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Theory of polar substorms Atkinson, Gerald 1967

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A THEORY OP POLAR SUBSTORMS by GERALD ATKINSON B . A . S c , U n i v e r s i t y o f B r i t i s h Columbia, i960 M.Se., U n i v e r s i t y of B r i t i s h Columbia, 1964 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of GEOPHYSICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA AUGUST 1 9 6 6 0 Gerald Atkinson 1967 In presenting t h i s t h e s i s in. p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l able f o r reference and study, I fur t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date 1 3 ^ i ABSTRACT The magneto-sphere may be c o n s i d e r e d as c o n s i s t i n g o f two r e g i o n s , t h e t a i l and the r e g i o n o f c l o s e d l i n e s o f f o r c e . The i n t e r c h a n g e o f f i e l d l i n e s between t h e s e two r e g i o n s i s i m p o r t a n t i n m a g n e t o s p h e r i c p r o c e s s e s . T r a n s p o r t o f magnetic f i e l d l i n e s f r o m the c l o s e d r e g i o n i n t o the t a i l may o c c u r by Dungey's mechanism o r by v i s c o u s i n t e r a c t i o n o f the magnetosphere w i t h the s o l a r wind. T r a n s p o r t f r o m t h e t a i l t o t h e c l o s e d r e g i o n o c c u r s by r e c o m b i n a t i o n t h r o u g h t h e n e u t r a l s h e e t . C o n v e c t i v e f l o w w i t h i n t h e c l o s e d r e g i o n i s c o n t r o l l e d by t h e " f o o t d r a g g i n g " e f f e c t which a r i s e s f r o m the d i s c h a r g i n g a c t i o n o f the i o n o s p h e r e on f l u x t u b e s . The model of p o l a r substorms p r e s e n t e d i s a f l o w o r time sequence as f o l l o w s : (1) F i e l d l i n e s a r e dragged f r o m t h e c l o s e d r e g i o n i n t o the t a i l by the s o l a r wind, w i t h a r e s u l t i n g s t o r a g e of p o t e n t i a l e n e rgy i n the t a i l . (2) The p o l a r substorm b e g i n s when t h e f i e l d l i n e s recombine i n an i m p l o s i v e f a s h i o n a t the n e u t r a l sheet, r e l e a s i n g the s t o r e d p o t e n t i a l energy. (3) The recombined f l u x t u b e s are added t o the n i g h t s i d e o f the c l o s e d r e g i o n as a g i a n t b u l g e . (4) The bul g e d r i v e s a r e t u r n f l o w of f l u x t u b e s towardsthe d a y s i d e i n the c l o s e d r e g i o n . i i I t i s l i k e l y t h a t r e c o m b i n a t i o n i s i n i t i a t e d by t h e f o r m a t i o n o f a n e u t r a l p o i n t a t about 13 o r 14 e a r t h r a d i i i n the a n t i s o l a r d i r e c t i o n . , and o c c u r s a c r o s s a w i d t h 8 o f t a i l of about 6 o r 7 e a r t h r a d i i , and t h a t 10 webers are a n n i h i l a t e d i n a time o f about 1/2 h o u r . The r e c o m b i n a t i o n i s p r o b a b l y s t o p p e d by t h e b u i l d - u p o f a g i a n t b u l g e on the n i g h t s i d e o f t h e c l o s e d r e g i o n , which maps to' t h e e a r t h ' s s u r f a c e a l o n g f i e l d l i n e s as the a u r o r a l break-up b u l g e , and which, as i t s p r e a d s out o v e r the n i g h t s i d e o f the r e g i o n of c l o s e d f i e l d l i n e s , c a u s e s t h e o b s e r v e d a u r o r a l e f f e c t s . The P e d e r s e n c u r r e n t i s n o t e x p e c t e d t o produce s i g n i f i c a n t magnetic e f f e c t s a t the s u r f a c e o f t h e e a r t h except a t a n o m a l i e s i n i o n o s p h e r i c c o n d u c t i v i t y . Such an anomaly a l o n g the a u r o r a l a r c s can e x p l a i n t h e westward e l e c -t r o j e t . However, i t seems p r o b a b l e t h a t t h e r e m a i n d e r o f the p o l a r cap c u r r e n t system i s t h e r e s u l t o f H a l l c u r r e n t s . i l l TABLE OF CONTENTS CHAPTER I CHAPTER I I THE MORPHOLOGY OF POLAR SUBSTORMS a. A u r o r a l break-up b. Geomagnetic bays c. Other phenomena d. S a t e l l i t e o b s e r v a t i o n s THE MAGNETOSPHERE a. The s t e a d y - s t a t e magnetosphere b. I n t e r c h a n g e o f f i e l d l i n e s between the r e g i o n s c. C o n v e c t i o n w i t h i n t h e c l o s e d r e g i o n CHAPTER I I I THE MODEL OF POLAR SUBSTORMS CHAPTER IV TRANSPORT INTO THE TAIL a. Dungey's mechanism b. V i s c o u s f o r c e s CHAPTER V FIELD LINE RECOMBINATION a. T r i g g e r i n g mechanism b. C h o k i n g mechanism c. E n e rgy of r e c o m b i n a t i o n CHAPTER V I BULGE ADDED ON THE NIGHTSIDE OF THE CLOSED REGION Page 1 1 4 7 7 10 10 12 14 24 26 26 27 33 33 35 36 3 8 i v Page CHAPTER V I I RETURN PLOW TO THE DAYSIDE OF THE CLOSED REGION CHAPTER IX DISCUSSION BIBLIOGRAPHY APPENDIX I Symbols and c o n v e n t i o n s u s e d i n t h i s t h e s i s 44 44 a. The f l o w b. An approximate f l o w e q u a t i o n and s o l u t i o n s c . The b u l g e shape 52 d. The westward t r a v e l l i n g surge 5 3 e. EGO s a t e l l i t e o b s e r v a t i o n s 56 CHAPTER V I I I MAGNETIC EFFECTS 5 9 a. P e d e r s e n c o n d u c t i v i t y 59 b. H a l l c o n d u c t i v i t y c. C u r r e n t systems i n the i o n o s p h e r e d. westward e l e c t r o j e t 65 66 62 62 6 9 71 APPENDIX I I Mapping f a c t o r s f o r a d i p o l e f i e l d 74 V FIGURES Page 1 Development o f the a u r o r a l substorm. A f t e r A k a s o f u , 1964. 2 2 Example o f a n e g a t i v e bay. 5 3 Bay c u r r e n t systems. 6 4 EGO magnetic f i e l d measurements, S e p t . 28-29, 1964. A f t e r Heppner, 1965. 9 5 The open t a i l e d model o f t h e magnetosphere. 11 6 I n t e r c h a n g e o f f i e l d l i n e s between t h e r e g i o n s . 13 7 G r o s s s e c t i o n o f a c o n v e c t l n g plasma s l a b . 16 8 Shear f l o w a t the boundary o f t h e magneto-s p h e r e . 29 9 B u l g e on the n i g h t s i d e o f the c l o s e d r e g i o n . 39 10 Flow p a t t e r n d u r i n g the c o l l a p s i n g phase. 40 11 C o o r d i n a t e system. .46 12 Geometry f o r e q u a t i o n (24). 46 13 Asymmetry of the b u l g e . 54 14 P l o w i n the t a i l i n t o the r e g i o n of a n n i h i l a t i o n . Page 54 15 Symmetry of the Pedersen c u r r e n t s y s t e m . 6 l 61 16 E f f e c t o f H a l l c u r r e n t s on the magnet ic f i e l d l i n e s . 17 P l o w of f l u x tube f e e t f o r a n o n r o t a t i n g e a r t h . 63 18 P l o w o f f l u x t u b e s i n t o the b u l g e . 63 19 Mapping f a c t o r s f o r a d i p o l e f i e l d . 75 TABLE 1 P r o p e r t i e s o f the magnetosphere. Page 18 v i i ACKNOWLEDGMENT I am d e e p l y i n d e b t e d t o Dr. T. Watanabe b o t h f o r h i s p a t i e n t s u p e r v i s i o n , and f o r h i s many h o u r s of d i s c u s s i o n of t h e i d e a s e x p r e s s e d i n t h i s t h e s i s . I would a l s o l i k e t o thank P r o f e s s o r J . A. J a c o b s f o r h i s encourage-ment d u r i n g the p r e p a r a t i o n o f t h i s work, and f o r h i s g u i -dance i n i t s f i n a l f o r m u l a t i o n . T h i s work was s u p p o r t e d by the O f f i c e of N a v a l R e s e a r c h under c o n t r a c t Nonr 3116(00). 1 CHAPTER I THE MORPHOLOGY OF POLAR SUBSTORMS P o l a r substorm i s a term used t o d e s c r i b e s e v e r a l o b s e r v e d geomagnetic and a e r o n o m i c a l phenomena which o c c u r s i m u l t a n e o u s l y , and are t h e r e f o r e assumed t o r e s u l t f r o m a common ca u s e . E a c h f a c e t of the p o l a r substorm i s d i s c u s s e d i n t u r n below. a. A u r o r a l break-up ( a u r o r a l substorm) The morphology of the a u r o r a l substorm has been d i s c u s s e d by A k a s o f u (1964). He p r e s e n t e d i t as a time sequence, the e s s e n t i a l s of which w i l l be r e v i e w e d h e r e . The development i s shown I n F i g . 1 ( f o r t h e n o r t h e r n hemi-sphere) . The q u i e t phase (T = 0) When the a u r o r a s i n the p o l a r r e g i o n are f r e e f r o m a c t i v i t y f o r a few hours, those n e a r m i d n i g h t t e n d t o take the f o r m of q u i e t homogeneous a r c s , l y i n g a l o n g geomagnetic l a t i t u d e c i r c l e s around 60° - 65°. The e x p a n s i v e phase (1) T = 0 - 5 min The f i r s t i n d i c a t i o n of t h e s t a r t of an a u r o r a l substorm i s the sudden b r i g h t e n i n g of one of the a r c s . 2 P i g . 1 . The time development of the auroral substorm - a f t e r Akasofu, 1 9 6 4 . View from above the north pole. Noon i s at the top of each diagram. 3 (2 ) T = 5 - 10 mln There i s a r a p i d p oleward m o t i o n of the a u r o r a n e a r m i d n i g h t , r e s u l t i n g i n t h e f o r m a t i o n o f a b u l g e . The b u l g e t e n d s t o expand northward, e a s t w a r d and westward as shown i n P i g . 1. The p o l e w a r d e x p a n s i o n r a t e i s u s u a l l y of the o r d e r o f 20 - 100 km m i n " 1 . The b u l g e i s u s u a l l y c e n t e r e d w i t h i n an hour o f t h e m i d n i g h t m e r i d i a n . (3) T = 10 - 30 min D u r i n g t h i s t i m e, the bands r e a c h t h e i r most n o r -t h e r l y p o s i t i o n (70° - 80° geomagnetic l a t i t u d e ) , and those w i t h i n the b u l g e are e x t r e m e l y a c t i v e . P a t c h e s appear f u r -t h e r south, and a l l the i r r e g u l a r s t r u c t u r e e x h i b i t s an eastward d r i f t on t h e morning s i d e o f m i d n i g h t and a west-ward d r i f t on the e v e n i n g s i d e . The w e s t e r n s i d e o f the b u l g e expands r a p i d l y westward a t v e l o c i t i e s o f 10 - 100 km m l n - 1 . T h i s f e a t u r e i s c a l l e d a westward t r a v e l l i n g s u r g e . The i n t e n s i t y o f the substorm d e t e r m i n e s the e x t e n t of the northward motion, the a m p l i t u d e of the surge, and t h e l o n g i t u d i n a l e x t e n t o v e r which th e d i s t u r b a n c e s a r e seen. The r e c o v e r y phase (1) T = 30 min - 1 h r In t h i s phase, the a u r o r a s r e t u r n southward, u s u a l -l y a t speeds l e s s t h a n the poleward motion. The westward surge d e c r e a s e s I n v e l o c i t y and d e g e n e r a t e s i n t o s m a l l s c a l e 4 i r r e g u l a r f o l d s o r l o o p s . These d r i f t westward a t 30 km m i n - 1 , and may t r a v e l g r e a t d i s t a n c e s ( ^ 1000 km). The b r i g h t n e s s o f the a r c s d e c r e a s e s , aid on the morning s i d e the arc. s t r u c t u r e d e g e n e r a t e s i n t o p a t c h e s which d r i f t e a s t -ward a t 20 km m i n - 1 . (2) T = 1 - 2 h r T h i s p e r i o d i s c h a r a c t e r i z e d by equatorward m o t i o n o v e r a wide range o f l a t i t u d e and l o n g i t u d e and f u r t h e r f a d -i n g o f t h e a r c s . (3) T = 2 - 3 h r The slow equatorward motion o f f a i n t a r c s c o n t i n u e s and no eastward o r westward motion of a u r o r a i s o b s e r v e d . b. Geomagnetic bays The phenomenon known as a n e g a t i v e geomagnetic bay o c c u r s on the n i g h t s i d e o f the e a r t h , and i s s t r o n g e s t i n the a u r o r a l zones, where the d e c r e a s e i n the h o r i z o n t a l component of the magnetic f i e l d i s u s u a l l y i n the range from 100 t o 1000f . The d i s t u r b a n c e i s seen w i t h l e s s e r i n t e n s i t y i n n e a r l y a l l l a t i t u d e s , p e r s i s t i n g g e n e r a l l y f o r one o r two hour3. An example i s shown i n F i g . 