Anisotropic velocity distributions of protons have long been considered as free energy sources for exciting electromagnetic ion cyclotron (EMIC) waves in the Earth's magnetosphere. Here we rigorously calculated the proton anisotropy parameter using proton data obtained from Van Allen Probe-A observations. The calculations are performed for times during EMIC wave events (distinguishing the times immediately after and before EMIC wave onsets) and for times exhibiting no EMIC waves. We find that the anisotropy values are often larger immediately after EMIC wave onsets than the times just before EMIC wave onsets and the non-EMIC wave times. The increase in anisotropy immediately after the EMIC wave onsets is rather small but discernible, such that the average increase is by ~15% relative to the anisotropy values during the non-EMIC wave times and ~8% compared to those just before the EMIC wave onsets. Based on the calculated anisotropy values, we test the criterion for ion cyclotron instability suggested by Kennel and Petschek (1966, https://doi.org/10.1029/JZ071i001p00001) by applying it to the EMIC wave events. We find that despite the weak increase in anisotropy, the majority of the EMIC wave events satisfy the instability criterion. We suggest that the proton distributions often remain close to the marginal state to ion cyclotron instability, and consequently, the proton anisotropy values should often be observed near threshold values for ion cyclotron instability. Additionally, we demonstrate the usefulness and limitation of the instability criteria expressed in the form of an inverse relation between the anisotropy and plasma beta.
- Proton anisotropy is often larger immediately after EMIC wave onsets than the times before EMIC wave onsets and non-EMIC wave times
- The amount of anisotropy increase is small; nevertheless, a large fraction of EMIC waves satisfies the ion cyclotron instability criterion
- Proton distributions often remain near a marginal state, with anisotropies close to threshold values for ion cyclotron instability
In the Earth's magnetosphere, electromagnetic ion cyclotron (EMIC) waves have long been considered to be among the important waves that can precipitate energetic charged particles into the atmosphere through pitch angle scattering (Chen et al., 2011; Cornwall et al., 1970; Jordanova et al., 2008; Kersten et al., 2014; Kubota et al., 2015; Li et al., 2007, 2014; Lorentzen et al., 2000; Meredith et al., 2003; Miyoshi et al., 2008; Ni et al., 2015, 2018; Shprits et al., 2009, 2013; Summers & Thorne, 2003; Thorne & Kennel, 1971; Usanova et al., 2014). EMIC waves are also known to cause heating of He+ ions (e.g., Gendrin & Roux, 1980; Young et al., 1981; Yuan et al., 2016; Zhang et al., 2010, 2011) and to suffer from Landau damping due to cold electrons (e.g., Cornwall et al., 1971; Thorne & Horne, 1992; Yuan et al., 2014; Zhou et al., 2013). All these interactions will lead to redistribution of different plasma populations in the magnetosphere.
EMIC waves are known to be excited with left-hand polarization at frequencies below the local proton cyclotron frequency through the growth of ion cyclotron instability. It is generally accepted that the waves are preferably generated at the weakest magnetic field regions along the magnetic latitude such as near the magnetic equator on the nightside and somewhat off the equator on the dayside, while the main region of EMIC wave occurrence is the dayside at large L-shell and the nightside region close to the plasmapause (Anderson et al., 1990; Cornwall, 1965; Fraser & McPherron, 1982; Kennel & Petschek, 1966; Meredith et al., 2003).
According to the classical linear perturbation theory of low frequency plasma waves, an anisotropic velocity distribution of energetic ions is an important parameter for ion cyclotron resonant instability (Chen et al., 2011; Denton et al., 1993, 1994; Gary et al., 1976, 1994a, 1994b, 2012; Hu et al., 1990; Kennel & Petschek, 1966; Kozyra et al., 1984; Lee et al., 2017; Silin et al., 2011). Kennel and Petschek (1966) have shown that the proton anisotropy is the only parameter that determines whether an ion cyclotron wave with a given frequency grows or damps. The growth rates of ion cyclotron waves are determined by combinations of the anisotropic velocity distributions of protons and the number of energetic protons participating in resonance interactions with L mode seed waves.
Ion cyclotron instability has also been examined based on an inverse relation of ion anisotropy with parallel plasma beta β∥ ( ) (Allen et al., 2016; Anderson et al., 1994; Blum et al., 2009, 2012; Gary et al., 1976; Gary et al., 1994a, 1994b; Lin et al., 2014; Phan et al., 1994; Spasojevic et al., 2011). The inverse relations are given by T⊥/T∥ − 1 = Q/β∥α, where the subscripts ∥and ⊥ refer to the directions parallel and perpendicular to the background magnetic field, respectively, and α and Q are the parameters that define specific models of inverse relationships. For example, in the inverse relation model of Blum et al. (2009), α and Q are a function of the ratio between hot proton density and cold proton density (an additional discussion is provided later in section 5).
Observational tests of ion cyclotron resonant instability are non-trivial or often limited. Nevertheless, there have been some attempts to test it using recent satellite observations. For example, a comprehensive statistical test was undertaken by Lin et al. (2014) using Cluster satellite observations. They identified time periods showing hot proton temperature anisotropy, for which they tested the threshold conditions for ion cyclotron resonant instability given by Kennel and Petschek (1966) and Blum et al. (2009). They concluded that the proton anisotropy is necessary but not sufficient alone for excitations of ion cyclotron waves and noted limitations of predicting the instability using the Blum et al.'s inverse relation. Blum et al. (2012) identified EMIC waves from geosynchronous observations for which they tested the inverse relation model of Blum et al. (2009). They reported a strong agreement between EMIC wave occurrence and the prediction from the inverse relation. Spasojevic et al. (2011) found a good correspondence between predictions from the Blum et al. inverse relation and subauroral proton precipitation, which is thought to result from interactions with EMIC waves. Zhang et al. (2014) also tested Blum et al.'s inverse relation for an EMIC wave event interval identified from satellite observations of the Van Allen Probes mission, which was formerly known as the Radiation Belt Storm Probes (RBSP) mission (Kessel et al., 2013; Mauk et al., 2013). They reported that the inverse relation predicts the excitation of waves with elevated thresholds. The recent work by Allen et al. (2016) paid special attention to the spatial distributions of the inverse relation model in Blum et al. (2009) as well as those of various plasma conditions for EMIC wave observations from the Cluster satellite.
