Journal of Geophysical Research: Space Physics

Electron Phase-Space Holes in Three Dimensions:Multispacecraft Observations byMagnetospheric Multiscale

J. C. Holmes1,2 , R. E. Ergun1,2 , D. L. Newman1,3 , N. Ahmadi1 , L. Andersson1 ,O. Le Contel4 , R. B. Torbert5 , B. L. Giles6 , R. J. Strangeway7 , and J. L. Burch8

1Laboratory for Atmospheric and Space Physics, Boulder, CO, USA, 2Department of Astrophysical and Planetary Sciences,University of Colorado Boulder, Boulder, CO, USA, 3Department of Physics, University of Colorado Boulder, Boulder, CO,USA, 4Laboratoire de Physique des Plasmas, CNRS/Ecole Polytechnique/Sorbonne Université/Université de ParisSud/Observatoire de Paris, Paris, France, 5Department of Physics, University of New Hampshire, Durham, NH, USA, 6NASAGoddard Space Flight Center, Greenbelt, MD, USA, 7Department of Earth, Planetary, and Space Sciences, University ofCalifornia, Los Angeles, CA, USA, 8Southwest Research Institute, San Antonio, TX, USA

Abstract Electron phase-space holes are kinetic plasma structures commonly observed in spaceplasmas on Debye length scales. Near the Earth’s duskside flank at 10 Earth radii, a series of 32 electron holes(EHs) are detected within a 1-s window on all four Magnetospheric Multiscale spacecraft. The spacecraftseparation of <7 km is similar to the expected EH size in this region. Length, width, amplitude, and relativepositions are determined for individual EHs using a cylindrically symmetric model fit to MagnetosphericMultiscale E field measurements. The model shows good agreement with observed E fields far from the EHcenter. Deviations in E⟂ from the model are present near the center, indicating observed EHs have complex,sometimes irregular, internal structure. Perturbation magnetic fields 𝛿B modeled assuming an E × B0

electron current reproduce the measured parallel perturbation in most cases, although there is a systematicvariation due to geometric and finite gyroradius effects. Many EHs in this event have large amplitude fortheir size, reaching the theoretical lower limit in length parallel to the background magnetic field, whichrequires the electron phase-space density to approach 0 in the center. It is possible that EHs of this type haverecently formed, eventually weakening or becoming longer over time. This study provides the most detailedmeasurements of EHs to date. Their derived properties are largely in agreement with expectations fromprevious research. It remains unclear whether the few notable differences are due to rapid time evolution orare specific to the local environment.

1. Introduction

Electron phase-space holes are a prominent member of a family of coherent, kinetic-scale plasma structuresknown as electrostatic solitary waves. They are characterized by a bipolar electric field parallel to the magneticfield with a positive electric potential, caused by a self-consistent decrease in density of electrons trappedin that potential. Electron holes (EHs) typically form from unstable counterstreaming electron beams or therelated bump-on-tail instability. Both of these mechanisms are a first step in thermalizing an unstable elec-tron distribution, often in the context of turbulence, mixing between two different plasma populations, oran accelerated beam driven by, for example, a plasma double layer (Ergun et al., 2003, 2004; Newman et al.,2001), magnetic reconnection (Drake et al., 2003; Egedal et al., 2015; Lapenta et al., 2011), or even relativisticplasmas (Eliasson & Shukla, 2005a, 2005b) such as astrophysical jets (Broderick et al., 2012).

Once created, phase-space holes can have an impact on their surrounding environment. They have beenproposed as particle accelerators, either by direct wave-particle interaction (Mozer et al., 2016) or as a seriesof small asymmetric potentials adding up to a large energy (McFadden et al., 2003; Temerin et al., 1982). Theinfluence of EHs is a collective kinetic process whose large-scale effects may be dependent on the detailedmicrophysics involved, particularly in the electromagnetic case.

Early models of phase-space holes are primarily electrostatic (Bernstein et al., 1957; Muschietti et al., 1999,2002; Schamel, 1982), focusing on structure parallel to the direction of flow. In the magnetized case,

RESEARCH ARTICLE10.1029/2018JA025750

Key Points:• For the first time, electron

phase-space hole measurements areunambiguously correlated across fourclosely spaced spacecraft

• Perpendicular electric fields can beunexpectedly weak inside electronholes, implying irregular interiorstructure

• Estimates of interior phase-spacedensity approach 0 in many cases,possibly representing an early stageof electron hole evolution

Correspondence to:J. C. Holmes,justin.holmes@lasp.colorado.edu

Citation:Holmes, J. C., Ergun, R. E.,Newman, D. L., Ahmadi, N.,Andersson, L., LeContel, O., et al.(2018). Electron phase-space holesin three dimensions: Multispacecraftobservations by MagnetosphericMultiscale. Journal of GeophysicalResearch: Space Physics,123, 9963–9978.https://doi.org/10.1029/2018JA025750

Received 1 JUN 2018

Accepted 30 OCT 2018

Accepted article online 11 NOV 2018

Published online 4 DEC 2018

©2018. American Geophysical Union.All Rights Reserved.

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EH velocity is expected primarily parallel to the background magnetic field B0. Analytic studies and simula-tions in 1-D and 2-D magnetized plasmas create remarkably stable structures, which can last for many plasmaperiods (Miyake et al., 1998; Omura et al., 1996; Oppenheim et al., 1999), rapidly traveling up and down fieldlines. This feature led to several theories on their ability to transport energy over long distances, heat plasmas(Miyake et al., 1998; Vasko et al., 2016), facilitate plasma mixing, or even contribute to coherent radio emission(Goodrich & Ergun, 2015; Treumann et al., 2011, 2012).

