Volume 126, Issue 10 e2020JA028842
Research Article
Free Access

Drift Resonance Between Particles and Compressional Toroidal ULF Waves in Dipole Magnetic Field

Li Li

Li Li

School of Earth and Space Sciences, Peking University, Beijing, China

Research Institute for Sustainable Humanosphere, Kyoto University, Kyoto, Japan

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Xu-Zhi Zhou

Corresponding Author

Xu-Zhi Zhou

School of Earth and Space Sciences, Peking University, Beijing, China

Correspondence to:

X.-Z. Zhou,

[email protected]

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Yoshiharu Omura

Yoshiharu Omura

Research Institute for Sustainable Humanosphere, Kyoto University, Kyoto, Japan

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Qiu-Gang Zong

Qiu-Gang Zong

School of Earth and Space Sciences, Peking University, Beijing, China

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Robert Rankin

Robert Rankin

Department of Physics, University of Alberta, Edmonton, Alberta, Canada

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Xing-Ran Chen

Xing-Ran Chen

School of Earth and Space Sciences, Peking University, Beijing, China

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Ying Liu

Ying Liu

School of Earth and Space Sciences, Peking University, Beijing, China

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Chao Yue

Chao Yue

School of Earth and Space Sciences, Peking University, Beijing, China

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Sui-Yan Fu

Sui-Yan Fu

School of Earth and Space Sciences, Peking University, Beijing, China

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First published: 07 October 2021
Citations: 11


We examine the drift-resonant particle dynamics for toroidal ultralow frequency (ULF) waves in a pure dipole background geomagnetic field. We confirm that the resonant condition originally believed to apply only for poloidal ULF waves, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0001, also applies for compressional toroidal waves. The predicted phase relationships have been confirmed from Van Allen Probes observations. Their good agreement provides the first observational evidence for the drift resonance condition controlled by the compressional toroidal ULF wave. Moreover, we extend the drift resonance theory into a nonlinear regime. The resulting particle motion can be described by a modified pendulum equation with solutions depending on the wave number urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0002. For high-urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0003 toroidal waves, the resonant islands become asymmetric to perturb the particle trajectories within each potential well and consequently increase the trapping widths in both energy and L-shell. We further carry out test-particle simulations to show the evolution of electron distribution functions when they interact with either toroidal or poloidal waves. These findings demonstrate that toroidal ULF waves, like their poloidal counterparts, play an important role in magnetospheric particle dynamics.

Key Points

  • Compressional toroidal ultra low frequency (ULF) waves have the same drift resonance condition as poloidal waves in a pure dipole model

  • Drift resonance between electrons and compressional-toroidal ULF waves is identified from Van Allen Probes observations

  • A modified pendulum equation is derived to describe the nonlinear drift resonance between toroidal waves and charged particles

Plain Language Summary

In the magnetosphere, ultralow frequency (ULF) electromagnetic waves in the mHz range are usually categorized into poloidal and toroidal modes. It has been widely accepted that poloidal ULF waves, characterized by electric field oscillations in the azimuthal direction, play a key role in accelerating and transporting charged particles in the inner magnetosphere. On the other hand, toroidal waves have long been considered incapable of accelerating particles since their radial electric field is perpendicular to the azimuthal drift velocity of inner magnetospheric particles unless their drift paths are significantly distorted by dayside compression of the magnetosphere. Here we show that even in a pure dipole field, the toroidal waves can still resonate with energetic particles because of the nonzero curl of the wave electric field in association with the compressional magnetic field oscillations, and the resonant condition remains the same as its poloidal counterpart. The predictions of compressional toroidal wave-particle drift resonance show good agreement with spacecraft observations. Further analysis shows that the particle nonlinear behavior in the toroidal wave field can be described by a modified pendulum equation. Therefore, it is important to consider not only the poloidal but also the toroidal modes in the study of ULF wave-particle interactions.

1 Introduction

Ultralow-frequency (ULF) waves with frequencies in the 2–22 mHz range have a significant influence on particle dynamics in the Earth's radiation belts (Claudepierre et al., 2013; Liu et al., 2016; Mann et al., 2013; Sarris et al., 2017; Zhou et al., 20152016; Q. Zong et al., 2017). Drift resonance theory proposed by Southwood and Kivelson (19811982) suggests that efficient energy exchange between poloidal ULF waves (with azimuthal electric field) and charged particles occurs when particle drift velocity matches the wave phase velocity in the azimuthal direction, which yields a resonance condition urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0004. Here urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0005 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0006 are the wave number and the particle's azimuthal drift angular frequency, respectively, and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0007 is the wave angular frequency. According to Southwood and Kivelson (1981), the observable signatures of poloidal wave-particle drift resonance include: the flux of resonant particles oscillates in antiphase with the azimuthal electric field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0008; for particles at higher and lower energies, the flux oscillations are urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0009 out of phase with urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0010. These characteristic phase relationships have been confirmed from observations (Claudepierre et al., 2013; Dai et al., 2013; Foster et al., 2015; Kokubun et al., 1977; Q. Zong et al., 2007; Zong et al., 2009), providing clear evidence of drift resonance between ULF waves and charged particles in the inner magnetosphere. The drift resonance theory has been further developed (A. Degeling & Rankin, 2008; Hao et al., 2017; Li et al., 2017a2020) and some predicted features have also been identified from spacecraft and ground-based observations (Li et al., 2017b; Zhou et al., 2016).

