Mirror-mode structures at the Galileo-Io flyby: Instability criterion and dispersion analysis

D. E. Huddleston,^1,2 R. J. Strangeway,¹ X. Blanco-Cano,³ C. T. Russell,^1,4 M. G. Kivelson,^1,4 and K. K. Khurana¹

Abstract. The mirror mode is typically excited in high-beta plasmas when there is a significant pressure anisotropy, with the greatest plasma pressure perpendicular to the magnetic field, B. Large-amplitude mirror-mode structures were identified in the Galileo magnetic field data on the edges of the cold Io wake. Here, despite the high ambient B field and low plasma beta, the enhanced perpendicular pressure due to the ring-type velocity distributions of heavy Iogenic pickup ions overcomes the instability threshold for the mirror mode and provides the free energy. In the center of the Io wake, the local pickup velocities are very low and hence the perpendicular pressure contribution is small, while farther from Io in the torus, the corotating isotropic plasma dominates and the mirror mode is not unstable but instead ion cyclotron waves grow. Thus the spatial region in which the mirror mode dominates is narrow. A warm plasma dispersion analysis is performed for the multispecies plasma conditions appropriate to the edges of the Io wake. In a multispecies plasma, multiple ion cyclotron modes are possible, each with a growth rate dependent upon the anisotropy and free energy provided by the specific gyroresonant ion species component. However, the mirror mode can dominate, since its growth rate depends on the combined anisotropic pressure of all ions present.

1. Introduction

The Io-Jupiter interaction is unique in our solar system. The continual supply of volcanic materials from Io (most notably sulfur dioxide, oxygen, sulfur, hydrogen, and sodium) to the Io torus and beyond is well known to be the major source of plasma in the Jovian magnetosphere. The torus plasma consists predominantly of sulfur and oxygen ions [e.g., Bagenal, 1994], consistent with rapid dissociation of sulfur dioxide [e.g., Smyth and Marconi, 1998, and references therein] on timescales much shorter than the corotation time but closer to the ion pitch angle scattering timescale [Huddleston et al., 1998a]. In the near-Io torus, in common with other pickup environments (such as comets; see, for example, reviews by Galeev [1991] and Lee [1992]), newly ionized particles are picked up into ring or ring-beam type velocity distributions which are highly anisotropic and unstable to wave generation. During the Galileo spacecraft's close approach to Io in December 1995, several wave phenomena were observed by the magnetometer instrument [e.g., Kivelson et al., 1996; Russell et al., 1999]. Ion cyclotron waves near the sulfur dioxide gyrofrequency are dominant in the near-Io torus [Kivelson et al., 1996; Huddleston et al., 1997, 1998a; Warnecke et al., 1997]. On the edges of the cold Io wake, where pickup ion densities are increasing rapidly near the Io source [Frank et al., 1996; Bagenal, 1997], large-amplitude mirror-mode structures have been identified. Figure 1 shows a schematic overview of the location of these observations along the Galileo flyby trajectory. The observed mirror structures are highly com-pressional, with wave normals at large angles to B, and their detailed properties are discussed in a companion paper [Russell et al., this issue]. The focus of the present paper is to explain the appearance of these structures by considering the instability criterion and by evaluating dispersion solutions for different values of the pressure anisotropy for the ion species components present near Io.

Figure 1. Overview of the Galileo-Io flyby geometry showing the locations of the mirror mode and ion cyclotron wave observations. (Figure adapted from Russell et al. [1997], and velocity vectors taken from Frank et al. [1996].)

The mirror mode is typically excited in high-beta plasmas ( is the ratio of plasma to magnetic pressures) when there exists a significant pressure anisotropy, P / P_// > 1, such as that created by pickup ion ring distributions. The mirror mode [Chandrasekhar et al., 1958; Barnes, 1966; Hasegawa, 1975] is a nonpropagating mode, with zero real frequency. It is often thought to be a fluid instability (in the long-wavelength limit), but Southwood and Kivelson [1993] have pointed out that the instability mechanism requires the inclusion of kinetic effects. Both fluid and kinetic theories give the same instability threshold, but an analysis of the growth rates should use a kinetic approach, as we do here. Maximum growth rates for the mirror mode occur at oblique angles ( _kB) to B, in contrast to the ion cyclotron mode, whose maximum growth is at _kB = 0^o.

