SOME BASIC CONCEPTS OF -PARTICLE INTERACTIONS IN COLLISION LESS PLASMAS

Bruce T. Tsurutani and Gurbax S. Lakhina Space PhysicsElement Jet PropulsionLaboratory California Instituteof Technology,Pasadena

Abstract. The physicalconcepts of wave-particleinter- the losscone, normal and anomalouscyclotron resonant actionsin a collisionlessplasma are developedfrom first interactions,pitch angle ,and cross-fielddiffu- principles.Using the Lorentz force, starting with the sion are developed.To aid the reader, graphicillustra- conceptsof particle gyromotion,particle mirroring and tions are provided.

1. INTRODUCTION quent. This statementis correctprovided that there are no wave-particleinteractions. Wave-particleinteractions play crucial roles in many The presenceof wavescan introduce finite dissipation phenomenaoccurring in the laboratory[Gill, 1981] and in a collisionlessplasma. Charged particles are scattered in spaceplasmas [Gary, 1992]. In laboratory plasmas, by the wave fields, and the particles' momenta and wave-particleinteractions come into play in severalim- energieschange through this process.The interaction portant applications,including beat wave acceleration, between a wave and a chargedparticle becomesstrong heating by radio at ion and electron cy- when the streamingvelocity of the particle is suchthat clotron frequencies,and transport lossesdue to edge the particlesenses the Doppler-shiftedwave at its cyclo- turbulence.In spaceplasmas, wave-particle interactions tron frequency or its harmonics.This is the so-called are thought to be important for the formation of the cyclotronresonance interaction between the wavesand magnetopauseboundary layer, generation of electro- particles.The specialcase of the Doppler-shiftedwave magnetic outer zone chorus and plasmaspherichiss frequencybeing zero (i.e., zero harmonicsof the cyclo- emissions,precipitation of particlescausing auroras, etc. tron frequency)corresponds to the well-knownLandau Further, low-frequencywaves can interactwith charged resonance.Landau [1946] showedthat plasmawaves in particlesover long spatial scalelengths and within the unmagnetizedcollisionless plasmas suffer dampingdue magnetospherecan transportenergy from one regionto to wave-particleinteractions, or "Landaudamping." The another. For example, the interaction of ion cyclotron physicalmechanism of Landau dampingcan be under- and whistler mode waveswith Van Allen belt particles stood as follows:at Landau resonancethe particlesdo can scatterenergetic protons and electronsinto the loss not see a rapidly fluctuatingelectric field of the wave, cone and thus lead to the ring current decayduring a and hence they can interact stronglywith the wave. magnetic storm recovery phase. Similarly, pitch angle Those particleshaving velocities slightly less (greater) scatteringresulting from cyclotronresonance between than the phase velocity of the wave are accelerated outer zone whistler mode chorus and 10- to 100-keV (decelerated)by the waveelectric field to movewith the trapped substormelectrons can lead to the lossof elec- wave phasevelocity. Thus the groupof particlesmoving trons by precipitation. These precipitating electrons slightlyslower (faster) than the phasevelocity gain en- causeionospheric phenomenon such as diffuseaurorae, ergy from (lose energyto) the wave. In a collisionless enhancedionization in the ionosphericD andE regions, plasmacharacterized by a Maxwelliandistribution func- and bremsstrahlungX rays. tion, the number of slower particles (in any interval In spaceplasmas the collisiontime betweencharged aroundthe phasevelocity) is greaterthan the numberof particlesis generallyvery long in comparisonwith the faster particles, as is shown in Figure la. Therefore characteristictimescales of the system,namely, the in- energygained from the wavesby slowerparticles is more verse of the plasmafrequency or cyclotronfrequencies, than the energy given to the wavesby faster particles, and thereforethe plasmacan be treated as collisionless. thusleading to net dampingof the waves.Consequently, This would imply that there is virtually no dissipationin Landau dampingprovides dissipation for a collisionless spaceplasmas, as particle-particlecollisions are infre- plasma. In a non-Maxwellian plasma, for example, a

Copyright1997 by the American GeophysicalUnion. Reviewsof Geophysics,35, 4 / November 1997 pages 491-502 8755-1209/97/97RG-02200515.00 Paper number 97RG02200 ß 491 ß 492 ß Tsurutani and Lakhina: WAVE-PARTICLEINTERACTIONS 35, 4 / REVIEWSOF GEOPHYSICS

f(v) teractions are derived. We assume that the electron plasmafrequency 12pe = (4•rNq2/m-)1/2 is greater than •'• / slowerparticlesthe electron cyclotronfrequency, 12-, where N is the articles electron number densityand m- is the electron mass.

