UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Nonlinear stability and statistical equilibrium of forced and dissipated flow Zou, Jieping 1991

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1991_A1 Z68.pdf [ 7.72MB ]
Metadata
JSON: 831-1.0053236.json
JSON-LD: 831-1.0053236-ld.json
RDF/XML (Pretty): 831-1.0053236-rdf.xml
RDF/JSON: 831-1.0053236-rdf.json
Turtle: 831-1.0053236-turtle.txt
N-Triples: 831-1.0053236-rdf-ntriples.txt
Original Record: 831-1.0053236-source.json
Full Text
831-1.0053236-fulltext.txt
Citation
831-1.0053236.ris

Full Text

NONLINEAR STABILITY AND STATISTICAL EQUILIBRIUM OF FORCED AND DISSIPATED FLOW by J i e p i n g Zou B. Sc. i n Atmospheric Sciences Program Shandong C o l l e g e of Oceanography, Qingdao, China, 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of Oceanography) accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA October 1991 © J i e p i n g Zou, 1991 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l l m e n t of the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y of B r i t i s h C o l u m b i a , I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the head of my department or by h i s or h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f Oceanography The U n i v e r s i t y of B r i t i s h C o l u m b i a 6270 U n i v e r s i t y B o u l e v a r d V a n c o u v e r , Canada 1W5 1Z4 D a t e : ABSTRACT A g l o b a l a n a l y s i s f o r t h e h y d r o d y n a m i c a l s y s t e m d e f i n e d f o r a h o m o g e n e o u s , i n c o m p r e s s i b l e l a y e r o f f l u i d on t h e 0 - p l a n e i s p e r f o r m e d i n b o t h i n f i n i t e a n d f i n i t e f u n c t i o n s p a c e . I t s a p p l i c a t i o n t o g l o b a l s t a b i l i t y h a s y i e l d e d an a l g o r i t h m f o r c h a r a c t e r i z i n g f l o w s b a s e d o n t h e e x i s t e n c e of i n i t i a l l y g r o w i n g p e r t u r b a t i o n s as o p p o s e d t o t h e n o r m a l mode a n a l y s i s ; i t s a p p l i c a t i o n t o t h e s e a r c h f o r o p t i m a l i n i t i a l p e r t u r b a t i o n s h a s l e d t o t h e l e a s t u p p e r b o u n d o f e n e r g y g r o w t h r a t e ; i t s a p p l i c a t i o n t o m u l t i p l e e q u i l i b r i a h a s g i v e n r i s e t o a n e c e s s a r y c o n d i t i o n f o r t h e i r e x i s t e n c e ; i t s a p p l i c a t i o n t o t h e s t u d y o f t h e r e l a t i o n s h i p of m o d a l to n o n m o d a l g r o w t h r a t e s h a s u n c o v e r e d t h e c a u s e u n d e r l y i n g many a s p e c t s o f t h e l i m i t a t i o n o f t h e m o d a l s t a b i l i t y a n a l y s i s i n c l u d i n g t h e f a i l u r e t o p r e d i c t t r a n s i e n t g r o w t h of d i s t u r b a n c e s i n s t a b l e f l o w s a n d t h e u n d e r e s t i m a t i o n of t h e i n t e n s i t y o f i n i t i a l d e v e l o p m e n t o f i n s t a b i l i t y i n u n s t a b l e f l o w s . N u m e r i c a l i l l u s t r a t i o n s made f o r some s p e c i f i c f l o w s h a v e s t r e n g t h e n e d t h e g e n e r a l r e s u l t s , s u g g e s t i n g t h a t a s t a b i l i t y a n a l y s i s o f a h y d r o d y n a m i c a l s y s t e m w i t h o u t a g l o b a l a n a l y s i s i s l i k e l y t o b e l i m i t e d i n many i m p o r t a n t a s p e c t s . T h e l o c a l a n a l y s i s o f a s y m p t o t i c b e h a v i o r o f n o n m o d a l d i s t u r b a n c e s t o h y p e r b o l i c e q u i l i b r i a o f t h e s y s t e m h a v e e s t a b l i s h e d : a) f o r a n y s u b c r i t i c a l f l o w o u t s i d e of i i m o n o t o n i c , g l o b a l s t a b i l i t y r e g i m e , t h e r e e x i s t s a f i n i t e n e i g h b o r h o o d a r o u n d t h e o r i g i n o f RM s u c h t h a t a d i s t u r b a n c e i n i t i a l i z e d i n t h i s n e i g h b o r h o o d w i l l u l t i m a t e l y d e c a y t o z e r o a f t e r i t e x h i b i t s O r r ' s t e m p o r a l a m p l i f i c a t i o n ; b) f o r a n y s u p e r c r i t i c a l f l o w , t h e r e e x i s t s a f i n i t e n e i g h b o r h o o d a d j a c e n t t o t h e o r i g i n o f RM s u c h t h a t a d i s t u r b a n c e i n i t i a l i z e d i n t h i s n e i g h b o r h o o d w i l l p e r s i s t a s t - x » ; a n d c) t h e n a t u r e o f t h e p e r s i s t e n t d i s t u r b a n c e s i s r e l a t e d t o t h e n a t u r e o f t h e n o n h y p e r b o l i c p o i n t i n p a r a m e t e r s p a c e o f i n t e r e s t . T h e n u m e r i c a l e x p e r i m e n t s a r e s e e n t o c o n f i r m t h e s e p r e d i c t i o n s . C l o s u r e m o d e l i n g o f f o r c e d - d i s s i p a t e d s t a t i s t i c a l e q u i l i b r i u m o f p e r t u r b e d f l o w s a r i s i n g f r o m i n i t i a l l y u n i f o r m z o n a l f l o w s o v e r r a n d o m t o p o g r a p h y i s d o n e w i t h s p e c i a l r e g a r d t o t h e c o r r e l a t i o n b e t w e e n d i s t u r b a n c e a n d u n d e r l y i n g t o p o g r a p h y a n d t h e r e s u l t i n g s t r e s s . S u c h a n e x e r c i s e h a s l e d t o , o n one h a n d , t h e n u m e r i c a l r e s u l t s f o r t o p o g r a p h i c a l s t r e s s s u g g e s t i n g c l e a r l y t h e s i g n i f i c a n c e o f t h i s f o r c e i n o v e r a l l momentum b u d g e t o f l a r g e s c a l e o c e a n c i r c u l a t i o n s . On t h e o t h e r . h a n d , i t h a s l e d t o an a p p r e c i a t i o n t h a t t h e d e t a i l e d c o n s e r v a t i o n o f e n e r g y a n d p o t e n t i a l e n s t r o p h y , w h i c h h o l d s r e g a r d l e s s o f t h e p r e s e n c e o f d i s s i p a t i o n i n t h e s y s t e m , p r o v i d e s a means f o r s y s t e m a t i c i n v e s t i g a t i o n o f n o n l i n e a r t r a n s f e r o f t h e t h e s e q u a n t i t i e s among i n t e r a c t i n g t r i a d s , a n a r e a n o t a c c e s s i b l e t o o t h e r a p p r o a c h e s . i i i T A B L E OF CONTENTS A B S T R A C T i T A B L E O F C O N T E N T S i i i L I S T OF F I G U R E S v i i ACKNOWLEDGEMENTS x i C H A P T E R 1 I N T R O D U C T I O N 1.1 Overview 1 1.2 Objective 5 1.3 Preview of subsequent chapters 6 C H A P T E R 2 G L O B A L A N A L Y S I S I : I N F I N I T E D I M E N S I O N A L S Y S T E M , with a p p l i c a t i o n to: global s t a b i l i t y , optimal perturbation, multiple e q u i l i b r i a and r e l a t i o n of i n i t i a l modal to nonmodal growth rates 2.1 Introduction 8 2.2 IBVP For S t a b i l i t y and P r e d i c t a b i l i t y 9 2.3 Symmetrized Energy (Error) Equation 13 2.4 Generalized Rayleigh Quotient and i t s properties .... 14 2.5 An Optimal Problem for the Generalized Rayleigh Quotient 18 2.6 Application I: global s t a b i l i t y and optimal perturbation 22 2.7 Application I I : multiple e q u i l i b r i a 29 i v 2.8 Application I I I : i n i t i a l modal vs. nonmodal growth rate . ... 35 2.9 Concluding Remarks 43 Appendix 2A A v a r i a t i o n a l approximation method for the optimal value of generalized Rayleigh quotient p.. 45 CHAPTER 3 GLOBAL ANALYSIS I I : FINITE DIMENSIONAL SYSTEM, with a p p l i c a t i o n to: global s t a b i l i t y and optimal perturbation 3 . 1 Introduction 4 6 3.2 Governing Equations i n RM 4 8 3.3 Global analysis i n RM 4 9 3.3.1 Disturbance equation 4 9 3.3.2 Symmetrized energy equation 51 3.3.3 Generalized Rayleigh P r i n c i p l e 54 3.4 Application to: global s t a b i l i t y and optimal perturbation 5 6 3 . 5 Numerical i l l u s t r a t i o n s 58 3 . 5 . 1 Set-up of the numerical experiments 58 3.5.2 Numerical r e s u l t s 59 3.6 Concluding Remarks 63 Appendix 3A Algebraic properties of the set {s^} 64 Appendix 3B Physical properties of the set {s.} 65 v CHAPTER 4 FINITE AMPLITUDE NONMODAL DISTURBANCE I: INITIAL BEHAVIOR, i t s r e l a t i o n t o i n i t i a l intense development of disturbances 4.1 Introduction 7 4 4.2 Nonmodal disturbance over transient period 75 4.2.1 Modal vs. nonmodal growth rate at i n i t i a l instant 75 4.2.2 Explosive development of nonmodal disturbance .. 77 4.3 Numerical i l l u s t r a t i o n s 78 4.4 Concluding Remarks 82 Appendix 4A Modal growth rate expressed i n terms of p .... 83 Appendix 4B Fundamental properties of nonmodal disturbance 84 CHAPTER 5 FINITE AMPLITUDE NONMODAL DISTURBANCE I I : ASYMPTOTIC BEHAVIOR, i t s r e l a t i o n t o b i f u r c a t i o n and multiple e q u i l i b r i a 5.1 Introduction 95 5.2 Asymptotic decay as t -» » i n s u b c r i t i c a l flow 96 5.3 Persistence as t -> oo i n s u p e r c r i t i c a l flow 99 5.4 P e r s i s t e n c e , c r i t i c a l i t y and s u p e r c r i t i c a l b i f u r c a t i o n 104 5.5 Numerical i l l u s t r a t i o n s 108 5.6 Concluding remarks 113 Appendix 5A The di r e c t method of Liapunov 115 Appendix 5B A u x i l i a r y lemmas 116 Appendix 5C Real canonical theory of l i n e a r operator .... 119 Appendix 5D Liapunov i n s t a b i l i t y function 121 v i A p p e n d i x 5E A l g o r i t h m s f o r b i f u r c a t i o n a n a l y s i s 123 CHAPTER 6 CLOSURE MODELING: FORCED-DISSIPATED STATISTICAL EQUILIBRIUM OF LARGE SCALE QUASI-GEOSTROPHIC FLOWS OVER RANDOM TOPOGRAPHY 6.1 I n t r o d u c t i o n 134 6.2 C l o s u r e F o r m u l a t i o n 135 6 . 2 . 1 A s e l f - c o n s i s t e n t m o d e l 135 6 .2 .2 Moment E q u a t i o n s 136 6 . 2 . 3 C l o s u r e H y p o t h e s i s a n d M a s t e r E q u a t i o n s 139 6 .3 N u m e r i c a l r e s u l t s a n d c o m p a r i s o n w i t h DNS 144 6 . 3 . 1 S o l u t i o n m e t h o d 144 6 . 3 . 2 M o d e l p a r a m e t e r s 14 6 6 . 3 . 3 N u m e r i c a l r e s u l t s 147 6 .4 C o n c l u d i n g r e m a r k s 155 A p p e n d i x 6A E x p r e s s i o n s f o r k , S 2 ^ a n d k . . . . 157 A p p e n d i x 6B C o n s e r v a t i o n p r o p e r t i e s o f t h e c l o s u r e m o d e l . . 1 5 8 CHAPTER 7 CONCLUSIONS 169 REFERENCES 174 v i i LIST OF FIGURES Figure Page 3.1 Streamfunctions of three representative equilibrium states for U* = 22.m/s and h^ = 500 m. (a) 2 / r = 2 9 days; (b) l/r=ll days; (c) 2/r= 5 . 5 days. The rest of the parameters are given i n the text 67 3.2 S t a b i l i t y regime diagram. The s o l i d l i n e for the f i n i t e s t a b i l i t y measure (t) . The dash l i n e s r = r = 0.17/day for the MGS boundary , r=r = N J L 0.075/day for the l i n e a r s t a b i l i t y boundary and r=r i s for the diagonal dash l i n e which i s used for t e s t i n g the condition $) -r > 0 68 3.3 An example of MGS. The equilibrium state ^ i s (c) i n F i g 3.1. The i n i t i a l perturbations $ are randomly generated with the r a t i o 5 of t h e i r i n i t i a l k i n e t i c energy to the energy i n $ ranging from 0 . 2 to 1.0 69 3.4 Existence of i n i t i a l l y growing nonmodal S m to e q u i l i b r i a $ (r) i n s u b c r i t i c a l regime, i . e . , region (II) i n F i g 3 . 2 70 3.5 The s p a t i a l configurations of f i v e i n i t i a l l y growing nonmodal perturbations to the basic state (b) i n F i g 3 . 1 71 3.6 The energy time series for f i v e disturbances i n i t i a l i z e d from the shown i n F i g 3 . 5 72 3.7 The growth rates of disturbances i n i t i a l i z e d from nonmodal perturbations over an i n i t i a l growing period. The basic state ^ i s F i g 3 . 1 (a) 73 v i i i 4.1 Maximum i n i t i a l m o d a l v e r s u s n o n m o d a l g r o w t h r a t e , i . e . , r ($) - r v s . r $) - r , f o r t h e s e t o f e q u i l i b r i a {^(C7 ) } o v e r z o n a l w a v e n u m b e r - 1 t o p o g r a p h y o f h e i g h t 1000 m. T h e Ekman d a m p i n g c o e f f i c i e n t r = 2 / ( 1 5 d a y s ) i s i n d i c a t e d b y t h e h o r i z o n t a l d a s h l i n e . T h e t h r e e t v e r t i c a l d a s h l i n e s a r e f o r t h e MGS b o u n d a r y U = £ 7 = 6 . 6 m / s , N a n d f o r l i n e a r s t a b i l i t y b o u n d a r i e s U* = U^= 1 0 . 5 m / s a n d U = CT = 2 6 . 3 m / s , w h e r e U a n d a r e L N L L o b t a i n e d a s r o o t s t o t h e e q u a t i o n s , r (t(U*))-r=0 a n d r (${U ))-r=0, r e s p e c t i v e l y 88 L 4.2 (a) T o p o g r a p h y c o n t o u r s ; ( b ) - ( f ) s t r e a m f u n c t i o n s o f s e v e r a l r e p r e s e n t a t i v e s t a t e s f r o m t h e s^et o f {^(C7 ) } u s e d i n F i g 4 . 1 , c o r r e s p o n d i n g t o U-6.20 m / s i n ( I ) , 9.12 m / s i n ( I I ) , 11.33 a n d 17.85 m / s i n ( I I I ) a n d 40.13 m / s i n ( I V ) , r e s p e c t i v e l y . . . . 89 4.3 T h e same a s F i g 4 . 1 e x c e p t t h a t t h e s e t o f e q u i l i b r i a {^(C7 ) } u s e d h e r e c o r r e s p o n d s t o t h e z o n a l w a v e n u m b e r - 2 t o p o g r a p h y 90 4.4 T h e same a s Fiq 4 . 1 e x c e p t t h a t t h e s e t o f e q u i l i b r i a (C7 ) } u s e d h e r e c o r r e s p o n d s t o 1/r = 30 d a y s 91 4.5 B a s i c s t a t e s , n o n m o d a l a n d m o d a l i n i t i a l p e r t u r b a t i o n s f o r t h e n u m e r i c a l e x p e r i m e n t s , (a) f o r s t r e a m f u n c t i o n o f t h e e q u i l i b r i u m s t a t e t a k e n f r o m F i g 4 . 4 c o r r e s p o n d i n g t o U=16.3 m / s , (b) a n d (c) f o r i t s f a s t g r o w i n g i n i t i a l n o n m o d a l a n d m o d a l p e r t u r b a t i o n s , r e s p e c t i v e l y 92 4.6 E v o l u t i o n o f t h e g r o w t h r a t e w i t h t i m e . T h e o p t i m a l n o n m o d a l i n i t i a l p e r t u r b a t i o n S m ( see F i g 4 . 5 ( b ) ) h a s g r o w t h r a t e a- (s ; ! ) = 2 / ( 5 . 2 5 d a y s ) , N M w h e r e a s t h e f a s t g r o w i n g m o d a l p e r t u r b a t i o n (FGMP) ( s e e F i g 4 . 5 ( c ) ) h a s Re (o-) = 1/(9.52 d a y s ) . T h e i n i t i a l p e r t u r b a t i o n s a r e i n d i c a t e d i n F i g 4 . 5 (a) a n d (b) 93 4 . 7 E v o l u t i o n o f d i s t u r b a n c e e n e r g y w i t h t i m e f o r t h e c a s e shown i n F i g 4 . 6 . T h e i n i t i a l p e r t u r b a t i o n e n e r g y i s 20 % o f t h e e n e r g y i n t h e e q u i l i b r i u m s t a t e s t o e n s u r e f i n i t e a m p l i t u d e f o r t h e i n i t i a l p e r t u r b a t i o n s . 94 i x 5.1 S t a b i l i t y regime diagram f o r a f a m i l y o f e q u i l i b r i a {t{r) } w i t h l/r r a n g i n g from 29.5 days t o 3.5 days, U = 22.0 m/s and topography b e i n g o f z o n a l wavenumber-1 and o f h e i g h t 500 m. The t o p c u r v e and t h e c u r v e abed a r e i t s (*) and r ($) , r e s p e c t i v e l y , w i t h r = r =1/(5.71 days) and L N r=r =1/(13.33 days) as i t s MGS boundary and l i n e a r s t a b i l i t y boundary. The c u r v e efgb i s r {t ) f o r t h e s e t o f e q u i l i b r i a {f (r) } b i f u r c a t i n g a t c r i t i c a l i t y r=r from t h e p r i m a r y b r a n c h { ^ ( r ) } . r=r'=l/(18.5 days) i s t h e l i n e a r s t a b i l i t y boundary f o r t h e s e t ( r ) } . 12 6 5.2 A s y m p t o t i c n o n v a n i s h i n g s t e a d y s t a t e s o f nonmodal d i s t u r b a n c e s . The u n d e r l y i n g e q u i l i b r i u m s t a t e s a r e t h o s e l o c a t e d on t h e p a r t (f ->b) o f t h e p r i m a r y b r a n c h ( c f . F i g 5.1) f o r v a l u e s o f r g i v e n i n t h e f i g u r e 127 5.3 S t r e a m f u n c t i o n s f o r b i f u r c a t i o n o f an e q u i l i b r i u m s t a t e (a) i n t o a new st e a d y f l o w ( f ) . The sn a p s h o t s ( c ) - ( f ) a r e from t h e exper i m e n t f o r l/r=17.1 days ( c f . t h e t h i c k s o l i d l i n e i n F i g 5.2) 128 5.4 L o c a l u n i q u e n e s s o f a s y m p t o t i c s t e a d y s t a t e o f nonmodal d i s t u r b a n c e s . The u n d e r l y i n g e q u i l i b r i u m s t a t e i s t h e same as one i n F i g 5 . 2 f o r expe r i m e n t o f 2/r=16".4 days. The i s o b t a i n e d from s c a l i n g s such t h a t t(t;s') a t t=0 has 10 % M M o f t h e b a s i c s t a t e energy 12 9 5.5 P e r i o d i c l i m i t i n g s t a t e s o f nonmodal d i s t u r b a n c e s , w i t h p e r i o d s 46.3 days f o r s o l i d l i n e and 85.8 days f o r dash l i n e , r e s p e c t i v e l y . The u n d e r l y i n g e q u i l i b r i a a r e l o c a t e d on t h e u n s t a b l e s e c t i o n o f t h e s t a t i o n a r y b i f u r c a t i o n b r a n c h ( c f . F i g 5.1), w i t h t h e v a l u e s o f r as i n d i c a t e d 130 5.6 S t r e a m f u n c t i o n s f o r b i f u r c a t i o n o f an e q u i l i b r i u m s t a t e (a) i n t o a p e r i o d i c f l o w . The sno p s h o t s ( b ) - ( f ) a r e t a k e n from t h e exp e r i m e n t f o r l/r=24.8 days ( c f . t h e s o l i d l i n e i n F i g 5.5) over a c y c l e o f o s c i l l a t i o n , w i t h t'=336.9 days and T=49.3 days 131 x 5.7 Repeated s u p e r c r i t i c a l b i f u r c a t i o n for the primary branch of e q u i l i b r i a ( c f . F i g 5.1). The point marked by x on the primary branch i s a stationary b i f u r c a t i o n point whereas the symbol + indicates the Hopf b i f u r c a t i o n point. The l i n e s drawn with dash corresponds to unstable equilibrium states 132 5.8 Nonmodal versus modal i n i t i a l perturbations i n t r a n s i t i o n to a periodic state. The basic state i s from the stationary b i f u r c a t i o n branch (r)} with l/r=18.8 days (cf . F i g 5.1), Jocated near the secondary b i f u r c a t i o n point r=r'. The i n i t i a l growth rate of ~$(t;a ) and $ (t;Re (t)) are 1/(5.08 M days) and 1/(400.0 days), respectively. 133 6.1 Topographic stress x as. a function of 17. The parameters are (h ,p,r) = (4.0, 0.8, 0.12). The m a x s o l i d l i n e i s for the closure re s u l t s and symbols for the dns data. The resonant point corresponds to U = 1.0 163 r 6.2 Streamf unction (0 = -rjy+$) for the two representative flows at t=20 (or t=230 days), with parameters as the same as i n F i g 6.1. (a): the subresonant flow with U = 0.25; (b) : the superresonant flow with U = 2.75. The dash contours are for negative values 164 6.3 Enstrophy (a), topographic stress (b) and vorticity-topography c o r r e l a t i o n (c) spectra ( s o l i d lines) for the subresonant flow case U =0.25, with parameters the same as i n F i g 6.1. The symbols are for the f i v e dns ensemble data. 165 6.4 V o r t i c i t y (a) and Topography (b) for the superresonant flow shown i n F i g 6.2(b) 166 6.5 The same as i n F i g 6.3 but for U = 2.75 167 6.6 Topographic stress x as a function of r. The parameters are (h ,p, U) = (6.2.0, 0.8, 3.0). m a x The s o l i d l i n e i s for the closure re s u l t s and symbols for the dns data 168 x i ACKNOWLEDGMENT I would l i k e t o thank D r . W i l l i a m H s i e h , my s u p e r v i s o r , f o r g i v i n g me t h e o p p o r t u n i t y t o roam f a r and wide i n s e a r c h o f t h e u l t i m a t e theme o f t h e t h e s i s work as w e l l as f o r h i s u n f a i l i n g s u p p o r t and c o n s t a n t encouragement a l o n g t h e way. I would a l s o l i k e t o acknowledge D r . Greg Hol loway f o r h i s l o n g term c o a c h i n g o f my s t u d y o f t u r b u l e n c e t h e o r y and f o r h i s s u p e r v i s i o n o f my work on c l o s u r e m o d e l l i n g , and D r . John F y f e f o r h i s e n l i g h t e n i n g s u g g e s t i o n s i n my s t a b i l i t y s t u d i e s and f o r h i s k i n d l y p r o v i d i n g me the /3-plane c h a n n e l model w h i c h has y i e l d e d many v i v i d i l l u s t r a t i o n s . Thanks are a l s o due t o D r . P e t e r B a r t e l l o f o r h i s h e l p i n s o l v i n g t h e c l o s u r e e q u a t i o n s , t o M r . D a v i d Ramsden f o r h i s t w o - l a y e r m o d e l . The f i n a n c i a l s u p p o r t from t h e U n i v e r s i t y G r a d u a t e F e l l o w s h i p , and from NSERC g r a n t s v i a W. H s i e h and from ONR g r a n t s v i a G . Hol loway i s d e e p l y a p p r e c i a t e d . On a more p e r s o n a l n o t e , I must admit t h a t I would not be a b l e t o endure t h e d i f f i c u l t i e s i n my f i v e - y e a r o v e r s e a s l i f e w i t h o u t t h e c o n s t a n t s u p p o r t and u n d e r s t a n d i n g o v e r the y e a r s f rom my s i s t e r , L i h u a Zou, t o whom I am d e e p l y i n d e b t e d . x i i CHAPTER 1 INTRODUCTION 1 . 1 Overview D e t e r m i n i n g t h e t e m p o r a l e v o l u t i o n o f d i s t u r b a n c e s i n a g i v e n h y d r o d y n a m i c a l s y s t e m h a s b e e n o f much i n t e r e s t t o o c e a n o g r a p h e r s a n d m e t e o r o l o g i s t s , s i n c e i t i s g e n e r a l l y b e l i e v e d t h a t many o b s e r v e d p h e n o m e n a i n c l u d i n g t h e d e v e l o p m e n t o f m e s o s c a l e e d d i e s i n t h e o c e a n s a n d c y c l o n e w a v e s i n t h e w e s t e r l i e s c a n b e t r a c e d t o some k i n d o f i n s t a b i l i t y . M a t h e m a t i c a l l y , t h e t a s k a m o u n t s t o f i n d i n g a s o l u t i o n t o a n i n i t i a l b o u n d a r y v a l u e p r o b l e m (IBVP) p o s e d f o r t h e d i s t u r b a n c e . I n p r a c t i c e , t h e IBVP i s o f t e n l i n e a r l i z e d a r o u n d t h e g i v e n f l o w a n d i t s s o l u t i o n o f t e n s o u g h t i n a s p e c i a l f o r m ( i . e . , m o d a l f o r m ) . T h i s a p p r o a c h , r e f e r r e d t o a s modal analysis h e r e a f t e r f o r s i m p l i c i t y , h a s o c c u p i e d a p r o m i n e n t p o s i t i o n i n t h e l i t e r a t u r e . M u c h o f t h e p r e s e n t u n d e r s t a n d i n g o f h y d r o d y n a m i c a l s t a b i l i t y owes i t s o r i g i n t o t h i s a p p r o a c h . I t h a s l o n g b e e n a s s o c i a t e d w i t h t h e t h e o r y o f C h a r n e y (1947) a n d E a d y (1949) o n b a r o c l i n i c i n s t a b i l i t y o f t h e w e s t e r l y w i n d . H o w e v e r , some a s p e c t s o f t h e t e m p o r a l e v o l u t i o n o f d i s t u r b a n c e s f a l l o u t s i d e t h e s c o p e o f m o d a l a n a l y s i s . T h i s i s e v e n t r u e o v e r a n i n i t i a l p e r i o d o f e v o l u t i o n o f s m a l l d i s t u r b a n c e s , w h i c h i s g e n e r a l l y c o n s i d e r e d a d e q u a t e f o r 1 l i n e a r t h e o r y . F o r example, i t has been d e m o n s t r a t e d u s i n g l i n e a r i n i t i a l v a l u e problems t h a t (a) p r o p e r l y c o n f i g u r e d p e r t u r b a t i o n s e x h i b i t l a r g e t r a n s i e n t growth i n some s u b c r i t i c a l f l o w s d e s p i t e t h e absence o f g r o w i n g modal p e r t u r b a t i o n s ( c f . O r r ; 1907; Rosen, 1971; F a r r e l l , 1982; Boyd,. 1983; O ' B r i e n , 1990); and (b) t h e i n s t a n t a n e o u s growth r a t e i s o f t e n c o n s i d e r a b l y l a r g e r t h a n t h e maximum modal growth r a t e o v e r t h e i n i t i a l p e r i o d ( c f . F a r r e l l , 1982; 1988; 1989a; Boyd, 1983; O' B r i e n , 1990) . F u r t h e r l i m i t a t i o n s o f t h e modal a n a l y s i s a r e n o t i c e d i n a p p l y i n g t h e modal growth r a t e t o account f o r t h e development o f m e s o s c a l e e d d i e s i n t h e oceans and s y n o p t i c s c a l e d i s t u r b a n c e s i n t h e atmosphere. F o r example, t h e growth r a t e s o f m e s o s c a l e o c e a n i c e d d i e s from modal a n a l y s i s a r e o f t h e o r d e r o f one y e a r (Schulman, 1967) whereas t h e o b s e r v a t i o n d a t a i n Crease (1962), Swallow (1971) and K o s h l y a k o v and Grachev (1973) suggest t h a t i t i s o f o r d e r o f a few months. A n o t h e r example i s found i n t h e s t u d y o f c y c l o n e g e n e s i s . I t i s w e l l known t h a t t h e t y p i c a l p e r i o d o f deepening o f o b s e r v e d c y c l o n e s i s between 12 and 48 hours ( c f . Roebber, 1984; Sanders, 1986) , c l e a r l y l a r g e r t h a n any found i n modal a n a l y s i s o f b a r o c l i n i c i n s t a b i l i t y , say, 133 hours i n V a l d e s and H o s k i n s (1988). W h i l e t h e r e i s l i t t l e doubt t h a t t h e s t u d y based on t h e 2 l i n e a r IBVP (e.g., Case, 1960; Rosen, 1971) y i e l d s more accurate description of the onset and early evolution of i n f i n i t e s i m a l disturbances than modal analysis by allowing for a more general type of i n i t i a l perturbation, the asymptotic behavior of disturbances as t •> » i s somewhat beyond the capacity of the l i n e a r IBVP approach, e s p e c i a l l y when the flow under concern i s subject to growing modal disturbances. In t h i s case, arises the problem of nonlinear saturation which i s concerned with the maximum amplitude attained by growing disturbances (Shepherd, 1988, 1989); or the problem of nonlinear e q u i l i b r a t i o n which deals with the mechanisms responsible for the arrest of exponentially growth (Pedlosky, 1970, 1981; Salmon, 1980; Mak, 1985). Consideration of asymptotic behavior of disturbances also arises when one wonders what happens to the i n i t i a l l y growing disturbances af t e r transient growth (cf. Rosen, 1971; Boyd, 1983; Shepherd, 1985) ; or how a t r a n s i t i o n from one equilibrium state to another takes place i n the presence of multiple e q u i l i b r i a (Charney & Devore, 1979; Vickroy & Dutton, 1979; Wiin-Nielsen, 1979a; Kallen, 1981; Legras & G i l l , 1983; P r o e f s c h r i f t , 1989). The asymptotic behavior of nonmodal disturbances i s not only of t h e o r e t i c a l i n t e r e s t but also of p r a c t i c a l value. Viewing the synoptic scale disturbances as transients superposed on the planetary scale westerlies, long range 3 forecast could benefit from the basic understanding of the long term behavior of nonmodal disturbances i n model atmosphere flows such as those considered i n Charney and Devore (1979). Taking the view that the climate i t s e l f i s a question of d i s t r i b u t i o n among a l l possible equilibrium states reachable by the atmosphere-oceans system and that climate change i s a matter of the r e d i s t r i b u t i o n under the influence of a changing boundary (Charney and Devore, 1979), climate modelers are e s s e n t i a l l y facing the same problem as considered here (e.g., Marotzke, 1989). Situations may arise i n which the t-*» behavior of persistent disturbances (or the r e s u l t i n g flows) can perhaps be u s e f u l l y characterized only i n a s t a t i s t i c a l sense. This w i l l be the case when s p e c i f i c forms of i n i t i a l perturbations to the hydrodynamical system under concern (or. equivalently when i n i t i a l conditions for the perturbed system) are not pr e c i s e l y known i n a deterministic sense but rather given i n terms of some p r o b a b i l i s t i c measure. S t a t i s t i c a l description of spectral behavior for the conservative quasi-geostrophic system on 0-plane i s obtained using equilibrium s t a t i s t i c a l mechanics by Salmon, Holloway and Hendershott (1976), on a rotat i n g sphere by Frederiksen and Sawford (1980), for dissipated-forced flows on an f-plane using closure theory by Herring (1976) and Holloway (1978), and on a 0-plane with large scale zonal flow component by Holloway (1987) . 4 1.2 Objective This thesis i s to es t a b l i s h some basic properties regarding the temporal evolution of disturbances i n a r e l a t i v e l y simple nonlinear hydrodynamical system, taking the view that the analysis for such a system would y i e l d some insight into the temporal behavior of disturbances i n more r e a l i s t i c systems such as those modelling the Gulf Stream in the A t l a n t i c Ocean and the westerlies at midlatitudes. The s p e c i f i c system considered here i s the one defined for a homogeneous, incompressible layer of f l u i d on a /3-plane, i . e . , fa/at;v20 + J O , v20 +py+{h/H)fQ) = -rCv2i/» - v V ; , (1.2.1) subject to v = d\lt/dx = 0, at y = 0 and d , \li(x+l,y,t) =>l>(x,y,t), B.C. (1.2.2a) or \li(x,y,t) = ip (x,y+d,t), ty(x,y,t) = \p (x+l,y,t), B.C. (1.2.2b) I.C. ili(x,y,0) = il>Q(x,y), (1.2.3) where \ji i s the streamfunction; V the two-dimensional Laplacian; J the Jacobian; p the beta parameter; f the C o r i o l i s parameter at reference l a t i t u d e e Q; h(x,y) the bottom topography; H the mean depth of the f l u i d , r the Ekman damping c o e f f i c i e n t and an externally prescribed f o r c i n g function; 1 the periodic length i n east-west d i r e c t i o n (x-axis); d i n (1.2.2a) the channel width, or i n (1.2.2b) the periodic length i n north-south d i r e c t i o n (y-axis). S p e c i f i c steps taken to achieve t h i s objective i s de t a i l e d i n subsequent preview of the t h e s i s . 1.3 Preview of subsequent chapters In chapter 2, we develop a global analysis for the system (1.2 .1)-(1.2.3) i n a Hil b e r t space consisting of a l l kinematically admissible functions (defined as those s a t i s f y i n g B.C.(1.2.2a) or (1.2.2b)), with no assumption made on the nature of the flow except that i t i s governed by the system. The re s u l t s are applied to a number of geophysical f l u i d dynamical problems ranging from global s t a b i l i t y to the r e l a t i o n of modal to nonmodal i n i t i a l growth rates. In chapter 3, a f i n i t e dimensional version of the global analysis i s made for flows subject to the channel condition (1.2.2a) and forced by an external zonal momentum source = -U*y. The r e s t r i c t i o n to t h i s s p e c i f i c type of forcing i s by no means es s e n t i a l but rather for the convenience of numerical experiments. Numerical calculations are made to i l l u s t r a t e the basic notions put forward i n t h i s and previous chapters. 6 E x t e n d i n g t h e s t u d y o f t h e r e l a t i o n o f i n i t i a l n o n m o d a l t o m o d a l p e r t u r b a t i o n s i n t h e H i l b e r t s p a c e , we g i v e a f i n i t e d i m e n s i o n a l a c c o u n t f o r t h e s u b j e c t i n c h a p t e r 4 . S p e c i f i c a l l y , we e x t e n d t h e c o n c l u s i o n s s t r i c t l y t r u e o n l y a t i n i t i a l i n s t a n t f r o m t h e g l o b a l a n a l y s i s t o an i n i t i a l p e r i o d o f t i m e b y e s t a b l i s h i n g some f u n d a m e n t a l p r o p e r t i e s o f n o n m o d a l d i s t u r b a n c e s . I n c h a p t e r 5 , a l o c a l a n a l y s i s i n s t a t e s p a c e i s made on t h e a s y m p t o t i c b e h a v i o r o f n o n m o d a l d i s t u r b a n c e s a s t -» oo, a n d o n i t s r e l a t i o n t o b i f u r c a t i o n o f one e q u i l i b r i u m f l o w i n t o a n o t h e r . T h e r e s u l t s f r o m b o t h n u m e r i c a l e x p e r i m e n t s a n d b i f u r c a t i o n a n a l y s i s c o n f i r m t h e t h e o r e t i c a l p r e d i c t i o n s . I n c h a p t e r 6, we u s e a f i n i t e d i m e n s i o n a l v e r s i o n o f ( 1 . 2 . 1 ) , ( 1 . 2 . 2 b ) t o s t u d y t h e s t a t i o n a r y s t a t i s t i c s o f a s y m p t o t i c s t a t e s o f p e r t u r b e d f l o w s a r i s i n g f r o m i n i t i a l l y u n i f o r m z o n a l f l o w s o v e r r a n d o m t o p o g r a p h y , w i t h f o c u s on t h e c o r r e l a t i o n b e t w e e n t o p o g r a p h y a n d v o r t i c i t y a n d t h e r e s u l t i n g t o p o g r a p h i c s t r e s s . T h e wor k i s b a s e d o n a c l o s u r e t h e o r y ( H o l l o w a y , 1987) a n d d i r e c t n u m e r i c a l s i m u l a t i o n o f f l o w s . T h e n u m e r i c a l r e s u l t s a r e o b t a i n e d f o r p a r a m e t e r s r e l e v a n t t o t h e m i d o c e a n e n v i r o n m e n t . Some b r i e f c o n c l u d i n g r e m a r k s o n t h e r e s u l t s o f t h e p r e s e n t s t u d y a n d p o s s i b l e e x t e n s i o n s f r o m t h e p r e s e n t w o r k a r e g i v e n i n c h a p t e r 7 . 7 CHAPTER 2 GLOBAL ANALYSIS PART I: INFINITE DIMENSIONAL SYSTEM, with a p p l i c a t i o n to: global s t a b i l i t y , optimal perturbation, multiple e q u i l i b r i a and r e l a t i o n of i n i t i a l modal to nonmodal growth rates 2.1 Introduction S i n c e Charney's s t r i k i n g t h e o r y on t h e i n s t a b i l i t y of w e s t e r l y winds i n t h e atmosphere ( c f . Charney, 1947), t h e normal modes s t a b i l i t y a n a l y s i s ( h e r e a f t e r r e f e r r e d t o as modal analysis f o r s i m p l i c i t y ) has been p l a y i n g , and w i l l c o n t i n u e t o p l a y , a v i t a l r o l e i n g e o p h y s i c a l f l u i d dynamics. However, some s h o r t c o m i n g s o f t h e a n a l y s i s have come i n t o l i g h t i n v a r i o u s c o n t e x t s o f hydrodynamics ( c f . Rosen, 1971; F a r r e l l , 1982; Boyd, 1983; Shepherd, 1985; O ' B r i e n , 1990) . I n t h i s c h a p t e r , we p r e s e n t an a n a l y s i s which c i r c u m v e n t s some o f t h e s e s h o r t c o m i n g s and, t o g e t h e r w i t h t h e modal a n a l y s i s , w i l l y i e l d a f u l l e r d e s c r i p t i o n o f t h e s t a b i l i t y p r o p e r t i e s o f a h y d r o d y n a m i c a l system. F o r t h e r e a s o n t h a t w i l l become a p p a r e n t , t h e a n a l y s i s i s h e r e a f t e r r e f e r r e d t o as global analysis as opposed t o t h e modal a n a l y s i s . The g l o b a l a n a l y s i s p e r f o r m e d here makes no r e f e r e n c e t o any s p e c i f i c f l o w e x c e p t t h a t i t i s governed by t h e continuum system (1.2.1)-(1.2 . 