vlasov simulation of langmuir wave packets

Nonlin. Processes Geophys., 14, 671–679, 2007www.nonlin-processes-geophys.net/14/671/2007/© Author(s) 2007. This work is licensedunder a Creative Commons License.

Nonlinear Processesin Geophysics

Vlasov simulation of Langmuir wave packets

T. Umeda

Solar-Terrestrial Environment Laboratory, Nagoya University, Nagoya, Aichi 464-8601, Japan

Received: 14 May 2007 – Revised: 3 August 2007 – Accepted: 18 October 2007 – Published: 25 October 2007

Abstract. Amplitude modulation and packet formation ofLangmuir waves are commonly observed during a nonlinearinteraction between electron beams and plasmas. In this pa-per, we briefly review the history of Langmuir wave packetsas developed by recent spacecraft observations and computersimulations. New one-dimensional electrostatic Vlasov sim-ulations are performed to study their formation processes. Itis found that the formation of Langmuir wave packets in-volves both an incoherent turbulent process and a coherentnonlinear trapping process. Existence of cold ions does notaffect nonlinear processes of the weak-electron-beam insta-bility in which the ion distribution is hardly modified by theexcited Langmuir wave packets.

1 Introduction

Electron-beam-plasma interactions are one of the most fun-damental processes in space plasmas. It is well known thatelectron beam instabilities develop into nonlinear waves andturbulence. Electron phase-space-density holes (e.g.Berkand Roberts, 1967) or electrostatic solitary waves (Mat-sumoto et al., 1994) are coherent nonlinear electrostaticstructures, while harmonic Langmuir waves and Langmuirwave packets have an incoherent quasi-power-law wavenum-ber spectrum which indicate a turbulent feature (Yoon et al.,2003; Gaelzer et al., 2003; Umeda et al., 2003; Umeda, 2006;Silin et al., 2007). The present study is aimed at the genera-tion of amplitude-modulated Langmuir waves and Langmuirwave packets which is called a Langmuir turbulence in spaceplasmas.

The amplitude modulation and packet formation of Lang-muir waves were commonly observed in self-consistent ki-netic simulations of electron beam instabilities (Muschiettiet al., 1995, 1996; Akimoto et al., 1996; Matsukiyo et al.,2004; Usui et al., 2005; Umeda, 2006; Silin et al., 2007).Strongly modulated waveforms and packets of Langmuir

Correspondence to:T. Umeda([email protected])

waves were also observed in various regions of the magneto-sphere, such as in the auroral ionosphere (Ergun et al., 1991;Stasiewicz et al., 1996; Bonnell et al., 1997; Pottelette et al.,1999; Lizunov et al., 2001), in the solar wind (Gurnett et al.,1993; Bale et al., 1996; Kellogg et al., 1999b), in the electronforeshock region (Kellogg et al., 1996, 1999a; Soucek et al.,2005), and in the magnetotail (Kojima et al., 1997).

Several nonlinear theories were adopted to explain mech-anisms for the amplitude modulation and packet formationof Langmuir waves. The first is the parametric decay ofLangmuir waves into ion acoustic waves (e.g.Cairns andRobinson, 1992; Cairns et al., 1998; Pottelette et al., 1999;Matsukiyo et al., 2004; Soucek et al., 2005) or lower hy-brid waves (e.g.Stasiewicz et al., 1996; Bonnell et al., 1997;Lizunov et al., 2001). Note thatCairns et al.(1998) demon-strated that electron-beam-driven Langmuir wave packetsobserved in the magnetosphere lie well outside the region ofparameter space for which the modulational instability canproceed.

