Wave Composition, Propagation, and Polarization of MHD Turbulence within 0.3AU as Observed by PSP
Xingyu Zhu, Jiansen He, Daniel Verscharen, Die Duan, Stuart D. Bale
DDraft version September 9, 2020
Typeset using L A TEX default style in AASTeX63
Wave Composition, Propagation, and Polarization of MHD Turbulence within 0.3AU as Observed byPSP
Xingyu Zhu , Jiansen He , Daniel Verscharen ,
2, 3
Die Duan , and Stuart D. Bale
4, 5, 6, 7 School of Earth and Space Sciences, Peking UniversityNo.5 Yiheyuan Road, Haidian DistrictBeijing, 100871, China Mullard Space Science Laboratory, University College London, Dorking RH5 6NT, UK Space Science Center, University of New Hampshire, Durham NH 03824, USA Physics Department, University of California, Berkeley, CA 94720-7300, USA Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA The Blackett Laboratory, Imperial College London, London, SW7 2AZ, UK School of Physics and Astronomy, Queen Mary University of London, London E1 4NS, UK
Submitted to ApJLABSTRACTTurbulence, a ubiquitous phenomenon in interplanetary space, is crucial for the energy conversionof space plasma at multiple scales. This work focuses on the propagation, polarization and wavecomposition properties of the solar wind turbulence within 0.3AU, and its variation with heliocentricdistances at MHD scales (from 10s to 1000s in the spacecraft frame). We present the probability den-sity function of propagation wavevectors (PDF( k (cid:107) , k ⊥ )) for solar wind turbulence winthin 0.3 AU forthe first time: (1) wavevectors cluster quasi-(anti-)parallel to the local background magnetic field for kd i < .
02, where d i is the ion inertial length; (2) wavevectors shift to quasi-perpendicular directionsfor kd i > .
02. Based on our wave composition diagnosis, we find that: the outward/anti-sunwardAlfv´en mode dominates over the whole range of scales and distances, the spectral energy density frac-tion of the inward/sunward fast mode decreases with distance, and the fractional energy densitiesof the inward and outward slow mode increase with distance. The outward fast mode and inwardAlfv´en mode represent minority populations throughout the explored range of distances and scales.On average, the degree of anisotropy of the magnetic fluctuations defined with respect to the mini-mum variation direction decreases with increasing scale, with no trend in distance at all scales. Ourresults provide comprehensive insight into the scenario of transport and transfer of the solar windfluctuations/turbulence in the inner heliosphere. INTRODUCTIONThe solar corona dynamically expands into interplanetary space in the form of the continuous solar wind (Parker1958; Coleman 1966), the birth and the acceleration mechanism of which are still not well understood (Tu et al. 2005;He et al. 2007; Cranmer & Winebarger 2019). The solar wind flowing into interplanetary space carries informationabout its source region, and on the other hand, involves diversified nonlinear physical processes (Tu & Marsch 1995;Bruno & Carbone 2013). It is essential to investigate the nature of near-sun fluctuations in order to analyze andunderstand these nonlinear physical processes as well as the heating and acceleration mechanisms of solar wind.The statistical properties of the solar wind generally vary with speed, location and type of source region andheliocentric distance (Bavassano et al. 1982; Tu et al. 1989; He et al. 2013; Matteini et al. 2014; Stansby & Horbury2018; Horbury et al. 2018; Wang et al. 2019; Perrone et al. 2019; Bandyopadhyay et al. 2020; Chen et al. 2020; Chhiberet al. 2020; Duan et al. 2020; Qudsi et al. 2020). Tu et al. (1989) contrast the properties of MHD turbulence between
