Diffusion Of Magnetospheric Protons By Turbulent Drift Waves

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Diffusion Of Magnetospheric Protons By Turbulent Drift Waves

Item Type Thesis

Authors Maggs, James Edward

Download date 31/08/2021 20:52:06

Link to Item http://hdl.handle.net/11122/9237

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72-13,059

I MAGGS, James Edward, 1943-

f;

DIFFUSION OF MAGNETOSPHERIC PROTONS BY TURBULENT DRIFT WAVES.

f

I University of Alaska, Ph.D., 1971

| Geophysics

I University Microfilms, A XEROX Com pany, Ann Arbor, Michigan ;

I THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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DIFFUSION OF MAGNETOSPHERIC PROTONS

BY TURBULENT D RIFT WAVES

A

DISSERTATION

P resented t o th e F a c u lt y o f the

U n iv e r s it y o f A la s k a in p a r t i a l f u l f i l l m e n t

o f th e re q u ire m e n ts

f o r th e d egree o f

DOCTOR OF PHILOSOPHY

by

James E . Maggs, B . S . , M .S.

C ol le g e , A la s k a

May 1971

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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DIFFUSION OF MAGNETOSPHER1C PROTONS

BY TURBULENT DRIFT WAVES

APPROVED:

0 , date : ^ !~l< ! l {

Dean o f th e C o lle g e o f M ath em atics, P h y s ic a l S cie nce s and E n g in e e rin g

^

V ic e P r e s id e n t f o r Research and Advanced Study

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PLEASE NOTE:

Some pages have indistinct print. Filmed as received.

UNIVERSITY MICROFILMS.

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A b s tr a c t

The d if f u s io n o f charged p a r t ic le s by t u r b u le n t e l e c t r o s t a t i c

o s c i l l a t io n s in th e magnetosphere is in v e s tig a te d as a p o s s ib le

mechanism f o r e n e r g iz in g and t r a n s p o r tin g r in g c u r r e n t p ro to n s . A

sim ple model o f th e magnetosphere is used t o c a r r y o u t th e i n v e s t i

g a t io n . In th e model th e geom agnetic f i e l d is taken to be an

a z im u th a lly sym m etric a x ia l f i e l d , th e ionosphere is taken t o be

p e r f e c t ly c o n d u c tin g , and the r in g c u r r e n t and plasmasphere plasmas

a re assumed t o have M axw ellian d is t r ib u t io n s o f v e l o c i t i e s . The

d if f u s io n process is assumed t o p re s e rv e th e m agnetic moment o r f i r s t

a d ia b a t ic in v a r ia n t o f th e p a r t i c l e s . The b e h a v io r o f low freq uency

(<< ion g y r o f re a u e n c y ), long wave length (>> ion g y r o ra d iu s )

e l e c t r o s t a t i c o s c i l l a t io n s is s tu d ie d . The e l e c t r o s t a t i c o s c i l l a t io n s

are t r e a te d as n a tu ra l modes o f th e m agnetospheric c a v i t y d r iv e n by

f lu c t u a t io n s in th e e l e c t r i c p o t e n t ia l on th e magnetopause, th e

m agn etospheric b o undary. I t is concluded t h a t d if f u s io n o f r in g

c u r r e n t p ro to ns by t h i s process is not im p o rta n t in th e ma.gnetosphere.

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ACKNOWLEDGEMENTS

I would lik e t o e xp re ss my deepest g r a t it u d e t o P ro fe s s o r D. W.

S w if t f o r both h is guidance and s u p p o rt, e s p e c ia lly in c r i t i c i s i n g and

re v ie w in g t h i s t h e s is . I acknowledge my Committee Members f o r t h e i r

gen era l s u p p o rt.

I would lik e to thank the personnel o f th e G e o p h ysic a l I n s t i t u t e

D r a f tin g O f f ic e and Steno Pool f o r t h e i r su sta in e d e f f o r t s d u rin g th e

in e v it a b le la s t minute ru sh .

My s in c e r e s t thanks goes t o my w if e , Jo a n n ie , who p a t i e n t l y t o l e r a t

ed a l l o f th e t r i a l s and t r i b u l a t i o n s which accompanied th e w r i t i n g o f

t h i s t h e s is .

T h is work was supported by th e N a tio n a l S cie n ce F oundation G ra nts

GA-77I and GA-4597.

iv

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T a b le o f C o ntents

Chap+c ' I .

A , r i e f d is c u s s io n o f th e aims o f th e paper and th e

■'-"hod t o be used in s o lv in g the problem . A d is c u s s io n of th e req uirem ents necessa ry f o r th e mechanism w hich s u p p lie s r in g c u rre n t p a r t ic le s and a re vie w o f some o f th e d i f f u s i o n processes a lre a d y in v e s tig a t e d and t h e i r . cone I us io n s .

C h a p te r 2. A Model M agnetosphere

A g e n e ra l d is c u s s io n o f seme o f th e known fe a tu re s and p a r t i c l e p o p u la tio n s o f th e m agnetosphere. A model magnetosphere which in c o rp o ra te s th e r in g c u r r e n t , plasm asphere, magnetopause and ionosp h ere is d e s c rib e d .

C h a p te r 3. The D if f u s io n E qu atio n

A b r i e f o u t lin e o f th e method used t o develo p a d i f f u s i o n e q u a tio n f o r th e phase space d e n s it ie s o f g u id in g c e n t e r s . A d e s c r ip t io n o f th e s t a t i s t i c a l model used t o re p re s e n t tu rb u le n c e >n the model magnetosphere. An e x p r e s s io n f o r th e d if f u s io n c o e f f ic i e n t .

C h a p te r 4. D r i f t Modes in th e Model Magnetosphere A d i f f e r e n t i a l eq u a tio n f o r th e e l e c t r i c p o t e n t ia l in th e model magnetosphere is g iv e n and d is c u s s e d . The method f o r s o lv in g th e e q u a tio n and r e s t r i c t i o n s on th e s o lu t io n s are g iv e n .

C h a p te r 5. The S o lu tio n s

A d is c u s s io n o f th e computer program used t o s o lv e th e d i f f e r e n t i a l e q u a tio n and th e r e s u lt s o f th e co m pu ta tio ns are review ed in the l i g h t o f th e r e s t r i c t i o n s g iv e n in c h a p te r 4. The s o lu t io n s are d is c u s s e d .

C h a p te r 6. R e s u lts and C o n c lu s io n s

The d if f u s io n c o e f f ic i e n t f o r r a d ia l d if f u s io n is c a lc u la t e a f o r the exam ple g iv e n in c h a p te r 5 in w hich energy can be tr a n s p o r te d in t o most o f th e r in g c u r r e n t re g io n . The in e f f e c t iv e n e s s o f p a r t i c l e d if f u s io n in the magnetosphere by t u r b u le n t e l e c t r o s t a t i c o s c i l l a t io n s is d is c u s s e d .

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Append ic e s

• 'Page

A p p e n d ix A - I . 73

The d evelopm ent o f the H a m ilto n ia n f o r p a r t i c l e motion in th e model magnetosphere under th e assum ption t h a t y is c o n se rv e d .

A p p e n d ix A-1 I . 80

The d e t a i ls o f th e d e r iv a tio n o f th e d if f u s io n eq u a tio n s u it a b le f o r th e model m agnetosphere.

A p e n d ix A-1 I I . 85

The d e t a i ls o f th e c a lc u la t io n o f th e d if f u s io n c o e f f ic i e n t .

A p p e n d ix B - l . 97

The developm ent o f th e d i f f e r e n t i a l e q u a tio n f o r th e e l e c t r i c p o t e n t ia l in th e model m agnetosphere.

A p p e n d ix B - l I . 106

A b r i e f re v ie w o f th e W .K .B. method.

A p p e n d ix C . • 112

The com puter programs used to s o lv e th e d i f f e r e n t i a l . e q u a tio n f o r th e e l e c t r i c p o t e n t ia l are g iv e n in d e t a i l.

vi

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L i s t o f F ig u re s

Page

F ig u re 2 .1 . 19

A diagram o f th e model used t o re p re s e n t th e magnetosphere.

