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Title Significance of Wave-Particle Interaction Analyzer for direct measurements of nonlinear wave-particle interactions Author(s) Katoh, Y.; Kitahara, M.; Kojima, H.; Omura, Y.; Kasahara, S.; Hirahara, M.; Miyoshi, Y.; Seki, K.; Asamura, K.; Takashima, T.; Ono, T. Citation Annales Geophysicae (2013), 31(3): 503-512 Issue Date 2013-03-19 URL http://hdl.handle.net/2433/193729 Right © Author(s) 2013. This work is distributed under the Creative Commons Attribution 3.0 License. Type Journal Article Textversion publisher Kyoto University

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Page 1: Title Significance of Wave-Particle Interaction …repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/...Title Significance of Wave-Particle Interaction Analyzer for direct measurements

Title Significance of Wave-Particle Interaction Analyzer for directmeasurements of nonlinear wave-particle interactions

Author(s)Katoh, Y.; Kitahara, M.; Kojima, H.; Omura, Y.; Kasahara, S.;Hirahara, M.; Miyoshi, Y.; Seki, K.; Asamura, K.; Takashima,T.; Ono, T.

Citation Annales Geophysicae (2013), 31(3): 503-512

Issue Date 2013-03-19

URL http://hdl.handle.net/2433/193729

Right © Author(s) 2013. This work is distributed under the CreativeCommons Attribution 3.0 License.

Type Journal Article

Textversion publisher

Kyoto University

Page 2: Title Significance of Wave-Particle Interaction …repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/...Title Significance of Wave-Particle Interaction Analyzer for direct measurements

Ann. Geophys., 31, 503–512, 2013www.ann-geophys.net/31/503/2013/doi:10.5194/angeo-31-503-2013© Author(s) 2013. CC Attribution 3.0 License.

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DiscussionsSignificance of Wave-Particle Interaction Analyzer for directmeasurements of nonlinear wave-particle interactions

Y. Katoh1, M. Kitahara 1, H. Kojima 2, Y. Omura2, S. Kasahara3, M. Hirahara 4, Y. Miyoshi4, K. Seki4, K. Asamura3,T. Takashima3, and T. Ono1

1Department of Geophysics, Graduate School of Science, Tohoku University, Miyagi 980-8578, Japan2Research Institute for Sustainable Humanosphere, Kyoto University, Kyoto 611-0011, Japan3ISAS, JAXA, Kanagawa 229-8510, Japan4Solar-Terrestrial Environment Laboratory, Nagoya University, Aichi 464-8601, Japan

Correspondence to:Y. Katoh ([email protected])

Received: 26 October 2012 – Revised: 24 January 2013 – Accepted: 18 February 2013 – Published: 19 March 2013

Abstract. In the upcoming JAXA/ERG satellite mission,Wave Particle Interaction Analyzer (WPIA) will be installedas an onboard software function. We study the statistical sig-nificance of the WPIA for measurement of the energy trans-fer process between energetic electrons and whistler-modechorus emissions in the Earth’s inner magnetosphere. TheWPIA measures a relative phase angle between the wavevector E and velocity vectorv of each electron and com-putes their inner productW , whereW is the time variationof the kinetic energy of energetic electrons interacting withplasma waves. We evaluate the feasibility by applying theWPIA analysis to the simulation results of whistler-modechorus generation. We computeW using both a wave elec-tric field vector observed at a fixed point in the simulationsystem and a velocity vector of each energetic electron pass-ing through this point. By summing upWi of an individualparticle i to give Wint, we obtain significant values ofWintas expected from the evolution of chorus emissions in thesimulation result. We can discuss the efficiency of the energyexchange through wave-particle interactions by selecting therange of the kinetic energy and pitch angle of the electronsused in the computation ofWint. The statistical significanceof the obtainedWint is evaluated by calculating the standarddeviationσW of Wint. In the results of the analysis, positiveor negativeWint is obtained at the different regions of ve-locity phase space, while at the specific regions the obtainedWint values are significantly greater thanσW , indicating ef-ficient wave-particle interactions. The present study demon-strates the feasibility of using the WPIA, which will be on

board the upcoming ERG satellite, for direct measurementof wave-particle interactions.

