n rate would be much greater than that of the c r i t i c a l height; t h i s suggests that may be much, greater than unity. The assumptions imply that the mean wind and turbulence f i e l d s adjust i n order to produce a strong i n t e r a c t i o n with, the wave-induced f i e l d . Using (6.6.14), equation (6.6.18) may be written as U = U n (r /k) o 1 o (6.6.17) k(c - U ) / ( r - r ) = H-a n o c o 3 n (6.6.18) 1 - V C n = V o ^ V ' n ) ' U > (6.6.19) 121. where x2 = ^K±^2Vo^ ^ ^S m n c ^ S r e a t e r than unity, but where i s of order unity. Putting (6.6.19) into (6.6.16), the f r a c t i o n a l rate of increase i n wave energy per radian i s found to be S = ( P a / p w ) [ x 2 ( u * / c n ) " 1 J x 3 t a n 9 o ' (6.6.20) where X-> = ^„H r • N ° w the "roughness" height z = r /k v a r i e s j I 3 o o o -2 -1 as o while the c r i t i c a l height z = r /k varies as o n c c n Thus, as a decreases, z increases f a s t e r than z increases, n o c This implies that there i s a low frequency cut-o f f f or the generation process at = X 2^( u*/§) where the roughness height becomes as large as the c r i t i c a l height; f o r lower frequencies there i s no e f f e c t -ive c r i t i c a l height and the present generation mechanism cannot occur. The preceding analysis i s v a l i d only for a c e r t a i n range of values of u-'-/cn • Since e has been used as an ordering parameter, 'we require that e « 1 . (6.6.21) Also, the expression (6.5.23) for ¥ i s v a l i d i f r « 1 and r e 2 / 3 < 1 . (6.6.22) c c From (6.6.14) and (6.6.19), the i n e q u a l i t i e s (6.6.21) and (6.6.22) become 122, U * / C n ^ X 3 / ( x 3 X 2 " H 2 ) , 4/3 -1/3 -2/3 - n ~ 1 2 3 2 (6.6.23) Therefore, the condition (6.6.21) places a lower l i m i t on U J . / C N ( i . e . a low frequency c u t - o f f ) , while the condition (6.6.22) that the • c r i t i c a l layer must extend to the a i r - s e a i n t e r f a c e places an upper l i m i t on xxJc^ . Thus, a high frequency cut-of f i s obtained because the c r i t i c a l layer decreases i n thickness f a s t e r than the c r i t i c a l height approaches the sea surface. We have assumed that x^  a n <3 are of order one and that x2 » 1 ; thus, the conditions (6.6.23) may be w r i t t e n as -1 , -2/3 x2 K < U*/Cn <~ x2 (6.6.24) 6.7 Comparison with Observation From equations (6.6.16) and (6.6.20), i t i s seen that S depends upon three independent parameters, v i z . Xi> X 2 a n d * While we expect x^  a n < 1 X 3 t 0 be- of order unity, x2 1 S P r°bably much greater than one. An obvious choice for the parameter x2 1 S -1/2 C^ , where C^ i s the drag c o e f f i c i e n t f o r wind passing over waves. Then the low frequency cut-off i n (6.6.20) corresponds with the wave phase speed j u s t matching the mean wind speed. Now 123. where U,. i s the mean winch speed 5 metres, say, above the mean sea l e v e l . For, at l e a s t , moderate values of U , the drag c o e f f i c i e n t - 3 appears to be approximately constant and equal to 1.2x10 (Miyake et a l (1970)). Thus, taking X, = C " 1 / 2 = 29 , (6.7.1) the condition (6.6.24) becomes i . e . 3.5 x 1 0 - 2 << u./c < 10 1 n ~ 1 << TT/c < 3 5 n ~ (6.7.2) The i n e q u a l i t i e s (6.7.2) define the range of values of c^ over which the r e s u l t s of §6.6 are v a l i d . Because von Karman's constant K i s approximately equal to 0.42, we have that X 1 = 2 . 4 ( H 1 / f f 2 ) 1 / 3 c o s But and should both be of order unity, and hence (HJH^) i s d e f i n i t e l y of order unity. Therefore, 1/3 X l - 2.4 (6.7.3) 124, From (6.7.2) and (6.7.3), the lower l i m i t on x^C-U^/c^) 1 S approx-3 imately equal to 0.48. Now Table I I I shows that f or X - j _ ( u A / c n ) greater than 0.48, the functions tan 0 , s i n 0 and cos 0 are a o • o o approximately constant. Thus, we take tan 0 = 3.2 , o s i n 0 = 0.95 , o cos 0 = 0.3 o (6.7.4) If i t i s further assumed that Y0 = 1/tan 0 = 0.31 A3 o • (6.7.5) then (.6.6.20) and (6.7.1) - (6.7.5) imply that the f r a c t i o n a l increase i n wave energy, per radian i s S = V ' w * < V c n " 1} (6.7.6) Equation (6.7.6) i s the growth law observed by Dobson (1969) The f l u c t u a t i n g pressure at the sea surface, x^ = n , i s p(x,n(x,t),t) = n(x,n,t) - <n(x,n,t)> , n(x,o,t) - <n(x,o,t)> +n ||- - <n|^ -> + .. P ( ^ ° ' t ) + ^ + ( r T i xT 3 - < T 5 ^ > + ' 125. where x = (x » x ) . Thus, the wave-induced pressure at the sea surface i s given by (P/P ) , = (P/P ) - gn + 0(k 2A 2) r a r=kn a r=o From (6.3.4), (6.3.5) and (6.6.9), we f i n d that j iO i k ( x -ct) (P/P ) , = gA[(l - U /c )Ve' ° - l]e a r=kn o n c i . e. 19 ( p ) r = k r i = P &gn Re S , (6.7.7) where R 2 = 1 + (1 - U /c ) V 2 - 2(1 - U /c ) V cos 0 , o n e o n e o tan 0 = -(1 - U /c ) V s i n 0 /II - (1 - U /c ) 2U' cos 0 ] s o n e o o n e o Equation (6.7.7) implies that the pressure f l u c t u a t i o n at the sea surface leads (in space) the wave by the angle 0 g which l i e s i n the second quadrant. Using (6.6.4) and (6.6.19), tan 0 g may be wri t t e n as tan 0 = -x,(x0u.,/c - l ) s i n 0 / [1 - xQ(Xou.-Vc - l ) c o s 0 ] . s 3 I '• n o 3 2 " n o Hence, from (6.7.1), (6.7.4) and (6.7.5), tan 0 = -0.3(U c/c - 1)/U - 0.093(U c/c - 1)] . (6.7.8) s 5 n b n TABLE IV Mean wind condition f o r measurements of Dobson (1969) Run U r (cm.'/sec.) c /Uc a (rad./sec. ) a (rad./sec.) 5 p 5 p 1 1 220 1.685 2.65 4.46 2b 310 0.887 3.57 3.17 2a 320 0.833 3.63 3.02 3 340 0.661 4.35 2.88 4b 700 0.435 3.21 1.40 4a 800 0.332 3.7 1.23 t ' Frequency at peak of wave spectrum, t t Frequency at which c /IL = 1 . Now Dobson (1969) has made: measurements of the pressure at the a i r - s e a i n t e r f a c e by means of a sensor attached to a buoy which i s allowed to move v e r t i c a l l y only. Although eight sets of data are presented, only s i x of them are considered here. In Dobson's run number 5, the pressure s i g n a l i s p a r t i c u l a r l y noisy and hence t h i s set of data i s neglected. Run 6 i s also neglected because the instrument c a l i b r a t i o n i s not known r e l i a b l y i n t h i s case (Dobson, p r i v a t e communication) and the phase angle 0^ va r i e s anomalously. For the other runs, the r a t i o of U,. to the phase speed at the peak of the wave spectrum c i s shown i n Table IV. In Figure 15, the P v a r i a t i o n of the r e l a t i v e phase angle (180 - 0 )° with U r/c r s 5 n as given by (6.7.8) i s compared with that found by Dobson. While there i s a considerable scat t e r i n the data, the observed phase appears to be consistent with the predicted curve. 127. BIBLIOGRAPHY Abramowitz, M. and Stegun. I.A. (1965) "Handbook of Mathematical Functions". Dover. Barnett, T. P. and Wilkerson, J.C. (1967) On the generation of ocean wind waves as i n f e r r e d from airborne radar measurements of f e t c h - l i m i t e d spectra. J . Mar. Res. 25_, 292. Batchelor, G.K. (1953) "The Theory of Homogeneous Turbulence". Cambridge Uni v e r s i t y Press. Benjamin, T.B. (1959) Shearing flow over a wavy boundary. J . F l u i d Mech. 6_, 161. Benjamin, T.B. and F e i r , J.E. (1967) The d i s i n t e g r a t i o n of wave tra i n s on deep water. Part 1. Theory. J . F l u i d Mech. 2_7, 417. Bracewell, R. (1965) "The Fourier Transform and i t s Ap p l i c a t i o n s " . McGraw-Hill. Charnock, H. (1955) Wind-stress on a water surface. Quart. J. Roy. Met. Soc. 81, 639. Cole, J.D. (1968) "Perturbation Methods i n Applied Mathematics". B l a i s d e l l . Conte, S.D. and Miles, J.W. (1959) On the numerical i n t e g r a t i o n of the Orr-Sommerfeld equation. J. Soc. Indust. Appl. Math. _7, 361. Dobson, F.W. (1969) Observation of normal pressure on wind-generated sea waves. Ph.D. Thesis, U n i v e r s i t y of B r i t i s h Columbia. Favre, A., G a v i g l i o , J . and Dumas, R. (1957) Space-time double c o r r e l a t i o n s i n a turbulent boundary layer. J . F l u i d Mech. _3, 313. Groen, P. and Dorrestein, R. (1958) Zeegolven. Kon. Ned. Met. Inst, report no. 11. Helmholtz, H. von (1868) liber d i s c o n t i n u i r l i c h e F l l i s s i g k e i t s -Bewegungen. Akad. Wiss., B e r l i n , Monatsber. 2_3, 215. Hinze, J.O, (1959) "Turbulence". McGraw-Hill. H o l s t e i n , H. (1950) liber die aussere und innere Reibungsschicht b e i Stb'rungen laminar Stromungen. Z. angew. Math. Mech. _30_, 25. Ince, E.L. (1956) "Ordinary D i f f e r e n t i a l Equations". Dover. 128. Lamb, H. (1932) "Hydrodynamics". Dover. L i g h t h i l l , M.J. (1962) "Introduction to Fourier Analysis and Generalised Functions". Cambridge University Press. L i g h t h i l l , M.J. (1962) P h y s i c a l i n t e r p r e t a t i o n of the mathematical theory of wave generation by wind. J. F l u i d Mech. JL4_, 385. Lin, C.C. (1945) On the s t a b i l i t y of two-dimensional p a r a l l e l flows. Parts I, I I , I I I . Quart. Appl. Math. 3, 117, 218, 277. L i n , C.C. (1953) On Taylor's hypothesis and the a c c e l e r a t i o n terms i n the Navier-Stokes equations. Quart. Appl. Math. 10, 295. Lin, C.C. (1955) "The Theory of Hydrodynamic S t a b i l i t y " . Cambridge Un i v e r s i t y Press. Longuet-Higgins, M.S. (1963) The generation of c a p i l l a r y waves by steep gravity waves. J. F l u i d Mech. JL6_, 138. Longuet-Higgins, M.S. (1969a) A nonlinear mechanism for the generation of sea waves. Proc. Roy. Soc. A311, 371. Longuet-Higgins, M.S. (1969b) On wave breaking and the equilibrium spectrum of wind-generated waves. Proc. Roy. Soc. A310, 151. Longuet-Higgins, M.S. and Stewart, R.W. (1960) Changes i n the form of short g r a v i t y waves on long waves and t i d a l currents. J . F l u i d Mech. 8, 565. Miles, J.W. (1957) On the generation of surface waves by shear flows. J . F l u i d Mech. 3, 185. Miles, J.W. (1959) On the generation of surface waves by shear flows. Part 2. J . F l u i d Mech. 6_, 568. Miyake, M. , Donelan, M., McBean, G., Paulson, C , Badgley, F. and L e a v i t t , E. (1970) Comparison of turbulent fluxes over water determined by p r o f i l e and eddy c o r r e l a t i o n techniques. Quart. J. Roy. Met. Soc. 96, 132. National P h y s i c a l Laboratory (1961) "Modern Computing Methods". Her Majesty's Stationery O f f i c e . P h i l l i p s , O.M. (1957) On the generation of waves by turbulent wind. J. F l u i d Mech. 2, 417. P h i l l i p s , O.M. (1958) The equilibrium range i n the spectrum of wind-generated waves. J. F l u i d Mech. h_, 426. P h i l l i p s , O.M. (1960) f i n i t e amplitude. On the dynamics of unsteady gravity waves of Part I. J . F l u i d Mech. 9_, 193. 129. P h i l l i p s , O.M. (1966) "Trie Dynamics of the Upper Ocean". Cambridge Un i v e r s i t y Press. Rayleigh, Lord (1880) On the s t a b i l i t y , or i n s t a b i l i t y , of c e r t a i n f l u i d motions. S c i e n t i f i c Papers, JL, 474. Cambridge Uni v e r s i t y Press. Rotta, J . (1951) S t a t i s t i s c h e Theorie nichthomogener Turbulenz. Z. Physik 129, 547. Snyder, R.L. and Cox, C.S. (1966) A f i e l d study of the wind generation of ocean waves. J . Marine Res. 24_, 141. Sokolnikoff, I.S. (1964) "Tensor Anal y s i s " . Wiley. Squire, H.B. (1933) On the s t a b i l i t y of the three-dimensional disturbances of viscous flow between p a r a l l e l w alls. Proc. Roy. Soc. AJL42, 621. Stewart, R.W. (1961) The wave drag of wind over water. J . F l u i d Mech. 10, 189. Stewart, R.W. (1967) Mechanics of the a i r - s e a i n t e r f a c e . Phys. Fluids Supplement, S47. Tollmien, W. (1929) Uber die Entstehung der Turbulenz. Nachr. Ges. Wiss. G'dttingen, Math.