Subproton-scale Intermittency in Near-Sun Solar Wind Turbulence Observed by the Parker Solar Probe
Rohit Chhiber, William H. Matthaeus, Trevor A. Bowen, Stuart D. Bale
DDraft version February 23, 2021
Typeset using L A TEX twocolumn style in AASTeX63
Subproton-scale Intermittency in Near-Sun Solar Wind Turbulence Observed by the Parker SolarProbe
Rohit Chhiber,
1, 2
William H. Matthaeus, Trevor A. Bowen, and Stuart D. Bale
3, 4 Department of Physics and Astronomy & Bartol Research Institute, University of Delaware, Newark, DE, USA Heliophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD, USA Space Sciences Laboratory, University of California Berkeley, Berkeley, CA, USA Physics Department, University of California Berkeley, Berkeley, CA, USA
ABSTRACTHigh time-resolution solar wind magnetic field data is employed to study statistics describing inter-mittency near the first perihelion ( ∼ . (cid:12) ) of the Parker Solar Probe mission. A merged datasetemploying two instruments on the FIELDS suite enables broadband estimation of higher order mo-ments of magnetic field increments, with five orders established with reliable accuracy. The duration,cadence, and low noise level of the data permit evaluation of scale dependence of the observed inter-mittency from the inertial range to deep subproton scales. The results support multifractal scaling inthe inertial range, and monofractal but non-Gaussian scaling in the subproton range, thus clarifyingsuggestions based on data near Earth that had remained ambiguous due to possible interference of theterrestrial foreshock. The physics of the transition to monofractality remains unclear but we suggestthat it is due to a scale-invariant population of current sheets between ion and electron inertial scales;the previous suggestion of incoherent kinetic-scale wave activity is disfavored as it presumably leadsre-Gaussianization which is not observed. INTRODUCTIONIntermittency is an important feature in the theoryof fluid and plasma turbulence (Sreenivasan & Antonia1997; Matthaeus et al. 2015), and has gained increasingattention in the study of space plasmas, including thecorona, the magnetosphere, and the solar wind (Abra-menko et al. 2008; Chhiber et al. 2018; Bruno 2019). Ineach of these venues the emergence of localized stronggradients is a consequence of the turbulent cascade ofenergy. The resulting coherent structures, of electriccurrent density, vorticity, or density, are likely sites ofenhanced kinetic dissipation, and heating (e.g., Osmanet al. 2012). Therefore intermittency is crucial in ter-minating the cascade and heating the plasma. Thesecoherent structures also compartmentalize the plasma,producing a distinctive flux tube “texture” that orga-nizes quantities such as temperature, density, magneticintensity, and energetic particles (Borovsky 2008; Tes-sein et al. 2013). Coherent structure forms in similarways in hydrodynamics (Sreenivasan & Antonia 1997),magnetohydrodynamics (Wan et al. 2009), and plas-mas (Burlaga 1991; Sorriso-Valvo et al. 1999; Matthaeus
Corresponding author: Rohit [email protected] et al. 2015), with important differences, particularly ap-proaching kinetic scales. Statistics in the plasma ki-netic range provide insight regarding physical mecha-nisms responsible for dissipation (e.g., Goldstein et al.2015; Chen 2016; Matthaeus et al. 2020), thus address-ing fundamental questions related to coronal heatingand acceleration of the solar wind (Fox et al. 2016).It remains unclear whether statistics at subprotonscales remain strongly intermittent and multifractal,or become scale-similar and monofractal (Kiyani et al.2009; Leonardis et al. 2013, 2016; Wan et al. 2016), oreven return to Gaussianity (Koga et al. 2007; Wan et al.2012; Wu et al. 2013; Chhiber et al. 2018; Roberts et al.2020). These questions persist in part because of thescarcity of high time-resolution data at locations wellseparated from the terrestrial bow shock. Here we ad-dress these issues by employing high-resolution measure-ments of the magnetic field made by the