In this paper, we revisit the problem of observational tests for ion cyclotron resonant instability. Our work is distinguished from the previous works mentioned above in the following aspects. First, our main emphasis is on the test of the instability criterion given by Kennel and Petschek (1966), although we also test and discuss the Blum et al. inverse relation. Second, for testing the Kennel-Petschek instability criterion, we rigorously calculate the original formula of the proton anisotropy parameter defined by Kennel and Petschek (1966) by using realistic proton distributions obtained from observations of Van Allen Probe-A. Third, we apply the anisotropy calculations to the EMIC wave times identified from Van Allen Probe-A observations and compare them to those undertaken for the times when no EMIC waves are seen (non-EMIC wave times). For a meaningful comparison, the non-EMIC wave times are carefully selected on the same L and MLT locations either just before or after the orbit where EMIC waves are identified. Fourth, we use proton data over a wider energy range than in the previous works so that any higher energy contributions to the anisotropy and other plasma parameters can be correctly included. Finally, we provide an interpretation of the test results on ion cyclotron resonant instability that differs from the previous works.
The present paper is organized as follows. In section 2, we provide descriptions of the data and wave identification methods used here. Section 3 introduces the calculation method of the anisotropy parameter. In section 4, we present the statistical results of the calculated anisotropy parameter, on the basis of which we further test the instability condition of Kennel and Petschek (1966). In section 5, we examine the inverse relations between the anisotropy and plasma beta, present the test results, and discuss their implications. Finally, we present the relevant discussions and conclusions in section 6.
2 Data and Identification of EMIC Waves
2.1 Van Allen Probe-A Observations
The Van Allen Probes satellites were launched on 30 August 2012. Their orbital inclinations are ±10°, they cover radial distances of ~500 to ~30,600 km, and their orbital periods are ~9 hr (Kessel et al., 2013; Mauk et al., 2013). The two identical satellites have the same orbit but different transit times. The line of apsides precesses 210°/year. In this work, we use data from the fluxgate magnetometer for EMIC wave identification, two particle detectors for examining proton distributions, and two wave instruments for determining cold electron density onboard Van Allen Probe - A.
Specifically, we use the magnetic field data from the Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) MAG (Kletzing et al., 2013) for EMIC wave identification. The sampling cadence and frequency coverage of the instrument are 64 Hz and 0–30 Hz, respectively. The maximum resolution of the data is 0.001 nT.
We use the data from the Helium Oxygen Proton Electron plasma spectrometer (HOPE), which is one part of the particle detectors of the Energetic particle, Composition, and Thermal plasma (ECT) suite and covers the low energy part among the three plasma instruments (Funsten et al., 2013). It covers ions and electrons in the energy range from ~1 eV to ~52 keV and pitch angles of 0 to 180°. In this study, we used proton L3 data, which provided differential directional fluxes every 22 s.
We also use the particle data from the Radiation Belt Storm Probes Ion Composition Experiment (RBSPICE), which provides differential directional fluxes for protons, helium, oxygen and electrons (Mitchell et al., 2013). We use the L3 TOFxEH proton flux data provided in the energy range from 37 to 488 keV at ~10-s resolution.
For the analysis performed in sections 3 to 5, we use the proton flux data by combining the HOPE and RBSPICE data. The combination is done by applying a temporal interpolation to the two data sets such that the combined data have the time resolution of 20 s.
The plasma density is obtained from both EMFISIS and the Electric Field and Waves (EFW) suite in different ways. The cold electron density of EMFISIS is calculated by estimating an upper hybrid frequency (Kurth et al., 2015). The upper hybrid frequency is not clearly identified at disturbed times. The data from EFW are calculated from spacecraft potential. The method from EFW is reliable in dense plasma regions, for example, inside the plasmasphere. For this research, we basically use the density data sets that are provided from the official EMFISIS products and those from the official EFW products when EMFISIS density data are not available. Both data sets are available from the NASA cdaweb site, ftp://cadweb.gsfc.nasa.gov/.
2.2 Wave Event Selection
From the observed three axis magnetic field data, we obtain EMIC wave events in the following way. First, the magnetic field fluctuation (δB) is obtained by subtracting a sliding averaged value of each X, Y, and Z component from the original magnetic field data in Geocentric Solar Ecliptic (GSE) coordinates. The fluctuation in the magnetic field is then transformed to the field-aligned coordinates (FAC), (XFAC, YFAC, ZFAC), where ZFAC is defined as a local mean field direction, YFAC = ZFAC × XGSE, and XFAC completes the right-handed orthogonal coordinate system. Second, we conduct a short-term Fourier transform to obtain a dynamic spectrum of the power spectral density (PSD) of the fluctuations in frequency-time space. The time window used in this process is set to ~1 min (4,000 points) and slides every 30 s. In all the procedure described here and below, we used the data only for L > 2.
- First, a median PSD in the common logarithmic form, log10(PSDm), is estimated over each orbital period of the satellites (for L > 2).
- By subtracting this from the logarithm of the original PSD, log10(PSD), we define the median-subtracted PSD in the logarithmic form, δ[log10(PSD)] = log10(PSD) − log10(PSDm). To remove unphysical spikes in the data, we apply a smoothing procedure to δ[log10(PSD)] over 5 points (2 min 30 s).
- To define an EMIC wave event, we only take an interval where δ[log10(PSD)] exceeds unity; that is, the original PSD is 10 times larger than the median PSD, PSDm.
- From the obtained wave spectra above, we exclude broadband wave events whose powers extend across more than two wave bands, which are determined by local ion cyclotron frequencies, and spiky events based on visual inspection.
- Finally, an event satisfying all these criteria is regarded to be independent when it is separated by 30 min or longer from adjacent events.
In addition, for the EMIC wave events identified from the above procedure, we impose an additional requirement that the proton data from both HOPE and RBSPICE are available so that the analysis of the proton distribution anisotropy can be undertaken over a wide energy range, as reported in the next section. Finally, we identify a total of 302 EMIC wave events from 1 March 2013 to 30 November 2015. Out of this, 125 events are identified only at H+ band, 107 events occur only at He+ band, and the remaining 70 events exhibit EMIC waves at both bands; this means that the number of H+ band waves is 195 (125 + 70) and that of He+ band waves is 177 (107 + 70). The durations of the identified EMIC wave events range from 30 s to ~271 min, with an average duration of ~44 min. Figure 1a shows the identified 302 EMIC wave locations on the L –MLT plane (left panel) and their statistics in L, MLT, and MLAT (right three panels). As widely known in previous studies, the majority of the EMIC waves are located in the afternoon sector, and readers are referred to previous reports for detailed statistical features of EMIC waves (Allen et al., 2015; Anderson et al., 1992; Fraser & Nguyen, 2001; Halford et al., 2016; Keika et al., 2013; Li et al., 2007; Lorentzen et al., 2000; Meredith et al., 2003, 2014; Min et al., 2012; Saikin et al., 2015; Tetrick et al., 2017; Usanova et al., 2013; Wang et al., 2015). Figure 1b is a similar plot for the 113 events selected from the 302 events, which satisfy an additional requirement on the proton flux data availability (this is described in detail in section 3.1).
