Examining Coherency Scales, Substructure ... - Wiley Online Library

PUBLICATIONS Journal of Geophysical Research: Space Physics RESEARCH ARTICLE 10.1002/2017JA024474 Special Section: Magnetospheric Multiscale (MMS) mission results throughout the first primary mission phase Key Points: • Cross-correlation analysis was used to calculate the wave normal vector for eight rising tone lower band chorus elements and several subelements • Substructure and wave phase were consistently coherent at all four MMS; plane wave approximation valid for six of eight elements up to 70 km scale • Elements observed at L~8.2, MLT~06:53, MLAT~
31°; k was consistently oriented earthward and significantly oblique to B0 and S

Correspondence to: D. L. Turner, [email protected]

Citation: Turner, D. L., Lee, J. H., Claudepierre, S. G., Fennell, J. F., Blake, J. B., Jaynes, A. N., … Santolik, O. (2017). Examining coherency scales, substructure, and propagation of whistler mode chorus elements with Magnetospheric Multiscale (MMS). Journal of Geophysical Research: Space Physics, 122, 11,201–11,226. https://doi. org/10.1002/2017JA024474 Received 13 JUN 2017 Accepted 7 OCT 2017 Accepted article online 13 OCT 2017 Published online 14 NOV 2017

Examining Coherency Scales, Substructure, and Propagation of Whistler Mode Chorus Elements With Magnetospheric Multiscale (MMS) D. L. Turner1 , J. H. Lee1 , S. G. Claudepierre1 , J. F. Fennell1, J. B. Blake1 , A. N. Jaynes2 T. Leonard2 , F. D. Wilder2 , R. E. Ergun2 , D. N. Baker2 , I. J. Cohen3 , B. H. Mauk3 , R. J. Strangeway4 , D. P. Hartley5 , C. A. Kletzing5 , H. Breuillard6, O. Le Contel6 , Yu. V. Khotyaintsev7 , R. B. Torbert8,9 , R. C. Allen9 , J. L. Burch9 , and O. Santolik10,11

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1

The Aerospace Corporation, El Segundo, CA, USA, 2Laboratory for Atmospheric and Space Physics, University of Colorado Boulder, Boulder, CO, USA, 3Johns Hopkins University Applied Physics Laboratory, Laurel, MD, USA, 4Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, CA, USA, 5Department of Physics and Astronomy, University of Iowa, Iowa City, IA, USA, 6CNRS/Ecole Polytechnique/UPMC Université Paris 06/University ParisSud/Observatoire de Paris, Paris, France, 7Swedish Institute of Space Physics, Uppsala, Sweden, 8Institute For the Study of Earth, Oceans, and Space, University of New Hampshire, Durham, NH, USA, 9Southwest Research Institute, San Antonio, TX, USA, 10Department of Space Physics, Institute of Atmospheric Physics, Czech Academy of Sciences, Prague, Czech Republic, 11 Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

Abstract

Whistler mode chorus waves are a naturally occurring electromagnetic emission observed in Earth’s magnetosphere. Here, for the first time, data from NASA’s Magnetospheric Multiscale (MMS) mission were used to analyze chorus waves in detail, including the calculation of chorus wave normal vectors, k. A case study was examined from a period of substorm activity around the time of a conjunction between the MMS constellation and NASA’s Van Allen Probes mission on 07 April 2016. Chorus wave activity was simultaneously observed by all six spacecraft over a broad range of L shells (5.5 < L < 8.5), magnetic local time (06:00 < MLT < 09:00), and magnetic latitude (
32° < MLAT <
15°), implying a large chorus active region. Eight chorus elements and their substructure were analyzed in detail with MMS. These chorus elements were all lower band and rising tone emissions, right-handed and nearly circularly polarized, and propagating away from the magnetic equator when they were observed at MMS (MLAT~
31°). Most of the elements had “hook”-like signatures on their wave power spectra, characterized by enhanced wave power at flat or falling frequency following the peak, and all the elements exhibited complex and well-organized substructure observed consistently at all four MMS spacecraft at separations up to 70 km (60 km perpendicular and 38 km parallel to the background magnetic field). The waveforms in field-aligned coordinates also demonstrated that these waves were all phase coherent, allowing for the direct calculation of k. Error estimates on calculated k revealed that the plane wave approximation was valid for six of the eight elements and most of the subelements. The wave normal vectors were within 20–30° from the direction antiparallel to the background field for all elements and changed from subelement to subelement through at least two of the eight elements. The azimuthal angle of k in the perpendicular plane was oriented earthward and was oblique to that of the Poynting vector, which has implications for the validity of cold plasma theory.

1. Introduction

©2017. The Authors. This is an open access article under the terms of the Creative Commons Attribution-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited and no modifications or adaptations are made.

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Whistler mode waves are naturally occurring electromagnetic emissions in plasma and can be categorized into several distinct types within Earth’s inner magnetosphere. Broadband, incoherent, right-handed whistler mode emissions regularly observed in the hundreds of hertz frequency range within Earth’s plasmasphere or high-density plasma plumes are known as hiss (e.g., Bortnik, Thorne, & Meredith, 2008; Meredith et al., 2006; Thorne et al., 1973). Outside of the plasmasphere, in the hotter, more tenuous plasma regime found there, electron whistler mode emissions are regularly observed as hiss-like emissions (e.g., Gao et al., 2014a, 2014b; Li et al., 2012) or discrete elements with often large amplitudes with respect to the background fields and rising or falling tone frequencies within two distinct bands; those emissions are known as chorus waves (e.g., Burtis & Helliwell, 1969; Burton & Holzer, 1974; Li et al., 2009, 2016; Meredith, Horne, & Anderson, 2001; Santolik et al., 2003; Taylor & Gurnett, 1968; Tsurutani & Smith, 1974). Chorus waves have been studied extensively due in part to their potential role: in the source and loss of hundreds of keV to MeV electrons in

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Earth’s outer radiation belt (e.g., Chen, Reeves, & Freidel, 2007; Horne & Thorne, 1998; Meredith et al., 2003; Reeves et al., 2013; Shprits, Meredith, & Thorne, 2007; Summers, Thorne, & Xiao, 1998; Thorne et al., 2013; Turner et al., 2013), as a source of plasmaspheric hiss (e.g., Bortnik et al., 2008; Bortnik et al., 2009; Chum & Santolik, 2005; Meredith et al., 2013; Santolik et al., 2006), and in scattering the electrons responsible for generating diffuse and pulsating aurorae (e.g., Nishimura et al., 2010; Thorne et al., 2010). Chorus is regularly observed throughout Earth’s inner magnetosphere, typically at L shells outside of the plasmasphere, in the premidnight to afternoon magnetic local time (MLT) sectors, and at magnetic latitudes (MLATs) ranging from the equatorial plane to |MLAT| > 45° (e.g., Agapitov et al., 2013; Li et al., 2009, 2011; Macúšová et al., 2015; Meredith et al., 2003, 2012; Santolík et al., 2014; Taubenschuss et al., 2016; Tsurutani & Smith, 1977). Chorus is thought to be generated near the magnetic equatorial plane, where the electron cyclotron frequency (fce) is minimized (along any particular L shell) enabling optimal conditions for a sufficiently anisotropic distribution of energetic (~keV to tens of keV) electrons to cause resonant wave growth (e.g., Omura, Katoh, & Summers, 2008). Mourenas et al. (2015) and Gao et al. (2016) presented theory and observational evidence showing how a combination of cyclotron and Landau resonance might explain the generation of more oblique whistler waves. The extent of the source region parallel to the background field has been estimated at being up to 3000 to 5000 km from the magnetic equatorial plane (Santolik et al., 2004; Taubenschuss et al., 2016). Chorus can be generated at very oblique angles near the resonance cone (e.g., Li et al., 2016; Santolik et al., 2009). After being generated, chorus waves propagate in either direction away from the source region (e.g., Santolik et al., 2010), and as they propagate, the wavefronts refract based on local plasma conditions (e.g., Agapitov et al., 2011; Bortnik, Thorne, & Meredith, 2007; Chen et al., 2013) and can further interact with energetic electrons along their raypaths (e.g., Mourenas et al., 2015), as proven with direct evidence of discrete chorus emissions interacting with ~20 keV electrons by Fennell et al. (2014). This gradual refraction can also lead to the so-called “high-latitude reflection” of the waves due to lower hybrid resonance (Breuillard et al., 2013, 2014; Lyons & Thorne, 1970; Smith & Angerami, 1968). As previously mentioned, chorus is typically observed in two frequency bands: a lower band at 0.1 to 0.5 times the equatorial fce (referred to from here on as just fce) and an upper band at 0.5 to 1.0 fce (e.g., Li et al., 2013, and references therein). Occurrence rates of upper versus lower band chorus (e.g., Li et al., 2011) show that lower band chorus occurs more frequently than upper band chorus, particularly for lower wave amplitudes. The occurrence and average amplitudes of both bands of chorus correlate with auroral activity as quantified with the AE index (e.g., Li et al., 2009, 2011; Meredith et al., 2001). Li et al. (2012) showed that for any particular range of wave amplitudes, rising tone chorus emissions typically occur 10 to 100 times more frequently than falling tone chorus. Macúšová et al. (2010) studied statistics of frequency sweep rates of rising and falling tone chorus elements and found that mean and median rates for rising tone emissions were on the order of ~1 kHz/s and that the sweep rate increased (up to >10 kHz/s) with increasing plasma density. Sweep rates up to 20 kHz/s were observed for large-amplitude (20 mV/m) chorus waves during a geomagnetic storm by Santolik et al. (2004). It has also long been recognized that individual chorus elements contain complex substructure, characterized by subelements separated by a few milliseconds in time (Santolik et al., 2004; Santolík et al., 2014), that evolve on time scales less than 1 min as the waves propagate (Gurnett et al., 1979). Crabtree et al. (2017) found that chorus subelements can be composed of nearly linear waves with continuously changing wave quantities, including frequency and wave normal vector; they suggest that the physics between subelements might be dominated by wave-wave or nonlinear wave-particle interactions (e.g., Agapitov, Krasnoselskikh, et al., 2015; Artemyev et al., 2014; Gao, Lu, & Wang, 2017), which might play a role in setting the saturation of the whistler mode instability responsible for chorus wave growth. Concerning the propagation of chorus, Li et al. (2013) conducted a statistical study of the Poynting flux and wave normal vectors of chorus waves using data from NASA’s Time History of Events and Macroscale Interactions during Substorms (THEMIS) mission. They found that the majority of chorus waves were propagating away from the magnetic equator, providing additional evidence that the source region is near the magnetic equatorial plane. They also found that the wave normal angles for lower band chorus peaked close to the local magnetic field line direction, while for upper band chorus, the wave normal angle distribution was much broader. Those results followed after a similar study was conducted by Santolik et al. (2010) using observations from the Polar mission. They too found that propagation of chorus was typically away from the magnetic equator. They also showed that chorus waves can be reflected within the magnetosphere, likely when the waves encounter boundaries with very sharp density gradients. Taubenschuss et al. (2016)

