wave-particle energy exchange directly observed in … · wave-particle energy exchange directly...

ARTICLE

Received 1 Sep 2016 | Accepted 20 Jan 2017 | Published 31 Mar 2017

Wave-particle energy exchange directly observedin a kinetic Alfven-branch waveDaniel J. Gershman1,2, Adolfo F-Vinas2, John C. Dorelli2, Scott A. Boardsen2,3, Levon A. Avanov1,2,

Paul M. Bellan4, Steven J. Schwartz5, Benoit Lavraud6,7, Victoria N. Coffey8, Michael O. Chandler8,

Yoshifumi Saito9, William R. Paterson2, Stephen A. Fuselier10, Robert E. Ergun11, Robert J. Strangeway12,

Christopher T. Russell12, Barbara L. Giles2, Craig J. Pollock2, Roy B. Torbert13,14 & James L. Burch10

Alfven waves are fundamental plasma wave modes that permeate the universe. At small

kinetic scales, they provide a critical mechanism for the transfer of energy between

electromagnetic fields and charged particles. These waves are important not only in planetary

magnetospheres, heliospheres and astrophysical systems but also in laboratory plasma

experiments and fusion reactors. Through measurement of charged particles and electro-

magnetic fields with NASA’s Magnetospheric Multiscale (MMS) mission, we utilize Earth’s

magnetosphere as a plasma physics laboratory. Here we confirm the conservative energy

exchange between the electromagnetic field fluctuations and the charged particles that

comprise an undamped kinetic Alfven wave. Electrons confined between adjacent wave peaks

may have contributed to saturation of damping effects via nonlinear particle trapping. The

investigation of these detailed wave dynamics has been unexplored territory in experimental

plasma physics and is only recently enabled by high-resolution MMS observations.

DOI: 10.1038/ncomms14719 OPEN

1 Department of Astronomy, University of Maryland, College Park, Maryland 20742, USA. 2 NASA Goddard Space Flight Center, Greenbelt, Maryland 20771,USA. 3 Goddard Planetary Heliophysics Institute, University of Maryland, Baltimore County, Maryland 21250, USA. 4 Division of Engineering and AppliedScience, California Institute of Technology, Pasadena, California 91125, USA. 5 Blackett Laboratory, Imperial College London, London SW7 2AZ, UK.6 Institut de Recherche en Astrophysique et Planetologie, Universite de Toulouse, Toulouse F-31400, France. 7 Centre National de la Recherche Scientifique,UMR 5277, Toulouse F-31400, France. 8 NASA Marshall Space Flight Center, Huntsville, Alabama 35808, USA. 9 JAXA Institute of Space and AstronauticalScience, Sagamihara, Kanagawa 252-5210, Japan. 10 Southwest Research Institute, San Antonio, Texas 78238, USA. 11 Astrophysical and Planetary Sciences,University of Colorado, Boulder, Colorado 80305, USA. 12 Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, California90095, USA. 13 Physics Department, University of New Hampshire, Durham, New Hampshire 03824, USA. 14 Southwest Research Institute Durham,Durham, New Hampshire 03824, USA. Correspondence and requests for materials should be addressed to D.J.G. (email: [email protected]).

NATURE COMMUNICATIONS | 8:14719 | DOI: 10.1038/ncomms14719 | www.nature.com/naturecommunications 1

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The Alfven wave is a ubiquitous plasma wave mode whereinions collectively respond to perturbations in the ambientmagnetic field direction1. No net energy is transferred

between the field and the plasma particles in ideal Alfven waves.However, ion motion decouples from electron motion when wavedynamics are faster than ion orbital motion around the localmagnetic field or are on scales smaller than the ion orbit size,defined by the gyrofrequency (oci) and gyroradius (ri),respectively. When the perpendicular spatial scale of an Alfvenwave approaches ri, the wave can support significant parallelelectric and magnetic field fluctuations that enable net transfer ofenergy between the wave field and plasma particles via Landau ortransit–time interactions2–4.

The transition of an ideal fluid-scale Alfven wave to a kinetic-scale Alfven wave (KAW) occurs at k?riB1 and k?4k||, where kis the wavevector and ‘?’ and ‘||’ are defined with respect to thelocal magnetic field direction. These KAWs are essential forenergy transfer processes in plasmas. Broadband KAWs havelong been associated in space physics with turbulent heating inthe solar wind and magnetosheath5–7 and are also thought toaccount for a substantial amount of the energy input into Earth’sauroral regions that can drive charged particle outflow andatmospheric loss8–13. In the laboratory, KAWs can transportenergy away from the core regions of fusion plasmas, resulting inthe unwanted deposition of energy at the reactor edges14,15.Understanding kinetic-scale wave generation, propagation andinteraction with charged particles is critical to unraveling andpredicting the relevant physics of these fundamental processes.

Alfven wave theory predicts that transverse fluctuations in thecurrent density (J) and electron-pressure-gradient-driven electricfield (Ep¼ �=�Pe/(nee)) are 90� out of phase with one another,such that the plasma heating term, D J?Ep?

� �, can be instanta-

neously non-zero but averages to zero over a wave period1. Insuch an undamped wave, power sloshes back and forth betweenthe wave field and particles with no net energy transfer. There areno corresponding fluctuations in DEp|| and DJ|| in an ideal Alfvenwave. For kinetic-scale Alfven waves, however, non-zero DEp||

fluctuations enable the Landau resonance, where particles withV||Bo/k|| can gain or lose energy through interaction with thewave field. These interactions, combined with an imbalance in thenumber of particles that are moving faster than or slower than thewave, result in net plasma heating or cooling4. Here, fluctuationsin DJ|| and DEp|| become in-phase such that the wave-averagedD(J||Ep||) is non-zero3,16. Likewise, fluctuations in DB|| result intransit-time damping effects, the magnetic analog of Landaudamping, where the magnetic mirror force takes the placeof Ep

2,4. For nonlinear KAWs, parallel fluctuations can besufficiently large in amplitude to trap electrons between adjacentwave peaks. The oscillatory bounce motion of these electronsproduces equal numbers of particles moving faster than or slowerthan the wave, limiting the effects of Landau and transit-timedamping, and enabling stable wave mode propagation4,17.

