UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

A finite element model of ocean circulation Bermejo-Bermejo, Rodolfo 1986

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

Item Metadata

Download

Media
831-UBC_1987_A6_7 B47_5.pdf [ 5.34MB ]
Metadata
JSON: 831-1.0053197.json
JSON-LD: 831-1.0053197-ld.json
RDF/XML (Pretty): 831-1.0053197-rdf.xml
RDF/JSON: 831-1.0053197-rdf.json
Turtle: 831-1.0053197-turtle.txt
N-Triples: 831-1.0053197-rdf-ntriples.txt
Original Record: 831-1.0053197-source.json
Full Text
831-1.0053197-fulltext.txt
Citation
831-1.0053197.ris

Full Text

A F I N I T E E L E M E N T M O D E L O F O C E A N C I R C U L A T I O N By RODOLFO BERMEJO-BERMEJO Naval A r c h t . and Marine Eng. The P o l i t e c h n i q u e U n i v e r s i t y of M a d r i d , 1977 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department o f Oceanography We accept t h i s t h e s i s as conf o r m i n g to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA December 1986 ® Ro d o l f o Bermejo-Bermejo, 1986 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. The University of British Columbia 1956 Main Mall Vancouver, Canada Department V6T 1Y3 DF-fifVft-M - i i A B S T R A C T P r e l i m i n a r y r e s u l t s o f a t w o - l a y e r q u a s i - g e o s t r o p h i c box model of a w i n d - d r i v e n ocean are p r e s e n t e d . The new a s p e c t s of t h i s work i n r e l a t i o n w i t h c o n v e n t i o n a l eddy models are a f i n i t e element f o r m u l a t i o n o f the q u a s i - g e o s t r o p h i c e q u a t i o n s and the use of n o - s l i p boundary c o n d i t i o n on the h o r i z o n t a l s o l i d b o u n d a r i e s . In c o n t r a s t to eddy r e s o l v i n g models t h a t u t i l i z e f r e e - s l i p boundary c o n d i t i o n s our r e s u l t s suggest t h a t the o b t e n t i o n of ocean eddies w i t h the n o - s l i p c o n s t r a i n t s r e q u i r e s a more r e s t r i c t e d range of p a r a m e t e r s , i n p a r t i c u l a r much lower h o r i z o n t a l eddy v i s c o s i t y eddy c o e f f i c i e n t s Aft and h i g h e r Froude numbers F^ and F 2 . We show e x p l i c i t l y t h a t a g i v e n range of parameters, which i s eddy g e n e r a t i n g when the f r e e - s l i p boundary c o n d i t i o n i s used, l e a d s to a q u a s i - l a m i n a r f l o w i n b o t h , upper and lower, l a y e r s . An a n a l y t i c a l model to i n t e r p r e t the n u m e r i c a l r e s u l t s ' i s p u t f o r t h . I t i s an e x t e n s i o n of an e a r l i e r model of I e r l e y and Young (1983) i n t h a t the r e l a t i v e v o r t i c i t y terms are o f p r i m a r y importance f o r the dynamics. Thus, i t i s shown t h a t the boundary l a y e r dynamics i s a c t i v e i n the i n t e r i o r of the second l a y e r , and i t can be concluded from our method t h a t f o r g i v e n F-| and F 2 such t h a t the lower l a y e r g e o s t r o p h i c c o n t o u r s a r e c l o s e d , to the e x i s t e n c e o f the w e s t e r n boundary l a y e r w i l l p r e v e n t the h o m o g e n i z a t i o n of the p o t e n t i a l v o r t i c i t y so long as Ay i s l a r g e enough to s t a b i l i z e the n o r t h w e s t e r n u n d u l a t i o n s o f the f l o w . - i v -T A B L E OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS . . i v LIST OF FIGURES , v ACKNOWLEDGEMENT v i i CHAPTER 1 - INTRODUCTION 1 CHAPTER 2 - EQUATIONS 10 2.1 E q u a t i o n of P o t e n t i a l V o r t i c i t y i n a S t r a t i f i e d Mid-Ocean Gyre 10 2.2 Boundary C o n d i t i o n s 15 2.3 E q u a t i o n s o f a Layered Ocean 21 CHAPTER 3 - NUMERICAL FORMULATION 35 3.1 B a s i c N o t a t i o n s and D e f i n i t i o n s 36 3.2 V a r i a t i o n a l F o r m u l a t i o n 39 3.3 G a l e r k i n - F i n i t e Element Method 44 3.4 Time D i s c r e t i z a t i o n 55 3.5 S t a b i l i t y A n a l y s i s 61 CHAPTER 4 - THE EXPERIMENTS 77 4.1 The Model and I t s Parameters 77 4.2 R e s u l t s 81 4.3 T h e o r e t i c a l A n a l y s i s 85 4.3.1 Boundary Layer E q u a t i o n s 92 CONCLUSIONS 106 REFERENCES 108 APPENDICES , 112 LIST OF FIGURES F i g u r e Page 1 The v e r t i c a l f i n i t e d i f f e r e n c e g r i d d i s c u s s e d i n S e c t i o n 2.3 21.1 2 Support of the space M n d i s c u s s e d i n S e c t i o n 3.3 54.1 3 Lower b a s i n g r i d i n Exp. 3. The upper b a s i n g r i d i s symmetric 79.1 4 Stream f u n c t i o n d i s t r i b u t i o n of the upper l a y e r i n Exp. 1 a t T = 200 days. P o s i t i v e v a l u e s i n f u l l l i n e , n e g a t i v e v a l u e s i n broken l i n e 81.1 5 Stream f u n c t i o n d i s t r i b u t i o n o f the lower l a y e r i n Exp. 1 at T = 200 days. P o s i t i v e v a l u e s i n f u l l l i n e , n e g a t i v e v a l u e s i n broken l i n e . 81.2 6 Stream f u n c t i o n d i s t r i b u t i o n o f the upper l a y e r i n Exp. 2 a t T = 200 days. P o s i t i v e v a l u e s i n f u l l l i n e , n e g a t i v e v a l u e s i n broken l i n e 82.1 7 Stream f u n c t i o n d i s t r i b u t i o n o f the lower l a y e r i n Exp. 2 at T = 200 days. P o s i t i v e v a l u e s i n f u l l l i n e , n e g a t i v e v a l u e s i n broken l i n e 82.2 8 R e l a t i v e v o r t i c i t y d i s t r i b u t i o n of the upper l a y e r i n Exp. 2 a t T = 200 days. P o s i t i v e v a l u e s i n f u l l l i n e , n e g a t i v e v a l u e s i n broken l i n e 82.3 9a,b Stream f u n c t i o n d i s t r i b u t i o n of the upper and lower l a y e r s , r e s p e c t i v e l y , i n Exp. 3 a t T = 130 days. P o s i t i v e v a l u e s i n f u l l l i n e s , n e g a t i v e v a l u e s i n broken l i n e 83.1,2 10,a,b Stream f u n c t i o n d i s t r i b u t i o n of the upper and lower l a y e r s , r e s p e c t i v e l y , i n Exp. 3 at T = 300 days. P o s i t i v e v a l u e s i n f u l l l i n e s , n e g a t i v e v a l u e s i n broken l i n e . . . 83.3,4 11a,b Stream f u n c t i o n d i s t r i b u t i o n of the upper and lower l a y e r s , r e s p e c t i v e l y , i n Exp. 3 a t T = 400 days. P o s i t i v e v a l u e s i n f u l l l i n e s , n e g a t i v e v a l u e s i n broken l i n e . . 83.5,6 - v i -Page 12,13,14 R e l a t i v e v o r t i c i t y d i s t r i b u t i o n o f the upper l a y e r i n Exp. 3 at T = 130, 300, 400 days, r e s p e c t i v e l y . P o s i t i v e v a l u e s i n f u l l l i n e , n e g a t i v e v a l u e s i n broken l i n e 84.1,2,3 15 P l o t of k i n e t i c energy per u n i t of a r e a o f the upper l a y e r . Exp. 3 84.4 16. The time dependent upper l a y e r stream f u n c t i o n at x = 100 km, y = 3 9 0 km... 84.5 - v i i -A C K N O W L E D G E M E N T S I w i s h to thank my s u p e r v i s o r , P r o f e s s o r L.A. Mysak f o r h i s guidance and a d v i c e d u r i n g t h i s s t u d y . I am a l s o v e r y g r a t e f u l f o r the support and encouragement r e c e i v e d from Dr. M. Foreman and Dr. W. H s i e h . T h i s work was supported by the g e n e r o s i t y of Dr. Mysak who a p p o i n t e d me as a Research A s s i s t a n t under an O f f i c e o f Naval Research G r a n t . - 1 -CHAPTER 1 INTRODUCTION D u r i n g the past f i f t e e n y e ars or so, oceanographers have made g r e a t p r o g r e s s i n the development and u n d e r s t a n d i n g of l a r g e - s c a l e o c e a n i c phenomena. T h i s p r o g r e s s i s the r e s u l t o f a b a l a n c e d s c i e n t i f i c approach i n v o l v i n g b e t t e r o b s e r v a t i o n a l t e c h n i q u e s as w e l l as more s o p h i s t i c a t e d t h e o r e t i c a l models. The impetus f o r the development of eddy r e s o l v i n g g e n e r a l c i r c u l a t i o n models (EGCMs) came from the need to s i m u l a t e n u m e r i c a l l y and to understand a p o s t e r i o r i the n a t u r e of the eddies which were the most o u t s t a n d i n g o b s e r v a t i o n a l f e a t u r e s of f i e l d programs such as POLYGON, MODE, and POLYMODE. The EGCMs, which i n c l u d e eddy motions a t l e a s t on some space-time s c a l e s , r e q u i r e much g r e a t e r h o r i z o n t a l r e s o l u t i o n to take i n t o account the i n s t a b i l i t y p r o p e r t i e s of the l a r g e r - s c a l e c i r c u l a t i o n , which y i e l d s mesoscale eddies on the s p a t i a l s c a l e of Rossby r a d i u s of d e f o r m a t i o n . Thus, i n c o n t r a s t w i t h the e a r l i e r a n a l y t i c a l models and n u m e r i c a l models w i t h c o a r s e r e s o l u t i o n and h i g h v i s c o s i t y which produced an inadequate r e p r e s e n t a t i o n of the p r o c e s s i n v o l v e d i n the ocean dynamics, EGCMs d e p i c t a c i r c u l a t i o n t h a t i s s t r o n g l y t u r b u l e n t and time-dependent. The f i r s t experiments of t h i s k i n d were c a r r i e d out by H o l l a n d and L i n (1975) i n which they showed t h a t i n s t a b i l i t y - 2 -i n the s w i f t boundary c u r r e n t s and t h e i r i n t e r t i a l r e t u r n f l o w s l e d to s i g n i f i c a n t eddy energy p r o d u c t i o n and t h a t the e d d i e s then m o d i f i e d the l a r g e - s c a l e f l o w i n a fundamental way. The success of t h i s e a r l y work l e d to f u r t h e r e d d y - r e s o l v i n g c a l c u l a t i o n s t h a t c o n t i n u e d to e x p l o r e the d e t a i l s of eddy p r o d u c t i o n and eddy i n t e r a c t i o n w i t h the mean f l o w and the e f f e c t s of the i n c l u s i o n of thermodynamics (Robinson et a l . 1977, Semtner and M i n t z , 1977). A l l t h e s e models were de s i g n e d to s i m u l a t e the mid-ocean gyre dynamics of the N o r t h A t l a n t i c Ocean by u s i n g the p r i m i t i v e e q u a t i o n s (PE) and as ocean domain an i d e a l i z e d box w i t h o u t bottom topography or w i t h a v e r y s i m p l e s h e l f and s l o p e . The f o r c i n g wind s t r e s s was a steady s i n u s o i d a l s i g n a l . I f one t h i n k s t h a t the main dynamical v a r i a b l e of a b a r o c l i n i c m i d - l a t i t u t d e ocean i s the p o t e n t i a l v o r t i c i t y q n = + t) • Vp q P where i s the e a r t h ' s r o t a t i o n v e c t o r , t i s the r e l a t i v e v o r t i c i t y and p a c o n s e r v a t i v e f l u i d p r o p e r t y , then i t seems n a t u r a l to s i m u l a t e the eddy dynamics by making use of the q u a s i - g e o s t r o p h i c a p p r o x i m a t i o n . S e v e r a l m o d e l l e r s w o r k i n g w i t h an E G C M - b a r o c l i n i c ocean have f o l l o w e d t h i s approach p a r t i c u l a r l y s i n c e 1978 when H o l l a n d (1978) i n i t i a t e d an e x t e n s i v e s e t of n u m e r i c a l experiments w i t h a two l a y e r box ocean and the q u a s i - g e o s t r o p h i c a p p r o x i m a t i o n to the PE. - 3 -L a t e r on, H o l l a n d has extended these 1978 experiments by i n c l u d i n g more l a y e r s (up to 8, 1986 p e r s o n a l communication) and e n l a r g i n g the g e o m e t r i c a l dimensions of the box ocean. The improvement i n the r e s o l u t i o n of the v e r t i c a l s t r u c t u r e y i e l d s such i n t e r e s t i n g r e s u l t s as the h o m o g e n i z a t i o n of p o t e n t i a l v o r t i c i t y i n the i n t e r m e d i a t e l a y e r s — a r e s u l t p r e d i c t e d t h e o r e t i c a l l y by Rhines and Young (1982). The advantages of a q u a s i - g e o s t r o p h i c model over a PE model are m a i n l y c o m p u t a t i o n a l (Semtner and H o l l a n d 1978) so l o n g as we assume t h a t the m i d - l a t i t u t d e ocean c i r c u l a t i o n i s w i t h i n a range of parameters. How r e p r e s e n t a t i v e such a parameter range i s of the r e a l ocean i s a d e b a t a b l e m a t t e r . More a c c u r a t e o b s e r v a t i o n a l and n u m e r i c a l r e s u l t s and f u r t h e r t h e o r e t i c a l developments show t h a t the q u a s i - g e o s t r o p h i c a p p r o x i m a t i o n i s i n s u f f i c i e n t to d i s p l a y some impo r t a n t f e a t u r e s o f the ocean dynamics such as the o u t c r o p p i n g of the i s o p y c n a l s a l l o w i n g the v e n t i l a t i o n o f s u b s u r f a c e l a y e r s . The l a t t e r i s b e l i e v e d to p l a y a fundamental r o l e i n the dynamics of p o t e n t i a l v o r t i c i t y ( L u y t e n et a l . 1983). Any n u m e r i c a l model designed to e l u c i d a t e the r o l e o f c e r t a i n mechanisms of the dynamics of the ocean c i r c u l a t i o n r e q u i r e s the a d o p t i o n of compromises i n c a r r y i n g out the c a l c u l a t i o n s . Two compromises of t h i s k i n d are the p a r a m e t e r i z a t i o n of the s u b - g r i d s c a l e motion and the c h o i c e o f the boundary c o n d i t i o n s . As f o r the f i r s t one, g e o s t r o p h i c t u r b u l e n c e t h e o r y (Charney, 1971) shows a cascade - 4 -o f energy to l a r g e r l e n g t h s c a l e s w h i l e the e n s t r o p h y cascades towards the s m a l l e r ones. T h i s e n s t r o p h y cascade, on s u b - d e f o r m a t i o n r a d i u s s c a l e s , must be r e p r e s e n t e d w i t h a p h y s i c a l l y m o t i v a t e d p a r a m e t e r i z a t i o n . At p r e s e n t , t h i s i s an u n s o l v e d problem i n ocean dynamics; however, two e x p l i c i t p a r a m e t e r i z a t i o n s are p o p u l a r i n EGCMs. The f i r s t one i s the c l a s s i c L a p l a c i a n f r i c t i o n i n which the l a t e r a l d i s s i p a t i v e terms of the r e l a t i v e v o r t i c i t y e q u a t i o n are r e p r e s e n t e d as V • (Ay V S ) , where Ay i s a d i f f u s i o n c o e f f i c i e n t ; the second one i s known as b i h a r m o n i c f r i c t i o n because the e x p l i c i t d i s s i p a t i o n i s of the form B H v4£, where B H i s a c o n s t a n t c o e f f i c i e n t . The p h y s i c a l m o t i v a t i o n of both p a r a m e t e r i z a t i o n s or c l o s u r e s has no s o l i d ground. The L a p l a c i a n f r i c t i o n i s a m o l e c u l a r - t y p e d i f f u s i o n o p e r a t o r , u s u a l i n most s u b s o n i c f l u i d dynamics c o m p u t a t i o n s . I t has v e r y poor damping s e l e c t i v e p r o p e r t i e s , which n e c e s s i t a t e k e e p i n g Ay as low as p o s s i b l e i n o r d e r not to damage s e v e r e l y the energy spectrum of the m o t i o n . The b i h a r m o n i c f r i c t i o n i s h i g h l y s e l e c t i v e so i t damps v e r y e f f i c i e n t l y the s m a l l wavelength s c a l e s and, t h e r e f o r e , the e n s t r o p h y can be n o n - i n t r u s i v e l y removed ( H o l l a n d , 1978). A p o s s i b l e i n c o n v e n i e n c e of t h i s type of c l o s u r e i s t h a t i t r e q u i r e s boundary c o n d i t i o n s t h a t are somewhat a r t i f i c i a l i n the sense t h a t v o r t i c i t y d e r i v a t i v e s of o r d e r h i g h e r than 2 are a r b i t r a r i l y set to zero i n o r d e r to c l o s e the m a t h e m a t i c a l problem. Other a l t e r n a t i v e c l o s u r e s have been suggested - 5 -r e c e n t l y (Basdevant et a l . ,< 1978 and Sadourny and Basdevant, 1981). The c h o i c e of boundary c o n d i t i o n s , p a r t i c u l a r l y on the h o r i z o n t a l s o l i d b o u n d a r i e s , i s a m a t t e r t h a t i s not y e t s e t t l e d . There are s e v e r a l p o s s i b i l i t i e s such as s l i p , n o - s l i p and o t h e r l e s s c o n v e n t i o n a l c h o i c e s . The l a t t e r ones, such as those used by M a r s h a l l (1982, 1984), are so m a t h e m a t i c a l l y c o n d i t i o n e d t h a t i t seems u n r e a l i s t i c t h a t the r e a l ocean obeys such c o n d i t i o n s . Most of EGCM m o d e l l e r s eager to get an ocean f u l l of eddies ( w i t h r e a s o n a b l e computer expenses) have used the s l i p boundary c o n d i t i o n ; however, i t i s not c l e a r whether t h a t c o n d i t i o n i s more r e a l i s t i c than the n o n - s l i p one. I f one f o l l o w s S t e w a r t ' s (1964) argument t h a t the ocean water i s a r e a l f l u i d and, t h e r e f o r e , adheres to the s o l i d b o u n d a r i e s so t h a t the v e l o c i t y must be zero on them, then i t seems t h a t the n o n - s l i p boundary c o n d i t i o n i s the proper one. Other arguments s u p p o r t i n g the c h o i c e of t h i s boundary c o n d i t i o n are shown i n S e c t i o n 2.1 of t h i s t h e s i s . However, i f one l o o k s at the n o - s l i p c o n s t r a i n t from a c o m p u t a t i o n a l p o i n t - o f - v i e w , the r e s u l t s are somewhat d i s c o u r a g i n g , because, as the author of t h i s t h e s i s has l e a r n e d i n the c o u r s e of these p r e l i m i n a r y e x p e r i m e n t s , such a boundary c o n d i t i o n i m p l i e s a l a r g e r amount of d i s s i p a t i o n of energy by the l a t e r a l f r i c t i o n terms than i n the case of the f r e e - s l i p c o n s t r a i n t . ( I e r l e y and Young, 1986, p e r s o n a l communication, - 6 -have noted t h a t the energy d i s s i p a t i o n of a b a r o t r o p i c model w i t h the n o - s l i p c o n d i t i o n i s 30 p e r c e n t l a r g e r than w i t h the f r e e - s l i p one.) T h i s means t h a t i f a g i v e n range of parameters i s v a l i d f o r the o b t e n t i o n of eddies w i t h the f r e e - s l i p c o n d i t i o n , the same range may become l a m i n a r by u s i n g the n o - s l i p c o n s t r a i n t . ( T h i s has been our e x p e r i e n c e i n these p r e l i m i n a r y experiments w i t h our model.) T h e r e f o r e , a d r a s t i c change i n some of the parameters such as A H (which has to be c o n s i d e r a b l y r e d u c e d ) , i s n e c e s s a r y i n o r d e r to s i m u l a t e the eddy dynamics of the ocean; such an a l t e r a t i o n of the v a l u e s of the parameters i n v o l v e s a g r e a t i n c r e a s e i n the c o m p u t a t i o n a l burden. One i m p o r t a n t aspect of any n u m e r i c a l model to which l i t t l e a t t e n t i o n has been p a i d by many ocean m o d e l l e r s i s the a c c u r a c y of the method. Most of the GCMs and EGCMs are f i n i t e d i f f e r e n c e models of second o r d e r i n time and space a c c u r a c y . Very few attempts to use h i g h e r o r d e r methods have been done so f a r ; however, we f e e l i t i s time to move to more a c c u r a t e methods because the p r e s e n t computer t e c h n o l o g y a l l o w s l a r g e - s c a l e s i m u l a t i o n s at r e a s o n a b l e p r i c e . The purpose of our work, of which t h i s t h e s i s i s a f i r s t s t e p , i s t w o f o l d ; f i r s t , the development of an eddy model u s i n g f i n i t e element as a n u m e r i c a l t e c h n i q u e because i t i s w e l l known t h a t f o r r e g u l a r g r i d s (see F i x , 1975, S t a n i f o r t h and M i t c h e l l , 1977, H a i d v o g e l et a l . , 1980) the a c c u r a c y of l i n e a r f i n i t e element i s much h i g h e r than t h a t of s t a n d a r d - 7 -f i n i t e d i f f e r e n c e methods which are commonly used i n the p r e s e n t o p e r a t i o n a l ECGM; s e c o n d l y , the s y s t e m a t i c study o f EGCM dynamics, under s e v e r a l parameter regimes, when the n o - s l i p c o n d i t i o n i s used as a l a t e r a l c o n s t r a i n t . We b e l i e v e t h a t the f i r s t stage of our p r o j e c t e d t a s k has been f u l f i l l e d i n the sense t h a t we have developed a f i n i t e element model to i n t e g r a t e n u m e r i c a l l y a s e t of cou p l e d p a r a b o l i c - e l l i p t i c e q u a t i o n s , and we have found a range of parameters which produces eddies w i t h the f r e e - s l i p boundary c o n d i t i o n but which y i e l d s a q u a s i - l a m i n a r dynamics w i t h the n o - s l i p one. An a n a l y t i c a l boundary l a y e r method to understand the n u m e r i c a l r e s u l t s has a l s o been d e v e l o p e d . The c o n t e n t s o f the t h e s i s are d i v i d e d i n t o c h a p t e r s , and the l a t t e r i n t o s e c t i o n s as f o l l o w s : Chapter 2 - In S e c t i o n 2.1 , we o b t a i n the e q u a t i o n of the p o t e n t i a l v o r t i c i t y f o r mid-ocean gyres based on the r e l a t i v e v o r t i c i t y e q u a t i o n and the 3 - p l a n e - q u a s i - g e o s t r o p h i c paradigm o f mid-ocean dynamics. In S e c t i o n 2.2, i t i s e s t a b l i s h e d t h a t the a p p l i c a t i o n of the n o - s l i p c o n d i t i o n on the s o l i d b o u n d a r i e s to the 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 removes M c W i l l i a m s 1 i n c o m p a t i b i l i t i e s of those e q u a t i o n s and the f r e e - s l i p c o n d i t i o n . I n S e c t i o n 2.3, a method i s d e s c r i b e d to d i s c r e t i z e the v e r t i c a l s t r u c t u r e of the 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 i n t o a f i n i t e number of l a y e r s . Chapter 3 - T h i s c h a p t e r d e s c r i b e s the procedure to c o n s t r u c t a f i n i t e element v e r s i o n of the p o t e n t i a l