Shear-Driven Transition to Isotropically Turbulent Solar Wind Outside the Alfven Critical Zone
D. Ruffolo, W. H. Matthaeus, R. Chhiber, A. V. Usmanov, Y. Yang, R. Bandyopadhyay, T. N. Parashar, M. L. Goldstein, C. E. DeForest, M. Wan, A. Chasapis, B. A. Maruca, M. Velli, J. C. Kasper
DDraft version September 15, 2020
Typeset using L A TEX preprint style in AASTeX63
Shear-Driven Transition to Isotropically Turbulent Solar Wind Outside the Alfv´enCritical Zone
D. Ruffolo, W. H. Matthaeus, R. Chhiber,
2, 3
A. V. Usmanov,
2, 3
Y. Yang , R. Bandyopadhyay, T. N. Parashar,
2, 6
M. L. Goldstein, C. E. DeForest, M. Wan , A. Chasapis, B. A. Maruca, M. Velli, and J. C. Kasper
11, 12 Department of Physics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand Department of Physics and Astronomy and Bartol Research Institute, University of Delaware, Newark, DE 19716,USA Heliophysics Science Division, NASA Goddard Space Flight Center, Greenbelt MD 20771, USA Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen,Guangdong 518055, People’s Republic of China Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA School of Chemical and Physical Sciences, Victoria University of Wellington, Wellington 6012, New Zealand Goddard Planetary Heliophysics Institute, University of Maryland Baltimore County, Baltimore, MD 21250, USA Southwest Research Institute, 1050 Walnut Street, Suite 300, Boulder, CO 80302, USA Laboratory for Atmospheric and Space Physics, University of Colorado Boulder, Boulder, CO 80303, USA Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, CA 90095, USA Climate and Space Sciences and Engineering, University of Michigan, Ann Arbor, MI 48109, USA Smithsonian Astrophysical Observatory, Cambridge, MA 02138 USA
ABSTRACTMotivated by prior remote observations of a transition from striated solar coronalstructures to more isotropic “flocculated” fluctuations, we propose that the dynamicsof the inner solar wind just outside the Alfv´en critical zone, and in the vicinity of thefirst β = 1 surface, is powered by the relative velocities of adjacent coronal magnetic fluxtubes. We suggest that large amplitude flow contrasts are magnetically constrained atlower altitude but shear-driven dynamics are triggered as such constraints are releasedabove the Alfv´en critical zone, as suggested by global magnetohydrodynamic (MHD)simulations that include self-consistent turbulence transport. We argue that this dy-namical evolution accounts for features observed by Parker Solar Probe ( PSP ) nearinitial perihelia, including magnetic “switchbacks”, and large transverse velocities thatare partially corotational and saturate near the local Alfv´en speed. Large-scale mag-netic increments are more longitudinal than latitudinal, a state unlikely to originatein or below the lower corona. We attribute this to preferentially longitudinal veloc-ity shear from varying degrees of corotation. Supporting evidence includes comparisonwith a high Mach number three-dimensional compressible MHD simulation of nonlinearshear-driven turbulence, reproducing several observed diagnostics, including character-istic distributions of fluctuations that are qualitatively similar to
PSP observations near [email protected] a r X i v : . [ phy s i c s . s p ace - ph ] S e p the first perihelion. The concurrence of evidence from remote sensing observations, insitu measurements, and both global and local simulations supports the idea that thedynamics just above the Alfv´en critical zone boost low-frequency plasma turbulence tothe level routinely observed throughout the explored solar system. INTRODUCTIONThe solar atmosphere originates in the highly dynamic photosphere and expands outward, gener-ating the magnetically dominated corona. The outward acceleration eventually causes the velocityto exceed the local Alfv´en speed, and in that super-Alfv´enic regime, embedded magnetic fluctuationswill only propagate outward. Consequently, a feature distinguishing the inner corona from the super-Alfv´enic solar wind is that magnetohydrodynamic (MHD) fluctuations can propagate both upwardsand downwards in the inner corona, but in the solar wind such signals cannot propagate back intothe corona. The hypothetical boundaries between these regions are the Alfv´en critical surface, wherethe Alfv´en speed of magnetic fluctuations equals the solar wind speed, and the sonic critical surface,where the speed of sound equals the solar wind speed. In view of the highly dynamic, or turbulent,nature of both the solar wind and corona, these boundaries are almost certainly better describedas critical zones (DeForest et al. 2018). For many years there has been discussion and speculationregarding what happens near and at these zones. In the simplest picture (the surface version) Alfv´enor sound waves can propagate only outward at the surface. Downward propagating fluctuationsbelow the Alfv´en surface cannot reach the solar wind. Downward propagating fluctuations abovethe Alfv´en critical zone in the solar wind cannot propagate back into the sub-Alfv´enic corona. Itwould not be unreasonable to imagine that with such stagnation of downward-moving fluctuationsand their interaction with upward-moving fluctuations, turbulence levels build up in the critical zone,a possibility that has also been suggested based on remote sensing observations (Lotova et al. 1985;Lotova et al. 2011). This reasoning has also long been offered as explaining why the inner solarwind is dominated by a broad band spectrum of outward traveling waves (Belcher & Davis Jr. 1971).This “Alfv´enic” property of the fluctuations is characteristic of the inner heliosphere, where it formsa power-law inertial range observed from the correlation scale to the ion inertial scale (Bavassanoet al. 1982; Bruno & Carbone 2013). The corona is presumed to be turbulent and the solar windis observed to be turbulent from the distance of the latest
