Abstract
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 highresolution MMS observations.
Introduction
The 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 (ω_{ci}) and gyroradius (ρ_{i}), respectively. When the perpendicular spatial scale of an Alfvén wave approaches ρ_{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 fluidscale Alfvén wave to a kineticscale Alfvén wave (KAW) occurs at ρ_{i}∼1 and >k_{}, 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 kineticscale 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 electronpressuregradientdriven electric field (E_{p}=−∇·_{e}/(n_{e}e)) are 90° out of phase with one another, such that the plasma heating term, Δ, can be instantaneously nonzero 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 ΔE_{p} and ΔJ_{} in an ideal Alfvén wave. For kineticscale Alfvén waves, however, nonzero ΔE_{p} fluctuations enable the Landau resonance, where particles with V_{}∼ω/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 ΔJ_{} and ΔE_{p} become inphase such that the waveaveraged Δ(J_{}E_{p}) is nonzero^{3,16}. Likewise, fluctuations in ΔB_{} result in transittime 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 transittime damping, and enabling stable wave mode propagation^{4,17}.
The detailed properties of KAWs (for example, ΔJ, ΔE_{p}, k) have been difficult to characterize due to their small spatial and temporal scales with respect to the capabilities of laboratory or onorbit plasma instrumentation. Accurate estimates of current density and the characterization of particle populations require full threedimensional 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 interspacecraft 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 ΔJ ·ΔE, we provide a direct measurement of the conservative energy exchange between the wave’s electromagnetic fields and particles. A perpendicular spatial scale of ρ_{i}∼1, nonzero ΔE_{p} and ΔJ_{} fluctuations, and a parallel wave speed close to the local Alfvén speed confirm that the wave packet is an ionscale 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, −6.9, 0.9] R_{e} (1 R_{e}=1 Earth radius=6,730 km). Magnetic reconnection at the magnetopause boundary^{19,20} generated a southward flowing exhaust at ∼22:25 UT denoted by a −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 (∼1 Hz) waves were observed in the exhaust in a ∼4 min interval localized to a region of strong proton temperature anisotropy (/T_{H+}∼2). MMS partially crossed the magnetopause into the magnetosheath for the first time at ∼22:35 UT (not shown) at [8.0, −6.9, 0.9] R_{e}. For the subsequent ∼2 h, multiple magnetopause crossings resulted in the MMS spacecraft sampling both +V_{z} and −V_{z} jets, that is, above and below the reconnection site. However, ∼1 Hz waves were only observed in the short interval shown in Fig. 1. The MMS observatories were in a tetrahedron configuration (quality factor^{21} ∼0.9) separated by ∼40 km, a distance which corresponded to a local thermal ion gyroradius (ρ_{i}=35 km).
The reconnection exhaust plasma consisted of mostly H^{+} and some He^{2+} with number density ratio n_{He2+}/n_{H+}<0.02 throughout the interval. The local ratios of ion thermal parallel and perpendicular pressure to magnetic pressure were β_{}≈0.2 and ≈0.5, respectively. In addition, the average plasma flow velocity during this interval was V_{o}=[−17, 73, −183] km s^{−1}. This velocity corresponded to a jet flowing nearly antiparallel to the background magnetic field ([0.10, −0.52, 0.85] direction) with speed ∼0.5 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+}=10 cm^{−3} and B=55 nT, the local Alfvén speed was estimated to be 380 km s^{−1}. Variations were observed in the number density (Δn), bulk velocity (Δv_{e}), temperature (ΔT_{}, Δ) of both ions and electrons, and in the electric (ΔE) and magnetic fields (ΔB) (see Fig. 2). The amplitude of these ∼1 Hz fluctuations were nonlinear with Δn_{H+}/n_{H+}∼0.2. The magnetic field fluctuations exhibited both lefthanded and righthanded polarization (see Supplementary Fig. 1). Finally, bursts of electron phase space holes measured in the total parallel electric field (ΔE_{}) 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 nontrivial and often relies on multispacecraft 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) kvector estimation from current and magnetic field fluctuations measured in the spacecraft frame^{23,24}, (3) comparison of spacecraftmeasured gradients with their corresponding spacecraftaveraged 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 fourspacecraftaveraged electron velocity and magnetic field fluctuations. Alfvénbranch waves have parallel wave speeds close to the local Alfvén speed, that is, ω/k_{}≈V_{A} and correlated transverse fluctuations^{16}, =−(ω/k_{})Δ/B. Positively correlated (R^{2}=0.92) Δ and Δ indicated that ω/k_{}=−1.15±0.03 V_{A}, that is, the wave propagated antiparallel to the background magnetic field near the Alfvén speed (see Supplementary Fig. 2). Although qualitatively similar ∼1 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 ∼0.5 V_{A} and the anticorrelation 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 kvector was derived directly from fluctuations in ΔJ and ΔB 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 fourspacecraft average of ΔB and the average ΔJ determined from magnetometer data using the fourspacecraft ‘curlometer’ technique^{27}. The value of k at the frequency of maximum spectral power, 0.9 Hz, was k=[7.1 × 10^{−3}, −2.0 × 10^{−2},−2.2 × 10^{−2}] km^{−1}, which corresponded to a wavevector angle (θ) of ∼100° with respect to the background magnetic field and ρ_{i}∼1.0.