Parker SolarProbe ( PSP ) in near-Sun solar wind (Fox et al. 2016). THEORETICAL BACKGROUNDIn turbulence, considerable information is contained inthe statistics of fluctuations and increments of the prim-itive variables. These are velocity in hydrodynamics,velocity and magnetic fluctuations in magnetohydrody-namics (MHD), density for compressible flows, and addi- a r X i v : . [ phy s i c s . s p ace - ph ] F e b Chhiber et al. tional variables for complex fluids and plasmas. The ba-sic second-order statistics include two-point correlations,their Fourier transforms, i.e., wavenumber ( k ) spectra(Matthaeus & Goldstein 1982), and the second-orderstructure functions (Burlaga 1991). These and otherrelevant statistics are moments of the underlying jointprobability distributions functions (PDFs; e.g., Monin& Yaglom 1971). Second-order moments describe thedistribution of energy over spatial scales (cid:96) ∼ /k . Todescribe the spatial concentration of energy in inter-mittent structures, we go beyond second-order statisticsand consider higher-order moments of PDFs (e.g., Frisch1995).The original K41 similarity hypothesis (Kolmogorov1941) postulates the statistical behavior of longitudinalvelocity increments at spatial lag (cid:96) , namely δu (cid:96) = (cid:96) · [ u ( x + (cid:96) ) − u ( x )]. K41 asserts that δu (cid:96) ∼ (cid:15) / (cid:96) / where (cid:15) is the total dissipation rate and isotropy is assumed.Thus, for an appropriate averaging operator (cid:104) . . . (cid:105) , all in-crement moments are determined as the structure func-tions S ( p ) = (cid:104) δu p(cid:96) (cid:105) = C p (cid:15) p/ (cid:96) p/ , a form that includesthe second-order law S (2) = C (cid:15) / (cid:96) / and (formally)the exact third-order law S (3) = − (4 / (cid:15)(cid:96) as specialcases. The refined similarity hypothesis (Kolmogorov1962, K62) takes into account intermittency, averagingthe local dissipation rate (cid:15) (cid:96) over a volume of linear di-mension (cid:96) and introducing this as an additional randomvariable. Incorporating the suggestion (Oboukhov 1962)that such irregularity of dissipation changes scalings ofincrements with (cid:96) , the refined K62 hypothesis becomes δu (cid:96) = A ( ∗ ) (cid:15) / (cid:96) (cid:96) / . Here A ( ∗ ) is a random functionthat depends on local Reynolds number, but not on (cid:15) (cid:96) or (cid:96) separately, and takes on a unique form at infiniteReynolds number. For moments S ( p ) = (cid:104) δu p(cid:96) (cid:105) , the hy-pothesis implies S ( p ) = C p (cid:104) (cid:15) p/ (cid:96) (cid:105) (cid:96) p/ = (cid:15) p/ (cid:96) p/ µ ( p ) (1)where µ ( p ) is a measure of the intermittency. OUTSTANDING OBSERVATIONAL QUESTIONSThe solar wind magnetic field spectrum in the inertialrange admits a power law over several decades (Cole-man 1966; Matthaeus & Goldstein 1982), although dis-cussion persists concerning exact spectral indices andanisotropy (e.g., Tessein et al. 2009). PDFs of magneticincrements exhibit non-Gaussian features that are in-creasingly prominent at smaller lags within the inertialrange (Sorriso-Valvo et al. 1999). As such, it is un-derstood that the scale-dependent kurtosis κ ( (cid:96) ) (SDK;see Section 5) increases with decreasing (cid:96) in the inertialrange, while higher-order exponents exhibit multifractalscaling (Frisch 1995). Overall, this picture is consis-tent with expectations from MHD (Carbone et al. 1995; Politano et al. 1998) which in turn are consistent