-phys. Klasse, 21. U r s e l l , F. (1956) Wave generation by wind. "Surveys i n Mechanics." (ed. G.K. Batchelor), 216. Cambridge U n i v e r s i t y Press. Wasow, W. (1958) The complex asymptotic theory of a fourth order d i f f e r e n t i a l equation of hydrodynamics. Ann. Math. (2) 49_, 852. Willmarth, W.W. and Wooldridge, C.E. (1962) Measurements of the f l u c t u a t i n g pressure at the w a l l beneath a t h i c k turbulent layer. J . F l u i d Mech. 14, 187. Corrigendum: I b i d . 21, (1965), 107 130. APPENDIX: FORMULATION OF LAMINAR INSTABILITY PROBLEM IN  CURVILINEAR COORDINATES We consider two-dimensional f l u i d motion i n the semi-i n f i n i t e domain R^ : - m < x,t < 0 0 and n < z < 0 0 . Here, (x,z) are c a r t e s i a n coordinates such that x i s the h o r i z o n t a l coordinate and z increases v e r t i c a l l y upward; t i s time; z = 0 i s the undisturbed surface of the water. A p e r i o d i c disturbance i s i n t r o -duced so that the actual water surface i s given by z = n = A exp[ik(x - ct)] , (Al) where the maximum surface slope i s small, i . e . kA << 1 . We take 1 2 new ca r t e s i a n coordinates (x ,x ) = (x - ct,z) . Thus we have made 1 2 a G a l i l e a n transformation so that the (x ,x ) coordinate system moves at the phase speed of the wave. Benjamin (1959) defines the coordinate transformation T : y 1 = x^ - i A exp[-k(x 2 - i x 1 ) ] ,^ 2 2 2 1 y = x - A exp[-k(x - i x )] . 2 Cl e a r l y , the l i n e y = 0 corresponds with the surface z = r\ to the f i r s t order i n kA . The Jacobian of the transformation i s t The superscript 1 or 2 on a vector quantity denotes a contra-v a r i a n t component (not a power) of the quantity. 131. &-r\ = 1 + 2kA exp[-k(x 2 - i x 1 ) ] 9x J (A2) To the f i r s t order i n kA , the inverse transformation T ^ i s x = y + lAe , (A3) 2 2 0 x = y + Ae , 2 1 where 0 = -k(y - i y ) . The metric tensor of t h i s transformation i s defined as (e.g. see Sokolnikoff 1964, §29) 3X 1 Bx 1 mn „ m „ n 3y 3y (A4) From (A2) - (A4), i t i s found that 0 , (A5) where J = 1 + 2kAe + 0(kA)' 1 2, Since g = 0 f o r i ^ j , the coordinate system (y ,y ) i s orthogonal For an i d e a l incompressible f l u i d , the E u l e r i a n equations of motion and the equation of con t i n u i t y are (Sokolnikoff 1964, §124) 132. 9v. . , ,, — ^ v. .v^ = F. - i ^ -9t i , j i p 9 y i v i = G - i / 2 9 (G 1 / 2 V 1 ) = Q 3Y 1 (A6) i i where v = dy /dt i s the contravariant v e l o c i t y tensor; p i s the s t a t i c pressure; p i s the a i r density; F^ i s the g r a v i t a t i o n a l body force; G = |g J + 0(kA) . Now, the covariant deriv-a t i v e of v. i s defined by l 9v. v. i»3 9y 1 r m , — - { . . > v J i J m (A7) where {.m.} i . 3 9y 9y 9y are the Chris t o f f e l 3-index symbols of the second kind. The funda-i k i i . i i i i i ,  mental tensor g i s constructed such that g g^j = °v ' 1 , e from (A5) , ( g 1 3 ) = rJ , 0 0 , J (A8) From equations (A5) - (A8), the equations of motion are therefore 9v. l 9t + v 9v. j I 9y-+ TV JV. J 7-2 3 r, 1 J 9y F. -P 9Y 1 (A9) 9(J V ) sy1 = 0 . 133, Equation (A9b) implies that a s c a l a r streamfunction may be intr o -duced inhere v = J — ~ and v = - J — — . (A10) 3y 3y The associated covariant v e l o c i t y components are given by v. = g. . v-1 ; l bX3 hence v 1 = ~ j and v„ = - -~- . ( A l l ) ay dyl The p h y s i c a l components v"~ of the v e l o c i t y vector v are given by the r e l a t i o n - i v = v e. , . ~ i where the e.'s are unit vectors c o d i r e c t i o n a l with the base ~ i vectors; i . e . the v ^ s represent the components of v acting t a n g e n t i a l l y to the coordinate l i n e s . I t follows that - i , V L / 2 i , • V v = (g..) v (no summation). Therefore, from (A5) and (A10), the p h y s i c a l components of the v e l o c i t y vector are -1 _L/2 3ij; , -2 1/2 d±_ ..,9, v = J and v = - J —^-r . (A12) 3y 3y 1 2 I f the v e l o c i t y components i n the (x ,x )-coordinate system are 1 2 (u ,u ), then from equation (A3), the r e l a t i o n between these and 1 2 the contravariant v e l o c i t y components i n the (y ,y )-coordinate system i s 134. u 1 = (1 - kAe")v" - i k A e V " + O(kA)" , 0s 1 0 2 2 0 ? 0 1 7 u = (1 - kAe )v + ikAe v + 0(kA) (A13) We now expand the streamfunction ij; , pressure p and Jacobean J i n asymptotic power s e r i e s i n kA , i . e . 00 oo oo * = I (kA ) V n ) , p = I ( k A ) n P ( n ) ,and J = 1 + \ ( k A ) n J ( n ) n=o n=o n-1 Benjamin assumes that = <J^ 0 /(y^) > i . e . that the primary flow i s steady i n t h i s coordinate system and the streamlines follow the wavy surface. Thus, the primary covariant v e l o c i t y components are (o) 8 i f J ( o ) ... 2. , (o) 'K = ••-„-- a U(y ) - c and v' 3y 1 0 . (A14) Far above the surface, U corresponds with the mean h o r i z o n t a l v e l o c i t y i n the stationary c a r t e s i a n (x,z) coordinate system; however, near the surface, U corresponds with a p e r i o d i c primary v e l o c i t y i n the (x,z) coordinate system. Su b s t i t u t i n g f or ty, p and J i n (A9a) and equating the c o e f f i c i e n t s of l i k e powers of kA to zero, the f i r s t order equations of motion are found to be (U - c) (U - c) ,2,(1) 3y 18y 2 3y 13y 1 (1) 3y + j ( U - c) 2 3 J( 1 ) 1 3 P ( 1 ) r(U - c) (1) 2 3J V ' _ 1 _3p 3y (1) 3y 3y 3y I (A15) 135. 2 where the motion i s assumed to be steady and U' = dU/dy Eli m i n a t i n g p ^ from (A15) y i e l d s the f i r s t order v o r t i c i t y equation Dy 9y 9y 9y 9y 9y 9y (A16) (1) 8 where from (A5), J = 2e C l e a r l y , the required motion i s p e r i o d i c i n y"^" , and hence a s o l u t i o n i s sought of the form = [F(y 2) + k\v - c ) e - k y 2 ] e i k y l . (A17) The transformation (A17) reduces (A16) to the f a m i l i a r homogeneous Rayleigh equation (Benjamin 1959); v i z . (U - c ) ( F " - k 2F) - U"F(y 2) = 0 . (A18) (1) 2 The streamfunction 4> must be such that the water surface y = 0 i s a streamline and such that the perturbation dies out f a r above the surface. Therefore, from (A17), the boundary conditions assoc-i a t e d with (A18) are F = k 1 ( c - U) on y 2 = 0 2 and F ->• 0 as y -> °° (A19) The system (A18) - (A19) i s seen to be i d e n t i c a l with the system (5.2.3), which describes the wave-induced streamfunction in car t e s i a n 136. coordinates. From equations (A15a) and (A17), the pressure i s given by (1) 2 N iky , p^ = pP(y )e (A20) where P(y 2) = U'F - (U - c)F'-(y 2) Comparing t h i s equation with (5.2.15), i t i s seen that the pressure 2 c a l c u l a t e d by the former at y i s equal to that c a l c u l a t e d by the l a t t e r at z ; i . e . to the f i r s t order i n kA , the pressure along 2 a wavy y -coordinate l i n e i s equal to that at the mean h o r i z o n t a l z-coordinate. Putting equations (A14) and (A17) into (A12), the ph y s i c a l components of the v e l o c i t y vector are found to be v 1 = U-c + k A [ F ' e l k y + k V e 9 ] + 0(kA) 2 , v 2 = - i k A [ k F e l k y + (U - c ) e 8 ] + 0(kA) 2 . \ (A21) 1 2 From (A13), the v e l o c i t y components i n the (x ,x ) coordinate system are ., 1 u = U - c + k A [ F ' e l k y + k 1U'e e] + 0(kA) 2 , u 2 = - i k 2 A F e l k y + 0(kA) 2 (A22) -1 1 Comparing (A21) and (A22), i t i s seen that, whereas v = u to the f i r s t order i n kA , the " v e r t i c a l " v e l o c i t y component v -2 137. 1 2 2 i n the (y ,y ) system i s equal to the actual v e r t i c a l v e l o c i t y u plus a term which accounts f o r the curvature of the coordinate l i n e s , The t h i r d term i n (A22a) i s required because U(y 2) = U(x 2) + ( y 2 - x 2)U'(x 2) + 0( k A ) 2 , = U(x 2) - aU'(x 2)e° + 0( k A ) 2 , from (A3); i . e . t h i s term allows for the p e r i o d i c i t y of U i n the (x ,x ) system. In §5.3, i t i s shown that closed streamlines or "cat's-eyes" 2 appear at the c r i t i c a l height y = z^ where Utz^) = c . I f i s l e s s than the wave amplitude A , then the streamline pattern near the surface i s not w e l l defined from the analysis of §5, which was c a r r i e d out i n c a r t e s i a n coordinates. To examine the influence of the r a t i o A/z on the streamline pattern, .we consider the most c simple mean p r o f i l e that has a c r i t i c a l height, v i z . the l i n e a r mean p r o f i l e U(y 2) - c = (c - U ) ( y 2 / z c - 1) (A23) 2 where U Q = U(0) . With U(y ) given by (A23), the system (A18) -(A19) i s r e a d i l y solved and the p h y s i c a l components of the v e l o c i t y are found to be, from (A21), v 1 = (c - U ) [ ( y 2 / z - 1) + (A/z )(1 - kz ) e 6 ] , o c c c v 2 = -ikA(c - U ) ( y 2 / z ) e 9 o c (A24) 138. From equation (A20), the f l u c t u a t i n g pressure i s p = p(A/z ) ( c - U q ) 2 [ 1 + k ( y 2 - z ) ] e 6 . (A25) We f i r s t note that equations (A24) and (A25) suggest that an order-ing parameter should be A/z^ i n addition to kA ; on the other hand, (c - U ) should decrease as z decreases so that there i s o c no s i n g u l a r i t y as z^ approaches zero. The sign of the pressure and the " h o r i z o n t a l " v e l o c i t y f l u c t u a t i o n at the surface ( r e l a t i v e to the surface displacement) i s seen to be equal to the sign of (1 - kz ) ; i . e . f o r large values of kz the pressure i s i n a n t i -c c phase with the surface displacement, while they are i n phase when kz^ i s small. When the c r i t i c a l height i s much la r g e r than the wave amplitude ( i . e . A/z << 1), the cat's-eye i n the streamline configuration l i e s f a r above the wave trough, as shown i n fi g u r e 16a. Now the closed streamline loops of the cat's-eye are formed because the amplitude of the f l u c t u a t i n g part of the " h o r i z o n t a l " v e l o c i t y i s greater than the mean v e l o c i t y near the c r i t i c a l height. Thus, the cat's-eye i s characterised by three stagnation points where -1 -2 v = 0 = v . There i s one at the centre of the cat's-eye, a distance of approximately z^ + A ( l - kz ) above the wave trough. When z^ > A ( l - kz^) , the two stagnation points at the corners of the cat's-eye are located a distance of z - A ( l - kz ) above the c c wave c r e s t s . However, when the c r i t i c a l height i s less than the wave amplitude ( i n p a r t i c u l a r , z^ < A ( l - kz^)) > the cat's-eye i s attached to the wave surface (fi g u r e 10b) and the corners are at the points where z = A ( l - kz ) cos ky^ . Thus, as z -> 0 , the c c c 139. cat's-eye centre approaches y = A and the corners approach points midway between the wave crest and trough, such that the three stag-nation points l i e on the same h o r i z o n t a l l i n e . We also note that when A/z > 1 (f i g u r e 16b), the motion above the cat's-eye contains an apparent harmonic because the cat's-eye " o v e r - f i l l s " the wave trough. When considering the problem i n cartesian coordinates, t h i s motion near the wave surface i s found by expanding the exponential -kz e , which appears i n the expression f o r the streamfunction, i n a power se r i e s i n kz . 0.5 0.4 u | 0.3 j§ O o. o x 0 2 x x X X X 1 0.5 Cp/U 5 XX FIGURE I Relationship between wave height and phase speed at peak of wave spectrum: data points from Stewart (1961). 