Toroidal waves, however, receive less attention since particle drift paths in the azimuthal direction are perpendicular to the radial electric field of toroidal waves. By taking into account the magnetospheric dayside compression, Elkington et al. (1999) proposed that the particle's drift velocity has a radial component in the dawn and dusk sectors of the magnetosphere, which can be aligned with the radial electric field of toroidal waves to enable a resonant interaction. The corresponding resonance condition is urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0011, where the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0012 factor originates from the noon-midnight asymmetry of the compressed dipole. In other words, the toroidal waves are believed to be able to accelerate particles only in a specific background field (e.g., under strong solar wind pressure), and this effect is usually considered a high-order effect. As is shown in Ukhorskiy et al. (2005), even when the background field is highly asymmetric, the diffusion rates caused by the toroidal wave-carried electric field in the radial direction are small. In other words, earlier studies led to the conclusion that toroidal ULF waves are relatively unimportant in the acceleration of particles in the magnetosphere. This conclusion is not accurate, since the wave-carried compressional magnetic field is not considered in these studies. However, the missing compressional field would violate the Faraday's law, unless the azimuthal wave number is assumed to be zero (e.g., Lee & Lysak, 1989; Singer et al., 1981). This assumption is usually unrealistic, and numerous observations of toroidal ULF waves have shown the existence of magnetic field oscillations in the background field direction (e.g., W. J. Hughes et al., 1978; Shi et al., 2013; Shen et al., 2015; Takahashi et al., 1984). Moreover, the drift resonance proposed by Elkington et al. (1999) does not have a clear phase relationship between particle response and wave signals, and therefore can hardly be recognized from observation. In this paper, we will show theoretically that the compressional magnetic field carried by toroidal ULF waves contributes to the acceleration of electrons under a pure dipole field, and provide observational evidence for the presence of such wave-particle interactions based on Van Allen Probes data.

We will also extend the theory of toroidal wave-particle drift resonance into the nonlinear regime, which is essential when the particle energy change is comparable to its initial energy due to large wave amplitude or long-duration interaction. Similar studies have been carried out previously for poloidal ULF waves (A. Degeling et al., 2007; A. W. Degeling et al., 2008; Wang et al., 2018). Especially, Li et al. (2018) proposed that the nonlinear particle behavior in the poloidal wave field can be described by a pendulum equation, which enables the first observational identification of the nonlinear ULF wave-particle resonance. This analysis also provides a theoretical framework for understanding the effects of inhomogeneity factors (such as magnetospheric convection, see Li et al., 2020) in the nonlinear dynamics of wave-particle interactions. However, these nonlinear analyses have never been extended into toroidal ULF waves (due to the common understanding that toroidal waves can only have weak interactions with drifting particles even if the magnetosphere is highly compressed), and it is our goal in this paper to theoretically explore the nonlinear particle interactions with toroidal ULF waves.

In Section 2, we demonstrate that drift resonance between particles and toroidal ULF waves can occur even without the noon-midnight asymmetry of the background magnetic field. This effect originates from the wave-carried magnetic field oscillations, which turn out to play a role in the energy exchange between toroidal ULF waves and charged particles. We predict the observable signatures of compressional toroidal wave-particle interactions, which are then compared with the data from Van Allen Probes to provide the observational identification of compressional toroidal wave-particle drift resonance in Section 3. We accordingly develop the nonlinear drift resonance theory in Section 4 with a modified pendulum equation. In Section 5, we perform test particle simulations solving the full Lorentz motion to analyze the particle behavior in toroidal and poloidal waves.

2 Drift Resonance for Toroidal Waves

We give an overview of the possible representation of ULF wave fields taken by Wang et al. (2018), which provides a decoupled solution of the poloidal and toroidal modes from Ampere's law, the cold plasma form of the linearized fluid momentum equation, and the Ohm's law. Figure 1 shows the ULF wave field, in which subscripts t and p represent toroidal and poloidal waves (also distinguished by the red and blue colors). The dipole coordinates urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0013, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0014, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0015 denote directions along the background magnetic field, perpendicular to field line and in the azimuthal direction, respectively. Figure 1a schematically shows the directions of toroidal and poloidal ULF waves at and off the magnetic equator. Figures 1b and 1c show the temporal evolution (indicated by the colored lines) of electric and magnetic field profiles of fundamental mode toroidal and poloidal waves on an urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0016 dipole field line within one wave cycle, respectively. This figure clearly shows that the compressional magnetic field always exists in association with the nonzero curl of the electric field (unless m equals zero).

