equator-s observations of drift mirror mode waves in the … · particles become trapped in the...

Equator-S observations of drift mirror mode waves in the dawnside

magnetosphere

I. Jonathan Rae,1 Ian R. Mann,1 Clare E. J. Watt,1 Lynn M. Kistler,2

and Wolfgang Baumjohann3

Received 12 September 2006; revised 11 May 2007; accepted 11 July 2007; published 6 November 2007.

[1] The mirror mode is a plasma instability that is typically excited in high-beta plasmaswhere there is significant pressure anisotropy and is most commonly observed in themagnetosheath. However, it is possible under sufficiently anisotropic conditions togenerate the mirror instability inside the magnetosphere, though as yet there are fewexamples. We present an extended interval of �7 hours of mirror mode activity on19 March 1998 when the Equator-S spacecraft was traversing the dawnsidemagnetosphere above the equatorial plane at radial distances of up to L � 11 andencountered quasi-monochromatic �2-min oscillations in magnetic field and ion numberdensity, temperature, and velocity. The magnetic field strength and number densities werein antiphase, and the plasma and magnetic field pressure perturbations were also inantiphase while the observed total pressure remained constant. This is consistent withexcitation via a mirror instability. We are able to discern that the mirror activity must beconfined to approximately within ±20� of the equatorial plane through a conjunction withthe Geotail spacecraft. We find that the condition for mirror mode waves to grow isstrongly met throughout the interval (Hasegawa, 1969). We believe that this is an excellenthigh temporal resolution example of the mirror instability exciting ULF waves inside theEarth’s magnetosphere. Given the coexistence of toroidal oscillations at almost the samefrequency, we suggest coupling between the mirror mode and local standing Alfvenwaves. Our observations hence add to the understanding of how energy can be transferredfrom hot plasma into ULF waves in the magnetosphere before being dissipated in theionosphere.

Citation: Rae, I. J., I. R. Mann, C. E. J. Watt, L. M. Kistler, and W. Baumjohann (2007), Equator-S observations of drift mirror mode

waves in the dawnside magnetosphere, J. Geophys. Res., 112, A11203, doi:10.1029/2006JA012064.

1. Introduction

[2] Compressional Pc5 (150–600 s period) pulsationswith large azimuthal wave numbers, m = 10–100 are oftenobserved in the Earths magnetosphere at large radial dis-tances (dipole L values greater than L = 8) from the Earth[e.g., Takahashi et al., 1987, 1990;Anderson et al., 1990; Zhuand Kivelson, 1991, 1994; Lessard et al., 1999]. Compres-sional high-m pulsation generation bymechanisms internal tothe magnetosphere can be attributed to a number of mecha-nisms including plasma pressure anisotropy [Hasegawa,1969] and ion drift bounce and bounce resonance includingthe drift effects of pressure gradient and gradient curvature ofthe magnetic field [Southwood et al., 1969; Southwood,1976; Cheng and Qian, 1994; Chen and Hasegawa, 1991].

Sonnerup et al. [1969] proposed that many high-m pulsationscan be attributed to drifting plasma pockets or to slowmagnetosonic modes, whileHasegawa [1969] suggested thatthese pulsations could be drift mirror mode driven. High-mwaves may also be excited within regions of high velocityshears associated with low-m transverse Alfven waves [Allanand Wright, 1997].[3] The mirror mode is a compressible slow mode typi-

cally excited in high-b plasmas, as is the drift compressionalmode [e.g., Rosenbluth, 1981; Crabtree and Chen, 2004].Unlike the drift compressional mode, however, the mirrormode requires significant perpendicular pressure anisotropy.Mirror modes may grow if the free energy of the perpen-dicular pressure anisotropy is sufficiently large [e.g., Vaivadset al., 2001]. The mirror instability is a macroinstabilitywhich generates waves which preferentially grow in thenearly perpendicular direction. Both electron and ion pres-sure anisotropies contribute to the linear instability condi-tion, but the species that has the strongest anisotropy willdominate (see section 4). However, the magnitude of thelinear growth rate for the mirror instability is typicallydominated by ion anisotropy and proportional to the parallelion thermal velocity. Observational evidence for the exis-

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1Department of Physics, University of Alberta, Edmonton, Alberta,Canada.

2Department of Physics, University of New Hampshire, Durham, NewHampshire, USA.

3Space Research Institute, Austrian Academy of Sciences, Graz,Austria.

Copyright 2007 by the American Geophysical Union.0148-0227/07/2006JA012064$09.00

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tence of these mirror mode structures are mainly concen-trated in the dayside magnetosheath [e.g., Tsurutani et al.,1982; Hubert et al., 1989; Hill et al., 1995; Schwartz et al.,1996; Lucek et al., 1999a, 1999b, 2001], where the neces-sary temperature anisotropies can develop. They have alsobeen observed during bow shock crossings [e.g., Lacombe etal., 1992; Czaykowska et al., 1998, 2001], but rarely insidethe magnetosphere, for example, in the magnetotail duringsubstorms [e.g., Haerendel, 2000], in the ring currentregion [e.g., Woch et al., 1988], in the dawnside plasmasheet [e.g., Vaivads et al., 2001], and on the magneto-spheric flanks [e.g., Zhu and Kivelson, 1991, 1994]. Astabilizing effect on this temperature anisotropy occurs dueto competition between mirror mode growth and otherkinetic instabilities arising from anisotropic plasma pres-sure such as electromagnetic ion cyclotron (EMIC) waves,which can also consume some of the free energy in thesystem [Treumann and Baumjohann, 1997].[4] Figure 1 shows a schematic (adapted from Treumann

