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TRITA-EPP-72-07
POTENTIAL DOUBLE LAYERS
IN THE IONOSPHERE
Lats P. Block
•
April 1972
Ihié report i* submitted for publicationin COsmic Electrodynamics, volume 3. Itreplaces the earlier report TRITA-ÉPP-71-14
Author's address:Department of Plasma Physicsox Department of MechanicsRoyal Institute of TechnologyS-100 44 Stockholm, Sweden
POTENTIAL DOUBLE LAYERS IN THE IONOSPHERE
Abstract.
In this paper Langmuir's old theory of double layers
is reviewed and extended to ionospheric conditions, including
effects of gravity ard expansion in diverging geomagnetic
flux tubes. Bohm's self-consistency condition is refined by
including the temperature of "the beam particles. Chances
for stable sheath? are greatest at 1000 - 3000 km altitude
on closed field lines, or higher on open plasma sheet field
lines, and near \he F-region maximum for upward currents,
according to observations. Randomly arising and disappearing
double layers may cause anomalous resistivity in regions
vhere the plasma is subject to rarefaction instabilities,
namely in the polar wind for upward currents, below the
F-region maximum for downward currents, and in the topside
ionosphere in a limited region at perhaps 1000 - 5000 km
altitude. When double layers occur plasma is accelerated
upwards, and the topside electron density may be reduced
considerably within minutes. At the same time the electric
fields in the double layers accelerate large fluxes of
particles, both downwards as precipitation, and upwards as
energetic particles, preferably, observed in the midnight
sector, e.g. at synchronous altitude. However, in spite of
the very large fluxes there is no particle depletion problem,
since current systems are always closed. Electrostatic
parallel electric fields can also be constructed, such that
both electrons and protons precipitate at the same place.
2.
!• Introduction.
experiments with gaseou- discharges in the laboratory
have shown that the current carrying capacity of a plasma is
limited (Langmuir, 1929; Tonks, 1937; Schönhuber, 1958;
Torvén, 1968; Andersson et.al., 1969). If the current density
exceeds a certain critical value the plasma breaks down and a
thin potential double layer appears, where quasi-neutrality is
not valid.
Double layers are often referred to as sheaths, in parti-
cular in caseous discharge physics. The characteristics of
uoubJe layers are:
a) The charged particle density is much lower than in the
surrounding plasma.
b) There is a substantial deviation from quasi-neutrality.
c) The total net space charge integrated over the double
layer volume is very small or negligible.
d) The thickness of the double layer is much less than a
mean free path of the charged particles (transverse to the
current the layer may extend over many mean free paths). Usu-
ally the thickness is of the order of one or several Debye
lengths.
e) The electric field within the double layer is very much
stronger than any fields that can possibly exist in the un-
disturbed plasma. It is directed along the smallest dimension
(the thickness) of the layer.
f) The potential difference across the layer is of the order
of kT/e or larger, T being the temperature of the plasma.
Double layers must not necessarily be associated with a
high current. They may also occur as boundaries between
plasmas with different temperatures and densities, where they
prevent a current that would otherwise flow between the two
plasmas, due to their different diffusion rates.
In space several sharp boundary regions between different
types of plasmas are known: the shock front separating the
solar wind from the magnetosheath, the magneto pause,. the
inner bondary of the plasma sheet, and the plasmapause.
Calvert (1966) has discovered very steep horizontal electron
3.
density gradients in the topside ionosphere. Drawing from
laboratory experience it may be concurred that a sharp
boundary, in the form of a double layer, may also separate
the cold ionospheric plasma from tho sometimes extremely hot
plasma sheet. In fact, it was suggested long ago by Alfven
(1958) that "a plasma sheath may be situated very high up in
the ionosphere, or perhaps in space far above the ionosphere,
and from this sheath the ionosphere will be bombarded by
positive ions and electrons".
A double layer of the kind proposed here is perpendicul
to the magnetic field, in contrast to the boundary regions
known before (magnetopause, etc.).
Jacobsen and Carlqvist (1964) proposed that solar flares
may be caused by circuit interruptions giving rise to extrem-
ely strong double layers with potential drops of the order of
10 - 10 volts for great flares. A similar mechanism was
proposed as an explanation of the explosive phase of an
auroral substorm (Akasofu, 1969). See also Mcllwain (196 ).
Many recent observations of auroral particle spectra have
indicated acceleration processes associated with parallel
electric fields. (For a review, see Block, 1972.) It is possible
that these fields are concentrated in double layers, as is in
fact the most likely explanation of the observations by Albert
and Lindström (1970).
In the present paper the most pertr.nent parts of the
double layer theory is reviewed, extended in some respects,
and applied to ionospheric conditions.
2. Qualitative structure of a sheath
The first self-consistent but very simplified theory of
a double layer was given by Langmuir (1929), To give some
feeling for the nature of a double layer before the more
detailed study of it in the following sections Langmuir's
theory will be outlined here. It is assumed that the layer
is infinitely extended in the x- and y-directions, that the
plasmas on both sides are cold, and that the layer is confined
between the planes 2 = 0 and z = z. > 0 . Because of
the negligible plasma temperatures, these planes
4.
are infinitely sharp divisions between plasma and double
layer.
If a current flows through the layer in the positive
z-direction, 2 ~ 0 is a source of ions with negligible
initial velocity, plasma electrons from z < 0 , being
reflected there by the layer electric field. At 2 = 2
the same occurs with the roles of ions and electrons being
reversed.
Flux and energy conservation require
n . u. ~ n u = o .1 1 0 0 1
n u = n u = oe e 1.1
2m. u. / v
" V 1 = e(vo - v)2
m ue e = e V
where V is the potential at 2 = 0 and V = 0 at 2 = 2
The meaning of the other symbols is obvious.
The Poisson equation together with the requirement of
zero electric field at 2 = z may be integrated once to
give
(1)
where E is the electric field and
.2 2= —-\/2e m (la)
(lb)
If E = 0 at V = VQ (z = 0) , equation (l) shows that
<j> = u<p. , which is the Langmuir condition (Cf section 6).
