Alfvénic versus non-Alfvénic turbulence in the inner heliosphere as observed by Parker Solar Probe
Chen Shi, Marco Velli, Olga Panasenco, Anna Tenerani, Victor Réville, Stuart D. Bale, Justin Kasper, Kelly Korreck, J. W. Bonnell, Thierry Dudok de Wit, David M. Malaspina, Keith Goetz, Peter R. Harvey, Robert J. MacDowall, Marc Pulupa, Anthony W. Case, Davin Larson, J. L. Verniero, Roberto Livi, Michael Stevens, Phyllis Whittlesey, Milan Maksimovic, Michel Moncuquet
AAstronomy & Astrophysics manuscript no. 39818corr © ESO 2021January 28, 2021
Alfvénic versus non-Alfvénic turbulence in the inner heliosphereas observed by Parker Solar Probe
Chen Shi , Marco Velli , Olga Panasenco , Anna Tenerani , Victor Réville , Stuart D. Bale , , , , Justin Kasper ,Kelly Korreck , J. W. Bonnell , Thierry Dudok de Wit , David M. Malaspina , Keith Goetz , Peter R. Harvey ,Robert J. MacDowall , Marc Pulupa , Anthony W. Case , Davin Larson , J. L. Verniero , Roberto Livi , MichaelStevens , Phyllis Whittlesey , Milan Maksimovic , and Michel Moncuquet Earth, Planetary, and Space Sciences, University of California, Los Angeles, Los Angles, California, USAe-mail: [email protected] Advanced Heliophysics Department of Physics, University of Texas at Austin, Austin, Texas, USAe-mail: [email protected] IRAP, Université Toulouse III—Paul Sabatier, CNRS, CNES, Toulouse, France 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 University of Michigan, Ann Arbor, MI, USA Smithsonian Astrophysical Observatory, Cambridge, MA, USA LPC2E, CNRS and University of Orléans, Orléans, France Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, Colorado, USA School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA Solar System Exploration Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, F-92195Meudon, France
ABSTRACT
Context.
Parker Solar Probe (PSP) measures the magnetic field and plasma parameters of the solar wind at unprecedentedly closedistances to the Sun. These data provide great opportunities to study the early-stage evolution of magnetohydrodynamic (MHD)turbulence in the solar wind.
Aims.
In this study, we make use of the PSP data to explore the nature of solar wind turbulence focusing on the Alfvénic characterand power spectra of the fluctuations and their dependence on the distance and context (i.e., large-scale solar wind properties), aimingto understand the role that di ff erent e ff ects such as source properties, solar wind expansion, and stream interaction might play indetermining the turbulent state. Methods.
We carried out a statistical survey of the data from the first five orbits of PSP with a focus on how the fluctuation propertiesat the large MHD scales vary with di ff erent solar wind streams and the distance from the Sun. A more in-depth analysis from severalselected periods is also presented. Results.
Our results show that as fluctuations are transported outward by the solar wind, the magnetic field spectrum steepens whilethe shape of the velocity spectrum remains unchanged. The steepening process is controlled by the “age” of the turbulence, which isdetermined by the wind speed together with the radial distance. Statistically, faster solar wind has higher “Alfvénicity,” with a moredominant outward propagating wave component and more balanced magnetic and kinetic energies. The outward wave dominancegradually weakens with radial distance, while the excess of magnetic energy is found to be stronger as we move closer toward theSun. We show that the turbulence properties can significantly vary from stream to stream even if these streams are of a similar speed,indicating very di ff erent origins of these streams. Especially, the slow wind that originates near the polar coronal holes has muchlower Alfvénicity compared with the slow wind that originates from the active regions and pseudostreamers. We show that structuressuch as heliospheric current sheets and velocity shears can play an important role in modifying the properties of the turbulence. Key words. plasma turbulence – solar wind
1. Introduction
Turbulence is a ubiquitous phenomenon in nature, arising in neu-tral fluids such as in Earth’s atmosphere and ocean as well as inastrophysical plasmas. The study of plasma turbulence is of greatnecessity because it is deeply related to fundamental nonlinearplasma physics and is crucial in understanding various important processes in astrophysics, such as the heating and acceleration ofthe solar wind and the acceleration of high-energy particles.In the solar wind, direct measurements have shown that fluc-tuations in the velocity and magnetic field display properties ofwell-developed turbulence (e.g., Coleman 1968). One importantfeature of these fluctuations that appears to be contradictory witha well-developed turbulence is the high Alfvénicity, that is to
Article number, page 1 of 13 a r X i v : . [ a s t r o - ph . S R ] J a n & A proofs: manuscript no. 39818corr say the strong correlation between velocity and magnetic fieldfluctuations invariably displaying the properties of Alfvén wavespropagating away from the Sun (e.g., Belcher & Davis Jr 1971),even though the solar wind is propagating much faster than anywave speed and should therefore advect fluctuations outward ir-respective of the direction of their propagation. This Alfvénicturbulence is most prevalent in high-speed solar wind streams,and the Alfvénic property appears to decay with distance fromthe Sun and survive out to distances much greater than 1 AUonly in the polar heliosphere at solar minimum (Bavassano et al.1998). Alfvénic turbulence is also nearly incompressible. Ra-dial evolution of the power spectra and other quantities, such ascross-helicity (defined below), seems