Parker Solar Probe (PSP) (Fox et al.2016) perihelia measurements out to the boundary of the heliosphere beyond 100 au. But how thisturbulence changes in character across the critical zones is not well understood. Furthermore, thenature of the transitions in all plasma properties from coronal to solar wind conditions remains to bediscovered. Fortunately,
PSP data are revealing these properties at progressively lower altitudes, andmore information will soon be forthcoming from the recently launched
Solar Orbiter (M¨uller et al.2013) mission. Together, these two missions are expected to unravel many mysteries of the innersolar wind and outer corona, including the issues we investigate here.Initial results from
PSP have revealed magnetic reversals and velocity spikes (Bale et al. 2019;Kasper et al. 2019; Dudok de Wit et al. 2020) similar to previous observations at 0.3 AU and beyond(Michel 1967; Kahler et al. 1996; Balogh et al. 1999; Crooker et al. 2004; Borovsky 2016; Horbury et al.2018; Lockwood et al. 2019). One explanation is that the reversals arise from outward propagationof large amplitude remnants of magnetic reconnection that occurred at lower altitudes (Axford &McKenzie 1992; Axford et al. 1999; Samanta et al. 2019; Fisk & Kasper 2020). Such a mechanism isplausible and difficult to rule out. However, another possibility is that the reversals reflect an onsetof strong shear-driven turbulence that began just outside the Alfv´en critical zone where the solarwind speed first exceeded the Alfv´en speed. Such shears could produce magnetic reversals throughlarge-scale perturbations of the flow. For example, such perturbations could result from excitationof the Kelvin-Helmholtz instability (Malagoli et al. 1996).This scenario is consistent with a suite of observable effects already apparent in imaging (DeForestet al. 2016) and in situ datasets (Borovsky 2016; Horbury et al. 2018). In particular, DeForestet al. (2016) interpreted the transition from elongated striae to relatively isotropic flocculae as asignature of the onset of shear-driven turbulent activity some 20-80 solar radii from the photosphere,where the magnetic field ceases to be a dominant constraint on transverse motions. In the presentwork, following DeForest et al. (2016), we refer to this process as flocculation. This interpretationis supported by results from turbulence-driven global simulations of the solar wind (Chhiber et al.2018). The presence of velocity shears is also strongly suggested by coronal imaging at lower altitudes(DeForest et al. 2018).Here we use
PSP observations in its first two orbits, along with supporting simulations, to examinethe character of the plasma dynamics in the solar wind at heliocentric distance r as low as 36 R (cid:12) (0.17au), along with evidence that PSP approached the Alfv´en critical zone. Our presentation incorporatesglobal heliospheric MHD simulations, local three-dimensional (3D) compressible MHD simulation,and
PSP observations. The goals are to understand the region where flocculation is believed to start,to identify signatures of the process of flocculation in
PSP data, and to evaluate the hypothesis thatthe transition from striation to flocculation is a consequence of velocity shears. We are thereforeled to consider physics related to nonlinear shear instabilities, essentially nonlinear Kelvin-Helmholtzdynamics or mixing layer dynamics, appropriately generalized to an MHD or plasma environment.While we cannot definitively resolve these questions based on the current observational evidence,global simulation, and supporting simulation of local physics, we nonetheless are able to concludethat the available evidence is consistent with our hypothesis of the role of velocity shear in the innersolar wind. MOTIVATION AND HYPOTHESIS2.1.
Heliospheric Imaging from STEREO
DeForest et al. (2016) analyzed image sequences recorded by the inner Heliospheric Imager in-strument on board the
Solar-Terrestrial Relations Observatory ( STEREO /HI1) in December, 2008.The analysis covered angular distances of approximately 4 ◦ to 24 ◦ from the center of the Sun. Anobserved systematic transition in the images was noted that consisted of anomalous fading of theradial striae that characterize the corona, along with an anomalous relative brightening of locallydense puffs of solar wind, which were described as “flocculae.” This transition was interpreted as theonset of dynamical activity associated with velocity shear present in the nascent solar wind plasmacoming from near-radial corotating flux tubes in the corona. The flux tubes confine the plasma,magnetic structures, and fluctuations that were injected at lower altitudes. Moving radially outward,the magnetic field progressively loses control of the plasma, which allows for additional physical pro-cesses to dominate, including those that give rise to the striation-flocculation transition. Significantstages of this transition are indicated by passage through regions where the flow speed exceeds theAlfv´en speed (the Alfv´en critical zone) and where the mechanical pressure approaches or exceeds themagnetic pressure (first β = 1 zones). Figure 1.
