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Double Hopf bifurcations in two geophysical fluid dynamics models Lewis, Gregory M. 2000

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DOUBLE HOPF BIFURCATIONS IN TWO GEOPHYSICAL FLUID DYNAMICS MODELS ' by G R E G O R Y M . L E W I S B . S c . (Phys ics) M c G i l l Un ive r s i t y , 1991 M . S c . (Phys ics ) M c G i l l Un ive r s i t y , 1993 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S D e p a r t m e n t of M a t h e m a t i c s Ins t i tu te o f A p p l i e d M a t h e m a t i c s W e accept th is thesis as c o n f o r m i n g to the requi red s t a n d a r d T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A • A p r i l 2000 © G r e g o r y M . L e w i s , 2000 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract W e ana lyze the double H o p f b i furca t ions w h i c h occur i n two geophys ica l f l u id d y n a m i c s mode l s : (1) a two- layer quasigeost rophic p o t e n t i a l v o r t i c i t y m o d e l w i t h forc ing a n d (2) a m a t h e m a t i c a l m o d e l of the dif ferent ia l ly heated r o t a t i n g annulus exper iment . T h e b i fu rca-t ions occur at the t r a n s i t i o n between a x i s y m m e t r i c s teady solu t ions a n d n o n - a x i s y m m e t r i c t r a v e l l i n g waves. F o r b o t h models , the results ind ica te tha t , close to the t r ans i t i on , there are regions i n paramete r space where there are m u l t i p l e s table waves. Hysteres is o f these waves is p red ic ted . F o r each m o d e l , center m a n i f o l d r educ t ion a n d n o r m a l f o r m theory are used to deduce the l o c a l behav iour o f the fu l l sys tem of p a r t i a l different ial equat ions f rom a l o w - d i m e n s i o n a l sys tem of o r d i n a r y differential equat ions . In each case, i t is not possible to compute the relevant eigenvalues a n d eigenfunct ions a n a l y t i c a l l y . Therefore , the l inear pa r t of the equat ions is d i sc re t ized a n d the eigenvalues a n d eigenfunct ions are a p p r o x i m a t e d f rom the resu l t ing m a t r i x eigenvalue p r o b l e m . H o w -ever, the p ro j ec t i on onto the center m a n i f o l d a n d r educ t ion to n o r m a l f o r m c a n be done a n a l y t i c a l l y . T h u s , a c o m b i n a t i o n of a n a l y t i c a l a n d n u m e r i c a l me thods are used to o b t a i n n u m e r i c a l a p p r o x i m a t i o n s of the n o r m a l fo rm coefficients, f rom w h i c h the d y n a m i c s are deduced . T h e first m o d e l differs f rom those p rev ious ly s tud ied w i t h b i fu rca t ion analys is since i t suppor t s a s teady so lu t ion w h i c h varies non l inea r ly w i t h l a t i t ude . T h e results i nd ica t e tha t the fo rc ing does not q u a l i t a t i v e l y change the behav iour . However , the f o r m of the b i f u r c a t i n g so lu t i on is affected. T h e second m o d e l uses the Navie r -S tokes equat ions i n the Bouss inesq a p p r o x i m a t i o n , i n c y l i n d r i c a l geometry. In a d d i t i o n to the double H o p f b i fu rca t ion analys is , a de ta i l ed a x i s y m -m e t r i c to n o n - a x i s y m m e t r i c t r a n s i t i o n curve is p roduced f rom the c o m p u t e d eigenvalues. A quan t i t a t i ve c o m p a r i s o n w i t h expe r imen ta l d a t a finds tha t the c o m p u t e d t r a n s i t i o n curve, c r i t i c a l wave number s a n d dr i f t rates o f the b i fu rca t ing waves are reasonably accura te . T h i s ind ica tes tha t the ana lys is , as we l l as the a p p r o x i m a t i o n s w h i c h are made , are v a l i d . i i C o n t e n t s Abstract i i Table of Contents i i i List of Figures v List of Tables v i i 1 Introduction 1 1.1 G e n e r a l c i r c u l a t i o n 2 1.2 E x p e r i m e n t s 5 1.3 A n a l y s i s 7 1.3.1 A d y n a m i c a l systems approach 9 1.4 O u t l i n e of thesis 14 2 The Models 16 2.1 T h e general equat ions of m o t i o n 17 2.1.1 T h e Rossby number : large scale flow 17 2.1.2 T h e equat ions of m o t i o n 18 2.1.3 T h e C o r i o h s a n d centr i fugal forces 21 2.1.4 B a r o c l i n i c i n s t a b i l i t y 24 2.1.5 T h e general equat ions of m o t i o n : s u m m a r y 26 2.2 T h e two-layer quas igeost rophic po t en t i a l v o r t i c i t y equat ions 26 2.3 T h e di f ferent ia l ly heated r o t a t i n g annulus exper iment 31 2.3.1 T h e exper iments 32 2.3.2 T h e m a t h e m a t i c a l m o d e l 35 3 Bifurcations in the two-layer model 40 3.1 T h e two-layer m o d e l a n d l inear s t a b i l i t y 40 3.2 H o p f b i fu rca t ion 46 i i i 3.2.1 T h e center eigenspace 46 3.2.2 T h e center m a n i f o l d 47 3.2.3 T h e n o r m a l fo rm 49 3.2.4 C o m p u t a t i o n a n d Resu l t s 51 3.3 D o u b l e H o p f b i fu r ca t i on i n the two- layer m o d e l 57 4 Double Hopf bifurcation in the differentially heated rotating annulus 72 4.1 T h e analys is 74 4.1.1 S teady a x i s y m m e t r i c so lu t i on 75 4.1.2 T h e p e r t u r b a t i o n equat ions 76 4.1.3 T h e eigenvalue p r o b l e m 77 4.1.4 T h e adjoint eigenvalue p r o b l e m 78 4.1.5 N o r m a l f o r m coefficients 79 4.2 N u m e r i c s 82 4.2.1 T h e o rde r ing of the unknowns 82 4.2.2 T h e mesh: non -un i fo rm spac ing 84 4.2.3 S o l u t i o n techniques 85 4.2.4 D i s c u s s i o n o f convergence 87 4.3 Resu l t s 88 4.3.1 T h e a x i s y m m e t r i c so lu t ion 89 4.3.2 N e u t r a l s t a b i l i t y a n d t r a n s i t i o n curves 90 4.3.3 D o u b l e H o p f n o r m a l f o r m coefficients: hysteresis 91 4.3.4 T h e eigenfunct ions: b i fu r ca t i ng wave fo rm 94 5 Conclusion 101 Bibliography 104 A Center manifolds for partial differential equations 109 B Outline of derivation of two-layer quasigeostrophic potential vorticity equations H I C Normal form formulae 118 i v L i s t o f F i g u r e s 1.1 The absorption of radiation on the earth's surface 3 1.2 A possible planetary convection cell 4 1.3 The differentially heated rotating annulus experiment 6 1.4 A schematic diagram depicting general experimental results for the differentially heated rotating annulus experiment 8 2.1 The Coriolis force 21 2.2 The centripetal force 23 2.3 Spherical coordinates 24 2.4 The mechanism of baroclinic instability 25 2.5 A two-layer fluid in a periodic channel of width L and height 2D 27 2.6 The fluid velocity ui in the first layer in the longitudinal (x) direction 32 2.7 The differentially heated rotating annulus with circular cylindrical coordinates 36 3.1 The neutral stability curves for the two-layer model for the azimuthal wave numbers m = 2,3,4 43 3.2 A schematic diagram of a center manifold 48 3.3 The form of xi the bifurcating stream function in the upper layer 68 3.4 A torus 69 3.5 The two-dimensional bifurcation diagram 70 3.6 The one-dimensional bifurcation diagram depicting the bifurcation observed along the path indicated with the dotted line in Figure 3.5 71 3.7 The hysteresis loop for the path indicated with the dotted line in Figure 3.5 71 4.1 The transformation from a uniform to a non-uniform grid 85 4.2 The axisymmetric solution for the differentially heated rotating annulus 89 4.3 Neutral stability curves for the wave numbers m = 3 to m = 8 96 4.4 Transition curves for theory and experiment delineating the axisymmetric from the non-axisymmetric regimes 97 4.5 Neutral stability curves: upper transition 98 v 4.6 Theoretical and experimental drift rates of bifurcating waves close to bifurcation point. 98 4.7 Theoretical transition curve between the axisymmetric and the non-axisymmetric regimes with the double Hopf bifurcation points (critical wave number transitions) 99 4.8 An example of the radial and vertical dependence of an eigenfunction 100 B . l A two-layer fluid in a periodic channel of width L and height 2D 112 vi L i s t o f T a b l e s 3.1 Numerical results for two-layer model 56 3.2 Numerical results for the mi = 2, m 2 = 3 double Hopf bifurcation point denoted by p23 on Figure 3.1 63 3.3 Numerical results for the mi = 3, m 2 = 4 double Hopf bifurcation point denoted by pzi on Figure 3.1 63 3.4 Numerical results for the mi = 4, ni2 = 5 double Hopf bifurcation point 64 3.5 Numerical results for the mi = 5, m 2 = 6 double Hopf bifurcation point 64 3.6 Estimation of the order of the numerical approximation 67 4.1 The annulus geometry and fluid properties used in the analysis 88 4.2 Numerical results for the mi = 5, m 2 = 6 double Hopf bifurcation point 92 4.3 Numerical results for the mi = 6, m 2 = 7 double Hopf bifurcation point 92 4.4 Numerical results for the mi = 7, m 2 = 8 double Hopf bifurcation point 92 4.5 Numerical results for the mi = 8, m 2 = 7 double Hopf bifurcation point 93 4.6 Numerical results for the mi = 7, m 2 = 6 double Hopf bifurcation point 93 4.7 Numerical results for the mi = 6, m 2 = 5 double Hopf bifurcation point 93 4.8 Numerical results for the mi = 3, m 2 = 4 double Hopf bifurcation point 94 4.9 Numerical results for the mi = 4, m 2 = 5 double Hopf bifurcation point 94 vii C h a p t e r 1 I n t r o d u c t i o n I n t h i s thesis, two m a t h e m a t i c a l mode l s f rom geophys ica l fluid d y n a m i c s are s t ud i ed . O n e poss ib le de f in i t i on o f geophys ica l fluid d y n a m i c s is tha t i t is the s t u d y o f r o t a t i n g a n d s t ra t i f i ed fluids, or m o r e precisely, fluid flows i n w h i c h r o t a t i o n a n d s t r a t i f i c a t i o n p l a y a n i m p o r t a n t role. T h e goa l of s t u d y i n g such fluid systems is the u n d e r s t a n d i n g o f the large scale d y n a m i c s o f the a tmosphere a n d ocean. In pa r t i cu l a r , i t is a search for the p h y s i c a l processes or mechan i sms w h i c h are most i m p o r t a n t to the charac ter of the large scale d y n a m i c s . T h u s perhaps a more appropr i a t e def in i t ion was pu t fo rward by P e d l o s k y [49]: " G e o p h y s i c a l fluid d y n a m i c s is the subject whose concerns are the s t u d y of the fundamen ta l d y n a m i c a l concepts essential to an unde r s t and ing o f the a tmosphere a n d ocean." T h e s t a r t i n g po in t o f th i s thesis is a sys tem of evo lu t i ona ry equat ions w h i c h descr ibes the d y n a m i c s of a fluid subject to r o t a t i o n . T h e premise is tha t a n u n d e r s t a n d i n g of geophys ica l flows c a n be ga ined by a careful s t u d y o f these equat ions (i.e. t ha t the equat ions accu ra t e ly represent the d y n a m i c s ) . T h i s is not an obvious po in t , s ince the n u m b e r of factors w h i c h con t r i bu t e to the d y n a m i c s of the a tmosphere and ocean is enormous . T h e y encompass a l l areas o f science, f rom physics to chemis t ry to b io logy. If a set of equat ions w h i c h i n c o r p o r a t e d a l l the poss ible factors c o u l d be w r i t t e n d o w n , the i r c o m p l e x i t y w o u l d be so great t ha t ana lys i s o f any k i n d w o u l d be imposs ib l e or at least unrevea l ing . It is therefore necessary to res t r ic t ou r a t t en t ion to t e m p o r a l a n d spa t i a l scales o n w h i c h i t is poss ib le to argue t h a t m a n y or mos t o f the factors or processes are u n i m p o r t a n t . T h e hope is t ha t the s i m p l i f i e d equat ions tha t we s tudy cap ture the character o f the rea l sys t em w h i l e b e i n g t r ac t ab le for m a t h e m a t i c a l ana lys is . I f we w i s h to accura te ly represent the rea l sys tem, s impl i f i ca t ions mus t be m a d e sys t em-a t i c a l l y a n d w i t h c a u t i o n . O n e poss ible m e t h o d of s i m p l i f i c a t i o n is c a l l ed s ca l i ng ana lys i s . Essen t i a l ly , a n e s t i m a t i o n is m a d e o f the magn i tudes o f the t e rms i n the d y n a m i c a l equa-t ions . T h e processes associa ted w i t h the te rms w i t h s m a l l m a g n i t u d e are j u d g e d to have l i t t l e effect a n d are therefore neglected, thus s i m p l i f y i n g the equat ions . A n o t h e r m e t h o d 1 of s i m p l i f i c a t i o n is to res t r ic t the s t udy to mo t ions w h i c h can be observed i n con t ro l l ed l a b o r a t o r y exper iments . In th is s i t u a t i o n , there are s ign i f ican t ly less factors to consider . W e s t u d y two geophys ica l fluid d y n a m i c s mode ls : (1) a m o d e l w h i c h is de r ived u s i n g s c a l i n g ana lys i s a n d (2) a m a t h e m a t i c a l m o d e l of a con t ro l l ed l a b o r a t o r y expe r imen t . T h e sys tems o f equat ions w h i c h resul t f rom these s impl i f i ca t ions are a n a l y z e d m a t h e m a t i c a l l y . W h e n n u m e r i c a l a p p r o x i m a t i o n s are made d u r i n g the ana lys is , we a t t e m p t to j u s t i fy the i r accuracy. In th is way, i t m a y be a rgued tha t the discrepancies between the ana lys i s a n d the rea l flow are no t an ar t i fac t of the n u m e r i c a l a p p r o x i m a t i o n s . F u r t h e r m o r e , s p e c u l a t i o n is no t used i n the ana lys i s . T h a t is , the d y n a m i c s are p red ic t ed a n d not p resumed . Specif-i ca l ly , a b i f u r c a t i o n analys is is per formed. T h e power o f th is m e t h o d is tha t i t finds ' new ' s teady ( t ime- independent ) or t i m e - p e r i o d i c so lu t ions o f the d y n a m i c a l equa t ions . T h e a n a l -ysis also gives the s t a b i l i t y o f each s o l u t i o n a n d the l o c a t i o n i n the pa rame te r space where t r ans i t i ons between so lu t i on types occur . If the sys tem ' l ives ' close to these t r ans i t i ons , th i s m e t h o d w i l l be m a x i m a l l y useful. However , even i f i t does not , the ana lys i s w i l l l ead to a be t te r u n d e r s t a n d i n g o f the d y n a m i c a l equat ions w h i c h , as m e n t i o n e d before, we assume represents the rea l flow. 1.1 General circulation W h e n rays of solar r a d i a t i o n reach the E a r t h , they are essent ia l ly pa r a l l e l . S ince the surface of the E a r t h at the equator is less ob l ique to the sun, more r a d i a t i o n is absorbed per u n i t a rea here t h a n at the poles. See F i g u r e 1.1. T h i s causes the dif ferent ia l h e a t i n g between the equator a n d the poles w h i c h , aside f rom l u n a r t ides, is u l t i m a t e l y respons ib le for a l l m o t i o n s i n the a tmosphere a n d ocean [49]. C o n s i d e r for a m o m e n t the a tmosphere o n a n o n - r o t a t i n g p lane t where the di f ferent ia l h e a t i n g is a x i s y m m e t r i c ( invar iant under r o t a t i o n abou t the p o l a r ax i s ) . A s a ca r i ca tu re , assume tha t there is a heat source at the equator a n d a heat s ink at the poles. T h e h e a t i n g at the equator w o u l d cause the s u r r o u n d i n g a i r to become less dense a n d therefore rise, w h i l e the c o o l i n g at the poles w o u l d cause the a i r to become more dense a n d therefore s ink . C o n t i n u i t y o f mass w o u l d close the c i r cu i t : the a i r aloft w o u l d move t o w a r d the poles a n d the a i r a l o n g the g r o u n d w o u l d t end to the equator . See F i g u r e 1.2. A l t e r n a t i v e l y , one c o u l d i m a g i n e t ha t the different ia l hea t ing o n the surface w o u l d cause a c o r r e s p o n d i n g n o r t h -s o u t h gradien t o f the dens i ty of the a i r a n d therefore a n o r t h - s o u t h pressure gradien t a l o n g the surface w h i c h induces equa to r -ward m o t i o n . T h e convergence at the equator causes a i r t o rise, w h i c h i n t u r n causes a pressure gradient aloft i n d u c i n g p o l e - w a r d m o t i o n . T h e a i r s inks at the poles c los ing the c i r cu i t . In ei ther v i ew, the resul t is an a x i s y m m e t r i c convec t ion 2 S u n ' s r a y s F i g u r e 1 . 1 : The absorption of radiation per unit area, 5, on the earth's surface, is proportional to the surface area of Aj, which is the surface which results from the projection of S onto a plane perpendicular to the direction of the rays. ce l l i n w h i c h fluid flow is a l igned a l o n g l ines of cons tant long i tude . If the gradients i n d u c e d by the dif ferent ia l hea t ing were large enough, p r e s u m a b l y more c o m p l i c a t e d flow pa t te rns c o u l d emerge. N o w consider the same scenario, bu t o n a r o t a t i n g p lanet where the di f ferent ia l h e a t i n g is s y m m e t r i c abou t the axis o f r o t a t i o n . If observed f rom a fixed l o c a t i o n o n the r o t a t i n g p lane t , objects i n m o t i o n w h i c h are not subject to ex te rna l fo rc ing seem to fo l low c u r v e d t ra jector ies ( this is a consequence o f the C o r i o l i s effect, see be low) . In a s i m i l a r manne r , the fluid m o t i o n i n d u c e d by the different ia l hea t ing is deflected p e r p e n d i c u l a r l y to the d i r e c t i o n of m o t i o n , the d i r ec t i on o f the def lect ion depend ing on the sense o f r o t a t i o n a n d the m a g n i t u d e of the def lect ion depend ing on the m a g n i t u d e of the v e l o c i t y a n d r o t a t i o n rate . T h i s i m p l i e s t ha t the fluid can no longer s tay i n a l o n g i t u d i n a l p lane . However , i f the ra te o f r o t a t i o n is s m a l l , the r e s u l t i n g flow p a t t e r n m a y s i m p l y be a t i l t e d ( b o t h v e r t i c a l l y a n d h o r i z o n t a l l y ) vers ion of the convec t ion ce l l of the n o n - r o t a t i n g p lane t , w i t h a m i r r o r s y m m e t r y between the hemispheres . T h a t is , the a x i s y m m e t r y w o u l d be m a i n t a i n e d . It w o u l d not be s u r p r i s i n g if, as the parameters (the r o t a t i o n rate a n d the di f ferent ia l hea t ing) were increased, the t i l t i n g w o u l d become severe enough so tha t the a x i s y m m e t r i c flow p a t t e r n w o u l d become uns table , l e ad ing to the deve lopment o f a different, pe rhaps m o r e c o m p l i c a t e d , flow pa t t e rn . In fact, the r o t a t i o n can lead to i n s t a b i l i t y o f the a x i s y m m e t r i c convec t ion p a t t e r n at s ign i f ican t ly lower values o f the different ia l h e a t i n g t h a n for a n o n -r o t a t i n g sys tem. 3 North Pole South Pole F i g u r e 1.2: A possible convection cell, where the depicted motion is axisymmetric and there is mirror symmetry with the southern hemisphere. A n a d d i t i o n a l in te res t ing feature is tha t the effects of r o t a t i o n are felt different ly at different l oca t ions o n the p lanet . F o r an observer at the equator , the r o t a t i o n is felt m o s t l y as a r e d u c t i o n o f gravi ty , w h i l e for an observer at the poles, essential ly, the r o t a t i o n is felt o n l y i n the h o r i z o n t a l . D u e to the consequences th is has on the d y n a m i c s , i t is useful to cons ider a l o c a l 'effective' r o t a t i o n rate, w h i c h has a m i n i m u m at the equa tor a n d a m a x i m u m at the poles. It is also poss ible to argue tha t there is a v a r i a t i o n o f the di f ferent ia l h e a t i n g w i t h l a t i t u d e (due to the v a r i a t i o n o f the ra te of change of absorbed solar r a d i a t i o n w i t h respect to l a t i t ude ; see F i g u r e 1.1). Therefore , since b o t h the l o c a l r o t a t i o n rate a n d di f ferent ia l h e a t i n g change w i t h l a t i tude , i t leads one to imag ine t ha t different bands of l a t i t u d e m i g h t favor flows of different character . Indeed th is is the case o n the E a r t h . N e a r the equator , there is a large scale a x i s y m m e t r i c c i r c u l a t i o n p a t t e r n ca l l ed the H a d l e y ce l l , w h i c h , o n average, the w i n d a p p r o x i m a t e l y fol lows. It is charac te r ized by u p w a r d m o t i o n at the equator , subsidence i n the sub- t ropics (near 30° l a t i tude) a n d pers is tent e q u a t o r -eas tward w i n d s (ca l led the t rade winds ) at the surface. T h i s def lec t ion of the surface w i n d s is p r e s u m a b l y caused by the effect of ro t a t i on . In the m i d - l a t i t u d e s , o s c i l l a t o r y non-a x i s y m m e t r i c pa t te rns are often observed w h i c h propagate to the east. T h e m o t i o n i n th i s r eg ion tends to be p r e d o m i n a t e l y h o r i z o n t a l a n d tends to fol low cu rved t ra jec tor ies . 4 In the thesis, we w i l l t r y to capture , not the coexis tence of such pa t te rns , bu t the changes o f such pa t te rns as the r o t a t i o n rate a n d different ia l h e a t i n g (the parameters ) are va r i ed . In p a r t i c u l a r , we w i l l be concerned w i t h the t r ans i t ions f r o m a x i s y m m e t r i c to n o n - a x i s y m m e t r i c flow pat terns . I f d i f ferent ia l h e a t i n g a n d r o t a t i o n define the character of large scale geophys ica l flows, t h e n i t is useful to s t u d y l a b o r a t o r y exper iments w h i c h a t t e m p t to isolate these effects. In p a r t i c u l a r , i f the f o r m of different ia l hea t ing , the geomet ry of the sys tem, p roper t i es o f the fluid, a n d b o u n d a r y cond i t ions p l ay a secondary role, the flows observed i n expe r imen t s m a y c o n t a i n the essential character of the i r geophys ica l counterpar t s . A l t e r n a t i v e l y , i f expe r imen t s are pe r fo rmed w i t h var ious conf igura t ions , the features w h i c h are c o m m o n to a l l the exper imen t s m a y be cons idered those inherent to d i f fe rent ia l ly hea ted r o t a t i n g sys tems. In th is sec t ion we discuss some general e x p e r i m e n t a l resul ts a n d i n so d o i n g , present some of the m a i n features of the general c i r c u l a t i o n o f the a tmosphere a n d ocean w h i c h our mode l s reproduce . M a n y different exper iments have been per fo rmed i n an a t t e m p t to deve lop a n under-s t a n d i n g o f d i f ferent ia l ly heated r o t a t i n g f lu id systems (see e.g. [27], [44] a n d [24]). T h e exper imen t s often take the f o r m of s t u d y i n g fluid flow i n a r o t a t i n g c y l i n d r i c a l annu lus , where the dif ferent ia l hea t ing is o b t a i n e d ei ther by keeping the inner a n d outer wa l l s of the annu lus at different tempera tures or by an i n t e rna l hea t ing of the fluid [27]. See F i g u r e 1.3. M a n y o ther conf igura t ions have been s tud ied a n d are m e n t i o n e d i n Sec t i on 2.3. T h e ex-pe r imen t s consis t of finding the var ious s table flow pa t te rns w h i c h occu r at different values of the r o t a t i o n rate a n d different ial hea t ing . U s u a l l y , the results are g iven i n a d i a g r a m where the t r ans i t ions between the different flow types are p l o t t e d i n pa ramete r space. T h e mos t i m p o r t a n t d imens ionless parameters were j u d g e d to be the T a y l o r n u m b e r T , a n d the t h e r m a l R o s s b y n u m b e r TZ [26]. T h e T a y l o r n u m b e r is a measure of the re la t ive i m p o r t a n c e of r o t a t i o n to viscosi ty , where Q is the ra te of r o t a t i o n , R — r& — ra is the difference between the outer a n d inner r a d i i o f the annu lus , v is the k i n e m a t i c v i scos i ty of the fluid a n d D is the d e p t h of the fluid. T h e t h e r m a l R o s s b y 1 . 2 Experiments r = 4Q2R5 v2D n u m b e r 11 = agDAT WR2 5 F i g u r e 1.3: The differentially heated rotating annulus experiment, where the annulus is rotated at rate CI and the inner wall is held at the fixed temperature Ta and the outer wall at temperature Tb, creating a differential heating. ra and rb are the radii of the inner and outer cylinders, R = rb - ra and D is the height of the annulus. is a measure of the re la t ive i m p o r t a n c e of r o t a t i o n to the dif ferent ia l hea t ing , where A T = Tb—Ta is the i m p o s e d h o r i z o n t a l t empera tu re gradient , a is the coefficient o f t h e r m a l e x p a n s i o n a n d g is the g r a v i t a t i o n a l acce le ra t ion . N o t e tha t , i f a l l o the r pa ramete r s are he ld fixed, there is a one-to-one re l a t ionsh ip between the d imens ionless parameters (ther-m a l R o s s b y n u m b e r a n d the T a y l o r number ) a n d the p h y s i c a l parameters (the di f ferent ia l h e a t i n g a n d rate o f ro t a t i on ) . If the parameters are he ld fixed at s m a l l values a n d the flow is a l lowed to equ i l i b r a t e ( t ransients are a l lowed to pass), a s table a x i s y m m e t r i c flow p a t t e r n is observed. B y s table , i t is meant observable: a s m a l l p e r t u r b a t i o n does not cause the pa t t e rn to d i sappear . If the parameters are inc remented s lowly a n d at each i n c r e m e n t a t i o n the flow is a l l owed to equ i l ib ra te , the reg ion i n pa ramete r space where th i s flow is s table m a y be m a p p e d out . I f the pa ramete r s are inc remented past ce r t a in c r i t i c a l values, the a x i s y m m e t r i c flow becomes uns tab le a n d a n o n - a x i s y m m e t r i c pa t t e rn arises. T h i s pa t t e rn is u sua l ly a t r a v e l l i n g wave whose wave leng th depends o n the paramete r values a n d the e x p e r i m e n t a l conf igura t ion . 6 T h e set o f c r i t i c a l pa ramete r values is ca l led the t r a n s i t i o n curve a n d i t forms the b o u n d -a ry between a x i s y m m e t r i c a n d n o n - a x i s y m m e t r i c flow regimes. If the same procedure is pe r fo rmed for the s table wave m o t i o n , t r ans i t ions to other , more c o m p l i c a t e d , flow regimes m a y be found . M o s t o f the exper iments find four m a i n flow regimes i n different regions o f pa rame te r space (see F i g u r e 1.4): Axisymmetric Flow. T h i s flow is charac te r ized by i ts a z i m u t h a l invar iance . Steady Waves. T h e flow i n th i s region is n o n - a x i s y m m e t r i c a n d resembles a r o t a t i n g wave w i t h cons tant a m p l i t u d e a n d phase. Different wavelengths are seen i n different subregions . Vacillation. In th i s region, the s t ruc ture or a m p l i t u d e or wave leng th o f the observed wave varies p e r i o d i c a l l y i n t i m e . Irregular Flow. T h i s region is charac te r ized by i ts i r regu la r na tu re i n b o t h space a n d t i m e . A l l of the observed flows have the i r counterpar t s i n the a tmosphere . T h e a x i s y m m e t r i c flow resembles the H a d l e y ce l l w h i c h is observed i n the a tmosphere near the equa tor where the ' l o c a l ' r o t a t i o n rate a n d different ial hea t ing are r e l a t ive ly s m a l l (see Sec t i on 2.1). G i v e n a coun te r -c lockwise r o t a t i o n a n d a pos i t ive t empera tu re gradient be tween the outer a n d inner c y l i n d e r , the e x p e r i m e n t a l flow i n th is region rises at the outer c y l i n d e r , s inks at the inner , w i t h def lect ion of the r a d i a l mo t ions to the r igh t . T h a t is , a t i l t e d convec t ion ce l l is observed. I n m i d - l a t i t u d e regions o f the E a r t h , the flow somet imes has wave charac te r i s t i cs tha t resemble the s teady waves a n d vac i l l a t ions seen i n the exper iments . In th i s reg ion i n the exper imen t , the flow t rajector ies are cu rved a n d v e r t i c a l m o t i o n is i n h i b i t e d . 1.3 Analysis N u m e r o u s m e t h o d s of ana lys is have been used i n the s t udy of geophys ica l fluid d y n a m i c s mode l s . O n e of the mos t i m p o r t a n t was pioneered by C h a r n e y [5] a n d E a d y [13] i n the la te 1940's. T h e y p o s t u l a t e d tha t m i d - l a t i t u d e cyclones (waves) are generated by the g r o w t h o f s m a l l d i s tu rbances o n a 'bas ic s tate ' , a s teady s o l u t i o n of the govern ing equat ions . T h e bas ic s ta te was t aken to be a z o n a l flow (a flow a l o n g isol ines o f l a t i t ude ) t ha t depended o n l y o n the v e r t i c a l coord ina te . A cen t ra l theme i n the ana lys is is tha t the ' i n s t a b i l i t y ' , or g r o w t h of d i s tu rbances , is i n d u c e d by the c o m b i n a t i o n o f the v e r t i c a l shear of the bas ic state a n d the s t r a t i f i ca t ion o f the fluid. T h i s i m p o r t a n t m e c h a n i s m is c a l l ed ' b a r o c l i n i c i n s t a b i l i t y ' . 7 upper symmetric log(Taylor number) F i g u r e 1.4: A schematic diagram depicting general experimental results. See e.g. [26]. To the left of all the curves is the axisymmetric regime which is separated into three (dynamically similar) regions: lower symmetric, knee, and upper symmetric. To the right of the curve is the non-axisymmetric regime which is separated into three dynamically distinct regimes: steady waves, vacillation and irregular flow. T h e ana lys i s consists of c o m p u t i n g the l inear s t a b i l i t y o f the basic state a n d finding the fastest g r o w i n g uns tab le mode , o r d i s tu rbance . E a d y found tha t , i n h i s m o d e l [13], t h i s m o d e has a n a z i m u t h a l (zonal) wavelength comparab l e to those o f the finite a m p l i t u d e waves observed i n the a tmosphere . I m p l i c i t i n the ana lys is are the a s sumpt ions tha t (1) the uns tab le wave c o u l d g row to finite a m p l i t u d e w i t h o u t b e i n g affected by non l inea r processes, (2) the g r o w t h stops w h e n some finite a m p l i t u d e is reached a n d (3) the g r o w t h is r e s t r i c t ed to th i s m o d e a lone (i.e. the g r o w t h o f the o ther uns tab le modes are somehow suppressed) . These a s sumpt ions were not verif ied, nor was i t shown w h y the t ime-dependent flow w o u l d app roach a reg ion close enough to the uns tab le basic state where the l inea r ana lys i s w o u l d be a p p r o x i m a t e l y v a l i d . M a n y s tudies have a p p l i e d th is idea to mode ls w i t h var ious features. T h e f o l l o w i n g l i s t is far f r o m comple te . B a r c i l o n [3] i n t r o d u c e d E k m a n (bounda ry ) layers . O t h e r s [42], [2], [6] used var ious basic states to inc lude h o r i z o n t a l shear (var ia t ions i n the m e r i d i o n a l d i r ec t i on ) , w h i l e o ther s tudies [1] were pe r fo rmed o n the ' be t a p lane ' , where the v a r i a t i o n 8 of the effective r o t a t i o n rate w i t h l a t i t ude is assumed to be l inear a n d a l l o ther cu rva tu re effects are neglected. M o s t s tudies used the quas igeos t rophic equat ions (see Sec t i on 2.2) i n three spa t i a l d imens ions , bu t some used the two-layer vers ion [49], w h i l e others used m o r e c o m p l e x mode l s [2]. T h e r e were also s tudies pe r fo rmed o n different geometr ies , i n c y l i n d r i c a l or spher ica l coordina tes . Some o f the s tudies were done a n a l y t i c a l l y a n d some n u m e r i c a l l y . However , they a l l s t ud ied the l inear s t a b i l i t y o f some bas ic state. A n o t h e r m e t h o d of ana lys is c o u l d be ca l l ed ' n u m e r i c a l e x p e r i m e n t a t i o n ' , s ince the p ro -cedure o f the l a b o r a t o r y exper iments is s i m u l a t e d o n a n u m e r i c a l m o d e l . T h i s m e t h o d is often used to a t t e m p t a r e p r o d u c t i o n of the flows observed i n the l a b o r a t o r y exper imen t s . T h e p rocedure often begins by finding a n a x i s y m m e t r i c s teady s o l u t i o n e i ther a n a l y t i c a l l y or n u m e r i c a l l y . T h i s so lu t i on , to w h i c h s m a l l a m p l i t u d e r a n d o m p e r t u r b a t i o n s are added to s imu la t e n a t u r a l fluctuations, is used as the i n i t i a l c o n d i t i o n for a n u m e r i c a l i n t eg ra t i on , or t ime- s t epp ing . If the t ime- s t epp ing produces so lu t ions w h i c h s tay near the s teady so lu -t i o n , t hen the s teady so lu t i on is l abe l l ed as s table i n the s tudies . T h e pa ramete r s are t hen v a r i e d a n d the process is repeated. I f the t ime- s t epp ing produces a s o l u t i o n w h i c h evolves away f r o m the s teady so lu t i on , i t is l abe l l ed uns table . In th i s case, the t i m e - s t e p p i n g is con t inued u n t i l t rans ient behav iour d isappears a n d the sys tem appears to reach a different s table s teady state, p e r i o d i c so lu t i on , or a more c o m p l i c a t e d s o l u t i o n . A g a i n , m a n y different mode ls have been s tud ied us ing n u m e r i c a l e x p e r i m e n t a t i o n . M o et al. [45] fo l lowed the above procedure o n a th ree -d imens iona l quas igeos t roph ic m o d e l w i t h E k m a n layers at the top and b o t t o m . H i g n e t t et al. [29], J ames et al. [30], M i l l e r a n d B u t l e r [43] a n d L u et al. [38] per formed n u m e r i c a l exper iments w i t h a Nav ie r -S tokes m o d e l i n c y l i n d r i c a l geomet ry for d i rec t c o m p a r i s o n to l a b o r a t o r y exper iments . K w a k a n d H y u n [33] pe r fo rmed the ana lys is o n a m o d e l o f the H a t h a w a y a n d F o w l i s expe r imen t [24], a n d C o l l i n s a n d J ames [8] s tud ied a s impl i f i ed g loba l c i r c u l a t i o n m o d e l . L e w i s [36] a n d M u n d t et al. [47] also pe r fo rmed t ime- in t eg ra t i on o f basic states o n different two- layer mode l s . 