Concerning the generation of geomagnetic giant

pulsations by drift-bounce resonance ring current

instabilities

Karl-Heinz Glassmeier, S. Buchert, U. Motschmann, A. Korth, A. Pedersen

To cite this version:

Karl-Heinz Glassmeier, S. Buchert, U. Motschmann, A. Korth, A. Pedersen. Concerning thegeneration of geomagnetic giant pulsations by drift-bounce resonance ring current instabilities.Annales Geophysicae, European Geosciences Union, 1999, 17 (3), pp.338-350. <hal-00316550>

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Concerning the generation of geomagnetic giant pulsationsby drift-bounce resonance ring current instabilities

K.-H. Glassmeier1, S. Buchert1�, U. Motschmann1, A. Korth2, A. Pedersen3

1 Institute for Geophysics and Meteorology, Technical University of Braunschweig, Germany2 Max-Planck-Institute for Aeronomy, Katlenburg-Lindau, Germany3 European Space Research and Technology Centre, Noordwijk, The Netherlands

Received: 7 July 1997 / Revised: 19 August 1998 /Accepted: 21 August 1998

Abstract. Giant pulsations are nearly monochromaticULF-pulsations of the Earth's magnetic ®eld withperiods of about 100 s and amplitudes of up to 40 nT.For one such event ground-magnetic observations aswell as simultaneous GEOS-2 magnetic and electric ®elddata and proton ¯ux measurements made in thegeostationary orbit have been analysed. The observa-tions of the electromagnetic ®eld indicate the excitationof an odd-mode type fundamental ®eld line oscillation.A clear correlation between variations of the proton ¯uxin the energy range 30±90 keV with the giant pulsationevent observed at the ground is found. Furthermore, theproton phase space density exhibits a bump-on-the-tailsignature at about 60 keV. Assuming a drift-bounceresonance instability as a possible generation mecha-nism, the azimuthal wave number of the pulsation wave®eld may be determined using a generalized resonancecondition. The value determined in this way,m � ÿ21� 4, is in accord with the value m � ÿ27� 6determined from ground-magnetic measurements. Amore detailed examination of the observed ring currentplasma distribution function f shows that odd-modetype eigenoscillations are expected for the case@f =@W > 0, much as observed. This result is di�erentfrom previous theoretical studies as we not only considerlocal gradients of the distribution function in real space,but also in velocity space. It is therefore concluded thatthe observed giant pulsation is the result of a drift-bounce resonance instability of the ring current plasmacoupling to an odd-mode fundamental standing wave.The generation of the bump-on-the-tail distributioncausing @f =@W > 0 can be explained due to velocitydispersion of protons injected into the ring current. Boththis velocity dispersion and the necessary substormactivity causing the injection of protons into thenightside magnetosphere are observed.

Key words. Magnetospheric physics (energetic particles,trapped; MHD waves and instabilities) á Space plasmaphysics (wave-particle interactions).

1 Introduction

Giant pulsations (Pg) are ULF-pulsations predominant-ly occurring in the morningside magnetosphere. Theyare characterised by their large amplitudes, and theirstriking quasi-monochromatic wave form with periodsof typically 100 s. Giant pulsations have been studiedintensively in the past by e.g. Annexstad and Wilson(1968), Green (1979), Rostoker et al. (1979), Glassmeier(1980), and more recently by Chisham and Orr (1991),Chisham et al. (1992), Takahashi et al. (1992), Chishamand Orr (1994), Chisham (1996), and Chisham et al.(1997).

These studies show that giant pulsations exhibit astrongly localised pulsation wave ®eld extending only200±300 km in the north-south direction, and 600±700km in the east-west direction at the surface of the Earth.Furthermore, Pgs are found to be characterised by rapideast-west phase variations with the azimuthal wavenumber, m, reaching values up to m � ÿ40, where thenegative sign indicates westward phase propagation.These characteristics led several authors to concludethat giant pulsation wave ®elds are propagated in theguided poloidal mode, ®rst described in Dungey's (1954)seminal work on ULF-pulsation theory. Although Pgground amplitudes are often smaller than those of Pc4-5pulsations the ionospheric magnetic ®eld of Pgs is muchlarger due to the screening e�ect of the atmospherewhich scales exponentially with the azimuthal wavenumber (e.g. Hughes and Southwood, 1976; Glassmeier,1984). This justi®es calling this type of pulsation giantpulsations.

However, there is still considerable debate about thepossible generation mechanism of Pgs. The fast

Correspondence to: K.-H. Glassmeier

*Present address: Solar-Terrestrial Environment Laboratory,Nagoya University, Japan

Ann. Geophysicae 17, 338±350 (1999) Ó EGS ± Springer-Verlag 1999

azimuthal phase variation and indications toward anodd-mode wave structure led Green (1979), Takahashiet al. (1992), and others to suggest that a drift-waveinstability of the magnetospheric ring current (e.g.Hasegawa, 1971) is the dominant plasma instabilitygenerating giant pulsations. Other authors such asGlassmeier (1980), Poulter et al. (1983), or Chishamand Orr (1991) favour the proton drift-bounce reso-nance instability (e.g. Southwood et al., 1969; South-wood, 1976) as a generation mechanism, and givearguments for an even-mode wave structure alonggeomagnetic ®eld lines. However, no ®rm conclusionon the generation mechanism has been reached as yet.

Any future theory on the generation mechanism ofgiant pulsations requires solution of the following threeproblems: ®rst, identi®cation of the instability mecha-nism; second, identi®cation of the mechanism generat-ing the unstable plasma situation, and providing theplasma free energy to feed the instability; third, identi-®cation of the wave mode the unstable plasma couplesto, and by which the disturbance is propagated withinthe magnetosphere, especially to the ground. Further-more, any theory should be able to predict and explainthe striking quasi-monochromatic wave form, the fastazimuthal phase variation, the strong spatial localisa-tion, and the predominantly morningside occurrence.

Using correlated ground-magnetic and geostationarysatellite observations of a giant pulsation event whichoccurred on October 4, 1978, 0645-0830 UT (0915-1100

MLT), these mentioned problems will be tackled and aconjecture is formulated which explains many typicalcharacteristics of giant pulsations.