Early observations of magnetic field perturbations in the Earth’s magnetosheath were interpreted as either very low frequency slow-mode waves or plasma condensations driven unstable by a mirror instability [Kaufmann et al., 1970]. Such structures were subsequently attributed to the mirror mode [Tsurutani et al., 1982; Hubert et al., 1989]. In addition to Earth magnetosheath observations, mirror mode structures have now been identified at comet Halley [Russell et al., 1991, and references therein; Glassmeier et al., 1993], in the solar wind [Winterhalter et al., 1994; Tsurutani et al., 1992], and in the magnetosheaths of Saturn [Tsurutani et al., 1982; Violante et al., 1995; Bavassano-Cattaneo et al., 1998] and Jupiter [e.g., Tsurutani et al., 1982; Balogh et al., 1992; Tsurutani et al., 1993]. In many of these environments, a critical question is where and how the mirror mode can dominate over the ion cyclotron mode. In the presence of ring-type pickup ion distributions, the gyroresonant modes are generally more unstable than the mirror mode, which has a higher anisotropy threshold, and in any given plasma the fastest growing mode will be observed.

At Io one might not immediately expect the mirror instability because the high ambient B of Jupiter's magnetosphere at the location of Io's orbit dictates that the plasma is generally low in the torus. However, kinetic theory and numerical simulations for an electron-proton plasma [McKean et al., 1992] have found that for moderate (rather than very high) ion plasma and anisotropies, the ion cyclotron and mirror modes can produce similar turbulence levels, while at very high values of either parameter the ion cyclotron mode dominates. Also, models and numerical simulations [Price et al., 1986; Gary, 1992; Gary et al., 1993; Anderson and Fuselier, 1993] have found that the addition of helium ions to a proton-electron plasma reduces the proton ion cyclotron growth without significant effect on the mirror mode, and linear theory can account for the transition between cyclotron and mirror-mode dominance under these conditions [Gary et al., 1993]. This suggests to us that the multicomponent plasma composition might be the key to understanding the appearance of mirror-mode structures at Io.

The investigation of pickup ion instabilities has been investigated in some detail at comets. Brinca [1991] provides a useful review of many of the features of cometary pickup ion instabilities, including a discussion of the strong dependence on composition. For example, Brinca and Tsurutani [1987] discuss the effects of pickup oxygen ions, finding growth near the oxygen gyrofrequency. Furthermore, Brinca and Tsurutani [1989] show that multiple pickup ion species can act in concert to drive fluid-like instabilities, while leaving the growth near each species gyrofrequency relatively unaffected, unless the different species have similar masses. As will be shown here, similar effects are found at Jupiter, but the results are further complicated by the presence of thermalized pickup ions that act to damp the cyclotron modes of the dissociated species. Additionally, once developed, the spatial structure of the mirror mode itself may cause the ion cyclotron resonance condition to become position-dependent and thus inhibit the resonance efficiency, as suggested by Southwood and Kivelson [1993]. The nonlinear model of Pantellini [1998] finds that for < 10, the evolved mirror-mode structures take the form of narrow, deep magnetic wells, as has been observed. In this paper we examine the instability criterion and perform a warm plasma dispersion analysis for conditions appropriate to the location of the mirror-mode observations near Io and discuss why they are confined to the narrow wake-edge regions.

2. Observations

Magnetic field measurements from the Galileo magneto-meter at the Io flyby are presented in Figure 2 [Kivelson et al., 1996]. The mirror-mode structures are seen as a series of dips in the field magnitude (and in B , the north-south component), observed between approximately 1744 and 1745:35 UT inbound and between 1747 and 1748:40 UT outbound along the spacecraft path. These are compressional structures, in contrast to the ion cyclotron waves more distant from Io, which are predominantly transverse (B_r , B components). In Figure 3 an example spectrum is given for the mirror-mode observations on the inbound pass (middle plot) and compared with examples from the "background" torus and central wake regions. The power spectra are steep in the presence of the mirror-mode structures, with emphasis toward the low wavelengths.

Figure 2. Magnetometer data from the Io flyby [Kivelson et al, 1996] in right-handed System III (1995) coordinates. The expanded panels show in detail the mirror-mode structures seen on the edges of the Io wake. These appear as a series of dips in the magnetic field magnitude and in B , where the north-south (approximately field-aligned) direction in this coordinate system. Br is radially outward from Jupiter, and B is azimuthal.