1.3.1 A dynamical systems approach In th i s thesis, a d y n a m i c a l systems approach w i l l be t aken . T h e u n d e r l y i n g p r i n c i p l e is t ha t a knowledge o f the so lu t i on s t ruc ture of the equat ions w i l l l ead to a be t te r u n d e r s t a n d i n g of the sys t em we are t r y i n g to m o d e l . W e do not seek a t ime-dependent s o l u t i o n g iven a s ingle i n i t i a l c o n d i t i o n , bu t ra ther we a t t empt to de te rmine w h a t the d y n a m i c s w i l l be, at least qua l i t a t i ve ly , g iven any i n i t i a l c o n d i t i o n . W h e n the sys t em depends o n parameters , such as the s t reng th of different ial hea t ing , the goal is to de te rmine the s o l u t i o n s t ruc tu re for a l l relevant pa ramete r values. A l t h o u g h i t is r a re ly poss ib le , i f we c a n find a l l the i nva r i an t sets o f the equat ions (the s teady so lu t ions , pe r iod i c o rb i t s , chao t i c i nva r i an t sets, 9 etc.) a l o n g w i t h the i r s t ab i l i ty , then th is gives an i dea of w h a t the l o n g - t i m e b e h a v i o u r w i l l be, regardless o f the i n i t i a l cond i t ions . T h a t is , t ime-dependent so lu t ions 'move away ' f rom uns tab le inva r i an t sets and ' app roach ' s table inva r i an t sets i f enough t i m e is a l lowed to pass. T h i s m e t h o d , however, often does not p rov ide sufficient i n f o r m a t i o n to deduce w h i c h s table inva r i an t sets w i l l be approached for a g iven i n i t i a l c o n d i t i o n a n d i t m a y no t be able to de te rmine how l o n g the approach w i l l take. Moreove r , i t is poss ib le t ha t the n a t u r a l f luc tua t ions present i n a l l phys i ca l systems c o u l d i n h i b i t the convergence to a s table i nva r i an t set. W e i m p l i c i t l y assume that , the t ime-dependent so lu t ions of the equat ions o f o u r m o d e l exis t a n d are un ique for a l l the i n i t i a l cond i t ions of interest . It is poss ib le to prove th i s for o r d i n a r y dif ferent ia l equa t ion mode l s p rov ided ce r t a in general c o n d i t i o n s are sat isf ied (see e.g. [56]). However , for the p a r t i a l different ial equa t ion mode l s we use, t h i s has no t been e x p l i c i t l y shown. Y e t , since the th ree -d imens iona l incompress ib le Nav ie r -S tokes equa-t ions a n d the t w o - d i m e n s i o n a l quas igeos t rophic (baro t ropic) p o t e n t i a l v o r t i c i t y equa t ions [9] e x h i b i t l o c a l existence a n d uniqueness of so lu t ions , i t is reasonable to assume tha t i n the pa ramete r range a n d i n i t i a l cond i t i ons of interest , the mode l s we s t u d y do as w e l l . B i f u r c a t i o n ana lys i s is a p a r t i c u l a r l y powerful m e t h o d o f f i nd ing i nva r i an t sets. G i v e n a k n o w n s teady so lu t i on , or o ther invar ian t set, for a g iven set o f the parameters of the sys tem, b i fu r ca t i on me thods can not o n l y be used to t race out how th i s s o l u t i o n changes as the parameters change, bu t i t c an also be used to uncover p rev ious ly u n k n o w n inva r i an t sets w h i c h 'b i furca te ' f r o m the k n o w n so lu t i on . B y b i fu rca t i on we m e a n a q u a l i t a t i v e change i n the s o l u t i o n s t ruc ture of the equat ions as the parameters are v a r i e d past some c r i t i c a l values. T h e l o c a t i o n i n pa ramete r space where a b i fu rca t ion occurs is ca l l ed a b i f u r c a t i o n p o i n t . U n d e r ce r t a in generic cond i t ions , i t c a n be shown tha t close to ce r t a in types of b i f u r c a t i o n po in t s , m u l t i p l e invar ian t sets exis t . F o r example , w h e n the l i n e a r i z a t i o n abou t a s teady s o l u t i o n (see Sec t ion 3.1) has a c o m p l e x conjugate p a i r of eigenvalues whose rea l pa r t s cross the i m a g i n a r y axis as a pa ramete r of the sys tem is va r i ed , a p e r i o d i c s o l u t i o n appears . T h i s is ca l l ed a H o p f b i fu rca t ion . A b i fu rca t ion ana lys i s can no t o n l y show the existence of the new invar ian t set bu t can also prov ide the s o l u t i o n f o r m of the i nva r i an t set a n d i t s s t ab i l i t y . F o r the b i fu rca t ion of a s teady so lu t i on , different forms o f the eigenvalues (e.g. rea l or c o m p l e x conjugate pai rs) a n d different numbers o f the eigenvalues w h i c h cross the i m a g i n a r y ax i s cor respond to different types of b i furca t ions , i n w h i c h different k i n d s of i nva r i an t sets m a y appear . T h e other k inds of invar ian t sets have analogous , bu t more c o m p l i c a t e d s i tua t ions w h i c h give rise to b i furca t ions . Theo re t i c a l l y , the n e w l y d iscovered i nva r i an t sets c a n also be t racked t h r o u g h paramete r space u n t i l subsequent b i fu rca t ions occur . 10 A s p r e v i o u s l y men t ioned , b i fu rca t ion analys is a t t empt s to find the s o l u t i o n s t ruc ture for a l l re levant pa ramete r values. A t the mos t basic level , the m o t i v a t i o n is tha t th i s leads to a be t t e r u n d e r s t a n d i n g o f the u n d e r l y i n g p r inc ip l e s w h i c h are b e i n g m o d e l l e d . F o r e x a m p l e , i t is useful to k n o w for w h a t values of the different ial hea t ing a n d r o t a t i o n rate a ce r t a in t ype o f flow p a t t e r n is observed. F r o m such a n analys is , i t c a n be learned , for example , t ha t a x i s y m m e t r i c flow is expected for s m a l l values o f the parameters , w h i l e wave m o t i o n is expec ted for larger values. Therefore , near the equator , one m a y look for an a x i s y m m e t r i c flow p a t t e r n w h i l e i n m i d - l a t i t u d e s , wave m o t i o n s h o u l d be seen. In t h i s l i gh t , b i f u r c a t i o n ana lys i s a n d the l a b o r a t o r y exper iments have the same goa l . However , i n a p a r t i c u l a r nat -u r a l l y o c c u r r i n g fluid flow, i t m a y be argued tha t the pa ramete r values are cons tan t a n d thus i t is necessary to make a c a l c u l a t i o n for o n l y one value of each pa ramete r . In fact , the parameters of m a t h e m a t i c a l mode ls often arise due to s imp l i f i ca t ions of the rea l sys tem. F o r ins tance , the different ial h e a t i n g o f the annulus exper imen t can be w r i t t e n as the t e m -pera tu re difference between the inner a n d outer annulus wal l s , however, i n the a tmosphere , th i s fo rc ing is m u c h more c o m p l i c a t e d , w i t h t e m p o r a l a n d spa t i a l dependence. In such a s i t u a t i o n , i t cannot be de t e rmined w h a t value of the pa ramete r w i l l best r ep roduce the b e h a v i o u r o f the rea l sys tem. Therefore , a knowledge o f the s o l u t i o n s t ruc tu re for a range of pa rameters w i l l more l i ke ly va l ida te a m o d e l . A l s o , i n some s i tua t ions , the pa rame te r m a y depend o n the state o f the sys tem a n d so i t m a y va ry i n t i m e . I f the pa rame te r changes s m o o t h l y a n d s lowly i n c o m p a r i s o n to the d y n a m i c s , perhaps i t c o u l d be assumed tha t the t i m e dependence of the paramete r s i m p l y changes the l o c a t i o n i n pa rame te r space w i t h o u t affecting the so lu t i on s t ruc ture . W e therefore t h i n k o f the a tmosphere as e x i s t i n g at different l oca t ions i n pa ramete r space at different t imes and , the ana lys i s does not t r y to address w h y a p a r t i c u l a r flow pa t t e rn is observed at a g iven t i m e . T h i s p o i n t is mos t i m p o r t a n t w h e n r e l a t i n g the results to the mid - l a t i t udes , where flows of different charac te r m a y be observed at different t imes . Ideal ly , i t w o u l d be possible to find a l l the invar ian t sets of the sys t em for a l l pa rame te r values of interest . However , th i s c a n be a fo rmidab le task even for l o w - d i m e n s i o n a l o r d i n a r y d i f ferent ia l equat ions , let a lone for the i n f in i t e -d imens iona l p a r t i a l d i f ferent ia l equa t ions we are in teres ted i n s t u d y i n g . O n e m e t h o d of s i m p l i f i c a t i o n is to generate a l o w - d i m e n s i o n a l o r d i n a r y di f ferent ia l equa-t i o n ( O D E ) f r o m the p a r t i a l different ia l equa t ion ( P D E ) m o d e l . T h i s c a n be achieved w i t h a t r u n c a t e d spec t r a l expans ion . T h i s m e t h o d consists of e x p a n d i n g the dependent var iab les o f the P D E s as a series o f comple t e o r t hogona l spec t r a l funct ions o f the s p a t i a l var iab les , w h i c h results i n a n in f in i t e -d imens iona l sys tem of o r d i n a r y dif ferent ia l equa t ions i n the t ime-dependen t coefficients o f the spec t ra l funct ions . T h e a p p r o x i m a t i o n resul ts by c o n s i d -11 e r ing o n l y a s m a l l n u m b e r of t e rms i n the expansions . A l t h o u g h th is m e t h o d is ve ry useful for the s t u d y o f the m a t h e m a t i c a l mechan isms w h i c h poss ib ly exis t i n the P D E s (see e.g. [37], [53], a n d also [19]), i t is , i n general , diff icul t to k n o w i f the b e h a v i o u r i n the O D E corresponds to tha t o f the o r i g i n a l equat ions . D u e to the presence o f the non l inea r t e rms i n the o r i g i n a l equat ions , inter-scale in te rac t ions are fundamen ta l to the d y n a m i c s a n d the low-order t r u n c a t i o n , i n effect, ignores the in te rac t ions between a l l bu t f in i t e ly m a n y scales. T h e advantage is t r a c t ab i l i t y . It is often poss ible to pe r fo rm a t h o r o u g h b i f u r c a t i o n ana lys i s , somet imes even a n a l y t i c a l l y [37]. N u m e r i c a l l y , a software package such as A U T O [10] can be used for l o w - d i m e n s i o n a l O D E s to f ind b i furca t ions a n d to fo l low s teady a n d p e r i o d i c so lu t ions t h r o u g h pa ramete r space. In order to i nc lude more of the inter-scale in te rac t ions , i t is poss ib le to p e r f o r m a h igher -order spec t r a l t r u n c a t i o n or use another m e t h o d of d i s c r e t i z a t i o n . T h e h igher-d i m e n s i o n a l sys tem, however, becomes diff icul t to ana lyze a n d the inter-scale in te rac t ions w h i c h are i gno red m a y s t i l l no t be negl ig ib le . D u e to the size of the d i sc re t i zed sys tems, the preferred m e t h o d of ana lys is is n u m e r i c a l expe r imen ta t i on . A l t h o u g h such ana lys i s can l ead to in te res t ing discoveries, the d y n a m i c a l behav iour can o n l y be s t u d i e d for a r e l a t i ve ly shor t i n t eg ra t i on t i m e a n d for o n l y re la t ive ly few i n i t i a l cond i t ions . It c a n be diff icul t to loca l i ze b i f u r c a t i o n po in t s or to de te rmine w h e n the t ime-dependent s o l u t i o n is close to a s table i nva r i an t set. In fact, th i s m e t h o d cannot show the existence of i nva r i an t sets; the existence mus t be p resumed. F o r example , to de te rmine whe ther a p e r i o d i c o rb i t has been approached , a j u d g m e n t mus t be made tha t the t ime-dependent s o l u t i o n has r e tu rned c lose ly enough to a p rev ious state a n d th is r e tu rn must occur m a n y t imes so t ha t a p e r i o d i c o rb i t is not m i s t a k e n l y p resumed i n place of a more c o m p l i c a t e d t ra jec tory . P r o b l e m s arise also i f the p e r i o d of the o rb i t is l ong . T h i s becomes even more di f f icul t , or p r o h i b i t i v e , for more c o m p l i c a t e d invar ian t sets. Fu r the rmore , errors are i n t r o d u c e d at each t i m e step a n d l o n g t i m e in tegra t ions are necessary to find the behav iou r o f interest . Q u a n t i t a t i v e j u s t i f i c a t i o n tha t these errors are not i m p o r t a n t m a y not be poss ib le . B i f u r c a t i o n ana lys i s is u s u a l l y not pe r fo rmed o n such systems, a l t hough Legras a n d G h i l [34] pe r fo rmed a de t a i l ed inves t iga t ion of a h ighe r -d imens iona l spec t ra l t r u n c a t i o n m o d e l of b a r o t r o p i c flow over topography . In th i s thesis, we choose to d i r e c t l y s tudy the p a r t i a l different ia l equat ions at the cost o f r e s t r i c t i ng the ana lys i s to i so la ted loca t ions i n pa ramete r space where a p a r t i c u l a r t ype of b i f u r c a t i o n occurs . In pa r t i cu l a r , we s tudy double H o p f b i fu r ca t i on po in t s w h i c h occu r w h e n the l i n e a r i z a t i o n abou t a s teady a x i s y m m e t r i c so lu t i on has two pa i rs of c o m p l e x conjugate eigenvalues t ha t s imu l t aneous ly cross the i m a g i n a r y axis as the parameters are v a r i e d . C l o s e to such po in t s , in te res t ing behav iou r m a y be discovered tha t c o u l d o therwise have been 12 missed . Cen t e r m a n i f o l d r educ t ion is used to f ind the d y n a m i c s o f the p a r t i a l d i f ferent ia l equa t ions close to the b i fu rca t ion po in t . T h i s is a m e t h o d of s i m p l i f y i n g the equat ions i n a way w h i c h takes i n to account a l l the non l inear inter-scale in te rac t ions . T h e r e d u c t i o n m e t h o d is desc r ibed i n de t a i l i n Sec t ion 3.2. T h e results are v a l i d for pa ramete r values close to the b i f u r c a t i o n po in t a n d w h e n the b i fu rca t ing s o lu t i on ( invar ian t set) is close to the a x i s y m m e t r i c s o l u t i o n . T h i s m e t h o d is somet imes referred to as w e a k l y non l inea r ana lys i s , s ince the non l inea r te rms i n the equat ions are assumed to be s m a l l bu t no t neg l ig ib le . T h e o r e t i c a l l y , i t is poss ible to wr i t e the b i fu rca t ing s o lu t i on as a series i n a s m a l l pa rame te r a n d thus e x p a n d the range of va l id i ty . In ac tua l i ty , th i s is ra re ly p r a c t i c a l a n d the s o l u t i o n is o n l y e s t ima ted to first order i n the s m a l l parameter . Essen t i a l ly , the technique is able to show the existence a n d s t a b i l i t y of the b i fu rca t ing s o lu t i on a n d give a first-order e s t imate o f the s o l u t i o n itself, bu t i t is not able to de te rmine i f the s o lu t i on persis ts for values o f the pa ramete r s far f rom the b i fu rca t ion po in t . T h i s type of b i fu rca t ion analys is has been successful i n var ious app l i c a t i ons . Pe rha ps the mos t w e l l - k n o w n is the onset o f convec t ion i n a f l u id heated f rom be low, the R a y l e i g h -B e n a r d p r o b l e m (see [51]). A n o t h e r a p p l i c a t i o n of note is the T a y l o r - C o u e t t e p r o b l e m (see [7] a n d the references con ta ined there in) , w h i c h is a fluid annu lus expe r imen t w i t h o u t d i f ferent ia l h e a t i n g where the inner a n d outer cy l inde r s ro ta te at different rates genera t ing a shear flow i n the in te r ior of the annulus . A r i c h var ie ty of b e h a v i o u r was uncovered u s ing b i f u r c a t i o n analys is , m u c h o f w h i c h was conf i rmed by e x p e r i m e n t a l resul ts . In the geophys ica l fluid d y n a m i c s l i t e ra ture , an a s y m p t o t i c m e t h o d , f o r m a l l y equivalent to center m a n i f o l d r educ t ion , was used to ana lyze 'weak ly non l inea r ' wave-wave in te rac t ions (double H o p f b i furca t ions) i n the two- layer quas igeos t rophic p o t e n t i a l v o r t i c i t y equat ions w i t h the b e t a effect (see Sec t i on 2.2) a n d a l inear basic state i n [46], [39], a n d [54]. A l s o , H a r t [22] s tud ied s i m i l a r equat ions w i t h no b e t a effect a n d D r a z i n [12] s t ud i ed E a d y ' s m o d e l [13], b o t h u s i n g s i m i l a r methods . T h e results i n d i c a t e d m u l t i p l e s table wave so lu t ions a n d hysteresis o f these so lu t ions . M o r o z and H o l m e s [46] found s table quas i -pe r iod i c d y n a m i c s close to one o f the b i fu rca t ion po in t s . However , our r e p r o d u c t i o n of the ana lys i s found t ha t due to a mi s t ake i n the f o r m u l a for one of the n o r m a l f o r m coefficients, th i s resul t was i n error . F o r a l l o f these models , i t is possible to find the results a n a l y t i c a l l y . In the field o f geophys ica l fluid d y n a m i c s , few mode l s exist w h i c h can be s t ud i ed p u r e l y a n a l y t i c a l l y . S ince the mode ls s tud ied i n th is thesis do not f a l l i n to th i s category, we use a c o m b i n a t i o n of weak ly non l inea r ana lys is w i t h n u m e r i c a l me thods . T h e resul t is a n a n a l y t i c a l - n u m e r i c a l h y b r i d ana lys is technique. A n a l y t i c a l l y , i t is poss ib le to reduce the t ime-dependent non l inea r P D E s to a series of s teady l inear P D E p rob lems . These l inea r sys tems are t hen solved numer ica l ly . N o t o n l y are the l inear p rob l ems less di f f icul t 13 to n u m e r i c a l l y a p p r o x i m a t e , bu t also the v a l i d i t y of the a p p r o x i m a t i o n s is more easi ly ver i f ied. E s s e n t i a l l y equivalent me thods are used i n the C o u e t t e - T a y l o r p r o b l e m [7] a n d i n [20], where a doub le H o p f b i fu rca t ion was ana lyzed i n a ba ro t rop i c quas igeos t rophic m o d e l . It s h o u l d be res ta ted tha t a priori i t is not k n o w n i f th i s ana lys i s w i l l l ead to the d is-covery o f p h y s i c a l l y rea l izab le invar ian t sets. F o r example , i n Po i seu i l l e flow, e x p e r i m e n t a l observat ions i nd i ca t e tha t i n s t a b i l i t y sets i n for s ign i f i can t ly lower pa rame te r values t h a n the l inea r s t a b i l i t y ana lys is predic ts [40] a n d the wave so lu t i on p red ic t ed by the b i f u r c a t i o n ana lys i s is not observed. T h e cause of th is has been a t t r i b u t e d to the n o n - o r t h o g o n a l i t y of the e igenfunct ions [14], due to the non-self-adjoint l i n e a r i z a t i o n a b o u t the s teady so-l u t i o n (see Sec t i on 3.1). In th is case, i t is poss ible tha t even w h e n the s teady s o l u t i o n is l i n e a r l y s table , s m a l l pe r tu rba t ions can grow to apprec iab le size before t hey u l t i m a t e l y decay. T h u s , s m a l l pe r tu rba t ions m a y take the flow in to the non l inea r reg ime where the s t a b i l i t y c a l c u l a t i o n is not v a l i d . A l t e r n a t i v e l y , cons ide r ing tha t s m a l l p e r t u r b a t i o n s w i l l a lways be present, there w i l l a lways be g r o w t h a n d thus the s o l u t i o n w i l l never converge to the s table flow pa t t e rn , i.e. the ' t rans ients ' w i l l never d isappear . It s h o u l d be n o t e d t ha t no t a l l non-self-adjoint systems w i l l have th is p rope r ty (e.g. the C o u e t t e - T a y l o r p r o b l e m [7], where l inea r s t a b i l i t y was used to reproduce the e x p e r i m e n t a l l y observed t r ans i t i ons be tween different flow regimes) . However , since the equat ions we w i s h to s t u d y have n o n -self-adjoint l inear par ts , th i s p r o b l e m m a y arise for our mode l s . It is therefore useful t o ana lyze a m o d e l where the results can be q u a n t i t a t i v e l y verif ied by the resul ts o f l a b o r a t o r y exper imen t s . If the results of the ana lys is can be conf i rmed , then no t o n l y c a n s o m e t h i n g be learned o f the s t ruc ture of the so lu t i on space, bu t i t also va l ida tes the m e t h o d for these a n d s i m i l a r mode l s . 1.4 Outline of thesis In th i s thesis, we s t u d y the t r ans i t ions f rom a x i s y m m e t r i c s teady so lu t ions to n o n - a x i s y m -m e t r i c t r a v e l l i n g waves i n two geophys ica l fluid d y n a m i c s mode ls : (1) a two- layer quas i -geos t rophic p o t e n t i a l v o r t i c i t y m o d e l w i t h a non l inear basic state a n d (2) a m o d e l o f the d i f fe rent ia l ly heated r o t a t i n g annulus exper iment . U s i n g the a n a l y t i c a l - n u m e r i c a l center m a n i f o l d r e d u c t i o n at the double H o p f b i fu rca t ion po in t s w h i c h occu r at th i s t r a n s i t i o n , i t is s h o w n tha t there are regions, i n the respect ive paramete r spaces, w h i c h suppo r t m u l t i p l e s table wave so lu t ions a n d hysteresis of these so lu t ions . T h e first m o d e l differs f rom those p rev ious ly s tud ied w i t h b i f u r c a t i o n ana lys i s i n t ha t i t con ta ins a l a t i t u d i n a l l y v a r y i n g forc ing t e r m w h i c h leads to a basic state w h i c h is non l inea r . T h e m o d e l has been p roposed to s tudy the i n t e rna l l y heated r o t a t i n g annu lus expe r imen t , 14 as w e l l as the general c i r c u l a t i o n of the great p lanets [36]. A s a consequence o f the f o r m of the bas ic state, the l i n e a r i z a t i o n abou t the a x i s y m m e t r i c s o l u t i o n is non-self-adjoint a n d resul ts canno t be c o m p u t e d a n a l y t i c a l l y . T h e resul ts i n d i c a t e t ha t the fo r c ing t e r m i n the equa t ion does not q u a l i t a t i v e l y affect the behav iour . However , there are some m i n o r differences w h i c h are of interest , i n pa r t i cu l a r , the f o r m of the b i f u r c a t i n g s o l u t i o n . T h e second m o d e l uses the Nav ie r -S tokes equat ions i n the B o u s s i n e s q a p p r o x i m a t i o n , i n c y l i n d r i c a l geometry. T h e d imens ions of the d o m a i n a n d the p roper t i es o f the f l u id are chosen to i m i t a t e the di f ferent ia l ly heated r o t a t i n g annulus expe r imen t o f F e i n [15]. T h e t r a n s i t i o n curves are a p p r o x i m a t e d by c a l c u l a t i n g the eigenvalues of the d i s c r e t i z a t i o n of the l i n e a r i z a t i o n abou t the a x i s y m m e t r i c so lu t ions . T o m y knowledge , t h i s is the first t i m e a de t a i l ed inves t iga t ion o f th is k i n d has been per fo rmed for the d i f fe ren t ia l ly heated r o t a t i n g sys tem. D o u b l e H o p f po in t s are loca ted a n d ana lyzed , a n d an a p p r o x i m a t i o n o f the reg ion o f hysteresis is also found . A g a i n , the l i nea r i za t ions are non-self-adjoint a n d n u m e r i c a l me thods mus t be i m p l e m e n t e d . T h e t r a n s i t i o n curve, c r i t i c a l wave n u m b e r a n d dr i f t ra te o f the wave are a l l found to reproduce we l l the e x p e r i m e n t a l observa t ions . T h e type o f ana lys i s tha t has been employed i n th is thesis has been p r e d o m i n a t e l y over-l o o k e d i n the geophys ica l f l u id d y n a m i c s c o m m u n i t y . Pe rh ap s th i s is due t o the d i f f icu l ty o f the ana lys i s . However , the work i n th is thesis indica tes tha t a b i f u r c a t i o n ana lys i s of th i s k i n d can l ead to v a l i d results a n d thus suppor t s the further a p p l i c a t i o n to other , s i m i l a r , p rob lems . T h e u b i q u i t y of the region c o n t a i n i n g m u l t i p l e s table waves leads to the specu-l a t i o n t ha t the existence of the region m a y be a fundamen ta l p r o p e r t y o f a l l d i f ferent ia l ly heated r o t a t i n g systems i n the paramete r ranges app l i cab le to geophys ica l f luids . In C h a p t e r 2, the mode l s w h i c h w i l l be ana lyzed are presented. A n overv iew of the bas ic concepts a n d a p p r o x i m a t i o n s l ead ing to the d y n a m i c a l equat ions is i n c l u d e d i n th i s chapter . In C h a p t e r 3, the a n a l y t i c a l - n u m e r i c a l center m a n i f o l d r e d u c t i o n is o u t l i n e d u s i n g the s i m p l e r e x a m p l e o f a H o p f b i fu rca t ion i n the two- layer m o d e l . I n the second pa r t o f C h a p t e r 3 a n d C h a p t e r 4, the results of the doub le H o p f b i fu rca t ions a p p l i e d to the two mode l s are presented. A conc lus ion fol lows. 15 C h a p t e r 2 T h e M o d e l s T h e a tmosphere is a very c o m p l e x sys tem. Its d y n a m i c s are inf luenced by the mo i s tu r e ca r r i ed by the a i r (e.g. v a p o u r a n d c louds) a n d the fact tha t the oceans, w h i c h are t h e m -selves c o m p l e x d y n a m i c a l systems, are i n t r i c a t e l y coup led to the a tmosphere . T h e r e are va r i a t i ons o f t h e r m a l fo rc ing over m a n y space and t i m e scales w h i c h are themselves the resul t of m a n y factors, a n d the boundar ies i n t roduce a d d i t i o n a l c o m p l e x i t y . F u r t h e r m o r e , there are non l inea r processes w h i c h cannot be ignored . T h e n o n l i n e a r i t y couples the s m a l l a n d large scales a n d since the a tmosphere a n d oceans e x h i b i t i m p o r t a n t d y n a m i c s at a l l scales ( they are t u rbu len t ) , a l l scales have to be we l l resolved i f we w i s h to r ep roduce a l l the flow features. A t present, the mos t powerful compute r s cannot c o m p u t e the range of scales necessary for such a r e p r o d u c t i o n a n d i t w i l l be a very l o n g t i m e , i f ever, before one is b u i l t . Because of the complex i t y , i t is useful, or necessary, to s t udy s i m p l i f i e d mode l s of the sy s t em tha t isolate p a r t i c u l a r flow features and tha t are also t r ac tab le to ana lys i s . I f the s imp l i f i ca t i ons are a r r ived u p o n sys temat ica l ly , then m u c h c a n be learned f rom these mode l s . A s a s i m p l e example , i f a m o d e l neglects a l l bu t one p h y s i c a l process a n d is s t i l l able to reproduce a n observed flow pa t t e rn , then i t is p laus ib le to pos tu la t e t ha t i t is th i s p h y s i c a l process w h i c h is mos t i m p o r t a n t i n the genera t ion of th i s flow. T h e fact t ha t such m e t h o d s have been very successful i n the field of geophys ica l f l u i d d y n a m i c s gives ev idence t ha t there are a r e l a t i ve ly s m a l l n u m b e r o f d o m i n a n t processes t ha t shape the m a i n s t ruc ture of the flows of interest . Here we are interested i n s t u d y i n g two p a r t i c u l a r s imp le mode l s for the large scale d y n a m i c s o f the a tmosphere (or ocean) . F i r s t , we w i l l s t u d y the two- layer quas igeos t rophic p o t e n t i a l v o r t i c i t y equat ions . T h e cen t ra l a s s u m p t i o n of th i s m o d e l is t ha t the charac te r i s t i c l e n g t h a n d ve loc i ty of the flow are such tha t the E a r t h ' s r o t a t i o n becomes a d o m i n a n t effect. T h e m o t i v a t i o n for s t u d y i n g th is m o d e l is tha t i t is the mos t s i m p l e m o d e l ava i lab le for s t u d y i n g b a r o c l i n i c effects (see be low) . T h e m o d e l we ana lyze differs f r o m the s t a n d a r d 16 m o d e l i n tha t i t conta ins a basic state w h i c h not o n l y has v e r t i c a l shear bu t also non l inea r h o r i z o n t a l shear. W e also s t u d y a con t ro l l ed l a b o r a t o r y exper imen t i n w h i c h the changes i n the flow pa t te rns i n a d i f ferent ia l ly heated r o t a t i n g annulus are observed as the i m p o s e d t empera tu re grad ien t a n d rate of r o t a t i o n are va r ied . W e a t t e m p t to reproduce q u a n t i t a t i v e l y some o f the observed resul ts u s ing an accura te m a t h e m a t i c a l m o d e l . T h e s t u d y o f th i s sy s t em w i l l not o n l y va l ida t e the accuracy of the ana lys is , bu t w i l l he lp develop an u n d e r s t a n d i n g o f the mechan i sms at w o r k i n flows w i t h different ial hea t ing a n d r o t a t i o n . These mechan i sms w i l l i n v a r i a b l y be present i n a l l large scale geophys ica l flows. 2.1 The general equations of motion Before the s imp l i f i ed mode l s are presented, i t is useful to discuss the general equat ions of m o t i o n for a r o t a t i n g fluid o f va r i ab le densi ty subject to g r av i t y (referred to be low as a s t ra t i f ied fluid). It is f r o m these equat ions tha t m a n y geophys ica l fluid mode l s are de r ived , i n c l u d i n g those o f interest here. However , the general equat ions a l r eady c o n t a i n i m p o r t a n t s imp l i f i ca t i ons . In th i s sec t ion , we wr i t e d o w n the general equat ions a n d discuss some of the re la ted fundamen ta l concepts , i n pa r t i cu l a r , the aspects tha t charac ter ize large scale flow. W e s tar t w i t h a de f in i t i on of ' large scale ' , w h i c h leads to the cons ide ra t ion o f s t u d y i n g the flow i n a r o t a t i n g frame of reference. T h i s w i l l lead to a d d i t i o n a l t e rms i n the equat ions o f m o t i o n o f the fluid. S ince these te rms are essential for cha rac t e r i z ing large scale flow, a d i scuss ion of t he i r influence o n the flow w i l l be g iven . Af t e rwards , the i m p o r t a n t concept o f b a r o c l i n i c i t y w i l l be i n t r o d u c e d . T h i s feature o f the flow has been a t t r i b u t e d w i t h caus ing the i n s t a b i l i t y w h i c h leads to the t r a n s i t i o n f rom a x i s y m m e t r i c to n o n - a x i s y m m e t r i c flow. 2.1.1 The Rossby number: large scale flow W e are in teres ted i n large scale flows o n the E a r t h , w h i c h have the feature tha t r o t a t i o n is a d o m i n a n t factor i n i ts cha rac te r i za t ion . A useful measure of the i m p o r t a n c e of r o t a t i o n is the R o s s b y n u m b e r where V a n d L are the charac ter i s t ic ve loc i ty a n d l eng th scales of the flow a n d Q, is the r o t a t i o n rate o f the E a r t h . It s h o u l d be no ted tha t V is measured i n a f rame o f reference r o t a t i n g at a ra te of fi; i t is the charac te r i s t i c ve loc i ty re la t ive to the surface of the E a r t h . W h e n the R o s s b y number e is s m a l l , r o t a t i o n is i m p o r t a n t . O n e way to see th is is to l ook 17 at e as a r a t i o o f t i m e scales. I f a f lu id element is m o v i n g w i t h speed V, t h e n to move a d i s tance L w o u l d take a t i m e in t e rva l L/V. T h u s , a charac te r i s t i c t i m e scale o f the f l u id m o t i o n is TM = L/V. T h e p e r i o d of r o t a t i o n is p r o p o r t i o n a l to rR — 1 / (20), so the R o s s b y n u m b e r is the r a t i o o f the p e r i o d of r o t a t i o n to the charac te r i s t i c t i m e scale o f the m o t i o n : e = TR/TM (the factor of 1 / 2 i n TR is there by convent ion , see be low) . I f the R o s s b y n u m b e r e is ve ry s m a l l , t hen the t i m e scale o f the fluid m o t i o n is m u c h longer t h a n the p e r i o d of r o t a t i o n . T h i s i m p l i e s tha t r o t a t i o n c o u l d p l ay an i m p o r t a n t role, s ince the r o t a t i o n has t i m e to influence changes i n the m o t i o n . It is possible to define flows as ' large scale ' i f t hey have e < 1 , a l t h o u g h we w i l l be cons ide r ing flows w i t h e < < 1 . A n o t h e r w a y to l ook at t h i s is to no t ice tha t , i f L is the same order o f m a g n i t u d e as the r ad iu s o f the E a r t h , t hen QL is the same order of m a g n i t u d e as the v e l o c i t y o f the surface of the E a r t h . T h u s i f the Rossby number is s m a l l , the re la t ive v e l o c i t y (measured i n a f rame of reference r o t a t i n g w i t h the E a r t h ) is s m a l l c o m p a r e d to the v e l o c i t y of the surface of the E a r t h . F o r th is reason, a n d because i t is the dev ia t ions f r o m s o l i d - b o d y r o t a t i o n tha t are in te res t ing , i t is useful to s t udy large scale flows i n a r o t a t i n g reference frame. 2.1.2 The equations of motion T h e s t a r t i n g p o i n t w i l l be N e w t o n ' s second l aw for a fluid c o n t i n u u m , i n a reference frame r o t a t i n g u n i f o r m l y at a rate f2: P + 2 n x u + n x ( n x r ) j = - V p + pv$ + T ( 2 . 