STEREO
Heliospheric Imager snapshot from 16 December 2008, analyzed as in DeForest et al.(2016) and Chhiber et al. (2018), showing the transition with radial evolution from a highly anisotropicstriated conformation closer to the Sun, to a more isotropic flocculated conformation at greater distances.Also indicated are first β = 1 surfaces (green curves) from a global heliospheric MHD simulation, based ona magnetogram corresponding to December 2008. For this simulation, the Alfv´en surface lies below 15 R (cid:12) (see Chhiber et al. 2018). From right to left, circled symbols indicate the Helios perihelion ( ⊕ ) and the first,fourth, and sixth PSP perihelia ( ⊗ ). The white plus sign shows the location of enhanced turbulence inferredby Lotova et al. (1985). In the present paper we develop the hypothesis that this transition is fueled bynonlinear shear instability outside the Alfv´en critical zone. Figure 1 illustrates a frame of the DeForest et al. (2016) analysis, in which the transition from stri-ation to flocculation is clear. The tendency for the structures in the image to become more isotropicwith increasing heliocentric distance was quantified by DeForest et al. (2016) by computations ofradial and transverse-to-radial second-order structure functions of the signal. Closer to the Sun, thestriation is due to more intense gradients in the transverse direction and weaker radial gradients, in-dicated by values of transverse structure functions greater than radial structure functions at a givenlag. Moving outward, the corresponding values of the two structure functions become more equal,indicating an evolution towards isotropy.Figure 1 is also annotated with approximate equivalent positions of the first, fourth, and sixth
PSP perihelia, and the perihelion of the
Helios mission at 0.29 au. Based on global MHD simulations thatinclude turbulence transport (Chhiber et al. 2019a), the figure also shows the first surfaces where theplasma beta is unity when considering protons and electrons ( β p + e = 1) and when considering onlyprotons ( β p = 1). For the simulation considered here, the Alfv´en surface lies below 15 R (cid:12) , to the leftof the view in this image. -100-50050100 B ( n T ) R |B| V p r o t on ( k m s - ) R T N R / R -100-50050100 B ( n T ) T N
Figure 2.
Overview of part of the first
PSP encounter. Top panel: Radial component (black) and magnitude(red) of the magnetic field; occurrence of numerous switchbacks (radial component reversals) is evident,although the magnitude exhibits much less variation. Second panel: Tangential (blue) and normal (red)components of the magnetic field. Third panel: Proton velocity components, with the same colors. Thedata plotted in (a)-(c) all have a cadence of 1 NYs ≈ PSP fromthe center of the Sun in units of solar radii.
Parker Solar Probe: Review of Prior Results
During its first few orbits, the
Parker Solar Probe ( PSP ) mission has made pioneering observationsin the inner solar wind that bear directly on the questions we explore here.
PSP is currently in itssixth orbit and may be approaching the Alfv´en critical zone, and will thus be directly examiningthe region of interest. We will delve into the observations in more detail below, but to begin thediscussion, Figure 2 illustrates some of the important and relevant measurements made by
PSP from2018 Oct 31 to Nov 11, an 11-day period surrounding its first perihelion at 35.7 solar radii (0.17 au)on 2018 Nov 6 at 0327 UT. Cartesian components of magnetic and velocity fields and the density areshown for this period at a cadence of 1 “New York second” (NYs) ≈ .
87 s, the fundamental cadenceof solar wind velocity measurements. These data will be discussed in greater detail below, and ourhypothesis will be evaluated in terms of a number of details of these observations. For the momentwe wish to call attention to a particular feature that has been written about in a number of theearly
PSP publications (Bale et al. 2019; Kasper et al. 2019; Mozer et al. 2020; Dudok de Wit et al.2020), viz. , the phenomenon of switchbacks , which has particular relevance to our proposed model.
PSP observations during most time periods near the first and second perihelia indicated a meanmagnetic field that was nearly radially inward. However, the data frequently indicate weakening andsometimes reversals (i.e., switchbacks) in the radial magnetic field B R . The weakening is accompaniedby the appearance of substantial transverse components, i.e., T and N components in the standardspacecraft-centered orthogonal RTN coordinate system, where +R is radial (antisunward), +T istangential (toward increasing heliolongitude), and +N is normal (toward increasing heliolatitude).Note that vector velocities in this frame are measured relative to the fixed stars, that is, the spacecraftvelocity has been subtracted out of the solar wind velocity measurement. These features are apparentin the corresponding panels of Figure 2 and are discussed further below.In what follows we will argue that switchbacks and related features of the complex dynamicsobserved by
PSP in this region can be explained by in situ shear-driven dynamics and are alsoconsistent with the striation-flocculation transition described in DeForest et al. (2016). Previousarguments in favor of an in situ origin of switchbacks and large-amplitude magnetic fluctuations weremade by Squire et al. (2020), based on expanding-box compressible MHD simulations, and Macneilet al. (2020), based on
Helios observations of switchbacks, which were more frequent at greaterdistance from the Sun. 2.3.