Details are in the caption following the image

(a) Schematic showing the directions of electric fields and compressional magnetic fields of toroidal and poloidal waves at and off the magnetic equator. Red and blue colors indicate the toroidal and poloidal ultra low frequency (ULF) waves, respectively. Latitude-dependent structures of (b) toroidal and (c) poloidal waves at urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0017 using a wave model based on Wang et al. (2018). The colored lines indicate the temporal evolution of ULF waves within one complete wave cycle.

Before we proceed, it is important to note that the decoupled model in Wang et al. (2018) is only a possible solution of the ULF wave field in the magnetosphere. It is generally accepted in the field line resonance theory that the poloidal and toroidal waves are coupled with each other (Allan & Poulter, 1992; Lee & Lysak, 1989). One may have even more complicated solutions after taking into account the pressure anisotropy in the hot plasma environment (e.g., Klimushkin, 1998). Therefore, it is not straightforward to specify the most realistic form of the ULF wave field. Here we simply select an asymptotic case in the absence of the poloidal wave field (radial magnetic field and azimuthal electric field) to highlight the role of the toroidal waves in the wave-particle interactions. One should keep in mind that in reality, the toroidal waves may not be fully decoupled with the poloidal waves, and it is important to consider their combined effects.

In this study, we mainly focus on the perpendicular-moving electron dynamics in the equatorial plane (marked by dotted lines in Figure 1), which indicates that the radial and azimuthal magnetic field components can be safely neglected (see Figures 1b and 1c). We also assume, for simplicity, an uniform Alfven speeds over a certain range of L-shells, the model therefore can be further simplified, which will be introduced in the following. Note that the transverse magnetic field perturbations (urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0018, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0019) are zero at the equator. The wave magnetic field to be discussed below is the compressional component urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0020. A pure dipole geomagnetic field in the equatorial plane is given by
where urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0022 is the equatorial magnetic field strength on the Earth's surface, and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0023 is the L-shell parameter. The toroidal wave-carried electric field is given by
where constant factor urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0025 represents the wave amplitude, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0026 is the azimuthal wave number, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0027 is magnetic longitude, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0028 is the initial phase, and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0029 is the wave angular frequency. For simplicity, here we assume that the wave angular frequency is not a function of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0030, although in reality (and in the Wang et al., 2018 model) the wave angular frequency could change slightly with urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0031 near the field line resonance location. According to Faraday's law, the wave carries magnetic field in the background field (urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0032) direction, which is given by
where urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0034 is the Earth's radius. The azimuthal gradient of this wave-carried magnetic field contributes to a radial drift velocity of an equatorial particle with charge urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0035, which is given by
where urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0037 is the particle's rest mass, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0038 is the relativistic Lorentz factor, and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0039 is the total magnetic field including the dipole field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0040 and the wave magnetic field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0041. The variation of the particle kinetic energy urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0042, based on Northrop (1963), is given by
where urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0044 is the particle's magnetic moment (the first adiabatic invariant). The second term on the right hand side represents the induction effect, caused by the curl of wave electric field acting on the particle gyromotion. Substituting Equations 2 urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0045 4 into 5, we have
where the first term on the right hand side (representing the particle's kinetic energy variations associated with its radial drift motion) shows energy oscillations at twice the wave frequency. In order to obtain the particle's energy change over many gyrations (in the linear regime), we integrate Equation 6 along the particle's unperturbed orbit back in time. Here we follow Southwood and Kivelson (1981) to assume a gradual growth of the wave amplitude, represented by a small imaginary part of the wave frequency, that is, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0047, so that the integration can be taken to the time when the wave amplitude is negligibly small. The integration result is given by
where urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0049. Note that the first term on the right-hand side of Equation 7 is relatively small and can often be neglected (see Section 4). One finds a resonance condition for drift-resonant acceleration

When Equation 8 holds the particles see the greatest increase in energy. Therefore, the toroidal ULF waves can interact with particles even if there is no compression of the dipole magnetic filed. It is the magnetic field carried by toroidal ULF waves that leads to this resonance condition, which has been often neglected in previous studies (Elkington et al., 19992003).

To better understand the phase relationship between particle energy gain and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0051, we now rewrite Equation 7 (with the first term on the right-hand side neglected) in the form,

When drift resonance condition urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0053 applies, the remaining term urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0054 in the denominator of Equation 9 indicates a urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0055 phase difference between urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0056 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0057. In contrast, at lower and higher energies, the denominator in Equation 9 is dominated by a real number (assuming a smaller urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0058 than urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0059), which indicates that the corresponding urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0060 oscillations would have smaller amplitude and be either in phase or antiphase with urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0061. Given the antiphase relationship between urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0062 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0063 (compare Equations 2 and 3), the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0064 oscillations are urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0065 out of phase with the electric field oscillations for particles at resonant energies. Such characteristics can be used as a diagnostic of compressional toroidal wave-particle drift resonance, especially in the case without significant interference from the poloidal waves.