Keywords. Magnetospheric physics (Plasma waves and in-stabilities) – Space plasma physics (Wave-particle interac-tions; Instruments and techniques)

1 Introduction

Wave-particle interactions in space plasmas occur withintime scales characteristic of plasma particles, such as the gy-roperiod and plasma oscillation period. These time scales are,especially for electrons in the magnetosphere, too short todetect wave-particle interactions in situ using conventionalplasma instruments. As an example, we focus on cyclotronresonant interactions of energetic electrons with a coherentwhistler-mode wave in the magnetosphere. The time scaleassociated with the motion of resonant electrons nonlinearlytrapped by a coherent whistler-mode wave is characterisedby the oscillation period of electrons within the trapping re-gion in velocity phase space (e.g., Matsumoto, 1985), andis one or two orders of magnitude longer than the gyrope-riod of electrons. The trapping oscillations of resonant elec-trons occur within a limited region of velocity phase spacedetermined by the cyclotron resonance condition and withina certain relative phase range with respect to the wave mag-netic field vector. Oscillations of resonant electrons result ina relaxation of the velocity distribution within the trappingregion and saturation of the interactions within time peri-ods of hundreds or thousands of electron gyroperiods (e.g.,

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

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504 Y. Katoh et al.: Statistical significance of WPIA

Katoh and Omura, 2004) corresponding to tens of millisec-onds. Using plasma instruments on board spacecraft, whichmostly have time resolutions of a few tens of millisecondsor less, it is impossible to directly measure relaxation of theelectron velocity distribution or energy exchange processesduring wave-electron interactions.

One observation method that enables wave-particle inter-actions to be investigated is a comparison of the observedwave phase and particles. There have been many attemptsto measure interactions between electron beams and electro-static waves, while previous instruments counted the num-ber of particles by referring the observed wave phase andmeasured the electron distribution in the wave phase space(Ergun et al., 1991, 1998; Gough et al., 1995; Buckley etal., 2000). Recently Fukuhara et al. (2009) proposed a newtype of instrument for direct and quantitative measurementsof wave-particle interactions, which is referred to as “Wave-Particle Interaction Analyzer (WPIA).” The WPIA uses all ofthe three-dimensional properties of the observed waveformsand particles and quantifies the kinetic energy flow by mea-suring the inner product of the observed instantaneous waveand the particle velocity vectors. To apply the WPIA to spaceplasma measurement, it is necessary to obtain both the wavefield vector and the velocity of each incoming particle on atime scale sufficient to resolve variations in the wave phaseand the gyro-motion of particles. Since the sampling clockin wave and plasma instruments can be adjusted to the orderof microseconds, measurement of the relative phase can becarried out using state-of-the-art instruments. It is not neces-sary for all of waveform and particle data to be transferredto the ground in order to apply the WPIA. Accumulationof the inner product can be conducted onboard, so that thetelemetry data size can be optimised to a realistic size bysending only the results of the accumulation. In the upcom-ing JAXA satellite mission: Exploration of energization andRadiation in Geospace (ERG) project (Miyoshi et al., 2012),WPIA measurements will be carried out using Software-typeWPIA (S-WPIA). One of the primary targets of this missionis to clarify the interaction of relativistic electrons with cho-rus emissions. During the mission, the onboard S-WPIA willbe used to investigate whistler-mode wave-particle interac-tions for the first time.

Recently, the gyroresonant interaction of relativistic elec-trons with whistler-mode chorus emissions has been investi-gated as a possible generation mechanism for MeV electronsin the outer radiation belt (e.g., Shprits et al., 2008; Thorne,2010; Ebihara and Miyoshi, 2011, for review). Observationshave revealed the possible roles of chorus in controlling theflux of radiation belt electrons (e.g., Miyoshi et al., 2003,2007; Horne et al., 2003; Kasahara et al., 2009). As theo-retical and simulation studies have clarified, nonlinear wave-particle interactions are essential when considering the gen-eration process of coherent chorus elements with rising tones(Nunn, 1974; Nunn et al., 1997; Trakhtengerts, 1995, 1999;Katoh and Omura, 2007a, 2011; Omura et al., 2008, 2009;

Hikishima et al., 2009; Omura and Nunn, 2011) and withfalling tones (Nunn and Omura, 2012) and rapid energisa-tion of relativistic electrons by chorus emissions (Katoh andOmura, 2007b; Omura et al., 2007; Summers and Omura,2007; Furuya et al., 2008; Katoh et al., 2008). In particular,the formation of an electromagnetic electron hole in velocityphase space is a key process related to nonlinear interactionswith chorus emissions. The electromagnetic electron hole isproduced by depletion of velocity phase space due to non-linear interactions between a coherent whistler-mode waveand resonant electrons moving along the field lines aroundthe magnetic equator. The presence of the hole results in theformation of nonlinear resonant currents that contribute tothe growth of coherent wave elements with a specific phasevariation with significantly large growth rate rather than thelinear growth rate. In addition, a fraction of the resonant elec-trons trapped in the hole undergo very efficient acceleration,while the majority which are outside the hole and are un-trapped by wave elements lose their kinetic energy in gener-ating chorus emissions. Based on the results of previous the-oretical and simulation studies on chorus emissions, the non-linear theory is promising and can be verified by obtainingevidence of an electromagnetic electron hole in the equato-rial region of the inner magnetosphere during typical chorusevents. However, the greatest difficulty with regard to directmeasurements is that the hole is formed in a specific phaserange relative to the wave magnetic field vector rotating withthe wave frequency, which is several kilohertz in the innermagnetosphere. This is a crucial problem for conventionalplasma instruments, but can be overcome by the WPIA.

