GEOPHYSICAL RESEARCH LETTERS, VOL. 34, L03102, doi:10.1029/2006GL028594, 2007
Computer simulation of chorus wave generation in the Earth’s inner magnetosphere Yuto Katoh1,2 and Yoshiharu Omura1 Received 28 October 2006; revised 16 December 2006; accepted 2 January 2007; published 3 February 2007.
[1] A self-consistent particle simulation with a dipole magnetic field model is carried out, reproducing chorus emissions with rising tones successfully. We assume energetic electrons forming a highly anisotropic velocity distribution in the equatorial region. No initial wave is assumed except for electromagnetic thermal noise induced by the energetic electrons. In the early stage of the simulation, coherent whistler-mode waves are generated from the equator through an instability driven by the temperature anisotropy of the energetic electrons. During the propagation of the whistler-mode waves, we find formation of a narrowband emission with negative frequency gradient (NEWNFG) in the spatial distribution of the frequency spectrum in the simulation system. The trailing edge of NEWNFG is continuously created at increasing frequencies in the region close to the equator. Observed at a fixed point, the NEWNFG shows a frequency variation of a typical chorus emission. Citation: Katoh, Y., and Y. Omura (2007), Computer simulation of chorus wave generation in the Earth’s inner magnetosphere, Geophys. Res. Lett., 34, L03102, doi:10.1029/2006GL028594.
1. Introduction [2] The generation mechanism of chorus emissions has been an enigma in space physics. Results of in situ observation have revealed that whistler-mode chorus emissions [Tsurutani and Smith, 1974] are observed in the dawn side of the Earth’s magnetosphere [Tsurutani and Smith, 1977; Meredith et al., 2001; Kasahara et al., 2005]. Detailed analyses of satellite observations have shown that the activity of whistler-mode chorus enhances during geomagnetically disturbed periods and clarified that whistlermode chorus often consist of narrow band emissions with rising tones. They have also revealed that the source region of whistler-mode chorus is close to the magnetic equator at the outside of the plasmapause [Lauben et al., 2002; Inan et al., 2004]. Especially, the Poynting vector direction of chorus emissions reverses during the passage of the equatorial region of the inner magnetosphere showing that the chorus generation region is close to the magnetic equator [Nagano et al., 1996; LeDocq et al., 1998; Santolı´k et al., 2003]. The waveform analysis made by Cluster satellites has revealed that the amplitude of the wave electric field of chorus emissions observed in the equatorial region of the 1 Research Institute for Sustainable Humanosphere, Kyoto University, Kyoto, Japan. 2 Now at Planetary Plasma and Atmospheric Research Center, Graduate School of Science, Tohoku University, Miyagi, Japan.
Copyright 2007 by the American Geophysical Union. 0094-8276/07/2006GL028594
magnetosphere typically exceeds 10 mV/m, which corresponds to the wave magnetic field greater than 100 pT [Santolı´k et al., 2003, 2004]. Such a large wave amplitude implies strong nonlinear effects in the generation process. Theoretical analyses have also suggested that the nonlinear cyclotron resonant interaction is essential in the generation process [Nunn, 1974; Matsumoto, 1979]. Although many models have been proposed for several decades to explain these observations [Omura and Matsumoto, 1982; Nunn et al., 1997; Trakhtengerts, 1999], the spontaneous successive generation of the narrowband rising tones has been left as a mystery. Details of observations and theories are reviewed by Omura et al. [1991] and Sazhin and Hayakawa [1992]. [3] Meanwhile, in recent years it has been widely accepted that the role of whistler-mode chorus is highly important in acceleration and loss processes of the Earth’s outer radiation belt electrons during a geomagnetic storm [Summers and Ma, 2000]; seed electrons radially diffused into the inner magnetosphere become resonant with whistlermode waves through cyclotron resonance [Summers et al., 1998; Katoh and Omura, 2004] and are eventually accelerated to relativistic energies [Horne et al., 2005]. The cyclotron resonant interaction is a crucial key to understanding of the acceleration process of radiation belt electrons as well as the generation mechanism of whistler-mode chorus emissions. [4] In the present study, we performed a self-consistent particle simulation with a dipole magnetic field model. The simulation reproduced chorus emissions with rising tones and revealed that the generation region of whistler-mode chorus is located close to the magnetic equator.