2 ) i.e. the change i n m o m e n t u m equals the force per un i t v o l u m e , where u = u(r,£) is the fluid ve loc i ty , r is the p o s i t i o n vector , t is t ime , p = p(r, t) is the pressure, p = p(r, t) is the dens i ty o f the fluid, V is the usua l th ree -d imens iona l gradient opera tor , pV$ is the b o d y force (e.g. g r a v i t y ) , T are the non-conservat ive forces (e.g. f r i c t i o n a l forces) a n d x is the vec tor cross p r o d u c t . T h e vector quan t i t y f2 represents the ra te of r o t a t i o n f2 m u l t i p l i e d b y the u n i t vec tor t ha t is p o i n t i n g a l o n g the ax i s o f r o t a t i o n such t h a t the r o t a t i o n is coun te r -c lockwise w i t h respect to the un i t vector . T h e equa t ion presented is i n vec tor f o r m (the b o l d t ype represents th ree -d imens iona l vectors) . T h e f o r m tha t the equa t ions take w h e n expanded w i l l depend on the coord ina te sys t em w h i c h is used. S ince the equa t ions descr ibe the change i n m o m e n t u m of fluid elements, they are somet imes referred to as the m o m e n t u m equat ions . T h e m a t e r i a l de r iva t ive d d also ca l l ed the subs tan t ive or t o t a l der iva t ive , represents the rate o f change of the g iven q u a n t i t y (e.g. u) t ha t a fluid element experiences as i t moves w i t h the flow. T h a t is , i t 18 is the ra te of change fo l lowing the flow. T h e p a r t i a l de r iva t ive d/dt is the ra te o f change of the q u a n t i t y a t a fixed l o c a t i o n i n space. T h e changes at a fixed l o c a t i o n are due to changes effected by ' ex te rna l ' var iables a n d the amoun t of the q u a n t i t y w h i c h is advec ted to (car r ied to) the l o c a t i o n by the flow. T h e t e r m u • V represents the advec t ion , w h i c h is the source o f the n o n l i n e a r i t y of the equat ions . T h e t e r m 2f2 x u i n the m o m e n t u m equat ions is the C o r i o l i s acce le ra t ion a n d ft x (f2 x r) is the cent r i fuga l acce lera t ion . These are the e x t r a te rms tha t arise due to the a p p l i c a t i o n of N e w t o n ' s l aw i n a n o n - i n e r t i a l frame of reference. M u l t i p l y i n g these te rms by the dens i ty p gives the associa ted forces, w h i c h are often referred to as i m a g i n a r y forces. These t e rms are respons ib le for the un ique character of geophys ica l flows a n d w i l l be d iscussed i n more d e t a i l be low. F o r a N e w t o n i a n fluid, the f r i c t i ona l force can be w r i t t e n as . F = / / V 2 u + ^ V ( V • u ) (2.3) where p is the m o l e c u l a r v iscosi ty . T h i s is exact o n l y i f p does not depend o n the v e l o c i t y o f the fluid (more precisely, when p, does not depend o n the ra te-of-s t ra in tensor; see [51]). In general , the m o l e c u l a r v i scos i ty p m a y be a func t ion o f t empera tu re , dens i ty a n d pressure. O n e o f the diff icul t ies w i t h cons ide r ing large scale flows is the non l inea r t e r m u • V u w h i c h is con ta ined i n the m a t e r i a l der iva t ive dn/dt. I f th i s t e r m was not present, i.e. i f the equat ions were l inear , then different scales w o u l d not influence each o ther a n d therefore c o u l d be so lved for independent ly . In the non l inea r case, th is is no t so, a n d the effects tha t the s m a l l scales have on the large scales cannot be ignored . A t t e m p t i n g to account for the effects of the s m a l l scale flow on the large scale flow is the so-ca l led t u rbu l en t c losure p r o b l e m a n d is one of the great unso lved p rob lems i n geophys ica l fluid d y n a m i c s a n d i n tu rbu lence m o d e l l i n g . T h e i dea is t o separate the large scales by w r i t i n g the s m a l l scale effects i n t e rms of large scale var iables . See [49] for a d i scuss ion . It is c o m m o n to i n c l u d e these effects w i t h the non-conservat ive f o r c e s T , w h i c h , therefore, w i l l not necessar i ly have the above f o r m (2.3). A l s o needed is the equa t ion for mass conserva t ion , ca l l ed the c o n t i n u i t y equa t ion : ^ + V - ( p u ) = | + p ( V - u ) = 0. (2.4) T h i s equa t ion states tha t the rate of change of dens i ty o f a fluid element is b a l a n c e d by the divergence o f the ve loc i ty o f the fluid at the l o c a t i o n of the element . I f the dens i ty is not constant , the first l aw of t h e r m o d y n a m i c s gives ano ther equa t ion , w h i c h after the a s s u m p t i o n t ha t p = p(p,T) a n d some m a n i p u l a t i o n (see [49]), becomes 19 Cp"4r--a^ = -^T + Q (2.5) dT Tdp_ dt p dt p where T = T ( r , t) is the t empera tu re , Cp is the specific heat at cons tant pressure, i (dP\ is the coefficient of t h e r m a l expans ion (the subscr ip t p ind ica tes t ha t the de r iva t ive is t aken at cons tan t pressure) , k is the t h e r m a l c o n d u c t i v i t y (assumed to be cons tant ) a n d Q is due to the i n t e rna l heat sources. T h e a s s u m p t i o n tha t p = p(p, T) is v a l i d for fluids cons i s t i ng of a s ingle substance. W i t h th is a s sumpt ion , i t is not possible to take i n to account s a l i n i t y effects i n the ocean or water vapour i n the a tmosphere . A t present there are s ix unknowns (u , T , p a n d p ) , bu t o n l y five equa t ions . Therefore , to f o r m a wel l -posed p r o b l e m , the equa t ion p = p(p, T ) , desc r ib ing the t h e r m o d y n a m i c proper t ies o f the fluid, mus t be specif ied. In s u m m a r y , the evo lu t i ona ry equat ions for a s t ra t i f ied fluid i n a r o t a t i n g frame of reference are p ^ + 2ft x u + Q x (n x r ) ^ = - V p + p V $ + T (2.6) ! + „ V . « = 0 (2.7) c>§ --4 = - v 2 r + « <2-8> at p dt p p = p(p,T) ' (2.9) where the different ia l hea t ing m a y be i n t r o d u c e d ei ther via the i n t e rna l heat source t e r m Q o r the b o u n d a r y cond i t i ons . A s yet , the d o m a i n or the b o u n d a r y cond i t i ons have not been discussed. C o n s i d e r i n g re-a l i s t i c d o m a i n s a n d b o u n d a r y cond i t ions w i l l g rea t ly increase the c o m p l e x i t y of the p r o b l e m a n d so, s imp l i f i ca t i ons of these are often cen t ra l features of a m o d e l . A t th i s t i m e , we defer fur ther d i scuss ion to the sections be low, where we present the p a r t i c u l a r s imp l i f i c a t i ons m a d e i n the mode l s o f interest . Here , i t w i l l s i m p l y be m e n t i o n e d tha t , usua l ly , t o p o g r a p h y is i gnored (or o n l y a very s imp le u n d u l a t i n g surface is considered) a n d the c o u p l i n g o f the a tmosphere a n d ocean are ignored (unless th is i t se l f is be ing s tud ied) . T h e choice o f coord ina te sys tem also m i g h t a d d an element of c o m p l e x i t y . T o cons ider the fluid flow on the surface of a r o t a t i n g planet , spher ica l coord ina tes are often used (see be low) . However , when the equat ions are w r i t t e n i n expanded f o r m , the use o f sphe r i ca l coord ina tes leads to a d d i t i o n a l (curvature) t e rms i n the equat ions . Therefore , the equat ions 20 are more eas i ly deal t w i t h when considered i n a C a r t e s i a n coo rd ina t e sys tem, as l o n g as there are a c c o m p a n y i n g a p p r o x i m a t i o n s . 2.1.3 The Coriolis and centrifugal forces W e are cons ide r ing the m o t i o n o f a fluid i n a r o t a t i n g sys t em. A s anyone w h o has a t t e m p t e d to w a l k on a m e r r y - g o - r o u n d w o u l d attest , the same laws of m o t i o n do not a p p l y to a r o t a t i n g sys t em as to a sys t em w h i c h is at rest. It is the i m a g i n a r y forces (the C o r i o l i s a n d cent r i fuga l forces) tha t arise f rom w r i t i n g the equat ions o f m o t i o n i n a n o n - i n e r t i a l reference frame, w h i c h lead to the u n i n t u i t i v e behav iour of a r o t a t i n g sys t em. F o r th is reason, i t is useful to examine t h e m further . I f the C o r i o l i s t e r m is b rought to the r i g h t - h a n d side of the equa t ion (2.6), i t becomes —p ( 2 0 x u), a n d c a n be in te rpre ted as a force affecting the acce le ra t ion of the fluid. N o t e t ha t th i s force is o n l y felt w h e n the ve loc i ty is non-zero, a n d tha t i t acts at r igh t angles to the ve loc i ty . T h i s p a r t i c u l a r effect leads to su rp r i s i ng behav iour . F o r the m o m e n t , consider a C a r t e s i a n coord ina te sys tem w h i c h has i ts v e r t i c a l (z) ax is c o i n c i d i n g w i t h the r o t a t i o n vector . In th i s case, for objects i n m o t i o n , the force w i l l be exc lus ive ly i n the h o r i z o n t a l a n d w i l l t end to push t h e m to the r igh t . See F i g u r e 2.1. T h u s , objects i n m o t i o n , w h i c h are no t sub jec ted to ex te rna l forces, w i l l appear to fo l low cu rved t ra jector ies w h e n observed i n a r o t a t i n g reference frame. See F i g u r e 2.1. F i g u r e 2.1: The Coriolis force. A particle moving with velocity u , in a frame of reference rotating counter-clockwise at rate fl, will feel a Coriolis force C pulling it to its right. A possible path the particle may follow is indicated with a dashed line. It is poss ib le to re interpret the Rossby n u m b e r e u s ing the C o r i o l i s acce le ra t ion . G i v e n V a n d L as above, a charac te r i s t i c scale for the acce lera t ion of the fluid (see equa t i on (2.6)) is V2/L (ve loc i ty over t ime) a n d for the C o r i o l i s acce le ra t ion is 2Q,V. T h u s , the R o s s b y z 2 1 n u m b e r is the r a t i o o f the re la t ive acce lera t ion to the C o r i o l i s acce le ra t ion (note the factor o f 1/2 i n the R o s s b y n u m b e r is now present) . A n d so, i f the R o s s b y n u m b e r is s m a l l , t h e n i t is the C o r i o l i s force w h i c h a p p r o x i m a t e l y balances the forces o n the r i g h t - h a n d side of equa t i on (2.6). I f a l l forces except the pressure t e r m are assumed to be o f order e or smal l e r , t hen the C o r i o l i s force a p p r o x i m a t e l y balances the pressure gradient . T h i s is ca l l ed the geos t rophic a p p r o x i m a t i o n , a n d is the fundamen ta l a s s u m p t i o n of the quas igeos t rophic p o t e n t i a l v o r t i c i t y equat ions discussed i n the next sec t ion . T h e cent r i fuga l force — p [fl x (ft x r)] acts to p u l l an object away f r o m the axis o f r o t a t i o n . C o n s i d e r an object t r a v e l l i n g w i t h u n i f o r m speed o n a c i r c u l a r p a t h a b o u t the o r i g i n , such t ha t i t s angu la r ve loc i ty is f). I f we observe the object f rom a s t a t i o n a r y ( iner t ia l ) reference frame, i t is c lear tha t a force, a c t i n g towards the o r i g i n , is needed to keep the object o n th i s p a t h ( i f there is n o force, there is no change i n m o m e n t u m a n d the object w o u l d t r ave l i n a s t ra ight l ine ) . See F i g u r e 2.2. S u c h a force is ca l l ed a c e n t r i p e t a l force. N o w , i f we look at the object i n a frame of reference r o t a t i n g at ra te Q, t h e n the object w i l l appear to be s ta t ionary . However , the force a c t i n g o n the object i n the i n e r t i a l f rame w i l l s t i l l be a c t i n g i n the r o t a t i n g frame. In order for the object to be s ta t ionary , by N e w t o n ' s second l aw, the s u m of the forces mus t be zero. N e w t o n ' s laws a p p l y i n i n e r t i a l frames o f reference, so i f we w o u l d l ike to use these laws i n the n o n - i n e r t i a l ( ro ta t ing) frame, i t is necessary to a d d an i m a g i n a r y force w h i c h balances the cen t r i pe t a l force. T h i s a d d i t i o n a l force is the cent r i fugal force. T h u s an object i n a r o t a t i n g reference frame a lways feels the cen t r i fuga l force t r y i n g t o p u l l i t away f r o m the ax i s o f r o t a t i o n (i.e. a n a p p l i e d force is needed to keep an object s t a t i ona ry ) . T h e cent r i fugal force is greater the fur ther the object is away f rom the axis o f r o t a t i o n a n d the h igher the r o t a t i o n rate a n d i t does no t depend o n the ve loc i ty of the object . It is i n s t ruc t i ve to wr i t e out the C o r i o l i s a n d centr i fugal t e rms e x p l i c i t l y i n sphe r i ca l co-ord ina tes , w h i c h is the most app l i cab le geomet ry for m o t i o n o n the E a r t h . I n th i s way, some of the i m p o r t a n t effects of the curva tu re can be seen, perhaps the mos t i m p o r t a n t b e i n g the va r i a t i ons o f the i m a g i n a r y forces as a func t ion o f l a t i t ude . A c o m m o n a p p r o x i m a t i o n assumes t ha t th i s v a r i a t i o n o f the C o r i o l i s force is the mos t i m p o r t a n t effect assoc ia ted w i t h the curva tu re of the planet . T h e sphe r i ca l coord ina tes w i l l be denoted (cp,6,r), see F i g u r e 2.3, where <p G [0,27r) is the l o n g i t u d i n a l coord ina te ( a z i m u t h a l angle) , 9 G [—7r/2, TY/2] is the l a t i t u d i n a l coo rd ina t e (equals 7r/2 m i n u s the p o l a r angle) a n d r G [0, oo) is the r a d i a l coo rd ina t e (dis tance f r o m the o r i g i n , the center o f the sphere) . W r i t e u = (u, v, w) = u(p + vO + wf, where the ha t s represent u n i t vectors i n the coord ina te d i rec t ions . T h e n the C o r i o l i s acce le ra t ion is 2ft x u = 2fl [(wcos6-vsin9)(p+(usm6)8- ( u c o s 0 ) f ] . (2.10) 22 F i g u r e 2.2: The centripetal force. A particle travelling in circle needs a force F acting radially to keep it on the circle. This force is real and is called the centripetal force. Gene ra l l y , the v e r t i c a l ve loc i ty w is assumed to be an order o f m a g n i t u d e s m a l l e r t h a n the h o r i z o n t a l ve loc i t ies (u a n d v), so the C o r i o l i s acce le ra t ion i n the h o r i z o n t a l is t hen a p p r o x i m a t e l y A t the equator , where 0 = 0, the h o r i z o n t a l C o r i o l i s acce le ra t ion is zero, w h i l e at the poles (9 = ± 7 r / 2 ) i t reaches a m a x i m u m . F o r th i s reason, i t is useful to define the local R o s s b y number , " l i <™> where / = 2fl s i n 9 is the C o r i o l i s parameter , a n d V a n d L are the charac te r i s t i c v e l o c i t y a n d l eng th scales of the flow. T h i s l o c a l pa ramete r is a more accura te measure , for ins tance , o f the v a l i d i t y o f the geost rophic a p p r o x i m a t i o n , w h i c h , i t c a n been seen, canno t h o l d near the equator . In sphe r i ca l coord ina tes , the cent r i fugal t e r m is also dependent o n l a t i t ude . A t the equator , i t acts pu re ly i n the v e r t i c a l w h i l e at the poles , i t is zero. U s u a l l y , the l a t i t u d i n a l componen t is neglected a n d the v e r t i c a l c o m p o n e n t is i n c l u d e d as a r e d u c t i o n i n gravi ty , since i t can be w r i t t e n as a b o d y force. ft x (ft x r ) = rfl2 [(sin 0 cos 0) § - ( cos 2 6) f (2.12) 23 r/ „ * \ / -©-F i g u r e 2.3: Spherical coordinates, r is the radial coordinate, (j) is the azimuthal coordinate and 0 is the latitudinal coordinate. 2.1.4 Baroclinic instability S t r a t i f i c a t i o n c o m b i n e d w i t h the v e r t i c a l shear o f the a x i s y m m e t r i c s o l u t i o n has been s t ip -u l a t e d to be the o r i g i n of the i n s t a b i l i t y w h i c h causes the t r a n s i t i o n f r o m a x i s y m m e t r i c to n o n - a x i s y m m e t r i c flow i n s t ra t i f ied r o t a t i n g systems. T h e v e r t i c a l shear i m p l i e s the existence of a h o r i z o n t a l t empera tu re gradient w i t h i n the fluid (via the geos t rophic app rox-i m a t i o n ) , therefore, the m e c h a n i s m has been ca l l ed b a r o c l i n i c i n s t ab i l i t y . A l t e r n a t i v e l y , i t is a m e c h a n i s m of interest since i ts existence depends on the presence o f r o t a t i o n a n d a h o r i z o n t a l t empera tu re gradient (wh ich m a y i m p l y different ia l hea t ing) . Before the presenta t ion o f the p a r t i c u l a r mode l s , we w i l l make a shor t d igress ion to discuss th i s i m p o r t a n t top ic . A fluid is ca l l ed b a r o c l i n i c i f surfaces of cons tan t dens i ty are no t a l i gned w i t h surfaces of constant pressure. It s h o u l d be no t ed t ha t for a b a r o t r o p i c fluid, these surfaces co inc ide , i n w h i c h case the effects of s t r a t i f i ca t ion m a y be neglected. B e l o w , we give a s imp l i f i ed e x p l a n a t i o n of the basic m e c h a n i s m of b a r o c l i n i c i n s t a b i l i t y w h i c h was o r i g i n a l l y a rgued by E a d y [13]. See also [49] or [19] for more d e t a i l . C o n s i d e r a s t ra t i f ied fluid i n e q u i l i b r i u m such tha t there is l igh te r fluid over heavier fluid. I f a fluid e lement (a s m a l l v o l u m e o f fluid) is a d i a b a t i c a l l y d i sp laced v e r t i c a l l y u p w a r d , t h e n the fluid element w i l l find i t se l f su r rounded by l igh ter fluid. D u e to gravi ty , the heavier fluid w i l l s ink back i n the d i r e c t i o n i t came, t e n d i n g to decrease the d i sp lacement . T h i s p r o p e r t y suggests tha t s m a l l pe r tu rba t ions f rom the o r i g i n a l fluid s t ruc ture w i l l no t s ign i f i can t ly change the s t ruc ture . T h u s , the e q u i l i b r i u m can be ca l l ed s table . T h i s a rgumen t is re levant , 24 for example , for Bouss inesq fluids, where the dens i ty does not depend o n the ambien t pressure. A m o d i f i c a t i o n is necessary for a tmospher i c a p p l i c a t i o n , however, the i dea is i d e n t i c a l . P = Pi F i g u r e 2.4: The mechanism of baroclinic instability. The isolines of constant density p of a fluid are tilted by an angle xjj from the horizontal. If a fluid element at position A is moved to position B, the particle will move further away from A. If the angle of displacement 4> is smaller than t/>, the same instability will be observed. N o w cons ider a f l u id i n an e q u i l i b r i u m , where the l igh te r fluid lies above the heavier fluid, b u t the surfaces o f constant dens i ty are t i l t e d by an angle ip i n the m e r i d i o n a l p lane . See F i g u r e 2.4. S u c h a s i t u a t i o n can arise, for ins tance, i n a geos t rophic z o n a l flow w i t h v e r t i c a l shear where the h o r i z o n t a l dens i ty gradient is needed to sat isfy the t h e r m a l w i n d equa t ions (see [49]). If a fluid element is d i sp laced f rom p o s i t i o n A i n F i g u r e 2.4, to p o s i t i o n B , i t w i l l be su r rounded by fluid w h i c h is heavier a n d w i l l feel a force w h i c h w i l l accelerate i t u p w a r d . T h i s tends to push the fluid element fur ther f rom i ts o r i g i n a l p o s i t i o n , i.e. there is an i n s t a b i l i t y . I f the angle o f d i sp lacement , (j), is sma l l e r t h a n tp. a n d above the h o r i z o n t a l , the i n s t a b i l i t y w i l l be observed. T h i s a rgument does not discuss under w h a t cond i t i ons there w i l l be p e r t u r b a t i o n s o f t h i s t ype w h i c h l ead to the i n s t a b i l i t y of the e q u i l i b r i u m . In fact, i f th i s i d e a is correct , these pe r tu rba t i ons cannot be present (or at least effective) i n a l l flows w i t h t i l t i n g dens i ty curves , s ince there ex is t s table flows w i t h th i s proper ty . F i n a l l y , the m e c h a n i s m depends c r u c i a l l y o n the existence of t i l t i n g surfaces o f cons tan t densi ty, w h i c h can be rea l ized i n a r o t a t i n g fluid i f the e q u i l i b r i u m flow has v e r t i c a l shear. 25 2.1.5 The general equations of motion: summary T h e genera l equat ions o f m o t i o n have been presented for a s t ra t i f ied fluid subjec t t o ro t a -t i o n a n d there has been a d i scuss ion o f some of the m a i n features o f such flows, namely , the p o s s i b i l i t y o f b a r o c l i n i c i n s t a b i l i t y a n d the C o r i o l i s a n d cent r i fuga l te rms. T h e equa-t ions a n d the concepts they incorpora te are the core of geophys ica l fluid d y n a m i c s a n d are present i n mos t mode l s . Ideal ly, i t w o u l d be possible to w o r k d i r e c t l y w i t h these equat ions , bu t r ea l i s t i ca l ly , i t is necessary to work w i t h s imp l i f i ed mode l s . B e l o w , different types o f s imp l i f i ca t i ons are used to fo rmula te two different models . O n e uses s ca l i ng a rguments a n d the o ther m o d e l s tudies a more s imp le phys i ca l sys tem w i t h o u t s i m p l i f y i n g the equat ions s igni f icant ly . It is hoped , however, tha t w i t h the s imp le mode ls s o m e t h i n g c a n be l ea rned of the genera l sys tem. 2.2 The two-layer quasigeostrophic potential vorticity equations T h e fundamen ta l a s s u m p t i o n o f the quas igeos t rophic p o t e n t i a l v o r t i c i t y equa t ions is tha t the flows b e i n g m o d e l e d have a s m a l l l o c a l Ros sby n u m b e r e. T h a t is , the effects o f r o t a t i o n are i m p o r t a n t . W i t h the l o c a l Ros sby number as the s m a l l pa rameter , the equa t ions c a n be de r ived f r o m the general govern ing equat ions (2.6) - (2.9) u s ing a f o r m a l a s y m p t o t i c expans ion . T h e expans ion results i n evo lu t iona ry equat ions for the zero th-order s t r e a m func t ions ( f rom w h i c h the zero th-order ve loc i t ies c a n be ca l cu l a t ed ) . T h e zero th-order dependent var iables are accurate to 0 ( e ) , where the n o t a t i o n O(-) i nd ica tes equa l i t y to a m u l t i p l i c a t i v e cons tant i n the l i m i t tha t e goes to zero. T h e two- layer vers ion of the m o d e l assumes tha t the fluid consists of two layers, where each layer has cons tan t dens i ty and there is no m i x i n g of the layers. T h e r a t iona le b e h i n d th is a s s u m p t i o n is tha t i t leads to the mos t s imp le sys tem i n w h i c h to s t u d y b a r o c l i n i c i n s t a b i l i t y ( w h i c h requires v e r t i c a l shear a n d s t r a t i f i ca t ion) . T h e s i m p l i f i c a t i o n resul ts i n the e l i m i n a t i o n o f the v e r t i c a l dependence of the dependent var iab les . T h e two- layer quas igeos t rophic p o t e n t i a l v o r t i c i t y equat ions have been shown to be a ve ry useful t o o l , a n d have been used by m a n y researchers i n c l u d i n g P e d l o s k y [48], L o r e n z [37], L e w i s [36], M o r o z a n d H o l m e s [46], A n d e r s o n [1] a n d W h i t a k e r and B a r c i l o n [55]. A n o u t l i n e o f the d e r i v a t i o n of the two-layer m o d e l is presented be low. F o r more d e t a i l see A p p e n d i x B or [49]. Here , we w i l l also discuss the a p p r o x i m a t i o n s i nvo lved a n d a t t e m p t to descr ibe the i r v a l i d i t y a n d usefulness. In th i s m o d e l , i t is assumed tha t the fluid consists of two i m m i s c i b l e layers a n d tha t 26 the fluid i n each layer is incompress ib le . T h e fluid is assumed to be con ta ined i n a p e r i o d i c channe l o f w i d t h L a n d height 2D (flat upper a n d lower boundar i e s ) . See F i g u r e 2.5. T h e dependent var iab les are funct ions of x, y a n d t. T h e x -coo rd ina t e gives the the z o n a l d i r e c t i o n a n d is analogous to the long i tude on the E a r t h . T h e y - coo rd ina t e defines the m e r i d i o n a l va r i ab le a n d is analogous to the l a t i t ude , z is the v e r t i c a l coo rd ina t e a n d t is t i m e . y-l a y e r 1 p l a y e r 2 ^2 h z=2D z=D z=0 F i g u r e 2.5: A two-layer fluid in a periodic channel of width L and height 2D. h = h(x,y,t) gives the height of the interface between the layers and pn is the constant density in the nth layer. It is a ssumed tha t the C o r i o l i s pa ramete r is / = / 0 + 30y, where / 0 = 2f2sin# 0> Po = ( 2 Q / r 0 ) cos#o> ^ is the r o t a t i o n rate of the E a r t h , r 0 is the rad ius o f the E a r t h , a n d 90 is a reference l a t i t u d e (wh ich de termines y = 0) . T h a t is , the C o r i o l i s force is a s sumed to va ry linearly w i t h l a t i t ude . It can be a rgued tha t th is channe l flow, c a l l ed the /3-plane a p p r o x i m a t i o n , mode l s the flow of a fluid on a sphere con ta ined between two l a t i t udes as l o n g as L/rQ is s m a l l (range o f l a t i t ude is s m a l l ) . T h a t is , i t is a ssumed tha t i n th i s case a l l cu rva tu re effects are negl ig ib le except the v a r i a t i o n of the C o r i o l i s force w i t h l a t i t u d e . T h e b o u n d a r y cond i t i ons at the upper a n d lower surfaces are d e t e r m i n e d t h r o u g h a m a t c h i n g w i t h a n E k m a n layer, w h i c h is a b o u n d a r y layer w h i c h assumes t ha t the R o s s b y n u m b e r e is s m a l l , f r i c t i ona l forces are i m p o r t a n t a n d the fluid v e l o c i t y is zero (no-s l ip) at the bounda ry . It is assumed tha t there are no edge effects at the l a t e r a l boundar i e s (y = ±L/2), a n d t ha t there is n o flow t h r o u g h these boundar ies . P e r i o d i c i t y i s r equ i r ed i n the ^ - d i r e c t i o n . T h e d e r i v a t i o n s tar ts w i t h the general equat ions (2.6) - (2.9) where the dependent 27 var iab les are funct ions of x, y, z a n d t and the non-conservat ive forces are g iven by T h a t is , the u sua l coefficient of v i scos i ty is replaced w i t h two la rger 'coefficients of v i scos i ty ' : A H for the h o r i z o n t a l va r i a t ions a n d Ay for the ve r t i ca l va r i a t ions [49], w h i c h are cons idered as parameters tha t depend o n the character is t ics of the s m a l l scale flow. T h i s a s s u m p t i o n is m a d e to a l l ow the dependent var iables to be in te rpre ted as large scale, where the effects o f the tu rbu len t sma l l e r scales are assumed to be pa rame te r i zed by the increased v iscos i ty . A l t h o u g h th i s is an ad hoc a s sumpt ion , i t is c o m m o n to make a s i m p l i f i c a t i o n o f th i s k i n d to o b t a i n a t r ac t ab le p r o b l e m . It is hoped tha t th is a s s u m p t i o n br ings the so lu t ions in to a l a m i n a r as opposed to a t u rbu l en t reg ime. S ince there is l i t t l e s m a l l scale v a r i a t i o n i n a l a m i n a r flow, i t c a n be hoped tha t e s t i m a t i o n techniques w i l l p roduce accura te resul ts . A f t e r a r e sca l ing o f the equat ions , i t is assumed tha t the dependent var iab les c a n be e x p a n d e d i n a series of powers of e. A t th is po in t several a s sumpt ions are m a d e conce rn ing the m a g n i t u d e of some of the d imensionless parameters . In p a r t i c u l a r , the effective h o r i -z o n t a l R e y n o l d s number , Re = VL/An, and the va r i a t ions of the height o f the interface are b o t h assumed to be of order e. T h e h o r i z o n t a l l eng th scale L is a ssumed to be m u c h larger t h a n the v e r t i c a l l eng th scale D a n d the h o r i z o n t a l ve loc i t ies are assumed to be m u c h larger t h a n the v e r t i c a l ve loci t ies . T h e h o r i z o n t a l pressure gradient is a ssumed to be o f the same order as the C o r i o l i s pa ramete r a n d the v e r t i c a l pressure grad ien t is a s sumed to be g iven by the hyd ros t a t i c a p p r o x i m a t i o n . F i n a l l y , the v e r t i c a l coefficient o f v i scos i ty Ay is a ssumed to be 0 ( e 2 ) , see [49]. C o n s i d e r i n g a l l these assumpt ions , the te rms i n the equat ions w h i c h are 0 ( 1 ) (zeroth-order) are co l lec ted to give the geost rophic a p p r o x i m a t i o n , where the C o r i o l i s force balances the pressure gradient . T h i s , however, does not give evo lu t i ona ry equat ions for the ze ro th -order dependent var iables a n d so i t is necessary to col lec t the 0 ( e ) t e rms . Here , i t is poss ib le to e l i m i n a t e the first-order dependent var iables . U p o n i n t eg ra t i on i n the v e r t i c a l , a n d a p p l i c a t i o n o f the cond i t i ons at the boundar ies , the layered quas igeos t rophic p o t e n t i a l v o r t i c i t y equat ions are ob t a ined . T h e equat ions descr ibe the t i m e e v o l u t i o n o f the ze ro th -order s t r e a m func t ion i n each layer (from w h i c h the flow v e l o c i t y can be ca l cu l a t ed ) , w h i c h are independent o f the v e r t i c a l coord ina te z. N o t e tha t i t is poss ib le to i n t r o d u c e the two-layer a p p r o x i m a t i o n o n l y at th i s last step, w i t h the v e r t i c a l i n t eg ra t ion b e i n g pe r fo rmed o n the t h ree -d imens iona l quas igeos t rophic p o t e n t i a l v o r t i c i t y equat ions . C o n s i d e r the two-layer , n o n - m i x i n g a s sumpt ion . In the annu lus expe r imen t , mos t o f the v e r t i c a l m o t i o n occurs close to the annulus boundar ies , a n d the flow i n the in t e r io r is p r e d o m i n a n t l y h o r i z o n t a l . Observa t ions show tha t large scale m i d - l a t i t u d e flows i n the (2.13) 28 a tmosphere are also p r e d o m i n a n t l y h o r i z o n t a l . T h u s , i t m a y be a rgued t ha t the v e r t i c a l m o t i o n is mos t i m p o r t a n t for the c rea t ion of the v e r t i c a l s t ruc ture o f the fluid a n d o therwise has secondary effects. F u r t h e r m o r e , i n P e d l o s k y [49], i t is shown tha t there is a 'one-to-one ' r e l a t i onsh ip between the layered m o d e l a n d a f ini te difference a p p r o x i m a t i o n i n the v e r t i c a l . These a rguments , however, w o u l d seem more p laus ib le i n the case w h e n there are m u l t i p l e layers. W e w i l l s i m p l y restate tha t the purpose of the two-layer a p p r o x i m a t i o n is to develop the mos t s i m p l e m o d e l poss ible i n w h i c h b a r o c l i n i c effects can be s tud ied . U p o n c o m p a r i n g the resul ts of, for ins tance , B a r c i l o n [3] a n d P e d l o s k y [48], i t c a n be seen t ha t the two- layer m o d e l con ta ins some o f the same character is t ics o f the t r a n s i t i o n f r o m a x i s y m m e t r i c to n o n - a x i s y m m e t r i c flow as the th ree -d imens iona l m o d e l . T h u s , i t is reasonable to assume tha t resul ts f r o m a two-layer m o d e l w i l l be q u a l i t a t i v e l y useful. A l t h o u g h m a n y assumpt ions r ega rd ing the m a g n i t u d e of the parameters are made , these are chosen w i t h the a i d of observat ions [49]. E v e n so, i t c an be a rgued t ha t the m o d e l re-produces flows w h i c h are i n the region o f paramete r space where the a s sumpt ions are v a l i d . I f we w i s h to observe flows outs ide th is region, the d e r i v a t i o n has to be repea ted w h i l e cons ide r ing new assumpt ions . F o r th i s reason, i t is necessary to r emember w h i c h approx-i m a t i o n s are m a d e so tha t we do not i nadve r t en t ly move ' too far ' f r o m the range where the a p p r o x i m a t i o n s are reasonable. It shou ld also be no ted tha t the two- layer equat ions are o n l y for the zeroth-order dependent var iables a n d therefore the best we c a n hope to have is a n 0 ( e ) a p p r o x i m a t i o n . F o r t y p i c a l large scale a tmospher i c flows o f interest , the observed R o s s b y number is a p p r o x i m a t e l y 0.1 [49]. Therefore , the errors r ega rd ing the a s sumpt ions of the size of the parameters seem acceptable i n r e l a t i o n to the p o t e n t i a l er-rors due to the a s s u m p t i o n tha t the flow can be descr ibed by two layers of a n o n - m i x i n g incompres s ib l e fluid, a n d the unprovab le a s s u m p t i o n tha t the s m a l l scale t u rbu l en t m o t i o n c a n be accoun ted for w i t h the coefficients AH and Av. C o n s i d e r i n g the assumpt ions tha t have been made , the m o d e l is not expec ted to give q u a n t i t a t i v e l y accura te results . However , the purpose of the m o d e l is t o p rov ide a s i m p l e t o o l w i t h w h i c h to q u a l i t a t i v e l y s t udy ce r t a in charac ter i s t ics of geophys ica l fluid d y n a m i c s . I n fact, th i s m o d e l exh ib i t s so lu t ions w h i c h c a n be associa ted w i t h observed a t m o s p h e r i c flows [48], [36]. T h e p a r t i c u l a r m o d e l s tud ied i n th is thesis conta ins a s inuso ida l fo rc ing t e r m w h i c h mode l s , i n a s i m p l e way, the l a t i t u d i n a l va r i a t ions of r ad i a t ive hea t ing . T h e fo rc ing resul ts i n a bas ic state (steady so lu t ion) w h i c h has s inuso ida l h o r i z o n t a l shear as w e l l as v e r t i c a l shear. T h e h o r i z o n t a l shear is a n a d d i t i o n a l p o t e n t i a l source o f i n s t a b i l i t y . It s h o u l d be n o t e d tha t an a l t e rna t ive v i ew is often taken; the s inuso ida l f o r m of the bas ic state is a s sumed a n d the fo rc ing is p rescr ibed so tha t the basic state satisfies the equat ions . L e w i s 29 [36] s t ud i ed th i s m o d e l u s ing n u m e r i c a l expe r imen t a t i on , a n d a rgued t ha t i t c o u l d be used to s t u d y the a tmospheres o f the g iant planets a n d the i n t e r n a l l y hea ted r o t a t i n g annulus expe r imen t . T h e two- layer quas igeos t rophic p o t e n t i a l v o r t i c i t y equat ions are g iven by 9V2a + 8^ + J [a, V V ) + J (d, V V ) = -r~x V V + Re'1 V V (2.14) dt dx ^ ( V 2 - 2F) 5 + 0^'+J V 2 5 ) + J (5, W ) + 2FJ (5, a) -r-1V28 + Re-1V2(y2-2F)5-Q (2.15) where Q = -4TT 2 ( r _ 1 + . R e - 1 (4TT2 + 2 F ) ) cos 2ity . T , . du dv du dv . . . . I n . . . . a n d J (u , v) = — — - — — , er = cr(x, y, t) = ( x i + X 2 ) / 2 , d = 5{x, y, t) = (xi - X2)/2 (where \n I S the s t r eam func t ion i n the n t h layer , n = 1,2), V 2 is the L a p l a c i a n i n two d imens ions a n d r, Re, 3 a n d F are rea l parameters . T h e pa ramete r r = eD (2fo/Av)* is the inverse of a rescaled E k m a n number , Re is the effective R e y n o l d s number , 3 = 8QL/ (e/o), is the scaled 80, a n d F = f^L2/ (g (Ap/p0) D) is the i n t e rna l r o t a t i o n a l F r o u d e number , where Ap — px — p 2 , Pn b e ing the dens i ty of the f lu id i n the n t h layer, p 0 is the average dens i ty a n d g is the g r a v i t a t i o n a l acce lera t ion . See A p p e n d i x B . T h e d o m a i n is g iven by: —oo < x < oo, 0 < y < 1, where x and y have been scaled a n d y has been shif ted for convenience. T h e rescaled E k m a n number r can be in te rpre ted as represent ing the i m p o r t a n c e o f the ' v i scous ' effects tha t are i m p o s e d via the E k m a n layer. A s r decreases, the assoc ia ted d i s s i p a t i o n increases. In fact, th i s pa ramete r is more i m p o r t a n t t h a n the effective R e y n o l d s number , w h i c h represents the v iscous effects w i t h i n the layer. T h e F r o u d e n u m b e r F is a n o n - d i m e n s i o n a l measure o f the re la t ive i m p o r t a n c e of r o t a t i o n a n d s t r a t i f i ca t i on a n d c a n be w r i t t e n i n te rms of a B u r g e r n u m b e r [49]. If the dens i ty difference between the layers is assumed to be generated by different ia l hea t ing , i t c a n be in te rp re ted as a measure o f the re la t ive i m p o r t a n c e of r o t a t i o n to different ia l hea t ing . T h e b o u n d a r y cond i t i ons are: (i) a(x,y,t) = a(x + -y,y,t) a n d 8{x, y, t) = 5 (x + 7, y, t), i. e. a a n d 5 are pe r iod i c i n x w i t h p e r i o d 7, w h i c h is a parameter . .... da 05 n (11) — = — = 0 at y = 0 a n d y = 1. T h i s ensures no t angen t i a l ve loc i ty at the l a t i t u d i n a l boundar i e s . 