Download Wave behaviour near critical frequencies in cold bi-ion

Planet. Space Sci., Vol. 43, No. 5, pp. 625-634, 1995
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0032-0633(94)00197-9
Wave behaviour near critical frequencies in cold bi-ion plasmas
P. Thompson, M. K. Dougherty and D. J. Southwood
Space & Atmospheric Physics, Imperial College, London SW7 2BZ, U.K.
Received 25 April 1994 ; revised 5 August 1994 ; accepted 13 September 1994
Introduction
We present
a discussion on some potentially
important
features of wave propagation
in a cold plasma containing
more than one species of ion. In practice, few plasmas
actually contain a single type of ion. Laboratory plasmas
are almost inevitably contaminated to some degree by
material from the container walls and in planetary
environments, the ionosphere as well as the magnetosphere also contains a variety of ion species, depending
on the composition of the planetary atmosphere. Differing
scale heights, ionization rates and other processes may
lead to a dominance of a particular ion species at a particular height or latitude, but in general there will be a
mixture of ion species at any particular point. Planetary
magnetospheres, which were once believed to be predominantly populated by protons, are now known to be
populated by a variety of species of heavy ions. For, e.g.
Jupiter’s magnetosphere contains a variety of heavy ions
of which its moon IO is the source (see for e.g. Bagenal
and Sullivan, 1981 ; Bagenal et al., 1980 ; Shemansky,
Correspondence to : P. Thompson
1980; Brown et al., 1983). In the Earth’s magnetosphere,
the substantial presence of heavy ions was a fairly late
discovery (Johnson et al., 1974). The terrestrial magnetospheric heavy ions are largely of terrestrial origin and
are brought from the planetary ionosphere by anomalous
heating processes in the topside aurora1 ionosphere.
More recently, in space plasma physics, the study of
comets (both real and artificial) has produced important
new motivation for the interest in waves in multi-fluid
plasmas. In cometary environments, the solar wind
plasma and ionized cometary material interact directly
with each other. The interaction of heavy ions with the
solar wind was also studied by means of the AMPTE
releases of barium (Valenzuela et al., 1986) in the magnetosheath and the solar wind. In either of the latter cases
there is relative motion between the different ion species
and as a result plasma waves may be driven unstable.
Indeed, both the cometary magnetic field environment
and the AMPTE ion releases were marked by the presence
of low frequency noise in the magnetic field measurements
(Ltihr et al., 1986 ; Gleaves et al., 1988 ; Smith et al., 1986 ;
Riedler et al., 1986; Neubauer et al., 1986).
Recently the existence of heavy ions and heavy charged
dust particles was revealed within the Martian orbit.
Heavy ions were detected by the Phobos spacecraft in
Martian orbit (see for example, Verigin et al., 1991 ; Lundin et al., 1990). The dust is likely to originate from
the Martian satellites, Phobos and Deimos, small heavily
cratered objects with low escape velocities. However, these
satellites do lie within the strong gravitational field of
Mars and material which escapes from them will be held
by the Martian field. This results in the neighbourhood of
their orbits potentially having a high dust concentration
and therefore a high concentration of heavy charged
material. A dust/gas torus appears to reside along the
orbit of the satellite Phobos as a result of the outgassing of
matter from Phobos and Deimos (Ip and Banaszkiewicz,
1990 ; Dubinin et al., 1990). The idea of a Phobos gas/dust
torus had been previously postulated by Soter (197 1) and
Ip (1988). Observations of solar wind plasma and magnetic field disturbances aboard the Phobos spacecraft in
626
the vicinity of the Phobos and Deimos orbits can be interpreted as signatures of solar wind interaction with the
dust/gas torus (Sauer et al., 1993). Since both satellites
spend substantial periods in the solar wind as they orbit
Mars, there is the possibility that strong wave-plasma
interactions will result in the excitation of multi-fluid
phenomena. The heavy Martian ions will form a cold
beam within the shocked solar wind, which could lead to
instabilities (Sagdeev et al., 1990).
In this paper we examine some simple properties of low
frequency wave propagation in a plasma containing a
mixture of light (electrons and protons) and heavy charged material (heavy ions or even charged dust particles).
Unlike many previous studies, we use the cold plasma
limit rather than the electrostatic limit. As the disturbances in both the AMPTE releases and in the vicinity
of the Phobos orbit were detected in the magnetic field,
the cold plasma limit is well suited for the examination of
electromagnetic disturbances. Both positively- and negatively-charged dust particles are to be found in dusty plasmas (Horanyi et al., 1990) and the effect of both of these
cases needs to be studied. We consider wave properties in
cold multi-ion plasmas, the new critical frequencies which
occur and the behaviour of the electrical current at these
critical frequencies. The mass density of any dust is likely
to be very much greater than the mass density of the
plasma. However, such a condition does not preclude a
strong electromagnetic interaction ; a significant effect can
be expected as long as the charge densities of the dust and
the ions of the local plasma environment are comparable.
The study of plasma waves in the presence of more than
one species of ion originated with the work of Buchsbaum
(1960), Yakimenko (1962), Smith and Brice (1964) and
D’Angelo et al. (1966). More recent studies have included
work by Young et al. (1981) and Roux et al. (1982) who
analysed signals associated with the presence of heavy
ions in the ULF wave data recorded on the GEOS 1 and
2 spacecraft, and work by Barbosa (1982) which examined
the dispersion properties of electrostatic waves in a multiion plasma. Recently Sauer et al. (1990) have reviewed the
wave mode structure in a multi-fluid plasma in connection
with their one-dimensional simulations of shocks in multifluid plasmas. Song et al. (1989) and D’Angelo (1990)
have examined electrostatic waves in dusty low frequency
multi-ion plasmas, while Rao (1993) has studied low frequency waves in magnetized dusty plasmas. Much theoretical work on the collective properties of dust particles
in plasmas has also been conducted by Havnes et al.