to confirm ongoing nonlin-ear dynamics (Bavassano et al. 1982, 1998), which for incom-pressible Alfvénic turbulence requires the interaction betweencolliding counter-propagating wave packets (Iroshnikov 1964;Kraichnan 1965).Much theoretical work has been devoted to understand-ing the nature of nonlinear interactions in incompressible mag-netohydrodynamic (MHD) turbulence. The early statisticallyisotropic phenomenological models of Iroshnikov (1964) andKraichnan (1965) were extended to include parallel and perpen-dicular wave-number anisotropy by Goldreich & Sridhar (1995);the concept of dynamical alignment (Dobrowolny et al. 1980)was introduced to explain the dominance of outwardly propa-gating Alfvénic turbulence in the solar wind as a nonlinear phe-nomenon, and this was shown to lead to di ff erent spectra for in-ward and outward fluctuations by Grappin et al. (1990). A phe-nomenology of anisotropic turbulence with a preferred sense ofpropagation was presented in Lithwick & Goldreich (2003).The above models predict di ff erent spectral slopes and en-ergy cascade rates, each under specific assumptions. However,solar wind observations cannot be solely explained by any oneof the models, and this may be due to the inapplicability of as-sumptions such as homogeneity and incompressibility: The windexpands spherically, slowing nonlinear interactions and provid-ing a quasi-scale free energy loss in the turbulence; di ff erent ve-locity streams with significant shear are present at meso-scales;and compressible processes such as parametric decay may occur(Primavera et al. 2019; Tenerani et al. 2020), not to mention thepotential role of particle distribution function anisotropies.Two important problems that stand out are the di ff erent spec-tral slopes for the magnetic field and velocity (e.g., Grappin et al.1991; Boldyrev et al. 2011; Chen et al. 2013) and the observedexcess of magnetic energy over the kinetic energy (e.g., Robertset al. 1987b; Marsch & Tu 1990; Grappin et al. 1991). It is ob-served beyond 0.3AU that the velocity spectrum, whose slopeis around − /
2, is shallower than the magnetic field spectrum,whose slope is close to − /
3. Evidence shows that beyond 1AU, the velocity spectrum steepens toward a − / ∼ . R s ) to the Sun in encounters (E) E4 and E5, which ismuch closer than the previous record held by Helios B at ∼ . R s . Thus, its data provide a unique opportunity to study solarwind turbulence in its early stage of evolution. Initial PSP datahave revealed many interesting phenomena, among which theomnipresence of the so-called magnetic switchbacks may be es-pecially important (Bale et al. 2019; Dudok de Wit et al. 2020;McManus et al. 2020; Tenerani et al. 2020). There are fluctu-ations in the solar magnetic field of a su ffi cient magnitude toinvert the local direction with respect to the Sun, that is to saythey switch the field backward locally into a fold. Intriguingly,these folds retain some features typical of Alfvénic turbulence,among which the strong correlation of velocity to magnetic fieldfluctuations as well as a nearly constant magnitude of the totalmagnetic field. The velocity-magnetic field correlation and theoutward sense of propagation from the Sun reveal themselvesthrough the presence of radial velocity outward jets superim-posed on the background solar wind flow (Matteini et al. 2014;Kasper et al. 2019; Horbury et al. 2020).Réville et al. (2020), via MHD simulations compared to thePSP data, show that the Alfvénic fluctuations provide su ffi cientpower to accelerate the measured slow solar wind streams. Chenet al. (2020) surveyed the PSP data from the first two orbits andanalyzed the Alfvénicity of the MHD turbulence in the solarwind. They show that the dominance of the outward-propagatingwave decreases with radial distance to the Sun, which is consis-tent with previous observations made beyond 0.3 AU (Robertset al. 1987a; Bavassano et al. 1998). In addition, a steepeningof the magnetic field spectrum from a slope around − . − .
67 is also observed.In this study, we make use of the PSP data from the first fiveorbits and conduct a statistical analysis of the MHD fluctuationsin the solar wind. We show how the properties of the turbulencevary with both radial distance to the Sun and the wind speed.The wind speed in combination with the radial distance controlsthe turbulence spectra via the useful concept of turbulence “age”(Grappin et al. 1991). The Alfvénicity has a complicated behav-ior. In general, the fast wind is more Alfvénic than the slowwind and the Alfvénicity, if defined by the relative amplitudeof the outward and inward propagating Alfvén waves, graduallydecreases with radial distance. However, the magnetic energyseems to be much larger than the kinetic energy close to the Sunand gradually relaxes to similar levels as the wind propagates. Inaddition, Alfvénicity of streams of a similar speed can be verydi ff erent. We discuss several factors that possibly influence theturbulence properties, including fast-slow stream shears, the he-liospheric current sheet, and the di ff erent origin of the solar windstreams at the Sun.
2. Instruments & data processing
There are four instrument suites onboard PSP. Here we makeuse of the Level-2 magnetometer (MAG) data from Fields Ex-periment (FIELDS) and Level-3 Solar Probe Cup (SPC) datafrom Solar Wind Electrons Alphas and Protons investigation(SWEAP). We refer to the five orbits of PSP as “Encounter 1,2, 3, 4, 5,” respectively, or “E 1, 2, 3, 4, 5” for short, as highresolution data are only produced near perihelions of the orbits( R ≤ . − . Article number, page 2 of 13hi et al.: Alfvénic vs. non-Alfénic turbulence in the inner heliosphere
Fig. 1.
