Properties of Intense Field-Aligned Lower-Band Chorus Waves: Implications for Nonlinear Wave-Particle Interactions

Abstract

Resonant interactions between electrons and chorus waves are responsible for a wide range of phenomena in near-Earth space (e.g., diffuse aurora and acceleration of > 1 MeV electrons). Although quasi-linear diffusion is believed to be the primary paradigm for describing such interactions, an increasing number of investigations suggest that nonlinear effects are also important in controlling the rapid dynamics of electrons. However, present models of nonlinear wave-particle interactions, which have been successfully used to describe individual short-term events, are not directly applicable for a statistical evaluation of nonlinear effects and the long-term dynamics of the outer radiation belt, because they lack information on the properties of intense (nonlinearly resonating with electrons) chorus waves. In this paper, we use the Time History of Events and Macroscale Interactions during Substorms and Van Allen Probes data sets of field-aligned chorus waveforms to study two key characteristics of these waves: effective amplitude (nonlinear interaction can occur when ) and wave packet length β (the number of wave periods within it). While as many as 10–15% of chorus wave packets are sufficiently intense ( –3) to interact nonlinearly with relativistic electrons, most of them are short (β < 10) reducing the efficacy of such interactions. Revised models of nonlinear interactions are thus needed to account for the long-term effects of these common, intense but short chorus wave packets. We also discuss the dependence of , β on location (MLT and L-shell) and on the properties of the suprathermal electron population.

Key Points

• The occurrence rate of lower-band chorus waves interacting nonlinearly with electrons is estimated
• Statistics on the fraction of intense lower-band chorus wave packets are given
• The important role of suprathermal electrons for intense chorus wave packet characteristics is shown

1 Introduction

Whistler mode chorus waves, one of the most intense electromagnetic emissions in the inner magnetosphere, are traditionally considered to be an important contributor to electron energization and scattering into the atmosphere (e.g., Andronov & Trakhtengerts, 1964; Dungey, 1963; Kennel & Petschek, 1966). Historically, the development of our understanding of how radiation belt dynamics is affected by whistler mode waves (see, e.g., first reviews by Kennel, 1969; Thorne, 1972) overlapped with the acceptance in popularity of the quasi-linear approach for the long-term description of wave-particle interactions (see original publications and their extensions to relativistic plasmas in Drummond & Pines, 1962; Lerche, 1968; Vedenov et al., 1962). Early applications of the quasi-linear diffusion theory on whistler mode waves provided remarkably good results (see, e.g., Kennel & Petschek, 1966; Lyons et al., 1972) leading to sustained use of this approach for the following 50 years.

Although strict applicability of quasi-linear theory necessitates that the waves are low-amplitude, broadband emissions with random phases (see, e.g., discussions in Albert, 2001; Karpman, 1974; Le Queau & Roux, 1987), these requirements are often relaxed in a realistic environment. First, inhomogeneities of the background plasma and magnetic field often result in stochastization of particle motion, even for electron interaction with a narrowband wave (see, e.g., Albert, 1993, 2010; Karimabadi et al., 1990; Karpman et al., 1975; Shklyar, 1981). Second, parasitic instabilities arising during wave-particle interaction (e.g., the side-band instability, see Dowden, 1982; Nunn, 1986), or the development of nonresonant background fluctuations (see, e.g., Artemyev, Mourenas, et al., 2015; Bortnik et al., 2015; Brinca, 1980), can also destroy the sensitive wave-particle resonance and emulate interaction with a broadband low-amplitude wave spectrum. Third, a strong amplitude modulation of wave packets can reduce the effective operation time scale of the resonant interaction and bring it closer to quasi-linear (random) diffusion (e.g., Tao, Bortnik, Albert, & Thorne, 2012). Therefore, these effects, taken together, can still justify the applicability (and explain the successes) of the quasi-linear approach in most environments, despite the fact that much of the time the actual population of lower-band chorus waves consists of coherent narrowband waves.

The increased availability of high-cadence waveform measurements from the inner magnetosphere over the past decade has provided us with unprecedented information on the internal structure of whistler mode wave packets in the radiation belts. A large population of very intense chorus waves was discovered (amplitudes up to 1% of the background magnetic field; see Agapitov et al., 2014; Cattell et al., 2008; Cully et al., 2008; Santolík et al., 2014; Wilson et al., 2011). This elevated the importance of nonlinear effects for wave-particle interactions and radiation belt dynamics (see discussion in Albert et al., 2013; Omura et al., 2013). Several models of nonlinear chorus wave interaction with electrons have already been relatively successful at reproducing specific event observations (e.g., Agapitov et al., 2015a, 2016; Bortnik et al., 2008; Demekhov et al., 2017; Foster et al., 2017; Mourenas et al., 2016; Nakamura et al., 2016; Shklyar, 2017). However, to date no statistical evaluation of the global importance of nonlinear effects has been performed, partly due to the lack of comprehensive statistical evaluation of the occurrence rate and spatial extent of the intense waves. The present study aims to fill this gap and help assess the global, long-term effects of nonlinear interactions in the outer radiation belt.