In the WPIA,W(t0) = qE(t0)·v(t0) is computed using thewave electric field vector measured by the plasma wave in-strument and the velocity vector of an energetic electron de-tected by the particle instrument att = t0. Considering thetime variation of the kinetic energy, we obtain

dK

dt= m0v ×

d(γ v)

dt= qE × v = W, (1)

whereK = m0c2(γ − 1) is the kinetic energy of a charged

particle including relativistic effects,m0 andq are the restmass and charge of a charged particle, respectively,c is thespeed of light, andγ is the Lorentz factor given byγ =

[1− (v/c)2]−1/2. Since Eq. (1) indicates thatW represents

the time variation of the kinetic energy of a charged particledue to wave-particle interactions, by summing upWi of an

individual particlei to giveWint =

N∑i=1

Wi , we can measure

the amount of energy transferred between waves and parti-cles in a volume filled byN plasma particles. In the presenceof efficient wave-particle interactions, the value ofWint canbe either positive or negative corresponding to particle ac-celeration or wave generation, respectively. In practice, how-ever, the limited data availability (e.g., finite particle count)

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Y. Katoh et al.: Statistical significance of WPIA 505

restricts the accuracy of the calculations, so that the statisticalsignificance of the obtainedWint must be carefully evaluated.

In the present study, to evaluate the statistical significanceof WPIA measurements of wave-particle interactions, theWPIA method is applied to dataset of the simulation resultsreproducing whistler-mode chorus generation (e.g., Katohand Omura, 2007a). The inner productW is computed fora wave electric field observed at a fixed point in the simula-tion system and the velocity vector of each energetic electronpassing through that point. We propose that the statistical sig-nificance can be evaluated based on the standard deviationσW of Wint and discuss the feasibility quantitatively by ap-plying the method to the simulation result. The outline ofthis paper is as follows. The meaning of the quantityW isdiscussed in Sect. 2. In Sect. 3, we briefly describe the sim-ulation model and results reproducing the chorus generationprocess. We show results of the WPIA analysis in Sect. 4 anddiscuss the statistical significance in Sect. 5. Section 6 givessummary of the present study.

2 WPIA

We examine the meaning ofW in detail. In the present study,ions are assumed to be an immobile neutralising compo-nent. We assume that the current densityJ consists of a coldplasma componentJ c and a hot componentJ R, namely,J =

J c+J R. The cold componentJ c is given byJ c = −eNcuc,whereNc and uc are the number density and velocity ofthe cold plasma component, respectively. By taking the in-ner product ofE andJ , we obtain

J · E = J c · E + JR · E. (2)

The equation of the motion of the cold plasma component isgiven by

∂uc

∂t= −

e

me(E + uc × B0) , (3)

whereme is the mass of an electron andB0 is the backgroundmagnetic field. By taking the inner product ofuc with bothsides of Eq. (3), we obtain

uc ·∂uc

∂t= −

e

meuc · E. (4)

Using Eq. (4), we find that the time variation of the kineticenergy of the cold plasma component is given by

∂t

(1

2Ncmeu

2c

)= Ncmeuc ·

∂uc

∂t= J c · E. (5)

From the divergence of the Poynting vectorS, we obtain

∇ ·S =∂

∂t

(ε0

2E2

+1

2µ0B2)

+ J · E, (6)

whereε0 andµ0 are the permittivity and the magnetic per-meability in vacuum, respectively. From Eqs. (2) and (5) weobtain

∇ ·S =∂U

∂t+ J R · E, (7)

where the variableU is the energy density of the plasmawaves andU =

ε02 E2

+1

2µ0B2

+12Ncmeu

2c. Let us consider

a wave generation regionV , and integrate Eq. (7) over thevolumeV , given by∫V

∇ ·SdV =∂

∂t

∫V

UdV +

∫V

J R · EdV . (8)

On the surfaceA of the generation regionV , we have∫A

∇ ·SdA =∂

∂t

∫V

UdV +

∫V

J R · EdV (9)

and∫A

∇ ·SdA > 0. (10)

From Eqs. (9) and (10), we obtain

∂t

∫V

UdV > −

∫V

J R · EdV . (11)

If∫V

J R · EdV < 0, we have

∂t

∫V

UdV > 0, (12)

which suggests the plasma wave energy is growing in time,i.e., an absolute instability such as generation of chorus istaking place in the volumeV .

