department of plasma physics.pdf

8/19/2019 Department of Plasma Physics.pdf

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TRITA-EPP-78-08

ON THE PARALLEL ELECTRIC FIELD ASSO-

CIATED WITH MAGNETIC MIRRORING OF

AURORAL ELECTRONS - SOME BASIC PHY-

SICAL PROPERTIES

Walter Lennartsson

May 1978

Department of Plasma Physics

Royal Institute of Technology

100 44 Stockholm, Sweden


2/67

ON THE PARALLEL ELECTRIC FIELD ASSOCIATED WITH MAGNETIC MIRRORING

OF AURORAL ELECTRONS - SOME BASIC PHYSICAL PROPERTIES

Walter Lennartsson

Royal Institute of Technology, Department of Plasma Physics

Stockholm 70, Sweden

Abstract

Increasing the precipitation of hot magnetospheric electrons into

the ionospnere by means of a parallel electric field requires a

large total potential difference AV along the magnetic field lines.

This is a consequence of the magnetic mirroring of the electrons.

Given an isotropic Maxwellian distribution of electrons of temoe-

rature T an increase of the precipitation flux by a factor • ,

as compared to the field-free precipitation, requires AV i. (*-l)

k T / e .

Thi* result is obvious in the case of scat ter-free motion

of the e'* trons but remains valid also in the presence of random

pitch-ai : > scattering and provides a simple basic explanation of

the obs.-;

r

^i eneroy spectrum of auoral electrons.

During


3/67

1. Introduction

In spite of their great complexity the various auroral observations

seem to allow a fairly simple interpretation in certain respects.

A steadily increasing number of observations of different kinds

thus seem to indicate that the intense precipitation of electrons

in visible auroras is associated with an electrostatic accelera-

tion of the electrons along the magnetic field lines at higher

altitudes.

The presence of a potential gradient along the magnetic field is

commonly inferred from the electron energy spectrum which con-

sistently shows a pronounced high-energy peak when observed above

auroral forms (e.g. Ackerson and Frank, 1972; Evans, 1974; Burch

et alL, 1976a; Lundin, 1976) . Also the common alignment along the

magnetic field of the electron precipitation flux is often ascribed

to such a potential gradient (Ackerson and Frank, 1972; Arnoldy

et al, 1974; Evans,

1974).

Recent observations at high altitudes

confirm that the precipitation of auroral electrons is associated

with a field-aligned outflux of energized positive ions from the

ionosphere (Shelley et


4/67

trons.

Tne arguments used in this step are based entirely on

the dynamics of the electrons and do not consider the quasi-

neutral i ty of the total particle population. Therefore the

actual distribution of the potential V is left essentially

undetermined (cf Knight, 1973, and Lemaire and Scherer,

1974).

2. It is shown that the problem of finding V from quasi-neutrali-

ty at every point along a magnetic field line in general is

overdetermined. That is, an otherwise continous solution V is

generally possible only if quasi-neutrality is violated at some

point and the potential distribution has a "discontinuity"

there. It is suggested, as a physical interpretation, that the

potential "discontinuity" is associated with a "double layer"

structure and, hence, that the formation of such a structure

is a necessary condition for a quasi-steady state to exist

in general when a discharge of hot electrons is set up by an

external voltage source. A closer study of existence criteria

shows that "double layer" formation is indeed a natural conse-

quence of the magnetic mirrorin


5/67

2. Current-Voltage Characteristics

Figure 1 is a schematic illustration of a raagnetosphere-ionosphere

current loop, where the two points P

{

and P

%

ar e located at a high

altitude in the isagnctcsphcrc an d P^ ond P arc the respective

ionospheric foot points

of the

magnet ic field lines through

P

t

and P

fc

. The magne tic field direction is assumed downwar d and the

length coordinate along the field dir ection is denoted by s. The

trans verse magnet ospher ic current from

P

a

to P

t

presumably flows

in a

wide altitude region, associated with a differential drift of hot

ions a nd electrons (Lennartsson, 1976 and

1977b),

b ut is drawn a s

a line current here

for

easy reference.

The

field-aligned curren t

front P

2

to Pj is assumed ass ociated wi th a discharge cf hot elec-

trons from Pj to ?,, whereas the field-aligned current from P^

to P

s

is assumed due to a released escape flux of cold electrons

from P

3

{Lennartsson, 1976 and 1977b) .

Given the phase space density f (v) of the hot electrons at P

the conservation of phas e space density and magnetic moment along

the field line imposes

an

upper limit

on the

field-aligned current

density i

e

at P :

\ Vmax <

f

e,'

.' V

AV)

'

where B and D are the respective magnetic field strengths and

AV is the potential difference. To illustrate this within the

nonrelativistic range the phase spaco density, magnetic moment and

total energy are assumed conserved:

f_ (s, v (s)) = f . (v (s )) , (1)

2 B (s)

- eV (s ) =»

~

v

2

(s ) - eV^ , (3)

where s refers to an arbitrary point along P -P and s refers to

P .

i

._


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In

a

formal sense these assumptions require

a

strictly fisld-

aligned particle drift, which implies in particular

a

zero trans

verse electric field. They are, however, still applicable with

a

lavrto tranevorco narH îo ri f*- »c 1 nna S SOatiAl aradiftntS Of

* * * *

z>

•"- ~ ~ " ~ —

•

w

" *

w

—

jr — -* •

f,

B

and

V

are negligible along the transverse drift direction.

Furthermore the electron source flux is assumed isotropic:

ei


7/67

member of (7) represents a complete precipitation of elec trons

with K* B eAV/(B - B ) at P . Such a complete precipitati on

occurs if and only if (8) is true . The second term in (7) re -

pj.t:i>fc;iilö Lite IIIOAUUUU: p o s s i b l e p t ö c i p i t a t i o n o f e l e c t r o n s w i t h

higher initial energies, given AV. The differential precipita-

tion flux of these elect rons at energy K + eAV at P is max imu m

when the entire pitch-angle range u < 90° is filled. This

occurs if and only if (8) is true.

The meaning of (8) is illustrated in Figure 2a where the mag-

netic field strength B is used as spatial coordinate instead of

s. A potential di strib ution V = V(B ) that satisf ies (8) is every-

wher e above or on a straight line between the two poi nts (B , V )

and (B , V ) .

2 2

Relation (7) takes on a part icul arly simple form if F

e j

(K)

is Maxwellian (c.f. Knight, 1973), that is if

f m

e

& . ̂ K_

F

ei

W =

M 2 7 k T .

As long as

eAV


8/67

A much larger increase requires

in particular since V(s) rooy not fulfill relation (8) .

It is important to keep in mind that (7) and (11) only refer to

electrons. Hence, these relations do not explicitly involve local

quasi-neutral ity but only show the maximum possible current

density due to hot electrons, given the total potential drop

AV along the current path. In order for AV to remain finite,

however, the electron number density has to be nearly the sane

as the number density of positive ions everywhere along electric

field lines, except maybe for small regions. This constitutes

a strong constraint on the potential distribution which

depends on the velocity distribution of all the various par-

ticles, as discussed below.

The positive ions not only serve to neutralize the electrons

but they also carry a certain amount of current. In a typical

electron precipitation event the hot positive ions carry

a negligible current (e.g. Burch et a_l,

1976a),

whereas the

simultaneous escape flux of positive ions from the ionosphere

may carry a significant current (cf Rassbach, 1973; Shelley et

al, 1976). The total current density at P

?

may then be written as

i = i

e

+ i

x

(12)

H 2

II

It?

where i

a

is carried by positive ions from the topside iono-

sphere and, hence, is limited in a steady state by the thermal

escape flux, i

x

& 10~ A/m

2

(Lemaire and Scherer, 1974, c.f.

n

?.

i

also

below).

O n closed field lines i may be reduced by a

II

7

return flux of ions from the conjugate hemisphere (c.f. Alfvén

and Fälthammar,

1963).

The cold electrons of ionospheric origin do not contribute to

i in the above idealized model. The reason for this is that

*2

the magnetospheric current P - P in Fig. 1 is assumed due to hot

particles only and therefore no negative charge can be removed

from P to P by means of cold electrons in a steady state. This

is not true, however, in ;i more realistic model, where the initial


9/67

transient state may have a considerable duration and the current

P - P flows in a wide altitude region. As noted below i nay

1 * »2

in a transient state be due in part to accelerated hot electrons

from high altitudes, in accordance with (7), and in part to

drifting cold electrons from lower altitudes. In reality a ty-

pical precipitation event may be in a transient state, of course

(e.g. by moving relative to the cold plasma).


