The Electromagnetic Signature of Outward Propagating Ion-Scale Waves
Trevor A. Bowen, Stuart D. Bale, J. W. Bonnell, Davin Larson, Alfred Mallet, Michael D. McManus, Forrest Mozer, Marc Pulupa, Ivan Vasko, J. L. Verniero
DDraft version May 25, 2020
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The Electromagnetic Signature of Outward Propagating Ion-Scale Waves
Trevor A. Bowen, Stuart D. Bale,
1, 2, 3, 4
J. W. Bonnell, Davin Larson, Alfred Mallet, Michael D. McManus,
1, 2
Forrest Mozer,
1, 2
Marc Pulupa, Ivan Vasko, and J. L. Verniero (The PSP/FIELDS and PSP/SWEAP Teams) Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA Physics Department, University of California, Berkeley, CA 94720-7300, USA The Blackett Laboratory, Imperial College London, London, SW7 2AZ, UK School of Physics and Astronomy, Queen Mary University of London, London E1 4NS, UK
ABSTRACTFirst results from the Parker Solar Probe (PSP) mission have revealed ubiquitous coherent ion-scalewaves in the inner heliosphere, which are signatures of kinetic wave-particle interactions and fluid-scale instabilities. However, initial studies of the circularly polarized ion-scale waves observed by PSPhave only thoroughly analyzed magnetic field signatures, precluding a determination of solar-windframe propagation direction and intrinsic wave-polarization. A comprehensive determination of wave-properties requires measurements of both electric and magnetic fields. Here, we use full capabilities ofthe PSP/FIELDS instrument suite to measure both the electric and magnetic components of circularlypolarized waves. Comparing spacecraft frame magnetic field measurements with the Doppler-shiftedcold-plasma dispersion relation for parallel transverse waves constrains allowable plasma frame polar-izations and wave-vectors. We demonstrate that the Doppler-shifted cold-plasma dispersion has a max-imum spacecraft frequency f ∗ sc for which intrinsically right-handed fast-magnetosonic waves (FMWs)propagating sunwards can appear left-handed in the spacecraft frame. Observations of left-handedwaves with | f | > f ∗ sc are uniquely explained by intrinsically left-handed, ion-cyclotron, waves (ICWs).We demonstrate that electric field measurements for waves with | f | > f ∗ sc are consistent with ICWspropagating away from the sun, verifying the measured electric field. Applying the verified electricfield measurements to the full distribution of waves suggests that the vast majority of waves propagateaway from the sun (in the plasma frame), indicating that the observed population of coherent ion-scalewaves contains both intrinsically left and right hand polarized modes. INTRODUCTIONUnderstanding observational signatures of energytransfer between electromagnetic waves and particle dis-tribution functions will constrain the kinetic processescontributing to heating and acceleration in coronal andsolar wind plasma. Instabilities in the solar wind arethought to drive coherent circularly polarized waves atboth ion and electron scales (Gary 1992, 1993; Garyet al. 2000; Kasper et al. 2002; Hellinger et al. 2006;Podesta & Gary 2011b; Verscharen et al. 2016; Yoon2017; Klein et al. 2018; Verscharen et al. 2019; Tong etal. 2019a,b; Verniero et al. 2020). At ion scales, the coldplasma approximation allows two circularly polarizedtransverse electromagnetic (EM) modes: left-hand po-
Corresponding author: Trevor A. [email protected] larized ion cyclotron waves (ICWs) and the right-handedpolarized fast-magnetosonic waves (FMWs) (Stix 1992;Gary 1993). However, spacecraft observations of trans-verse waves are often limited to single point magneticfield measurements, precluding determination of intrin-sic plasma-frame polarization. Single point magneticfield measurements only estimate the handedness offluctuations in the spacecraft frame (Narita et al. 2009;Howes, & Quataert 2010).In the solar wind, large (supersonic) flow speeds signif-icantly modify spacecraft frame observations of ion-scaleelectromagnetic waves through Doppler shift (Fredricks,& Coroniti 1976; Klein et al. 2014). While the spacecraftframe polarization may be determined through waveletor Fourier methods, the wave propagation direction inthe plasma frame may be difficult to infer from theDoppler-shifted observations (He et al. 2011; Podesta &Gary 2011a; Klein et al. 2014b; Wicks et al. 2016; Wood-ham et al. 2019). Wave vectors of transverse waves are a r X i v : . [ phy s i c s . s p ace - ph ] M a y commonly determined using a minimum variance anal-ysis (MVA) of the magnetic field, which gives the di-rection of minimum variance as the wave-vector prop-agation direction ˆ k (Sonnerup, & Cahill 1967; Means1972; Santol´ık et al. 2003; Jian et al. 2009; Vernieroet al. 2020). However, MVA analysis, and eigenvec-tor/eigenvalue determinations, cannot distinguish wavepropagation direction parallel or anti-parallel the mini-mum variance direction (i.e. ˆ k from -ˆ k ) through mag-netic field observations alone. Together, the degeneratedetermination of sunward/anti-sunward propagation di-rection and the presence of large Doppler shifts precludeknowledge of the plasma frame polarization from singlepoint magnetic field measurements. Rigorous constraintof dynamical processes, which generate and govern theevolution of these waves, inevitably requires an abilityto discern FMW from ICW modes as well as the wave-vector propagation direction.Including electric field measurements significantlyincreases the feasibility of wave-mode identification(Santol´ık et al. 2003; Bellan 2012, 2016). Bale et al.