Wave-particle energy exchange directly observed in a kinetic Alfvén-branch wave

Alfvén 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 electromagnetic 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 Alfvén 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.

T he Alfvén wave is a ubiquitous plasma wave mode wherein ions collectively respond to perturbations in the ambient magnetic field direction 1 . No net energy is transferred between the field and the plasma particles in ideal Alfvén waves. However, ion motion decouples from electron motion when wave dynamics are faster than ion orbital motion around the local magnetic field or are on scales smaller than the ion orbit size, defined by the gyrofrequency (o ci ) and gyroradius (r i ), respectively. When the perpendicular spatial scale of an Alfvén wave approaches r i , the wave can support significant parallel electric and magnetic field fluctuations that enable net transfer of energy between the wave field and plasma particles via Landau or transit-time interactions [2][3][4] .
The transition of an ideal fluid-scale Alfvén wave to a kineticscale Alfvén wave (KAW) occurs at k ? r i B1 and k ? 4k || , where k is the wavevector and '?' and '||' are defined with respect to the local magnetic field direction. These KAWs are essential for energy transfer processes in plasmas. Broadband KAWs have long been associated in space physics with turbulent heating in the solar wind and magnetosheath [5][6][7] and are also thought to account for a substantial amount of the energy input into Earth's auroral regions that can drive charged particle outflow and atmospheric loss [8][9][10][11][12][13] . In the laboratory, KAWs can transport energy away from the core regions of fusion plasmas, resulting in the unwanted deposition of energy at the reactor edges 14,15 . Understanding kinetic-scale wave generation, propagation and interaction with charged particles is critical to unraveling and predicting the relevant physics of these fundamental processes.
Alfvén wave theory predicts that transverse fluctuations in the current density (J) and electron-pressure-gradient-driven electric field (E p ¼ À =P e /(n e e)) are 90°out of phase with one another, such that the plasma heating term, D J ? E p? À Á , can be instantaneously non-zero but averages to zero over a wave period 1 . In such an undamped wave, power sloshes back and forth between the wave field and particles with no net energy transfer. There are no corresponding fluctuations in DE p|| and DJ || in an ideal Alfvén wave. For kinetic-scale Alfvén waves, however, non-zero DE p|| fluctuations enable the Landau resonance, where particles with V || Bo/k || can gain or lose energy through interaction with the wave field. These interactions, combined with an imbalance in the number of particles that are moving faster than or slower than the wave, result in net plasma heating or cooling 4 . Here, fluctuations in DJ || and DE p|| become in-phase such that the wave-averaged D(J || E p|| ) is non-zero 3,16 . Likewise, fluctuations in DB || result in transit-time damping effects, the magnetic analog of Landau damping, where the magnetic mirror force takes the place of E p 2,4 . For nonlinear KAWs, parallel fluctuations can be sufficiently large in amplitude to trap electrons between adjacent wave peaks. The oscillatory bounce motion of these electrons produces equal numbers of particles moving faster than or slower than the wave, limiting the effects of Landau and transit-time damping, and enabling stable wave mode propagation 4,17 .
The detailed properties of KAWs (for example, DJ, DE p , k) have been difficult to characterize due to their small spatial and temporal scales with respect to the capabilities of laboratory or on-orbit plasma instrumentation. Accurate estimates of current density and the characterization of particle populations require full three-dimensional distribution functions of both electron and ions on timescales faster than the wave frequency in the observation frame of reference. In addition, estimates of pressure gradients and wavevectors rely on multiple observation points being available within a single wave peak. However, NASA's recently launched Magnetospheric Multiscale (MMS) mission 18 consists of four identical observatories deployed in a tetrahedron configuration that measure charged particle and electromagnetic fields orders of magnitude more quickly than previous space missions. This increased temporal sampling combined with a small MMS inter-spacecraft separation enables plasma parameters and their spatial gradients to be determined at kinetic scales.
Here we use observations from MMS to characterize the microphysics of a monochromatic Alfvén wave. Through the calculation of DJDE, we provide a direct measurement of the conservative energy exchange between the wave's electromagnetic fields and particles. A perpendicular spatial scale of k ? r i B1, non-zero DE p|| and DJ || fluctuations, and a parallel wave speed close to the local Alfvén speed confirm that the wave packet is an ion-scale KAW. Finally, analysis of the velocity distribution function of electrons reveals a population that is nonlinearly trapped within the wave's magnetic minima. These trapped electrons may have enabled nonlinear saturation of damping processes, resulting in marginally stable wave propagation and providing evidence in support of early analytical theories of wave-particle interactions in collisionless plasmas.

