Anisotropy of Solar-Wind Turbulence in the Inner Heliosphere at Kinetic Scales: PSP Observations
Die Duan, Jiansen He, Trevor A. Bowen, Lloyd D. Woodham, Tieyan Wang, Christopher H. K. Chen, Alfred Mallet, Stuart D. Bale
DDraft version March 1, 2021
Typeset using L A TEX twocolumn style in AASTeX63
Anisotropy of Solar-Wind Turbulence in the Inner Heliosphere at Kinetic Scales: PSP Observations
Die Duan, Jiansen He, Trevor A. Bowen, Lloyd D. Woodham, Tieyan Wang, Christopher H. K. Chen, Alfred Mallet, and Stuart D. Bale
2, 6, 3, 5 School of Earth and Space Sciences, Peking University, Beijing, 100871, China Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA The Blackett Laboratory, Imperial College London, London, SW7 2AZ, UK RAL Space, Rutherford Appleton Laboratory, Harwell Oxford, Didcot OX11 0QX, UK School of Physics and Astronomy, Queen Mary University of London, London E1 4NS, UK Physics Department, University of California, Berkeley, CA 94720-7300, USA (Received; Revised; Accepted)
ABSTRACTThe anisotropy of solar wind turbulence is a critical issue in understanding the physics of energytransfer between scales and energy conversion between fields and particles in the heliosphere. Using themeasurement of
Parker Solar Probe ( PSP ), we present an observation of the anisotropy at kinetic scalesin the slow, Alfv´enic, solar wind in the inner heliosphere. A steepened transition range is found betweenthe inertial and kinetic ranges at all the directions with respect to the local background magnetic fielddirection. The anisotropy of k ⊥ (cid:29) k (cid:107) is found evident in both transition and kinetic ranges, with thepower anisotropy P ⊥ /P (cid:107) >
10 in the kinetic range leading over that in the transition range and beingstronger than that at 1 au. The spectral index varies from α t (cid:107) = − . ± . α t ⊥ = − . ± . α k (cid:107) = − . ± . α k ⊥ = − . ± .
07 in the kinetic range. The correspondingwavevector anisotropy has the scaling of k (cid:107) ∼ k / ⊥ in the transition range, and changes to k (cid:107) ∼ k / ⊥ in the kinetic range, consistent with the kinetic Alfv´enic turbulence at sub-ion scales. Keywords:
Space plasmas(1544) — Solar wind(1534) — Interplanetary turbulence(830) INTRODUCTIONMagnetic field fluctuations in the solar wind are highlyturbulent. The measured power spectral density (PSD)of fluctuating magnetic field always exhibits power laws k − α , where k is the wavenumber, and α is the spec-tral index. A single spacecraft measures the PSD asa function of f − α in the frequency domain, which canbe converted to the spatial domain under the TaylorHypothesis. According to the physical processes at dif-ferent scales, the PSD in the solar wind can be dividedinto several segments, which can be fitted with different α . The inertial range, which is dominated by magneto-hydrodynamic (MHD) turbulence, follows the cascademodels with spectral indices α i from around 3/2 to 5/3(Bruno & Carbone 2013; Chen et al. 2020). The PSDsbecome steepened below the ion scales (ion thermal gy- Corresponding author: Jiansen [email protected] roradius ρ i or ion inertial length d i ), where kinetic mech-anisms should be taken into account. Sometimes a sharptransition range is observed with α t ∼ α k ∼ /
3, which can be explainedas the MHD Alfv´enic turbulence developing into a typeof kinetic wave turbulence, e.g., kinetic Alfv´en waves(Bale et al. 2005; Chen et al. 2013) or whistler waves(Saito et al. 2008). Intermittency in the kinetic rangecould lead to an 8/3 spectrum (Boldyrev & Perez 2012;Zhao et al. 2016). The spectral indices increase againbeyond the electron kinetic scales, indicating the turbu-lence energy dissipates to electrons (Sahraoui et al. 2009; a r X i v : . [ phy s i c s . s p ace - ph ] F e b Duan et al.