(1987), Rosenberg (1993) and Shukla (1992). We draw
on some of the discussion of these papers, in examining
the general wave properties in a low frequency cold
plasma. Our main purpose is to discuss some of the physical processes associated with the critical frequencies introduced by the presence of heavy charge carriers.
The heavy dust grains can be charged either positively
or negatively. The charge on a dust grain can best be
described in terms of its corresponding potential, @, with
respect to the surrounding medium (Whipple, 1981;
Whipple et al., 1985). There are many different processes
which contribute to the charging of a dust grain; we
mention the two most important ones. In the first instance,
if the primary electrons in the plasma are energetic
P. Thompson et al. : Wave behaviour in cold bi-ion plasmas
enough, they will release secondary electrons which cause
the surface potential to become positive. Absorption of
plasma ions has the same effect. The second case depends
on photoelectron emission. Absorption of solar ultraviolet radiation releases photoelectrons, which constitutes
a positive charging current; in the case where photoelectron emission is unimportant the charging current will
be negative. We shall not consider the possibility of the
charge being modified by the actual plasma wave itself
(see e.g. Li et al., 1994).
Theoretical development
A cold plasma model is used in which there is no relative
streaming between species in the background state. This
is a simplifying approximation. In fact it does not best
describe the interaction of the solar wind plasma and
either ions from a planetary source, a dust ring in the
Martian environment, or indeed the AMPTE ion release.
In all the examples there would be a significant nett velocity difference between the solar wind and the heavy
charged material which in practice would be likely to
be very important in generating electromagnetic noise.
However, limiting as this approximation may be, we feel
that insight into the wave properties of the plasma can be
obtained from our approach.
The dispersion relation in a cold plasma, written in
terms of the refractive index I [Stix (1962), Swanson
(1989) and McKenzie (1978)] is given by the familiar
form :
[S sin’ 0 + P cos’ B]r4
-[RLsin28+PS(l+cos2~)]r2+PRL=0,
(1)
where
S = ;(R+L)
The plasma frequency of the specific species (denoted by
1) is given by IX,‘,= qfn,/eom,, the gyro-frequency by
Q, = q,B/m,, and E, = + 1 for ions and electrons, respectively, where q, is the charge, n, the number density, m, the
mass and s0 the permittivity. The angle between the wave
propagation vector, k, and the background magnetic field
B, is 19.In this paper, we concentrate on low frequency
modes, i.e. where w c ape, the (electron) plasma
frequency. The large inertia of the ions relative to the
electrons means that the ions dynamically dominate in the
low frequency modes ; the approximation is equivalent to
ignoring electron inertial effects. This assumption simplifies the analysis considerably and allows the dispersion
relation (1) to be solved analytically. We allow 1P ( + co
and can then write the dispersion relation as follows,
(co? 8)r4 -s
(1 + cos* 812 + RL = 0.
(3)
P. Thompson ef al. : Wave behaviour in cold bi-ion plasmas
627
Propagation
of waves may occur at any angle 8,
although it is often useful, initially, to examine specific
modes of perpendicular and parallel propagation. The
low frequency dispersion rel.ation for all &’will briefly be
consideied later. -Writing Y,,= Y cos 8, rI = r sin 13:we
have
rI----
(4 -W(rf -L)
(Sri)
(4)
R
This dispersion relation is entirely equivalent to that
derived from the Smith and Brice (1964) alternative formulation which is written in terms of the phase velocity,
W = o/k = c/r, whereupon ,the dispersion relation in the
low frequency regime becomes,
(W2-WW,Z)W2sinZ8+(W2--Wi)(WZ-Wt)cos2tl=0,
Fig. 1. The dispersion properties for a single species plasma,
with rp = (c/WJ = (~“=/a).
Solid lines show parallel
(0 = 0) propagation, and broken lines perpendicular (0 = 7c/2)
propagation
(5)
where WR,L is the phase velocity of the parallel propagating (6’ = 0) right- (R) and left-hand (L) circularly
polarized modes, respectively. The phase velocity of the
transverse (6’= 7c/2)extraordinary e mode is given by W,,
where W,Z = l/2( Wi + Wg). The phase velocities are
related to the coefficients in (equation (2) by
and the R mode becomes the whistler mode at the higher
frequencies. It can be seen that in the o -+ 0 limit, the
three critical phase velocities of the different modes tend
to the (proton) AlfvCn speed, W, = c/r,.
Two ion-species plasma
(6)
Single ion-species plasma
Before considering a two ion species plasma, let us recall
what occurs in the case of .a single species plasma, consisting of electrons (denoted by subscript e) and protons
(denoted by subscript p). The electrons and protons are
singly charged and charge neutrality yields that
Extending the analysis to a two ion-species plasma (Smith
and Brice, 1964), we take account of protons (denoted by
subscript p) and a heavy ion species or dust particles
(denoted by subscript h). The effect of both positivelyand negatively-charged
heavy material needs to be
analysed. The electrons and protons are singly charged,
i.e. lqpl = )qJ = e, whereas the heavies (dust grains or
ions) we allow to be multiply charged, i.e. lq,, = Ze.
