Full particle simulation of whistler-mode rising chorus emissions in the magnetosphere

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114,, doi: /2008ja013625, 2009 Full particle simulation of whistler-mode rising chorus emissions in the magnetosphere M. Hikishima, 1 S. Yagitani, 1 Y. Omura, 2 and I. Nagano 3 Received 20 July 2008; revised 30 September 2008; accepted 6 November 2008; published 13 January [1] We perform an electromagnetic full particle simulation to study the generation mechanism of VLF whistler-mode chorus emissions in the equatorial region of the magnetosphere. Parabolic variation of the static magnetic field is assumed as a model for the dipole magnetic field in the vicinity of the equator. We have cold thermal electrons and relatively low anisotropic hot electrons as plasma particles. In the initial phase, the amplitude growth of the incoherent whistler-mode waves is determined by the linear growth rate. When the wave amplitude reaches a certain level, it begins to grow more rapidly with a series of rising tone emissions consisting of coherent phase structures in the vicinity of the magnetic equator, and their wave packets propagate away form the magnetic equator. The frequency sweep rates of the excited rising tone elements decrease gradually. We find a distinct threshold for such a nonlinear wave growth generating the rising chorus-like elements. The relation between the wave amplitude and the frequency sweep rate of each element found in the simulation fully supports the nonlinear wave growth theory of chorus emissions. Citation: Hikishima, M., S. Yagitani, Y. Omura, and I. Nagano (2009), Full particle simulation of whistler-mode rising chorus emissions in the magnetosphere, J. Geophys. Res., 114,, doi: /2008ja Introduction [2] VLF chorus emissions are associated with the injection of anisotropic energetic electrons toward the dawnside magnetosphere during geomagnetospheric disturbances. It is generally considered that they are excited through nonlinear wave-particle interaction between anisotropic energetic electrons from kev to tens of kev and the whistler-mode waves propagate along the geomagnetic field line. Many chorus emissions have been observed in situ by many scientific satellites since early observations [e.g., Oliven and Gurnett, 1968; Burtis and Helliwell, 1969; Lauben et al., 1998; Gurnett et al., 2001; Horne et al., 2005]. It has also been revealed that the enhanced chorus emissions are often detected near the magnetic equator [Tsurutani and Smith, 1974, 1977]. Furthermore, Nagano et al. [1996] and LeDocq et al. [1998] have reported the evidence that the chorus emissions are induced in the vicinity of the equator and propagate toward both hemispheres by estimating their wave normal vectors and Poynting vectors along the ambient magnetic field line near the equatorial region. More recent satellite observations have shown that chorus source regions are located quite close to the magnetic equator by 1 Graduate School of Natural Science and Technology, Kanazawa University, Kanazawa, Japan. 2 Research Institute for Sustainable Humanosphere, Kyoto University, Kyoto, Japan. 3 Department of Electrical and Computer Engineering, Kanazawa University, Kanazawa, Japan. Copyright 2009 by the American Geophysical Union /09/2008JA using the detailed data obtained by simultaneous multipoint measurement of chorus elements in the inner magnetosphere [Santolik et al., 2003, 2004a]. [3] The early study of chorus generation by the cyclotron resonance assumes the electron phase bunching [Brice, 1963]. The contributions of the phase correlated particles and the stimulated transverse resonant current via the cyclotron resonant interaction have been examined in numerical simulations [Nunn, 1971, 1974; Helliwell and Crystal, 1973]. A numerical study of nonlinear frequency shift has been performed by Vlasov hybrid simulation technique [Nunn, 1990; Nunn et al., 1997], where a coherent wave was a priori assumed, and the large frequency variation of chorus emissions emerging from incoherent thermal noise was not studied sufficiently. Katoh and Omura [2006, 2007] have reproduced such a large frequency variation of rising chorus by using an electron-hybrid code where hot resonant electrons are treated as particles and cold thermal electrons are treated as a fluid. However, the effects of full particle kinetics were not included and highly anisotropic energetic electrons much greater than those observed were assumed. [4] In the present study, to figure out precisely the essential physical mechanisms of chorus emissions we perform an electromagnetic full particle simulation (KEMPO code) [Omura, 2007]. The particle code treats both cold and hot electrons as particles with a realistic temperature anisotropy of hot resonant electrons, and it includes effects of adiabatic mirror motion and the loss cone distribution in the nonuniform dipole field to model particle dynamics in the Earth s magnetosphere [Katoh and Omura, 2006, 2007]. We confirm the nonlinear wave 1of11

