Simultaneous occurrence of four magnetospheric wave modes and the resultant combined scattering effect  on radiation belt electrons

1.   Introduction

The Earth’s radiation belts, which are also called the Van Allen belts, consist of energetic electrons trapped by the geomagnetic field (e.g., Baker et al., 1986; Li XL et al., 1997), in which electrons undergo gyration, bounce, and drift motions. The energetic electron fluxes are affected by the combination of acceleration, transportation, and loss processes within the magnetosphere (Reeves et al., 2003; Ma X et al., 2020). Wave–particle interactions are important for radiation belt dynamics and can significantly change the pitch angle distributions (PADs) and energy spectrum of electrons (Li LY et al., 2005; Summers et al., 2007a, b; Thorne, 2010; Ni BB et al., 2015).

Several typical plasma wave modes observed in the outer radiation belt (3 < L < 6) are believed to contribute to scattering energetic electrons. Electrostatic electron cyclotron harmonic (ECH) waves are usually observed outside the plasmapause close to the magnetic equator, propagating almost perpendicularly to the ambient magnetic field in bands between the harmonics of the electron gyrofrequency (${{f}}_{\text{ce}}$) with the dominant wave power centered around (n + 1/2)${{f}}_{\text{ce}}$ (Kennel et al., 1970; Roeder and Koons, 1989; Meredith et al., 2009; Ni BB et al., 2017a, b; Lou YQ et al., 2018; Guo DY et al., 2021). Electron cyclotron harmonic waves can resonate with plasma sheet electrons at hundreds of electron volts to ~10 keV and scatter them into the loss cone, resulting in the generation of the nightside diffuse aurora (Lui et al., 1977; Meng et al., 1979; Ni BB et al., 2008, 2016; Frey et al., 2019; Zhang XJ et al., 2019). Electron cyclotron harmonic waves also contribute to the pulsating aurora at higher L values.

Plasmaspheric hiss is a kind of incoherent, broadband whistler mode emission generally confined within the dense plasma environment, such as the plasmasphere and plasmaspheric plumes, with a frequency range of ~100 Hz to a few kilohertz (Thorne et al., 1973; Chappell, 1974; Carpenter et al., 1993; Laakso et al., 2015; Yu J et al., 2017; Su ZP et al., 2018; Guo DY et al., 2021; Li HM et al., 2022; Yang L et al., 2022). However, hiss captured outside the plasmapause, which is called exohiss, can primarily be observed in the dayside high-latitude ($\lambda$ > 35°) region (Russell et al., 1970; Thorne et al., 1973). Statistical results indicate that plasmaspheric hiss is a major source of exohiss waves outside the plasmapause (Wang JL et al., 2020). Plasmaspheric hiss is thought to be the major generating mechanism of the slot region that occurs between the inner and outer radiation belts by pitch-angle scattering of energetic electrons at a wide energy range from a few kiloelectron volts to >1 MeV (Lyons and Thorne, 1973; Meredith et al., 2006; Summers et al., 2008; Ni BB et al., 2013; Thorne et al., 2013; Zhu H et al., 2015; Xiang Z et al., 2020a; Xiang Z et al., 2020b), and can form a reversed energy spectrum (Ni BB et al., 2019).

Magnetosonic (MS) waves (also called equatorial noise) are highly oblique electromagnetic emissions frequently observed within 5° of the magnetic equator, both inside and outside the plasmasphere (Russell et al., 1970; Santolík et al., 2004; Němec et al., 2005, 2006; Li LY et al., 2017a; Liu B et al., 2018; Fan F et al., 2021). Magnetosonic waves are generated in the frequency range between the local proton gyrofrequency (${{f}}_{\text{ce}}$) and the lower hybrid resonance frequency (${{f}}_{\text{LHR}}$) by unstable proton ring distributions (Perraut et al., 1982; Laakso et al., 1990; Santolík et al., 2002; Horne et al., 2007; Chen LJ et al., 2010; Jordanova et al., 2012; Ma QL et al., 2014). Magnetosonic waves can scatter energetic electrons, resulting in the formation of butterfly distributions (Shprits et al., 2009; Ma QL et al., 2015, 2016; Li JX et al., 2016b; Li LY et al., 2017b; Li HM et al., 2020).