F ig u re 5 .1 . 42

The geom agnetic f i e l d es measured by s a t e l l i t e is compared t o th e a p p ro xim a tio n l/ r ^ .

F ig u re 5 .2 . 45

The f u n c tio n s U ( r ) and q ( r ) f o r a la rg e va lu e o f m a n d -th e e xp o n e n tia l number d e n s ity model.

F ig u re 5 .3 . 47

The fu n c tio n s U ( r ) and q ( r ) w ith a t u r n in g p o in t nea r the magnetopause and the e xp o n e n tia l number d e n s ity m odel.

F ig u re 5 .4 . 52

The fu n c tio n s U ( r ) and q ( r ) f o r a power law number d e n s ity model o f th e r in g c u r r e n t.

F ig u re 5 .5 . 55

The plasma d is p e r s io n fu n c tio n f o r re a l v a lues o f th e argum ent.

F ig u re 5 .6 . 58

The graphs o f th e r a d ia l wave fu n c tio n s f o r an e x p o n e n tia l and power law number d e n s it y .

F ig u re 5 .7 . 60

The im pulse response o f th e model m agnetospheric c a v i t y .

F ig u re 5 .8 . 62

The wave len gth s o f the s lo w ly d eca yin g ra d ia l wave f u n c tio n s a t 8 R .

F ig u re 5 .9 .

A com parison o f th e models used t o re p re s e n t th e r in g c u r r e n t number d e n s it y .

vii

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D if f u s io n o f M agnetospheric

Protons by T u rb u le n t D r i f t

Waves

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C hapter I

The purpose o f t h i s paper is t o in v e s tig a t e th e d if f u s io n o f

p ro to n s in t u r b u le n t d r i f t waves as a p o s s ib le mechanism f o r

e n e r g iz in g and t r a n s p o r tin g r in g c u rre n t p a r t i c l e s . S ince the

r in g c u r r e n t seems t o p la y a b a s ic r o le in th e m agn etospheric

substorm as both a p a r t i c l e and energy r e s e r v o ir , an u n d ersta nd ing

o f th e mechanism which s u p p lie s r in g c u r r e n t p a r t i c l e s and en erg y

is b a s ic t o an und erstand ing o f th e m agnetospheric sto rm . A d is c u s s io n

o f th e m agnetospheric storm and th e concept o f a substorm is g iv e n by

S . - I . Akasofu (1968). Much o f th e f o llo w in g g enera l m a te ria l on

m agnetospheric storms is drawn from t h i s r e fe re n c e .

The m agnetospheric storm is a p p a re n tly th e p ro d u c t o f th e in t e r

a c tio n o f the s o la r wind w ith th e m agnetosphere. The m agnetospheric

storm can be d iv id e d in t o two p a r t s , an i n i t i a l phase and a number cpf

r e l a t i v e l y s h o rt m agnetospheric substorm s. The r in g c u r r e n t p la y s a

c e n tr a l r o le in the m agnetospheric storm in t h a t th e phase o r p ro g re s s

o f a storm can be r e la te d to th e b e h a v io r o f th e p a r t ic le s in th e r in g

c u r r e n t . O r i g i n a l l y , th e r in g c u r r e n t was co n cieved as a c u r r e n t

system su rro u n d in g th e e a rth o u ts id e th e ionosnhere which accounted

fo r th e main phase decrease o f th e m agnetospheric sto rm . The r in g

c u r r e n t was consid ere d a m a th e m a tic a lly c o n v e n ie n t r e p re s e n ta tio n o f

th e main phase decrease mechanism and not ne c e ss a ri ly a p h y s ic a l

r e a l i t y . However, a p h y s ic a l e n t i t y c a lle d th e storm tim e r a d ia tio n

b e lt which behaves v e ry much lik e th e r in g c u r r e n t has been d etected

by s a t e l l i t e measurements. The storm tim e r a d ia t io n b e lt c o n s is t s

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p r i m a r i l y o f 1-50 Kev p ro to n s . The term " r i n g c u r r e n t " is now

g e n e r a lly used in te rc h a n g e a b ly w it h th e term "s to rm tim e r a d ia t io n

b e l t " t o r e f e r t o t h i s group o f m a gn eto sph eric p ro to n s . The

d i s t r i b u t i o n and t o t a l in t e n s it y o f r in g c u r r e n t p a r t i c l e s in th e

magnetosphere is g r e a t l y e ffe c te d by th e number and d u r a tio n o f

p re v io u s m a gn e to sph eric substorm s.

The co ncep t o f th e m agnetospheric substorm encompasses a number

o f a p p a re n tly in t e r r e la t e d phenomena w hich o c cu r s p o r a d ic a lly and

la s t f o r 1-3 h o u rs. The a u ro ra l su bsto rm , p o la r m a gn etic su b s to rm ,

m ic r o p u ls a tio n su bsto rm , io n o s p h e ric su bsto rm , x - r a y su bsto rm ,

p ro to n a u ro ra substorm and th e VLF em is sio n substorm are a l l c o n s id e re d

t o be p a r t o f a la r g e r w hole, the m a gn eto sph eric su bsto rm . D u rin g

a m a gn eto sph eric substorm th e t o t a l number o f p ro to n s w it h e n e r g ie s

between 1-50 Kev in c re a se s in s id e th e t r a p p in g r e g io n . The in c re a s e

in th e t o t a l number o f p a r t ic le s is n o t u n ifo rm in lo c a l t im e .

Indeed, th e e f f e c t o f th e m a gn eto sph eric substorm seems t o be t o

in c re a s e both th e asymmetry and th e t o t a l number o f r i n g c u r r e n t

p a r t i c l e s .

D u rin g th e co urse o f a m a gn eto sph eric sto rm , th e t o t a l number o f

p ro to n s in th e r in g c u r r e n t b u ild s up f a i r l y r a p i d l y v ia th e mechanism

o f th e m a gn eto sph eric substorm and then decays s lo w ly back t o p r e -s u b

sto rm l e v e ls . F o r a moderate storm th e b u ild up o f r in g c u r r e n t p a r t i c l e s

la s t s about 10 h o u rs. The decay ta ke s a bout 5-15 days depen din g upon

th e s iz e o f th e sto rm . The decay in th e t o t a l number o f r i n g c u r r e n t

p a r t i c l e s may be in t e r r u p t e d by th e o c cu rre n c e o f a d d itio n a l su b s to rm s .

The fo rm a tio n o f th e storm tim e r a d ia t io n b e l t in v o lv e s th e t r a n s p o r t

and /o r e n e r g iz a tio n o f la rg e numbers o f p ro to n s in th e m agnetosphere

on th e tim e s c a le o f th e m agn eto sph eric su bstorm . ■

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3

Many t r a n s p o r t and e n e r g iz a tio n mechanisms t h a t m igh t be fu n c t io n in g

in the m agnetosphere have been in v e s t ig a t e d . Some o f the mechanisms

proposed and s fu d ie d a r e : m agnetohydrodynam ic c o n v e c tio n o f p a r t ic le s

due to a v is c o u s in t e r a c t io n between th e s o la r wind and th e magneto

sphere (A x f o r d and H ines 1961); c o n v e c tio n o f p a r t i c l e s due t o a s t a t i c

e l e c t r i c f i e l d a cro ss th e m agnetosphere (K e llo g o . 1959; T a y lo r and

Hones, 1965); r o t a t io n a l and g r a d ie n t d r i f t s ( S w i f t , 1971); magneto

s p h e ric t a i l i n s t a b i l i t y (A x f o r d 1967, Dungey 1968, P id d in g to n 1968);

d if f u s io n o f p a r t i c l e s due t o s t o c h a s t ic changes in th e geom agnetic

f i e l d (P a rk e r I9 60, Nakada and Mead 1965). None o f the mechanisms

proposed t o d ate have been s u c c e s s fu l in e x p la in in g th e fo rm a tio n

o f the storm tim e r a d ia t io n b e l t . However, some o f th e mechanisms

s tu d ie d u n d o ub ted ly o p e ra te t o some degree d u rin g a m agnetospheric

substorm . Most l i k e l y , th e m a g n eto sph eric substorm invokes a

complex c o m b in a tio n o f t r a n s p o r t mechanisms.