By individually examining each of the |B| dips using a triple cross product method, Russell et al. [1999] find that the observed mirror-mode k-vectors are aligned with _kB between 54^o and 85^o to the field. The duration of the |B| dips (Figure 2) is of the order of 3 s, during which time Galileo travels ~60 km relative to Io (approximately perpendicular to the flow), while the plasma travels of the order of between 180 and 60 km (slowing down adjacent to wake) in the plane approximately perpendicular to the B direction. This gives an idea of the scale size of the structures. The Larmor radius _L (= mv / qB) of SO₂⁺ in this region is ~33 km. The gyro-orbit path lengths 2 _L are ~200 km for SO₂⁺, 100 km for S⁺, and 50 km for O⁺, and are therefore of the same order as the mirror-mode structure scale size. This scale size is considerably smaller than those reported in planetary magnetosheath regions (between the magnetopause and bow shock), because of the high ambient field strength (hence small _L ) at the location of Io deep in the Jovian magnetosphere. Io orbits at a distance of 5.9 R_J, where the ambient field is ~1800 nT, while the Jovian magnetopause stands off ~60 R_J at the subsolar point depending on external solar wind pressure and internal magnetodisk mass loading [e.g., Slavin et al., 1985; Khurana, 1997; Huddleston et al., 1998b, and references therein].

Figure 3. Examples of power spectra observed by Galileo in the Io torus and wake regions. The observed large-amplitude mirror-mode structures are approximately linearly polarized, with k vectors aligned 60o-85o to B [Russell et al, this issue].

The mirror-mode waves near Io were observed in regions of steep gradients in the plasma parameters, as seen in Figure 4, occurring approximately where the corotating torus plasma flow is deflected around Io, adjacent to the cold, stagnant wake plasma. Here the field strength depression toward an average ~1200 nT is seen (see also Figure 2), the plasma velocity falls from an enhanced value of ~80 km/s down to nearly zero in the center of the cold wake [Frank et al., 1996], and ion densities are increasing rapidly due to pickup of ionized volcanic gases in close proximity to Io. The middle panel of Figure 4 shows an estimate of the source pickup density obtained by Bagenal [1997] using a total density profile derived from both plasma analyzer (PLS) [Frank et al., 1996] and plasma wave subsystem (PWS) [Gurnett et al., 1996] observations and subtracting an average torus background density contribution. Plotted also is an SO₂⁺ component density estimated from the observed ion cyclotron wave power, using a free-energy analysis of the ion pitch angle scattering [Huddleston et al., 1998a].

Figure 4.Averaged profiles of plasma parameters from the Galileo-Io flyby. From top to bottom the panels show magnetic field magnitude [Kivelson et al., 1996], ion velocity, and temperature from the plasma analyzer instrument [Frank et al., 1996]; total Iogenic source density estimated by Bagenal [1997] (see text) and SO2+ component pickup density estimated by Huddleston et al. [1998a]; and comparison of the RMS amplitudes of compressional and transverse components of B.

In the bottom panel of Figure 4 we compare the RMS amplitudes of compressional and transverse fluctuations through the flyby region. This clearly shows that the regions in which the mirror mode dominates are very narrow. Also, note that transverse power is still evident within the "quiet" wake region where there are no mirror-mode structures observed (Figure 4, bottom panel), and thus it is unlikely that transverse fluctuations through these regions are merely due to "nonideal" alignments of mirror-mode phase fronts for _kB < 90^o.

3. Mirror-Mode Instability Criterion

The mirror mode arises in high-beta plasmas when there is a pressure anisotropy, with P >> P_//. If there are inhomogeneities in the plasma, where the magnetic tension forces cannot contain the perpendicular plasma pressure, the instability leads to magnetic "bottle" structures. Figure 5 is an idealized schematic of the mirror mode. A spacecraft passing through these structures will see the magnetic field strength and plasma density varying in antiphase. By conservation of particle magnetic moment, the plasma is "squeezed" from strong field regions into weak field regions by the mirror force (while the particles' total energy remains unchanged). Since the mirror mode has zero phase velocity (nonpropagating) in the plasma frame, particles with V_// = 0 can be termed "resonant" with the zero-frequency "wave" [Southwood and Kivelson, 1993]. Nonresonant particles will respond to variations in B along the field lines due to their parallel motion, while resonant particles will see mostly temporal changes in B. In addition, strong gradients in the ambient field strength, such as those seen close to Io, may be important if B significantly changes over the scale size of the ion gyro-orbit.