Electromagnetic ion cyclotron (EMIC) waves are observed with frequencies below the distinct ion gyrofrequency (${{f}}_{{\text{H}}^{{+}}}$, ${{f}}_{{\text{He}}^{{+}}}$, and ${{f}}_{{\text{O}}^{{+}}})$. Rapid losses of radiation belt relativistic electrons and >100 keV ring current protons can be caused by cyclotron-resonant interactions with EMIC waves (Thorne and Kennel, 1971; Summers et al., 2004, 2007a, b; Liu KJ et al., 2012; Shprits et al., 2013; Usanova et al., 2014; Ni BB et al., 2015; Yu J et al., 2015; Li LY et al., 2016; Shprits, 2016; Xiang Z et al., 2018, 2021; Xue ZX et al., 2021; Cao X et al., 2020).

Although different wave modes have different excitation mechanisms, they are all associated with particle distributions and are more likely to be observed after active geomagnetic states, which makes this simultaneously observation of four wave modes possible. Electron cyclotron harmonic waves are thought to be generated by a loss-cone distribution (Ashour-Abdalla and Kennel, 1978; Horne et al., 1981). Plasmaspheric hiss can be excited by the anisotropic instability of hot electrons (Chen LJ et al., 2012; Tang RX and Summers, 2019). Exohiss is suggested to be a production of the leakage of plasmaspheric hiss (Thorne et al., 1979; Zhu H et al., 2015). Magnetosonic waves and EMIC waves are generated in the inner magnetosphere and mainly arise from the ring current protons near the equatorial plane (Kennel and Engelmann, 1966). Magnetosonic waves are driven by energetic protons at energies of tens of kiloelectron volts with a positive energy gradient (Curtis and Wu CS, 1979; Perraut et al., 1982; Chen LJ et al., 2010; Liu KJ et al., 2011; Xiao FL et al., 2013; Ma QL et al., 2014), and EMIC waves are attributed to the cyclotron resonance instability of thermally anisotropic ring current protons with energies in the range of ~1–100 keV (Cornwall, 1965; Kennel and Petschek, 1966; Gary et al., 1995; Chen LJ et al., 2009). Previous studies have demonstrated that electrons can be scattered by two wave modes simultaneously. Xiao FL et al. (2015) suggested that the combined acceleration by chorus and magnetosonic waves can explain the electron flux evolution in both the energy and butterfly PAD. Ni BB et al. (2017b) investigated the competitive effects on electron dynamics attributable to plasmaspheric hiss and MS waves. Teng SC et al. (2021) and Zhou RX et al. (2022) analyzed the cooperation between EMIC waves and MS waves upon electron loss at several megaelectron volts. The competitive influences of different plasma waves on radiation belt electrons depend on the density of ambient plasmas and the relative intensity of the waves (Li LY et al., 2017a, 2021, 2022; Yu J et al., 2020)

In this study, we report a representative case of four concurrent wave modes at L ≈ 5.0, namely, ECH waves, exohiss, MS waves, and EMIC waves, observed by Van Allen Probe A on October 15, 2015. To quantify the scattering effects of simultaneously observed multimode waves on energetic electrons both separately and jointly, the diffusion coefficients of electrons induced by these waves are calculated and two-dimensional (2-D) Fokker–Planck diffusion simulations are performed. The satellite observations are shown in Section 2. Section 3 presents the results of the diffusion coefficient calculations and the 2-D Fokker–Planck diffusion simulations. Finally, the conclusions are summarized in Section 4.