and Baumjohann [1997]) of the structure of a simple mirrormode structure in a straight field line topology, with nocoupling to the drift mode or drift Alfven ballooning mode[e.g., Chen and Hasegawa, 1991]. Particles become trappedin the magnetic mirrors under the action of large perpen-dicular plasma pressure and the magnetic field inflateslocally. In the magnetic bottles, particles mirror betweenthe local magnetic field maxima and particles stream intothe mirror during instability growth. Spatial structures thusdevelop that have maxima in magnetic field strength at thesame locations where there are minima in plasma density. Inthis schematic the mirror mode is a zero frequency standingmode; any oscillatory frequency observed by a spacecraft isgenerated by the motion of the spacecraft through spatialstructures perpendicular to the background field.

[5] The drift mirror instability takes into account theeffects of particle drifts in curved field line topologies butis generated by the same pressure anisotropy and adheres tothe same threshold conditions as the mirror instability[Hasegawa, 1969] under spatial homogeneity. However,in the case of the drift mirror instability there is a realoscillation frequency associated with the diamagnetic driftfrequency generated via coupling to drift waves. On curvedquasi-dipolar field lines in the Earth’s magnetosphere,coupling between drift mirror instability and the transverse(shear) Alfven wave must also be taken into account [e.g.,Chen and Hasegawa, 1991, 1994]. Including both strongmagnetic field line curvature and plasma density inhomo-geneity in the derivation of the dispersion relation canreduce the instability threshold for ring current plasmas[e.g., Woch et al., 1990]. Observations indicate that themirror effect often dominates the collective particle re-sponse [e.g., Baumjohann et al., 1987; Zhu and Kivelson,1994].[6] Lucek et al. [1999a, 1999b] used Equator-S magnetic

field data to classify magnetosheath mirror waves as large-amplitude regular fluctuations in magnetic field strength,with a small angle between maximum variance and theaverage field direction. Treumann and Baumjohann [2000]and Treumann et al. [2004] studied how mirror modes couldbe trapped inside magnetic boundaries, such as the magne-tosheath. In these papers, the authors suggested that themirror mode is rarely observed in its linear state, that themagnetic perturbations tend to be comparable to the totalmagnetic field strength (dB/B � 1). Baumjohann et al.[2000] identified intervals of magnetospheric ‘‘lion roar’’;that is, near-monochromatic bursts of electron whistlerwaves that in the magnetosheath are well correlated withmirror wave activity. In their study, Baumjohann et al.[2000] found that these whistler waves were not onlyabundant in the magnetosheath but also occurred in theequatorial dawnside magnetosphere, perhaps indicative ofmirror mode activity inside the magnetosphere on a regularbasis. Haerendel et al. [1999] studied the entirety of theEquator-S database and showed that approximately onethird of the spacecraft orbits in the morningside (between1100 and 0200 MLT) contained periods of high thermal tomagnetic pressures (high plasma beta), and named thesefeatures ‘‘plasma blobs.’’ Furthermore, the plasma andmagnetic pressures were in antiphase, such that the totalpressure remained approximately constant. Although nodefinitive classification of these plasma blobs as beingmirror mode waves is reached in the Haerendel et al.[1999] study, the inference is that mirror waves may bemore prevalent in the magnetosphere than previouslythought.[7] In this paper we present an interval of compressional

Pc5 perturbations observed by the Equator-S spacecraft inthe dawnside magnetosphere. We identify that the compres-sional perturbations are associated with the mirror instabil-ity and that the linear mirror instability criteria is stronglysatisfied throughout the interval. We find that there is noclear energy dependence to the particle response [cf. Zhuand Kivelson, 1994], the symmetric enhancements in parti-cle flux perpendicular to the field as a function of energyand pitch angle suggesting that the waves were excited by

Figure 1. A schematic representation of (top) the spatialstructure across a mirror-unstable region and (bottom)hypothetical satellite measurements across this region(after Treumann and Baumjohann [1997], reprinted bypermission).

A11203 RAE ET AL.: MAGNETOSPHERIC MIRROR MODE WAVES

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the mirror instability rather than the classic drift bounceresonance mechanism [cf. Takahashi et al., 1990].

2. Instrumentation

[8] The Equator-S spacecraft was launched into an iner-tial, eccentric, near-equatorial orbit in December 1997. Inthis paper we use prime parameter data at 1.5 s resolutionfrom the magnetic field instrument (MAM) [Fornacon etal., 1999]. The Ion Composition Instrument (ESIC) [Kistleret al., 1999] measures the three-dimensional (3-D) distribu-tion functions of the major ion species, H+, He+, He++, andO+ in the energy range 15 eV to 40 keV. In this paper weutilize proton densities, temperatures, and 3-D velocitymoments at spacecraft spin resolution (�1.5 s). The mag-netic field (MGF) [Kokubun et al., 1994] and plasmamoment data (LEP) [Mukai et al., 1994] on board theGeotail spacecraft were utilized at 3 s and 12 s, respectively.

