Dependence of Quasi‐Electrostatic Magnetosonic Wave Generation on Plasma Density and Suprathermal Protons

Magnetosonic (MS) wave is one of the most common electromagnetic emissions in the magnetosphere, while quasi‐electrostatic magnetosonic (QEMS) waves are seldom observed. Here we present an interesting QEMS event detected by Van Allen Probe A on 9 January 2014, in which the presence of multi‐band QEMS emissions shows evident correlations with relatively low plasma densities (∼10 cm−3) and enhancements of suprathermal (∼10 − 100 eV) proton fluxes. Based on the observed proton distributions, simulations demonstrate very similar profiles to the observed QEMS in the wave spectral characteristics. Parametric studies indicate that the QEMS growth rate can increase by ∼20 times with the plasma density decreasing from 50 to 10 cm−3. Effective growth rates of higher‐band QEMS occur when the suprathermal proton population is sufficiently large. This study reveals that low densities and large suprathermal proton populations are favorable for generating distinct multi‐band QEMS in the magnetosphere.

10.1029/2023GL103083 2 of 8 MS waves are generally electromagnetic fluctuations with banded-structures Meredith et al., 2008;Min et al., 2018;Perraut et al., 1982;Yuan et al., 2018Yuan et al., , 2019. The magnetic component of MS waves can be quite weak when the frequency approaches the lower hybrid frequency (Yuan et al., 2017;X. Yu et al., 2021). Recently, quasi-electrostatic magnetosonic (QEMS) waves only occurring in the first few harmonic bands are reported by Z. Gao et al. (2021). Their simulation demonstrates that proton ring-like distributions are the free energy source of QEMS as well. However, the favorable conditions of QEMS generation in the terrestrial magnetosphere are still unrevealed. That is the main goal of this study.

Observations
Here we use the three-component magnetic field data detected by the fluxgate magnetometer (MAG) of the Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS)  and the two-component electric field data in mGSE coordinate system measured by Electric Field and Wave (EFW) instrument (Wygant et al., 2013). A fast Fourier transform (FFT) with a 1024-point sliding window is performed on the electromagnetic field data in the time domain to obtain the wave spectra. The hot proton differential fluxes j are collected by the Helium, Oxygen, Proton, and Electron (HOPE) Mass Spectrometer (Funsten et al., 2013), and converted to phase space density (PSD) F by using F = j/p 2 (p is the particle momentum). The electron density n e is inferred from the upper hybrid resonance frequency detected by the High Frequency Receiver (HFR) of the EMFISIS Waves instrument (Kurth et al., 2015).
As shown in Figure 1, the event was observed during 17:45-19:20 UT on 9 January 2014 when Van Allen Probe A traveled through the equatorial region (MLAT = −0.4° to −0.2°) with L = 5.6 − 5.7 and magnetic local time (MLT) = 13.4 − 14.4. Figure 1a plots the background plasma density which varies between ∼10 and 50 cm −3 , indicating that the satellite is outside the plasmapause. Figure 1b displays the omni-directional differential fluxes of protons with the energy range 10 eV-30 keV, showing that a pronounced proton ring-like distribution occurs around 10 keV. Figures 1c-1d exhibit the magnetic ( Figure 1c) and electric ( Figure 1d) field spectrums. Distinct electrostatic emissions occur just below the harmonics of the proton cyclotron frequency f cp , showing spectral characteristics in coincidence with QEMS features (Z. Gao et al., 2021). A very interesting thing here is that the occurrence of QEMS waves is exactly corresponding to both the presence of relatively low plasma density (n e ∼ 10 cm −3 ) and the enhancement of suprathermal (10 − 100 eV) proton flux as highlighted by the gray shaded area, while the proton ring-like distribution is barely changed during the whole interval. This leads to a reasonable speculation that the plasma density and the suprathermal proton population can significantly affect the generation of QEMS.

