We perform test particle simulations assuming whistler mode chorus wave packets that are generated at the geomagnetic equator propagate away from the equator in both poleward directions. While electrons in the energy range 10–100 keV are primarily responsible for the generation of chorus waves through pitch angle diffusion into the loss cone, it has been found that a fraction of the higher-energy electrons from a few hundred keV to a few MeV are effectively accelerated by chorus due to a special nonlinear trapping process called relativistic turning acceleration (RTA). This mechanism has been recently demonstrated for a coherent whistler mode wave packet with a constant amplitude and constant frequency. In the present study we confirm that the RTA process takes place for a wave packet with variable frequency such as that occurring in a rising tone of chorus emissions. We study the efficiency of the RTA process for different particle energies. A Green's function method is used to describe the evolution of the particle energy distribution function. The RTA process due to chorus emissions creates a high-energy tail in the electron energy distribution function. The shape of the high-energy tail is determined by the distribution function of the seed electrons in the lower-energy range. RTA can accelerate electrons in a much shorter timescale than that estimated by quasi-linear diffusion theory, e.g., it typically takes tens of minutes to hours for ∼100 keV seed electrons to be accelerated to energies of a few MeV by RTA.
 Gyroresonant interaction with whistler mode chorus can generate relativistic electrons in the Earth's radiation belt [Summers et al., 1998, 2002, 2004; Roth et al., 1999; Summers and Ma, 2000; Meredith et al., 2002; Albert, 2002; Miyoshi et al., 2003; Horne et al., 2005; Varotsou et al., 2005; Summers, 2005; Shprits et al., 2006]. Timescales for electron acceleration by chorus diffusion can be estimated by using quasi-linear theory, (e.g., see Summers , Summers et al. [2007a, 2007b], and references therein). According to quasi-linear diffusion theory, it typically takes one day to a few days for seed electrons of energies of order 100 keV to be accelerated to energies of a few MeV. Recently Omura et al.  discovered a very efficient acceleration mechanism for accelerating relativistic electrons by chorus-mode waves called relativistic turning acceleration (RTA). Omura et al.  derived the relativistic second-order resonance condition for whistler mode wave-electron interaction in an inhomogeneous magnetic field, which is fundamental to the RTA process. Resonant electrons satisfying the second-order resonance condition undergo acceleration when the particle is interacting with a whistler mode wave propagating away from the equator. Since the accelerated electrons become progressively heavier because of the relativistic effect, the cyclotron frequency becomes progressively smaller and approaches the wave frequency. There then occurs a simple cyclotron resonance without any Doppler-shift, i.e., the resonance velocity becomes zero, resulting in a change of direction of the particle motion. When this turning occurs before the resonant electrons reach the equator, the acceleration mechanism becomes particularly efficient. Omura et al.  referred to this acceleration process as relativistic turning acceleration (RTA). Omura et al.  derived a simple analytical formula for the energy increase achieved by the RTA process. This formula was verified by Omura et al.  using a test particle simulation devised by Omura and Summers . The timescale to accelerate ∼100 keV seed electrons to energies of a few MeV by RTA is typically of order tens of minutes to hours. Chorus emissions are observed as a series of coherent rising tones [e.g., Santolik et al., 2004; Kasahara et al., 2005]. The generation process of chorus emissions has been reproduced in a self-consistent particle simulation by Katoh and Omura [2007a]. The simulation clarifies that each chorus element is continuously generated at the equator with a gradual frequency variation in the range 0.1-0.5 ΩEQ, where ΩEQ is the equatorial cyclotron frequency. In the simulation model of Katoh and Omura [2007a] the chorus emissions grow spontaneously from electromagnetic thermal noise. This is different from the previous simulation studies that assume an initial coherent whistler mode wave [Nunn, 1974; Omura and Matsumoto, 1982; Nunn et al., 1997; Katoh and Omura, 2004, 2006a, 2006b].