2.   Event Overview and Model Adoption

2.1   Instrumentation and Event Overview

The twin-satellite Van Allen Probes mission was launched into an equatorial elliptical orbit with a distance range from ~1.1 ${{R}}_{\text{E}}$ (radius of the Earth) to ~6.5 ${{R}}_{\text{E}}$ on August 30, 2012, providing high-quality data sets that included the ambient magnetic field, background plasma density, radiation belt particle fluxes, and electromagnetic field and waves (Mauk et al., 2013). In this study, the Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS; Kletzing et al., 2013) is used to provide both electric and magnetic field data. The EMFISIS suite contains the high-frequency receiver (HFR) and the waveform frequency receiver (WFR), which can measure the wave properties in a wide frequency range from ~10 to ~500 kHz. The Level 4 data sets of the Van Allen Probes are applied to acquire the ambient plasma density information, which is extracted from the trace of the upper hybrid resonance frequency measured by the HFR (Kurth et al., 2015). Helium, Oxygen, Proton, and Electron (HOPE) instrument (Funsten et al., 2013) measurements are used to obtain the hot electron distribution.

Figure 1 shows a simultaneously occurring multimode wave event from 04:00 to 05:00 universal time (UT) on October 15, 2015. Four wave modes are observed in this event: ECH waves, exohiss, MS waves, and EMIC waves. Especially for the ECH waves and the EMIC waves, the two wave modes appear and disappear almost simultaneously despite their different excitation mechanisms. Figure 1a presents the auroral electrojet (AE) index (<300 nT during the selected time period) and the SYM-H index (less than −15 nT during the selected time period), indicating a relatively quiet geomagnetic environment. Meanwhile, both the small value of electron density in Figure 1b and the occurrence of the ECH waves in Figure 1c suggest that this event occurred outside the plasmapause. The exohiss waves observed by the WFR instrument in Figures 1d–1g are marked by a red arrow. The MS waves can be clearly observed, according to Figures 1d–1g, by the enhancement of the magnetic field power spectral density at frequencies below the lower hybrid resonance frequency (${{f}}_{\text{LHR}}$, blue solid line in Figure 1e). Although parts of the MS waves overlap the exohiss, the two wave modes can be identified by the ellipticities and wave normal angles shown in Figures 1f and 1g. Figures 1h–1j display the H⁺ band EMIC waves with frequencies between the proton gyrofrequency (${{f}}_{\text{cp}}$, magenta dashed line in Figure 1h) and the helium gyrofrequency (${{f}}_{{\text{He}}^{{+}}}$, magenta solid line in Figure 1h).

Figure  1.  Plasma wave observations from Van Allen Probe A from 03:00 to 06:00 UT on October 15, 2015. (a) Auroral electrojet (AE) and SYM-H indices. (b) Background electron density. (c) Spectrograms of the electric field power spectral intensity measured by the high-frequency receiver (HFR). (d, e) Spectrograms of the electric and magnetic field power spectral intensity measured by the waveform frequency receiver (WFR). The white curves represent the equatorial electron gyrofrequency ${{f}}_{\text{ce}}$ (solid), 0.5 ${{f}}_{\text{ce}}$ (dotted and dashed), and 0.1 ${{f}}_{\text{ce}}$ (dashed), and the blue curve indicates the lower hybrid resonance frequency ${{f}}_{\text{LHR}}$ (solid), respectively. (f, g) Ellipticity and wave normal angle corresponding to the WFR measurements. (h) Power spectrum density derived from EMFISIS (Electric and Magnetic Field Instrument Suite and Integrated Science). The dashed line shows the local gyrofrequency of protons, whereas the solid line illustrates the local gyrofrequency of helium. (i, j) Wave ellipticity and wave normal angle of electromagnetic ion cyclotron (EMIC) waves. The gray-shaded bar (a–j) indicates the time period of the multimode wave event. (k, l, o, p) Filtered waves. (m, n, q, r) Wave spectral intensities. The gray curves show the wave spectral intensities, and the thick red curves represent the mean profiles of the power spectral densities derived from the shaded time period in (k), (l), (o), and (p) of 04:21–04:36 UT. ECH, electron cyclotron harmonic waves; Hiss, exohiss; MS, magnetosonic waves.

