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The generation of unstable waves and the generation of transverse upwelling : two problems in geographical… Lee, Clinton Arthur 1975

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THE GENERATION OF UNSTABLE WAVES AN D THE GENERATION OF TRANSVERSE UFREL-IIKG: TWO PROBLEMS IN GEOPHYSICAL FLUID DYNAMICS CLINTON ARTHUR LEE B.A., U n i v e r s i t y cf C a l i f o r n i a , l a J o l l a , 1968 M.Sc. , U n i v e r s i t y of B r i t i s h Columbia, 1970 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOB OF PHILOSOPHY in the I n s t i t u t e cfOceanography and the I n t i t u t e of Applied Mathematics and S t a t i s t i c s accept t h i s t h e s i s a;s ^ o n ^ r m i n g to th-^ r e q u i r e d standards THE UNIVERSITY OF BRITISH COLUMBIA February, 1975 In presenting th i s thesis in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th i s thes i s for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th i s thes is fo r f inanc ia l gain sha l l not be allowed without my writ ten permission. Department of The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada Date k f r J ? f l / l ? ? f ABSTRACT In t h i s thesis two d i f f e r e n t problems i n geophysical f l u i d , dynamics ars studied. In Part A we consider the general problem of the generation of unstable shelf waves, a problem that re l a t e s to the generation of the meanders i n the Gulf Stream. In Part B we consider the generation df transverse motions of the thermocline i n long two layer bodies of water by general wind stresses. The i n s t a b i l i t y of f l u i d systems has been studied for a long time, but the problem of how the i n s t a b i l i t i e s are generated and grow has been neglected. Recently, however, in the study of the in t e r a c t i o n of plasmas with electron streams, techniques have been developed to study the.growth of i n s t a b i l i t i e s . These techniques can i n p r i n c i p l e be applied to any unstable l i n e a r system that i s excited by stationary forcing. In Part A we describe these techniques and axtend them to cover the case of moving forcing e f f e c t s . He then use the r e s u l t s to study the generation of unstable shelf waves. Long barotropic waves trapped on an abrupt change i n bottom topography are shelf waves; the presence of l a t e r a l shear i n the persistent ocean currents gives r i s e to unstable shelf wavas. The possible presence of such waves in the Gulf Stream system could explain the generation of the meanders in the Gulf Stream. Thus we study the response of a model Gulf Stream that supports unstable s h e l f waves to the wind. Only c u r l free wind stress i s considered, mainly f o r convenience. I t i s found that only on the offshore side of the stream, where the response i s always larger than on the inshore side, are the unstable waves always dominant. A aind system moving slowly i n the d i r e c t i o n of the stream i s the most e f f i c i e n t at generating the unstable waves, but i t s e f f i c i e n c y i s affected guite d r a s t i c a l l y by the duration of the iisturbance. A wind system moving counter to the stream i s less e f f i c i e n t by about a factor of eight, but i s p r a c t i c a l l y unaffected by a change i n duration. In Part B we consider a stable system. We use simple Fourier expansion to study the motions of the thermocline i n an i n f i n i t e l y long two layer body of water generated by both long axis and cross channel winds. It i s found that i n a aide l.ake; such as Lake Michigan, cross channel winds are more e f f i c i e n t i n generating transverse motions which i n a l l cases are multi-modal. Here wide means wide i n comparison to the Rossby radius of curvature. These r e s u l t s agree only q u a l i t a t i v e l y with observations made on Lake Michigan. In f a c t reasonable wind stresses only give responses about one-tenth of those observed. In a narrow lake i t i s found that the response to a long axis wind i s larger than that to a cross channel wind, both giving a uni-modal response. I l l Observations of wind and thermocline depth taken at Babine Lake i n northern B. C. agree quite well with the theory. TABLE OF CONTENTS ABSTRACT . . . . . i TAELE OF CONTENTS . . i v LIST OF FIGURES FOR FART A . . . v i LIST OF FIGURES FOR PART B i X ACKNOWLEDGEMENTS . . . x i PART A: The Genera t i on Of Uns tab le Waves 1 CHAPTER I: I n t r o d u c t i o n 2 CHAETER II: Uns tab le S h e l f Waves 17 SECTION 1. I n t r o d u c t i o n 17 SECTION 2. The Equat ions . . . 1 7 S E C T I O N 3. T h e T r a n s f c r o i e d E q u a t i o n s .25 SECTION 4. The I n v e r s i o n I n t e g r a l 28 CHAPTER I I I: Asya ip tot i c s . . . 3 3 SECTION 1. I n t r o d u c t i o n To The Method For F ( k , s ) And A(k,s) E n t i r e . . . . . . . 3 3 • SECTION 2. E x t e n s i o n To &,(k,s) And F (k , s ) With Branch P o i n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^1 SECTION 3. T r a n s i t i o n V e l o c i t i e s 45 SECTION 4. Moving D i s t u r b a n c e s 48 SECTION 5. Numer ica l Methods . . . . 5 2 CHAPTER IV: D i s c u s s i o n . And R e s u l t s . . . . . 5 8 SECTION 1. I n t r o d u c t i o n . . . . . . . . . 5 8 SECTION 2. The D i s p e r s i o n R e l a t i o n 59 SECTION 3. Response To A S t a t i o n a r y D i s tu rbance .65 V SECTION 4 . . Mcving D i s t u r b a n c e s ......67 SECTION 5. D i s c u s s i o n 73 CHAPTER V: C o n c l u s i o n . .76 FIGURES FOR PART A 80 REFERENCES .TOR PART A 105 PART B: T ransver se Upwel l rng In A Long Narrow Lake . . . 108 CHAPTER I : I n t r o d u c t i o n ....1.09 CHAPTER I I : Fo rmu la t i on 111 CHAPTER i l l : The S o l u t i o n s . . . . . . . . . . 1 1 4 SECTION A. Lona Ax i s Wind 114 SECTION B. C r e s s Channel Wind 119 CHAPTER I V : Computed R e s u l t s 122 SECTION A. B a h i n e La'kfc . . . . . . . . . . . . . . . . . 1 2 3 SECTION B. L a k e M ich igan 126 CHAPTER V: Comparison Viith O b s e r v a t i o n s For Eabine Lake 130 CHAPTER VI: C o n c l u s i o n 133 FIGURES FCR FART B 134 REFERENCES FOR PART B 157 v i LIST OF FIGURES FOR PART A Figure II.1 The depth and depth averaged v e l o c i t y p r o f i l e representative of the Gulf Stream over the Bla ke Plataeu. Figure I I . 2 The simple model, Figure III.1 The branch l i n e s of G(y,s) are due to k (s) crossing the r e a l k a x i s . Figure III.2 Indenting the Fourier contour around encroaching k(s). Figure III.3 S i n g u l a r i t y in G(y,s ) when k +(s) and k_(s) merge across the r e a l k axis to k o = k + (s„)=k_ ( s 0 ) . Figure III.4 Steepest descent: interpreta tion of the c r i t e r i o n . (a) The saddle point contributes since ±» are i n opposite valleys. (b) The saddle point does not contribute since + » i s on a h i l l , and -«© i s in a v a l l e y . Figure III.5 Transition v e l o c i t y at V=VX due to i n t e r a c t i o n of saddle points at A and B. Figure IV. 1 The dispersion curve Im(s) vs. Ra (k) with lm(k)=0 for ^(k,s) given i n (11.38) with > =.22 and p =2.8. Figure IV.2 Re (k) v s V found from (III.36«) and (III.37') for stable waves. Figure IV.3 (a) Re (k) vs V for unstable waves. (b) Im(k) vs V f o r unstable waves. (c) Re (s) vs V for unstable waves. (d) Im(s) vs V f o r unstable wavas (C) and f o r stable waves. The branch (A) i s for DB branch v i i Figure iv.4 Figure iv.5 Figure iv.6 Figure iv.7 Figure iv.8 Figure IV.9 Figure IV.10 Figure IV.11 Figure IV.12 Figure IV. 13 Figure iv.14 i n Figure IV. 2 ana the (B) i s for EC. Interaction cf saddle points at V=.00470 26252. (a) V=.005-both contribute (b) V=.0047026252-interaction (c) V = . 004-only the unstable wave contributes. Response to impulsive stationary forcing for x=1,0 with t=100 and 200 as a function of y. Response to impulsive stationary f o r c i n g for y=4.0 with t=200 as a function of x. Response to impulsive stationary f o r c i n g for x=2.0 with t=100 and 200 as a function of y. Response to impulsive stationary forcing for y = 20 as a function of t with (a) x=1 and (b) x=2. V=Ve i s a t r a n s i t i o n v e l o c i t y f o r a moving disturbance i f the saddle point i s contributing. V=V0 i s not a t r a n s i t i o n v e l o c i t y i f the saddle point i s not contributing. V=VT found by C3 i s a t r a n s i t i o n v e l o c i t y i f Re(s, ) >Re(s 0) . (a) ^ T ( 2 , y , 1 0 0 ) and %{2, y, 2 0 0 ) , (b) 1^(2,y,100) and >vF(2,y,200) , (s) t T ( 1 r y # 1 0 0 ) and *M1,y, 2 0 0 ) , for V* = .03. «Vx(2,y,100) and *VT(2,y,200) for V* = -.1. (a) tT(2,y,100) and ^(2,7, 200) for V* = .03 and s*=-.06. V l l l (b) *VF(2,y,100) ana ^(2^,200) f o r V*=. 03 ana s*=-. 06. . Figure IV. 15 Mv(2,y f100) and ^2,yt200) f o r V*=-.1 and s*=-. 06. ix LIST OF FIGURES FD3 PART B Figure 1 Cross section of the model. Figure 2 P r o f i l e s of depth anomaly f o r upper layer, h, -H, , aft e r a 1 dyne/cm2 long shore wind s t r e s s has acted for 22 hrs, for A=10s, 10* and 10 3 cm 2/sec, for Babine Lake model. Figure 3 P r o f i l e s of long shore v e l o c i t y i n the upper layer, v, , a f t e r a 1 dyne/cm2 long shore wind stress has acted for 22 hrs, for A=105, 10 3 cm 2/sec, for Babine Lake modal. 10* and Figure 4 Maximuoa value of h, -H, for a 1 dyne/cm2 long shore wind stress as a function of time with A=10s, 10* and 10 3 cm 2/sec, for Babine Lake model. Figure 5 Maximum value of v, for a 1 dyne/cm2 long shore wind stress as a function of tima with A=10s and 10* cm 2/sec, for Babine Lake model. Figure 6 Maximum value of cross stream v e l o c i t y , u,, for a 1 dyne/cm2 long shore wind stress as a function of time with A=10s and 10* cm 2/sec, for Babine Lake model. Figure 7 Maximum value cf h, -H, as a function of time for a 1 dyne/cm2 cross channel wind stress as a function of time with A=105 and 10* cm 2/sec, for Babine Lake model. . Figure 8 P r o f i l e of h,-H, a f t e r a 1 dyne/cm2 cross channel wind stress has acted for 22 hrs, with A=105 and 10* cm 2/sec, for Babine Lake model. Figure 9 P r o f i l a of h, -H, after a 1 dyne/cm2 cross cheannel wind stress has acted f o r 1,2 and 3 days with A=106 cm 2/sec, for model Lake Michigan. Figure 10 P r o f i l e of h, -H, after 1 dyne/cm2 long shore wind stress has acted f o r 10, 20 and 30 days with A=106 cm 2/sec, for model Lake Michigan. Figure 11 P r o f i l e of v, af t e r a 1 dyne/cm2 long shore wind s t r s s has acted for 10 days with A=106 cm 2/sec, X for molal Lake Michigan. Figure 12 Map of Babine Lake with locations of transects and anemometer shown. Figure 13 Plots of isotherms for 14:30-16:30 PST f 21 August 1973 (a,b,c) and for 8: 00- 10: 00 PST, 24 August 1973 (d,e,f) at Babine Laka. Transect N was done f i r s t followed by M and S, with each transect iniatated on the Pinkut Craek side of the lake. Figure 14 Figure 15 Plots of wind stress f o r 21 August. Plots of wind stress for 22-24 August. x i ACKNOWLEDGED ENTS I am indebted to Dr. L. A, Mysak for h i s advice and sncouragemant i n the writing of thi s thesis and the work that led up to i t . I would also l i k e to thank Dr. D. M. Farmar f o r giving me the opportunity to spend two weeks-at Babine Lake during which time I made some of the observations reported here. During part of the time I was working on t h i s thesis I was supported by a National Research Council Postgraduate Scholarsphip,, and during the *hole period I was given the opportunity b y the Mathematics Department at U. B. C. to obtain extra support by le c t u r i n g . F i n a l l y I would l i k e to thank my wife Harion. Without her help and encouragement I would not have been able to complete t h i s task. PART A The Generation of Unstable Haves with Applications to the Meanders i n the Gulf Stream 2 CHAPTER I - Introduction A common problem in many branches of physical science i s to determine under what conditions or parameter ranges, i f any, a physical system i s unstable, when perturbed by a cer t a i n class of time dependent disturbances. Dne technigue ised to solve t h i s problem i s that developed i n l i n e a r hydrodynamic s t a b i l i t y theory (Lin, 1955; Chandrasekhar, 1961; Ried, 1967). For a cne-dimensiona1 system the scalar or vector valued function d>(x,t) describing the perturbations to the system due to the disturbance i s written as a t r a v e l l i n g wave with j>U,t )* AeiplLtcut-fcx)] . (1.1) Introducing (1.1) into the equations for ^(x, t) that describe the system i t i s assumed that the amplitudes of the perturbations are small enough so that any products of perturbations are n e g l i g i b l e . Thus a set of l i n e a r equations for the amplitudes A i s obtained that has a solution only i f a certa i n r e l a t i o n between k and to, i s s a t i s f i e d . This r e l a t i o n i s c a l l e d the dispersion r e l a t i o n and the graph of uj=(o{k) obtained by solving (1.2) for u> with k r e a l i s c a l l e d the disperion curve. If for some r e a l k a solu t i o n w(k) of (1.2) i s complex with IDI[OJ(1C) }<0, the system i s said to be unstable since the wave ifith wavenumber k grows exponentially i n time. In most problems t h i s i s as far as the study of s t a b i l i t y i s taken. The i n t e r v a l s on the raal k axis for which there are unstable roots of the dispersion r e l a t i o n are determined. A l t e r n a t i v e l y , w i s assumed r e a l and roots of the disparsion r e l a t i o n with k complex are sought. In the former case the system has a temporal i n s t a b i l i t y , and i n the l a t t e r a s p a t i a l i n s t a b i l i t y . Another approach to the i n s t a b i l i t y problem that i s useful i n systems with • damping i s through "neutral s t a b i l i t y " curves (or surfaces) along which Im{oj(k) } = 0, in a suitable parametar space. For sxample, i n the s t a b i l i t y of p a r a l l e l flows (Betchov and Griminale, 1967) neutral s t a b i l i t y curves are drawn i n the (k,R) plane. Here R i s the Reynolds number. Such curves represent t r a n s i t i o n s from s t a b i l i t y to i n s t a b i l i t y . These approaches have yialded much useful information about many int e r e s t i n g and important systems (Chandrasekhar, 1961). In most cases i t i s d i f f i c u l t to go beyond t h i s "simple" s t a b i l i t y analysis. In f a c t , i t i s often impossible even to obtain a closed form expression for the dispersion r e l a t i o n (Reid, 1967) . However, i f A(k,w) can be expressed i n closed form and i s a simple enough function, i t would be useful to solve the 4 i n i t i a l value problem for <£(x,t) i n order to see how the unstable waves a c t u a l l y grow. This would serve at least as a check on the assumption i m p l i c i t i n a l l the approaches to the s t a b i l i t y problem mentioned above, i . e . i f the system has a temporal i n s t a b i l i t y i t w i l l eventually dominate the response of the system. As w i l l be seen, i t i s possible for the unstable waves to propagate out of a f i n i t e geophysical system before i t grows to s i g n i f i c a n t s i z e . This problem has been neglected, e s p e c i a l l y in geophysical problems, aainly due to the d i f f i c u l t i e s mentioned above. However, CTriminale and Kovasznay (1962) have solved the problem of the growth of an i n i t i a l disturbance i n a laminar boundary layer. Also, i n the study of the i n t e r a c t i o n of electron streams with plasmas the growth of unsatble waves i s important, and i n that f i e l d techniques have been developed to f i n d the asymptotic behaviour f o r large time of an unstable system (Briggs, 1964), In the study of unstable plasma systems i t i s useful to c l a s s i f y the i n s t a b i l i t y of a system into two types, absolute and convective. Both types are unstable by the c r i t e r i o n given above, i . e. Im{cu(k) }<0 for some r e a l k. In a system with an absolute i n s t a b i l i t y an observer at any point sees exponential growth i n time. In a system with Dnly convective i n s t a b i l i t i e s only observers moving with non-zero v e l o c i t i e s i n a certain range see exponential growth. Thus i n a convectively unstable system a stationary observer at a distance from some i n i t i a l disturbance sees growth with time i n i t i a l l y , but eventually he must see the perturbations decay with time as the i n s t a b i l i t y i s "convected" past him. I t i s important to be able to distinguish between convective and absolute i n s t a b i l i t i e s i n designing plasma systems because i n a convectively unstable systam i t i s possible for a potentialy dangerous i n s t a b i l i t y to convect out of the f i n i t e system before i t has a chance to grow to destructive proportions (Hall and Heckrotte, 19 68) . Another way to state the d i s t i n c t i o n between absolute and convective i n s t a b i l i t i e s i s in terms of the group ve l o c i t y . Recall that the group v e l o c i