Wave Normal Angle Research Articles

Simultaneous Observations of Mirror Mode Structures and Electromagnetic Ion Cyclotron Waves in the Earth's Outer Magnetosphere

AbstractElectromagnetic ion cyclotron (EMIC) waves and mirror mode structures respectively produced by the cyclotron and mirror instabilities are both generated by the temperature anisotropy of ions. However, whether these two instabilities can grow simultaneously in the magnetosphere is still unclear, as well as how they distribute if they can. In this paper, we first report an event of simultaneous observation of mirror mode structures and EMIC waves and then perform a statistical survey in the magnetosphere based on the measurements from Magnetospheric Multiscale (MMS) satellites. The event is observed in the duskside (MLT ∼ 18 hr) of Earth's outer magnetosphere (L ∼ 12) near the magnetic equator (MLAT ∼ 12°). The further linear instability analyses demonstrate that mirror mode structures and EMIC waves can be simultaneously and locally generated in the magnetosphere. The statistical results show that simultaneous mirror mode structures and EMIC waves are mainly distributed in the dawnside and duskside of the outer magnetosphere with slightly larger numbers in the dawnside but no significant differences in the proton number density, and plasma beta, proton temperature anisotropy, power‐weighted wave frequency, and wave normal angle.

Statistical Properties of Quasi‐Periodic Electromagnetic Ion Cyclotron Waves: ULF Modulation Effects

AbstractElectromagnetic ion cyclotron (EMIC) waves effectively scatter relativistic electrons in Earth's radiation belts and energetic ions in the ring current. Empirical models parameterizing the EMIC wave characteristics are important elements of inner magnetosphere simulations. Two main EMIC wave populations included in such simulations are the population generated by plasma sheet injections and another population generated by magnetospheric compression due to the solar wind. In this study, we investigate a third class of EMIC waves, generated by hot plasma sheet ions modulated by compressional ultra‐low frequency (ULF) waves. Such ULF‐modulated EMIC waves are mostly observed on the dayside, between magnetopause and the outer radiation belt edge. We show that ULF‐modulated EMIC waves are weakly oblique (with a wave normal angle ) and narrow‐banded (with a spectral width of of the mean frequency). We construct an empirical model of the EMIC wave characteristics as a function of ‐shell and MLT. The low ratio of electron plasma frequency to electron gyrofrequency around the EMIC wave generation region does not allow these waves to scatter energetic electrons. However, these waves provide very effective (comparable to strong diffusion) quasi‐periodic precipitation of plasma sheet protons.

Simultaneous Occurrence of Electromagnetic and Quasi‐Electrostatic Magnetosonic Waves

AbstractMagnetosonic (MS) waves usually manifest as electromagnetic fluctuations, while quasi‐electrostatic magnetosonic (QEMS) waves have been recently reported. Here we present an intriguing coexistence of low harmonic electromagnetic magnetosonic (EMMS) waves and QEMS waves within the same band. Ion Bernstein instabilities are calculated based on the observed proton distribution. Effective growth rates can occur in the first‐fourth harmonic bands with a broad wave number range when the wave normal angle is sufficiently large. In each harmonic band, the MS waves would transform from electromagnetic to electrostatic as the wave number increases. This study reveals that the simultaneous occurrence of EMMS and QEMS arises from the expansive excitation range of MS waves in ‐space, thereby implying the potential resonant interactions between MS and particles in a broad energy spectrum.

Prevalence and Propagation of Lightning‐Generated Whistlers in Van Allen Probes EFW Burst Data

AbstractWe assess the prevalence of ducted and non‐ducted whistler propagation using burst mode data from the Van Allen Probes Electric Field and Waves instrument (EFW). We have identified burst periods containing lightning‐generated whistlers (LGWs), resulting in a data set available for future use. The entire burst data set is filtered through an analysis of the search coil magnetometer (SCM) noise, identifying signals in frequency space that exceed an adaptive noise threshold. We implement DBSCAN (Density‐Based Spatial Clustering of Applications with Noise) to identify individual whistlers and clusters of whistlers. With magnetic spectral analysis, we calculate the mean wave normal angle (WNA) for each LGW group. We use ray tracing to estimate the expected WNA distributions for non‐ducted LGWs to compare to the data and determine methods for identifying potentially ducted LGWs. The ray‐tracing results provide a clear threshold in WNA for ducted whistlers. Using this threshold, we estimate that at least of LGWs in this data set are ducted. We find that the majority of potentially ducted whistlers are below and nearly half of LGWs below and within 9–18 MLT may be ducted.

