NONLINEAR STABILITY AND STATISTICAL EQUILIBRIUM OF FORCED AND DISSIPATED FLOW by Jieping B. Shandong Zou Sc. i n Atmospheric Sciences College o f Oceanography, Program Qingdao, China, 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department accept this thesis o f Oceanography) as c o n f o r m i n g t o t h e r e q u i r e d THE UNIVERSITY OF BRITISH COLUMBIA O c t o b e r 1991 © Jieping Zou, 1991 standard In presenting requirements British freely that this for an Columbia, available permission thesis I for for advanced agree may department o r by h i s or her shall that extensive purposes be partial degree reference scholarly that in the and Department Date: Canada of by representatives. Oceanography 6270 U n i v e r s i t y B o u l e v a r d 1W5 1Z4 study. thesis be a l l o w e d w i t h o u t my w r i t t e n of the Library granted The U n i v e r s i t y o f B r i t i s h C o l u m b i a Vancouver, at copying c o p y i n g or p u b l i c a t i o n of t h i s not fulfillment I It for the University of shall it make further this the of agree thesis head is of for my understood f i n a n c i a l gain permission. ABSTRACT A for global a analysis for homogeneous, 0-plane is space. hydrodynamical incompressible performed Its the in application both to infinite global algorithm for characterizing initially growing perturbations analysis; its application perturbations growth rise rate; to its a to of the including stable initial Numerical global to is of and likely the of existence of normal bound of energy given existence; underlying instability in suggesting system to in many growth many of the of unstable specific hydrodynamical to analysis underestimation some its modal of transient limited mode initial stability results, be an optimal cause for the function e q u i l i b r i a has modal made general a the on yielded the their predict flows to relationship the development the analysis analysis of illustrations strengthened stability the uncovered failure intensity have has of the upper for fluid has for multiple condition rates in search defined finite on opposed least to and based as the the study disturbances flows. to limitation the of to of stability flows application the growth aspects led necessary application nonmodal has layer system flows that a without a important aspects. The local disturbances established: analysis to a) of hyperbolic for any asymptotic behavior equilibria subcritical i i of the flow of nonmodal system outside have of monotonic, global neighborhood initialized zero any it in nature nature of of interest. the the Orr's R there such M will R a neighborhood w i l l persistent The n u m e r i c a l that persist disturbances nonhyperbolic point experiments a finite such M that a in are is decay for a disturbance as t-x»; related to b) to neighborhood parameter seen finite disturbance amplification; exists of exists ultimately temporal there origin this of neighborhood flow, the regime, origin exhibits to initialized the this supercritical adjacent the around in after stability and to the space confirm c) of these predictions. Closure modeling equilibrium of zonal flows regard to topography to, on stress hand, other . hand, system, it nonlinear an a transfer area not of has regardless provides between of of of the is done Such an results led scale to an energy and presence systematic the these quantities i i i to other exercise of has led topographical this force in circulations. On that potential for special underlying appreciation of uniform with and for ocean means accessible statistical initially significance large the from disturbance stress. numerical clearly conservation holds triads, the arising topography resulting momentum b u d g e t detailed which correlation suggesting forced-dissipated flows random and the one overall the perturbed over the of the enstrophy, dissipation in the investigation of among interacting approaches. TABLE OF CONTENTS ABSTRACT TABLE LIST i OF OF CONTENTS i i i FIGURES vii ACKNOWLEDGEMENTS CHAPTER xi 1 INTRODUCTION 1.1 Overview 1 1.2 Objective 5 1.3 Preview of subsequent CHAPTER 6 2 GLOBAL ANALYSIS with chapters I: application INFINITE to: global multiple equilibria DIMENSIONAL stability, and r e l a t i o n SYSTEM, optimal perturbation, of i n i t i a l modal t o nonmodal growth r a t e s 2.1 Introduction 8 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 9 2.3 Symmetrized 2.4 G e n e r a l i z e d R a y l e i g h Quotient and i t s p r o p e r t i e s 2.5 An Optimal Energy Problem (Error) Equation f o r the 13 .... Generalized Rayleigh Quotient 2.6 2.7 Application 14 18 I: global stability and optimal perturbation 22 Application I I : multiple equilibria 29 iv 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 r a t e 2.9 Concluding Appendix 2A . ... 43 Remarks A 35 variational approximation method o p t i m a l value of g e n e r a l i z e d R a y l e i g h for q u o t i e n t p.. the 45 CHAPTER 3 GLOBAL ANALYSIS I I : FINITE DIMENSIONAL SYSTEM, w i t h 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 and o p t i m a l perturbation 3.1 Introduction 3.2 Governing Equations i n R 48 3.3 Global analysis i n R 49 3.4 46 M M Disturbance equation 3.3.2 Symmetrized energy equation 51 3.3.3 Generalized Rayleigh P r i n c i p l e 54 Application to: global and 3.5 3.6 49 3.3.1 stability optimal p e r t u r b a t i o n 58 Numerical i l l u s t r a t i o n s 3.5.1 Set-up of the numerical 3.5.2 Numerical r e s u l t s Concluding 56 58 experiments 59 63 Remarks Appendix 3A A l g e b r a i c p r o p e r t i e s of the set Appendix 3B P h y s i c a l p r o p e r t i e s of the set v {s^} {s.} 64 65 CHAPTER 4 FINITE AMPLITUDE NONMODAL DISTURBANCE I : INITIAL BEHAVIOR, its relation to i n i t i a l i n t e n s e development o f d i s t u r b a n c e s 4.1 Introduction 74 4.2 Nonmodal d i s t u r b a n c e over t r a n s i e n t p e r i o d 75 4.2.1 Modal v s . nonmodal at initial growth rate instant 75 4.2.2 E x p l o s i v e development o f nonmodal d i s t u r b a n c e .. 77 4.3 Numerical i l l u s t r a t i o n s 78 4.4 C o n c l u d i n g Remarks 82 Appendix 4A Modal growth r a t e expressed i n terms o f p .... 83 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 84 CHAPTER 5 FINITE AMPLITUDE NONMODAL DISTURBANCE I I : ASYMPTOTIC BEHAVIOR, its relation t o bifurcation and m u l t i p l e equilibria 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 P e r s i s t e n c e as t -> oo i n s u p e r c r i t i c a l 99 5.4 Persistence,criticality 5.5 Numerical i l l u s t r a t i o n s 108 5.6 C o n c l u d i n g remarks 113 flow 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 Appendix 5A The d i r e c t method o f Liapunov 115 Appendix 5B Auxiliary 116 Appendix 5C Real c a n o n i c a l theory o f l i n e a r o p e r a t o r Appendix 5D Liapunov i n s t a b i l i t y f u n c t i o n lemmas vi .... 119 121 Appendix 5E Algorithms for bifurcation analysis 123 CHAPTER 6 CLOSURE MODELING: FORCED-DISSIPATED STATISTICAL EQUILIBRIUM OF LARGE SCALE QUASI-GEOSTROPHIC FLOWS OVER RANDOM TOPOGRAPHY 6.1 Introduction 134 6.2 Closure 135 6.3 6.4 Formulation 6.2.1 A self-consistent 6.2.2 Moment E q u a t i o n s 6.2.3 Closure Hypothesis Numerical results S o l u t i o n method 6.3.2 Model 6.3.3 Numerical results 135 136 and Master and comparison 6.3.1 Concluding model Equations with 139 DNS 144 144 parameters 14 6 147 remarks 155 Appendix 6A E x p r e s s i o n s for Appendix 6B C o n s e r v a t i o n p r o p e r t i e s k , S of 2 ^ the and k closure . . . . model 157 ..158 CHAPTER 7 CONCLUSIONS 169 REFERENCES 174 vii LIST OF FIGURES Figure 3.1 Page Streamfunctions of three representative 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 h^ = 5 0 0 m. (a) 2 / r = 2 9 days; (b) l/r=ll days; (c) 2 / r = 5 . 5 days. The r e s t of the parameters are g i v e n i n the 67 text 3.2 S t a b i l i t y regime diagram. The s o l i d l i n e f o r 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 f o r the MGS boundary , r=r = L J N 0.075/day f o r the l i n e a r s t a b i l i t y boundary and r=r i s f o r the d i a g o n a l dash l i n e which i s used f o r t e s t i n g the c o n d i t i o n $) -r > 0 3.3 3.4 3.5 An example of MGS. The e q u i l i b r i u m s t a t e ^ i s (c) in F i g 3.1. The initial perturbations $ 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 $ ranging from 0 . 2 t o 1.0 Existence of initially growing nonmodal equilibria $ (r) in subcritical r e g i o n (II) i n F i g 3 . 2 regime, The s p a t i a l c o n f i g u r a t i o n s of growing nonmodal p e r t u r b a t i o n s state five to S m 68 69 to i.e., 70 initially the b a s i c (b) i n F i g 3 . 1 71 3.6 The energy time s e r i e s f o r 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 r a t e s o f d i s t u r b a n c e s i n i t i a l i z e d from nonmodal p e r t u r b a t i o n s over an i n i t i a l growing p e r i o d . The b a s i c s t a t e ^ i s F i g 3 . 1 (a) 73 viii 4.1 Maximum rate, initial i . e . , r ($) modal versus - r vs. r $) nonmodal - r, growth f o rthe set of equilibria {^(C7 ) } over zonal wavenumber-1 topography o f height 1000 m . T h e E k m a n d a m p i n g coefficient r = 2/(15days) i s i n d i c a t e d b yt h e horizontal dash line. The t h r e e vertical dash l i n e s a r e f o r t h e MGS b o u n d a r y U = £ 7 = 6 . 6 m / s , N t 4.2 4.3 4.4 4.5 4.6 and f o rlinear m/s a n dU = stability boundaries U* = U^= 10.5 CT = 2 6 . 3 m / s , where U a n d are L N L L obtained as roots t o t h e equations, r (t(U*))-r=0 and r (${U ))-r=0, respectively L (a) T o p o g r a p h y c o n t o u r s ; (b)-(f) streamfunctions of several representative 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 in ( 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 . . . . 88 89 T h e same a s F i g 4 . 1 e x c e p t that the set of equilibria {^(C7 ) } u s e d h e r e corresponds to the z o n a l wavenumber-2 t o p o g r a p h y 90 T h e same equilibria 30 d a y s 91 a s Fiq 4.1 except that the set of (C7 ) } u s e d h e r e c o r r e s p o n d s t o 1/r = Basic states, nonmodal a n d modal initial perturbations f o r the numerical experiments, (a) for streamfunction o f t h e equilibrium state taken 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 modal 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 Evolution o f t h e growth rate with time. The o p t i m a l nonmodal i n i t i a l p e r t u r b a t i o n S (see F i g m 4.5(b)) 4.7 h a s growth rate a- ( s ; ! ) = 2 / ( 5 . 2 5 d a y s ) , N M whereas the fast growing modal perturbation (FGMP) (see F i g 4.5(c)) h a s R e (o-) = 1/(9.52 d a y s ) . The 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 E v o l u t i o n o f disturbance energy with time 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 perturbation 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 states t o ensure f i n i t e amplitude f o r t h e i n i t i a l perturbations. 94 ix 5.1 Stability regime diagram f o r a family of 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 f r o m 29.5 d a y s t o 3.5 d a y s , U = 22.0 m/s a n d t o p o g r a p h y b e i n g o f z o n a l wavenumber-1 a n d o f h e i g h t 500 m. The 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 (*) a n d r ($) , respectively, with L r = r =1/(5.71 days) a n d N r=r =1/(13.33 days) a s i t s MGS b o u n d a r y a n d linear s t a b i l i t y b o u n d a r y . The c u r v e efgb i s r {t ) f o r the set of equilibria { f (r) } bifurcating branch at criticality {^(r)}. stability r=r'=l/(18.5 r=r from t h e primary days) boundary f o r t h e s e t i st h e l i n e a r (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 disturbances. The u n d e r l y i n g equilibrium states are those located 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 in the figure 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 state (a) i n t o a new s t e a d y flow (f). The snapshots ( c ) - ( f ) a r e from t h e experiment f o r l/r=17.1 days (cf.the t h i c k s o l i d l i n e i n F i g 5.2) 128 5.4 Local uniqueness o f asymptotic steady state 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 equilibrium state i s t h e same a s one i n F i g 5 . 2 f o r e x p e r i m e n t o f 2/r=16".4 d a y s . The i s obtained from s c a l i n g s such t h a t M of t h e basic t(t;s') a t t=0 h a s 10 % M s t a t e energy 12 9 5.5 Periodic limiting states of nonmodal disturbances, w i t h p e r i o d s 46.3 d a y s f o r s o l i d l i n e a n d 85.8 d a y s f o r d a s h l i n e , respectively. The u n d e r l y i n g equilibria are located on t h e unstable section of t h e stationary bifurcation branch (cf.Fig 5.1), with t h e values o f r as indicated 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 state (a) i n t o a p e r i o d i c f l o w . The s n o p s h o t s (b)-(f) a r e taken from t h e experiment f o r l/r=24.8 d a y s ( c f . t h e s o l i d l i n e i n F i g 5.5) o v e r 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 a n d T=49.3 days 131 x 5.7 Repeated supercritical bifurcation for the primary branch of e q u i l i b r i a ( c f . F i g 5.1). The p o i n t marked by x on the primary branch i s a 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 whereas the symbol + i n d i c a t e s the Hopf b i f u r c a t i o n p o i n t . The l i n e s drawn with dash corresponds to unstable equilibrium states 5.8 Nonmodal versus modal i n i t i a l 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 t o a p e r i o d i c s t a t e . The b a s i c s t a t e i s from the s t a t i o n a r y b i f u r c a t i o n branch (r)} w i t h l/r=18.8 days ( c f . F i g 5.1), J o c a t e d near the secondary b i f u r c a t i o n p o i n t r=r'. The initial growth r a t e of ~$(t;a ) and $ (t;Re (t)) are 1/(5.08 M days) and 1/(400.0 6.1 132 days), r e s p e c t i v e l y . 133 Topographic s t r e s s x as. a f u n c t i o n of 17. parameters are (h ,p,r) = (4.0, 0.8, 0.12). The The max s o l i d l i n e i s f o r the c l o s u r e r e s u l t s and symbols f o r the dns data. The resonant p o i n t corresponds to U = 1.0 163 Streamf u n c t i o n (0 = -rjy+$) f o r the two r e p r e s e n t a t i v e flows at t=20 (or t=230 days), w i t h 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 n e g a t i v e v a l u e s 164 Enstrophy (a), t o p o g r a p h i c stress (b) and vorticity-topography correlation (c) spectra ( s o l i d l i n e s ) f o r the subresonant flow case U =0.25, with parameters the same as i n F i g 6.1. The symbols are f o r the f i v e dns ensemble data. 165 6.4 Vorticity superresonant 166 6.5 The 6.6 Topographic s t r e s s x as parameters are (h ,p, U) r 6.2 6.3 (a) and Topography (b) flow shown i n F i g 6.2(b) same as i n F i g 6.3 for the but f o r U = 2.75 a = 167 f u n c t i o n of r . The (6.2.0, 0.8, 3.0). max The s o l i d l i n e i s f o r the symbols f o r the dns data xi closure results and 168 ACKNOWLEDGMENT I for of would like to g i v i n g me t h e the ultimate unfailing term like coaching supervision Fyfe and for for which has Thanks solving Graduate of William of to the thesis and c o n s t a n t to acknowledge my s t u d y my work of on roam turbulence kindly providing due t o closure model. me Fellowship, to financial able to without endure the the f r o m my s i s t e r , theory Lihua for and his way. his and /3-plane I long for Dr. his John studies channel Mr. David support I must for from grants deeply admit support Zou, to the via model W. help for in his University Hsieh and appreciated. that and u n d e r s t a n d i n g whom I am d e e p l y xii his Ramsden d i f f i c u l t i e s i n my f i v e - y e a r constant as along the Bartello and f r o m NSERC note, well search i n my s t a b i l i t y the Dr. Peter f r o m ONR g r a n t s v i a G . H o l l o w a y i s On a more p e r s o n a l as in illustrations. equations, The supervisor, and wide modelling, his also my D r . Greg Holloway f o r e n l i g h t e n i n g suggestions are far encouragement closure y i e l d e d many v i v i d Hsieh, work his the two-layer theme of Dr. opportunity support would a l s o thank I would not overseas over the indebted. be life years CHAPTER 1 1.1 INTRODUCTION Overview Determining given the hydrodynamical oceanographers believed waves and that development in solution an initial linearlized around sought special in a a prominent theory this of instability Charney disturbances even true disturbances, some fall It (1947) over aspects outside an which flow modal i n the (IBVP) Eady often approach, Much associated (1949) on a often simplicity, stability of posed is This literature. has l o n g been and finding solution form). for cyclone to IBVP its hereafter has of the owes its with the baroclinic wind. of the the temporal scope initial is and and the kind problem the to some amounts a generally including to task in interest is traced hydrodynamical of the westerly However, is approach. it oceans value (i.e., of much the practice, position understanding to the modal analysis occupied be given form to origin as the referred present can In of since in boundary disturbance. disturbances phenomena eddies westerlies of been observed Mathematically, to the has meteorologists, mesoscale the evolution system many of instability. for temporal of period generally 1 modal of evolution analysis. evolution considered of adequate of This small for linear theory. F o r example, linear initial v a l u e problems perturbations subcritical exhibit flows perturbations i t has been that large despite demonstrated (a) p r o p e r l y transient the ( c f . O r r ; 1907; Rosen, configured growth absence of using in some growing modal 1971; F a r r e l l , 1982; Boyd,. 1 9 8 3 ; O ' B r i e n , 1 9 9 0 ) ; a n d (b) t h e i n s t a n t a n e o u s g r o w t h rate i s often considerably larger growth r a t e over t h e i n i t i a l period 1989a; Boyd, Further than t h e maximum (cf. Farrell, modal 1 9 8 2 ; 1988; 1 9 8 3 ; O ' B r i e n , 1990) . limitations o f t h e modal analysis are noticed 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 of mesoscale eddies in the oceans d i s t u r b a n c e s i n t h e atmosphere. of mesoscale order data i n Crease Grachev (1973) Another example is well observed 1984; known (Schulman, (1962), suggest that modal (1971) synoptic scale t h e growth rates analysis 1967) w h e r e a s Swallow are of the the observation and K o s h l y a k o v and i t i s o f o r d e r o f a few months. i s found i n the study of cyclonegenesis. I t that the typical c y c l o n e s i s between Sanders, analysis F o r example, o c e a n i c e d d i e s from o f one y e a r and 1986) , c l e a r l y of baroclinic period of 12 a n d 48 h o u r s deepening of ( c f . Roebber, l a r g e r t h a n any f o u n d i n modal instability, s a y , 133 h o u r s i n Valdes and H o s k i n s (1988). While there i s little doubt 2 that t h e study based on t h e linear IBVP accurate (e.g., Case, d e s c r i p t i o n of the infinitesimal disturbances for general a more 1960; than asymptotic behavior beyond the c a p a c i t y of the when the flow disturbances. saturation a t t a i n e d by the under is linear a n a l y s i s by as the with growing d i s t u r b a n c e s t •> » to is the especially modal of nonlinear maximum amplitude (Shepherd, problem of n o n l i n e a r e q u i l i b r a t i o n the somewhat growing problem of allowing perturbation, subject arises more evolution IBVP approach, is concerned yields early initial disturbances case, 1971) and modal of concern In t h i s which onset type of Rosen, 1988, 1989); which d e a l s with or the mechanisms r e s p o n s i b l e f o r the a r r e s t of e x p o n e n t i a l l y growth (Pedlosky, 1970, Consideration of 1981; Salmon, 1980; behavior of asymptotic Mak, 1985). disturbances a r i s e s when one wonders what happens t o the i n i t i a l l y disturbances 1983; a f t e r t r a n s i e n t growth Shepherd, equilibrium multiple Dutton, Gill, only 1979; Proefschrift, Viewing the superposed on the place & Devore, 1979a; 1971; transition takes of interest synoptic a growing Boyd, from one i n the presence of 1979; Vickroy & Legras & Kallen, 1981; 1989). behavior theoretical how (Charney Wiin-Nielsen, asymptotic of or s t a t e t o another equilibria 1983; The 1985) ; ( c f . Rosen, also nonmodal but scale planetary also disturbances of disturbances scale 3 practical as westerlies, is not value. transients long range forecast long c o u l d b e n e f i t from term behavior atmosphere Devore (1979). question states flows of such Taking nonmodal as those change by understanding disturbances considered among i s a matter in model itself a l l possible t h e atmosphere-oceans of the i n Charney t h e view t h a t t h e c l i m a t e distribution reachable climate of the b a s i c and is a equilibrium system o f the r e d i s t r i b u t i o n and t h a t under the i n f l u e n c e o f a changing boundary (Charney and Devore, c l i m a t e modelers a r e e s s e n t i a l l y f a c i n g t h e same problem as c o n s i d e r e d here Situations persistent be (e.g., Marotzke, 1989). may arise disturbances usefully in which the (or t h e r e s u l t i n g c h a r a c t e r i z e d only t-*» t h e hydrodynamical when initial concern f o r the perturbed of can perhaps in a statistical system under conditions behavior flows) w i l l be t h e case when s p e c i f i c forms o f i n i t i a l to 1979), sense. This perturbations (or. e q u i v a l e n t l y system) are not p r e c i s e l y known i n a d e t e r m i n i s t i c sense but r a t h e r g i v e n i n terms o f 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 d e s c r i p t i o n of spectral system behavior on 0-plane f o r the conservative i s obtained using mechanics by Salmon, Holloway rotating Frederiksen sphere by quasi-geostrophic equilibrium and Hendershott and Sawford statistical (1976), (1980), on a for d i s s i p a t e d - f o r c e d flows on an f - p l a n e u s i n g c l o s u r e theory by Herring (1976) and Holloway (1978), and on a 0-plane l a r g e s c a l e z o n a l flow component by Holloway 4 (1987) . with 1.2 Objective This thesis regarding the is to temporal establish some evolution of basic properties disturbances in a r e l a t i v e l y simple n o n l i n e a r hydrodynamical system, t a k i n g the view t h a t insight t h e a n a l y s i s f o r such into realistic behavior of d i s t u r b a n c e s systems such as those m o d e l l i n g the A t l a n t i c specific t h e temporal a system would y i e l d i n more t h e G u l f Stream i n Ocean and t h e w e s t e r l i e s at m i d l a t i t u d e s . The system homogeneous, some considered here incompressible i s t h e one d e f i n e d layer of fluid on a for a /3-plane, i.e., fa/at;v 0 + J O , v 0 2 +py+{h/H)f ) 2 Q = -rCv 2 i/» - vV;, (1.2.1) subject t o v = d\lt/dx = 0, B.C. \li(x+l,y,t) at y = 0 and d , (1.2.2a) =>l>(x,y,t), or B.C. I.C. \li(x,y,t) = ip ty(x,y,t) = \p ili(x,y,0) where Coriolis bottom (1.2.2b) (x+l,y,t), = il> (x,y), (1.2.3) Q \ji i s Laplacian; (x,y+d,t), the streamfunction; J t h e Jacobian; parameter topography; V p t h e beta at r e f e r e n c e H t h e mean the parameter; latitude depth two-dimensional e ; Q f h(x,y) of t h e f l u i d , the the r the Ekman damping forcing coefficient function; direction 1 (x-axis); (1.2.2b) the (y-axis). Specific the an e x t e r n a l l y periodic length periodic length steps taken in prescribed in d i n (1.2.2a) t h e channel d e t a i l e d i n subsequent 1.3 and width, north-south t o achieve t h i s east-west or i n direction objective i s preview o f t h e t h e s i s . Preview o f subsequent chapters In chapter 2, we develop a g l o b a l a n a l y s i s f o r t h e system (1.2 .1)-(1.2.3) kinematically in a Hilbert admissible space functions consisting (defined of a l l as those s a t i s f y i n g B.C.(1.2.2a) or (1.2.2b)), with no assumption on t h e nature o f t h e flow except system. The r e s u l t s are a p p l i e d that i t i s governed made by t h e t o a number of g e o p h y s i c a l f l u i d dynamical problems ranging from g l o b a l s t a b i l i t y t o the r e l a t i o n o f modal t o nonmodal i n i t i a l growth r a t e s . In chapter 3, a f i n i t e analysis dimensional v e r s i o n of t h e g l o b a l i s made f o r flows s u b j e c t t o t h e channel condition (1.2.2a) and f o r c e d by an e x t e r n a l zonal momentum source -U*y. The r e s t r i c t i o n t o t h i s no means numerical essential experiments. but s p e c i f i c type of f o r c i n g rather Numerical f o r the = i s by convenience calculations of are made t o i l l u s t r a t e t h e b a s i c n o t i o n s put forward i n t h i s and p r e v i o u s chapters. 6 Extending modal the study perturbations of in dimensional account Specifically, we e x t e n d initial of instant time by nonmodal In and on its bifurcation the the of initial space, we subject conclusions global local behavior The nonmodal give in strictly analysis some to an a to finite chapter 4. true at only initial fundamental 6, period properties states zonal correlation resulting to of flows of between theory (Holloway, flows. The relevant to on extensions perturbed numerical the the midocean results from the arising and are present the work 7 statistics of with based obtained Some present are of from numerical environment. of is given and version initially focus vorticity work direct results experiments dimensional flows The and flow predictions. random t o p o g r a p h y , stress. as on equilibrium stationary topography 1987) one made -» oo, theoretical the is t numerical finite study over topographic a space disturbances of from both use state nonmodal confirm the we in bifurcation results (1.2.2b) asymptotic analysis of to analysis chapter (1.2.1), a relation another. remarks Hilbert for from the 5, asymptotic uniform relation disturbances. chapter In the establishing the into the and on a brief study the the closure simulation for in on of parameters concluding and chapter possible 7. CHAPTER 2 GLOBAL ANALYSIS PART I : INFINITE DIMENSIONAL SYSTEM, with application multiple growth t o : global stability, optimal perturbation, e q u i l i b r i a and r e l a t i o n o f i n i t i a l modal t o nonmodal rates 2.1 I n t r o d u c t i o n Since Charney's westerly winds normal modes modal analysis light stability some properties will as global specific system analysis dynamics. of the analysis 1985; analysis a fuller description as opposed flow except system. the analysis performed that here O'Brien, which into 1971; 1990). circumvents with t h e modal of the stability F o r t h e reason i s hereafter makes come ( c f . Rosen, t o t h e modal that referred to analysis. The no r e f e r e n c e t o any i t i s governed by t h e continuum ( 1 . 