DownLoad: Full-Size Img PowerPoint

We pick up four waves modes (Figures 1k, 1i, 1o, 1p) as well as the corresponding mean profiles of wave spectral intensities from 04:21 to 04:36 UT, as indicated by the shaded regions in Figures 1m, 1n, 1q, and 1r when L = 5 ± 0.1 (N_e = 16.8 cm⁻³, MLT = 16.1°, LAT = 1°, and B_eq = 218 nT). The ECH waves have a typical structure that the first band waves—between ${{f}}_{\text{ce}}$ (electron resonance frequency) and ${2}\;{{f}}_{\text{ce}}$, having most of the energy according to the amplitudes —denoted in Figure 1m. Note that Figure 1k is derived from a combination of HFR and WFR instrument spectra because these two instruments both capture part of the ECH waves. The exohiss waves in Figure 1l occur at frequencies ranging from 200 to 2,000 Hz, and the amplitude is 12.9 pT, as marked in the upper right corner of Figure 1n. The amplitude of the MS waves shown in Figure 1o is 104.2 pT, as derived from Figure 1q. For the H⁺ band EMIC waves, the averaged amplitude is 316.6 pT in the specific period, as shown in Figure 1r.

2.2   ECH Wave Model

To obtain the bounce-averaged diffusion coefficients induced by ECH waves, the information on wave propagation characteristics, such as the wave normal angle and the wave power spectrum, are required (Ni BB et al., 2011; Lou YQ et al., 2021). In this study, the WHAMP (waves in homogeneous, anisotropic, multicomponent plasmas) code (Rönnmark, 1982) is adopted to evaluate the scattering effects of the ECH waves. The WHAMP calculates the dispersion relationship of waves with the assumption of homogeneous, anisotropic, magnetized plasma. As the inputs of the WHAMP code, the phase space density (PSD) of energetic electrons is expressed by the sum of the subtracted Maxwellian distribution given by Ashour-Abdalla and Kennel (1978):

where the subscript i represents the ith component of energy electrons, f is the PSD of electrons, n is the electron density, ${\alpha}_{\Vert}$ and ${\alpha}_{\bot}$ denote the thermal velocities parallel and perpendicular to the ambient magnetic field, and $\Delta$ and $\beta$ determine the depth and width of the loss cone, respectively.

To obtain the precise dispersion relation of ECH waves, the averaged electron PSD derived from the HOPE instrument from 04:21 to 04:36 UT with an energy range from ~25 eV to ~50 keV (the full 72 energy channels) is fitted by summing the nine components with the subtracted Maxwellian distribution. The PSDs shown in Figure 2 are induced from the flux data inferred by HOPE during the investigated time period. Figure 2a shows the observation (solid lines) and simulation (dashed lines) of the electron PSDs by taking the natural logarithm, whereas Figure 2b shows the PSDs as a function of the pitch angles. The parameters of the fitted electron components are shown in Table 1. Figures 1c–1e are normalized frequencies, wavenumbers, and growth rates as a function of the wave normal angles, respectively.

Figure  2.  (a) Contours of the electron phase space density (PSD) in electron velocity (${v}_{\| },{v}_{\bot }$) space (or in kinetic energy (${{E}}_{\|}$, ${{E}}_{\bot}$) space, marked by the red axis) from 04:21 to 04:36 UT. The dashed curves illustrate the observational electron PSD obtained from the HOPE (Helium, Oxygen, Proton, and Electron) instrument, and the solid curves denote the distribution of energetic electrons modeled by the sum of the subtracted Maxwellian components. (b) Observed and fitted averaged electron PSD as a function of the pitch angle. (c) Simulated frequencies normalized to ${{f}}_{\text{ce}}$ from 1.3 to 1.9 as a function of the wave normal angle (WNA). (d) Wavenumber k normalized to the gyroradius ${\rho}_{\text{e}}$ of 1 eV component. (e) The corresponding normalized growth rates $\dfrac{\gamma}{{{ \varOmega }}_{\text{ce}}}$ of the wave frequency at 1.3–1.9 $\dfrac{{f}}{{{f}}_{\text{ce}}}$.

DownLoad: Full-Size Img PowerPoint

Table  1.  The nine electron components used to fit the measured electron phase space density from 04:21 to 04:36 UT.

