On the Scaling Properties of Magnetic Field Fluctuations Through the Inner Heliosphere
Tommaso Alberti, Monica Laurenza, Giuseppe Consolini, Anna Milillo, Maria Federica Marcucci, Vincenzo Carbone, Stuart D. Bale
aa r X i v : . [ phy s i c s . s p ace - ph ] J un Draft version June 8, 2020
Typeset using L A TEX default style in AASTeX63
On the Scaling Properties of Magnetic Field Fluctuations Through the Inner Heliosphere
Tommaso Alberti, Monica Laurenza, Giuseppe Consolini, Anna Milillo, Maria Federica Marcucci, Vincenzo Carbone, and Stuart D. Bale INAF - Istituto di Astrofisica e Planetologia Spaziali, via del Fosso del Cavaliere 100, 00133, Roma, Italy Universit`a della Calabria, Dip. di Fisica, Ponte P. Bucci, Cubo 31C, 87036, Rende (CS), Italy Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA Physics Department, University of California, Berkeley, CA 94720-7300, USA (Received; Revised; Accepted)
Submitted to ApJLABSTRACTAlthough the interplanetary magnetic field variability has been extensively investigated in situ by means of data coming from several space missions, the newly launched missions providing high-resolution measures and approaching the Sun, offer the possibility to study the multiscale variabilityin the innermost solar system. Here by means of the Parker Solar Probe measurements we investigatethe scaling properties of solar wind magnetic field fluctuations at different heliocentric distances. Theresults show a clear transition at distances close to say 0 . f − / frequency power spectra and regular scaling properties, while for distances larger than 0 . f − / and are characterized by anomalous scalings. Theobserved statistical properties of turbulence suggests that the solar wind magnetic fluctuations, in thelate stage far form the Sun, show a multifractal behaviour typical of turbulence and described throughintermittency, while in the early stage, when leaving the solar corona, a breakdown of these propertiesare observed, thus showing a statistical monofractal global self-similarity. Physically the breakdownobserved close to the Sun should be due either to a turbulence with regular statistics or to the presenceof intense stochastic fluctuations able to cancel out correlations necessary for the presence of anomalousscaling. Keywords:
Sun: magnetic fields — Sun: solar wind — methods: data analysis — methods: statistical— turbulence INTRODUCTIONSince the 70s several space missions have been launched to provide new insights into the solar phenomena and solarwind properties (e.g., Helios, Ulysses, Wind, ACE) allowing us to collect a wide amount of data about the processesthat cause the solar wind formation and evolution throughout the interplanetary space (e.g., Rosenbauer et al. 1977;Denskat & Neubauer 1982; Grappin et al. 1990). Among other topics (e.g., Burlaga et al. 1982; McComas et al. 1995;Marsch 2018), a wide attention has been paid to turbulence in the solar wind by investigating the scaling behavior ofboth velocity and magnetic field components (e.g., Dobrowolny et al. 1980; Matthaeus & Goldstein 1982; Tu & Marsch1990; Bruno & Carbone 2013; Alberti et al. 2019a, and references therein). Indeed, solar wind magnetic field fluctu-ations around the large-scale mean field, usually described within the magnetohydrodynamic (MHD) framework, arecharacterized by scale-invariant features over a wide range of scales (e.g., Bruno & Carbone 2013). At 1 au, this rangeof scales, known as inertial range (Kolmogorov 1941; Frisch 1995), is dominated by Alfv´enic fluctuations (Belcher 1971;Bruno & Carbone 2013) mixed with slow mode compressive ones (Howes et al. 2012; Klein et al. 2012; Verscharen et al.
Corresponding author: Tommaso [email protected]
Alberti et al.