withhydrodynamic scaling (Sreenivasan & Antonia 1997).The situation is less clear when comparing solar windstatistics in the kinetic range with either MHD or plasmastudies. A major issue is the evidence that solar windsubproton-scale kurtosis decreases in the kinetic range(Koga et al. 2007; Wan et al. 2012; Chhiber et al. 2018).This is partially at odds with kinetic (Leonardis et al.2013) and MHD simulation (Wan et al. 2012) as well asobservations in the terrestrial magnetosheath (Chhiberet al. 2018). A putative decrease may be due to inter-ference by incoherent waves from foreshock activity, ornoise of instrumental or numerical origin (Chian & Mi-randa 2009; Wu et al. 2013), while a constant SDK maysignify a physically relevant transition to monofractalscaling (Kiyani et al. 2009; Leonardis et al. 2016). If in-coherent plasma waves are the culprit then proximity tothe terrestrial bowshock may play a role, and there aresome suggestions to this effect in contrasting ACE and
Cluster observations (Wan et al. 2012). These issues areresolved below in the
PSP observations that we present. PSP
OBSERVATIONS IN NEAR-SUN SOLARWINDWe examine higher-order inertial and kinetic scalestatistics in a region of young solar wind explored forthe first time recently by
PSP (Fox et al. 2016), usingmeasurements of the magnetic field from the FIELDS in-strument (Bale et al. 2016). We focus on a 1-hour inter-val near first perihelion from UTC 2018-11-06T02:00:00to 2018-11-06T03:00:00, when
PSP was at ∼ . (cid:12) .We use the SCaM data product, which merges fluxgateand search-coil magnetometer (SCM) measurements bymaking use of frequency-dependent merging coefficients,thus enabling magnetic field observations from DC to 1MHz with an optimal signal-to-noise ratio (Bowen et al.2020). Solar Probe Cup (SPC) data from the SWEAPinstrument (Kasper et al. 2016; Case et al. 2020) provideestimates of bulk plasma properties.For the interval used here, the SCaM data-set is re-sampled to 0.0034 s time cadence. Time series of he-liocentric RT N components (Fr¨anz & Harper 2002) ofthe magnetic field are shown in Figure 1. SPC measure-ments of ion density, velocity, and thermal speed areresampled to 1 s resolution and cleaned using a time-domain Hampel filter (e.g., Pearson 2002). The generalproperties of the plasma during the interval are listedin Table 1. The radial velocity V R during this inter-val indicates a slow wind, with V R (cid:46)
450 km/s. Threeprominent reversals, or switchbacks (Dudok de Wit et al.2020), of the radial magnetic field are present (see Foot-note 1). A high degree of correlation, or Alfv´enicity, of inetic-scale intermittency in Near-Sun Solar Wind Turbulence - PSP Observations Time on 2018-11-06 (cid:104) V (cid:105) (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) b (cid:105) (cid:104) V A (cid:105) β i UTC 02:00:00 - 03:00:00 343 km/s 52 km/s 3 . ° × K 304 cm −
13 km 99 nT 63 nT 124 km/s 0.4
Table 1.
Bulk plasma parameters. Shown are the average values of proton speed (cid:104) V (cid:105) ≡ (cid:104) (cid:112) V R + V T + V N (cid:105) , rms velocityfluctuation (cid:104) v (cid:105) ≡ (cid:112) (cid:104)| V − (cid:104) V (cid:105)| (cid:105) , ion temperature (cid:104) T i (cid:105) , ion density (cid:104) n i (cid:105) , ion inertial scale d i , magnetic field magnitude (cid:104) B (cid:105) ≡(cid:104) (cid:112) B R + B T + B N (cid:105) , rms magnetic fluctuation (cid:104) b (cid:105) ≡ (cid:112) (cid:104)| B − (cid:104) B (cid:105)| (cid:105) , Alfv´en speed (cid:104) V A (cid:105) ≡ (cid:104) B (cid:105) / (cid:112) πm i (cid:104) n i (cid:105) , and ion beta.Averaging is performed over the entire interval. Figure 1.
Top : Time series of heliocentric RTN componentsof magnetic field, and radial ion velocity.