v o r t i c i t y - 8 -e q u a t i o n s i n a two l a y e r ocean. A f t e r the i n t r o d u c t i o n i n S e c t i o n 3.1 of some n o t a t i o n and b a s i c d e f i n i t i o n s , s t a n d a r d i n f i n i t e element l i t e r a t u r e , we set up the v a r i a t i o n a l f o r m u l a t i o n of the t w o - l a y e r 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 i n S e c t i o n 3.2. I n S e c t i o n 3.3, we d e s c r i b e the G a l e r k i n t e c h n i q u e to d i s c r e t i z e the v a r i a t i o n a l e q u a t i o n s . S p e c i a l c a r e has been taken t o d e r i v e the d i s c r e t e n o - s l i p boundary c o n d i t i o n . Time d i s c r e t i z a t i o n of the e q u a t i o n s of S e c t i o n 3.3 and t h e i r s t a b i l i t y a n a l y s i s i s taken up i n S e c t i o n 3.4. We have not p r e s e n t e d any e r r o r e s t i m a t e a n a l y s i s of the d i s c r e t e e q u a t i o n s because such m a t t e r s w i l l be r e p o r t e d e l s e w h e r e . Chapter 4 - T h i s c h a p t e r p r e s e n t s the r e s u l t s of t h r e e n u m e r i c a l e x p e r i m e n t s . We have chosen f o r t h i s stage of our st u d y the same paramters H o l l a n d used i n h i s experiment of 1978 w i t h the a d d i t i o n o f l i n e a r bottom f r i c t i o n . The rea s o n s of c h o o s i n g such a range of parameters was to compare c o n s i s t e n t l y our r e s u l t s w i t h those o f H o l l a n d . I n S e c t i o n 4.2, we develop an a n a l y t i c a l model to i n t e r p r e t the n u m e r i c a l r e s u l t s of the ex p e r i m e n t - 3 . T h i s a n a l y t i c a l model i s based on boundary l a y e r methods and has as n o n - l i n e a r i t i e s the v o r t e x s t r e t c h i n g and r e l a t i v e v o r t i c i t y terms. A n o n - d i m e n s i o n a l a n a l y i s r e v e a l s t h a t the upper and lower l a y e r s dynamics are uncoupled i n the w e s t e r n boundary because the s t r e t c h i n g terms are much s m a l l e r than the o t h e r terms o f the e q u a t i o n s ; however, the downward - 9 -momentum f l u x produced by the s t r e t c h i n g terms i n the Sverdrup i n t e r i o r i s the d r i v i n g mechanism of the motion i n the lower l a y e r . Our a n a l y t i c a l model i s a g e n e r a l i z a t i o n of I e r l e y and Young's (1983) model because i t i n c l u d e s the i m p o r t a n t c o n t r i b u t i o n of the r e l a t i v e v o r t i c i t y a t the s o l i d b o u n d a r i e s . The c o n c l u s i o n s and the l i s t of r e f e r e n c e s t e r m i n a t e the t h e s i s . The o b t e n t i o n of the r e l a t i v e v o r t i c i t y e q u a t i o n i s c a r r i e d out i n the appendix. - 10 -CHAPTER 2 2.1 E q u a t i o n o f P o t e n t i a l V o r t i c i t y i n a S t r a t i f i e d  M i d - O c e a n G y r e L e t us c o n s i d e r a bounded s t r a t i f i e d o c e a n i c domain D c e n t e r e d at a m i d - l a t i t u d e 9g The e q u a t i o n of the r e l a t i v e v o r c i t y c i n such domain i s (see A p p e n d i x ) : § ^ + u • Vf = fW z + A R v 2 , + |_ ( v a ? ) , (2.1) where u ( x , y , z ) = (u,v) i s the h o r i z o n a l v e l o c i t y v e c t o r , w i s the v e r t i c a l component of the v e l o c i t y , 2Q C O S 6 f = f 0 + By, 3 = — , f Q = 20, s i n e Q , A^ i s the l a t e r a l eddy v i s c o s i t y c o e f f i c i e n t , T i s the e x t e r n a l f o r c i n g f u n c t i o n , D' 3 ^ > TTt = 8t + u * V S i n c e our main i n t e r e s t i s the d e s c r i p t i o n of motions which take p l a c e at m i d - l a t i t u d e s and such t h a t t h e i r m e t r i c can be a c c u r a t e l y d e s c r i b e d by the tangent plane a p p r o x i m a t i o n , we i n t r o d u c e the st a n d a r d s c a l i n g v a r i a b l e s f o r a m i d - l a t i t u d e 3-plane. The s c a l i n g parameters are ( P e d l o s k y 1979) - 11 -P = P s ( z ) (1 + e F p ' ) , U = Rossby Number, e = T ' » L N 2 ( 2 . 2 ) U where F = L i s a t y p i c a l l o n g i t u d e s c a l e of the motion D i s the mean depth of the ocean U i s a t y p i c a l v e l o c i t y s c a l e of the f l o o r p'(x,y) i s a p e r b u r b a t i o n d e n s i t y i n excess o f P g ( z ) For ocean gyre s c a l e s , the f o l l o w i n g p a r a m e t r i c r e l a t i o n s h o l d ( P e d l o s k y , 1979). g » 1 , F » 0 ( 1 ) , e s = 0(1) , 6 e = •(£-) < 1 , 0 - 1 2 -S e » 1 , ( 2 . 3 a , b , c , d ) where rg i s the r a d i u s o f the e a r t h . The e q u a t i o n ( 2 . 1 ) i s up to o r d e r 0 ( 1 ) v 3 Q = f w z > ( 2 . 4 ) o r , e q u i v a l e n t l y , the a d v e c t i o n of p l a n e t a r y v o r t i c i t y i n the i n t e r i o r o f the mid-ocean gyres i s b a l a n c e d by v o r t e x - t u b e s t r e t c h i n g . The v e r t i c a l i n t e g r a l o f ( 2 . 4 ) from z = -h, where i t i s assumed t h a t the v e l o c i t y i s z e r o , up to the s u r f a c e , where an Ekman l a y e r type i s a l l o w e d , g i v e s the c l a s s i c Sverdrup r e l a t i o n : The g e o s t r o p h i c e q u i l i b r i u m a l l o w s the i n t r o d u c t i o n of a stream f u n c t i o n i() a s a f u n c t i o n of the p e r t u r b a t i o n p r e s s u r e p 1 to the h y d r o s t a t i c p r e s s u r e P s. Thus, we d e f i n e as: 3 Q S_h vdz = f [ w ] ^ h = ( ( V A ) z T ) , ( 2 . 5 ) where t i s the s u r f a c e wind s t r e s s . ( 2 . 6 ) The h y d r o s t a t i c r e l a t i o n i s now w r i t t e n as P = - 8z V 9 p s g 3Z - 13 -3 p o because - 5 — i s of the o r d e r o f 0(10 ) . w i s g i v e n as a Z Z (see Appendix) D't *sz K s z D' f 9 _ , p s f 0 _3*\i + i _ / G . D^T L 3 z ^ g p g z 8 z ; j + 3z ^ p s z ; = _ r 9 _ ,£p_ 3JK, J _ ,_G, , . rPT L 3 z \ T2 3 z ; j 8z Q p ; ' N sz where G i s d e f i n e d i n ( A . 1 2 ) . S u b s t i t u t i n g (2.7) i n t o (2.1) and a r r a n g i n g terms y i e l d s to the c l a s s i c e q u a t i o n o f the p o t e n t i a l v o r t i c i t y : = f h <?r-> + H v2« + h «™>z ?>- <2-8a> sz The heat e q u a t i o n (A.12) becomes i n terms o f ty = Z o Z D'l|» M 2 9 o ^ + | _ w = K ^ + £g Q > ( 2 ^ g b ) where q i s now g i v e n by - 14 -At p l a n e t a r y s c a l e s ( g y r e - s c a l e s ) and o u t s i d e c e r t a i n narrow areas a t t a c h e d to the b o u n d a r i e s , the p o t e n t i a l v o r t i c i t y q can be aproximated by whereas, f o r the motions of the mesoscale, q i s g i v e n by ( 2 . 9 a ) . Hence, we can co n c l u d e t h a t so l o n g as — < 1, no r 0 2 2 t h e r m o c l i n e p r o c e s s e s , and N = N (z) e q u a t i o n s (2.8) are v a l i d e i t h e r f o r s y n o p t i c s c a l e s or g y r e - s c a l e s . At t h i s p o i n t , i t i s n e c e s s a r y to s t r e s s the fundamental r o l e p l a y e d by e q u a t i o n s (2.4) and (2.5) to e x p l a i n why t h e r e must e x i s t z o n a l v e l o c i t i e s i n the ocean o r , i n o t h e r words, why the ocean c i r c u l a t i o n i s composed o f g y r e s . For an homogeneous ocean, i t i s obvious from the Sverdrup r e l a t i o n t h a t t h e r e e x i s t s a r e t u r n f l o w a l o n g the western b o u n d a r i e s of the oceans and, hence, the e x i s t e n c e of z o n a l v e l o c i t i e s . I n the case of a b a r o c l i n i c ocean where the t h e r m o c l i n e e f f e c t s as w e l l as the h o r i z o n t a l s v a r i a t i o n s of the i s o p y c n a l s are n e g l e c t e d , the d r i v i n g mechanism i s the v e r t i c a l v e l o c i t y from the d i v e r g e n t Ekman l a y e r at the s u r f a c e . A s c a l e a n a l y s i s of (2.4) l e a d s to vw ;1 D - 15 -I f we now assume t h a t the c i r c u l a t i o n i s t w o - d i m e n s i o n a l i n m e r i d i o n a l p l a n e s , t h a t i s , t h a t the d i v e r g e n c e of the Ekman l a y e r y i e l d s to c l o s e d c e l l c i r c u l a t i o n s i n m e r i d i o n a l p l a n e s , the mass c o n s e r v a t i o n e q u a t i o n r e q u i r e s t h a t Vy + w z = 0. A new s c a l e a n a l y s i s of the l a t t e r r e l a t i o n y i e l d s to the new r a t i o (-), - ^ *V2 D ' where L v i s the n o r t h - s o u t h s c a l e of the f l o w . The r a t i o o f these e s t i m a t e s g i v e s : ^ 1 : ^ 2 = IE" = TT > 1 y y So, t h e r e must be a z o n a l v e l o c i t y f l o w i n g i n t o the m e r i d i o n a l p l a n e s . 2.2 Boundary C o n d i t i o n s T h i s i s one of the most c o n f l i c t i n g i s s u e s of the q u a s i - g e o s t r o p h i c models, and t h e r e f o r e , one o f the sh o r t c o m i n g s of the q u a s i - g e o s t r o p h i c dynamics. The s e n s i t i v i t y of these models to the type of boundary c o n d i t i o n s i s w e l l known. B l a n d f o r d (1971) shows how the s o l u t i o n s of the v o r t i c i t y e q u a t i o n of a b a r o t r o p i c ocean can be a l t e r e d by changing the boundary c o n d i t i o n s . I e r l e y and - 16 -Young ((1986) p e r s o n a l communication) r e a s s e r t B l a n d f o r d ' s c o n c l u s i o n s f o r a b a r o c l i n i c ocean. M c W i l l i a m s (1977) shows t h a t the a p p l i c a t i o n of f r e e s l i p boundary c o n d i t i o n on the b o u n d a r i e s of a m u l t i p l y connected c l o s e d domain l e a d s to i n c o m p a t i b i l i t i e s between the o r d e r of the 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 and the number of c o n s t r a i n t s r e q u i r e d f o r the problem to be w e l l - p o s e d . To remove both i n c o m p a t i b i l i t i e s , M c W i l l i a m s c o n c l u d e s t h a t n e i t h e r h o r i z o n t a l d i f f u s i o n o f t emperature nor v e r t i c a l d i f f u s i o n of momentum are p r o c e s s e s to be r e p r e s e n t e d by the q u a s i - g e o s t r o p h i c dynamics. However, i f one c o n s i d e r s t h a t the ocean water i s a r e a l f l u i d , then the c o n d i t i o n r e q u i r e d on the s o l i d b o u n d a r i e s i s t h a t the t a n g e n t i a l v e l o c i t y be z e r o . I t i s t h i s c o n d i t i o n , t o g e t h e r w i t h the k i n e m a t i c c o n d i t i o n ( t h a t the normal component of the v e l o c i t y be zero on the s o l i d b o u n d a r i e s ) t h a t i s r e q u i r e d to remove M c W i l l i a m s 1 i n d e t e r m i n a n e i e s w i t h o u t r u l i n g out any d i f f u s i v e p r o c e s s from the q u a s i - g e o s t r o p h y dynamics. I t i s worth n o t i n g t h a t the i m p o s i t i o n o f the n o - s l i p boundary c o n d i t i o n on the bottom and a s l i g h t m o d i f i c a t i o n of i t at the f r e e s u r f a c e a l l o w s the e x i s t e n c e of Ekman boundary l a y e r s ( P e d l o s k y , 1979). The boundary c o n d i t i o n s f o r the v e l o c i t y a r e : a. s o l i d b o u n d a r i e s •*• * n u • n = 0 on o u n H (2.10) u • t = 0 B - 1 7 -where &H denotes the h o r i z o n t a l s o l i d w a l l s and fig stands *>- -> f o r the bottom, n and t are the outward normal v e c t o r and c l o c k w i s e tangent v e c t o r , r e s p e c t i v e l y , on such b o u n d a r i e s . b. f r e e s u r f a c e The r i g i d l i d c o n d i t i o n and the c o n t i n u i t y of f r i c t i o n a l s t r e s s a c r o s s the s u r f a c e are the c o n s t r a i n t s used at the f r e e s u r f a c e . As f o r buoyancy, we use K o t c h i n e ' s theorem ( K r a u s s , 1 9 7 3 ) and assume t h a t the s o l i d b o u n d a r i e s are i n s u l a t e d , whereas, at the f r e e s u r f a c e , t h e r e i s an a t m o s p h e r i c buoyancy f l u x which i s communicated to the ocean v i a boundary l a y e r p r o c e s s on a g e o s t r o p h i c s c a l e s . Such a f l u x i s g i v e n go Q A by T J T n where H 1 i s the t h i c k n e s s o f the uppermost H u p 0 C p u l e v e l of p e n e t r a t i o n of a t m o s p h e r i c heat Q^. We f u r t h e r assume t h a t t h e r e are no i n t e r n a l p r o c e s s e s of g e n e r a t i o n / s i n k i n g of buoyancy. Thus, we have K H V^p • n = 0 on 2^ ( K v h ~ KH V H B * V> p = 0 o n % K v T z = 0 a t t h e f r e e s u r f a c e . ( 2 . 1 1 ) where Hg(x,y) i s the bottom p r o f i l e . C o n d i t i o n s ( 2 . 1 0 ) and ( 2 . 1 1 ) can be w r i t t e n i n terms of the - 18 -g e o s t r o p h i c stream f u n c t i o n , through a l e a d i n g o r d e r i n e, as * - « t > . | i - 0, K„ | j - 0 on a H , 2 w = Ug • VH g - y 7 % , K v A_| = 0 on fiR , 9 z (VA) ft2 . w = — i r - ^ x w , K -2-4 = 0 at z = 0 (2.12a,b,c) r 0 V 3 z Z where ((VA) t w ) denotes the z-component of the c u r l o f the wind s t r e s s . C ( t ) i s a time dependent c o n s t a n t on the p e r i m e t e r of the domain. u B i s the h o r i z o n t a l v e l o c i t y at the edge o f the bottom Ekman l a y e r . y i s a bottom f r i c t i o n c o e f f i c i e n t . N o t i c e t h a t the c o n d i t i o n ( n o - s l i p ) = 0 i m p l i e s t h a t t h e r e be no f l u x e s of buoyancy by Newtonian p r o c e s s e s through the h o r i z o n t a l s o l i d w a l l s . So, we have j u s t removed one of M c W i l l i a m s " i n c o m p a t i b i l i t i e s , t h a t i s , we can i n c l u d e Newtonian d i f f u s i o n of buoyancy i n the h o r i z o n t a l d i r e c t i o n w i t h o u t demanding any a d d i t i o n a l boundary c o n d i t i o n . The c o n d i t i o n s (2.12b) and (2.12c) can be w r i t t e n w i t h the h e l p of (2.8a) as K A_| = o on fiR (2.13) v 3 z 2 - 19 -and D't + H ' p0 C p gQ Ao = 0 K v (2.14) a t z = 0. These boundary c o n d i t i o n s are not i n c o m p a t i b l e w i t h the d i f f e r e n t i a l o r d e r of (2.8) w i t h r e s p e c t to the z - c o o r d i n a t e . Hence, the s t r i c t a p p l i c a t i o n o f the n o - s l i p boundary c o n d i t i o n to the Q.G.E. a l l o w s the i n c l u s i o n o f Newtonian f r i c t i o n a l and d i f f u s i v e mechanisms i n both v e r t i c a l and h o r i z o n t a l d i r e c t i o n s on the a g e o s t r o p h i c s c a l e s . A s i m p l e s c a l e a n a l y s i s shows t h a t f o r t y p i c a l v a l u e s of K H and K V ( K H » 1 0 6 K y ) , the l a t e r a l d i f f u s i o n of buoyancy i s dominant w i t h r e s p e c t to the v e r t i c a l one, so the l a t t e r p rocess i s not i n c l u d e d i n many q u a s i - g e o s t r o p h i c models. The u s u a l way of computing the s o l u t i o n of (2.8) i s by d i s c r e t i z i n g the v e r t i c a l s t r u c t u r e of the ocean i n a g i v e n number of l a y e r s of c o n s t a n t d e n s i t y . As we w i l l see i n the next s e c t i o n , i n doing so, the v e r t i c a l d i f f u s i o n of momentum appears as a drag term of the form Av2j(i|^ +^ - ^ ) , where i denotes the i - t h l a y e r , A i s a drag c o e f f i c i e n t equal to - 20 -A v / H i and i s the t h i c k n e s s of the i - t h l a y e r . The drag term i s s i m i l a r to the form of the l a t e r a l d i f f u s i o n o f buoyancy. A g a i n , a s c a l e a n a l y s i s shows t h a t the l a t e r a l d i f f u s i o n o f buoyancy i s much l a r g e r than the drag term. The s i m i l a r i t y between both p r o c e s s e s goes f u r t h e r than a pure m a t h e m a t i c a l e x p r e s s i o n i n the c o n t e x t of 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 , f o r as i t has been mentioned i n the p r e v i o u s s e c t i o n , the p o t e n t i a l v o r t i c i t y i n the i n t e r i o r of m i d - l a t i t u d e ocean gyes i s g i v e n by f — , i n a l a y e r e d 3-plane ocean by f + - ^ ) » so the l a t e r a l d i f f u s i v i t y o of p o t e n t i a l v o r t i c i t y i n such an ocean i s ^ ^ i + l ~ ^ i ^ * T h i s means t h a t the r o l e p l a y e d by the h o r i z o n t a l d i f f u s i o n of buoyancy and the v e r t i c a l d i f f u s i o n of momentum i n the q u a s i - g e o s t r o p h i c dynamics i s e q u i v a l e n t to the l a t e r a l d i f f u s i o n of p o t e n t i a l v o r t i c i t y . Rhines and Young (1982a) and de Szoeke (1985) show t h a t the h o m ogeneization of p o t e n t i a l v o r t i c i t y i n s i d e c l o s e d g e o s t r o p h i c c o n t o u r s at i n t e r m e d i a t e l a y e r s i s o n l y p o s s i b l e i f the dominant d i s s i p a t i v e mechanism i s l a t e r a l d i f f u s i o n of p o t e n t i a l v o r t i c i t y by mesoscale e d d i e s . I t t u r n s out t h a t a good p a r a m e t e r i z a t i o n , at l e a s t q u a l i t a t i v e l y , o f the d i f f u s i v e e f f e c t s of the mesoscale eddies i s g i v e n by a term of the form V • ( V(\j>^+.| - ij>p ( a c o e f f i c i e n t ) which i s s i m i l a r to the l a t e r a l d i f f u s i o n of momentum i n the case t h a t i s t a ken as c o n s t a n t . - 21 .-The q u a s i - g e o s t r o p h i c e q u a t i o n s ( 2 . 8 ) , (2.13) and (2.14) are c o n s i s t e n t through l e a d i n g o r d e r i n e, but they are t h emselves i n s u f f i c i e n t to a s s u r e an a s y m p t o t i c a l l y c o r r e c t s o l u t i o n to the p r i m i t i v e e q u a t i o n s . In p a r t i c u l a r , some a d d i t i o n a l c o n s t r a i n t s are to be imposed to i n s u r e c o r r e c t i n t e g r a l budgets of energy, c i r c u l a t i o n and mass c o n s e r v a t i o n . As M c W i l l i a m s (1977) shows, the mass c o n s e r v a t i o n c o n s t r a i n t i n a c l o s e d domain r e q u i r e s t h a t a t each l e v e l - i J S J wda = 0. (2.15) T h i s c o n s t r a i n t w i l l be used to determine the c o n s t a n t C ( t ) o f the stream f u n c t i o n on the p e r i m e t e r of the domain. 2.3 Equations of a Layered Ocean L e t us c o n s i d e r t h a t the v e r t i c a l s t r u c t u r e of the ocean domain D i s composed of N l a y e r s of c o n s t a n t d e n s i t y and depth Hj c(k = 1,2,...,N). We w i s h to o b t a i n the new form of e q u a t i o n s (2.8) f o r such an ocean. S i n c e the p e r t u r b a t i o n d e n s i t y pk i s c o n s t a n t w i t h i n each l a y e r , the v e r t i c a l v a r i a t i o n of the stream f u n c t i o n i^k i s c o n t r o l l e d by the upper and lower d e n s i t y jumps of each l a y e r , t h a t i s to say, w i t h i n each l a y e r the T a y l o r -Proudman theorem a p p l i e s i f the f l u i d s were i n v i s c i d , b u t , i t i s the presence of d e n s i t y jumps t h a t p e r m i t s jumps i n the v e l o c i t y . - 21.1 -V e r t i c a l f i n i t e d i f f e r e n c e g r i d . - 22 -The d e r i v a t i v e s w i t h r e s p e c t to the z - c o o r d i n a t e i n e q u a t i o n s (2.8) are computed by u s i n g c e n t e r e d d i f f e r e n c e s . Thus, f o r i n s t a n c e , the z - d e r i v a t i v e of a g i v e n magnitude A at the l e v e l j + 1/2 (see F i g . 1) i s : 0 . + , = ^ 7 7 ^ • ( 2 . 1 6 ) The v a l u e a s s i g n e d to the p o t e n t i a l v o r t i c i t y i n each l a y e r i s i t s v a l u e taken at the m i d - l e v e l . M i d - l e v e l s are denoted by i n t e g e r numbers i n F i g u r e 1. The i n t e r f a c e s between a d j a c e n t l a y e r s are l a b e l l e d w i t h i n t e g e r numbers p l u s 1/2. So, the p o t e n t i a l v o r t i c i t y o f the k - t h l a y e r i s g i v e n by 2 f 0 V H ~ *k *k " *k-1 ^k = ? k + f + I T [ " - — — 1 ' ( 2 ' 1 7 ) where g 1 = g r , (2.18) k + 1 pk+1 and (2.16) has been a p p l i e d t w i c e . L i k e w i s e , the v e r t i c a l d e r i v a t i v e s of f - ^ (——) and (VA) T a t the k - t h l a y e r a r e p s z z g i v e n by - 23 -i f i _ (JL_)} 1 3z l p ' sz 2 K 2 p s z ' k + 1 sz k - j (2.19a) N e g l e c t i n g the v e r t i c a l d i f f u s i o n of buoyancy G = K H VH *z + C" 8 Q. P (2.19a) becomes 72 r f 0 *k+1 K H VH I H T ( - g -k & 1 k + 2 g k-1 n ( k ) — ) ] + f r S-•0 H, (2.19b) where Q<k> = Q* k + j- k - ± k + ^ p = «_ H k + H k + 1/2 Qk + 1/2 ( V A ) Z T = k + I ( 7 A ^ k + 1 12 ~ <VA*> k - 1 12 Jk + 1/2 ( V A T ) 1 + 1 / 2 'k - 1/2 T ( D (2.19c,d) T ( k ) , k # 1,N 'k + 1/2 ( V A T ) N - 1/2 = T ( N - 1/2) JN + 1/2 ~ ZN - 1/2 (2.20) - 24 -A s t a n d a r d p a r a m e t e r i z a t i o n of (VA) t i s < V A > z t = A v ! l - < 2' 2 1> By u s i n g (2.21) and ( 2 . 1 6 ) , the formulae (2.20) become 2 T ( k ) = [ 2 Ck+1 " _ 2 g k " g k - 1 j H k Hk+1 + H k " H k + H k - 1 T ( N ) = _ 2 [ N "N-1 ] t H N + HN +1 ^ T h i s p a r a m e t e r i z a t i o n i s e q u i v a l e n t to a drag i n t e r f a c e f r i c t i o n . T h i s form of c o u p l i n g between l a y e r s has been used by Rhines and Young (1982) to p a r a m e t e r i z e the v e r t i c a l t r a n s f e r o f momentum by eddy a c t i v i t y . P a r t i c u l a r l y i n t e r e s t i n g i s the f a c t t h a t (2.22) i s the o n l y drag f r i c t i o n f o r m u l a t i o n which a l l o w s the h o m o g e n i z a t i o n of p o t e n t i a l v o r t i c i t y i n the i n t e r m e d i a t e l a y e r s because i n 2 the q u a s i - g e o s t r o p h i c a p p r o x i m a t i o n £^ = VH^k s o t ^ l a t (2.22) i s e q u i v a l e n t to a l a t e r a l d i f f u s i o n of p o t e n t i a l v o r t i c i t y . When l a t e r a l d i f f u s i o n of buoyancy i s i n c l u d e d and p a r a m e t e r i z e d a c c o r d i n g to (2.19b) a comparison between (2.22) and (2.19b) r e v e a l s t h a t l a t e r a l d i f f u s i o n of p o t e n t i a l v o r t i c i t y by buoyancy d i f f u s i v e p r o c e s s e s i s - 25 -2 H k K H f 0 2 "T j « 0(10 ) times g r e a t e r than the i n t e r f a c i a l drag v sk+1/2 f r i c t i o n , so i n t h i s case i t i s j u s t i f i e d to n e g l e c t i n t e r f a c i a l s t r e s s as a way of p a r a m e t e r i z i n g the v e r t i c a l t r a n s f e r of momentum by e d d i e s . T y p i c a l v a l u e s of the c o e f f i c i e n t s , Ky and Ay, i n m i d - l a t i t u d e ocean are K H * 1 0 2 m 2 s " 1 , Ay » 10" 3 m 2 s _ 1 f o r an Ekman l a y e r depth &e = ( 2 A v / f Q ) 1 ^ 2 of the o r d e r 5-10 m. S u b s t i t u t i n g ( 2 . 