To compare these predictions with the observational data, the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0066 oscillations should be transformed into urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0067, which represents the gyration-averaged change of the particle phase space density and can be directly measured by an actual particle detector. Assuming that only the third adiabatic invariant is violated by ULF waves (Southwood & Kivelson, 1981), urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0068 can be written as
which shows that urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0070 is directly proportional to urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0071 with a ratio depending on the gradient of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0072 over energy and/or space. Therefore, the phase relationship between ULF wave fields and the particle flux variations should be the same with that between wave fields and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0073 oscillations.

One should note that this diagnostic of compressional toroidal wave-particle drift resonance is different from that with poloidal ULF wave, although there is a common feature of a urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0074 phase shift in particle flux oscillations between the lowest and the highest energies. The difference between them is summarized below. For poloidal ULF waves, the flux of resonant particles oscillates in antiphase with the azimuthal electric field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0075; for particles at higher and lower energies, the flux oscillations are urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0076 out of phase with urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0077. For compressional toroidal ULF waves, the flux oscillations of resonant particles are urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0078 out of phase with the compressional magnetic field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0079 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0080 with the radial electric field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0081; for particles at higher and lower energies, the flux oscillations are in phase or antiphase with urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0082 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0083, respectively. The characteristic signatures for compressional toroidal ULF wave are to be compared in the next section with Van Allen Probes observations.

3 Observations

The ULF wave event and the energetic electron response on September 18, 2016 are investigated based on Van Allen Probes observations, including electric field data from Electric Fields and Waves (EFW) (Wygant et al., 2013), magnetic field data from Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) (Kletzing et al., 2013), and electron data from Magnetic Electron Ion Spectrometer (MagEIS) (Blake et al., 2013). The geomagnetic index data used for this study are available from the OMNI database.

Figure 2a presents a weak substorm that occurred after urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0084 UT, with an enhancement of the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0085 index to a peak of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0086 nT and a minimum urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0087 of approximately −100 nT, while the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0088 index show a nonstorm condition during a period of relatively steady solar wind (not shown). The two vertical blue lines mark the time period when RBSP B detected the ULF wave event over 0600 UT to 0630 UT. Figures 2b and 2c show the magnetic and electric field measurements, which clearly indicate excitation of ULF waves with the period of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0089 s. Figure 2d shows the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0090 pitch angle electron fluxes in each energy channel from 54 to 143 keV. The electron fluxes show gradual enhancements in association with the substorm injection process, and they also oscillate at about the same frequency as the ULF waves. To distinguish different wave modes, the electric and magnetic fields are projected to a local mean field-aligned (MFA) coordinate system (e.g., see Takahashi et al., 1990), in which the parallel direction urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0091 is determined by 15-min sliding average EMFISIS magnetic field data, the azimuthal direction urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0092 is parallel to the vector product of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0093 and the spacecraft geocentric position vector, and the radial direction urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0094 completes the triad. This decomposition allows us to divide the wave fields into different components, including azimuthal magnetic field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0095, radial electric field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0096, radial magnetic field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0097, azimuthal electric field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0098 and parallel component urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0099.

Details are in the caption following the image

Overview of the September 18, 2016 ultralow frequency wave event. (a) AE and AL indices. (b) The Van Allen Probe B observations of the magnetic field in GSM coordinates. (c) The electric field measured in mGSE coordinates. (d) The urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0100 pitch angle electron fluxes at multiple energy channels. (e) The compressional magnetic field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0101 in mean-field aligned (MFA) coordinates. (f) The azimuthal magnetic field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0102 and radial electric field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0103 in MFA coordinates. (g) The radial magnetic field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0104 and azimuthal electric field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0105 in MFA coordinates. (h) The residual electron fluxes urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0106 derived from (d).

These ULF oscillations in MFA coordinates are shown in Figures 2e–2g. The field perturbations associated with the toroidal waves (urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0107 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0108, see Figure 2f) are much stronger than the poloidal wave field (urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0109 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0110, see Figure 2g). One may also find the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0111 phase difference between the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0112 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0113 perturbations, which indicates the dominance of the toroidal-mode standing waves (W. J. Hughes et al., 1978) over the poloidal waves. The toroidal waves are also associated with significant oscillations in the compressional magnetic field (see Figure 2e). One may note that the compressional urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0114 in Figure 2e experiences a rapid growth stage until urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0115 UT when it starts to decay from its peak, whereas in Figure 2f, the toroidal-mode wave field continues to increase until urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0116 UT. This could be a signature of energy transfer from compressional to toroidal waves with the same frequency via field line resonance (Southwood, 1974). Figure 2h shows the residual electron flux urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0117, in which urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0118 is the flux observed in Figure 2d and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0119 is a 10 min sliding average of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0120. It is clear that the phases of electron flux oscillations at higher energies lead those at lower energies, which indicates that the drift resonance may occur.

