Volume 46, Issue 15 p. 8624-8632
Research Letter
Free Access

Combined Effects of Equatorial Chorus Waves and High-Latitude Z-Mode Waves on Saturn's Radiation Belt Electrons

J. Yu

Corresponding Author

J. Yu

Space Science Institute, Macau University of Science and Technology, Macao, China

Now at School of Atmospheric Sciences, Sun Yat-sen University, Zhuhai, China

Correspondence to: J. Yu,

[email protected]

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L.Y. Li

L.Y. Li

School of Space and Environment, Beihang University, Beijing, China

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J. Cui

J. Cui

Space Science Institute, Macau University of Science and Technology, Macao, China

School of Atmospheric Sciences, Sun Yat-sen University, Zhuhai, China

Key Laboratory of Lunar and Deep Space Exploration, Chinese Academy of Sciences, Beijing, China

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J.B. Cao

J.B. Cao

School of Space and Environment, Beihang University, Beijing, China

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J. Wang

J. Wang

Space Science Institute, Macau University of Science and Technology, Macao, China

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First published: 19 July 2019
Citations: 18

Abstract

In this paper, we examined the combined effects of equatorial chorus waves and high-latitude Z-mode waves on the energy and pitch angle distribution of Saturn's radiation belt electrons. Our simulation results show that these two types of waves at different latitudes jointly control the fluxes of Saturn's radiation belt electrons, which is very different from the individual effects of each kind of waves. The presence of Z-mode waves can efficiently inhibit the reversed energy spectrum of electrons driven by chorus waves, whereas the presence of chorus waves can further accelerate and scatter the relativistic electrons accelerated by Z-mode waves toward the larger and smaller pitch angles. Our findings provide a new insight on how different types of waves at different latitudes jointly affect the radiation belt electron dynamics, including, but not limited to, Saturn.

Key Points

  • Chorus waves and Z-mode waves at different latitudes jointly control the fluxes of relativistic electrons in Saturn's radiation belt
  • Z-mode waves can efficiently inhibit the reversed energy spectrum of electrons driven by chorus waves
  • Chorus waves can further accelerate and scatter relativistic electrons accelerated by Z-mode waves toward larger and smaller pitch angles

1 Introduction

The presence of the radiation belt in Saturn's magnetosphere was first confirmed by the measurements from the flyby of Pioneer 11 (Simpson et al., 1980). Similar to the Earth's radiation belts, radial diffusion and local acceleration/loss (e.g., cyclotron resonant interactions between particles and plasma waves) are two main physical processes responsible for the dynamics of the Kronian radiation belts (Lorenzato et al., 2012). Z-mode waves and whistler mode chorus waves are two main potential candidates accounting for the local acceleration and loss of Saturn's radiation belt electrons (Gu et al., 2013; Shprits et al., 2012; Woodfield et al., 2018).

Whistler mode chorus waves at Saturn were first observed by the Voyager spacecraft (Gurnett et al., 1981; Scarf et al., 1982). The most common Saturn chorus waves are observed below 0.5fce (fce is the electron gyrofrequency), occur at almost all local times, and are mostly amplified near the magnetic equator in the range of 5 < L < 8 (Hospodarsky et al., 2008; Menietti et al., 2014). The most intense Saturn chorus waves usually occur at latitude of ~8° and at frequency of ~0.3fce (Menietti et al., 2014). Unlike the case on Earth (Li et al., 2005, 2017; Thorne et al., 2013), whistler mode chorus waves were found to be capable of rapidly scattering keV electrons but unable to efficiently accelerate MeV electrons in Saturn's outer magnetosphere at L = 6 (Shprits et al., 2012). The weak acceleration of MeV electrons is probably caused by the high ratio of plasma frequency to electron gyrofrequency (fpe/fce), the relatively weak wave intensity, and the narrow wave latitude distributions (Woodfield et al., 2018).