10/67

3. Quasi-Neutral i ty in the Absence of Scattering

Consider Fig. 1 again. In view cf the above discussion it is

cunciudec chat the plasma between P and P is a conductor

with finite real impedance. That is, given an external sta-

tionary voltage source a large potential difference is main-

tained between P and P because of the maqnetic mirroring of

negative charges from P . This, however, is only a description

i

of the current-voltage characteristics of the conductor and it

does not reveal the spatial distribution of the potential along

the current path. The potential distribution is the solution

of Poisson's equation in three dimensions,

jfv = e"

1

e (n - n.) , (13)

o e i

where J& denotes the Laplace operator and n and n. denote the

number densitites of electrons and ions, respectively. It is

assumed here and in the following that all ions are positive

and singly charged. If n and n- are completely known ét priori

as functions ot the potential distribution Equation (13) can

De solved for V, in principle, with suitable boundary conditions.

Within certain limits the distribution of V can be found without

actually solving (13) . This is because c,' o n

i

is typically a

very lar^e quantity compared to an aven ge value of X'v and, hem:",

\n - n.

| ••<

n almost everywhere, in order for V to remain

"finite".

This is called the quasi-neutrality condition. Provi-

ded that n and n. are continuous and linearly independent single-

valued functions of both the potential and the spatial coordinates

it is thus possible to determine V at a given point by requring

n

e

= n^ (14)

In general n and n may also be functions of the spatial

derivatives of V and (14) is then a differential equation that

must be solved in a region. In a strict sense this equation is

in conflict with (13), however, and it should therefore be used

with great care. It is important to keep in mind that Equation

(13) does not imply quasi -neutrality at every point but the

actual solution ot (J3)

n\ny

well require jn_- n.| ~n in certain

small regions.


11/67

3.1 Simplifying Assumptions

Al. The transverse electric field component E^ is assumed neg-

ligible along the magnetic flux tube under consideration. This

is a ratner restrictive assumption vis-a-vis the ions and some

consequences of a large E, are nentioned later on.

A2. Tne magnetic field strength B is known a priori at every

point and B(s) ̂ 3(s'} if s > s . ence, the agnetic field

strengtn

may be used as

spatial coordinate along

the

field

line instead of s,

s = s (B) (15)

A3. The phase space density f, the magnetic moment and

the total energy are conserved along the field line between P

and

P for all

particle species involved,

f

v

(s, v (sj) = f.' (s', v (s'j) , (16)

m vl (s) -T v i (3

v L -- _i~i , ,-,nd (17)

2B 2B--

— v

2

is; • a v i s . ••-- -i: •'• •.* ) - i. ' - ' : * • > , (1 8>

2

where s ,»ncl A- i:M"tr t ,•

(

ir:^ - i r iry r>.\ in',?< •«jj»:' iibl» -:: a q i ven

pa r t icl e of specip.s v ar.d

z'm

romai

-wnq

aynbolo are convtjitiond'

A 4. The pa ral lel ej fc tr i. r fieJ'.J E ,\l a iilven point alonci

P . - P_ is either uj-.'*ard •.;•:. zero,

tip < 0 (19)

Aj>. During the initial ostabl ishin-; of the current P - P the

parallel electric field strength at any point has been growing

mnnotonically as a function of tin»i t ,

;E

B

(t) > E

fl

(t

r

; I , t t ' (20)

A6. The current system has been "turned on" for a sufficiently

long tine for a quasi.-steady state to bo established.

In discussing the electrostatic potential distribution the

following definition?; are needed:


12/67

%

Dl. Given the spatial distribution V = V( s) , that is V = V(B ) ,

I then

the set of all values V(B)

s

for B''


13/67

n

where B* and V" refer to any point between P and P'. Electrons

contributing to n at a point with magnetic field strength B

are either arriving from P or returning to P after mirroring at

B

x

> B. Tnese two populations have identical energy distributions

except for the electrons that are lost at P

2

. The downflowing

electrons are thus selected according to {v(B , B)} whereas the

upflowing are selected according to the complete potential distri-

bution -fv(B , B )]• , where the definition (21) has been used.

Analogously, the electrons contributing to n

e 2

are either arriving

from P , selected according tc{v(B, B )} , or returning from

2

2

r i

higher altitudes selected according to |V(Bj, B )t

Hence,

in general,

n

e i

= n

e

,

( B

' {v (B , B }} ) and (26a)

n

e 2

= n

e 2

( B , { v ( 1 3

j f

B

2

; } ) (26b)

With an arbitrary source distribution at P , for instance, the

number of independent variables are reduced to a minimum, that

is

n

«*,

= R

o (B, V - v , V - V ) , and (27a)

n

e

?

.

= n

e 2

l B

'

V

V

'

V

2

" V '

U 7 b )

if and only if the following two conditions are fulfilled every-

wnere:

V - V

V' - V > — L ( B' - B ) , and (28a)

1

B - Bj i

V - V > - 1 _ ~ (B* - B) , (28b)

B - B

2

where V and B refer to the arbitrarily given point P, v'and

B'

re-

fer to points between P

i

and P and V'and

B*

refer to

points

between P and P . These conditions follow from (25a) and (25b)

and are equivalent to

d

2

v

< 0


14/67

throughout the interval, B

1


15/67

3 . 2 P ro b le m S t r u c t u r e

Wi th t h e a s su m pt ion s (15) - ( 20) E qua t i on (14) may be w r i t t e n a s

n

e

+ n

e 2

+ n

e i

a B

Z t n

V l

- n

v

, , ( 3 4 )

v

where tne summation refers to the various ion species.

Given ther

a) the respective source distributions at P and P ,

b) the potential V at P , for instanc e, and

c) the local magne tic field strength B, B


16/67

problem to be physically meaningful it is thus necessary to

specify as a boundary condition

d) the total potential difference AV = V. - V .

With all four conditions a through d giv-en independently Equation

(34) evidently represents a mathematically overdetermined problem.

To some extent a and d may well be coupled but there is no obvious

physical reason why this coupling should be the particular one

that warrants a con tinuous solution of (34 ). Such a coupling

is certainly not required by Poisson's Equation (13). O ne there-

fore has to conclude that quasi-neutrality in general must be

violated scmewhcre between P and P ,allowing V(B) to have a

"discontinuity" thore. it is illustrated below how such a "dis-

continuity" may actually arise physically.

In reality there are also particles trapped between P and P .

As long as (19) und (20) are valid, and in the absence of scatte-

ring, these particles are electrons originating from ?

]

only.

Trapped particles are difficult tc handle quantitatively because

they result from a violation of the constants of motion (violation

of (18) when no scattering is present). Of particular interest here

is the effect of trapped particles on a potential "discontinuity".

Now it can be seen qualitatively that trapping ho , electrons from

P cetween a potential "discontinuity" at one altitude and the

magnetic mirror below in fact acts to reinforce -the "discontinuity".

This is seen below to be particularly important when pitch-angle

scattering is present.


17/67

4. Some Expected Properties, of the Potential Distribution

4.1. Maxwellian Particle Sources

With reference to Fig. i the following assumptions are made

about the particle sources.

A7.

The hot particle source at ? emits electrons and one kind

" 1

of singly charged positive ions isotropica]ly in the pitch angle

interval a


18/67

The

hot

electron density

n

(B, V ) ,

on the

ther hand, tends

to increase with increasing B and V, within certain limits.

If an energy distribution function F (K) is defined according

to

(5) and (D) the

density

is

found

to be

e i

= — I F ( K) / i c + éT v - V }' CL (K)dK

(37a)

whereil

(K) is the

sclid angle occupied

by

electrons with energy

K and

£1

(K)

= A

e i

< K ,

B,

X L

e i

(K )

=.

B ,

V -

B

)}) in

general

and

f

V - V) , i f

the relations (28a-b) hold. As lonq as the down flowing electrons

almost fill

a

solid angle

of

2TT their density increases with

V because of the term e(V - V ) inside the square root. This

may also give a net increase in the total uensity n

el

. In par-

ticular, if the. potential distribution satisfies (8) and

e(V

2

- v

(

)

:

; stveral times tĥ . avti.ragf initlai energy at P

it follows that

(3 7b)

despite the fact that n (P- ) "ontain.s no rnirrorcvi electrons.

ei

2

A consaquence of (36bj and (3 7b) is that cold ions from P

must play an important role in neutralizing the ot electrons

at iower altitudes. If the effect: of gravity on the ions is

included in a formal fashion the density n.,(B, V) is seen to

vary roughly as

'

v )

)• foi

( 3 8 )

where

n.

(B,

v )

includes

the

variation with altitude

due to

1 2

;>

-V) •£ cT, . At higher

the function n. (B, V ) is

I

2

7

gravity

at low

altitudes, where

0 i e

altitudes,

where e\'J -V) •• kT.

a constant and serves as the appropriate source density. If

I*:,, I >> ^g/e,

g

being

the

gravitational acceleration,

n.

(B,

V )

is a constant regardless of the value of V -V, of course. At


19/67

altitudes where e(V.-V)

;

- kT, the ions from P

>

have a strongly

field-aligned distribution.