2. BASIC CONCEPTS

qT•'l •

v Equation (1) below is the Lorentz force in centime- ph a) ter-gram-second(cgs) units. A particle with chargeq moving with velocity V across a magnetic field of strengthB 0 experiencesa force, the well known Lorentz f(v) force, Fi•, which is orthogonalto both V and B0, slower s FL = qc V XS 0 (1)

where c is the speed of light. Figure 2 illustratesthis situationfor a positivelycharged particle (e.g., a proton) i 2• moving exactly perpendicular to a uniform magnetic + + v field B0. Sincein a uniform field, the Lorentz force can v v ph 0 changeonly the directionof the particle'svelocity vector b). Vñ perpendicularto B0, the chargedparticle will exhibit a circularmotion aboutthe magneticfield B0. The radius Figure 1. Schematicof a group of particlesinteracting reso- r of this , known as the particle gyroradius,can be nantlywith wavesin an unmagnetizedplasma. (a) Maxwellian calculatedby balancingthe magnitudeof the Lorentz plasma. The energy gained from the waves by the slower particlesis more than the energy given to the wavesby the forceFi• = (qVñBo/c)with the centrifugal force mV2•/r, faster particles. (b) Beam-plasmasystem where the phase where m is the massof the particle. velocityof the wave is lessthan the beam speedVo. Equatingthe Lorentz and centrifugedforces and solv- ing for r, one getsr = m l/ñc/qB o. Further, the angular frequencyof motion,dO/dt = I/z/r, is equal to qBo/mc, the cyclotron(or Larmor) frequency12 of the charged beam-plasmasystem, one can createa situationwhere in particle. a givenvelocity interval aroundthe phasevelocity of the Figure 3 illustratesthe concept of a particle pitch wave, there are a greater number of fasterparticles than angle.For thisparticular example we assumethe particle of slower particles. Such a case is shown in Figure lb. chargeis positive(positive ion). In a uniform magnetic This situation correspondsto inverseLandau damping field the angle that the instantaneousparticle velocity or plasma(Cherenkov) instability, as the wavesgrow by makes relative to the magneticfield vector is constant gaining energy from the particles.For this latter situa- and is called the pitch angle.The particlevelocity vector tion, one can saythat there is "free energy"available for wave growth. Similarly, the cyclotronresonant interac- tions between the wavesand the particlesgive rise to a damping or instability phenomenon which is akin to Landau dampingor instability[Stix, 1962]. v Space plasmas are magnetized and can support a variety of plasma waves. The resonantinteraction be- tween electromagneticwaves and particles has been • Bo studiedin detail [Kenneland Petschek,1966; Lyons and Williams,1984]. The interactingparticles undergo pitch angle ,which causesthem to be scatteredinto / F, the atmosphericloss cone, or undergoenergy diffusion, / \ which resultsin a harder spectrumfor the trapped par- ticles. \ / In this review we have tried to explain some funda- \ / mental conceptsof wave-particleinteractions involving •'"' - - "• protoncyclotron motion gyromotion electromagneticwaves. The Lorentz force playsa crucial Left-hand rotation role in the resonant interactionsbetween electromag- netic waves and particles. Analytical expressionsfor Figure 2. The Lorentz force and a positivelycharged particle pitch angle diffusion due to resonantwave-particle in- gyromotionin a uniform magneticfield. 35, 4 / REVIEWSOF GEOPHYSICS Tsurutani and Lakhina: WAVE:PARTICLE INTERACTIONS ß 493

can be broken down into two orthogonalcomponents, oneparallel to B0,Vii, and the other perpendicular to B0, V•_, suchthat v: + v, (2) I • Bo where b = B0/B0. The pitch angle ot of the particle is definedas ot = sin-l(V•_/V)as shown in Figure3. Since there are no forcesexerted on the particle in the parallel directionin a uniform B0,the particlemoves Figure 4. Schematic illustrating the mirror force. The unimpededwith a constantvelocity Vii along B0. There is Lorentz force actsperpendicular to V, so it doesno work. The a cyclotronmotion associatedwith the V•_velocity com- mirroringprocess transforms particle parallel energyinto per- ponentas shownabove. Although the directionchanges, pendicularenergy with total energyE r conserved. the magnitudeof V•_ remainsunchanged. Thus the pitch angle, or,will be constantin a uniform B0. A positively chargedparticle thus movesin a left-hand spiralmotion perpendicular(to the field) plane. Owing to the conver- along the magneticfield. This handednessis important genceof the magneticfield lines, the Lorentz force has for resonant interactions, as will be illustrated later. a componenttoward the left, i.e., oppositeto the mirror Positiveions gyrate in a left-hand senserelative to B0, point, leading to particle accelerationin a direction independentof whether they are moving along B0 or oppositeto the gradient, and thus "reflection." antiparallelto B0. The central field line about which the Since the Lorentz force operatesin a direction or- particle gyratesin Figure 3 is calledits guidingcenter. If thogonal to the velocity vector, there is no work done. the field oscillatesslowly, the particle will follow the The total energyof the particleremains constant. Squar- guidingcenter accordingly. ingequations (2)and multiplying by«m, we get Because electrons and negative ions have negative 1 1 1 charge,the V x B Lorentzforce is oppositelydirected to ET= •mV2= •mV• + •mV• = Ell+ E•_ (3) ihat of positivelycharged ions. Thus electrons and neg- whereEll and E•_ are the paralleland perpendicular ative ionsgyrate about the magneticfield in a right-hand kinetic energiesof the chargedparticle. For a particle corkscrewsense, opposite to that shownin Figure 3. movingfrom left to right in a constantmagnetic field, If there is a strongmagnetic field gradient,the parti- both Ell and E•_ are constantvalues. However, for a cles can be "mirrored," or reversedin directionby the particle moving from left to right in a magnetic field Lorentz force. We showa particle at its mirror point in gradient,as shownin Figure4, Ell decreasesas E•_ Figure 4 to illustratethis as a consequenceof propaga- increases,keeping ET constant.The mirror point is tion in a nonuniformmagnetic field. Although Figure 4 reachedwhen E •_ = ET. The particlethen startsto move indicates a one dimensionalgradient with a positive to the left,with Ell increasingand Eñ decreasing. sense,i.e., the gradientof IBI increasingto the rightat A magnetic"bottle" is depictedat the top of Figure 5. the mirror point, the reader shouldimagine this to be a The magneticfield line (flux) is pinchedat two endsand two-dimensionalgradient where similar field line con- expandedin the center.It hasa positivegradient on the vergenceoccurs into and out of the paper aswell. At the moment in time when the particle is being mirrored, Vii = 0, andV = Vñ, i.e., all of its velocityis in the Magnetic Bottle