2. The whole magnetic d i s t u r b a n c e I s c a l l e d a magnetic substorm ( a f t e r Chapman -Akasofu, 1964). I t i s u s u a l t o r e p r e s e n t the magnetic p e r t u r b a t i o n s by an e q u i v a l e n t i o n o s p h e r i c c u r r e n t system, based on a v e r a g e s over many e v e n t s . Some of th e s e are shown i n 5 r t H ^ AGIN dAY 1 COUR 3. \9i T 53 H = I 5.40 r/mrr D-—v 1 D= 1 .05 '/mm t Z •—1 0 6 Z = J 5.30 Vmnr I i 1 2 P i g . 2. Example o f a n e g a t i v e bay. A g i n c o u r t , May 13, 1963. 6 A B ® 2 1 1 5 U T A P R 3 0 1933 P i g . 3. Bay c u r r e n t systems. (a) Older models . (b) Model o f A k a s o f u , Chapman and Meng (1965). (c) O b s e r v a t i o n a l r e s u l t s of V e s t i n e . (d) O b s e r v a t i o n s of Pukushima (1953) of an i n d i v i d u a l b a y . 7 F i g . 3- Fukushima (1953) s u g g e s t s t h a t the bay c u r r e n t system i s s i m i l a r t o a two d i m e n s i o n a l d o u b l e t a l i g n e d a l o n g the a u r o r a l zone, and t h a t diagrams such as t h o s e i n f i g u r e 3 are a s u p e r p o s i t i o n o f many such d o u b l e t s , r e f l e c t i n g p o s i t i o n changes o f t h e b a s i c d o u b l e t , r a t h e r t h a n t h e c u r -r e n t systems o f i n d i v i d u a l b a y s . c. Other phenomena Other phenomena a s s o c i a t e d w i t h p o l a r substorms i n c l u d e :* ( i ) P i 2 m i c r o p u l s a t i o n s . These are i r r e g u l a r p u l s a t i o n s o f the geomagnetic f i e l d w i t h p e r i o d s i n the range 40 - 150 s e c . They c o i n c i d e w i t h the on s e t of a geo-magnetic bay, and are a n i g h t s i d e phenomenon. ( i l ) X - r a y b u r s t s . X - r a y b u r s t s are o b s e r v e d a t b a l l o o n a l t i t u d e s under a c t i v e a u r o r a . They a r e b e l i e v e d t o be due t o e l e c t r o n s (E ^ 25 kev) bombarding the atmos-phere . d. S a t e l l i t e o b s e r v a t i o n s ( i ) O ' B r i e n and T a y l o r (1964), I n an experiment on I n j u n 3 s a t e l l i t e a t an a l t i t u d e o f 950 km, r e p o r t e d an i n c r e a s e i n the number of m i r r o r i n g e l e c t r o n s when the f l u x of p r e c i p i t a t i n g e l e c t r o n s i n c r e a s e d (E ^ 40 k e v ) . T h i s i m p l i e s an i n j e c t i o n o f p a r t i c l e s i n t o the magnetic f l u x tube a t t i m e s of a c t i v e a u r o r a . 8 ( i i ) D a v i s (1966) r e p o r t e d l a r g e jumps i n the p r o t o n d e n s i t y (E =5- 100 kev) on the n i g h t s ! d e o f the e a r t h i n t h e t r a p p i n g r e g i o n , s i m u l t a n e o u s w i t h the o b s e r -v a t i o n o f geomagnetic bays. ( i i i ) Heppner (1966), u s i n g the EGO r e s u l t s , ob-s e r v e d a drop i n the magnetic f i e l d s t r e n g t h i n the t a i l o f the magnetosphere about 15 min a f t e r the bay commencement at ground s t a t i o n s . These r e s u l t s a r e shown i n P i g . 4. 9 21*04" Satellite Local Time 2'17" 21*31" 21*47" 22*07" 22*37" 23*25" 13.3 Geog. Lot. 11.6 9.6 7.1 45 -0.6 -7.9 4.1 Mag. Lot. 6.9 9.6 11.0 10.5 7.6 0.5 16.1 L 14.2 12.4 10.4 8.3 6.2 4.1 r-4— Re 14 1 1? 10 i 8 r 6 i 4 1 s IU(6k-Wn--40ii__|7li Ms 40s |gk 20s 40s |gk 20s 40* ggk 20s 40s gk 20* 40s 20s 40* 2 J k 20* 40» Qk 20* 40m Qf JOmJOm^fcl ~~~~ U.T,SEPT. 28, 1964 SEPT. 29, 1964 Kiruna, Sweden North Component Local Standard Time U.T. 6" 12* 15* 18* 21* 0* 6* 9* F i g . 4 . EGO magnet ic f i e l d measurements, S e p t . 28-29, 1964 - a f t e r Heppner (1966) , showing decreases i n the t a i l f i e l d s t r e n g t h d u r i n g a geomagnetic bay . 10 CHAPTER II THE MAGNETOSPHERE a. The s t e a d y - s t a t e magnetosphere I n t h e open t a i l e d model of the magnetosphere, as d e f i n e d by P i d d i n g t o n (i960), A x f o r d , P e t s c h e k and S l s c o e ( 1965) , and Ness ( 1965) , two main r e g i o n s o f space may be d e f i n e d . (1) The r e g i o n c o n s i s t i n g of c l o s e d l i n e s o f f o r c e . (See P i g . 5 b . ) T h i s r e g i o n i s doughnut-shaped w i t h the e a r t h i n the c e n t r a l h o l e of t h e doughnut. I t w i l l h e n c e f o r t h be r e f e r r e d t o as the c l o s e d r e g i o n o f t h e mag-n e t o s p h e r e , o r , f o r b r e v i t y , as t h e c l o s e d r e g i o n . The r o t a t i o n of t h e e a r t h c a u s e s t h e doughnut t o r o t a t e about the a x i s AA i n P i g . 5 b . ( i i ) The r e g i o n c o n s i s t i n g o f l i n e s o f f o r c e not c l o s e d i n t h e v i c i n i t y of the e a r t h , (See F i g . 5 c ) T h i s i s commonly r e f e r r e d t o as the t a l l o f the magnetosphere, or the t a i l f o r b r e v i t y . I t may be v i s u a l i z e d as two "sau-sages", l i n k e d t o g e t h e r a t the e a r t h , t h r o u g h the h o l e o f the doughnut-shaped r e g i o n . The f o r c e of the s o l a r wind bends t h e s e " s a u s a g e s " i n t h e a n t i - s o l a r d i r e c t i o n so t h a t t h e y wrap around the doughnut i n t h e s u r f a c e CB o f F i g . 5 and a r e f l a t t e n e d a g a i n s t each o t h e r i n the p l a n e AB. 11 P i g . 5 . The open t a i l e d model of the magnetosphere-s c h e m a t i c . (a) S e c t i o n t h r o u g h the 1200-2400 LT m e r i d i a n . (b) The doughnut-shaped r e g i o n of c l o s e d f i e l d l i n e s . (c) The t a i l , o r r e g i o n of open f i e l d l i n e s . 12 The r o t a t i o n o f t h e e a r t h t e n d s t o cause t h e s e two " s a u s a g e s " t o r o t a t e i n o p p o s i t e d i r e c t i o n s about t h e i r own axes. b. I n t e r c h a n g e o f f i e l d l i n e s between the r e g i o n s There are two mechanisms by which l i n e s o f f o r c e may move f r o m t h e c l o s e d r e g i o n i n t o t h e t a i l . ( i ) Dungey's mechanism. I t has been s u g g e s t e d by Dungey (1958) t h a t an o p p o s i t e l y a l i g n e d i n t e r p l a n e t a r y mag-n e t i c f i e l d l i n e can combine w i t h f i e l d l i n e s i n the c l o s e d r e g i o n o f the magnetosphere a t a n e u t r a l , p o i n t , and d r a g i t i n t o t h e t a i l r e g i o n . T h i s p r o c e s s I s I l l u s t r a t e d I n F i g . 6a. ( i i ) C l o s e d t u b e s o f f o r c e may be d i s t e n d e d , and dragged b y t h e s o l a r wind from t h e f r o n t and s i d e s of t h e c l o s e d r e g i o n i n t o t h e t a i l . T h i s must o c c u r by f r i c t i o n o r v i s c o u s l i k e i n t e r a c t i o n s between the s o l a r wind and the mag-n e t o s p h e r e . The f r i c t i o n a l f o r c e c o u l d be o f many k i n d s , i n c l u d i n g wave t r a n s m i s s i o n a c r o s s the boundary, p a r t i c l e d i f f u s i o n a c r o s s the boundary, o r i n s t a b i l i t i e s s i m i l a r t o R a y l e i g h - T a y l o r , o r K e l v l n - H e l m h o l t z i n s t a b i l i t i e s which may grow t o the e x t e n t t h a t f l u x t u b e s o r i g i n a l l y b e l o n g i n g t o t he c l o s e d r e g i o n c o u l d become caught up i n the s o l a r wind, and swept i n t o t h e t a i l . T h i s i s i l l u s t r a t e d i n F i g . 6b. The f r i c t i o n a l p r o c e s s e s and i n s t a b i l i t y growth have been r e v i e w e d by P i d d i n g t o n (1964). 13 P i g . 6. I n t e r c h a n g e o f f i e l d l i n e s between the r e g i o n s . (a) Dungey's mechanism - the numbered l i n e s r e p r e s e n t s u c c e s s i v e p o s i t i o n s o f a f i e l d l i n e . (b) V i s c o u s and f r i c t i o n a l f o r c e s . Loops of f i e l d l i n e s o r i g i n a l l y i n the c l o s e d r e g i o n are extended by the s o l a r wind. (c) Sweet's mecha-n i s m - the arrows a r e s t r e a m l i n e s o f the f l o w . 14 The r e t u r n of the f i e l d l i n e s f r o m the t a i l r e g i o n t o the c l o s e d r e g i o n i s a c c o m p l i s h e d by Sweet's mechanism o c c u r r i n g i n the p l a n e AB o f P i g , 5 ( t h e n e u t r a l s h e e t ) . Sweet (1958) c o n s i d e r e d t h e r e c o m b i n a t i o n o f o p p o s i t e l y a l i g n e d f i e l d s t h r o u g h a n e u t r a l s h e e t . P e t s c h e k (1964) has m o d i f i e d t h i s p r o c e s s t o a l l o w f o r the e f f e e t s o f h y d r o -magnetic waves i n t h e removal of gas f r o m the n e u t r a l s h e e t . A x f o r d , P e t s c h e k and S i s c o e (1965) have a p p l i e d t h i s t h e o r y t o the t a i l of t h e magnetosphere, assuming a s t e a d y s t a t e r e c o m b i n a t i o n , and f o u n d t h a t t h e f l o w towards th e n e u t r a l sheet i s ^ 1/10 where V A I s the A l f v e n v e l o c i t y i n the u n p e r t u r b e d t a i l . The f l o w a l o n g the n e u t r a l sheet towards the c l o s e d r e g i o n i s of the o r d e r o f V^. The f l o w p a t t e r n s a r e i l l u s t r a t e d i n P i g . 6c. M a g n e t i c f i e l d i s an-n i h i l a t e d i n t h e n e u t r a l sheet, and i t s e n ergy g i v e n t o the plasma i t c o n t a i n e d . c. C o n v e c t i o n w i t h i n t h e c l o s e d r e g i o n . The c o n v e c t i o n of f l u x t u b e s w i t h i n the magneto-sphere has been d i s c u s s e d by Gold (1959). F°r c o n v e c t l v e f l o w t o o c c u r , the d r i v i n g f o r c e s f o r the f l o w must cause a charge s e p a r a t i o n , which c r e a t e s an e l e c t r i c f i e l d a l l o w -i n g plasma d r i f t a c r o s s the magnetic f i e l d , as viewed f r o m a s t a t i o n a r y r e f e r e n c e frame. Because o f t h e h i g h conduc-t i v i t y p a r a l l e l t o t h e f i e l d , f i e l d l i n e s are assumed t o 15 be e q u i p o t e n t i a l s , i n which case one c a n speak o f the f l o w of f l u x t u b e s , as each tube p r e s e r v e s i t i d e n t i t y i n such a m o t i o n . V i s c o u s f o r c e s i n the magnetosphere are s m a l l , so t h a t c o n v e c t i o n can o c c u r f a i r l y f r e e l y e x c e p t f o r t h e d i s c h a r g i n g a c t i o n o f the i o n o s p h e r e on the f l u x t u b e s . Gold p r o p o s e d a model i n which t h e r e was a sharp t r a n s i t i o n f r o m f r o z e n f i e l d c o n d i t i o n s t o z e r o c o n d u c t i v i t y . I n t h e r e a l c a s e , t h e r e s h o u l d be a d r a g on t h e moving f i e l d l i n e s due t o the P e d e r s e n c o n d u c t i v i t y of t h e i o n o s p h e r e (see P i g . 