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Wave propagation in cold multi-fluid plasmas
Adv. Space Res. Vol. 13, No. 10, pp. (10)305—(10)308, 1993 Printed in Great Britain. All rights reserved.
0273—1177/93 $24.00 Copyright © 1993 COSPAR
WAVE PROPAGATION IN COLD MULTI-FLUID PLASMAS M. K. Dougherty and D.
J.
Southwood
Space and Atnwspheric Physics, Imperial College, London, U.K.
ABSTRACT We discuss wave propagation in a cold plasma which contains heavy ion species or charged dust grains. Our analysis reveals a modification in the behaviour of low frequency (hydromagnetic) modes for the two cases of positively or negatively charged dust or heavy ions. When positively charged heavy material is present with charge density comparable to the local light ion charge density the well-known phenomena associated with the presence of a cross-over frequency are found. The bi-ion hybrid and cut-off frequencies also occur and the behaviour of the electrical current at these critical frequencies is discussed. The resultant wave modes which arise may be pertinent in describing observations of wave activity seen during the Phobos mission. INTRODUCTION In this paper we discuss some of the potentially important features of electromagnetic wave propagation in a cold plasma containing more than one species of ion. We discuss the new critical frequencies which occur and the behaviour of the electrical current at these critical frequencies. The recent exploration of the Martian system by the Phobos mission revealed the existence of heavy ions and heavy charged dust particles within the Martian orbit. The wave modes described in the following analysis may well be relevant to the description of magnetic disturbances seen by the spacecraft on crossings of the Phobos orbit /1/.
THEORY A cold plasma model is used in which there is no relative streaming between the species in the background state. This approximation enables us to gain an insight into the wave properties of the plasma. We concentrate on low frequency modes in this analysis, where w <
(w2
—
w:) w2
w~)
w~)
where WRL is the phase velocity of the parallel propagating right (R) and left (L) circularly polarized modes respectively. W~is the phase velocity of the transverse extraordinary mode where W = ~ (W~+ W~). SINGLE SPECIES PLASMA It is useful to recall the results which arise in the case of a single species plasma, consisting of electrons (subscript ‘e’) and protons (subscript ‘p’), where charge neutrality yields that n~= m~,
(10)305
(2)
M. K. Dougherty and D.J. Southwood
(10)306
where n denotes the number density. The dispersion properties can be seen in Figure 1, where the phase velocity is plotted versus frequency. The L mode has a resonance at the proton gyrofrequency, ~l,,, and is simply the well-known electromagnetic ion-cyclotron mode. The transverse e mode has a resonance at the lower hybrid frequency, Wi,, and the R mode becomes the whistler mode at higher frequencies. It can be seen that in the ~4, —~ 0 limit, the phase velocities of the three modes tend to the Alfvén speed. Vph
v~ A
0p
Fig. 1. Phase velocity versus frequency for a single species plasma. TWO SPECIES PLASMA The analysis is then extended to that of a two species plasma,/3/, where protons and heavy ions or dust particles are taken into account. The electrons and protons are assumed to be singly charged and the heavies (subscript ‘h’) multiply charged (with the charge given by Z). The heavy dust grains can be positively or negatively charged depending on which region ofthe plasma they are found in /4/. Charge neutrality yields; n~=n,±Zn,,
(3)
the upper sign denoting postively charged heavies and the lower negatively charged heavies. The resulting dispersion properties for a two species plasma are shown in Figure 2(a) for positively charged heavies and Figure 2(b) for negatively charged heavies. Figure 2(a) reveals that since there are no negatively charged particles in the plasma that the R mode has no cut-offs or resonances. At higher frequencies, below the electron cyclotron frequency this mode develops into the whistler mode. The L mode has resonances at each of the ion gyrofrequencies, with ~ acting as an upper frequency cut-off. Another cut-off occurs at an intermediate frequency w~resulting in a stop band in the frequencies where Il,,
~
Zm~n, > ~ which for heavy material is rather unlikely.