3. Observations of 19 March 1998

3.1. Spacecraft Location

[9] This interval was one of the identified Geotail/Equator-S conjunctions that occurred before the failureof the Equator-S satellite. Figure 2 shows the positions ofboth the Equator-S and Geotail spacecraft in the GSM X-Zplane (Figure 2a) and X-Y plane (Figure 2b) between 1700and 2400 UT on 19 March 1998; the satellite symbol

indicates the position at the start of the interval. Alsoplotted in Figure 1 is a model magnetopause [Shue et al.,1997] for perspective for the average IMF and solar windconditions during the interval detailed in section 3.2. FromFigure 2 it can be seen that Equator-S started the intervalclose to apogee on the dawnside of the magnetosphere.Equator-S was situated close to the magnetic equator at ageocentric distance between 10 and 11.5 RE. During thisperiod, Geotail crossed the Equator-S satellite path in theGSM X-Y plane but was situated some 3–5 RE below theequatorial plane and mapped to a larger geocentric dis-tances of up to 12.5 RE at closest approach to Equator-S.

3.2. Solar Wind

[10] In this paper we monitor the solar wind conditionsusing data from the magnetometer (MAG) [Smith et al.,1998] and the solar wind proton alpha monitor (SWEPAM)[McComas et al., 1998] on the Advanced CompositionExplorer (ACE) spacecraft. Figure 3 shows the unlaggedupstream solar wind conditions for 19 March 1998, 2000–2400 UT measured by ACE which was situated around[XGSE, YGSE, ZGSE] � [243, �20, 16] RE around this time.

Figure 2. Location of the Equator-S and Geotail space-craft in Geocentric Solar Magnetic (GSM) coordinates in(a) X-Y plane and (b) the X-Z plane between 1700 and2400 UT on 19 March 1998, with the satellite symbolindicating the position of each spacecraft at the start of thetime interval. Also plotted in Figure 2 are the positions of theShue et al. [1997] magnetopause for the relevant inter-planetary magnetic field (IMF) and solar wind conditions.

Figure 3. ACE magnetic field and plasma data from 19March 1998, 2000–2400 UT in GSM coordinates. From topto bottom, the following parameters are plotted: IMF Bx, By,Bz, BT, solar wind number density (n), velocity (Vsw), anddynamic pressure (Psw). ACE was situated around [XGSE,YGSE, ZGSE] � [243, �20, 16] RE around this time and nolag has been added to the data.


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During the interval the IMF Bx was first negative andaround �2 to �3 nT before it increased from 2110 UT toaround 2200 UT where it turned positive and reached amaximum value of around 4 nT. The IMF By was predom-inantly negative and decreased from around�4 nT to�6 nTin strength, apart from a positive excursion between�2205 and 2220 UT, and the IMF Bz was predominantlynegative and up to �5 nT in strength apart from a positiveexcursion between �2040 and 2110 UT. The magnetic fieldstrength varied between �3 nT at the start of the interval

and �7 nT at the end. The solar wind number density ishigh and variable between 8 and 16 particles cc�1, the solarwind velocity slow and relatively constant (<330 km s�1),and the solar wind dynamic pressure variable between 1.3and 2.6 nPa. In summary, the solar wind and IMF con-ditions are unremarkable, lacking those conditions usuallyassociated with enhanced ULF wave power inside themagnetosphere, that is, high solar wind speed flow, solarwind buffeting, or quasi-periodic variations in the interplan-etary medium.

3.3. Equator-S Plasma and Magnetic Field Data

[11] Figures 4a–4e show the GSM Bx, By, Bz, andmagnetic field magnitude BT from the Equator-S MAMinstrument, together with the parallel (black trace) andperpendicular (grey trace) proton temperatures between2100 and 2400 UT on 19 March 1998. Figures 4f–4j showexactly the same parameters for the Geotail spacecraft forcontext. There are clear perturbations in all three compo-nents of the magnetic field observed at Equator-S, as well asthe total magnetic field of varying periodicity but approx-imately �2 min duration. The magnetic perturbations aresmall (�4 nT peak-to-peak maximum) in the GSM xdirection (�azimuthal in this interval), and large in the y(�radial) and z (�field-aligned) directions, with maximumpeak-to-peak fluctuation amplitudes of around �10 nT and�15 nT in the y and z directions, respectively. Also obviousfrom Figure 4 is that the By and Bz components of themagnetic field are very strongly and clearly anticorrelatedthroughout the interval, and the Bz and total magnetic fieldsare extremely well correlated. The magnetic field magnitudeincreased from �15 nT at the start of the interval to �30 nTat 2300 UT, perhaps suggesting that the spacecraft wasapproaching the magnetopause. The temperature anisotropyis large; the perpendicular proton temperatures observedwere up to twice as large as their parallel counterparts. TheGeotail data show longer period magnetic perturbationswhich are strongest (�4 nT peak-to-peak) in the GSMy direction. Contrary to the Equator-S observations, themagnetic field observed by Geotail is dominated by theGSM y component, indicative of the fact that Geotail isbelow the equatorial plane. Geotail proton temperatures areup to �20% larger in the perpendicular direction as in theparallel direction, a ratio an order of magnitude lower thanthe anisotropy observed by Equator-S.[12] Figure 5 shows the Equator-S magnetic field data in

a field-aligned coordinate system. To calculate the directionof the measured fields in a coordinate system aligned withthe ambient magnetic field, the data was transformed from[XGSM, YGSM, ZGSM] to [XFA, YFA, ZFA], where ZFA is field-aligned. Here, XFA lies in a plane defined by ZFA and thegeocentric radius vector to the spacecraft and is perpendic-ular to ZFA, and YFA completes this right-handed orthogonalset. This orthogonal coordinate system was derived using arunning mean of the ambient magnetic field of 15 minduration, and is the same coordinate system used by Raeet al. [2005]. In this interval XFA is approximately radialand YFA is approximately azimuthal. From top to bottom,Figure 5 shows BXFA (Figure 5a), BYFA (Figure 5b), andBZFA (Figure 5c) magnetic field components, togetherwith three component ion velocity VXFA, VYFA, and VZFA