Corresponding author: Jiansen [email protected] a r X i v : . [ phy s i c s . s p ace - ph ] S e p Zhu et al. high speed and low speed solar wind at 0.3AU using the spectra of Els¨asser variables, cross helicity, residual energy,Alfv´en ratio and Els¨asser ratio. They consider that, compared to the high speed wind, the turbulence evolves in anadvanced state in slow wind, due to the longer expansion time. The z + and z − are close to a balanced state with anapproximate -1.67 spectral index. The mode composition therein is dominated by the Alfv´en mode and slow mode inthe limit of incompressibility (Dobrowolny et al. 1980). Bavassano et al. (1982) study the variation of the nature ofthe fluctuations with heliocentric distance and scale in the trailing edge of a stream interaction region. The anisotropydefined with respect to the direction corresponding to the minimum eigenvalue decreases as the heliocentric distanceincreases and the scale decreases. The magnetic field closer to the sun is more compressed. However, Chen et al.(2020) report the evolution of solar wind turbulence from 0.17AU to 1AU, recently, and find at 0.17AU that: (1) thespectra of magnetic field, velocity and Els¨asser variables present a -3/2 slope at MHD scales; (2) the magnetic fieldis less compressed and (3) the outward propagating Alfv´en waves are more dominant than at 1AU. Fast solar wind ischaracterised by highly Alfv´enic fluctuations, although a new type of Alfv´enic slow solar wind, possibly coming fromquiet-Sun regions or coronal-hole boundaries, has been reported at distances from 0.3 AU to 1 AU. (D’Amicis & Bruno2015; Wang et al. 2019; Perrone et al. 2020; Parashar et al. 2020). It is of interest to study this kind of solar wind, onaccount of its distinct properties, which differ from the classical slow solar wind.Even if we relax the assumption of a pure superposition of linear waves, nonlinear turbulent fluctuations still retaincertain polarization and correlation properties of linear modes (Tu & Marsch 1995). When we use the term ’wave’,we refer to the mode composition of the fluctuations within this wave-turbulence paradigm. The composition of wavemodes in the solar wind near 1 AU has been extensively studied and controversially discussed. There are many meansto diagnose the wave modes: correlation analysis between velocity and magnetic field fluctuations (Wang et al. 2012;ˇSafr´ankov´a et al. 2019), cross helicity analysis (Roberts et al. 1987), comparison of the MHD dispersion relationsderived from measurements with theory predictions (Shi et al. 2015), and mode recognition methods (Glassmeieret al. 1995; Narita & Marsch 2015; Chaston et al. 2020). According to these studies, non-compressive outward Alfv´enmodes dominate the fluctuations in the solar wind especially in fast streams (Bruno & Carbone 2013). Compressivewaves likely suffer strong Landau damping (Barnes 1966), resulting in their suppression in the overall fluctuations.Correlations among variables (e.g. magnetic pressure, thermal pressure, density and temperature), show that thecompressive component simultaneously exists of magnetosonic waves and pressure balanced structures (PBSs) (Kellogg& Horbury 2005; Yao et al. 2011; Yang et al. 2017). The majority of the compressive fluctuations is slow-mode-likerather than fast-mode-like in polarization (Howes et al. 2012; He et al. 2015; Shi et al. 2015). In situ observationsshow that the Alfv´enicity decreases with heliocentric distance, which might be caused by the increased contributionof inward propagating Alfv´en waves or the compressive fluctuations (Bruno & Bavassano 1993).To further comprehend the underlying multi-scale nature and evolution of near-sun turbulence, we systematicallystudy the variation of the fluctuations’ properties with scale and heliocentric distance within 0.3AU. The propertiesinclude the propagation direction of the wave, the mode composition, and the characteristic of anisotropy on average. InSection 2, we briefly introduce the data sets we use. In Section 3, we present our methods and analysis results, and giveour summary and conclusions in Section 4. Our observations can provide observational evidence for the verification ofexisting theoretical models at closer heliocentric distances, and also impose constraints on the improvement of existingtheoretical models and the proposal of new models. DATA SETS AND DATA DEDUCTIONWe conduct our analysis using the data obtained from