0.5 Cp/U 5 FIGURE 2 Relationship between wind duration and phase speed at peak of wave spectrum: data points from Stewart (1961). FIGURE 3 Closed streamline cat's-eyes at c r i t i c a l height FIGURE 5 General behaviour of F (z) as r -> 0 FIGURE 6 General behaviour of (a) F (z) and (b) x r ( z ) a s ^ increases when J(z) i s monotonic increasing Co) o__ 4^ P (2) CO \ \ / w{z) ( b ) 0 / P (2) / / / CO o W (2) FIGURE 7 Phasor diagrams for p, w and n when J(z) i s monotonic increasing: (a) V i s small, and (b) r i s large .tv i s Shape of J(z) FIGURE 9 for a logarithmic mean v e l o c i t y p r o f i l e FIGURE 1 0 General behaviour of (a) F (z) and (b) X^(z) for small values F when J i s given by (5.3.6) FIGURE 11 General behaviour of (a) F (z) and (b) X (z) for r = T r s FIGURE 12 General behaviour of (a) F^_(z) and (b) X (z) for V > T r s V. \ X \ \ \ co / w(z) FIGURE 13 Phasor diagram f o r p, w a n d n when J(z) i s given by (5.3.6) and when r i s large FIGURE 14 Streamline configuration when J(z) i s given by (5.3.6) and when Y i s large FIGURE 15 V a r i a t i o n of r e l a t i v e phase of pressure at surface with U^/c : data points from Dobson (1969) FIGURE 16 Streamline configuration for a l i n e a r mean p r o f i l e when (a) kz > 1 and (b) z . < A (1 - kz ) c c c PART II PROGRESSIVE INTERNAL WAVES WATER OF VARIABLE DEPTH 140. 1. introduction This work i s e s s e n t i a l l y i n the form o r i g i n a l l y submitted for p u b l i c a t i o n and i t was w r i t t e n i n close c o l l a b o r a t i o n with L. A. Mysak. In p a r t i c u l a r however, Dr. Mysak found the s o l u t i o n (3.4) f o r waves propagating over a l i n e a r depth p r o f i l e , and he suggested the a p p l i c a t i o n of a general i n t e g r a l transformation to equation (3.2) to obtain the wave s o l u t i o n f o r an a r b i t r a r y depth p r o f i l e . The two-dimensional propagation of time-periodic, small-amplitude i n t e r n a l waves i n an i n v i s c i d , s tably s t r a t i f i e d f l u i d whose depth varies only i n one h o r i z o n t a l d i r e c t i o n has been the sub-t j e c t of several recent i n v e s t i g a t i o n s (Sandstrom 1966; Wunsch 1968, 1969; Longuet-Higgins 1969; Mooers 1969). With the exception of Wunsch, each author used the method of c h a r a c t e r i s t i c s to construct the s o l u t i o n corresponding to a p a r t i c u l a r topography or class of topographies. Wunsch, on the other hand, showed that for a Boussinesq f l u i d with constant Brunt-Vaisala frequency and l i n e a r l y varying depth, simple a n a l y t i c eigenmode solutions e x i s t . In view of the rather ad hoc method Wunsch used to construct these s o l u t i o n s , i t i s relevant to ask whether exact solutions e x i s t for any other smoothly varying topographies, and i f so, whether there i s any systematic method which can be used to construct these s o l u t i o n s . t A f t e r t h i s work was completed, two papers on i n t e r n a l waves i n water of' v a r i a b l e depth have appeared ( K e l l e r and Mow 1969 ; Hurley and Imberger 1969). However, i n both of these papers, only s p e c i a l asymptotic s o l u t i o n s are obtained which are not relevant to the c e n t r a l theme of t h i s work. 14.1. For a Boussinesq f l u i d with, a constant Brunt-V'ais'ala frequency and r i g i d surface, we show that the problem of f i n d i n g the wave eigenmodes corresponding to a given depth p r o f i l e reduces to f i n d i n g the s o l u t i o n set of a c e r t a i n f u n c t i o n a l equation (equation (3.2) i n §3). To derive (3.2) we ex p l o i t the homogeneous boundary conditions and hyperbolic nature of the governing d i f f e r e n t i a l equation f o r the s p a t i a l dependence of the wave stream function. For a l i n e a r depth p r o f i l e , (3.2) i s simple to solve, and the s o l u t i o n set indeed agrees with that found by Wunsch. For a more general smooth p r o f i l e , we also deduce an a n a l y t i c expression f o r each eigen-mode s o l u t i o n to (3.2). However, these expressions are not i n closed form: each s o l u t i o n consists of products of trigonometric functions whose arguments involve i n f i n i t e s e r i e s that converge provided the depth v a r i a t i o n s are s t r i c t l y transmissive ( i . e . the magnitude of the bottom slope i s everywhere s t r i c t l y l e s s than the magnitude of the c h a r a c t e r i s t i c s l o p e ) . For the case of a slowly varying depth (§4), we obtain an asymptotic representation of each eigenmode s o l u t i o n to the governing d i f f e r e n t i a l equation and boundary conditions by means of a two-scale perturbation expansion. The f i r s t few terms of each eigenmode s o l u t i o n agree with those of each eigen-mode s o l u t i o n to (3.2) i n the l i m i t of small bottom slope. In §5 we discuss some of the p h y s i c a l implications of the solutions to (3.2). In p a r t i c u l a r , the e f f e c t s of the bottom slope and curvature on the l o c a l wavelengths are determined, and expressions f o r the mean l o c a l k i n e t i c energy and shear s t r e s s are derived. In §6 we present pl o t s of the streamlines and wave p r o f i l e s i n order to i l l u s t r a t e q u a n t i t a t i v e l y the e f f e c t s of v a r i a b l e topography on the amplitude, 142. wavelength and phase of the waves. F i n a l l y i n §7 we discuss a class of closed-form solutions to (3.2) which represent "tuned" propagating waves. That i s , each s o l u t i o n set corresponds to a depth p r o f i l e which i s a function of the wave frequency. P h y s i c a l l y these solutions represent waves propagating over depth p r o f i l e s that are marginally transmissive. 2. Equations of Motion We consider the time-dependent motion of an i n v i s c i d , stably s t r a t i f i e d , uniformly r o t a t i n g ocean within the domain R : -h(x) < z < 0 , -K> < y < oo and a l l x such that h > 0 . Here x,y are the h o r i z o n t a l c a r t e s i a n coordinates and z i s the v e r t i c a l coordinate. The mean sea l e v e l i s at z = 0 and the ocean 00 bottom topography i s s p e c i f i e d by z = -h(x) , where h e C We r e s t r i c t our a t t e n t i o n to wave propagation that i s two-dimensional only and therefore assume a l l dynamic v a r i a b l e s are independent of y . Then the l i n e a r i s e d perturbation equations are given by u t - f v + P x/p Q = 0 , v + f u = 0 , t w + P Z/P Q + gP/P 0 = 0 ' (2.1) p + wp = 0 t 'oz (2.2) u + w = 0 , x z 143. where u,v,w are the perturbation v e l o c i t y components i n the x,y,z d i r e c t i o n , r e s p e c t i v e l y , p i s the density perturbation about the mean f i e l d p (z) , and p i s the perturbed pressure f i e l d . The o constants f and g are the i n e r t i a l frequency and ac c e l e r a t i o n of gravity, r e s p e c t i v e l y . In the abov^e we assume that o and p ' o oz are constant (the usual Boussinesq approximation). Equation (2.3) permits the in t r o d u c t i o n of a s t r e am function such that u = ij; , w = -ij; . Then (2.1) and (2.2) can be reduced to + i|> ) + N 24J + f2i> = 0 , (2.4) X X Z Z t t X X zz 1/2 where N = (-gp /p )" i s the (constant) Brunt-Vaisala frequency. oz o n J Upon assuming that — tot iKx,z,t) = §(x,z)e , o > 0 , equation (2.4) takes the form E> _ w2 $ = o , (2.5) ZZ X X 2 2 2 2 2 whe re w — (N — o )/(o — f ) > 0 . Thus, wi t h i n the domain R , the s p a t i a l dependence of the wave motion i s governed by a hyperbolic equation with c h a r a c t e r i s t i c s x + wz = constant. On the boundaries of R , the following conditions are imposed: )(x,0) = 0 (2.6) 144. $(x,~h(x)) = 0 . (2.7) Equation (2.6) states that the mean sea surface i s a streamline; i . e . the i n t e r a c t i o n between surface and i n t e r n a l gravity waves i s neglected. P h i l l i p s (1966 s §5.2) shows that i n the ocean t h i s approximation i s usually v a l i d . Equation (2.7) implies that the ocean f l o o r i s also streamline and that there i s no instantaneous net flow. 3. Solution for transmissive depth p r o f i l e s : f u n c t i o n a l equation  approach. The propagation of two-dimensional, p e r i o d i c i n t e r n a l waves i n a Boussinesq f l u i d with a v a r i a b l e bottom topography i s described mathematically by (2.5) - (2.7). The general s o l u t i o n to (2.5) i s where f and j. are a r b i t r a r y functions of class C . The boundary •conditions (2.6), (2.7) imply that $(x,z) = f ( x + u z ) + j (x - ujz) ,2 f(x) + j(x) = 0 , and f (x - wh) + j (x + ,o)h) = 0 , r e s p e c t i v e l y . Hence we can write the s o l u t i o n i n the form 145. $(x,z) = f ( x + oiz) - f ( x - a)z) , (3.1) where f ( x + uih'(x)) = f ( x - wh(x)) . (3.2) Thus, f or a given smooth depth p r o f i l e , the problem simply reduces to f i n d i n g the s o l u t i o n set of the f u n c t i o n a l equation (3.2). Although the general s o l u t i o n of (3.2) i s by no means obvious, i t may be solved d i r e c t l y f o r some simple topographies. For example, f o r the l i n e a r depth p r o f i l e h = h Q + sx , x > -h /s (h , s > 0) , (3.2) reduces to o o f ( ( l + sw)x + wh ) = f ( ( l - SOJ)X - coh ) o o On s e t t i n g x = y - h^/s , we obtain f ( ( l + sco)y - h /s) = f ( ( l - sto)y - h /s) Now l e t f ( 6 ) = F (6 + h Q/s) and n = (1 - sto)y ; thus we have F(n) = F(An) , (3.3) where X = (1 + sw)/(l - sto) . F i n a l l y , under the transformation n = e^ and X = e™ and upon defining g(<)>) = F(e^) , (3.3) implies that g(C) = g(? + m) 146. Therefore, g i s any p e r i o d i c function with period m ; c l e a r l y a l l functions s a t i s f y i n g this condition can be constructed from the complete set g n(p) = exp(i2Trp/m) , n e I = {0,+l,+2,... } Thus we f i n d that f ( r) = exp[i2mra l o g ( r + h /s)] , where a = l / l o g [ ( l + soj)/(l - soo)] . Hence, from (3.1) the simplest eigenwave solutions are given by $ (x,z) = exp[i2mra log(x + co z + h /s)] n o - exp[i2nira log(x - wz + tWs) ] , (3.4a) which can also be written as x-to z+h Is 2 2 2 $ (x,z) = - 2 i sin[nira log — j — ]exp{ infra log [ (x+h /s) z ]} . n ° x+toz+ho/s r ° o (3.4b) From (3.4b) i t i s obvious that * = 0 for z = 0 , -h - sx , as n o required. Further, f or the s p e c i a l case h^ = 0 (corresponding to h = sx) , (3.4) reduces to the solutions found by Wunsch (1968, 1969) 'by inspection'. This s o l u t i o n has also been found by Balazs (1961), who considered the analogous problem of the o s c i l l a t i o n of a s t r i n g 147. with moving boundaries. F i n a l l y i t can be v e r i f i e d that upon m u l t i -p l y i n g (3.4b) by exp[-i2niTa log(h /s)] and then taking the l i m i t s 0 , we obtain the f a m i l i a r s o l u t i o n set f o r a f l u i d of constant depth h , v i z . o $ (x . z ) = -2i sin ( n i r z/h ) exp (imrx/ioh ) . (3.5) n o o It i s also easy to show that (3.5) can be obtained d i r e c t l y from (3.1) and (3.2). The s o l u t i o n given by (3.4) suggests that f o r a general smooth topography h , (3.2) might also be s a t i s f i e d by a denumerable set of l i n e a r l y independent functions. Under t h i s assumption we now develop a method f o r s o l v i n g (3.2) f o r a general h . We suppose that f(x) can be w r i t t e n i n the form co f(x) = / dk F(k) exp[i k g(x)] . —CO I t follows that s u f f i c i e n t conditions f o r (3.2) to be s a t i s f i e d are that g(x + a)h,w) - g(x - o)h,co) = y(co) (3.6) and F(k) = fi(k - k ) , where k (to) = 2nTr/vCa)) , n e I , and <5(k) i s the Dirac d e l t a n function. The parameter to has been introduced as an e x p l i c i t 148. independent v a r i a b l e of g since, i n general, h i s independent of a) . Hence the s o l u t i o n set of (3.2) i s given by f (x) = f (X;OJ) = exp[ik g(x,io)] , n e I n n n From (3.1) i t thus follows that the s o l u t i o n set of (2.5) - (2.7) i s given by $ n(x,z) = $ (x,z;w) = exp [ (i2mr/y)g(x + wz,w)] - exp[(i2nir/Y)g(x - (DZ,W)] , n e I . (3.7) Equation (3.6) suggests the p o s s i b i l i t y that g(x,to) i s separable i n the sense that g(x + o)h(x)) - g(x - uh(x)) = y(uj) . (3.8) We now assume that, f o r some co > 0 , g(x) i s a n a l y t i c at some point x , about which a Taylor ser i e s can be constructed with radius of convergence greater that coh . (T h e o r e t i c a l l y , the s o l u t i o n may be eventually extended by a n a l y t i c continuation to allow f o r the case of a r b i t r a r y co .) Thus (3.8) becomes V w 2 n + 1 ,2n+l. \ (2n+l), , _ 1 (x) g (x) = — y(w) , L (2n+l)! 