A Parametric Study of Locally Generated Magnetosonic Waves by Ring‐Beam Hot Protons in the Martian Heavy Ion‐Rich Environment

AbstractMagnetosonic (MS) waves with frequencies above the proton gyrofrequency can be locally generated by ring‐beam protons in the Martian heavy ion‐rich ionosphere. In this study, we conduct a parametric analysis to investigate the effects of heavy ion concentrations, energy (Erb), and angle (αrb) of ring‐beam protons, and wave normal angle on the excitation features of Martian ionospheric MS waves. We find the growth rates and frequency range of MS waves decrease by including O+ and O2+ ions but are insensitive to their relative concentrations. With increasing Erb or αrb, the growth rates of MS waves show a general dropping tendency. Meanwhile, their frequency and wavenumber range are almost unaffected by increasing Erb but shrink to a narrower range mainly distributed in high frequencies by increasing αrb. Unstable MS waves expand to a wider wavenumber range but shrink to a narrower frequency range as they become more oblique.

Beam‐Driven Electron Cyclotron Harmonic and Electron Acoustic Waves as Seen in Particle‐In‐Cell Simulations

AbstractRecent study has demonstrated that electron cyclotron harmonic (ECH) waves can be excited by a low energy electron beam. Such waves propagate at moderately oblique wave normal angles (∼70°). The potential effects of beam‐driven ECH waves on electron dynamics in Earth's plasma sheet is not known. Using two‐dimensional Darwin particle‐in‐cell simulations with initial electron distributions that represent typical plasma conditions in the plasma sheet, we explore the excitation and saturation of such beam‐driven ECH waves. Both ECH and electron acoustic waves are excited in the simulation and propagate at oblique wave normal angles. Compared with the electron acoustic waves, ECH waves grow much faster and have more intense saturation amplitudes. Cold, stationary electrons are first accelerated by ECH waves through cyclotron resonance and then accelerated in the parallel direction by both the ECH and electron acoustic waves through Landau resonance. Beam electrons, on the other hand, are decelerated in the parallel direction and scattered to larger pitch angles. The relaxation of the electron beam and the continuous heating of the cold electrons contribute to ECH wave saturation and suppress the excitation of electron acoustic waves. When the ratio of plasma to electron cyclotron frequency ωpe/ωce increases, the ECH wave amplitude increases while the electron acoustic wave amplitude decreases. Our work reveals the importance of ECH and electron acoustic waves in reshaping sub‐thermal electron distributions and improves our understanding on the potential effects of wave‐particle interactions in trapping ionospheric electron outflows and forming anisotropic (field‐aligned) electron distributions in the plasma sheet.

Statistical Observations of Proton‐Band Electromagnetic Ion Cyclotron Waves in the Outer Magnetosphere: Full Wavevector Determination

AbstractElectromagnetic Ion Cyclotron (EMIC) waves mediate energy transfer from the solar wind to the magnetosphere, relativistic electron precipitation, or thermalization of the ring current population, to name a few. How these processes take place depends on the wave properties, such as the wavevector and polarization. However, inferring the wavevector from in‐situ measurements is problematic since one needs to disentangle spatial and time variations. Using 8 years of Magnetospheric Multiscale (MMS) mission observations in the dayside magnetosphere, we present an algorithm to detect proton‐band EMIC waves in the Earth's dayside magnetosphere, and find that they are present roughly 15% of the time. Their normalized frequency presents a dawn‐dusk asymmetry, with waves in the dawn flank magnetosphere having larger frequency than in the dusk, subsolar, and dawn near subsolar region. It is shown that the observations are unstable to the ion cyclotron instability. We obtain the wave polarization and wavevector by comparing Single Value Decomposition and Ampere methods. We observe that for most waves the perpendicular wavenumber (k⊥) is larger than the inverse of the proton gyroradius (ρi), that is, k⊥ρi > 1, while the parallel wavenumber is smaller than the inverse of the ion gyroradius, that is, k‖ρi < 1. Left‐hand polarized waves are associated with small wave normal angles (θBk < 30°), while linearly polarized waves are associated with large wave normal angles (θBk > 30°). This work constitutes, to our knowledge, the first attempt to statistically infer the full wavevector of proton‐band EMIC waves observed in the outer magnetosphere.