Figure 5. Schematic illustration of the warped magnetic field structure of the mirror mode and idealized anticorrelated field magnitude and density variations along a hypothetical spacecraft path through these structures. [Adapted from Treumann and Baumjohann, 1996; Kivelson and Southwood, 1996].

The instability criterion [e.g., Hasegawa, 1969; Siscoe, 1983] can be expressed

(1)

These estimations are shown in Figure 6b. Note that for two points near 1751 UT (outbound), the estimated total pickup ion perpendicular pressure is too high to have been included in the P_i measured by the PLS instrument. Here P_// cannot be calculated from (3), so instead we extrapolate P_// values for these two points.

Figure 6. (a) Comparison of the observed magnetic field and ion plasma pressure [Kivelson et al., 1996; Frank et al., 1996] and the estimated pickup ion perpendicular pressure. (b) The inferred perpendicular and parallel components of ion pressure (see text). (c) Perpendicular plasma beta calculated from P and overall effective beta (from observed Pi ). (d) Evaluation of the mirror-mode instability criterion (left- and right-hand sides of equation (1)), which is satisfied where the lines overlap, at ~1744 UT.

Figure 6c shows the perpendicular ion plasma calculated from P , which is of the order of 0.05 in the torus and increases to ~0.7 on the edge of the Io wake. The instability criterion is evaluated in Figure 6d, indicating instability where the traces of (1+1/ ) and P /P_// overlap. Because of the uncertainties in the estimated parameters, these numbers should be taken to be illustrative rather than exact. In fact, while we are successful in finding the criterion satisfied near 1744 UT, we are not so successful near 1748 UT (outbound pass). However, the important result is that conditions adjacent to the Io wake are most appropriate for mirror-mode growth, where |B| is depressed and high pickup ion densities provide a large pressure anisotropy that can approach and exceed the threshold for instability. This is indeed the region in which the mirror-mode structures were observed. Near the center of the wake, low plasma velocities imply low pickup speeds (i.e., low V_ring), and hence no significant perpendicular pressure enhancement is provided by the pickup ions there (see Figure 6). Farther from Io, where the ion cyclotron waves were seen, ~0.05, and the background torus plasma dominates over the new source contribution; hence P P_// , and the mirror mode is stabilized. Thus the mirror mode is confined to narrow spatial regions near Io.

4. Dispersion Analysis

A linear, warm plasma dispersion analysis is performed for multiple ion species plus electrons, each described using bi-Maxwellian anisotropic distributions, for the conditions appropriate to the mirror-mode-dominated regions adjacent to the Io wake. Bi-Maxwellians are appropriate for our purposes since it is the anisotropy that drives both the ion cyclotron and mirror-mode growth. We chose average values of B=1200 nT (see Figure 2) and total density N = 10,000 /cm³ for both the ions (summed over species) and electrons to preserve quasi-neutrality. The background density components (N_bg for the corotating torus plasma) are based on the torus species composition observed by Voyager [Bagenal, 1994] and wake observations by Galileo [Frank et al., 1996], for a plasma density of the order of 4000/cm³ (as was used by Huddleston et al. [1997] and Warnecke et al. [1997]). The enhancement in densities observed above background near the Io wake (i.e., ~6000/cm³) are the new pickup ions added into ring distributions, which we include as N_ring while maintaining the appropriate species composition ratios. These densities are listed in Table 1.

For each ion species, an approximate effective perpendicular temperature T is the calculated average over background and ring density contributions, assuming realistic torus background temperatures of T _bg = T_// bg = T_bg = 100 eV and ring T _ring calculated from V_ring for the appropriate species ion mass. Thus T is obtained from

(4)

As an initial step, we investigate the effect of the plasma anisotropy on the growth rates of the mirror mode and the SO₂⁺ ion cyclotron mode, which is the dominant wave in the torus. Figure 7a shows how the growth rate and propagation angle (at peak growth rate) vary with anisotropy if the anisotropy is the same for all species present, while Figure 7b investigates the effect of varying only the SO₂⁺ anisotropy (with the remaining species anisotropies as listed in Table 1, case A). It is clear from Figure 7 that the growth rate of the mirror mode depends on the effective anisotropy of all the species present, while varying just the SO₂⁺ anisotropy (a minor component by number density) has little effect. The SO₂⁺ ion cyclotron mode, however, clearly depends very strongly on the SO₂⁺ anisotropy in both Figures 7a and 7b, as expected, and will grow most rapidly when the SO₂⁺ anisotropy is high but that of the other component species is kept low. In both cases, the mirror-mode propagation angle decreases with increasing anisotropy.