The t r a n s p o r t o f p a r t i c l e s in th e m agnetosphere due t o d if f u s io n in

random e l e c t r i c and m a gn etic f i e l d s has been e x t e n s iv e ly in v e s tig a t e d .

Many in v e s t ig a t o r s (P a r k e r I960, D a vis and Chang 1962, Dungey 1965,

Nakada and Mead 1965, Faltham mar 1965, 1968; C o n ra th 1967, S c h u ltz

and E v ia t a r 1969) have s tu d ie d d i f f u s i o n d r iv e n by p re s su re changes

in th e s o la r wind w hich s u b s e q u e n tly cause changes in th e p o s itio n o f

the magnetopause. R a d ia l d if f u s i o n o f p a r t i c l e s o c cu rs th ro ugh v i o l a t i o n

o f th e t h i r d ( f l u x ) a d ia b a t ic i n v a r i a n t . The d if f e r e n c e s in the t r e a t

ment o f th e problem a re u s u a lly th e method f o r o b ta in in g th e d if f u s io n

e q u a tio n and th e c o n d itio n s imposed upon th e m a g n eto sph eric plasma. A

d is c u s s io n o f th e w ork o f Nakada and Mead (1965) and S c h u lz and E v ia t a r

(1969) i l l u s t r a t e s th e d i f f e r e n t approaches t o th e problem .

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Nakada and Mead use th e Mead (1964) model o f th e geom agnetic

f i e l d . The e f f e c t on th e main f i e l d o f the p o s it io n o f th e magneto

pause is taken in to a ccount in t h i s model o f the f i e l d . The d if f u s io n

mechanism is assumed t o work in the f o llo w in g manner. A com pression

o f the magnetosphere ( i . e . an inward motion o f th e magnetopause) is

assumed t o take pla ce on a tim e s c a le s h o r t in com parison w it h th e tim e

i t ta ke s an e q u a t o r ia lly tra p p e d (J=0 ) p a r t i c l e t o d r i f t around th e

e a rth . Depending upon the p a r t i c l e 's p o s it io n in lo c a l tim e when the

com pression o c c u rs , p a r t i c l e s which were d r i f t i n g on the same s h e ll

o f c o n s ta n t B w i l l f o llo w th e lin e o f fo r c e t o a d i f f e r e n t v a lu e

o f B and d r i f t on d if f e r e n t s h e lls o f c o n s ta n t B - thus v i o l a t i n g

th e t h i r d a d ia b a tic in v a r ia n t , i f the decay o f th e f i e l d t o i t s

o r i g i n a l c o n fia u r a tio n is slow in com parison w ith th e tim e i t ta k es a

p a r t i c l e t o d r i f t around th e e a r t h , th e t h i r d in v a r ia n t w i l l be conserved

and th e p a r t i c l e w i l l remain on i t s new s h e ll o f c o n s ta n t B. The t o t a l

e f f e c t is t o spread o u t p a r t ic le s on n e ig h b o rin g s h e lls o f c o n s ta n t

B. T h a t i s , th e p a r t ic le s undergo r a d ia l d i f f u s i o n .

Such sudden com pressions in the f i e l d w ith slo w re c o v e ry do

indeed o c c u r. They a re c a lle d sudden im pulses o r sudden commencements.

Nakada and Mead c a lc u la te d a d is p e rs io n c o e f f ic i e n t by u s in g an

ensemble o f these impulses com piled from a c tu a l o b s e r v a t io n s .

Nakada and Mead use a F o k k e r-P la n k e q u a tio n t o d e s c rib e the

e v o lu t io n o f th e d i s t r i b u t i o n o f p a r t i c l e s tra p p e d in th e magneto

sp h e re. P a r t ic l e en ergy lo ss e s due t o Coulomb s c a t t e r in g and charge

exchange c o l l is i o n s w ith n e u tr a ls a re taken in t o account in the

s o lu t io n o f th e e q u a tio n . The c o e f f ic i e n t o f dynam ical f r i c t i o n is

assumed to be p ro p o rtio n a l t o th e d is p e r s io n c o e f f ic i e n t . The v a lu e used

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f o r th e d is p e rs io n c o e f f ic ie n t was .031 (|J^/L® ) (e a r th r a d ii) ^ / d a y .

Lq is the d is ta n c e from the c e n te r o f th e e a rth to th e magnetopause

a lo n g th e ea rth sun l i n e . The eq ua tio n was so lv e d u sin g measured .

v a lu e s o f p a r t i c l e f lu x e s > 100 Kev f o r an i n i t i a l d i s t r i b u t i o n .

A lth o u g h some q u a l i t a t i v e agreement w ith measured e q u ilib r iu m p a r t i c l e

f lu x e s was ob ta in e d f o r high energy p a r t ic le s (>50 K e v ), a la r g e r d is

p e rs io n c o e f f ic i e n t than the one c a lc u la te d is needed t o o b ta in q u a n ti

t a t i v e agreement between the th e o ry and o b s e r v a tio n .

I t was i m p l i c i t in the d is c u s s io n o f th e d if f u s io n mechanism

above t h a t the com pression o f the magnetopause was so sudden t h a t

a l l p a r t ic le s on a f i e l d lin e moved w ith i t , re g a rd le s s o f t h e i r

y v a lu e (y is th e magnetic moment). T h is , o f c o u rs e , is no t s t r i c t l y

t r u e and a d if f u s io n equation and d if f u s io n c o e f f ic i e n t f o r which

t h i s assum ption is n o t made has been d e riv e d by S ch ulz and E v ia t a r

(1969). They do, however, assume J=0 and E*B=0. The e v o lu t io n o f th e

d i s t r i b u t i o n f u n c tio n is given by a d if f u s io n eq ua tio n r a t h e r than the

F o k k e r-P la n k e q u a tio n . The d if f u s io n eq u a tio n is d e riv e d u s in g the

Mead (1964) model o f th e f i e l d , but the unperturbed d r i f t o r b i t s o f the

p a r t i c l e around th e e a rth are assumed t o be n o n - c ir c u la r . The d if f u s io n

c o e f f ic i e n t depends upon the power spectrum o f the f lu c t u a t io n s in

th e s t a n d -o f f d is ta n c e o f the magnetopause (L Q) . I f an in v e rs e

power law is assumed f o r the power spectrum the e xp re s s io n f o r th e

d if f u s io n c o e f f ic i e n t is s im ila r t o Nakada and Mead’ s f o r L<<Lo .

A d i f f e r e n t mechanism f o r d if f u s io n o f p a r t ic le s in th e magneto

sphere has been in v e s tig a te d by Birmingham (1969). He su gg ests

t h a t d if f u s io n is not d riv e n by th e induced e l e c t r i c f i e l d caused .

by f lu c t u a t io n s in th e geom agnetic f i e l d b u t by v a r ia t io n s in th e 5

(17)

e l e c t r i c p o t e n t ia l f ie ld s (Vx?=0) governed by +he dynamics o f low energy

(<8Kev) trapped p a r t i c l e s . The p o te n tia l nas wave le n g th s on th e o rd e r o f

the dim ensions o f the magnetosphere. T h a t i s , Birmingham assumes th e low

energy component o f trapped p a r t i c l e f lu x e s produces an e l e c t r i c p o t e n t ia l

V such t h a t ? = - W = - ~ x ^ where v is th e h ydrom agnetic flo w v e l o c i t y o f the

low energy plasma. Birmingham uses the c u t - o f f energy o f 6 Kev because

p a r t ic le s below t h i s energy are dominated by e l e c t r i c f i e l d d r i f t s o f

" t y p i c a l " f i e l d s found in the magnetosphere w h ile p a r t ic le s above 8 Kev

a re dominated by g ra d ie n t and c u rv a tu re d r i f t s . .