Overview of Encounter 01, 04, and 05. Row (a) shows the magnetic field with blue, orange, and green curves being radial, tangential,and normal components (RTN coordinates) and the black curve being the magnitude. Row (b) shows the radial ion flow speed (blue) and the ionthermal speed (orange). Row (c) shows the ion density (blue) and radial distance of PSP to the Sun (orange). Row (d) shows the spectral slopes ofthe magnetic field in Alfvén speed (blue) and velocity (orange). The two dashed lines mark the values 3 / / σ c , σ r , E b , and E v , respectively, as defined in Section 2. In each panel of these four rows, three curves are plotted and they correspond to wave band2 (blue), 5 (purple), and 8 (yellow), respectively. All quantities were averaged or calculated through Fourier analysis in the 2048 × . ≈ & A proofs: manuscript no. 39818corr lution of FIELDS is smaller than 13.7ms. The exact time periodsthat are analyzed in this study are listed in Table 1.We first resampled the measurements of the magnetic field,proton density, velocity, and thermal speed to a time resolution of0.874s, which is enough for the purpose of analyzing MHD tur-bulence. Then we binned the data into 2048-point time windowsand filled the data gaps using linear interpolation. Windows witha data gap ratio larger than 10% were discarded. We determinedthe polarity of the radial magnetic field by averaging B r insideeach time window and defined the two Elsässer variables Z o , i = U ∓ sign( B r ) B √ µ ρ (1)where subscripts “o” and “i” represent “outward” and “inward,”respectively, and B r is the averaged B r . We note that to havewell-defined outward and inward propagating waves, the anglebetween the background magnetic field and the radial directionshould not be too large. One can estimate that for a solar windspeed of 300 km / s, the spiral angle of the magnetic field is ap-proximately 20 degrees at 60 solar radii, which is su ffi cientlysmall. In Eq (1), the density is the averaged value in each half-hour window. As is shown in Fig. 4, the relative density fluctu-ation ∆ n / n is mostly small with values around 0.05-0.10. Thisdensity fluctuation introduces a small, negligible uncertainty,around (2 . − U , V A = B / √ µ ρ , and Z o , i to obtain power spectra. We then fit the power spectra overmodes 5-60, which correspond to periods T ∈ [30 s , s ], whichare within the inertial range of the turbulence. Similar to Grap-pin et al. (1991), we divided the Fourier modes into ten loga-rithmic bands, such that band i includes modes [2 i − , i ). Insideeach band, integrated wave energies E b , E v , E o , and E i were cal-culated. Then we defined the normalized cross helicity σ c = E o − E i E o + E i , (2)which measures the relative amplitude of outward and inwardAlfvén wave energies, and the normalized residual energy σ r = E v − E b E v + E b , (3)which measures the relative amplitude of kinetic and magneticenergies. We note that σ c = ± σ r = ± σ c = σ r =
3. Results
In Fig. 1 we present the overview plot of Encounter 1 (left), 4(middle), and 5 (right). We did not plot Encounter 2 & 3 dueto the limited figure size and less data coverage during the twoencounters. All quantities were calculated, either averaged orFourier-analyzed, in the 2048 × . ≈
30 min time window asdescribed in Section 2. Consequently, the magnetic switchbacks,whose typical time scales are several minutes long (Dudok deWit et al. 2020), are absent from the magnetic field plot. We notethat the large gaps in the last four rows ( σ c , σ r , E b , and E v ) ofthe middle and right columns do not mean that the original SPCand MAG data have large gaps. The reason is, as mentioned in Fig. 2.
Measurements of radial magnetic field (top panel), radial flowspeed (second panel), and proton number density (third panel) duringEncounter 4. The values were plotted on a radius-(Carrington longitude)grid, i.e., in the reference frame corotating with the Sun. The bottompanel shows PSP’s orbit with the z -axis being the Carrington latitude.We note that the variation in latitude is small. The colors represent timesuch that PSP moves from the light-colored end to the dark-colored end. Section 2, that we discarded the half-hour time windows withdata gap ratios larger than 10%. So these large gaps are actuallya result of frequent small data gaps in Encounters 4 & 5.There are several points that are worth underscoring here.(1) From Row (g)&(h), we can see that both magnetic and ki-netic energies in the waves decrease as we move away from theSun. This is a natural result, mainly of the spherical expansion ofthe solar wind but also of the energy cascade of the turbulence.A similar trend of E o and E i was reported in (Chen et al. 2020).(2) The streams measured by PSP during Encounters 4&5 are Article number, page 4 of 13hi et al.: Alfvénic vs. non-Alfénic turbulence in the inner heliosphere
Table 1.
Time periods selected for analysis of the PSP data. The third column shows PSP perihelion dates and the fourth column shows the distanceof each perihelion to the Sun.
Encounter R s R s R s R s R s Fig. 3.
Top: Radial solar wind speed varying with Carrington longitudeof PSP measured during Encounter 4. The colors represent the timeand PSP travels from the light-colored end toward the dark-colored endas indicated by the red arrows. Red circles connected by dashed linesmark several selected structures that were observed at di ff erent radialdistances. Bottom: Radial wind speed as a function of radial distanceto the Sun. Each line corresponds to one single structure marked by theconnected red circles in the top panel. The structures are numbered as1, 2, 3, 4, 5, and 6, as annotated in the top panel. mostly of a very low speed (Row (b)). As is shown in Section3.2, the streams are actually still accelerating radially. (3) Theion thermal speed (Row (b)), or equivalently the square-root ofion temperature, is highly correlated with the radial flow speed,which is a well-known phenomenon that has already been ob-served by other satellites (Grappin et al. 1990; Démoulin 2009)and the PSP measurements show that this correlation exists atradial distances even down to ∼
28 solar radii. A statistical anal-ysis of this point is presented in Section 3.3. (4) The densityprofile (Row (c)), except for a slow variation with the radial distance, shows strong structures near the perihelion during En-counters 4&5. For example, a short plasma sheet crossing wasobserved on January 30, 2020 and a long plasma sheet cross-ing was observed on June 8, 2020. These structures have sig-nificant impacts on the turbulence properties as is discussed indetail in Section 4. (5) The slopes of the magnetic field and ve-locity spectra (Row (d)) highly fluctuate and show a dependenceon the stream properties. It can be observed from Fig. 1 thatthe magnetic field spectrum is usually steeper than the veloc-ity spectrum, especially far from the Sun. Near perihelion, thetwo slopes seem to be close to each other. A statistical surveyof spectral variability is presented in Section 3.4. (6) Usually,the normalized cross helicity σ c (Row (e)) is close to 1, imply-ing a status of dominating outward-propagating Alfvén waves.However, there are periods when σ c oscillates and becomes neg-ative, for example, November 10, 2018, January 17-21, 2020,and June 8, 2020. As is shown in Section 4, these periods corre-spond to PSP observing heliospheric large-scale inhomogeneousstructures, such as velocity shears and the heliospheric currentsheet. Furthermore, σ c is also found to be significantly smallerthroughout E5 when compared to E1& E4 and the reason forthis is also discussed in Section 4. (7) We note that σ r (Row(f)) shows interesting behavior: During Encounter 1, its valueis very close to zero, indicating balanced magnetic and kineticenergies, which are expected for Alfvénic turbulence. However,during Encounters 4&5, most of the time it is negative, espe-cially close to perihelion. This suggests the possibility that theturbulence is magnetically-dominated at its origin inside certaintypes of streams. Statistical analyses are presented later in Sec-tion 3.4. As PSP travels to a su ffi ciently low altitude above the Sun, its rel-ative longitudinal speed to the rotating solar surface changes signwhen it crosses a critical height. That is to say, in the referenceframe corotating with the Sun, PSP moves toward the west firstas its altitude lowers, then it retrogrades to the east near the per-ihelion and finally moves back toward the west as it goes awayfrom the perihelion. This unique feature of PSP’s orbit makes itpossible to conduct a better analysis of the spatial structures inthe solar wind as PSP may measure streams from the same re-gion on the solar surface for two or three times at di ff erent radialdistances to the Sun during one encounter.In Fig. 2, we show the measurements of the radial magneticfield, radial flow speed, and proton number density during En-counter 4 in the top three panels. Instead of plotting these quan-tities against time, we plotted them on a radius-(Carrington lon-gitude) grid so that the projection of the curves on the grid isapproximately the trajectory of PSP in the reference frame coro-tating with the Sun. We note that the inclination of PSP’s orbitis very low: The Carrington latitude of PSP during Encounter4 varies between ∼ ± ◦ . In the bottom panel, we plotted the Article number, page 5 of 13 & A proofs: manuscript no. 39818corr
Fig. 4.
