Nonlinear whistler instability driven by a beamlike distribution of resonant electronsMartin Lampe, Glenn Joyce, Wallace M. Manheimer, and Gurudas Ganguli Citation: Physics of Plasmas (1994-present) 17, 022902 (2010); doi: 10.1063/1.3298733 View online: http://dx.doi.org/10.1063/1.3298733 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/17/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Nonlinear electron acceleration by oblique whistler waves: Landau resonance vs. cyclotron resonance Phys. Plasmas 20, 122901 (2013); 10.1063/1.4836595 Nonlinear electron magnetohydrodynamics physics. IV. Whistler instabilities Phys. Plasmas 15, 062109 (2008); 10.1063/1.2934680 Precipitation of trapped relativistic electrons by amplified whistler waves in the magnetosphere Phys. Plasmas 14, 062903 (2007); 10.1063/1.2743618 Computer simulations of relativistic whistler-mode wave–particle interactions Phys. Plasmas 11, 3530 (2004); 10.1063/1.1757457 Interaction of suprathermal electron fluxes with lower hybrid waves Phys. Plasmas 11, 3165 (2004); 10.1063/1.1715100

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Nonlinear whistler instability driven by a beamlike distributionof resonant electrons

Martin Lampe,1 Glenn Joyce,2 Wallace M. Manheimer,3 and Gurudas Ganguli11Plasma Physics Division, Naval Research Laboratory, Washington, DC 20375-5346, USA2University of Maryland, College Park, Maryland 20740, USA3Icarus Research Inc., Bethesda, Maryland 20814, USA

�Received 25 September 2009; accepted 11 December 2009; published online 5 February 2010�

Theory and simulation are used to study the instability of a coherent whistler parallel-propagatingin a simplified model radiation belt with a background of cold electrons, as well as a ringdistribution of energetic electrons. A nonlinear instability is initiated at the location z+, where theelectrons are cyclotron resonant with the wave, on the side of the equator �z=0� where the wave ispropagating away from the equator. The instability propagates backward toward the equator,growing both spatially and temporally. As the instability develops, frequency falls in such a way asto keep the electrons nearly resonant with the waves over the entire region 0�z�z+. The instabilitycauses a sharp drop in the pitch angle of the resonant electrons and eventually saturates with peakamplitude near the equator. © 2010 American Institute of Physics. �doi:10.1063/1.3298733�

I. INTRODUCTION

It has long been known1–5 that a powerful coherent whis-tler of angular frequency �0, propagating along a geomag-netic field line in the magnetosphere, can “trigger” secondaryemissions, which begin at frequency �0 but then rise or fallin frequency by substantial factors, while growing to ampli-tudes up to 30 dB above that of the trigger wave. The strongfrequency variation is a clear indication that this is a nonlin-ear phenomenon, not just a linear instability. There is a largebody of theoretical analysis and simulation6–27 of these situ-ations and related issues in chorus generation, and severalpossible explanations have been proposed for instabilitieswith rising frequency, known as risers. However there doesnot appear to be any explanation in the literature for insta-bilities with falling frequency, known as fallers.

Trakhtengerts and co-workers23,24 have argued that trig-gering could in some cases be due to the development of asingularity in the parallel velocity distribution of fast elec-trons, resulting from the action of the trigger wave itself orsome external mechanism. For example, a beamlike distribu-tion of parallel velocities v�, or a sharp edge in the distribu-tion of v�, allows many electrons to participate coherently ininstability growth and perhaps even more important, singlesout a particular spatial point where the electrons are in cy-clotron resonance with the wave. The purpose of this paper isto demonstrate that an instability of this type does in factoccur and to explore in some detail the physical mechanismsthat underlie the instability. We therefore consider here onlythe simplest case, a delta-function distribution �also called abeam or ring distribution�,

f�z,v�,v�� = 2���v�2 − v�0

2 B0�z�B0�0��

���v� −�v�02 + v�0

2 1 −B0

2�z�B0

2�0�� , �1�

where z is the coordinate along a field line, and v�0 and v�0

are the electron velocity at the equator �z=0� perpendicularand parallel to the magnetic field B0�z�. Note that the distri-bution �1� is a function of the two unperturbed constants ofthe motion, the kinetic energy