(2005) use electric field measurements to demonstratethe Alfv´enic nature of solar wind turbulence. Salemet al. (2012) additionally use the anisotropy of thespacecraft frame measurements of E/B to suggest thatsolar wind turbulence is consistent with a cascade ofquasi-perpendicular Alfv´enic turbulence. Stansby etal. (2016) use electric field measurements to determinewave-vectors and frequencies in the solar wind frameto provide an empirical determination of the whistlerdispersion relation. In the magnetosphere, observationsof E and B have long been used simultaneously to studycoherent wave phenomena (Cattell et al. 1991; Chastonet al. 1998, 2002).First results from PSP have revealed the presenceof quasi-parallel propagating ion-scale waves with bothleft and right hand polarizations, which are preferen-tially observed with radial alignments of the magneticfield (Bale et al. 2019; Bowen et al. 2020a). Vernieroet al. (2020) show the simultaneous existence of coher-ent circularly polarized waves with unstable 3D ion-distributions in the PSP/SPAN-ion data. While numer-ical solutions to warm-plasma dispersion suggest thatboth FMW and ICW are potentially driven by insta-bilities, the specific mode composition of the observedwaves has yet to be rigorously constrained (Verniero etal. 2020).The circular polarized events in the PSP data sharequalities with wave phenomena observed in the so-lar wind at 1 AU by the STEREO spacecraft: quasi-parallel propagation, mixed handedness at ion scales,and preference for radially aligned mean fields (Jian et al. 2009). Previous inner-heliosphere measurements ofcoherent ion-scale waves by MESSENGER and
Helios revealed significant scaling in the amplitudes and occur-rence rates of ion-scale waves (Jian et al. 2010; Boardsenet al. 2015).Jian et al. (2010) propose that populations of left andright-handed waves correspond to ICW modes propa-gating in opposite directions; the transit time differencebetween inward (sunward) and outward (anti-sunward)propagating waves are due to solar wind expansion, re-sulting in different radial scalings for each polarizationsignature. Boardsen et al. (2015) note that left-handedion-scale waves are more common than the right-handedwaves, and that the amplitude of left-hand waves scaleswith δB ∼ r − , consistent with WKB-like propaga-tion suggested by Hollweg (1974). Meanwhile, the right-handed waves show significantly shallower radial scalingthan the left-handed waves (Boardsen et al. 2015).Coherent waves at ion-scales are generated by in-stabilities associated with deviations of particle veloc-ity distribution functions from a Maxwellian distribu-tion (Marsch 2006). The generation of left-handedion-cyclotron waves are commonly associated with a T ⊥ /T (cid:107) > T (cid:107) /T ⊥ > T ⊥ /T (cid:107) > T ⊥ /T (cid:107) <
1, drives intrinsically right-handedwaves in-wards (Podesta & Gary 2011b). Telloni et al.(2019) show strong correlations with the proton tem-perature anisotropy and polarization, suggesting thatdamping of MHD turbulence leads to anisotropic distri-butions, which subsequently emit parallel propagatingICWs.Temperature anisotropy alone may not produce co-herent wave signatures seen in the solar wind (Gary etal. 2016; Verniero et al. 2020). The presence of proton(or α -particle) beams introduce extra asymmetry intothe velocity distribution, and subsequently more free en-ergy, which may drive waves (Gary 1991; Marsch 1991,2006; Podesta & Gary 2011a). Differential flow speedsbetween the beam and the core (as well as density ratiosand beam temperature anisotropies) provide additionalconstraints on the instability thresholds leading to wavegeneration (Verscharen et al. 2013; Verniero et al. 2020).There is significant observational evidence for the im-pact of the proton beam on wave growth. Gary et al.(2016) demonstrate that theoretical growth rates fromobserved proton distribution functions may be domi-nated by the effect of the beam. Wicks et al. (2016)show that the beam drift correlates well with the am-plitude of coherent-wave “storms” at 1 au. Zhao et al.(2019) suggest, based on statistical observations, thatboth temperature anisotropy and core-beam drift con-tribute to the growth of left-handed waves at 1 au. InPSP data, Verniero et al. (2020) demonstrate correla-tions between the presence of a proton beam and signif-icant ion-scale wave activity (Bowen et al. 2020a).PSP provides electromagnetic measurements throughthe FIELDS instrument (Bale et al. 2016). Magneticfields measurements are made through both a low-frequency flux-gate magnetometer and a high frequencysearch coil magnetometer (Bowen et al. 2020b). Ad-ditionally, the Digital FIELDS Board (DFB) samplesthe four FIELDS antenna in the plane of the spacecraftheat-shield, producing both single ended and differentialvoltage waveforms at a survey cadence of up to 292.969Sa/Sec (Malaspina et al. 2016). Mozer et al. (2020)outline the calibration of the electric field E sc from dif-ferential measurements, where a frequency dependenteffective length α ( f ) with E sc = ∆ V ij /α is constructedusing observations of coherent waves at ion-scales andabove. However, knowledge of the wave phase speed,and thus a specific wave mode and polarization, is re-quired to perform a precise correction.The linearized Faraday equation provides the funda-mental relation between E and B for electromagneticoscillations. In the solar wind plasma frame, E sw = − v ph ˆ k × B (1)and v ph = ω/ | k | , such that measurements of B andthe E sw components perpendicular to k uniquely deter-mine both the wave phase speed and