Relative density fluctuation ∆ n / n (left) and ion temperature (right) expressed in thermal speed squared as functions of radial solar windspeed. Each dot corresponds to a single half-hour window and the colors represent the radial distance to the Sun. Squares on solid curves aremedian values of the dots binned according to V r , and squares on dashed curves are the other two quartiles. trajectory of PSP for reference purposes with z -axis being theCarrington latitude. The colors of the curves represent the timesuch that PSP travels from the light-colored end toward the dark-colored end. From the V r plot, we can clearly see that the streammeasured near perihelion (the dark branch of the curve) containssimilar structures observed in the stream further away from theSun (the light branch of the curve). This similarity can also beseen in the B r and n p plots, implying that the satellite observedstreams coming from the same region of the Sun twice as it trav-eled inward and outward during the encounter. In the top panelof Fig. 3, we show a 2D V r -longitude plot so that we can bettercompare the measurements made as PSP traveled back-and-forthin longitude. Despite some deformation, the two curves showvery similar variations. We marked the identified similar struc-tures at di ff erent radial distances by the red circles connected bydashed lines and the radial wind speed at these circles is plottedagainst the radial distance in the bottom panel, where each linecorresponds to one single structure, which is numbered as 1, 2,3, 4, 5, and 6 as annotated in the top panel. We can see the windis accelerated at a rate of 1 − / s / R s for most of these struc-tures and overall the streams are accelerated from ∼ (200 − / s near perihelion ( ∼ R s ) to ∼ (300 − / s beyond60 R s . Thus, these kinds of measurements made uniquely by PSPcan be used to quantify the acceleration of the solar wind in thefuture with increasing data volume. A couple of caveats shouldbe noted here. First, in the corotating frame of the Sun, there isan additional longitudinal speed of the wind. Thus, the streammeasured by PSP at a certain longitude should come from a re-gion located at a larger longitude on the solar surface. For ex-ample, a 300 km / s wind drifts along longitude by ∼ ◦ after itpropagates 50 R s . Apparently, streams of di ff erent velocities andmeasured at di ff erent distances should drift by di ff erent amountsin longitude. In addition, considering the wind is accelerating, itis even more complicated to estimate this longitudinal drift. Sec-ond, the variation of the latitude of PSP, though not very large,may also account for the deformation of the longitudinal profilesof the measured streams. In Fig. 4, we show the scatter plot of the relative density fluc-tuation ∆ n / n and temperature V th versus the radial flow speed V r . Each single dot represents a half-hour time window and thecolor of each dot shows the radial distance to the Sun. We notethat V r , V th , and n are averaged over the time window while ∆ n is the standard deviation of n inside the window. To better showthe trend, we binned the dots with 50 km / s V r intervals and cal-culated the three quartiles inside each bin, which are shown asthe blue squares.Although the value of ∆ n / n is pretty scattered, it is in gen-eral small, mostly smaller than 0.2, and it decreases with V r . Wenote that the rise of the blue squares at V r ∈ [450 , / sis very likely a result of a lack of data points. From the colorsof the dots, we cannot see a clear relation between ∆ n / n and R .Thus, we conclude that the density fluctuation is larger in theslow streams than the fast streams and it does not evolve signif-icantly as the solar wind propagates. The ion temperature is lessscattered than ∆ n / n and the V th − V r relation shows very goodlinearity, as already mentioned in Section 3.1. This strong T − V correlation is a well-known phenomenon observed at 1 AU (e.g.,Elliott et al. 2005; Matthaeus et al. 2006; Démoulin 2009) andPSP data show that this correlation is already well established asclose as 30 R s . This may be a clue as to the origin of this T − V correlation. Matthaeus et al. (2006) proposed that this correla-tion is a result of the fact that the transport equation of temper-ature with a constant radial speed V has a solution of the form T = T ( R / V ). Thus, in supposing T is a decreasing function, weexpect that a larger radial speed leads to a slower decay of T with R , resulting in the observed positive T − V correlation. Onthe other hand, Démoulin (2009) argued that this correlation is arequirement by the momentum equation as a higher temperatureis needed to accelerate the solar wind to a higher speed. Sincethe measurements made during the encounters of PSP are likelyin the accelerating solar wind streams as pointed out in Section3.2, it is reasonable to say that the origin of the positive T − V correlation is related to the acceleration mechanism of the solarwind. A better modeling of the solar wind heating and acceler- Article number, page 6 of 13hi et al.: Alfvénic vs. non-Alfénic turbulence in the inner heliosphere
Fig. 5.