2 Ground-magnetic observations

During the period of the International MagnetosphericStudy, 1976±1979, the University of MuÈ nster operatedthe Scandinavian Magnetometer Array (Fig. 1; forfurther details see KuÈ ppers et al., 1979), whose Earthmagnetic ®eld observations will be used in the presentstudy. Figure 2 displays bandpass ®ltered(60 s < T < 300 s) records of the D-component of theEarth magnetic ®eld variations in the time intervalOctober 4, 1978, 0645-0830 UT along a north-southpro®le, pro®le 4 (see Fig. 1) of the Scandinavianmagnetometer array (SMA). The station SOY at about71� geomagnetic latitude is the northernmost, and OULat about 63� latitude the southernmost station along thepro®le; station separation is about 120 km.

From the data in Fig. 2 it can be seen that clear Pg-activity with a period of about 100 s was recorded inNorthern Scandinavia in the time intervals 0650-0718UT ( MLT � UT � 2:5 h), and 0735-0818 UT. ThePg is most prominent at the stations MUO and PEL,where peak-to-peak amplitudes reach values of up to 20nT. To the north and south of these two stations the Pgbecomes less visible and more irregular, especially at the

Fig. 1. Locations of the magnetometerstations of the Scandinavian magnetome-ter array in geographic coordinates. En-circled numbers with arrows denotedi�erent N-S pro®les (after KuÈ ppers et al.,1979)

K.-H. Glassmeier et al.: Concerning the generation of geomagnetic giant pulsations 339

northern stations. This hints toward the usually ob-served strong NS-localisation of the Pg-wave ®eld (e.g.Glassmeier, 1980).

The observations indicate a clear amplitude modula-tion, or better, a double-event characteristic of the giantpulsationwith amplitudemaxima at 0658UT and around0755 UT. Between the two wave groups a short-durationPc5 event may be identi®ed at about 0720 UT.We do notregard this Pc5 event as associated with the Pg-phenom-ena and do not discuss it any further. The clear amplitudemodulation allows us to determine the horizontal groupvelocity. However, in contrast to earlier work by Glass-meier (1980) or Poulter et al. (1983), where westwardvelocities in the Earth's frame of reference of the order of200 m/s were found, inspection of the Pg signal alongeast-west pro®les AND-MAT-VAD, FRE-EVE-ROS-MIE-SKO, and GLO-RIJ-KIR-MUO (see Fig. 1) indi-cates no appreciable east-west group motion in theEarth's frame of reference for the Pg event studied.

The azimuthal phase velocity can be determined asdisplayed in Fig. 3, where records of the D-componentare shown along an east-west station pro®le RIJ-KIR-MUO (see Fig. 1) for the time interval 0652-0658 UT.The pro®le is located at about 65� geomagnetic latitude.The oscillation maximum is ®rst seen at the easternmoststation MUO and about 30 s later at the westernmoststation RIJ. Thus, phase propagation is from east towest in the Earth's frame of reference. The apparenthorizontal phase speed is vph � ÿ0:135 deg/s, where thenegative sign denotes westward propagation, that isvariations in azimuth are counted positive eastward.This corresponds to an apparent angular phase velocity

around the Earth of XA � ÿ2:4� 10ÿ3 rad/s. With thePg-frequency x � 2p=100s this gives an azimuthal wavenumber m � ÿ26� 7, with the error estimate resultingfrom a sampling period of 10 s for the magnetic ®eldobservations used.

3 Satellite observations

For the giant pulsation event analysed satellite obser-vations of the magnetospheric electromagnetic ®eld aswell as electron and ion ¯ux measurements madeonboard the European geostationary satellite GEOS-2are available. GEOS-2 was launched into a geostation-ary orbit on July 14, 1978. During the time intervaldiscussed the satellite was located at 37� easterngeographic longitude; local time onboard GEOS-2corresponds to UT+150 min. Field line tracing undermoderate Kp conditions with a Tsyganenko-magneto-spheric magnetic ®eld model gives a nominal footpointof the GEOS-2 ®eld line close to the SMA station ROS(see Fig. 1). Thus the SMA and GEOS-2 are undernominal conjugate conditions. The geomagnetic longi-tude of GEOS-2 in the equatorial plane is determined as104�.

Having available satellite electric and magneticobservations is a signi®cant achievement in trying toclarify the Pg generation mechanism. For example,inspection of the electromagnetic ®eld observationsallows us to determine the mode of oscillation of theobserved event. Previous studies reached con¯ictingresults with respect to this question. Green (1979), andmore recently Takahashi et al. (1992) favour an odd-mode, fundamental oscillation with anti-symmetric(symmetric) transverse magnetic ®eld (electric ®eld and®eld line displacement) variations. On the other hand,Glassmeier (1980), and Chisham and Orr (1991) givearguments for and strongly favour an even-mode,second harmonic standing ®eld line oscillation withsymmetric (anti-symmetric) transverse magnetic ®eld(electric ®eld and ®eld line displacement) variations. An

Fig. 2. Magnetic ®eld variations recorded along pro®le 4 of theScandinavian magnetometer array for the time interval October 4,1978, 0645-0830 UT (0845-1030MLT). The data have been band-pass®ltered in the period range 60±300 s

Fig. 3. Magnetic ®eld variations recorded along the three east-westaligned stations RIJ, KIR, and MUO. Geomagnetic latitude of thethree stations is 64:8� � 1�. The dashed line connects correspondingmaxima at the di�erent stations

340 K.-H. Glassmeier et al.: Concerning the generation of geomagnetic giant pulsations

even mode (odd mode) can thus be identi®ed in themagnetospheric equatorial plane as an oscillation ex-hibiting a node (an anti-node) of the electric ®eldoscillation and an anti-node (a node) of the transversemagnetic ®eld variation.