 In this paper, we study the dynamics and acceleration of relativistic electrons in the presence of chorus waves near the equator. To incorporate into the previous test particle simulations [Omura and Summers, 2006] the effect of chorus elements generated at the equator, we set up a model of chorus wave generation and propagation near the equatorial region by solving the wave equation for a whistler mode wave packet. Since the RTA theory presented by Omura et al.  assumed a whistler mode wave with a constant frequency, we need first to confirm that RTA is possible for a whistler mode chorus element with a variable frequency. Having demonstrated that the RTA process is effective for waves of variable frequency, we then determine the time evolution of the electron energy distribution function. We do this over a long period of repeated interactions between the chorus elements and seed electrons. We thereby demonstrate the formation of a high-energy tail in the electron distribution function.
 In section 2 we describe the chorus element model assumed in the test particle simulations, and confirm that the RTA process takes place for a chorus wave packet. We also compare the efficiency of RTA for the cases of constant-frequency whistler mode waves and a time-varying wave packet. In section 3 we describe our numerical method to simulate the evolution of the distribution function of resonant electrons interacting with a chorus element. The method is based on the use of a Green's function technique. In section 4 we discuss the results of the present study.
2. Relativistic Turning Acceleration With a Variable Frequency Wave
 The relativistic turning acceleration (RTA) process has been studied for a wave packet with a fixed frequency [Omura et al., 2007]. We now examine if the RTA process can take place for a chorus element with a rising frequency. We assume that the wave amplitude Bw (t, h) and wave frequency ω (t, h) satisfy the wave equations [Omura et al., 1991],
Here, Vg is the group velocity given by
c is the speed of light, ξ is a parameter which is a function of ω(t, h) and Ωe (h) given by
and ωpe is the electron plasma frequency, assumed constant in the present study.
 Observations by Cluster [Parrot et al., 2003] confirm that chorus waves propagate away from the geomagnetic equator into the northern and southern hemispheres. Moreover, the source region of chorus emissions is observed to be located close to the geomagnetic equatorial plane. To simplify the simulation model, we assume that wave packets are generated symmetrically with respect to the geomagnetic equator due to the nonlinear whistler mode instability occurring at the equator. We assume that the wave amplitude Bw = 100 pT, and that the wave frequency ω changes from 0.1 to 0.45 ΩEQ (1.4 to 6.3 kHz) with a constant sweep rate dω/dt = 10 kHz/s, with ωpe = 2.0 ΩEQ. The frequency sweep rate is a typical value at L = 4 as observed by Cluster [Santolik et al., 2004; Breneman et al., 2007]. We assume that instability continues until t = t1 < t2, and that the waves are generated at the equator. During the time interval t1 < t < t2, we assume that the waves are linearly attenuated to zero, i.e.,
where T is the time when the wave tail reaches the absorbing boundary. We call this period one cycle. We set t1 = 0.50 s and t2 = 0.56 s. The wave frequency at the equator ω(t, 0) is assumed to increase linearly in time until t = t2, namely, we put ω(t1, 0) = 0.45 ΩEQ and ω(t2, 0) = 0.5 ΩEQ.
 We now describe the method to update Bw, ω, k, and Vg defined at each grid point. Converting (1) and (2) into difference equations using an upwind method, we obtain,
where the subscript “m” denotes a grid point, and the superscripts denotes times. From (6) and (7) we obtain the wave amplitude and frequency at position hm and time tn. It should be noted that Δh and Δt are chosen as 1.0 and 0.25, respectively, to satisfy the Courant condition Vg < Δh/Δt. Substituting into the whistler mode dispersion relation
and formula (3), we obtain the wave number and group velocity
 The wave phase is first advanced at the equator h = 0 by using
where ωEQ(tn) is the equatorial wave frequency. The phases on the spatial grids are determined by the recursive formula of phase integration given by
After interpolating the wave number linearly, we integrate the result, and hence obtain the wave phase at the particle position h as
The wave electric field is obtained from the Maxwell equation k × Bw = ωEw. Figure 1 shows a wavefield model in which wave packets of chorus elements are generated at the equator and propagate in both northward and southward directions.