7). T h i s d i s s i p a t i o n was d i s c u s s e d by A x f o r d and H i n e s (1961). Steady s t a t e c o n v e c t i o n r e q u i r e s t h a t t h e e l e c t r i c f i e l d be m a i n t a i n e d . T h e r e f o r e , t h e r e must be a charge s e p a r a t i o n c u r r e n t p r o d u c e d i n the c o n v e c t l n g r e g i o n t o o f f -s e t the d i s c h a r g i n g a c t i o n of the i o n o s p h e r e . I f , i n t h e i o n o s p h e r e , E i s assumed t o be h o r i -z o n t a l , and v e r t i c a l l y u n i f o r m , and B i s assumed t o be v e r t i c a l , t h e n the h e i g h t i n t e g r a t e d d i s c h a r g e c u r r e n t ( i n the i o n o s p h e r e ) i s J . = cr^ E, where o x i s the h e i g h t i n t e g r a t e d P e d e r s e n c o n d u c t i v i t y . E a c h f l u x tube e x p e r i e n c e s a d r a g g i n g f o r c e jj" ( ^ x B ) d l dA = jjf j B dA d l . F o r a f l u x tube J' B dA i s c o n s t a n t and J3 = -_v x B. Thus the f o r c e on a f l u x tube o f u n i t a r e a a t t h e i o n o s p h e r e i s 16 P i g . 7. C r o s s s e c t i o n of a c o n v e c t i n g plasma s l a b . The plasma c o n v e c t i o n v. , r e q u i r e s the e l e c t r i c f i e l d E = - v x Bp . T h i s c a u s e s a d i s c h a r g e c u r r e n t J ± i n t h e ~ i o n o s p h e r e which must be b a l a n c e d by a charge s e p a r a t i o n c u r r e n t j , i n the magnetosphere. 17 where f , used as a s u b s c r i p t , d e n o t e s th e v a l u e a t t h e f e e t of t h e f i e l d l i n e s . The f a c t o r 2 i s i n s e r t e d because each f l u x tube has two f e e t . C o n s i d e r t h e N a v i e r S t o k e s e q u a t i o n i n t e g r a t e d o v e r t h e f r o z e n f i e l d r e g i o n o f a f l u x t u b e . J I s the t o t a l c u r r e n t f l o w i n g , and i n c l u d e s j , the con-t r i b u t i o n o f t h e " f o o t d r a g g i n g " term. The meaning of t h e v a r i o u s symbols I s g i v e n i n Appendix I . The o r d e r o f magni-tude o f the v a r i o u s terms I s c o n s i d e r e d based on v a l u e s o f the p a r a m e t e r s f o r t h e magnetosphere g i v e n i n T a b l e I . I n a f l u x tube B x A i s c o n s t a n t , where A i s the c r o s s s e c t i o n a l a r e a o f t h e t u b e . Thus i f a mapping f a c t o r , g, a l o n g f l u x t u b e s ( e q u a t o r t o f e e t ) i s d e f i n e d by: (2) 2 t h e n (3) 18 TABLE I PROPERTIES OP THE MAGNETOSPHERE • T a i l Magneto- F e e t sphere ( i o n o -( o u t s i d e sphere) knee) Magn e t i c f i e l d ( 7 ) 30 50 50,000 Number d e n s i t y (m~ 3 ) 2 x I O 6 2 x I O 6 I O 1 2 Mass d e n s i t y (kg m~ 3) 2 x 10~ 2 1 2 x I O - 2 1 10~ 1 5 Mapping f a c t o r -g 4 l 32 1 E f f e c t i v e l e n g t h (m) ? I O 8 I O 6 A l f v e n v e l o c i t y 450 750 1050 (km s e c - 1 ) where d, E and v denote distances, e l e c t r i c f i e l d s and v e l o c i t i e s perpendicular to the f i e l d l i n e s . In actual f a c t , the mapping f a c t o r s f o r distances measured r a d i a l l y ( g p ) and i n the angular d i r e c t i o n ( g 0 ) may be d i f f e r e n t (see Appendix I I and F i g . 1 1 ) . However, the use of g = j &R &Q t s s u f f i c i e n t l y accurate f o r the purposes of the f o l l o w i n g c a l c u l a t i o n s . For a f l u x tube of unit area i n the equatorial plane, the foot dragging term becomes: F = x B dV = j JB dV = AB J" j d l but 1 j d l = ^ F - 2 ^ f B f _ _ = 2 ^ V B 2 ( 4 a ) S u b s t i t u t i n g = 10 m U B = 5 x 1 0 " u weber m~2 F = 5 x 1 0 - 1 ^ v newton (4b) _1 where v i s the v e l o c i t y (m sec ) of the f l u x tube i n the equatorial plane, 20 On t h e a s s u m p t i o n t h a t the term on the r i g h t hand s i d e of e q u a t i o n (2) c o n t a i n s terms of the s i z e of ( 4 ) , the energy-a s s o c i a t e d w i t h the v a r i o u s terms can be compared. I t i s n e c e s s a r y to compare e n e r g i e s , r a t h e r t h a n f o r c e s , because of the d i f f e r e n t v e l o c i t i e s o f d i f f e r e n t p a r t s of the f l u x t u b e . I t i s f o r the same r e a s o n t h a t (4a) v a r i e s f r o m (1) by -3 - 2 a f a c t o r o f g -J r a t h e r t h a n g as would be e x p e c t e d f r o m a r e a c o n s i d e r a t i o n s a l o n e . I n o t h e r words e q u a t i o n (2) s h o u l d be an e q u a t i o n i n a n g u l a r momentum. I n t r e a t i n g i t as an e q u a t i o n in- l i n e a r momentum as i s done here , i t i s n e c e s s a r y to m u l t i p l y f o r c e s a t the f e e t by a "moment arm" o f g - 1 . The f i r s t two terms of e q u a t i o n (2) are o f t h e f o r m { p A r e p r e s e n t i n g power I p A ^ Thus the r a t i o of the c o n t r i b u t i o n s near the f e e t , and i n the o u t e r r e g i o n s of the magneto sphere a r e : I f pf Af V f 2 = &f |QP (5) I p A V 2 4 f> g 4 S u b s t i t u t i o n of v a l u e s f r o m t a b l e I, i n d i c a t e s t h a t the i n e r -t l a l e f f e c t s of the o u t e r magnetosphere dominate the i n e r t i a l e f f e c t s of the f e e t by a f a c t o r cf 1 0 2 . S i m i l a r l y the power of the f o u r t h and f i f t h terms of e q u a t i o n (2) i s — JLu A. The r a t i o of power nea r the f e e t t o the o u t e r r e g i o n s i s : I t seems u n l i k e l y t h a t the r a t i o px/p > \(J , and the main c o n t r i b u t i o n p r o b a b l y comes from the o u t e r r e g i o n s . 21 Thus the assumption i s made t h a t the f l o w i s p r o d u c e d by f o r c e s i n t h e o u t e r r e g i o n s o f the magnetosphere, and equa-t i o n (2) i s v a l i d i f the i n t e g r a l s are r e s t r a i n e d t o t h i s r e g i o n . Once t h i s I s assumed, I t I s s u f f i c i e n t t o compare f o r c e s i n e q u a t i o n (2) t o determine which terms dominate the f l o w . C o n s i d e r the f i r s t term i n e q u a t i o n ( 2 ) . I t c a n be approximated by: F, = £ f>^ - I O " ' * -%r newton (7) f o r the v a l u e s i n T a b l e I. Thus t h i s term i s n o t i m p o r t a n t f o r f l o w s w i t h a time s c a l e l o n g e r t h a n 1 sec, and w i l l have l i t t l e e f f e c t on the l a r g e s c a l e f l o w s o f the magnetosphere. S i m i l a r l y , the second t e r m o f e q u a t i o n (2) i s of o r d e r : F a . - ^ r° "3"" - 1 0 ~ 1 3 i f n a u , t o f V ( 8 ) T h i s term i s o f the same o r d e r as F f o r J£ ~ \ ^ec""' ( 9 ) d T h i s c o n d i t i o n i s n o t s a t i s f i e d f o r l a r g e s c a l e f l o w s i n the magnetosphere, and t h u s F dominates t h i s term. A s i d e r e s u l t o f t h i s i s t h a t c o n n e c t i v e t u r b u l e n c e d r i v e n by the main f l o w s h o u l d not o c c u r i n the magnetosphere. C o n s i d e r the f i f t h term i n e q u a t i o n ( 2 ) . The k i n e m a t i c v i s c o s i t y i n t h e ' o u t e r magneto sphere i s p r o b a b l y of the o r d e r of 10^ sec""1" ( t h i s i s d i s c u s s e d i n more d e t a i l i n C h a p t e r I V ) . U s i n g the v a l u e s : 22 £ ~ I O8 m p ~ I O kg m ^ j£ ^ \0<i TA~ sec the f i f t h term i s of order; P5 = £ jj -^g- 10 —p; newton (io) comparable with P only when d < 10 m„ This point w i l l be dealt with more s p e c i f i c a l l y i n Chapter IV i n e s t a b l i s h -in g a boundary l a y e r thickness. The f o u r t h term or c o m p r e s s i b i l i t y term i s assumed small as ^ ^ p . This makes the dimensional a n a l y s i s the same as the f i f t h term. Thus equation (2) can be approximated by: J" v p JV = fjx B dV P This can be w r i t t e n : (11) r ( & - ' - S ) < J V = 0 (12) 23 w h e r e I s t h e M a x w e l l s t r e s s t e n s o r . T h u s l a r g e s c a l e c o n v e c t i v e f l o w w i t h i n t h e m a g n e t o s p h e r e i s c o n t r o l l e d b y g a s p r e s s u r e and m a g n e t i c s t r e s s e s . 24 CHAPTER I I I THE MODEL OP POLAR SUBSTORMS The model o f p o l a r substorms t h a t t h e a u t h o r w i s h e s t o p r e s e n t i s as f o l l o w s : (1) F i e l d l i n e s are dragged f r o m the c l o s e d r e g i o n i n t o the t a i l by the s o l a r wind, by e i t h e r o r b o t h of the mechanisms a l r e a d y d i s c u s s e d , r e s u l t i n g i n an i n c r e a s e o f the t a i l f i e l d s t r e n g t h , and a s t o r i n g of p o t e n -t i a l e n e r gy. (2 ) The p o l a r substorm b e g i n s when f i e l d l i n e s recombine i n an i m p l o s i v e f a s h i o n a t t h e n e u t r a l sheet, i n the manner i n d i c a t e d by P e t s c h e k ( 1 9 6 4 ) . T h i s i m p l i e s the r e l e a s e o f s t o r e d p o t e n t i a l energy. (3) The recombined f l u x t u b e s are added t o the n i g h t s i d e o f the c l o s e d doughnut-shaped r e g i o n , as a g i a n t b u l g e , c a u s i n g the a u r o r a l e f f e c t s . ( 4 ) The f l u x t u b e s f l o w around the c l o s e d r e g i o n towards the d a y s i d e c a u s i n g the magnetic substorm, and f u r t h e r a u r o r a l e f f e c t s . The above o r d e r can be thought of as e i t h e r a f l o w sequence o r a time sequence. However, i t i s l i k e l y t h a t some of the 25 s t a g e s w i l l be o c c u r r i n g s i m u l t a n e o u s l y . Step (1) i s p r o -b a b l y a more o r l e s s c o n t i n u o u s p r o c e s s , d e p e n d i n g on the s o l a r wind and i n t e r p l a n e t a r y f i e l d c o n d i t i o n s . Steps ( 2 ) , (3), and (4) c o u l d and p r o b a b l y do t o some e x t e n t p r o c e e d s i m u l t a n e o u s l y , a l t h o u g h step ( 2 ) must be i n i t i a t e d b e f o r e s t e p (3) can o c c u r , and s i m i l a r l y w i t h s t e p s (3) and (4). Each s t e p w i l l be d i s c u s s e d i n t u r n i n the f o l l o w -i n g c h a p t e r s . 