To analyze the phase relationship between the wave field and the electron flux oscillations, we carry out a Morlet wavelet coherence analysis (Grinsted et al., 2004) between urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0121 (on each energy channel) and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0122, as is shown in Figures 3b–3e. For comparison, the wavelet power spectrum of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0123 is given in Figure 3a. One may find from Figures 3b–3e that the compressional magnetic field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0124 from 0613 to 0623 UT is well correlated with the electron residual fluxes urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0125, with the wavelet coherence of more than 0.8 at the period of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0126 s. Similar conclusions can be also made between urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0127 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0128 (see Figures 3h–3k, also with high coherence of over 0.8), while the coherence between azimuthal electric field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0129 and electron residual fluxes urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0130 (see Figures 3n–3q) is very low to indicate a weak influence of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0131 on electron flux oscillations. In other words, the injected electron fluxes are modulated by the compressional toroidal ULF waves.

Details are in the caption following the image

The wavelet power spectrum of (a) the compressional magnetic field component urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0132, (g) the radial electric field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0133, and (m) azimuthal electric field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0134. The wavelet coherence between electron residual fluxes urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0135 in the 54–132 keV energy channels and (b)–(e)urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0136, (h–k)urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0137 and (n–q)urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0138, respectively. (f and l) The phase differences of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0139 (at multiple energy channels) from urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0140 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0141 at the dominant ultralow frequency wave period of 100 s.

Figures 3f and 3l show the phase differences of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0142 from urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0143 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0144 when the corresponding wavelet coherence is greater than 0.5, respectively. At 75 keV, the phase difference between urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0145 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0146 is close to urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0147 (green line in Figure 3f) and the modulation amplitude of electron flux peaks (see yellow line in Figure 2h), which can lead to a conclusion that 75 keV electrons are in drift resonance with the observed ULF waves. Similar conclusions can be also made from the phase difference of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0148 between urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0149 at 75 keV and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0150 (see Figure 3l). This resonant energy corresponds to the wave number of 60 according to the resonance condition (8). The electron flux oscillations at 54, 102, and 132 keV are slightly weaker (Figure 2h) than those at 75 keV. The total phase shift across energies, according to Figures 3f, is ∼urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0151. These signatures are consistent with the theoretical predictions in the previous section, and therefore provide a strong evidence for the presence of wave-particle drift resonance dominated by compressional toroidal ULF waves in the inner magnetosphere.

4 Modified Pendulum Equation

We next formulate the nonlinear particle trapping in a toroidal wave-carried potential well. For ease of calculation, we set the exponential function of the waves used above in the sinusoidal form in the following. Using the first adiabatic invariant urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0152, we obtain from Equation 4 the L-shell changes at the rate
The equatorially mirroring particle also follows the grad urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0154 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0155 drifts to move in the azimuthal direction, with a velocity
where the first term on the right hand side is the magnetic gradient drift velocity of the particle in the pure magnetic dipole urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0157 (Northrop, 1963), and the second term is due to the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0158 gradient in the radial direction urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0159. The third term denotes the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0160 drift motion in association with the wave electric field in Equation 2. Noting that the particle's relativistic kinetic energy urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0161, we have
which is used in the following derivation. We next follow Li et al. (2018) to define
which represent the phase of the particle location in the wave rest frame and its time derivative, respectively. Note that the drift resonance condition of Equation 8 is equivalent to the first-order resonance in this system, with urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0164. Using the angular drift frequency urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0165, we obtain the time derivative of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0166 as
Substituting Equations 11 urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0168 13 into 15, we have

Note that urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0176, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0177, and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0178 all undergo minor variations due to the time variations of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0179 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0180 (see Equations 11 and 13). If we neglect these minor variations, Equation 16 suggests that the particle motion can be approximately described by a modified pendulum equation with the modification coefficients urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0181 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0182. When urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0183 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0184 are extremely small, the pendulum equation suggests that near-resonant particles can be trapped in the wave-carried potential well centered at urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0185, with the trapping frequency urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0186. Also, by definition, the second-order resonance condition is satisfied when urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0187. This simple picture, however, can become more complicated if urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0188 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0189 are not negligible. Therefore, it is important to evaluate the relative magnitudes of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0190, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0191, and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0192, which is relatively straight-forward since the terms in the brackets of Equations 20 urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0193 22 are dimensionless quantities.