For the Langmuir decay instability, the amplitude of pri-mary Langmuir waves must be high enough to modify iondistributions. The observations in the magnetosphere haveshown that the amplitude of Langmuir wave packets some-times exceeds several hundred mV/m. The observed fre-quency spectra also show apparent double peaks around theelectron plasma frequency with a small low-frequency com-ponent (Kellogg et al., 1996; Bale et al., 1996; Soucek et al.,2005), which indicates the Langmuir decay instability. Onthe other hand, The GEOTAIL spacecraft observation in themagnetotail has shown that the amplitude of Langmuir wavepackets is several hundredµV/m, which is too small forthe Langmuir decay instability. The computer simulationshave also demonstrated that amplitude-modulated Langmuirwaves with a very small amplitude can be generated dur-ing a very-weak-electron-beam instability without ion dy-namics (Akimoto et al., 1996; Usui et al., 2005; Umeda,2006; Silin et al., 2007), which is consistent with the GEO-TAIL observation. These observations and simulations sug-gest that Langmuir wave packets with a small amplitude

Published by Copernicus Publications on behalf of the European Geosciences Union and the American Geophysical Union.

672 T. Umeda: Langmuir wave packets

Fig. 1. Time evolution of the wavenumber spectrum for Runs A andB. The wave intensity is normalized bymeωpeVte/e.

and a large amplitude are generated by different mechanismswith small amplitude packets generated by electron dynam-ics while large amplitude packets likely involve ion dynamicsas well.

A possible mechanism for the formation of small-amplitude Langmuir wave packets is the “kinetic localiza-tion” (Muschietti et al., 1995, 1996), in which Langmuirwaves are modulated in space due to electron bunching inthe position-velocity phase space. Later, this mechanism de-veloped into the nonlinear trapping theory based on particle-in-cell simulation results (Akimoto et al., 1996; Usui et al.,2005). In the nonlinear trapping theory, Langmuir waves aremodulated in space due to nonlinear trapping of electrons bythe electrostatic potential of the Langmuir waves. However,a recent Vlasov simulation of a very-weak-electron-beam in-stability has demonstrated that Langmuir waves are not di-rectly modulated by the nonlinear trapping but are modulatedby nonlinear interaction between the most unstable primaryLangmuir mode and its sideband modes (Umeda, 2006).The amplitude-modulated Langmuir waves have a broadbandwavenumber spectrum, which indicates a turbulent featurerather than a coherent feature. The electron phase-spacedistribution function of the amplitude-modulated Langmuirwaves also shows a strong modification of untrapped back-ground electrons but not any phase-space vortex associ-ated with trapped beam electrons, implying that it is mostlythe untrapped background electrons, not the trapped beamelectrons, which are responsible for the wave modulations(Umeda, 2006; Silin et al., 2007).

2 Simulation with periodic boundary

To demonstrate whether the coherent nonlinear trapping pro-cess can really generate the spatial modulation of Langmuirwaves, a one-dimensional Vlasov simulation has been carriesout (Umeda, 2006). The Vlasov simulation code uses a stan-dard time-advance scheme call the splitting scheme (Cheng

and Knorr, 1976), while a conservative and non-oscillatorycubic polynomial interpolation scheme (Umeda et al., 2006)is adopted for stable time-integration of phase-space distri-bution functions. The simulation domain is taken along anambient magnetic field. We assume that a very weak elec-tron beam is drifting against the major background electronsand background ions. The density ratio of the beam compo-nentR=nb/(ne+nb) is set as 0.1%, where the subscripts “b”and “e” represent beam electrons and background electrons,respectively. We assume that the beam and background elec-trons have the equal thermal velocityVte=Vtb=1.0. The totalelectron plasma frequency is assumed asωpe=1.0. The beamdrift velocityVd is set as 8.0Vte. Mobile ions are also evolvedin order to confirm the absence of consequences of ion dy-namics on the weak-beam instability. The ion-to-electronmass ratio is set asmi/me=1836, and the ion-to-electrontemperature ratio is set asTi/Te=0.1, where the subscripts“ i” represents background ions. The density of ions isgiven by ni=ne+nb. The number of spatial grid cells isNx=16 384. The number of velocity grid cells isNvx =4096over a velocity range fromvmax=24.0Vte to vmin=−16.0Vte

for electrons and fromvmax=13.0Vt i to vmin= − 13.0Vt i forions. The grid spacing is equal to1x=0.4λe (λe≡Vte/ωpe),and the time step is equal toωpe1t = 0.005. It is noted thatε0=1 is used for simplicity. We imposed the open bound-ary condition in thevx direction and the periodic boundarycondition in thex direction. Here we show two simulationresults; One is started with a coherent-single-wave-mode per-turbation (Run A), and another is started with a white-noiseperturbation (Run B).