As a proxy of∫V

J R · EdV , we measure1W(r, t)

emerged during a time interval1t , given by

1W(r, t) =

t+1t∫t

E(r, t ′) · J R(r, t ′)dt ′. (13)

The current density of the hot componentJ R is given by

J R(r, t) =

∫∫∫qvf (r,v, t)dv, (14)

wheref is the phase space density of the hot plasma compo-nent. Substituting Eq. (14) into Eq. (13), we obtain

1W(r, t) =

t+1t∫t

∫∫∫qE(r, t ′) · vf (r,v, t ′)dvdt ′. (15)

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506 Y. Katoh et al.: Statistical significance of WPIA

(a)

(b)

(c)

Fig. 1.Schematic illustration of the WPIA for whistler-mode wave-particle interactions.(a) and(b) represent the relation between theperpendicular component of the velocity of an energetic electron,and the wave electric (EW ) and magnetic (BW ) fields of whistler-mode waves propagating parallel to the background magnetic fielddirection for the cases of(a) W < 0 and(b) W > 0. (c) Distribu-tion of energetic electrons as a function ofW for an example of anegativeWint corresponding to wave generation. The total numberof energetic electrons isN0, while N+ andN− are the numbers ofenergetic electrons having positive and negativeW , respectively.

Since measurement of the phase space densityf is con-ducted for discrete times, we re-express1W(r, t) as a sum-mation ofqE · v measured over the time interval1t , givenby

1W(r, t) =

N∑t≤ti≤t+1t

qE(r, ti) · vi, (16)

whereN denotes the number of particles detected during the1t interval, the subscripti designates quantities of thei-th particle, andti is the timing of the detection of thei-thparticle. We note thatti can be identical for differenti. Wealso note that the expression of1W given by Eq. (16) is analternative representation of the quantity “I ” introduced byFukuhara et al. (2009). To obtain1W , we should determine

the time interval1t of the accumulation from the time-scaleof wave-particle interactions. The time resolution ofti shouldbe shorter than the time-scale of wave-particle interactions,but 1t should be sufficiently longer than the time-scale ofwave-particle interactions to observe energy exchange be-tween waves and particles. In addition, since we use a wave-form and a finite number of particles obtained during a finitetime interval, statistical fluctuation should be considered inthe measurement of this energy exchange.

Figure1 shows a schematic illustration of the WPIA forinteractions between energetic electrons and whistler-modewaves propagating parallel to the background magnetic fielddirection. Figure1a and b show the relation between the per-pendicular component of the velocity of an energetic elec-tron, and the wave electric (EW ) and magnetic fields (BW )of whistler-mode waves for the cases ofW < 0 andW > 0.The sign ofW is determined by the relative angleθ betweenv⊥ andEW . Here we assume that a finite volume is filledby N0 plasma particles and that a fraction of these particlesresonate with whistler-mode waves propagating in the vol-ume. Since we can evaluate the time variation of the kineticenergyWi of each particle, the net energy exchange betweenthe particles and waves in the volume can be evaluated by ac-cumulatingWi over allN0 particles to obtainWint. Figure1cshows a distribution of energetic electrons in the volume asa function ofW for an example with a negativeWint, whichcorresponds to wave generation. We denote the numbers ofenergetic electrons having positive and negativeW as N+

andN−, respectively. If efficient energy exchange occurs inthe volume, then the net energy exchange originates in ener-getic electrons satisfying the cyclotron resonance conditionwith whistler-mode waves. In such a case,N+ andN− shouldbe significantly different from each other, while similarN+

and N− would be obtained if only non-resonant electronswere present. Figure1c corresponds to the case of efficientwave-particle interactions resulting in the wave generation,whereN− is larger thanN+ to satisfy thatWint is negative.As shown, the differenceδN betweenN+ and N− makesthe distributionn(W) asymmetric. IfδN was negligible andthe resultantWint was almost zero, essentially no net energyexchange would occur: i.e., the waves would be merely prop-agating in the volume without growing nor damping.