Bottom : Tracemagnetic field power spectral density × / f ci , and inverse of ion and elec-tron inertial lengths (1 /d i and 1 /d e ) on wavenumber axis.Equal ion and electron densities are assumed to compute d e . velocity and magnetic field is observed (Kasper et al.2019; Chen et al. 2020).The correlation time (Matthaeus & Goldstein 1982)is ∼
450 s, corresponding to a correlation length of ∼ . × km, using Taylor’s frozen-in approximation(Taylor 1938) with a mean speed of 340 km/s. Taylor’shypothesis has reasonable validity during the first PSP orbit (Chhiber et al. 2019; Chen et al. 2020); here it isreaffirmed by noting from Table 1 that (cid:104) v (cid:105) / (cid:104) V (cid:105) ∼ . (cid:104) V A (cid:105) / (cid:104) V (cid:105) ∼ . RT N magnetic field components. Similar
PSP spectra have been reported previously (e.g., Chenet al. 2020). We find an inertial range that extendsmore than two decades in wavenumber. Above the iongyrofrequency the spectrum steepens to a ∼ − / S/N ) at high (kinetic range)frequencies, where the relevant instrumental noise floor is that of the SCM (Bowen et al. 2020), shown in Figure1. Clearly
S/N ≥
100 up to 100 Hz, and remains ≥ INTERMITTENCY OBSERVED BY
PSP
We define increments of magnetic-field components attime t as δB i ( t, τ ) = B i ( t + τ ) − B i ( t ) , (2)where i ∈ { R, T, N } and τ is a temporal lag. To converttemporal lags to spatial lags we use the Taylor approx-imation, wherein the spatial lag corresponding to τ is (cid:96) = (cid:104) V R (cid:105) τ (see Chhiber et al. 2020), with mean radialsolar-wind speed (cid:104) V R (cid:105) ∼
335 km/s here. In this way weobtain spatial increments δB i ( t, (cid:96) ) using Equation (2).The magnitude of the vector magnetic increment is then δB ( t, (cid:96) ) ≡ ( δB R + δB T + δB N ) / .The p -th order structure functions of δB are S ( p ) B ( (cid:96) ) = (cid:104) [ δB ( t, (cid:96) )] p (cid:105) T , (3)where the (cid:104) . . . (cid:105) refers to averaging over the time interval T (cid:29) τ . Similarly, for for each component B i , S ( p ) B i ( (cid:96) ) = (cid:104) [ δB i ( t, (cid:96) )] p (cid:105) T . (4)The accuracy of computed higher-order moments is af-fected by sample size; a rule of thumb is that the highestorder that can be computed reliably is p max = log N − N is the number of samples (Dudok de Wit et al.2013). With N ∼ . × for the present interval weget p max = 5. Statistics of higher order than this areinterpreted with some reservation.The top panel of Figure 2 shows S ( p ) B ( (cid:96) ) for p rang-ing from 1 to 8, and spatial lags (cid:96) ranging from ∼ . d i , deep within the kinetic range, to 10 d i , closeto the energy-containing scales (the correlation lengthis ∼ . × d i ). The slopes of S ( p ) vs. (cid:96) are largerat kinetic scales, indicating the presence of relativelystronger gradients. Structure functions for individualcomponents (Equation (4)) are very similar (not shown).Next we investigate the slopes of the structurefunctions in greater detail. For Gaussian and non-intermittent statistics consistent with K41 (see §
2) one
Chhiber et al.
Figure 2.
Top : Structure functions for δB (Equation (3)) vs. temporal ( τ ) and spatial ( (cid:96) ) lags. A reference (cid:96) / curve (dashed, purple) is shown. Shaded region (cream) (cid:96) = 10 − d i demarcates inertial-range. Shaded region(blue) (cid:96) = 0 . − d i demarcates kinetic-range. Middle : Scal-ing exponents ζ ( p ) vs. p for inertial, kinetic, and interme-diate ranges. Dashed line: K41 prediction ζ ( p ) = p/
3. Mo-ments not determined with reliable accuracy: grey-shadedregion. 1 σ uncertainty estimates for straight line fits to de-termine ζ ( p ) are shown, but are generally smaller than thesymbols. Bottom : Same as middle, but using ESS; scalingexponents for each range of lags are divided by ζ (3) for therespective range. Kinetic-range curve (blue triangles) over-laps the K41 curve. expects S ( p ) ( (cid:96) ) ∝ (cid:96) ζ ( p ) with ζ ( p ) = p/