t y , (k) , i s given by VjCIO-cu'U). (1.3) An observer moving at the v e l o c i t y (k) sees a wave of the form e xp{ i[w (k) t-kx ] }. This i s derived using the method of stationary phase (e.g. L i g h t h i l l , 1965). An absolutely unstable systam i s ona for which there i s an unstable wave that has zero group ve l o c i t y ; a convectivaly unstable system i s ona for which thara are no unstable waves with zero group v e l o c i t y . Theoretical plasma ph y s i c i s t s hava developed methods fo r determining whether an unstable system has an 6 absolute i n s t a b i l i t y or only convective i n s t a b i l i t i e s (Briggs, 1964 and D e r f l e r , 1970) and the v e l o c i t y ranges i n which growth w i l l be observered for both absolutely and convectively unstable systems (Hall and Heckrotte, 1968). The r e s u l t s referred to above can be used to obtain e x p l i c i t asymptotic expressions f o r the response of an unstable system to a stationary disturbance. In Part A of this thesis these r e s u l t s are used to study the growth of unstable waves in a geophysical system. Since i t i s of interest to find the response f o r t r a v e l l i n g disturbances, the r e s u l t s are also extended to cover t h i s case. To my knowledge t h i s i s the f i r s t time that such methods have bean used to study the response of an unstable geophysical system. The system that we study here i s a l a t e r a l l y sheared flow over a variable bottom topography. The flow i s assumed to ba v e r t i c a l l y homogeneous and uniformly r o t a t i n g . With or without a basic flow, low frequency waves with periods of the order of days propagate as perturbations on the f l u i d v e l o c i t i e s i n a d i r e c t i o n perpendicular to the depth gradient. Such waves are c a l l e d shelf waves and are said to be "trapped" by the depth gradient (Robinson, 1964; Mysak, 1 967; Buchwald and Adams, 1968). N i i l e r and aysak (1971), hereafter refer r e d to as NM, have found that the presence of the l a t e r a l shear gives r i s e to unstable waves propagating 7 i n thai- direction, of the flow, these are unstable shelf /rfaves. Here the generation of such waves by a wind stress acting at the surface i s studied. In the case of stable shelf wavas t h i s problem has been studied by Adams and Buchwald ( 1969) . This problem i s of i n t e r e s t i n that i t re l a t e s to the problem of the meanders i n the Gulf Stream. For many years i t has been known that tha Gulf Stream changes position ovar periods of days. Such changes have been termed meanders. The Gulf Stream i s a phenomenon that i s s imilar to phenomena seen on the western side of most of the ocean basins of the world. I t i s a region of an intense polaward surface current that i s the return flow of the o v e r a l l surface c i r c u l a t i o n of the North A t l a n t i c . The flow i s concentrated strongly along the coast due to the v a r i a t i o n of the C o r i o l i s paramter with la t i t u d e (Stommel, 1948 and Munk, 195D ; for a physical explanation see Stewart, 1964) .. Associated with the Gulf Stream are large surface temperature gradients with temperature increasing away from the coast. It i s found that these large gradients coincide with tha maximum surface current v e l o c i t i e s (Stommel, 1966). Thus the location of Gulf Stream can be accomplished by determining tha position of the large surface temperature gradients. Observations of the position of the Gulf Stream over many yaars have revealed that i t s position i s not at a l l steady; in fact t h i s meandering seems to be an i n t r i n s i c 8 property of the Gulf Stream and other western i n t e n s i f i e d currents. Haurwitz and Panofsky (1950) were the f i r s t to suggest that the meanders might be the r e s u l t of low frequency unstable waves propagating along the Gulf Stream. Their model includes f a i r l y r e a l i s t i c piecewise continuous approximations to the l a t e r a l l y sheared flow of the Gulf Stream but does not include the e f f e c t s of bottom topography. Since v e r t i c a l variations, of density and velocity are also neglected i t i s c a l l e d a barotropic model. With t h i s simple modal they predicted that t h e system w i l l support unstable waves i f the region of l a t e r a l shear i s far enough from ths boundary. The unstable waves predicted by their model have properties s i m i l a r to the wave properties of the meanders i n the Gulf Stream. Since the work of Haurwitz and Panofsky many models that are applicable to the Gulf Stream and support unstable rfaves have been studied. Most of these models have included the most important feature of the Gulf Stream ignored by Hauraitz and Panofsky--the v e r t i c a l variation of density and veloci t y . Such moials are c a l l e d b a r o c l i n i c . Two methods for modeling such v a r i a t i o n s have been used. The simplest i s to assume the flow to be divided into horizontal layers in which there are no v e r t i c a l v a r iations; properties change discontinuous! y across the i n t e r f a c e between layers. Two-9 layar models in which the lower layer i s stationary and the ipper layer i s l a t e r a l l y sheared have been studied by Stern (1961), Lipps ( 1963), Jacobs (1971) and Sela and Jacobs (1971). Orlanski (1969) has studied two layer models that include the dynamics of the lower layer i n order to examine the a f f e c t of bottom topography on the s t a b i l i t y of the flow. V e r t i c a l structure may also be modeled by continuous v e r t i c a l changes. This has been done by Pedlosky (1964 a and b) who considered both two-layer and continuous variation models. A l l of the models mentioned above are complicated enough that, even though they by no means include a l l of the physics of the Gulf Stream, i t i s possible to only do the "simple" s t a b i l i t y analysis described above. Much work has been expended in determining the parameter ranges i n which the models are unstable and the simplest properties of the unstable waves, e.g. growth rate, wavelength, frequency and phase speed. The baratropic model of NM on the other hand i s simple enough that the asymptotic techniques developed f o r unstable plasma systems are e a s i l y applied. Thus the response of the model to various disturbances can be calculated. The model i s similar to that of Haurwitz and Panofsky except that i t includes bottom topography. In Part A of t h i s thesis we calculate the response of t h i s model to c u r l free wind systems both stationary and moving. This may be regarded as 10 a f i r s t step in the problem of c a l c u l a t i n g the response of mora complicated models to mere general wind systems. Also, as w i l l ba saan, tha response to a c u r l free wind stres s c e r t a i n l y gives a lower bound to the t o t a l response of the systam. Further, t h i s problem also serves the purpose mentioned abova of checking the basic assumption of s t a b i l i t y theory that in an unstable system the i n s t a b l i t y eventually dominates the response. I t i s already apparent that i n a convectively unstable system, as the NM model i s , this assumption must be treated with care, but e x p l i c i t c alculations indicate furthar ramifications. For axampla, i t is found that in the NM model while the response on the offshora side of tha stream is.always dominated by the unstable waves, that on the inshore side i s not. F i n a l l y , again to my knowledge t h i s i s the f i r s t time that the e x p l i c i t asymptotic response of an unstable geophysical systam has been calculated, i n s p i t e of the large number of such systems that have been studied. I t might ba useful here to give a b r i e f history of t h i s project. When f i r s t proposed i t was intended as a qualifying problem for the I n s t i t u t e of Applied Mathematics at U. B. C. to be completed i n a couple of months. The i n i t i a l boundary value problem for the response of the NM nodel was to be solved using Fourier-La place transforms with tha resulting inversion i n t e g r a l s . 11 '-00 CO ( 1 . 4 ) to be evaluated using standard asymptotic techniques, such as the method of steepest descent. Using t h i s approach the Laplace inversion i n t e g r a l i s evaluated f i r s t by summing residues, and then the Fourier inversion i n t e g r a l i s avaluatad asymptotically for large time and small y by the method of steepest descant. In order to extend the re s u l t s to f i n i t e y l e t y=y0+vt and do the asymtotics for each V of in t e r e s t . In tha mathod of steepest descent i n t e g r a l s of the form are to be evaluated asymtotically for large t by deforming the path of integration i n the complex k plana i n t o a path decreases most r a p i d l y . Obviously the most important contribution to tha i n t e g r a l comes from the point on the new path of integration at which Re[f(k)} i s a maximum. There are only two p o s s i b i l i t i e s . Either the maximum occurs at an endpoint A or B, or the maximum occurs at an i n t e r i o r point, k 0, at which f ' (kJ^O. The point k Q i s a saddle point of f( k ) . Only in the second case does the saddle point contribute to tha aysmptotic development of I (t). The form (I. 5) of stsepest descant, i . a. a path along which Re{f(k)} 12 of the asymptotic expansion depends on which of the two p o s s i b i l i t i e s arise (Sirovich, 1971 or J e f f r e y s and J e f f r e y s , 1956). In evaluating the asymptotic behaviour of the Fourier inversion i n t e g r a l f o r the response of the NM model by the method of steepest descent problems arose. The function &(k,s) i s quadratic i n s but i s a complicated transcendental function of k. Thus finding the paths of steepest descent in the k plane along which Im[s} i s constant seamed impossible. Thus deciding between the two p o s s i b i l i t i e s given above would also be impossible. Therefore, at that point i t seemed advisable, i f the problem rf as indeed to be finished in a shor-t time, just to evaluate the Fourier i n t e g r a l numerically i n order to g.et some idea of the growth of the unstable waves. Unfortunately the numerical integrations turned out to be quite expensive, due mainly to the large number of evaluations of the integrand required f o r convergence. For t h i s reason the i n t e g r a l was evaluated only at one point i n the stream which was chosen to b5 the point of velocity maximum. The expected growth of the unstable waves was not seen. Shortly af t e r t h i s rather discouraging r e s u l t was obtained the work of the plasma physicists came to my attention through Briggs (1964). The method used there i s e s s e n t i a l l y the same as the method of steepest descent, but the point of view i s changed. In t h i s approach the Fourier 13 i n t e g r a l i s evaluated f i r s t by summing residues. Then causality in the form of the requirement that the integrand i n the Laplace inversion i n t e g r a l must be a n a l y t i c to the right of the Laplace inversion path i s used to f i n d a c r i t e r i o n f o r deciding whether a given saddle point contributes or not. The c r i t e r i o n i s that i f the paths of steepest descent approach a saddle point from opposite sides of the r e a l k axis the saddle point contributes, otherwise i t does not and the asymptotic representation i s exponentially small compared to the contribution from the saddle point. Apparently the c r i t e r i o n given above s t i l l requires that paths of steepest descent be drawn and saddle points be determined for each V. However, i n a paper by Hall and Seckrotte (1963) a method i s given for determining i n t e r v a l s for V within which saddle points contribute and outside of which they do not. It i s then only necessary to find the endpoints of these i n t e r v a l s and to test one saddle point on each side. In the same paper i t i s pointed out that the problem of determining saddle points as a function of V i s the same as solving the system of equations A k ( ^ ) + wA 5a 1 s) --o , J which can be recast as a d i f f e r e n t i a l equation i n V to be (1.6) 14 solved numerically. It was found to be easier, however, to solve (1.6) d i r e c t l y using Newton's method. I t i s s t i l l necessary to solve the dispersion r e l a t i o n i n order to draw steepest descent paths and to f i n d s t a r t i n g values for Newton's method. This problem i s solved by a method due to Delvas and Lynass (1967). Using the conglomeration of methods described b r i e f l y abovs i t was possible to calc u l a t e the response of the NM aodel to c u r l free wind stresses. Of course, the f i r s t thing to do was to v e r i f y the r e s u l t s with the numerical integrations. The calculations agreed quits well, so further c a l c u l a t i o n s were made using the asymptotic methods. It was found that on the offshore side of tha stream the unstable wavas are always dominant, but on the inshore side th i s i s not always the case. This i s due to the f a c t that on tha inshora sida the integrand i s small in the i n t e r v a l of i n s t a b i l i t y , compared to i t s value i n the stable regime, wheraas on the offshore side i t i s of the same order i n the stable and unstable regimes. This explained the r e s u l t s of the numerical integration. The c o l l e c t i o n of r e s u l t s used to solve the problem considered hare should in p r i n c i p l e be applicable to any i n i t i a l value problem that can be solved using Fourier-Laplace transforms. Thus any l i n e a r system that can be considered i n f i n i t e i n one d i r e c t i o n can be handled by t h i s 15 method. The main problem i s to solve the dispersion r e l a t i o n . Since in many cases i t i s impossible even to ifrite a closed form expression for the dispersion r e l a t i o n this can be a very d i f f i c u l t problem. However, since the dispersion relation can be analysed once and for a l l , and the r e s u l t s then be used f o r any forcing function, i t i s possible that i t would be f e a s i b l e to consider such d i f f i c u l t systems. This could require quite extensive, numerical work however. In the remainder of t h i s introduction we give a b r i e f outline of Part A of t h i s thesis. In Chapter II the NM model i s described i n more d e t a i l , and the equations of motion governing i t are derived. These equations are solved by Fourier-La place transforms subject to the i n i t i a l condition that at t = 0 a general wind s t r e s s i s applied. The inversion i n t e g r a l s are written out e x p l i c i t l y only for a c u r l free wind stress. This i s done mainly for mathematical convenience. However, the methods developed and used here can be extended to cover more general wind stress models. Further, these c a l c u l a t i o n s should only be regarded as a f i r s t step i n solving more complicated problems. F i n a l l y , the c a l c u l a t i o n s cannot be expected to be very good representations of the response of the Gulf Stream to a general wind stress as the unstable waves that are generated are of a large enough amplitude 16 that a l i n e a r modal ..would. not r e a l l y apply. In Chapter III the r e s u l t s developed by the plasma physicists that ara relevant to the problem at hand are presented. These are then extended to cover the case of a traveling disturbance, a problem that has not been considered before i n the case of unstable systems. In Chapter IV the re s u l t s of Chapter III are applied to the problem of c a l c u l a t i n g the response of the model presented i n Chapter II to c u r l free wind stresses. I t i s found that on the coastal side of the stream the unstable waves do not appear u n t i i - a f t e r they have t r a v e l l e d well out of the system. On the offshore side of the stream, however, the unstable waves c e r t a i n l y do dominate the response. k disturbance moving slowly in the d i r e c t i o n of the stream i s the most e f f i c i e n t at generating the unstable waves. A. disturbance moving upstream i s much les s e f f i c i e n t . 