26 CHAPTER IV TRANSPORT INTO THE TAIL OP THE MAGNETOSPHERE The f i r s t s t a g e o f t h e model i s t h e d r a g g i n g o f f i e l d l i n e s f r o m t h e c l o s e d r e g i o n i n t o the t a i l o f the magnetosphere by t h e s o l a r wind. As the f i e l d l i n e s are s t r e t c h e d I n t o t h e t a i l , p o t e n t i a l e n e r g y i s s t o r e d , which can l a t e r be r e l e a s e d as a p o l a r substorm. a. Dungey's mechanism F i r s t c o n s i d e r Dungey's mechanism. When t h e t r a n -s i t i o n r e g i o n magnetic f i e l d i s a n t i p a r a l l e l t o the e a r t h ' s f i e l d , t he a n a l y s i s o f P e t s c h e k (1964) c a n be used, and c o m b i n a t i o n o c c u r s a t 1/10 V A ( A x f o r d , P e t s c h e k and S i s c o e , 1965) . A f i e l d of 2 O 7 and a p r o t o n number d e n s i t y of 10' m~3 i s perhap s a r e a s o n a b l e e s t i m a t e f o r c o n d i t i o n s i n the t r a n s i t i o n r e g i o n at the s u b s o i a r p o i n t . (A d e n s i t y jump by a f a c t o r of 3 o r 4 over I n t e r p l a n e t a r y c o n d i t i o n s i s e x p e c t e d - S p r e i t e r e t a l . , 1966.) Thus V A & 140 km s e c - 1 , and magnetic f l u x i s t r a n s p o r t e d i n t o the t a i l a t a r a t e o f 2 x 10^ webers s e c " 1 , assuming t h a t c o m b i n a t i o n o c c u r s a c r o s s a r e g i o n 10 m wide a t the f r o n t o f the 27 magnetosphere. I t w i l l be shown i n l a t e r c h a p t e r s t h a t o p o l a r substorms i n v o l v e s t h e c o l l a p s e o f 10 webers f r o m the t a i l I n t o t h e n i g h t s i d e o f the c l o s e d doughnut-shaped r e g i o n . At t h e above r a t e , s u f f i c i e n t f i e l d c o u l d be t r a n s -f e r r e d t o the t a i l f o r a substorm i n 1 1/2 h o u r s . I n a c t i v e t i m e s , substorms o c c u r e v e r y few h o u r s ( A k a s o f u , 1964). The above r a t e i s e x p e c t e d t o be an o v e r e s t i m a t e , and c o u l d be out by o r d e r s of magnitude. The e r r o r a r i s e s p r i n c i p a l l y f r o m the assumption of a n t i p a r a l l e l f i e l d s . I n f a c t , t h e o r y p r e d i c t s ( P a r k e r , 1963), and s a t e l l i t e o b s e r v a -t i o n s c o n f i r m (Ness e t a l . , 1965) t h a t the i n t e r p l a n e t a r y f i e l d l i n e s l i e p r i m a r i l y on the s u r f a c e o f cones o f c o n s t a n t s o l a r l a t i t u d e , o r a p p r o x i m a t e l y p a r a l l e l t o t h e s o l a r e q u a t o r i a l p l a n e a t the e a r t h ' s o r b i t . However, t h e r e a r e many t e m p o r a l f l u c t u a t i o n s , and t h i s mechanism c o u l d c o n t r i -b ute s i g n i f i c a n t l y t o t h e t r a n s p o r t of f l u x t u b e s i n t o the t a l l . b . V i s c o u s f o r c e s A n o t h e r way i n which f i e l d l i n e s may be t r a n s p o r t e d i n t o the geomagnetic t a i l i s by v i s c o u s - l i k e I n t e r a c t i o n a t the o u t e r boundary of the c l o s e d doughnut-shaped r e g i o n . The r a t e o f t r a n s p o r t I n t o the t a i l s h o u l d be d e t e r m i n e d by the v i s c o s i t y o f the medium. A x f o r d (1964) p r e d i c t e d 28 a k i n e m a t i c v i s c o s i t y o f 10^ mc sec -" 1" t o a c c o u n t f o r f l o w p a t t e r n s w i t h i n the magnetosphere, and t h e n showed t h a t sound waves c r o s s i n g the boundary c o u l d produce t h i s . P a r k e r (1958) o b t a i n e d the same f i g u r e assuming t h a t t h e v i s c o s i t y was p r o d u c e d by the s c a t t e r i n g o f p a r t i c l e s by magnetic f i e l d i n h o m o g e n e i t i e s . I n C h a p t e r I I , an o r d e r of magnitude c a l c u l a t i o n i n d i c a t e d t h a t v i s c o s i t y was Important o n l y f o r f l o w d i s t a n c e s c a l e s -< 10 m. A more e x a c t boundary a n a l y s i s i s c a r r i e d out below. C o n s i d e r a f l u x tube shear f l o w i n v o l v i n g a t h i n s u r f a c e l a y e r of the c l o s e d r e g i o n (assumed c y l i n d r i c a l l y symmetric, and s t e a d y - s t a t e f l o w ) . The f l o w i s i n the 6 d i r e c t i o n , and i s d r i v e n by a f l o w o u t s i d e the boundary. (See F i g . 8a.) C u r v a t u r e e f f e c t s are n e g l e c t e d . E q u a t i o n (2) becomes, f o r f o r c e s i n the 0 d i r e c t i o n : i / ° v ^ =• n < " d s ( i 3 ) where n i s the u n i t normal v e c t o r t o the s u r f a c e . F o r a tube 1 m by 1 m a t t h e e q u a t o r , e q u a t i o n ( 1 3 ) can be appro-ximated by: 29 Shear flow at the boundary of the magnetosphere. Idealized problem. .Shear flow Into the t a i l . 3 0 3 * (1^) The t e r m on the r i g h t hand s i d e i s the " f o o t d r a g g i n g " term, which i s t h e shear s t r e s s a c r o s s t h e f e e t o f the t u b e s . Symmetry r e q u i r e s t h a t t h e Maxwell s t r e s s e s on the o t h e r s u r f a c e s c a n c e l . The s o l u t i o n f o r v i s e x p o n e n t i a l , d e c r e a s i n g w i t h d e p t h t o 1/e o f i t s v a l u e i n a d i s t a n c e th (15) u s i n g v a l u e s o f the p a r a m e t e r s : g = 3 2 n = i n 6 m - 3 1 0 m i = Q 2 1 0 ^ m sec I - 1 0 8 m 1 0 mho B f - 5 0 , 0 0 0 7 The c o r r e s p o n d l n i the n i g h t s i d e i s 2 5 0 webers sec"" 1, assuming t h a t B = 507, and t h a t the boundary l a y e r i s dragged a l o n g a t a v e l o c i t y 31 o f IO-3 m sec~x by the s o l a r wind. T h i s i m p l i e s a time of 10 sec t o b u i l d up enough energy f o r a substorm. T h i s i s a f a c t o r o f 10 t o o l o n g , but c o n s i d e r i n g t h e n a t u r e o f the c a l c u l a t i o n s , and the u n c e r t a i n t y i n V (the v a l u e of which c o u l d v a r y by 2 o r more o r d e r s o f magnitude), t h i s p r o c e s s cannot be r u l e d out as the mechanism of t r a n s p o r t i n t o t h e t a i l . w i l l f l o w towards the n i g h t s i d e , and t h e n be s t r e t c h e d i n t o the t a i l . C o n s i d e r the f l o w d u r i n g the s t r e t c h i n g p r o c e s s . (See P i g . 8b.) The f i e l d l i n e s , due t o t e n s i o n e f f e c t s , f e e l a f o r c e a p p r o x i m a t e l y i n the s o l a r d i r e c t i o n of magni-tude B 2 yp 0 R where R i s t h e i r r a d i u s of c u r v a t u r e . I f t h e d r a g g i n g f o r c e i s v i s c o u s , the e q u a t i o n f o r t h e f l o w i s : I f t h i s mechanism i s dominant, the f i e l d l i n e s (16) A s o l u t i o n t o t h i s e q u a t i o n i s : V 32 u s i n g v a l u e s : B = 30 7 1 = 10 9 m 2 n = I O 6 n f R — 10 8 m and t h e boundary c o n d i t i o n v (x = 0) = 10^ m sec J". P r e s s u r e - l i k e terms i n t h e Maxwell s t r e s s t e n s o r g i v e a f o r c e o f t h e o r d e r ^Q L_ (the p r e s s u r e g r a d i e n t ) . As t h e f l o w i s b e l i e v e d t o e x t e n d t o v e r y l a r g e d i s t a n c e s i n the a n t i s o l a r d i r e c t i o n , l~ » R , and t h e s e terms can be n e g l e c t e d . T h i s s o l u t i o n f a l l s t o 1/e of i t s v a l u e i n .9 x 10-* m. Thus the r a t e o f f l o w i n t o t h e t a i l by v i s c o u s mechanisms i s p r o b a b l y c o n t r o l l e d by the a b i l i t y o f f l u x t u b e s t o c o n v e c t around the s i d e s of the magneto-sphere, r a t h e r t h a n by t h e i r s t r e t c h i n g I n t o the t a i l . 33 CHAPTER V FIELD LINE RECOMBINATION The r e c o m b i n a t i o n of f i e l d l i n e s a c r o s s a n e u t r a l sheet has been d i s c u s s e d by Sweet (1958) and m o d i f i e d by P e t s c h e k (1964) f o r the s t e a d y s t a t e c a s e . A maximum f l o w towards the n e u t r a l sheet o f 1/10 V. was e s t i m a t e d f o r the magnetosphere by A x f o r d , P e t s c h e k and S i s c o e (1965). The f l o w t a k e s t h i s maximum v a l u e when the geometry i s as shown i n F i g . 6c. However, the f l o w can be c o n t r o l l e d by o t h e r f a c t o r s such as t h e b u i l d up of a back p r e s s u r e p r e v e n t i n g the o u t f l o w a l o n g the n e u t r a l sheet of F i g . 6c, o r the hampering o f t h e i n f l o w i f the f i e l d l i n e s i n the plasma f l o w i n g towards the n e u t r a l sheet d e v e l o p a g r e a t e r c u r v a -t u r e ( i . e . magnetic t e n s i o n e f f e c t s r e s t r i c t the f l o w ) . F o r r e c o m b i n a t i o n t o p r o c e e d i m p u l s i v e l y , as r e q u i r e d by the model, a t r i g g e r i n g and a c h o k i n g mechanism are needed. a. T r i g g e r i n g mechanism The t r i g g e r i n g i s p r o b a b l y d e t e r m i n e d by c o n d i -t i o n s i n the n e u t r a l sheet which s h o u l d be i n f l u e n c e d by the f i e l d s t r e n g t h i n the t a i l - a s t r o n g e r f i e l d p r o d u c i n g a g r e a t e r s t r e s s a c r o s s the n e u t r a l s h e e t . T h i s i s p r o b a b l y 34 t h e cause o f t h e h i g h e r o c c u r r e n c e - f r e q u e n c y of substorms d u r i n g geomagnetic storms, when the t a i l f i e l d s t r e n g t h i s h i g h e r . F i g . 4 shows a g r a d u a l d e c r e a s e i n f i e l d s t r e n g t h f u r t h e r back I n the t a i l ( e x c e p t f o r t e m p o r a l f l u c t u a t i o n at 1937 and 2139 UT). T h i s i s due t o the e x p a n s i o n of t h e t a i l i n the a n t i s o l a r d i r e c t i o n . F i g . 4 shows v a l u e s of 40 - 50T around 9 t o 13 e a r t h r a d i i , but o n l y 25 - 3 O 7 beyond 14 e a r t h r a d i i . T h i s i s c o n s i s t e n t w i t h the Imp I r e s u l t s (Ness e t a l . , 1965) which i n d i c a t e a t a i l r a d i u s i n c r e a s e o f 2 o r 3 e a r t h r a d i i i n g o i n g f r o m 10 t o 20 e a r t h r a d i i i n t h e a n t i s o l a r d i r e c t i o n . S i m i l a r l y t h e r e s h o u l d be a d e c r e a s e i n t o t a l f i e l d as the cusp i s approached, j u s t o u t s i d e the c l o s e d r e g i o n . The l o c a t i o n of the cusp i s perha p s b e s t i n d i c a t e d by t h e Imp I p a r t i c l e e x p e r i m e n t s . Anderson (1965) p l a c e s the l i m i t of the s t a b l y t r a p p e d r e g i o n at 8 e a r t h r a d i i , and c a l l s a r e g i o n o f plasma e x t e n d i n g f r o m 8 t o 14 e a r t h r a d i i the cusp. T h i s i s presumably plasma t e m p o r a r i l y t r a p p e d i n the r e g i o n of d e p r e s s e d magnetic f i e l d around the cusp. Thus the t a i l f i e l d p r o b a b l y e x e r t s the maximum s t r e s s a c r o s s the n e u t r a l sheet at about 13 or 14 e a r t h r a d i i i n the a n t i s o l a r d i r e c t i o n . T h i s i s the mo3t l i k e l y p l a c e f o r the n e u t r a l p o i n t of Sweet's mechanism to form, 35 and f o r t h e r e c o m b i n a t i o n t o be t r i g g e r e d . Once a normal component i s e s t a b l i s h e d , hydromagnetic f o r c e s sweep a s i d e the n e u t r a l sheet plasma, and the r e c o m b i n a t i o n p r o c e e d s i m p l o s i v e l y . As t h e geometry i s at f i r s t s i m i l a r t o t h a t o f P e t s c h e k (1964), the r e c o m b i n a t i o n s h o u l d o c c u r i n i t i a l l y a t 1/10 V A . b. C h o k i n g mechanism The c h o k i n g mechanism i s p r o b a b l y a hydromagnetic back p r e s s u r e . I t i s e x p e c t e d t h a t a l a r g e b u l g e ( d i s c u s s e d f u r t h e r i n C h a p t e r VI) w i l l b u i l d up on the n i g h t s i d e o f the e a r t h , and w i l l r e s t r i c t t h e f l o w p a t t e r n of P i g . 6c. The bu l g e i s e x p e c t e d t o ex t e n d s e v e r a l e a r t h r a d i i on the n i g h t -s i d e of the c l o s e d r e g i o n , and w i l l move the boundary o f t h e c l o s e d r e g i o n , and t h e cusp outward by a s i m i l a r amount. T h i s p l a c e s the o r i g i n a l n e u t r a l p o i n t w e l l i n s i d e t h e cusp, where r e c o m b i n a t i o n i s hampered by the c u r v a t u r e o f the f i e l d l i n e s , and consequent lower s t r e s s a c r o s s the n e u t r a l sheet, and perhaps a l s o by the back p r e s s u r e f r o m the o u t f l o w i n g gas. I t i s p o s s i b l e t h a t the n e u t r a l p o i n t moves outward ahead o f the cusp. M o t i o n of t h i s type would b r i n g i t i n t o a r e g i o n of weaker f i e l d , f u r t h e r i n the t a i l , w i t h a consequent s l o w i n g down o f the r e c o m b i n a t i o n r a t e . However, f o r r e c o m b i n a t i o n t o stop e n t i r e l y , i t seems l i k e l y t h a t t h e r e must be a gas p r e s s u r e b u i l d up a t the n e u t r a l p o i n t . 