Figure 4 displays magnetic and electric ®eld obser-vations made onboard GEOS-2 in the time intervalOctober 4, 1978, 0600-0800 UT. Magnetic ®eld data aretaken from the GEOS-2 ¯uxgate magnetometer (Can-didi et al., 1974) and are represented in a dipolecoordinate system. Thus, Br denotes the radial compo-nent, counted positive outward, B/ the azimuthalcomponent, counted positive to the east, and Bhcompletes the right-hand system. In all three magnetic®eld components no clear signal resembling, in any way,the characteristics of the giant pulsation event recordedat the ground can be identi®ed. The regular, about 1 minoscillations in the Br and B/ components are instru-mental and due to a beat between the spin rate of 6 s andthe data readout time of 5.5 s. These oscillations do notindicate any natural variation. It is worthwhile notingthat only the magnetic ®eld magnitude increases (seen inthe Bh component) at about 0615 and 0630 UT.

The electric ®eld observations, however, displayclearly the giant pulsation seen at the ground. Thecoordinate system used is again a dipole coordinate

system with Eh approximating the magnetic ®eld-alignedcomponent. It does not completely vanish due todeviations from the h-direction from the actual magnetic®eld direction. Spiky perturbations at about 4 min pasteach hour are due to calibration sequences. The main Pgsignal is recorded in the azimuthal component. At about0630 UT a wave packet is seen lasting until about 0700UT, when activity ceases for about 10 min. From 0707UT until 0737 UT the event is seen again. Some activityis also seen in the radial component. Spectral analysisreveals a pulsation period of T � �110� 5� s.

The correspondence between the period of thepulsation seen at the ground and in space as well asthe clear packet structure of the event with almost thesame packet length (see Figs. 2 and 4) strongly suggeststhat the same giant pulsation event has been observed.Moreover, it should be noted that the event at GEOS-2already starts at 0630 UT, that is about 15 min earlierthan at the ground. This is somewhat surprising as thenominal footpoint of the GEOS-2 ®eld line is coveredby the SMA viewing area. Thus, simultaneous onset ofthe perturbation was expected. We have, at present, noother explanation for this di�erence in onset time thanquestioning the validity of the ®eld line tracing methodused. As can be seen from Fig. 4 the magnetosphericmagnetic ®eld signi®cantly changed at about 0630 UT,

Fig. 4. Magnetic and electric ®eld measurements made onboard the GEOS-2 satellite in the time interval October 4, 1978, 0600±0800 UT. Theobservations are represented in a dipole coordinate system

K.-H. Glassmeier et al.: Concerning the generation of geomagnetic giant pulsations 341

a temporal variation which cannot be covered by anymeans with a Tsyganenko model. As the geographicposition of GEOS-2 is about 20� east of Scandinavia itis justi®ed assuming that the true footpoint of theGEOS-2 ®eld line is also located east of Scandinavia.Glassmeier (1980) and Poulter et al. (1983) reportedwestward group velocities of 200±300 m/s. Thus adelay time of 15 min between ground and satellite caneasily be explained by an eastward shift of thefootpoint of a few hundred kilometres. If the truefootpoint is located east and if an azimuthal westwardgroup velocity of a few hundred metres is assumed adelay of 15 min is easily explained.

The observation of transverse electric ®eld variationsin the magnetospheric equatorial plane and the absenceof any correlated magnetic ®eld variations indicates thepresence of an odd mode standing ®eld line oscillation.Whether the fundamental or any odd harmonic oscilla-tion has been excited can be decided on grounds of thewave period. The electron density was measured atabout 3 cmÿ3; assuming a plasma predominantlyconsisting of protons gives an equatorial mass densityof 5� 10ÿ21 kg/m3. The observed large azimuthal wavenumber m suggests that a guided poloidal mode hasbeen excited. For this type of mode Cummings et al.(1969) have calculated the eigenperiods for di�erentharmonics and density indices n for density pro®lesq�r� � q0�r0=r�n along ®eld lines passing the geostation-ary orbit, r0 � 6.6 RE. With this value for q0 � 5� 10ÿ21kg/m3 and n = 1 eigenperiods of T = 127 s and T = 23s for the fundamental and the third harmonic guidedpoloidal mode, respectively, result. We may concludethat the observed electric ®eld observations are due to afundamental mode poloidal eigenoscillation. The ob-served predominantly azimuthal electric ®eld variationcorresponds to a predominantly radial magnetic ®eldperturbation. Taking into account an ionospheric rota-tion by 90� (e.g. Hughes, 1974; Glassmeier, 1984) aground-magnetic perturbation dominating the D-varia-tion is expected. This is indeed observed, which supportsour conjecture of a guided poloidal mode having beenexcited.

However, the observed electron density may be toolow because only part of the distribution function hasbeen measured. Furthermore, the magnetospheric plas-ma is not a pure electron-proton plasma, but can alsoexhibit a signi®cant oxygen component. Wedeken et al.(1984), for example, found a ratio of oxygen to protonions of 1/15 at the geostationary orbit. Singer et al.(1979) report even higher ratios. Using the Wedekenet al. (1984) value a mass density of q0 � 6:4�10ÿ21 kg/m3 results. This corresponds to poloidal modeeigenperiods of 145 s, 41 s, and 26 s for the fundamental,second harmonic, and third harmonic mode, respec-tively. A clear identi®cation of the mode structure is thusnot possible due to a lack of suitable mass densitymeasurements.

Electron and proton observations made onboard theGEOS-2 satellite have been studied to ®nd correlativesignatures with the ground magnetic and the satelliteelectric ®eld oscillations, identi®ed as giant pulsations.

The particle measurements were performed with amagnetic spectrometer to separate electrons from ions(Korth and Wilken, 1978; Korth et al., 1978). Electronenergy spectra are determined in 16 channels between 16keV and 213 keV. Ions are analysed in 10 energychannels between 28 and 402 keV. Full energy spectraare obtained every 5.5 s in various pitch angle ranges.No identi®cation of di�erent ion species is possible. Forthe data interpretation we assume that most of thedetected ions are protons.