Results
Event overview. On 30 December 2015, the four MMS observatories were near the dayside magnetopause, that is, the interface between the interplanetary magnetic field and the Earth's internal magnetic field, at [7.8, À 6.9, 0.9] R e (1 R e ¼ 1 Earth radius ¼ 6,730 km). Magnetic reconnection at the magnetopause boundary 19,20 generated a southward flowing exhaust at B22:25 UT denoted by a À V z jet, an increase in plasma density, and a decrease in plasma temperature (see Fig. 1). There was no discernable rotation in the magnetic field suggesting that the spacecraft constellation remained inside the Earth's magnetosphere throughout this interval. Low frequency (B1 Hz) waves were observed in the exhaust in a B4 min interval localized to a region of strong proton temperature anisotropy (T H þ ? /T H þ || B2). MMS partially crossed the magnetopause into the magnetosheath for the first time at B22:35 UT (not shown) at [8.0, À 6.9, 0.9] R e . For the subsequent B2 h, multiple magnetopause crossings resulted in the MMS spacecraft sampling both þ V z and À V z jets, that is, above and below the reconnection site. However, B1 Hz waves were only observed in the short interval shown in Fig. 1. The MMS observatories were in a tetrahedron configuration (quality factor 21 B0.9) separated by B40 km, a distance which corresponded to a local thermal ion gyroradius (r i ¼ 35 km).
The reconnection exhaust plasma consisted of mostly H þ and some He 2 þ with number density ratio n He2 þ /n H þ o0.02 throughout the interval. The local ratios of ion thermal parallel and perpendicular pressure to magnetic pressure were b || E0.2 and b ? E0.5, respectively. In addition, the average plasma flow velocity during this interval was V o ¼ [ À 17, 73, À 183] km s À 1 . This velocity corresponded to a jet flowing nearly anti-parallel to the background magnetic field ([0.10, À 0.52, 0.85] direction) with speed B0.5 V A , where V A is the Alfvén speed, that is, the characteristic speed in which information can be transferred along a magnetic field. For this interval, with n H þ ¼ 10 cm À 3 and B ¼ 55 nT, the local Alfvén speed was estimated to be 380 km s À 1 . Variations were observed in the number density (Dn), bulk velocity (Dv e ), temperature (DT || , DT ? ) of both ions and electrons, and in the electric (DE) and magnetic fields (DB) (see Fig. 2). The amplitude of these B1 Hz fluctuations were nonlinear with Dn H þ /n H þ B0.2. The magnetic field fluctuations exhibited both left-handed and right-handed polarization (see Supplementary Fig. 1). Finally, bursts of electron phase space holes measured in the total parallel electric field (DE || ) were bunched with the wave in locations of strong electron pressure gradients.
Wave properties. Accurate determination of the wavevector (k) was critical to identify the observed wave mode. In situ estimation of k, especially for broadband wave spectra, is non-trivial and often relies on multi-spacecraft techniques 22 . Fortunately, the monochromatic nature of the observed wave enabled the application of several independent methods of wavevector determination. Here we utilized four methods to provide a robust estimate of k: (1) parallel component of the wavevector derived from the correlation between velocity and magnetic field fluctuations 16 , (2) k-vector estimation from current and magnetic field fluctuations measured in the spacecraft frame 23,24 , (3) comparison of spacecraft-measured gradients with their corresponding spacecraft-averaged quantities, that is, the planewave approximation 4 , and (4) phase differencing of the magnetic field fluctuations between each spacecraft 25 .
In the first method, we estimated the parallel component of the wavevector through comparison of four-spacecraft-averaged electron velocity and magnetic field fluctuations. Alfvén-branch waves have parallel wave speeds close to the local Alfvén speed, that is, |o/k || |EV A and correlated transverse fluctuations 16 , DV e? ¼ À (o/k || )DB ? /B. Positively correlated (R 2 ¼ 0.92) DV e? and DB ? indicated that o/k || ¼ À 1.15 ± 0.03 V A , that is, the wave propagated anti-parallel to the background magnetic field near the Alfvén speed (see Supplementary Fig. 2). Although qualitatively similar B1 Hz fluctuations have been observed near Earth's bow shock that are more consistent with magnetosonic wave modes 26 , a parallel phase speed well above the local sound speed of B0.5 V A and the anti-correlation between density and magnetic field fluctuations were inconsistent with slow and fast magnetosonic wave modes, respectively.