At sufficiently low altitudes n

i 2

• n

i

and the dominant nege-

tive constituents are the electrons from P . If the density

c

n is divided into a cold electron density n

g 2

and a hot elec-

tron density n ,

J

e i'

n = n

c

+ n

h

, (39a)

e2 e2 e2 '

it is seen that

e(V,-V)

n

e

C

2

(B,V)~e

kT

e> n j (B, V J , (39b)

where n

c

(B, V ) accounts for the effect of outward diverging

magnetic field lines. O bviously, only a negligible number of

cold electrons can penetrate to altitudes where e(V - V )

> > k T

e 2

>

starting at P . Because of the assumptions (19) and (20) there

can be no isolated "islands" of cold electrons left at higher

altitudes in a steady state. Thh2 hot electron density n is

also decaying with increasing value of V - V although much

more slowly because of (3S) .

As mentioned above there is also, in general, a trapped

electron component, original ?.y froin P , that does not have

access to either P or P . The density of this component n' ,

however, cannot be simply related to the electron source at P by

means of (16)-(18) and (21) and is therefore treated as an

essentially unknown variable here.

4 . 2 C ause and Effect

The above expressions ta>-.sr Uvje';hor reflect an intrinsic causa-

lity between the eLcctrostntic potential distribution and the

phase space distribution of trie particles. It folLows from (36a,

1

and (37a) that the density ratio n (4>)/n. {i) of down flowing

particles from ?

t

js increasing with increasing D and V, except

maybe for short intervals with a 3trong potential gradient,

whereas the ratio n

(

(t)/n. (f) of upflowing (mirrored) par-

ticles from P is decroafirvq v i t.h increasing B and V. As a con-

U-


20/67

18

sequence the potential di' fcrcn:e bot'.-'O-on C

and ?

2

which

satisfies V, < V\ , cannot be produced originally by the motion

along P - P_ of the hot particles iron P, but requires that

an external voltage source be applied to increase tn.3 "loss

cone

1

' of electrons and decrease that of

ions.

The net negative

charge required at high altitudes of course may be accomplished

by a slight excess of hot electrons at P

}

but the corresponding

net positive charge at low alcitudes must be due to the cold

ions from P, (cf (36b) ard (37b) and n. (P.) «•• n (P,)).

According to (38) the density n. ,, is indeed an increasing

function of b and V. It can be seen that n

Q ]

=

n

i t

+ n

i

2

^

o r

the more complete Equation (34)) does have continuous solutions

V(B) with 3V/ÖB > 0 above a certain altitude, given a potential

difference V. - V - 0 . That is, the velocity distribution of

the total particle population at P, is made consistent with a

large potential difference V., - v by the application of an

external voltage source.

The two expressions (:36a) and (37a) are certainly based on the

assumption A7 but a tendency tor n

._

to exceed n. at B >> B

may be expected also with more general source distributions,

unless the electrons have a considerably more perpendicular

pitch-angle distribution than the ions. In particular if (8)

is satisfied it may bt exported that n

o

(i

J

) '- n. (P 1 ,

or even n^ (P ) -••-

n

. (p ) _ This is apparently consistent

c 1 i' 1(2

with typical auroral particle uata (cf e.q. Sharber

to P . An upward direction

of i

u

means a positive value of this .integral i.,, - E

(|

ds, which

implies a net trfiji&ler of electrostatic enerqy into particle

kinetic energy. Tnat in, i

)(

• E

(|

ds - 0 implies a reshaping

of the velocity distributions along the particle trajectories

at the expense of electrostatic energy. The loss of electro-

static energy between P and P must be compensated for in a

(quasi-) steady state by a net gain in soir.e other section of

the current loop, presumably between P and P , where the

reversed process is taking place and i * E < 0. This active

section

( dynamo )

must therefore be what maintains V - V ' 0

2 1

in accordance with some current-voltage relation Like (11), for

instance (c.:i. Lennartf-son, L977b) .


21/67

4.3. Iliu£tra_t_ion_ of_._a__£oteritj^ 1 .^Pisc

The notion "discontinuity" here refers to a considerable change

of V with B nearly constant.

It is fir.'t noted that the sprtial distributions of the various

particle species are sensitive to electric fields of vastly

different magnitudes. Roughly speaking the suatiaJ distribution

of a certain particle species is defined via the function

U B + q V, where M is a characteristic (average; magnetic

moment and q. = e for ions and q ~ -e for electrons (c.f.

(16-(18)). Hence, the hot electron component n

(kT

j

dB/ds "- k.T

n

/e R

p

for instance,

where

s only affected by

IE,,

the radius

of the

Earth,

R. , is

used

as a

characterist ic length

of the Earth's niac.net.ic field. With

i

E

«

•kT

/eR the spatial

1 Jj

variation

of n is

accordingly aiso characterized

by R

£

. As

a contrast the cold electron component n^ changes appreciably

e ;

over

a

distance

of one R

with jE

}J

j

~ k T /eR^ . In the

unperturbed

cold plasma the quasi-neutrality, of course, ±s being maintained

by E . ~ - m.g/e, where g is the gravitational acceleration. If

m. is the proton mass and ;;T •»» 1 eV, for instance, m.g/e ̂ k T / e R ^

in

the

vicinity

of the

Earth.

The rot.

electron component

n is

affected

by

electric fields

of

magnitude

JE I *

K^/eRg.

Figure

3a

depects

in a

qualitative fashion

the

variation

of n

with distance

s

along

B in tl 2

presence

of £„

~ -

kT /eR

E

as

well

a s the

variation

of n

p

in the

presence

o f E^**/ -

m.g/e.

The

stronger field,~10~

V/m

with

kT ~ 1 keV , is

required

to

balance

the magnetic mirror force

on the hot

electrons

in

order

for

these

to carry

an

increased current,

in

accordance with

(11) . The

weaker field,

̂ 1 0"

only field that allows

a

quasi-neutral distribution

of the

cold

unperturbed plasma. These

two

plasma distributions

are

obviously

incompatible

and

hence there must

be a

transition somewhere,

as

indicated

by the

dashed curve segment between

s' and s" .

That

is

a

transition between

a

high-aLMtude region v/ith strong elec-

tric field

and

onJy

hot

electrons

and a

low-altitude region with

weak electric field

und

cold elect rons oresent .

V/m with

m.

beinq

the

proton mass,

is the

suppose that. Kquat'.on ' \] car; be ^at isf iel at V with the

a s s u m p t i o n s

A 7 - / 9

,.ni.«i ; i ] . K i \

t h p t o u

• - o L c i t i.-ii :I i

f

f o r e n c o

L V

~ V - V S i t t l s f y i f i q

( J . i ; . S u p p o s e f u r t i u r t h a t

( 3 4 ) i s

s a t i s -

f i e d

a l s o

i'ji.

m i : r -•:•:,;:•.,'..:•.• M

v ; i ; h : • •

r

••.•>;•.; i

n p o f . ( T f i . ; , i d i

u t r i . b u

V

--

V J B )

.

i r,

•.•iic

:-.i'--...'r

•:•-. : a 3

f l

V J

•

o

/i • y \ < >

1

. -

n '•

- ; j •

i v

i . - . - . v r < \ \ < A , ;


22/67

function of H , c.:\

long as ;.; (V

-V kT,

the

i

or,

according

to

ar;

be matched

by

.̂Lections ,u t; present in 'c.hi.s hi.jh-

As e (V

_

-V) uiroroacho ̂ kT the varia-

density n, is a relatively slow function of V

(3 8 ) ,

and 'xtrioe the spatial '•.jriatior of n . „

the hot par'ticleo. \ o:;.t

tion

of

n. with

V

bo-.'r:?nes faster than t.h" variation with

V of

any

of

the hot components, including the hot electron component

n

i

(c.f. (35) and '•? „ *'-V

̂

. Since n. ,

is

also proportiorujl

to

B at

constant V, which

is

true also

of ,

the spatial

kT.

, eventually exceeds

rowth rate

of n., as e (V

- V

what

can be

compensated

for

by

the ho t

parti cles. According

to

(35) this critical point is reached whil e e(V

-

V) is still

large compared to kT.

,

if T. _ «•»

T .

Yet the cold electr on

density

n „

does

not

become s ignificant until

e(V \ -V) ,c

a

few

times kT^, , according to

(39b).

As

a

con sequence quasi.-neutrality

must break down

in the

transit ion region

and

a

region

of

unbalan-

ced positive charge is formed between s' and s

by

the upward

acceleratiny tons from

P,

as illustrated in F ie. 3b.

It

is

important

to

otice that this conclusion

is

reached without

any

a

priori assumption about the electric field inside the

transition region nor the potential difference across this re-

gion, V(s")

-

vfs

7

) .