Equation (l) may be integrated numerically once more to
give V as a function of 2 • Fig. 1 illustrates qualitativ-
ely the result. At small V the two square-roots containing
V nearly cancel, and it is then found that
V -4/3 2/3 1/3
e e(lc)
which is equivalent to the famous Child* s law for c. diode
(See e.g. Hemenway et.al., 1967, p 109). Similarly, when
V « V,
(V - V) ~4/3 2/3 1/3 (Id)
Thus, a double layer may be described as approximately
equivalent to two simple diodes in series, one electron and
one ion diode (with an ion emitting anode instead of an
electron emitting cathode) or one more complex diode with an
electron emitting cathode and an ion emitting anode. Due to
the Langmuir condition cp. m. = <p m equations (lc) and (id)
are symmetric.
Note that the particle emission powers of the diode
electrodes or plasma boundaries are assumed to be infinite in
the theory, the current being limited by space charge effects.
3. Basic equations of field-aligned currents.
It is expected that ionospheric double layers usually
exist in the presence of field-aligned currents in the auroral
zones. A reasonably complete study of double layers should
include the transition (in time and space) from the plasma
state to the double layer. Although the present study is by
no means complete, field-aligned currents will be considered,
both in a plasma and in double layers.
To avoid unnecessary complications the following assump-
tions are made
(i) All ions are singly charged and of identical mass, m. .
(ii) The field-aligned current is vertical in a magnetic flux
tube with height-dependent cross-section A.
(iii) All velocity distributions are thermal in a frame of
6 .
reference following the average field-aligned velocity
of the particles considered, or their macroscopic
properties of importance here can at least be expressed
in terms of an equivalent temperature.
The following notations are used
nr.
E
Index
Index
T
Q
R
altitude above ground
earth's radius
vertical coordinate, positive upwards, with
z - 0 at h = h,
o0
lenotes value at z = 0
'X - e, i denotes electrons or ions.
Ten perature in energy units
ionizations per unit volume and time
recombinations per unit volume and time
force per particle of species ct , due to collisions
with other particle species
acceleration of gravity
proton charge
Other notations are self-evident.
e. = -e = ei e
Conservation of momentum and particles then give
ru*5 du
0 Va «a O =(2)
(& (A nauaa
FT (3)
A = A J -7-r0
r = rE + h = rE + h0 (4)
v2(5)
7.
Equations (2) and (3) give
• = n_i e E - F_ -\J - (6)
with
= Ta " Va' (7)
2 1 m u_
*a = W a (9)
Both double layer and plasma solutions of these equations
will be studied. The Poisson equation with n. * n musti e
be used to obtain a double layer solution. For the plasma
solution the Poisson equation is replaced by the quasi-neutral-
ity condition n. = n = n .
4. Stationary double layers.
Since quasi-neutrality is not valid in a double layer, it
can be thought of as a vacuum region that separates two plasmas
(sometimes with different properties) from one another.
Electrons from one of the plasmas are reflected by the layer,
and ions are accelerated to a beam into the other plasma. The
roles of particles from the other plasma are reversed.
If the total potential drop VQ across the layer is of
the same order as T/e of the plasma the layer is said to be
weak. If V » T/e the layer is strong.
In this section a strong time-independent double layer of
infinite extent transverse to the z-direction (magnetic field)
will be studied. Since the layer thickness is less than a
mean free path all collision terms of equation (8) can be
neglected. The current is assumed to be upward if the opposite
8.
is not specifically mentioned.
Choose the altitude h = h such that 2 = 0 at the
lower edge of the layer. At the upper edge 2 = 2 and
V = 0 . Index 0 refers to plasma properties at 2 = 0 or
to particles coming from that plasma. Index 1 is the same for
z = z . Index (3 is a general symbol for 0 or 1 .
Index ctp refers to particles a (e,i) originating in
plasma {3 (0,1). For example, u 1 (which is a function of
2) is the linear mass velocity of beam electrons originating
above the layer. Symbols with index 0 or 1 only are
always constants. Symbols with two indices (oc(3) depend on 2 .
The total current is
ni0UiO > 0 (10)
Particles of categories (eO) and (il) do not contribute
since they are reflected. Of course, particles in the high
energy tail of the distributions can overcome the layer
potential,but this may be taken care of by regarding a suit-
able number of the beam particles as really reflected (beyond
the layer). This point of view greatly simplifies the theory.
It makes it possible to regard particles retarded in the layer
as belonging to stationary populations. Hence, u _ and ueO
vanish at the layer»
Thus, there are four groups of particles present in the
layer, namely electrons and ions from below and above. The
time-independent version of (6) applied to these particles
gives for reflected particles
dn
eO dzeO -eE - FeO (Ha)
dn.- Fu) (lib)
and for beam particles
9.
weldnel -eE - Fel
dT
dz"el (12a)
d ni0wiO dz
= nio( dz(12b)
Equations (lla, b) do not contain any temperature gradient
terms since the temperature is not affected by a reflecting
potential. However, in a layer where collisions are unimport-
ant the "parallel" temperature of accelerated particles
approximately obeys the adiabatic law
dT dn dnui an / - \ un-~- = K = (.Y - 1)T n n
(13)
where n = 2 for one degree of freedom.
In the special case when F = m g (somewhat unrealisticex a
ally assuming other terms in (8) to be negligible) these equat-
ions may easily be integrated analytically, yielding exponent-
ial decrease of reflected particle densities with respect to
potential (including gravity potential) and approximately
V and (vo ~ V) ' density decrease for beam electrons
and ions, respectively. The results are illustrated in
principle in figure 2.
The Poisson equation is
dE f n - n - n )i l eO e l / (14)
Multiplication with E , substitution of n fleE obtained from
(ll) and (12), and integration yields
T
where
f«
+ " i 0m i u i 0
10.
In the limit of" infinitely strong double layers all
T _ may be neglected. The same is valid for F and F.
since the electric field provides the dominating force.
Equation (15) then becomes identical with (l).
The main deficiency of this theory is the assumption
of thermal distributions. These cannot be maintained under
electric field acceleration in the absence of collisions.
However, thermal distributions are often assumed, and the
results describe the layer interior quite well, but for a
detailed analysis of the plasma-double layer transitions
the Boltzmann equation must be used. No satisfactory theory
of this transition region has yet been formulated. For
further details spe Torvén (1969). Exact solutions of the
collision-les? Boltzmann equation (electrostatic waves) have
been obtained by Bernstein et.al. (1957).
