Clustering of Intermittent Magnetic and Flow Structures near Parker Solar Probe's First Perihelion -- A Partial-Variance-of-Increments Analysis
Rohit Chhiber, M. Goldstein, B. Maruca, A. Chasapis, W. Matthaeus, D. Ruffolo, R. Bandyopadhyay, T. Parashar, R. Qudsi, T. Dudok de Wit, S. Bale, J. Bonnell, K. Goetz, P. Harvey, R. MacDowall, D. Malaspina, M. Pulupa, J. Kasper, K. Korreck, A. Case, M. Stevens, P. Whittlesey, D. Larson, R. Livi, M. Velli, N. Raouafi
DDraft version December 10, 2019
Typeset using L A TEX twocolumn style in AASTeX63
Clustering of Intermittent Magnetic and Flow Structures near Parker Solar Probe’s First Perihelion –A Partial-Variance-of-Increments Analysis
Rohit Chhiber,
1, 2
M L. Goldstein,
3, 4
B. A. Maruca,
1, 5
A. Chasapis,
1, 6
W. H. Matthaeus,
1, 5
D. Ruffolo, R. Bandyopadhyay, T. N. Parashar, R. Qudsi, T. Dudok de Wit, S. D. Bale,
9, 10, 11
J. W. Bonnell, K. Goetz, P. R. Harvey, R. J. MacDowall, D. Malaspina, M. Pulupa, J. C. Kasper,
14, 15
K. E. Korreck, A. W. Case, M. Stevens, P. Whittlesey, D. Larson, R. Livi, M. Velli, and N. Raouafi Department of Physics and Astronomy, University of Delaware, Newark, DE, USA Code 671, NASA Goddard Space Flight Center, Greenbelt, MD, USA Code 672, NASA Goddard Space Flight Center, Greenbelt, MD, USA University of Maryland Baltimore County, Baltimore, MD, USA Bartol Research Institute, University of Delaware, Newark, DE 19716, USA Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO, USA Department of Physics, Faculty of Science, Mahidol University, Bangkok, Thailand LPC2E, CNRS and University of Orl´eans, Orl´eans, France Space Sciences Laboratory, University of California, Berkeley, CA, USA Physics Department, University of California, Berkeley, CA, USA The Blackett Laboratory, Imperial College London, London, UK School of Physics and Astronomy, University of Minnesota, Minneapolis, MN, USA Code 695, NASA Goddard Space Flight Center, Greenbelt, MD, USA Climate and Space Sciences and Engineering, University of Michigan, Ann Arbor, MI, USA Smithsonian Astrophysical Observatory, Cambridge, MA, USA Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, CA, USA Johns Hopkins University Applied Physics Laboratory, Laurel, MD, USA
ABSTRACTDuring the Parker Solar Probe’s (PSP) first perihelion pass, the spacecraft reached within a heliocen-tric distance of ∼ R (cid:12) and observed numerous magnetic and flow structures characterized by sharpgradients. To better understand these intermittent structures in the young solar wind, an importantproperty to examine is their degree of correlation in time and space. To this end, we use the well-testedPartial Variance of Increments (PVI) technique to identify intermittent events in FIELDS and SWEAPobservations of magnetic and proton-velocity fields (respectively) during PSP’s first solar encounter,when the spacecraft was within 0.25 au from the Sun. We then examine distributions of waitingtimes between events with varying separation and PVI thresholds. We find power-law distributionsfor waiting times shorter than a characteristic scale comparable to the correlation time, suggesting ahigh degree of correlation that may originate in a clustering process. Waiting times longer than thischaracteristic time are better described by an exponential, suggesting a random memory-less Poissonprocess at play. These findings are consistent with near-Earth observations of solar wind turbulence.The present study complements the one by Dudok de Wit et al. (2019, present volume), which focuseson waiting times between observed “switchbacks” in the radial magnetic field. INTRODUCTIONTurbulence has diverse effects in fluids, magnetoflu-ids, and plasmas such as the interplanetary medium andthe solar wind (Pope 2000; Biskamp 2003; Matthaeus &Velli 2011). While far less is understood about the latter
Corresponding author: Rohit [email protected] case compared to the two fluid cases, plasma turbulenceapparently shares with classical turbulence its capacityto greatly enhance transport. This includes the trans-port of energy across scales, suggested by the presence ofcharacteristic second-order statistics such as wavenum-ber spectra (Bruno & Carbone 2013), as well as third-order statistics (Politano & Pouquet 1998) which quanti-tatively (and in some cases, approximately; see Hellingeret al. 2018) characterize the rate of cascade across scales. a r X i v : . [ phy s i c s . s p ace - ph ] D ec Turbulence also produces intermittency (Sreenivasan &Antonia 1997; Matthaeus et al. 2015), and the associ-ated coherent structures, including current sheets andvortices (Zhdankin et al. 2012; Parashar & Matthaeus2016), are responsible for spatial concentration of phys-ical processes observed in heliospheric plasmas, such asheating, heat conduction, temperature anisotropies, andlocal particle energization (Osman et al. 2011; Grecoet al. 2012; Karimabadi et al. 2013; Tessein et al. 