Charge neutrality implies the following relation :
n, = n,+Zn,
(7)
which is equivalent to
n, := np
which is equivalent
to
Figure 1 shows the dispersion properties for a single
species plasma where the refractive index is plotted against
frequency. The extreme cases of perpendicular and parallel propagation have been sketched together, although
they cannot propagate simultaneously. The L mode
(0 = 0), which has a resonance (r -+ co) at the proton
gyro-frequency, Sz,, is simply the well-known electromagnetic ion-cyclotron mode ; the transverse e mode
(0 = n/2) has a resonance at the lower hybrid frequency,
w,~, where
the upper sign denoting the case for positively-charged
material and the lower sign that for negatively-charged
material.
The low frequency limit, P + co, implies that EIB,
where say B = (O,O,B) with E = (E,,E,.,O) and
k = (k,,O,k,). The polarization of the electric field can be
written as
i-E,_-PCr2-D
E,
S
2(WkWf-W2Wz)
W’(WZ-
w;>
.
(9)
This reveals that the polarization of the parallel propagating R and L modes is right- and left-hand circularly
polarized, respectively, and that for rR x rL, i.e.
w; X5 Wl, linear polarization results. This particular case
P. Thompson et al. : Wave behaviour in cold bi-ion plasmas
628
occurs for low frequencies, when the phase velocity tends
to the (heavy) Alfvtn speed, B/[~l~(n,m,+n~rn~)]“~, but it
can also occur elsewhere as we shall see below.
We can now examine the dispersion curves for low
frequency wave propagation in a cold plasma for two
heavy
particular cases ; that of positively-charged
material, and that of negatively-charged heavy material,
both in addition to protons. For a detailed analysis of
certain wave properties evoked by the presence of positively-charged ions in space, see Horne and Thorne (1993)
and Rauch and Roux (1982). We need to keep in mind in
the dispersion curves discussed below, that although all
three wave modes are shown, they do not exist simultaneously, and that they in fact describe the two limits
of, in one instance, the parallel propagating R and L
modes, and in the other, the perpendicular e mode.
Positively-charged heavy material
The dispersion curves for the R, L and e modes present
in a proton plasma with an additional positively-charged
ion species are shown in Fig. 2. For the parallel propagating L mode (6 = 0), two resonances result at
o = Q,,R,, the gyro-frequencies of the two different ion
species. Resonant frequencies are important because they
reveal the frequency at the boundary of a stop band and
furthermore, in warm plasma theory, wave-particle interactions are close to these resonant frequencies. Waves
at resonant frequencies are typified by the fact that the
components of the magnetic field become small and hence
waves here are dominated by the electric field and are
termed electrostatic waves. Two resonances also result
in the case of the transverse e mode. The lower hybrid
resonance, which is also to be found in a single species
plasma, which for w,~>>R,, is given by
0;
(10)
=
--+&
1m,Z
and a new resonance, the bi-ion hybrid resonance, which
is located
between
the two ion gyro-frequencies
fib < o&h < R,, which for mr,mh >> m, is given by
(11)
Another critical frequency which occurs is that of the
cut-off frequency, at which the refractive index tends to
zero, i.e. the phase velocity W + GO.A cut-off frequency
then separates a frequency band of propagation from a
band of evanescence, for one wave mode. Depending on
the charge of the heavies, one mode cannot continue to
propagate at frequencies below the cut-off frequency (for
all values of 8, as will be seen below). For the case of
positively-charged heavies (dust or ions), this occurs for
the parallel propagating L mode and for the transverse
propagating e mode at
(12)
As a result of the particular resonances and cut-off
frequency, the dispersion curve for positively-charged
heavies shows a stop band in frequency for the L mode
for !& < w < oC, and a stop band for the e mode in the
range abh < w < 0,. The R mode goes on to become the
whistler mode at higher frequencies.
Finally, an interesting phenomenon arises when R = L,
which implies polarization reversal. At this frequency, the
magnitude of the refractive indices, or phase velocities of
the R and L modes (0 = 0), and the e mode (6’= 742)
are all identical, i.e. re = rL = rR = c/W, = c/W,_ = cl W,.
This ‘common’ phase velocity is approximately the proton
Alfvtn velocity and this phenomenon will be discussed
in some detail later. For positively-charged heavies, this
frequency is given by
e
UW
Negatively-charged heavy material
?h
Fig. 2. The dispersion properties for a two species plasma,
containing
positively-charged
heavy material, with r, =
(c/W,) = (c p,,(m,n, +m,n,)/B).
Solid lines show parallel
(0 = 0) propigation, and broken lines perpendicular (0 = 7c/2)
propagation
For the case of negatively-charged heavies, the dispersion
curves are shown in Fig. 3. Resonances occur at the two
gyro-frequencies, w = !&,flp as before, but this time, it is
the R mode which resonates at ah. Once again, the e mode
also resonates at the lower hybrid frequency ml,, and at
the bi-ion hybrid frequency e&h, given by equations (10)
and (11). The negative material introduces a cut-off frequency, given by equation (12) for the R mode and the e
mode. The effect is to produce a stop band in frequency
for the R mode for R, < o -Cw,, and for the e mode in
the range o&h < o < 0,. Again, the R mode becomes the
whistler mode at higher frequencies. Finally, concerning
the crossover frequency, where polarization reversal
occurs, we can write an expression for the case of negatively-charged heavies as
P. Thompson et al. : Wave behaviour in cold bi-ion plasmas
629
7000
6000
5000
f 4000
L 3000
2000
1000
0
‘h
Obh
Fig. 3. The dispersion propertie:s for a two species plasma, containing negatively-charged heavy material, with the same rA as
in Fig. 2. Solid lines show parallel (0 = 0) propagation, and
broken lines perpendicular (0 = 7c/2) propagation
(13b)
However, for negatively-charged
quency will only occur if
JZI
dust,
a crossover
fre-
(z*>;
)
which for heavy material is rather unlikely. An interesting
feature which arises from tlhis treatment of the various
critical frequencies is that a careful interpretation of the
frequency spectrum in different planetary environments
could lead to a great deal of information on which species
are present within the plasma and more specifically on
the relevant abundance of each species. Although these
critical frequencies are already well known (e.g. Budden,
1985), they are included here for completeness.