Comparison Between the Simulations and Observations
In order to investigate the effects of plasma density and suprathermal protons on QEMS excitation, three typical time intervals are selected for the simulations, as labeled in Figure 1. At 18:00 UT (case A), the QEMS waves were pronounced (>10 −2 mV 2 m −2 Hz −1 ) with a clear multi-band structure, while n e ∼10 cm −3 and the suprathermal proton flux was enhanced. At 19:20 UT (case B), the QEMS was weak with electric field spectral power density ∼10 −3 mV 2 m −2 Hz −1 and the wave only occurred in the first harmonic band, when n e remained ∼10 cm −3 but the enhancement of suprathermal proton flux decayed. At 19:30 UT (case C), QEMS disappeared as n e increased to ∼30 cm −3 and the enhancement of suprathermal proton flux faded away. Actually, the selection of case C can be arbitrary. Here case C is selected because it is just 10 minutes after case B when the location is still very close to cases A and B with the deviation of L-shell<0.2.
The particle PSD are modeled by a sum of subtracted Maxwellian components where N is the number of proton components; for the ith component, n i is the density, T i is the parallel temperature, th = √ 2 ∕ is the parallel thermal velocity. α i denotes the temperature anisotropy, and β i and Δ i characterize the size and depth of the loss cone feature.
The electron distribution F e is assumed to be a single Maxwellian component with the temperature of 1 eV. The electron densities n e are from the observations as mentioned earlier. Four components (N = 4) are used to describe the proton distributions. The first component (cold) with temperature T c = 1 eV controls the cold protons, the second component (h1) with T h1 = 63 eV mainly represents the suprathermal protons, the third component (h2) is used to adjust and smooth the fitting, and the fourth component (r) with T r = 9.253 keV dominates the ring distribution. Here n h1 , n h2 , and n r are obtained by fitting the observations and n c = n e − (n h1 + n h2 + n r ) to keep the electric neutrality. Considering that the proton ring distribution is stable during the entire event, the third (n h2 ) and fourth (n r ) components keep unchanged for all the three cases. Table 1 lists the parameters for modeling the particle distributions. Figure 2 plots the comparison between the modeling and observations. Proton PSD at pitch angle α = 90° (left column) shows that a clear bump in the energy range ∼2-10 keV is observed (dots). This indicates that a pronounced proton ring-like distribution occurs, which can provide free energies for the excitation Electron ( of ion Bernstein mode. The modeled PSD at α = 90° (solid lines) fits the observed ring-like distribution very well. The modeled ring-like distribution in the energy E k and pitch angle α domains (right column) roughly agrees with the observational data (middle column), indicating that the temperature anisotropy of ring protons is reasonably characterized as well.
The ion Bernstein dispersion relation is solved by using the Waves in Homogeneous, Anisotropic, Multicomponent Plasmas (WHAMP) code (Ronnmark, 1982) in the wave normal angle θ range of 85°-89.95° with a step of 0.05°. Since the variation of L−shell is less than 0.2 for the three cases, we set the ambient magnetic field as 144.5 nT in the simulations, which is the in-situ measurement at 18:00 UT, to focus on the influence of n e and suprathermal protons. The growth rates for the frequencies below 5f cp are calculated. The dispersion curves of the peak growth rates are shown in Figure 3, with the corresponding normal angle labeled. For example, in case A, the maximal growth rate occurs at θ = 89.9° for the first harmonic band but at θ = 89.95° for the higher harmonic bands. The peak growth rates occur at very perpendicular directions with θ ≥ 89.9°, and the dispersion relations exhibit quite similar patterns to the results in Z. Gao et al. (2021).
For case A, significant growth rates γ/Ω cp >10 −4 occur for all the four bands at frequencies marginally below the multiples of f cp and the growth rates gradually decrease as the harmonic number increases. For case B, effective growth rates only occur in the first band with the maximum growth rate slightly decreasing. For case C, only weak growth rates appear in first band, with the values approximately one order of magnitude lower than these of cases A and B. The simulation results of all the three cases demonstrate very similar profiles to the observations in the wave amplitude and frequency spectral characteristics. Besides, there is no effective growth rate close to f cp , while weak fluctuations are observed near the fundamental harmonic. It implies that those weak fluctuations are unlikely to be excited locally. X. Gao et al. (2018) suggested that the nonlinear wave-wave interactions between higher-QEMS harmonics can be responsible for the weak fluctuations. In order to focus on the effect of the density (n e ) and the suprathermal proton populations (n h1 ), we use fixed anisotropy parameters for the suprathermal protons in the simulations. Considering that the growth rate may depend on the particle anisotropy, we model a proton distribution with beams for case A, which well reproduces the temperature anisotropy below 1 keV, as shown in Figure S1 and Table S1 in Supporting Information S1. The corresponding growth rate ( Figure S2 in Supporting Information S1) shows very similar profiles to that in Figure 3a, indicating that the anisotropy of suprathermal protons affects the QEMS growth rate little.