3) . I t i s t h u s e x p e c t e d t h a t t h e g l o b a l a n a l y s i s and t h e r e s u l t i n g c o n c l u s i o n s a p p l y t o any f l o w 8 r e a l i z e d by (1.2.1)-(1.2.3). The chapter i s organized as follows. The global analysis i s constructed step by step from § 2.2 to § 2.5, with § 2.2 devoted to the formulation of an i n i t i a l boundary value problem (IBVP) for the s t a b i l i t y (or p r e d i c t a b i l i t y ) of a flow 0(x,t;^ o) governed by (1.2.1)-(1.2.3); § 2.3 to derivation of a symmetrized k i n e t i c energy (or error) equation for disturbance (or e r r o r ) ; § 2.4 to introduction of the generalized Rayleigh quotient into the k i n e t i c energy (or error) equation, and to i t s basic properties; § 2.5 to construction of an algorithm for finding the extreme value of the generalized Rayleigh quotient i n some relevant function space W. In the remainder of the chapter, the global analysis i s applied to monotonic, global s t a b i l i t y (MGS), to a search for optimal i n i t i a l perturbations i n § 2.6; to multiple e q u i l i b r i a and p r e d i c t a b i l i t y i n § 2.7; to r e l a t i o n of i n i t i a l modal to nonmodal growth rate i n § 2.8, followed by concluding remarks i n § 2.9. 2 . 2 IBVP f o r s t a b i l i t y and p r e d i c t a b i l i t y The f i r s t step i n the global analysis i s to formulate an IBVP for disturbances (or errors) i n a given flow. Let 0(x,t;0 Q ) denote a flow governed by (1.2.1)-(1.2.3) and i n i t i a l i z e d from 0 Q(x,y). Consider another flow ^(x,t;0o+#o) which i s r e a l i z e d under the same conditions as those for 9 0 ( x , t ; 0 Q ) e x c e p t f o r i t s i n i t i a l f i e l d : 0O+0O- To a d d r e s s t h e q u e s t i o n i f a n d u n d e r what c o n d i t i o n t h e two f l o w s s t a y t o g e t h e r ( r o u g h l y s p e a k i n g , s t a b l e o r p r e d i c t a b l e ) o r s t a y a p a r t ( u n s t a b l e o r u n p r e d i c t a b l e ) , we c o n s i d e r t h e d i f f e r e n c e f i e l d «>, g i v e n b y <f>(x, t) • 0Cx,t/^ o+ 4>0) ~ 0 f x , t ; 0 o ; ( 2 . 2 . 1 ) T h e s y s t e m d e s c r i b i n g t h e d y n a m i c s o f <p i s o b t a i n e d a f t e r s u b t r a c t i n g t h e g o v e r n i n g e q u a t i o n s f o r ip (x, t ;^ o+# o) f r o m t h o s e f o r tfr(x,t;^Q) (d/dt)V 4> + £[<t>;ip] + N[<t>] = 0, B.C. (Ill) : • o r B.C. (IV) : I.C. 8<f>/dx = 0, for y = 0,d and 0*xslf <f>(x,y,t) - 4> (x+l,y,t), for Osyzd, <p(x,y,t) = <f> (x,y+d, t) , for 0*x*l, <(>(x,y,t) = <j>(x+l,y,t) , for Ozysd, <p(x,tQ) = 0Q (x). ( 2 . 2 . 2 ) ( 2 . 2 . 3 a ) ( 2 . 2 . 3 b ) ( 2 . 2 . 4 ) w h e r e £ a n d N : H -> IH a r e r e s p e c t i v e l y l i n e a r a n d n o n l i n e a r d i f f e r e n t i a l o p e r a t o r s , a n d a r e g i v e n b y ( i n i n d i c i a l n o t a t i o n w i t h s u m m a t i o n c o n v e n t i o n a p p l i e d t o t h e r e p e a t e d i n d i c e s ) ( 2 . 2 . 5 ) MM - e.jkz .a^a.aman*. ( 2 . 2 . 6 ) 10 8^ s a/dx^; (x1,x2,x3) s (x,y,z), (2.2.7a) (zvz2,z3) = (0, 0,1), (2.2.7b) e . = alternating tensor, (2.2.7c) IJK Q • 3 i 8 i 0 + py + (h/H)fQ = p o t e n t i a l v o r t i c i t y i n 0 (2.2.8) w i t h IH b e i n g t h e s e t o f a l l k i n e m a t i c a l l y a d m i s s i b l e f u n c t i o n s f o r wh i c h t h e e v o l u t i o n problems (1.2.1)-(1.2.3) and ( 2 . 2 . 2 ) - ( 2 . 2 . 4 ) a r e w e l l d e f i n e d . I n p a r t i c u l a r , t h e eleme n t s from IH s a t i s f y t h e f o l l o w i n g r e q u i r e m e n t s : ( i ) B.C. ( I l l ) (or B.C. (IV)) (2 .2.9) ( i i ) smoothness s u f f i c i e n t t o a s s u r e t h a t t h e s p a t i a l d e r i v a t i v e s i n (1.2.1)-(1.2.3) (or (2.2.2)-(2.2.4) a r e w e l l d e f i n e d . (2.2.10) From now on. we assume t h a t IH i s a Hilbert Space. T h i s means, among t h e o t h e r t h i n g s , t h a t i t i s a v e c t o r space over R w i t h an i n n e r p r o d u c t <a,b> • f ab ; ||a|| e < a , a i / 2 , V a and b e H, (2.2.11) where t h e o v e r b a r denotes t h e complex c o n j u g a t e which i s i r r e l e v a n t f o r t h e elements from H but i s needed f o r t h e eleme n t s from IH. The l a t t e r i s t h e c o u n t e r p a r t o f IH over C. The symbol n denotes t h e a r e a e n c l o s e d by t h e p e r i o d i c c h a n n e l ( c f . ( 2 . 2.3a)) o r t h e doubl e p e r i o d i c c e l l ( c f . ( 2 . 2.3b) depending on t h e model geometry. Remark 2 . 2.1 The system ( 2 . 2 . 2 ) - ( 2.2.4) admits two 11 i n t e r p r e t a t i o n s . I f <pQ i s t a k e n a s a n i n i t i a l p e r t u r b a t i o n t o t h e f l o w 0 (x,t;tpQ) a t t = t Q , t h e n t h e s y s t e m g o v e r n s t h e s u b s e q u e n t e v o l u t i o n o f t h e i n i t i a l p e r t u r b a t i o n c6Q, t h e r e b y l e a d i n g t o t h e s t a t e m e n t o n s t a b i l i t y o f t h e f l o w \j> (x, t ; ^ o ) . On t h e o t h e r h a n d , i f <PQ i s v i e w e d a s a n e r r o r i n t r o d u c e d i n a s s i g n i n g i n i t i a l s t a t e 0Q t o t h e f l o w <p (x, t;ip ), t h e s y s t e m (2 .2 .2)-(2 .2 .4) t h e n d e s c r i b e s g r o w t h o r d e c a y o f t h e i n i t i a l e r r o r t h u s y i e l d i n g i n f o r m a t i o n o n p r e d i c t a b i l i t y o f t h e f l o w 0(x,t;0Q) . I n what f o l l o w s , n o a s s u m p t i o n on t h e n a t u r e o f 4>Q i s made u n l e s s s p e c i f i e d e x p l i c i t l y , t h e r e b y u n i f y i n g t h e t r e a t m e n t o f s t a b i l i t y a n d p r e d i c t a b i l i t y w i t h i n one f r a m e w o r k . I t i s o b v i o u s t h a t i f <Po = c o n s t a n t , t h e n <f> s 0 V t>0 ( c f . (2 .2 .2 ) ) . I t i s n o t d i f f i c u l t t o s e e t h a t <f> = 0 i s a n u l l s o l u t i o n o f (2 .2 .2) - (2.2.4) s u b j e c t t o #o= c o n s t a n t f o r a l l x € n. M o r e o v e r , t h e s t a b i l i t y o f <j> = 0 i m p l i e s t h e s t a b i l i t y o f t h e f l o w \j> (x, t;ifi ) g o v e r n e d b y (1 .2 .1) - (1 .2 . 3) a s s e e n f r o m (2 .2 .1) . T h i s c o r r e s p o n d e n c e l e a d s u s t o s t u d y t h e s t a b i l i t y p r o b l e m f o r f l o w 0(x,t;0Q) v i a t h e one f o r <p = 0 . T h e l a t t e r i s d e f i n e d b y (2 .2 .2) - (2 .2 . 4) w i t h <pQ* c o n s t a n t f o r xen. W i t h t h i s n o t e i n m i n d , we s t a t e t h e r e s u l t s b e l o w o n l y i n t e r m s o f t h e f l o w \ji (x,t;\j) ) f o r t h e s a k e o f s p a c e . A l s o n o t e t h a t t h e r e s t r i c t i o n o f <b t o a n o n - c o n s t a n t f i e l d o i s n o t e s s e n t i a l h e r e b u t i s p h y s i c a l l y m o t i v a t e d , s i n c e <j> = c o n s t a n t c o r r e s p o n d s t o a p e r t u r b a t i o n w i t h k i n e t i c e n e r g y equal to zero, or an error with variance being zero. 2.3 Symmetrized energy (error) equation The second step i n the global analysis i s to symmetrize the l i n e a r operator £ (cf. ( 2 . 2 . 5 ) ) , i . e . , to remove the asymmetric constituents i n £ (e.g., the second term i n £) such that the time tendency i n the disturbance energy equation (or error equation) i s expressed i n terms of a s e l f - a d j o i n t operator (cf., Reddy, 1986). We s t a r t with taking the inner product of (2.2.2) with <j> € W, followed by d i r e c t l y evaluating i n d i v i d u a l terms and using the adjoint operator £*of £ defined by <£a,b>=<a,£*b>, V a,b e IH (cf., chapter '8 i n Reddy, 1986). Omitting the algebraic d e t a i l s involved, we give the r e s u l t i n g equation below (d/dt) <di<t>, 8±<t>> = <</>,(£+£)<(>>, (2.3.1) - djfdjU^^) - a ^ u ^ ^ ^ r (2.3.2) (£+£) [<t>;\l>] = - d^djUjd^) - BjUfdjdjt + 2rd±d±(t>r (2.3.3) Uim eijkZjdk*> <2'3-4) where a l l t h e symbols here a r e t h e same as t h o s e i n ( 2 . 2 . 2 ) - ( 2 . 2 . 8 ) . S e v e r a l remarks on (2 . 3.1)-(2.3.4) a r e i n o r d e r . 13 Remark 2 .3.1 T h e s e l f - a d j o i n t n e s s o f £ + £ * c a n b e e a s i l y v e r i f i e d b y n o t i n g t h a t <a, (£+£*) b> = <(£+£) a,b>, V a , b € w. T h i s m a t h e m a t i c a l p r o p e r t y w i l l b e e s s e n t i a l t o o u r a p p r o a c h . Remark 2.3.2 I n a d d i n g t h e a d j o i n t £ to £ t o f o r m t h e o p e r a t o r £ + £ * , t h e p h y s i c a l p r o c e s s e s c o r r e s p o n d i n g t o t h e n o n s e l f - a d j o i n t c o n s t i t u e n t s i n £ a r e a n n i h i l a t e d f r o m (2.3.1) a s d e s i r e d . F o r e x a m p l e , t h e a d v e c t i o n o f p o t e n t i a l v o r t i c i t y Q i n f l o w \ji b y cfr, a s r e p r e s e n t e d b y t h e s k e w - s y m m e t r i c o p e r a t o r £ [<f>;\p] = e . .,z .8,<f>d,.Q, i s e l i m i n a t e d f r o m £ + £ * . Remark 2 .3.3 I n t h e s y m m e t r i z a t i o n l e a d i n g t o (2 . 3.1) - (2 .3 . 4) , t h e n o n l i n e a r t e r m f/[<p] v a n i s h e s e x a c t l y w i t h o u t i n v o k i n g l i n e a r i z a t i o n h y p o t h e s i s . T h u s , a n y s t a t e m e n t s w h i c h f o l l o w f r o m (2.3.1) h o l d f o r a r b i t r a r y <f> e H r e g a r d l e s s o f i t s s i z e . 2.4 Generalized Rayleigh Quotient and i t s properties T h e n e x t two c r u c i a l s t e p s i n t h e g l o b a l a n a l y s i s a r e : t o i n t r o d u c e a f u n c t i o n a l s u c h t h a t i t i s b o u n d e d i n H, a n d t o c o n s t r u c t a n a l g o r i t h m f o r o b t a i n i n g t h e e x t r e m e v a l u e o f t h e f u n c t i o n a l i n W. T h e o b j e c t i v e o f t h i s s e c t i o n i s t o a c c o m p l i s h t h e f o r m e r . We d o t h i s b y a f u r t h e r m a n i p u l a t i o n o f t h e o p e r a t o r £ + £ . W h i l e t h e m a t h e m a t i c a l m a n i p u l a t i o n i n v o l v e d i s e l e m e n t a r y , we m u s t n o t e t h e f o l l o w i n g two c r u c i a l p h y s i c a l f a c t s . F i r s t , t h e v e l o c i t y g r a d i e n t d j U ± ^ n t h e f l o w ip (x, t;\pQ) ( c f . ( 2 . 3 . 3 ) ) c a n b e d e c o m p o s e d i n t o t h e r a t e o f s t r a i n t e n s o r f c ^ j ] a n d t h e v o r t i c i t y t e n s o r [ v ^ . ] , i . e . , [ 8 ^ ] = [d±j] + [v±j], ( 2 . 4 . 1 a ) d ± & (1/2) (d±Uj+ 8M±) , ( 2 . 4 . 1 b ) v±jm (1/2) (d±Uj- BjU±) . ( 2 . 4 . 1 c ) S e c o n d , t h e s p i n n i n g o f <f> b y \p, a s s o c i a t e d w i t h [v^_.] , i s c o n s e r v a t i v e a s f a r a s t h e k i n e t i c e n e r g y o f <j> i s c o n c e r n e d , w h i c h c a n b e r e a d i l y shown b y n o t i n g t h e a s y m m e t r i c n a t u r e o f t h e t e n s o r [v^ . ] . O m i t t i n g t h e a l g e b r a i c d e t a i l s , we g i v e t h e r e s u l t i n g f o r m (d/dt) (l/2)<di4>,di(p> = <d14>ld±(p> (p((f>;ili)-r), ( 2 . 4 . 2 ) p(</>;tp) * <<f>,d(l>>/<<t>,%<(>>, ( 2 . 4 . 3 ) d[<p;ip] s - d 1 d J j S 3 4 , ( 2 . 4 . 4 ) s/>J • - a 1 a 1 » > / ( 2 . 4 . 5 ) w h e r e p(<p;\p): IH -» IR i s t h e g e n e r a l i z e d R a y l e i g h q u o t i e n t , a n d S B : IH - » IH a r e d i f f e r e n t i a l o p e r a t o r s . Remark 2 . 4 . 1 K i n e m a t i c a l l y , <<p,A4>> r e p r e s e n t s t h e s t r a i n i n g o f <f> b y 0 v i a t h e p r e s e n c e o f r a t e o f s t r a i n t e n s o r id^jl o f 15 0 i n (2.4.4). In terms of energetics, i t serves as an energy source for the development of 0, or more pr e c i s e l y as an energy conversion between 0 and 0. On the other hand, <0, £0> i s the t o t a l k i n e t i c energy i n <f>. Note that i f r * 0, the t o t a l Ekman damping on <f> i s r<0, £0>. Thus, the generalized Rayleigh Quotient p(<t>;*p) can be p h y s i c a l l y interpreted as the r a t i o of energy conversion between 0 and <f> to the energy in 0, or to the Ekman d i s s i p a t i o n i f r * 0 (or the r a t i o of the error generation to the e r r o r ) . The following properties of operators A and S are used throughout the chapter. Lemma 2.1 Let A and S be given by (2.4.4) and (2.4.5). Then d and £ are self-adjoint on H, viz. (L) A* = A, (2.4.6) f i i ; B* = SB, <a,5Ba> >0, V a € IH with a * constant. (2.4.7) Proof: (i) for any a,b e IH, we have d . (bd . .8 .a) - a .bd . .8 .a I IJ j I ij j d <=* j) (since d. . = d . .) IJ ii' d±(adi.a.b) - adidi.8.b\ = <a,Ab>. 16 On t h e o t h e r h a n d , we h a v e <Aa,b>=<a,A*b> b y d e f i n i t i o n ( c f . , R e d d y , 1 9 8 6 ) , w h i c h t o g e t h e r w i t h t h e l a s t l i n e i n t h e a b o v e c a l c u l a t i o n o f <Aa,b> e s t a b l i s h e s t h e s e l f - a d j o i n t n e s s o f A. (ii) f o l l o w s s i m i l a r l y . B Lemma 2 . 2 L e t p(<l>;ip) be defined by (2.4.3). Then: \p(<p;ij>)\ & max max (ky\2), V <f>eW, ( 2 . 4 . 8 ) X € f i where (i=l,2) are principal values of the rate of strain tensor a t x e fi. Proof: F i r s t , we c a n p u t p(<p;ip) i n t h e f o r m p(4>;$) = f d.<l>d..d.<p A d.<t>a.<p , 4 e IH ( 2 . 4 . 9 ) fi •*• J J J n w i t h d i = . d e f i n e d b y ( 2 . 4 . 1 b ) . S i n c e t h e r a t e o f s t r a i n t e n s o r [d^j] i s r e a l a n d s y m m e t r i c , t h e r e t h u s e x i s t s a s e t o f p r i n c i p a l a x e s a t e a c h p o i n t x € fi w i t h r e s p e c t t o w h i c h ^ c f ^ h a s a d i a g o n a l f o r m . I n f a c t , s u c h a d i a g o n a l f o r m i s a c h i e v e d l o c a l l y ( i . e , a t a n y x e fi) b y p e r f o r m i n g a n o r t h o g o n a l s i m i l a r i t y t r a n s f o r m a t i o n lc±jf [d±j] [c±j] = diag (\,\). ( 2 . 4 . 1 0 ) w h e r e \^(i=l,2) a r e t h e e i g e n v a l u e s o f d e t ([d^^J-XI) = 0, I s a u n i t matrix e R 2 x 2 , ( 2 . 4 . 1 1 ) a n d h e n c e a r e t h e p r i n c i p a l v a l u e s o f [d. .] a t x, a n d t h e 17 column vectors of ^c±j^ e r2*2 a r e ^~he corresponding eigenvectors and thus constitute a set of p r i n c i p a l axes at x for the rate of s t r a i n tensor [d^..] . It i s clear that [Cj_j] constructed with the normalized eigenvectors of (2.4.11) enjoys the following orthogonal property [ c i j 1 [ c i j ] T = [ c i j ] T [ c i j ] = 1 '* [ c i j ] ~ l = [ c i j ] T ' (2.4.12) An estimate of the numerator i n (2.4.9) thus follows from (2.4.10) and (2.4.12) | J n a . * d . y y | s j j a ^ ([a. 1 [c..]T [d..] [a..] [a..]' ^ | s^max(\i,\2)di£l 8^ with d±Z = [c±j]'18^ smax max (X , X )\ 8 8 .£ X€Q 1 2 J n 1 1 =max max (\ ,X )\ 8 ,<f> 8 .<f> xefi 1 2 J n 1 1 from which (2.4.8) i s obtained. a 2.5 An optimal problem f o r the generalized Rayleigh quotient As the very f i n a l step i n the global analysis, we demonstrate that the extreme value of p i n H can be obtained v i a a boundary eigenvalue problem (BEP). Lemma 2.3 If 0 e IH maximizes the generalized Rayleigh Quotient p(<p;ip), then <j> solves the boundary eigenvalue problem: 18 A<f> = XB0, ( 2 . 5 . 1 ) 0 satisfies the B.C. (Ill) (or B.C. (IV) ), ( 2 . 5 . 2 ) with X = X[0] s p(4>;\l>) = max p(<t>;^i) ( 2 . 5 . 3 ) 0eH and A, S given by (2.4.4) and (2.4.5), respectively. Proof: F i r s t , we a s s o c i a t e w i t h t h e d o m a i n H o f t h e R a y l e i g h q u o t i e n t p ( 0 ; 0 ) , a s e t IM o f c o m p a r i s o n f u n c t i o n s s u c h t h a t f o r a n y 0 e IH a n d a n y £ e W, 0 + e £ € IH f o r a n y e € R. T h i s i s d o n e b y c h o o s i n g W = W . Now, d e f i n e l i n e a r f u n c t i o n a l s §: M -> R b y 9[+] • <4>,A<p>, ( 2 . 5 . 4 ) &/-0J = < 0 , £ 0 > , ( 2 . 5 . 5 ) r e s p e c t i v e l y , T h e n , f o r p(0) = 9 [<p] [<p] (where f o r s i m p l i c i t y o f n o t a t i o n , 0 a s an a r g u m e n t i s s u p p r e s s e d ) , 6 p ( 0 ; € ) = (1/V[<t>]2){$[<!>]8V[<t>;V - 88[</>]}. I f 0 i n IH m a x i m i z e s p , t h e n 5 p ( 0 ; £ ) = 0, o r &[4>]8$- 8&[4>;%]9[4>J = 0 f o r a l l £ e M. ( 2 . 5 . 6 ) I t i s f o u n d f r o m lemma 2 . 1 t h a t 8^[4>;V = 2<£,A4>>, ( 2 . 5 . 7 a ) S&/"0 ;£ J = 2 < £ , £ 0 > . ( 2 . 5 . 7 b ) I n t r o d u c i n g ( 2 . 5 . 7 ) i n t o ( 2 . 5 . 6 ) y i e l d s <€, M - X 2 ) 0 > = 0, V a n y € e M, 19 with x = X s yftJ/'Sti] = max V [<t>][<t>], «>€lH which implies (d-XB)4> = 0, thus completing the proof. a While lemma 2.3 provides a systematic means of solving the optimal problem max p i n IH, i t s p r a c t i c a l implementation may not turn out to be simple i n the case of zonally asymmetrical flow 0 which i n generally leads to nonseparable c o e f f i c i e n t s i n (2.5.1). However, the lemma i s c r u c i a l to our subsequent development of the theory, as we w i l l see.soon. In appendix 2A, we present a Rayleigh-Ritz procedure for the numerical solution of the problem: max p i n IH. Remark 2.5.1 As seen i n writing X[\p] , X depends on the state of the flow 0(x,t;0Q) and hence i s a function of time t i f 0Q i {*}, where {*} denotes the set of equilibrium states of (1.2.1)-(1.2.3). However, i t i s clear that A[0] = X[0 ] = a constant if ipQ e {*} . ( 2 . 5 . 8 ) For p r a c t i c a l concerns, the following lemma assures us some desired properties of the eigensystem ( 2 . 5 . 1 ) - ( 2 . 5 . 2 ) . Lemma 2.4 For the eigensystem (2.5.1) - (2.5.2), it holds that 20 (i.) all eigenvalues A are real; (ix) X is bounded; f i i i ) i f Xi* X2, then the corresponding eigenfunctions c^ , #2 are orthogonal in the sense that <8^</>^f8^<t>2> = 0. Proof: (i) L e t (X,<p) b e d e t e r m i n e d f r o m ( 2 . 5 . 1 ) . C o n s i d e r <A4>,4>> = <<f>,d*<p> = <<p,d<l>> (by Lemma 2.1(D) ( 2 . 5 . 9 ) I t t h u s f o l l o w s f r o m ( 2 . 5 . 1 ) a n d ( 2 . 5 . 9 ) t h a t <XB0,c>> = <«>, A53#> =* X<£<f>,4>> = X<(p,S(p> = X<£<t>,4>> (by Lemma 2.1(H)) w h i c h h o l d s o n l y i f A = A, i . e , A i s r e a l . f i i ) T h e b o u n d e d n e s s o f A f o l l o w s f r o m lemma 2 . 2 . ( i i i ) l e t A^ a n d <f>^(i=l,2) b e s t a t e d a b o v e . Now, c o n s i d e r «t>itd<t>2> = <d4>if<t>2> * «t>i,x2s<i>2> = <xis<t>i,<t>2> * (xi-x2)<di<i>i.di<t>2> = o w h i c h l e a d s t o <8 .d> ,8 .<f> > = 0 i f A * A j T l 12 1 2 • Remark 2 . 5 . 2 T h e p h y s i c a l i n t e r p r e t a t i o n o f ( i i i ) i n lemma 2 . 4 i s t h a t t h e v e l o c i t y f i e l d s a s s o c i a t e d w i t h d i f f e r e n t A a r e o r t h o g o n a l w . r . t t h e i n n e r p r o d u c t ( 2 . 2 . 1 1 ) . T o s u m m a r i z e , t h e g l o b a l a n a l y s i s i n c l u d e s : 1) t h e s y m m e t r i z e d e n e r g y ( o r e r r o r ) e q u a t i o n ( 2 . 4 . 2 ) ; 2) an 21 algorithm for optimizing the generalized Rayleigh quotient p, i. e . , lemma 2.3. In view of the fact that a bounded functional i s introduced i n § 2.4 and a v a r i a t i o n a l argument i s invoked to rel a t e the extreme problem to an Euler-type boundary eigenvalue problem, the global analysis developed here may f a l l into the category of v a r i a t i o n a l energy method (Serrin, J., 1959a, 1959b; Joseph, D.D., 1966, 1976). In the remainder of the chapter, we s h a l l explore the res u l t s of the global analysis ( i . e . , lemma 2.1-2.4) i n various contexts. F i r s t , we note from remark 2.2.1 that our re s u l t s can be stated i n terms of the s t a b i l i t y of the flow \ji fx, t;\po), or equally well i n terms of i t s p r e d i c t a b i l i t y , depending on our int e r p r e t a t i o n of the i n i t i a l data 0Q . Except for a few occasions i n the subsequent developments, however, much of argument i s presented only i n terms of the s t a b i l i t y f or the sake of space. 2.6 App l i c a t i o n I: Global s t a b i l i t y and optimal perturbation F i r s t , we give the d e f i n i t i o n of global s t a b i l i t y for the system (1.2.1)-(1.2.3) D e f i n i t i o n 2.1 (global stability) Let <f>(x,t;<f>o) satisfy (2.2.2;-(2.2.4; . Then: 22 (i) a flow 0Cx,t/0 o ; governed by ' (1.2.1) - (1.2.3) is said to be asymptotically stable if there exists a finite number 6 such that V a) € IH o <8 .6 ,8.6 >s6 =» <8 .6, 8 .6>/<8 .6 , 8 .<f> >-» 0 as t -» oo. ( 2 . 6 . 1 ) ( i i ; a flow ip(xrt;\fio) is said to be globally stable if the limit ( 2 . 6 . 1 ) holds when 8 -» oo. (iii) a flow ip(x,t;<jtQ) is said to be monotonically, globally stable if it is globally stable and, in addition, (d/dt)<8 ^6,8 ^6^0, for t>0. Remark 2 . 6 . 1 The 5 here i s r e f e r r e d t o as an a t t r a c t i n g r a d i u s o f t h e f l o w \p (x,t;^iQ) i n IH s i n c e 5 d e f i n e s a sub s e t o f IH i n wh i c h t h e a c c l a i m e d p r o p e r t i e s o f 0 (x,t;\p ) h o l d . I n t h e c a s e o f g l o b a l s t a b i l i t y , 6 = oo and hence 0 (x,t;\p ) has an i n f i n i t e l a r g e a t t r a c t i n g domain i n IH. D e f i n i t i o n 2 . 2 The finite stability measure of a flow 0(x,t;0 ), denoted by r (\p), is defined as 0 K r (iji) s sup %, ( 2 . 6 . 2 ) t > t o where X is given by (2.5.3). Theorem 2 . 1 A flow 0(x,t;0Q) is monotonic, globally stable if and only if r (\b) - r < 0. ( 2 . 6 . 3 ) N Proof: ( i ) s u f f i c i e n c y . From (2.4.2) and (2.6.2) (d/df ) (1/2) <d ,<t>,d .<p> * <8. (j>,8 .</>> (r (ip)-r). (2.6.4) J . J . J . J . N I n t e g r a t i n g (2.6.4) w i t h r e s p e c t t o t ' from t t o t g i v e s <d±4>,d±<l>> s <a i0 o,3 i0 o>exp {2(r^ (I/J) -r) (t-tQ) }, V t±tQ (2.6.5) where 0Q i s i n i t i a l p e r t u r b a t i o n . T a k i n g t h e l i m i t t -» » i n (2.6.5) under t h e c o n d i t i o n (2.6.3), we e s t a b l i s h t h e s u f f i c i e n c y . ( i i ) t h e c o n v e r s e ( p r o o f by c o n t r a d i c t i o n ) . Suppose t h a t 0 i s MGS b u t r (ip) - r a 0. (2.6.6) N Then, t a k e 4> ( c f . lemma 2.3) as an i n i t i a l p e r t u r b a t i o n , i . e . , <t>(x,t;<j>) =0 , a t t = t Q , and e v a l u a t e t h e i n i t i a l t endency o f k i n e t i c energy o f <(>(x, t;<j>) (d/dt) (l/2)<di<t>,di<t>>\^ = <S i0, a±4>> (p -r) (by (2.4.2)) o =<d .0,9,4> (r ft) - r) i 0, (by (2.6.6) ) a s t a t e m e n t c o n t r a d i c t i n g t o MGS o f 0(x,t ; 0 Q ) . T h i s completes t h e p r o o f . a The concept o f g l o b a l s t a b i l i t y o c c u p i e s a prominent p o s i t i o n i n t h e g l o b a l t h e o r y o f s t a b i l i t y o f t h e N a v i e r - S t o k e s e q u a t i o n ( c f . S e r r i n , J . , 1959a, 1959b; Joseph, 24 D.D, 1976) and t h e B o u s s i n e s q e q u a t i o n ( c f . Joseph, D . D , 1966). However, i t has b a r e l y appeared i n g e o p h y s i c a l f l u i d dynamics (GFD). The e x p l i c i t account o f g l o b a l s t a b i l i t y i n GFD were t a k e n i n V i c k r o y and Dut t o n (1979) and K a l l e n and W i i n - N i e l s e n (1980) f o r a t h r e e d i m e n s i o n a l system. The paper by C r i s c i a n i and M o s e t t i (1990) e s s e n t i a l l y d e a l s w i t h t h e a s p e c t o f g l o b a l s t a b i l i t y f o r w i n d - d r i v e n ocean c i r c u l a t i o n . No o p t i m i z a t i o n i s made i n t h e s e e a r l i e r p a p e r s . The l a c k o f i n t e r e s t i n t h i s a s p e c t o f s t a b i l i t y may a r i s e from t h e c o n s i d e r a t i o n t h a t any g e o p h y s i c a l l y r e l e v a n t f l o w s can h a r d l y be g l o b a l l y s t a b l e . G i v e n some element o f t r u t h f u l n e s s i n t h e above c o n s i d e r a t i o n , we w i l l see t h a t t h e g l o b a l a n a l y s i s o r i g i n a t e d from t h e st u d y o f g l o b a l s t a b i l i t y has a p p l i c a t i o n s i n a number o f i m p o r t a n t problems i n GFD such as m u l t i p l e e q u i l i b r i a , - bounds on t h e growth r a t e o f d i s t u r b a n c e s i n s u p e r c r i t i c a l f l o w s and t r a n s i e n t growth i n s u b c r i t i c a l f l o w s . I t i s c l e a r from t h e above p r o o f t h a t theorem 2.1 can a l s o be s t a t e d i n terms o f t h e e x i s t e n c e o f i n i t i a l l y g r o w i n g p e r t u r b a t i o n s (or e r r o r s ) t o a g i v e n f l o w 0 ( x,t ; 0 Q ) a t i n s t a n t t . Thus, we have C o r o l l a r y 2.1 At any instant t' t ), there exists at least one initially growing perturbation (error) to the state 0 (x, t' ;\I>Q) if and only if X C0 (x, t' ;\j)^)) - r > 0, where X is given by (2.5.3). 25 Corollary 2.1 covers an important s p e c i a l case where t' = t and the i n i t i a l state e {*}. In t h i s s p e c i a l case, we have C o r o l l a r y 2.2 Let 0Q € Then: X (\pQ) -r>0 <=> existence of initially growing <po to \pQ, or r^frliQ)-r>0 «=» existence of initially growing 6q to 0Q. Identifying i n i t i a l l y growing perturbations to 'a given flow, a subject with a long history s t a r t i n g from Orr's work on plane Couette flow (1907) , has been the focus of several recent papers on i n v i s c i d constant shear flows (Boyd 1983; Shepherd 1985 and F a r r e l l 1987), on a s p e c i f i c zonally asymmetric flow ( F a r r e l l 1989c), on the Charney and Eady models with Ekman damping ( F a r r e l l 1982, 1984 and 1989a) and on the Green model (O'Brien, 1990). Directing attention to such perturbations i s natural to synoptic meteorologists since forecast rules often emphasize the roles of these properly configured perturbations (Palmen and Newton, 1969). For zonal shear flows, i t has been known since Orr (1907) that a perturbation with i t s phase l i n e s oriented against the mean shear exhibits temporal amplification. , The forgoing r e s u l t s ( i . e . , Corollary 2.1 and 2.2) may be regarded as a continued e f f o r t of the above c i t e d works in the context of time-dependent, nonlinear and zonal asymmetric 26 f l o w s . I n d e e d , t h e c o r o l l a r i e s t o g e t h e r w i t h lemma 2 . 3 c o n s t i t u t e a n a l g o r i t h m f o r a s y s t e m a t i c a l s e a r c h o f i n i t i a l l y g r o w i n g p e r t u r b a t i o n s t o a g i v e n f l o w . A s o p p o s e d t o t h e common n o t i o n o f " l e a n a g a i n s t s h e a r " , i t i s n o t d i f f i c u l t t o s e e f r o m ( 2 . 4 . 2 ) a n d ( 2 . 4 . 3 ) t h a t f o r i n s t a n t a n e o u s g r o w t h o f a p e r t u r b a t i o n 0(x) t o a g i v e n f l o w 0(x,t;0 ) , i t s u f f i c e s t o r e q u i r e t h a t t h e r a t i o o f t h e e n e r g y c o n v e r s i o n b e t w e e n <f> a n d 0 t o t h e e n e r g y i n 0 a t t h a t i n s t a n t b e l a r g e r t h a n t h e Ekman d a m p i n g r . I t i s s i g n i f i c a n t t h a t i n t h e c a s e o f z o n a l l y a s y m m e t r i c f l o w s , what m a t t e r s f o r a p e r t u r b a t i o n t o g r o w i s n o t s i m p l y t h e e n e r g y c o n v e r s i o n r a t e b e t w e e n 0 a n d 0 b u t r a t h e r t h e g e n e r a l i z e d R a y l e i g h q u o t i e n t ( i . e . , t h e e n e r g y c o n v e r s i o n r a t e p e r u n i t d i s t u r b a n c e e n e r g y ) , a s c e n a r i o n o t o b v i o u s t o o u r i n t u i t i o n . T o g i v e t h e n o t i o n o f o p t i m a l i n i t i a l p e r t u r b a t i o n ( o r w o r s t e r r o r ) a q u a n t i t a t i v e m e a s u r e , we i n t r o d u c e Defini t ion 2 . 3 The instantaneous growth rate cr ($;$) of a N disturbance (or error) <p(x,t;<f>Q) in a flow 0(,x,t/0Q/) is defined as <r C0;0; s (1/2) (d/dt) ln{<d .0,3 .$>} . ( 2 . 6 . 7 ) N J. J. Theorem 2 . 2 A t any instant t, among all possible perturbations (or errors) 0 in IH to a flow 0 Cx, t;0j, 0 27 (cf.lemma 2.3) is the optimal one (or the worst one) in the sense that er (<(>;&) t <r (<i>;ip), V 6 e IH. N N Proof: from (2.4.2) and (2.6.7) % = P C«>;0) - r * p(i>;^>) - r (2 .6 .8) (by lemma 2.3) = \(\l>) - r (by (2.5.3)) where X (ty; i s g i v e n by (2.5.3), which completes t h e p r o o f . B The i d e a o f o p t i m a l p e r t u r b a t i o n emerged i n t h e s e a r c h f o r f a v o r a b l y c o n f i g u r e d p e r t u r b a t i o n and i t s growth r a t e t o e x p l a i n t h e e x p l o s i v e growth o b s e r v e d i n c y c l o n e g e n e s i s (Roebber, 1984; Sanders, 1986) or i n model s t u d i e s o f i n i t i a l v a l u e problems ( F a r r e l l , 1988, 1989a; O ' B r i e n 1990). The o p t i m a l p e r t u r b a t i o n s o b t a i n e d i n e a r l y s t u d i e s have found o t h e r a p p l i c a t i o n s . F o r example, t h o s e d e t e r m i n e d f o r b a r o t r o p i c c o n s t a n t s h e a r f l o w are used i n o p t i m a l e x c i t a t i o n o f n e u t r a l Rossby waves ( F a r r e l l , 1988). The o p t i m a l p e r t u r b a t i o n s found i n t h i s s t u d y d i f f e r from t h o s e i n F a r r e l l (1988, 1989b) and O ' B r i e n (1990) i n s e v e r a l a s p e c t s . F i r s t , our s e a r c h i s c a r r i e d out w i t h i n a f u l l y n o n l i n e a r c o n t e x t whereas t h e o t h e r s were made u s i n g l i n e a r models. Second, t h e f u n c t i o n a l t o be o p t i m i z e d i s t h e 28 generalized Rayleigh quotient i n our case but i s the energy norm or the £-2 norm i n other studies. Third, the optimization i n t h i s study i s taken with respect to a l l possible kinematically admissible functions whereas the same process i n previous studies i s c a r r i e d out conditionally, i . e . , subject to some constraints which are p h y s i c a l l y sound but e s s e n t i a l l y subjective. 2.7 Ap p l i c a t i o n I I : multiple e q u i l i b r i a . The notion of multiple e q u i l i b r i a was f a i r l y recently introduced into geophysical f l u i d dynamics i n an e f f o r t to account f o r the v a c i l l a t i o n between the low index ("blocking pattern") and high index flows i n the atmosphere (Charney & Devore, 1979) and for nonlinear e f f e c t s on p r e d i c t a b i l i t y (Vickroy & Dutton, 1979). The concept has since been extended to increasingly sophisticated physical models, y i e l d i n g much insight into the v a r i a b i l i t y i n large scale atmospheric flows (Kallen & Wiin-Nielsen, 1980; Kallen; 1981; Legras & G i l l , 1983; Rambaldi & Mo, 1984; Tung & Rosenthal, 1985), recurrence of weather systems (Proefschrift, 1989) and climate changes (Marotzke, 1989). The d i f f i c u l t y with nonlinearity has led the early investigators to low order spectral models (Charney and Devore, 1979; P r o e f s c h r i f t , 1989) or numerical models (Kall§n, 1982, 1985) . In the former case, the question 29 n a t u r a l l y a r i s i n g i s : are multiple equilibrium states simply the consequence of severe truncation to the o r i g i n a l i n f i n i t e dimensional systems ? or do multiple e q u i l i b r i a e x i s t i n the o r i g i n a l continuum models? In the l a t t e r case, given an equilibrium state obtained from numerical integration of the model equations, one wants to know i f there ex i s t s another equilibrium state. In t h i s section, we address the above questions i n the context of system (1.2.1)-(1.2.3) based on the global analysis. We do so by considering the problem of uniqueness of the equilibrium state. It i s important to dis t i n g u i s h t h i s uniqueness problem from the one for IBVP. In t h i s problem, one seeks m u l t i p l i c i t y of equilibrium states for a given set of external and internal model parameters. In contrast, the one for IBVP i s concerned with how many flows can evolve from a given i n i t i a l state. We s h a l l show below that the uniqueness of IBVP holds unconditionally whereas the uniqueness of equilibrium state may break down for some values of model parameters, thus implying the p o s s i b i l i t y of multiple e q u i l i b r i a . F i r s t , we have Theorem 2.3 (uniqueness of IBVP (1.2.1) - (1.2.3)) The Velocity field of a flow 0 (x,t;\fiQ) is continuous w.r.t its initial condition and hence there can be only one flow evolving from a given initial velocity field. Proof: Consider another , flow 0 (x, t ; 0 Q +0Q j of (1.2 .1) - (1.2 . 3) r e a l i z e d under the same conditions as those for 0 (x, t;0 ) except i t s i n i t i a l state 0o+0Q. It thus follows from (2.4.2) and (2.6.2) that the k i n e t i c energy of the difference f i e l d 0(x,t ;0 Q ) s a t i s f i e s (d/dt)<d.<t>,d.<l» s 2<8.<p,d.<p>(r(ip)-r) J. J. J. J. N which implies that for t st's t o 0 * J f c / (d/df) <a i0, e±f>-2 fr N (x\>) -r) <d±*, d±<t>>\e 2 (rv W r ) t ' dt' = Kdf.B^e-2***™-***' f . (2.7. la) o Now, l e t the difference between the i n i t i a l v e l o c i t y f i e l d s , as measured by ||v0 fl, approach to zero. Then, we have from (2.7.1a) that at any fixe d t * t Osc2||70||2= c2<di4>fd±<t>>s <a i 0 o ,a i 0 Q > = ||V0o||2 -» 0, (2.7.1b) with C 2 H e " 2 ^ 0, A t=t-t , which leads to ||70|| -» 0 as ||v"0o|| -> 0 and hence the continuity of v e l o c i t y f i e l d w.r.t i t s i n i t i a l data. The uniqueness of the IBVP follows from (2.7.1b) aft e r s e t t i n g ||V0 fl =0 i n (2.7.1b), which establishes the desired r e s u l t s . -Viewing the prediction of motions i n the atmosphere or oceans as an i n i t i a l value problem, we ar r i v e at the notion, o n t h e b a s i s o f t h e o r e m 2 . 