The In Situ Signature of Cyclotron Resonant Heating

Trevor A. Bowen,1, ∗ Jonathan Squire,2 Stuart D. Bale,1, 3 Ben Chandran,4 Die Duan,5 Kristopher G. Klein,6 Davin

Larson,1 Alfred Mallet,1 Michael D. McManus,1, 3 Romain Meyrand,2 J.L. Verniero,7 and Lloyd D. Woodham8

1Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA2Department of Physics, University of Otago, 730 Cumberland St., Dunedin 9016, New Zealand

3Physics Department, University of California, Berkeley, CA 94720-7300, USA4Department of Physics & Astronomy, University of New Hampshire, Durham, NH 03824, USA

5School of Earth and Space Sciences, Peking University, Beijing, 100871, China6Department of Planetary Sciences & Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721

7NASA Goddard Space Flight Center8Department of Physics, The Blackett Laboratory,Imperial College London, London, SW7 2AZ, UK

The dissipation of magnetized turbulence is an important paradigm for describing heating and en-ergy transfer in astrophysical environments such as the solar corona and wind; however, the specificcollisionless processes behind dissipation and heating remain relatively unconstrained by measure-ments. Remote sensing observations have suggested the presence of strong temperature anisotropyin the solar corona consistent with cyclotron resonant heating. In the solar wind, in situ magneticfield measurements reveal the presence of cyclotron waves, while measured ion velocity distributionfunctions have hinted at the active presence of cyclotron resonance. Here, we present Parker So-lar Probe observations that connect the presence of ion-cyclotron waves directly to signatures ofresonant damping in observed proton-velocity distributions. We show that the observed cyclotronwave population coincides with both flattening in the phase space distribution predicted by resonantquasilinear diffusion and steepening in the turbulent spectra at the ion-cyclotron resonant scale. Inmeasured velocity distribution functions where cyclotron resonant flattening is weaker, the distribu-tions are nearly uniformly subject to ion-cyclotron wave damping rather than emission, indicatingthat the distributions can damp the observed wave population. These results are consistent withactive cyclotron heating in the solar wind.

PACS numbers:

Introduction Observations of the solar corona revealplasma that is millions of degrees hotter than the black-body temperature of the solar surface. It is known thatthe energy required to heat the corona, and acceleratethe solar wind, most likely originates from solar mag-netic fields; however, the specific pathways to heating andparticle acceleration remain elusive [1]. The dissipationof Alfvenic turbulence has become a common paradigminvoked in coronal heating and solar wind acceleration[2–5]. Though the specific mechanisms governing dissi-pation remain under debate, inter-particle collisions inthe solar wind and corona are infrequent, implying thatthe processes must be collisionless in nature [6].

Possible mechanisms for dissipating Alfvenic turbu-lence at kinetic scales include damping through Landauor cyclotron resonance [7–11], stochastic heating [12], ormagnetic reconnection [13, 14]. It is additionally un-clear whether turbulent dissipation deposits energy di-rectly into ions directly, or if secondary processes relatedto kinetic [15–17] and fluid [18–21] instabilities, or in-termittent effects and coherent structures [22–25] playa significant role. Furthermore, the portion of energydeposited by these processes at ion scales, versus thatwhich is subject to a kinetic cascade and dissipated byelectrons, remains an open question [11, 26–29].

It is well known that the observed ion temperature pro-files in the solar wind require significant perpendicular

heating [19, 30], which is likely initiated at ion kineticscales, where particles interact efficiently with electro-magnetic waves [10, 28, 31–35]. Cyclotron resonant cou-pling of electromagnetic fluctuations with ion gyromotion[36], has received particular attention as a potential coro-nal heating mechanism [37–41].