2 . 1 ) - ( 1 . 2 . 3) . I t i s t h u s and t h e r e s u l t i n g have and, t o g e t h e r o f a hydrodynamical analysis t o as role i ngeophysical f l u i d shortcomings yield referred the and w i l l c o n t e x t s o f hydrodynamics become a p p a r e n t , global has been c h a p t e r , we p r e s e n t a n a n a l y s i s will 1947), of playing, shortcomings o f these analysis, the instability (hereafter 1982; Boyd, 1983; Shepherd, In t h i s some a vital on ( c f . Charney, analysis f o r simplicity) i n various Farrell, theory i n t h e atmosphere continue t o play, However, striking expected that conclusions apply 8 the global t o any f l o w r e a l i z e d by The is (1.2.1)-(1.2.3). chapter i s organized as f o l l o w s . The g l o b a l a n a l y s i s c o n s t r u c t e d s t e p by step from devoted t o the formulation problem (IBVP) governed o derivation § 2.2 an value f o r the s t a b i l i t y 0(x,t;^ ) flow of § 2.2 t o § 2.5, w i t h of a by symmetrized equation f o r disturbance initial boundary (or p r e d i c t a b i l i t y ) (1.2.1)-(1.2.3); § kinetic (or energy of a 2.3 to error) (or e r r o r ) ; § 2.4 t o i n t r o d u c t i o n of 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 i n t o t h e k i n e t i c energy (or error) equation, and to i t s basic properties; § 2.5 t o c o n s t r u c t i o n o f an a l g o r i t h m f o r f i n d i n g t h e extreme v a l u e of the generalized Rayleigh quotient i n some r e l e v a n t f u n c t i o n space W. In t h e remainder o f t h e chapter, the g l o b a l a n a l y s i s is a p p l i e d t o monotonic, g l o b a l s t a b i l i t y for optimal equilibria initial initial and perturbations predictability (MGS), t o a search 2.6; to multiple in § in § 2.7; to relation of modal t o nonmodal growth r a t e i n § 2.8, f o l l o w e d by c o n c l u d i n g 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 IBVP first f o r disturbances 0(x,t;0 ) Q denote initialized which step i n t h e g l o b a l a n a l y s i s i s t o formulate an from i s realized a (or e r r o r s ) flow 0 (x,y). Q under governed Consider t h e same 9 in a by given flow. L e t (1.2.1)-(1.2.3) another flow conditions and ^(x,t;0 +# ) o o as those f o r 0(x,t;0 ) except question i f together (roughly Q apart initial what speaking, 0 +0 - field: O condition stable or unpredictable), the or To a d d r e s s O two flows predictable) we c o n s i d e r the or the stay stay difference «>, g i v e n b y The 0 o system subtracting those (2.2.1) • 0 C x , t / ^ + 4> ) ~ 0 f x , t ; 0 ; <f>(x, t) for (d/dt)V B.C. and under (unstable field for its describing the t h e dynamics governing o f <p i s equations for obtained ip ( x , t ; ^ + # ) o o after from tfr(x,t;^ ) Q 4> + £[<t>;ip] (Ill) o :• + N[<t>] = 0, (2.2.2) 8<f>/dx = 0, for <f>(x,y,t) 4> (x+l,y,t), - y = 0,d and 0*xsl for f (2.2.3a) Osyzd, or B.C. (IV) : <p(x,y,t) = <f> (x,y+d, t) , <(>(x,y,t) = <j>(x+l,y,t) , I.C. where <p(x,t ) = 0 Q £ and N : differential notation with summation for 0*x*l, (2.2.3b) Ozysd, (2.2.4) (x). Q H -> IH a r e operators, for respectively and are convention linear given applied by to and nonlinear (in indicial the repeated indices) (2.2.5) MM - e. z jk (2.2.6) .a^a.a a *. m n 10 8^ s a/dx^; (x ,x ,x ) 1 2 3 e. s (x,y,z), 3 (2.2.7a) = (0, 0,1), (z z ,z ) v 2 = alternating (2.2.7b) tensor, (2.2.7c) IJK Q • 3 8 0 + p y + (h/H)f i with i Q IH b e i n g functions the set of f o rwhich ( 2 . 2 . 2 ) - (2.2.4) and = potential vorticity i n 0 a l l kinematically the evolution are well problems defined. (2.2.8) admissible (1.2.1)-(1.2.3) In particular, the e l e m e n t s f r o m 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 ) (ii) smoothness derivatives (2.2.9) ( o r B.C. ( I V ) ) sufficient to i n (1.2.1)-(1.2.3) assure that the (or (2.2.2)-(2.2.4) spatial are well defined. (2.2.10) F r o m now on. we assume t h a t among t h e o t h e r t h i n g s , IH i s a Hilbert that Space. This means, i t i s a vector space over R w i t h , V a a n d b e H, (2.2.11) an i n n e r p r o d u c t <a,b> • f a b ; ||a|| e < a , a i where t h e overbar irrelevant elements The channel denotes t h e complex f o r t h e elements from symbol / 2 n (cf. IH. The l a t t e r denotes (2.2.3a)) from t h e area 2.2.1 The enclosed o r t h e double by 11 for the o f IH o v e r C. the periodic periodic cell (cf. geometry. (2.2.2)-(2.2.4) system which i s H b u t i s needed i sthe counterpart (2.2.3b) d e p e n d i n g on t h e model Remark conjugate admits two interpretations. the If taken at = Q 0 (x,t;tp ) flow <p i s Q subsequent evolution leading the On the to other assigning hand, (2.2.2)-(2.2.4) thus flow 0(x,t;0 ) of Q 4> i s the i f <P made then the initial perturbation system is an e r r o r viewed to as the governs perturbation of Q flow growth the c6 , f low decay follows, no specified \j> (x, t ; ^ ) stability and . o ), the of the assumption explicitly, the introduced in <p (x, t;ip or to thereby Q on t h e thereby predictability system initial i n f o r m a t i o n on p r e d i c t a b i l i t y what of , describes unless treatment Q Q an i n i t i a l on s t a b i l i t y 0 yielding In t the state then . Q of statement initial error t as of the nature unifying within one framework. It is obvious (2.2.2)). It solution € n. of of is from flow <P = the difficult problem The latter is for xen. With this only in terms of Also note that of for by note the that to <j> = governed # = leads via flow we \ji (x,t;\j) restriction of is a for to a as study x <p * the results Q sake seen the <p = 0 . for with the null stability to one (cf. for a l l the us the state ) <b = 0 ( 1 . 2 . 1 ) - ( 1 . 2 . 3) (2 .2 .2) - (2 .2 . 4) i n mind, <f> implies by Q <f> s 0 V t > 0 constant o 0 0(x,t;0 ) flow then see correspondence defined the to subject stability This constant, o \j> (x, t;ifi ) (2.2.1). stability not i f ( 2 . 2 . 2 ) - (2.2.4) Moreover, the that constant below of space. non-constant field o is not e s s e n t i a l constant here corresponds but i s to a physically perturbation motivated, with since kinetic <j> energy = equal t o zero, or an e r r o r with v a r i a n c e b e i n g z e r o . 2.3 Symmetrized The the second linear energy step (error) i n the g l o b a l operator £ constituents i n £ such the time equation (or e r r o r s e l f - a d j o i n t operator analysis (cf. (2.2.5)), asymmetric that equation (e.g., tendency equation) i s t o symmetrize i . e . , t o remove the the second i n the disturbance i s expressed ( c f . , Reddy, term i n £) energy i n terms of a 1986). We s t a r t with t a k i n g the i n n e r product o f (2.2.2) with <j> € W, f o l l o w e d by d i r e c t l y e v a l u a t i n g i n d i v i d u a l terms and u s i n g the a d j o i n t o p e r a t o r £*of £ d e f i n e d by <£a,b>=<a,£*b>, e IH ( c f . , chapter '8 i n Reddy, 1986). a,b V O m i t t i n g the a l g e b r a i c d e t a i l s i n v o l v e d , we g i v e the r e s u l t i n g e q u a t i o n below (d/dt) <d <t>, 8 <t>> = <</>,(£+£)<(>>, i (£+£) i U m djfdjU^^) [<t>;\l>] e (2.3.1) ± - a^u^^^r = - d^djUjd^) (2.3.2) - BjUfdjdjt + 2rd d (t> ± ± (2.3.3) r ijk j k*> where Z <'- d 2 a l l t h e symbols (2.2.2)-(2.2.8). Several here a r e t h e same r e m a r k s on order. 13 as those (2 . 3.1)-(2.3.4) 3 4) in are in Remark 2 . 3 . 1 verified This The by noting mathematical Remark 2.3.2 operator non self-adjointness In adding £+£*, as the skew-symmetric from will the b> be For operator £ <(£+£) a,b>, £ to cfr, are = a,b V € our approach. £ to form the to the annihilated of from potential represented e . .,z w. to advection as [<f>;\p] easily corresponding £ the be by .8,<f>d,.Q, i s the eliminated £+£*. 2.3.3 Remark In (2 . 3.1) - (2 . 3 . 4) , without the the invoking statements which regardless of nonlinear hypothesis. (2.3.1) from The next two hold crucial a functional construct an algorithm in accomplish the the operator W. for to exactly Thus, arbitrary any <f> e H size. introduce functional leading f/[<p] v a n i s h e s term linearization follow its symmetrization 2.4 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 and i t s of can essential in by \ji = processes example, flow £+£* adjoint constituents in Q (£+£*) physical desired. vorticity <a, property self-adjoint (2.3.1) that of The former. £+£ . steps such for in that it global is o b t a i n i n g the objective We the do this of by properties bounded extreme this a analysis in H, value section further are: and to to of the is to manipulation While the elementary, facts. mathematical we m u s t First, ip (x, t;\p ) the (cf. Q strain tensor [ 8 ^ ] = note manipulation the f o l l o w i n g two c r u c i a l velocity (2.3.3)) gradient can be + ±j d j ± ^ U decomposed fc^j] and the v o r t i c i t y [d ] involved tensor n into is physical t h the [v^.], flow e rate of i . e . , (2.4.1a) [v ], ±j d & ± (1/2) (d Uj+ 8M ) , (2.4.1b) v jm (1/2) (d Uj- BjU ) . (2.4.1c) ± ± ± Second, the conservative which the ± ± s p i n n i n g o f <f> b y \p, as f a r as can be r e a d i l y tensor resulting (d/dt) [v^ . ] . the kinetic with o f <j> is the asymmetric Omitting the algebraic details, [v^_.] , concerned, nature we g i v e of the form (l/2)<d 4>,d (p> i i 1 l ± d[<p;ip] - d d jS 4, s 1 • -a a »> 1 1 J (2.4.2) = <d 4> d (p> (p((f>;ili)-r), * <<f>,d(l>>/<<t>,%<(>>, where energy shown b y n o t i n g p(</>;tp) s/>J associated (2.4.3) (2.4.4) 3 (2.4.5) / p(<p;\p): a n d S B : IH - » IH a r e IH - » IR i s the generalized differential Rayleigh operators. Remark 2 . 4 . 1 K i n e m a t i c a l l y , <<p,A4>> represents of of rate of <f> b y 0 v i a t h e p r e s e n c e quotient, 15 strain the tensor straining id^jl of is 0 i n (2.4.4). source In terms o f e n e r g e t i c s , i t serves as an energy precisely as an energy c o n v e r s i o n between 0 and 0. On t h e other hand, <0, £ 0 > is f o r t h e development the t o t a l total kinetic Ekman damping R a y l e i g h Quotient ratio o f 0, or more energy i n <f>. Note t h a t on <f> i s r<0, £0>. p(<t>;*p) Thus, can be p h y s i c a l l y i f r * 0, the the generalized i n t e r p r e t e d as the o f energy c o n v e r s i o n between 0 and <f> t o t h e energy i n 0, o r t o t h e Ekman d i s s i p a t i o n i f r * 0 (or t h e r a t i o o f the e r r o r generation t o the e r r o r ) . The f o l l o w i n g p r o p e r t i e s of o p e r a t o r s A and S are used throughout t h e chapter. Lemma 2.1 Let A and S be given and £ are self-adjoint by (2.4.4) and (2.4.5). on H, viz. (L) A* = A, fii; Proof: Then d (2.4.6) B* = SB, <a,5Ba> >0, V a € IH with a * constant. (2.4.7) (i) f o r any a,b e IH, we have d . (bd . .8 .a) - a .bd . .8 .a I d <=* IJ ± i d. . = d . .) IJ ii' - ad d .8.b\ i I j) (since d (ad .a.b) j i 16 = <a,Ab>. ij j On t h e o t h e r Reddy, hand, 1986), calculation which of follows (ii) Lemma 2 . 2 we h a v e together with the last <Aa,b> e s t a b l i s h e s similarly. & max max the line (cf., i n the self-adjointness above o f A. B L e t p(<l>;ip) \p(<p;ij>)\ <Aa,b>=<a,A*b> b y d e f i n i t i o n be defined by (2.4.3). Then: (2.4.8) V <f>eW, (ky\ ), 2 X€fi where (i=l,2) tensor at First, Proof: p(4>;$) = with d i = . defined the rate there thus exists respect such a diagonal to performing [d ] where \^(i=l,2) ±j ±j [c ] ±j hence are the rate of strain i n the (2.4.9) (2.4.1b). strain a set which form is = diag tensor [d^j] = 0, I s a is real of principal axes ^ c f ^ h a s diagonal achieved a at locally similarity and symmetric, each point form. (i.e, at In any x unit principal x € fi fact, e fi) transformation (2.4.10) (\,\). are the eigenvalues the form d.<t>a.<p , 4 e IH an orthogonal d e t ([d^^J-XI) and by of with of x e fi. f d.<l>d..d.<p A fi •*• J J Jn lc f values we c a n p u t p(<p;ip) Since by are principal matrix values 17 of e R of 2 x 2 , [d. .] a t (2.4.11) x, and the column vectors ^ ±j^ of c * e r2 2 a r ^~he corresponding e e i g e n v e c t o r s and thus c o n s t i t u t e a s e t o f p r i n c i p a l axes at x [d^..] . I t i s c l e a r t h a t for the rate of s t r a i n tensor constructed with the normalized eigenvectors [Cj_j] o f (2.4.11) enjoys t h e f o l l o w i n g orthogonal p r o p e r t y [ c ij An 1 [ c ij ] T = estimate (2.4.10) [ c ij ] T [ c ij ] = '* 1 [ c ij ~ ] o f t h e numerator l = [ c ij ] T i n (2.4.9) ' (2.4.12) thus f o l l o w s from and (2.4.12) |J a.*d.yy n |s j j a ^ ([a. 1 s^max(\ ,\ )d £ i 2 i [c..] [d..] [a..] [a..]' ^ | with d Z = [c ]' 8^ 8^ l smax max (X , X )\ 8 X€Q n 1 T 2 J 1 ± ±j 1 8 .£ 1 =max max (\ ,X )\ 8 ,<f> 8 .<f> xefi from which 1 2 J n 1 (2.4.8) i s o b t a i n e d . 1 a 2.5 An o p t i m a l problem f o 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 As t h e very final step i n the global analysis, we demonstrate t h a t t h e extreme value o f p i n H can be o b t a i n e d v i a a boundary e i g e n v a l u e problem Lemma 2.3 Quotient If 0 e IH maximizes p(<p;ip), then <j> solves problem: 18 (BEP). the generalized the boundary Rayleigh eigenvalue A<f> = XB0, 0 satisfies (2.5.1) 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) Proof: First, quotient for is done Now, &/-0J IM o f comparison linear + H of the Rayleigh functions = functionals §: M -> R b y (2.5.5) for of notation, p(0) = 9 [<p] 0 as an argument (1/V[<t>] ){$[<!>]8V[<t>;V p, then 5p(0;£) 8&[4>;%]9[4>J = 0 from lemma 2.1 = is [<p] (where for suppressed), 88[</>]}. 2 0 i n IH m a x i m i z e s i s found This (2.5.4) Then, &[4>]8$- that W= W. = <0,£0>, 6p(0;€) such e £ € IH f o r a n y e € R . <4>,A<p>, • simplicity It set by choosing respectively, If a respectively. w i t h t h e domain e IH a n d a n y £ e W, 0 define 9[+] we a s s o c i a t e p(0;0), any 0 and (2.4.5), 0, or f o r a l l £ e M. (2.5.6) that 8^[4>;V = 2<£,A4>>, (2.5.7a) S&/"0;£J (2.5.7b) = 2<£,£0>. Introducing <€, M - X 2 ) 0 > (2.5.7) = 0, into (2.5.6) V a n y € e M, 19 yields with x = X s yftJ/'Sti] which = max V [<t>][<t>], «>€lH implies (d-XB)4> = 0, thus completing t h e p r o o f . a While lemma 2.3 p r o v i d e s a s y s t e m a t i c means of s o l v i n g the o p t i m a l problem max p i n IH, i t s p r a c t i c a l implementation may not t u r n out t o be simple i n t h e case o f z o n a l l y asymmetrical flow 0 which coefficients in i n g e n e r a l l y leads t o nonseparable (2.5.1). However, t h e lemma development i s crucial o f t h e theory, as we w i l l 2A, we p r e s e n t t o our subsequent see.soon. a R a y l e i g h - R i t z procedure In appendix f o r t h e numerical s o l u t i o n o f t h e problem: max p i n IH. Remark 2.5.1 As seen i n w r i t i n g X[\p] , X depends on the s t a t e of t h e flow 0(x,t;0 ) Q i {*}, where and hence i s a f u n c t i o n o f time t i f 0 {*} denotes t h e s e t of e q u i l i b r i u m s t a t e s of (1.2.1)-(1.2.3). However, i t i s c l e a r t h a t A[0] = X[0 ] = a For p r a c t i c a l constant if ipQ e {*} . concerns, t h e f o l l o w i n g d e s i r e d p r o p e r t i e s o f t h e eigensystem Lemma 2.4 For the eigensystem that 20 (2.5.8) lemma assures us some (2.5.1)-(2.5.2). (2.5.1) - (2.5.2), it holds Q (i.) A are all eigenvalues (ix) X is fiii) are bounded; if X* i X, then 2 orthogonal the corresponding in the sense (i) L e t Proof: real; f <A4>,4>> = <<f>,d*<p> = <<p,d<l>> It thus follows from (2.5.1) # 2 <8^</>^ 8^<t> > = 0. that be d e t e r m i n e d (X,<p) c^, eigenfunctions 2 from (2.5.1). Consider (2.5.9) (by Lemma 2.1(D) and (2.5.9) that <XB0,c>> = <«>, A53#> =* X<£<f>,4>> = X<(p,S(p> = X<£<t>,4>> which holds let (iii) A = it 2 if i i i i leads 2 from stated lemma above. 2.2. consider Now, to Remark 2 . 5 . 2 is that are orthogonal the A * A 12 1 physical interpretation velocity w.r.t summarize, symmetrized 0 if <8 .d> ,8 .<f> > = The 2.4 2 = o jTl To A follows real. 2 i (x -x )<d <i> .d <t> > which A is = <x s<t> ,<t> > * 2 2 i.e, A ^ a n d <f>^(i=l,2) b e «t> ,x s<i> > 2 of A, 2.1(H)) = <d4> <t> > * «t> d<t> > i if The boundedness fii) i only (by Lemma energy the the (or fields inner 2 21 (iii) with in lemma different A (2.2.11). analysis error) of associated product global • equation includes: (2.4.2); 1) 2) the an a l g o r i t h m f o r o p t i m i z i n g the g e n e r a l i z e d R a y l e i g h i.e., lemma 2.3. In view introduced relate J., of the fact that a bounded i n § 2.4 and a v a r i a t i o n a l t h e extreme eigenvalue fall problem, problem to the global i n t o t h e category functional i s argument i s invoked t o an Euler-type boundary a n a l y s i s developed here may o f v a r i a t i o n a l energy method (Serrin, 1959a, 1959b; Joseph, D.D., 1966, 1976). In t h e remainder results o f t h e chapter, of the global contexts. results can be s t a t e d i n terms \ji fx, t;\p ), o depending Except First, analysis various or equally shall explore the lemma 2.1-2.4) i n we note from remark 2.2.1 t h a t our well of the s t a b i l i t y i n terms o f t h e flow of i t s p r e d i c t a b i l i t y , on our i n t e r p r e t a t i o n o f t h e i n i t i a l f o r a few o c c a s i o n s stability we (i.e., i n t h e subsequent however, much o f argument i s presented 2.6 q u o t i e n t p, only data 0 . Q developments, i n terms of the f o r the sake o f space. A p p l i c a t i o n I : G l o b a l s t a b i l i t y and o p t i m a l p e r t u r b a t i o n First, system we g i v e t h e d e f i n i t i o n o f g l o b a l s t a b i l i t y f o r the (1.2.1)-(1.2.3) Definition 2.1 (global stability) (2.2.2;-(2.2.4; . Then: 22 Let <f>(x,t;<f> ) o satisfy (i) a flow 0Cx,t/0 ; governed by ' (1.2.1) - (1.2.3) o be asymptotically stable if there exists is said to a finite number 6 such that V a) € IH o <8 .6 ,8.6 >s6 =» <8 .6, 8 .6>/<8 a flow ip(x t;\fi ) (ii; r limit >-» 0 as t -» oo. to be globally (2.6.1) stable if it is (d/dt)<8 ^6,8 ^6^0, globally o f t h e f l o w \p IH i n w h i c h case i s referred and, in addition, t o as an a t t r a c t i n g i n IH s i n c e (x,t;^i ) Q t h e acclaimed ofglobal properties stability, 5 defines o f 0 (x,t;\p ) a subset o f hold. In 6 = oo a n d h e n c e 0 (x,t;\p ) has l a r g e a t t r a c t i n g d o m a i n i n IH. an i n f i n i t e 2.2 Definition stable for t>0. Remark 2 . 6 . 1 The 5 h e r e radius the globally Q stable if -» oo. a flow ip(x,t;<jt ) is said to be monotonically, (iii) the , 8 .<f> is said o holds when 8 (2.6.1) .6 The finite stability measure of a flow 0(x,t;0 ), denoted by r (\p), is defined as 0 r (iji) s K sup %, (2.6.2) t > t o where X is given by (2.5.3). Theorem 2 . 1 A flow 0(x,t;0 ) Q if r globally stable and only if (\b) N is monotonic, - r < 0. ( 2 . 6 . 3 ) Proof: (i) sufficiency. From (2.4.2) (d/df ) (1/2) a n d (2.6.2) <d ,<t>,d .<p> * <8. (j>,8 .</>> (r J. J. J. <d 4>,d <l>> s <a i 0 o ,3 i 0 >exp {2(r^ ± (I/J) o where 0 i s initial (2.6.5) under Q (2.6.4) (2.6.4) w i t h r e s p e c t t o t ' f r o m t Integrating ± (ip)-r). N J. the perturbation. -r) (t-t ) }, Q V t±t we (2.6.5) Q Taking the l i m i t (2.6.3), condition to t gives t -» » i n establish the sufficiency. (ii) the converse 0 i s MGS b u t r ( p r o o f by c o n t r a d i c t i o n ) . Suppose that (ip) - r a 0. (2.6.6) N T h e n , t a k e 4> ( c f . lemma 2.3) <t>(x,t;<j>) and =0, evaluate a s an i n i t i a l perturbation, i . e . , at t = t , Q the initial tendency of kinetic energy of <(>(x, t;<j>) (d/dt) (l/2)<d <t>,d <t>>\^ i i = <S 0, a 4>> (p i -r) ± (by ( 2 . 4 . 2 ) ) o =<d .0,9,4> (r i 0, a statement the proof. The position ft) - r) (by (2.6.6) ) c o n t r a d i c t i n g t o MGS o f 0(x,t;0 ) . T h i s Q completes a concept in of the global global stability theory Navier-Stokes equation (cf. Serrin, 24 occupies of a prominent stability J . , 1 9 5 9 a , 1959b; of the Joseph, D.D, 1976) a n d t h e B o u s s i n e s q 1 9 6 6 ) . However, dynamics GFD i t has barely (GFD). The e x p l i c i t were t a k e n i n Vickroy by andMosetti No optimization interest (1979) aspect and K a l l e n and s y s t e m . The p a p e r essentially deals f o r w i n d - d r i v e n ocean that of stability with the circulation. The l a c k o f may a r i s e any g e o p h y s i c a l l y fluid stability i n i s made i n t h e s e e a r l i e r p a p e r s . i n this consideration i n geophysical dimensional (1990) Joseph, D . D , (cf. of global andDutton (1980) f o r a t h r e e aspect o fglobal s t a b i l i t y appeared account Wiin-Nielsen Crisciani equation relevant from t h e flows can h a r d l y b e g l o b a l l y s t a b l e . G i v e n some e l e m e n t o f t r u t h f u l n e s s in t h e above analysis consideration, originated from we w i l l t h e study see that of global the global 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 p r o b l e m s i n GFD s u c h a s multiple equilibria,- disturbances subcritical bounds i ns u p e r c r i t i c a l flows. on flows I t i s clear t h e o r e m 2.1 c a n a l s o b e s t a t e d initially 0(x,t;0 ) Q growing perturbations growth rate and t r a n s i e n t from t h e above i n terms of growth i n proof that of the existence of (or e r r o r s ) t o a given flow a t i n s t a n t t . T h u s , we h a v e 2.1 At any instant Corollary one initially 0 (x, t' ;\I> ) Q given by the growing if t' t ), perturbation and only if there exists (error) X C0 (x, t' ;\j)^)) (2.5.3). 25 to - r at least the > 0, state where X is C o r o l l a r y 2.1 covers and t h e i n i t i a l C o r o l l a r y 2.2 state Let 0 e {*}. In t h i s s p e c i a l case, € Q X (\p ) -r>0 <=> existence r^frli )-r>0 «=» existence Q Identifying flow, recent papers (Farrell models with Ekman damping t h e Green model perturbations since forecast zonal shear Q t o 'a given from Orr's on a flows work of several (Boyd 1983; specific zonally 1989c), on t h e Charney (Farrell 1982, 1984 and 1989a) and often and Eady 1990). D i r e c t i n g a t t e n t i o n t o i s natural flows, shear 1987), (O'Brien, rules to 0 . perturbations constant t o synoptic emphasize properly configured perturbations For q (1907) , has been t h e focus on i n v i s c i d flow Q growing 6 growing 1985 and F a r r e l l asymmetric such o of initially initially Couette flow Shepherd on growing <p to \p , or a s u b j e c t with a l o n g h i s t o r y s t a r t i n g on p l a n e we have Then: of initially Q s p e c i a l case where t' = t an important meteorologists the r o l e s of (Palmen and Newton, i t has been known s i n c e these 1969). O r r (1907) t h a t a p e r t u r b a t i o n with i t s phase l i n e s o r i e n t e d a g a i n s t t h e mean shear e x h i b i t s temporal a m p l i f i c a t i o n . , The f o r g o i n g r e s u l t s regarded as a continued the context (i.e., effort C o r o l l a r y 2.1 and 2.2) may be o f t h e above c i t e d o f time-dependent, n o n l i n e a r and zonal 26 works i n asymmetric flows. Indeed, the constitute an initially growing to the common difficult energy ), i t i n the for a Rayleigh case rate of the error) Definition against require flow. to to 0(x) that is the a scenario of error) It is rather to Q that for given flow of the that matters the energy generalized rate per unit our i n t u i t i o n . perturbation (or introduce growth rate <p(x,t;<f> ) not significant the initial we is what simply not obvious measure, i t i n 0 at conversion optimal of ratio flows, not the energy search opposed a the energy and 0 but 0 2.3 (2.4.3) asymmetric grow lemma As shear", and The instantaneous (or given <f> a n d 0 t o a quantitative disturbance a perturbation zonally notion 2.3 to t h e Ekman d a m p i n g r . (i.e., energy), give a with systematical (2.4.2) to between quotient disturbance worst than a "lean between perturbation conversion of suffices be l a r g e r that of from growth together for perturbations see conversion instant To algorithm notion to instantaneous 0(x,t;0 corollaries in a cr ($;$) of a N flow 0( x,t/0 ) is , Q/ defined as <r C0;0; s (1/2) (d/dt) ln{<d J. N Theorem .0,3 .$>} 2.2 perturbations At any (or errors) . (2.6.7) J. instant 0 in 27 t, IH to among a flow all possible 0 Cx, t ; 0 j , 0 (cf.lemma sense 2.3) is the one (or the worst one) in the that er (<(>;&) t <r (<i>;ip), V 6 e IH. N N Proof: from % = P C«>;0) The (2.4.2) and (2.6.7) (2.6.8) r - * p(i>;^>) - (by lemma r = (by w h e r e X (ty; \(\l>) - r i s g i v e n by 2.3) (2.5.3)) (2.5.3), which completes the proof. B 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 favorably explain value configured the (Roebber, other 1984; barotropic constant of n e u t r a l Rossby nonlinear models. First, context Second, i t s growth in 1989a; O'Brien in early example, studies those (Farrell, (1988, our found is whereas t h e the 1990). have The found determined for excitation in this study d i f f e r O'Brien (1990) carried others functional 28 initial 1988). 1989b) and search to cyclonegenesis shear flow are used i n optimal waves rate o r i n model s t u d i e s o f obtained For and observed 1988, perturbations in Farrell aspects. 1986) (Farrell, perturbations optimal growth Sanders, applications. The perturbation explosive problems optimal those optimal to out i n several within w e r e made u s i n g be from optimized a fully linear is the generalized norm or Rayleigh optimization possible process i.e., but £-2 the quotient norm in this i n previous subject in study kinematically i n our case but i s the energy other i s taken admissible studies studies. with respect functions i s carried t o some c o n s t r a i n t s Third, the to all whereas the same out c o n d i t i o n a l l y , which are p h y s i c a l l y sound essentially subjective. 