3 , t h a t t h e f l o w 0 (x,t;{fiQ) g o v e r n e d b y ( 1 . 2 . 1 ) - (1 .2 .3) i s p e r f e c t l y p r e d i c t a b l e i n t h e s e n s e t h a t i t s i n i t i a l s t a t e 0Q u n i q u e l y d e t e r m i n e s t h e s u b s e q u e n t e v o l u t i o n o f t h e f l o w e v o l v i n g f r o m t h e g i v e n 0 q . I n p r a c t i c e , t h i s p o i n t o f v i e w h a s d i f f i c u l t i e s , a s t h e " p e r f e c t p r e d i c t i o n " m e n t i o n e d a b o v e d e p e n d s on t h e p e r f e c t k n o w l e d g e o f b o u n d a r y c o n d i t i o n s (1 .2 .2) a n d i n i t i a l c o n d i t i o n ( 1 . 2 . 3 ) , w h i c h i s n o t h u m a n l y a t t a i n a b l e . T h i s , t o g e t h e r w i t h t h e f a c t t h a t u n d e r c e r t a i n c o n d i t i o n s ( c f . c o r o l l a r y 2 . 1 ) , i n i t i a l e r r o r s i n B . C . o r I . e . t e n d t o a m p l i f y a s t i m e a d v a n c e s , i m p l i e s t h a t p r e d i c t a b i l i t y p r o b l e m s r e m a i n d e s p i t e t h e u n i q u e n e s s o f I B V P . T a k i n g t h e p o i n t o f v i e w t h a t t h e a t m o s p h e r e o r o c e a n s a c t a s d y n a m i c s y s t e m s v a c i l l a t i n g among t h e p o s s i b l e e q u i l i b r i u m s t a t e s , l i m i t c y c l e s a n d s t r a n g e a t t r a c t o r s , i t a p p e a r s t h a t t h e u n i q u e n e s s p r o b l e m o f t h e e q u i l i b r i u m s t a t e i s more r e l e v a n t t o t h e p r e d i c t i o n p r o b l e m s . T h i s p o i n t o f v i e w may b e t r a c e d b a c k t o L o r e n z ' s w o r k (1963, 1969) . I n c o n n e c t i o n t o c l i m a t e c h a n g e s , i t i s r e f l e c t e d i n t h e r e m a r k t h a t c l i m a t e i t s e l f i s a q u e s t i o n o f d i s t r i b u t i o n among t h e p o s s i b l e e q u i l i b r i u m s t a t e s , a n d c l i m a t e v a r i a t i o n a m a t t e r o f how t h e c h a n g e d b o u n d a r y a l t e r s t h e d i s t r i b u t i o n ( C h a r n e y a n d D e v o r e , 1979) . I t i s c l e a r t h a t f o r s u c h an a p p r o a c h t o b e s u c c e s s f u l , i t i s i m p o r t a n t t o h a v e a n a c c u r a t e c o u n t o f m u l t i p l e e q u i l i b r i a a v a i l a b l e t o s y s t e m s f o r g i v e n 32 conditions. Unfortunately, no general algorithm for accomplishing t h i s task has come into l i g h t . The d i f f i c u l t i e s i n determining whether a system has other equilibrium states other than the known one arise frequently i n pr a c t i c e . The following r e s u l t provides a l i m i t e d solution to the problem. Theorem 2.4 (necessary condition for multiple equilibria) Let 0Q € If the system has an equilibrium state other than 0Q , then r^ty^-r^O. (2.7.2) Proof: (by contradiction) Suppose that the system has an equilibrium state other than 0Q but r (0 )-r <0. (2.7.3) N O Let 0' denote the other equilibrium state, 6 the difference o o f i e l d between 0 and 0', i . e . , <p = 0 - 0 ' . Now, consider two r 0 0 0 0 0 r e a l i z a t i o n s of (1 .2 .1) - (1 .2 .3) , 0(x,t ;0 ) and 0(x,t ;0^), i n i t i a l i z e d from 0q and 0^, respectively, and the difference f i e l d o>(x,t;0o). By the assumption that 0q and 0^  are two d i s t i n c t e q u i l i b r i a of (1.2 .1)-(1.2.2) and the uniqueness of the two r e a l i z a t i o n s (cf. theorem 2.3) , we have WM, = V*0\* V t * t Q . (2.7.4) On the other hand, we apply the energy inequality (2.6.5) over the i n t e r v a l [ t , t Q ] to have Ni l a \V*\oexp{2(rvW-r) (t -tQ)}, V t * t Q 33 < | V 0 | O (by (2.7.3)) contradicting (2.7.4). This completes the proof. B To the extent that the system (1.2.1)-(1.2.3) governs the motions i n the atmosphere or the oceans, the forgoing results indicate that the phenomenon of multiple equilibrium states as observed i n Charney and Devore (1979) and i n many subsequent works (e.g., Kall§n, 1981; P r o e f s c h r i f t , 1989) may not a r i s e from the truncations invoked i n these studies but rather occurs under cert a i n external and i n t e r n a l conditions for which (2.7.2) or some corresponding form of (2.7.2) i s met. We w i l l provide numerical evidence to strengthen t h i s point i n the next chapter. By now, there appears to be l i t t l e doubt on the existence of multiple e q u i l i b r i a at mathematical l e v e l . The remaining concern i s how relevant the parameter values for which multiple e q u i l i b r i a are observed are to the r e a l atmosphere and climate system (Tung & Rosenthal, 1985). For the p r a c t i c a l concerns mentioned at the onset of the section, the condition (2.7.2) with **N(0O) determined from the global analysis ( i . e . from (2.5.1) and (2.5.2)) provides a quick check on the p o s s i b i l i t y of multiple e q u i l i b r i a . It i s important to note that when (2.7.2) as a necessary condition i s s a t i s f i e d a d i r e c t c a l c u l a t i o n i s needed before concluding the existence of multiple e q u i l i b r i a . However, a 34 v i o l a t i o n o f (2.7.2) c l e a r l y r u l e s out such a p o s s i b i l i t y . 2 . 8 A p p l i c a t i o n I I I : i n i t i a l modal vs. nonmodal growth rate The l i m i t a t i o n s o f t h e modal growth r a t e i n a c c o u n t i n g f o r some o b s e r v e d o c e a n i c and a t m o s p h e r i c phenomena have been known f o r some t i m e s . F o r example, t h e growth r a t e s o f mes o s c a l e o c e a n i c e d d i e s from modal a n a l y s i s a r e o f t h e o r d e r o f one y e a r (Schulman, 1967) whereas t h e o b s e r v a t i o n d a t a i n Cre a s e (1962), Swallow (1971), and K o s h l y a k o v and Grachev (1973) s u g g e s t t h a t i t i s o f o r d e r o f a few months. T h i s s i t u a t i o n has l e d t o o p t i m i z a t i o n o f t h e modal growth r a t e e x p r e s s i o n w.r.t. model parameters such as 3 and t o p o g r a p h i c h e i g h t i n o r d e r t o match t h e o b s e r v e d p o p u l a t i o n o f e d d i e s i n t h e oceans (Cf. Robinson and M c w i l l i a m s , 1974). A n o t h e r example i s found i n t h e s t u d y o f c y c l o n e g e n e s i s . I t i s w e l l known t h a t t h e t y p i c a l p e r i o d o f deepening o f o b s e r v e d c y c l o n e s i s between 12 and 48 hours ( c f . Roebber, 1984; Sanders, 1986), c l e a r l y l a r g e r t h a n any found i n modal a n a l y s i s o f b a r o c l i n i c i n s t a b i l i t y , e.g., 133 hours i n V a l d e s and H o s k i n s (1988). F u r t h e r examples may be found i n s t a b i l i t y s t u d i e s o f h y d r o d y n a m i c a l f l o w s v i a i n i t i a l v a l u e p r o b l e m s . I t i s o b s e r v e d t h a t t h e maximum modal growth r a t e i s o f t e n c o n s i d e r a b l y l o w e r t h a n t h e i n s t a n t a n e o u s growth r a t e d u r i n g t h e i n i t i a l p e r i o d , and t h i s happens even when t h e f l o w s u p p o r t s s t r o n g modal i n s t a b i l i t y ( c f . F a r r e l l , 1982; Boyd, 1983; O ' B r i e n , 1990). 35 Despite the increasing evidence for t h i s defect and the suggestion that the neglect of the continuous spectrum i n the modal analysis i s responsible for the flaw, the following questions remain: 1) does the modal growth rate in e v i t a b l y underestimate the growth rate of disturbances to a given flow? 2) i f so, what i s the underlying cause? 3) i s there any systematic procedure to overcome the defect? Our objective of t h i s section i s to provide e x p l i c i t answers to the above questions i n the context of barotropic,. nonlinear and zonally varying flows. We do t h i s by an e x p l i c i t comparison of BEP ( 2 . 5 . 1 ) - ( 2 . 5 . 2 ) a r i s i n g i n the global analysis with the BEP a r i s i n g from the modal analysis. The BEP a r i s i n g i n the modal analysis i s obtained aft e r assuming that the solution to the l i n e a r i z e d version of (2.2.2) - (2.2.4) has the modal form e £,(x), where o- € C and %(x) € W (cf. section 2.2), and introducing i t into ( 2 . 2 . 2 ) and (2.2.3), <rv2£ + £[%;*] = 0, ( 2 . 8 . 1 ) where € s a t i s f i e s B.C.(Ill) or (IV), with £ given by ( 2 . 2 . 5 ) . To make the subsequent discussions precise, we introduce 36 the following d e f i n i t i o n s D e f i n i t i o n 2.4 A solution <f>(x,t;4>Q) to (2.2.2) - (2 .2 .4) i s r e f e r r e d to as a finite amplitude nonmodal disturbance (or simply a disturbance); a solution <p(x,t;<pQ) to the l i n e a r i z e d version of (.2 .2 .2) - (2 .2 .4) as a linear disturbance; a li n e a r <rt disturbance i n the form </> (x, t;%,<r) = £e , with (%,<r) determined from ( 2 . 8 . 1 ) , as a modal disturbance. Furthermore, </>o i s sa i d to be a modal initial perturbation i f <6q = Re (£) or Im(£) with £ € the eigenspace {£} of (2.8.1), s a t i s f y i n g ( 2 . 8 . 1 ) , otherwise as a nonmodal initial perturbation. A growth rate i s said to be modal growth rate i f i t i s obtained as the r e a l part of the eigenvalue of (2.8.1) ( i . e . , Re (<r) ) , otherwise nonmodal growth rate. To setup a stage for comparison of the two BEPs we f i r s t e s t a b l i s h Lemma 2.5 Let * be an equilibrium state of (1.2.1) - (1.2.3) . Then, its modal disturbance e £(x) has the growth rate Re(<r) = p f £ / * ; " r, (2.8.2a) pCC/*; = < £ , ^ > / « ; , B € > , (2.8.2b) where A and £ are defined by (2.4.4) and (2.4.5) . Proof: l e t <• € {£}. Taking the inner product of (2.8.1) with € and adding i t to i t s complex conjugate, we have 37 ( 2 . 8 . 3 ) where £ i s the adjoint o f ! 2 given by ( 2 . 3 . 2 ) . Since £ i s a s e l f - a d j o i n t on S 2 {£[} (cf. lemma 2 . 1 ( i i ) ) , <££,€> i s thus r e a l on {?}. It hence follows from ( 2 . 8 . 3 ) that •* -2He (<r) <£, + <£, (2+ £ * ) € > = 0 , (by self-adjointness of £+£*) which immediately leads to (2.8.2). B Next, we introduce the notion of l i n e a r s t a b i l i t y measure of an equilibrium i n D e f i n i t i o n 2 . 5 The linear stability measure of an equilibrium state ¥, denoted by r (y), is defined as L r C* ; • max p (£;*). ( 2 . 8 . 4 ) It then follows from ( 2 . 8 . 2 ) and ( 2 . 8 . 4 ) that max Re (cr) = r f*; - r, ( 2 . 8 . 5 a ) Xi as opposed to the maximum nonmodal growth rate (cf. ( 2 . 6 . 8 ) ) max <r„C0;0; = r (9) - r, ( 2 . 8 . 5 b ) N N where r N (*) i s defined by ( 2 . 6 . 2 ) . The answer to the question 1 ) raised above l i e s i n the r e l a t i v e s i z e of r (<b) to r (*), which i s s e t t l e d i n =» -2Re (<r) <£ , B€> + 2<£, (4-z3)Z> = 0, (by ( 2 . 3 . 3 ) ) N L 38 Lemma 2 .6 (comparison of stability measures) For an equilibrium state * of (1.2.1) - (1.2. 3), it holds that r (*; s r m . (2 .8 .6) Li N T o p r o v e t h e l e m m a , we n e e d t h e f o l l o w i n g i n e q u a l i t y : Let {x } and {y } be two real finite sequences and y >0 for n n n any n. Then Y x ft y s max(x /y ) . (2 .8 .7) Lt n Lt n n n n n n T o s e e ( 2 . 8 . 7 ) , i t s u f f i c e s t o n o t e t h a t J x ft y = J (x y )/y ft y Lt n Lt n Lt n n n (L. n n n n n w h i c h i m m e d i a t e l y g i v e s ( 2 . 8 . 7 ) . Now, we t u r n t o Proof: ( i ) L e t £ e { £ } a n d b e w r i t t e n £ = E x p r e s s i n g p(%;V) i n t e r m s o f w e h a v e P(V*> = {l <K^n>}{l <Sn'**A- ( 2 - 8 . 8 ) V-n= r , i ' Vr>= r , i ' A s a n e i g e n s t r u c t u r e , £ ( x)*0 f o r x e P.. F u r t h e r m o r e , l e t £(x) * c o n s t a n t ( s i n c e k i n e t i c e n e r g y o f a n o r m a l mode w i t h c o n s t a n t s t r u c t u r e i s z e r o , t h u s c o r r e s p o n d i n g t o a c a s e o f l i t t l e i n t e r e s t p h y s i c a l l y ) . T h e l a t t e r c o n s t r a i n t i m p l i e s t w o c a s e s : ( i ) b o t h a n d £ a r e n o t c o n s t a n t a n d ( i i ) one o f t h e m i s c o n s t a n t . F o r c a s e ( i ) , e i t h e r t e r m i n t h e 39 denominator of (2.8.8) i s s t r i c t l y p o s i t i v e (cf. (2.4.7)) and hence (2.8.7) i s applicable to (2.8.8) , which y i e l d s p f€/tf; =s max j<€ >]•/(<? r « C >]• s max \<4>,M>\i\<<bi %<P>\ = r (¥) . (2.8.9) <6elH I J I i N For case ( i i ) , w.l.o.g., suppose that %r*0 and ^=0. Then, only the term for 6^ appears i n (2 .8 .8) . Thus, we have (2.8.9) without appealing to (2 .8 .7) . To t h i s end, taking the maximum on the two side of (2.8.9) over {£} proves the a s s e r t i o n . a The fundamental property (2.8.6) i n conjunction with (2.8.5) y i e l d s an e x p l i c i t affirmative answer to question 1 ) raised above. To see t h i s , we note that for any ( c r , £ ) determined from (2 .8.1) , i t holds that fie(«r) = p(€/*; - r, (by (2.8,2)) s rh(9) -r, (by (2.8.4)) s r (*) - r , (by (2.8.6) ) N = X (9) - r, (by lemma 2.3) N = cr (by theorem 2.2) N which allows for the conclusion: Theorem 2.5 (initial nonmodal and modal growth rate) Let * be an equilibrium state of (1.2.1) - (1.2.3) . Then, it has at 40 least one nonmodal initial perturbation which has a growth rate not less than the maximum modal growth rate. Moreover, the interpretation of the property (2.8.9) reveals the answer to question 2) : the ratio of energy conversion between a basic flow and a disturbance to disturbance energy in the eigenspace {%} e S is always less than or equal to the extreme value of the same ratio in the space of all real kinematically admissible disturbances (i.e., in M) . The global analysis ( i . e . , lemma 2.1-2.3) together with the comparison of the s t a b i l i t y measures (2.8.6) constitutes a general procedure for circumventing the underestimation o f growth rate of disturbances by the modal analysis, thus cl o s i n g question 3 ) above. Further applications of the global analysis and the property (2.8.6) can be found, for example, i n explaining why there exists transient growth of disturbances in subcritical flows, or equivalently why the modal analysis inevitably fail to predict the temporal amplification phenomenon in stable flows. To see t h i s , consider a s u b c r i t i c a l flow ^ o f (1.2.1)-(1.2.3) . It i s evident from (2.8.5a) that r (*)-r<0 for the *. However, by the comparison of s t a b i l i t y measure (2.8.6), i t i s s t i l l possible to have r (*)-r>0 i f the * i s N o u t s i d e o f t h e MGS r e g i m e , a n d h e n c e t o h a v e a n i n i t i a l g r o w i n g n o n m o d a l p e r t u r b a t i o n . On t h e o t h e r h a n d , i f r N ( * ) - r < 0 , t h e n t h e * h a s n o i n i t i a l g r o w i n g n o n m o d a l d i s t u r b a n c e . M o r e o v e r , b y t h e c o m p a r i s o n o f t h e s t a b i l i t y m e a s u r e s , i t h a s no g r o w i n g m o d a l d i s t u r b a n c e e i t h e r . To s u m m a r i z e : Theorem 2 . 6 (initially growing nonmodal perturbation in subcritical flows) If an equilibrium state * of (1.2.1) - (1.2.3) has no initially growing nonmodal perturbation, then it does not have any growing modal perturbation; the converse is false. T h e p h e n o m e n o n o f t r a n s i e n t g r o w t h o f d i s t u r b a n c e s i n f l o w s s u p p o r t i n g no g r o w i n g m o d a l d i s t u r b a n c e s was o r i g i n a l l y r e p o r t e d b y O r r (1907) i n h i s w o r k o n p l a n e C o u e t t e f l o w . I t h a s b e e n d o c u m e n t e d i n s e v e r a l r e c e n t w o r k s on i n i t i a l v a l u e p r o b l e m s o f t h e C h a r n e y a n d E a d y m o d e l s ( F a r r e l l , 1985) , o f b a r o t r o p i c c o n s t a n t s h e a r f l o w s on j 3 - p l a n e ( B o y d , 1 9 8 3 ; F a r r e l l , 1 9 8 7 ) , a n d o f t h e G r e e n m o d e l ( O ' B r i e n , 1 9 9 0 ) . I t i s f r e q u e n t l y f o u n d i n t h e s e s t u d i e s t h a t f o r a r e a l i s t i c v a l u e o f t h e Ekman d a m p i n g t h e m o d e l f l o w s h a v e d i s t u r b a n c e s on t h e s y n o p t i c c y c l o n e s p a t i a l s c a l e s w h i c h e x h i b i t l a r g e t r a n s i e n t g r o w t h d e s p i t e t h e a b s e n c e o f , o r p r e s e n c e o f o n l y w e a k l y u n s t a b l e , m o d a l p e r t u r b a t i o n s . T h i s o b s e r v a t i o n h a s l e d t o t h e a r g u m e n t t h a t t h e m o d a l i n s t a b i l i t y c a n n o t i n g e n e r a l s e r v e t o e x p l a i n t h e o c c u r r e n c e o f c y c l o n e s c a l e d i s t u r b a n c e s i n m i d l a t i t u d e b u t r a t h e r t h e y a r i s e p e r h a p s p r e d o m i n a n t l y f r o m t h e r e l e a s e o f mean f l o w p o t e n t i a l e n e r g y b y p r o p e r l y c o n f i g u r e d i n i t i a l n o n m o d a l p e r t u r b a t i o n s ( F a r r e l l , 1 9 8 5 ) . G i v e n t h e p o s s i b i l i t y t h a t c y c l o n e s c a l e d i s t u r b a n c e s a r e i n i t i a t e d b y f a i r l y l a r g e a m p l i t u d e p e r t u r b a t i o n s as e m p h a s i z e d i n s y n o p t i c e x p e r i e n c e s ( c f . P e t t e r s s e n , 1 9 5 5 ; P a l m e n a n d N e w t o n , 1 9 6 9 ) , we b e l i e v e t h a t a s u c c e s s f u l i n c l u s i o n o f b a r o c l i n i c i t y i n t o t h e p r e s e n t f o r m a l i s m may r e s u l t i n a f r a m e w o r k w h i c h a l l o w s f o r f i n i t e a m p l i t u d e p e r t u r b a t i o n s a n d h e n c e more s u i t a b l e f o r t h e s t u d y o f c y c l o n e g e n e s i s . 2.9 Concluding remarks We h a v e p e r f o r m e d a g l o b a l a n a l y s i s o f t h e s y s t e m ( 1 . 2 . 1 ) - ( 1 . 2 . 3 ) , w i t h r e s u l t s a p p l i e d t o a n u m b e r o f G F D p r o b l e m s . When a p p l i e d t o t h e g l o b a l s t a b i l i t y , t h e a n a l y s i s h a s y i e l d e d a s y s t e m a t i c p r o c e d u r e f o r c h a r a c t e r i z i n g s t a b i l i t y o f f l o w s o n t h e b a s i s o f w h e t h e r t h e r e e x i s t s an i n i t i a l l y g r o w i n g p e r t u r b a t i o n r e g a r d l e s s o f i t s a m p l i t u d e a n d s p a t i a l s t r u c t u r e , a s o p p o s e d t o t h e m o d a l a n a l y s i s w h i c h c h a r a c t e r i z e s s t a b i l i t y a c c o r d i n g t o w h e t h e r t h e r e e x i s t s an e x p o n e n t i a l l y g r o w i n g p e r t u r b a t i o n . T h e same p r o c e d u r e c a n a l s o b e u s e d t o i d e n t i f y an o p t i m a l i n i t i a l n o n m o d a l p e r t u r b a t i o n t o a g i v e n f l o w . When a p p l i e d t o t h e s t u d y o f r e l a t i o n o f i n i t i a l m o d a l t o n o n m o d a l g r o w t h r a t e s , t h e g l o b a l a n a l y s i s , t o g e t h e r w i t h t h e f u n d a m e n t a l c o m p a r i s o n p r o p e r t y ( c f . , lemma 2 . 6 ) , h a s u n c o v e r e d t h e c a u s e u n d e r l y i n g t h e l i m i t a t i o n o f m o d a l g r o w t h r a t e a s a n i n d i c a t o r o f p o t e n t i a l d e v e l o p m e n t o f d i s t u r b a n c e s , a n d t h e c a u s e u n d e r l y i n g p h e n o m e n a s u c h a s t h e w e l l known t r a n s i e n t g r o w t h i n s u b c r i t i c a l f l o w s a n d e x p l o s i v e d e v e l o p m e n t i n w e a k l y u n s t a b l e f l o w s . T h e g l o b a l a n a l y s i s h a s a l s o f o u n d a p p l i c a t i o n i n t h e s t u d y o f m u l t i p l e e q u i l i b r i u m s t a t e s , r e s u l t i n g i n a n e c e s s a r y c o n d i t i o n f o r ( o r a s u f f i c i e n t c o n d i t i o n f o r r u l i n g o u t ) t h e e x i s t e n c e o f s u c h p h e n o m e n o n . I t i s e x p e c t e d t h a t t h e s e r e s u l t s h o l d f o r a l l f l o w s g o v e r n e d b y ( 1 . 2 . 1 ) - ( 1 . 2 . 3 ) . I n t h e f o l l o w i n g two c h a p t e r s , we w i l l p r e s e n t a n a l y s e s w h i c h a r e p a r a l l e l t o , a n d e x t e n d , t h o s e i n t h i s c h a p t e r , b u t a r e p e r f o r m e d i n a f i n i t e d i m e n s i o n a l f u n c t i o n s p a c e (or e q u i v a l e n t l y i n RM) a n d f o r a more s p e c i f i c p h y s i c a l s y s t e m . M o r e o v e r , we w i l l c o m p l e m e n t t h e a n a l y s e s t h e r e w i t h n u m e r i c a l e x p e r i m e n t s . 44 Appendix 2A A v a r i a t i o n a l a p p r o x i m a t i o n m e t h o d f o r t h e o p t i m a l v a l u e o f g e n e r a l i z e d R a y l e i g h q u o t i e n t p I n t h i s a p p e n d i x , we p r e s e n t a R a y l e i g h - R i t z p r o c e d u r e A ( c f . R e d d y , 1986) f o r o b t a i n i n g t h e a p p r o x i m a t e s o l u t i o n 6 t o t h e o p t i m a l p r o b l e m max p(<p;\p) f o r 0e(H. L e t u , u u 1 2 M d e n o t e M l i n e a r l y i n d e p e n d e n t f u n c t i o n s i n H a n d w r i t e i = cjui ( 2 A . 1 ) • w h e r e t h e s u m m a t i o n c o n v e n t i o n i s a p p l i e d t o t h e r e p e a t e d i n d i c e s , w i t h (i=l,2,.. M) b e i n g c o n s t a n t s t o be A d e t e r m i n e d . E v a l u a t i n g p(<p;\ft) a t </>, we h a v e A Ti(ClfC2, . . .,CH)mp(A)=C(C1, C2, CM)/D(Cif . . , C J , ( 2 A . 2 ) C(c , c , ...,c)m <u.,du.>c.c.r ( 2 A . 3 ) D(c, c , . . , c )= <u . , 8 u .>c .c ., ( 2 A . 4 ) 1 Z M -L J X J w h e r e <•> d e n o t e s t h e i n n e r p r o d u c t i n H ( c f . ( 2 . 2 . 1 1 ) ) , w i t h t h e o p e r a t o r s d a n d SS g i v e n b y ( ( 2 . 4 . 4 ) a n d ( 2 . 4 . 5 ) , r e s p e c t i v e l y . T h e o p t i m i z i n g c o n s t a n t s c=(c ,c , ...,c^) i n ( 2 A . 1 ) t h e n m u s t s a t i s f y 8Tl/dc^=0 (i=l,2,..M), w h i c h l e a d s t o dC/ac±=Ti a D / a c i , i=i,2,...M ( 2 A . 5 ) o r i n t e r m s o f m a t r i x e i g e n v a l u e f o r m [<uifdUj>}c = u [ < u i , S u : , > ] c , ( 2 A . 6 ) w h e r e u B 11(C). T h e l a r g e s t e i g e n v a l u e o f ( 2 A . 6 ) i s t h e n t h e R a y l e i g h - R i t z a p p r o x i m a t i o n t o t h e o p t i m a l v a l u e o f p(6;ifi) i n IH, a n d i t s c o r r e s p o n d i n g e i g e n v e c t o r c w i l l y i e l d an o p t i m a l A d i s t u r b a n c e 6 v i a ( 2 A . 1 ) . 4 5 CHAPTER 3 GLOBAL ANALYSIS II: FINITE DIMENSIONAL SYSTEM, with application to: global s t a b i l i t y and optimal perturbation 3.1 Introduction No g e n e r a l t h e o r y o f g l o b a l a n a l y s i s f o r any h y d r o d y n a m i c a l s y s t e m i n f i n i t e d i m e n s i o n a l f u n c t i o n s p a c e ( o r e q u i v a l e n t l y i n RM) h a s b e e n k n o w n , e x c e p t f o r a few a t t e m p t s made t o l o w o r d e r s y s t e m s ( c f . V i c k r o y a n d D u t t o n , 1 9 7 9 ; K a l l § n a n d W i i n - N i e l s e n , 1 9 8 0 ) . T h e o b j e c t i v e o f t h i s c h a p t e r i s t o p e r f o r m s u c h an a n a l y s i s . T o f a c i l i t a t e t h e s u b s e q u e n t n u m e r i c a l e x p e r i m e n t s , t h e a n a l y s i s i s d o n e f o r a s y s t e m d e f i n e d b y ( 1 . 2 . 1 ) , ( 1 . 2 . 2 a ) ( p e r i o d i c c h a n n e l g e o m e t r y ) a n d ( 1 . 2 . 3 ) w i t h t h e e x t e r n a l f o r c i n g 0* a c t i n g a s u n i f o r m z o n a l momentum s o u r c e , i . e . , 0* = -Uy ( 3 . 1 . 1 ) w h e r e C7* i s a p a r a m e t e r d e s c r i b i n g t h e f o r c i n g s t r e n g t h . We n o t e t h a t t h e r e s t r i c t i o n o f 0* t o t h e f o r m ( 3 . 1 . 1 ) i s n o t e s s e n t i a l t o t h e a n a l y s i s . B e f o r e c o n c l u d i n g t h e i n t r o d u c t i o n , l e t u s make a s i m p l e p h y s i c a l o b s e r v a t i o n ( e . g . , G r a v e l , 1988) t h a t t h e s t r e a m f u n c t i o n 0 c a n b e d e c o m p o s e d i n t o two p a r t s : one a r i s i n g d i r e c t l y f r o m t h e u n i f o r m z o n a l momentum f o r c i n g 4 6 ( 3 . 1 . 1 ) a n d o n e r e p r e s e n t i n g t h e d e v i a t i o n , i . e . , >l>(x,y, t)=-U*y + *(x,y, t) . ( 3 . 1 . 2 ) T h u s , i t f o l l o w s f r o m i n t r o d u c i n g ( 3 . 1 . 2 ) i n t o ( 1 . 2 . 1 ) , ( 1 . 2 . 2 a ) a n d ( 1 . 2 . 3 ) t h a t (8/at)V29 + J(-Uy+*, V29+py+ (h/H) f ) = -zM2<t>, ( 3 . 1 . 3 ) d<b/dx=0, at y=0,d and Osxsl, ( 3 . 1 . 4 a ) 9(l,y,t)=9(0,y,t), at O s y s d , ( 3 . 1 . 4 b ) *(x,y, 0;=*o (x,y), ( 3 . 1 . 5 ) . w h e r e QQ(x,y) i s s u c h t h a t \p(x,y, 0) =-U* y+$o (x, y) . A s o l u t i o n t o t h e s y s t e m ( 3 . 1 . 3 ) - ( 3 . 1 . 5 ) t h u s d e t e r m i n e s t h e r e s p o n s e o f t h e o r i g i n a l s y s t e m t o t h e e x t e r n a l f o r c i n g ( 3 . 1 . 1 ) . F u r t h e r m o r e , s i n c e t h e u n i f o r m z o n a l p a r t -U* y o f \JJ ( c f . ( 3 . 1 . 2 ) i s t i m e i n d e p e n d e n t , t h e s t u d y o f t h e s t a b i l i t y o f 0 i s t h e n e f f e c t i v e l y r e d u c e d t o t h a t f o r T h e c h a p t e r i s o r g a n i z e d a s f o l l o w s . T h e s y s t e m ( 3 . 1 . 3 ) - ( 3 . 1 . 5 ) i s c a s t i n t o a s p e c t r a l f o r m i n § 3 . 2 . A g e n e r a l p r o c e d u r e f o r t h e g l o b a l a n a l y s i s i n RM i s p r e s e n t e d i n § 3 . 3 , w i t h r e s u l t s f r o m i t s a p p l i c a t i o n t o g l o b a l s t a b i l i t y a n d s e a r c h f o r i n i t i a l g r o w i n g p e r t u r b a t i o n s a n d o p t i m a l i n i t i a l n o n m o d a l p e r t u r b a t i o n s p r e s e n t e d i n § 3 . 4 . N u m e r i c a l i l l u s t r a t i o n s a r e made u s i n g a s e t o f e q u i l i b r i a d e p e n d i n g o n r i n § 3 . 5 , w i t h a c o n c l u d i n g r e m a r k i n § 3 . 6 . 47 3.2 Governing equations i n RM By v i r t u e of the channel boundary condition (3.1.4), we, following M i t c h e l l and Derome (1983) , introduce a set of orthonormal functions G s {G^}, with G^ given by G± • G{m,n) = ' 21/2 cos (nny/d) if lsisl 2sin (2nmx/l) sin (nny/d), if I+l*isI+I*J (3.2.1) 2cos (2nmx/l) sin (nny/d), if I+I*J+lsi*I+2l*J m=l,2, . . . ,J; n=l,2, . . ., I; M=I+2I*J. where the index i for G\ increases based on the columnwise convention for the two-dimensional array G(m,n). Taking (3.2.1) as a basis for IH (cf. § 2.2), we expand a l l dependent variables i n the form M (9,h) = £ (^irh±)G± (3.2.2) i and i n s e r t them into (3.1.3) to obtain the spectral version of (3.1.3)-(3.1.5) below (d/dt)t = F &;*), (3.2.3) t(0) = (3.2.4) where and fi are column vectors i n RM, e.g., t=[i ]T 1 ' 2 M and F : RM -» RM i s an Af-dimensional nonlinear vector f i e l d , with the i t h component given by 48 + 1 I c i ^ j [ ^ o / w v 4 $ j c ] fi I b.A.-ral*. y ( 3 . 2 . 5 ) + j w i t h a2±, bj^.., c±jk g i v e n b y s a (m, n) s ' (nn/d) , i f l s i < i (2nm/l)2 + (nn/d)2, If I+lsl*I+I*J ( 3 . 2 . 6 ) r27rnj/i;2+ frcn/d; 2 , if I+l+I*Jsi*I+2l*J c.jk , ^ . J f G . . , ^ = c J J f e l = = - c j i J k / , ( 3 . 2 . 7 ) b i j S < G ± d G j / 8 x > = ~hj±- ( 3 . 2 . 8 ) F i n a l l y , a i n ( 3 . 2 . 3 ) r e p r e s e n t s a n y m o d e l p a r a m e t e r whose e f f e c t o n t h e s t a b i l i t y o f e q u i l i b r i u m s t a t e s o f ( 3 . 2 . 3 ) i s u n d e r c o n c e r n . 3.3 Global analysis in RM W h i l e t h e d e t a i l s o f t h e g l o b a l a n a l y s i s i n RM i s n o t i c e a b l y d i f f e r e n t f r o m i t s c o u n t e r p a r t i n IH, t h e b a s i c i d e a s a r e e s s e n t i a l l y t h e s a m e . I n l i g h t o f t h i s , d e s c r i p t i o n o f t h e m o t i v a t i o n s b e h i n d t h e s u b s e q u e n t m a t h e m a t i c a l m a n i p u l a t i o n s i s c o n s i d e r a b l y s i m p l i f i e d o r n e g l e c t e d ( see c h a p t e r 2 f o r d e t a i l s ) . 3.3.1 Disturbance equation L e t t(t;tQ) d e n o t e t h e s o l u t i o n o f ( 3 . 2 . 3 ) i n i t i a l i z e d 49 from tQ. Consider another r e a l i z a t i o n <f(t;^o+<lo) of ( 3 . 2 . 3 ) made under the same condition as those for t(t;tQ) except i t s I.B.: tQ+$Q. As i n the section 2 . 2 , the s t a b i l i t y problem for t(t;tQ) can be investigated by considering the temporal evolution of the difference t*t{t;tQ+$o) - *<t;S o), ( 3 . 3 . 1 ) which i s governed by the i n i t i a l value problem (d/dt)% = t($,4,oi), ( 3 . 3 . 2 a ) 1(tQ) = $ 0 , ( 3 . 3 . 2 b ) where f : IRM -» R M i s an M-dimensional vector f i e l d a r i s i n g from the expansion of the nonlinear vector f i e l d F a s a Taylor series around the motion t, defined b y f($,4,cc) m T^tftf) + a/2)Ffttf\$\$) + 0(\~$\2), ( 3 . 3 . 3 a ) F|(^|^) e aw($+6xt/tfa)/881\s = Q , ( 3 . 3 . 3 b ) F ^(^|2) • 82T ($+8 1+8 1;%,a) /88 88 \ , ( 3 . 3 . 3 c ) 1 2 A d i r e c t evaluation of ( 3 . 3 . 3 b ) and ( 3 . 3 . 3 c ) y i e l d s T$($\t) = *$, ( 3 . 3 . 4 a ) F f t < * l $ | £ > = ZBffi' ( 3 . 3 . 4 b ) or i n component form {F|(^|^)} i = I A..*., ' ( 3 . 3 . 4 c ) j OfctflJtfni - 2l I B l j k * j * k ' ( 3 . 3 . 4 d ) 3 k 50 w i t h A s [ A . . ] , A. . s (l/a2.)(u*b. .{-a2.) + pjb. . - ra2.8 . . xj i \ ij J i J J- 13 + Y \c .$,(-a2.)+c . .,(f h,/H L |_ lkj k j ijkx 0 k k Note t h a t t h e l i n e a r o p e r a t o r A : RM-> R M and b i l i n e a r o p e r a t o r B : R M x R M -» R M a r e t h e c o u n t e r p a r t s o f £ and # ( c f . 2.2.5) and (2.2.6)) i n R M , r e s p e c t i v e l y . A l s o , note t h a t t h e h i g h e r o r d e r terms i n (3.3.3) v a n i s h i d e n t i c a l l y s i n c e F i s a p o l y n o m i a l o f degree 2 i n I ( c f . ( 3 . 2 . 5 ) ) . To t h i s end, we have a c l o s e d form f o r f f.= I Aij*j + I I Bijk*j*k' ( 3 ' 3 ' 5 a ) j J k or £ = &$ + B$$. (3.3.5b) ]}' (3.3.4e) (3.3.4f) 3.3.2 Symmetrized energy equation The n e x t s t e p i s t o put t h e d i s t u r b a n c e energy e q u a t i o n i n t o a form such t h a t i t s t i m e tendency can be e x p r e s s e d i n terms o f a s e l f - a d j o i n t o p e r a t o r i n R M and hence i n terms o f a bounded f u n c t i o n a l . To a c c o m p l i s h t h i s , we f i r s t s t a t e a r e s u l t from m a t r i x a l g e b r a 51 Lemma 3 . 1 Let P = [P±j] e R M X M be a symmetric matrix and Q = CC^ j ] e R M X M be a skew-symmetric matrix. Then PQ - I I P ± j Q ± j = 0. ( 3 . 3 . 6 ) i 3 W i t h (3.3.5), (3.3.2) can be w r i t t e n i n component form (d/dt)4>± - [ A±.<p. + [ [ B . ^ . ^ . (3.3.7) j 7 * 2 1 Now, we m u l t i p l y (3.3.7) w i t h a±4>£ and sum o v e r i t o have ( d / d t ; { I < l / 2 ) a > J } - [ [ ^ A . ^ + £ [ £ a2.B..k 0.4,.^. i i j i j k (3.3.8) Note t h a t t h e b i l i n e a r o p e r a t o r a ,B. c o r r e s p o n d s t o a skew-symmetric m a t r i x w i t h r e s p e c t t o i n d i c e s i, j by v i r t u e o f (3.2.7) and (3.3.4f) whereas c>.<>.<6. i s o b v i o u s l y symmetric 1 3 K w.r.t i,j. I t t h u s f o l l o w s from lemma 3.1 t h a t t h e second summation i n (3.3.8) v a n i s h e s e x a c t l y . S i m i l a r l y , t h e symmetric p r o p e r t i e s (3.2.7) and (3.2.8) demand t h a t t h e terms i n v o k i n g b. . and c . ., i n t h e l i n e a r o p e r a t o r a2.A . , ( c f . ^ i j ijk ^ i ij (3.3.4e) make z e r o c o n t r i b u t i o n t o t h e f i r s t summation i n (3. 3 . 8 ) . W i t h t h e s e o b s e r v a t i o n s , (3.3.8) i s re d u c e d t o ( i n terms o f i n n e r p r o d u c t on RM) (d/dt) (1/2)<%,V%> =<<J,A#>-r<$,D#>, (3.3.9) A . [A..], A.. S I c.kj$k(-a2.), (3.3.10) k 52 D = [D±j] = DT, D±j s a\ 8 ± j . (3.3.11) It i s clear from (3.3.10) that (3.3.9) has not yet achieved the desired form due to the non self-adjointness of the operator A. To remove the non s e l f - a d j o i n t constituent in A, we observe that Lemma 3.2 Let P € RM X M. Then, there exists a decomposition P = P + + P ~ , with T T p + = p • (1/2) ( T + p T ; , p " =-p"=- (i/2) ( P - P T ; , such that <%,V%> = <$,l?+'$>, V % e RM. _ T _ The lemma follows from lemma 3.1 after noting P = - P . Applying t h i s observation to (3.3.9), we obtain (d/dt) (1/2)<%,T>%> =<^,A +£> - r<%,V%>, (3.3.12a) or i n terms of the Generalized Raleigh Quotient p(~$;t) (d/dt) (1/2)<%,*>%>= <$,D$> (p$;t) - r), (3.3.12b) where A + i s given by A + s (1/2) (kT+ A), (3.3.13) p($;t)'m <$,k+%>/<$,$$>. (3.3.14) Remark 3.3.1 It follows from (3.3.11) and (3.3.13) that the operators A and D are s e l f - a d j o i n t i n IR , i . e . , (i) A +=(A +;*, (3.3.15) ( i i ) D = D*and D is positive definite, (3.3.16) 5 3 w h e r e ( )* d e n o t e s t h e a d j o i n t o f ( ) . T h e a b o v e two p r o p e r t i e s a r e t h e r e m i n i s c e n t o f t h o s e f o r t h e d i f f e r e n t i a l o p e r a t o r s A a n d SB i n W ( c f . ( 2 . 4 . 6 ) a n d ( 2 . 4 . 7 ) ) 3 . 3 . 3 Generalized Rayleigh P r i n c i p l e A s i n t h e c a s e o f t h e c o n t i n u u m s y s t e m , t h e c r u c i a l s t e p s i n t h e g l o b a l a n a l y s i s a r e t o i n t r o d u c e a b o u n d e d f u n c t i o n a l a n d t o c o n s t r u c t a n a l g o r i t h m f o r o p t i m i z i n g t h e f u n c t i o n a l . A s s u g g e s t e d b y ( 3 . 3 . 1 2 b ) , we c o n s i d e r t h e e x t r e m e v a l u e s o f p ( c f . ( 3 . 3 . 1 4 ) ) i n R M . F o r t h i s , i t i s i m p o r t a n t t o r e c o g n i z e t h a t p($;t) • <%,2L"$>/<%,!>%> i n ( 3 . 3 . 