Hypothesis of Shear Driving: Cartoon
Our initial hypothesis is an extrapolation of the ideas in DeForest et al. (2016), in which themorphological transition between striation and flocculation that is apparent in the STEREO imagesreflects a transition from largely collimated elongated structures, relatively slowly varying in radius,to more disordered shapes suggestive of a more isotropic distribution of fluctuations. This wasattributed to an isotropization of turbulence as the magnetic field above the Alfv´en critical zone(and later above a first β = 1 zone) gives up much of the control over the plasma that it maintainedin the highly magnetized sub-Alfv´enic corona.Here we pursue a particular form of that hypothesis, in which above the Alfv´en critical surface orzone, the velocity differences between adjacent flux tubes may be tapped to supply energy for a moreisotropic form of turbulence. The physical picture we have in mind involves essentially nonlinearmagnetized Kelvin-Helmholtz dynamics, perhaps better described as a magnetized mixing layer. Ahighly simplified sketch of the scenario we propose is provided in Figure 3.2.4. Hypothesis of Shear Driving: Expectations and Background
In examining whether shear-driven dynamics are responsible for prominent features observed by
PSP near perihelion, we are led to consider basic physics suggested in Figure 3 that is similar to theclassic hydrodynamic problem of a mixing layer (Rogers & Moser 1992). When two co-linear streamshaving different velocities come into contact, the early part of the dynamics can resemble a linearKelvin-Helmholtz instability, quickly evolving into a nonlinear mixing layer characterized by vortexrollup.The addition of a uniform parallel magnetic field into the problem presents the complication thatthe transverse displacements needed to produce rollup are inhibited by magnetic field line tension.The linear theory of stability of planar MHD shear layers – a magnetized Kelvin-Helmholtz instability– was considered by Chandrasekhar (1981), who stated the important condition that the instabilityis suppressed when the velocity contrast does not exceed the Alfv´en speed, that is when ∆
V < V A .This instability was subsequently considered in greater detail by Lau & Liu (1980) and Miura &Pritchett (1982) who refined instability criteria for particular assumptions. More generally one doesnot expect that the mixing layer dynamics will be described by linear theory, particularly in cases inwhich the initial state is not an equilibrium and turbulence, possibly broadband, is present withinthe velocity streams. See the SWEAP Data User’s Guide.
B BBCoronaV A > V Solar WindV A < VAlfven critical“surface”V A = V higher speed lower speedlower speed SHEAR LAYERInstability if (cid:39)
V > V A SHEAR LAYER
Instability if (cid:39)
V > V A Figure 3.
Sketch describing our proposed hypothesis. In the corona, the strong magnetic field regulatesthe dynamics of the nascent solar wind. Each flux tube may contain differing radial speeds and differentradial field strengths due to processes at lower altitudes. Beyond the Alfv´en critical zone the magnetic fieldis no longer capable of constraining the dynamics and the energy in the velocity contrasts becomes availableto drive nonlinear magnetized Kelvin-Helmholtz-like dynamics, including magnetic field amplification anddirectional change, with associated deflection of velocities into the transverse directions. This may explainthe transition from striation to flocculation in
STEREO images such as that in Figure 1, and in the presentwork we point out characteristics of
Parker Solar Probe data that are consistent with this picture of howshear-driven dynamics at and above the Alfv´en critical zone boosts low frequency turbulence to the levelsobserved throughout the heliosphere.
To examine the nonlinear evolution of MHD mixing layer/Kelvin-Helmholtz dynamics, Miura (1982)appealed to numerical simulation of compressible MHD. In the nonlinear regime rollup of both vorticesand magnetic field occurs with a substantial component of transient amplification of magnetic energy.Similar configurations were investigated using an incompressible MHD model (Goldstein et al. 1987,1989a) with the goal of understanding magnetospheric boundary effects and reduction of cross helicity(Alfv´enicity) in the solar wind (Roberts et al. 1992). Further study using a compressible model(Malagoli et al. 1996) revealed additional details of the rollup process, which involves strong couplingbetween flows and magnetic field structure. In particular, for cases in which the magnetic field isnot strong enough to stabilize the dynamics, the fully developed state is substantially influenced bythe presence of the (amplified) magnetic field, which exhibits distinctive structure within the vortexrollups.We should note that related studies (Dahlburg et al. 1998; Einaudi et al. 1999) employed anincompressible MHD model to examine the linear and nonlinear evolution of a radial jet confinedwithin a neutral sheet. This system, in effect a simplified model of a coronal streamer, also exhibitsKelvin-Helmholtz-like dynamics at distances large enough that the jet speed exceeds a multiple ofthe Alfv´en speed. Instability and topological changes in the magnetic field at the tip of such astreamer were considered in Rappazzo et al. (2005). The effect of shear and expansion on a spectrumof Alfv´enic fluctuations, previously examined by incompressible simulation (Roberts et al. 1992) andin turbulence transport theory (Breech et al. 2008), was recently considered using expanding boxsimulation in the context of
Parker Solar Probe data in Shi et al. (2020).Below we will examine our hypothesis by comparison of features of
PSP observations near perihelionto computed features of 3D compressible MHD mixing layer simulations. We note that similarcomparisons were employed to explain polarity reversals seen in data from the
Ulysses spacecraft(Landi et al. 2005; Landi et al. 2006). METHODS3.1.