From the data analysis of the particle instrumentfollows that only the ion observations signi®cantlycorrelate with the giant pulsation event observed. Figure5 displays proton ¯ux measurements in six di�erentenergy channels from 28 keV up to 114 keV in the pitchangle range 25� ÿ 35�. The ¯ux shows appreciablevariations in the di�erent energy ranges. The mostpronounced variations are increases of the ¯ux ataround 0700 UT and 0800 UT (see, for example,channel 5, 59±75 keV, in Fig. 5). These ¯ux intensi®ca-tions parallel the amplitude modulation of the Pg asobserved in the ground-magnetic ®eld and the GEOS-2electric ®eld observations. This indicates a possiblerelationship bewteen the modulation and ¯ux increases.A more detailed interpretation, however, is not yetpossible as the ¯ux intensi®cations are local e�ects whilethe wave activity is caused by local plasma instabilitiesas well as waves which propagate in from other regionsof the magnetosphere to the points of observations.Azimuthal Poynting ¯ux observations are required toelucidate this point further. They are not available forthe present study.

The proton ¯uxes also show velocity dispersion, thatis the higher-energy protons arrive earlier than thelower-energy ones. This points toward the protoninjection not occurring locally, but at some distancefrom the satellite. It should be noted that this modula-tion feature of the proton ¯uxes as well as the velocitydispersion can be observed for protons with pitch-anglesup to 40�, but not at higher pitch-angles, for which nocorrelation between proton ¯ux and ground-magnetic®eld variations has been found. Also, the proton ¯uxobservations do not indicate the presence of a signi®cantproton temperature anisotropy which could give rise toa drift-mirror instability.

A possible injection mechanism for the observedproton clouds is substorm activity in the nightsidemagnetosphere. Figure 6 displays ground-magnetic ®eldvariations observed at the Canadian station Fort Prov-idence (located at 292� geomagnetic longitude) during theinterval October 4, 1978, 0400±0900 UT (2100±0200MLT), and indicates several substorms or substormintensi®cations in the nightside magnetosphere at thesame time the giant pulsation event is observed in themorningside (see Figs. 2 and 6). Substorm activity istherefore a candidate for the observed proton injections.Therefore, a relation between substorm activity and Pgs issuggestive, and such a connection has also been discussedby Glassmeier et al. (1992) and Chisham et al. (1992).

The observed velocity dispersion in the di�erentenergy channels can be used to determine the drift time

342 K.-H. Glassmeier et al.: Concerning the generation of geomagnetic giant pulsations

and also the drift periods of the protons. If / is theactual longitude a proton has reached after the time t atthe injection point /0, the relation

/�t� � /0 � v/ � t �1�holds where v/ is the azimuthal drift velocity. Thisvelocity is negative for protons due to our de®nition ofazimuthal variations counted positive eastward. Ac-cording to Schulz and Lanzerotti (1974) v/ / W , theparticle energy. With the di�erence in arrival time,dt � ti ÿ tj, observed at the same place between theparticles in energy channels i and j, from the earlierequation

tj � Wi � dtWj ÿ Wi

�2�

where for the particle energies Wj > Wi. The velocitydispersion as seen in Fig. 5 is emphasized most in theenergy channels 3±6. With Eq. (2) a drift time of e.g.t5 � 2935 s for the 59±75 keV protons results.

Assuming an injection point at around midnight (thatis when Fort Providence exhibits substorm activity) theprotons observed at GEOS-2 have thus drifted adistance of about ÿ188� (the distance between FortProvidence and GEOS-2 in geomagnetic longitude)within this drift time. Accordingly a drift frequencyXD;5 � ÿ�1:1� 0:2� � 10ÿ3 radsÿ1 may be determinedfrom the particle observations. Here we allowed for anerror of about 20% in the determination of XD toaccount for the uncertainty of the determination of theinjection point.

In a dipole ®eld geometry gradient and curvaturedrift for 67 keV protons would result in a drift frequencyXD � ÿ0:8� 10ÿ3 radsÿ1. This value corresponds to the

measured one. The proton ¯ux measurements can beused to derive phase space densities as well. Figure 7exhibits phase space densities at times 0655:18 and0700:21 UT for the pitch angle range 5� ÿ 15�. The mostremarkable feature is a shoulder at an energy of about60 keV, i.e. the energy range of channel 5. A similarshoulder is also seen at the same energy in other pitchangle ranges below 40�. We interpret this shoulder asindicating an unstable, bump-on-the-tail distribution.Thus, not only the correlation between Pg amplitudemodulation and proton injection, but also the phasespace densities point toward a causal relationshipbetween the observed giant pulsations and protoninjection.

4 Drift-bounce resonance instabilityof the ring current plasma

Hughes et al. (1978) reported a quasi-monochromaticwave event observed onboard the satellite ATS 6 in theafternoon sector of the magnetosphere and concurrentmeasurements of bump-on-the-tail protons phase spacedistributions. They convincingly argue that this unstabledistribution via the bounce-resonance instability givesrise to the observed quasi-monochromatic wave. How-ever, their bump-on-the-tail was observed at an energyof about 10 keV, and the wave was not identi®ed as agiant pulsation.

Nevertheless, it is tempting to discuss the describedGEOS-2 proton observations in terms of a drift-bounceresonance instability of the ring current plasma (South-wood et al., 1969). This instability might be discussed asan analogy of the cyclotron instability. In a mirror ®eld

Fig. 5. Proton-Flux measurements madeonboard the geostationary satellite GEOS-2in the pitch-angle range 25� ÿ 35� on October4, 1978, 06±09 UT (upper panel), and Earthmagnetic ®eld variations of the D-componentat the station MUO (pro®le 4), close to thenominal footpoint of the GEOS-2 ®eld line(bottom panel). The ¯ux measurements areshown for six di�erent energy channels;channel 2: 28±36 keV; 3: 36±49 keV; 4: 49±59keV; 5: 59±75 keV; 6: 75±98 keV; 7: 98±114keV. The magnetic ®eld observations havebeen ®ltered in the range 60±300 s

K.-H. Glassmeier et al.: Concerning the generation of geomagnetic giant pulsations 343

geometry a charged particle performs a bounce motionbetween the mirror points and an azimuthal drift motionin addition to the gyromotion around the local magnetic®eld direction. While the cyclotron instability is aresonant interaction between a plasma wave propagat-ing along the ambient magnetic ®eld and the gyratingparticles, the drift-bounce resonance describes a reso-nant interaction between the drift-bounce motion of theparticles and an azimuthally propagating wave. Thechange of the energy, W , of the particle with time isgiven by

dWdt� q vk Ek � l

@bk@t� q~E �~vD �3�

where q and l are the electric charge and the magneticmoment of the particles, respectively, and where vk; bk,

and Ek denote the particle velocity along the ambientmagnetic ®eld as well as the wave magnetic and electric®eld perturbations along the background ®eld, and~vD isthe unperturbed particle drift velocity. If the particleinteracts with a purely transverse magnetohydrodynam-ic wave the ®rst two terms in Eq. (3) vanish, and theinteraction is via the third term only, i.e. in aninhomogeneous background ®eld the azimuthal gradientand curvature drifts cause an interaction with theazimuthal wave electric ®eld.