In the next section, we start with a discussion of existing statistical data sets of chorus wave properties and explain why such data sets cannot be used for calculations of nonlinear wave-particle interactions (section 1.1). Then, we describe what wave characteristics are needed for most models of nonlinear wave-particle interaction (section 1.2). The methods and data sets used in the present study to explore these required wave characteristics are discussed in section 2. Section 3 contains our main statistical results on intense lower-band chorus wave characteristics. Section 4 investigates the effects of the supraprathermal electrons on chorus wave properties. Section 5 provides estimates of typical time scales of nonlinear wave-particle interactions based on the wave characteristics obtained. Our conclusions are summarized in section 6.

1.1 Lower-Band Chorus Wave Statistics and Knowledge Gaps

Several important characteristics of chorus waves have been previously investigated observationally: the location (L-shell and MLT) of the most intense wave activity (e.g., Agapitov et al., 2013; Li et al., 2011; Meredith et al., 2003, 2012), the average wave intensity (e.g., Agapitov et al., 2015b, 2018; Li et al., 2012; Spasojevic & Shprits, 2013), the wave propagation angles (e.g., Agapitov et al., 2013; Artemyev, Agapitov, et al., 2015; Haque et al., 2010; Li et al., 2013; Li, Santolík, et al., 2016), and distributions of the wave intensity along magnetic field lines (e.g., Agapitov et al., 2015b, 2018; Bunch et al., 2012). All these characteristics are typically parameterized by geomagnetic activity, using AE (e.g., Shprits et al., 2007), K_p or Dst indices (e.g., Agapitov et al., 2015b, 2018), or solar wind conditions. Once the distribution of wave intensity in geometrical space (L-shell, MLT, and magnetic latitude) and the distribution of wave propagation angles are known, the local quasi-linear diffusion coefficients can be calculated (Kennel & Engelmann, 1966; Lerche, 1968) and averaged over the 3-D spatial domain of the radiation belts (e.g., Albert, 2008; Artemyev, Agapitov, et al., 2013; Horne et al., 2013; Glauert & Horne, 2005; Mourenas, Artemyev, Agapitov & Krasnoselskikh, 2014; Shprits & Ni, 2009). Such diffusion coefficients (expressed as functions of L-shell, MLT, and geomagnetic activity/solar wind conditions) are central to state-of-the-art diffusion models describing radiation belt dynamics—that is, relativistic electron acceleration (e.g., Horne et al., 2005; Li et al., 2014; Mourenas et al., 2014; Thorne et al., 2013), electron precipitation (e.g., Ni et al., 2016; Thorne et al., 2010), and electron transport to lower L-shells (together with radial diffusion, e.g., Ma et al., 2016). Because a wide range of phenomena observed in the radiation belts can be described relatively successfully by quasi-linear diffusion models (see also Albert et al., 2009; Drozdov et al., 2015; Glauert et al., 2014; Ma et al., 2015; Su et al., 2010), most past investigations of chorus waves have focused on the aforementioned wave parameters/characteristics needed to evaluate the quasi-linear diffusion coefficients.

The spatial distribution (in L-shell, MLT, and magnetic latitude) of the waves is actually of general interest, as it can be used not only in quasi-linear diffusion models but also in those including the effects of nonlinear wave-particle interactions. However, the concept of average wave intensity is not universally applicable. This concept assumes that the resonant interaction between particles and waves takes place over a sufficiently long period and is sufficiently slow that a typical particle can interact with a large number of waves (from the wave ensemble) before a significant change in the particle distribution can occur. In such a case, individual wave characteristics are not very important and averaged characteristics (e.g., mean intensity) can be safely used (quasi-linear diffusion coefficients directly depend on the averaged wave intensity—e.g., see reviews by Lyons & Williams, 1984; Schulz & Lanzerotti, 1974; Trakhtengerts & Rycroft, 2008, and references therein). But when averaging the wave intensity over a sufficiently long time interval, any information on the distribution on wave amplitudes is lost—For instance, after this averaging, one cannot distinguish anymore a single, rarely observed, but very intense wave from a group of less intense waves occurring over longer time interval. This irreversible loss of information about the detailed characteristics of the wave electromagnetic field forbids us to use the existing chorus wave databases for an accurate evaluation of nonlinear wave-particle interactions. New statistical wave models must be constructed to provide such information.

1.2 Nonlinear Wave-Particle Interactions: Critical Wave Properties

Wave-particle resonant interactions are characterized by a change of the particle transverse and parallel velocity components (Δv_⊥ and Δv_∥), or equivalently, by a change of particle energy and pitch angle (ΔE and Δα). For fixed wave characteristics, there is a direct correlation between Δv_⊥ and Δv_∥ (or between ΔE and Δα) due to energy conservation in the wave reference frame (e.g., Summers et al., 1998, and references therein). Therefore, to describe the main properties of the resonant interaction, we can operate with ΔE only (assuming that Δα is a known function of ΔE).