Since the number of particles in the volume is finite, theremust be fluctuation ofWint over time. This fluctuation orig-inates from thermal fluctuation of energetic electrons andfluctuation of both wave electric field amplitude and relativeangleθ . The value ofWint should be sufficiently larger thanthe fluctuation, particularly for the case of efficient wave-particle interactions. Based on the central limit theorem, wecan evaluate the statistical significance of the obtainedWintby computing the standard deviationσW . Here we considerthe integration ofW(t) over a time intervaldT for the caseof N energetic particles. In this case, we can evaluateσW of

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Y. Katoh et al.: Statistical significance of WPIA 507

the obtainedWint by computing

σW =

√√√√ N∑i=1

(qEW (ti) · vi)2−

1

N

(N∑

i=1

qEW (ti) · vi

)2

.

(17)

Here the first and second terms on the right-hand side ofEq. (17) correspond to the width and the centre of the dis-tribution of qEW (t) · v, respectively. In the WPIA, we canidentify the energy exchange between waves and particlesif we can obtain aWint sufficiently larger thanσW . The in-tegration ofW(t) over time should be conducted until theobtainedWint exceeds the value ofnσW corresponding tothe required statistical significance: assuming a Gaussian dis-tribution, n = 1.64 for a statistical significance of 90 % andn = 1.96 for 95 %.

3 Simulation of chorus generation

To evaluate the feasibility of the WPIA method, we anal-yse simulation results reproducing chorus emissions. We usean electron hybrid code with a dipole magnetic field (Ka-toh and Omura, 2006, 2007a, b, 2011; Omura et al., 2008),in which cold electrons are treated as a fluid and energeticelectrons as particles using the particle-in-cell (PIC) methodincluding fully relativistic effect. We assume a spatially one-dimensional system along the background magnetic field di-rection, withh as the distance from the magnetic equator.Although the simulation system is one-dimensional, we takeinto account the mirror motion of energetic electrons in thedipole magnetic field by using a cylindrical field model (Ka-toh and Omura, 2006). We solve Maxwell’s equations withthe current densities computed from the motion of electronsof both cold and energetic components. We treat transverseelectromagnetic waves propagating parallel to the dipolemagnetic field, while we neglect the longitudinal componentof the electric field.

The time step and the grid spacing of the simulation are0.01�−1

e0 and 0.06c�−1e0 , respectively, where�e0 is the elec-

tron gyrofrequency at the magnetic equator. The simula-tion system is along the field line and the number of gridpointsLh = 16384. The number of particles initially placedin the simulation system representing energetic electrons is4096×Lh. We assume that the plasma frequency of the coldelectronsωpe at the equator is 4�e0, while the number den-sity of cold electronsNe,cold is constant in space and time.Because of limited computational resources, we assume thatthe spatial inhomogeneity of the background magnetic fieldis larger than the realistic dipole field ofL = 4 of the Earth’smagnetosphere. Approximating the spatial inhomogeneity ofthe dipole magnetic field in the equatorial region of the mag-netosphere as�e(h) = �e0(1+ ah2), the constanta is givenby 4.5/(LRE)2 for the Earth’s magnetosphere, while we as-sume 4.5(3.5/LRE)2

= 55/(LRE)2 in the simulation.

(a)

( )

-6

B0

-5

-4log10 E/

1.0

0.8

0.6

0.4

0.2

0.0

ω

0.5 1.0 1.5 2.00.0

h = 92(b)

10 4 ( ) B 0 log 10 B / w

(c)

Fig. 2. Simulation results of the electron hybrid code used forthe pseudo-observations of the WPIA.(a) The initial velocity dis-tribution of energetic electrons assumed at the magnetic equator.Dashed and dash-dotted lines respectively denote resonance ellipsesof ω = 0.2 and 0.5�e0 (after Summers et al., 1998). The five solidsemicircles represent constant energy contours of 1 keV, 10 keV,100 keV, and 1 MeV, and the speed of light.(b) Spectrogram ob-served at the fixed pointh = 92c�−1

e0 in the Northern Hemisphereof the simulation system.(c) Time evolution of the spatial profile ofthe wave magnetic field amplitude.

We assume an anisotropic velocity distribution with aloss cone for the initial velocity distributionf (v‖,v⊥) ofenergetic electrons at the magnetic equator, given by

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508 Y. Katoh et al.: Statistical significance of WPIA

Region ARegion C

Region B

Fig. 3. Distribution of Wint in velocity space computed ath =

+200c�−1e0 over the time interval from 0 to 20 000�−1

e0 . Dashedlines denote resonance ellipses of forward propagating whistler-mode waves ofω = 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6�e0 (after Sum-mers et al., 1998). Coloured rectangles represent the energy andpitch angle ranges of Regions A, B, and C given in Table1.

f (v‖,v⊥) = A0exp

(−

v2‖

2V 2t‖

)g(v⊥) (18)

and

g(v⊥) =1

1− β

{exp

(−

v2⊥

2V 2t⊥

)− exp

(−

v2⊥

2βV 2t⊥

)}, (19)

whereA0 is a constant,β = 0.5, andVt‖ andVt⊥ are thermalvelocities of energetic electrons in the directions parallel andperpendicular to the background magnetic field, respectively.We assume that the thermal velocitiesVt‖ andVt⊥ are 0.225cand 0.6c, respectively, and that the perpendicular componentof the effective thermal velocity<Vt⊥> is

√2(1+ β)Vt⊥.