� � 12m�v�

2 + v�2 �

and the magnetic moment mv�2 /B0, and thus represents an

adiabatic equilibrium in the presence of the spatially varyingmagnetic field B0�z�. In order to focus clearly on the physicalmechanisms, we follow in detail the evolution of a singlesimulation. We do not argue that a ring distribution of elec-trons is likely to occur in any particular situation of interest,but only that this is the best and simplest illustration of theessential features of this “doubly coherent” instability,wherein a single large-amplitude wave is interacting coher-ently with a single value of v� selected by the distributionfunction. We believe that the case of a step discontinuity inthe distribution function is of greater physical interest, asdistribution function discontinuities can form at the edge ofthe trapping region of a large amplitude coherent wave. Wehave in fact seen, in other simulations, similar but somewhatweaker instabilities driven by a descending step discontinuityin the distribution of v�. However, the elucidation of thephysics is considerably more complicated in that case, andwill be deferred to a future publication.

II. SIMULATION MODEL

Both experiment3–5 and theory6–27 clearly indicate thatthe instabilities considered here depend strongly on the non-uniformity of B0�z�, even though the scale length for varia-tion of B0�z� is much longer than the whistler wavelength.Similarly, the instability evolution depends strongly on theduration of the injected wave packet, even though this is longcompared with the whistler period. Finally, the inflow of un-perturbed electrons from regions outside the wave packetstrongly influences the evolution. For all of these reasons,

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one cannot base any of the nonlinear analysis on the fre-quently used model of a plane wave propagating in an infi-nite uniform medium.

Our simulation was performed with HEMPIC,28 a two-dimensional hybrid simulation code �cold fluid bulk elec-trons plus PIC fast electrons�, which assumes quasineutralityand neglects both the longitudinal and transverse compo-nents of the displacement current. These simplifications toMaxwell’s equations greatly increase numerical efficiency byeliminating the plasma oscillation time scale �p

−1 �typicallymuch shorter than the whistler period� and the Courant con-dition associated with the speed of light c �irrelevant sincewhistlers propagate at speed much less than c�, and by elimi-nating the need for enormous numbers of simulation par-ticles to calculate the small difference ne−ni. The code isfully relativistic, although relativistic effects play no role inthe case considered here. In the present study, we retain onlyone spatial dimension, the distance z along the field line, plusthree particle velocity components. The curvature of the geo-magnetic field line is ignored in the code, but thez-dependence of the magnitude B0�z� is properly modeled byincluding a mirror force. A very important point is that theboundaries of the simulation are open: electrons are deletedif they cross the simulation boundary, and fresh electronscontinually flow in from the boundary. In the simulationshown in this paper, the PIC fast electrons are initially loadedin pairs with zero net current, and similarly electrons come infrom the boundaries in zero-current pairs, so there is no par-ticle noise. The electromagnetic disturbance arises entirelyfrom a wave which we initiate by modulating the cold fluidvelocity at t=0.

Although the HEMPIC model is greatly simplified, ascompared with full-Maxwell PIC codes such as that of

Omura et al.,27 we believe that HEMPIC includes all of thephysics necessary to accurately simulate whistler instabili-ties. This point is discussed at some length in Ref. 28. Thebenefit of using the reduced simulation model is a very sub-stantial gain in numerical efficiency. The simulation shownhere uses about 320 000 particles and runs in 40 min on aPC, whereas full Maxwell PIC simulations typically usemany million particles and run on supercomputers.27

III. SIMULATION RESULTS

Our simulation domain extends �2875 km on each sideof the equator. It represents conditions along the field lineL=4.9, appropriate for the HAARP experiments,5 exceptthat the scale length for geomagnetic field variation is shrunkby an order of magnitude to improve numerical efficiency�as is often done in magnetospheric simulations27�. Thegeomagnetic field near the equator is set to B0�z�=370��1+ �z /1220 km�2� nT. The cold plasma density is200 cm−3 and the energetic electron density is 10−3 cm−3,both assumed uniform. We set the energetic electron velocityto v�0=−105 and v�0=4.5�104 km /s. At t=0, we initiate avery nearly flat-topped wave packet, centered at the equator,with angular frequency �0=10 kHz and phase velocity+1.49�104 km /s, as shown in Fig. 1�a�. The wave is initi-ated by modulating the cold fluid bulk electric velocityv�c= �vxc,vyc� at a spatially varying wave number k0�z�, cho-sen to satisfy the whistler dispersion relation, i.e., at t=0 weset

vxc�z,0� = v�c0 sin�0

z

dz�k0�z�� , �2a�

a

c

d

e

f

g

h

i

j

k

l

b

FIG. 1. ��a�–�d�� Wave magnetic field B��z , t� at times t=0.5, 2.0, 4.5, and 9.0 ms. ��e�–�h�� Fast electron distribution in �z ,v�� at the same four times. ��i�–�l��Fast electron distribution in �z ,v�� at the same four times.