2.7 Application . The notion introduced I I : multiple of multiple into geophysical equilibria equilibria fluid was fairly dynamics recently i n an e f f o r t t o account f o r the v a c i l l a t i o n between the low index pattern") and high index flows i n the atmosphere on (Charney & Devore, 1979) and f o r n o n l i n e a r (Vickroy & Dutton, 1979). The concept has s i n c e been extended to effects ("blocking predictability i n c r e a s i n g l y s o p h i s t i c a t e d p h y s i c a l models, y i e l d i n g much i n s i g h t i n t o t h e v a r i a b i l i t y i n l a r g e s c a l e atmospheric flows (Kallen 1983; & Wiin-Nielsen, Rambaldi recurrence climate The of (Kall§n, Mo, 1984; weather changes to 1979; 1982, with low & & Rosenthal, (Proefschrift, nonlinearity order In 1981; Legras Gill, 1985), 1989) and 1989). spectral Proefschrift, 1985) . Tung systems (Marotzke, difficulty investigators Devore, & 1980; K a l l e n ; 1989) the 29 former has l e d the models or (Charney numerical case, early the and models question naturally a r i s i n g i s : are m u l t i p l e the consequence equilibrium o f severe t r u n c a t i o n t o t h e o r i g i n a l d i m e n s i o n a l systems ? o r do m u l t i p l e original models? continuum equilibrium states equilibria In t h e l a t t e r infinite e x i s t i n the case, g i v e n an s t a t e o b t a i n e d from numerical i n t e g r a t i o n o f the model e q u a t i o n s , one wants t o know i f t h e r e e x i s t s equilibrium In this context simply another state. section, we address o f system t h e above (1.2.1)-(1.2.3) a n a l y s i s . We do so by c o n s i d e r i n g questions i n based on t h e problem the the global o f uniqueness of t h e e q u i l i b r i u m s t a t e . I t i s important t o d i s t i n g u i s h t h i s uniqueness one of problem from seeks m u l t i p l i c i t y external t h e one f o r IBVP. of equilibrium and internal model In t h i s states parameters. problem, f o r a given set In c o n t r a s t , t h e one f o r IBVP i s concerned w i t h how many flows can e v o l v e from a given initial uniqueness of uniqueness state. IBVP We holds of equilibrium shall show below unconditionally state may break that the whereas t h e down for some v a l u e s o f model parameters, thus i m p l y i n g t h e 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 field of a flow condition a given 2.3 (uniqueness 0 (x,t;\fi ) Q and hence initial of IBVP there velocity (1.2.1) - (1.2.3)) is continuous can be only field. The Velocity w.r.t its initial one flow evolving from Proof: realized except and another , flow 0 (x, t ; 0 +0 j Consider under Q t h e same s t a t e 0 +0 . its initial (2.6.2) t h a t 0(x,t;0 ) conditions o o f (1.2 .1) - (1.2 . 3) Q as those f o r 0 (x, t;0 ) I t thus f o l l o w s Q from (2.4.2) t h e k i n e t i c energy o f t h e d i f f e r e n c e field satisfies Q (d/dt)<d.<t>,d.<l» J. s 2<8.<p,d.<p>(r(ip)-r) J. J. J. N which i m p l i e s t h a t f o r t st's t o 0 * J f c / (d/df) <a 0, e f>-2 f r i ± = Kdf.B^e- ***™-***' (x\>) -r) <d *, d <t>>\e N ± ± f . 2 v 2 (r W r ) t ' dt' (2.7. la) o Now, as l e t the d i f f e r e n c e between the i n i t i a l by ||v0 fl, approach t o zero. measured velocity fields, Then, we have from (2.7.1a) t h a t a t any f i x e d t * t Osc ||70|| = c <d 4> d <t>>s < a 0 , a 0 > = ||V0 || -» 0, 2 2 with C 2 2 i f i 2 i Q o 2 (2.7.1b) t o ||70|| -» 0 as ||v"0|| -> 0 and hence the c o n t i n u i t y o of v e l o c i t y f i e l d IBVP o 0, A t = t - t , He " ^ which l e a d s the ± follows w.r.t i t s i n i t i a l from (2.7.1b) data. after The uniqueness o f setting ||V0fl=0 i n (2.7.1b), which e s t a b l i s h e s the d e s i r e d r e s u l t s . - Viewing the prediction oceans as an i n i t i a l o f motions i n t h e atmosphere or v a l u e problem, we a r r i v e a t the n o t i o n , on the basis by ( 1 . 2 . 1 ) - (1.2.3) its o f theorem initial of practice, this "perfect the together amplify as problems remain the limit relevant be to to traced climate vacillating problem to changes, i t itself possible equilibrium and be is how t h e c h a n g e d Devore, 1979) . successful, multiple i t among the a is states, is equilibria is of This clear important to available 32 appears point in more o f v i e w may connection In the remark that among variation f o r such act that is a the distribution that to equilibrium distribution alters (cf. tend state 1969). and climate This, o r oceans i t equilibrium reflected initial IBVP. attractors, (1963, question boundary It work perfect predictability the possible problems. Lorenz's climate of of I.e. that of the conditions t h e atmosphere and strange the prediction back implies as and B . C . or In q attainable. certain in that 0 . on t h e (1.2.2) that subsequent given depends the uniqueness o f view cycles uniqueness errors the the humanly under advances, despite systems not that i n t h e sense difficulties, above is initial time has conditions fact the point dynamic mentioned from governed Q determines view which the 2.1), Taking of 0 (x,t;{fi ) predictable evolving boundary with corollary states, flow (1.2.3), the flow uniquely Q point of condition as 0 prediction" knowledge that is perfectly state evolution 2.3, the matter (Charney an approach have an a c c u r a t e to systems count for to of given conditions. Unfortunately, no general algorithm for a c c o m p l i s h i n g t h i s t a s k has come i n t o l i g h t . The d i f f i c u l t i e s i n d e t e r m i n i n g whether a system has other e q u i l i b r i u m other than t h e known one a r i s e frequently states i n p r a c t i c e . The f o l l o w i n g r e s u l t p r o v i d e s a l i m i t e d s o l u t i o n t o the problem. Theorem 2.4 0 Let (necessary € Q condition If the system for multiple has an equilibrium equilibria) state than 0 , then r^ty^-r^O. (2.7.2) Q Proof: (by c o n t r a d i c t i o n ) r Suppose s t a t e other than 0 an e q u i l i b r i u m Q that t h e system Let (2.7.3) 0' denote t h e other e q u i l i b r i u m o field between 0 r realizations initialized field and 0', 0 0 0 (1.2.1)-(1.2.3), of from 0 q and 0^, equilibria respectively, o f (1.2 . 1 ) - ( 1 . 2 . 2 ) = V* \* 0 On t h e other V 0 and 0^ a r e Q hand, v two (2.7.4) t *t . we apply \V*\ exp{2(r W-r) o q we have t h e energy inequality Q a 0(x,t;0^), and the uniqueness of over t h e i n t e r v a l [ t , t ] t o have Nil and and the d i f f e r e n c e that the two r e a l i z a t i o n s ( c f . theorem 2 . 3 ) , WM, Now, c o n s i d e r two 0 0(x,t;0 ) By t h e assumption o 6 the difference o state, i . e . , <p = 0 - 0 ' . 0 o>(x,t;0 ). distinct has but (0 )-r <0. N O other (t -t )}, Q 33 V t *t Q (2.6.5) < |V0| O contradicting (by (2.7.3)) (2.7.4). T h i s completes the p r o o f . To t h e e x t e n t t h a t the system motions i n the atmosphere indicate as that observed governs the or the oceans, the f o r g o i n g Charney subsequent works not a r i s e (1.2.1)-(1.2.3) the phenomenon of m u l t i p l e in B and Devore (e.g., Kall§n, equilibrium (1979) and 1981; P r o e f s c h r i f t , from the t r u n c a t i o n s results states in many 1989) may invoked i n these s t u d i e s but r a t h e r occurs under c e r t a i n e x t e r n a l and i n t e r n a l conditions for (2.7.2) which met. We (2.7.2) will or some c o r r e s p o n d i n g form p r o v i d e n u m e r i c a l evidence t o p o i n t i n the next c h a p t e r . By now, of is strengthen t h i s t h e r e appears t o be little doubt on the 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 at mathematical level. The remaining concern i s how relevant v a l u e s f o r which m u l t i p l e e q u i l i b r i a r e a l atmosphere and c l i m a t e system For the p r a c t i c a l section, the the g l o b a l a n a l y s i s ( i . e . from important to note condition i s s a t i s f i e d that (2.5.1) and when a direct concluding the existence N determined O (2.7.2) calculation from (2.5.2)) p r o v i d e s of m u l t i p l e of m u l t i p l e 34 at the onset of the ** (0 ) a q u i c k check on the p o s s i b i l i t y is (Tung & Rosenthal, 1985). with (2.7.2) parameter are observed are t o the concerns mentioned condition the e q u i l i b r i a . It as a necessary i s needed b e f o r e equilibria. However, a violation 2.8 some limitations observed known r u l e s out such a p o s s i b i l i t y . i n i t i a l modal v s . nonmodal o f t h e modal growth oceanic f o r some mesoscale clearly III: Application The of o f (2.7.2) rate i n accountingf o r and atmospheric times. growth r a t e F o r example, phenomena h a v e the growth been rates of 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 one y e a r (Schulman, Crease (1962), (1973) suggest situation 1967) w h e r e a s t h e o b s e r v a t i o n d a t a i n Swallow that (1971), and Koshlyakov i t i s of order has l e d t o o p t i m i z a t i o n w.r.t. expression of a and Grachev few months. o f t h e modal This growth rate such as 3 and t o p o g r a p h i c model parameters h e i g h t i n o r d e r t o match t h e observed p o p u l a t i o n o f e d d i e s i n the oceans (Cf. Robinson and Mcwilliams, 1974). Another example i s found i n the study of cyclonegenesis. I t i s well known typical that the cyclones i s between Sanders, 1986), period 12 and clearly of deepening 48 hours larger than of observed ( c f . Roebber, any found in 1984; modal analysis of baroclinic i n s t a b i l i t y , e . g . , 133 h o u r s i n Valdes and Further found Hoskins (1988). stability studies problems. I t i s observed is considerably often rate the 1982; of hydrodynamical during the i n i t i a l flow supports examples that lower Boyd, 1983; O ' B r i e n , be flows v i a i n i t i a l t h e maximum m o d a l than period, strong may modal 1990). 35 growth the instantaneous and t h i s happens instability even in value rate growth when (cf. Farrell, Despite t h e i n c r e a s i n g evidence for this defect s u g g e s t i o n t h a t the n e g l e c t o f the continuous modal analysis i s responsible questions 1) and the spectrum i n the f o r t h e flaw, the following remain: does t h e modal growth rate inevitably growth r a t e o f d i s t u r b a n c e s t o a g i v e n underestimate t h e flow? 2) i f so, what i s the u n d e r l y i n g cause? 3) i s t h e r e any s y s t e m a t i c procedure t o overcome the d e f e c t ? Our i s t o provide objective of t h i s section answers t o t h e above q u e s t i o n s nonlinear explicit and z o n a l l y comparison i n t h e context varying o f BEP flows. We (2.5.1)-(2.5.2) explicit of barotropic,. do this by an arising in the g l o b a l a n a l y s i s with the BEP a r i s i n g from the modal a n a l y s i s . The BEP a r i s i n g assuming that i n t h e modal the solution a n a l y s i s i s obtained t o the l i n e a r i z e d (2.2.2) - (2.2.4) has t h e modal form %(x) € W ( c f . s e c t i o n 2.2), and e £,(x), after v e r s i o n of where o- € C and and i n t r o d u c i n g i t i n t o (2.2.2) (2.2.3), <rv £ + £[%;*] 2 = 0, (2.8.1) where € s a t i s f i e s B . C . ( I l l ) o r (IV), with £ g i v e n by ( 2 . 2 . 5 ) . To make t h e subsequent d i s c u s s i o n s p r e c i s e , we i n t r o d u c e 36 the f o l l o w i n g Definition definitions 2.4 A solution t o as a finite referred simply a disturbance); <f>(x,t;4> ) amplitude nonmodal disturbance (or a solution <p(x,t;<p ) t o the l i n e a r i z e d Q o f (.2 .2 .2) - (2 .2 .4) as a linear version t o (2.2.2) - (2 .2 .4) i s Q disturbance; a linear <rt disturbance determined in from the form (2.8.1), </> (x, t;%,<r) = £e , as a modal disturbance. Furthermore, </> i s s a i d t o be a modal initial perturbation or {£} of (2.8.1), o Im(£) w i t h £ € t h e eigenspace otherwise (2.8.1), as a nonmodal initial (%,<r) with i f <6 = Re (£) q satisfying perturbation. A growth r a t e i s s a i d t o be modal growth rate i f i t i s o b t a i n e d as t h e r e a l p a r t o f t h e e i g e n v a l u e o f (2.8.1) (i.e., Re (<r) ) , otherwise nonmodal growth rate. To setup a stage f o r comparison of t h e two BEPs we first establish Lemma 2 . 5 Then, its Re(<r) = pCC/*; = Let * be an equilibrium modal disturbance pf£/*; " e £(x) state of (1.2.1) - (1.2.3) . has the growth rate r, (2.8.2a) (2.8.2b) <£,^>/«;,B€>, where A and £ are defined by (2.4.4) and (2.4.5) . Proof: l e t <• € {£}. Taking t h e i n n e r product o f (2.8.1) with € and adding i t t o i t s complex conjugate, we have 37 (2.8.3) where £ i s the adjoint self-adjoint o f 2 g i v e n by ( 2 . 3 . 2 ) . on S 2 {£[} (cf. lemma 2 . 1 (ii) ) , r e a l on {?}. I t hence f o l l o w s from ( 2 . 8 . 3 ) <££,€> i s thus that + <£, ( 2 + £ * ) € > = 0 , (by s e l f - a d j o i n t n e s s o f £+£*) •* -2He (<r) <£, =» -2Re (<r) < £ , B€> + 2<£, (4-z3)Z> = 0, (by which immediately l e a d s t o ( 2 . 8 . 2 ) . (2.3.3)) B Next, we i n t r o d u c e t h e n o t i o n o f l i n e a r of Since £ i s a ! stability measure an e q u i l i b r i u m 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) I t then f o l l o w s from ( 2 . 8 . 2 ) max Re (cr) and ( 2 . 8 . 4 ) that (2.8.5a) = r f*; - r , Xi as opposed t o t h e maximum nonmodal max <r„C0;0; = r N where The N (*) i s d e f i n e d by answer (cf. ( 2 . 6 . 8 ) ) (9) - r, N r growth r a t e (2.8.5b) (2.6.2). t o the question r e l a t i v e s i z e of r N (<b) t o r L 1 ) raised above lies (*), which i s s e t t l e d i n 38 in the 2.6 Lemma (comparison equilibrium r (*; stability measures) For an state * of (1.2.1) - (1.2. 3), it holds that s r Li of m . (2.8.6) N To p r o v e Let t h e lemma, we n e e d t h e f o l l o w i n g {x } and {y } be two real n finite inequality: sequences and y >0 for n n any n. Then Y x ft y s max(x /y ) . Lt n n Lt n To see J n (2.8.7), J x ft y = Lt n n Lt n which n i t n n (i) = Let p(%;V) i e two (2.8.7). {£} and Now, we t u r n to be written £ of w e = have (2-8.8) n ' (since structure interest cases: them that {l <K^ >}{l <Sn'**A- constant little t o note n n terms Vr>= r , i (i) i s ' £(x)*0 an e i g e n s t r u c t u r e , constant of n (L. n £ V-n= r , i * suffices (x y )/y ft y Lt n Expressing As (2.8.7) n immediately gives Proof: P(V*> n n for kinetic is zero, constant. thus and £ For e P.. F u r t h e r m o r e , energy physically). both x of a normal corresponding The l a t t e r 39 (i), mode a constraint are not constant case to let either £(x) with case of implies a n d ( i i ) one term i n the denominator o f (2.8.8) hence (2.8.7) p f€/tf; =s max j<€ i s strictly positive i s a p p l i c a b l e t o ( 2 . 8 . 8 ) , which y i e l d s >]•/(<? r « C >]• s max \<4>,M>\i\<<bi %<P>\ = r <6elH I For case only J I suppose (2.8.9) that % *0 and ^=0. r i n (2.8.8). f o r 6^ appears on t h e two s i d e assertion. Thus, we Then, have (2.8.9) of over {£} proves t h e a fundamental y i e l d s an e x p l i c i t raised above. To = p(€/*; s r (9) h s r (*) - r, (by -r, i n conjunction with a f f i r m a t i v e answer t o q u e s t i o n 1 ) see t h i s , (2.8.1), determined from (2.8.6) property (2.8.5) fie(«r) (¥) . N without a p p e a l i n g t o ( 2 . 8 . 7 ) . To t h i s end, t a k i n g the maximum The i w.l.o.g., (ii), t h e term (2.8.9) ( c f . (2.4.7)) and we note that f o r any (cr,£) i t holds that (2.8,2)) (by (2.8.4)) - r , (by (2.8.6) ) N = X (9) - r, (by lemma 2.3) N = cr (by theorem 2.2) N which a l l o w s f o r t h e c o n c l u s i o n : Theorem 2.5 (initial an equilibrium state nonmodal and modal growth rate) Let * be of (1.2.1) - (1.2.3) . 40 Then, it has at least one nonmodal initial perturbation which has a growth rate not less than the maximum modal growth rate. Moreover, reveals the interpretation t h e answer 2) : t o question a basic of conversion between flow disturbance energy in the eigenspace the property the and ratio a (2.8.9) of energy disturbance {%} e S is always less than or equal to the extreme value of the same ratio space of (i.e., in M) . The the all real global kinematically analysis (i.e., rate of disturbances c l o s i n g question Further (2.8.6) by t h e modal of the global analysis, analysis transient growth of disturbances or equivalently why the modal analysis the flows. see To temporal amplification this, consider ( 1 . 2 . 1 ) - ( 1 . 2 . 3 ) . I t i s evident t h e *. However, (2.8.6), constitutes thus 3 ) above. applications to predict for measures with and the (2.8.6) can be found, f o r example, i n e x p l a i n i n g why there exists flows, disturbances procedure f o r circumventing t h e u n d e r e s t i m a t i o n o f a general property in the lemma 2.1-2.3) t o g e t h e r comparison o f t h e s t a b i l i t y growth admissible to i tis still a from in subcritical inevitably phenomenon in subcritical flow (2.8.5a) t h a t by t h e comparison of s t a b i l i t y possible t o have r (*)-r>0 N fail stable ^ of r (*)-r<0 measure i f the * i s outside of growing the MGS r e g i m e , nonmodal r (*)-r<0, perturbation. then N disturbance. measures, the * Moreover, i t and hence has no has by On no the growing to have the other initial initial hand, growing comparison modal an of nonmodal the disturbance i f stability either. To summarize: (initially 2.6 Theorem subcritical flows) (1.2.1) - (1.2.3) perturbation, flows If has then perturbation; The of does has documented been problems barotropic Farrell, constant growth 1987), frequently found i n several growth cyclone despite unstable, argument spatial the modal on perturbations. that the modal have or This flow. on i n i t i a l (Boyd, 1990). realistic presence large It value 1985) , disturbances which e x h i b i t of, Couette (Farrell, for a in was o r i g i n a l l y (O'Brien, that flows scales absence of nonmodal disturbances j3-plane and o f t h e Green model t h e Ekman d a m p i n g t h e m o d e l synoptic works models flows studies of on p l a n e recent and Eady i n these * any growing modal disturbances i n h i s work shear in false. transient o f t h e Charney state growing not have s u p p o r t i n g no growing modal b y O r r (1907) the equilibrium initially the converse is phenomenon nonmodal perturbation an no it reported of growing of 1983; It is value on t h e transient of only weakly observation has l e d to in general instability cannot serve in to explain midlatitude from the but release configured Given the occurrence of initial the rather mean by fairly emphasized in synoptic inclusion result Newton, of in flow possibility and they that 1969), and which hence (cf. believe into the more properly 1985). disturbances are perturbations as Petterssen, that a present allows by (Farrell, amplitude we predominantly energy scale experiences framework perturbations cyclone disturbances perhaps potential large baroclinicity a arise scale nonmodal p e r t u r b a t i o n s initiated Palmen of cyclone for suitable 1955; successful formalism finite for may amplitude the study of cyclonegenesis. 2.9 C o n c l u d i n g remarks We have performed (1.2.1)- (1.2.3), problems. has When yielded with applied a of initially growing spatial flows structure, exponentially growing used perturbation to to the to a as analysis applied global of according to to perturbation. identify given flow. an a system number ofG F D for there of its t h e modal whether The analysis the an amplitude which exists procedure initial to exists analysis there same applied the characterizing whether optimal When the stability, regardless opposed of to procedure basis perturbation stability be results on t h e characterizes also global systematic stability and a an can nonmodal study of relation global of analysis, property the (cf., unstable flows. application in resulting in condition for by disturbances, as explosive the results two chapters, hold indicator the of cause growth development weakly in has also found equilibrium for existence these underlying known t r a n s i e n t multiple condition an and analysis of necessary that the well global out) rate the comparison as study ruling fundamental growth and The the a i s expected (or of a states, sufficient such phenomenon. f o r a l l flows governed will analyses (1.2.1)-(1.2.3). In which are flows rates, the cause of such the growth has uncovered modal phenomena nonmodal with 2.6), of subcritical to together development underlying It modal lemma limitation potential in initial the following are p a r a l l e l performed equivalently Moreover, numerical in to, a in R ) we M we and extend, finite those dimensional a n d f o r a more will complement experiments. 44 in this chapter, function specific the present space physical analyses there but (or system. with Appendix 2A value A variational of generalized In this approximation Rayleigh quotient appendix, we p r e s e n t a method f o r the optimal p Rayleigh-Ritz procedure A (cf. Reddy, the 1986) optimal denote for obtaining problem M linearly the approximate max p(<p;\p) independent solution u , f o r 0e(H. L e t 1 functions • where u u M 2 i n H and write i = cu j 6 to (2A.1) i the indices, summation convention (i=l,2,.. with is M) applied being to the constants repeated to be A determined. p(<p;\ft) Evaluating at </>, we h a v e A Ti(C C , lf . . .,C )mp(A)=C(C , 2 H C(c , c D(c, c where the , ...,c)m , . . , operators (2A.1) then dC/ac =Ti ± or i n terms if where J -L the inner d must , i and SS satisfy (2A.3) X (2A.4) J product given i n H (cf. by (2.2.11)), ((2.4.4) constants and c=(c ,c , 8Tl/dc^=0 (i=l,2,..M), Rayleigh-Ritz to form (2A.6) : The l a r g e s t approximation eigenvalue of (2A.6) is value o f p(6;ifi) i n to the optimal eigenvector A 6 via in (2A.5) eigenvalue and i t s corresponding (2.4.5), which leads = u[<u ,Su ,>]c, i with ...,c^) i=i,2,...M of matrix u B 11(C). disturbance . . , CJ , (2A.2) if <u . , 8 u .>c .c ., The o p t i m i z i n g aD/ac [<u dUj>}c IH, )= denotes respectively. M r M <•> C )/D(C 2 <u.,du.>c.c. c Z 1 C, 1 (2A.1). 45 c will yield then the an o p t i m a l 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 No general hydrodynamical (or system equivalently attempts 1979; made Kall§n chapter to in R ) is (periodic forcing global finite has M low order done channel 0* a c t i n g such known, systems for function except note Vickroy that Before physical a the subsequent for a system few and Dutton, The o b j e c t i v e numerical defined and as u n i f o r m z o n a l parameter by (1.2.3) of this experiments, the (1.2.1), with momentum describing restriction to the (1.2.2a) the source, external i . e . , the observation 0 directly of 0* to the forcing the form strength. (3.1.1) is We not analysis. concluding streamfunction arising a (3.1.1) C7* i s essential space for = -Uy where any analysis. the geometry) (cf. 1980). an analysis dimensional been and W i i n - N i e l s e n , facilitate analysis of in i s to perform To 0* theory can from introduction, (e.g., be the let Gravel, decomposed uniform 46 into zonal us make 1988) two a simple that the parts: one momentum forcing (3.1.1) and one r e p r e s e n t i n g the deviation, i . e . , >l>(x,y, t)=-U*y + *(x,y, t) . Thus, i t follows from (1.2.2a) and (1.2.3) (8/at)V 9 + J(-Uy+*, 2 at d<b/dx=0, (3.1.2) introducing (3.1.2) into (1.2.1), that V 9+py+ (h/H) f ) = -zM <t>, 2 (3.1.3) 2 and Osxsl, y=0,d 9(l,y,t)=9(0,y,t), at (3.1.4a) Osysd, (3.1.4b) *(x,y, 0;=* (x,y), (3.1.5). o Q (x,y) where to is Q t h e system the system since i s time is effectively The chapter in § procedure 3.3, stability optimal with =-U* y+$ (x, y) o determines external zonal . A solution the response forcing part the study organized cast -U* y of (3.1.1). of \JJ ( c f . of the s t a b i l i t y into as follows. a spectral form for the global analysis i n R results i t s and search initial uniform 0) of 0 reduced to that f o r is is thus the independent, (3.1.3) - (3.1.5) general to the (3.1.2) then \p(x,y, that (3.1.3)- (3.1.5) original Furthermore, such from for initial nonmodal Numerical illustrations depending on r i n § 3 . 5 , with i n M growing using § system 3.2. A presented to global perturbations and presented in § a equilibria set of a c o n c l u d i n g remark 47 is application perturbations a r e made The 3.4. i n § 3.6. 3.2 Governing equations i n R M By v i r t u e following o f t h e channel boundary c o n d i t i o n Mitchell and Derome (1983) , (3.1.4), we, introduce a set of orthonormal f u n c t i o n s G s {G^}, with G^ g i v e n by ' 2 G ± • G{m,n) = 1/2 cos (nny/d) lsisl 2sin (2nmx/l) sin (nny/d), if 2cos (2nmx/l) sin (nny/d), if m=l,2, . . . ,J; where t h e index convention if I+l*isI+I*J (3.2.1) I+I*J+lsi*I+2l*J n=l,2, . . ., I; M=I+2I*J. i f o r G\ i n c r e a s e s based on t h e columnwise f o r t h e two-dimensional array G(m,n). Taking (3.2.1) as a b a s i s f o r IH ( c f . § 2.2), we expand a l l dependent v a r i a b l e s i n t h e form M (9,h) = £ (^ h )G ir ± (3.2.2) ± i and i n s e r t of them i n t o (3.1.3) t o o b t a i n t h e s p e c t r a l version (3.1.3)-(3.1.5) below (d/dt)t = F &;*), (3.2.3) = t(0) (3.2.4) and fi are column v e c t o r s i n R , e.g., where M ] t=[i 1' 2 T M and F : R M -» R i s an Af-dimensional M w i t h t h e i t h component given by 48 nonlinear vector field, + 1I c i ^ j [ ^ o / + fi I b.A.-ral*. w v 4 $ j c ] (3.2.5) y j a , bj^.., c j with 2 ± ± given by k ' (nn/d) , a (m, n) s (2nm/l) s i f lsi<i + (nn/d) , If I+lsl*I+I*J 2 2 r27rnj/i; + f r c n / d ; , 2 c. , jk b ij S ^ . J f G . . , ^ ± < G d G Finally, whose 8 x > a is = in effect (3.2.3) under While the noticeably ideas the represents stability details from the equilibrium global its chapter details). behind considerably analysis counterpart t h e same. is Let model parameter states of in is M of manipulations 3.3.1 of any concern. different for (3.2.7) (3.2.8) motivations 2 , j i J k / h are e s s e n t i a l l y the (3.2.6) I+l+I*Jsi*I+2l*J ~ j±(3.2.3) on = -c = J J f e l Global analysis i n R 3.3 of j/ = c if 2 In l i g h t the in IH, of t h i s , subsequent simplified or the R M basic description mathematical neglected (see Disturbance equation t(t;t ) Q denote the solution 49 of (3.2.3) initialized from t . C o n s i d e r another Q realization <f(t;^ +<l ) o o (3.2.3) of made under t h e same c o n d i t i o n as those f o r t(t;t ) except i t s Q I.B.: t +$ . As i n t h e s e c t i o n 2 . 2 , t h e s t a b i l i t y problem f o r Q Q t(t;t ) can be Q investigated by c o n s i d e r i n g t h e temporal evolution of the d i f f e r e n c e - *<t;S o ), t*t{t;t +$ ) Q o (3.3.1) which i s governed by t h e i n i t i a l v a l u e problem (d/dt)% = t($,4,oi), 1(t ) = $ , Q (3.3.2b) 0 f : IR -» R where from (3.3.2a) M i s an M-dimensional M t h e expansion of the nonlinear vector vector field field arising F as a T a y l o r s e r i e s around t h e motion t, d e f i n e d b y m f($,4,cc) F|(^|^) F + T^tftf) a/2)Ffttf\$\$) e aw($+6 t/t a)/88 \ x ^(^|2) • 1 f s = Q + 0(\~$\ ), (3.3.3a) 2 , (3.3.3b) 8 T ($+8 1+8 1;%,a) /88 88 \ 1 2 2 A direct evaluation of ( 3 . 3 . 3 b ) , and ( 3 . 