36 c. E n e r g y o f r e c o m b i n a t i o n The magnetic f i e l d e n e r gy t h a t i s a n n i h i l a t e d i n the r e c o m b i n a t i o n i s g i v e n p r i m a r i l y t o the p a r t i c l e s which i t c o n t a i n e d . These cause the a u r o r a l b r i g h t e n i n g a s s o c i a t e d w i t h a substorm. I f the t a i l f i e l d s t r e n g t h i s 3 O 7 , and a number d e n s i t y of 10 m i s assumed, t h e n f r o m the r e l a t i o n s h i p p — =. E n > the e n e rgy (E) g i v e n 2. H o t o the p a r t i c l e s i s 4 kev, i n r e a s o n a b l e agreement w i t h the p a r t i c l e e n e r g i e s c a u s i n g v i s u a l a u r o r a . An e s t i m a t e of the energy f l u x of a u r o r a l e l e c -t r o n s i s 400 e r g s cm f o r a b r i g h t a u r o r a (Chamberlain, 1961) . I f a f l u x o f t h i s o r d e r l a s t s f o r l C p sec (the d u r a t i o n of the a u r o r a l substorm e x p a n s i v e phase) on an 20 a r e a 100 km by 1000 km, the t o t a l e n e r gy I s 3 x 10 e r g s . The volume o f t a i l f i e l d t h a t must be a n n i h i l a t e d t o p r o -duce t h i s e n e r gy has a l i n e a r d i m e n s i o n of 4 x 10' m, o r 6 1/2 e a r t h r a d i i . T h i s Is i n good agreement w i t h the l e n g t h of the t a i l i n v o l v e d i n the r e c o m b i n a t i o n p r e d i c t e d e a r l i e r i n t h i s c h a p t e r . (The l e n g t h w i l l i n c l u d e f i e l d f r o m the boundary of the c l o s e d r e g i o n out t o the n e u t r a l p o i n t or r e g i o n of maximum s t r e s s . ) I f the d i m e n s i o n s o f the c r o s s s e c t i o n a l a r e a are the same, and the f i e l d 7 i s 307, t h i s r e p r e s e n t s a f l u x of 5 x 10 webers, i n c l o s e g agreement w i t h the v a l u e of 10 webers assumed h e r e i n . 37 The r e c o m b i n a t i o n r a t e i s i n i t i a l l y 1/10 V"A, —1 8 o r 45 km s e c " (see T a b l e I ) . To a n n i h i l a t e 10 webers t h r o u g h a s l o t 4 x 10^ m wide a t t h i s r a t e t a k e s 30 min, which i s the d u r a t i o n of the e x p a n s i v e phase o f t h e a u r o r a l substorm ( A k a s o f u , 1964), 38 CHAPTER V I BULGE ADDED ON THE NIGHTSIDE OP THE CLOSED REGION The r e c o m b i n a t i o n of f i e l d l i n e s a t t h e n e u t r a l sheet r e s u l t s i n the a d d i t i o n of f l u x t u b e s t o t h e n i g h t -s i d e of the c l o s e d doughnut-shaped r e g i o n . The i n f l o w i n t o t h e n i g h t s i d e o c c u r s i n i t i a l l y a t a r a t e of V f l, whereas the o u t f l o w I s much slower because o f the f o o t d r a g g i n g e f f e c t . Thus a l a r g e b u l g e i s e x p e c t e d t o form, as shown i n P i g . 9 . The f l o w p a t t e r n d u r i n g the c o l l a p s i n g phase i s shown I n F i g . 10. I f the l o c a t i o n o f the a u r o r a i s the map a l o n g f i e l d l i n e s o f the i n n e r edge o f the n e u t r a l sheet, t h e n t h e a d d i t i o n of a b u l g e t o the n i g h t s i d e of the c l o s e d r e g i o n s h o u l d r e s u l t i n a northward b u l g e o f the a u r o r a l a r c s i n t o the p o l a r cap (southward at t h e s o u t h e r n p o l e ) , as i s seen i n F i g . 1c. The dim e n s i o n s of the bul g e i n F i g . l c a r e 1000 km by 1000 km, and the f i e l d i s 50,0007, g i v i n g a f l u x o f 5 x 10 webers i n the b u l g e , i n agreement w i t h the 10 webers e a r l i e r a s s o c i a t e d w i t h a substorm. As g = 32, t h i s maps as a bul g e o f 5 e a r t h r a d i i on the n i g h t s i d e o f the c l o s e d r e g i o n . I t cannot, however, be 3 9 1200 1200 P I K 9 B u l g e on the n l g h t s i d e of the c l o s e d r e g i o n (a) e q u a t o r i a l s e c t i o n . (b) p i c t o r i a l view P i g . 10. Flow p a t t e r n d u r i n g the c o l l a p s i n g phase. 41 assumed t h a t shapes w i l l be e x a c t l y p r e s e r v e d i n mapping a l o n g f i e l d l i n e s . T h i s would o n l y be t h e c a s e i f the f i e l d l i n e s were c o n s t r a i n e d t o l i e i n m e r i d i a n p l a n e s . The a c t u a l b u l g e i s l i k e l y t o be somewhat f l a t t e r i n the e q u a t o r i a l p l a n e , t h a n a t t h e f e e t , as t h e f l u x t u b e s bend away f r o m a h i g h p r e s s u r e r e g i o n . (Hydromagnetic p r e s s u r e P = B2/2|>0 -+ p •) The n o r t h w a r d e x p a n s i o n r a t e of A k a s o f u ' s b u l g e can be p r e d i c t e d f r o m th e model. R e c o m b i n a t i o n o c c u r s i n i t i a l l y a t 1/10 V A (45 km s e c " 1 . See T a b l e I.) T h i s maps as 1 km s e c - 1 i n t h e a u r o r a l zone. A k a s o f u (1964) 3 3 - 1 quotes 0 . 3 x 10^ t o 1.7 x 10 m sec as b e i n g t y p i c a l o b s e r v e d v a l u e s . I t s h o u l d be emphasized t h a t t h i s does no t I n v o l v e s i g n i f i c a n t m o t i o n o f the f e e t o f the f i e l d l i n e s , o n l y o f the mapping of the i n n e r edge o f the neu-t r a l s h e e t , and t h e r e i s t h e r e f o r e no " f o o t d r a g g i n g " e f f e c t r e s t r i c t i n g t h e northward e x p a n s i o n o f the b u l g e . The l a r g e f l u x tube added t o the n i g h t s i d e con-t a i n s f i e l d l i n e s which have j u s t recombined, and which, t h e r e f o r e , c o n t a i n e n e r g e t i c plasma. Thus, I n s i d e the b u l g e s h o u l d be an a c t i v e a u r o r a l r e g i o n , and t h i s i s i n f a c t i n d i c a t e d by A k a s o f u ' s diagrams ( F i g . 1 . ) . Many a u r o r a l p a t c h e s are o b s e r v e d i n s i d e t h i s r e g i o n . 42 The P i 2 type m i c r o - p u l s a t i o n a c t i v i t y at t h i s time can be u n d e r s t o o d i n terms of the " c l o s i n g " f l u x t u b e s bombarding th e n i g h t s i d e of the magnetosphere a t h i g h v e l o -c i t i e s . T r a n s v e r s e waves are e x c i t e d and g u i d e d i n t o the a u r o r a l zones by the f i e l d l i n e s . The o b s e r v a t i o n s o f I n j u n I I I ( O ' B r i e n and T a y l o r , 1964) add s u p p o r t t o t h i s p i c t u r e , i n t h a t an i n c r e a s e o f the t r a p p e d e l e c t r o n f l u x (E 40 kev) I s o b s e r v e d on the n i g h t s i d e a t t i m e s o f enhanced a u r o r a l a c t i v i t y . D a v i s (1966) r e p o r t e d a d e p l e t i o n of e n e r g e t i c p r o t o n s i n t h e c l o s e d r e g i o n on the n i g h t s i d e a t t i m e s o f a geomagnetic storm. The r e c o v e r y phase c o n t a i n e d a s e r i e s of p o l a r substorms, and w i t h each substorm t h e r e was a sudden s t e p - l i k e i n c r e a s e i n the p r o t o n count, i m p l y i n g i m p u l s i v e i n j e c t i o n of e n e r g e t i c p a r t i c l e s , as would be e x p e c t e d from the model. The measurements were f o r p r o t o n s of energy > 100 kev. At the time of the maximum northward e x t e n t o f the b u l g e , the f l o w i n t o the b u l g e must e q u a l the f l o w o u t . The f l o w i n t o the b u l g e i s 54 , 0 0 0 webers sec ( I . e . the r e c o m b i n a t i o n r a t e a t the n e u t r a l sheet at 45 km s e c - 1 , t h r o u g h a s l o t 4 x 10^ m wide, and a t a l l f i e l d s t r e n g t h of 3 O 7 ) . An i n s p e c t i o n of F i g . 1 shows t h a t the o u t f l o w 43 f r o m the b u l g e e x t e n d s o v e r a range o f about 15° i n l a t i -tude, o r 1500 km. I f t h e o u t f l o w o f f l u x t u b e s t h r o u g h m e r i d i a n s on each s i d e of m i d n i g h t i s c o n s i d e r e d , an o u t -f l o w v e l o c i t y o f 350 m s e c - 1 i s p r e d i c t e d , i n agreement w i t h o b s e r v e d e a s t w a r d and westward d r i f t r a t e s . I n t h i s c a l c u l a t i o n , t h e assumption has been made t h a t t h e recombina-t i o n r a t e h as n o t slowed down a t t h e time o f maximum n o r t h -ward e x t e n t , and t h i s d r i f t r a t e s h o u l d t h e r e f o r e be a maxi-mum v a l u e . 44 CHAPTER V I I RETURN PLOW TO THE DAYSIDE a. The f l o w Once the b u l g e has been d e p o s i t e d on the n i g h t s i d e of the e a r t h , i t may be e x p e c t e d t o d r i v e a r e t u r n f l o w towards t h e d a y s i d e i n t h e o u t e r r e g i o n s o f t h e c l o s e d doughnut-shaped r e g i o n . T h i s t y p e of m o t i o n appe a r s i n P i g s . 1 c, d, e, f . The a u r o r a l p a t c h e s and i r r e g u l a r s t r u c -t u r e are o b s e r v e d t o d r i f t e a s t w a r d on the e a s t s i d e o f the m i d n i g h t m e r i d i a n , and westward on the west s i d e , i n d i c a t i n g o u t f l o w f r o m the m i d n i g h t m e r i d i a n . The g i a n t b u l g e i s ex-p e c t e d t o s p r e a d out o v e r the s u r f a c e of the c l o s e d r e g i o n as p a r t o f t h i s g e n e r a l f l o w . b. An approximate f l o w e q u a t i o n and s o l u t i o n When the b u l g e i s f i r s t d e p o s i t e d , t h e r e s h o u l d be an Immediate d e f o r m a t i o n of the f l u x t u bes, at A l f v e n v e l o c i t i e s , w i t h n e g l i g i b l e f o o t m o t i o n. F u r t h e r f l o w must t h e n o c c u r by the d r a g g i n g of the f e e t t h r o u g h the i o n o s p h e r e T h i s f l o w s h o u l d be much slower than the A l f v e n v e l o c i t y . To p u t t h i s on a mathem a t i c a l b a s i s , and o b t a i n some i d e a of the flo w , the assumption i s made t h a t f l u x 45 t u b e s a r e c o n f i n e d t o l i e a p p r o x i m a t e l y i n m e r i d i a n p i i n t h e i r c o n v e c t i o n w i t h i n t h e magnetosphere ( i . e . B$. -and m i r r o r symmetry e x i s t s i n the e q u a t o r i a l p l a n e . T f o r f l o w i n the0direction, e q u a t i o n 12 g i v e s : fe - WJV - O (18) The c o o r d i n a t e system I s shown I n P i g , 11. XQ i s used r a t h e r t h a n 6 t o i n d i c a t e t h a t t h e system w i l l be t r e a t e d as C a r t e s i a n , c u r v a t u r e e f f e c t s i n the 6 d i r e c t i o n b e i n g I g n o r e d . E q u a t i o n (l8) c a n be i n t e g r a t e d o v e r a tube lm by lm a t t h e e q u a t o r t o g i v e the approximate r e l a t i o n : £ytt + ^ T ^ U p J * 3* -O (19a) or — ~~ A VQ (19b) where A „ 2 61 6>-f and P = p + B /2p 0 I s the hydromagnetic p r e s s u r e . The t h i r d term o f e q u a t i o n (19a) i s the f o o t d r a g g i n g 46 P i g . 11. C o o r d i n a t e system. F i g . 12. Geometry f o r e q u a t i o n ( 2 4 ) . 