Birmingham uses a d if f u s io n equa tio n developed by Birm ingham ,

N o rth ro p and Faltnammar (1967). The e q ua tio n is v a lid f o r p a r t ic le s w ith

a r b i t r a r y va lu es o f p and J . A ga in d if f u s io n o ccu rs as a r e s u lt o f

v i o l a t i o n o f the f l u x in v a r ia n t . A sim ple model is used t o d e r iv e an

e x p l i c i t e xp re ss io n f o r th e d if f u s io n c o e f f ic i e n t s . A d ip o le m agnetic f i e l d is assumed and a p o t e n t ia l o f th e form V = A ( t ) r sin<f>/sin 20 is

used t o d e s c rib e the e l e c t r i c p o t e n t ia l in s p h e ric a l c o o r d in a te s . The

flo w p a tte rn o f th e low energy component o f the plasma Is s i m i l a r t o the

c o n v e c tiv e flo w p a tte rn s envisaged by Levy e t a l . (1964) and, t o some

e x t e n t , o th e r c o n v e c tiv e flo w models ( e . g . A x fo rd and Hines 1961). I t

is shown t h a t the d if f u s io n c o e f f ic i e n t is p ro p o r tio n a l t o and the

v a lu e o f the power spectrum o f the tim e dependent c o e f f ic i e n t A ( t )

e va lu a te d a t th e azim uthal d r i f t fre q u e n c y, i f th e a u t o c o r r e la t io n

fu n c tio n o f A ( t ) ( i . e . th e in v e rs e F o u r ie r tra n sfo rm o f the power spectrum ) is assumed to have the form C exp ( . - t 2 2 where t c is

th e c o r r e la t io n tim e , then the d if f u s io n c o e f f ic i e n t is p ro p o r tio n a l

(18)

+o L^Tc exp (-W p F o r p a r t i c l e s w it h azim uth a l d r i f t fre q u e n c ie s

nip much le ss than — i t is p r o p o r tio n a l t o and th u s independant

o f y and J .

Birm ingham s o lv e s a one dim ensional d i f f u s i o n e q u a tio n w it h no

lo s s term s u s in g th e a p p ro xim a tio n t o th e d i f f u s i o n c o e f f i c i e n t w it h UpTc « I . 2 2 Thus the r e s u lt s a re a p p lic a b le t o p a r t i c l e s w it h

d r i f t p e rio d s 2ir (.= 4.4*10^ CL x [e n e rg y in K ev^)- - m in u te s) much

“ D

la r g e r th a n th e c o r r e la t io n tim e . Birmingham uses a v a lu e o f one h o ur f o r th e c o r r e la t io n tim e . C h oosing a v a lu e o f (2x10 ^ v/m )^ f o r th e

c o e f f ic i e n t C in th e c o r r e la t io n fu n c t io n , Birm ingham shows t h a t an

i n i t i a l d e lt a fu n c tio n d i s t r i b u t i o n in number d e n s it y a t 8 R (e a r th

' e

r a d i i ) w i l l y i e l d a s i g n i f i c a n t number o f p a r t i c l e s a t 5 Rg in a bout

18 h ours (see Birmingham 1969, f i g u r e 1 ). A s i m i l a r a n a ly s is

u s in g th e d if f u s io n c o e f f ic i e n t g iv e n by Nakada and Mead ta k e s o v e r

100 days t o produce the same r e s u lt s . Thus a s u b s ta n t ia l in c re a s e

in th e d i f f u s i o n ra te s can be o b ta in e d by u s in g th e v a r i a t i o n s in

th e c o n v e c t iv e e l e c t r i c f i e l d a s s o c ia te d w ith th e flo w o f th e low

energy component o f the m agn eto sph eric plasm a. The d i f f u s i o n r a te s

a pp ea r, how ever, t o be s t i l l to o slo w t o t r a n s p o r t p a r t i c l e s s e v e ra l

e a rth r a d ii on th e tim e s c a le o f th e m agn eto sph eric su b sto rm . I f

d i f f u s i o n p rocesses are t o be im p o rta n t d u rin g a su bstorm an in c re a s e

in th e d i f f u s i o n ra te is n e c e ss a ry.

Most o f th e s tu d ie s o f d i f f u s i o n processes in th e m agnetosphere

have c o n c e n tra te d on v e ry low fre q u e n c y v a r ia t io n s in th e m a g n etic

f i e l d . The fre q u e n c ie s a re low enough t h a t th e second a d ia b a t ic

in v a r ia n t o f th e p a r t ic le s ( J ) can be assumed t o be c o n s e rv e d .

T h is s tu d y w i l l co n ce n tra te on e l e c t r o s t a t i c o s c i l l a t i o n s whose

- 7

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8

fre q u e n c ie s a re near th e bounce freq u en cy o f a t y p ic a l r in g c u r r e n t

p ro to n b ut much le s s than i t s g y r o fre c u e n c y . The in t e r a c t io n o f a

p ro to n w ith e l e c t r o s t a t i c o s c i l l a t io n s in t h i s fre q u e n c y range w i l l

v i o l a t e th e second ( lo n g i t u d i n a l ) and t h i r d a d ia b a t ic in v a r ia n t s o f

th e p a r t i c l e . The f i r s t in v a r ia n t , y , is assumed t o be c o n se rv e d .

Such e l e c t r o s t a t i c o s c i l l a t io n s which can e x i s t in plasmas w ith

number a n d /o r te m p e ra tu re g r a d ie n t s , such as th e m agn eto sph eric

plasm a, a re g e n e r a lly c a lle d d r i f t waves. D r i f t waves c h a r a c t e r

i s t i c a l l y have wave le n g th s p a r a lle l t o th e m agnetic f i e l d , 3, much l a r g e r than t h e i r wave le n g th s p e rp e n d ic u la r to B. T h is is an

Id e a l p r o p e r t y in th e re sp e ct t h a t th e req uirem ent o f c o n s e rv in g th e

f i r s t a d ia b a t ic in v a r ia n t demands t h a t th e p a r a l l e l e l e c t r i c f i e l d

be sm all b u t f o r s tro n g d if f u s io n th e e l e c t r i c f i e l d p e r p e n d ic u la r

t o th e m agn etic f i e l d must be la rg e so t h a t th e ExB d r i f t o f th e

p a r t i c l e s is la r g e . The e q u a tio n f o r th e e l e c t r i c p o t e n t ia l a s s o c ia te d

w ith th e d r i f t waves w i l l be d eveloped under th e assum ption t h a t th e

wave le n gth p e r p e n d ic u la r to is la r g e r than an ion g y r o ra d iu s and th e

wave fre q u e n c y is le s s than th e ion g y ro fre q u e n c y b u t g r e a t e r than

U is th e a zim u th a l wave number in t h a t is th e wave le n g th o f

th e o s c i l l a t i o n in th e azim uth al d ir e c t io n and is th e a zim u th a l ion 2t

d r i f t fre q u e n c y s in c e — is th e tim e i t ta kes an io n t o d r i f t around

UD '

th e e a r t h . ) The en e rgy source f o r th e d if f u s io n is assumed t o be near

ste a d y ( i n tim e ) f lu c t u a t io n s in th e e l e c t r i c p o t e n t ia l a t th e magnet

o s p h e r ic b o u n d a ry, th e magnetopause. Thus th e s o lu t io n s o f p rim a ry

in t e r e s t w i l l be undamped o r o n ly s l i g h t l y damped o s c i l l a t io n s which

have s i g n i f i c a n t e l e c t r i c f i e l d s th ro u g h o u t a la rg e p a r t o f th e mag

n e to sp h e re .

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Filmed as received without page(s) 9

UNIVERSITY MICROFILMS.

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C h a pter 2

A Model Magnetosphere

The in t e r a c t io n o f the s o la r wind w ith th e e a r t h 's m agnetic f i e l d

c re a te s a m agnetic c a v i t y . The confinem ent o f th e e a r t h 's f i e l d by

th e s o la r plasma was in v e s tig a te d t h e o r e t ic a l l y as e a r ly as 1931 by

Chapman and F e rra ro . W ith th e advent o f th e space age and s a t e l l i t e

borne magnetometers the confinem ent o f th e e a r t h 's f ie ld - in space

was dem onstrated.