(Figures 5d–5f). Figures 5g and 5h show the ion number

Figure 4. Equator-S and Geotail magnetic field andplasma data from 2100 to 2400 UT on 19 March 1998.From top to bottom, the following parameters are plotted:Equator-S (a) GSM Bx, (b) By, (c) Bz, and (d) BT, and(e) parallel (black) and perpendicular (grey) ion tempera-tures. (f)–(j) As in Figures 4a–4e but for Geotail GSM.


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density and (parallel, denoted by black trace, and perpen-dicular, denoted by grey trace, components of) ion temper-ature, respectively, as derived from the ESIC instrument.Figure 5i shows the mirror instability condition, G = 1 +b?(1 � T?/Tk) [Hasegawa, 1969], where b? = nikBTi?/(B2/2m0). We concentrate mainly on the compressional(BZFA) magnetic field component and the two azimuthalcomponents of the ESIC ion velocity vXFA and vYFA. Theseparameters show that there are significant �2 min periodmagnetic field perturbations (>10 nT peak-to-peak) parallelto the background magnetic field direction, indicative ofcompressional wave activity. Also the ion velocities in bothcomponents transverse to the magnetic field are much largerand more oscillatory than the velocities parallel to theambient magnetic field. The ion number density rangesfrom 0.4 to 0.8 cm�3 and also oscillates during the interval.Furthermore, both the parallel and perpendicular iontemperatures also oscillate in phase with the ion numberdensity for extended periods during the interval, this beingmost clearly observed between 2100 and 2400 UT betweenthe ion number density and the perpendicular iontemperature. Interestingly, throughout most of the interval,the mirror instability criterion is strongly satisfied, incontrast to previous studies [e.g., Zhu and Kivelson, 1994;Vaivads et al., 2001] where even close to the magneticequator, the instability criteria was only more sporadicallysatisfied.[13] Figure 6 shows the relationship between the com-

pressional component of the magnetic field together (blackline) with the ion number densities (grey line) during theinterval 2100–2400 UT on 19 March 1998; the ion densitiesare high-pass filtered at 1 mHz to remove any long-termtrends and plotted on the right-hand y axis scale forcommon reference. It is clear from Figure 6 that thecompressional magnetic field perturbations and densitiesare unequivocally in antiphase. In almost every case, thecompressional components enhancements (reductions) areaccompanied by density reductions (enhancements). Thisbehavior is characteristic of a slow mode wave or mirrormode wave activity and is discussed further in section 5below.

Figure 5. Equator-S magnetic field and plasma datatransformed into field-aligned coordinates [XFA, YFA, ZFA](see text for details). This orthogonal coordinate set isdetailed by Rae et al. [2005], using a running mean of theambient field of 15 min duration. From top to bottom,Figure 5 shows the wave dBXFA, dBYFA, dBZFA, dVXFA,dVYFA, dVZFA, parallel (black line) and perpendicular (greyline) temperature, number density, and mirror instabilitycriterion for the interval 1900–2400 UT.

Figure 6. Figure 6 shows the relationship between thecompressional component of the magnetic field (blacktrace) and ion number densities (grey trace) during theinterval 2100–2400 UT on 19 March 1998.


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[14] Figure 7 shows the calculated magnetic (B2/2mo) andperpendicular ion thermal (nikBT?) and total (B2/2mo +nikBT?) pressures from the Equator-S spacecraft for asubsection of the interval 2136–2248 UT. From Figure 7we can see that the plasma pressure and the magneticpressure oscillate in anti-phase such that the total pressure,although rising slightly, remains approximately constant.More importantly, the perturbations that are present in boththe plasma and magnetic terms are not present in thecalculated total pressure. This remarkable total pressureperturbation cancellation occurs despite the fact that the ionplasma pressure (ESIC) and magnetic pressure (MAM) aremeasured independently by completely different instru-ments, and despite the fact that the perpendicular pressure iscalculated for a single perpendicular ion temperaturederived only from measurements in the ESIC energy rangeusing an ideal gas law and that ion number density isderived only using ions within the energy range of the ESICinstrument. Figure 5 shows that the maximum temperaturein this interval is �7 keV, and so the ESIC instrumentshould measure the majority of thermal ions thus yieldingan accurate ion pressure measurement. We also do not takeinto account the contribution from electron thermal pressureeven though this may contribute around 1/5–1/7 of the totalthermal pressure [e.g., Baumjohann et al., 1989].[15] Figure 8 shows ion spectra and magnetic field data

from the Equator-S ESIC and MAG instruments from a

smaller subsection of the event interval, from 2145 to2220 UT. From top to bottom, Figure 8 shows an energyspectrogram of omnidirectional ion fluxes between 10 and30 keV, the compressional magnetic field variation, andintegrated ion fluxes between 15 and 25 keV as a functionof pitch angle. Omnidirectional flux enhancements in Figure8a are clearly associated with local magnetic field minima(Figure 8b), and the local magnetic field enhancements areassociated with the minima in ion flux, that is, the magneticfield magnitude is anticorrelated with ion flux. Figure 8cshows the pitch angle of the 15–25 keVenergy ions, where 0�and 360� are parallel to the magnetic field, 180� is antipar-allel, and 90� and 270� are perpendicular to the magneticfield. We can see that there are symmetric flux enhancementsin the field perpendicular direction. In summary, magnetic

Figure 7. Calculated total, magnetic (B2/2mo denoted bydashed black trace), perpendicular thermal (nikBTf denotedby grey trace) and total (thick black) proton pressuresbetween 2136 and 2248 UT.