A numerical plot of the dispersion curves for 8 = 0 --f
n/2 is shown in Fig. 4, where the refractive index is plotted
against frequency normalized to the proton gyrofrequency. This is included to give some idea of how the
dispersion curves of the two extreme cases of parallel
and perpendicular propagation are linked to produce a
dispersion surface. The parameters were chosen to show
Ok,,, IS, and o, clearly, and were B = 3 nT, m,, = 4m,,
np = lo5 me3 = lOn, and Z = + 1. In this cold, low frequency regime there are, in general, two modes, except
at 0 = n/2, where one of the modes vanishes. It is also
interesting to note how the resonance at R, for 0 = 0
changes into the resonance at w,,~ for 0 = x/2, and that
the cut-off frequency, w,, occurs for all values of 8. Finally,
please note that the two solutions intersect only at one
point, which is given by the crossover frequency at 0 = 0.
Fig. 4. The dispersion properties for a two species plasma, containing positively-charged heavy material for values of propagation angle 0 = 0 -+ 42 with B = 3 nT, m,, = 4m,, np = 10’
mm3 = 10~ and 2 = + 1. One of the modes vanishes at 0 = n/2
asv+co
field in the z-direction. Maxwell’s equations give a wave
equation relating E and J ;
Vx(VxE)=
-po;-$$.
(14)
Assuming plane wave solutions and using J = aE for
an anisotropic medium allows us to write equation (14)
as
E,,
where 6,, is the unit tensor and caS the conductivity tensor
for the medium. Keeping the notation of the previous
section, the elements of O,~are
g=
-icoEo
is;
;;
;,:I
which allows us to write down an expression for the
current,
The behaviour of the electric current
J, = -ioso[(S-
It is interesting to examine the behaviour of the electrical
Jy = - ioeo[iDEx + (S-
current of
indeed the
the critical
behaviour.
J, = -iws,,[(P-
the wave at the various critical frequencies ;
major message ‘of this paper is that each of
frequencies exhibits unique electrical current
The analysis is done as before with a static B
l)E,-DC,]
l)Ez].
l)E,]
(15)
In the limit which we are considering, E, = E,, -+ 0,
although J,, may be finite, as V * J --) 0 in the wave for
et al. : Wave behaviour
P. Thompson
630
the low frequencies we are considering. Let us, however,
consider the JI components which, drawing on the convention of a collisional plasma, we split into the Pederson
current, JP, and the Hall current, JH. This can be compared to writing the total electrical current in the form
(Akasofu and Chapman, 1972),
(BxE)
J = @+o,E,+~,~.
(16)
Table 1. Electrical
critical frequencies
Case
current
Electrical
A
B
C
direction
current
in cold bi-ion plasmas
for waves oscillating
at the
Critical frequency
J,+O
J,-+O
JL + 0*
Wbh
0,
w,
*For one mode.
where
E = -kx(kxE)
L
k2
mode at this frequency. We can summarize these results
in Table 1. The cases are :
’
with E = (I&,I$, 0) and k = (k,, 0, kz). If we then examine
the current in the Pederson direction, J,, and in the Hall
direction, JH; we may write,
JP = -ios,(S-
l)E,fi
(17)
JH = we,DEL6,
where
The (S- 1) term in the Pederson current can be written
as S in this analysis as we shall ignore the displacement
current, which can be neglected at these low frequencies
(m <<Wpe).
Let us now consider three special cases, namely where
JP + 0, JH + 0 and J, -+ 0. The current in the Pederson
direction, JP tends to zero when S + 0, or R = -L. This
condition, which we call case A, implies,
(A) The bi-ion hybrid frequency, o,,~, results at the
intermediate resonance of the e mode where r + co. For
this wave mode, the propagation vector k = (k,, 0,O). The
polarization condition implies that E, + 0, i.e. the electric
field component in the direction of propagation dominates. Hence, the resulting waves are electrostatic in
nature, as is evident from one of Maxwell’s equations,
wb = k x E. This can be confirmed by an examination of
the electrical current behaviour at this critical frequency,
where the resulting current is purely in the Hall direction,
i.e. JP -+ 0.
(B) The crossover frequency, w,, results where the magnitude of the refractive index is the same for all three wave
modes. The polarization condition reveals that the waves
are linearly polarized, i.e. iE,/E, + 0. The perpendicular
electrical current is parallel to E, i.e. J, + 0, which implies
that the electric field components become small, resulting
in electromagnetic wave modes. When mh >>mp, and
nh cc np, the phase velocity at this frequency is given to a
good approximation by the proton AlfvCn velocity, i.e.
w = WR,L,e=
dh
which is simply equation (1 l), i.e. the bi-ion hybrid
frequency.
Case B is where JH + 0. One has D -+ 0, or R = L,
which implies,
i.e. equation (13), the crossover frequency.
We have one more special frequency to examine, the
cut-off frequency, which will be our study for case C. For
a plasma containing positively-charged dust L + 0 at w,,
which means that S = D, or
iJp
-_=I
JH
(18)
&=
wY
The behaviour of the wave modes at this critical crossover
frequency are of particular interest and shall be discussed
in detail in the following section.