Figure 7. Variation of peak growth rate and peak propagation angle with (a) the anisotropy of all ion species and (b) the anisotropy of the SO2+ component only, while maintaining the remaining species anisotropies at the values of Table 1, case A.

It is unfortunate that the actual ion distribution anisotropies near Io at the time of the Galileo flyby are not known. (These parameters have not been made available from the plasma instruments, at least not to date). For our dispersion analysis, the idealized pickup ion "ring" geometry is assumed, and the relative contribution of the thermalized background torus plasma can only be estimated. A total torus background density of ~4000 ions/cm³ (all species) is expected to be appropriate, at least throughout the region in which ion cyclotron waves were observed and in toward the edge of the wake. However, this thermalized component density is expected to decline in the regions sampled as Galileo moves behind Io into the wake. The transition will most likely be occurring across the wake edge, our region of interest, while the pickup ion density is simultaneously increasing rapidly and the plasma velocity is beginning to ramp down (see Figure 4). We will therefore consider three cases of effective anisotropies in order to investigate the wave dispersion.

Figure 8 presents the dispersion relation solutions for the anisotropy values denoted "case A" in Table 1. In this case, we use the observed ion component densities and our best estimates of the temperature anisotropies as listed in the table, with T values calculated from (4), assuming the background component is present at 4000/cm³ as appropriate to the outermost edge of the region of observed mirror-mode growth. In order to limit the number of species in our code, the O⁺ and S⁺⁺ densities are combined as a single O⁺ component since the gyroresonance is the same for both, (i.e., m/q=16). We compare the resulting SO₂⁺ ion cyclotron wave mode with the mirror mode. For the case A conditions, the mirror mode-growth peaks at _kB = 62^o. The SO₂⁺ ion cyclotron mode has the dominant growth rate for this case, while the cyclotron resonant modes of O⁺ and S⁺ are damped. This is basically the same situation as observed in the regions of the near-Io torus, where SO₂+ ion cyclotron waves dominate (see Figure 1), i.e., from just outside the wake edge out to more than 10 R_Io.

Figure 8. Solutions of the dispersion relation for the parameters given in Table 1, with the anisotropies of case A. Frequency and growth rate are normalized to the SO2+ gyrofrequency (1.8 rad/s for |B|= 1200 nT). In case A, the S+ and O+ ion cyclotron modes are damped (not shown).

Next, for illustrative purposes we investigate the relative growth rates that would result in a hypothetical case where all species have the same anisotropy, while we maintain the observed density ratios (Table 1, third column from left) for the Io wake edge. In case B we arbitrarily choose a high anisotropy (T /T_// =10) for all heavy ion species present, for which the dispersion solutions are presented in Figure 9. In this case, the overpowering density of the O⁺, S⁺⁺ (combined) component prevails, despite the lower ion mass, and produces the dominant ion cyclotron mode. The mirror-mode growth rate peaks at a propagation angle of 45^o and is 80% of the dominant mode peak growth.

Figure 9. Dispersion relation solutions for case B (see Table 1), with T /T// =10 for all the heavy ion species. For the mirror mode, this case has the highest effective anisotropy (over all species present).

As discussed above, there are no observations to tell us how the contribution of the isotropized background torus plasma declines toward the innermost (Io side) of the wake edge in the region of observed mirror-mode growth; however, cases A and B give us a good indication of the conditions that must exist in order for the mirror mode to become dominant there. For the mirror mode to dominate, the SO₂⁺ cannot be the only highly anisotropic species component (as in case A). On the other hand, sufficient isotropic torus background must still be present in order to reduce the effective anisotropies of O⁺, S⁺⁺, and S⁺ to less than that of the SO₂⁺component and hence partially quench the ion cyclotron growth rates (of O⁺ in particular) to below that of case B.