6 v y 2 n=o , (m) ,m..,,m 0 . , r . m . n where g (x) = d g(x)/dx . Since the functions 03 and w , 149. m 4 n > are l i n e a r l y independent, we must have ,2n+l. , (2n+l), \ _ t h (.x) g (x) = a2 +i » a constant , or (2n+l) , . _ . 2n+l, , g ( x ) = a 2 n + 1 / h C x ) D i f f e r e n t i a t i n g the l a t t e r equation twice, we f i n d that h(x) s a t i s i f e s the d i f f e r e n t i a l equation hh" - (2n + 2)h' 2 = -a_ ,,/(2n+l)a 0 2n+3 2n+l Hence dh/dx = [ C 2 h 4 n + 4 + a. (2n+l)(2n+2)a„ . J 1 / 2 zn+j zn+l where C i s an a r b i t r a r y constant. C l e a r l y h = h(x,n) f o r C 4 0 , so that the only allowable s o l u t i o n i s 1/2 dh/dx = [a 3 / ( 2 n + l ) ( 2 n + 2 ) a 2 n + 1 ] ' = s , a constant. Hence h(x) = h + sx and o (a 1/s)log(h o+sx) , s#) , g(x) = /(a 1/h(x))dx = a,x/h , s=0 ^ 1 o From (3.8) we thus f i n d 150. ( a ^ s ) log[(l+u>s)/(l-u)s) ] , s ^ 0 y((jj) = 2a, co p . = 0 With these expressions f o r g and Y > (3.7) y i e l d s , to within a m u l t i p l i c a t i v e constant, the s o l u t i o n set {$ } found above for n h = h Q + sx , v i z . (3.4b) and (3.5) f o r s + 0 and s = 0 , r e s p e c t i v e l y . Thus, we conclude that the most general topography which allows g(x,w) to be separable i s that with a constant slope. We now consider the more general problem where g = g(x,to) and g s a t i s f i e s the f u n c t i o n a l equation (3.6). Setting x = 0 i n (3.6) i t i s seen that where h(x) i s defined so that x = 0 i s a regular point. We assume that g(£,<*)) i s a n a l y t i c at t, = x and to = 0 , and that for some to > 0 , the r a d i i of convergence are greater than toh(x) and to , r e s p e c t i v e l y . Then a double Taylor s e r i e s expansion may be constructed about the point (x,0) . Hence we have Y(CO) = g(coh(0),to) - g(-toh(0) ,to) , (3.9) CO g(x + toh.to) = Y (ojha + to3 ) ng(x,0)/n! U "V M (3.10) n=o Using the binomial expansion and defining 151. (3.10) reduces to CO ft g(x + toh,co) = -l w n[ I h S ( x ) M ( s ) ( x ) / s ! ] . (3.11) n-s n=o s=o Therefore, using (3.9) and (3.11), (3.6) takes the form 00 ft 1 { I [ l - ( - l ) S ] [ h S ( x ) M < S > ( x ) - h S(0)M ( s )(0)]/s!}co n = 0 . (3.12) n=o s=o n s Since to and to , n ^ s , are l i n e a r l y independent, each c o e f f i c i e n t of to11 i n (3.12) must vanish, v i z . I [ l - ( - l ) S ] [ h S ( x ) M ( s ^ ( x ) - h S ( 0 ) M ^ ( 0 ) ] / s ! = 0 , (3.13) n s IT s s=o fo r a l l n e W = {0,1,2,...} . Because a l l the terms f o r £ven s vanish, (3.13) may be separated into two d i s t i n c t s e r i e s f o r , s e W and M2s+1 ' s £ ^ ' ^ n p a r t i c u l a r , f o r s e W , h(x)M£>2oo = h(0 ) : 4 1| 2(0) s+1 - n h 2 " ' + 1 ( , ) M < ^ « 2 ( x ) - h 2 - \ o ) „ « - « 2 ( o , ] / ( 2 „ + i ) ! m=l (3.14a) where h(x)M <^ 1 ) (x) = h ( 0 ) M ( 1 ) ( 0 ) , and o o h(x ) I 4 1| 3(x) = h(0)>41|3(0) s+1 - Z [ h ^ 1 ( x ) M 2 ^ » 3 ( x > - h ^ 1 ( 0 ) « < ^ 3 ( 0 ) 1/(2^1)! , m=l (3.14b) where h ( x ) M ^ (x) = h(0)M^(0) . C l e a r l y , by s e t t i n g (0) = aM^^ (0) , where a i s an a r b i t r a r y constant, i t follows that ^2s+l^ = a M 2 s ^ x ' ) > f o r a 1 1 s e W • Therefore, g(x,co) = I w n M n ( x ) = (1 + a u ) I a) M 2 n(x) n=o n=o I t i s evident from (3.6) that we may take a = 0 without any loss of ge n e r a l i t y ; thus we consider the set {M2 (x) : n e W} only. Since (3.14a) s p e c i f i e s M^^(x) , the function M 2 (x) i t s e l f i s determined to within an a r b i t r a r y constant. This follows n e c e s s a r i l y from (3.6), which i s s a t i s f i e d by g(x , io) = g 1 (x,w) + o (to) where g also s a t i s f i e s (3.6) and S i s a r b i t r a r y . Without loss of g e n e r a l i t y , we may therefore s e l e c t ^ 2 s ^ > s e W , such that M^ 1 )(0)h(0) = 1 , s+L h ( 0 ) M 2 s + 2 ( 0 ) + I h 2 m + 1 ( 0 ) M 2 s - 2 m + 2 ( 0 ) / ( 2 m + 1 ) ! = 0 * . m=l Hence, from (3.14a) and (3.15), a s o l u t i o n of (3.6) i s given by (3.15) 9 g(x,oi) = I w M (x) , (3.16a) where 2n n=o M^ 1 }(x) = l/h(x) , (3.16b) 2+1 £> ( X) = - l h 2 ^ x ) M 2 ^ 2 ( x ) / ( 2 m + l ) ! . (3.16c) M 2s+2 m=l 153. By repeated s u b s t i t u t i o n of the expression for (x) in t o i t s e l f , i t may be shown that, f o r s e N = {1,2,...} , M 9 ( 1 )(x) = - h 2 3 M ( 2 s + 1 ) + S I 1 h 2 m ( h 2 s - 2 V 2 s - 2 m + 1 ) ) ( 2 m ) / ( 2 m + l ) ! 2s o L T o m=l - S f S " f 1 h2m [ h2n ( h2s-2m-2n M(2s-2m-2n+l) ) (2n) ] (2m) / ( 2 m + 1 ) , ( 2 n + 1 ) , m=l n=l + ... (3.17) The function y ( w ) m a Y be found by expanding (3.9) i n a double Taylor s e r i e s about x = 0 = co ; i . e . from (3.9) and (3.11) we f i n d 0 0 n ( \ Y(o)) = I o)n{ I [ l - ( - l ) S h S ( 0 ) M ^ ( 0 ) / s !} . n=o s=o But M. ' (x) = 0 , s e W , so that zs+1 co n Y(io) = 2 I w 2 n + 1 [ I h 2 s + 1 ( 0 ) M ^ ) ( 0 ) / ( 2 s + l ) ! ] n=o s=o CO = 2h(0)M ( 1 )(0)orf 2 Y J n + 1 [hCCOM^ (0) o u. 2n n=l + I h 2 s + 1 ( 0 ) M 2 ( 2 ^ ) ( 0 ) / ( 2 s + l ) ! ] s=l 2 IT t '1 However, from (3.15) i t follows that each c o e f f i c i e n t of to n z N , i s zero. Therefore Y(CO) = 2 to (3.18) 154. Using (3.16a) we now expand g(x + O)Z,(D) i n a Taylor s e r i e s about the point (x,0) aand c o l l e c t the terms i n v o l v i n g even and odd powers of z . Thus we assume that the convergent serie s f o r g(x + c o Z j O ) ) can be rearranged to give two d i s t i n c t convergent s e r i e s as follows: oo g ;(x+ Wz ,co) = I JS[ I z 2 V 2 M J (x)/(2m)!] — u u zs-zm s=o m=o ± I 0 3 2 S + 1 [ I z 2 m + 1 l 4 2 ^ 1 ) ( x ) / ( 2 m + l ) ! ] . (3.19) 2s-2m s=o m=o Upon s u b s t i t u t i n g (3.18) and (3.19) in t o (3.7) and r e c a l l i n g that x z — i z s i n z = (e - e ) / 2 i , we obtain & (x,z) = 2 i exp [ (imr/w) S^(x, z ; c o ) ] s i n [nuS^ (x,z ;w) ] , (3.20) where CO g S, = I o> 2 s [ I z 2 n V ^ ™ 2 m ( x ) / ( 2 m ) ! ] , s=o m=o CO g S 2 - I JS[ I z 2 m f l M ^ 2 ^ ) ( x ) / ( 2 m + l ) ! ] s=o m=o The set {M„ ; m E W} i s determined from .(3.16b,c) or (3.17), so 2m that „ r \ T ' f i n i r f 1 dx 1- 2 r, 3z N dh „ t /dh.2 dx , J> n (x,z) = 2 i e x p { - — [ / ^ + j a, [(1 - - j ) — - 2/ (—) ^ ] - K h 2 2 . ,nTrz r i . 1 2. z w u d h 0 , d h N 2 , , , x s l n { h T x T [ 1 + 6 w ( 1 " 72?( ~7~2 ~ 2 ( d ^ ) ) + " - ] } • h dx (3.21) 155. Equation (3.21) represents the complete s o l u t i o n set {* : n E 1} to (2.5) subject to the boundary conditions (2.6) - (2.7), provided that the ser i e s and converge. On introducing the assumed time dependence in t o (3.20), we can define txvo independent sets of r e a l s olutions of (2.4), v i z . c s { i l l } and {il> } , where n n \b = (coA /x4n-rr) ($ e - e ) , r n n n n di = (-coB /4niT) ($« e + $ e ) n n n n These reduce to i i i (x,z,t;cj) = (coA /n-rr) cos [(rm/ai)S n - at] sin(mrS_) n n 1 z g i> (x,z,t;ta) = (uB /n-rr) s i n [ (nTr/w) S., - at] sin(mrS_) . n n 1 2 (3.22) Because nonlinear terms have been neglected i n the equations of motion (2.1) - (2.3), the solutions (3.22) are v a l i d only when 2 2 I n-rr A /ah I << 1 and Imr B /ah I << 1 . 1 n 1 1 n 1 We now e s t a b l i s h the convergence of the s e r i e s and i n (3.20) for the c l a s s of topographies for which there e x i s t s a 6 > 0 such that, for a l l s e W and {m : 0 <_ m <_ s} , i M p ^ ^ x ) ! .< a ( 2 m ) ! 6 2 s / h 2 m + 1 ( x ) , (3.23) 1 zs-zm 1 — s-m 156. where a = 1 and a = ( 4 / 3 ) S / 4 f o r s > 0 . That (3.23) o s r e s t r i c t s the r e l a t i v e magnitudes of the de r i v a t i v e s of h(x) i s seen by putti n g s = m ; i . e . from (3.16b,c) and (3.23), | M ( 2 s + l ) ( x ) i = | ( 1 / h ) ( 2 s ) , < ( 2 s ) ! 6 2 s / h 2 s + 1 . ( 3 . 2 4 ) C l e a r l y , the e q u a l i t y sign i s required i n (3.24). and hence i n (3.23), when s = 0 . The i n e q u a l i t y (3.23) must also be consistent with (3.16c). Putting the former into the r i g h t hand side of the l a t t e r , we obtain |M<»faO| - | - i h % < ^ » < x , / ( 2 * l ) ! | m=l 9 s S < (5 /h) y a /(2m+l) — , s-m m=l 9 s < (6 Z S/3h) 7 a — L^ s-m m=l However, from the d e f i n i t i o n of a , we have that s s a = (1/3) y a . (s > 0) (3.25) s u - s-m m=l Therefore, |M 9 ( 1 )(x)| < a 6 2 s/h(x) . Z S o Since the above i n e q u a l i t y i s also obtained by l e t t i n g m = 0 i n (3.23) the i n e q u a l i t y i s indeed consistent with (3.16c). 157. Using (3.23), an upper bound can now be established f o r S | . We have 0 0 s S - ( x , Z ; U ) | < I JS[ I h 2 m + 1 | M 9 ( 2 m ^ 1 ) | / ( 2 m + l ) ! ] (since | z\<h) 2 s-2m s=o m=o 00 ? s < I (u5) I a /(2m+l) (by (3.23)) — u u s-m s=o m=o < 2 £ ( a i 6 ) 2 s a s (by (3.25)) s=o 2 + (1/2) I (u>6) 2 s(4/3) S s=l 2 + 2(to6) 2/3[l - 4(co6) 2/3] , f o r (w<5)2 < 3/4 Thus the s e r i e s S 2(x, z;co) converges uniformly and absolutely i n R provided 106 < /3/2 . S i m i l a r l y , we can also e s t a b l i s h an upper bound f o r I 9 S, I . We have 1 x I 1 3 xS Cx,z; U)| < I co 2 s[ I h 2 m | M ^ ) | / ( 2 m ) ! ] s=o m=o 0 0 s < y ( W 6 ) 2 s y a /h s=o m=o = (4/h) I a (eo6) 2 s ^ s s=o ( 4 / h ) [ l + (1/4) I (cu6) 2 s(4/3) 8] s=l (4/h){l+(o)6) 2/3[l-4(o)6) 2/3} , for (w<5)2<3/4 a A/h 158. Thus h9 S converges uniformly and absolutely i n R provided X X w<5 < /3/2 . We now l e t h_ = min h(£) , where E, e P - [x ,x] Then we have m p o' S 1(x,z;u) - S 1(x o,z;a))| =|/ x o X 1 / I V l ' ^ X ? o x , < A / h X ( O d £ X o < (x - x )Ah 1 — o m But, since S (x,z;w)| - |S (x , z;o))j < |S (x,z;to) - S (x ,z;w)| , 1 1 1 1 o ' — ' 1 l o ' we thus have S, (X,Z;OJ)| < |S n(x ,z;oi)| + (x - x )Ah ^  1 ' — 1 1 o o m Therefore, provided w6 < /3/2 and i s bounded at some point x^ for a l l z e (-h,0) , then i s bounded f o r a l l ( f i n i t e ) x > X Q and z e (-h,0) . With an obvious modification i n the hypotheses, i t follows that S, i s also bounded f or a l l x < x and z e (-h,0) 1 o provided w6 < /3/2 . Thus, the s e r i e s and converge i n R provided that 159. w<5 < /3/2 . (3.26) The p h y s i c a l i m p l i c a t i o n of (3.26) i s understood by noting, from (3.20), that the h o r i z o n t a l wavelength, I , of the o s c i l l a t i o n i s p r oportional to w , while the h o r i z o n t a l length scale, L , of the topography varies i n v e r s e l y with 6 . In p a r t i c u l a r , . I ^ ,wh/nn and L ^ h/6 , Thus, (3.26) implies that the wavelength of the motion must be less than the h o r i z o n t a l scale of the depth v a r i a t i o n s . Moreover, the magnitude of the bottom slope must be somewhat smaller than the magnitude of the slope of the wave c h a r a c t e r i s t i c s , which i s 1/to . That i s , i n the terminology of Sandstrom (1966), the depth p r o f i l e must be s t r i c t l y transmissive. 4. Solution for slowly varying depth p r o f i l e s : multiple scale  approach In t h i s s ection we construct an approximate s o l u t i o n to (2.5) - (2.7) f o r the case when the depth p r o f i l e h i s a slowly varying function. More s p e c i f i c a l l y , we assume that h(x) i s of the form h(x) = H(ex) , (4.1) where 0 < e << 1 and H' = 0(1) . Also, f or the purpose of ordering i n a subsequent perturbation a n a l y s i s , we assume that x and z have been nondimensionalized by some average depth h and that 160. to = 0(1) . Following a procedure s i m i l a r to that used by C a r r i e r (1966) in. connection with a surface g r a v i t y wave problem, we assume that the stream function may be w r i t t e n i n the form i x $(X,Z;W,E) = G(x*, z;u>, s)e , (4.2) x where x* - ex and x = / k(to,ex')dx' are new independent v a r i a b l e s x o t and k i s a l o c a l wave number to be determined. From (4.1), (4.2) and (2.5) - (2.7), i t follows that G s a t i s f i e s the equation G + co 2k 2G = ieio 2(2kG . + k .G) + E 2OJ 2G , , (4.3) zz v x* x* x*x* with G = 0 at z = 0 , -H . We now seek a perturbation expansion for G such that CO G(x*,z;o3,e) = £ e mG (X*,Z;LO) . (4.4) m=o Upon s u b s t i t u t i n g (4.4) i n t o (4.3) and equating the c o e f f i c i e n t s of powers of e to zero, we obtain an i n f i n i t e set of equations for the set {G : m = 0,1,2,...