Parker Solar Probe (PSP), which is the closest human-builtsatellite to the sun up to now (Fox et al. 2016). We use the Level-2 magnetic field data supplied by the Flux-gateMagnetometer (MAG; Bale et al. (2016)) and the Level-3i proton data provided by the Solar Probe Cup (SPC; Kasperet al. (2016)). The time interval investigated spans from UTC2018-10-31/20:00:00 to UTC2018-11-10/15:00:00 inwhich period PSP cruised between 0.166AU (35.78 solar radii)and 0.243AU (56.37 solar radii). The interval we chooseis shorter than the high-cadence interval around the first perihelion, because there are several sampling gaps longerthan 30 minutes in the other intervals from which the SPC data are unavailable. We analyze time periods of thefluctuations in the range from 10s to 1000s, corresponding to MHD scales in the plasma frame. We do not exclude theso-called ‘switchback’ patterns that exist among various scales (see Bale et al. (2019); Kasper et al. (2019); Dudok deWit et al. (2020)).For the analysis of propagation direction and fluctuation anisotropy, we use the Singular Value Decomposition (SVD)method to resolve the frequencies and wavevectors of the waves based on Faraday’s law. We estimate the three singular
HD waves in inner heliosphere E = − V i × B (Shi et al. 2015), where V i is the proton bulk velocity obtained from SWEAP/SPC. Note that only one wavevector is solved for every specificfrequency with the SVD method. Therefore, the resolved frequency and wavevector can be regarded as the frequencyand wavevector of the major wave mode. In reality, it is possible that multiple wave modes exist in the turbulence atthe same time and scale. For the mode composition analysis, we use the method suggested by Glassmeier et al. (1995),and get the contributions of the six MHD modes (parallel and anti-parallel propagating Alfv´en mode, fast mode andslow mode) to the fluctuations. We estimate the spectral energy density of each mode as e Ti S ( f sc , t ) e i , where S ( f sc , t )is the spectral density matrix as defined by Glassmeier et al. (1995) and e i is the eigenvector of the correspondingmode. ANALYSIS RESULTSWe present an overview of the observed magnetic field and plasma measurements in Figure 1. To highlight thecorrelated fluctuations of the variables over such a long duration of about 10 days, we smooth all the measurementswith a running window of 30 min. Figure 1(a) shows that the proton density ( N p ) and the magnetic field strength( | B | ) decrease with increasing heliocentric distance. The three components of the magnetic field ( B R , B T , B N ) and theproton velocity ( V R , p , V T , p , V N , p ) in the RTN coordinates are positively correlated, respectively, which suggests thatthe large-scale outward-propagating Alfv´enic fluctuations are dominant during this encounter. The proton thermalvelocity ( W p ) varies between 50 km/s and 100 km/s and there is no global correlation between the proton density andthe thermal velocity. The plasma beta ( β p ) is around 2, which does not exhibit a significant variation with heliocentricdistance ( R ).We solve the wavevector, k ( τ, R ( t )), at different heliocentric distances ( R ( t )) and periods ( τ ), where R is a function oftime ( t ). The local background magnetic field, B ( τ, R ( t )), is acquired by Gaussian-weighting of the magnetic-field timeseries at time t , where the width of the Gaussian profile is defined by the period τ (Podesta 2009). We then calculatethe angles between k and B , θ k , B ( τ, R ( t )). Figure 2 (a1), (b1) and (c1) show the probability distribution functions(PDFs) of wavevectors in three distance ranges. For kd i > .
02, the wavevectors cluster around the quasi-perpendiculardirection. For kd i < .
02, the most probable wavevectors are quasi-parallel, relative to the local background magneticfield. Figure 2 (a2), (b2) and (c2) show the PDFs of θ k , B depending on kd i , in three different distance ranges. Thepropagation angles are close to 160 ◦ for kd i < .
02, and close to 90 ◦ for kd i < .