Z-mode waves are electromagnetic emissions confined in the frequency range between the cutoff frequency ( ) and the upper hybrid resonance frequency ( ). However, Saturn Z-mode waves predominantly occur as two narrowband emissions centered near 5 and 20 kHz in low plasma density or strong magnetic field regions (Gu et al., 2013; Menietti et al., 2015; Ye et al., 2010). In these regions, the plasma frequency is smaller than the electron gyrofrequency (fpe/fce < 1). Therefore, these emissions at Saturn are primarily distributed at most latitudes at low L shells (L < 4) and at high latitudes at large L shells (L > 4; Menietti et al., 2015). Similar to the case on Earth (Glauert & Horne, 2005; Xiao et al., 2012), recent studies show that Z-mode waves can efficiently accelerate electrons in Saturn's inner magnetosphere (L < 4) and are responsible for the formation of Saturn's radiation belts inside the orbit of Enceladus (Gu et al., 2013; Woodfield et al., 2018).

It is worthwhile to point out that previous studies only considered the effect of one type of waves (chorus waves or Z-mode waves) on Saturn's radiation belt electrons. However, statistical studies have shown that chorus waves occur at low latitude while Z-mode waves can occur at high latitude at the same L shell (e.g., 5 < L < 7, Menietti et al., 2014, 2015). Thus, radiation belt electrons will encounter and interact with these two types of waves at different latitudes during their bounce motions. However, this combined diffusion of electrons by chorus waves and Z-mode waves remains unknown. This is the primary goal of the current study. Although the present study focuses on the combined effects of chorus waves and Z-mode waves on Saturn's radiation belt electrons, the results have potential application to the combination of other plasma waves at other planets, such as the combined diffusion by equatorial magnetosonic waves and high-latitude superluminous waves at Earth.

2 Plasma Density and Wave Models

First, we adopt the Persoon's plasma density model (Persoon et al., 2013). In their model, the plasma density (Ne) varies with both latitude (λ) and L shell at Saturn:

According to Persoon et al. (2013), the scale height (H) is ~0.85RS, and the equatorial plasma density (Ne0) is ~36.7 cm−3 at L = 6.

Second, we adopt a dipole magnetic field to approximate the Saturn's background magnetic field (Gu et al., 2013; Woodfield et al., 2018).

The global distributions of chorus waves and Z-mode waves have been comprehensively investigated in Saturn's magnetosphere (e.g., Menietti et al., 2014, 2015). The statistical results show that the intensity of chorus waves and Z-mode waves follow Gaussian distributions with latitude (λ) and frequency (ω = 2πf).
(1)
(2)
where
(3)

Bw is wave amplitude, λm and δλ are the latitude of peak power and bandwidth, λmin and λmax are the lower- and upper-latitude cutoffs, ωm = 2πfm and δω = 2πδf are the frequency of peak power and bandwidth, and ωmin and ωmax are the lower- and upper-frequency cutoffs.

Additionally, chorus waves are found to be an order of magnitude stronger in amplitude than Z-mode waves, and both waves can occur at all local times (e.g., Menietti et al., 2014, 2015). Thus, the amplitudes of chorus waves and Z-mode waves are assumed to be 100 pT and 10 pT, respectively, whereas their MLT percentages are assumed to be 100% (i.e., waves are distributed at all MLT). The tangent of wave normal angles (X = tan θ) of emissions are assumed to follow a Gaussian distribution (Menietti et al., 2014; Ni et al., 2011; Shprits et al., 2012; Woodfield et al., 2018).
(4)

Xm =  tan θm and δX =  tan δθ is the peak and angular width, Xmin =  tan θmin and Xmax =  tan θmax are tangents of the lower- and upper-wave normal angle cutoffs.

According to previous studies (Menietti et al., 2014, 2015; Shprits et al., 2012; Woodfield et al., 2018), the adopted wave model parameters are shown in the Table 1.

Table 1. Parameters of Wave Properties for Chorus and Z-mode Waves
Waves Amplitude Bw MLT percentage Spectral properties Propagation angle Latitude distribution
Chorus 100 pT 100% fm = 0.3fce θm = 0° λm = 8.541
δf = 0.186fce δθ = 30 δλ = 5.226
fmin = 0.05fce θmin = 0 λmin = 0
fmax = 0.5fce θmax = 70 λmax = 20
5 kHz band Z-mode 10 pT 100% fm = 5 kHz θm = 21.88 λm = 25
δf = 1.1 kHz δθ = 17.18 δλ = 14.142
fmin = 3.5 kHz θmin = 0 λmin = 20
fmax = 7.5 kHz θmax = 56.24 λmax = 60
20 kHz band Z-mode 10 pT 100% fm = 17 kHz θm = 21.88 λm = 25
δf = 3 kHz δθ = 17.18 δλ = 14.142
fmin = 10 kHz θmin = 0 λmin = 20
fmax = 26 kHz θmax = 56.24 λmax = 60