4 7 term, and I s the Maxwell s t r e s s ( i n the Q d i r e c -t i o n on the f e e t of t h e t u b e . I t i s assumed t h a t the f o o t d r a g g i n g t e r m i s t h e o n l y s t r e s s on the f e e t i n the 8 d i r e c -t i o n . The o t h e r two terms a r e the gas p r e s s u r e and Maxwell s t r e s s on the s i d e s of the t u b e . F o r t h o s e who f i n d the concept of Maxwell s t r e s s u n s a t i s f a c t o r y , e q u a t i o n (11) can be w r i t t e n : o r a p p r o x i m a t e l y : , A ^ ABA. ftjaa* d l P°3 where f x and f g denote the two f e e t of the tube. I t I s shown i n C h a p t e r V I I I , e q u a t i o n (32) t h a t B 0 f l ^ - =^  «3h_ B>r and e q u a t i o n (19) f o l l o w s . The e r r o r i n a p p l y i n g e q u a t i o n (19) t o f l u x t u b e s not c o n f i n e d t o m e r i d i a n p l a n e s , assuming the a n g u l a r depart-ure i s n o t t o o l a r g e , a r i s e s not so much from the f a c t t h a t t h e y a r e a t an angle t o the m e r i d i a n p l a n e s , as from the 48 f u r t h e r d i s t o r t i o n o r shape change t h a t such a tube under-goes i n t h e f l o w ( i . e . the g r e a t e s t e r r o r i s p r o b a b l y due t o changes i n the e n e r g y o f the magnetic f i e l d i n the f l u x t u b e ) . A c o n s i s t e n t assumption i n c o n s i d e r i n g f l o w i n the r d i r e c t i o n i s t h a t the " f o o t d r a g g i n g " t e r m a c t s as a p e r t u r b a t i o n on t h e shape o f the f l u x t u b e . Then e q u a t i o n (12) can be w r i t t e n : I ^ o o r ^ U x r \ 2 | i . y **~ ^ V f o R / 3* u (20) where R i s an a p p r o p r i a t e average r a d i u s o f c u r v a t u r e o f the f l u x tube i n the u n p e r t u r b e d s t a t e ( <5j_ = 0 ). The t h i r d term i n e q u a t i o n (20) a r i s e s f r o m the s t r e s s t e n s o r component: y° ^ ~ r°R ( 2 1 ) The j u s t i f i c a t i o n f o r t h i s i s as f o l l o w s : I f a f i e l d l i n e i s l o c a l l y approximated by a c i r c u l a r a r c , t h e n i t obeys the e q u a t i o n : x 2 + y 2 - R 2 49 i n an x, y plane with the o r i g i n at the center of curvature Then dy x ^ x dx y R f o r small x. Therefore B ^ _ B x R dx R and equation (21) f o l l o w s . A comparison of the t h i r d and f o u r t h terms of equation (20) using the f o l l o w i n g values f o r the parameters: = 10 mho 1 = I O 8 m R = 10 7 m B = 50 7 B|> = 50,000 7 6 = 32 v f r 10 m sec (22) i n d i c a t e s that the t h i r d term dominates the f o u r t h by 3 orders of magnitude. An in s p e c t i o n of P i g . 1 shows l i t t l e rthward or southward motion of f l u x tube f e e t , i . e . the no 50 a u r o r a l d r i f t m o t i o n s are p r e d o m i n a n t l y eastward o r westward, v- i s t h e r e f o r e s m a l l , and the a p p r o x i m a t i o n : can be used I n s t e a d o f ( 2 0 ) . P h y s i c a l l y e q u a t i o n s (19) and (23) i m p l y t h a t c u r v a t u r e f o r c e s i n the x r d i r e c t i o n cause p r e s s u r e g r a d i e n t s , and t h e r e f o r e f l o w i n the x 0 d i r e c t i o n , opposed by the f o o t d r a g g i n g e f f e c t . A s o l u t i o n t o e q u a t i o n s (19) and (23) f o r the geo-metry of F i g . 12, i s : f r l £ _ ^ alt (23) Thus f r o m e q u a t i o n ( 1 9 ) : J Q L Ak o r s u b s t i t u t i n g v a l u e s i n ( 2 2 ) : V© = - $>x\O5 din ^ s e c (25) or 10' m s e c -I (26) I f i t i s assumed t h a t f i g u r e 1 i s a map of the b u l g e shape, s l o p e s o f o r d e r 0.1 e x i s t on t h e morning s i d e , i m p l y i n g f o o t f l o w v e l o c i t i e s of s e v e r a l hundred m s e c - 1 i n agree-ment w i t h the o b s e r v e d d r i f t v e l o c i t i e s of a u r o r a l p a t c h e s ( A k a s o f u , 1964). An attempt I s now mads t o get a wave e q u a t i o n f o r the f l o w . I f the f l u i d i s assumed i n c o m p r e s s i b l e , t h e e q u a t i o n of c o n t i n u i t y , d l v v = 0, can be r e p l a c e d by: S u b s t i t u t i n g (25) i n t o t h i s : I f e q u a t i o n (28) i s l i n e a r i z e d by w r i t i n g h = h + h (h c o n s t a n t ) : o K o 1 (27) (28) — - • v v m sec (2Q) T h i s i s the d i f f u s i o n e q u a t i o n , and the s u b s t i t u t i o n of a s m a l l a m p l i t u d e s i n u s o i d g i v e s a phase v e l o c i t y of £> x lo'^k -i 7 m sec (assuming h = 10'm - the depth of p e n e t r a t i o n of the f l o w ) , o r a phase v e l o c i t y at the f e e t o f : C s . ^ x IO kf m s e c 1 (30) 52 where k i s the wave number. I n s p e c t i o n of P i g . 1 shows t h a t the dominant wavelength I s of the o r d e r 10^m. Thus c ^  4 x l o \ i s e c - 1 . T h i s i s a f a c t o r of 40 f a s t e r t h a n the o b s e r v e d v e l o c i t y of the westward surge, which must, t h e r e -f o r e , v i o l a t e t h e assumptions i f i t i s a s u r f a c e wave on t h e c l o s e d s e c t i o n o f t h e magnetosphere. c. The b u l g e shape The shape of the bulge to the east I s s i m i l a r t o t h a t e x p e c t e d f r o m the d i f f u s i o n e q u a t i o n , but the westward t r a v e l -l i n g surge i s q u i t e d i f f e r e n t f r o m t h a t p r e d i c t e d by e q u a t i o n ( 2 9 ) . A n u m e r i c a l t r e a t m e n t of the n o n - l i n e a r e q u a t i o n (28) g i v e s the same o r d e r o f magnitutde f o r the v e l o c i t i e s , and t h e wave shape i s s i m i l a r t o t h a t o b s e r v e d i n h e a t f l o w problems. Thus the shape of the westward t r a v e l l i n g surge cannot be accounted f o r by e q u a t i o n ( 2 8 ) . The b a s i c e a s t - w e s t asymmetry i s p r o b a b l y a r e s u l t of the e a r t h ' s r o t a t i o n . The v e l o c i t y of the f e e t due t o r o t a t i o n I s ~ 130 m sec -" 1", whereas the o b s e r v e d d r i f t v e l o -c i t i e s are 300 m sec . Thus at any g i v e n time, a u r o r a l p a t c h e s and i r r e g u l a r i t i e s to the east, i n P i g . 1, are two o r t h r e e t i m e s f u r t h e r away f r o m the m i d n i g h t m e r i d i a n t h a n those t o the west. An asymmetry i n the shape c f the b u l g e i s e x p e c t e d 5 3 i f i t i s assumed t h a t t h e r e c o m b i n a t i o n s l o t remains f i x e d i n l o c a l t i m e, w h i l e t h e e a r t h and c l o s e d r e g i o n r o t a t e under i t . P i g . 13 shows the e x p e c t e d shape of the b u l g e a f t e r 1/2 h r , a l l o w i n g f o r a d d i t i o n on an a r e a s i x e a r t h r a d i i wide, and a l l o w i n g e a s t - w e s t s p r e a d at t h e r a t e o f —1 -1 10 km sec (the map of 300 m sec ) f r o m the time of d e p o s i t i o n o f a p a r t i c u l a r l a y e r o f the b u l g e . d. The westward t r a v e l l i n g surge A l t h o u g h t h e e x i s t e n c e o f an e a s t - w e s t asymmetry c a n be e x p l a i n e d as above, i t s t i l l o f f e r s no e x p l a n a t i o n of the s t e e p f r o n t o f the westward t r a v e l l i n g s u r g e . S e v e r a l p o s s i b i l i t i e s a r e d i s c u s s e d here which might, by t h e m s e l v e s , or i n c o m b i n a t i o n cause such a shape: (1) I t has been assumed t h u s f a r t h a t the r e c o m b i n a t i o n s l o t has a w i d t h of about 6 e a r t h r a d i i . I f t h i s w i d t h i n c r e a s e d as t h e 3ubstorm p r o g r e s s e d , t h e surge would be the map of the edge of t h e s l o t , the northward motion o f the a u r o r a l a r c s b e i n g o f the same n a t u r e as the northward motion n e a r m i d n i g h t , i . e . the mapping of the outward m o t i o n of the i n n e r edge of the n e u t r a l sheet.. (2) The f i e l d l i n e s i n the t a i l c a n f l o w i n f r o m the s i d e s , as w e l l as the top, t o r e p l a c e the a n n i h i l a t e d f i e l d i n the t a i l (see P i g . 14). These f i e l d l i n e s can then recombine t h r o u g h the s l o t , be added to the b u l g e , and r e t u r n t o t h e i r 5 4 P i g . 13. Asymmetry of the b u l g e . I t i s assumed t h a t recombined f i e l d l i n e s a r e added t h r o u g h a s l o t 6 e a r t h r a d i i wide, f o r 1/2. h r w h i l e the e a r t h r o t a t e s under the s l o t . Each l a y e r i s assumed t o s p r e a d a t 10 km s e c - 1 f r o m the time of i t s d e p o s i t i o n . P i g . 14. Flow i n the t a i l I n t o the r e g i o n of a n n i h i l a t i o n . T h i s i s a c r o s s s e c t i o n of the t a i l . 55 o r i g i n a l l o n g i t u d e w i t h l i t t l e o r no m o t i o n o f t h e i r f e e t , i . e . the f l o w t a k e s p l a c e by d i s t o r t i o n of t h e f i e l d l i n e s . As seen a t the e a r t h , t h i s would be s i m i l a r i n appearance t o (1) above, and would account f o r the f e a t u r e s of the westward s u r g e . (3) The f e e t of the o u t e r f i e l d l i n e s a r e a t h i g h e r l a t i t u d e s . T h i s e f f e c t was i g n o r e d due t o the C a r t e s i a n c o o r d i n a t e system a p p r o x i m a t i o n . I f a f i e l d l i n e i n the t o p o f the surge ( c o -l a t i t u d e 16° - see f i g . 1 d) i s compared w i t h one at the base ( c o l a t i t u d e 21°), the r a t i o of e n e r g y l o s s e s (F . v,) f o r the same a n g u l a r v e l o c i t y i s the r a t i o o f t h e i r v f squa-re d , o r (16/21)2 = .58. Thus the t o p of the b u l g e s h o u l d t r a v e l f a s t e r t h a n i s i n d i c a t e d by e q u a t i o n (28), l e a d i n g t o a s t e e p e r wave f r o n t , and a p o s s i b l e e x p l a n a t i o n of the westward s u r g e . (4) I n a d d i t i o n t o the above f l o w p a t t e r n s , a number of magnetic f i e l d d i s t o r t i o n s o c c u r r i n g i n the f l o w c o u l d change the b a s i c wave shape f r o m t h a t p r e d i c t e d by e q u a t i o n (28). F o r i n s t a n c e , i f f l u x t u b e s A, B and C i n F i g . 14, l y i n g e a s t - w e s t i n the t a i l , and anchored e a s t - w e s t i n the p o l a r cap, a r e d e p o s i t e d on t o p of each o t h e r i n the b u l g e , as might be e x p e c t e d f r o m the f l o w , t h e n t h e r e I s a t w i s t I n the f l u x tube which w i l l r e s u l t i n t o r s i o n a l o n g a f l u x t ube. With t h i s , and o t h e r t y p e s of d i s t o r t i o n t h a t c o u l d o c c u r , i t i s q u i t e p o s s i b l e t h a t t h e shape o f the f l u x tube 56 t h a t makes up the b u l g e i s q u i t e d i f f e r e n t a t t h e e q u a t o r and a t t h e f e e t . I n F i g s . 1 c and d t h e r e i s a h i g h l a t i t u d e a u r o r a l a r c which moves northward j u s t ahead o f the westward su r g e . T h i s i s c o n s i s t e n t w i t h the i d e a of a s p r e a d i n g b u l g e , as t h e r e must be northward movement of the t a i l f i e l d l i n e s ' f e e t i n o r d e r t o accommodate the b u l g e . T h i s f a v o u r s (3) and (4) above, as does the e x i s t e n c e o f westward d r i f t . However, ( l ) and (2) cannot be d i s c o u n t e d e n t i r e l y on t h e s e grounds, as the r e s u l t a n t f l o w s t i l l depends on whether h i n e q u a t i o n (24) i s g r e a t e r a t the m i d n i g h t m e r i d i a n o r a t t h e s u r g e . As the b u l g e c o n t i n u e s t o s p r e a d i n the r e c o v e r y phase, t h e r e i s a g e n e r a l southward motion of the n o r t h e r n edge of the a u r o r a l a r c s ( F i g s . 