Observations of perpendicular electric fields attributed to phase-space holes have been made in the auroralregion (Ergun et al., 1998; Singh et al., 2011) and magnetotail (Cattell et al., 2005; Omura et al., 1999), appearingas unipolar spikes indicative of ellipsoidal 3-D structure. Simulations have shown that the perpendicular shapeof EHs varies depending on many factors, including magnetization (Lu et al., 2008; Oppenheim et al., 1999,2001; Wu et al., 2010, 2011). Laboratory experiments have also resolved the perpendicular electric field (Foxet al., 2012), indicating that 3-D EHs are scalable to very high densities.

In order to calculate EH velocity, single-spacecraft measurements have previously used probe-to-probeinterferometry (Cattell et al., 2003; Graham et al., 2015) or inferred a velocity from the relativistic Lorentztransformation of the hole’s electromagnetic field (Andersson et al., 2009; Tao et al., 2011). Electromagnetic,three-dimensional models for EHs (Andersson et al., 2009; Chen et al., 2004; Tao et al., 2011; Treumann &Baumjohann, 2012) also predict a dipole-like perturbation magnetic field created by electrons drifting aroundthe hole due to E × B0 motion. Tao et al. (2011) used the perpendicular magnetic field to determine the holevelocity but could not completely separate the two magnetic field sources. In cases with strong ΔE⟂, the rel-ativistic contribution dominated, providing accurate velocities, with the residual magnetic field serving as anestimate of the dipole component. A recent study in the magnetotail (Le Contel et al., 2017) captured an elec-tromagnetic phase-space hole on three Magnetospheric Multiscale (MMS) spacecraft, computing the velocityusing both time delays and the relativistic component. They found that the relativistic magnetic signature cansignificantly overestimate the velocity when the rest frame magnetic perturbation is neglected.

Studies of electron phase-space hole shape are largely consistent for any number of spacecraft. Franz et al.(2005) performed a statistical study for phase-space holes using Polar data with a large number of EHs(N ∼ 3, 000). This study was able to neglect the relativistic contribution since any resolved solitary waveswere moving at under ∼ 3, 000 km/s, and the electric field was too weak to produce a measurable induced Bfield. Based on the relative strength of the perpendicular and parallel electric field components, most struc-tures were found to be oblate and passing by the spacecraft to one side. Simultaneous, in situ observationsfrom multiple spacecraft are an excellent tool for pursuing a detailed understanding of three-dimensionalphase-space hole dynamics (Pickett et al., 2004). Previous measurements of solitary waves on two spacecraft(Holmes et al., 2018; Norgren et al., 2015; Pickett et al., 2008) show improvement in determining individualstructure properties such as perpendicular distance to the structure center. More points of reference reduceuncertainty in which signatures correlate with one another, particularly for closely spaced solitary waves. In afour-satellite formation, EHs detected on three spacecraft and a nondetection on the fourth have been usedto limit the perpendicular scale (Le Contel et al., 2017). A well-resolved detection on all four spacecraft wouldbe able to determine EH size and shape with little ambiguity.

This paper addresses the three-dimensional structure of electron phase-space holes as observed by the MMSmission (Burch et al., 2016) for a 1-s time period in Earth’s magnetosphere. All four MMS spacecraft measurea series of ≈ 50 electromagnetic structures with high time resolution. Electric fields are measured using a setof spin plane and axial double probes (Ergun et al., 2016; Lindqvist et al., 2016), and high-frequency magneticfield data are gathered from a search coil magnetometer instrument (Le Contel et al., 2016). The MMS forma-tion is on a scale similar to the EH size, allowing for an accurate determination of velocity using the parallelelectric field and time delays between spacecraft. By converting time series data to spatial coordinates, theelectric and magnetic fields are recreated from a 3-D model including perpendicular and parallel size. Mod-eled EH properties are then compared on a statistical basis with theories of hole structure such as Franz et al.(2000) and Chen and Parks (2002).

The remainder of this paper is organized as follows: Section 2 presents the observations of EHs streamingpast all four MMS spacecraft and the time delay analysis used to determine their velocities. In section 3, the3-D model for electromagnetic EHs is derived. Section 4 describes the regression used to fit the model to

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Figure 1. Data from MMS1 over roughly 30 min detailing an encounter with a patch of plasma. Where applicable, alldata are in GSE coordinates. The dashed vertical line marks the time of electron hole observations: 27 September 2017,01:18:21 UTC. (a) Ion energy spectrogram. (b) Electron energy spectrogram. (c) Magnetic field. (d) Ion velocity. (e) Electricfield. (f ) Map of MMS location relative to Earth at the time of electron hole observations. An estimate of themagnetopause location is shown in the upper right with colors representing ±𝜎 boundaries (Shue et al., 1998). (g) MMStetrahedron formation. GSE = geocentric solar ecliptic; MMS = Magnetospheric Multiscale.

data and presents a pair of example fits with different properties. Section 5 contains a statistical analysis forselected EHs, finding agreement with analytic predictions and previous studies. Concluding remarks are madein section 6.