Finally, the latitude range, where chorus waves and Z-mode waves propagate, can be obtained by the plasma density model, background magnetic field model, and wave spectral model. Figure 1 shows the latitude effect on wave propagation at L = 6 at Saturn based on the above three models. The wave propagation areas are marked by color-coded shadowing. Figure 1 indicates that chorus waves can only propagate near the magnetic equator (λ < ~20°), whereas Z-mode waves can only propagate at high latitudes (λ > ~20°). Thus, we assume that chorus waves are confined in the latitude range of 0° < λ < 20° and Z-mode waves are confined in the latitude range of 20° < λ < 60°, as shown in the last column of Table 1.

Details are in the caption following the image
The latitude effect on the propagation of chorus and Z-mode waves at Saturn at L = 6. The black, blue, green, and red curves denote the electron gyrofrequency, the electron plasma frequency, the cutoff frequency, and the upper hybrid resonance frequency, respectively. The wave propagation areas are marked by color-coded shadowing.

3 Electron Scattering Rates

The quasi-linear bounce-averaged diffusion coefficients in a dipole magnetic field are given by (Glauert & Horne, 2005; Lyons, 1974)
(5)
(6)
(7)
where λm is the mirror latitude and T(αeq) = 1.30 − 0.56 sin αeq. αeq and α are the equatorial and local pitch angle, respectively. Dαα, Dαp⟩, and Dpp are bounce-averaged pitch angle, cross, and momentum diffusion coefficients, respectively, Dαα, Dαp, and Dpp are the corresponding local diffusion coefficients which are written as
(8)
with
(9)
(10)

The local diffusion coefficients are evaluated at the resonant wave number ki and the resonant frequency ωi, which are obtained by simultaneously solving the resonance condition and the wave dispersion relation (Glauert & Horne, 2005). The term n,k|2 depends on the wave dispersion relation, and N(ω) is a normalization factor. The expressions of n,k|2 and N(ω) are given by Lyons (1974).

Based on the models described above, we obtain the bounce-averaged diffusion coefficients driven by plasma waves through the quasi-linear theory at L = 6 (Glauert & Horne, 2005). In this region, chorus waves are mainly near the equator, Z-mode waves are confined at high latitudes, and both emissions are intense, according to the statistical studies (Menietti et al., 2014, 2015, 2018). Figures 2a–2f show the chorus and Z-mode wave-driven bounce-averaged pitch angle (<Dαα>/p2), cross (|<Dαp>|/p2), and momentum (<Dpp>/p2) diffusion coefficients, respectively. Their combined diffusion coefficients are shown in Figures 2g–2i. The equatorial confined chorus waves can scatter almost all pitch angle and energy electrons. Chorus wave-driven pitch angle scattering rates are several orders of magnitude larger than the momentum scattering rates, particularly for <1 MeV electrons near the loss cone (Figures 2a and 2c), indicating that these emissions can cause rapid loss of electrons below 1 MeV on the time scale of several hours. As electron energy increases, the wave-driven scattering rates decrease rapidly at small pitch angles, indicating that the wave-driven loss of high-energy electrons (>1 MeV) is not as efficient as that of low-energy electrons (<1 MeV). Unlike chorus waves, Z-mode waves only interact with electrons at pitch angles smaller than 49° due to the high-latitude confinement of these waves. In contrast to chorus waves, the Z-mode wave-driven momentum scattering rates are at least an order of magnitude larger than the pitch angle scattering rates. Thus, the energy diffusion of electrons is dominant when they resonate with Z-mode waves at Saturn. As shown in Figures 2g–2i, the combined scattering rates are very different from the scattering rates driven by each kind of waves. Chorus waves dominate in the pitch angle scattering at most pitch angles and in the momentum diffusion at large pitch angles, whereas Z-mode waves dominate in the momentum diffusion at small pitch angles. Thus, the electron diffusion simultaneously by both chorus waves and Z-mode waves play a rather different role in radiation belt dynamics, compared with each wave individually.