1 2 b ) w o u l d r e d u c e t o t h e s t a n d a r d R a y l e i g h q u o t i e n t i f D was a u n i t o p e r a t o r I o f o r d e r M, a n d t h a t i n t h a t c a s e p w o u l d b e b o u n d e d a b o v e b y t h e l a r g e s t e i g e n v a l u e o f A + a n d b e l o w b y t h e s m a l l e s t a c c o r d i n g t o R a y l e i g h p r i n c i p l e ( c f . K r e y s z i g , 1978). M o t i v a t e d b y t h i s o b s e r v a t i o n , we s h a l l show b e l o w t h a t g i v e n p d e f i n e d b y ( 3 . 3 . 1 4 ) w i t h A + a n d D s a t i s f y i n g t h e p r o p e r t i e s ( 3 . 3 . 1 5 ) a n d ( 3 . 3 . 1 6 ) , i t i s p o s s i b l e t o g e n e r a l i z e t h e s t a n d a r d R a y l e i g h p r i n c i p l e s u c h t h a t p i s b o u n d e d i n DRM a n d t h e d i s t u r b a n c e $ i n IRM r e a l i z i n g t h e e x t r e m e v a l u e s o f p c a n b e o b t a i n e d s y s t e m a t i c a l l y . T h i s i s made p o s s i b l e b y e s t a b l i s h i n g lemma 3 . 3 ( see A p p e n d i x 3 A ) . Lemma 3.3 {Algebraic properties of the {s^}) Let A + and D be given by (3.3.13) and (3.3.11), respectively. 54 Then, the eigensystem A +s=ADs, (3.3.17) has the following properties (i) all X± are real, (3.3.18) (ii) (3.3.11) has M linearly independent eigenvectors s^, (iii) let S and A be given by S = [ s . s . . . . , s 7 , (3.3.19) 1 z M A = diag[X.], with A s A < X , (3.3.20) ^ 2 1 2 M then S TA + S = A , (3.3.21a) STDS = I, (3.3.21b) where I is a unit matrix of order M. As an a s i d e , we observe t h e s t r i k i n g resemblance o f t h e m a t r i x e i g e n v a l u e p r o blem (MEP) (3.3.17) t o t h e BEP ( 2 . 5 . 1 ) - ( 2 . 5 . 2 ) . Based on lemma 3.3., we can show t h a t Lemma 3.4 (Generalized Rayleigh principle) Let p($;$) be given by (3.3.14). Then: (i) X£ p($;$) s A , v'$ € IRM, (3.3.22) 1 M (ii) min p($;t) = p(sif4) = Xitf), (3.3.23) 4>eJR (Hi) max p($;$) = p(a:t) = \&)m r tf), (3.3.24) M M N 0 € R M where s ,s X and X are determined from (3.3.11). 1 M 1 M 5 5 P r o o f : We use t h e n o n s i g u l a r i t y o f S t o d e f i n e a l i n e a r M M mapping S: R R 1 — S<^ . I t t h u s f o l l o w s from (3.3,.20) and (3.3.21) t h a t i n terms of % and hence t h a t V % € RM p($;$) - X = | [ ( V V ^ A J ^ A " ° ' ( b y < 3- 3- 2 0>) i i w h i c h p r o v e s t h a t p i s bounded above by X^. A s i m i l a r argument e s t a b l i s h e s t h e l e f t s i d e o f (3.3.22). U s i n g t h e f a c t t h a t (i=l,Af) s a t i s f y (3.3.17) and t h a t X^ are r e a l ( c f . ( 3 . 3 . 1 8 ) ) , we e v a l u a t e p($;f) a t s and s t o o b t a i n M 1 (3.3.23) and (3.3.24), which completes t h e g e n e r a l i z a t i o n ^ I n summary, t h e g l o b a l a n a l y s i s i n RM f o r t h e M- d i m e n s i o n a l system (3.2.3) i n c l u d e s : 1) symmetrized d i s t u r b a n c e energy e q u a t i o n (3.3.12); 2) t h e e i g e n s y s t e m (3.3.17); and 3) t h e g e n e r a l i z e d R a y l e i g h p r i n c i p l e . 3.4 Application to : global s t a b i l i t y and optimal perturbation By arguments p a r a l l e l t o t h o s e i n § 2.6, we o b t a i n t h e f o l l o w i n g r e s u l t s w i t h t h e p r o o f o m i t t e d (see § 2.6 f o r p a r a l l e l i s m ) 56 Theorem 3.1 (monotonic, global s t a b i l i t y ; Let r^ (t) denote the finite stability measure of <f and be defined by ry(t) m max p($,4) =\ (3.4.1) N M with X given by (3.3.24). Then, an equilibrium state $ of M (3.2.3) is MGS iff r ($) - r s 0. (3.4.2) N C o r o l l a r y 3.1 There exists at least one initially growing disturbance to an equilibrium state ¥* of (3.2.3) if and only if r tf) -r>0 (3.4.3) N Theorem 3.2 (optimal initial perturbations) Let the growth rate <r ($,4) of a disturbance $ to a given flow t be given by N <r (%,4) • (1/2) (d/dt) In (<$,H%>) . (3.4.4) N Then, among all the possible initial perturbations $o e IRM to a given equilibrium state is the optimal one in the sense that (i) the growth rate of $(t;s ) at t=t is the largest in RM M 0 <r^(SH,4) * V 2o€ (RM, (3.4.5) (ii) the value cr (s ,4) is the least upper bound on the N M growth rate of all %(t,4Q) V $ Q e RM for t s t . Further properties of the set {s^} determined from (3.3.17) as nonmodal i n i t i a l perturbations are given in Appendix 3B. 57 3.5 Numerical illustrations The o b j e c t i v e o f t h i s s e c t i o n i s t o i l l u s t r a t e some o f c o n c e p t s , p u t f o r w a r d here and i n c h a p t e r 2, r e g a r d i n g g l o b a l s t a b i l i t y , i n i t i a l l y g r o wing p e r t u r b a t i o n s and o p t i m a l p e r t u r b a t i o n s by a p p l y i n g them t o p l a n e t a r y s c a l e a t m o s p h e r i c f l o w s . 3.5.1 Set-up of the numerical experiments The geometry o f t h e f l o w f i e l d i s t h e same as i n F y f e and Derome (1987), i . e . , t h e 0-plane r e - e n t r y c h a n n e l c e n t e r e d a round t h e m i d l a t i t u d e , c h a r a c t e r i z e d by 1 = 2nRcos (6Q), d= 4xl06m, H — 1.x 10Am, (3.5.1a) f Q = 2a sin(eo), p - 2Qcos(eo)/R, (3.5.1b) where Q, R and e a r e t h e r o t a t i o n a l r a t e o f t h e e a r t h , t h e o r a d i u s o f t h e e a r t h and t h e r e f e r e n c e l a t i t u d e , r e s p e c t i v e l y , t a k i n g t h e v a l u e s o f 7.292xl0" 5/s, 6.37xl0 6m and 45°N, r e s p e c t i v e l y . The l o w e r boundary o f t h e geometry i s assumed t o have a s i n u s o i d a l form h (x,y) =h^2sin (2nmx/l) sin (nny/d) , (3.5.2) w i t h jn = 1, n = 1 and h^ = 500 m. The p h y s i c a l p arameters o f t h e system (3.2.3) a r e t h e u n i f o r m z o n a l momentum f o r c i n g s t r e n g t h C7* and t h e Ekman damping c o e f f i c i e n t r . F o r i l l u s t r a t i o n p u r p o s e , we s e t CJ* = 22 m/s based on t h e 58 p r e v i o u s s t u d i e s ( e . g . , T u n g a n d R o s e n t h a l , 1 9 8 5 ) , a n d t a k e r a s t h e c o n t r o l l i n g p a r a m e t e r , i . e . , s e t a = r ( c f . s e c t i o n 3 . 2 ) , w i t h r r a n g i n g f r o m 1/(1 day) to 1/(35 days). T h e n u m e r i c a l e x p e r i m e n t s a r e p e r f o r m e d v i a t h e i n i t i a l v a l u e p r o b l e m s d e f i n e d b y t h e s y s t e m ( 3 . 2 . 3 ) - ( 3 . 2 . 4 ) . T h e s y s t e m i s s o l v e d n u m e r i c a l l y w i t h t h e n o n l i n e a r t e r m s c a l c u l a t e d v i a F F T a n d t h e t i m e s t e p p i n g t a k e n v i a l e a p - f r o g s c h e m e i n c o n j u n c t i o n w i t h t h e R o b e r t f i l t e r ( s e e F y f e , 1989 f o r m o r e d e t a i l s ) . We e m p h a s i z e t h a t w h i l e t h e o r e m 3 . 1 a n d 3 . 2 h o l d r e g a r d l e s s o f t r u n c a t i o n , a s u f f i c i e n t r e s o l u t i o n i s r e q u i r e d t o o b t a i n c o n v e r g e n t n u m e r i c a l s o l u t i o n s . I n what f o l l o w s , a r e l a t i v e l y l o w t r u n c a t i o n a t Af=15 w i t h I =2 a n d J=3 ( c f . ( 3 . 2 . 1 ) ) i s m a d e . I t m u s t b e s t r e s s e d t h a t w h i l e some c h a n g e s a r e o b s e r v e d i n r u n s ( n o t shown) w h e r e M i s i n c r e a s e d f r o m M=15 t o Af=210, q u a l i t a t i v e l y s p e a k i n g , t h e n u m e r i c a l r e s u l t s a r e u n c h a n g e d . 3.5.2 Numerical r e s u l t s a) equilibrium states A f a m i l y o f e q u i l i b r i u m s t a t e s o f ( 3 . 2 . 3 ) , d e n o t e d b y {^ ( r ) } , i s o b t a i n e d n u m e r i c a l l y u s i n g t h e c o n t i n u a t i o n a l g o r i t h m ( c f . K e l l e r , 1 9 7 8 ) . I n F i g 3 . 1 , we p r e s e n t t h r e e r e p r e s e n t a t i v e e q u i l i b r i a c o r r e s p o n d i n g t o (a) weak d i s s i p a t i o n ( r = 1 / 2 9 d a y s ) , (b) m o d e r a t e d i s s i p a t i o n (r = 1 / 1 1 days) and (c) s t r o n g d i s s i p a t i o n (r = 1/5.5 d a y s ) , r e s p e c t i v e l y . I t i s seen, as e x p e c t e d , t h a t s t r o n g wavy m o t i o n has d e v e l o p e d as t h e Ekman damping d e c r e a s e s . b) optimal MGS boundary and MGS To o b t a i n t h e o p t i m a l MGS boundary i n r - s p a c e , we s o l v e t h e g e n e r a l i z e d m a t r i x e i g e n v a l u e p r o blem (3.3.17) f o r each t e {¥*(r) } t o o b t a i n r tf) u s i n g t h e E i s p a c k r o u t i n e s (Garbow, N 1 9 7 4 ) , w i t h r e s u l t s shown i n F i g 3.2. I t i s seen t h a t r tf) N m o n o t o n i c a l l y d e c r e a s e s as a f u n c t i o n o f r and i n t e r s e c t s w i t h t h e dashed l i n e r = r a t r = r . I t i s c l e a r from theorem N 3 . 1 t h a t t h e l i n e r = r d e f i n e s t h e MGS boundary. A l s o shown i n F i g 3.2 i s t h e normal mode s t a b i l i t y boundary r=r , where r L i s n u m e r i c a l l y d e t e r m i n e d from t h e u s u a l modal s t a b i l i t y a n a l y s i s o f t h e s e t {¥*(r)}. I t i s e v i d e n t from F i g 3.2 t h a t t h e two c r i t i c a l v a l u e s o f Ekman damping, i . e . , and r , d i v i d e t h e whole s t a b i l i t y regime o f t h e f a m i l y {¥*(r)} i n t o t h r e e d i s t i n c t subregimes: s u p e r c r i t i c a l regime ( I ) , s u b c r i t i c a l regime ( I I ) w i t h i n i t i a l l y g r o w i n g p e r t u r b a t i o n and MGS regime ( I I I ) . To i l l u s t r a t e MGS o f t h e e q u i l i b r i a *\ we p e r f o r m e d f i v e n u m e r i c a l e x p e r i m e n t s c o r r e s p o n d i n g t o f i v e d i f f e r e n t i n i t i a l c o n d i t i o n s o f t h e form t (0) =<$Q=$+$o where $ i s t h e e q u i l i b r i u m s t a t e shown i n F i g 3 . 1 (c) and $ i n each e x p e r i m e n t i s a randomly g e n e r a t e d i n i t i a l p e r t u r b a t i o n , w i t h the r a t i o 6 of i n i t i a l energy i n ~$o to the energy i n ^ varying from 0.2 to 1 i n these experiments. F i g 3 . 3 shows the time series of disturbance k i n e t i c energy (scaled by p^Hdl, with the density of a i r and H, d and 1 the s p a t i a l scales defined before). It i s seen i n F i g 3 . 3 that even for ~$o as large as t ( i . e . , with 5 = 1), the predicted monotonic decay to zero i s observed. c) I n i t i a l l y transient growth in subcritical flows The existence of i n i t i a l l y growing nonmodal perturbations to those t i n regime (II) (cf. F i g 3.2) i s i l l u s t r a t e d i n F i g 3.4-6. In each of the six experiments reported i n F i g 3.4, we set t(0) to be 1(0) = t = +^s , where s i s obtained from 0 m m solving (3.3.17) using ^ i n regime (II). The existence of such perturbations i s c l e a r l y demonstrated even for the equilibrium state t(r) with l/r=6 days, which i s just across the MGS boundary. The growth rates of the six i n i t i a l nonmodal perturbation are found to be 0.207/day, 0.204/day, 0.2/day, 0.197/day and 0.187/day, respectively. It i s seen that these disturbances i n i t i a l i z e d from these perturbations undergo transient growth over a period as long as 5 days, saturate at an energy l e v e l as high as roughly 60 % more of t h e i r i n i t i a l values, then asymptotically decay to zero. The existence of more than one such perturbation to a given s u b c r i t i c a l flows i s i l l u s t r a t e d for the equilibrium state t shown i n F i g 3.1(b). In F i g 3.5, we show the sp a t i a l configuration of f i v e such nonmodal perturbations s^, with F i g 3.6 showing the time series of k i n e t i c energy of the disturbances started from them. It i s i n t e r e s t i n g to note in F i g 3.5 that with a decrease i n t h e i r i n i t i a l growth rate (from the top panel to the bottom one), s ^  appear to be more elongated i n the zonal d i r e c t i o n . As for the temporal evolution of these ^ ( t ; s i ) , much of the observations for Fig 3.4 can be c a r r i e d over to F i g 3.6. The transient growth has been reported i n a variety of model flows (cf. Orr, 1907; Rosen, 1971; Boyd, 1983; etc; . Its relevance to cyclonegenesis has been strongly suggested i n F a r r e l l (1985; 1989a). As a cautionary note, we stress that the asymptotic decay to zero revealed i n F i g 3.5 and Fig 3.6 does not constitute a proof for estab l i s h i n g t h i s as a generic phenomenon i n s u b c r i t i c a l flows. We w i l l return to t h i s aspect i n chapter 5. d) optimal nonmodal perturbations in supercritical regime In terms of growth rate, the superior status of S m to a l l i n i t i a l disturbances ~$Q € IRM i s guaranteed by theorem 3.2. Here, we report the res u l t s from f i v e numerical experiments designed to i l l u s t r a t e the conclusion. Each of the experiments s t a r t s from one of the following i n i t i a l states t(0) = I = t + S m , i=0,l, . . , 4 , U M — 1 62 where i s the equilibrium state shown i n F i g 3.1(a), and are i t s i n i t i a l l y growing nonmodal perturbations determined from (3.3.17). In F i g 3.7, we show the time series of growth rate <TN over the f i r s t 15 days. The i n i t i a l growth rate of $( t ; s .) M - i (i=0,l,..,4) are found to be 0.185/day, 0.182/day, 0.144/day, 0.113/day and 0.028/day, respectively, i n contrast to the maximum modal growth rate 0.070/day, which c l e a r l y demonstrate the optimal status of as an i n i t i a l perturbation to We note that i d e n t i f y i n g favorably configured perturbations i n models other than ours has been a subject of several recent works ( F a r r e l l , 1988, 1989a; O'Brien, 1990). The p l a u s i b l e role of these perturbations i n explosive development of cyclone waves i n midlatitude has been suggested, for example, i n F a r r e l l (1985). 3.6 Concluding remarks A reduction i n dimension of the system has made the global analysis i n RM a procedure amenable to standard matrix eigenvalue c a l c u l a t i o n package such as Eispack (cf. Garbow, 1974). The numerical examples given i n § 3.5 c l e a r l y indicate that a s t a b i l i t y analysis of a hydrodynamical system without a global analysis i s l i k e l y to be li m i t e d . 63 Appendix 3A A l g e b r a i c p r o p e r t i e s o f t h e s e t {s^} Proof: The r e a l i t y o f A f o l l o w s from t h e s e l f - a d j o i n t n e s s o f A + and D ( c f . (3.3.15) - (3.3.16)) . To see t h e e x i s t e n c e o f M l i n e a r l y i ndependent s^, we c o n s i d e r t h e p o s i t i v e square r o o t D 1 / 2 o f t h e o p e r a t o r D and t h e l i n e a r mapping d e f i n e d by - 1 / 2 _M _M D : IR -» IR S = D 1 / 2 8 (3A.1) w i t h w hich, (3.3.17) y i e l d s As =• As, with A given by (3A.2) A = (Dl,2r1k+(D1/2)-\ where t h e e x i s t e n c e o f t h e i n v e r s e o f D 1 / 2 i s a s s u r e d by D b e i n g p o s i t i v e d e f i n i t e . From (3.3.16), i t i s not d i f f i c u l t t o see t h a t D 1 / 2 i s d i a g o n a l and hence A i s s e l f - a d j o i n t . I t f o l l o w s t h e n t h a t t h e e i g e n v a l u e problem (3A.2) has an o r t h o n o r m a l s e t o f Af e i g e n v e c t o r s s\, i . e . , As..= A..S.., V i=l,2,...rM (3A.3) S 1 s ] = 8 1 J / V i , 3=1,2, . , , M (3A.4) w h i c h y i e l d , v i a t h e l i n e a r mapping (3A.1), A + S i=A iDs i/ V i = 1, 2, M. (3A.5) To see t h a t S and A c o n s t r u c t e d a c c o r d i n g t o (3.3.20) s a t i s f y (3.3.21), i t s u f f i c e s t o note t h a t (3A.1) and (3A.4) y i e l d (3.3.21b), and t h a t summing up (3A.5) over i g i v e s 64 I • • I - , S m ] - D [ 8 L F 8 2 STDSA = A / • • / S ]A M as desired. Appendix 3B Physical Properties of the set {s^} C o r o l l a r y 3.1 Let s^. be determined by (3.3.17). Then, as a nonmodal i n i t i a l perturbation has the properties: (i) the i n i t i a l growth rate of ~$(t;s^) is independent of the size of a ., i.e., ( i i ) the velocity fields associated with different X^ are orthogonal in sense that where v(s^) and v (sy are velocity fields of and s respectively, and <.,.> is the inner product in H (cf. (2.2.11)) . Proof: (i) The proof for (i) follows from ( 3 . 3 . 1 2 b ) , (3 .3.17) and (3.4.4) . ( i i ) F i r s t note that O^Ds^ (X±- X..) = 0. (by (3.3.16) and (3.3.17)) (3B.3) Next, l e t l(a^) and l(s^) denote the streamfunctions generated using and as expansion c o e f f i c i e n t s according to (3.2.2). It thus follows that the inner product of v(s^) and v f s y w.r.t the (2.2.11) 65 <v (s±) ,v (s .) > = 0 if X±- X.*0, (3B .2 ) <v(s .),v(s .)> = <V<t>(s.),V<t>(3.)> = V s .a 2s ( 3B .4 ) J J- J OCX CC CC J cc which i s i d e n t i c a l to < s ^ , D S j > i f the inner product i n RM i s used to represent the summation i n ( 3B .4 ) , where s . and s , ai aj (<x=l,2,..,M) are the the components of and s ^ . The orthogonality of v(a^) and v f s y then follows from (3B.3) and ( 3 B . 4 ) . B 66 P i g 3.1 S t r e a m f u n c t i o n s o f t h r e e r e p r e s e n t a t i v e e q u i l i b r i u m s t a t e s f o r U* = 22.m/s and = 500 m. (a) 2/r--29 d a y s ; (b) l / r = l l days; (c) l / r=5 .5 d a y s . The r e s t o f the p a r a m e t e r s are g i v e n i n the t e x t . 67 F i g 3.2 S t a b i l i t y regime diagram. The s o l i d l i n e for the f i n i t e s t a b i l i t y measure r tf). The dashed l i n e s r=r = J N N 0.17/day for the MGS boundary , r=r = 0.075/day for the l i n e a r s t a b i l i t y boundary and r=r i s for the diagonal dashed l i n e which i s used for t e s t i n g the c o n d i t i o n r tf) N -r > 0. 68 TIME (day) F i g 3.3 An example of MGS. The e q u i l i b r i u m s t a t e ^ i s (c) i n F i g 3 .1 . The i n i t i a l p e r t u r b a t i o n s % are randomly generated w i t h the r a t i o 5 of t h e i r i n i t i a l k i n e t i c energy t o the energy i n t ranging from 0.2 t o 1 . 0 . 6 9 0 .9 r C M M M n E cn UJ 2 UJ m c r (— to t—i o l / r = 10 days l / r = 9 days l / r = 8 days l / r = 7 days -l / r = 6.5 days — — l / r = 6 days — 0 .0 6 .0 12 .0 18 .0 24 .0 30 .0 TIME (day) F i g 3.4 Existence of i n i t i a l l y growing nonmodal s to e q u i l i b r i a t(r) i n s u b c r i t i c a l regime, i . e . , region (II) in F i g 3 . 2 . 70 F i g 3 .5 The s p a t i a l configurations of f i v e i n i t i a l l y growing nonmodal perturbations to the basic state (b) i n F i g 3.1. 71 F i g 3 . 6 The energy time series for f i v e disturbances i n i t i a l i z e d from the s . shown i n F i g 3 .5. 72 -0.10 I 1 1 1 I I 0.0 3.0 6.0 9.0 12.0 15.0 TIME (day) F i g 3.7 The growth rates of disturbances i n i t i a l i z e d from nonmodal perturbations s i over an i n i t i a l growing period. The basic state t i s F i g 3 . 1 (a). 73 CHAPTER 4 FINITE AMPLITUDE NONMODAL DISTURBANCE I: INITIAL BEHAVIOR i t s r e l a t i o n to i n i t i a l intense development of disturbances 4 .1 Introduction The l i m i t a t i o n o f t h e growth r a t e o f t h e most u n s t a b l e (or t h e l e a s t damped) normal mode as an i n d i c a t o r o f p o t e n t i a l development o f d i s t u r b a n c e s t o a g i v e n f l o w has l e d one t o s e a r c h f o r f a v o r a b l y c o n f i g u r e d p e r t u r b a t i o n o f nonmodal s t r u c t u r e as a l t e r n a t i v e s ( c f . , F a r r e l l , 1988; O ' B r i e n , 1990; o r § 2.6 and § 3.4 i n t h i s t h e s i s ) . G i v e n t h a t such a p e r t u r b a t i o n i s found by some means, e.g., by t h e g l o b a l a n a l y s i s , i t remains n e c e s s a r y t o e s t a b l i s h t h a t t h e r e e x i s t s an i n i t i a l p e r i o d o v e r which t h e a m p l i f i c a t i o n t r i g g e r e d by t h i s p e r t u r b a t i o n p r o c e e d s more i n t e n s e l y t h a n t h a t due t o any modal perturbation i n o r d e r t o a s c r i b e (or r e l a t e ) t h e e x p l o s i v e development o b s e r v e d , f o r example, i n c y c l o n e g e n e s i s ( c f . Roebber, 1984; Sanders, 1986) t o t h e p r o p e r l y c o n f i g u r e d nonmodal p e r t u r b a t i o n s . A common approach t o demonstrate t h i s i s based on n u m e r i c a l s o l u t i o n s o f i n i t i a l v a l u e p roblems, as done i n § 3.5. W h i l e t h e r e s u l t s from such a approach (e.g., F a r r e l l , 1989b, o r F i g 3.7 i n t h e p r e v i o u s c h a p t e r ) suggest a f f i r m a t i v e answer t o t h e above c o n c e r n , no d i r e c t p r o o f has been known t o d a t e . 74 The o b j e c t i v e o f t h i s c h a p t e r i s t o show t h a t t h i s i s i n d e e d t h e case i n § 4.2. T h i s i s done i n p a r t by e x t e n d i n g some r e s u l t s i n § 2.8 t o t h e f i n i t e d i m e n s i o n a l system ( 3 . 2 . 3 ) , and i n p a r t by e s t a b l i s h i n g some b a s i c p r o p e r t i e s o f f i n i t e a m p l i t u d e nonmodal d i s t u r b a n c e s (here d e f i n e d as s o l u t i o n s t o t h e n o n l i n e a r system ( 3 . 2 . 3 ) - ( 3 . 2 . 4 ) , h e r e a f t e r r e f e r r e d t o as nonmodal d i s t u r b a n c e s f o r s i m p l i c i t y ) i n Appendix 4B. N u m e r i c a l examples a r e g i v e n i n § 4.3, f o l l o w e d by c o n c l u d i n g remarks i n § 4.4. 4 . 2 Nonmodal disturbance over t rans ien t per iod The approach t a k e n here i s t o compare t h e i n i t i a l b e h a v i o r o f nonmodal d i s t u r b a n c e s s t a r t i n g from o p t i m a l nonmodal p e r t u r b a t i o n s w i t h t h e i n i t i a l b e h a v i o r o f t h o s e i n i t i a l i z e d f rom any o t h e r p e r t u r b a t i o n , i n c l u d i n g t h o s e o f modal s t r u c t u r e . 4 . 2 . 1 modal v s . nonmodal growth rate at i n i t i a l ins tant F i r s t , we e s t a b l i s h t h e o p t i m a l s t a t u s f o r t h e nonmodal p e r t u r b a t i o n s o b t a i n e d as s o l u t i o n s t o t h e MEP (3.3.17) w.r.t t h e most u n s t a b l e normal mode a t i n i t i a l i n s t a n t t . F o r t h e o sake o f space, o n l y t h e o u t l i n e o f t h e t r e a t m e n t i s g i v e n h e r e (see § 2.8 f o r t h e c o u n t e r p a r t i n IH) . To o b t a i n t h e MEP f o r t h e modal a n a l y s i s o f (3.2.3), we c o n s i d e r t h e l i n e a r i z e d v e r s i o n o f (3.3.2) 7 5 (d/dt)% = *$, (4.2.1) V(tQ) = VQ (4.2.2) where A: CM-> C M i s given by (3.3.4e), with $ and y Q e C M. Introducing a modal form solution t(t;t,<r) = fe 0"', f o r f e CM and <r e C (4.2.3) into (4.2.1), we have the standard matrix eigenvalue problem (A - <rl)f = 0, (4.2.4) where I i s the unit operator of order M. With the derivation directed to Appendix 4A, we state the resu l t i n Lemma 4.1 For any modal disturbance lp (t;t,<r) = f e c r t to given equilibrium state t of (3.2.3), with f and <r determined from (4.2.4), i t s growth rate i s given by Re(<r(v,4)) = p(t,4) - r (4.2.5) where p i s generalized Rayleigh Quotient as given by (3.3.14). Defining the l i n e a r f i n i t e s t a b i l i t y measure by r ($) s max p (t;t) (4.2.6) with p being the generalized Rayleigh quotient and {f} c C M denoting the set of eigenvectors of (4.2.4), we can show, using the properties (3.3.15) and (3.3.16), that Lemma 4.2 (comparison of stability measures) Let r^rf) and r be given by (3.3.24) and (4.2.6), respectively. Then: L a 76 r tf) s r tf) . L N ( 4 . 2 . 7 ) I n t e r m s o f r tf) a n d r tf), we c a n w r i t e L N max Re(<r) = r tf) - r ( 4 . 2 . 8 ) Li a s o p p o s e d t o t h e maximum n o n m o d a l g r o w t h r a t e max <r ($,tf) = r tf) - r ( 4 . 2 . 9 ) N ^ N w h e r e we h a v e u s e d ( 3 . 3 . 1 2 b ) , ( 3 . 3 . 2 4 ) a n d ( 3 . 4 . 4 ) t o g e t ( 4 . 2 . 9 ) . F r o m ( 4 . 2 . 7 ) , we t h e n h a v e ( c f . t h e o r e m 2 . 5 i n H) Theorem 4 . 1 For any equilibrium state t of (3.2.3), it has at least one nonmodal initial perturbation which has a growth rate not less than its maximum modal growth rate. 4 . 2 . 2 Explosive development of nonmodal disturbance T o d e m o n s t r a t e t h e d e s i r e d a s s e r t i o n f r o m t h e o r e m 4 . 1 , s t r i c t l y s p e a k i n g , we n e e d t h e c o n t i n u i t y o f s o l u t i o n ~$(t;~$o) o f ( 3 . 3 . 2 ) i n t i m e t , a n d t h e c o n t i n u i t y o f t h e i n s t a n t a n e o u s g r o w t h r a t e cr $,4) i n $ f o r % i n some o p e n s e t o f RM, b o t h N o f w h i c h a r e e s t a b l i s h e d i n A p p e n d i x 4 B . G i v e n t h e s e a u x i l i a r y p r o p e r t i e s o f n o n m o d a l d i s t u r b a n c e s , we p r o c e e d b y c o n t r a d i c t i o n , v i z . , s u p p o s i n g t h a t f o r a s u f f i c i e n t l y s m a l l i n i t i a l p e r i o d [tQ,tQ+T) a n d V t = t Q +t ' w i t h t ' s T, cr $ f t + f ; s ) , 4 ) < o-v($(t+ t';1n),4), ( 4 . 2 . 1 0 ) N U N N U U w h e r e $ Q = Re(f) o r = Im(f), w i t h f t h e f a s t e s t g r o w i n g m o d a l p e r t u r b a t i o n (FGMP) d e t e r m i n e d f r o m ( 4 . 2 . 4 ) . T a k i n g l i m t'->0+ 77 i n ( 4 . 2 . 1 0 ) a n d u s i n g t h e c o n t i n u i t y o f <r i n ~$ a n d o f N $(t;$o) i n t ( c f . A p p e n d i x 4B) , we h a v e v($(t+0+; su),4) s *($(t+0+; %n) tf) . ( 4 . 2 . 1 1 ) N U M N O U A g a i n , b y c o n t i n u i t y o f %(t;a^) a n d '$(t;'$o) a t t , we i d e n t i f y $(t+0+;a ) i n ( 4 . 2 . 1 1 ) w i t h %(t ;s ), a n d s i m i l a r l y 0 M 0 M % (tQ+0+;$o) w i t h $(tQ;$Q), t h u s l e a d i n g t o a s t a t e m e n t c o n t r a d i c t i n g t h e e s t a b l i s h e d f a c t t h a t <r (s ;$) * <r ($ ;t) 3 N M N ^ O f o r ^ q e RM ( c f . ( 3 . 4 . 5 ) ) . S i m i l a r l y , i t c a n b e shown t h a t <r ($(t;a ) ;$) i s n o t l e s s t h a n max Re (a-) V t = t +t ' € [tQ,tQ+T) b a s e d o n t h e o r e m 4 . 1 . To s u m m a r i z e , we h a v e Theorem 4 . 2 There exists an initial period [t ,t +T) over which the initial development of at least one nonmodal initial perturbation to a given equilibrium state of (3.2.3) will proceed at a rate not less than the maximum modal growth rate or not less than that of nonmodal disturbances initialized from modal initial perturbations. 4.3 Numerical i l l u s t r a t i o n s T h e p u r p o s e o f t h i s s e c t i o n i s t w o f o l d : f i r s t t o s u p p l e m e n t some n u m e r i c a l c a l c u l a t i o n s t o t h e d i s c u s s i o n o f t h e r e l a t i o n o f i n i t i a l m o d a l t o n o n m o d a l g r o w t h r a t e s ( c f . § 2 . 8 a n d § 4 . 2 . 1 ) , a n d s e c o n d t o p r o v i d e f u r t h e r n u m e r i c a l i l l u s t r a t i o n s f o r t h e o r e m 4 . 2 a s s u p p l e m e n t a r y t o F i g 3 . 7 . 78 We d o s o b y c o n s i d e r i n g t h e p r o b l e m o f p l a n e t a r y s c a l e a t m o s p h e r i c f l o w o v e r t o p o g r a p h y , w i t h t h e m o d e l g e o m e t r y b e i n g t h e same a s i n § 3 . 5 . T h r e e s e t s o f e q u i l i b r i a o f ( 3 . 2 . 3 ) a r e u s e d h e r e , e a c h o f w h i c h i s o b t a i n e d n u m e r i c a l l y u s i n g t h e c o n t i n u a t i o n a l g o r i t h m ( K e l l e r , 1978) w i t h C7* as t h e c o n t r o l l i n g p a r a m e t e r r a n g i n g f r o m 5m/s t o 5 0 m / s , a r a n g e c o v e r i n g t h e t y p i c a l w i n d v e l o c i t y i n t h e u p p e r p a r t o f t r o p o s p h e r e , w i t h t h e r e s t o f m o d e l p a r a m e t e r s s p e c i f i e d as n e e d e d . (a) I n i t i a l modal vs. nonmodal growth rates F r o m ( 4 . 2 . 8 ) a n d ( 4 . 2 . 9 ) , i t s u f f i c e s t o c o m p a r e r tf) w i t h r (^) i n o r d e r t o s e e t h e u n d e r e s t i m a t i o n o f p o t e n t i a l a m p l i f i c a t i o n o f d i s t u r b a n c e s b y t h e m o d a l a n a l y s i s ( c f . , t h e o r e m 4 . 1 ) . H e r e , r tf) i s c a l c u l a t e d a s i n § 3 . 5 . 2 ( b ) , N w i t h r tf) o b t a i n e d f r o m max Re (cr) v i a ( 4 . 2 . 8 ) a f t e r s o l v i n g t h e s t a n d a r d e i g e n v a l u e p r o b l e m ( 4 . 2 . 4 ) , (A - <rl)f = 0. F i g 4 . 1 shows t h e r e s u l t s f r o m one c a s e o f c o m p a r i s o n . I t i s s e e n t h a t t h e c u r v e f o r tf) i s a l w a y s a b o v e t h e one f o r r tf) o v e r t h e w h o l e r a n g e o f C7*, t h u s c o n f i r m i n g t h e p r e d i c t e d p o t e n t i a l l y more i n t e n s e i n i t i a l d e v e l o p m e n t o f n o n m o d a l d i s t u r b a n c e s . T h e m o d e l p a r a m e t e r s f o r t h e s e t o f e q u i l i b r i a i$(U) ) a r e g i v e n i n t h e c a p t i o n , w i t h some r e p r e s e n t a t i v e s t a t e s f r o m t h e s e t {¥*( [ /* ) } shown i n F i g 4 . 2 . 79 V i e w i n g t h e s t a b i l i t y o f t h e s e t {¥*([ /*) } a s a f u n c t i o n o f t h e e x t e r n a l f o r c i n g U , i t i s s e e n t h a t i n t h e d i r e c t i o n o f i n c r e a s i n g U*, t h e {¥*(£/*) } f i r s t l o s e s m o n o t o n i c , g l o b a l s t a b i l i t y (MGS) a t U*=U^=6.6 m/s, t h e n l o s e s m o d a l s t a b i l i t y o n c e a c r o s s C7*=C7L =10.5m/s f r o m b e l o w , a n d r e c o v e r s t h e m o d a l L s t a b i l i t y f o r C7* l a r g e r t h a n U2= 26.3 m/s ( c f . F i g 4 . 1 ) . L C o r r e s p o n d i n g t o t h e e x c h a n g e o f s t a b i l i t y o f {U) } i s t h e l o s s a n d r e g a i n o f z o n a l i t y i n e q u i l i b r i u m f l o w s i n p h y s i c a l s p a c e ( c f . F i g 4 . 2 ) . T h e p h y s i c a l m e c h a n i s m u n d e r l y i n g t h e s u p e r c r i t i c a l r e g i m e (III) i s f r e q u e n t l y a s s o c i a t e d w i t h t h e s o - c a l l e d t o p o g r a p h i c i n s t a b i l i t y , w h i c h h a s r e c e i v e d much a t t e n t i o n i n t h e e f f o r t t o e x p l a i n t h e f o r m a t i o n , m a i n t e n a n c e a n d b r e a k down o f " b l o c k i n g p a t t e r n s " i n a t m o s p h e r i c f l o w s ( e . g . , C h a r n e y a n d D e v o r e , 1 9 7 9 ) . T h e c o n s e q u e n c e o f t h e u n d e r e s t i m a t i o n v a r i e s f r o m one s t a b i l i t y r e g i m e t o a n o t h e r . I n t h e m o n o t o n i c g l o b a l s t a b i l i t y r e g i m e ( i . e . , r e g i o n (I) i n F i g 4 . 1 ) , i t i s p h y s i c a l l y i n s i g n i f i c a n t s i n c e no d i s t u r b a n c e i s a l l o w e d t o g r o w b y ( 4 . 2 . 8 ) a n d ( 4 . 2 . 9 ) a n y w a y ; i n t h e s u b c r i t i c a l r e g i m e s ( i . e . , r e g i o n s (II) a n d (IV) i n F i g 4 . 1 ) , i t l e a d s t o t h e l o s s o f t e m p o r a l a m p l i f i c a t i o n p h e n o m e n o n ; i n t h e s u p e r c r i t i c a l r e g i m e ( i . e . , r e g i o n (III), i t i m p l i e s p o t e n t i a l e x p l o s i v e i n t e n s i f i c a t i o n o f n o n m o d a l d i s t u r b a n c e s . T h e c o n s e q u e n c e i s more d r a m a t i c when t h e b a s i c s t a t e s { ¥ * ( [ / ) } a r e o n l y w e a k l y u n s t a b l e o r s t a b l e , a s shown i n F i g 80 4 . 3 . I t i s s e e n t h a t o v e r t h e e n t i r e r a n g e o f if, t h e p o t e n t i a l i n i t i a l a m p l i f i c a t i o n s o f d i s t u r b a n c e s t o $(lf) s o l e l y a r i s e f r o m p r o p e r l y c o n f i g u r e d n o n m o d a l p e r t u r b a t i o n s s i n c e m o d a l g r o w t h r a t e i s n e g a t i v e e v e r y w h e r e . C o n s i d e r t h e c h a n g e o f p h y s i c a l p a r a m e t e r s i n t h e d i r e c t i o n o f p r o m o t i n g t h e m o d a l g r o w t h r a t e t o s e e how t h i s w i l l a f f e c t t h e r e l a t i o n o f i n i t i a l m o d a l t o n o n m o d a l g r o w t h r a t e . T h i s i s d o n e f o r t h e s e t o f {^ (C7*) } d e t e r m i n e d u s i n g t h e same s e t o f p a r a m e t e r s a s i n F i g 4.1 e x c e p t t h e r e d u c e d Ekman d a m p i n g t i m e s c a l e 1/r- = 30 d a y s . I n F i g 4 .4 , w h i l e t h e p o r t i o n o f U*-parameter s p a c e s u p p o r t i n g t h e m o d a l g r o w t h i s e n l a r g e d c o m p a r e d t o r e g i o n (III) i n F i g 4 . 1 , t h e c o n c e r n f o r t h e u n d e r e s t i m a t i o n o f p o t e n t i a l a m p l i f i c a t i o n o f d i s t u r b a n c e s u s i n g t h e m o d a l g r o w t h r a t e r e m a i n s a s b e f o r e . (b) Intense development of i n s t a b i l i t y over an i n i t i a l period T o s e e t h e d e v e l o p m e n t o f n o n m o d a l d i s t u r b a n c e s a t a r a t e l a r g e r t h a n w o u l d b e e x p e c t e d f r o m t h e maximum m o d a l g r o w t h r a t e o v e r some i n i t i a l p e r i o d , i t s u f f i c e s t o c o n s i d e r i f i t h a p p e n s when t h e u n d e r l y i n g e q u i l i b r i u m s t a t e s u p p o r t s s t r o n g l y u n s t a b l e m o d a l p e r t u r b a t i o n s . N u m e r i c a l e x p e r i m e n t s b a s e d o n t h e i n i t i a l v a l u e p r o b l e m ( 3 . 2 . 3 ) a r e p e r f o r m e d as i n § 3 . 5 . T h e e q u i l i b r i u m s t a t e f u s e d h e r e i s shown i n F i g 4 . 5 ( a ) , c o r r e s p o n d i n g t o t h e p e a k on t h e r^(t) c u r v e i n F i g 4 . 