Drift Resonance Between Particles and Compressional Toroidal ULF Waves in Dipole Magnetic Field
Abstract
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, 
, 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 
. For high-
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., 2015, 2016; Q. Zong et al., 2017). Drift resonance theory proposed by Southwood and Kivelson (1981, 1982) 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 
. Here 
and 
are the wave number and the particle's azimuthal drift angular frequency, respectively, and 
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 
; for particles at higher and lower energies, the flux oscillations are 
out of phase with 
. 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., 2017a, 2020) 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 
, where the 
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 
, 
, 
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 
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).
(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 
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.
, 
) are zero at the equator. The wave magnetic field to be discussed below is the compressional component 
. A pure dipole geomagnetic field in the equatorial plane is given by
(1)
is the equatorial magnetic field strength on the Earth's surface, and 
is the L-shell parameter. The toroidal wave-carried electric field is given by
(2)
represents the wave amplitude, 
is the azimuthal wave number, 
is magnetic longitude, 
is the initial phase, and 
is the wave angular frequency. For simplicity, here we assume that the wave angular frequency is not a function of 
, although in reality (and in the Wang et al., 2018 model) the wave angular frequency could change slightly with 
near the field line resonance location. According to Faraday's law, the wave carries magnetic field in the background field (
) direction, which is given by
(3)
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 
, which is given by
(4)
is the particle's rest mass, 
is the relativistic Lorentz factor, and 
is the total magnetic field including the dipole field 
and the wave magnetic field 
. The variation of the particle kinetic energy 
, based on Northrop (1963), is given by
(5)
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 
4 into 5, we have
(6)
, so that the integration can be taken to the time when the wave amplitude is negligibly small. The integration result is given by
(7)
. 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
(8)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., 1999, 2003).
, we now rewrite Equation 7 (with the first term on the right-hand side neglected) in the form,
(9)When drift resonance condition 
applies, the remaining term 
in the denominator of Equation 9 indicates a 
phase difference between 
and 
. In contrast, at lower and higher energies, the denominator in Equation 9 is dominated by a real number (assuming a smaller 
than 
), which indicates that the corresponding 
oscillations would have smaller amplitude and be either in phase or antiphase with 
. Given the antiphase relationship between 
and 
(compare Equations 2 and 3), the 
oscillations are 
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.
oscillations should be transformed into 
, 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), 
can be written as
(10)
is directly proportional to 
with a ratio depending on the gradient of 
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 
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 
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 
; for particles at higher and lower energies, the flux oscillations are 
out of phase with 
. For compressional toroidal ULF waves, the flux oscillations of resonant particles are 
out of phase with the compressional magnetic field 
and 
with the radial electric field 
; for particles at higher and lower energies, the flux oscillations are in phase or antiphase with 
and 
, 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 
UT, with an enhancement of the 
index to a peak of 
nT and a minimum 
of approximately −100 nT, while the 
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 
s. Figure 2d shows the 
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 
is determined by 15-min sliding average EMFISIS magnetic field data, the azimuthal direction 
is parallel to the vector product of 
and the spacecraft geocentric position vector, and the radial direction 
completes the triad. This decomposition allows us to divide the wave fields into different components, including azimuthal magnetic field 
, radial electric field 
, radial magnetic field 
, azimuthal electric field 
and parallel component 
.
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 
pitch angle electron fluxes at multiple energy channels. (e) The compressional magnetic field 
in mean-field aligned (MFA) coordinates. (f) The azimuthal magnetic field 
and radial electric field 
in MFA coordinates. (g) The radial magnetic field 
and azimuthal electric field 
in MFA coordinates. (h) The residual electron fluxes 
derived from (d).