022902-2 Lampe et al. Phys. Plasmas 17, 022902 �2010�

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vyc�z,0� = v�c0 cos�0

z

dz�k0�z�� , �2b�

where

k0�z� ��p

c��z�

�0− 1−1/2

, �3�

and we choose v�c0=1.5 km /s. Note that the wave propa-gates slowly to the right, while the electrons propagate muchfaster to the left, as is required for cyclotron resonance.The fast electrons interact strongly with the wave only atcyclotron resonance, which initially occurs at two discretepoints z�= �875 km where �0−k0�z�v��z�+��z�=0. Here��z��eB0�z� /m is the spatially varying electron cyclotronfrequency.

Figures 1�a�–1�d� show the wave amplitude B� at foursuccessive times, t=0.5, 2.0, 4.5, and 9.0 ms. At early time,Figs. 1�a� and 1�b�, the wave begins to grow at both reso-nance points z�. This is the linear instability,28 driven by thehighly anisotropic and singular nature of the distribution �1�.In the present case, the wave starts at a rather large initialamplitude and the linear stage ends after about one order ofmagnitude of growth. As regards the later nonlinear develop-ment, the essential point is that the instability is initiated bya single coherent wave rather than growing out of noise. Inour simulation, with its quiet start properties, we can if wechoose initiate a coherent wave at smaller initial amplitude,which simply results in a longer period of �rather uninterest-ing� linear growth, localized at the two resonant points, fol-lowed by exactly the same nonlinear behavior. In real life,the dominance of a single coherent wave usually means thatthe wave amplitude is initially large. At the end of the linearstage, the instability at z− saturates, while the instability at z+

continues to grow. The wave packet propagates slowly�barely visibly� to the right, but the instability grows back-ward from z+ toward the equator, Fig. 1�c�. At late time,Fig. 1�d�, the instability saturates with peak amplitude atz 700 km and the waves propagate slowly to the right,creating a broad area of large amplitude with no furthergrowth.

Figures 1�e�–1�h� show the distribution of simulationparticles in the �z ,v�� section of phase space at the samefour times. Notice that the electrons are perturbed upon pass-ing through each of the two resonance points. At early times,Fig. 1�f�, the spread in v� is present at both z+ and z− and isnearly symmetric about the unperturbed distribution, charac-teristic of the linear instability. However, at later time, Fig.1�g�, strong perturbation occurs only downstream of z+ andv� decreases for most electrons, i.e., the electrons are scat-tered toward lower pitch angle � tan−1�v� /v��. In the fullydeveloped instability, Fig. 1�h�, v� drops nearly to zero forsome electrons in this single pass through the waves.

It can be shown29 that an electron that is scatteredtoward smaller pitch angle, by a coherent whistler with���, also loses kinetic energy �,

1

�

d�

d=

2� cos sin

� − � sin2 . �4�

Figures 2�a�–2�d� show the distribution of simulation par-ticles in �z ,�� and Figs. 2�e�–2�h� show the average energy �̄as a function of z at the same four times. We see that thetypical energy loss, upon passing through the resonant pointz+, is �1%. It is this energy loss which drives the growth ofthe waves.

Figures 3�a�–3�d� show the local wave frequency ��z�at the same four times. �It should be explained that fre-quency is here defined as �=�w /�t, where w�z , t�� tan−1�By�z , t� /Bx�z , t�� is the phase of the wave. This defi-nition gives a clear result for monochromatic circularly po-larized whistlers, but can look noisy when there is more thanone wave present, or a band of waves. Fourier transforms arenot easily used to define the frequency spectrum because thetime scale for changes in frequency is not that much longerthan the wave period itself.� The dashed line is the driverfrequency �0=10 kHz. In Fig. 3�a� we see that the waveinitially has �=�0, which is not entirely trivial since weinitiate the wave by spatially modulating the cold fluid ve-locity at the spatially varying wave number k0�z� needed togenerate a uniform-�0 wave. As the system evolves, ��z , t�falls, with this perturbation to � propagating to the left fromthe resonance point z+ along with the perturbed electrons thathave passed through z+, and along with the growth of thewave. �Compare with Fig. 1.� � reaches a minimum�2 kHz on the equator.