02. This indicates that the propagationangles are scale-dependent and turn from quasi-parallel at large scales to quasi-perpendicular at small scales.We carry out a mode composition diagnosis (Glassmeier et al. 1995) and directly obtain the fractions of the six MHDwave modes, at different heliocentric distances and periods. According to the radial component of the local backgroundmagnetic field, B r = B · ˆ e r , we transform the parallel and anti-parallel modes into outward/anti-sunward modeswhen k · B >
0, and inward/sunward modes when k · B <
0, respectively. The variation results of the averagedfractions of the transformed MHD modes are shown in Figure 3. The upper three panels show the variation of thefractions of the MHD modes with period averaged over the distance ranges of 0.180AU-0.185AU, 0.209AU-0.214AUand 0.238AU-0.243AU, respectively. The wave mode occupying the highest spectral proportion is the outward Alfv´enmode at most scales in these three R -intervals. The outward fast mode, the inward Alfv´en mode and the outwardslow mode represent the modes with the lowest fractional proportions throughout the whole MHD range at thesedistances. The fractional proportions of these three modes slightly increase with increasing distance. On average, from0.180AU to 0.185AU, the inward fast mode is the second-most abundant mode, while the inward slow mode is in thirdplace. From 0.209AU to 0.214AU, the inward slow mode and the inward fast mode have approximately equivalentproportions. From 0.238AU to 0.243AU, the inward slow mode is in second place, followed by the inward fast mode.We also find this change of mode composition with distance in the radial variation of the period-averaged fraction ofmode compositions (see lower panel of Figure 3). The outward Alfv´en mode dominates throughout the whole near-Sunregion under investigation. The fractional contribution of the fast mode decreases with increasing distance, while thecontribution from inward slow modes increase with distance.To further verify this composition diagnosis results, we reconstruct the dispersion relations of Alfv´en waves andslow waves, as shown in Figure 4. We first demonstrate a benchmark test to verify the ability of the SVD methodto resolve the MHD dispersion relations. The preset basic parameters are: bulk velocity, V = 400km / s, backgroundmagnetic field, B = 90nT, proton number density, n p = 300cm − , proton thermal velocity, 60km / s, θ k , B = 20 ◦ .Based on the polarization relations of the Alfv´en mode and slow mode, we set up the corresponding magnetic field Zhu et al.
Figure 1.
Time sequences overview of magnetic and plasma measurements during PSP’s first encounter. Panel (a): mag-netic field strength ( | B | ) and proton density ( N p ). Panel (b&c&d): magnetic fields ( B R , B T , B N ) and proton bulk velocities( V R , p , V T , p , V N , p ) in RTN coordinates. Panel (e): proton density ( N p ) and thermal velocity ( W p ). Panel (f): plasma beta ( β )and heliocentric distance ( R ) of spacecraft’s position. HD waves in inner heliosphere Figure 2.
Panel (a1&b1&c1): Probability distribution functions of the wavevector in | k (cid:107) d i | − | k ⊥ d i | space in the range of0.180AU-0.185AU, 0.209AU-0.214AU, and 0.238AU-0.243AU, respectively. The white solid lines represent the relation between k (cid:107) and k ⊥ as predicted from the phenomenology of critical balance, k (cid:107) ∼ k ⊥ k , where k is the wavenumber of the outer scale.Panel(a2&b2&c2): PDFs of the propagation angle ( θ k , B ) at differing scales ( kd i ), in the corresponding distance ranges. and velocity disturbances of the two modes respectively, and use these disturbances as an artificial data input of theSVD method. In order to test the robustness of the SVD method, we also add 0.1% level of noise for each wave at allscales to the virtual data input. As a result, we obtain a solution in terms of the wavevector ( kd i ) at every frequency( ω/ω ci ) and during every local time interval, and ω ci is the ion cyclotron frequency. Furthermore, we construct thePDF( kd i , ω/ω ci ) statistically based on the information of kd i ( ω/ω ci , t ). The PDFs( kd i , ω/ω ci ) for the benchmark testsof the Alfv´en mode and slow mode are illustrated in Figure 4(a) and 4(b). The dispersion relations as indicated bythe ridges with high PDF values are fully consistent with the theoretical dispersion relations, which means that theSVD method is well able to resolve the MHD dispersion relations from our observations. Thereafter, we apply theSVD method to the observational measurements to examine whether the dispersion relations of Alfv´en waves and slowwaves prevail. The results are shown in Figure 4(c) and Figure 4(d). The green patches corresponding to high levelsof PDF concentrate on and around the theoretical dispersion relations of MHD Alfv´en and slow modes. These resultsfurther confirm the existence of incompressible Alfv´en waves (the most prevalent component) and compressible slowwaves (the sub-dominat component).Lastly, we investigate the variation of the fluctuation anisotropy with period and distance in Figure 5. The ratioof the middle and maximum singular values of the spectral matrix (Eq.