It is

quite realistic, however,

to

assume

3 priori that e(V{s ) - V(s')) •> kT.

,,.

This

is

ecause s' must

b

at a sufficiently low altitude for the total potential difference

Ay

= V

2

- V to

satisfy (1.1) . More specifically, with

e

IK

M

| **M

ei

dB/ds

~

it follows that

^

/B

dB/ds for the high a ltit ude field

V(s')

- Vi


23/67

must exceed by far the cold plaswa field m.g/e /- kT

e

,/eR_ ~

kT /eR ~n this enhanced electric field the spatial decay of

1 2 '

E*

the cold electron density n_^ is , accordinq to i39b> , iv.uoh

faster than the spatial variation of any of the hot particle

components and also much fastor than the spatial decay ci

n. for e(V,- V; > kT . Hence, it must, again be concluded

that a region of unbalanced positive charge is formed between

s' and s" , as illustrated in Fig. 3b.

The positive space charge in F ig. 3b is qualitatively consistent

with a transition between the stronger electric field to the

left and the (much) weaker (3*0) field to the right. It is also

consistent witt» the above finding that the positive charge

required at low altitudes to maintain V, ' V, must be due to cold

ions from P. , in gener al. This is, howev er, not a quan titative

consistency, which car: be seen by inspection of the amount of

positive cnanje in Fig. 3b. The space charge densit y is roughly

of the order of magnitude 0.2 e n ',' («5 }, as seen from (38) and

( 3 9 D ) , and henc e, if ;JK

M

| -̂ ;E

|

it follows that

I E J ~ 0 . 2 e n

c

(s*J (s* - S' J/L, ,

where t^is the vacuum dielectric constant . Trie potential diffe

V(s") -

V(£i'>

*•-.

(-,» -

s' )

jE

n

j

is •

, accorvlinq fj 0 9 b ) , a nd , he n c e,

rence V(s") - V(£i'> *•-.

( - , -

s')jE

n

| is at leant

sever.T 1

times

e

i

i - || ;

* . I I r *

t

t

~:

? n

and

L b i s a w r y l o i r ' ; t -

. m c o m p - i r € ; d t o k T ̂

/ e l ; , , .

W i t h

n.^"

[.; )

a s ä n u i j :

3

jj

c m "

J

,

n v/, ( c . F . . M o z c i

of: a J ,

I'j77)

a n d k T '. r.-v

< l

:

s i ^ c M i : f i r . l.d i s s t i l l y 0 . ^ V /: i •• :• k T / e R

. L".' I T

^ ]

V,"--., >,

• t a '-• _ '- k ' V .

T : : ; c -


24/67

i f J E J

> >

i

E

n

i

J

i n

order

for the

associated pot ential variation

to

b e

compatible with available part icle en ergi es.

A s a

conse-

quence, a

region

o f

negat ive spac e charge

1

must

b e

present

a t

s

< s' to

natch

the

excess positive charge

att s > s' .

That

is the

transition region mus t have

a

structure

similar

t o

tiie "double layer*

or

"electrostatic sh ock" struct ures

previously studi.-d

b y e.g.

Block

(1972 and

1976)

and

Swift

(1975

and 1 9 7 6 ) . O n the

spatial scale

of B it is

therefore

appropriate

to use the

notion "discontin uity"

for the

potential

change accross

the

transition region.

Fig. 3c is a.

schematic illustration

of

self consis tent parti cle

and charge distributions

in the

transition region.

T h e

negative

space cnaxge

nay be due

essentially

to hot

electrons moving

upward (leftward)

and

overshooting

t he

ions from

P in

density

before they

a r e

reflected

by the

very stro ng elect ric field.

This

is

assumed

to be the

case

i n F i g . ic. In

principle

a

negative space charge

way

also

be-

accomplished

by

reflection

of downward no-inq ions

"rom P,, Th :^

n>ny

b e o*

particular

i m -

portance

on

closed n.^qnetic field linos wh or e

a

large fra ction

of the» dcwr.f iowir.g ion*.- c,*y

̂

i;o >->r».

;

Ln^te from

P, or

from

the

conjugate ionospheric point.

In a

vy

case the

negative charge

requires

3

V 3V

(40)

at

t n e

upper edge

o f the

transition region,

at s° in F ig . 3c.

If

the

negative charge

is

essentially

due to

overshooting

ener getic electr ons from belo*/ whic h have

a

characteristic

energy

K' it is

thus neces sary that

V(s") - V(s°) > K' /G

which is also consistent with a large reduction of n. across

the transition region if K.' -- kT. , c.f. (38) . In analogy with

tne cold electrons these electrons should be reflected

within

a distance roughly defined as

with n'

(

being the density of these electrons. It is shown in the


25/67

next section that trapped , or rather qvutsi-'rapped, electrons

must play a central role ir

t:ii£

respect. That is, the neqative

cnarge may be largely due to electrons temporarily trapped

between the potential barrier and t;;e magnetic mirror below.

Figure 4 is a qualitative sketch of two potential distributions

V = V( B) , part a, which includo a discontinuity and the associated

plasma density distributions n =

r.

(B) , part b. The dotted line

in Fig. 4a represencs a potential distrabution with d V/dB = 0.

The potential distribution represented by the solid contour

satisfies the condition (8) and therefore produces the maximum

field-aligned current density (that is equality in (7) and approxi-

mate equality in (il); with a given total potential difference

AV = V - V or , equivalently, allows the minimum &V at a given

2 1

current densi ty. Since (8) also implies an isotropic precipi tation

flux at B of all electrons wich initial energy K ^ B eAV/(B - 3 ) ,

2 ] 2 1

given an isotropic source flux at B , the solid contour is also

i

consistent with "maxiruin isotropy" of the precipitation flux.

The potential distribution represented by toe dashed contour

violates the condition (8) and produces a smaller current. On

the other hand this potential distribution produces a strong

field-alignment of the precipitation flux at B .

2

It must be kept in mind that t..e electric field also has a trans-

verse component, which rreans that V(B) differs f oir. one magnetic

field line to tne other. A twc-dircensionul picture of equipoten-

tial contours may have the general structure of Fin, 5a. for

instance (c.f. Gurnett, 1972; Lennartsson, 1973 nnd 1977a; Swift

et a_l, 1976) . The "discontinuity" in V is marked here by a

close spacing of the equipotentials. It is noted here that the

transition region between the hot and cold plasmas must not be

completely equipotential. Some equipotential contours must turn

downwards and reach the ionosphere to ensure current continuity

(Lennartsson, 197 3 and 1977a; Burch et al_, 1976b) . The distri-

bution V - V(I3) along the two dashed lines is sketched in Fig. 5b.


26/67

5. Discussion

5.1.Formal Mathematical Aspects of a Potential "Discontinuity"

Regardless of the precise physical mechanism it must be concluded

that the potential distribution along the magnetic field, V =

V(B),

in the absence of collisions most likely has a discontinuity"

at the interface between the hot plasma at high altitudes and the

mainly cold plasma below. That is a discontinuity on the spa-

tial scale of B. This is evidently a necessary condition for a

(quasi-)steady state to be at all possible with general velocity

distributions of the hot magnetospheric particles at P in Fig. 1.

In mathematical terms this is due to an intrinsic overdetermina-

tion (actually a special case of this) of the problem of finding

V = V(B) from quasi-neutrality, as discussed above, and may be

briefly described as follows. The hot and cold particle densities,

n. (B, V) and n (B, V) resp., are quite different functions of V.

Yet,

in order for quasi-neutrality to hold continuously it is

necessary that n, « n„ in a finite interval AB, which in general

implies an overlapping of two different electric fields.

From a formal mathematical standpoint a discontinuity in V (B)

at some point along a magnetic field line is d

1

so a sufficient

condition for a steady-state solution of Equation (34) at other

points on the same field line. This can be seen to hold during

rather genrrai conditions. Givrn the source particle distributions

at T and P in Fig. 1 and the total potential difference V -V ,

Equation (34) may be solved for V = V(B) in a high-altitude region

at B < B B' (see (38) and (39b)). Th.is is provided, of course, that

the cold plasma can indeed neutralize the charge imbalance among

the hot particles within it.

The mere existence oi a potential discontinuity does not per so

imply a particular location

n or a

particular potential step,

however. In the simplest possible configuration (19) is valid and

hence V(B) has the general shape of either of the two broken

contours in F ig, 4u. The a.izocio Ltd equipot.cnt.ia.L contours in

two dimensions may have the general topology of Fig. 5a (cf

curl E = 0 ) .


27/67

5.2.

hysical Interpretation :._j

From a physical standpoint tha problem is considerably more

complex, in particular vis-a-vis the sufficient conditions for

a steady o- quasi-steady state It is quite possible that some

of the simplifying assumptions used here, like (19) and (20)

for instance, are incompatible with a steady state. The assumption

of a completely collisionless plasma is certainly unrealistic but

it is beyond the scope of this paper to investigate in detail

the consequences of wave-particle interactions and binary colli-

sions. Only a few important remarks on this problem are made

below.