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revisited chorus statistics from the THEMIS mission, and they found Poynting vectors that were strongly field aligned and that the most oblique wave normal vectors were confined to the magnetic equatorial plane and had azimuthal angles (in the plane perpendicular to B0, the background magnetic field) that pointed away from Earth. They also found that for the majority of lower band chorus waves, the Poynting and wave normal vectors were coplanar with B0, indicating a dispersion relation consistent with that of cold plasma theory, and the wave normal vectors became more field aligned with increased latitude. That latter point is consistent with results from the Cluster mission (Santolík et al., 2014) but contradictory to the results from theoretical ray tracing such as Bortnik et al. (2011) and Breuillard et al. (2012), which indicate that chorus wave normal angles should become more oblique with increasing latitude. Using Van Allen Probes observations, Li et al. (2016) found that a nonnegligible amount of lower band chorus waves had large wave normal angles close to the resonance cone. With the recent availability of multispacecraft observations at various spatial scales, research efforts to estimate the spatial scales of the chorus active and source regions have become more common. Santolik and Gurnett (2003) used the European Space Agency’s Cluster mission to study the coherency of lower band chorus waves; they found that the correlation was significant at spacecraft separtation distances of 60–260 km parallel to B0 and 7 to 100 km perpendicular to B0. Those results were consistent with a chorus source with a Gaussian peak of radiated power with a half-width of 35 km perpendicular to B0 (Santolik, Gurnett, & Pickett, 2004). Agapitov et al. (2010) used the THEMIS mission to study the same chorus elements using multiple spacecraft. Using phase correlation analysis and a different definition compared to Santolik and Gurnett (2003), Agapitov et al. (2010) estimated that the characteristic spatial-scale perpendicular to B0 for individual elements was in the range of 2800 to 3200 km. Agapitov et al. (2017) presented results from lapping events of the two Van Allen Probes spacecraft, and they found that individual chorus elements have spatial scales of ~600 km perpendicular to B0 for upper band chorus and up to 800 km for lower band chorus at larger amplitudes. Li et al. (2017) presented results from Van Allen Probes of coherently modulated chorus and hiss waves over large distances, up to 4.3 Earth radii (RE); however, those waves examined were not the same individual elements but were from modulation of the chorus active region, encompassing many individual elements, rather than extremely large individual chorus elements and respective source region. There is no clear consensus yet for the coherency scale of individual chorus elements or the greater chorus active region, and these are topics of ongoing study since those properties of whistler mode chorus are important for modeling how chorus waves affect energetic electrons in the outer radiation belt. In this study, we performed the first multipoint analysis of individual chorus elements using data from NASA’s latest Heliophysics mission: Magnetospheric Multiscale (MMS). The results presented here are from a single case study and are intended to detail the analysis method and present preliminary results from that case. With MMS, we show here how the chorus wave normal vectors can be calculated directly from the four-point observations of the wave electric and magnetic field vectors without relying on minimum variance analysis. The paper is laid out as follows. Section 2 reviews the data used for this study, and section 3 overviews the details of the case study event itself, providing an orientation of the event for greater context. The methods employed for this study are reviewed in section 4, with the details of the analysis and results presented in section 5. A discussion of the key results and how they fit into context with the existing literature follows in section 6, and the paper is summarized and concluded in section 7.

2. Data Wave and particle data from NASA’s MMS (Burch et al., 2016) and Van Allen Probes (Mauk et al., 2013) missions were used for this study. Critical to the analysis here, the MMS mission consists of four identically instrumented spacecraft that are held in tight formation (interspacecraft separations less than 100 km) throughout most of their highly eccentric, near-equatorial orbits around Earth. From MMS, burst resolution data were analyzed from the FIELDS (Torbert et al., 2016) and Energetic Particle Detectors (EPD) (Mauk et al., 2016) instrument suites. MMS FIELDS consists of data collected using search coil (Le Contel et al., 2016) and fluxgate (Russell et al., 2016) magnetometers (SCM and FGM, respectively) as well as spin-plane and axial double probes (Ergun et al., 2016; Lindqvist et al., 2016) and an electron drift instrument (Torbert et al., 2016). The MMS EPD suite consists of the Energetic Ion Spectrometers (Mauk et al., 2016) and the Fly’s Eye Energetic Particle Spectrometers (FEEPS) (Blake et al., 2016). For this study, we used the following

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Figure 1. (a and b) MMS (black) and Van Allen Probes (RBSP-A in red and RBSP-B in green) positions in the GSM coordinate frame at 01:25:33 UT on 07 April 2016. From the Tsyganenko and Sitnov (2005) field model, field lines that map to each spacecraft and L = 6 at dawn and midnight MLT are also plotted, with solid lines in the x-y plane being above the spacecraft and dashed lines being below the spacecraft (or the equatorial plane in the case of L = 6 lines). (c) A zoom-in on the MMS constellation showing interspacecraft separations. MMS-1, MMS-2, MMS-3, and MMS-4 are shown with the black square, red circle, green cross, and blue triangle, respectively. The origin for this plot is centered at the MMS-1 location, and note that the axis units are kilometers here. Dashed lines project each spacecraft position onto the three principal planes (x-y, x-z, and y-z) for clarity in this three-dimensional plot.

data: three-axis survey (62.5 ms resolution) magnetic fields from the FGM, three-axis burst AC (8192 Hz sampling) magnetic fields from the SCM, three-axis burst DC (8192 Hz sampling) electric field data from the double probes, and all-sky burst (0.3 s resolution) energy spectra and pitch angle distributions of ~30 keV to 600 keV electrons from FEEPS. From Van Allen Probes, we used Electric and Magnetic Fields Instrument Suite for Integrated Science (EMFISIS) (Kletzing et al., 2013) power spectral densities from magnetic and electric wavefields and Magnetic Electron and Ion Spectrometers (MagEIS) (Blake et al., 2013) burst pitch angle distributions for 30 keV to ~200 keV electrons. For this study, MMS is used for the majority of the analysis, while Van Allen Probes data are used mostly for context of the event.

3. Event Orientation We examined a short period of MMS burst data from a conjunction event between MMS and Van Allen Probes-B (RBSP-B) that occurred around 01:25 UT on 07 April 2016. Figure 1 shows the satellite locations and MMS interspacecraft configuration for this event. From Figures 1a and 1b, all six spacecraft (four MMS plus RBSP-A and RBSP-B) were located in the dawn and morning local time sectors. RBSP-A and RBSP-B were straddling either side of their respective lines of apsides, with RBSP-A being inbound from apogee and RBSP-B being outbound during the period of interest. At 01:25 UT, the MMS constellation was moving inbound along its orbit and was located around L = 8.2 at
31° magnetic latitude and 06:53 magnetic local time (MLT). Figure 1c shows the configuration of the MMS constellation around 01:25 UT. The four spacecraft were separated by distances ranging from 30 km (MMS-3 to MMS-4) to 70 km (MMS-2 to MMS-3), and the constellation was in a good tetrahedron-like configuration, making it ideal for multispacecraft analyses. The nearest spatial conjunction between MMS and RBSP-B occurred around ~01:20 UT, when the separation between the MMS constellation and RBSP-B got down to within 0.9 RE. However, the closest magnetic conjunction between the two missions did not occur until ~02:05 UT due mostly to the difference in latitude between MMS and RBSP-B.