E.L g = -- = constant B So

'-Bo (first adiabatic invariant)

sinc•= V. V Earth's Radiation Belts

• B0 MagneticMirror•.•"'• Earth•1•

Left-hand corkscrew motion spiral

Figure 3. "Pitch"angle ct of a positivelycharged particle. Figure 5. Magneticbottles for plasmaparticles. 494 ß Tsurutani and Lakhina' WAVE-PARTICLE INTERACTIONS 35, 4 / REVIEWS OF GEOPHYSICS

Loss Cone commonlybelieved that the neutronsproduced during the interactionof cosmicray particleswith upper atmo- a) sphere atoms and molecules decay into protons and V.L electrons (called cosmic ray albedo neutron decay

'Ioss cone • (CRAND) particles)within the magnetosphere,thus populatingthe belts. Interaction of whistler waveswith V• the energetic electrons may cause important lossesof Aurora Borealis and Australis trappedelectrons in the Van Allen belts[Tsurutani et al., 1975;Walt et al., 1996].Particles of lowerenergy (10-300 keV) can be also injected into the radiation belt by substormsand magneticstorm electric fields [Chen et al., 1997; Wolf et al., 1997]. Particleslosses during magnetic stormsare discussedby Sheldonand Hamilton [1993] and Kozyraet al. [1997]. The "loss cone" is the cone of pitch angleswithin which particles are lost to the upper atmosphere.The particle mirror points are deep in the atmosphere,and Figure 6. (a) Equatorialloss cone, and (b) aurorasassoci- the particles thus lose their energy by collisionswith ated with particlepitch anglescattering into the losscone. atmospheric/ionosphericatoms and moleculesand thus do not return to the magnetosphere.Consequently, the magnetosphericequatorial phase space (pitch angle) right (as one goes from left to right) and a negative distributionhas signaturesthat lookslike Figure 6a. On gradienton the left. As a consequence,the Lorentz force the other hand, the precipitatingparticles, i.e., the par- at both mirror pointsis directedtoward the center,i.e., ticles that are scattered into the loss cone, lose their away from both right and left mirror points. Particles energyto the neutral atomsand molecules.These atoms with large pitch anglesare "trapped" by the two mirror and moleculesare excited to higher energy states and points and will bounce back and forth between them. produce auroral line and band emissionsas they decay. However, particlesthat have pitch anglesof 0ø or close This light is the aurora borealis(northern hemisphere) to 0ø will mirror at only extremely high field strengths and auroraaustralis (southern hemisphere) (Figure 6b). and may escapeout the ends of the bottle. The size of the losscone can be calculatedby assum- If one bendsthe lines of force, to a shapeof a dipole ing constancyof the first adiabaticinvariant Ix = E ñ/B o. field (Figure 5, bottom), we have the generalshape of This assumptionis valid when the magneticfield changes planetarymagnetospheric fields. Particle radiation, such slowlyrelative to the Larmor period and Larmor radius. as the Van Allen radiation belts, are trapped on these For the dipole field shownin Figure 6b, we calculatethe field lines [VanAllen, 1991].The particlesgyrate about value below.As was mentionedpreviously, at the mirror the magnetic fields and also bounce back and forth point the particle's perpendicularkinetic energyEñ is betweentheir mirror points.The particlesalso undergo equal to the total kinetic energyET. Thus we can write a drift in the azimuthal direction around the Earth in the Ix asET/Bmirror at the mirror point. At the equator, IX is equatorial region due to the curvature and magnetic equalto E_l_/Beq.Equating these two values,we have field gradientsin the radial direction (this drift is not ET/Bmirror= Eñ/Beq.We rearrangethis as Eñ/ET - shown).Because the senseof drift is dependenton the Beq/Bmirror.From .previous discussions we knowthat signof the chargeon the particle, protonsand electrons Eñ/ET = •mVñ/•mV] 2 I 2 = sin2 ao, wherea 0 is the drift in opposite directions.These different drifts con- particle pitch angle at the equator. stitutea "ringof current"(ring current)which intensifies Thus for the loss cone we have during magneticstorms (owing to injectionand energi- zation of ring currentparticles). The injectionof plasma sin2 ao - Beq/Bmirror (4) causesdecreases in the magneticfield measuredat the Earth's surfacenear the equator.This is calledthe storm Thevalues for Beq and Bmirror can be calculatedassum- main phase. The loss of these particlesthrough wave- ing a dipolefield dependencewith distance,B/r 3 = particle interactionsand other processes[see Kozyra et const.At the Earth the surface equatorial field is ap- al., 1997] leads to a decreasein the ring current and an proximately-0.3 G. Thus for any dipole field line, the increasein the field at the equator. This is called the losscone can be easilycalculated. For any particle with storm recoveryphase. It has been shown [Desslerand pitch anglesat the equator with a < a 0, suchthat the Parker, 1959; Sckopke,1966] that the magnitudeof the height of the mirror point is within the upper atmo- field decreasein the main phaseis directlyrelated to the sphere,the particlesare lost by collisionswith neutrals. total energy of particlesin the ring current. In expression(4), a 0 is the pitch angleat the edgeof the How one gets energetic(energies of-MeVs) parti- loss cone. cles on these trapped is another problem. It is The Earth's field is not a pure dipole. There are 35, 4 / REVIEWSOF GEOPHYSICS Tsurutaniand Lakhina:WAVE-PARTICLE INTERACTIONS ß 495 variationsin the local surfacefield strength.One area, Electromagnetic Wave Polarizations called the Brazilian anomaly (previouslycalled the SouthAtlantic Anomaly,but this magneticregion has recentlydrifted inland) is a regionof low magneticfield strength.In this regionthe magneticfields are weaker, Left-hand ' ' "• Bo and therefore the mirror points are shifted to lower altitudes.The particlesthat are normallyjust outside the ,,o,ar,z.d '- losscone will mirror deeperin the atmospherethan at ion cyclotron (high to) other longitudesand are lostby collisionswith the ion- Alfv&n mode (low to) ospheric/atmosphericatoms and molecules.A satellite passingjust abovethe ionospherewould seemore par- • V•,h ticle flux comingdown (or higher radiationdoses) in Right-hand • Bo comparisonwith the regionoutside the Braziliananomaly. polarized