2013).Coherent current and field structures can also signifi-cantly influence the transport of field lines and chargedparticles, affecting distributed transport and accelera-tion (Ruffolo et al. 2003; Zank et al. 2014; Tooprakaiet al. 2016). There are, therefore, numerous applica-tions that provide motivation for better understandingof the occurrence rate and distribution of intensities ofcoherent structures such as current sheets.Traditionally, sharp changes in the magnetic field havebeen classified as various type of “discontinuities”, whichare convected or propagated as approximate solutions oflinear ideal magnetohydrodynamics (MHD) (Burlaga &Ness 1969; Tsurutani & Smith 1979; Neugebauer et al.1984; Neugebauer & Giacalone 2010). The ongoingrecognition that these structures are generated rapidlyand generically by turbulence changes the nature of theirstudy at a level of basic physics. As a consequence ofturbulence, coherent structures are a manifestation ofnonlinear dynamics and intermittency, a direct reflec-tion of the higher-order correlations that are implied bythe cascade process itself (Oboukhov 1962; Frisch 1995).Such higher-order statistical correlations are routinelyobserved in space observations in plasmas such as thesolar wind (Horbury & Balogh 1997; Sorriso-Valvo et al.1999; Chhiber et al. 2018).For these reasons, as Parker Solar Probe (PSP; Foxet al. 2016) explores regions of the heliosphere previ-ously inaccessible to in-situ observation, several base-line questions arise concerning the observed coherentstructures, whether one chooses to call them disconti-nuities, structures, or current sheets and vortex sheets.Since these coherent structures are routinely observedat 1 au and elsewhere and are often found to be relatedto flux-tube structures, (Borovsky 2008; Neugebauer &Giacalone 2015; Zheng & Hu 2018; Pecora et al. 2019), adescription of their statistical distribution along PSP’sorbits becomes a matter of theoretical interest as well asconsiderable import in the various heliospheric plasmaphysics applications alluded to above. Here we makeuse of a simple and well-studied method for character-izing statistics of coherent structures, namely the Par-tial Variance of Increments (PVI) method (Greco et al.2009a, 2018), and apply it to characterize coherent mag- netic field and velocity field structures during the firstPSP solar encounter.The paper is organized as follows – Section 2 definesthe PVI measure and provides some background; in Sec-tion 3 we describe the data used and its processing; Sec-tions 4 and 5 describe the results of the analyses of themagnetic and velocity PVI, respectively; we concludewith a discussion in Section 6; Appendix A discussesthe association between power-law waiting times andprocesses that can be described as a Cantor set. BACKGROUNDThe Partial Variance of Increments (PVI) is a well-tested measure of the occurrence of sharp gradients inthe magnetic field – discontinuities, current sheets, sitesof magnetic reconnection, etc. See Greco et al. (2018)for a review of applications. Such discontinuities arebelieved to play a key role in enhanced dissipation, andparticle heating (e.g., Chasapis et al. 2015) and energiza-tion (e.g., Tessein et al. 2013) in space plasmas. If weconsider a given time lag between measurements, for lagsmuch larger than the correlation time, measurementsof turbulent velocity and magnetic fields are typicallyuncorrelated and distributions of increments are Gaus-sian. However, for time lags corresponding to distanceswithin the inertial range, probability density functions(PDFs) of increments have “fatter” non-Gaussian tailsand are fit better by stretched exponential, lognormal,truncated L´evy-flight, and kappa distributions (Kailas-nath et al. 1992; Sorriso-Valvo et al. 1999; Bruno et al.2004; Pollock et al. 2018). Distributions of waiting timesbetween high PVI “events” typically exhibit power lawsat inertial range lags and exponential behavior at longer,uncorrelated lags Greco et al. (2009b, 2018). Power-law behavior indicates a correlated “clustering” processwhich is statistically self-similar in time and possesses“memory”, as opposed to a random Poisson processwhich results in exponential waiting-time distributions.Waiting-time analyses have been employed to make thisdistinction in diverse fields of study: space physics (Bof-fetta et al. 1999; Lepreti et al. 2001; D’Amicis et al. 2006;Greco et al. 2009b), geophysics (Carbone et al. 2006),laboratory materials (Ferjani et al. 2008), and seismol-ogy (Mega et al. 2003), to name a few.Other studies analyzing PSP data (this volume) haverevealed many sharp jumps in magnetic field measure-ments by FIELDS and proton velocity measurements bySWEAP during PSP’s first solar encounter (Bale et al.2019; Horbury et al. 2019). Dudok de Wit et al. (2019)have examined statistical distributions of events identi-fied by inspection as “switchbacks” or “jets”. Examina-tion of the same data using the PVI method representsa complementary approach, since the PVI is not tai-lored to a specific type of discontinuity but is instead ageneral tool for identifying a broad class of intermittentstructures (Greco et al. 2018). Another motivation forthis study lies in the fact that the PVI technique is usedin two other concurrent studies submitted to the PSPSpecial Issue ApJ – (1) Bandyopadhyay et al. (2019a)examine the association of energetic-particle fluxes fromIS (cid:12)