(C) At the bi-ion cut-off frequency, Ok, Y-+ 0. It can be
seen from equation (9) that waves at this frequency are
right hand circularly polarized for the case of a plasma
containing
positively-charged
heavy material,
i.e.
iEJEy = 1, and are left hand circularly polarized for a
plasma containing negatively-charged
material, i.e.
iEJE, = - 1. We find that depending on the charge of the
dust, the perpendicular current associated with one mode
becomes zero here. This is because one of the wave modes
becomes evanescent at the cut-off frequency, and the current is carried by the only remaining mode which can
propagate to lower frequencies. A cold plasma theory,
then, predicts total reflection of one of the wave modes
from a dusty region at the ambient cut-off frequency, 0,.
’
This implies that the direction of the current is rotating
in a right-handed circular fashion, simply following the
polarization of the right-hand polarized wave, as the lefthanded wave is cut off here. So, with positively-charged
dust, J, --*0 for the left-handed mode at w,. In a plasma
with negatively-charged dust, J, + 0 for the right-handed
Special feature of the crossover frequency
The special behaviour of waves propagating at the crossover frequency is very interesting. We have already looked
at one property an electromagnetic wave has-its polarization, and shown that at o,, waves are linearly polar-
P. Thompson et al. : Wave behaviour in cold bi-ion plasmas
ized. Now, we shall consider another property of waves ;
their refractive index, r = c/U’. In order to gain an insight
into the type of wave behaviour to be expected at this
frequency, we examine wave modes propagating through
a plasma in which a portion of the plasma contains some
heavy charged material, e.g. the dust/gas torus along the
Phobos orbit. Suppose we have an interface separating a
medium 1, which consists of a single species plasma, from
medium 2, a two species plasma containing heavy positively-charged material. We shall assume that the changes
in the refractive index at this interface occur over a distance much shorter than the wavelength of the incident
wave. We consider a wave incident on the boundary with a
propagation vector making an angle 4 with the boundary
normal and an electric field vector parallel to the plane of
incidence. Subscript ‘i’ denotlcs the incident wave, ‘r’ the
reflected wave and ‘t’ the transmitted wave. Assuming
continuity of the electric and magnetic fields of the wave
across the boundary, we can derive Fresnel’s equations :
E,
4
-=
ri
rl COS
4, - r2 COS pi
Cj$+ r2 COS 4i
2r, COSpi
-6
E,-
COS
rl
COS
& + r2 COS 4i ’
(191
Our examination of wave modes propagating at the
crossover frequency has revealed that these modes are
linearly polarized, have a phlase velocity approximately
given by the proton Alfven speed and are electromagnetic
in nature. If the phase velocity of any wave mode at o,
was given exactly by the proton Alfven velocity, this would
imply that these waves were essentially unaffected by the
presence of heavy charged material and would propagate
as if they were in a single species plasma. In this case,
r, = r2, and the Fresnel equations would then yield,
E,
- 4.0
E,
Et
Jy+
1.
(20)
That is, waves very near the crossover frequency, which
propagate through a region containing heavy charged
material, would not ‘see’ the heavy material at all. The
waves would be totally transmitted through the interface
and no reflection would occur. This would also be the case
for the Fresnel equations derived with the electric field
perpendicular to the plane of incidence. In order to quantify the matching of refractive indices across such an interface, we shall look in moire detail at our reflection
coefficient. For relative algebraic convenience, we shall
choose B and k to both be parallel to the normal of
the boundary between the single and two species plasma
regions. The assumption is also made that the proton
number density is continuous across the boundary. The
model consists of an incident linearly polarized wave in
medium 1, ‘splitting’ into left- and right-hand polarized
waves, as these are the only supportable parallel modes in
the two species plasma, in medium 2. A linearly polarized
reflected wave also exists in medium 1. Our Fresnel equation becomes,
631
E
2=R=2rl-(r~+r~l
2r, + (rR +
4
rL)
’
(211
where
Note that for o, <<L$, we have assumed that r, is a
constant. A sketch of the shape of the curve of R which is
produced is shown in Fig. 5 for mh >>mp. It is apparent
that the reflection coefficient rises steeply at frequencies
below w,, z o,, where R = 0 at cuO.Above o,,, the gradient
of R is rather more shallow. It should also be noted that
for all B, the line (rR + rL) crosses 2r,, which implies a zero
value for R at some frequency. In fact, the crossover
frequency is still important for all 8, as it is always the
frequency at which polarization reversal occurs. In order
to quantify the gradient of R in this area, we expand it
around the point oO, i.e.
R
=
2rl -@R+d
= R(o,)+(w-cow,)
2rl + @R+ rL)
After some algebra, we find that
coo a n:”
(22)
and
a n;“.
(231
00
So we see that w0 varies with IE,,in the same manner as
o i.e. as n:” for nh <<np. A characteristic bandwidth for
which the reflection coefficient is arbitrarily low is of some
importance to know, and is simply given by the reciprocal
of the gradient of R at oO, i.e.