Parker Solar Probe (PSP),
BepiColombo , and
Solar Orbiter , andthe in situ orbiting ones, e.g., ACE, Wind, and STEREO, offer the unique opportunity of multi-spacecraft combinedobservations of the interplanetary medium variability, the evolution of turbulence and solar wind structures at differentdistances from the Sun, the interaction between the solar wind plasma and planetary environments, and so on (e.g.,Milillo et al. 2010; M¨uller et al. 2013; Howard et al. 2019; Kasper et al. 2019; McComas et al. 2019). Recently, inthe framework of solar wind turbulence Chen et al. (2020) investigated the behavior of the power spectral density atdifferent heliocentric distances by means of the first two orbits of the Parker Solar Probe spacecraft showing that thepower-law spectral index moves from α B ∼ -3/2 to α B ∼ -5/3 when passing from r ∼ r ∼ q − order scaling features of magnetic field components at different heliocentricdistances (Section 3). In Section 4, the results show that the inertial range scaling properties significantly change whenmoving from closer to farther the Sun, with intermittency completely emerging at distances larger than 0.4 au. Indeed,scaling exponents show a linear behavior at smaller heliocentric distances, while larger exponents, being characterizedby a nonlinear convex behavior with the statistical order q , are found at r > DATAFor this study we use solar wind magnetic field components in the heliocentric RTN reference frame (R=radial,T=tangential, N=normal) as measured by the PSP magnetometer. The PSP magnetic field data are taken by theoutboard FIELDS Fluxgate Magnetometer (MAG) (Bale et al. 2016, 2019) and are averaged to 1-s cadence from theirnative 4 samples per cycle cadence (Fox et al. 2016). Data were freely retrieved from the Space Physics Data Facility(SPDF) Coordinated Data Analysis Web (CDAWeb) interface at https://cdaweb.gsfc.nasa.gov/index.html/.For investigating the evolution of the interplanetary magnetic field we used the first and the second orbit of PSPtowards the Sun, only considering adjacent temporal measurements during which no data gaps were found (i.e., thebest time coverage of the FIELDS instrument). These corresponds to the period between 15 October and 04 December,2018, and between 16 March and 10 April, 2019, for the first and the second orbits, respectively. During the intervalsof investigation the solar wind speed was between 250 km/s and 650 km/s and the proton density ranged between n ∼
10 cm − (at 0.7 au) and ∼
400 cm − (at 0.17 au). Figure 1 shows the three components of the interplanetarymagnetic field (at 1-s resolution) and the PSP radial distance from the Sun (at 1-hr resolution). caling Laws Through the Inner Heliosphere -1000100 Encounter 1 -1000100-1000100 290 300 310 320 330
DOY 2018
Encounter 2
75 80 85 90 95 100
DOY 2019
Figure 1. (From top to bottom) The three components of the interplanetary magnetic field (at 1-s resolution), and (lower panel)the PSP radial distance from the Sun (at 1-hr resolution). The blue, orange, and yellow lines refer to the radial, tangential, andnormal components, respectively. The right and the left panels show measurements during the first and the second PSP orbitsapproaching the Sun, respectively.