3 Estimation of the Kennel-Petschek Anisotropy Parameter
3.1 Computation of the Anisotropy Parameter
Here the subscripts ∥and ⊥ indicate the parallel and perpendicular directions to the background magnetic field, and refers to the resonant velocity. Consequently, equation 2 implies that the specific shape of the velocity distribution function of particles f determines whether a wave of a given frequency grows or not. Note that Kennel and Petschek (1966) did not assume a specific distribution function in the derivation of the growth rate expression 1. If the distribution function is a bi-Maxwellian distribution with temperatures T⊥ and T∥, the KP anisotropy can be simplified to which is independent of energy and which we call “temperature moment anisotropy” in this paper. The use of the temperature moment anisotropy may be inappropriate when the particle distribution is not represented by a single bi-Maxwellian. Therefore, in this paper, we aim to evaluate the KP anisotropy by using observed proton velocity distributions for a more accurate determination of ion cyclotron wave growth.
There are a few issues in evaluating Equation 2. First, the integration in 2 is to be conducted at the resonant parallel velocity. This means that only the particles that participate in resonant interactions determine the KP anisotropy parameter. Determining a resonant velocity requires a precise treatment of the full kinetic dispersion relation for ion cyclotron waves. It has been reported that the solution of the ion cyclotron wave dispersion relation depends sensitively on several key parameters such as heavy (both cold and hot) ion contents and temperatures (Chen et al., 2011; Gary et al., 2012; Lee et al., 2017; Silin et al., 2011). Observational information on such key parameters for individual EMIC wave events is highly limited. Therefore, for this work, instead of attempting to determine a precise resonant parallel velocity for the individual EMIC wave events, we rather calculate the KP anisotropy for various parallel energies between 4 and 300 keV separately (as shown in next section), expecting the actual parallel resonant energy to lie within that range for most situations (Cho et al., 2016; Meredith et al., 2003).
Second, equation 2 requires integrations over v⊥, which is in principle to be done for velocities ranging from 0 to infinity. However, actual particle observations are usually limited to some finite energy extent. Thus, evaluations of KP anisotropy by using observed particle distributions may be erroneous unless the energy coverage is sufficiently wide. We determine an acceptable energy range in the integration for which the estimated KP anisotropy is reliable in the following way. For that purpose, we take a bi-Maxwellian distribution for which a specific temperature moment anisotropy, A0, is given from the first. We then numerically compute the KP anisotropy in 2 and compare it with A0. The integration of 2 is performed by varying the upper limit of the integrations in the perpendicular velocity direction. We aim to determine an approximate upper limit of the integrations for which the difference between the numerically calculated A and the prescribed solution A0 is less than 1%. Table 1 shows the results for the selected values of A0 = 1, 2, and 4. A higher A0 requires a higher upper limit to meet the error limit of 1%. Because we believe that a case of A0 > 4 is unusual in the inner magnetosphere, the estimated upper limit of 360 keV is high enough to give a reliable integration result. This energy limit is well covered by the RBSPICE measurements onboard the Van Allen Probes.
|Ao = 1 case||~4||~160 keV|
|Ao = 2 case||~5||~250 keV|
|Ao = 4 case||~6||~360 keV|
- Note. Where Ao is a prescribed value for a bi-Maxwellian distribution and the corresponding energy (obtained by assuming a thermal energy of 10 keV).
Lastly, particle observations from satellites are usually provided in kinetic energy-pitch angle coordinates. For the calculation of the KP anisotropy in 2, the original data are transformed into the parallel and perpendicular velocity coordinates. Because the integration over the perpendicular velocity is to be done along the parallel resonant velocity lines, data on perpendicular velocities for a fixed kinetic energy become less available for higher parallel resonant energies. Therefore, in practice, for a fixed kinetic energy, we should undertake the integrations in the perpendicular velocity with different available data for different parallel resonant velocities. We have tested the effect of this limitation on the accuracy of evaluating the KP anisotropy in the following way. We assume a bi-Maxwellian distribution within the kinetic energy range from 0 to 488 keV, beyond which particles are assumed to not exist. We then perform the integrations at various parallel resonant velocities (thus with different ranges of perpendicular velocity). We finally compare the obtained KP anisotropy values with the prescribed A0 from the bi-Maxwellian distribution. We find that the estimated KP anisotropy values are underestimated if the perpendicular velocity data are less available for higher parallel resonant energies. Nevertheless, we found that the error in A compared to that in A0 is less than ~1.5% as long as the data availability for the perpendicular velocity direction is 60% or larger. Therefore, for the results presented in the following sections, we take the integration results only when the data availability in the perpendicular velocity is 60% or larger, with a choice of 488 keV as the upper limit of the integrations in 2. This requirement leads us to a total of 113 EMIC wave events (66 in H+ band and 47 in He+ band) out of the original 302 events obtained in section 2.2. The spatial distribution of these events is shown in Figure 1b, which is similar to that of the original 302 events in Figure 1a.
3.2 Observed Proton Distribution Functions
For the KP anisotropy calculations, we use both the HOPE and RBPICE proton data. We first perform temporal and pitch angle interpolations to match the HOPE and RBSPICE measurements. For fixed times and pitch angles, we then combine the 52–488 keV proton flux data from RBSPICE with the 1 eV–52 keV proton fluxes from HOPE (see more discussion on this issue in section 6.1). Next, the phase space density (PSD) of the particles is calculated from the combined directional flux. Finally, we transform the coordinates from energy-pitch angle space to perpendicular-parallel velocity space where the integrations in 2 are performed. The use of both HOPE and RBSPICE covers a wide energy range so that a possible proton anisotropy at high energies can be treated (Wang et al., 2012).
where n is the number density, θ⊥, θ∥ are the thermal velocities, Γ is the gamma function, and κ is the index that determines the relative importance of the high-energy tail of the distribution. In the special case where κ is infinity, the bi-Lorentzian distribution reduces to the bi-Maxwellian distribution. We find that (i) a double bi-Lorentzian distribution represents the median PSD profiles much better than when using a single bi-Maxwellian distribution and (ii) perfect fits for individual events are not realized, however. Thus, in the subsequent sections, we will compute the anisotropy parameter using the actually measured distributions rather than using any fitted distribution models.