Measurements of coronal ion temperature anisotropiesby ultraviolet spectroscopy suggest large T⊥/T‖, consis-tent with heating through cyclotron damping [39, 42–44].In the solar wind, observations of magnetic helicity at ionscales have been interpreted as evidence for active cy-clotron damping of quasi-parallel Alfvenic fluctuations,which contribute to turbulent heating [9, 10, 45, 46].Theoretical signatures of cyclotron resonance in particledistribution functions have been studied in the frame-work of quasilinear diffusion [27, 47, 48, 50]. Observa-tional evidence for cyclotron resonant diffusion has beenfound in signatures of the proton velocity distributionfunctions fp(v) observed by Wind and Helios [49, 51, 52].

The presence of ion-cyclotron waves has been welldocumented throughout the heliosphere both as soli-tary waves and as part of the background spectrum offluctuations [53–59]. While the generation of cyclotronwaves through instabilities has been widely discussed[56, 57, 59–61] and signatures of cyclotron resonant dis-sipation have been observed [6, 9, 10, 49, 52, 62, 63],definitive signatures of cyclotron resonant heating suffi-

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FIG. 1: (a) Magnetic field measurements from PSP/FIELDS.(b) Velocity measurements from PSP/SPANi. (c) Spectra ofmagnetic field data. Vertical lines at fΩ and fρ as well asspectral indices at -3/2,-4 and -2.7 are shown. Helicity σb isshown in blue. (d) Joint distribution of helicity with θvB withleft/right handed helicity in blue/red. Frequencies fΩ and fρand the approximate sign change in helicity f∗ are shown.

cient to power the solar wind have not been observed.In this Letter, we use quasilinear theory [47, 48] to

provide evidence for resonant cyclotron heating in thesolar wind. Through comparing theoretical signatures ofcyclotron damping with observed ion distributions andmeasurements of helicity and ion-scale waves, a com-pelling picture of ion-heating in the solar wind emerges.

Data The Parker Solar Probe (PSP) mission [64] at-tempts to answer many fundamental questions surround-ing coronal heating and solar wind acceleration. Recentwork from PSP has revealed both prevalent ion-scale elec-

tromagnetic waves [58, 59] and ion-distributions out ofthermal equilibrium [65, 66], and resonant wave-particleinteractions consistent with quasilinear theory [67].

PSP provides measurements of the inner heliosphereusing the electromagnetic FIELDS [68] and Solar WindElectron Alpha and Proton (SWEAP, [69]) instruments.We study a stream from PSP perihelion 4 from 2020-01-31/00:00-04:00 with well resolved core and beamobservations. Magnetic field data are obtained fromPSP/FIELDS [58]. We use merged search coil and flux-gate magnetometer data [70] enabling measurement ofthe inertial, transition, and kinetic scales of turbulence;the merged data set only has two axes available, thus weuse vector-fluxgate magnetometer data to study wave-polarization. Figure 1(a) shows B in the spacecraftframe. Proton velocity distribution functions fp(v) areobtained from the PSP/SWEAP Solar Probe ANalyzer(SPANi). Bi-Maxwellian fits to a proton core and field-aligned beam provide estimates of bulk velocity and tem-perature u, T‖,⊥, and the beam to core nb/nc protondensity ratio [66, 69]. Figure 1(b) shows measurementsof u in the spacecraft frame. The density of fp(v) iscalibrated to quasi-thermal noise from FIELDS [68, 71].

The stream is relatively slow with an average speed of∼200 km/s, and moderately Alfvenic with cross helicityof ∼0.85. The mean magnetic field was directed sunward,with an Alfven speed of 60 km/s. The mean proton den-sity is np = 1100/cm3, SPANi gives an average beam tocore density ratio of 0.48. The core/beam have T⊥ of17 eV/20 eV and T‖ of 12 eV/23 eV. The average driftof the beam relative to the core is 80 km/s. The indi-vidual core and beam have βc = 0.65 and βb = 0.92. Asingle component bi-Maxwellian proton population withequivalent macroscopic thermodynamic properties to thedrifting bi-Maxwellian fit has parallel and perpendiculartemperatures of 29 eV and 18 eV, with T⊥/T‖ < 1 andβ = 0.96 [66].