‘bandwidth a n; ‘12
(24)
that is, if the number density of the heavies were quadrupled, the frequency at which the reflection coefficient is
Fig. 5. The reflection coefficient plotted
region of the crossover frequency
against frequency
in the
P. Thompson et al. : Wave behaviour in cold bi-ion plasmas
632
zero would be doubled, but the range in frequency for
which the reflection coefficient is arbitrarily small would
be halved. At low frequencies (o <<Q,), the gradient of R
can be written to first order as,
The last term in brackets is in fact rather small, giving an
uncharacteristically broad bandwidth for a small reflection coefficient, e.g. bandwidth ~$12 for R = 0.1. This
is because the gradient of R becomes rapidly shallower
near w,,, in the region of expansion. This broad bandwidth
may seem appropriate as nh -+ 0 and the boundary
becomes less important, but it is also consistent with
increasing &. In this case w. and o, would both move
towards the proton gyro-frequency, R,, and r, can no
longer be treated as a constant as it now represents the
ion cyclotron mode. A similar analysis to the above can
still be made, by writing the average of the refractive
indices of the possible modes as a linearly polarized wave
undergoing Faraday rotation in an active medium. The
implications are the same: one sees that the refractive
indices of the wave either side of the boundary are closely
matched for frequencies above the crossover frequency,
which corresponds to the wide frequency bandwidth of a
low reflection coefficient.
So, we can say that the crossover frequency marks the
frequency region where the heavies no longer dominate
the dynamics of the plasma, but only produce a much
smaller electrostatic effect, as essentially n, # np. Rewriting the Fresnel equation, given by equation (19) in terms
of a characteristic
impedance for each medium,
X = E/H = cp/r gives
E
T
_
X*COSe,-X,
Ei - X,
COS8i
cos et + XI cos Bi
(25)
If near the crossover frequency, no reflection occurs
then equation (25) implies that the impedances for each
medium are ‘matched’ and hence one would expect the
optimum power transfer from medium 1 to medium 2, or
vice versa. In the extreme case which has been described
in detail where the cloud is quite uniform but has well
defined boundaries, one would expect that waves near
the crossover frequency are completely unaffected by the
presence of the heavy charged material and will propagate
through the boundary as if they were in a single species
plasma. Hence, at the ambient crossover frequency, which
is transparent to polarization and where the phase velocity
of the wave modes is given approximately by the proton
Alfven velocity, we would expect for a localized cloud of
heavy positively-charged material that strong electromagnetic coupling can be achieved between oscillations
in the interior of the cloud and waves exterior to it, as the
crossover frequency represents the lowest frequency range
for good coupling of electromagnetic waves across a dust
boundary.
In many circumstances in the solar system the light
and heavy charged material may also have a significant
velocity difference. In practice the velocity difference may
be a source of energy and further analysis of specific
circumstances
should take account of such effects.
Whereas the sheer velocity difference, and not the crossover frequency, may be the generator of wave energy, our
analysis seems to show that if a system of regions of finite
scale length can radiate at the crossover frequency, then
it seems to be the most favourable channel of inter-region
coupling between the cloud of heavies and the surrounding plasma. We have also restricted our analysis to
cold plasmas, and therefore have not considered absorption of the wave energy. No cold modes exist, in this
frequency regime, where absorption is apparent, although
one could speculate on the role of the bi-ion hybrid frequency, for example, in setting up electrostatic oscillations
in a warm plasma theory. However, concerning the crossover frequency, we are primarily interested in any form
of electromagnetic coupling of wave energy across a dusty
plasma/proton plasma boundary. Since the phase velocity
of waves at the crossover frequency is near the proton
Alfven speed, one may feel fairly confident that we are
not in a region where cold theory becomes inaccurate.
As remarked earlier, low frequency electromagnetic disturbances have been detected in the vicinity of the Phobos
orbit (Dubinin et al., 1991a) and furthermore both
AMPTE Lithium and Barium releases exhibited low frequency oscillatory behaviour (Ltihr et al., 1986; Gleaves
et al., 1988). As the latter paper shows, in the barium
release the magnetic oscillations were coherent between
the two spacecraft near the release; in all likelihood the
cloud as a whole was shaking or oscillating with a period
of order some tens of seconds. Our analysis indicates that
the coupling of signals from a region of heavy material to
its surroundings is very good in the vicinity of the crossover frequency. At this frequency, the heavies have no
effect on the polarization and furthermore there is likely
to be a good impedance match across the boundary of the
region where the heavies are present. We also propose
that electromagnetic disturbances may be preferentially
generated at this frequency in the vicinity of a dust/gas
torus such as that believed to be present in Mars orbit
(Dubinin et al., 1991b).
It is also tempting to speculate on the significance of
the crossover frequency in such locations as the environment of a conducting obstacle in a plasma flow, when the
gyroradii of any heavy ions present are comparable to the
length scale of the obstacle. We find such a situation in
the moons of Titan (Neubauer et al., 1984) and Triton
(Neubauer et al., 1991), for example, which lie in the
magnetospheres of Saturn and Neptune, respectively. In
both cases, we find that a characteristic frequency based
on the size of the obstacle and speed of the flow would be
in approximately the same frequency range as the gyrofrequencies of the ambient charged heavy ions, such as
N’, and possibly the critical frequencies described in this
paper. The precise effect of these ions on the current
structure generated at the obstacle and surrounding
plasma is, as yet, unknown. However, at the crossover
frequency, not only do these waves have a velocity similar
to the proton Alfvtn speed, but they are linearly polarized,
and furthermore, have a zero Hall component of perpendicular current. These magnetohydrodynamic characteristics at a frequency above the magnetohydrodynamic
regime could possibly enable the waves generated by the
P. Thompson et al. : Wave behaviour in cold bi-ion plasmas
obstacle to carry current
straightforward
fashion.
around
a large current
loop in a
Summary and conclusions
The major new results of this paper concern the nature of
the electrical current flow at low frequencies,
i.e. frequencies below the proton gyro-frequency.