}. With the notation t In the beginning of h i s a n a l y s i s , C a r r i e r does not assume an e x p l i c i t form such as (4.2) f o r the v e l o c i t y p o t e n t i a l , cb , but simpl that <j> = <j)(x*, x, z; a), E) . However^ he shows that i n order to avoid — i x secular terms ( i . e . terms l i k e xe ) i n a subsequent perturbation expansion, the p o t e n t i a l must i n fa c t be of the form (4.2). 161. 2 2 2 L = 3 + co k , z (4.5) M = 21c3x5, + k 2 A , the f i r s t three of these equations are L(G ) = 0 , (4.6) o L(G ) = ito2M(G ) , (4.7) 1 ' o L(G ) = co2[G + 1M(G )] . (4.8) The boundary conditions associated with (4.6), (4.7) and (4.8) are G = 0 at z = 0 , - H (m = 1,2,3) . m C l e a r l y , once G i s known, G (m > 1) can be determined i n an o m — i t e r a t i v e manner. The solutions to (4.6) which s a t i s f y the boundary conditions are given by G n = A n(x*) sin(wk z) , n e I , (4.9) o o n where k = nTr/<jjH(x") . (4.10) n The function A n must be determined from the s o l u t i o n of equation o (4.7). The s u b s t i t u t i o n of (4.9) and (4.10) i n t o (4.5) and (4.7) now y i e l d s 162. L^CG^) = a sin(nTrz/H) - bz cos(mrz/H) , (4.11) 2 2 where L = 3 + (UTT/H) , n z a = i(n™/H~)[2H(A nr - H'An] o o b = i2to(rrrr) 2 H 'H 3 A n The general s o l u t i o n of (4.11) i s given by G? = [A?(x*) - i ^ wnTr H'(z/H) 2A n] sin(mrz/H) 1 1 / o [B?(x*) - iaiz(A n) '] cos(n7Tz/H) . (4.12) 1 o The boundary conditions immediately give = 0 and A n = constant. (4.13) 1 o Hence the s o l u t i o n f o r G n i s completely determined. To determine o A^ , and hence G^ completely, the s o l u t i o n of (4.8) must be examined. Su b s t i t u t i n g the solutions f o r G^ and G^ in t o (4.8), we obtain an equation f o r G^ s i m i l a r to (4.11). The a p p l i c a t i o n of the boundary condition G^ (x*,-H ;to) = 0 to the general s o l u t i o n of th i s equation gives, a f t e r much algebra, (A?)' = i(n™/6H)[HH" - 2(H') 2]A n (4.14) 1 o Thus, from equations (4.2), (4.5), (4.10) and (4.12) - (4.14), we 163. 2 f i n a l l y f i n d that to 0(e") , cb = A n { l +' i ( e m r / 6 ) [ ( l - 3z 2/H 2)H' - H'(x*) n o o - 2 / H _ 1(u) [H'(u)] 2du] } s i n (mrz/H) x* o x -1 x exp[i(nTr/ai) / H '(eu)duj . (4.15) x o Equation (4.15) shows that, whereas the magnitude of the zeroth order term G i s f i x e d by A n , the f i r s t order perturbation o o G., depends upon both A n and the a r b i t r a r y constant x* . The 1 r o o cause of t h i s ambiguity i s found by considering the formal s o l u t i o n (3.20). This equation involves an exponential function whose argu-ment contains a r b i t r a r y a d d i t i v e constants, which simply set the i n i t i a l phase of the s o l u t i o n 0^ . However, to compare th i s s o l u t i o n with equation (4.15), the exponential function, with h(x) = H(ex) , must be expanded i n a power se r i e s and only terms of 0(e) retained. By j u d i c i o u s l y choosing the additive constants i n the exponential, equation (3.20) indeed reduces to (4.15). The ambiguity may be eliminated when there e x i s t s a point x* where the topograph}' i s p e r f e c t l y f l a t ( i . e . H^n^ (x*) = 0 for a l l n ) . At that point, a l l terms of 0(e) must be zero so that $^ i s then determined uniquely. F i n a l l y , we note that equation (4.15) i s a uniformly v a l i d approximation to $ with respect to e . However, i t i s not a p a r t i c u l a r l y u seful representation of the s o l u t i o n f o r the higher mode waves, i . e . f o r values of n such that emra)/6 > 1 . 164. 5. Discussion of s o l u t i o n From equation (3.7), i t i s seen that each eigenfunction of the system (2.5) - (2.7) consists of two i n d i v i d u a l waves with wave-number vectors k ^ = [(2mr/Y)g''(x + coz) , + (2mrco/6) g'(x + coz) ] ~n — ~~ — The c h a r a c t e r i s t i c s of equation (2.5) are x + coz = constant and x - coz = constant; vectors which are p a r a l l e l to these f a m i l i e s are (+) (-) C = (io,-l) and C = (to,l) , r e s p e c t i v e l y . Hence, k ( ± ) • = 0 ; ~n i . e . the wavenumbers k ^ and k^ ^ are perpendicular to the ~n ~n c h a r a c t e r i s t i c s x + coz = constant and x - coz = constant, resp-e c t i v e l y . Using (3.20) the above wavenumber vectors may be a l t e r -n a t i v e l y w r i t t e n as k ( ± ) = (mr/a>)(S.. + OJS. , S _ + coS„ ) ~n 1, x — 2, x 1, z — 2, z We now consider the l o c a l wavenumber vector k which ~n consists of the superposition of k ^ and k^ ^ . Thus, from . ~n ~n (3.20), we obtain k = (k ,k ) = (mr/co)(S.. ,S1 ) ~n x z l , x l , z 165. as the l o c a l wavenumber . vector associated with each eigenfunction $ . From (3.20) we f i n d that k = (mrAohHl + (oa 2/6)(l - 3z 2/h 2) [h - 2 A 2 ] + . . . } ' x , I dx dx k = ~ ( n T r / ( J : i ) ( z / h ) { ~ + ... } z dx \ (5.D In general we note that bottom slope ( p o s i t i v e curvature) causes k to be an in c r e a s i n g (decreasing) function of depth. . The leading 2 terms of equation (5.1) imply that, f o r (z/h) < 1/3 , the presence of a s l o p i n g bottom tends to reduce the l o c a l h o r i z o n t a l wavenumber (or increase the wavelength) as compared with that f o r a f l a t topo-graphy. On the other hand, the curvature of the bottom also a f f e c t s the h o r i z o n t a l wavenumber: a ridge i n the ocean f l o o r tends to increase k , whereas a v a l l e y tends to decrease i t . However, below x 2 the depth where (z/h) = 1/3 , these e f f e c t s of the bottom slope and curvature on the h o r i z o n t a l wavenumber are reversed. From (5.1) i t i s also noted that the v e r t i c a l wavenumber k^ increases l i n e a r l y with depth and i s pro p o r t i o n a l to the bottom slope. c c The v e l o c i t y components u and w corresponding to the n n streamfunction ijj are obtained from (3.12) by d i f f e r e n t i a t i o n : C C *\ u (x,z,t) = ib = coA. S cos [(mr/aOS - at] cos(n-rrS0) n Tn, z n 2, z 1 2 - A S sin[(nTr/w)S - a t ] sin(.mrS ) , n _L j z x \ (5.2) c c w (x,z,t) = -Th = A S. sin[(mT/w)S. - at] sin(nirS„) n" rn,x n l , x 1 I -toA S cos [(n-rr/aj) S. - at] cos(nirS ) . n z j x x A. 166, I f the wave frequency i s much, greater than the i n e r t i a l frequency (cr/f >> .1) , then the v e l o c i t y component v (see (2.1)) may be neglected i n computing the mean l o c a l k i n e t i c energy per u n i t mass; to the second, order i n the wave amplitude, this energy i s given by E C(x,z) = i <(u C ) 2 + (w C) 2> n 2 n n = (1/4) A 2 [S 2 + S 2 ] sin 2(n7fS.J n l , x l , z z + (1/4) A 2 [S 2 + S 2 ] c o s 2 ( m i S j , (5.3) n I ,x I, z 2 where < > denotes the average over an i n t e g r a l number of periods. From (3.11) and (5.3), the k i n e t i c energy at the ocean floor, (z = -h(x)) i s found to be E^(x,-h) = (coA n/2h) 2[l + (1 + 4 a 2 / 3 ) ( | | ) 2 - (2w 2/3)h — - + . . . ] . (5.4) dx Thus, a sloping bottom tends to increase the mean k i n e t i c energy above that f o r a f l a t bottom. A p o s i t i v e curvature, however, decreases the energy, so that there i s more motion i n a v a l l e y than around the top of a ridge at the. same mean depth. This suggests that s l i g h t i r r e g u l a r i t i e s i n the bottom topography may be rei n f o r c e d by i n t e r n a l wave a c t i v i t y which tends to scour depressions more than ridges. Equations (5.2) show that a v a r i a b l e topography induces a c Reynolds shear s t r e s s T such that n 167. c c c X = - <u w > n n u 1 2 2 2 = r u A S„ S_ cos CnTrS„) 2 n 2,x 2,z 2 + A 2 S_ S, sin 2(mTS„) . 2 n l , x 1,z 2 That i s x C = -2(wA / 2 h ) 2 ( z / h ) ^ +••• (5-5) n n dx c c From (5.4) and (5.5), i t i s seen that T /E i s of order dh/dx , n n the bottom slope. On the continental slope, where dh/dx can be of order 0.1 , this shear st r e s s may lead to secondary motions and so hasten the degeneration of i n t e r n a l waves i n t o turbulence. 6. Solutions f o r s p e c i f i c depth p r o f i l e s > In this s e c t i o n , the e f f e c t of a v a r i a b l e bottom topography on i n t e r n a l waves i s i l l u s t r a t e d by three s p e c i f i c depth p r o f i l e s : ( i ) one which, increases monotonically from zero at x = 0 and which asymptotically approaches a constant h^ as x ->- 0 0 , ( i i ) one which has a symmetric v a l l e y centred at x = 0 and which asymptotically approaches a constant h^ at |x| -> <» , and ( i i i ) one which has a symmetric ridge centred at x = 0 and which asymptotically approaches a constant h as I x l -> 0 0 . Each depth p r o f i l e h i s characterised by a parameter E such that h. = h(Ex/h ) . I t i s seen from equation o (3.24) that z may be compared with the slope parameter 6 . 168. Therefore, the wave frequency parameter to was selected i n a l l examples so that toe < /3/2 , i n accordance with (3.26). Diagrams of the instantaneous wave streamfunction and wave p r o f i l e (as calc u l a t e d from equations (3.22) and (5.2), r e s p e c t i v e l y ) demonstrate the general features of the s o l u t i o n s . The wave p r o f i l e i s defined to be (Thorpe 1968) t n(x,z,t) = ~p/p Q z = / w(x,z,t')dt' . Thus, r\ represents the v e r t i c a l displacement of the surface n = z due to the i n f i n i t e s i m a l wave motion. The f i r s t example has a depth p r o f i l e given by h => h [1 - exp(-ex/h.') ] , x > 0 . (6.1) o o Figure 1 depicts the instantaneous streamlines and wave p r o f i l e s n.(x,zo,0) (-z /h = 0.1(0.1)0.9) corresponding to a f i r s t mode wave propagating over t h i s topography when e = 1 and to = y . The streamlines form the f a m i l i a r c e l l s associated with propagating i n t e r n a l waves. The net wavelength of the motion (or extent of the c e l l s ) i s seen to decrease markedly as the depth approaches zero. This decrease i n wavelength i s accompanied by an increase i n wave amplitude. As the wave approaches the o r i g i n (x = 0) , the bottom slope and curvature cause the l o c a l wavelength to decrease with depth (see §5).. Thus the phase speed i s greater near the surface than that near the bottom. This, together with the incr e a s i n g wave 169. amplitude, produces steepening of the wave fro n t , and hence a wave i s expected to break as i t approaches the o r i g i n . We note . however that the l i n e a r model used here becomes inadequate w e l l before breaking occurs. The streamlines and "wave p r o f i l e s f o r the second mode of the above wave are shown i n Figure 2. In this case, the l o c a l wavelength i s h a l f that of the f i r s t mode wave. There are two complete streamline c e l l s at each x - p o s i t i o n , with a surface 1 of no normal motion at z = —-h and t h i s surface moves upwards as the depth decreases. Although the asymptotic (x ->- °°) amplitude of thi s wave i s equal to that of the f i r s t mode wave, the smaller l o c a l wavelength causes the former wave to break down as i t approaches the o r i g i n before the l a t t e r wave. For the second example, the depth p r o f i l e i s described by h = h [1 + 1 / U + (ex / h ) 2 } ] , (6.2) o o where e i s taken equal to — . The instantaneous streamlines and wave p r o f i l e s i n Figure 3 are f o r the lowest mode wave with co = Y . As a wave passes over the v a l l e y centred at x = 0 , the net wavelength i s seen to increase and the amplitude of the wave p r o f i l e decreases, approximately as l/h(x) . Equation (5.1) implies that the l o c a l h o r i z o n t a l wavelength of the motion depends 2 2 2 upon the quantity a = h(d h/dx ) - 2(dh/dx) . For i 1/2 x/h I > 2(2/3) , a i s greater than zero and so the l o c a l h o r i -o zontal wavelength increases with depth. Consequently, the h o r i -zontal l o c a l phase speed i s greater near the bottom of the ocean than that near the surface and the streamline c e l l s are s l i g h t l y skewed. 