Excitation of extremely low‐frequency chorus emissions: The role of background plasma density

Low‐frequency chorus emissions have recently attracted much attention due to the suggestion that they may play important roles in the dynamics of the Van Allen Belts. However, the mechanism (s) generating these low‐frequency chorus emissions have not been well understood. . In this letter, we report an interesting case in which background plasma density lowered the lower cutoff frequency of chorus emissions from above 0.1 f ce (typical ordinary chorus) to 0.02 f ce (extremely low‐frequency chorus). Those extremely low‐frequency chorus waves were observed in a rather dense plasma, where the number density N e was found to be several times larger than has been associated with observations of ordinary chorus waves. For suprathermal electrons whose free energy is supplied by anisotropic temperatures, linear growth rates (calculated using in‐situ plasma parameters measured by the Van Allen Probes) show that whistler mode instability can occur at frequencies below 0.1 f ce when the background plasma density N e increases. Especially when N e reaches 90 cm–3 or more, the lowest unstable frequency can extend to 0.02 f ce or even less, which is consistent with satellite observations. Therefore, our results demonstrate that a dense background plasma could play an essential role in the excitation of extremely low‐frequency chorus waves by controlling the wave growth rates.


Introduction
Chorus waves, one of the most common whistler modes observed in the inner magnetosphere, have been studied for several decades through satellite data (e.g. Gurnett and O'Brien, 1964;Tsurutani and Smith, 1974;Santolík et al., 2003a;Bell et al., 2009;Li W et al., 2011Li W et al., , 2012Li W et al., , 2013Meredith et al., 2014) and theoretical research (e.g. Kennel, 1966;Omura et al., 2008;Jordanova et al., 2010;Chen LJ et al., 2013;Kim et al., 2015). Chorus waves are typically divided into two distinct bands: the upper band (frequencies 0.5-0.8 f ce ) and the lower (0.1-0.5 f ce ) (Tsurutani and Smith, 1974;Santolík et al., 2003a). Both chorus wave bands are usually characterized by continually repeating rising-tone or/and fallingtone elements, which are considered to result from nonlinear wave growth (e.g. Omura et al., 2008). It has been demonstrated that, before evolving into the nonlinear growth phase, chorus emissions progress through their linear growth phase sufficiently for the chorus wave amplitude to reach the threshold of second order resonance (Katoh and Omura 2011;Summers et al., 2012). During linear wave growth, the free energy required is deemed to be supplied by suprathermal electrons (kinetic energy E k~s everal 10 keV) with anisotropic temperatures (Kennel，1966; Jordanova et al., 2010).
Chorus emissions have attracted more and more attention since suggestions that these waves make significant contributions to the dynamics of the Van Allen Belts (e.g. Shprits et al., 2006Shprits et al., , 2008Chen Y et al., 2007;Su ZP et al., 2011Thorne et al., 2013;Huang J et al., 2018). Quasi-linear resonant diffusion coefficient calculations by Summers et al. (2007) have demonstrated that chorus waves can exert effective energy diffusion on relativistic electrons via cyclotron resonance, which confirms that chorus can be a viable mechanism for generating relativistic electrons. Moreover, seed electrons (E k~s everal 100 keV) can be accelerated to relativistic energies through second order resonance, which is also called relativistic turning acceleration (RTA) in Omura et al. (2007).
Recently, low-frequency chorus emissions, whose frequencies can fall below 0.1 f ce , have been reported (Cattell et al., 2015). Gao ZL et al. (2016) have studied the detailed structures of low-frequency chorus waves and their role in radiation belt electron evolution. Different from typical chorus waves, the low-frequency chorus waves are found to cause rapid precipitation of the seed electrons via pitch angle scattering. However, just how these low-frequency chorus waves are excited has not been well understood. In this letter, we report an interesting case of extremely low-frequency chorus emissions observed by the Van Allen Probes mission; we focus on the role played by background plasma density in the excitation of these emissions.