(8) of Santol´ıK et al. (2003)), λ mid /λ max , isadopted to represent the anisotropy of the magnetic field fluctuations in the plane perpendicular to the propagationdirection, which we take to be oriented along the direction with the minimum singular value λ min . λ mid /λ max is alsoknown as the ellipticity. The ratio increases from around 0.3 to over 0.37 as the wave period increases, throughout Zhu et al. F r a c t i on F r a c t i on F r a c t i on F r a c t i on Outward Alfven Outward Fast Outward SlowInward Alfven Inward Fast Inward Slow
Figure 3. (Top) The period-depending variation of the spectral fractions of the six MHD modes i.e., outward/anti-sunward( solid line ) and inward/sunward ( dashed line ) propagating Alfv´en modes ( green ), fast modes ( blue ) and slow modes ( red ), asaveraged over different distance ranges: 0.180AU-0.185AU (
Left ), 0.209AU-0.214AU (
Middle ) and 0.238AU-0.243AU (
Right ),respectively. (Bottom) The heliocentric distance variation of the spectral fractions of the six MHD modes as averaged over thetime scale (period) from 10 to 1000s. The lime shadow sections correspond to the distance ranges used for the averaging of theintervals in the upper three panels. the distance range under investigation. According to the above analysis, the dominant Alfv´en mode increases in itsdegree of circular or arc polarization with increasing period. CONCLUSIONThe diversity, complexity and evolution of solar wind turbulence have always been important research topics inheliospheric physics. Hence, we statistically study mode propagation, mode composition, and fluctuation anisotropyof the solar wind MHD turbulence as measured by PSP. We find that:(1) At 0.166AU < R < θ k , B ) of wave-like turbulent fluctuations for kd i < .
02 aregreater than 135 ◦ , mainly concentrating around 160 ◦ , while the distribution gradually shifts its center to θ k , B ∼ ◦ for 0 . < kd i < . HD waves in inner heliosphere Figure 4.
Panel (a)&(b): The PDFs of the normalized wavenumbers, kd i , for Alfv´en waves and slow waves, at each normalizedangular frequency, ω/ω ci in the plasma frame, resolved by a benchmark test of the SVD method, with MHD Alfv´en-mode andMHD slow-mode fluctuations. The theoretical MHD dispersion relations of Alfv`en mode, fast mode and slow mode are plottedin black, blue and red solid lines, respectively. Panel(c)&(d): The PDFs of kd i for Alfv´en waves and slow waves, at each ω/ω ci ,obtained from application of the SVD method to the magnetic and velocity measurements from PSP in [20:00, 21:00] UT on2018-11-05 (panel c) and [18:20, 18:25] UT on 2018-11-04 (panel d), consistent with the dispersion relations of Alfv´en and slowmodes, respectively. Unlike in panels (a) and (b), we use the averaged plasma parameters over the corresponding time intervalsin panels (c) and (d). Figure 5. (a) The distance profiles of the ratio between the middle and maximum singular values of the magnetic spectralmatrix ( λ mid /λ max ) for the magnetic fluctuations at different periods from 10 to 1000 s. (b) The variations of λ mid /λ max withperiod for the magnetic fluctuations at different distances from 0.17 to 0.24 AU. (2) The distance variations of the scale-averaged fractions of the MHD modes show that: (a) the outward/anti-sunward propagating Alfv´en mode dominates the mode composition throughout the whole investigated range of dis-tances, while the outward slow mode, the inward/sunward Alfv´en mode and the outward fast mode represent the threesmallest proportions; (b) the fraction of the inward fast mode decreases with distance, whereas the fraction of the Zhu et al. inward slow mode increases with distance; (c) at 0.166AU < R < < R < k (cid:107) V A ∼ k ⊥ δv ) in strong MHD turbulence with balanced Els¨asser fluxes. However, in solar wind turbulence withimbalanced fluxes dominated by outward Alfv´en waves, our probability distribution function of wave propagation in k (cid:107) − k ⊥ space (see color maps in Figure 2)) is inconsistent with this prediction of critical balance theory (see whitesolid lines in Figure 2). For kd i < .