Figure 4�a� shows on a linear scale the temporal growthof the wave amplitude �specifically, the largest amplitude atany spatial point, at given time t, as a function of t�, whileFig. 4�b� shows the same plot on a semilog scale. Note that

a

b

c

d

f

g

h

e

FIG. 2. ��a�–�d�� Distribution of particles in position z and kinetic energy �at times t=0.5, 2.0, 4.5, and 9.0 ms. ��e�–�h�� Average electron kinetic en-ergy �̄�z , t� at the same four times.

022902-3 Nonlinear whistler instability… Phys. Plasmas 17, 022902 �2010�

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the amplitude grows exponentially with time in the initiallinear stage, but roughly linearly with time in the large-amplitude regime.

IV. ANALYSIS

To summarize the remarkable features of this instability:�1� the instability propagates backward as compared with thewave propagation and resides in the region between theequator and the initial resonant point z+. It is clearly drivenby the modulated wake of electrons that have passed throughz+. It thus resembles a feedback-driven absolute instability,although this terminology is normally applied to linear insta-bilities. �2� The electrons interact strongly with the wavethroughout the region 0�z�z+ not just at the two initialcyclotron resonance points z�. �3� � falls sharply as the in-stability grows and propagates. �4� In the nonlinear stage, theamplitude grows proportionally to time and saturates at verylarge amplitude, �4–5 nT in this case. �5� Most of the reso-nant electrons are driven to sharply lower pitch angles, withsignificant pitch angle spread.

The strong wave-particle interaction over an extendedspatial region indicates that electrons stay close to cyclotronresonance long after passing through the initial resonancepoint z+, even though � depends strongly on z. It is naturalto think this might be due to phase trapping of the electronsby the wave,7–9,12,13 which can strongly modify v� in such away as to keep the electrons in resonance. However, it turnsout that this is not the case here. Rather, it is the waveswhose wave number k�z,t� and frequency ��z , t� are stronglyperturbed, in just such a way as to keep the waves resonantwith the only slightly perturbed v� of the particles. Figures

3�i�–3�l� show plots of ��z� and the distribution among elec-trons of the Doppler-shifted frequency �̂0��0−k�z , t�v� ateach of the four times. The two original resonances z� arethe points where the ��z� curve and the �̂0 distribution�which at this time is a well-defined curve� cross in Fig. 3�i�.As the instability develops, k�z,t� falls, in just such a way asto make the �̂0 distribution nearly overlap the ��z� curve.Plots of k�z,t�, defined as �w /�z and shown in Figs.3�e�–3�h�, show that the evolution of the �̂0 distribution isdue primarily to the change in k�z,t�. The distribution of fastelectrons in �z ,v�� is shown in Figs. 1�i�–1�l� and it is evidentthat the perturbations to v� are relatively small. Let us first tryto explain this.

Initially a large amplitude wave with constant �0 andspatially varying k0�z� is present over an extended spatialregion, but the energetic electrons interact strongly with thiswave only at the discrete points z�. Electrons that passthrough z+ are phase-bunched by the wave, creating a trans-verse current J�0 exp�−i�0t� at z=z+ that rotates temporallyat frequency �0. Assume for the moment that the electronsno longer interact with the wave after they pass through z+,and simply free-stream to the left at velocity v��z�, whilerotating at ��z�. This then will create a current patternJ��z , t�=J� exp�i�z+

z dz�k�z��−i�0t�, where k�z� satisfies�0−k�z�v��z�−��z�=0. Since ��z� decreases as the elec-trons approach the equator, k�z��k0�z�. A little to the left ofz+, this current will drive a new wave, with wave numberk�z�. The frequency �0 and the new wave number k�z� do notquite satisfy the linear whistler dispersion relation �3�, butthere is enough frequency spread so that J��z , t� can drive awhistler at wave number k�z� and at the frequency ��z� for

a

b

c

d

e

f

g

h

j

j

k

l

FIG. 3. ��a�–�d�� Wave frequency ��z� at times t=0.5, 2.0, 4.5, and 9.0 ms. The dashed line is the driver frequency �0=10 kHz. ��e�–�h�� Wave number k�z�at the same four times. ��i�–�l�� Dot distribution: �0−kv� vs z for the electrons at times t=0.5, 2, 4.5, and 9 ms. Dashed curve: cyclotron frequency � vs z.