Alexandrova et al. 2012; Chen et al. 2019) or transitionsto a further dispersive cascade (Chen & Boldyrev 2017).Because of the background interplanetary magneticfield (IMF), the turbulence in the solar wind isanisotropic. At the MHD scales, the energy transferrate depends on the angle θ kB between the wavevector k of fluctuations and the background magnetic field (Gol-dreich & Sridhar 1995). The anisotropic energy cascadecould lead to the anisotropy of power level and spectralindex (Chen et al. 2010b), which is observed in the solarwind turbulence (Horbury et al. 2008; Podesta 2009).Goldreich & Sridhar (1995) also predicts a critical bal-anced wavevector anisotropy of k (cid:107) ∼ k / ⊥ and Boldyrev(2006) predicts k (cid:107) ∼ k / ⊥ . Here k ⊥ is the wavevectorperpendicular to the background magnetic field direc-tion, and k (cid:107) is the wavevector along the parallel direc-tion. He et al. (2013) found that turbulent power isenhanced along a ridge at k ⊥ > k (cid:107) in the 2D wavevectorspace. Moreover, it is argued that other possible reasonscould lead to the observed anisotropy, such as intermit-tency (Wang et al. 2014), solar wind expansion (Ver-dini et al. 2019) and non-stationarity of the backgroundmagnetic field (Wu et al. 2020). How the MHD-scaleanisotropy rises in the solar wind is still a challengingquestion.In the kinetic range, the fluctuations remainanisotropic. Theoretically, the specific form of wavevec-tor anisotropy will depend on the nature of the fluctu-ations. The kinetic Alfv´enic wave (KAW) turbulencemodels predict k (cid:107) ∼ k / ⊥ (Howes et al. 2008; Schekochi-hin et al. 2009). The intermittent KAW model givesthe scaling of k (cid:107) ∼ k / ⊥ (Boldyrev & Perez 2012).The tearing-instability-mediated-turbulence model pre-dicts from k (cid:107) (cid:46) k / ⊥ to k (cid:107) (cid:46) k ⊥ (Boldyrev &Loureiro 2019). In observations, the power along quasi-perpendicular directions are found dominant via thestructure function approach (Chen et al. 2010a) andthe k -filtering technique (Sahraoui et al. 2010). Thewave modes are also anisotropic, as He et al. (2011)and Huang et al. (2020) found that the ion-scale tur-bulence contains quasi-parallel Alfv´en-cyclotron waves(ACWs) and quasi-perpendicular KAWs. The numer-ical kinetic simulation is another way to explore thephysics of anisotropy, and different scalings are reached,for example, k (cid:107) ∼ k ⊥ (Arzamasskiy et al. 2019; Landiet al. 2019), k (cid:107) ∼ k / ⊥ (Groˇselj et al. 2018) and k (cid:107) ∼ k / ⊥ (Cerri et al. 2019).The previous studies are mainly based on measure-ments in the vicinity of 1 au. The Parker Solar Probe(PSP) spacecraft (Fox et al. 2016), which has reached aperihelion at 0.1 au, could shed light on the physics of nascent solar wind in the inner heliosphere. In this pa-per, the anisotropy of the magnetic field turbulence atthe kinetic scales is investigated, which could be helpfulto understand the origin and evolution of the solar windturbulence in the inner heliosphere. Section 2 describesthe data and method used in this work. Section 3 showsthe result of the anisotropy. Section 4 is the conclusionand discussion. DATA AND METHODThe data of the
PSP at its first perihelion (0.17 au)are used in this study. The FIELDS and Solar WindElectron Alpha and Proton (SWEAP) instruments pro-vide the in situ measurements of the inner-heliosphericsolar wind (Bale et al. 2016; Kasper et al. 2016).We use a merged data set from flux-gate magnetome-ter (FIELDS/MAG) and search coil (FIELDS/SCM)measurements (both operate at 293 Hz), resolving thefull range from MHD to kinetic scales simultaneously(Bowen et al. 2020b). The plasma measurements arefrom the Solar Probe Cup (SWEAP/SPC) (Case et al.2020). During the perihelion,