3.3 An Example of the Calculated Anisotropy Parameter
Figure 3 shows an example of the KP anisotropy parameter computed following all the procedures described above for an EMIC wave event. The top panel shows the magnetic PSD, where we identify the EMIC wave event that starts at 20:04:53 UT and ends at 20:40:49 UT, as marked by the two vertical lines. Following the criterion described in section 2.2, we regard the wave activities in this interval as one event although the event is characterized by breaks in wave activity. The wave occurred in both the H+ and He+ bands at different times (the curves of the local ion gyrofrequencies are shown for reference).
The middle two panels in Figure 3 show the evolution of the calculated KP anisotropy parameter at selected eight parallel energies (colored lines) and that of the temperature moment anisotropy obtained from the moment calculations of the HOPE and RBSPICE observations for reference (gray lines). Note that the presentation of the eight energy channels is divided into two panels for visual clarity, and in each of the two panels the gray line is repeated for reference. The calculations of the KP anisotropy are shown for the interval from ~19:50 UT (~15 min prior to the EMIC wave onset) to ~20:30 UT (~10 min prior to the EMIC wave end) (After 20:30 UT, the requirement of proton data availability up to 488 keV for a reliable calculation of the KP anisotropy, as demonstrated in section 3.1, is not satisfied). The definition of the temperature moment anisotropy (gray lines) used in this figure is based on the use of T⊥ = P⊥/n and T∥ = P∥/n where P⊥, P∥, and n refer to the total pressures and density, respectively, obtained by combining the moments from the HOPE and RBSPICE measurements.
The bottom two panels shown side by side in Figure 3 show the 10-min averaged phase space density of protons at the EMIC wave onset time in the energy-pitch angle space (left) and velocity space (right), respectively. Although the data coverage in the velocity space is incomplete, it still satisfies our 60% criterion mentioned in section 3.1. From these two bottom panels, we can identify the energy regime between 10 and 100 keV where the distribution is anisotropic (between the two white vertical lines in the left panel and equivalently between the two white circles in the right panel, also marked by the two red arrows).
We pay a particular attention to the KP anisotropy shown in the middle two panels. The second panel in Figure 3 indicates that the 50 keV KP anisotropy parameter shows a large increase at the H+-band EMIC wave onset at ~20:04 UT: the anisotropy increases from ~0.27 to ~0.94 when averaged over 10 min around the onset time. It also shows that those of 30 and 70 keV are also enhanced but less significantly (from ~0.42 to ~0.57 and from ~0.34 to ~0.39, respectively) and at slightly different timings near the wave onset. However, the KP anisotropy at higher energies shown in the third panel in Figure 3 does not show any significant response with H+-band wave onset. The KP anisotropy parameters for 130 and 150 keV increased from ~0.30 to ~0.53 and ~0.32 to ~0.55, respectively, near the time of the delayed He+-band wave onset at ~20:20 UT, but the changes are gradual. Incidentally, we also observe that the profile of the temperature moment anisotropy obtained from the moment calculations of the HOPE and RBSPICE measurements (gray lines) differs from those of the KP anisotropy at different energies, although it is closest to the 10 keV KP anisotropy. The temperature moment anisotropy does not change much throughout the EMIC wave interval. Therefore, this example demonstrates that a reliable examination of the anisotropy should be based on each energy separately, the result of which can lead to a different conclusion from what would be obtained by simply relying on the temperature moment anisotropy obtained from the moment calculations.
4 Application to the Observed EMIC Wave Events
4.1 Anisotropy Calculation Results
We first perform the estimation of the KP anisotropy at the times near the EMIC wave onsets. Specifically, the estimation is undertaken for times 20 s before and after each EMIC wave event onset. With this distinction of two adjacent times, we intend to see how the KP anisotropy evolves across the onset of the wave growth. Because we require that the minimum data coverage in the perpendicular velocity is 60%, as discussed in section 3.1, we find that this requirement at the two adjacent times is satisfied for 113 EMIC wave events out of the 302 events that are identified in section 2.2. The calculation results of the KP anisotropy are summarized in the middle and right panels of Figure 4 for various parallel kinetic energies from 4 to 280 keV.
In addition, in order to compare this result with the times when no EMIC wave events are identified, we define the “non-EMIC wave times” in the following way. For each EMIC wave event, we check whether there is any wave activity (including any broadband emissions) at the same L and MLT location on the previous orbital path as that where the EMIC wave is identified. If no wave activity is observed, we define this as a non-EMIC wave time. In cases where wave activity is present, we instead check one orbit after the orbit where the EMIC wave is identified. It should be mentioned that the satellite does not travel exactly the same orbit between successive orbits, and therefore, there is a difference in location between the EMIC wave times and the corresponding non-EMIC wave times. However, the difference is sufficiently small so that an identification of a non-EMIC wave time corresponding to a given EMIC wave time can be made to a reasonable degree: The average differences in L, MLT, and MLAT between the EMIC wave times and the corresponding non-EMIC wave times are 0.02, 0.18 hr, and 2.66°, respectively. In that way, 48 events out of the 113 events are excluded due to the presence of wave activities and the remaining 65 events are selected for the non-EMIC wave times. For those non-EMIC wave times, we estimate the KP anisotropy, the result of which is shown in the left panel of Figure 4. Therefore, using Figure 4, we intend to check the statistical evolution of the KP anisotropy from the non-EMIC times, through the times just before EMIC wave onset to the times just after EMIC wave onset.
