The effects of amplitude modulation on nonlinear interactions between a parallel propagating chorus wave packet and electrons in a dipole field are investigated in this work using a test particle code. Previous research used a single wave to model a chorus wave element, leading to nonlinear processes such as phase trapping and bunching when the wave amplitude was large enough. However, high resolution observations of chorus wave packets show a modulation of the wave amplitude, forming the so-called chorus subpackets or subelements. Here we first extend a previous method to model a realistic chorus packet. Using the modeled chorus packet, we demonstrate directly that including the realistic amplitude modulation could significantly affect the behavior of resonant electrons. Our results suggest that the amplitude modulation should be considered in the quantitative treatment of interactions between electrons and chorus waves.
 Whistler mode chorus waves are electromagnetic emissions with discrete rising or falling tones. These waves have been shown to be very important in both the acceleration [e.g., Summers et al., 1998; Horne et al., 2005a] and the precipitation [Thorne et al., 2005] of relativistic radiation belt electrons. Chorus waves have also been demonstrated to be the dominant wave mode for scattering plasma sheet electrons into the atmosphere to form the diffuse aurora [Thorne et al., 2010], the pulsating aurora [Nishimura et al., 2010] and to create the pitch angle distributions of energetic electrons observed in the inner magnetosphere [Tao et al., 2011a].
 The effects of chorus waves on energetic electrons have been mainly modeled by quasilinear theory, which assumes wideband and small amplitude wave fields [e.g., Horne et al., 2005b]. High resolution observations of chorus waves, however, have revealed the discrete and coherent nature of chorus wave packets [see, e.g., Santolík et al., 2004; Tsurutani et al., 2009]. Tsurutani et al.  also showed that the amplitude of a chorus packet is generally larger than the averaged amplitude that has been used in the quasilinear modeling of radiation belts. The large amplitude and the discrete and coherent nature of chorus waves have led to the question of the applicability of quasilinear theory to the interactions between electrons and chorus waves.
 Nonlinear theories have also been developed to describe, in an inhomogeneous field, interactions between charged particles and a single coherent wave [Dysthe, 1971; Albert, 1993], which might be a better approximation to a chorus packet. These single-wave nonlinear theories predict nonlinear phase bunching and phase trapping of resonant particles, which have been suggested to play an important role in the radiation belt electron dynamics [Albert, 2002; Furuya et al., 2008; Bortnik et al., 2008]. However, the single-wave theories do not take into account the modulation of chorus amplitude, which was shown recently by high resolution observations [Santolík et al., 2004; Tsurutani et al., 2009]. The modulation of the wave amplitude leads to structures called chorus subpackets [Santolík et al., 2004; Tsurutani et al., 2009], whose effects on linear pitch angle scattering have been studied by Lakhina et al. . We consider in this work the effects of including subpackets on the nonlinear interactions between electrons and chorus waves using a test particle simulation.
 We first introduce an extended method to model a parallel propagating chorus wave packet with rising frequency and realistic amplitude modulation in Section 2. We then use the model to demonstrate that including the amplitude modulation can significantly affect the nonlinear interactions between electrons and a chorus wave packet in Section 3. Our work is summarized in Section 4.