Van Allen Probe Observations
The Van Allen Probes mission, consisting of two identical satellites, was launched on 30 August 2012. Several precision instruments on these satellite provide high-resolution data that allow us to study mechanisms of wave excitation. In this work, wave spectral matrices in the frequency range from ~ 10 Hz to 12 kHz are attained from the Waveform Receiver (WFR) in the Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) onboard Van Allen Probe A (Kletzing et al., 2013). We utilize these spectral matrices to extract relevant information, including wave spectral intensity, elliplicity, and wave normal angles (WNA) of whistler waves through Singular Value Decomposition (Santolík et al., 2003b;Zhou QH et al., 2014). The Electric Field and Waves (EFW) Instruments (Wygant et al., 2013) measure the spacecraft potential, which can be used to infer the background plasma number density N e , and burst mode data (up to ~16.4 samples per second) of 3-dimensional electric fields; these data are used to investigate the fine structures of chorus waves. Furthermore, particle flux attained from the Helium, Oxygen, Proton, and Electron (HOPE) mass spectrometer (Funsten et al., 2013) are applied to obtain the phase space distributions which would be input parameters for calculations of linear growth rates. Figure 1a illustrates the trajectory of the Van Allen Probe A (thin blue curve) between 2014 August 11, 23:30 UT, and 2014 August 12, 02:00 UT, and the plasmapause (green curve) derived from plasmapause test particle (PTP) simulations (Goldstein et al., 2014). The thick red curve denotes the time interval of interest (2014 August 12, 00:30-01:30 UT), during which the low-frequency chorus waves are observed. It is easily seen that, during this time interval of interest, Van Allen Probe A is flying outside the plasmapause. Plasma densities attained from EFW instruments are exhibited by the black upper triangle symbols in Panel (b). Since density data are nowhere available, we have estimated the electron density by performing a best fit of the available electron density data to the spacecraft potentials by the form of N e = (aU) b , following Moullard et al. (2002). As a result, a = 0.0926 and b = -2.2143 are obtained and then the electron density profiles are derived and shown by the red curve. It is indicated that at about 00:20 UT, Van Allen Probe A moves outside the plasmapause.
An overview of the low-frequency chorus emissions observed by Van Allen Probe A during the time interval of interest is presented in Figure 2. In Panel (a) of Figure 2, both electron densities N e attained from EFW instruments and spacecraft potential modeling are shown in the same form. The magnetic field spectral density, electric field spectral density, ellipticity ε, and WNA are displayed in Panels (b)-(e), respectively. The three white curves in Panel (b) denote the frequencies at 0.5 f ce , 0.1 f ce , and 0.02 f ce , respectively. In Panels (b) and (c), during 00:40 UT-01:30 UT, the typical twoband structures of usual chorus waves with frequencies of 0.1-0.5 f ce and 0.5-0.8 f ce are recognized. During this time interval, the socalled low-frequency chorus waves with frequencies extending below 0.1 f ce are clearly seen at several time intervals denoted by the red arrows on the top of Panel (a). As shown in Panels (d) and (e), these low-frequency chorus waves are right-hand polarized (ε = 1) and propagate nearly along the background magnetic field (WNA < 30°). The magnetospheric coordinates (L, MLT, MLAT) on the bottom of Panel (e) demonstrate that these waves occurred in the morning sector (MLT≈ 5 hours) near the equatorial plane (MLAT≈ 4°) in the inner magnetosphere (L≈ 5), indicating that these chorus waves might be locally excited (LeDocq et al., 1998;Meredith et al., 2003;Li W et al., 2011). It is worth noting that the low-frequency chorus waves are observed in a quite dense plasma (N e ≥ 50 cm -3 ), which is several times larger than densities associated with typical chorus emissions (e.g., Li W et al., 2015).