1 7 ) , (2.19) and (2.22) i n t o (2.8a) y i e l d s : D T ^k - *n V I H T ( — i i ] + H V H ? K z 8k+1/2 8k-1/2 + ^r- Q ( k ) + T ( k ) > (2.23) ^k where n 1 k = JL_ + i _ + 9 D£ 8t u k 3x v k 3y 9 *k 9 *k U k = - ay"' V k = ax" < 2* 2 4> Thus, e q u a t i o n (2.23) i s the e q u a t i o n of p o t e n t i a l v o r t i c i t y f o r an ocean whose v e r t i c a l s t r u c t u r e i s composed of N - l a y e r s of c o n s t a n t d e n s i t y . To s o l v e ( 2 . 2 3 ) , i t i s n e c e s s a r y to a p p l y the boundary c o n s t r a i n t s ( 2 . 1 2 a ) , (2.13) and (2.14) i n a form s u i t a b l e f o r a l a y e r e d ocean. In doing so, one can - 26 -o b t a i n another v e r s i o n of ( 2 . 2 3 ) , which i s more c o n v e n i e n t f o r n u m e r i c a l i m p l e m e n t a t i o n by i n c o r p o r a t i n g the s u r f a c e and bottom c o n s t r a i n t s i n t o the 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 and thus l e a v i n g as boundary c o n d i t i o n s to be imposed the l a t e r a l c o n d i t i o n s ( 2 . 1 2 ) . The v e r t i c a l v e l o c i t y o f the K-th l a y e r i s g i v e n by the d i f f e r e n c e between the v e r t i c a l v e l o c i t i e s of the upper and lower i n t e r f a c e s . Thus, W k = Wk-1/2 - Wk+1/2- (2.25a) From (2.8b) one o b t a i n s J D ' / f0 , N f 0 K H _2. ag f 0 n l w k - i r F T ( ^ V - —ZT V z - C ^k+1/2 IN N p N I t i s now c l e a r , by u s i n g (2.16) and t a k i n g ,D' = _9_ 3 \ ± 1/2 _9_ 8 \ ± 1 12 _9_ ^D't;k±112 9t " 9y 9x 9x 9y ' t h a t (2.25b) becomes: - 27 -+ f 0 T / x f 0 g t - j - J ( * k + 1 ~ V *k+1/2 } - g ^ J<*k ' * k - V +k-1/2 ) k + 2 k~2 „ 7 2 r f 0 r V l " *k *k " i , f 0 n ( k ) , 9 0 , , - K H V l Z [ 1 1 H HT Q ' ( 2 . 2 5 c ) K 8k+1/2 8k-1/2 fc where j / \ N N 9 A 8 B 3 A 3 B . _ u T , . _ , J(A,B) = 9 x 9 y - 8 y 8 x i s the J a c o b i a n . The stream f u n c t i o n s +^1/2 a r e t a k e n a s t n e i n t e r p o l a t i o n v a l u e s of ^k+1 ' ^k a n <^ ^k 1 " Thus, f o r example, *k±1/2 = Ck±1/2 *k+1 " ( 1 _ Ck+1/2 } V (2.26a,b) ' k + 1 / 2 H k + H k + 1 I t i s a s i m p l e e x e r c i s e to show, by u s i n g ( 2 . 2 6 ) , t h a t f 0 f 0 J ( V n - K> - TTT J(K - K_i . K_i /o) g 1 " " ^ k + l *V "VM' i 1 ~ J V H , k " vk-1 ' ^k-1/2 k + 2 k " 2 ^k+1 " ^k ^k " +k-1 J [ f 0 ( - ^ ^ - -|, p - ) , + k ] . (2.27) Hence, (2.25c) becomes: - 28 -D ' k r . ,*k+1 ~ \ \ - *k-1„ k + ^ k - 7 2 *k+1 " *k *k " *k-1 f0 Q ( k ) - K H V H tV-nr—r ' i )] + ~^k~' ( 2 , 2 8 ) k + 2 k " 2 A f t e r combining the l a s t e q u a t i o n and (2.25) we can w r i t e (2.23) as: ( ? 1 + f ) = A R V 2 5 1 . ^ w l + 1 / 2 + x<1> D' f D-T ( ? k + f ) = A H V H ^ k + < Wk-1/2 - W k + 1/2> + T ( k ) ' 1 < k < N D^T <?N + F ) = A H V H % + WN-1/2 + ^ ( N ) ' < 2' 2 9> where x ^ ^ and x^ N^ are g i v e n by the f o l l o w i n g e x p r e s s i o n s x<1> = w + T ( 1 )  T H 1 W 1 / 2 + i _(N) _ f o (N) _ wN+1/2 + T * (2.30a,b) and T^N^ are the i n t e r f a c e s t r e s s at the upper and bottom l a y e r s , r e s p e c t i v e l y , and t h e i r f o r m u l a t i o n s i n terms o f the r e l a t i v e v o r t i c i t y are g i v e n by ( 2 . 2 8 ) . w-| +1/2 - 29 -and wn + 1/2 are the v e r t i c a l v e l o c i t i e s at the s u r f a c e and on the bottom, r e s p e c t i v e l y , t h a t e n t e r the problem as the boundary c o n d i t i o n s ( 2 . 1 2 b , c ) . So, w i t h the h e l p of (2.22) and (2.12b,c) x ( D and xW can be w r i t t e n as (1) < 7 A r \ + A v ^2 - g1 x = T; + TT- n : — r r — H-j + H 2 ' * W " " ^ (3» • ™ B + " N > + ^ < ^ % J ) - ( 2 - 3 ' a ' b ) (k) T v y , the i n t e r f a c e s t r e s s at the K-th l a y e r , i s g i v e n by (2.1 2 b ) . The c o e f f i c i e n t y i n (2.31b) i s expressed i n terms of the eddy v i s c o s i t y c o e f f i c i e n t A v and the parameter f n as ( P e d l o s k y , 1979): . 1 , 2 A v . 1/2 Y = j ( f — > Z r 0 The v e r t i c a l v e l o c i t y at the l e v e l j + 1/2 ( j = 1,2...N-1) i n (2.29) are computed from (2.8b) by u s i n g (2.16) as w j + i / 2 = g ' ° 1 i l t (^j+1 " * j ) " J ^ j + 1 " *j» *. . ^ j + T J 2 + K H V 2 - + ^ Q ( j ) . (2.32) - 30 -In g e n e r a l , Q ( j ) i s taken to be zero f o r j > 1. For i = 1 p0 C p H 1 (2.33) where Q s i s the a m p l i t u d e of the s u r f a c e heat f l u x . Now, the boundary c o n d i t i o n s of the problem (2.29) and (2.32) are the l a t e r a l boundary c o n d i t i o n s and the mass c o n s e r v a t i o n c o n s t r a i n t ( 2 . 3 1 ) . As f o r the l a t e r a l c o n s t r a i n t s , we have c a r r i e d out experiments w i t h the f r e e - s l i p and n o n - s l i p boundary c o n d i t i o n s , i . e . , F r e e - s l i p ^ ( x . y ) C k ( t ) , (x,y) £ a 0, rC XJ J 2 • • • N (2.34a,b) Non-slip * k(x,y) (x,y) £ 8 B = 0 or ? k # 0, k = XJ y 2 • • » N (2.35a,b) 3 n As a l r e a d y d i s c u s s e d i n the p r e v i o u s p a r agraph, the f r e e - s l i p c o n d i t i o n l e a d s to i n c o m p a t i b i l i t i e s i n e q u a t i o n s (2.29) u n l e s s the buoyancy d i f f u s i v e p r o c e s s e s a re n e g l e c t e d . Such - 31 -i n c o m p a t i b i l i t i e s do not e x i s t when n o n - s l i p c o n s t r a i n t s are used. Other more s o p h i s t i c a t e d forms of boundary c o n d i t i o n s have been used by M a r s h a l l (1982) and I e r l e y and Young ( ( 1 9 8 6 ) , p e r s o n a l communication). The e q u a t i o n s f o r a t w o - l a y e r ocean w i t h no ther m a l p r o c e s s e s , no s t r e s s at the i n t e r f a c e and c o n s t a n t depth H are r e a d i l y o b t a i n e d from (2.29 - 2.32) as 3? 1 3 ^ 2 f Q ( V A T W ) Z a t " " J ( ? 1 ' * 1 } - 3 a i " + A H V H <M ~ H7 W,1 + H 7 2 1 (2.36a,b) 3 ? 9 a^ 9 9 f n a t " " J ( ? 2 ' *2> - 3 3 i T + A H V H «2 - H T W , 1 " £ ?2 ' z 1 2 W f 0 , 3 '2 1 j 2 (2.36c) f 0 Y where e = — ^ — i s the l i n e a r bottom f r i c t i o n c o e f f i c i e n t . H 2 E q u a t i o n s (2.36a,b) are the r e l a t i v e v o r t i c i t y e q u a t i o n s 2 2 which are c l o s e d w i t h the r e l a t i o n s £^  = V ^ i ^ , ? 2 = V^^. In s o l v i n g (2.36) the d i f f i c u l t i e s a r i s e from the term w . . 4 A way of overcoming such a problem, p a r t i c u l a r l y when a time e x p l i c i t scheme i s used, i s by fo r m i n g the modal e q u a t i o n s . - 32 -The e q u a t i o n f o r the b a r o c l i n i c mode ¥ = - ^ 2 i s determined by s u b t r a c t i n g (2.36b) from (2.36a) and u s i n g ( 2 . 3 6 c ) . Thus (V 2 - X 2) v = Y a , in D, (2.37a) where Y a = » * i > - J ( ? 2 , ^ 2 ) ~ ^ ( t1 " V 2 T , , , , x , ^ ' z , A „ 2 X JCipj-il^, * ^ + g + A HV R C - C2) + E ? 2 1 i 1 X2 = Hf 2/HiH 2g' . (Note that X"1 i s the internal radius of deformation). The mass conservation constraint (2.15) together with (2.36c) y i e l d // 1=- (?) dx dy = 0. (2.37b) D From the boundary conditions either (2.34a) or (2.35b) we have that ( v ) = C ( t ) on n R . ( 2 . 3 8 ) To solve (2.37a) and (2.37b) we l e t - 33 -I i = 3 t ^ + ^ 3 ^ ' ^ 2 - 3 9 ) where 2 2 3 ? a <VH - X > TtT = Y a i n D> I t *a * °« ° n Q B 9 /3 m \ 0 if N.S.C. n , n N 3h <3t V = free if F . S . C ° n V ( 2 ' 4 0 a ) and It *b • ^ o n V 3 ,3 . . 0 i f N-S.C. . / 0 / n u N 3r7 <at * a ) = f r e e i f F - S . C o n V (2.40b) The c o n s t a n t C ( t ) i s determined by u s i n g (2.37b) and (2.39) as 3¥ // dx dy C ( t ) = - — ^ . (2.41) // dx dy H i|> + H ?* 9 The e q u a t i o n f o r the b a r o t r o p i c mode $ = —— !— 5—^—=-i s V H ( | | ) - Y b, (2.42a) - 34 -where 1 ( V A T K Y b = [tL, J ( ? 1 ,1^) + H 2 J ( C 2 ' + 2 ) ] i r + H + I V H ( H I * I + H 2 + 2 > J i " ir 4-The boundary c o n d i t i o n s of (2.41a) are I I - 0, on n H , 3 /3 x\ _ 0 i s N-S.C. ^„ / 0 /ou\ 3n ( 3 t ^ " f r e e i f F - S . C o n H * (2.42b) - 35 -CHAPTER 3 NUMERICAL FORMULATION In t h i s c h a p t e r , we d e s c r i b e d the n u m e r i c a l f o r m u l a t i o n o f the q u a s i - g e o s t r o p h i c e q u a t i o n s of a t w o - l a y e r ocean by f i n i t e element. E x t e n s i o n to a n - l a y e r ocean i s o n l y a m a t t e r of adding more e q u a t i o n s t o be s o l v e d ; however, the n u m e r i c a l b a s i s remains u n a l t e r e d . Our n u m e r i c a l f o r m u l a t i o n uses f i n i t e element to d i s c r e t i z e the continuum e q u a t i o n s i n space and f i n i t e d i f f e r e n c e s to step forward i n time the c o m p u t a t i o n s . We f i r s t b e g i n i n t r o d u c i n g some n o t a t i o n s and d e f i n i t i o n s , which are s t a n d a r d i n the c o n t e x t of f i n i t e element t h e o r y . Then we t r a n s f o r m the e q u a t i o n s i n t o an i n t e g r a l form which p e r m i t s o b t a i n i n g the weak s o l u t i o n o f such e q u a t i o n s . The next s t e p c o n s i s t s of p a s s i n g from a continuum f o r m u l a t i o n of the e q u a t i o n s to a d i s c r e t e one which i s s u i t a b l e f o r n u m e r i c a l c o m p u t a t i o n . T h i s i s c a r r i e d out by G a l e r k i n ' s method and the f i n i t e element t e c h n i q u e o f p a r t i t i o n i n g the domain D i n t o a f i n i t e number E of almost d i s j o i n t subdomains T e and i n t e r p o l a t i n g any f u n c t i o n d e f i n e d i n D. A key r o l e i n our method i s p l a y e d by the boundary spaceMh which a l l o w s the c o r r e c t i m p o s i t i o n of the n o - s l i p boundary c o n d i t i o n . To c o n s t r u c t such a space, we d e f i n e our s o l u t i o n space i n the form suggested by G l o w i n s k i et a l . (1985) to study the steady s t a t e stream - 36 -f u n c t i o n - v o r t i c i t y f o r m u l a t i o n o f the N a v i e r - S t o k e s e q u a t i o n s . 3.1 Basic Notations and Definitions B e f o r e embarking i n the c o n s t r u c t i o n of the v a r i a t i o n a l f o r m u l a t i o n o f the e q u a t i o n s to compute t h e i r weak s o l u t i o n , i t i s u s e f u l to w r i t e down the d e f i n i t i o n s of m a t h e m a t i c a l spaces on which we a r e o p e r a t i n g and some n o t a t i o n s t h a t we use throughout t h i s c h a p t e r . Q denotes an open bounded domain i n the n - d i m e n s i o n a l E u c l i d e a n space R n w i t h boundary r . We always assume t h a t Q i s "w e l l - b e h a v e d " and t h a t r i s at l e a s t L i p s c h i t z i a n . r i s L i p s c h i t z i a n i f i n a neighborhood of a g i v e n p o i n t x 1 € r , |x - x 1 j < ^  , r can be r e p r e s e n t e d as x^ = F(x^ , x ^ ) , F b e i n g c o n t i n u o u s mapping and ^ s u f f i c i e n t l y s m a l l . Q T = Q U [ 0 , T ] where [ 0 , T ] i s a time i n t e r v a l . P o i n t s i n Q or R n are denoted by x = (x.j , x 2 , . . . ,x ) whereas p o i n t s b e l o n g i n g to Q T are w r i t t e n as ( x , t ) . An element of s u r f a c e o f e i t h e r Q or Q-p i s denoted by dx and an element of a r c by ds . Le t u be a smooth f u n c t i o n d e f i n e d i n Qx- We sometimes use m u l t i - i n d e x n o t a t i o n to r e p r e s e n t the d e r i v a t i v e s of u; t h a t i s , l e t a = ( a-| , a 2 , . . ., a n) be an n - t u p l e of n o n - n e g a t i v e i n t e g e r s and denote - 37 -j a j = a^ + a 2 + ..• + aN« Then by D a u, we s h a l l mean the a-th d e r i v a t i v e of u d e f i n e d by 3 H u D u = - a1 a 2 aN 3 x ^ 3^ 2 • • » 3 C M (Q T) = the l i n e a r space c o n s i s t i n g of a l l f u n c t i o n s u w i t h p a r t i a l d e r i v a t i v e s D au, 0 _< j a j _< m c o n t i n u o u s on Qx. C°° (Q-j.) = the l i n e a r space of f u n c t i o n s i n f i n i t e l y d i f f e r e n t i a b l e i n Q-p . C Q ( Q t ) , C ™ ( Q t ) = l i n e a r subspaces of C M ( Q T ) and CO C ( Q T ) , r e s p e c t i v e l y , such t h a t i f u € C M ( Q T ) , C Q ( Q t ) then u ( x , t ) = 0 V- ( x , t ) € C M ( Q T ) , C " ( Q T ) . C M (Q T) = l i n e a r subspace of CM ( Q ) such t h a t i f u C M ( Q ) then D | a | u i s bounded and u n i f o r m l y c o n t i n u o u s i n Q^. P L (Q.J.) = space of f u n c t i o n s u d e f i n e d on such t h a t J Q | U | P dx < 0 < t < T p The norm of L (Q T) i s d e f i n e d as , | U , IP = ^ q | u | P d x > 1 / P ' 1 < p < 0 0 ' - 38 -2 For p = 2, L ( Q T ) i s a H i l b e r t space w i t h i n n e r p r o d u c t : ( u , v ) 2 = J Q U v dx. L e t m be a n o n - n e g a t i v e i n t e g e r and l e t p s a t i s f y 1 <. P <. °°« The Sobolev space VjP ( Q ^ , ) o f o r d e r (m,p) i s d e f i n e d as: W P = {u|D au e L p ( Q T ) , 0 < a < m} . The spaces are g e n e r a l l y endowed w i t h the norm Bull 0 jP = Cf E l D a u l P ^ 1 / P 1 < P < m'P'Q T J Q 0 < |1| < m I " I } ' t e [O.T] W £ « T ) C ( Q T ) = {u € W P | u | r = 0} f o r p = 2. V T = H M and W Z = H ™ r m m 0 H m and H™ are s e p a r a b l e H i l b e r t spaces w i t h i n n e r p r o d u c t (u,v) = J n E D u D v dx 4 0 < I a | < m and w i t h the a s s o c i a t e d norm lull n = null n n m,Q T m,2 , Q T (See Adams, 1 975). - 39 -3.2 V a r i a t i o n a l Formulation For the sake of completeness, we w r i t e down a g a i n the e q u a t i o n s (2.37) and (2.41) whose weak s o l u t i o n s we w i s h to compute. S i n c e the v o r t i c i t y and modal e q u a t i o n s are l i n k e d t o g e t h e r by the c o r r e s p o n d i n g (ip,c) P o i s s o n e q u a t i o n s , we need to set up an a s s o c i a t e d v a r i a t i o n a l f o r m u l a t i o n of (2.37) and (2.41) of mixed type i n o r d e r to use l i n e a r f i n i t e e l ements. A c t u a l l y , we c o u l d compute the weak s o l u t i o n s from the modal e q u a t i o n s a l o n e , but i n t h i s c a s e , we s h o u l d use f i n i t e elements o f c l a s s (Q T ) which are e x p e n s i v e from a c o m p u t a t i o n a l p o i n t of view, a l t h o u g h the s o l u t i o n i s more a c c u r a t e ; hence, a mixed type v a r i a t i o n a l f o r m u l a t i o n r e p r e s e n t s a compromise between c o m p u t a t i o n a l burden and n u m e r i c a l a c c u r a c y . An a s s o c i a t e d v a r i a t i o n a l f o r m u l a t i o n (or more p r o p e r l y a f i n i t e element method) i s of mixed type whenever independent a p p r o x i m a t i o n s are used f o r both the dependent v a r i a b l e and i t s d e r i v a t i v e s . In our p a r t i c u l a r problem, the independent v a r i a b l e i s the stream f u n c t i o n ( i = 1 , 2 ) ; however, we chose to s u b s t i t u t e v2i|)^ = ^ and the r e s u l t i s two new e q u a t i o n s ( v o r t i c i t y and modal e q u a t i o n s ) of lower o r d e r which are approximated i n d e p e n d e n t l y . Le t us c o n s i d e r the v o r t i c i t y - m o d a l e q u a t i o n s f o r the t w o - l a y e r ocean - 40 -I t " = J ( ? i ' +i> - 0 air + nf wz + A H V H <=i + 7 (1 + (-D1) ec. + \ (1 - > ( 3 > 1 ) i n Q T { ] Z . , Z. = { ( x , y , H.(x,y))£ R j } , i=1 ,2 where f Q 3 ( ^ 1 - i |> 2 ) w 2 = g"1- i 3T + J ( * 1 " *2' *1 1 / 2 ^ ? i | r = °iCx,y,t), c i1 1 = 0 = 0. (3.2) ai can be zero ( f r e e - s l i p boundary c o n d i t i o n ) or non-zero ( n o n - s l i p boundary c o n d i t i o n ) , or zero on some r - | C r and d i s t i n c t from zero on T2 C r such t h a t r i U T2 = r . ( V 2 - X 2 ) V = T A i n Q T , (3.3) * a | r - °> * a | t - o = ° ' ( 3 ' 4 ) ( V 2 - X 2 ) T b = 0 i n Q T , (3.5) * b | r * 1 ' ( 3 ' 6 ) f = V'a + C ( t ) f b , (3.7) / o r a d x = - ll f • dx> <3'8> - 41 -where ¥ ' - It: <*1 - *2>> X 2 J ( * 1 - *2- * P + A H V H ( ? 1 - ?2> - ^2 + TT !2 1 VH = Y b i n Q T (3.9) 0 *'|t=0 = °' (3.10) where 9 9t ^ H ^' Y b = rl ( H 1 J ^ 1 »*1> + H 2 J ^ •+2 ) ) - B h ( H L * 1 H H ^ 2 ) 2 r H 1 g 1 + H 2 % H 2 " F + AH VH ( i q > " e H - ^2 H * F o l l o w i n g G l o w i n s k i et a l . (1985), we d e f i n e two new NS FS spaces W and W as f o l l o w s : r a a W^S = {(6,*) e H 1 ( Q T ) x H 1 ( Q ? ) , t | r = C ( t ) - J Q T V ' V H * D X = / Q t 9 q d x ~ / R - § H D S ' ^ € H 1 < Q T ^ - 42 -,NS W*'S = ( ( e , t ) e Hj ( Q T ) x H 1 ( Q T ) , * | r = C ( t ) - J r | 1 d s , Vq 6 H 1 ( Q T ) } VJ'^ ° i s t h e s p a c e o f s o l u t i o n s f o r t h e n o n - s l i p b o u n d a r y c o n d i t i o n w h i l e W ' i s f o r t h e f r e e - s l i p c o n d i t i o n . An a s s o c i a t e d m i x e d v a r i a t i o n a l f o r m u l a t i o n o f e q u a t i o n s ( 3 . 1 - 3 . 1 0 ) i s : F i n d 6 Wl<, (K = N S , FS) s u c h t h a t ¥ ( 9 , * ) e HQ X HQ , (jT-> 9 ) = ( J C C i - ^ ) , e) - (B ^ i , 0 ) + ( ( - 1 ) 1 £o ^ 2 , 9 ) + ( A H V 2 C . , 6) + 1 (1 + ( c . f e ) + C 1^ 1^ 1 1-) ( j j - , e) i n Q T / l Z i (1-1,2) c i r = ° i ( V H ^a - X ?~ •> = < V • ) , i n Q , R N Z . Y' =() a r ' ( V H y b " x 2 * b ' * } = °> i n Q T ft ^ V,' = 1 b r ( V H * ' . * ) = ( Y b , * ) , i n g T Z 4>' r = o, - 43 -w h e r e ( u , v ) = / Q u • v dx . I n t e g r a t i n g by p a r t s t h o s e t e r m s o p e r a t e d by V 2 y i e l d s 3 c i ^ i i f 0 ( T F ~ ' 0 ) = ( J ( c i - * i> . o) - (P J F 1 , e) + ( ( - 1 ) ^ w 2, 8) - ( A H V H C l , V H 0) + 1 (1 + ( - 1 ) 1 ) ( ? . , 9) + ° " ^ 1 ) L 0 ) , ( 3 . 1 1 ) L I r i ' - ( V H V R 4 ) - ( A 2 4) = ( Y a , 4) , * ; | r = 0 , ( 3 . 1 2 a ) - ( V H H'^ , V H 4 ) - ( A 2 * a > 4) = 0 , ^blr = 0 , ( 3 . 1 2 b ) - ( V H *' , V H 4 ) = ( Y b , *) , | r = 0 , ( 3 . 1 3 ) N o t i c e t h a t t h e l a r g e s t o r d e r o f the d e r i v a t i v e s i n ( 3 . 1 1 ) ( 3 . 1 3 ) i s f i r s t o r d e r i n c o n t r a s t w i t h t h e s e c o n d o r d e r d e r i v a t i v e s o f t h e o r i g i n a l e q u a t i o n s ( 3 . 1 ) - ( 3 . 1 0 ) ; t h i s means t h a t we can r e l a x t h e s m o o t h n e s s r e q u i r e m e n t s o f t h e s o l u t i o n s o f ( 3 . 1 1 ) - ( 3 . 1 3 ) w i t h r e s p e c t to t h o s e r e q u i r e m e n t s o f t h e c l a s s i c a l s o l u t i o n s o f ( 3 . 1 ) - ( 3 . 1 0 ) . T h u s , w h e r e a s t h e c l a s s i c a l s o l u t i o n s a r e to be o f c l a s s - 44 -C ( Q T)» t n e s o l u t i o n of the v a r i a t i o n a l f o r m u l a t i o n belongs t o H 1 (Q T ) , but C 2 (Q T ) C H 1 (Q T ) , so we have expanded the space of f u n c t i o n s t h a t can be s o l u t i o n of (3.1) - ( 3 . 1 0 ) . I t i s c l e a r t h a t i f an i n i t i a l boundary v a l u e problem of e l l i p t i c a l and p a r a b o l i c type has a c l a s s i c s o l u t i o n i t a l s o has a weak s o l u t i o n because every c l a s s i c s o l u t i o n i s a l s o a weak s o l u t i o n ; however, t h e r e may be weak s o l u t i o n s which are not c l a s s i c a l s o l u t i o n s , ' a l t h o u g h e q u a t i o n ( 3 . 1 1 ) , which i s p a r a b o l i c , and e q u a t i o n s (3.12) and ( 3 . 1 3 ) , both of which are e l l i p t i c , have unique weak s o l u t i o n s which, under c e r t a i n r e g u l a r i t y r e q u i r e m e n t s f o r the c o e f f i c i e n t s and f o r c i n g f u n c t i o n F, c o i n c i d e w i t h the c l a s s i c s o l u t i o n s . (See V.P. M i j a i l o v , 1977). Bennet and Kloeden (1981) have shown the e x i s t e n c e and uniqueness o f the c l a s s i c a l s o l u t i o n o f the d i s s i p a t i v e q u a s i - g e o s t r o p h i c e q u a t i o n s w i t h p e r i o d i c boundry c o n d i t i o n s . 