We show that
at the bi-ion cut-off frequency, the total electrical current
is zero for one mode. This frequency is the upper limit of
the stop band for the parallel propagating
L mode and
the transverse e mode in the case of a plasma containing
positively-charged
heavy material, and for the R and e
modes for a plasma containing
negatively-charged
heavy
material. The wave modes at this frequency are circularly
polarized. At the bi-ion hybrid frequency, a resonance for
the e mode, the current in the wave is strictly perpendicular
to the electric field, i.e. the current is purely in the Hall
direction. The wave modes at this frequency are electrostatic in nature.
The third critical frequency, is the crossover frequency,
and properties of waves at this frequency are particularly
important in multi-fluid plasmas. The electrical current is
strictly parallel to the electric field, i.e. the Hall current is
zero. This leads to small elelctric field components
and
hence the waves are electromagnetic
in nature.
Polarization reversal occurs at this frequency so the waves
are linearly polarized,
i.e. this particular
frequency
is
transparent
to polarization.
The magnitude
of the phase
velocities of the parallel propagating
R and L modes and
the transverse e mode are equal and are in the vicinity
of the proton Alfvtn speed. This implies that near this
frequency the waves may behave as if no heavy material
were present at all. Electromagnetic
waves will tend to
radiate at this frequency and in particular,
a localized
region of heavy ions could preferentially
radiate at the
local crossover frequency.
References
Akasofu,
S. and Chapman, S., Solar Terrestrial Physics. The
Clarendon Press, Oxford, 19’72.
Bagenal, F. and Sullivan, J. D., .Direct plasma measurements in
the 10 torus and inner magnetosphere of Jupiter. J. geophys.
Res. 86 8447, 1981.
Bagenal, F., Sullivan, J. D. and Siscoe, G. L., Spatial distribution
of plasma in the 10 torus. Geophys. Res. Lett. 7,41, 1980.
Barbosa, D. D., Multi-ion resonances in finite temperature
plasma. Astrophys. J. 254, 376, 1982.
Brown, R. A., P&her, C. A,. and Strobel, D. F., Spectrophotometric studies of the 10 torus, in Physics ofthe Jooian
Magnetosphere (edited by A. J. Dessler). Cambridge Uni-
versity Press, New York, 198’3.
Buchsbaum, S. J., Resonance in a plasma with two ion species.
Phys. Fluids 3,418, 1960.
Budden, K. G., The Propagation of Radio Waves, p. 57 + chapter.
Cambridge University Press, Cambridge, 1985.
D’Angelo, N., Low-frequency electrostatic waves in dusty plasmas. Planet. Space Sci. 38, 1143, 1990.
D’Angelo, N., Goeler, S. V. and Ohe, T., Propagation and damping of ion waves in a plasma with negative ions. Phys. Fluids
4, 1605, 1966.
633
Dubinin, E. M., Lundin, R., Pissarenko, N. F., Barabash, S.
V., Zakharov, A. V., Koskinen, H., Schwingebschuh, K. and
Yeroshenko, Ye. G., Indirect evidences for a gas/dust torus
along the Phobos orbit. Geophys. Res. Lett. 17, 861, 1990.
Dubinin, E. M., Pissarenko, N. F., Barabash, S. V., Zacharov, A.
V., Lundin, R., Pellinen, R., Schwingenschuh, K. and Yeroschenko, Ye. G., Plasma and magnetic field effects associated
with Phobos and Deimos tori. Planet. Space Sci. 39, 113,
1991a.
Dubinin, E. M., Pissarenko, N., Barabash, S. V., Zacharov, A.
V., Lundin, R., Koskinen, H., Schwingenschuh, K. and Yeroshenko, Ye. G., Tails of Phobos and Deimos in the solar wind
and in the Martian magnetosphere. Planet. Space Sci. 39,
123, 1991b.
Gleaves, D., Southwood, D. J., Dunlop, M. W. and MierJedrzejowicz, W. A. C., Low frequency magnetic wave spectra
associated with the AMPTE barium release of 27 December
1984. Adv. Space Res. 8, 181, 1988.
Havnes, O., Goertz, C. K., Mortill, G. E., Griin, E. and Ip, W.,
Dust charges, cloud potential and instabilities in a dust cloud
embedded in a plasma. J. geophys. Res. 92,2281, 1987.
Horanyi, M., Burns, J. A., Tatrallyay,
M. and Luhmann, J. G.,
Toward understanding the fate of dust lost from the Martian
satellites. Geophys. Res. Lett. 17, 853, 1990.
Horne, R. B. and Thorne, M., On the preferred source location
for the convective amplification of ion cyclotron waves. J.
geophys. Res. 98,9233, 1993.
Ip, W.-H., On a hot oxygen corona of Mars. Icarus 76, 135,
1988.
Ip, W.-H. and Banaszkiewicz, M., On the dust/gas tori of Phobos
and Deimos. Geophys. Res. Lett. 17, 857, 1990.
Johnson, R. G., Sharp, R. D. and Shelley, E. G., The discovery
of energetic He+ ions in the magnetosphere. J. geophys. Res.
79,3135, 1974.
Li, F., Havnes, 0. and Melandso, F., Longitudinal waves in a
dusty plasma. Planet Space Sci. 42,401, 1994.
Liihr, H., Southwood, D. J., KlScker, N., Acuiia, M., Hlusler,
B., Dunlop, M. W., Mier-Jedrzejowicz, W. A. C., Rijnbeek,
R. P. and Six, M., In situ magnetic field measurements during
the AMPTE solar wind lithium releases. J. geophys. Res. 91,
1261, 1986.
Lundin, R., Zakharov, A., Pellinen, R., Barabasj, S. W., Borg,
H., Dubinin, E. M., Hultqvist, B., Koskinen, H., Liede, I.
and Pissarenko, N., Aspera/Phobos measurements of the ion
outflow from the Martian ionosphere. Geophys. Res. Lett.