Figure 8. From top to bottom are shown Equator-S IonComposition Instrument (ESIC) Omnidirectional Ion FluxSpectrogram between 10 and 30 keV, together with thecompressional component of the magnetic field and pitchangle information of the ESIC ion fluxes. The dotted linesindicate 90� and 270�; that is directions perpendicular to thebackground magnetic field.


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field enhancements (depressions) are associated with perpen-dicular ion flux reductions (enhancements).

4. Mirror Mode Instability Criteria

[16] From linear kinetic theory, we can obtain the mirrorinstability condition for a uniform magnetic field and thedrift mirror instability condition in curved field line geom-etry [Hasegawa, 1969]:

Xs

b2s?bsk

> 1þXs

bs? ð1Þ

where

bs?;k¼

nskBTs?;k

B2=2m0

� �

in the limit k? kk where kk,? are the parallel andperpendicular wave numbers. Here, B is the ambientmagnetic field strength, Tk,? are the parallel and perpendi-cular temperatures, bk,? are ratios of the parallel andperpendicular plasma pressures to the magnetic fieldpressure, and s indicates a summation over species. In thispaper we concentrate on the ion temperature anisotropies,and so from now on bsk,? are understood to be theperpendicular and parallel ion b derived from the ESICdata. If the condition in equation (1) is satisfied then the

mirror instability is able to extract energy out of the ambientplasma. However, if kk � k? then the temperature anisotropyneeds to be much higher in order for mirror mode instabilityto occur [Vaivads et al., 2001]. Figure 9 shows the solutionsof equation (1) resulting from bk and b? calculated usingthe proton densities and temperatures derived from theEquator-S ESIC and MAM data (Figure 9a) and the GeotailLEP and MGF data (Figure 9b). The grey line in Figures 9aand 9b shows the mirror mode growth criteria with equalityin equation (1). The black dots denote the values of the leftside of the equation calculated from the ion density and iontemperature and magnetic field parameters measured by theEquator-S ESIC and MAM instruments, and for context,Geotail LEP and MGF data, respectively, between 1700 and2400 UT on 19 March 1998. Figure 9a clearly shows thatthis criteria for mirror mode growth is strongly satisfiedthroughout the entire interval, since the black dots all lieabove the grey line. This is in contrast to some previousstudies such as Vaivads et al. [2001] where during aninterval identified as mirror mode activity in the magneto-sphere, the same criteria for mirror mode growth was onlysporadically satisfied. Figure 9b shows that the mirrorinstability criterion is not satisfied at the location of theGeotail spacecraft, since the black dots lie below well thegrey line.[17] An approximate linear mirror mode growth rate, gmi,

can be calculated using the expression derived by Treumannand Baumjohann [1997] from the linear kinetic dispersionrelation, and using the assumptions used in equation (1),that is, in the limit k? kk, and without summing over allion species:

gmi ¼ffiffiffi2

p

rbik

b2i?

bi?bi?bik

� 1

! !� 1

" #kk�thik : ð2Þ

Here, w is the wave frequency, Wi = qB/m is the ion

gyrofrequency, k = (k?2 + kk

2)1=2 , vthi,k =

2kBTikmi

�1=2is the

parallel ion thermal speed. Zhu and Kivelson [1994] andVaivads et al. [2001] identified that mirror mode activitywas limited to within ±20� of the magnetic equator in themagnetosphere. We can verify this by utilizing the conjunc-tion between Geotail and Equator-S. Around the point ofclosest approach, the two spacecraft were separated by�3.6 RE in the ZGSM direction, and Geotail was around[XGSM, YGSM, ZGSM] = [�2, 9, �4] RE and did notobserve mirror activity. This gives an estimate that Geotaillies within �24� of the magnetic equator. We use the ±20�limit to estimate a minimum value of kk assuming that theobserved waves correspond to the fundamental mode. Theobservations presented in this paper are taken at a distanceof �11 RE from the Earth. Assuming a dipolar fieldgeometry, the arc length, ds, of a field line is

ds ¼Zq2q1

req 1þ 3 sin2 q �1=2dq ð3Þ

Here, req is the equatorial crossing distance of a dipolarfield line, and q is the north-south angle with respect to themagnetic equator. By numerical integration of this

Figure 9. Mirror mode instability criteria for both the (a)Equator-S and (b) Geotail observations from 1700 to2400 UT on the 19 March 1998. In both Figures 9a and 9b,the grey line represents the solution to the right-hand side ofequation (1) while the points denote the values of the left-hand side of equation (1) calculated from the plasma andmagnetic field parameters measured by the Equator-S ESICand magnetic field instrument (MAM) and Geotail plasmamoment data (LEP) and magnetic field data (MGF),respectively.