3.3 G a l e r k i n - F i n i t e Element Method G a l e r k i n 1 s method c o n s i s t s of a p p r o x i m a t i n g the s o l u t i o n s o f the e q u a t i o n s (3.11) - (3.13) through the s o l u t i o n s C i h , <Kh °f t n e p r o j e c t i o n s of (3.11) - (3.13) onto the f i n i t e d i m e n s i o n a l space We a s s i g n a s u b s c r i p t h to the f i n i t e d i m e n s i o n a l subspaces and d i s c r e t e v a r i a b l e s to imply t h a t t h e i r p r o p e r t i e s g e n e r a l l y depend on some r e a l parameter h (such as mesh s i z e ) such t h a t , as h d e c r e a s e s , - 45 -the f i n i t e d i m e n s i o n a l spaces tend to f i l l up the c o r r e s p o n d i n g i n f i n i t e s paces. To be more s p e c i f i c , assume t h a t { e . ( x , t ) } €. C ° ( Q T ) i s a complete set of l i n e a r l y 1 i=1 ,2 1 d i m e n s i o n a l subspace, say V H ( Q T ) C L ( Q T ) , G a l e r k i n ' s method i s to seek a f u n c t i o n w^(x,t) 6 V^(Q T)» which i s the o r t h o g o n a l p r o j e c t i o n of u « ( Q T ) on V n, such t h a t wh(x,t) s a t i s f i e s the o r i g i n a l p a r t i a l d i f f e r e n t i a l e q u a t i o n i n Vh. as w e l l as the i n i t i a l and boundary c o n d i t i o n s p r o j e c t e d from ( Q T ) onto Vh« T h i s p r o c e d u r e cannot be a g e n e r a l method f o r c o n s t r u c t i n g the f i n i t e d i m e n s i o n a l subspaces. I t i s at t h i s j u n c t u r e t h a t the f i n i t e element method (f.e.m.) p l a y s i t s r o l e as an a p p r o x i m a t i o n method, f i r s t by p a r t i t i o n i n g the domain Q i n t o a f i n i t e number E of almost d i s j o i n t subdomains Te, and s e c o n d l y , by a p p r o x i m a t i n g any f u n c t i o n d e f i n e on Q from i t s v a l u e s at the v e r t i c e s of the subdomains, Te. Thus, c o n s i d e r t h a t the bounded domain Q has been d i v i d e d i n t o a f i n i t e number of almost d i s j o i n t subdomains Te. These subdomains form f o r a g i v e n h ( t h e mesh s i z e ) a f i n i t e f a m i l y {^h} such t h a t independent f u n c t i o n s i n ( Q T ) t h a t g e n e r a t e a f i n i t e E ( i ) Q = connected model of Q = U Te, e Q T = Q U [ 0 , T ] ; - 46 -( i i ) V Te, Te' e 5 " . , ^, Te # Te' , we have e i t h e r Te H Te 1 = (empty s e t ) , or Te, Te 1 have o n l y one v e r t e x i n common, or Te, Te' have o n l y a whole edge i n common. Once the continuum domain 07 = QUr has been p a r t i t i o n e d , 1 1 NS FS we approximate the spaces H (Q T) , H Q ( Q T ) , W Q , W o by the f i n i t e d i m e n s i o n a l s p a c e s . H n = K K e C° ( V ' v n | T e € P K ' ¥ T e £ ^ n H 0 h = H h ° H o ( Q T } = { v h l v h e E l v h | r = °}» W a h = f < V V € H 0 h x Hh> *h|r - C h ^ ) 3$, ds " ' H \ * V h " /" V h " ' r -JT- • * % e Hh>-Pk. i s the space o f p o l y n o m i a l s i n (x,y) of degree _< K, so Vh|T ^ ^k m e a n s t n a t t n e f u n c t i o n s of the d i s c r e t e spaces are p o l y n o m i a l s i n (x,y) of degree < K on the subdomains Te. I t f o l l o w s t h a t N Dh = number of v e r t i c e s of 5~h which are i n t e r i o r to Q T > i s the dimensi o n of H Q ^ w h i l e the dimension of i s eq u a l to the number of of v e r t i c e s of 9"^  which b e l o n g to Q T« - 47 -C ^ ( t ) i s a c o n v e n i e n t a p p r o x i m a t i o n to C ( t ) i n H h ( Q T ) • I n s p i r e d by G l o w i n s k i and P i r o n n e a u (1979), we b u i l d up an a d d i t i o n a l f i n i t e d i m e n s i o n a l space M^ i n o r d e r to impose p r o p e r l y the n o - s l i p boundary c o n d i t i o n on the v o r t i c i t y e q u a t i o n . Mft i s the complementary space o f H^ h i n (Q T) , i . e . , M h C H1 ( Q T ) , H h = H 0 h © M h ' Hence, V C i h € H,* ( Q T ) , ( i = 1,2), which v e r i f i e s t h a t ' i h c n h ^T' ? i h - ( 4 h - ? i h > + where Hh ^ Mh' * i h " Hh € Hoh' So i s the component o f the r e l a t i v e v o r t i c i t i e s ? ^ i n NS M^. The r e s t r i c t i o n of the i n t e g r a b i l i t y p r o p e r t y of W^ ^ i n Mh i s r1 „1 V (6ih^ih) £ Hh x Hh> ^ Mh - '~ VH +ih * VH ^h = / ~ T 9 i h *h " ' r I F " *h d s ( i = 1 ' 2 ) « (3.14) - 48 -C o n f i n i n g o u r s e l v e s to Lagrange-type elements o f o r d e r (K=1 i n P^ .) we generate the g l o b a l b a s i s f u n c t i o n s v^ e (Q^.) from l o c a l a p p r o x i m a t i o n v i a the f o r m u l a E Ne ( e^N , v v, . = U Z a. v J . e J (x) h i e = 1 N = 1 .] where j = 1,2...Nh Ne i s the number of v e r t i c e s of a g i v e n Te (e) v 'N i s the Boolean t r a n s f o r m a t i o n m a t r i x f o r the j element Te, v^ e^ i s the l o c a l i n t e r p o l a t i o n f u n c t i o n c o r r e s p o n d i n g to the element Te. The G a l e r k i n a p p r o x i m a t i o n s , a r e w r i t t e n as N h ? i h ^ - Jt = 1 ? i n A ( t ) v h * ( x > ' x * H h N h * i h - ^ * i n t ^ ) V h ^ x > ' < i = 1> 2> < 3' 1 5> R e c a l l i n g the r e l a t i o n s h i p s among b a r o t r o p i c and b a r o c l i n i c modes and stream f u n c t i o n s , ty-\ and ^ 2, as w e l l as r e l a t i o n s h i p s among ty^ and ^  and i|> ^  we can w r i t e 12" - 49 -N h % h ( x , t ) = z , . h i ( t ) V l u ( x - ) , N h _ H 2 »1 h + H1 *2h * 4 h — — • ( 3 - 1 6 ) N h The v a r i a t i o n a l f o r m u l a t i o n (3.11) - (3.13) i s now approximated as: F i n d { c i h , * l h } € W a h such t h a t V * h e Hh« ^ i h - ? I h £ H o h ' 4 e V 3 ? i h H - u ( —> V = <J ( 5 l h . * i h ) , « h ) - (3 — 1 £ , $ h ) f + ( ( - D L H 7 w 2 h ' *h> - < A H V l h ' V H V + i ( 1 + ( " 1 ) i > < e «1h' V + ( 1 - ( - 1 > L ) ( F _ , #h) i n Q T h a Z., (3.17a) - 50 -" /" VH * i h * V h = j~ C i h vn + 'r I n " ^ h d s> ^Th ^Th (3.17b) ' a h k = °> *ah 11-0 = °> < 3- 1 8 a> h " ( V H ^bh' VH V " < X* *bh' V = 0 i n Z i > * b h | r = 1, (3.18b) h " ( VH *h- V r l V = <* b h, V i n Q Th/1 Z i > *h|ih= ° > *h|t=0 = °' ( 3 - 1 9 ) Upon s u b s t i t u t i n g (3.15) and (3.16) i n t o (3.17) -( 3 . 1 9 ) , we o b t a i n the f o l l o w i n g i m p l i c i t systems o f o r d i n a r y d i f f e r e n t i a l e q u a t i o n s f o r the weight f u n c t i o n s : * i h = U l h 4 } . * i h = 1 W « (£=1,2,...N h). ( i ) R e l a t i v e v o r t i c i t y e q u a t i o n s : M ^ i h " J h < * i h ' * i h ' B ) - A H K ^ i h + \ (1 + (-1) 1) e M l i h i n Q T h H V = a.u. (3.20) T , I _ 1 h rh - 51 -( i i ) B a r o c l i n i c mode e q u a t i o n s : - K ? a h - X 2 M $ a h = T a h i n Q T h H Z l f ^ a h l r h = ° - < 3' 2 1 a> - K l b h - X 2 M \ h = 0 i n Q o h n \ , *bhL - 1- ^ 3- 2 1 b) r h ( i i i ) B a r o t r o p i c mode e q u a t i o n s : - K $ h - ^ b h i n Q T H f\ Zt l h | r = 0 (3.22) An upper dot on the v a r i a b l e s stands f o r t i m e - d e r i v a t i v e , the "mass m a t r i x " M has the e n t r i e s / Q v h k v h j d x ( k » J = 1 ' 1 • • • N h ) ' and the e n t r i e s of the " s t i f f n e s s m a t r i x " K are g i v e n by /Q VH v h k * VH v h j d x> ( k , j = 1 , 2 , . . . N h ) . M i s symmetric and p o s i t i v e d e f i n i t e , whereas K i s symmetric o n l y . J ( ? ^ b , 1 ^ , 3) i n (3.20) i s the m a t r i x which i n c l u d e s the c o n t r i b u t i o n s of the J a c o b i a n and 3 terms of the e q u a t i o n s . Depending on the time scheme used to step forward i n time the e q u a t i o n s J can-be e i t h e r a column v e c t o r m a t r i x - 52 -or a skew symmetric square m a t r i x . J<*h i s a (2 x N n) column v e c t o r whose e n t r i e s a r e : lq fff w 2 h + 17 ( M ^ ) ' v h j d x = »h + TT V (J=1,2,.,.N h). The r i g h t - h a n d s i d e s Yah a n d Ybh °f the e q u a t i o n s (3.21a) and (3.21b), r e s p e c t i v e l y , are m a t r i c e s which came from the terms (yah» $ h ) i n (3.18a) and (ybh» $h) i n ( 3 . 1 9 ) . An i m p o r t a n t a s p e c t t h a t s t i l l remains to be e l u c i d a t e d i s the computation of i n ( 3 . 2 0 ) . I f we assume, a p r i o r i , the v a l i d i t y of the f r e e - s l i p boundary c o n d i t i o n on the s o l i d b o u n d a r i e s then we take equal to zero at the p o i n t s o f such b o u n d a r i e s and we go on w i t h the computation o f the s o l u t i o n s of (3.20); however, i f we wish to impose on the s o l i d b o u n d a r i e s the n o - s l i p c o n d i t i o n , then are unknown v a l u e s at the p o i n t s s i t u a t e d on the s o l i d b o u n d a r i e s t h a t have to be computed from the r e l a t i o n (3.17b) which i s h i g h l y dependent on the p r o p e r t i e s of the s p a c e M ^ , so we sh o u l d chose/H^ such t h a t the v a l u e s o f on the s o l i d b o u n d a r i e s were easy to compute from (3.17b). Let us f i r s t d e f i n e g e n e r a l p r o p e r t i e s t h a t we w i s h M n to have: ( i ) S i n c e M h i s a complementary space of H Q ^ , then we take dim (M h) = N, = No. boundary p o i n t s . - 53 -( i i ) We wi s h t h a t the support of Hh be as s m a l l as p o s s i b l e . A c o n v e n i e n t support would be M h * Hh = > y h | T e = 0 ¥ T e e f " h s u c h t h a t Te H r h = (empty s e t ) . ( i i i ) The above p r o p e r t i e s ( i ) and ( i i ) suggest how to d e f i n e a c a n o n i c a l b a s i s o f the space A l ^ . h BM = i W £ M £ = 1.2,...,Nb) such t h a t ( i i i - 1 ) W^C ( i i i - 2 ) w £ ( x \ ) = 6 £ j i f (it.) r , w A ( x . ) = o i f 5 i r . P r o p e r t i e s ( i ) to ( i i i ) are q u i t e g e n e r a l and do not depend on how the boundary and an i n t e r i o r narrow s t r i p o f a d j a c e n t to the boundary are d i s c r e t i z e d . They are a l s o independent of the e x i s t e n c e of t a n g e n t i a l v e l o c i t y on t h e boundary. In our p a r t i c u l a r problem, the t a n g e n t i a l v e l o c i t y i s zero on the s o l i d b o u n d a r i e s , so i f we r e q u i r e t h a t V - Te e T~h3 Te 0 r * (empty s e t ) , e i t h e r one s i d e o f Te i s normal to the boundary w i t h a v e r t e x on i t , or - 54 -two v e r t i c e s o f Te, one on the boundary another i n the i n t e r i o r , can be connected a l o n g the outward normal to the boundary. We can d e f i n e another p r o p e r t y such as ( i v ) — = 0 V x e r , where t i s the c l o c k w i s e tangent v e c t o r on the boundary. N o t i c e t h a t ( i v ) i s a p a r t i c u l a r p r o p e r t y i n the sense t h a t i t depends, f i r s t , on the t o p o l o g y of the d i s c r e t e space Q b c l o s e to the boundary and, s e c o n d l y , on the v a n i s h i n g of the t a n g e n t i a l v e l o c i t y on the boundary. The support of Mft i s then composed of the segments B i Pj Pj normal to the b o u n d a r i e s . See F i g . 2. In t h a t f i g u r e { P B } j denotes boundary p o i n t s and { P 1 } j are the i n t e r i o r p o i n t s c l o s e s t to the b o u n d a r i e s a l o n g the normal d i r e c t i o n . I t i s c l e a r t h a t (3.14) becomes J f i N 3 n 3 n d*h S Cih uh d V whe B „ I i h r e n„ = { P B P 1} h, By u s i n g L a g r a n g i a n l i n e a r b a s i s f u n c t i o n s , the above i n t e g r a l becomes - 54 . 1 -F i g . 2 S u p p o r t o f t h e s p a c e d i s c u s s e d i n S e c t i o n Mn 3 . 3 - 55 -'l,w = - T2 <*i,w - * l f I ) " . (3.23 *h where c . and i|>. are the v o r t i c i t y and stream f u n c t i o n v a l u e s , r e s p e c t i v e l y , at a g i v e n boundary p o i n t s ; C. T and ip. j are the same q u a n t i t i e s but taken at the 1,1 I , I c o r r e s p o n d i n g i n t e r i o r p o i n t s ; 1^ i s the d i s t a n c e a l o n g the normal d i r e c t i o n between a boundary p o i n t and i t s c o r r e s p o n d i n g i n t e r i o r p o i n t . E q u a t i o n (3.23) i s Woods' f o r m u l a t i o n f o r w a l l v o r t i c i t y , a commonly used second o r d e r a c c u r a c y a p p r o x i m a t i o n . Roache (1972) has p o i n t e d out t h a t such f o r m u l a t i o n f o r the w a l l v o r t i c i t y can y i e l d i n s t a b i l i t i e s at h i g h Reynolds numbers (Re >_ 300) . T h i s i s not our case because as we w i l l see l a t e r our Reynolds number i s Re = 50. The same f o r m u l a has been d e r i v e d by B a r r e t (1978) u s i n g a v a r i a t i o n a l p r i n c i p l e of Bateman's type f o r the stream f u n c t i o n v o r t i c i t y f o r m u l a t i o n of the N a v i e r - S t o k e s e q u a t i o n s w i t h n o - s l i p c o n d i t i o n s i n c o n j u n c t i o n w i t h f i n i t e element. 3.4 Time D i s c r e t i z a t i o n In o r d e r to c a r r y out the time i n t e g r a t i o n of e q u a t i o n s (3.20) - ( 3 . 2 2 ) , we use the l e a p f r o g scheme w i t h the - 56 -L a p l a c i a n f r i c t i o n a l terms lagged by one time s t e p and the bottom f r i c t i o n terms taken as the average of the forward and backward time steps to a v o i d l i n e a r n u m e r i c a l i n s t a b i l i t y . R i c h t myer (1967). The reasons f o r u s i n g such a scheme a r e : ( i ) I t i s e x p l i c i t so i t i s easy to implement, ( i i ) I t i s second o r d e r a c c u r a c y i n t i m e , ( i i i ) I t i s n o n - d i s s i p a t i v e . The l a t t e r r e a son i s more t h e o r e t i c a l r a t h e r than p r a c t i c a l , because the e x i s t e n c e of n u m e r i c a l phase d i s p e r s i o n , which i s u n a v o i d a b l e , i s e q u i v a l e n t to a damping mechanism (Roache (1 972 , pp. 56-60 and 80-81). A d i s a d v a n t a g e of the l e a p f r o g scheme i s i t s c o m p u t a t i o n a l mode (Arakawa-Messinger, 1976) which l e a d s , on one s i d e , to the s p l i t t i n g of the s o l u t i o n s and, on the o t h e r s i d e , i t a c t s as a t r i g g e r mechanism f o r the appearance of n o n - l i n e a r i n s t a b i l i t y , B r i g g s et a l . (1983). A l l these t r o u b l e s can be m i t i g a t e d by u s i n g every 50 time steps a r e s t a r t c o n s i s t i n g of t a k i n g (n+1/2) _ u< n + 1> + u<n> u 2 (n-1/2) _ u ( " ) + u ( n - 1 ) u 2 > where n i s the number of time s t e p , and then to c o n t i n u e the computation i n the u s u a l way. Sadourny (1975) r e p o r t s v e r y good r e s u l t s w i t h such a r e s t a r t , as f o r the n o n - l i n e a r i n s t a b i l i t y c o n c e r n s , i n the i n t e g r a t i o n of the - 57 -q u a s i - g e o s t r o p h i c e q u a t i o n s w i t h f i n i t e d i f f e r e n c e p o t e n t i a l v o r t i c i t y q u a s i - c o n s e r v a t i v e schemes i n space and l e a p f r o g scheme i n t i m e . K i l l w o r t h (1985) showed the good c h a r a c t e r i s t i c s of the above r e s t a r t when i t i s compared, i n terms of an e q u i v a l e n t f i l t e r , w i t h o t h e r s t a n d a r d r e s t a r t s i n g e o p h y s i c s . For every time s t e p , the e q u a t i o n s t h a t have to be s o l v e d a r e : M ( ? ^ + 1 ) - = 2 A t * ( n ) (n+1) (n+1) C = o (3.24) i h ' i h r *ah° |r = °* ( 3 * 2 5 ) I f n=1 we s o l v e (3.21b) .. " K ""h " Y b h • ->• ? £ n ) | r = 0. (3.26) *(n) = *(n) + £(n) h bh ah f$ _ $ N(n+D _ /* _ * x(n-1) Q M h T 2 h ; ^ 1 h *2h ; i ( n ) _ (Mlh + M2h)("+1> - ( H ^ 1 h + H ^ 2 h ) 2 At (n-1) -"1 y h 1 L 1 2 Y 2 h y 2At (3.27) - 58 -+ (n+1) -*-(n+1) t(n+1) _ H 2 ^1h + H 2 *hh 12 h w<J> = - - J(|{»> - + (n) ( 3 > 2 8 ) where t(n+1) + t ( n - 1 ) _•_ 1 / i / n i \ w / i h i h x . >(n) + j ( 1 + ) e M ( 2 ^ °P h ' (3.29) S i n c e the l e a p f r o g scheme i s a t h r e e - l e v e l one, we cannot s t a r t the computations w i t h i t , so we need another scheme, say, a t w o - l e v e l one. The most common scheme used f o r t h i s purpose i s the E u l e r forward scheme g i v e n by u(n+1) = u ( n ) + A t R ( n ) ^ which i s h i g h y u n s t a b l e i f i t were used to c a r r y out the time i n t e g r a t i o n of the e q u a t i o n s f o r many time s t e p s . The i n i t i a l v a l u e s of a l l v a r i a b l e s are equal to z e r o . As we see from (3.24) - (3 . 2 6 ) , we have to s o l v e f o u r systems of e q u a t i o n s every time s t e p . Two systems f o r the - 59 -v o r t i c i t y e q u a t i o n s and o t h e r two f o r both the b a r o c l i n i c and b a r o t r o p i c modes. T h i s i s a d i s a d v a n t a g e of f.e compared t o second o r d e r f i n i t e d i f f e r e n c e s where o n l y the modal e q u a t i o n s are to be s o l v e d . One form of remedying t h i s i s t o lump the mass m a t r i x M by adding a l l the elements o f a g i v e n row onto the d i a g o n a l o f the row and then s e t the o f f -d i a g o n a l terms to z e r o . That i s M. . = Z M.. i = i M i : j = 0 i * j L a t e r on, we w i l l g i v e the main reason to lump the m a t r i x M. An a l g o r i t h m of the computations i s as f o l l o w s : (1) Compute the element components of the m a t r i c e s M and K and s t o r e them i n t o one d i m e n s i o n a l compact v e c t o r s . T h i s compact form of s t o r i n g the m a t r i c e s i s p a r t i c u l a r l y c o n v e n i e n t f o r the Conjugate G r a d i e n t Method and f o r s o l v i n g the a l g e b r a i c e q u a t i o n s , because r a t h e r than o p e r a t i n g w i t h m a t r i c e s and v e c t o r s , we o n l y o p e r a t e w i t h o n e - d i m e n s i o n a l v e c t o r s t h a t r e p r e s e n t f a s t e r computations even on s c a l a r computers as the one we have used. Another remarkable f e a t u r e of the compact s t o r i n g i s the s u b s t a n t i a l s a v i n g of computer memory (up to 60 p e r c e n t ) i n r e l a t i o n w i t h more s t a n d a r d t e c h n i q u e s used i n f i n i t e element codes which s o l v e the system of e q u a t i o n s by d i r e c t methods. The computations of M and K are done once f o r a l l . - 60 -(2) I f n = 1 set c | 0 ) = $ 1 ( 0 ) and = 0 (i=1,2) 1 2 and ( i ) s o l v e ( 3 . 2 5 ) , (3.21b) and (3.26); ( i i ) compute a n d ^ r o m (3.22) by u s i n g the E u l e r forward scheme r a t h e r than the l e a p f r o g one; ( i i i ) compute w£ 1 ^  from (3.28) and R^ 1) from ( 3 . 2 9 ) ; ( i v ) i f the n o n - s l i p c o n d i t i o n s are a p p l i e d on the v o r t i c i t y e q u a t i o n s and the m a t r i x M i s r e t a i n e d , then compute the boundary c o n d i t i o n s ; o t h e r w i s e , go on to s o l v e ( 3 . 2 4 ) ; (v) s o l v e e q u a t i o n s (3.24) and impose the boundary c o n d i t i o n s ; i f the m a t r i x M i s lumped and the n o n - s l i p c o n s t r a i n t s imposed, then the computation and i m p o s i t i o n s of the boundary c o n d i t i o n s are made t o g e t h e r . For n >^ 2, steps ( 2 - i ) to (2-v) are re p e a t e d s u c c e s s i v e l y , but u s i n g the l e a p f r o g scheme i n s t e a d o f the E u l e r forward one. Observe t h a t we have to s o l v e one H e l m h o l t z e q u a t i o n f o r the b a r o c l i n i c mode (two e q u a t i o n s when n=1 ) and one P o i s s o n e q u a t i o n f o r the b a r o t r o p i c mode. The n u m e r i c a l experiments have shown t h a t f o r the v a l u e s of X2 and the g r i d s p a c i n g - 61 -used i n our co m p u t a t i o n s , the Hel m h o l t z m a t r i x i s w e l l -c o n d i t i o n e d w h i l e the P o i s s o n one i s o n l y f a i r - c o n d i t i o n e d r e s u l t i n g , t h e r e f o r e , i n a computation of the b a r o c l i n i c mode which i s r e l a t i v e l y cheap and f a s t , whereas the b a r o t r o p i c mode i s e x p e n s i v e and l e n g t h y of computing. The m a t r i x M i s v e r y w e l l c o n d i t i o n e d , so the s o l u t i o n of (3.24) i s o b t a i n e d r e l a t i v e l y f a s t (few i t e r a t i o n s ) ; however, i f the experiments are to be c a r r i e d out f o r a l a r g e number of time s t e p s , then the p r o c e s s of s o l v i n g two a d d i t i o n a l systems can become burdensome. T h i s i s an argument t h a t many a n a l y s t s used to j u s t i f y the lumping of the m a t r i x M so lon g as the l o s s of a c c u r a c y i s not b a d l y damaged. 