Test Particle Simulations of Interaction Between Monochromatic Chorus Waves and Radiation Belt Relativistic Electrons

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1 DOI /s ORIGINAL ARTICLE Test Particle Simulations of Interaction Between Monochromatic Chorus Waves and Radiation Belt Relativistic Electrons Zhonglei Gao Hui Zhu Lewei Zhang Qinghua Zhou Chang Yang Fuliang Xiao Received: 4 January 2014 / Accepted: 18 February 2014 Springer Science+Business Media Dordrecht 2014 Abstract Chorus waves have been suggested to be effective in acceleration of radiation belt electrons. Here we perform gyro-averaged test-particle simulations to calculate the bounce-averaged pitch angle and energy diffusion coefficients for parallel-propagating monochromatic chorus waves, and perform a comparison of test-particle TP) model with quasi-linear QL) theory to evaluate the influence of nonlinear processes. For small amplitude chorus waves, the diffusion coefficients of TP and QL models are in good agreement. As the wave amplitude reaches a threshold value, two nonlinear processes phase trapping and phase bunching) start to occur, especially at large equatorial pitch angles. Phase trapping yields rapid increases in pitch angle and kinetic energy. In contrast, phase bunching causes overall decreases in pitch angle and kinetic energy. For the waves with amplitudes slightly above the threshold value, the average behavior is dominated by the phase trapping, and TP diffusion coefficients are larger than QL ones. As wave amplitude increases, TP diffusion coefficients become smaller than QL ones, indicating that phase trapping gradually reduces the dominance over phase bunching. Keywords Nonlinear wave-particle interaction Test-particle simulation Radiation belts Electron acceleration Z. Gao L. Zhang Q. Zhou C. Yang F. Xiao B) School of Physics and Electronic Sciences, Changsha University of Science and Technology, Changsha, Hunan, , China flxiao@126.com H. Zhu CAS Key Laboratory of Geospace Environment, Department of Geophysics and Planetary Sciences, University of Science and Technology of China, Hefei, Anhui, , China 1 Introduction Whistler-mode chorus waves are important electromagnetic waves in the inner magnetosphere dynamics Horne et al. 2003, 2005; Summers et al. 2004, 2007a; Thorne 2010;Xiao et al. 2010; Su et al. 2011b). These waves are usually distributed outside the plasmasphere and in the duskside region, present as a combination of coherent, discrete monochromatic elements Santolik et al. 2004). Their typical angular frequency is about several khz Meredith et al. 2000), slightly lower than electron gyrofrequency. Statistics results indicate that the typical amplitude of chorus waves is pt Meredith et al. 2001, 2003). Previous works suggest chorus waves to be responsible for precipitation of lowenergy electrons Ni et al. 2008; Suetal.2009d, 2010a). Chorus waves Albert 2005; Summers et al. 2007b;Suetal. 2009b,c; Xiao et al. 2009), together with other VLF Su et al. 2011a; Su and Zheng 2009a) and ULF Zong et al. 2007, 2009) waves, have been suggested to control dynamics of radiation belt energetic electrons. Quasi-linear theory QL) has been widely used to describe these wave-particle interactions Kennel and Engelmann 1966; Horne and Thorne 1998; Summers et al. 1998). Several quasi-linear kinetic radiation belt models have been build, such as RAM Beutier and Boscher 1995), RBE Jordanova and Miyoshi 2005), VERB Shprits et al. 2009) and STEERB Su et al. 2010b; Xiao et al. 2010b). One of the fundamental assumptions of QL theory is that electromagnetic waves have small amplitudes. For waves with large amplitudes, nonlinear processes occur and QL theory becomes inappropriate to describe wave-particle interaction. Both the test-particle TP) method and Hamilton theory have suggested two types of nonlinear processes Albert 2002; Bortnik and Thorne 2010; Liu et al. 2012), phase trapping and phase bunching. The Electromagnetic ion cyclotron EMIC) waves with typical amplitude 1 10 nt can