17 CHAPTER II - Unstable SheIf Haves Section 1. Introduction In t h i s chapter we study the generation of low frequency waves i n a rotating, l a t e r a l l y sheared flow over a changing bottom topography. In a study of the propagation of such waves NM found that the presence of the l a t e r a l shear in the flow gives r i s e to unstable waves that propagate i n the d i r e c t i o n of the stream. It i s our purpose here to solve the i n i t i a l boundary value problem for the generation of these waves from an equilibrium configuration in order to study the growth of the unstable waves. The approcah used har i s to derive the l i n e a r i z e d long wave equations for perturbations to a l a t e r a l l y shaared flow and then solve tha aquations using Fourier-Laplaca transforms for general fo r c i n g . The inversion i n t e g r a l s are written out a x p l i c i t l y , howavar, only for s p e c i a l c u r l free wind stresses. Section 2. Tha Equations Consider a l a t e r a l l y sheared flow p a r a l l e l to a long straight coast, running north-south, i n a uniformly rotating ocean with the bottom topography varying only in a d i r e c t i o n perpendicular to tha coast. It i s assumed that the flew i s 18 depth independent and that the density i s a constant, i . e . the flow i s barotropic. In studies of more general models that include continuous s t r a t i f i c a t i o n i t i s found that the higher b a r o c l i n i c modes have slower growth rates than "the barotropic and f i r s t b a r o c l i n i c modes. Thus i t i s reasonable to consider a barotropic model. Introduce a right-handed rectangular coordinate system with x eastward, y northward and z v e r t i c a l l y upward. The or i g i n i s on the coast at the l e v e l the free surface would assume i f there uere no flow. The bottom i s at z=-h(x). At time t=0 there i s an equilibrium state with the flow northward with velocity p r o f i l e Ve (x) . At t h i s time a surface wind stress t = ( t v , t ^ ) i s applied. Hare the equations governing the response of t h i s system to t are derived. The bottom topography and observed depth averaged v e l o c i t y p r o f i l e representative of the Gulf Stream over the Blake Plateau off South Carolina are shown i n Figure II. 1. Let u, v and w be the t o t a l v e l o c i t y i n the x, y and z di r e c t i o n respectively and u, v and w the perturbation to the eguilibrium v e l o c i t y . Further, l e t P c and 7. be the equilibrium pressure and surface displacement above z-Q and p and ^  the t o t a l pressure and surface displacements. Then i l ' U ^ - . V . t V ^ e V ^ . P , ^ , +•»] \ (II. 1) i f we put (II. 1) into the Havier-Stokes equations for a 19 aniformly rotating system, and by the usual method of l i n e a r hydrodynamic s t a b i l i t y theory we drop a l l products of perturbations, then a set of l i n e a r equations for the perturbations i s obtained. The equilibrium state i s geostrophic and hydrostatic, so that ( V ( » ^ > (II. 2) tfhere p^ i s the atmospheric pressure, assumed uniform. We suppose that the shelf waves are long compared to the depth so that the v e r t i c a l v e l o c i t i e s and accelerations are a l l negligible. Then the v e r t i c a l equation of motion implies that the t o t a l pressure i s hydrostatic, so that (II.1 ) and (II.3) give p - m . * ( I I - 4 ) Since V0 depends only on x, (II. 2) and (II. 3) show that depends only on x and p o depends only on x and z. This, along with the long wave hypothesis, gives the following equations of motion in the x and y d i r e c t i o n s : . ' i ! i + v.22.-? v..-,22L +2 .Hr, . ( I I . 5 ) i r + V r f i : + ( t < 4 S u . . . 2 L + i » l ' > . ( I I . 6 ) where f=2Hsin^> ( p l a t i t u d e and ft=angular v e l o c i t y of the 20 Earth's rotation) i s the C o r i o l i s parameter, here assumed constant. A l l f r i c t i o n a l e f f e c t s are neglected except for the wind stress at the surface which i s modeled as a body force. For a good discussion of the j u s t i f i c a t i o n for t h i s see P o l l a r d ( 1970) . The boundary condition at the bottom, z=-h(x), i s vr+ — u- = 0 ,, «t i = -U*). (II«7.) A* This i s just a statement of the fact that the v e l o c i t y normal to the bottom i s zero. The kinematic boundary condition at the free surface, z= ^ (x, y, t ) , i s 2 ! + V, p - -r y at •+ ). (II. 8) It d <j d * ' Now integrate the continuity equation v e r t i c a l l y using (II.7) and (II.8) . . I f we further assume that ^<<h we get Note that the assumption ^ « h i s not equivalent to the l i n e a r hydrodynamic s t a b i l i t y procedure as ^ i s the t o t a l surface displacement. However, from (II.2) that * ~ -Fir0L * T where L i s a c h a r a c t e r i s t i c horizontal length scale. From Figure II. 1 we see that Loi5x1D6 cm and v o W 1 0 2 cm/sec. Then \ *• 5*0 c>n <:< ( 0 f c ^ , . 21 ifhich concurs with the approximation 7 « h . F i n a l l y suppose that the r i g h t hand side of (II.9) can be ignored, i . e . the motions are h o r i z o n t a l l y non-divergent. This assumption i s j u s t i f i e d i f the time scale of the motions i s long compared to L/^ghcjl 0 min. Since the shelf waves have periods of the order of days t h i s assumption i s cer t a i n l y j u s t i f i e d . By ignoring the right hand side of (II.9) we are f i l t e r i n g out the surface gravity waves. In li g h t of t h i s assumption a transport stream function, ^ ( X / Y / t ) * i - s introduced such that U : - — O L f t A ' W'-^-. (11.10) Introducing (11.10) into (II. 5) and (II.6) and' i n t g r a t i n g v e r t i c a l l y , remembering that ^<<h, we obtain He have assumed' that u and v are independent of z, a consequence of the long wave hypothesis. He can eliminate ^ by d i f f e r e n t i a t i n g (11.11) with respect t o y and (11.12) with respect to x. Since f i s independent of x and y, t h i s gives t ''J 22 This i s the equation of motion for the model that we wish to solve. This i s the v o r t i c i t y equation, and the term on the right hand side, e s s e n t i a l l y the c u r l of the wind stress, i s the source of v o r t i c i t y . Note that i n the absence of c u r l in the wind stress, however, the slope of the bottom s t i l l gives a forcing term. Before (11.13) can be solved we must specify a form for h (x) and V e(x). We choose the simple form shown in Figure II.2 which i s the simplest that includes a l l of the features of the Gulf Stream that we wish to include. One major advantage gained from using discontinuous depth and shear p r o f i l e s i s that they give r i s e to only one mode, whereas using a continuous p r o f i l e r e s u l t s i n an i n f i n i t e number of modes, as i n Buchwald and Adams (1968). For appropriate parameters t h i s i s an^adequate model of the Gulf Stream over the Blake Plateau. For the configuration shown in Figure II.2 equation (II.3) becomes in each of the regions marked 1,2 and 3. The boundary condition at the coastal edge of region 1 i s (II. 14) (II. 15) 23 since the coast must be a streamline. Matching conditions at x=L and x=2L are required. There the surface' displacements and the mass transport perpendicular to the disc o n t i n u i t y must be continuous. The second of these requires that d*r"/c>y be continuous from (11.10), which means ?«H>O^* = 0 a t X= L arxA X=2L. (II. 16) Here If ^ i s continuous at x=L and x=2L then so i s c^/c; y. Then (11.12) gives From (11.14) we see that only one i n i t i a l condition i s needed and i t i s ^L%^o) =0, ( i i . 18) since at t=0 the perturbations are a l l zero. Befpre proceeding we non-dimensionalize the equations as follows: j>* \ A?j> * (11.19) 24 Then equation (11.14) i s written (11.20) here = v 0 / L f . We have dropped the * above as we w i l l do i n the sequel and remember that the variables are now non-dimensionalized. The boundary and matching conditions are (11.21) 1 W l <>V T> ic V A*. ; 1^ (11.22) = °, Q t (II. 23) (11.24) Now f o r the model i n Figure II.2 we have 0 2 £ * • (II. 25) and kt>0 = 1 O i X U j (II. 26) 25 where JA =1+D/d>2. S e c t i o n 3. The Transformed Eguations The problem s t a t e d i n the l a s t s e c t i o n i s an i n i t i a l boundary value problem f o r *f (x, y, t ) . The domain i s -oo < Y < c o i 0<x< °» and 0<t<oo. Such a problem l e n d s i t s e l f to transform methods, where we take a La p l a c e t r a n s f o r m i n time and a F o u r i e r t r a n s f o r m i n y. A f t e r we have taken the F o u r i e r - L a place transform ( h e r e a f t e r denoted by FLT) we are l e f t with an o r d i n a r y d i f f e r e n t i a l e q u a t i o n i n x. Define the FLT of <f (x,y,t) as .00 rco e ^ U . ^ t ) ^ .(11.27) and a p p l y i t to (11.20) and (11.23). T h i s g i v e s ( $ - L W > ) ( ^ ~ - = H*,k,S) (.11.28) ^•ICs-jAvJog + l ( u ^ ) ^ _ = ° °* * = \,^ (11.29) where F(x,k,s) i s the FLT of (curlT) • k and T ^ ( x f k f s ) i s the FLT of .T"»(*» y,t) . We denote the s o l u t i o n of. (11.28) i n r e g i o n i by -J^ . . I t i s p o s s i b l e to f i n d p a r t i c u l a r s o l u t i o n s of (11.28) t h a t s a t i s f y 26 % ? Lojte.s} * * p Cv,k,«) ^ O.W/O - i ^ U j k . S ^ ^ Cz.W.s)- o, (II. 30) an d (11.31) They are given by J o \S - A > W V„(.=n] (II. 32) — AX , (11.33) C O (11.34) Note that for (11.34) to s a t i s f y (11.31) F(x,k,s) must be bounded as x -eo. Now the solutions of (11.28) that are continuous at x=1 and x=2 are 27 r (11.35) The f u n c t i o n s A(k,s) and E(k,s) can fce found ty a p p l y i n g the matching c o n d t i c n s (11.29) at x=1 and x=2. The equations t h u s d e r i v e d are a ^T^v,W,S) + ^ ^ | ^ t \ ^ ) - ^ ^ n , t e > S ^ j . (11.36) (11.37) These are of the form and the solutions are A * - I b , ^ - bic*^^ / A. , & = C V u ~ V>, , 28 where (31.38) The equation ( 1 3 . 3 . 9 ) i s the dispersion r e l a t i o n as defined in (1 .1) with s = ioj. In Figure IV.2 we plot Im{s} as a function of k fcr r e a l k. Over a certain range cf k the Tm(s} curves coalesce sc that i n t h i s range there are twc complex solutions cf (11.39) that form a compex conjugate pair for i s . Thus there i s one f o r which Re{s}>0 and hence the system i s unstable. Section 4. The Inversion Integral The inversion theorems for Fourier and Laplace transforms give where the Laplace inversion path (LIP), Re{s}-T, must be to the r i g h t of a l l s i n g u l a r i t i e s . Here we w i l l write cut these inversion integrals for the wind stress models we wish to consider. 29 F i r s t note that even for the simplest form of F(x,k,s) the i n t e g r a l s i n (11.32) and (11.33) are not expressible i n terms of elementary functions and that % f and \^ are multi-valued i n the complex k and s planes due to the (s-ikV 0 (^)) i n the denominator of the i n t e g r a l s . Thus in order to make the problem of evaluating the inversion i n t e g r a l s as easy as possible l e t F(x,k,s) = 0. Thus we consider only c u r l free wind stresses. There i s s t i l l a response, as wesee from (11.13), due to the d i s c o n t i n u i t y of the bottom topography. As we pointed out e a r l i e r we do not expect to get r e s u l t s from our c a l c u l a t i o n s that can be compared d i r e c t l y with data due to the fact that once the unstable waves have grown to appreciable size tha l i n e a r model no longer applies. Thus i t makes very l i t t l e sense to try to do c a l c u l a t i o n s f o r r a a l i s t i c wind stresses. What our c a l c u l a t i o n s should t e l l us i s whether surface wind stresses are e f f i c i e n t i n generating the unstable waves and what p a r t i c u l a r forms of tha wind stress are most e f f i c i e n t . The problems involved in c a l c u l a t i n g the response for wind stress with c u r l have been looked at and are c e r t a i n l y not insurmountable. It i s possible that t h i s could be the t o p i c of a future paper. In the case of a c u r l free wind strass the c o e f f i c i e n t s A (k,s) and B (k,s) are of the form 30 Notice that for a c u r l free wind stress T ^ f X f k f S ) i s independent of x and so we have written i t as T^ ( k,s). Then from (11.35) and (II.41) we have *i*,1iO = — e as E JLW q ^ V.oo J-=o It MW.S) ' (11.44) 1 ' : ' ' o " * , K e s cas • ~ : e '^Aw, (li.4 5) 7 S - A . C O — £ 7 elW. (11.46) notice that the i n t e g r a l s in (II.44)- (II.46) w i l l appear whether or not the wind stress i s c u r l f r ee. Thus the r e s u l t s f o r a c u r l free wind stress can be considered to be a lower bound for the response to general wind stresses, unless i t turns out that the response from the c u r l of the wind cancels the c u r l free response. F i n a l l y the two forms of T'1(k,s) for which e x p l i c i t 31 calculations w i l l be done i n Chapter IV are given. The f i r s t i s an impulsive l i n e source of wind stress at y=0 for which t H M . t ^ Sop-StO > (11.47) and thus T'Hw;o*l. (11.48) The response calculated for thi s case w i l l represent a Green's function from which the response to any other kind of wind strass in p r i n c i p l e can be calculated using convolution techniques. The second form of T^(k,s) i s that f o r a moving l i n a source of wind stress t r a v e l i n g with velocity V and having tima behaviour exp(s t) . Then t 1 ^ V ^ s S L v v * ° eS**Hl.-0 , (11.49) where H(t) i s the Heavyside step function. Then we have T U \ ^ ) = .U-s+-iWV*r». (11.50) Re could of course compute the response for t h i s T^)(k,s) using the Green's function found i n tha f i r s t problem, but i t i s easier to treat t h i s case separately. The wind stress given i n (11.49) i s a more r e a l i s t i c representation of the wind stress acting on tha Gulf Stream than the stationary one given i n (11.47) and wa w i l l f i n d some int e r e s t i n g 32 r e s u l t s for the moving wind stress. Now we must turn to the problem of evaluating the i n t e g r a l s in (II .44 ) - (II. 46 ) , to which we devote the next chapter. 3 3 CHAPTER III - Asj;m£totics Section 1. Introduction to the Method for FJk xs]_ and AiiSxSl Entire In the previous chapter we have expressed the solution of the i n i t i a l boundary value problem for the unstable shelf waves in terms of i n t e g r a l s of the form This chapter w i l l be devoted to the problem of evaluating *V(y,t) asymptotically for large t and arb i t r a r y y. I t i s to be expected in a system that supports unstable waves that eventually the "most unstable wave" w i l l dominate the the "simple" s t a b i l i t y analysis described in the introduction. Here we present a method for determining the t o t a l , both stable and unstable, response cf the system i n order to make t h i s rather vague statement more precise. In the next chapter we apply the method develcped here to the unstable shelf wave problem. As an introduction to the method we outline the approach for the simplest possible case, in which both F (k,s) and &(k,s) are entire functions of k and s. In t h i s form the r e s u l t s are mainly due to Driggs (1964) and Hall and Heckrotte (1968) who investigated such problems i n (III. 1) response. This i s e s s e n t i a l l y the basic assumption behind 34 connection w.ith unstable waves i n plasmas. In the case of stable waves si m i l a r techniques have been used by Adams (1972). In (III. 1) we evaluate the k integral f i r s t . If F(k,s) i s entire in the complex k plane and | F (k,s)/& (k,s) h 0 as )k|^co in the complex k plane sc that Jordan's lemma holds then, f o r a given s, the value of the Fourier i n t e g r a l is determined e n t i r e l y by the rccts cf Atte^ s o (3 31.2). i n the complex k p l a n e . W = close the ccntcur in the -tipper or lower k p l a n e a c c c r - i i n g a s y<0 or y>0. lor y>0 we have by evaluating r e s i d u e s 0.1. k J1-ict> where for any s on the LIP such that £(k,s) has cnly simple zeroes. The sum extends over a l l the zeroes of o(k,s), k<">, that l i e in the lower half k plane. A s i m i l a r expression holds for y<0 with k<^ > replaced by k< ^ >, vihere kfv\> a r e the zeroes cf &(k,s) that l i e in the upper half k plane. For y<0 there i s no minus sign in front cf the sum in (III.4) . It i s necessary for c a u s a l i t y , i.e. ^ ( y , t ) = 0 f c r t<0, 35 that G(y,s) be analytic to the rig h t of the LIP. Thus we must investigate the s i n g u l a r i t i e s of G(y,s). If we l e t s vary the zeroes k<A> and kf n>, which depend on s through (ITI.2), vary too. It i s possible that as s varies a zero kc*> or k< n > w i l l cross the r e a l k axis. When t h i s happens G (y,s) as defined by (III.4) undergoes a jump as one term on the r i g h t side of (III.4) i s l o s t or gained. Thus the images of the r e a l k axis under the mapping (III.2) represent branch lines of G(y,s) i n the complex s plane. See Figure I I I . 1 . Thus caus a l i t y requires that the LIP be to the right of a l l such branch l i n e s . However, G(y,s) 'can be continued through these branch l i n e s . Suppose G(y,s) i s to bs evaluated at s which i s to the l e f t cf a branch l i n e . Choose s, on the LIP with Im( s, }=Im{s0}. For s, A(k,s) has zeroes k<«-> and k_<i>. We l e t s vary along the l i n e Im{s }=Im{s. } from s, to s e . As s varies at le a s t one of the k^^-or- k< ^  > crosses the re a l k axis since s„ i s tc the l e f t of a branch l i n e . When t h i s happens we indent the Fourier contour around the encroaching k^ *• > and k_U' so that i t i s s t i l l included or excluded when evaluating residues; see Figure III.2. Thus we define ^ « J , S ) - ^ J S f ^ ) e • 3 a k , . (III.5) c where C i s the indented Fourier contour. The G(y,s) defined in (III.5) agrees with that defined i n (III.4) f o r s tc the 36 right of the LIP. Thus i t i s an an a l y t i c continuation cf G(y,s) as defined i n (III. 4), and i t no longer has jumps across the branch l i n e s . When we refer to G(y,s) from now on i t i s to be understood as the an a l y t i c continuation defined in ( I I I . 