Spectral slopes of the magnetic field in Alfvén unit B / √ µ ρ (top left), velocity (top right), outward Elsässer variable (bottom left), andinward Elsässer variable (bottom right) as functions of the radial solar wind speed V r and radial distance to the Sun R . The data points were binnedaccording to V r and R , and the median value inside each bin was calculated, which is reflected in the colors and written in the plot. The bracketednumbers in the plots are the number of data points inside each bin. Bins with no more than 15 data points were discarded. ation is necessary to fully understand this issue. Last, from theright panel of Fig. 4, it seems that very close to the Sun (lightyellow dots), the slope of the T − V relation is larger than thatfurther away from the Sun (dark red dots). If it is true that the T − V slope changes radially, it implies that the adiabatic coolingrate is a function of the solar wind speed, which is true for elec-trons (Maksimovic et al. 2020). However, we should be cautiousin making this conclusion because during di ff erent encounters,the solar condition might be very di ff erent. As already described in Section 2, we calculated the spectralslopes over a period range T ∈ [30 s , s ] for the magnetic fieldin Alfvén speed unit, velocity, and outward and inward Elsässervariables. The statistical results of these slopes are presented inFig. 5. We binned the data points according to the radial solarwind speed V r and the radial distance to the Sun and then cal- culated the median value inside each bin. The median values arereflected by the colors of the blocks in Fig. 5 and are also writtenin the blocks. The bracketed numbers in the plots are the numberof data points and we discarded the bins with no more than 15data points (values were set to N / A).We first compared the top two panels of Fig. 5, that is tosay the spectral slopes of the magnetic field ( S b ) and velocity( S v ). There is no clear V r -dependence of S v , while a negative S b - V r correlation is observed in the range of R ∈ [35 , R s .Close to the Sun ( R < R s ), the di ff erence between S b and S v is small. Both of the magnetic field and velocity spectra areflatter than the Kolmogorov’s prediction -5 / / Article number, page 7 of 13 & A proofs: manuscript no. 39818corr
Fig. 6.
Averaged power spectra of magnetic field (in Alfvén speed) and velocity for di ff erent R and V r . Left: 35 ≤ R / R s ≤
45 and 300km / s ≤ V r ≤ / s. Middle: 65 ≤ R / R s ≤
75 and 300km / s ≤ V r ≤ / s. Right: 35 ≤ R / R s ≤
45 and 200km / s ≤ V r ≤ / s. The spectra were fittedover 2 . × − s − ≤ f ≤ . × − s − as shown by the dotted (for magnetic field) and dashed (for velocity) lines and the fitted slopes are writtenin the legend. Fig. 7.
Left: Spectral slope of outward Elsässer variable S o as a function of the spectral slope of magnetic field S b . Right: Spectral slope of velocity S v as a function of the spectral slope of magnetic field S b . Vertical lines mark S b = /
3. The horizontal line in the left panel marks S o = / S v = /
2. The colors represent the radial speed of the solar wind. range. We fit the spectra over the period range T ∈ [30 , s and the fitted slopes are written in the plot. The left panel is for R ∈ [35 , R s and V r ∈ [300 , / s and the two spectra havenearly identical slopes close to − .
5. The middle panel is for R ∈ [65 , R s and V r ∈ [300 , / s and it shows that as R in-creases to around 0.3 AU, the magnetic field spectrum becomesclose to the Kolmogorov’s spectrum while the velocity spectrumis still the Iroshnikov-Kraichnan spectrum. The right panel is for R ∈ [35 , R s and V r ∈ [200 , / s and by comparing itwith the left panel, we can see that at the same radial distance R , the slower wind has a steeper magnetic field spectrum. It hasbeen long observed outside 0.3 AU that the magnetic field spec-trum is steeper than the velocity spectrum (e.g., Grappin et al.1991). Figures 5&6 suggest that very close to the Sun, the twospectra may have the same slope. The anticorrelation between S b and V r and the positive correlation between S b and R imply theexistence of a “turbulence age” which determines the level of theturbulence development. A recent work analyzing Helios, Wind,and Ulysses data reveals similar “aging” of turbulence radiallybeyond 0.3 AU (Weygand & Kivelson 2019). From the bottomtwo panels of Fig. 5, the spectral slope of Z o shows a similarevolution with that of the magnetic field, but it is shallower. Thespectral slope of Z i , on the other hand, resembles the velocity,that is to say it does not show significant radial evolution and itis even smaller than the velocity slope. In Fig. 7 we show the cor-relation between S o and S b in the left panel and the correlationbetween S v and S b in the right panel. Both of the two correla-tions are high, especially that between S o and S b . The S v − S b correlation is weaker than the S o − S b correlation due to the factthat S b varies with V r and R , while S v is quite constant. For ref- Article number, page 8 of 13hi et al.: Alfvénic vs. non-Alfénic turbulence in the inner heliosphere
Fig. 8.
Normalized cross helicity σ c (left) and normalized residual energy σ r (right) of wave band 5 ( T ≈ −
56 s) as functions of the radialdistance to the Sun R and the radial speed of solar wind V r . The colors of each block represent the median values of the binned data. Text on eachblock shows the value of the block and the number of data points (in brackets) in the block. Bins with no more than 15 data points were discarded. erence purposes, we marked S b = / S o = / S v = / S o is close to S b , though slightly smaller, while S v isclearly smaller than S b such that S b = / S v around 1.55-1.6. This result is similar to that reported by Grap-pin et al. (1991) (see their Figure 7), although the data used hereare mainly within 0.3 AU, while Grappin et al. (1991) analyzedHelios data that were collected outside 0.3 AU.Figures 5&6 reveal that in the very young solar wind, themagnetic field and velocity spectra have the same slope; fur-thermore, as the turbulence evolves, the magnetic field spectrumsteepens while the velocity spectrum has an invariant slope. Thisposes a challenge in understanding the nature of the MHD tur-bulence in the solar wind. Most of the turbulence theories (e.g.,Kraichnan 1965; Goldreich & Sridhar 1995; Lithwick & Gol-dreich 2003; Zank et al. 2017) describe the