IS with intermittent magnetic structures, as identi-fied by the PVI technique; (2) Qudsi et al. (2019) studythe association of high proton-temperatures measuredby SWEAP with high magnetic-PVI values.The PVI is essentially the magnitude of the (vector)increment in a field at a given lag, normalized by thevariance of the field. Note that the increment of a tur-bulent field has long occupied a central role in turbu-lence research, with particular importance having beengiven to moments of the increment, the so-called struc-ture functions (e.g., Monin & Yaglom 1971; Tu & Marsch1995). The PVI is related to the first-order structurefunction, but is distinct in that it is a pointwise (not-averaged) measure. For the magnetic field B the PVIat time s is defined, for lag τ in time, as (Greco et al.2018): PVI s,τ = | ∆ B ( s, τ ) | / (cid:112) (cid:104)| ∆ B ( s, τ ) | (cid:105) , (1)where the (cid:104) . (cid:105) denotes averaging over a suitable interval(see Isaacs et al. 2015; Krishna Jagarlamudi et al. 2019).The increment is defined as ∆ B ( s, τ ) = B ( s + τ ) − B ( s ). The velocity PVI is defined similarly. To computethe variance we use a moving average over a windowten times larger than the correlation time of magneticor velocity fluctuations, as appropriate (see Sections 4and 5, below). Note that PVI captures gradients ineach vector component of B. In the following figures, wewill denote the magnetic and velocity PVI as PVI B andPVI V , respectively.Values of PVI > > > DATAWe use magnetic-field data from the flux-gate mag-netometer (MAG) aboard the FIELDS instrument suite(Bale et al. 2016) and proton-velocity data from the So-lar Probe Cup (SPC) on the SWEAP instrument suite(Kasper et al. 2016; Case & SWEAP 2019), covering aperiod of about 10 days centered on the first perihelion.The magnetic field data used span the full range fromUTC time 2018-11-01T00:00:00 to 2018-11-09T23:59:59,and have been resampled to 1-second cadence using lin-ear interpolation. Note that data gaps are not an issue inMAG measurements during the period considered here.The resampled magnetic data in heliocentric RTN co-ordinates (Fr¨anz & Harper 2002) are shown in the toppanel of Figure 1. For a detailed description of theseobservations, including the large “switchbacks” in theradial magnetic field, see other papers in the presentvolume (Bale et al. 2019; Horbury et al. 2019, Dudok deWit et al. 2019).We use proton velocity measurements at 0.87-secondresolution from the SPC, which are then processed to re-move spurious or artificial spikes. These data are shownin heliocentric RTN coordinates in the bottom panel ofFigure 1. Despite the numerous fluctuations, the bulkflow is fairly steady and well-described as slow wind( V R <
500 km s − ) for most of the period consideredhere. During the last day PSP may have passed overa small coronal hole and sampled relatively fast windabove 600 km s − . Data gaps are a more significantissue in SPC measurements during the first encounter(compared to MAG measurements), and we have usedthe following procedure to prepare the data for our anal-yses. We first split the time series of velocity mea-surements from 2018 November 1 to 2018 November 10into 8-hour sub-intervals. We then discard sub-intervalsthat have data gaps larger than 10 seconds. The re-maining sub-intervals have gaps with an average du-ration of about 1.5 s, and linear interpolation is usedover these gaps. This procedure produces three periods– (i) 2018-11-01T00:00:03 to 2018-11-03T08:00:02; (ii)2018-11-05T16:00:03 to 2018-11-07T00:00:03; (iii) 2018-11-08T00:00:04 to 2018-11-10T08:00:03, within each ofwhich we have continuous time series of velocity mea-surements at 0.87 s cadence. The PVI waiting-timeanalyses are performed separately within each of thesethree periods, and the results are then accumulated toobtain improved statistics (see details in Section 5, be-low). Note that reliable waiting-time estimation pre-cludes the use of intervals with large data gaps. Bulkplasma properties over the encounter are shown in Table1. PVI ANALYSIS OF THE MAGNETIC FIELDAs mentioned in Section 2, to compute the PVI time-series we need estimates of the correlation time. We usethe Blackman-Tukey technique (see Matthaeus & Gold-stein 1982) with an averaging interval of 24 hours tocompute the autocorrelation of the magnetic field. Thecorrelation time is estimated as the time at which