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 kvector determined from the phase differencing of the B_{X}, B_{Y} and B_{Z} fluctuations (using MMS3 as a reference) 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. Although similar phase shifts were observed in all components of ΔB 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 ρ_{i}∼1 and the phase differencing of B_{X} and B_{Y} components, those with the largest fluctuation power, both produced ω/k_{}=−1.1 V_{A}.
Finally, in the fourth method, the small MMS spacecraft separations and highquality 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 planewave approximation (that is, ‘∇·’≈ik and ‘∇ × ’≈ik × 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}−V_{e}), (2) magnetic field ‘curlometer’^{27}, that is, ∇ × B/μ_{o}, and (3) the planewave approximation, that is, ik × B/μ_{o}. All three estimates of ΔJ are shown in Fig. 4. k_{y} and k_{z} most strongly influenced the planewavederived currents such that this intercomparison was relatively insensitive to errors in the determination of k_{x}. The electronpressuregradientdriven electric field determined from four spacecraft measurements (that is, −∇·_{e}/(n_{e}e)), when compared with its planewave approximated value (that is, −ik·_{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 Xcomponent comparison demonstrates that k_{x} is of the correct sign but may underestimate the fourspacecraft gradient.
We adopted the kvector derived using the Bellan^{23,24} method k=[7.1 × 10^{−3}, −2.0 × 10^{−2}, −2.2 × 10^{−2}] km^{−1} because it simultaneously leveraged data from all four spacecraft and all components of the magnetic field. Allowing for ∼30% (3σ level) uncertainty in each individual component, we found ρ_{i}=1.02±0.07 with wavevector angle 104±4° from the magnetic field. The 0.9 Hz peak observed in the spacecraft frame (ω_{sc}) was then Dopplershifted by ω=ω_{sc}−k·V_{o} to obtain a frequency of ω/ω_{ci,H+}=0.61±0.08 in the plasma frame. We conclude that multiple independent methods indicated that MMS resolved a kineticscale Alfvénbranch wave.
Modelled wave growth rates
Growth rates (γ=Im{ω/ω_{ci}}) and polarization (Re{iE_{y}/E_{x}}) solutions along the Alfvénbranch dispersive surface were estimated using a linear dispersion solver and are shown as a function of θ in Fig. 5. The dispersion solver predicted that the large ion temperature anisotropy of /T_{i}∼2 produced a nearly monochromatic ion cyclotron wave mode that propagated parallel/antiparallel to the background magnetic field (θ=0°, 180°) with ω/ω_{ci}∼0.5, kρ_{i}∼0.4 and lefthanded 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 Alfvénbranch dispersion curves are shown in Fig. 5 as a function of kρ_{i} and θ. The observed KAW mode (ω/ω_{ci}=0.6, kρ_{i}=1, θ=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 {ω/ω_{ci}=0.3, kρ_{i}=1, θ=100°}, {ω/ω_{ci}=0.6, kρ_{i}=1.6, θ=100°} and {ω/ω_{ci}=0.6, kρ_{i}=1, θ=110°}. All of these nearby solutions were weakly damped (γ∼10^{−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 θ 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 planewave approximation for ΔE_{p}, the electronpressuregradientdriven electric field was estimated at a single spacecraft, for example, MMS4, using −ik ·_{e}/(n_{e}e). Fluctuations of ΔE_{p} and ΔJ 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 ΔE_{p} further confirmed the presence of kineticscale effects. These parallel fluctuations were an order of magnitude smaller than those in Δ as expected from KAW theory^{16}. Furthermore, fluctuations in all components of ΔJ and ΔE_{p} (both perpendicular and parallel) were each ∼90° out of phase with one another. These phase differences resulted in a nonzero instantaneous value of Δ(J · E_{p}) with ΔJ · E_{p}_{max}≈50 pW m^{−3} and nearzero waveaveraged Δ and Δ(J_{}E_{p}) quantities. These data demonstrated the conservative energy exchange between the particles and fields that comprise an undamped KAW.
Because ρ_{e}<<1, 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 ∼90° 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 ∼50% of the density fluctuations within the KAW. Although these electrons also resulted in a ∼20% increase in , they were not indicative of heating but rather of a nonlinear capture process.