 We choose a group of electrons with the initial energy K = 2715 keV (γ = 6.32) located at the initial position h = 320 ΩEQ/c (1091 km). We solve the relativistic equations of motion for electrons assuming a dipole magnetic field (L = 4) and the coherent wavefield described above. The numerical model and scheme have already been described by Omura and Summers . Figure 2a shows the variation of kinetic energy K of electrons in the (h, K)-phase space. One out of 16 electrons was trapped by a wave packet propagating toward the north and this electron is accelerated by the RTA process. The maximum energy increase is 150 keV within 0.13 s. The kinetic energy at the turning point is 2780 keV (γ = 6.45) and the corresponding wave frequency is approximately 0.155 ΩEQ. We compare this result with the constant frequency case with ω = 0.1 ΩEQ shown in Figure 2b. These results imply that the frequency increase of chorus emissions makes the turning of trapped particles more likely, even though the number of trapped electrons decreases. In the time-dependent frequency case, VR varies with position and time, as described by the resonance condition,
Since the wave frequency ω(h,t) seen by the trapped electrons increases in time, the absolute value of VR becomes small by (12). Furthermore, for relativistic trapped electrons (γ > 1), the absolute value of VR decreases rapidly to zero, as described by Omura et al. . This corresponds to the situation in which the value of γ at the turning point (γ = Ωe/ω) decreases as the wave frequency ω increases, and RTA takes place in a lower energy range. RTA makes the duration of the resonant trapping longer, and at the same time it keeps the resonance region on one side of the magnetic equator where the trapped electron always sees the accelerating electric field; a trapped-but-not-turning electron only sees the electric field for a shorter period of time, and moves to the other side of the equator where it no longer sees the wave.
 However, the frequency increase in chorus emissions makes wave trapping difficult. As discussed by Omura and Summers , wave trapping is controlled by the inhomogeneity ratio S which is a function of the spatial inhomogeneity of the static magnetic field and a time derivative of the wave frequency as seen by a particle. If ∣S∣ < 1, then wave trapping of a resonant particle becomes possible. A positive value of dω/dt makes the absolute value of S large, and the trapping region becomes smaller as the frequency rises. Hence the duration for trapping in the RTA process becomes smaller. From a comparison of Figures 3a and 3b, we find that the kinetic energy gain is about half of the value by RTA with a fixed frequency. We conclude that once electrons are trapped, RTA can take place more frequently, even though the acceleration achieved by a single RTA encounter becomes smaller.
 Since the wave frequency varies in position and time, we uniformly distributed 1500 electrons in the simulation field from h = 60 ΩEQ/c to 1500 ΩEQ/c, with a spatial interval Δh = 60 ΩEQ/c. We also uniformly distributed electrons in gyrophase from 0 to 2π with Δϕ = 6°. To ensure that all electrons are trapped by the static magnetic field, we distributed particles at the mirror points (90 degree pitch angle) at different locations along the magnetic field. Through the adiabatic motion in the dipole magnetic field, the particles take pitch angles less than 90 degrees. By solving the relativistic equations of motion for electrons, we study the acceleration of particles during one cycle of chorus element propagation from the equator.
 In Figure 3a, we show the simulation results for electrons interacting with chorus wave packets. We find that electrons are accelerated by RTA, and that the maximum energy increase is about 300 keV and occurs within 0.2 s. The large energy increase takes place at approximately t = 0.3 s and electrons undergo the turning motion at t = 0.4 s. From Figure 1 we observe that the corresponding frequency ω at the equator is approximately 0.4 ΩEQ. The corresponding value of γ at the turning point (VR = 0) is 3.1 from (12).
 For comparison with Figure 3a, we have plotted trajectories of electrons for a fixed frequency case with ω = 0.28 ΩEQ in Figure 3b. The value of γ at the turning point is 3.6 corresponding to 1.3 MeV. As seen in the right panels, the number of trapped electrons significantly increases for the fixed frequency case. Only four of the trapped electrons, however, undergo turning motion. Some of the trapped electrons undergo efficient acceleration but reach the equator before the turning and the acceleration stops about halfway. The maximum increase in energy of the turning electrons is about 600 keV, in agreement with the analytical formula given by Omura et al. .
 We examined the acceleration process by starting the test particle simulations at different initial energies for a highly relativistic energy range around 3 MeV, a relativistic energy range around 1 MeV, and a low-energy range around 0.2 MeV. The results are plotted in Figures 4a, 4b, and 4c, respectively. We find that the trajectories vary greatly depending on the initial energy ranges. For the relativistic ranges in Figures 4a and 4b, the RTA process is clearly taking place, while for the lower-energy range in Figure 4c, the increase in energy of trapped electrons are much less significant. Untrapped resonant electrons are decelerated because they move outside the trapping potential of the chorus wave in the velocity phase space, as demonstrated by Omura and Summers .