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equation between ±20�, one finds that the maximum extentof a half-wavelength of this fundamental mode is �8 RE.Assuming that the previous observations are correct, thisprovides a minimum kk of around 6.2 � 10�8 m�1. Usingequation (3), we can thus calculate an approximate lineargrowth rate of the mirror instability. We estimate theaverage linear growth rate to be 5.3 � 10�3 s�1, whichcorresponds to an e-fold increase of the perturbationamplitude in 189 s, which is of the order of the waveperiod of the observed perturbations. Note that the ratio ofdB/B is large (0.2–0.3) during this event, which meansthat a linear approach may be insufficient in diagnosingthe behavior of this instability. Pokhotelov et al. [2004]demonstrated that the mirror instability criteria can bestrongly modified by the inclusion of finite gyroradiuseffects and that it is critically dependent on k?

2 ri2.

Although it is difficult to estimate k? from single pointmeasurements, it is sometimes possible to estimate aperpendicular wave number using finite gyroradiusmethods [e.g., Su et al., 1977]. However, these methodsrely on a phase difference between back-to-back iondetectors, which unfortunately in this instance does notoccur; the perpendicular ion enhancements are symmetricalin both field-perpendicular directions (cf. Figure 9b).[18] Interestingly, Figure 5 shows that at times during the

mirror interval large amplitude oscillations with similarfrequencies to the mirror mode structures are seen in theperpendicular plasma velocity, especially vYFA. Accordingto simulations by Gary [1992] and Yoon [1992] and theanalysis of Pokhotelov et al. [2004], the mirror mode mayhave a maximum growth rate of k?ri � 0.7. Thiscorresponds to a perpendicular wavelength l? � 9ri. Usinga typical perpendicular plasma velocity of ±70 km/s (seeFigures 5d and 5e), a ULF wave with a 2 min period wouldhave a perpendicular displacement z of ±1300 km. For a20 keV plasma, the H+ ri � 1000 km in a 20 nT backgroundfield. Consequently, since the ratio z/l? � 0.14, formaximum growth k?ri � 0.7, the advection of the plasmaacross the spacecraft by the v? field of the coincident ULFmode should lead to the quasi-periodic mirror modeperturbations observed. Of course, the local structure ofthe mirror mode means the waveforms in v? and bk will notbe identical, just as observed. Indeed, as discussed by Chenand Hasegawa [1991, 1994], the mirror and Alfven modesthemselves may also be coupled.

5. Discussion

[19] The mirror mode is in practice difficult to distinguishfrom the perpendicularly propagating slow mode, particu-larly from single-point spacecraft measurements. Through-out this interval, the Equator-S observations show that themirror mode instability criteria are satisfied, and the lineargrowth rate is large and positive over a �7 hour period. Inthis case, we rule out slow mode activity, as a slow modemight be expected to be heavily ion-Landau damped in ahigh-b plasma [e.g., Krauss-Varban et al., 1994].[20] Other potential generation mechanisms include ex-

ternal energization of the magnetospheric cavity or interac-tion of the wave electric field with a drifting particlepopulation. The magnetospheric cavity may be energizedby either high solar wind speeds [e.g., Southwood, 1974;

Chen and Hasegawa, 1974; Mann et al., 1999; Mann et al.,2002; Rae et al., 2005] or by changes in the properties ofthe solar wind and/or IMF [Allan et al., 1986; Wright, 1994;Wright and Rickard, 1995;Mathie and Mann, 2000], neitherof which are present in the observed interplanetary con-ditions. As discussed in section 3.2, the solar wind speed isslow (<330 km s�1) and the interplanetary conditionsunremarkable other than a relatively high proton numberdensity. The generation mechanisms described above arenot typically observed under these interplanetary condi-tions; the solar wind driven energy input into the magneto-sphere is not expected to vary substantially. The driftbounce resonance mechanism [Southwood et al., 1969] isnot likely to be responsible for the generation of thecompressional perturbations, as the resonant particlesshould oscillate 90� out of phase with the magnetic pertur-bations, rather than the 180� observed in this case. If theseperturbations were driven via drift bounce resonance aphase change of 180� should also be observed across theenergy ranges immediately above and below the resonantparticle energy [cf. Takahashi et al., 1990]. However, in thiscase, there is no distinct energy-dependent phase responseof the ions, at least within the energy range of the ESICinstrument. The response of the particles is such that thecompressional wave influences the ions at all energies at thesame phase 180� out of phase with the compressionalmagnetic component of the wave.[21] Zhu and Kivelson [1994] summarized results of

particle flux modulation by compressional ULF waves bynoting that energetic proton flux (>27 keV) oscillated inantiphase with magnetic pressure variations [e.g., Kremseret al., 1981]. Zhu and Kivelson [1994] analyzed eight eventsfrom ISEE-1 and -2 data to determine the structure andgeneration mechanism of compressional ULF waves frommagnetic field and plasma diagnostics. In their paper, Zhuand Kivelson [1994] found an energy dependence in theparticle response and that compressional waves were ob-served where b > 1. In particular, these authors concludedthat the energy response of >10 keV ions varied in antiphasewith the compressional magnetic variations, consistent withmirror mode activity. In this paper the authors also con-cluded that if there were compressional waves present in theouter magnetosphere, the criteria for mirror instability wasfrequently found to be satisfied near the magnetic equator.The interval presented in this paper is consistent with theconclusions of Zhu and Kivelson [1994] on both counts.The ion fluxes >10 keV shown in Figure 8 vary in antiphasewith the compressional component of the magnetic field.The orbit of the Equator-S spacecraft was designed to spendthe majority of its lifetime in the equatorial plane, and in thisinterval we find that the mirror instability condition isgenerally satisfied throughout the 1700–2400 UT interval.Further, the off-equatorial measurements from Geotail con-firm that these drift mirror observations are spatially local-ized; the perturbations observed by Geotail �1.5 RE furtherout in L (and �4 RE apart in ZGSM) have a longer period,there is a smaller temperature anisotropy (20% c.f. >200%),and the mirror instability criterion is not satisfied during thesame interval. The particle response to these compressionalwaves is also discussed by Southwood and Kivelson [1993]and Kivelson and Southwood [1996]. These authors foundthat there was a strong pitch angle dependence on particle