For our purposes the deviations from thermal distribut-
ions cause no serious problems.
5. Self-consistency conditions.
Consider equations (l2a, b). Due to (13) these can be
transformed into
dnW elel dz
= nel(-eE - P ) (16a)
dn. ..w 12 - nlO dz ~ ni0 eE - F io)
where
W.
(16b)
(17)
The boundaries (z = 0 and z = z ) of a strong double
layer will now be considered. Only in this case are the
boundaries relatively well defined.
11.
At the boundaries charge neutrality is valid. Near
we have n = ne eO
and n. ~ n._i IO since nel
and
n. have decreased very much due to acceleration and
reflection in the layer. Hence, the contribution of neland n... to the space charge near 2 = 0 may be neglected
in the case of a strong double layer. Differentiating the
Poisson equation (14) with respect to z and using (lla)
and (l6b) then yields
d2E
dz iO
eE + Fe0 e0>
(18)
Since n = n. at 2 = 0 equation (18) shows that E cane i
only increase with increasing z (into the layer, that is)
if
2m.u .-i i0 e0
(19)
z = 0
It is easily seen from (lla) and (l6b) that
dn /dz = dn. /dze i
at z = 0 if (19) is just satisfied.
The same arguments applied to the region near
in the layer gives
z = z.
eE+Fm u 7 ̂e el el + Til \ eE-F,
el (20)il
z - z.
These are necessary but not sufficient conditions for
the stability of a double layer. Before beam particles (in
being) reach the layer they must be preaccelerated to
approximately sonic velocity in order to maintain the necess-
ary space charge within the layer. If their velocity is
lower, any tendency for a rising electric field will be
counteracted by the resulting space charges.
12.
Bohm (1949) developed a similar criterion for wall
sheaths
m .u .** > T
neglecting the influence of the ion temperature. Persson
(1968, 1969) has pointed out that the Bohm criterion for ions
and a similar criterion for electrons, namely
2m u > • .e e i
must be valid for free sheaths. If the influence of the beam
particle temperatures is accounted for, the Bohm criteria are
replaced by the self-consistency conditions (19) and (20).
See also Andrews :>nd Allen (1969), and Torvén (1969).
If the double layer is not very strong all the four
categories of particles must be included in the analysis.
Furthermore, the trick of assuming the reflected populations
to be at rest cannot be upheld, which means that retarded but
not reflected particles will be subject to the adiabatic law.
The result of these facts is that the limiting kinetic energies
may differ from those given by (19) and (20). However, that
is a much more complicated problem.
The self-consistency conditions are further analyzed in
section 8.
6. Pressure balance and the Langmuir condition.
Equation (15) describes the pressure balance through the
double layer. The integral /n F 6z is usually very small
and can be neglected. In a strong layer the terms in
P(z) - P(o) are of four different orders of magnitude. At
z = 0 the P(0) hierarchy is
il xl nel el neO eO.ÄT.rtiO xO
2 2.-m.u.- « n ,m u .xO x xO el e el
(21)
13.
provided all temperatures are of the same order of magnitude.
The third inequality is due to the self-consistency criter-
ion (19).
At 2 = 2 the corresponding hierarchy is obtained if
e and i , and O and 1 are interchanged.
If E = E equation (l5) gives
P(O) = P( 2 l
implying that
n . _ m . u. )lO i lO
= z.
~ In .m u )\ el e el/z = 0
(22)
(23)
Since these beam particles habe been accelerated through
the s«
equal
the same potential drop V their kinetic energies must be
( m.u.V i i
2iO 2 = 2.
Hence (»J2 = 2. (»J
2 = 0
and2 = 0
n . _ u . _ I. ,lO lO i /_ /_1 = —— = ( m /m.
n _u , I_ V e' i
1/2
(24)
where I. and I are the ion and electron current densities,i e '
respectively, through the sheath.
Equation (24) is the Langmuir condition which was derived
for an extreme case (zero temperatures) in section 2. However,
here it is seen that (24) is valid only for relatively strong
double layers, i.e. when eVQ » all T . If eV is of
the same order as some T/vO (21) and hence also (23) are notap
necessarily valid. In practical cases the validity of the
Langmuir condition can only be tested by analysis of the
pressure balance (15).
14.
7. Plasma rarefaction and compression instabilities»
In this section a plasma (acoustic) instability will be
considered, that should tend to develop into double layers.
Since a plasma is considered here, n. = n = n .
Equations (6), (13) and (17) applied to electrons and
ions give
w3n 3*
it
Wi 32 ~ s 1 / 3t
The sum of these may be written
W 3n3z
= -nF - a*3t
(25)
with
W = w + w.e i
= r(? + T.} -1 \ e 1/
2 2m u - m.u.
e e i i
F = F + F.e i
+ # . = n ( m u + m . u . )i s e e i i /
(25a)
(25b)
(25c)
F and F. are given by (8).
Consider a disturbance r\ in the equilibrium charged
particle density n so thats
n = n + TIs ' (26a)
Since F and W may depend on25a-c)
F = F + dFs
n (cf equations 8,
(26b)
W = W + dWS
(26c)
15.
The stationary solution is given by
Wdn
s 02- n F
s s(27)
Introducing (26a-c) into (25), regarding (27) and
omitting second order terms in the disturbance quantities,
yields
dnTT=-T)F - n dF - W ̂ - dW - :3t ' s s 3z 3z
(28)
A density instability will occur if 3$/3t enhances
a density deviation r\ •
Case 1 F = dF = 0 . According to (27) this gives
3n./3z = O . Equation (28) is then3̂
= - w3t
(29)
Figure 3a shows a density depression and a density enhancem-
ent. The arrows at the slopes of these indicate the direction
of mass flow acceleration 3$/3t when W < O . For example,
in the depression (where TJ < 0) 3$/3t is negative to the
left where 3T)/3Z < 0 , giving an outflux in the frame of
reference moving with the plasma, so that $ = O . Thus, in
this frame the center position of the disturbance region will
be at rest, but the disturbance amplitude will increase if
W < 0 (current associated with supersonic particle flow).