whistler mode {high to) 3. RESONANT WAVE-PARTICLE INTERACTIONS magnetosonic mode (low

Previously,we showedthat chargedparticles have a Figure 7. Left-handand right-handparallel propagating cir- circular(cyclotron) motion about the ambientmagnetic cularlypolarized electromagnetic waves. field (gyromotion)plus a translationalmotion along the magneticfield. When a particlesenses an electromag- neticwave Doppler-shifted to itscyclotron frequency (or the wavephase speed can be approximatedby the local its harmonics),it can interactstrongly with the waves. Alfv6nspeed V A = [B2/4'rrp]•/2, where p isthe ambient The conditionfor this cyclotronresonance between the plasmamass density in cgsunits. wavesand the particlescan be written as Figure 7 illustratesthe spatialvariation of the wave to - k. V = nil (5) (perturbation)magnetic vector as a functionof distance along the magneticfield. Here we illustratecircularly In expression(5), toand k are thewave frequency (taken polarized,parallel-propagating electromagnetic waves. as positivehere) and wavevector, and n is an integer There are two basictypes of polarization,right-handed equalto 0, _+1,_+2,.... The caseof n = 0 corresponds and left-handed.Elliptical or linear polarizationsare to the Landau resonancediscussed previously. When combinationsof these two fundamentalpolarizations. condition(5) is satisfied,the wavesand particles remain The polarizationof wavesis definedby the senseof in phase,leading to energyand momentumexchange rotation of the wave field with time at a fixed location. between them. The senseis with respectto the ambientmagnetic field For illustrativepurposes, we first describethe n = 1 and is independentof the directionof propagation. (fundamental)resonance for electromagneticwaves In a magnetizedplasma where llpe > D-, left-hand- propagatingeither parallelor antiparallelto the mag- polarizedwaves can existat frequenciesup to the ion neticfield direction, i.e., we takek = kll b. cyclotronfrequency. At the high end of the frequency Thus (5) simplifiesto range,this mode is calledan ion cyclotronwave. At low to- kllVii= 11 (6) frequenciesthis mode maps into the Alfv6n mode branch.Right-hand waves can existup to the electron If the frequencyof the waveand the localgyrofrequency cyclotronfrequency. These waves are dispersive(in this of the particle are known,then the particle resonance case,higher frequencies have higher phase velocities as energycan be calculated.From (6) we havethe parallel longas tois sufficientlybelow D-; as toapproaches D-, resonancespeed the phasevelocity decreases with increasingto, but the wave suffersheavy cyclotrondamping and ceasesto (•o- fi) exist).When these right-hand waves travel any substan- tial distance,the highest-frequencycomponent arrives first. Lightning-generatedelectromagnetic noise travel- Then the parallel kinetic energyof resonantparticles can be written as ing within plasma "ducts"(field-aligned density en- hancementsor depletionswith Ap/p> 5%) throughthe • 2 • (to- 1/)2 magnetospherefrom one hemisphereto the other ends up havinga whistlingsound; thus the name "whistler EliR: •mVii R:•m kl• =«mV•h(1 --11/00) 2 (7) mode." The whistlingsound starts at high frequencies whereVph .= to/kllis the parallelphase speed of the and descendsto lower frequencies.At low or magneto- wave. For the case in which the resonant waves are at hydrodynamicfrequencies, this wave is the magne- frequenciesmuch less than the ion cyclotronfrequency, tosonic mode. 496 ß Tsurutani and Lakhina' WAVE-PARTICLE INTERACTIONS 35, 4 / REVIEWSOF GEOPHYSICS