30 1 rx d2a 1 fx d25 (iii) l i m —- / -rr-rrdx = l i m —- / Tr-zrdx = 0 at y = 0 a n d y = 1. v ' x^oo 2X J-x dydt x-+oo 2X J-x dydt T h i s means there is no change i n c i r c u l a t i o n abou t the l a t i t u d i n a l boundar i e s . W i t h Q chosen as above, a s teady so lu t ion is: 6 = - cos {2ny) (2.16) a = 0. (2.17) T h i s s o l u t i o n corresponds to a s t r eam func t ion i n the first layer Xi = ~ c o s fay) a n d the oppos i t e s ign i n the second layer. T h e zona l fluid ve loc i ty un a n d the m e r i d i o n a l ( l a t i t u d i -nal ) v e l o c i t y vn, i n each layer (ro = 1,2), can be ca l cu l a t ed f r o m the s t r e a m func t ions Xn us ing un = -dxn/dy a n d vn = dx/dx. T h e s teady s o l u t i o n is therefore g iven by u\ = — 2-7T s in (27n/) u2 = 27rsin(27ry) (2.18) w i t h no flow i n the l a t i t u d i n a l d i r e c t i o n i n b o t h layers (vn = 0, ro = 1,2), where un is the fluid v e l o c i t y a l o n g the channe l i n the roth layer. T h i s is a p u r e l y z o n a l flow w h i c h is independen t o f x a n d has non l inea r l a t i t u d i n a l shear. T h e r e are two oppos i t e l y d i r ec t ed ' jets ' i n each layer a n d also oppos i t e ly d i rec ted i n the v e r t i c a l . See F i g u r e 2.6. 2.3 The differentially heated rotating annulus exper-iment A s i m p l e e x p e r i m e n t a l sys tem w h i c h has been used to s t u d y r o t a t i n g s t ra t i f ied flow is the di f ferent ia l ly heated r o t a t i n g c y l i n d r i c a l annulus . In th is sys tem, the annu lus is p l aced o n a t u rn - t ab l e w i t h fluid filling the space between the c y l i n d e r wa l l s . T h e different ia l h e a t i n g is i m p o s e d by m a i n t a i n i n g the s ide-wal ls of the annulus at different t empera tu res . See F i g u r e 2.7. T h e deta i l s of the flow are observed as the t empera tu re difference a n d r o t a t i o n ra te o f the tu rn - t ab l e are va r i ed . Here , we are interested i n s t u d y i n g a m a t h e m a t i c a l m o d e l w i t h the intent of q u a n t i t a t i v e l y r e p r o d u c i n g the observat ions of the l a b o r a t o r y expe r imen t . F o r th i s reason, we t r y to keep the a p p r o x i m a t i o n s to a m i n i m u m . B e l o w , we descr ibe the equat ions o f the m o d e l w h i c h are very s i m i l a r to the general equat ions (2.6) - (2.9). However , to beg in , we w i l l discuss the l a b o r a t o r y exper iments i n more d e t a i l a n d i n so d o i n g , descr ibe some of the results we hope to reproduce . 31 y o o x b) 0.9 0.8 -0.7 -0.6 • =~OS 0.4 0.3 -0.2 -0.1 -o> • ' ' ' 1 ' ' ' ' O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x F i g u r e 2.6: The fluid velocity ui in the first layer in the longitudinal (x) direction. The vertical axis is in units of 2n. (a) a surface plot; (b) a contour plot which shows that there is no variation of the fluid velocity in the longitudinal direction. 2.3.1 The experiments M a n y e x p e r i m e n t a l conf igura t ions have been used to s t u d y s t ra t i f ied fluids subjec t t o ro-t a t i o n . See [26] or [52] for a review. M a n y of the first expe r imen t s were no t i n c y l i n d r i c a l a n n u l i , bu t i n a r o t a t i n g d i sh pan , where the edge of the p a n was hea ted ( E x n e r , 1923 vis. [18], [17]). A l t h o u g h the geomet ry seems to be more a p p l i c a b l e to the a tmosphere a n d ocean (smal le r height to w i d t h r a t io as we l l as no inner b o u n d a r y ) , some o f the flow features were no t r ep roduc ib l e due to expe r imen ta l diff icul t ies [26]. T h e r e has also been a n expe r imen t w h i c h has s i m u l a t e d the differential h e a t i n g by s t u d y i n g a two- layer fluid where the flow was generated by r o t a t i n g the t op a n d b o t t o m surfaces at different rates [22]. Here , we concent ra te o n the dif ferent ia l ly heated r o t a t i n g annu lus expe r imen t s . E v e n w i t h i n th i s class o f exper iments , m a n y var ia t ions are poss ib le . H i d e a n d M a s o n [27] used i n t e r n a l hea t ing via an e lec t r i ca l current to di f ferent ia l ly heat the fluid w i t h the i dea tha t i t m o d e l s so lar h e a t i n g m o r e rea l i s t i ca l ly . A l s o , for s i m i l a r reasons, e x p e r i m e n t s have been done w i t h d i f ferent ia l h e a t i n g f rom be low [24]. However , mos t often, the di f ferent ia l hea t ing 32 is p r o d u c e d by m a i n t a i n i n g the inner and outer c y l i n d e r wa l l s at different t empera tu res . M a n y exper imen t s of th i s k i n d have been per formed w i t h different e x p e r i m e n t a l var iab les (e.g. different fluid proper t ies a n d a n n u l i of different d imens ions) w h i c h , we w i l l see, has a n effect o n the results . U s u a l l y , the outer c y l i n d e r is wa rmer t h a n the inner c y l i n d e r . However , the oppos i te was used to s t udy the effects of the cent r i fuga l b u o y a n c y [31] (see be low) . A n i m p o r t a n t d i s t i n c t i o n between some o f the exper iments is i n the n a t u r e o f the uppe r bounda ry . Some exper iments a l lowed the uppe r surface to be free (no l i d ) , i n w h i c h case t racers were i n t r o d u c e d in to the l i q u i d a n d the observat ions were m a d e v i s u a l l y by the exper imen te r [16], [11]. O t h e r exper iments h a d a r i g i d l i d [29], [30], [32], [35], where the flow was observed via t empera tu re fluctuations measured by the rmocoup le s p l aced at va r ious loca t ions ins ide the annulus . U s i n g th is technique, the v e r t i c a l s t ruc tu re o f the flow is more eas i ly observed by p l a c i n g the rmocoup les at var ious depths . However , the t he rmocoup le s are invasive . A d iscuss ion of the poss ible effects of the t he rmocoup le s is g iven by F e i n [15]. W i t h the use of m u l t i - l e v e l t racer s t reak p h o t o g r a p h y [11], i t is also poss ib le to o b t a i n the v e r t i c a l s t ruc ture w i t h o u t the rmocouples . However , i t seems tha t t h i s technique has not been w i d e l y used. F e i n [15] gives a n e x p e r i m e n t a l c o m p a r i s o n of the r i g i d versus free uppe r boundary . In the free upper b o u n d a r y exper imen t s , the uppe r surface o f the fluid tends to have a p a r a b o l i c shape w i t h concav i ty d e p e n d i n g o n the ra te o f r o t a t i o n . It has been a rgued tha t the p a r a b o l i c upper surface s imula tes a r a d i a l v a r i a t i o n o f the r o t a t i o n rate s i m i l a r t o the 3 effect [41], [15]. T h e v a r i a t i o n o f the u p p e r surface i n the free uppe r b o u n d a r y case makes the m a t h e m a t i c a l ana lys i s more dif f icul t t h a n for the r i g i d l i d case. However , i f the va r ia t ions i n height are neg l ig ib le , the a s s u m p t i o n o f a flat uppe r surface w i t h stress free b o u n d a r y cond i t i ons at the t o p of the annu lus m a y be reasonable . See, for ins tance , M i l l e r a n d B u t l e r [43] w h o per fo rmed n u m e r i c a l expe r imen t s u s ing th i s a s s u m p t i o n . T h e r e are also va r i a t ions i n the expe r imen ta l procedures , e.g. the i n i t i a l i z a t i o n o f the flow. S o m e exper imen t s pe r fo rm a ' sp in -up ' f rom rest to the des i red r o t a t i o n rate [29]; t rans ients are a l lowed to pass before the flow p rope r ty is de t e rmined . O t h e r s inc rement the pa ramete r s s lowly , a l l o w i n g e q u i l i b r a t i o n to take p lace before c h a n g i n g the parameters . In th i s manne r , the t r a n s i t i o n curves are more easi ly t raced out a n d the b i f u r c a t i n g flow m a y be observed to g row s lowly as the parameters are increased. T h i s is the p rocedure w h i c h is mos t l ike the b i fu r ca t i on analys is w h i c h we w i l l pe r fo rm. It s h o u l d be no t ed however t ha t theore t ica l ly , i f o n l y the i nc remen ta l approach is used, t hen a l l the poss ib le l o n g t i m e b e h a v i o u r m a y no t be observed. T h e idea of the ' sp in -up ' is t ha t the noise i n the i n i t i a l c o n d i t i o n s m a y cause the flow to approach different e q u i l i b r i a . A n o t h e r m e t h o d w h i c h has 33 been used i n a n a t t e m p t to reproduce a l l possible so lu t ions is the m i x i n g m e t h o d [30]. Here , the f l u i d is a l l owed to equ i l ib ra t e a t a g iven r o t a t i o n ra te a n d t e m p e r a t u r e di f ferent ia l . I t is t hen t h o r o u g h l y m i x e d a n d the resu l t ing flow is observed. T h e hope is t ha t the m i x i n g creates a va r i e ty of i n i t i a l cond i t i ons w h i c h are 'different ' enough tha t a l l the poss ib le e q u i l i b r i a (stable a t t rac tors ) m i g h t be found . A s discussed i n C h a p t e r 1, the results are u sua l ly presented i n a log - log p lo t o f the T a y l o r n u m b e r T = 4Q2R5/(u2D) versus the t h e r m a l R o s s b y n u m b e r K = agDAT/(02R2). 0 is the rate of r o t a t i o n , R = — ra is the difference between the outer a n d inner r a d i i o f the annulus , v is the k i n e m a t i c v i scos i ty of the fluid a n d D is the d e p t h of the fluid, A T = T j , — T 0 is the i m p o s e d h o r i z o n t a l t empera tu re gradient , a is the coefficient of t h e r m a l expans ion a n d g is the g r a v i t a t i o n a l acce lera t ion . A l t h o u g h there have been m a n y conf igura t ions a n d d imens ions o f the r o t a t i n g annu-lus exper iments , m a n y of the observat ions are s t r i k i n g l y s i m i l a r . I n p a r t i c u l a r , general ly , the expe r imen t s find four m a j o r flow regimes: a x i s y m m e t r i c , s teady wave, v a c i l l a t i o n a n d i r regu la r . See F i g u r e 1.4 for a schemat ic p lo t of a t y p i c a l exper imen t . T h e shape o f the t r a n s i t i o n curve between the a x i s y m m e t r i c a n d n o n - a x i s y m m e t r i c regimes is genera l ly o f the same shape as w e l l . T h e r e is a c r i t i c a l value w h i c h the T a y l o r n u m b e r mus t exceed for the a x i s y m m e t r i c s o l u t i o n to be unstable . F o r values o f the T a y l o r n u m b e r less t h a n th i s va lue , the a x i s y m m e t r i c so l u t i o n is s table a n d for values greater, there is a range o f the t h e r m a l R o s s b y n u m b e r for w h i c h i t is uns table . T h i s leads t o the knee shape o f the t r a n s i t i o n curve . T w o other i m p o r t a n t c o m m o n features are (1) the v a r i a t i o n o f the c r i t i c a l wave n u m b e r (the wave number o f the b i fu rca t ing wave) a long the t r a n s i t i o n curve be tween the a x i s y m m e t r i c a n d n o n - a x i s y m m e t r i c regimes a n d (2) the hysteresis of s teady waves o f different wave numbers i n the s teady wave regime. T h e wave n u m b e r gives the n u m b e r o f s p a t i a l u n d u l a t i o n s i n a u n i t d is tance (e.g. the n u m b e r of crests per u n i t d i s tance) . A l t h o u g h the different d imens ions of the e x p e r i m e n t a l annulus lead to s i m i l a r shaped reg ime d i ag rams , there are signif icant differences i n the c r i t i c a l wave numbers a n d the loca t i ons i n pa ramete r space where the t rans i t ions occur . T h i s c a n be seen by c o m p a r i n g the resul ts o f James et al. [30] a n d those o f F e i n [15]. Different exper iments have revealed hysteresis i n var ious regions. T h e mos t m a r k e d , however, seems to be i n the s teady wave regime; different wavelengths are seen i n the same subregions . See [28], [32] a n d [29]. T h e hysteresis is u s u a l l y s t ud i ed by a l l o w i n g e q u i l i b r a t i o n of a s table wave so l u t i o n of a ce r t a in wave n u m b e r for a g iven value o f the parameters . O n e of the parameters is inc remented u n t i l the wave becomes uns tab le a n d another wave o f a different wave number is a l lowed to equ i l ib ra te . T h e process is t h e n reversed u n t i l the second wave becomes uns tab le and the o r i g i n a l wave is aga in observed. 34 Hysteres i s occurs w h e n the fo rward a n d b a c k w a r d t r ans i t ions between the two waves do not occu r at the same l o c a t i o n i n paramete r space. In the annulus exper iments , the hysteresis o f the waves occurs i n a l l regions of the s teady wave regime, a n d seems to be assoc ia ted w i t h the c r i t i c a l wave n u m b e r t r ans i t ions a l o n g the a x i s y m m e t r i c to n o n - a x i s y m m e t r i c t r a n s i t i o n curve [32]. W e hope the analys is w i l l shed l igh t o n the m e c h a n i s m o f th is hysteresis . 2.3.2 The mathematical model In m o d e l i n g the exper iment d i rec t ly , we choose the same annulus d imens ions a n d fluid p roper t ies tha t were used i n the exper iments of F e i n [15]. I n order to keep the a p p r o x i -m a t i o n s to a m i n i m u m , we s tar t w i t h the general equat ions (2.6) - (2.9) where we assume the non-conserva t ive forces T are g iven by equa t ion (2.3). W e also adop t the B o u s s i n e s q a p p r o x i m a t i o n . I n th i s a p p r o x i m a t i o n , va r i a t ions of a l l fluid proper t ies are as sumed to be neg l ig ib le a n d the effects o f the v a r i a t i o n of the dens i ty o f the fluid are assumed to be evident o n l y i n the buoyancy forces. In pa r t i cu l a r , the equa t ion o f state o f the fluid is assumed to be p = Po(l-a(T-T0)) (2.19) where p is the dens i ty of the fluid, T is the tempera tu re , a is the coefficient o f t h e r m a l expans ion , po is the dens i ty at a reference t empera tu re To a n d a (T — To) is a s sumed to be s m a l l c o m p a r e d to the q u a n t i t y one. W i t h th is a p p r o x i m a t i o n , the fluid can be cons idered incompress ib l e as l o n g as the t empera tu re gradients are not too large. T h e b o u n d a r y cond i t i ons w h i c h are chosen require the ve loc i ty o f the fluid t o be zero o n a l l boundar i e s (no-s l ip cond i t i ons ) , a n d we consider the t empera tu re at the inner a n d outer c y l i n d e r he ld f ixed at Ta a n d Tb, respect ively, w h i l e the b o t t o m a n d t o p are i n s u l a t i n g ( this requires the der iva t ive of the t empera tu re w i t h respect to the v e r t i c a l coo rd ina t e to v a n i s h at the boundar ies ) . W e consider the fluid i n a reference frame r o t a t i n g w i t h the cy l i nde r , w h i c h gives rise to te rms i n the equa t ion w h i c h represent the C o r i o l i s a n d cent r i fuga l forces. C i r c u l a r c y l i n d r i c a l coord ina tes are used, where the r a d i a l , a z i m u t h a l a n d v e r t i c a l (or ax i a l ) coord ina tes w i l l be denoted r, (p a n d z, respect ively, w i t h u n i t vectors ?, (p a n d A; (see F i g u r e 2.7). In the B o u s s i n e s q a p p r o x i m a t i o n , the nonconservat ive forces are g iven by T = p V 2 u , a n d the c o n t i n u i t y equa t ion is reduced to the i n c o m p r e s s i b i l i t y c o n d i t i o n V • u = 0, where u = ui + vtp + wk is the ve loc i ty o f the f lu id . U p o n d i v i d i n g by p, the vec to r f o r m of the general equat ions (2.6) - (2.9) i n c y l i n d r i c a l coord ina tes becomes : = uV2u -2Qkxu+ (gk - Q?ri) a(T - T 0 ) - — Vp - ( u • V ) u (2.20) 35 D J a. R fluid F i g u r e 2.7: The differentially heated rotating annulus. Circular cylindrical coordinates are used: the radial coordinated r is the distance from the cylinders' axes, <p is the azimuthal coordinate and the vertical coordinate z is the distance from the floor of the cylinders. The annulus is rotated at rate Ct and the inner wall is held at the fixed temperature Ta and the outer wall at temperature Tt, creating a differential heating. The inner and outer radii of the annulus are at r = ra and r = rj. The width of the annulus is R = r\) — ra and D is the height of the annulus. dt v ' V - u = 0 where p is the pressure d e v i a t i o n f rom pQ = pog(D - z) + \pQQ,2r2, u = p/p0 is the k ine -m a t i c v iscos i ty , K = k/(p0Cp) is the coefficient of t h e r m a l di f fusivi ty , g is the g r a v i t a t i o n a l acce le ra t ion a n d the d o m a i n is ra < r < r^, 0 < ip < 2ir, 0 < z < D. T h e k i n e m a t i c v i scos i ty v a n d the coefficient of t h e r m a l d i f fus iv i ty K are assumed to be constants . W i t h the r o t a t i o n vector a l igned w i t h the ve r t i ca l ax is , the C o r i o l i s acce le ra t ion 2Q,k x u a n d cen t r i fuga l acce le ra t ion — Q,2ra (T — T 0 ) i act o n l y i n the h o r i z o n t a l . S ince the cons tan t dens i ty pa r t o f the cent r i fugal force has been e l i m i n a t e d by a s s u m i n g the f o r m of po, the r e m a i n i n g pa r t depends o n l y o n the v a r i a t i o n of the fluid dens i ty a n d is therefore often ca l l ed the cent r i fuga l buoyancy t e r m . T h e other b o d y force is g iven by the g r a v i t a t i o n a l b u o y a n c y t e r m ga ( T — T 0 ) j. T h e b o u n d a r y cond i t i ons are u = 0 T = Ta (2.21) (2.22) o n r = ra, ri, a n d z = 0, D o n r = ra 36 T = Tb on r = rb dT — = 0 on z = 0,D (2.23) a z w i t h 2TT p e r i o d i c i t y i n <p. In order t o general ize the p r o b l e m a n d s i m p l i f y the b o u n d a r y cond i t i ons , we make a change o f var iables : r = Rr' z = Dz' (2.24) a n d w r i t e T = T' + ATr' - A T ^ + Ta (2.25) where R — rb — ra a n d A T = Tb—Ta. U p o n d r o p p i n g the p r imes , the equat ions i n e x p a n d e d f o r m become du /„, u 2 dv\ n _ 1 dp i2 ' * i ? 2 ' s R? cP_ ld_ dr2 r dr r2 dip2 82 dz2 if _ v2 \ 1 f du v du 1 du v2' r 1 / « f \ 1 / dv v dv 1 dv uv R\ s r J R\ dr r dip 5 dz r (2.26) dt " V ^ P o ^ o V -tfRarT - Q2Rar (ATr - A T ^ + T a - T 0 ) - JV r dv f„9 v 2 du\ ^ 1 dp A r / r i _ = j _ M „ _ ^ _ A f , (2.27) (2.28) c)T A T A T = K S V 2 T + K S — - ^ u - N T (2.29) a t r R £ + - + sr + j7r = ° ( 2 3 0 ) dr r dip d dz where u = m + v<p + IUK, O = — , vs = — , K S = — a n d V 2 = — + 4 - - — + (2 31) 1 , . 1 ( dw vdw 1 dw\ 37 T h e b o u n d a r y cond i t i ons are now u = 0 on T 7 , T; A N D z = 0,l T dT dz~ 0 0 o n o n r = ra H W R R' R 0,1 (2.33) w i t h 2ir p e r i o d i c i t y i n <p. It is now in s t ruc t i ve to non-d imens iona l i ze the sys tem. W e rescale var iab les as R2 Tt, T = V U T Vu, V = ( A T ) T v p ->• Pp, P = R (2.34) where R, r , V a n d P are the charac te r i s t i c l eng th scale, t i m e scale, v e l o c i t y a n d pressure of the f low. T h e charac te r i s t i c l eng th has been chosen as the gap w i d t h R a n d the v i scous t i m e scale has been chosen. T h e charac te r i s t i c ve loc i ty m a y be in te rp re ted as a m i x t u r e of a diffusive ve loc i ty (u/R) a n d the annulus ve loc i ty (RQ). N o t e t ha t the s p a t i a l coord ina tes have a l r eady been scaled u s ing R a n d D (see above) . W i t h th i s s ca l i ng the equat ions become du di dv di dw Tt dT dt ^ 2 U 2 dv r2 dip v 2 du r 2 r 2 dip + Vfv | - ( a A T r ) f + 4 i V r /= I dp T VTu- —iV v r dip 5dz T 4 T. KV2T + R - - ^ - U - ^-NT s r 4 4 (2.35) (2.36) (2.37) (2.38) where T = T + r - rJR + ( T a - T 0 ) / A T , the def ini t ions of V 2 a n d N. a n d the i n c o m -pres s ib i l i t y equa t i on are unchanged , a n d T = K = 5 = A02R4 gaAT WR R D are the T a y l o r number , t h e r m a l Rossby number a n d aspect r a t i o , respect ively . These parameters , a l o n g w i t h R = K/V a n d a A T , are the d imensionless parameters o f the p r o b l e m 38 (the parameters con ta ined i n the def in i t ion o f T can be shown to be ins ign i f ican t for our purposes) . It is poss ible to wr i t e a A T as TIT \y21 (4gR3>) }, where the express ion ins ide the c u r l y bracket is another d imens ionless parameter . M o s t e x p e r i m e n t a l resul ts are presented as reg ime d i a g r a m s w i t h T a y l o r number o n the h o r i z o n t a l ax is a n d t h e r m a l R o s s b y n u m b e r o n the v e r t i c a l ax is . F o r a g iven exper iment (f luid proper t ies a n d annu lus geomet ry fixed), these two parameters have a one-to-one r e l a t ion to f2 a n d A T , the p h y s i c a l pa rameters w h i c h are va r i ed . T h u s , as l o n g as the other d imensionless parameters are equiva len t , the resul ts s h o u l d be i den t i c a l f rom exper imen t to exper iment . However , c h a n g i n g the aspect r a t i o , for ins tance , w i l l change the regime d i a g r a m . T h i s is i m m e d i a t e l y obv ious w h e n c o m p a r i n g the results of F e i n [15] to those o f James et al. [29]. A l t h o u g h the shape o f the t r a n s i t i o n curves are s i m i l a r , the loca t ions o f the t r ans i t ions as w e l l as the c r i t i c a l wave number s are different. T h e def ini t ions o f the T a y l o r a n d t h e r m a l R o s s b y number s somet imes c o n t a i n the aspect r a t io (see Sec t ion 1.2), however, i t is not poss ib le to der ive a s i m i l a r i t y p r i n c i p l e f rom the equat ions w i t h these def in i t ions . A l s o , the ad jus ted def in i t ions do not reconci le the differences between the e x p e r i m e n t a l results m e n t i o n e d above. S ince the d imens ionless parameters are no t a l l independen t a n d because a n i m p o r t a n t pa ramete r appears i n the non l inea r par t , the ana lys i s w i l l be easier o n the d i m e n s i o n a l equat ions . 39 C h a p t e r 3 B i f u r c a t i o n s i n t h e t w o - l a y e r m o d e l In th i s chapter , we show how center m a n i f o l d r e d u c t i o n is used to find the b e h a v i o u r close to the b i f u r c a t i o n po in t s . T h e m e t h o d produces , f rom the p a r t i a l d i f ferent ia l equa t ions , a finite-dimensional sys tem w h i c h re ta ins a l l the i m p o r t a n t l o c a l d y n a m i c s . See A p p e n d i x A , w h i c h con ta ins a b r i e f d i scuss ion of some general theory. F o r de ta i l s see H a s s a r d et al. [23]. O n c e the finite-dimensional sys tem is ob ta ined , i t c a n be conver ted to n o r m a l f o r m , f r o m w h i c h the d y n a m i c s m a y be more r ead i ly deduced . T h e a p p l i c a t i o n of i m m e d i a t e interest is the two-layer m o d e l . F i r s t , a n ove rv iew of the center m a n i f o l d r e d u c t i o n w i l l be g iven . T h e n the example of a H o p f b i f u r c a t i o n w i l l be g iven i n some d e t a i l . F o l l o w i n g is the a p p l i c a t i o n of the r e d u c t i o n at a doub le H o p f p o i n t . In th i s l a t t e r desc r ip t ion , less de t a i l w i l l be g iven since the fundamenta l s are the same a n d the h igher d i m e n s i o n o f the equat ions o n the center m a n i f o l d makes the de ta i l s cumbersome . T h e ana lys i s w i l l be pe r fo rmed o n the two-layer quas igeos t rophic p o t e n t i a l v o r t i c i t y equa-t ions : | - ( V 2 -2F)5 + 3~ + J (a, V25) + J (5, W ) + 2FJ (S, a) 3.1 The two-layer model and linear stability (3.2) where 40 T dudv dudv . . , . I n . r, ^ , \ / n / J (M, v) = — — - — — , a = <r(a;, y , t) = ( x i + X2) A 5 = <J(x, y , t) = ( x i - X2) / 2 (Xn is the s t r e a m func t ion i n the n t h layer , n = 1,2), V 2 is the L a p l a c i a n i n two d imens ions a n d r, Re, 8 a n d F are pos i t ive rea l parameters . See Sec t ion 2.2. T h e b o u n d a r y cond i t i ons are: ( i ) a(x,y,t) = o(x + -y,y,t) a n d 6(x,y,t) = 8(x + 7,y,t) i.e., a a n d 8 are pe r iod i c i n x w i t h p e r i o d 7, w h i c h is a paramete r . ( i i ) — = — = 0 at y = 0 a n d y = 1. T h i s ensures no t angen t i a l ve loc i ty at the l a t i t u d i n a l boundar ies . ( i i i ) l i m -\- f ^ ^ dx - l i m — / v ' x^oo 2X J-x dudt x^oo 2X J-1 rx d26 _  / jr^-dx = 0 at y = 0 a n d y = 1. dydt x^oo 2X J-x dydt T h i s means there is no change i n c i r c u l a t i o n abou t the inner a n d outer b o u n d a r y . W i t h Q chosen as above, a s teady s o l u t i o n is: 5 = — cos 2ny o = 0. (3-3) (3.4) See Sec t i on 2.2 for the f lu id ve loc i ty to w h i c h th i s s o l u t i o n corresponds . S e t t i n g 8 = — cos 2^y + 8 a n d 0 — 0 a n d s u b s t i t u t i n g these in to the above equat ions , the ' p e r t u r b a t i o n equat ions ' for a a n d 8 are ob ta ined . D r o p p i n g the t i ldes the equat ions c a n be w r i t t e n as: where AU = LU + N(U, U) U = (3.5) (3.6) V 2 0 0 V 2 - 2F (3.7) 1 _ r - i v 2 + Re^V4 - 3^-ox d 2nsm 2iry— (47T 2 + V 2 ) KJJU d \ 2 7 r s i n 2 7 r 2 / ^ (4TT2 + 2F + V 2 ) - r ^ V 2 + Re~LV2 ( V 2 - 2F) - 3— j (3.8) 41 J(a,V2o) - J(5, V 2 5 ) \ N(U,U) (3.9) v -J(a,V25)- J(5,V2o)-2FJ(5,o) J T h e same b o u n d a r y cond i t ions a p p l y a n d U = 0 is now a s teady s o l u t i o n . T h e non l inea r par t , N(U, U), is w r i t t e n as such to emphas ize the fact tha t i t is a q u a d r a t i c non l inea r i ty . N o t e also t ha t U is a vector of funct ions a n d tha t L a n d A are ma t r i ces o f l inea r d i f ferent ia l opera tors . N e u t r a l s t a b i l i t y curves (or surfaces, depend ing o n the d i m e n s i o n o f pa ramete r space) are the boundar ie s between the regions i n pa ramete r space where the s teady s o l u t i o n o f interest is l i n e a r l y s table a n d those regions where i t is l i n e a r l y uns table . I n our case, we have a n e u t r a l s t a b i l i t y curve for each zona l (or a z i m u t h a l ) wave n u m b e r (every in teger) . F i g u r e 3.1 shows the neu t r a l s t a b i l i t y curves for the present m o d e l w h e n the pa rame te r space is cons idered two-d imens iona l (r a n d F are considered the parameters w h i l e a l l o ther pa ramete r s are cons idered cons tan ts ) . I n the l i nea r i zed equat ions , to the left o f each curve a l l s m a l l pe r tu rba t ions o f the g iven wave number decay to zero, whereas to the r igh t there is a p e r t u r b a t i o n w h i c h grows exponen t ia l ly . I n the reg ion o f pa rame te r space w h i c h is to the left o f all the n e u t r a l s t a b i l i t y curves, the so l u t i on is l i n e a r l y s table . I n the reg ion to the r igh t o f any o f the curves, the so l u t i on is uns table . I f the parameters are v a r i e d such t ha t there is a c ross ing f rom the s table reg ion to the uns tab le region, we expect a b i f u r c a t i o n to take place. T h e l inea r s t a b i l i t y of a s teady so l u t i on is defined i n te rms of the eigenvalues of the sys t em l inea r i zed abou t tha t so lu t i on . I f the rea l par ts o f all the eigenvalues are negat ive , t h e n the s o l u t i o n is sa id to be l i n e a r l y s table . I f any of the eigenvalues has pos i t ive rea l par t , t h e n the s o l u t i o n is l i n e a r l y uns tab le . T o see th is , wr i t e where we w i l l assume tha t U is some s m a l l p e r t u r b a t i o n f rom the t r i v i a l s o l u t i o n . I f we subs t i tu t e th i s in to the l inear pa r t of equa t ion (3.5), we get i.e. a genera l ized eigenvalue p r o b l e m , where A are the eigenvalues. Presen t ly , we are concerned w i t h how a l l s m a l l pe r tu rba t ions abou t the g iven s teady s o l u t i o n behave. If we p e r t u r b i n the d i r e c t i o n o f a n e igenfunct ion ( m u l t i p l i c a t i o n b y a s m a l l cons tan t ) , t h e n the c o r r e s p o n d i n g eigenvalue w i l l give us the l inear t i m e dependence via equa t i on (3.10). I f the opera tors A a n d L have ce r t a in proper t ies , then the eigenfunct ions f o r m a comple t e U(x,y,t) = extU(x,y) (3.10) \AU = LU (3.11) 42 70 r 60 50 40 LU 30 h 20 10 0 ' — 0.02 P 3 4 * . \ m=4 23 m=3 m=2 0.04 0.06 0.08 0.1 0.12 0.14 0.16 F i g u r e 3.1: The neutral stability curves for the two-layer model for the azimuthal wave numbers m = 2,3,4. To the left of all the curves, the solution is linearly stable; to the right of any of the curves, the solution is linearly unstable. See Section 3.2.4 and Section 4.2.3 for details of the computation of the neutral stability curves. The points where the curves cross, p 23 and pu, are double Hopf bifurcation points and are discussed in Section 3.3. basis , a n d any func t ion can be w r i t t e n as a l inear c o m b i n a t i o n of the e igenfunct ions . If the pe r tu rba t i ons U are expanded i n such a way, t hen i t can be seen how the set of eigenvalues give the t i m e behav iou r of a l l pe r tu rba t ions . T h a t is , i f a l l eigenvalues have negat ive rea l pa r t , t hen i n equa t ion (3.10), the n o r m of each t e r m i n the expans ion of any p e r t u r b a t i o n decays e x p o n e n t i a l l y since each t e r m is an e igenfunct ion m u l t i p l i e d by a cons tan t . T h u s , ( the n o r m s of) a l l pe r t u rba t i ons even tua l ly decay e x p o n e n t i a l l y as l o n g as the p e r t u r b a t i o n s are s m a l l enough tha t the non l inea r te rms are negl ig ib le . N o t e t ha t the i m a g i n a r y par t causes o s c i l l a t i o n , no t g r o w t h or decay. If o n l y one eigenvalue has pos i t i ve rea l pa r t , the pa r t o f a p e r t u r b a t i o n w h i c h is i n the d i r ec t i on of the co r r e spond ing e igenfunc t ion w i l l g row e x p o n e n t i a l l y ( u n t i l n o n l i n e a r i t y becomes i m p o r t a n t ) . T h u s , i f a l l eigenvalues have negat ive rea l par t , the s o l u t i o n is ca l l ed l i nea r ly s table . If there is any eigenvalue w i t h pos i t i ve rea l 43 par t , t hen the s o l u t i o n is l i n e a r l y uns table . If there are o n l y eigenvalues w i t h zero rea l pa r t a n d negat ive rea l pa r t i t is ca l l ed n e u t r a l l y s table , since there is ne i ther g r o w t h nor decay o f the p a r t o f the p e r t u r b a t i o n w h i c h is i n the d i r ec t i on o f the e igenfunc t ion c o r r e s p o n d i n g to the eigenvalue w i t h zero rea l par t . I n the present case, for the pa rame te r values to the left o f the neu t r a l s t a b i l i t y curves, the eigenvalues co r r e spond ing to the g iven wave n u m b e r a l l have negat ive real par t , a n d to the r igh t at least one has pos i t i ve rea l par t . T h i s de f in i t ion o f s t a b i l i t y says n o t h i n g abou t t rans ient behav iour . F o r th is reason l inea r s t a b i l i t y is referred to as the long- t ime behav iour . L i n e a r s t a b i l i t y ana lys i s is jus t i f i ed by the H a r t m a n - G r o b m a n theo rem (see e.g. [56] for a precise s ta tement o f the theorem) . A h y p e r b o l i c s teady s o l u t i o n is one whose l i n e a r i z a t i o n has no eigenvalues w i t h zero rea l par t . T h e H a r t m a n - G r o b m a n theo rem states t ha t i f the s teady s o l u t i o n is h y p e r b o l i c , then the l i n e a r i z a t i o n abou t the s o l u t i o n has q u a l i t a t i v e l y the same b e h a v i o u r as the fu l l non l inea r sys tem. T h i s imp l i e s tha t i f the s o l u t i o n is l i n e a r l y s table , t hen i t is indeed a s table so lu t i on . M o r e precisely, i f a s o l u t i o n is linearly s table , t hen there is a n e i g h b o u r h o o d abou t the so l u t i on where a l l i n i t i a l cond i t i ons for the nonlinear sys t em w i l l t end to the so lu t i on as t —> oo. However , the theo rem does not s tate the size o f the n e i g h b o u r h o o d abou t the s o l u t i o n , and so i t c o u l d be tha t the n e i g h b o u r h o o d is ve ry s m a l l . T h i s , however, depends o n the specific equat ions a n d s o l u t i o n . If the parameters are va r i ed (e.g., the paramete r r i n F i g u r e 3.1) such t ha t there is a t r a n s i t i o n f r o m the s table region to the uns tab le region, the s teady s o l u t i o n is expec ted to undergo a b i fu rca t ion . T h e f o r m of the b i fu r ca t i on depends o n the t ype a n d n u m b e r of eigenvalues w h i c h cross the i m a g i n a r y axis , o n the eigenfunct ions, a n d o n the non l inea r t e rms i n the d y n a m i c a l equat ions . In th i s p r o b l e m , each of the s t a b i l i t y curves i n F i g u r e 3.1 cor responds to two eigenvalues, w h i c h f o r m a c o m p l e x conjugate p a i r (one each for the pos i t ive a n d negat ive integer wave number ) . S ince the o r i g i n a l equat ions are rea l , the eigenvalues w i l l be rea l , or c o m p l e x conjugate pa i rs . If the t r a n s i t i o n between the regions occurs at a l o c a t i o n where two neu t r a l s t a b i l i t y curves cross (see po in t s p 2 3 a n d P34 o n F i g u r e 3.1), t hen at th i s l o c a t i o n there are a t o t a l o f four eigenvalues w i t h zero rea l pa r t ( two c o m p l e x conjugate pa i r s ) . A t th is po in t a doub le H o p f b i f u r c a t i o n occurs . T w o parameters are requ i red to descr ibe a l l the possible behav iou r near th i s b i f u r c a t i o n (here we use r a n d F) a n d so i t is a c o d i m e n s i o n two b i fu rca t ion . It is abou t these l oca t i ons t ha t we are able to show tha t hysteresis of the b i fu r ca t i ng so lu t ions c a n occur . If the t r a n s i t i o n between regions occurs at any other l o c a t i o n , a s ingle c o m p l e x conjugate pa i r o f eigenvalues crosses the i m a g i n a r y axis . Here a H o p f b i fu r ca t i on occurs a n d o n l y a s ingle pa rame te r is necessary to descr ibe a l l l o c a l behav iour . Before we proceed, a few def ini t ions are necessary. A s m e n t i o n e d above, the l inea r 44 stability problem for the trivial solution amounts to solving a linear eigenvalue problem. That is, the eigenvalues, Xj, and eigenfunctions, tyj, satisfy L¥,- = AjAv&j (3.12) with the same boundary conditions as above. Also, define the adjoint eigenfunction ^ as a vector function which satisfies L ^ ^ A * * * (3.13) where L* and A* are the adjoints of L and A respectively and the over-line represents the operation of complex conjugation. The adjoints satisfy (LV,W) = (V,L*W) (3.14) and (AV,W) = (V,A*W) (3.15) for all functions V and W. The inner product, denoted by the angled braces, is defined as (V, W) = f1 [* (V • W) dx dy (3.16) Jo where the dot represents the usual vector dot product (in particular, there is complex con-jugation of the components of W). The eigenvalues of the adjoint problem are the complex conjugates of those of the original problem (equation (3.12)), so the adjoint eigenvalues and eigenfunctions have been labeled in equation (3.13) to match their corresponding solutions of equation (3.12). The adjoint operators L* and A* can be calculated from equations (3.14) and (3.15), using Green's identity and integration by parts. It turns out that A* = A and L* = / ft ft ft% \ -r-1^2 + Re~lV4 + 6— -2nsm2ny— (2F + V 2 ) - 8TT 2cos2TT?