The anisotropic velocity distribution of energetic electronsdrives a linear instability generating a band of whistler-modewaves.

Figure 2 summarises the simulation results reproducingthe generation process of chorus emissions with rising tones(Katoh and Omura, 2007a). In the early stage of the simu-lation (t = 0–2500�−1

e0 ), a band of whistler-mode waves isgenerated through the instability driven by the temperatureanisotropy of energetic electrons. The time scale and the fre-quency range of the evolution of whistler-mode waves areconsistent with the growth rate predicted by the linear the-ory including the relativistic effect (Xiao et al., 1998; Katohand Omura, 2011). As shown in Fig.2a, aftert = 2500�−1

e0 ,we observe the generation of coherent wave elements withrising tones. Details of the generation process reproduced inthe simulation results have been discussed in previous stud-ies (Katoh and Omura, 2007a, 2011; Omrua et al., 2008,2009; Omura and Nunn, 2011). Coherent wave elementsare successively generated from the magnetic equator, whilethe wave intensity of each element is significantly amplifiedduring its propagation away from the equator, as shown inFig. 2b.

Table 1.Definition of Regions A, B, and C.

Energy range Pitch angle Sign ofrange [deg.] Wint

Region A 200–400 keV 100–110 negativeRegion B 1–4 MeV 85–90 positiveRegion C 100–300 keV 85–90 –

4 WPIA in the simulation results

Using the simulation results for the chorus generation, weconduct a WPIA analysis. We computeW(t) = qEW (t)·v(t)

from the wave electric fieldEW observed at a fixed point inthe simulation system and the velocity vectorv of each en-ergetic electron passing through the observation point. ThecomputedW(t) is integrated over time to obtainWint. Fig-ure 3 shows the distribution of the obtainedWint in the ve-locity space ath = +200c�−1

e0 for the time interval from 0to 20 000�−1

e0 . Since chorus emissions generated from themagnetic equator propagate away from the equator, energeticelectrons of tens or hundreds of keV moving toward the equa-tor satisfy the cyclotron resonance condition; the resonantelectrons have negative values ofv‖ ath = +200c�−1

e0 . Fig-ure3 reveals that the kinetic energy of electrons having largepitch angles andv‖ opposite to the wave normal direction istransferred to waves in significant amount, resulting in theefficient wave growth as shown in Fig.2b.

For the discussion of the statistical significance ofWint ob-tained by simulation, we specify three regions in the velocityphase space as given in Table1 and shown in Fig.3. WeevaluateWint and σW separately for each region by usingthe energetic electrons in the corresponding kinetic energyand pitch angle ranges. Figure4a shows the time history ofWint computed in Region A of velocity space. We find thatWint decreased over time, demonstrating the growth of cho-rus emissions. Using the dashed lines included in Fig.4a,which show the valuesWint ± 1.96σW , we can verify thatthe kinetic energy of the energetic electrons in Region A de-creased to a statistically significant degree; this differencewas transferred to plasma waves contributing the generationof chorus emissions. Figure4b shows the results computedfor Region B. The value ofWint increased over time and be-came larger than the values of 1.96σW aftert = 16000�−1

e0 .Based on the results shown in Fig.4b, we can verify with suf-ficient statistical certainty that relativistic electrons obtainedkinetic energy from plasma waves in Region B. Finally, weanalyseWint obtained in Region C. Since energetic electronsin Region C do not satisfy the cyclotron resonance condi-tion with whistler-mode waves generated in the simulationresult, we expect that efficient wave-particle interactions willnot be found in Region C. Figure4c shows thatWint in Re-gion C fluctuated over time, but not significantly relative to

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Y. Katoh et al.: Statistical significance of WPIA 509

-30

-25

-20

-15

-10

-5

0

5

0 5000 10000 15000 20000

[ ]Time

200 - 400 keV, Pitch angle = 100 - 110 deg.(a)

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 5000 10000 15000 20000[ ]Time

1 - 4 MeV, Pitch angle = 85 - 90 deg.(b)

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0 5000 10000 15000 20000

[ ]Time

100 - 300 keV, Pitch angle = 85 - 90 deg.(c)

Fig. 4. Time histories ofWint in (a) Region A,(b) Region B, and(c) Region C of velocity space. Energetic electrons passing throughthe observation point toward the magnetic equator are used in thecomputation. Dashed lines denoteWint ± 1.96σW .