2 growth theory for chorus emissions [Omura et al., 2008] by analyzing the simulation results. 2. Simulation Model [5] Maxwell s equations and the relativistic equations of motion for a large number of electrons are solved by the centered difference scheme. The cold thermal electrons and hot electrons are assumed in the system as a model of the inner magnetospheric plasma environment at the time of magnetic disturbance, while ions are treated as a neutralizing background. In solving motion of electrons, the relativistic effect is retained. We assume a purely transverse whistler-mode wave propagating along the ambient magnetic field line where we neglect electrostatic field to suppress unnecessary diffusion due to enhanced thermal fluctuations. The plasma frequency of electrons is constant throughout the simulation system. The open boundary regions are attached to both ends of the system with their lengths being approximately ten times of the wavelength of the dominant whistler-mode waves. In the open boundary regions, outgoing whistler-mode waves are absorbed by applying a smoothly damping parabolic function. [6] The simulation system is based on a one-dimensional model with the x-axis taken along the ambient magnetic field line near the equator. We extend it to a cylindrical model of the magnetic flux tube, incorporating the adiabatic motion of charged particles along a nonuniform magnetic field [Katoh and Omura, 2006]. We assume a plane wave propagating along the magnetic field line. The transverse components of electromagnetic field are assumed to be uniform over the gyroradius of a gyrating particle at a fixed point on the x-axis. We assume a magnetic field variation B 0x = B 0eq (1 + ax 2 ) to model the ambient dipole magnetic field line. The center of the system (x = 0) corresponds to the magnetic equator, and B 0eq is the magnitude at the equator. The magnitude of B 0eq is determined from the electron gyrofrequency with an arbitrary choice of the electron charge-to-mass ratio e/m. For simplicity we assume e/m = 1, and thus B 0eq = 1 when the electron gyrofrequency at the equator is chosen to be unity. We assume the parabolic coefficient a = for a distance x normalized by cw 1 e0 where c and W e0 are the speed of light and the electron gyrofrequency at the equator, respectively. It is noted that we treat only the equatorial region of the dipole magnetic field which is well approximated by the parabolic variation. We assume the parabolic coefficient much greater than the real magnetosphere to reduce the size of the simulation region for computational efficiency. The radial component B r of the ambient magnetic field is given by B r ¼ r 0x ; where r g = gv? /W e, and g =[1 (v k 2 + v? 2 )/c 2 ] 1/2 with particle velocity components v k and v? parallel and perpendicular to the ambient magnetic field, respectively. The local gyrofrequency of electrons is denoted by W e. The maximum strength of the ambient magnetic field B 0x at both ends of the system is 1.1 times larger than that at the ð1þ equator, which corresponds to a few degrees of the magnetic latitudes in the vicinity of the equator. In the nonuniform magnetic field, the force of F x = m(@b 0x /@x) acts on electrons in addition to their gyromotion, where m = m 0 gv? 2 / 2B 0x with the electron rest mass m 0. Energetic electrons with large pitch angles are magnetically trapped in the vicinity of the equator in the system. [7] As for plasma particle treatment in the simulation system, we use two species of particles, cold thermal electrons and hot energetic electrons. The velocity distribution function of the cold thermal electrons is an isotropic Maxwellian, while the energetic hot electrons form an anisotropic modified-maxwellian for a loss cone in the relativistic momentum space (u k, u? ), where u k = gv k and u? = gv?, respectively, with g =[1 (v k 2 + v? 2 )/c 2 ] 1/2 =[1+(u k 2 + u? 2 )/c 2 ] 1/2. The momentum distribution of the hot electrons is given by n h f u k ; u? ¼ ð2pþ 3=2 U thk Uth? b exp u2? 2U 2 exp u2 k 2U 2 th? thk! exp u2? 2bUth? 2 ; ð2þ where n h, U thk, U th? are density, parallel and perpendicular components of thermal momentum per unit mass, respectively. The expression of loss cone distribution originally comes from Ashour-Abdalla and Kennel [1978]. The b means a coefficient that determines a loss cone angle. When increasing the b, the loss cone angle becomes larger. We assume the loss cone depth b = 0.3 in the simulation. [8] For the cyclotron interaction between counter-streaming electrons and the whistler-mode waves propagating parallel to the ambient magnetic field line, the resonance condition is expressed as a function of the frequency w [Omura et al., 2008], V R ¼ cdx 1 W e ; ð3þ gw where x and d are dimensionless parameters defined by x 2 ¼ w ð W e wþ ; ð4þ w 2 pe d 2 ¼ 1 w2 c 2 k 2 ¼ 1 1 þ x 2 ; ð5þ where w pe is the plasma frequency of the cold electrons and k is the wavenumber. We assume a typical chorus frequency range w = W e0 and w pe =5W e0. Based on the resonance condition, we determine the parallel thermal momentum U thk = 0.20 c at the equator so that the distribution of anisotropic hot electrons can cover the resonant velocity range. We assume the perpendicular thermal momentum U th? = 0.33 c which gives a typical magnetospheric anisotropy observed in the inner magnetosphere [Burton and Holzer, 1974; Kaye et al., 1978]. Since the Lorentz factor g equals to 1.12 with these initial parallel 2of11