4 . T h e i n s t a n t a n e o u s g r o w t h r a t e s f r o m t h r e e e x p e r i m e n t s 81 are shown i n F i g 4.6, with the corresponding energetics given i n F i g 4.7. It i s seen that the response of $ to the nonmodal i n i t i a l perturbations s and s i s noticeably more intense M M - 1 than to the fastest growing modal i n i t i a l perturbation f. S p e c i f i c a l l y , the i n i t i a l growth rates of $(t;s .) (i=0,l) are nearly 50 % larger than the maximum modal growth rate and are noticeably higher than that of $(t;1Q) with ~$q = Re (f) over a period of two days. In terms of energetics, the energy l e v e l of ~$(t;s .) (i=0,l) i s considerably higher than that of ~$(t;Re(t)) for more than f i v e days (cf. F i g 4.7). 4.4 Concluding remarks The global analysis (cf. § 2 . 2 - 2 . 5 and § 3.3) provides a systematical algorithm for the search of favorably configured nonmodal perturbations. The study of the r e l a t i o n of i n i t i a l modal to nonmodal growth rate reveals the cause responsible for the l i m i t a t i o n of modal analysis and provides the ratio n a l e f or phenomena such as the transient growth in s u b c r i t i c a l flows. The investigation of i n i t i a l behavior of nonmodal disturbances i n i t i a l i z e d from those favorably configured nonmodal perturbations i s a key to understanding the rapid development of disturbances observed, for example, i n the'atmosphere (Sanders, 1 9 8 6 ) . 82 Appendix 4A M o d a l g r o w t h r a t e e x p r e s s e d i n t e r m s o f p P r o o f : F i r s t , we p u t t h e s t a n d a r d e i g e n v a l u e p r o b l e m ( 4 . 2 . 4 ) i n t o a n e q u i v a l e n t f o r m A| = crDf; A = DA, ( 4 A . 1 ) w h e r e A a n d D a r e g i v e n b y ( 3 . 3 . 4 e ) a n d ( 3 . 3 . 1 1 ) , r e s p e c t i v e l y . T h e e q u i v a l e n c e f o l l o w s f r o m t h e f a c t t h a t D i s p o s i t i v e d e f i n i t e i n R ( c f . ( 3 . 3 . 1 1 ) ) a n d h e n c e i n v e r t i b l e . A t t h i s p o i n t , i t i s c r u c i a l t o n o t e f r o m ( 3 . 3 . 4 e ) , . ( 3 . 3 .11 ) a n d ( 4 A . 1 ) t h a t A c a n b e d e c o m p o s e d a s A = A + + A", ( 4 A . 2 ) w i t h A + s A - rD, ( 4 A . 3 a ) A s [A, . ] ; A. . s V c , ( - a 2 . ) , ( 4 A . 3 b ) 1 ij1 ij L lkj k y ' k A — A — A — rp A H [A±j] = -A T , ( 4 A . 4 a ) Kj ^ b i j + l cijk<fA/H ~ aPk>• ( 4 A - 4 b ) k T a k i n g i n n e r p r o d u c t o f ( 4 A . 1 ) w i t h r e s p e c t t o an a r b i t r a r y e i g e n v e c t o r ^ o f ( 4 . 2 . 4 ) g i v e s <trkt> = v<t,Di>, ( 4 A . 5) w h e r e t h e o v e r b a r d e n o t e s t h e c o m p l e x c o n j u g a t e . A d d i n g t h e c o m p l e x c o n j u g a t e o f (4A-.5) t o i t s e l f , we o b t a i n , w i t h t h e u s e o f t h e d e f i n i t i o n s o f A a n d A , 2Re{<r}<i,T>t> = <|,ATf> + <|,A|> T = <%,k+1> + <i,k+t> 83 = 2<t,k% - 2r<i,Di>, (4A.6) A — where t h e c o n t r i b u t i o n from A t o t h e RHS o f (4A.6) c a n c e l s out by v i r t u e o f t h e f a c t (4A.4a), w i t h A + d e f i n e d by (3.3.13). The r e a l n e s s o f t h e RHS o f (4A.6) i s e n s u r e d by the s e l f - a d j o i n t n e s s o f A + and D ( c f . (3.3.13) and ( 3 . 3 . 1 1 ) ) . To t h i s end, i n t r o d u c i n g t h e d e f i n i t i o n o f p i n t o (4A.6) l e a d s t o t h e a s s e r t i o n . B Appendix 4B Fundamental p r o p e r t i e s o f nonmodal d i s t u r b a n c e B - l Existence, uniqueness and continuity of ~$(t;~$Q) The c o n t i n u i t y o f nonmodal d i s t u r b a n c e ~$(t;~$o) i n t i m e t as w e l l as o t h e r fundamental p r o p e r t i e s such as e x i s t e n c e and uni q u e n e s s o f s o l u t i o n s t o (3.3.2) can be e a s i l y o b t a i n e d v i a a s i m p l e a p p l i c a t i o n o f t h e P i c a r d - L i n d e l o f theorem ( c f . theorem 3.1 i n H a l e , 1963). The e s s e n t i a l c o n d i t i o n a s s u r i n g t h e s e p r o p e r t i e s i s t h a t t h e v e c t o r f i e l d a) g i v e n by (3.3.3) i s l o c a l l y L i p s c h i t z i a n i n %, as s t a t e d i n Lipschitzianism of the vector f i e l d f(~$;t,a) Let fC$;t,a) be defined by ( 3 . 3 . 3 ) and let QJ be any bounded, closed set in RM . Then, £$;$,a.) is locally Lipschitzian in ~$ e Qj. Proof: The boundedness and c l o s e d n e s s o f s e t QJ i m p l i e s t h e e x i s t e n c e o f a c l o s e d b a l l B(o,d) o f r a d i u s d c e n t e r e d a t 0, t h e o r i g i n o f RM, such t h a t QJ £ 6(0, d) . Now, t a k e any $, % 84 e (U a n d c o n s i d e r ||f ( $ ;$ , * ) - f t f , 4 , a ; | =s |A||? - $\\ + \\B$(t-h I + \B ($-"$) i\f ( 4 B . 1 ) w h e r e we h a v e u s e d ( 3 . 3 . 5 ) f o r f , w i t h A a n d B g i v e n b y ( 3 . 3 . 4 e ) a n d ( 3 . 3 . 4 f ) , r e s p e c t i v e l y , a n d | || b e i n g t h e £ n o r m . N o t e t h a t e v e r y l i n e a r o p e r a t o r i n £(JRH) , a v e c t o r s p a c e o f l i n e a r o p e r a t o r s on RM, i s b o u n d e d ( c f . t h e o r e m 2 . 7 - 9 i n K r e a y s z i g , 1 9 7 8 ) . I n f a c t , b y t h e S c h w a r z i n e q u a l i t y , we o b t a i n HA« * {[ I A i j ) 1 / 2 ' <4B-2) i 3 w h e r e A^_. i s e n t r y o f A ( c f . ( 3 . 3 . 4e) ) . F o r t h e two t e r m s i n v o l v i n g B i n ( 4 B . 1 ) , we h a v e \*$a~ h\\ * d { [ £ l B . 2 . ^ ^ - 1\, ( 4 B . 3 ) i j k \B# s d {[ I I B i j j k } 1 / 2 | ^ - * l - ( 4 B - 4 ) i j k I n t r o d u c i n g ( 4 B . 2 ) - ( 4 B . 4 ) i n t o ( 4 B . 1 ) , we g e t {£($;$,a) - f ($;t,a)\ =s L(V) [$ - %\, w h e r e L(U) i s L i p s c h i t z i a n c o n s t a n t o f s e t QJ L A U = « { {l I 2, [in B.^} 1' 2} . i j i j ^ T h i s c o m p l e t e s t h e p r o o f . B 85 B y v i r t u e o f t h e P i c a r d - L i n d e l o f t h e o r e m , we h a v e Existence, uniqueness and continuity Let QJ be any bounded, closed set in R M and let E c RxR M denote the domain of definition for the solution %(t;~$Q) of (3.3.2) E = | r t ^ o ; | a($Q) =s t s b($Q); tQ e U c R M j , ( 4 B . 5 ) with (a $Q) ,b ($)) c IR being the maximum interval of existence of %(t;%Q). Then, for any $q e QJ, the system (3.3.2) has a unique solution %-(t;"$ ) through ~$q. Moreover, $(t;$Q) is continuous in E. B - 2 : Continuity of cr ($,4) i n % N T h e c o n t i n u i t y o f <r i n % f o r % i n some o p e n s e t o f N R M i s g i v e n b y : Continuity of <r $;$) i n % Let the growth rate cr ($;$) of N N $(t;~$Q) be defined by (3.4.4), and Let QJ be any bounded, closed set in R M . Then, <r : QJ -> R i s c o n t i n u o u s . N Proof: F r o m ( 3 . 3 . 1 2 b ) , ( 3 . 3 . 2 4 ) a n d ( 3 . 4 . 4 ) , i t s u f f i c e s to show t h a t p : QJ -» R i s c o n t i n u o u s . R e c a l l t h a t p i s the g e n e r a l i z e d R a y l e i g h Q u o t i e n t d e f i n e d b y ( 3 . 3 . 1 4 ) . F o r t h i s , we o b s e r v e t h a t f o r a n y %', % e QJ c B ( 0 , d ) \<$',k+%'> - < ^ A + ^ > | s 2d | |A + ||||0 > ' - $\\, ( 4 B . 6 ) 86 \<$' ,D$'> - <$,Ti$>\s c^?j|0|| ||^ ' - |||, (4B.7) where d i s radius of the closed b a l l 1 (0 ,d) chosen to enclose U, with A + and D given by (3.3.13) and (3.3.11), respectively. It i s evident from (4B.6)-(4B.7) that both <$,&+$> and <$,!>$> are continuous i n a neighborhood of % defined by the open b a l l B(^;S), with 8 given by min {l/(2dfA + f l ) , l/(2d\D\)}, from which we conclude that p{%;^) (s <$fK+'$>/<$fD$>) and hence <r ($4) are continuous i n some N neighborhood of $ not larger than B($;8) , u 87 0.5E+00 tr id - 0 . 5 E - 0 1 cc to •X m in 0 . 5 E - 0 2 r 0 . 5 E - 0 3 (15days) 0.5E+01 0 .5E+02 ZONAL MOMENTUM FORCING (m/s) F i g 4 .1 Maximum i n i t i a l modal versus nonmodal growth rate, i . e . , r L (t) - r vs. ($) - r, for the set of e q u i l i b r i a {t{U) } over zonal wavenumber-1 topography of height 1000 m. The EJcman damping c o e f f i c i e n t r = 2/(I5days) i s indicated by the horizontal dashed l i n e . The three v e r t i c a l dashed l i n e s are for the MGS boundary U = U^=6.S m/s, and for l i n e a r s t a b i l i t y boundaries U = tV= 10.5 m/s and U - U2= 26.3m/s, where U and CT1, C72 are obtained as L N L L roots to the equations, r ($(U*))-r=0 and r (t(u'))-r=0, N L respectively. 88 ( a ) (b) Ic) (d) (e) If) ~<-eo ~ N  20 F i g 4.2 (a) Topography contours; (b)-(f) streamfunctions of several representative states from the set of {t(U) } used i n Fig 4.1, corresponding to 13=6.20 m/s i n (I), 9.12 m/s i n (II), 11.33 and 17.85 m/s i n (III) and 40.13 m/s i n (IV), respectively. 89 0.5E-03 J 0.5E+01 0.5E+02 ZONAL MOMENTUM FORCING (m/s) Fig 4.3 The same as F i g 4.1 except that the set of e q u i l i b r i a {^(C7*)} used here corresponds to the zonal wavenumber-2 topography. 90 0.5E+00 r IS -o - 0.5E-01 UJ tr J 0.5E-02 CQ <£ in 0.5E-03 =5.Om/s w ( I D to \ E II 11 i II *s» 1 r - 1/(30days) ( I I I ) 0.5E+01 0.5E+02 ZONAL MOMENTUM FORCING Im/s) Fig 4.4 The same as F i g 4.1 except that the set of e q u i l i b r i a {t(U)) used here corresponds to l/r = 30 days. 91 F i g 4 . 5 Basic states, nonmodal and modal i n i t i a l perturbations for the numerical experiments. (a) for streamfunction of the equilibrium state taken from F i g 4 . 4 corresponding to U=16.3 m/s, (b) and (c) for i t s fast growing i n i t i a l nonmodal and modal perturbations, respectively. 92 0.24 T3 (— CC 3 O CC C J 0.18 -0.13 0.07 -0.01 -0.04 --0.10 ' \ \ max Re (<r) =0.11/day 0.0 1.0 2.0 3.0 TIME (day) 4.0 5.0 Fig 4.6 Evolution of the growth rate with time. The optimal nonmodal i n i t i a l perturbation s (see F i g 4 .5(b)) M has growth rate cr (s ;**) = 1/(5.29 days), whereas the fast growing modal perturbation (FGMP) (see F i g 4 . 5(c)) has Re (cr) = 1 / ( 5 . 5 2 days). The i n i t i a l perturbations are indicated i n the F i g 4 .5 (a) and (b). 93 3 0 . 0 0.0 1.0 2 .0 3 .0 4 . 0 5 . 0 TIME (day) Fig 4.7 Evolution of disturbance energy with time for the case shown in Fig 4 . 6 . The i n i t i a l perturbation energy is 20 % of the energy in the equilibrium states to ensure the f i n i t e amplitude for the i n i t i a l perturbations. 94 CHAPTER 5 FINITE AMPLITUDE NONMODAL DISTURBANCE I I : ASYMPTOTIC BEHAVIOR i t s r e l a t i o n to b i f u r c a t i o n and multiple e q u i l i b r i a 5.1 Introduction Consideration of the asymptotic behavior of disturbances arises i n a variety of circumstances. When a modal analysis reveals the existence of exponentially growing perturbations to a given flow, one wants to determine the maximum amplitude that can be attained by the growing disturbances, a problem of nonlinear saturation (Shepherd, 1988, 1989); or to inquire about the arrest of the exponential growth, a problem of nonlinear e q u i l i b r a t i o n (Pedlosky, 1970, Salmon, 1980; Mak, 1985). On the other hand, the existence of i n i t i a l l y growing nonmodal perturbations uncovered by . such analysis as optimization procedures ( F a r r e l l , 1988; O'Brien, 1990; chapter 2 of t h i s thesis) raises the question: what happens to these disturbances a f t e r the i n i t i a l transient growth. Further, i d e n t i f y i n g multiple e q u i l i b r i a from a b i f u r c a t i o n analysis (Vickroy and Dutton, 1979) leads one to wonder how a t r a n s i t i o n from one equilibrium state to another takes place. The answers to these problems or questions l i e i n the study of the asymptotic behavior of f i n i t e amplitude nonmodal disturbances. The objective of t h i s chapter i s to e s t a b l i s h some basic 95 f a c t s r e g a r d i n g t h e t-*» b e h a v i o r f o r t h e system (3.2.3)-(3.2.4) . The d i s c u s s i o n o f a s y m p t o t i c decay i n s u b c r i t i c a l f l o w s i s g i v e n i n § 5.2, w i t h t h e t r e a t m e n t f o r p e r s i s t e n t d i s t u r b a n c e s i n s u p e r c r i t i c a l f l o w s i n § 5.3. The a priori d e t e r m i n a t i o n o f t h e n a t u r e o f t h e a s y m p t o t i c s t a t e s o f nonmodal d i s t u r b a n c e s i s a d d r e s s e d i n § 5.4. N u m e r i c a l i l l u s t r a t i o n s a r e g i v e n i n § 5.5, f o l l o w e d by c o n c l u d i n g remarks. 5.2 Asymptotic decay as t -> w i n s u b c r i t i c a l flow I n t h i s and next s e c t i o n s , we c o n f i n e t h e d i s c u s s i o n t o a g e n e r i c case where t h e u n d e r l y i n g e q u i l i b r i u m s t a t e t o f (3.2.3) i s hyperbolic, i . e . , Re {<r^  (A tf) ) } * 0, i = 1 , 2 , . . . M , where . A<$) i s t h e l i n e a r p a r t o f t h e v e c t o r f i e l d F (or f ) ( c f . ( 3 . 2 . 3 ) o r (3.3.3a)) e v a l u a t e d a t ^  (or a t $ = 0 ) . The re a s o n f o r t h e r e s t r i c t i o n t o h y p e r b o l i c i t y i s more t h a n f o r c o n v e n i e n c e . I n f a c t , i t i s m o t i v a t e d by t h e f a c t t h a t h y p e r b o l i c i t y i s a g e n e r i c p r o p e r t y o f l i n e a r o p e r a t o r s on R . More p r e c i s e l y , t h e s e t o f h y p e r b o l i c o p e r a t o r s i s dense and open i n £(\RK), w h i c h means t h a t any o p e r a t o r P e £(\RK) w i t h Re(<r^(P)) = 0 f o r some i can be ap p r o x i m a t e d a r b i t r a r i l y c l o s e l y by some Q e £(IRM) s a t i s f y i n g Re (o-^ (Q) ) * 0 f o r any i ( c f . theorem 3, pp.157 i n H i r s h and Smale, 1976). However, v i o l a t i o n o f h y p e r b o l i c i t y o f t e n o c c u r s a t a c r i t i c a l e q u i l i b r i u m s t a t e o f (3.2.3) which i n t u r n c o r r e s p o n d s t o a b i f u r c a t i o n p o i n t i n some u n d e r l y i n g p h y s i c a l parameter space, which i s the subject of § 5.4. The main r e s u l t of t h i s section i s theorem 5.1. The analysis leading to t h i s r e s u l t i s based on the di r e c t method of Liapunov (Hahn, 1967; Hale, 1969; Hirsh and Smale, 1976; Verhulst, 1985) . The basic ideas of the method are given in Appendix 5A for reference. For a better focus on the main development, a u x i l i a r y lemmas are directed to Appendix 5B. Theorem 5.1 Suppose an equilibrium state t of (3.2.3) is hyperbolic for which the following hold: (i) r^tf)-r<0; (ii) r tf)-r>0, where r tf) and r tf) are given by (3.3.24) and N N L (4.2.6), respectively. Then, there exists a neighbourhood QJ of the null solution % = 0 of (3.3.2) such that any nonmodal disturbance %(t;%^) to ¥* approaches 0 as t -» co whenever ^q € QJ. Proof: Hyperbolicity of ¥* and the condition (i) implies that Re(<r(A)) < 0. It thus follows from lemma 5.2 (cf. Appendix 5B) that there exists a po s i t i v e d e f i n i t e Q e R M x M as a solution to Liapunov matrix equation A T Q + QA = - I , I a unit matrix of order M (5.2.1) where we have set R i n (5B.3) to - I . Put V($) = <$,Q$>. We assert that such a chosen V i s a Liapunov function of the nonlinear system (3.3.2) at ~$ = 0 (cf. Appendix 5A) . It i s 9 7 obvious that V(0) = 0 and V(%) = <$,&$> >0 in RK-{0} (5.2.2) with the l a t t e r property due to the p o s i t i v e d e f i n i t e n e s s of Q. Now, consider the o r b i t a l d e r i v a t i v e of V along the t r a j e c t o r y of any disturbance '$(t;'$Q) to ^ V = <(d/dt)'$,Q$> + <$,Q(d/dt)%> = <A$,Q$> + + <B$$,Q$> + <$,QB$$> (by (3.3.5b)) • V C$) + w$) (5.2.3) where V2($) corresponds to the f i r s t two terms i n (5.2.3) , and w($) to the l a s t two ones. Moreover, V2($) = <$, (ATQ + QAj|> = -\$\\2 (by(5.2.1)) and hence i s negative d e f i n i t e i n RM-{0}. For w {$) , we have the f o l l o w i n g estimate \w($)\ s |<B#,0$>| + \<$,QB$$>\ s ||B^||||QJ|| + ||^ ||||Q||B^|| (by Schwarz inequal i ty ) s 2c||Q|||^ ||3. (by lemma 5.3 (cf. Appendix 5B) ) It thus fol lows from lemma 5.1 (cf.Appendix 5B) that there e x i s t s a neighbourhood (U of the n u l l s o l u t i o n ~$ = 0 of (3.3.2) such that V s a t i s f i e s (5A.4). This l a s t property of V together with (5.2.2) proves that V i s a Liapunov funct ion f o r the n u l l s o l u t i o n , which y i e l d s the des i red r e s u l t by Liapunov's c r i t e r i o n for asymptotic s t a b i l i t y . a 98 Remark 5 . 2 . 1 I t f o l l o w s t h a t < £ ( t ; s ^ ) -» 0 a f t e r a n i n i t i a l t r a n s i e n t g r o w t h i f s^ € U, w h e r e i s a n i n i t i a l l y g r o w i n g n o n m o d a l p e r t u r b a t i o n d e t e r m i n e d f r o m ( 3 . 3 . 1 7 ) a n d a s s u r e d b y t h e c o n d i t i o n ( i i ) . T h i s r e s u l t p r o v i d e s a t h e o r e t i c a l b a s i s f o r t h e a s y m p t o t i c d e c a y i n g t o z e r o s e e n i n F i g 3 . 4 a n d F i g 4 . 6 . I t h a s w e l l b e e n known f o r l i n e a r m o d e l f l o w s ( O r r , 1907 , R o s e n , 1 9 7 1 ; B o y d , 1 9 8 3 ; F a r r e l l , 1987) t h a t f o l l o w i n g t h e t r a n s i e n t g r o w t h a s s o c i a t e d w i t h t h e c o n t i n u o u s s p e c t r u m i s a n a s y m p t o t i c d e c a y t o z e r o . By i m p o s i n g h y p e r b o l i c i t y a n d c o n d i t i o n s ( i ) - ( i i ) o n ¥*, we h a v e e s s e n t i a l l y m a n a g e d t o d e m o n s t r a t e t h a t t h i s i s a l s o t r u e f o r t h e n o n l i n e a r s y s t e m ( 3 . 2 . 3 ) . N o t e t h a t t h i s e x t e n s i o n r e m a i n s l o c a l as l o n g as t h e n e i g h b o r h o o d dl i s f i n i t e . 5.3 Persis tence as t -> « i n s u p e r c r i t i c a l flow T o c o n d u c t t h e s u b s e q u e n t d i s c u s s i o n i n a m o s t t r a n s p a r e n t m a n n e r , we a s s u m e t h a t t h e t u n d e r c o n c e r n s a t i s f i e s (a) t i s h y p e r b o l i c , ( 5 . 3 . 1 a ) (b) Refer.. (A) ; < 0 , i = 1 , 2 , . . . , p , ( 5 . 3 . 1 b ) (c) Refr^OD) > 0, i = p+l,p+2, . . . , s , s * p ( 5 . 3 . 1 c ) w h e r e A i s g i v e n b y ( 3 . 3 . 4 e ) e v a l u a t e d a t w i t h t h e o\£ (X) =ot.j+i&^ (/3^ t 0), i=l,2,...,s, b e i n g i t s d i s t i n c t e i g e n v a l u e s o f a l g e b r a i c m u l t i p l i c i t y n . . T h e c o n d i t i o n s * p 99 i s meant to exclude the case discussed i n § 5.2. It i s c l e a r that t h i s ¥* has i n i t i a l l y growing modal perturbations by (5.3.1c) and hence nonmodal perturbations by (4.2.7)-(4.2.9). From e a r l i e r works though for other model flows (e.g., Stuart, 1960; Pedlosky, 1970, 1981; Mak, 1985), i t may be expected that the disturbances started from unstable normal modes would be ultimately e q u i l i b r a t e d v i a mechanisms such as nonlinear cascade (Salmon, 1980). It remains unclear to those disturbances a r i s i n g from perturbations of nonmodal structure. This concern i s strengthened by the argument that the observed rapid deepening of synoptic scale disturbances perhaps predominantly arises from the release of the mean flow p o t e n t i a l energy by favorably configured nonmodal perturbations ( F a r r e l l , 1985). The main re s u l t of t h i s section ( i . e . , theorem 5.2) may be regarded as an e f f o r t to address the concern. E s s e n t i a l l y , we f i r s t f i n d a new basis for the space of a l l the kinematically admissible disturbances ( i . e . , for RM) according to the real canonical theory of l i n e a r operator (Hirsh & Smale, 1974), and then divide the disturbances into two subset R+ and R (cf. lemma 5.4 i n Appendix 5C). Next, we show that under the conditions (5.3.1) there exists a Liapunov i n s t a b i l i t y function V + ($) i n the neighbourhood QJ+ of $=0 whose p r o p e r t i e s a l l o w u s t o d e t e c t t h e t -» » a s y m p t o t i c b e h a v i o r o f a n y d i s t u r b a n c e i n i t i a l i z e d i n (U+n R+-{0} ( c f . lemma 5 . 5 i n A p p e n d i x 5D f o r p r o p e r t i e s o f V+). Now, we s t a t e Theorem 5.2 Let the equilibrium state $ of (3.2.3) satisfy (5.3.1). Then, any nonmodal disturbance ~$(t;~$Q) to f will persist as t -» « if 0q € II n R -{0}, where R and U a r e t h e same as in lemma 5.4 and lemma 5.5, respectively. This is particularly true for %(t;a^) when is contained in U + n R+-{0}, the nonmodal initial perturbation from (3.3.17). Proof: U n d e r t h e h y p o t h e s e s ( 5 . 3 . 1 ) on ^ , t h e p r o p e r t y (a) o f f u n c t i o n V+ h o l d s ( c f . A p p e n d i x 5D) . T h u s , f o r a g i v e n c o n s t a n t a>0, t h e r e e x i s t s a c l o s e d b a l l B(0,jb) s u c h t h a t | V+($) | s a for $ e B CO,b) . ( 5 . 3 . 2 ) M o r e o v e r , b y c o n t i n u i t y o f V+ a t 0, i t i s p o s s i b l e t o c h o o s e b t o e n s u r e t h e c l o s e d b a l l i n ( 5 . 3 . 2 ) t o s a t i s f y B(0,b) e (U + . ( 5 . 3 . 3 ) We a s s e r t t h a t a n y n o n m o d a l d i s t u r b a n c e ~$(t;~$o) t o ^ w i l l l e a v e t h e b a l l i n f i n i t e t i m e i f ~$o € B(0,£>) n U + n \R+-{0}. We p r o v e t h i s b y c o n t r a d i c t i o n , a s s u m i n g t h a t ^ ( t ; | o ) S 1(0,i>) f o r t a t i f $Q e B(0,jb) n U + n R+-{0}. ( 5 . 3 . 4 ) F r o m p r o p e r t y (c) o f V+ i n lemma 5 . 5 , we c h o o s e ~$ s u c h t h a t 101 $ o € 6(0,23) n R+-{0}, i . e . , V+$Q) > 0. (5.3.5) Since f o r t a t o V+($(t,4Q)) - V+ ($q) = \l V+> 0 (5.3.6) V + i s nondecreasing along the orbit c ^ ( t ; ^ Q ) , where the p o s i t i v e sign i n (5.3.6) follows from (b) i n lemma 5.5, (5.3.3) and (5.3.5). Put y £ | % | V+($(t/$0)) > V+$Q); 1(t,4Q) e B(0,b) j . It i s obvious from (5.3.5), and (5.3.6) that the set y & 0, • + and hence from (b) of lemma 5.5 that ii = i n f V > 0. Continuing (5.3.6) leads to V+ ($(t;%o)) - V+ a u(t-tQ) ^ co as t -> oo, a statement contradicting (5.3.2), thus completing the proof. Remark 5.3.2 It i s clear from lemma 5.4 and 5.5 (cf. Appendices 5C and 5D) that the set V+r\ JR+-{0} i s not empty under the condition (5.3.1), implying that there always exist perturbations leading to persistent disturbances to a flow ^ for which (5.3.1) holds, and that a t r a n s i t i o n from the given I to a new state i s p o t e n t i a l l y i n e v i t a b l e . We consider the forgoing re s u l t useful primarily for i t s capacity to cope with perturbations not r e s t r i c t e d to any 102 s p e c i a l type and basic flows not necessarily zonally sheared, and f o r i t s assurance of the existence of a set of perturbations capable of t r i g g e r i n g a t r a n s i t i o n from a given flow to another d i s t i n c t state, but not for i t s prediction of d e t a i l s of l i m i t i n g status. In fact, i t y i e l d s no information on the nature of persistent disturbances ( i . e . , steady or p e r i o d i c or chaotic), as opposed to weakly nonlinear theory (e.g., Stuart, 1960; Watson, 1960; Pedlosky, 1970, 1981). Further, i t i s s i l e n t about the mechanism responsible for the e q u i l i b r a t i o n which the persistent evolving disturbances are expected to experience ultimately (Pedlosky, 1970; Salmon, 1980), and about the saturation l e v e l (Shepherd, 1988, 1989). It i s important to note that a perturbation superimposed onto a ^ for which (5.3.1) i s s a t i s f i e d may not survive in time, i . e . , i t may decay to 0 as t ^ oo, despite the s u p e r c r i t i c a l nature of ^. Indeed, by a s i m i l a r argument, we can have Remark 5.3.2 Under the hypothesis on f i n (5.3.1), there e x i s t s a function V : (U— -> R (UJ a neighbourhood of ~$ = 0) defined by V~ = <$,Q~ £>, Q~= diag [ Q 1 , - Q 2 ] (5.3.7) s a t i s f y i n g (a) V~(0) = 0 (5.3.8a) 103 (b) V~ < 0 in V~-{0}, ( 5 . 3 . 8 b ) (c) V~ (%) > 0 V % € ffT-fO/ ( 5 . 3 . 8 c ) w h e r e % € RM i s w r i t t e n w . r . t t h e b a s i s & ( 5 C . 7 ) , Q i a n d Q 2 t h e s o l u t i o n s t o (5D.5 ) a n d ( 5 D . 6 ) , r e s p e c t i v e l y . F u r t h e r m o r e , I t f o l l o w s f r o m ( 5 . 3 . 8 ) a n d lemma 5 . 4 ( c f . A p p e n d i x 5C) t h a t a n y n o n m o d a l d i s t u r b a n c e % (t;$Q) t o ^ w i l l a p p r o a c h t o 0 a s t » i f l o € U — n o f -{0}, e s p e c i a l l y i f ~$o = e i f " n R — -{0}, w h e r e i s d e t e r m i n e d f r o m ( 3 . 3 . 1 7 ) . The s e t U — rs R~-{0} a n d s e t QJ+n E+-{0} a r e r e m i n i s c e n c e s - o f t h e s t a b l e a n d u n s t a b l e m a n i f o l d s o f % = 0 a s s o c i a t e d w i t h ( 3 . 3 . 2 ) , r e s p e c t i v e l y ( c f . K e l l e y , 1 9 6 7 ; H a l e , 1 9 6 9 ) . 5.4 Pers is tence , c r i t i c a l i t y and s u p e r c r i t i c a l b i f u r c a t i o n G i v e n t h e e x i s t e n c e o f p e r s i s t e n t ^$(t;'$o) t o a s u p e r c r i t i c a l ^ ( c f . t h e o r e m 5 . 2 ) , i t r e m a i n s t o d e t e r m i n e t h e p r e c i s e n a t u r e o f t h e p e r s i s t e n t n o n m o d a l d i s t u r b a n c e s ( i . e . , s t a t i o n a r y , p e r i o d i c o r c h a o t i c ) . F r o m t h e b i f u r c a t i o n t h e o r y o f N a v i e r - S t o k e s e q u a t i o n ( J o s e p h a n d S a t t i n g e r , 1 9 7 2 ; J o s e p h , 1 9 7 6 ) , i t w o u l d b e e x p e c t e d t h a t t h i s n a t u r e i s r e l a t e d t o t h e p r o p e r t i e s o f principal eigenvalue or ( d e f i n e d a s t h e o n e w i t h t h e l a r g e s t r e a l p a r t ) o f t h e l i n e a r p a r t o f f ( ^ ; f ( a ) ; a ) a t some c r i t i c a l v a l u e s o f a . T h e g o a l o f t h i s s e c t i o n i s t o v e r i f y t h i s e x p e c t a t i o n f o r t h e s y s t e m ( 3 . 3 . 2 ) . I t c a n b e a r g u e d t h a t t h e l i n e a r s t a b i l i t y b o u n d a r y a=^ L 104 f o r the set e q u i l i b r i a } of (3.2.3) i n a-parameter space i s a c r i t i c a l i t y ( r e c a l l that r i s the root to the equation, L r (¥*(a))-r = 0) . To see t h i s , we f i r s t note that Re (cr) I - = L ' 1 a=r L r (¥*(r ) ) - r = 0 by (4.2.8), which implies that **(r ) i s L L L nonhyperbolic. Next, under the hyperbolicity condition, the asymptotic behavior of %(t;~$Q) undergoes q u a l i t a t i v e changes upon crossing a = r (cf. theorem 5.1 and theorem 5.2). L Further, the existence of persistent ~$(t;%o) to a s u p e r c r i t i c a l ¥*(a) with oc i n some neighborhood of cx—r indicates that a=i~L i s a s u p e r c r i t i c a l b i f u r c a t i o n point of (3.3.2) since i n the s u p e r c r i t i c a l side of « - = ^ h i (3.3.2) also has %=0 as i t s solution. To see the l i n k between the property of c ( r L ) and the nature of persistent $(t;$Q), we s h a l l focus on some neighborhood of a=o~L and r e s t r i c t discussion to the case in which the following holds: (a) cr (i- ^ ) (= aw) i s a simple eigenvalue of at ($,a) = (0,r ); furthermore, f-» at (0,r ) has no eigenvalue of L (p L the form imu> with m e 1- {1, -1} ; (5.4.1) (b) f at (0,r ) e range f 4 at (0,r j (5.4.2) ( X L CO L (c) the loss of s t a b i l i t y of {^(a) } at a=i"L i s s t r i c t i n the sense that (d/dt)Re (a-) \ _- > 0. (5.4.3) L It i s of in t e r e s t to note that while the above assumptions are made for s i m p l i c i t y , i t turns out that these conditions 105 a r e w e l l s a t i s f i e d i n n u m e r i c a l c a l c u l a t i o n s s u c h a s t h o s e r e p o r t e d i n t h e n e x t s e c t i o n . We f i r s t c o n s i d e r t h e c a s e o f <r(r^)=0 ( i . e . u =0 b y ( 5 . 4 . 1 ) ) . I t i s c r u c i a l t o n o t e t h a t t h e s t a t i o n a r y s o l u t i o n s o f ( 3 . 3 . 2 ) t r a c e o u t a c u r v e i n s t a t e - p a r a m e t e r s p a c e ( i . e . , i n RMxlR) a s t h e p a r a m e t e r a v a r i e s i n some p h y s i c a l l y r e l e v a n t r a n g e . We s h a l l show n e x t t h a t u n d e r c o n d i t i o n s (a) a n d ( b ) , t h e r e a r e two s u c h c u r v e s p a s s i n g t h r o u g h t h e p o i n t ( 0 , r L ) , o n e f o r t h e n u l l s o l u t i o n ~$=0 a n d one f o r t h e s t a t i o n a r y p e r s i s t e n t s t a t e o f '$(t;'$Q). We b e g i n w i t h t h e o b s e r v a t i o n t h a t d i m { r a n g e f | (Q;t ( r j , r j } = d i m { r a n g e A ( ^ ( r L ) ) ; (by ( 3 . 3 . 5 ) ) = M-l (by a- (r )=0 a n d c o n d i t i o n (a) ) ( 5 . 4 . 4 ) L Now, d e f i n e D f : R M + 1 -» RM b y D f E [f^, f^] . T h e n , c o n d i t i o n (a) a n d (b) t o g e t h e r w i t h ( 5 . 4 . 4 ) i m p l y t h a t d i m { n u l l Df}I -1 a=r L = ( d i m { d o m a i n Df} - d i m { r a n g e c f H | - = 2. ( 5 . 4 . 5 ) L N e x t , s e t (d/dt)% = 0 f o r a n y s t a t i o n a r y l i m i t i n g s t a t e % o f ( 3 . 3 . 2 ) t o h a v e ; a ) = 0. ( 5 . 4 . 6 ) F o r t a n g e n t (d$,da.) a t p o i n t ($, a) o n a n y s o l u t i o n c u r v e o f ( 5 . 4 . 6 ) , we t a k e t h e d i f f e r e n t i a l o n b o t h s i d e s o f ( 5 . 4 . 6 ) t o g e t 106 f £ ( ? ; 3 ( a ) ;cc) d$ + f a (a) ;a) da = 0, or Z)f [d0\da] T = 0. (5.4.7) It then follows from (5.4.5) that the solution space of (5.4.7) i s of dimension 2, and hence that there exists two d i s t i n c t tangents at (0,r ). L For the case <r (r ) =iu with u * 0, we appeal to the well L known Hopf theorem (cf. Marsden & McCracken, 1976) and note that the conditions (5.4.1)- (5.4.3) coincide with those leading to Hopf theorem. Thus, without further e f f o r t , we summarize the discussion for both u = 0 and w * 0: Theorem 5.3 (Asymptotic state and criticality) Let f (%4(<*•) ;a) be given by (3.3.5), and satisfy the conditions (5.4.1)-(5.4.3) at ($,a) = (0,r ) . Then, generically, for an equilibrium state ¥* (a) of (3.2.3) for which a is in a small neighborhood of r , its nonmodal disturbance %(t;%) L 0 approaches an unique stationary state as t -» to if <r (r^) = 0, or a periodic limiting state if <r (r^) = iu with w * 0. Remark 5.5.1 The foregoing results y i e l d no information on the d i r e c t i o n i n which the expected b i f u r c a t i o n branches o f f . A determination of the bi f u r c a t i o n d i r e c t i o n requires, in general, higher order derivatives of the vector f i e l d f (cf. Guckenheimer & Holmes, 1983). 107 Remark 5.5.2 By d e f i n i t i o n (3.3.1), %(t;%Q) i s t h e d i f f e r e n c e between t h e p e r t u r b e d f l o w and t h e o r i g i n a l e q u i l i b r i u m s t a t e . I t t h e n f o l l o w s t h a t under t h e c o n d i t i o n s o f theorem 5.2 and theorem 5.3, s u p e r c r i t i c a l b i f u r c a t i o n o c c u r s a t c r i t i c a l i t y a = r . More p r e c i s e l y , out of L i n s t a b i l i t y o f a g i v e n e q u i l i b r i u m s t a t e $ (a) t o an i n i t i a l p e r t u r b a t i o n ^ , a new st e a d y motion ^(a) +~$(t;$Q) o f (3.2.3) emerges if <r (r ) = 0, o r a p e r i o d i c f l o w b i f u r c a t e s from t (a) L if <r (r ) - iw w i t h u * 0. L Remark 5.5.3 L e t t h e c o n d i t i o n s (5.4.1)-(5.4.3) be s a t i s f i e d . A n e c e s s a r y c o n d i t i o n f o r s u b c r i t i c a l b i f u r c a t i o n b r a n c h i n g o f f a t a = i s v i o l a t i o n o f h y p e r b o l i c i t y f o r a i n t h e s u b c r i t i c a l s i d e o f some s u f f i c i e n t l y s m a l l n e i g h b o r h o o d o f a = . Note t h a t t h e above r e s u l t s e s s e n t i a l l y s o l v e t h e e x i s t e n c e problem. A c t u a l l y o b t a i n i n g t h e s e l i m i t i n g s t a t e s o f nonmodal d i s t u r b a n c e s r e q u i r e s , i n g e n e r a l , f u r t h e r n u m e r i c a l b i f u r c a t i o n a n a l y s i s (see Appendix 5E f o r d e t a i l s ) . 5.5 Numerical i l l u s t r a t i o n s a) Primary e q u i l i b r i u m branch and i t s s t a b i l i t y To i l l u s t r a t e t h e main r e s u l t s o f t h i s c h a p t e r , we c o n s i d e r p l a n e t a r y s c a l e a t m o s p h e r i c f l o w s o ver t o p o g r a p h y . I n F i g 5.1, we show t h e modal and g l o b a l s t a b i l i t y a n a l y s i s r e s u l t s f o r a s e t o f such f l o w s {t(r)} o b t a i n e d as e q u i l i b r i u m s t a t e s 108 o f ( 3 . 2 . 3 ) ( s e e t h e c a p t i o n f o r m o d e l p a r a m e t e r s ) . T h e t o p c u r v e a n d t h e c u r v e abed a r e i t s l i n e a r a n d n o n l i n e a r s t a b i l i t y m e a s u r e r ($) a n d r ( ¥ * ) , r e s p e c t i v e l y . I t i s n o t N L d i f f i c u l t t o s e e f r o m ( 4 . 2 . 8 ) - ( 4 . 2 . 9 ) t h a t r = r a n d r = r N L s e r v e a s t h e m o n o t o n i c g l o b a l s t a b i l i t y b o u n d a r y a n d t h e l i n e a r s t a b i l i t y b o u n d a r y o f t h e s e t { ¥ * ( r ) } , r e s p e c t i v e l y . T h e c u r v e efgb i s r^tf ) f o r t h e s e t o f e q u i l i b r i a {¥* (r) } w h i c h b i f u r c a t e s a t t h e c r i t i c a l i t y r = r L f r o m t h e p r i m a r y b r a n c h { ¥ * ( r ) } a n d i s o b t a i n e d v i a a n u m e r i c a l b i f u r c a t i o n a n a l y s i s ( s e e A p p e n d i x 5E f o r d e t a i l s ) , w i t h r=r^ a s i t s l i n e a r s t a b i l i t y b o u n d a r y . F o r i l l u s t r a t i o n o f t h e m a i n r e s u l t s , we w i l l f o c u s o n r e g i m e s (I) a n d (II) . See F i g 3 . 3 - 3 . 6 f o r a s y m p t o t i c d e c a y t o z e r o i n t h e s u b c r i t i c a l r e g i m e . b) Steady l i m i t i n g states and s t a t i c b i f u r c a t i o n M o v i n g a l o n g t h e p r i m a r y b r a n c h o f e q u i l i b r i a {^(r)} i n t h e d i r e c t i o n o f d e c r e a s i n g r , we c r o s s t h e l i n e a r s t a b i l i t y b o u n d a r y r=r^=l/(13.33 day) a n d s t e p i n t o t h e s u p e r c r i t i c a l r e g i m e . I t i s e x p e c t e d f r o m t h e o r e m 5 . 