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response. Kivelson and Southwood [1996] found that par-ticles with pitch angles around 90� give up their energy tothe instability, whereas particles with close to 0� pitchangles can pass directly through the series of magneticmirrors easily. Southwood and Kivelson [1993] describedthese two particle populations as resonant and nonresonantdistributions, respectively. These authors found that plasmaand magnetic field variations were in phase for resonantpopulations, whereas plasma and magnetic field variationswere in antiphase for nonresonant (or bulk) particle pop-ulations. From Figure 7, it can be seen that the latter case isdominant in this case study. The ion number density andmagnetic field strength observed by Equator-S are in anti-phase, further indication of a mirror mode source, which isshown in Figure 1. The plasma and magnetic field varia-tions are in antiphase, such that the perpendicular ionpressure (nikBT?) derived from ESIC ion density andtemperature and the magnetic pressure (B2/2mo) derivedfrom the MAM instrument on Equator-S and shown inFigure 7 are also anticorrelated. The result is that althoughthe total pressure increases slightly (from �0.7 to 0.9 nPa),the total pressure remains approximately constant. Remark-ably though, the oscillatory behavior of both the plasma andmagnetic pressures are not evident in the combined ion total(plasma plus magnetic) pressure. Haerendel et al. [1999]presented several intervals with the same characteristics;that is plasma and magnetic pressure perturbations that werein antiphase. In absence of the mirror instability analysispresented in this paper, these authors termed these features‘‘plasma blobs.’’ However, these plasma blobs are mostlikely mirror instability related. This pressure relationship isexpected for the mirror mode dominated by a nonresonantparticle population. The study presented in this paper is arare observation of long-lasting (�7 hours), drift mirrormode wave activity inside the magnetosphere.[22] Overall, the data presented in this interval provide

compelling evidence in support of mirror mode activity inthe near-equatorial dawnside magnetosphere. The linearmirror mode instability criteria (derived for kk � k?[Hasegawa, 1969]) is satisfied strongly for most of theinterval 1700–2400 UT. In comparison, both Zhu andKivelson [1994] and Vaivads et al. [2001] presentedintervals where this same linear drift mirror instabilitycriteria (equation (1)) was only sporadically satisfied, evenwhen relatively close to the magnetic equator. In ourinterval, there are few data points that even approach thelinear instability condition given in equation (1). Duringthis event, the magnetospheric plasma was highlyanisotropic, such that the linear instability criterion wasfulfilled with ease. Recent work by Walker [2005] hasshown that when wave-particle interactions are taken intoaccount, the instability criteria may be changed, such thatthe grey solid line shown in Figure 9a would have agradient of six, rather than one. Under these circumstances,we would not expect to see growth during this interval, butmore likely a saturated instability. In our interval, the vastmajority of points lie between the linear instabilitycondition described in equation (1) and a line with twicethe gradient. The most complete kinetic analysis so far[Pokhotelov et al., 2004] indicates that the fastest growingwave mode satisfies k?ri < 1.

[23] We estimate the average linear growth rate during2140–2204 UT to be 5.3 � 10�3 s�1, which is remarkablyfast. This means that the growth rate is therefore a sizeablefraction of the real observed frequency, and the instabilitywould be expected to saturate quickly with respect to theduration of the observations presented here. We concludethat the linear dispersion relation is likely to be inadequateto describe this system. One would expect that as the waveextracts energy out of the plasma, the temperature anisot-ropy would decrease to bring the plasma closer to equilib-rium. However, Figure 9 shows that even after a prolongedperiod, the linear mirror instability criterion is still satisfied.Southwood and Kivelson [1993] identified that the lineargrowth rate was inversely proportional to the number ofresonant particles (particles with low parallel velocity)available inside the mirror region. Figure 8 (bottom) showsthe pitch angle distribution of energy flux integrated be-tween 15 and 25 keV. It is clear from Figure 8 that thegreatest energy fluxes for this energy range are concentratedin the perpendicular directions (90� and 270� denoted by thedashed lines). On the contrary, there are significantly lowerfluxes in the parallel direction (0� and 180�), indicating thatthere are few resonant particles observed in this region.From the work of Southwood and Kivelson [1993], wemight expect the growth rate to be lower than the estimateof 5.3 � 10�3 s�1 given above.[24] The interval presented in this paper shows large-