These ULF oscillations in MFA coordinates are shown in Figures 2e–2g. The field perturbations associated with the toroidal waves (
and 
, see Figure 2f) are much stronger than the poloidal wave field (
and 
, see Figure 2g). One may also find the 
phase difference between the 
and 
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 
in Figure 2e experiences a rapid growth stage until 
UT when it starts to decay from its peak, whereas in Figure 2f, the toroidal-mode wave field continues to increase until 
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 
, in which 
is the flux observed in Figure 2d and 
is a 10 min sliding average of 
. 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 
(on each energy channel) and 
, as is shown in Figures 3b–3e. For comparison, the wavelet power spectrum of 
is given in Figure 3a. One may find from Figures 3b–3e that the compressional magnetic field 
from 0613 to 0623 UT is well correlated with the electron residual fluxes 
, with the wavelet coherence of more than 0.8 at the period of 
s. Similar conclusions can be also made between 
and 
(see Figures 3h–3k, also with high coherence of over 0.8), while the coherence between azimuthal electric field 
and electron residual fluxes 
(see Figures 3n–3q) is very low to indicate a weak influence of 
on electron flux oscillations. In other words, the injected electron fluxes are modulated by the compressional toroidal ULF waves.
The wavelet power spectrum of (a) the compressional magnetic field component 
, (g) the radial electric field 
, and (m) azimuthal electric field 
. The wavelet coherence between electron residual fluxes 
in the 54–132 keV energy channels and (b)–(e)
, (h–k)
and (n–q)
, respectively. (f and l) The phase differences of 
(at multiple energy channels) from 
and 
at the dominant ultralow frequency wave period of 100 s.
Figures 3f and 3l show the phase differences of 
from 
and 
when the corresponding wavelet coherence is greater than 0.5, respectively. At 75 keV, the phase difference between 
and 
is close to 
(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 
between 
at 75 keV and 
(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 ∼
. 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 obtain from Equation 4 the L-shell changes at the rate
(11)
and 
drifts to move in the azimuthal direction, with a velocity
(12)
(Northrop, 1963), and the second term is due to the 
gradient in the radial direction 
. The third term denotes the 
drift motion in association with the wave electric field in Equation 2. Noting that the particle's relativistic kinetic energy 
, we have
(13)
(14)
. Using the angular drift frequency 
, we obtain the time derivative of 
as
(15)
Note that 
, 
, and 
all undergo minor variations due to the time variations of 
and 
(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 
and 
. When 
and 
are extremely small, the pendulum equation suggests that near-resonant particles can be trapped in the wave-carried potential well centered at 
, with the trapping frequency 
. Also, by definition, the second-order resonance condition is satisfied when 
. This simple picture, however, can become more complicated if 
and 
are not negligible. Therefore, it is important to evaluate the relative magnitudes of 
, 
, and 
, which is relatively straight-forward since the terms in the brackets of Equations 20 
22 are dimensionless quantities.
Figure 4 shows the values of 
, 
, and 
with various parameters including the wave electric field amplitude 
, wave period 
, and L-shell values, as functions of azimuthal wave number 
for resonant electrons. Among the three parameters 
, 
, and 
, we fix two of them in each column to show the variations of 
, 
, and 
with the remaining parameter. One may find from case (I) and (III) of Figure 4 that the absolute values of 
, 
, and 
increase as the wave electric field amplitude 
or L-shell increases for an intermediate or high wave number 
, whereas for case (II), the 
value hardly changes with 
. The absolute value of 
increases inversely with the wave period 
, while the absolute value of 
is proportional to 
. It is clear that although the absolute value of 
increases slightly with wave number 
, the value of 
is considerably lower than 
and 
, and therefore can usually be neglected in the modified pendulum Equation 16. Also we note that for low-
ULF waves, the 
value is always close to zero regardless of the 
, 
, and 
selections, and the dominance of the 
term suggests that the particle behavior can be approximately described by a simple pendulum equation. However, the 
term becomes increasingly important in the modified pendulum equation as 
increases, which can even exceed the 
term in certain cases (especially those with higher values of 
, 
and/or 
).