Observations by the Parker Solar Probe
In the present paper we make use of publicly available data from the first two orbits of
Parker SolarProbe , from two instrument suites, FIELDS (Bale et al. 2016) and SWEAP (Kasper et al. 2016). Weuse Level 2 magnetic field data from FIELDS, which typically have a data rate of 299 Hz, and Level 3plasma data from SWEAP, with solar wind speed typically available at a cadence of 1 NYs. We thenusually resample both types of data to either 1-NYs or 1-s cadence, and plasma data from SWEAPare processed to remove spurious spikes. The latter procedure makes use of a time-domain Hampelfilter (Davies & Gather 1993), with a filtering interval of 120 seconds and outliers identified as valuesmore than three times larger than the local standard deviation. We designate the inner segments ofthese orbits (about two months surrounding perihelion) as the first (solar) encounter (E1), roughlyduring 2018 October-November, and the second encounter (E2), roughly during 2019 March-April.Observations during E1 were well described in the first results papers (Bale et al. 2019; Kasper et al.2019) and in relevant papers in the special Astrophysical Journal Supplement issue (e.g., Dudok deWit et al. 2020; Mozer et al. 2020).3.2.
Global Solar Wind Simulations
We employ global 3D MHD modeling to compare with several features of
PSP observations thatwill be discussed in following sections. The fully 3D model that we employ is based on mean-field (Reynolds-averaged) solar wind equations, which are solved simultaneously with turbulencetransport equations (Usmanov et al. 2018). The equations also include electron heat conduction,Coulomb collisions, Reynolds stresses, and electron/proton heating by a turbulent cascade. Theresolved equations and unresolved (subgrid-scale) turbulence equations are solved self-consistently. The turbulence model includes three equations: for turbulence energy, normalized cross helicity, andcorrelation length.The current computation has evolved from previous work (Usmanov et al. 2014, 2016, 2018) andis carried out in four regions: (1) corona, 1-20 R (cid:12) , (2) inner heliosphere, 20 R (cid:12) -5 au, (3) middleheliosphere, 5-40 au, and (4) outer heliosphere, 40-1200 au. Boundary conditions are specified at thecoronal base (just above the transition region) using ADAPT (Air Force Data Assimilative Photo-spheric Flux Transport) solar synoptic magnetic field maps (Arge et al. 2010; Hickmann et al. 2015),in which flux evolution models for the photospheric magnetic field are assimilated with photosphericmagnetic field observations. We use the ADAPT map, which is based on the GONG (Global Oscilla- All data were downloaded from https://cdaweb.gsfc.nasa.gov/pub/data/psp/ We use Reynolds averaging based on an ensemble average (cid:104) . . . (cid:105) (McComb 1990). One decomposes a variable, such asthe velocity u , as u = (cid:104) u (cid:105) + u (cid:48) where (cid:104) u (cid:105) is the mean and u (cid:48) is the unresolved or fluctuating component. Reynoldsaveraging of nonlinear terms involves contributions from fluctuations such as (cid:104) u (cid:48) i u (cid:48) j (cid:105) , known as the Reynolds stress,which is particularly prominent when fluctuations are incompressible. tion Network Group) magnetogram, with the central meridian time 2018 November 6 at 12:00 UTC.The ADAPT map values are scaled by a factor of 2 and are smoothed using a spherical harmonicdecomposition up to 15th order.This global solar wind simulation model has been under continual development (Usmanov et al.2018) and was recently employed to provide context for the STEREO observations described in § PSP mission (Chhiber et al. 2019a,b).3.3.
Compressible MHD Simulations of Mixing Layer Dynamics
To demonstrate the basic physics associated with our hypothesis, we have carried out a series ofnonlinear simulations of fluid-scale turbulence triggered by strong velocity shear in the presence ofa moderately strong uniform DC magnetic field. We solve the compressible MHD equations via ahybrid compact-weighted essentially non-oscillatory (WENO) scheme, which couples a sixth-ordercompact scheme for smooth regions and a fifth-order WENO scheme for shock regions. The timemarching is performed by the third-order Runge-Kutta scheme.The numerical simulations are conducted either in two dimensions (2D) in a (2 π ) domain with256 resolution, or in three dimensions (3D) in a (2 π ) domain with 256 resolution, all with periodicboundary conditions. For simplicity, equal viscosity and resistivity are used, i.e., the magnetic Prandtlnumber is set equal to unity and an ideal gas equation of state is adopted. While we have variedsome initial parameters to investigate the robustness of our conclusions, here we report results fromone representative 3D simulation. Initially flow reversal occurred across two thin layers at y = L y / y = 3 L y /
4, where L y = 2 π . Initially the velocity and magnetic field are only in the x -direction.The x -direction velocity is given by u x = U (cid:20) − tanh (cid:18) y − L y / d (cid:19) + tanh (cid:18) y − L y / d (cid:19)(cid:21) , (1)where U = 0 .