3 Figure 1. Hot electron distributions at time (left) t =0W 1 e0 and (right) t = 10,000 W 1 e0 in phase spaces (a) (x, v k ), (b) (x, v? ), and (c) (v k, v? ) at the magnetic equator. and perpendicular thermal momentums, the relativistic effect is not significant for most of the particles in the simulation. [9] We allocate the cold thermal electrons and the hot electrons over the whole system region assuming adiabatic motion in the nonuniform magnetic field. In the left panels of Figure 1, we plot the spatial distributions and the velocity distribution of the hot electrons at the magnetic equator at the initial simulation time t =0W 1 e0. The right panels show those at the end of the simulation run t = 10,000 W 1 e0. [10] As for the treatment of particles in the simulation, we do not inject new particles with arbitrary pitch angles into the system. Instead, we allow that the outgoing particles from the system outside the loss cone are bounced back with constant pitch angles at the boundaries of the system. This is because the outgoing particles outside the loss cone will eventually be bounced back at somewhere out of the simulation system. The particles inside the loss cone are not reflected at the boundaries of the system because we assume that they are precipitated into the ionosphere. Therefore we remove the particles with pitch angles inside the loss cone angle of 20 degrees at the boundaries so as to maintain the initial loss cone distribution in the vicinity of the equator. [11] In the initial stage of whistler-mode wave generation from thermal noise, the linear wave growth should take place before any nonlinear processes develop. Using the nonrelativistic linear theory of Kennel and Petschek [1966], we estimate the liner growth rates of whistler-mode waves and show dependence on the simulation parameters. Since we use weakly relativistic hot electrons, we use the nonrelativistic resonant velocity V R0 by assuming g = 1 in (3). The linear growth rates w i of a parallel propagating whistlermode wave are given by w i ¼ pw e 1 w 2 hðv R0 Þ½AV ð R0 Þ A c Š; ð6þ W e where h(v R0 ), A(V R0 ), A c represent electron density, temperature anisotropy of resonant electrons, and critical temperature anisotropy defined, respectively, by h ðv R0 Þ ¼ 2p W e w k AðV R0 Þ ¼ Z 1 0 Z 1 0 v? Fdv? j vk ¼V v 2 v k k v Z k 1 ; 2 v? Fdv? vk ¼V R0 0 ð7þ ð8þ 3of11

4 (6)(9), and are plotted as a function of wave frequency below the electron gyrofrequency. We vary each of the key parameters in each panel shown in Figure 2. Especially, the ratio of the cold plasma frequency to the electron gyrofrequency sensitively determines the range of unstable frequencies (Figure 2a). Increase of the cold plasma frequency generates the whistler-mode waves at lower frequencies, and the linear growth rates are enhanced because the density of hot electrons increases under the constant density ratio of the hot electrons to the cold electrons. By increasing the temperature anisotropy of hot electrons (Figure 2b), we increase the perpendicular thermal velocity while keeping the parallel thermal velocity constant. Increase of anisotropy corresponds to increase of the linear growth rates as expected from (6), while the unstable frequency range extends toward slightly higher frequencies. We also find that increase of the hot electron density to the cold thermal electron density only results in magnitude variation of the linear growth rate in the same frequency range (Figure 2c). [13] The solid lines in Figure 2 correspond to the parameters used in the present simulations, and they are listed in Table 1. We used approximately 8 million superparticles for cold electrons and 67 million superparticles for hot electrons. The computation took 240 hours with 8 processors (3.2 GHz clock). Figure 2. The linear growth rates as a function of wave frequency near the magnetic equator at t =0W 1 e0. Each panel indicates dependence of the linear growth rates on (a) cold plasma frequency of electrons, (b) temperature anisotropy, and (c) density ratio of hot electrons to cold thermal electrons. The parameter values are shown in the top right corner of each panel. The solid lines correspond