@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix skos: . vivo:departmentOrSchool "Science, Faculty of"@en, "Earth, Ocean and Atmospheric Sciences, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Atkinson, Gerald"@en ; dcterms:issued "2012-03-06T22:26:01Z"@en, "1967"@en ; vivo:relatedDegree "Doctor of Philosophy - PhD"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """The magneto-sphere may be considered as consisting of two regions the tail and region of closed lines of force. The interchange of field lines between these two region is important in magnetospheric processes. Transport of magnetic field lines from the closed region into the tail may occur by Dungey's mechanism or by viscous interaction of the magnetosphere with the solar wind. Transport from the tail to the closed region occurs by recombination through the neutral sheet. Convective flow within the closed region is controlled by the “foot dragging” effect which arises from the discharging action of the ionosphere on flux tubes. The model of polar substorms presented is a flow or time sequence as follows: (1) Field lines are dragged from the closed region into the tail by the solar wind with a resulting storage of potential energy in the tail. (2) The polar substorm begins when the field lines recombine in an implosive fashion at the neutral sheet, releasing the stored potential energy. (3) The recombined flux tubes are added to the nightside of the closed region as a giant bulge. (4) The bulge drives are turn flow of flux tubes towards the dayside in the closed region. It is likely that recombination is initiated by the formation of a neutral point at about 13 or 14 earth radii in the antisolar direction and occurs across a width of tail of about 6 or 7 earth radii and that 10⁸ webers are annihilated in a time of about 1/2 hour. The recombination iis probably stopped by the build-up of a giant bulge on the nightside and the closed region which maps to the earth’s surface along fieldlines as the auroral break-up bulge and which, as i t spreads out over the nightside of the region of closed fieldlines causes the observed auroral effects. The Pedersen current is not expected to produce significant magnetic effects at the surface of the earth except at anomalies in ionospheric conductivity. Such an anomaly along the auroral arcs can explain the westward electrojet. However, it seems probable that the remainder of the polar cap current system is the result of Hall currents."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/41187?expand=metadata"@en ; skos:note "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 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