Figure 1 shows time evolution of the wavenumber spec-trum for Runs A and B. The wave intensity is normalized

by meωpeVte/e=

√nemeV

2te and is plotted on a log scale. In

Run A, an initial perturbation is imposed only at the mostunstable wavenumber in order to achieve the coherent non-linear trapping by a single sinusoidal wave mode. The mostunstable primary Langmuir mode (kxλe∼0.144) grows fromthe imposed initial noise level (∼10−9), while other unstablemodes grow from the round-off noise level of the double-precision computation (∼10−15). The primary mode satu-rates atωpet∼1250, while there appear an upper and a lowersideband mode atkxλe∼ 0.135 and 0.154, respectively, fromωpet∼2000. Although we have started Run A with a sin-gle mode perturbation, the final wavenumber spectrum (atωpet=4000) has become broadband with several discretestructures.

In contrast to Run A, there appears a broadband wavenum-ber spectrum fromωpet∼1200 in Run B, because all theunstable modes grow from the same initial noise level of∼10−9. Although the two simulation runs have been startedwith different initial wavenumber spectra, the resulting finalwavenumber spectra look similar to each other. The finalspectrum in Run B also shows a broadband feature with sev-eral discrete peaks at quasi-random wavenumbers. There are

Nonlin. Processes Geophys., 14, 671–679, 2007 www.nonlin-processes-geophys.net/14/671/2007/

T. Umeda: Langmuir wave packets 673

also several differences. Firstly, the most dominant mode inRun B (kxλe∼0.142) is not the most unstable primary mode(kxλe∼0.144). Secondly, the wavenumber spectra have dis-crete peaks at different wavenumbers. Thirdly, the genera-tion of the sideband modes is not apparent in Run B.

The spatial modulation of Langmuir waves in Run A islikely due to the discrete structures in the wavenumber spec-trum. Does the nonlinear trapping process generate the spa-tial modulation of Langmuir waves? To answer this question,we show the energy history of the total electric field for RunA in the top panel of Fig. 2. We also show the histories ofwave intensity for primary mode (kxλe=0.144), and the mostdominant two sideband modes (kxλe= 0.135 and 0.154) inthe middle panel. The electric field amplitude and energyare normalized bymeωpeVte/e andnemeV

2te=m2

eω2peV

2te/e

2,respectively. One can see that there exist two stages inRun A. In the first stage, the primary mode linearly growsand saturate atωpet∼1250. After the saturation of theprimary mode, the total electric field energy oscillate at0.5ωb, whereωb is the bounce/trapping frequency given byωb≡

√ekL|Ex(kL)|/me∼0.048ωpe. HerekLλe∼0.144 and

|Ex(kL)|∼0.016meωpeVte/e denote the wavenumber andwave amplitude of the primary mode, respectively. However,there is no spatial modulation of Langmuir waves, imply-ing that the spatial amplitude modulation process is differentfrom the conventional nonlinear frequency shift by the trap-ping process. As seen in Fig. 2b, the growth rates of thetwo sideband modes slightly change after the saturation ofthe primary mode. This is because the velocity distributionfunction is modified by nonlinear trapping by the primarymode. However, the upper and lower sideband modes con-tinue to grow linearly even after the saturation of the primarymode.

The saturated primary mode keeps almost the same ampli-tude untilωpet∼2000, while the two sideband modes con-tinuously grow. The second stage starts when the amplitudeof the sideband modes reaches a certain level (∼10−2.5) atwhich the sideband modes can modify the velocity distri-bution. During the growth and saturation of the sidebandmodes, the primary mode becomes unstable again and is am-plified. The interaction between the primary and sidebandmodes results in the strong spatial modulation of Langmuirwaves. The spatial scale of the amplitude modulation is givenby 2π/1k, and the wavenumber difference between the pri-mary and sideband modes can be estimated as1k∼2ωb

vp

(Umeda, 2006), wherevp denotes the phase velocity of theprimary mode. In the present simulation, we can obtain amore exact value of the bounce frequency asωb/ωpe∼0.048,because the coherent nonlinear trapping is achieved by thesingle mode perturbation. The wavenumber difference is ob-tained as1kλe∼0.01.