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Between 00:00 and 02:30 UT, a complex series of energetic particle injections were observed, enveloping the period we are examining here, and lower band chorus waves were also observed during this period. Figure 2 provides a summary of those chorus wave observations from all four MMS spacecraft and the two Van Allen Probes. The clear majority of the chorus waves observed in this event were lower band chorus, falling between 0.1 and 0.5 of the equatorial electron cyclotron frequency from mapping in the Tsyganenko and Sitnov (2005) model, as indicated with the magenta dashed lines in Figure 2. The strongest chorus activity occurred after around 01:40 UT and was associated with the drifting 0.98 are shown in Figures 3d–3g.

For the results in Figure 3, we employed the wave analysis techniques used by Lee and Angelopoulos (2014), Li et al. (2013), Santolik et al. (2003), Santolik et al. (2010), Bortnik et al. (2007), Samson and Olson (1980), and Means (1972). These techniques rely on fast Fourier transforms (FFTs) of the B wave data to generate dynamic spectrograms and calculate wave polarization quantities. Using those data with high degree of polarization, the magnetic divergence equation can be invoked with minimum variance analysis to estimate the direction of the wave normal vector. In Figure 3 and the following analysis, the wave ellipticity, the Poynting vector (S) normal angle (θS; the angle between B0 and S) and azimuthal angle (ΦS; the angle from the x axis in the FAC perpendicular plane), and the wave normal angle (θk) were calculated using the wavefields in FAC and are shown only for polarization ratios >0.98, where the polarization ratio is the ratio of polarized wave power to total power (Bortnik et al., 2007; Li et al., 2013). Such wave analysis results in a 180° ambiguity in the direction of the wave normal vector, but the Poynting vector can also be used to eliminate the ambiguity by assuming that k and S are in the same hemisphere with respect to B0 (i.e., their normal angles differ by less than 90°). With that assumption and the wave normal angle, θ, from 0 the polarization analysis, we defined θk = 180°
θ as shown in Figures 3 and 10. From Figure 3, note that this analysis results in ellipticity that is near +1 for these waves, indicating they are righthanded, nearly circularly polarized waves with energy propagating away from the magnetic equator (θS > 90°), as is expected for whistler mode chorus waves.

The analysis technique to estimate the direction of k described above uses only single-point observations of the wave electric and magnetic fields and does not provide the magnitude of k. However, with the close four-point observations provided by MMS, we can also calculate k directly using multipoint data analysis. Following Balikhin et al. (2003) and Pinçon and Glassmeier (2008), four-point observations of electromagnetic wavefields can be used to calculate k using the observed phase difference between different sets of spacecraft. Assuming a sinusoidal wave function,
 Δφ ¼ ðk·Ra –ωt Þ–ðk·Rb –ωtÞ ¼ jk jjRab j cosðθkR Þ ¼ k·Rab ¼ k x ðRax
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Rbz Þ (1) where Δφ is the wave phase difference observed between two spacecraft, a and b; k is the wave normal vector; Rab is the separation vector between the two spacecraft; and θkR is the angle between k and Rab. With four noncoplanar spacecraft (MMS-1, MMS-2, MMS-3, and MMS-4) and the phase difference observed between each pair of them, k can then be solved using the following set of linear equations: 2 32 3 2 3 R12x R12y R12z kx Δφ12 6R 76 7 6 7 (2) 4 13x R13y R13z 54 k y 5 ¼ 4 Δφ13 5 R14x

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Figure 4. Zoom-in on the eight chorus elements analyzed for this study. Note that the time ranges are slightly different for each element shown here. For each element, the B wave power spectra and amplitudes are plotted. The power spectra are from MMS-4, while the amplitudes are shown from each of the four MMS spacecraft using the colors corresponding to those for each spacecraft in Figure 1c. Note that three elements, 5–7, are shown in Figure 4e. Also, the magenta box plotted over Figure 4d shows the range of time shown in Figure 5.

at least 16 nonsingular solutions of the wave normal vector, thus also providing standard deviation as an estimate of error on the average of the resulting vectors. The phase difference between each pair of spacecraft, Δφ, can be calculated using cross-correlation analysis (i.e., the correlation between the two signals as a function of a relative temporal shift, or lag, between the two

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signals), an example of which is shown in Figure 8. With the three components of both the electric and magnetic fields of the wave, it is possible to calculate six different cross correlations, again allowing for some estimate of the accuracy of the results. For example, in Figure 8a, the cross correlation for the parallel and two perpendicular components of B wave between MMS-2 and MMS-4 is plotted versus lag time for the chorus subelement shown in Figure 7 (i.e., 42, meaning chorus element 4, subelement 2). The peak in the cross correlation, ~0.99, occurred at a lag time of approximately +1.0 ms in all three of the B wave components. The analysis from the E wave components for the same subelement is shown in Figure 8b, which reveals consistent results (maximum XCorr of >0.98 at lag of ~+1.0 ms). The maximum cross correlation from all six wave components was 0.989, with an average lag time of +1.06 ms. This means that this subelement’s phase fronts were first observed at MMS-2, approximately 1.06 ms before the same wave’s phase fronts were observed at MMS-4. The period of the wave is also provided by the cross-correlation analysis: it is the periodicity in the cross-correlation curve as a function of lag time, since the wave components go into and out of phase with each other at integer values of the wave period as the data are shifted over a range of different lag times. The correlation drops at higher lag times due to the shift in the wave’s frequency over the test period. For this subelement, the observed period was 3.66 ms. The phase difference between the two spacecraft can then be calculated as

Figure 5. Waveform data in the FAC system for (top) E wave, (middle) B wave, and (bottom) S = (E × B)/μ0 from MMS-4. This period corresponds to chorus element 4 and the magenta box plotted in Figure 4d. For each plot here, the field-aligned direction (par) is shown in red, the first perpendicular direction (Perp1) in blue, and the second perpendicular direction (Perp2) in green. The magenta boxes plotted over this indicate the periods used for subelements 41, 42, 43, and 44, respectively and as labeled.

Δφ ¼ 2π

Δt T

(3)

where Δt is the lag time corresponding to the peak cross correlation and T is the observed period of the wave. So for clarity in this example, Rab = R42 = R4
R2 and Δφ = Δφ42 = 2π*(1.06/3.66). Figures 8c and 8d show the same results but for MMS-3 and MMS-4. This pair of spacecraft revealed a maximum correlation of 0.992 at an average lag time of
0.61 ms, meaning that the wave was first observed at MMS-4 and then 0.61 ms later at MMS-3. The results of this cross-correlation analysis from each pair of spacecraft for each of the eight elements from Figure 4 plus multiple subelements are shown in Table 1, which will form the core for the discussion points in the following sections.

5. Analysis and Results Examining the details of the power spectral densities and derived quantities in Figure 3 revealed results that are strongly consistent with whistler mode chorus waves. The individual wave elements were observed as rising tone emissions at frequencies 0.1 fce < f < 0.5 fce, with fce here and shown in Figures 2 and 3 being the equatorial electron cyclotron frequency, and the waves were clearly electromagnetic in nature. The ellipticity showed that the waves were strongly right handed and near circularly polarized. The Poynting vector revealed that the energy from these waves was propagating predominantly in the antiparallel field-aligned direction, which in this geometry was away from the magnetic equatorial plane. Finally, the wave normal angles were predominantly field aligned in the antiparallel direction. Each of these features is entirely consistent with lower band whistler mode chorus waves (e.g., Li et al., 2013; Santolik et al., 2003). Finer details of the individual chorus elements can be gleaned from the plots in Figure 4. Each rising tone element had a frequency sweep rate in the range of 0.3 to 0.8 kHz/s, which is below yet similar to the mean and median sweep rates of ~1 to 2 kHz/s reported by Macúšová et al. (2010). Interestingly, most of the chorus elements observed during this period had “hook”-like features (e.g., Burtis and Helliwell, 1976) observed immediately following the initial rising tone emission; these “hooks” are manifest as the flat or falling tone TURNER ET AL.