Our visual inspection of Figure 4 indicates that for all three groups, the anisotropy values are mostly positive over the entire energy range. More importantly, for many of the events and at wide energies, the anisotropy values are slightly larger for the times 20 s after EMIC wave onset than for the other two times. (Although the results in Figure 4 are shown without distinguishing the H+- and He+-band waves, a similar trend is maintained for each group of the two bands.) We present this feature in another way in Figure 5, which shows the statistics of the KP anisotropy values at selected energies, with comparisons among the three times. It is clear in Figure 5 that despite the rather dispersed nature of the distributions, the anisotropy values at all the energies shown are often larger for the times 20 s after the EMIC wave onsets (red) than for the other two times (black and blue), although the differences are small. This trend is confirmed in Table 2, which presents a summary of median values of the KP anisotropy parameter at selected energies for the three groups of the times. Based on those median values, the increase in anisotropy at times 20 s after the EMIC wave onsets, as averaged over the entire energies, is by ~15% relative to the non-EMIC wave times and ~8% relative to the time 20 s before the EMIC wave onsets. Therefore, we conclude that there is a statistical trend such that the KP anisotropy increases from the non-EMIC wave times, through just prior to the EMIC wave onsets, to just after the EMIC wave onsets. Although the amount of that increase is small, the average trend is discernable. In the next section, we address the implication of this small increase in the anisotropy.
|Epar (keV)||Non-EMIC event||20 s before EMIC onset||20 s after EMIC onset|
4.2 Test of the Kennel-Petschek Instability Criterion
Proton cyclotron instability is expected when condition 4 is satisfied. The criterion depends purely on the anisotropy for a given wave frequency normalized by proton gyrofrequency, although the wave frequency is in turn related to the plasma conditions via the dispersion relations. Therefore, using the calculated KP anisotropy values above and the identified wave frequencies from the observations, we calculate the left-hand side of 4 as a function of the energies. Herein, we use the frequency values identified at maximum PSD within the observed EMIC wave band. In this section, we aim to determine the extent to which the instability criterion is satisfied throughout the entire period of each EMIC wave event. This is distinguished from the way done in section 4.1 where we focused on the anisotropy differences only at two adjacent time points across the wave onset (as well as at the non-EMIC times). Accordingly, the calculation here is performed for the entire data points during each of the 302 EMIC wave events as long as they meet the condition of the minimum data coverage (60%) in the perpendicular velocity. The results are shown in Figure 6, where the number of data points is color-coded in the space defined by the instability discriminant, namely, the left-hand side of 4, and energy. The horizontal gray line corresponding to unity is the threshold value for instability.
It is seen in Figure 6 that the majority of the data points are concentrated near the instability threshold value (namely, discriminant = 1) for the energy range up to <150 keV. For the sake of convenience, we refer to the points in the region of the discriminant from ~0.6 to ~1.6 as “the primary group.” There is a “secondary group” of much fewer data points in the distribution in the range of the discriminant between ~2.0 and ~2.8 and in the energy range up to ~100 keV (the blue area in the left half side of the panel). This separation of the data point distribution into two groups is closely related to the fact that the tested events consist of the two EMIC wave bands, that is, the 195 number of H+ band waves and 177 number of He+ band waves (see section 2.2). Clearly, it is more advantageous for a He+ band EMIC wave to satisfy the instability criterion 4 due to its lower wave frequency than for a H+ band EMIC wave if the anisotropy A is given the same. That is, it is most often the case that the instability discriminant is larger for the He+ band waves than for the H+ band waves. Consequently, the secondary group is mainly the He+ band waves (95.1% of the data points), whereas the majority of the primary group are the H+ band waves (88.4% of the data points). There are exceptional cases that a high-frequency H+ band wave in an EMIC wave event is associated with a sufficiently large amount of the anisotropy, resulting in a high value of the discriminant, whereas a low-frequency He+ band wave in a different EMIC wave event is associated with a small amount of the anisotropy, giving a small value of discriminant.
Incidentally, the secondary group (mostly He+ band waves) is concentrated around the discriminant value of ~2.4 and the primary group (mostly H+ band waves) is centered around the discriminant value of ~1. The ratio of these central values is different from the ratio of two ions' gyrofrequencies. This is not surprising because it is the wave frequency that determines the instability criterion 4 (not gyrofrequency), and in reality, the H+ band waves often occur at a frequency well below H+ gyrofrequency to meet the resonance condition with protons, whereas the He+ band waves occur rather close to He+ gyrofrequency.
In addition, ~30% of the total points in Figure 6 belong to the discriminant value range of 0.8 to 1.2. Out of this, ~44% lie above the threshold value. We do not preclude the possibility that the calculated discriminant values for these border line events may suffer from any measurement errors that may exist in the proton flux data.
Table 3 summarizes the percentages of unstable events satisfying 4 for selected energies up to 100 keV. The percentages of unstable cases are ~73% for 10 keV and ~63% for 100 keV. Therefore, we conclude that even the small amount of anisotropy found in the previous section can still be sufficient for satisfying the instability criterion for many events, the extent of which depends on resonant parallel energies. We will discuss the implication of this conclusion in section 6.
|Epar (keV)||Percentage of unstable events|
5 Assessment of the Instability Criteria Expressed in Terms of Plasma Beta
The primary goal in this work is to test the anisotropy parameter and associated instability condition of Kennel and Petschek (1966), as presented in the previous two sections. On the other hand, it has long been known that threshold conditions for some instabilities can be expressed in terms of plasma beta. In particular, inverse relations between proton anisotropy and parallel plasma beta β∥ have long been recognized for ion cyclotron instabilities in the solar wind and magnetosphere (Anderson et al., 1994; Gary et al., 1994a, 1994b; Phan et al., 1994). Compared to the original expression for the instability criterion given by Kennel and Petschek (1966), such inverse relations have an advantage such that wave frequency does not appear explicitly because it is transformed into a form expressed in terms of plasma parameters (as described below in detail). Therefore, they are convenient in the sense that one can apply the inverse relation forms not only to the EMIC wave times but also to non-EMIC wave times when a specific wave frequency is not defined (Note that the test of the Kennel and Petschek criterion in the previous section was done only for the times of EMIC wave activities, not for the times prior to the wave onsets). However, we demonstrate below that the inverse relations are still an approximation to the original expression of threshold conditions for ion cyclotron instability. Therefore, in this section, as a subsidiary work, we attempt to assess the usefulness and/or limitations of using the plasma beta as a key parameter in determining ion cyclotron instability.
This is equivalent to equation 6 in Gary and Lee (1994). Therefore, the inverse relation expression in Gary and Lee (1994) is simply another way of expressing the same threshold condition originally from the Kennel-Petschek criterion, although Gary and Lee (1994) also derived similar expressions in two other ways. In addition, we emphasize that although Gary and Lee (1994) derived this starting from the dispersion relation with a bi-Maxwellian distribution, we derived the same expression above from the original Kennel-Petschek expression without requiring any particular distribution function. The additional effort that Gary and Lee (1994) made further is an explicit determination of and ω/kvA in 6 by numerically solving the bi-Maxwellian-based dispersion relation with assumed and fixed growth rates. They found a β∥ dependence of the two factors ( is independent of β∥, and ω/kvA is roughly proportional to β∥0.1) and finally determined anisotropy A as a power law of β∥ alone.