5 ) . The function G(y,s) s t i l l has s i n g u l a r i t i e s . I f two zeroes k<*> and kf "i > - merge across the re a l k axis then i t i s no longer possible to indent the Fcurier ccntour around either of them. In t h i s case the s at which t h i s merging takes place i s a s i n g u l a r i t y of G(y,s); see Figure III.3. We w i l l now investigate the nature of t h i s s i n g u l a r i t y . - If two zeroes of A(k,s) merge at k 0 f c r s=s e then we must have A* (teo.S.^ = O. (III. 6 ) Expanding A(k,s) about (k<,,s0) gives whereA < ? >= (k.,s„), etc. Remember that Ate{e,5=0 since k 0 i s a double zero of A(k,s), If A s < o )^0, the two leading terms i n (III.7) indicate that We can find the c o e f f i c i e n t s in (111*8) by substituting in (III.7) and equating powers of (s-s 0)' /' i. The f i r s t two c o e f f i c i e n t s are ACfe,s} = ts-s.) * i A'£ (te-voV-. • (III.7) (III.8) 37 Thus i f we denote the two zeroes of A(k,s) near kfl by k, and k 2 we have fe, - k 0 * «•>Os-s.ilz + a. x(.<>-•>«,•) t - - -(111.10) If k, and coalesce access the rea l k axis then only cne of them gives a contribution when evaluating the i n t e g r a l in ( I I I . 5 ) . Suppose that i t i s k,. Then the contribution frcir. the residue at k , i s expanded about s Q as e - * ^ F ' 0 ) ,/ Thus i f k, and k., coalesce across the rea l k axis G(y,s) has a ( s - s 0 ) / z s i n g u l a r i t y at s 0 . However, i f . k ( and k z coalesce frcm the same side cf the real k axis they both contribute to the i n t e g r a l in ( I I I . 5 ) . In t h i s case i t i s net hard to show that the (s-s0)"»' , n=0,1,2,..., terms in the expansion abcut s0 a l l cancel, and hence that G(y,s) i s analytic at s e . Thus the only s i n g u l a r i t i e s cf G(y,s) are these s»s that correspond to double zeroes of A(k,s) that coalesce across the r e a l k axis. He find the asymptotic behaviour for large t cf *V(y,t) (III.9) 38 by applying Laplaces method (see C a r r i e r , et a l , 1966) to the i n t e g r a l in (III.3). In order to apply t h i s method we must have |G(y,s) |-*0 as |s|~«> so that Jordan's lemma applies. If we can insure that Jordan's lemma holds by taking the l i m i t as y-0. Then the f i r s t term of the asymptotic expansion for f( y , t ) at y=0 Is ^ I O ^ ^ - A - L ^ ^ e ? a t F"» , (III.11) where the sum extends over a l l double zeroes of A( K» £)» ( k p , s 0 ) , at which k, and k z coalesce across the k axis. Note that we choose a, in (III.11) so that k, approaches the r e a l k axis from below. Note that i f one of the saddle points (k 0,s 0) has Re{so}>0 then the system i s absolutely unstable. This i s because there i s exponential grcwth in time i n a neighborhood of the o r i g i n . Notice that (III.6) implies that k„ i s a saddle pcint of the function s (k) defined i m p l i c i t l y by (III.2), i . e . k s a t i s f i e s S'^0) = o. (III. 12) For t h i s reason we c a l l the k e»s saddle points and the k 0's for which k, and kj coalesce across the rea l k axis pinching saddle points. Sometimes we w i l l also refer tc the pair 39 (k 0,s 0) as a saddle point. This nomenclature points out the rela t i o n s h i p between the method given here and the method of steepest descent as described i n the introduction. In fact the above re s u l t can ba derived d i r e c t l y from the method of steepest descent. Recall that to use the method of steepest descent the'Laplace i n t e g r a l i s evaluated f i r s t , and then an asymptotic representation of tha Fourier i n t e g r a l i s obtained by deforming the path of integration into a staapast descant path. If a saddle point i s of the pinching type, then the endpoints cf the path of integration, t^o t H e in d i f f e r e n t valleys (Sirovich, 1971), and the path cf steepest descent must go through the saddle point. Cn the othar hand, i f the saddle point i s not of the pinching type one of the endpoints l i a s in a va l l e y , and the othar l i e s on a h i l l ; sea Figure III.4. So far we are limited tc finding the asymptotic behaviour of (III..1) at y=0. However, we can get arcund t h i s l i m i t a t i o n in the following way. Let y=Vt+y0 and make the changa of variable i n (III. 1) (III. 13) Then (III.1) becomes Hnv \ . ]_„ A U , s V ' ( i l l . 1 4 ) 40-where ( I I I . 15) Then we proceed with (III. 14) the same way we did with (III.1). For each V we find the saddle points for &(k,£) and determine which of them are of the pinching type. Then we form the sum as in (III.11) with yo-*0 for the pinching saddle points. In t h i s way we can find the asymptotic behaviour for any value of y. Note that saddle points are those points at which =o-. ( i j i . 1 6 ) But by (III.15) t h i s i s eguivalant tc v , ; Ai. e (III.17) A s ctk In words t h i s says that an observer moving with v e l o c i t y V sees a wave with wavenumber ke and frequency - i s Q that has grcup ve l o c i t y V. This i s as we expected. Recall the d e f i n i t i o n of group v e l o c i t y in (1.3). However, net a l l such waves are observed but only those that correspond to a pinching saddle point. The general procedure for determining the asymptotic behaviour of (III.1) for large t i n the case considered here i s the following. For each V determine the saddle points 41 for &(k,s) . For a system that supports unstable and/cr decaying waves thi s i s not necessarily straight forward. The methods used to solve t h i s problem are discussed in section 5. Then we must determine which of the saddle points are of the pinching type. This i s net as bad as i t sounds, as there are i n t e r v a l s for V in which the saddle points are pinching and outside of which they are net. There are c r i t e r i a that can be used to find the endpoints cf these i n t e r v a l s , c a l l e d t r a n s i t i o n v e l o c i t i e s . F i n a l l y , for each V the asymptotic behaviour i s given as a sum in the form of (III. 11) . The case considered here, in which A(k,-s) and F(k,s) .are entire,functions, unfortunately dees net ccver the problem of unstable shelf waves as can he seen from (11.44) - (11.46) . In t h i s case both F(k,s) and A(k,s) have branch points at k=0 due to the presence of |k|. This i s the problem we investigate in the next section. Section 2. Extension to A i k ^ s l and FJk xs]_ with Branch Points As pointed out above we must now consider the case in which A(k,s) and/or F(k,s) have branch points at k=0. We w i l l find that the re s u l t s are the same as before. The r e s u l t s given here are due tc Derfler (1970) and were developed to study the growth of i n s t a b i l i t i e s in hot 42 plasmas. He s t a r t with equation ( I I I . 3 ) where new we define ^ T T T T ^ - ( I I I . 18) Since the branch point at k=0 i s due to the presence of |k| i n both F(k,s) and Mk,s) we s p l i t the in t e g r a l in ( I I I . 18) at k=0. Raking the change of variables k=-k in the int e g r a l over (-co,0) , we have A^,S>' V ITT J . t*,S) -ZTT J„ A.c«,S) 1 ( I I I . 19) The function A +(k,s) i s the same as A(k,s) with |k| replaced by k, and A_(k,s) i s the same as A(-k,s) with |k| replaced by k. Simi l a r l y for F +(k,s) and F_(k,s). Now A+, A., F + and F_ are a l l entire functions. Now expand i n p a r t i a l f r a c t i o n s ^ ( I I I . 20) Of!' 1 Ke can do t h i s i f we can f i n d a sequence cf contours C^ on which F+/A^ (or F_/A_) i s bounded such that R m=clcsest distance of to the o r i g i n becomes i n f i n i t e as m-*«>, since F+./A+ and F_/A- are analytic except at the zerces cf A+ and 43 where k<"> and k<"> are the roots of + A ^ S ^ O o-wck A-Ch,sb=.o, (III.21) respectively, and the sum extends over a l l such r c c t s . Thus for G f(y,s) we have Note that where ^(1,1.; z) i s the confluent hypergeometric function of the second kind with 03 (III. 24) f and c=.e (tf=. 5772.. . i s Euler's constant). See E r d e l y i , et a l (1953) equations 6.5(12) and 6.7.1(13). The function t (1,1 ; - i k ^ 5y) i s a multi-valued function, and as defined in (III.23) i s discontinuous along the images in the s-plane of the posi t i v e r e a l k axis under the.mapping in (III.21). Thus these image curves are branch curves of G (y,s) as defined i n (III.22). Again continue G (y,s) through the branch l i n e s by indenting the Fourier contour around any encroaching kj*>; see Figure III.3. In t h i s way we define 44 an analytic continuation of G v(y,s) as defined i n (III.22) that i s not discontinuous across the branch l i n e s . This an a l y t i c continuation i s defined as in (111.22) with the path of integration in the in t e g r a l s replaced by the indented paths cn. He have \!TZriy " * - ^ ' ,131.25, i f k<n> crosses the postive r e a l k axis from the lcwer half plane into the upper half plane as s varies along l i n e s Im{s }=constant from right to l e f t ; see Figure III.3. Again G +(y,s) s t i l l has s i n g u l a r i t i e s even after the analy t i c continuation has been carreid out. The s i n g u l a r i t i e s are now the saddle points k 0 of A +(k,s) that pinch the £ositive r e a l k axis. Near k e there are two 'solutions- k, and k^ of (III.21), just as in (III. 10). If only one of k ^  and k 1 cross the posit i v e r e a l k axis then only one term i n (III.22) w i l l have the extra term as i n (III.25). In t h i s case G + (y,s) has a {s-s0)~'^z s i n g u l a r i t y just as before and the leading term in the asymptotic expansion i s as given in (111.11). If k, and k z both cross the p o s i t i v e r e a l k axis or neither dc then we can show that a l l of the i r r a t i o n a l terms i n the expansion of G(y,s) about s e cancel.and thus that G+.(y,s) i s ana l y t i c at s=s c. Notice that there i s no problem with a logarithmic s i n g u l a r i t y at s=s 0 unless k =0. This point w i l l be taken up l a t e r . 45 Exactly the same arguments as above apply i n considering the contribution from G_(y,s). Thus the f i r s t term i n the asymptotic expansion for .(III. 1) i n the case considered here i s Now the sums extend ever the saddle points of A* and A- that pinch the £ostive r e a l k axis. we extend the r e s u l t s to non-zero y just as we did in section 1. Section 3. l£ansiticn V e l o c i t i e s In the la s t twe sections we have found asymptotic expressions for the i n t e g r a l in (III . 1 ) for large t and arbi t r a r y y. In order to evaluate these expressions i t i s necessary to find the saddle points of A(k,s) that pinch the (pestive) r e a l k axis for each value of V. This seeics to be a rather formidable task. F i r s t we must find the saddle points for each V, then for each saddle point we must find the images of Im{s }=Im{s0} to determine i f the twe zerces cf A(k,s) near ke coalesce from opposite sides of the (positive) r e a l k axis. Even given an e f f i c i e n t methed of solving A(k,s)=0 th i s could be very time consuming. In thi s section we introduce a method for streamlining t h i s process. The r e s u l t s given here are due mainly to H a l l and Heckrotte (III.26) 46 (1968) . As V varies the saddle point k 0 ( V ) also varies. For some i n t e r v a l I, k Q ( V ) i s of the pinching type i f veI and i s not of the pinching type i f V^I. We c a l l the endpcints of any such i n t e r v a l t r a n s i t i o n s v e l o c i t i e s . Notice that a l l the i n t e r v a l s I must be bounded since otherwise there would be an i n f i n i t e propagation v e l o c i t y . There are three f a i r l y simple c r i t e r i a for determining when a given velocity i s a t r a n s i t i o n v e l o c i t y . They are: C1) I f , as V-Voo, there i s a k 0(V) such that | k e-(V) | - t « > , then Voo i s a t r a n s i t i o n velocity. C2) If at V=V^ thsre i s a k Q(V) such that then V K i s a t r a n s i t i o n v e l o c i t y . Note that i n general t h i s i s equivalent to solving the system of equations which i s a set of six re a l equations (A i s a complex valued function) in the fi v e r e a l unknowns , k r , s ^ , s ^ and V. Here any solutions. However, when there are stable, non-decaying waves, i . e . when there i s an i n t e r v a l J such that k r e a l and (.11-1.27) (III. 28) k=kr + ik; and s=s r+is^ Thus in general there w i l l net be 47 k e j implies that any root of 4 (k, s) =0 i s imaginary, then V found from A b (.te,? )'- A b (b,s+ik v) + ^ V Asdte,s+ik V) = 0 (.111.29)' i s automatically r e a l . This i s because (III.29) gives V - — ' -1 — - A? t k,S+ i.Uv) c\fe which i s r e a l . Thus the system (III.28) i s a set cf three equations in three unknowns when the waves are stable and non-decaying. In t h i s case V K i s a maximum or minimum value for V as (III.27) requires that wnen V— a'* Then the point y=V^t i s the leading or t r a i l i n g edge cf the wave. C3) I f for V=Vlf there are two saddle points (k<°-> rsJ- <x->) and (kc^>,s<^>) such that Im{ s^* > }=Im{ sD<k >), then Vx i s a t r a n s i t i o n v e l o c i t y , Figure III,5 i l l u s t r a t e s ttie i n t e r a c t i o n of the two saddle points that takes place here. For VXVj, say, both the saddle points A and B contribute with branch I coming from the upper half plane and branch II going into the lower half plane, see Figure 111.5(a). For V=Vj the two saddle points " i n t e r a c t " and branch I and II coalesce. See Figure 111.5(b). The curve Im{s]=Im{s0 } goes 48 through B f i r s t then A since B l i e s tc the right cf A in the complex s plane. Fcr V>VI only B contributes and now branch I comes in from the lower half plane and branch II gees into the upper half plane, see Figure 111.5(c). The branches I and II have exchanged positions. Note that i t i s only the saddle point with the smaller,Re{s} that can be l e s t i n th i s way. Note also that t h i s c r i t e r i o n only makes sense in a system that supports unstable or decaying waves sc that a l l of the saddle points do-not l i e along the imaginary s axis. The procedure then for determining the asymptotic development for (III.1) i s to find the saddle points f cr each V and then use- the c r i t e r i a above tc find the t r a n s i t i o n v e o c i t i e s . Then it- i s necessary only tc check one v e l o c i t y in each i n t e r v a l to see, i f the saddle points ar.e pinching or not i n the whole i n t e r v a l . Note that we must check both sides of the t r a n s i t i o n v e l o c i t i e s since the c r i t e r i a above are not s u f f i c i e n t conditions to insure a t r a n s i t i o n . Section 4. Moving Disturbances In sections 1 and- 2 we have assumed that F(k,s) has nc poles. This corresponds to finding the response to a stationary disturbance that i s f i n i t e in time and space. Now we extend the r e s u l t s given above to the case in which F(k,s) has a simple pole which i s due to a moving and/cr 49 o s c i l l a t i n g disturbance. l i g h t h i l l (1967) has considered the problem of moving disturbances i n stable systems and Briggs (1964) has touched cn the problem of o s c i l l a t i n g disturbances in unstable systems. However, no one has worked out the d e t a i l s of the response of an unstable system to moving o s c i l l a t i n g disturbances. As we see from (11.50) a disturbance moving with v e l o c i t y V and with time dependence exp (s t) gives r i s e to an i n t e g r a l of the form ^Cvt^M 6,-t^=—- e ^ t s j TT—T — ; (III.30) He know from our previous results that the asymptotic behaviour of (III.30) i s governed by the pinching saddle points of the denominator. These are of two types here: 1) the saddle points of A(k.,s) ; and 2) the points (k„,s0) such that i - ^ - ; f e 8 ( v t v ) =o , (III.31) A ^ o , ? . "> = 0 . We have already treated the saddle points of A(k,s) in sections 1 and 2, and we need only replace F(k,s) by F (k,s) /[ s-s* - i k (V*- V) ] i n (III.11). We c a l l t h i s the transient response, iff. Ncte that i f (III.31) i s s a t i s f i e d at a saddle point then t h i s i s unbounded. Special 5b consideration can be given to such points but the d e t a i l s are not given here. We now determine the response due to the points given by (III.31), which we w i l l c a l l the forced response, ^ F . Given an s near s e there are two k's, k t and k z l near kQ that contribute to the Fourier i n t e g r a l in (III.30). These are given by U,.ko= - -r- ( H I . 