The detailed properties of KAWs (for example, DJ, DEp, k)have been difficult to characterize due to their small spatial andtemporal scales with respect to the capabilities of laboratory oron-orbit plasma instrumentation. Accurate estimates of currentdensity and the characterization of particle populations requirefull three-dimensional distribution functions of both electron andions on timescales faster than the wave frequency in theobservation frame of reference. In addition, estimates of pressuregradients and wavevectors rely on multiple observation pointsbeing available within a single wave peak. However, NASA’srecently launched Magnetospheric Multiscale (MMS) mission18

consists of four identical observatories deployed in a tetrahedronconfiguration that measure charged particle and electromagneticfields orders of magnitude more quickly than previous space

missions. This increased temporal sampling combinedwith a small MMS inter-spacecraft separation enables plasmaparameters and their spatial gradients to be determined atkinetic scales.

Here we use observations from MMS to characterizethe microphysics of a monochromatic Alfven wave. Throughthe calculation of DJ�DE, we provide a direct measurement of theconservative energy exchange between the wave’s electromagneticfields and particles. A perpendicular spatial scale of k?riB1,non-zero DEp|| and DJ|| fluctuations, and a parallel wave speedclose to the local Alfven speed confirm that the wave packet is anion-scale KAW. Finally, analysis of the velocity distributionfunction of electrons reveals a population that is nonlinearlytrapped within the wave’s magnetic minima. These trappedelectrons may have enabled nonlinear saturation of dampingprocesses, resulting in marginally stable wave propagation andproviding evidence in support of early analytical theories ofwave–particle interactions in collisionless plasmas.

ResultsEvent overview. On 30 December 2015, the four MMSobservatories were near the dayside magnetopause, that is, theinterface between the interplanetary magnetic field and theEarth’s internal magnetic field, at [7.8, � 6.9, 0.9] Re (1 Re¼ 1Earth radius¼ 6,730 km). Magnetic reconnection at themagnetopause boundary19,20 generated a southward flowingexhaust at B22:25 UT denoted by a �Vz jet, an increase inplasma density, and a decrease in plasma temperature (see Fig. 1).There was no discernable rotation in the magnetic fieldsuggesting that the spacecraft constellation remained inside theEarth’s magnetosphere throughout this interval. Low frequency(B1 Hz) waves were observed in the exhaust in a B4 mininterval localized to a region of strong proton temperatureanisotropy (THþ?/THþ ||B2). MMS partially crossed themagnetopause into the magnetosheath for the first time atB22:35 UT (not shown) at [8.0, � 6.9, 0.9] Re. For thesubsequent B2 h, multiple magnetopause crossings resulted inthe MMS spacecraft sampling both þVz and �Vz jets, that is,above and below the reconnection site. However, B1 Hz waveswere only observed in the short interval shown in Fig. 1.The MMS observatories were in a tetrahedron configuration(quality factor21 B0.9) separated by B40 km, a distance whichcorresponded to a local thermal ion gyroradius (ri¼ 35 km).

The reconnection exhaust plasma consisted of mostly Hþ andsome He2þ with number density ratio nHe2þ /nHþo0.02throughout the interval. The local ratios of ion thermal paralleland perpendicular pressure to magnetic pressure were b||E0.2and b?E0.5, respectively. In addition, the average plasma flowvelocity during this interval was Vo¼ [� 17, 73, � 183] km s� 1.This velocity corresponded to a jet flowing nearly anti-parallel tothe background magnetic field ([0.10, � 0.52, 0.85] direction)with speed B0.5 VA, where VA is the Alfven speed, that is, thecharacteristic speed in which information can be transferredalong a magnetic field. For this interval, with nHþ ¼ 10 cm� 3

and B¼ 55 nT, the local Alfven speed was estimated to be380 km s� 1. Variations were observed in the number density(Dn), bulk velocity (Dve), temperature (DT||, DT?) of both ionsand electrons, and in the electric (DE) and magnetic fields (DB)(see Fig. 2). The amplitude of these B1 Hz fluctuations werenonlinear with DnHþ /nHþB0.2. The magnetic field fluctuationsexhibited both left-handed and right-handed polarization (seeSupplementary Fig. 1). Finally, bursts of electron phase spaceholes measured in the total parallel electric field (DE||) werebunched with the wave in locations of strong electron pressuregradients.

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14719

2 NATURE COMMUNICATIONS | 8:14719 | DOI: 10.1038/ncomms14719 | www.nature.com/naturecommunications


Wave properties. Accurate determination of the wavevector (k)was critical to identify the observed wave mode. In situ estimationof k, especially for broadband wave spectra, is non-trivialand often relies on multi-spacecraft techniques22. Fortunately, themonochromatic nature of the observed wave enabled theapplication of several independent methods of wavevectordetermination. Here we utilized four methods to provide arobust estimate of k: (1) parallel component of the wavevectorderived from the correlation between velocity and magnetic fieldfluctuations16, (2) k-vector estimation from current and magneticfield fluctuations measured in the spacecraft frame23,24,(3) comparison of spacecraft-measured gradients with theircorresponding spacecraft-averaged quantities, that is, the plane-wave approximation4, and (4) phase differencing of the magneticfield fluctuations between each spacecraft25.

In the first method, we estimated the parallel component of thewavevector through comparison of four-spacecraft-averagedelectron velocity and magnetic field fluctuations. Alfven-branchwaves have parallel wave speeds close to the local Alfven speed,that is, |o/k|||EVA and correlated transverse fluctuations16,DVe? ¼ � (o/k||)DB?/B. Positively correlated (R2¼ 0.92) DVe?and DB? indicated that o/k||¼ � 1.15±0.03 VA, that is, thewave propagated anti-parallel to the background magnetic fieldnear the Alfven speed (see Supplementary Fig. 2). Although

qualitatively similar B1 Hz fluctuations have been observed nearEarth’s bow shock that are more consistent with magnetosonicwave modes26, a parallel phase speed well above the local soundspeed of B0.5 VA and the anti-correlation between density andmagnetic field fluctuations were inconsistent with slow and fastmagnetosonic wave modes, respectively.

In the second method, we combined fluctuations ofcurrent and magnetic field in the spacecraft frame to estimate kas a function of frequency using spectral techniques recentlydeveloped by Bellan23,24. Here the k-vector was derived directlyfrom fluctuations in DJ and DB measured in the spacecraft frame(see Fig. 3). Although this technique could have been appliedto data from a single spacecraft, in order to maximizespectral resolution we used the four-spacecraft average of DBand the average DJ determined from magnetometer datausing the four-spacecraft ‘curlometer’ technique27. The value ofk at the frequency of maximum spectral power, 0.9 Hz, wask¼ [7.1� 10� 3, � 2.0� 10� 2,� 2.2� 10� 2] km� 1, whichcorresponded to a wavevector angle (y) of B100� with respectto the background magnetic field and k?riB1.0.