(3)
Wave Propagation
VF,t,
Vph
(a)
(b)
0~
O~h
0)c
(0,,
Op
(Oth
0)
(10)307
Oh
(0th
0),,
Op
O~h
(0
Fig. 2. A two species plasma with (a) positively and (b) negatively charged heavies. BEHAVIOUR OF THE ELECTRICAL CURRENT Each of the three new critical frequencies, that of the cut-off frequency, w,,, the bi-ion hybrid frequency, wb,, and the cross-over frequency w~are characterised by special physical behaviour of the waves. In this section we shall discuss the behaviour of the electrical current. The detailed analysis is carried out in /2/. The total electrical current of the wave can be written in the form; (B x E) J=o~
0E11+o0Ei+o2 B (4) We concentrate on the J1 components which, drawing on convention of a collisional plasma, we split into the Pederson current, the second term in equation (4) and the Hall current, JH, the third term. When the total perpendicular current is zero, the wave frequency which results is that ofthe cut-off frequency given by,
Jp
(5) and on examination of waves at this frequency it is found that the resultant modes are electrostatic in nature. When
Jp
=
0, the bi-ion hybrid frequency occurs where,
2 — ~ ri9m~+ m~m~ 2n,,m~ — ~ 6 npmh + Z Again, on closer analysis of the wave behaviour we find that the resulting waves modes are electrostatic in nature. Finally, when Jff = 0, the cross-over frequency results, where
.
=
±Z~.
(7)
The wave modes which arise at this frequency are found to be electromagnetic in nature. This cross-over frequency occurs when all three wave modes have the same phase velocity and one indirect consequence of this is that the polarization is linear for all three modes. Furthermore it can be shown that the phase velocity in question is that given by the proton Alfvén velocity i.e. the phase velocity of low frequency waves outside any cloud of charged dust or heavy material; VA
B
(8)
(10)308
M. K.Dougherty and D. J. Southwood
It follows that where there is a localized cloud of heavy positively charged material we can expect that in the vicinity of the mean cross-over frequency for the cloud, strong coupling can be achieved between oscillations in the interior of the cloud and waves exterior to the cloud. In the extreme case where the cloud is quite uniform but with well defined boundaries one expects that waves at the cross-over frequency are completely unaffected by the presence of the heavy charged material and will be able to propagate through the boundary as if they are in a single species plasma. Where there is a localized region of heavy ions or dust particles, the plasma at this frequency will be transparent to the external medium and as a result dust or other heavies would be expected to preferentially oscillate as electromagnetic waves at the cross-over frequency. DISCUSSION Low frequency waves and more specifically weak L mode waves, /5/, have been observed at the proton gyrofrequency in the magnetosheath region of Mars. These types of waves are consistent with our model which predicts that for positively or negatively charged dust particles that L waves will be found near which acts as a resonance for this particular wave mode. Low frequency electromagnetic waves less than the local gyrofrequency as well as intense electrostatic waves have also been observed, /6/. From the above analysis we would expect to find electrostatic mode waves near the bi-ion hybrid frequency which acts as a resonance for the transverse e mode. It has also been suggested that in a plasma containing positively charged heavy material one would expect to see electromagnetic radiation preferentially emitted at the local cross-over frequency. It is also tempting to speculate that the magnetic pulsing of the heavy ion releases made during the AMPTE space mission were due to this cause, /7/. REFERENCES 1. E.M. Dubinin, R. Lundin, N.F. Pissarenko, S.V. Barabash, A.V. Zakharov, H. Koskinen, K. Schwingenshuh and Ye.G. Yeroshenko, Indirect Evidences of a Gas/Dust Torus at the Phobos Orbit, Geophys. Rea. Lett., # 17, 861-864 (1990). 2. M.K. Dougherty and D.J. Southwood, Wave Propagation in Cold Multi-Fluid Plasmas, paper in preparation (1992). 3. R.L. Smith and N. Brice, Propagation in Multicomponent Plasmas, J. Geophys. Res., # 69, 5029-5040 (1964). 4. M. Horanyi, J.A. Burns, M. Tatrallyay and J.G. Luhmann, Toward understanding the fate of dust lost from the Martian satellites, Geophys. Res. Lelt., # 17, 853-856 (1990). 5. C.T. Russell, J.G. Luhmann, K. Schwingenshuh, W. Riedler and Ye. Yeroshenko, Upstream Waves at Mars : Phobos Observations, Geophys. Rca. Leti., ~ 17, 897-900 (1990). 6. R. Grard, C. Nairn, A. Pedersen, S. Klimov, S. Savin, A. Skaisky and J.G. Trotignon, Plasma and Waves around Mars, Planet. Space Sci., # 39, 89-98 (1991). 7. H. Liihr, D.J. Southwood, N. Kl6cker, M.W. Dunlop, W.A.C. Mier-Jedrzejowicz, R.P. Rijnbeek, M. Six, B. Häusler and M. Acuna, In situ magnetic field observations of the AMPTE artificial comet, Nature, ~ 320, 708-711 (1986).