Component	Ne (m−3)	${{T} }_{\| }$ (eV)	${{T} }_{\bot }$ (eV)	${\Delta }$	$\beta$
1	$1.2\times {10}^{7}$	1.0	1.0	1.0	0.5
2	$4.5\times {10}^{6}$	12.2	13.7	1.0	0.0
3	$2.2\times {10}^{5}$	64.0	64.0	0.4	0.4
4	$5.3\times {10}^{4}$	256.0	256.0	0.4	0.7
5	$4.0\times {10}^{4}$	1,648.6	2,057.0	0.3	0.7
6	$3.1\times {10}^{4}$	5,215.0	5,162.3	0.1	0.7
7	$5.4\times {10}^{3}$	3,200.0	6,100.0	0.5	0.0
8	$4.2\times {10}^{3}$	15,350.1	28,450.0	0.0	0.2
9	$1.3\times {10}^{4}$	77,102.7	25,600.0	0.6	0.4

 | Show Table

DownLoad: CSV

Figures 2a and 2b clearly show that our fitted results of the electron PSDs match the observations well. Loss-cone distributions, which are important for the excitation of ECH waves, can be clearly observed in Figure 2a. In Figure 2b, an isotropic distribution is shown below ~100 eV, and the pancake PADs can be recognized at higher energies. On the basis of the modeled electron distributions, we calculate the dispersion relation of seven specific frequencies, which are adopted from 1.3 ${{f}}_{\text{ce}}$ to 1.9 ${{f}}_{\text{ce}}$ with an increment of 0.1 ${{f}}_{\text{ce}}$; the results are demonstrated in Figures 2c–2e. The simulated growth rates of ECH waves shown in Figure 2e have a peak at 1.6 $\dfrac{{f}}{{{f}}_{\text{ce}}}$, which matches the satellite observation well. In addition, the maximum growth rate for 1.5–1.7 $\dfrac{{f}}{{{f}}_{\text{ce}}}$ is likely to arise at the wave normal angles around 88.5°, whereas it is > 89° for wave frequencies at 1.3, 1.4, 1.8, and 1.9 $\dfrac{{f}}{{{f}}_{\text{ce}}}$. Consequently, the farther away from the central frequency of 1.6 $\dfrac{{f}}{{{f}}_{\text{ce}}}$, the larger is the wave normal angle corresponding to the peak growth rate for a certain frequency.

3.   Simulation Results

3.1   Wave-Induced Electron Scattering Rates

To quantify the combined scattering effects on electrons by the four wave modes, the bounce-averaged electron diffusion coefficients induced by each wave mode are evaluated individually. The Full Diffusion Code (FDC) is adopted to compute the quasilinear bounce-averaged electron diffusion coefficients by exohiss, EMIC waves, and ECH waves. Their resonance harmonics |N| ≤ 10 are considered (Ni BB et al., 2008, 2022; Liu YXZ et al., 2022). Because of the different propagation characteristics of these three wave modes, we adopt different latitude ranges for exohiss (|LAT| ≤ 40°), EMIC waves (|LAT| ≤ 40°), and ECH waves (|LAT| ≤ 3°). Exohiss takes a wave normal angle model following Ni BB et al. (2013), EMIC waves adopt the wave normal angle model introduced in Ni BB et al. (2015) and assume the ionic components of ${\eta }_{{\mathrm{H}}^{+}}$= 85%, ${\eta }_{{\mathrm{H}\mathrm{e}}^{+}}$ = 10%, ${\eta }_{{\mathrm{O}}^{+}}$ = 5% in the same energy range. For ECH waves, we focus on the first harmonic band because the wave power of the first harmonic band dominates over the other harmonic bands in the selected ECH wave event. We adopt the wave frequency spectrum from the observation frequencies (~1.3–1.9 ${{f}}_{\text{ce}}$) with an increment of 0.1 ${{f}}_{\text{ce}}$. In particular, the scattering effect of ECH waves simulated by the FDC is conducted on the energy range from 10 eV to 100 keV because of the wave–particle resonance energy range. For MS waves, we adopt the test particle code to simulate the electron scattering effect of Landau resonances and transit-time diffusion with |LAT| ≤ 3° and pitch angle $\theta$ = 89° following previous studies (Horne et al., 2007; Ni BB et al., 2017a, 2018). The main parameters required are summarized in Table 2.