2 t h a t f (r) w i t h r i n r e g i o n (I) a n d (II) h a s p e r s i s t e n t n o n m o d a l d i s t u r b a n c e s ~$(t;$Q). F u r t h e r , t h e n a t u r e o f t h e n o n v a n i s h i n g l i m i t i n g s t a t e s o f ~$(t;~$J a s t •» » i s r e l a t e d t o t h e p r o p e r t i e s o f f^ a t (0,r ) when t h e u n d e r l y i n g t(r) h a s r c l o s e t o r . The L L d i r e c t c a l c u l a t i o n s i n d i c a t e t h a t t h e c o n d i t i o n s l i s t e d i n ( 5 . 4 . 1 ) - ( 5 . 4 . 3 ) a r e s a t i s f i e d a t t h e c r i t i c a l i t y r = l / ( 1 3 . 3 3 d a y ) w i t h w =0. T h u s , t h e l o n g t e r m b e h a v i o r o f %(t;%^) t o $(r) f o r r i n t h e s u p e r c r i t i c a l s i d e o f a s m a l l n e i g h b o r h o o d o f r i s e x p e c t e d t o b e s t a t i o n a r y a n d u n i q u e ( c f . t h e o r e m L 5 . 3 ) . T o v e r i f y t h e s e e x p e c t a t i o n s , n u m e r i c a l e x p e r i m e n t s b a s e d o n ( 3 . 3 . 2 ) a r e p e r f o r m e d , w i t h t h e r e s u l t s shown i n F i g 5 . 2 . T h e u n d e r l y i n g e q u i l i b r i a a r e t a k e n f r o m t h e p a r t ( f ' - » b ) on t h e p r i m a r y b r a n c h t o e n s u r e t h a t t h e f o u r t(r) a r e c l o s e e n o u g h t o t h e c r i t i c a l i t y r (= 13.33 d a y s ) . I t i s c l e a r l y L s e e n t h a t a f t e r t h e t r a n s i e n t g r o w t h , t h e d i s t u r b a n c e s s e t t l e down t o some s t e a d y s t a t e s . A s an e x a m p l e , we show i n F i g 5 . 3 t h e s t r e a m f u n c t i o n s f o r b i f u r c a t i o n o f an e q u i l i b r i u m s t a t e F i g 5 . 3 ( a ) i n t o a new s t e a d y f l o w F i g 5 . 3 ( f ) . T h e s n a p s h o t s ( c ) - ( f ) a r e t a k e n f r o m t h e e x p e r i m e n t f o r l / r = l l . 1 days ( c f . t h e t h i c k s o l i d l i n e i n F i g 5 . 2 ) . F i g 5 . 4 i n d i c a t e s t h a t t h e l i m i t i n g s t a t e o f 1f>(t;~$Q) i s i n d e p e n d e n t o f i n i t i a l c o n d i t i o n s . T h i s l o c a l u n i q u e n e s s i s t h e c o n s e q u e n c e o f t h e f a c t t h a t { N u l l JDf} | - =2 ( c f . ( 5 . 4 . 5 ) ) i n t h e p r e s e n t c a s e . L T h e e x p e r i m e n t i n i t i a l i z e d f r o m s'^ s u g g e s t s t h a t t h e r e g i o n o v e r w h i c h t h e u n i q u e n e s s h o l d s i s f a i r l y l a r g e . c) p e r i o d i c states and Hopf b i f u r c a t i o n We s h a l l u s e t h e s e t {^ ( r ) } b i f u r c a t i n g f r o m t h e p r i m a r y b r a n c h a t r = r =1/(13.33 XJ d a y ; f o r i l l u s t r a t i o n o f t h e s e c o n d p a r t o f t h e o r e m 5 . 3 . I t 110 i s seen from r tf ) (cf. curve efg i n F i g 5.1) that the L exchange of s t a b i l i t y for the set {¥* (r) } occurs at r=r'=l/(18.5 days). Moreover, a di r e c t c a l c u l a t i o n reveals that r=r' i s a Hopf b i f u r c a t i o n point, i . e . , the conditions L (5.4.1)-(5.4.3) hold with u * 0. It i s thus anticipated that nonmodal disturbances %(t;%0) to those t (r) with r i n the s u p e r c r i t i c a l side of some neighborhood of r=r' w i l l tend to asymptotic periodic states. A set of numerical experiments confirms t h i s a n t i c i p a t i o n . The results from two of these experiments are shown i n F i g 5.5 and F i g 5.6 , with the former displaying the time series of disturbance energy and the l a t t e r for the streamfunctions of b i f u r c a t i o n of a given steady flow shown i n F i g 5.6 (a) (corresponding to the thick curve i n F i g 5 .4) into a periodic one shown i n F i g 5.6 (b ) - ( f ) . d) Repeated s u p e r c r i t i c a l b i f u r c a t i o n In agreement with the forgoing numerical experiments, i t i s found from the bi f u r c a t i o n analysis (cf. Appendix 5E) that two stationary solution curves of (3.3.2) emanates from ^'rj) s u p e r c r i t i c a l l y . The one for steady l i m i t i n g state of ~$(t;~$o) plus the underlying primary e q u i l i b r i a {^(r)} y i e l d s the stationary b i f u r c a t i o n branch as seen i n F i g 5.7. A modal s t a b i l i t y analysis of t h i s branch {¥* (r) } indicates that when the difference between ¥* (r) and ¥*(r) as measured by the amplitude of the corresponding nonmodal disturbance ~$(t;~$) i s small ^ (r) i s l i n e a r l y stable but exchanges the s t a b i l i t y when the b i f u r c a t i n g flows deviate further away from the o r i g i n a l ones (cf. F i g 5.1), a resu l t v i r t u a l l y true to a l l s u p e r c r i t i c a l b i f u r c a t i o n (e.g., Joseph, 1976). Moreover, the c r i t i c a l i t y on the stationary b i f u r c a t i o n (r) } i s a Hopf b i f u r c a t i o n point from which a periodic branch of %(t;%Q) bifurcates and i s numerically obtained (cf. Appendix 5E). The l a t t e r plus the underlying (r) } gives r i s e to the periodic branch (r) } as shown i n F i g 5.7. It i s remarkable that each time out of i n s t a b i l i t y of a given flow to persistent nonmodal disturbances, a new flow emerges with a higher l e v e l of energy norm, which manifests as an increase i n the complexity l e v e l of temporal behavior (e.g., from steady to periodic) and s p a t i a l structure (e.g., from predominant zonal to wavy motion). The repeated s u p e r c r i t i c a l b i f u r c a t i o n such as the one seen i n F i g 5.7 has been a prominent topic i n the subject of t r a n s i t i o n to turbulence since Landau's conjecture (1944) on t h i s subject (cf., Joseph, 1976). The present treatment of the repeated b i f u r c a t i o n could be c a r r i e d out further by inqu i r i n g the s t a b i l i t y of the periodic branch (r)} v i a Floquet theory (cf. Arnol'd, 1983; Guckenheimer & Holmes, 1983) , which i s c e r t a i n l y necessary to study t r a n s i t i o n to turbulence or chaotic behavior i n the deterministic system (3.2.3). 112 e) T r a n s i t i o n due t o modal v s . nonmodal i n i t i a l p e r t u r b a t i o n s G i v e n t h e i n c r e a s i n g e v i d e n c e f o r a p l a u s i b l e r o l e o f n o n m o d a l i n i t i a l p e r t u r b a t i o n s i n e x p l o s i v e d e v e l o p m e n t o f d i s t u r b a n c e s o b s e r v e d i n t h e a t m o s p h e r e ( e . g . , S a n d e r s , 1986) a n d i n m o d e l s t u d i e s ( e . g , F a r r e l l , 1 9 8 5 ; F i g 3 . 7 a n d F i g 4 . 6 a n d 4 . 7 i n t h i s t h e s i s ) , i t i s o f i n t e r e s t t o s e e t h e r o l e p l a y e d b y t h e two t y p e s o f p e r t u r b a t i o n s i n t r a n s i t i o n f r o m o n e s t a t e o f f l o w t o a n o t h e r . F i g 5 . 8 d i s p l a y s t h e r e s u l t s f r o m two e x p e r i m e n t s , one t r i g g e r e d b y t h e o p t i m a l n o n m o d a l p e r t u r b a t i o n s a n d one b y t h e f a s t g r o w i n g n o r m a l m o d e l M Re (if) . I t i s s e e n c l e a r l y t h a t t h e f o r m e r i s much more e f f e c t i v e a s a t r i g g e r i n g a g e n t f o r t r a n s i t i o n t h a n t h e l a t t e r . 5 .6 C o n c l u d i n g remarks I n t h i s c h a p t e r , we h a v e a n a l y z e d t h e a s y m p t o t i c b e h a v i o r o f n o n m o d a l d i s t u r b a n c e s %(t;%Q) f o r a g e n e r i c c a s e w h e r e t h e p e r t u r b e d s t a t e i s h y p e r b o l i c u s i n g t h e d i r e c t m e t h o d o f L i a p u n o v a n d t h e l i k e . S p e c i f i c a l l y , we h a v e shown t h a t u n d e r c e r t a i n c o n d i t i o n s ( c f . t h e o r e m 5 . 1 ) t h e t r a n s i e n t g r o w t h a s s o c i a t e d w i t h f i n i t e a m p l i t u d e n o n m o d a l d i s t u r b a n c e s w i l l u l t i m a t e l y d i m i n i s h a s t -> » , a p h e n o m e n o n w h i c h h a s l o n g b e e n known f o r some l i n e a r m o d e l s ( e . g . , P e d l o s k y , 19 6 4 ) . We h a v e a l s o e s t a b l i s h e d t h a t a t r a n s i t i o n f r o m a g i v e n f l o w t o a n o t h e r d i s t i n c t s t a t e i s i n e v i t a b l e when t h e f l o w u n d e r c o n c e r n h a s g r o w i n g m o d a l p e r t u r b a t i o n s ( c f . , t h e o r e m 5 . 2 ) . Further, we have demonstrated that the nature of asymptotic behavior of persistent disturbances i s related to the nature of the neighboring nonhyperbolic point i n some underlying parameter space. As a f i n a l note, the r e s u l t s here are l o c a l i n the state space RM or i n the underlying parameter space. 114 Appendix 5A The di r e c t method of Liapunov The method i s widely available i n textbooks on dynamical systems (see, e.g., Hirsh and Smale, 1976; Verhulst, 1985, for more extensive treatment). To state the basic features of the subject, we consider the system (d/dt)x = q(x), x(tQ) = X q (5A.1) where g: ID -> RM i s a C 1 vector f i e l d on an open set D c R M . Let x be an equilibrium of (5A.1) . Let V : y •> R be a e d i f f e r e n t i a b l e function defined i n a neighbourhood HI c D of x . An orbital derivative of function V along the solution e x(t,xQ) of (5A.1), denoted by V, i s defined by V = W . g , where v the gradient operator i n R M . V i s said to be a Liapunov function i f i t s a t i s f i e s : (a) V(x ) = 0 and V(x) > 0 i n Ii - {x }, (5A.2) e e (b) V * 0 i n u - {x }. (5A.3) e I f V meets the condition (a) and (c) V < 0 i n U - {x }, (5A.4) e i t i s c a l l e d a strict Liapunov function. Theorem (Liapunovf s criterion for stability). An equilibrium state x^ of (5A.1) is stable if there exists a Liapunov function V(x) in a neighbourhood U c D of x ; It is e asymptotically stable if there exists a strict Liapunov function V in U. 115 Appendix 5B Au x i l i a r y lemmas The following lemmas are developed to f a c i l i t a t e the proof of the main r e s u l t s i n § 5.2 and § 5.3 and included here for convenience of reference. Some variants of lemma 5.1 and 5.2 may be found i n Hale (1969) or i n Hirsh and Smale (1974). Lemma 5.1 Let f: RM -> IR be written as f (x) = f (x) + w(x) p such that (i) f (x) is a negative (positive) definite p homogeneous polynomial of degree p; (ii) w(x) = o f|x||p) as |x|| -> 0. Then, f (x) is negative (positive) definite in U-{0}, OJ a neighbourhood QJ of 1 = 0. Proof: (for negative d e f i n i t e case) By ( i ) , we put max fp(x) = -a < 0 (5B.1) | X | = 1 and hence have, for x € IRM-{0} f (x) = \\xff fx/||x||p; s -a\\x\\p. (by (5B.1) and (i) ) Note that condition ( i i ) implies that for a given c (>0) , there e x i s t s a closed b a l l 1(0,5) such that \w(x) \ s e||x|p for x € B(0,8). (5B.2) Now, s e t t i n g c=(l/2)a i n (5B.2) and then combining (5B.1) and (5B.2) y i e l d f(x) £ -(a/2) ||x||p < 0, for x e B(0,8)-{0}, a desired r e s u l t . Identifying §(0 ,5) with U completes the proof 116 Lemma 5 . 2 Let P , Q , R e £ ( I R M ) . The Liapunov matrix equation P T Q + OP = R (5B .3 ) has a positive definite Q as a solution for any negative definite R if and only if Re (<r CP)) < 0. Proof: We p r o v e t h e c o n v e r s e o f lemma 5 . 2 , w h i c h i s u s e d s u b s e q u e n t l y . F o r a n y g i v e n n e g a t i v e d e f i n i t e R, we c o n s t r u c t a p o s i t i v e d e f i n i t e Q a c c o r d i n g t o Q B ^exp(PTt) (-R) expfPt) dt ( 5 B . 4 ) w h e r e expfPt) (s Y^^CPt)1/!!) i s t h e e x p o n e n t i a l o f o p e r a t o r P t . T h e c o n v e r g e n c e o f t h e i n t e g r a n t i n ( 5 B . 4 ) f o l l o w s f r o m t h e f a c t t h a t t h e r e e x i s t p o s i t i v e c o n s t a n t s c. a n d a , , c l I i=l,2, s u c h t h a t ||expfPt;|| s c^xp f - c ^ t ; , ( 5 B . 5 a ) \\exp(2Tt)\\ * c2exp(-a2t) ( 5 B . 5 b ) i f Re (a-CP)) <0. T h u s , Q i s w e l l d e f i n e d . To s e e Q s a t i s f i e s ( 5 B . 3 ) , c o n s i d e r (d/dt)^exp (P Tt) (-R;expfPt;J-= P T e x p f P T t ; f - R ; e x p ( P t ; + exp(PTt) (-R)exp(Pt)P ( 5 B . 6 ) w h e r e t h e f a c t t h a t Pexp(Pt)P 1=exp (Pt) i s u s e d . I n t e g r a t i n g ( 5 B . 6 ) f r o m 0 t o » , we o b t a i n (5B .3 ) w i t h t h e u s e o f t h e e s t i m a t e 0s||expfPTt; C -R;expfPt;|| s||exp CPTt) 11 C - R ; 11exp (Pt) || S S | | ( - R ; I | C expC-a t) c exp (-at) -» 0 as t -» ». (by (5B.5)) " u 1 1 2 2 and the corresponding l i m i t exp(PTt) (-R) exp (Bt) •> 0 as t •» ». T h i s establishes the converse of lemma 5.2. B Lemma 5.3 The bilinear operator B defined by (3.3.4f) is bounded such that |Bxy|| s cflx||||y||, for any x and y e RM, ( 5 B . 7 a ) where c i s given Jby ( 5 B . 7 b ) with c i a p o s i t i v e c o n s t a n t . Proof: F i r s t , by norm e q u i v a l e n c e i n R M ( c f . K r e y s z i g , p p . 9 6 ) , t h e r e e x i s t s a c o n s t a n t c i(>0) t h a t ||ac|| s c i ||x||, for any x e RM ( 5 B . 8 ) where norm fl. fl i s d e f i n e d , as t h e sum o f a b s o l u t e v a l u e s of a l l components o f x. Now, f o r any x and y i n IRM, c o n s i d e r i j k i j k i -1 j k 118 s c 2 H 2 ||y| 2 <b v (5B . 8 ) ) w h i c h l e a d s t o (5B.7a) i m m e d i a t e l y , w i t h c a c o n s t a n t g i v e n by (5B.7b). H Appendix 5C R e a l c a n o n i c a l t h e o r y o f l i n e a r o p e r a t o r We a p p l y t h e r e a l c a n o n i c a l t h e o r y o f l i n e a r o p e r a t o r s on RM (e.g. H i r s h and Smale, 1974) t o f i n d a new b a s i s f o r R M w.r.t w h i c h t h e l i n e a r p a r t A ( c f . (3.3.4a) o f t h e v e c t o r f i e l d f (3.3.3a) t a k e s a s p e c i f i c form. Lemma 5.4 Let the equilibrium state t of (3.2.3) satisfy (5.3.1). Then, RM has a basis h such that (i) RM = R~® R +, dim R~=Af-k and dim R + = k , (5C.1) with k equal to the sum of algebraic multiplicity for eigenvalues with positive real parts; (ii) A in the basis & has a matrix of form A & = diag[K_,K+J (5C.2) where A_ e £(R ) and A + € £(R+), satisfying Re(<r(AJ) < 0 and Re(<r(A+)) > 0, (5C.3) respectively. Proof: L e t E^(<r^) c C denote t h e g e n e r a l i z e d e i g e n s p a c e of n . A b e l o n g i n g t o <r^(A), i . e . , E^ = N u l l (A-cr^ (A;) 1 , where n^ i s a l g e b r a i c m u l t i p l i c i t y o f c r ^ . Put cr^ = + i p ^ . I f tr^ (A) i s r e a l ( i . e . , 0 ^ = 0 ) , by t h e S-N d e c o m p o s i t i o n ( c f . theorem 2 , pp.129 i n H i r s h and Smale, 1974), E^ has a b a s i s &^  g i v i n g t h e r e s t r i c t i o n o f A t o E.^ ( i . e . , A | E ^ ) a m a t r i x i n t h e r e a l c a n o n i c a l f o r m (*xUx = d ± a g ^ ] 1 ) , A | 2 ) . . . ( A | f l i ) 7 ( 5 C . 4 ) w h e r e e a c h o f d i a g o n a l b l o c k s i n ( 5 C . 4 ) h a s a f o r m 1 a a . 1 i-(5.C.5) w i t h t h e g e o m e t r i c m u l t i p l i c i t y o f (= d i m N u l l (A-cr^) ) . I f o\£ (A) i s c o m p l e x w i t h 0^ > 0, er^ (A) i s a l s o a n e i g e n v a l u e o f A a s A e JE(IRM). M o r e o v e r , T a k i n g t h e u n i o n o f r e a l p a r t s a n d i m a g i n a r y p a r t s o f t h e b a s i s e l e m e n t s o f E^(c^) y i e l d s a b a s i s A. f o r E I c " E . (cr ,) ©E . (cr .) X X x x' w h i c h h a s d e c o m p l e x i f i c a t i o n E . ( H E . n Xz X c ?M) c RM o f d i m e n s i o n 2 n A g a i n b y t h e S - N d e c o m p o s i t i o n , A | E ^ c u n d e r t h e b a s i s &^ h a s a m a t r i x o f f o r m ( 5 C . 4 ) b u t w i t h e a c h b l o c k s c h a r a c t e r i z e d b y t h e f o r m I a D . 211 2 I D . 2 X D . = x * , - & ? ] • N o t e t h a t t h e d i a g o n a l b l o c k s o f f o r m ( 5 C . 5 ) o r ( 5 C . 6 ) i n ( 5 C . 4 ) a p p e a r a s many t i m e s a s w h e r e a s cr^ o r D ^ a p p e a r s i n ( 5 C . 4 ) a s o f t e n a s n ^ . I t i s c l e a r t h a t t h e s e b a s e s {&^} f o r m a l i n e a r l y i n d e p e n d e n t s e t a n d h e n c e t h e u n i o n o f &. p r o v i d e s a b a s i s &. f o r R 120 & = &, u &0 v...& u u ( 5 C . 7 ) 1 2 p P+l S T h e d i r e c t sum d e c o m p o s i t i o n ( 5 C . 1 ) o f IRM f o l l o w s i m m e d i a t e l y f r o m s u b d i v i d i n g t h e b a s i s A ( 5 C . 7 ) i n t o two p a r t s , one c o n s i s t i n g o f t h e b a s e s &^ (i=l,2, . . .,p) o f E^(<r^) w i t h Re(<r^(A.)) < 0 a n d o n e c o m p o s e d o f t h e r e m a i n i n g &^ o f E^(cr^) w i t h Re(<r.(R.)) > 0. S i n c e V x e E ^ ( o r E^) , i t h o l d s t h a t (K-a.(A)) 1Ax = 0, E . ( o r E . ) i s t h u s i n v a r i a n t u n d e r A . J. J. X r H e n c e , A u n d e r t h e b a s i s A h a s a m a t r i x r e p r e s e n t a t i o n o f f o r m A & = dxag [(^1)k , (K2)h (A )k ] ( 5 C . 8 ) 1 2 s w h i c h y i e l d s t h e a s s e r t e d f o r m ( 5 C . 2 ) a f t e r g r o u p i n g t h e f i r s t p b l o c k s i n ( 5 C . 8 ) i n t o A _ a n d t h e r e m i n d e r i n t o A . T h e p r o p e r t y ( 5 C . 3 ) o f A + a n d A _ f o l l o w s f r o m t h e f a c t t h a t t h e e i g e n v a l u e s <r. o f A i n E . (o r E . ) a r e as t h e same a s X X X r t h o s e o f ( A ^)^ b y ( 5 C . 4 ) - ( 5 C . 6 ) . T h i s c o m p l e t e s t h e p r o o f . B i A p p e n d i x 5D L i a p u n o v i n s t a b i l i t y f u n c t i o n E x i s t e n c e o f a f u n c t i o n V+ s a t i s f y i n g t h e p r o p e r t i e s ( a ) - ( c ) i s t h e k e y s t e p t o e s t a b l i s h t h e o r e m 5 . 2 . T h e c o n s t r u c t i o n o f s u c h V+ f o r t h e s y s t e m ( 3 . 3 . 2 ) i s m o t i v a t e d b y t h e t r e a t m e n t i n H a l e ( 1 9 6 9 ) . Lemma 5 . 5 L e t the equilibrium state $ of (3.2.3) satisfy (5.3.1). Then, there exists a C 1 function V+: QJ+ c RM -» R, V+ 121 the neighborhoods of the null solution ~$ = 0 of (3.3.2), such that (a) V+ ($) = 0 as % -> 0, (b) V+ > 0 in U + - {0}, (c) V+ C$ ) > 0, V l | I c R + with % -» 0 as n co, n I nj n where R + is determined in Lemma 5.4. Proof: We pro v e t h e lemma by c o n s t r u c t i n g a f u n c t i o n V+ w h i c h meets t h e c o n d i t i o n s l i s t e d i n (5D.1 )-(5D.3). F i r s t , we not e t h a t m a t r i c e s r e p r e s e n t i n g a g i v e n o p e r a t o r i n d i f f e r e n t bases a r e s i m i l a r . I t t h e n f o l l o w s t h a t t h e r e e x i s t s a n o n s i n g u l a r m a t r i x P e i£(RM) such t h a t P_1AP = A & = diag[K_,K+], (5D.1) where A i s t h e l i n e a r p a r t o f t h e v e c t o r f i e l d f ( c f . , ( 3 . 3 . 3 ) - (3.3.5)) and A^ i s i t s c o u n t e r p a r t w.r.t. t h e b a s i s & ( c f . , lemma 5.4). I n f a c t , P i t s e l f d e f i n e s a map from R M w i t h t h e b a s i s & ( c f . (5C.7)) t o R M w i t h t h e s t a n d a r d b a s i s . S u b s t i t u t i n g % = V%. (5D.2) i n t o ( 3 . 3 . 2 ) , we have t h e g o v e r n i n g e q u a t i o n w r i t t e n w.r.t t h e b a s i s & (d/dt)% = diag[K_,K+]% + BP^P^ ( 5 D . 3 ) where B i s t h e same as b e f o r e , g i v e n by ( 3 . 3 . 4 f ) . Now, d e f i n e t h e f u n c t i o n V+ by 122 V+= <$,Q+!$>, Q+= diag [-Q^QJ, (5D.4) w h e r e € £(RK~k) a n d Q2 € £ (Rk) a r e t h e s o l u t i o n s t o + Q A = -I , (5D.5) 1 1 M-k C-A^Q2 + Q2(-A+) = ( 5 D . 6 ) I a n d I t h e u n i t o p e r a t o r s o f o r d e r M-k, k, r e s p e c t i v e l y . T h e e x i s t e n c e o f Qi a n d Q2 a r e g u a r a n t e e d b y lemma 5.2. The p r o p e r t y (a) f o l l o w s i m m e d i a t e l y f r o m t h e d e f i n i t i o n . (5D.4) . T o s e e ( c ) , we u s e t h e f a c t (5C.1) t o w r i t e % i n t h e f o r m $ = [ u , v ] T , u e IR a n d v e IR +. T h e n , f o r a n y {~$ } c R + w i t h $ -» n n 0 a s n -> co, (5D.4) y i e l d s V+(% ) = <v ,Qv > > 0. (5D.7) n n 2 n T h e p o s i t i v e n e s s i n (5D.7) r e s u l t s i n f r o m t h e same p r o p e r t y o f Q 2 . We t u r n t o (b) . A d x r e c t e v a l u a t i o n o f V a l o n g t h e o r b i t o f ~$(t;%o) o f (5D.3) g i v e s , V+= diag[-AlQx - Q2A_, A*Q2 + Q 2A +7*> + o (\\-$f) = |*|2 + o([$f) . (5D.8) w h e r e we h a v e u s e d a n a r g u m e n t s i m i l a r t o lemma 5.3 t o g e t t h e e s t i m a t e o f t h e h i g h o r d e r t e r m s i n (5D.8). A p p l y i n g lemma 5.1 r e s u l t s i n ( b ) , w h i c h c o m p l e t e s t h e p r o o f . B Appendix 5E A l g o r i t h m f o r b i f u r c a t i o n a n a l y s i s H e r e , we o n l y b r i e f l y o u t l i n e t h e n u m e r i c a l p r o c e d u r e s f o r f i n d i n g t h e s t a t i o n a r y o r p e r i o d i c l i m i t i n g s t a t e s o f 123 nonmodal d i s t u r b a n c e s governed by (3.3.2a) (see t h e r e f e r e n c e s g i v e n below f o r a d e t a i l e d a c c o u n t ) . F o r s t e a d y s t a t e s , t h e f o l l o w i n g s e v e r a l s t e p s a r e i n v o l v e d : 1) t o l o c a t e t h e s t a t i o n a r y b i f u r c a t i o n p o i n t ; 2) t o f i n d t h e t a n g e n t s on t h e b i f u r c a t i n g b ranches emanating from (0,r^) (which a r e t h e s o l u t i o n s t o (5 . 4.7) and t h e n use them as a p r e d i c t o r t o o b t a i n an approximate s t e a d y s o l u t i o n t o (3 . 3 . 2 a ) ; and 3) t o a p p l y t h e Newton-Ramphson a l g o r i t h m ( c f . P a r k e r and Chua, 1989) as a c o r r e c t o r t o improve i t t o get t h e t r u e s t e a d y s o l u t i o n s o f (3.3.2a); 4) t o use t h e c o n t i n u a t i o n a l g o r i t h m ( K e l l e r , 1978) t o t r a c e out t h e r e m a i n i n g b r a n c h . F o r 1 ) , Det (VL) i s i n t r o d u c e d as a t e s t f u n c t i o n t o l o c a t e t h e s t a t i o n a r y b i f u r c a t i o n p o i n t , w i t h M d e f i n e d by M s ( 5 E . 1 ) dc/tj da where f i s g i v e n by (3.3.3a). I t can be shown t h a t a s i g n change i n det (M) d u r i n g t h e p r o c e s s o f c o n t i n u a t i o n w.r.t. parameter a i n d i c a t e s t h a t a b i f u r c a t i o n p o i n t i s p a s s e d over ( c f . P a r k e r and Chua, 1989). An a p p l i c a t i o n o f b i s e c t i o n a l g o r i t h m t o det (M) y i e l d s t h e b i f u r c a t i o n p o i n t . F o r s t e p 2, (5 . 4.7) i s s o l v e d f o r t h e n u l l space o f SDf u s i n g t h e s i n g l e v a l u e d e c o m p o s i t i o n (SVD) ( c f . Nobel and D a n i e l , 1988) . The b a s i s elements o f t h e n u l l space o f Df are th e n u s e d as a p p r o x i m a t e t a n g e n t s . More s o p h i s t i c a t e d b r a n c h i n g s w i t c h t e c h n i q u e s a r e a v a i l a b l e but i n c r e a s e t h e amount of computation s i g n i f i c a n t l y (cf. Kubicek and Marek, 1983). For periodic asymptotic states, we convert the i n i t i a l value problem (3.3.2) into a two point boundary value problem (d/dt)[ I ] = [ f (5E.2) subject to the boundary conditions = 0 (5E.3) 1(0) - %(T) •fkC$(0)rt,a) where T i s the period and the ($ (0) ,4, a) = 0 i s introduced into (5E.3) as a phase condition, with k a r b i t r a r i l y f i x e d to between 1 and M (cf. Seydel, 1988) . Other types of phase conditions are also possible (Seydel, 1988) . The standard shooting method (cf. Ascher, et a l , 1988) i s used for solutions of (5E.2)-(5E.3). 125 0 . 3 E - 0 1 0 .3E+00 EKMAN DAMPING (1/day) F i g 5.1 S t a b i l i t y regime diagram for a family of e q u i l i b r i a {^(r) } with 1/x ranging from 29.5 days to 3.5 days, U = 22.0 m/s and topography being of zonal wavenumber-1 and of height 500 m. The top curve and the curve abed are i t s r ($) and x (t), respectively, with N L r-r =1/(5.71 days) and r=x^~l/(13.33 days) as i t s MGS boundary and li n e a r s t a b i l i t y boundary. The curve efgb i s r ($ ) for the set of e q u i l i b r i a {t (r) } b i f u r c a t i n g at Li c r i t i c a l i t y r =^' L from the primary branch {¥*(r)). r=x'=l/(18.5 days) i s the li n e a r s t a b i l i t y boundary for the set {** (r) }. 126 CM M « \ CM •* •K E 19 CC b J O Z <C CO cc I— CO 10.0 r 8 .0 6 .0 4 . 0 -2 .0 -0 .0 0 .0 100 .0 200 .0 300 .0 400 .0 TIME IcUy) Fig 5 .2 A s y m p t o t i c n o n v a n i s h i n g s teady s t a t e s o f nonmodal d i s t u r b a n c e s . The u n d e r l y i n g e q u i l i b r i u m s t a t e s are those l o c a t e d on the p a r t (f -»b) o f the p r i m a r y b r a n c h (cf .Fig 5.1) f o r v a l u e s o f r g i v e n i n the f i g u r e . 127 U) basic s t a t e (b) s tc) t~BO days td) t=ltTO days (e) t=320 days (f) t=400 days N L \ \ \ \ Ai i • ' 8 B \ \ \ > s y J IV)) I O , Fig 5.3 Streamfunctions f o r b i f u r c a t i o n of an e q u i l i b r i u m s t a t e (a) i n t o a new steady flow (f) . The snapshots ( c ) - ( f ) are from the experiment f o r 2/r=17.1 days ( c f . t h e t h i c k s o l i d l i n e i n F i g 5.2). 128 CM * CM X * E CD CC L d U J CJ 2 «S m to i—i Q 2 0 . 0 16.0 - 12 .0 8 .0 4 . 0 0 .0 I \ s ' H -1 M-2 J L. 0 .0 100.0 200 .0 300 .0 400 .0 TIME (day) F i g 5 . 4 L o c a l uniqueness o f asymptotic steady s t a t e of nonmodal d i s t u r b a n c e s . The u n d e r l y i n g e q u i l i b r i u m s t a t e i s the same as one i n F i g 5.2 f o r the experiment of l/r=16.4 days. The s' i s obtained from s c a l i n g s such t h a t %(t;s') M • M M at t=0 has 10 % of the b a s i c s t a t e energy. 129 CM M * in \ CM M X e >-cs cc UJ z UJ CJ z e t m cc t-to » — < a 40.0 r 32.0 -- 24.0 UJ 16.0 -8.0 -0.0 r=l/( ;24.8 d a y s ) r=2/C15.1 days) 0.0 100.0 200.0 300.0 400.0 TIME (day) Fig 5 .5 Periodic l i m i t i n g states of nonmodal disturbances, with periods 46.3 days for the s o l i d l i n e and 85.8 days for dahsed l i n e , respectively. T h e underlying e q u i l i b r i a are located on the unstable section of the stationary b i f u r c a t i o n branch (cf.Fig 5.1), with the values of r as indicated. 130 (») basic s t a t e (b) t=V Ic) t = t ' + T / 4 ( d ) t=t'+2T/4 ( e ) (f) \ — - x \ K y \ -- —_^-sa. - -4 .50 ~ - « s I ' / - . — -S-°& - — \ r 50 • ~ ^ -4. SO F i g 5 .6 Streamfunctions f o r b i f u r c a t i o n of an e q u i l i b r i u m state (a) into a p e r i o d i c f low. The snopshots (b)-(f) are taken from the experiment for l/r=24.8 days (cf . the s o l i d l i n e i n F i g 5.5) over a cycle of o s c i l l a t i o n , with t'=336.9 days and 1=43.3 days. 131 1 6 0 . 0 r 1 5 0 . 0 | 1 4 0 . 0 o cc lx) z 1 3 0 . 0 -1 2 0 . 0 0 . 2 E - 0 1 primary branch s t a t i o n a r y branch o o o o P e r i o d i c branch _! ' ' ' ' 0 . 2 E + 0 0 E K M A N D A M P I N G ( 1 / d a y ) F i g 5.7 Repeated s u p e r c r i t i c a l b i f u r c a t i o n for the primary branch of e q u i l i b r i a ( c f . F i g 5.1). The point marked by x on the primary branch i s a stationary b i f u r c a t i o n point whereas the symbol + indicates the Hopf bi f u r c a t i o n point. The l i n e s drawn with dahsed corresponds to unstable equilibrium states. 132 5.0 r 0.0 100.0 200.0 ; 300.0 400.0 TIME (d*y) F i g 5 . 8 Nonmodal versus modal i n i t i a l perturbations in t r a n s i t i o n to a periodic state. The basic state i s from the stationary b i f u r c a t i o n branch (r) } with l/r=18.8 days ( c f . F i g 5.1), located near the secondary b i f u r c a t i o n point r=r'. The i n i t i a l growth rate of %(t;s ) and $ (t;Re (t) ) are L M 1/(5.08 days; and 1/(400.0 days;, resp e c t i v e l y . 133 CHAPTER 6 CLOSURE MODELING: FORCED-DISSIPATED STATISTICAL EQUILIBRIUM OF LARGE SCALE QUASI-GEOSTROPHIC FLOWS OVER RANDOM TOPOGRAPHY 6.1 Introduction T h e s t a t i s t i c a l e q u i l i b r i u m a c h i e v e d b y q u a s i - g e o s t r o p h i c f l o w s o v e r a r a n d o m t o p o g r a p h y h a s b e e n i n v e s t i g a t e d b y many a u t h o r s . S a l m o n , H o l l o w a y a n d H e n d e r s h o t t ( 1 9 7 6 ; h e r e a f t e r r e f e r r e d t o a s SHH) o b t a i n e d t h e i n v i s c i d - u n f o r c e d s t a t i s t i c a l e q u i l i b r i u m ( a l s o r e f e r r e d t o a s t h e a b s o l u t e e q u i l i b r i u m ) f o r o n e - a n d t w o - l a y e r f l o w s u s i n g t h e m e t h o d o f c l a s s i c a l s t a t i s t i c a l m e c h a n i c s . T h e y f o u n d t h a t a t a b s o l u t e e q u i l i b r i u m , t h e l o w e r - l a y e r f l o w i s a n t i - c y c l o n i c c i r c u l a t i o n a r o u n d h i l l s . F u r t h e r m o r e , f o r s c a l e s l a r g e r t h a n t h e i n t e r n a l d e f o r m a t i o n r a d i u s , t h e f l o w c o r r e l a t i o n w i t h t o p o g r a p h y was shown t o e x t e n d t h r o u g h o u t t h e d e p t h o f f l u i d . W h i l e t h e a b s o l u t e e q u i l i b r i u m p r o v i d e s v a l u a b l e i n s i g h t s i n t o t h e d y n a m i c s o f a p r o b l e m , o m i s s i o n o f f o r c i n g a n d d i s s i p a t i o n l e a d s t o t h e r e s u l t s w h i c h h a s l i t t l e t o do w i t h v i s c o u s f l o w s i n many a s p e c t s o f d y n a m i c s . To c o p e w i t h t h e s e a s p e c t s , H e r r i n g (1977) a n d H o l l o w a y (1978) d e v e l o p e d s t a t i s t i c a l t h e o r i e s f o r f o r c e d - d i s s i p a t e d f l o w s on t h e f - p l a n e u s i n g t h e d i r e c t i n t e r a c t i o n a p p r o x i m a t i o n ( K r a i c h n a n , 1 9 6 7 ) a n d t h e t e s t f i e l d m o d e l (TFM) ( K r a i c h n a n , 1971) . R e c e n t l y , some e f f o r t h a s b e e n made t o e x t e n d t h e 134 f - p l a n e f o r m u l a t i o n t o t h e 0 - p l a n e c a s e w i t h a n o n v a n i s h i n g d o m a i n a v e r a g e d z o n a l v e l o c i t y c o m p o n e n t ( H o l l o w a y , 1987 ; h e r e a f t e r r e f e r r e d t o H87) T h e o b j e c t i v e o f t h i s w o r k i s t o s t u d y t h e s t a t i s t i c a l e q u i l i b r i u m e s t a b l i s h e d when a f l o w i s f o r c e d o v e r a r a n d o m t o p o g r a p h y b y a n e x t e r n a l u n i f o r m z o n a l momentum s o u r c e . The s t u d y i s b a s e d o n a c l o s u r e m o d e l (H87) a s w e l l a s o n d i r e c t n u m e r i c a l s i m u l a t i o n ( d n s ) . One a s p e c t o f t h e e q u i l i b r i a i s s i n g l e d o u t f o r c o n s i d e r a t i o n , i . e . , t h e v o r t i c i t y - t o p o g r a p h y c o r r e l a t i o n a n d t h e r e s u l t i n g t o p o g r a p h i c s t r e s s a c t i n g on t h e e q u i l i b r i u m f l o w s . I n § 6 . 2 , t h e c l o s u r e f o r m u l a t i o n i s o u t l i n e d a n d two i n v a r i a n t s o f m o t i o n f o r t h e p h y s i c a l s y s t e m u n d e r c o n c e r n a r e b r i e f l y d i s c u s s e d . I n § 6 . 3 , we s o l v e n u m e r i c a l l y t h e c l o s u r e e q u a t i o n s f o r p a r a m e t e r s r e l e v a n t t o m i d o c e a n e n v i r o n m e n t , w i t h r e s u l t s c o m p a r e d t o t h o s e f r o m e n s e m b l e d n s . A b r i e f summary a n d d i s c u s s i o n o f t h e r e s u l t s a r e p r e s e n t e d i n § 6 . 4 . 6.2 Closure formulation 6.2.1 A s e l f - c o n s i s t e n t model C o n s i d e r a f l o w g o v e r n e d b y ( 1 . 2 . 1 ) , f o r c e d b y an e x t e r n a l z o n a l momentum s o u r c e ( i . e . , 0*=-C7*y i n ( 1 . 2 . 1 ) ) a n d b o u n d e d b y a d o u b l e p e r i o d i c c e l l ( I . e . , s a t i s f y i n g B . C . ( 1 . 2 . 2 b ) ) . 135 F u r t h e r , we w r i t e t h e s t r e a m f u n c t i o n a s ( c f . H87) ip(x,y,t) = -U(t)y + *(x,y,t), ( 6 . 2 . 1 ) t h e n a s e l f - c o n s i s t e n t s y s t e m i s o b t a i n e d ( c f . H87) 3 . V 2 * + J C * -Uy, V 2 * + p y + h) = - D V 2 * , ( 6 . 2 . 2 ) (d/dt)U = r(U* - U) + hd*/dx, ( 6 . 2 . 3 ) * ( x + l , y , t ) = * C x , y , t ; , <b(x,y+l,t) = <b(x,y,t), ( 6 . 2 . 4 ) w h e r e (h/H)fQ i n o u r e a r l i e r n o t a t i o n ( c f . ( 1 . 2 . 1 ) ) i s s i m p l y w r i t t e n a s h t o b e more c o n s i s t e n t w i t h t h e n o t a t i o n c o m m o n l y u s e d i n c l o s u r e f o r m a l i s m , a n d D i n ( 6 . 2 . 2 ) d e n o t e s t h e d i s s i p a t i o n o p e r a t o r i n c l u d i n g t h e Ekman a n d b i h a r m o n i c d i s s i p a t i o n s . T h e t e n d e n c y e q u a t i o n ( 6 . 2 . 3 ) may b e o b t a i n e d f r o m c o n s i d e r i n g t h e x - d i r e c t e d momentum e q u a t i o n ( e . g . , H a r t , 1 9 7 9 ) . T h e t e r m i n ( 6 . 2 . 3 ) w i t h o v e r b a r i s r e f e r r e d t o a s t h e topographic stress x a r i s i n g f r o m t h e f l o w - t o p o g r a p h y i n t e r a c t i o n , w h e r e t h e o v e r b a r d e n o t e s t h e d o m a i n a v e r a g e . I n t h e n e x t s e c t i o n , t h e p r o b l e m d e f i n e d b y ( 6 . 2 . 2 ) - ( 6 . 2 . 4 ) i s p r o j e c t e d o n t o a F o u r i e r s p a c e , g i v i n g a s e t o f c o u p l e d n o n l i n e a r 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 s f o r w h i c h t h e c o r r e s p o n d i n g s t a t i s t i c a l p r o b l e m i s f o r m u l a t e d v i a moment e q u a t i o n s . 6.2.2 Moment equations B y v i r t u e o f ( 6 . 2 . 4 ) , we c h o o s e t h e b a s i s f u n c t i o n s , w h e r e x = ( x , y ) a n d k e K d e f i n e d b y 136 K • -|k = (m,n)2n/L \ (m,n) = 0, ± l,±2,..,ls |k| s j , and expand *, h i n the form (*,h) = [ <*K'hK)e±}:"*' (6.2.5) k where the summation i s over a l l k € DC. After introducing (6.2.5) into (6.2.2) and (6.2.3), and using the orthonormal property of set G, we obtain the spectral version of (6.2.2)-(6.2.3), i n terms of < k (= ~k2\) (d/dt)^ = -i* xC7h k - a U k + v k K k " [ \ p q + ( 6 . 2 . 6 ) A (d/dt)U = r(U - U) - Imj £ k ^ h ^ / k 2 \, ( 6 . 