PSP encountered a slow( V SW <
400 km/s), but highly Alfv´enic solar wind( σ c ∼ . . < f < ∼
30 Hz, so the kinetic rangeis defined as 30 Hz < f <
100 Hz. A short-time-Fourier-transfrom method is used to remove the artificial spikes(Bowen et al. 2020c) (Woodham et al., in preparation).We avoid fitting f > θ BV ( f, t ) are calcu-lated. To estimate the angular distribution of PSD,we partition θ BV ( f, t ) into 9 angle bins from θ BV ∈ (90 ◦ , ◦ ] to θ BV ∈ (170 ◦ , ◦ ]. The PSD is averagedover each bins as P ( f, θ i ) = 1 N f,i (cid:88) P ( f, t ) | θ i <θ BV ( f,t ) ≤ θ i +10 ◦ , (1) nisotropy of Inner-Heliospheric Kinetic Turbulence θ i = 10 ◦ i + 80 ◦ , i = 1 , , ..., N f,i is thenumbers of points in each bin at each frequency (Podesta2009). RESULTSFigure 1 shows an example interval from 14:30 to 15:30on Nov 5, 2018. The merged data set is in the Radial-Tangential-Normal (RTN) coordinate system, where B R is the radial component of the magnetic field along theSun-spacecraft line. The amplitude of the magneticfield keeps constant as | B | ∼ ρ p ∼
316 cm − , the solar wind speed is V sw ∼ w p ∼
61 km/s, yielding the Alfv´en speed v A ∼ ρ p ∼ . d p ∼
13 km. The plasma β p is 0.3. The θ BR covers the range from 90 ◦ to 180 ◦ ,allowing to estimate the anisotropy. The inertial, tran-sition and kinetic ranges are observed distinctly in theaveraged trace PSD. At the inertial range, the spectralindex α i is -1.56, similar to the statistical result of Chenet al. (2020) at 0.17 au. Then the PSD sharply decreaseswith α t = − .
77 at the transition range. In the ki-netic range, the spectral index increases to α k = − . θ BR is close to 90 ◦ , the RH modes dominate around 4to 20 Hz, which could be the quasi-perpendicular kineticAlfv´enic waves (Huang et al. 2020).The angular distribution of the PSDs P ( f, θ i ) areshown in Figure 2. From the bottom to top, the differ-ent curves correspond to different angular bins from theparallel to the perpendicular directions. The weakestamplitude of the PSD at the parallel direction is largerthan the noise level of the SCM, indicating the vadil-ity of the measurement. The PSDs for the remainingangular bins have been offset by 10 for easier viewing.The small bump around 2 Hz in the bottom spectrum isthe power of the ion-cycolotron waves along the paralleldirection. We demonstrate for the first time that thetransition range exist in all of the directions in the inner heliosphere. The break between the inertial range andthe transition range f it is around 2 Hz, and the breakbetween the transition and kinetic ranges f tk is near6 to Hz. Using Taylor’s Hypothesis, we calculate theDoppler frequency corresponding to the scales of ρ i and d i in the spacecraft frame. We find that the frequenciesof f di = V sw / πk with kd i ∼ f ρ i = V sw / πk with kρ i ∼ f it and f tk . Taylor hypothesis has been shown tohold in the inertial range for the early PSP orbits (Chenet al. 2021). The spectrum with 120 ◦ < θ BV ≤ ◦ is lifted and has more power than the other directionsbelow 20 Hz, which may be enhanced by the reactionwheels of the spacecraft.Figure 2 (b) shows the spectral anisotropy for thethree ranges. The spectral indexes α of each range allhave a decreasing trend from the quasi-parallel direc-tion to the quasi-perpendicular direction. The spectralindex α i is -1.4 along the perpendicular direction and α i ∼ − . α t ∼ − . α t ∼ − . α k increase to -2.8 at parallel direction along theparallel direction and -2.6 along the perpendicular direc-tion, which is consistent with the anisotropy of B ⊥ spec-tra from the Cluster observation (Chen et al. 2010a).We define the