3 . 3 c ) (3.3.3c) yields T$($\t) = *$, F ft<*l$|£> or (3.3.4a) ZBffi' = (3.3.4b) i n component form {F|(^|^)} i = I A..*., j lI OfctflJtfni - 2 ' B lj *j* ' k (3.3.4c) (3.3.4d) k 3 k 50 with A s [A. .], A. . s (l/a .)(u*b. .{-a .) + pjb. . - ra .8 . . xj i \ ij J i J J- 13 2 2 2 + Y \c .$,(-a .)+c . .,(f h,/H L |_ lkj k j ijk 0 k k 2 ]}' x (3.3.4e) (3.3.4f) N o t e t h a t t h e l i n e a r o p e r a t o r A : R -> R a n d b i l i n e a r M B :R xR M -» R M are the counterparts i n R , respectively. (2.2.6)) order M polynomial i n (3.3.3) o f degree vanish 2 in I operator o f £ a n d # ( c f . 2.2.5) Also, M terms M note that identically (cf. (3.2.5)). and t h e higher since To t h i s F i sa end, we have a c l o s e d form f o r f .= I ij*j j f A + I I ijk*j*k' J k B ( 3 ' ' 3 5 a ) or £ = &$ + B$$. 3.3.2 Symmetrized energy equation The into (3.3.5b) next step i s t o put the disturbance a form such t h a t i t s time terms o f a s e l f - a d j o i n t a bounded functional. r e s u l t from m a t r i x operator tendency inR To a c c o m p l i s h algebra 51 M energy equation can be expressed i n and hence i n terms o f this, we f i r s t state a Let P = [P±j] e R Lemma 3 . 1 = CC^j] e R PQ - I I P i be a skew-symmetric matrix. Then M X M Q ± j be a symmetric matrix and Q M X M = 0. ± j (3.3.6) 3 With (3.3.5), (d/dt)4> (3.3.2) c a n b e w r i t t e n - [ A .<p. + [ [ B . ^ . ^ . ± (3.3.7) ± j 7* 2 Now, we m u l t i p l y 1 (3.3.7) w i t h a±4>£ a n d sum o v e r I <l/2)a>J (d/dt;{ i n component f o r m } - [ i [ ^ A . ^ + i j a .B.. £ [ £ i k j i t o have 2 k 0.4,.^. (3.3.8) Note that thebilinear skew-symmetric of matrix with respect 1 i,j. summation symmetric terms I t thus follows i n (3.3.8) properties from vanishes (3.2.7) corresponds t o indices 3 (3.3.4e) ij make lemma 3.1 t h a t exactly. a n d (3.2.8) ijk zero (3.3.8). With these observations, demand that t h e operator a .A . , ( c f . t othefirst (3.3.8) t h e second Similarly, the ^ contribution symmetric K i n v o k i n g b. . a n d c . ., i n t h e l i n e a r ^ toa i, j b y v i r t u e c>.<>.<6. i s o b v i o u s l y (3.2.7) a n d ( 3 . 3 . 4 f ) w h e r e a s w.r.t a ,B. operator 2 i ij summation i n i s reduced t o (in terms o f i n n e r p r o d u c t on R ) M (d/dt) A . (1/2)<%,V%> =<<J,A#>-r<$,D#>, [A..], A.. S I c. $ (-a .), kj k (3.3.9) (3.3.10) 2 k 52 D = [D ] = D, D T ±j It i s clear achieved s a\ 8 ±j from ± j . (3.3.11) (3.3.10) that (3.3.9) has not yet t h e d e s i r e d form due t o t h e non s e l f - a d j o i n t n e s s of the o p e r a t o r A. To remove the non s e l f - a d j o i n t c o n s t i t u e n t i n A, we observe t h a t Lemma 3.2 Let P € R P MXM . Then, there exists a decomposition = P + P ~ , with + p T + = p • (1/2) ( T + p ; , T T p " =-p"=- (i/2) ( P - P T ; , such that <%,V%> = <$,l? '$>, V % e R . + M _T The lemma Applying follows from t h i s observation lemma 3.1 a f t e r noting (3.3.12a) + or i n terms o f the G e n e r a l i z e d Raleigh (d/dt) (1/2)<%,*>%>= <$,D$> (p$;t) A + + Quotient p(~$;t) - r), (3.3.12b) i s given by s (1/2) (k + A), (3.3.13) T p($;t)'m <$,k %>/<$,$$>. (3.3.14) + Remark 3.3.1 operators A I t follows from (3.3.11) and (3.3.13) t h a t t h e and D are s e l f - a d j o i n t i n IR , (i) A = ( A ; * , + =-P. t o (3.3.9), we o b t a i n (d/dt) (1/2)<%,T>%> =<^,A £> - r<%,V%>, where A P _ (3.3.15) + ( i i ) D = D*and D is positive i.e., definite, 53 (3.3.16) where ()* denotes the adjoint are the reminiscent and SB i n W ( c f . 3.3.3 to As p (2.4.6) i n the case the global and p($;t) of the continuum are to by (3.3.12b), properties operators we c o n s i d e r i n R . For this, i f order and that i n that case eigenvalue p by generalize extreme i n + + (3.3.14) and standard DR and the M of and (cf. A A lemma {Algebraic properties and D be given by (3.3.13) 54 recognize the show i t in to to the I of above by smallest below that satisfying the possible to is such IR M that p realizing systematically. 3.3 of 1978). principle obtained by e s t a b l i s h i n g by and D $ values bounded shall + functional. operator Kreyszig, disturbance p can be be steps functional reduce unit below (3.3.16), Rayleigh would a we with the important p would observation, the values Lemma 3.3 by (3.3.15) possible Let A this defined properties bounded A is was D to Rayleigh principle Motivated given of bounded the extreme (3.3.12b) quotient according it M Rayleigh largest a the crucial for optimizing • <%,2L"$>/<%,!>%> i n M, system, introduce standard made two (2.4.7)) an a l g o r i t h m (cf.(3.3.14)) the The above f o r the d i f f e r e n t i a l and analysis construct suggested that of those (). Generalized Rayleigh P r i n c i p l e As in of This is the is (see Appendix 3 A ) . of the {s^}) and (3.3.11), respectively. Then, the eigensystem A s=ADs, (3.3.17) + has the following properties (i) all X are real, (3.3.18) ± (ii) (3.3.11) has M linearly (iii) independent eigenvectors s^, let S and A be given by S = [s.s. 1 ...,s7, z (3.3.19) M A = diag[X.], with A s A < ^ 2 1 X, (3.3.20) M 2 then S A S = A, (3.3.21a) S DS = I , (3.3.21b) T + T where I is a unit matrix of order M. As an a s i d e , matrix we o b s e r v e t h e s t r i k i n g eigenvalue problem resemblance (MEP) (3.3.17) to of the the BEP ( 2 . 5 . 1 ) - ( 2 . 5 . 2 ) . B a s e d o n lemma 3.3., we c a n show t h a t Lemma 3 . 4 (Generalized given by (3.3.14). Rayleigh principle) Let p($;$) be Then: (i) X£ p($;$) s A , v'$ € IR , (3.3.22) M 1 (ii) M min p($;t) = p(s 4) if = X tf), (3.3.23) i 4>eJR (Hi) max p($;$) = p(a:t) M 0€R tf), N M M where s ,s 1 = \&)m r M X and X are determined from 1 M 55 (3.3.11). (3.3.24) Proof: We use t h e n o n s i g u l a r i t y M t o define a linear M m a p p i n g S: R 1 of S R — S<^. I t thus f o l l o w s from (3.3,.20) a n d (3.3.21) t h a t hence t h a t V % € R and | p($;$) - X= which proves argument fact (cf. M [(VV^AJ^A i p establishes that " °' i that i s bounded the left (i=l,Af) (3.3.18)), i n terms of % satisfy we e v a l u a t e side 3 above of (3.3.17) p($;f) < - - >) ( b y 3 20 X^. by (3.3.22). similar Using the X^ a r e r e a l and t h a t at s A and s to obtain 1 M (3.3.23) a n d ( 3 . 3 . 2 4 ) , w h i c h c o m p l e t e s t h e g e n e r a l i z a t i o n ^ In summary, M-dimensional disturbance the system energy global analysis (3.2.3) includes: equation (3.3.12); ( 3 . 3 . 1 7 ) ; a n d 3) t h e g e n e r a l i z e d Rayleigh in 2) R f o r the M 1) symmetrized the eigensystem principle. 3.4 A p p l i c a t i o n t o : global s t a b i l i t y and optimal perturbation By arguments following parallel results with t o those the proof parallelism) 56 i n § 2.6, we o b t a i n t h e omitted (see § 2.6 f o r Theorem 3.1 (monotonic, the finite r (t) stability m max p($,4) y global stability; Let r^ (t) denote measure of <f and be defined by =\ N (3.4.1) M with X given by (3.3.24). Then, an equilibrium state $ of M (3.2.3) is MGS iff r ($) - r s 0. (3.4.2) N Corollary 3.1 disturbance if There exists at least to an equilibrium state one initially ¥* o f (3.2.3) growing if and only r tf) -r>0 (3.4.3) N Theorem 3.2 (optimal initial perturbations) rate <r ($,4) of a disturbance Let the growth $ 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 a given equilibrium initial state perturbations is sense that (i) the growth rate of $(t;s <r^(S ,4) H (ii) is the largest o cr (s ,4) N is M in R M (3.4.5) M the value e IR to 0 V 2 € (R, * o the optimal one in the ) at t=t M $ the least upper bound on the M growth rate of all %(t,4 ) V $ e R for t s t . Q Further (3.3.17) properties as nonmodal M Q o f t h e s e t {s^} determined initial perturbations Appendix 3B. 57 from are given i n 3.5 Numerical The illustrations objective concepts, of this put forward stability, section i st o illustrate here and i n chapter initially growing some o f 2, r e g a r d i n g perturbations p e r t u r b a t i o n s b y a p p l y i n g them t o p l a n e t a r y global and scale optimal atmospheric flows. 3.5.1 Set-up o f the numerical experiments The geometry o f t h e flow f i e l d Derome (1987), i s t h e same a s i n F y f e a n d i . e . , t h e 0-plane re-entry channel centered around 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 (6 ), d= 4xl0 m, H — 1.x 10 m, 6 Q f = Q 2a sin(e ), (3.5.1a) A p - 2Qcos(e )/R, o (3.5.1b) o where Q, R a n d e a r e t h e r o t a t i o n a l o rate 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 latitude, respectively, taking t h e values respectively. o f 7.292xl0" /s, 5 The l o w e r b o u n d a r y of the earth, the 6.37xl0 m 6 o f t h e geometry and 45°N, 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 a n d h ^ = 500 m. The p h y s i c a l p a r a m e t e r s o f the system strength (3.2.3) a r e t h e uniform C7* a n d t h e Ekman illustration purpose, we damping s e t CJ* = 58 zonal momentum coefficient 22 m/s b a s e d forcing r. on For the previous as the 3.2), studies controlling with The value (e.g., r problems system is defined calculated via leap-frog scheme (see Fyfe, theorem 1989 3.1 sufficient must in runs Af=210, and in at be f o r more 3.2 In stressed with that shown) qualitatively the time where to a n d J=3 some speaking, initial . The nonlinear the terms taken Robert of via filter that while truncation, obtain (cf. (3.2.1)) changes are from numerical a convergent relatively increased the (cf.section v i a the a r days). We e m p h a s i z e follows, M is r stepping with required while = and take (3.2.3)-(3.2.4) regardless =2 a 1/(35 system what I set to details). is 1985), are performed the hold Af=15 w i t h (not day) conjunction solutions. truncation 1/(1 numerically FFT and i . e . , by the resolution numerical It from experiments solved and Rosenthal, parameter, ranging numerical Tung is low made. observed M=15 results to are unchanged. 3.5.2 Numerical r e s u l t s a) e q u i l i b r i u m A family {^(r)}, is algorithm states of equilibrium obtained (cf. Keller, representative dissipation (r numerically 1978). equilibria = 1/29 states days), In of (3.2.3), using F i g 3.1, corresponding (b) moderate the denoted continuation we p r e s e n t to by (a) dissipation three weak (r = 1/11 days) and respectively. (c) It strong is dissipation seen, as (r expected, 1/5.5 = that days), strong wavy m o t i o n h a s d e v e l o p e d a s t h e Ekman d a m p i n g d e c r e a s e s . b) o p t i m a l MGS To the e obtain boundary and the optimal generalized matrix MGS boundary eigenvalue r {¥*(r) } t o o b t a i n MGS tf) using i n r-space, problem (3.3.17) the Eispack we solve f o r each routines t (Garbow, N 1974), with r e s u l t s shown i n F i g 3.2. I t i s seen that tf) r N monotonically with the decreases dashed line as a function of r and r=r at r=r . I t i s c l e a r intersects from theorem N 3.1 that in F i g 3.2 r i s numerically L the analysis the three To determined critical t h e MGS subregimes: (II) regime regime regime (III) illustrate with equilibrium state of the modal , where stability f r o m F i g 3.2 i.e., family supercritical of the e q u i l i b r i a experiments corresponding of r=r shown and r , {¥*(r)} i n t o regime growing that (I), perturbation . MGS conditions the usual initially Also boundary o f Ekman d a m p i n g , stability distinct from boundary. {¥*(r)}. I t i s e v i d e n t values t h e whole numerical defines i s t h e n o r m a l mode s t a b i l i t y subcritical a n d MGS r=r of the set two divide line the experiment i s a randomly we performed Q in Fig five to five different i n i t i a l t (0) =<$ =$+$ w h e r e form shown *\ 3.1 is the in each perturbation, with o (c) generated i n i t i a l and $ $ the 6 ratio of initial energy i n ~$ to o the energy in ^ v a r y i n g from 0.2 t o 1 i n these experiments. F i g 3 . 3 shows the time series with of disturbance k i n e t i c ( s c a l e d by p^Hdl, t h e d e n s i t y o f a i r and H, d and 1 the s p a t i a l scales defined before). l a r g e as t to (i.e., in Fig 3.3 that even f o r ~$ as o w i t h 5 = 1), the p r e d i c t e d monotonic decay zero i s observed. c) I n i t i a l l y The to I t i s seen energy transient growth i n s u b c r i t i c a l flows existence of i n i t i a l l y those t i n regime growing nonmodal p e r t u r b a t i o n s (II) ( c f . F i g 3.2) i s i l l u s t r a t e d in Fig 3.4-6. In each o f the s i x experiments r e p o r t e d i n F i g 3.4, we t(0) set = t t o be 1(0) = ^+s (3.3.17) solving such perturbations equilibrium the using MGS s t a t e t(r) boundary. that 0.197/day i n regime clearly The growth demonstrated days, which rates 0.187/day, these d i s t u r b a n c e s i n i t i a l i z e d undergo transient growth over i s o b t a i n e d from ( I I ) . The are found t o be and s m w i t h l/r=6 nonmodal p e r t u r b a t i o n 0.2/day, ^ is , where m 0 of existence even of f o r the i s just across the six 0.207/day, respectively. initial 0.204/day, I t i s seen from these p e r t u r b a t i o n s a period as long as 5 days, s a t u r a t e at an energy l e v e l as h i g h as roughly 60 % more of their i n i t i a l The given v a l u e s , then a s y m p t o t i c a l l y decay t o zero. existence o f more subcritical flows than one such i s illustrated perturbation f o r the to a equilibrium s t a t e t shown i n F i g 3.1(b). configuration 3.6 Fig of f i v e showing such the time In F i g 3.5, we show the s p a t i a l nonmodal p e r t u r b a t i o n s s^, series of k i n e t i c energy with of the d i s t u r b a n c e s s t a r t e d from them. I t i s i n t e r e s t i n g t o note i n 3.5 Fig that with a decrease i n their initial growth rate (from t h e t o p panel t o the bottom one), s ^ appear t o be more elongated in the zonal direction. e v o l u t i o n o f these ^ ( t ; s ) , i 3.4 model in transient flows growth temporal much o f the o b s e r v a t i o n s f o r F i g has been relevance t o cyclonegenesis Farrell (1985; 1989a). As phenomenon i n a variety 1971; Boyd, has been 1983; e t c ; . strongly a cautionary of suggested note, we stress decay t o zero r e v e a l e d i n F i g 3.5 and F i g does not c o n s t i t u t e generic reported ( c f . Orr, 1907; Rosen, t h a t t h e asymptotic 3.6 f o r the can be c a r r i e d over t o F i g 3.6. The Its As a proof for establishing in subcritical flows. We this will as a return to t h i s aspect i n chapter 5. d) optimal nonmodal perturbations i n s u p e r c r i t i c a l regime In terms o f growth r a t e , the s u p e r i o r s t a t u s of S d i s t u r b a n c e s ~$ initial Q € IR Here, we r e p o r t the r e s u l t s designed to experiments t(0) = I illustrate i s guaranteed from the five numerical conclusion. , i=0,l, . . , 4 , m M— 1 62 to a l l by theorem 3.2. experiments Each s t a r t s from one o f the f o l l o w i n g i n i t i a l = t + S U M m of the states where are 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(a), and its initially from growing nonmodal p e r t u r b a t i o n s determined (3.3.17). In F i g 3.7, we show t h e time s e r i e s o f growth r a t e <T N over t h e f i r s t 15 days. The i n i t i a l growth r a t e of $ ( t ; s .) M -i (i=0,l,..,4) a r e found t o be 0.185/day, 0.182/day, 0.144/day, 0.113/day maximum and 0.028/day, modal demonstrate growth the respectively, rate optimal i n contrast 0.070/day, status of t o the which as clearly an initial perturbation to We note that identifying favorably configured p e r t u r b a t i o n s i n models other than ours has been a s u b j e c t of several The r e c e n t works plausible development role of (Farrell, of cyclone 1988, 1989a; O'Brien, these perturbations waves suggested, f o r example, i n F a r r e l l in in midlatitude 1990). explosive has been (1985). 3.6 Concluding remarks A r e d u c t i o n i n dimension of t h e system has made the g l o b a l analysis eigenvalue in R M a procedure calculation amenable package such to standard as E i s p a c k ( c f . Garbow, 1974). The n u m e r i c a l examples g i v e n i n § 3.5 c l e a r l y that a stability a n a l y s i s o f a hydrodynamical 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 be l i m i t e d . 63 matrix system indicate without Appendix 3A Proof: A The r e a l i t y and D + 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 ( c f . (3.3.15) - ( 3 . 3 . 1 6 ) ) . To s e e t h e e x i s t e n c e linearly D 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^} i n d e p e n d e n t s^, we c o n s i d e r t h e p o s i t i v e o f t h e operator 1 / 2 -1/2 _M D and t h e l i n e a r of M square mapping root defined by _M : IR -» IR D S = D 1 / 2 8 (3A.1) with which, (3.3.17) As =• As, with yields A given by (D r k (D )-\ A = l,2 1 + 1/2 where t h e e x i s t e n c e being to (3A.2) positive see that D follows then As..= A..S.., 1 1 definite. 1 / 2 i s diagonal that J / yield and hence A i s s e l f - a d j o i n t . I t problem s\, (3A.2) (3.3.21), S h a s an i.e., (3A.3) (3A.4) v i a t h e l i n e a r mapping see that by D i t i s not d i f f i c u l t V i , 3=1,2, . , , M i satisfy i s assured 1 / 2 r + To (3.3.16), t h e eigenvalue A S = A D s / V i = 1, 2, i of D V i=l,2,... M which y i e l d , i From s e t o f Af e i g e n v e c t o r s orthonormal S s] = 8 of the inverse (3A.1), M. (3A.5) and A c o n s t r u c t e d i t suffices according t o note t h a t to (3A.1) (3.3.20) a n d (3A.4) ( 3 . 3 . 2 1 b ) , a n d t h a t summing up (3A.5) o v e r i g i v e s 64 I • -• ,IS ] - D [ 8 m L F 8 2 / • • / S ]A M S DSA = A T as d e s i r e d . P h y s i c a l P r o p e r t i e s o f t h e s e t {s^} Appendix 3B C o r o l l a r y 3.1 a nonmodal (i) L e t s^. be determined initial the i n i t i a l growth size of a ., i.e., (ii) the velocity orthogonal perturbation fields in sense ± v(s^) and respectively, has the of Then, properties: ~$(t;s^) is independent associated as with different of the X^ are that <v (s ) ,v (s .)=> 0 if where rate (3.3.17). by X± v (sy and (3B.2) X.*0, are velocity <.,.> is the fields of inner product s and in H (cf. (2.2.11)) . Proof: and ( i ) The p r o o f f o r ( i ) f o l l o w s (3.3.12b), (3.3.17) (3.4.4) . (ii) First O^Ds^ Next, note t h a t (X - X..) = ± l e t l(a^) generated u s i n g to from (3.2.2). and v f s y 0. and and (by (3.3.16) l(s^) denote and (3.3.17)) the streamfunctions as expansion c o e f f i c i e n t s I t thus f o l l o w s t h a t t h e i n n e r product w.r.t t h e (2.2.11) 65 (3B.3) according o f v(s^) <v(s .),v(s .)> = J <V<t>(s.),V<t>(3.)> = V s .a s J- J OCX CC CC J cc which i s i d e n t i c a l t o < s ^ , D S j > i f the i n n e r product i n R i s M used t o r e p r e s e n t t h e summation i n ( 3 B . 4 ) , (<x=l,2,..,M) are the orthogonality of v(a^) and v f s y (3B.4). (3B.4) 2 the components B 66 of where s . and s , ai aj and then f o l l o w s from s^. The (3B.3) and Pig 3.1 equilibrium 2/r--29 of days; Streamfunctions states (b) the parameters for l/r=ll of U* = days; are given three 22.m/s (c) i n the 67 and l/r=5.5 text. representative = 500 days. m. The (a) rest Fig 3.2 finite Stability stability regime diagram. measure linear for the stability The dashed line lines for dashed l i n e which i s and , r=r = 0.075/day r=r is used f o r t e s t i n g for the the for the diagonal condition r tf) N -r > = N MGS boundary boundary the r=r N J 0.17/day tf). r The s o l i d 0. 68 TIME (day) Fig 3.3 in Fig An example 3.1. The o f MGS. The initial perturbations generated with the r a t i o to the energy in t equilibrium 5 of t h e i r r a n g i n g f r o m 0.2 69 initial to 1.0. state % ^ is are kinetic (c) randomly energy 0.9 r l/r l/r l/r l/r l/r l/r CM M M n E cn UJ 2 UJ = = = = = = 10 days 9 days 8 days 7 days 6.5 days — 6 days — — m cr — ( to t—i o 0.0 6.0 12.0 18.0 TIME Fig 3.4 Existence equilibria in F i g t(r) of initially in subcritical 3.2. 70 24.0 30.0 (day) growing regime, nonmodal i . e . , region s to (II) Fig 3.5 The spatial configurations growing nonmodal p e r t u r b a t i o n s in Fig to 3.1. 71 of five the basic initially state (b) Fig 3.6 The energy time series for i n i t i a l i z e d from the s . shown i n F i g 72 five 3.5. disturbances -0.10 I 1 1 1 0.0 3.0 6.0 9.0 I I 12.0 15.0 TIME (day) Fig 3.7 The growth r a t e s o f d i s t u r b a n c e s nonmodal p e r t u r b a t i o n s s i over an i n i t i a l The b a s i c s t a t e t i s F i g 3 . 1 ( a ) . 73 initialized growing from period. CHAPTER 4 FINITE AMPLITUDE NONMODAL DISTURBANCE I : INITIAL BEHAVIOR its r e l a t i o n t o i n i t i a l i n t e n s e development o f d i s t u r b a n c e s 4.1 Introduction The the l i m i t a t i o n o f t h e g r o w t h r a t e o f t h e most u n s t a b l e ( o r least damped) development search normal mode a s a n i n d i c a t o r o f p o t e n t i a l of disturbances f o r favorably t o a given configured flow perturbation § 2.6 and § perturbation analysis, an any i s found period perturbation modal proceeds configured value more i n order that such this intensely than t o ascribe observed, exists on that for example, suggest Farrell, A common a p p r o a c h numerical solutions of the results 1989b, o r F i g 3.7 i n t h e affirmative 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 h a s b e e n known t o d a t e . 74 in 1986) t o t h e a s done i n § 3.5. W h i l e (e.g., due t o (or relate) the 1984; Sanders, i s based problems, chapter) a e.g., by t h e g l o b a l nonmodal p e r t u r b a t i o n s . from such a approach previous means, Given 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 ( c f . Roebber, demonstrate initial over development cyclonegenesis to thesis) . b y some perturbation explosive properly i n this nonmodal i t remains necessary t o e s t a b l i s h t h a t there initial this 3.4 of 1988; O ' B r i e n , 1990; 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 , or h a s l e d one t o The objective of this indeed some t h e case results chapter i n § 4.2. T h i s in § 2.8 i s t o show that i s done i n p a r t b y to the finite this i s extending dimensional system ( 3 . 2 . 3 ) , a n d i n p a r t b y 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 finite amplitude nonmodal disturbances solutions t o t h e n o n l i n e a r system referred to as nonmodal A p p e n d i x 4B. N u m e r i c a l by (here defined (3.2.3)-(3.2.4), disturbances hereafter f o r simplicity) examples a r e g i v e n as in i n § 4.3, f o l l o w e d c o n c l u d i n g r e m a r k s i n § 4.4. 4.2 Nonmodal The of over t r a n s i e n t period a p p r o a c h t a k e n h e r e i s t o compare t h e i n i t i a l nonmodal perturbations from disturbance any disturbances starting with the i n i t i a l other from behavior perturbation, optimal of those including behavior nonmodal initialized those of modal structure. 4.2.1 modal v s . nonmodal growth r a t e a t i n i t i a l First, we e s t a b l i s h the optimal status 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 a s s o l u t i o n s t o t h e MEP t h e most u n s t a b l e n o r m a l mode a t i n i t i a l instant (3.3.17) w.r.t instant t . For the o sake o f space, here ( s e e § 2.8 f o r t h e c o u n t e r p a r t i n IH) . To only o b t a i n t h e MEP the outline o f t h e treatment f o r t h e modal a n a l y s i s o f (3.2.3), 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) 75 i s given we (d/dt)% V(t ) Q = *$, = V (4.2.1) (4.2.2) Q where A: C-> C M i s given M by (3.3.4e), with $ and y e C . M Q I n t r o d u c i n g a modal form s o l u t i o n t(t;t,<r) into = fe "', f o r f e C 0 and <r e C M (4.2.3) (4.2.1), we have t h e standard m a t r i x e i g e n v a l u e problem (A - <rl)f = 0, (4.2.4) o p e r a t o r o f order M. With t h e d e r i v a t i o n where I i s t h e u n i t d i r e c t e d t o Appendix 4A, we s t a t e t h e r e s u l t i n Lemma F o r any 4.1 given equilibrium from (4.2.4), Re(<r(v,4)) where t state its = p(t,4) p is modal growth disturbance of rate ($) s generalized with max p being denoting is given with be Rayleigh Quotient stability as given determined (3.3.14). by measure by p (t;t) (4.2.6) t h e g e n e r a l i z e d R a y l e i g h q u o t i e n t and {f} c C (comparison given toa by of (4.2.4), u s i n g t h e p r o p e r t i e s (3.3.15) and (3.3.16), r <r c r t (4.2.5) the set of eigenvectors Lemma 4.2 f and fe - r Defining the l i n e a r f i n i t e r (3.2.3), lp (t;t,<r) = by of stability (3.3.24) and L 76 measures) (4.2.6), we M can show, that Let r^rf) and respectively. Then: r (4.2.7) s r tf) . tf) L N In terms o f r tf) a n d r tf), we c a n w r i t e L max Re(<r) = r N tf) - r (4.2.8) Li as o p p o s e d t o t h e maximum n o n m o d a l g r o w t h max <r ($,tf) = r tf) - r N ^ where we h a v e 4.1 least rate (3.3.12b), (4.2.7), we (3.3.24) then have For any equilibrium one nonmodal not less 4.2.2 than initial and (3.4.4) ( c f . theorem state t its maximum modal which growth 2 . 