(Normal) Cyclotron Resonance and structureless,having falling tonesand having"rising hook" emissions.Figure 9 showsan example of two- frequency rising-tone chorus detected by OGO-5 on August 15, 1968, at L = 5.9 (for a dipole field, the L value is the distance in Earth radius that the field line ion Left- hand wave crossesthe magneticequator). This event had an aver- •-•.•:a+ agepeak power of 9 x 10-7 (nT)2 Hz-•. Oneemission • + k,V,: • + band is at ---700Hz, the base frequencyfor tones rising

The relative motion between the wave and from 700 to 1000 Hz, while a secondthin band occursat particle•Doppler shifts the wave up to the ion ---1150Hz and consistsof short (•0.1-s duration), dot- cyclotron frequency. • like emissions. Chorus has been detected at all local times and at L Vph I I I I valuesbetween the plasmapauseand the magnetopause. electrons II ,i ;i i • • Bø However,it occurspredominantly between midnight and ,LLL' LLLL' 1600 local time (LT). Chorusoccurs principally in two electron Right-hand wave magneticlatitude regions,namely, the equator(equato- _ • + k,V,: • rial chorus), and at latitudes above 15ø (high-latitude chorus)as shown schematically in Figure 10. The density Figure 8. Normal first-order cyclotron resonancebetween of the dotsindicates the regionswhere chorusis likely to electromagneticcircularly polarized waves and charged particles. be generated,with higher densitiesindicating greater probabilityof generation.Equatorial chorusoccurs pri- marily duringsubstorms, whereas the high-latitudecho- 3.1. Normal Resonance rus often occursduring quiet periods. Many observed The normal cyclotronresonance between waves and featuresof equatorialchorus can be explainedby the chargedparticles is diagrammedin Figure 8. For this cyclotronresonance condition (equation (5)) between resonantinteraction the wavesand particlespropagate the whistler mode wavesand energetic(10-100 keV) towardseach other. Left-handpositive ions interact with electronsinjected by substormelectric fields. As can be left-handedwaves, and correspondingly,right-hand elec- seenfrom (6)-(7), for a particularfrequency wave, the trons interact with right-hand waves. Since the waves lowest-velocity(or resonantenergy) electron will be in and particlesapproach each other, k. V has a negative cyclotronresonance at the equator, where the gyrofre- sign.Thus the Doppler shift term (-k' V) in equation quencyis minimum.Because the typicalmagnetospheric (5) is a positiveone. The relativemotion of the waves electron spectrumhas more particlesat lower energies a.ndparticles causesa Doppler shift of the wave fre- than at higher energies,wave-particle interactions will quencyto up to the particle cyclotronfrequency 12. be most intense at the equator, and this can lead to a One plasma instability generating these waves in rapid wavegrowth provided the energeticelectrons have planetarymagnetospheres is the "losscone instability." losscone distributions[Kennel and Petschek,1966; Tsu- Thisinstability occurs when conditions rñ/rll > 1 exist rutani and Smith, 1977]. [Kenneland Petschek, 1966]. Tii(Tñ) is the ion or elec- As energetic electrons are convectedearthward by tron temperatureparallel (perpendicular)to B0, assum- substormelectric fields, the particle motion in the pres- ing the plasmahas a "bi-Maxwellian"distribution. Elec- ence of magneticfield gradientsand the magneticfield tron loss cone instabilities generate whistler mode curvature causesthe electronsto drift azimuthally to- emissionsdescriptively called auroral zone "chorus" ward dawn. This leads to energetic electron flux en- [Tsurutaniand Smith, !974, 1977; Kurth and Gumett, hancementand precipitationprincipally in the postmid- 1991] and "plasmaspherichiss" [Thorne et al., 1973; night sector. Owing to compressionof the dayside Tsurutaniet al., 1975;Kurth and Gumett, 1991] because magnetosphere,the azimuthallydrifting electronsend of the sound they make when played through a loud- up on higherL values.This is the so-calleddrift shell- speaker(the wavesin the outer magnetospheredo not splitting effect [Roederer,1970]. This can explain the bounce several times like lightning whistlers,and fre- chorus asymmetrynear midnight and the spread in L quency-timestructures are due to intrinsic generation with local time. mechanisms). There is an increasein chorusactivity at dawn and the Extremelylow frequency(ELF) chorusis a common dawn-to-noonsector. Enhanced wave-particleinterac- naturally occurring, intense electromagneticemission tions occur as a result of the loweringof resonantener- observedin the Earth's magnetosphere[Russell et al., giesin the presenceof increasedambient plasma density 1969; Dunckel and Helliwell, 1969; Burton and Holzer, in those sectors.The latter effect is caused by iono- 1974; Burtis and Helliwell, 1976; Anderson and Maeda, sphericheating due to solar irradiation at these local 1977; Comilleau-Wehrlin et al., 1978; Inan et al., 1983; times.Although energeticelectrons continue to precip- Goldsteinand Tsurutani,1984; Alford et al., 1996]. The itate as the cloud drifts in longitude from midnight to frequency-timecharacteristics of choruscan be banded dawn, this effect leads to a remarkable increase in cho- 35 4 / REVIEWS OF GEOPHYSICS Tsurutani and Lakhina: WAVE-PARTICLE INTERACTIONS ß 497