Fig. 6. Magnetic ®eld observations, recorded on October 4, 1978, 04±09 UT, at Fort Providence, Canada (data courtesy G. Rostoker)

Fig. 7. Phase space density of energetic protons recorded onboardGEOS-2 in the pitch-angle range 5�±15� on October 4, 1978, 0655:18and 0700:21 UT

344 K.-H. Glassmeier et al.: Concerning the generation of geomagnetic giant pulsations

The resonance condition is given by (e.g. Southwood,1976)

x ÿ mXD ÿ NXB � 0 �4�where x is the wave frequency, m an azimuthal wavenumber, and XD and XB denote the particle drift andbounce frequencies, respectively. According to previousstudies on the bounce resonance instability by e.g.Southwood et al. (1969), Southwood (1976), Glassmeier(1980), Southwood and Kivelson (1982), or Takahashiet al. (1992) N is an integer. However, this is only true ifthe transverse component of the electric ®eld perturba-tion is symmetric or antisymmetric about the magneticequator. Any asymmetry allows N to take any realvalue. Such an asymmetry can be due to asymmetry ofthe ionospheric Pedersen conductivity between theNorthern and Southern Hemisphere. This occurs, forexample, at the solstices, when one polar ionosphere isilluminated for a long time and the other is correspond-ingly in darkness. In this case the resulting standingwave has a quarter-wave form along the ®eld line withclear asymmetry (Allan and Knox, 1978).

In more general terms the wave ®eld along theambient magnetic ®eld must be regarded as a superpo-sition of the ®elds of a standing wave and a travellingwave. Finite ionospheric conductivity causes a ®niteJoule heating in and thus a Poynting ¯ux into theionosphere (e.g. Allan and Knox, 1979; Glassmeieret al., 1984). If the Pedersen conductivities are equal inboth hemispheres the Poynting ¯ux into the iono-spheres, integrated along the ®eld line, must also beequal. This requires the existence of a null-point, wherethe local Poynting ¯ux vanishes (Allan, 1982). Forsymmetric or antisymmetric electric and magnetic ®eldperturbations this null-point is located right at themagnetic equator. However, if the ionospheric conduc-tivities di�er from each other this null-point is signi®-cantly shifted towards that ionosphere where theconductivity is larger. This implies a break-down ofany symmetric or antisymmetric behaviour of the wave®elds along the ®eld line with respect to the magneticequator. The null-point may be regarded as the newpoint of symmetry.

The e�ect of asymmetries on the resonance condition(4) can be demonstrated by the following simpli®edcomputation of the energy exchange per time betweenthe wave ®eld and a bouncing particle. The energychange of a particle due to its interaction with theelectromagnetic ®eld of the unstable wave is given by

d _WB�s� � qE/�s� vD�s� exp�i�m/ÿ xt�� �5�where q is the electric charge of the particle, E/�s� thearc length s dependant azimuthal electric ®eld compo-nent of the wave, vD�s� the arc length dependantazimuthal drift velocity, and / the azimuth angle orazimuthal drift phase. The electric ®eld may be writtenas

E/�s� � ÿiEo exp

�iapL

s�

�6�

This expression allows for asymmetric electric ®eldvariations with respect to the equatorial plane. Param-eters a and L describe this asymmetry and the arc lengthbetween the northern and southern footpoint of theoscillating ®eld line, respectively. The arc length iscounted positive from the northern mirror point. Fora � 1 this functional dependance of E/ allows todescribe the odd-mode fundamental eigenoscillation.The quarter-wave is described by a � 1=2, while e.g.a � 2=3 describes an electric ®eld variation which isasymmetric with respect to s � L=2 and exhibits di�er-ent values of the electric ®eld at conjugate ionospheres.Such a mode would allow us to describe a situation withthe null-point shifted towards one hemisphere.

For a particle bouncing in a dipole ®eld one has

s � l cos XBt �7�with l the half-bounce path length, that is the recti®edpath distance between the northern and southern mirrorpoints of the bouncing particle (e.g. Roederer, 1970,p.35); s � 0 for the northern mirror point. This relationbetween arc length of the bouncing particle and timere¯ects the harmonic motion of the particle in a dipole®eld, provided cos2 h� 1, where the angle h � XBtdenotes the particle bounce phase (e.g. Schulz, 1991, p.203). The general expression relating arc length andbounce phase is a more complicated nonlinear relationgiven by Eq. (150) of Schulz (1991). With this expressionfor the arc length s the variation of the electric ®eld asseen by a bouncing particle can be approximated by

E/�s� � ÿiEo exp iaplL

cosXBt� �

�8�

The arc length dependant drift velocity may be approx-imated by its bounce averaged value via

vD�s� � hvD�s�i � vD �9�and the drift phase is given by /�t� � XDt.

With these de®nitions one has

d _WB � ÿiEo vD exp�i�mXD ÿ x�t � i

aplL

cosXBt��10�

In this expression wave frequency x and drift frequencyXD appear in a linear relationship with time t, while thebounce frequency XB is related to time in a nonlinearcosine function term. To obtain a similar linear relationbetween bounce frequency and time as one has betweendrift frequency and time the expression exp�i apl

L cosXBt�may be expanded into a Fourier series as done by e.g.Southwood (1976):

exp�iaplL

cosXBt��

XN��1

N�ÿ1cN exp�iNXBt� �11�

It should be noted that Southwood (1976) actuallyexpands the bounce phase h. The expansion coe�cientscN denote the spectral weight of each bounce phaseharmonic function.