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Perp2

0 -2 -4 -6 6 4

Perp1

MMS-2

Par

2 Perp2

0 -2 -4 -6 6 4

Perp1

MMS-3

Par

2 Perp2

0 -2 -4 -6 6 4

Perp1

MMS-4

Par

2 Perp2

0 -2

Perp1

-4 -6

.000 Seconds 2016 Apr 07 01:25:33

.200

.400

SCM B-wave FAC [nT]

2

SCM B-wave FAC [nT]

Par

SCM B-wave FAC [nT]

All MMS: B-wave

MMS-1

SCM B-wave FAC [nT]

EDP E-wave FAC [mV/m]

EDP E-wave FAC [mV/m]

EDP E-wave FAC [mV/m]

EDP E-wave FAC [mV/m]

All MMS: E-wave 6 4

10.1002/2017JA024474

0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15

MMS-1

Par Perp2 Perp1

MMS-2

Par Perp2 Perp1

MMS-3

Par Perp2 Perp1

MMS-4

Par Perp2 Perp1

Seconds .000 2016 Apr 07 01:25:33

.200

.400

Figure 6. Waveform data in the FAC system for (left column) E wave and (right column) B wave for MMS-1, MMS-2, MMS-3, and MMS-4, respectively, in the rows from top to bottom. The coordinate colors correspond to those from Figure 5. The magenta box plotted over the data here represents the period for chorus subelement 42 shown in Figure 7.

-4 -6 6 4

Perp1

Par

MMS-2

2 Perp2

0 -2 -4 -6 6 4

Perp1

Par

MMS-3

2 Perp2

0 -2 -4 -6 6 4

Perp1

MMS-4

Par

2 Perp2

0 -2

Perp1

-4 -6

.020 Seconds 2016 Apr 07 01:25:33

.040

.060

.080

.100

SCM B-wave FAC [nT]

Perp2

0 -2

SCM B-wave FAC [nT]

Par

MMS-1

2

SCM B-wave FAC [nT]

All MMS: B-wave

SCM B-wave FAC [nT]

EDP E-wave FAC [mV/m]

EDP E-wave FAC [mV/m]

EDP E-wave FAC [mV/m]

EDP E-wave FAC [mV/m]

All MMS: E-wave 6 4

0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15

Par

MMS-1

Perp2 Perp1

Par

MMS-2

Perp2 Perp1

Par

MMS-3

Perp2 Perp1

MMS-4

Seconds .020 2016 Apr 07 01:25:33

Par Perp2 Perp1 .040

.060

.080

.100

Figure 7. Waveform data in the FAC system for E wave and B wave from all four MMS spacecraft shown in the same format as in Figure 6 but zoomed in on chorus subelement 42.

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MMS OBSERVATIONS OF CHORUS ELEMENTS

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Journal of Geophysical Research: Space Physics a)

c)

B-wave

0.5

MMS-2 to MMS-4 01:25:33.01 to 01:25:33.10 UT

+ Perp1 Perp2 Parallel

0.0

-0.5

0.0

-1.0 -20

-10

0 Lag [ms]

10

20

E-wave

b) 1.0

-20

d)

MMS-2 to MMS-4 01:25:33.01 to 01:25:33.10 UT

X-Correlation

X-Correlation

0.5

MMS-3 to MMS-4 01:25:33.01 to 01:25:33.10 UT

-0.5

-1.0

0.5

B-wave 1.0

X-Correlation

X-Correlation

1.0

10.1002/2017JA024474

0.0

-0.5

-10

0 Lag [ms]

10

20

10

20

E-wave 1.0

0.5

MMS-3 to MMS-4 01:25:33.01 to 01:25:33.10 UT

0.0

-0.5

-1.0

-1.0 -20

-10

0 Lag [ms]

10

20

-20

-10

0 Lag [ms]

Figure 8. Example results from the cross-correlation analysis. (a) The cross correlation versus lag time between the three components of B wave in FAC from MMS-2 and MMS-4. Results for the Perp1, Perp2, and parallel FAC components are shown with black plus sign, blue cross, and red asterisk, respectively. The time range used for this analysis is printed on each plot, with the examples shown being from 01:25:33.01 to 01:25:33.10 UT, corresponding to chorus subelement 42. (b) The same as in Figure 8a but for E wave. (c and d) The results for B wave and E wave from MMS-3 and MMS-4.

frequency portion at the later part of the elements, after the peak frequency. The wave amplitudes in Figure 4 also indicate the complex substructure within each of these chorus elements. In each of the elements, the peak wave amplitude occurred near the peak frequency, but the substructure was somewhat bursty, with a single element being made up of multiple subelements of enhanced wave power. Typically, the amplitude of the hook features decayed over time and was significantly lower than the peak frequencies and the average amplitude of the rising tone. The wave amplitudes plotted in Figure 4 show results from all four MMS spacecraft. There are some differences in the wave amplitudes observed at the different spacecraft, but those differences were almost entirely within 50% of the mean amplitude from all four spacecraft. In general, for each of these chorus elements, very similar substructure was observed at all four spacecraft, indicating that whatever causes that substructure was relatively stable over the separation times (on the order of ~1 ms) and distances (tens of kilometers) between the spacecraft. Since all four spacecraft observed amplitudes at similar magnitude, the spatial extent of these chorus element substructures must have been larger than the separation scales between MMS for this event. Zooming in on the largest-amplitude chorus element from this period, element 4 is shown in greater detail in Figure 5. The four different subelements that we analyzed within element 4 occurred in the time ranges marked with the magenta boxes in Figure 5, labeled 41, 42, 43, and 44 for subelements 1 through 4, respectively. Figure 5 shows how the wave amplitude increased with increasing frequency in the rising tone portion (subelements 41 and 42), then the wave amplitudes increased and then decreased nonlinearly during the period of nearly flat frequency around the peak time in subelement 43 and then decreased more linear in time during the hook feature in subelement 44. Note that the Poynting flux remained predominantly in the antiparallel direction through all four subelements. Figure 6 shows that all four MMS spacecraft observed consistent results for the substructure of this chorus element.

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Element 1

01:25: 09.400

01:25: 09.520

01:25: 09.360

3

Full element

01:25: 09.360

2

1

MMS OBSERVATIONS OF CHORUS ELEMENTS

01:25: 09.590

01:25: 09.590

01:25: 09.490

01:25: 09.400

1

4

3

2

1

4

3

2

1

4

3

2

1

Start Stop Chorus time time element Subelement (hh:Mm: (hh:Mm: MMS (#) (#) Ss.Fff) Ss.Fff) (s/c ID)

[7,210.685,
37,980.293,
898.163]

[7,210.857,
37,979.916,
898.124] [7,165.011,
37,993.728,
901.071] [7,201.580,
37,934.063,
893.700] [7,195.007,
37,960.449,
906.330]

[7,210.685,
37,980.293,
898.163] [7,164.838,
37,994.105,
901.110] [7,201.407,
37,934.441,
893.739] [7,194.834,
37,960.827,
906.369]

[7,210.598,
37,980.482,
898.182] [7,164.752,
37,994.293,
901.129] [7,201.321,
37,934.630,
893.758] [7,194.748,
37,961.015,
906.388]

MMS position, GSE (km)

0.977 0.907

2 to 4 3 to 4

0.968

0.976 0.954

2 to 4 3 to 4 1 to 2

0.932

0.987

1 to 4

2 to 3

0.942

1 to 3

0.978

0.956

2 to 3

1 to 2

0.995

0.955

0.977 0.923 0.979

0.930

0.982

0.916

0.972

1 to 4

1 to 3

2 to 4 3 to 4 1 to 2

2 to 3

1 to 4

1 to 3

1 to 2


0.57 3.05

1.10
0.51

0.44

0.43

1.16


0.65 3.17

1.04
0.49


1.54

0.46


1.95

1.18
0.98
1.06 2.95


1.90

0.73


2.38


0.69 3.80

MMS pair Lag XCorr (IDa Maximum time period to IDb) XCorr (#) (ms) (ms)

Table 1 Results of the Cross-Correlation Analysis for Chorus Elements and Subelements

|k| (rad/km)

k, FAC (rad/km)

5.38 ± 9.16 × 1e
2

[
1.17, 1.07, 24
5.14] ± [3.16, 7.84, 3.53] × 1e
2

[3.39,
1.82, 8.92 ± 22.77 [
2.98, 7.99, 75
8.04] ± [4.65, × 1e
2
2.60] ± [5.39,

[0.46, 3.76,
3.82] ± [2.80, 2.16, 8.45] × 1e
2

[5.60,
4.26, 12.99 ± 28.87 [
4.68, 11.96, 77
10.92] ± [6.03, × 1e
2
1.93] ± [6.98, 25.15, 12.34] 5.52, 27.69] × × 1e
2 1e
2

[3.69,
3.53, 9.82 ± 30.32 [
2.94, 9.27, 96
8.39] ± [6.23, × 1e
2
1.37] ± [7.23, 5.80, 29.11] × 26.45, 12.95] 1e
2 × 1e
2

k, GSE (rad/km)

[0.1, 0.4, 175 [29.1,
4.6] ×
148.9, 1e
3 105.8]

107 [0.5, 0.1, 168 [29.1,
2.4] ×
148.9, 1e
3 105.8]

163 [0.2, 0.0, 171 [29.1,
1.3] ×
148.9, 1e
3 105.8]

99

98 [0.1,
0.1, 164 [29.1,
0.5] ×
148.9, 1e
3 105.8]