Parametric Study
As the components of the modeled proton distribution are specified, it is convenient for us to distinguish the influences of the plasma density and suprathermal proton population on the QEMS generation by adjusting the values of n e and n h1 . For the plasma density, n e is set as 10, 30, and 50 cm −3 , where 10 cm −3 (cases A and B) and 30 cm −3 (case C) are the realistic values and 50 cm −3 is chosen to make the values of n e an arithmetic progression for comparison. For the suprathermal proton population, n h1 is respectively set as 0.75, 0.08, and 0.02 cm −3 , which are the fitting values for the three observed cases as shown in Table 1. In total, we can get nine cases with different permutations and combinations of n e and n h1 . The simulation results together with the selected n e and n h1 are shown in Figure 4. Accordingly, Figures 4a, 4d, and 4h are respectively corresponding to the three typical intervals, which are the same as in Figure 3.
We can find that the peak growth rate has monotonic dependence on n e . When n e increases from 10 to 50 cm −3 , the peak growth rate decreases by about 20 times. Besides, with an increasing n e , the growth rates occur at smaller wave number k with a narrower range, suggesting that the generated emissions tend to follow the cold plasma dispersion relations of MS mode (red dashed lines in Figure 4) (Min & Liu, 2015). The three rows in Figure 4 are calculated by using three different densities of the suprathermal protons n h1 , which are respectively 0.75, 0.08, 0.02 cm −3 as labeled. As n h1 decreases to ≤0.08 cm −3 , effective growth rates only occur in the first band, and the changes in the morphology and the maximum value of the first band growth rates are imperceptible. It is suggested that the suprathermal proton population is beneficial for generating the higher bands and thus forming the multi-band structure of QEMS.

Summary
In this study, we report a QEMS event observed by Van Allen Probe A on 9 January 2014. The presence of pronounced multi-band QEMS emissions are strongly associated with the existence of low plasma density and the flux enhancement of suprathermal protons with energies ∼10-100 eV. We conduct simulations based on fully kinetic theory to investigate the effects of plasma density and suprathermal protons on the QEMS generation. The main results are summarized as follows.
1. Based on the observational data, the simulations based on fully kinetic theory can well explain the variations of wave features in the wave intensity and the frequency pattern. 2. Parametric studies show that the peak QEMS growth rate increases about 20 times as the plasma density n e decreases from 50 to 10 cm −3 , suggesting that the tenuous cold plasma is favorable for generating distinct QEMS. 3. Effective growth rates extend to the second to fourth bands when the suprathermal proton density n h1 increases to 0.75 cm −3 , while the morphology and the maximum value of the first band growth rates are barely changed. This suggests that a large suprathermal proton population plays a critical role on generating the higher bands and thus forming the multi-band structure of QEMS.
This study enriches our understanding of QEMS that the low plasma density and high suprathermal proton fluxes provide favorable conditions of generating distinct QEMS with multi-band structures. Compared with the electromagnetic MS waves, the requirement of these additional conditions, besides the free energies from the proton ring-like distribution, may lead to the much lower occurrence of QEMS in the terrestrial magnetosphere.