In the second method, we combined fluctuations of current and magnetic field in the spacecraft frame to estimate k as a function of frequency using spectral techniques recently developed by Bellan 23,24 . Here the k-vector was derived directly from fluctuations in DJ and DB measured in the spacecraft frame (see Fig. 3). Although this technique could have been applied to data from a single spacecraft, in order to maximize spectral resolution we used the four-spacecraft average of DB and the average DJ determined from magnetometer data using the four-spacecraft 'curlometer' technique 27 . The value of k at the frequency of maximum spectral power, 0.9 Hz, was k ¼ [7.1 Â 10 À 3 , À 2.0 Â 10 À 2 , À 2.2 Â 10 À 2 ] km À 1 , which corresponded to a wavevector angle (y) of B100°with respect to the background magnetic field and k ? r i B1.0.
In the third method, we used the phase difference 25 measured between each pair of MMS spacecraft for each component of the magnetic field to derive additional estimates of k. At the spectral peak of 0.9 Hz, the k-vector determined from the phase   differencing of the B X , B Y and B Z fluctuations (using MMS3 as a reference) were: [ À 7.4 Â 10 À 5 , À 8.5 Â 10 À 3 , À 1.5 Â 10 À 2 ], [2.9 Â 10 À 2 ,4.7 Â 10 À 3 , À 1.1 Â 10 À 2 ], and [2.3 Â 10 À 2 , À 3.5 Â 10 À 3 , À 1.0 Â 10 À 2 ] km À 1 , respectively. Although similar phase shifts were observed in all components of DB between MMS2, MMS3 and MMS4, there were significantly different shifts of MMS1 with respect to the other observatories for each component (see Supplementary Fig. 3). These differences demonstrated that this wave packet was not truly planar and exhibited spatial structure on the order of an ion gyroradius. Because MMS1 was farthest from the magnetopause (that is, the X direction), the k X component was most strongly affected by this structure. Despite this discrepancy, all determinations of k result in k ? r i B1 and the phase differencing of B X and B Y components, those with the largest fluctuation power, both produced Finally, in the fourth method, the small MMS spacecraft separations and high-quality tetrahedron formation enabled gradients of particle and field quantities to be estimated directly from the MMS data. These gradients were compared with those predicted by the plane-wave approximation (that is, 'r'Eik and 'r Â 'Eik Â at a single frequency 4 ) to both evaluate the validity of this approximation to the observed wave packet and to provide further validation of k (see Fig. 4). The current was calculated from three methods: (1) direct particle observations, that is, en e (V i À V e ), (2) magnetic field 'curlometer' 27 , that is, r Â B/m o , and (3) the plane-wave approximation, that is, ik Â B/m o . All three estimates of DJ are shown in Fig. 4. k y and k z most strongly influenced the plane-wave-derived currents such that this intercomparison was relatively insensitive to errors in the determination of k x . The electron-pressure-gradient-driven electric field determined from four spacecraft measurements (that is, À rP e /(n e e)), when compared with its plane-wave approximated value (that is, À ikP e /(n e e)), provides further confidence in the determination of k (see Fig. 4). Here all three components of k contributed to this result. The X-component comparison demonstrates that k x is of the correct sign but may underestimate the four-spacecraft gradient. We adopted the k-vector derived using the Bellan 23,24 method k ¼ [7.1 Â 10 À 3 , À 2.0 Â 10 À 2 , À 2.2 Â 10 À 2 ] km À 1 because it simultaneously leveraged data from all four spacecraft and all components of the magnetic field. Allowing for B30% (3-s level) uncertainty in each individual component, we found k ? r i ¼ 1.02 ± 0.07 with wavevector angle 104 ± 4°from the magnetic field. The 0.9 Hz peak observed in the spacecraft frame (o sc ) was then Doppler-shifted by o ¼ o sc À kV o to obtain a frequency of o/o ci,H þ ¼ 0.61±0.08 in the plasma frame. We conclude that multiple independent methods indicated that MMS resolved a kinetic-scale Alfvén-branch wave.