2. MMS ObservationsOn 27 September 2016 at 01:18:21 UTC, MMS encountered a patch of plasma moving through Earth’s magne-tosphere on the duskside flank. Figure 1 shows observations by MMS1 of this event. Plasma measurements ofcold ions in Figure 1a and electrons in Figure 1b prior to encountering the patch are characteristic of mange-tospheric plasma, whose estimated location is shown in Figure 1f. At roughly 01:20 UTC, a magnetic fieldrotation is seen in Bx (Figure 1c), accompanied by a sudden increase in density and a transition from weak tail-ward flow to fast sunward ion velocity (Figure 1d). A strong burst of nonlinear electrostatic waves (Figure 1e)is also observed inside the patch. It is likely a structure, which became detached from the magnetopause andis diffusing into the magnetosphere. Mixing of plasma populations along the edge of this patch could lead tothe formation of electrostatic waves observed on the dashed line at 01:18:21 UTC.

At this time, all four MMS spacecraft flying in a close tetrahedron formation (Figure 1g) observed a series ofelectron phase-space holes with parallel electric fields over 100 mV/m (shown for MMS1 in Figure 2a). Thespacecraft potential in Figure 2b varies along with the electric fields, indicating strong variation in local elec-tron density. Figures 2c–2e show the reduced, smoothed electron distribution functions with an enhancedcomponent traveling antiparallel to B0 at approximately −10, 000 km/s. Figure 3 shows a 10-ms sample ofthis event, which spans 700 ms in total. Two electromagnetic structures labeled Example 1 and Example 2 arefound within this time range, to be referenced in later sections. The three components of the high-frequencyelectric field E measured by all MMS spacecraft are contained in Figures 3a–3c. The magnetic perturbationdBin Figures 3d–3f is from the alternating current coupled (AC-coupled) search coil magnetometer data(8,192 samples per second, with 0.15-pT resolution at 1 kHz) and high-pass filtered above 100 Hz. All field dataare shown in field-aligned coordinates with the first perpendicular axis “⟂1” in the B0× Y geocentric solar eclip-tic coordinates direction and the final axis “⟂2” completing a right-handed coordinate system. In this region,

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Figure 2. MMS1 data for the entire range sampled (700 ms). Plot (a) shows high time resolution E∥. MMS encounters a sudden burst of electron holes, which laterbecome less well-defined nonlinear waves. The spacecraft potential (plot [b], 8-kS/s resolution) increases sharply with each passing EH, indicating a significantdecrease in electron density. In some instances, a small dip at the peak of these potential spikes is visible. Plots (c)–(e) are reduced, 30-ms electron distributionstaken in 30-ms time intervals beginning 21.1277, 21.2177, and 21.3977 s after 27 September 2016 01:18. They show relatively strong drifting electron featureswith velocities similar to the electron holes. GSE = geocentric solar ecliptic; MMS = Magnetospheric Multiscale; EDP = Electric field Double Probe; SC = spacecraft.

the magnitude of the background magnetic field B0 is 63 nT distributed nearly evenly in the (−X, +Y , and +Z)geocentric solar ecliptic coordinates direction with negligible variation (such that B0 × Y is well defined).

Each EH is well separated and highly correlated between spacecraft in E∥, though the direction of the perpen-dicular component varies significantly, suggesting each hole has a radially confined structure with its centernearby or passing through the MMS tetrahedron formation. Each detection is roughly 1 ms in duration andhas a maximum time delay between any two spacecraft of 2 ms. There is a small but significant magnetic fieldcomponent for many of these structures, also with varying amplitude and direction.

Based on the distribution functions of Figure 2 and intervening data (not shown), electron density and tem-perature average roughly ne = 0.1 cm−3 and Te = 400 eV, respectively, varying by up to a factor of 2 due tolow particle count. The derived plasma frequency 𝜔pe = 1.78 × 104 rad/s is close to the cyclotron frequencyΩce = 1.1× 104 rad/s, and the electron Debye length 𝜆D,e = 0.47 km. Electron phase-space holes are typicallyno larger than 10𝜆D,e (Franz et al., 2005, and references therein), roughly the same scale as the MMS formation,favorable conditions for studying the 3-D structure of moderately magnetized EHs.

2.1. Data SelectionTo closely study individual EH properties, a simple algorithm is used to identify EHs and divide the time seriesdata. Parameter choice is primarily motivated by limiting selections to single EHs and discounting other typesof structures.

1. The minimum and maximum E∥ must fall within a window of no less than 0.1 ms and no more than 2 ms oneach spacecraft.

2. The range of E∥ must be greater than 30 mV/m on each spacecraft.

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Figure 3. Zoomed-in sample of alternating current coupled electric field (a–c) and magnetic field (d–f ) data inmagnetic field-aligned coordinates for all MMS spacecraft. ⟂1 is aligned with ‚B0 × ‚Y geocentric solar eclipticcoordinates, and ⟂2 completes a right-handed coordinate system. Electron hole waveforms are consistently delayedfrom MMS2 and MMS3 to MMS4 and MMS1. MMS = Magnetospheric Multiscale EDP = Electric field Double Probe; SCM= Search Coil Magnetometer.

3. Conditions #1 and #2 must be satisfied on all four spacecraft within the same 4-ms window centered on themidpoint between the minimum and maximum E∥ on MMS1.

4. Structures matching previous items, which have error in model fit parameters Φ0, L∥, or L⟂ consistent with0 (within 3𝜎) as determined in section 4 are retroactively excluded.