turbulence based onthe two Elsässer variables, thus they cannot directly capture thedi ff erential evolution of the magnetic field and velocity spectra.Boldyrev et al. (2011) conducted 3D incompressible MHD sim-ulations based on the reduced equation set of Elsässer variablesand they reproduced the di ff erent magnetic field and velocityspectra statistically. However, how the final status is establishedis still unknown from the simulations. In addition, PSP data showthat the steepening of the magnetic field spectrum is quite slow.The top-left panel of Fig. 5 implies the steepening from 3 / / R ∼ R s to R ∼ R s .This is much longer than the nonlinear time, that is the “eddy-turnover” time or the Alfvén crossing time, of the turbulence.Thus, it is possible that in the solar wind, the di ff erential evo-lution of B and V is controlled by some external mechanisms,such as stream shears and the spherical-expansion e ff ect, whichleads to di ff erent decay rates of the magnetic energy and kineticenergy (Grappin & Velli 1996).In Fig. 8, we present the ( V r , R ) variation of the normalizedcross helicity σ c (left panel) and the normalized residual energy σ r (right panel), in a similar manner as we do for Fig. 5. Here thevalues were calculated for the wave band 5, that is correspond-ing to wave period T ∈ [112 , σ c , an overall positive σ c - V r correlation is observed, at least for R ≤ R s , indicatingthat the fast wind is generally more Alfvénic than the slow wind.The lack of a definite σ c − V r correlation for R > R s might bedue to the lack of data points so that the value in one single blockmainly reflects the turbulence property inside one stream in-stead of multiple streams, increasing the uncertainty. The σ c − R correlation is clearly negative in the range of R ≥ R s and V r ∈ [300 , / s, implying that the dominance of the out-ward propagating wave declines with the radial distance, whichwas already reported in previous works (e.g., Chen et al. 2020).But this correlation is not well-defined in other parametric re-gions. Especially, for measurements made below 35 R s and forvery slow wind ( V r ≤ / s), σ c is much lower comparedwith the neighboring blocks in V r − R space. This is caused by thenon-Alfvénic, or low-Alfvénic, slow wind measured by PSP dur-ing Encounter 5 (see right column of Fig. 1). For σ r , we can seethat it is in general negative, that is to say the magnetic energyexceeds the kinetic energy, which is a well-known phenomenonthat is not fully understood yet. For R ≤ R s , σ r is also pos-itively correlated with V r . That is to say, in the fast wind, themagnetic and kinetic energies are more balanced, which is con-sistent with the high σ c values which imply a highly Alfvénicstatus. The radial evolution of σ r , however, shows a surprisingresult as it is clear that inside 65 R s , σ r increases with radial dis-tance, meaning that the turbulence is relaxing from a magnetic-dominating status toward a more balanced status. Actually, byexamining the middle and right columns of Fig. 1, one can findthat σ r is clearly an increasing function of R from January 21-29,2020 and from June 1-6, 2020, which is consistent with the sta-tistical result here. Even for Encounter 1 (left column of Fig. 1),a slight increase in σ r with R is observed from November 6-9,2018. Outside 65 R s , the evolution is not very clear but it seemsthat σ r may start to drop with R . Similar to σ c , the values of σ r are extremely low for R ≤ R s and for V r ≤ / s. As men-tioned before, this region in the parameter space corresponds tothe very low Alfvénic streams observed during Encounter 5. Article number, page 9 of 13 & A proofs: manuscript no. 39818corr
Fig. 9.
Blow-ups of the time periods marked by the shaded regions in Fig. 1. Row (a) shows the radial magnetic field B r (blue) and the magnitudeof magnetic field | B | (black). Row (b) shows the radial flow speed V r (blue) and the ion thermal speed V th (orange). Row (c) shows the ion density n p . Row (d) shows the relative ion density fluctuation ∆ n / n (blue) and the plasma beta β (orange), defined as the ion thermal pressure p th = n p m i V th divided by magnetic pressure p mag = B / µ . Row (e) shows the spectral slopes of magnetic field in Alfvén speed (blue) and velocity (orange).The two dashed lines mark 3 / / σ c (blue) and σ r (orange). Row (g) shows the energies in magnetic fieldfluctuations E b (blue) and velocity fluctuations E v (orange).
4. Discussion
From Section 3.4, we conclude the following points: (1) Dur-ing the evolution of the solar wind turbulence, the magnetic fieldspectrum steepens from a − / − / − /
2. (2) The fast solarwind is in general more Alfvénic than the slow solar wind, with σ c closer to 1 and σ r closer to 0. However, we should empha-size here that the “fast” solar wind in this study is not the typi-cal fast wind that originates from large-scale polar coronal holeopen regions because during the first five encounters, PSP didnot observe any long-lasting fast solar wind of this type. Thus, itis more likely that the “fast” winds here should probably be clas-sified as examples of “faster” Alfvénic slow wind (D’Amicis &Bruno 2015; Panasenco et al. 2020). (3) Closer to the Sun, σ c increases toward 1, confirming that the turbulence is dominatedby outward propagating Alfvén waves in the young solar wind. However, there are periods where σ c is quite low even at veryclose distances to the Sun (below 35 R s ). (4) For some solar windstreams, for example, those observed during Encounters 4&5, σ r evolves from negative values toward 0 at close distances, sug-gesting that the turbulence is actually magnetic-dominated at itsorigin and then gradually relaxes to a more balanced status inthese streams.The above conclusions are based on the statistical resultsusing all high-resolution data from PSP’s first five encounters.While they help us depict an average picture of the evolution ofsolar wind turbulence, it is still necessary to examine the turbu-lence from di ff erent time periods so that we can have deeperinsights on how the turbulence varies in di ff erent streams. Infact, as one may have noticed in Fig. 1, fluctuations in streamsof a similar radial wind speed can have significantly di ff erentAlfvénicity. For example, during E1 from November 3-7, the so- Article number, page 10 of 13hi et al.: Alfvénic vs. non-Alfénic turbulence in the inner heliosphere
Fig. 10.
