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Particle Interactions in Collisionless Plasmas SOME BASIC CONCEPTS OF WAVE-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 scattering,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 introducefinite 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 plasma heating by radio waves 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 orbit, 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 diffusion,which causesthem to be scatteredinto / F, the atmosphericloss cone, or undergo energy 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
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