Figure 4 shows the values of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0194, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0195, and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0196 with various parameters including the wave electric field amplitude urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0197, wave period urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0198, and L-shell values, as functions of azimuthal wave number urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0199 for resonant electrons. Among the three parameters urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0200, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0201, and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0202, we fix two of them in each column to show the variations of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0203, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0204, and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0205 with the remaining parameter. One may find from case (I) and (III) of Figure 4 that the absolute values of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0206, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0207, and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0208 increase as the wave electric field amplitude urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0209 or L-shell increases for an intermediate or high wave number urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0210, whereas for case (II), the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0211 value hardly changes with urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0212. The absolute value of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0213 increases inversely with the wave period urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0214, while the absolute value of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0215 is proportional to urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0216. It is clear that although the absolute value of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0217 increases slightly with wave number urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0218, the value of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0219 is considerably lower than urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0220 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0221, and therefore can usually be neglected in the modified pendulum Equation 16. Also we note that for low-urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0222 ULF waves, the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0223 value is always close to zero regardless of the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0224, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0225, and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0226 selections, and the dominance of the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0227 term suggests that the particle behavior can be approximately described by a simple pendulum equation. However, the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0228 term becomes increasingly important in the modified pendulum equation as urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0229 increases, which can even exceed the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0230 term in certain cases (especially those with higher values of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0231, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0232 and/or urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0233).

Details are in the caption following the image

urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0234, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0235, and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0236 as a function of wave number urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0237 for (I) urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0238 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0239s, (II) urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0240 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0241mV/m and (III) urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0242s and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0243mV/m.

One may also describe the particle orbits in the (urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0244) phase space, by eliminating the time variable urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0245 from Equations  14 and 16,
which can be integrated into
where urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0248 is a constant that corresponds to a specific trajectory. To better understand the different effects of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0249 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0250 on particle motion, we decompose Equation 24 with the different terms on the left hand side into

The corresponding electron trajectories (denoted by the colored lines of different urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0253, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0254 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0255 values) are presented in Figure 5 for Equations 25 (case (I)), 26 (case (II)), and 24 (case (III)), all with the same parameters (urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0256 mV/m, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0257 s, and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0258). For each case, a number of electrons are launched at urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0259 with their initial phases distributed from urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0260 to urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0261 in the wave-carried potential well at their respective resonant energies (urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0262). The top three panels show the three-dimensional electron trajectories in (urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0263) space and the bottom panels show their projection into the (urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0264) space. The red and black asterisks are used as references linking the top and bottom panels to better illustrate the resonant island topology of these electron trajectories. One may note that the electron trajectories appear to intersect one another in the bottom panels, but this is only a projection effect with different urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0265 values at the intersection points.

Details are in the caption following the image

Contour maps of Equations 25-24 for urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0266 toroidal ultralow frequency wave. Top panels and bottom panels show the three- and two-dimensional electron trajectories, respectively. Values of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0267, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0268 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0269 are indicated by different colors. The red and black asterisks are used as a reference in each case to show the topology of electron trajectories.

In case (I), the bottom panel has a similar configuration of particle trapping in potential wells carried by many plasma waves (e.g., see Li et al., 2018, Figure 1), although a slight asymmetry with respect to urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0270 exists due to the minor variations of the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0271 value with energy and L-shell. In case (II), the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0272 term results in two resonance islands with one complete and two half-islands within one wavelength as shown in Figure 5. The corresponding electron trajectories within each island are also quite different from case (I), with strong asymmetry with respect to urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0273. This asymmetry can be understood by the large variations of the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0274 term with urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0275 for a fixed, high-urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0276 value in Figure 4(III). Although the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0277 term is included in the analysis, case (III) can be regarded as the superposition of case (I) and (II) since urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0278 is very small as can be seen in Figure 4. The configuration of the compound island is very similar to the island in case (I) since the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0279 range is much wider than the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0280 range, except that the island extends to more negative urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0281 values due to the island overlap near urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0282 and becomes more asymmetric with respect to urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0283. The centers of the additional two half-islands in case (II) with relatively small urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0284 values merge with the saddle points (marking a separatrix between neighboring islands at urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0285, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0286) in case (I), so that these half-islands do not affect the island topology in case (I). Note that we only show here an example with high urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0287 values (urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0288). For lower urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0289 values, the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0290 term becomes very small, and the configuration of the electron trajectories becomes essentially the same as in Figure 5(I).

5 Particle Behavior in Toroidal and Poloidal ULF Waves

From Sections 2 and 3, we know that particles can resonantly interact with toroidal ULF waves in a background dipole geomagnetic field, which means that the impact of toroidal ULF waves on particles is not limited to the outer magnetosphere as previously understood, and they can play a role in particle acceleration in the inner magnetosphere. Therefore, it would be natural to analyze the contributions of toroidal and poloidal ULF waves to the acceleration of magnetospheric particles.