Modelled wave growth rates. Growth rates (g ¼ Im{o/o ci }) and polarization (Re{iE y /E x }) solutions along the Alfvén-branch dispersive surface were estimated using a linear dispersion solver and are shown as a function of y in Fig. 5. The dispersion solver predicted that the large ion temperature anisotropy of T i? /T i|| B2 produced a nearly monochromatic ion cyclotron wave mode that propagated parallel/anti-parallel to the background magnetic field (y ¼ 0°, 180°) with o/o ci B0.5, kr i B0.4 and left-handed polarization. At increasingly oblique wavevector angles, the predicted wave growth was substantially reduced. There was no slow or fast magnetosonic wave growth predicted for the measured plasma parameters. Several Alfvén-branch dispersion curves are shown in Fig. 5 as a function of kr i and y. The observed KAW mode (o/o ci ¼ 0.6, kr i ¼ 1, y ¼ 100°) was close to but not precisely on the solution surface. Nearby Alfvénic solutions to the measured data (matching two of the three wave parameters) were All of these nearby solutions were weakly damped (|g|B10 À 2 ) such that local generation of the observed KAW was not predicted by linear wave theory. However, local spatial gradients of plasma density may have increased the y of the ion cyclotron mode during its propagation, converting it into an oblique Alfvén wave 4 . Furthermore, nonlinear effects and parametric forcing (for example, magnetopause motion) were not taken into account by the homogenous dispersion solver, yet may have played a role in the evolution of the observed KAW.
Wave-particle interactions. Given the demonstrated validity of the plane-wave approximation for DE p , the electron-pressuregradient-driven electric field was estimated at a single spacecraft, for example, MMS4, using À ikP e /(n e e). Fluctuations of DE p and DJ in magnetic coordinates on MMS4 are shown in Fig. 6. In addition to the transverse electric field fluctuations expected for all Alfvén waves, fluctuations in DE p|| further confirmed the presence of kinetic-scale effects. These parallel fluctuations were an order of magnitude smaller than those in DE p? as expected from KAW theory 16 . Furthermore, fluctuations in all components of DJ and DE p (both perpendicular and parallel) were each  (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 k ¼ [7.1 Â 10 À 3 , À 2.0 Â 10 À 2 , À 2.2 Â 10 À 2 ] km À 1 (red). (d-g) MMS-averaged DE p and D(JE p ) 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 differences resulted in a non-zero instantaneous value of D(JE p ) with D|JE p | max E50 pW m À 3 and near-zero wave-averaged D J ? E p? À Á and D(J || E p|| ) quantities. These data demonstrated the conservative energy exchange between the particles and fields that comprise an undamped KAW.
Because k ? r e o o1, electrons should have remained magnetized throughout the wave packet. Close examination of the electron velocity distribution function in the parallel wave frame revealed three distinct populations of electrons in the wave packet: (1) an isotropic thermal core, (2) suprathermal beams counterstreaming along the magnetic field, and (3) trapped particles with near B90°magnetic pitch angles (Fig. 7). Thermal and counterstreaming electrons are commonly observed in the magnetopause boundary layer in the absence of analogous wave activity 28 . However, trapped electron distributions are atypical of ambient boundary layer plasmas. Furthermore, these trapped electrons were dynamically significant: they accounted for B50% of the density fluctuations within the KAW. Although these electrons also resulted in a B20% increase in T e? , they were not indicative of heating but rather of a nonlinear capture process. The depth of the parallel potential well estimated from DE p|| and k || was found to be B10 V (Fig. 7). In addition, the parallel magnetic field of the wave generated a mirror force that resulted in a kinetic-scale magnetic bottle between successive wave peaks. This mirror force supplemented the force from the wave's parallel electric field, enabling trapping of electrons with magnetic pitch angles between B75°and B105°(B min /B max ¼ 0.96). To understand the combined effects of these forces, electrons measured in the magnetic minima were Liouville-mapped to other locations along the wave using various parallel potential well depths (Fig. 8). The full-width at half maximum distance along the wave at a pitch angle of 90°was calculated for each potential and compared with the measured data. The best match between measured and Liouville-mapped distributions was found for a potential well depth of |F max | ¼ 10 V. Such agreement provided additional validation of DE p|| and k || . In addition, these distributions demonstrated that the effect of the parallel electric field was to confine magnetically trapped electrons closer to magnetic minima.