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this k�z� according to the dispersion relation �3�, i.e., at ��z�slightly less than �0. This wave, at z slightly to the left of z+

will then further modulate the electron stream at frequency��z�, and in this way a wave can be driven whose wavenumber k�z� and frequency ��z� continuously change in justsuch a way as to satisfy the dispersion relation �3� and nearlysatisfy the cyclotron resonance condition at all locations z.

If this were the whole story, the current carried by thewake electrons would simply drive waves to the left of bothz− and z+ at about the same amplitude as the disturbance atthe original resonance point. However, at z+, where the elec-tron is moving toward a decreasing ��z�, there is in additionan unstable interaction between the waves at frequency ��z�and the resonant electrons, which transfers energy from theelectrons to the wave. This can be shown by plotting, forindividual electrons, the phase difference ��t���t�−w�z , t� between the electron perpendicular velocity phase,defined by tan �t��vy�t� /vx�t� and the wave magneticfield phase, defined by tan w�t��By�t� /Bx�t�. For a whis-tler, −ev�•Ew�0 and thus the electron loses energy, ifsin ��0. It can be shown formally27 that the energy lost bythe electrons goes into wave growth. Note that

�̇ � � − � + kv� , �5�

and thus �̇=0 corresponds to resonance. A plot of ��t� isshown in Fig. 5, for a typical electron that begins at z�z+

with �̇�0, as indicated by Eq. �5� and Fig. 5. Exact reso-nance does not occur for this electron at z+, since � hasdecreased from �0 by the time it reaches z+; however, theelectron is close to resonance at z+ and approaches closer toresonance as it propagates to the left of z+. ��t� thereforecontinues to increase, but slowly. Eventually the electronpasses through resonance, i.e., ��t� reaches a maximum �res

and begins to decrease. Since �by definition� �̇=0 at reso-nance, the electron dwells for a long time near �=�res, andthus the energy exchange between the electron and the waveis dominated by this one point. Notice from Fig. 5 that sin�res�0, so the electron should lose energy. Figure 5 alsoshows the kinetic energy ��t� of this electron �dashed curve,right scale�, and we see that ��t� takes a single big drop asthe electron passes through resonance. Nearly all the elec-trons to the right of the equator show this type of behavior,and that is why the wave grows in this region. Let us nowexplain why electrons moving toward the equator lose en-ergy on passing through resonance.

After many approximations, the exact equations of mo-tion for an electron, under the influence of the mirror force aswell as the electric and magnetic fields of a wave with speci-fied k�z,t� and ��z , t�, can be reduced to a simple self-contained equation27 for ��t�,

d2�

dt2+

d ���d�

= 0, ��� � �T2 cos � + S� , �6�

where �T��kv��w�1/2 is the phase trapping frequency,�w�eB� /m, and S depends on both the spatial gradientB0��z� and the time derivative d� /dt as seen by the movingelectron. In our case, the B0� contribution is dominant. Equa-tion �6� has the form of Newton’s equation of motion for a“particle” moving in � under the influence of a potentialwhich is the sum of a sinusoid and a ramp, as shown inFig. 6. To the extent that �T can be regarded as a constant,30

H���̇2 /2�+ ��� is a constant of the motion and the equa-tion can be solved by quadrature. H plays �formally� the roleof energy, but it is not a physical energy but rather is aquantity that depends on the initial phase of an electron; thusthere is a distribution of H among different electrons. For an

electron beginning at z�z+, �̇�0 initially and the ramp

FIG. 4. �a� Linear plot of maximum wave magnetic field as a function oftime. �b� Semilog plot of the same curve.

FIG. 5. Velocity phase � relative to the wave �solid curve� and kineticenergy �dashed curve� for a single energetic electron.

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slope S is positive. Thus the electron must eventually

“bounce off” the potential, as shown in Fig. 6, i.e., �̇ de-creases to zero and then becomes negative; in other words,the electron passes through resonance. The value �=�res,where this occurs depends on H, i.e., on the initial phase. Forthe same reason, the spatial location z where any given elec-tron passes through resonance depends on its initial phase.Say �res lies between the �n−1�th maximum of ��� at �n−1

and the nth maximum at �n. Then ��n−1��H= ��res�� ��n�. Thus �n−1� ��res��n, where �n−1� is the pointshown in Fig. 6, where ��n−1� �= ��n−1�. If �as is the casein our simulation� S /�T

2 �1, both �n and �n−1� lie close to themaximum of the cos � function in Eq. �6� and one can showby Taylor expanding that �n−1� 2�n− �2�S /�T

2�1/2, while�n 2�n+S /�T

2. The point �=2�n thus lies between �n−1�and �n, but much closer to the latter. Those electrons forwhich �res�2�n will have sin �res�0 and thus will gainenergy on passing through resonance, but the much morenumerous class of electrons with �res�2�n will havesin �res�0 and thus will lose energy, thereby drivinggrowth of the wave. The opposite is true on the left side ofthe equator, where the electrons are moving toward increas-ing B0�z�. Thus nonlinear instability does not occur in thatregion.