OGO-5 REV 63 MLAT: -4.2 ø 15 AUGUST, 1968 L:5.92R 0118 LOCAL TIME B0: 93),e

I I i ! i

i i i i i :30

1500

N

I I I lsec

Figure 9. Two-frequencyrising-tone chorus. One emissionband is at •700 Hz, the base frequency for chorusrising from 700 to 1000 Hz, and a secondband at •1150 Hz, consistingof •0.1s duration a dotlike emissions.In the blown-uppart of the figure, the higher-frequencydots are seen to be the high-frequency portionsof the risingtones with a strongextinction in the frequencyrange from 1000 to 1100 Hz (•0.5•-). From Tsurutaniand Smith [1974, Figure 6]. rus intensity and electron precipitationbetween dawn erating suchwaves include the ion beam instabilityin and noon. - planetaryforeshocks (the magneticallyconnected region There is no single mechanismfor the high-latitude upstreamof shocks)[Hoppe et al., 1981; Smith et al., chorus.Some events appear to be generatedby a loss 1985; Goldsteinet al., 1990; Gary, 1991; Verheestand cone instability.This local generationwould occur in Lakhina, 1993; Lakhina and Verheest,1995] and ion minimum B pockets,which are regionsof local mini- pickup around comets [Tsurutani and Smith, 1986; mum magneticfield betweenmagnetic latitudes of 20ø Thorne and Tsurutani, 1987; Brinca, 1991; Tsurutani, and 50ø formed by the compressionof the daysidemag- 1991; Neubauer et al., 1993; Glassmeier et al., 1993; netosphere[Roederer, 1970; Tsurutaniand Smith, 1977]. Mazelle and Neubauer,1993]. The ion beam generates Some other high-latitude chorus events appear to be right-handmagnetosonic waves. In the foreshockcase equatorialchorus that has simplypropagated to higher the sourceof the ionsis eithershock-reflected (1-5 keV) magneticlatitudes. solarwind particlesor ions streamingfrom the magne- tosheathwith energiesup to •40 keV. In the cometary 3.2. Anomalous Resonance case, neutral molecules/atoms sublimate from the nu- There is anothertype of resonance,called anomalous cleus as the comet approachesthe Sun. This neutral cyclotronresonant interactions.This is shown for the cloudcan be •10 6 km in radius.The photoionization caseof positiveions in Figure 11. Positiveions interact and charge exchangeof cometaryH20 group neutrals with right-handwaves. They do so by overtakingthe (H20 , OH, O) lead to the formationof a "beam"in the waves(Vii > Vph)SO that the ionssense the wavesas solar wind frame. For instabilitythe typical kinetic en- left-handpolarized. Because left-hand ions interact with ergy of the ions relative to the solar wind plasma is right-handwaves, this interactionis called"anomalous." 30-60 keV. From the expressionin the resonancecondition, the The same anomalouscyclotron resonant interactions Doppler shift decreasesthe wave frequencyto that of occur between electrons and left-hand-mode waves. the cyclotronfrequency. Examples of the instabilitygen- However, since the left-hand waves are at frequencies 498 ß Tsurutani and Lakhina: WAVE-PARTICLE INTERACTIONS 35, 4 / REVIEWS OF GEOPH.YSICS

particle. For ease of visualizationwe separate particle Vñ andVii components. Clearly the resonant interaction of particles with arbitrary pitch angles will include a combinationof the two velocitycomponents. In Figure 12a, we showthe casewhen the interactionis through Vñ. Since a constantB is imposedon the particle, the Lorentz force is in the B0 direction. If the particle is propagatingtoward the right, the pitch angle will be decreased,and if the particle is travelingto the left, it will be increased.However, we have arbitrarily chosen the B to be in the upward directionin the figure. If the relative phase between the wave and particle were shiftedby 180ø such that B waspointing downward, all of the resultsstated previously would be reversed. Figure 12b showsthe particle interaction due to the Figure 10. Schematicrepresentation of the regionsin which parallel component of particle velocity. Here the chorusis generated.The figure showsthe magneticfield in the Lorentz force is in a direction oppositeto that of the noon-midnightmeridian plane based on the magnetosphere gyromotionof the left-hand ion. Therefore the interac- field model of Mead and Fairfield [1975].The regionsin which chorus is thought to be produced are noted by dots. Near tion decreasesVñ (E ñ) and decreasesthe pitch angleof midnight,chorus is generatedclose to the magneticequator by the particle. If the phase of the wave were different by substorm-injectedelectrons from the plasma sheet. Chorus 180ø, such that B was directed downward, F L would continuesto be generatednear the magneticequator as the acceleratethe particle in Eñ, and the pitch anglewould electronsdrift around from dawn to noon. On the daysideat be increased. large L values, chorus is also generatedover a much larger Resonant wave-particleinteractions occur on time- span of magnetic latitudes. Within 1-2 R•r of the magneto- scalessmall in comparisonwith the cyclotron period, pause,chorus may originatein minimumB pockets,which are thusthe first adiabaticinvariant !* is not conserved(it is local regionsof minimum magneticfield strengththat occurat "broken").In the inertial frame, the total energyof the high latitudesas a result of the solarwind compressionof the particle is not conservedduring the wave-particleinter- dayside magnetosphere.From Tsurutani and Smith [1977, action. However, the energy of the particle in the rest Figure 14]. frame of the wave is conservedas shownby the following physicalargument. Let us assumethat duringwave par- ticle interaction a particle gains a quantum of energy, belowthe ion cyclotronfrequency (a value far belowthe AE, from a wave. Then AE = hto/2,r, where to is the electron cyclotronfrequency), resonant electronsare wave frequencyand h is the Planck'sconstant. The gain typicallyrelativistic (Ell > MeV) for thisinteraction to in the parallel momentumof the particle will be take place. Even so, it is speculatedthat suchan insta- bility is occurring upstream of the Jovian magneto- sphere,perhaps as a resultof leakageof Jovianradiation mAVII= hkll 2'rr- klltoAE. belt electrons[Smith et al., 1976; Goldsteinet al., 1985]. The actualphysical mechanism for particle pitch an- If the energygain is smallcompared to the total particle gle scattering due to electromagneticwaves is the energy,then Lorentz force.This is illustratedin Figure 12 for positive ions. At cyclotronresonance, the particle experiences Wave-Particle (Cyclotron) Interaction the wave magneticfield B gyratingin phase with the a) a• _ b) v,