Sub‐MeV Electron Precipitation Driven by EMIC Waves Through Nonlinear Fractional Resonances

AbstractElectromagnetic ion cyclotron waves in the Earth's outer radiation belt drive rapid electron losses through wave‐particle interactions. The precipitating electron flux can be high in the hundreds of keV energy range, well below the typical minimum resonance energy. One of the proposed explanations relies on nonresonant scattering, which causes pitch‐angle diffusion away from the fundamental cyclotron resonance. Here we propose the fractional sub‐cyclotron resonance, a second‐order nonlinear effect that scatters particles at resonance order n = 1/2, as an alternate explanation. Using test‐particle simulations, we evaluate the precipitation ratios of sub‐MeV electrons for wave packets with various shapes, amplitudes, and wave normal angles. We show that the nonlinear sub‐cyclotron scattering produces larger ratios than the nonresonant scattering when the wave amplitude reaches sufficiently large values. The ELFIN CubeSats detected several events with precipitation ratio patterns matching our simulation, demonstrating the importance of sub‐cyclotron resonances during intense precipitation events.

Large Amplitude Whistler Waves in Earth's Plasmasphere and Plasmaspheric Plumes

AbstractWhistler mode waves in the plasmasphere and plumes drive significant losses of energetic electrons from the Earth's radiation belts into the upper atmosphere. In this study, we conducted a survey of amplitude‐dependent whistler wave properties and analyzed their associated background plasma conditions and electron fluxes in the plasmasphere and plumes. Our findings indicate that extremely large amplitude (>400 pT) whistler waves (a) tend to occur at L > 4 over the midnight‐dawn‐noon sectors and have small wave normal angles; (b) are more likely to occur during active geomagnetic conditions associated with higher fluxes of anisotropic electrons at 10 s keV energies; and (c) tend to occur at higher latitudes up to 20° with increasing amplitude. These results suggest that extremely large amplitude whistler waves in the plasmasphere and plumes could be generated locally by injected electrons during substorms and further amplified when propagating to higher latitudes.

Bounce Resonance Between Energetic Electrons and Magnetosonic Waves: A Parametric Study

AbstractMagnetosonic (MS) waves are electromagnetic emissions from a few to 100 Hz primarily confined near the magnetic equator both inside and outside the plasmasphere. Previous studies proved that MS waves can transport equatorially mirroring electrons from an equatorial pitch angle of 90° down to lower values by bounce resonance. However, the dependence of the bounce resonance effect on wave or background plasma parameters is still unclear. Here, we applied a test particle simulation to investigate electron transport coefficients, including diffusion and advection coefficients in energy and pitch angle, due to bounce resonance with MS waves. We investigate five wave field parameters, including wave frequency width, wave center frequency, latitudinal distribution width, wave normal angle root‐mean‐square of wave magnetic amplitude, and two background parameters, L‐shell value and plasma density. We find different transport coefficient peaks resulting from different bounce resonance harmonics. Higher‐order harmonic resonances exist, but the effect of fundamental resonance is much stronger. As the wave center frequency increases, higher‐order harmonics start to dominate. With wave frequency width increasing, the energy range of effective bounce resonance broadens, but the effect itself weakens. The bounce resonance effect will increase when we decrease the wave normal angle, or increase the wave amplitude, latitudinal distribution width, L‐shell value, and plasma density. The parametric study will advance our understanding of the favorable conditions of bounce resonance.

Statistical Distribution of Whistler Mode Waves in the Martian Induced Magnetosphere Based on MAVEN Observations

AbstractWhistler mode waves are a common type of electromagnetic waves in the Martian induced magnetosphere. Using high‐resolution magnetic field data from the Magnetometer (MAG) instrument onboard Mars Atmosphere and Volatile Evolution (MAVEN) from October 2014 to November 2022, we perform a detailed analysis of the statistical distribution of the occurrence rate, averaged amplitude, peak frequency, wave normal angle and ellipticity of left‐hand and right‐hand polarized whistler mode waves in the Martian induced magnetosphere. Our results show that whistler mode waves are mainly observed in the subsolar and magnetic pileup region, with the occurrence rate of right‐hand mode waves higher than that of left‐hand mode waves. The averaged wave amplitude ranges from 0.02 to 0.13 nT and peak wave frequency ranges from 2 to 9 Hz. We also find that the wave normal angles for both left‐hand and right‐hand polarized whistler waves are relatively larger in the subsolar region and magnetic pileup region where the corresponding wave ellipticity is closer to the linear polarization. Our results are valuable to in‐depth understanding of the generation mechanism of whistler mode waves as well as their contributions to the electron dynamics in the Martian induced magnetosphere.