/—— ' ox ox oxoy . - 2 7 r s i n 2 7 n / — V 2 - 8TT2 cos 27n/-^-— -r~lV2 + tfe^V2 ( V 2 - 2F) + 3— \ ox oxoy ox ) Since L* ^ L, L is not self-adjoint. This is a consequence of the nonlinear steady solution. Because L is not self-adjoint, the adjoint eigenfunctions are needed to project an arbitrary vector function in the direction of an eigenfunction. In particular, it can be shown that iii ^ j , then (* i ,A*^) = 0. (3.17) This means that an adjoint eigenfunction is orthogonal to the span of all eigenfunctions except its corresponding eigenfunction. This important property will be used below to define a projection which will lead to a simplification of the dynamical equations. 45 3.2 Hopf bifurcation Suppose we fix a l l parameters except r , w h i c h we consider as the b i f u r c a t i o n parameter . Def ine r 0 as the value o f r for w h i c h there is a s ingle pa i r of c o m p l e x conjugate eigenvalues w i t h zero rea l pa r t ( A i = itjQ a n d A 2 = A i = —iuo), w h i l e a l l the o ther eigenvalues have negat ive rea l par t . Therefore , as r is va r i ed past ro, the t r i v i a l s o l u t i o n loses i t s s t a b i l i t y as a s ingle c o m p l e x conjugate p a i r o f eigenvalues crosses the i m a g i n a r y ax is , a n d a H o p f b i f u r c a t i o n i s expec ted to occur . However , i n order to see the de ta i l s o f the b i fu r ca t i on , the n o r m a l f o r m coefficients mus t be c o m p u t e d . 3.2.1 The center eigenspace Define $ 0 as a vec tor func t ion w h i c h satisfies L $ 0 = i w 0 A $ 0 (3.18) w h e n r = r 0 . Its c o m p l e x conjugate $ 0 is a vec tor func t ion w h i c h satisfies L $ 0 = - i w 0 A $ 0 (3.19) w h e n r = TQ. A n adjoint e igenfunct ion $Q satisfies L * $ * = - M , A * $ * (3.20) w h e n r = r 0 , where L * a n d A * are the adjoints of L a n d A respec t ive ly (see equat ions (3.14) a n d (3.15)). W e n o r m a l i z e $Q a n d $ 0 so tha t ($o,A*$S) = l (3.21) a n d < $ o , A * $ ; ) = l . (3-22) T h e r e is also a n o r m a l i z a t i o n constant associa ted w i t h <&0, bu t i t is a rb i t r a ry . Def ine the center eigenspace, Ec = span {<P0, $ 0 } , a n d the s table eigenspace, Es, as the span o f a l l o ther e igenfunct ions (those co r re spond ing to eigenvalues w i t h negat ive rea l p a r t ) . W e m a y decompose any vector func t ion U i n to a pa r t w h i c h is i n Ec a n d a pa r t w h i c h is i n Es. T h a t is , we can wr i t e U = z<f>0 + z$o + C (3-23) where z is some c o m p l e x number , a n d C = U — z$0 — ^$o- T h i s m a y be done by t a k i n g z = (U, A * $ o ) 5 t hen by the proper t ies o f the adjoint , z $ 0 + z$0 e Ec a n d C € E3. 46 3.2.2 The center manifold T h e ana lys i s o f the sys tem w o u l d be grea t ly s impl i f i ed i f the s table a n d center par t s c o u l d be decoup led . In fact, u s ing center m a n i f o l d r educ t ion , th i s c a n essent ia l ly be done close to the b i f u r c a t i o n po in t . Define the p ro j ec t i on P of a vector func t ion U as PU=(U,V0)$Q + (U,$'0)$0.' (3.24) T h i s p ro j ec t i on is chosen since i t decouples the linear pa r t o f the d y n a m i c a l equat ions i n to a pa r t w h i c h is i n the center eigenspace a n d a pa r t w h i c h is i n the s table eigenspace. T h u s , t a k i n g the p ro jec t ion P of equa t ion (3.5), we get i $ o + = iwo*$o ~ ^ 0 2 $ 0 + (N, $*0)$o + (N, $*0)$o (3.25) a n d so z = iuoz+(N,&0). (3.26) However , (N, $ 5 ) s t i l l depends o n the s table space var iables a n d so the sy s t em has not yet been decoupled . T h i s is done u s ing the center m a n i f o l d theorem. T h e center m a n i f o l d theo rem states tha t , g iven ce r t a in t echn ica l c o n d i t i o n s o n L a n d A , there exists a differentiable center m a n i f o l d for equa t ion (3.5) : W[oc = {U = z$0 + z$o + H ( z$„ , z$o) , l k $ 0 + * $ o | | s m a l l , H : Ec Es) (3.27) w h i c h is l o c a l l y invar ian t , tangent to Ec at 0 (z = 0 a n d r = r0) a n d l o c a l l y e x p o n e n t i o n a l l y a t t r a c t i n g (see A p p e n d i x A or [23]). In (3.27), || • || is the n o r m tha t cor responds to the inner p r o d u c t (3.16). W e w i l l assume, w i t h o u t proof, t ha t the c o n d i t i o n s of the t heo rem are sat isf ied. It has been shown tha t the incompress ib le Nav ie r -S tokes equat ions a n d the Nav ie r -S tokes equat ions i n the Bouss inesq a p p r o x i m a t i o n do sat isfy the c o n d i t i o n s (see [25] a n d [7]), therefore i t m a y not be unreasonable to assume tha t they are also sat isf ied by the quas igeos t rophic equat ions (see also [9]). Because the center m a n i f o l d is l o c a l l y exponen t i a l l y a t t r a c t i n g , any t r a j ec to ry whose i n i t i a l c o n d i t i o n is close to the center m a n i f o l d w i l l app roach i t e x p o n e n t i a l l y q u i c k l y as t —> oo. T h i s i m p l i e s tha t the long- t ime d y n a m i c s of the fu l l s y s t em of equa t ions are deduc ib l e f rom the d y n a m i c s on the center m a n i f o l d . If there are any fixed po in t s or p e r i o d i c o rb i t s t ha t exis t close to the b i fu r ca t i on po in t , they mus t exis t o n the center m a n i f o l d . B y close, i t is meant tha t b o t h | | z $ 0 + 2$o | | a n d \r - r0\ are s m a l l . Therefore , i f we find an equa t ion w h i c h describes the d y n a m i c s o n the center m a n i f o l d , we have found a l ower -d imens iona l sys t em f rom w h i c h we can deduce the d y n a m i c s . T o find t h i s equa t ion , 47 note t ha t the center m a n i f o l d theory states tha t on the center m a n i f o l d , £ = H(z,z) for some s m o o t h func t ion H. T h a t is , there is a one-to-one cor respondence between the coo rd ina t e z a n d the center m a n i f o l d . See F i g u r e 3.2. So i f H(z,z) is used i n p lace o f ( i n the z equa t i on (3.26), then the equa t ion w i l l descr ibe the d y n a m i c s o n the center m a n i f o l d a n d the o n l y dependent va r iab le w i l l be z. T h i s is a consequence of the inva r i ance p r o p e r t y ( w h i c h means t ha t i f a t ra jec tory or ig inates on the center m a n i f o l d i t w i l l s tay o n the center m a n i f o l d ) . T h e invar iance c a n also be used to ca lcu la te a T a y l o r series a p p r o x i m a t i o n to the func t i on H. N o t e tha t the tangency c o n d i t i o n a l ready i m p l i e s t ha t H = O (\z,z\2). C =H(z,z) real(z) F i g u r e 3.2: A schematic diagram of a center manifold. The center eigenspace is the horizontal plane and £ is written as a function of z, the coordinate on the center eigenspace. T h u s , C " i a y be w r i t t e n as C = H{z,z)=0 (\z, z\2) = H20z2 + Hnzz + H02z2 + ... (3.28) where l S - i + t i H » = w l ^ m o ) (3-29) are the T a y l o r series coefficients of H(z, z), a n d close to the b i fu r ca t i on po in t , the in te res t ing d y n a m i c s of equa t ion (3.5) are con ta ined i n z = iu0z + G(z,z) (3.30) where G(z,z) = (N ( z $ 0 + z$o + H) ,$S> = G20z2 + Guzz + G02z2 + G2iz2z + ... (3.31) 48 a n d Gij are the T a y l o r series coefficients of G(z, z) as i n (3.29). T h a t is , Gij = {Nij,^l) (3.32) where N (z, z) = N20z2 + Nnzz + N02z2 + N2lz2z + ... (3.33) a n d N (z, z) is the non l inea r pa r t (3.9) w r i t t e n i n te rms o f z u s ing the d e c o m p o s i t i o n o f U i n equa t ion (3.23) w i t h equa t ion (3.28) subs t i tu ted . 3.2.3 The normal form W e wan t to reduce equa t ion (3.30) to n o r m a l fo rm. T h e r e d u c t i o n to n o r m a l f o r m is a series o f near - iden t i ty t r ans format ions w h i c h t r ans fo rm the equat ions to a coo rd ina t e sy s t em where they take on the most s imp le fo rm possible . M o r e specif ical ly , the non l inea r pa r t is expanded i n a T a y l o r series (as G above) a n d the t r ans fo rmat ions e l i m i n a t e as m a n y of the t e rms as poss ible . T h e t r u n c a t i o n occurs when i t can be shown tha t (for s m a l l z) the a d d i t i o n o f e x t r a te rms w i l l not q u a l i t a t i v e l y effect the so lu t ions o f the equat ions ; by th i s we m e a n tha t the a d d i t i o n o f e x t r a te rms does not a l te r the existence a n d s t a b i l i t y o f the so lu t ions (see e.g. [56]). W r i t i n g z = pe10, where p is the m a g n i t u d e o f z i n the c o m p l e x p lane a n d 9 is the phase of z, the n o r m a l fo rm of equat ions (3.30) i n the p o l a r coord ina tes (p, 9) are p = B(r-r0)p + Rp3 + O(p') 9 = uj0 + E(r-r0)+Kp2 + O(p3) { 6 ' 6 V where B,R,E,K € 9ft are the n o r m a l f o r m coefficients. If B ^ 0 a n d R ^ 0, t h e n for a l l p a n d | r — J*O| sufficiently s m a l l , we have 0 / 0 , a n d there exists a p e r i o d i c s o l u t i o n o f equa t i on (3.34). T h u s , there is a H o p f b i fu rca t ion at r = r 0 . I f 9 ^ 0 (for a l l t i m e ) , p = 0, a n d p = p p ^ 0, t hen p = ' pp describes the pe r iod i c o rb i t . I f B ^ 0, R ^ 0 a n d p ^ 0, t hen i t c a n be shown tha t the 0 ( p 4 ) te rms i n the p equa t ion a n d the 0(p2) a n d 0(\r — r 0 | ) t e rms i n the 9 equa t ion m a y be neglected when cons ide r ing the existence a n d s t a b i l i t y of so lu t ions close to the b i fu r ca t i on po in t . If the r e d u c t i o n to n o r m a l f o r m is per formed for a general equa t ion , the f o l l o w i n g fo rmulae c a n be o b t a i n e d (3.35) B = - ^ R e a l ( A i ) dr r=ro a n d fi = R e a l { G ' 2 1 + ^ - ( G ' 2 o G ' 1 1 - | G 1 1 | 2 - ^ | G ' o 2 | 2 ) } • (3.36) where ' R e a l ' means ' the rea l pa r t o f a n d A i is the eigenvalue w h i c h has rea l pa r t zero w h e n r = r 0 . 49 In the present application, it can be easily shown that B > 0 (which implies that the steady solution is stable for r < r 0 ) , so the sign of R will indicate the direction of the bifurcation. If R is positive, there will be a sub-critical Hopf bifurcation with an unstable periodic orbit (i.e. there exists an unstable orbit for r < ro). If i? is negative, the bifurcation will be supercritical with a stable periodic orbit (i.e. for r > r 0 ) . In order to compute R, we need compute only G20, Gn, G02 and G2i- Thus, the JVy of interest in 3.33 are: where N(-, •) on the right hand side is N(U,U) of (3.9). See Appendix C. It can be seen that G20, Gn, and G02 depend only on the eigenfunctions $0, $0 and $0, while G2\ depends only on Hn, H2Q and the eigenfunctions. This is a consequence of the fact that N is quadratic. Now we use the invariance property of the center manifold and apply (I — A P ) to equation (3.5) to find Hn and H20. Taking (/ — A P ) of equation (3.5), we get where equations (3.23), (3.17) and (3.24) have been used to eliminate the ^-dependence in the linear part. The invariance of the center manifold gives N21 N20 Nn N(<J>0,®Q) A T ( $ o , $ o ) + i V ( $ o , $ o ) N($0, Hn) + N(Hn, $0) + N($0, H20) + N(H2Q, $0) (3.37) (/ - A P ) AC = (/ - A P ) L( + (/ - A P ) N (3.38) C = DZH (z, z)z + DzH (z, z) i, (3.39) where Dz indicates derivative with respect to z. So, A C = ADZH (z, z)z + ADZH (z, z) t (3.40) and using equations (3.26) and (3.28) we get AC - A (2zH20 + zHn + • • •) (ico0z + ...) + A (zHn + 2zH02 + ...) (-iu0z + ...) = 2iuoAH20z2 + iuoAHnZZ — iuoAHnZZ + ... thus, (3.41) 50 Substituting (3.41) into equation (3.38) and collecting terms of order z2, it can be seen that H20 satisfies (/ - A P ) (2ZCJ0A - L) # 2 0 = (/ - AP) N20. (3.42) Collecting terms of order zz in a similar way gives the equation (/ - A P ) LHn = - (/ - A P ) Nn (3.43) from which Hn can be calculated. To recapitulate, the following equations must be solved for r 0 , Ai = IOJ0, $ 0 , $01 -^20 and Hn: L $ 0 = « w 0 A $ 0 (3-44) L*$; = -iuj0A*<f>*0 (3.45) (/ - A P ) (2zu;0A - L) H20 = (I - AP) N20 (3.46) (/ - A P ) LHU = -(I- A P ) Nn (3.47) and their solutions substituted into the formulae for G20, Gn, G02 and G2\. The equations (3.44) - (3.47) are linear partial differential equations. By using this method, the dynamics near the bifurcation point for the full time-dependent nonlinear partial differential equations can be found by solving steady linear problems (but still partial differential equations). These equations are general and apply to any Hopf bifurcation analysis. In the present application, we can simplify things a little more by using some information specific to the problem. In particular, the equations can be reduced to a series of ordinary differential equations by assuming a form of the ^-dependence for the eigenfunctions. However, it is not possible to analytically solve for the y-dependence, and so this is done numerically. Using a numerical approximation of the relevant inner products leads to numerical approximations of the normal form coefficient, R. 3.2.4 Computation and Results We look for eigenfunctions of the form = i,(y)eikmx (3.48) where k = 271-/7 a n d m is an integer which gives the number of crests of the wave along the channel. Technically, the wave number is given by km. However, since m is a conceptually more appealing means of distinguishing between the different waves, we use m to unam-biguously label the wave number with the understanding that the wave number is actually km. 51 U p o n s u b s t i t u t i o n o f th is f o r m (3.48) i n the eigenvalue p r o b l e m (3.12), the p r o b l e m is reduced to a set o f eigenvalue p rob lems i n one spa t i a l va r iab le : L H f W = ^ A ( m ) * ( m ) (3.49) (one equa t ion mus t be solved for every integer ra), where l / m ) = e~lhmx (Lelkmx^ ( l ikewise for A ( m ) ) . I n prac t ice , for each ra, r = is found such tha t a l l eigenvalues o f (3.49) have rea l pa r t negat ive except a finite number (here a c o m p l e x conjugate p a i r ) . T h e n ro is t a k e n as m i n ( r ^ ) . T h i s separa t ion of the ^-dependence is the reason t ha t w h e n two paramete r s are cons idered , there are different neu t r a l s t a b i l i t y curves for each wave n u mb er . It is necessary to look at the b o u n d a r y cond i t i ons carefully. G i v e n so lu t ions as i n equa t i on (3.48), b o u n d a r y c o n d i t i o n (i) w i l l a lways be satisf ied. However , the o ther c o n d i t i o n s w i l l be different for different values o f ra. F o r m ^ O , (iii) w i l l a lways be satisfied. T o satisfy (ii), we mus t have: = 0, at y = 0 , 1 . (3.50) F o r ra = 0, (ii) w i l l a lways be sat isf ied a n d we mus t have dvj>(m) — — = 0, at y = 0,1 (3.51) dy so t ha t (iii) is sat isf ied. T h i s is the case for the exact so lu t i on . It is easy to show a n a l y t i c a l l y tha t for ra = 0, a l l eigenvalues w i l l be negat ive as l o n g as r > 0, Re > 0 a n d F > 0. F o r ra ^  0, the eigenvalues a n d e igenfunct ions mus t be found n u m e r i c a l l y . T h e p r o b l e m , however, is unde rde t e rmined (the o n l y b o u n d a r y cond i t i ons are g iven i n equa t i on (3.50)). T o comple te the p r o b l e m , the c o n d i t i o n — 7 1 - = ° ' a t y = 0 ' 1 ( 3 - 5 2 ) dy2 w i l l be chosen. T h i s corresponds to the c o n d i t i o n o n the m e r i d i o n a l fluid v e l o c i t y t ha t d2v/dy2 = 0 at y = 0 , 1 . T h i s choice is diff icul t to argue o n p h y s i c a l g rounds , a n d so i t w i l l be cons idered as an a d d i t i o n a l a p p r o x i m a t i o n of the m o d e l . T h e c r i t i c a l pa ramete r value r 0 c an be found by s o l v i n g the non l inea r equa t i on R e a l ( A i ( r ) ) = 0 (3.53) where A i is the eigenvalue w i t h the largest rea l par t , regardless o f ra. D e n o t e mc as the ra t o w h i c h A i cor responds . T h e d i s c r e t i z a t i on o f the opera tors i n equa t i on (3.49) a n d the eigenvalue a n d e igenfunct ion a p p r o x i m a t i o n are discussed la ter . 52 Write ikmcx I Akmcx $ o = <$>{y)elkm°x = V e V 02 and $* = $ * ( y ) e - i f c m c I . (3.54) It is $ and $* which must be calculated numerically. For the present application, it can be shown that G2o, Gn, G02 = 0 because none of the second-order terms of the expansion of the nonlinear part (Nij with i + j = 2) will have integer mc in their (exponential) x-dependence, which means that they will be orthogonal to the adjoint eigenfunction. Thus, the transformation to normal form will affect only the fourth-order terms. In addition to the eigenfunctions, the coefficient G2\ depends only on the following two single-variate (in y) functions: Hn and H2lmc\ where H-™^ is defined by Hij(x,y) = Y:H\jl\y)ei^ (3.55) m and Hij is defined by equation (3.28). To see this, expand H2Q(x,y) = ^ H^\y)eikmx and write m L # 2 0 = Yjeikmxt{m)H{^) (3.56) m and similarly for AH20, LHn and AHn. Then, P L W ^ e t e = 0 and PA^H^)eikmx = 0 for all m ^ mc. Also note that PN20 = 0 and PNn = 0. So, collecting terms of e2ikmcx m e q U a t i o n (3.42), we get: (t^ - 2uj0A<-2m°>) H £ T C ) = NSMC) (3.57) and collecting terms of e° in equation (3.43) we get: LWffJ? = (3.58) Solving the last two equations give H$> and The boundary conditions for equation (3.57) are the same as those for equation (3.49) with m ^ 0, while the boundary conditions for equation (3.58) are also the same as those for equation (3.49) but with m = 0. For the latter case, the problem is underdetermined. 53 Therefore , as we d i d for the case m / 0, we make the same a rgumen t a n d choose an a d d i t i o n a l b o u n d a r y c o n d i t i o n . T h e c o n d i t i o n is: d3H^ dy3 = 0 at y = 0,1. (3.59) T h e coefficient G2\ = ( - ^ 2 1 , $ * ) c a n be w r i t t e n as an in tegra l i n y w h i c h depends o n l y o n Hn, H2lmc\ * > a n ( l ® a n d s o m e ° f t n e i r der iva t ives . A l l o f these func t ions a n d the in t eg ra l mus t be ca l cu la t ed numer ica l ly . T h e t r a p e z o i d a l ru le is used to a p p r o x i m a t e the in tegra l : (3.60) where 6 = Vi ^ s t n e p o s i t i o n of the ith g r i d po in t , — = /1 = l / ( i V + 1), for a l H , N is the n u m b e r of u n i f o r m l y spaced in te r io r g r i d po in t s a n d the bounda r i e s are at y0 = 0 a n d yN+i = 1. C e n t e r e d finite-differencing is used i n the d i sc re t i za t i on of the eigenvalue a n d eigenfunc-t i o n p r o b l e m , i n the so lu t i on of Hn a n d H2lmc\ a n d also i n c a l c u l a t i n g the der iva t ives i n v o l v e d i n G2i. T h e four th der iva t ives o f funct ions f(y) are a p p r o x i m a t e d u s i n g dy4 y=yi 6+2 - 4£ + i + 66 - 4^-1 + 6-2 ft4 + O (h2) a n d the t h i r d der iva t ives w i t h a dy3, i> \y=yi a n d second der iva t ive w i t h * i dy2 a n d the first de r iva t ive w i t h 6+2 ~ 26+i + 2^-1 — 6-2 2h3 + y=yi 6+1 ~ 26 + 6-1 + O (h2) (3.61) (3.62) (3.63) dy y=yi 6+1 ~ 6-1 2/ i + O (h2) . (3.64) F o r m ^ 0, i n order to satisfy the b o u n d a r y cond i t ions , we mus t have t ha t 6) = 0 a n d 6v+i = 0 a n d dH dy2 y=yo 6-1-26,+ 6 = 0 dy2 y-yN 6v ~ 2^Af+l + 6v+2 h2 = 0. (3.65) (3.66) 54 Therefore we take £ _ ! = — 6 a n d 6 v + 2 = — 6 v , w h i c h are needed for the four th-order der iva t ives at yi a n d y^-i- S i m i l a r cond i t i ons a p p l y for the m = 0 case. U p o n d i s c r e t i za t i on , the cont inuous eigenvalue p r o b l e m L W | ( m ) — A A ( m ) * ( m ) (3.67) w i t h * M = 0, - \ T 2 - = 0, at y = 0 , l (3.68) dyz m = ± 1 , ± 2 , i s reduced to the genera l ized m a t r i x eigenvalue p r o b l e m L{™]X = A A i m ) X (3.69) (one p r o b l e m for each m ) , where L J ^ and AJ™^ are 2 i V x 27V rea l -va lued ma t r i c e s a n d X, w h i c h is o f size 2N, is the discrete s o lu t i on vector of the e igenfunct ions . T h e first N componen t s o f X are the a p p r o x i m a t e values of the first c o m p o n e n t of the u n k n o w n vec tor funct ions (e.g. cf>i) at the N g r i d po in t s a n d the last N componen t s o f X are the a p p r o x i m a t i o n s for the second componen t (e.g. fa)- It is not k n o w n i f th i s m e t h o d for c a l c u l a t i n g eigenvalues converges i n general . However , i t is reasonable to assume tha t i t does (see e.g. [7]). See C h a p t e r 4 for further d iscuss ion . S i m i l a r l y , d i s c r e t i z a t i o n o f the p rob lems (3.57) a n d (3.58) for a n d H^71^ l ead to sys tems of l inear a lgebra ic equat ions w h i c h can be r ead i ly solved. T h e o n l y excep t i ona l note is tha t i n the c o m p u t a t i o n of H[1\ the s o lu t i on is no t k n o w n at yo a n d yjv+i (since th i s f unc t i on satisfies the b o u n d a r y cond i t i ons for m = 0) a n d therefore, a n a p p r o x i m a t e s o l u t i o n mus t be found at these po in ts . T h e b o u n d a r y cond i t i ons are used to find £ _ 2 , £N+2 a n d £N+3 i n t e rms of in te r io r po in t s (as i n equat ions (3.65) a n d (3.66)), s ince these are needed for der iva t ives at y0, yi, yN-i a n d y^. F i n a l l y , i n the c o m p u t a t i o n of the i V y , der iva t ives o f the a p p r o x i m a t e d funct ions mus t be ca l cu l a t ed . It c a n be shown tha t the a p p r o x i m a t i o n s of these der iva t ives are s t i l l 0(h2). See C h a p t e r 4 for fur ther d i scuss ion . A l l the c o m p u t a t i o n s were pe r fo rmed i n M a t l a b p r o g r a m m i n g env i ronment . T h e parameters are set at F = 5.5, 3 = 0.5, Re = 240 a n d 7 = \/l2, w h i l e r is cons id -ered the p r i m a r y b i fu rca t i on parameter . T h e results for a n e x a m p l e o f a H o p f b i f u r c a t i o n at different n u m b e r o f g r i d po in t s , N, are s u m m a r i z e d i n T a b l e 3.1. A few r emarks shou ld be made . F i r s t , the n o r m a l f o r m coefficient R is negat ive a n d therefore a s u p e r c r i t i c a l H o p f b i fu rca t ion occurs at r 0 . T h a t is , there is a s table p e r i o d i c o r b i t for values of r > r 0 , for r close enough to ro- T h e convergence o f the m e t h o d w i l l no t be proven , m o s t l y because the accuracy of the eigenvalues is no t k n o w n . It w i l l be 55 N R To m c 20 -50.95 0.1219 2 40 -55.15 0.1215 2 80 -56.26 0.1212 2 160 -56.53 0.1212 2 T a b l e 3.1: Numerical results. N is number of grid points, R is the normal form coefficient, r0 is the value of r where the bifurcation occurs and mc gives the value of m corresponding to the zero eigenvalue. no ted tha t the d i sc re t i za t ions tha t are used a n d the n u m e r i c a l i n t eg ra t i on are 0(h2), a n d so i t is reasonable to expect tha t the m e t h o d is convergent . T h i s seems to be w h a t is observed, s ince the differences between the values of R o b t a i n e d at the var ious N appea r to decrease l ike h2. T h e n o r m a l f o r m coefficient R was also c a l c u l a t e d at several o ther po in t s i n pa ramete r space, where o n l y a single pa i r o f eigenvalues crossed the i m a g i n a r y axes as r was increased. It was found tha t R was a lways negat ive. F r o m equa t ion (3.34), the p e r i o d i c o rb i t , to lowest order i n r — ro, is g iven by IB (r - rQ) _ . , P = \l _R +O(r-r0) 9 = u0t + O(r-rQ) (3.70) where z = pe10, or i n terms o f z: z - —K (3.71) w h i c h describes a near -c i rcu la r pe r iod i c o rb i t . F i n a l l y , i n the o r i g i n a l var iab les , where to first-order, U = z$0 + z<fr0 = R e a l (z$o), the pe r iod i c s o l u t i o n is g iven by : U = R e a l B (r '"o) t^t^ t f c m c x -R + O (r - r 0 ) IB(r-r0) - R [ $ r cos (kmcx + ut) - $ i s in (kmcx + cut) +0 (r - r 0 ) (3.72) where l> r a n d $ j are the rea l a n d i m a g i n a r y par t s of respect ively. T h i s is a t r a v e l l i n g wave o f a z i m u t h a l (zonal) wave n u m b e r mc, m o v i n g i n the x d i r e c t i o n w i t h the s t ruc tu re o f the l a t i t u d i n a l dependence g iven by $ a n d w i t h m a g n i t u d e g r o w i n g as y/r - r 0 . See F i g u r e 3.3 for an example of the wave f o r m w i t h mc = 2. 56 3.3 Double Hopf bifurcation in the two-layer model In th i s sec t ion , we discuss an ex tens ion of the ana lys is of the prev ious sec t ion . In p a r t i c u l a r , the resul ts w i l l be shown for the double H o p f b i furca t ions t ha t occu r i n the two- layer m o d e l . W e show tha t there is a region i n paramete r space where there are two s i m u l t a n e o u s l y s table p e r i o d i c o rb i t s w h i c h cor respond to two wave so lu t ions . Hys teres i s o f these so lu t ions is p r ed i c t ed . A g a i n we s tar t w i t h the equat ions of the two-layer m o d e l (3.2) w r i t t e n as AU = LU + N(U, U) (3.73) where (3.74) a n d A , L a n d N(U,U) are g iven i n equat ions (3.7), (3.8) a n d (3.9). R e c a l l t ha t these are the p e r t u r b a t i o n equat ions abou t the a x i s y m m e t r i c s o l u t i o n 5 = - c o s 2 7 n / (3.75) a = 0. (3.76) A doub le H o p f b i fu r ca t i on (somet imes referred to as a H o p f - H o p f b i fu rca t ion) is charac-t e r i zed by two c o m p l e x conjugate pa i rs of eigenvalues s imu l t aneous ly c ross ing the i m a g i n a r y axis as the parameters are va r i ed . It is a c o d i m e n s i o n two b i fu r ca t i on , w h i c h means t h a t t w o paramete r s are needed to descr ibe a l l the poss ib le l o c a l behav iour . T h e t w o pa rame-ters t ha t we use are r a n d F. See Sec t ion 2.2. T h i s choice fol lows w o r k by P e d l o s k y [48], M a n s b r i d g e [39], M o r o z a n d H o l m e s [46], a n d L e w i s [36]. M o s t o f the ana lys i s is s i m i l a r to the H o p f b i fu r ca t i on case g iven i n d e t a i l i n the p rev ious sec t ion . F o r th i s reason, o n l y the m a i n po in t s a n d the differences i n n o t a t i o n are g iven . In p a r t i c u l a r , the center m a n i f o l d r educ t ion is the same i n p r i n c i p l e , except t ha t the d i m e n s i o n of the center m a n i f o l d is four ins tead of two. T h i s leads to a n o r m a l f o r m equa t i on w h i c h is a fou r -d imens iona l o r d i n a r y different ia l equa t ion . T h u s , m u c h more c o m p l i c a t e d d y n a m i c s are poss ib le . T h e first s tep is to p lo t the neu t r a l s t a b i l i t y curves i n the t w o - d i m e n s i o n a l pa r ame te r space. A s desc r ibed i n Sec t ion 3.1, there w i l l be one s t a b i l i t y curve for each a z i m u t h a l wave number . T h e results have a l ready been presented i n F i g u r e 3.1. S ince each curve cor responds to a c o m p l e x conjugate pa i r of eigenvalues w i t h zero rea l par t , doub le H o p f b i f u r c a t i o n po in t s occur at intersect ions of the curves. T h e po in t s o f interest are o n the r i gh t -mos t b o u n d a r y o f the s table regime (a l l o ther curves are to the r i gh t ) . These po in t s 57 are indicated in the figure with the corresponding wave numbers of the neutrally stable waves. It must be mentioned that it is impossible to calculate the neutral stability curves for all wave numbers. It is possible to show [3], [28] that in certain quasigeostrophic dynamics with dissipation, there is a critical wave number where perturbations of higher wave number cannot be unstable in the relevant parameter range. Although it cannot be shown that this will follow in the present application, we quote this result and show that as the wave number increases, the stability curves monotonically shift to higher parameter values (see Figure 3.1). Label the double Hopf bifurcation point as (ro,F0). That is, at r = r 0 and F = F0 the linear eigenvalue problem XAU = LU (3.77) has two complex conjugate pairs of eigenvalues with zero real part (for a total of four), which will be denoted by: Ai = pi + iuJi, Ai, A 2 = A*2 + iuJ2, A 2, (3.78) where at ro and F0, p,\ = fJ-2 = 0. Assume also that all other eigenvalues have negative real part. The eigenfunctions corresponding to the above eigenvalues are written as $, so that A x A $ = L $ (3.79) and X2A^ = Ltf. (3.80) Note the change in notation from the previous section: \I> now refers to a single eigenfunction corresponding to an eigenvalue with zero real part. Define the projection PU = ( £ / , $ * ) $ + + + {U,V*)V (3.81) where $* and \f* are the adjoint eigenfunctions which satisfy the adjoint eigenvalue problem associated with Ai and A 2 respectively (see equation (3.13)) and the angled braces are the inner product defined in equation (3.16). Writing U = z$ + z§ + w$ + wty + C (3.82) 58 where z = z(t) = ([/, A*$* ) and u; = w(t) = (£/, A*\I>*) are complex numbers and z $ + z<I> + w^S + w$ G Ec, and ( e Es (E° and 2?" are the center and stable eigenspaces, respectively, discussed in the previous section). Taking the projection of equation (3.73), we get, 10 = \2w + {N,V*). (3.83) The center space is now four-dimensional where previously it was two-dimensional. Now assume that the center manifold theorem applies (see Section 3.2). That is, there is a locally invariant, locally exponentially attracting differentiable center manifold which is tangent to the center space at 0 (r = r0, F = F0, z = 0 and w = 0). Then on the center manifold, we can write C = H(z,z,w,w) = 0(\z,z,w,w\2). (3.84) Expand H in a Taylor series as H (z, z, w, w) = H2oooz2 + Hnoozz + +H0020W2 + Hoonww +Hwwzw + HW01zw + cc. + 0(3) (3.85) where H i j k l = iijiAU! dz*dPdw"dtfH{0' °' °'0) are the Taylor series coefficients, cc. are the complex conjugates of the terms which are written explicitly, and 0(n) = 0(\z, z, w, w\n). Also write N(z, z, w, w) in the same form, where N(z, z, w, w) is the nonlinear term (3.9) written in terms of z,z,w and w, using the decomposition of U given in equation (3.82), with £ written using equations (3.84) and (3.85). See Appendix C for the formulae for the coefficients of N. The normal form equations for the non-resonant case (see below) are z = \xz 4- Gnz2z + G12zww + (9(4) w = X2w + G2izzw + G22W2w + 0(A) (3.86) where G^i are the normal form coefficients, Xj=\j(r, F) are the eigenvalues and it can be shown that truncating the 0(4) terms does not qualitatively change the dynamics close to the origin. This normal form requires the condition that there is no resonance. This simply means that the imaginary parts of the eigenvalues (wi ,^) , must satisfy mwi + nu)2 ^ 0, |m| + \n\ < 4. If there is resonance, then there are extra terms in the normal form. See e.g. [56] or [21]. 59 T h e d y n a m i c s are more r e a d i l y deduced w h e n the n o r m a l f o r m equat ions are w r i t t e n i n p o l a r coord ina tes (i.e. the magn i tudes pj a n d phases 6j of the c o m p l e x dependent va r iab les ) . W r i t e z = pi^1 /\j\Grn\ a n d w = p2el62 /\J\G22\, where G\j is the rea l pa r t o f Gij, a n d subs t i tu te these i n equa t ion (3.86). T h e n i n p o l a r coord ina tes , the t r u n c a t e d n o r m a l f o r m equat ions are Pi = Pi (pi + ap\ + bpf) P2 = P2 (pi + cp\ + dpfj e\ = OJI 92 = u2 (3.87) where = ± 1 , b = Gu_ \G\i\ 12_ i c y r - ^21 \Gny d = TSK = ± 1 , (3-88) \G r 221 Xj = pj + iuj, the 0(\pi,p2\A) t e rms are ignored i n the pj equat ions , a n d the 0(\pi,p2\2) t e rms are ignored i n the Oj equat ions . T h i s choice o f coord ina tes e l imina te s the d iscrep-ancies i n the n u m e r i c a l values of the coefficients tha t can arise due to different choices of the a r b i t r a r y n o r m a l i z a t i o n c o n d i t i o n (see Sec t ion 3.2.1). B e l o w , the paramete r s w i l l be cons idered to be p\ a n d p2 w i t h the u n d e r s t a n d i n g tha t they are dependent u p o n r a n d F. T o see the poss ible behav iou r tha t these equat ions c o u l d descr ibe , l o o k for f ixed po in t s o f the reduced sys t em Pi = Pi (pi + apl + bpj) p 2 = P2 (pi + cp\ + dp2) . (3.89) T h a t is , f ind the px a n d p2 such tha t pi = p2 = 0. T h e t r u n c a t e d equat ions for the 6j i m p l y t h a t 6j increases or decreases l i n e a r l y ( and so m o n o t o n i c a l l y ) i n t i m e . T h i s adds a con t inuous r o t a t i o n to the d y n a m i c s of the reduced sys tem. Therefore , f ixed po in t s of the reduced sys tem (3.89) cor respond to one o f three different types o f so lu t ions for the equat ions (3.87), depend ing o n the values of pi a n d p2. I f pi = p2 = 0 a n d 1. pi = p2 — 0, t hen th is is also a fixed po in t for equat ions (3.87). 