Table  2.  Parameters required for calculating the diffusion coefficients.^a

Parameter	Wave mode
	ECH	Exohiss	MS	EMIC
Amplitude	0.023 mV/m	12.9 pT	104.2 pT	316.6 pT
LAT	≤ 3°	≤ 40°	≤ 3°	≤ 40°
|N|	≤ 10	≤ 10	0	≤ 10
WNA	Table 1	following Ni et al. (2013)	89°	following Ni et al. (2015)
Other	L = 5, Ne = 16.8 cm−3, Beq = 218 nT
aECH, electron cyclotron harmonic waves; MS, magnetosonic waves; EMIC, electromagnetic ion cyclotron waves; LAT, latitude; N, resonance harmonics; WNA, wave normal angle.

 | Show Table

DownLoad: CSV

Figure 3 displays the 2-D bounce-averaged pitch angle, cross diffusion (pitch angle, momentum), and momentum diffusion coefficients at L = 5 (corresponding to the gray shaded region in Figure 1). For exohiss, the pitch angle diffusion rates in Figure 3a show an effective diffusion on hundreds of kiloelectron volt electrons with a time scale of approximately one day. Two regions show strong diffusion rates. The part near the high pitch angle is due to Landau resonance, whereas the region corresponding to a wider pitch angle range results from the cyclotron resonances. The coefficients of cross diffusion and momentum diffusion by exohiss are also considerable, with maximum diffusion rates of ~10⁻⁵ s⁻¹ for electrons at ~100 keV with pitch angles ranging from 40° to ~90°. The diffusion coefficients of MS waves are mainly in the range of 100 keV–1 MeV, with pitch angles ranging from 50° to 80° in Figures 3d–3f (Bortnik and Thorne, 2010; Fu S et al., 2019). The momentum scattering rates are comparable to the pitch angle diffusion rates for MS waves, suggesting a feasible acceleration mechanism. According to the quasilinear theory, the amplitudes of waves have a quadratic effect on the calculation result of the diffusion coefficient. Although the amplitude of MS waves in this case is several times greater than that of exohiss, to show the transit-time diffusion, we adapt the test particle simulations instead of the FDC, whose combined scattering rates are generally an order of magnitude weaker because the electrons are moved out of the Landau resonance by the advective effect of the bounce resonance (Fu S et al., 2019). In contrast, EMIC waves show a significant diffusion effect on electrons above 5 MeV with a time scale of minutes, which is more remarkable than that of exohiss because of the much larger amplitude. Regarding the combined scattering effect of concurrent exohiss and MS waves, Hua M et al. (2019) suggested that when exohiss is comparable to or stronger than MS waves, the combined scattering effect is mainly featured by exohiss—induced pitch angle scattering, which can produce butterfly PADs for less than ~1 MeV electrons by strong exohiss, top—hat PADs for less than ~150 keV electrons, and flat—top PADs for greater than ~300 keV electrons by both weak and strong exohiss. Pitch angle diffusion by intense exohiss can also drive a considerable PSD valley between ~50 keV and 1 MeV. Because of the small amplitude, ECH waves have only weak effects on electrons from tens of electron volts to tens of kiloelectron volts at small pitch angles (< 20°). According to the diffusion energy range of the four wave modes, exohiss and MS waves are likely to have a combined scattering effect on electrons at hundreds of kiloelectron volts, but this is far away from the main energy range of strong diffusion coefficients induced by EMIC waves and ECH waves.

Figure  3.  Two-dimensional plots of the bounce-averaged pitch angle, cross, and momentum diffusion coefficients (from left to right: <D_αα>, |<D_αp>|, <D_pp>) as a function of the equatorial pitch angles and electron kinetic energies ${E}_{k}$ at L = 5 induced by (a–c) exohiss, (d–f) magnetosonic (MS) waves, (g–i) electromagnetic ion cyclotron (EMIC) waves, and (j–l) electron cyclotron harmonic (ECH) waves.