The depth of the parallel potential well estimated from ΔE_{p} and k_{} was found to be ∼10 V (Fig. 7). In addition, the parallel magnetic field of the wave generated a mirror force that resulted in a kineticscale 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 ∼75° and ∼105° (B_{min}/B_{max}=0.96). To understand the combined effects of these forces, electrons measured in the magnetic minima were Liouvillemapped to other locations along the wave using various parallel potential well depths (Fig. 8). The fullwidth 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 Liouvillemapped distributions was found for a potential well depth of Φ_{max}=10 V. Such agreement provided additional validation of ΔE_{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_{}, that is, θ∼90°. This plasma heating was accompanied by significant reductions in field fluctuation power. The wave presented here had a somewhat higher frequency (ω_{ci,He2+}<ω<ω_{ci,H+}) than those considered in these previous KAW studies (ω<<ω_{ci,H+,} ω_{ci,He2+}). Furthermore, its comparatively nonperpendicular wavevector (θ≈100°) and large scale (ρ_{i}≈1) indicated that the observed wave was close to the transition point between ideal and kinetic regimes. Nonetheless, the wave had nonzero ΔJ_{} and ΔE_{p} fluctuations, confirming that it contained kineticscale structure not present in an ideal Alfvén wave. These observations demonstrated that the mere presence of a KAW or parallel electric field fluctuations do not necessarily imply heating via Landau damping. Only inphase fluctuations in ΔJ and ΔE_{p} result in such net transfer of energy from the wave field to the plasma particles.
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 transittime 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, outofphase ΔE_{p} and ΔJ_{} fluctuations and a finite wave amplitude for several wave periods (that is, γ<<1) indicated the absence of strong wave growth or damping. Although a hot core population (V_{th,e}≫ω/k_{}) does not lead to strong damping (Fig. 5), the velocity distribution function of electrons was not directly sampled at energies corresponding to V_{}∼ω/k_{} (that is, ∼0.5 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_{}∼ω/k_{}, amplifying damping rates. However, nonlinear KAW theories have predicted that trapped electrons with V_{}∼ω/k_{} lead to wave stabilization if their bounce frequency (ω_{B}) is significantly faster than the damping or growth rate, that is, ω_{B}/ω_{ci}≫γ^{4,17,32}. We estimated ω_{B}/ω_{ci}∼1 for this wave, consistent with such a criterion. Therefore, the presence of trapped electrons here could have contributed to nonlinear instability saturation in a singlemode wave even if there were low energy structure in the electron distribution function that was not resolved by MMS.
Finally, at higher frequencies (∼1 kHz), fluctuations in the total parallel electric field ΔE_{} 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 electronscale 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 ionscale ΔE_{p} fluctuations. Furthermore, electron holes may have been responsible for higher frequency contributions to Δ(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 ΔJ and ΔE_{p} were 90° out of phase with one another, leading to instantaneous nonzero Δ(J · E_{p}). Furthermore, we have discovered a significant population of electrons trapped within adjacent wave peaks by the combined effects of the parallel electronpressuregradientdriven electric field and the magnetic mirror force. In addition to contributing ∼50% of the density fluctuations in the wave, these trapped electrons may have provided nonlinear saturation of Landau and transittime 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 kineticscale 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 righthanded 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} × B_{3} direction and B_{2} completed the righthanded coordinate system, that is, B_{1} × B_{2}=B_{3}.
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+} is the mass of H^{+}. The ion gyrofrequency was calculated using,
The plasma thermal pressure was calculated using n_{H+}k_{B}T_{H+}. The magnetic pressure was calculated using B^{2}/2μ_{o} where μ_{o} is the magnetic permeability of free space. Finally, the Alfvén speed was calculated using
All calculations were done in SI units.
ΔV_{e}–ΔB correlations
The comparison of ΔV_{e} and ΔB was done in the direction of minimum current density fluctuations ([0.93, 0.32, 0.18]) such that ion and electron velocities were approximately equal. This minimum variance direction was nearly perpendicular to the background magnetic field direction b=[0.10, −0.52, 0.85].
Electric field measurements
The electric field in the electron frame was defined as E+V_{e} × B, where E was the measured electric field in the spacecraft frame^{23}. Since J is frame independent, this electronframe electric field is conveniently used for estimates of energy transfer, that is, plasma heating occurs when J ·( E+V_{e} × B)>0. 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+V_{e} × B at low frequencies. The individual amplitudes of E and V_{e} × B were measured to be on the order of several mV m^{−1}. Systematic uncertainty in both particle and fields measurements would have led to a challenging recovery of E+V_{e} × B because E+V_{e} × B<<E,V_{e} × B. Therefore, accurate direct estimates of J ·( E+V_{e} × 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}=−∇·_{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 E≈E_{p}.