3. Evolution of the Resonant Electron Distribution due to Interaction With Chorus Elements
 The acceleration and scattering of electrons by a chorus element involves both trapped and untrapped resonant electrons, and can be described by the equations of motion and wave equations given in the previous section. In the present model, we assume that a pair of chorus elements are generated at the magnetic equator, and propagate in opposite directions to the higher latitude regions. Through the propagation of the chorus elements some of the resonant electrons are trapped and accelerated, while others are scattered and decelerated. To study the evolution of the resonant electron distribution, we assume that a group of electrons form a delta function distribution δ(K − Ki) at a specific kinetic energy Ki. Within this group, electrons are placed at different locations and have different phases with respect to the chorus elements. For simplicity, however, we only consider an average particle distribution function in the system as a function of kinetic energy K, neglecting the systematic changes of pitch angle associated with the changes of kinetic energy. Following interaction with the chorus elements, we determine the distribution of electrons G(K, Ki) that have evolved from the delta function. We express the particle scattering process by
where is the particle scattering operator due to a pair of chorus elements. We assume that all particles forming the delta function remain in the system after the interaction, being trapped by the dipole magnetic field. Therefore the total number of particles are unchanged, which is expressed by
The operator represents a test particle simulation in which motions of particles are independent of each other. Thus we assume that possesses the basic properties of a linear operator, namely,
where F1 and F2 are any particle distribution functions and a is a constant. We multiply (13) by a specific distribution function Fn(Ki) evaluated at time tn, and we carry out integration with respect to Ki (Ki 0). Hence
The right-hand side of (18) reduces to [Fn(K)] which is the distribution function Fn+1(K) at the time tn+1 after interaction with the chorus elements. Therefore we obtain the result,
where G plays the role of a Green's function for the process. Discrete versions of equations (19) and (14) are used in the following as part of the numerical scheme to model the evolution of the particle distribution function.
 We calculate an energy distribution function for a group of electrons with identical kinetic energy Ki. For each energy Ki, we assume that 600 electrons are distributed at specified gyrophases and positions between h = 100 c/ΩEQ and h = 1000 c/ΩEQ. After one time cycle in which the chorus elements propagate through the simulation region, we find that the electrons forming a delta function initially undergo a spread in energy and form a distribution function g(K, Ki). In calculating the energy distribution function g(K, Ki), we make use of a linear weighting function to distribute the particle density at a specific energy over discrete values adjacent to Ki. We normalize the density so that g(K, Ki) dK = 1. We vary the Lorentz factor γ over the range 1.10 ∼ 13.08 with an interval Δγ = 0.02. This corresponds to an energy range from 51 keV to 6161 keV with an interval 10.2 keV. As can be observed in Figure 4c, however, the number of trapped electrons strongly fluctuates depending on slight changes in the parameters. This is caused by a cluster effect due to a non-uniform distribution of the particle trajectories near the resonance velocity in the phase space [Matsumoto and Omura, 1981]. Therefore we take an average over a small range of energy, i.e., we set
where ΔK = m0c2Δγ. We plot the Green's functions G(K, Kj) for j = 1, 2, 3,…, 60 in Figure 5, where Kj = m0c2 (0.2 j − 0.01). The shapes of the Green's functions vary smoothly with the energy K. Since we have only determined the Green's functions at discrete values corresponding to the interval 10 ΔK = 0.2 m0c2, we extend the functions to arbitrary values by interpolation. The Green's function G(K, Kj) may be regarded as the output distribution corresponding to the delta function source δ(K − Kj). Defining the difference Kd = K − Kj, we re-write the Green's function Hj(Kd) for a delta function source at a specific energy Kj as
We plot Hj for the energy values 0.1, 0.4, 1.0, 2.0, 3.0, 4.0, 5.0, and 6.0 MeV in Figure 6. We determine the Green's function i for a delta function source at i,where Kj ≤ i < Kj+1, by the linear interpolation,
where q = (i − Kj)/(10ΔK).