In the third method, we used the phase difference25 measuredbetween each pair of MMS spacecraft for each component of themagnetic field to derive additional estimates of k. At the spectralpeak of 0.9 Hz, the k-vector determined from the phase

Mag

neto

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Figure 1 | MMS observations of a reconnection exhaust. (a) Illustration of the MMS constellation near the dayside magnetopause on 30 December 2015.

MMS entered a southward flowing reconnection exhaust in the separatrix region on the magnetospheric (msp) side of the magnetopause. (b–i) Plasma

parameters from MMS4 across the jet are shown from 22:23 to 22:30 UT. The density increased to B10 cm� 3 (d) and �Vz increased by B200 km s� 1

(e). No rotation in the magnetic field (B) indicated that the spacecraft remained inside the magnetosphere during this time period. Approximately 1 Hz

waves (h,i) were observed to be localized in a region of enhanced ion temperature anisotropy, with T?/T||B2. Hþ dominated the ion composition during

this time period.

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14719 ARTICLE



differencing of the BX, BY and BZ fluctuations (using MMS3 as areference) were: [� 7.4� 10� 5, � 8.5� 10� 3, � 1.5� 10� 2],[2.9� 10� 2,4.7� 10� 3, � 1.1� 10� 2], and [2.3� 10� 2,� 3.5� 10� 3, � 1.0� 10� 2] km� 1, respectively. Althoughsimilar phase shifts were observed in all components of DBbetween MMS2, MMS3 and MMS4, there were significantlydifferent shifts of MMS1 with respect to the other observatoriesfor each component (see Supplementary Fig. 3). These differencesdemonstrated that this wave packet was not truly planar andexhibited spatial structure on the order of an ion gyroradius.Because MMS1 was farthest from the magnetopause (that is, theX direction), the kX component was most strongly affected by thisstructure. Despite this discrepancy, all determinations of k resultin k?riB1 and the phase differencing of BX and BY components,those with the largest fluctuation power, both producedo/k||¼ � 1.1 VA.

Finally, in the fourth method, the small MMS spacecraftseparations and high-quality tetrahedron formation enabledgradients of particle and field quantities to be estimated directly

from the MMS data. These gradients were compared with thosepredicted by the plane-wave approximation (that is, ‘r�’Eik and‘r� ’Eik� at a single frequency4) to both evaluate the validityof this approximation to the observed wave packet and to providefurther validation of k (see Fig. 4). The current was calculatedfrom three methods: (1) direct particle observations, that is,ene(Vi�Ve), (2) magnetic field ‘curlometer’27, that is, r�B/mo,and (3) the plane-wave approximation, that is, ik�B/mo. Allthree estimates of DJ are shown in Fig. 4. ky and kz most stronglyinfluenced the plane-wave-derived currents such that thisintercomparison was relatively insensitive to errors in thedetermination of kx. The electron-pressure-gradient-drivenelectric field determined from four spacecraft measurements(that is, �r�Pe/(nee)), when compared with its plane-waveapproximated value (that is, � ik�Pe/(nee)), provides furtherconfidence in the determination of k (see Fig. 4). Here all threecomponents of k contributed to this result. The X-componentcomparison demonstrates that kx is of the correct sign but mayunderestimate the four-spacecraft gradient.

ΔB (

nT)

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MMS1

MMS4MMS3MMS2

22:26:34UT

Figure 2 | MMS observations of a KAW packet. Plasma parameters measured by the four MMS observatories on 30 December 2015 in a KAW

packet. (a,b) Compressive fluctuations are observed in anti-correlated electron density (Dne) and magnetic field magnitude (DB) measurements.

(c,d) Positively correlated fluctuations are observed in near-transverse components of the magnetic field (DBX) and electron bulk velocity (DVex).

(e–h) Fluctuations in both parallel and perpendicular temperature of both electrons (DTe) and ions (DTi) are shown, with the strongest relative fluctuations

(B10%) observed in the perpendicular electron temperature. (i) Bursts of electron-scale phase space holes measured in the parallel electric field (DE||) are

bunched with the ion-scale KAW wave and correspond to some of the gradients in the measured electron pressure.




We adopted the k-vector derived using the Bellan23,24 methodk¼ [7.1� 10� 3, � 2.0� 10� 2, � 2.2� 10� 2] km� 1 because itsimultaneously leveraged data from all four spacecraft and allcomponents of the magnetic field. Allowing for B30% (3-s level)uncertainty in each individual component, we foundk?ri¼ 1.02±0.07 with wavevector angle 104±4� from themagnetic field. The 0.9 Hz peak observed in the spacecraftframe (osc) was then Doppler-shifted by o¼osc� k�Vo toobtain a frequency of o/oci,Hþ ¼ 0.61±0.08 in the plasmaframe. We conclude that multiple independent methods indicatedthat MMS resolved a kinetic-scale Alfven-branch wave.

Modelled wave growth rates. Growth rates (g¼ Im{o/oci}) andpolarization (Re{iEy/Ex}) solutions along the Alfven-branch dis-persive surface were estimated using a linear dispersion solverand are shown as a function of y in Fig. 5. The dispersion solverpredicted that the large ion temperature anisotropy of Ti?/Ti||B2produced a nearly monochromatic ion cyclotron wave mode thatpropagated parallel/anti-parallel to the background magnetic field(y¼ 0�, 180�) with o/ociB0.5, kriB0.4 and left-handed polar-ization. At increasingly oblique wavevector angles, the predictedwave growth was substantially reduced. There was no slow or fastmagnetosonic wave growth predicted for the measured plasmaparameters. Several Alfven-branch dispersion curves are shown inFig. 5 as a function of kri and y. The observed KAW mode(o/oci¼ 0.6, kri¼ 1, y¼ 100�) was close to but not preciselyon the solution surface. Nearby Alfvenic solutions to themeasured data (matching two of the three wave parameters) were{o/oci¼ 0.3, kri¼ 1, y¼ 100�}, {o/oci¼ 0.6, kri¼ 1.6, y¼ 100�}and {o/oci¼ 0.6, kri¼ 1, y¼ 110�}. All of these nearby solutionswere weakly damped (|g|B10� 2) such that local generation ofthe observed KAW was not predicted by linear wave theory.However, local spatial gradients of plasma density may haveincreased the y of the ion cyclotron mode during its propagation,converting it into an oblique Alfven wave4. Furthermore,nonlinear effects and parametric forcing (for example,

magnetopause motion) were not taken into account by thehomogenous dispersion solver, yet may have played a role in theevolution of the observed KAW.