02, the most probable wavevector is more parallel, while for kd i > .
02, the mostprobable wavevector is closer to the quasi-perpendicular direction. This observational result will help to enlighten andpromote the theory of turbulence anisotropy characterized by a transition of propagation direction from quasi-parallelto quasi-perpendicular with a large angular jump at a certain scale. After integrating the ideas of both ”slab+2D”and ”critical balance” scenarios, an upgraded turbulence phenomenology in Fourier space was proposed to involve”quasi-parallel wavelike fluctuations” and ”quasi-2D fluctuations” as well as energy transfer between them and withinthemselves (Oughton et al. 2015). The observed transition from quasi-parallel to quasi-perpendicular propagation withincreasing wavenumber shows a way how to improve the turbulence model in the future.The transition of the dominant outward Alfv´en mode from θ k , B ∼ ◦ to θ k , B ∼ ◦ , as the period decreases from1000s to 10s, may also indicates the geometry of the Alfv´en waves at kinetic scales in the near-sun solar wind. Quasi-perpendicular Alfv´en waves are more likely to dominate at scales closer to ion scale. Accordingly, quasi-perpendicularmodes (e.g. kinetic Alfv´en waves) may participate in the turbulent cascade and further dissipation, energizing andshaping the non-thermal ion distributions, which may develop temperature anisotropic and feed back to excite theion-cyclotron waves reported during this interval (Bowen et al. 2020). In the future, we will study such chain ofenergy conversion process: damping of quasi-perpendicular kinetic waves −→ energization of particles −→ growth ofquasi-parallel waves.In some respect, our mode composition diagnosis results differ from the results of Chaston et al. (2020). They studythe spectral energy density fractions of six MHD modes inside and outside the field reversal regions, separately. Theyreport that the three outward (anti-sunward) modes are dominant at MHD scales on average. This difference may liein the calculation of the propagation angle, which is an input parameter of the mode-recognition method (Glassmeieret al. 1995). Chaston et al. (2020) obtained the propagation direction using the spectral matrix of the magnetic fieldonly as suggested by Samson & Olson (1980), while we use both the magnetic and the electric field based on theFaraday’s law (Santol´ıK et al. 2003). This aspect may lead to the different results of mode composition.The ellipticity serves here as an indicator to distinguish if the polarization is circular ( λ mid /λ max ∼ <λ mid /λ max < λ mid /λ max ∼ θ k , B ∼ ◦ . ACKNOWLEDGEMENTSThis work at Peking University (PKU) is supported by NSFC under contracts 41574168, 41674171, 41874200, and41421003. The team from PKU is also supported by CNSA under contract Nos. D020301 and D020302. D.V. issupported by the STFC Ernest Rutherford Fellowship ST/P003826/1 and STFC Consolidated Grant ST/S000240/1.S.D.B. acknowledges the support of the Leverhulme Trust Visiting Professorship program. The authors acknowledgethe contributions of the Parker Solar Probe mission operations and spacecraft engineering teams at the Johns HopkinsUniversity Applied Physics Laboratory as well as the FIELDS and SWEAP teams for use of the data. PSP data isavailable on SPDF (https://cdaweb.sci.gsfc.nasa.gov/index.html/).REFERENCES
Bale, S. D., Goetz, K., Harvey, P. R., et al. 2016, SSRv,204, 49, doi: 10.1007/s11214-016-0244-5 Bale, S. D., Badman, S. T., Bonnell, J. W., et al. 2019,Nature, 576, 237, doi: 10.1038/s41586-019-1818-7