2 nonlinearly interact with radiation belt electrons Albert and Bortnik 2009) and ring current ions Zhu et al. 2012), and the corresponding TP transport coefficients have been found to significantly deviate from the prediction of QL theory Su et al. 2012, 2013). Recently, numerous events of chorus waves with large amplitudes have been reported Parrot and Gaye 1994; Cattell et al. 2008; Cully et al. 2008), and nonlinear effects of large amplitude chorus waves on radiation belt electron have been demonstrated Bortnik and Thorne 2008). Using TP method, Zheng et al. 2012) focused on the cumulative nonlinear effect of chorus wave over multiple electron bounce periods and calculated the transport coefficients at indicated energy and pitch angle. In their extended work Zheng et al. 2013), they considered the obliquely propagating chorus waves and found that the corresponding high-order resonances can have significant impacts on the major resonance. Moreover, two mechanisms are proposed for accelerating high-energy electrons by coherent whistler mode waves: relativistic turning acceleration RTA) Omura et al. 2007) and ultra-relativistic acceleration URA) Summers and Omura 2007). However, comprehensive and quantitative evaluations on nonlinear interactions between monochromatic chorus waves and radiation belt electrons has seldom been reported so far. In this study, we adopt TP method to investigate the nonlinear mechanisms of relativistic electrons driven by monochromatic parallelpropagating chorus waves. Firstly, we shall exhibit three cases of electrons trajectories, which correspond to linear behavior, phase trapping and phase bunching, respectively. Secondly, we shall calculate the statistical diffusion coefficients via TP method, and compare with those from QL theory. Finally, we shall discuss the dependence of TP and QL diffusion coefficients on the wave amplitude. 2 Model 2.1 Test-Particle Model In this study, we adopt a dipole background model to investigate nonlinear wave-particle interactions at the center of outer radiation belt L = 5 in a simple hydrogen plasma. We choose the typical value ω p0 = 3 Ω e0, where, ω p0 and Ω e0 denote the equatorial plasma frequency and the gyrofrequency of electrons. We use a field-aligned electron density model n e = 5.45 cos 4 λcm 3 Denton et al. 2002). The monochromatic parallel-propagating chorus wave with frequency ω = 0.15 Ω e0 is assumed to propagate with constant amplitude from the equator towards higher latitude λ < 35 ) Xiao et al. 2009, 2010b;Suetal.2011b). Following previous works Zhu et al. 2012; Suetal. 2012, 2013), gyro-averaged equations of motions for electrons with the rest mass m e and the charge q = e can be written as dp dp = qb w p sin η ω = qb w k p ω k p dη = qb w p p2 B s, 1) 2 B ) sin η + p p 2 B ) cos η + B s, 2) ω kp Ω ) e, γ 3) ds = p, 4) where γ is the relativistic factor, p and p denote the parallel and perpendicular components of electron momentum, η is the wave-particle phase, Ω e is the local gyrofrequency of electrons, s is the field line length, k is wave number, B w and B represent the wave amplitude and background geomagnetic field strength. Combining Eqs. 1) and 2), the change rates of equatorial pitch angle α eq and kinetic energy E k can be obtained Zhu et al. 2012;Suetal.2013) dα eq de k = qb w p 2 = qb w ω k tan α eq tan α [ ω k p ) ] p p2 sin η, 5) p sin η. 6) When the righthand of Eq. 3) equals zero and the term on the order of B w /B is ignored, we can obtain the cyclotron resonance condition ω kp = Ω e γ. 7) At the resonance location, the equatorial pitch angle change becomes Zhu et al. 2012;Suetal.2013) dαeq ) R = qb w tan α eq γp 2 Ω ) e p + p2 sin η. 8) tan α k m e Base on Eqs. 1) 4), initializing the equatorial pitch angle α eq, kinetic energy E k and wave-particle phase η 0,the electron trajectories can be calculated. At any values of α eq and E k, test electrons with η 0 uniformly distributed in the range [0, 360 ] are launched from the northern hemisphere mirror points toward the equator. Following the electron trajectories, the transit time t TP from the mirror point to equator, about one quarter of the bounce period), the final equatorial pitch angle αeq TP, the final energy ETP, can be obtained. Using above quantities, the statistical expressions of bounce-averaged TP pitch angle and energy diffusion co-

3 Fig. 1 In the case of initial α eq = 10 and E k = 1MeV, evolution of a) equatorial pitch angle α eq and b) kinetic energy E k along the latitude λ;the dependence of net changes of pitch angle α eq and kinetic energy E k on η 0 ; variation of e) wave-particle phases η with λ.infigs.1c d, the dotted and dashed lines represent the mean changes and the mean changes ± one standard deviation. In Figs. 1e, the dashed line represent the linear resonance locations determined by Eq. 7). Colors are coded according to values of η 0 efficients can be written as Su et al. 2012) α eq α eq = D TP DEE TP = α TP eq αtp eq 2 t TP ) 2 ) 2 E TP E TP 2 t TP E TP2, 9), 10) where the... represents the mean value over the initial phase. 2.2 Quasi-Linear Model Following the previous work Su et al. 2012), we obtain the bounce-averaged QL pitch angle and energy diffusion coefficients for the monochromatic chorus wave: Dα QL eq α eq = π Ω e 2 cos α cos 7 λ 4γ 2 Tα eq ) cos 2 α eq B2 w B 2 1 γm ) e ω 2 p k cos α D QL EE =π Ω e 2 kp λ + Ω e γ 1 + 3sin 2 λ cos λγ + 1) 2 4γ 4 Tα eq ) cos α ) 1, 11) ) B2 w γme ω 2 B 2 p k sin α kp + Ω ) e 1, 12) λ γ