Discussion
KAWs in turbulent space plasmas are thought to account for heating of plasmas at kinetic scales [5][6][7] . In previous studies 29, 30 , such waves were found to have k ? ck || , that is, yB90°. This plasma heating was accompanied by significant reductions in field fluctuation power. The wave presented here had a somewhat higher frequency (o ci,He2 þ oooo ci,H þ ) than those considered in these previous KAW studies (oo oo ci,H þ , o ci,He2 þ ). Furthermore, its comparatively non-perpendicular wavevector (yE100°) and large scale (k ? r i E1) indicated that the observed wave was close to the transition point between ideal and kinetic regimes. Nonetheless, the wave had non-zero DJ ||  igure 5 | Modelled dispersion curves for the local plasma environment. (a) The real part of o/o ci , that is, the wave oscillation frequency, (b) the imaginary part of o/o ci , that is, the wave growth/damping rate and (c) the real part of iE x /E y , that is, the polarization of the wave, as a function of scaled wavevector magnitude kr i . Coloured curves correspond to solutions of a linear dispersion relation solver taken along the Alfvén 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, y ¼ 0°, 180°) at o/o ci B0.5 and kr i B0.4 and is left-hand polarized (that is, Re{iE x /E y }o0). A transition to right-hand polarization (that is, Re{iE x /E y }40) occurred at yB130°. No strong growth or damping was predicted for the observed KAW In linear KAW theory, the electrostatic field formed by parallel gradients in electron pressure enables the energization of particles via the Landau resonance 4,13,16 . Similarly, the transit-time resonance becomes relevant for systems where there are parallel gradients in magnetic field magnitude. Despite the presence of these field gradients in the observed KAW, out-of-phase DE p|| and DJ || fluctuations and a finite wave amplitude for several wave periods (that is, |g|o o1) indicated the absence of strong wave growth or damping. Although a hot core population (V th,e c|o/k || |) does not lead to strong damping (Fig. 5), the velocity distribution function of electrons was not directly sampled at energies corresponding to V || Bo/k || (that is, B0.5 eV). Electrons at these low energies are often present as they serve to neutralize a ubiquitous population of 'hidden' cold ions that flow out from the ionosphere 31 . Such ionospheric electrons may have added structure to the velocity distribution function near V || Bo/k || , amplifying damping rates. However, nonlinear KAW theories have predicted that trapped electrons with V || Bo/k || lead to wave stabilization if their bounce frequency (o B ) is significantly faster than the damping or growth rate, that is, o B /o ci c|g| 4,17,32 . We estimated o B /o ci B1 for this wave, consistent with such a criterion. Therefore, the presence of trapped electrons here could have contributed to nonlinear instability saturation in a single-mode wave even if there were low energy structure in the electron distribution function that was not resolved by MMS.
Finally, at higher frequencies (B1 kHz), fluctuations in the total parallel electric field DE || associated with electron phase space holes 33 were bunched in phase with the low frequency wave packet (Fig. 1). Because these structures persisted outside of the KAW interval (not shown), it is unlikely that they were related to its initial generation. However, the location of these electron-scale structures within the wave was coincident with the location of electron pressure gradients, suggesting that they could have contributed, in an average sense, to some of the observed ion-scale DE p|| fluctuations. Furthermore, electron holes may have been responsible for higher frequency contributions to D(J || E || ) in the form of nonlinear and turbulent terms in the electron momentum equation 34 .
Using MMS data, we have experimentally confirmed the conservative energy exchange between an undamped kinetic Alfvén wave field and plasma particles: fluctuations of all three components of DJ and DE p were 90°out of phase with one another, leading to instantaneous non-zero D(JE p ). Furthermore, we have discovered a significant population of electrons trapped within adjacent wave peaks by the combined effects of the parallel electron-pressure-gradient-driven electric field and the magnetic mirror force. In addition to contributing B50% of the density fluctuations in the wave, these trapped electrons may have provided nonlinear saturation of Landau and transit-time damping. The monochromatic nature of the wave enabled a direct comparison of observations with linear and nonlinear KAW theories. It is crucial to understand these dynamics to predict the evolution of kinetic-scale waves in laboratory fusion reactors, planetary magnetospheres and astrophysical plasmas.

Methods
Coordinate systems. The coordinate system used in this study (unless otherwise noted) was the Geocentric Solar Ecliptic (GSE) coordinate system, where the X direction pointed towards the Sun along the Earth-Sun line, the Z direction was oriented along the ecliptic north pole and the Y direction completed the right-handed coordinate system 35 . Local 'magnetic coordinates' were derived from GSE vectors where B 3 was parallel to the local magnetic field direction, B 1 was in the X GSE Â B 3 direction and B 2 completed the right-handed coordinate system, that is, Pressuregradient-driven electric field quantities were inferred from the k-vector and electron pressure tensor from MMS4 using the plane-wave approximation (that is, E p ¼ À ikP e /n e e). 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(JE p ), which provided confirmation of the conservative energy exchange between the wave field and plasma particles. The amplitude of D J ? E p?