K.-H. Glassmeier et al.: Concerning the generation of geomagnetic giant pulsations 345

Of particular interest is the bounce integrated energytransfer, that is;

dWB �Z TB

0

d _WB dt �12�

with the bounce period TB � 2p=XB. Southwood's(1976) expansion results in:

dWB � qEovDXN��1

N�ÿ1cN

exp�i�mXD ÿ x� NXB� � t�xÿ mXD ÿ NXB

�TB

0

;

�13�an expression approximated by Southwood [1976] as:

dWB � qEo vD d�xÿ mXD ÿ NXB�� exp�i�mXD ÿ x� NXB�t�

�TB

0: �14�

It is at this point where the condition N 2N hasbeen introduced by truncating the series over N . Thistruncation implies the following approximation:

exp iaplL

cosXBt� �

� exp�iNXBt�: �15�

A di�erent approximation allows relaxation of thecondition N 2N. The cosine dependence s(t) may beapproximated by a triangle function via:

s�t� �lp XBt if 0 � t � TB=2lp �1ÿ XBt� if TB=2 � t � TB:

(�16�

For t � TB=2 this gives, for example, s�t � TB=2� � l:This triangular approximation provides one with alinear relationship between s and t. It thus equals thealready used relations between frequency and time aswell as drift frequency and time. No expansion as usedby Southwood (1976) is required. With this approxima-tion the bounce integrated energy change reads:

dWB � ÿiqEo vD

�Z TB=2

0

exp�i�mXD � nXB ÿ x� � t� dt

�Z TB

TB=2

exp�i�mXD ÿ nXB ÿ x� � t�

� exp�in� dt

�17�

where n � al=L has been introduced. Integration of the®rst term on the right hand side of Eq. (17) givesZ TB=2

0

exp�i�mXD � nXB ÿ x� � t� dt

� i

xÿ mXD ÿ nXB� exp�i�mXD � nXB ÿ x� � t�

�TB=2

0

:

�18�This expression maximizes, if xÿ mXD ÿ nXB � 0.Integration of the second term on the right hand sideof Eq. (17) gives

Z TB

TB=2

exp�in� exp�i�mXD ÿ nXB ÿ x� � t� dt

� i exp�in�xÿ mXD � nXB

� exp�i�mXD ÿ nXB ÿ x� � t��TB

TB=2

:

�19�This expression maximizes, if xÿ mXD � nXB � 0.Thus, the resonance condition reads

xÿ mXD ÿ nXB � 0 �20�with n 2 R. This is a proper generalization of theSouthwood (1976) condition. Eqs. (4) and (20) agree, ifL � l and if a � �1;�2 . . .This relation can be used to determine the azimuthalwave number

m � xÿ nXB

XD�21�

where m > �<�0 corresponds to an eastward (westward)propagating wave and a suitable value for n has to beassumed. The protons under consideration have a pitchangle of about 30�, that is they mirror at an equatorialdistance of about 35� or l/2 = 0.25 L. Thus, we havel=L � 0:5.

To estimate the value a is more di�cult. It requiresknowledge of the ionospheric conductivity at the foot-points of those ®eld lines perturbed by the Pg pulsationconsidered. Measurements of the conductivities are notavailable. However, using a Tsyganenko 89 magnetic®eld model to determine the ionospheric footpoints andthe IRI 90 ionosphere model values for the Pedersenconductance of 3.3 S and 8.4 S in the Northern andSouthern Hemisphere, respectively, are found. The cleardi�erence in hemispheric conductances indicates a clearshift of the null-point toward the Southern Hemisphere.This asymmetry of the Pg electric ®eld variation may bedescribed by a value a � 0:8. Note, that a � 1 wouldcorrespond to a fundamental mode oscillation. Withthese estimates for l=L and a we have n � 0:4.

The bounce frequency may be computed using therelation (Schulz and Lanzerotti, 1974)

XB � p��������2mp

pL RE

�����Wp

� 1

T �y� �22�

where mp is the particle mass, RE = 6371 km the Earthradius, L the McIlwain parameter, W the particle energy,and T �y� a factor taking into account the pitch angle awith y � cos a. For the pitch angles discussed hereT �y� � 1:3. Thus with W � 67keV, a valueXB � 10ÿ1radsÿ1 results. With x � 2p=100 s, whereT � 100s is the observed giant pulsation period, andXD � ÿ�1:1� 0:2� � 10ÿ3radsÿ1, the empirically deter-mined channel 5 proton drift frequency, we receiveazimuthal wave numbers

m� � ÿ21 � 4

mÿ � ÿ93 � 20

346 K.-H. Glassmeier et al.: Concerning the generation of geomagnetic giant pulsations

for n � �0:4, respectively. The error estimate is basedon the error in the determination of XD. Uncertainties inthe determination of n have not been considered. Theywill give rise to larger error limits for the wave numbers.Within the error limit the n � 0:4-value of the azimuthalwave number, m�, agrees with that one determinedusing ground-based magnetic ®eld observations. Thisagreement strongly favours a drift-bounce resonanceinstability with n � �0:4 as a source mechanism for theobserved giant pulsation. Any waves generated inn � ÿ0:4 resonance would barely be observable at theground due to the large m-value causing a large spatialattenuation of the wave signal below the ionosphere(e.g. Hughes and Southwood, 1976).

However, resonance condition (20) is only a neces-sary condition for instability. A more detailed investi-gation requires consideration of the physics of low-frequency ring current instabilities. General approachesto this problem have been provided by Southwood(1976) and Chen and Hasegawa (1991). While South-wood (1976) has studied the problem using a drift-kinetic approach, Chen and Hasegawa (1991) make useof the more detailed gyro-kinetic approximation. In thefollowing we shall argue that the observed giantpulsation is generated by the unstable high-energybump-on-the-tail part of the proton distribution. Forthese resonant protons and an azimuthal wave numberof the order jmj = 30 the ratio k? q � 0:1 (where k? isthe transverse wave number and q the resonant protongyroradius). Thus a drift-kinetic approximation as usedby Southwood (1976) seems appropriate.