Nonlinear electron scattering by electrostatic waves in collisionless shocks

We present a theoretical analysis of electron pitch-angle scattering by ion-acoustic electrostatic fluctuations present in the Earth's bow shock and, presumably, collisionless shocks in general. We numerically simulate electron interaction with a single wave packet to demonstrate the scattering through phase bunching and phase trapping and quantify electron pitch-angle scattering in dependence on the wave amplitude and wave normal angle to the local magnetic field. The iterative mapping technique is used to model pitch-angle scattering of electrons by a large number of wave packets, which have been reported in the Earth's bow shock. Assuming that successive electron scatterings are not correlated, we revealed that the long-term dynamics of electrons is diffusive. The diffusion coefficient depends on the ratio $\varPhi _0/W$ between the wave packet amplitude and electron energy, $D\propto (\varPhi _0/W)^{\nu }$ . A quasi-linear scaling ( $\nu \approx 2$ ) is observed for sufficiently small wave amplitudes, $\varPhi _0\lesssim 10^{-3}W$ , while the diffusion is nonlinear ( $1<\nu <2$ ) above this threshold. We show that pitch-angle diffusion of ${\lesssim }1$ keV electrons in the Earth's bow shock can be nonlinear. The corresponding diffusion coefficient scales with the intensity $E_{w}^{2}$ of the electrostatic fluctuations in a nonlinear fashion, $D\propto E_{w}^{\nu }$ with $\nu <2$ , while its expected values in the Earth's bow shock are $D\sim 0.1\unicode{x2013}100$ $(T_{e}/W)^{\nu -1/2}\,{\rm rad}^{2}\,{\rm s}^{-1}$ . We speculate that in the Earth's quasi-perpendicular bow shock the stochastic shock drift acceleration mechanism with pitch-angle scattering provided by the electrostatic fluctuations can contribute to the acceleration of thermal electrons up to approximately 1 keV. The potential effects of a finite perpendicular coherence scale of the wave packets on the efficiency of electron scattering are discussed.

Two‐Dimensional Hybrid Simulation of the Second‐Harmonic Generation of EMIC Waves in the Inner Magnetosphere

AbstractTwo‐dimensional (2‐D) hybrid model is developed to investigate the second harmonic (SH) generation of electromagnetic ion cyclotron (EMIC) waves. Applying the singular value decomposition method to simulated fields, we show that the SH exhibits wave properties analogous to typical EMIC waves generated by ion cyclotron instabilities, that is, left‐hand polarization and small wave normal angle. However, the bicoherence index inferred from simulated fields reflects a strong phase coupling between the fundamental wave (FW) and the SH, illustrating the nonlinear generation of the SH by the FW. The necessary conditions, especially for the wave vector relation, are further verified from a 2‐D perspective. The simulated amplitude ratios well meet the theoretical results only in the SH saturation stage, while the necessary conditions remain satisfied almost throughout the simulation. This study provides a comprehensive analysis of the SH excitation in a 2‐D simulation domain, contributing to a deeper understanding of EMIC wave nonlinear generation.

Long‐Term Variations of Energetic Electrons Scattered by Signals From the North West Cape Transmitter

AbstractVery‐low‐frequency (VLF) signals emitted from ground‐based transmitters for submarine communication can penetrate the ionosphere and leak into the magnetosphere, leading to electron precipitation via wave‐particle interaction and thereby providing a potential means for radiation belt remediation. In this study, we systematically analyze the dependence of quasi‐trapped electron fluxes scattered by signals from the North West Cape (NWC) transmitter on electron energy, L‐shell, and geomagnetic activity (i.e., the Dst index) using long‐term measurements from the DEMETER satellite. Considering potentially changed theoretical cyclotron resonant condition, we find that the variations of wave normal angle (WNA) of NWC transmitter signals or of the background electron density can explain the variated “wisp” positions in energy versus L plane. The long‐term data analyzation suggests that the energy‐dependences increases can help to distinguish the different source mechanisms of quasi‐trapped electrons. The enhancement of quasi‐trapped electron fluxes induced by NWC transmitter signals is more obvious at L = 1.8 than L = 1.6 due to higher trapped flux levels and strong pitch angle diffusion induced by transmitter signals.