27 and d = 0 . L y is half the thickness of the shear layer. Similarly, the x -directionmagnetic field is given by B x = C (cid:20) − tanh (cid:18) y − L y / d (cid:19) + tanh (cid:18) y − L y / d (cid:19)(cid:21) + C , (2)where C = 0 .
05 and C = 0 .
13 in our simulation. Therefore, the velocity streams have u x = + U or u x = − U , while the magnetic field in the same stream regions has B x = C + C = 0 .
18 or B x = C − C = 0 .
08, respectively, in Alfv´en speed units. The magnetic field is initially entirelytoward + x , stronger in the top and bottom regions and weaker in the middle region. In this arrange-ment, qualitatively consistent with the diagram in Figure 3, the current layers between flux tubesare collocated with the vorticity layers separating streams.Some of the parameters of the representative 3D simulation are shown in Table 1. The simulationinitially has uniform density ρ = 1 and a velocity difference between streams of ∆ U/V A = 2 U /V A = 3,where V A is derived from the strong field region. The plasma beta, i.e., the ratio of plasma pressure tomagnetic field pressure, is β = 1. The initial turbulent Mach number is M t = ∆ U/c s = 3 . c s . The initial temperature pattern is set to achieve uniform total pressure. The value of thepolytropic index is γ = 1 .
4, a value for which extensive testing of the code was carried out for highMach number MHD turbulence as described by Yang et al. (2016). Details of the implementation,transport coefficients, and other numerical details are given in the same reference.0
Table 1.
Initial parameters for shear-driven 3D compressible MHD simulation by methods of Yang et al.(2016) ∆
U/V A β M t u x ± B x (strong region) 0.18 B x (weak region) 0.08Code resolution 256 RESULTS:
PSP
OBSERVATIONS4.1.
Expectation of High Turbulence Levels Near the Alfv´en critical zone
Global simulations of the same type as those described in § δB/B , is expected to be near unity or even above in this zone (Chhiber et al. 2019b). Thisis the same region in which remote radio observations have found enhanced scattering, from whichenhanced turbulence levels are inferred (Lotova et al. 2011).Further evidence is presented by new global simulation results shown in Figure 4 that address morespecifically the likelihood of large polarity reversing magnetic fluctuations. Such simulations providevaluable insights into the behavior of the solar wind velocity and magnetic field components alongthe first three PSP orbits.At the cadences shown in Figure 4, neither the observations nor the simulations capture the fullfluctuation amplitudes (which can be seen in Figure 2). The global code includes a self-consistenttransport model for the turbulence amplitude, which can be used to estimate a likely range ofturbulent fluctuations. This range of turbulence values, when superposed on the resolved simulationvariables, provides an estimate of the full range of likely magnetic and velocity components along the
PSP trajectory during the first encounter. The resulting range of predicted values agrees reasonablywell with the range of fluctuations suggested by the averaged
PSP data. For example, focusing onthe radial component of B in Figure 4, the range of expected values accounts for the possibility ofnumerous switchbacks.Based on these considerations and prior evidence, one may anticipate that the fluctuations becomelarge in and near the Alfv´en critical zone. Sufficiently large fluctuations, particularly in the Alfv´enmode (Matteini et al. 2018) can produce large deflections including reversals of magnetic polarity,i.e., switchbacks. Previous studies focused mainly on the presence of large amplitude fluctuations,while here we point out a number of other characteristics of PSP data that require further study.It is also useful to take a closer look at the
PSP data near perihelion to motivate the more detailedanalysis that follows. Figure 5 provides an example of magnetic field data for one hour near the1 -40-2002040-100-500 -40-2002040 Br , nT B φ , nT B θ , nT u r , km/s u φ , km/s u θ , km/s -100-50050100 Figure 4.
Magnetic field (top) and velocity (bottom) components (left: radial, center: azimuthal, right:meridional) measured by
PSP during the 1st encounter, shown as hour averages (blue line). Also shown areresolved (mean field) solutions for the corresponding magnetic field components from the Usmanov et al.(2018) global simulation employing an ADAPT magnetogram corresponding to the period of the encounter.Finally the shaded background is an envelope corresponding to an estimate of the turbulence amplituderelative to the mean field computed in the simulation using self-consistent turbulence transport equationsas explained in the text. -100-80-60-40-20020406080100 B ( n T ) B R B T B N |B| Figure 5.