1 e and f ) , as must be ex-p e c t e d i f t h e b u l g e s p r e a d s w i t h o u t f u r t h e r a d d i t i o n of f l u i d . T h i s m o t i o n does o f c o u r s e i n v o l v e f o o t d r a g g i n g , and i s c o n s e q u e n t l y s l o w e r t h a n the o r i g i n a l northward e x p a n s i o n . e. EGO s a t e l l i t e o b s e r v a t i o n s Now c o n s i d e r F i g . 4, the r e s u l t s of Heppner, I965. EGO s a t e l l i t e o b s e r v e d sudden d e c r e a s e s i n f i e l d s t r e n g t h a t 1937 and 2139 U.T. Two p o s s i b l e e x p l a n a t i o n s of t h i s are o f f e r e d : 57 (1) 10 webers o f t a l l f i e l d a r e a n n i h i l a t e d , c a u s i n g a d e c r e a s e i n t h e t a i l f i e l d s t r e n g t h . There are two o b j e c -t i o n s t o t h i s . The approximate time f o r an e x p a n s i o n wave to c r o s s the t a i l i s about 5 min, so t h a t i t i s r a t h e r d i f f i -c u l t t o ac c o u n t f o r the bay commencement, o b s e r v e d a t t h e e a r t h , o c c u r r i n g 15 min b e f o r e the o b s e r v e d d e c r e a s e a t the s a t e l l i t e . The second o b j e c t i o n i s one o f magnitude. The a r e a of f i e l d a n n i h i l a t e d i s about 1 . 6 x l O 1 ^ m 2 i n a c c o r d -ance w i t h t h e model. The a r e a o f the upper h a l f t a i l i s l 6 2 1 . 5 x 10 m (15 e a r t h r a d i i r a d i u s ) . Thus the e x p e c t e d d e c r e a s e i s 10$ o r ky, whereas the o b s e r v e d d e c r e a s e i s 40$ or 157. F o r t h e s e r e a s o n s t h e a u t h o r p r e f e r s t h e e x p l a n a t i o n g i v e n below. (2) T h i s i n v o l v e s the westward s u r g e . The s a t e l l i t e l o c a l time a t t h e s e e v e n t s i s 2130. The westward t r a v e l l i n g surge can be seen t o p a s s t h i s m e r i d i a n i n P i g . 1 d sometime i n the p e r i o d 10 t o 30 min a f t e r the substorm commencement, a c c o u n t i n g f o r the 15 min observed d e l a y at the s a t e l l i t e . The a m p l i t u d e of the surge i s 500 km, c o r r e s p o n d i n g t o a bu l g e of about 2 1/2 e a r t h r a d i i . Thus, on the n i g h t s i d e , the boundary o f the c l o s e d r e g i o n , and the cusp, o r r e g i o n of d e p r e s s e d magnetic f i e l d , may be e x p e c t e d t o move outward by about t h i s amount as the surge p a s s e s . As the s a t e l l i t e was at low l a t i t u d e s , and a t g e o c e n t r i c d i s t a n c e s 10 and 58 13 e a r t h r a d i i , t h i s movement s h o u l d p l a c e the s a t e l l i t e f u r t h e r i n the c usp. The f i e l d s t r e n g t h i n the cusp v a r i e s f r o m z e r o t o the t a i l f i e l d s t r e n g t h , and the d e c r e a s e c o u l d be q u i t e l a r g e . 59 CHAPTER V I I I MAGNETIC EFFECTS a. P e d e r s e n c o n d u c t i v i t y As was d i s c u s s e d i n C h a p t e r I I , and i l l u s t r a t e d i n F i g . 7J s t e a d y s t a t e c o n v e c t i o n i s accompanied by a P e d e r s e n c u r r e n t i n t h e i o n o s p h e r e which c o m p l e t e s i t s c i r c u i t by f l o w i n g up t h e f i e l d l i n e s , and t h e n a c r o s s them, i n e s s e n c e f o r m i n g a c u r r e n t l o o p . I n any s t e a d y -s t a t e f l o w , the f e e t o f the f i e l d l i n e s must t r a c e out c l o s e d l o o p s i n the i o n o s p h e r e . C o n s i d e r i n p a r t i c u l a r a plasma c o n v e c t i o n i n a c i r c u l a r p a t t e r n o v e r an Iono-sphere o f u n i f o r m c o n d u c t i v i t y . Assume f o r s i m p l i c i t y t h a t the f i e l d l i n e s are v e r t i c a l . The r e s u l t i s a s e r i e s of c u r r e n t l o o p s whose normals f o r m c i r c l e s c o a x i a l w i t h t h e c o n v e c t i o n , and the c u r r e n t f l o w I s s i m i l a r t o t h a t i n a t o r o i d a l c o i l . A t o r o i d a l c o i l p r o d u c e s l i t t l e or no e x t e r i o r magnetic f i e l d . Thus such a f l o w s h o u l d produce l i t t l e magnetic e f f e c t a t the e a r t h ' s s u r f a c e . Mathemati-c a l l y i t can be s t a t e d : The l i n e i n t e g r a l i s t a k e n a l o n g a c i r c l e on the e a r t h ' s s u r f a c e d i r e c t l y u n d erneath and c o a x i a l w i t h the c o n v e c t i n g (31) 60 f l o w (see P i g . 15). J t h r o u g h such a c i r c l e i s z e r o , and symmetry r e q u i r e s t h a t B be the same everywhere i . e . z e r o s i n c e t h e R.H.S. o f e q u a t i o n (31) i s z e r o . C o n s i d e r the case where the c o n v e c t i n g s l a b i s not of u n i f o r m t h i c k n e s s . The h o r i z o n t a l f i e l d component j u s t above the i o n o s p h e r e due t o the Pe d e r s e n c u r r e n t l o o p s i s &>U — |U0T — po <5x Vsp. Bx and hence BkW — po cs~x_ Vr. Br. W (32) where w i s the t h i c k n e s s o f the s l a b . F o r a s t e a d y - s t a t e f l o w , v f B f w i s a c o n s t a n t a t any c r o s s s e c t i o n , and t h u s t h e t o t a l h o r i z o n t a l magnetic f l u x remains c o n s t a n t w i t h i n a c o n v e c t i n g slab., I m p l y i n g t h a t the f i e l d l i n e s due t o the Ped e r s e n c u r r e n t t e n d t o s t a y i n s i d e the s l a b , e x c e p t perhaps at p l a c e s where t h e r e are sharp changes i n geometry. Thus one can say t h a t i n g e n e r a l , a s t e a d y - s t a t e P e dersen c u r r e n t produced by c o n v e c t i o n p roduces magnetic e f f e c t s much s m a l l e r t h a n those p r e d i c t e d by the sheet c u r r e n t a p p r o x i m a t i o n i f the P e d e r s e n c o n d u c t i v i t y o f the i o n o s p h e r e i s r e a s o n a b l y u n i f o r m . 6 1 P i g . 1 6 . E f f e c t of H a l l c u r r e n t s on the magnetic f i e l d l i n e s . 62 b. H a l l c o n d u c t i v i t y The H a l l c o n d u c t i v i t y e x i s t s because the e l e c t r o n s s t a y f r o z e n t o lower l e v e l s t h a n do t h e p r o t o n s , and can c o n s e q u e n t l y f o l l o w the motion of the f l u x tube f e e t a t lower l e v e l s . E l e c t r o n m o t i o n i n the d i r e c t i o n o f f l u x tube f l o w i s a c u r r e n t i n the o p p o s i t e d i r e c t i o n . The r e s u l t I n a c l o s e d s t e a d y - s t a t e c o n v e c t i o n f l o w i s t h a t the f e e t of f i e l d l i n e s are crowded t o g e t h e r , o r d i m i n i s h e d i n s i d e the H a l l c u r r e n t l o o p s , as shown i n P i g . 16. c. C u r r e n t systems i n the Ionosphere Most e s t i m a t e s o f i o n o s p h e r i c c o n d u c t i v i t i e s y i e l d v a l u e s f o r the r a t i o o f the h e i g h t i n t e g r a t e d Peder-sen c o n d u c t i v i t y t o t h e h e i g h t i n t e g r a t e d H a l l c o n d u c t i v i t y i n the range f r o m 1:10 t o 1:1. However, the c o n s i d e r a t i o n s a t the b e g i n n i n g o f t h i s c h a p t e r show t h a t the Pedersen c o n d u c t i v i t y s h o u l d have l i t t l e magnetic e f f e c t a t the e a r t h ' s s u r f a c e , so t h a t much of the o b s e r v e d e f f e c t I s p r o b a b l y due t o H a l l c u r r e n t s . These o c c u r at an a l t i t u d e of about 100 km. As the s p a t i a l v a r i a t i o n on the e a r t h ' s s u r f a c e u s u a l l y i n v o l v e s d i s t a n c e s g r e a t e r t h a n t h i s , the sheet c u r r e n t a p p r o x i m a t i o n sh o u l d be r e a s o n a b l e . (B = p QJ/2) Thus the f l o w p a t t e r n of f l u x tube f e e t i n the i o n o s p h e r e may be ex p e c t e d t o produce the same p a t t e r n f o r the c u r r e n t s , but w i t h the d i r e c t i o n r e v e r s e d . P i g . 17 63 F i g . 1 8 . Flow of f l u x t u b e s i n t o the b u l g e . 64 shows the type of s t e a d y - s t a t e f l o w which might be e x p e c t e d f o r a n o n - r o t a t i n g e a r t h , w i t h s t e a d y - s t a t e c o n v e c t i o n and i n t e r c h a n g e o c c u r r i n g as d e s c r i b e d i n e a r l i e r c h a p t e r s . The c u r r e n t p a t t e r n produced by t h i s has some f e a t u r e s i n common w i t h t h o s e i n F i g . 3. The r o t a t i o n of the e a r t h , and t h e non s t e a d y - s t a t e t h a t e x i s t s i n a substorm would be e x p e c t e d t o m o d i f y t h i s p a t t e r n somewhat. However, i t seems u n l i k e l y t h a t i t i s c a p a b l e of a c c o u n t i n g f o r the s t r o n g westward e l e c t r o j e t t h a t i s o b s e r v e d on the n i g h t s i d e C o n s i d e r the c r o s s p o l a r cap f l o w produced by a p o l a r substorm. F o r f i e l d l i n e s i n the t a i l , m o t i on i s c o n t r o l l e d by hydromagnetic wave v e l o c i t i e s . However, the m o t i o n of t h e i r f e e t i s c o n t r o l l e d by the f o o t d r a g g i n g e f f e c t o f the i o n o s p h e r e , and may t h u s be e x p e c t e d t o be s l o w e r . The a n n i h i l a t i o n of f i e l d r e s u l t s i n f l o w toward the n e u t r a l sheet at 1/10 of the A l f v e n v e l o c i t y . The c o r -r e s p o n d i n g m o tion of f l u x tube f e e t a c r o s s the p o l a r cap towards the n i g h t s i d e w i l l l a g t h i s . Thus the f e e t f l o w a c r o s s the polar cap d u r i n g a bay, s h o u l d e q u a l the f l u x a n n i h i l a t e d a t the n e u t r a l sheet d u r i n g the i m p l o s i o n . The d u r a t i o n of a bay i s 4000 sec, and the c o r r e s p o n d i n g p o t e n t i a l a c r o s s the p o l a r cap i s 10 webers/4000 sec = 2.5 x 10^ v o l t s . T h i s i s of the c o r r e c t o r d e r of magnitude f o r f i e l d s c a l c u l a t e d f r o m the bay c u r r e n t system ( e . g . T a y l o r and Hones, 1965). Thus the H a l l c u r r e n t can account f o r t h e c r o s s p o l a r cap f l o w . 65 d. Westward e l e c t r o j e t I t has been i n d i c a t e d (Kim and Kim, 1963) t h a t the h e i g h t i n t e g r a t e d c o n d u c t i v i t i e s i n an a u r o r a e x c i t e d i o n o s p h e r e might be as much as two o r d e r s of magnitude g r e a t e r t h a n t h a t i n a q u i e t i o n o s p h e r e . Thus, a h i g h c o n d u c t i v i t y s t r i p might be e x p e c t e d a l l a l o n g the a u r o r a l a r c s . The a d d i t i o n o f f l u i d i n t o t h e b u l g e , and the r e t u r n f l o w toward t h e d a y s i d e means t h a t t h e r e i s a southward f l o w of f l u x tube f e e t a c r o s s the a u r o r a l a r c s , as shown i n P i g . 18. T h i s f l o w s h o u l d cause a l a r g e P e d e r s e n c u r -r e n t i n the h i g h c o n d u c t i v i t y s t r i p , which b e i n g l o n g and t h i n , g i v e s a c u r r e n t s i m i l a r t o a s i n g l e l o o p , r a t h e r t h a n the t o r o i d a l c u r r e n t system d i s c u s s e d e a r l i e r . The p r e v i o u s p a r a g r a p h p r e d i c t e d a c r o s s p o l a r cap v o l t a g e o f 4 -1 2.5 x 10 v o l t , or an e l e c t r i c f i e l d of 0 . 