2. Simulation Model [5] A promising model of the generation mechanism has been proposed by Nunn [1974] who has pointed out the importance of the resonant currents formed by nonlinearly trapped resonant electrons including the effect of the mirror force due to an inhomogeneity of the background magnetic field. He also predicted that the distribution function in the resonant particle trap would be depressed. The depression, which is recently interpreted as an electromagnetic electron hole in the wave phase space by Omura and Summers [2006], results in resonant currents inducing emissions with a rising frequency. The simulation study performed by Katoh and Omura [2006] has confirmed the model and has reproduced a coherent emission with a rising tone triggered by a constant frequency wave pulse. To explain the generation mechanism of chorus emissions by the same model, we need to start the simulation with the thermal noise. One of the problems in the previous studies of chorus generation is that a coherent whistler-mode wave is assumed
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Table 1. Simulation Parametersa Parameter
Value
Time step Dt Grid spacing Dz Number of grid points Nz Number of superparticles representing energetic electrons Parallel perpendicular thermal velocities (Vth,k, Vth,?) Loss cone parameter b Density ratio between cold a nd energetic electrons at the magnetic equator Plasma frequency of the cold electrons Pe
0.01 We1 0.06 cWe1 16384 4096 Nz (0.225 c, 0.6 c) 0.5 8 104 4 We
a
The velocity distribution function of energetic electrons is given as in the work by Katoh and Omura [2006].
a priori as a seed of a chorus element for computational efficiency. The recent progress of computational resources, however, enables us to perform a large scale numerical simulation which solves Maxwell’s equations for electromagnetic field directly along with equations of motion for one hundred million particles representing energetic electrons. We use an electron hybrid code with a dipole magnetic field model [Katoh and Omura, 2006]. In this model, we assume spatially one-dimensional simulation system aligned with the external magnetic field direction with a variable z as a distance from the equatorial plane. The dipole magnetic field is locally expanded by the circular cylindrical geometry (Br, Bf, Bz) where r is a gyroradius of an electron with its guiding center on the z-axis. Assuming Bf = 0, we determine Br by Br = r/2(@Bz/@z) so as to satisfy r B = 0. We assume a constant distribution of the number density of cold electrons in space and time. To
Figure 1. (a, b) Initial distribution functions of energetic electrons in (vk, z) and (v?, z), respectively. Dashed and dash-dotted lines in Figure 1a denote resonance velocities, vr = ±(w eBz/me)/k, for frequencies w = 0.2 and 0.5 We, respectively. (c) Initial velocity distribution function in (vk, v?) at the magnetic equator. Dashed and dash-dotted lines represent resonance ellipses of w = 0.2 and 0.5 We, respectively. Five solid semicircles denote constant energy contours of 1, 10, 100 keV and 1 MeV, and the speed of light, respectively. The color scale in each plot is shown in an arbitrary unit.
reproduce whistler-mode waves propagating parallel to the dipole magnetic field, we solve purely transverse electromagnetic waves by Maxwell’s equations with the current density computed from the cold electron fluid and the energetic electrons, neglecting the longitudinal electrostatic field. The parameters used in the present simulation is given in Table 1, where We and c are the electron gyrofrequency at the equator and the speed of light, respectively. In the present simulation we assume the energetic electrons forming a highly anisotropic velocity distribution in the equatorial region, which is typical in the magnetosphere during a disturbed period. No initial wave is assumed except for electromagnetic thermal noise induced by the energetic electrons. The spatial inhomogeneity of Bz is 3.5 times larger than that of the realistic dipole field corresponding to that of L = 4 of the Earth’s magnetosphere, because we need to scale down the simulation system in order to reduce the computational time. The computational time used in the present study is two weeks with 512 CPUs of supercomputers (Fujitsu PRIMEPOWER HPC2500) of Kyoto University. [6] As for the initial settings of the simulation, we load energetic electrons having an anisotropic loss cone distribution as assumed by Katoh and Omura [2006], which drives an instability generating narrow band whistler-mode
Figure 2. Spatial distribution of frequency spectra of the wave electric field at different times. The arrow in magenta shows the edge of the same NEWNFG emerging at a higher frequency in each plot.