3.5 S t a b i l i t y A n a l y s i s I t i s customary to choose the g r i d s p a c i n g from the b e g i n n i n g based on the p h y s i c s o f the phenomena one wants to s i m u l a t e n u m e r i c a l l y , u n l e s s a moving mesh or a u t o m a t i c r e f i n e m e n t mesh method be used. Once the g r i d s p a c i n g has been f i x e d , the time i n t e r v a l i s determined by the s t a b i l i t y c o n d i t i o n of the d i s c r e t i z e d e q u a t i o n s . The study of the s t a b i l i t y c r i t e r i o n of the d i s c r e t i z e d e q u a t i o n s i s r e l e v a n t because as the Lax e q u i v a l e n c e theorem s t a t e s : i f a l i n e a r d i f f e r e n t i a l problem which i s w e l l posed i s approximated by a c o n s i s t e n t d i f f e r e n c e scheme, then the r e s u l t a n t - 62 -a p p r o x i m a t i o n converges to the t r u e s o l u t i o n f o r a l l i n i t i a l d a t a as the space-time mesh i s r e f i n e d i f and o n l y i f the d i f f e r e n c e scheme i s s t a b l e f o r t h i s r e f i n e m e n t ( R i c h t m y e r , R.D. and K.W. Morton, 1967). A consequence o f t h i s theorem i s the c l o s e d r e l a t i o n s h i p between the w e l l - p o s e d n e s s o f the continuum problem and the s t a b i l i t y o f the d i s c r e t i z e d one; a c t u a l l y , f o r any continuum (even n o n - l i n e a r ) p a r t i a l d i f f e r e n t i a l e q u a t i o n , the common way of d e t e r m i n i n g the uniqueness o f the s o l u t i o n i s v i a energy norm. The s o l u t i o n of the d i s c r e t e a p p r o x i m a t i o n of an i n i t i a l boundary v a l u e problem ( e i t h e r l i n e a r or n o n - l i n e a r ) i s s t a b l e i f and o n l y i f V e > 0, 5 > 0 such t h a t l l u ^ 0 ) II2, < 6 => sup l l u ( n ) II 2 < e. h 2 M The above d e f i n i t i o n of s t a b i l i t y i s the \2- norm s t a b i l i t y . For l i n e a r problems, and assuming t h a t the domain Q T i s i n f i n i t e o r the problem has p e r i o d i c boundary c o n d i t i o n s , F o u r i e r a n a l y s i s i s the t e c h n i q u e which g i v e s s t a b i l i t y c r i t e r i a t h a t most of the times are e q u i v a l e n t to the L^-norm s t a b i l i t y c r i t e r i o n (see K r e i s s , 1978). I f the e q u a t i o n s are o f v a r i a b l e c o e f f i c i e n t s i t i s s t i l l p o s s i b l e to a p p l y F o u r i e r a n a l y s i s by " f r e e z i n g " c o e f f i c i e n t s and the l o c a l s t a b i l i t y c r i t e r i o n o b t a i n e d i n t h i s form can g i v e an - 63 -e s t i m a t e of the o v e r a l l s t a b i l i t y i n cases of d i s s i p a t i v e systems ( K r e i s s , 1978); o t h e r w i s e , the e s t i m a t e i s o n l y i n d i c a t i v e of how t h i n g s can t r a n s p i r e . I f the e q u a t i o n s are n o n - l i n e a r , as the q.g.e. a r e , the p r o p e r t i e s of the l i n e a r i z e d e q u a t i o n s are not at a l l s u f f i c i e n t f o r d e t e r m i n i n g n u m e r i c a l s t a b i l i t y because the L^-norm o f the l a t t e r does not imply the L z-norm of the f u l l n o n - l i n e a r e q u a t i o n s ( P h i l l i p s , 1959). The boundedness of the L^-norm i m p l i e s t h a t c e r t a i n q u a d r a t i c q u a n t i t i e s are to be bounded ( i f t h e r e i s no d i s s i p a t i o n , they are to be c o n s e r v e d ) ; t h e r e f o r e , the f i r s t c o n d i t i o n we have to r e q u i r e from our d i s c r e t i z e d scheme i s to keep bounded ( o r to conserve) such q u a d r a t i c q u a n t i t i e s . I n q u a s i - g e o s t r o p h i c dynamics, i t t u r n s out t h a t the q u a d r a t i c q u a n t i t i e s , which are bounded or conserved i f d i s s i p a t i o n i s n e g l e c t e d , a re the k i n e t i c energy and the e n s t r o p h y . The c o n s e r v a t i o n of such q u a n t i t i e s i m p l i e s the c o n s e r v a t i o n of the average wave number and s p e c i f i e s the d i r e c t i o n of the n o n - l i n e a r energy t r a n s f e r s . The energy i s t r a n s f e r r e d towards lower wave numbers w h i l e the p o t e n t i a l e n s t r o p h y i s i n the o p p o s i t e d i r e c t i o n (Charney, 1971). T h e r e f o r e , i t i s c l e a r t h a t the d i s c r e t e schemes of the q.g.e. which are a b l e to conserve the k i n e t i c energy and the p o t e n t i a l e n s t r o p h y can reproduce the exchanges o f energy among the modes b e t t e r than those which are n o t . A f i r s t success i n t h i s d i r e c t i o n was a c h i e v e d by - 64 -A r k a v a i n 1966 when he produced a p a r t i c u l a r form of the c o n v e c t i v e terms of the v o r t i c i t y e q u a t i o n which i n terms o f e n s t r o p h y i s q u a d r a t i c a l l y s e m i - c o n s e r v a t i v e . F i x (1975) shows t h a t f . e . d i s c r e t i z a t i o n o f the b a r o t r o p i c q u a s i - g e o s t r o p h i c e q u a t i o n s are p o t e n t i a l v o r t i c i t y s e m i - c o n s e r v a t i v e and k i n e t i c energy and p o t e n t i a l e n trophy q u a d r a t i c a l l y s e m i - c o n s e r v a t i v e . See Appendix B f o r d e f i n i t i o n s . In f a c t , J e s p e r s e n (1979) shows t h a t A rakawa 1s scheme i s a f i n i t e element scheme. The t r o u b l e i s t h a t n e i t h e r f i n i t e element nor Arakawa's scheme are a b l e to g e n e r a t e schemes t h a t i n the absence of d i s s i p a t i o n are k i n e t i c energy and p o t e n t i a l e n s t r o p h y q u a d r a t i c a l l y c o n s e r v a t i v e ; at most, they can y i e l d M |< n + 1> J< n> - Cte m - 1 "m m^ m=l i f l e a p f r o g scheme i s used. where t i s p o t e n t i a l v o r t i c i t y , n i s the number of time s t e p , m denotes the number of the g r i d p o i n t s . The l o s s o f the s t r i c t c o n s e r v a t i o n of the q u a d r a t i c p r o p e r t i e s when the complete d i s c r e t i z a t i o n o f the e q u a t i o n s has been c a r r i e d o u t , can make the n u m e r i c a l s o l u t i o n go u n s t a b l e . Basdevant and Sadourny (1975) show t h a t i n a d d i t i o n to the k i n e t i c energy and p o t e n t i a l e n s t r o p h y , h i g h e r o r d e r moments have to be c onserved by any complete - 65 -scheme of the n o n - l i n e a r e q u a t i o n s i f one wants to a v o i d the onset o f n o n - l i n e a r i n s t a b i l i t y . At p r e s e n t , i t seems d i f f i c u l t to d e v i s e h i g h e r o r d e r moment c o n s e r v a t i v e schemes which may be c o m p u t a t i o n a l l y f e a s i b l e , so we w i l l have to l i v e w i t h the t h r e a t o f the n o n - l i n e a r i n s t a b i l i t y and to combat i t by u s i n g r e c i p e s as the one we use i n our c o m p u t a t i o n s . The main mechanism i n p r o v o k i n g n o n - l i n e a r i n s t a b i l i t y i n those schemes which are n e i t h e r q u a d r a t i c a l l y s e m i - c o n s e r v t i v e nor q u a d r a t i c a l l y c o n s e r v a t i v e i s a l i a s i n g ( P h i l l i p s , 1959). However, w i t h schemes such as the Arakawa scheme or f i n i t e element method, i n which the a l i s a i n g problem i s almost r u l e d o u t , v e r y l i t t l e i s known about the s u b t l e mechanisms r e s p o n s i b l e f o r the onset o f the n o n - l i n e a r i n s t a b i l i t y . T h i s i n s t a b i l i t y i s v e r y d i f f e r e n t from the one p r e s e n t i n l i n e a r e q u a t i o n s when the s t a b i l i t y c r i t e r i o n i s v i o l a t e d . I t s p e c u l i a r f e a t u r e s i n those schemes which are q u a d r a t i c a l l y s e m i - c o n s e r v a t i v e a r e : ( i ) I t shows up a f t e r s e v e r a l hundred time steps o f s u c c e s s f u l c o m p u t a t i o n s , ( i i ) As a t h r e s h o l d i s reached, i t s t a r t s l o c a t i n g at a few p o i n t s o f the g r i d , then f o c u s e s and l a t e r on, a f t e r a few s t e p s of i n t e g r a t i o n , extends v e r y f a s t a l l over the domain. At t h i s p o i n t , the computations blow up. ( i i i ) P o s s i b l e t r i g g e r mechanisms are ( B r i g g s et a l . , 1983) the c o m p u t a t i o n a l mode i f the l e a p f r o g scheme - 66 -i s used, the s p u r i o u s o s c i l l a t i o n s produced e i t h e r at boundary p o i n t s which are not w e l l - r e s o l v e d , o r a t d i s c o n t i n u i t y i n t e r f a c e s and o t h e r s . At t h i s p o i n t , i t seems obv i o u s to ask about the r e l e v a n c e of c a r r y i n g out a l i n e a r s t a b i l i t y a n a l y s i s by " f r e e z i n g " the c o e f f i c i e n t s o f the n o n - l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n . The answer may be g i v e n by assuming t h a t the f i r s t s t ages of the n o n - l i n e a r n u m e r i c a l i n s t a b i l i t y a re v e r y s i m i l a r to those o f the n o n - l i n e a r hydrodynamic s t a b i l i t y . In doi n g so, we can determine which are the most u n s t a b l e modes on one s i d e , and an assessment of the n o n - l i n e a r i n s t a b i l i t y t h r e s h o l d i n terms of the v a l u e s o f the l i n e a r s t a b i l i t y c r i t e r i o n on the o t h e r s i d e ( B r i g g s et a l . 1 983) . As i s common p r a c t i c e i n the t h e o r y of the hydrodynamic s t a b i l i t y ( P e d l o s k y , 1970, 1971), we s t a r t by decomposing the stream f u n c t i o n s i ^ i as * . ( x , y , t ) = *<°> + *'f ( . = 1 > 2 ) such t h a t (0) (0) = v ; (0) ay i where u. (0) and v: (0) are the b a s i c f l o w v e l o c i t i e s . In our i i - 67 -c o n t e x t , they can be i n t e r p r e t e d as the Sverdrupean v e l o c i t i e s ( i n the f i r s t l a y e r ) i n the i n t e r i o r o f the ocean and the j e t stream v e l o c i t i e s i n the western boundary l a y e r . ( x , y , t ) r e p r e s e n t s the p e r t u r b a t i o n to the b a s i c f l o w . With such a d e c o m p o s i t i o n , the J a c o b i a n terms of the v o r t i c i t y and modal e q u a t i o n s (3.1 - 3.10) become, upon l i n e a r ! z a t i o n , J U t , *t) * - v[ 0 ) • V c. , i = 1 ,2, where = (u£^ , v £ ^ ) . The (') s i g n has been dropped from the v o r t i c i t y of the r i g h t - h a n d s i d e . The s t a b i l i t y of the l i n e a r i z e d e q u a t i o n s (3.24) - (3.29) w i l l be d i c t a t e d by the s t a b i l i t y of the f i r s t l a y e r v o r t i c i t y e q u a t i o n because i t i s i n t h i s l a y e r where the v e l o c i t i e s ( U ^ ^ , ) a t t a i n the h i g h e s t v a l u e s . I t i s w e l l known t h a t the s t a b i l i t y of the homogeneous e q u a t i o n i m p l i e s the s t a b i l i t y o f the inhomogeneous one p r o v i d e d the f o r c i n g term i s bounded ( K r e i s s , 1978). T h e r e f o r e , we have to study the l i n e a r s t a b i l i t y o f the f o l l o w i n g e q u a t i o n : >(n+1) _ ? ( n - 1 ) +(n) - A R K (3.30) The e q u a t i o n e q u i v a l e n t to (3.30) at the node (pAx, qAy) i s - 68 -3el ^p,q p.q+1 p.q-1 p+1.q p-1,q ( CP+1 ,q+i + S+Uq-1 + S-1 ,q+i + Vi,q-1 ) } u<0> 1 1 TT^Ax ( Cp+1,q+1 " ?p-1 ,q+1 + ?p+1 ,q-1 " ?p-1,q-1 ^ v}°> + 12^Ax ( 5p+1,q+1 " Cp+1 ,q-1 + 5p-1 ,q+1 " Cp-1 ,q-1 + 4 « p . q + 1 " t p , q . 1 » $ n ) I - ^ Cp+1 ,q+1 " ?p,q+1 + S-1 ,q+1 + S+1 ,q " 8 ^P,  + ' ;p-1 ,q + V 1 ,q-1 + CP,q-1 + S-1,q-lJ(n"1>' (3'31) where u j * ^ and are assumed to be c o n s t a n t s , ( (n+1) ( n - 1 ) N Assume t h a t C p " q = "I e x P [ ^ k x p + * V ] x p = pAx, y q = qAy, 0 < p < P, 0 < q < Q PAx = L , QAy = Ly - 69 -A.j i s an a m p l i f i c a t i o n f a c t o r E q u a t i o n (3.31) g i v e s (0) 2 , , r , . ^ r U 1 /1 - x 2 H + i 6 L At { - i / • o /~M \ -i. (y + 2) 1 1 1 Ax 4 + 2 (x+y) + xy w 7 v, 7 1 2 Ax 4 + 2 (x+y) + xy ^ x + z ; ' A,. 48 At ( 1 " H A x Ay ( 4 + 2 X^x+y) +*xy) * " °' (3.33) where x = coskAx, y - cosJIAy, -1 < x,y _< 1 . E q u a t i o n (3.31) i s s t a b l e i f and o n l y i f X1 | < 1 . (3.34) From (3.33) we o b t a i n t h a t X1 = i 3 AtB ± (- 9 At 2 B 2 + 1 - A ) 1 / 2 , (3.35a) where u}°) /1 - x 2 v<°> /I - y 2  B Ax 2 + x + Ay 2 + y ' A = 4 8 A H A t (4 - ( x + y ) - 2 x Y ) ( h ) A Ax Ay C (2 + x) (2 + y) ; * (3.35b; - 70 -T h e n |A1 | 2 = 1 - A, f o r 3 A t B < (1 - A ) 1 / 2 . T a k i n g i n t o a c c o u n t t h a t -1 < x , y _< 1 t h e n 1 9 2 A R A t m a x A = —- - — , A x A y ' A > 0, u ( 0 ) v ( 0 ) 1 , U1 , V1 I f A x = A y a n d u < ° > = v < ° > = 1- , t h i s g i v e s /2 192A A t max A = • h 2 (3.36) max B = — (—! + — ) /o o - 7 „ \ ^2 A x A y ; * (3.37a) A ^ °» (3.37b) max B = /|. V. By u s i n g (3.37b) the s t a b i l i t y c r i t e r i o n (3.36) become; 192A HAt 0 1 - — ^ - < M < 1. - 71 -f o r 192A uAt < 1 , and 192A uAt 1/2 SZ V < (1 - (3.38a,b,c) where h i s the s m a l l e s t v a l u e of the g r i d s p a c i n g , and V i s the h i g h e s t p o s s i b l e v e l o c i t y . C o n d i t i o n s (3.38a) and (3.38b) are q u i t e a severe s t a b i l i t y c r i t e r i o n f o r the g r i d s i z e and A H t h a t we have to use i n o r d e r to make f e a s i b l e the c o m p u t a t i o n s . N o t i c e t h a t (3.35b) i s the s t a b i l i t y c r i t e r i o n f o r the h y p e r b o l i c p a r t of the e q u a t i o n s . From (3.36a) we can show (see Appendix C) t h a t the scheme i s d i s s i p a t i v e of o r d e r 2 a c c o r d i n g to K r e i s s 1 d e f i n i t i o n . (See K r e i s s ( 1 9 7 8 ) ) . The term r e s p o n s i b l e o f the h a r s h n e s s o f (3.38) i s the c o n s i s t e n t mass m a t r i x M t h a t makes i m p l i c i t i n space the time d e r i v a t i v e o p e r a t o r . A s t a n d a r d t e c h n i q u e f o r c o n v e r t i n g the i m p l i c i t d i s c r e t i z a t i o n i n t o an e x p l i c i t one i s the lumping of the mass m a t r i x M. Thomee (1985) shows t h a t the a p p r o x i m a t i o n e r r o r of the lumped scheme i s o f the same o r d e r as the c o n s i s t e n t scheme; however, the e v o l u t i o n a r y e r r o r of the lumped scheme i s o f second o r d e r i n c o n t r a s t w i t h the f o u r t h o r d e r e r r o r t h a t can be a c h i e v e d by u s i n g the c o n s i s t e n t scheme. The most s t a n d a r d f i n i t e d i f f e r e n c e models used i n oceanography, such -li-as H o l l a n d (1978) and Bryan-Cox's models are second o r d e r a c c u r a c y t o o . S i n c e i t i s not the purpose of t h i s t h e s i s to d i s c u s s the p o s i t i v e p r o p e r t i e s of c o n s i s t e n t G a l e r k i n - f i n i t e element method i n r e l a t i o n to the e v o l u t i o n a r y e r r o r , we do not go f u r t h e r on such t o p i c . R e s u l t s on e v o l u t i o n a r y e r r o r , phase p r o p a g a t i o n and group v e l o c i t y of the f i n i t e element, a p p l i e d to the q u a s i - g e o s t r o p h i c e q u a t i o n s , w i l l be r e p o r t e d e l s e w h e r e . As we have a l r e a d y e x p l a i n e d , the way of lumping the m a t r i x M i s by adding a l l the elements of a g i v e n row onto the d i a g o n a l of the row and then s e t the o f f - d i a g o n a l terms to z e r o . In d o i n g so, the term on the l e f t - h a n d s i d e of (3.31) becomes whereas the terms on the r i g h t - h a n d s i d e remain u n a l t e r e d . The e q u a t i o n f o r the a m p l i f i c a t i o n f a c t o r s X] is (n+1) _ (n-1) p.q p>q 2At u (0) v (0) + i 2 i - j A 1 A t [ A x • 1-x2 (2 + y) + A y / 1 - y2 (2 + x ) ] 3 A x A y (4 - (x + 1) (y + 1)} = 0. (3.39) - 73 -Hence X1 - 1 f B1 +~ <- ^ B i + 1 - A ^ 1 7 2 , (3.39) where A. A H A t A 1 = l 4 ( 4 " ( X + y ) - 2 x y ) ' u ( 0 ) v ( 0 ) B 1 = /1 - x 2 (2 + y) + - l y - M ~ J 2 (2 + x) . (3.40a,b) The s t a b i l i t y c r i t e r i o n i s A i | 2 = 1 - V f o r A t B 1 1/2 " J - < d " A-j ) , (-1 < x, y, < 1 ) , ( 3 . 4 1 ) and A 1 > 0. At (x,y) = - (1/2, 1/2) A 1 has a r e l a t i v e maximum i n the i n t e r v a l -1 < x, y < 1 . A T J A t max A . = 18 1 1 0 A x A y ' - 74 -I f A x = A y h a n d u | 0 ) = v j 0 ) = V _ , t h e n • I m a x B n = 3*^ 2 — 1 /2 a n d A H A t m a x = 18 — y ~ T h u s (3.41) b e c o m e s 1 - 18 -JJ2- < < 1, f o r 18 A u A t H < 1 h z a n d ft h z w h e r e h a n d V r e p r e s e n t s , a s b e f o r e , t h e s m a l l e s t v a l u e t h e g r i d s p a c i n g a n d t h e h i g h e s t p o s s i b l e v e l o c i t y , r e s p e c t i v e l y . - 75 -By comparing (3.38) and (3.42) two obvious c o n c l u s i o n s emerge: ( i ) The c o n s t r a i n t imposed by the d i f f u s i v e terms i s much l e s s severe i n (3.42) than i n (3.38) because max A « 10 max A - | . On the o t h e r hand, f o r the s m a l l e s t v a l u e s A x , A y , A H (3.42b) g i v e s A t A H -V1 < 0.55 which i s not d i f f i c u l t to s a t i s f y f o r the u s u a l p a r a m e t e r s , A H , A t , h used i n eddy models. ( i i ) The c o n d i t i o n (3.43c) to be s a t i s f i e d by the h y p e r b o l i c terms i s p r a c t i c a l l y the same as t h a t o f the f . d . f o r m u l a t i o n and i s , o f c o u r s e , l e s s s e v e r e than (3.38c) because f o r g i v e n v a l u e s o f A H » A t and h2 the r i g h t - h a n d s i d e of the i n e q u a l i t i e s i s much l a r g e r i n (3.43c) than i n ( 3 . 3 8 c ) . The scheme i s a g a i n d i s s i p a t i v e of second o r d e r . C o n c l u s i o n s ( i ) and ( i i ) have been the main reasons f o r the lumping of the mass m a t r i x . From t h i s s i m p l e study o f the s t a b i l i t y p r o p e r t i e s e x h i b i t e d by two s t a n d a r d d i s c r e t i z a t i o n s o f the mass m a t r i x M of the f i n i t e e l e m e n t - - l e a p f r o g scheme a p p l i e d to the q u a s i - g e o s t r o p h i c e q u a t i o n s , i t i s c l e a r t h a t to e x p l o i t the f u l l advantages of the i m p l i c i t n e s s of the f i n i t e element, i t i s b e t t e r to use i m p l i c i t schemes to t i m e - d i s c r e t i z e the e q u a t i o n s . The main problem w i t h such schemes i s t h a t the - 76 -a l g e b r a i c sysems are not symmetric any more and are time-dependent a l s o , so s p e c i a l c a r e has to be taken t o compute the s o l u t i o n i n these c a s e s . Work i n t h i s d i r e c t i o n i s c u r r e n t l y under way to s o l v e the n o n - l i n e a r q u a s i -g e o s t r o p h i c e q u a t i o n by u s i n g a more p o w e r f u l v e r s i o n of the Conjugate G r a d i e n t Method i n c o n j u n c t i o n w i t h l e a s t squares t e c h n i q u e s . - 77 -CHAPTER 4 THE EXPERIMENTS 4.1 The Model and Its Parameters We h a v e c a r r i e d o u t t h r e e e x p e r i m e n t s . T w o s i n g l e g y r e e x p e r i m e n t s d e n o t e d a s E x p . 1 ( f r e e - s l i p b o u n d a r y c o n d i t i o n ) a n d E x p . 2 ( n o - s l i p b o u n d a r y c o n d i t i o n ) . T h e s y m m e t r i c d o u b l e g y r e e x p e r i m e n t w i t h n o - s l i p b o u n d a r y c o n d i t i o n i s d e n o t e d a s E x p . 3 . T w o m o d e l b a s i n s a r e u s e d . T h e s i n g l e g y r e m o d e l b a s i n i s a s q u a r e o f L x = L y = 1 , 0 0 0 k m a s h o r i z o n t a l d i m e n s i o n s , a u n i f o r m d e p t h H = 5 k m a n d i s c e n t e r e d a t 3 5 ° N . T h e d o u b l e g y r e m o d e l b a s i n h a s 1 ^ = 1 , 0 0 0 k m , L y = 2 , 0 0 0 k m a n d H = 5 k m a n d i s c e n t e r e d a t 4 0 ° N . S u b g r i d - s c a l e h o r i z o n t a l d i f f u s i o n o f m o m e n t u m i s p a r a m e t e r i z e d b y a c o n s t a n t e d d y d i f f u s i o n c o e f f i c i e n t A H = 3 3 0 m 2 s " 1 . N o t h e r m a l f o r c i n g n o r t h e r m a l d i f f u s i o n a r e a l l o w e d i n t h e s e p r e l i m i n a r y e x p e r i m e n t s . T h e m o d e l i s f o r c e d b y a s i n u s o i d a l w i n d s t r e s s T = - T Q C O S ( i T y / L y ) . T h e n u m e r i c a l g r i d i s c o m p o s e d o f e i t h e r s q u a r e s o r r e c t a n g l e s w i t h v a r i a b l e A x a n d A y . T h e r a n g e o f v a l u e s o f A x a n d A y i s f r o m 2 0 k m u p t o 4 0 k m . A d j a c e n t t o t h e - 78 -b o u n d a r i e s t h e r e are s t r i p s w i t h A x = A y = 20 km. Thus f o r i n s t a n c e , the w e s t e r n s t r i p i s 200 km wide, then the g r i d s i z e i n c r e a s e s p r o g r e s s i v e l y w i t h the sequence (25 km, 30 km, 35 km) up to A x = A y = 40 km a t a p p r o x i m a t e l y 350 km from the w e s t e r n c o a s t ; t h i s s i z e i s kept from t h i s p o i n t up to x = 880 km where a g a i n the g r i d i s decreased p r o g r e s s i v e l y . A s i m i l a r p a t t e r n i s repeated a l o n g the n o r t h and south b o u n d a r i e s . See F i g u r e 3. The v e r t i c a l r e s o l u t i o n c o n s i s t s of two l a y e r s w i t h depths H-| = 1000 m and H.2 = 4000 m, r e s p e c t i v e l y . The c h o i c e of these c o n d i t i o n s and q u a n t i t i e s deserves e x p l a n a t i o n . The time s t e p A t i s , a c c o r d i n g to ( 3 . 3 6 b ) , e q u a l to 4 h o u r s , | v ( 0 ) | = 1 ms"1 and h = A x = 20 km. N o t i c e t h a t i n o r d e r to c a l c u l a t e A t we have used the s m a l l e s t v a l u e of A x and m u l t i p l y the r e s u l t by a f a c t o r o f 0.8 which i s an e m p i r i c a l c r i t e r i o n i n d e a l i n g w i t h n o n - l i n e a r or v a r i a b l e c o e f f i c i e n t e q u a t i o n s . The p h y s i c a l parameters Aft, T Q and the v e r t i c a l r e s o l u t i o n are the same as those of H o l l a n d (1978) so t h a t we can compare the r e s u l t s o f our model w i t h those o f H o l l a n d ' s model. On the o t h e r hand, A H cannot be s m a l l e r than 200 m 2 s - 1 because, o t h e r w i s e , the western boundary l a y e r w i d t h , as we w i l l see l a t e r , i s not r e s o l v e d p r o p e r l y by a g r i d w i t h A x = 20 km and, t h e r e f o r e , the s o l u t i o n e x h i b i t s o s c i l l a t i o n s a t the boundary which propagate to the i n t e r i o r (see Gresho and Lee - 79 -1981). The h o r i z o n t a l g r i d d i s t r i b u t i o n was determined by the need of k e e p i n g as low as p o s s i b l e the computer s t o r a g e r e q u i r e m e n t s c o m p a t i b l e w i t h the r e s o l u t i o n needed at the w e s t e r n boundary and p o s s i b l e eddy a r e a s . The l a r g e s t v a l u e of Ax, Ay i s s m a l l e r than the i n t e r n a l d e f o r m a t i o n r a d i u s , so even i n the c e n t r a l r e g i o n we s t i l l keep a r e s o l u t i o n h i g h enough to r e s o l v e the e d d i e s . I t i s , of c o u r s e , the n o n - d i m e n s i o n a l statement of the model and i t s parameters which are p h y s i c a l l y m e a n i n g f u l . The s t a n d a r d n o n - d i m e n s i o n a l i z a t i o n scheme i s ( H o l l a n d , 1978): (x,y) = L ( x * , y * ) , t = ( 3 L ) " 1 t * , * * T H 2 T Q * (u,v) = U(u , v ) , w = ( f H L ) w , where the s t a r q u a n t i t i e s are d i m e n s i o n a l ones, U = ( — ^ — j - ) i s a t y p i c a l h o r i z o n t a l v e l o c i t y . U s i n g such a scheme, we d e f i n e the f o l l o w i n g n o n - d i m e n s i o n a l parameters: = o ^ (Rossby number) , U H, g z L J F i g . 3 L o u « r B a s i n | r l d . E x p . 