5 i n H) (3.2.3), of perturbation to get it has has a growth rate. E x p l o s i v e development o f nonmodal d i s t u r b a n c e To demonstrate strictly of used From Theorem (4.2.9) N (4.2.9). at rate speaking, (3.3.2) growth the we n e e d i n time rate desired t, from the continuity andthe continuity i n $ cr $,4) assertion f o r % i n some theorem 4.1, o f s o l u t i o n ~$(t;~$ ) o of the open instantaneous set of R , M both N of which are auxiliary established properties in o f nonmodal contradiction, v i z . , supposing initial [t ,t +T) cr period $ f t + N where f;s),4) Q = v N Re(f) perturbation or = (FGMP) that Q < o- ($(t+ for a Given these we p r o c e e d b y sufficiently with small t ' s T, t';1 ),4), (4.2.10) n U Im(f), 4B. disturbances, a n dV t=t +t' Q N U $ Q Appendix U with determined f from 77 the fastest (4.2.4). growing Taking modal l i m t'->0+ in (4.2.10) and using the continuity o f <r i n ~$ and of N $(t;$ ) in t o A p p e n d i x 4B) , s ),4) v($(t+0+; N (cf. s *($(t+0+; u U M Again, by N ) 0 % (t +0+;$ ) Q %(t;a^) (4.2.11) and o %(t ;s with 0 contradicting thus $(t ;$ ), Q the Q established that ^ e q R <r ($(t;a Q based Q 4.2 Theorem which the initial will proceed or not than 4.1. exists less to a given not less than and § some of this numerical of i n i t i a l 4.2.1), illustrations ;t) shown t=t that +t' € we h a v e [t ,t +T) over least one nonmodal state of (3.2.3) than the maximum modal growth that of nonmodal disturbances perturbations. illustrations purpose relation V period equilibrium <r ($ N ^ O Re (a-) at statement * can be max of from modal initial supplement it an initial a M To summarize, development at a rate Numerical The 2.8 There Similarly, less on theorem initial initialized the not perturbation rate 4.3 (3.4.5)). is ) ;$) [t ,t +T) to N (cf. M we and s i m i l a r l y <r (s ;$) 3 for ), t , M leading fact at '$(t;'$ ) M with o n U of in (4.2.11) % )tf). O continuity $(t+0+;a identify we h a v e section calculations modal and second f o r theorem is 4.2 twofold: to t o nonmodal to as 78 provide the first discussion growth rates further supplementary (cf. to of § numerical to F i g 3.7. We do so by atmospheric being using the flow the (3.2.3) considering same are the over as used in § here, 3.5. each typical troposphere, with of the is velocity rest of geometry to with 50m/s, the C7* a upper parameters of numerically 1978) 5m/s scale equilibria obtained in model model of (Keller, from planetary the sets which ranging wind of with Three algorithm c o n t r o l l i n g parameter the problem topography, continuation covering the as range part of specified as needed. (a) I n i t i a l modal v s . nonmodal growth r a t e s From with (4.2.8) r (^) in and order amplification of theorem Here, 4.1). (4.2.9), to see it the tf) by is to compare underestimation disturbances r suffices the modal calculated of r potential analysis as in tf) § (cf., 3.5.2(b), N with the r r obtained standard Fig is tf) 4.1 seen tf) eigenvalue shows that over predicted nonmodal from the the the curve whole representative ) problem for The are states via one is tf) of model given in set 79 (A - case of always C7*, intense from the (4.2.8) (4.2.4), from range more disturbances. i$(U) Re (cr) results potentially equilibria max <rl)f = thus parameters 0. the one confirming It for the development for the caption, {¥*([/*) } solving comparison. above initial the after shown set with in Fig of of some 4.2. Viewing the the external increasing stability once stability forcing U*, U, the C7*=C7 L i t is {¥*(£/*) } {¥*([/*) } seen that first =10.5m/s from function of i n the direction of loses U*=U^=6.6 m/s, (MGS) a t across of the set a monotonic, then below, as loses global modal and recovers stability the modal Fig 4.1). L stability for C7* l a r g e r than U= 26.3 2 m/s (cf. L Corresponding loss of (cf.Fig 4.2). supercritical so-called attention (e.g., of stability by (i.e., loss regions of supercritical potential The explosive are only is In the much maintenance varies the flows (I) in (4.2.9) anyway; amplification dramatic Fig i t 4.1), is the unstable 80 or allowed (III), o f nonmodal stable, the as is to subcritical it leads phenomenon; when one global in region intensification from monotonic no d i s t u r b a n c e (i.e., weakly with atmospheric underestimation since more the received (II) a n d (IV) i n F i g 4 . 1 ) , regime the underlying has in is i n physical the formation, region temporal consequence {¥*([/) } and flows associated which patterns" another. insignificant {U) } 1979). the (i.e., (4.2.8) frequently to explain to of mechanism instability, of regime physically regimes (III) i s and Devore, regime stability i n equilibrium "blocking consequence stability the regime down of The p h y s i c a l i n the effort Charney The zonality topographic break grow t h e exchange and regain space and to it in to the implies disturbances. basic shown states in Fig 4.3. It is potential solely since seen initial arise from modal direction of the This same Ekman is set of enlarged the rate is done the for of U*-parameter using = 1/r- underestimation disturbances as region rate modal of in Fig 4.1 30 days. In how this see larger rate the than over development would some Fig 4.1, the when strongly unstable based the in § 3.5. 4.5(a), 4.4. The The the from period, it underlying modal initial reduced 4.4, while modal growth the equilibrium corresponding instantaneous to the remains as state the f for here the from at a is r^(t) three rate growth consider are of period modal Numerical used on is before. state (3.2.3) rates 81 to equilibrium peak growth maximum suffices problem the concern disturbances perturbations. value using amplification rate nonmodal expected initial happens on be of growth determined the Fig the nonmodal except potential growth see in (b) Intense development o f i n s t a b i l i t y over an i n i t i a l To $(lf) perturbations to to {^(C7*) } (III) i n modal to parameters supporting of the nonmodal the if, disturbances growth set of everywhere. initial space to range physical the scale of modal parameters compared entire negative of relation of the configured change damping time portion properly promoting affect rate. the the over amplifications growth Consider will that if it supports experiments performed as shown in Fig curve in Fig experiments are shown i n F i g 4.6, with t h e corresponding energetics given i n F i g 4.7. I t i s seen t h a t t h e response o f $ t o t h e nonmodal initial perturbations s and s M than t o the fastest Specifically, i s n o t i c e a b l y more M- growing the i n i t i a l intense 1 modal growth initial perturbation f. o f $(t;s rates .) (i=0,l) are n e a r l y 50 % l a r g e r than t h e maximum modal growth r a t e and are noticeably higher than that of $(t;1 ) with Q ~$ = Re (f) q over a p e r i o d o f two days. In terms o f e n e r g e t i c s , the energy o f ~$(t;s level of 4.4 .) ~$(t;Re(t)) (i=0,l) i s considerably f o r more than f i v e days global analysis systematical algorithm nonmodal p e r t u r b a t i o n s . (cf. § 2 . 2 - 2 . 5 that ( c f . F i g 4.7). the limitation rationale nonmodal configured flows. The study of t h e r e l a t i o n of i n i t i a l of reveals modal such analysis and responsible provides as the t r a n s i e n t initialized nonmodal p e r t u r b a t i o n s (Sanders, 1986). 82 from growth i n behavior those the of favorably i s a key t o understanding r a p i d development o f d i s t u r b a n c e s i n the'atmosphere t h e cause The i n v e s t i g a t i o n o f i n i t i a l disturbances a configured f o r phenomena subcritical and § 3.3) p r o v i d e s f o r the search of f a v o r a b l y modal t o nonmodal growth r a t e the than Concluding remarks The for higher observed, f o r example, Appendix 4 A Proof: into Modal First, growth rate we p u t t h e an e q u i v a l e n t expressed standard i n terms eigenvalue of p problem form (4A.1) A | = crDf; A = DA, where A and respectively. positive At this and A given by The e q u i v a l e n c e definite (4A.1) = A are D point, in i t that is R follows crucial to and from the (cf.(3.3.11)) from (3.3.11), fact and hence note can be decomposed A (3.3.4e) that is D invertible. (3.3.4e), .(3.3.11) as (4A.2) + A", + (4.2.4) with A s + A s A - rD, (4A.3a) ij ij L k [A j] = ± Kj ^ i j b + (4A.3b) 2 A —rp A— A— A H lkj k ( - a y. ) ,' A. . s V c , [A, .1 ] ; 1 -A T , (4A.4a) l ijk< A c f ~ Pk>• /H a ( 4 A - 4 b ) k Taking inner eigenvector <trkt> = where use ^ of of (4.2.4) (4A.1) with respect to an ( 4 A . 5) overbar conjugate denotes of of the definitions 2Re{<r}<i,T>t> the (4A-.5) of A complex to itself, and A , = <|,A f> + <|,A|> T T = arbitrary gives v<t,Di>, the complex product <%,k 1> + + <i,k t> + 83 conjugate. we Adding obtain, with the the = 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 out by v i r t u e (3.3.13). (4A.4a), with A cancels defined + by 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 b y t h e self-adjointness this of the fact t o t h e RHS o f (4A.6) o fA + a n d D ( c f . (3.3.13) a n d ( 3 . 3 . 1 1 ) ) . To 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 to t h ea s s e r t i o n . Appendix 4B of p into (4A.6) leads B Fundamental p r o p e r t i e s o f nonmodal disturbance B - l E x i s t e n c e , uniqueness and c o n t i n u i t y o f ~$(t;~$ ) 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 as w e l l a s o t h e r ~$(t;~$ ) i n t i m e t o fundamental p r o p e r t i e s such as e x i s t e n c e and u n i q u e n e s s o f s o l u t i o n s t o (3.3.2) c a n b e e a s i l y a simple application t h e o r e m 3.1 i n H a l e , these properties of the Picard-Lindelof obtained v i a theorem ( c f . 1 9 6 3 ) . The e s s e n t i a l c o n d i t i o n i s that the vector assuring a) g i v e n b y field (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 %, a s s t a t e d i n L i p s c h i t z i a n i s m o f t h e v e c t o r f i e l d f(~$;t,a) defined R . M by Then, Proof: and let (3.3.3) £$;$,a.) QJ be any bounded, closed is locally The b o u n d e d n e s s Let fC$;t,a) Lipschitzian and closedness set be in in ~$ e Qj. 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 ofR , M s u c h t h a t QJ £ 6(0, d) . Now, t a k e 84 a n y $, % e (U a n d c o n s i d e r ||f ( $ ; $ , * ) - ftf,4,a;| =s |A||? - $\\ + where we (3.3.4e) have and norm. Note space of 2.7-9 I 3 A i A^_. involving \*$a~ operators ij) 1 / 2 is LAU s This operator on R , is M 1978). (4B.1), {[ d of £ l j I In | in || b e i n g £(JR ) , H bounded fact, given by a (4B. by the £ vector (cf. theorem the Schwarz ( c f . ( 3 . 3 . 4e) ) . B . . ^ ^ 2 I i j completes the two - 1\, B ijjk} 1 / 2 terms (4B.3) |^ - * l - ( k into (4B.1), 2, [in i proof. we g e t [$ - %\, L i p s c h i t z i a n constant { {l the we h a v e ( 4 B . 2 ) - (4B.4) L(U) i s For 2) k I j A - f ($;t,a)\ =s L(V) = « and and B 4B i where A f < - i Introducing with i\ ' entry B in \B# f, \B ($-"$) + we o b t a i n h\\ * d { [ {£($;$,a) linear Kreayszig, A for respectively, every linear inequality, H « * {[ (3.3.5) (3.3.4f), that in where used \\B$(t-h I of s e t QJ B.^} ' } . 1 j B 85 ^ 2 4 B - 4 ) By virtue Existence, closed definition with in R and let M ; | c Q existence $(t;$ ) M denote t Q (3.3.2) has a unique solution (4B.5) M the for Q domain of e Uc R j, IR being of %(t;% ). Then, the maximum interval any $ of e QJ, the system q %-(t;"$ ) through ~$ . Moreover, q is continuous in E. Q B-2: RxR Q Q Q c we h a v e Let QJ be any bounded, %(t;~$ ) of a($ ) =s t s b($ ); (a $ ) ,b ($)) (3.3.2) E for the solution o theorem, uniqueness and c o n t i n u i t y set E = | r t ^ of the Picard-Lindelof C o n t i n u i t y o f cr ($,4) i n % N The continuity of <r in % for % in some open set of N R M is given by: C o n t i n u i t y o f <r $;$) in% Let the growth rate cr ($;$) of N N $(t;~$ ) be defined by (3.4.4), Q closed set in R M . Then, <r : and Let QJ be any bounded, QJ -> R i s continuous. N Proof: show From that generalized we observe (3.3.12b), p: QJ -» R Rayleigh that (3.3.24) is continuous. Quotient f o r a n y %', + + defined % e QJ c \<$',k %'> - < ^ A ^ > | s 2 d | | A | | | | 0 ' + and (3.4.4), > - $\\, 86 Recall by i t suffices to that (3.3.14). p is For the this, B(0,d) (4B.6) - <$,Ti$>\s ^c?j|0|| ||^' \<$' ,D$'> where d i s r a d i u s of the c l o s e d b a l l U, with A and + respectively. <$,& $> and defined by + {l/(2dfA fl), + (s It D is <$,!>$> the + neighborhood evident are open l/(2d\D\)}, <$ K '$>/<$ D$>) and f given by from continuous ball from (4B.7) |||, - B(^;S), which 1 ( 0 , d ) chosen t o e n c l o s e (3.3.13) (4B.6)-(4B.7) in a with we <r ($4) are N o f $ not l a r g e r than B($;8) , f (3.3.11), that neighborhood 8 given conclude hence 87 and that continuous u by both of % min p{%;^) i n some 0 . 5 E + 0 0 tr (15days) id - 0.5E-01 cc to •X 0.5E-02 r m in 0.5E-03 0.5E+01 0.5E+02 ZONAL MOMENTUM FORCING (m/s) F i g 4 . 1 Maximum i n i t i a l modal versus nonmodal growth r a t e , i.e., r (t) - r v s . L ($) - r , f o r the s e t of e q u i l i b r i a {t{U) } over zonal wavenumber-1 topography o f height 1000 m. The EJcman indicated by damping coefficient the horizontal r dashed = 2/(I5days) i s line. The three v e r t i c a l dashed l i n e s are f o r the MGS boundary U = U^=6.S and f o r l i n e a r s t a b i l i t y boundaries U = tV= 10.5 m/s m/s, and U - roots U = 26.3m/s, where U and CT, C7 are obtained as L N L L t o t h e equations, r ($(U*))-r=0 and r (t(u'))-r=0, 2 1 2 L N respectively. 88 (a) (b) ~<-eo ~~ Ic) N (d) 20 (e) If) F i g 4.2 of (a) Topography several representative used i n F i g 4.1, m/s contours; states (b)-(f) from the streamfunctions set of corresponding t o 13=6.20 m/s i n ( I I ) , 11.33 and 17.85 m/s (IV), r e s p e c t i v e l y . 89 {t(U) i n (I), i n (III) and 40.13 m/s } 9.12 in 0.5E-03 J 0.5E+01 0.5E+02 ZONAL MOMENTUM FORCING (m/s) Fig 4.3 equilibria The same {^(C7*)} as F i g 4.1 except used here that corresponds wavenumber-2 topography. 90 the set of t o t h e zonal 0.5E+00 r IS -o - 0.5E-01 r UJ 1/(30days) - tr J CQ <£ =5.Om/s w (ID 0.5E-02 in 11 \ to (III) E II II i *s» 1 0.5E-03 0.5E+01 0.5E+02 ZONAL MOMENTUM FORCING Im/s) Fig 4.4 equilibria The same {t(U)) as F i g 4.1 except that used here corresponds t o l/r 91 the s e t o f = 30 days. Fig 4.5 Basic states, modal streamfunction o f the e q u i l i b r i u m s t a t e taken from F i g 4 . 4 growing initial U=16.3 m/s, nonmodal experiments. initial for to numerical and perturbations corresponding the nonmodal (b) and and respectively. 92 (a) f o r (c) f o r i t s f a s t modal perturbations, 0.24 0.18 T3 (— CC 3 O CC - 0.13 0.07 ' \ \ max Re (<r) =0.11/day - 0.01 CJ -0.04 - -0.10 0.0 1.0 2.0 3.0 TIME Fig 4.6 Evolution optimal of nonmodal i n i t i a l the 4.0 5.0 (day) growth rate perturbation s with time. (see F i g The 4.5(b)) M has growth r a t e cr (s ;**) = 1/(5.29 growing Re (cr) = modal perturbation 1/(5.52 days). i n d i c a t e d i n the F i g 4.5 The days), (FGMP) (see initial (a) and 93 (b). whereas the fast 4.5(c)) has perturbations are Fig 30.0 0.0 1.0 2.0 3.0 4.0 5.0 TIME ( d a y ) F i g 4.7 Evolution of disturbance energy with time for the case shown i n F i g 4 . 6 . The i n i t i a l perturbation energy i s 20 % of the energy i n the equilibrium states to ensure the f i n i t e amplitude for the i n i t i a l 94 perturbations. CHAPTER 5 FINITE AMPLITUDE NONMODAL DISTURBANCE I I : ASYMPTOTIC BEHAVIOR its r e l a t i o n t o b i f u r c a t i o n and m u l t i p l e e q u i l i b r i a 5.1 Introduction Consideration arises o f t h e asymptotic i n a v a r i e t y o f circumstances. r e v e a l s the e x i s t e n c e to a g i v e n flow, that behavior of disturbances When a modal a n a l y s i s o f e x p o n e n t i a l l y growing perturbations one wants t o determine the maximum amplitude can be a t t a i n e d by t h e growing d i s t u r b a n c e s , a problem of n o n l i n e a r s a t u r a t i o n (Shepherd, 1988, 1989); or t o i n q u i r e about the a r r e s t nonlinear of the e x p o n e n t i a l equilibration (Pedlosky, growth, 1970, Salmon, 1985). On t h e other hand, the e x i s t e n c e nonmodal perturbations optimization 2 of t h i s chapter to these uncovered procedures (Farrell, after multiple by . such 1988; the i n i t i a l equilibria 1980; Mak, of i n i t i a l l y growing analysis O'Brien, t h e s i s ) r a i s e s the q u e s t i o n : disturbances a problem of 1990; what happens transient from as growth. Further, identifying a bifurcation analysis (Vickroy and Dutton, 1979) leads one t o wonder how a t r a n s i t i o n from one e q u i l i b r i u m s t a t e t o another takes The of answers t o these the asymptotic problems or questions behavior of finite place. l i e i n the study amplitude nonmodal disturbances. The objective of t h i s chapter 95 i s t o e s t a b l i s h some b a s i c facts regarding the t-*» behavior (3.2.3)-(3.2.4) . The d i s c u s s i o n subcritical i s given persistent a priori of flows of f o r the asymptotic determination o fthe nature disturbances illustrations are given decay i n i n § 5.2, w i t h t h e t r e a t m e n t f o r disturbances i n s u p e r c r i t i c a l nonmodal system i n § 5.3. o fthe asymptotic i s addressed i n§ flows The states i n § 5.4. N u m e r i c a l 5.5, f o l l o w e d by concluding remarks. Asymptotic decay as t -> w i n s u b c r i t i c a l 5.2 In this generic and next case 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 where the underlying (3.2.3) i s hyperbolic, w h e r e . A<$) reason convenience. restriction In fact, hyperbolicity open i n £(\RK), w i t h Re(<r^(P)) of the vector t ohyperbolicity i t i s motivated i sa generic R . More p r e c i s e l y , and part i = 1,2,...M, field F (or property i s more t h a n by t h e f a c t of linear means t h a t for that o p e r a t o r s on t h e set o f hyperbolic operators which t of e v a l u a t e d a t ^ ( o r a t $ = 0 ) . The o r (3.3.3a)) f o r the equilibrium state i . e . , Re {<r^ (A tf) ) } * 0, i sthe linear f)(cf.(3.2.3) flow P e any o p e r a t o r = 0 f o r some i c a n b e a p p r o x i m a t e d i s dense £(\R ) K arbitrarily c l o s e l y b y some Q e £(IR ) s a t i s f y i n g Re (o-^ (Q) ) * 0 f o r a n y i M (cf. theorem violation equilibrium bifurcation 3, pp.157 i nHirsh of hyperbolicity a n d Smale, often s t a t e o f (3.2.3) w h i c h point i n some occurs 1976). However, at critical a i n t u r n corresponds underlying physical t oa parameter space, which i s the s u b j e c t o f § 5.4. The main result of this section i s theorem 5.1. The a n a l y s i s l e a d i n g t o t h i s r e s u l t i s based on the d i r e c t method of Liapunov (Hahn, 1967; Hale, Verhulst, 1985) . The b a s i c Appendix 5A f o r r e f e r e n c e . 1969; H i r s h ideas and Smale, 1976; o f t h e method a r e given i n For a better focus on t h e main development, a u x i l i a r y lemmas are d i r e c t e d t o Appendix 5B. Theorem Suppose an equilibrium 5.1 state hyperbolic for which the following r where r tf) and r tf) are given tf)-r>0, N N (4.2.6), solution disturbance %(t;%^) of (3.2.3) is (i) r^tf)-r<0; (ii) by (3.3.24) and L respectively. of the null hold: t Then, there exists % = 0 of (3.3.2) a neighbourhood QJ such that any nonmodal to ¥* approaches 0 as t -» co whenever ^ € q QJ. Proof: H y p e r b o l i c i t y o f ¥* and the c o n d i t i o n Re(<r(A)) 5B) that < 0. I t thus there exists follows from a positive lemma (i) i m p l i e s that 5.2 ( c f . Appendix definite Q e R M x M as a s o l u t i o n t o Liapunov matrix equation A Q T + Q A = - I , I a unit matrix of order M where we have s e t R i n (5B.3) assert that nonlinear such system t o - I . Put V($) = <$,Q$>. We a chosen V i s a Liapunov (3.3.2) a t ~$ = 0 9 7 (5.2.1) function ( c f . Appendix of 5A) . I t the is obvious t h a t V(0) = 0 and V(%) = <$,&$> >0 in R -{0} (5.2.2) K with the l a t t e r property Q. Now, c o n s i d e r the due t o the p o s i t i v e orbital t r a j e c t o r y o f any d i s t u r b a n c e derivative definiteness of V along of the '$(t;'$ ) t o ^ Q V = <(d/dt)'$,Q$> + <$,Q(d/dt)%> = <A$,Q$> + + <B$$,Q$> + <$,QB$$> (by (3.3.5b)) • V C$) + w$) where V ($) 2 (5.2.3) corresponds t o the f i r s t and w($) t o t h e l a s t two ones. V ($) 2 = <$, (A Q + QAj|> = -\$\\ T and hence the i s negative two terms in (5.2.3), Moreover, (by(5.2.1)) 2 definite i n R -{0}. F o r w {$) , we have M f o l l o w i n g estimate \w($)\ s |<B#,0$>| + \<$,QB$$>\ s + ||^||||Q||B^|| ||B^||||QJ|| s 2c||Q|||^|| . 3 It thus follows (by lemma 5.3 from lemma 5.1 exists a (3.3.2) such t h a t V s a t i s f i e s together for neighbourhood with the n u l l (by Schwarz (5.2.2) proves solution, Liapunov's c r i t e r i o n (U o f ( c f . Appendix 5B) ) (cf.Appendix the null (5A.4). that f o r asymptotic 5B) t h a t solution ~$ = there 0 of This l a s t property V i s a Liapunov which y i e l d s 98 inequality) the d e s i r e d stability. a of V function result by Remark 5 . 2 . 1 transient nonmodal It growth follows i f s^ perturbation the condition for the € U, where determined ( i i ) . This asymptotic <£(t;s^) that decaying is from result -» 0 a f t e r zero initial an i n i t i a l l y (3.3.17) provides to an a seen growing and a s s u r e d by theoretical i n F i g 3.4 basis and F i g 4.6. It has well Rosen, 1971; transient an been Boyd, growth asymptotic 1983; decay (i)-(ii) demonstrate that the Note Persistence To conduct manner, on ¥*, is this we a s s u m e (b) Refer.. ( A ) ; where A is 0, the that true extension following continuous imposing have ( O r r , 1907, hyperbolicity essentially f o r the remains the spectrum managed nonlinear local as i discussion the t under i n a most is and to system long as concern transparent satisfies (5.3.1a) = 1,2, . . . , p , (5.3.1b) > 0, i = p+l,p+2, . . . , s , s * p given (X) =ot.j+i&^ (/3^ eigenvalues also 1987) flows finite. that t is hyperbolic, (c) Refr^OD) we the subsequent < By model as t -> « i n s u p e r c r i t i c a l flow (a) o\£ zero. this that with to n e i g h b o r h o o d dl i s 5.3 Farrell, associated conditions (3.2.3). known f o r l i n e a r by t (3.3.4e) 0), of algebraic evaluated i=l,2,...,s, (5.3.1c) at being with its distinct m u l t i p l i c i t y n . . The c o n d i t i o n 99 the s * p i s meant t o exclude It is clear that p e r t u r b a t i o n s by it (e.g., may unstable From 1960; expected normal mechanisms of potential growing the energy from by (Farrell, for 1970, other 1981; first cascade 1985), started (Salmon, disturbances structure. admissible that scale the the 1980). release and then the function of d i v i d e the conditions V + is rapid perhaps mean configured flow nonmodal 1985). may be Essentially, we b a s i s f o r the space of a l l the k i n e m a t i c a l l y theory ( c f . lemma 5.4 from observed of It concern disturbances favorably disturbances canonical from arising This as an e f f o r t t o address the concern. f i n d a new by model Mak, The main r e s u l t of t h i s s e c t i o n ( i . e . , theorem 5.2) regarded modal ultimately equilibrated via argument synoptic though disturbances be those arises perturbations the nonmodal of predominantly works nonlinear to by deepening that as unclear strengthened initially Pedlosky, modes would such perturbations ¥* has earlier Stuart, be remains this 5.2. (5.3.1c) and hence nonmodal p e r t u r b a t i o n s (4.2.7)-(4.2.9). flows the case d i s c u s s e d i n § (i.e., linear for R) M operator disturbances according (Hirsh i n t o two i n Appendix 5C). Next, we (5.3.1) there ($) the in exists a neighbourhood & to the Smale, subset R + real 1974), and show t h a t under Liapunov QJ + of R the instability $=0 whose properties of allow any disturbance in Appendix (5.3.1). + o f V ). state + Proof: Under function constant | V ($) V a>0, Moreover, (5.3.1) a closed contained from on ^ , 5D) . ball in U n + (3.3.17). the property Thus, for a B(0,jb) such the ball prove this of V a t 0, + i t i n (5.3.2) i s possible to to choose satisfy (5.3.3) any nonmodal i n finite for t a t disturbance time by contradiction, S 1(0,i>) given (5.3.2) + leave (a) that $ e B CO,b) . the closed ball that will This is e (U . assert ^(t;| ) exists to f Q is ( c f . Appendix by continuity b t o ensure B(0,b) holds there | s a for + o + ~$(t;~$ ) perturbation t h e hypotheses satisfy respectively. when the nonmodal initial R -{0}, 5.5 where R and U a r e t h e q for %(t;a^) lemma $ of (3.2.3) any nonmodal disturbance true behavior N o w , we s t a t e + as t -» « if 0 € II n R -{0}, particularly (cf. + as in lemma 5.4 and lemma 5.5, same of -» » a s y m p t o t i c i n (U n R -{0} Let the equilibrium Then, persist the t initialized 5D f o r p r o p e r t i e s Theorem 5 . 2 We us t o detect i f ~$ € o assuming i f $ Q ~$(t;~$ ) o B(0,£>) to ^ will n U n \R+-{0}. We + that e B(0,jb) n U n + R -{0}. + (5.3.4) From property (c) of V + i n lemma 101 5 . 5 , we c h o o s e ~$ such that $ € 6(0,23) n R -{0}, i.e., + o V$ ) + > 0. Q (5.3.5) Since f o r t a t o V ($(t,4 )) + V - V ($q) = \l V > i + + + Q s nondecreasing the p o s i t i v e 0 (5.3.6) along the orbit s i g n i n (5.3.6) f o l l o w s from c^(t;^ ), where Q (b) i n lemma 5.5, (5.3.3) and (5.3.5). Put y £| It % | V ($(t/$ )) + i s obvious 0 from > V $ ); + Q (5.3.5), 1(t,4 ) e B(0,b) Q j . and (5.3.6) t h a t t h e s e t y & 0, • + and hence Continuing from (b) o f lemma 5.5 a u(t-t ) + o Q inf V > 0. ^ co as t -> oo, a statement c o n t r a d i c t i n g (5.3.2), 5.3.2 Appendices ii = (5.3.6) l e a d s t o V+ ($(t;% )) - V Remark that It i s clear 5C and 5D) t h a t under t h e c o n d i t i o n (5.3.1), perturbations from thus completing lemma 5.4 and 5.5 t h e s e t V r\ JR -{0} + the proof. + (cf. i s not empty i m p l y i n g t h a t t h e r e always e x i s t leading t o p e r s i s t e n t disturbances t o a flow ^ f o r which (5.3.1) holds, and t h a t a t r a n s i t i o n from t h e given I to a new s t a t e i s p o t e n t i a l l y inevitable. We c o n s i d e r t h e f o r g o i n g r e s u l t capacity t o cope with useful primarily f o r i t s perturbations 102 not r e s t r i c t e d t o any s p e c i a l type and b a s i c flows not n e c e s s a r i l y z o n a l l y sheared, and for its assurance p e r t u r b a t i o n s capable of the existence d e t a i l s of l i m i t i n g s t a t u s . In f a c t , the nature periodic (e.g., of Further, set i t y i e l d s no 1960; information (i.e., steady opposed t o weakly n o n l i n e a r Watson, 1960; Pedlosky, 1970, 1981). i t i s s i l e n t about the mechanism r e s p o n s i b l e f o r the to 1980), and It experience ultimately (Pedlosky, about the s a t u r a t i o n l e v e l i s important t o note t h a t onto a ^ f o r which time, or theory e q u i l i b r a t i o n which the p e r s i s t e n t e v o l v i n g d i s t u r b a n c e s expected of f o r i t s p r e d i c t i o n of p e r s i s t e n t disturbances or c h a o t i c ) , as Stuart, a of t r i g g e r i n g a t r a n s i t i o n from a given flow t o another d i s t i n c t s t a t e , but not on of i.e., it may s u p e r c r i t i c a l nature (5.3.1) to a perturbation 0 as of ^. Indeed, by Salmon, (Shepherd, 1988, is satisfied decay 1970; may t ^ a similar 1989). superimposed not oo, are survive despite in the argument, we can have Remark 5.3.2 exists Under the hypothesis a f u n c t i o n V : (U — -> R (UJ on f in (5.3.1), a neighbourhood there of ~$ = 0) d e f i n e d by V~ = <$,Q~ £>, Q~= diag[Q 1 ,-Q ] (5.3.7) 2 satisfying (a) V~(0) = 0 (5.3.8a) 103 (b) V~ < 0 in (c) V~ (%) V~-{0}, > 0 V %€ where %€ R the solutions M to It 5C) t h a t approach t o 0 as t i f " n R -{0}, U rs R~-{0} 5.4 the precise nature stationary, (cf. Kelley, of equation would to the properties section It at some can be argued that persistent ( 3 . 3 . 