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Planet. Space Sci., Vol. 43, No. 5, pp. 625-634, 1995 Elsevier Science Ltd Printed in Great Britain 00324633/95 $9.50+0.00 Pergamon 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. 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