perpendicular and parallel power spec-tra as P ⊥ ( f ) = P ( f, ◦ < θ BV ≤ ◦ ) and P (cid:107) ( f ) = P ( f, ◦ < θ BV ≤ ◦ ). Figure 2 (c) shows the powerspectra ratio ( P ( f, θ BV ) /P (cid:107) ( f ), including P ⊥ ( f ) /P (cid:107) ( f ))at three selected frequencies in the three ranges. Thepower anisotropy P ⊥ /P (cid:107) is around 4 at 0.5 Hz in theinertial range, and increases to 6 at 4 Hz in the transi-tion range. At the kinetic scale, P ⊥ /P (cid:107) reaches 20 at 60Hz, which is much larger than 5 measured by structurefunction in Chen et al. (2010a). It reveals that below thetransition range, the power anisotropy at kinetic scalein the inner heliosphere is stronger than at 1 au.Using the method introduced by Wang et al. (2020),the structure functions SF ( l ) along the parallel andperpendicular directions are calculated and shown inFigure 3 to explore the wavevector anisotropy, here l = 1 /k is the spatial displacement. The spectral indexes α SF of the structure functions are consistent with the in- Duan et al. dexes α P SD from the PSD as | α SF | + 1 = | α P SD | (Chenet al. 2010a). By equating SF ( l (cid:107) ) and SF ( l ⊥ ), theanisotropy relation between l (cid:107) and l ⊥ is estimated. Dif-ferent ranges manifests different anisotropy. In the tran-sition range (along the perpendicular direction) we get l (cid:107) ∼ l . ⊥ , which is close to k (cid:107) ∼ k / ⊥ , the predictionof critical balance (Goldreich & Sridhar 1995) or the in-termittent KAW turbulence (Boldyrev & Perez 2012).In principle, l and k have the same anisotropy scaling.Note, however, that ion cyclotron waves in the k (cid:107) spec-trum may be contributing to this measurement. Below d i , l (cid:107) ∼ l . ⊥ at the kinetic range, similar to the pre-diction of critical-balance KAW turbulence of k (cid:107) ∼ k / ⊥ (Schekochihin et al. 2009).Statistical analysis of the wavevector anisotropy isperformed by dividing the data during 00:00:00 Nov 5-00:00:00 Nov,7 into one-hour intervals with 50% overlap-ping. The SCM was operating at a 293 Hz sample rate.There are 90 intervals in total. Here we only considerthe transition and kinetic ranges, because several direc-tions do not have enough samples (counts < θ BV > ◦ is considered. Intervalsthat do not have enough samples in the perpendicular orparallel directions to provide spectra in both directionsare also excluded.Figure 4(a) exhibits the statistical results of the spec-tral anisotropy. The parallel direction has the steepestindexes, with α t (cid:107) = − . ± .
3, and α k (cid:107) = − . ± . α t ⊥ = − . ± . α k ⊥ = − . ± .
07 are observed along the perpendic-ular direction. This result confirms the existence of atransition range signature in all directions, with a trendthat the spectra get steeper from the perpendicular di-rection to the parallel direction.Figure 4(b) shows the histograms of the scalings ofthe wavevector anisotropy. In the transition range alongthe perpendicular direction, the median scaling is l t (cid:107) ∼ l . ± . t ⊥ , in line with the prediction of 2/3. The scalingin the kinetic range along the perpendicular direction is0 . ± .
07, following the relation of l k (cid:107) ∼ l / k ⊥ . CONCLUSION AND DISCUSSIONIn this letter, we present a statistical study of theanisotropy in the kinetic-scale range in the inner helio-sphere. By measuring the power spectra along different θ BV , the anisotropy of spectral index and wavevector inthe transition range and the kinetic range are investi-gated. We show that the transition range and the ki-netic range have different scalings of anisotropy. Thespectral indexes varies from α t (cid:107) = − . ± . α t ⊥ = − . ± . α k (cid:107) = − . ± . α k ⊥ = − . ± .