To study the detailed structure of the low-frequency chorus waves, burst mode data from EFW instruments around 01:17:36-01:18:05 UT, denoted by the black rectangle in obtain the dynamic spectrum. The dynamic spectrum in the frequency range 100 Hz to 8 kHz is displayed in Figure 3e; the corresponding electron densities are zoomed in and shown in Figure 3d. The abscissa is the time lag (∆t) in seconds from 01:17:36.322 UT. As seen in these two panels, at a tenuous background plasma (N e ≤ 10 cm -3 ), usual chorus emissions occur with typical frequency characteristics: two-band structures with a lower cutoff frequency of about 0.1 f ce and rising-tone sequences (marked by thin black lines). When the background plasma density increases to about 70 cm -3 , the lower cutoff frequency is much less than 0.1 f ce . Especially, as shown in Figure 2b, when N e reaches 90 cm -3 , the lower cutoff frequency is about 0.02 f ce . It is a remarkable fact that for the low-frequency chorus waves shown in our case, the portions with frequencies above 0.1 f ce are fine structured while those with frequencies below 0.1 f ce are unstructured ("hiss-like"). Gao ZL et al. (2016) and Xiao FL et al. (2017) have reported that there are two types of low-frequency chorus waves: fine structured and hiss-like. The case of low-frequency chorus waves shown in Figure 2 is of the latter type. Li W et al. (2012) have also reported hisslike emissions, but with frequencies above 0.1 f ce . However, the excitation mechanism of these emissions has not been understood. In this paper, we use the in situ particle data of Van Allen Probe A to study the role played by density in the excitation of low-frequency chorus waves with the lower cutoff frequency extending to 0.02 f ce .
In Figure 3, the electron densities and electric field wave amplitudes are displayed in Panel (a) as red and blue curves, respectively, while the electric field spectral densities are illustrated in Panel (b), and the observed electron fluxes provided by HOPE instruments are exhibited in Panel (c). Here the wave amplitude is obtained by integrating the spectral density during frequencies from 100 Hz to 5 kHz. The suprathermal electrons, which are generally considered to be the source of free energy for whistler wave growth, can be seen to have only slight variations. To show this more clearly, in Panel (f) we overlay the parallel f (v ⊥ = 0) and perpendicular f (v // = 0) components of the electron phase space distribution functions during 00:30:00 UT-01:30:00 UT as light gray and dark gray curves. The thick black and red curves denote the averaged parallel and perpendicular phase space distribution functions. Panel (f) also shows that these suprathermal electrons with energies up to 30 keV have anisotropic temperatures; this observation suggests local growth of the observed chorus waves.
Since only small changes are observed in all physical parameters of the plasma environment except the background electron density, it is suggested that electron density may play an important role in the excitation of low-frequency chorus waves.

Discussion
Reports of low-frequency chorus waves (Cattell et al., 2015;Gao ZL et al., 2016) have raised considerable interest in the mechanisms that might generate them. Xiao FL et al. (2017) have proposed that high-energy relativistic electrons with a pitch angle anisotropy can provide the free energy for low-frequency chorus wave growth. Unfortunately, the background plasma density data have been unavailable and thus only the wave excitation in several low densities (N e < 10 cm -3 ) has been analyzed in their studies. On the other hand, Omura et al., (2015) have proposed in theoretical calculations that a higher plasma density will result in lower-frequency whistler waves. Assuming a certain model of source electrons, Woodroffe et al. (2017) also argued that the generation of   Earth and Planetary Physics doi: 10.26464/epp2019001 3 low-frequency whistler emission might be due to enhanced background electron densities. In our work, however, we implemented investigation of the excitation of low-frequency chorus emissions using simultaneously satellite wave observations and in situ plasma environment data, in addition to data regarding source electrons and background plasma densities. As other physical quantities changed only gently, during the time period studied, in comparison to background plasma density, data from this particular interval of interest provide a good opportunity to study the role played by background electron density in the excitation of low-frequency chorus emissions.
With the physical parameters of the plasma environments acquired from the satellite in situ observations, we used the linear growth theory to investigate the role of background electron density. These physical parameters are displayed in detail in Table 1.