σW , indicating that neither systematic wave-generation norelectron acceleration takes place in the Region C.

We expect that the enhancement ofWint shown in Fig.4ais a signature of the nonlinear wave growth of chorus ele-ments. Based on the nonlinear growth theory (Omura et al.,2008), the explosive amplification of the wave fields can beexplained by the contribution of nonlinear resonant currents

0 90 180 270 360

0

(a)

3.5

3.6

3.7

3.8

0 90 180 270 360

×105

[deg.]

Num

ber o

f cou

nted

par

ticle

s

(b)

Fig. 5. (a)Trajectories of resonant electrons inv‖ − ζ phase spacefor the inhomogeneity factorS = −0.4. (b) The accumulated distri-bution inζ space of energetic electrons in Region A. The estimatedstatistical noiseσN (ζ ) at eachζ bin is also shown.

arising from the nonlinear wave-particle interactions. Theformation process of the nonlinear resonant currents has beenexplained in terms of the concentration of nonlinear trajecto-ries of resonant electrons in the specific phase range of ve-locity phase space due to the presence of an electromagneticelectron hole. Since the optimum condition for the nonlinearwave growth of chorus elements is satisfied during the timeinterval of the simulation, an electromagnetic electron holeis formed in thev‖−ζ space as shown in Fig.5a correspond-ing to the conditionS = −0.4, whereζ is the relative phaseangle between the wave magnetic fieldBw andv⊥ of an en-ergetic electron, andS is the inhomogeneity factor (Omuraet al., 2008). Based on Fig.5a, we expect a depletion of thedistribution of energetic electrons in velocity phase space inthe energy and pitch angle range showing largeWint values.In the WPIA, the measurement of instantaneous wave andvelocity vectors of electrons enables us to obtain the distri-bution of energetic electrons in wave phase space. Figure5bshows the accumulated distribution inζ space of energeticelectrons moving toward the magnetic equator with kinetic

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510 Y. Katoh et al.: Statistical significance of WPIA

200 - 400 keV

Pitch angle [deg.]

-16

-14

-12

-10

-8

-6

-4

-2

0

2

0 30 60 90 120 150 180

(a)

1 - 4 MeV

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 30 60 90 120 150 180

Pitch angle [deg.]

(b)

Fig. 6.Wint computed(a) in the energy range from 200 to 400 keVand(b) from 1 to 4 MeV for the time interval from 0 to 20 000�−1

e0in pitch angle intervals of 5 degrees. The computed 1.96σW of eachpitch angle bin is also shown.

energies from 200 to 400 keV and pitch angles from 100to 110 degrees. We divided the wave phase angle into 24bins, and the accumulation is conducted over the time in-terval from 0 to 20 000�−1

e0 , the same as that used in Fig.3.We find a clear depletion in the distribution inζ space cor-responding to the region where the electromagnetic electronhole is formed under the optimum condition for nonlinearwave growth. Assuming that the detection (count) of par-ticles N follows a Poisson distribution in each bin of thewave phase angleζ , the standard deviationσN is equal tothe square root of the particle counts (Knoll, 2000):

σN (ζ ) =

√N(ζ). (20)

We show the estimatedσN (ζ ) as vertical bars in Fig.5band find that the size of the depletion is statistically signif-icant. This result clarifies that nonlinear resonant currentscontributing to the large values ofWint are maintained bythe asymmetry of the electron distribution in velocity phasespace due to the presence of the electromagnetic electronhole.

5 Discussion

We obtained a largeWint with sufficient statistical signifi-cance to indicate wave generation in Region A and electronenergisation in Region B, as shown in Fig.4a and b. We alsofound fluctuations ofWint in velocity phase space beyond justRegions A and B, as shown in Fig.3. As an example, wefound positiveWint in the vicinity of Region B, which rep-resents energisation of relativistic electrons. An evaluationof the statistical significance of theseWint values is shownin Fig. 6. Specifically, we showWint computed in the en-ergy range from 200 to 400 keV (Fig.6a) and that from 1 to4 MeV (Fig.6b) for the time interval from 0 to 20 000�−1

e0 inpitch angle intervals of 5 degrees. The computed 1.96σW ofeach pitch angle bin is also shown. According to the resultsshown in Fig.6a, statistically significantWint representingwave generation were obtained not only Region A but pitchangles from 85 to 115 degrees. This result indicates that, inthe energy range from 200 to 400 keV, energetic electrons ofa wide range of pitch angles (from 85 to 115 degrees) con-tributed to the chorus generation, while energetic electronsin the pitch angle range from 100 to 110 degrees were partic-ularly effective in the simulation. In Fig.6b, a positiveWintlarger than 1.96σW is observed only for the pitch angle rangefrom 85 to 90 degrees, while a statistically significant nega-tive Wint was obtained in the pitch angle range from 90 to100 degrees. Based on the consideration of the statistical sig-nificance ofWint, energy exchange between waves and parti-cles is not evident in other pitch angle ranges.