170. 1' 2 Hoxjever, f o r | x/h | < 2(2/3) ' , a i s negative and the l o c a l h o r i z o n t a l wavelength becomes a decreasing function of depth. Figure 4 shows the instantaneous streamlines and wave p r o f i l e s f o r a wave with ^ = \ propagating through an ocean given by h = h Q { l - + ( e x / h ) 2 ] } , (6.3) where e = -j ; t h i s i s the inverse of the second example. In this 1/2 case, the quantity a i s p o s i t i v e f o r l x / ^ 0 l < (2/3) and i t i s 1/2 negative f o r l x / n 0 l > (2/3) . Thus, away from the centre of the ridge, the l o c a l h o r i z o n t a l phase speed i s a decreasing function of depth. 7. Sp e c i a l class of exact solutions In §3 the eigenfunctions of the system (2.5) - (2.7) were shown to be given by (3.7) i n which g was a s o l u t i o n of the f u n c t i o n a l equation (3.6). Assuming that h = h(x) , a class of solutions was subsequently found. However equation (3.6) also admits solutions when h = h(x,to) ; i n p a r t i c u l a r , we consider h = h(x/co) and g = (x/cu) so that (3.6) becomes g(x + Kx)) " g(x ~ h(x)) = y » a constant, (.7.1) where x = x/co . This implies that the topography allows only waves 171. of a given frequency to propagate; i . e . the waves are 'tuned' to the topography. Since the depth p r o f i l e h depends upon X/OJ , i t i s c l e a r that a> tends to s t r e t c h the h o r i z o n t a l coordinate. However, the r e a l ocean f l o o r generally varies quite slowly. There-fore, tuned waves would be associated with large values of OJ , which implies, from equation (2.5), that the wave frequency approaches the i n e r t i a l frequency. We also note that while the solutions obtained i n §3 correspond to purely transmissive topographies . (0 (dh/dx) < a) ) , equation (7.1) represents topographies f o r which 0(dh/dx) = to ^  ; i . e . i t represents the l i m i t i n g class of trans-missive topographies. Solutions to (7.1) may be found by considering the !inverse problem'. That i s , the function g i s s e l e c t e d such that the l o c a l h o r i z o n t a l wavenumber, which v a r i e s l i k e dg/dx , has the expected general behaviour f o r a given topography. Then the ac t u a l ocean depth h(x) i s determined from (7.1), and the corresponding eigenfunctions are found from (3.7). For example, a wave propagat-ing over a bottom that asymptotically approaches a constant depth h^ as x -<» and zero depth as x -> +*> , must have a wavenumber that approaches a constant value as x ->- -°° and i n f i n i t y as x ->• +<*>. A s u i t a b l e form f o r g i s , i n t h i s case g(x) = x/ho + a exp(gx/ho) '> a>3 p o s i t i v e constants. Thus, from equation (7.1), i t i s found that the ocean bottom h i s given by 172. X/h = 6 l o g { ( l - h/h )/[« s i n h(gh/h )']} , ( 7 . 2 ) o o o where the constant y i s determined from the condition that h/h^ 1 as x ~ ra> which implies that Y = 2 . Figure 5 shoxvs the streamlines and wave p r o f i l e s associated with a f i r s t mode wave propagating over t h i s topography. As the ocean depth decreases, the wave behaves i n a manner s i m i l a r to that shown i n Figure 1 (§6); i . e . the wave amplitude increases and the h o r i z o n t a l phase speed becomes a decreasing function of depth. A wave propagating up a slope and in t o a region of asymptotically constant depth h Q i s described by g( X ) = log{a + exp ( 3 X/b )} , and hence the act u a l topography i s X/h = -h/h - 3 _ 1 l o g { [ l - exp{26(l - h/h )}] o o o / [ a ( e 2 6 - 1)]} , (7.3) where y i s found to be equal to 2 3 • We note that h ^ -x/io as x -> -oo ; i . e . f o r -x/h >> 1 , the ocean f l o o r corresponds to o a c h a r a c t e r i s t i c of the equation of motion ( 2 . 5 ) . Thus, t h i s ocean represents an example of a l i m i t i n g transmissive slope. A time sequence of the streamlines associated with a f i r s t mode wave are shown i n Figure 6. At time at = 0 , a streamline c e l l of i n f i n i t e extent l i e s along the slope of the ocean f l o o r . Such motion arises 173. because, as x ~y ~m > exp(x + z) ~y 0 while exp(X - z) remains f i n i t e i n the neighbourhood of the ocean f l o o r where X % z • Hence, only one of the component waves (see equation (3.7)) contributes s i g n i f i c a n t l y to the o v e r a l l motion and the c h a r a c t e r i s t i c s x ~ z = constant are e s s e n t i a l l y l i n e s of constant phase. As at increases, the head of the c e l l moves i n t o the s h e l f region and the t a i l contracts so that the streamlines are closed as they enter the s h e l f . At the same time, another c e l l propagates downstream to replace the contract-ing c e l l . Figure 7 depicts the wave p r o f i l e s at time at = 0 . Away from the ocean bottom, there i s almost no motion i i r the slope region. However, near the bottom, the elongated streamline c e l l produces a wave of s i g n i f i c a n t amplitude which could appear as an i n t e r n a l t i d e i n the r e a l ocean. The wave f o r which g(x) = x/h- + a sin(Bx/h- o) , corresponds to the p e r i o d i c topography given by X/h = f 1 c o s _ 1 { ( l - h/h )/[a sin(3h/h )]} . (7.4) o o o In t h i s case, y = 2 , so that h^ i s the average ocean depth. The instantaneous streamlines and wave p r o f i l e s f o r a f i r s t mode wave are i l l u s t r a t e d i n Figure 8. S i m i l a r l y , other examples of topographies that are e x p l i c i t functions of co may be constructed. Now i f equation (7.1) i s such 174. that, given h(x) , the function g(x) i s determined to xvithin an a r b i t r a r y a d d i t i v e constant, then the solutions obtained above are unique. Hence, oceans where h = h(x/co) permit the existence of waves of a given frequency only. 175. BIBLIOGRAPHY Balazs, N.L. (1961) On the s o l u t i o n of the wave equations with moving boundaries. J . Math. Anal. Appl. _3, 472. C a r r i e r , G.F. (1966) Gravity waves on water of v a r i a b l e depth. J . F l u i d Mech. _24, 641. Hurley, D.G. and Imberger, J. (1969) Surface and i n t e r n a l waves i n a l i q u i d of v a r i a b l e depth. B u l l . A u s t r a l . Math. Soc. 1, 29. K e l l e r , J.B. and Mow, V.C. (1969) Int e r n a l wave propagation i n an inhomogeneous l i q u i d of nonuniform depth. J . F l u i d Mech. 38, 365. Longuet-Higgins, M.S. (1969) On the r e f l e c t i o n of wave c h a r a c t e r i s t i c s from rough surfaces. J . F l u i d Mech. _37_, 231. Mooers, C.N.K. (1969) The i n t e r a c t i o n of an i n t e r n a l t i d e with the f r o n t a l zone i n a coa s t a l upwelling region. Ph.D. d i s s e r t a t i o n , Oregon State U n i v e r s i t y , C o r v a l l i s . P h i l l i p s , O.M. (1966) "The Dynamics of the Upper Ocean". Cambridge Uni v e r s i t y Press. Sandstrom, I-I. (1966) The importance of topography i n generation and propagation 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 j San Diego. Thorpe, S.A. (1968) On the shape of progressive i n t e r n a l waves. P h i l . Trans. Roy. Soc. A263, 563. Wunsch, C. (1968) On the propagation of i n t e r n a l waves up a slope. Deep-Sea Res. 15_, 251. Wunsch, C. (1969) Progressive i n t e r n a l waves on slopes. J . F l u i d Mech. 35, 131. Streamlines (a) and wave p r o f i l e s (b) for mode n = 1 corresponding to depth p r o f i l e (6.2) Streamlines (a) and wave p r o f i l e s (b) for mode n = 1 corresponding to depth p r o f i l e (7.2) i n which ct = 1 = g PART I I I THE DIFFRACTION OF INTERNAL WAVES BY A SEMI-INFINITE BARRIER 176. 1. Introduction This work i s e s s e n t i a l l y i n the form submitted f o r p u b l i -cation and i t was w r i t t e n i n close c o l l a b o r a t i o n with L.A. Mysak. In p a r t i c u l a r , however, Dr. Mysak recognized the mathematical corres-pondence between the problem of the d i f f r a c t i o n of long surface waves and that of the d i f f r a c t i o n of i n t e r n a l waves, such that the l a t t e r reduces to a two-dimensional problem f o r a given wave mode. The d i f f r a c t i o n of long g r a v i t y waves by a s e m i - i n f i n i t e v e r t i c a l b a r r i e r i n a uniformly r o t a t i n g system has been considered by Crease (1956) f o r the case of normal incidence and l a t e r by Chambers (1964), for a r b i t r a r y incidence. One of the remarkable r e s u l t s of t h e i r work was that the presence of r o t a t i o n gives r i s e to a K e l v i n wave which propagates x^ithout attenuation i n t o the region behind the b a r r i e r . The d i r e c t i o n of propagation of t h i s wave i s along the b a r r i e r and i t s amplitude f a l l s o ff exponentially i n the d i r e c t i o n normal to the b a r r i e r . Further, for c e r t a i n ranges of the wave frequency and i n c i d e n t angle, the amplitude at the b a r r i e r exceeds that of the i n c i d e n t wave f i e l d . We now consider an analogous d i f f r a c t i o n problem f o r i n t e r n a l waves i n a uniformly r o t a t i n g , stably s t r a t i f i e d f l u i d of constant depth and Brunt-Vaisal'a frequency. We show that the boundary value problem f o r the h o r i z o n t a l s p a t i a l dependence of the nth mode of the d i f f r a c t e d wave f i e l d , i|; , i s formally equivalent to that derived by Crease and Chambers f o r the s p a t i a l dependence of the d i f f r a c t e d long wave, t, . However, because of the existence of 177. two frequency passbands f o r i n t e r n a l waves, the solutions f o r ^ and t, are not i d e n t i c a l i n form. For the case £ < <r < N , . where f,a and N are r e s p e c t i v e l y the i n e r t i a l , wave and Brunt-V a i s a l a frequencies, .the solutions are i d e n t i c a l i n form. . For the case N < a < f , , there arises a wave which travels normal to the b a r r i e r without attenuation and which has an amplitude that f a l l s o f f exponentially i n the d i r e c t i o n along the b a r r i e r . 2. Formulation of boundary value problem We consider the time-dependent motion of a uniformly rotat-i n g , i n v i s c i d and incompressible f l u i d i n the domain R : - 0 0 < x, • y < 0 0 , -h < z < 0 e x t e r i o r to the s e m i - i n f i n i t e s t r i p y = 0 , x < 0 . Here x,y,z form a right-handed ca r t e s i a n system with z as the v e r t i c a l coordinate. The mean sea l e v e l i s at z = 0 and the ocean bottom at z = -h , a constant. The l i n e a r i s e d equations f o r the conservation of momentum, mass and volume are u - f v + p /p = 0 , t x o v + f u + p la = 0 , t y ro w t + P z/p o + og/p o = 0 , _ (2.1) p + p w = 0 , r t ' oz u + v + w =0 x y z (2.2) (2.3) 178. Here u,v,w are the perturbation v e l o c i t y components i n the x,y,z d i r e c t i o n s r e s p e c t i v e l y ; p i s the perturbed pressure; P G ( Z ) a n ^ p are the mean (stable) and perturbed density f i e l d s r e s p e c t i v e l y ; f i s the i n e r t i a l frequency and g i s the a c c e l e r a t i o n of gravity. Applying the Boussinesq approximation (see Phil l i p s . 