DownLoad: Full-Size Img PowerPoint

3.2   Evaluation of the PSD Evolution of Energetic Electrons

With the acquisition of individual bounce-averaged diffusion rates induced by ECH waves, exohiss, MS waves, and EMIC waves, we perform a 2-day wave-induced evolution simulation of the energetic electron PSD by numerically solving the 2-D Fokker–Planck equation:

where ${\alpha }_{\text{eq}}$ is the equatorial pitch angle, $S\left({\alpha }_{\text{eq}}\right)$ denotes the normalized bounce time, where ${S}\left({\alpha}_{\text{eq}}\right){\thickapprox 1.38-0.32} \left[{\text{sin}}\;{\alpha}_{\text{eq}}{+}\right. \left. {\left({\text{sin}}\;{\alpha}_{\text{eq}}\right)}^{\frac{{1}}{{2}}}\right]$, and ${G=}{{p}}^{{2}}{S}\left({\alpha}_{\text{eq}}\right){\text{sin}}\;{\alpha}_{\text{eq}}{\text{cos}}\;{\alpha}_{\text{eq}}$. In addition, f is the electron PSD, which is related to electron flux j and electron momentum p by $f=\dfrac{{j}}{{{p}}^{{2}}}$. And $\left\langle{{{D}}_{{\alpha}_{\text{eq}}{\alpha}_{\text{eq}}}}\right\rangle$, $\left\langle{{{D}}_{{pp}}}\right\rangle{,}$ and $\left\langle{{{D}}_{{\alpha}_{\text{eq}}{p}}}\right\rangle=\left\langle{{{D}}_{{p}{\alpha}_{\text{eq}}}}\right\rangle$ demonstrate the bounce-averaged diffusion coefficients in pitch angle, momentum, and cross pitch angle momentum (Summers et al., 2007a; Xiao FL et al., 2009). We compute the electron PSDs at all pitch angles with the initial condition derived from satellite observations. For the boundary condition in the energy space, we assume that the electron PSDs are constant at ${E}_{k}$ = 1 keV and 10 MeV with f = 0 at 10 MeV (Summers et al., 2004; Li JX et al., 2016a). For the boundary condition in the equatorial pitch angle space, we take f = 0 inside the loss cone and $\dfrac{{\partial}f}{{\partial}{\alpha}_{\text{eq}}}$ = 0 at ${\alpha}_{\text{eq}}$ = 90°.

Figure 4 presents the simulated temporal evolution of radiation belt electron PSDs at L = 5 under the influence of (a–e) exohiss, (f–j) MS waves, (k–o) exohiss combined with MS waves, (p–t) the combination of exohiss, MS waves, and EMIC waves, and (u–y) ECH waves. The five columns indicate wave–particle interaction time stamps of 2 days, with an increment of 12 hours. As illustrated in Figure 4, (1) exohiss can cause electron PSD decay at a wide energy range below megaelectron volts, especially around 100 keV electrons, resulting in the formation of a reversed energy spectrum, as shown in Figures 4d–4e; (2) the scattering effect of MS waves is most efficient in hundreds of kiloelectron volts, which can scatter particles at pitch angles above 60°, forming butterfly distributions; (3) compared with exohiss scattering alone, the overall PSD decay from the combination of exohiss and MS waves is enhanced, but the butterfly distributions caused by MS waves are inhibited by exohiss; (4) when EMIC waves are included in the diffusion coefficients because of the combination of exohiss and MS waves, no clear difference occurs because of the high resonance energy of EMIC waves (> 5 MeV); (5) ECH waves can scatter < 20° electrons at a 0.1–10 keV energy range into loss cones effectively.

Figure  4.  Two-dimensional temporal evolution of electron phase space density (PSD) under the impact of (a–e) exohiss, (f–j) magnetosonic (MS) waves, (k–o) a combination of exohiss and MS waves, (p–t) a combination of exohiss, MS waves, and electromagnetic ion cyclotron (EMIC) waves, and (u–y) electron cyclotron harmonic (ECH) waves in a time span of 2 days.