Simulation study on “Wisp” electron spectra generated by NWC very low frequency transmitter signals

Very low frequency (VLF) signals emitted by worldwide spread ground-based man-made transmitters mainly propagate in Earth-ionospheric waveguides and are used for submarine communication. A portion of these signals penetrate the ionosphere and leak into the magnetosphere when the ionospheric electron density decreases on the nightside due to the attenuated sunlight. The VLF transmitter signals in the magnetosphere can scatter electrons with energy of 100 Kev in the inner radiation belt into the drift loss cone through cyclotron resonance. This is an important loss mechanism for electrons in the inner radiation belt and plays an important role in transferring energy and mass from magnetosphere to ionosphere. Electrons scattered by transmitter signals exhibit a “wisp” characteristic in <i>L</i>-<i>E</i><sub><i>k</i></sub> spectrum, satisfying the first-order cyclotron resonance relationship between the electrons and the transmitter signals. The “wisp” spectrum can be clearly observed by low earth orbit satellites, presenting opportunities to study wave-particle interactions in near-Earth space. In this study, using the drift-diffusion-source model, we reproduce the “wisp” spectrum formed by scattering effects of NWC transmitter signals observed by DEMETER satellite on March 19, 2009. Our simulation results suggest that the equatorial pitch angle of electrons, observed by DEMETER, varies with the longitude, resulting in distinctions in the observed “wisp” spectrum along different longitudes. Specifically, as the satellite approaches South Atlantic Anomaly (SAA) region, both the energy range and flux level of the observed “wisp” spectrum gradually increase. When the previously studied wave normal angle model (with a central wave normal angle of 60°) and the background electron density model are used, the energy range of the simulated “wisp” spectra is higher than the observed value. Adjusting the central wave normal angle to 40° or increasing the background density by a factor of 1.3, the simulated results accord well with the observations. Our results elucidate the scattering effect of NWC transmitter signals on electrons in the radiation belt, and emphasize the importance of analyzing the formation of “wisp” spectrum for understanding wave-particle interactions in near-earth space. Additionally, the drift-diffusion-source model can be used to study wave-particle interactions in the inner radiation belt, providing an important basis for developing radiation belt remediation technology.

New Chorus Diffusion Coefficients for Radiation Belt Modeling

AbstractWhistler mode chorus is an important magnetospheric wave emission playing a major role in radiation belt dynamics, where it contributes to both the acceleration and loss of relativistic electrons. In this study we compute bounce and drift averaged chorus diffusion coefficients for 3.0 < L* < 6.0, using the TS04 external magnetic field model, taking into account co‐located near‐equatorial measurements of the wave intensity and fpe/fce, by combining the Van Allen probes measurements with data from a multi‐satellite VLF wave database. The variation of chorus wave normal angle (WNA) with spatial location and fpe/fce is also taken into account. We find that chorus propagating at small WNAs has the dominant contribution to the diffusion rates in most MLT sectors. However, in the region 4 ≤ MLT < 11 high WNAs dominate at intermediate pitch angles. In the region 3 < L* < 4, the bounce and drift averaged pitch angle and energy diffusion rates during active conditions are primarily larger than those in our earlier models by up to a factor of 10 depending on energy and pitch angle. Further out, the results are similar. We find that the bounce and drift averaged energy and pitch angle diffusion rates can be significantly larger than the new model in regions of low , where the differences can be up to a factor of 10 depending on energy and pitch angle.