A self-consistent treatment of the two- or three-dimensional

character of the potential structure also necessitates a proper

consideration of the transverse electric field component E,. When

the potential structure has th«? same spatial scale size as a

discrete auroral form the E. component has a strongly disturbing

effect on the magnetic /noment of at least the ions (cf Swift,

1975 and

1976).

As a consequence the cold-ion density i? no

longer approximated by (38) but in a more complicated function

of V involving spatial derivatives of V also. This is briefly

commented on at the end of this tec ion.

In physical terras a "discontinuity" in V( B) , on the spatial scale

of B, is actually a finestract ire in a continuous potential

distribution

"(s),

with s beina a true spatial coordinate. In

this context the term "double layer" is nore appropriate iv r«.;-

ferring to two closely spaced thin spacecharge layers of opposite

polarity. This "double layer" is , of course, subject to various

criteria for its own existence and stability and can be handled

mathematically only if the etude concept of qua:-;.\-neutrality is

replaced by Poissons't; equation (13) ,

5.3. The Existence or' a Itonzero Parallel Electric Fl.qjLd

Regarding the existence of general rolutions of (13) , dependent

or independent of time, thn loll owing is noted. The two current-

voltage relations (7) and ( H ) are derived under the assumption

of an isotrop.Lc influx cf pVucit.rons at P, (

>- i 90°) . If the

downward flux is isotropic at all lowt..- altitude*, too, the in-

clusion of ela.s'.ic: random .~ca.-. I..'..-ring or natch angles only serves

-


28/67

to reduce the current by scattering electrons into the empty

"loss cone" in the return flux from below. If the flux is field-

aligned below some altitude some electrons will also be scatte-

red into the "forbidden" velocity region around

a

~

:

90 . This

leads, as shown below, to a transient current of capacitive nature

This current may, in prinicple, increase the total current

transiently but it is then also accompanied by a decreasing con-

duction current at constant AV . Anyway, the fact that electrons

are filling this "forbidden" region means that isotropy is being

restored. As a consequence this capacitive current can be of

nearly the same magnitude as the conduction current in (7) and

(11) only on a time scale defined by the transit time of precipi-

tating electrons through the spatal region with anisotropic flux,

that is typically a second or less. Hence, on a time scale

of several seconds or longer (7) and (11) are certainly valid

for hot electrons even in the presence of strong pitch-angle

scattering, as far as this is elastic and random.

Inelastic scattering, on the other hand, may chanqe the current-

voltage characteristics but the effect is twofold. Electrons

tnat gain energy at random pitch angles tend to oppose the

current (by transferring electrons from the low-energy integral

to tne high-energy inteural in (7) ), whereas electrons losing

energy randomly tend to promote the current. Obviously, only a

systematic redaction of the magnetic moment of electrons may be

expected to pose a severe limit on the parallel electric field

strength (Lennartsscn, 1976) .

5.4.Existence Criteria for a "Double Layer"

5.4.1. Boundary__Condi tions_on_the_Cold-Plasma_Side_ JPosi tive)_

Now consider the "double layer" structure in Fig. 3c.

A closer study of the variation with V of the cold particle densi-

ties n, and n at e(V -V) £> kT. shows that the criterion (40)

• *• 6 2 2 X 2

is not satisfied at point

s

in Fig. 3c if the cold ions and elec-

trons have similar pitch-angle distributions and temperatures.

That is, the initial "infinitesimal" density changes as the cold

particles enter an electric field i E

f(

I

>

m.g/e are inconsistent

with the electric field direction in terras of div E ~ e(n^- "

e

) /

This is valid for upward as well as downward E

ff

. Consider ( 38) ,


29/67

for instance. Although this expression is a rather crude approxi-

mation ofc' n. wtion G (V - V)

kT. it can he used to show for

instance that an. ,/

V

}V -*"

as V,

- V -

0+ if B •- D,

chat is if

the ions n ve an isotropic dis'.ribution. Tf the electrons also

nave an isotropic distr ibution


30/67

where

(v,,) . = / ii- T v

Dh

(42b)

n

i

i

j m

i

max

is approximately the average parallel velocity of cold ions upon

entering tne layer, as seen in the rest frame of the leiyer, and

t$V is the potential change through the layer (c.f. Fig.4a). It

is assumed here that the cold electrons exchange a negligible

amount of momentum with other particles as compared to the cold

ions. Tne density of trapped electrons n' is included here and

discussed below.

From (42a~b) it is seen that momentum balance becomes increasingly

restrictive as the double layer moves downward. At very high

altitude the unperturbed cold plasma density is comparable to

n

a

(see e.g. Gurnett and rank, 1974) and hence v „/- l/2e $V/m, I

At low altitudes, however, the cold plasma much exceeds the hot

plasma in density and v '_ ' ' •' |/2ejv/r?.. . For reasonable values

ofSv, sayAV ;• 10 kV (o.f. Shelley e t a_l, 1976), the double layer

thus cannot move veiy much fastar H)an the thermal speed of the

cold ions,

]/'k"'

:

?

/ITI. , and eventually the motion must stagnate.

When that,happens tne

splf-ccjis:

stency criterion (40) may no

longer be satisfied, nrd thv electric field perhaps becomes highly

irregular ind fluctuating.

Whether or not a rroving double layer structure is compatible

with a steady or

ri

qua HI-steady " potential distribution V = V(B)

Is a question of definition. Considering the large dimensions of d.o

current synlem in Fi(,'. 1 a potential discontinuity moving at

about tiie cold-ion thermal velocity ( •10 km/s) may seem compa-

tible with a quasi ,steady state on a time scale of minutes, at.

least.

It is noted that a aovnwuro' motion of a double Jayer structure

is tantamount tc an increase of thy upward tield-alignod current

carried by 'ons .jbovo the double layer . Below the lay«r this cur-

rent increment is carried by the cold electrons v.-hich evidently

obtain a downward Dilk n j*:.i.on r-ia elastic collisions with the

moving layer (c.f., the last paragraph in Section 2). This means

that a moving double layer extracts more power from the external


31/67

circuit tiian a statio nary ev..-. 'V".-?

veT.

•. :x\v -ji '• "io 1

.

;

.-;L'

i. ,

howe ver, limited by mo men tun balan ce, as c.'s-.*urf?od ibcvr-.

It is also n ote d tnat a downw ard mot ion of the "do.-tie la-'er

1

'

st ru ct ur e i.i Fig . 3c may be vi e/ ed uyor; a^ t'K- .To.iiis by which, the

hi gh- alt it ude ele ct ric field (at s -. s°) .:at.s into the cold pla sm a

It appears natu ral, from an intuitive st andpoint , that this pro-

cess should be associated with an increased power consumption.

5 .4 .

2.

A Chargir.2_Mechanism: _Traj

::

^in2_of _ElectroijS

Now recall the brief discussio n above of trapped elect ron ? in

conn ectio n with (29)-(31) and cons ider the solid contou r in F ig.

4a. Evidently t he part of V(B) between P and B, must be ^n

efficient trap for electrons from P, duri ng the initial gro wth

of V

2

- V, . Vet this is considerin o only a coliisionl ess pla sma .

In the dens e cold plasma at H > B ' collisio ns and w ave-p art icle

interactio ns are probably impor tant (c.f. e.g. Kindel and Ken nel ,

1971).

Sin ce the elect ron ; from P, attain :\ 'n.o\-e or le ss fi el d-

aligned dist ri bu ti on ai- they en ter civ .vid plosrna ret-ion they

ar e su scep tib le to pi tc h an gle scat'corlsv.-j wr.j«-h na y t er vo a s a

continuous source for temporarily trapped -", ".'K: :•; L3


32/67

the trap per elect ron s bou-.ee trt

their respective position:?

in the Layer and the mag ne ti c

rirrcr

uel ow. Morf/.:r, dur ina the

major portion of their bouncing notion these electrons stay within

the cold plas ma b elo w an d, he nc e, K̂-_>y expe l cold ele ct ro ns to

the io nos phere (point 1' in F ig . 1) . j.'his can be seen to p rovide

2

an equally large current from the ionosphere to the lower posi-

tive Hide of tne "do uble j.ay?r" (the cold plas ma being inf ini tel y

conducting). That is, the .icat Kiinj of primacy electrons into

the "fo rbidden " region pr ovid es in :i self- con sist ent ma nne r a

chargin g cur rent to the "do uble laye r" . This is illustrat ed in

Fig. 7.