Figure 2a and b suggest that the spatial modulation ofLangmuir waves is not directly generated by the nonlineartrapping. However, the nonlinear trapping does play a role to

Fig. 2. Nonlinear evolution of the beam instability for RunA: (a) The energy history of the total electric field normalizedby nemeV

2te=m2

eω2peV

2te/e

2. (b) The histories of wave inten-sity for waves modes atkxλe= 0.144, 0.135, and 0.154 normal-ized by meωpeVte/e. (c) The x−vx phase-space electron den-sity atωpet=1250. (d) Thex−vx phase-space electron density atωpet=3500.

“filter” the sideband modes. It is well known that the weak-electron-beam instability is unstable over a wide wavenum-ber range aroundkL and that the most unstable primaryLangmuir mode grows fastest and saturates earliest. By thesaturation of the primary mode, there appear vortices in theposition-velocity phase space of electrons as seen in the bot-tom panel of Fig. 2. The velocity space around the phase ve-locity of the primary modevp is stabilized by a strong nonlin-ear trapping in the phase space. As a result, unstable modeswith phase velocities inside the trapping velocity range arestabilized by the nonlinear trapping, while other modes withphase velocities outside the trapping velocity range can con-tinue to grow. In other words, the nonlinear trapping processworks as a bandstop filter (Umeda, 2006).

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Fig. 3. Nonlinear evolution of the beam instability for RunB: (a) The energy history of the total electric field normalizedby nemeV

2te=m2

eω2peV

2te/e

2. (b) The histories of wave inten-sity for waves modes atkxλe= 0.144, 0.135, and 0.154 normal-ized by meωpeVte/e. (c) The x−vx phase-space electron den-sity atωpet=1250. (d) Thex−vx phase-space electron density atωpet=3500.

Does this story apply to the realistic case with a broadbandinitial noise? In Fig. 3, we show the temporal development ofthe weak-beam instability for Run B, in which a white noisewith uniformly-distributed random phases is imposed as aninitial perturbation. The histories of the total electric field,wave intensity for waves modes atkxλe= 0.144, 0.135, and0.154, and thex−vx phase-space electron density are shownwith the same format as Fig. 2. The two-step evolution ofthe weak-beam instability seen in Run A is not so apparentin Run B, because many wave modes are excited in a widewavenumber range. The primary mode (kxλe= 0.144) sat-urates earlier (ωpet∼1100) at a lower saturation level. Theother two modes (kxλe= 0.135 and 0.154) also saturate atωpet∼1100 with the primary mode. Linearly unstable wavemodes around the primary mode saturate with the primary

mode in Run B. Thus the excitation of sideband modes isnot identified. Note that several simulation runs with differ-ent random phases of the initial white noise are performed toexamine the final wavenumber spectrum. The results showthat the final wavenumber spectrum in each run has discretepeaks at different wavenumbers. This means that it dependson the initial perturbation which modes dominate in the finalwavenumber spectrum.