07 in the kinetic range. The wavevectoranisotropy exhibits the feature of the KAW turbulence,with the scaling of k (cid:107) ∼ k . ⊥ in the transition range andchanging to k (cid:107) ∼ k / ⊥ in the kinetic range.Similar to the observations at 1 au (Chen et al. 2010a),the anisotropy of the spectral index at 0.17 au showsthe trend of the critical balance prediction k ⊥ (cid:29) k (cid:107) ,but the substantial difference is the anisotropy in thetransition range. The observed transition range couldnot fit the cascade model of the kinetic waves as -7/3or -8/3. However, according to the “inertial range -ion dissipation range - electron inertial range” model(Sahraoui et al. 2010), the anisotropy of the spectral in-dex implies the different dissipation mechanisms in theparallel and perpendicular directions. As the KAWsexist along the perpendicular direction (Huang et al.2020), the perpendicular spectra P ⊥ may be interpretedas the Landau damping P ⊥ ∼ k − of KAW in transitionrange (He et al. 2020) and the intermittent KAW cas-cade model P ⊥ ∼ k − . in the kinetic range (Boldyrev &Perez 2012). Woodham et al. (2021, in prep.) show thatthe transition range and kinetic range magnetic helicityand magnetic compressibility are also consistent withthe transition to KAW turbulence at small scales. Forthe parallel direction, the -6 spectra is steeper than theprediction of -5 from the critical balance KAW model(Chen et al. 2010b; Schekochihin et al. 2009) and -7/2from the intermittent KAW model (Boldyrev & Perez2012). The existence of ion cyclotron waves could ex-plain the extreme steepening by enhancing the power atthe onset of the transition range and dissipating signifi-cantly to the protons via cyclotron damping (Woodhamet al. 2018; He et al. 2019; Bowen et al. 2020c). Belowthe transition range, the -3 parallel spectra is similar tothe simulation of Landi et al. (2019), which proposes a2D intermittent model at the sub-ion scales. However,the k (cid:107) ∼ k / ⊥ anisotropy scaling is inconsistent with theintermittent model.The ion-scale coherent structures also may contributethe anisotropy. Boldyrev & Loureiro (2019) predictsthat the ion-scale current sheets from the tearing in-stability could mediate the kinetic Alfv´enic turbulenceto the scalings of k (cid:107) ∼ k / ⊥ or k (cid:107) ∼ k ⊥ . However, the ki-netic tearing instability is predicted to rise significantlyaround the electron scale, which is difficult to measurewith PSP .In addition, the existence of the transition range indi-cates a strong ion heating in the inner heliosphere. Ac-cording to Equation (7) from Bowen et al. (2020a), theheating ratio from the cascade energy flux at the tran-sition range could be estimated under the assumption nisotropy of Inner-Heliospheric Kinetic Turbulence > ∼ A. PIECEWISE PSD FITTING FOR TRANSITION RANGESTo determine the frequency range of the transition range, we divide the PSD(f) into three sections, which connectone another at the points of f and f , respectively. The inertial range is from 0.1 Hz to f , the transition range isfrom f to f , and the kinetic range is from f to 100 Hz. We implement linear fitting to each range and get a piecewiselinear fitting function in the log-log space:log PSD fit ( f ; f , f ) = α i log f + b i . < f ≤ f α t log f + b t f < f ≤ f α k log f + b k f < f ≤
100 Hz (A1)Then we compute the deviation function:Dev( f , f ) = n (cid:88) i =1 [log PSD fit ( f i ; f , f ) − log PSD( f i )] ,n is the total number of the frequencies. We search the best f and f to minimize the deviation function and finallywe get the frequency range [ f , f ] as the transition range.ACKNOWLEDGMENTSWe thank the NASA Parker Solar Probe
Mission and the FIELDS and SWEAP teams for use of data. D.D. andJ.S.H. are supported by NSFC under 41874200 and CNSA under D020301 and D020302. L.D.W. was supported bythe STFC consolidated grant ST/S000364/1 to Imperial College London. C.H.K.C. is supported by STFC ErnestRutherford Fellowship ST/N003748/2 and STFC Consolidated Grant ST/T00018X/1. The FIELDS and the SWEAPexperiment on the Parker Solar Probe spacecraft was designed and developed under NASA contract NNN06AA01C.The authors acknowledge the extraordinary contributions of the Parker Solar Probe mission operations and spacecraftengineering teams at the Johns Hopkins University Applied Physics Laboratory. PSP data is available on SPDF(https://cdaweb.sci.gsfc.nasa.gov/index.html/). REFERENCES
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