the au-tocorrelation falls to 1 /e . Note that the correlation timedoes not change significantly on using a 12-hour averag-ing interval instead of 24 hours. The paper by Parasharet al. in the present special issue shows correlation timescomputed in this way for the encounter. See also Smithet al. (2001), Isaacs et al. (2015), Krishna Jagarlamudiet al. (2019) and Bandyopadhyay et al. (2019b) (presentvolume) for discussions of subtleties and potential is-sues in accurate determination of correlation times. Analternative estimate of the correlation time may be ob-tained from the break frequency between the “1 /f ” andinertial ranges in the power spectrum of the magneticfluctuations (Chen et al. 2019); we confirmed that thisestimate is comparable to the correlation time we usehere. Furthermore, the PVI does not appear to dependsensitively on the averaging interval used.During the period analyzed here the magnetic correla-tion time varies from about 1000 s to 350 s. Accordingly,assuming an average correlation time of 600 s for the en-counter, we use a rolling boxcar average over a window10 ×
600 seconds in duration to estimate the variance[the denominator in Equation (1)] for the computationof the magnetic PVI. The resulting time series is shownin Figure 2 for three different lags: τ = 1, 10, and 100seconds. The 1 and 10 seconds lags lie well within theinertial range (the ion inertial scale corresponds to anapproximate temporal lag of 0.05 s [Parashar et al. 2019,this issue]), while the 100 s lag is comparable to the cor-relation time. Note that as the lag τ is increased we stillsample over times with 1-second cadence.It is clear from Figure 2 that at smaller lags the PVImeasure captures highly intermittent events, while suchevents are relatively rare in the time series computedusing a 100-second lag. This is reinforced by Figure3, which shows histograms of PVI for the three lags.The most probable value of PVI is about 0.5 for allthree cases, and corresponds to the majority of events,that are, by definition, non-intermittent. While all threelags capture a large number of non-Gaussian (PVI > > >
8) are detected with 1-second lag.In Figure 4 we present the main results of this work –PDFs of waiting times (WT) between intermittent PVIevents with varying lag and threshold. Here the waitingtime between two events is defined as the time betweenthe end of the first event and the beginning of the secondevent. Note that the events may have finite duration;i.e., if the PVI stays above the threshold for consecu-tive times then these times are regarded as part of thesame event. Power-law and exponential fits (based onchi-squared error minimization) to the PDFs are alsoshown, and the average waiting times computed fromthe distribution are indicated as (cid:104) WT (cid:105) . Uncertaintiesin fit parameters and goodness-of-fit estimates are alsoreported in the caption.It is apparent from all four panels of Figure 4 that thedistribution of waiting times is well described as a powerlaw for events whose temporal separation is smaller thanthe correlation time, suggesting strong correlation andclustering. For events that are farther apart in time, thedistribution is better fit by an exponential, indicativeof a random Poisson-type process (Greco et al. 2009b).In fact, the break between the power-law and exponen-tial regimes is associated with the average waiting time.While acknowledging that these power-law distributionslack a well-defined average (Newman 2005), we might in-terpret WT < (cid:104) WT (cid:105) to be an intracluster waiting time,and WT > (cid:104) WT (cid:105) as an intercluster waiting time. Thelatter is consistent with an exponential, so WT between clusters is governed by a uniform random-Poisson pro-cess. Within clusters, there is strong correlation. Weremark here that truncated L´evy-flight (TLF) distribu-tions include both a power-law range along with an ex-ponential cutoff (Bruno et al. 2004), and in future workit would be worth examining the present results in thecontext of such TLFs.Another feature of interest in Figure 4 is the smallspike in the PDF near WT = τ , suggesting that thePVI at a given lag may preferentially pick out eventswith a characteristic waiting time equal to the specifiedlag. We also note that the magnitude of the slope α of the power law systematically increases with increas- This appears to be a general property of the PVI measure thathas been seen in previous work (e.g., see Figure 4 in Greco et al.2009b), but has either not been noticed or not remarked uponuntil now. As a tentative explanation, imagine a single data-pointwith a strong fluctuation relative to its neighbors, and supposethe lag is 100 s. Then, 100 s before this point there is a stronglikelihood of a PVI event, and of another PVI event at the timeof this point. This could lead to an increased chance of WT = τ compared to neighboring values of WT. Figure 1.