The fact that the trapped electrons stay witain the quasi-neutra]

cold plasma during the major part of a boun ce period means that

the "doable layer" structure in Fla. 3c is indeed a capacitor with

respect to negat ive charge s from P (each elect ron con trib utes

a s mall c narge jqj


33/67

in the

f

ieid -al i n:;io;u

::as-:.'

the p has e ;pj< o C uu si ty m: . ido the

trappinq region

1,

•-

\

•-'. 180

1

' - .1 nay b^ expeot td to still

bo fairly Iirc.e. ii^ut , a typical aur nv ji part's cle observation

at low altitude may show 3 strcigly enhanced electron flux at

pitch angles a < 30 °, say, a region of weake r flux for

30° < a -' 30° and a con sider able enha ncem ent at 80°< 1 < 100

c

As pointed out in the next section this type of observation is in-

deed common.

n

What may be expected vis-a-vis the charging current density i

|(

~ V

That depends on trie scatt ering pr obab ility . Su ppos e for instance

that tne "average" primary electron in scattered through o total

1

anv ' 0 £'•* > 30° wit h a pro babi lit y p wh il e tr aver sin g the cold

plas ma. If the precipi tatio n curren t dens ity is i^ the initial

chargin g curren t density is roughly given by i.. > p i^. Hence

if i

t

, - 10 ~

6

A/ n

2

'it the al tit ude H of the lay er and p

'-10^,

say,

then i.® -i, 10" ' A/in

2

, while tne "fcrbidden" r̂ yton is still parti-

ally filled only , and the "double layer" section con sidered above

is being cnargccl at. a rat e of i kV/ s, at legist.

The scattering of primary electrons irto the "forbidden" region

in F ig. 6 not only provid es a charging curr ent to the "double

layer" but also greatly help s ratisfying the self-con sist ency

criter ion (40) on the negat ive side of the layor . In fact, this

mechan ism is suffici ent to satisfy (40) at s° in F in. 3c initially

whil e still ,a, at the average energy . If the "for bidden" re-

gion is at least partia lly filled according to the dashed con -

tours in Fig. 6, that

is

if J f/) m ;•. 0 at

:t

---••

a

;

it is seen (c.f.

(37a)) that the total electron density n

__ ?>,as a

positive drriativc

i

/3V at s ° in Fig.

ic

;

f

< .-

'-* 1

wh er e V, as usual is the Pot en tial at. P -n F iq. I. 1'he

•> ''

sign here accounts for backscatterfJ oloct^ons L ow,

>

in Fig. 1.

Tnis relat ion is quite inseasillve to the actual orie ntat ion of

the"douD.le Layer " st ru ct ur e w;.th r espe ct to the inaanotic field

dir ect io n, bec aus e the tncrnia

1

velocity of hot elect ron s is typi-


34/67

caliy large coir.pared to iac F: :•: 3 drift velocity even if Ej_ — V/m.

The same is not true or the cold ions from P but n. is sensitive

to strong iniomogeneities in ir^. \t a firs , approximation (38) may

be used, co. aspondig to iE i ;•, i i-,

.- ,

from which it is seen that:

i < £

h (

A A

•>

o V 'v „ — V i > }

where V- refers to P

2

in Fig. 1. T?ie "' : " si.cn here accounts for

the presence of positive ions from P

i

According to (44a-b) it is possible to satisfy the self-consistency

criterion (40) on the negative äLOG of the "double layer" structure

in Fi g. 3c, at s , merely by scattering primary electrons into the

"forbidden" region in Fig. 6, provided

V - V(s°) •• V(s°) - V + kT /e (45)

2 ; er

that is provided the potential step Sv at the "discontinuity" is

a sufficiently large fraction ' •

l,s.)

of the total potential dif-

ference AV. This la assuming that the negative .-:pace charge in

Fig. 3c is entirely due to trapped electrons from P and that

the cole

1

ions from P are the only ions piesenu. Iv. eality there

are always backscattered electrons from F increasing the value

of 9n

i

(s°)/ 3V as well as ions from P reducing the value of

3n.(s°)/'jV, and therefore the condition (40) may be satisfied

with a relatively smaller potential difference across the "double

layer" than suggested by (45).

On the other hand (44a) is no longer automatically accomplished by

pitch-angle scattering when the electron precipitation becomes

field-aligned, that is when u •'- a. at average ener« y, Also , when

j E,| >>

|

E..

|

the polarization drift of the upf lowing ions from I',,

may invalidate (44b) ( cf Swift, 1975 ; S K O also

below).

Hence,

in order for the self-consistency criterion (40) to be satisfied

on tne negative (upper) side of a "double layer" structure, it is

generally necessary

t-.hat

the potential difference

'

E

(

, there (compare the two V (B)-prof lle-3 in Fig. 5b) .


35/67

Whether the "do uble iay-rr" str uctur e in Fi a. Tc i? nov ing do wn -

ward at about the cold-ion t hermal velo city , in orde r for (40)

to be fulfilled at s" , is presumably of mir.or importance vis-a-vif.

1

the trapjjeu c ie ct ro ns . .U: any the effect s houl d be in favour of

(40) Ly mean s of compr ess ion of the tri,veci popu lat ion .

5 . 5 . ̂ i i o _ i i ' l £ o £

t

- 4

n

i ^ i l l i l i J ^ ' i r A ^ i i

4

' S c a t t e r

in.':

As far as the forna tion and mai nta inin g -;f a "do_ble layer" s tr uc-

ture is concerned the fre aen co of randou y i

I

cii--~nale scattering

on the low-altitude side is apparently v-iy favourable:, perhaps

even crucial. Above a. ring :- "douh i.e la yer" the influx of el ec -

trons most likely is inolropic or has -j aJ iqnt enhancement at large'

pitch angles (on o 1

o3:_>•;

field lir.es) . •*.£ a co ns eq ue nce the in -

clusion of random pitch-arKj - scattering of elections above the

"double layer" has little '- r no ef fe ct on the i.r/:;r. Th e ions, on

the ot her han d, hdv e a 3t7.org ly f ield -al .i.gnod co cp en en t abo ve the

layer and are ther efore sensi tive to pi ich-angJe scatter ing. If

no ne of trie up flo win g .ions ul urr ;" vcpen tield lin es) this, of

course, does no t afreet the "double laye r". On closed field iinei'

these ions ret urn to sonie ex te nt , ana may affect, the "doub le

layer"

forma tio n. Tn t:.ii ''ide^]" case wit.'i no s cat te ri ng and wit h

compl ete hemisp heri c aod aiiinn;thai sym met ry *-he retur ning com -

ponent is, of cour se, +he mirror vcano with respect to Vj

(

= 0.

In reality the return ing compo nent is probabl y disper sed in phas e

space and may facilitate (40) on the negat ive

r;ide

by con tribu-

ting a negat ive term to rin./)V.

5.6 Internal Charge Balance^

Tne condition thai. (-10) be satisfied on either side of the "double

layer"

is a necessary but not per so a sufficient condition for a

self-consistent soati.al structure. It is also necessary that the

net surface charge of the layer be much less than either of the

internal surfac e char ges but slightly pos iti ve (in the simples t con -

figuration) in order to match the hLqh-31t itude electric field

(upward). To determine whether or not this is satisfied winb a

preconceived set of pjrticlea jnu other boundary conditio ns

the method is to expr ess the various part icle dens i it es r»

v

(V) as


36/67

functions of rrw potential oisi.ribut.ion and ther solve Poisson's

Equction (13; in a two- or three-dimensional geometry. In general

n (V) is an operato r itself because the mag net ic moment may not

v

be a constant of the motio n, in particular not for tne ions,

where f

1JL

* 0

(c.f. Swift ,

1 9 7 5 ) .

Provided the self-cons isten cy criterio n (40) is satisfied on

either side of the layer and the various n 's are continuo us pe-

rators (functions}of V it is certain that n = n. at ome point

s' inside the layer, as illustrated in Fi g. 3c. Therefor e the

prool em is mathematica lly well defined and simple in prin ciple.

To carry out this problem quantitatively is a rather tedious

task, nowever. Also, the potential distribution inside the layer

has to be scug'T- ^long with the high-altitude distribution, be-

cause the quasi-neutvaiity conditions at higher altitudes, n

general, are cont ingen t upon the location of the layer and

the J ayer potent ial oV (note (28;i-b) are not satisfied) as well

as the velocity of a moving layer. In reality a moving layer can

match the positive and negative char ges, within certain limit s,

by adjustinq its velocity (of Block, 1 9 7 2 ) .