Details are in the caption following the image
Bounce-averaged pitch angle, cross, and momentum diffusion coefficients driven by chorus waves only (a–c), Z-mode waves only (d–f), and combined scattering (g–i).

4 Electron Distribution Evolution

Based on the scattering rates, the temporal evolution of electron pitch angle distributions can be obtained through performing the 2-D Fokker-Planck diffusion simulations (Yu, Li, et al., 2019). Following previous studies (e.g., He et al., 2016; Li et al., 2017; Shprits et al., 2012; Yu, Wang et al., 2019), the initial distributions of electrons are assumed to be F = e−(E − 0.02)/0.1(sinαeq −  sin αLC)/(pc)2 (E is the electron energy in MeV, αLC is the loss cone), and the boundary conditions are taken as follows: F = 0 at αeq = αLC, ∂F/∂αeq = 0 at αeq = 90°, F = constant at E = 10 keV and F = 0 at E = 10 MeV. Figure 3 shows the electron temporal evolution diffused by chorus waves only (a–d), Z-mode waves only (e–h), and combined diffusion (i–l) at the indicated times at L = 6 at Saturn. Chorus waves can cause rapid scattering loss of energetic electrons in the energy range from tens of keV to ~500 keV at all pitch angles but accelerate relativistic electrons (>0.7 MeV) to higher energies at large pitch angles (40–90°). Consequently, a reversed energy spectrum is formed which is similar to the observations in Earth's radiation belt caused by plasmaspheric hiss (Ni et al., 2019; Zhao et al., 2019). As the time accumulates, the loss of energetic electrons and the acceleration of relativistic electrons are more pronounced. Unlike chorus waves, Z-mode waves mainly accelerate relativistic electrons at small pitch angles (< 50°). Z-mode waves can also lead to efficient scattering loss of energetic electrons near the loss cone (<30°). However, the loss of energetic electrons due to Z-mode waves is not as efficient as that due to chorus waves. When chorus waves and Z-mode waves simultaneously occur, the temporal evolution of electron pitch angle distributions driven by these two types of waves are significantly different from that by each kind of waves. The combined diffusion effect can significantly slow down the scattering loss of energetic electrons due to chorus waves and the acceleration of relativistic electrons at small pitch angles due to Z-mode waves. On the other hand, the combined diffusion effect can strongly speed up the relativistic electron energization at large pitch angles due to chorus waves. Moreover, the reversed energy spectrum disappears once the effects of Z-mode waves are taken into consideration.

Details are in the caption following the image
The electron temporal evolution due to the diffusion by chorus waves only (a–d), Z-mode waves only (e–h), and combined scattering (i–l) at the indicated times at L = 6 at Saturn.

Figure 4 shows the line plots of temporal evolution of electron pitch angle distributions driven by chorus waves only (a–d), Z-mode waves only (e–h), and combined diffusion (i–l) at different electron energies. Chorus waves can cause the rapid and considerable scattering loss of 100 keV electrons over all pitch angles but significantly increase the phase space density of higher-energy electrons over a very broad pitch angle range. As the electron energy increases from 0.8 to 2.0 MeV, the increase of the electron phase space density is more pronounced, and the pitch angle coverage of electron acceleration gradually narrows. Similar to chorus waves, Z-mode waves can also lead to the loss of 100 keV electrons and the remarkable acceleration of higher-energy electrons. However, the only difference is that the phase space density of 100 keV electrons increases in the pitch angle range of 35–50°. Compared with chorus waves, Z-mode waves lead to a larger growth of the relativistic electron (>0.8 MeV) phase space density on the same time scale. When chorus waves and Z-mode waves simultaneously interact with electrons, the combined diffusion greatly weakens the loss of 100 keV electrons but remarkably enhances the acceleration of relativistic electrons driven by chorus waves. On the time scale of a day, the loss efficiency of 100 keV electrons reduces by about 2 orders of magnitude, whereas the acceleration efficiency of >1.5 MeV electrons increases by more than an order of magnitude. Moreover, the presence of chorus waves can further accelerate and scatter the relativistic electrons accelerated by Z-mode waves toward the larger and smaller pitch angles, and thus result in a flatter pitch angle distribution at small pitch angles.