These results show that the sideband interaction is appar-ent in the coherent weak-beam-plasma interaction but not inthe incoherent case. Then, what is the role of coherent non-linear trapping in the weak-beam instability with a broadbandinitial noise? In the bottom panel of Fig. 3, we show theposition-velocity phase space of electrons atωpet=1250. Itis obvious that the saturation of the primary mode involvesthe phase-space vortices. However, the trapping velocityrange in Run B is wider than that in Run A because of the ex-citation of various unstable wave modes. To understand thesaturation process we show in Fig. 4 the temporal develop-ment of the velocity distribution function averaged overx forRun B. Atωpet=1000 there is no modification of the veloc-ity distribution function. However there exists spatial mod-ulation of wave amplitude due to the initial white noise (notshown). Atωpet=1250 we found the plateau formation in anarrow velocity range around the phase velocity of primarymode. However, it is obvious that wave modes with a phasevelocity vp/Vte∼ 6 and 8 are unstable. The velocity dis-tribution is strongly modified atωpet=1500. However, thevelocity distribution is still unstable, and wave modes with ahigher phase velocity can be excited at a smaller wavenumberwhich was not unstable in the linear stage. Atωpet=1750 wefound the plateau formation in a wide velocity range. How-ever, there exists a positive gradient in the velocity distribu-tion function, and thus wave modes with large wavenumbers(vp/Vte<6) become unstable. Langmuir waves are excitedin a broadband wavenumber range by the modification ofvelocity distribution function, which is similar to the side-band interaction. Thus the coherent nonlinear trapping playsa role in the generation of a turbulent (incoherent) spectrum.However, there is not any coherent nonlinear structures is thesteady state as seen in Figs. 2d and 3d. Note that a quasi-linear approach taken bySilin et al. (2007) is useful to un-derstand which mode can grow in the nonlinear stage. How-ever, neither linear nor quasilinear approaches can tell whichmodes dominate in the final spectrum. In other words, thelinear and quasi-linear approaches cannot tell the exact satu-ration level of each wave mode. The amplitude modulationof Langmuir waves is very complex involving both coherentnonlinear trapping and quasilinear modification of velocitydistribution function. Thus a self-consistent simulation is aunique way to analyze the spatial scale of wave packets (e.g.Silin et al., 2007).



Fig. 4. Electron velocity distribution functions averaged overx at different times for Run B. The velocity is normalized byVte. Thedistribution functions are normalized to unity.

3 Simulation with open boundary

We extend the one-dimensional simulation in the previoussection to a more realistic model (Run C). We adopted aone-dimensional open system in which an electron beam isinjected into the system from a boundary while outgoingplasmas are absorbed without reflection at boundaries (e.g.Umeda et al., 2002). As the initial condition, we assume thatthe background plasma exists uniformly in the simulation do-main without the electron beam. When a computer simu-lation is started, the electron beam is continuously injectedfrom the left boundary into the background homogeneousplasma. The injected electron beam and the background elec-trons form an unstable velocity distribution function. In thepresent study, we inject the electron beam with a constantflux. The background electrons and ions are assumed to becontinuous at the right boundary as if there is no boundary.The background ions are also assumed to be continuous atthe left boundary, while the flux of background electrons atthe left boundary is modified to keep the charge neutrality.We used the same simulation parameters as Run B to make adirect comparison between the periodic and open systems.

The right panel of Fig. 5 shows the energy density of elec-tric field |Ex(x, t)|2 for Run C. In the left and middle panels,plots of the energy density of electric field for Runs A andB are also shown as references. The magnitude is normal-ized bynemeV

2te=m2

eω2peV

2te/e

2. Note that the entire simu-lation domain is not shown in Fig. 5. In the present opensystem where a very weak electron beam is continuously in-jected from the boundary, the beam instability develops inboth space and time (e.g.Umeda et al., 2002). Initially,the electron beam is injected into the unperturbed plasma,making the velocity distribution function unstable. However,the sudden injection of the electron beam at the onset alsogenerates an impulsive strong perturbation at a high level of∼10−6, which is a much higher level than that of the initialperturbation (∼10−10). Thus the weak-beam instability inRun C saturates earlier (atωpet∼600) than in Run B.

There also exist other differences. In Runs A and B, wavepackets are observed in an almost random manner in bothspace and time. The temporal waveform of the Langmuirwave packets atx/λe∼2000 in Run C is similar to that inRun B. On the other hand, the temporal waveform of theLangmuir wave packets atx/λe=500∼1000 becomes quasi-periodic fromωpet∼2400 in Run C. The period of the gen-eration of wave packets ist∼450/ωpe, which is longer thanthe ion plasma period, i.e.,t=2π

√mi/me/ωpe∼270/ωpe,

implying the absence of ion dynamics. The nonlinear beam-plasma interaction takes place uniformly in space in RunsA and B, while the nonlinear interaction results in the for-mation of a single Langmuir wave packet in a localized re-gion close to the beam source (x/λe∼500) in Run C. Thewave packet propagates at the group velocity of the primaryLangmuir mode, which is much slower than the beam ve-locity. Since the electron beam is continuously injected inRun C, the free energy source is supplied to Langmuir wavesand positive gradient in the velocity distribution exists for alonger time. Thus the amplitude of excited Langmuir modein Run C becomes much higher than in Runs A and B.