Obviously, W < 0 can only occur if the current density
is sufficiently large. Zero current gives W = Y(T 6+T^) > 0
in the frame moving with the plasma. When W > 0 (subsonic
flow) the arrows in figure 3a are reversed as shown in figure
3b and all disturbances are counteracted.
Thus, the instability condition is
(30)
16.
where2 T = T + T.
e i(31)
It should be pointed out that
2 2m.u. << m u1 1 e e
by a factor m /m. in the frame moving with the plasma so
that the term m.u. is actually meaningless within the
accuracy of the present theory.
The condition (30) was originally derived by Carlqvist
(1972), using a different but equivalent formalism. The
equality sign in (30) implies W = 0 . The plasma is then
unstable for rarefactions since W becomes smaller as the
rarefaction developes. However, compressions can only be
unstable when W < 0 , initially. The growth of a compress-
ion instability cannot proceed beyond the density correspond-
ing to W = 0 . If W = 0 initially, a compression will make
W > 0 so that all instabilities are counteracted.
The case when W < 0 is discussed further in section
10 f.
Case 2 F is given by equations (8) and (25b) but
with all particle encounters excluded except Coulomb collisions,
The latter contribute nothing to F but they help maintain
thermal distributions. Thus,
/ 2 2 \ 1 dA= mg - ( m u + m . u . ) — —
* \ e e i i / A d z
(32)
where
m = m + m.e i
In this case the dependence of
evaluated. Introduce
2 2U = m u + m.u.e e ii
u = « ( m u + m.u . )m \ e e i i/
v = u. - ui e
I = e n v
(33)
F and W on n must be
(34)
(35)
(36)
(37)
17.
The purpose of the present calculation is twofold:
a) to obtain an instability condition, and b) to find
the direction of plasma motion in rarefaction and compression
regions. It can be expected that the right hand side of (28)
will contain (to first order) some terms proportional to r\ ,
and other terms proportional to 9r)/3z . The latter should
give the instability condition, since the r̂ -terms are anti-
symmetric with respect to the center of the disturbance region
and thus deduct as much influx on one side as they add on the
other, with no net stabilizing or destabilizing effect. On
the other hand, the r\-terms give a finite 8$/Bt in the middle
where dr\/dz = 0 , thus accelerating the disturbance one way
or another.
Equations (34) - (37) give
2 2U = mu + j/n
J =m m.Ie 3.
2me
(38)
(39)
It is convenient to work in the frame where u = 0 •
The result of the analysis will then show whether the dis-
turbance tends to move faster or slower than the stationary
plasma. Equation (38) may be differentiated to give
dU = - 2 Jn/n3 = - 2 U -n
(40)
where r\ = dn according to (26a). Now, (25a), (31), and
(34) give
dW = 2 Y dT - dU = 2(yxT + U) —n
Equations (32) and (30) yield
dF = An dz
Now, (32) with F(28) to give
(42)
= F , (4l) and (42) may be introduced in
18.
= Wat w 32
s
L = Fs[3U + 2(*2- 1)T] - 2Ws 2 S
(43)
(44)
Hence, the instability condition (30) is valid also in
this case.
To determine the direction of plasma motion within the
disturbances the sign of the quantity L must be found.
Assume that a certain instability region 2 < 2 < 2
can be defined such that W < 0 there. Equation (27) then
implies that F < 0 there, provided 3n /32 < 0 . This iss s
expected in the ionosphere, of course. Furthermore, W = 0 ,F = 0 and hence L = 0 at 2 = 2. and 2 = 2_ . Tos 1 2determine the sign of L in the instability region we express
L as a function of U , noting that T and W are also
functions of U . Equations (13) and (40) may be integrated
to give
UT = U T- = constant (45)
provided x = 2 . Equation (44) may then be written
L = - i m U2 + 3 rag U + 36 mg
1 8 dAd2 (46)
Noting that tL = 2 y T- •• = 6 T it is then found that
dL _ d^LdU ~
= 0 (47)
U3= - 6 mg/U < 0 (48)
at U = t L . Hence, L behaves in principle as shown in
figure 4. It has an inflexion with zero derivative at U = IL ,
being positive for U < U- and negative in the instability
region where U > U- .
19.
Two cases may now be distinguished, according to the
sign ox 3n/dz , that is above or below the F -maximum.
Equation (27) shows that if 8n/dz > 0 (below F -max) then
F and W have opposite signs. At the boundary between
stable and unstable plasma W = 0 and hence F = 0 . Ons s
the other hand, inspection of (25a) and (32) shows that W
and F must then have the same sign if T is unaffected by
channes in U , or if it obeys the adiabatic law (13). This
paradox is resolved b\ noting that all collisional terms have
been left out in (32). The recombination term is particularly
important here. Thus, the present theory does not cover this
case.
The other case is the topside ionosphere, where Bn/3z< 0.Equation (27} ;. lien shows that W and F have the sameM ' s ssign in agreement with (32). Since L < 0 in the instability
region it is seen from (43) that plasma in compressions is
accelerated upwards (in the direction of F since 3$/Öt> O)
and downwards in rarefactions. It is physically sensible that
dense regions should be accelerated in the direction of F
and dilute regions in the opposite direction.
8. The transition from instabilities to double layers.
Carlqvist (1972) shows that a plasma rarefaction instab-
ility of the kind studied in the previous section tends to
develop into a double layer. Here the relationship between
the double layer self-consistency conditions (19) and (20)
and the plasma instability condition will be considered.
Suppose that the electron velocitv is such that
m u = YT + XT.e e ' e i (49)
while the ions are at rest. If the arbitrary constant X is
put equal to y an instability occurs. Equations (lib) and
(16a) with (4 9) inserted now shows that
20.
eE +
eE -el
= \ (50)
Z = 2.
when n ., = n., and dn ^/dz = dn.,/dz . Hence, the selfel il el il
consistency condition (20) seems to be fulfilled for any
value of \ . However, (20) may be written
2m u Ae el
YTel1 +
eE-F.(51)
due to (25b). If \ < 1 (51) cannot be satisfied above a
certain maximum electric field. Hence, the growth of E will
be limited. On the other hand, if X £ 1 there is no limit-
ation on the growth of E as long as (l8) is valid. Hence
a sheath of unlimited strength can only exist if
m u ,e el 'el il
(52)
It is easily shown that (19) may similarly be reduced to
miui0 e0 (53)
These two conditions are necessary for the existence of strong
sheaths.