For our final case, we consider the following: As discussed previously, on crossing the edge of the wake inbound toward Io, both the addition of highly anisotropic new ions and the decline of background torus contributions will begin to increase the S⁺ and O⁺ anisotropies above the values of case A. In the central wake itself we envisage that the background component is excluded entirely, having been deflected around Io, but flow velocities are observed to be stagnant there (Figures 1 and 4) and hence there is little pickup energy imparted to new ions in this region (ring V = Vplasma is small), as evidenced by the low level of wave activity observed there. Nevertheless, on the wake-edge transition region of interest the flow velocity is significant and the estimated pickup ion P is considerable (section 3 and Figure 6). Since the transition in the contribution ratio of the thermalized component across the wake edge has not been characterized observationally, in our dispersion analysis we have undertaken a parameter study to find a representative intermediate case. Case C is such an example in which the three cyclotron mode growth rates are more similar to each other, and for which the mirror mode dominates. These dispersion solutions are shown in Figure 10 and are obtained by doubling the anisotropies of O⁺, S⁺⁺, and S⁺ in case C compared with those represented in case A. Note that since there is no significant SO₂⁺ component in the thermalized background torus plasma, the SO₂⁺ anisotropy is unchanged from that of case A and remains the most highly anisotropic component species.

Figure 10. Dispersion relation solutions for case C (see Table 1). In this case, O+ and S++ species are included separately in the dispersion but produce a combined mode (the gyroresonance is for the same mass/q). The mirror mode dominates. Parameters from this result are listed in Table 2.

The total possible ion cyclotron wave energy available in case C is divided between the individual modes (of different gyroresonances) corresponding to the multiple species anisotropic distributions present. Thus, while each of the ion cyclotron modes depends upon the anisotropy of one particular gyroresonant ion species only, the total pressure anisotropy of all species contributes to the mirror-mode instability (see also Figure 7). Table 2 presents the wave parameters obtained from case C (Figure 10). For the mirror mode at peak growth, the dominant wavenumber k = 0.07/km corresponds to a scale size
= 2 /k = 90 km, which is comparable to the observed scale size of the |B| dips (section 2).

5. Conclusions

The present paper has addressed the issues of (1) how the mirror mode can grow in a high |B|, low- plasma and (2) under what circumstances the mirror mode dominates over ion cyclotron modes near Io. The mirror mode is prominent only in a narrow spatial region on the edges of the Io wake. Farther from Io in the torus, the corotating "background" plasma dominates the plasma pressure, i.e., P P_// , and the SO₂⁺ ion cyclotron mode is dominant. The corotating background (thermalized) torus plasma is deflected around Io, and in the center of the cold Io wake the plasma is stagnant (® 1 km/s flow velocity observed). Hence local pickup velocities are very low, the perpendicular pressure contribution from locally picked up ions is small, and there is very little energy for pickup ion wave generation in the central wake. However, at the edges of the wake where |B| is depressed and pickup ion density is increasing, injection V_ring is significant and the P /P_// ratio is then sufficient to overcome the mirror-mode instability threshold (as seen in Figure 6).

The more species there are in the plasma, the more different ion cyclotron modes are possible, but the individual growth rates for these modes will be lower compared with those for a single-species plasma of the same total N. On the other hand, the mirror mode is fed by the combined anisotropy of all the ion species present and can therefore dominate in a multispecies plasma. At Io, the mirror mode is observed in a region where there is a significant deflected plasma flow velocity, an increased density of pickup ions, and declining contributions from the isotropic background torus plasma. However, it is important that there is still enough isotropized O⁺, S⁺, and S⁺ to partially quench the ion cyclotron modes of these species. We have been successful in obtaining a dominant mirror-mode solution to the dispersion (in case C, Figure 10) for realistic parameters in the near-Io environment.

Once mirror-mode structures begin to form, the large spatial gradients in |B| created at the "dips" (and those gradients already present in N due to the Io pickup source) will act to suppress the ion cyclotron resonance if finite V_// takes ions along B where they drop out of resonance. Additionally, if the scale length over which the resonance condition varies is smaller than the heavy ion gyro-orbit, then the resonance would become "confused." The gradients in the ambient plasma parameters will also affect the dispersion for the mirror mode. For the future, a nonlinear analysis of the dispersion at Io may help to determine the importance of these effects.

Acknowledgments. We are indebted to the Galileo magnetometer team and to all those responsible for the success of the Galileo mission. We thank L.A. Frank and the PLS team for the use of the published PLS data. This work was supported by the National Aeronautics and Space Administration through Jet Propulsion Laboratory grants JPL 958510 and JPL 958694. R.J.S. and D.E.H. acknowledge support of grant NAG 5-6965. The travel of X.B.C. to collaborate at UCLA is supported by the UC MEXUS-CONACYT program. This is UCLA-IGPP publication 5072.

Michel Blanc thanks Michele K. Dougherty and another referee for their assistance in evaluating this paper.

References

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