Cyclotron Waves A Morlet wavelet transform is ap-plied to the vector magnetic field data [72] and rotatedinto field aligned coordinates. Signatures of circularlypolarized waves can be found in the magnetic helicity

σB(f, t) = −2Im(B⊥1B∗⊥2)/(B2

⊥1 +B2⊥2), (1)

with left/right handed waves corresponding to posi-tive/negative helicity [54, 55, 73, 74].

Figure 1(c) shows the magnetic field spectra alongsidethe average σB over the measured interval; the measure-ments of σB are consistent with previous PSP observa-tions [75]. A steep transition range is observed at ion-kinetic scales [76–78]. We consider frequencies (assumingTaylor’s hypothesis) corresponding to the ion gyroscaleand cyclotron resonance of the thermal speed [32]

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mpv⊥pth(2)

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ΩpvA + v‖pth

. (3)

Helicity is measured in the spacecraft frame, such thatthe measured sign may not correspond to the innateplasma frame polarization [73]. A sign change in σB oc-curs if the wave is Doppler shifted to negative frequenciesin the spacecraft frame. However, it has been demon-strated that the majority of waves are outward propagat-ing; thus, Doppler shift does not change their handednesswhen observed in the spacecraft frame [79].

Figure 1(d) shows the joint distribution helicityσ(f, θvB), revealing the distribution of helicity as a func-tion of sampling direction. Both left- and right-handedmodes are evident at θvB ∼ 0 with right-handed modesoccurring at higher frequencies. The helicity inverts atapproximately f∗ = 3.6 Hz. At θvB ∼ 0 the mea-sured fluctuations correspond to k‖ fluctuations, whileat oblique θvB the measurements are dominated by k⊥modes [80, 81]. Observation of parallel propagating cir-cular polarized waves may be inhibited when the back-ground turbulent amplitude is sufficiently large or whenthe angle between the solar wind and the mean magneticfield is oblique such that polarization plane is out of theplane sampled by the spacecraft [73]. The lack of circularpolarization signatures at ion scales when θvB ∼ 1 rad isconsistent with sampling effects of quasi-parallel waves atoblique angles in anisotropic turbulence [59], suggestingthat ion-scale waves may persist at oblique θvB .Distribution Functions Figure 2 shows a distribution

function from the stream at 2020-01-30/03:54:55. Fig-ure 2(a) shows a interpolation of the 3D measurementsin the v⊥ − v‖ plane constructed by identifying values ofv⊥ and v‖ for each 3D energy bin assuming gyrotropy. Adrifting bi-Maxwellian fit to the data, without assuminggyrotropy, is shown in Figure 2(b). Figure 2(c) showsthe single bi-Maxwellian with macroscopic np, P‖, andP⊥ equal to the drifting bi-Maxwellian fit [66]. The dis-

tributions are shown in field aligned coordinates with B0

in the vertical direction. The drifting bi-Maxwellian fitsprovide good approximation of core and beam properties,the true distribution function is not exactly parameter-ized by the fit, as evidenced by the interpolated fp(v)falling to zero more slowly than the fit fp(v).

Resonant interaction of ion scale waves with particlescan significantly affect the shape of the distribution func-tion. The ion cyclotron resonance condition is

ω(k) = ±Ω + kv‖, (4)

such that outward propagating cyclotron waves resonatewith the inward propagating portion of the distributionfunction. The evolution of the distribution function in

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FIG. 2: (a) Gyrotropic interpolation of fp(v) from SPANi 3Ddata in the v⊥ − v‖ plane; data are in magnetic field alignedcoordinates, with the mean field pointing vertically. Pointsshow locations of SPANi measurements in the v⊥ > 0 half ofthe plane. The minimum v∗‖ resonant speed and a set resonantcyclotron diffusion contours are plotted. (b) Correspondingdrifting bi-Maxwellian fit to fp(v). (c) Corresponding sin-gle bi-Maxwellian fit with equivalent macroscopic thermody-namic properties. (d) Contours of fp(v) determined by inter-polating the gyrotropic distribution along diffusion contoursof parallel cyclotron resonance from quasilinear theory. (e-f) Contours of fp(v) from drifting and single population bi-Maxwellian fits evaluated along cyclotron resonance contours.Colored + marks v∗⊥ corresponding to v∗‖ , the minimal speedneeded for cyclotron resonant interaction for each contour;constant values of fp(v) are expected for v⊥ < v⊥∗.