MMS4 Use? MMS4 Savg, Bavg, (yes: blue θ_kS θ_kB FAC and no: θ_SB GSE 2 (deg) (deg) (uW/m ) (deg) (nT) red)

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Element 2

01:25: 01: 17.160 25:17.190

01:25: 01: 17.060 25:17.500

2

Full element

01:25: 17.115

01:25: 17.060

1

MMS OBSERVATIONS OF CHORUS ELEMENTS 3

2

1

4

3

2

1

4

3

2

1

4

3

2

Start Stop Chorus time time element Subelement (hh:Mm: (hh:Mm: MMS (#) (#) Ss.Fff) Ss.Fff) (s/c ID)

Table 1. (continued)

[7,221.406,
37,956.897,
895.746] [7,175.566,
37,970.715,
898.694]

[7,221.406,
37,956.897,
895.746] [7,175.566,
37,970.715,
898.694] [7,212.137,
37,911.017,
891.322] [7,205.561,
37,937.419,
903.943]

[7,221.233,
37,957.275,
895.785] [7,175.392,
37,971.092,
898.732] [7,212.051,
37,911.206,
891.341] [7,205.388,
37,937.796,
903.982]

[7,164.838,
37,994.105,
901.110] [7,201.407,
37,934.441,
893.739] [7,194.834,
37,960.827,
906.369]

MMS position, GSE (km)

0.983 0.992 0.958

2 to 4 3 to 4 1 to 2

0.992

0.953

1 to 3

1 to 4

0.889

0.956 0.979

2 to 4 3 to 4 1 to 2

0.923

0.991

2 to 3

1 to 4

0.973

0.962

2 to 3

1 to 3

0.997

0.991

1 to 4

1 to 3

0.983

0.974 0.845

2 to 4 3 to 4 1 to 2

0.935

0.991

0.936

2 to 3

1 to 4

1 to 3

0.51

1.20


0.89 4.15

1.57
0.73

2.23

0.49

1.22

1.44
0.73
1.06 4.03

2.89

0.51

1.26


0.93 4.27

1.06
0.61


0.92

0.46


1.25

MMS pair Lag XCorr (IDa Maximum time period to IDb) XCorr (#) (ms) (ms)

[1.71, 4.49,
2.04] ± [0.13, 0.11, 0.57] × 1e
2

[2.29, 4.80,
2.43] ± [0.10, 0.08, 0.37] × 1e
2

[2.09, 5.10,
1.17] ± [1.11, 0.86, 3.64] × 1e
2

4.29, 21.87] × 1e
2

k, GSE (rad/km)

5.22 ± 0.60 × 1e
2

5.84 ± 0.39 × 1e
2

5.63 ± 3.90 × 1e
2

|k| (rad/km)

[
2.53,
0.67, 64
4.52] ± [0.15, 0.52, 0.26] × 1e
2

[
3.15,
0.46, 59
4.90] ± [0.11, 0.34, 0.16] × 1e
2

[
3.01,
1.68, 74
4.45] ± [1.25, 3.35, 1.56] × 1e
2

19.84, 9.78] × 1e
2

k, FAC (rad/km)

150 [1.1, 0.3, 146 [28.6,
1.7] ×
149.4, 1e
3 106.1]

147 [2.1, 0.0, 154 [28.6,
4.3] ×
149.4, 1e
3 106.1]

142 [3.6, 1.0, 143 [28.6,
4.9] ×
149.4, 1e
3 106.1]

Use? MMS4 MMS4 Bavg, (yes: blue Savg, FAC θ_SB GSE and no: θ_kS θ_kB 2 red) (deg) (deg) (uW/m ) (deg) (nT)

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Element 3

01:25: 24.070

01:25: 24.125

3

01:25: 23.970

2

1

MMS OBSERVATIONS OF CHORUS ELEMENTS

01:25: 24.220

01:25: 24.125

01:25: 24.070

4

3

2

1

4

3

2

1

4

3

2

1

4

Start Stop Chorus time time element Subelement (hh:Mm: (hh:Mm: MMS (#) (#) Ss.Fff) Ss.Fff) (s/c ID)

Table 1. (continued)

[7,231.001,
37,935.938,
893.583] [7,185.165,
37,949.761,
896.530] [7,221.826,
37,889.843,
889.139]

[7,231.001,
37,935.938,
893.583] [7,185.079,
37,949.950,
896.550] [7,221.740,
37,890.032,
889.158] [7,215.160,
37,916.449,
901.771]

[7,230.828,
37,936.316,
893.622] [7,184.992,
37,950.139,
896.569] [7,221.567,
37,890.410,
889.197] [7,214.987,
37,916.827,
901.810]

[7,212.137,
37,911.017,
891.322] [7,205.561,
37,937.419,
903.943]

MMS position, GSE (km)

0.988

1 to 4

0.903

0.978

1 to 3

2 to 3

0.953

0.979 0.967

2 to 4 3 to 4 1 to 2

0.922

0.992

0.980

2 to 3

1 to 4

1 to 3

0.963

0.959 0.972

2 to 4 3 to 4 1 to 2

0.922

0.988

2 to 3

1 to 4

0.985

0.897 0.945 0.943

2 to 4 3 to 4 1 to 2

1 to 3

0.798

2 to 3

1.73

0.47

1.10


0.63 3.54

1.06
0.43

1.50

0.57

0.98


0.46 4.03


1.54
0.83


2.29

0.59

1.42

1.36
0.65
1.04 4.76

2.12

MMS pair Lag XCorr (IDa Maximum time period to IDb) XCorr (#) (ms) (ms) |k| (rad/km)

4.96 ± 0.26 × 1e
2

[
1.37, 0.44, 31
4.75] ± [0.11, 0.22, 0.09] × 1e
2

[3.06,
6.09, 139 4.36] ± [7.59, 15.90, 6.77] × 1e
2

k, FAC (rad/km)

[1.21, 4.61, 5.15 ± 0.65 × [
2.05,
0.86, 35
1.96] ± [0.10, 1e
2
4.65] ± [0.12, 0.09, 0.64] × 0.56, 0.31] × 1e
2 1e
2

[0.67, 3.83,
3.08] ± [0.10, 0.07, 0.23] × 1e
2

[
2.99,
0.66, 8.09 ± 18.87 7.49] ± [6.79, × 1e
2 4.91, 16.91] × 1e
2

k, GSE (rad/km)

[0.1, 1.0, 159 [28.4,
2.6] ×
149.5, 1e
3 106.3]

154 [1.3, 1.3, 171 [28.4,
11.0] ×
149.5, 1e
3 106.3]

163 [2.8, 1.8, 162 [28.4,
10.5] ×
149.5, 1e
3 106.3]

57

Use? MMS4 MMS4 Bavg, (yes: blue Savg, FAC θ_SB GSE and no: θ_kS θ_kB 2 red) (deg) (deg) (uW/m ) (deg) (nT)

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Element 4

MMS OBSERVATIONS OF CHORUS ELEMENTS

1

Full element

4

01:25: 32.950

01:25: 23.970

01:25: 24.250

01:25: 33.005

01:25: 24.450

01:25: 24.450

4

3

2

1

4

3

2

1

4

3

2

1

Start Stop Chorus time time element Subelement (hh:Mm: (hh:Mm: MMS (#) (#) Ss.Fff) Ss.Fff) (s/c ID)

Table 1. (continued)

[7,243.272,
37,909.108,
890.815] [7,197.356,
37,923.127,
893.782] [7,234.021,
37,863.170,
886.390] [7,227.437,
37,889.605,
898.992]

[7,231.001,
37,935.938,
893.583] [7,185.165,
37,949.761,
896.530] [7,221.826,
37,889.843,
889.139] [7,215.160,
37,916.449,
901.771]

[7,231.174,
37,935.560,
893.544] [7,185.252,
37,949.572,
896.511] [7,221.913,
37,889.654,
889.119] [7,215.333,
37,916.071,
901.732]

[7,215.160,
37,916.449,
901.771]

MMS position, GSE (km)

0.908

1 to 2

0.937

2 to 3

0.947

0.980

1 to 4

2 to 4

0.978

1 to 3

0.960

0.918 0.950

2 to 4 3 to 4 1 to 2

0.776

0.988

2 to 3

1 to 4

0.957

0.940 0.966

2 to 4 3 to 4

1 to 3

0.865

0.986

0.973

0.969 0.971 0.934

2 to 3

1 to 4

1 to 3

2 to 4 3 to 4 1 to 2

1.12

1.56

0.51

1.15


0.53 4.51

1.18
0.59

1.77

0.55

1.10


0.69 3.78

1.08
0.57

1.63

0.51

1.08

1.14
0.57
0.57 3.54

MMS pair Lag XCorr (IDa Maximum time period to IDb) XCorr (#) (ms) (ms) |k| (rad/km)

k, FAC (rad/km)