32) and l ^ - l e , c u t s - ? . ) + < x 3 . C ? - % y S •• • . (II I. 33) We assume here that Ak(k„ , s 0) /0, i . e . k0 i s . not a saddle point for A(k,s), as we mentioned above. From the arguments in-section 1 and 2 we know that i f k, and k t coalesce across the (postive) r e a l k axis, then s „ i s s i n g u l a r i t y cf <*<.1.,s>= — J_Lli£ L. dk. (III.34) We must investigate t h i s s i n g u l a r i t y . The contribution to G(y,s) from k,, assuming that y>0 so that we close below and that Im{k, }<0, i s and from k 2, under the same assumptions, i s 51 Now where V =iA<°VA| 0 ) i s the group ve l o c i t y cf ( k 0 , s 0 ) . Also since a, =-i/(V 0-V) . Now A * t l e ^ ? ^ - * A " 1 t v 0 - V ) so that Therefore, the contributions from k ( and k z are expanded abcut s„ as N * - v e } A . ; o > 15-Si,) and LVo-v* ) A ^ ) \ " ' respectively. The rest of the terms in the expansion are anal y t i c at s 0 . Thus we see that i f k , and k z approach the r e a l k axis from the same side, then the singular terms cancel. On the other hand i f they appraoch from cppcsite sides only one contributes and there i s no cancelation. Note that we have assumed here that V* >V so that k, 52 approaches the r e a l k axis form the lower half plane. If A (k,s) have branch points as considered in section 2 then the r e s u l t s are extended as they were there. The r e s u l t s given above show that the leading term i n the asymptotic expansion of the forced response i s given by where the sum extends over a l l solutions of (III.31) that i r e of the pinching type. The t o t a l response i s % + Vf.. The c r i t e r i a given in section 3 can be applied here to determine the t r a n s i t i o n v e l o c i t i e s . Note that V=V* i s a t r a n s i t i o n velocity since from (III. 32) as V -V changes sign so does the d i r e c t i o n that k , moves as s moves toward s 0 from the right . A t r a n s i t i o n by C2 comes about when the soluti o n of (III.31) i s a saddle point. When C3 i s applied some rather interesting results arise. This w i l l be pursued further i n Chapter IV. Section 5. Numerical Methods As we have seen the problem of determining the asymptotic behaviour of the i n t e g r a l i n (III.1) reduces to the problem of finding the solutions (k e >,s 0) qf V* <V then the signs must be reversed. I f F(k rs) and/or (III. 3 5) (111,36) 53 (III.37) where A(k,s) i s defined i n (III.15). This system i s equivalent to the system (III.36') (III.37') After we have solved either of these systems for a given .V we must map the l i n e s Im(s}=Im{s0} into the complex k plane by (III. 36) in order to determine whether (k0,s„) i s a pinching saddle point. This requires that we able tc solve (III.36) for k as afunction of s. In finding the response to a moving disturbance we are led to the problem of evaluating the aymptotic behaviour of the i n t e g r a l in (III.30). This problem reduces to that of solving (III.36) and (III.37) again, and the system (III.31). Solving (III.31) i s equivalent to solving the single equation This equation needs only be solved once for a given s and V +. C a l l the solution k,. However, we s t i l l need to map ACie,s<+^ kv*) = o. (III. 38) the l i n e s Im{ s }=Iiu{ s*+ik , (V*-V) }• for each V in order tc determine the pinching behaviour of the solutions of 54 (III. 31) . The basic method'that we use tc selve (III. 36') and (III.37') i s Newton's method for a system of equations. We make an i n i t i a l guess, (k,,s,), for the solution cf the system. Then we calculate a new guess, (k i,s l.), by A C S " ( LL£T'A V A ? ^ - A ' " ( C * j V A M \ , (III.39) A" ( aff » i v - A^  C ^  * , v &r) These expressions are obtained by expanding (III.36') and (III.37')"about the unknown point tc f i r s t order. Then.we assume that ( k 1 ( s j i s indeed a solution of the system. This leads to a linear set of equations fcr (k,_,Sj.) whose solution i s given i n (III.39). Continue in t h i s way tc generate a sequence cf (k,s) pairs. This sequence w i l l converge i f the i n i t i a l guess was close enough tc the solution of the system. Cnce we have solved t h i s system f c r one value of V we can use this solution as an i n i t i a l guess for a s l i g h t l y d i f f e r e n t V. In t h i s way we can generate a l l of the branches of (k 0(V), s 0 (V)) given by (III.36') and (III.37'). A l l we need i s a f i r s t guess for each cf the branches. These are obtained by solving (III.36') f c r s a with k<> given, say, and then computing the V 0 for which (III.37') i s s a t i s f i e d . Then (k 0,s e) can be used as an i n i t i a l guess for a V close to V 0. If k 0 i s i n the range of 55 i n s t a b i l i t y , i . e . s 0 complex, then V 0 may te complex. In that case the f i r s t step i s to use (III.39) to f i n d the solution for V With Be{V)=Be{V0} and Im{V]=0. After we have found (k <> (V), s „ (V)) as described above we must apply the c r i t e r i a given i n section 3 to find the t r a n s i t i o n v e l o c i t i e s . To apply C1 and C3 i s straight forward. For C1 we just plot Re[k 0(V)} and Im{k.0(V)] and look for v e r t i c a l asymptotes. For C3 we plot Im{s0(V) ) for each branch. Points of i n t e r s e c t i o n cf twc branches are t r a n s i t i o n v e l o c i t i e s . For C2 we must finds points where (III.27) i s s a t i s f i e d . Note that t h i s requires no extra computation since when (III.36') and (III.37') are s a t i s f i e d then the denominators in (III.39) are just Thus we can compute A'tetas we compute (k 0(V) , s 0 (V)) at no extra expense and thus applying C2 i s also easy. F i n a l l y we must determine the pinching behaviour cf the saddle points (k Q(V) ,s c(V)) on either side of the t r a n s i t i o n v e l o c i t i e s . This requires that we solve (III.36) f c r k as a function of s, near a double zero. Newton's method applied to the single equation (III.36) converges slowly, i f at a l l , at such points and even i f i t does converge d e f l a t i o n must be used to find both solutions. A method due to Celves and Lyness (1967) can be used to solve t h i s problem. 56 He use the following theorem from the theory of complex variables. If f(z) i s ana l y t i c i n a closed bounded region .of the z-plane and has zeroes z^ , , i=1 ,n inside C-3C then t h i s formula which i s just a consequence of Cauchy's theorem. For r = 0 the value of the i n t e g r a l i n (III.40) i s just the number, n, of zeroes within C. Using the s r for r=1,...,n i t i s possible t c construct a polynomial of degree n that has the same zeroes as the function f (z) inside G by means of the so-called Newton r e l a t i o n s . There are powerful techniques f o r solving polynomials numerically so we can consider the problem solved when we find the polynomial. Anyway, i n the problem we are considering we are interested only in cases where there are only two or three zeroes i n the region of inte r e s t . In order to determine the pinching behaviour of a . saddle point (k 0,s 0) we map the l i n e s lm{s}=Im{s0} into the complex k plane by (III.36). We choose an s on thi s l i n e near s 0 and apply the above method to f i n d the two k's close to k e by l e t t i n g i n (III.40). Wa choose C to be a small c i r c l e around k . (III.40) A multiple zero i s counted according to i t s m u l t i p l i c i t y i n 57 The contour i n t e g r a l in (III.-40) i s computed numerically. The rapid convergence of numerical ccntour integrations i s discussed by lyness and Delves (1967) in a companion paper to t h e i r one on root finding. Once we have found the two k's near k 0 we can use Newton's method on the single equation (III.36) t c trace the l i n e s Im{s}=Im{s0} awaj from the saddle point. F i n a l l y we note that in order to solve (111.38) we can use the method of Delves and Lynass. 58 CHAPTER IV - Discussion and Results Section 1. I n t r o d u c t i o n In NM i t i s suggested that the unstable shelf waves studied there might be the o r i g i n cf the meanders i n the Gulf Stream and i n p a r t i c u l a r that a fa s t moving disturbance might be most, e f f i c i e n t in generating the unstable waves. We are now in a position to make a f i r s t step in checking t h i s hypothesis. In NM two models are considered, cne that i s applicable to the r e g i o n in which the Gulf Stream flews over the Blake Plateau and one that i s a p p l i c a b l e t c the region n o r t h e a s t of Cape Hatteras a f t e r the Gulf Stream has detached from the continental shelf. Here we consider the f i r s t model, which has been described in Chapter II. Of course, our l i n e a r model i s net applicable tc large amplitude waves and cannot be expected to test the hypothesis in NM i n any d e t a i l . Fcr example, i t would be u n r e a l i s t i c to compute the detailed response of cur model to ° complicated, though r e a l i s t i c , wind stresses and expect the calculated response to resemble the actual behaviour cf the Gulf Stream. Hence we have r e s t r i c t e d ourselves tc considering the simple c u r l free wind stress models described in section 4 of Chapter I I . Dsing these mcdels alcne i t i s possible, for example, to determine hew the e f f i c i e n c y of the generation of the unstable waves varies 59 with the speed and duration of the disturbance. In order to apply the methods of Chapter III to obtain these results we must f i r s t study the dispersion r e l a t i o n . This we do in section 2; In section 3 we calcu l a t e the response for an impulsive l i n e source of wind s t r e s s , mainly in order to gain experience i n using the methods without the additional complications introduced by moving disturbances. In section 4 we turn our attention tc the moving distutbance problem. Section 2. The Cis^ersion Relation The actual bottom topography and depth averaged current p r o f i l e of the Gulf Stream over the Blake Plateau are shown in Figure II.1. The best f i t to t h i s by the model shewn in Figure II.2 i s given by X-.2-2. ^. = 2.8 with D=800m, L=50km and f=10-*sec- 1. He now study the dispersion r e l a t i o n A(te,s)=o (IV. 1) where Mk,s) i s given in (11.38), for these values of \ and M • Before we proceed we note that A(k,s) i s of the fcrm studied in section 2 of Chapter III and thus we must consider the two functions 60 A- tfe,-*i'> = -S2$Ck> i.S'^ tte') J r>(k), (IV.2) From (III.38) we see that f, g and h are r e a l when k i s r e a l and that (IV.3) where denotes the complex conjugate. This allows us to write A_ (>,S W A + I W , S ) . From (11.44) - (ii.U6) we see that F +(k,s) and F_(k,s) also s a t i s f y t h i s r e l a t i o n . Thus we have so- that and the t o t a l response i s just given by 1 a iu Hence in the following we consider only A+(k,s) which here we write as A ( k , s ) . As we see from the results of Chapter III i t i s f i r s t necessary to solve (IV.1) for s with k taking r e a l values. This gives the branch l i n e s of G(i,S) in the complex s-61 plane. Since A(k,s) i s quadratic i n s t h i s i s net hard tc do and the results are given i n Figure IV.1 where we plot Im(s} for k r e a l . For most r e a l values of k there are two pure imaginary solutions of (IV.1). However, in a small i n t e r v a l the solutions are complex with solutions s and -s. This i s the " i n t e r v a l of i n s t a b i l i t y " since one cf the solutions has positive r e a l part. Next for each V we must fi n d the saddle pcints (k 4,s 0) that are solutions cf the system (III.36') and. (III.37 1). Outside of the i n t e r v a l of i n s t a b i l i t y this i s straight foward and we do not need the methods described in section III.5. By (IV.2) and- (IV.3) we have that is. r e a l when k i s r e a l and s i s pure imaginary. Thus when we solve (IV.1) for s e with k e r e a l and outside the i n t e r v a l of i n s t a b i l i t y , the point (k 5,s,) i s a saddle point f c r v. - + • Then graphically we can invert and get k a as a function of V 0. Within the i n t e r v a l of i n s t a b i l i t y we must use Newton's method as described i n section III.5. As a s t a r t i n g point we use the r e a l k 0 at which Ra[s,T[ i s a maximum. At t h i s 62 point V0 =iA^ o V&s( 0 ) i s r e a l . 1 In Figure IV. 2 we have plotted Re{k) as a function of V for the stable saddle points. In Figure IV.3 we have plotted Re{k}, Im{k}, Ee{s) and Im{s} as a function of V for the unstable saddle points. The saddle points found i n t h i s way are the only cnes that can be of the pinching type. This i s because i f any other saddle points are found for some r e a l V they cannct be of the pinching type since neither of the paths cf steepest descent through k 0 can have crossed the r e a l k axis. . once we have found a l l of the saddle points that can be of the pinching type we must determine which of these r e a l l y are pinching saddle points. A glance at Figure IV.2 shews us that there c e r t a i n l y must be t r a n s i t i o n v e l o c i t i e s , since otherwise we would have signals traveling at a r b i t r a r i l y large v e l o c i t i e s . This physical c r i t e r i o n i s a geed check on the mathematical r e s u l t s that we obtain. In crder tc f i n d the t r a n s i t i o n v e l o c i t i e s we apply the c r i t e r i o n given i n section III.3. Recall that we do not expect C2 tc apply i n the i n t e r v a l of i n s t a b i l i t y and that C3 can only apply there as i t requires two saddle points with d i f f e r e n t Re{s}. From Figure IV.3 (a) we see that by C1 the unstable i This i s because ds/dk=- and when Re[s} i s a maximum dRe{s}/dk=0. Thus Im{ i Ak/&<, }=0. 63 waves only contribute oyer a f i n i t e i n t e r v a l with endpoints given by 2 Thus the system i s convectively unstable. We w i l l r e f e r to t h i s i n t e r v a l as the (velocity) i n t e r v a l of i n s t a b i l i t y (dropping the v e l o c i t y when no confusion w i l l a r i s e ) . In Figure IV.2 we have marked the points where t r a n s i t i o n s take place by C2 and C3 by the number of the c r i t e r i o n used to find i t . To find t r a n s i t i o n s by C2 we need only find the maxima and minima i n the graph of V as a function of r e a l k. In t h i s way we f i n d a t r a n s i t i o n at labeled A in Figure IV.2. To f i n d the t r a n s i t i o n s given by C3 we must plot Im[ s} for a l l the branches of (k 0( V) , s 0 (V) ) i n the i n t e r v a l of i n s t a b i l i t y and any points of in t e r s e c t i o n of these curves w i l l be t r a n s i t i o n v e l o c i t i e s . In Figure IV.3(d) we have plotted Im{s} for the unstable saddle points, branch C, and Im(s} f o r the two sets of stable saddle points, branches A and B. There are two points of i n t e r s e c t i o n at 2 A l l of the v e l o c i t i e s are given i n dimensional units i n which the velocity of the stream i s X=.22=77km/day, since in dimensional units s/k=Lf s*/k*«> s*/k* = X s / k = v G . 64 labeled B and C respectively in Figure IV.2. In Figure IV.4 we show how the t r a n s i t i o n takes place f o r one of these' t r a n s i t i o n v e l o c i t i e s . F i n a l l y , since we are dealing with & + (k,s) only a saddle point that arises from a merging across the negative r e a l k axis does not contribute. Thus from Figure IV.2 we see that D and. E are also t r a n s i t i o n points., with V r , U V = - .H44-" > r e s p e c t i v e l y . At t h e s e v e l o c i t i e s (k 0, s e) = (0 , 0 ) , a s i t u a t i o n t h a t requires special t r e a t m e n t . A t these p o i n t s we see f r o m F i g u r e IV . 1 t h a t (k a ,s 0) = (0 ,0 ) , and f r o m (III.22) o r (III. 24 ) i t s-s-sms • tha t t h i s i n t r o d u c e s a logarithmic s i n g u l a r i t y w h i c h w o u l d giva r i s e t o a r e s p o n s e for a l l V. However, a d e t a i l e d e x p a n s i o n o f G(y,s) about s=0 indicates that "the singular terms cancel out so that the two v e l o c i t i e s given above are indeed t r a n s i t i o n v e l c c i t i e s . The expansions are quite messy and have not bean worked out completely so are not given here. When we check which i n t e r v a l s contain pinching saddle points we find that besides' the i n t e r v a l of i n s t a b i l i t y the sections AF, BD and EC i n Figure IV.2 contain pinching saddle points. A l l of the rest are non-pinching saddle points. This i s as we should expect since otherwise i n f i n i t e signal v e l o c i t i e s would have resulted. One very i n t e r e s t i n g consequence of the above r e s u l t s i s 65 that no wave traveling i n the downstream d i r e c t i o n t r a v e l s faster that the maximum speed of the basic flow. Also, waves traveling upstream can t r a v e l twice as f a s t as these going downstream. However, the unstable waves only t r a v e l downstream. From Figure IV.3 (c), where we have plotted Re(s} as a function of V, we see that the fastest growing wave travels at a spaed V^A/2=.11. The e-folding time cf the f a s t e s t growing wave i s about 81.3 i n dimensionless units or about 9.4 days.' Section 3. !