Wave–particle interactions. Given the demonstrated validity ofthe plane-wave approximation for DEp, the electron-pressure-gradient-driven electric field was estimated at a single spacecraft,for example, MMS4, using � ik�Pe/(nee). Fluctuations of DEp

and DJ in magnetic coordinates on MMS4 are shown in Fig. 6. Inaddition to the transverse electric field fluctuations expected forall Alfven waves, fluctuations in DEp|| further confirmed thepresence of kinetic-scale effects. These parallel fluctuations werean order of magnitude smaller than those in DEp? as expectedfrom KAW theory16. Furthermore, fluctuations in all componentsof DJ and DEp (both perpendicular and parallel) were each

Frequency (Hz)

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Figure 3 | Wavevector estimated from current density fluctuations.

(a) Power spectral density of MMS-averaged magnetic field magnitude

from 22:26:28.18–22:26:35.83 UT, (b) the imaginary part of the Fourier

amplitudes of fluctuations in MMS-averaged J�B and (c) corresponding

components of k(o) derived using the Bellan23,24 technique. At the spectral

peak of B0.9 Hz, k¼ [7.1� 10� 3, � 2.0� 10� 2, � 2.2� 10� 2] km� 1.

This wavevector yielded k?riB1 and an angle of B100� with respect to the

background magnetic field.ΔJ

x (n

Am

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ne(vi-ve)

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252015105

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22:26:28 22:26:32 22:26:36

0

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0

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Figure 4 | Comparison of current and electric field estimates.

(a–c) MMS-averaged current fluctuations (DJ) derived from the

curlometer technique (blue), four-spacecraft-averaged particle

observations (black) and four-spacecraft-averaged plane-wave

approximation using k¼ [7.1� 10� 3, � 2.0� 10� 2, � 2.2� 10� 2] km� 1

(red). (d–g) MMS-averaged DEp and D(J�Ep) derived from the divergence

of the electron pressure tensor (blue) and from the plane-wave

approximation (red). Agreement between all quantities provides additional

confidence in the estimation of k.




B90� out of phase with one another. These phase differencesresulted in a non-zero instantaneous value of D(J�Ep)with D|J�Ep|maxE50 pW m� 3 and near-zero wave-averagedD J?Ep?� �

and D(J||Ep||) quantities. These data demonstrated theconservative energy exchange between the particles and fields thatcomprise an undamped KAW.

Because k?reoo1, electrons should have remained magnetizedthroughout the wave packet. Close examination of the electronvelocity distribution function in the parallel wave frame revealedthree distinct populations of electrons in the wave packet: (1) anisotropic thermal core, (2) suprathermal beams counterstreamingalong the magnetic field, and (3) trapped particles with nearB90� magnetic pitch angles (Fig. 7). Thermal and counter-streaming electrons are commonly observed in the magnetopauseboundary layer in the absence of analogous wave activity28.However, trapped electron distributions are atypical of ambientboundary layer plasmas. Furthermore, these trapped electronswere dynamically significant: they accounted for B50% of thedensity fluctuations within the KAW. Although these electronsalso resulted in a B20% increase in Te?, they were not indicativeof heating but rather of a nonlinear capture process.

The depth of the parallel potential well estimated from DEp||

and k|| was found to be B10 V (Fig. 7). In addition, the parallelmagnetic field of the wave generated a mirror force that resultedin a kinetic-scale magnetic bottle between successive wave peaks.This mirror force supplemented the force from the wave’s parallelelectric field, enabling trapping of electrons with magnetic pitch

angles between B75� and B105� (Bmin/Bmax¼ 0.96). To under-stand the combined effects of these forces, electrons measured inthe magnetic minima were Liouville-mapped to other locationsalong the wave using various parallel potential well depths(Fig. 8). The full-width at half maximum distance along the waveat a pitch angle of 90� was calculated for each potential andcompared with the measured data. The best match betweenmeasured and Liouville-mapped distributions was found for apotential well depth of |Fmax|¼ 10 V. Such agreement providedadditional validation of DEp|| and k||. In addition, thesedistributions demonstrated that the effect of the parallel electricfield was to confine magnetically trapped electrons closer tomagnetic minima.

DiscussionKAWs in turbulent space plasmas are thought to account forheating of plasmas at kinetic scales5–7. In previous studies29,30,such waves were found to have k?ck||, that is, yB90�. Thisplasma heating was accompanied by significant reductions in fieldfluctuation power. The wave presented here had a somewhathigher frequency (oci,He2þooooci,Hþ ) than those consideredin these previous KAW studies (ooooci,Hþ , oci,He2þ ).Furthermore, its comparatively non-perpendicular wavevector(yE100�) and large scale (k?riE1) indicated that the observedwave was close to the transition point between ideal and kineticregimes. Nonetheless, the wave had non-zero DJ|| and DEp||

k�i

Re{

�/�

ci}

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iEy/E

x}Im

{�/�

ci}

1.2

1.0

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0.0200.0150.0100.0050.000

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20015010050

–50–100–150–200

0.2 0.4 0.6 0.8 1.8 1.2 1.4 1.6 1.8

0

a

b

c

Observed waveNearby solutions

179°170°160°150°140°130°120°110°100°91°

Figure 5 | Modelled dispersion curves for the local plasma environment. (a) The real part of o/oci, that is, the wave oscillation frequency, (b) the

imaginary part of o/oci, that is, the wave growth/damping rate and (c) the real part of iEx/Ey, that is, the polarization of the wave, as a function of scaled

wavevector magnitude kri. Coloured curves correspond to solutions of a linear dispersion relation solver taken along the Alfven branch for different

wavevector angles (y) relative to the background magnetic field. The fastest growing wave mode has a wavevector parallel/anti-parallel to the background

magnetic field (that is, y¼0�, 180�) at o/ociB0.5 and kriB0.4 and is left-hand polarized (that is, Re{iEx/Ey}o0). A transition to right-hand polarization

(that is, Re{iEx/Ey}40) occurred at yB130�. No strong growth or damping was predicted for the observed KAW (y¼ 104±4�, o/oci¼0.61±0.08 and

k?ri¼ 1.02±0.07), indicated with the shaded area in (a). The dimensions and colour of the shaded area correspond to the reported uncertainties of the

measured o/oci and k?ri parameters and yE100�, respectively Nearby solutions that match two of the measured {o/oci, kri, y} parameters (but not all

three) are shown as solid circles. The colour of each circle corresponds to the wavevector angle.




fluctuations, confirming that it contained kinetic-scale structurenot present in an ideal Alfven wave. These observationsdemonstrated that the mere presence of a KAW or parallelelectric field fluctuations do not necessarily imply heating viaLandau damping. Only in-phase fluctuations in DJ and DEp resultin such net transfer of energy from the wave field to the plasmaparticles.