Each resonant interaction results in a change of the particle energy (ΔE), depending on wave characteristics and initial particle coordinates in velocity space (initial energy, pitch angle, and gyrophase, ζ₀). Energy and pitch angle (calculated at some point of the particle trajectory, e.g., at the equatorial plane) are conserved in the absence of a wave (i.e., each time a particle crosses the equatorial plane, the pitch angle remains the same), whereas the particle gyrophase is fast oscillating and can be excluded from the system by a gyroaveraging procedure (i.e., by introducing the adiabatic invariant [magnetic moment]; see Landau & Lifshitz, 1988; Northrop, 1963). Therefore, the gyrophase is generally ignored in calculations of unperturbed (in the absence of waves) particle trajectories, whereas the dependence ΔE(ζ₀) is considered as a dependence on a random variable ζ₀. Averaging over a range of ζ₀ values is equivalent to averaging over a particle ensemble (with the same initial energy and pitch angle) and such averaged variables, 〈ΔE〉 and 〈Δα〉, can be used to study the evolution of the particle distribution function in a system including many resonant interactions. In addition to 〈ΔE〉, one can define the variance 〈(ΔE)²〉 and use these two terms to construct a Fokker-Planck equation for the particle distribution (e.g., Van Kampen, 2003). The main differences between quasi-linear resonant electron interaction with low-amplitude waves and nonlinear resonant interaction with intense waves depend on two factors (Karpman, 1974; Le Queau & Roux, 1987; Shapiro & Sagdeev, 1997): (i) which is the more important term among 〈ΔE〉 and 〈(ΔE)²〉? and (ii) how do 〈ΔE〉 and 〈(ΔE)²〉 depend on wave intensity normalized to the background magnetic field ?

Under the approximation of low wave amplitudes (quasi-linear theory), we have 〈ΔE〉=0 and (e.g., Kennel & Engelmann, 1966, and references therein). For intense waves, one can obtain and (Albert, 1993; Karpman et al., 1974), where the index scat means that 〈ΔE〉_scat is calculated over a ζ₀ range corresponding to particle nonlinear scattering (also called phase bunching; Omura et al., 1991). There is also 〈ΔE〉_trap for trapped particles, and 〈ΔE〉_trap is not proportional to in any degree (see examples of 〈ΔE〉_trap evaluation in, e.g., Artemyev et al., 2014, and references therein). In the absence of an average value (〈ΔE〉=0), diffusion ∼〈(ΔE)²〉 becomes the dominant process. But if 〈ΔE〉_scat,trap≠0, nonlinear scattering and trapping dominate the evolution of the particle distribution function. This is because is much larger than the diffusion term for intense waves (note that , since ; the magnetic field of the most intense chorus waves always remains much smaller than the background geomagnetic field).

The transition between quasi-linear (diffusion ) and nonlinear (drift ) resonant interaction is defined by a critical wave intensity level : For , nonlinear interaction plays the dominant role. This critical level is determined by the competition between the wave electromagnetic field force and generalized inertial forces acting on charged particles in an inhomogeneous magnetic field (a combination of inertia and mirror force; see, e.g., Bell, 1984; Karpman et al., 1974; Nunn, 1971, 1974). The full expression of such generalized inertial forces (and, thus, of ) has to be derived for each particular plasma system (see, e.g., Agapitov et al., 2014; Bell, 1986; Crabtree et al., 2017; Omura et al., 2007, 2008; Tao et al., 2013), but their order of magnitude is given by the product of the magnetic moment and the field-aligned gradient of the background magnetic field, ∼μ∇_∥B₀. For relativistic electrons interacting with lower-band chorus waves at magnetic latitudes ≈5–15° in a dipolar geomagnetic field, it reduces to a dimensionless parameter 1/η ∼ m_ec²/B_eqR ∼ c/RΩ_eq, where R = R_EL is the spatial scale of geomagnetic field inhomogeneity and Ω_eq>0 is the equatorial electron gyrofrequency. Although lower-band chorus waves are usually generated near the magnetic equator, they indeed reach significant amplitudes only several degrees away from the equator at least (e.g., Omura et al., 2008, 2013). Accordingly, we assume here that the main nonlinear interaction occurs around magnetic latitudes ≈5–15° to provide a rough estimate of the threshold wave amplitude for nonlinear interaction.

To characterize wave-particle interaction, we can therefore introduce a dimensionless parameter , ratio of the peak wave amplitude to a critical wave amplitude for nonlinear interaction of the order of B₀/η. This gives an approximate threshold (corresponding to a critical intensity ) for electron energy ∼0.3–0.5 MeV at L ∼ 4–5 and equatorial pitch angles of ∼50–70° in an approximately dipolar magnetic field, where cyclotron resonance with parallel lower-band chorus waves occurs at geomagnetic latitudes ≈5–15° (Agapitov et al., 2014; Artemyev et al., 2014; Tao & Bortnik, 2010). For the above parameters, corresponds to a threshold wave amplitude of about ≈120–160 pT. Of course, the threshold only provides a very simplified and approximate estimate of the minimum wave amplitude required for nonlinear interaction with relativistic electrons: A complex multiplicative factor in the full expression of should be evaluated in each considered situation, using the precise characteristics of electrons, plasma, magnetic field, and wave frequency—which leads, for example, to an increase of the threshold wave amplitude at higher electron energy or smaller equatorial pitch angles (Agapitov et al., 2014; Tao & Bortnik, 2010; Tao et al., 2013). For the sake of simplicity, however, we shall hereafter use both the parameter (the product of the normalized wave magnetic field magnitude, B_w/B₀, and the normalized wave number, kR, where B₀ and R are the typical magnitude and spatial scale of the background magnetic field) to characterize the amplitude of observed field-aligned chorus waves, and the approximate threshold as an estimate of the minimum wave amplitude required for significant nonlinear effects during resonant interactions with relativistic electrons in the outer radiation belt.