0273—1177/93 $24.00 Copyright © 1993 COSPAR
WAVE PROPAGATION IN COLD MULTI-FLUID PLASMAS M. K. Dougherty and D.
J.
Southwood
Space and Atnwspheric Physics, Imperial College, London, U.K.
ABSTRACT We discuss wave propagation in a cold plasma which contains heavy ion species or charged dust grains. Our analysis reveals a modification in the behaviour of low frequency (hydromagnetic) modes for the two cases of positively or negatively charged dust or heavy ions. When positively charged heavy material is present with charge density comparable to the local light ion charge density the well-known phenomena associated with the presence of a cross-over frequency are found. The bi-ion hybrid and cut-off frequencies also occur and the behaviour of the electrical current at these critical frequencies is discussed. The resultant wave modes which arise may be pertinent in describing observations of wave activity seen during the Phobos mission. INTRODUCTION In this paper we discuss some of the potentially important features of electromagnetic wave propagation in a cold plasma containing more than one species of ion. We discuss the new critical frequencies which occur and the behaviour of the electrical current at these critical frequencies. The recent exploration of the Martian system by the Phobos mission revealed the existence of heavy ions and heavy charged dust particles within the Martian orbit. The wave modes described in the following analysis may well be relevant to the description of magnetic disturbances seen by the spacecraft on crossings of the Phobos orbit /1/.
THEORY A cold plasma model is used in which there is no relative streaming between the species in the background state. This approximation enables us to gain an insight into the wave properties of the plasma. We concentrate on low frequency modes in this analysis, where w <
(w2
—
w:) w2
w~)
w~)
where WRL is the phase velocity of the parallel propagating right (R) and left (L) circularly polarized modes respectively. W~is the phase velocity of the transverse extraordinary mode where W = ~ (W~+ W~). SINGLE SPECIES PLASMA It is useful to recall the results which arise in the case of a single species plasma, consisting of electrons (subscript ‘e’) and protons (subscript ‘p’), where charge neutrality yields that n~= m~,
(10)305
(2)
M. K. Dougherty and D.J. Southwood
(10)306
where n denotes the number density. The dispersion properties can be seen in Figure 1, where the phase velocity is plotted versus frequency. The L mode has a resonance at the proton gyrofrequency, ~l,,, and is simply the well-known electromagnetic ion-cyclotron mode. The transverse e mode has a resonance at the lower hybrid frequency, Wi,, and the R mode becomes the whistler mode at higher frequencies. It can be seen that in the ~4, —~ 0 limit, the phase velocities of the three modes tend to the Alfvén speed. Vph
v~ A
0p
Fig. 1. Phase velocity versus frequency for a single species plasma. TWO SPECIES PLASMA The analysis is then extended to that of a two species plasma,/3/, where protons and heavy ions or dust particles are taken into account. The electrons and protons are assumed to be singly charged and the heavies (subscript ‘h’) multiply charged (with the charge given by Z). The heavy dust grains can be positively or negatively charged depending on which region ofthe plasma they are found in /4/. Charge neutrality yields; n~=n,±Zn,,
(3)
the upper sign denoting postively charged heavies and the lower negatively charged heavies. The resulting dispersion properties for a two species plasma are shown in Figure 2(a) for positively charged heavies and Figure 2(b) for negatively charged heavies. Figure 2(a) reveals that since there are no negatively charged particles in the plasma that the R mode has no cut-offs or resonances. At higher frequencies, below the electron cyclotron frequency this mode develops into the whistler mode. The L mode has resonances at each of the ion gyrofrequencies, with ~ acting as an upper frequency cut-off. Another cut-off occurs at an intermediate frequency w~resulting in a stop band in the frequencies where Il,,
~
Zm~n, > ~ which for heavy material is rather unlikely.