60 2. pi = 0 a n d p2 ^ 0, or px ^ 0 a n d p 2 t ions (3.87). = 0, t hen there is a p e r i o d i c o r b i t for equa-3. p\ 7 ^ 0 a n d p2 ^ 0, t hen there is an inva r i an t torus for equat ions (3.87). T h e p e r i o d i c o rb i t occurs because the rad ius is fixed, and so the r o t a t i o n w i l l necessar i ly b r i n g an i n i t i a l p o i n t back to i t se l f after one p e r i o d of r o t a t i o n . T h e torus resul ts since there are two fixed r a d i i a n d abou t each, a cont inuous r o t a t i o n . A l t h o u g h , th i s canno t be p i c t u r e d i n four d imens ions , an ana logy i n three d imens ions is the surface o f a doughnu t . C o n s i d e r m o t i o n o n a c i rc le w i t h (constant) r ad ius r\, where the p o s i t i o n a l o n g the c i rc le is g iven by the angle ipi. A t 7*1, i p i , a d d another c i rc le , of rad ius r2 ( w i t h o r i g i n at r\, <pi), i n a p e r p e n d i c u l a r p lane . See F i g u r e 3.4. T h e p o s i t i o n a long th is second c i rc le is g iven by the angle ip2. T h e n i f r i a n d r2 are fixed a n d <pi a n d ip2 are va r i ed , the t r a j ec to ry w i l l f a l l o n the surface o f a doughnu t . T h i s is the same s i t u a t i o n as above (fixed p o i n t 3) , w i t h a different i n t e rp re t a t i on for the var iables . A note s h o u l d be m a d e t ha t the torus is i nva r i an t a n d t ha t i f ui/u>2 is i r r a t i o n a l , t hen no t ra jec tory w i l l ever pass t h r o u g h i t s i n i t i a l p o s i t i o n . However , i t w i l l come a r b i t r a r i l y close an inf in i te number o f t imes as t —>• 00. T h a t is , there w i l l be quas i -pe r iod ic d y n a m i c s . However , i f UJI/UJ2 is r a t i o n a l , t h e n there w i l l be a n inf in i te n u m b e r of p e r i o d i c so lu t ions o n the torus a n d every po in t o n the torus w i l l be o n a p e r i o d i c o rb i t . It c an be shown tha t the i n c l u s i o n of the h igher-order t e rms compl i ca t e s the behav iou r , bu t , close to the b i fu rca t ion po in t , does not affect the existence a n d s t a b i l i t y o f the so lu t ions desc r ibed above. See G u c k e n h e i m e r a n d H o l m e s [21] for a s u m m a r y of the different behav iour s found for a l l poss ib le c o m b i n a t i o n s o f the values of the n o r m a l f o r m coefficients. T h i s inc ludes the regions i n pa ramete r space where the different types of so lu t ions are observed, as w e l l as the i r l i nea r s t ab i l i t i e s . B e l o w we brief ly discuss the specific b e h a v i o u r t ha t is observed i n the two- layer m o d e l . In the case o f the two-layer m o d e l the f o r m of the x-dependence o f the e igenfunct ions is k n o w n . W e use th i s knowledge to wr i t e t h e m as where the <j>j, for j = 1,2 are funct ions o n l y o f the y, k = 271-/7, a n d rrij is a n integer w h i c h we use to l a b e l the wave n u m b e r krrij. (3.90) (3.91) 61 The normal form coefficients are given by G n = (iV21oo,$*>, G12 = (Nmu**), G2i = (/Vino,**),-G 2 2 = (/Voo2i,**>, (3.92) where Nijki are the Taylor coefficients of N(z,z,w,w) (see Section 3.2.2). In general, the formulae also depend on the terms that are quadratic in z and w (e.g. N2ooo)- However, here these terms will vanish in the projection (3.83) since they are orthogonal to the adjoint eigenfunctions (due to their rr-dependence). Also, in the same manner (see the previous section), it can be shown that to find (iV~2ioo,$*), only N^o is needed, where N2W0 = N2?o0(y)eikmx, since all terms with a m factor elkmx, m / mi will be orthogonal to $* and therefore vanish in the inner product. The only possible pairs of z<&, and zlz^wkwlHijki whose product results in a z2z term with a factor eikmiX are z$, zzH^ and 2 $ , z 2 # £ o o ° - Therefore, only H^w and #2000^ appear in the formula for N^ioo- This is due to the specific form of nonlinearity and the fact that derivatives leave the form of elhmx invariant. Thus, in addition to the eigenfunctions, • to compute (A^ioo, $*) , we only need Tfnoo and #2000^ Similarly, • to compute {Nwn,$*), we only need H $ n , H ^ ' ^ and i f & + m 2 ) , • to compute (JV*Uio, * * ) , we only need //{So, i ?o iu»~ m i ) and H[Zo+m2\ • to compute ( A r o o 2 i , w e only need #0011 a n d #0020^ • The equations satisfied by the Hijki are derived using the invariance of the center man-ifold at various orders in z and w. The derivations are exactly analogous to those of equations (3.42) and (3.43) in the previous section. The relevant H^ki satisfy (I - AP) [2AiA - L] #2ooo = ( ' - A P ) iV2000 ( J - A P ) L # U O o = -(I -AP)NUoo ( 7 - A P ) [2A 2 A - L] #0020 = ( / - AP) N0020 (I- AP)LHoon = - A P ) N o o n (I-AP) [(A1 + A 2 ) A - L ] i f 1 o i o = ( / - AP)NWW ( 7 - A P ) [ ( A 1 + A 2 ) A - L ] / f 1 o o i = ( / - AP)NWoi- (3.93) 62 T h e ( / — A P ) o n the r igh t h a n d side are reminders tha t the so lu t ions Hijki mus t be i n the correct func t ion space. Subsequent ly , the ha t t ed Hijki satisfy r r ( 2 m i ) _ •"2000 — [ 2 A i A( 2 m i ) - L ( 2 m i ) f ( 0 ) jfir(O) _ -"1100 — 2 A 2 A ( 2 " 1 2 ) - L ( 2 m 2 ) ] # £ 2 ) L ^ f f i i [(Ai 4- A 2) A ^ m i + m 2 ^ — L ( m i + T O 2 ) j j ^ ( " i i + m 2 ) [(Ai + A 2 ) A ( m i - m a ) - I / ™ ™ ) ] H A/-(2mi) i v 2 0 0 0 - 7 V ( 0 ) i v 1 1 0 0 _ A r ( 2 m 2 ) — -'v0020 = - i V , ( 0 ) [1010 0011 7 C r ( m i + m 2 ) -<v1010 r ( m i - m 2 ) '1001 ( m i — m 2) 1001 (3.94) Therefore , apar t f rom the eigenfunct ions ( $ , <!*, \I>*), a n a d d i t i o n a l s ix func t ions (mi+ma) „(0) O T 1 ^ £ ( 2 m 2 ) N o t e ^ mus t be ca l cu l a t ed : # i i 0 0 , #2000 >^ ^ 1 0 0 1 a n c * ^ioio ^ 0 0 1 1 a n d ^ 0 0 2 0 A-(m 2-mi) -"0110 — ff&~ma)- A l l these are funct ions o f y o n l y a n d are the so lu t ions o f o r d i n a r y di f ferent ia l b o u n d a r y value p rob lems . These equat ions cannot be so lved a n a l y t i c a l l y , there-fore n u m e r i c a l m e t h o d s are employed . T h e de ta i l s are g iven i n the p rev ious sec t ion . Here the resul ts w i l l be quo ted w i t h a d iscuss ion to fol low (see Tab les 3.2 - 3.5 ). N mi m 2 ro Po 0J2 a b c d 20 2 3 0.09807 11.66 0.04963 0.05163 -1 -1.234 -2.494 -1 40 2 3 0.09715 11.82 0.04949 0.05153 -1 -1.160 -2.563 -1 80 2 3 0.09692 11.86 0.04945 0.05150 -1 -1.141 -2.591 -1 T a b l e 3.2: Numerical results for the mi = 2, m 2 = 3 double Hopf bifurcation point denoted by p 2 3 on Figure 3.1. AT is number of grid points, (r0,Fo) is the location in parameter space where the bifurcation occurs, UJI, u)2 are the imaginary parts of the eigenvalues at (r0,F0) and a,b,c and d are the normal form coefficients in (3.87). N mi m 2 F0 CJ2 a b c d 20 3 4 0.04278 44.75 0.03403 0.03444 -1 -1.640 -1.808 -1 40 3 4 0.04260 45.04 0.03397 0.03439 -1 -1.632 -1.813 -1 80 3 4 0.04255 45.12 0.03395 0.03438 -1 -1.630 -1.815 -1 T a b l e 3.3: Numerical results for the mi = 3, m 2 = 4 double Hopf bifurcation point denoted by p 3 4 on Figure 3.1. N is number of grid points, (TQ,FQ) is the location in parameter space where the bifurcation occurs, u i , 0J2 are the imaginary parts of the eigenvalues at (ro,F0) and a,b,c and d are the normal form coefficients in (3.87). 63 N m i m 2 TO F0 tu2 a b c d 20 4 5 0.024670 95.445 0.023802 0.024225 -1 -1.6460 -1.8045 -1 40 4 5 0.024613 95.915 0.023770 0.024194 -1 -1.6435 -1.8059 -1 80 4 5 0.024598 96.039 0.023762 0.024185 -1 -1.6429 -1.8064 -1 T a b l e 3.4: Numerical results for the mi = 4, mi = 5 double Hopf bifurcation point. N is number of grid points, (ro,F0) is the location in parameter space where the bifurcation occurs, u>i, u>2 are the imaginary parts of the eigenvalues at (ro,-Fo) and a,b,c and d are the normal form coefficients in (3.87). TV m i m 2 ro F0 co2 a b c d 20 5 6 0.016344 168.04 0.017621 0.018121 -1 -1.6517 -1.8574 -1 40 5 6 0.016327 168.70 0.017600 0.018099 -1 -1.6488 -1.8595 -1 80 5 6 0.016323 168.87 0.017594 0.018093 -1 -1.6480 -1.8601 -1 T a b l e 3.5: Numerical results for the mi = 5, m,2 = 6 double Hopf bifurcation point. N is number of grid points, (ro,Fo) is the location in parameter space where the bifurcation occurs, UJI, u)2 are the imaginary parts of the eigenvalues at (TO,FQ) and a,b,c and d are the normal form coefficients in (3.87). F o r a l l doub le H o p f po in t s , the results are t ha t the coefficients a, b, c, a n d d are a l l negat ive a n d the i r re la t ive magn i tudes i m p l y tha t there is a reg ion i n pa rame te r space i n w h i c h the two b i fu r ca t i ng pe r iod i c o rb i t s are b o t h s table . T h e q u a l i t a t i v e b e h a v i o u r o f the so lu t ions i n the different regions i n paramete r space is presented i n F i g u r e 3.5. A s d iscussed above, the results fo l low f rom an inves t iga t ion o f the fixed po in t s o f the reduced n o r m a l f o r m equat ions (3.89), w h i c h occur when b o t h pi = 0 a n d p2 = 0. Here , there are fixed po in t s when 1. Pi = Pi = 0 2. p2 = 0 a n d px = pp = 3. pi = 0 a n d p2 = pq = (T) l-dpi + bp2 (T) Icpi ~ o,p2 4- Pi = Pi' = y j a n d p2 = p\} = W where A = ad —be a n d w i t h the c o n d i t i o n tha t the quant i t ies ins ide the square roo t signs mus t be pos i t ive . F i x e d p o i n t (1.) is a fixed po in t o f the fu l l n o r m a l f o r m equa t ions for a l l values o f the parameters . In th i s case, i t is f a i r ly easy to see tha t for s m a l l pi a n d p2, pi a n d p2 w i l l 64 have the same s ign as p i a n d p 2 , respect ively. T h i s means tha t s o l u t i o n (1.) w i l l be s table i f b o t h pi a n d p 2 are negat ive a n d uns tab le i f e i ther one is greater t h a n zero. F i x e d po in t s (2.) a n d (3.) co r respond to p e r i o d i c so lu t ions o f the fu l l n o r m a l f o r m equat ions a n d exis t w h e n pi > 0 a n d p 2 > 0, respec t ive ly (when the q u a n t i t y ins ide the r a d i c a l is pos i t ive ) . T h e to rus (4.) exists when (—dpi + bp2) /A > 0 and (cpx — ap2) /A > 0. It c a n be shown tha t i f A < 0, a < 0, b < 0, c < 0 a n d d < 0 as is the case for a l l resul ts , t hen the to rus exists i n the wedge, dpi/b < p2 < cpi/a, pi > 0, p2 > 0. See F i g u r e 3.5. In order to dec ipher the s t a b i l i t y regions for the p e r i o d i c o rb i t s , first , for the p e r i o d i c o r b i t (2.), consider , P i = P i ( p i + ap\ + bpf) (3.95) I f p2 is s m a l l , t hen i f pi is larger (smal ler) t h a n pp, t hen the q u a n t i t y i n the bracke t w i l l be negat ive (pos i t ive) , a n d so pi w i l l be negat ive (pos i t ive) . T h i s i m p l i e s t ha t p 2 = 0 is pa r t o f the s table m a n i f o l d o f fixed po in t (2.) for a l l pi > 0. N o w consider p 2 = P2 ( p 2 + cp\ + dpi) (3.96) a n d w r i t e p2 = cpi/a + e M , pi = pp + £ i a n d p 2 = e. T h e n p 2 = e ( e „ + 1cppsi + 0(e2)) (3.97) since the pi t e r m cancels p 2,. T h e n for any £ M we can choose \ei\ s m a l l enough such t ha t the s ign of p 2 is g iven by the s ign of e^. T h i s means t ha t for px > 0, w h e n p 2 < (>) cpx/a, t h e n p 2 < (>) 0 a n d so fixed po in t (2.) is s table (unstable) . S i m i l a r l y , i t c a n be shown tha t , for pi > 0, fixed po in t (3.) is s table (unstable) w h e n p2 > (<) dpi/b. Therefore , since ad < be (A < 0) , there is a reg ion g iven by dpi/b < p2 < cpi/a, pi > 0, p2 > 0 where there are two s table p e r i o d i c o rb i t s . R e c a l l t ha t a p e r i o d i c so lu t i on o f the n o r m a l f o r m equat ions corresponds to a t r a v e l l i n g wave s o l u t i o n for the P D E . See the previous sec t ion for an example . T h e l inea r s t a b i l i t y for the torus can be found by f ind ing the eigenvalues of the l i n -e a r i z a t i o n o f the reduced equat ions abou t the fixed po in t (4.). F o r a l l cases, the torus is uns tab le . T h e phase po r t r a i t s i n the var ious regimes are g iven i n F i g u r e 3.5. N o t e t ha t since o n l y one s o l u t i o n loses s t a b i l i t y o n each of the edges of the wedge, there is hysteresis , a n d the l ines p 2 = dpi/b and p2 = cpi/a give the boundar ies of the hysteresis reg ion . See F i g u r e 3.6 for a b i fu rca t i on d i a g r a m o f a poss ib le one pa rame te r t ransect o f the p a r a m e t e r space s h o w i n g the hysteresis. T h e pa ramete r s c o u l d be ei ther r or F or a f u n c t i o n o f b o t h , d e p e n d i n g o n the p a r t i c u l a r c i rcumstances . A n example o f a poss ib le p a t h is i n d i c a t e d o n 65 Figure 3.5. The bifurcation point so is a Hopf bifurcation from a stable fixed point to a stable periodic orbit. As the parameter s is increased, there is a bifurcation at si from the unstable fixed point to a different, unstable periodic orbit. At s 2, the second periodic orbit becomes stable and an unstable invariant torus bifurcates. Finally, the invariant torus is annihilated at s 3, where the first periodic orbit loses stability. The hysteresis occurs in the region between si and s 2. See Figure 3.7. Imagine an experiment (laboratory or numerical) which starts at a stable fixed point. If the parameter is slowly incremented, the solution will stay near the fixed point until s0 is reached. As s0 is passed, the solution will move away from the unstable fixed point and come to rest on the stable periodic orbit. Now if the parameter is increased, the experimenter will not observe the bifurcations at s x and s 2, and the solution will reside near the periodic orbit until it becomes unstable at S3. At this point, the solution will move away from the unstable periodic orbit and (after transients die down) come to reside near the second (now stable) periodic orbit. If the parameter is now decreased, the solution will reside near the second periodic orbit until it becomes unstable at s 2, where it will move back to the first periodic orbit. The transition from the first periodic orbit to the second does not occur at the same value of the parameter as the reverse transition, and so this is hysteresis. In a comparison of the results presented above to the previously published results on the two-layer quasigeostrophic equations without the forcing [48], [46], [39], some interesting points arise. The qualitative dynamical behaviour with and without forcing is identical. That is, the analysis of both models indicates that there are several regions in parameter space where there are two stable waves. The shape of the axisymmetric to non-axisymmetric transition curves and the wave numbers at transition are also similar. These observations support the idea that it is the baroclinic as opposed to the barotropic effects that are most important in the realization of the dynamics. A n interesting difference, however, is in the form of the bifurcating wave. When the forcing is omitted, the basic state is linear and the y-dependence of the bifurcating wave is simply a sine function of wave number one. The form of the bifurcating wave with forcing is shown in Figure 3.3. In this wave, the oscillatory part of the wave is more dominant for higher latitudes than for the lower latitude where the wave is closer to being axisymmetric. In the lower (second) layer the opposite occurs. For the linear basic state, there is no such latitudinal variation. It is difficult to say exactly which feature of the forcing led to the observed wave. However, investigating the form of this wave for different forms of forcing would be an interesting future study. In Table 3.6, the numerical differences between the results at the different levels of discretization (from Tables 3.2 - 3.5 ) are listed as evidence of the convergence of the method. If the sequence of differences of the results from adjacent discretization levels 66 (e.g. results for JV = 20 and N = 40), converges to zero, th i s ind ica tes convergence of the n u m e r i c a l a p p r o x i m a t i o n . In the l i m i t as the g r i d spac ing h goes to zero, the ra te at w h i c h th i s convergence takes p lace depends o n the order o f the m e t h o d . B y e s t i m a t i n g the ra te of convergence f rom two of the n u m e r i c a l differences, an a p p r o x i m a t i o n o f the order c a n be c a l c u l a t e d . m i m 2 NUN2 bdiff cdiff 2 3 20,40 0.074 -0.069 2 3 40,80 0.019 -0.028 2 3 order 2 1.3 3 4 20,40 0.008 -0.005 3 4 40,80 0.002 -0.002 3 4 order 2 1.3 4 5 20,40 0.0025 -0.0014 4 5 40,80 0.0006 -0.0005 4 5 order 2 1.5 5 6 20,40 -0.0029 -0.0019 5 6 40,80 -0.0008 -0.0006 5 6 order 1.9 1.7 T a b l e 3.6: Estimation of the order of the numerical approximation. iVi and i V 2 are the number of grid points at the two levels which are being compared, mi and m 2 indicate the wave numbers of the bifurcating waves at the double Hopf point and a,b,c and d are the normal form coefficients. T h e results i nd i ca t e convergence since the n u m e r i c a l differences are s ign i f i can t ly sma l l e r for the Ni = 40, N2 = 80 difference. However , i t is hoped tha t , s ince the d i s c r e t i z a t i o n formulae were second-order , the results c o u l d be accura te to second-order . In fact, the resul ts i nd i ca t e tha t the a p p r o x i m a t i o n of the n o r m a l f o r m coefficient b is o f order 2, w h i l e for c, i t seems to be a p p r o x i m a t e l y 1.5. It is not k n o w n w h a t causes th i s . It is poss ib le t h a t Ni a n d N2 are not large enough to b r i n g us in to the a s y m p t o t i c range, a n d so we canno t accu ra t e ly es t imate the order of convergence. However , the results i n d i c a t e convergence of at least order 1.5. A m i n i m u m requirement for the accuracy of the resul ts is t h a t the m a x i m u m error s h o u l d be too s m a l l to effect the associa ted q u a l i t a t i v e d y n a m i c s , w h i c h are deduced f r o m the signs of a, b, c, d a n d ad — be. Here , i t is qu i te reasonable to assume tha t th i s requi rement is satisfied. 67 b) 1 1 1 1 \ i i 0 1 1 0 i 1 / 1 1 1 -11 U I I U 1 U 1 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x Figure 3.3: The form of x i the bifurcating stream function in the upper layer. The actual solution is a travelling wave of this form whose amplitude grows as y/r - r0 • a) Surface plot, b) contour plot (lines of constant value of the stream function); the fluid tends to follow these lines. 68 Figure 3.4: A torus. If 7*1 and r 2 are fixed, then <j>i and <fo give a position on the torus. An example of a dynamical path is also plotted. 69 P2 /b Figure 3.5: The two-dimensional bifurcation diagram. The dynamical behaviour corresponding to the calculated normal form coefficients is shown. The regions of different character are separated by solid lines. In each region, the corresponding phase portrait is drawn showing the behaviour in the region. The dotted line indicates a possible one-parameter path which will lead to hysteresis of the wave solutions. The bifurcation points along this path are indicated by SQ, SI, s2 and S3 (see Figure 3.6). 70 • stable periodic orbit (wave) 0 unstable periodic orbit (wave) A unstable torus — stable steady solution . . . . unstable steady solution • bifurcation point A • • • • \\u\\ 0 • • 0 • 0 • 0 m » '1 F i g u r e 3.6: The one-dimensional bifurcation diagram depicting the bifurcation observed along the path indicated with the dotted line in Figure 3.5. The bifurcation points are labelled as s0, si, s2 and s3. See text for details. \\U\\ is a measure of the size of the solution and s is the bifurcation parameter. F i g u r e 3.7: The hysteresis loop for the path indicated with the dotted line in Figure 3.5. ||?7|| is a measure of the size of the solution and s is the bifurcation parameter. The arrows show how the solution will change as the parameter is increased or decreased. 71 C h a p t e r 4 D o u b l e H o p f b i f u r c a t i o n i n t h e d i f f e r e n t i a l l y h e a t e d r o t a t i n g a n n u l u s B e l o w , we present the ana lys is for double H o p f b i fu rca t ion po in t s w h i c h occu r i n a rea l -i s t i c m o d e l o f the d i f fe rent ia l ly hea ted r o t a t i n g annulus expe r imen t . A l t h o u g h p r e v i o u s l y w r i t t e n , for completeness a n d easy referral , the m o d e l is w r i t t e n d o w n aga in here. T h e d y n a m i c a l equat ions of the m o d e l are essent ia l ly the general m o m e n t u m equa t ions of f l u id m o t i o n i n a r o t a t i n g reference frame (2.6) - (2.9) s imp l i f i ed u s ing the B o u s s i n e s q ap-p r o x i m a t i o n . In p a r t i c u l a r , we cons ider the va r i a t ions o f a l l fluid p roper t i es t o be neg l ig ib le a n d the equa t i on of s tate o f the f lu id is assumed to be where p is the dens i ty of the fluid, T is the t empera tu re , a is the coefficient o f t h e r m a l expans ion , p0 i s the dens i ty at a reference t empera tu re T0 a n d a (T — T 0 ) is a s sumed to be s m a l l . A s ignif icant s i m p l i f i c a t i o n due to the Bouss inesq a p p r o x i m a t i o n is t ha t the fluid c a n be cons idered a p p r o x i m a t e l y incompress ib le . T h e boundar ies are the i nne r a n d outer wa l l s o f the c y l i n d r i c a l annulus as w e l l as a r i g i d flat t op a n d b o t t o m . A t the boundar i e s , the no - s l ip c o n d i t i o n is i m p o s e d o n the fluid, a n d the t empera tu re is Ta a n d TJ, at the i nne r a n d outer wa l l s , respect ively. T h e b o t t o m a n d top are t h e r m a l l y i n s u l a t i n g . T h e equat ions are w r i t t e n i n c i r c u l a r c y l i n d r i c a l coord ina tes i n a frame of reference r o t a t i n g w i t h the annu lus (at ra te Q). T h e r a d i a l , a z i m u t h a l a n d ve r t i ca l (or ax ia l ) coord ina tes w i l l be deno ted r , <p a n d 2, respec t ive ly w i t h u n i t vectors i, (p a n d k. T h e equa t ions d e s c r i b i n g the e v o l u t i o n o f the vector fluid ve loc i ty , u = u(r, (p, z, t) = ui + vcp + wk a n d the t empera tu re o f the fluid, T = T(r, <p, z, t) are as fol lows: p = Po(l-a(T-T0)) (4.1) du dt dT ~di z A 7 2 u - 2Qkx u + [gk - tfri) a(T- T0) V p - (u • V ) u (4.2) Po 72 V - u = 0 (4.4) where p is the pressure deviation from p0 = pog(D — z) + po£l2r2/2, v is the kinematic viscosity, K is the coefficient of thermal diffusivity, g is the gravitational acceleration and the domain is r G (ra, rb), <p G [0, 2TV), Z G (0, D). See Figure 2.7. The boundary conditions are u = 0 on r = ra,rb and z — 0, D T = Ta on r = ra T = Tb on r = r6 dT — = 0 on z = 0,D (4.5) with 27r-periodicity in ip. Making a change of variables: r Rr' z -> Dz' (4.6) and writing T ->T' + ATr' - A T ^ + T a (4.7) R where R — rb- ra and AT = Tb - Ta, then upon dropping the primes, the equations in scalar form become du f„9 u 2 dv \ n n I dp -Q2RarT - Q2Rar (&Tr - A T ^ + Ta - T 0) - Nr (4.8) _ = „. I V l v _ _ + _ _ \ _ j f t , _ _ _ - JV„ (4.9) ^ = , . V J » + S . T + 5 Q ( A T r - A T ^ + T . - T 0 ) - - N, (4.10) ^ = K . V ' T + K , — - ^ - u - N T (4 .11) du u dv ldw _ ^ dr r d<p 8 dz where u = ui + v<p + wk, 8 = D / R , vs = v/R2, ns = K / R 2 and „ 2 d2 i o i a 2 1 a 2 V : 1 1 h oY2 r or r 2 6V2 <52 dz2 73 N = U - V ^ \ --( — -— - — - — \ R R \ s r J R \ dr r dp 5 dz r) , T 1 / _ uv\ 1 / dv v dv 1 dv uv\ , , 1 , _ N 1 f dw vdw 1 dw\ NT = - ( u . V . T ) = - ^ - + + (4.13) T h e b o u n d a r y cond i t i ons are now: dz w i t h 27r-periodicity i n (p. u = 0 o n r = ^ a n d 2 = 0 ,1 T = 0 r = P ' P u = 0 on z = 0 , l (4.14) 4.1 The analysis T h e r o t a t i o n ra te Q, a n d the t empera tu re difference between the inner a n d outer annu-lus wa l l s A T are the parameters of interest . These are the p h y s i c a l quan t i t i e s (ex te rna l var iables) w h i c h are easi ly va r i ed i n an exper iment . T h e other parameters descr ibe the geomet ry o f the annulus or proper t ies of the fluid. A n o t h e r poss ib le choice is to use the d imens ionless parameters , the T a y l o r number T a n d the t h e r m a l R o s s b y n u m b e r K (see Sec t i on 1.2), w h i c h have a one-to-one correspondence w i t h 0 a n d A T . T h e resul ts are quo ted i n t e rms o f these parameters since e x p e r i m e n t a l results are u s u a l l y presented o n a l og - log p lo t o f T versus 7c. However , the ana lys is was ca r r i ed ou t u s i n g the d i m e n s i o n a l pa rameters Q a n d A T , since n o n - d i m e n s i o n a l i z a t i o n d i d not s i m p l i f y the equa t ions (see Sec t i on 2.3). In p r i n c i p l e , the choice o f parameters w i l l not change the p rocedure or the resul ts . A shor t s u m m a r y of the m a i n steps of the ana lys is are as fol lows: 1. P l o t the n e u t r a l s t a b i l i t y curves, by (a) c a l c u l a t i n g the s teady a x i s y m m e t r i c s o lu t i on at a p a r t i c u l a r l o c a t i o n i n p a r a m -eter space, (b) s o l v i n g the eigenvalue p r o b l e m for th i s s o l u t i o n to find i t s l i nea r s t ab i l i t y , 74 (c) r epea t ing steps (a) a n d (b) at var ious loca t ions i n pa ramete r space to find the pa ramete r values where the s o lu t i on is n e u t r a l l y s table . 2. L o c a l i z e the p o i n t i n pa ramete r space where the double H o p f b i f u r c a t i o n occurs (the in tersect ions of the neu t r a l s t a b i l i t y curves) . 3. C a l c u l a t e the eigenvalues a n d eigenfunct ions at the b i fu r ca t i on po in t . 4. C o m p u t e the appropr i a t e n o r m a l fo rm coefficients, w h i c h involves (a) c a l c u l a t i n g the adjoint e igenfunct ions, (b) c a l c u l a t i n g the center m a n i f o l d coefficients. T h e procedure is a lmos t the same as tha t for the two- layer m o d e l , w h i c h was presented i n the p rev ious chapter . O n e m a i n difference is tha t an a n a l y t i c a l f o r m for the s teady a x i s y m m e t r i c s o l u t i o n is not k n o w n a n d so th is too has to be a p p r o x i m a t e d n u m e r i c a l l y . In the ana lys i s , th i s is deal t w i t h by l eav ing the a x i s y m m e t r i c s o l u t i o n unreso lved w h e n the p e r t u r b a t i o n equat ions are w r i t t e n d o w n . T h e n , for the n u m e r i c a l a p p r o x i m a t i o n o f the e igenfunct ions a n d center m a n i f o l d coefficients, the values o f the a x i s y m m e t r i c s o l u t i o n are o n l y needed at specific loca t ions (the g r i d poin ts ) a n d n u m e r i c a l a p p r o x i m a t i o n s are used. O t h e r differences occur o n l y i n the deta i l s of the c o m p u t a t i o n a l p rocedure . F o r th i s reason, i t is the differences tha t are emphas ized . F o r a more de t a i l ed e x p l a n a t i o n o f the center m a n i f o l d r e d u c t i o n a n d n o r m a l fo rm equat ions , see the p rev ious chapter . 4.1.1 Steady axisymmetric solution T h e ana lys i s begins w i t h the c o m p u t a t i o n o f a s teady a x i s y m m e t r i c s o l u t i o n . T h a t is , we l o o k for so lu t ions o f equat ions (4.8) — (4.12), w i t h the b o u n d a r y c o n d i t i o n s (4.14), i n the f o r m u = u(r,z), v = v(r,z), w = w(r,z), T = T(r, z), (4-15) i.e. the dependent var iab les are independent of ip ( a x i s y m m e t r i c ) a n d t ( s teady) . N o t e t ha t the so lu t ions also depend o n the parameters . S t r e a m funct ions are used to solve for the a x i s y m m e t r i c so lu t ions . W i t h the f o r m of the so lu t ions as above, the i n c o m p r e s s i b i l i t y equa t ion (4.12) becomes ^ + - + ^ = °. (4.16) or r o dz I f u a n d w are w r i t t e n i n t e rms of a s t r eam func t ion £, u = ~7T r oz 75 w = -S-^ (4.17) r or t h e n the i n c o m p r e s s i b i l i t y c o n d i t i o n (4.16) is a u t o m a t i c a l l y sat isf ied. A f t e r u s i n g (4.17) to replace u a n d w i n the a x i s y m m e t r i c equat ions , the pressure t e rms can be e l i m i n a t e d by t a k i n g X d 1 d - — [equation (4.10)1 - — — [equation (4 .8)] . (4.18) Ror Doz T h i s results i n three equat ions i n the three u n k n o w n funct ions v, £ a n d T . T h e equat ions were c o m p u t e d s y m b o l i c a l l y u s ing M a p l e , a n d are sufficiently c o m p l i c a t e d t ha t no ins igh t is ga ined by e x p l i c i t l y w r i t i n g t h e m here. T h e b o u n d a r y cond i t i ons are the same for v a n d T a n d the no-s l ip c o n d i t i o n s o n u a n d w become - = - = 0 o n r = - , - a n d z = 0 , l . T h i s c o n d i t i o n i m p l i e s tha t £ is cons tant on the boundar ies a n d since there is a f reedom to choose £ up to an add i t i ve constant , the a d d i t i o n a l b o u n d a r y c o n d i t i o n is chosen to be £ = 0 on r = -^:~B a n d z = 0 , 1 . 4.1.2 The perturbation equations T h e nex t set of equat ions tha t are needed are the p e r t u r b a t i o n equat ions . It is th i s sys t em o n w h i c h the center m a n i f o l d r educ t ion is pe r fo rmed . T h e p e r t u r b a t i o n equat ions are o b t a i n e d by w r i t i n g the so lu t ions i n the fo rm U - fu + u', V = fv + V1, W = fw + w', T = fT + r, P - U+P', (4.19) where ( / u , fv, fw, fT, fp) is a s teady so l u t i on of the a x i s y m m e t r i c equat ions . U p o n s u b s t i t u t i o n of (4.19) i n to equat ions (4.8) - (4.12), a n d d r o p p i n g p r imes , the p e r t u r b a t i o n equat ions are ob ta ined : I ' u 2 dv\ I dp _ 2 n ^ _ = , s ^ - - - _ - j + 2 ^ - — --VBarT -^Fu-Vsu + u-Vsfu-2^j-Nr (4.20) 76 ®v — y (TJ*V V + 2 <9w\ 1 dp dt S \ s r2 r2 dip ) poRr dip -\ (FU -VSV + U. VJv + f^ + f-^-Nv (4.21) ftjL = vsV2w + gaT--^^-^(Fu-Vsw + u-Vsfw)-Nz (4.22) dT A T 1 — = K S V 2 T - — u - - ( F „ • V S T + u • V . / r ) - NT (4.23) du u 1 dv ldw „ ,, _ ^ - + - + - ^ - + ^ ^ - = 0 (4.24) dr r r dip o dz d \ d X ^  where FU = L z + /„<£ + V 5 = i— + <2>-— + k- — a n d the non l inea r t e rms TV. aga in dr r dip 6 dz are g iven by the formulae i n equa t ion (4.13). 4.1.3 The eigenvalue problem I f we assume tha t the a l l the unknowns m a y be w r i t t e n as u = u(r, ip, z, t) = e A *u m ( r , z)eim,p, w i t h m a n integer, a n d the p e r t u r b a t i o n equat ions (4.24) are l i nea r i zed , t h e n a l inea r eigen-value p r o b l e m is o b t a i n e d for each a z i m u t h a l wave n u m b e r m. F r o m these equa t ions the l inea r s t a b i l i t y of the a x i s y m m e t r i c so lu t i on is found (see Sec t ion 3.1). N o t e t ha t the f o r m of the a z i m u t h a l dependence is a resul t o f the 27r-per iodici ty i n ip. T h e eigenvalue p r o b l e m is: Xum = V s ( V m u m - ^ - ^fvm j + 2Qvm ^ -tfRarfm -^(FU- V m « m + u m • Vsfu - 2^^j (4.25) . „ / _ o „ vm 2im „ \ im ^  Xvm = Us I V m V m - —.+ -^-Um I - 2VtUm - ^J^Pm ~ ( ^ « - V m ^ + ^ . V J B + ^  + ^  (4.26) X dp X Xwm = usV2mwm + gafm - - ^ - ^ - ^ (Fu • V m w m + u m • Vsfw) (4.27) AT, ro KsV2sfm - ^ - u m ~ ^ { F u - V m T m + u m • V s / T ) (4.28) dum um im^ 1 dwm . , ~^— + 1  Vm + 1-^— = 0 I 4 ' 2 9) dr r r o dz where „ 2 m 2 a 2 i a 1 <92 V = 1 1 1 r2 dr2 rdr 62 dz2 77 a n d U m ~ iS^rni ^mi ^TO) • I f m ^ 0, i t is poss ible to e l im ina t e pm a n d vm. F r o m the i n c o m p r e s s i b i l i t y equa-t i o n (4.29), we o b t a i n ir (du vm = 1 dWr, m m \ dr a n d f r o m equa t ion (4.26) we have r ^ S dz (4.30) Pn —X-.poRr m -Xvm + vs \ V2mvm 1 vm 2im„ \ —k- -\ 7T Um ~  2i"^ j-i T-7 - . r 7 t i fvum . fu^n -E\Fu- V m w m + u m • V s / „ H 1 R \ r r (4.31) T h e r e s u l t i n g three equat ions i n the three r e m a i n i n g u n k n o w n s um, wm, Tm m a y be w r i t t e n i n the f o r m XA.mUm — TjmUm (4.32) where U„ ( Um \ V Tm J a n d A m a n d L m are 3 x 3 mat r ices o f l inear opera tors . If m = 0, a s t r eam func t ion m e t h o d c a n be used i n e x a c t l y the same m a n n e r as i n the c a l c u l a t i o n o f the a x i s y m m e t r i c s o l u t i o n . A g a i n the equat ions were ca l cu l a t ed us ing the M a p l e s y m b o l i c c o m p u t a t i o n package a n d are t o o c o m p l i c a t e d to wr i t e here. T h e inner p r o d u c t of two vector funct ions U = (u,v,w,T) a n d U' = (u',v',w',T') is t aken to be (U, U')= I* (uu1 + vv1 + wwI + X T ' ) r dr dip dz (4.33) E q u a t i o n s (4.32) are solved n u m e r i c a l l y a n d equa t ion (4.30) is used to restore the m i s s i n g c o m p o n e n t . 4.1.4 The adjoint eigenvalue problem F i n a l l y , the adjo in t equat ions are necessary to ca lcu la te the adjoint e igenfunct ions . T h e equa t ions are o b t a i n e d i n the same manne r as i n Sec t ion 3.3. N o t e t ha t i n the d e r i v a t i o n o f the ad jo in t opera tors , i t is poss ib le to e l i m i n a t e the pressure t e r m u s i n g the incompress -i b i l i t y c o n d i t i o n . T h e adjoint eigenvalue p r o b l e m is 78 •y * (-2 * u* 2dv*\ 1 dp* AT +UF»-v°u*-v:-df-f-f)  ( 4 - 3 4 ) , 2 . . . ^ . 2 5 w * \ 1 5 ^ r 2 r 2 <9<p / poRr dip +UK.Vy^V:.