Unlike quasi-linear diffusion, nonlinear resonant interactions lead to particle drift (due to nonlinear scattering) and fast particle transport (due to trapping) in phase space (see reviews by Albert et al., 2013; Artemyev et al., 2016; Shklyar & Matsumoto, 2009). Scattering, even nonlinear, is a local process, and 〈ΔE〉_scat depends only on the wave characteristics at the resonance (at one particular point along a particle trajectory). Trapping, on the other hand, can lead to a significant change in the particle motion, when particles locked in the resonance by the wavefield move rapidly over a long distance, of the order of ≈LR_E (see examples in Artemyev, Vasiliev, et al., 2013; Bortnik et al., 2008; Demekhov et al., 2006; Hsieh & Omura, 2017a; Omura et al., 2015). Therefore, 〈ΔE〉_trap depends on the distribution of wave characteristics along particle trajectories (along geomagnetic field lines) and on the shape of the wave packet (e.g., Artemyev et al., 2012; Tao, Bortnik, Thorne, et al., 2012). The most important characteristics of a wave packet are its amplitude and its length (duration). Resonant electrons interacting with field-aligned waves move in the opposite direction to the wave propagation (except for very high energy particles; see Omura et al., 2007), and thus, the time scale of trapped particle acceleration is defined by how long particles can move through the wave packet. Therefore, in this study we use a relative scale of wave packet, β, defined as the number of wave periods observed within one wave packet, to characterize this effect.

In summary, we use two main parameters to describe how the properties of observed waves, and β, affect the efficiency of nonlinear wave-particle interaction: tells us if the wave is sufficiently intense to trap electrons into resonance (when ) and β indicates if the wave packet length is sufficient to provide a relatively prolonged nonlinear resonant interaction (when β > 50 typically; see details in the section 5). These parameters provide simplified and not complete characterizations of the efficiency of the nonlinear resonant interactions, which also depend on other (secondary) factors: other wave characteristics (frequency and angle of propagation), resonant particle energy and pitch angle, magnetic latitude of resonance, etc. (see reviews by Albert et al., 2013; Artemyev et al., 2016; Omura et al., 1991; Shklyar & Matsumoto, 2009, and references therein). However, as a first step toward a full evaluation of nonlinear effects, we shall examine here the broad characteristics over all L and MLT regions and neglect (in a first-order approximation) the effects of wave frequency and electron energy (which define the resonance condition), considering fixed (narrowband) lower-band chorus frequencies ω/Ω_eq∼0.25–0.35. In such a case, the distribution of these two main parameters provides first-order estimates of the number of observed waves that potentially interact nonlinearly with electrons. This information is important to assess whether we need to include nonlinear effects into global models of radiation belt dynamics (e.g., Albert et al., 2009; Drozdov et al., 2015; Glauert et al., 2014; Ma et al., 2015; Su et al., 2010) or whether nonlinear effects are of more limited interest, restricted to the description of short-term events observed under particular conditions.

2 Data Set and Methodology

We used data from five Time History of Events and Macroscale Interactions during Substorms (THEMIS) probes (Angelopoulos, 2008) and two Van Allen Probes (Mauk et al., 2013). We utilized the measurements of the THEMIS search coil magnetometer (Le Contel et al., 2008) providing magnetic field fluctuations and waves in three orthogonal directions over a frequency range from 0.1 Hz to 4 kHz and measurements from the THEMIS Electric Field Instrument providing waveforms in three orthogonal directions from DC up to ∼16 kHz (Bonnell et al., 2008). The THEMIS Fluxgate Magnetometer (Auster et al., 2008) data are also used for the determination of the background magnetic field. For the investigation of whistler mode chorus waves observed by Van Allen Probes, we used measurements from the Electric and Magnetic Field Instrument Suite and Integrated Science Waves instrument (Kletzing et al., 2013) providing all components of the wave electric and magnetic fields. To exclude measurements within the plasmasphere, we employed the electron plasma frequency deduced from the upper hybrid resonance frequency line in the 10-400 kHz range (Kurth et al., 2015). To determine the electron β_e parameter (the electron total thermal pressure to magnetic field pressure ratio), we used measurements provided by the Helium Oxygen Proton Electron (HOPE) instrument from the Energetic Particle Composition and Thermal Plasma Suite (Funsten et al., 2013; Spence et al., 2013). Electron thermal pressure is calculated by integrating the electron distribution function within the energy range 0.2–40 keV.

We surveyed 1.5 years of Van Allen Probes observations (covering a full MLT cycle) and 5 years of THEMIS spacecraft observations, when waveforms are available during the burst mode. For the Van Allen Probes data set, we used waveforms during continuous burst operations and rotated them into the magnetic field-aligned coordinate system (originally in the spacecraft science coordinate), which provide a fairly good coverage (with no bias in the wave amplitude). Contrary to the burst mode of the Van Allen Probes, the burst mode of THEMIS data is triggered during most years by magnetic field B_z (in geocentric solar magnetospheric coordinates) dipolarization, or, during some seasons, by wave power increases in the whistler mode frequency range. Thus, in the THEMIS data set, we mainly deal with chorus waves observed during particle injections from the magnetotail. We focused on lower-band chorus waves with frequency ω∈[0.1,0.5]Ω_ce (Ω_ce being the equatorial electron gyrofrequency), with a relatively moderate frequency variation: Δω/ω < 0.5 during each selected wave packet. We also limit our study to the outer radiation belt region, outside of the plasmasphere but inside the typical flow-breaking region, that is, radial distances R from Earth center between 4 and 10R_E. Finally, we elected to the present study only field-aligned chorus waves (most effective in nonlinear interaction; see, e.g., Shklyar & Matsumoto, 2009), by requiring that the wave normal angle averaged over a wave packet remain less than 25^∘.