In te n s iv e in v e s tig a t io n o f the p a r t ic le s and f i e l d s su rro u n d in g

th e e a rth has revealed most o f th e s a lie n t fe a tu re s o f th e magneto

s p h e ric c a v i t y . However, th e re is s t i l l disagreem ent o v e r whether

th e c a v it y is open (geom agnetic f i e l d lin e s connected to in t e r p la n e t a r y

f i e l d lin e s ) o r closed (no c o n n e c tio n ). F u r t h e r , some re g io n s o f th e

m agnetosphere have y e t t o be in v e s tig a t e d . S evera l general fe a tu re s

o f th e magnetosphere have been f ir m ly e s t a b lis h e d , however.

The magnetosphere is g e n e r a lly d iv id e d up in t o v a rio u s re g io n s

such as the m a g n e to ta il, the tr a p p in g r e g io n , th e pIasm asphere,

e t c . Regions in th e magnetosphere are i d e n t i f i e d by changes in the

c h a r a c te r o f th e magnetic f i e l d o r by v a r io u s c h a r a c t e r is t ic s o f the

lo c a l p a r t i c l e po p u la tion o r energy spectrum . S u rro u n d in g the

magnetosphere is a regio n c a lle d the m agnetosheath. The magnetosheath

Is e s s e n t i a l l y a t r a n s it io n between the re g io n dominated by the

geom agnetic f i e l d and th e o rd ered fle w o f th e s o la r w in d . The

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11

magnetosheath is c h a ra c te riz e d by t h e .t u r b u le n t n a tu re o f the m agnetic

f i e l d whose f lu c tu a tio n s can be as la rg e as 100$ o f th e average f i e l d

(Sm ith and D a v is , 1970). The bow shock marks th e o u t e r boundary o f

th e magnetosheath on the sunward sid e o f the m agnetosphere. The bow

shock is an e s s e n t ia lly permanent, col I i s i o n l e s s , s ta n d in g shock

wave whose th in n e s s (much le ss than an ion c y c lo tr o n ra d iu s (Heppner

e t a l . , 1967)) su ggests th a t i t is an e l e c t r o s t a t i c a l l y generated

phenomenon. The in n e r boundary o f th e m agnetosheath, th e magnetopause,

s e p a ra tes the d iso rd e re d f i e l d o f th e magnetosheath from th e ord ered

f i e l d o f th e e a r t h . On the day sid e o f the magnetosphere the

magnetopause is e a s i l y id e n t if ie d by a change in f i e l d m agnitude

and d ir e c t io n o v e r a d ista n c e o f a few hundred k ilo m e t e r s . The

magnetopause is more d i f f i c u l t t o id e n t i f y in th e t a i l re g io n o f

th e magnetosphere.

The geom agnetic f i e l d c lo s e t o the e a rth is alm ost d ip o la r but

d e v ia te s c o n s id e ra b ly from a d ip o le f i e l d near the magnetopause

and in the m a g n e to ta il. On th e day sid e the f i e l d is compressed and

lin e s o f c o n sta n t 3 in the e q u a to ria l plane are d is p la c e d outward

(away from the e a rth ) from those o f a d ip o le f i e l d . In th e a n t is o la r

d ir e c t io n th e f i e l d is in f la t e d and lin e s o f c o n sta n t B are d is p la c e d

inward from those o f a d ip o le f i e l d . Near the m agnetospheric

e q u a to r ia l plane th e f ie ld changes d ir e c t io n a b r u p t ly fo rm in g a

r e g io n c a lle d th e ne u tra l s h e e t. The n e u tra l sheet exten d s acro ss

most o f th e m agnetospheric t a i l in th e m agnetospheric e q u a to r ia l

p la n e . The in n e r edge o f th e n e u tra l sheet b eg in s a t about 10-15 Rg

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in th e a n t i s o l a r d ir e c t io n . The h ig h la t it u d e f i e l d lin e s a l l e xte n d

in th e a n t i s o l a r d ir e c t io n and c o n s t it u t e most o f th e m a g n etic f i e l d

in th e t a i I . '

P a r t ic l e p o p u la tio n s in and n ea r th e m agnetosphere are id e n t i f i e d

by th e shape o f t h e i r energy s p e c tra and t h e i r number d e n s it ie s .

U s u a lly , the in te rn a l energy o f a plasma can be used to d e fin e a

• 3kT

te m p e ra tu re ( - ^ — = in te rn a l e n e r g y ). The a c tu a l p a r t i c l e d i s t r i b u t i o n

f u n c t io n can then be a pproxim ated by a M a xw e llia n d i s t r i b u t i o n o f

v e l o c i t i e s c o rre sp o n d in g t o t h i s te m p e ra tu re . The a c tu a l measured

plasma v e l o c i t y d is t r ib u t io n can th en be compared t o th e a s s o c ia te d

M a xw ellia n d i s t r i b u t i o n . F o r e x a m p le ,-th e magnetosheath e le c tr o n

v e l o c i t y d is t r i b u t i o n is b ro a d e r and has few er low e n e rg y p a r t i c l e s

than i t s a s s o c ia te d M a xw e IIia n d i s t r i b u t i o n . The magnetosheath p ro to n

d i s t r i b u t i o n s i m i l a r l y la cks low en ergy p a r t i c l e s and a ls o has a

sm all peak in the number o f p a r t i c l e s lo cated a t about 2 -3 tim e s th e

a verage p a r t i c l e e n e rg y. The average p ro to n e n e rg y is a few hundred eV

and th e average e le c tro n energy 1-2 hundred eV (M ontgom ery e t a l . , 1970).

The t r a n s i t i o n from the s o la r wind plasma to th e m agnetosheath plasma

which o c cu rs a t o r near th e bow shock is a ls o d isc u ss e d by Montgomery

e t a l .

Frank (1970) has id e n t i f i e d two c h a r a c t e r i s t i c p ro to n e n e rg y

s p e c tra in th e magnetotai I . One spectrum is q u it e s i m i l a r t o t h a t

.o f magnetosheath pro to ns w hich s u g g e sts t h a t m agnetosheath p a r t i c l e s

have access t o the t a i l . The o th e r group o f p a r t i c l e s id e n t i f i e d

by Frank has a much bro a de r c h a r a c t e r i s t i c sp ectrum w it h an a vera ge

• 12

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13

e n e rg y nea r 5 Kev. Both o f th ese gro up s o f io n s a re g e n e r a lly taken

t o be members o f th e plasma s h e e t. The plasma sh eet is a p a r t i c l e

p o p u la tio n cen tered about th e n e u tra l s h e e t. In th e t a i l re g io n i t

is ab o ut 4-6 Rg t h i c k , measured p e rp e n d ic u la r t o th e n e u tra l s h e e t.

I t e xte n d s near th e e a r t h in t o th e dawn and dusk s e c to r s where i t is

6-12 Rg t h i c k . The in n e r edge o f th e plasma sh e e t is u s u a lly

id e n t i f i e d by th e b e h a v io r o f th e e le c t r o n s . The e le c tr o n v e l o c i t y

d i s t r i b u t i o n is ro u g h ly M a xw e llia n but has a h ig h e n e rg y t a i l . The

a ve ra g e e le c tr o n en ergy is near 1 Kev b ut can range up o r down by an

o r d e r o f magnitude C V e tte , 19705. The b e s t in d ic a t o r o f th e in n e r

edge seems t o be a d ecre ase in th e t o t a l e n e rg y d e n s it y o r a decrease

in th e number d e n s it y o f a l l p a r t ic le s w ith e n e rg ie s g r e a t e r than

700 eV. The d e te rm in a tio n o f th e in n e r edge depends upon th e energy

o f th e e le c tr o n s o b served (S c h ie ld and F ra n k , 1970). The in n e r edge

o f th e plasma sheet has not been o b served t o p e n e tra te th e plasmasphere

even d u rin g d is tu rb e d tim e s . However, th e c h a r a c t e r o f th e e le c tr o n

en e rg y d e n s it y and th e p o s it io n o f th e in n e r edge is a p p a r e n tly q u ite

d i f f e r e n t d u rin g q u ie t and d is tu rb e d tim e s . V a s y lin n a s CI968) pu ts

th e in n e r edge o f th e plasma sh eet d u rin g q u ie t tim e s a t I I + I Rg .