(3)
Wave Propagation
VF,t,
Vph
(a)
(b)
0~
O~h
0)c
(0,,
Op
(Oth
0)
(10)307
Oh
(0th
0),,
Op
O~h
(0
Fig. 2. A two species plasma with (a) positively and (b) negatively charged heavies. BEHAVIOUR OF THE ELECTRICAL CURRENT Each of the three new critical frequencies, that of the cut-off frequency, w,,, the bi-ion hybrid frequency, wb,, and the cross-over frequency w~are characterised by special physical behaviour of the waves. In this section we shall discuss the behaviour of the electrical current. The detailed analysis is carried out in /2/. The total electrical current of the wave can be written in the form; (B x E) J=o~
0E11+o0Ei+o2 B (4) We concentrate on the J1 components which, drawing on convention of a collisional plasma, we split into the Pederson current, the second term in equation (4) and the Hall current, JH, the third term. When the total perpendicular current is zero, the wave frequency which results is that ofthe cut-off frequency given by,
Jp
(5) and on examination of waves at this frequency it is found that the resultant modes are electrostatic in nature. When
Jp
=
0, the bi-ion hybrid frequency occurs where,
2 — ~ ri9m~+ m~m~ 2n,,m~ — ~ 6 npmh + Z Again, on closer analysis of the wave behaviour we find that the resulting waves modes are electrostatic in nature. Finally, when Jff = 0, the cross-over frequency results, where
.
=
±Z~.
(7)
The wave modes which arise at this frequency are found to be electromagnetic in nature. This cross-over frequency occurs when all three wave modes have the same phase velocity and one indirect consequence of this is that the polarization is linear for all three modes. Furthermore it can be shown that the phase velocity in question is that given by the proton Alfvén velocity i.e. the phase velocity of low frequency waves outside any cloud of charged dust or heavy material; VA
B
(8)
(10)308
M. K.Dougherty and D. J. Southwood
It follows that where there is a localized cloud of heavy positively charged material we can expect that in the vicinity of the mean cross-over frequency for the cloud, strong coupling can be achieved between oscillations in the interior of the cloud and waves exterior to the cloud. In the extreme case where the cloud is quite uniform but with well defined boundaries one expects that waves at the cross-over frequency are completely unaffected by the presence of the heavy charged material and will be able to propagate through the boundary as if they are in a single species plasma. Where there is a localized region of heavy ions or dust particles, the plasma at this frequency will be transparent to the external medium and as a result dust or other heavies would be expected to preferentially oscillate as electromagnetic waves at the cross-over frequency. DISCUSSION Low frequency waves and more specifically weak L mode waves, /5/, have been observed at the proton gyrofrequency in the magnetosheath region of Mars. These types of waves are consistent with our model which predicts that for positively or negatively charged dust particles that L waves will be found near which acts as a resonance for this particular wave mode. Low frequency electromagnetic waves less than the local gyrofrequency as well as intense electrostatic waves have also been observed, /6/. From the above analysis we would expect to find electrostatic mode waves near the bi-ion hybrid frequency which acts as a resonance for the transverse e mode. It has also been suggested that in a plasma containing positively charged heavy material one would expect to see electromagnetic radiation preferentially emitted at the local cross-over frequency. It is also tempting to speculate that the magnetic pulsing of the heavy ion releases made during the AMPTE space mission were due to this cause, /7/. REFERENCES 1. E.M. Dubinin, R. Lundin, N.F. Pissarenko, S.V. Barabash, A.V. Zakharov, H. Koskinen, K. Schwingenshuh and Ye.G. Yeroshenko, Indirect Evidences of a Gas/Dust Torus at the Phobos Orbit, Geophys. Rea. Lett., # 17, 861-864 (1990). 2. M.K. Dougherty and D.J. Southwood, Wave Propagation in Cold Multi-Fluid Plasmas, paper in preparation (1992). 3. R.L. Smith and N. Brice, Propagation in Multicomponent Plasmas, J. Geophys. Res., # 69, 5029-5040 (1964). 4. M. Horanyi, J.A. Burns, M. Tatrallyay and J.G. Luhmann, Toward understanding the fate of dust lost from the Martian satellites, Geophys. Res. Lelt., # 17, 853-856 (1990). 5. C.T. Russell, J.G. Luhmann, K. Schwingenshuh, W. Riedler and Ye. Yeroshenko, Upstream Waves at Mars : Phobos Observations, Geophys. Rca. Leti., ~ 17, 897-900 (1990). 6. R. Grard, C. Nairn, A. Pedersen, S. Klimov, S. Savin, A. Skaisky and J.G. Trotignon, Plasma and Waves around Mars, Planet. Space Sci., # 39, 89-98 (1991). 7. H. Liihr, D.J. Southwood, N. Kl6cker, M.W. Dunlop, W.A.C. Mier-Jedrzejowicz, R.P. Rijnbeek, M. Six, B. Häusler and M. Acuna, In situ magnetic field observations of the AMPTE artificial comet, Nature, ~ 320, 708-711 (1986).