There are several biases associated with this selection. Specifying a time window (condition #3) is beneficialto the time delay analysis in the next section, since it defines an exact window for cross correlation. In com-bination with condition #1, structures with low velocities or abnormally large widths will be overlooked. Thesecond excludes weak holes, which cannot be accurately fit due to background fluctuations. Requiring allspacecraft to meet this condition excludes a number of holes, which pass far to the side of the formation such

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Figure 4. Histogram of parallel velocities for 32 electron holes. Thedistribution has an average of −9,650 km/s and a standard deviation of2,440 km/s.

that the farthest spacecraft has a weak signal. The final item biases theresults toward clean, cylindrically symmetric structures. Of 47 detectionssatisfying the first three conditions, 15 are excluded by condition #4. Thereare three cases of plausible EHs, which are poorly fit, three instances ofmultiple EHs found within the same window that cannot be reliably fit orcross-correlated, and nine instances of nonideal E∥ profiles on at least onespacecraft that are inappropriate for this study.

2.2. Time Delay AnalysisFor the selected time period, MMS spacecraft are arranged in a tetra-hedron formation with less than 7-km separation. The delay betweenmeasurements is determined by cross-correlating E∥ of each hole fromone spacecraft to the next with lags ranging between ±2 ms, choosingthe delay with the highest correlation coefficient. This yields a total of sixΔt per hole, corresponding to the six vector offsets between each pair ofspacecraft.

EHs are expected to move primarily along magnetic field lines, orientedalong an axis of symmetry z parallel to B0 and the phase velocity v. There-fore, the EH velocity v ≈ v∥z. For each structure, v∥ (Figure 4) is determinedusing a least squares linear fit to the six time delays Δtij and parallel dis-tances Δzij between spacecraft (numbered by subscripts i and j), with

weighting factor 1∕√|Δzij| such that large separations with lower relative uncertainty in time delay are

favored by the fit. In cases where alignment along z is not satisfied, measured time delays are fit poorly by alinear model and uncertainty in v∥ increases appropriately.

Calculated uncertainties in velocity of ∼10% are consistent with instrumental timing errors. Clock drift of theultrastable oscillators onboard MMS is expected to be 216 μs per 65 hr, with upkeep performed near perigeeroughly every 24 hr during this phase of the mission (Tooley et al., 2016). Relative timing will at most be offsetby 50 μs at apogee for each spacecraft—under 10% for a 1-ms delay.

Perpendicular motion of B0-aligned EHs is independent of the parallel velocity and assumed to be relativelysmall. Testing of randomly generated, simulated data (not shown) using the above technique confirms thatv⟂ is not accurate for holes oriented along B0. Simulated EHs with v⟂ > 1, 000 km/s do not significantly impactthe determination of v∥, which is accurate to within 10% with or without perpendicular drift. In principle, v⟂could be estimated by including it as a free parameter in the fits of section 4, which would account for possible

Figure 5. Diagram showing the 3-D structure of an electron phase-spacehole, including the relativistic contribution 𝛿Brel due to the EH’s motion.The magnetic field in the center-bottom image is only due to the electrondrift current, which connects into a ring above and below the page. Aspacecraft passing to the side will observe a rotating perpendicular B fieldcoplanar with the spacecraft position and electron hole velocity vector, plusa unipolar out of plane feature due to a change of reference frame(illustrated on the right).

EH drift and tilt at the cost of added complexity in the fit. For the sakeof simplicity, reliability, and consistency with most previous studies, weassume v⟂ = 0.

3. Analytic Model

To fit MMS data, we adopt a model (schematic, Figure 5) derived froma cylindrically symmetric Gaussian potential, given by equation (1). Theprimary axis (z) is chosen parallel to B0. This particular model has theadvantages of straightforward integration and distinct parallel and per-pendicular length scales and has been successfully applied to both theoryand observations in several instances (Andersson et al., 2009; Chen et al.,2004, 2005; Norgren et al., 2015; Tao et al., 2011), so those results can becompared easily.

Φ(r, z) = Φ0e− r2

2L2⟂ e

− z2

2L2∥ (1)

Taking the gradient of Φ, the electric field in the hole frame is

E = Φ(r, z) rL2⟂

r + Φ(r, z) zL2∥

z, (2)

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where Φ0 is the strength of the phase-space hole in volts. Width parameters L∥ and L⟂, where the electricfield peaks on their respective axes, are in units of kilometers.

We calculate a perturbation magnetic field from the modeled electric field due to two effects. The first comesfrom the assumption that electrons are subject to an E×B0 drift in and around the hole, while the ions are rela-tively stationary. This will create an azimuthal current J perpendicular to B0, and a resulting induced magneticfield 𝛿B′ given by the Biot-Savart law (equation (3); Jackson, 1998; Tao et al., 2011), which must be computednumerically for an arbitrary position, such as the path of a spacecraft through an EH. For a cylindrically sym-metric model with v⟂ = 0, J is entirely in the �� direction (equation (4)). The magnetic field is calculated byintegrating over a grid of current loops embedded in a 2-D plane. The final 𝛿B′ vector is found by a rotationback to a common coordinate system.