Figure 6 shows time evolution of the wavenumber spec-trum for Run C. The wave intensity is normalized by

meωpeVte/e=

√nemeV

2te and is plotted on a log scale. We

found several discrete peaks atkxλe∼ 0.125, 0.144, and0.154 in the wavenumber spectrum atωpet∼600. The sat-uration level of these modes is∼10−2. Although this satu-ration level is close to the saturation level in the runs withperiodic boundaries (Runs A and B), the wave amplitude inRuns C and B (or A) is very different. The Langmuir wavesare uniformly excited in Runs A and B, while a single wavepacket is excited in a localized region in Run C. Since theFourier transformation of a spatially-localized wave packetis taken in Fig. 6, the actual amplitude of the wave packet ismuch higher (∼0.3meωpeVte/e).

The discrete structure in the wavenumber spectrum in RunC is similar to that in the run with a uniform and coherentinitial perturbation (Run A), which is likely because the co-



Fig. 5. Electric field energy density as a function of time and position for Runs (left) A , (middle) B, and (right) C. The electric field energydensity is normalized bynemeV

2te.

Fig. 6. Time evolution of the wavenumber spectrum for Run C.The wave intensity is normalized bymeωpeVte/e.

herent behavior of the beam-plasma interaction is enhancedby a high noise level (∼10−6) due to the sudden injection ofelectron beam. However, these modes are not dominant forωpet>1000. Other wave modes with a broader wavenumberband appear inkxλe=0.14∼0.16 atωpet=1200. This may

correspond to the break of the first wave packet and the for-mation of the second packet in the left panel of Fig. 5. Theamplitude of the wave packet exceeds 0.6meωpeVte/e but islower than 1.0meωpeVte/e, implying that the parametric in-stabilities due to ion dynamics are not essential. The two-stepevolution of the weak-beam instability in Run C is similar tothat in Run A.

For ωpet>2000 we found three discrete peaks in thewavenumber spectrum. It is unclear whether these are anartifact of the present boundary condition or a real physics.However, a typical wavenumber difference of the discretestructure is about1kλe∼0.023 atωpet=2500, which corre-sponds to the spatial scale length of wave packetsL/λe∼270.From Fig. 5 the group velocity of the wave packets is es-timated asvg∼0.6Vte. Thus a typical timescale of ampli-tude modulation is estimated ast=L/vg∼450/ωpe, whichis consistent with the time interval of the wave packets atx/λe=500∼1000. Forωpet>3000 we also found severaldiscrete structures in the wavenumber spectrum. These struc-tures are expected to be the incoherent behavior of the beam-plasma interaction as seen in Run B.

In order to analyze the effect of coherent nonlinear trap-ping process, phase-space distribution functions of electronat different times are shown in Fig. 7. At all the time there ex-ist coherent electron phase-space vortices in the region closeto the beam source, which is different from Run B, in which



electron phase-space vortices modulated by an initial pertur-bation appear only at the saturation state (see Fig. 3c). Theeffect of the initial perturbation is likely suppressed by thespatial development of the beam-plasma instability, becausean envelope of a wave packet is formed by the spatial devel-opment. It is also noted that the size of vortices in Run Cbecomes much larger than that in Runs A and B (see Figs. 2cand 3c).

At ωpet=600 we found electron phase-space vortices withpropagation velocitiesvp/Vt∼ 6 and 8 forx/λe=600∼800.These vortices correspond to the wavenumber enhancementat kxλe∼ 0.125 and 0.155, implying that there is a processsimilar to the sideband interaction as seen in Run A. Atωpet=1200 there are very large vortices forx/λe=600∼800.Later these vortices split into two wave packets as seen in theleft panel of Fig. 5. Atωpet=2000 we found small vorticeswith a slow propagation velocity (vp/Vt∼5) between thefirst and second packets (atx/λe∼1100). Such vortices arealso found atωpet=2500 andx/λe∼1000, suggesting thatthe quasi-periodic amplitude modulation would be gener-ated by the interaction between the primary Langmuir modeand wave modes with a slow phase velocity (vp/Vt∼5).It is also noted that the phase-space distribution functionfor x/λe=800∼1000 is different from that forx/λe>1200where the phase-space distribution function shows a moreturbulent feature. In other words, the incoherent process isdominant forx/λe>1200, while the coherent nonlinear trap-ping process is likely dominant in the region close to thebeam source. Forx/λe=800∼1000, Langmuir wave pack-ets are formed by the interaction between two or three wavemodes with different phase velocities, which is similar to thesideband interaction.