to the values used in the simulation. and A c ¼ 1 W e =w 1 : In the linear theory above, the contribution of the displacement current is neglected. Notice that the velocity distribution function F is normalized to unity. In case that the velocity distribution forms a simple bi-maxwellian, the temperature anisotropy is expressed by the well-known formula A = T? /T k 1, where T k and T? are parallel and perpendicular temperatures, respectively. [12] We calculate the linear growth rates for different physical parameters as shown in Figure 2. These linear growth rates depend on the cold plasma frequency of electrons, the temperature anisotropy, and the density ratio of hot electrons to cold thermal electrons. The linear growth rates are calculated by integrating the initial distribution function numerically at the resonance velocity using ð9þ 3. Simulation Results 3.1. Linear Growth Phase [14] The linear growth phase of whistler-mode wave generation is initially observed in the simulation prior to the subsequent nonlinear growth phase. We calculate spatial dependence of the linear growth rates to estimate the initial growth of the whistler-mode waves at each location in the simulation region. The calculation of the linear growth rates is performed in the same manner as in the case of Figure 2. The result is shown in Figure 3. The center position x =0 cw 1 e0 corresponds to the magnetic equator, and the positive and negative x correspond to the northern and southern hemispheres, respectively. The linear growth rates are calculated from the initial particle distributions in the negative velocity range for the positive k of the whistlermode waves at each location at the initial time t =0W 1 e0. For the positive velocities and the negative k we obtain essentially the same result because of the symmetry of the velocity distribution function. The whistler-mode waves Table 1. Simulation Parameters Parameter Value Time step W e0 Grid spacing cw e0 Number of grids 32,768 Plasma frequency of 5 W e0 cold electrons: w pe Total number of hot electrons 67,108,864 Total number of cold electrons 8,388,608 Thermal momentum of hot electrons 0.20 c, 0.33 c at the equator: U thk, U th? Thermal momentum of cold electrons 0.01 c, 0.01 c at the equator Density ratio of hot electrons to cold electrons: N h /N c Coefficient of parabolic magnetic field: a (c 1 W e0 ) 2 4of11

5 Figure 3. Spatial profile of the linear growth rates at t =0 W 1 e0. The linear growth rates on both hemispheres are estimated from the distribution in the negative velocity range corresponding to the whistler-mode waves with positive k vectors. The positive and negative x correspond to the northern and southern hemispheres, respectively. have significant linear growth below half the gyrofrequency. The linear growth rates are particularly enhanced at the equator, and we can see the maximum linear growth rate at approximately w =0.3W e0. This implies a possibility of enhanced whistler-mode wave generation close to the equator. The increase of the unstable frequencies off the equator is caused by the increases in both the gyrofrequency and the anisotropy. [15] Diagnostics of the initial linear phase of the whistlermode waves focused mostly on the time period t < 800 W e0 1 are shown in Figure 4. Figure 4a shows the waves propagating as a function of time in the overall region. At the initial phase, the whistler-mode noise generated from the thermal fluctuations due to the local anisotropic hot electrons. The initial amplitude originating from the thermal noise corresponds to approximately B w = B 0eq, being averaged near the equator. The linear growth of the whistler-mode waves occurs steadily all over the region. The growth is particularly enhanced near the equator as expected from the spatial variation of the linear growth rates in Figure 3. Figure 4b shows the transverse components of the magnetic wave amplitude as a function of time near the equator. The amplitudes are averaged over x = cw 1 e0. It shows an exponential growth as a function of time up to t = 400 W 1 e0. Thereafter, the wave amplitude undergoes significant amplification with a larger growth rate than the linear growth rate. In Figure 4c, we give the spectral intensities of the growing waves at the equator. In the initial phase of the interaction (t = W 1 e0 ), incoherent whistler-mode waves grow according to the linear growth rate. It is clear that the initial development of the whistlermode waves follows the linear growth represented by the solid lines of Figure 2. Incoherent whistler-mode waves corresponding to the estimated frequency band width w = W e0 grow gradually as time elapses. It has the largest growth at approximately w =0.3W e0 corresponding to the maximum linear growth rate. Figure 4. The linear phase of whistler-mode wave generation. (a) The early phase of the spatiotemporal variation of the wave amplitude. (b) The amplitude of the whistler-mode waves near the equator. The amplitude is averaged over x = cw 1 e0. The dashed lines indicate the time and the threshold for the nonlinear growth. (c) The dynamic spectra of the wave magnetic field at the equator in the early phase. 5of11