In linear KAW theory, the electrostatic field formed byparallel gradients in electron pressure enables the energization

of particles via the Landau resonance4,13,16. Similarly, thetransit-time resonance becomes relevant for systems wherethere are parallel gradients in magnetic field magnitude.Despite the presence of these field gradients in the observedKAW, out-of-phase DEp|| and DJ|| fluctuations and a finite waveamplitude for several wave periods (that is, |g|oo1) indicated theabsence of strong wave growth or damping. Although a hot corepopulation (Vth,ec|o/k|||) does not lead to strong damping(Fig. 5), the velocity distribution function of electrons was notdirectly sampled at energies corresponding to V||Bo/k|| (that is,B0.5 eV). Electrons at these low energies are often present as theyserve to neutralize a ubiquitous population of ‘hidden’ cold ionsthat flow out from the ionosphere31. Such ionospheric electronsmay have added structure to the velocity distribution functionnear V||Bo/k||, amplifying damping rates. However, nonlinearKAW theories have predicted that trapped electrons withV||Bo/k|| lead to wave stabilization if their bounce frequency(oB) is significantly faster than the damping or growth rate,that is, oB/ocic|g|4,17,32. We estimated oB/ociB1 for this wave,consistent with such a criterion. Therefore, the presence oftrapped electrons here could have contributed to nonlinearinstability saturation in a single-mode wave even if there were lowenergy structure in the electron distribution function that was notresolved by MMS.

Finally, at higher frequencies (B1 kHz), fluctuations in thetotal parallel electric field DE|| associated with electronphase space holes33 were bunched in phase with the lowfrequency wave packet (Fig. 1). Because these structurespersisted outside of the KAW interval (not shown), it isunlikely that they were related to its initial generation.However, the location of these electron-scale structureswithin the wave was coincident with the location of electronpressure gradients, suggesting that they could have contributed,in an average sense, to some of the observedion-scale DEp|| fluctuations. Furthermore, electron holes mayhave been responsible for higher frequency contributions toD(J||E||) in the form of nonlinear and turbulent terms in theelectron momentum equation34.

Using MMS data, we have experimentally confirmed theconservative energy exchange between an undamped kineticAlfven wave field and plasma particles: fluctuations of allthree components of DJ and DEp were 90� out of phase withone another, leading to instantaneous non-zero D(J�Ep).Furthermore, we have discovered a significant population ofelectrons trapped within adjacent wave peaks by the combinedeffects of the parallel electron-pressure-gradient-driven electricfield and the magnetic mirror force. In addition to contributingB50% of the density fluctuations in the wave, these trappedelectrons may have provided nonlinear saturation of Landauand transit-time damping. The monochromatic nature of thewave enabled a direct comparison of observations with linearand nonlinear KAW theories. It is crucial to understandthese dynamics to predict the evolution of kinetic-scale wavesin laboratory fusion reactors, planetary magnetospheres andastrophysical plasmas.

MethodsCoordinate systems. The coordinate system used in this study (unless otherwisenoted) was the Geocentric Solar Ecliptic (GSE) coordinate system, where theX direction pointed towards the Sun along the Earth–Sun line, the Z directionwas oriented along the ecliptic north pole and the Y direction completed theright-handed coordinate system35. Local ‘magnetic coordinates’ were derived fromGSE vectors where B3 was parallel to the local magnetic field direction, B1 was inthe XGSE�B3 direction and B2 completed the right-handed coordinate system,that is, B1�B2¼B3.

1.51.00.5

–0.5–1.0–1.5

0.08 200150

50

–50–100–150–200

0

100

100

50

0

–50

–100100

50

0

–50

–100

0.060.040.02

–0.02–0.04–0.06

642

–2–4

–1

0

1

2

–2

–1

0

1

2

–2

0

–0.08

0.00

0.0

hh:mm:ss

ΔEp|

| (m

V m

–1)

a

b

c

d

e

f

ΔB (

nT)

Δ (J

||Ep|

|)(p

Wm

–3)

ΔEp⊥

1(m

V m

–1)

ΔEp⊥

2(m

V m

–1)

ΔJ⊥

2(p

A m

–2)

ΔJ⊥

1(p

A m

–2)

Δ (J

⊥E

p⊥)

(pW

m–3

)

150

500

–50–100–150

22:26:28 22:26:32 22:26:36

100

ΔJ|| (

pA m

–2)

Figure 6 | Current and electric field fluctuations in a KAW. Fluctuations in

(a) magnetic field magnitude DB, (b) parallel electric field DEp|| and parallel

current DJ||, (c) D(J||Ep||), (d,e) perpendicular electric fields (DEp?1 and

DEp?2) and current (DJ?1 and DJ?2) and (f) D J?Ep?ð Þ observed by MMS4

on 30 December 2015 between 22:26:27 and 22:26:37 UT. Pressure-

gradient-driven electric field quantities were inferred from the k-vector and

electron pressure tensor from MMS4 using the plane-wave approximation

(that is, Ep¼ � ik�Pe/nee). Current densities were derived directly from

MMS4 particle observations. Current density and electric field fluctuations

were 90� out of phase in both the perpendicular and parallel directions,

resulting in non-zero instantaneous D(J�Ep), which provided confirmation

of the conservative energy exchange between the wave field and plasma

particles. The amplitude of D J?Ep?ð Þ was an order of magnitude higher than

D(J||Ep||). The wave-averaged D(J�Ep) was approximately zero, indicating

that the wave was in a marginally stable state, that is, was neither growing

nor damping. Quantities are shown in magnetic coordinates.




Calculation of plasma parameters. The thermal gyroradius was calculated using

ri¼mHþ

ffiffiffiffiffiffiffiffiffiffiffiffikB THþ?

mHþ

qeB

ð1Þ

where kB is Boltzmann’s constant, e is the elementary charge and mHþ is the massof Hþ . The ion gyrofrequency was calculated using,

oci¼eB

mHþð2Þ

The plasma thermal pressure was calculated using nHþkBTHþ . The magneticpressure was calculated using B2/2mo where mo is the magnetic permeability of freespace. Finally, the Alfven speed was calculated using

VA¼Bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

monHþmHþp ð3Þ

All calculations were done in SI units.