Let us first compare the different particle behavior in the poloidal and toroidal ULF waves. Readers may refer to Li et al. (2018) for the relevant theory for poloidal ULF waves. The particle motion in the poloidal ULF waves can be described by a pendulum equation, in which the particle behaves similarly to the case (I) of Figure 5, while the motion of a particle interacting with toroidal ULF waves is more complicated due to the modification factor urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0291 as shown in case (III) of Figure 5. The urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0292 value is found to increase with the wave number urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0293 (Figure 4), which suggests a strong urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0294-dependence of the particle behavior near the drift-resonant energy. For low-urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0295 toroidal ULF waves, the features of electron behavior are very similar to those for poloidal waves. For intermediate-urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0296 and high-urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0297 cases, the high asymmetries of electron trajectories caused by large urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0298 value result in the increase of trapping widths in urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0299, which in turn increase the trapping widths in energy and L shell.

In order to confirm our analysis and gain a better understanding of the difference in particle acceleration between toroidal and poloidal ULF waves, we next utilize test particle simulations under models of azimuthal-propagating toroidal or poloidal ULF waves, in the presence of a dipole magnetic field. In our previous paper (Li et al., 2020), the motion of the guiding center was solved, but here we trace the gyromotion of electrons by integrating the relativistic equation of motion using the Buneman-Boris method (Buneman, 1993). The toroidal mode wave model is constructed based on Equations 1 urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0300 3, and the poloidal mode model is set up with the radial electric field in Equation 2 replaced by azimuthal electric field
To satisfy the Faraday's law, the compressional magnetic field for poloidal waves is given by

Here we discuss the acceleration processes with two different wave numbers, namely, urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0303, respectively. The wave angular frequency and electric field amplitude are the same as in Figure 5. The test electrons are distributed uniformly all around the Earth at the L-shell of 5, with the initial resonant energies of 2 MeV for urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0304 (low-urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0305) ULF wave and 45.6 keV for urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0306 (high-urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0307) case, respectively. The evolution of the electron distributions in energy and L-shell as they experience 0, half, one, and two trapping periods are shown in the left panels of Figures 6 and 7, for low-urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0308 and high-urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0309 waves, respectively. The middle panels show the electron positions in the equatorial plane. In the right panels of each case, we display the time evolution of the electron positions in the (urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0310) phase space, which is selected from the region within a single wave length in the middle panels. The instantaneous energy and L-shell are indicated by the colors of the dots in the middle and right panels, respectively. We should point out that the trapping periods given here are theoretical values of electrons near the well center.

Details are in the caption following the image

The evolution of the distribution function of resonant electrons in energy and L-shell for urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0311 toroidal ultra low frequency (ULF) wave (left column of (I)) and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0312 poloidal ULF wave (left column of (II)) at 0, half, one and two trapping periods. The white dot in each panel shows the initial energy of 2 MeV. The middle columns of each case show corresponding positions of electrons in the equatorial plane. The red dashed lines show the initial positions of electrons, where the L-shell is 5. The electrons initially at resonant energy are uniformly distributed along the dashed line. Right columns of (I) and (II) show the electron positions in (urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0313) space within one wave potential well from the middle column.

Details are in the caption following the image

The same as Figure 6 but the initial energy 45.6 keV and the wave number urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0314.

One can clearly find from Figures 6 and 7 that in all cases, the electron distribution functions evolve from delta functions to a wide range of L-shells and energies (left panels of each case). Since urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0315 is conserved, the electrons can move radially inward or outward within their respective potential wells to form coherent structures whose number is equal to the corresponding wave number. These structures can gradually roll up with time and become significant over half of the particle trapping period (Figures 6b6f7b, and 7f). The smaller trapping period suggests a larger roll up speed (Li et al., 2018). Comparing Figures 6b and 6f, we find that the electrons roll up faster interacting with poloidal waves (case (II)) because it takes a shorter time period for these electrons to develop to the same stage than those interacting with toroidal waves (case (I)). Also, electrons in case (II) are distributed in the wider range of energy, L-shell and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0316 within a single wave potential well. Therefore, nonlinear acceleration induced by poloidal ULF waves is slightly more efficient than toroidal waves for low-urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0317 cases. Note that the trapping period of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0318 toroidal case is 5,208 s, which in many cases may exceed the ULF wave lifetime. Therefore, the scenario only applies to long-lived ULF waves (see W. Hughes et al., 1979; Yang et al., 2011), which can last for up to 2 h. For high-urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0319 ULF waves (Figure 7), the particle behavior differs from Figure 6 in that the electron distribution becomes more extended on L-shell, energy and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0320, and the formation time of the 55 coherent structures reduces. The reduced formation time is caused by the shorter trapping period of electrons in the high-urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0321 wave field, and therefore this scenario applies even for short-lived ULF waves.