ions ,.,., , • ,, ß • \\

ion Right-hand wave • B0 1,, Bo Figure 11. Schematicillustrating anomalouscyclotron reso-

nancebetween electromagnetic circularly polarized waves and Ion positivelycharged particles. The left-hand ion overtakesthe example right-handwave (V•i > Vph)and senses it left-handpolarized. The anomalouscyclotron resonance occurs when the condition Figure 12. Pitch angle scatteringby resonant electromag- to - kllV•l= -12 + is satisfied.Note that the relativemotion neticwaves for (a) Vñ and (b) Vii components.The ¾ x B between the particle and the wave Doppler shifts the wave interactionwith the waveB•o increases or decreasesthe paral- down to the gyrofrequency.The interaction is "anomalous" lel velocityof the particle,depending on the particle'sdirection becauseright-hand waves interact with left-hand ions. of propagation. 35, 4 / REVIEWSOF GEOPHYSICS Tsurutaniand Lakhina:WAVE-PARTICLE INTERACTIONS ß 499

mo) : m(V,aV, + vav) : av,: mVp.aV,(8) Figure 13. Pitch angle scat- Integrating(8), we get tering by the electric compo- 1 1 • Bo nent of resonant electromag- •mV2•+ • re(V,- Vp,)2=const (9) netic or electrostatic waves. which showsthat particle energyin the wave frame is conserved. Equation (8) indicates that the particle energy changes,on the basis of thesign of/X Vii for a givenphase velocity(Vp, > 0 takenhere). Particles that increase Vii through resonantinteractions increase energy and ab- sorbwave energy, and those that decreasein Viilead to D• 2At = 2 At (12) the generationof waveenergy. The thermalbackground The time At is the time needed for a particle Ak/2 out plasma,which is out of resonancewith the waves,does of resonanceto change its phase by 1 tad, or At • not exchangeenergy during resonantwave particle in- teractions.In general,if one startswith a highly aniso- 2/akV,. We now get tropicpitch angle distribution(say T_L>> Tii), one exciteswave growth by the loss cone instability. The B 2/iXk fl B 2/A k k wavesin turn scatterthe particlesand "fill" the losscone D•fi B• •=•Vcos• B• cos• (13) to further reducethe free energyavailable in the aniso- tropicpitch angle distribution until one getsto the stably Again, assuminglarge pitch angle particles, trappedlimit of Kenneland Petschek[1966]. For waveswith electricfield amplitudesE, the parti- D=fi • (14) cle'sperpendicular kinetic energyincreases or decreases dependingon the phaseof the wavewith respectto the where• = (fi/AkVii) is the fractionalamount of time particle. The situation for increasedE_L is shown in that the particlesare in resonancewith the waves. Figure 13. Analogousarguments can be made for wave- Particle transport acrossthe magneticfield can be particleinteractions due to electrostaticwaves having a calculatedonce the mobili• of the chargedparticles in componentE perpendicularto Bo, or the electriccom- the direction perpendicular to the ambient magnetic ponent of electromagneticwaves. field, the so-called Pederson mobiliS, is known. The Pedersonmobili• •x of particlesin the directionper- pendicularto Bo [Schultzand Lanzerotti,1974] is 4. PITCH ANGLE SCATTERING = (C/o)md[1 + (me,,) The overallparticle pitch anglescattering rates due to where •eff is the effective time be•een wave-particle electromagneticor electrostaticwaves have been de- "collisions." finedby Kennel and Petschek [1966] and haveempirically The ma•mum cross-field diffusion occurs when the been shown to be valid for the rate of scattering of particlesare scatteredat a rate equal to their •roffe- electronsin the outer magnetosphere.Here we derive quencies,or •½• • eBo/mc(Bohm diffusion). A spatial similarpitch angle scatteringrates from simplephysical diffusioncoefficient derived by Roseand Clark [1961] is arguments.We havetan et = V•_/Vii,and for largepitch angle particleswhere V•_ • V, we have D• = (•)2/2A• = (mV•/2e)• (16) Aot= - AV•i/V, (10) For Bohm [1949] diffusion, The maximum change in the parallel velocity of a D • = E•c/2eBo = Din= (17) charged particle interacting with our electromagnetic wave is given by For conditionswhere fi%ff >> 1 and %ff • l/D, Tsum- tani and Thorne [1982] have used (13) and (16) to determine the cross-fielddiffusion rate due to the mag- netic componentof electromagneticwaves by

whichwith the help of (10) can be written as E•c 1 D = e - (18) eV• 1 B --BAt -- = fiat (11) mc V• Bo Similarly for electrostaticwaves, we get