[0.60, 3.63,
1.95] ± [0.22, 0.17, 0.73] × 1e
2

[1.24, 4.47,
2.43] ± [0.10, 0.08, 0.35] × 1e
2

4.16 ± 0.78 × 1e
2

5.23 ± 0.37 × 1e
2

[
1.25,
0.39, 31
3.95] ± [0.25, 0.67, 0.31] × 1e
2

[
2.05,
0.39, 35
4.80] ± [0.11, 0.32, 0.15] × 1e
2

[1.25, 3.73, 6.06 ± 11.96 [
1.92, 1.81, 32
4.60] ± [1.82, × 1e
2
5.45] ± [2.41, 3.32, 11.34] × 10.97, 4.10] × 1e
2 1e
2

k, GSE (rad/km)

162 [0.9,
0.3, 166
3.7] × 1e
3

157 [0.8, 0.8, 167
5.1] × 1e
3

149.7, 106.6]

[27.7,

149.5, 106.3]

[28.4,

154 [0.5, 0.3, 166 [28.4,
2.4] ×
149.5, 1e
3 106.3]

Use? MMS4 MMS4 Bavg, (yes: blue Savg, FAC θ_SB GSE and no: θ_kS θ_kB 2 red) (deg) (deg) (uW/m ) (deg) (nT)

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01:25: 33.128

01:25: 33.185

01:25: 32.950

4

Full element

01:25: 33.010

3

2

MMS OBSERVATIONS OF CHORUS ELEMENTS

01:25: 33.390

01:25: 33.390

01:25: 33.182

01:25: 33.100

2

1

4

3

2

1

4

3

2

1

4

3

2

1

Start Stop Chorus time time element Subelement (hh:Mm: (hh:Mm: MMS (#) (#) Ss.Fff) Ss.Fff) (s/c ID)

Table 1. (continued)

[7,243.445,
37,908.730,
890.776]

[7,243.531,
37,908.541,
890.756] [7,197.702,
37,922.371,
893.704] [7,234.367,
37,862.413,
886.312] [7,227.696,
37,889.038,
898.933]

[7,243.445,
37,908.730,
890.776] [7,197.616,
37,922.560,
893.723] [7,234.280,
37,862.602,
886.331] [7,227.610,
37,889.227,
898.952]

[7,243.358,
37,908.919,
890.795] [7,197.443,
37,922.938,
893.762] [7,234.107,
37,862.981,
886.370] [7,227.523,
37,889.416,
898.972]

MMS position, GSE (km)

0.969 0.974 0.972

2 to 4 3 to 4 1 to 2

0.991

0.938

2 to 3

1 to 3

0.972

0.980

1 to 4

1 to 3

0.956

0.991 0.992

2 to 4 3 to 4 1 to 2

0.972

2 to 3

0.995

0.994

1 to 3

1 to 4

0.990

0.989 0.992

2 to 4 3 to 4 1 to 2

0.977

0.993

0.994

0.962 0.987

2 to 3

1 to 4

1 to 3

3 1 to 4 2

1.08

1.04
0.67
0.59 3.54

1.69

0.47

1.10


0.59 3.66

1.04
0.63

1.67

0.43

1.04


0.63 3.29

1.06
0.61

1.69

0.49

1.10


0.54
0.57 3.66

MMS pair Lag XCorr (IDa Maximum time period to IDb) XCorr (#) (ms) (ms)

[0.99, 4.49,
1.20] ± [0.04, 0.03, 0.15] × 1e
2

[0.91, 4.40,
0.81] ± [0.06, 0.05, 0.25] × 1e
2

[1.26, 4.68,
1.09] ± [0.04, 0.03, 0.16] × 1e
2

[0.88, 4.44,
1.40] ± [0.04, 0.03, 0.12] × 1e
2

k, GSE (rad/km)

4.75 ± 0.16 × 1e
2

4.56 ± 0.26 × 1e
2

4.97 ± 0.16 × 1e
2

4.74 ± 0.13 × 1e
2

|k| (rad/km)

[
1.79,
1.45, 31
4.16] ± [0.04, 0.14, 0.07] × 1e
2

[
1.70,
1.72, 31
3.87] ± [0.07, 0.23, 0.11] × 1e
2

[
2.09,
1.62, 34
4.21] ± [0.04, 0.14, 0.07] × 1e
2

[
1.67,
1.27, 29
4.25] ± [0.04, 0.11, 0.05] × 1e
2

k, FAC (rad/km)

149.7, 106.6]

[27.7,

149.7, 106.6]

[27.7,

149.7, 106.6]

[27.7,

151 [1.9,
1.7, 172 [27.7,
18.3] ×
149.7, 1e
3 106.6]

148 [0.6,
1.0, 166
4.8] × 1e
3

148 [3.6,
4.3, 170
33.4] × 1e
3

154 [4.9,
2.9, 174
55.4] × 1e
3

Use? MMS4 MMS4 Bavg, (yes: blue Savg, FAC θ_SB GSE and no: θ_kS θ_kB 2 red) (deg) (deg) (uW/m ) (deg) (nT)

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Full element

Full element

1

Element 5

Element 6

Element 7

MMS OBSERVATIONS OF CHORUS ELEMENTS

01:25: 41.043

01:25: 40.750

01:25: 40.450

01:25: 41.013

01:25: 40.400

01:25: 39.980

3

2

1

4

3

2

1

4

3

2

1

4

3

Start Stop Chorus time time element Subelement (hh:Mm: (hh:Mm: MMS (#) (#) Ss.Fff) Ss.Fff) (s/c ID)

Table 1. (continued)

[7,254.417,
37,884.721,
888.300] [7,208.507,
37,898.746,
891.267]

[7,253.640,
37,886.423,
888.475] [7,207.729,
37,900.448,
891.442] [7,244.397,
37,840.458,
884.050] [7,237.809,
37,866.909,
896.643]

[7,252.949,
37,887.936,
888.631] [7,207.124,
37,901.772,
891.579] [7,243.705,
37,841.972,
884.206] [7,237.118,
37,868.423,
896.799]

[7,197.616,
37,922.560,
893.723] [7,234.280,
37,862.602,
886.331] [7,227.610,
37,889.227,
898.952]

MMS position, GSE (km)

1 to 4

1 to 3

0.978

0.976

0.959

0.895 0.931

2 to 4 3 to 4 1 to 2

0.808

2 to 3

0.923

0.884

1 to 3

1 to 4

0.869

0.854 0.927

2 to 4 3 to 4 1 to 2

0.824

0.888

0.856

2 to 3

1 to 4

1 to 3

0.845

0.981 0.985

2 to 4 3 to 4 1 to 2

0.956

0.987

2 to 3

1 to 4

0.51

0.98


0.61 3.54

1.16
0.63

1.71

0.51

1.18


0.63 3.66

1.10
0.56

1.55

0.51

1.12


0.73 3.90

1.04
0.63

1.67

0.47

MMS pair Lag XCorr (IDa Maximum time period to IDb) XCorr (#) (ms) (ms)

5.09 ± 0.61 × 1e
2

4.68 ± 1.22 × 1e
2

|k| (rad/km)

[
1.82,
1.03, 31
4.64] ± [0.20, 0.52, 0.24] × 1e
2

[
1.81,
0.45, 39
4.29] ± [0.56, 1.02, 0.38] × 1e
2

k, FAC (rad/km)

36 [0.73, 2.17, [
0.97, 1.22, 3.93 ± 24.02 × 1e
2
3.20] ± [7.37,
3.61] ± [6.54, 20.66, 9.79] × 5.12, 22.54] × 1e
2 1e
2

[0.98, 4.65,
1.82] ± [0.18, 0.14, 0.56] × 1e
2

[1.09, 4.04,
2.09] ± [0.50, 0.37, 1.05] × 1e
2

k, GSE (rad/km)

144 [
0.8, 1.5, 127 [27.9,
1.3] ×
149.9, 1e
3 106.8]

156 [
0.1, 0.4, 166 [28.0,
1.6] ×
150.0, 1e
3 106.8]

157 [0.1, 0.4, 158 [28.0,
1.0] ×
149.9, 1e
3 106.8]

Use? MMS4 MMS4 Bavg, (yes: blue Savg, FAC and no: θ_SB GSE θ_kS θ_kB 2 red) (deg) (deg) (uW/m ) (deg) (nT)

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Element 8

01:25: 43.330

01:25: 43.480

1

2

Full 01:25: element 40.850

MMS OBSERVATIONS OF CHORUS ELEMENTS

01:25: 43.580

01:25: 43.470

01:25: 41.300

4

3

2

1

4

3

2

1

4

3

2

1

4

Start Stop Chorus time time element Subelement (hh:Mm: (hh:Mm: MMS (#) (#) Ss.Fff) Ss.Fff) (s/c ID)

Table 1. (continued)

[7,257.786,
37,877.344,
887.540] [7,211.964,
37,891.183,
890.487] [7,248.546,
37,831.368,
883.114]