3 - 80 --1 A H -1 Re" = -jjj- (Reynolds number) , * = H i e Lg H These n o n - d i m e n s i o n a l parameters can be used to r e l a t e the v a r i o u s boundry l a y e r w i d t h s to L: £ * Wg = -g = e L Stommel L a y e r ; WM = ( ^ ) 1 / 3 = ( ^ ) 1 / 3 L Munk L a y e r ; e W I = ( I ) 1 / 2 = r 1 / 2 L I n t e r t i a l L a y e r . The v a l u e s o f the n o n - d i m e n s i o n a l parameters and the boundary w i d t h s i n a l l of our experiments a r e : U = 0.0157 m s " 1 R Q = 8 x l O " 4 , R e = 5 0 , E * = 4 x 10~ 3 Wg = 5 km, WM = 25 km, Wj. = 28 km N o t i c e t h a t a c c o r d i n g to the v a l u e s o f Wx and W M , the model i s a compromise between a n o n - l i n e a r and a f r i c t i o n a l model. - 81 -4.2 Results The main o b j e c t i v e of Exp. 1 was to check the i n t e r n a l l o g i c of our model by comparing the f i n i t e element r e s u l t s w i t h those g i v e n by f i n i t e d i f f e r e n c e models w i t h the same parameters as H o l l a n d (1978). The o n l y d i f f e r e n c e between our experiment and H o l l a n d ' s experiment No. 1 i s the i n c l u s i o n of bottom f r i c t i o n . Exp. 1 was c a r r i e d out f o r 200 days. F i g u r e 4 shows the upper l a y e r stream f u n c t i o n p a t t e r n at T = 200 d. The dominant f e a t u r e s a r e : ( i ) a western boundary c u r r e n t ; ( i i ) a n o r t h e r n w a l l boundary c u r r e n t f l o w i n g to the e a s t ; ( i i i ) a r e c i r c u l a t i o n system f l o w i n g p r e d o m i n a n t l y to the west; ( i v ) a mid-ocean c i r c u l a t i o n dominated by c l o s e d c i r c u l a t i o n systems w h i c h , at t h i s s t a g e , i s q u a l i t a t i v e l y s i m i l a r to the t r a d i t i o n a l Svendrup i n t e r i o r . A l t h o u g h t h e r e are no eddies y e t , the ocean should s t a r t shedding them from about 400th day; a c c o r d i n g to H o l l a n d ( 1 9 7 8 ) . The f e a t u r e s o f our s o l u t i o n agree w e l l w i t h the average p a t t e r n of H o l l a n d ' s r e s u l t s . T h e r e f o r e , we assume t h a t the l o g i c of our model i s c o r r e c t . The n e x t c o m p u t a t i o n , Exp. 2, d i f f e r s from the p r e v i o u s one i n t h a t n o - s l i p boundary c o n d i t i o n s are used on the s o l i d - 01.1 -- 81.2 -- 82 -b o u n d a r i e s . I n s o f a r as the a u t h o r ' s knowledge i s concerned, no experiment w i t h a b a r o c l i n i c ocean and such boundary c o n d i t i o n has been r e p o r t e d . F i g u r e s 6 and 7 show the upper and lower l a y e r stream f u n c t i o n p a t t e r n s a f t e r 200 days of i n t e g r a t i o n . The dominant f e a t u r e s of F i g u r e 6 a r e : ( i ) a weste r n boundary; ( i i ) a s e p a r a t i o n o f the f l o w from the w e s t e r n boundary p a s t the p o i n t of maximum a m p l i t u d e of wind s t r e s s ; ( i i i ) the e n t r y o f the f l o w to the i n t e r i o r v i a u n d u l a t i o n s ; ( i v ) a mid-ocean c i r c u l a t i o n dominated by a c i r c u l a t i o n system i n Sverdrup b a l a n c e . The d i f f e r e n c e s between t h i s p a t t e r n and t h a t o f Exp. 1 are the absence of the no r t h w e s t r e c i r c u l a t i o n and the s e p a r a t i o n o f the f l o w from the westen boundary. The f l o w p a t t e r n o f the f i r s t l a y e r i n Exp. 2 i s v e r y s i m i l a r to t h a t o f the Bryan (1963) b a r o t r o p i c ocean w i t h comparable Re and Ro and n o - s l i p boundary c o n d i t i o n on the West and East b o u n d a r i e s . The u n d u l a t i o n s of the stream f u n c t i o n i n the n o r t h - w e s t e r n p a r t o f the b a s i n are more conspicuous i n the v o r t i c i t y p l o t s shown i n F i g u r e 8. R e s u l t s o f the symmetric two-gyre experiment w i t h n o - s l i p boundary c o n d i t i o n are shown i n F i g u r e s 9 to 11; they r e p r e s e n t upper and lower l a y e r s stream f u n c t i o n d i s t r i b u t i o n s at i n s t a n t s T = 130 days, 300 days and 400 - 82 . 1 -Strc.-im Function Distribution 1st Lav (CI 600 m' s"1) - 8 2 . 2 -F i g . 7 Stream F u n c t i o n D i s t r i b u t i o n 2nd Layer ( CI 60 m' S ' [ ) - a . J -- 83 -days, r e s p e c t i v e l y . We observe the absence of the upper l a y e r f r e e j e t at m i d - l a t i t u d e which i s a c h a r a c t e r i s t i c f e a t u r e when f r e e - s l i p boundary c o n d i t i o n i s a p p l i e d w i t h the same and wind s t r e s s d i s t r i b u t i o n ( H o l l a n d and L i n ( 1 9 7 5 ) . T h i s f a c t i s due to the s e p a r a t i o n of the f l o w from the w e s t e r n boundary. The c i r c u l a t i o n o f the upper l a y e r i s e s t a b l i s h e d at T = 130 days and from t h a t i n s t a n t on t h e r e e x i s t s a b a l a n c e d boundary l a y e r - i n t e r i o r Sverdrup p a t t e r n even when the ocean i s b e i n g spun up as F i g s . 15 ( k i n e t i c energy per u n i t o f a r e a o f the upper l a y e r ) and 16 (stream f u n c t i o n at x = 700 km, y = 390 km) show. T h i s c i r c u l a t i o n p a t t e r n i s i n d i c a t i v e o f the b a r o t r o p i c i t y o f the regime o f the f i r s t l a y e r . As f o r the lower l a y e r , i t takes l o n g e r to re a c h a r e g u l a r p a t t e r n , w hich i s s i m i l a r to t h a t of the upper l a y e r out of the lower southern/upper n o r t h e r n c o r n e r s o f the ch a n n e l where an eddy i s s e t t i n g up. T h i s eddy, which matches w i t h the upper/lower i n t e r i o r dynamics, s t a y s at the same l o c a t i o n and i n c r e a s e s i t s i n t e n s i t y a t the same r a t e o f the boundary l a y e r c i r c u l a t i o n . The eddy r e g i o n i s j u s t below the r e g i o n of the upper l a y e r where the western boundary i s more i n e r t i a l . We have not attempted to p r o v i d e a t h e o r e t i c a l a n a l y s i s o f t h e eddy dynamics because our a n a l y t i c a l approach i s a b l e to e x p l a i n the boundary l a y e r - S v e r d r u p i n t e r i o r of the l o w e r l a y e r as w e l l as the upper l a y e r dynamics. The lower l a y e r g y r e s are d i s p l a c e d poleward i n agree-ment w i t h e m p i r i c a l o b s e r v a t i o n s , Rhines and Young(1980). Stream ( C I . Function D i s t r i b u t i o n 1st Laye F i H >>n 500 - 1 m s 100 m3 s 1 ) . T = 130 d clays Stream Function D i s t r i b u t i o n 1st Laye ) T = 300 days ( C . I . ± 900 m' s-1 T / i | l , n i iooqO"' / / H / I I I Hi t / II M ) / /III/ ' I / I Scream Function D i s t r i b u t i o n 1st Layer ( C . I . 1000 m ' s" 1) T = 400 days I- i « . I 1 a I ' . i g . l IN Stream Fu n c t i o n D i s t r i b u t i o n 2nd Layer ( C I . 100 nT s" 1) T = 400 days - 84 -The n u m e r i c a l r e s u l t s f o r both upper and lower l a y e r s can be c o n c e p t u a l l y e x p l a i n e d by S t e w a r t ' s (1964) arguments. In the w e s t e r n boundary l a y e r the r e l a t i v e v o r t i c i t y i s zero b e f o r e the c u r r e n t reaches the s e p a r a t i o n p o i n t ; but the wind s t r e s s i n p u t s c l o c k w i s e r e l a t i v e v o r t i c i t y and the f l u i d has no r e l a t i v e v o r t i c i t y when i t e n t e r s i n the boundary l a y e r . Thus, the o n l y p o s s i b i l i t y i s f o r c o u n t e r - c l o c k w i s e v o r t i c i t y to form at the b o u n d a r i e s and to d i f f u s e i n t o the c u r r e n t l a t e r a l l y . F u r t h e r n o r t h of the s e p a r a t i o n p o i n t , and as l o n g as the i n t e r i o r c i r c u l a t i o n i s i n Sverdrup b a l a n c e , t h e stream l i n e s t u r n seaward, but the r e l a t i v e v o r t i c i t y i n the i n t e r i o r of the ocean i s n e g l i g i b l e , so the water which l e a v e s the boundary to r e - e n t e r the i n t e r i o r has to c a n c e l i t s r e l a t i v e v o r t i c i t y by a new mechanism. T h i s new mechanism i s the u n d u l a t i o n s o f the stream f u n c t i o n s which d i s s i p a t e e i t h e r p o s i t i v e or n e g a t i v e r e l a t i v e v o r t i c i t y coming out from the boundary l a y e r . Thus, the r o l e of the u n d u l a t i o n s i s to a l l o w time f o r the excess of the r e l a t i v e v o r t i c i t y t o be d i s s i p a t e d b e f o r e the boundary f l o w matches the i n t e r i o r one. F i g u r e s 12 to 14 show the adjustment o f the r e l a t i v e v o r t i c i t y . I f the f l o w i s such t h a t the d i s s i p a t i v e terms are l a r g e r than the i n e r t i a l ones, then the c a n c e l l a t i o n of boundary v o r t i c i t y w i l l be f a s t e r and, hence, the u n d u l a t i n g r e - e n t r y w i l l be l e s s c o n s p i c u o u s , o r , i n the absence o f n o n - l i n e a r i t i e s , t h e r e w i l l not be such a r e - e n t r y at a l l (Munk's d y n a m i c s ) . Now, upon comparing the p a t t e r n F i g . 12 V o r t i c i t y D i s t r i b u t i o n 1st Layer T = 130 day (CI = 1 0 _ 8 S - 1 ) V o r t i c i t y D i s t r i b u t i o n 1st Layer T = 400 days ((; j x K ; " 8 s " 1 ) CD CO 0 - 0 1 0 . 0 2 0 . 0 3 0 . 0 40 0 DAYS. CX 1 0 1 j F i g . 1 5 . i n e t i c E n e r g y p e r u n i t o f a r e a o f t h e u p p e r l a y e r . E x p . 3 . o - 85 -of the lower l a y e r w i t h t h a t of the f i r s t l a y e r ( F i g s . 10-11) we see t h a t the former has more o f a d i s s i p a t i v e c h a r a c t e r ; t h i s i s so because the d r i v e n mechanism of the second l a y e r i s the weak v o r t e x s t r e t c h i n g terra w h i l e the c o e f f i c i e n t A H i s the same at both l a y e r s . On the o t h e r hand, the c i r c u l a t i o n p a t t e r n o f the f i r s t l a y e r resembles the eddy-f r e e p a t t e r n s reached by b a r o t r o p i c models which have the same Re and Ro. Bryan (1963) and I e r l e y and Young ( ( 1 9 8 6 ) , p r i v a t e communication). In the next s e c t i o n , we d e s c r i b e an a n a l y t i c a l model to i n t e r p r e t the n u m e r i c a l r e s u l t s . 4 . 3 Theoretical Analysis In the l a s t s e c t i o n we have d i s c u s s e d the r e s u l t s o f our e x p e r i m e n t s , e m p h a s i z i n g those r e s u l t s o f Exp. 3. Our purpose now i s to d e s c r i b e an a n a l y t i c a l method to e x p l a i n the main f e a t u r e s of Exp. 3 and to i s o l a t e the mechanisms t h a t c o n t r o l the dynamics when the n o - s l i p boundary c o n d i t i o n i s imposed. S p e c i f i c a l l y , we w i s h to c l a r i f y the r o l e s p l a y e d by the western boundry l a y e r and the s t r a t i f i c a t i o n . L e t us w r i t e the continuum e q u a t i o n s o f the f i r s t and second l a y e r s i n terms of the p o t e n t i a l v o r t i c i t i e s , q-| , q2 as J ( q i , * 1) + A H V 2 ? 1 + c u r l T q 2t 2 J ( q , \p ) + A V x, -H2 2 H H 2 E 7 » H 2 (4.1a,b) - 86 -where q 1 = V 2 ^ + f + F, (ip 2 - * 1 ) , q 2 = VH *2 + f + F 2 (*1 " V ' (4.2a,b) f 2 f 2 We form the e q u a t i o n f o r the b a r o t r o p i c mode, v i z . , B H H1 H ? by adding ^- (4.1a) to ^  ( 4 . 2 b ) . Thus, H B 1 r„ T, . x . „ T , . M » 3 * B 9t = H <H1 J ( « 1 ' V + H 2 J ( C 2 , ^ 2 ) } - 3 3 x ,2 _ H 2 „2 , , C u r l x " A H VH ?B " TT E VH *2 + H — > < 4' 4 a> Hi Ci + H 9 5 B = - H * ( 4 " 4 b ) Note t h a t the v o r t e x s t r e t c h i n g terms, ± F^ (ty^  - ip 2) have c a n c e l l e d . In the Sverdrup i n t e r i o r , o u t s i d e the f r i c t i o n a l w e s t e r n boundary l a y e r , the J a c o b i a n and f r a c t i o n a l terms can be n e g l e c t e d . A l t h o u g h the ocean i s s t i l l s p i n n i n g up, F i g s . 9-11 show t h a t t h e r e i s a b a l a n c e between p l a n e t r y v o r t i c i t y and the c u r l of wind s t r e s s i n the i n t e r i o r o f the ocean - 87 -so we can n e g l e c t ^ t h e r e . Hence, the e q u a t i o n i n the i n t e r i o r i s . 9l^B c u r l T . / . , / / C N B 15T = R—• 6 < x < V <4'5> where 6 denotes the w i d t h of the western boundary l a y e r . The s o l u t i o n o f (4.5) i s *B = $B f ( y ) 0 " I ~ } ' ( 4 ' 6 ) where x 0 B H 3 L ' y f ( y ) = s i n (4.7) y The p o t e n t i a l v o r t i c i t y o f the second l a y e r i n the Sverdrup i n t e r i o r i s w r i t t e n i n terms o f and i|>2 as q 2 , I = f + F (*B " *2> F* = g - F 2 . (4.8) U s i n g (4.8) one can w r i t e (4.1b) i n the Sverdrup i n t e r i o r as J ( q , + 2) + 0 ( A H , E) = 0, (4.9a) - 88 -where q = f + F* * B (4.9b) r e p r e s e n t s the g e o s t r o p h i c c o n t o u r s of the second l a y e r . Use has been made of the i d e n t i t y J(^2» +2) = 0 to o b t a i n (4.9a) upon s u b s t i t u t i n g (4.8) i n t o ( 4 . 1 a ) . The s o l u t i o n o f (4.9) up to zero o r d e r i s * 2 = G ( q ) . ' (4.10) The g e o s t r o p h i c c o n t o u r s q w i l l c l o s e i n the northw e s t p a r t of the b a s i n i f * F * x 3y - H g < 0, or F 7T T Y = o — — > 1 (4.11) H 3 L y In our e x p e r i m e n t s : T q = 1 0 - l + , F = 6.25 x 1 0 " 1 0 , Ly = 1 0 6 m, H = 1 0 5 m and hence y = 0.2. Thus the g e o s t r o p h i c c o n t o u r s s h o u l d not c l o s e . However, t h e r e e x i s t c l o s e d s t r e a m l i n e s ^2 t h a t go through a w e s t e r n boundary l a y e r and o t h e r ones t h a t e x h i b i t eddy n a t u r e . Rhines and Young (1982) and Young and Rhines (1982) show, by a p p l i c a t i o n of the B a t c h e l o r -- 89 -P r a n d t l theorem, t h a t i n s i d e the c l o s e d g e o s t r o p h y c o n t o u r s the p o t e n t i a l v o r t i c i t y i s homogenized whereas o u t s i d e such c o n t o u r s the whole r e g i o n i s threaded by g e o s t r o p h i c c o n t o u r s t h a t h i t the e a s t e r n boundary where ^ 2 = 0. I t i s t h i s boundary c o n d i t i o n t h a t f o r c e s the f u n c t i o n G of (4.10) to be i d e n t i c a l l y z e r o . The parameter y i s g r e a t e r than 1.0 i f t o g e t h e r . Homogenization of p o t e n t i a l v o r t i c i t y arguments r e q u i r e the e x i s t e n c e o f f r i c t i o n a l f o r c e s , no m a t t e r how weak they a r e , w i t h no a d d i t i o n a l h y p o t h e s i s on the e x i s t e n c e of w e s t e r n boundary l a y e r s . Indeed, Young and Rhines (1982) c l o s e t h e i r t h e o r y by appending a western boundary l a y e r i n the c l a s s i c sense. I e r l e y and Young (1983) show, however, t h a t the presence of a Stommel f r i c t i o n a l boundary l a y e r type p r e v e n t s the h o m o g e n e i z a t i o n of the p o t e n t i a l v o r t i c i t y i n s i d e the c l o s e d g e o s t r o p h i c c o n t o u r s of the second l a y e r . In our n u m e r i c a l e x p e r i m e n t s , the v a l u e of F i s so s m a l l t h a t we can c o n s i d e r q = gy + 0(F ) i n the i n t e r i o r w h i l e the e x i s t e n c e o f a weak lower l a y e r f l o w (1^2 # U ) must be due to the a c t i o n o f the weak v o r t e x s t r e t c h i n g terms and the eddies t h a t have j u s t s t a r t e d s e t t i n g up i n the lower l e f t c o r n e r of the second l a y e r . However s m a l l t h i s f l o w i s , the p r e s ence of a western boundary l a y e r , where r e l a t i v e v o r t i c i t y i s g e n e r a t e d , i s so i n f l u e n t i a l on the o v e r a l l c i r c u l a t i o n t h a t t h e r e are c l o s e d stream f u n c t i o n s ip2 g o i n g e i t h e r the f o r c i n g TQ i s s t r o n g enough or or both - 90 -t h r o u g h the boundary l a y e r and r e t u r n i n g w e l l i n t o the i n t e r i o r , i n c o n t r a p o s i t i o n to RY t h e o r y . T h i s f e a t u r e w i l l be dominant on the lower l a y e r dynamics ( a t l e a s t i n the range of parameters we use and w i t h the n o n - s l i p boundary c o n d i t i o n ) even i n the presence of e d d i e s . To proceed f u r t h e r i n our a n a l y s i s , l e t us w r i t e the e q u a t i o n f o r the b a r o c l i n i c -mode ¥. S i n c e ^ = - ^ 2 » s u b s t r a c t i n g (4.1b) from (4.1a) y i e l d s F 1 J ( * 1 , * 2) - F 2 J (^ , ^ ) + g |_ (^ - ty2) = A R V 2 ( ? 1 - ? 2 ) + e v.2 ^ + c u r l T ^ (4.12) where the terms have been i g n o r e d . To study the Sverdrupean dynamics of ( 4 . 1 2 ) , we a g a i n assume t h a t the r e l a t i v e v o r t i c i t y i s v e r y s m a l l i n the i n t e r i o r . The d i m e n s i o n a l arguments such as *1 = u*1 L *-| *2 = U 2 L * 2 H1 where U 2 < 0 (^— ) and parameter v a l u e s F 2 = 1 .25 x 1 0 - 1 0 , F ] = 5 x 1 0 - 1 0 l e a d to F 1 U 1 I + 3 fx" " £ ^ ~ 1 + °< F2 J < V * 2 » + °( AH' e>' (4.13) - 91 -Hence, the i n t e r i o r dynamics o f the b a r o c l i n i c mode i s s i m i l a r to the f i r s t l a y e r dynamics. We should p o i n t out t h a t f o r l a r g e r v a l u e s o f F-| and F 2 ( t h a t i s , i n c r e a s i n g the s t r e n g t h o f the v o r t e x s t r e t c h i n g terms by i m p r o v i n g the v e r t i c a l r e s o l u t i o n o f the model), the c h a r a c t e r of the i n t e r i o r dynamics o f the few f i r s t b a r o c l i n i c modes as w e l l as of the upper l a y e r s would be l e s s of Sverdrup type and more n o n - l i n e a r . The s o l u t i o n o f (4.13) up to f i r s t o r d e r i n ( •) i s : V = Q (1 - x-) f ( y ) (4.14) where n = - (1 - 1 •) + 0 (-4-^) + 0 (A„, e) (4.15) and f ( y ) i s g i v e n by (4.7b) . S i n c e $2 1 S g i v e n i n terms o f the b a r o c l i n i c and b a r o t r o p i c modes as *2 = *B " H *1 = *B + (4.16) - 92 -then from ( 4 . 1 5 ) , ( 4 . 6 ) , and (4.16) one o b t a i n s the f o l l o w i n g e x p r e s s i o n o f i|>2 i n the i n t e r i o r . , 1 1 X 0 / 1 X v /T*Y\ f I 1 -7 \ *2,I = 1 H3Ly- ( 1 " I T ) S L n ( L > * ( 4 ' 1 7 a ) ' x y A n a l o g o u s l y , X 0 /, X v „. /TTV, 1 x y T h i s e x p r e s s i o n y i e l d s a c i r c u l a t i o n i n the second l a y e r t h a t i s , a t l e a s t , 1/13 times weaker than t h a t i n the f i r s t l a y e r and w i t h the same s i n u s o i d a l p a t t e r n . Both f e a t u r e s are q u a l i t a t i v e l y c o n f i r m e d by the n u m e r i c a l experiments except i n the l e f t lower/upper c o r n e r s where eddies are s e t t i n g up w i t h i n t e n s i t y g r e a t e r than i|>2,1 • See F i g s . 10b and 11b. 4 . 3 . 1 B o u n d a r y L a y e r E q u a t i o n s I n t h i s s u b s e c t i o n , we c o n f i r m a n a l y t i c a l l y the f e a t u r e s e x h i b i t e d by the upper and lower f l o w s i n the western boundary l a y e r . As p o i n t e d out p r e v i o u s l y , we b e l i e v e t h a t S t e w a r t ' s arguments about the r o l e of the r e l a t i v e v o r t i c i t y i n the w e s t e r n boundary are a b l e to e x p l a i n c o n c e p t u a l l y the f e a t u r e s of the n u m e r i c a l e x p e r i m e n t s . Our approach to examining the b e h a v i o u r o f the n o n - l i n e a r terms o f the e q u a t i o n s (4.1a,b) i n the presence of l a t e r a l f r i c t i o n w i l l - 93 -be i n the l i n e o f Moore's (1963) h e u r i s t i c model of a damped s t a t i o n a r y Rossby wave support e d by the eastward e f f l u x from the w e s t e r n boundry. A l t h o u g h such a model i s a s y m p t o t i c a l l y c o r r e c t at the o u t e r edge of the boundary l a y e r and agrees w i t h Munk's model when the boundary l a y e r Reynolds number tends to z e r o , i t has the d e f i c i e n c y o f r e p l a c i n g the t a n g e n t i a l and normal a d v e c t i o n of r e l a t i v e v o r t i c i t y by the normal a d v e c t i o n o f r e l a t i v e v o r t i c i t y at the o u t e r edge o f the l a y e r . T h i s s h o r t c o m i n g can be e x p l a i n e d by the main h y p o t h e s i s o f the boundary l a y e r t h e o r y , i . e . , 3 x ^ 8 y and by the l a c k of knowledge of the s t r u c t u r e of the u-component o f the v e l o c i t y w i t h i n the boundary l a y e r . S i m i l a r c r i t i c i s m can be made of the i n e r t i a l t h e o r i e s of the w e s t e r n boundary dynamics. As we have a l r e a d y p o i n t e d out i n S e c t i o n 4.1, the s t r u c t u r e o f the wester n boundary i n our experiments i s a bl e n d of i n e r t i a l and Munk's boundary l a y e r t y p e s . The Stommel l a y e r i s r e l a t i v e l y u n i m p o r t a n t so we n e g l e c t i n the s e q u e l the bottom f r i c t i o n terms. Our a n a l y s i s assumes t h a t the time dependence of the v a r i a b l e s i s i n c o n s e q u e n t i a l f o r the s t r u c t u r e of the boundary l a y e r because the eddy v i s c o s i t y c o e f f i c i e n t A H i s so l a r g e t h a t we a l l o w enough - 94 -time f o r the e x i s t e n c e of a b a l a n c e among the g e n e r a t i o n , a d v e c t i o n and d i s s i p a t i o n o f r e l a t i v e v o r t i c i t y i n the boundary l a y e r at any i n s t a n t . T a k i n g f o r gra n t e d t h a t 3 9 "3x ^ Ty w i t n : i - n t n e boundary l a y e r we w r i t e (4.1) as H i , ! X + ! ! i + H i l i ! ! ? _ ^ !jh + , c u n x N 3 ( 9 X 3 > 3x 3 9x = 3 9 X 4 3 H, » U 9 33T|>9 3T|>9 U 9 F 9 3 ^ A P 3 % 9 + 9 ^ + " ¥ a5T = T • (4.18a,b) 9i|>. where we have r e p l a c e d by - u \ ( x , y ) , ( i = 1,2), which are the i n t e r i o r v e l o c i t i e s i n the upper and lower l a y e r s . I t i s c o n s i s t e n t w i t h the i n t e r i o r f i e l d f l o w to assume a l s o t h a t i n the boundary l a y e r 1^ > \|>2. U F U F S i n c e - 5 — - - 0 ( 1 0 _ 1 ) and - § - ^ « 0 ( 1 0 ~ 2 ) , i n our p P e x p e r i m e n t s , then we can approximate (4.18a,b) as 3 4 Ui 3 J i 3 ^ \ J^h_ / C u r l T\ 0 C 9 x 3 } 9 X ~ » 3 x 4 * H 1 9 ^ 9 ^ 3 9x 3 9x ; ' U 2 9 3i|; 2 8 U 2 F 2 3^ 34i|>2 The symbol < > means t h a t the c u r l T i s not an a c t i v e term o f (4.19a) . - 95 -The above e q u a t i o n s show t h a t up to o r d e r U 1 F 1 0 ( — r — - — ) the upper l a y e r i s uncoupled from the lower one, but i t i s not obvious t h a t the lower l a y e r be independent of the upper one because we do not know ye t U 2 F 2 3 the r e l a t i v e magnitude of the term — - r — w i t h r e s p e c t to the terms of (4.19b). Thus, l e t us c o n s i d e r as t e n t a t i v e the f o l l o w i n g s c a l i n g scheme: Upper Layer L l f i = ( U . / B ) 1 / 2 , x = L^ g X, * 1 = U 1 L 1 B (4.20a) Lower Layer L 2 f i = ( U 2 / B ) 1 1 1 , x = L 2 B X , ^ 2 = U 2 L 2 B *2 (4.20b) L-|g and L 2 1 3 a r e tbe i n e r t i a l boundry l a y e r w i d t h s of the upper and lower l a y e r s , r e s p e c t i v e l y . T y p i c a l v a l u e s of L-| g i n our experiments are L-| g = 28 km, L 2 B = 14 km. The n o n - d i m e n s i o n a l e q u a t i o n s are ( d r o p p i n g a s t e r i s k s ) - 96 -3 2ip a ^ 1 a V - V - + air = zr + 0 ( } 3 aX Re, 4 * 3*2 . 