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Figure 3. Chorus emissions in the simulation result. Spectrograms observed at fixed point (z = 103 and +92 cWe1) in (a) the northern hemisphere and (b) the southern hemisphere in the simulation. Several chorus emissions are generated in both hemispheres in the simulation system. The lower cutoff frequency of the emissions in the present simulation result is attributed to the lower limit of the linear growth of whistler-mode waves under the present parameters. The generation of chorus emissions is gradually quenched because the anisotropic distribution of energetic electrons is relaxed through repeated generation processes of whistler-mode waves.
waves, and supply fresh electron fluxes from the boundaries so as to realize an open system aligned with a geomagnetic field line. In Figure 1, we show initial spatial distributions of energetic electrons in (vk, z) and (v?, z) and the initial velocity distribution function at the magnetic equator. In the absence of waves, the trajectory of the energetic electrons follows adiabatic motion due to the mirror force in the inhomogeneous magnetic field. The linear theory gives the positive growth rate g > 103We within the frequency range w = 0.14 0.5We under the present parameters of energetic electrons at the magnetic equator. The maximum linear growth rate of 2.67 103We is found at the frequency of 0.27We. For the background magnetic field intensity at the magnetic equator of L = 4 (486 nT), the linear growth
rate corresponds to 1982 dB/s. We should note that we have to assume a higher temperature anisotropy to obtain faster growth of whistler-mode waves in the rescaled system.
3. Generation Process of Whistler-Mode Chorus [7] In the early stage of the simulation run, the narrow band whistler-mode waves are generated in the equatorial region of the simulation system, propagating away from the equator along the magnetic field line. The generated whistler-mode waves interact with the counter-streaming energetic electrons. The time series of spatial distributions of wave spectra is shown in Figure 2. In Figure 2, we find a characteristic spectrum structure showing a narrowband
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Figure 4. Spatial profile of wave amplitude BW, normalized by the external magnetic field at the equator B0, and its time evolution. Wave packets of whistler-mode waves are successively generated from the equator with their BW growing through propagation in both hemispheres. We also find that narrowband emissions of small BW, seeds of NEWNFGs, appear from slightly upstream of the equator.
emission with negative frequency gradient (NEWNFG). As denoted by arrows in Figure 2, the new edge of the NEWNFG is continuously generated at increasing frequencies in the region very close to the magnetic equator. The frequency of the generated NEWNFG is almost unchanged during its propagation, indicating frequency rising occurs close to the magnetic equator. The frequency variation of the NEWNFG can be described by [after Omura et al., 1991] @w @w m @ vg JB ; þ vg ¼ 0 2 @t BW @t @z
ð1Þ
where vg is the group velocity of the whistler-mode wave. The frequency variation becomes small after the wave growth due to the large wave amplitude BW. Each NEWNFG is periodically generated with a frequency gap dw. The typicalpamount of dw is 0.1 We while the trapping frequency ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wT (= kv?0 eBW =me ) for an electron having 90 keV energy is estimated to be 0.026We, where k, v?0, e, and me are the wave number, the perpendicular velocity, the charge and the rest mass of a trapped electron, respectively. [8] In Figures 3a and 3b we show the wave spectrum observed at z = 103 and +92 cW1 e . We find that multiple narrowband emissions with rising tones are observed in both hemispheres in the simulation system. The typical frequency sweep rate is estimated to be 4.98 105 W2e , corresponding to 9.23 kHz/s. Figure 4 shows that the generation region of the narrowband emissions is located close to the magnetic equator and that the emissions are propagating away from the magnetic equator. We also find in Figure 4 that small amplitude narrowband emissions which become seeds of NEWNFGs appear from slightly upstream of the magnetic equator, showing two oppositely traveling waves coexist around the magnetic equator. These results are consistent with recent observations by Cluster satellites showing that the propagation direction of chorus emissions reverses at the region close to the magnetic equator [Santolı´k et al., 2003, 2004].The wave magnetic