[7,257.613,
37,877.723,
887.579] [7,211.705,
37,891.750,
890.545] [7,248.373,
37,831.747,
883.153] [7,241.784,
37,858.204,
895.742]

[7,254.158,
37,885.289,
888.358] [7,208.334,
37,899.125,
891.306] [7,244.915,
37,839.322,
883.933] [7,238.328,
37,865.774,
896.525]

[7,245.175,
37,838.753,
883.874] [7,238.587,
37,865.206,
896.466]

MMS position, GSE (km)

0.886 0.879 0.946

2 to 4 3 to 4 1 to 2

2 to 3

0.784

0.959

0.926

1 to 3

1 to 4

0.797

0.944 0.976

2 to 4 3 to 4 1 to 2

0.933

0.980

2 to 3

1 to 4

0.980

0.768

2 to 3

1 to 3

0.910

1 to 4

0.890

0.966 0.968 0.869

2 to 4 3 to 4 1 to 2

1 to 3

0.920

2 to 3


2.01

0.45

1.02


0.54 3.66

1.12
0.67


0.69

0.45

1.14

1.13
0.41
0.67 3.78

1.38

0.54

1.08

1.10
0.94
0.61 3.66


1.69

MMS pair Lag XCorr (IDa Maximum time period to IDb) XCorr (#) (ms) (ms)

5.73 ± 1.54 × 1e
2

|k| (rad/km)

[
1.71, 0.86, 11
5.40] ± [0.60, 1.31, 0.55] × 1e
2

k, FAC (rad/km)

[
1.16, 0.81, 5.52 ± 23.45 [0.99, 3.79, 53
5.33] ± [6.65, × 1e
2
3.89] ± [7.48, 5.15, 21.89] × 20.13, 9.42] × 1e
2 1e
2

[
0.26, 2.14, 4.23 ± 15.40 [
0.14, 1.75, 35
3.64] ± [4.44, × 1e
2
3.85] ± [4.99, 3.43, 14.34] × 13.21, 6.14] × 1e
2 1e
2

[0.96, 4.18,
3.80] ± [0.53, 0.41, 1.39] × 1e
2

k, GSE (rad/km)

135 [0.0,
0.1, 172 [28.0,
149.9,
0.7] × 106.9] 1e
3

155 [0.4,
0.9, 169 [28.0,
5.3] ×
149.9, 1e
3 106.9]

160 [
0.2, 0.2, 171 [27.9,
1.7] ×
149.9, 1e
3 106.8]

Use? MMS4 MMS4 Bavg, (yes: blue Savg, FAC θ_SB GSE and no: θ_kS θ_kB 2 red) (deg) (deg) (uW/m ) (deg) (nT)

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1.12
0.69 0.902 0.958

4

3

2

2 to 4 3 to 4


0.71 0.895 2 to 3

0.45 0.974 1 to 4

1.14 0.971

01:25: 43.580

1

[7,257.786,
37,877.344,
887.540] [7,211.964,
37,891.183,
890.487] [7,248.546,
37,831.368,
883.114] [7,241.957,
37,857.826,
895.703]

1 to 3

1.06
0.63
0.67 3.78 0.805 0.911 0.917 01:25: 43.240

[7,241.957,
37,857.826,
895.703]

2 to 4 3 to 4 1 to 2 Full element

Start Stop Chorus time time element Subelement (hh:Mm: (hh:Mm: MMS (#) (#) Ss.Fff) Ss.Fff) (s/c ID)

Table 1. (continued)

MMS position, GSE (km)

MMS pair Lag XCorr (IDa Maximum time period to IDb) XCorr (#) (ms) (ms)

k, GSE (rad/km)

|k| (rad/km)

k, FAC (rad/km)

[1.01, 2.64, 5.58 ± 16.96 [
1.48, 2.56, 43
4.81] ± [6.85, × 1e
2
4.73] ± [7.46, 3.97, 15.01] × 13.8, 6.45] × 1e
2 1e
2

148 [0.2,
0.4, 169 [28.0,
2.4] ×
149.9, 1e
3 106.9]

Use? MMS4 MMS4 Bavg, (yes: blue Savg, FAC θ_SB GSE and no: θ_kS θ_kB 2 red) (deg) (deg) (uW/m ) (deg) (nT)

Journal of Geophysical Research: Space Physics

10.1002/2017JA024474

Zooming in yet again on subelement 42, Figure 7 shows the waveforms from this subelement observed by all four MMS spacecraft. As discussed in the previous section, these four-point wave observations were clearly phase coherent between all four spacecraft and can be used to calculate the wave normal vector, k; the results of that analysis are in Table 1. For subelement 42, the cross-correlation analysis resulted in k = ([
1.67,
1.27,
4.25] ± [0.04, 0.11, 0.05]) × 10
2 rad/km in FAC. Figure 9 summarizes the results of the analysis for subelement 42. In Figure 9, the MMS constellation is shown in the same format as in Figure 1c (though here rotated and in GSE coordinates), but the maximum cross correlations and lag times are shown for each pair of spacecraft. Unit vectors for the background magnetic field, Poynting, and wave normal vectors (B0, S, and k, respectively, in Figure 9) are also displayed at the top (again in GSE coordinates) and plotted originating from the origin location at MMS-1. As can be seen there and in Table 1, nearly 30° separated S from k in subelement 42, and while S was strongly field aligned (θS = 174°), k (with θk = 154°) was very far from the whistler mode resonance cone (~75° and 105° in this case), so the waves can be considered as quasiparallel. Note too that the plane wave assumption for subelement 42 was valid over the scale of the tetrahedron, given that the various lag times add up between different sets of spacecraft (e.g., lag for wave from MMS-2 to MMS-4 is equivalent within error to lag from MMS-2 to MMS-1 plus lag from MMS-1 to MMS-4). That result was expected based on the low error estimate calculated for this subelement. Also, the wave normal vector result is consistent with the observed propagation of this subelement from MMS-2 to MMS-1 to MMS-4 to MMS-3. Combined with the results from the other subelements of element 4, all of which had very low error in the results for k, the subelements within chorus element 4 trended to be more oblique over time: θk = 162° for subelement 41 and θk = 148° for subelements 43 and 44. That result is also consistent with the low-error results of k from element 3: θk = 163° for subelement 32 and θk = 154° for subelement 33. The key results from Table 1 are summarized in Figure 10. Figure 10a shows the parallel and perpendicular magnitudes and directions for k calculated from each element and subelement with error of |k| < 100% from Table 1. Note that the circle symbol at the end of each k plotted here is sized inversely proportional to the error estimate; that is, a large (small) circle at the end of the vector means the error was small (large). Results of k for full elements are plotted with solid lines and are color coded, while different types of dashed/dotted lines are used for the different subelements as described in the caption. Also plotted in Figure 10a are the unit vectors for the corresponding average Poynting vector (using asterisks at the ends of those vectors) for each element and B0, all normalized to a magnitude of 0.05 on this scale. This plot shows that for most of these chorus elements, the Poynting vector was more field aligned than the wave normal vector, with element 2 being the only exception. Elements 1 and 8 were not included here due to their large error estimates on the results for k. Also plotted in Figure 10 are comparisons between the wave normal angles from the crosscorrelation analysis and the polarization analysis (Figure 10b) and the differences between the azimuthal angle (i.e., in the perpendicular plane) of k and S calculated directly from the cross product of the waveform data (ΦS in Figure 10c) and from the FFT analysis (ΦSF in

MMS OBSERVATIONS OF CHORUS ELEMENTS

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Journal of Geophysical Research: Space Physics

MMS-1 MMS-2 MMS-3 MMS-4

B0 = [0.15, -0.80, 0.57] S = [-0.24, 0.81, -0.53] k = [0.19, 0.94, -0.30]

2 to 3: 2 to 1:

5

4 to 3:

0.98 1.69ms

B0

0.99 0.57ms

0.99 0.61ms

1 to 3: 0.99 1.10ms

0

k

-10 -40

-20

X-GSE [km]

S 0

0.99 0.49ms

20

0 -20

60 40 20

[km ]

1 to 4:

0.99 1.06ms

Figure 10d), which were calculated with independent algorithms for this study. Results for the wave normal angle compared well between the two methods, with the differences being less than ~15° for all of the elements and subelements with low estimated error. The azimuthal angles between S and k showed similar differences between the two analysis methods (comparing Figure 10c to 10d), as expected. However, there is an interesting implication to those differences: based on cold plasma theory, k and S can have different wave normal angles but should be coplanar with respect to B0 for right-handed whistler mode waves, meaning that the difference between Φk and ΦS should be either 0° or 180° (e.g., Taubenschuss et al., 2016). From both Figures 10c and 10d, it is clear that for most of the elements examined, the difference in the azimuthal angles was near neither 0° nor 180°, the implications for which will be discussed in the next section.