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 VlasovMaxwell 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 ω and k. The plasma parameters used as input to the dispersion solver (assuming stationary plasma) were n_{e−}=10 cm^{−3}, B=55 nT, =T_{e}=35 eV, T_{H+}=175 eV and =350 eV. Wave polarization was calculated using the simulated electric field fluctuations as Re{iE_{x}/E_{y}}. Lefthand and righthand polarization corresponded to Re{iE_{x}/E_{y}}<0 and Re{iE_{x}/E_{y}}>0, respectively^{4}. No growth was observed for the slowmode or fastmode magnetosonic branches of the dispersion relation. Additional simulations were run to evaluate the influence of He^{2+} on the observed instability. Increased n_{He2+}/n_{H+} ratios up to 0.02 with T_{He2+}=550 eV reduced the maximum wave growth but did not alter the sharpness of the peak in kspace. No new wave modes appeared to be introduced into the system from the presence of the local He^{2+} 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=B_{min}/B_{max}=0.96, and a sinusoidal profile of electric potential Φ 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 ‘+’ and ‘−’ branches of equation (4) correspond to the sign of v_{}. For each (v_{,} ) point in the reconstructed skymap, equations (4 and 5) provided a point (v_{o,} ) that was used to map a phase space density in the reference distribution, that is, f(v_{,} )=f_{o}(v_{o,} ).
In the magnetic minimum (D=λ_{}/2), =1 and Φ=Φ_{o}=0. At the magnetic maximum (D=0, λ_{}), and Φ=−Φ_{max}, that is,
Finally, bounce frequencies (ω_{B}=1/τ_{B}) for trapped electrons were estimated using
where R was defined as the reflection point along the wave (that is, v_{} (R)=0). Electrons with pitch angles 75–90° and energies 100–400 eV produced bounce frequencies of 1.4±0.3 Hz (that is, ω/ω_{ci}=1.6±0.3) in a λ_{}=830 km wave with M=0.96.
MMS data sources and processing
Particle, magnetic field and electric field data were measured by the Fast Plasma Investigation^{39} (FPI), the Fluxgate Magnetometers^{40} and Electric Field Double Probe^{41} instruments, respectively. Corresponding composition data at ∼10 s time resolution was obtained from the Hot Plasma Composition Analyzer^{42}. Time series data were highpass filtered with a fifthorder digital Butterworth IIR filter with coefficients b=[0.85850229, −4.29251147,8.58502295, −8.58502295, 4.29251147, −0.85850229] and a=[1.0, −4.69504063,8.82614592, −8.30396669, 3.90989399, −0.73702619], where b and a correspond to the filter’s numerator and denominator polynomials listed in increasing order. This filter had an effective cutoff frequency of 0.5 Hz and no discernable effect (<1%) on the amplitude or phase of a 0.9 Hz input signal.
Data availability
Data used for this study is available to download from the MMS Science Data Center (https://lasp.colorado.edu/mms/sdc/) or from the corresponding author upon request.
Additional information
How to cite this article: Gershman, D. J. et al. Waveparticle energy exchange directly observed in a kinetic Alfvénbranch wave. Nat. Commun. 8, 14719 doi: 10.1038/ncomms14719 (2017).
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Acknowledgements
We thank the members of the FPI ground operations and science team for their feedback and support and Lynn Wilson for his insights about wave properties. This research was supported by the NASA Magnetospheric Multiscale Mission in association with NASA contract NNG04EB99C and by NSF award 1059519, AFOSR award FA95501110184 and DOE awards DEFG0204ER54755 and DESC0010471. S.J.S. is grateful to the Leverhulme Trust for its award of a Research Fellowship.
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D.J.G. conducted the majority of the scientific and data analysis and was responsible for initial preparation of the manuscript text. A.FV. assisted with the interpretation of wave signatures, 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 and with the preparation of the manuscript text. S.J.S. assisted with the Liouville mapping of electron data and preparation of the manuscript text. B.L., V.N.C., M.O.C., Y.S. and W.R.P. provided and ensured quality of highresolution plasma data and assisted with the preparation of the manuscript text. S.A.F. provided and ensured the quality of the plasma composition data. R.E.E. and R.B.T. provided and ensured the quality of highresolution electric field data and assisted with the preparation of the manuscript text. R.J.S. and C.T.R. provided and ensured the quality of highresolution fluxgate magnetometer data. B.L.G., C.J.P. and J.L.B. provided institutional and missionlevel support of the analysis and ensured overall quality of MMS and FPI data.
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Gershman, D., FViñas, A., Dorelli, J. et al. Waveparticle energy exchange directly observed in a kinetic Alfvénbranch wave. Nat Commun 8, 14719 (2017). https://doi.org/10.1038/ncomms14719
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