1 7 ) . The that with bifurcation ^$(t;'$ ) to o a t o determine the Sattinger, this (i.e., theory 1972; nature or of the linear of a. the 1969). eigenvalue part) values The goal expectation f o r t h e system the linear stability 104 o From t h e b i f u r c a t i o n expected real will i f ~$ = nonmodal d i s t u r b a n c e s and 2 (cf. associated 0 1967; Hale, o f principal critical i s to verify this from % = (Joseph be t h e one with t h e l a r g e s t f(^;f(a);a) especially 5 . 2 ) , i t remains or chaotic). 5.4 Q and s u p e r c r i t i c a l of the persistent i t of Q respectively. a r e reminiscences - o f + (cf.theorem periodic 1976), related determined and i % (t;$ ) t o ^ disturbance manifolds Q a n d lemma — criticality Navier-Stokes Joseph, as ^ (5D.6), (5.3.8) i s existence & (5C.7), € U n o f -{0}, o + Persistence, supercritical of » i f l respectively Given from any nonmodal and unstable (3.3.2), and a n d s e t QJ n E -{0} — stable (5D.5) where — (5.3.8c) w . r . tthe basis follows Appendix set ffT-fO/ i s written Furthermore, e (5.3.8b) is (defined part of of this (3.3.2). boundary a=^ L for the set e q u i l i b r i a is a criticality } o f (3.2.3) i n a-parameter space ( r e c a l l that r i s the r o o t t o the equation, L r L r (¥*(a))-r = 0) . To see t h i s , we f i r s t ' 0 (¥*(r ) ) - r = L by (4.2.8), which Re (cr) I note t h a t 1 L implies that **(r ) i s L L nonhyperbolic. asymptotic upon - = a=r Next, under the h y p e r b o l i c i t y c o n d i t i o n , the o f %(t;~$ ) undergoes q u a l i t a t i v e changes behavior crossing Q a = r ( c f . theorem 5.1 and theorem 5.2). L Further, the supercritical indicates that existence of ¥*(a) w i t h a=i~ oc i n some of a cx—r b i f u r c a t i o n p o i n t of (3.3.2) s i n c e i n the s u p e r c r i t i c a l s i d e of has to o neighborhood i s a supercritical L ~$(t;% ) persistent « - = ^ h i (3.3.2) a l s o %=0 as i t s s o l u t i o n . To see the l i n k nature of between which t h e f o l l o w i n g cr (i- ^) (= we Q and r e s t r i c t L (a) $(t;$ ), persistent neighborhood o f a=o~ the p r o p e r t y of c ( r ) and the L shall focus discussion on t o the case i n holds: aw) is a ($,a) = (0,r ); furthermore, simple eigenvalue of (p L the form imu> with m e 1- {1, -1} ; (5.4.1) a t (0,r ) e range f 4 at (0,r j ( X CO L (d/dt)Re (5.4.2) L (c) t h e l o s s o f s t a b i l i t y sense t h a t at f-» at (0,r ) has no e i g e n v a l u e of L (b) f some o f {^(a) } at a=i" L i s strict (a-) \ _- > 0. i n the (5.4.3) L It are i s o f i n t e r e s t t o note t h a t made f o r s i m p l i c i t y , while i t turns 105 the above out t h a t these assumptions conditions are well satisfied reported We i n the next first (5.4.1) ). of It is R xlR) relevant range. and there (0,r ), one L stationary the the for of curve show two s u c h the null <r(r^)=0 that parameter persistent observation note a We s h a l l are case to out as M calculations such as those section. crucial trace (i.e., in (b), numerical consider (3.3.2) dim in the in a of some under passing solution state in ~$=0 '$(t;'$ ). solutions space physically conditions through and We Q by =0 state-parameter that curves u stationary varies next (i.e. the one begin (a) point for the with the that {range f | (Q;t ( r j , r j } = d i m {range A ( ^ ( r ) ) ; (by L = M-l ( b y a- (r )=0 and c o n d i t i o n (3.3.5)) (a)) (5.4.4) L Now, (a) define and dim = (b) {null Df : R together Df}I 1 a=r (dim {domain -» R M + 1 M with by Df E (5.4.4) [f^, f^] . imply Then, condition that L Df} - d i m {range c H| f - = 2. (5.4.5) L Next, set (3.3.2) For (d/dt)% to have ;a) = 0. tangent (5.4.6), = for any s t a t i o n a r y limiting state % of (5.4.6) (d$,da.) we t a k e 0 at point ($, a) the d i f f e r e n t i a l get 106 on any on b o t h solution sides of curve of (5.4.6) to f £ ( ? ; 3 ( a ) ;cc) d$ + f Z)f[d0\da] It then (a) ;a) da = 0, o r a = 0. T (5.4.7) follows from (5.4.5) (5.4.7) i s o f dimension that the solution 2, and hence t h a t there space o f e x i s t s two d i s t i n c t tangents a t (0,r ) . L t h e case <r (r ) =iu with u * 0, For we appeal t o the well L known Hopf theorem that ( c f . Marsden & McCracken, the conditions leading t o Hopf (5.4.1)- (5.4.3) theorem. Thus, 1976) and note coincide without with further those effort, we summarize t h e d i s c u s s i o n f o r both u = 0 and w * 0: Theorem (Asymptotic 5.3 f (%4(<*•) ;a) be (5.4.1)-(5.4.3) equilibrium given at by ¥* (a) of r , of and (3.3.5), ($,a) = (0,r state neighborhood state ). (3.2.3) its criticality) and satisfy Then, for nonmodal the generically, which Let a is conditions for in disturbance a or an unique a periodic limiting Remark 5.5.1 small %(t;%) 0 L approaches an stationary state The f o r e g o i n g state if <r (r^) as t -» to if = iu with results yield <r (r^) = 0, w * 0. no i n f o r m a t i o n on the d i r e c t i o n i n which t h e expected b i f u r c a t i o n branches o f f . A determination general, higher of the bifurcation order direction d e r i v a t i v e s of the vector Guckenheimer & Holmes, 1983). 107 requires, i n field f (cf. Remark 5.5.2 By definition (3.3.1), the perturbed flow difference between equilibrium state. of 5.2 a n d t h e o r e m theorem occurs I t then follows at criticality a = that i s %(t;% ) Q the and the o r i g i n a l under t h e c o n d i t i o n s 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 r . More precisely, out of L instability of a given perturbation equilibrium ^ , a new s t e a d y motion <r (r ) = 0, o r a p e r i o d i c e m e r g e s if $ (a) state ^(a) t o an i n i t i a l o f (3.2.3) +~$(t;$ ) Q flow bifurcates f r o m t (a) L if <r (r ) - i w w i t h u * 0. L Remark 5.5.3 satisfied. branching in A necessary off these a) for side = bifurcation of some . Note that t h e above problem. Actually the existence states o f nonmodal sufficiently disturbances be for a small results obtaining requires, i n (see Appendix details). Numerical i l l u s t r a t i o n s Primary planetary for f o rs u b c r i t i c a l further numerical b i f u r c a t i o n analysis illustrate 5.1, solve limiting general, 5.5 of a (5.4.1)-(5.4.3) i s v i o l a t i o n of hyperbolicity the subcritical essentially condition at a = neighborhood 5E L e t t h e conditions equilibrium t h e main scale branch results atmospheric of this flows we show t h e m o d a l a n d g l o b a l a s e t o f such flows {t(r)} and i t s stability chapter, over we topography. stability analysis o b t a i n e d as e q u i l i b r i u m 108 To consider In Fig results states of (3.2.3) curve (see t h e c a p t i o n and the stability curve measure r f o r model abed ($) are and r N difficult to see i t s (¥*), parameters) . linear The and respectively. nonlinear It i s not L from (4.2.8) - (4.2.9) that r=r and r=r L N serve as linear The the curve which monotonic stability efgb bifurcates branch {¥*(r)} analysis global boundary i s r^tf at results, 3.3-3.6 for the obtained 5E focus boundary {¥*(r)}, of r v i a a = r For decay respectively. from L to the numerical zero i n primary r=r^ of (I) a n d {¥* ( r ) } bifurcation with illustration regimes and the equilibria for details) , on asymptotic set criticality boundary. we w i l l stability f o r the set the and i s stability of ) (see Appendix linear top the (II) . the as its main See F i g subcritical regime. b) Steady along limiting the direction primary of boundary r=r^=l/(13.33 It region (I) a n d states at of (0,r ) the ~$(t;~$J when r, cross and step theorem of 5.2 the t(r) {^(r)} the that stability f (r) with r in disturbances nonvanishing limiting to the properties has r close to r off^ . The L calculations (5.4.1)-(5.4.3) the supercritical L direct in linear nonmodal •» » i s r e l a t e d the underlying the into persistent nature as t equilibria we from (II) h a s Further, Q of day) i s expected Moving and s t a t i c b i f u r c a t i o n branch decreasing regime. ~$(t;$ ). states indicate are satisfied that at the conditions the c r i t i c a l i t y listed in r=l/(13.33 day) with $(r) f o r r of r w =0. Thus, the long term i n the supercritical i s expected to side behavior o f %(t;%^) t o of a small be stationary neighborhood and unique (cf.theorem L 5.3). To on verify these are performed, (3.3.2) The underlying the primary enough expectations, with equilibria branch to numerical the results a r e taken ensure to the criticality experiments that r (= from shown 13.33 i n F i g the part the four It 5.2. (f'-»b) t(r) days). based are on close i s clearly the disturbances settle L seen that after down t o some the transient steady the streamfunctions Fig 5.3(a) into states. a new (cf.the thick solid line limiting state of fact that taken steady are conditions. This {Null A s an example, for bifurcation (c)-(f) the growth, from local JDf} | flow F i g5 . 3 ( f ) . 1f>(t;~$ ) i s Q state l/r=ll.1 days indicates that independent of initial i s t h e consequence =2 ( c f . ( 5 . 4 . 5 ) ) - 5.3 The snapshots for F i g5 . 4 5.2). uniqueness i n Fig o f an e q u i l i b r i u m t h e experiment i n Fig we s h o w ofthe i n the present case. L The experiment over which c) p e r i o d i c {^ (r)} initialized t h e uniqueness states bifurcating from holds s'^ s u g g e s t s i s fairly and Hopf b i f u r c a t i o n from the primary that the region large. We s h a l l branch at usethe r=r set =1/(13.33 XJ day; f o r illustration o f t h e second 110 part o f theorem 5 . 3 . It is seen r tf ) from ( c f . curve efg i n F i g 5.1) t h a t t h e L exchange of r=r'=l/(18.5 days). r=r' that stability f o r t h e s e t {¥* (r) } Moreover, a direct i s a Hopf b i f u r c a t i o n point, occurs calculation at reveals i . e . , the conditions L (5.4.1)-(5.4.3) nonmodal 0 asymptotic side o f some neighborhood periodic this experiments former I t i s thus d i s t u r b a n c e s %(t;% ) t o those supercritical confirms h o l d with u * 0. states. displaying the l a t t e r (r) with o f r=r' The r e s u l t s from that r in will A s e t of numerical anticipation. a r e shown t anticipated the tend t o experiments two o f these i n F i g 5.5 and F i g 5.6 , with t h e t h e time series o f d i s t u r b a n c e energy and f o r the streamfunctions of b i f u r c a t i o n of a given steady f l o w shown i n F i g 5.6 (a) (corresponding t o t h e t h i c k curve i n F i g 5.4) into a periodic one shown in Fig 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 forgoing numerical bifurcation solution experiments, analysis curves i t i s found ( c f . Appendix of (3.3.2) In agreement with t h e 5E) t h a t emanates from two s t a t i o n a r y the underlying stationary bifurcation s t a b i l i t y analysis the difference amplitude primary equilibria branch as seen o f t h i s branch between ¥* (r) o f the corresponding ^' j) from s u p e r c r i t i c a l l y . The one f o r steady l i m i t i n g s t a t e plus the {^(r)} r of ~$(t;~$ ) o y i e l d s the i n F i g 5.7. A modal {¥* (r) } i n d i c a t e s t h a t when and ¥*(r) as measured nonmodal disturbance by the ~$(t;~$) i s s m a l l ^ (r) i s l i n e a r l y s t a b l e 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 original ones flows deviate further ( c f . F i g 5.1), a r e s u l t supercritical bifurcation virtually from the true to a l l (e.g., Joseph, 1976). Moreover, the criticality on t h e s t a t i o n a r y b i f u r c a t i o n bifurcation point from away which a p e r i o d i c branch b i f u r c a t e s and i s n u m e r i c a l l y o b t a i n e d l a t t e r plus the underlying (r) } i s a Hopf %(t;% ) of Q ( c f . Appendix 5 E ) . The (r) } g i v e s r i s e t o the p e r i o d i c branch (r) } as shown i n F i g 5.7. I t i s remarkable each time out o f i n s t a b i l i t y o f a given flow to persistent nonmodal d i s t u r b a n c e s , a new flow emerges with a h i g h e r of energy norm, complexity level which manifests as o f temporal behavior an increase (e.g., that from level i n the steady t o p e r i o d i c ) and s p a t i a l s t r u c t u r e (e.g., from predominant zonal t o wavy motion). The repeated supercritical bifurcation as t h e one seen i n F i g 5.7 has been a prominent t o p i c such i n the s u b j e c t o f t r a n s i t i o n t o t u r b u l e n c e s i n c e Landau's c o n j e c t u r e (1944) on treatment further this subject ( c f . , Joseph, o f the repeated by inquiring the s t a b i l i t y (r)} v i a F l o q u e t theory Holmes, 1983) , transition to bifurcation which could is certainly or (3.2.3). 112 The present be c a r r i e d out o f the p e r i o d i c (cf. Arnol'd, turbulence d e t e r m i n i s t i c system 1976). branch 1983; Guckenheimer & necessary chaotic behavior to in study the e) Transition due t o Given increasing the nonmodal initial modal v s . observed and i n model studies and 4.7 played one by state from two this i n the (e.g, two of flow to is of one and explosive 5.8 triggered one by the of development of Sanders, Fig interest Fig role (e.g., 1985; of perturbations plausible perturbations another. experiments, s in a atmosphere it types for Farrell, thesis), the perturbation evidence perturbations disturbances in nonmodal i n i t i a l 3.7 to in and F i g see the fast 4.6 the role transition from displays by 1986) the results optimal growing nonmodal normal model M Re (if) . It is effective as seen a clearly that triggering the agent former for is much transition more than the latter. 5.6 Concluding In of this remarks chapter, we h a v e nonmodal d i s t u r b a n c e s perturbed Liapunov certain state is and the associated with ultimately diminish been known for have also (cf. as distinct concern has that state growing -> is modal », a models a asymptotic generic the inevitable perturbations where the method transient which Pedlosky, from when (cf., a growth will has long 19 6 4 ) . given the of under disturbances phenomenon transition case shown t h a t the nonmodal (e.g., behavior direct we h a v e 5.1) amplitude t a using theorem linear established another Q Specifically, finite some %(t;% ) f o r hyperbolic like. conditions analyzed the flow We to flow under theorem 5.2). Further, we have demonstrated b e h a v i o r of p e r s i s t e n t of the neighboring that the nature of asymptotic d i s t u r b a n c e s i s r e l a t e d t o the nonhyperbolic point in some u n d e r l y i n g parameter space. As a f i n a l note, the r e s u l t s here are i n the s t a t e space R M or i n the u n d e r l y i n g parameter 114 nature local space. Appendix 5A The d i r e c t method o f Liapunov The method i s widely systems a v a i l a b l e i n textbooks on dynamical (see, e.g., H i r s h f o r more e x t e n s i v e and Smale, treatment). 1976; V e r h u l s t , 1985, To s t a t e the b a s i c f e a t u r e s o f the s u b j e c t , we c o n s i d e r the system (d/dt)x = q(x), where g : Let x x(t ) = X Q ID -> R isa C M (5A.1) q vector 1 be an e q u i l i b r i u m field of on an open s e t D c R . M (5A.1) . L e t V : y •> R be a e differentiable x . An orbital function defined derivative of i n a neighbourhood HI c D of function V along the s o l u t i o n e x(t,x ) o f (5A.1), Q where denoted v the gradient Liapunov function by V, i s d e f i n e d operator in R . V by V = W . g , i s said M i f i t satisfies: (a) V(x ) = 0 and V(x) > 0 i n Ii - {x }, e (5A.2) e (b) V * 0 i n u - {x }. (5A.3) e I f V meets t h e c o n d i t i o n (c) V < 0 i n U - {x }, (a) and (5A.4) e it t o be a i s c a l l e d a strict (Liapunov s Theorem f equilibrium state Liapunov function Liapunov criterion x^ of V(x) function. (5A.1) is for stability). stable if there An exists in a neighbourhood U c D of x ; It a is e asymptotically function stable if there V in U. 115 exists a strict Liapunov Appendix 5B A u x i l i a r y lemmas The f o l l o w i n g lemmas a r e developed t o f a c i l i t a t e t h e p r o o f o f the main r e s u l t s i n § 5.2 and § 5.3 and i n c l u d e d convenience o f r e f e r e n c e . may be found i n Hale Lemma 5.1 L e t f: here f o r Some v a r i a n t s o f lemma 5.1 and 5.2 (1969) o r i n H i r s h and Smale R > IR be written M as f (x) (1974). = f (x) + w(x) p that (x) is homogeneous polynomial of such (i) f a negative (positive) definite p |x|| -> 0. degree p; Then, f (x) is negative (ii) (positive) w(x) = o f|x|| ) as p definite in U-{0}, OJ a neighbourhood QJ of 1 = 0. Proof: max | X ( f o r n e g a t i v e d e f i n i t e case) By ( i ) , we put f (x) p = -a < 0 (5B.1) | =1 and hence have, f o r x € IR -{0} M f (x) = \\xff fx/||x|| ; s -a\\x\\ . p Note that condition ( i i ) implies e x i s t s a c l o s e d b a l l 1(0,5) there \w(x) \ s e||x| p f(x) a that f o r a given c (>0) , such t h a t f o r x € B(0,8). Now, s e t t i n g c=(l/2)a (5B.2) (by (5B.1) and (i) ) p (5B.2) i n (5B.2) and then combining (5B.1) and yield £ -(a/2) desired ||x|| < 0, f o r x e p result. Identifying proof 116 B(0,8)-{0}, §(0,5) with U completes t h e Let Lemma 5 . 2 P Q M equation + OP = R T (5B.3) has a positive definite R if Proof: We positive prove For expfPt) fact Re (<r CP)) converse any g i v e n for any lemma 5.2, negative definite which R, that is 1 of the there exist such ||expfPt;|| s construct the exponential integrant in positive (5B.4) constants of operator follows c. and a,, I that c^xp f-c^t;, T Re (a-CP)) <0. (5B.3), from l (5B.5a) \\exp(2 t)\\ * c exp(-a t) if used to c i=l,2, we is (5B.4) Y^^CPt) /!!) (s negative < 0. of Q according The c o n v e r g e n c e the Q as a solution (-R) expfPt) dt T where the definite Q B ^exp(P t) Pt. definite and only if subsequently. a The Liapunov matrix P , Q , R e £(IR ) . (5B.5b) 2 2 Thus, Q is well defined. To see Q satisfies consider (d/dt)^exp (P t) (-R;expfPt;JT = P e x p f P t ; f - R ; e x p (Pt; T where (5B.6) + T the fact from 0 that to », exp(P t) (-R)exp(Pt)P Pexp(Pt)P we 0s||expfP t; C-R;expfPt;|| T s||exp CP t) 11 C - R ; 11exp (Pt) || exp (Pt) 1= obtain estimate T (5B.6) T (5B.3) is used. with the Integrating use of the expC-a t) c exp (-at) SS||(-R;I|C " 1 1 u 2 -» 0 as t -» ». (by (5B.5)) 2 and t h e c o r r e s p o n d i n g limit exp(P t) (-R) exp (Bt) •> 0 as t •» ». T T h i s e s t a b l i s h e s the converse The bilinear Lemma 5.3 o f lemma 5 . 2 . operator B B defined by (3.3.4f) is bounded such that |Bxy|| s cflx||||y||, for any x and y e R , (5B.7a) M where c i s given Jby (5B.7b) with c Proof: a positive i First, b y norm e q u i v a l e n c e there exists ||ac|| s c where all i constant. a constant ||x||, for c (>0) i any x e R i n R (cf. Kreyszig, pp.96), M that (5B.8) M normfl.fli s d e f i n e d , a s t h e sum o f a b s o l u t e values of c o m p o n e n t s o f x. Now, f o r a n y x a n d y i n IR , c o n s i d e r M i i j k i jk - 1 j k 118 sc2 which by H ||y| 2 2 < (5B.8)) bv l e a d s t o (5B.7a) i m m e d i a t e l y , (5B.7b). with c a constant 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 We a p p l y t h e r e a l R w.r.t which the linear 1974) t o f i n d part A Let (5.3.1). (i) of the vector state t of (3.2.3) satisfy M M (ii) M Then, R has a basis h such that + k equal eigenvalues & equilibrium f o rR form. R = R~® R , dim R~=Af-k and dim R with A the o p e r a t o r s on a new b a s i s ( c f . (3.3.4a) f i e l d f (3.3.3a) t a k e s a s p e c i f i c Lemma 5.4 operator canonical theory o f l i n e a r ( e . g . H i r s h a n d Smale, M given to the sum of with positive real + = k , algebraic (5C.1) multiplicity for parts; A in the basis & has a matrix of form = diag[K_,K J (5C.2) + where A_ e £(R ) and A Re(<r(AJ) € £(R ), + + satisfying < 0 and Re(<r(A )) > 0, (5C.3) + respectively. Proof: L e t E^(<r^) A belonging is c C denote t h e g e n e r a l i z e d eigenspace o f n. t o <r^(A), i . e . , E^ = N u l l (A-cr^ 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 ^ . P u t cr^ = is real pp.129 ( i . e . , 0 ^ = 0 ) , b y t h e S-N d e c o m p o s i t i o n i n H i r s h a n d Smale, 1974), (A;) 1 , where n ^ + i p ^ . I f tr^ (A) ( c f . theorem 2, E^ h a s a b a s i s &^ g i v i n g the restriction canonical ( *xUx 1 A to E.^ (i.e.,A|E^) a matrix in the real form = where of d ± a each ^] g of ,A| 1 ) 2 ) ... A| ( diagonal (5C.4) i 7 f l ) blocks in (5C.4) has a form a (5.C.5) a . i- 1 with the If o\£ (A) of A and geometric is as A complex e JE(IR ). M imaginary A. basis a by the 0^ > 0, Moreover, of for the matrix with parts E decomplexification Again multiplicity the Taking form (5C.4) the X X E . n ? ) M X c of but x' c R with of eigenvalue real E^(c^) of blocks parts yields which under c each M N u l l (A-cr^) ) . an union x A|E^ dim also elements S-N decomposition, of is E . (cr ,) ©E . (cr .) (H Xz (= er^ (A) basis Ic" E. of has dimension the a basis 2n &^ has characterized by form I D. a D . x 1 2 1 2 I D . 2 Note that (5C.4) in form a provides *,-&?]• X the appear (5C.4) = as diagonal as many often linearly a basis as blocks times n^. It independent &. f o r of as form whereas is clear set and R 120 (5C.5) that hence cr^ or or D^ these the (5C.6) appears bases union in {&^} of &. & = &, u & 1 2 The v...& 0 direct from u p consisting of Re(<r^(A.)) < 0 the the A Ax = 0, (or E. under = dxag [(^ ) the into two e ) the E^ is parts, E^(<r^) w i t h of remaining (or E^) , thus one it &^ o f E^(cr^) holds that invariant under A. Xr J. A of V x immediately M &^ (i=l,2, . . .,p) Since E. 1 o f IR f o l l o w s (5C.7) a n d one composed J. Hence, (5C.1) basis bases Re(<r.(R.)) > 0. (K-a.(A)) (5C.7) S sum d e c o m p o s i t i o n subdividing with u P+l basis A has a ) ] matrix representation of form A & 1 , k (K ) 2 1 which first yields p The property the eigenvalues those of 5D Existence (a)-(c) is construction by i 5.5 (5.3.1). (5C.8) by of A of A form into + (5C.2) of and A _ f o l l o w s in E. (or the of Let function key such step V + i n Hale V This ) from are fact the the A . that same as proof. B function establish f o r the the into the as completes satisfying + to E. grouping reminder Xr X (5C.4)-(5C.6). a after A _ and the Liapunov i n s t a b i l i t y the treatment Lemma <r. X A Appendix in (5C.8) s asserted (5C.3) ( ^)^ k 2 the blocks (A h system the properties theorem (3.3.2) is 5.2. The motivated (1969). the equilibrium Then, there exists state $ of a C function 1 121 (3.2.3) satisfy V : QJ c R -» R, V + + M + the neighborhoods of the null solution ~$ = 0 of such (3.3.2), that (a) V ($) = 0 as % -> 0, + (b) V + (c) + > 0 in U - {0}, + V C$ ) > 0, V l | I c R n where R I is + Proof: + with % -» 0 as n nj determined in Lemma 5.4. We p r o v e t h e lemma b y c o n s t r u c t i n g 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 note that matrices bases P AP = A _1 where representing are similar. nonsingular A & matrix I t then a given follows P e i£(R ) s u c h M operator + that i n different there exists a that = diag[K_,K ], (5D.1) + i s the linear lemma V a function i n ( 5 D . 1 ) - ( 5 D . 3 ) . F i r s t , we part o f t h e vector ( 3 . 3 . 3 ) - ( 3 . 3 . 5 ) ) a n d A^ i s i t s c o u n t e r p a r t (cf., co, n 5.4). I n fact, P itself w i t h t h e b a s i s & ( c f . (5C.7)) t o R M f(cf., w.r.t. t h e b a s i s & defines with field a map f r o m t h e standard R M basis. Substituting % = V%. into (3.3.2), (5D.2) we h a v e the governing equation written w.r.t the b a s i s & (d/dt)% = diag[K_,K ]% + + BP^P^ w h e r e B i s t h e same a s b e f o r e , the (5D.3) g i v e n b y ( 3 . 3 . 4 f ) . Now, d e f i n e f u n c t i o n V by + 122 V= <$,Q $>, + Q= +! where € £(R ~ ) K + QA 1 = -I + I and the The existence I property (c), solutions to (5D.5) = ( 5 D . 6 ) unit Q operators and i follows we u T the , of (a) [u,v] , are k M-k 2 see (5D.4) € £ (R ) 2 + Q (-A ) 2 [-Q^QJ, and Q k 1 C-A^Q To diag + e use IR are Q 2 fact v e order from (5C.1) t o IR . Then, + M-k, k, guaranteed immediately the and of by the lemma any % in {~$ } the c R (5D.4) . form with + n 0 as n -> co, of n We 2 orbit of V= to of ~$(t;% ) o where we have estimate lemma 5.1 Here, finding we the A dxrect in from evaluation (5D.3) g i v e s , - Q A_, A*Q + Q A 7*> 2 the + 2 + o same of V property along (5D.8) used of an the in argument high (b), Algorithm only the (\\-$f) . results Appendix 5 E (5D.7) r e s u l t s 2 + o([$f) 2 (5D.7) (b) . x = |*| the turn in diag[-AlQ + -» n 2 positiveness Q . $ = n > > 0. + The $ (5D.4) y i e l d s V (% ) = <v ,Qv n The 5.2. definition. write for respectively. which for briefly stationary order similar to terms in completes bifurcation outline or the 5.3 the proof. to get Applying (5D.8). B analysis numerical periodic 123 lemma procedures limiting states for of nonmodal disturbances references states, given the locate (which below f o r a following the tangents governed stationary on (3.3.2a) detailed several steps bifurcation to obtain to an 2) the and Chua, true steady continuation remaining function defined M s 1989) and t h e n approximate algorithm branch. F o r 1 ) , Det to locate 1978) i s given i n det Parker algorithm (M) single and t o det (5.4.7) by (3.3.3a). during value switch as Chua, to algorithm (cf. 4) to i t t o get to use the out the trace as bifurcation point, the process shown a test with M 1989). An that elements approximate (SVD) space tangents. are available More but of ( c f . Nobel of the n u l l space of w.r.t. over bisection For step SDf u s i n g the and o f Df sophisticated increase sign i s passed application f o r the n u l l a of continuation a bifurcation point decomposition techniques I t c a n be (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 . i s solved 1988) . The b a s i s used as a (5E.1) parameter a i n d i c a t e s that (cf. (0,r^) by f change the solution (VL) i s i n t r o d u c e d the stationary to find from t o improve (3.3.2a); (Keller, steady 1) dc/tj da where 2, of the use them steady as a c o r r e c t o r solutions to emanating ( 3 . 3 . 2 a ) ; a n d 3) t o a p p l y t h e Newton-Ramphson Parker For involved: point; (5.4.7) (see account). are the b i f u r c a t i n g branches are the solutions predictor by Daniel, are then branching t h e amount of computation s i g n i f i c a n t l y For periodic ( c f . Kubicek and Marek, 1983). asymptotic states, we convert the initial v a l u e problem (3.3.2) i n t o a two p o i n t boundary v a l u e problem I ] = [ (d/dt)[ (5E.2) f s u b j e c t t o t h e boundary c o n d i t i o n s 1(0) - %(T) •f C$(0)rt,a) = 0 (5E.3) k where T i s t h e p e r i o d and t h e into ($ (0) ,4, a) = 0 i s introduced (5E.3) as a phase c o n d i t i o n , with k a r b i t r a r i l y f i x e d t o between 1 and M ( c f . Seydel, conditions shooting are also method possible 1988) . Other (Seydel, ( c f . Ascher, s o l u t i o n s o f (5E.2)-(5E.3). 