DVe–DB correlations. The comparison of DVe and DB was done in the directionof minimum current density fluctuations ([0.93, 0.32, 0.18]) such that ion andelectron velocities were approximately equal. This minimum variance direction

was nearly perpendicular to the background magnetic field direction b¼ [0.10,� 0.52, 0.85].

Electric field measurements. The electric field in the electron frame wasdefined as EþVe�B, where E was the measured electric field in the spacecraftframe23. Since J is frame independent, this electron-frame electric field isconveniently used for estimates of energy transfer, that is, plasma heating occurswhen J�(EþVe�B)40. At the scales relevant for this KAW packet, electronsremained magnetized such that electron inertia and anomalous resistivitycontributions to the electric field were neglected and the pressure gradient termshould have been the dominant contributor to EþVe�B at low frequencies. Theindividual amplitudes of E and Ve�B were measured to be on the order of severalmV m� 1. Systematic uncertainty in both particle and fields measurements wouldhave led to a challenging recovery of EþVe�B because |EþVe�B|oo|E|,|Ve�B|.Therefore, accurate direct estimates of J�(EþVe�B) were not recovered forthis event. Instead, here we focussed on effects of the electric field generated bythe divergence of the electron pressure tensor, that is, Ep¼ �r�Pe/(nee) andvalidated the measurement using multiple methods. In the electron frame, theelectrons are not moving so there is no magnetic term in the electron equationof motion giving EEEp.

V⊥

V||

800 = �||6005004003002001000

D (km)10 eV14 eV17 eV23 eV29 eV37 eV48 eV62 eV79 eV102 eV131 eV168 eV216 eV277 eV355 eV455 eV585 eV750 eV962 eV1,223 eV

D = �||/2D = 0

0 45 90 135 180 0 45 90 135 180

–10

10–25

10–26

10–27

10–28

10–29

10–30

10–31

0–0.04

0.04

0.00

535455

35

40

8

10

� (V

)

Ep|

|(m

V m

–1)

B(n

T)

Te⊥

(eV

)n e

(cm

–3)

Thermal coreCounterstreaming

Trapped

a

f h

Pha

se s

pace

den

sity

(s3 c

m–6

)

Magnetic pitch angle (°)

700

b

c

d

e

g

Figure 7 | Structure inside of a KAW packet. Profile of (a) density ne, (b) perpendicular electron temperature Te? , (c) magnetic field magnitude B,

(d) parallel electric field DEp|| inferred from electron pressure gradients and (e) parallel potential F integrated from DEp|| as a function of position D in the

wave for MMS4 from 22:26:29.94 to 22:26:30.90 UT. The reference value for the potential (F¼0) was taken at the centre of the wave, that is, at the

magnetic minimum. The wave had a parallel wavelength of l||B830 km or B20 ri. The ratio of the minimum to maximum magnetic field magnitude was

Bmin/Bmax¼0.96, which was sufficient to trap electrons with magnetic pitch angles between B75� and B105�. Phase space density as a function of energy

and magnetic pitch angle are shown at the magnetic (f) maximum (D¼0) and (g) at the magnetic minimum (D¼ l||/2) in the wave frame of reference

(that is, all measured velocities shifted by �VA along the magnetic field direction). An illustration of three corresponding populations of electrons is shown

in V||�V? space in panel (h). Thermal (energies below TeE35 eV) electrons have nearly isotropic pitch-angle distributions (blue contours). Suprathermal

(energies above Te) electrons were observed as peaks in the phase space density at pitch angles near 0� and 180� (red contours). Finally, a trapped

population with energies above Te is shown between the dashed vertical lines (purple contours). These trapped electrons were responsible for the

increased perpendicular temperature at the magnetic minima and accounted for B50% of the increase in density.




Linear instability analysis. To determine the properties of kinetic modes thatinteract with ions and electrons at their respective scales, we used the lineardispersion solver PLADAWAN36 (PLAsma Dispersion And Wave ANalyzer) tosolve the linearized Vlasov-Maxwell system for arbitrary wavevector directions.Using measured plasma parameters of ions and electrons, the dispersion solverproduced growth rates and wave properties as functions of o and k. The plasmaparameters used as input to the dispersion solver (assuming stationary plasma)were ne� ¼ 10 cm� 3, B¼ 55 nT, Te? ¼Te||¼ 35 eV, THþ ||¼ 175 eV andTHþ? ¼ 350 eV. Wave polarization was calculated using the simulated electric fieldfluctuations as Re{iEx/Ey}. Left-hand and right-hand polarization corresponded toRe{iEx/Ey}o0 and Re{iEx/Ey}40, respectively4. No growth was observed for theslow-mode or fast-mode magnetosonic branches of the dispersion relation.Additional simulations were run to evaluate the influence of He2þ on the observedinstability. Increased nHe2þ /nHþ ratios up to 0.02 with THe2þ ¼ 550 eV reducedthe maximum wave growth but did not alter the sharpness of the peak in k-space.No new wave modes appeared to be introduced into the system from the presenceof the local He2þ population.

Liouville mapping and electron bounce motion. Under the assumption thatelectron phase space density f(v) was conserved along particle trajectoriesthroughout the wave interval (that is, Liouville’s theorem), we used f(v) measuredin the magnetic minimum, defined as fo(v), a sinusoidal profile of the magneticfield strength B with M¼Bmin/Bmax¼ 0.96, and a sinusoidal profile of electricpotential F to infer the velocity distribution along the wave37,38. Velocity space wastransformed using equations

v jj o¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2? Dð Þ 1� Bo

B Dð Þ

� �þ v2

jj Dð Þ� 2eme

F Dð Þ

sð4Þ

and

v?o¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2?ðDÞ

Bo

B Dð Þ

� �s; ð5Þ

where the ‘o’ subscripts denote values at the magnetic minimum of the wave.The ‘þ ’ and ‘� ’ branches of equation (4) correspond to the sign of v||.For each (v||, v?) point in the reconstructed skymap, equations (4 and 5) provided apoint (v||o, v?o) that was used to map a phase space density in the referencedistribution, that is, f(v||, v?)¼ fo(v||o, v?o).