the presence of resonant interactions can be describedby quasilinear diffusion theory [47, 48]. In a stationaryframe moving with the wave, the electric field of the waveis zero and particles conserve kinetic energy as they scat-ter off waves, tracing contours in v⊥ and v‖ defined by thewave dispersion relation and resonance condition [40, 47–

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49]. Defining y = k‖vA/Ω, the contours for parallel cold-plasma ion cyclotron waves [48, 82],

v2⊥v2a

+1

y2− sinh−1y/2 + lny = const, (5)

with y(v‖) defined implicitly through Equation 4, areoverlaid on the distribution functions in Figure 2(a-c).If fp(v) is a decreasing function along the scatteringcontours, diffusion heats the plasma in the range of fre-quencies where there are resonant waves. Conversely,if fp(v) is an increasing along the scattering contours,the distribution function will become unstable, leadingto the growth of outward propagating cyclotron wavesand cooling the plasma. Cyclotron resonant equilibriumcorresponds to a flattening of fp(v) along the contours.Figure 2(d-f) shows fp(v) evaluated along the quasilin-ear diffusion contours, parameterized on v⊥, for the in-terpolated data as well as the drifting bi-Maxwellian, andsingle bi-Maxwellian fits. The interpolated and driftingbi-Maxwellian fp(v) are characteristically flat along con-tours, suggesting the presence of cyclotron resonance.

The lack of left hand polarized waves above f∗ in-dicates a minimum v∗‖ for particles to interact reso-nantly with outward propagating cyclotron waves. Par-ticles with v < v∗‖ do not have a resonance at wave-numbers with significant power in the cyclotron mode;accordingly, quasilinear diffusion cannot heat the plasmathrough cyclotron resonance and the distribution func-tion should not flatten against the diffusion contours forv < v∗‖ . The minimum v∗‖ corresponding to f∗ is shown

in the distribution functions in Figure 2(a-c). The corre-sponding v∗⊥ for each contour is shown in Figure 2(d-f).

A statistical analysis of flattening is performed by com-puting the values of the distribution function evaluatedon contour paths, fp(v)|C , for 2048 measurements overthe studied interval. For each contour, the average valueof the derivative along the diffusion contour 〈∂vfp(v)|C〉is computed for both the resonant, v‖ > v∗‖ , and non-resonant, v‖ < v∗‖ portions of the contour. The analy-sis is performed for both the drifting bi-Maxwellian fitand the single bi-Maxwellian fit. Figure 3(a) shows thedistribution of 〈∂vfp(v)|C〉, over the resonant portion ofthe contour for the drifting bi-Maxwellian. The sign of〈∂vfp(v)|C〉 is near uniformly negative indicating thatthe curvature along the distributions should damp theobserved waves. Additionally, Figure 3(b) shows the ra-tio of the average derivative 〈∂vfp(v)|C〉 for the resonantpart of the diffusion contour to the non-resonant part ofthe diffusion contour. For the drifting bi-Maxwellian fit,the gradient in fp(v) along the contour is typically muchlarger in the non-resonant portion of the diffusion con-tour, indicating relative flattening in the cyclotron reso-nant portion of the contour. Figure 3(c,d) show corre-sponding data for the single bi-Maxwellian fit, which hasmuch larger curvature than the drifting bi-Maxwellian,

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FIG. 3: (a) Distribution of 〈∂vfp(v)|C〉 for the cyclotron reso-nant portion of the quasilinear diffusion contours for driftingbi-Maxwellian fits; black line shows 〈∂vfp(v)|C〉=0, dottedlines show values corresponding to the distribution in Fig-ure 2. (b) Distribution of ratios of 〈∂vfp(v)|C〉 of resonantv‖ > v∗‖ to non-resonant v‖ < v∗‖ portion of the diffusion con-tours; black line shows unity. (c-d) Corresponding data forsingle proton population bi-Maxwellian fits with equivalentthermodynamic properties.

and has steepest gradients along the v‖ > v∗‖ portion ofthe contour, contrasting signatures of resonance evidentin the two-component fit and interpolated fp(v).