3. Figure 2 (mid-dle panel) shows the scaling exponents ζ ( p ) vs p , com-puted separately for the inertial (10 − d i ) and kinetic(0 . − d i ) ranges, as well as an intermediate range(2 − d i ). The exponents are computed by using chi-squared error minimization to fit straight lines to ln S ( p ) vs ln (cid:96) . Inertial range exponents (red circles) begin to di-verge from the K41 curve beyond p = 3, with higher or-ders showing larger departures, indicating strong inter-mittency with multifractal statistics (see Equation (1)).The kinetic-range curve (blue diamonds) also lies farfrom the K41 prediction, but is rather close to a straightline, suggesting monofractal and scale-similar but non-Gaussian statistics (e.g., Frisch 1995; Kiyani et al. 2009).Exponents for the intermediate range show a transitionfrom inertial to kinetic range behavior.The bottom panel of Figure 2 employs the ExtendedSelf Similarity (ESS) hypothesis (Benzi et al. 1993),which posits that scalings of structure functions at eachorder are related to that of other orders. In particularthe scaling of S ( p ) ( (cid:96) ) with order p > S (3) ( (cid:96) ) than to the lag (cid:96) itself. Ac-cordingly we proceed by dividing ζ ( p ) for the differentlag ranges by ζ (3) for the respective range. This rescal-ing does not affect the inertial range result significantly.Remarkably, the kinetic-scale exponents collapse almostperfectly to the K41 line. As far as we are aware, this hasnot been previously demonstrated for magnetic fluctua-tions in the solar wind. Similar use of ESS has been ap-plied in kinetic simulations (Wan et al. 2016; Leonardiset al. 2016) and to solar wind density fluctuations atsubproton scales (Chen et al. 2014). The intermediaterange once again exhibits transitional behavior.To further investigate near-Sun kinetic scale intermit-tency we examine PDFs of increments of B R at lagsranging from near energy-containing scales, through theinertial range, down to subproton scales. We first nor-malize increments (Equation (2)) at each lag by the cor-responding standard deviation, and then compute PDFsby calculating the relative frequency of occurrence of in-crements within designated bins and dividing these fre-quencies by the bin width to obtain probability densi-ties. The resulting PDFs (Figure 3) are compared witha Gaussian PDF for reference. Increments at (cid:96) = 5000 d i measure structures at scales of about half a correlationlength, and these non-uniform, “system-size” structuresexhibit a highly irregular PDF, which nevertheless hasthe narrowest tails of all. PDFs for the two inertial rangelags (100 and 10 d i ) show wide, super-Gaussian tails,signifying the presence of outlying “extreme” events andintermittency. The 10 d i lag has slightly wider tails, con-sistent with the well known property of stronger inter- inetic-scale intermittency in Near-Sun Solar Wind Turbulence - PSP Observations Figure 3.
PDFs of δB R normalized by their standard de-viation σ . Gaussian PDF shown for reference (dashed line). Inset : PDFs of unnormalized δB R ; lags from 1 d i (outermostcurve, green) to 0 . d i (innermost, red). Collapse of greenand red curves after rescaling is evident in the main graphic.All PDFs include bins with population ≥ mittency at smaller inertial-range scales (e.g., Sorriso-Valvo et al. 1999; Chhiber et al. 2018).Moving on to kinetic-range lags (1 and 0 . d i ), we seesuper-Gaussian tails in PDFs, indicating the continuedpresence of intermittent structures at these scales. How-ever, the widths of these tails are comparable to (per-haps even slightly narrower than) the 10 d i case, sug-gesting a saturation of the level of intermittency at pro-ton scales (see also Figure 4, below). Furthermore, thescale similarity suggested by the investigation of scalingexponents in the kinetic range (Figure 2) is reaffirmedby the fact that PDFs of the 1 and 0 . d i lags over-lap to large degree. To emphasize this “monoscaling”,the inset in Figure 3 shows PDFs of increments of B R for (cid:96) = { , . , . , . . , . } d i , not normalized by therespective standard deviations as in the main graphic.The outermost (green) curve is for (cid:96) = 1 d i and the in-nermost (red) curve is for (cid:96) = 0 . d i . The scale-similarmonoscaling of the PDFs is demonstrated by the factthat these PDFs collapse on to each other after beingrescaled by their standard deviations (c.f. Kiyani et al.2009; Osman et al. 2015). PDFs of δB T and δB N behavesimilarly.The final diagnostic of intermittency we examine isthe SDK, a normalized fourth-order moment that em-phasizes the tails of PDFs presented previously: κ ( (cid:96) ) = S (4) ( (cid:96) ) (cid:2) S (2) ( (cid:96) ) (cid:3) , (5) Figure 4.