6 Summary and Discussion

The toroidal ULF waves have long been thought incapable of interacting with particles under an uncompressed background dipole magnetic field, in which particles drifting predominantly in the azimuthal direction cannot interact with the radial electric fields of toroidal waves. According to Elkington et al. (1999), the impact of the toroidal ULF waves on the radiation belt particles depends on the degree of asymmetry of the background magnetic field. The compression of the dipole field can contribute to the radial motion, resulting in the drift-resonant interaction with the resonance condition urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0322.

In the Elkington et al. (1999) scenario, however, the effects of the wave-carried magnetic field have been neglected. After taking into account the wave-carried magnetic field, we have verified that the drift resonance between particles and toroidal ULF waves can occur under a pure dipole magnetic field, and we have obtained a new resonance condition urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0323. The newly predicted signature of compressional toroidal wave-particle drift resonance is that the resonant electron flux would be in quadrature with the toroidal wave-carried compressional magnetic field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0324, and be in phase or antiphase with urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0325 for higher or lower energies. Because of the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0326 phase difference between urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0327 and radial electric field urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0328 (Wang et al., 2018), the phase relationship between urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0329 and electron flux would be opposite to that between urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0330 and electron flux. These predicted signatures are indeed observed by Van Allen Probes, which provides the first observational evidence of the compressional toroidal wave-particle drift resonance.

We have also derived a modified pendulum equation describing the nonlinear trapping of charged particles in the toroidal wave field, with solutions highly dependent on the relative values of urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0331 and urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0332 in Equation 24. The term urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0333 is dominant for low-urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0334 waves and accordingly determines a nearly symmetric resonance island (similar to case (I) of Figure 5) of particle trajectories in (urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0335) phase space. At higher urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0336 values, the urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0337 term becomes increasingly important, which can increase the scaling of the trapping width in case (I) of Figure 5 and result in the formation of a compound asymmetric island ((III) of Figure 5).

The picture of nonlinear drift resonance has been reproduced by a test-particle simulation that resolves the relativistic equation of gyromotion, which has also been utilized to analyze the electron behavior in toroidal and poloidal ULF waves under the dipole magnetic field. In all cases, we find formation of coherent structures that gradually roll up with time. For cases with low-urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0338 numbers, toroidal ULF waves accelerate particles less efficiently than (yet still comparable to) poloidal ULF waves, with a longer formation time of the coherent structures associated with the nonlinear trapping and less efficient transportation in energy and L-shell. On the other hand, the acceleration efficiency gradually increases with increasing wave amplitude. In the case of high-urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0339 ULF waves, toroidal ULF waves enable significant energy transfer between the wave and particles with a broad distribution range of particles in L-shell. In other words, the toroidal ULF wave effects on particles in the magnetosphere have long been underestimated. Even if toroidal ULF waves are usually believed to have relatively low wave number urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0340 (Sarris et al., 2013; Zong et al., 2007), their acceleration efficiency, in this case, is still comparable to that of poloidal ULF waves as shown in Figure 6.

Finally, we should point out that our analysis is based on a simple model, with the wave-carried electromagnetic field in Equations 2-27 and 28. Different wave profiles would correspond to different acceleration efficiencies. The relative importance of poloidal and toroidal waves in accelerating magnetospheric particles may also differ. Future work will take into account the noon-midnight asymmetry with the solar wind compressing the magnetic field on the dayside. Two resonance islands that satisfy the resonance condition urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0341 can be excited in this case (Elkington et al., 2003) and coexist with the resonance island satisfying urn:x-wiley:21699380:media:jgra56797:jgra56797-math-0342. One can expect that if the boundaries of the resonant islands begin to overlap, the particle behavior would become chaotic. This noon-midnight asymmetry can also lead to a location dependence of the particle drift speed, which could be equivalently understood in the wave-particle interaction framework as the monochromatic waves becoming broad-band (Shklyar & Matsumoto, 2009).


This work was supported by NSFC grant 41774168 and JSPS KAKENHI grant JP17H06140. Li Li's visit to Kyoto University was sponsored by the Chinese Scholarship Council. We acknowledge the usage of OMNI data available from NASA's Space Physics Data Facility (SPDF) at https://omniweb.gsfc.nasa.gov/. The authors also acknowledge NASA's Van Allen Probes, especially the EMFISIS, MagEIS and EFW teams for usage of observational data, which are available online at: http://emfisis.physics.uiowa.edu/Flight/, https://www.rbsp-ect.lanl.gov/ and http://www.space.umn.edu/rbspefw-data/, respectively.

    Data Availability Statement

    The simulation codes are available online (via https://doi.org/10.5281/zenodo.3928114).