§s£onse to a Stationary Disturbance Here we present the results of calculations made using the above results for tha impulsively applied stationary l i n e source of wind stress given i n (11.47). This i s new s t r a i g h t forward using the r e s u l t s given i n Chapter III. In Figure IV.5 we plot ^ (1,y,100) and ^(1,y,200) as found using the asymptotic solution given i n Chapter III. This corresponds to a snapshot of the response response taken at times t=100 and t=200, i . e . 16 and 32 days after the wind stress was applied, at the center cf the stream. The important fact to notice here i s that the unstable waves are c e r t a i n l y not dominant. At t=200 the most unstable wave has propagated 1100km, the length of the region in which the model i s applicable. This r e s u l t seems to indicate that the i n s t a b i l i t y i s convected out of the systam before i t grows 66 to s i g n i f i c a n t proportions. Note that no wave trav e l s faster than about . 22, so that beyond t h i s point no waves appear. That i s why in t h i s and the following figures the curves are cut off at a point corresponding to V^.22. Actually there should be a smooth exponential decay to zero as shown by an Airy phase analysis, but t h i s was net dene here. However, i n Figure IV.6 we plot i-(x, 4, 200) f c r 0<x<2, and we see that the response at x=2, the off shore, edge of the steeam, i s much larger than the response at x=1, the point of maximum v e l o c i t y . The p r o f i l e i s s i m i l a r at a l l points along the stream. in Figure IV.7 we plot t(2,y,100) and ^/(2,y, 200), and we see that not only i s the response larger than at x=1, but also i t has a d i f f e r e n t form. Here the unstable waves are completely dominant. Thus we see that the stationary c u r l free wind stress generates the unstable waves much more e f f i c i e n t l y on the offshore side cf the stream than on the inshore side. If we look at (11.44) and (11.45) along with Figures IV. 3 (a), (c) and (d) the reason for t h i s i s c l e a r . In the i n t e r v a l of i n s t a b i l i t y Im{ s}cs>,/2 = . 11 , Re[s}^. 01 and Re{k}«3. Thus here sinhk*.. 5exp(k) so that the integrand i n (11.44) i s only of order Re{s}».01 while the integrand in (11.45) i s cf crder Im{s}-x..11. This argument in fact shows that the "free waves" have t h i s behaviour so that we can expect that f c r 67 a l l wind stresses the unstable waves w i l l be generated most e f f e c i e n t l y on the offshore side of the stream. In Figure IV.8 we plot ^(x,20,t) as a. function of t for (a), x=1, and (b), x=2, for 100<t<350. Here the difference i n tha behaviour between x=1 and x=2 i s again apparent. Bote that when computing ^ (x,y,t) as a function of t for a fixad value of y i t i s s and not s that determines the response. This i s because the exponent i n tha asymptotic formula i s and y i s a constant. F i n a l l y we estimate the current v e l o c i t i e s generated i n th i s case. Ha are mainly interested i n the cross stream v e l o c i t y , u, as t h i s w i l l t e l l us how far we can expect the stream to be displaced due to tha wave motions. By (11.10) we must find tha long shore derivative of ^ in order tc find a. From Figure IV.7 we estimate u f o r x=2 and t=200 at the position of tha f a s t e s t growing wave to be about 1 km/day for a 1 dyne/cm2 wind s t r e s s . Sinca f o r the f a s t e s t growing wave Im{s}^.11 tha period i s approximately 6.7 days. Hence a as estimated above would r e s u l t i n displacements of about 4 km, which i s small compared to the observed Gulf Stream meanders. 68 Section 4. Moving Disturbances Now we present r e s u l t s for the moving l i n e source of wind stress as given i n (11.49). To calculate the transient response, *Vr, we proceed exactly as we did f o r the stationary wind stress except we replace F (k, s) by F(k,s) /[s-s* -ik(V*- V) ]. To calculate the forced response, 'Vp , i t i s necessary to solve (III. 38) for the values of F and s that are of interest. I t i s found by solving (III.38) for various values of s* imaginary and V* that only r e a l values of k arise. Thus moving and/or o s c i l l a t i n g disturbances do not force the unstable waves d i r e c t l y , i . e . through ."V?. Also, as w i l l be- seen l a t e r a time-decaying disturbance, Re{s }<0, only gives r i s e to a time decaying «VF . To begin we w i l l consider the problem i n the case for which s*=0. Then since there are no complex solutions cf (III. 38), solving i t i s equivalent to finding the intersections of the curves in Figure IV.1 with the li n e Im{s}=ikV*, i . e . we find the wave that has phase v e l o c i t y V*. Note that we can also treat the case of s* pure imaginary t h i s way too. when stated t h i s way the problem i s f a i r l y simple. Note f i r s t that k=0 i s always a solut i o n of (III. 38) for s* = 0. Again we note that t h i s gives r i s e to a logarithmic s i n g u l a r i t y . In t h i s case, with a moving disturbance with no exponential time behaviour, the 69 logarithmic s i n g u l a r i t y can give r i s e to a small a d d i t i o n a l response i n a f i n i t e v e l o c i t y i n t e r v a l . Again t h i s r e s u l t i s based on a detailed expansion of G(y,s) about s=0 i n which i t i s seen that the singular terms cancel outside of certain f i n i t e v e l o c i t y i n t e r v a l s . Next note that for -.<V <0 there i s only one non-zero solu t i o n and that for 0<V*r<.22 there are always twe non-zero sloutions. Otherwise, k=0 i s the only s o l u t i o n . After we have found k 0 for a given V from (III.38) we must determine over what range of V the point (k 0,s o) obtained from (III.31) i s cf the pinching type. F i r s t we know that V* i s a t r a n s i t i o n v e l o c i t y . Next when V=VG, the group ve l o c i t y at k 0, the point (k 0,s 0) i s a saddle point of &(k,s+ikV 0). If t h i s saddle point i s of the pinching, type then we have a s i t u a t i o n as shown i n Figure IV.9. As V goes through V e, the point (k 0,s 0) changes from pinching tc ncn-pinching. Thus in this case V=VC i s at r a n s i t i o n v e l o c i t y . However, i f the saddle point at (k 0,s 0) i s not of the pinching type, then we see from Figure IV. 10 that V=V0 i s not a t r a n s i t i o n v e l o c i t y . There must be another t r a n s i t i o n v e l o c i t y , since otherwise we would have i n f i n i t e signal v e l o c i t i e s . Note that the c r i t e r i o n that V=Ve i s a tr a n s i t i o n velocity i s equivalent to C2 as given in section III.3, since at (k 0,s 0) when V=V( o • 70 What about C3? Consider the s i t u a t i o n in Figure IV.11. Here we have (k 0,s o) i n t e r a c t i n g with a saddle pcint {k,,s () f o r which Im[ s D }=Im[ s, }, at V=V£. As V goes through V-j the point (k0 fs6') changes from pinching to non-pinching when Re{s,-}>Re{ s0.}. Otherwise there i s no t r a n s i t i o n at V x (just change the d i r e c t i o n of the arrows through the saddle pcint k 0 in Figure IV.11) . Thus i f Ve i s not a t r a n s i t i o n v e l o c i t y we should be able to f i n d a t r a n s i t i o n v e l o c i t y , V T, by C3. We see that f c r the forced response the group vel o c i t y of the excited wave i s not necessarily the si g n a l v e l o c i t y . We now apply the above remarks to the problem of finding the response for pa r t i c u l a r moving disturbances. F i r s t wa consider a slow downstream moving disturbance with V =.03. We choose t h i s value for V ^ a s i t i l l u s t r a t e s the remarks above about C3 and as i t happens i s one of the most e f f i c i e n t disturbance v e l o c i t i e s f o r generating the unstable waves. Solving (111.38) for V* = .03 gives two solutions with From Figure IV.1 we see that the saddle points f o r these group v e l o c i t i e s are non-pinching. Thus we must apply C3. Again we r e c a l l that C3 cnly applies i n the i n t e r v a l of i n s t a b l i l i t y . Hence i f we plot Iw»{sf.= fe.(v*-v) as a function of V for each of the k„'s above along with 71 Im{s} for tha unstable waves i n the i n t e r v a l of i n s t a b i l i t y , tha points of int e r s e c t i o n w i l l ba t r a n s i t i o n v e l o c i t i e s . In t h i s way we find t r a n s i t i o n v e l o c i t i e s at V= All. -Cov b,.--2.tl ' ' Thus f o r the forced response we have the following s i t u a t i o n : for V<.00081663 and V>.1815881 only the transient response i s observed, for .00081663<V<.03 the forced response has wavenumber k0=3.67 and f o r .03<V<. 1 815881 i t has wavanumber k6=2.61. In Figure IV. 12 we plct ^(x, y, 100) and ^ (x, y, 200) for V = .03. In (a) and (b) wa plot ^ r and *YF, respectively, for x=2.. We sae that behind tha disturbance the transient and forced responses are cf about the same magnitude. However, in. front of the disturbance the transient response dominates since here the unstable waves are important. In (c) and (d) we plot "VT and respectively, for x=1. Here we see that though the response i s s t i l l much smaller than at x=2 the unstable waves are now dominant, as they were not in the case of the stationary disturbance. This i s because the phase speed, -is / k , of the unstable wavas i s very clcse (in the compex plane) tc tha ve l o c i t y of tha disturbance so that (s-ikv ) _ l i s very large i n the asymptotic formula. Thus the unstable waves are amplified. A comparison of Figures IV.12 (a) and IV.7 shows that tha amplification factor i s 72 about 40. Hence we see that a slow downstream moving disturbance i s very e f f i c i e n t at generating the unstable waves. In fact we f i n d that the maximun amplification i s produced by a disturbance moving with V*=.03902. In contrast to the l a s t example we now consider a fast upstream moving disturbance with-V =-. 1. In t h i s case there i s only one solution to (III.38} with Since the saddle point for t h i s V 0 i s a pinching one V 0 i s a t r a n s i t i o n v e l o c i t y . Thus f c r -. 1<V<. 141 there i s a forced response with wavenumber k =1.45 and outside t h i s i n t e r v a l thc-re is only the transient response. In Figure IV. 13 we plot %.(2,y,100) and ^ T(2,y,200). we ignore the forced response since, as in the la s t example, i t i s small ecupared to. the transient response in the region of wave growth. Here we see that although the unstable waves are amplified, the amplification i s only about one-fifth that i n the previous case. The d i s c o n t i n u i t i e s i n the plots of *¥T(x,y,t) occur at V=Ve which, as was pointed out i n Chapter I I I , i s a point at which the asymptotic formulas do net aPPly« We w i l l see below that introducing damping into the wind stress removes the d i s c o n t i n u i t i e s . Now we consider the above examples i n the case that the disturbances are decaying in time with s*=-.06. This corresponds to an e-fcldir.g time of about two days which i s a reasonable time scale f cr weather systems. Now when solving (III.38) both the k 0»s and V 0 * s obtained are complex, so that now in both cases the t r a n s i t i o n v e l c c i t i e must be found using C3. The k e's, V0* s and t r a n s i t i o n v e l o c i t i e s f or V*=.03 are now and for V*=-. 1 are It i s important tc note that the t r a n s i t i o n v e l o c i t i e s are such that Ee{ s*+ik ( v * - V ) ]<0 for a l l V s for which there i s forced response. In Figure IV. 14 wa show ^ ( 2 ,y,t) and "^ f (2,y,t) for V=.03 i n t h i s case. We see that here the amplification of the unstable waves i s not as great as with the non-decaying disturbance since the transient response i only about one-half of what i t i s i n that case. The forced response now decays from the t r a n s i t i o n v e l o c i t y . Hence we can see the si g n i f i c a n c e cf the t r a n s i t i o n v e l o c i t y as the signal velocity. In Figure IV. 15 we plot AV T(x,y,t) f c r V"*=-.1. We see that in t h i s case the decay of the disturbance has no eff e c t on the transient response, ether than smoothing out the d i s c o n t i n u i t i e s . Again the forced response i s s i g n i f i c a n t only near the t r a n s i t i o n v e l o c i t y , 7a and aven here i t i s completely n e g l i g i b l e . Section 5. Discussion In t h i s section we w i l l discuss the application of the r e s u l t s prasantad above to tha problem of meanders in the Gulf Stream. As has been pointed out i n NM the wavelength of the" unstable waves i s about 140 km and the period abc'ut 10 days, which corresponds wall to the wave-like c h a r a c t e r i s t i c s of the meanders i n the Gulf Stream (see Hansen, 1970). Thus i t i s reasonable to consider the wind generation problem. A l l of the re s u l t s given above indicate that the unstable waves are generated on tha o f f shora side of tha v e l o c i t y maximum by any kind of wind, since the c u r l free component studied here would contribute i n any case. The displacements of the s t r e a i that can ba expected to r e s u l t from these waves can be estimated from estimates cf the cross stream v e l o c i t y by multiplying the maximum cross stream ve l o c i t y by one-half the period of the waves. This was done above for the stationary wind s t r s s . For the wind strass moving with V"* = .03( 10 km/day) the response calculated i s about 40 times that f o r the stationary disturbance and thus so are the displacements. The displacements for the upstream moving disturbance with V =-.1 are about 8 times those for the stationary 75 disturbance. Thus displacements of the order of 150 km can be accounted for as unstable waves generated by moving disturbances. A result that could be important to the expermimental problem of observing unstable shelf waves propagating along the Gulf Stream i s contained i n the r e s u l t s given above. The method used for detecting shelf waves has been to analyse tide gauge records taken along the coast. Non-barometric responses are attributed tc shelf waves propagating along the coast {see Mysak and Hamcn, 1969). Our r e s u l t s indicate that the unstable waves have small' amplitudes on the inshore side of the stream, and that in some cases they are cf abcut the same amplitude as the stable waves. Thus i t i s questionable that the unstable waves could be observed in t h i s way. Note that though our r e s u l t s have been derived f c r c u r l free wind stresses i t i s expected that the re s u l t s for wind' stress with c u r l would agree at l e a s t q u a l i t a t i v e l y because of the structure of the "free waves". Before considering the problem of more r e a l i s t i c wind stresses i t would be mere important to try to look at a more r e a l i s t i c model. For example we have completely ignored the s t r a t i f i c a t i o n and v e r t i c a l shear in the stream. F i n a l l y , though, we point out again that the methods developed here apply only f c r li n e a r models that cannot hope tc cope accurately with f i n i t e amplitude phenomena such as the meanders in the Gulf Stream. 