2 . 7 ) k ^ = - p/k2)k , ( 6 . 2 . 8 ) A k p q s Z- (k x p ) / p 2 = 2- (p x q ) / p 2 = Z- Cq x k) / p 2 , ( 6 . 2 . 9 ) where ()* denotes the complex conjugate of quantity (); the symbol £ represents the summation over wavevectors p , q such A that k + p + q = 0; i s the l i n e a r Rossby wave frequency for mode k and A. i s the int e r a c t i o n c o e f f i c i e n t for the kpq t r i a d ( k , p , q ) ; v>k i s the d i s s i p a t i o n operator i n the spectra space; f i n a l l y , Z i s the unit vector along the z-axis. Consider a phase space r spanned by the re a l and imaginary part of a l l < k and h f c plus U ( i . e . , Re (Cfc) ,Im(Cjc) ,Re (h^) ,Im(hk) ,17, where the symbols Im, Re denote the imaginary and re a l part of the quantity) . Then, 137 o n e s i n g l e r e a l i z a t i o n o f (6 .2 .6) a n d (6 .2 .7) y i e l d s a t r a j e c t o r y i n r, a n d a n e n s e m b l e o f s u c h r e a l i z a t i o n s c o r r e s p o n d s t o t h e e v o l u t i o n o f a c l u s t e r o f p h a s e p o i n t s J^Re (C^), Im (t^) .Re (hyj . Im (hy), t/| i n r. A c o m p l e t e s t a t i s t i c a l d e s c r i p t i o n o f t h e e n s e m b l e r e q u i r e s t h e k n o w l e d g e o f p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n P (Re {C,^), imU;^), Re(h^), Im(hy).U.t) w h i c h e x i s t s i n p r i n c i p l e a s t h e s o l u t i o n t o t h e L i o u v i l l e e q u a t i o n i m p l i e d b y (6 .2 .6) a n d ( 6 . 2 . 7 ) , o r e q u i v a l e n t l y , r e q u i r e s t h e j o i n t moments o f a l l o r d e r s b e t w e e n t h e p h a s e p o i n t s -JRe (<k), Im (<k) ,Re (h^), Im (h^), C/|. H o w e v e r , t h e d e t e r m i n a t i o n o f P c a n r a r e l y b e c a r r i e d o u t i n t h e p r e s e n c e o f e x t e r n a l f o r c i n g U* a n d d i s s i p a t i o n D , w h i l e i t i s p o s s i b l e t o s e e k t i m e i n d e p e n d e n t P a s a f u n c t i o n a l o f t h e i n v a r i a n t s o f (6 .2 .6) a n d (6 .2 .7) f o r t h e c a s e w h e r e C7* = D = 0 ( e . g . , S H H ; C a r n e v a l e & F r e d e r i k s e n , 1987) . I n t h i s s t u d y , we r e s t r i c t o u r s e l f t o t h o s e s t a t e s o f t h e e n s e m b l e f o r w h i c h t h e e n s e m b l e s t a t i s t i c s i s t i m e - i n d e p e n d e n t . We r e f e r t o t h o s e s t a t e s a s forced-dissipative statistical equilibrium a s o p p o s e d t o t h e a b s o l u t e e q u i l i b r i u m . F u r t h e r m o r e , we a r e o n l y i n t e r e s t e d i n two s e c o n d o r d e r moments o f t h e e q u i l i b r i a , i . e . , s <CJCCJc> a n d <C^hj c >, w h e r e t h e a n g l e b r a c k e t d e n o t e s t h e e n s e m b l e a v e r a g e . P h y s i c a l l y , t h e f i r s t r e p r e s e n t s t h e m o d a l s p e c t r a l f o r t h e v o r t i c i t y a n d t h e s e c o n d f o r t h e m o d a l s p e c t r a l o f t h e v o r t i c i t y - t o p o g r a p h y c o r r e l a t i o n . 138 I n w h a t f o l l o w s , we o u t l i n e t h e c l o s u r e p r o c e d u r e a t t h e l e v e l Z k a n d C ^ . F r o m ( 6 . 2 . 6 ) , we h a v e (d/dt)Zk = 2k lmrc f c; - 2vyZ^ ~ I 2J1kpq{Rfi<W*> + R e <Wk >}' ( 6 . 2 . 1 0 ) A (d/dt)C^ = ikxUH^ + ( i w k - v k ) C k w h e r e Hy& <h^h^>. I t i s c l e a r t h a t t h e s o l u t i o n s o f ( 6 . 2 . 1 0 ) , a n d ( 6 . 2 . 1 1 ) r e q u i r e t h e k n o w l e d g e o f t h e t h i r d o r d e r moments o f t h e f o r m s < « h > a n d <h£h> w h i c h a r e i n t u r n d e p e n d o n t h e f o u r t h o r d e r m o m e n t s , a n d s o o n , l e a d i n g t o a u n c l o s e d h i e r a r c h y o f moment e q u a t i o n s . A t t h i s s t a g e , some c l o s u r e h y p o t h e s i s o n t h e r e l a t i o n s among d i f f e r e n t o r d e r moments a r e n e c e s s a r y i n o r d e r t o c l o s e t h e h i e r a r c h y a t t h e l e v e l o f Z k a n d t? k . 6.2.3 Closure hypothesis and master equations F o r t h i s p u r p o s e , we f i r s t a s s u m e t h a t t h e e n s e m b l e s t a t i s t i c s o f t o p o g r a p h y lnK} ^ s G a u s s i a n . T h e d i r e c t c o n s e q u e n c e o f t h i s a s s u m p t i o n i s t h e v a n i s h i n g o f o d d moments o f t h e f o r m <hhh>. N e x t , we a s s u m e : ( i ) t h e r o l e o f t h e f o u r t h o r d e r c u m u l a n t s , i n t r o d u c e d i n e x p r e s s i n g f o u r t h o r d e r moments i n t e r m s o f p r o d u c t s o f t h e s e c o n d o r d e r 139 moments Z k a n d ffk, i s t o p r o v i d e d a m p i n g a c t i o n l e a d i n g t o t h e s a t u r a t i o n o f t h i r d o r d e r m o m e n t s ; ( i i ) t h e c h a r a c t e r i s t i c r e l a x a t i o n t i m e f o r t h i r d o r d e r moments i s s h o r t e r t h a n t h o s e f o r t h e s e c o n d o r d e r moments ( i . e . , 2 ^ ) . V i e w e d s i m p l y , t h e h y p o t h e s i s ( i ) p e r m i t s r e p l a c e m e n t o f t h e f o u r t h o r d e r c u m u l a n t s i n t h e t h i r d o r d e r moment e q u a t i o n s b y a l i n e a r d a m p i n g t e r m w i t h c h a r a c t e r i s t i c e d d y - d a m p i n g r a t e M^pq f o r t h e t r i a d (k,p,q). T h e h y p o t h e s i s ( i i ) a l l o w s u s t o o b t a i n t h e q u a s i - s t a t i o n a r y t r i p l e c o r r e l a t i o n s f r o m t h e t h i r d o r d e r moment e q u a t i o n s <C_C C, > = -0' 1 1 \A , Z Z. + A , Z Z. + A. Z Z spsqsk kpq \ pqk q k qpk p k kpq p q + A . Z Ci + A . Z.C* + A . Z C* + A . Z.C* pqk q k pkq k q qpk p k qkp k p + Avr>„ z « c « + \ m z „ c « l ' ( 6 . 2 . 1 2 a ) kpq p q kqp q pj <VqV = - e k ; q { A p q k z q c k + Z p c k + w Z q H k + Aqpk V* + Vq + V? CkCp}' ( 6 . 2 . 1 2 b ) = -ekPq {^pqk CqCk+ APqk V* + Apkq V,}' ( 6 . 2 . 1 2 c ) S^cpq S Akpq + Akqp' Apqk S Apqk + Apkq' Aqpk S Aqpk + Aqkp' ( 6 . 2 . 1 2 d ) w h e r e © i f 1 ^ a r e c h a r a c t e r i s t i c o f r e l a x a t i o n ( t o w a r d s 140 quasi-equilibrium) by the nonlinear transfer and external d i s s i p a t i o n . S p e c i f i c a t i o n of the three arrays ejj.pq (i=l,2,3) are subject to various constraints resulted from the consideration of conservation requirements on the closure model and the s t a t i s t i c a l r e a l i z a b i l i t y ( i . e . , a 0 V k € K and t s 0) (cf., H78) , among which are the following (i) 6 j [ p q = e k p q s 0, for i = 1, 2, 3 and k,p,q e K, (6.2.13a) ( i i ) Sfcpq i s invariant to permutation to k,p,q, '(6.2.13b) where e K p q i s a r e a l array to be sp e c i f i e d below. T h e a c t u a l e l e m e n t s o f e, „ may b e o b t a i n e d v i a v a r i o u s kpq t h e o r i e s ( e . g . , TFM a n d EDQNM ( c f . L e s i e u r , 1 9 8 7 ) ) . I n T F M , e, i s c a l c u l a t e d f r o m a n a u x i l i a r y p r o b l e m i n w h i c h e. i s kpq 2 c kpq r e l a t e d t o t h e r a t e a t w h i c h a d v e c t i o n o f a t e s t f i e l d d u e t o t h e t u r b u l e n c e f i e l d <; i n t r o d u c e s t h e e x c h a n g e b e t w e e n t h e s o l e n o i d a l a n d c o m p r e s s i v e p a r t s o f t h e t e s t f i e l d ( K r a i c h n a n , 1971) . T h e c a s e o f 2 - D t u r b u l e n c e w i t h R o s s b y w a v e s b u t w i t h o u t t o p o g r a p h y h a s b e e n e x a m i n e d b y H o l l o w a y a n d H e n d e r s h o t t ( 1977 ; h e r e a f t e r r e f e r r e d t o as H H 7 7 ) . G e n e r a l i z i n g t h e r e s u l t i n HH77 t o t h e p r e s e n t s t u d y , we h a v e 'W^kpq7 ^ kpq+4pq; ' ( 6 . 2 . 1 3 c ) ^ P q S M k + W ( 6 . 2 . 1 3 d ) w k p q Hu k +w p +t<) q , (6 . 2 . 1 3 e ) 141 A V + Hn/(k2q2) + Rp(2/(q2k2) -l/(p2q2) -l/(p2k2))^, < 6 . 2 . 1 3 f ) w h e r e g i s a p h e n o m e n o l o g i c a l c o n s t a n t o f o r d e r u n i t y . W i t h ( 6 . 2 . 1 2 ) - ( 6 . 2 . 1 3 ) , t h e moment e q u a t i o n s ( 6 . 2 . 1 0 ) - ( 6 . 2 . 1 1 ) a n d z o n a l f l o w t e n d e n c y e q u a t i o n ( 6 . 2 . 7 ) a r e c l o s e d a t t h e l e v e l Zy a n d Cy, l e a d i n g t o (d/dt) Zy=2kUIy-2VyZy+ T ^ y + 2* y, ( 6 . 2 . 1 4 ) (d/dt) Cy=lkxUHy+ (iUy-Vy) Cy + ^ 2 f ^ + ^ ^  y, ( 6 . 2 . 1 5 ) (d/dt) U=r (U-U)-^ kjy/k2, ( 6 . 2 . 1 6 ) k w h e r e Iy i s t h e i m a g i n a r y p a r t o f Cy , a n d T. ^ ( i = 1 ,2 ) a n d 5 . ( i = 2 , 3 ) a r e t h e e n s t r o p h y t r a n s f e r a n d t h e c o r r e l a t i o n i , JC p r o d u c t i o n , r e s p e c t i v e l y , w i t h t h e s u b s c r i p t s c o r r e s p o n d i n g t o t h e t h r e e d i s t i n c t t h i r d o r d e r moments o f f o r m < < « > , <CCn> a n d <C,hh> i n ( 6 . 2 . 1 0 ) a n d ( 6 . 2 . 1 1 ) . T h e d e t a i l e d e x p r e s s i o n s f o r T a n d S i n t e r m s o f Zy a n d Cy a r e o b t a i n e d u s i n g ( 6 . 2 . 1 2 ) a n d l i s t e d i n t h e A p p e n d i x 6 A . I n t h e a b s e n c e o f n o n c o n s e r v a t i v e e f f e c t s s u c h a s Ekman d a m p i n g r, t h e c o n t i n u o u s s y s t e m ( 6 . 2 . 1 ) - ( 6 . 2 . 4 ) h a s two i n v a r i a n t s o f t h e m o t i o n , i . e . , t h e t o t a l k i n e t i c e n e r g y a n d t o t a l p o t e n t i a l e n s t r o p h y 142 EW = (l/L)2\(l/2) | v ^ | 2 d n J a = (1/L)2[ (1/2) | V * | 2 d Q + (1/2)1?), ( 6 . 2 . 1 7 ) 0(1//) • (1/L)2\ (1/2) (V 20 + h ; 2 d n + 317, ( 6 . 2 . 1 8 ) w h e r e \fi i s g i v e n b y ( 6 . 2 . 1 ) a n d P. i s t h e d o u b l e p e r i o d i c c e l l o f l e n g t h 1. T h e e x i s t e n c e o f t h e two i n v a r i a n t s c a n be r e a d i l y o b t a i n e d f r o m ( 6 . 2 . 2 ) a n d ( 6 . 2 . 3 ) b y a p p l y i n g t h e d i v e r g e n c e t h e o r e m a n d t h e b o u n d a r y c o n d i t i o n ( 6 . 2 . 4 ) ( c f . , C a r n e v a l e a n d F r e d e r i k s e n , 1 9 8 7 ) . F o r t h e t r u n c a t e d s y s t e m ( 6 . 2 . 6 ) a n d ( 6 . 2 . 7 ) , i t c a n b e shown t h a t t h e s y s t e m h a s t h e t r u n c a t e d v e r s i o n o f ( 6 . 2 . 1 7 ) a n d ( 6 . 2 . 1 8 ) , d e n o t e d b y E ( N ) (N) i . i a n d Q , a s i t s i n v a r i a n t s . M o r e o v e r , t h e c l o s u r e m o d e l ( 6 . 2 . 1 3 ) - ( 6 . 2 . 1 6 ) c o n s e r v e s £ ( N ) a n d Qm i n t h e s t a t i s t i c a l s e n s e , i . e . , i t c o n s e r v e s <E> a n d <Q> ( see A p p e n d i x 6B f o r p r o o f ) <E> m < £ < N ) > = £ (1/2) Z^/k2+ (1/2) C72, ( 6 . 2 . 1 9 ) K <Q> = <QW> = £ (1/2) (Z^ + 2R^) + fiU, ( 6 . 2 . 2 0 ) k I t i s n o t e d i n e s t a b l i s h i n g ( 6 . 2 . 1 9 ) a n d ( 6 . 2 . 2 0 ) t h a t t h e i n v a r i a n t s , <E> a n d <Q>, r e s u l t f r o m some d e l i c a t e c a n c e l l a t i o n o f t e r m s i n t h e e n s t r o p h y t r a n s f e r ( i . e . , T . , T . ) a n d t h e c o r r e l a t i o n p r o d u c t i o n 1 / JC 2 f JC ( i . e . , 5 V , S . ) ( c f . A p p e n d i x 6A) . I t i s t h u s e x p e c t e d t h a t 2 / JC 3 / JC n e g l e c t i n g some t e r m s i n t h e t r a n s f e r f u n c t i o n s w h i l e 1 4 3 r e t a i n i n g o t h e r s may r e s u l t i n t h e l o s s o f <E> a n d <Q>, as i s f o u n d i n H 8 7 . 6.3 Numerical results and comparison with DNS W i t h t h e s e c o n s e r v a t i o n p r o p e r t i e s a s i d e , we p r o c e e d t o u s e t h e c l o s u r e m o d e l t o s t u d y t h e f o r c e d - d i s s i p a t i v e s t a t i s t i c a l e q u i l i b r i u m . F o r t h i s o b j e c t i v e , i t s u f f i c e s t o s o l v e t h e s e t ( 6 . 2 . 1 4 ) - ( 6 . 2 . 1 6 ) f o r i t s s t a t i o n a r y s o l u t i o n s , t h o u g h t h e s e t i s a p p r o p r i a t e t o t h e s t u d y o f q u a s i - s t a t i o n a r y s t a t i s t i c s o f t h e e n s e m b l e r e a l i z a t i o n s o f ( 6 . 2 . 6 ) - ( 6 . 2 . 7 ) . We s t a r t w i t h a b r i e f d e s c r i p t i o n o f t h e s o l u t i o n m e t h o d a n d t h e p h y s i c a l p a r a m e t e r s , f o l l o w e d b y a p r e s e n t a t i o n o f n u m e r i c a l r e s u l t s . 6.3.1 solution method A t t h e f o r c e d - d i s s i p a t i v e s t a t i s t i c a l e q u i l i b r i u m , t h e t i m e t e n d e n c y f o r Z ^ , a n d U i n ( 6 . 2 . 1 4 ) - ( 6 . 2 . 1 6 ) v a n i s h e s , t h u s l e a v i n g ( 6 . 2 . 1 4 ) - ( 6 . 2 . 1 6 ) a s a s e t o f c o u p l e d n o n l i n e a r a l g e b r a i c e q u a t i o n s f o r t h e m o d a l s p e c t r a Z ^ , c ? k a n d f o r U. T h e t a s k o f f i n d i n g t h e s t a t i o n a r y v a l u e s f o r Z ^ , C f c a n d U i s t h e n e q u i v a l e n t t o t h e one o f f i n d i n g t h e r o o t s t o t h a t s e t o f e q u a t i o n s . T h e r e e x i s t a t l e a s t two ways t o a c c o m p l i s h t h i s t a s k : o n e i n w h i c h ( 6 . 2 . 1 4 ) - ( 6 . 2 . 1 6 ) a r e s i m u l t a n e o u s l y s o l v e d f o r ^ 2 ^ , C ^ , C/j- w i t h a g i v e n s e t o f 17* a n d r ; a n d one i n w h i c h Z . a n d C. a r e f i r s t o b t a i n e d b y s o l v i n g 144 (6.2.14)-(6.2.15) for some prescribed values of U and r, with the value of U* necessary to achieve such stationary Z^, and U then found from (6.2.16). In t h i s study we adopt the second approach mainly because of the consideration that the physi c a l basis for assigning values to U* i s unclear whereas the range of mean v e l o c i t y for geophysical relevant flows i s better documented i n the l i t e r a t u r e (for oceans, e.g., Crease, 1962; Swallow, 1971) . The actual numerical scheme used here i s the one used by B a r t e l l o and Holloway (1991) (hereafter referred to as BH) i n t h e i r study of passive scalar transport i n p-plane turbulence. The algorithm i s i t e r a t i v e , with the d i s s i p a t i o n terms treated pseudo-analytically (for d e t a i l s see the Appendix i n BH) . Based on the same consideration, the d i r e c t numerical simulation of (6.2.6) i s ca r r i e d out with the uniform zonal flow U set to some prescribed value. As i n the closure case, the external momentum forcing U* necessary for maintaining the forced-dissipative equilibrium may be found from (6.2.7). The dns i s done i n a spectral domain truncated i s o t r o p i c a l l y at k with the inte r a c t i o n terms among t r i a d s (k,p,q) in m a x (6.2.6) calculated by the dealiased pseudo spectral method (Orszag, 1971), the d i s s i p a t i o n terms evaluated a n a l y t i c a l l y , and the time derivative approximated by the leapfrog scheme with Robert f i l t e r (for d e t a i l s see Ramsden, et a l , 1985). 145 6.3.2 Model parameters A l l t h e n u m e r i c a l r e s u l t s r e p o r t e d b e l o w a r e e x p r e s s e d i n t e r m s o f t h e m o d e l u n i t s U = 0.05m/s; L = l/2n; => T = 10es 0 0 0 w h e r e I , t h e l e n g t h o f t h e p e r i o d i c c e l l , i s s e t t o 320 km. T h e c h o i c e o f t h e m o d e l u n i t s , t o some e x t e n t , i s a r b i t r a r y , a n d i s made m a i n l y f o r t h e c o n v e n i e n c e o f p r e s e n t i n g n u m e r i c a l r e s u l t s . T h e m o d e l p a r a m e t e r s ( e . g . , 0, r a n d h ) a r e s e t t o r m s t h o s e r e p r e s e n t a t i v e o f t h e m i d l a t i t u d e d e e p o c e a n e n v i r o n m e n t . I n p a r t i c u l a r , we s e t (3 = 1 . 6 x l 0 " 1 1 m " 1 s " 1 , a t y p i c a l v a l u e f o r p l a n e t a r y 0 a t m i d l a t i t u d e . We t a k e v^=r + i> 4 k 4 , w h e r e i s i n t r o d u c e d t o r e m o v e t h e e n s t r o p h y p i l e d up n e a r k d u e t o f i n i t e t r u n c a t i o n . F o r k = 15 , we s e t v max m a x 4 = 6 x l 0 " 5 t o h a v e p r o p e r d a m p i n g a t h i g h e r k. A s f o r r , we t a k e r = 0 . 1 2 ( o r 1 / ( 1 0 0 d a y s ) ) , a v a l u e c o r r e s p o n d i n g t o weak d a m p i n g f o r m o s t o f o u r n u m e r i c a l c o m p u t a t i o n s . T h e t o p o g r a p h y {h^} f o r d n s a n d {H^} f o r t h e c l o s u r e m o d e l a r e g e n e r a t e d a c c o r d i n g t o t h e i s o t r o p i c t o p o g r a p h y v a r i a n c e s p e c t r a (H87) H(k) = hQ/(3.+k)2-5 ( 6 . 3 . 1 ) w i t h t h e p h a s e o f {h^} g e n e r a t e d r a n d o m l y s a t i s f y i n g u n i f o r m d i s t r i b u t i o n o v e r ( 0 , 2 i r ) . We c h o s e h i n ( 6 . 3 . 1 ) s u c h t h a t h i s e q u a l t o 4.0 (or 4.0xl0~6/s) c o r r e s p o n d i n g t o 200m rms bumps i n a 5000m deep ocean. Such roughness o f our "ocean f l o o r s " may be c o n s i d e r e d as r e p r e s e n t a t i v e o f r e a l ocean f l o o r s . As f o r U, We c o n s i d e r t h e e q u i l i b r i u m f o r which U v a r i e s from 0 t o 5 (or from 0 t o 0.25m/s), a range around t h e t y p i c a l mean v e l o c i t y f o r f l o w s below t h e main t h e r m o c l i n e : 0.05 - 0.1 m/s (Crease, 1962). To o b t a i n a sense o f how t h e n u m e r i c a l r e s u l t s depend on th e model p a r a m e t e r s , we c o n s i d e r c a s e s where t h e Ekman damping c o e f f i c i e n t r v a r i e s from 0.12 (1/(100 days)) t o 1.16 (1/(10 d a y s ) ) , w i t h h =2.0 and 0 = 0.8. max A t e s t o f convergence o f t h e s o l u t i o n t o t h e c l o s u r e model (6.2 .15) - (6.2 .16) w i t h r e s p e c t t o k i s made f o r k =5, ma x m a x 15 and 30v. The degrees o f freedom r e t a i n e d i n t h e s e t r u n c a t i o n s a r e 96, 748 and 2932, r e s p e c t i v e l y . The t e s t r e s u l t s show t h a t r e l a t i v e improvement from k =15 t o k = max max 30 i s i n s i g n i f i c a n t f o r t h e range o f U and t h e parameters mentioned above. F o r example, a t U =0.25, t h e r e l a t i v e change o f t o p o g r a p h i c s t r e s s i s l e s s t h a n 1 % but t h e cpu t i m e i n c r e a s e s an o r d e r o f magnitude. Thus, t h e t r u n c a t i o n k m a x 15 i s r e t a i n e d f o r t h e subsequent c a l c u l a t i o n s . 6 . 3 . 3 Numerical r e s u l t s I n t h e f o l l o w i n g p r e s e n t a t i o n o f our n u m e r i c a l r e s u l t s , we f i r s t make t h e o b s e r v a t i o n s b a s e d o n t h e c l o s u r e r e s u l t s a n d t h e n make r e m a r k s o n t h e d e g r e e t o w h i c h t h e s e o b s e r v a t i o n s a g r e e w i t h d n s e n s e m b l e d a t a . T h e m o d e l p a r a m e t e r s w h i c h a r e u s e d b o t h i n d n s a n d i n c l o s u r e c a l c u l a t i o n s d e s c r i b e d i n (a) a n d (b) a r e s e t t o r = 0 . 1 2 , h = 4 . 0 a n d (3 = 0 . 8 rms w i t h t h e v a l u e s f o r U s p e c i f i e d a s n e e d e d . (a) Topographic stress as a function of U and r T h e f i r s t s e t o f c a l c u l a t i o n s i n v o l v e d a f a m i l y o f f o r c e d - d i s s i p a t i v e s t a t i s t i c a l e q u i l i b r i a f o r w h i c h U v a r i e s f r o m 0 . 0 o r 5 . 0 . T h e s t a t i o n a r y moments Zy a n d Cy o f t h e s e e q u i l i b r i a a r e o b t a i n e d f r o m ( 6 . 2 . 1 4 ) - ( 6 . 2 . 1 5 ) u s i n g t h e m e t h o d d e s c r i b e d a b o v e . T h e t o p o g r a p h i c s t r e s s T a c t i n g on t h e e q u i l i b r i u m f l o w s i s t h e n c a l c u l a t e d a c c o r d i n g t o t h e s e c o n d t e r m o n t h e r i g h t h a n d s i d e o f ( 6 . 2 . 1 6 ) u s i n g t h e s t a t i o n a r y Cy, w i t h t h e r e s u l t s shown i n F i g 6 . 1 i n a s o l i d l i n e . I t i s s e e n t h a t T a s a f u n c t i o n o f U f i r s t r a p i d l y i n c r e a s e s when U moves away f r o m z e r o u n t i l t h e r e s o n a n t U r i s r e a c h e d ( f o r t h e p a r a m e t e r s u s e d , U = 1.0) . T h e n , T m o n o t o n i c a l l y d e c e a s e s a s U a p p r o a c h e s l a r g e v a l u e s . T h e r e s o n a n t b e h a v i o r may b e a n t i c i p a t e d f r o m t h e m o d a l s p e c t r a l o f t h e t o p o g r a p h i c s t r e s s T ( see ( 1 6 ) - ( 1 8 ) i n H87 f o r t h e c o m p l e t e f o r m ) 148 r k « tfk/ (w2 + £ k) (6.3.2) with i > k a term which may not concern us for the present discussion. F i r s t r e c a l l that u. s (U-p/k2)k . It i s thus JC x expected that x k reaches a maximum for (U,k) such that U -p/k2 = 0. Next, note that for the topographic stress spectrum (6.3.1) and the parameters used here, the dominant contribution to x i s found from those x k with |k| (=k) = 1 (cf. F i g 6.3 (b) and F i g 6.5 (b) ) . It then follows from (6.3.2) and 0 = 0.8 that x reaches i t s maximum around U = 1.0, i n agreement with the numerical observation U =1.0. For the i d e n t i c a l parameters, f i v e ensemble dns runs are performed for some representative values of U = 0.25, 1.0, 1.75, 2.75 and 3.75, each of which consists of 5 experiments corresponding to f i v e random r e a l i z a t i o n s of {h^}. The quantities of intere s t are then calculated i n each experiment as time-averages over s t a t i s t i c a l stationary period. The data points f o r the topographic stress i n F i g 6.1 are obtained from evaluation of the second term on RHS of (6.2.7). It i s seen i n F i g 6.1 that the agreement between the t h e o r e t i c a l values and dns data i s better for those e q u i l i b r i a with U away from the resonant U ( = 1.0). It i s r noted that over the range of U where the considerable discrepancy occurs, dns data exhibit noticeable variations 149 from one r e a l i z a t i o n o f (6.2.6) t o a n o t h e r , w h i c h i m p l i e s t h a t x i s s e n s i t i v e t o t h e d e t a i l s o f arrangement of t h e s e l i t t l e bumps on t h e "ocean" f l o o r s . G i v e n t h a t t h e p arameters used i n t h i s s t u d y are r e p r e s e n t a t i v e o f m i d l a t i t u d e deep ocean environment, and g i v e n t h a t t h e p r e s e n t s t a t u s o f ocean c u r r e n t s i s at s t a t i s t i c a l e q u i l i b r i u m , i t t h e n f o l l o w s from F i g 6.1 t h a t t h e c u r r e n t w i t h t y p i c a l mean U = 0.25 ( o r 1.25cm/s) i s s u b j e c t t o | T | = 0.233 (or 0.06 N/m ), a v a l u e comparable w i t h t h e mean wind s t r e s s o f o r d e r O(10 _ 1N/m 2), where t h e ocean d e p t h Ho = 5000 m and seawater d e n s i t y p = 10 3 kg/m 3 are assumed. Even f o r c u r r e n t s w i t h U away from t h e r e s o n a n t r e g i o n , our s t u d y i n d i c a t e s t h a t T i s not a n e g l i g i b l e f a c t o r i n t h e momentum budget. From t h e a n a l y s i s o f t h e s t a t i o n a r y . Z f c and and t h e ensemble dns f o r U e (0., 5.0), i t i s n o t e d t h a t t h e s t a t i s t i c a l e q u i l i b r i a f o r which U < U share much of r c h a r a c t e r s o f t h o s e w i t h U near . T h i s may be a n t i c i p a t e d from F i g 6.1 s i n c e t h e s u b r e s o n a n t r e g i o n i s v e r y narrow and a l s o c l o s e t o t h e . Moreover, t h e s e e q u i l i b r i a d i f f e r i n some a s p e c t s o f s p e c t r a l b e h a v i o r from t h o s e w i t h U l o c a t e d i n t h e s u p e r e s o n a n t r e g i o n (CJ > CM but away from u. I t t h u s appears t o be i n s t r u c t i v e t o c o n s i d e r two r e p r e s e n t a t i v e c a s e s , s e p a r a t e l y . (b) Two representative cases ( i ) Case 1 : subresonant flow (U = 0.25) F i r s t , we p r e s e n t F i g 6 . 2 (a) t o show a r e p r e s e n t a t i v e s u b r e s o n a n t f l o w i n p h y s i c a l s p a c e b e f o r e d i s c u s s i o n i n s p e c t r a l d o m a i n . T h e f l o w f i e l d shown i n F i g 6 .2 (a) i s o b t a i n e d f r o m one o f t h e r a n d o m r e a l i z a t i o n o f ( 6 . 3 . 1 ) . I t i s s e e n i n (a) t h a t a t t =20 .0 ( o r a t t = 230 d a y s ) w h i c h i s w e l l i n t o t h e s t a t i s t i c a l s t a t i o n a r y p e r i o d , t h e f l o w h a s d e v e l o p e d a s t r o n g m e r i d i o n a l c i r c u l a t i o n on t h e w e s t s i d e o f t h e d o m a i n b u t a c o n c e n t r a t e d z o n a l j e t i n t h e c e n t r a l p a r t w i t h p r o n o u n c e d wavy m o t i o n s o n t h e s c a l e k s 2 l o c a t e d t o i t s s o u t h a n d a l a r g e s t a g n a n t r e g i o n t o i t s n o r t h . T h r e e s p e c t r a o f t h e e q u i l i b r i a a r e s i n g l e d o u t f o r d i s c u s s i o n . T h e y a r e e n s t r o p h y v a r i a n c e Z ( k ) , t o p o g r a p h i c s t r e s s x(k) a n d v o r t i c i t y - t o p o g r a p h y c o r r e l a t i o n R(k), shown i n F i g 6 . 3 i n ( a ) , (b) a n d (c) , r e s p e c t i v e l y . H e r e , Z(k), x (k) a n d R(k) a r e b a n d - a v e r a g e d s p e c t r a o b t a i n e d f r o m c o r r e s p o n d i n g m o d a l s p e c t r a : Z . , x. - -k I . / k 2 a n d R. = Re JC JC x JC JC {C^} a c c o r d i n g t o () (k) — (2nk/N(k)) [ ()k k - 1 / 2 S |k|£ k + 1 / 2 w h e r e t h e s u m m a t i o n i s t a k e n o v e r a l l t h e modes ( ) k c o n t a i n e d w i t h i n t h e b a n d d e f i n e d b y k - l / 2 s | k | s k + 1 / 2 , w i t h N(k) b e i n g 151 t h e number o f modes i n t h e kth band. The c o r r e s p o n d i n g dns s p e c t r a r e p r e s e n t e d by t h e symbols a r e o b t a i n e d i n a 5-dns ensemble i n t h e same manner a c c o r d i n g t o t h e s p e c t r a l v e r s i o n s o f C C , hd$/8x and 75c, where t h e o v e r b a r denotes t h e domain average as b e f o r e . I t i s seen i n F i g 6.3 (a) t h a t Z(k) has a b r o a d peak around k = 3, wh i c h may account f o r t h e dns o b s e r v a t i o n (e.g., F i g 6.2(a)) t h a t t h e dominant eddy a c t i v i t y i n s u b r e s o n a n t f l o w s i s on t h e s c a l e k * 2. I t i s a l s o n o t e d from t h e s o l u t i o n s t o (6.2.14)-(6.2.15) f o r U s U t h a t t h e r peak w i l l s h i f t t o some h i g h e r wave number and become s h a r p e r when U moves away from U towards U = 0, which i s r c h a r a c t e r i s t i c o f t h e e q u i l i b r i a i n t h e s u b r e s o n a n t regime. As f o r s p e c t r a l d e c o m p o s i t i o n o f t h e t o p o g r a p h i c s t r e s s , i t i s n o t e d i n (b) t h a t x w i l l m a i n l y be f e l t by t h e f l o w on t h e k a 3. P r a c t i c a l l y , t h e c o n t r i b u t i o n s t o x from h i g h e r wavenumbers a r e n e g l i g i b l e . The c o r r e l a t i o n R(k) i s seen t o have a b r o a d peak around k = 2 and 3 and d i m i n i s h as k approaches k . Note t h a t t h e V-shaped c o r r e l a t i o n s p e c t r a max c h a r a c t e r i z e s a l l t h e subres o n a n t f l o w s . The above o b s e r v a t i o n s a r e seen t o be c o n f i r m e d by t h e dns d a t a i n F i g 6.3, a t l e a s t q u a l i t a t i v e l y , though t h e e x t e n t t o wh i c h t h e two c a l c u l a t i o n s agree w i t h each o t h e r v a r i e s from one q u a n t i t y t o a n o t h e r and from one s c a l e t o 152 another. Comparison of the s o l i d curves with data points indicates that the closure predictions tend to overestimate at low wave numbers, and that the discrepancy becomes smaller as k approaches k for x (k) and R(k) but p e r s i s t s for Z (k). m a x It i s important to note that the discrepancy varies from one r e a l i z a t i o n to another, as seen i n F i g 6.3. To assess the s t a t i s t i c a l s i g n i f i c a n c e of the comparison, s t a t i s t i c a l analysis of the ensemble data at in d i v i d u a l k i s performed, with r e s u l t s i n d i c a t i n g that the worst discrepancy occurs around the k where the standard deviation of dns data reaches i t s maximum. For example, i t i s noted from the analysis for R(k) that the standard deviation take i t s maximum at k = 3 whereas the difference between closure and dns reaches maximum at k = 2. In r e l a t i o n to previous work, our results for T appear to support the observation (Treguier, 1989) that topographic stress i s dominated by i t s component at the largest available scale, though for subresonant flows the present study suggests that T i s also f e l t by the motions on the scales s l i g h t l y smaller than the largest one (e.g., F i g 6.3). ( i i ) Case 2: superesonant flows (U = 2.75) As i n the previous case, we f i r s t present a representative superesonant flow i n the physical domain. F i g 6.2 (b) displays i t s streamfunction, with i t s v o r t i c i t y i n F i g 6.4(b) 153 a n d t h e t o p o g r a p h y r a n d o m l y r e a l i z e d i n t h i s e x p e r i m e n t i n F i g 6 . 4 ( a ) . A s s e e n i n F i g 6 . 2 ( b ) , t h e s u p e r e s o n a n t f l o w e x h i b i t s e d d y a c t i v i t y o n t h e l a r g e s t r e s o l v e d s c a l e (k=l) i n c o n t r a s t t o t h e p r e v i o u s c a s e w h e r e n o t i c e a b l e e d d y s c a l e i s a r o u n d k = 2 o r e v e n 3 ( c f , F i g 6 . 3 ( a ) ) . I n d e e d , i t i s f o u n d f r o m t h e d n s e n s e m b l e m e n t i o n e d i n (a) t h a t w i t h a f u r t h e r i n c r e a s e i n U, t h e s t r e a m f u n c t i o n f i e l d s a t t h e s t a t i s t i c a l e q u i l i b r i u m a r e c h a r a c t e r i z e d b y a s t r o n g u n i f o r m z o n a l c o m p o n e n t -Uy a n d a r e l a t i v e weak e d d y m o t i o n * . A s f o r t h e v o r t i c i t y f i e l d ( c f . F i g 6 .4 (a) a n d ( b ) ) , t h e m i r r o r i m a g e r e l a t i o n b e t w e e n h a n d C i n t h i s c a s e i s m o s t c l e a r l y s e e n a t t h e l a r g e s t r e s o l v e d t o p o g r a p h i c s c a l e . F i g 6 . 5 shows t h e s p e c t r a Z(k) , x(k) a n d R(k) f o r t h i s c a s e . I n s t e a d o f g o i n g t o make d e t a i l e d comments on t h e s p e c t r a l d e c o m p o s i t i o n , we s i m p l y p o i n t o u t t h e d i f f e r e n c e b e t w e e n t h e c u r r e n t c a s e a n d t h e p r e v i o u s c a s e . F i r s t , Z(k) i s s e e n t o p e a k a t k = 1 ( c f , (a) ) , w h i c h i s c o n s i s t e n t w i t h t h e d n s o b s e r v a t i o n b a s e d o n F i g 6 . 2 ( b ) . S e c o n d , t h e c o n t r i b u t i o n t o T e s s e n t i a l l y comes f r o m one s i n g l e s c a l e , t h a t i s , k = 1, i n a s h a r p c o n t r a s t t o t h e n e a r r e s o n a n t c a s e w h e r e x a c t s a l s o s i g n i f i c a n t l y o n t h e i n t e r m e d i a t e s c a l e m o t i o n s , s a y , k = 2 ( c f , F i g 6 . 3 (b) ) . T h i r d , i n s t e a d o f a V - s h a p e d f o r m f o r R(k) a s s e e n i n t h e s u b r e s o n a n t f l o w s ( e . g . , F i g 6 . 3 (c) ) , R(k) e x h i b i t s m o n o t o n i c b e h a v i o r ( see (c) ) w i t h c h a n g e s i n k. I n p a r t i c u l a r , t h e l a r g e s t 154 c o r r e l a t i o n t a k e s p l a c e a t k = 1 i n a g r e e m e n t w i t h d n s o b s e r v a t i o n ( c f . F i g 6 . 4 ) . A s f o r t h e a g r e e m e n t b e t w e e n c l o s u r e a n d d n s a t i n d i v i d u a l k, much o f t h e r e m a r k s f o r t h e p r e v i o u s c a s e a p p l i e s t o t h e c u r r e n t c a s e . (c) Topographic s t ress as a f u n c t i o n of r T o h a v e a s e n s e o f d e p e n d e n c e o f t h e a b o v e n u m e r i c a l r e s u l t s o n t h e m o d e l p a r a m e t e r s , we h a v e t a k e n t h e Ekman c o e f f i c i e n t r a s a n e x a m p l e a n d c o n s i d e r e d t h e c a s e w h e r e r v a r i e s f r o m 0.12 (1/(100 d a y s ) ) t o 1 . 1 6(1 / 1 0 d a y s ) ) , w i t h U= 3.0, h =2.0 a n d 0 a s t h e same a s a b o v e . T h e r e s u l t s f r o m max t h e s e c a l c u l a t i o n s a r e p r e s e n t e d f o r t h e t o p o g r a p h i c s t r e s s i n F i g 6 . 6 . I t i s s e e n t h a t t h e s t r e s s i n c r e a s e s w i t h i n c r e a s e d r a n d t h e c l o s u r e p r e d i c t i o n i s i n a g o o d a g r e e m e n t w i t h t h e e n s e m b l e mean o f t h e e x p e r i m e n t d a t a . N o t e t h a t t h e i n c r e a s e o f x w i t h r a s s e e n i n F i g 6 . 6 i s n o t a l w a y s t h e c a s e . I t i s n o t e d i n o u r t e s t r u n s ( n o t shown h e r e ) t h a t f o r t h o s e s t a t i s t i c a l e q u i l i b r i u m f l o w s w i t h U i n t h e s u b r e s o n a n t r e g i m e , t h e d e p e n d e n c e o f x o n r w i l l b e o p p o s i t e t o t h a t s e e n i n F i g 6 . 6 6.4 Concluding remarks T h e c l o s u r e m o d e l i s u s e d , t o g e t h e r w i t h d n s , t o s t u d y t h e f o r c e d - d i s s i p a t e d s t a t i s t i c a l e q u i l i b r i u m a p p r o a c h e d b y t h e m o m e n t u m - d r i v e n t o p o g r a p h i c f l o w s . A p a r t i c u l a r a t t e n t i o n i s d i r e c t e d t o t h e t o p o g r a p h i c s t r e s s w h i c h a c t s o n t h e u n i f o r m 155 z o n a l component o f t h e e q u i l i b r i u m f l o w s . The r e s u l t s i n d i c a t e s t h a t t h e s t r e s s e x h i b i t s r e s o n a n t b e h a v i o r f o r t h o s e e q u i l i b r i a w i t h U near t h e r e s o n a n t v a l u e [7 .