It should be observed that the manipulations made here
are not obviously allowed if F = F. + F = 0 since then the
density gradients vanish in the plasma and hence also at
2 = 0 and 2 = z1 • However, equation (18) and its correspond-
ence valid at z = z are so reduced that (52) and (53)
follow at once when F = 0 .
Thus, it has been shown that when a plasma rarefaction
instability occurs the double layer self-consistency cond-
itions are amply satisfied.
Laboratory experiments corroborate the view that rare-
faction instabilities develop into double layers. Torvén
and Babic (1972) have found potential peaks in a low-density
discharge just before the current has reached the critical
value for a double layer. This was predicted by Alfvén and
21.
Carlqvist (1967) and Carlqvist (1972). Torvén (private
communication) has found noise in the typical frequency
range of ion-acoustic waves when double layers are formed.
9. Stability of double layers.
Although conditions (52) and (53) are necessary for
the existence and stability of double layers they are not
sufficient . No sufficient conditions are known at present.
Only a few remarks will be made here.
In the laboratory it is observed that quite stable double
layers occur only at constrictions in the discharge tube.
In space a diverging magnetic field may perhaps serve the same
purpose. The gravitational force may also help to anchor a
sheath at some altitude.
However, it is also observed in the laboratory that
double layers often appear and disappear at a regular frequency
(10 - 100 kHz) at the same place so that it looks quite
stable to the bare eye. This may be explained as a wall effect.
When the layer appears the walls are bombarded with energetic
particles that release neutral molecules from the walls. When
these are ionized the current-carrying capacity of the plasma
increases and the layer breaks down. Then no energetic
particles reach the wall, no more neutral particles can be
ionized and the layer reappears (Torvén, 1968).
In space no such effect can occur, so the double layers
may be more stable.
Remembering that in the frame moving with the plasma
n.u. « n uii e e
(30) may be written
2 2and m_.u_. « m u the instability conditioni i
l o m DA = An T $ - ~ -
2e Y
if equation (10) is used.
(54)
22.
In the polar wind ionosphere A is varying with altitude,
qualitatively as shown in figure 5. (The situation when there
is no polar wind is discussed in section 10.) To the left is
shown a case when the field-aligned current is upward.. The
plasma above a topside ionospheric double layer would be un-
stable, because if (54) is just fulfilled at that layer, it
is amply fulfilled above it. However, below a layer in the
F-region the plasma can be stable since the current is carried
by the beam electrons there. The plasma electrons do not
have to carry any current to speak of.
The conclusion is that for upward electric fields stable
double layers would only exist in the F-layer. For downward
electric fields stable double layers may exist in the topside.
This agress both with laboratory experiments (Torvén
and Babic, private communication) and with observations in
space. Measurements of parallel electric fields have only
been made so far at F-region altitudes. Only downward electric
fields have been found with the double probe technique (see
e.g. Mozer and Fahleson, 1970). These would occur in the
unstable plasma where resistivity would be anomalously high.
Furthermore, the only observations that may more directly
be interpreted in terms of double layers (Albert and Lindström,
1970) clearly indicated upward electric field within the layer.
The stability of a double layer may be further enhanced
by plasma convection around the field-aligned currenc-carrying
plasma. As pointed out by Block (1969) and by Carlqvist and
Boström (1970) the electrostatic equipotentials in a double
layer must turn upwards to become field-aligned around the
field-aligned current (figure 6). This means that there is
convection around the field-aligned currents. The inertia of
this convection should tend to prevent break-down of a double
layer. The existence of such convection is strongly indicated
by the field-reversals observed by Cauffman and Gurnett (1971).
The Langmuir condition requires supply of electrons and
ions in the right proportion from both sides to the double
layer. If that is not possible the layer will be charged.
The resulting external electric field will accelerate it to
23.
an equilibrium velocity such that the Lanqmuir condition is
fulfilled in the frame of the layer. For example, if the
high potential plasma is unable to supply sufficiently many
ions the layer moves towards that plasma, increasing the ion
velocity and flux relative to the layer.
Note that this effect is also of importance for the
self-consistence conditions.
Moving sheaths may live as long as they can move un-
hindered by for example a too dense plasma, where they may
stop and perhaps become stationary.
Moving double layers (usually called striations) are
extremely common in laboratory discharges (see Olesen and
Cooper, 1968).
It is observed in the laboratory that when a double layer
has appeared somewhere, there is a strong tendency for another
layer to appear at about a thermal relaxation mean free path
of the beam electrons towards the anode. This distance may
be typically 10 - 50 cm. The presence of a b^am of supra-
-thermal particles creates good conditions for another11 layer-creating" instability when these particles are thermal-
i2ed and become part of the background plasma.
The long thermal relaxation distance, mentioned above,
is calculated assuming binary collisions. However, wave-
particle interactions may shorten this distance considerably.
A striking example of this has been reported by Morgulis
at.al. (1967, 1968) who also found that the dispersion of
the beam electrons was accompanied by noise in the GHz range.
In any case, when the beam particles, sooner or later,
are thermalized and dispersed through interaction with the
background plasma, this plasma must take up the beam pressure.
This is greater than the plasma pressure already when the
particles enter the double layer, due to the self-consistency
conditions. Hence, after acceleration in the layer it is
still greater. If the interaction distance is short, the
plasma must then be pushed back so that the layer is widened.
Then the layer potential rises, the beam pressure increases
still more, and so on. The situation is unstable and the
24.
double layer roust break down very soon, perhaps to reappear
again and go through the same instability. However, if the
interaction distance is very long (as for auroral particles
where the layer may be hundreds or thousands of kilometers
above the aurora) the plasma or neutral pressure may be
sufficiently high to balance the beam pressure so the double
layer may be stable.