Time series of the heliocentric RTN components (blue, red, and green curves, respectively) of the magnetic andvelocity fields during the period considered here are shown in the top and bottom panels, respectively. Note that the R-componentis plotted using a thicker curve than the other two components.Time (cid:104) V (cid:105) (cid:104) T i (cid:105) (cid:104) n i (cid:105) d i (cid:104) B (cid:105) (cid:104) δB (cid:105) (cid:104) V A (cid:105) β i − . × K 215 cm −
17 km 70 nT 50 nT 100 km s − Table 1.
Bulk plasma parameters for PSP’s first solar encounter. Shown are average values of proton speed (cid:104) V (cid:105) ≡(cid:104) (cid:112) V R + V T + V N (cid:105) , proton temperature (cid:104) T i (cid:105) , proton density (cid:104) n i (cid:105) , proton inertial scale d i , magnetic field magnitude (cid:104) B (cid:105) ≡(cid:104) (cid:112) B R + B T + B N (cid:105) , the rms magnetic fluctuation (cid:104) δB (cid:105) ≡ (cid:112) (cid:104)| B − (cid:104) B (cid:105)| (cid:105) , Alfv´en speed (cid:104) V A (cid:105) , and proton beta β i . Averaging isperformed over the entire duration of UTC time 2018-11-01T00:00:00 to 2018-11-10T23:59:59. ing lag, indicating a weakening of the clusterization (seeAppendix A). Furthermore, for τ = 1, the slope is shal-lower for the case of the PVI > > > − forthe encounter. Note that the Taylor hypothesis hasbeen found to have reasonable validity during the firstencounter (see Chen et al. 2019, Chasapis et al. 2019,and Parashar et al. 2019 in the present volume), con-sistent with predictions based on turbulence modeling Since PSP and the solar wind plasma were in near-corotationnear perihelion (Kasper et al. 2019), we reasoned that the plasmawas convecting past the spacecraft primarily in the radial direc-tion, and therefore used the radial speed of the solar wind whileemploying the Taylor hypothesis. of the solar wind (Chhiber et al. 2019). However, thedistances shown in Table 2 should be considered crudeestimates, since the radial velocity varies by up to a fac-tor of 2 relative to the mean used here. Note that thisconstant radial speed is used only in estimations of char-acteristic distances, and plays no role in our temporalanalyses and conclusions. PVI ANALYSIS OF VELOCITYNext we present the results of the PVI waiting-timeanalysis for the proton velocity. As discussed in Sec-tion 3, data gaps are a more significant issue for ve-locity measurements by the SPC, compared with MAGdata. Velocity data selection and processing is describedin Section 3. The resulting subsamples (i), (ii), and (iii)have velocity correlation times of 1700, 325, and 700 sec-onds, respectively, and a rolling average over an intervalten times larger than these times is employed for com-puting the PVI time-series [Equation (1)]. The time se-ries for the second subsample (near perihelion) is shownin Figure 5, for three different lags. As in the case ofthe magnetic field, smaller lags detect more intermit-tent events, although there appear to be relatively fewerevents with very high PVI. This finding is reinforced bythe histograms shown in Figure 6, with the caveat thatthe volume of data used in the analyses for the velocityis smaller than that for the magnetic field, since in the (a)(b)
Figure 2. (a) PVI (with lag τ equal to 1, 10, and 100 s) time-series for magnetic field during the first encounter. (b) The sametime series for about 15 minutes on 2018 November 5. In both panels the 10 s case is shown as a thicker line compared to theother two. PVI B > B > τ in s (km) α (cid:104) WT (cid:105) in s (km) (cid:104) T dur (cid:105) in s (km) α (cid:104) WT (cid:105) in s (km) (cid:104) T dur (cid:105) in s (km)1 (350) − .
83 67.6 (23,000) 1.2 (420) − .
65 719.5 (252,000) 1.0 (350)10 (3500) − .
95 152.6 (53,000) 2.7 (945)100 (35,000) − .