propagation vector.However, in a frame moving relative to the solar wind,i.e. the spacecraft frame, the electric field E sc includescontribution both from the solar wind electric field E sw and convection: E sc = E sw − V sw × B . (2)For a transverse quasi-parallel electromagnetic wave,the spacecraft frame electric field is given by, E sc = − v ph ˆ k × B − V sw × B (3) E sc = − ( v ph ˆ k + V sw ) × B (4) E sc = − V eff × B , (5)with the effective velocity V eff corresponding to thetotal wave speed (convected plus phase speed) measuredin the spacecraft frame. In addition to in situ electromagnetic field measure-ments, PSP makes measurements of the plasma distri-bution functions with the Solar Wind Electron Alphaand Proton (SWEAP) instrument suite (Kasper et al.2016). SWEAP can determine the bulk plasma flow V sw as well as thermodynamic moments of the plasma,such as density n and temperature T . Integrating mea-surements from FIELDS and SWEAP provide a set ofmeasurements capable of constraining the observed dis-tributions of wave-modes and propagation directions ofcoherent polarized waves at ion scales. DATAData is obtained from PSP FIELDS and SWEAP onNov 04, 2018. A continuous wavelet transform, similarto Bowen et al. (2020a), is used to locate circularly po-larized ion-scale events. Magnetic and electric fields areconvolved with a set of wavelets ψ ( ξ ) normalized to unitenergy W j ( s, τ ) = N − (cid:88) i =0 ψ (cid:18) t i − τs (cid:19) B j ( t i ) (6)with the un-normalized Morlet wavelet ψ ( ξ ) defined as ψ ( ξ ) = π − / e − iω ξ e − ξ (7)where ω = 6, with the relationship between waveletscale and spacecraft frequency approximated as f ≈ ω πs f s . The local, scale dependent, mean magnetic fieldis computed using a Gaussian envelope of each wavelet(Bowen et al. 2020a; McManus et al. 2020). The scaledependent mean field determines a local right-handedfield aligned coordinate system for each scale ˆ W =( ˆ B ⊥ , ˆ B ⊥ , ˆ B ).The polarization relative to B is computed as σ B ⊥ = (cid:104)− B ⊥ B ∗⊥ ) (cid:105)(cid:104) B ⊥ + B ⊥ (cid:105) (8)where (cid:104) ... (cid:105) denotes time averaging over a period de-fined by the e -folding of the wavelet (i.e. the cone ofinfluence (Torrence & Compo 1998)).Measurement of a left-hand circularly polarized wavegives σ B ⊥ = 1, while a right-hand circularly polarizedwave gives σ B ⊥ = −
1. For each wavelet scale, we iden-tify a polarized event if | σ B ⊥ | ≥ .
95, where the startand end times of the event are defined at the transition | σ B ⊥ | = 0 . x - y plane–where z is approx-imately sun pointing. Once wave events are identifiedin the magnetic field, in the mean-field aligned system( ⊥ , ⊥ , (cid:107) ), the subsequent analysis is performed in thespacecraft coordinate system ( x , y , z ) in order to facili-tate comparison with electric field measurements.As electric field measurements are limited to thespacecraft x - y plane, we recompute the magnetic he-licity measured along the spacecraft x and y axes, σ Bxy ,for each interval where σ B ⊥ meets the condition for cir-cular polarization. Differences in σ B ⊥ between σ Bxy re-sult from variations in the angle between the magneticfield and solar wind flow θ BV ; accordingly, not all inter-vals which meet the σ B ⊥ polarization condition main-tain helicity in σ Bxy . To account for this difference, weonly consider intervals with at least half the magneticfield measurements with a helicity of | σ Bxy | > .
85. Theelectric field polarization σ Exy is additionally computed.Figure 1 shows four sample events (one event per row)with helical electromagnetic fields. The first columnshows hodograms of the wavelet filtered magnetic fieldin the x − y plane. The second column shows the cor-responding wavelet filtered electric field in the x − y plane. The third and fourth columns show measured E x vs B y and E y vs B x . For transverse circularly polar-ized waves travelling in the z direction, Equation 1, indi-cates that a linear relationship with E x /B y ≈ v phz and E y /B z ≈ − v phz , where v phz is in the measured (space-craft) frame; v ph ( x,y ) and B z are taken as small terms.Respective root-mean-square (rms) ratios E rmsx /B rmsy ,e.g. E rmsx = (cid:113) (cid:104) E x (cid:105) , are plotted, in addition to the linecorresponding to v ph = V sw . In these several cases, themeasured phase speed of the waves is larger than thelocal solar wind speed. Additionally the sign of slope ofthe line indicates outward propagation in the spacecraftframe.We note that by measuring helicity in the spacecraftcoordinate system, we have removed the dependence on B , which is necessary to determine the plasma-framepolarization. However, during the first perihelion, PSPwas connected to a small coronal hole of negative polar-ity, with a field in the + ˆz spacecraft direction (the solarwind flow is approximately in the − ˆz direction) (Bale etal. 2019; Badman et al. 2020). Since the measurementsare taken with a single (positive) background-field direc-tion, no rectification is necessary to account for sectorstructure associated with heliospheric polarity (Wood-ham et al. 2019; McManus et al. 2020; Badman et al.2020). However, deviations from a + ˆz oriented meanfield occur during large-scale magnetic field switchbacks,in which the mean field deflects, and the polarizationplane of the quasi-parallel waves is no longer alignedwith the x - y plane: i.e. σ Bxy significantly deviates from σ B ⊥ (Dudok de Wit et al. 2019). Circularly polarizedevents occurring within switchbacks are then typically excluded by the condition σ Bxy > .