𝛿B′(x) =𝜇0

4π ∫ J(x′) × x − x′

|x − x′|3d3x′ (3)

In the case of this specific model, J is

J(r, z) =−eneΦ0

B0

rL2⟂

e− r2

2L2⟂ e

− z2

2L2∥ ��. (4)

One caveat of this model is that it assumes J is at its ideal, static value at all points within the hole with uniformelectron density. However, all electrons entering the hole do not contribute to E×B0 equally. If they have a highenough velocity compared to the size of the hole, their crossing time t will be shorter than the time needed tofully accelerate and will not produce the predicted current. Tao et al. (2011) used particle simulations to testthe assumption that electrons are able to contribute to the E × B0 drift current given that they only spend ashort period of time within the EH. They found that for transit times> 1 tc, where tc is the electron gyroperiod1∕fce, the theoretical maximum current is an appropriate approximation for the drift-induced current. For thisevent, the gyroperiod of 0.56 ms is comparable to the average crossing time of 0.35 ms (computed from thefit parameters in section 4), so our current density model may not be accurate in all cases. Norgren et al.(2015) also used particle simulations to test finite gyroradius (𝜌e) effects on the magnetic field, finding thatfor L⟂∕𝜌e ≈ 2 the 𝛿B∥ was reduced by up to a factor of 2. We will later show that some combination of theseeffects is nonnegligible away from the center of the phase-space hole in the perpendicular direction.

The other source of perturbation magnetic field is the relativistic transformation of the phase-space hole. Inthe spacecraft frame, the structures in this study are traveling up to 5% the speed of light (see Figure 4). Thetotal EH magnetic field including this effect is found using a Lorentz transformation:

𝛿B = 𝛾

(𝛿B′ − 1

c2v × E

), (5)

where 𝛿B′ is the magnetic field calculated from equation (3); v and E are both measured quantities infield-aligned coordinates. We have neglected that some portion of the measured E in the perpendicular direc-tion comes from the relativistic transformation of the hole’s current-induced magnetic field 𝛿B′. For this event,the relativistic contribution to E⟂ is on the order of 10 μV/m and can be safely disregarded. Additionally, at thevelocities considered here (< 0.05c), 𝛾 = 1∕

√1 − v2∕c2 is close to 1.

4. Model Fitting Results

In order to determine EH properties, we take the straightforward approach of directly fitting equation (2) toelectric field data. Velocities found in section 2.2 are first used to convert the time domain measurementsto spacecraft-relative positions. Since the model is ellipsoidal, the r and z components of the electric fieldare coupled in r and z. When performing a fit, the vector components cannot be considered independentlyof any variable. All model parameters depend on the full-position coordinates. Since the spacecraft are notnecessarily aligned radially, we return to a Cartesian basis by setting r2 = x2 + y2. Additionally, we allow fora position offset (x → x − x0, y → y − y0, z → z − z0), where x0, y0, and z0 represent the center position ofthe phase-space hole in each coordinate relative to MMS1 positioned at the origin. In total, there are six freeparameters x0, y0, z0, L∥, L⟂, and Φ0. Each phase-space hole selection covers 4 ms or 263 data points perspacecraft per component for a total of 3,156 degrees of freedom.

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Figure 6. Model fitting results for electron hole example 1. MMS data (solid black lines) have been converted to spatialfield-aligned coordinates using the velocity computed in section 2.2. Fits to the E field are red dashed lines. The dashedred lines in 𝛿B plots are a direct calculation of equations (3)–(5) using the E field fit. Each curve contains 263 data points,with 𝛿B interpolated up from 33 points. MMS1 timestamp: 27 September 2016/01:18:21.212. MMS = MagnetosphericMultiscale.

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Figure 7. Model fitting results for electron hole example 2. MMS data (solid black lines) have been converted to spatialfield-aligned coordinates using the velocity computed in section 2.2. Fits to the E field are red dashed lines. The dashedred lines in 𝛿B plots are a direct calculation of equations (3)–(5) using the E field fit. Each curve contains 263 data points,with 𝛿B interpolated up from 33 points. MMS1 timestamp: 27 September/01:18:21.216. MMS = MagnetosphericMultiscale.

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Figure 8. Smoothed slice through the perpendicular plane in the center of example 1. Vectors show the perpendicular electric field averaged over a smallwindow, taken at the tail of the vector, and scaled to MMS1 values (black, centered). The gradient is the model fit perpendicular electric field in millivolts permeter. Generally, vectors in darker areas should be longer than those in lighter areas. MMS1 timestamp: 27 September 2016/01:18:21.212.MMS = Magnetospheric Multiscale.

All fits are performed using a nonlinear least squares fitting program (Markwardt, 2009; Marquardt, 1963;Moré, 1978; Levenberg, 1944). Error bars for each fit parameter are determined by scaling the returned sigmavalues to the 𝜒2 per degree of freedom, which for a good fit is a reasonable representation of uncertainty.

We emphasize that these fits are performed on all components of E on all four spacecraft at once. Hence, themodel is geometrically constrained. In some cases, it is impossible to successfully fit all the components, suchas when the perpendicular field does not point radially or varies dramatically in amplitude. While some degreeof deviation is expected, exceptionally poor fits are excluded by selection criterion #4 defined in section 2.1.

4.1. Example FitsTwo examples from this time period are shown in Figures 6–9. Figures 6 and 7 contain plots of electric andmagnetic field data for a 4-ms time period converted to spatial, field-aligned coordinates. Solid curves areMMS data, while red, dashed curves are fits or model results. The magnetic perturbation (𝛿B) model in theright column (dashed red) is derived from the electric field fit in the left column (also dashed red). Curves in𝛿B have been interpolated up to the electric field resolution.