4 Conclusion and discussion

In this paper we have briefly reviewed mechanisms for theamplitude modulation and packet formation of Langmuirwaves by performing one-dimensional Vlasov simulationswith different initial and boundary conditions.

In the run with a uniform and coherent initial perturba-tion, there are two stages for the nonlinear developmentof the weak-electron-beam instability. In the first stage, aweak-electron-beam instability excites Langmuir waves overa wide wavenumber range which is essentially a linear pro-cess. The primary Langmuir mode saturates by the nonlineartrapping of electrons, which involves electron-phase-spacevortices. The nonlinear trapping process filters the unsta-ble modes with phase velocities inside the trapping veloc-ity range. In the second stage, the sideband modes growto a level comparable to the saturation level of the primarymode. The sideband modes modify the phase-space distribu-tion function, making the primary mode unstable again. As aresult, the primary mode and the sideband modes dominate.Langmuir waves are modulated in space by an interaction

Fig. 7. The x − vx phase-space electron density atωpet = 600,1200, 2000, 2500, and 3500.

between the primary and sideband modes. The sideband in-teraction is an extension of the kinetic localizationMuschi-etti et al.(1995, 1996) and is a modification of the nonlineartrapping theory (Akimoto et al., 1996; Usui et al., 2005). Thenonlinear trapping process does not directly generate the spa-tial modulation of Langmuir waves, but plays a role to filterwave modes with wavenumbers close to the most unstablewave number. A turbulent wavenumber spectrum with sev-eral discrete peaks results in the spatial modulation of Lang-muir waves. In contrast, the coherent process is not apparentin the run with an incoherent white-noise perturbation. Aquasilinear modification of velocity distribution functions ismore essential than in the run with the coherent initial per-turbation (Silin et al., 2007).

In the run with open boundaries where an electron beam isinjected from a boundary, nonlinear evolution of the beam-plasma instability is drastically modified from that of the runwith periodic boundaries. The coherent nonlinear trappingprocess is enhanced by the localized injection of an elec-tron beam. The result again looks similar to that of the runwith uniform and coherent initial perturbation. Because ofthe spatial development of the beam-plasma instability, thesaturation level also changes. The result suggests that beam-plasma interactions in nonuniform systems become differ-



ent from the temporal evolution in uniform periodic sys-tems when the propagation velocity of waves and free-energysource is different.

In conclusion, both coherent nonlinear process and inco-herent turbulent process are important for the formation ofLangmuir wave packets. In the present study we have alsosolved ion distributions. However, the effect of ion dynam-ics is absent, because the amplitude of Langmuir wave pack-ets is smaller than 1.0meωpeVte/e. Simulation runs with ahigh-density electron beam is left as a future work, in whichparametric instabilities due to ion dynamics would play animportant role in the formation of Langmuir wave packets.

Acknowledgements.This work was supported by Grant-in-Aid forYoung Scientists (Start-up) #19840024 and in part by Grant-in-Aidfor Creative Scientific Research #17GS0208 “The Basic Studyof Space Weather Prediction” from the Ministry of Education,Science, Sports, Technology, and Culture of Japan. The computersimulations were performed on the Fujitsu PRIMEPOWER HPC2500 at Kyoto University and Nagoya University. The computingresources are provided by Research Institute for Sustainable Hu-manosphere, Kyoto University and Solar-Terrestrial EnvironmentLaboratory, Nagoya University.

Edited by: A. C. L. ChianReviewed by: D. L. Newman and another anonymous referee

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vlasov simulation of langmuir wave packets

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