Energy Transfer Between Various Electron Populations Via Resonant Interaction With Whistler Mode Wave

AbstractElectron energization in the Earth's radiation belts caused by resonant interaction with whistler mode waves is currently under intense investigation. When the waves are excited due to cyclotron instability in collisionless plasma, that is, when the energy source for the waves is the free energy of unstable electron distribution, particle energization by excited waves is nothing but energy transfer from one group of electrons to another mediated by the waves. An example of such a process is considered in which a quasi‐monochromatic whistler mode wave packet with the frequency and the wave normal angle corresponding to the maximum growth rate is excited at the equator. Since the maximum growth rate corresponds to parallel propagating wave, only the first cyclotron resonance particles play a part in this excitation. While propagating from the equator toward the Earth, the wave normal vector becomes more and more oblique, and all cyclotron resonances, in particular, Landau resonance come into play. Wave‐particle interaction at Landau resonance leads to the wave damping and the corresponding particle energization on the average. Moreover, we show that the mean square variation of resonant particle energy greatly exceeds the average value, thus, the energy increase of some particles is much larger than the Landau resonance particles get from the wave on the average. This means that the exchange of energy between different groups of particles through a wave is a more efficient process than the amplification or damping of a wave due to its resonant interaction with particles.

Characteristics and Occurrence of Magnetosonic Waves in Plasmaspheric Plumes

AbstractMagnetosonic waves, or equatorial noise, are whistler‐mode emissions distributed near the Earth's magnetic equator between proton cyclotron frequency and lower hybrid resonance frequency. Their origin and characteristics inside and outside the plasmasphere have been investigated due to their potential role in scattering energetic electrons and protons. However, their characteristics in plasmaspheric plumes remained undocumented. This study, for the first time, statistically investigated magnetosonic waves in the plasmaspheric plumes based on the entire mission period (2012–2019) of Van Allen Probes A and B. Results showed that the occurrence rate of magnetosonic waves in plumes was 25%, with an average amplitude and wave normal angles of 44 pT and 84°–88°, respectively. Increased geomagnetic activity enhanced its amplitude and ratio of plasma frequency to electron cyclotron frequency. Approximately 78% of selected magnetosonic wave events were simultaneously observed with plasmaspheric hiss that is most effective in scattering electrons in the plume.

Variation of Electron Lifetime Due To Scattering by Electrostatic Electron Cyclotron Harmonic Waves in the Inner Magnetosphere With Electron Distribution Parameters

AbstractThrough resonant wave‐particle interactions with electrostatic electron cyclotron harmonic (ECH) waves, low‐energy plasma sheet electrons can be scattered into the atmospheric loss cone and precipitate. This process can be investigated by calculating bounce‐averaged quasi‐linear diffusion coefficients. However, the numerical calculation of ECH wave‐induced scattering rates requires the specification of several parameters, including the properties of the hot plasma sheet electrons responsible for the wave excitation. The dependence of the diffusion coefficients on these parameters and the exact contribution of scattering by ECH waves to diffuse auroral precipitation is still poorly understood and has not been carefully quantified yet. In this study, we calculate bounce‐averaged quasi‐linear scattering rates and compare the resulting lifetimes to the strong diffusion limit. We vary the temperature, plasma density, and loss cone parameters of the hot component in the electron loss cone distribution over a large range based on previous studies and observations. By adopting a new model that derives the wave normal angle from the midpoint and the angular width of the growth rate profile, our findings suggest that the electron lifetime near the loss cone is not significantly affected by the hot electron temperature and loss cone parameters. However, there is an increase in electron lifetimes with increasing plasma density. For an electric field amplitude of 1 mV/m, the electron lifetime is below or equal to the lifetime calculated using the strong diffusion limit up to E = 0.25 keV when n = 1.5 cm−3, and up to E = 0.22 keV when n = 9 cm−3.

Observational Evidence of the Four‐Wave Interaction Between Magnetospheric Whistler Mode Waves

AbstractWhistler mode wave is a crucial emission which can significantly affect the electron dynamics in the magnetosphere. The nonlinear three‐wave interaction between whistler mode waves has been frequently observed, while the four‐wave interaction between these waves is seldom reported. Here, we present two multiband whistler mode wave events by Van Allen Probes, in which the relatively weak wave bands occur exactly corresponding to the strong wave bands. We find that the frequencies of the weak and strong bands satisfy the frequency matching conditions of the four‐wave interaction, and the wave normal angles of the weak bands are almost within the possible ranges calculated from the wave vector matching conditions. Additionally, the cross‐correlation coefficients between the amplitudes of the weak bands and that of strong bands approach 0.80. These results indicate that the weak bands are very likely to be produced by the four‐wave interaction between whistler mode waves. Similar to the three‐wave interaction, the four‐wave interaction can extend the frequency of whistler mode waves from close to 0.5fce to a wider range in both the lower (0.26fce) and upper (0.78fce) bands, and thus potentially play an important role in the electron dynamics.