3*2 . F 2 U1 L 1 B 1 3 S 7T + aX" + B L 0 B = Re7 - J " + °< >> (4.21a,b) ax" u " p " 2 " " c 2 ax where Re, = u f 2 and R e 2 = » (4.22a,b) A H 3 are the boundary l a y e r Reynolds numbers i n the f i r s t and second l a y e r s , r e s p e c t i v e l y . In o r d e r to know whether the s c a l i n g scheme (4.20a,b) y i e l d s a system of uncoupled e q u a t i o n s , we have to v e r i f y t h a t the t h i r d term of the l e f t - h a n d s i d e of (4.21b) i s much s m a l l e r than the o t h e r terms o f the e q u a t i o n ; t h i s i s t r u e i n our experiments because F-IL L 1 R II II . The boundary c o n d i t i o n s a s s o c i a t e d w i t h (4.21 a,b) are - 97 -and *1 ' *2 = C1 , 2 ( t ) ^ ± = ^ = 0 a t X = 0 , (4.23) *1 = * 1 , I *2 = *2 I a t X + », x + 0. (4.24) N o t i c e t h a t (4.23) i n c l u d e s the n o n - s l i p c o n d i t i o n at the w e s t e r n boundary o f the b a s i n , whereas (4.24) r e p r e s e n t s the t y p i c a l m atching c o n d i t i o n s between the boundary l a y e r s o l u t i o n s and the i n t e r i o r s o l u t i o n s . The d i f f e r e n c e between (4.21 a,b) and Munk's model l i e s 33<p. i n the term — ^ — which a r i s e s from the J a c o b i a n o f the 3 X J r e l a t i v e v o r t i c i t y terms. The g r e a t e r Re-| (Re2) i s , the more i n e r t i a l the f l o w becomes, e v e n t u a l l y as Re^ (Re2) •*• 0 0 the f l o w becomes F o f o n o f f ' s i n e r t i a l f l o w . On the c o n t r a r y , the s m a l l e r Rei (Re2) becomes, the more f r i c t i o n a l the f l o w i s ; t h u s , as Re-| (Re2) + 0, the f l o w reduces to Munk's s o l u t i o n because the o n l y n o n - t r i v i a l c o m b i n a t i o n o f (4.21) i s , f o r t h i s c a s e , between the r i g h t - h a n d s i d e and the second term of the l e f t - h a n d s i d e . S o l u t i o n s to (4.21 a,b) are of the form ^ j B (x,y) = A e w X , i P 2 > B (x,y) = B e a X , (4.25a,b) - 98 -S u b s t i t u t i n g (4.25) i n t o (4.21) y i e l d s u>3 - Re 1 (to 2 + 1) = 0, u»0 = 0, a 3 - R e 2 ( a 2 + 1) = 0, a Q = 0. (4.26a,b) The above e q u a t i o n s are two c u b i c e q u a t i o n s . The n a t u r e of t h e i r r o o t s depends on the v a l u e s o f the c o e f f i c i e n t s (Abramowitz and Stegun, 1964). In our e x p e r i m e n t s , when the v a l u e s of U-| (y) and U2(y) are ( y ) , U 2 ( y ) > 0 or i f n e g a t i v e , | u i ( y ) | . |u 2<y) | < ( f 1 ) 1 7 3 3 1 / 3 A 2 / 3 the r o o t s of (4.26) are a).] = pi + ib-j , <i>2 = p 2 + i b 2 , tog > 0, a1 = p 2 + ^°2> a2 = p2 + ^ 2 ' a3 ^ 0 ' (4.27a,b) where p. = -{{ [ ( a i + c . ) 1 ' 3 + ( a . - c . ) 1 / 3 ] - ^ i } , Re. b. = / 3 [ ( a . + c . ) 1 / 3 - (a. - C i ) 1 / 3 ] , (i=1,2) For p o s i t i v e v e l o c i t i e s U-) (y) and U2(y) - 99 -1 R e i a i = R e i ^2 + TT* ' Re? c i = R e i (1 + 2 7 L ) 1 / 2 ' (i=1,2). (4.29) I f (y) and ^ ( y ) are n e g a t i v e 1 R e i a i = R e i ( 2 " TT> ' Re? c. = Re. (\ - T f 2 L ) U 2 , (i=1,2). (4.30) The s o l u t i o n s ty] t% and i f>2 ,B a r e , t h e n , of the form 1 _ 2 x x =— x U). L L 3 L *1 B = A 0 + A 1 6 1 B + A 2 6 1 B + A 3 e 1 B a1 a2 T X T X a ^ T + 2 fi = B Q + B 1 e ^ D + B 2 e + B 3 e z . (4.31) x By imposing the boundary c o n d i t i o n s (4.23) and (4.24) and t a k i n g i n t o account t h a t the u n i f o r m s o l u t i o n v a l i d f o r t h e whole domain i s g i v e n as ty (x,y) = ty ( i n t e r i o r ) + ty (boundary) - ty (matching) y i e l d s - 100 -x - 1» j [e l v. J + e J ] , (.1=1 , 2 ) , (4.32) where v . = b x e . = t a n J -1 J J B + i T ( x»y) a r e g i v e n by ( 4 . 1 7 ) . When the v e l o c i t i e s U-| (y) and U2(y) are p o s i t i v e the s o l u t i o n s (4.32) are v a l i d f o r the no r t h w e s t r e g i o n of the b a s i n where the f l o w l e a v e s the boundary and r e - e n t e r s the i n t e r i o r . The wavy n a t u r e of (4.32) i n t h i s r e g i o n i n c r e a s e s w i t h the v a l u e taken by Re^ and r e p r e s e n t s a damped s t a n d i n g wave whose r o l e , as we have a l r e a d y mentioned, i s to c o n t r i b u t e to the d i s s i p a t i o n o f the p o s i t i v e r e l a t i v e v o r t i c i t y t h a t the f l o w owns when i t l e a v e s the boundary. A way to v e r i f y q u a l i t a t i v e l y the n u m e r i c a l r e s u l t s i s t o computing the damping s c a l e LD^, the wave-length A ^ and the number of the u n d u l a t i o n s t h a t the s t a n d i n g wave undergoes b e f o r e b e i n g c o m p l e t e l y damped. - 101 -I n our experiment, we have the f o l l o w i n g v a l u e s Upper L a y e r Lower L a y e r Re 1 = 1.333 Re 2 = 0.167 LD.| « 130 km L j ^ * 63 km X1 « 208.6 km X2 = 1 8 6 * 3 ^ N o t i c e t h a t the wavelengths o f the s t a n d i n g waves i n the upper and lower l a y e r s are comparable; however, L ^ « 2 L ^ , so the damping of the lower s t a n d i n g wave w i l l be much s t r o n g e r than t h a t of the upper l a y e r , a f a c t t h a t the n u m e r i c a l r e s u l t s c o n f i r m . I f the wavelength i s g r e a t e r than the damping s c a l e , the number of u n d u l a t i o n s w i l l depend on the r a t i o — - Thus, we have, ^- = 1 .6, e " 1 * 6 - 20% LD1 so i t w i l l be p o s s i b l e to observe a second c r e s t o f the s t a n d i n g wave; t h i s means t h a t Nu-| = 2 as we can see i n F i g u r e s 9-11. For the lower l a y e r ~ - = 2.96, e ~ 2 , 9 6 - 5%, LD1 so a second c r e s t of the wave w i l l be d r a s t i c a l l y damped ( F i g s . 9-11). - 102 -We s h o u l d p o i n t out t h a t f o r a s y m p o t o t i c v a l u e s o f Re^, i . e . , R e i > 1 and Re^ < 1, the r e s u l t s of eqn. (4.32) a c c o r d w i t h those g i v e n by P e d l o s k y (1979), s e c t i o n s 5.7 and 5.8. We have j u s t shown t h a t a s i m p l e a n a l y t i c a l approach based on the boundary l a y e r t h e o r y and the Sverdrup regime i n the i n t e r i o r i s a b l e to e x p l a i n , a t l e a s t q u a l i t a t i v e l y , the dynamics of our n u m e r i c a l experiments w i t h the n o - s l i p c o n d i t i o n a t the s o l i d b o u n d a r i e s . The main p o i n t of our a n a l y s i s l i e s i n the f a c t t h a t weak v o r t e x s t r e t c h i n g terms and l o c a l boundary s c a l e a n a l y s i s p e r m i t s us to uncouple the dynamics of b o t h l a y e r s and, t h e r e f o r e , to study the m o t i o n o f each l a y e r u s i n g the boundary l a y e r t h e o r y f o r a homogeneous ocean. In the case where the v o r t e x s t r e t c h i n g terms were l a r g e enough to y i e l d a s t r o n g c o u p l i n g between the l a y e r s , we s t i l l can perform a l o c a l boundary s c a l e a n a l y s i s to i s o l a t e the boundary l a y e r dynamics o f each l a y e r and then to s o l v e a system of f o u r t h o r d e r p a r t i a l d i f f e r e n t i a l e q u a t i o n s i n one v a r i a b l e t h a t would r e p l a c e (4.21 a,b). In t h i s case, e q u a t i o n (4.11) a l s o p r e d i c t s t h e e x i s t e n c e of c l o s e d g e o s t r o p h i c c o n t o u r s i n the northwest r e g i o n of the second l a y e r and a c c o r d i n g to YR (1982) the h o m o g e n i z a t i o n o f p o t e n t i a l v o r t i c i t y w i t h i n those c o n t o u r s ; however, w i t h base on our method to o b t a i n the a n a l y t i c a l s o l u t i o n s o f the 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 , we f e e l t h a t - 103 -the f u n c t i o n G(q) i n (4.10) i s so dependent on the western boundary dynamics t h a t i t p r e v e n t s the h o m o g e n i z a t i o n of t h e p o t e n t i a l v o r i t i c i t y . In t h i s r e s p e c t , we can c o n s i d e r our method as a g e n e r a l i z a t i o n of t h a t of I e r l e y and Young's (1983) which was used to study the i n f l u e n c e o f the boundary l a y e r on the d i s t r i b u t i o n of p o t e n t i a l v o r t i c i t y i n the i n t e r i o r . They choose a Stommel boundary l a y e r c l o s u r e i n a l i n e a r i z e d problem ( t h e i r n o n - l i n e a r i t i e s are s t r o n g v o r t e x s t r e t c h i n g terms which they l i n e a r i z e d by u s i n g an " a n s a t z " such as i|>2 = k ^B> k a c o n s t a n t ) and n e g l e c t the r e l a t i v e v o r t i c i t y a l l over the whole domain. However, the h o m o g e n i z a t i o n of p o t e n t i a l v o r t i c i t y has r e c e n t l y been observed by McDowell et a l . (1982), Coats (1981), e t c . , so any r e l i a b l e n u m e r i c a l model has to be a b l e to reproduce t h i s f e a t u r e ( H o l l a n d , 1985). We b e l i e v e t h a t a p o s s i b l e way to o b t a i n t h a t i n our model i s : (1) by i n c r e a s i n g the v e r t i c a l r e s o l u t i o n , p a r t i c u l a r l y i n the upper t h e r m o c l i n e , i n o r d e r to f o r c e the e x i s t e n c e o f c l o s e d g e o s t r o p h i c c o n t o u r s which h e r a l d the onset of b a r o c l i n i c i n s t a b i l i t y ; (2) by d e c r e a s i n g the v a l u e o f Aj|, say Aj^ = 50 m2 s - 1 i n s t e a d of A^ = 200-300 m 2 s " 1 , and i n c r e a s i n g the f o r c i n g term up t o a more r e a l i s t i c v a l u e . The consequences of (2) are a r e d u c t i o n i n the d i s s i p a t i o n time and an i n c r e a s e i n the amount of p o s i t i v e r e l a t i v e v o r t i c i t y g e n e r a t e d at the boundary, so the s t a n d i n g waves might become - 104 -u n s t a b l e and shed eddies to d i s s i p a t e the r e l a t i v e v o r t i c i t y coming from the boundary. I t i s p o s s i b l e that d i s t u r b a n c e s am p l i f y i n t h i s r e g i o n and e v e n t u a l l y e q u i l i b r a t e by inducing a mean flow i n the lower l a y e r which removes the r e v e r s a l i n s i g n of the mean p o t e n t i a l v o r t i c i t y . A n a l y t i c a l d e t a i l s of weakly n o n - l i n e a r i n s t a b i l i t y in the presence of f r i c t i o n can be seen i n Pedlosky (1970, 1971 and 1972) and Newell (1972). To accomplish t h i s e q u i l i b r a t i o n , a l a r g e f r a c t i o n of the t o t a l t r a n s p o r t must migrate down to the lower l a y e r . This process may make the p o t e n t i a l v o r t i c i t y g r a d i e n t equal to' zero; i f t h i s i s so, then the b a r o c l i n i c i n s t a b i l i t y , l o c a l i z e d i n the northwest, produces a source r e g i o n of uniform p o t e n t i a l v o r t i c i t y . In order to confirm these somewhat s p e c u l a t i v e arguments, new numerical experiments are underway. The f i r s t group of such experiments deals with the inc r e a s e of v e r t i c a l r e s o l u t i o n i n the upper thermocline, with the r e s t of parameters ( i . e . , Ax, Ay, AR. C u r l x, etc.) being s i m i l a r to those we have used so f a r , and i t i s intended to extend the computations up to at l e a s t f i v e years. The main purpose these f i r s t experiments i s to compare t h e i r r e s u l t s with the present ones and to v a l i d a t e our a n a l y s i s . The second group of experiments w i l l i n c l u d e (1) and (2) together. This experiments are co m p u t a t i o n a l l y very demanding because, i n a d d i t i o n being s t r o n g l y n o n - l i n e a r , a very narrow boundary - 105 -l a y e r d e v e l o p s a t t h e w e s t e r n s i d e t h a t r e q u i r e s A x « 5 k m i n a s t r i p a d j a c e n t t o t h a t s i d e a n d , t h e r e f o r e , A t < 30 m i n u t e s i f a n e x p l i c i t s c h e m e i s t o b e u s e d . T h e o n l y r a t i o n a l w a y o f o v e r c o m i n g s u c h a s e v e r e p r o b l e m i s b y u s i n g a t i m e i m p l i c i t s c h e m e , w h i c h a l l o w s a A t « 6 t o 8 h o u r s , i n c o n j u n c t i o n w i t h a l e a s t s q u a r e s - c o n j u g a t e g r a d i e n t m e t h o d t o r e d u c e t h e f u l l n o n - l i n e a r p r o b l e m t o a s e q u e n c e o f P o i s s o n p r o b l e m s a t e v e r y t i m e s t e p . I n o r d e r t o s o l v e a s e f f i c i e n t l y a n d e c o n o m i c a l l y a s p o s s i b l e t h i s s e q u e n c e , w e a r e p r e s e n t i m p l e m e n t i n g a m o r e p o w e r f u l v e r s i o n o f t h e c o n j u g a t e g r a d i e n t m e t h o d . - 106 -CONCLUSIONS The numerical model co n s t r u c t e d here i s the f i r s t step of a longer s e r i e s of numerical s t u d i e s on ocean c i r c u l a t i o n u s i n g higher accuracy models. We have shown that a f i n i t e element f o r m u l a t i o n with lumping and v a r i a b l e g r i d spacing of the d i s s i p a t i v e q u a s i - g e o s t r o p h i c equations i s able to d e l i v e r r e l i a b l e s o l u t i o n s . To take advantage of the i m p l i c i t n e s s of the method and i t s higher accuracy to r e p r e s e n t a wave, a time i m p l i c i t scheme has to be used. I f one c o n s i d e r s that the ocean water i s a r e a l f l u i d then the i m p o s i t i o n of the n o - s l i p c o n s t r a i n t on the s o l i d boundaries removes the i n c o m p a t i b i l i t i e s between c e r t a i n p h y s i c a l processes and the q u a s i - g e o s t r o p h i c equations which a r i s e when f r e e - s l i p c o n s t r a i n t s are used (McWillians 1977). However, f o r a given range of parameters such as , Ax, Ay, F^ and F^, which i s eddy generating as the l a t t e r c o n s t r a i n t i s imposed, the flow becomes quasi-laminar i n both upper and lower l a y e r s with the n o - s l i p c o n d i t i o n . Thus, i t i s concluded that the g e n e r a t i o n of eddies with t h i s c o n s t r a i n t r e q u i r e s a much lower A^ and higher F-| and F 2 . The western boundary l a y e r i s a c t i v e i n c o n t r o l l i n g the dynamics of the second l a y e r , as i s seen in the a n a l y t i c a l model, and i t i s expected that even at higher F-| and F 2 - 107 -such t h a t the lower g e o s t r o p h i c c o n t o u r s are c l o s e d the w e s t e r n boundary l a y e r w i l l p r e v e n t the h o m o g e n i z a t i o n o f the p o t e n t i a l v o r t i c i t y i n the second l a y e r u n l e s s A H i s s m a l l enough to a l l o w the appearance o f i n s t a b i l i t i e s i n the n o r t h w e s t e r n meanders. - 108 -REFERENCES Abramowitz, M. and Stegun, I.A. 1964. Handbook o f M a t h e m a t i c a l F u n c t i o n s . N a t i o n a l Bureau o f S t a n d a r d s . Adams, R.A. 1975. Sobolev Spaces. Academic P r e s s , New York. B a r r e t , K.E. 1978. A v a r i a t i o n a l p r i n c i p l e f o r the stream f u n c t i o n - v o r t i c i t y f o r m u l a t i o n of the N a v i e r - S t a t e s e q u a t i o n s i n c o r p o r a t i n g n o - s l i p s c o n d i t i o n s . J . Comp. P h y s i c s , 153-161. Basdevant, C , M. L e s i e u r , and R. Sadoury. 1978. S u b g r i d - s c a l e m o d e l i n g of en s t r o p h y t r a n s f e r i n tw o - d i m e n s i o n a l t u r b u l e n c e . J . Atm. S c i . 35, 1028-1042. Basdevant, C , and R. Sadourny. 1 975. E r g o d i c p r o p e r t i e s o f i n v i s c i d t r u n c a t e d models of t w o - d i m e n s i o n a l i n c o m p r e s s i b l e f l u i d s . J . F l u i d Mech., 69: 673-688. Be n n e t t , A.F., and P.E. K l o e d e n . 1981. The d i s s i p a t i v e q u a s i g e o s t r o p h i c e q u a t i o n s . Mathematika. 28, 265-285. B l a n f o r d , R. 1971. Boundary c o n d i t i o n s i n homogeneous ocean models. Deep-Sea, Res. 18, 739-751. B r i g g s , W.L., N e w e l l , A.C. and S a r i e , T. 1983. F o c u s i n g : A mechanism f o r i n s t a b i l i t y o f n o n - l i n e a r f i n i t e d i f f e r e n c e e q u a t i o n s . J . Comp. P h y s i c s , 51, 83-106. Bryan, K. 1963. A n u m e r i c a l i n v e s t i g a t i o n of a n o n - l i n e a r model o f a wind d r i v e n ocean. J . Atmos. S c i . 20, 594-606. Coa t s , D.A. 1981. An estime of a b s o l u t e g e o s t r o p h i c v e l o c i t y from the d e n s i t y f i e l d i n the n o r t h e a s t e r n P a c i f i c Ocean. J . Geophy. Res., 86, 8031-8036. Charney, J.G. 1971. G e o s t r o p h i c t u r b u l e n c e . J . Atm. S c i . 28: 1087-1095. deSzoeke, R.A. 1985. W i n d - d r i v e n mid-ocean b a r o c l i n i c g y r e s over topography: A c i r c u l a t i o n e q u a t i o n e x t e n d i n g the Sverdrup r e l a t i o n . J . Mar. Res. 43, 793-824. F i x , G.J. 1975. F i n i t e element models f o r ocean c i r c u l a t i o n problems. SIAM J . A p p l . Math. 29, 371-387. F o f o n o f f , N.P. 1954. Steady f l o w i n a f r i c t i o n l e s s homogeneous ocean. J . Marine Res. 13, 254-2 62. - 109 -G l o w i n s k i , R. and P i r o n n e a u , 0. 1979. N u m e r i c a l methods f o r the f i r s t b i h a r m o n i c e q u a t i o n and f o r the tw o - d i m e n s i o n a l Stokes problem. SIAM Re. 2121, 2, 167-212. G l o w i n s k i , R., K e l l e r , H.B. and R e i n h a r t , L. 1985. C o n t i n u a t i o n - C o n j u g a t e g r a d i e n t methods f o r the l e a s t squares s o l u t i o n of n o n - l i n e a r boundary v a l u e prolems. SIAM J . S c i . S t a t . Comput. 6,4, 793-832. Gresho, P.M. and R.L. Lee. 1981. Don't suppress the w i g g l e s — they are t e l l i n g you something! Comp. and F l u i d s . 9, 223-253. H a i d v o g e l , D.B., et a l . 1980. The a c c u r a c y , e f f i c i e n c y , and s t a b i l i t y o f t h r e e n u m e r i c a l models w i t h a p p l i c a t i o n t o open ocean problems. J . Comp. Phys. 34, 1-53. H o l l a n d , W.R. 1985. S i m u l a t i o n o f mesoscale ocean v a r i a b i l i t y i n m i d - l a t i t u d e g y r e s . In Adv i n Geophy. V o l . 28A, 439-523. H o l l a n d , W.R. 1978. The r o l e of mesoscale eddies i n the g e n e r a l c i r c u l a t i o n of the ocean n u m e r i c a l e x p e r i m e n t s u s i n g a wind d r i v e n q u a s i - g e o s t r o p h i c model. J . Phys. Oceanog. 8, 363-392. H o l l a n d , W.R., and L.B. L i n . 1975a. On the o r i g i n o f mesoscale eddies and t h e i r c o n t r i b u t i o n to the g e n e r a l c i r c u l a t i o n of the ocean. I . A p r e l i m i n a r y n u m e r i c a l e x p e r i m e n t . J . Phys. Oceanogr. 5, 642-657. H o l l a n d , W.R., and L.B. L i n . 1975a. On the o r i g i n o f mesoscale e d d i e s and t h e i r c o n t r i b u t i o n to the g e n e r a l c i r c u l a t i o n of the ocean. I I . A parameter s t u d y . J . Phys. Oceanogr. 5, 658-669. I e r l e y , G.R. and Young, W.R. 1983. Can the western boundary l a y e r a f f e c t the p o t e n t i a l v o r t i c i t y d i s t r i b u t i o n i n the Sverdrup i n t e r i o r o f a wind gyre? J . Phys. Oceanog. 13, 1753-1763. J e s p e r s e n , D.C. 1974. Arakawa's method i s a f i n i t e element method. J . Comput. Phys. 16, 383-390. K i l l w o r t h , P.D. 1985. A note on smoothing t e c h n i q u e s f o r l e a p f r o g t i m e - i n t e g r a t i o n schemes. Ocean Model 60, 5-8. K r a u s s , W. 1973. Dynamics o f the Homogeneous and the Quasi-homogeneous Ocean. Gerbruder B o r n t r a e g e r , B e r l i n . - 110 -K r e i s s , H.O. 1978. N u m e r i c a l Methods f o r S o l v i n g Time-Depending Problems f o r P a r t i a l D i f f e r e n t i a l E q u a t i o n s . Les P r e s s e s de L ' U n i v e r s i t e de M o n t r e a l . L u y t e n , J.R., J . P e d l o s k y , and H. Stommel. 1983. The v e n t i l a t e d t h e r m o c l i n e . J . Phys. Oceanogr. 13, 292-309. M a r s h a l l , J.C. 1982. On the treatment of the l a t e r a l b o u n d a r i e s i n q u a s i - g e o s t r o p h i c ocean models. U n p u b l i s h e d M a n u s c r i p t . M a r s h a l l , J.C. 1984. Eddy-mean f l o w i n t e r a c t i o n i n a b a r o t r o p i c ocean model. Q u a r t . J.R. Met. Soc. 110, 573-590. McDowell, S., R h i n e s , P. and K e f f e r , T. 1982. N o r t h - A t l a n t i c p o t e n t i a l v o r t i c i t y and i t s r e l a t i o n to the g e n e r a l c i r c u l a t i o n . J . Phys. Oceaog. 