DownLoad: Full-Size Img PowerPoint

To display the contributions by the different wave modes on the PSD evolution more clearly, Figure 5 shows the temporal evolution of the simulated electron PSDs at the color-coded electron energies within 48 h under the impact of exohiss and MS waves individually and in combination. As shown in Figures 5a–5e, the scattering effects of electrons by exohiss on 100 keV are significantly stronger than those of other energies, causing the PSD to decrease by more than one order of magnitude in 2 days for electrons at pitch angles less than ~88°, forming top-hat PADs. Magnetosonic waves alone can scatter α_eq > 60° hundreds of kiloelectron volt electrons within 1 day, leading to the butterfly distributions in Figures 5f–5j. We found that the top-hat distribution induced by exohiss and the butterfly distribution induced by MS waves are both mainly caused by their cross-diffusion coefficients but that the former is positive and the latter is negative. Thus, when considering the combination of those two cross-diffusion coefficients, we can see that the butterfly distributions caused by MS waves are substantially inhibited by exohiss and that the top-hat distributions caused by exohiss are inhibited by MS waves (Figures 5k–5o).

Figure  5.  The simulated evolution of electron pitch angle distributions at selected energies over 2 days. (a–e) Exohiss (Hiss) waves, (f–j) magnetosonic (MS) waves, (k–o) a combination of exohiss and MS waves. PSD, phase space density.

DownLoad: Full-Size Img PowerPoint

Because of the distinct resonant regions between EMIC waves (> 5 MeV), ECH waves (< 100 keV), exohiss, and MS waves (~100 keV–1 MeV), we display the PSD evolution caused by ECH waves and EMIC waves individually in Figure 6. In Figures 6a–6e, because of the rapid scattering in the order of minutes caused by EMIC waves on energetic electrons > 5 MeV with a pitch angle < 50°, electron PSDs reduce by 3 orders of magnitude within 12 hours and by 4 orders of magnitude within 48 hours, causing top-hat PADs. As shown in Figures 6f–6j, ECH waves can scatter ~100 eV to ~10 keV electrons with pitch angles <20° into loss cones.

Figure  6.  The simulated evolution of electron pitch angle distributions at selected energies over 2 days. (a–e) Electromagnetic ion cyclotron (EMIC) waves, and (f–j) electron cyclotron harmonic (ECH) waves.

DownLoad: Full-Size Img PowerPoint

4.   Conclusions

In this study, we have reported a specific case of four concurrent magnetospheric wave modes, namely ECH waves, exohiss, MS waves, and EMIC waves, that were observed at L ≈ 5.0 by Van Allen Probe A on October 15, 2015. The wave-induced quasilinear diffusion coefficients are calculated on the basis of the FDC and test particle simulations. By simulating the net electron diffusion process in terms of the 2-D Fokker–Planck diffusion simulations, we have investigated in detail the combined electron scattering effects of the four wave modes, in association with their competition and cooperation, upon the temporal evolution of the radiation belt electron distribution.

The main conclusions are summarized as follows:

(1) A representative case of simultaneously observed ECH waves, exohiss, MS waves, and EMIC waves at L ≈ 5.0 is reported from 04:00 to 05:00 UT on October 15, 2015. The ECH waves and EMIC wave appeared and disappeared almost simultaneously despite their different excitation mechanisms.

(2) In this case, the scattering effects induced by the four wave modes are distinct: ECH waves mainly scatter low-pitch-angle (< 20°) electrons at 0.1–10 keV; exohiss can significantly scatter hundreds of kiloelectron volt electrons to form a reversed energy spectrum; MS waves mainly affect high-pitch-angle electrons (> 60°); and EMIC waves scatter only > 5 MeV electrons.

(3) Regarding the combined scattering effect, the butterfly distributions caused by MS waves are substantially inhibited by exohiss, and the top-hat distributions caused by exohiss are inhibited by MS waves. Electron cyclotron harmonic waves and EMIC waves have a slight combined scattering effect with other waves.