The pitch angle diffusionrate is thus D = (19) 500 ß Tsurutani and Lakhina: WAVE-PARTICLE INTERACTIONS 35, 4 / REVIEWS OF GEOPHYSICS

possiblenew ground-baseddiagnostic of the outer high- latitude magnetosphere,J. Geophys.Res., 101, 83, 1996. Anderson, R. R., and K. Maeda, VLF emissions associated with enhancedmagnetospheric electrons, J. Geophys.Res., 82, 135, 1977. Bohm, D., Quantitative description of the arc plasma in a magneticfield, in Characteristicsof ElectricalDischarges in MagneticFields, edited by A. Guthrie and R. Walkerling,p. 1, McGraw-Hill, New York 1949. Figure 14. Particle cross-field diffu- Brinca,A. L., Cometarylinear instabilities:From profusionto sionby resonantinteractions with waves. perspective,in CometaryPlasma Processes, edited by A.D. Johnstone,Geophys. Monogr. Ser., vol. 61, p. 211, AGU, Washington,D.C., 1991. Burtis, W. J., and R. A. Helliwell, Magnetosphericchorus: Occurrence patterns and normalized frequency, Planet. SpaceSci., 24, 1007, 1976. Burton,R. K., and R. E. Holzer, The origin and propagationof chorus in the outer magnetosphere,J. Geophys.Res., 79, 1014, 1974. Chen, M. W., M. Schulz,and L. R. Lyons, Modeling of ring current formation and decay:A review, in MagneticStorms, Figure 14 showsthe processof cross-fielddiffusion due Geophys.Monogr. Ser., vol. 98, edited by B. T. Tsurutani, to resonantwave-particle interactions. B0 is the original W. D. Gonzalez, Y. Kamide, and J. K. Arballo, p. 173, guiding center magnetic field line. After pitch angle AGU, Washington,D.C., 1997. scattering,the guiding center lies on the B[ magnetic Cornilleau-Wehrlin, N., R. Gendrin, F. Lefeuvre, M. Parrot, field line. The particlehas diffused across magnetic field R. Grard, D. Jones, A. Bahnsen, E. Ungstrup, and W. Gibbons, VLF electromagenticwaves observed onboard lines. GEOS-1, SpaceSci. Rev., 22, 371, 1978. Using the measuredwave amplitudesobserved by Dessler, A. J., and E. N. Parker, Hydromagnetictheory of ISEE 1 and 2 at the magnetopause,Thorne and Tsuru- magneticstorms, J. Geophys.Res., 64, 2239, 1959. tani [1991]showed, using expressions (18) and (19), that Dunckel, N., and R. A. Helliwell, Whistler mode emissionsand magnetosheathplasma can diffuse at one tenth of the the OGO satellite,J. Geophys.Res., 74, 6371, 1969. Gary, S. P., Electromagneticion/ion instabilitiesand their Bohm diffusion limit. This rate is adequate to account consequencesin spaceplasmas; A review, SpaceSci. Rev., for the formationand maintenanceof the magnetopause 56, 373, 1991. boundarylayer. Gary, S. P., What is a plasmainstability?, Eos Trans.AGU, 73, 529, 1992. Gill, R. D., Plasma Physicsand Nuclear Fusion Research,Ac- 5. FINAL COMMENTS ademic,San Diego, Calif., 1981. Glassmeier, K.-H., U. Motschmann, C. Mazelle, F. M. We have tried to give simple explanationswith illus- Neubauer, K. Sauer, S. A. Fusilier, and M. H. Acufia, trations to explain the fundamentalsof wave-particle Mirror modes and fast magnetoacousticwaves near the interactions.Clearly, more complexinteractions and sec- magneticpileup boundaryof comet P/Halley, J. Geophys. Res., 98, 20,955, 1993. ond-order effects, which indeed are present, have not Goldstein, B. E., and B. T. Tsurutani, Wave normal directions been includedin this basicdescription. of chorusnear the equatorial sourceregion, J. Geophys. Res., 89, 2789, 1984. Goldstein,M. L., H. K. Wong, A. F. Vifias, and C. W. Smith, ACKNOWLEDGMENTS. B.T.T. wishes to thank B. Buti Large-amplitudeMHD wavesupstream of the Jovianbow for the invitation to lecture on the topic of wave-particle shock:Reinterpretation, J. Geophys.Res., 90, 302, 1985. interactionsat the Trieste, Italy, SummerSchool, sponsored by Goldstein, M. L., H. K. Wong, and K.-H. Glassmeier,Gener- UNESCO/Italy. He would like to particularlythank the Trieste ation of low-frequencywaves at comet Halley, J. Geophys. 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