A more intuitive representation of E⟂ is shown in Figures 8 and 9. The vectors show measured E⟂ in a planebisecting the hole, scaled to the measurement on MMS1, and averaged in the parallel direction from−1 < z <1 km. Spacecraft are located at the tail of each arrow in the perpendicular plane (z = 0). Modeled magnitude

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Figure 9. Smoothed slice through the perpendicular plane in the center ofexample 2. Vectors show the perpendicular electric field averaged over asmall window, taken at the tail of the vector and scaled to MMS1 values(black, centered). The gradient is the model fit perpendicular electric field inmillivolts per meter. Generally, vectors in darker areas should be longer thanthose in lighter areas. MMS1 timestamp: 27 September 2016/01:18:21.216.MMS = Magnetospheric Multiscale.

of E⟂ at z = 0 is plotted as a density gradient. In both cases, the arrowspoint away from the hole center and have amplitudes in rough agreementwith the model.

Electric field fits overall provide a good estimate of electron phase-spacehole properties. Parallel electric fields E∥ are fit well by the model, with aslightly wider profile than the observations. Uncertainty in L∥ is dominatedby the velocity error of ∼ 10%. However, there are significant deviationsfrom the model in E⟂. In a number of instances, the unipolar Gaussianprofile does not fit the general shape of the data. One reason could beunaccounted for perpendicular velocity or EH tilt. In that case, E⟂ wouldbe stronger from one side to the other or possibly reverse. Though E⟂ ismostly unipolar in example 2, there is a hint of reversal on MMS2. In exam-ple 1, E⟂2 on MMS3 and MMS4 has a dimple shape (labeled bottom left ofFigure 6) where the field becomes weaker in the center, which requires adifferent explanation.

A perpendicular dimple cannot be accounted for by the chosen model,since on a straight path E⟂ explicitly falls off with distance. Several EHsin this data set contain flattening or dimpling, most often on spacecraftpassing near the hole center. EHs may be more flat than expected or haveirregular interior structure. For instance, a surplus of trapped electrons inthe center of the hole could explain a dimpled E⟂, similar to structures pro-posed by Muschietti et al. (2002). However, a significant flattening of theparallel field is also required, which is not observed here.

Another possible explanation for flattened E⟂ is resonant bunching of electrons trapped in the phase-spacehole potential. Given the model of equation (1), trapped electrons bounce parallel to B0 within the electro-

static potential of the hole at the bounce frequency𝜔B =√

eΦ0∕meL2∥. They also are subject to perpendicular

motion at the cyclotron frequency Ωce = 1.1 × 104 rad/s and a position-dependent E × B0 drift frequency𝜔d ∝ Φ0e−r2∕B0 about the hole center. If resonant bunching can occur, the radial dependence of 𝜔d couldlead to electrons gathering in a ring or cylinder within the EH, creating the increased electron density requiredto match observations.

Resonance typically requires matching at least two of the relevant frequencies in integer multiples. Choosingaverage EH size parameters L∥ = 3.5 km and L⟂ = 4.0 km, we can solve for the potential Φ0 where 𝜔d is res-onant with Ωce or 𝜔B in the limit of r → 0 at z = 0. Cyclotron resonance occurs for Φ0 = 11.2 kV and bounceresonance at Φ0 = 14.6 kV. For resonance at 2𝜔d these values change to 5.6 and 3.7 kV, respectively. The max-imum Φ0 found in this study is below 1.3 kV. For most EHs in this study, resonant bunching of this type is notlikely. For smaller-sized EHs (L ≈ 2 km), the required potential decreases to values consistent with observa-tions. However, dimpled E⟂ is observed in both large and small EHs. Rather than assuming average (thermal)electron parameters, a small subset of low-energy, high-pitch angle electrons can satisfy the resonance con-ditions or have chaotic orbits capable of particle bunching. There are many possibilities, which may require3-D particle tracking or even self-consistent particle-in-cell (PIC) simulations to determine if and how electronbunching within an EH potential might occur.

Despite interior structure, fits in E⟂ outside the EHs reasonably match the measured width and ampli-tude. It is likely that L⟂ is a reasonable estimate of phase-space hole width, although a triaxial model maybe more appropriate. It is also possible that we are only estimating the average width of a more amor-phous or time-varying structure. This is plausible for holes with perpendicular size larger than the electrongyroradius—in this case about 3 versus 0.75 km, respectively.

So far, only the electric field has been considered. The EH magnetic field was computed at each spacecraftposition by numerically integrating equation (3) using the current J (equation (4)) determined by the fit param-eters and adding the relativistic contribution of equation (5) using the model E as an input. The 𝛿B model(dashed red lines) is plotted along with data (solid black line) in the right column of Figures 6 and 7. Since thisis not a direct fit, some discrepancies between the two are not surprising.

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Figure 10. Comparison between maximum observed 𝛿B∥ and the model foreach spacecraft and electron hole. The solid line is plotted for equalitybetween the model and observations, and the dashed line is a linear fit. Onaverage, the model is within 0.01 nT of the data. Observed values are offsetby an effective noise floor, while the model is able to predict very low fields.At higher values, 𝛿B∥ is smaller than predicted.

A number of effects should be noted when analyzing the magnetic sig-nature. First, the magnetometer data contain undersampled fluctuationson the order of 0.01 nT near the upper hybrid frequency. These couldbe the result of weak electromagnetic waves and effectively serve as anoise floor for determining the EH 𝛿B. AC-coupled measurements alsohave ambiguity in offset and slope for short time ranges, making a precisedetermination of 𝛿B difficult.