12, 1417-1436. M c W i l l i a m s , J.C. 1977. A note on a c o n s i s t e n t q u a s i - g e o s t r o p h i c model i n a m u l t i p l e connected domain. Dyn. Atmos. Ocean, 1, 427-441 . M i j a i l o v , P.V. 1978. P a r t i a l D i f f e r e n t i a l E q u a t i o n s ( E n g l i s h T r a n s l a t i o n ) , M i r . Moscow. Moore, D.W. 1963. Rossby waves i n ocean c i r c u l a t i o n . Deep-Sea Res. 10, 735-747. Munk, W.H. 1950. On the wind d r i v e n c i r c u l a t i o n . J . Meteor. 7, 79-93. N e w e l l , A.C. 1972. The p o s t b i f u r c a t i o n stage o f a b a r o c l i n i c i n s t a b i l i t y . J . Atm. S c i . 29, 69-76. P e d l o s k y , J . 1979. G e o p h y s i c a l F l u i d Dynamics, s p r i n g e r - V e r l a g , N.Y. P e d l o s k y , J . 1972. L i m i t c y c l e s and u n s t a b l e b a r o c l i n i c waves. J . Atm. S c i . 29, 53-63. P e d l o s k y , 1971. F i n i t e - a m p l i t u d e b a r o c l i n i c waves w i t h s m a l l d i s s i p a t i o n . J . Atm. S c i . 28, 587-597. P e d l o s k y , J . 1970. F i n i t e - a m p l i t u d e b a r o c l i n i c waves. J . Atm. S c i . 27, 15-30. P h i l l i p s , N.A. 1959. An example o f n o n - l i n e a r c o m p u t a t i o n a l i n s t a b i l i t y . The Atmosphere and the Sea i n M o t i o n , Rossby Memorial Volume. R o c k e f e l l e r I n s t i t u t e P r e s s , 501-504. - 111 -R h i n e s , P.B. and W.R. Young, 1982. Mid-Ocean Gyres. J . Mar. Res. 40 ( s u p p . ) , 559-596. R i c h t m y e r , R.D. and K.W. Morton. 1967. D i f f e r e n t methods f o r i n i t i a l v a l u e problems. I n t e r s c i e n c e P u b l i s h e r s . Roache, P.J. 1972. C o m p u t a t i o n a l F l u i d Dynamics. Hermosa P r e s s , A l b u r q u e r q u e , N.M. Robinson, A.R., D.E. H a r r i s o n . , Y. M i n t z , and A . J . Semtner. 1977. Eddies and the g e n e r a l c i r c u l a t i o n o f an i d e a l i z e d o c e a n i c g y r e : A wind and t h e r m a l l y d r i v e n p r i m i t i v e e q u a t i o n n u m e r i c a l experiment. J . Phys. Oceanogr. 7, 182-207. Sadourny, R. and C. Basdevant. 1981. Une c l a s s e d ' o p e r a t e u r s adaptes a l a m o d e l i s a t i o n de l a d i f f u s i o n t u r b u l e n t e en d i m e n s i o n deux. C R . Acad. S c i . P a r i s . 292 I I , 1061-1064. Sadourny, R. 1975. The dynamics o f f i n i t e d i f f e r e n c e models of the s h a l l o w water e q u a t i o n s . J . Atmos. S c i . 32, 680-689. Semtner, A . J . , and Y. M i n t z . 1977. N u m e r i c a l s i m u l a t i o n o f the G u l f Stream and mid-ocean e d d i e s . J . Phys. Oceanogr. 7, 208-230. Semtner, A . J . and W.R. H o l l a n d . 1978. I n t e r - c o m p a r i s o n o f q u a s i - g e o s t r o p h i c s i m u l a t i o n s o f the w e s t e r n N o r t h A t l a n t i c c i r c u l a t i o n w i t h p r i m i t i v e e q u a t i o n r e s u l t s . J . Phys. Oceanogr. 8, 735-754. S t a n i f o r t h , A.N. and H.L. M i t c h e l l . 1977. A s e m i - i m p l i c i t f i n i t e element b a r o t r o p i c model. Month Weath. Rev. 105, 154-169. S t e w a r t , R.W. 1964. I n f l u e n c e of f r i c t i o n on i n e r t i a l models i n o c e a n i c c i r c u l a t i o n . I n : S t u d i e s on Oceanography: Papers d e d i c a t e d to P r o f e s s o r H i d a k a i n commemoration of h i s 60th b i r t h d a y , K. Y o s h i d a , e d i t o r . Tokyo U n i v e r s i t y , G e o p h y s i c a l I n s t i t u t e . Stommel, H. 1948. The westward i n t e n s i f i c a t i o n of w i n d - d r i v e n ocean c u r r e n t s . Trans. Amer. Geophys. Un i o n . 99, 202-206. Thomee, V. 1985. G a l e r k i n F i n i t e Element Methods f o r P a r a b o l i c Problems. L e c t u r e Notes i n Math. No. 1054. S p r i n g V e r l a g , B e l i n . Young, W.R. and P.B. R h i n e s . 1982. A t h e o r y of the w i n d - d r i v e n c i r c u l a t i o n T I gyres w i t h w e s t e r n boundary l a y e r s . J . Mar. Res. 40, 849-872. - 112 -APPENDIX A Equation of The Relative V o r t i c i t y i n a S t r a t i f i e d Mid-Ocean  Gyre C o n s i d e r a r e s t r i c t e d o c e a n i c d o m a i n D . T h e d y n a m i c a l v a r i a b l e s u s u a l l y r e q u i r e d t o d e s c r i b e t h e m o t i o n i n s u c h a s y s t e m a r e t h e d e n s i t y p , t h e p r e s s u r e p , t h e v e c t o r v e l o c i t y u = ( u , v , w ) , a n d c e r t a i n f u r t h e r t h e r m o d y n a m i c v a r i a b l e s a s t h e t e m p e r a t u r e T , t h e i n t e r n a l e n e r g y p e r u n i t o f m a s s e a n d t h e s p e c i f i c e n t r o p y s . I n c e r t a i n s i t u a t i o n s , d e p e n d i n g o n t h e p h y s i c a l n a t u r e o f t h e f l u i d , a d d i t i o n a l v a r i a b l e s s u c h a s s a l i n i t y m a y b e r e q u i r e d , o r i n c a s e s w h e n t h e t h e r m o d y n a m i c r e l a t i o n s c a n b e s i m p l i f i e d , s o m e s t a t e v a r i a b l e s c a n b e d i s p e n s e d . I n a n E u l e r i a n d e s c r i p t i o n o f t h e m o t i o n , t h e d y n a m i c a l v a r i a b l e s a r e f u n c t i o n s o f t h e v e c t o r p o s i t i o n x = ( x , y , z ) a n d o f t i m e t . T h e e q u a t i o n s o f m o t i o n o f a B o u s s i n e s q f l u i d i n a r o t a t i n g c o o r d i n a t e s y s t e m a r e : C o n s e r v a t i o n o f m a s s D p T 5 t + V • p u = 0 o r D p t i t + p V • u = 0 (A.1) M o m e n t u m e q u a t i o n + 2ftAtf = - 1 V p + VO + JF (A.2) H e a t e q u a t i o n - 113 -° | = < V 2 T + £ - (A.3) where -jy^ - = + u • V i s the m a t e r i a l d e r i v a t i v e & i s the speed of r o t a t i o n of the e a r t h , = 7.2910" 5 s~l K i s the c o e f f i c i e n t of t h e r m a l d i f f u s i v i t y Cp i s the s p e c i f i c heat a t c o n s t a n t p r e s s u r e Q i s the r a t e of heat a d d i t i o n per u n i t o f mass by heat s o u r c e s . To complete the system, f u r t h e r thermodynamic s t a t e r e l a t i o n s e x p r e s s i n g the p h y s i c s n a t u r e of the f l i d are r e q u i r e d . For s a l t water p = p(p,T,S) (A.4) I f the e f f e c t s of c o m p r e s s i b i l i t y are m i n o r , then (A.4) can be w r i t t e n as p = p Q[1 - a(T - T Q ) ] (A.5) Hence, e q u a t i o n (2.3) can be w r i t t e n e n t i r e l y i n terms o f the d e n s i t y as (A.6) - 114 -I t i s worth n o t i n g t h a t e q u a t i o n s (A.1) and ( A . 6 ) , a l t h o u g h b o t h g i v e n i n terms of d e n s i t y , are r e l a t e d to d i f f e r e n t p h y s i c a l p r i n c i p l e s ; the former ex p r e s s e s mass c o n s e r v a t i o n and the l a t t e r , f o r a l i q u i d , e x p r e s s e s energy c o n s e r v a t i o n . I f we assume t h a t the ocean water i s an i n c o m p r e s s i b l e l i q u i d , or v e r y n e a r l y so, d e n s i t y d i f f e r e n c e s a re so s l i g h t t h a t t h e i r e f f e c t s on the mass b a l a n c e can be n e g l e c t e d , so t h a t E q u a t i o n (A.7) does not mean = 0. Indeed, ^ i s g i v e n by (A. 6) i n terms of i n t e r n a l h e a t i n g and heat d i f f u s i o n . jj£ i s i d e n t i c a l l y zero i f the f l u i d i s i n c o m p r e s s i b l e and the m o t i o n i s a d i a b a t i c . body f o r c e and the n o n - c o n s e r v a t i v e f o r c e s r e s p e c t i v e l y . $ i s the p o t e n t i a l o f the c o n s e r v a t i v e f o r c e s and i s , i n t h i s s t u d y , the f r i c t i o n a l f o r c e a c t i n g on the f l u i d which i s g i v e n i n terms o f the v e c t o r v e l o c i t y as V • u = 0 (A.7) The terms, V$ e q u a t i o n (2.2) r e p r e s e n t the (A.8) - 115 -A J J and Ay are the h o r i z o n t a l and v e r t i c a l eddy v i s c o s i t y c o e f f i c i e n t s , r e s p e c t i v e l y , which are taken as c o n s t a n t . The l a r g e - s c a l e gyre ocean motions are p r i m a r i l y c h a r a c t e r i z e d by the f o l l o w i n g n o n - d i m e n s i o n a l parameters Rossby Number e = j ^ j - << 1 (A. 9a) B u r g e r ' s Number S = g A£ — D = ( ^ 2 < < ; } (A. 9b) p 4 n z l / L V e r t i c a l R a t i o Number 6 = °- « 1 (A. 9c) where U,L,D denote t y p i c a l s c a l e s o f v e l o c i t y , h o r i z o n t a l l e n g t h and depth o f the m o t i o n , r e s p e c t i v e l y . The s m a l l v a l u e o f the Rossby Number e i m p l i e s t h a t the C o r i o l i s f o r c e w i l l be dominant over most p a r t of the ocean g y r e s except i n narrow areas where the i n e r t i a l and the f r i c t i o n a l terms o f the momentum e q u a t i o n s can become i m p o r t a n t . On the o t h e r hand, the s t r o n g s t r a t i f i c a t i o n o f the ocean lea d s t o l a r g e - s c a l e o c e a n i c motions which are n e a r l y h o r i z o n t a l . The h y d r o s t a t i c b a l a n c e o f the motion i s guaranteed by 6 < 1. So as a f i r s t a p p r o x i m a t i o n the dynamics of the l a r g e - s c a l e o c e a n i c motions can be d e s c r i b e d by the g e o s t r o p h i c and h y d r o s t a t i c e q u a t i o n s ( P e d l o s k y , 1979). N a t u r a l l y , as i s e v i d e n t from the p a r a m e t r i z a t i o n i n v o l v e d , the g e o s t r o p h i c - 116 -a p p r o x i m a t i o n breaks down i n the v i c i n i t y o f the equator and i n h i g h l a t i t u d e s ; i n the l a t t e r a r e a , t h e r e i s no way w i t h the g e o s t r o p h i c r e l a t i o n s o n l y t o c a l c u l a t e the p r e s s u r e f i e l d . As a d i a g n o s t i c t o o l , the g e o s t r o p h i c e q u a t i o n s i n f o r m us t h a t i f the v e l o c i t y f i e l d s l o w l y v a r i e s i n t i m e , then the C o r i o l i s f o r c e w i l l be b a l a n c e d by the e v o l v i n g p r e s s u r e g r a d i e n t ; however, no i n f o r m a t i o n about the p r e s s u r e f i e l d i s conveyed by such an a p p r o x i m a t i o n . In f a c t , any p r e s s u r e f i e l d w i t h a s m a l l Rossby number i s a c c e p t a b l e because the c o n t i n u i t y e q u a t i o n (A.7) i s always s a t i s f i e d up to 0 ( e ) . T h i s i n d e t e r m i n a n c y of the g e o s t r o p h i c a p p r o x i m a t i o n can be removed by t a k i n g i n t o account the e f f e c t s o f s m a l l d e p a r t u r e s from g e o s t r o p h y , i n o t h e r words, by i n t r o d u c i n g h i g h e r o r d e r dynamics. In doing so, we s h o u l d n o t i c e t h a t by a p p l y i n g the c u r l o p e r a t o r to the g e o s t r o p h i c e q u a t i o n s the c o n t r i b u t i o n of the C o r i o l i s and p r e s s u r e g r a d i e n t f o r c e s to the v o r t i c i t y b a l a n c e v a n i s h , so t h a t i t i s i n terms of the h i g h e r dynamics t h a t the v o r t i c i t y b a l a n c e can be a c h i e v e d . The f o r c e s which determine such a dynamics are u s u a l l y the s m a l l n o n - g e o s t r o p h i c f o r c e s as: a. t h e i n e r t i a l f o r c e s which y i e l d to n o n - g e o s t r o p h i c v e l o c i t i e s o f o r d e r 0 ( e ) ; b. the v o r t e x tube s t r e t c h i n g by the Ekman sucti o n / p u m p i n g v e l o c i t i e s at the b o t t o m / s u r f a c e , - 117 -a c t i n g on the p l a n e t a r y v o r t i c i t y f i l a m e n t s , w i l l produce a v a r i a t i o n o f the r e l a t i v e v o r t i c i t y p r o p o r t i o n a l t o 0 ( E y 2 ) . (E = Ik^/lttD2) . These n o n - g e o s t r o p h i c e f f e c t s a r i s e from the v e r t i c a l f r i c t i o n and g i v e p l a c e both to an Ekman bottom l a y e r which a c t s as a s i n k o f v o r t i c i t y and a s u r f a c e Ekman l a y e r at the b a s i s of which the e x t e r n a l m e c h a n i c a l f o r c e s i n p u t r e l a t i v e v o r t i c i t y ; c. h o r i z o n t a l d i f f u s i o n of momentum which i s 2A E p r o p o r t i o n a l to 0(E„). (E„ = — — = 2R~ 1 ) ; 2f t L Z e e d. s o u r c e s , s i n k s and d i f f u s i o n of h e a t . The a b s o l u t e v o r t i c i t y i n our r o t a t i n g ocean system i s t + 2$, t = VAu i s the r e l a t i v e v o r t i c i t y and 2$, i s the p l a n e t a r y v o r t i c i t y which dominates \ because E « 1. The dominancy o f the p l a n e t a r y v o r t i c i t y i n t r o d u c e s some d i s t i n c t i v e p e c u l i a r i t i e s i n t o the v o r t i c i t y dynamics of a r o t a t i n g system. Thus, f o r i n s t a n c e , the c l a s s i c e f f e c t o f v o r t e x l i n e s t r e t c h i n g o f t h r e e d i m e n s i o n a l n o n - r o t a t i n g f l u i d s i s absent h e r e . I t i s not u s e f u l now to take % as a t r a c e r o f the motion because the p r o d u c t i o n j s | 2 and the a c c e l e r a t e d d i s s i p a t i o n o f k i n e t i c energy cannot occur at p l a n e t a r y s c a l e s a t which R e i s l a r g e . I n s t e a d , the p o t e n t i a l v o r t i c i t y q w i l l be a good t r a c e l i k e s c a l a r on s u r f a c e s of c o n s t a n t p o t e n t i a l d e n s i t y . D e f o r m a t i o n o f - 118 -c o n t o u r s o f q l e a d s to d i s s i p a t i o n of q 2 i t s e l f . The e q u a t i o n o f the p o t e n t i a l v o r t i c i t y q as a conserved p r o p e r t y i s g i v e n by E r t e l ' s theorem. F o l l o w i n g P e d l o s k y (1979) we take the c u r l o f (A.2) and o b t a i n P From (2.1) V • u = — Bp-p Dt Hence, V • u can be e l i m i n a t e d from (2.10) and then DT = ' v ) u + h ( V p A V p ) + 7 ( ^ f ) ' ( A - 1 1 ) p where | = t + 2^ a I n the absence of f r i c t i o n a l e f f e c t s and f o r a d i a b a t i c motions the conserved magnitude X which l e a d s to the c o n s e r v a t i o n o f p o t e n t i a l v o r t i c i t y i n the ocean i s the d e n s i t y p. In a more g e n e r a l c o n t e x t , where d i s s i p a t i v e e f f e c t s are to be taken i n t o account to remove the i n d e t e r m i n a n c y of the m o t i o n , i t i s s t i l l c o n v e n i e n t to keep p as a v a l i d c h o i c e o f a s c a l a r p r o p e r t y o f the f l u i d t h a t t o g e t h e r w i t h % can y i e l d to a u s e f u l v e r s i o n o f E r t e l ' s - 119 -theorem a p p l i e d to the ocean c i r c u l a t i o n , L e t us w r i t e e q u a t i o n (A.6) as §f = G = K V 2p - Q, (A.12) and c o n s i d e r the f o l l o w i n g i d e n t i t i e s : ->• ? a DVp , ? a r,\ Dp „ , ? a „ >N — • D t ^ (— • V) - Vp • (— • V u) , n t t Vp A Vp V p ' DT ( ^ } = V p * [ ( / * V ) * ] + V p ' [ 3 ] P (A.13a,b) The sum o f (A.13a) and (A.13b) then y i e l d s w i t h (A.12) g- . V p } = vG • t±-2? + ^ • [ ™ ? ] . (A.14) Dt 1 p J p p p v ' E q u a t i o n (A.14) i s the most g e n e r a l e q u a t i o n of the p o t e n t i a l t + 2& v o r t i c i t y q = - — - — — • V p ; but i t i s not i n t h i s form as t h i s e q u a t i o n i s used i n oceanography. F u r t h e r m a n i p u l a t i o n s i n (A.14) u s i n g the r e l a t i o n s o f the g e o s t r o p h y dynamics and B o u s s i n e s q a p p r o x i m a t i o n y i e l d a more s u i t a b l e form of the e q u a t i o n of p o t e n t i a l v o r t i c i t y . Thus, the h o r i z o n t a l components o f r e l a t i v e v o r t i c i t y which are produced m a i n l y by the buoyancy t w i s t i n g term, Vp A V p , are computed from the t h e r m a l wind r e l a t i o n ( P e d l o s k y , 1979); whereas, the v e r t i c a l - 120 -components o f the a b s o l u t e v o r t i c i t y and of the g r a d i e n t o f d e n s i t y are the components which a c t u a l l y are used i n ( 2 . 1 4 ) . T h e r e f o r e , t h i s e q u a t i o n can be w r i t t e n as D' X f r j_ N p z , 3G , £ + . p z . „2 ITT [ ( F + ? ) 7T ] = 17 ( — ^ "p~" H ? p z 3 >N + t f i ( ( V A ) z T>' < A' 1 5> where r r r = IT + 5 H * V H « 5 H = ( U > V ) > V H = <lx > l y 0 ' t are the s t r e s s e s which a c t on the water column, Z i s the v e r t i c a l component o f the r e l a t i v e v o r t i c i t y , f = 2|& | s i n e . I t i s c o n v e n i e n t to w r i t e e q u a t i o n (A.12) i n terms o f the o p e r a t o r ijjr^ as g ^ = - w | f + G . (A.16) The e q u a t i o n s (A.15) and (A.16) form the b a s i s f o r the s t u d y o f o c e a n i c motions which extend from the s y n o p t i c s c a l e s to the p l a n e t a r y s c a l e s . However, i n o r d e r to f a c i l i t a t e t he t h e o r e t i c a l and n u m e r i c a l a n a l y s i s o f any dynamical s t a t e , i t i s n e c e s s a r y to i s o l a t e the r e l e v a n t mechanism of such a s t a t e a t a g i v e n s c a l e . A d e t a i l e d d e s c r i p t i o n o f such - 121 -mechanisms as w e l l as of t h e i r i n t e r a c t i o n s w i t h o t h e r s c a l e s of the g l o b a l d y namical s t a t e r e q u i r e s the i n t r o d u c t i o n o f complementary and p h y s i c a l l y and m a t h e m a t i c a l l y c o n s i s t e n t a p p r o x i m a t i o n s to the e q u a t i o n s (A.15) and ( A . 1 6 ) . S i n c e our main i n t e r e s t i s the d e s c r i p t i o n o f motions which take p l a c e at m i d - l a t i t u d e s and such t h a t t h e i r m e t r i c can be a c c u r a t e l y d e s c r i b e d by the tangent plane a p p r o x i m a t i o n , we now i n t r o d u c e the s t a n d a r d s c a l i n g v a r i a b l e s f o r a m i d - l a t i t u d e 3-plane. The s c a l i n g parameters are ( P e d l o s k y , 1979): p P s ( z ) (1 + e F p 1 ) , e U S (A.17) 2 N 2 B r u n t - V a r s a l a f r e q u e n c y , f f 0 U - 122 -f§L 2 where F = ^ n , f g = 2|ft|sin9Q, 6Q i s a r e f e r e n c e l a t i t u d e . N o t i c e t h a t the buoyancy f r e q u e n c y i s taken to v a r y o n l y w i t h d e p t h . A l t h o u g h t h i s i s an u n r e a l i s t i c assumption f o r the r e a l ocean we take i t as v a l i d because i n our computations we e x c l u d e any dynamical change caused by h o r i z o n t a l v a r i a t i o n s of the d e n s i t y . I t i s easy to o b t a i n from (A.15) and (A.16) the e q u a t i o n f o r the r e l a t i v e v o r t i c i t y : ^ + u • V H f = f w z + A H V.2C + | _ ( V A ? ) . (A.18) - 123 -APPENDIX B C o n s i d e r t h a t the magnitude t, obeys the e q u a t i o n | | + u • V c = 0, V • u = 0. We c l a s s i f y a n u m e r i c a l scheme as : ( i ) S e m i - C o n s e r v a t i v e I f n e g l e c t i n g e r r o r s i n time d i f f e r e n c i n g It / D C dx - 0. The i n t e g r a l has been r e p r e s e n t e d by a summation f o r m u l a the same o r d e r as the d i s c r e t i z a t i o n method. ( i i ) C o n s e r v a t i v e I f the v a l u e s of z, g i v e n by the n u m e r i c a l s o l u t i o n /p ? dx = c o n s t . ( i i i ) Q u a d r a t i c a l l y S e m i - c o n s e r v a t i v e I f n e g l e c t i n g e r r o r s i n the time d i f f e r e n c i n g \t / D dx = 0. - 124 -( i v ) Q u a d r a t i c a l l y C o n s e r v a t i v e I f the v a l u e s of g i v e n by the n u m e r i c a l s o l u t i o n s a t i s f y 2 /_ ? dx = Const. - 125 -APPENDIX C We prove here t h a t the a p p r o x i m a t i o n (3.31) i s d i s s i p a t i v e o f second o r d e r f o r e i t h e r a c o n s i s t e n t mass m a t r i x f o r m u l a t i o n or a lumping mass m a t r i x f o r m u l a t i o n . We use K r e i s s 1 d e f i n i t i o n o f d i s s p a t i v e n e s s (see K r e i s s (1978). Thus, the a p p r o x i m a t i o n (3.31) i s d i s s i p a t i v e o f or d e r 2r i f t h e r e e x i s t s an a s such t h a t Kj i s the modulus of the l a r g e s t e i g e n - v a l u e o f the d i s c r e t e o p e r a t o r o f the a p p r o x i m a t i o n . In our case |< i s the modulus of the a m p l i f i c a t i o n f a c t o r . 6 > 0 i s some L e t us c o n s i d e r , f o r example, the e q u a t i o n ( 3 . 3 6 ) ; t h a t i s K(S h) < e (1 - 6 | ? | Z r ) , |?| < M, where c o n s t a n t i n d e p e n d e n t o f h a n d A t . = 1 ( C 1 ) where x = c o s k A x , y = c o s £ A y . For k A x and ihy s m a l l , 4 - (x + y) - 2xy = (1 - x) + (1 - y) + 2(1 - xy) - 126 -b e c o m e s (1 - c o s 2 e 1 ) + (1 - c o s 2 e 2 ) + 2(1 - cos2e-| c o s 2 e 2 ) = 2 s i n 2 e 1 + 2 s i n 2 e 2 + 2(1 - cos2e, c o s 2 e 2 ) , (C.2) w h e r e k A x _ £ A y e1 2 ' e2 2 B y e x p a n d i n g i n T a y l o r s e r i e s t h e r i g h t h a n d s i d e o f (C.2) a n d r e t a i n i n g u p t o 2nd o r d e r t e r m s y i e l d s 4 - (x + y) - 2xy - 6 ( e 2 + e 2) = 6 1 e 1 2 (C.3) L i k e w i s e , t h e n u m e r a t o r (2 + x ) (2 + y ) i s e q u i v a l e n t t o 4 f o r k A x a n d £ A y v e r y s m a l l . T h e r e f o r e , w e c a n w r i t e (C.1) a s o A i r A t 9 X l | * 1 - 7 2 A ^ A y | £ | ( G ' 4 ) A R A t 1 s i n c e A x A y - T92 a c c o r d i n 8 t o (3.38b) t h e n A1 I 2 < 1 - 6 I e l 2 , ( C . 5 ) w h e r e 8 = 72/192. - 127 -S i m i l a r l y , we can show t h a t f o r the lumped mass m a t r i x f o r m u l a t i o n M 2 < 1 " 6IH2> 5 =TIT 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0053197/manifest

Comment

Related Items