amplitude (dB/B � 0.2–0.3) mirror mode waves. Theseobserved waves are of 2–3 min duration and predomi-nantly compressional in nature. However, between 2145and 2230 UT, there is evidence of significant magneticfield perturbations transverse to the background field(�4–6 nT peak-to-peak), along with compressionalperturbations (�15 nT peak-to-peak). Further, the perpen-dicular ion velocities (±100 km s�1 peak-to-peak) weredominant compared to the field-aligned velocities and showcoherent oscillatory behavior in specific intervals (seeFigure 5e). In the interval 2100–2400 UT the frequenciesof the perpendicular azimuthal velocities (v?y) and com-pressional magnetic field perturbations are not generallyrelated. However, during the two intervals with clearand strong azimuthal ion velocity perturbations (2205–2225 UT and 2330–2340 UT), the azimuthal velocitiesand compressional magnetic field perturbations are of thesame frequency and are 90� out of phase. This suggests thatduring these two intervals, the oscillations are not solelydrift mirror waves, but rather that the v?y perturbationsmay be bending the local field line structure and theazimuthal v?y signature is the signature of a standingAlfven wave coupled to the mirror mode, that is, there mayan Alfvenic component standing along the field lines.However, we note that the treatment of the mirror mode byGenot et al. [2001] and Pokhotelov et al. [2004] reveals thatthe mirror mode may also generate an Alfvenic-liketransverse magnetic field component simply by incorporat-ing finite gyroradius effects. Unfortunately, during thisinterval both the Equator-S and Geotail spacecraft were notmagnetically conjugate, and furthermore, the magnetic footpoints of both spacecraft mapped to a region north of theRussian Coast that had few ground magnetometer stations,and so we could not confirm whether there was a groundsignature of a standing Alfven wave with sufficiently low k?


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to be observed on the ground [e.g., Hughes and Southwood,1976]. We do however suggest that the Alfvenic waveoscillation in dv?y is an important part of the plasmaresponse, as our observations are consistent with a coupleddrift Alfven-mirror mode. This is perhaps to be expected oncurved quasi-dipolar field lines where perpendiculargradient-curvature drift of ions providing the thermalpressure at energies at �20 keV is likely to be important.We expect this observational interval to promote furtherstudies of drift Alfven-mirror mode coupling.

6. Conclusions

[25] We present a long-lasting (�7 hour) interval of �2–3 min period drift mirror mode waves observed by theEquator-S MAM and ESIC instruments on 19 March 1998.We identify the compressional perturbations as likely driftmirror waves for a number of reasons:[26] 1. The proton number density and temperatures are

correlated, while both density and temperature are anticor-related with magnetic field strength.[27] 2. The oscillatory perpendicular ion pressure

(nikBT?) and magnetic pressure (B2/2mo) are anticorrelated,such that the ion total (plasma plus magnetic) pressureremains approximately constant; the oscillations in plasmapressure are cancelled perfectly by the magnetic pressureperturbations.[28] 3. The linear instability criterion [Hasegawa, 1969]

is strongly satisfied throughout the �7 hour interval and thatthe necessary plasma anisotropy is larger than the requiredinstability threshold.[29] 4. The linear growth rate is positive and large

throughout the entire �7 hour interval, such that a lineare-fold increase in perturbation amplitude is predicted tooccur on minute timescales (gmi � 5.3 � 10�3s�1).[30] From these measurements we determine that the

equatorial dawnside magnetosphere (confined to within�20� of the magnetic equator) was likely unstable to mirrormode growth for the entire period of interest. However, wenote that as the compressional perturbations are large (dB/B�0.2–0.3), the linear dispersion relation may be insufficientto fully describe the instability.[31] During the intervals 2205–2225 UT and 2330–

2340 UT, the magnetic field and velocity components are90� out of phase, a component of the activity relating to thesignature of a standing Alfven wave. We postulate that thismay be an indication that this event is better described as asignature of the drift Alfven-mirror mode. Given the curvedfield geometry, coupling between the drift mirror mode andthe standing Alfven mode on the curved field line geometryon the dawn flank is not unreasonable.[32] There are several remarkable aspects to this event:

that drift mirror mode wave activity is observed for such aprolonged period of time and that two instruments measur-ing completely different quantities can be used in combi-nation to fully characterize the waves. We believe that this isthe longest duration mirror or drift mirror wave reported todate inside the magnetosphere. It provides an excellentopportunity to study a fundamental plasma instability inspace, while also revealing important physics about themechanisms by which hot plasma energy can be dissipatedvia the excitation of ULF waves. Future work will involve

an investigation of the linear dispersion characteristics ofthis drift mirror instability incorporating finite gyroradiuseffects.

[33] Acknowledgments. The authors would like to thank N. Ness(Bartol Research Institute) and D. J. McComas, for the use of ACE MFI andSWEPAM data, respectively, as well as CDAWeb. I.J.R and C.E.J.W arefunded by the Canadian Space Agency (CSA). IRM is supported by aCanadian NSERC Discovery Grant. Geotail magnetic field data and plasmadata were provided by T. Nagai, and H. Hayakawa and T. Mukai,respectively, through DARTS at the Institute of Space and AstronauticalScience (ISAS) in Japan.[34] Zuyin Pu thanks Elizabeth Lucek, Andris Vaivads, and another

reviewer for their assistance in evaluating this paper.

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��W. Baumjohann, Space Research Institute, Austrian Academy of

Sciences, Schmiedlstr. 6, A-8042 Graz, Austria.L. M. Kistler, Department of Physics, University of New Hampshire,

Durham, NH 03824, USA.I. R. Mann, I. J. Rae, and C. E. J. Watt, Department of Physics,

University of Alberta, Edmonton, Alberta, Canada T6G 2J1. ([email protected])


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