1 9 6 6 , §2.4), p^ and p are now treated as constants. On the boundaries of R oz we impose the conditions w = 0 on z = 0 , -h (2.4) v = 0 on y = 0 , x < 0 . The boundary condition at z = 0 implies that there i s no v e r t i c a l motion at the sea surface; i . e . coupling between surface and i n t e r n a l waves i s ignored i n t h i s a n a l y s i s . From (2.1) - (2.3) i t follows that the v e r t i c a l v e l o c i t y component s a t i s f i e s the equation Aw + f 2w + N 2A„w = 0 (2.5) t t zz II 2 2 2 2 where ATT = 3 +3 , A = ATT + 3 and N = -gp /p i s the Brunt-H x y II z oz Va i s a l a frequency, assumed to be constant. The h o r i z o n t a l v e l o c i t y components are r e l a t e d to the v e r t i c a l v e l o c i t y component by the equations u = L(w + fw ) , zt xt y ' v = L(w - fw ) , zt yt x (2.6) 179. 2 2 - 1 2 2 where L = (3 + f ) (3 + N ) t t Upon, assuming a p e r i o d i c time dependence w(x,y,z,t) = W(x,y,z) exp(-iat) , a > 0 , (2.7) and def i n i n g 2 2 2 2 2 w = (N Z - a )/(a - f Z ) , (2.8) equation (2.5) becomes co2A W - W = 0 . (2.9) H zz From (2.4), (2.6) and (2.7) we f i n d that the boundary conditions for W are W = 0 on z = 0 , -h , (2.10a) W - i YW = 0 on y = 0, x < 0 , (2.10b) y x where y = f/a Let W be a wave approaching the b a r r i e r at an angle 0 ( 1 0 1 < IT) to the x-axis (see f i g u r e 1). We assume W has the o form W = Y s exp l - i k (x cos 0 + y s i n 0 ) 1 sin(mrz/h) , (2.11) o 'j n n n=l 180. 2 2 where k~ = (mr/a)h) > 0 ; c l e a r l y W , being a superposition of normal modes, s a t i s f i e s (2.9) and (2.10a). The c o e f f i c i e n t s s , n which are assumed known, determine the shape of the i n c i d e n t wave. 2 The condition k > 0 together with (2.8) implies that two frequency passbands are p h y s i c a l l y r e a l i s a b l e : f < a < N and N < a < f . Since the b a r r i e r gives r i s e to a d i f f r a c t e d wave f i e l d which must also s a t i s f y (2.10a), we take the t o t a l wave f i e l d W to have the form W = I s i|>T(x,y) sin(mrz/h) , (2.12) n=l T where ^ = exp[-ik^(x cos G + y s i n 0)] + i^(x,y) . Hence (x,y) represents the h o r i z o n t a l s p a t i a l dependents of the nth mode of the d i f f r a c t e d wave f i e l d . The s u b s t i t u t i o n of (2.12) into (2.9) and (2.10b) gives, upon dropping the subscript n , ( A „ + k 2 H = 0 , (2.13a) ii (x,0) - iy\b (x,0) = a exp (-ilex cos 0) , x < 0 , (2.13b) y x where a = k ( i s i n 6 + y cos 0) . To determine a unique s o l u t i o n to (2.13a,b) we also specify that ( i ) ib . s a t i s f i e s the Sommerfeld r a d i a t i o n c o n d i t i o n , ( i i ) \b i s bounded every\-7here i n R and i|* (0+,0) and \p (0+,0) are integrable and ( i i i ) ij; and x y ii< - iyip are continuous across y = 0 for x > 0 and a l l x y x r e s p e c t i v e l y . Condition ( i i i ) , v;hich ensures that w and v are continuous across y = 0 for x > 0 and a l l x r e s p e c t i v e l y , also ensures that u i s continuous across y = 0 for x > 0 . 181. From (2.8) and (2.11) we see that the wave eigenmodes s a t i s f y the dispersion r e l a t i o n K 2 = ( a 2 - f 2 ) / ( N 2 - a ) , (2.14X where K = kh/nT . Equation (2.14) implies that a = o(k) , i . e . that a depends upon the magnitude of the h o r i z o n t a l wavenumber vector (k^jk^) . Thus, the wave group v e l o c i t y components c _^  are given by - 9 o n / i \ d a c . = — = (k./k) -rr-g i dk x dk Now the wave phase v e l o c i t y components c^ are defined as c. = k.a/k2. , x 1 and hence, from (2.14), the sc a l a r product of the phase and group v e l o c i t i e s i s c c - | f f - h V - f 2 ) . ( 2.15) ~ g k 3k , . 2 ... , 2, 2 6 (mr) (1+K ) This implies that whereas the phase and group v e l o c i t i e s are i n the same d i r e c t i o n when f < a < N , the group v e l o c i t y has the opposite sign to that of the phase v e l o c i t y when N < a < f • Thus, i n order that the vector i n f i g u r e 1 represents the d i r e c t i o n of incoming wave energy, we take k > 0 for f < a < N and k < 0 for N < o < f i n (2.11) - (2.13a,b). Equation (2.15) must also be taken into consideration when applying the Sommerfeld r a d i a t i o n condition, namely, that the d i f f r a c t i o n wave f i e l d i> contains out-ward propagating energy only. We note here that equations (2.13a,b) with k > 0 also govern the behaviour of the d i f f r a c t e d wave due to a long gravity wave incident upon a s e m i - i n f i n i t e v e r t i c a l b a r r i e r i n a r o t a t i n g system (see Crease 1956, Chambers 1964). In that case ^ represents the s p a t i a l dependence of the sea surface e l e v a t i o n and 2 2 2 k = (a - f")/gh > 0 i s the wave number of a f r e e l y propagating long wave. Further, f o r long waves i n a r o t a t i n g f l u i d i t follows that c*c = gh > 0 . Thus the two d i f f r a c t i o n problems are mathe-m a t i c a l l y equivalent when y < 1 i n the sense that both can be reduced to the same boundary value problem (BVP) which i n turn reduces to the BVP associated with the c l a s s i c a l Sommerfeld d i f f r a c t i o n problem when y = 0 (no r o t a t i o n ) . 3. I n t e g r a l representation of s o l u t i o n f o r f < a < N To construct the s o l u t i o n to (2.13a,b) (with 6 = TT/2) , Crease (1956) f i r s t converts the BVP into an equivalent i n t e g r a l equation. From the l a t t e r he derives a Wiener-Hopf i n t e g r a l equation fo r a function m(x) (see (3.5) below) whose kernel consists of the appropriate Green's function f o r the BVP. He then uses the Wiener-Hopf method to solve t h i s second i n t e g r a l equation f o r m(x) . The s u b s t i t u t i o n of m(x) into the o r i g i n a l i n t e g r a l equation then y i e l d s , 183. a f t e r some manipulations, a Fourier i n t e g r a l f o r the d i f f r a c t e d wave. Crease's approach i s unnecessarily complicated, however, since (2.13a,b) can be solved d i r e c t l y by the Wiener-Hopf method without f i r s t recasting the BVP in t o an i n t e g r a l equation. We b r i e f l y o u t l i n e this a l t e r n a t i v e approach (see C a r r i e r et a l 1966, §8.1) below. Crease's problem (for a r b i t r a r y 6) has also been solved by Chambers (1964) who used a method previously developed by him-s e l f (Chambers 1954). Chambers assumes that the d i f f r a c t e d wave i s equal to a l i n e a r combination of " d i f f r a c t i o n functions" each of which i s proportional to a c e r t a i n complex Fresnel i n t e g r a l . The unknown c o e f f i c i e n t s i n t h i s equation are then determined by appro-p r i a t e boundary conditions. While Chambers' approach i s f a i r l y elegant, i t does not appear to be any simpler than the method we now give. Upon applying to (2.13a) the Fourier transform with respect to x as defined by 00 F(A,y) = / e i A x F(x,y)dx , —00 we obtain i|>yy - U 2 - k 2 H = 0 . (3.1) The s o l u t i o n to (3.1) which i s bounded as |y| -> °° i s 184. |A(A) exp[y(A 2 - k 2 ) 1 / 2 ] , y < 0 kB(X) exp[-y(A 2 - k 2 ) 1 7 2 ] , y > 0 , _ (3.2) 2 2 1/2 where the branch of (A - k ) i s chosen so that 2 2 1/2 arg (A" - k ) -> 0 as A ->- + « . I t follows that the r a d i a t i o n condition i s s a t i s f i e d i f k i s assumed to have a small p o s i t i v e imaginary part, v i z . k = k Q + i e (k > 0 , E > 0) . Thus we draw the branch cuts from A = + k i n the A-plane as shown i n f i g u r e 2. The continuity of \}> - i Y ^ x across y = 0 implies ,,2 .2.1/2 B(A) = ( Y X " U ? ~ V 1 / ? ) A ( A ) yX + (A Z - k Z ) ± / Z (3.3) To determine A(A) , and hence B(A) , we now introduce two half-known functions g(x) and m(x) by the equations * ( x,0) - i Y * x ( x , 0 ) = < rg(x) , x > 0 , La exp(-ikx cos 0) , x < 0 , (3.4) i K x,0+) - i | » ( x,0-) = m(x) , (3.5) where ;(x) = , x > 0 , (3.6) 0 , x < 0 185. r0 , x > 0 , m(x) = j V (3.7) 1 , x < 0 . Transforming (3.4) and using (3.2) (for y < 0) and (3.6) we obtain [ ( x2 _ k 2 ) 1 / 2 - Y X ] A(X) = ( r — - " - i 3 — - ) _ + , (3.8) where the subscript - denotes a minus function, which i s a n a l y t i c i n the lower h a l f plane (LHP) Im X < e cos 6 and the subscript + denotes a plus function, which i s assumed to be a n a l y t i c i n the UHP Im X > - e. Transforming (3.5) and using (3.2) and (3.7) we obtain B(A) - A(X) = m_(X) , (3.9) where m_ i s assumed to be a n a l y t i c i n the LHP Im X < e . Using (3.3) i n (3.9) and s u b s t i t u t i n g the r e s u l t f o r A in t o (3.8) we obtain the Wiener-Hopf equation 2 2 2 -[k - (1 - y )X ]m(X) = - i a , ~ m 2 ( A 2 _ k 2 j l / 2 X - k cos 9 & K J ' which can be rewritten i n the form (X 2-1)(X-X )m(X) . , , fl.l/2 r _ o i a ( k + k cos 8) , 2 ( X - k ) 1 / 2 ( A o + k C O S 0 ) . ( X " k c o s S ) - i a [ ( X + k ) 1 / 2 ( X +k cos G) + (k+k cos 0) 1 / 2(X+X )] , ^ l X l / 2 - , l N _ r o • o (X+k) g(A)-. '• (X+X Q)(X-k cos G)(X Q+k cos 6) ( A +V (3.10) 186. where A = k ( l - Y 2) 1 / 2 The l e f t hand side of (3.10) i s a n a l y t i c f o r Im X < e cos 6 , and the r i g h t hand sid e i s a n a l y t i c f o r Im X > -e . Thus (3.10) holds in. the s t r i p -e < Im X < e cos 6 and defines an e n t i r e function E(A) which i s a n a l y t i c i n the whole A-plane by a n a l y t i c continuation. Since \b(0-,0) i s bounded and ^(0+,0) , <J> (0+,0) are integrable, i t follows that m = 0(X ) as | A | ->- 0 0 i n the LHP and g = 0(A ) (6 > 0) as [ A | -> 0 0 i n the UHP r e s p e c t i v e l y . Therefore each side of (3.10) tends to zero as | A | °° i n the s t r i p , and, by L i o u v i l l e ' s theorem E(A) = 0 . Thus, upon s e t t i n g the l e f t hand side of (3.10) equal to zero, we obtain •Mv -2ia(A~k) 1 / 2(k+k) cos 6 ) 1 / 2 m(.A) = ^ . (.3.11) (Y -D(A +k cos 6) (A-A ) (A-k cos 0) o o Taking the inverse Fourier transform of (3.2) i n which A and B are now determined from (3.3), (3.9) and (3.11), we obtain 2 2 1 / ? ,/ V b YX-sgn y ( A - k ) ' r .. • ,,.2.2,1/2, i(;(x,y) = Tr- JdX J ^ — j-pr exp[-iAx - |y|(A -k ) ] 1 1 r (A-A )(A-k cos 0)(A+k)- L / Z o (3.12) 1/2 2 where b = k ( s i n 0 - i y cos 0)(k+k cos 0) /(1-y )(X +k cos 0) and o the i n v e r s i o n path T l i e s i n the s t r i p -e < Im A < e cos 0 , as shown in f i g u r e 2. I t can be v e r i f i e d that (3.12) indeed s a t i s f i e s (2.13a) and the associated boundary conditions. Further, f o r 0 = IT/2, 187. equation (3.12) reduces to Crease's Fourier i n t e g r a l representation of the s o l u t i o n f o r the d i f f r a c t e d wave. 4. Asymptotic s o l u t i o n f o r f < a < N To determine the general nature of the d i f f r a c t e d wave (3.12), we now consider the asymptotic form of <jj f o r large 2 2 1/2 kr = k(x + y ) . However, i n contrast to Crease (1956) who f i r s t transforms the s o l u t i o n i n t o complex Fresnel i n t e g r a l s before determining the asymptotic behaviour, we apply the method of steepest descent d i r e c t l y to (3.12). L e t t i n g x = r cos , y = r s i n <j> , where -TT < <j> < TT , and X = k£ , equation (3.12) becomes 1/2 ,, „ s (sin9 - iycos 0)(1 + cos 0) r r . <jj(r,0) = ^ — / dt, exp[-kr ( i c cos 2TT(1 - y ) (£ + cos 0) T* 4 - l c , - „ A\fr2 -\\ll2l [YC-sgn cb ( C 2 - D 1 / 2 n + |sm cb|(c; - l ) J r-7y , (4.1) (C-e 0)(?