In the electron hybrid code used in the present study, weassumed a large spatial inhomogeneity and neglected the par-allel component of the electromagnetic wave field. These as-sumptions were necessary to reduce computational costs andare not critical for the comparison with theoretical studies;however, the necessary computation costs prevent us fromconsidering realistic plasma parameter values, namely, thoseobserved in the magnetosphere. In the simulation, we as-sumed energetic electrons having relatively higher kineticenergy and larger temperature anisotropy compared with typ-ical observation results. As shown in Figs.3 and4, the en-ergy range that mainly contributed to the wave growth in thesimulation result was a few hundred keV, while previous ob-servations reported that the typical energy range of electronsfor chorus generation is a few or tens of keV in the innermagnetosphere. As for a future study, it is necessary to carryout simulations under realistic initial conditions of the back-ground magnetic field as well as velocity distribution func-tion of energetic electrons and to conduct the WPIA in the re-sults so as to evaluate feasibility of the WPIA quantitatively.

The quantityW(t) actually has both parallel and per-pendicular components with respect to the backgroundmagnetic field. Since the longitudinal component is ne-glected in the electron hybrid code, we only obtainedthe perpendicular componentsW⊥(t) = qE⊥(t) · v⊥(t) inthe pseudo-observations. Although the assumption of the

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Y. Katoh et al.: Statistical significance of WPIA 511

parallel propagation of chorus is acceptable in the generationregion close to the magnetic equator (e.g., Li et al., 2011),recent observations reported chorus propagating obliquelyfrom the source region (Santolık et al., 2009; Chum et al.,2009). In addition, theoretical studies of the chorus gener-ation process have suggested possible important roles forthe parallel component of the wave electric field in under-standing nonlinear wave damping at half the gyrofrequency(Omura et al., 2009). Interaction of energetic electrons withobliquely propagating whistler-mode waves is an importantresearch target of the WPIA and, therefore, evaluation ofW‖(t) is necessary in future study by simulation on solvingthe electromagnetic wave field in three dimensions.

6 Summary

By applying the WPIA to simulation results, we confirmedthe effectiveness of the WPIA for studying the energy trans-fer process between relativistic electrons with whistler-modechorus emissions. Using the simulation results, we computedW(t), the inner product of the wave electric field observedat a fixed point and the velocity vector of each energeticelectron passing through the fixed point along the field line,whereW(t) is the time variation of the kinetic energy of elec-trons interacting with plasma waves. By integratingW(t)

over time, we obtainedWint. We verified that large valuesof Wint, representing efficient wave-particle interactions, oc-curred in the simulation, which is consistent with previousstudies (e.g., Katoh and Omura, 2007a; Omura et al., 2008,2009; Katoh et al., 2008). We also derived an expression forthe standard deviationσW of Wint and proposed that the sta-tistical significance of the obtainedWint be evaluated by com-paring it with σW . We showed that statistically significantWint can be used to identify wave generation and accelerationof relativistic electrons. We also conducted an analysis to ob-tain the distribution of energetic electrons inζ space and clar-ified that a deviation of the distribution inζ space occurred invelocity phase space corresponding to the largeWint values.This deviation in theζ space distribution was consistent withthe formation of nonlinear resonant currents proposed by re-cent theoretical studies on the generation mechanism of cho-rus emissions. The result of the present study demonstratesthat the WPIA is useful for analysing the energy exchangebetween waves and particles and suggests that the statisticalsignificance of the WPIA can be evaluated using the standarddeviationσW of Wint.

Acknowledgements.The computations described herein were per-formed using supercomputers at the Research Institute for Sustain-able Humanosphere and Academic Center for Computing and Me-dia Studies of Kyoto University, and the Solar-Terrestrial Environ-ment Laboratory of Nagoya University. This work was supported byGrants-in-Aid for Scientific Research S (23224011) and for YoungScientists A (22684025) from the Ministry of Education, Science,

Sports and Culture of Japan. This work was supported by the GlobalCOE program “Global Education and Research Center for Earth andPlanetary Dynamics” of Tohoku University.

Topical Editor R. Nakamura thanks D. Nunn and one anony-mous referee for their help in evaluating this paper.

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