Comparison between the ion and electron densities during En-counter 5. Blue: Ion density measured by the Faraday cup (SPC). Or-ange: Electron density calculated using the quasi thermal noise (QTN)measurements made by the Radio Frequency Spectrometer Low Fre-quency Receiver (RFS / LFR). lar wind speed is around 300km / s and the fluctuations are highlyAlfvénic, while during E5 from June 1-6, the solar wind speedis also around 300km / s but the Alfvénicity of the fluctuations isquite low. A more detailed analysis is presented later in this sec-tion. In Fig. 9, we present the blow-ups of Fig. 1 over three shorttime periods marked by the shades in Fig. 1. Compared with Fig.1, the top three rows of Fig. 9 present data at a time resolutionof 0.874s instead of a half hour. In addition, in the bottom tworows of Fig. 9, σ c , σ r , E b , and E v were calculated by integratingover all wave modes except mode 0, that is to say the backgroundfield. The left column of Fig. 9 shows the time period from 12:00November 9 to 00:00 November 11, 2018 during Encounter 1.Before 08:00 November 10, PSP was inside a fast stream witha radial speed of V r ∼ − / s. Between 08:00 and13:00 November 10, PSP crossed a fast-slow stream shear re-gion, marked by the shaded region, after which the wind speeddropped to less than 400km / s. Inside the fast stream, a largeamount of switchbacks were observed with nearly constant | B | and n p , as well as σ c ≈ σ r ≈
0. These parameters im-ply that the turbulence is highly Alfvénic, with very little inwardpropagating wave component. Inside the shear region, a decreasein σ c and increase in σ r were observed and the wave energieswere dissipated right after the shear. From Panel (a1), we cansee that inside and shortly after the shear region, no switchbacksare observed, implying a strong dissipation of the wave ener-gies. These results are consistent with the 2D MHD simulations(Roberts et al. 1992; Shi et al. 2020), which showed that near thefast-slow stream interaction region, the wave energy is dissipatedquickly because the shear transfers energies from long wave-lengths to short wavelengths rapidly. They also found that in-side the stream interaction region, the outward wave dominanceis destroyed and kinetic energy exceeds the magnetic energy atsmall scales, which is consistent with the drop in σ c and increasein σ r observed by PSP. The positive σ r indicates that the veloc-ity shear e ffi ciently transfers kinetic energies from large to small scales. Thus, the velocity shear may play an important role inthe turbulence evolution and is a good candidate to explain theobserved negative σ c − R relation and positive σ r − R relation asdiscussed in Section 3.4. The middle column of Fig. 9 shows the time period from 12:00June 5 to 12:00 June 6, 2020 during Encounter 5. During thistime period, and for most of Encounter 5 shown in Fig. 1, theturbulence property is “abnormal.” From Panel (a2), we can seethat the magnetic field strength | B | is quite constant and a lotof switchbacks are present. In addition, Panel (c2)&(d2) showthat the plasma density is quite constant with very small fluctu-ations. These features normally indicate a highly Alfvénic sta-tus of the turbulence. However, we can see from Panel (f2) that σ c is systematically small, around 0.5, and as is σ r , which isaround -0.75. That is to say, in this time period, there is a non-negligible amount of inward propagating wave component whilemagnetic energy significantly exceeds the kinetic energy, despitethe near incompressibility. One can see from Fig. 1 that actuallyduring most of Encounter 5, the turbulence has low Alfvénic-ity and the wind speed is slow. In examining the middle columnof Fig. 1, we noticed that in Encounter 4 after the heliosphericcurrent sheet crossing on February 1 until February 4, the so-lar wind was also quite slow and had relatively low σ c and σ r ,which is similar to what PSP observed in Encounter 5. Thus,the observed non-Alfvénic, or low-Alfvénic, turbulence is pos-sibly related with the sources of the very slow solar wind. Onething that we should point out is that the ion density measured bythe Faraday cup (SPC) seems to be lower than the electron den-sity derived using the quasi thermal noise (QTN) measurementsmade by the Radio Frequency Spectrometer Low Frequency Re-ceiver (RFS / LFR) (Moncuquet et al. 2020). In Fig. 10, we plot-ted these two quantities for Encounter 5, where blue is the SPCion density n p and orange is the QTN electron density n e . We cansee that n p is systematically lower than n e and the di ff erence canbe as large as ∼
30% for some time periods. As we expect thatthe QTN measurements are more accurate than the SPC mea-surements, this indicates that the real ion density is larger thanthe SPC data used in the current study. As a result, the mag-netic energy density E b = b /µ ρ calculated here is larger thanreal, leading to an overestimate of the magnetic energy excessover the kinetic energy. Thus, we used the QTN-derived densityto reconduct the calculation of the magnetic energy, σ c and σ r .The result is not presented here but we confirm that the e ff ect ofthis density di ff erence is not significant and does not change thelow-Alfvénicity in E5.In Fig. 11, we show the SDO / HMI image of the whole diskof the Sun taken on June 16, 2020. During most of Encounter5, PSP was flying over this side of the Sun, which is very quietas can be seen from the image. We note that this image was nottaken during the period that PSP data were analyzed (May 30-June 13, 2020) since PSP was not on the Sun-Earth line duringE5 so there is a time lag between the encounter and when SDOwas looking at the solar surface over which PSP flew by. In theright panel of Fig. 11, we show the map of magnetic pressureat R = . R s , which was calculated using the PFSS model withthe source surface set to R ss = . R s and the SDO / HMI mea-surements as input. The blue diamond is the direct radial pro-jection of PSP to the source surface and the blue crosses are thefoot points of the magnetic field lines connected to PSP on thesource surface. Di ff erent crosses correspond to a prediction us-ing varying wind speeds, from 230-80km / s to 230 + / s. The Article number, page 11 of 13 & A proofs: manuscript no. 39818corr
Fig. 11.