-cps6) (1+0 where r = X k 1 and the path of i n t e g r a t i o n T* i s as shoxvn i n o o f i g u r e 3. We have taken e = 0 , because i t has served i t s purpose of determining the correct:contour. From the argument of the exponential function, i n (4.1), we f i n d that the saddle point i s located at ? g = ~ c o s <t> a n d that the corresponding path of steepest descent, T g , i s given by 188. (C + cos tj>)(C cos <ji + 1) 3 i 2 ^ 1 / 2 ' " i n (j>|[(C + cos <j>) " + s i n " <j>] where £ , £. are r e s p e c t i v e l y the r e a l and imaginary parts of r, . r i Thus, three cases f o r T g are possible, v i z . cos <j> > 0 , cos <j> = 0 and cos <*> < 0 , as shown i n figu r e 3. The con t r i b u t i o n to (4.1) from the saddle point i s 1 / 9 i. ( rh) [ s i n (j)[ ( i s i n <j> - ycos <j>) ( s i n 6 - i y cos 6) (1+cos 9) V r 2 1/2 (1-Y )(C Q+cos 0)(? Q+cos (j))(cOS <f) + cos 6)(l-cos <J>) i(kr-7r/4) (2iTkr) (kr) Equation (4.2) i s not v a l i d , however, f or cos <j> - - cos 6 or cos 6 = 1 . From f i g u r e 3 i t i s cl e a r that the contour r* can be deformed i n t o T without capturing any poles provided that cos <J) > - cos 0 and cos <j> > - l / ? 0 • Then, by Cauchy's theorem, (4.2) i s the leading term of the asymptotic s o l u t i o n f o r if> . But when cos <j> < - cos 0 or cos d> < , there are a d d i t i o n a l o contributions from the poles at £ = cos 0 or r, = £ , re s p e c t i v e l y . T Thus, as kr ->• 0 0 , the t o t a l wave mode $ can be wri t t e n i n the T form \p = ip (r,<f>) + [H(cos <b + cos 0) + H(-cos 6 - cos 0) s 2 ~ T T / ± • n\ i r -w n , ^ \ i , ( s i n 0 - i y cos 6) x H(sgn <j> sxn 0)] exp[-ik(x cos 0 + y s i n 0)] + ^ ! ~ — (Y cos 0 + s i n 0) x H(-cos (j) - cos 0)H(sgn <J> s i n 0) exp[-ik(x cos 0 - y s i n 0)] 2(C - D 1 / 2 ( l + cos 9 ) 1 / 2  + - ( Y ° C O S 0 - i s i i T i ) H(-y)H(C o + 1/cos <j>)exp[ ^ k ( y y - i x ) ] , (4.3) 189. where t, = (1 - y ) and H(x) i s the Heaviside u n i t step function. The f i r s t term i n (4.3) represents a modulated c y l i n d r i c a l wave propagating out from the o r i g i n ; i t i s given e x p l i c i t l y by equation (4.2). The incoming free wave i s given by the second term i n (4.3). It e x i s t s i n the regions I and II shown i n f i g u r e 4. The s e m i - i n f i n i t e b a r r i e r gives r i s e to a r e f l e c t e d free wave i n the region I I (figu r e 4); th i s i s described by the t h i r d term i n (4.3). Although the incoming and r e f l e c t e d waves are of the same amplitude, they are not i n phase at the b a r r i e r unless ei t h e r y - 0 or cos G = 0 , i . e . unless there i s no r o t a t i o n or the incoming wave propagates normal to the b a r r i e r . This phase change i s a con-sequence of the mixed boundary condition (2.13b). Behind the b a r r i e r , there i s a 'shadow zone' (region III) i n which no free wave e x i s t s . The l a s t term i n (4.3) describes an i n t e r n a l K e l v i n wave propagating away from the o r i g i n with the b a r r i e r to i t s r i g h t (f > 0) ; i . e . i t appears i n region IV of figu r e 4. In f i g u r e 5, the amplitude of the Ke l v i n wave at the h a r r i e r i s p l o t t e d as a function of y for fi x e d values of 6. As in d i c a t e d by Crease (1956), this wave can have a greater amplitude than that of the incoming wave. Because the amplitude i s independent of k , i t i s also independent of N , the Brunt-Vais.ala frequency. When y > 0 and 0 > 0 , as shown i n figu r e 4, the Ke l v i n wave appears i n the shadow zone behind the b a r r i e r . However, when one of the parameters y and 0 i s negative, the wave e x i s t s on the same side of the b a r r i e r as the incoming wave. 190. 5. Asymptotic s o l u t i o n f o r N < g < f For the passband N < a < f , the waves described by (2.13a,b) are not p h y s i c a l l y equivalent to long surface waves because 2 i n the l a t t e r case, k < 0 when a < f and hence free waves of the type (2.11) do not e x i s t . The waves of (2.13a,b) are not s t r i c t l y i n e r t i a l i n t e r n a l waves. I t was shown i n §2 that to consider the d i f f r a c t i o n problem f o r the case N < a < f which i s p h y s i c a l l y s i m i l a r to that f o r the case f < a < N , we must replace k by -k i n (2.11) - (2.13). Now the Sommerfeld r a d i a t i o n condition i s s a t i s f i e d by assuming i n (3.2) that k has a small negative imaginary part. Thus, the X-plane f o r th i s case i s given by the r e f l e c t i o n of f i g u r e 2 across the Im X-axis, except that the poles at X = +_ X q are moved to X = + i | X I . The d i f f r a c t e d wave ib i s f i n a l l y found to be given — o 2 -1/2 by (3.12) with k and X q replaced by -k and i k ( y - 1) , re s p e c t i v e l y ; the contour r now passes below the s i n g u l a r i t i e s at X = -k and -k cos 0 and passes above the branch point at X = k . To determine the asymptotic form of \b as kr 0 0 , we again apply the method of steepest descent. The saddle point i s now located at £ g = cos <b , and the possible paths of steepest descent are given by the r e f l e c t i o n of fig u r e 3 about the Im £-axis, with the pole at r moved to ± I r 1 The contribution to ' 'o 1 o 1 \b from the saddle point i s 191 . '1/9 \h (r cb) = ls:i-n $\ s j - n $ " Ycos cb) ( s i n 9 - i y cos 6) (1 + cos 0)  S ( y 2 - 1) (±Z1 - cos 0)(i£ - cos cb) (cos cb + cos 9)(l-cos cb) 1 7' 2 -i(kr-rr/4) x ~ 7 7 T i 7 T - - + 0 [ 7 r f 3 7 2 ] ' ( 5 - 1 } (2-rrkr) (kr) 2 -1/2 where £ = (y - 1) . Thus, using Cauchy's theorem, the t o t a l T wave mode \p has the asymptotic form ,T , , r T T / . , . ,,, , ,, T 1 / , , ik(xcos6+ysin6) \b - IJJc + [H(cos<b + cos0) + Il(-coS(j> - cos9)H(sgn<j> sgn0)]e 2 , (sinu - iycos0) . . . , . ik(xcos0 ~y sin0) — 2 2 — 2~ ^ " C C " 3 * ~ cos0)H(sgncb sgn0)e J (y COS 9 + s i n 0) 1/2 1/2 2 Y ' (1 + c o s e ) 1 ^ r. -1,2 ..-1/2,.,, . Y Yfl exp[i tan (y -1) ]H(-y) (Y ""I) (ycosG - i s i n 0 ) H(-coscb - (Y 2-D 1 / 2 I s i n $ \) exp [ ( y 2 - l ) " 1 / 2 k ( x + i y y ) ] , (5.2) where ty^ i s given by (5.1). C l e a r l y , the (x,y)-plane may be divided as i n f i g u r e 4, except that the Stokes l i n e bordering region IV at 2 —1/2 cos cb = -1/C i s replaced by one at cos cb =-(y - 1) |sin <j>| . The amplitudes of the waves i n region I to III are exactly the same as for the case f < a < N ; however, the phase of each wave i s now propagating i n the opposite d i r e c t i o n . The l a s t term i n (5.2) represents an i n e r t i a l - i n t e r n a l wave propagating i n the region IV away from and normal to the b a r r i e r and decaying i n the negative x-d i r e c t i o n . The maximum amplitude of th i s wave i s shown as a function of y f ° r f i x e d values of 0 i n figu r e 5; as for the case f < a < N , i t can also be greater than the amplitude of the incoming 192. wave. However, since these waves only appear at large distances from the o r i g i n ( i . e . large kr) , t h e i r actual amplitude i s somewhat less than that shown i n fi g u r e 5. On the other hand, since the r a t i o of the decay length of the wavelength i s y > the attenuation i n the negative x - d i r e c t i o n i s small over one wavelength f o r very low frequencies ( i . e . f o r a « f) . 6. Discussion of i n t e r n a l K e l v i n waves The existence of i n t e r n a l waves of K e l v i n type i n region IV of f i g u r e 4 i s understood by considering normal mode solutions of (2.9) and (2.10). These equations admit solutions of the form I7 tlx +my . , , . rr-w W = e ' sin(mrz/h) , (6.1) where £ 2 = k 2 / ( l - y 2 ) = (mr/h) 2a 2/(N 2 - a 2) and m = - y l . The comparable normal mode long surface gravity wave problem (see §2) has the s o l u t i o n i£x + my . . ijj = e (6.2) 2 2 2 2 where I = k /(1-y ) = a /gh > 0 and m = ~ y l . Equation (6.2) represents the f a m i l i a r K e l v i n wave s o l u t i o n i n which the wave i s trapped against the b a r r i e r ; i . e . the wave t r a v e l s i n the x-d i r e c t i o n and i t s amplitude decays exponentially away from the b a r r i e r . 193. From (6.1), i t i s seen that the i n t e r n a l wave analogue of this wave 2 occurs when a < N , i . e . when l" > 0 . Applying the r a d i a t i o n condition that energy must be propagating away from the o r i g i n , we f i n d that f o r y > 0 Kelvin waves can e x i s t i n the t h i r d quadrant only. Since they do not s a t i s f y a con t i n u i t y condition at the plane x = 0 , they cannot e x i s t alone. In f a c t , free waves present along the plane x = 0 (y < 0) should tend to ex c i t e K e l v i n waves which are "natural o s c i l l a t i o n s " i n t h i s region. I f the semi-i n f i n i t e b a r r i e r i s replaced by a i n f i n i t e one, then K e l v i n waves are exact solutions to the equations of motion. The wavenumbers of both types of Ke l v i n waves are independent of y and the r a t i o of the decay length to the wavelength i s y ^ i n each case. The waves can e x i s t f o r a l l y ( ^ l ) , but they co-exist with free waves 2 only when y < 1 , i . e . when k •> 0 . We note that whereas long surface gravity Kelvin waves are non-dispersive, i n t e r n a l K e l v i n waves are d i s p e r s i v e . When a > N , equation (6.1) describes a wave f o r which 2 £ < 0 . Thus, this i n e r t i a l - i n t e r n a l wave propagates normal to the b a r r i e r and i t s amplitude decays along the b a r r i e r . As before, the r a d i a t i o n condition implies that the wave can e x i s t only i n the t h i r d quadrant of the (x,y)-plane. However, because the amplitude of t h i s wave increases exponentially i n the x - d i r e c t i o n , i t cannot e x i s t alone, even i f the b a r r i e r spans the whole x-axis. The disper-sion r e l a t i o n f o r the wave i s , from (6.1), ? 7 ? ? ? in = (nu/h)V/(a - N Z) 194. The wave i s c l e a r l y d i s p e r s i v e , and i t i s essentiality an i n e r t i a l wave because N may be set equal to zero without a f f e c t i n g the mathematics (or the physics) of the problem. This wave i n region IV described i n §5 might thus be c a l l e d e i t h e r a degenerate i n t e r n a l K e l v i n wave or an i n e r t i a l K e l v i n wave. We f i n a l l y note that although there i s an energy f l u x (TJT Re(pu*)) along the b a r r i e r associated with a steady-state K e l v i n wave (f < a < N) , there i s no energy f l u x associated with a (steady-state) i n e r t i a l K e l v i n wave (N < a < f ) . On the other hand, by allowing the wave amplitude to vary slowly i n time ( i . e . by allowing the frequency a to have a small p o s i t i v e imaginary part, e) , i t i s found that an i n e r t i a l K e l v i n wave i s maintained by an energy f l u x along the b a r r i e r that i s proportional to e . 195. BIBLIOGRAPHY Ca r r i e r , G.F., Krook, M. and Pearson, C.E. (1966) "Functions of a Complex Variable". McGraw-Hill. Chambers, Ll.G. (.1954) D i f f r a c t i o n by a half - p l a n e . Proc. Edinb. Math. Soc. 10, 92. Chambers, Ll.G". (1964) Long waves on a r o t a t i n g earth i n the presence of a s e m i - i n f i n i t e b a r r i e r . Proc. Edinb. Math. Soc. 14, 25. Crease, J . (1956) Long waves on a r o t a t i n g earth i n the presence of a s e m i - i n f i n i t e b a r r i e r . J . F l u i d Mech. 1_, 86. P h i l l i p s , O.M. (1966) "The Dynamics of the Upper Ocean". Cambridge Univ e r s i t y Press. FIGURE 1 Plan view of wave approaching s e m i - i n f i n i t e b a r r i e r y = 0 , x < 0 FIGURE 2 The X-plane corresponding to (3.10) for f < a < N FIGURE 3 The C-plane corresponding to (4.1) when f < a < N for the cases (a) cos <j> > 0 , (b) cos 6 = 0 and (c) cos <f> < 0 . The path of steepest descent i s denoted by ^ s FIGURE 4 Regions around b a r r i e r i n which terms of asymptotic s o l u t i o n (equation (4.3)) become important f o r the case f < a < N and where (a) 0 < 0 < TT/2 and (b) TT/2 < 9 < TT FIGURE 5 Maximum amplitude of i n t e r n a l K e l v i n wave 

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