6 Figure 5. Spatial profile of the amplitude of the transverse wave magnetic field and its time evolution. The center position x =0cW 1 e0 corresponds to the magnetic equator. The dynamic frequency spectra at the white dashed lines are indicated in Figure 6. The positive and negative x correspond to the northern and southern hemispheres, respectively Nonlinear Growth Phase [16] In Figure 5, we show spatial profiles of the amplitude of whistler-mode waves in the simulation as a function of time after the linear growth phase in Figure 4a. Starting from t = 400 W 1 e0, a series of intense discrete wave packets are generated successively from the equatorial region. The whistler-mode wave packets are progressively amplified through the resonant interaction with counter-streaming anisotropic hot electrons. The occurrences of wave packets near the equator gradually decrease. By continuously generating wave packets through the cyclotron resonant inter- 6of11

7 1 Figure 6. Dynamic frequency spectra of transverse wave magnetic field at x = 0, 30, 70, 140 cw e0 corresponding to the positions of the white dashed lines in Figure 5. actions with anisotropic electrons, the whole anisotropy in the system is gradually relaxed due to pitch angle diffusion. The reduction in the anisotropy eventually terminates the linear wave growth of the incoherent whistler-mode waves. [17] We observe frequency characteristics of the waves at several points in the simulation region for the wave packets propagating northward. The wave spectral intensities are shown in Figure 6. We have selected four fixed positions x = 0, 30, 70, 140 cw 1 e0, which are indicated by the white dashed lines in Figure 5. We can find fine rising tone emissions at all the positions along the magnetic field line. The rising tone structures at each position are mainly separated into four discrete elements. The third element has typical features of chorus elements observed near the magnetic equator (see Figure 4 of Santolik et al. [2003]). At the start of the rising tone element, the frequency sweep rate increases with the increasing wave amplitude. The first 1 rising tone element appears clearly around t = 1000 W e0 after the linear growth phase period. The lower limit of frequency of rising tone emissions corresponds to the lowest unstable frequency w =0.1W e0 determined by the linear growth theory, while the upper limit is due to the nonlinear frequency development. In the nonlinear phase, a coherent wave with rising frequency gradually emerges as time goes on at the equator. It is observed that the frequency sweep rates of the four rising tone elements steadily decrease with time. These frequency sweep rates are roughly estimated = , , , W 2 e0 for the first, second, third and fourth elements, respectively. Note that the rising chorus-like elements are already formed near the magnetic equator. The elements propagating away from the equator are amplified with almost the same frequency variations. [18] In Figure 7, we show a transverse component of the magnetic wave field as a function of time. The amplitudes of transverse magnetic wave components are averaged over x = cw 1 e0 where rising tone emissions are generated. When the generation process of the first rising tone element undergoes a transition from the linear phase to the nonlinear phase, the amplitude dramatically grows, and reaches to a certain saturation. Thereafter the wave amplitude gradually decreases by the lower anisotropy due to strongly diffused hot electrons by the enhanced rising tone emissions. After the transition to the nonlinear phase (vertical dashed line in Figure 7), we find the generation of chorus-like elements near the equator as we find in Figures 5 and 6. The time period is characterized by the wave amplitude greater than the threshold level B w =7 7of11

8 Figure 7. The wave amplitudes averaged over x = cw 1 e0. The horizontal dashed line indicates the threshold level for the generation of rising chorus-like emissions and the vertical dashed line indicates the starting time B 0eq (horizontal dashed line). After the amplitude reaches this threshold, the electromagnetic hole [Omura et al., 2008] causing the nonlinear wave growth is formed. It is noted that the nonlinear trapping time is inversely proportional to the wave amplitude, while the trapping zone near the equator depends on the inhomogeneity of the magnetic field. Therefore the threshold should depend on the wave amplitude and the spatial inhomogeneity. However, such a parametric study and theoretical analysis are out of the scope of the present paper Frequency Sweep Rate [19] We