29 599.2 (210,000) 4.5 (1575)
Table 2.
Power-law indices α of fits to WT distributions, mean waiting times (cid:104) WT (cid:105) in s, and mean durations (cid:104) T dur (cid:105) in s, fordifferent PVI lags ( τ ) and thresholds, for the magnetic field. Times have been converted to approximate characteristic distances(shown in km in parentheses) assuming Taylor’s hypothesis with an average radial speed of 350 km s − . For reference, themean correlation time (distance) for magnetic fluctuations during the encounter is about 600 s (200,000 km). Above (cid:104) WT (cid:105) thewaiting times depart from a power law and follow an exponential distribution. former case only intervals that survive the data-selectionprocedure (Section 3) are used. Nevertheless, we do findthousands of non-Gaussian events (PVI >
3) and morethan a hundred possible current sheets (PVI > (cid:104) WT (cid:105) , and randomPoisson intercluster processes. The magnitude of the slope α of the power law increases with increasing lag,and, for the 0.87 s lag, the slope of the PVI > > τ isalso seen here.Table 3 shows power-law slopes, average waitingtimes, and average durations of PVI events for the var-ious lags and thresholds considered. Times have beenconverted to approximate characteristic distances as-suming Taylor’s hypothesis with a constant speed of 350km s − , as in the magnetic case. PVI V > V > τ in s (km) α (cid:104) WT (cid:105) in s (km) (cid:104) T dur (cid:105) in s (km) α (cid:104) WT (cid:105) in s (km) (cid:104) T dur (cid:105) in s (km)0.87 (305) − .
72 108.4 (38,000) 0.98 (343) − .
35 1696.2 (594,000) 0.89 (312)8.7 (3045) − .
99 155.9 (55,000) 2.3 (805)87 (30,450) − .
38 445.1 (156,000) 3.2 (1120)
Table 3.
Power law indices α of fit to WT distributions, mean waiting times (cid:104) WT (cid:105) in s, and mean durations (cid:104) T dur (cid:105) in s,for different PVI lags ( τ ) and thresholds, for the proton velocity. Times have been converted to distances (shown in km inparentheses) assuming Taylor’s hypothesis with an average radial speed of 350 km s − . For reference, the mean correlation time(distance) for velocity fluctuations near perihelion is about 325 s (114,000 km). Above (cid:104) WT (cid:105) the waiting times depart from apower law and follow an exponential distribution. Figure 3.
Histograms (showing frequency of occurrence, ornumber of counts) of PVI values for different lags τ , for themagnetic field during the first encounter. Note the elevatedlikelihood of large PVI values at shorter lags, indicative ofenhanced small-scale intermittency, typical of non-Gaussianprocesses and turbulence.6. CONCLUSIONS AND DISCUSSIONIn this paper we have employed the PVI methodologyto provide a baseline statistical characterization of the“roughness”, or intermittency, of the observed magneticand velocity field during the first solar encounter of thePSP. Quantification of roughness using the PVI tech-nique has the dual advantages of being closely related toturbulence intermittency statistics, while also being re-lated to classical-discontinuity identification procedures(Greco et al. 2009b, 2018). The present work extends in By “classical discontinuity” we are referring to the traditionalinterpretation of (mostly magnetic) discontinuities in the solarwind as members of a class of MHD stationary convected struc-tures (such as tangential discontinuities) or propagating rota-tional discontinuities, which are viewed as static solutions of theideal MHD equations (e.g., Neugebauer & Giacalone 2010). a natural way analogous studies carried out at 1 au andbeyond (Greco et al. 2018). Values of PVI above appro-priate thresholds have been found to be related to clas-sical discontinuities (Greco et al. 2008, 2009b), intermit-tency and current sheets (Greco et al. 2009a; Malaspinaet al. 2013), particle energization (Tessein et al. 2013;Tessein et al. 2015; Tessein et al. 2016), kinetic effectssuch as elevated temperature and high degrees of non-Guassianity in the velocity distribution function (Os-man et al. 2011, 2012; Servidio et al. 2015; Qudsi et al.2019), and, at high PVI, likelihood of magnetic recon-nection (Servidio et al. 2011). In this sense it is a nat-ural follow-on to examine whether those tendencies ex-tend further into the inner heliosphere than has beenpreviously explored. However, additional motivation isobtained through early reports that the magnetic andvelocity fields near PSP perihelion exhibit strong “jets”or “switchbacks” that may suggest enhanced, episodic,and large-amplitude quasi-discontinuous jumps in theplasma conditions (see several papers in this special edi-tion). PVI seems to be an appropriate general tool forbroadly identifying and quantifying such intermittentstructure. Note that further detailed study of specifictypes of structures, such as the observed “switchbacks”,requires a more specialized approach (see Dudok de Witet al. 2019, current issue).Our main results are summarized in the Tables. Dur-ing the first Parker Solar Probe encounter, the fluctua-tions of both the magnetic field and velocity field exhibitstatistical features, specifically the inter-event waiting-time distributions, that suggest the appearance of bothcorrelated as well as random or Poissonian events. Suchevents are interpreted as non-Gaussian coherent struc-tures, consistent with current sheets and vortex sheets.The presence of these signals may be related to inter-pretations based on intermittent turbulence, althoughthe method itself is also sensitive to classical discontinu-ities. For waiting times shorter than about a correlationscale, the presence of power-law distributions indicatescorrelations and is suggestive of clustering. In Appendix (a) (b)(c) (d)
Figure 4.