In the magnetic minimum (D¼ l||/2), BoB Dð Þ ¼ 1 and F¼Fo¼ 0. At the magnetic

maximum (D¼ 0, l||), BoB Dð Þ ¼M and F¼ � |Fmax|, that is,

Bo

B Dð Þ¼Mþ 1�Mð Þsinpl jj

D

� �ð6Þ

F Dð Þ¼� Fmaxj j2

1þ cos2pl jj

D

� �� : ð7Þ

Finally, bounce frequencies (oB¼ 1/tB) for trapped electrons were estimatedusing

tB¼4Z R

l jj =2

dDv jj Dð Þ ; ð8Þ

where R was defined as the reflection point along the wave (that is, v|| (R)¼ 0).Electrons with pitch angles 75–90� and energies 100–400 eV produced bouncefrequencies of 1.4±0.3 Hz (that is, o/oci¼ 1.6±0.3) in a l||¼ 830 km wave withM¼ 0.96.

MMS data sources and processing. Particle, magnetic field and electric fielddata were measured by the Fast Plasma Investigation39 (FPI), the FluxgateMagnetometers40 and Electric Field Double Probe41 instruments, respectively.Corresponding composition data at B10 s time resolution was obtained from theHot Plasma Composition Analyzer42. Time series data were high-pass filtered witha fifth-order digital Butterworth IIR filter with coefficients b¼ [0.85850229,� 4.29251147,8.58502295, � 8.58502295, 4.29251147, � 0.85850229] anda¼ [1.0, � 4.69504063,8.82614592, � 8.30396669, 3.90989399, � 0.73702619],where b and a correspond to the filter’s numerator and denominator polynomialslisted in increasing order. This filter had an effective cutoff frequency of 0.5 Hzand no discernable effect (o1%) on the amplitude or phase of a 0.9 Hz inputsignal.

Data availability. Data used for this study is available to download from theMMS Science Data Center (https://lasp.colorado.edu/mms/sdc/) or from thecorresponding author upon request.

Phase space density ×10–27 (s3 cm–6)

D (km)

Data

|�|max = 0 V

|�|max = 5 V

|�|max = 10 V

|�|max = 15 V

|�|max = 20 V

|�|max = 25 V

700600 500400300200100

Mag

netic

pitc

h an

gle

(°)

0

0

90

180

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.0

0

90

180

0

90

180

800 = �||

a

b

c

d

e

f

g

0

90

180

0

90

180

0

90

180

0

90

180

Figure 8 | Liouville-mapped electrons in a KAW. Measured phase space

densities from MMS4 as a function of magnetic pitch angle and position

in the wave, D, between successive magnetic field maxima in the KAW

packet from Fig. 3 (22:26:29.94–22:26:30.90 UT) for 132 eV electrons.

Liouville-mapped distributions are shown for |F|max¼0, 5, 10, 15, 20 and

25 V (a–g). These distributions were constructed using measured phase

space densities at the magnetic minimum (that is, D¼ l||/2). The mirror

ratio of Bmin/Bmax¼0.96 confined particles to pitch angles between 75�and 105� in all cases. The parallel potential formed from DEp|| provided

additional spatial localization of the trapped population within the wave

minima. Vertical dashed lines denote the full-width at half-maximum along

D at a pitch angle of 90�. The best agreement with the measured data

occurred for the distribution mapped using |F|max¼ 10 V, which was

consistent with independent estimates of k|| and DEp||.



https://lasp.colorado.edu/mms/sdc/


References1. Alfven, H. Existence of electromagnetic-hydromagneticwaves. Nature 150,

405–406 (1942).2. Barnes, A. Collisionless damping of hydrodynamic waves. Phys. Fluids 9, 1483

(1966).3. Hasegawa, A. & Chen, L. Parametric decay of ‘kinetic Alfven wave’ and its

application to plasma heating. Phys. Rev. Lett. 36, 1362–1365 (1976).4. Stix, T. H. Waves in Plasmas 566 (American Institute of Physics, 1992).5. Rezeau, L., Morane, A., Perraut, S., Roux, A. & Schmidt, R. Characterization of

Alfvenic fluctuations in the magnetopause boundary layer. J. Geophys. Res. 94,101–110 (1989).

6. Leamon, R. J., Smith, C. W., Ness, N. F., Matthaeus, W. H. & Wong, H. K.Observational constraints on the dynamics of the interplanetary magnetic fielddissipation range. J. Geophys. Res. 103, 4775–4787 (1998).

7. Sahraoui, F., Goldstein, M. L., Robert, P. & Khotyaintsev, Y. V. Evidence of acascade and dissipation of solar-wind turbulence at the electron gyroscale. Phys.Rev. Lett. 102, 231102 (2009).

8. Louarn, P. et al. Observation of kinetic Alfven waves by the Freja spacecraft.Geophys. Res. Lett. 21, 1847–1850 (1994).

9. Chaston, C. C. et al. Drift-kinetic Alfven waves observed near a reconnection Xline in the Earth’s magnetopause. Phys. Rev. Lett. 95, 065002 (2005).

10. Seyler, C. E., Wahlund, J.-E. & Holback, B. Theory and simulation of lowfrequency plasma waves and comparison to Freja satellite observations.J. Geophys. Res. 100, 453 (1995).

11. Ergun, R. E. et al. Localized parallel electric fields associated with inertial Alfvenwaves. Phys. Plasmas 12, 072901 (2005).

12. Chaston, C. C. et al. Ionospheric erosion by Alfven waves. J. Geophys. Res. 111,A03206 (2006).

13. Artemyev, A. V., Rankin, R. & Blanco, M. Electron trapping and acceleration bykinetic Alfven waves in the inner magnetosphere. J. Geophys. Res. Space Phys.120, 305–10,316 (2015).

14. Wong, K.-L. A review of Alfven eigenmode observations. Plasma Phys.Controlled Fusion 4, R1–R56 (1999).

15. Belova, E. V., Gorelenkov, N. N., Fredrickson, E. D., Tritz, K. & Crocker, N. A.Coupling of neutral-beam-driven compressional Alfven eigenmodes to kineticAlfven waves in NSTX Tokamak and energy channeling. Phys. Rev. Lett. 115,015001 (2015).

16. Hollweg, J. V. Kinetic Alfven wave revisited. J. Geophys. Res. 104, 14811–14819(1999).

17. O’Neil, T. M. Collisionless damping of nonlinear plasma oscillations. Phys.Fluids 8, 2255 (1965).

18. Burch, J. L., Moore, T. E., Torbert, R. B. & Giles, B. L. MagnetosphericMultiscale overview and science objectives. Space Sci. Rev. 199, 5–21 (2015).