focus on the gradually decreasing wave amplitudes after the saturation (t > 1300 W 1 e0 ). Omura et al. [2008] theoretically derived a relation between the frequency sweep rate of a chorus element and the wave amplitude (see their equation (50)). We give the ¼ 0:4 d V?0 gx c w W e0 1 V 2 R B w W 2 e0 V g B ; 0eq ð10þ where V?0 is the averaged perpendicular velocity on the velocity distribution of electrons, V g is the group velocity of the chorus emissions. We focus on the variable wave amplitude B w and the wave frequency w as well as the averaged perpendicular velocity V?0 as dominant parameters in the above formula of the frequency sweep rate. In Figure 7, we find a slow decrease of the magnetic wave amplitude after the saturation of the wave amplitude. As expected from equation (10), the decreasing wave amplitude causes decrease of the frequency sweep rates for the successive rising tone elements as seen in Figure 6. We apply a numerical approach to estimate a validity of the theoretical frequency sweep rates. We integrate the equation (10) numerically with a discrete time step Dt for the individual rising tone element. The numerical integration is performed as follows. w ðt þ DtÞ ¼ w ðþþ ðw ðþ; t B w ðþ; t V?0 ðþ t ÞDt: The frequency is updated every time step Dt = 0.08 W 1 e0. In Figure 8a, we give the comparison of theoretical frequency variations (black solid lines) by equation (11) with the rising tone elements observed in the simulation (Figure 6, top left). For precise integration, we need to evaluate the detailed amplitude variations of the individual rising tone elements. We clip out each enhanced rising tone element in the frequency-time domain, and obtain accurate temporal variation of the wave amplitude by performing the inverse Fourier transform for the clipped element. The clipped spectra and corresponding wave amplitudes are given in the left and right panels of Figures 8b 8e, respectively. Variations of V?0 of the hot electrons are also evaluated. We determine the starting frequency and time to estimate the theoretical frequency variation on the basis of the phase variation of the wave form of each rising tone element by using the threshold value of the wave amplitude estimated in Figures 4b and 7. From the careful evaluation and calculation, we find that the theoretically reproduced frequency variations closely represent the frequency variations of the chorus-like elements in the simulation. Based on (10) the decrease of frequency sweep rate is explained by the decreasing wave amplitude in the simulation. This result demonstrates that the wave amplitude of a chorus element is the key parameter controlling the frequency variation. The excellent agreement between the theoretical evaluation and the simulation demonstrates the validity of the nonlinear theory of the generation of chorus emissions [Omura et al., 2008]. 4. Summary and Discussion [20] We have performed an electromagnetic full particle simulation to study the generation of chorus emissions in the magnetosphere. The simulation system is assumed to be a symmetrical nonuniform model under the dipole magnetic field expanded in a parabolic form near the magnetic equator along the ambient magnetic field line. We treated only purely transverse whistler-mode waves propagating parallel to the ambient magnetic field line. Treatment of plasma particles allows existence of almost stationary cold thermal electrons and hot electrons with relatively low temperature anisotropy. The velocity distribution function of hot electrons was assumed as a bi-maxwellian with a weak loss cone representing energetic electrons injected into the inner magnetosphere. The simulation model is quite close to the actual magnetospheric condition generating chorus emissions. [21] We calculated the linear growth rates from the initial particle distribution for different simulation parameters to clarify the characteristics of the initial linear phase leading to the subsequent nonlinear evolution of chorus emissions. We have found the following: [22] 1. The ratio of the cold plasma frequency to the electron gyrofrequency mostly determines the frequency range of the initial growth of the whistler-mode wave. The smaller plasma frequency enhances the whistler-mode waves at higher frequencies, and vice versa, thus controlling the frequency band formation of rising tone emissions. [23] 2. Increase of the temperature anisotropy gives an enhancement of the linear growth rates, slightly extending the frequency range to a higher frequency. 8of11