2. Method of Modeling a Realistic Chorus Packet
 Chorus waves are generally believed to be generated near the equatorial plane, and then propagate toward higher latitudes [e.g., Helliwell, 1967]. Two main features of chorus waves are that they are coherent and the central frequency of the wave packet varies with time if observed at a given latitude. In this work, the frequency ω and the wave phase φ of a parallel propagating (wave normal angle equals 0) chorus wave packet are described by the following wave equations, as suggested by Furuya et al. ,
for z ≠ 0, and for z = 0, ∂ω/∂t = Γ and ∂φ/∂t = −ω, where z the distance along the background magnetic field line from the equatorial plane. Here vg(ω) is the group velocity at frequency ω, and k is the wave number from cold plasma wave theory [Stix, 1992]. The frequency sweep rate Γ of a chorus packet is assumed to be independent of the wave frequency in this work for simplicity and has a value of (fmax − fmin)/τ, where fmax is the maximum frequency, fmin is the minimum frequency, and τ is the duration of the chorus packet observed. Numerically, the advection equation for ω in Equation (1) is solved by the method of characteristics. We select a number of frequencies between ωmin (≡ 2πfmin) and ωmax (≡ 2πfmax), and solve for their z coordinates according to dz/dt = vg after they are generated at z = 0, because the wave frequency ω is constant along its characteristic curve. The frequency at any given value of z at a given time is then obtained from the coordinates of the selected frequencies by interpolation. All other equations for ω and φ are solved by normal finite difference methods. The wave magnetic field (Bw) and electric field (Ew) of a parallel propagating chorus at a given latitude is then given by Bw = Bxwcosφex − Bywsinφey and Ew = −Exwsinφex − Eywcosφey, where Bxw, Byw, Exw, and Eyw are the corresponding field amplitudes and are related by cold plasma wave theory [Tao and Bortnik, 2010].
 Another important feature of a chorus wave packet, which has never been considered by previous single-wave models as far as we are aware, is the variation of the wave amplitude inside the main packet. We show inFigure 1 (left) the power spectrogram and the waveform of a typical chorus packet observed by THEMIS D satellite [Angelopoulos, 2008]. To model the amplitude modulation of this packet, we first obtain the envelope of the y-component magnetic fieldBy(t) of the chorus packet as shown by the dark red line in Figure 1 (middle left). Because the central frequency of the packet also varies with time, we combine the observed frequency ω(t) with the obtained envelope By(t) to obtain the wave amplitude Byw as a function of the wave frequency ω. The amplitudes of other field components (Bxw, Exw and Eyw) are then obtained from the cold plasma wave theory. In this work, we assume for simplicity that Byw for a given frequency ω is constant along the magnetic field line.
3. Effects of Amplitude Modulation on Nonlinear Interactions
 We simulate the electron interaction with the chorus wave packet shown in Figure 1 by solving the relativistic Lorentz equation for each test particle [Tao et al., 2011b]. The background field B0 used in our model is a simplified dipole field without field line curvature [Bell, 1984]. We use a Cartesian coordinate system and choose the z-component of the background magnetic fieldB0z(z) to be a function of z only, where z is the distance along the field line from the equatorial plane. Latitude (λ) is also used below to facilitate the interpretation of our simulation results, with dz = LRE(1 + 3sin2λ)1/2cosλdλ and . Here L is the L-shell value andRE is the Earth radius. In this section we choose L = 6.4, which is the L-shell where the chorus packet inFigure 1 was observed. The x and y components of B0 are chosen so that ∇ ⋅ B0 = 0 is satisfied. In this work, we simply use B0x = −x(dB0z/dz)/2 and B0y = −y(dB0z/dz)/2. The cold electron density is given by ne = ne0cos−4λ, following Denton et al. , with ne0 = 10 cm−3.
 We consider now the interaction between electrons and the chorus packet shown in Figure 1. The modeled chorus wave packet using the method described in the previous section is shown in Figure 1 (right). The overall agreement with observation shown in Figure 1(left) is excellent. The small differences are mainly due to two reasons: first, the observed wave packet is only quasi-parallel with an averaged wave normal angle of about 15°; second, we assumed a constant frequency sweep rate when modeling the packet. However, these small differences should not affect our conclusions below.
 To demonstrate the effects of amplitude modulation on nonlinear interactions between electrons and the chorus packet, we perform two sets of runs. In the first set, we use a constant wave magnetic field amplitude for all frequencies, correspondingly the wave packet contains no subpackets or amplitude modulation. In the second set, we use the realistic amplitude modulation as shown in Figure 1. The effects of amplitude modulation can thus be clearly seen by comparing the two sets of results.