125 et types 1988) . The a l , 1988) of phase standard i s used f o r 0.3E-01 0.3E+00 EKMAN DAMPING (1/day) Fig 5.1 equilibria days, U Stability = abed 22.0 diagram 1/x r a n g i n g {^(r) } with wavenumber-1 curve regime m/s =1/(5.71 and o f height a r e i t s r ($) days) boundary and l i n e a r r ($ ) and from 29.5 and topography family of days t o 3.5 being of zonal 500 m. The t o p curve and t h e and x (t), N r-r for a respectively, with L r=x^~l/(13.33 days) as i t s MGS s t a b i l i t y boundary. The curve efgb i s f o r 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 criticality r=x'=l/(18.5 the r = ^' L days) s e t {** (r) }. from the primary i s the l i n e a r 126 branch stability {¥*(r)). boundary f o r 10.0 CM M « r 8.0 \ CM •* •K E 6.0 19 CC bJ O Z <C CO 4.0 - 2.0 - cc I— CO 0.0 0.0 100.0 200.0 300.0 400.0 TIME IcUy) Fig 5.2 Asymptotic disturbances. located 5.1) nonvanishing The u n d e r l y i n g on t h e part (f -»b) f o r values of r given steady states equilibrium of the i n the 127 states primary figure. of nonmodal are those branch (cf .Fig N U) basic state L \ \ \ \ Ai i • IO s y J IV)) '8B\\\> , (b) s t~BO tc) days t=ltTO td) days t=320 (e) days t=400 (f) days F i g 5.3 state (a) (c)-(f) thick Streamfunctions f o r b i f u r c a t i o n into a r e from solid line a new steady t h e experiment i n F i g 5.2). 128 flow o f an (f) . The equilibrium snapshots f o r 2/r=17.1 days (cf.the 20.0 CM 16.0 * s' CM X H-1 * -E 12.0 M-2 CD CC Ld I \ 8.0 UJ CJ 2 «S m to 4.0 i—i Q 0.0 L. J 0.0 100.0 200.0 300.0 400.0 TIME ( d a y ) Fig 5.4 Local uniqueness nonmodal d i s t u r b a n c e s . the o f asymptotic The u n d e r l y i n g state of equilibrium state i s same a s one i n F i g 5.2 f o r t h e e x p e r i m e n t d a y s . T h e s' i s o b t a i n e d from s c a l i n g M • at steady t=0 h a s 10 % o f t h e b a s i c s such t h a t M state 129 of energy. l/r=16.4 %(t;s') M 40.0 M * CM 32.0 in \ r r=l/(;24.8 days) r=2/C15.1 days) - M X CM e - 24.0 >cs cc UJ z UJ UJ CJ 16.0 - tto a 8.0 - z et m cc »—< 0.0 0.0 100.0 200.0 300.0 400.0 TIME (day) Fig 5.5 Periodic disturbances, and 85.8 underlying of limiting with p e r i o d s days days f o r dahsed equilibria the stationary 46.3 line, are located bifurcation states 130 nonmodal f o r the s o l i d line respectively. The on t h e u n s t a b l e branch the v a l u e s o f r as i n d i c a t e d . of (cf.Fig section 5.1), with basic (») state (b) — - t=V \ x \ y \ - - —_^-sa. K Ic) t=t'+T/4 --4.50 ~-«s I ' /- — \ . — (d) t=t'+2T/4 -S-°& - r (e) 50 • ~ ^ -4. SO (f) F i g 5.6 state (a) Streamfunctions f o r b i f u r c a t i o n into a periodic taken from the line in Fig t'=336.9 days flow. experiment f o r 5.5) and over 1=43.3 a The snopshots l/r=24.8 cycle days. 131 of an e q u i l i b r i u m of days (b)-(f) (cf.the oscillation, are solid with 160.0 r primary branch 150.0 stationary branch o o o o | 140.0 Periodic branch o cc lx) z 130.0 - _! ' ' ' ' 120.0 0.2E-01 0.2E+00 EKMAN Fig 5.7 primary marked Repeated branch by x of on b i f u r c a t i o n point DAMPING supercritical equilibria the primary (1/day) bifurcation for the (cf.Fig 5.1). The branch is stationary a point whereas the symbol + i n d i c a t e s the Hopf b i f u r c a t i o n p o i n t . The l i n e s drawn with dahsed corresponds to unstable equilibrium states. 132 5.0 r 0.0 100.0 200.0 TIME 5.8 Fig Nonmodal transition stationary (cf.Fig r=r'. to a periodic bifurcation 5.1), The versus located initial modal state. near growth rate basic (r) } the ) of %(t;s state 133 in i s from the l/r=18.8 days bifurcation point and M 1/(5.08 days; and 1/(400.0 days;, 400.0 perturbations with secondary L 300.0 (d*y) initial The branch ; respectively. $ (t;Re (t) ) are CHAPTER 6 CLOSURE MODELING: FORCED-DISSIPATED STATISTICAL EQUILIBRIUM OF LARGE SCALE QUASI-GEOSTROPHIC FLOWS OVER RANDOM TOPOGRAPHY 6.1 Introduction The statistical flows over authors. a to statistical classical for the topography the was aspects, f-plane using (Kraichnan,1967) 1971) . Recently, for the and throughout of and some that effort depth of dynamics. To c o p e Holloway (1978) has model been than of fluid. insights to do with and with these developed flows on the approximation (TFM) made with forcing little interaction 134 larger valuable omission field of absolute correlation the which has method anti-cyclonic forced-dissipated test absolute at scales provides direct the the is flow aspects theories found radius, results (1977) as for the many hereafter using the flow problem, the i n many Herring statistical to to flows They by inviscid-unforced Furthermore, extend a (1976; the referred equilibrium of quasi-geostrophic Hendershott mechanics. shown t o leads and and t w o - l a y e r absolute flows investigated been lower-layer dynamics dissipation has (also deformation the by obtained around h i l l s . internal viscous one- statistical circulation into SHH) equilibrium equilibrium, While Holloway as equilibrium) achieved random t o p o g r a p h y Salmon, referred the equilibrium to (Kraichnan, extend the f-plane formulation domain averaged hereafter The objective topography study is singled In work when a § closure closure A presented closure motion equations with brief 6.2 C l o s u r e nonvanishing (Holloway, flow study is the forced 1987; momentum (H87) as of i . e . , the statistical over zonal well the a random source. as on The direct equilibria is vorticity-topography topographic formulation for the In for and is physical § 6.3, we parameters results summary in § to One a s p e c t resulting discussed. environment, a stress acting on flows. the of is model (dns). and the 6.2, with component uniform out f o r consideration, briefly dns. this simulation invariants are of on a equilibrium case t o H87) by an e x t e r n a l correlation the 0-plane velocity established based numerical the zonal referred equilibrium to outlined system solve under to discussion those of concern numerically relevant compared a n d two the to midocean from ensemble the results are 6.4. formulation 6.2.1 A s e l f - c o n s i s t e n t model Consider a flow governed zonal momentum by double a source periodic (i.e., cell by (1.2.1), 0*=-C7*y (I.e., 135 in forced b y an (1.2.1)) satisfying external and bounded B.C.(1.2.2b)). Further, we w r i t e the streamfunction ip(x,y,t) = -U(t)y + then + J C * -Uy, V 2 (d/dt)U where Q in closure operator Hart, 1979). the interaction, In The term where the next ( 6 . 2 . 2 ) - (6.2.4) of coupled the projected (1.2.1)) (6.2.2) Ekman is (6.2.3) with denotes the onto ordinary statistical a is referred problem defined space, differential is functions, (6.2.4), where we c h o o s e x = (x,y) 136 by giving equations a for formulated v i a equations of to average. equations. virtue (e.g., flow-topography t h e domain problem the obtained equation the Fourier commonly biharmonic may b e overbar from simply denotes and momentum x arising section, nonlinear in D equation the overbar the corresponding 6 . 2 . 2 Moment basis is and (6.2.3) stress (cf. with the notation x-directed in (6.2.4) <b(x,y,t), notation including the topographic = consistent The tendency considering By (6.2.2) 2 (6.2.3) formalism, from moment ( c f . H87) -DV *, <b(x,y+l,t) i n our e a r l i e r dissipations. which i s obtained * + p y + h) = a s h t o b e more dissipation set system = *Cx,y,t;, (h/H)f written as (6.2.1) = r(U* - U) + hd*/dx, *(x+l,y,t) used 2 ( c f . H87) *(x,y,t), a self-consistent 3.V * as the and k e K d e f i n e d by K • -|k = (m,n)2n/L \ (m,n) = 0, ± l,±2,..,ls |k| s j , and expand *, h i n t h e form (*,h) = [ <* ' ) "*' K h K (6.2.5) e±}: k where t h e summation (6.2.5) into property a l l k € DC. A f t e r i s over (6.2.2) and (6.2.3), o f s e t G, we obtain (6.2.2)-(6.2.3), i n terms o f < (d/dt)^ = -i* C7h x - k a U k + and u s i n g v K k introducing t h e orthonormal the spectral version of (= ~k \) 2 k " [ k \ p + q (6.2.6) A (d/dt)U = r(U - U) - Imj £ k ^ h ^ / k \, 2 (6.2.7) k ^ A = k p - p/k )k 2 , (6.2.8) s Z- (k x p ) / p q where 2 = 2- (p x q ) / p ()* denotes t h e complex 2 = Z- Cq x k) / p , conjugate symbol £ r e p r e s e n t s t h e summation (6.2.9) 2 of quantity (); the over wavevectors p , q such A k + p + q = 0; that mode k and A. for i s the l i n e a r Rossby wave frequency i s the i n t e r a c t i o n coefficient f o r the kpq triad (k,p,q); space; f i n a l l y , v> i s t h e d i s s i p a t i o n k operator i n the spectra Z i s the u n i t v e c t o r along t h e z - a x i s . C o n s i d e r a phase space r spanned by t h e r e a l and imaginary part of a l l < and k h Re (C ) ,Im(Cj ) ,Re (h^) ,Im(h ) ,17, where denote t h e imaginary part fc c k and r e a l 137 f c plus U t h e symbols (i.e., Im, Re of the quantity) . Then, one single realization of ensemble of i n r, corresponds to the evolution of a J^Re (C^), Im (t^) . R e (hyj . Im (hy), t/| in r. of probability the ensemble distribution Im(hy).U.t) which Liouville equation exists requires between phase the the it presence of external i s possible the invariants D = 0 study, for (e.g., SHH; to the equilibrium as Furthermore, we moments where are of the equilibria, the Physically, vorticity angle bracket the first and the vorticity-topography and moments k of Re(h^), to the (6.2.7), of a l l or orders k be c a r r i e d P as a f u n c t i o n a l o f f o r t h e case & Frederiksen, to those states is to the of the i . e . , denotes for correlation. 138 in s <CJ CJ C the modal c > C7* = this ensemble We statistical equilibrium. two a second n d ensemble t h e modal the In time-independent. absolute interested where 1987). forced-dissipative as out i n U* a n d d i s s i p a t i o n D , w h i l e represents second knowledge as t h e s o l u t i o n (6.2.6) statistics only statistical -JRe (< ), Im (< ) , R e (h^), Im (h^), C/|. Carnevale opposed the a points (Re {C,^), imU;^), P a n d (6.2.7) states phase complete independent ourself yields realizations of of P can rarely time ensemble those A joint forcing (6.2.6) we r e s t r i c t which refer t o seek of by points the determination such cluster i n principle the (6.2.7) requires function implied equivalently, However, an and trajectory description and (6.2.6) spectral spectral < order C^hj >, c average. f o r the of the In what level Z (d/dt)Z k follows, and C ^ . From = 2k k we o u t l i n e lmrc ; - A (iw x (6.2.11) moments depend - v )C k require moments hierarchy order of hypothesis on are necessary of Z Wk }' > that knowledge and moments, moment the consequence moments the of fourth order <h£h> third on, order among in leading At this close (6.2.10), which are and so relations to the of turn to stage, some different the hierarchy a order at the a n d t? . k k this statistics of equations. i n order (6.2.10) the solutions 6.2.3 C l o s u r e h y p o t h e s i s and master For the k <«h> on t h e f o u r t h closure the o f t h e forms unclosed level k >+R e < Hy& <h^h^>. I t i s c l e a r where at y Rfi< (d/dt)C^ = ik UH^ + procedure we h a v e kpq{ W* 2J1 closure 2v Z^ fc ~ I and (6.2.6), the purpose, of this form order moments first topography of the we in l } n K assumption <hhh>. N e x t , cumulants, terms of equations assume ^ s is the in of ensemble The direct vanishing we a s s u m e : products the Gaussian. introduced 139 that (i) the of role expressing the odd second of fourth order moments the Z and k ff , saturation characteristic shorter than Viewed the fourth (ii) by the a allows to obtain pqk , Z Z. + \A kpq \ pqk q k s + Z Ci + . A pqk «q«k vr>„ kpq p q + A <VqV = -e ; k q z c {A p q k z c q qpk V* V q + + A moments replacement z k term with characteristic (k,p,q). quasi-stationary moment Z + p c ZZ kpq p q . A + k Z C* + A w Z q C P C + A P H C + . Z.C* qk pk p (6.2.12a) V ? k p}' C triple equations k (6.2.12b) = - k q {^pqk q k qk V* pkq V,}' e moment The h y p o t h e s i s c + of order Z.C* + m is (i.e.,2^). pkq„ k« lq' qpk p k kqp q pj \ + the third qpk p k . A moments , Z Z. + A. A (ii) to the the order leading order permits the t r i a d from the t h i r d 1 1 (i) damping M^pq f o r third order in action moments; for hypothesis cumulants us damping order time linear rate <C_C C, > = - 0 ' s third f o r the second order correlations provide relaxation those eddy-damping to of simply, equations s is k A (6.2.12c) ^Scpq kpq kqp' pqk pqk pkq' qpk qpk qkp' S A + A A S A + A A S A +A (6.2.12d) where ©if ^ 1 are characteristic 140 of relaxation (towards quasi-equilibrium) by t h e n o n l i n e a r transfer and e x t e r n a l d i s s i p a t i o n . S p e c i f i c a t i o n o f t h e t h r e e a r r a y s jj.pq (i=l,2,3) are from the e subject to consideration various of conservation model and t h e s t a t i s t i c a l t s 0) and (i) 6j[ constraints resulted requirements realizability on t h e c l o s u r e a 0 V k € K (i.e., ( c f . , H 7 8 ) , among which are t h e f o l l o w i n g = e p q s 0, f o r i = 1, 2, 3 and k,p,q e K, k p q (6.2.13a) ( i i ) Sfcpq i s i n v a r i a n t t o permutation t o k,p,q, where pq is e r e a l a r r a y t o be s p e c i f i e d below. a K The '(6.2.13b) actual elements of e, „ may b e obtained via various kpq theories is e, (e.g., T F M a n d EDQNM calculated from (cf. an a u x i l i a r y kpq 2 related the to the rate turbulence solenoidal waves and field and (Kraichnan, but at without Hendershott Generalizing The result 'W^kpq ^kpq 4pq ^Pq w kpq S M + introduces case topography (1977; the 7 <; ; has hereafter i n HH77 to ' i n w h i c h e. TFM, is of a test exchange of the examined referred the present due t o between test turbulence been field to field with by as study, the Rossby Holloway HH77). we have (6.2.13c) k W (6.2.13d) + p In kpq the 2-D Hu +w +t<) , k problem parts of 1987)). c which advection compressive 1971) . Lesieur, (6 . 2 . 1 3 e ) q 141 A V + H /(k q ) 2 n where g + R (2/(q k ) 2 is a flow Zy a n d Cy, -l/(p q ) 2 2 phenomenological (6.2.12)-(6.2.13), zonal 2 p leading T ^y + Cy (iUy-Vy) x (d/dt) U=r (U-U)-^ (6.2.7) <6.2.13f) 2 order unity. With (6.2.10)-(6.2.11) are closed at the and level to y (d/dt) Cy=lk UHy+ of equations equation (d/dt) Zy=2kUIy-2 yZ + V 2 constant t h e moment tendency -l/(p k ))^, 2 + y, 2* ^ ^ 2f + (6.2.14) ^ ^ y, (6.2.15) kjy/k , (6.2.16) 2 k where 5 . i , Iy (i i s the imaginary = 2,3) o f Cy , are the enstrophy a n d T. ^ transfer (i = and the 1,2) and correlation JC production, to part the <CCn> three and expressions using In the r, invariants distinct <C,hh> for (6.2.12) damping total respectively, in with third order (6.2.10) and T and S and l i s t e d absence the of i n terms subscripts of moments corresponding of (6.2.11) . Zy a n d Cy form <<«>, The detailed are obtained i n t h e Appendix 6A. nonconservative continuous of the motion, potential the system i . e . , enstrophy 142 effects such (6.2.1)-(6.2.4) the total kinetic as Ekman has two energy and = EW (l/L) \(l/2) J 0(1//) a (1/L) [ (1/2) |V*| dQ + • (1/L) \ (1/2) (V 0 + h ; d n + 317, 2 2 \fi i s given length readily 1. from version and the , as i . e . , it of and 1987). two (6.2.17) invariants. and by condition For periodic invariants (6.2.3) c a n b e shown its the that can applying (6.2.4) truncated the system (6.2.18), cell be the (cf., system has the denoted by E ( N ) i conserves it the boundary . (6.2.13)-(6.2.16) sense, (6.2.2) i (N) Q of and Frederiksen, truncated (6.2.18) a n d P. i s t h e d o u b l e existence and (6.2.7), (6.2.17) 2 (6.2.1) theorem Carnevale (6.2.6) 2 The (1/2)1?), 2 by obtained divergence and 2 = where of |v^| dn 2 Moreover, £ conserves ( N <E> ) the and Q in m a n d <Q> closure (see the model statistical Appendix 6B f o r proof) <E> m <£ < N ) > = £ (1/2) Z^/k + 2 (1/2) C7 , (6.2.19) 2 K <Q> = <Q > W =£ (1/2) (Z^ + + 2R^) (6.2.20) fiU, k It is noted invariants, in <E> cancellation (i.e.,T establishing . , T .) of and result <Q>, terms and (6.2.19) in the the and (6.2.20) from some enstrophy correlation that the delicate transfer production 1 / JC 2 f JC (i.e.,5 , 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 neglecting some terms in the transfer functions while V 143 retaining found others may r e s u l t i n the loss o f <E> a n d <Q>, as is i n H87. 6.3 Numerical r e s u l t s and comparison with DNS With use these the closure statistical solve conservation set quasi-stationary For this is We method aside, the start and the for its of the with a proceed to forced-dissipative it suffices to stationary solutions, to study the ensemble brief physical of numerical we objective, appropriate statistics (6.2.6)-(6.2.7). presentation study ( 6 . 2 . 1 4 ) - (6.2.16) the solution to equilibrium. the set though model properties of realizations description parameters, of followed of the by a results. 6.3.1 solution method At the forced-dissipative time tendency thus leaving algebraic The task then of which ( 6 . 2 . 1 4 ) - (6.2.16) as f o r t h e modal of finding the stationary equivalent task: solved and U i n equations equations. this for Z^, statistical to the There one o f exist one i n which f o r ^ 2 ^ , C ^ , C/j- w i t h Z. and C. at equilibrium, (6.2.14)- (6.2.16) vanishes, a nonlinear set of spectra least coupled Z^, values finding the (6.2.14)-(6.2.16) are are first 144 of k roots ways set c ? and f o r for Z^, C two a given the to to and U i s that set accomplish simultaneously 17* a n d r ; obtained f c U. a n d one i n by solving (6.2.14)-(6.2.15) f o r some p r e s c r i b e d v a l u e s o f U and r , with o f U* necessary the value and U then second found from t o achieve (6.2.16). such In t h i s stationary study Z^, we adopt the approach mainly because o f t h e c o n s i d e r a t i o n t h a t the p h y s i c a l b a s i s f o r a s s i g n i n g v a l u e s t o U* i s u n c l e a r whereas the range o f mean v e l o c i t y better documented Crease, used referred transport iterative, t o as BH) with on simulation by B a r t e l l o i n p-plane pseudo-analytically Based literature ( f o r oceans, 1971) . The a c t u a l i s t h e one used (hereafter scalar i n the 1962; Swallow, here f o r g e o p h y s i c a l r e l e v a n t flows i s the e.g., numerical scheme and Holloway (1991) i n their turbulence. study The dissipation of p a s s i v e algorithm i s terms treated ( f o r d e t a i l s see the Appendix i n BH) . t h e same c o n s i d e r a t i o n , the d i r e c t o f (6.2.6) i s c a r r i e d numerical out with the uniform zonal flow U s e t t o some p r e s c r i b e d v a l u e . As i n the c l o s u r e case, the e x t e r n a l momentum forcing U* necessary f o r maintaining the f o r c e d - d i s s i p a t i v e e q u i l i b r i u m may be found from The at dns i s done i n a s p e c t r a l domain t r u n c a t e d k with the i n t e r a c t i o n terms among (6.2.7). isotropically triads (k,p,q) i n max (6.2.6) calculated by the d e a l i a s e d pseudo spectral (Orszag, 1971), the d i s s i p a t i o n terms e v a l u a t e d and t h e time w i t h Robert derivative filter approximated method analytically, by the l e a p f r o g scheme ( f o r d e t a i l s see Ramsden, et a l , 1985). 145 6.3.2 Model parameters All the numerical terms o f t h e model results reported The the choice and is The of mainly 0 the periodic o f t h e model numerical in e 0 length made expressed L = l/2n; => T = 10 s 0 I, are units U = 0.05m/s; where below units, for to cell, some the is set extent, to is convenience 320 k m . arbitrary, of presenting results. model parameters (e.g., 0, r and h ) are set to rms those representative environment. typical value i> k , where near k 4 4 In of the particular, we for planetary 0 is to introduced due t o finite midlatitude at set (3 = truncation. the 6xl0" take weak to = have 0.12 model (or are We t a k e enstrophy = 15, a 1 v^=r + piled up we s e t v 4 2 the phase distribution days) ) , higher a value for dns according and to the k. As for r, we corresponding to computations. {H^} for the isotropic closure topography (H87) H(k) = h /(3.+k) Q at of our numerical {h^} generated spectra damping 1/(100 topography variance with proper damping f o r most The 1 max 5 r 1 1 For k max = ocean 1. 6xl0" m" s" , midlatitude. remove deep (6.3.1) 5 of over {h^} generated (0,2ir). randomly We c h o s e h in satisfying (6.3.1) uniform such that h i s e q u a l t o 4.0 ( o r 4.0xl0~ /s) corresponding t o 6 200m rms bumps i n a 5000m deep ocean. Such floors" may b e c o n s i d e r e d floors. A s f o r U, We c o n s i d e r roughness o f our as representative the "ocean ofreal ocean f o r which U equilibrium v a r i e s f r o m 0 t o 5 ( o r f r o m 0 t o 0.25m/s), a r a n g e a r o u n d t h e typical mean v e l o c i t y f o r flows below t h e main thermocline: 0.05 - 0.1 m/s ( C r e a s e , 1 9 6 2 ) . To the obtain model a sense o f how t h e n u m e r i c a l r e s u l t s parameters, we c o n s i d e r cases where depend on the Ekman d a m p i n g c o e f f i c i e n t r v a r i e s f r o m 0.12 (1/(100 d a y s ) ) t o 1.16 (1/(10 d a y s ) ) , w i t h h =2.0 and 0 = 0.8. max A test of convergence (6.2 .15) - (6.2 .16) with of the s o l u t i o n t o the closure respect t ok i s made for k ma x 15 a n d 30 . T h e d e g r e e s truncations results are show t h a t 96, 748 a n d 2932, relative retained respectively. from k improvement i n these The t e s t =15 t o k max 30 i s insignificant f o r the mentioned above. F o r example, of topographic stress increases range o f U and an o r d e r o f magnitude. than the 1 % but Thus, = max parameters a t U =0.25, t h e r e l a t i v e i s less =5, max o f freedom v model the change cpu time the t r u n c a t i o n k max 15 i s r e t a i n e d f o r t h e s u b s e q u e n t 6.3.3 calculations. 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 o u r n u m e r i c a l r e s u l t s , we first make then the make agree used remarks with both and observations dns the ensemble i n dns (b) a r e on set and based degree data. in on to the closure which these The model closure results observations parameters calculations and which described are in (a) family of to = 0.12, r h =4.0 a n d (3 = 0.8 rms with the values for U specified as (a) T o p o g r a p h i c s t r e s s as a f u n c t i o n The first set of forced-dissipative from 0.0 or 5.0. equilibria method the are The equilibrium second term stationary line. It Cy, is increases flows on the with seen when from above. The is the that U moves moments away which and a stress calculated hand as Zy topographic side results T for a Cy (6.2.14)-(6.2.15) then right involved equilibria stationary obtained described o f U and r calculations statistical needed. function from zero of these using T Fig of 6.1 U until on to the using the in a first the the acting (6.2.16) in varies according of shown U solid rapidly resonant U r is reached monotonically The spectral for the (for deceases resonant of the the complete parameters as behavior used, U approaches may topographic be 148 = large anticipated stress form) U T (see 1.0) . Then, T values. from the (16)-(18) modal in H87 r « tf / (w + £ ) 2 k k i> with k (6.3.2) k a term discussion. which First may recall not concern that us f o r t h e present (U-p/k )k . I t i s thus u. s 2 x JC x p/k Next, note t h a t f o r t h e t o p o g r a p h i c s t r e s s spectrum and dominant = 0. (6.3.1) the parameters t o x i s found contribution (cf. k F i g 6.3 from (b) and F i g 6.5 (6.3.2) and 0 = 0.8 1.0, a maximum used here, those x (b) ) . x reaches that k such U - that 2 reaches f o r (U,k) expected that the |k| (=k) with I t then = 1 follows i t s maximum from U = around i n agreement with t h e numerical o b s e r v a t i o n U =1.0. For t h e i d e n t i c a l performed 1.75, parameters, five ensemble f o r some r e p r e s e n t a t i v e v a l u e s dns runs are o f U = 0.25, 1.0, 2.75 and 3.75, each o f which c o n s i s t s of 5 experiments corresponding to five random realizations {h^}. of q u a n t i t i e s o f i n t e r e s t a r e then c a l c u l a t e d i n each as time-averages points over s t a t i s t i c a l f o r the topographic The experiment s t a t i o n a r y p e r i o d . The data stress i n F i g 6.1 are o b t a i n e d from e v a l u a t i o n o f t h e second term on RHS of (6.2.7). It i s seen theoretical equilibria i n F i g 6.1 values with and that dns U away from t h e agreement data is t h e resonant between the better U for ( = 1.0). those It i s r noted that discrepancy over the occurs, range dns data of U where exhibit 149 the considerable noticeable variations from one that x realization i s sensitive little bumps on t h e Given that given that the the H = o assumed. region, details "ocean" floors. parameters present stress 5000 Even m for and N/m order are Fig p away and is 6.1 at that 1.25cm/s) comparable 2 density U these study ( o r O(10 N/m ), with of currents from 0.25 _ 1 implies environment, ocean s t u d y i n d i c a t e s t h a t T i s not our this ), a v a l u e seawater currents in follows U = mean which arrangement ocean of i t then of of used status typical another, deep | T | = 0.233 ( o r 0.06 to to the midlatitude with mean w i n d depth (6.2.6) equilibrium, current subject of the statistical to the representative of where the = kg/m 10 from 3 the is with ocean are 3 resonant a negligible factor i n t h e momentum b u d g e t . From the ensemble analysis dns for statistical U of e equilibria the stationary . Z (0., 5.0), for which i t U is < and fc and noted U that share the the much of r characters of from Fig also close 6.1 i n the appears cases, since to some a s p e c t s U near those with the of the subresonant . Moreover, spectral be instructive separately. (CJ > CM to may region these behavior superesonant region to . This consider anticipated i s very equilibria from those but be with away f r o m u. two narrow differ U and in located I t thus representative (b) Two r e p r e s e n t a t i v e cases (i) Case 1 : subresonant f l o w (U = 0.25) First, we present subresonant spectral flow Fig in domain. 