10. Double layer altitudes.
Double layers should preferably occur where A given
by (54) exhibits a minimum. Such minima may exist when the
relatively dense and hot plasma sheet is in contact with the
ionosphere, and when there is no polar wind, as shown below.-4 2
Field-aligned current densities up to about 10 A/m
have been observed (Zmuda et.al. 1970) at altitudes above
1000 km. Assuming the maximum current density that ever-3 2
occurs is of the order of 10 A/m it may be concluded
from (54) that the maximum electron density where a- double
layer can exist is about 10'- 10 electrons/cm provided the
temperature is of the order of 1000 K. However, still higher
densities may be conceivable if there are particles trapped
between the layer and the magnetic mirror below, as observed
by Albert and Lindström (1970). This may be explained as
follows.
Assume that a double layer is created because of an in-
stability at minimum A . Assume also that the current is
upward. Energetic electrons with large pitch-angles may then
be trapped below the layer as described by Albert and Lindström,
If the "parallel" kinetic energy of these electrons is compar-
able to the potential drop across the layer they may contribute
significantly to the space charge within the layer, since they
penetrate deep into it before being reflected. Their number
may transiently increase at first, thus increasing the negative
space charge region within the layer. As a compensation the
25.
positive ion flux through the layer must increase. Hence, the
layer must move downwards (cf. section 9) towards more dense
plasma, until the number of trapped particles is saturated due
to equal scattering of electrons into and out of the trapping
pitch-angle cone. During the downward movement of the layer
the trapped electrons may be Fermi-accelerated but that is
probably a small effect.
Equation (54) may be regarded both as a plasma instabil-
ity condition and as an approximate condition for double layer
self-consistency. The latter concerns the space charges within
an existing (but possibly moving) layer. Then, the direction
of motion of the charged particles through the layer does not
matter but only the time they spend in the layer. The self-
consistency conditions (5 2) and (53) were derived assuming
that the energies of practically all reflected particles are
small relative to the layer potential. If not, the charges
of reflected particles from the opposite side must be consid-
ered. The trapped particles were assumed to have non-negligr-
ible energy. Their contribution may be accounted for in (54)
by a larger value for the current j , thus including a proper
fraction of the flux of trapped electrons, even though these
do not contribute to the actual current, of course. Hence,
for a given current the electron density n may be larger than
given by (54) if electrons are trapped below the layer. This
may explain the low double layer altitudes found by Albert and
Lindström (250 - 300 km).
If the "parallel" energy of all the trapped electrons is
exactly equal to the potential drop across the double layer
the current j in the formula (54) for A should be replaced,
by
= 3 2e (55)
where (p is the one-way total flux of trapped electrons
within the flux tube and j is the real net current.
Obviously the space charge balance in a double layer
depends on the velocity distributions of the particles involved.
26.
Addition of energetic trapped particles may be equivalent to
raising the corresponding temperature of ambient plasma part-
icles. According to laboratory experience weak layers are
most stable, whereas strong layers are usually explosive, i.e.
their voltage increases explosively. The high double layer
potentials ( - 100 volts) reported by Albert and Lindgtrom
(1970) correspond to very strong layers in terms of normal
ionospheric temperatures. However, the trapped energetic
particles may make them equivalent to weak layers, which may
explain their good stability.
Bearing in mind that double layers apparently may appear
at quite low altitudes, the most likely altitude may neverth-
eless be much higher up where A is minimum. The altitude
where that occurs depends strongly on the ionospheric-exospheric
model. The simplest model would be a stationary isothermal-2ionosphere with gravity decrease as r
3crease as r . This gives a A-minimum at
and flux tube in-
ho = ro1250 M - 1 (56)
6 T
where r = earth's radius, M = ion mass in atomic mass units,
and T, = average ion-electron temperature in degress K .
If there is an outflux of ionospheric particles, such
as in the polar wind, the situation is different. Using the
exospheric polar model of Lemaire and Scherer (1970) the
critical current j in a flux tube of one m cross-section
at 2000 km altitude is given as a function of altitude in
figure 7. The current j is calculated from equation (54)
for each altitude, using the temperature and density given by
Lemaire and Scherer. The corresponding current density I
at 2000 km is then calculated and given in the figure. It is
seen that quite low current densities give plasma instabilities
above a few thousand kilometers. However, there is no minimum
Ic (corresponding to minimum A ) in a polar wind flux tube.
This is natural since the pressure tends to zero at infinite
altitude.
In figure 8 the well-known solutions of Bank's and
Holzer's (1968) polar wind equations are shown, in principle.
27.
These equations are equivalent to the time-independent
equation (25) of the present paper, with F given by (8) and
(25b), and with 2ero current. Curve A - A of figure 8 is
the polar wind solution, which qualitatively best resembles
Lemaire's and Scherer's model.
However, there are also breeze solutions given by curves
of type C - C . These may exist on closed field-lines if
there is magnetospheric convection that is continually remov-
ing plasma through field-line merging at the day-side magneto-
pause, or if there is plasma with finite pressure on open
field lines, for example the plasma sheet. In such a case A
is minimum near maximum Mach number on curve C • If a
sufficiently strong current is turned on a switch to a curve
of type A' - P - B occurs with minimum A at P . A layer
will then appear there, with stable plasma on both sides
regardless of the current direction. The current densixy I
will be limited to
= e n ( (57)
where index P denotes values at minimum A .
This is perhaps the most likely place for stable double
layers. The theoretical solutions then predict double layer
altitudes of about 1000 - 3000 km, depending on the ion
composition and the temperatures at these altitudes.
11. Anomalous resistivity through random unstable double layers.
It has often been argued that E caused by anomalous
resistivity could be responsible for so called "monoenergetic"
particle fluxes, double peaked velocity distributions of
auroral particles, and other similar phenomena. Anomalous
resistivity is often thought of as being due to turbulence.
However, it is difficult to see how such electric fields could
accelerate particles, since the turbulence must impede the
acceleration through an increased effective collision frequency.
28.
That is the mechanism through which the resistivity is
made anomalously high. In any case it could not cause extra
velocity distribution peaks or unusually sharp high or low
energy cat-off (Block, 1972).
Double layers, on the other hand, are limiting the
current through space charge effects. They are also laminar.
All particles getting through the layer are accelerated. Any
desired velocity distribution can be explained by a large
number of layers with suitable injection of particles into the
flux tube between them.