Southwood's (1976) general approach to low-fre-quency instabilities in the ring current provides anelegant framework for studying wave generation in thering current. His central Eq. (22) may be written as (seealso Chen and Hasegawa, 1991)

x2 dK � 2 � dU � dM � � ixdR �23�where x is the wave frequency, x2dK the change in ¯uxtube integrated plasma kinetic energy, dU the change inplasma bulk energy, and dM the perturbation ofmagnetic energy. The term ixdR describes energychanges due to resonant wave-particle interactions withdR given in a dipole ®eld topology by

dR �X

n

ZdsZ

2p2 j~Bj dldWvk

janj2

� d�xr ÿ mXD ÿ nXB� dfdW

: �24�

Here the integration is over the ¯ux tube volumede®ning the interaction region, ~B is the backgroundmagnetic ®eld, f the particle distribution function, Wthe particle energy and l its magnetic moment. Sum-mation is over the di�erent wave modes. The coe�cientan describes the Fourier expansion of the particlebounce motion along ®eld lines (Southwood, 1976).The delta-distribution describes the resonance characterof the interaction.

Equations such as Eq. (23) are very suitable toanalyse a plasma's stability. For example, withoutresonant particles dR � 0, we have

x2 dK � 2 �dU � dM� �25�and the plasma is unstable, if dU � dM < 0, as in thiscase x becomes an imaginary number. The mirror and®rehose modes are instabilities of this kind, i.e. they arepurely growing modes (Kruskal and Oberman, 1958;Chen and Hasegawa, 1991). In more general cases withx � xr � ic from the imaginary part for the growth ratewe assume:

c � dR2dK

: �26�

Thus an instability may be driven by resonant wave-particle interaction, provided dR > 0 and dK > 0.

The original treatment by Southwood (1976) discuss-es the e�ect of an adiabatically injected backgroundplasma population and concludes that mainly particlesin n � ÿ1 resonance destabilise the plasma, while then � �1 particles have a stabilising in¯uence. Thus,Southwood (1976) as well as Green (1979), Hughes etal. (1978), Glassmeier (1980), Poulter et al. (1983), orChisham and Orr (1991) assumed that only even-modeharmonics with an asymmetric electric ®eld variationcan be excited. As some observations suggest that giantpulsations are odd-mode ®eld line eigenoscillations(with symmetric electric ®eld variation) the bounce-resonance instability has been ruled out as a viablegeneration mechanism for these peculiar wave modes(Green 1976; Takahashi et al., 1992). However, in viewof the clear correlation found between the October 4,1978 giant pulsation event and substorm associatedproton injection as well as the observation of a bump-on-the-tail distribution of the ring-current plasma asomewhat di�erent interpretation emerges, if one care-fully considers the assumptions on which the even-modehypothesis is based.

In particular, Southwood (1976) assumed an adia-batically injected background plasma distribution as heis mainly interested in studying the stability of thebackground plasma against bounce resonance interac-tions. In the particular case we are analysing thebackground plasma distribution has been changed bysubstorm injected protons. The total distribution func-tion may thus be approximated as

f �l;W � � fB�l;W � � fI�l;W � �27�where indices `B' and `I ' denote the backgrounddistribution and the substorm injected particles, respec-tively. The gradient df =dW may be estimated as

dfdW

� @fB

@Wjl;L �

@fI

@Wjl;L �

@L@W

@fB

@Ljl;W �28�

Here, the variation of the substorm injected particledistribution fI�l;W � with radial distance has beenneglected as we regard its e�ect in the followinginstability considerations as minor.

K.-H. Glassmeier et al.: Concerning the generation of geomagnetic giant pulsations 347

Due to a lack of detailed information about thesubstorm injected distribution we assume that it varieswith energy W in a manner proportional to the variationof the background plasma. This ansatz

@fI

@Wjl;L � g � @fB

@Wjl;L �29�

for the phase space gradient of the bump-on-the-tailpart of the proton distribution is very convenient as itallows formulation of the gradient in terms of the driftfrequency XD and via the resonance condition in termsof the bounce frequency XB, the mode number n, andthe azimuthal wave number m. The proportionalitybetween between the background and bump-on-the-tailparts is described by the factor g.

The phase space gradient for the resonant particles ofthe background plasma can be approximated as (South-wood et al., 1969)

@fB

@W� nXB

xr

@fB

@W

���l;L; �30�

and the total phase space gradient is approximatelygiven by the expression,

dfdW

��g � nXB

xr

� @fB

@W

���l;L: �31�

The growth rate Eq. (26) then reads

cdK �X

n

ZdsZ

p2 j~Bj dldWvk

janj2�

d�xr ÿ mXD ÿ nXB��g � nXB

xr

� @fB

@W

���l;L: �32�

Under normal adiabatic conditions @fB=@W < 0, andthe ring current plasma becomes unstable againstresonant wave-particle interaction if

g � nXB

xr< 0 �33�

If there are no substorm injected particles present, i.e.g � 0,

nXB

xr< 0 �34�

results, which gives one n < 0 for instability. Particles inn > 0 resonance are obviously stabilising the plasma.However, if g 6� 0, as expected for a bump-on-the-taildistribution, we have

g < ÿnXB

xr�35�

for instability. Particles in n > 0 resonance now con-tribute to instability provided jgj > jn XB

xrj, that is if the

bump-on-the-tail distribution phase space gradient issu�ciently large. The interpretation of this ®nding isstraightforward. Instability is due to two di�erentgradients, the one in L-space and the one in W -space.With respect to the L-space gradient background plasmaparticles in n > 0 resonance stabilise the plasma, whilebump-on-the-tail n > 0 resonant particles destabilise

due to their positive W -space gradient. If the laterparticles outweigh the stabilising e�ect of the back-ground particles, that is if the bump-on-the-tail is strongenough, the plasma becomes unstable.