19. Cowley, S. W. H. in Magnetic Reconnection in Space and Laboratory PlasmasVol. 30 (ed. Hones, E.W.) 375–378 (American Geophysical Union,Washington, D.C. 1984).

20. Paschmann, G. et al. Plasma acceleration at the Earth’s magnetopause: evidencefor magnetic reconnection. Nature 282, 243–246 (1979).

21. Fuselier, S. A. et al. Magnetospheric Multiscale science mission profile andoperations. Space Sci. Rev. 199, 77–103 (2016).

22. Pincon, J. L. & Glassmeier, K.-H. in Multi-Spacecraft Analysis MethodsRevisited (European Space AgencyPublications, 2008).

23. Bellan, P. M. Improved basis set for low frequency plasma waves. J. Geophys.Res. 117, A12219 (2012).

24. Bellan, P. M. Revised single-spacecraft method for determining wave vector kand resolving space-time ambiguity. J. Geophys. Res. Space Phys. 121,8589–8599 (2016).

25. Balikhin, M. A., Woolliscroft, L., Alleyne, H., Dunlop, M. & Gedalin, M. A.Determination of the dispersion of low-frequency waves downstream of aquasiperpendicular collisionless shock. Ann. Geophys. 15, 143 (1997).

26. Wilson, III L. B. et al. Low-frequency whistler waves and shocklets observedat quasi-perpendicular interplanetary shocks. J. Geophys. Res. 114, A10106(2009).

27. Dunlop, M. W., Balogh, A., Glassmeier, K.-H. & Robert, P. Four-point clusterapplication of magnetic field analysis tools: the curlometer. J. Geophys. Res. 107,1384 (2002).

28. Hall, D. S., Chaloner, C. P., Bryant, D. A., Lepine, D. R. & Tritakis, V. P.Electrons in the boundary layers near the dayside magnetopause. J. Geophys.Res. 96, 7869–7891 (1991).

29. Sahraoui, F., Goldstein, M. L., Belmont, G., Canu, P. & Rezeau, L. Threedimensional anisotropic k spectra of turbulence at subproton scales in the solarwind. Phys. Rev. Lett. 105, 131101 (2010).

30. Salem, C. S. et al. Identification of kinetic Alfven wave turbulence in the solarwind. Astrophys. J. Lett. 745, L9 (2012).

31. Andre, M. & Cully, C. M. Low-energy ions: a previously hidden solar systemparticle population. Geophys. Res. Lett. 39, L03101 (2012).

32. Manheimer, W. M. Instabilities in a plasma coexisting with a large-amplitudewave. Phys. Rev. A. 3, 1402 (1971).

33. Ergun, R. E. et al. Debye-scale plasma structures associated withmagnetic- field-aligned electric fields. Phys. Rev. Lett. 81, 826 (1998).

34. Che, H. et al. A current filamentation mechanism for breaking magnetic fieldlines during reconnection. Nature 474, 184–187 (2011).

35. Hapgood, M. A. Space physics coordinate transformations: a user guide. PlanetSpace Sci. 40, 711 (1992).

36. Vinas, A., Wong, H. K. & Klimas, A. J. Generation of electron suprathermaltails in the upper solar atmosphere: implications for coronal heating. Astrophys.J. 528, 509 (2000).

37. Olsen, R. C., Scott, L. J. & Boardsen, S. A. Comparison between Liouville’stheorem and observed latitudinal distributions of trapped ions in theplasmapause region. J. Geophys. Res. 99, 2191–2203 (1994).

38. Chisham, G., Schwartz, S. J. & Burgess, D. The anisotropy variations of electrondistributions in the terrestrial ion foreshock. J. Geophys. Res. 101, 445–455(1996).

39. Pollock, C. J. et al. Fast plasma investigation for Magnetospheric Multiscale.Space Sci. Rev. 199, 331–406 (2016).

40. Russell, C. T. et al. The Magnetospheric Multiscale magnetometers. Space Sci.Rev. 199, 189–256 (2014).

41. Ergun, R. E. et al. The axial double probe and fields signal processing for theMMS mission. Space Sci. Rev. 199, 167–188 (2014).

42. Young, D. T. et al. Hot plasma composition analyzer for the MagnetosphericMultiscale mission. Space Sci. Rev. 199, 407–470 (2014).

AcknowledgementsWe thank the members of the FPI ground operations and science team for their feedbackand support and Lynn Wilson for his insights about wave properties. This research wassupported by the NASA Magnetospheric Multiscale Mission in association with NASAcontract NNG04EB99C and by NSF award 1059519, AFOSR award FA9550-11-1-0184and DOE awards DE-FG02-04ER54755 and DE-SC0010471. S.J.S. is grateful to theLeverhulme Trust for its award of a Research Fellowship.

Author contributionsD.J.G. conducted the majority of the scientific and data analysis and was responsible forinitial preparation of the manuscript text. A.F-V. assisted with the interpretation of wavesignatures, plasma wave modeling and with the preparation of the manuscript text.J.C.D., S.A.B. and L.A.A. assisted with the interpretation of plasma wave signatures,detailed analysis of plasma data and with the preparation of the manuscript text.P.M.B. assisted with the implementation of the wavevector determination method andwith the preparation of the manuscript text. S.J.S. assisted with the Liouville mapping ofelectron data and preparation of the manuscript text. B.L., V.N.C., M.O.C., Y.S. andW.R.P. provided and ensured quality of high-resolution plasma data and assisted withthe preparation of the manuscript text. S.A.F. provided and ensured the quality ofthe plasma composition data. R.E.E. and R.B.T. provided and ensured the quality ofhigh-resolution electric field data and assisted with the preparation of the manuscripttext. R.J.S. and C.T.R. provided and ensured the quality of high-resolution fluxgatemagnetometer data. B.L.G., C.J.P. and J.L.B. provided institutional and mission-levelsupport of the analysis and ensured overall quality of MMS and FPI data.

Additional informationSupplementary Information accompanies this paper at http://www.nature.com/naturecommunications

Competing interests: The authors declare no competing financial interests.

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How to cite this article: Gershman, D. J. et al. Wave-particle energy exchangedirectly observed in a kinetic Alfven-branch wave. Nat. Commun. 8, 14719doi: 10.1038/ncomms14719 (2017).

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