4 Fig. 2 Same as Fig. 1 except for the initial α eq = 25. Note: the evolution of η in the region λ<25 is shown only for the trapped electrons where all quantities with subscript eq are evaluated at the equator, and the other quantities are evaluated at the resonance location. 3 Result 3.1 Phase Trapping and Phase Bunching Here, we primarily simulate 1 MeV electron motions for the wave amplitude B w = 1 nt for three indicated initial pitch angles: α eq = 10, 25 and 70. Then we demonstrate the corresponding variation of α eq and E k with the latitude λ, the dependence of net changes of α eq and E k on η 0, and the evolution of wave-particle phase η. As shown in Fig. 1 the initial α eq = 10 ), the pitch angle and energy of electrons are scattered when electrons go through the resonance location, and net changes of the equatorial pitch angle α eq and the kinetic energy E k vary with η 0 in a sine form. The actual resonance locations where dη/ = 0) and the linear one determined by Eq. 7) are well coincident with each other, and the trajectories in the η λ plane are quite uniformly distributed. These results indicate that this case can be considered as a normal diffusion process, consistent with the description of QL theory. Figure 2 shows the case with initial α eq = 25.Most of electrons experience the phase bunching with resonance phases around 40, whose pitch angles and energies show a systematic decrease. The electron with η 0 = 180 experiences the phase trapping with resonance phases in the range [100, 300 ], whose pitch angle and energy exhibit a rapid increase. Obviously, two different nonlinear mechanisms lead to opposite changes of α eq and E k. Moreover, phase trapping causes large standard deviations of α eq and E k, leading to large diffusion coefficients see Eqs. 9) and 10)). Figure 3 presents the case with initial α eq = 70. Similarly, all the test electrons experience phase bunching with

5 Fig. 3 Same as Fig. 1 except for the initial α eq = 70 the bunching phases in the range [0, 180 ]. Therefore, the net changes α eq and E k are negative. Clearly, the small deviations of α eq and E k induced by phase bunching correspond to small pitch angle and energy diffusion coefficients, respectively. 3.2 Diffusion Coefficients of TP and QL Models After obtaining the trajectories, we shall calculate the diffusion coefficients of TP and QL models to quantify the influence of nonlinear interaction. Variations of diffusion coefficients of TP and QL models with initial α eq at different energies and wave amplitudes are presented in Fig. 4. The black and red dashed lines represent the diffusion coefficients of TP and QL models for B w = 0.01 nt chorus wave, respectively. Clearly, TP diffusion coefficients are consistent with QL diffusion coefficients at any energy, indicating that the chorus wave is too weak to produce nonlinear processes. The black and red solid lines show the diffusion coefficients of TP and QL models for B w = 1 nt chorus, respectively. For E k = 500 kev, as the pitch angle increases, TP diffusion coefficients decrease with oscillation, while QL diffusion coefficients monotonously increase with pitch angle. In the pitch angle range [10, 42 ], the TP diffusion coefficients are mostly higher than QL ones due to the dominance of phase trapping, but the situation is opposite in the pitch angle range [43, 80 ] since phase trapping occurs more rarely at higher initial α eq. The variation trends of the diffusion coefficients at E k = 1 MeV are similar to those at E k = 500 kev except that TP and QL diffusion coefficients are still close in the pitch angle 15. This indicates the absence of nonlinear processes in the range. In particular, at E k = 2 MeV, energy diffusion coefficients show larger oscillations than pitch angle diffusion coefficients probably due to the RTA effect Omura et al. 2007).