 Since each electron trajectory is independent in the test particle simulation, the energy distribution functions for specific groups of electrons are also independent. Therefore any linear combinations of solutions to the acceleration and scattering processes are also solutions. We evaluate the convolution integral (19) numerically by taking the summation,
Thereby, we can determine the evolution of the energy distribution function over the time step Δt during which the group of trapped electrons interact with two chorus elements propagating in opposite directions away from the magnetic equator in the dipole field.
 Since the resolution interval of the Green's functions is 10 ΔK = 100 keV, we assume a constant step-like distribution function f(t, K) as the boundary condition over the energy range 0–100 keV. We assume that electrons in this energy range are constantly maintained by particle injection into the inner equatorial region by means of radial diffusion from an outer source-region. We repeat the convolution summation (23) over many time steps to find the long-time evolution of the energy distribution function. We plot the evolution of the distribution functions using the logarithmic scales in Figure 7 after 10, 100, 1000, 2000, 3000, 10,000, 20,000, and 30,000 repeated cycles. If we assume that two chorus elements are excited at the equator at each second, the repetition time corresponds to the time in seconds. We do not find a power law distribution as found in the quasi-linear diffusion model of Ma and Summers .
 We traced a second case of the evolution of the particle distribution function using a step-like seed distribution function over the energy range 0–400 keV. Since the lower-energy distribution is constantly maintained in this range, a substantial number of electrons are accelerated to higher energies due to the RTA process. Comparing the shape of the Green's functions for 0.1 MeV and 0.4 MeV in Figure 6, we observe that the particle scattering and acceleration are much greater at 0.4 MeV.
 In the cases of fixed distributions of seed electrons, we find that the distribution approaches an asymptotic form after many cycles of resonant interaction. This asymptotic form corresponds to the steady state distribution for the particle scattering process induced by the chorus elements.
4. Discussion and Conclusions
 Based on a recent self-consistent simulation reproducing chorus emissions [Katoh and Omura, 2007a], we have constructed a wavefield model in which whistler mode waves are generated at the geomagnetic equator with a variable frequency ω/ΩEQ = 0.1 ∼ 0.45. By increasing the frequency from 0.1 ΩEQ to 0.45 ΩEQ gradually, we can generate coherent chorus elements propagating northward and southward from the equator. Utilizing such chorus elements, we have confirmed that the relativistic turning acceleration (RTA) process takes place just as in the case of a wave packet with a constant frequency [Omura et al., 2007]. For the variable frequency case, the energy gain of a turning particle is roughly one half of that for a constant frequency. This is because the increasing frequency ω makes the Lorentz factor γ decrease due to the resonance condition γ = Ωe/ω at the turning point where VR = 0 (see (12)). The Lorentz factor γ determines the kinetic energy K = m0c2(γ − 1).
 To find the contribution of the RTA process to the evolution of the electron energy distribution function, we have performed a series of test particle simulations to trace the electron dynamics. We have employed a delta function in energy K with different phase angles corresponding to perpendicular velocities and positions uniformly distributed in the phase space. We started the calculation of the particle motion from the mirror points by assuming a pitch angle of 90 degrees initially so as to ensure that these electrons are trapped by the dipole magnetic field near the equator. We calculated the energy distribution function after a period of interaction in which the chorus elements excited at the equator propagate through the simulation region. The simulation region is taken to cover the wave trapping zone where resonant wave trapping and RTA are possible. Solving the cyclotron resonance condition (12) for γ, we find
At the turning point of the RTA process, where VR = 0, condition (24) gives γ = Ωe/ω. Assuming Ωe ∼ ΩEQ, we find that the frequency range of the chorus element ω = 0.1–0.45 ΩEQ can cause RTA for resonant particles with Lorentz factor γ = 1.22–10.0, which corresponds to the energy range 622–4590 keV. We observe very efficient particle acceleration in the energy range 1–4 MeV in Figure 6.
 The particle scattering in the energy range <1 MeV is due to the interaction for kVR < 0, i.e., the particles with negative parallel velocity interact with a wave with positive phase velocity ω/k > 0 in the northern hemisphere (h > 0), and particles with positive parallel velocity interact with a wave with negative phase velocity ω/k < 0 in the southern hemisphere (h < 0). For the chorus elements generated at the equator, electrons satisfying the second-order resonance condition [Omura et al., 2007] are trapped by the wave in the vicinity of the equator, and are always accelerated regardless of the sign of the resonance velocity. The particle scattering in the higher energy range >4 MeV is due to the interaction for kVR > 0, i.e., the particles with positive parallel velocity interact with a wave with positive phase velocity ω/k > 0 in the northern hemisphere, and particles with negative parallel velocity interact with a wave with negative phase velocity ω/k < 0 in the southern hemisphere. We refer to this process as ultra-relativistic acceleration (URA) [Summers and Omura, 2007].