^ + 2h±-m (4.35) Ft V r dip r r I A T * = nsV2sT*-tfRaru*+ gaw*+ ^ -Fn-VsT* (4.37) du* u* Idv* ldw* „ „„. ^ - + - + - — + - r ^ = 0 4.38 a r r r dip o dz where V* = (u*,v*,w*,T*), Fa = (fu, fv, fw, IT), and where, for example , V* . ^  = u*— + v*— + w*— + T * — a dr dr dr dr dr A g a i n so lu t ions are sought i n the f o r m u*(r,ip,z) = u m ( r , z)etmip, a n d aga in i t is poss ib le to e l i m i n a t e v*m a n d p*m. 4.1.5 Normal form coefficients In th i s sec t ion the def in i t ions needed for the n o r m a l f o r m coefficient formulae w i l l be g iven . T h i s sec t ion is essent ia l ly an overv iew of Sec t ion 3.3, except t ha t the m a t e r i a l is presented i n the contex t o f the r o t a t i n g annulus m o d e l . D e t a i l s are kept to a m i n i m u m a n d the reader is referred to Sec t i on 3.3 for more de t a i l . A s s u m e tha t the b i fu rca t i on o f interest occurs at = fi0 a n d A T = A T 0 . T h a t i s , at Q.Q a n d A T 0 , the l inear eigenvalue p r o b l e m given i n equat ions (4.25) — (4.29) has two c o m p l e x conjugate pa i r s of eigenvalues w i t h zero rea l pa r t (for a t o t a l o f four) , w h i c h w i l l be deno ted by A i = Pi + itoi, A i , A 2 = P2 + iv2, A 2 , (4.39) where at Q0 a n d A T 0 , p\ = P2 = 0. A s s u m e also tha t a l l o ther eigenvalues have negat ive r ea l pa r t . T h e e igenfunct ions co r r e spond ing to the above eigenvalues w i l l be w r i t t e n as $, $, tt, 79 where they have the f o r m $ = <l(r, z)eim^ = (<j>u, <j>v, <f>w, <fr) eimi*, w i t h m i b e i n g the a z i m u t h a l wave number co r r e spond ing to $ . \I> has the same f o r m w i t h vn,2 l a b e l i n g i t s wave number . T h e adjoin t e igenfunct ions co r re spond ing to $ a n d ^ are deno ted by <J>* a n d re-spect ively , where $* a n d \J>* are found f rom equat ions (4.34) - (4.38), w i t h the i r respect ive wave numbers . T h e n o r m a l i z a t i o n w i l l be chosen such tha t ( $ , $ * ) = 1. (4.40) T h i s is s l i g h t l y different t h a n p rev ious ly (equat ion (3.21) ), s ince i n the present case A = I (where I is the i den t i t y m a t r i x ) . See equat ions (4.25) — (4.29). W e w r i t e the vector of unknowns as U = (u,v,w,T) = z $ + z $ + w # + + C (4.41) where z = z(t) — (U, $*) a n d w = w(t) = (U, are c o m p l e x numbers a n d z§ 4- z $ + wty + wty G Ec, a n d ( G Es (where Ec a n d Es are the center a n d s table eigenspaces, r espec t ive ly ) . N o t e t ha t the the def ini t ions of z a n d w are s l i g h t l y different aga in since A = 1 O n the center m a n i f o l d , we can wr i t e ( = H(z,z,w,w) =0(\z,z,w,w\2) (4.42) a n d e x p a n d H i n a T a y l o r series as H(Z, Z, W, W) = #2000<2 2 + HN00ZZ + +H002oW2 + HQOHWW +Hwwzw + Himizw + cc. + 0 ( 3 ) (4.43) where Hijkl are the T a y l o r coefficients of H(z, z, w, w), 0(n) = 0(\z, z, w,w\n) a n d c c . are the c o m p l e x conjugates of the te rms tha t are w r i t t e n e x p l i c i t l y . W e also w r i t e N(z, z, w, w) i n the same f o r m , where N(z, z, w, w) is the non l inea r t e r m (4.13) w r i t t e n i n t e rms o f z, z, w a n d w, u s i n g the d e c o m p o s i t i o n of U g iven i n equa t ion (4.41), w i t h ( w r i t t e n u s i n g (4.42) a n d (4.43). See A p p e n d i x C for formulae for the coefficients of N. T h e n o r m a l f o r m equat ions for the non-resonant case are z = Xiz + Gnz2z + Gi2zww + 0 ( 4 ) (4.44) to = \2w + G2\zzw + G22w2w + 0 ( 4 ) (4.45) 80 where Xj=Xj(£l, AT), a n d the n o r m a l fo rm coefficients Gki are g iven by <?ii = <iV 2 ibo,**>, G12 = ( / V i o i i , $ * ) , G2i = ( N i m , * * ) , G22 = <JVoo2i,**>. (4-46) W e w r i t e z = p\el6xl\j\G\x \ a n d w = P2&1621 \J\G22\, where G[j is the rea l pa r t o f Gij a n d subs t i tu t e these expressions i n to (4.45). In these p o l a r coord ina tes , the t r u n c a t e d n o r m a l f o r m equat ions are Pi = Pi (pi + ap\ + bpt) P2 = P2 (pi + cp\ + dpfj 9\ = coi 62 = cu2 (4.47) where fir b = c = sir ^12 \Gr GT 2 2 1 T 21 IG11I d = T ^ = ±1> (4-48) \G •r 221 a n d Xj = pj+iuj. T h e 0(\p\, p^4) t e rms are ignored i n the pj equat ions a n d the 0 ( | p i , p 2 | 2 ) t e rms are i gno red i n the Oj equat ions . G i v e n mx a n d m2, i t c an be shown tha t the n o r m a l f o r m coefficients a,b,c,d c a n be w r i t t e n i n t e rms of the fo l l owing funct ions , w h i c h are a l l funct ions of r a n d z o n l y : • the e igenfunct ions <l, $ * , if, • the coefficients of the center m a n i f o l d : £V(0) rV(2mi) r r ( m i - m 2 ) r V(mi+ro 2) fr(0) , r r ( 2 m 2 ) •"1100) " 2 0 0 0 1 "1001 ) -"1010 > "0011 d I l u -"0020 > 81 where H, = ^ H^e%m'p. T h e center m a n i f o l d coefficients are found f rom equat ions (3.94), m w i t h A ( m ) = I a n d where l / m ) is the l inear pa r t i n equat ions (4.25) - (4.29). F o r m ^ 0, the same s o l u t i o n m e t h o d tha t was used for the eigenvalue p r o b l e m can be used here (i.e. e l i m i n a t i o n o f the pressure t e r m a n d one ve loc i ty c o m p o n e n t ) . F o r m = 0, the s t r eam func t i on m e t h o d (as for the a x i s y m m e t r i c so lu t ion) c a n be used. 4.2 Numerics A n a l y t i c so lu t ions for the u n k n o w n funct ions (the a x i s y m m e t r i c s o l u t i o n , e igenfunct ions a n d center m a n i f o l d coefficients) are not possible , therefore t hey are a p p r o x i m a t e d numer -i ca l ly . A s i n the ana lys i s of the two-layer m o d e l , centered f ini te d i f ferencing is used to discre t ize the s p a t i a l der iva t ives . U p o n d i s c r e t i z a t i on , the a x i s y m m e t r i c so lu t i on is a p p r o x i m a t e d f rom a sys t em o f n o n -l inea r a lgebra ic equat ions , the p a r t i a l different ial eigenvalue p rob lems become m a t r i x eigen-va lue p rob l ems a n d the p a r t i a l different ia l b o u n d a r y value p rob lems for finding the center m a n i f o l d coefficients become systems of l inear equat ions . 4.2.1 The ordering of the unknowns po in t s i n the in te r io r o f the d o m a i n defined by: r = rk, 1 < k < N a n d z where N, k, I are pos i t ive integers, where r 0 = r ^ + i = -TZ, Z0 = 0 a n d z^+i W e a p p r o x i m a t e the value of the u n k n o w n funct ions at the loca t ions o f the N x N g r i d zhl<l<N, 1. Def ine ^(rk,zi) = £hi, a n d s i m i l a r l y for the other u n k n o w n funct ions . I n the n u m e r i c a l procedures , i t is necessary to arrange the unknowns , (n-h m vec tor f o r m , as opposed to the g r i d d e d n o t a t i o n . W e order the u n k n o w n s i n the concep tua l l y mos t s i m p l e way a n d w r i t e the vector o f u n k n o w n s as 61 V 6VJV J xf (4.49) 82 where X% is o f size N2. T h a t is, we stack the rows (constant k) o f the u n k n o w n s ( t ransposed) o n t op of each other . T h i s c o u l d also be done such tha t the c o l u m n s are s tacked, however, s ince the order o f the der iva t ives w i t h respect to r a n d z are the same, i t is u n l i k e l y tha t one o r d e r i n g is be t te r t h a n the other . In fact, s ince the o r i g i n a l p rob lems are systems of p a r t i a l d i f ferent ia l equat ions , there are three u n k n o w n funct ions w h i c h must be w r i t t e n i n vector fo rm. A g a i n the mos t s i m p l e m e t h o d o f d o i n g th i s is to s tack the three vectors o f the f o r m (4.49). F o r example , the vec tor of u n k n o w n s for equa t ion (4.32) is where each XUm a n d XWm are o f size N2, w h i l e XTm is size N(N + 2) ( this is due to the i n s u l a t i n g b o u n d a r y cond i t i ons at z = 0 , 1 ; the values of T at these g r i d po in t s mus t also It is easy to see how the d i s c r e t i za t i on reduces the p rob lems to those o f l i nea r a lgebra . W h e n the der iva t ives at a p a r t i c u l a r g r i d l o c a t i o n are a p p r o x i m a t e d w i t h the di f ferencing formulae , an a lgebra ic equa t ion i n te rms of (\M is fo rmed, since the £t/ are s i m p l y rea l var iab les . If t h i s is done for every g r i d po in t , a sys tem of a lgebra ic equa t ions i n the (iki is p r o d u c e d . It is useful to wr i t e the l inear pa r t of th is equa t ion i n t e rms of the coefficient m a t r i x , where the entries of the m a t r i x are the coefficients o f the (\ki i n the equat ions (each row corresponds to an equa t ion a n d each c o l u m n to the £ w , i.e. w h e n the m a t r i x is m u l t i p l i e d to the vector of unknowns , the l inear pa r t of the sys t em of equat ions resul ts ) . G i v e n tha t Um is a vector o f size 3N2 + 2N, th i s i m p l i e s tha t the ma t r i ce s w i l l be o f size (3N2+ 2N) x (3N2+ 2N). T h e ma t r i ce s associa ted w i t h the chosen o rde r ing of the u n k n o w n s is sparse, however, t he i r b a n d w i d t h are r e l a t ive ly large. T h i s comes f rom the fact tha t the centered di f ferencing fo rmulae for the der iva t ives w i t h respect to r at ( r ^ , ^ ) depend o n ne ighbours a l o n g a c o l u m n o f the square g r i d (the u n k n o w n s at (r^—i, zi), {rk+i, zi), etc.). These u n k n o w n s are separa ted by AT entries i n the vector of unknowns (see equa t ion (4.49)), a n d therefore, so are the associa ted entries i n the coefficient m a t r i x . Therefore , w i t h t h i r d - a n d four th -order der iva t ives i n r , the b a n d w i d t h of the m a t r i x w i l l be abou t 4N. However , the l inea r solver t ha t was used o n l y takes advantage of the sparseness p r o p e r t y a n d no t the s t ruc tu re . S ince th i s is adequate for our purposes , the o rde r ing is not cons idered fur ther . T h e eigenvalue solver , w h i c h also does not take advantage of the s t ruc ture , is d iscussed be low. (4.50) be cons idered as unknowns ) . T h u s , Um is a vector of size 3N2 + 2N. 83 4.2.2 The mesh: non-uniform spacing W i t h the c o m b i n a t i o n o f the no-s l ip b o u n d a r y cond i t i ons a n d the s m a l l pa r ame te r (u) m u l t i p l y i n g the second der iva t ive t e r m , b o u n d a r y layers i n the s o l u t i o n of the a x i s y m -m e t r i c p r o b l e m are expec ted . T h e naive approach w o u l d ignore th i s fact a n d a t t e m p t to a p p r o x i m a t e the s o l u t i o n at loca t ions o n a u n i f o r m g r i d . F o r s m a l l values o f the adjus table parameters , th i s m a y be sufficient. However , as the parameters increase, n o n - p h y s i c a l h i g h frequency osc i l l a t ions are observed near the b o u n d a r y i f the mesh size is no t sufficient to resolve the steep gradients i n the b o u n d a r y layer. T h e op t ions are to increase the n u m b e r o f g r i d po in t s , to use non-centered differencing or to use a n o n - u n i f o r m g r i d . U n f o r t u n a t e l y , g iven the ava i lab le resources, the first is not sufficient to solve the p r o b l e m . T h e second o p t i o n was also passed over since non-centered differencing resul ts i n a lower order accura te s o l u t i o n . It was therefore necessary t o look for a m e t h o d o f choos ing a n o n - u n i f o r m g r i d . S ince the large changes i n the s o l u t i o n o c c u r a t the boundar ies , a h ighe r dens i t y o f g r i d po in t s s h o u l d be loca t ed here. T h e so lu t i on has slower changes away f r o m the bounda ry , so i t w o u l d be inefficient to have too m a n y po in t s i n the in te r io r . T h e op t ions are t o m a n u a l l y choose the g r i d po in t s or to use a sca l ing m e t h o d . A n example of the first o p t i o n w o u l d be to m o d i f y a u n i f o r m g r i d by h a l v i n g the mesh size for po in t s w i t h i n a c e r t a i n d i s tance o f the bounda ry . T h i s was not done since the t h i r d a n d four th-order der iva t ives w o u l d make the c o d i n g for the coefficient m a t r i x very l abor ious . A s c a l i n g m e t h o d was therefore chosen. T h i s m e t h o d consists o f m a k i n g a change o f coord ina tes a n d c a l c u l a t i n g the so lu t ions on a u n i f o r m g r i d i n the new coord ina tes . T h a t is , the inverse o f the t r a n s f o r m a t i o n takes a u n i f o r m g r i d to a g r i d w i t h m a n y po in t s near the bounda ry . T h e t r a n s f o r m a t i o n w h i c h takes (r, z) to the new coord ina tes (x,y), is _ t a n - 1 (TJX) 1 ra . V = 2 t a n " 1 (J) + 2 + R z = + i (4.51) 2 t a n - i ( § ) 2 where r] is a s ca l i ng factor w h i c h de termines the m a g n i t u d e of compress ion near the b o u n d -ary. See F i g u r e 4 .1 . T h e d o m a i n r € [ra/R,rb/R], z G [0,1] goes t o a; 6 [ - 1 / 2 , 1 / 2 ] , ye [ - 1 / 2 , 1 / 2 ] . T h e equat ions are t r ans fo rmed s i m p l y by w r i t i n g u(r, z) — u'(x, y) ( l ikewise for o ther funct ions) a n d us ing the cha in ru le to wr i t e the equat ions i n t e rms o f der iva t ives w i t h respect to x a n d y. F o r example , £=¥•%• <4-52> or ox or 84 a) 0.5 >- 0 -0.5 -0.5 0.5 F i g u r e 4.1: The transformation of the grid points, (a) A uniform grid (equally spaced grid points) of N = 20 (b) the grid obtained by applying the change of coordinates (4.51) with T? = 6; it is on this non-uniform grid which the solutions are approximated. where, here, | = ^ — ( | ) ( l + , V ) . (4.53) A note s h o u l d be m a d e tha t the b o u n d a r y layers observed i n the so lu t ions o f the eigen-va lue p r o b l e m are not as severe as those i n the a x i s y m m e t r i c so lu t ions . I n fact , s igni f icant errors are i n t r o d u c e d i n the eigenvalues a n d eigenfunct ions i f the po in t s i n the in te r io r are t o o sparse. T h i s occurs even i f the a x i s y m m e t r i c so lu t ions appear to be w e l l represented. T h i s p r o b l e m suggests tha t different sca l ing factors s h o u l d be used for the a x i s y m m e t r i c a n d eigenvalue p rob lems . However , the errors i n t r o d u c e d i n the i n t e r p o l a t i o n seemed to negate the benefit of u s ing m u l t i p l e sca l ing factors. In the ca l cu l a t i ons presented, the sca l -i n g fac tor 77 = 6 is the smal les t va lue poss ib le w h i c h gives q u a l i t a t i v e l y g o o d resul ts for the a x i s y m m e t r i c p r o b l e m w h e n TV = 20. It seems tha t smal l e r rj (for th i s N) do not resolve the b o u n d a r y layer w e l l enough a n d larger r\ do not con t a in enough in te r io r po in t s to descr ibe the e igenfunct ions w e l l enough. N o t e however, t ha t for larger values o f JV the resul ts are consis tent . 4.2.3 Solution techniques F o r the c o m p u t a t i o n of the a x i s y m m e t r i c so lu t i on , N e w t o n ' s m e t h o d is used [4]. T h i s m e t h o d is ve ry effective since a c o n t i n u a t i o n technique can be used to generate a ve ry g o o d guess of the s o l u t i o n for any paramete r value. It is k n o w n tha t i f fl = A T = 0 t h e n the 85 t r i v i a l s o l u t i o n satisfies the a x i s y m m e t r i c equat ions . T h u s for Q, a n d A T s m a l l , the zero vec tor is a g o o d guess of the s o l u t i o n . F o r s m a l l increments f r o m these pa ramete r values, the p rev ious s o l u t i o n is a g o o d guess. A s l o n g as the i n c r e m e n t a t i o n o f the parameters is s m a l l enough , a s o l u t i o n c a n be o b t a i n e d for any pa ramete r va lue . I n a d d i t i o n , be tween increments , a secant l ine a p p r o x i m a t i o n can be made so tha t the guess for the next s tep is even bet ter . F o r example , g iven a s o lu t i on at s — 1, Xa-i a n d s, Xs, a guess for the s o l u t i o n at s + 1 c o u l d be Xs + (Xs — Xs-i) / A s , where A s is the change i n the pa rame te r va lue . T h i s guess is expec ted to be be t ter t h a n Xs. T h e n e u t r a l s t a b i l i t y curves are found us ing a n i t e ra t ive secant m e t h o d . V a r y i n g one pa ramete r w h i l e the o ther is he ld fixed, the largest rea l pa r t o f the eigenvalues is found at two different loca t ions . C o n s i d e r i n g the largest real pa r t as a func t ion o f the pa ramete r , the l ine pass ing t h r o u g h b o t h po in t s is ca l cu la t ed . A hopefu l ly i m p r o v e d guess o f the p o i n t o n the n e u t r a l s t a b i l i t y curve is the l o c a t i o n where the l ine passes t h r o u g h zero. T h e largest rea l pa r t is c a l cu l a t ed at the new guess a n d a l ine is d r a w n between th is p o i n t a n d the be t te r guess of the two prev ious po in t s . T h i s is repeated u n t i l the des i red accu racy is o b t a i n e d . T h e doub le H o p f b i fu r ca t i on po in t s occur at in tersect ions o f two n e u t r a l s t a b i l i t y curves . Therefore , the fo l l owing procedure is used to loca te t h e m . T w o po in t s are found o n each n e u t r a l s t a b i l i t y curve a n d a l ine i n paramete r space is d r a w n t h r o u g h each pa i r . T h e in te r sec t ion o f these l ines is an i n i t i a l guess of the l o c a t i o n o f the b i fu r ca t i on p o i n t . F r o m th i s guess, one pa ramete r is va r i ed w h i l e the o ther is fixed so tha t a new p o i n t o n each of the n e u t r a l s t a b i l i t y curves can be found . F o r each curve, a new p a i r o f po in t s is fo rmed w i t h the new po in t a n d one of the previous po in t s . T h e in te r sec t ion o f the l ines d r a w n t h r o u g h the new pai rs is (hopeful ly) a bet ter guess. T h i s is repeated u n t i l the m a g n i t u d e s o f the rea l par t s o f the two relevant eigenvalues are less t h a n a specif ied to lerance . A f t e r d i s c r e t i z a t i on , the eigenvalue p r o b l e m (4.32) is so lved u s ing the M a t l a b c o m m a n d 'eigs ' . T h i s m e t h o d computes a g iven number of eigenvalues (and c o r r e s p o n d i n g eigenvec-tors) u s ing A r n o l d i i t e r a t i on [50]. In pa r t i cu l a r , i t is able to ca lcu la te the eigenvalues w i t h smal les t m a g n i t u d e . A l t h o u g h we are interested i n the eigenvalues w i t h largest r ea l p a r t (when th i s is close to zero) , i t was j u d g e d adequate since e x p e r i m e n t a l results have shown the phase speed o f the observed waves to be very s low (see be low) . T h e phase speed is g iven a p p r o x i m a t e l y by the i m a g i n a r y pa r t of the c r i t i c a l eigenvalue, w h i c h i m p l i e s t ha t the m a g n i t u d e o f the c r i t i c a l eigenvalue is s m a l l . T h i s hypothes i s was tes ted o n the N = 20 case b y c o m p u t i n g a l l the eigenvalues. In a l l cases, i t was found t h a t the e igenvalue w i t h smal les t m a g n i t u d e was indeed the eigenvalue w i t h largest rea l par t . T h i s does not guar-antee t ha t for sma l l e r g r i d spacings th is is s t i l l t rue, however, the correspondence between the different g r i d spacings suggests tha t the a s s u m p t i o n is v a l i d . A l s o , to reduce the r i sk 86 of error , not o n l y the smal les t m a g n i t u d e eigenvalues were c o m p u t e d , bu t the p smal les t , where i n mos t cases p = 12. T h e 'eigs ' f unc t i on is able to use the sparseness proper t ies o f the mat r i ces . However , for the genera l m a t r i x eigenvalue p r o b l e m , (XBv = Av, where A a n d B are ma t r i ces , v is the eigenvector , A is the eigenvalue) , A r n o l d i i t e r a t i on needs B to be s y m m e t r i c a n d pos i t ive defini te . Unfo r tuna t e ly , th i s is not the case w i t h the present p r o b l e m , so i t is necessary to inver t B to o b t a i n the u sua l m a t r i x eigenvalue p r o b l e m (i.e. Xv = B~lAv). T h i s destroys mos t o f the sparseness of the mat r ices . U s i n g the sparseness feature, however, s t i l l saves a factor o f two i n memory . It s h o u l d also be no ted tha t the n o n - s y m m e t r i c p r o p e r t y o f B is not due to the c o m p u t a t i o n on the n o n - u n i f o r m g r i d , s ince there are va r i ab le coefficients i n the pre- t ransformed equat ions . T h e d i sc re t i zed ( t ransformed) equat ions a n d the entries o f the coefficient ma t r i ce s were c o m p u t e d s y m b o l i c a l l y u s ing M a p l e . T h e results were t hen t ransferred to M a t l a b , to find the n u m e r i c a l a p p r o x i m a t i o n s . 4.2.4 Discussion of convergence A couple o f notes s h o u l d be made conce rn ing the a p p r o x i m a t i o n by centered finite differ-enc ing . T h e formulae tha t were used are der ived us ing t r u n c a t e d T a y l o r series expans ions . T h e error i n the value of the der iva t ive at any g r i d p o i n t (ca l led the l o c a l t r u n c a t i o n error) is 0(h2) (i.e. a p p r o x i m a t e l y a constant t imes h2, as h —r 0), where h is the ( local ) mesh size. T h i s means t ha t the error w o u l d be expected to decrease by a factor o f a p p r o x i m a t e l y four w h e n h is ha lved . T h e constant m u l t i p l y i n g the h2 depends o n h igher -order der iva t ives t h a n the de r iva t ive tha t is b e i n g a p p r o x i m a t e d . F o r ins tance , for the a p p r o x i m a t i o n o f the four th-order der iva t ive , the constant depends o n the s ix th -order de r iva t ive . M o r e specif-i ca l l y , cons ider the first-order der iva t ive of the func t ion £ ( r ) at r = rk. T h e t r u n c a t i o n h2 d3E\ error for the a p p r o x i m a t i o n o f ^f-dr ' 1 8 12 dr3 r=rk , where rk-i < C < ^fc+i- T h a t is , the cons tan t depends on the th i rd -o rde r der iva t ive eva lua ted at some p o i n t i n the g iven in t e rva l . A n uppe r b o u n d on the error depends o n the m a x i m u m value o f the t h i r d - o r d e r de r iva t ive i n the in t e rva l . It s h o u l d be no ted tha t the cons tant is o n l y ' cons tan t ' l oca l l y . T h a t is , i t varies as the g r i d po in t varies. C o n s i d e r i n g a l l th i s , i m p l i c i t l y we are a s s u m i n g t ha t the so lu t ions we are a p p r o x i m a t i n g are s m o o t h enough tha t these der iva t ives exis t everywhere a n d are s m a l l enough tha t the errors are not too large. It c a n be shown tha t der iva t ives of the a p p r o x i m a t e d so lu t ions are also 0 ( / i 2 ) , i f the a p p r o x i m a t e s o l u t i o n a n d the differencing scheme for the de r iva t ive are 0(h2). T h i s is because the ' cons tan t ' i n the 0(h2) of the a p p r o x i m a t e s o l u t i o n is a f unc t i on o f l o c a t i o n . 87 T h u s , the de r iva t ive w i l l be 0(h2), bu t w i t h a different ' cons tant ' . C o n s i d e r i n g the above d iscuss ion , i t seems reasonable to assume tha t g iven a n 0(h2) l o c a l t r u n c a t i o n error i n the der ivat ives , then the a p p r o x i m a t i o n w o u l d be convergent (the error goes to zero as h goes to zero) and tha t the accuracy o f the a p p r o x i m a t i o n w i l l a lso be 0(h2). I n fact, th i s can be shown to be t rue for the b o u n d a r y va lue p rob lems , bu t not for the eigenvalue p r o b l e m . It is obvious tha t the m a t r i x a p p r o x i m a t i o n to the eigenvalue p r o b l e m w i l l not be able to con t a in a l l the so lu t ions of the con t inuous p r o b l e m . (There are at mos t n eigenvectors o f an n x n m a t r i x where there are an inf in i te n u m b e r of e igenfunct ions o f the cont inuous p r o b l e m ) . S ince we are e x p e c t i n g the e igenfunct ions to be of r e l a t i ve ly low wave number , we hope tha t i t is these funct ions w h i c h are a p p r o x i m a t e d by the m a t r i x p r o b l e m . T h i s seems reasonable since the d i s c r e t i z a t i o n is not able t o resolve the h i g h wave numbers (wh ich are h i g h l y osc i l l a to ry ) a n d i f the e igenfunct ions are o f l ow wave number , t hen the errors i n the differencing are r e l a t ive ly s m a l l . B e l o w , the resul ts i n d i c a t e tha t the m e t h o d seems to be convergent . However , i t seems tha t h c o u l d no t be t a k e n s m a l l enough to o b t a i n an accurate es t imate o f the order . 4.3 Results T h e resul ts o f the analys is are presented i n th is sec t ion . T h e geomet ry o f the annu lus a n d fluid proper t ies are l i s t ed i n T a b l e 4.1. These values co r re spond to the expe r imen t s pe r fo rmed by F e i n [15]. O u r results are c o m p a r e d w i t h those o b t a i n e d i n t ha t s tudy. 3.48 c m n 6.02 c m R 2.54 c m D 5 c m V l . O l e - 2 c m 2 / s e c K 1.41e~ 3 c m 2 / s e c a 2 . 0 6 e - 4 1/° C Po 0.998 g m c m 3 T0 20.0 ° C 9 980 g m / c m 3 T a b l e 4 .1: The annulus geometry and fluid properties used in the analysis, after [15]. See text for definitions of symbols. 88 4.3.1 The axisymmetric solution An example of the axisymmetric solution is plotted in Figure 4.2. Qualitatively, the form of the solution is the same for all values of the parameters. Figure 4.2: The axisymmetric solution: Fluid velocity in the (a) radial (b) azimuthal (c) vertical directions and (d) temperature of fluid. The temperature is the deviation from ATr - AT^- + T„. This solution is for the N = 25 case and is observed at the m\ = 7, = 8 double Hopf point, where Q = 0.8169 and AT = 0.3820. The figure shows that the fluid velocity in the interior of the fluid is predominantly in the azimuthal direction. The radial velocity is almost zero everywhere except at the upper and lower boundaries, where it is negative and positive, respectively. The vertical velocity is largest at the inner and outer walls, where there is rising at the outer wall and sinking at the inner wall. The azimuthal velocity exhibits an almost linear shear in the vertical in 89 the in t e r io r w i t h a pos i t i ve ve loc i ty i n the upper h a l f o f the annu lus a n d negat ive v e l o c i t y i n the lower half . T h e r e su l t ing c i r c u l a t i o n is a convec t ion ce l l w h i c h is t i l t e d f rom the r a d i a l p lane such tha t , at the uppe r a n d lower boundar ies , the i n w a r d a n d o u t w a r d m o t i o n is deflected to the r igh t . 4.3.2 Neutral stability and transition curves T h e n e u t r a l s t a b i l i t y curves are presented i n F i g u r e 4.3. T h e r e is a separate curve for each a z i m u t h a l wave number . T h e curves are the po in t s i n the pa ramete r space where , for the g iven wave number , there are eigenvalues w i t h zero rea l pa r t w h i l e a l l o ther eigenvalues assoc ia ted to t h a t wave n u m b e r have negat ive rea l pa r t . F i g u r e 4.3 shows the n e u t r a l s t a b i l i t y curves for the wave numbers m = 3 to 8 a n d for c la r i ty , F i g u r e 4.5 shows the 'knee ' a rea a n d the uppe r t r a n s i t i o n (h igh t h e r m a l R o s s b y number , see Sec t i on 1.2). N = 25 for a l l curves shown . T h e other wave numbers w h i c h were ca l cu l a t ed , m = 2 , 9 a n d 10, were found to be to the r igh t o f at least one other curve. T h a t is , the c r i t i c a l wave n u m b e r is never m = 2, 9 or 10, where the c r i t i c a l wave n u m b e r is the wave n u m b e r o f the b i f u r c a t i n g wave at the a x i s y m m e t r i c to n o n - a x i s y m m e t r i c t r a n s i t i o n . It is no t poss ib le to ca l cu la t e the n e u t r a l s t a b i l i t y curves o f a l l wave numbers , a n d so we repeat the a rgumen t o f C h a p t e r 3 w i t h a d d i t i o n a l j u s t i f i c a t i on c o m i n g f rom c o m p a r i s o n w i t h the e x p e r i m e n t a l resul ts . T h e s t a b i l i t y curves i n the lower t r a n s i t i o n reg ion are so c losely g r o u p e d t ha t a p lo t o f th i s r eg ion does no t reveal any more d e t a i l . In F i g u r e 4.4, the curve w h i c h delineates the a x i s y m m e t r i c f rom the n o n - a x i s y m m e t r i c regimes is p l o t t e d . A l o n g th is curve i t can be seen tha t there are t r ans i t i ons o f the c r i t -i c a l wave number . These t rans i t ions occur at in tersect ions of the n e u t r a l s t a b i l i t y curves a n d co r r e spond to the double H o p f b i fu rca t ion po in t s . A l s o p l o t t e d o n F i g u r e 4.4 is the e x p e r i m e n t a l l y observed t r a n s i t i o n curve t aken f rom F e i n [15], w i t h c r i t i c a l wave n u m b e r t r ans i t ions . T h i s is the curve where a t r a n s i t i o n f rom the a x i s y m m e t r i c to s teady wave flow was observed. A l l curves are p l o t t e d on a log- log g r a p h o f T a y l o r n u m b e r T versus t h e r m a l R o s s b y n u m b e r 7c. T h i s is so tha t the results c o u l d be easi ly c o m p a r e d to the exper imen t s . A n u m b e r o f observat ions can be made . • T h e r e is a g o o d correspondence between the n u m e r i c a l a n d e x p e r i m e n t a l resul ts . • T h e uppe r t r a n s i t i o n seems to be bet ter represented t h a n the lower t r a n s i t i o n . H o w -ever, the e x p e r i m e n t a l errors are m u c h higher for the lower pa r t o f the t r a n s i t i o n curve t h a n for o ther regions. Because of the l og scale, s m a l l errors i n a n d A T , e i ther n u m e r i c a l or expe r imen ta l , w i l l s t and out more i n the lower t r a n s i t i o n . See F e i n [15]. T h i s alone m a y account for the discrepancies . 90 • T h e r e is c u s p i n g a long the uppe r t r a n s i t i o n curve, associa ted w i t h changes i n the c r i t i c a l wave number , i n b o t h the expe r imen ta l a n d n u m e r i c a l resul ts . • T h e r e is a l o c a l m a x i m u m of c r i t i c a l wave n u m b e r ( m = 8) a l o n g b o t h o f the lower t r a n s i t i o n curves. • It seems tha t the discrepancies i n the wave n u m b e r t r ans i t i ons a l o n g the s t a b i l i t y curve are r e l a t ive ly large. T h i s c o u l d be due to the d i f f icu l ty i n l o c a t i n g these t r a n -s i t ions , b o t h n u m e r i c a l l y a n d exper imen ta l ly . • T h e theore t i ca l lower t r a n s i t i o n curve is not l inear o n the g r aph . F e i n [15] be l i eved t ha t his e x p e r i m e n t a l d a t a showed evidence (a lbei t inconc lus ive) o f th i s c l a i m . In fact, a l l the m a i n features of the t r a n s i t i o n curves observed i n the expe r imen t s are r ep l i ca t ed w i t h the n u m e r i c a l results . N e i t h e r the c u s p i n g a long the uppe r t r a n s i t i o n nor the c r i t i c a l wave n u m b e r m a x i m u m of m = 8 a long the lower t r a n s i t i o n has been p r e d i c t e d before i n such a rea l i s t ic m o d e l . M i l l e r a n d B u t l e r [43], u s ing n u m e r i c a l e x p e r i m e n t a t i o n (see Sec t i on 1.3), d i d not loca te enough po in t s a l o n g the t r a n s i t i o n curve to reproduce the curve , a n d so c o u l d not make these pred ic t ions . In fact, the i r results showed a c r i t i c a l wave n u m b e r m a x i m u m of m = 7. A l s o , the 'd r i f t rates ' ujd = Uj/rrij at the a x i s y m m e t r i c to n o n - a x i s y m m e t r i c t r a n s i t i o n are p l o t t e d i n F i g u r e 4.6, where the dr i f t rate is the frequency tha t fu l l wavelengths dr i f t pas t a f ixed p o i n t o n the annulus . T h e s o l i d l ine i n the figure is a l i ne w h i c h is consis tent w i t h e x p e r i m e n t a l d a t a [15]. A g a i n there is g o o d correspondence. T h e e x p e r i m e n t a l results do not cover the who le t r a n s i t i o n curve since some o f the wave speeds were j u d g e d to be too s low to measure accura te ly [15]. 4.3.3 Double Hopf normal form coefficients: hysteresis T h e resul ts of the l o c a t i o n of the double H o p f b i fu rca t ion po in t s a n d n o r m a l f o r m coeffi-cients , as w e l l as the i m a g i n a r y par t s o f the eigenvalues, are presented i n Tab le s 4.2 - 4.7. T h e doub le H o p f po in t s are l abe l l ed i n te rms of the associa ted c r i t i c a l wave number s ( m i a n d m 2 ) . F o r a l l doub le H o p f po in t s , a l l the n o r m a l f o r m coefficients a, b, c a n d d are negat ive a n d also satisfy ad — be < 0. T h i s i m p l i e s tha t near the b i f u r c a t i o n p o i n t there is a r eg ion where the two b i fu rca t ing waves are stable, a n d hysteresis o f these waves is p r ed i c t ed . T h e resul ts also ind ica t e the existence of an uns tab le inva r i an t torus . See Sec t i on 3.3 for a de t a i l ed desc r ip t i on a n d b i fu rca t ion d i a g r a m (F igu re 3.5) showing the regions i n pa rame te r space where the different behav iours are observed. In F i g u r e 4.7, the a p p r o x i m a t e b o u n d -aries of the reg ion of hysteresis are d r a w n , for the double H o p f po in t s for w h i c h they were 91 ca l cu l a t ed . These boundar ies were ca l cu la t ed f rom the c o n d i t i o n dpi/b < p 2 < cpi/a, Hi > 0, fj,2 > 0 (see Sec t ion 3.3). N m i m 2 A T o w x • OJ2 b c 20 5 6 0.6354 1.543 - 7 . 9 4 6 • 1 0 ~ 3 - 1 . 0 3 9 - IQ-'2 -1.360 -2.134 25 5 6 0.6171 1.489 - 8 . 4 6 7 • 1 0 ~ 3 - 1 . 1 0 1 • l O " 2 -1.430 -2.173 30 5 6 0.6102 1.490 - 8 . 4 6 2 • l O " 3 - 1 . 0 9 1 • l O " 2 -1.470 -2.187 T a b l e 4.2: Numerical results for the mi = 5, m 2 = 6 double Hopf bifurcation point, a = -1 and d = — 1 for all iV. N is number of grid points on one side, Qo, ATo is the location in parameter space where the bifurcation occurs, u>i and u2 are the imaginary parts of the eigenvalues at fi0, A T 0 and a,b,c and d are the normal form coefficients. N m i m 2 A T b U)2 b c 20 6 7 0.6117 0.6972 - 3 . 7 1 1 • l O " 3 - 4 . 9 6 0 • l O " 3 -1.473 -2.253 25 6 7 0.5927 0.6950 - 3 . 9 5 3 • l O " 3 - 5 . 2 8 9 • l O " 3 -1.506 -2.246 30 6 7 0.5838 0.6944 - 4 . 0 4 6 • l O " 3 - 5 . 3 9 8 • l O " 3 -1.532 -2.256 T a b l e 4.3: Numerical results for the mi = 6, m 2 = 7 double Hopf bifurcation point, a = — 1 and d = — 1 for all N. N is number of grid points on one side, fl0,ATo is the location in parameter space where the bifurcation occurs, ui and w2 are the imaginary parts of the eigenvalues at Clo, ATo and a,b,c and d are the normal form coefficients. N m i m 2 0^ A T o 0>2 b c 20 7 8 0.8699 0.3959 - 8 . 5 8 2 • 1 0 " 4 - 1 . 1 6 1 • l O " 3 -1.628 -2.433 25 7 8 0.8169 0.3820 - 9 . 0 1 7 - l O " 4 -1.237-10-3 -1.614 -2.412 30 7 8 0.7925 0.3758 - 9 . 2 9 4 • l O " 4 -1.283-10-3 -1.616 -2.408 T a b l e 4.4: Numerical results for the mi = 7, m 2 = 8 double Hopf bifurcation point, a = -1 and d = -1 for all N. N is number of grid points on one side, f i 0 , ATo is the location in parameter space where the bifurcation occurs, wi and OJ2 are the imaginary parts of the eigenvalues at fi0, A T 0 and a,b,c and d are the normal form coefficients. T h e n u m e r i c a l differences between the n o r m a l f o r m coefficients at the different levels o f d i s c r e t i z a t i o n , decrease w i t h inc reas ing d i s c r e t i z a t i on leve l . T h i s is a n i n d i c a t i o n o f convergence. However , i t appears tha t TV is not large enough to make a n e s t i m a t i o n of the order of convergence. Y e t , the differences i n the n o r m a l f o r m coefficients at different N are qu i te s m a l l . I n pa r t i cu l a r , for the cases on the lower pa r t o f the t r a n s i t i o n curve they are a p p r o x i m a t e l y 1%, w h i c h ind ica tes tha t these results are p r o b a b l y at least q u a l i t a t i v e l y 92 N mi m 2 A T o 0>2 b c 20 8 7 1.603 0.4692 - 4 . 0 1 0 • 1 0 ~ 4 - 3 . 4 9 3 • 1 0 " 4 -2.309 -1.748 25 8 7 1.622 0.4613 - 3 . 8 4 9 • 1 0 ~ 4 - 3 . 3 6 8 • l O " 4 -2.281 -1.728 30 8 7 1.635 0.4581 - 3 . 7 4 8 • l O " 4 - 3 . 2 8 4 • 1 0 ~ 4 -2.274 -1.723 T a b l e 4.5: Numerical results for the m,\ = 8, m 2 = 7 double Hopf bifurcation point, a = - 1 and d = - 1 for all TV. JV is number of grid points on one side, f20, A T 0 is the location in parameter space where the bifurcation occurs, ui and 0J2 are the imaginary parts of the eigenvalues at ficbATo and a,b,c and d are the normal form coefficients. N mi m 2 A T o 0>2 b c 20 7 6 2.226 0.4625 - 1 . 5 5 2 9 • l O " 4 - 1 . 