ð Þwas an order of magnitude higher than D(J || E p|| ). The wave-averaged D(JE p ) 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. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14719 ARTICLE Calculation of plasma parameters. The thermal gyroradius was calculated using where k B is Boltzmann's constant, e is the elementary charge and m H þ is the mass of H þ . The ion gyrofrequency was calculated using, The plasma thermal pressure was calculated using n H þ k B T H þ . The magnetic pressure was calculated using B 2 /2m o where m o is the magnetic permeability of free space. Finally, the Alfvén speed was calculated using All calculations were done in SI units. Electric field measurements. The electric field in the electron frame was defined as E þ V e Â B, where E was the measured electric field in the spacecraft frame 23 . Since J is frame independent, this electron-frame electric field is conveniently used for estimates of energy transfer, that is, plasma heating occurs when J(E þ V e Â B)40. At the scales relevant for this KAW packet, electrons remained magnetized such that electron inertia and anomalous resistivity contributions to the electric field were neglected and the pressure gradient term should have been the dominant contributor to E þ V e Â B at low frequencies. The individual amplitudes of E and V e Â B were measured to be on the order of several mV m À 1 . Systematic uncertainty in both particle and fields measurements would have led to a challenging recovery of E þ V e Â B because |E þ V e Â B|o o|E|,|V e Â B|. Therefore, accurate direct estimates of J(E þ V e Â B) were not recovered for this event. Instead, here we focussed on effects of the electric field generated by the divergence of the electron pressure tensor, that is, E p ¼ À rP e /(n e e) and validated the measurement using multiple methods. In the electron frame, the electrons are not moving so there is no magnetic term in the electron equation of motion giving EEE p . The reference value for the potential (F ¼ 0) 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 r i . The ratio of the minimum to maximum magnetic field magnitude was B min /B max ¼ 0.96, 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 (D ¼ 0) and (g) at the magnetic minimum (D ¼ l || /2) in the wave frame of reference (that is, all measured velocities shifted by À V A along the magnetic field direction). An illustration of three corresponding populations of electrons is shown in V || À V ? space in panel (h). Thermal (energies below T e E35 eV) electrons have nearly isotropic pitch-angle distributions (blue contours). Suprathermal (energies above T e ) 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 T e 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 that interact with ions and electrons at their respective scales, we used the linear dispersion solver PLADAWAN 36 (PLAsma Dispersion And Wave ANalyzer) to solve the linearized Vlasov-Maxwell system for arbitrary wavevector directions. Using measured plasma parameters of ions and electrons, the dispersion solver produced growth rates and wave properties as functions of o and k. The plasma parameters used as input to the dispersion solver (assuming stationary plasma) were n e À ¼ 10 cm À 3 , B ¼ 55 nT, T e? ¼ T e|| ¼ 35 eV, T H þ || ¼ 175 eV and T H þ ? ¼ 350 eV. Wave polarization was calculated using the simulated electric field fluctuations as Re{iE x /E y }. Left-hand and right-hand polarization corresponded to Re{iE x /E y }o0 and Re{iE x /E y }40, respectively 4 . No growth was observed for the slow-mode or fast-mode magnetosonic branches of the dispersion relation. Additional simulations were run to evaluate the influence of He 2 þ on the observed instability. Increased n He2 þ /n H þ ratios up to 0.02 with T He2 þ ¼ 550 eV reduced the 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 presence of the local He 2 þ population.
Liouville mapping and electron bounce motion. Under the assumption that electron phase space density f(v) was conserved along particle trajectories throughout the wave interval (that is, Liouville's theorem), we used f(v) measured in the magnetic minimum, defined as f o (v), a sinusoidal profile of the magnetic field strength B with M ¼ B min /B max ¼ 0.96, and a sinusoidal profile of electric potential F to infer the velocity distribution along the wave 37,38 . Velocity space was transformed using equations and where the 'o' subscripts denote values at the magnetic minimum of the wave. The ' þ ' and ' À ' branches of equation (4) Finally, bounce frequencies (o B ¼ 1/t B ) for trapped electrons were estimated using where R was defined as the reflection point along the wave (that is, v || (R) ¼ 0  Liouville-mapped distributions are shown for |F| max ¼ 0, 5, 10, 15, 20 and 25 V (a-g). These distributions were constructed using measured phase space densities at the magnetic minimum (that is, D ¼ l || /2). The mirror ratio of B min /B max ¼ 0.96 confined particles to pitch angles between 75°a nd 105°in all cases. The parallel potential formed from DE p|| 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| max ¼ 10 V, which was consistent with independent estimates of k || and DE p|| .