The layers must not necessarily be stable. Fluctuating
or flickering layers, appearing and disappearing at random,
will also accelerate particles. A region with a large number
of such unstable layers could have an essentially constant
or slowly varying total potential drop.
The theory of plasma instabilities presented in section
7 is one-dimensional, and as such it can only handle laminar
flow. Carlqvist's (1972) theory of the development of rare-
faction instabilities into double layers is also one-dimension-
al. It is not clear if a complete 3-dimensional theory would
predict turbulence or laminar flow. However, if nature is
kind enough to maintain laminar flow, it seems that double
layers would inevitably appear and disappear in a disordered
random way in an extended unstable plasma region. This would
then cause anomalous resistivity that is not turbulent and
that could explain observed auroral spectra.
Figure 9 illustrates how anomalous resistivity could
develop in a region around a A-minimum, (for example around
the altitude given by (56), which may be somewhere between
about 1000 and 5000 km) provided no turbulence destroys the
laminar flow. Suppose the current is rising gradually. A
rarefaction first develops at minimum A when W = 0 there.
The plasma from the rarefaction wells up on both sides
(figure 9a). This causes A to increase a little there,
allowing the current to rise a little more. The rarefaction,
of course, becomes a double layer but the minimum value of A
in the plasma has increased, and that determines the plasma
29.
instability condition. When the current has risen W = 0 in
a small region around the layer so that one or two new rare-
factions can develop, more plasma wells up and the minimum
A is increased still more, corresponding to a still higher
current in the flux tube. The region W = 0 gets still
larger and so on. The end result is a large number of layers
with W = O in all the plasmas between them (figure 9b).
This could be characterized as a region with average W < O
in terms of the plasma distribution before the instabilities
developed.
If the double layers are unstable, some are breaking
down while others are built up, all the time.
As was pointed out above a large: number of double layers
may produce any spectral distribution of precipitating particles,
However, it is also of interest to compare energy distributions
produced by laboratory double layers with observed auroral
distributions. Figure 10 shows an energy distribution obtained
from one weak double layer in a laboratory discharge (Andersson
et.al. 1969). It is remarkably similar to the spectra
observed by Westerlund (1969) at 400-800 km altitude. The
high energy peak in figure 10 is due to beam electrons from
the double layer, and the low energy peak is plasma electrons
trapped between the layer and a reverse electric field at the
anode.
This, of course, is just an example of what double layers
can do. Westerlund*s low energy electrons may for example
ha ,-s- been injected between a strong upper layer and a weaker
lower layer further down.
30.
12. Depletion of plasma in the upper ionosphere.
Block and Fälthammar (1968) have shown that a field-
aligned current can more or less deplete the topside ion-
osphere if the ratios between electron and ion current at
the top and bottom (F-region) of the flux tube differ from
one another. The stationary electron density altitude
distribution is such that the net outflux of plasma exactly
covers the height integrated difference between ionization
and recombination.
If a double layer exists high up in the topside
ionosphere the plasma current just below the layer is carried
by one kind of particles only. If the beam current is not
dispersed by wave-particle interaction the beam particles
will not be stopped until they reach the E-region, and they
do not contribute to the topside electron density. In the
E-region the rate of recombination is high, so the precipit-
ation will not very much influence the topside density.
Hence, for upward current the net outflux of plasma
from the topside ionosphere is given by the upward ion flux
at the double layer minus the negligible upward ion flux in
the F-layer. According to Block and Fälthammar (1968) the
most drastic reduction of topside density occurs if the
current carried by the upward moving ions at the double layer
is about 3 y 10 A/m , provided the ionization rate is as
high as that due to daylight solar UV radiation. In case
the ions are protons at the double layer the total current,—4 2
including the beam electrons, would be about 10 A/m for
the most drastic density reduction. Of course, some reduct-
ion effect occurs also for weaker currents. At night, the
reduction at stationary state is larger at a given current,
because of the lower rate of ionization.
For downward currents the density reduction is less
than for upward currents by about a factor three or four
according to the theory by Block and Fälthammar (1968).
The time constant for reaching stationary state after
a depleting current is switched on is of the order of a few
31.
hours. However, most of this time is used up for reduction
of the lowest parts near the F-region maximum. Reducing the
topside above, say 500 km, takes only a few minutes.
If there is an extended instability region and not
just a double layer the same reduction will, of course, also
occur below the region of plasma instability or afiomalous
resistivity. However, in this case a net upward flux will
also occur within the instability region, due to the upward
acceleration of compression regions according to the theory
in section 7.
During the removal of plasma from the flux tube (before
stationary state has been reached) the total potential dröp'
must increase or the current must decrease due to the diminish*
ing apparent conductivity. However, complete depletion of
plasma can never occur since a current is always closed. New
particles must be carried dqto the flux tube due to perpendic-
ular currents even if there is no convection whatsoever.
Thus, there is no particle depletion problem, contrary to
what has been claimed by e.g. O'Brien (1970). Such a problem
exists only in models where auroral particles arc assumed to
be squeezed out of the flux tube, or in "leaky bucket models".
If the field-aligned current (which may be in the form
of a current sheet above an auroral arc, see figure 11) is
moving sideways, for example north-south, the topside
ionosphere may be depleted over a large region. This may
be important for the explanation of the nightside ionospheric
ttough..
13. Concluding remarks.
Many authors have considered plasma instabilities
associated with field-aligned currents in the upper ionosphere
(Swift, 1965, 1970; Kindel and Kennel, 1971). A rarefaction
instability has been considered here and Carlqvist (1972) has
shown that this instability leads to formation of double
32.
layers. When this double layer is formed certain self-
consistency conditions for double layers are already amply
fulfilled.
The most likely altitude for stable double layers is
1000 - 3000 km on closed field lines and perhaps a few
thousand kilometers higher up on open field lines, if they
reach the hot and dense, expanded plasma sheet before
substorms. The stability of the double layers should be
enhanced by the plasma convection around the field-aligned
current. This kind of convection has been observed as
electric field reversals by Cauffman and Gurnett (1971).