9 Figure 8. (a) Theoretically estimated frequency variations (black solid lines) plotted on the rising chorus-like elements at the equator in Figure 6. (b) (e) (left) Clipped spectra and (right) the estimated wave amplitudes of the rising chorus-like elements. The time spans of the wave amplitude variations correspond to those of the theoretical curves in Figure 8a. [24] 3. The density ratio of the hot electrons to the cold thermal electrons only affects the magnitudes of the linear growth rates. [25] We found formation of coherent rising tone emissions from fluctuation of thermal electron noise. We summarize the simulation result in the following: [26] 4. Rising tone emissions are reproduced by the fullparticle simulation with realistic temperature anisotropy of hot energetic electrons. [27] 5. Rising tone emissions are generated near the magnetic equator through the nonlinear wave growth mechanism after the linear growth of incoherent waves attains a certain threshold of the total wave amplitude. [28] 6. The frequency sweep rates of rising tone emissions gradually become small as the total wave amplitude near the equator decreases. [29] 7. The theoretical relation between the wave amplitude and the frequency sweep rate [Omura et al., 2008] faithfully reproduces frequency variations of rising tone elements in the simulation. [30] 8. Rising tone elements generated at the magnetic equator are amplified through propagation from the equator to the higher latitude region. [31] With the full particle simulation we have reproduced rising chorus-like emissions, which are consistent with those in the electron-hybrid simulation by Katoh and 9of11

10 Omura [2006, 2007]. Since the full particle simulation requires more computational resources than the electronhybrid simulation, we reduced the spatial scale of the system by enhancing the inhomogeneity of the magnetic field. The reduced system requires a larger growth rate and a larger threshold for the chorus emissions to occur in a shorter time scale. However, we have confirmed that the fundamental physical processes of the generation of rising tone emission can be reproduced by the full particle simulation. This is an important step for realizing a more realistic velocity distribution function of the cold thermal electrons that is connected smoothly to the hot energetic component, which is not possible in the electron-hybrid simulation. [32] We have shown variations of the initial growth rate with the plasma frequency in Figure 2. We have assumed that the plasma frequency is constant along the ambient magnetic field in the simulation. Actually the density can vary proportional to the local magnitude B 0 /B 0eq [Summers and Ni, 2008]. As the simulation is restricted in the vicinity of the equator, the maximum magnitude of the ambient magnetic field B 0 /B 0eq = 1.1, which corresponds to a small variation of the plasma frequency less than 5%. [33] In the right panels of Figure 1, we showed distributions of the hot electrons at the time t = 10,000 W 1 e0 when the generated rising tone emissions almost disappeared from the system. The shape of the distribution at the equator indicates a consequence of consecutive diffusions by the rising tone emissions. The electrons diffused by the generation of rising tone emissions lose the initial pitch angles, while a fraction of the electrons undergo accelerations based on RTA (relativistic turning acceleration) and URA (ultrarelativistic acceleration) processes [Omura et al., 2007; Summers and Omura, 2007]. [34] Omura et al. [2008] derived the spatial range where nonlinear growth is effectively maintained, based on the nonlinear wave growth theory. The spatial range predominantly depends on the frequency sweep rate and on the wave frequency. Assuming the frequency w = W e0 of the rising tone emissions with our simulation parameters, the spatial extent from the equator is roughly x = cw 1 e0. It implies the effective nonlinear wave growth due should occur in the whole region of the simulation. [35] In the presented rising tone elements, particularly in the last two elements, the frequency sweep rates gradually increase with time. Such a feature of frequency variation has often been found in rising chorus elements observed by many satellites such as GEOTAIL [Nagano et al., 1996] and CLUSTER [Santolik et al., 2003, 2004b]. This feature is in agreement with the nonlinear wave growth theory as demonstrated in the present study. Other types of chorus emissions with different frequency sweep rates and even with falling tones and hooks have also been observed. Simulations and theoretical analyses of various types of chorus emissions are left as future studies. [36] Acknowledgments. Part of the computation in the present study was performed with the KDK system of RISH at Kyoto University. This work was partially supported by Grant-in-Aid and 17GS0208 for Creative Scientific Research The Basic Study of Space Weather Prediction of the Ministry of Education, Science, Sports and Culture of Japan. [37] Amitava Bhattacharjee thanks Manuel Platino and another reviewer for their assistance in evaluating this paper. References Ashour-Abdalla, M., and C. F. Kennel (1978), Nonconvective and convective electron cyclotron harmonic