85, as the polariza-tion plane is then perpendicular to the spacecraft z -axis.Figure 2(a) shows the distribution of measured phasedifference between spacecraft x and y magnetic fieldcomponents for events with σ Bxy > .
85. A value of+90 ◦ corresponds to left-handed polarization and − ◦ corresponds to right-handed polarization. Figure 2(b)shows the corresponding distribution for electric fieldmeasurements. While the electric field measurementsare similarly peaked at ± ◦ , corresponding to circu-lar polarization, the respective peaks are much broader.There are a significant number of measurements thathave helical magnetic fields, but no helical signaturein the electric field. This is possibly a result of in-creased noise in the electric field measurements arisingfrom either the innate non-orthogonality of the sensors,or the optimization routines which generate electric fieldfrom the measured differential potentials (Mozer et al.2020). Additionally, we do not exclude the possibil-ity that physical dynamics other than parallel propa-gating transverse waves, e.g. an electrostatic compo-nent or non-parallel propagation, may be relevant insome cases. We limit our study to measurements with | σ Exy | > .
5, respectively shown in red (blue) for left(right) handed polarizations, such that our discussion ofcircular polarization maintains connection with physi-cally realized measurements. In the magnetic field 7153events with σ Bxy > .
85 (over all scales) were measured, | σ Exy | > . DISPERSION & DOPPLER SHIFTThe cold plasma approximation, commonly used tomodel transverse electromagnetic plasma waves, has adispersion-relation with two branches corresponding todifferent plasma-frame polarizations of the electric fieldrelative to the mean background field: (cid:18) ω ± Ω i (cid:19) = (cid:20) kd i (cid:18)(cid:113) k d i + 4 ± kd i (cid:19)(cid:21) (9)where Ω i = eB /m i is the ion gyrofrequency and d i = V A / Ω i is the ion inertial length. The branchwith ω F M = ω + corresponds to the FMW wave withan intrinsic right-handed polarization, and the ω ICW = ω − branch corresponds to the left-handed ion-cyclotronwave; Figure 3(a) shows the two branches of the disper-sion relation. Note that we take an approximation with -10 0 10B x [nT]-10010 B y [ n T ] σ Bxy =-1.05.8Hz -5 0 5E x [10 -3 V/m]-505 E y [ - V / m ] σ Exy =-0.9
Ex X By = +Z -> (Inward) -10 0 10B y [nT]-505 E x [ - V / m ] E rmsx /B rmsy Ey X Bx = -Z (Outward) -10 0 10B x [nT]-505 E y [ - V / m ] -340km/sV swz E rmsy /B rmsx -10 0 10B x [nT]-10010 B y [ n T ] σ Bxy =-1.04.7Hz -5 0 5E x [10 -3 V/m]-505 E y [ - V / m ] σ Exy =-0.9
Ex X By = +Z -> (Inward) -10 0 10B y [nT]-505 E x [ - V / m ] E rmsx /B rmsy Ey X Bx = -Z (Outward) -10 0 10B x [nT]-505 E y [ - V / m ] -350km/sV swz E rmsy /B rmsx -10 0 10B x [nT]-10010 B y [ n T ] σ Bxy = 1.05.8Hz -5 0 5E x [10 -3 V/m]-505 E y [ - V / m ] σ Exy = 0.8
Ex X By = +Z -> (Inward) -10 0 10B y [nT]-505 E x [ - V / m ] E rmsx /B rmsy Ey X Bx = -Z (Outward) -10 0 10B x [nT]-505 E y [ - V / m ] -330km/sV swz E rmsy /B rmsx -20 0 20B x [nT]-20020 B y [ n T ] σ Bxy = 1.01.7Hz -10 0 10E x [10 -3 V/m]-10010 E y [ - V / m ] σ Exy = 0.9
Ex X By = +Z -> (Inward) -20 0 20B y [nT]-10010 E x [ - V / m ] E rmsx /B rmsy Ey X Bx = -Z (Outward) -20 0 20B x [nT]-10010 E y [ - V / m ] -284km/sV swz E rmsy /B rmsx Figure 1.