S c h ie ld and Frank p la c e th e in n e r edge 1-5 Rg from th e plasmapause.

The d if f e r e n c e in th e tw o r e s u lt s may be due t o a la t i t u d i n a l dependence

(.S c h ie ld and F ra nk, 1970).

The plasmasphere is a c o ld dense plasma (compared t o th e r in g

c u r r e n t ) t h a t c o ro ta te s w it h th e e a r t h . The o u t e r boundary o f th e plasma

sp h e re , th e plasmapause, is most e a s i l y i d e n t i f i e d b y an a b ru p t change in the

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14

number d e n s ity o f plasmasphere io n s . The change in th e number d e n s it y

is t y p i c a l l y o ve r an o rd e r o f m agnitude. Several examples o f plasmapause

c r o s s in g s and r e p re s e n ta tiv e number d e n s ity p r o f i l e s are g iv e n by T a y l o r

e t a l . (1968) and Chappell e t a l . (1970). The d e n s ity o f c o ld ions

3 4 3

(10 -10 °K) j u s t o u ts id e th e plasmapause is t y p i c a l l y I-I0 / c m and

in s id e th e d e n s ity r is e s a b r u p t ly t o 100-1000/cm^ and c o n tin u e s t o

in c re a s e a t a slo w e r ra te w ith d e c re a s in g L v a lu e . The s tu d ie s

by T a y l o r and Chappell both in d ic a te d th a t th e plasmapause

tended t o move inwards w ith in c re a s in g magnetic a c t i v i t y . R e c e n t ly ,

R u s se ll and Thorne (1970) have re p o rte d th a t th e plasmapause a c t u a lly

c o in c id e s v e ry c lo s e ly w ith the lo c a tio n o f th e maximum d e n s it y o f

th e storm tim e r a d ia tio n b e l t .

The development o f s e n s it iv e d i f f e r e n t i a l energy a n a ly s e rs by

L .A . Frank has enabled him t o stu d y v e ry low en e rg y pro to n s in the

e n e rgy range 200 eV to 50 Kev (F ra nk 1967, 1970). P ro to n s in t h is

e n e rgy range were su bse qu en tly found to be an im p o rta n t p a r t i c l e

p o p u la tio n in the magnetosphere. The storm tim e r a d ia tio n b e l t , p r im a r ily

com prised o f p a r t ic le s in the 1-50 Kev range, is la r g e ly re s p o n s ib le

f o r th e main phase decrease d u rin g m agnetic storms (F ra n k , 1967) and

th u s can be c lo s e ly i d e n t if ie d w ith the r in g c u r r e n t . The r in g c u rre n t

p ro to n s in the 1-50 Kev range a re re c o g n ize d by t h e i r c h a r a c t e r i s t i c

en e rgy spectrum and r a d ia l number d e n s ity p r o f i l e . The d i f f e r e n t i a l

e n e rg y spectrum hardens s l i g h t l y as L d ecre ases. The spectrum a t 6 o r

7 Rg is alm ost f l a t from 3 t o 30 Kev. The average en ergy is t y p i c a l l y

between 5-20 Kev. The number d e n s ity o f these low e n e rg y pro to n s

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15

depends upon th e magnetic a c t i v i t y and the lo c a l tim e . S a t e l l i t e

and ground based magnetometer measurements ( C u m m i n g s e t a l .1968) in d ic a t e

t h a t d u rin g a m a gn eto sph e ric. substorm th e re may be a decrease o f

r in g c u r r e n t p a r t ic le s in the m id n ig h t t o dawn s e c to r (2400-0500

lo c a l tim e ) w h ile in the e a r ly even in g t o m id n ig h t s e c to r (1500-2400

lo c a l tim e ) an increa se o f r in g c u r r e n t p a r t ic le s has been d i r e c t l y

measured by s a t e l l i t e (Frank and Owens, 1970). F u rth e rm o re , Frank

(1970) has measured the number d e n s ity a t d i f f e r e n t lo c a l tim es

d u rin g v a rio u s phases o f a magnetic storm and found th e re e x is t s an

asymmetry in th e number d e n s ity d u rin g a l l phases o f a storm . The

asymmetry appears t o be la rg e s t d u rin g th e e a r ly development o f th e

main phase, F ra nk and Owens (1970) Have in v e s tig a +ed th e proton

d i s t r i b u t i o n in th e m id nigh t s e c to r and found the p e r s is t e n t presence

o f a " q u ie t tim e r in g c u r r e n t" w ith peak pro to n energy d e n s itie s

near L=6.5. S everal ty p ic a l f l u x p r o f i l e s i . e . th e L dependence o f

th e f l u x are given by Frank (1967b) f o r v a rio u s energy ranges. These

p r o f i l e s a l l have th e same q u a lit a t iv e b e h a v io r as fu n c tio n s o f L . S ince

th e shape o f th e d i f f e r e n t i a l e nergy spectrum changes o n ly s l i g h t l y w ith

L and is n e a rly f l a t , the number d e n s ity as a fu n c tio n o f L wi 11 be

p ro p o r tio n a l to th e f l u x p r o f i l e f o r energy ranges near the average e n e rg y .

Thus the f l u x p r o f i l e f o r the energy range 16-25 Kev sh o uld have th e same

L dependence as th e number d e n s it y . Peak number d e n s it ie s in the

storm tim e b e lt v a ry w ith the m agnetic a c t i v i t y . Frank (1967)

re p o rts a measurement o f the peak number d e n s ity o f 8 + 2 protons/cm ^

a t m agnetic la t it u d e 27° f o r the main phase o f a moderate sto rm . A .

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16

s l i g h t l y la r g e r v a lu e is excected f o r th e e q u a to ria l number d e n s it y .

The e le c tr o n energy d e n s ity is o n ly about 7 5 % o f th e p ro to n e n e rg y -

d e n s ity in the storm tim e r a d ia tio n b e l t .

The upper re g io n s o f the e a r t h 's atmosphere are p a r t i a l l y io n iz e d

by s o l a r r a d ia tio n and e n e rg e tic p a r t i c l e bombardment fo rm in g a

co n d u ctin g la y e r known as th e io n o sp h ere. The c o n d u c t iv it y o f the

ionosphere must be expressed as a t e n s o r . The components o f the

c o n d u c t iv it y te n s o r are fu n c tio n s o f e le c tr o n d e n s it y , c o l l i s i o n fre q u e n c y,

g y ro fre q u e n c y , la t itu d e and lo c a l tim e . R a th er than use a complex model

o f th e io nosp h ere, i t w i l l be represented in th e model magnetosphere

as a p e r f e c t c o n d u cto r.

Id e a lly a model o f the magnetosphere should r e f l e c t a l l o f th e

s i g n i f i c a n t fe a tu re s o f the a ctu a l magnetosphere but s t i l l be sim ple

enough t o work w it h . Models o f the geom agnetic f i e l d in p o p ula r

use are formed by a d ip o le , image d ip o le and t a i l c u r r e n t and account

f o r th e gro ss fe a tu re s o f the m agnetospheric f i e l d , th e magnetopause

and th e m a g n e to ta il. However, such e f f e c t s as f i e l d i n f l a t i o n due

t o p a r t i c l e p o p u la tio n s are not taken in t o a cco u n t.

To f a c i l i t a t e the study o f d if f u s io n in the magnetosphere i t is

b e st to chose the sim p le s t model p o s s ib le which a d e q u a te ly d e s c rib e s

th e p h y s ic a l s i t u a t i o n . Thus Nakada and Mead as w e ll as S ch ulz and

E v ia t a r chose a model which accounts f o r th e magnetopause s in c e th e y

a re c o n s id e r in g d if f u s io n d riv e n by induced e l e c t r i c f i e l d s produced

by changes in the magnetopause p o s it io n . Birm ingham, how ever,

chooses a d ip o le f i e l d modol s in c e he is c o n s id e rin g d if f u s io n ‘

d r iv e n by an in t e r n a l l y generated e l e c t r i c p o t e n t i a l . The

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17

d e te rm in in g f a c t o r in c h o o s in g a mode I . sui tab le f o r t r e a t i n g c a v i t y

modes i s . t h e ease w ith which boundary c o n d itio n s can be s p e c if ie d .