3 6 1 3 - 1 0 - 4 -2.311 -1.734 25 7 6 2.224 0.4522 - 1 . 5 0 9 8 • l O " 4 - 1 . 3 0 2 9 - 1 0 - 4 -2.308 -1.723 30 7 6 2.231 0.4457 - 1 . 4 4 0 6 - 1 0 - 4 - 1 . 2 2 8 9 - 1 0 ~ 4 -2.309 -1.719 T a b l e 4.6: Numerical results for the mi = 7, = 6 double Hopf bifurcation point, a = — 1 and d = -1 for all JV. TV is number of grid points on one side, Q0,AT0 is the location in parameter space where the bifurcation occurs, u>i and u>2 are the imaginary parts of the eigenvalues at £lo,ATo and a,b,c and d are the normal form coefficients. accura te . T o say th is w i t h cer ta inty , the ana lys is mus t be pe r fo rmed o n a h igher level of d i s c r e t i z a t i o n . T h i s was not poss ible w i t h the ava i lab le resources. It s h o u l d be n o t e d t ha t i n n u m e r i c a l exper iments o n s i m i l a r systems (e.g. [43], [29]) u s u a l l y N = 25 was used. F i n a l l y , s ince the results accura te ly reproduce the e x p e r i m e n t a l resul ts , we conc lude t ha t the a p p r o x i m a t i o n s are sat isfactory. W e also i nc lude the incomple t e results for the mi = 3, m 2 = 4 a n d mi = 4, m 2 = 5 doub le H o p f po in t s i n Tables 4.8 a n d 4.9. F o r these wave n u m b e r pa i rs , the e igenfunct ions for the N = 20 case were not w e l l resolved a n d so the eigenvalues were inaccura te . F o r the mi = 3, m 2 = 4, N = 30 case, the i te ra t ive process to loca te the b i f u r c a t i o n p o i n t d i d no t N mi m2 ^ 0 A T o U>2 b c 20 6 5 3.843 0.2559 - 2 . 0 6 4 • 1 0 ~ 5 - 2 . 0 8 3 • 1 0 ~ 5 -2.376 -1.746 25 6 5 4.238 0.2087 - 2 . 2 6 2 • l O " 5 - 2 . 2 1 1 • 1 0 ~ 5 -2.361 -1.738 30 6 5 4.696 0.1718 - 2 . 2 7 8 • 1 0 ~ 5 - 2 . 1 9 8 • 1 0 ~ 5 -2.350 -1.733 T a b l e 4.7: Numerical results for the mi = 6, m,2 = 5 double Hopf bifurcation point, a = -1 and d = -1 for all N. N is number of grid points on one side, Vt0, A T 0 is the location in parameter space where the bifurcation occurs, uj\ and 0J2 are the imaginary parts of the eigenvalues at f2o, ATo a n d a,b,c and d are the normal form coefficients. 93 converge. A s an i n i t i a l guess for th is i t e r a t i on , we used the l o c a t i o n o f the b i f u r c a t i o n p o i n t for N = 25, a n d for th i s case, i t was not a g o o d enough guess. T h e increase i n n u m e r i c a l d i f f icu l ty seen as the different ial hea t ing is increased is not caused by the i n a b i l i t y to resolve the b o u n d a r y layer i n the a x i s y m m e t r i c so lu t ion , bu t ra ther by the i n a b i l i t y to resolve the e igenfunct ions i n the in te r io r of the d o m a i n . T h e results , however, are s t i l l consis tent w i t h the e x p e r i m e n t a l results a n d so are quo ted here. N mi m 2 A T o UJl OJ2 b c 25 3 4 1.062 14.25 -0.03019 -0.04416 -0.9754 -2.139 T a b l e 4.8: Numerical results for the mi = 3, m 2 = 4 double Hopf bifurcation point, a — -1 and d = -1 for all N. N is number of grid points on one side, fio, A T 0 is the location in parameter space where the bifurcation occurs, w\ and w2 are the imaginary parts of the eigenvalues at Do, ATo and a,b,c and d are the normal form coefficients. N m i m 2 O0 A T 0 Ul 0J2 b c 25 4 5 0.7383 3.785 -0.01525 -0.02060 -1.280 -2.236 30 4 5 0.7313 3.772 -0.01450 -0.01936 -1.332 -2.273 T a b l e 4.9: Numerical results for the mi = 4, m 2 = 5 double Hopf bifurcation point, a = -1 and d = -1 for all N. N is number of grid points on one side, fl0, A T 0 is the location in parameter space where the bifurcation occurs, u)\ and are the imaginary parts of the eigenvalues at CIQ, ATQ and a,b,c and d are the normal form coefficients. 4.3.4 The eigenfunctions: bifurcating wave form A n example o f a n e igenfunct ion is p l o t t e d i n F i g u r e 4.8. T h i s is the e igenfunc t ion w i t h m = 6 w h i c h is observed at the m x = 6, m 2 = 7 double H o p f po in t (see T a b l e 4.3). F r o m equa t i on (4.47), the p e r i o d i c o rb i t , to lowest order i n is g iven by Pi = ^ + 0{Hj) 9j = ujjt + Oipj) (4.54) or i n t e rms o f z: z = y ^ e ^ 4 + O fa) (4.55) 94 w h i c h descr ibes a nea r -c i rcu la r pe r iod i c o rb i t . F i n a l l y , for the mi wave i n the o r i g i n a l var iab les , where to first order , U = z$ + z $ = R e a l ( z $ ) , the p e r i o d i c s o l u t i o n is g iven by: U R e a l = cos (m xcp + u\t) - $ l s i n (mxip + uit) + O M (4.56) where $ = < l e i m i ¥ ' , l> r is the rea l pa r t of $ a n d <l l is the i m a g i n a r y pa r t o f $ . N o t e t ha t i f t is fixed, t hen at different <p, U is a different l inear c o m b i n a t i o n of $ r a n d a n d so the f o r m of the e igenfunct ion gives the fo rm of the b i fu r ca t i ng wave. T h e t empera tu re profi le o f the b i fu r ca t i ng wave is consis tent w i t h expe r imen t . Measu re -ments i n d i c a t e d t ha t the t empera tu re has a m a x i m u m at m i d - r a d i u s m i d - d e p t h [15]. O t h e r expe r imen t s [11] ( w i t h no uppe r l i d ) showed via s t reak pho tog raphy a decreased h o r i z o n t a l fluid speed at m i d - d e p t h , w h i c h is also consistent . T h e exper iments do no t resolve enough levels for the fine d e t a i l of the wave forms to be va l i da t ed . 95 101 Taylor number Figure 4.3: Neutral stability curves are plotted for the wave numbers m = 3 to m = 8. The curves are calculated by finding the parameter values where for each m, the eigenvalues of (4.32) all have negative real part except one with zero real part. The curves are plotted on a log-log graph of thermal Rossby number versus Taylor number. 96 10" jQ E ?10" 1 >. m (0 O DC 75 E v 10 -2 10" — theoretical transition curve W * theoretical wave number transitions — experimental transition curve + experimental wave number transitions • 1 i i— 10a 10" 10' Taylor number 10° Figure 4.4: Transition curves for theory and experiment delineating the axisymmetric from the non-axisymmetric regimes. The critical wave number transitions (double Hopf bifurcation points), labelled as (mi, 7712) , are also plotted along the curve. 97 10" Taylor number 107 Figure 4.5: Neutral stability curves: upper transition. Same as Figure 4.3, but only top part is shown. — experimental • theoretical x theoretical Figure 4.6: Theoretical and experimental drift rates of bifurcating waves close to bifurcation point, where the drift rate is the frequency that full wavelengths drift past a fixed point on the annulus. The solid line is consistent with the experimental results (within experimental error) of [15]. Theoretical results are calculated from the imaginary parts of the critical eigenvalues. aAT/Cl is the intensity of the 'thermal wind', see [15]. 98 T 1 1—I—I—r—|-10" XI i i o - 1 1 >> n in in O CC 15 £ <D r. *-> 10" 4,5 • X 3,4 5,6 7,8 v v 8,7 X 7,6 \ \ 10_°H _ " theoretical transition curve * theoretical critical wave number transitions — borders of region of hysteresis _j i i L I X 6,5 • 1 i — 103 10° 10' Taylor number 10° Figure 4.7: Theoretical transition curve between the axisymmetric and the non-axisymmetric regimes with the double Hopf bifurcation points (critical wave number transitions) plotted along the curve. The area between the solid lines attached to the double Hopf points is the region where there are multiple stable wave solutions. 99 F i g u r e 4.8: An example of the radial and vertical dependence of an eigenfunction with m = 6 at fi = 0.5927 and AT = 0.6950: (a) real part and (b) imaginary part of the radial component of the eigenfunction, (c) real part and (d) imaginary part of the azimuthal component of the eigenfunction, (e) real part and (f) imaginary part of the vertical component of the eigenfunction, (g) real part and (h) imaginary part of the temperature component of the eigenfunction. That is, the actual components of the eigenfunctions are the plotted functions multiplied by etmv'. 100 C h a p t e r 5 C o n c l u s i o n In th i s thesis, we s t u d y the t r ans i t ions f rom a x i s y m m e t r i c s teady so lu t ions to n o n - a x i s y m -m e t r i c t r a v e l l i n g waves i n two geophys ica l fluid d y n a m i c s mode ls : (1) a two- layer quas i -geos t rophic p o t e n t i a l v o r t i c i t y m o d e l w i t h a non l inea r basic state, a n d (2) a m o d e l of the d i f fe ren t ia l ly heated r o t a t i n g annulus exper iment . U s i n g the a n a l y t i c a l - n u m e r i c a l center m a n i f o l d r e d u c t i o n at the double H o p f b i fu rca t ion po in t s w h i c h occu r at th i s t r a n s i t i o n , i t is shown tha t there are regions i n the respect ive pa ramete r spaces w h i c h suppo r t m u l t i p l e s table wave so lu t ions . Hysteres is o f these so lu t ions is observed. T h e results o n the first m o d e l ind ica te tha t the fo rc ing t e r m i n the equa t i on does not q u a l i t a t i v e l y effect the behav iour . However , there are some m i n o r differences w h i c h are o f interest , i n p a r t i c u l a r i n the fo rm of the b i fu r ca t i ng s o l u t i o n . T h i s suppor t s the i dea t ha t the b a r o c l i n i c effects as opposed to the ba ro t rop i c effects are mos t i m p o r t a n t i n the r e a l i z a t i o n of the d y n a m i c s . T h e second m o d e l consists of the Nav ie r -S tokes equat ions i n the B o u s s i n e s q a p p r o x i -m a t i o n i n c y l i n d r i c a l geometry. T h e d imens ions of the d o m a i n a n d the proper t ies o f the f l u id are chosen to i m i t a t e the di f ferent ia l ly heated r o t a t i n g annu lus expe r imen t o f F e i n [15]. T h e a x i s y m m e t r i c to n o n - a x i s y m m e t r i c t r a n s i t i o n curve, c r i t i c a l wave number s a n d dr i f t rates of the waves are a l l found to we l l reproduce the e x p e r i m e n t a l observa t ions . T h i s suppor t s not o n l y the v a l i d i t y of the a p p r o x i m a t i o n s w h i c h are made , bu t also the v a l i d i t y o f the ana lys i s itself. T h i s is the first theore t i ca l s t udy o n a Nav ie r -S tokes m o d e l of the d i f fe rent ia l ly hea ted r o t a t i n g annu lus w h i c h uses eigenvalues to t race out the t r a n s i t i o n curve between the a x i s y m m e t r i c a n d n o n - a x i s y m m e t r i c regimes. A n u m b e r of features of the e x p e r i m e n t a l d a t a have been r ep roduced here for the first t ime : (1) the c u s p i n g o f the u p p e r t r a n s i t i o n curve associa ted w i t h c r i t i c a l wave number t r ans i t ions , (2) the r e p r o d u c t i o n o f m = 8 as the l o c a l m a x i m u m of the c r i t i c a l wave n u m b e r a long the lower t r a n s i t i o n curve , (3) a de t a i l ed r e p r o d u c t i o n o f the dr i f t rate of the b i fu rca t ing wave, a n d (4) the i n d i c a t i o n t ha t 101 the lower t r a n s i t i o n curve is not l inear , a specu la t ion t ha t F e i n felt h is d a t a s u p p o r t e d bu t c o u l d no t c o n f i r m . T h e b i fu rca t i on ana lys i s presented i n th is thesis is the first o f i t s k i n d w h i c h has been pe r fo rmed o n such a m o d e l . T h e results show tha t there are s table waves w h i c h b i furca te f r o m the a x i s y m m e t r i c so lu t i on a n d tha t hysteresis of the b i fu r ca t i ng waves occurs near c r i t i c a l wave n u m b e r t r ans i t ions . A s s o c i a t e d w i t h the hysteresis is the exis tence of an uns tab le torus . T h e results , w h i c h are ob t a ined by n u m e r i c a l l y a p p r o x i m a t i n g the n o r m a l f o r m coefficients, are consistent w i t h b o t h l a b o r a t o r y a n d n u m e r i c a l exper iments . A l t h o u g h there is evidence o f n u m e r i c a l convergence, due to the i n a b i l i t y to prove the convergence, i t is not poss ib le to prove comple t e ly r igourous ly tha t the p red ic t ed b e h a v i o u r occurs i n the fu l l p a r t i a l d i f ferent ia l equa t ion m o d e l . T h e m e t h o d does prove the exis tence a n d s t a b i l i t y o f the b i fu r ca t i ng so lu t ions i n the o r d i n a r y different ia l equat ions w h i c h are fo rmed by d i s c r e t i z i n g the o r i g i n a l m o d e l . T h e t y p e o f ana lys i s tha t has been employed i n th is thesis has no t been e x p l o i t e d by the geophys ica l fluid d y n a m i c s c o m m u n i t y . However , the w o r k i n th i s thesis i nd ica te s t ha t b i f u r c a t i o n ana lys i s can lead to v a l i d results and thus suppor t s the fur ther a p p l i c a t i o n to s i m i l a r p rob lems . T h e m e t h o d was able to h igh l igh t the d y n a m i c a l s i m i l a r i t y be tween two geophys ica l fluid d y n a m i c s mode ls of vas t ly different scales. T h i s not o n l y suppor t s the use o f the m e t h o d bu t also is an a rgument for the usefulness of s t u d y i n g these mode l s , i.e. t hey inco rpo ra t e the fundamen ta l proper t ies of d i f ferent ia l ly heated r o t a t i n g sys tems. A l t e r n a t i v e l y , i t c a n be a rgued t ha t the d y n a m i c s the mode l s share are the f u n d a m e n t a l charac te r i s t ics o f the flows i n different ial heated r o t a t i n g systems. T h e s t u d y presented here is a beg inn ing , a n d there are m a n y poss ib le d i r ec t ions future w o r k c o u l d take. In the ana lys is o f the annulus exper iment , the effect o f the u p p e r b o u n d a r y c o u l d be s t ud i ed u s ing the a s s u m p t i o n o f stress-free b o u n d a r y cond i t i ons (as i n [43]). In th i s case, perhaps the in te res t ing behav iour at the uppe r t r a n s i t i o n c o u l d be e x p l a i n e d . T h e r e is also the p o s s i b i l i t y of resonant behav iou r close to w h a t is ca l l ed the t r i p l e - p o i n t , w h i c h is the p o i n t where three regimes meet ( a x i s y m m e t r i c , wave a n d i r r egu la r ) . V e r y close to the e x p e r i m e n t a l l y l oca t ed t r i p l e -po in t , the i m a g i n a r y par t s of the two eigenvalues w i t h largest rea l pa r t are equa l . A l t h o u g h no double H o p f p o i n t was observed i n the v i c i n i t y , the d y n a m i c s found close to a resonant double H o p f b i fu r ca t i on m a y s t i l l be f o u n d here. T h i s m a y e x p l a i n the existence of the t r i p l e -po in t . A n o t h e r in te res t ing d i r ec t i on , w o u l d be to a t t e m p t to fo l low the b i f u r c a t i n g so lu t ions as the parameters move away f rom the b i fu rca t ion po in t . A n in te res t ing flow tha t is ob-served i n the annu lus ( b o t h e x p e r i m e n t a l l y a n d n u m e r i c a l l y ) is v a c i l l a t i o n . It has been hypo thes i zed [46] tha t the m e c h a n i s m responsible for v a c i l l a t i o n is the i n t e r a c t i o n of two 102 waves via a s table torus . V a c i l l a t i o n - t y p e behav iour is somet imes observed d u r i n g e q u i l i -b r a t i o n o f s teady waves. T h i s is consistent w i t h the existence of a n uns tab le to rus , w h i c h we have shown can occur i n the s teady wave regime. T h u s , i t is poss ib le tha t i f the uns tab le to rus c o u l d be fo l lowed further in to the s teady wave regime, a b i fu r ca t i on to a s table torus (and v a c i l l a t i o n ) m a y be discovered. A t the momen t , such a s t u d y is c o m p u t a t i o n a l l y pro-h i b i t i v e . However , i f the curva tu re o f the annulus is neglected, a s y m m e t r y o f the r e s u l t i n g sys t em leads to a b i fu rca t ion to a s teady so lu t i on as opposed to a p e r i o d i c o rb i t . I n t h i s case, the c o m p u t a t i o n m a y be poss ible . T h e c o m p a r i s o n o f theore t i ca l a n d expe r imen ta l results w h i c h t ook place i n the inves-t i g a t i o n o f the T a y l o r - C o u e t t e flow led to m a n y more discoveries a b o u t the sy s t em t h a n o therwise w o u l d have been poss ible . W o r k presented i n th is thesis s tar ts such a c o m p a r i s o n for the d i f fe rent ia l ly heated r o t a t i n g annulus flow. 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Sp r inge r -Ver l ag , N e w Y o r k , 1990. 108 Appendix A Center manifolds for partial differential equations Consider the nonlinear equation ii = Lu + N(u), ueV, u(o)=u0en (A.l) where ri is a Banach space, V C R, L : V —> %, generates a linear semigroup 1, N : 7i —¥ Ti is C2 with N(0) = 0, N'(0) = 0 and the dot represents differentiation with respect to time. See Hazard et al. [23]. Note that since N(0) = 0, u — 0 is a fixed point of equation A . l . Suppose L has some eigenvalues, A, with Real(X) = 0 and all the rest satisfying Real(X) < e < 0. Write ri = E C © E S where E C and E S are invariant under L, i.e. u G E A Lu G Ea,a = c,s and E C is finite dimensional. Denote B = L\E' (B : E C —>• E C , Bu = Lu if u G i? 0) and C = L |E s > then all the eigenvalues of B have zero real part and all eigenvalues of C have negative real part. There is a projection P which takes u to E C along E S (i.e. Pu G E C for all u and P(j> = $ for all </> <E £ c , Pip = 0 for all •0 G i? s). Then, any u £ H may be written uniquely as u = 4> + where </> = Pw G E C , tp = (L — P)u G E * . If u = u(t) is differentiable, then equation A . l may be written as, <f) + ijj = L<f) + LiP + N(<l) + iP). (A.2) Taking P and (I — P) of equation A.2, and noting that PL(j) = Lcf) = P^i and (7 — P ) L ^ = Lib = Cip, we get, j> = B(t> + PN((l> + ib) ( A , i) = Cip + {I-P)N{<j> + vl)). K " ' 1 A linear semigroup is a family, (Tt)t>o, of bounded linear operators from H to H, satisfying (i) To = I, (ii) Ts+t = TsTt for all s,t > 0, (iii) lim^o ||Ttwo - "oil = 0 for all u0 G H. A linear operator, L, generates the linear semigroup if Lu = limtj.0 T '"~" [23]. 109 N o w , g iven our assumpt ions , the center m a n i f o l d theo rem states t ha t there exists a different iable center m a n i f o l d for equa t ion A . l Wfoc = {u = 0 + h(<f>) : (f> G Ec, \\</>\\ s m a l l , h : Ec -> Es} ( A . 4 ) w h i c h is l o c a l l y invar ian t under the (nonl inear) l o c a l s emi f low 2 o f equa t ion A . l , t angent to Ec at 0 (h(0) = 0, h'(0) = 0), a n d l o c a l l y exponen t i a l l y a t t r a c t i n g . T h e l o c a l semif low is t o p o l o g i c a l l y equivalent to ( in p a r t i c u l a r , a n a s y m p t o t i c s table fixed po in t i n Ec, for the first equa t ion , cor responds to an a s y m p t o t i c s table fixed po in t i n H). N o t e tha t th is is now a decoup led sys t em. T h e equa t ion i n <fi is finite d i m e n s i o n a l and c a n be w r i t t e n as an o r d i n a r y di f ferent ia l equa t ion . T h e equa t ion i n is l inea r w i t h a l l eigenvalues o f C negat ive . T h i s is v a l i d for | | « | | suff icient ly s m a l l (i.e. u close to u = 0) a n d h(4>) c an be found us ing the inva r i ance o f '" loc 2($t)t>o is called a (nonlinear) local semiflow if the unique solution to equation A.l is u(t) = $t(i*o), where <J>0 = I, $s+t = °$t for all s, t > 0 (whenever both sides are defined) and limtj.0 ||$t(uo)— "oil = 0 for all u0 6 %• It can be proven that there is a semiflow for 0 < t < T for some T > 0 for all ||uo|| sufficiently small (i.e. UQ 'near' the fixed point u = 0) [23]. 110 A p p e n d i x B O u t l i n e o f d e r i v a t i o n o f t w o - l a y e r q u a s i g e o s t r o p h i c p o t e n t i a l v o r t i c i t y e q u a t i o n s Contained here is an outline of the derivation of the two layer quasigeostrophic potential vorticity model. See Pedlosky [49]. Assume a fluid consists of two layers and that the density is constant within each layer. See Figure B . l . The fluid occupies the region 0 < z* < 2D, 0 < y* < L and there is periodicity in x*, with period 7. There are rigid boundaries at z* = 0, 2D, y* = 0, L. The * indicates dimensional variables. Edge effects are neglected at y* = 0, L and for now (see below) at the interface between the two layers, but an Ekman layer will be included at z* = 0, 2D. It is assumed that there is no mixing of the two layers. The height of the interface will be given by z* = h*(x*,y*,t*). If the fluid is at rest, both layers have height D. That is, the interface between the two layers will be at z* = D. With the given assumptions, the boundary conditions are as follows: no flow through the boundaries at z* = h*, y* = 0, L ; periodicity in x* ; at z* = 0,2D there will be a vertical velocity determined by the Ekman layer. The velocity of the fluid in each layer, in the x*, y* and z* directions, respectively, is given by u*, v* and w*, n = 1,2. The velocities are all functions of t*, x*, y* and z*. The equations of motion in each layer are: 1 dp*n , Tl (B.l) Pn 9x* p*n 1 dp*n ^ T; (B.2) Pn ^ Pn 1 dp*n Pn (B.3) Pn dz* 111 y-layer 1 p layer 2 ^2 h z=2D z=D z=0 Figure B . l : A two-layer fluid in a periodic channel of width L and height 2D. du* dv* dwt where d* dt* dx* d_ dt* + dy* + dz* = 0 d d + <d^ + <dy-*Jr<dh (B.4) (B.5) Tl are the frictional forces and / is the Coriolis parameter. These equations are meant to give the evolution of a (( time-average of the dependent variables, )) where the effects of the turbulent smaller scales are assumed to be seen only in the frictional forces (see below). It is hoped that these equations will describe the large-scale behaviour of atmospheric flow. It will be assumed that f = f0 + day* where /o = 2fisin0 o and 80 = (2ficos0 o) /ro and Q is the rotation rate of the earth, r 0 is the radius of the earth, and 90 is a reference latitude (this determines the y = 0). This simplification is valid for a range of latitude about #o where it can be assumed that the Coriolis varies linearly with latitude (y small enough). Now the variables will be non-dimensionalized by rescaling them as follows: x* = Lx, y* = Ly, z* = Dz, u*n = Vun, v*n = Vvn, w*n = (D/L)Vwn and t* = (L/V)t, where V is a characteristic velocity. The scaling of i«* reflects the difference in the horizontal and vertical scales. The scaling of t* is chosen to ensure that the advective terms are of the same order of magnitude as the time evolution term. The pressure in the two layers will be scaled as p\ = p\g (2D — z*) + PifoVLpi pi = pigD + p2g (D — z*) + p2foVLp2, i.e. pn are the non-dimensional deviations in each layer 112 f r o m the hyd ros t a t i c a p p r o x i m a t i o n : = ~Pn9 (B .6) dz* w h i c h holds i n the absence of m o t i o n . T h e sca l ing o f the dev ia t ions , p „ , was chosen i n a n t i c i p a t i o n t ha t the pressure gradients are of the same order as the C o r i o l i s acce le ra t ion (/<;/<)• T h e f r i c t i o n a l t e rms w i l l be rescaled as fol lows : K_VA„(^ + l £ \ V £ / * g \ ( R 7 ) p*n L2 \ dx2 dy D2 [dz2, VAV (d2vn\ D2 {dz2 j VAV ( d2wn y~v VAn (d2vn d2vn , r *i.v , ^ ^n . , . TZ DAh (d2wn d2w, Pi L3 \ dx2 ' dy2 J + LD [ dz2 ) ( B ' 9 ) i.e. the m a g n i t u d e of the f r i c t i ona l forces is e s t ima ted i n te rms of ' the charac te r i s t i c t u r b u l e n t m i x i n g coefficients ' , where the molecu la r v i scos i ty is r e l a t i ve ly neg l ig ib le . T h i s comes f r o m a n a d hoc theory w h i c h assumes tha t the effect of the s m a l l scale m o t i o n o n the large scale m o t i o n can be pa ramete r i zed by an increased f r i c t i o n a l coefficient. N o t e , t ha t AH is a pa ramete r as opposed to a f ixed p rope r ty o f the f lu id F i n a l l y , we w i l l rescale the height of the interface as h* = Dh = D(l + Rrj) (B .10) where r\ is ca l l ed the surface deflect ion w h i c h is the n o n - d i m e n s i o n a l d e v i a t i o n of the i n -terface f r o m i t s height when the fluid is at rest (h* = D i m p l i e s h = 1) a n d R is a s ca l i ng factor chosen such tha t rj is order 1. T o de te rmine i ? , we w i l l use the requirement tha t the pressure mus t be con t inuous at the interface. C h o o s i n g R as R=P-~^L (B .11) (where p 0 = (Pi + P2) / 2 ) w i l l ensure tha t rj is of order 1. W i t h th i s choice of R we get t ha t foft-PiPO ( B 1 2 ) Po Define the R o s s b y n u m b e r to be e = V/ (foL) a n d the other d imens ionless pa ramete r s to be 5 = D/L, Re = (VL)/AH, w h i c h is the R e y n o l d s number , 8 = ( l / 2 /?o) /V, a n d EV = (2Ay) I ( / o £ > 2 ) - It w i l l be assumed tha t the charac te r i s t i c l eng th a n d v e l o c i t y of the flows o f interest are such tha t e = O (L/r0) <C 1, w h i c h i m p l i e s t ha t p = 0 ( 1 ) . T h i s 113 a s s u m p t i o n essent ia l ly means t ha t r o t a t i o n i n the f o r m of C o r i o l i s is i m p o r t a n t , bu t t ha t the charac te r i s t i c l eng th is s m a l l enough tha t there is o n l y s m a l l v a r i a t i o n o f the C o r i o l i s over the d o m a i n . If i t is assumed tha t Re = 0 ( 1 ) , 5 < 0(e), R = 0(e) a n d Ev = 0(e2) a n d t ha t the dependent var iab les can be expanded i n powers of e, for example : U n = uM + euW + . . . (B .13) t h e n the 0 ( 1 ) (zeroth-order) t e rms of the rescaled d imens ionless equat ions f o r m the geo-s t roph ic a p p r o x i m a t i o n : „£» = ^ (B .14) dy dz (B .15) = 0 (B .16) E q u a t i o n s (B .14 ) a n d (B .15) i nd i ca t e tha t to first order , the C o r i o l i s acce le ra t ion balances the pressure gradien t w h i c h is the geost rophic a p p r o x i m a t i o n . N o t e tha t p^ is independen t o f z a n d therefore, by equat ions (B.14) and ( B . 1 5 ) , so are uffl a n d v£\ T a k i n g the de r iva t ive o f equa t i on (B.14) w i t h respect to y a n d the der iva t ive o f equa t ion (B .15) w i t h respect to x, i t c an be seen t ha t dv.™ dvW ^ + ^y- = ° (B .18) a n d so, f r o m equa t ion (B.17) - ^ = 0. (B.19) T h e v e r t i c a l v e l o c i t y at z = 0 is 0 ( e ) , [49], w h i c h i m p l i e s tha t ,(o) (B .20) If the 0 ( e ) t e rms are co l lec ted and the first-order dependent var iab les (e.g. u$) are e l i m i n a t e d , the fo l l owing equa t ion can be ob ta ined : dt n dx + n dy + P n dz + Re\dx2 + dy2 J { ' where dv0 du(°) 114 Since a n d are independent of z, equa t ion (B.21) c a n be eas i ly in tegra ted w i t h respect to z. In t eg ra t ing equa t ion (B.21) w i t h n = 1 f rom z = h to z = 2, we o b t a i n : (2 - 1 - Rq) 3Ci (o) at + u (0) (0) (0) a 2ci 0 ) dx2 + z=2 z=h L i k e w i s e , i n t eg ra t i ng equa t ion (B.21) w i t h n = 2 f rom z = 0 to z = /A (B .23) 1 + Rq, we o b t a i n : (1 + Rrj) dd0) , .(o,ad0) at 5a: + •"2 <o)0do) ay i /Vd0) a 2d o r + ;=h 2=0 (B .24) N o w the b o u n d a r y cond i t i ons at z = 0, z = 2 a n d at the interface w i l l be used to e l i m i n a t e the v e r t i c a l ve loci t ies i n equa t ion (B.23) a n d ( B . 2 4 ) . T h e c o n d i t i o n t ha t there is no m i x i n g of the layers i m p l i e s tha t there is no flow t h r o u g h the interface. T h i s i n t u r n i m p l i e s t h a t a fluid element i n i t i a l l y adjacent to the interface w i l l never m o v e f r o m the interface (it w i l l r e m a i n on the interface for a l l t ime ) . It c a n be shown tha t the c o n d i t i o n s o n the first-order v e r t i c a l ve loc i ty of the fluid at the interface are g iven by z=h z=h V dt (o)dr){0) M U [ d x ~ + V <o)0i7 ( o )\ 1 dy ) \ dt where F = R dx /o 2^ 2Po dy (B.25) (B .26) (B .27) e gDAp Because i t has been assumed tha t J? = O(e ) , th i s imp l i e s tha t F = 0 ( 1 ) . T h e v e r t i c a l ve loci t ies at the upper a n d lower boundar ies w i l l be d e t e r m i n e d t h r o u g h a m a t c h i n g w i t h the E k m a n layer (a b o u n d a r y layer w h i c h assumes e is s m a l l , f r i c t i o n is i m p o r t a n t a n d no-s l ip at the b o u n d a r y is requi red) . T h e 0 ( e ) cond i t i ons are w (i) z~2 = -r-'<f «4» z=0 = r-'Cf where r x = El 2e = 0 ( 1 ) . If we w r i t e , (o) P i (B .28) (B .29) (B .30) (B .31) 115 a n d then a n d a n d ip2 = p[0) ftyn = (0) dx n d^n _ _ (0) A l s o , i f we assume tha t t h e n by equa t i on (B.12) dy Pn = PO + tpnl) •+ • • • B . 3 7 B . 2 9 ), the T h e n u p o n s u b s t i t u t i n g the cond i t i ons of equat ions ( B . 2 5 ) , ( B . 2 6 ) , (B .28) a n d in to equat ions (B.23) a n d (B.24) a n d neglec t ing te rms 0 ( e ) ( r emember ing R = 0 ( e equat ions for ipi a n d ip2 are ob ta ined : E q u i v a l e n t l y , X ( B . 4 0 F - (ih -tl>i) + J (rp2, V 2t/> 2) - FJ (</>!, ^) + = - r " 1 V 2 V 2 + ^ V ^ 2 " B . 4 1 McW _ dAdJJ dx dy dy dx D e f i n i n g ^ 1 + ^ 2 r tpl ~ ifo a = , o = 2 2 t h e n ipi = a + 6, ip2 = a - 8. B . 3 2 B . 3 3 B . 3 4 B . 3 5 B . 3 6 B . 3 8 B . 3 9 B . 4 2 B . 4 3 a c an be ca l l ed the ba ro t rop i c mode , since i t is an average of the s t r eam funct ions o f the two layers a n d 5 c a n be ca l l ed the first b a r o c l i n i c mode , since i t con ta ins the first a p p r o x i m a t i o n 116 of the s t r a t i f i ca t i on effects. S u b s t i t u t i n g th is in to equat ions (B.40) a n d (B.41) the equat ions for a a n d S c a n be ob ta ined : ^ + J (a, VV) + J (6, W ) + 8^ = - r _ 1 V V + ^ W (B .44) ^ - 2Fft + J (a, W ) + J (5, VV) - 2 F J (a, 5) + = - r " 1 W + i - W (B.45) where the b o u n d a r y cond i t ions are: (i) a(x) = a (x + 7) a n d 5(x) = 5 (x + 7) i.e. a a n d 5 are p e r i o d i c i n x w i t h p e r i o d 7, w h i c h w i l l be a pa ramete r . (ii) — = — = 0 a t y = 0 a n d y = l T h i s ensures no t angen t i a l ve loc i ty at the boundary . 1 rx d2a 1 fx d25 (iii) l i m — - / -^——dx = l i m -— / -^—-dx = 0 at y = 0 a n d y = 1. v y x ^ o o 2 X J-x dydt x-+oo 2X J-x dydt T h i s c o n d i t i o n ensures t ha t there is no change i n c i r c u l a t i o n abou t the i nne r a n d outer bounda ry , w h i c h comes f rom the fact tha t the ve loc i t ies are geos t rophic [49]. I f i t is assumed tha t the veloci t ies a long the interface i n the two layers need not be the same, yet there is a d r a g between the layers w h i c h is p r o p o r t i o n a l to the difference i n the ve loc i t ies at the interface ( w i t h a constant of p ropo r t i ona l i t y , F/Re), t hen i t c a n be shown tha t th i s results i n the a d d i t i o n o f the t e r m , (F/Re) V2<5 to equa t i on ( B . 4 5 ) . A fo rc ing t e r m was also added to the equat ions used for the ana lys i s i n th i s thesis . 117 A p p e n d i x C N o r m a l f o r m f o r m u l a e In t h i s a p p e n d i x , the general formulae for the center m a n i f o l d r e d u c t i o n are g iven . T h e specific formulae for the app l i ca t ions discussed i n th i s thesis are easi ly c o m p u t e d f r o m the genera l fo rmulae . T h e fu l l de ta i l s are not presented since they are ve ry l e n g t h y a n d d o not l ead to fur ther ins igh t . However , one of the s imples t f o r m u l a is w r i t t e n i n d e t a i l to c l a r i fy the n o t a t i o n . T h e doub le H o p f b i fu rca t ion ana lyzed i n th is thesis, the n o r m a l f o r m coefficients are g iven by G n - </V2ioo,<r>, G i 2 = (iwr), G21 = (JV1110,**), G22 = (M)021,**>, ( C l ) where Nijkl are the T a y l o r coefficients of N(z,z,w,w). T h a t is , ^ - ^ W ^ ^ ( a 2 ) where N(z,z,w,w) is the non l inea r pa r t N(U,U) w i t h U w r i t t e n as U = z$ + z$ + w$ + Wtt +z2H20oo + ZZHUQQ + w2H0002 + wwHoon + zwHWi0 + zwHW(n + c . c . + 0 ( 3 ) (C.3) where $ a n d tt are the eigenfunct ions, Hijki are the center m a n i f o l d coefficients a n d c.c. represents the c o m p l e x conjugates of the Hijki w h i c h are w r i t t e n e x p l i c i t l y . T h u s , the d e t a i l of the n o r m a l f o r m coefficients are con ta ined i n the Nijki. 118 F i r s t , we w i l l w r i t e the formulae for the relevant Nijki w h i c h are q u a d r a t i c i n z, z, w, w. N2000 = 7V($,$) ^1100 = iV(<S>,$) + N ( $ , $ ) NQ020 -^1010 -^1001 = + J V ( W , $ ) T h e r e m a i n i n g formulae for the quad ra t i c t e rms can be found by t a k i n g the c o m p l e x con -juga tes of these. These w i l l be necessary for the c o m p u t a t i o n o f the center m a n i f o l d coef-ficients. T h e formulae for the relevant coeffecients w h i c h are cub ic i n z,z,w,w are as fol lows: J V 2 1 0 0 = N ffnoo) + N (#1100, $) + N ($, H2000) + N (#2000, $ ) i V i o n = N(^,H0011) + N(H0011,^)+N(^,H101O)+N(H101O,^) +N(*,H10OI) + N(H100I,V) Nmo = N ffono) + N (HOUo, *) + N H10W) + N (HW10, $) +N(V,Hnoo) + N(Hnoo,*) iYoo2i = N Hoon) + N (#0011, * ) + N (*, #0020) + N (#0020, These formulae are the same for a l l non-resonant double H o p f b i fu rca t ions . I n each a p p l i c a t i o n , the formulae mus t be expanded us ing the app rop r i a t e def in i t ions o f the n o n -l inea r pa r t N(U, U). N o t e tha t for the two-layer m o d e l , U is a t w o - d i m e n s i o n a l vec tor of funct ions a n d therefore each $ , H ^ M and,N(U, U) are t w o - d i m e n s i o n a l . F o r the annu lus expe r imen t a p p l i c a t i o n , U is a fou r -d imens iona l vec tor of funct ions . Here we w i l l not e x p l i c i t l y wr i t e out a l l the formulae , however, the f o r m u l a for iVioio for the two- layer case w i l l be g iven as an example i n order to c la r i fy the n o t a t i o n . F o r the two- layer case, N(U, U) is g iven by : J (crx, V2a2) - J (Si, V2<52) \ N(UUU2) = ( C A ) JKW 2) J(Si,V2a2)-2FJ(5i,a2) where (C.5) J (u, v) du dv du dv a n d the subscr ip ts have been added to the U for c l a r i f i c a t i o n . dx dy dy dx 119 W r i t i n g * = ( ) (C-6) * = ( tm ) (C'7) we get tha t jvgl0 = - J ^ W . V V J - J ^ . V V 2 ' ) - j v 2 $ w ) - J (¥2\ v2$(2)) iVg}0 = - j ( $ ( 1 ) , v V 2 ) ) - j ( $ ( 2 ) , V 2 t t W ) - 2 F j ( $ ( 2 ) , t t ( 1 ) ) - J V 2 $ ( 2 ) ) - J (#(2), V2<£> )^ - 2 F J (tf( 2>, $ W ) (C.8) T h a t is , th i s is found f r o m i V i 0 i 0 = N($, \&)+JV(tt, $ ) , where , for N($, the s u b s t i t u t i o n s a i -> 5i ->• < & ( 2 ) , <72 - » a n d 5 2 - » tt(2) have been m a d e i n JV(t/i,[/ 2) a n d for 7V(*, $ ) , the subs t i t u t ions CTI ->• <Ji ->• * ( 2 ) , <r2 ->• a n d <52 ->• $ ( 2 ) have been m a d e i n N(UUU2). 120 

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