In the present case xr � 6:28� 10ÿ2 rad sÿ1,XB � 10ÿ1 rad sÿ1, and n � 0:45. Thus, a valueg < ÿ0:72 would be su�cient for instability of theodd-harmonic such as the n � 0:45 mode. Such agradient is compatible with the GEOS-2 particle distri-bution observations. This discussion of the su�cientinstability conditions therefore provides further supportof a bump-on-the-tail driven bounce-resonance instabil-ity driving the observed Pg.

A question remaining is the generation of the theunstable phase space density, that is the bump-on-the-tail distribution causing @f =@W > 0. Hughes et al.(1978) argue that unstable bump-on-the-tail distribu-tions are naturally generated due to the combined actionof gradient and curvature drift as well as ~E �~B drift dueto a solar wind induced dawn-dusk electric ®eld. The~E �~B drift is essentially energy independant and largerthan the gradient drift for energies below about 10 keV.Above 10 keV ions gradient drift from the afternoonside westward via the frontside to the morning side (seeFig. 8). The ~E �~B drift opposes the gradient drift ofprotons, giving rise to a velocity ®lter e�ect allowingonly protons above 10 keV to drift westward to themorning.

In the present case associated proton energies aremuch larger, and ~E �~B drift e�ects are essentiallyunimportant. Velocity dispersion will lead to unstabledistribution far away from the injection point asschematically shown in Fig. 8. Faster protons will reacha certain point earlier than lower energy particles. High-energy particles are then added to the local backgroundplasma at a higher rate than low-energy particles. Thiscauses the build-up of the bump-on-the-tail distribution.This distribution, however, is not stable as lower energyparticles follow up and the higher energy ones aremoving further. Only if the dispersion is large enoughand the injection of particles lasts for a long enough time

Fig. 8. Schematic representation of proton injection, velocitydispersion, and resulting unstable particle phase space densitydistribution in the dayside magnetosphere

348 K.-H. Glassmeier et al.: Concerning the generation of geomagnetic giant pulsations

can an unstable distribution develop giving rise to theobserved wave activity.

A detailed examination of how the unstable distribu-tion evolves is beyond the scope of the present work andwill be the subject of a future study on proton drifts in arealistic magnetic ®eld geometry. However, severalpeculiarities of giant pulsations may be explained by thissource hypothesis. For example, the instability can onlyset in at a great distance from the injection point, wherethe lifetime of the unstable distribution is larger than theinverse of the growth rate of the instability. The distri-bution is not only relaxing back to equilibrium by wave-particle interactions, but also due to the lower energyparticles arriving and ®lling the low energy part of thephase space. This condition for the instability to occur ®tsvery well with the observations that giant pulsations areusually observed in the early morning hours, that is at agreat distance away from the proton injection point.

5 Conclusions

A ground-satellite coordinated case study of a giantpulsation event has been presented, which demonstratesthat giant pulsations are associated with a driftingproton cloud injected into the nightside magnetosphericring current due to substorm activity. The followingconclusions are reached based on results from this studyand previous work:1. Substorm activity in the nightside magnetospherecauses injection of high-energy protons from the mag-netotail into the eveningside magnetosphere.2. Under the in¯uence of gradient and curvature driftsthese particles drift westward towards the dayside in thering current regime. Velocity dispersion occurs, much asobserved.3. This velocity dispersion can lead to bump-on-the-tailphase space distributions with the injected protonsoverlaying the ambient plasma. Further theoreticalwork and numerical studies are required to follow theevolution of the injected particle population in detail. Itmay be speculated that dispersion must be large enoughto result in distributions unstable against wave-particleinteraction. The predominantly morningside occurrenceof Pg's could be explainable in this way.4. The bump-on-the-tail distribution causes a positivephase space density gradient, that is @f =@W > 0.Under assumptions discussed in more detail earlier thispositive gradient causes the plasma to become unstableagainst the excitation of odd-mode type fundamental®eld line eigenoscillations via a drift-bounce resonance.The conclusion of an odd-mode being excited di�ersfrom previous studies (e.g. Southwood, 1976; Chen andHasegawa, 1991) where the unstable plasma has beenassumed to be an adiabatically generated type with@f =@W < 0. The addition of particles injected due tosubstorm activity changes the phase space gradient andallows the odd-mode to become unstable. For the ®rsttime this odd-mode structure has been con®rmedobservationally for the October 4, 1978 event, discussedin this study.

5. A resonant wave-particle interaction also allows us tounderstand the typical quasi-monochromatic wave formof Pgs. Furthermore, resonance requires a signi®cantDoppler shift, which causes a large azimuthal wavenumber, m, of any unstable wave. Observed andtheoretically deduced m-values for the event underdiscussion agree fairly well.6. A large azimuthal wave number, about m � ÿ26 in ourcase, requires the excited wave to propagate as a guidedpoloidal mode. These modes have a predominantly radialmagnetic ®eld perturbation as ®rst discussed by Dungey(1954). A radial magnetic ®eld variation is associatedwith an azimuthal electric ®eld oscillation. The observedelectric ®eld perturbation at GEOS-2 is mainly in theazimuthal direction, in accord with the hypothesis.Furthermore, at the ground a magnetic ®eld oscillationis expected which is rotated by about 90� against that inthe magnetosphere (e.g. Hughes, 1974; Glassmeier,1984). Thus the Pg oscillation should be largest in the Dcomponent at the ground. This is as observed.

Finally, we would like to mention that giant pulsa-tions are evidence of a very fundamental process for thegeneration of unstable particle phase space distributionsin the Earth's magnetosphere.

Acknowledgements. We are grateful to Gordon Rostoker forproviding us with Fort Providence magnetic ®eld observatories.The work by K.H.G. and S.B. was supported by the DeutscheForschungsgemeinschaft and the German Space Agency DARA(now DLR), the work by A.K. by the German BundesministeriumfuÈ r Wissenschaft und Technologie, and the work of U.M. througha Werner-Heisenberg-Stipendium of the Deutsche Forschungs-gemeinschaft. Support from Frank Budnik, Stefan Markgraf andNicole Cornilleau-Wehrlin is gratefully acknowledged.

The Editor in chief thanks T.K Yeoman and another referee fortheir help in evaluating this paper.

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