^ ( c f . , F i g 6.1). I t i s found t h a t t h e s p e c t r a l dynamics of t h e e q u i l i b r i a undergoes some q u a l i t a t i v e changes upon c r o s s i n g C7_(cf. F i g 6.3 and F i g 6.5). R e c a l l t h a t t h e p e r i o d i c c e l l l e n g t h 1 i s s e t t o 320 km, a s i z e c o r r e s p o n d i n g t o t h e g r i d s i z e i n a c o a r s e - r e s o l u t i o n ocaen c i r c u l a t i o n model, and t h a t U c o r r e s p o n d s t o t h e domain ave r a g e d mean. Thus, t h e t o p o g r a p h i c s t r e s s T may be viewed as t h e e f f e c t o f t o p o g r a p h i c f e a t u r e s o f s u b g r i d s c a l e on t h e U d e f i n e d a t c o a r s e g r i d s . W i t h t h i s p o i n t o f view, our c a l c u l a t i o n s seem t o s t r e n g t h e n t h e c o n c e r n , which has been r e c o g n i z e d f o r some t i m e , about t h e adequacy o f r e s o l u t i o n i n l a r g e s c a l e ocean m o d e l i n g . 156 Appendix 6A E x p r e s s i o n s f o r T T S„ . and S . 1 s JC 2 / JC 2 /JC 3 / J w Note t h a t t h e energy and p o t e n t i a l e n s t r o p h y t r a n s f e r T . i , JC and 5 . a r e t h e c l o s u r e a p p r o x i m a t i o n s t o t h e f o u r t r i a d i , X summations i n (6.2.10) and (6.2.11), r e s p e c t i v e l y . I t thus f o l l o w s u s i n g (6.2.12) t h a t T . = Y 2A. 9, \A , Z Z. + A . Z Z, + A, Z Z 1, k L c^pq kpq \ pqk q k qpk p k kpq p q A + ApqkVk + V q ^ q + AqpkVk + V p V p + V q V q + AkqpVp}' and s i m i l a r l y , T . = V 2A. e. AA , Z.R + A. Z R + A . Z.tf + A. Z fl 2, k L 'Tcpq kpq\ pkq k q kpq p q pkq k q kpq p q A + W V k + J q V + Akqp^Vp + V P ' } ' ( 6 A < 2 ) 5 2 , k - I ^ p q V q j W V k + AqpkZPCk + W V k + AqpkVk A + ApkqCkCq + VpCkCp}' ( 6 A ' 3 ) 5 3 . k " I Wkpq{A PqkCqCk + W V k + W V * q } ' < 6 A ' 4 ) where t h e i n t e r a c t i o n c o e f f i c i e n t s A, and A, a r e g i v e n by Tcpq kpq (6.2.9) and (6.2.12d) i n t h e t e x t . 157 Appendix 6B C o n s e r v a t i o n p r o p e r t i e s o f t h e c l o s u r e model The p r o o f o f (6 . 2.19) and (6 . 2 . 2 0 ) l i e s on somewhat more s t r i n g e n t c o n s e r v a t i o n p r o p e r t i e s o f t h e model, v i z . , t h e d e t a i l e d c o n s e r v a t i o n o f t h e energy and p o t e n t i a l e n s t r o p h y , w h i c h a r e summarized i n Detailed conservation Let k,p,q e K form a triad (k,p,<i) for which k + p + q = 0 . Let T (klp^; and T fk|p,q; be the e q symmetrized energy and potential enstrophy transfer densities at wavenumber k, defined by r e , k " ( T L K + S I *.fk|p,q;, ( 6 B . 1 ) A Tq .k + R 2 , k + ^ (S2.K + 5 3 , k ^ S I Vk| P,q,, ( 6 B ' 2 ) A where T T v , S . and S . are given by (6A. 1) - (6A. 4), 1 f JC 2 r JC 2 f JC 3 f JC respectively, with the symbol £ defined as in (6.2.6). Then, Te(k|p,q; and r flc|p,q; for the closure model (6.2.13)-(6.2.16) satisfy T <Tc|p,q; + T (-p|q,k; + T l q | k , P ; = 0, (6B .3) e 1 e ' e 1 T dc|p,q; + T {-p|q,k; + T (q|k,p; = 0. ( 6 B . 4 ) q ' q q Proof: S i n c e t h e method used t o prove ( 6 B . 4 ) p a r a l l e l s t o t h e one f o r e s t a b l i s h i n g (6B.3), t h e p r o o f g i v e n h e r e i s thus 158 made e x p l i c i t l y f or the energy argument (6B.3). F i r s t , we symmetrize the terms i n k <cf.(6B.l)) using (6A.1) and (6A.2) w.r.t. p, q to get :Mk|p,q) T(k|p,q) = T ' z z ) fk|P,q) + r ( Z R )(k| P,q) + rJ Z H ) <k|p,q) + T<R R + I I ) (k|p,q) , (6B.5) T ( Z Z ,(k| P ,q) = V q { V q W Z q Z k + \ q p A q p k 2 p 2 k ^kpqAkpqZp2q] ^ Z R ,(k| P ,q) = 0 k p q { \ p q V q Z k ^ q + W W V P + A k p q \ p q V q + AkqpAkqp VP + AkpqApqk V k + AkqpAqpk V k + AkpqApkqZk*q + A k q p A q k p W +K—K—Z~Z^/k2, (6B. 6 ) + A. kpqAqpkZp*k + AkqpApqkZq*k + A. A . Z, R + A, A , Z, R kpq qkp k p kqp pkq k q + A. A. Z R + A. A, Z R kpq kpq p q kqp kqp q p + A. A. Z jR + A. A. ZRX/K2, (6B.7) kpq kqp q p kqp kpq p qj rjZH)(k|p,q) = e k p q J A k p q A p k q Z k H q + A k q p A q k p Z k H p A k P q A k P q Z p^q + W V p } ^ ' ( 6 B ' 8 > y ( R R + n ) ( k | p / q ) _ e k p q { A k p q A p q k ^ + i q i k ; 159 where efcpq' A k p q a n d ^kpq' e t c ' a r e g i v e n by (6.2.9), (6.2.12d) and (6.2.13), r e s p e c t i v e l y . Next, note t h a t t o e s t a b l i s h (6B.3), i t s u f f i c e s t o show t h a t (6B.3) h o l d s f o r any one o f t h e components i n T (k|p,q) . To demonstrate t h a t e 1 (ZH ) t h e l a t t e r i s t r u e , we t a k e (k|p,q) as an example and show t h a t r ( Z H ) (k|p,q) + T < Z H ) (p|q,k) + T ( Z H ) (q|k,p) = 0. (6B.10) I n t e r c h a n g i n g k w i t h p . i n (6B.8) y i e l d s r^ZH) (p|q, k) . S i m i l a r l y , t h e o p e r a t i o n : K <==> q i n (6B.8) g i v e s T < Z H ) (q|k,p) . O m i t t i n g t h e l e n g t h y a l g e b r a , we s i m p l y p o i n t e 1 out t h a t (6B.10) f o l l o w s a f t e r i n t r o d u c i n g t h e two i n t o (6B.10) and c a n c e l i n g t h e terms p a i r w i s e w i t h t h e use o f t h e symmetric p r o p e r t i e s o f a n c * ^kpq' i ' e * ' * * p q - " A qpk - W** ( 6 B ' 1 1 ) and t h e symmetric p r o p e r t y o f 0 K p q ( c f - ( 6.2.13)). W i t h t h e arguments l e a d i n g t o (6B.10), i t t h u s f o l l o w s t h a t t h e d e t a i l e d c o n s e r v a t i o n o f t o t a l energy (6B.3) h o l d s f o r t h e c l o s u r e model ( 6 . 2 . 1 3 ) - ( 6 . 2 . 1 6 ) . 160 Now, we show t h a t t h e two i n v a r i a n t s <E> a n d <Q> g i v e n b y ( 6 . 2 . 1 9 ) a n d ( 6 . 2 . 2 0 ) a r e s i m p l y t h e c o n s e q u e n c e o f t h e d e t a i l e d c o n s e r v a t i o n p r o p e r t i e s ( 6 B . 3 ) a n d ( 6 B . 4 ) . S p e c i f i c a l l y , we h a v e Invariants of the motion. L e t <E> and <Q> given by (6.2.19)-(6.2.20) . Then <E> and <Q> are conserved by the nonlinear transfer in (6.2.13)-(6.2.16) in the sense I <Ti.k + r 2 , k > / * 2 " ° ' < 6 B ' 1 2 ) k k Moreover, <E> and <Q> are the invariants of the model in the absence of the dissipation. Proof: T h e c l o s u r e v e r s i o n s o f t h e t o t a l k i n e t i c e n e r g y a n d p o t e n t i a l e n s t r o p h y e q u a t i o n s f o l l o w f r o m ( 6 . 2 .1'4) - ( 6 . 2 .16) (d/dt) <E> = r(U - U)U - I V V * 2 + <1/2)1 (TI,K+ T2.K)/K2' ( 6 B . 1 4 ) (d/dt)<Q> = - [ v^(Z^ + R^) k + (1/2)I {(TLTK+ T2,K> + 2 ' R e < S 2 ,k + S3 . k > } ' k ( 6B .15 ) I n t e r m s o f T ( k | p , q ) a n d T ( k l p , q ) , ( 6B .14 ) a n d ( 6 B . 1 5 ) r e a d e 1 q ' 161 (d/dt)<E> = r(U - U)U - I VyZy/k + (l/2)£ [ [ T (k|p,q) SCk+p+q), k k p q (6B .16 ) (d/dt) <Q> = - I Vy(Zy + Ry) + (1/2)^ [ [ T q (k|p,q) 5 Ck+p+q) , k k p q ( 6 B . 1 7 ) w h e r e t h e s u m m a t i o n s a r e o v e r a l l k , p , q € K a n d 8Ck+p+q) = 1 when k+p+q = 0 o t h e r w i s e i t v a n i s h e s . I n t e r c h a n g i n g t h e dummy i n d i c e s i n (6B . 14) - (6B . 15) : k <=> p a n d k <=» q, a n d a d d i n g t h e r e s u l t i n g e q u a t i o n s t o ( 6 B . 1 6 ) a n d ( 6 B . 1 7 ) , r e s p e c t i v e l y , y i e l d (d/dt)<E> = r(U - U)U - £ v^zyJk k + (1/6) £ £ [ |r e ( k | P , q ) + Te<p|q,k) + T e ( q | k , p ) | 5 Ck+p+q;, k p q ( 6 B . 1 8 ) (d/dt) <Q> = - [ Vy(Zy + Ry) k + (1/6) H I |r q ( k | P / q ) + T q (p|q,k) + r q ( q | k , p ) | 5 Ck+p+q; , k p q ( 6 B . 1 9 ) T h e c o n d i t i o n s ( 6 B . 1 2 ) a n d ( 6 B . 1 3 ) i m m e d i a t e l y f o l l o w a f t e r c o m p a r i n g ( 6 B . 1 8 ) w i t h ( 6 B . 1 4 ) a n d ( 6 B . 1 9 ) w i t h ( 6 B . 1 5 ) a n d t h e n i n v o k i n g ( 6 B . 3 ) a n d ( 6 B . 4 ) . I n t r o d u c i n g ( 6 B . 1 2 ) a n d ( 6 B . 1 3 ) i n t o ( 6 B . 1 4 ) a n d ( 6 B . 1 5 ) shows t h a t <E> a n d <Q> a r e t h e i n v a r i a n t s o f t h e m o t i o n when no d i s s i p a t i o n i s i n t h e s y s t e m , w h i c h c o m p l e t e s t h e p r o o f . -162 cn cn U J cc t— cn x CL <c cc CD O CL O -0 .50 r -0 .43 --0.36 --0 .29 --0.21 -0.14 --0 .07 0.00 0.00 1.00 2.00 3.00 4.00 5.00 D O M A I N A V E R A G E D M E A N V E L O C I T Y U F i g 6.1 Topographic stress x as a function of L7. The parameters are (h ,0,r) = (4.0, 0.8, 0.12). The s o l i d max l i n e i s for the closure results and symbols for the dns data. The resonant point corresponds to U =1.0. 163 F i g 6.2 Streamf unction (tfr = -Uy+t) f o r the two representat ive flows at t=20 (or ts230 days), with parameters as the same as i n F i g 6.1. (a): the subresonant flow with U - 0.25; (b) : the superresonant flow with U = 2.75. The dash contours are f o r negative va lues . 164 0 . 2 E + 0 1 F (a) (b) (c) M >-X C L o cc I— to 3 to to L U C C I— to o CL o cc z o L U C C cc o o 0 . 2 E + 0 0 -0 . 2 E - 0 1 ^ I E - 0 1 0 . 1 E + 0 2 - 1 . 6 0 0 . 1 E + 0 1 0 . 1 E + 0 2 VAVENUMBER K F i g 6."3 Enstrophy (a), topographic stress (b) and vorticity-topography c o r r e l a t i o n (c) spectra ( s o l i d lines) fo r the subresonant flow case U =0.25, with parameters as the same as i n Fig 6.1. The symbols are for the f i v e dns ensemble data. 165 F i g 6.4 V o r t i c i t y (a) a n d T o p o g r a p h y (b) f o r t h e s u p e r r e s o n a n t f l o w shown i n F i g 6 . 2 ( b ) 166 (a.) 0 . 2 E + 0 2 Er M 0 . 2 E + 0 1 r >-X Q-O CC t -1/1 z LU LO LO ( b ) LU cc t— CO o 0-O cc z o t -< _J LU CC tr o o 0 . 2 E + 0 0 r 0 . 2 E - 0 1 - 6 . 0 0 0 . 1 E + 0 1 0 . 1 E + 0 2 W A V E N U M B E R K F i g 6.5 The same as i n F i q 6.3 but for U = 2.75 167 F i g 6.6 Topographic stress T as a function of r. The parameters are (h ,0,C7) = ( 6 . 2 . 0 , 0 . 8 , 3 . 0 ) . The s o l i d max l i n e i s for the closure results and symbols for the dns data. 168 CHAPTER 7 CONCLUSIONS We have analyzed the problem of temporal evolution of perturbations to the hydrodynamical system (1.2.1)-(1.2.3) in both i n f i n i t e and f i n i t e dimensional function spaces, and in both deterministic and p r o b a b i l i s t i c sense. In doing so, we have developed an algorithm for the global analysis of the system (1.2.1)-(1.2.3) as opposed to the usual modal analysis. The global feature arises from the fact that the analysis i s conducted i n a space of a l l kinematically admissible disturbances, as opposed to the modal analysis which takes care of only i n f i n i t e s i m a l disturbances with the modal form, or as opposed to weakly nonlinear theory which i s r e s t r i c t e d to a subset, consisting of f i n i t e amplitude disturbances, of the above space. This global nature of the analysis allows us to overcome several d i f f i c u l t i e s found i n the usual modal analysis. Its a p p l i c a t i o n to global s t a b i l i t y has yielded a systematic algorithm for separating flows with i n i t i a l growing perturbations from those without (cf. theorem 2.1 and 3.1). The same algorithm has also been used to f i n d an optimal nonmodal perturbation to a given flow (cf. theorem 2.2 and 3.2). The growth rate of such a perturbation i s shown 169 t o b e t h e l e a s t u p p e r b o u n d o n t h e g r o w t h r a t e . I t s a p p l i c a t i o n t o m u l t i p l e e q u i l i b r i a h a s l e d t o a n e c e s s a r y c o n d i t i o n f o r t h e e x i s t e n c e o f m u l t i p l e e q u i l i b r i a . I t s a p p l i c a t i o n t o t h e s t u d y o f r e l a t i o n o f m o d a l t o n o n m o d a l g r o w t h r a t e , i n c o n j u n c t i o n w i t h t h e c o m p a r i s o n o f t h e s t a b i l i t y m e a s u r e s , h a s u n c o v e r e d t h e c a u s e u n d e r l y i n g many s h o r t c o m i n g s o f m o d a l a n a l y s i s , o r e q u i v a l e n t l y h a v e r e s o l v e d s u c h p a r a d o x e s a s g r o w t h o f d i s t u r b a n c e s i n s u b c r i t i c a l f l o w s a n d e x p l o s i v e d e v e l o p m e n t o f i n s t a b i l i t y i n w e a k l y s u p e r c r i t i c a l f l o w s . N u m e r i c a l i l l u s t r a t i o n s made f o r some s p e c i f i c f l o w s h a v e s t r e n g t h e n e d t h e g e n e r a l r e s u l t s , s u g g e s t i n g t h a t a s t a b i l i t y a n a l y s i s o f a h y d r o d y n a m i c a l s y s t e m w i t h o u t a g l o b a l a n a l y s i s s u c h a s t h e o n e d e v e l o p e d h e r e i s l i m i t e d i n many i m p o r t a n t a s p e c t s . A l s o , t h e s e n u m e r i c a l e x a m p l e s h a v e c l e a r l y d e m o n s t r a t e d how t h e g l o b a l a n a l y s i s c a n b e s y s t e m a t i c a l l y i m p l e m e n t e d t o s p e c i f i c f l o w p r o b l e m s . I n o u r l o c a l a n a l y s i s o f a s y m p t o t i c b e h a v i o r o f n o n m o d a l d i s t u r b a n c e s t o h y p e r b o l i c e q u i l i b r i a o f t h e s y s t e m , we h a v e e s t a b l i s h e d : a) f o r a n y s u b c r i t i c a l f l o w o u t s i d e o f MGS r e g i m e , ' t h e r e e x i s t s a f i n i t e , t h o u g h p e r h a p s s m a l l , n e i g h b o r h o o d a r o u n d t h e o r i g i n o f RM s u c h t h a t a n o n m o d a l d i s t u r b a n c e i n i t i a l i z e d i n t h i s n e i g h b o r h o o d w i l l u l t i m a t e l y d e c a y t o z e r o a f t e r e x h i b i t i n g O r r ' s t e m p o r a l a m p l i f i c a t i o n 170 ( t h e o r e m 5 . 1 ) ; b) f o r a n y s u p e r c r i t i c a l f l o w , t h e r e e x i s t s a f i n i t e , t h o u g h p e r h a p s s m a l l , n e i g h b o r h o o d a d j a c e n t t o t h e o r i g i n o f R M s u c h t h a t a n o n m o d a l d i s t u r b a n c e i n i t i a l i z e d i n t h i s n e i g h b o r h o o d w i l l p e r s i s t s a s t-xn ( c f . t h e o r e m 5 . 2 ) ; a n d c) t h e n a t u r e o f t h e p e r s i s t e n t n o n m o d a l d i s t u r b a n c e s i s r e l a t e d t o t h e n a t u r e o f n o n h y p e r b o l i c p o i n t i n p a r a m e t e r s p a c e o f i n t e r e s t ( c f . t h e o r e m 5 . 3 ) . T h e n u m e r i c a l e x p e r i m e n t s a r e s e e n t o c o n f i r m t h e s e p r e d i c t i o n s . O u r p r o b a b i l i s t i c s t u d y o f f o r c e d - d i s s i p a t e d s t a t i s t i c a l e q u i l i b r i u m o f p e r t u r b e d f l o w s a r i s i n g f r o m i n i t i a l u n i f o r m z o n a l f l o w s o v e r r a n d o m t o p o g r a p h y i s d o n e w i t h s p e c i a l r e g a r d t o t h e c o r r e l a t i o n b e t w e e n d i s t u r b a n c e a n d u n d e r l y i n g t o p o g r a p h y a n d t h e r e s u l t i n g f o r c e . W h i l e many q u e s t i o n s r e m a i n t o b e a n s w e r e d , s u c h a n e x e r c i s e h a s l e d t o an a p p r e c i a t i o n a b o u t t h e m e t h o d i t s e l f as w e l l a s a b o u t t h e p r o b l e m o f f l o w - t o p o g r a p h y i n t e r a c t i o n . F o r t h e f o r m e r , we n o t e t h a t t h e d e t a i l e d c o n s e r v a t i o n o f e n e r g y a n d p o t e n t i a l e n s t r o p h y , w h i c h h o l d s r e g a r d l e s s o f t h e p r e s e n c e o f d i s s i p a t i o n i n t h e s y s t e m ( c f . A p p e n d i x 6B), p r o v i d e s a means f o r s y s t e m a t i c a l i n v e s t i g a t i o n o f n o n l i n e a r t r a n s f e r o f t h e t h e s e q u a n t i t i e s among i n t e r a c t i n g t r i a d s , a n a r e a n o t a c c e s s i b l e t o o t h e r a p p r o a c h e s . F o r t h e l a t t e r , n u m e r i c a l r e s u l t s f o r t o p o g r a p h i c a l s t r e s s c l e a r l y i n d i c a t e t h e s i g n i f i c a n c e o f t h i s f o r c e i n t h e o v e r a l l momentum b u d g e t o f l a r g e - s c a l e o c e a n c i r c u l a t i o n a n d s t r e n g t h e n t h e n e e d t o parameterize i t i n general c i r c u l a t i o n models. There are several possible extensions from the present work. F i r s t , the obvious l i m i t a t i o n of the system (1.2.1)-(1.2.3) as a model to study geophysically relevant flows i s the neglect of b a r o c l i n i c i t y , which has been known to be a primary source of i n s t a b i l i t y i n the atmosphere and oceans since Charney (1947) and Eady (1949). It i s clear that a global analysis for a system with t h i s e f f e c t included allows one to draw conclusions on the o r i g i n , development and decay of disturbances from model studies i n closer connection to such phenomena as synoptical scale disturbances in the westerly winds and mesoscale eddies i n the oceans. It i s expected from the work on the Boussinesq equation (cf. Joseph, 1966) that the physical p r i n c i p l e (cf. lemma 2.2) which allows for the present analysis w i l l hold while i t w i l l take a more general form involving the r a t i o of the conversion between the t o t a l energy ( i . e . , k i n e t i c energy plus p o t e n t i a l energy) i n a basic flow and i n a disturbance to the t o t a l d i s s i p a t i o n . It thus follows that an extension from the present barotropic system to a b a r o c l i n i c system i s ensured i n p r i n c i p l e while the analysis i n the b a r o c l i n i c case w i l l be f a r more involved than the present case. Second, an o r b i t a l s t a b i l i t y analysis v i a Floquet theory (Arnol'd 1983; Guckenheimer & Holmes, 1983) i s a necessary 172 n e x t s t e p t o s t u d y t h e t r a n s i t i o n f r o m p e r i o d i c f l o w s t o q u a s i - p e r i o d i c o n e s a n d e v e n t u a l l y t o c h a o t i c o r t u r b u l e n t f l o w s . T h i s w o u l d a l l o w u s t o c o m p l e t e t h e r e p e a t e d s u p e r c r i t i c a l b i f u r c a t i o n shown i n F i g 5 . 7 . F i n a l l y , a s t h e t o p o g r a p h i c s t r e s s a p p e a r s s i g n i f i c a n t f r o m t h e r o u g h c o m p a r i s o n b e t w e e n i t a n d t h e w i n d s t r e s s , an i m p o r t a n t t a s k i s t o b r i n g i t i n t o g e n e r a l o c e a n c i r c u l a t i o n m o d e l s t o s e e i f i t e n h a n c e s t h e i r p e r f o r m a n c e w.r.t. some s p e c i f i c f a i l u r e s o f t h e s e m o d e l s , e . g . , l a t i t u d e o v e r s h o o t o f t h e G u l f S t r e a m . T h i s e f f o r t , i n c o n j u n c t i o n w i t h H o l l o w a y ' s p r o p o s a l f o r Unprejudiced ocean circulation ( H o l l o w a y , 1 9 9 1 ) , may l e a d t o some p r a c t i c a l scheme t o p a r a m e t e r i z e t h e e f f e c t o f s u b g r i d - s c a l e t o p o g r a p h y on t h e t h e g r i d - s c a l e m o t i o n s . 173 R E F E R E N C E S A r n o l ' d , V . T . , 1 9 8 3 : Geometrical methods in the theory of ordinary differential equations, S p r i n g e r - V e r l a g , New Y o r k . A s c h e r , U . M . , R . M . M . M a t h e i l a n d R . D . R u s s e l l , 1 9 8 8 : Numerical solution for ordinary differential equations. P r e n t i c e H a l l , 595 p p . B a r t e l l o , P . a n d H o l l o w a y , G . , 1 9 9 1 : P a s s i v e s c a l a r t r a n s p o r t i n 0 - p l a n e t u r b u l e n c e . J. Fluid Mech., 223, 5 2 1 - 5 3 6 . B o y d , J . P . , 1 9 8 3 : T h e c o n t i n u o u s s p e c t r u m o f l i n e a r C o u e t t e f l o w w i t h b e t a e f f e c t . J. Atmos. Sci.,40, 2 3 0 4 - 2 3 0 8 . B r o w n , A . P . , a n d K . S t e w a r t s o n , 1 9 8 0 : On t h e a l g e b r a i c d e c a y o f d i s t u r b a n c e s i n a s t r a t i f i e d s h e a r f l o w . J. Fluid Mech.,100, 8 1 1 - 8 1 6 . C a r d , P . A . , a n d A . B a r c i l o n , 1 9 8 2 : T h e C h a r n e y s t a b i l i t y p r o b l e m w i t h a l o w e r Ekman l a y e r . J. Atmos. Sci., 39, 2 1 2 8 - 2 1 3 7 . C a r n e v a l e , G . F . a n d F r e d e r i k s e n , J . S . , 1 9 8 7 : N o n l i n e a r s t a b i l i t y a n d s t a t i s t i c a l m e c h a n i c s o f f l o w o v e r t o p o g r a p h y . J. Fluid Mech., 175, 1 5 7 - 1 8 1 . C a s e , K . M . , 1 9 6 0 : s t a b i l i t y o f i n v i s c i d p l a n e C o u e t t e f l o w . Phys. Fluids, 3, 1 4 3 - 1 4 8 . C h a r n e y , J . G . , 1 9 4 7 : T h e d y n a m i c s o f l o n g w a v e s i n a b a r o c l i n i c w e s t e r l y c u r r e n t . J. Meteor., 4, 1 3 5 - 1 6 2 . C h a r n e y , J . G . , a n d D e v o r e , J . G . , 1 9 7 9 : M u l t i p l e f l o w e q u i l i b r i a i n t h e a t m o s p h e r e a n d b l o c k i n g . J. Atmos. Sci.., 36, 1 2 0 5 - 1 5 1 6 C h a r n e y , J . G . , J . S h u k l a a n d K . C . Mo, 1 9 8 1 : C o m p a r i s o n o f a b a r o t r o p i c b l o c k i n g t h e o r y w i t h o b s e r v a t i o n . J. Atmos. Sci., 38, 7 6 2 - 7 7 9 . C r e a s e , J . , 1 9 6 2 : V e l o c i t y m e a s u r e m e n t s i n t h e d e e p w a t e r o f t h e W e s t e r n N o r t h A t l a n t i c . J. Geophy. Res., 67, 3 1 7 3 - 3 1 7 6 . C r i s c i a n i , F . a n d R . M o s e t t i , 1 9 9 0 : On t h e s t a b i l i t y o f w i n d - d r i v e n f l o w s i n o c e a n i c b a s i n s . J. Phys. Oceanogr. 20, 1 7 8 7 - 1 7 9 0 . E a d y , E . J . , 1 9 4 9 : L o n g waves a n d c y c l o n e w a v e s . Tellus, 1, 174 33-52. F a r r e l l , B.F., 1982: The i n i t i a l growth o f d i s t u r b a n c e s i n a b a r o c l i n i c f l o w . J. Atmos. Sci., 39, 1663-1686 , 1984: Modal and non-modal b a r o c l i n i c waves. J. Atmos. Sci., 41, 668-673. , 1985: T r a n s i e n t growth o f damped b a r o c l i n i c waves. J. Atmos., Sci., 42, 2718-2727. , 1987: D e v e l o p i n g d i s t u r b a n c e s i n a s h e a r . J . Atmos. Sci., AA, 2191-2199. , 198 8: O p t i m a l e x c i t a t i o n o f n e u t r a l Rossby waves. J.Atmos. Sci., 45, 163-172. , 1989a: U n s t a b l e b a r o c l i n i c modes damped by Ekman d i s s i p a t i o n . J. Atmos. Sci., 46, 397-401. , 1989b: O p t i m a l e x c i t a t i o n o f b a r o c l i n i c waves. J. Atmos. Sci., 46, 1193-1206. -, 1989c: T r a n s i e n t Development i n C o n f l u e n t and d i f f l u e n t f l o w : J. Atmos. Sci., 46, 3279-3288. F r e d e r i k s e n , J.S., and B.L. Sawford, 1980: S t a t i s t i c a l dynamics o f t w o - d i m e n s i o n a l i n v i s c i d f l o w on a sphere. J. Atmos. Sci., 37, 717-732. F y f e , J . and J . Derome, 1987: A b a r o t r o p i c s t a b i l i t y s t u d y of f r e e and f o r c e d p l a n e t a r y waves. J. Atmos. Sci., 43, 2162-2182. F y f e , J . , 1989: I n v i s c i d n o n l i n e a r s t a b i l i t y o f b a r o t r o p i c s t e a d y f l o w . Geophys. Astrophys. Fluid Dynamics, 46, 71-92. Garbow, B.S., 1974: E i s p a c k - a package o f m a t r i x e i g e n v a l u e system r o u t i n e s . Computer Physics Comm., 7, 179-184. G r a v e l , S., 1988: Steady and p e r i o d i c s o l u t i o n s o f t h e f o r c e d - d i s s i p a t e d p o t e n t i a l v o r t i c i t y e q u a t i o n . Ph.D t h e s i s , M c G i l l U n i v e r s i t y , 116 pp. Guckenheimer, J . and P . Holmes, 1983: Nonlinear oscillations, dynamical systems and bifurcations of vector fields, A p p l i . Math. S c i e n c e s 42, S p r i n g e r - V e r l a g , New York. Hahn, W., 1967: Stability of motion. S p r i n g e r - V e r l a g , B e r l i n H e i d e l b e r g . 175 Hale, J.K., 1969: Ordinary differential equation. Wiley-interscience, 332 pp. Hart, J.E., 1979: Barotropic quasi-geostrophic flow over anisotropic mountains. J. Atmos. Sci.f 36, 1736-1746. Herring, J.R., 1977: On the s t a t i s t i c a l theory of two-dimensional topographic turbulence. J. Atmos. Sci., 34, 1731-1750. Hirsch, M.W., and S. Smale, 1976: Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press. Holloway G. and Hendershott, M. C., 1977: Stochastic closure for nonlinear Rossby waves. J. Fluid Mech., 82, part 4, 747-765. Holloway, G., 1978: A spectral theory of nonlinear barotropic motion above i r r e g u l a r topography. J. Atmos. Sci., 41, 414-427. 1987: Systematic forcing of large-scale geophysical flows by eddy-topography i n t e r a c t i o n . J . Fluid Mech. 184, 463-476. 1991: Representing topographic stress for large scale ocean models: the Neptune e f f e c t , (preprint). Joseph, D.D, :1966: Nonlinear s t a b i l i t y of the Boussinesq equations by the method of energy. Arch. Rational Mech. Anal., 22, 163. Joseph, D.D. and D.H. Sattinger, 1972: Bi f u r c a t i n g time peri o d i c solutions and t h e i r s t a b i l i t y . Arch. Rational Mech. Anal. 45, 79. Joseph, D.D., 1976: Stability of fluid motion I . Springer-Verlag, 282 pp. Kalian, E., and A. Wiin-Nielson, 1980: Non-linear, low order i n t e r a c t i o n s . Tellus, 32, 393-409. Kallen, E. 1981: The nonlinear e f f e c t s of orographic and momentum forcing i n a low-order, barotropic model. J. •Atmos. Sci., 38, 2150-2163. , 1982: Bif u r c a t i o n properties of quasi-geostrophic, barotropic models and t h e i r r e l a t i o n to atmospheric blocking. Tellus, 34, 255-265. , 1985: On h y s t e r e s i s - l i k e e f f e c t s i n orographically forced models. Tellus, 37A, 249-257. 176 K e l l e r , H.B., 1978: Global homotopies and Newton methods. Nonlinear analysis, C. de Boor and G.H. Golub, Eds, Academic Press, 73-94. Kelley, 1967: The stable, center-stable, center, center-unstable and unstable manifolds. J. Diff. Egs., 3, 546-570. Koshlyakov, M.N., and Y.M. Grachev, 1973: Meso-scale currents at a hydrophysical polygon i n the t r o p i c a l A t l a n t i c . Deep Sea Res.r 20, 507-526. kraichnan, R.H., 1967: I n e r t i a l ranges i n two-dimensional turbulence. The Physics of Fluids, 10, No. 2, 1417-1423. , 1971: An almost-Markovian Galilean-invariant turbulent model. J. Fluid Mech., 47, 513-524. Kreyszig, E., 1978: Introductory functional analysis with applications. John, Wiley & Sons, 681 pp. Kubicek, M. and M. Marek, 1983: Computational methods in bifurcation theory and dissipatative systems, Springer-Verlag, New York. Legras, B., and M. G i l l , 1983: Persistent anomalies, blocking and v a r i a t i o n s i n atmospheric p r e d i c t a b i l i t y . J . Atmos. Sci., 42, 433-471. Lesieur, M., 1987: Turbulence in Fluids. Martin N i j h o f f . Lorenz, E.N., 1963deterministic nonperiodic flow. J. Atmos. Sci., 20, 130-141. — , 1969: Three approaches to atmospheric p r e d i c t a b i l i t y . B u l l . Amer. Meteorol. Soc. 50, 345-349. Mak, M., 1985: E q u i l i b r a t i o n i n nonlinear b a r o c l i n i c i n s t a b i l i t y . J. Atmos. Sci., 42, 2764-2781. Marsden, J.E., and McCracken, 1976: The Hopf bifurcation and its applications. Appl. Math. Sciences 19, Springer-Verlag, New York. Marotzke, J., 1989: I n s t a b i l i t y and multiple steady states of the thermohaline c i r c u l a t i o n . Oceanic circulation lation models: combining data and dynamics, D.L.T Anderson and J. Willebrand, eds., Kluer, 501-511. M i t c h e l l and Derome, 1983: Blocking-like solutions of po t e n t i a l v o r t i c i t y equation: Their s t a b i l i t y at 177 equilibrium and growth at resonance. J. Atmos. Sci., 40, 2522-2536. Noble, B. and J.W. Daniel, 1988: Applied linear algebra. 3rd, edi t i o n , Prentice H a l l . O'Brien, E., 1990: Optimal i n s t a b i l i t y i n the quasi-geostrophic i n i t i a l value problem. Submitted to Geophys. Astrophys. Fluid Dynamics. Orr , M c F . , 1907: S t a b i l i t y or i n s t a b i l i t y of the steady motions of a perfect l i q u i d . Proc. Roy. Irish Acad., 27, 9-69. Orszag, S.A., 1971: Numerical simulation of incompressible flows within simple boundaries I : Galerkin (spectral) representations. Stud, in Appl. Math., 50, 293-327.B Palmen, E., and C. Newton, 1969: Atmospheric Circulation Systems. Academic Press. 603 pp. Parker, T.S. and L.O. Chau, 1989: Practical Numerical Algorithms for chaotic systems. Springer-Verlag, pp. 341. Pedlosky, J., 1964: I n i t i a l value problem i n the theory of b a r o c l i n i c i n s t a b i l i t y . Tellus, 16, 12-17. , 1970: F i n i t e amplitude b a r o c l i n i c waves. J. Atmos. Sci., 27, 15-30. , 1981: The nonlinear dynamics of b a r o c l i n i c wave ensembles. J. Fluid Mech., 102, 169-210. Petterssen, S., 1955: A general survey of factors influencing development at sea l e v e l . J. Meteor., 12, 36-42. Pierrehumbert, R.T., 1983: Bounds on the growth rate of perturbations to non-parallel steady flow on the barotropic beta plane. J. Atmos. Sci., 40, 1207-1217. P r o e f s c h r i f t , 1989: V a c i l l a t i o n and p r e d i c t a b i l i t y properties of low-order atmospheric spectral models. Rambalda, S., and K. Mo, 1984: Forced stationary solutions i n a barotropic channel: Multiple e q u i l i b r i a and theory of nonlinear resonance. J. Atmos. Sci., 41, 3135-3146. Ramsden, D., Whitfield, D. and Holloway, G.,1985 Spectral Transform simulations of turbulent flows, with geophysical applications. Sidney, V i c t o r i a : I n s t i t u t e of Ocean Sciences. 178 Reddy, B.D., 1986: Functional analysis and boundary-value problem: an introductory treatment. Longman S c i e n t i f i c & T e c h n i c a l . R o b i n s o n , A.R., and J.C. M c w i l l i a m s , 1974:The B a r o c l i n i c i n s t a b i l i t y o f t h e open ocean. J. Phy. Ocgy. 4, 281-294. Roebber, P . J . , 1 9 8 4 : S t a t i s t i c a l a n a l y s i s and updated c l i m a t o l o g y o f e x p l o s i v e l y c y c l o n e s . Mon. Wea. Rev., 112, 1577-1589. Rosen, G., 1971: G e n e r a l s o l u t i o n f o r p e r t u r b e d p l a n e C o u e t t e f l o w . Phys. Fluids, 14, 2767-2769. Salmon, R., H o l l o w a y , G. and Hendershot, M . , 1976: The e q u i l i b r i u m s t a t i s t i c a l mechanics o f s i m p l e q u a s i - g e o s t r o p h i c models, J . Fluid Mech. 75, p a r t 4 691-703. Salmon, R., 1980: B a r o c l i n i c i n s t a b i l i t y and g e o s t r o p h i c t u r b u l e n c e . Geophys. Astrophys. Fluid Dyn., 15, 167-211. Sanders, F., and J.R. Gyakum, 1980: Sy n o p t i c - d y n a m i c c l i m a t o l o g y o f t h e "bomb". Mon. Wea. Rev., 108, 1589-1606. Sanders, F., 1986: E x p l o s i v e c y c l o g e n e s i s i n t h e W e s t - C e n t r a l N o r t h A t l a n t i c Ocean, 1981-84. P a r t I : composite s t r u c t u r e and mean b e h a v i o r . Mon. Wea. Rev., 114, 1781-1794. Schulman, E., 1967: The b a r o c l i n i c i n s t a b i l i t y o f a mid-ocean c i r c u l a t i o n . Tellus, 19, 292-305. S e r r i n , J . 1959a: M a t h e m a t i c a l p r i n c i p l e s o f c l a s s i c a l f l u i d mechanics.Handbuch d e r P h y s i k , Bd V I I I / 1 . Eds.: S . F l u g g e and C. T r u e s d e l l . B e r l i n - G o t t i n g e n - H e i d e l b e r g : S p r i n g e r . S e r r i n , J . 1959b: On t h e s t a b i l i t y o f v i s c o u s f l u i d m o t i o n s . Arch. Rational Mech. Anal. 3, 1. S e y d e l , R., 1988, From Equilibrium to Chaos, practical bifurcation and stability analysis. E l s e v i e r . Shepherd, T.G, 1985: Time development o f s m a l l d i s t u r b a n c e s t o p l a n e C o u e t t e f l o w . J. Atmos. Sci.,42, 1868-1871. , 1988: R i g o r o u s bounds on th e n o n l i n e a r s a t u r a t i o n of i n s t a b i l i t i e s t o p a r a l l e l s h ear f l o w s . J . Fluid Mech., 196, 291-322. 179 , 1989: N o n l i n e a r s a t u r a t i o n o f b a r o c l i n i c i n s t a b i l i t y . P a r t I I : c o n t i n u o u s l y s t r a t i f i e d f l u i d . J. Atmos. Sci., 46, 888-907. S t u a r t , J.T., 1960: On t h e n o n l i n e a r mechanics o f wave d i s t u r b a n c e s i n s t a b l e and u n s t a b l e p a r a l l e l f l o w s . P a r t 1. The b a s i c b e h a v i o r i n p l a n e P o i s e u i l l e f l o w . J. Fluid Mech, 9, 353-370. Swallow, J . C , 1971: The a r i e s c u r r e n t measurements i n t h e Western N o r t h A t l a n t i c . Philosophical Transactions of the Royal Society A, 270, 451-4 60. T r e g u i e r , 1989: T o p o g r a p h i c a l l y g e n e r a t e d s t e a d y c u r r e n t s i n b a r o t r o p i c t u r b u l e n c e . Geophys. Astrophys. Fluid Dynamics, 47, 43-68. Tung, K.K., and A . J . R o s e n t h a l , 1985: On t h e extended-range p r e d i c t a b i l i t y o f l a r g e - s c a l e q u a s i - s t a t i o n a r y p a t t e r n s i n t h e atmosphere. Tellus, 38A, 333-365. V a l d e s , P . J . , and B . J . H o s k i n s , 1988: B a r o c l i n i c i n s t a b i l i t y o f t h e z o n a l l y averaged f l o w w i t h boundary l a y e r damping. J. Atmos. S c i . , 45, 1584-1593. V e r h u l s t , F., 1985: Nonlinear differential equations and dynamical systems. S p r i n g e r - V e r l a g , 367 pp. V i c k r o y , J.G., and J.A. Du t t o n , 1979: B i f u r c a t i o n and c a t a s t r o p h e i n a s i m p l e , f o r c e d , d i s s i p a t i v e q u a s i - G e o s t r o p h i c f l o w . J. Atmos. Sci., 36, 42-52. Watson, J . , 1960: On t h e n o n l i n e a r mechanics o f wave d i s t u r b a n c e s i n s t a b l e and u n s t a b l e p a r a l l e l f l o w s . P a r t 2. The development o f a s o l u t i o n f o r p l a n e P o i s e u i l l e f l o w and p l a n e C o u e t t e f l o w . f l o w . J. Fluid Mech, 9, 371-389. W i i n - N i e l s e n , A., 1979a: Steady s t a t e s and s t a b i l i t y p r o p e r t i e s o f a l o w - o r d e r b a r o t r o p i c system w i t h f o r c i n g and d i s s i p a t i o n . Tellus 31, 375-386 W i i n - N i e l s e n , A., 1979b: On t h e a s y m p t o t i c b e h a v i o r o f a s i m p l e s t o c h a s t i c - d y n a m i c system. Geoph. and Astroph. Fluid Dynamics. 12, 295-311. 180 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0053236/manifest

Comment

Related Items