76 CHAPTER V - Conclusion In Part A of t h i s thesis we have studied the problem cf the generation of unstable shelf waves by the wind i n a barotropic model that could apply to the Gulf Stream ever the Blake Plateau. This was done i n order to test the hypothesis that the meanders observed in the Gulf Stream might originate as unstable shelf waves that extract energy from the l a t e r a l shear of the stream and are generated by the wind. The propagation of such waves was studied i n NM where i t was found that the unstable shelf waves have properties s i m i l a r tc the wavelika properties of the meanders. Here we have studied the generation of these waves in order to f i n d under what conditions the wind generates them most e f f i c i e n t l y . In order to solve t h i s problem we solved the i n i t i a l boundary value problem for the response of the model tc an applied wind stress by means of Fourier-Iaplace transforms. This leads us to the problem of evaluating the asymptotic behaviour for large t cf Fcurier-Iaplace inversion i n t e g r a l s f o r systems that support unstable waves. This problem has been worked out by plasma physicists studying the i n t e r a c t i o n of plasmas with electron streams fcr the case cf stationary forcing functions. An observer moving at vel o c i t y V w i l l see the wave exp (s„-ik 0y) i f (k 0,s 0) i s a 77 double zero of A(k,s+ikv) (a(k,s)=0 i s the dispersion r e l a t i o n ) , such that the steepest descent paths through k approach k 0 from opposite sides of the (positive) r e a l k axis. Such a k 0 i s c a l l e d a pinching saddle pcint. C r i t e r i a can be found to determine t r a n s i t i o n v e l o c i t i e s at which saddle points change from pinching tc non-pinching. These r e s u l t s are e a s i l y extended to the problem of determining the asymptotic response for a moving forcing function. The response i s the sum of two terms, the forced response and the transient response. The transient response i s determined exactly as the response f o r a stationary forcing function. The forced response i s a wave whose phase v e l o c i t y , - i s / k , i s the same as the velocity of the forcing function. It appears only i n a f i n i t e range of v e l o c i t i e s th-e endpoints of which may or may not be the group v e l o c i t y of the forced wave. We have applied the above techniques to the problem of the generation of unstable s h e l f waves by the wind for simple c u r l free wind stresses. The r e s u l t s Indicate that moving wind systems can be very e f f i c i e n t in generating the unstable waves. A.slow downstream moving system i s the most e f f i c i e n t at generating the unstable waves, though i t s e f f i c i e n c y i s affected quite considerably by the duration of the disturbance. A fast upstream moving wind system i s less e f f i c i e n t at generating the unstable waves, but the 78 e f f i c i e n c y here i s almost completely independent of the duration of the disturbance. It i s found in general f c r the c u r l free wind stresses that the unstable waves are much larger on the offshore edge of the stream than anywhere else. The r e s u l t s described above could be useful i n interpreting Gulf Stream meander data since now we can obtain a re l a t i o n s h i p between the velocity and time scale of a weather system, and i t s e f f i c i e n c y at generating the unstable waves. In solving the problem of the generation of the unstable shelf waves by the wind we have made many simplifying assumptions. The s t r a t i f i c a t i o n and v e r t i c a l shear in the Gulf Stream have been ignored. The model studied i s l i n e a r so that i t cannot describe the f i n i t e amplitude meanders accurately. The wind stresses that we have considered are c u r l free when almost a l l wind systems in .the r e a l world have non-zero c u r l . However, we have solved the problem we set out to do which was to gc beycnd the s t a b i l i t y analysis that i s usually done and to determine how much f a i t h can be put in t h e i r r e s u l t s . Our re s u l t s indicate that care should be exercised i n applying general s t a b i l i t y analysis to geophysical systems and that i f possible i t should be augmented by at least determining whether the system i s convectively or absolutely unstable. Further as we have seen i t i s possible f o r the i n s t a b i l i t y 79 to be of importance only i n a part of the system. Besides t h i s we have obtained r e s u l t s that can possibly be of use i n interpreting meander data. . The work presented here shows how methods developed i n one f i e l d can be used p r o f i t a b l y in another. We have solved a problem in geophysical f l u i d dynamics, not attempted before, by methods developed i n plasma physics. This wcrk at least indicates a d i r e c t i o n that i n s t a b i l i t y studies of more r e a l i s t i c models in geophysical f l u i d dynamics cculd take, since the methods gathered together here are s u f f i c i e n t l y general that they could te applied tc a large class of such models. Further, the results are of i n t e r e s t i n themselves as they indicate'that certain wind systeirs, in p a r t i c u l a r these moving slcwly in the d i r e c t i o n of the stream, are very e f f i c i e n t at generating the unstable waves, at least on the offshore side of the stream. 80 FIGURES FOR PART A 82 0 k-pWrve. 83 F I G U R E TU.<+ tb) F\<Tiuiie in. «• 85 86 87 P f S to < 88 89 f , l.o 94 Figure IV. 8 CO J 1 .—i i i S O 20 0 25" O 30 0 ISO T fe.-fta<\e S-pla^e F I SURE 'T3Z-. 1 100 Ff^ ure. TV. \lLc) 101 Co o 103 ? X o - I a O O O <3~ O 105 REFERENCES FOR PART A Adams, J. K. , 1972: Ph. D. Thesis. University of New South Wales. Adams, J. K. and v. T. Euchwald, 1969: The generation cf continental shelf waves. J. F l u i d Mech. , 35 , 815-826. Betchov, R. and W. 0. Critninale, 1967: S t a b i l i t y of F l u i d Flows. New York, Academic Press. Briggs, R. J . , 1964: Electrcn^Stream Interaction with Plasmas. Cambridge, Mass. , KIT Press. Buchwald, V. T. and J. K. Adams, 1968: The propagation of continental shelf waves. Proc. Roy. Soc. £. , 305, 235-250. Chandrasekhar, S., 1961: Hydrodvnamic and Hjdromagnetic §^§.hilit_y Theory, Oxford, Oxford Univ. press. Criminale, W. 0. and L. S . G. Kovasznay, 1962: the growth of lo c a l i z e d disturbances in a laminar boundary layer. J . F l u i d Mech., 14, 59-80. Delves, L. M. and J. N. Lyness, 1967: A numerical method for locating the zeroes of an a n a l y t i c function. Math. Comp., 21, 543-560. Der f l e r , H., 1967: Growing wave and i n s t a b i l i t y c r i t e r i a for hot plasmas. Proc. 8th Int. Conf. On Phen. In Ionized Gases, Vienna, Springer-Verlag, 294. Hall, L. S. and W. Heckrctte, 1968: I n s t a b i l i t i e s : Convective vs. Absolute. Phys. Rav. , 166(1), 120-128. Hansen, D. V., 1970: Gulf Stream meanders between Cape Hatteras and the Grand Banks. Deep-Sea Res., _17, 495-511. Haurwitz, D. and H. A. Panofsky, 1950: S t a b i l i t y and meandering of the Gulf Stream. Trans. Am. Geophys. Union, 31, 723-731. Jacobs, S. J . , 1971: Agecstrophic s t a b i l i t y of divergent j e t s . J . Atmos. S c i . , 28, 954-961. Jef f r e y s , H. and B. S. Jeffreys, 1 956: Methods cf 106 ?i3themat i c a l P h_y_sics. Cambridge Univ. press. L i g h t h i l l , M., 1965: Group v e l o c i t y . J . Inst. Mats. Aplics., j l , 1-28. ; . , 1967: on waves generated i n dispersive systems by t r a v e l l i n g forcing a f f e c t s , with applications tc the dynamics cf rotating f l u i d s . J . Flu i d Mech., 27, 725-752. Lin , C. C., 1955: The Theory of Hydrodynamic S t a b i l i t y , Cambridge Univ. Press. Lipps/ F. B., 1963: S t a b i l i t y of jets i n a divergent barotropic f l u i d . J . Atmos. S c i . , 20, 120- 129. Lyness, J. N. and L. K. Delves, 1967: Cn numerical contour integration round a closed contour. Math Comp., 21, 561-577. Munk, W. H. , 1950: On the wind-driven ocean c i r c u l a t i o n . J. Met., 7, 79-93. Mysak, L. A., 1967a: On the theory of continental shelf waves. J . Mar. Res., 25, 205-227. ._, 1967b: On the very low frequency spectrum of the sea l e v e l cn a continental shelf. J . Geophys. Res., 72(12) , 3043-3047. Mysak, L. A. and E. V. Hamon, 1969: Low-frequency sea l e v e l behaviour and continental shelf waves o f f North Carolina. J . Geophys. Res., 74(6), 1397-1405. N i i l e r , P. P. and L. A. Mysak, 1971: Barotropic waves along an eastern continental shelf. Geophys Fl u i d Dyn. , 2, 273-288. Reid, W. H., 1967: The s t a b i l i t y of f l u i d flows. Fro BP Basic Developments in Flu i d Djnamics, ed. Maurice Holt, New York, Acadenic PressJ 249- 307. Robinson, A. R., 1 964: Continental shelf waves "and the response of sea-level to weather systems. J . Geophys. Res., 69, 367- 368. Orlanski, I., 1969: The influence of bottom topography cn the s t a b i l i t y cf jets i n a barotropic f l u i d . J . Atmos. S c i . , 26, 1216-1232. 107 Pedlosky, J . , 1964a: The s t a b i l i t y of currents i n the atmosphere and the ocean: Part I. J. Atmos. S c i . , 2.1, 201-219. , 1964b: The s t a b i l i t y cf currents i n the atmosphere and the ocean: Part I I . J . Atmos. S c i . , 2 J , 342-353. Pol l a r d , R. T., 1970: On the generation of i n e r t i a l waves in the ocean. Deep-Sea Res., J 7 , 795-812. Sirovich, L., 1971: Techniques cf Asjmjptc t i c Analysis. B e r l i n , Springer-Verlag. Stern, M. E., 1961: The s t a b i l i t y of thermoclinic je t s . T e l l u s , 13, 503-508. Stewart, R. W. , 1964: The influence cf f r i c t i o n on i n e r t i a l models of oceanic c i r c u l a t i o n . Studies in Oceanography, ed. Kczo Yoshida, Univ. Of Washington Press, 3-9. Stommel, H., 1948: The westward i n t e n s i f i c a t i o n of wind-driven ocean currents. Trans. Am. Geophys. Union, 29 , 202-206. 1966: The Gulf Stream, 2nd Edition, Berkeley, Univ. C a l i f . Press, PART B Transversa Upwelling in a Long Narrow Lake with Applications to Babina Lake .-'0 109 CHAPTER I - Introduction Upwelling can be important to the b i o l o g i c a l processes i n any body of water. In p a r t i c u l a r i t can be a l i m i t i n g factor i n the productive capacity of a nutrient peer lake. Thus i t i s important to understand the processes responsible f o r the generation of the thermocline motions that would be associated with upwelling in lakes. I t i s well established that l o n g i t u d i n a l thermccline motions (longitudinal I n t e r n a l seiches) are generated in lakes and i n l e t s by winds p a r a l l e l to the long axis cf the lake or i n l e t (see Heaps and Ramsbcttom (1966) , Farmer (1 972 ) ) . Csanady (1973) has also found that transverse i n t e r n a l seiches are generated by winds i n large oblong lakes. His work indicates that winds perpendicular tc the long axis of the lake are recst e f f i c i e n t i n generating these motions, e s p e c i a l l y for narrow lakes. However, in long narrow g l a c i a l lakes the topography around the lake usually confines the winds to blow p a r a l l e l to the axis cf the lake. Thus we wish to know whether such winds can generate s i g i n f i c a n t transverse thermocline motions. Csanady's work does show that long axis winds do generate transverse motions that grow with time but no numerical r e s u l t s are given. Here we investigate the response of a long narrcw lake 110 to arb i t r a r y winds and compare the t h e o r e t i c a l r e s u l t s to observations made at Babine Lake i n northern B r i t i s h Columbia. We also compare our r e s u l t s to those given l y Csanady for Lake Michigan. The model that we use i s much the same as the i n f i n i t e l y long two layer Channel model used by Csanady. The main difference being that we include the ef f e c t s of horizontal turbulence whereas Csanady's model i s comletely i n v i s c i d . In the next section we describe the model i n mere d e t a i l and present the equations cf motion for a r b i t r a r y wind stress. Next we present the solutions for two di f f e r e n t impulsively applied wind stresses, a long axis wind and a cross channel wind,' both uniform i n s p a c e and time. We find that the inclusion cf eddy v i s c c s i t y improves the convergence of the serie s representing the solutions sc that i t i s p r a c t i c a l to sum them. Next we present calculated results for a model Eabine Lake and a model Lake Michigan. F i n a l l y we compare the re s u l t s from the model with data taken at Babine Lake by the author. 111 CHAPTER II - Formulation We consider a rotating i n f i n i t e l y long two layer channel with a f l a t bottom. Horizontal turbulence i s modeled through a constant c o e f f i c i e n t of eddy v i s c o s i t y . A, that i s assumed to be the same i n both layers. W*e ignore v e r t i c a l stresses, except for the wind stress at the surface. Thus both bottom f r i c t i o n and i n t e r f a c i a l f r i c t i o n are ignored. O'Brien and Hurlburt (1972) have found that these are neg l i g i b l e in the i r numerical models of. ocean upwelling. The li n e a r i z e d equations cf motion f c r thi s model are u,t-?v,--jU.+ vO* + ftu.,K* + , .. ( 1 a) v . f . C t L , - Av.^ + rsVj.W, > (1b) k,t + H,u,» « o , (1c) v^+^uj-- Av 7 X X ; (1e) The subscripts 1 and 2 refer to the upper and lower layers 112 respectively, and g 1 =g(j^-p, )/p^ • The C o r i o l i s parameter, f, i s assumed constant. The geometry i s shown i n Figure 1, where H i are the equilibrium depths of the la y e r s , assumed constant and the h^ are the perturbed depths. These equations include the assumptions that the bottom i s f l a t and that the channel i s i n f i n i t l e y long. This last assumption allows us tc assume no y dependence i f the wind stress i s uniform along the lake. Csanady has discussed the j u s t i f i c a t i o n for these assumptions. B r i e f l y , the i n f i n i t e channel approximation i s v a l i d for times less than the time required for internal waves to tr a v e l the length of the ' lake. In ignoring variations in bottom topography we are neglecting s i n g u l a r i t i e s in the equations cf motion that' r e s u l t from the depth going to zero at the side. Fcr a lake with steep sides and a shallow thermccline the possible e f f e c t s of neglecting t h i s s i n g u l a r i t y w i l l be confined tc a narrow layer at the sides. In l i n e a r i z i n g the equations we have also neglected.the eff e c t of any curvature that the lake might have. However, i t i s possible to estimate the importance cf the curvature by comparing v,/R with f . Here R i s the radius of curvature. After we have computed v, using the linear model above we w i l l make t h i s comparison to make sure that we were j u s t i f i e d i n neglecting curvature. F i n a l l y , i t has been assumed that the motion i s depth 113 independent and h o r i z o n t a l i n each l a y e r s e p a r a t e l y . Thus the wind s t r e s s a c t s as a body f o r c e i n t h e upper l a y e r and i s o n l y c o u p l e d t o the lower l a y e r by means of h y d r o s t a t i c e q u i l i b r i u m . The system of e q u a t i o n s (1) i s s o l v e d s u b j e c t t o the boundary c o n d i t i o n s (2) and the i n i t i a l c o n d i t i o n s (3) 114 CHAPTER III - The Solutions The system (1) i s solved by expanding In Fourier series i n x. This i s equivalent to expanding in terms cf the in t e r n a l and gravity wave modes. From the boundary conditions (2) we see that the expansions w i l l be cf- the form U.j.UiO U. i nU>/^^ , (4a) oo V-U,^-- 5- ^ t t \ W £ L * , • (4b) and then we must have (4c) u. Substituting these expressions into (1) gives a system cf f i r s t order ordinary d i f f e r e n t i a l equations in t . They are then solved subject to the i n i t i a l conditions ^(»\=^(^-^L0)=O. (5) We now present solutions obtained i n t h i s way f c r a uniform long axis wind and a uniform cross channel wind. Section A. Lohcj Axis Wind For the long axis wind we have f c r the wind stress 115 (6a) i t vo (6b) Then t ^ ^ t ) can be expanded i n a Fourier series as 00 (7) Thus we see immediately that for uniform wind only odd modes w i l l be excited. The Fourier c o e f f i c i e n t s for the upper layer fcr this wind stress are 3 (8) t9) (10) for odd n. They are a l l zero for n even. The expressions for the lower layer are s i c i l a r but are not given here. The subscripts I and E refer to the b a r o c l i n i c and tarctrop.ic modes, respectively. The c o e f f i c i e n t s in the above expressions are given by 116 ^ - -r^rrr. ~ r ~ > (11) r ^ ^ L - o ' 1 ( 1 2 ) ^ti r M ^ l i > (13) where j and k are the c y c l i c permutations of i . Identical expressions hold for the barotropic mcde with the positions of I and E interchanged. The exponents, s, in (8)- (10) s a t i s f y the cubic equation where for the s T , v= XT a n < 3 f o r the s e , X=\ E with V \ = ^ V u l [ ^ A ^ f ^ M ] , ^ - T r ^ C * \_(v^ra?r)/2] } (15) and Ii ^'W.rz l U»/(HA«0"] , *s » <}(W,+VV>$-Z. (16) The length scales f r and T e are the b a r o c l i n i c and barotropic Rossby r a d i i of curvature, respectively, as defined by O'Brien (1973). Further 117 where «• W* E. (18) Note that the coupling between the b a r o c l i n i c and the barotropic modes, which i s measured by £, i s small I f 'tfj^'frg. This i s always the case, since both i n the oceans and in lakes g'<<g. Further the barotropic mode i s less important than the b a r o c l i n i c since ^l^<?j f c r the same reason. Using (14) i t i s possible to shew that 5,- - A ^ L " 1 Cv\l- O , • (1 9) where in situations that are not dominated by f r i c t i o n , l.WnM"* . (20) These are the situ a t i o n s of interest tc us. Fcr small n, the other two solutions of (14) are a complex conjugate pair with ^ S ^ - - K ^ V - l ^ L * 2 C v ^ O > (21) and We w i l l see la t e r that the second term in (22) i s small f o r n small enough. Thus we see that at least the f i r s t few 118 modes have frequencies close to those given by the i n v i s c i d theory (see Csanady). When n becomes large, of course, the second term dominates and then (22) nc longer holds. When t h i s happens a l l of the solutions of (11) become r e a l and instead of (21) and (22) we have ^ - - A t f r H v ^ s M ) (23) S^ a: - 1 > A - » T ' 1 - ) } (24) where s=-s^-K. The t r a n s i t i o n comes when n=nf where nc i s given by n c * l \v LVATI1 x X > V ' •  (2 5) or V\6* vT"•£ L 1 / Anx % * <. \. (26) The important thing tc note i s that a l l cf tha modes have a time decaying component and a steady state component (in some cases the steady state component i s zero) and that up to a certain value cf n, the higher the mode number the faster the decay. Of course, aft e r a l l the roots turn real S j no longer increases with n, but we w i l l see that the contributions from such terms are ne g l i g i b l e . The steady state components can be summed exactly and the sums with the decaying components converga very rapidly. T.hus i t i s 119 p r a c t i c a l to sum the se r i e s numerically. The sums of the steady state part of the solutions are These are the asymptotic d i s t r i b u t i o n s f o r t-**> of the solutions i n (8)-(10). They may be obtained from |1) d i r e c t l y assuming steady state (and that h, (x) = 0 at x=l/2). Notice that there i s no coastal jet contained in these steady state solutions, as i s to be expected, since the coastal j e t i s a time dependent phenomenon, as has been pointed out by O'Brien and Hurlburt (1972) i n the case of coastal upwelling Section B. Cross Channel Wind For the cross channel wind stress we have (27) (2 8) U,LV> : 0 . (29) t "> o (30a). 