Left: SDO / HMI image taken on Jun 16, 2020, corresponding to Encounter 5 of PSP. The grid is in Carrington degrees. One can see thatduring Encounter 5, the visible side of the Sun was very quiet. Right: Magnetic pressure map at R = . R s calculated by the PFSS model with thesource surface at R ss = . R s and SDO / HMI data on Jun 16, 2020 as input. The blue diamond is the direct radial projection of PSP to the sourcesurface. The blue crosses are the foot points of the magnetic field lines connected to PSP on the source surface. Di ff erent blue crosses correspondto a prediction using varying wind speeds, from 230 − / s to 230 + / s. The blue circles are on the surface R = . R s and are magneticallyconnected to the blue crosses. The thick black lines are the neutral lines at R = . R s , and colored regions are the open magnetic field regions withblue being negative polarity and red being positive polarity. blue circles are on the surface R = . R s and are magneticallyconnected to the blue crosses according to the PFSS model re-sults. The detailed procedure to create this plot can be found inPanasenco et al. (2020) and Velli et al. (2021, this issue). We cansee that at this time period PSP was connected to the boundaryof the northern polar coronal hole without any activities nearby,neither active regions, nor pseudo-streamers, which are shownto be crucial in generating the Alfvénic slow wind observed inEncounter 1 (Panasenco et al. 2020). For most of E5, PSP wasmagnetically connected to the boundaries of either the northernor southern polar coronal hole (Velli et al., 2021, this issue). Thismay be relevant to explain why the slow wind observed duringEncounter 5 is non-Alfvénic despite of the quite incompressiblefluctuations. One possibility is the di ff erent ion compositions inthe slow wind originating from di ff erent regions. For example, ifthe slow wind that originates near the boundaries of polar coro-nal holes comprises more helium or heavier ions which are notconsidered in the current study, the real plasma density shouldbe larger than our estimate. As a result, the real magnetic energydensity should be smaller than our calculation. If so, σ r shouldbe closer to 0 and σ c should be closer to 1, that is the Alfvénicityof the wind should be larger than our estimate. Further analysisof the ion composition is necessary, but this is beyond the scopeof the current study. Other mechanisms are also possible. For ex-ample, if the Alfvén waves in the slow wind originating near thepolar coronal holes experience strong reflection due to large in-homogeneity of the background Alfvén velocity, the Alfvénicityis low. Modeling the propagation of Alfvén waves at di ff erentregions of the Sun will be a future topic. We conclude here thatthe coronal magnetic structures play a key role in the Alfvénicproperties of the solar wind. The right column of Fig. 9 shows the time period from 18:00June 7 to 00:00 June 9, 2020 during Encounter 5. In this time pe-riod, PSP crossed a plasma sheet, inside which the ion density,speed, and temperature were all enhanced while the magnetic field strength was weakened with multiple polarity reversals.These measurements imply that PSP crossed the heliosphericcurrent sheet, which is typically embedded inside a plasma sheet(Smith 2001), multiple times. The turbulence properties insidethis plasma sheet are very di ff erent compared with those in thenormal solar wind streams. First, the spectra of both the mag-netic field and velocity become steeper, with slopes close to − σ c is on aver-age close to 0, that is there are no well-defined Alfvénic fluc-tuations or the outward and inward propagating Alfvén wavesare strongly mixed. Third, σ r is close to -1, implying magnetic-dominant fluctuations. During Encounter 4, from January 17 toJanuary 20, 2020, PSP also crossed current sheets multiple timesand one can observe from the middle column of Fig. 1 that in thistime period, σ c was frequently negative and σ r was very low.These measurements suggest that current sheets may also playan important role in generating the low σ c and σ r fluctuationsobserved in the slow streams such as that shown in the middlecolumn of Fig. 9. Malara et al. (1996), via 2.5D MHD simula-tions of Alfvén waves on top of a current sheet, showed that theinitially large σ c is rapidly destroyed in the vicinity of the currentsheet, supporting our observation. In assuming that these fluctu-ations in the slow wind are strongly a ff ected by current sheetssuch that they are non-Alfvénic at their origins, then we need toexplain why the magnitude of magnetic field is still nearly con-stant. Firehose instability may play a key role in explaining thisas Tenerani & Velli (2018) show that magnetic field fluctuationsin high- β plasma naturally relax to a constant- | B | status due tothe firehose instability.
5. Conclusions
In this study, we have analyzed data from the first five orbitsof PSP. We focus on the properties of the MHD-scale turbu-lence and how they vary with the large-scale solar wind streams.A general nonlinear steepening of the magnetic field spectrumfrom a − / − / Article number, page 12 of 13hi et al.: Alfvénic vs. non-Alfénic turbulence in the inner heliosphere and the radial distance to the Sun, suggesting the existence of a“turbulence age” that controls the steepening process (see Fig.5). The slope of velocity spectrum, on the contrary, remains al-most constant at − /
2. The observed spectral evolution indicatesthat, on average, the magnetic field and velocity have similarspectra in the very young solar wind and their spectra evolve dif-ferently. Better theoretical models are still needed to explain thisdi ff erential evolution of velocity and the magnetic field and theywill be a future research topic. We investigated the Alfvénicityof the turbulence through two widely used diagnostics, namelythe normalized cross helicity σ c , which measures the relativeabundance of outward and inward propagating Alfvén wave en-ergies, and the normalized residual energy σ r , which measuresthe relative abundance of magnetic and kinetic energies. Statisti-cally, turbulence in fast solar wind is more “Alfvénic” than thatin slow wind as σ c is closer to 1 and σ r is closer to 0 in thefast wind. During radial evolution, in general, the dominance ofan outward propagating wave gradually weakens, manifested ina decreasing σ c (see left panel of Fig. 8). The magnetic-kineticenergy comparison is surprising as our result shows that the mag-netic energy significantly exceeds the kinetic energy close to theSun and gradually relaxes to a balanced status. This is in contrastto the commonly accepted idea that the magnetic energy excessis a result of the dynamic evolution of MHD turbulence (e.g.,Grappin et al. 1983). A similar result was reported by Bavas-sano et al. (1998), who analyzed Ulysses data and showed thatthe least evolved high-latitude stream has the strongest imbal-ance between magnetic and kinetic energies compared with moreevolved mid- and low-latitude streams. They attributed this phe-nomenon to the abundance of pickup ions in the polar region,which modifies the kinetic normalization of the Alfvénic unit.However, other mechanisms, such as the contribution of heavyions and the e ff ect of the velocity shears, may also play impor-tant roles.We note that the above results are all based on a statisticalanalysis. In practice, individual streams can be quite di ff erentfrom each other and one cannot simply infer the turbulence prop-erties from the wind speed. For example, from Fig. 1 & 9, weobserve that the slow streams with a similar speed ( ∼ / s)can be either highly Alfvénic (Encounter 1) or non-Alfvénic(Encounters 4&5). To fully understand the cause of these di ff er-ences, we must examine the origin of each individual solar windstream because the location of the origin can significantly im-pact the Alfvénicity of the slow wind (D’Amicis & Bruno 2015;Panasenco et al. 2020). In addition, it is possible that the large-scale structures, such as the heliospheric current sheets and ve-locity shears, greatly modify the turbulence properties at the veryearly stage (e.g., Roberts et al. 1992; Shi et al. 2020). Acknowledgements.
This research was funded in part by the FIELDS experimenton the Parker Solar Probe spacecraft, designed and developed under NASA con-tract NNN06AA01C and the NASA Parker Solar Probe Observatory Scientistgrant NNX15AF34G.
References