PDFs of waiting times (WT) between magnetic PVI > τ equal to (a) 1, (b) 10, and (c) 100 seconds.Panel (d) shows the PDF of waiting times between magnetic PVI > (cid:104) WT (cid:105) are also indicated, with downward arrows marking their location on the horizontal axes. Power-law fits ( βx α ) are shown as solidgreen lines and exponential fits ( γe − δx ) as dashed blue curves; here x refers to the waiting time. 1-sigma uncertainty estimatesfor fit parameters { β, α, γ, δ } for the four panels are, respectively: (a) { } ; (b) { } ; (c) { } ; (d) { } . The Pearson correlation-coefficients indicatinggoodness-of-fit for the power-law fits are above 0.95 for each panel. A we consider an analogy with generalized self-similarCantor sets, for which the power-law index α rangesfrom − − (a)(b) Figure 5. (a) PVI (with lag τ equal to 0.87, 8.7, and 87 s) time-series for the proton velocity from UTC 2018-11-05T16:00:03to 2018-11-07T00:00:03, including the first perihelion. (b) The same time series for about 15 minutes on 2018 November 6. Inboth panels the 8.7 s case is shown as a thicker line compared to the other two. Figure 6.
Histograms (showing frequency of occurrence, ornumber of counts) of PVI values for different lags τ , for theproton velocity during the first encounter. Note the elevatedlikelihood of large PVI values at shorter lags, indicative ofenhanced small-scale intermittency, typical of non-Gaussianprocesses and turbulence. rutani & Smith 1979; Bruno et al. 2001; Greco et al.2008). Note that other studies have found exponentialwaiting-time distributions for intermittent events in thenear-Earth (and beyond) solar wind (Tsurutani & Smith1979; Bruno et al. 2001), without a power-law regime.Interestingly, Hu et al. (2018) find power-law distribu-tions at longer waiting times ( >
60 minutes) and expo-nential behavior before that, for small-scale flux ropesidentified using a Grad-Shafranov reconstruction tech-nique with
WIND observations.Assuming wind speed as the sole criterion for clas-sification, the current PSP observations are mostly re-stricted to slow-wind conditions in the ecliptic duringsolar minimum. Future orbits are expected to sampleextended periods of fast wind as well, and it will beinteresting to compare waiting-time statistics betweenslow and fast wind in the near-Sun plasma. Fartheraway,
Helios observations find power-law behavior up tolonger waiting times in the case of slow wind comparedwith fast wind (D’Amicis et al. 2006). It would also beinteresting to use full-cadence MAG data (or search-coilmagnetometer measurements) from PSP to probe PVIevents at kinetic-scale lags.0 (a) (b)(c) (d)
Figure 7.