Four example intervals with significant helical signatures. Each row shows one interval. Left-most (first) columnshows the hodogram of magnetic fluctuations in x - y plane. The wavelet frequency and time-span of each event are noted inthe first column. Second column shows corresponding hodogram of electric-field fluctuations in x - y plane. Third column showsphase diagrams of E x and B y ; rms E x /B y is shown in a dashed green line, with the total rms E/B shown as solid purple line.Right-most (fourth) column shows phase diagrams of E y and B x ; the measured rms E y /B x is shown in a dashed green line,with the total rms E/B in purple. The measured solar wind speed along the z -direction is plotted in yellow. ω < Ω e , which is quite appropriate for the low-frequencywaves under consideration.In the spacecraft frame, these polarized transverseelectromagnetic waves appear at the Doppler-shifted fre-quency 2 πf = ω ( k ) + k · V sw . (10) For quasi-parallel propagation, characteristic of thetransverse ion-scale waves measured in the solar wind(Jian et al. 2010; Boardsen et al. 2015; Bowen et al.2020a), Equation 10 reduces to2 πf = ω ( k ) ± kV sw cosθ BV , (11)where θ BV is the angle between the mean magnetic fieldand solar wind flow. B Helicity -180 -90 0 90 180
Figure 2. (a) Distribution of phase difference between B x and B y for circularly polarized intervals. (b) Distribution ofphase difference between E x and E y for circularly polarizedintervals. Black curve shows the total measured distribu-tion for σ Bxy > .
85, while the red and blue distributionsshow the respective conditions of | ± σ Exy | > . ± corresponds to left/right (red/blue) helicity measured in thespacecraft frame. Observations from the solar wind in the inner-heliosphere and at 1 AU reveal a preference for occur-rence of waves when the mean field is aligned with thesolar wind flow direction, i.e. θ BV ∼ θ BV ∼ π (Jianet al. 2010; Boardsen et al. 2015; Bale et al. 2019). Fewwaves are observed when π/ < θ BV < π/
4, thoughthis is likely a sampling effect due to single spacecraftmeasurements of quasi-parallel waves in anisotropic tur-bulence Bowen et al. (2020a).The spacecraft frame frequency f sc (in units of Hz) isa positive definite quantity such that the observationsat Dopper-shifted frequencies f , or − f , are both ob-served at f sc = | f | . The plasma frame frequency, ω ( k ),is intrinsically positive; however, the Doppler-shifted fre-quency of a wave to a negative value of f , causes achange of sign in the measured helicity in the Doppler-shifted (spacecraft) frame. For the two parallel propa-gating ICW and FMW modes, observation of a wave atthe spacecraft frequency f sc most generally correspondsto one of eight cases ± πf sc = ω ICW ± kV sw cosθ BV (12) ± πf sc = ω F M ± kV sw cos θ BV , (13)where each ± sign on either side of the equations corre-spond to a set of two equations.Several cases with no real solutions and can be dis-carded a priori: with the convention of positive fre-quency ω ( k · V sw < k cannot produce a negative spacecraft frequency. Thus, outward propagating FMWand ICW ( k · V sw >
0) are never Doppler-shifted tonegative frequency, and always appear in the space-craft frame with their intrinsic plasma-frame polariza-tion. Additionally, consideration of the ICW dispersionin the low k limit gives a phase speed v ICWph = V A ,with v ph monotonically decreasing with increasing k .Under the condition V A < V sw , inward propagatingICWs are observed at negative spacecraft frequencies–with an observed right-hand polarization in the space-craft frame (we note that in future perihelion encoun-ters, the V A < V sw ordering may not hold and a moregeneral approach must be taken).Figure 3(b) shows the Doppler shifted cold plasmaICW and FMW dispersion equations with measured Ω i , d i , and V sw , from an interval with significant signa-ture of left-hand polarization at 09:28 on November, 042018. For a wave at a given spacecraft frequency f sc there are five possible combinations of Doppler shiftedwave-modes: outward (inward) propagating ICW ap-pearing at a positive (negative) f . Outward propagatingFMW waves appear at a positive f and inward propa-gating FMW waves may appear at either positive ornegative f . From Equation 10, the inward propagat-ing FMW appears right-handed as long as its plasmaframe frequency is larger than the Doppler shift, i.e. ω F M > | k · V sw | . The inward propagating FMW willoccur as a left hand wave in the spacecraft frame at afrequency if ω F M < | k · V sw | .Clearly, measurements of the spacecraft-frame helicityof the magnetic field do not uniquely determine wave-mode and propagation direction. However, in somecases the cold-plasma dispersion can constrain wave-modes within a range of frequencies. Specifically, onlytwo possible cases lead to left handed spacecraft-framepolarization, e.g. Figure3(b): outward propagating ioncyclotron waves and inward propagating FMW waveswith | k · V sw | > ω F M . However, outside of the limit k di << ω F M grows faster than k (e.g. Equation 9)while the Doppler shift term is proportionate with − k .Continuity of the dispersion relation imposes a criticalwave number at which the FMW phase speed is greaterthan the solar wind speed, such that Doppler