17, 873, 1990.
McKenzie, J. F., A summary of the propagation properties of
magnetosonic-whistler,
AlfvCn-ion cyclotron, ion-acoustic
(cyclotron) and Bernstein waves, in multi-component, low p
plasmas. Research report, Danish Space Research Institute,
1978.
Neubauer, F. M., Gurnett, D. A., Scudder, J. D. and Hartle, R.
E., Titan’s magnetospheric interaction, in Saturn (edited by
T. Gehrels and M. Shapley-Matthews), pp. 760-787. University of Arizona Press, Tucson, 1984.
Neubauer, F. M., Glassmeier, K. H., Pohl, M., Raeder, J., Acufia,
M. H., Burlaga, L. F., Ness, N. F., Musmann, G., Mariani,
F., Wallis, M. K., Ungstrup, E. and Schmidt, H. U., First
results from the Giotto magnetometer experiment at comet
Halley. Nature 321, 352, 1986.
Neubauer, F. M., Liittgen, A. and Ness, N. F., On the lack of a
magnetic signature of Triton’s magnetospheric interaction
on the Voyager 2 flyby trajectory. J. geophys. Res. 96, 19,171,
1991.
Rao, N. N., Low-frequency waves in magnetized dusty plasmas.
J. Plasma Phys. 49, 375, 1993.
Rauch, J. L. and Roux, A., Ray tracing of ULF waves in a
multicomponent magnetospheric plasma : consequences for
the generation mechanism of ion cyclotron waves. J. geophys.
Res. 87, 8191, 1982.
634
Riedler, W., Schwingenschuh, K., Yeroshenko, Ye. G., Styashkin,
V. A. and Russell, C. T., Magnetic field observations in comet
Halley’s coma. Nature 321 288, 1986.
Rosenberg, M., Ion- and dust-acoustic instabilities in dusty plasmas. Planet. Space Sci. 41,229, 1993.
Roux, A., Perraut, S., de Villedary, C., Gendrin, R., Kremser, G.,
Korth, A. and Young, D. T., Generation of ion cyclotron
waves and heating of He. J. geophys. Res. 87, 8 174, 1982.
Sagdeev, R. Z., Shapiro, V. D., Shevchenko, V. I., Zacharov, A.,
Kiraly, P., Szego, K., Nagy, A. F. and Gerard, R., Wave
activity in the neighbourhood of the bowshock of Mars.
Geophys. Res. Lett. 17, 893, 1990.
Sauer, K., Motschmann, U. and Roatsch, T., Plasma boundaries
at comet Halley. Ann. Geophys. 8, 243, 1990.
Sauer, K., Baumgartel, K. and Motschmann, U., Phobos events
as precursors of solar wind-dust interaction. Geophys. Res.
Lett. 20, 165, 1993.
Shemansky, D. E., Radiative cooling efficiencies and predicted
spectra of species of the 10 torus. Astrophys. J. 236, 1043,
1980.
Shukla, P. K., Low-frequency modes in dusty plasmas. Physica
Scripta 45, 504, 1992.
Smith, E. J., Tsurutani, B. T., Slavin, J. A., Jones, D. E., Siscoe,
G. L. and Mendis, D. A., International cometary explorer
encounter with Giacobini-Zinner
: magnetic field observations. Science 232, 382, 1986.
Smith, R. L. and Brice, N., Propagation in multicomponent
plasmas. J. geophys. Res. 69, 5029, 1964.
Song, B., Suszcynsky, D., D’Angelo, N. and Merlino, R. L.,
P. Thompson et al. : Wave behaviour in cold bi-ion plasmas
Electrostatic ion-cyclotron waves in a plasma with negative
ions. Phys. Fluids B1(12), 2316, 1989.
Soter, S., The dust belts of Mars. Center for Radiophysics and
Space Research, Cornell University. CRSR 462, 197 1.
Stix, T., The Theory of Plasma Waves. McGraw-Hill, New York,
1962.
Swanson, D. G., Plasma Waves. Academic Press, New York,
1989.
Valenzuela, A., Haerendel, G., Fi5pp1, H., Melzener, F., Neuss,
H., Rieger, E., Stiicker, J., Bauer, O., Hiifner, H. and Loidl,
J., The AMPTE Artificial Comet experiments. Nature 320,
700, 1986.
Verigin, M. I., Shutte, N. M., Galeev, A. A., Kotova, G. A.,
Remizov, A. P., Rosenbauer, H., Hemmerich, P., Richter, A.
K., Apathy, I., Szegii, K., Riedler, W., Schwingenschuh, K.
and Steller, M., Ions of planetary origin in the Martian
magnetosphere (Phobos 2/Taus Experiment). Planet. Space
Sci. 39, 131, 1991.
Whipple, E. C., Potentials of surfaces in space. Rep. Prog. Phys.
44,1197,1981.
Whipple, E. C., Northrop, T. G. and Mendis, D. A., The electrostatics of a dusty plasma. J. geophys. Res. 90, 7405, 1985.
Yakimenko, V. L., Oscillations in a cold plasma containing two
ion species. Sov. Phys. Tech. Phys. 7, 117, 1962.
Young, D. T., Perraut, S., Roux, A., De Villedary, C., Gendrin,
R., Korth, A., Kremser, G. and Jones, D., Wave-particle interactions near OMEGA He+ on GEOSl and 2. 1. Propagation
of ion cyclotron waves in He+ rich plasma. J. geophys. Res.
86,6755, 1981.