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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
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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
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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
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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
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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
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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
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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
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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).
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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.
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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:
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%
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''
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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
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throughout the interval, B
1
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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
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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.
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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
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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
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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-
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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) .
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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 , ;
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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
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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 -
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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
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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.
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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 ) .
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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
-
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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) ,
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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
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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
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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
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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
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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-
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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) .
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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
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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".
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- 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
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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
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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 .
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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
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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'
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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
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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
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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
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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
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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