 We use a resonant frequency fres = 980 Hz whose amplitude By = 0.30 nT, while the average amplitude of the whole chorus packet obtained from the averaged wave power spectral density is 0.11 nT. Both amplitudes are large enough to cause phase bunching and phase trapping of electrons in the runs with a constant amplitude wave. We show here only the results with By = 0.30 nT since that is the amplitude of the wave at resonance. We use 1600 electrons with their initial gyrophases evenly distributed between 0 and 2π. Electrons are launched from λ = 15° with velocity (v) toward the equatorial plane. We select three initial equatorial pitch angles (α0 = 10°, 20°, and 30°), and the energies (E = 7.8, 9.5, and 12.7 keV, respectively) of electrons are correspondingly determined to satisfy the cyclotron resonance condition ω − kv∥ = −Ωe/γ at λres = 10°. Here Ωe = qB0(λ)/mis the signed non-relativistic local electron cyclotron frequency, withq the charge, m the mass, γ the relativistic factor, and v∥ the velocity component parallel to the background magnetic field.
 We first show the detailed changes of α0 of selected 24 electrons whose gyrophases are roughly evenly distributed between 0 and 2π in Figure 2. The changes of E are similar and not shown here to simplify the illustration. In the first set of runs without the amplitude modulation (Figure 2, left), the electrons show phase bunching and phase trapping in all three cases with a wave packet having a finite frequency sweep rate, consistent with results of single-wave theories [Albert, 1993; Bortnik et al., 2008]. In Figure 2(right), we see that when including a realistic amplitude modulation, the electrons' behavior is significantly affected. In general, the amplitude modulation causes de-trapping of some initially phase-trapped electrons. The normalized distributions of equatorial pitch angle of all 1600 electrons after interactions are shown inFigure 3. In the case without amplitude modulation (Figure 3, left), the resulting electrons are clearly divided into two groups due to phase-bunching and phase-trapping [Albert, 1993]. On the other hand, the distributions of electrons are significantly changed when the realistic amplitude modulation of the chorus is included (Figure 3, right), and it may not be appropriate to separate the electron distributions into two groups.
 The main reason for the deviation of the electron distribution from previous single-wave theories is that chorus is not an ideal single wave, but has amplitude modulation. Incorporating the amplitude modulation into the quantitative determination of the long-term evolution of the energetic electron distribution is thus critically important, as shown inFigure 3. In future work, we hope to develop an analytical treatment to describe these effects, extending previous work for monochromatic waves [e.g., Albert, 2002].
4. Summary and Discussion
 In this work, we extended a method by Furuya et al. to model a chorus wave packet with realistic amplitude modulation from observation. The modeled chorus wave packet has a waveform that is consistent with observation, and it can be used to study in detail the interaction between electrons and realistic chorus packets. We then explored the effects of including chorus subpackets on nonlinear interactions between electrons and chorus waves by modeling an observed chorus wave packet and simulating the interactions between the packet and electrons using a test particle method. By comparing with simulations where we fixed the wave amplitude, we demonstrated directly that realistic chorus subpackets can significantly modify the way electrons interact with a large amplitude chorus wave packet predicted by single-wave theories. As far as we are aware, this is the first time that chorus subpackets are considered in nonlinear interactions between electrons and chorus waves.
 The subpacket structure is potentially important not just for the parallel propagating chorus packet we showed, but also for oblique chorus waves where Landau and cyclotron harmonic resonances might also be important. One main purpose of understanding the interaction processes between electrons and chorus waves is to quantify the effects of chorus waves on radiation belt electrons. Our results thus suggest including the chorus subpackets in calculation could significantly change the corresponding advection and diffusion coefficients from the single-wave nonlinear theories and should be considered in the numerical radiation belt models. Calculation of the advection and diffusion coefficients that could be used in a transport equation will be left for a future study.
 This research was supported at UCLA by NSF grant 0903802, which was awarded through the NSF/DOE Plasma Partnership program, and NASA grant NNX11AR64G. We acknowledge NASA contract NAS5-02099 and V. Angelopoulos for use of data from the THEMIS mission, specifically A. Roux and O. LeContel for use of SCM data.
 The Editor wishes to thank T. Paul O'Brien and an anonymous reviewer for their assistance evaluating this paper.