As discussed in section 1.2, we are interested here in two main characteristics of intense chorus waves: their relative amplitude and the wave packet size β. Accordingly, we need to separate the recorded wave signal into individual wave packets and then determine these two characteristics for each wave packet. Toward that goal, we investigated waves with time-averaged full amplitude B_w>10 pT, corresponding to relatively intense waves (B_w was calculated as the square root of the transverse wave intensity). For quasi-parallel whistler mode waves, B_w(t) represents the profile of the wave packet envelope (the intensity of the wave transverse magnetic field is constant for parallel waves). We further separated the waveforms into wave packets, by checking when B_w(t) decreased below 200 pT (a threshold chosen sufficiently high to allow safely separating intense wave packets reaching the nonlinear regime among a background of ∼50- to 100-pT weaker waves; see Figure 1a) and marking the closest troughs in the wavefield profile as the start and end of each packet. For each packet, we then calculated the wave peak intensity and the wave packet duration (see Figure 1 for and β definitions). We obtained β from the ratio of the wave packet duration and the average wave period within this packet. To determine , we used the equatorial geomagnetic field intensity and the L^∗ parameter provided by the Van Allen Probes data set. For THEMIS spacecraft measurements, often at large radial distances from the Earth where geomagnetic field is far from dipolar, we used the measured B_z field (in geocentric solar magnetospheric coordinate) to estimate the equatorial geomagnetic field intensity (since most THEMIS measurements were near the equatorial plane, one has ) and estimated the effective radial location as the distance L^∗, where the dipolar equatorial geomagnetic field is equal to the measured B_z. Using these parameters, we calculated both η = RΩ_eq/c (with R = R_EL^∗) and .

Figure 1

Lower-band chorus wave packets observed by the Van Allen Probe B on 3 January 2015. One of the two wave transverse magnetic field components is displayed together with the transverse wave intensity. (top) Identified wave packets are separated by vertical dashed lines, with β the number of wave periods inside each such packet. Stars indicate the peak wave magnetic field amplitude used for calculations. (bottom) Corresponding time-integrated wave spectrum: frequency normalized to the equatorial electron gyrofrequency, vertical lines showing the mean frequency of the identified wave packets.

3 Fraction of Intense Chorus Waves Interacting Nonlinearly With Electrons

Using the above lower-band chorus wave data sets from Van Allen Probes and THEMIS, we calculated the occurrence rate of intense waves (i.e., the percentage of time when intense wave packets are observed during the total time interval of wave measurements within the burst mode data set). However, the burst mode on THEMIS spacecraft was mainly switched on during dipolarizations, correlated with injections, contrary to the Van Allen Probes waveform measurements, which collected such waveforms without any preconditions. Therefore, to compare THEMIS and Van Allen Probes wave packet statistics, we restricted Van Allen Probes waveforms to relatively high AE (>500 nT) time periods (this selection is justified in the text below). Figure 2a shows the occurrence rate of intense chorus waves in this data set. As previously discussed, the waves were selected to be sufficiently intense ( ) to potentially trigger nonlinear effects and are plotted as a function of L^∗ for three different ranges of the β parameter, at geomagnetic latitudes within ±10^∘ of the equator. For β > 3 (roughly the lowest β level for waves able to participate in nonlinear wave-particle interactions and the largest, statistically most significant subset in our database), THEMIS and the Van Allen Probes give similar occurrence rates ∼10–15% for such intense waves, with almost no dependence on L^∗. This shows that during relatively high geomagnetic activity (AE > 500 nT) periods, there are a lot of intense chorus wave packets, from L ∼ 4.5 in the outer radiation belt outside the plasmasphere up to L ∼ 10 in the magnetotail. The occurrence rate of relatively long, intense wave packets with β > 10, is roughly half of β > 3 in Van Allen Probes statistics at L ∼ 4.5–6 as well as in THEMIS statistics at L≥6.6, and it decreases toward higher L^∗. This reveals that short wave packets with 3 < β≤10 have the highest occurrence rate among all intense chorus wave packets. During roughly half of the time intervals when intense chorus emissions with are observed, the waves are composed of short, intense wave packets. The occurrence rate of even longer wave packets (those with β > 50) is only ∼10–15% of the occurrence rate of all β > 3 wave packets (Figure 2a) and decreases quickly as L^∗ increases above 5–6.

Figure 2

(left) Occurrence rate of intense ( ) lower-band chorus wave packets at low latitudes, as a function of L^∗ based on Van Allen Probes and THEMIS statistics between 20 and 10 magnetic local time (MLT) where the most intense waves are observed (Agapitov et al., 2013; Li et al., 2011; Meredith et al., 2002). (right) Occurrence rate of intense ( ) lower-band chorus wave packets as a function of β and AE based on Van Allen Probes statistics between 20 and 10 MLT.