5 . 7 . £|£f*?c_r_3_o_£_ a Large Tran sver se Electric^ Pi old

Swift (1975 and L97&; has treated a two-dimensio nal solution f

Poisson's I-

V

{uation (13) in certain as pe ct s, emphasizing in par -

ticular the efiect of a strongly inhomogenecus transverse ele c-

tric field componen t |E,j '•> |c

(|

| . Assuming that an equipoten tial

structure similar to the one shown here in ig. 5a has a large

transverse spatial dimension as compared to a typical ion gy ro -

radius he finds that the polarization drift of the upflowing ion-

ospneric ions tends to cancel the applied space charges, that is

n

g

- n•, in consistency with the assumpt ion. In a further deve lop-

ment of his model he simply assumes "quas i-neutrality" n , = n.,

where n^, because of the polarization drift, is a function of the

first and second derivative of V with respect to the tran sver se

spatial coo rdinat e. iiy a suitable choice of boundary condition s

for the particles aid tht potential he is able to constrct s olu -

tions for V ~ V(r) tram r\

>

- r.. , which hav e an equipoten tial

structure somewhat sxniilai to Fig. 5a nd are associated with

kV-potenttal di fferences along the magnet ic field. The m agnetic

field is assumed ho mogen eous und hence the parallel electrostatic

force-; atp rjaj ancrc' pnt.jrc^y

hy

i;,aiticle

J

er »..ui.l ,'ovccs, as i.;; t.y-

piccil '.>'.

.i

'' J.. .I D

L>-.

iayci".


37/67

- 1 . »

Swift's calculations may illustrate soi.se of tue effects

of finite ion gyroradii on the transition region between the cold

plasma at low altitudes and the hot plasma above. It is seemingly

reasonable to expect the thickness of this transition region tc

be at least as large as the gyroradii of energetic ions (~km)

wherever IE. ( ä,

|E,,

j . Swift's model is a rather crude approxima-

tion, however, and does not allow a conclusive interpretation

with respect to the spatial finestructure. Also, where J E

X

| >> JE

|(

tne spatial variation of the magnetic field is not necessarily

negligible.

5.8. he Role of the Current Density Level.

In Swift's model the "doublo layer" structure is presumed to be

the end result of a local current-driven plasma instability. This

is in principle different from the present model where the

"double layer" formation does not require the current density to

exceed a local threshhold for instabilities. Here it is rather

the mean s by which the cold pasmn is able to screen out the

high-altitude electric field resulting from magnetic mirroring of

hot magnetospheric electrons and, hence, is related to the current

density only indirectly via (11) , for instance. It was found above

tnat pitch-angle scattering within the cold plasma may play a

central role in the formation of a "double layer" structure.

This,

of course, means that a plasma instability may be of importance by

means of promoting pitch-angle .scattering but it does not have to

be current-driven nor coincide spatially with the "double layer"

structure.

5.9 On the High-Altitude Electric Field

Chiu and Schulz (1978) have treated numerically the equation

n = n. in one dimension emphasizing the effect of unequal pitch-

angle distributions for the hot electrons and ions of magnetoapheric

origin. Specifically they .-.ssume the electrons to have a bi-Maxwel-

lian distribution with a transverse temperature about four times

nigher than the parallel temperature whereas the ions have an .iao-

tropic Maxwell tan distribution. Because of this imposed difference:

in tne pitch-angle distributions quasi-neutrality is possible only

with an upward parallel electric field being present. Chiu and


38/67

Schulz calculate the required potential distribution in the form

V = V(B) starting in the equatorial plane and stopping at an

altitude of 2000 km above the Earth, where particles of magneto-

spheric origin are assjmed to be lost.Particles of ionospheric origin

are also taken into account as well as backscattered and secondary

electrons. In order for n and n̂ ^ to be functions of the local

potential only (and the potentials at the endpoints) thus allowing

V to be determined independently at each point, they require V

to satisfy the "accessibility criterion" (28c). By varying a

number of particle parameters, such as the temperature of the hot

ions an d tne cold-ion densities at 2000 km altitude, for instance,

they are able to obtain in a trial and error fashion a continuous

potential distribution V = v(B) which both satisfies (28c) and

is consistent with a certain prerequisite value of the total

potential difference (~kV) . No restriction is set up for the

field-aligned current, however (c.f. Section 3 ) .

The calculations by Chiu and Schulz may illustrate how

the continuous high-altitude portion of V(B) can be constructed in

a specific case, even though they are based on a somewhat parti-

cular model. Since (28c) is required to hold the model is inherently

unable to reproduce a "discontinuity" ( double layer") in V(B) and,

nence, solutions are possible only with certain sets of velocity

distributions for the various particle species.

5.10. Electrostatic O scillations and Pitch-Angle Scattering

In concluding this section a final remark is made with respect

to the stability of any potential structure. Consider (27a-b) and

(33a-o), which represent the simplest possible case. Even then

n

and n^ are functions of the potentials and particle sources at

two points P , and P

which are widely separated in space. This

means that quasi-neutrality, n -«n., at


39/67

37

In reality the maint.iin.inq of quasi-neutrality is ever, more

complex, since the accessibility criteria (2ba-b), or (28c), are

most likely violated in certain reqicns ör.d hence n at one point

is otter» »- function of the potentials at several other points,

cf

(26a-b).

This is true in particular when V«B> has a "discon-

tinuity"

which

was found in Section 3 to be most likely, cf Fig.

4a (V{b) may have more than one "discontinuity") . Even when no

"discontinuity" is present V(B) nicy only barely satisfy

(28c),

as illustrated by the calculations by Chiu and Schulz

(1978),

and hence easily violate (28c) durinq any temporal change. In

general the maintaining of quasi-neutrality thus require

a temporal correlation of the conditions at different points which

is possible only within limits set by the finite speed of informa-

tion exchange. Therefore it may be expected that a

potential

structure is typically .ssociated with (small-scale) electro-

static oscillations superposed on an "average" (large-scale)

structure which is being maintained by the external voltage source

via a current-voltage relation liko (11) . For the sane reasons

it may then also be expected that sorso pitch-angle scattering

of electrons is naturally associated with any real potential

structure.

It does n ot, however, follow from these: considerations that the

parallel current density has *o exceed any particular threshold

for current-driven plasma iiiL.tabil.it.:us in order for the

oscillations to occur. On the other hand it is evident from

the considerations -n Section 2 that part of thu current density

i., is due to escaping col.i ions from the ionosphere,

i

,, ,

and hence that i i .


40/67

38

6. Comparison with Observa;:to

figure 8 is a schematic •; simplified.' representation of the phase

space derisi y f of hot electror. ;, as determined by a conventional

particle detector flown .ibovc a visible auroral form {e.g. Evans,

1974;

Burch et. al, 1976a». The solid curve, labelled "influx" is

a schematic graph of tiK' logarithm of f versus kinetic energy K,

as this appears to a detector looking upwards along the magnetic

field line. Provided the detector hat; a sufficiently small geometric

factor,

a given kinetic energy K can be identified as m v'/2 and

f (v

()

) = f (V^K/nTj- The solid curve labelled "outflux" represents

in an analogous fashion the phase space density seen by a detector

looking downwards along the field line.

As indicated in Fig. 8 by the sloping dashed lines the high-energy

portion of the 'influx" curve can be fairly well approximated by

a displaced Maxwellian. The displacement along the energy axis is

defined, within the accurary of the instrument, by the location

of the peak in log f (ct Evans, 1974). The thermal energy kT of

the source distribution is defined by the slope of log f above the

peak. In a typical case the energy K of the peak and the source

energy kT are related by K a 10 kT (Evans, 1974 ; Burch et al,

e e ——

1976a;

Lundin,

1976).

At energies far above K the observed phase

space density may have a more extended high-energy tail than a

plain Maxwellian distribution (cf Burch ot a_l, 1976a).

The "outflux" curve in Fig. 8 represents the backscattercd primary

electrons as well as the energetic secondaries. At energies below

K '• he "outflux" and "influx" curves coincide (cf Reasöner and

Cli-vpeLl,

1973; Kvans, 1974; Lundin, 1977). This is, of course, to

be expected if the upflowing electrons with K < K are being reflec-

ted by a potential barriär above the- particle detector position

and all primaries originate from above this potential barrier.

Basically the two so]id curves in Fig. 8 are thus both indicative

of an electrostatic potentidi difference AV - K/o between thy mag-

netospherit: particle source ami the location oil the particle detec-

tor .

When the ph;ase space dnnsity f is .s.rnpj

o.å.

at several different piLcii

angles a for i.'a< h energy, with a -~- 0 referring to downward motion,

it is typiciliy fo-ind tL


41/67

and f

(•••i)

_i i ( o ) fo r eiu.i' : r i . u : ( H i e r . ' n -

. -..'.'•.

i . ' : - .. i :-...•-. : . ' ' ; . , . •.. ,, -« V i o . n .; . 1" :•:

o c c u r s o v e r a h e ' i / • . • n ; . i s u s ; u . c o c - i " • • - • " . - • ( ; i l t ..••••••: .-• '

k i n

t . : ; .. t e w

K K J A c k c i s o i i , I S ) • ' I

; .