6 Fig. 4 Comparison among three types of pitch-angle diffusion coefficients top) and energy diffusion coefficients bottom) ate k = 500 kev left),1 MeVmiddle) and 2 MeVright). The colors are coded according to different models. The cyan lines represent mean TP diffusion coefficients over 10 points 3.5 ). The dashed and solid lines represent the diffusion coefficients for B w = 0.01 and 1 nt, respectively 3.3 Dependence on Wave Amplitude Figure 5 exhibits the averaged values of diffusion coefficients by TP circles) and QL dashed lines) models at three energies over the pitch-angle regions: 20 30,40 50 and Obviously, when the wave amplitude B w is relatively small, similar to QL cases, both pitch angle and energy diffusion coefficients grow linearly with B 2 w. When B w reaches a threshold value about 0.1 nt, nonlinear phase trapping starts and TP coefficients become higher than QL ones. When B w further increases, TP coefficients become smaller than QL ones for energies of 0.5 and 1 MeV. Those complex dependence results from the occurrences of phase trapping and phase bunching which have the opposite effect on the diffusion coefficients, viz., phase trapping dominates the average behavior for the wave amplitude slightly beyond the threshold but lose the dominance over phase bunching for the wave amplitude far above the threshold. However, for energy 2 MeV, similar to the case in Fig. 4, when B w increases beyond the threshold, energy diffusion coefficients display more dramatic oscillations than pitch angle diffusion coefficients potentially due to the RTA effect Omura et al. 2007). Furthermore, the threshold value decreases if the pitch angle range increases, implying that nonlinear mechanism tends to occur at high pitch angles. 4 Conclusion Quasi-linear chorus-electron interaction has been long considered to be efficient in acceleration of radiation belts electrons. Recent works have investigated the nonlinear interaction between chorus waves and electrons Bortnik and Thorne 2008; Omura et al. 2007). In this study, we calculate the diffusion coefficients via TP model and compare with those via QL model. Numerical results reveal that phase trapping can accelerate electrons more effectively than quasi-linear interaction, in the meanwhile, phase bunching can produce an overall decrease in pitch angle α eq and kinetic energy E k. For the small wave amplitude B w, TP and QL diffusion coefficient agrees well with each other, increasing linearly with B 2 w.forb w reaches a threshold value, nonlinear phase trapping occurs and TP coefficient starts to deviate from QL coefficients. The threshold value is smaller at larger initial α eq and smaller energy, suggesting

7 Fig. 5 Wave-amplitude dependence of bounce-averaged test-particle circles) and quasi-linear dashed lines) pitch angle diffusion coefficients top) and energy diffusion coefficients bottom) fore k = 500 kev left), 1 MeV middle) and 2 MeVright) electrons. The colors are coded according to the mean diffusion coefficients in different pitch angle ranges that nonlinear mechanism prefers to occur at high pitch angles. Furthermore, phase trapping dominates the average behavior over phase bunching if the wave amplitude is slightly above the threshold, allowing TP diffusion coefficients to be larger than QL ones. However, as wave amplitude continues to increase, phase trapping gradually reduces the dominance over phase bunching, allowing TP diffusion coefficients to be smaller than QL ones. Acknowledgements This work is supported by the National Natural Science Foundation of China grants , , , the Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province, and the Construct Program of the Key Discipline in Hunan Province. References Albert, J.M.: Geophys. Res. Lett. 298), ) Albert, J.M.: J. Geophys. Res. 110, A ) Albert, J.M., Bortnik, J.: Geophys. Res. Lett. 36, L ) Beutier, T., Boscher, D.: J. Geophys. Res. 100, ) Bortnik, J., Thorne, R.M.: J. Geophys. Res. 115, A ) Bortnik, J., Thorne, R.M.: Geophys. Res. Lett. 35, L ) Cattell, C., Wygant, J.R., Goetz, K., et al.: Geophys. Res. Lett. 35, L01, ) Cully, C.M., Bonnell, J.W., Ergun, R.E.: Geophys. Res. Lett. 35, L17S ) Denton, R.E., Goldstein, J., Menietti, J.D.: Geophys. Res. Lett. 2924), ) Jordanova, V.K., Miyoshi, Y.: Geophys. Res. Lett. 32, L ) Horne, R.B., Thorne, R.M.: Geophys. Res. Lett. 25, ) Horne, R.B., Glauert, S.A., Thorne, R.M.: Geophys. Res. Lett. 309), ) Horne, R.B., Thorne, R.M., Glauert, S.A., et al.: J. Geophys. Res. 110, A ) Kennel, C.F., Engelmann, F.: Phys. Fluids 9, ) Liu, K., Winske, D., Gary, S.P., et al.: J. Geophys. Res. 117, A ) Meredith, N.P., Horne, R.B., Johnstone, A.D., et al.: J. Geophys. Res. 105, ) Meredith, N.P., Horne, R.B., Anderson, R.R.: J. Geophys. Res. 106, ) Meredith, N.P., Horne, R.B., Thorne, R.M.: Geophys. Res. Lett. 30, ) Ni, B., Thorne, R.M., Shprits, Y.Y., et al.: Geophys. Res. Lett. 35, L ) Omura, Y., Furuya, N., Summers, D.: J. Geophys. Res. 112, A ) Parrot, M., Gaye, C.A.: Geophys. Res. Lett. 21, ) Santolik, O., Gernett, D.A., Pickett, J.S., et al.: Geophys. Res. Lett. 31, L ) Shprits, Y.Y., Subbotin, D., Ni, B.: J. Geophys. Res. 114, A ) Su, Z.P., Zheng, H.N.: Chin. Phys. Lett. 26, a)

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