According to the observations by Albert and Lindström
(1970) stable douole layers can also exist at F-region
altitudes if the current is upward. Measurements of E
in the F-region (Mozer and Fahleson, 1970) show only down-
ward electric fields» This agrees with theory and laboratory
experiments since the F-region plasma should be unstable if
double layers occur in the F-region at downward electric
field. This unstable plasma should then develop anomalous
resistivity. If this anomaly is caused by turbulence it
should not produce double peaked or sharply cut-off precipit-
ation spectra, It is even questionable if it can at all
accelerate auroral particles.
However, if laminar flow is maintained in the unstable
plasma, double layers should arise and disappear at random.
Any kind of precipitation spectrum may be produced, depending
on the details of the current system»
When double layers (stable or unstable) are formed, a
reduction of the topside ionospheric electron density should
occur, as described by Block and Fälthammar (1968). If there
is a region of anomalous resistivity in the topside, due to
randomly appearing and disappearing double layers, an
additional mechanism of topside density reduction sets in,
due to upward acceleration of the compressed plasma regions.
Thus, large amounts of plasma may suddenly move up from
the ionosphere into the magnerosphere when strong field-
aligned currents are switched on, for example near midnight
33.
during auroral breakup. Large fluxes of particles should
at the same time be accelerated through the double layers,
both upward and downward.
Finally, it is not difficult to understand how both
ions and electrons can be accelerated in the same direction
by a parallel electric field (see O'Brien, 1970). Figure 12
shows a simple potential distribution taylored for this
purpose. Block (1969) proposed that a potential asymmetry
between conjugate points can do that for one hemisphere.
This should occur in nearly all cases since perfect symmetry
should be rare. However, figure 12 shows how it is possible
to get simultaneous ion and electron precipitation at both
conjugate points, provided that the particles are not too
much dispersed by wave-particle interactions on their way
down towards the ionosphere.
Réme and Bosqued (1971) have suggested a model of this
kind where the parallel field is downward at low altitude,
due to the negative space charge deposited by high energy
electrons in the E-layer.
Acknowledgements.
The author is indebted to C.-G. Fälthammar and, in
particular, to P. Carlqvist for stimulating discussions and
positive criticism.
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3055.
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Lemaire, J., and Scherer, M., 1970, Planet.Space Sci.,
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Ionized Gases. Vienna, Austria.
Morgulis, M.D., Korchevoi, Yu. P., and Dudko, D.Ya., 1968,
Zhurnal Tekhnicheskoi Fisiki. 38. 1065.
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37.
Figure 1.
Figure captions.
Distributions of potential, electric field,
and space charge in a typical plane double
layer.
Figure 2.
Figure 3.
Approximate distributions of reflected and
beam particles in a double layer if the potent-
ial drop is about 100 kT/e and the particle
density at z = z is 80 % of that at z = 0 .
Development of rarefaction and compression
disturbances.
a) The instability condition is satisfied, and
plasma is accelerated from rarefactions and
towards compressions, as indicated by the arrows,
b) The instability condition is not satisfied
so the disturbances are suppressed by the plasma
flow.
Figure 4.
Figure 5,
The behaviour, in principle, of the quantity L
given by equation (46). L < O in the plasma
instability region U > U , showing that plasma
in compression instabilities is accelerated
upwards in the topside ionosphere.
Stable and unstable plasma regions. To the left
is shown that the topside polar wind ionosphere
is unstable above a certain altitude for upward
currents, but stable double layers and plasma
can be expected in the F-region. To the right is
shown how the stability conditions are reversed
for downward currents. If there is no polar wind
the topside is stable regardless of current
direction, (not shown in the figure).
38.
Figure 6. Equipotentials from a double layer turning
upwards around a field-aligned current sheet.
Figure 7. Critical current densities at 2000 km altitude,
for which double layers and plasma instabilities
can be expected, as a function of the altitude
where marginal instability occurs.
Figure 8.
Figure 9.
Solutions, in principle, of the polar wind and
field-aligned current equations (Banks and
Holzer, 1968). The wind solution is given by
A' - P - A , and the breeze solutions A' - P - B
and C - C . Stable double layers may be
expected at P when A' - P - B applies.
Unstable layers with anomalous resistivity is
expected on P - A in the polar wind.
Development of anomalous resistivity through
unstable double layers in a region around minimum
A , i.e. the weakest point of the plasma in the
flux tube.
Figure 10. Electron energy distributions observed in a
laboratory discharge with a double layer (sheath)
according to Andersson et.al. (1969). The big
peak around about 12 eV in the second lowest
curve is produced by the layer, but further down
the tube (+ 1 cm and + 6 cm) this peak is
gradually dispersed and the beam particles are
thermalized.
Figure 11. Depletion of topside electron density by a
north-south moving sheet current associated
with an auroral arc.
Figure 12. Potential distribution for a parallel electric
field that may simultaneously precipitate posit
ive and negative particles.
Electronenergy
distribution
2.1010+
1 10
eV
Discharge current 0.9 Amps
Pressure incathode vessel 0.87 m torr
(•6 cm)
(•1cm)
anode side ofsheath (tubeorifice 0cm)
cathode sideof sheath(1-2cm)
> | i i i I5 10 15
i i i i I i iI20 eV
Fig. 10.
Not yetdepletedtopside
Depletedtopside
Sheet current
Double layer
Auroral arc
Motion of sheet current andauroral arc.
Fig, 11.
TRITA-EPP-72-07
Royal Institute of Technology, Department of Plasma Physics,
Stockholm, Sweden
POTENTIAL DOUBLE LAYERS IN THE IONOSPHERE
Lars P. Block
April 1972, 51 p. incl. 12 illus., in English.
In this paper the acceleration of auroral particles in double
layers is considered. The theory of double layers, including
the instability leading to double layers, is presented. It
is shown that both stable and unstable double layers should
occur in the topside ionosphere above 1000 km altitude and
also in the F-region for observed field-aligned current
densities. The stability of double layers depends on the
current direction in some specified cases. Double layers
accelerate particles both towards and away from the earth.
Unstable layers may cause anomalous resistivity. When double
layers are formtd, plasma is accelerated upwards from the
topside ionosphere. This effect may be very important for the
explanation of the ionospheric trough.
Key words; Auroral particles, field-aligned current,
ionosphere, topside, F-region, parallel electric field,
anomalous resistivity, double layer, sheath, instability.