6.2 (a) physical The flow to show space field shown in f r o m one o f t h e random r e a l i z a t i o n seen in (a) well into developed the the its a strong Three stress Fig x (k) 6.3 and (or wavy They motions in the are (a), R(k) according enstrophy (b) are modal and within to spectra: Z., 6.2 days k s its is which is flow 2 I./k x of part located singled spectra JC side central Z(k), -k has to north. are - is It on t h e west out for topographic R(k), respectively. x. ) the i n the in (a) (6.3.1). correlation band-averaged shown Here, Z(k), obtained from a n d R. = Re 2 JC JC to — (2nk/N(k)) [ t h e summation i s t a k e n the 230 variance (c) , Fig of scale region discussion period, equilibria k - 1 / 2 where = jet on t h e stagnant of zonal JC () (k) t and v o r t i c i t y - t o p o g r a p h y corresponding {C^} at stationary concentrated spectra x(k) =20.0 meridional circulation and a large discussion. in a pronounced south t statistical domain but with at representative before obtained that a band defined by S () k |k|£ k over + 1/2 a l l t h e modes k-l/2s|k|sk+1/2, 151 () with k contained N(k) being the i n t h e kth number o f modes spectra band. r e p r e s e n t e d by t h e symbols ensemble in the same o f C C , hd$/8x versions manner a n d 75c, The c o r r e s p o n d i n g dns a r e o b t a i n e d i n a 5-dns according to the spectral where t h e o v e r b a r d e n o t e s t h e domain average as b e f o r e . It i s seen around k (e.g., Fig = i n F i g 6.3 3, subresonant which 6.2(a)) may account that the has a broad f o r t h e dns dominant k * f l o w s i s on t h e s c a l e from t h e s o l u t i o n s Z(k) (a) t h a t eddy 2. observation activity I t i s also for U s t o (6.2.14)-(6.2.15) peak U in noted that the r peak w i l l U when s h i f t t o some h i g h e r wave number a n d become s h a r p e r moves away U from U towards = 0, which is r characteristic As of the e q u i l i b r i a f o r spectral i n the subresonant regime. decomposition of the topographic stress, 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 b y t h e f l o w on t h e k to a 3. Practically, the contributions wavenumbers a r e n e g l i g i b l e . have a broad peak k approaches . around Note that The c o r r e l a t i o n k = 2 c h a r a c t e r i z e s a l l the subresonant dns above data extent t o which varies from at least t o another 152 R(k) and higher i s seen t o diminish correlation t o be as k spectra c o n f i r m e d by t h e qualitatively, t h e two c a l c u l a t i o n s one q u a n t i t y from flows. o b s e r v a t i o n s a r e seen i n F i g 6.3, 3 t h e V-shaped max The and x agree though with and from each one the other scale to another. Comparison indicates at o f the s o l i d curves with t h a t t h e c l o s u r e p r e d i c t i o n s tend data points t o overestimate low wave numbers, and t h a t t h e d i s c r e p a n c y becomes s m a l l e r as k approaches k f o r x (k) and R(k) but p e r s i s t s f o r Z (k). max It i s important realization t o another, statistical analysis with t o note t h a t t h e d i s c r e p a n c y v a r i e s from one as seen significance of the o f t h e ensemble data indicating comparison, k i s performed, at i n d i v i d u a l that occurs around t h e k where t h e standard d e v i a t i o n of dns data reaches maximum. F o r example, R(k) t h a t t h e standard whereas the t h e worst statistical discrepancy its results i n F i g 6.3. To assess the i t i s noted from i t s maximum at k = 3 d e v i a t i o n take difference between the analysis f o r closure and dns reaches maximum at k = 2. In r e l a t i o n t o p r e v i o u s work, our r e s u l t s f o r T appear t o support the observation (Treguier, 1989) t h a t topographic s t r e s s i s dominated by i t s component at the l a r g e s t scale, though f o r subresonant T i s also the by t h e motions present suggests that slightly s m a l l e r than t h e l a r g e s t one (e.g., F i g 6.3). ( i i ) Case 2: superesonant felt flows available flows study on t h e s c a l e s (U = 2.75) As i n t h e p r e v i o u s case, we f i r s t present a r e p r e s e n t a t i v e superesonant flow i n the p h y s i c a l domain. d i s p l a y s i t s s t r e a m f u n c t i o n , with i t s v o r t i c i t y 153 F i g 6.2 (b) i n F i g 6.4(b) and the topography Fig 6.4(a). As exhibits eddy contrast to around = from k the the 2 the are -Uy vorticity field the case. spectral is to that 1, k = x acts motions, say, (e.g., (c) ) h Fig with 6.3 Fig relative topographic case at k = 1 T = the based sharp on 2 for (cf, R(k) (c) ) , changes as R(k) in point which Fig on In 154 the the (b) ) . in exhibits k. is a found further statistical *. As zonal for mirror the image seen the for comments the case. at is on Z(k) consistent with Second, single near resonant Third, case scale instead subresonant of a flows behavior the the scale, intermediate particular, the difference one monotonic this First, 6.2(b). from to 6.3 seen scale is R(k) out previous contrast Fig in clearly and detailed comes significantly most (k=l) uniform the x(k) (a) ) , essentially a , simply (cf, is flow scale. Z(k) and the motion (b)), it with strong eddy make we Indeed, at case to eddy fields and C i n t h i s going noticeable that and spectra scale a in superesonant (a) by experiment resolved in weak 6.4 this 6.3(a)). (a) also form where mentioned the in k case (cf, observation to V-shaped a current contribution where largest streamfunction of peak dns is, on t h e decomposition, the seen the (cf.Fig shows between the and Instead 6.2(b), 3 resolved 6.5 Fig characterized between largest Fig even U, component in in ensemble equilibrium realized previous or in relation seen activity dns increase randomly (see largest correlation takes observation (cf. closure and dns previous (c) case place Fig at at k 6.4). to the in 1 As individual applies = for k, the current have results on coefficient varies sense the r of model as an f r o m 0.12 h 3.0, a example days)) (1/(100 and 0 =2.0 and as the the of remarks the above have for the the 1.16(1/10 as numerical taken considered same between r we to dns case. dependence parameters, with agreement much o f Topographic s t r e s s as a f u n c t i o n o f To agreement the case where days)), above. The Ekman with results r U= from max these in calculations Fig increased with the 6.6. It is r and the ensemble increase of case. It is those statistical regime, seen 6.4 are the in Fig x seen that closure mean with noted presented r in of seen our test equilibrium dependence of the the is in Fig runs (not flows with r will a good data. 6.6 is Note not U i n the with agreement that the always the shown h e r e ) be stress increases in experiment on x topographic stress prediction the as for that for subresonant opposite to that 6.6 C o n c l u d i n g remarks The closure model i s forced-dissipated used, together statistical with equilibrium dns, topographic flows. A particular directed topographic stress which the 155 study approached momentum-driven to to acts on by attention the the the is uniform zonal component indicates that of the equilibrium the stress flows. exhibits The resonant behavior f o r those e q u i l i b r i a with U near t h e resonant value 6.1). that It equilibria is found undergoes the spectral results [7^. (cf.,F i g dynamics some q u a l i t a t i v e c h a n g e s upon of the crossing C7_(cf. F i g 6.3 a n d F i g 6 . 5 ) . Recall that the periodic c e l l size corresponding t o the grid length size 1 i s s e t t o 320 km, a in a coarse-resolution U c o r r e s p o n d s t o t h e domain ocaen c i r c u l a t i o n model, and t h a t a v e r a g e d mean. T h u s , t h e t o p o g r a p h i c stress as t h e e f f e c t o f t o p o g r a p h i c of subgrid U defined at coarse calculations recognized grids. features With seem t o s t r e n g t h e n f o r some t i m e , this T may point the concern, be viewed s c a l e on t h e o f view, which our has been 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 modeling. 156 Appendix Expressions f o r T 6A Note t h a t 1 s JC S„ . a n d S T 2/JC 2 / JC t h e energy and p o t e n t i a l . 3/Jw T . enstrophy t r a n s f e r i , JC and 5 . i , are the closure follows . 1, k i n (6.2.10) using and (6.2.11), (6.2.12) respectively. I t thus that A A pqkVk V q ^ q + + A qpkVk + and similarly, T . = V L e. 2A. , AA W V k + J qV 2, k - I ^ p q V q j W V k + + Z R kpq p q A A VpVp + VqVq + A. Z.R 'Tcpq kpq\ pkq k q A + 5 triad = Y 2A. 9, \A , Z Z. + A . Z Z, + A, Z Z L ^cpq kpq \ pqk q k qpk p k kpq p q + 2, k t o the four X summations T approximations kqp^Vp qpk k Z C P + + A + A + kqpVp}' . Z.tf pkq k q VP'}' W V k + A. ( + A Z fl kpq p q 6 A < 2 ) qpkVk A + 5 pkq k q A C C + Vp k p}' C 3 . k " I Wkpq{ qk q k A P C C C + W V k + w h e r e 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, (6.2.9) a n d (6.2.12d) i n t h e t e x t . 157 WV*q}' Tcpq a n d A, kpq ( 6 A ' 3 ) < 6 A ' 4 ) a r e g i v e n by Appendix 6B The 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 o f (6.2.19) proof stringent conservation detailed and (6.2.20) lies o n somewhat p r o p e r t i e s o f t h e model, 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 more viz., the enstrophy, which a r e summarized i n Let k,p,q e K form a triad Detailed conservation k +p which + q = Let 0. (klp^; T and T e r " e,k ( T L K defined I S + enstrophy transfer q .k + R 2,k ^ + 2.K the densities *.fk|p,q;, (S be by (6B.1) A T fk|p,q; for q symmetrized energy and potential at wavenumber k, (k,p,<i) 3,k^ 5 + S I Vk| ,q,, ( 6 B P ' 2 ) A where T 1 f T JC 2 respectively, , v r JC e <Tc|p,q; + e dc|p,q; + T q ' Proof: T e 1 and S . JC 3 f are given T r for flc|p,q; (6A. 1) - (6A. 4), as in the (6.2.6). closure Then, model satisfy (-p|q,k; + T l q | k , ; = 0, (6B.3) P ' e 1 {-p|q,k; + T (q|k,p; = 0. q Since by JC with the symbol £ defined (6.2.13)-(6.2.16) T . 2f and T (k|p,q; S (6B.4) q t h e method u s e d t h e one f o r e s t a b l i s h i n g t o prove (6B.4) parallels to (6B.3), t h e p r o o f g i v e n here 158 i s thus made explicitly symmetrize f o r the energy t h e terms (6A.2) w.r.t. T' T(k|p,q) = p, q z z ) + rJ T (k| ,q) (ZZ, in (6B.3). <cf.(6B.l)) k using P (ZR) <k|p,q) + P T< (k|p,q) , RR+II) (6B.5) Z Z + A 2 2 2 ^ (k| ,q) = 0 P {\ Vq k^q k p q Z p q + A + A + A + A kpq pqk V k + A kpq pkq k*q + A A Z kpq qpk p*k A Z + = e A k | p / q ) A _ e k A. k p q A Z p q kqp qkpW A kqp pqk q*k A A Z jR { A p k q p k q kqpA,kqpZ q Rp + A. A. q A p q k A + ^ 159 ZRX/K , 2 kqp kpq p qj k q p A q k p W V p } ^ ' + k Z H Z , Z, R kqpA pkq kq A. JA P A + A. k q k q p^q P A kpqA.kpqZpR q + k p q kqp qpk V k A, kpq kqp q p ZH) A + A. + A. rj (k|p,q) kqp kqp VP . Z, R kpqA qkp kp + (6B. 6 ) W W V P + kpq\pqVq + A. ( and (k| ,q) +K—K—Z~Z^/k ^kpqAkpq, Zp2q] y(RR+n) (6A.1) we = Vq{VqW q k \qp qpk p k P ZR, First, :Mk|p,q) t o get fk| ,q) + r ZH) argument + i i ; q k Z H k (6B.7) p (6 B ' > 8 where e fcpq' A kpq (6.2.12d) and establish (6B.3), any a n ^kpq' d (6.2.13), e t ' c a r given e respectively. i t suffices one o f t h e components by Next, t o show t h a t (6.2.9), note that (6B.3) h o l d s f o r i n T (k|p,q) . To d e m o n s t r a t e e to that 1 (ZH ) the latter i s true, (k|p,q) we t a k e a s an e x a m p l e a n d show t h a t r (ZH) (k|p,q) + T k Interchanging Similarly, T < Z H ) the (q|k,p) . e <ZH) (p|q,k) + T p with = 0. . i n (6B.8) K operation: Omitting (q|k,p) < = = > t h e lengthy (6B.10) (p|q, k) . yields r^ in (6B.8) q algebra, ZH) we gives simply point 1 out that (6B.10) follows (6B.10) a n d c a n c e l i n g after **pq- " A qpk - a n c detailed leading to conservation c l o s u r e model * ^kpq' with t h e two into t h e use of t h e i ' * ' e W * * * t h e symmetric property arguments introducing t h e terms p a i r w i s e symmetric p r o p e r t i e s o f and ( Z H ) ' (6B 11) o f 0 pq K (6B.10), of total (6.2.13)-(6.2.16). 160 ( - (6.2.13)). cf i t thus energy follows (6B.3) holds With t h e that the for the Now, we s h o w t h a t (6.2.19) a n d (6.2.20) detailed Invariants nonlinear + 2 the properties consequence (6B.3) of the and (6B.4). L e t <E> and <Q> given o f t h e motion. Then <E> and <Q> are conserved transfer r simply <E> a n d <Q> g i v e n b y we h a v e (6.2.19)-(6.2.20) . T are conservation Specifically, I < i.k t h e two i n v a r i a n t s ,k>/* in (6.2.13)-(6.2.16) " 2 by by the in the sense ° ' < 6 B ' 1 2 ) k k Moreover, <E> and <Q> are the invariants absence of the Proof: dissipation. The closure potential of the model in the versions enstrophy (d/dt) <E> = r(U of the total equations - U)U - IV V * 2 + follow < 1 1/2) kinetic from ( 6 . 2 .1'4) - ( 6 . 2 . 1 6 ) I,K (T energy and T + 2.K )/K2 (6B.14) ' (d/dt)<Q> = - [ v^(Z^ + R^) k + (1/2) I { (T K + LT T 2,K> + 2 ' R e < S 2 ,k + S 3.k>}' k (6B.15) In terms o f T (k|p,q) e 1 andT (klp,q), q ' 161 (6B.14) a n d (6B.15) read (d/dt)<E> = r(U - U)U - I VyZy/k + (l/2)£ [ [ T (k|p,q) k SCk+p+q), k pq (6B.16) (d/dt) <Q> = - I Vy(Zy + Ry) + (1/2)^ [ [ k T ( k | p , q ) 5 Ck+p+q) , q k pq (6B.17) where t h e summations are over k+p+q = 0 o t h e r w i s e when indices i n resulting i t vanishes. ( 6 B . 14) - ( 6 B . 15) : equations a l l k , p , q € K a n d 8Ck+p+q) to Interchanging k <=> p a n d k <=» q , (6B.16) and = 1 t h e dummy and adding the (6B.17), respectively, yield (d/dt)<E> = r(U - U)U - £ v ^ yJ z k k + (1/6) £ £ [ | r ( k | , q ) e P + T <p|q,k) e + T ( q | k , p ) | 5 Ck+p+q;, e k p q (6B.18) (d/dt) <Q> = - [ Vy(Zy + Ry) k + (1/6) H I |r (k| q) q P/ + T (p|q,k) q + r ( q | k , p ) | 5 Ck+p+q; , q k p q (6B.19) The conditions comparing then the (6B.18) invoking (6B.13) into invariants system, (6B.12) which a n d (6B.13) with (6B.3) (6B.14) (6B.14) and a n d (6B.19) (6B.4). a n d (6B.15) o f the motion completes immediately the proof.- 162 with Introducing shows when follow that after (6B.15) (6B.12) and and <E> a n d <Q> a r e no d i s s i p a t i o n is i n the cn cn UJ cc t— cn -0.50 r -0.43 - -0.36 - -0.29 - x CL <c cc CD O -0.21 CL O -0.14 - -0.07 0.00 0.00 1.00 DOMAIN Fig 6.1 parameters Topographic are (h stress ,0,r) = 2.00 AVERAGED x as (4.0, 3.00 4.00 5.00 MEAN V E L O C I T Y U a of function 0.8, 0.12). The L7. The solid max line i s f o r the closure results and symbols d a t a . The resonant p o i n t corresponds t o U 163 f o r the =1.0. dns Fig 6.2 Streamf u n c t i o n representative flows at (tfr t=20 = -Uy+t) (or parameters as t h e same as i n F i g 6 . 1 . flow 2.75. with U - 0.25; ts230 for days), with ( a ) : t h e subresonant (b) : t h e superresonant flow with The dash contours are f o r n e g a t i v e v a l u e s . 164 the two U = (a) 0.2E+01 F 0.2E+00 - M >X CL o cc I— to 0.2E-01 ^IE-01 0.1E+02 3 (b) to to LU CC I— to o CL o cc z o (c) LU CC cc o o -1 .60 0.1E+01 0.1E+02 VAVENUMBER K Fig 6."3 Enstrophy vorticity-topography for t h e subresonant (a), t o p o g r a p h i c correlation stress (c) s p e c t r a (b) and (solid lines) flow case U =0.25, with parameters as the same as i n F i g 6.1. The symbols ensemble d a t a . 165 are f o r the f i v e dns Fig 6.4 Vorticity superresonant flow (a) shown and in Fig 166 Topography 6.2(b) (b) for the (a.) M 0.2E+02 Er 0.2E+01 r 0.2E+00 r >X QO CC t1/1 z LU 0.2E-01 (b) LO LO LU cc t— CO o 0O cc z o t- < _J LU CC tr o o -6.00 0.1E+01 0.1E+02 WAVENUMBER F i g 6.5 The same as i n F i q 6.3 167 but K for U = 2.75 Fig 6.6 parameters Topographic are (h stress ,0,C7) = T as a (6.2.0, function 0.8, 3.0). of The r. The solid max line i s f o r the c l o s u r e r e s u l t s and data. 168 symbols f o r the dns CHAPTER 7 We CONCLUSIONS have analyzed the problem o f temporal p e r t u r b a t i o n s t o the hydrodynamical system both i n f i n i t e and f i n i t e dimensional analysis (1.2.1)-(1.2.3) i n f u n c t i o n spaces, both d e t e r m i n i s t i c and p r o b a b i l i s t i c In doing e v o l u t i o n of and i n sense. so, we have developed an a l g o r i t h m f o r the g l o b a l o f the system (1.2.1)-(1.2.3) as opposed t o the u s u a l modal a n a l y s i s . The g l o b a l f e a t u r e a r i s e s from the f a c t that the analysis kinematically modal admissible analysis disturbances with nonlinear theory of finite global difficulties Its which in disturbances, takes care form, optimal The as of space of a l l opposed t o the only infinitesimal or as opposed disturbances, of the above of the a n a l y s i s allows t o weakly consisting space. This us t o overcome s e v e r a l found i n t h e u s u a l modal a n a l y s i s . algorithm to global for stability separating growing p e r t u r b a t i o n s from those without 3.1). a which i s r e s t r i c t e d t o a subset, application systematic conducted the modal amplitude nature is same algorithm has nonmodal p e r t u r b a t i o n 2.2 and 3.2). also has flows yielded with a initial ( c f . theorem 2.1 and been t o a given used flow to find an ( c f . theorem The growth r a t e o f such a p e r t u r b a t i o n i s shown 169 to be the least application condition to growth to rate, stability such and existence the study of paradoxes explosive supercritical such as aspects. disturbances to Its necessary equilibria. modal the rate. a to cause in Its nonmodal comparison of the underlying many have resolved subcritical instability there analysis of any exists around the initialized zero flow after some in is without limited analysis flows weakly can flows a stability a global in many analysis important have be have clearly systematically problems. asymptotic equilibria finite, origin in that examples this exhibiting 170 of behavior of subcritical a specific suggesting numerical global for for system here hyperbolic a) neighborhood the specific to established: decay to equivalently of made developed how local disturbance of the or results, these implemented to regime,' led disturbances hydrodynamical Also, our of general one demonstrated In with development the a the growth has relation analysis, growth multiple uncovered illustrations strengthened of of the flows. Numerical analysis has modal as of conjunction measures, on equilibria the in shortcomings bound multiple for application upper the flow though R M such neighborhood Orr's of nonmodal system, we have outside of MGS perhaps that will temporal a small, nonmodal ultimately amplification (theorem finite, origin this c) 5.1); b) though of R for perhaps such M any that the related space nature to the of to probabilistic zonal flows regard to over the remain be to appreciation that answered, flow-topography the detailed which holds dissipation i n the system for systematical these quantities accessible results to for significance large-scale among other of this ocean is of circulation in parameter numerical For the of has well of led the former, we and potential presence an latter, a of means of area the the not numerical indicate momentum strengthen an the transfer clearly to about provides triads, the questions as the special underlying many energy overall and and For uniform with While as statistical initial nonlinear stress in is done A p p e n d i x 6B), approaches. force from itself interacting topographical disturbances The exercise regardless investigation and predictions. force. an in 5.2); point interaction. (cf. theorem a the initialized disturbance conservation enstrophy, to 5.3). arising between method adjacent forced-dissipated such the exists nonmodal topography resulting about of of random the t-xn ( c f . theorem flows correlation and as nonhyperbolic study there disturbance confirm these perturbed topography note of (cf. seen of nonmodal persistent nature are equilibrium problem the flow, neighborhood persists interest experiments Our of small, a neighborhood w i l l supercritical budget need the of to parameterize There work. i t i n g e n e r a l c i r c u l a t i o n models. are several possible First, the (1.2.1)-(1.2.3) as obvious a model extensions from the present limitation of the system to study geophysically relevant flows i s the n e g l e c t of b a r o c l i n i c i t y , t o be a primary source oceans s i n c e Charney a global of i n s t a b i l i t y (1947) and Eady analysis for a system which has i n the (1949). with been known atmosphere It i s clear that this effect a l l o w s one t o draw c o n c l u s i o n s on the o r i g i n , such phenomena as synoptical scale the w e s t e r l y winds and mesoscale eddies expected Joseph, from 1966) the that work the on the included development decay of d i s t u r b a n c e s from model s t u d i e s i n c l o s e r to and and connection disturbances in i n the oceans. I t i s Boussinesq equation physical principle (cf. ( c f . lemma 2.2) which a l l o w s f o r the present a n a l y s i s w i l l h o l d while i t w i l l take a more conversion general between plus p o t e n t i a l t o the t o t a l form the involving total energy the (i.e., energy) i n a b a s i c flow and kinetic in a the energy disturbance extension from the p r e s e n t b a r o t r o p i c system t o a b a r o c l i n i c system i s in principle I t thus of f o l l o w s t h a t an ensured dissipation. ratio while the analysis in the case w i l l be f a r more i n v o l v e d than the present Second, (Arnol'd an 1983; orbital stability Guckenheimer case. a n a l y s i s v i a Floquet & Holmes, 172 baroclinic 1983) is a theory necessary next step to quasi-periodic flows. This from the models to ones and as to bring see if it enhances of these Gulf Holloway's (Holloway, Stream. proposal 1991), parameterize grid-scale the in it This for may effect lead of complete Fig it into e.g., in Unprejudiced 173 the repeated the significant wind stress, ocean w.r.t. some overshoot conjunction practical an circulation latitude ocean subgrid-scale motions. turbulent performance some to or appears general effort, to flows 5.7. and their models, periodic chaotic stress between is to to topographic task of the us shown comparison failures from eventually allow the specific the transition bifurcation rough important the would supercritical Finally, study with circulation scheme topography on to the REFERENCES Arnol'd, V . T . , ordinary York. Ascher, U . M . , R.M.M.Matheil solution Hall, Bartello, in Boyd, Geometrical methods in the theory of differential equations, S p r i n g e r - V e r l a g , New 1983: 595 for pp. and R . D . R u s s e l l , ordinary differential 1988: equations. Numerical Prentice P . and H o l l o w a y , G . , 1991: Passive scalar transport 0 - p l a n e t u r b u l e n c e . J. Fluid Mech., 223, 521-536. J.P., flow 1983: The 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 Couette with beta e f f e c t . J. Atmos. Sci.,40, 2304-2308. Brown, A . P . , and K. Stewartson, 1980: On t h e a l g e b r a i c of disturbances i n a s t r a t i f i e d shear flow. J. Mech.,100, 811-816. Card, P.A., and A.Barcilon, 1982: The p r o b l e m w i t h a l o w e r Ekman l a y e r . 2128-2137. Carnevale, G.F. and stability and t o p o g r a p h y . J. Case, K . M . , 1960: Phys. Fluids, J. Charney Charney, J.G., 1947: The dynamics 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. plane of Fluid stability Atmos. 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Nonlinear stability and statistical equilibrium of forced and dissipated flow Zou, Jieping 1991
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Title | Nonlinear stability and statistical equilibrium of forced and dissipated flow |
Creator |
Zou, Jieping |
Publisher | University of British Columbia |
Date Issued | 1991 |
Description | A global analysis for the hydrodynamical system defined for a homogeneous, incompressible layer of fluid on the β-plane is performed in both infinite and finite function space. Its application to global stability has yielded an algorithm for characterizing flows based on the existence of initially growing perturbations as opposed to the normal mode analysis; its application to the search for optimal initial perturbations has led to the least upper bound of energy growth rate; its application to multiple equilibria has given rise to a necessary condition for their existence; its application to the study of the relationship of modal to nonmodal growth rates has uncovered the cause underlying many aspects of the limitation of the modal stability analysis including the failure to predict transient growth of disturbances in stable flows and the underestimation of the intensity of initial development of instability in unstable flows. Numerical illustrations made for some specific flows have strengthened the general results, suggesting that a stability analysis of a hydrodynamical system without a global analysis is likely to be limited in many important aspects. The local analysis of asymptotic behavior of nonmodal disturbances to hyperbolic equilibria of the system have established: a) for any subcritical flow outside of monotonic, global stability regime, there exists a finite neighborhood around the origin of RM such that a disturbance initialized in this neighborhood will ultimately decay to zero after it exhibits Orr's temporal amplification; b) for any supercritical flow, there exists a finite neighborhood adjacent to the origin of RM such that a disturbance initialized in this neighborhood will persist as t→∞; and c) the nature of the persistent disturbances is related to the nature of the nonhyperbolic point in parameter space of interest. The numerical experiments are seen to confirm these predictions. Closure modeling of forced-dissipated statistical equilibrium of perturbed flows arising from initially uniform zonal flows over random topography is done with special regard to the correlation between disturbance and underlying topography and the resulting stress. Such an exercise has led to, on one hand, the numerical results for topographical stress suggesting clearly the significance of this force in overall momentum budget of large scale ocean circulations. On the other hand, it has led to an appreciation that the detailed conservation of energy and potential enstrophy, which holds regardless of the presence of dissipation in the system, provides a means for systematic investigation of nonlinear transfer of the these quantities among interacting triads, an area not accessible to other approaches. |
Subject |
Hydrodynamics |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-03-07 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0053236 |
URI | http://hdl.handle.net/2429/32124 |
Degree |
Doctor of Philosophy - PhD |
Program |
Oceanography |
Affiliation |
Science, Faculty of Earth, Ocean and Atmospheric Sciences, Department of |
Degree Grantor | University of British Columbia |
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UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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