It is clear that magnetic field fluctuations decrease with increasing heliocentric distance of about one order ofmagnitude (i.e., B ( r ) ∼ /r , Parker 1958). However, by simply looking at the time series it is not sufficient to clearlydiscriminate between the different dynamical regimes and their evolution at different heliocentric distances, that is acrucial point for correctly characterizing dynamical processes such as the evolution of turbulence and intermittency,the large-scale structures dynamics, the mean field approximation, and so on. METHODSInvestigating field fluctuations is usually one of the most important aspects of dealing with the existence of dynamicalprocesses and phenomena characterizing physical systems. Generally, this can be achieved by means of data analysismethods allowing us to extract embedded features from several kinds of data and by assuming some mathematicalassumptions (e.g., Huang et al. 1998). Obviously, a suitable and well-built data analysis method should require tominimize mathematical assumptions and numerical artifacts, trying to maximize its adaptivity to the data underinvestigation (e.g., Huang et al. 1998). A suitable method with the above characteristics is the well-known and well-established Hilbert-Huang Transform (HHT), firstly introduced by Huang et al. (1998) as an adaptive and a posterioridata analysis procedure, mainly based on two different steps: a decomposition method, known as Empirical ModeDecomposition (EMD), and a statistical spectral method, e.g., the HSA (e.g., Huang et al. 1998). Being B µ ( t ) the µ -th component of the interplanetary magnetic field, by means of the EMD and HSA we can write B µ ( t ) = N X k =1 A µ,k ( t ) cos [Φ µ,k ( t )] + R µ ( t ) , (1)being C µ,k ( t ) = A µ,k ( t ) cos [Φ µ,k ( t )] the k -th empirical mode, A µ,k ( t ) and Φ µ,k ( t ) its instantaneous amplitude andphase, respectively, and R µ ( t ) the residue of the decomposition, e.g., a non-oscillating function (e.g., Huang et al.1998). More details about the HHT can be found in Appendix A.Although the HHT is surely interesting for investigating the multiscale behavior of physical systems, a distinguish-ing attribute is its suitability for investigating spectral and scaling features from a statistical point of view (e.g., Alberti et al.
Huang et al. 2011). This can be done by defining the generalized marginal Hilbert power spectral density (gPSD) as S q ( f ) = Z T H q ( t ′ , f ) f dt ′ , (2)being T the time length and H q ( t ′ , f ) the generalized Hilbert-Huang spectrum accounting for the q − order amplitudedistribution over the time-frequency plane (cfr. Appendix A, and Huang et al. 2011). The scaling behavior of S q ( f )can be characterized by means of scaling exponents β q as S q ( f ) ∼ f − β q , (3)being β q related to the scaling exponents ζ q of the generalized structure functions S q ( τ ) = | B µ ( t + τ ) − B µ ( t ) | q ∼ τ ζ q as β q = ζ q + 1 (e.g, Huang et al. 2011; Carbone et al. 2018). However, due to its local nature, S q ( f ) allows to determinescaling properties by reducing the effect of the noise, large-scale structures and inhomogeneities, and sampling effects(e.g., Huang et al. 2011). RESULTS & DISCUSSIONIt has been widely shown that solar wind magnetic field fluctuations are characterized by a scaling law behaviorin a wide range of frequencies, supporting the existence of an inertial regime where energy is transferred throughan inviscid mechanism to higher frequencies (e.g., to smaller scales, Kolmogorov 1941; Iroshnikov 1965; Kraichnan1965; Bruno & Carbone 2013). As recently pointed out by Chen et al. (2020) spectral exponents move from α B ∼ -3/2 to α B ∼ -5/3 when passing from r ∼ r ∼ ζ q = β q − ζ (2) as a function of the heliocentric distance, together with the95% confidence level.Results clearly show a difference between the scaling exponents ζ for distance below 0.4 au with respect to thoseevaluated at larger distances (i.e., larger than 0.4 au). This difference suggests that magnetic field fluctuations followsa f − / scaling closer to the Sun, being ζ ≃ /