, 
, and 
as a function of wave number 
for (I) 
and 
s, (II) 
and 
mV/m and (III) 
s and 
mV/m.
) phase space, by eliminating the time variable 
from Equations 14 and 16,
(23)
(24)
is a constant that corresponds to a specific trajectory. To better understand the different effects of 
and 
on particle motion, we decompose Equation 24 with the different terms on the left hand side into
(25)
(26)The corresponding electron trajectories (denoted by the colored lines of different 
, 
and 
values) are presented in Figure 5 for Equations 25 (case (I)), 26 (case (II)), and 24 (case (III)), all with the same parameters (
mV/m, 
s, and 
). For each case, a number of electrons are launched at 
with their initial phases distributed from 
to 
in the wave-carried potential well at their respective resonant energies (
). The top three panels show the three-dimensional electron trajectories in (
) space and the bottom panels show their projection into the (
) 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 
values at the intersection points.
Contour maps of Equations 25-24 for 
toroidal ultralow frequency wave. Top panels and bottom panels show the three- and two-dimensional electron trajectories, respectively. Values of 
, 
and 
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 
exists due to the minor variations of the 
value with energy and L-shell. In case (II), the 
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 
. This asymmetry can be understood by the large variations of the 
term with 
for a fixed, high-
value in Figure 4(III). Although the 
term is included in the analysis, case (III) can be regarded as the superposition of case (I) and (II) since 
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 
range is much wider than the 
range, except that the island extends to more negative 
values due to the island overlap near 
and becomes more asymmetric with respect to 
. The centers of the additional two half-islands in case (II) with relatively small 
values merge with the saddle points (marking a separatrix between neighboring islands at 
, 
) 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 
values (
). For lower 
values, the 
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 
as shown in case (III) of Figure 5. The 
value is found to increase with the wave number 
(Figure 4), which suggests a strong 
-dependence of the particle behavior near the drift-resonant energy. For low-
toroidal ULF waves, the features of electron behavior are very similar to those for poloidal waves. For intermediate-
and high-
cases, the high asymmetries of electron trajectories caused by large 
value result in the increase of trapping widths in 
, which in turn increase the trapping widths in energy and L shell.
3, and the poloidal mode model is set up with the radial electric field in Equation 2 replaced by azimuthal electric field
(27)
(28)Here we discuss the acceleration processes with two different wave numbers, namely, 
, 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 
(low-
) ULF wave and 45.6 keV for 
(high-
) 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-
and high-
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 (
) 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.
The evolution of the distribution function of resonant electrons in energy and L-shell for 
toroidal ultra low frequency (ULF) wave (left column of (I)) and 
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 (
) space within one wave potential well from the middle column.
The same as Figure 6 but the initial energy 45.6 keV and the wave number 
.
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 
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 6b, 6f, 7b, 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 
within a single wave potential well. Therefore, nonlinear acceleration induced by poloidal ULF waves is slightly more efficient than toroidal waves for low-
cases. Note that the trapping period of 
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-
ULF waves (Figure 7), the particle behavior differs from Figure 6 in that the electron distribution becomes more extended on L-shell, energy and 
, 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-
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 
.
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 
. 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 
, and be in phase or antiphase with 
for higher or lower energies. Because of the 
phase difference between 
and radial electric field 
(Wang et al., 2018), the phase relationship between 
and electron flux would be opposite to that between 
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 
and 
in Equation 24. The term 
is dominant for low-
waves and accordingly determines a nearly symmetric resonance island (similar to case (I) of Figure 5) of particle trajectories in (
) phase space. At higher 
values, the 
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-
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-
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 
(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 
can be excited in this case (Elkington et al., 2003) and coexist with the resonance island satisfying 
. 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).
Acknowledgments
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.
Open Research
Data Availability Statement
The simulation codes are available online (via https://doi.org/10.5281/zenodo.3928114).