0 0 6 v m . F o r 2 a h e i g h t i n t e g r a t e d P e dersen c o n d u c t i v i t y of 2 x 10 mho i n the h i g h c o n d u c t i v i t y s t r i p (Kim and Kim, 1963) the h e i g h t i n t e g r a t e d P e dersen c u r r e n t i s 1 .2 amp m"1. U s i n g the sheet c u r r e n t a p p r o x i m a t i o n , t h i s g i v e s a magnetic f i e l d of 0OO7, and t h u s g i v e s an adequate e x p l a n a t i o n of the westward e l e c t r o j e t . 66 CHAPTER IX DISCUSSION The model of the magnetosphere as p r e s e n t e d h e r e i n c o n t a i n s many f e a t u r e s of e a r l i e r models, i n c l u d i n g t h o s e o f A x f o r d and H i n e s (1961), A x f o r d , P e t s c h e k and S l s c o e (1965), Dungey (1963J, Ness ( 1965 )0 The new f e a t u r e s of the p r e s e n t model a r e : (1) R e c o m b i n a t i o n o c c u r s i m p u l s i v e l y at the n e u t r a l sheet c a u s i n g p o l a r sub storms. (2 ) R e c o m b i n a t i o n i s l i m i t e d t o a w i d t h of 6 o r 7 e a r t h r a d i i i n t h e t a i l . (3) The " f o o t d r a g g i n g " e f f e c t i s found t o dominate the I n e r t i a l and v i s c o u s e f f e c t s i n the f l o w of f l u x t u b e s . T h i s i s t a k e n i n t o a ccount i n the development of the model. (4) A l a r g e b u l g e i s formed on the n i g h t s i d e , d r i v i n g the r e t u r n f l o w , and p r o b a b l y c r e a t i n g a w a v e - l i k e f e a t u r e on the n i g h t s i d e o f the c l o s e d region., (5) The e f f e c t o f the Pedersen c u r r e n t i s n e g l i g i b l e a t t h e s u r f a c e o f the e a r t h , e x c e p t f o r the c o n d u c t i v i t y ano-maly a l o n g the a u r o r a l a r c s , where i t c o u l d be the cause of the westward e l e c t r o j e t . 67 D u r i n g the l a s t few months of the p r e p a r a t i o n of t h i s t h e s i s , the a u t h o r has r e c e i v e d p r e p r i n t s f r o m W. I. A x f o r d and J . W. Dungey, b o t h of whom have advanced what i s e s s e n t i a l l y the same model, i . e . substorms are the r e s u l t s of I m p u l s i v e r e c o m b i n a t i o n i n the geomagnetic t a i l . However, the development i n t h e t h r e e c a s e s i s q u i t e d i f f e r e n t . There are a few o t h e r e x p e r i m e n t a l o b s e r v a t i o n s which might be e x p l a i n e d i n terms o f t h e model: (1) As the Anderson (1965) e l e c t r o n i s l a n d s s t a r t at the o u t s i d e of the cusp r e g i o n , and t h e i r e n e r g i e s are s i m i l a r to t h o s e c a u s i n g a u r o r a l X - r a y s , i t I s p o s s i b l e t h a t t h e y are the o t h e r h a l f of t h e r e c o m b i n a t i o n p r o c e s s p r o c e e d i n g outwards a l o n g t h e t a i l . I t would be i n t e r e s t i n g t o l o o k f o r a c o r r e s p o n d e n c e w i t h substorms. (2) The o b s e r v a t i o n of C a r p e n t e r (1966) o f a sudden i n c r e a s e i n the e q u a t o r i a l r a d i u s of the e l e c t r o n knee around 1800 LMT c o u l d be e x p l a i n e d i n terms of a type of s t a g n a t i o n p o i n t , as C a r p e n t e r s u g g e s t e d . T h i s would be an average s t a g n a t i o n p o i n t as the f l u i d added by s u c c e s s i v e substorms t r i e d t o f l o w upstream around the e v e n i n g s i d e . I t s s i z e and p o s i t i o n would depend on the s.ubstorm s i z e and o c c u r r e n c e f r e q u e n c y . C a r p e n t e r o b s e r v e d t h a t the knee becomes more symmetric w i t h d e c r e a s i n g geomagnetic a c t i v i t y , and t h a t the p o s i t i o n of the i n c r e a s e i n r a d i u s v a r i e s o v e r s e v e r a l hours i n l o c a l t i m e . I t would be i n t e r e s t i n g t o l o o k f o r 68 a c o r r e s p o n d e n c e between knee p o s i t i o n and s i z e , and west-ward surge e v e n t s , o r substorm f r e q u e n c y . Many f e a t u r e s o f the model can be t e s t e d by s a t e l -l i t e r e s u l t s . I t i s d e s i r a b l e t o see what i s happening at many p o s i t i o n s on the n i g h t s i d e d u r i n g substorms, b o t h t o the magnetic f i e l d , and t o the p a r t i c l e s . I n p a r t i c u l a r , more o b s e r v a t i o n s i n the range 7 t o 14 e a r t h r a d i i , between 1800 and 0600 LMT, and t h e i r c omparison w i t h p o l a r substorm e v e n t s are needed. 69 BIBLIOGRAPHY A k a s o f u , S..-I., The development o f the a u r o r a l substorm, P l a n e t . Space S c i . , 12, 273, 1964. A k a s o f u , S.-I., S. Chapman and C . I . Meng, J . Atmos. T e r r . Phys., 24, i n p r e s s , 1965. Anderson, K.A., E n e r g e t i c e l e c t r o n f l u x e s i n the t a i l of t h e geomagnetic f i e l d , J . Geophys. Res. 70, 4741, 1965. A x f o r d , W. I . , V i s c o u s i n t e r a c t i o n between t h e s o l a r wind and t h e e a r t h ' s magnetosphere, P l a n e t . Space S c i . 12, 45 , 1 9 6 4 . . A x f o r d , W.I., C. 0 . H i n e s , A u n i f y i n g t h e o r y of h i g h l a t i t u d e g e o p h y s i c a l phenomena and geomagnetic storms, Can. J . Phys., 39, 1433, 1961. A x f o r d , W.I., H.E. Pe t s c h e k , G. L. S i s c o e , T a i l o f the mag-n e t o s p h e r e , J . Geophys. Res. 70, 1231, 1965. C a r p e n t e r , D. L., W h i s t l e r s t u d i e s of the plasma-pause i n the magnetosphere, J . Geophys. Res. 71, 693, 1966. C h a m b e r l a i n , J . W., P h y s i c s of the a u r o r a and a i r g l o w , Aca-demic P r e s s , New York, 1961. D a v i s , L. R., Enhancement of t r a p p e d p r o t o n s d u r i n g the mag-n e t i c storm o f A p r i l 18, I 9 6 5 , Am. Geophys. Union, 47th Annual Meeting, Washington. Paper 57a . 1966. Dungey, J . W., Cosmic e l e c t r o d y n a m i c s , Cambridge U n i v . P r e s s , 1958. Dungey, J . W., S t r u c t u r e of the exosphere o r a d v e n t u r e s i n v e l o c i t y space, G e o p h y s i c s The E a r t h ' s Envirpnment, Gordon and B r e a c h , New York, 1963. Pukushima, N., P o l a r magnetic storms and geomagnetic bays, J . F a c . S c i . Tokyo U n i v . 8, 293, 1953-Gold, T., M o t i o n s i n the magnetosphere of the e a r t h . J . Geophys. Res. 64, 1219, 1959-70 Heppner, J . P., Recent measurements o f the magnetic f i e l d i n t he o u t e r magnetosphere and boundary r e g i o n s , NASA X-612-65-490. Kim, H. Y., J . S. Kim, Atmos p h e r i c c o n d u c t i v i t i e s over F o r t C h u r c h i l l , J . Atmos. T e r r . Phys., 25, 4 8 l , 1963. Ness, N. F., The E a r t h ' s magnetic t a i l , J . Geophys. Res. 70, 2989, 1965. Ness, N. F., C. S. S c e a r c e , J . B. Seek, J . M. W i l c o x , A summary of r e s u l t s f r o m t h e IMP-1 Magnetic f i e l d e x p e r i m e n t , NASA X-612-65-180, 1965. O ' B r i e n , B. J . , H. T a y l o r , H i g h - l a t i t u d e g e o p h y s i c a l s t u d i e s w i t h S a t e l l i t e I n j u n 3. 4. A u r o r a s and t h e i r e x c i t a t i o n , J . Geophys. Res. 69, 45, 1964. P a r k e r , E . N., I n t e r p l a n e t a r y d y n a m i c a l p r o c e s s e s , I n t e r -s c i e n c e , New York, I963. P a r k e r , E . N., I n t e r a c t i o n o f t h e s o l a r wind w i t h the geo-magnetic f i e l d , Phys. F l u i d s , 1, 171, 1958. P e t s c h e k , H.E., Magnetic f i e l d a n n i h i l a t i o n , A A S - N A S A Sympo-sium on the p h y s i c s of S o l a r F l a r e s , ed. by W. N. Hess, NASA SP-50, 1964. P i d d i n g t o n , J . H., Geomagnetic storm t h e o r y , J . Geophys. Res. 65, 93, I960. P i d d i n g t o n , J . H., Geomagnetic storms, a u r o r a s and a s s o c i a t e d e f f e c t s , Space S c i . Rev. 3, 724, 1964. S p r e i t e r , J . R., A. L. Summers, A. Y. A l k s n e , Hydromagnetic f l o w around the magnetosphere, P l a n e t . Space S c i . 14, 223, 1966. Sweet, P.A., The n e u t r a l p o i n t t h e o r y of s o l a r f l a r e s , I.A.U. Symposium No. 6., ed. by B. L e h r e s t , Cam-b r i d g e U n i v . P r e s s , 1958. T a y l o r , H. E., E. W. Hones, J r . , A ^ i a b a t i c motion of a u r o r a l p a r t i c l e s i n a model of the e l e c t r i c and magnetic f i e l d s s u r r o u n d i n g the e a r t h , J . Geophys. Res. 70, 3605, 1965. 71 APPENDIX I Symbols and C o n v e n t i o n s Used A C r o s s s e c t i o n a l a r e a o f f l u x t u b e . A l s o : c o n s t a n t i n (19b) B Magnetic f i e l d B^ H o r i z o n t a l component of magnetic f i e l d c Phase v e l o c i t y d D i s t a n c e s c a l e p e r p e n d i c u l a r t o the magnetic f i e l d E E l e c t r i c f i e l d . A l s o : E n e r g y of p a r t i c l e s f As a s u b s c r i p t d e notes the v a l u e a t t h e f e e t o f the f i e l d l i n e s P F o r c e g Mapping f a c t o r a l o n g a f l u x tube, f e e t t o e q u a t o r h H e i g h t as d e f i n e d by P i g . 12 j C u r r e n t i n the magnetosphere J C u r r e n t j P e d e r s e n c u r r e n t i n the i o n o s p h e r e 1 L e n g t h s c a l e of f i e l d l i n e s . A l s o : C o o r d i n a t e a l o n g f i e l d l i n e s . 72 n Number d e n s i t y of p a r t i c l e s p Gas p r e s s u r e P Magnetohydrodynamic p r e s s u r e (P = p + B /2JJ0 ) r C o o r d i n a t e r R a d i u s of the e a r t h o rQ R a d i u s o f a f i e l d l i n e i n the e q u a t o r i a l p l a n e R R a d i u s of c u r v a t u r e of a f i e l d l i n e , s Arc l e n g t h T,t Time v V e l o c i t y of f l u i d element V Volume V A A l f v e n v e l o c i t y i n the t a i l W T h i c k n e s s of the s l a b x C o o r d i n a t e y C o o r d i n a t e z C o o r d i n a t e r*. L a t i t u d e 7 3 L a t i t u d e o f the f e e t o f a f i e l d l i n e U n i t of magnetic f i e l d (10~^ gauss) A n g u l a r c o o r d i n a t e ( l o n g i t u d e ) A Lame c o n s t a n t Dynamic v i s c o s i t y (Lame c o n s t a n t ) K i n e m a t i c v i s c o s i t y Mass d e n s i t y H e i g h t i n t e g r a t e d P e d e r s e n c o n d u c t i v i t y Maxwell s t r e s s t e n s o r 74 APPENDIX I I Mapping F a c t o r s i n a D i p o l e F i e l d Throughout the t h e s i s i t has been assumed t h a t O^Q =<^R = Q ^ } whereas i n a c t u a l f a c t ^ \ gr c^ e • However, the a p p r o x i m a t i o n i s s u f f i c i e n t l y a c c u r a t e p r o v i d e d t h a t 9^//5© ~ ' * T h e m a P P i n S f a c t o r s f o r a d i p o l e f i e l d a re c o n s i d e r e d and i t i s shown t h a t t h i s i s t r u e f o r a r e a -sonable f i e l d c o n f i g u r a t i o n . The geometry i s shown i n F i g . 1 9 . I n t h e a n g u l a r d i r e c t i o n : AfS^ ~ c o s <=*-o F o r a f i e l d l i n e : M > r~1 , - Qa -I n the r a d i a l d i r e c t i o n : *~o =• W C O S 2 " c x l o (33) 75 F i g . 1 9 . Mapping f a c t o r s f o r a d i p o l e f i e l d . 76 D i f f e r e n t i a t i o n g i v e s d oCo _ COS OLc d Hence 2~ r e S I A C X O r 0 d<x c - 1 ^ r e . — 2. r« sin. o C o r 0 cos CX-o Prom e q u a t i o n s (33) and ( 3 4 ) , (34 ) 6 ' The t h e o r y i n v o l v e s f i e l d l i n e s f o r which 60° <C oc0 < 90° Thus 2 s i n o C 0 and hence -Q— v a r i e s between 1 and 2 i n t h e r e g i o n of i n t e r e s t f o r a d i p o l e f i e l d . 

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