Magnetic field components and magnitude as measured by
PSP /FIELDS during one hour nearfirst perihelion (2018 November 6, 0500-0600 UT) at radius 36 R (cid:12) , at the full sampling frequency of 299Hz. Short-term fluctuations are mostly Alfv´enic in the sense of conserving | B | . Domains of nearly constant | B | are often separated by minute-scale changes in | B | and sharp, major jumps in the components of B . Weargue that these separate domains of magnetic pressure-balanced Alfv´enic fluctuations could correspond toflocculation mixing layers. first perihelion. Here we see that the magnetic field components show large fluctuations. The radialcomponent is predominantly negative, but shows sporadically large clusters of fluctuations (Chhiberet al. 2020). Meanwhile, the magnetic field displays a structure consisting of regions of relativelyconstant magnetic field strength separated by sharp changes. At high temporal cadence we canidentify a few regions in which the main (radial) magnetic field changes polarity for brief times; theseare the switchbacks.2 Figure 6.
Alfv´en speed vs. time computed at four levels of coarse graining during parts of the (top) first and(bottom) second
PSP orbit. The data are plotted at 1-hr cadence in each case, while averaging is performedover a moving window of specified duration. The red vertical lines mark the respective perihelia, and thedashed vertical lines demarcate a period of 10 days centered on the perihelia. Selected heliocentric distancesof
PSP are marked above the upper horizontal axes.
Radial Velocity Shear, Alfv´en Speed and Conditions Near Shear Instability
A key element of the shear-driving hypothesis is the conversion of initially more ordered but inho-mogeneous flows into more randomized flows, a conversion that must occur in the presence of orderedmagnetic fields. This is a scenario that can explain the transition from striation to flocculation inthe
STEREO images (DeForest et al. 2016). As discussed in § PSP data,which we show for the first two encounters (E1 and E2) in Figure 6. Four levels of averaging areshown, over 1, 4, 12, and 24 hours. The salient feature is that there is considerable variation ofAlfv´en speed in both encounters, with values around 80 to 110 km s − near first perihelion, withhigher values close to 200 km s − near second perihelion, and with values as low as ∼
10 km s − atgreater distance from the Sun.As a next step we examine the radial velocity, normalized to the Alfv´en speed, for E1. This ismotivated by Chandrasekhar’s condition for suppression, V A > ∆ V , where ∆ V is an appropriatemeasure of the velocity contrast across a shear layer. (In the current complex environment, we viewthat a physically relevant value of the Alfv´en speed V A would be an appropriate regional average.)However we emphasize that here we are not looking for conditions for subsequent linear instability,3 Figure 7. (Top) Radial velocity of solar wind protons in Alfv´en speed units, near the first
PSP perihelion.Radial speed is sampled at a cadence of 1 NYs ≈ .
87 s, and normalized either by the 1 NYs Alfv´en speedor the 1-hour running average of the Alfv´en speed, as indicated. (Bottom) Red curve shows the absolutevalue of increments ∆ V R = V R ( t + τ ) − V R ( t ) of the radial velocity V R computed from 0.87 s data and fora time lag of τ = 10 min, approximately the correlation time in this part of the PSP orbit (see Parasharet al. 2020). Also shown is the 1-hour rms of ∆ V R (blue curve). The increments are normalized by the1-hour moving average of V A . It is apparent that there are many intermittently distributed V R -incrementsthat exceed the local (smoothed) Alfv´en speed. since the additional signatures we examine would indicate a past instability closer to the Sun ratherthan an imminent instability. Nevertheless, we do not rule out that subsequent instability may takeplace, particularly because, as mentioned above, polarity reversals are observed further from theSun and there is evidence that their frequency may actually increase with increasing radial distance(Macneil et al. 2020; Owens et al. 2020). The presence of velocity changes over relatively shortdistances that exceed the local Alfv´en speed is an indication that the criterion for Kelvin-Helmholtzrollup is likely to be reached in this region (or it may even be in progress as we observe it)—somethingthat cannot be ascertained with single-point measurements.In Figure 7 (top) we show the radial plasma velocity of solar wind protons. Two resolutions areshown, one in which V A is computed at 0.87-s resolution, and the other using one-hour smoothed(running averaged) V A . There are numerous variations of V R that are larger than one or a few Alfv´enspeeds, but we must ask at what scales these occur. To that end, we compute the increments ofthe observed radial component of plasma velocity, normalized in an analogous way to the 1-hourmoving average of V A . The increment is defined as ∆ V R = V R ( t + τ ) − V R ( t ) with time lag τ . Thevalue τ = 10 minutes is selected, corresponding to the typical measured correlation time in the firstencounter (Parashar et al. 2020), which is expected to be a typical large scale magnetic flux tubesize. Therefore, the measured increments are estimates of velocity contrasts ∆ U between adjacentflux tubes as suggested in Figure 3. We see that ∆ V R frequently exceeds V A . This is a way to assessthe likelihood of nearby nonlinear K-H activity. We conclude that the case for the development of amixing layer is reasonably well supported.4.3. Transverse Velocity and Fluctuation Components V R / V A | V T | / V A | V N | / V A - hr24 - hr Rigid Rotation
36 R ô
45 4560 60 80 100 150
Figure 8.