T h is f a c t o r e lim in a t e s the c o n s id e r a tio n o f num erica l models and

d isc o u ra g e s th e use o f complex g e o m e trie s .

The model t h a t w i l l be used f o r c a lc u la t in g th e f u n c t io n a l depen

dence o f th e c a v i t y modes is ro u g h ly e q u iv a le n t t o a t i n can f i l l e d

w ith a lo s s y d i e l e c t r i c . The model is b e st d e s c rib e d in term s o f th e

usual c y l i n d r i c a l c o o rd in a te s ( r , 0, z ) . An a z im u th a lly sym m etric

a x ia l m a gn etic f i e l d is assumed. The f i e l d is ta ken t o be in th e z

d ir e c t io n and th us i t s magnitude is a fu n c tio n o f r o n l y . The

magnetopause is represented by th e c y l i n d r i c a l s u r f a c e , r = 10 Rg (e a r t h r a d i i ) .

The iono sp h e re is represented by p e r f e c t ly c o n d u c tin g e n d p la te s

a t z = + A Rg . The plasmasphere is an in n e r c o re o f c o ld , M a x w e llia n ,

e le c tr o n -p r o t o n plasma w ith T . (th e ion te m p e ra tu re ) = T (th e

i e

4

e le c tr o n te m p e ra tu re ) = 10 °K. The storm tim e b e l t is re p re s e n te d

by a h o t, M a xw e llia n , e le c tr o n -p r o t o n plasma w ith T . = 10® °K.

The plasmapause is assumed t o be lo c a te d a t r = 5.9 Rg

and is re p re s e n te d by a sharp decrease in th e number d e n s it y

o f th e c o ld plasma. The hot r in g c u r r e n t plasma number d e n s it y

has a maximum va lu e a t 6 Rg . The b e h a v io r o f th e r i n g c u r r e n t

number d e n s it y on r between th e lo c a tio n o f th e peak number

d e n s ity (6 Rg ) and th e magnetopause (10 Rg ) is g iv e n by e i t h e r an e x p o n e n tia l model (k je x p - ( r -6Rg ) /a ) o r a power law model 2 2

( k | / ( r ^ ) ^ ) . The use o f these models is d isc u ss e d in C h a p te r 5.

A sch e m a tic r e p re s e n ta tio n o f th e b e h a v io r o f th e r i n g c u r r e n t

and plasm asphere number d e n s it ie s is g iv e n in F ig u r e 2 .1 . •

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F ig u r e 2.1 A diagram o f th e model used t o re p re s e n t

th e m agnetosphere. The model is a z im u th a lly sy m m e tric .

The f i e l d lin e s are s t r a i g h t and d ir e c te d p e rp e n d ic u la r

t o th e io n o sp h e re s. The ionospheres are re p re s e n te d by

p e r f e c t l y co n d u ctin g d is k s a t e it h e r end o f th e c y l i n d e r .

The o u t e r c y l i n d r i c a l s u rfa c e re p re s e n ts th e magnetopause

and th e in n e r c y l i n d r i c a l s u rfa c e s in d ic a t e th e lo c a t io n

o f th e plasmapause. The c a v i t y is 2A long and 20 Rg in

d ia m e te r. The schem atic dependence o f th e number d e n s it ie s

o f th e r in g c u rre n t and plasmasphere plasmas on r a d ia l

d is ta n c e a re g iv e n in s e m i-lo g p lo t s .

(30)

M ag ne top au se

Figure 2J

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20

The c y l i n d r i c a l geometry o f the model makes i t v e ry easy t o a p p ly

boundary c o n d itio n s . Since th e magnitude o f th e m agnetic f i e l d '

depends upon r o n ly , the g ra d ie n t d r i f t is in th e azim uthal d ir e c t io n

o n ly . Since the f i e l d lin e s are s t r a i g h t , th e re is no c u rv a tu re

d r i f t nor m ir r o r fo rc e on the p a r t i c l e s . The m ir r o r in g m otion o f

p a r t ic le s in the magnetosphere is sim u la ted by c o n s id e r in g the

ionosphere as p e r f e c t ly e l a s t i c . Th at i s , p a r t ic le s a re r e f le c te d a t

z = +_ A w ith no lo ss o f e n e rg y . The q u e s tio n o f w hether o r no t t h i s

model ad equ ately rep re s e n ts th e e s s e n tia l fe a tu re s o f any d if f u s io n

processes th a t might be d riv e n by d r i f t modes in th e magnetosphere is

b est taken up a f t e r t h i s sim ple model has been in v e s tig a t e d .

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C h a pter 3

The D if f u s io n Equation

D if f u s io n in th e magnetosphere is assumed t o r e s u lt from th e plasm a’ s

in t e r a c t io n w ith wave tu rb u le n c e p resen t in th e plasma. The tu rb u le n c e

in th e plasma is assumed to be weak enough th a t i t can be tre a te d as a

p e r tu r b a tio n e f f e c t on the p a r t i c l e 's b e h a v io r in th e absence o f tu rb u le n c e .

Changes in th e phase space d e n s ity o f p a r t ic le s due t o p e r tu r b a tio n s in

th e p a r t i c l e 's o r b it s can be expressed by means o f a d if f u s io n eq u a tio n

in a manner s i m ila r t o th a t used by Birmingham, N o rth ro p and Falthammar

(1967). The a ctu a l wave tu rb u le n c e in th e magnetosphere is represented

by a s t a t i s t i c a l model in which th e phases o f th e v a rio u s F o u r ie r com

ponents o f th e p e rtu rb a tio n wave f i e l d s a re assumed t o v a ry randomly w ith

tim e . The phase space o r b i t o f a t e s t p a r t i c l e is exp ressed in terms o f

th e a m p litu de s and phases o f th e p e rtu rb in g wave f i e l d s and th e d if f u s io n

c o e f f ic ie n t s used in th e d if f u s io n e q u a tio n can then be expressed in terms

o f th e random phase v a r ia b le s . A f in a l e xp re ss io n f o r th e d if f u s io n

c o e f f ic ie n t s is o b ta in e d by ensemble a v e ra g in g .

The d if f u s io n eq ua tio n w i l l be developed r e s t r i c t i n g th e d is c u s s io n

to th e m agnetic f i e l d o f the model magnetosphere and e l e c t r o s t a t i c o s c i l

la t io n s . The development o f th e equation which is based on Birm ingham,

N o rth ro p and FSIthammar (1967) is o u tlin e d here and th e d e t a ils a re g iv e n

in A ppen d ix A - l I . F i r s t the m agnetic f i e l d , B ( r ) T ^ , is exp ressed in th e

usual E u le r c o o rd in a te s a and 3: B ( r ) = Va x 73. The r e s t r i c t i o n t h a t

the m agnetic moment o f th e p a r t i c l e , u , be conserved is used t o o b ta in

the H a m ilto n ia n (see appendix A - l )

21

(33)

s is th e measure o f d is ta n c e a lo n g a f i e l d lin e fro m a f ix e d re fe re n c e

p la n e . The f i e l d lin e is g iv e n by th e in t e r s e c t io n o f s u rfa c e s o f

co n s ta n t a and 6. pg is th e momentum c o n ju g a te t o s . The c o n t i n u i t y

e q u a tio n in th e phase space ( a ,3, s , p s ,p ) f o r th e d e n s it y o f g u id in g c e n te rs Q

$ + 5 ^ ’ + & iSq> + & ,iQ> + s i K 5 ’ = 0 can be w r it t e n as a L i o u v i l l e eq u a tio n

i 32. + + c + n - 0

3t 3a e3S 3s Ps3ps

s in c e a , 3 and s , pg are ca n o n ica l v a r ia b le s (s e e a p p e n d ix A - l ) .