For EH velocity parallel to B0, the modeled 𝛿B∥ is only determined bythe drift current within the hole. The shape is generally unipolar in thedirection of the background B field but can be antiparallel at larger r, asillustrated in Figure 5. The amplitude of 𝛿B∥ matches well in the examplesshown, but on inspection of all sampled EHs, there is a systematic differ-ence. Figure 10 shows the peak modeled 𝛿B∥ compared to the observedmaximum 𝛿B∥ for every EH on each spacecraft. At the low end, there is anoise floor of ≈ 0.005 nT in the observations, causing an offset comparedto the model. The overestimate at high 𝛿B∥ is significantly larger than theuncertainty.

The most plausible explanation for overestimating 𝛿B∥ is that theconstant-density assumption of equation (4) is incorrect. As shown in thenext section, many of the sample EHs have an electron density in the cen-ter much lower than the background. EHs with the largest modeled 𝛿B∥typically have the largest potential and size and are most susceptible to anerror in density.

Even in cases where the peak 𝛿B∥ matches, the wings of the modeled 𝛿B∥are consistently too broad. This is likely due to a combination of finite gyro-

radius effects and EH size. At |z| and r> 0, both the perpendicular and parallel extent of an EH are smallerthan in the center. Therefore, the crossing time t decreases with r, and the perpendicular space within theEH decreases with |z|. As both of these values approach tc and 𝜌e, the particle simulations of both Tao et al.(2011) and Norgren et al. (2015) predict a falloff of current, which becomes significant when t∕tc < 0.7, andL⟂∕𝜌e < 2. This additional effect is dominated by larger guiding center current loops, whose magnetic fieldfalls off with parallel distance less than smaller ones. As a result, the peak in 𝛿B∥ is slightly sharper than in themodel.

The modeled perpendicular components of B, particularly 𝛿B⟂, do not reproduce the data well. Qualitatively,when the cross terms of E⟂ are small, the corresponding magnetic field component is bipolar—dominatedby the field driven by the electron drift current. For example, in Figure 7, MMS3 E⟂1 is weak, so the MMS3 𝛿B⟂2

component does not have a strong unipolar feature. Comparing E⟂2 and 𝛿B⟂1 shows the opposite: a unipolarfeature in both components, indicating that the relativistic contribution is significant. The model regularlypredicts a weaker B⟂ than the measurements, even when the (sometimes stronger) measured E⟂ is used inthe calculation instead of the model (not shown).

5. Statistical Analysis

Electron phase-space hole parameters determined using the 3-D model fit can be compared with theoreticalexpectations. A number of theories exist relating EH size, speed, and potential. For instance, based on thegyroradius of electrons trapped within the EH potential, the length-to-width ratio is predicted by Franz et al.(2000) to roughly follow the relation given by equation (6):

L∥∕L⟂ ≈ [1 + 𝜔2pe∕Ω

2ce]

−1∕2. (6)

A Bernstein-Greene-Kruskal model using the potential of equation (1) can be used to derive an inequality(equation (7)) satisfying the condition of a nonzero electron distribution within the EH center (Chen et al.,2005). This model assumes an unmagnetized plasma with no ion contribution and a Maxwellian velocity dis-tribution at zero potential. Dimensionless quantities are introduced for potential 𝜓 = eΦ0

Teand geometric

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Figure 11. Fit parameter results for selected electron holes. By default, curves are calculated using a temperature Te = 400 eV and density ne = 0.1 cm−3.(a) Comparison between L⟂ and L∥ . The dashed line is the expected ratio from equation (6). The red solid curve is a lower limit assuming the median potential.The leftmost black limit is the most strict, assuming the minimum observed potential. (b) Comparison between Φ0 and L∥ . The lowest, red curve uses theminimum observed L⟂ and a temperature of Te = 400 eV. The leftmost black and blue dash-dot curves take the limit of 𝛿r → ∞ for temperatures of 400 and600 eV, respectively. Roughly half the points lie near the most stringent limit, indicating many EHs have very low density trapped populations. (c) Comparison

between Φ0 and L⟂. The curve follows the minimum allowed 𝛿r =√

4e−𝜓√𝜓∕[1 − erf (

√𝜓)], for which 𝛿z is undefined.

parameters 𝛿r = L⟂∕𝜆De and 𝛿z = L∥∕𝜆De. Units of energy w = mev2∕2 − eΦ(z, r) are defined such that elec-trons with energy w < 0 are trapped within the EH potential. A more complete derivation of equation (7) isgiven in the appendix of Chen (2002).

𝛿z ≥√√√√ 2

√𝜓(4ln2 − 1)√

πe𝜓[1 − erf (

√𝜓)

]− 4

√𝜓∕𝛿2

r

(7)

Equation (7) can be rearranged as a function of any two parameters 𝜓, 𝛿z , and 𝛿r . These limits are plotted inphysical units as curves in Figures 11a–11c along with fit parameters for all selected EHs. With the exceptionof the dashed line in Figure 11a, curves are calculated from equation (7) by fixing the parameter left out ofeach plot, chosen such that the leftmost curve represents the most physically stringent limit. Solid lines havehad temperature fixed to Te = 400 eV.

The dashed line in Figure 11a is generated from equation (6). Most points lie below this line, indicating thatmany EHs are more elongated along B0 than expected. EHs range broadly from prolate to oblate. This may bebecause of the irregular structure inside some of the EHs altering their shape. If the trapped electrons havean enhancement, particularly near the center, then we expect L∥ to be enhanced as well.