Although Van Allen Probes measurements at L ∼ 5.5 give a similar occurrence rate of β > 3 wave packets as THEMIS, they give ≈2 (≈5) times lower occurrence rates than THEMIS for β > 10 (β > 50). These apparent discrepancies likely stem from (i) the different orbital coverages and measurement periods of THEMIS spacecrafts and the Van Allen Probes, (ii) the known bias of THEMIS waveform measurements during dipolarization/injections, and (iii) the efficient detection of waveforms by the burst collection of THEMIS. This is likely because burst mode wave collection on board THEMIS is performed in a localized region of activation (injection/dipolarization) where intense waves are also present, whereas for the Van Allen Probes we use a global index, AE, that is not as good a measure of activity when this activity is quite localized. Thus, the relative occurrence rate of β > 10 (β > 50), as compared with β > 3, wave packets in THEMIS statistics may be higher than in Van Allen Probes statistics, due to the anticipated increase of the relative occurrence rate of β > 10 (β > 50) wave packets as AE increases from AE < 100 nT to AE > 150 nT in Figure 2b. It is also possible that the chorus wave characteristics measured during injections (THEMIS) are different from the more usual chorus elements observed on Van Allen Probes in the radiation belts. Note also that both the Van Allen Probes and THEMIS satellites provide the occurrence rate of intense wave observations during the burst mode of wave measurements. However, the Van Allen Probe burst mode is not triggered by wave amplitude. The occurrence rate of intense waves derived from Van Allen Probes measurements is therefore expected to be close to the actual time-averaged occurrence rate. Comparisons with THEMIS results in Figure 2a indicate that statistics from THEMIS generally provide an overestimate of the actual occurrence rate of intense waves, except at low β < 10.

The Van Allen Probes data set is indeed sufficiently representative to provide the distribution of intense ( ) chorus wave packets as a function of the geomagnetic activity (AE index). Figure 2b shows that the occurrence rate of intense waves increases with AE for all β levels. Above AE ∼ 150 nT, the 1-D occurrence rate distributions with β have similar shape but only vary in magnitude under different AE conditions. Therefore, relative proportion of short (β ∼ 3–10) and long wave packets (β > 50) does not depend on AE when AE > 150 nT, but the occurrence rate of intense wave packets increases roughly linearly with AE.

Considering separately the Van Allen Probes and THEMIS data sets at low magnetic latitudes, the full distribution of intense wave packets in the space is displayed in Figure 3 for three L^∗-shell domains. This shows that wave packets are distributed over and β∈[2,100], that is, the collected data set consists of intense wave packets with a wide range of sizes (β parameter). There is no significant difference between distributions collected at different L^∗; however, more wave packets with large β are again observed at smaller L^∗. The main core of the wave packet distribution, with the highest occurrences, is confined within and β∈[2,10], demonstrating again that the main population of intense chorus wave packets consists of short packets.

Figure 3

Distribution of intense lower-band chorus wave packets in the space in three L^∗ ranges based on Van Allen Probes and THEMIS statistics at low latitudes between 20 and 10 MLT.

Interestingly, the increase of the wave intensity is accompanied by a global increase of wave packet length β. However, it is also worth noting that most of the highest intensity ( ) wave packets have β∈[2,20] (Figure 3) over all L^∗ domains. Thus, intense, short wave packets with β < 15 are much more common than intense, long packets with β > 40. Finally, because only near-equatorial observations from the Van Allen Probes and THEMIS spacecraft are plotted in Figure 3, it does not include short wave packets found at high latitudes >30^∘ (e.g., Tsurutani et al., 2011).

Actually, we chose to focus on the MLT region corresponding to the strongest population of chorus waves, to consider a relatively homogeneous wave population comprising the most intense waves generated by plasma sheet electrons injected near midnight. Thus, only chorus waves between 20 and 10 MLT have been plotted in Figures 2 and 3, because that is where the most intense chorus waves occur (e.g., see Agapitov et al., 2013; Li et al., 2011; Meredith et al., 2002). To support our MLT restriction, we plot the average and the wave packet occurrence rate in (MLT, β) space in Figure 4 using Van Allen Probes data. In Figure 4a, we see that wave packets of all intensities (including the most intense ones, ) are observed between 20 and 10 MLT. Also, in Figure 4b we see that wave packets of all sizes, including the longest ones (β > 20), exhibit the same longitudinal preference. It also corroborates the findings from Figure 3 that longer wave packets have higher average amplitudes (Figure 4a) and much smaller occurrence rates (Figure 4b). For instance, long wave packets with β > 50 have average amplitudes , 2–3 times larger than short wave packets with β < 10 but occurrence rates that are at least 20 times smaller. Figure 4 also indicates that the longest wave packets, those with β > 50, are mostly observed in the midnight/dawn sector with MLT∈[0,7] and have large average amplitudes, . Therefore, the most efficient, traditional nonlinear interaction between electrons and field-aligned chorus waves, involving long wave packets, is expected to occur in the MLT sector where chorus wave activity is typically high. Nevertheless, the vast majority of nonlinear interactions, even in this sector, involve short chorus wave packets of moderately high intensity, those with β < 2–10 (Figure 4b).

Figure 4

Distribution of lower-band chorus wave intensity and wave packet occurrence rate in the (MLT, β) space based on data from the Van Allen Probes.

4 Role of Background Hot Electrons

The HOPE instrument on board Van Allen Probes provides an opportunity to analyze the correlation between chorus wave activity and thermal/suprathermal/warm electron distributions (e.g., Fu et al., 2014; Li, Mourenas, et al., 2016; Ma et al., 2017). Using it, we estimate the equatorial electron β_e parameter, defined as the ratio of electron thermal pressure and magnetic field pressure. This parameter controls whistler mode wave propagation and damping or amplification (e.g., An et al., 2017; Ma et al., 2017; Yue et al., 2016, and references therein). We considered separately magnetic latitude ranges |λ|<5^∘ and |λ|>10^∘ to assess the effect of propagation (along magnetic field lines) of intense chorus wave packets, usually generated close to the equator (note, however, that magnetic latitudes rarely exceed 20^∘ due to the orbit of Van Allen Probes).