Scale-dependent kurtosis of magnetic field. where S ( p ) can be defined using either Equation (3) or(4). κ ( (cid:96) ) may be thought of as the inverse of the fillingfraction for structures at scale (cid:96) ; i.e., if κ ( (cid:96) ) increaseswith decreasing (cid:96) then the fraction of volume occupiedby structures at scale (cid:96) decreases with decreasing (cid:96) . Thescalar Gaussian distribution has κ = 3; a value κ > κ ( (cid:96) ) for individual components of δ B as well as it’s magnitude. All four cases behave simi-larly – the kurtosis is near Gaussian at the largest lags,increases to values between 10 and 25 as the lag is de-creased across the inertial range to ∼ d i , and thenstays roughly constant down to 0 . d i . Once again, thisindicates a saturation of the intermittency and scale-similar, monofractal behavior at kinetic scales. Thisresult is consistent with kinetic simulations and
Cluster observations in the solar wind presented by Wu et al.(2013). A likely candidate for producing monofrac-tal kinetic-scale kurtosis is a scale-independent frag-mentation of current structures between ion and elec-tron scales, as suggested by some kinetic simulations(Karimabadi et al. 2013). Note the marked contrastto Figure 8 of Chhiber et al. (2018), where SDK is re-Gaussianized at kinetic scales presumably due to ter-restrial foreshock activity and/or instrumental noise in
MMS measurements. DISCUSSION AND CONCLUSIONS To test the robustness of our results and their sensitivity to in-terval selection (stationarity), the analysis was repeated sepa-rately for the first and second halves of the interval, as well asthe “quiet” period between 02:20 and 02:50 (see Figure 1). Theresults were essentially unchanged, although SDK in the quietperiod flattens at relatively larger scales (tens of d i ). Chhiber et al.
In this paper we investigated intermittency in near-Sun solar wind observations of inertial and kinetic rangemagnetic turbulence, using standard measures includingSDK and scaling of higher-order moments up to eighth.Use of a unique
PSP
FIELDS dataset, merged fromfluxgate and search-coil magnetometer measurements(Bowen et al. 2020), enables study of high frequencieswell into the subproton scales, taking Taylor’s hypothe-sis into account. Our main results extended several priorstudies and clarified outstanding questions concerningsolar wind intermittency. First, we observed clearly amonofractal, non-Gaussian, subproton kinetic range. Inparticular, with
PSP data close to the sun at 36 R (cid:12) ,far from any foreshock activity, and measurements un-affected by noise (Koga et al. 2007; Chian & Miranda2009; Wan et al. 2012; Wu et al. 2013; Chhiber et al.2018), it is possible to establish clearly that the kur-tosis does not re-Gaussianize at sub-ion scales and thestatistics remain intermittent. Another major result ofinterest from the perspective of turbulence theory (Benziet al. 1993) is that Extended Self-Similarity works ex-tremely well for sub-ion scales – consistent with resultsreported for kinetic simulation (Wan et al. 2016). Asfar as we are aware this has not been previously demon-strated for magnetic fluctuations in the solar wind. Fi-nally, we report evidence that the magnetic field in near-Sun solar wind exhibits multifractal scaling in the in-ertial range (c.f., Zhao et al. 2020; Alberti et al. 2020),which is consistent with near-Earth observations (Bruno2019, and references therein), as well as kinetic simu-lations of turbulence (Leonardis et al. 2016; Wan et al.2016). Multifractal inertial-range scaling of higher-ordermoments is a familiar result in large turbulent MHD sys-tems (Politano et al. 1998; Wan et al. 2012).We emphasize that the present results are enabled bythe the unique orbital position of
PSP , along with the high-cadence low-noise character of the FIELDS/SCaMmagnetic field dataset. Even with the clarifications thisanalysis provides, there remain unanswered questions.One major outstanding issue is why the subproton rangebecomes monofractal. This implies self-similarity (or“rescaling”) of the underlying PDF over that range (e.g.,Kiyani et al. 2009). One possible interpretation is thatthe range between proton and electron scales is popu-lated by scale-invariant sheet-like concentrations of elec-tric current density. In fact, large numbers of highlydynamic subproton-scale current sheets are seen in ki-netic simulations (e.g., Karimabadi et al. 2013) and havebeen inferred in observations (e.g., Retin`o et al. 2007).This may be distinguished from an effect of incoherentlinear (noninteracting) waves, which may be expectedto produce a return to Gaussianity (Koga et al. 2007;Chhiber et al. 2018), and not an onset of monofrac-tal scaling. However, a rigorous connection of structurewith monofractality remains to be established and is de-ferred to future research.ACKNOWLEDGMENTSWe thank A. Chasapis for useful discussions. Thisresearch was supported in part by the
PSP mission un-der the IS (cid:12)
IS project (contract NNN06AA01C) and asubcontract to University of Delaware from Princeton(SUB0000165), and NASA HSR grant 80NSSC18K1648.We acknowledge the
PSP mission for use of the data,which are publicly available at the NASA Space PhysicsData Facility. FIELDS data are publicly available athttps://fields.ssl.berkeley.edu/data/.REFERENCES
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