shift can-not change the sign of the handedness in the spacecraftframe. Thus a minimum value exists f ∗ , i.e. a maxi-mum spacecraft frequency f ∗ sc = | f ∗ | at which, an in-ward propagating FMW can appear left-handed, e.g.Figure 3(b). Figure 3(c) shows the curve of f ∗ sc as afunction of V sw cos θ/V A , determined through Equations9 and 10. A left-handed wave observed at | f | > f ∗ sc isout of the possible frequency range for an inward prop-agating, Doppler shifted, FMW and is thus uniquelyexplained by an outward propagating ICW.As a specific example, the curves in Figure 3(b) showDoppler shifted cold dispersion equations with measuredΩ i , d i , and V sw . The minimum negative frequency fora FMW with left-handed spacecraft frame polarizationis approximately 2 Hz. However, left-hand circularlypolarized power extends up to ∼ | f | > f ∗ sc allow for a directcomparison of FIELDS observations of E sc , B , and thederived V eff with the values predicted by cold plasmadispersion. Figure 4(a) shows the number of circularlypolarized measurements on Nov, 04 2019 at each waveletscale. For each interval, the maximum spacecraft fre-quency that a FMW wave can appear left handed ( f ∗ sc )is determined using measured values of d i , V sw , and thecold plasma dispersion equation (Equation 9). The dis-tribution of all waves with | f | > f ∗ sc is shown in blue,and the distribution of left-hand polarized waves with | f | > f ∗ sc is additionally shown in red–we take this as ameasured distribution of outward propagating ICWs.For each of these ICW events, the minimum variancedirection of B is determined and taken as the wave prop-agation direction (Sonnerup, & Cahill 1967; Means 1972;Santol´ık et al. 2003; Verniero et al. 2020). The observa-tion of | f | > f ∗ sc and left handed polarization breaks thedegeneracy of the MVA determination, and the mini-mum eigenvector corresponding to outward propagationis chosen as k . The cold plasma phase speed corre-sponding to the intersection of a Doppler shifted ICWwith the wave frequency f ∗ sc is determined through aNewton-Raphson root finding algorithm. Estimation ofthe phase speed allows construction of a synthetic solarwind frame electric field ˜ E sw obtained from B as˜ E ICWsw = − v ICW ˆ k × B . Subsequently, a model spacecraft frame electric field isgiven as ˜ E sc = ˜ E ICWsw − V sw × B (14)˜ E sc = − ( v ICW ˆ k + V sw ) × B (15)˜ E sc = − ˜ V ICWeff × B (16)Knowledge that waves with left hand polarization inthe spacecraft frame with | f | > f ∗ sc must be on the ICWbranch, allows for comparison of the theoretical ˜ E sc tothe measured E sc . Equation 16, predicts the rms quan-tities measured in the spacecraft frame, i ω / Ω i LHRH
ICWFM/W a 0 1 2 3 4 5kd i -10-50510 f [ H z ] b1 2 3 4 5 6 7 8V sw cos θ /V Alf f * /f Ω i ω * sw / Ω i f* sc /f Ω i c0.01 0.10 1.00 10.00 100.00f Hz0.0000.01010.000 T r a c e B n T / H z -8 -7 -6 T r a c e E m V / H z d σ BxBy f sc σ BxBy
Figure 3. (a) ICW (solid) and FMW (dashed) cold-plasmadispersion curves in the solar wind frame. Red and Bluecorrespond to the left/right handed transverse wave polar-ization. (b) Doppler shifted cold plasma ICW and FMW dis-persion into the spacecraft frame using measured d i and V sw ;four curves are shown, outward propagating FMW and ICW(i,ii), and inward propagating FMW and ICW (iii,iv), eachcurve has been Doppler shifted outward by the solar wind.The minimum spacecraft frequency of the inward propagat-ing FMW is shown in orange. For | f | < f ∗ sc there are at mostsix possible intersections of the Doppler shifted dispersion re-lations with f sc . For | f | > f ∗ sc there are four possible inter-sections between | f sc | and the Doppler shifted Dispersion re-lation; of the four, only one is left hand polarized. (c) Showsthe curve of f ∗ sc /f Ω i as a function of the ratio V sw cos θ BV / V A for a parallel propagating wave; the corresponding curve inthe plasma frame is shown in purple ω ∗ sc / Ω i . (d) Shows acoherent circularly polarized wave packet with left-hand po-larization | f | > f ∗ sc . (d) Wavelet spectrogram of circularpolarization shows left-hand polarization with | f | > f ∗ sc in-dicating that the wave must be an ICW. C oun t s All CountsNo LH S/C Frame FM/WLH Wave + No LH S/C Frame FM/W a0 2 4 6 8 10f [Hz]0123 M ed i an E / E γ x γ y b ~ Figure 4. (a) Distribution of measured ion-scale waves on04 November, 2018 (black). The subset of measured eventswhich occur at a frequency | f | > f ∗ sc is shown in blue. Ad-ditionally, the subset of waves with frequency | f | > f ∗ sc andhave a left-handed polarization is shown in red. (b) Mean ra-tio between measured rms electric fields for left-handed waveswith | f | > f ∗ sc and theoretical rms electric fields, assumingoutward propagating ICW. At each wavelet scale; error barsshow one standard deviation. The x and y antennas areshown in purple/orange. ˜ V z + = (cid:115) (cid:104) ˜ E scx (cid:105)(cid:104) B scy (cid:105) ˜ V z − = (cid:115) (cid:104) ˜ E scy (cid:105)(cid:104) B scx (cid:105) . Additionally, we test the frequency dependent effec-tive length factors applied in Mozer et al. (2020) bycomparing the theoretical ICW electric fields with theempirical measurements through calculation ofusing γ x = (cid:115) (cid:104) E x (cid:105)(cid:104) ˜ E x (cid:105) (17) γ y = (cid:115) (cid:104) E y (cid:105)(cid:104) ˜ E y (cid:105) (18)(19)Figure 4(b) shows the mean γ x and γ y measured ateach wavelet scale, with error bars given by one standarddeviation; only scales with more than 10 counts wereevaluated. The correction factor γ x is systematicallylarger than γ y indicating that measured electric fields are systematically larger in the x direction than the y direction. The larger uncertainties at higher frequenciesare likely the result of measuring low amplitude polar-ized signals at the edge of broadband wave peak: Bowenet al. (2020a) show that the proton-gyroscale ( ∼ γ into the effective antenna lengths givecorrections α (cid:48) x = αγ x and α (cid:48) y = αγ y . A scale-by-scale gain correction is then applied to themeasured electric fields. As these correction factorsare of order unity, their omission does not significantlychange the subsequent results–or more importantly, ourconclusions.Application of γ x and γ y to waves with a left-handspacecraft frame polarization enables measurements ofrms speeds V z + = (cid:115) (cid:104) ˜ E scx (cid:105)(cid:104) B scy (cid:105) V z − = (cid:115) (cid:104) ˜ E scy (cid:105)(cid:104) B scx (cid:105) . which constrains the mode and wavevector compositionof the wave population. Figure 5 shows distributionsof synthetic ˜ V z ± and measured ˜ V z ± , normalized to V sw ,with correction factors γ x and γ y applied; as the correc-tion factors are of order unity, their omission does notsignificantly change the subsequent results.Figure 5(a) shows the distribution of measured V z + /V sw for waves with | f | > f ∗ sc , along with thesynthetic distribution ˜ V z + /V sw constructed from thetheoretical properties for outward propagating ICWs.Figure 5(b-c) shows the distribution for left handedwaves, but with | f | < f ∗ , for which inward propagatingFMW may appear left-handed. In Figure 5(b) we com-pare the measured distribution to synthetic quantities˜ V z + /V sw corresponding to outward propagating ICW;Figure 5(c) shows synthetic quantities corresponding toinward propagating FMW waves (two roots are gener-ally available). Notably, the distribution of measured V z + /V sw for frequencies | f | < f ∗ is similar to the syn-thetic distribution of outward-propagating ICW.Similar distributions for measured V z − /V sw and syn-thetic ˜ V z − /V sw are computed with identical results aspresented for V z + /V sw in Figure 5(a-c). Due to the re-dundancy in results producing an additional figure, weelect not to display both sets of distributions. Analy-sis of V z + and V z − both strongly suggest that observeddistribution of left-hand population of waves have aneffective speed similar to outward propagating ICWs.Figure 5(d-f) show measured distributions of V z + /V sw for all observed right-handed waves. Synthetic distri-butions ˜ V z + /V sw for outward propagating FMW wavesare shown in Figure 5(d), while synthetic distributionsfor inward propagating ICW are shown in Figure 5(e).Synthetic data corresponding to an inward- propagatingFM, which maintains right-hand polarization is shownin Figure 5(f). The measured distribution of data againlargely has V z + /V sw >
1, consistent with a dominantpopulation of outward propagating FMW. Careful in-spection of the measured distributions in Figure 5(d-f)reveals the presence of two peaks, which we identify ascorresponding to two separate wind streams in the data.Separate populations are plotted for V sw >
320 km/sand V sw <
320 km/s. The synthetic data does not re-solve two separate distributions. In either case, bothdistributions are roughly consistent with outward prop-agating FMW waves. Interestingly, no bi-modality asso-ciated with stream-speed is observed in the left-handedwaves.Again, data for V z − /V sw were similarly analyzed.However, identical results are obtained as that from Fig-ure 5(d-f), suggesting that the right-hand population ofwaves are propagating in the spacecraft frame with ef-fective speeds similar to outward propagating FMW. Toavoid redundancy, the distributions of V z − /V sw are notshown. DISCUSSIONPSP observations of circularly polarized waves in thespacecraft frame suggest that effective wave speeds aretypically larger than the bulk solar wind flow, indicat-ing that most waves are likely outward propagating.While sub-dominant populations are present with effec-tive speeds less than the solar wind speed, Figure 5(a-c),it is not clear whether this occurs as the result of sta-tistical noise, or whether a subdominant population ofinward propagating FMW is actually present. Furthercase studies may reveal the existence of individual in-ward propagating transverse waves.Outward propagating waves, which are Doppler-shifted to larger spacecraft frequencies by the solarwind, (with our convention of positive plasma framefrequency ω ), cannot occur at a negative frequency inthe spacecraft frame. This suggests that, typically, noinversion of helicity occurs due to Dopper shift: e.g.left-hand ICWs observed in the spacecraft frame retaina left handed polarization. Thus, for the vast majority of measured waves, it is likely that the measured helicitycorresponds to their intrinsic plasma-frame polarization.Doppler shifting the cold-plasma dispersion relationreveals a maximum frequency, f ∗ sc for which inward-propagating right-hand FMWs appear left handed inthe spacecraft frame. Left-handed waves above thisfrequency are uniquely described by ICWs propagatingoutward. The measured electric field of waves which areleft-handed and have f > f ∗ sc , are to within our cer-tainty, consistent with theoretical predictions for ICWmade by the cold plasma dispersion. This analysis ismade possible by an empirical verification of electricfield calibration presented in Mozer et al. (2020) inthe range of ion-scales.Many previous studies utilize the low wave-numberapproximation kd i (cid:28)