2, while a steeper scaling is found at larger distances ( ζ ≃ / r > . ζ observed near theSun could be related to a more steady-state nature of the inertial range, due to the large number of nonlinear times(Matthaeus & Goldstein 1982). Conversely, the larger values of ζ at r > . r increases as well as to the role of intermittency (Bruno & Carbone 2013). Both findingsare also well in agreement with predictions made by numerical simulations of Alfv´enic turbulence in homogeneousplasmas (Boldyrev 2006; Lithwick et al. 2007; Perez & Boldyrev 2009; Chandran et al. 2015; Mallet & Schekochihin2017), suggesting that the inertial range processes vary from purely nonlinear interacting components to less organizedfluctuations (Velli et al. 1989; Bruno & Carbone 2013). The transition from ζ ∼ / ζ ∼ / r decreasesgradually occurs and can be easily interpreted in the general framework of far-from-equilibrium complex systemsas the evidence of a sort of dynamical phase transition which is consistent with the observed decreasing trend ofpositive correlation and the increasing of the outer scale with r (Chen et al. 2020). However, it is not sufficient toconsider only one statistical moment of the probability distribution function to fully characterize solar wind turbulence.Indeed, since the pioneering work by Kolmogorov (1941) we know that turbulence is a phenomenon characterized by ahierarchy of scales whose statistics are scale-invariant (e.g., Kolmogorov 1941; Iroshnikov 1965; Kraichnan 1965; Frisch1995; Alberti et al. 2019a). The statistical scale-invariance implies that the scaling of field increments should occurwith a unique scaling exponent, thus implying that the statistical moments of the field increments should scale as S q ( τ ) ∼ τ q/D , being D = 3 for fluid turbulence (e.g., Kolmogorov 1941; Frisch 1995) and D = 4 for plasma turbulence(e.g., Iroshnikov 1965; Kraichnan 1965; Bruno & Carbone 2013). Nevertheless, there is considerable evidence thatturbulent flows deviate from this behavior, being the scaling exponents a nonlinear function of the order q (e.g.,Carbone et al. 1995), which point out an ”anomalous” scaling process and proves the appearence of intermittency(e.g., Frisch 1995; Bruno & Carbone 2013). For low orders the discrepancy with the linear behavior is very small, thusexplaining why the Kolmogorov spectrum is usually observed in turbulence (e.g., S ( τ ) ∼ τ / → S ( f ) ∼ f − / ).However, for high order statistics the difference is significant, and the breakdown of the statistical self-similarity is clear,thus questioning, in the modern theory of turbulence, what is really universal in the inertial range (e.g., Alberti et al. caling Laws Through the Inner Heliosphere Figure 2.
The behavior of the scaling exponents ζ for each magnetic field component at different heliocentric distances r ,together with the 95% confidence level. The blue, orange, and yellow symbols refer to the radial B R , tangential B T , and normal B N components, respectively. The continuous and dashed black lines are used as a reference to 2 / / r = 0 .
01 au (error bars are evaluated as the standard deviations). ζ q , q ∈ [0 , S q ( f ) (cfr. Section 3), at different heliocentric distances as shown in Figure 3.Firstly, a clear difference emerges from the scaling behavior for r < r > q , while the latter shows the typical convex nonlinear shape with q . The surprisingly behavior of scalingexponents near the Sun, suggesting a monofractal nature of field fluctuations within the inertial range, supports theassumptions of global statistical self-similar scale-invariance. Conversely, these assumptions break at 0.4 au, wherethe nonlinear convex behavior of scaling exponents, suggest a multifractal behavior of magnetic field fluctuations (e.g.,Bruno & Carbone 2013; Alberti et al. 2019a). This transition could be related to physical processes suppressing thescaling properties of the energy transfer rate close to the Sun, being consistent with the emergence of intermittencyin solar wind turbulence for r > β parameter, themagnetic compressibility, the expansion/correlation time of fluctuations within the inertial range, the slow-/Alfv´enic-mode variability within the heliosphere, the outward propagating Alfv´enic fluctuations (predominantly originatingfrom the Sun but undergoing a dynamical evolution due to nonlinear and velocity-shear), localized phenomena givingrise to intermittency, local changes in the cross-helicity, and so on (Denskat & Neubauer 1982; Bavassano et al. 1982;Matthaeus & Goldstein 1982; Tu & Marsch 1990; Marsch & Tu 1990; Grappin et al. 1990; Carbone et al. 1995; Marsch2018; Chen et al. 2020). Thus, the scaling exponents are not only a function of the statistical order q but they alsodepend on the radial distance r (i.e., ζ q ( r )) which is the reflection of both global evolving and local dynamical processes. Alberti et al.
Figure 3.