To see whether the results in section 4.2 can be understood better based on the plasma beta and related inverse relation, we first compare the parallel plasma beta β∥ between EMIC and non-EMIC wave time intervals. Here for the EMIC wave intervals, we use all the data points during the identified 302 EMIC wave intervals as shown in Figure 1a, and for the non-EMIC wave intervals, we use all the data points from 1 March 2013 to 30 November 2015, as long as an EMIC wave is not observed. For this estimation of parallel plasma beta β∥, we use the particle data from both HOPE and RBSPICE of Van Allen Probe-A, which thus covers the higher energy contribution to plasma temperature. Figure 7 shows the estimated results separately for the EMIC (red) and non-EMIC (blue) wave intervals. It is seen that for the EMIC wave intervals, the plasma beta is often larger, with its average and median values being larger by a factor of ~2 compared to those for the non-EMIC wave intervals. This enhancement in the plasma beta is due to both an increase in the plasma density and a decrease in the magnetic field intensity on average during the EMIC wave intervals compared to the conditions for the non-EMIC wave intervals, although the contribution by the magnetic field decrease is slightly larger than that by the density increase: the inverse of the squared average magnetic field (that is, 1/〈B〉2) during the EMIC wave intervals is larger by a factor ~2 than that for the non-EMIC wave intervals, whereas an average density during the EMIC wave intervals is greater by a factor of ~1.5 than that for the non-EMIC wave intervals. Therefore, we can conclude to some extent that statistically the occurrence of proton cyclotron instability prefers a higher plasma beta condition, which helps reduce the threshold amount of proton anisotropy required for the instability through the inverse relation. However, it is also obvious in Figure 7 that there is a significant overlap between the two distributions of plasma beta. This implies that the plasma beta alone is not sufficient for firmly determining (or predicting) the instability under the inner magnetospheric conditions, where other plasma conditions can also play a role in triggering ion cyclotron instability. Thus, one needs to include factors other than the plasma beta alone for a more sophisticated expression of the inverse relation.
Here the subscript h means the hot component of plasma and ne is electron density. Blum et al. (2009) suggested fitting parameters of σ0 = 0.429, σ1 = 0.124, σ2 = 0.0118, a0 = 0.409, a1 = 0.0145 and a2 = 0.00028. The fitting parameters were obtained by assuming a pure proton plasma due to lack of ion composition from LANL data.
We test the usefulness of the Blum et al. criterion above by applying it to the two groups used in Figure 7, namely, the EMIC wave and non-EMIC wave times. The calculated results are shown in Figure 8, where Σhis calculated as a function of nh/ne, and the number of events is color-coded in the space defined by Σh and nh/ne. The gray line refers to Sh as a function of nh/ne. The calculations of the temperature anisotropy in 7 are undertaken for different energies. For the plasma beta and hot proton densities, we use the measurements from both HOPE and RBSPICE. This provides overall higher plasma beta values than would be obtained using only HOPE measurements and thus contributes to Σh in 7. The percentages of unstable data points are stated on the upper left sides of each panel. It is clear from Figure 8 that many of the events are in the unstable domain where Σh > Sh. Specifically, the lower panels show the results for the EMIC wave events, which indicate that the percentages for satisfying the instability criterion ranges above 75%. However, the upper panels imply that ~60% of the non-EMIC times also satisfy the Blum et al. criterion. Clearly, the effects due to other factors that are not treated in their formula must be nonnegligible to better distinguish EMIC wave times and non-EMIC wave times. This sets a limitation on the usefulness of the Blum et al. expression, particularly for the purpose of predicting the occurrence of ion cyclotron instability, although it can still be useful for some purposes (Allen et al., 2016; Blum et al., 2012; Spasojevic et al., 2011).
In 10 the following notations are used: x = ω/Ωp, which can be in general complex, f = ωpe/|Ωe|, Ωp and Ωe are the proton and electron gyrofrequencies, respectively, and ωpe is the electron plasma frequency. The ε is the electron to proton mass ratio. The η with appropriate subscripts (e.g., p and hp referring to cold and hot protons, respectively) refers to ion density relative to cold electron density such that ηp + ηHe + ηhp + ηhHe = 1. The hot ions' temperature anisotropy is represented by A (= ) with appropriate subscripts. Z refers to the plasma dispersion function with argument, ζp and ζHe, for each hot ion population, respectively, as defined by ζp = (ω - Ωp)/[k(2T||p/mp)1/2] and ζHe = (ω − ΩHe)/[k(2T||He/mHe)1/2]. Since the wave frequency in the dispersion relation is generally complex, the solution of 10 is numerically obtained by letting ω = ωr + iγ, given the specific values of the set of the free parameters (η of ions, ε, f, Ahp, AhHe, kc/ωpe, T||p, T||He, and β||p = 2μoNhpT||p/B2, which appears when ζp and ζHe are expressed in a dimensionless form).
In solving 10, we aim to find a pair of Ahp and β||p at the marginal condition (that is, when satisfying γ = 0) for a given combination of the other parameters. To be practical, the determination of the marginal state is done by imposing a condition that the growth rate γ is 0.0001 times the proton gyrofrequency ωr (this is reasonable based on the reports from previous works (e.g., Chen et al., 2011; Gary et al., 2012; Lee et al., 2017). The final results obtained in this way lead to a relationship between Ahp and β||p at an instability threshold for a given set of the other parameters. This search is performed for various combinations of the parameters.