Discussion Previous studies have suggested the sig-nificant role that cyclotron resonance may play in shap-ing magnetic field and helicity spectra [9, 10, 34, 46, 62]as well as particle distribution functions [49, 52]. Re-cent work has additionally shown signatures of quasilin-ear resonant diffusion of the proton beam with outwardpropagating right-handed modes [67]. Here, we show sig-nificant statistical flattening of the observed proton dis-tribution functions along contours of cyclotron resonancepredicted by quasilinear diffusion [47, 48, 50]. Flatteningalong quasilinear diffusion contours implies an equilib-rium with respect to the Alfven/ion cyclotron instability;in our observations, flattening is only observed in the in-terpolated and drifting (two component) bi-Maxwellianapproximations to the data, indicating that correctly un-derstanding growth/damping rates through quasilinaercontours requires accurate descriptions of fp(v) that maynot be possible with a single particle population.

Similar results have been previously reported in Helios[49] and Wind data [52]; however, our observations cou-ple the presence of flattened fp(v) along the cyclotron

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resonance diffusion contours to an observed spectrum ofcyclotron waves. The left handed wave spectrum, withcutoff at f∗, corresponds well to observed flattening inthe proton distribution, which is a significant predictionof cyclotron resonant quasilinear diffusion [47, 48]. Whilemany of the studied ion distributions are significantly flatalong the diffusion contours, those with curvature almostuniformly occur with a negative pitch-angle gradient, in-dicating that the distributions damp the observed popu-lation of cyclotron waves and do not contribute to theirgrowth through instabilities. These observations providesignificant evidence for the mediation of ion-scale dissi-pation in the solar wind through cyclotron resonance.

The origin of the observed cyclotron waves remains animportant unresolved point. Our observations show that95% of the time helicical signatures with σ > 0.5 arepresent, a relatively moderate left-handed polarization,the curvature of the distribution functions is negative,indicating the absorption of cyclotron resonant waves,which specifically suggests that the observed signaturecyclotron resonant signature corresponds to a heatingmechanism. While we have focused on the signature ofparallel cyclotron waves, the damping of oblique wavesmay play a role in heating [83, 84]; the oblique diffusioncontours are steeper [27], indicating the observed distri-butions are likewise subject to oblique damping.

There are two main physical origins for theseAlfven/ion cyclotron waves. First, they may be ex-cited by beam instabilities [85], though recent work sug-gests that the dominant instability associated with thestrong beam corresponds to right-handed modes [65, 66].Second, they may be generated by turbulence, thoughcanonical theories of Alfvenic nonlinearity preferentiallytransport energy to large k⊥ but not large k‖ [86, 87], pos-ing a difficulty to the turbulent generation of cyclotronresonant waves. However, recent work suggests that im-balance, i.e. the dominance of the outward Alfven mode,may prevent energy from cascading to kinetic scales [88].Fully kinetic simulations in the presence of such a bar-rier [82] show the generation of quasi-parallel cyclotronmodes, similar to those observed in the solar wind. Thisbarrier to imbalanced energy flux [82, 88] provides a novelmethod for generating cyclotron waves that can dissipateimbalanced turbulence that is consistent with a varietyof observations [58, 59, 78, 89–93]. Further work is re-quired to understand the origin of cyclotron waves andtheir connection to turbulence.

This work explicitly shows that distribution functionsin the solar wind contain evidence of cyclotron resonantheating. Whether the observed heating is sufficient topower the expanding solar wind remains an outstandingquestion; however, these results are significant to under-standing the underlying physics of collisionless heating inastrophysical plasmas.

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