I f Q, , a, e t c . are c o n sid e re d as random v a r i a b l e s , th e y can be

c o n v e n ie n t ly w r it t e n in the form x = <x> + <5x. The b r a c k e ts < >

in d ic a t e th e ensemble average and 6 th e f lu c t u a t in g p a r t o f th e

v a r ia b le s . I f th e p e r tu r b a tio n s in th e p a r t i c l e ’ s phase space o r b i t s

d u rin g a tim e in t e r v a l which is le ss than o r equal t o th e c o r r e l a t i o n

tim e o f th e random process a re assumed smal I compared t o th e s c a le

s iz e o f th e a verage phase space d e n s ity <Q>, i t can be shown (see

a pp end ix A - l I ) t h a t <Q> s a t i s f i e s a d i f f u s i o n e q u a tio n :

4 4 4

3 <Q> V • ° <Q> _ v V 3 /n 3 <Q> * vF £ <* i > T x ~ " “ ay v x 3x

i= l 1 3xi i = l j= l 3 x i X i Xj 3xj

where th e n o t a t io n x (=a; x2=3; x^=:

phase space c o o r d in a te s . The d i f f u s i o n c o e f f ic i e n t s D a re i J g iv e n by th e e x p re s s io n

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23

Dv .

- < 1 ^ ' « i < « .6. s , p s , t )

The un p ertu rb ed o r ze ro o rd e r o r b i t s o f th e p a r t i c l e in phase space

as a f u n c t io n o f the param eter t ' - t = x which measures tim e "ba ck w a rd s"

a lo n g th e o r b i t a re denoted by w .. The w. must s a t i s f y c e r t a in

c o n d it io n s :

The n o t a t io n w in d ic a te s d i f f e r e n f i a t ion w ith re s p e c t t o th e tim e

argum ent. S in c e th e f lu c t u a t io n s 6x can be exp re ss e d in term s o f

th e Hami I to n i an ( f o r examp I e , 6 a = - J j y - j ^ <H>

J

th e d if f u s io n c o e f f ic i e n t s can be expressed in term s o f th e H a m ilto n ia n . B e fo re

p ro c e e d in g w it h th e c a lc u la t io n o f th e d if f u s io n c o e f f ic i e n t s , a

s t a t i s t i c a l model o f th e tu rb u le n c e must be d e v e lo p e d .

Suppose we a re c o n s id e r in g th e i n t e r a c t io n o f a group o f p a r t i c l e s

w ith a c o h e re n t e le c t r o s t a t i c pla ne wave o f wave le n g th X = 2m/k and

fre q u e n c y w. Suppose a ls o t h a t th e p a r t i c l e s have a v e l o c i t y

d i s t r i b u t i o n g ( v ) and g(to/k) i 0 . ' Then some o f th e p a r t i c l e s w i l l

undergo s u b s ta n t ia l tx 3 d r i f t s sin ce ' th e y see th e wave as an alm o st

c o n s ta n t e l e c t r i c f i e l d . A f t e r a tim e T , th e number o f p a r t i c l e s

t h a t undergo a s u b s ta n tia l d r i f t is ro u g h ly g(oi/k) X /J. So th e

number o f p a r t i c l e s t h a t see th e wave as c o n s ta n t becomes a r b i t r a r i l y

sm all a f t e r a s u f f i c i e n t l y long tim e . As a r e s u l t e s s e n t i a l l y no

p a r t i c l e s , on th e average, a re tra n s p o rte d by th e wave. T u rb u le n ce

e f f e c t i v e l y p u ts an upper l i m i t T c on T , so t h a t even f o r long tim es

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24

a group o f ro u g h ly g(^0 j ~ p a r t ic le s in t e r a c t s w ith th e wave and is c

tra n s p o rte d by i t . T h is e f f e c t can be reproduced by u s in g a s t a t i s t i c a l

model to re p re s e n t th e tu rb u le n c e . A common p r a c t ic e is t o assume t h a t

th e phase o f th e wave is changing random ly. An example o f a

re p re s e n ta tio n o f tu rb u le n c e w ith a random wave phase s t a t i s t i c a l

model is g iv e n in a d isc u ss io n o f plasma h e a tin g by s t o c h a s t ic e l e c t r i c

f i e l d s by P u ri (1966). P uri assumes t h a t th e wave phase can change

by an a r b i t r a r y amount a t in t e r v a ls o f tim e which a re randomly

d is t r ib u t e d . The s t a t i s t i c a l model employed here wi I I be s i m i l a r in

t h a t th e wave phase w i l l be co n sid ered a random v a r ia b le b u t th e phase

w i l l be assumed t o change o n ly a sm alI amount in a sm alI tim e .

I t w i l l be shown la t e r t h a t the e l e c t r i c p o t e n t ia l a s s o c ia te d w ith

th e e l e c t r o s t a t i c flu c tu a tio n s in the model magnetosphere can be

w r it t e n as

$ ( r ,0, z , t ) = £ A (£ ,m ,n ,r ) exp i (w t+jt,0-kz+i{i(ll,m ,n) ) 3-1

£,m,n

k = ^ ' s The p a r a lle l wave number, o = u)(£,m ,n) is th e wave freq uency

and i|i(2,,mf n) is th e phase o f th e wave. I t is c o n v e n ie n t t o in tro d u c e

th e shorthand n o ta tio n

£ f ( U and g (L ) f o r £ f(£ ,m ,n ) and g (£ ,m ,n )

L £,m,n

The phase 4i(L) is considered a random v a r ia b le . The random process can be

c o m p le te ly d e sc rib e d by g iv in g e x p l i c i t e xp re ss io n s f o r th e h ie r a r c h y o f

p r o b a b i l i t y d is t r ib u t io n s

W ( * , ( ! ) , < » , (2) ,111,(3) •••t( ; i|i2<l),i|i2( 2 ) ,*2( 3 ) - - - t 2;

♦n ( l ) , * n ( 2 ) ,* n( 3 ) - + n )

(36)

(see Wang and U hlenbeck, 1945 as a general r e f e r e n c e ). Aga in i t is

'c o n v e n ie n t t o in tro d u ce a s h o rt hand n o ta tio n and w i l l be used t o denote

« ( I ) , 4 (2 );* (3) m m m m( » )

I t w i l l be assumed, as is u s u a l, th a t th e e n t ir e process is

s t a t io n a r y and M a rk o ff. So th e h e ira rc h y o f p r o b a b i l i t y d e n s it ie s

can be e xpressed in term s o f th e second o r d e r p r o b a b i l i t y d e n s ity

|» 11 ; Rt/2, t 2) . A ls o , i t is assumed t h a t th e second o rd e r

p r o b a b i l i t y d e n s it y f o r th e whole process can be w r it t e n as a pro d u c t

o f th e second o rd e r p r o b a b ilit y d e n s it ie s f o r in d iv id u a l wave

components:

W2(R ^ |, t |; R<|>2, t 2) = 7T W2 ( iji j ( i ) , t ! ; ^ 2 ( i ) , t 2) 3-2

W2(i|i|( L ) , t1; ^2C L ) , t2)di|i|(L) dif»2( L ) is th e p r o b a b i l i t y t h a t th e phase o f

th e component has a value between tpj ( L ) and ij>| (L)+d<i>| (L ) a t th e tim e

t j and a v a lu e between anc*0 2^-)+diJJ2(L ) "*"'me ^2' P h y s ic a lly

th e assum ption 3-2 means the p a r t ic le s see th e phase o f each component

ind epe nd e ntly o f any o t h e r . L e t i|)|,^2» ’ denote th e phase a n g les

o f any p a r t i c u l a r component a t t| »+2’ "iV ‘ * r e s p e c t iv e ly . Then i t f o llo w s from assum ption 3-2 and th e assumption t h a t th e e n t ir e process is M a rko ff t h a t the process f o r an in d iv id u a l wave component is

a ls o M a rk o ff, sin c e

W3 (W W W = / W3(R ^ |, t |;R ^2, t2;R ^3, t 3)

= / W2 <R^j , t j ;R02 , t 2 )P2 (R^2, t 2 | R4»3,+ 3 ) - W2 |>t | ; ^2*^7) ?2(^2' t2 I ^3*^3^

where / denotes in t e g r a t io n o v e r the phase v a r ia b le s o f a l l

components e xc e p t | *^2 *^3 anc* ‘ s second o r d e r c o n d itio n a l