E

2 to 4:

GS

-5

Y-

Z-GSE [km]

10

10.1002/2017JA024474

6. Discussion

With the availability of multipoint observations from recent missions (e.g., Cluster, THEMIS, Van Allen Probes, and now MMS), the spatial Figure 9. Summary of the cross-correlation analysis results to calculate k for scales of chorus active regions and individual elements remain a topic chorus subelement 42. MMS positions are shown in three-dimensional space in the same format as in Figure 1c, though here in a rotated GSE coordinate form. of active research. Quantifying the distributions of these spatial scales is Overplotted are the B0, S, and k unit vectors, shown in cyan, yellow, and critically important for developing better models of how whistler mode magenta, respectively; those unit vector coordinates are also shown at the top of chorus waves interact with energetic (tens of keV to several MeV) electhe plot. Also printed on this plot are the maximum cross correlations and trons in Earth’s inner magnetosphere. MMS can further contribute to corresponding lag times between the various pairs of MMS spacecraft. For such studies by allowing for investigation of chorus elements with four example, for MMS-2 (red circle) to MMS-1 (black square), the maximum cross correlation between the six different FAC components of B wave and E wave was observatories separated at smaller scales than previously possible. In 0.99 using a lag time of 0.57 ms. This plot is intended as a way to visualize the this section, we will discuss some of the details and implications of results of the cross-correlation analysis shown in Table 1. our results concerning the phase coherency scale of individual chorus elements, the propagation of chorus elements at off-equatorial latitudes, chorus element substructure, and the validity of plane wave and cold plasma assumptions. First, concerning the phase coherency of individual chorus elements, the results presented in Table 1 show clearly that the eight chorus elements examined in detail here and their substructure were all phase coherent over spatial scales up to at least 38 km parallel to B0 and 58 km perpendicular to B0 (~70 km total separation distance). Note that the calculated wavelengths for these elements in the spacecraft frame were >100 km, which is larger than the interspacecraft separation and a condition for the multipoint analysis employed here. For the majority of chorus elements and subelements examined, the cross coherency of the wave electric and magnetic field components was >0.90 (>80% of cases) and often even >0.98 (~10%). To put these scales into context with the electrons that chorus waves interact with, a 10 keV electron mirroring locally at MMS had a gyroradius of ~2 km in the local field (B0 = 186 nT), while an electron at 10 keV mirroring at the magnetic equator on the same L shell (Beq = 41 nT) had a gyroradius of ~8 km. Those same numbers for a 100 keV electron were ~5 km in the local field and 23 km at the magnetic equator. Thus, these chorus elements were phase coherent over distances up to many times the size of ~1 keV to hundreds of keV (and even MeV) electron gyroradii, where the chorus elements were observed by MMS at
31° magnetic latitude. Assuming that the phase coherency scale did not somehow grow after the element was generated, this result was likely also true within the source region near the magnetic equator. With Van Allen Probes observations, Agapitov et al. (2017) found that the scale of individual lower band chorus elements in the direction perpendicular to B0 was ~800 km in the outer radiation belt. Using Cluster data, Santolik & Gurnett (2003) found that the amplitude coherency scale of lower band chorus elements within 1° of the magnetic equatorial plane and at L ~4.4 was ~100 km perpendicular to B0. Our results from MMS are consistent with both of those previous results, although it will be interesting in the future to test a different period when MMS were separated at greater distances to further test where in the hundreds of kilometer range the coherency is lost. From our calculations of S and k, the Poynting vector was more field aligned than the wave normal vector for most of the elements and subelements examined. In the plane perpendicular to B0, most of the elements TURNER ET AL.

MMS OBSERVATIONS OF CHORUS ELEMENTS

11,219

Journal of Geophysical Research: Space Physics b) 180 Element 2 Element 3 Element 4 Element 5 Element 6 Element 7

-0.02

160 140 120 100

-0.03

k

Pararllel [rad/km]

-0.01

)

0 B0

kF

a)

10.1002/2017JA024474

80 -0.04 -0.05 -0.06 0

60 40 S7

k7

20 0

0.005 0.01 0.015 0.02 0.025 0.03 0.035

21 22 2 32 33 3 41 42 43 44 4 5 6 7

Perpendicular [rad/km] 160

160

140

140

120

SF

S

)

d) 180

)

c) 180

100

Element (Sub)

120

k

k

100

80

80

60

60

40

40

20

20

0

21 22 2 32 33 3 41 42 43 44 4 5 6 7

0

21 22 2 32 33 3 41 42 43 44 4 5 6 7

Element (Sub)

Element (Sub)

Figure 10. (a) k and S vectors projected onto a two-dimensional perpendicular-parallel plane with respect to B0. This shows the various normal angles of k and S with respect to the background magnetic field (gray unit vector). S vectors are unit vectors normalized to a magnitude of 0.05 and are indicated with “asterisks” at their vector tips. The axes are not square, so a dashed black line at a magnitude of 0.05 has been added for clarity. k vectors are their true respective magnitudes in units of rad/km and are color coded by element, as shown in the legend. Elements and various subelements are shown with different line types: k calculated over the full element periods are shown with solid lines, while different types of dashed lines are used for subelement results. Circles at the vector tips further indicate k vectors, with the size of the circles being inversely proportional to the error estimated for each k. Bigger circles indicate smaller error. The S7 and k7 labels are intended for clarity. (b) Difference between the wave normal angles calculated using the two different analysis methods: cross-correlation analysis (θk) and the polarization and 0 FFT analyses (θkF). Results are plotted for each chorus element (circles with crosses) and subelement (plus signs) that had reasonably small error (see Table 1). (c) Difference between the azimuthal angles of k (Φk) and average of S (ΦS) from the cross product of B wave and E wave. (d) Difference between the azimuthal angles of k (Φk) and average of S from the FFT analysis (ΦSF).

with low error had wave normal components in the sunward and earthward directions, which is opposite of that expected from the statistics of Li et al. (2013), showing azimuthal components typically oriented away from Earth. However, ray tracing studies (e.g., Bortnik et al., 2011) have shown that chorus waves with earthward oriented azimuthal angles are more likely to propagate to higher latitudes since they experience less Landau damping as they propagate; that picture is consistent with our results. The evolution of k for subelements of elements 3 and 4 indicated that the subelements were becoming more oblique over time or that subelements at higher frequency were more oblique than those at lower frequency, consistent with Taubenschuss et al. (2014). This effect was previously observed by Van Allen Probes EMFISIS instruments, which recorded increasing instantaneous wave normal angles and decreasing amplitudes of wave subpackets during the evolution of intense chorus elements (Santolík et al., 2014). The magnitude and direction of k was similar for most of the elements and subelements with low error, implying that the elements must have had a similar origin location and similar plasma conditions along their path of propagation. Based on ray tracing theory (e.g., Bortnik et al., 2011; Breuillard et al., 2012), the wave normal angles of chorus waves should become more oblique at higher latitudes due to refraction along the path of propagation. From the eight elements examined in this event, the substructure of elements 2 through 7 was generally consistent at all four MMS spacecraft. For elements 1 and 8, however, there were larger differences between the

TURNER ET AL.

MMS OBSERVATIONS OF CHORUS ELEMENTS

11,220

Journal of Geophysical Research: Space Physics

10.1002/2017JA024474

wave amplitudes and substructure observed by the four spacecraft. For elements 2 through 7, the fact that the substructures were coherent implies that they were stable on time scales of at least a few milliseconds and over distances of at least tens of kilometers. The substructure of individual chorus elements is important for nonlinear wave-particle interactions between chorus waves and energetic electrons (e.g., Agapitov, Artemyev, et al., 2015; Albert, 2002; Artemyev et al., 2015; Bortnik, Thorne, & Inan, 2008; Mourenas et al., 2016; Osmane & Hamza, 2012; Tao et al., 2012). Any temporal stability of the substructure of these chorus elements ensures that energetic electrons can interact with and/or between chorus subelements over multiple gyroperiods. The scale of chorus active regions and individual elements is also critical to understanding the spatial scales of “microburst” electron precipitation events observed by spacecraft in low-Earth orbit (e.g., Anderson et al., 2017; Blake & O’Brien, 2016; Crew et al., 2016; Lorentzen et al., 2001; O’Brien, Looper, & Blake, 2004; Osmane et al., 2016). The assumption that these chorus elements and subelements had planar phase fronts (i.e., the plane wave approximation) was valid for elements 2 through 7 and the majority of their subelements that were analyzed. This validity of the plane wave approximation was evident from the low error estimates of k for these elements, calculated from the various triplets of spacecraft pairs, as well as the general agreement of the lag times between various pairs of spacecraft. However, elements 1 and 8 revealed very large errors in the estimate of k and lag times that were largely inconsistent between different pairs of spacecraft. Upon closer investigation of the phase coherency of elements 1 and 8, the waveforms were well structured at all four spacecraft, and the waves’ various electric and magnetic field components were apparently in phase. One possible reason for the disagreement in k calculated for these elements is that the plane wave assumption was not valid for elements 1 and 8. That result has interesting and important implications for future studies. First, it implies that variations in the plasma conditions between the source region and the spacecraft were significantly evolving on time scales