1, in which v ph ∼ V A , in order tostudy ion scale waves (Jian et al. 2009, 2014; Gary et al.2016; Wicks et al. 2016; Woodham et al. 2019; Bowenet al. 2020a). However, the low wave number approxi-mation is not broadly applicable, as these waves occurat ion-scales with kd i ∼
1; our consideration of the fullplasma dispersion presents a significant advancement inunderstanding ion-scale waves. Verniero et al. (2020)similarly pursued a more explicit estimate of k , throughDoppler-shifting the warm-plasma dispersion relation todetermine the intersection with spacecraft frame fre-quency. The method of Doppler-shifting the dispersionrelation curves is a promising method to infer intrinsicwave handedness, and has the advantage of being inde-pendent of the validity of the Taylor hypothesis.The FMW and ICW dispersion relations are both non-linear at kd i ∼
1, e.g. Figure 3, implying that theobserved ion-scale waves should be subject to disper-sive effects. However, broadband wave packets are com-monly observed, which contain a range of wave-phasespeeds, e.g. Figure 3(e). Over relatively short time-scales, broadband wave-packets at k di ∼ kd i = [0 . , β = 0 .
43, beam-to-core drift speed of 1.5 V A , and α -to-core drift speed of 0.7 V A , the ICW warm plasma0 V z+ Outward (LH) ICW (|f|>f *sc ) z+ /V sw C oun t s V z+ /V sw V z+ /V sw ICW a V z+ Outward (LH) ICW (|f|
320 km/s (green) and V sw >
320 km/s (teal) phase speed is 9%-33% lower than the ICW cold plasmaphase speed; likewise, the FMW phase speed is 1.3%-3.6% faster the FMW cold plasma phase speed (see Fig-ure 7(c) of Verniero et al. (2020)). These perturbationsare not significant enough to affect our interpretationof Figure 5; however, warm plasma effects may con-tribute to the dispersion in the measured distribution of V z + /V sw. We suggest that while solutions to the warm-plasma dispersion are needed to understand growth anddamping rates of these waves, propagation remains welldescribed by the cold-plasma approximation.Our measurements of effective phase speeds suggestthat the vast majority of waves are outward propagat-ing. Observations of left and right handed waves have attimes been interpreted as observations of counter prop-agating waves, e.g. Jian et al. (2009). The observed radial scaling of wave properties (e.g. amplitudes andoccurrence rates) has been attributed to differences intransit time (Jian et al. 2010; Boardsen et al. 2015). Weargue that radial scaling is more likely related to theheliospheric evolution of the ion distribution function(Hellinger et al. 2013; Huang et al. 2020).Observations and theoretical analysis of quasi-parallelpropagating ion-scale waves, suggest three general typesof ion-driven instabilities that result in either left orright-hand wave generation (Jian et al. 2009, 2010, 2014;Wicks et al. 2016; Gary et al. 2016; Verniero et al. 2020):(1) proton anisotropies with T ⊥ > T (cid:107) drive the ion-cyclotron instability Gary et al. (2001), resulting in left-handed ICWs, (2) proton anisotropies with T (cid:107) > T ⊥ drive the firehose instability, generating right-handedFMWs Hellinger et al. (2006), and (3) relative drift1speeds between ion components drive magnetosonic in-stabilities, also producing right-handed FMWs (Gold-stein et al. 2000; Gary et al. 2000, 2016). ThoughPodesta & Gary (2011b) suggest that FMW driventhrough temperature anisotropy should be inward prop-agating, observations of proton distribution functionsat 1 au suggest that both temperature anisotropy andcore-beam drift instabilities, are capable of driving out-ward propagating FMW (Gary et al. 2016).Furthermore,Verniero et al. (2020) show that waves observed by PSPare likely correlated with a proton beam. Our obser-vation of a mixed distribution of outward propagatingwave-modes suggests that a variety of instabilities areimportant in the young solar wind. Generally, Klein et al. (2018) suggest that the solarwind at 1 au is unstable approximately half of the time,with a vast range of both fluid and kinetic scale insta-bilities capable of converting free energy into electro-magnetic waves (Yoon 2017; Verscharen et al. 2019). Inthe inner-heliosphere, these instabilities are ever-morepresent (Klein et al. 2019). Our constraint of mode com-position and wave vector distribution of ion scale wavesin the solar wind through electric field measurements,provides a key step in understanding how energy in thewave-field is redistributed to the plasma. ACKNOWLEDGEMENTSThe FIELDS instrument was designed and developedunder NASA contract NNN06AA01C.REFERENCES
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