Figure 5 shows the occurrence rate of intense lower-band chorus wave packets (with ) in the (β_e,β) space. Close to the equator at |λ|<5^∘ (Figure 5a), intense waves are observed over a broad range of β_e∈[0.01,1] and most of the waves are observed when β_e>0.04. The existence of a significant population of near-equatorial wave packets in high β_e conditions may stem from the presence of high electron fluxes in the generation region, necessary to provide a sufficiently large wave growth rate (e.g., Demekhov et al., 2017; Omura et al., 2013, and references therein). Consistent with Figure 4a, Figure 5 also shows that intense chorus wave packets with the highest occurrence rates are short, β < 15. Long wave packets, with β > 50, have very low occurrence rates, especially in high β_e conditions. This suggests that, assuming that the generation of long wave packets involves a weakly anisotropic population of resonant electrons (to provide the free energy for weakly growing waves), the low background electron temperature (and β_e) may be necessary to reduce the Landau damping of these quasi-parallel waves (e.g., since Landau damping increases at higher electron density and temperature, corresponding to higher β_e, for finite wave normal angles, see Artemyev et al., 2016; Bortnik et al., 2007; Chen et al., 2013; Kennel, 1966). However, we should note that this suggestion is based on the assumption that the locally measured electron temperature is similar to the equatorial electron temperature (we use equatorial magnetic field estimated from L^∗ and local electron pressure to evaluate β_e). Moreover, the wave occurrence rates shown in Figures 5a and 5b can be affected by a nonuniform β_e distribution, since large β_e values are observed much less often than small β_e. Figures 5c and 5d show the wave occurrence rate normalized on the β_e occurrence rate (collected over 1.5 years of Van Allen Probe observations), demonstrating that the main conclusion derived from Figures 5a and 5b does not depend on the β_e distribution—Intense waves are generally observed near the geomagnetic equator for larger β_e than waves observed at |λ|>10^∘.

Figure 5

(left panels) Distribution of intense lower-band chorus wave packets in the (β_e,β) space for two separate latitude ranges: (a) |λ|<5^∘ and (b) |λ|>10^∘ (only Van Allen Probes data are used). (right panels) Distribution of intense lower-band chorus wave packets in the (β_e,β) space, normalized on the β_e occurrence rate.

In the off-equatorial region (|λ|>10^∘), intense chorus wave packets are observed for even lower β_e than at the equator, β_e<0.1. As in the near-equatorial region, long wave packets are observed only for low β_e. Moreover, off the equator long wave packets are typically of lower β_e than at the equator. These observations support the idea that intense waves propagating as long wave packets can be found mostly in the low electron temperature plasma, where Landau damping is significantly reduced. Similar occurrence rates in the near-equatorial and off-equatorial regions correspond to different ranges of β_e. We infer that most of the intense chorus waves generated for β_e>0.1 do not reach |λ|≥10^∘ (at least not with the similar intensity), because they are likely damped or lose their coherence. Conversely, waves generated close to the equatorial plane when β_e<0.1 experience less Landau damping and can more easily propagate to higher latitudes, where they represent the main population of intense wave packets.

5 Implications for Nonlinear Wave-Particle Interaction

Here we estimate the characteristic time scales of nonlinear wave-particle interactions using the observations described earlier. The efficiency of the wave-particle interaction depends on how long particles can stay in resonance with the waves, that is, how long a particle with velocity v can remain sufficiently close to the resonant velocity v_R. Particles can be assumed to be in resonance with a wave if |v − v_R|<Δv_R, where the resonance width Δv_R is determined by the wave dispersion (Δv_R is approximately equal to the difference between the group and phase velocities; Karpman, 1974), the background geomagnetic field inhomogeneity (for monochromatic waves Δv_R is proportional to the spatial gradient of v_R; Albert, 2010; Trakhtengerts & Rycroft, 2008), and the wave amplitude (for intense waves Δv_R is proportional to ; Karpman et al., 1974; Nunn, 1974). For low-amplitude waves ( ), particle trajectories are only slightly influenced by the wavefield and particles quickly pass through the resonance (see the schematic in Figure 6). For intense waves ( ), however, resonant particle trajectories are significantly affected by the wavefield, and particles are forced to spend a much longer time in the resonance. Nonlinear scattering can be characterized by the typical time scale of wave-particle resonant interaction, which can be approximated with the period of particle oscillation in the (v − v_R,ζ) plane (Figure 6b). Particle trapping can be characterized as an interaction over many such periods (Figure 6c).

Figure 6

A schematic view of the resonant particle trajectory in the phase plane (difference of particle velocity v and resonant velocity v_R, wave phase ζ) for three different regimes: quasi-linear scattering (low-amplitude waves, ), nonlinear scattering (phase bunching, ), and trapping ( ).