Many EHs appear to have very low interior density given their length and potential. A significant numberof points in Figure 11b have parallel size tracing the minimum limit line (black, leftmost curve), similar to aprevious study using Fast Auroral SnapshoT (FAST) data (Ergun et al., 1999). These EHs must have exceptionallylow electron phase-space density in the center to maintain their structure. Uncertainty in temperature maybe responsible for some of the points to the left of the solid, 400-eV limit. The hot background in the electrondistribution (Figure 2) suggests an effective Te > 400 eV, which lowers𝜓 and relaxes the limits in Figure 11. Forcomparison, the blue, dash-dot curve in Figure 11b shows the limit for Te = 600 eV, which is also a reasonableestimate for some of the electron distributions from this event.

The tendency of EHs toward minimum size (or maximum potential) may imply that under the right conditions,EHs preferentially form with nearly empty potential wells. This is possible with fast, cold beams as seen intwo-stream instability simulations. The alternative of time-dependent evolution to an empty interior seemsunlikely; EHs would require some mechanism to leak electrons out and increase their potential or shrink insize. Mergers between EHs are unlikely to contribute to this process, as they typically result in increased lengthand have been shown to heat the surrounding plasma (Miyake et al., 1998), which implies a decrease in thepotential rather than an increase.

Points in Figure 11b away from the lowest L∥ limit do not have any strong trend in potential. Unlike the strongEHs near the leftmost limiting curve, these weak EHs could be explained by recent or ongoing mergers, whichtend to increase EH size and may lead to the scatter at higher L∥. It is possible that the weak EHs are more

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evolved forms of strong EHs. Based on simple superposition of unipolar E⟂ signatures, merging could alsoexplain the dimple features in the interior E⟂. Complex structure in E⟂ is visible in some simulations of merging2-D EHs although it is unclear whether this is due the merging process or kinking of "phase-space tubes"(Oppenheim et al., 1999) due to the transverse instability (Muschietti et al., 2000).

In Figure 11c, the L⟂ lower limit is obeyed by nearly all EHs in this data set. There is no immediately apparenttrend in perpendicular size of electron phase-space holes with respect to potential.

Theoretical limits for EH properties derived in equation (7) do not account for hole speed, ion physics, ormagnetization. In most cases, including these effects would cause the limiting curves in Figure 11 to shiftto the right. For instance, increasing the EH Mach number M serves to increase the lower limit on 𝛿z for agiven potential. The effect becomes significant for M> 1 (Chen et al., 2005). In this study, the sound speedcs ≈ 1.4 × 104 km/s, so most EHs have M < 1. Similarly, taking simple ion dynamics into account adds anegative term into the denominator of equation (7), increasing the minimum 𝛿z (Chen et al., 2004). In lightof more complete physical models, a large portion of EHs in this study have more extreme parameters thanexpected.

6. Conclusions

Electron phase-space holes appear to have a more complex 3-D structure in reality than a smooth, symmetricmodel suggests. While the observed E∥ smoothly passes through 0 near the center of an EH, E⟂ often hasthe unexpected feature of a flattop or dimple, which cannot easily be explained by concentrating trappedelectrons in the hole center, since this would also appear in E∥. Some resonant bunching of electrons relatedto the electron drift frequency within smaller holes may be responsible. It is also possible that EHs are notcylindrically symmetric but better fit by a triaxial ellipsoid or a more irregular model.

Magnetic field perturbations 𝛿B are computed assuming ideal E × B0 electron drift and purely parallel EHvelocity. Though the modeled parallel component 𝛿B∥ fits reasonably well to observations, measured peaks in𝛿B∥ are more narrow than predicted and vary in amplitude due to finite gyroradius effects. For well-resolvedEHs not suffering from these effects, a magnetic model could be used as an independent estimate of electrondensity. Modeled relativistic 𝛿Brel and current-induced 𝛿B′

⟂ are of roughly the same amplitude for this event,depending on spacecraft position relative to the EH. Comparing the model with 𝛿B⟂ data shows significantdisagreement in some cases, indicating that important microphysics is not correctly predicted. A combina-tion of background fluctuations and the surprising internal structure of the EHs are likely responsible for themismatch.

Observations of larger and faster EHs like those found in Earth’s magnetotail would significantly improve themagnetic field measurements here. However, the density is also very low in that region, and EHs can travelat a significant fraction of the speed of light. Low-density, fast EHs will have a strong 𝛿Brel, potentially wash-ing out the contribution from internal electron current. Furthermore, the background magnetic field can beweaker in the tail, implying a larger EH perpendicular size and a yet larger gyroperiod for electrons. If the rel-ativistic contribution to 𝛿B⟂ does not dominate, it becomes more important to account for finite gyroradiuseffects in the tail, which in addition to low-density measurements, can be problematic for constraining ananalytic model.

Many of the EHs observed approach a theoretical lower limit for the amplitude-length ratio, implying struc-tures with electron phase-space density in the center approaching 0. There does not appear to be anothercommon factor between this subset of EHs. It is possible that they form with a maximized potential for theirsize and then dissipate over time, smoothing out the potential. The rarity of such a process could be deter-mined using both 3-D electromagnetic simulations and a more complete survey of MMS data. If strong EHswith efficiently packed electromagnetic energy are common, they may have a more significant interactionwith the local plasma environment than previously thought.

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