The behavior of the scaling exponents ζ q for each magnetic field component at different heliocentric distances r . Thedifferent colors correspond to different distances r as reported in the legend. The continuous and dashed black lines are used asa reference to q/ q/ caling Laws Through the Inner Heliosphere ζ , at different heliocentricdistances (e.g., Denskat & Neubauer 1982; Marsch & Tu 1990; Tu & Marsch 1990; Chen et al. 2020), there seems tobe a change as the Sun is approached, rather suddenly inside 0.4 au (Denskat & Neubauer 1982; Chen et al. 2020).Our findings not only strongly agree with seminal works when q = 2 is considered (e.g., Denskat & Neubauer 1982;Marsch & Tu 1990; Tu & Marsch 1990; Chen et al. 2020) but also allow, for the first time, to monitor the evolutionof the scaling properties at different locations for high-order statistics, showing that the solar wind nature movesfrom monofractal to multifractal near 0.4 au. This change can be directly observed by looking at the behavior ofsingularities on the topology of solar wind magnetic field by means of the singularity strengths α ( r ) = dζ µq ( r ) dq as usualin the multifractal approach (Frisch 1995; Bruno & Carbone 2013; Alberti et al. 2019a). In this way we can alsoprovide a sort of multifractal measure ∆ α ( r ) = max { α ( r ) } − min { α ( r ) } (although we can only access the left part ofthe usual singularity spectrum f ( α ) since q ≥ α ( r ) for each magnetic fieldcomponent at different heliocentric distances r as in Fig. 3, while the inset show the behavior of singularity strengths α ( r ) at different distances r . Figure 4.
The behavior of the multifractal width ∆ α ( r ) for each magnetic field component at different heliocentric distances r as in Fig. 3. The blue, orange, and yellow symbols refer to the radial B R , tangential B T , and normal B N components,respectively. Error bars show the 95% confidence level. The inset shows the behavior of α ( r ) at different distances r . We clearly observe a breakdown of the multifractal width ∆ α ( r ), moving from values closer to zero up to largervalues ∆ α ( r > . > .
2, thus suggesting the emergence of singularities as r increases. This is confirmed by lookingat the inset of Fig. 4 in which is easy to detect a spread in singularity strengths α ( r ) as r increases, with the transitionobserved near r ∼ . CONCLUSIONSIn this manuscript we dealt with the characterization of scaling features of magnetic field components as measuredby PSP at different locations. We showed that the inertial range dynamics moves from a monofractal behaviour anda power spectrum scaling f − / , at r < f − / , at r > Alberti et al. of its leaving the solar corona, show statistical self-similarity, while a breakdown of the statistical self-similarity forhigh-order statistics is found at a distance larger than 0.4 au from the Sun. In fact, we observed a roughly abrupttransition of the multifractal width ∆ α ( r ), moving from values closer to zero up to larger values ∆ α ( r > . > . r > .
4, also confirmed by the spread in singularity strengths α ( r ) as r increases, with a transition observed near r ∼ . q but theyalso depend on the radial distance r from the Sun, e.g., ζ q ( r ), moving from a linear to a nonlinear convex behavior as r increases.The observed transition could be related to something that suppresses the scaling properties of the energy transferrate through the inertial range and the phase-coherency across the cascade for fluctuations close to the Sun. Roughlyspeaking, when the magnetic field is strong enough, since the scaling of the power spectra for inwards/outwardsfluctuations are the same (Chen et al. 2020), the usual Iroshnikov-Kraichnan model suggests that fluctuations shouldscales as h (∆ b ) q i ∼ c A h ǫ q/ ℓ i ℓ q/ , instead of the usual Kolmogorov scaling h (∆ b ) q i ∼ h ǫ q/ ℓ i ℓ q/ (Bruno & Carbone2013). In both cases, anomalous scaling laws ζ q = hq + µ ( hq ), being h either h = 1 / h = 1 /