Taking into account the limited length β of intense parallel chorus wave packets, the typical time scale (Δt) of passage of a resonant electron through a given wave packet can be estimated as Δt≈2πβ/k|v_g−v_R|, where v_g=∂ω/∂k denotes the wave group velocity and v_R=(ω−Ω_ce/γ)/k the velocity of the particle in cyclotron resonance with the wave (Ω_ce>0 being the electron gyrofrequency, k the wavenumber, and γ the Lorentz factor). The minimum time interval of nonlinear wave-particle interaction can be estimated by the period of trapped particle motion Δt_tr≈2π/Ω_tr, where (e.g., Karpman et al., 1974; Nunn, 1971; Omura et al., 2007, 2008). As a result, resonant particles may interact with wave packets times, with . is an important parameter of nonlinear interaction, because it determines the efficiency of electron acceleration (e.g., see models from Foster et al., 2017; Hsieh & Omura, 2017b; Omura et al., 2015). Using a simplified approximation v_g≈ω/k (for parallel propagating waves the typical ratio of kv_g/ω is 1–2; see Stix, 1962), we obtain Δt≈2πβ/Ω_ce and

where the factor , which depends on the transverse velocity v_⊥ of the resonant particles, is ∼1 (∼1.4) for ∼300–500 keV (>2 MeV) electrons with equatorial pitch angles α₀∼50–70° interacting with typical parallel lower-band chorus waves with frequencies ∼(0.15–0.25)Ω_ce, and an electron plasma frequency to gyrofrequency ratio ∼3.5–4 at L = 4–5. The parameter is, therefore, mainly determined by the wave packet length β, the normalized wave amplitude , and the electron energy through γ. Figure 7 shows the distribution of the parameter for the observed intense chorus wave packets, assuming cyclotron resonance with ∼300-keV and 2-MeV electrons in various L regions. The majority of the wave packets fall into the regime when considering 300-keV electrons and only a very small percentage of wave packets have for both 300-keV and 2-MeV electrons, especially at L < 6 in the outer radiation belt.

Figure 7

Distribution of the nonlinear parameter in three L^∗ ranges for first-order cyclotron resonance with 300-keV (black) and 2-MeV (blue) electrons at L^∗∼4–5. Numbers indicate the proportion (normalized to 100%) of the distributions with .

6 Conclusions

In this paper, we have investigated the statistical properties of intense, quasi-parallel lower-band chorus wave packets in the magnetosphere. We focused on two main characteristics of wave packets, important for models of nonlinear wave-particle interaction: the normalized wave amplitude and the wave packet length β. Our analysis of THEMIS and Van Allen Probes observations revealed the following:

• The occurrence rate of intense ( ) wave packets reaches ∼20% during active geomagnetic conditions (AE > 500 nT), but the majority of the observed intense wave packets (even for very intense waves with ) are rather short, with β < 10. On average, only ≈5% of the intense wave packets are sufficiently long (β > 50) to justify the applicability of the typical nonlinear models of electron interaction with long wave packets for particle energies <0.5–1.0 MeV.
• The most intense wave packets with are concentrated around the midnight/dawn MLT sector.
• The suprathermal electron population likely controls the characteristics of intense chorus wave packets. Under β_e>0.01 conditions, intense wave packets are mostly found in the near-equatorial generation region, but not off the equator. Only for β_e<0.01, intense wave packets are observed in both regions. We caution that the above conclusions are based on statistical chorus wave characteristics, averaged over years of measurements. Long wave packets with β > 50 may still prevail during some limited time periods, especially during low β_e conditions.

Our statistical results on whistler wave characteristics indicate that new theoretical approaches are needed to investigate the effects of short, intense wave packets (being the prevalent population) on radiation belt dynamics, and, in particular, on electron acceleration. Moreover, even though long wave packets (β > 50) are relatively infrequent (∼5%), they are very efficient at scattering electrons, so the relative impact of both types of nonlinear interactions, those due to short and long wave packets, on the global radiation belt dynamics should be evaluated in the future. Finally, we note that, at very high electron energies (∼2 MeV), a finite proportion of intense wave packets may still undergo prolonged nonlinear resonant interactions (N_NL>10; see Figure 7) for which the usual processes of nonlinear scattering and trapping may still be significant, although cyclotron resonance then occurs at higher latitudes >20–30° and still remains <10 in general for less than 3-MeV electrons.

Acknowledgments

X. J. Z., A. V. A., and V. A. acknowledge NASA contract NAS5-02099 for use of data from the THEMIS mission, specifically J. W. Bonnell and F. S. Mozer for use of EFI data, A. Roux and O. LeContel for use of SCM data, and K. H. Glassmeier, U. Auster, and W. Baumjohann for the use of FGM data provided under the lead of the Technical University of Braunschweig and with financial support through the German Ministry for Economy and Technology and the German Center for Aviation and Space (DLR) under contract 50 OC 0302. The work of X. J. Z. was also supported by RBSP-EMFISIS and RBSP-ECT funding 443956-TH-81074 and 443956-TH-79425 under NASA's prime contract NNN06AA01C. J. B. would like to acknowledge support from NASA award NNX16AG21G, NSF/DOE basic plasma physics award DE-SC0010578, and AFOSR FA9550-15-1-0158. The research at The University of Iowa was supported by JHU/APL contract 921647 under NASA Prime contract NAS5-01072. Van Allen Probes EMFISIS data were obtained from https://emfisis.physics.uiowa.edu/data/index, and THEMIS data are from http://themis.ssl.berkeley.edu/. We also thank the World Data Center for Geomagnetism, Kyoto, for providing AE index, and the Space Physics Data Facility at the NASA Goddard Space Flight Center for providing the OMNI data used in this study.