Details are in the caption following the image
The temporal evolution of the pitch angle distribution of given energy electrons by chorus waves only (a–d), Z-mode waves only (e–h), and combined scattering (i–l). The color-coded curves denote the electron phase space densities at different interaction times.

5 Conclusions and Discussions

The current study aims to investigate the interactions between electrons and the equatorial chorus waves and high-latitude Z-mode waves in Saturn's outer magnetosphere at L = 6. The equatorial chorus waves are shown to scatter electrons at almost all pitch angles, whereas high-latitude Z-mode waves can only effectively diffuse electrons at small pitch angles. By performing the 2-D diffusion simulations, we find that chorus are mainly responsible for the rapid and considerable loss of <500 keV electrons and the acceleration of >500 keV electrons, whereas Z-mode waves mainly accelerate electrons. A reversed energy spectrum of electrons is formed due to the chorus waves. As the electron energy increases, the enhancements of the electron phase space density driven by each type of waves are more pronounced. The combined diffusion by the equatorial chorus waves and high-latitude Z-mode waves can considerably enhance the phase space density of relativistic electrons and facilitate the formation of the radiation belt in Saturn's outer magnetosphere. The presence of Z-mode waves strongly reduces the loss of energetic electrons (<500 keV) but remarkably enhances the acceleration of relativistic electrons (>0.7 MeV) driven by chorus waves. Moreover, the Z-mode waves efficiently inhibit the reversed energy spectrum of electrons driven by chorus waves. Meanwhile, chorus waves further accelerate and scatter the relativistic electrons accelerated by Z-mode waves toward the larger and smaller pitch angles, and thus results in a flatter pitch angle distribution at small pitch angles.

The present study focuses on the combined diffusion of electrons by chorus waves and Z-mode waves at Saturn. Other plasma waves, such as electron cyclotron harmonic (ECH) emissions, can also contribute to the dynamics of Saturn's radiation belt (Menietti et al., 2017); future simulations are required to consider their contributions. Although the pilot simulations are performed at L = 6, the method described here can be directly applied to other L shells. However, since the plasma environment, wave latitude distributions, and wave frequency spectra are very different at different L shells, the effects of combined diffusion on electrons may not simply vary with L shells linearly. It would be interesting to investigate how the electron diffusion simultaneously by different waves at different latitudes varies with L shells.

Our simulation results clearly demonstrate that the combination of different types of waves at different latitudes have very different effects on electron dynamics from each kind of waves when electrons bounce along the magnetic field line. This kind of combined diffusion can not only occur at Saturn but also at other planets, such as Earth. In the Earth's radiation belt, magnetosonic waves, ECH waves, and chorus waves are usually confined near the magnetic equator (Li et al., 2009; Lou et al., 2018; Ma et al., 2013; Yu, Li, et al., 2019; Zhima et al., 2013), whereas the superluminous waves usually propagate in the high latitudes (Xiao et al., 2007). When the Earth's magnetosphere is compressed, chorus waves, ECH waves, and electromagnetic ion cyclotron (EMIC) waves can also be easily excited in the dayside high latitudes where the magnetic field strength reaches a local minimum (Liu et al., 2012; Lou et al., 2018; Tsurutani & Smith, 1977). How these types of waves at different latitudes jointly scatter electrons should be required careful investigation in the future radiation belt modeling. Our results also show that, except for the equatorial waves, the high-latitude waves also play a crucial role in controlling the fluxes of relativistic electrons in planet's radiation belt. However, the effects of high-latitude waves have not been incorporated into the radiation belt modeling yet (e.g., Albert et al., 2016; Hua et al., 2018; Shprits et al., 2012; Su et al., 2010; Yu et al., 2015) due to the lack of systematic studies on the global distributions of these waves. Therefore, more efforts are required to address the global distributions of high-latitude waves to better understand their contributions to the radiation belt dynamics.

Acknowledgments

J. Y. and J. C. are supported by the National Natural Science Foundation of China (NSFC) through Grants 41525015 and 41774186. L. Y. L. is supported by National Natural Science Foundation of China (41874192 and 41431071). No data set is used in this study. The numerical data are available at figshare website (https://dx.doi.org/10.6084/m9.figshare.8217755).