V . ' - i

;

i e

}

j . s

• a i y i i i : ) i . T » t h i s i j j n r r . - r t . h > • • p h a s e

s p a c e dr. ' is i tv a t ;< _ K r;.-.-y ,-•;•...: i r> ru--.cii t.

:

,c

;

s îi .̂ ;> ;,, i. ion c.\ K - K

( e . q . U ur; :h et -ii , I '^'t-.-i.i, :n; : :oii l i tv; t t , : t on«. •'•e ,'.• 11 t e i throi ' .-j h a 3 pa t i al J . y

va r yi nc i A V. As di.scMJ--.s-.:-:: or ev . o u s l y ( U - nr a ; t s s o n , 1/J73 : nd 1977 ,3 )

t h i s k i iid of. sp i r i a l vj r i a ni o. i ol' '-.V iii ŷ be ex pect .? o if t h e e l e c -

t r i c .1 x ei d .;..•> de e t'.- .

;

ia t , je nqvir . ' i f . ion r at he r tha n e I ect ro -n i aqn et i c

i nd uc t i on . Thi s i s iU 'i sU -.i t/ . . ^ Kti> .•• by i iq.

r

, (co.Mjjr .-t: /••/ f i lonq t h e

two dusae d l i ne t . ) . Se ; a l t o G ur ne t ' ' i '»V2) and .SwiEt e t a l (1 97 6 ) .

A s n o t e d by L un di n (107*, and i.').'?) t hn

fc

ic-cL ron ene.r v \ ,M t h e k i s e ' i e e n e r qy w i t h i n t h e a c c e l e -

r a t i o n

yiy.qinn.

Th.i.: i s .i • :-.O i 'i J

: I : - : '

I ̂ I e-i '.,'/ K i ' j . B . J

t n e \ \\.\y r i ' - ' : - ; i iy

; : > > r- , : > ;

d̂ noter.

.-.-v ^

i' v'"j i riv-f--. from

P"

j

• T

. 8 tho'


42/67

40

e

' I I ;

, «nd

I ->

->

AV

as K = e AV becomes ir.uch larger them kT

̂ •

The current densit y

i,.̂ is usually seen to inc rea se along wit h A (e.g. Lu nd in , 1976

and

1977).

That is , it appears as if a hot electron population

is being pushed through the converging magn eti c flux tubes by an

electric field viiile py.chauginci only little energy with other

particles on uhe average.

in view of Section

2

the observed electron energy spectrum can

thus be given a an i tv- si/up] t; basic physical inter pretat ion. That

is, the electron precipitation represents an electr ic dischar ge

between the ionosphere and a rocrion of excessive negat ive charg e

in the magn eto sph er ic pJa?Tia. This re gion is void of cold ele c-

trons and the negative c'Kirjf

1

is due entirely to hot electrons .

It is maint ained at

u

larjo negative pot-ent.ial with respect to

the ionos phere by the owribi ned ac tion of an external, vol tag e

source and tht m^gre

:

;.r •;;i ior Lng of the negative- charges (cf

Lt-iinartsson,

I'v7u

an d l

l

>'7i;).

In a typical case t he hot R'M'rce elect ron s ma y hav e a roug hly

i sotropic Maxwel lian distr ibutio n (e.g. Ev an s, 197 4: Ijundin, 1976

and

1977).

As suggest ed by (.11

>

Lae dischar ge is then ass ociat ed

with a local increase of the ner field-aligned electron flux by,

at mo st , a factor e ;\V/kT , whe re e AV

jnfer red frorti the ene rgy spect rum.

T\ nd T

-- T are

ei e

Within certain, limits the precipit ating elec tro ns should also

carry information on the potential distribution along the mag-

netic field Li nes . This has been in vestigated by sever al auth ors

among ot her s C vnns (. 974). Kaufma nn e_t eij. (1976) ond Lu rd in (1976

and 19 77).

Tm*

cotmr-o

1

'. coll

i.'«w

f

.

ion of the. flux along the ma gn et ic

K

ines, at en c n i e p in the neighbourhood of K, is often

ascribed to parallel e

I

o

c

t

rosiati': erce.loraticn at alt itu des of

at most a few thous-ind kr> f-.? g. W'.a.l.e'. a;id M e Oi.arni.id, 1 972 ;

B o s q a e d e t a l , 1 r , 4 : h i v a n s , 1 9 7 4 ; .-. r .- di zi , 1 0 7 7 » .

d e ' , > / .c -;'. t h e " V ; c i . i o n s o u r c e i s a t

, 1

M

.: (>>.i i j a a t j o n i i i ip l i/> .y t h a t V ( B )

1):

i.errdf» of r h e p.: r

r

; e r t

i h ig

V i

ft

I t i f. . ie ( B

i e" - • ; •' • | - e ' , , r . i - ': " . '

1

' '

v.-u.

J

ri w i


43/67

41

in F ig. 2a . As lor.g as V(B) is every where above or on this straight

line there is some col lima t ion only very close to K, in the i nter-

val K i K < K B / ( B - B ) * K ( 1 + B / B ) , corresponding to the

i 2 I 1 2

low-energy integral in (7 ) . 01 the two pot enti al prof iles in

F i g . (4a) the dashed one must then be the most represent ative,

although the collimat ion doe s not per se imply a "discont inuity"

in V(B) (cf below).

O n the ot her han d it has been rioted by Lundin (private corrunun ca-

tion ) that t:he parallel curr ent densit y inferr ed iro n the number

fluxe s ir.ay oft en bo roug hly prop orti onal to K wit hin one and the

Ax

same "inverted-V" s truct ure, when K is considerably l.Mrgir than

kT . Thi s would then imply that ill) is valid with ' W arid hen ce

e

l

'

that V( B) does not *'all ver y far bel ow t he s tra ight line in Pig. a.

Also,

it implies in part icula r that the parallel electric

f.ield

does extend to high altitudes and, hence, that the electron

so ur ce is at a high alt.'.tu..lo. Tht: "con stant of pr op or ti on al it y"

has been seen to chun-;e abruptly whj 1


44/67

4 2

ture

and a cor.

tin.icis Ä it ci tr -l u.?

>\. >•

Iburicn

ä t

hiqli-.

altitudes,

as illustrated by i'ig. J.

In

t he

present model

a

'uov.b'e layer" structure

is

expected

to

form

a t

least

in the

transition region be tween

t he

cold plasma

at

l o w

altitudes

and the hot

plasma abov e.

It xs

(generally)

required

t o

enable quani-neutrality throughout

t h e

remainder

of

t he

magn eti c flux tube whan

h o t

electrons

a r e

being pushed

against

t h e

mirr or forces

by an

external volta ge sou rce.

A s

found

in t h e previous section the physical realization of the "double

layer" m a y well be by means of pitch-ang le sca tter ing, weak or

strong, within t he cold plasma belo w t h e layer. This scattering

generates a population of energetic elect rons each o f which is

temporarily trapped between t he layer a nd the magnetic mirror

below

a n d

contributes

a

negat ive space charge

t o the

layer,

a s

illustrated by Figures 6 an d 7. It is therefore o f great interest

to look fo r evidence? o f nuch a trapped populatio n among t he ob -

servational data.

Consider F ig . 6. If th e "forbidden" pitch-angle region

a < a < 1 80 ° - a is completel y f illed by scattering t he enclosed

population is evidently invisible to an observer below the

"double layer". H e wil l only ob serv e a n isot ropi c flux, except fo r

the empty loss con e, with n o sign o f a "double layer". However,

A

at ener gies suffici ently clos e to K the angle a is smalle r than

the loss-con e half-an gle u. and at these energi es ther e is a part

m

a < a.

of the "forbidden " region which cann ot hold tr apped

electrons. This region must appear as a depression in the pith-

angle distribution and is then an indirect evidence

of

trapped

electrons at a, <

i -'

130° --

a..

At lov; alti tude s w here


45/67

43

energies in the neighbourhood of K = K SslO keV. The trough marked

a demarkation between precipitating electrons at smaller pitch

angles and a mirroring population with pitch angles u^ < a


46/67

44

it only proves that V(B) is not an entirely convex function, see

conditions (30) and (31) . However, the presence of both a field-

aligned population and a trapped population may be considered

rather convincing evidence. Consider F igure 9a. The potential

profile marked (1) is consistent with the presence of a trapped

electron population near B but is inconsistent with a field-

aligned precipitation flux. This is realized from (30) and (31)

and from the discussion after (8), respectively. In the same

manner it is realized that the profile (2) is consistent with

field-alignment but inconsistent with trapping. In other words,

the potential profile needs to be both convex and concave, as

illustrated by the two dashed profiles in Fig. 9b. Now, the

dashed profile marked (1) can be excluded on the ground that it

allows trapped electrons only at high altitudes near B , where

it is concave. Hence, the simplest possible profile (with the

least numbe

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