The results shown in Figure 9 are an example of the calculations undertaken for eight sets of different combinations of the key parameters, a summary of which is given in Table 4. Runs 1 to 4 are distinguished by the relative importance among the populations of the cold protons, hot protons, cold He+, and hot He+: Run 1 is dominated by cold protons, Run 2 by hot protons, and Runs 3 and 4 are designed by adding substantial populations of cold He+ and hot He+, respectively. The other four sets of the runs (Runs 1.1, 2.1., 3.1, and 4.1) are basically the same set of the corresponding runs (that is, Runs 1, 2, 3, and 4), but with the difference that the He+ temperature is now 100 times higher. In all these runs, the He+ temperature is isotropic. It is clear that the trend of an inverse relation between A (referring to Ahp) and the parallel plasma beta β||p is well maintained for all of the cases shown. There is also an overall trend that the blue curves for the Runs with cold proton dominant conditions are lower than the red curves for the Runs with hot proton dominant conditions. The main point that we intend to emphasize in Figure 9 is that the threshold anisotropy can be different to a significant degree, depending on the specific sets of the various parameters. The significance of the differences can easily exceed the small differences in the observed A between the EMIC wave times and non-EMIC wave times, as presented in section 4.1. Therefore, the use of a simplified form in terms of limited plasma parameters can be often insufficient to predict occurrences of ion cyclotron instability. The basically same point was already suggested by Lin et al. (2014) in a different way.
- a The subscripts c and h refer to cold and hot plasmas, p and He indicate protons and He+, and .
6 Discussion and Conclusions
We demonstrate here distinctive aspects of our work in comparison with the previous work by Lin et al. (2014), who reported results of testing the instability criterion of Kennel and Petschek (1966) and the inverse relation formula of Blum et al. (2009). Their test was based on an evaluation of the proton anisotropy given by using hot proton temperature moments obtained from Cluster satellite observations. Our work is an improvement on the work by Lin et al. (2014) in three aspects. First, we have rigorously calculated the proton anisotropy by using the original definition of Kennel and Petschek (1966) with observed, realistic distributions of protons. Our treatment closely follows the linear theory of ion cyclotron resonant instability and thus is an improvement over simply computing the anisotropy by using temperature moments. Second, the proton energy range used in our calculation of KP anisotropy is much wider than that used in Lin et al. (2014) to estimate their temperature moment anisotropy. Lin et al. (2014) used proton temperature moments obtained in the energy range from 10 to 40 keV, but our calculations of KP anisotropy are performed over an energy range up to 488 keV. The use of a wide energy range allows us to include possible proton anisotropy at high energies (Wang et al., 2012). The different energy ranges can also affect the temperatures, which enter into β∥. Third, in testing the Kennel-Petschek criterion for ion cyclotron instability, Lin et al. (2014) compared two groups of hot proton temperature anisotropy intervals, one with EMIC wave occurrences and the other without EMIC waves (representing non-EMIC wave times). They showed that the ranges of proton temperature anisotropy are similar between the EMIC and non-EMIC wave times. On that basis, they suggested that hot proton temperature anisotropy is necessary but not sufficient for ion cyclotron instability. In contrast, our test of the Kennel-Petschek criterion is based on a more direct comparison of anisotropy between the EMIC wave times and non-EMIC wave times. For a meaningful comparison of anisotropy between two groups, the non-EMIC wave times in our work have been selected from the orbit in which no EMIC wave was observed just before or after the one where the EMIC wave was observed. This is distinguished from the rather random sampling of non-EMIC wave times of Lin et al. (2014). It allows us to examine a statistical trend of how the KP anisotropy changes from the times of no EMIC wave occurrences to the times of EMIC wave occurrences. Based on a rigorous calculation of the KP anisotropy, we demonstrated that although there is a significant overlap in the anisotropy ranges between the EMIC wave times and non-EMIC wave times, as in the case of Lin et al. (2014), we found that there is statistically an evolution in anisotropy from non-EMIC wave times to EMIC wave times. The evolution leads to a rather small increase in anisotropy, which we interpret to imply that the proton distributions often remain close to a marginal state, with anisotropy values close to threshold values for ion cyclotron instability. Consequently, it is not surprising that observations encounter anisotropy values within a similar range regardless of the existence of EMIC waves. Therefore, it can be observationally difficult to predict instability based on observed anisotropy values alone. However, we emphasize that this does not mean that proton anisotropy is necessary but not sufficient for instability triggering. Rather we suggest that theoretically proton anisotropy remains the main factor that determines instability as long as it exceeds threshold value, although determinations of threshold anisotropy are in practice complicated due to effects by other plasma conditions.
It should be emphasized that the analysis performed in section 4 was applied to the times 20 s before and after EMIC wave onsets as well as the non-EMIC wave times, but not to the entire period of each EMIC wave event. The anisotropy can evolve during EMIC wave periods in accordance with changes in EMIC wave activity: Examples can be found in the previous work in Cho et al. (2016). In particular, the proton anisotropy can decrease during EMIC wave duration as noted by Saikin et al. (2018). In fact, we find that on average in our events, there is ~3.5% decrease in the average KP anisotropy (in the energy range of 10 to 300 keV) over the EMIC wave intervals relative to the average KP anisotropy over 10-min interval prior to the EMIC wave onsets. Since detailed situations in the anisotropy evolution during EMIC wave periods may vary from event to event, a comprehensive analysis is required for individual events. We emphasize that in section 4 of the present study, we have focused on the evolution only over a short time interval (±20 s) across EMIC wave onsets and we have observed a small increase in the anisotropy on average, which is sufficient to trigger the ion cyclotron instability at the wave onset of many events.
A few issues are worthy of discussion in using the Kennel-Petschek instability criterion in equations 1 and 2. First, it should be noted from the growth rate expression 1 and anisotropy expression 2 that the precise treatment requires the use of the resonant parallel energy, whereas we tested various energies in the wide energy range that covers the expected resonant energies from previous reports. It is unclear to us to what extent our results can be affected if we use a precise resonant energy for individual events. Second, the growth rate expression 1 indicates that for a meaningful growth of seed waves, there needs to be a supply of a sufficient number of energetic protons that participate in resonance. Such a particle supply is known to be possible by substorm/storm injections and enhancements in the solar wind dynamic pressure, which can all provide increased number of energetic ions with anisotropy T⊥ > T∥. Lastly, the Kennel-Petschek instability criterion is based on the assumption of parallel propagating waves. The growth rate is expected to be largest for parallel propagation, and it decreases with increasing wave normal angle. For an oblique wave normal angle, the Landau damping effect takes place and the higher cyclotron harmonics produces resonance due to elliptical wave polarization. In this sense, the present study should be regarded as a test of the criterion for the most unstable fundamental mode, that is, the parallel propagating case. Observationally, though not presented here (but see in Figure 1 in Lee et al., 2018), we find that the statistical distribution of wave normal angle peaks at a rather moderate oblique angle, ~20°–30°, when the Van Allen Probes satellites were located near the equator.