120 (30b) Then i s expanded as i n (7) for I**! i n the previous case. The c o e f f i c i e n t s new are (31) (32) (33) where we have (34) Again an i d e n t i c a l expression holds for the b s; with I and I interchanged. Here the steady state responses are (35) (36) and we see that the thickening of the upper layer on the downwind side of the lake i s forced d i r e c t l y by the cross channel wind. Whereas for the long shore wind the 121 thickening on the side of the lake to the r i g h t of the wind (in the Nothern Hemisphere) i s forced i n d i r e c t l y by the wind through the C o r i o l i s force due to the earth's r o t a t i o n . Note that the series i n (31) -(33) would not converge at a l l r a p i d l y i f i t were not for the decrease in decay time with mode number. Ihis i l l u s t r a t e s the d i f f i c u l t i e s encountered i n trying to sum these seri e s in the i n v i s c i d case. 122 CHAPTER IV - Computed Results We now present numerical r e s u l t s obtained f o r the above model i n two d i f f e r e n t s i t u a t i o n s . The f i r s t i s f c r a model Babine Lake and the second i s f o r the model Lake Michigan considered by Csanady. The relevant parameters f cr the two models are given in Table I. Table I r — ., . , | Parameter | Babine Lake | Lake Michigan | H, I 10 m | 15 m H^  | 190 m | 60 m L I 2 kin | 120 km g' | 1 cm/sec 2 | 2 cm/sec 2 f | 10-* s e c - 1 | 10"* s e c - 1 | 3.1 km | 4.9 km | '450 km | 920 km Note that no values are given i n t h i s table f c r A. This i s because we knew very l i t t l e about the value of A for lakes except that i t c e r t a i n l y should be smaller than oceanic values since the largest horizontal length scales are smaller. Thus for Babine Lake r e s u l t s are presented f c r a range of values of A. I t i s found that for A<10* cm 2/ s e c the r e s u l t s are almost independent of A. For Lake Michigan we present r e s u l t s f cr only cne representative A. In order to calcu l a t e the response of the model we must sum the series in (4) with the c o e f f i c i e n t s given in (8)-(10) f o r long axis wind and (31)-(33) for cross channel 123 winds. As has beenpointed out before t h i s w i l l present nc problems for times longer than the decay times of the higher modes. This i s not the case for the i n v i s c i d problem where the c o e f f i c i e n t s in the series do not decay i n time. Thus introducing eddy v i s c o s i t y i s at least a computational convenience in that i t greatly improves the convergence of the seri e s . This i s e s p e c i a l l y true for the cross channel wind where the c o e f f i c i e n t s decrease only as L\\fiv?-)~v for the i n v i s c i d model (see Csanady (1973)). Section A. Babine Lake We w i l l present re s u l t s for A=10s, 10* and 103 Cm 2/sec. Using the values in Table I we have $i = l.oO , S E . = Tutor* , Thus we see immediately that the barotropic modes are negli g i b l e compared to the b a r o c l i n i c , since . Thus we only consider the b a r c c l i n i c mode i n the discussion below. Before we present the solutions we w i l l f i r s t point out the relevant time scales for the motion. From eguaticns (19) and (21) we have, s i n c e : Xi ^  j 1 24 up to n=nc. From (25) n c i s given by V\c* ^  L \or/,\) For n>n (23) and (24) give for n<n(L then, t h e important time scale i s F o r n>n c we must a l s o c o n s i d e r t h e time scale for the s 3 component, which i s Thus we see that the contributions from the s^ components fo r n>n c are unimportant for any times longer than one hour and, therefore, the T n are the important time scales. For A=105 cm 2/sec, T., = 11 hr and for A=103 cm 2/sec, T, = 1100 hr sc that for A = 10 5 cm 2/sec the system w i l l reach equilibrium afte r only about one day. Whereas for A=103 cm 2/sec equilibrium i s reached only a f t e r more than two or three months. However, even for A=103 cm 2/sec, TJ2= 1. 1 hr sc that even then not too many terms a ra needed to approximate the sums. F i n a l l y , the frequencies of the o s c i l l a t i n g mcdes are 125 The f i r s t mode has a period cf about 3.6 hr. The re s u l t s obtained for a long axis south wind are shown in Figures 2-6. In Figures 2 and 3 we show the depth anomaly of the upper layer and the long shore ve l o c i t y in the upper layer after a 1 dyne/cm2 wind stress has been applied for 22 hrs. Note that there i s l i t t l e d i fference between the p r o f i l e s for A-10 3 and A=104 cm 2/sec, except in a narrow viscous boundary layer. The depth p r o f i l e i s unimodal, as i s to be expected for a narrow lake. The difference in the depth cf the interface across the lake i s 12 m. As A 0 these p r o f i l e s approach those for the i n v i s c i d problem. In Figures 4 and 5 the maximum values of the depth anomaly and the long shore v e l o c i t y are plotted as functions of. time for a 1 dyne/cm2 wind stress. The depth anomaly o s c i l l a t e s s l i g h t l y at the frequency of the f i r s t i n t e r n a l mode but the vel o c i t y does net. Again we note that there i s l i t t l e difference between A=103 and A=104 cm 2/sec. In fact the curves for A=103 and A=10* cm 2/sec f c r v are i n d i s t i n g u i s h i b l e . Internal o s c i l l a t i o n s at the frequency of. the f i r s t i n t e r n a l mode dominate the cross stream vel o c i t y as i s seen i n Figure 6. Note that the lake i s tec narrow for i n e r t i a l o s c i l l a t i c n s . In Figures 7 and 8 we show some re s u l t s for a cress channel wind. Here we see that the i n t e r n a l o s c i l l a t i o n s 126 dominate the depth anomaly response, as seen i n Figure 7. Again the depth p r o f i l e i s unimodal and almost completely independent of A. One important difference between the response to a cross channel wind and long axis wind i s that the response to the former i s much faster. The maxirruic displacements are reached within the f i r s t i n t e r n a l wave period for the cross channel wind. The displacements f c r a long axis wind grow almost l i n e a r l y i n time. So that for short periods the cross channel response dominates, tut as i s seen by comparing Figures H and 7 the response due to a long axis wind eventually domiates, i n contrast tc what Csanady suggests. Section B. Lake Michigan For the case of a wide lake we expect the r e s u l t s to be quite d i f f e r e n t . Csanady has found that for the i n v i s c i d case the f i r s t few modes are almost of equal importance so that the response i s expected to be multi-modal instead of unimodal as for a narrow lake. This agrees with observations made at Lake Michigan (see Csanady (1973)). ,He w i l l present r e s u l t s only f o r A=106 cm 2/sec as t h i s i s as large as we expect A to be (it i s abcut the lowest value used for the ocean) and i t i s found that again the re s u l t s are pretty much independent of A for A<106 cm 2/sec. From Table I we find Xi- \/to > K\« 2CO/48 , 127 Thus again the b a r o c l i n i c mode dominates, i . e . , Sp^$ x. However, now >\i\ so that we are i n a d i f f e r e n t regime. Here we have u n t i l /^"V),* , i . e . , for r<7. For n>7 we have again Re^-Tv = «e{W ~ - AT.''VN* / 2 l * The time scales of the f i r s t few modes go as Thus the response time for the f i r s t mode i s about 30 years. Further r\(_ «• 2 0 O 0 and the time scale for the s ? mode for n>nc i s about 7 itins. The important time scales for n>7 then go as fo r n~63, T„ = 1 hr. So that again summation of the series i s tractable; however, here more terms are required tc obtain 128 s a t i s f a c t o r y accuracy for times of the order of hours. The frequencies of the f i r s t few modes are almost i n e r t i a l , i . e . , with periods of about 18 hrs. In Figures 9-11 we show some r e s u l t s obtained f c r the model Lake Michigan. In Figure 9 we show the depth anomaly as a function of x for times of one, two and three days f c r a wind stress of 1 dyne/cm2. we see that again the response i s very f a s t with the maximum response being reached within the f i r s t i n e r t i a l period. The p r o f i l e i s multi-modal as we expected. However, tha f u l l structure does not develop within the f i r s t o s c i l l a t i o n . Note that tha maximum displacement away from the sides i s of the order cf 1 rr. In Figure 10 we show the depth anomaly for a long axis wind with a 1 dyna/cm2 stress for times of 10, 20 and 30 days. We see that only the f i r s t and thi r d modes are i n evidence and that only f o r rather long times does the response tc a long axis wind dominate the response to a cross channel wind. In Figure 11 we show the long shore v e l o c i t y as a function of x for a 1 dyne/cm2 long axis wind. We note that the coastal j e t i s presant here. The r e s u l t s described above indicate that a steady wind strass i n an i n f i n i t e l y long channel model i s not s u f f i c i e n t to account for the observed displacements in Lake Kichigan which are of the order of 10 m (see Csanady (1973)). These larger displacements cannct be axplanied as a response of 129 the i n f i n i t e l y long channel to a long term long axis wind since the observed displacements d e f i n i t e l y have a multi-modal structure. Anyway, such long term winds would ce r t a i n l y require i n c l u s i o n cf end e f f e c t s . Two p o s s i b i l i t i e s that could be considered within a linear model are that either the responses are a resonance e f f e c t or that end e f f e c t s are important. Either of these p o s s i b i l i t i e s requires that .a closed basin mcdel be considered. 130 CHAPTER V - Comparison with Observations f c r Babine Lake Here we present some observations made at Eabine Lake by the author and compare them with the th e o r e t i c a l r e s u l t s given above. During 21-25 August, 1973 a series of temperature transects was made at Babine Lake. Three" transects were made at d i f f e r e n t locations twice a day, once i n the morning and once in late afternoon cr early evening. See Figure 12 for map of Babine Lake and locations of transects. During t h i s period wind data were being recorded by instruments placed by D. Farmer of the Marine Sciences Branch. The instrumentation was placed as part cf a larger program cf study being conducted by Farmer in conjunction with the Fisheries Research Board at Babine Lake. The purpose of doing the transects was to determine whether there were s i g n i f i c a n t transverse displacements cf the thermocline that could be correlated with winds. On two separate occasions there were s i g n i f i c a n t t i l t s i n the thermocline. These occurred on the afternccn run cf 21 August and the mcrning run cf 24 August, see Figure 13. On the 21st there was a large displacement•at two stations but none was observed on the other. The maximum displacement across the lake was 6 m. On the 24th displacements were observed cn a l l transects with a displacement of about 4 m across the lake. In both cases 131 the displacements were unimodal as predicted by the ircdel. It was found that the t i l t i n g observed on the 21st was observed just after a 5 hr period of rather strong ncrth winds, beginning at about 11:00 PST, see Figure 14. Note that the winds are in the right d i r e c t i o n to give the observed t i l t s . On the transect where there was no t i l t i n g observed cross channel winds were almost as large as the long shore wind. This cculd explain the lack of observed displacemants. At the other two transects the wind stress had an average value cf abcut 1 dyne/cm2 f c r the 5 hr period. Going to Figure 3 we see that we get- a displacement of about 3 m in t h i s case. The t i l t i n g of the isotherms observed on the 24th follows a period of abcut 2 days cf mild intermittent long shore north winds with a short period of stronger long shcre winds occurring just before the measurements were taken, see Figure 15. The average value of the wind stress f c r t h i s period was about .1 dyne/cn 2. Figure 3 gives a displacement of about 3 m across the lake for t h i s case. Even though the t h e o r e t i c a l results agree f a i r l y well with the observations we w i l l estimate the e f f e c t cf neglecting the curvature cf the lake to determine whether we are j u s t i f i e d by the success of the agreement. As can be seen from Figure 11 there i s appreciable curvature i n the area where the measurements were taken. The radius cf 132 curvature here i s about 5 km. As stated e a r l i e r we car estimate the importance cf the curvature by comparing v, /R with f . Now f o r both cases described above Figure 4 indicates that v, <. 5 m/sec. Thus v,/R<10-4 s e c - 1 , and we see that i n e r t i a l e f f e c t s c e r t a i n l y do not dominate but that they might modify the flow somewhat. F i n a l l y we mention that end effects are probably not important since i t reguires about 60 hrs for an i n t e r n a l wave to t r a v e l the length of the lake. 133 CHAPTER VI - Conclusion In part B of t h i s thesis we have presented a model that describes the generation of transverse motions of the interface i n an i n f i n i t e l y lcng two layer lake by uniform wind of a r b i t r a r y direction,* W'e have^ included/the effects" of horizontal eddy v i s c o s i t y . The main consequence of i t s incl u s i o n seems to be that the convergence of the series solutions for the model i s greatly improved and makes numerical summation practicable. A comparison of the computed results with data from Lake Michigan indicates that a cross channel wind generates the multi-modal structure observed there, but that the magnitude of the displacements are larger by a factor cf 10 than those predicted by the theory. long axis winds do not even generate motions of the observed form. It seems that a model including end e f f e c t s and/or non-linear e f f e c t s i s required. Applying the theory tc a narrower lake gives better r e s u l t s . Observations made by the author at Babine Lake show that in a long narrow lake s i g n i f i c a n t i n t e r f a c e displacements are generated by long shore winds. The model predicts displacements that agree both i n magnitude and form with those observed. I \ I-FIGURES FOR PART B 135 136 137 ( 35>s/ w, ) 'A 139 sf s- sv H O U, i LIT- cm/sec> -.1 .1 . .1 .1 .V .5 .± o 141 3 142 q 144 7i' fir 148 149 150 1 5 2 LONGITUDINAL W INLY'Si" .. FOR 8/21/7: 153 RESS 2 ° / A F i . I 4 1 1 1 — 1.3 • 10.0 12.0 TIME; : P S ; I —1 24. 0 2.0 4.0 6.0 14.0 .16.3 20.0 22.0 TRANSVERSE WIND STRESS FOR 8/21/73 2.0 v v 4.0 67o —I— 2.0 4.0 6.0 I— e.o 1 1— 10.0 12.0 TIME: (PSD TRANSVERSE WIND STRESS FOR 8/22/73 LONGITUDINAL WIND STRESS FOR 8/23/73 1 — - — — — • ; — ° — • — ' r i 1 1 1 .0 10.0 12.0 14.0 16.0 18.0 50.0 52.0 24.0 Figure \5 b 0.0 2.C 4.0 6.0 8.0 . 1J.0 . 12.0 TIME: : P S : J 14.0 1C.0 IS.3 23.0 22.0 24.0 ' TRANSVERSE WIND STRESS FOR 8/23/73 14.0 10.0 12.0 i i 10.0 12.0 riME (PSD i — 14.0 -1 ICO - 1 — Ift.O -) 24.0 0.3 2.0 4.0 6.0 0.0 20.0 22.0 LONGITUDINAL WIND STRESS . 156 FOR 8/24/73 -1 : 1 1 1 1 — 1 —1 1 1 8.0 10.0 12.0 14.0 16.0 16.0 20.0 22.0 24 .0 - 1 14.0 —I— 22.0 - 1 24 .3 4.0 6.0 8.0 10.3 12.0 riME ipsn i e . a i8 .o TRANSVERSE WIND STRESS FOR 8/24/73 — i — 2.0 - 1 6.0 —I e.o - i 1 — 13.0 12.0 TIME. (PSD -1 i c . o — i 24 .0 i e . 0 20.0 22.0 157 REFERENCES FOR PART B Csanady, G. T., 1973: Transverse I n t e r n a l Seiches i n Large Oblong Lakes and Marginal Seas. J. Phys. oceancgr., 3, 439-447. Farmer, D. M., 1972: The I n f l u e n c e of Wind on the Surface Waters of A l b e r n i I n l e t . T h e s i s , I n s t i t u t e c f Oceanography, U n i v e r s i t y of B r i t i s h Columbia. Heaps, N. S. And A. E. Ramsbcttorn, 1966:: Wind E f f e c t s on the Water i n a Narrow Two Layer Lake. P h i l . Trans. Roy. S_oc. London, 259, 391-430. O'Brien, 3. J. and H. E. U u r l b u r t , 1972: A Numerical Model of C o a s t a l Upwelling. . J. Phys. Cceancgr., 2,14-26. O'Brien, J . J . , 1973: A n a l y t i c a l S o l u t i o n s f c r Two-Layered Models of the Cnset of a C o a s t a l Upwelling Event. to appear i n J . Phys. Oceanogr. 

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