instabilities, J. Geophys. Res., 83(A4), Brice, N. (1963), An explanation of triggered Very-Low-Frequency emissions, J. Geophys. Res., 68(15), Burtis, W. J., and R. A. Helliwell (1969), Banded chorus - A new type of VLF radiation observed in the magnetosphere by OGO 1 and OGO 3, J. Geophys. Res., 74(11), Burton, R. K., and R. E. Holzer (1974), The origin and propagation of chorus in the outer magnetosphere, J. Geophys. Res., 79(7), Gurnett, D. A., et al. (2001), First results from the Cluster wideband plasma wave investigation, Ann. Geophys., 19, Helliwell, R. A., and T. L. Crystal (1973), A feedback model of cyclotron interaction between whistler-mode waves and energetic electrons in the magnetosphere, J. Geophys. Res., 78(31), Horne, R. B., R. M. Thorne, S. A. Glauert, J. M. Albert, N. P. Meredith, and R. R. Anderson (2005), Timescale for radiation belt electron acceleration by whistler mode chorus waves, J. Geophys. Res., 110, A03225, doi: /2004ja Katoh, Y., and Y. Omura (2006), A study of generation mechanism of VLF triggered emission by self-consistent particle code, J. Geophys. Res., 111, A12207, doi: /2006ja Katoh, Y., and Y. Omura (2007), Computer simulation of chorus wave generation in the Earths inner magnetosphere, Geophys. Res. Lett., 34, L03102, doi: /2006gl Kaye, S. M., C. S. Lin, and G. K. Parks (1978), Adiabatic modulation of equatorial pitch angle anisotropy, J. Geophys. Res., 83(A6), Kennel, C. F., and H. E. Petschek (1966), Limit on stably trapped particle fluxes, J. Geophys. Res., 71, 1. Lauben, D. S., U. S. Inan, T. F. Bell, D. L. Kirchner, G. B. Hospodarsky, and J. S. Pickett (1998), VLF chorus emissions observed by POLAR during the January 10, 1997, magnetic cloud, Geophys. Res. Lett., 25(15), LeDocq, M. J., D. A. Gurnett, and G. B. Hospodarsky (1998), Chorus source locations from VLF Poynting flux measurements with the Polar spacecraft, Geophys. Res. Lett., 25(21), Nagano, I., S. Yagitani, H. Kojima, and H. Matsumoto (1996), Analysis of wave normal and Poynting vectors of the chorus emissions observed by Geotail, J. Geomagn. Geoelectr., 48, 299. Nunn, D. (1971), A theory of VLF emissions, Planet. Space Sci., 19, Nunn, D. (1974), A self-consistent theory of triggered VLF emissions, Planet. Space Sci., 22, 349. Nunn, D. (1990), The numerical simulation of VLF nonlinear wave-particle interactions in collision-free plasmas using the Vlasov hybrid simulation technique, Comput. Phys. Commun., 60, 1. Nunn, D., Y. Omura, H. Matsumoto, I. Nagano, and S. Yagitani (1997), The numerical simulation of VLF chorus and discrete emissions observed on the Geotail satellite using a Vlasov code, J. Geophys. Res., 102(A12), Oliven, M. N., and D. A. Gurnett (1968), Microburst phenomena: 3. An association between microbursts and VLF chorus, J. Geophys. Res., 73(7), Omura, Y. (2007), One-dimensional electromagnetic particle code: KEMPO1, in Advanced Methods for Space Simulations, edited by H. Usui and Y. Omura, pp. 1 21, Terra Sci., Tokyo. Omura, Y., N. Furuya, and D. Summers (2007), Relativistic turning acceleration of resonant electrons by coherent whistler mode waves in a dipole magnetic field, J. Geophys. Res., 112, A06236, doi: / 2006JA Omura, Y., Y. Katoh, and D. Summers (2008), Theory and simulation of the generation of whistler-mode chorus, J. Geophys. Res., 113, A04223, doi: /2007ja Santolik, O., D. A. Gurnett, and J. S. Pickett (2003), Spatio-temporal structure of storm-time chorus, J. Geophys. Res., 108(A7), 1278, doi: /2002ja Santolik, O., D. A. Gurnett, and J. S. Pickett (2004a), A microscopic and nanoscopic view of storm-time chorus on 31 March 2001, Geophys. Res. Lett., 31, L02801, doi: /2003gl Santolik, O., D. A. Gurnett, and J. S. Pickett (2004b), Multipoint investigation of the source region of storm-time chorus, Ann. Geophys., 22, Summers, D., and B. Ni (2008), Effects of latitudinal distributions of particle density and wave power on cyclotron resonant diffusion rates of radiation belt electrons, Earth Planets Space, 60, 763. Summers, D., and Y. Omura (2007), Ultra-relativistic acceleration of electrons in planetary magnetospheres, Geophys. Res. Lett., 34, L24205, doi: /2007gl of 11

11 Tsurutani, B. T., and E. J. Smith (1974), Postmidnight chorus: A substorm phenomenon, J. Geophys. Res., 79(1), 118. Tsurutani, B. T., and E. J. Smith (1977), Two types of magnetospheric ELF chorus and their substorm dependences, J. Geophys. Res., 82(32), M. Hikishima and S. Yagitani, Graduate School of Natural Science and Technology, Kanazawa University, Kakuma-machi, Kanazawa , Japan. (hikisima@reg.is.t.kanazawa-u.ac.jp; yagitani@reg.is.t.kanazawau.ac.jp) I. Nagano, Department of Electrical and Computer Engineering, Kanazawa University, Kakuma-machi, Kanazawa , Japan. (nagano@ reg.is.t.kanazawa-u.ac.jp) Y. Omura, Research Institute for Sustainable Humanosphere, Kyoto University, Uji, Kyoto , Japan. (omura@rish.kyoto-u.ac.jp) 11 of 11