A conventional wisdom is that ion cyclotron instability occurs preferably at latitudinal locations where the magnetic field is weakest, near the equator on the nightside and somewhat off the equator on the dayside. Accordingly, it is desired to determine if the EMIC wave events studied in the present paper were actually at the source region or not. It should be noted that our event intervals were selected from observations on the Van Allen Probe-A satellite, which remained close to the equator (mostly within 10° of magnetic latitude as shown in Figure 1, for example). According to Loto'aniu et al. (2005), the Poynting flux vector is bidirectional both away and toward the equator for the EMIC waves observed below 11° |MLAT| and unidirectional away from the equator for all events above 11° |MLAT|. This implies that a broad source region is centered around the equator, although there is a report on an exceptional case (Allen et al., 2013). Therefore, although a detailed study of Poynting flux for individual events is desirable, we expect that most of our events were within the source region and any latitudinal effect on our results should be not significant.
Another effect that should be considered is the inclusion of heavy ions in the ion cyclotron instability criterion, noting that the Kennel-Petschek criterion includes only protons. Previous works emphasized the importance of including heavy ions in the dispersion relation (Kozyra et al., 1984; Hu et al., 1990; Hu & Fraser, 1994; Silin et al., 2011; Chen et al., 2011; Gary et al., 2012; Lee et al., 2017). As shown in Figure 9, we have shown that He+ conditions can affect the inverse relation between proton anisotropy and parallel plasma beta. Lee et al. (2017) demonstrated the effect of a hot anisotropic He+ distribution on EMIC wave growth. The He+ effect is complicated because its effectiveness is also affected by other parameters (e.g., H+ and He+ temperatures, hot and cold ion densities, and even cold electron density). The present study should be regarded as an attempt to determine the extent to which the proton anisotropy alone can be responsible for triggering ion cyclotron instability. Our future work will seek to test ion cyclotron instability with a more appropriate criterion that includes heavy ions.
In the present work we have used the HOPE and RBSPICE particle flux data without adjusting one data relative to the other one. Our calculation results may be subject to this uncertainty to some extent, which however we cannot precisely determine unless a reliable cross-calibration between the two instruments can be made. Nevertheless, we expect that this uncertainty will not seriously affect the main conclusions.
As a final remark, in the present study, we have used the data from the Van Allen Probe-A observations only. An analysis using the data from both spacecraft is certainly worthwhile to pursue. For example, a direct examination of anisotropy evolution can be plausible for individual events when both satellites are closely located to each other. However, an interpretation may be complicated by various situations such as the cases where an EMIC wave is observed by only one spacecraft, EMIC waves are seen by both spacecraft simultaneously, EMIC waves are seen by both spacecraft but with some time lag, or EMIC waves are observed by both spacecraft but with different polarization features and/or at different frequency bands (Blum et al., 2017; Cho et al., 2017). Based on a preliminary work, we find that if we require the condition that the spatial separation between two spacecraft is <0.5 L and <1 MLT, an EMIC wave is identified also by Van Allen Probe-B for ~81% out of the EMIC wave events that are identified by Van Allen Probe-A. (For reference, Blum et al., 2017, report that ~70% of EMIC waves were identified by both Van Allen Probes satellites when the spatial separation was within 0.5 RE.) A test of anisotropy evolution and related instability analysis for these events of ours requires availability of reliable particle data from both satellites. We leave this to a next project.
- The KP anisotropy is often larger at times immediately after EMIC wave onsets than at the times just before EMIC wave onsets and the non-EMIC wave times. This is the case for an energy range of ~10 keV to a few hundreds of keV, which covers the usual energy regime for resonant interactions with ion cyclotron waves. However, the increase in anisotropy during the EMIC wave times is small, such that on average it is by ~15% relative to that at the non-EMIC wave times and ~8% relative to that at the times just before the EMIC wave onsets.
- Despite the rather weak increase in anisotropy, the majority of the EMIC wave events (~60% to 70% at energies up to ~100 keV) satisfy the criterion for ion cyclotron instability given by Kennel and Petschek (1966).
- The results (i) and (ii) above imply that the proton distributions must often stay close to a marginal state, with anisotropy close to threshold values for ion cyclotron instability. This in turn implies that it is natural that observations often encounter proton anisotropy values within a similar range regardless of EMIC wave occurrence.
- Parallel plasma beta is on average higher by a factor of ~2 during the EMIC wave events than during non-EMIC wave times. However, there is a significant overlap in the plasma beta distributions for the EMIC wave times and non-EMIC wave times. This implies that while a higher beta is preferred for excitation of ion cyclotron instability (as is often the case for various plasma instabilities), it alone cannot be a solid determinant.
- Approximately 70% of the observed EMIC wave events satisfy the criterion for ion cyclotron instability expressed in the form of the inverse relation given by Blum et al. (2009), which contains the effect of hot proton density. However, more than 60% of the non-EMIC wave times also satisfy the same criterion.
- The results (iv) and (v) imply that while an inverse relation between proton anisotropy and plasma beta remains valid and is still useful for some purposes, its effectiveness can be limited for the purpose of predicting instability. This is theoretically confirmed by solving the fully kinetic dispersion relation, which indicates a rather wide range of threshold anisotropy for a given plasma beta that depends on the specific combination of other plasma parameters.
This research at Chungbuk National University was supported by the National Meteorological Satellite Center of Korea, by the Korea Astronomy and Space Science Institute under the R&D program supervised by the Ministry of Science, ICT and Future Planning, and by the National Research Foundation of Korea (2016M1A3A3A02017017). The work at the New Jersey Institute of Technology was supported by the NASA Van Allen Probes RBSPICE instrument project provided by JHU/APL subcontract 131803 under NASA prime contract NNN06AA01C. The Van Allen Probes data used for this paper can be obtained from NASA's CDAWeb FTP server, ftp://cdaweb.gsfc.nasa.gov, and from the EMFISIS website for the magnetic field data at http://emfisis.physics.uiowa.edu/, plasma density data at http://www.space.umn.edu/, and proton flux data from ECT-HOPE at https://rbsp-ect.lanl.gov/data_pub/ and those from RBSPICE at http://rbspicea.ftecs.com/. The data shown in Figure 9 is theoretical and can be obtained by solving equation 10 based on the method described in detail in the main text of section 5. The authors are grateful to C. A. Keltzing and Sheng Tian for checking the quality of the magnetic and electric field data. S. Noh is grateful to K. Min and L. W. Blum for their useful comments on the present work.
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