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field of the emissions in the present simulation is order of 103 of the background magnetic field intensity. [9] As the amplitudes of the whistler-mode waves grow during their propagation from the equator, a nonlinear effect arises and the trajectories of energetic electrons are significantly modified through the nonlinear resonant trapping. The nonlinear resonant trapping of each resonant electron is explained by the combined effect of the mirror force of the inhomogeneous magnetic field and the nonlinear Lorentz force expressed by cross product of the wave magnetic field and the perpendicular velocity of each resonant electron. This nonlinear effect forms an electromagnetic electron hole [Omura and Summers, 2006] in the velocity phase space of each coherent wave packet propagating away from the equator. Appearance of the electromagnetic electron hole in the phase space induces channelling of the trajectories of resonant electrons in the specific phase range with respect to the wave magnetic field resulting in formation of nonlinear resonant currents [Katoh and Omura, 2006]. The phase relation between the resonant currents and the wave electromagnetic field is fixed at the phase range inducing wave growth (due to negative E J) and frequency increase (due to negative B J) [Nunn, 1974; Omura et al., 1991]. The resonant currents result in a NEWNFG, which propagates away from the equator, with progressive tail formation with the increasing frequency in the region close to the equator. The tail formation stops when the frequency exceeds the range with positive linear growth rates. The formation process of a NEWNFG is repeated at the lower frequency with a certain frequency gap below the preceding NEWNFG. The whole generation process stops when the temperature anisotropy of the equatorial velocity distribution function becomes sufficiently small because of the pitch angle diffusion through the chorus wave generation. The correspatial scale of the generation region ±100 cW1 e sponds to ±350 km. Since the gradient of the magnetic field is 3.5 times larger than the Earth’s dipole model, the generation region is 3.5 times larger, i.e., 350 cW1 e 1200 km.
4. Summary [10] We reproduced whistler-mode chorus emissions with rising tones by a self-consistent particle code with an inhomogeneous magnetic field. The present study shows that NEWNFGs are formed in the simulation system and that the trailing edges of NEWNFGs evolve close to the magnetic equator during their propagation away from the magnetic equator toward both hemispheres. The generation process of whistler-mode chorus is explained by the role of nonlinear resonant currents generating an emission with a rising tone. The nonlinear effect is also important in understanding dynamics of high energy electrons in the Van Allen belts because the nonlinear trapping of resonant electrons by a coherent whistler-mode wave results in an effective acceleration of the trapped particles [Omura and Summers, 2006]. While the majority of resonant electrons lose energy in generating the whistler-mode waves, a fraction of particles are accelerated by the waves, contributing to formation of relativistic electron flux in the Van Allen belts.
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[11] Whistler-mode chorus emissions are commonly observed in the magnetospheres of the magnetized planets, such as Jupiter [Scarf et al., 1979] and Saturn [Gurnett et al., 1981]. Since the generation process of whistler-mode chorus emissions reproduced in the present study is related to the dynamics of mirroring electrons in a dipole magnetic field, the present simulation encompasses the fundamental physics of the wave generation process in a magnetized planet. [12] Acknowledgments. Computation in the present study was performed with the KDK system of RISH and ACCMS (Academic Center for Computing and Media Studies) at Kyoto University. This work was partially supported by Kyoto University Active Geosphere Investigations for the 21st Century Centers of Excellence Program (KAGI21) and Grantin-Aid 17340146 and 17GS0208 (for Creative Scientific Research ‘‘The Basic Study of Space Weather Prediction’’) of the Ministry of Education, Science, Sports and Culture of Japan. Y.K. is supported by a research fellowship of the Japan Society for the Promotion of Science for Young Scientists.
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Y. Katoh, Planetary Plasma and Atmospheric Research Center, Graduate School of Science, Tohoku University, Sendai, Miyagi 980-8578, Japan. (
[email protected]) Y. Omura, Research Institute for Sustainable Humanosphere, Kyoto University, Uji, Kyoto 611-0011, Japan.
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