In brief, we assume that the background plasma consists of only two species, electrons and hydrogen ions. The hydrogen ions are modeled to have a temperature-isotropy Maxwellian distribution, while the electrons are assumed to have four populations (a cold temperature-isotropy population, and two warm-and a hot-temperature-anisotropy populations), of which each component also has a bi-Maxwellian distribution with the detailed parameters shown in Table 1. The temperatures of the cold electrons and hydrogens are set following Olsen et al. (1987) and Horne and Thorne (1993). The parameters of the warm and hot electron distributions are obtained by performing a best fit to the averaged electron phase space distributions (Figure 3f). The fitted and averaged electron phase space distributions are shown in Figure 4a as curves and asterisks, where the red and black symbols (curves and asterisks) denote the parallel and perpendicular components, respectively. Moreover, the density of the cold hydrogen ions n p is set to maintain charge neutrality. Linear growth rates γ are calculated by solving the kinetic dispersion relation for a complex frequency (ω+iγ) and a real wave vector k, with the assumption that the imaginary frequency γ is much less than the real frequency ω (Kennel, 1966;Chen et al., 2010 ;Yu XD et al., 2016Yuan ZG et al., 2017. We have calculated the linear growth rates of the parallel whistler mode (WNA = 0°) for a set of different cold electron densities N e , and show in Figure 4b the results of five cases. The growth rates γ (normalized to the electron cyclotron frequency Ω e ) for five different densities (N e = 6, 20, 90, 300, and 800 cm -3 ), denoted by dark EFW Model blue, green, red, light blue, and purple curves, respectively, are exhibited as a function of the normalized real frequency ω/Ω e . As we focus on the low-frequency chorus emissions, growth rates are il-lustrated for ω/Ω e ≤ 0.5. As shown in Figure 4b, instabilities (γ > 0) can occur in the frequency interval 0.10 < ω/Ω e < 0.38 when the background electron density is low (N e = 6 cm -3 ); this is the typic- 2014-08-12 01:17:24 2014-08-12 01:20:00 al case for occurrence of the usual lower band chorus waves (Meredith et al., 2003). When the density increases, the lower cutoff frequency of the whistler instabilities tends to move toward a normalized frequency below 0.1. Especially, instabilities can extend to a normalized frequency of 0.02 (denoted by the dashed vertical line) when the density reaches 90 cm -3 or more; this result is consistent with our observations (Figure 2). It is suggested that if background electron densities increase further, the unstable normalized frequencies will become lower than 0.02 and then hiss wave instabilities will be expected to occur, as demonstrated in Su et al. (2017). We present in Panels (c) and (d), respectively, the calculated growth rates along with the observed electric field spectral densities of the low-frequency chorus waves (at 01:17:24 UT) and the usual chorus waves (at 01:20:00 UT). Note that positive growth rates indicate only the possibilities of wave excitation and growth, so that the comparison between calculated growth rates and observed wave spectrums is to show the possibility of the excitation of low-frequency chorus waves. To study the final wave intensity and particle distributions, quasilinear theory and/or particle simulations are needed (e.g. Tao X et al., 2017). The consistency between the growth rates and the observed spectral densities demonstrates that an adequately dense plasma could result in low-frequency whistler chorus emissions whose free energy is provided by temperature-anisotropic suprathermal electrons.

Conclusion
In conclusion, we have presented analysis of an event of extremely low-frequency chorus emissions observed by Van Allen Probe A on 12 August 2014. These chorus emissions have intense power in frequencies lower than those of usual chorus waves. Using the physical parameters obtained from in situ satellite observations, linear growth theory is utilized to investigate the role of background plasma density in the excitation of these low-frequency chorus emissions. The results show that in a certain dense background plasma (N e = 90 cm -3 in this case), the whistler instabilities can extend to a normalized frequency ω/Ω e of 0.02, which is consistent with our observations. It is demonstrated that the background plasma density might play an important role in the excitation of extremely low-frequency chorus emissions. Although the low-frequency chorus waves reported here are unstructured so that only linear growth theory is utilized, nonlinear processes proposed by Omura et al. (2008) should be applied to investigate the effects of high background plasma density on the fine structures of low-frequency chorus emissions, which is beyond the scope of this study but will be carried out in future work. Moreover, since these low-frequency chorus waves are observed in an electron density higher than expected, their effects on radiation belt electrons must be reconfirmed, which will be carried out in future work using quasi-linear resonant diffusion theory.