 The variation in height of the Green's functions at different energies shown in Figure 5 also confirms that the scattering and acceleration process becomes most effective in the energy range 1–4 MeV. Since each Green's function is appropriately normalized (as expressed by (14)), a smaller height of the Green's function implies a more effective scattering and energization of particles.
 As a result of the resonant interaction and the RTA process, the particles are effectively scattered in energy and accelerated, and we can determine the resulting evolution of the distribution function using delta functions. On the basis of a Green's function method, we obtained a convolution integral involving numerical forms of the Green's function and the energy distribution function fK,t in order to find the evolved distribution fK,t+Δt.
 By repeatedly computing the discrete form of the convolution integral, we estimated long-time evolutions of the distribution function assuming seed electrons in the ranges 0–100 keV and 0–400 keV, for successive generation of chorus elements. We found rapid generation of relativistic electrons in the energy range 1–3 MeV. The formation process strongly depends on the initial distribution of seed electrons. Because of the efficiency of the RTA mechanism, a steady supply of seed electrons of energy in the range of a few hundred keV can significantly contribute to the formation of relativistic electron flux in the outer radiation belt. We have traced the evolution of the particle distribution over a very long timescale, namely, repeating the resonance cycle 30,000 times. As shown in Figure 7, the particle distributions approach an asymptotic form in each case. If we assume that a pair of chorus emissions are generated at the equator during every second on average, the number of interaction cycles corresponds to the required acceleration time in seconds. It takes tens of minutes to hours for 1–100 keV seed electrons to be accelerated to energies of a few MeV by the RTA process. Although we have established the formation of a high-energy electron flux in the MeV range, this does not contribute to the whistler mode instability, because the phase space density of such an electron population is much lower than that in the lower energy range controlling the wave growth.
 The step function distribution of 0–400 keV electrons used in Figure 7b is not fully realistic, but it is useful to examine the effect of seed electrons in the range 100–400 keV in contrast to the case with 0–100 keV seed electrons. We find that electrons in the range 100–400 keV contribute substantially to the formation of MeV electron flux. Although a more realistic source distribution of seed electrons could be used, the structure of the asymptotic distribution functions in the high-energy range (1–6 MeV) is essentially the same, regardless of the initial distribution functions in the lower energy range. With the supply of 0–400 keV electrons, we find that a substantial flux of MeV electrons is formed after 3000 interaction cycles (Figure 7b). This could explain the short time-scale, less than 60 min, of flux enhancement of relativistic electrons in the inner (L < 3) magnetosphere reported by Nagai et al.  in association with large substorms during the main phase of a magnetic storm.
 A limitation of the present study is that it is a test particle approach and therefore not self-consistent. Self-consistent particle simulations to model chorus emissions with the free energy provided by seed electrons have recently been reported by Katoh and Omura [2007a]. They also found that RTA takes place during the generation process of chorus emissions [Katoh and Omura, 2007b]. To determine the formation of the high-energy tail due to RTA, however, we need to maintain the chorus wave emissions over a timescale much longer than the timescale of the generation of chorus wave elements. Therefore our current test particle simulation approach based on a Green's function method provides a very powerful tool to estimate the long-time evolution of the relativistic electron distribution. The method still needs to be further tested for various parameters and different forms of chorus emission. In the present test particle simulations, we have assumed a constant amplitude for the whistler mode wave packets. In a more realistic model, the wave packets excited at the equator may grow in amplitude as they propagate away from the equator. Such a variation in the wave amplitude may affect the efficiency of the RTA process. However, this is left as a future study.
 This work was partially supported by Grant-in-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, and International Communication Foundation. D. S. acknowledges support from the Natural Sciences and Engineering Research Council of Canada under grant A-0621.
 Zuyin Pu thanks the reviewers for their assistance in evaluating this paper.