Note that if an electron passes through resonance verynear the equator, S is very small, i.e., the ramp is quite flat,and in that case the electron passes just barely above one ormore maxima of ��� before reaching �res. While the elec-

tron is passing over each of these maxima, �̇ is positive butvery small and thus the electron spends considerable time ateach of the maxima. Since sin ��0 at the maxima, there canbe positive spikes in the electron energy before and after itreaches resonance. For this reason, energy exchangefrom the electrons to the wave is unpredictable and ineffi-

cient near the equator, and the wave amplitude peaks at aboutz=+700 km rather than at z=0.

We next consider the temporal growth of the instability.The electron kinetic energy change �� on passing throughresonance is proportional to the wave amplitude B�, to thedwell time � at resonance, which scales as B�

−1/2, and tosin �res. The average value �sin �res� is difficult to predict, asit depends on S /�T

2, but the empirical result from the simu-lations is that ���B�. Since the wave energy scales as B�

2 ,this leads to wave growth dB�

2 /dt�B�, for which the solu-tion is B�� t, as observed in the simulations.

Wave propagation has thus far been left out of the dis-cussion. It appears to play a small role during the growthphase of the instability, since the whistler phase and groupvelocities are much less than v� for the resonant electrons.However, for wave growth B�� t, the growth rate of waveenergy d�B�

2 /8��dt� t, while the loss rate of wave energy ata given location, due to wave propagation, is proportional toB�

2 � t2. Eventually growth and loss balance and wavegrowth must stop, as seen in Fig. 4. After that time, the peakwave amplitude broadens and propagates forward in z, asseen in Fig. 1�d�. Examination of individual electron orbitsindicates that even at late times, the electrons continue tolose energy to the waves upon passing through resonance,and thus to supply energy for the spatially broadened regionof large-amplitude waves.

V. CONCLUDING REMARKS

We have found that a coherent parallel-propagatinglarge-amplitude whistler in the presence of a beamlike distri-bution of fast electrons is subject to a powerful nonlinearabsolute instability, which originates at the cyclotron reso-nance point z+ located near but not at the equator. Althoughthe driver wave is, at this point, propagating away from theequator, the instability grows back toward the equator, reach-ing a maximum amplitude somewhere between the equatorand z+. The triggered wave has a spatially varying frequency��z� such that the electron beam remains close to cyclotronresonance everywhere between the equator and z+; hence �falls as the instability propagates toward the equator, i.e.,toward decreasing geomagnetic field B0.

Previous analyses of triggered waves25–27 have beenbased on the normal case of broad, smooth electron distribu-tions, in which case the basic driving mechanism is aniso-tropy of the distribution. Cyclotron resonance occurs �fordifferent electrons� over a broad spatial range, but the mostunstable spatial point is at the equator, where the geomag-netic field is slowly varying. Instabilities arise there and con-vect away from the equator with the waves, rising in fre-quency because of the increase in B0�z�. In the present study,we have begun with a beamlike fast electron distribution,which picks out the two off-equator cyclotron resonancepoints z� as the initial loci of instability. The growth of theinstability backward toward the equator results in waves withfalling frequency. The beamlike distribution probably doesnot occur in physical situations of interest, but the essentialaspect of picking out a single dominant value of v�, and

FIG. 6. Schematic showing how an electron of “energy” H bounces off thepotential ��� at �=�res.

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therefore a pair of discrete off-equator points of dominantcyclotron resonance, also occurs in the case of weaker dis-tribution function singularities, such as a step discontinuityin v�. We have in fact seen, in other simulations, that verysimilar instabilities occur when there is a descending stepdiscontinuity. In future work we intend to explore the forma-tion of distribution function discontinuities by large-amplitude waves, as well as the scaling of instability featureswith physical parameters, most notably the fast electron den-sity or the magnitude of the step.

ACKNOWLEDGMENTS

This work was supported by ONR. The authors appreci-ate suggestions from Professor K. �Dennis� Papadopoulos,especially stressing the transformative effects of distributionfunction discontinuities.

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