PDFs of waiting times between (proton) velocity PVI > > (cid:104) WT (cid:105) are also indicated, with downward arrows marking their location on the horizontal axes. Power-law fits ( βx α ) are shown as solid green lines and exponential fits ( γe − δx ) as dashed blue curves; here x refers to the waitingtime. 1-sigma uncertainty estimates for fit parameters { β, α, γ, δ } for the four panels are, respectively: (a) { } ; (b) { } ; (c) { } ; (d) { } . The Pearsoncorrelation-coefficients indicating goodness-of-fit for the power-law fits are above 0.95 for each panel. The dichotomy betweeen a strongly-correlated clus-tering process and a random Poissonian process may berelated to two contrasting views of the origin of mag-netic structures in the solar wind – in-situ generationvia turbulent cascade vs. passive advection from thesolar source. The strong clustering seen in our presentresults readily leads to the suggestion that these ob-served features may originate in a hierarchy of nonlinearprocesses that generate correlations of nearby structuresover a broad range of scales. Our preferred explanationis strong turbulence occurring in the corona and/or in-terplanetary medium. Turbulence is known to producefeatures of the type reported here, as has been observedroutinely in space plasmas including the solar wind andthe terrestrial magnetosheath (Yordanova et al. 2008; Matthaeus & Velli 2011; Bruno & Carbone 2013). Theunique feature of the present analysis is finding theseindicators of intermittency and turbulence at distancescloser to the Sun, and therefore closer to source andboundary surfaces, than has been accomplished in anyprevious space mission. This may eventually produceconstraints on how turbulence is initiated in the innerheliosphere, or how it is transmitted and propagatedfrom the corona into the super-Alfv´enic solar wind.Fully satisfactory answers to such questions will likelyrequire additional complementary observations by PSPin subsequent orbits, and by the upcoming Solar Orbitermission. Furthermore, it is likely that more completeinterpretations will require context support from globalheliospheric simulations to establish likely connections1between in-situ observation and remote sensing of theinner solar atmosphere, for example by Solar Orbiter orby the upcoming PUNCH mission. ACKNOWLEDGMENTSThis research as been supported in part by theParker Solar Probe mission under the IS (cid:12)
IS project(contract NNN06AA01C) and a subcontract to Uni-versity of Delaware from Princeton (SUB0000165).Additional support is acknowledged from the NASA LWS program (NNX17AB79G) and the HSR pro-gram (80NSSC18K1210 & 80NSSC18K1648), and grantRTA6280002 from Thailand Science Research and In-novation. Parker Solar Probe was designed, built, andis now operated by the Johns Hopkins Applied PhysicsLaborotary as part of NASA’s Living With a Star (LWS)program (contract NNN06AA01C). Support from LWSmanagement and technical team has played a criticalrole in the success of the Parker Solar Probe mission.APPENDIX A. WAITING TIMES FOR THE CANTOR SETHere we provide details of the association between power-law waiting times and processes or structures that can bedescribed by a Cantor set.For a given power-law distribution of waiting times, with PDF ∝ WT α , it may not be clear how to physicallyinterpret the power-law index α . Intuitively, it seems that a harder distribution should indicate stronger clusteringthan a softer distribution, i.e., α ≈ − α ≈ −
2, because a process with aharder waiting time distribution more frequently has a long hiatus followed by numerous events in rapid succession.To interpret α more quantitatively, and given that (statistical) self-similarity is a common feature of inertial-rangeturbulence, we consider the waiting-time distribution of the Cantor set (Smith 1874; Cantor 1883). Recall that thisset is defined as the points remaining after an infinite sequence of operations: At stage n = 0 we start with the set[0 , n = 1 we remove the middle 1/3 with 2 segments remaining at either side, and in each subsequentstage n we remove the middle 1/3 of each remaining segment, doubling the number of remaining segments to become2 n . If the “waiting time” T is defined as the distance between successive points in the Cantor set, then all waitingtimes are T n = 3 − n for some n ∈ { , , , . . . } , and the number of waiting times generated in stage n is N n = 2 n − .An unnormalized PDF of waiting times can be defined as N n / ( T n − T n +1 ), which results inPDF( T n = 3 − n ) = 2 n − (2 / − n = 92 6 n − . (A1)This implies that α = ln PDF( T n +1 ) − ln PDF( T n ) T n +1 − T n = − ln 6ln 3 ≈ − .
631 (A2)[The same power-law index results if we instead define the PDF from N n / ( T n − − T n ).] Remarkably, some of thepresent observational results for PVI events have α close to -1, implying that large field-increments in the solar windcan be more strongly clustered than the Cantor set. Similar slopes have been observed near 1 au (Greco et al. 2009b).As a generalization of the Cantor set, consider a set in which at each stage, instead of removing 1/3 of each segment,we remove a fraction f of the segment from the middle. As f →
1, more of the segment is removed and the remainingpoints are more clustered with wider gaps. Each remaining segment after n = 1 has a size (1 − f ) /
2, and after stage n the segment size is [(1 − f ) / n . Then T n = f [(1 − f ) / n − and we still have N n = 2 n − , soPDF( T n ) = 2 n − f [(1 + f ) / − f ) / n − = 2 f (1 + f ) (cid:18) − f (cid:19) n − (A3)and α = ln[4 / (1 − f )]ln[(1 − f ) /
2] = − − ln(1 − f )ln 2 − ln(1 − f ) . (A4)For 0 < f <
1, we have − < α < −
1, with α → − f →
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