Proton velocity components in (coarse-grained) Alfv´en speed units during the first
PSP orbit.Two levels of coarse-grained Alfv´en speed are used (4- and 24-hour moving averages), while the protonvelocities are plotted at 1-hour cadence. The brown curve in the middle panel shows the speed of rigidrotation in units of the 4-hour moving average of V A . This would be the tangential speed of the plasma ifit were corotating with the Sun, with angular speed corresponding to the sidereal rotation period of 24.47days. Selected heliocentric distances of PSP are marked above the upper horizontal axis.
The behavior of the transverse velocity components is also significant and may exhibit signaturesof plasma rollup, a process that also involves convection of the magnetic field. In the idealized case,the magnetic field resists the rollup, and is amplified as it is distorted by the velocity shear (Miura1982; Goldstein et al. 1989b; Roberts et al. 1992; Malagoli et al. 1996). If there were little or notransverse velocity initially, one would expect that the maximum excursion of the transverse velocitywould be, roughly speaking, bounded by the local (amplified) Alfv´en speed, according to the typicalcondition of equipartition of energy between magnetic and flow energies in the solar wind frame.To examine the excursion of the transverse velocities, Figures 8 and 9 show the three Cartesiancomponents of velocity normalized to the locally-averaged Alfv´en speed during the first and secondorbits, respectively. It is apparent that the two transverse components V T and V N are almost alwaysnicely bounded by the local Alfv´en speed. To be specific, among 1-s values of | V T | /V A and | V N | /V A from the first encounter, about 7% exceeded 1 and none exceeded 4. Note that a 2-hour movingaverage of V A was used to obtain these percentages.Figures 8(b) and 9(b) also indicate the speed of rigid rotation with the Sun at the sidereal rotationperiod of 24.47 d (P´ecseli 2020), which is plotted in units of the 4-h moving average of V A (browncurves). It can be seen that at times near both the first and second perihelia, V T was comparable tothe speed of corotation with the Sun.5 V R / V A | V T | / V A | V N | / V A - hr24 - hr Rigid Rotation
36 R ô
45 4560 6080 80100 100150 150
Figure 9.
Proton velocity components in (coarse-grained) Alfv´en speed units during the second
PSP orbit.See caption of Figure 8 for more details.
150 100 50 0 50 100 150 V T (km s ) P D F Mean:-25.38 (a)
150 100 50 0 50 100 150 V T (km s ) P D F Mean:62.43 (b)
150 100 50 0 50 100 150 V T (km s ) P D F Mean:5.94 (c)
Figure 10. (a) Probability distribution function (PDF) of longitudinal solar wind velocity V T (in km s − )as measured by PSP with a cadence of 1 NYs ( ≈ .
87 s) during 2018 Oct 31, 0800-1600 UT, about 6 daysbefore first perihelion. Vertical dashed line indicates mean value. During this and other time periods farfrom perihelion, V T is randomly distributed around zero, with sub-distributions around positive or negativevalues for up to a few hours (as seen here mostly at negative values). (b) Similar PDF for 2018 Nov 6,0000-0800 UT, including the time of first perihelion. Near perihelion, V T is clearly biased toward positivevalues, indicating partial corotation with the Sun. (c) Similar PDF for 2018 Nov 11, 0000-0800 UT, or 5days after first perihelion. Far from perihelion, V T is again randomly distributed around zero. The probability distributions of the longitudinal velocity component V T for E1 are shown at threepositions along the orbit in Figure 10. We note that the distribution for the inbound orbit, 6 daysprior to perihelion, shows a multi-component distribution with several distinct peaks (Figure 10(a)).Each peak covers a spread in V T that resembles a separate sub-distribution. From the time series (notshown) these are seen to result from time periods in which V T fluctuates about positive or negative6values for durations from a fraction of an hour to a few hours; longer durations are more common atgreater distance from the Sun. The 8-hour time period shown in Figure 10(a) happens to have morenegative values. Near perihelion, the distribution shows a strong bias towards positive V T (Figure10(b)) at speeds comparable to the corotation velocity of around 70 km s − . There is again a broaddistribution, in this case skewed towards smaller values. Five days after perihelion, for the exampleperiod shown in Figure 10(c), the distribution is centered roughly about V T = 0 with a single strongmaximum. The distributions of the latitudinal component V N for E1 (not shown) are qualitativelysimilar except they do not exhibit a bias toward positive values near perihelion.In the corona, the nascent solar wind is expected to be channeled along magnetic flux tubes thatcorotate with the Sun in the longitudinal direction. Thus the PSP observations near first perihelionare consistent with partial corotation in that the longitudinal solar wind velocity V T fluctuates aroundthe corotation speed while the latitudinal component V N fluctuates around zero. These observedpatterns are also evident in Figure 4, where the global simulation variables ( u θ , u ϕ ) correspond to PSP velocity components ( − V N , V T ). We refer to partial corotation because, according to Figure10(b), most of the solar wind has V T below the corotational value of ≈
70 km s − . Indeed, such“slippage” of solar wind elements from corotation is expected to occur beyond the Alfv´en criticalzone where the magnetic field no longer controls the solar wind flow. Therefore, we interpret theobservation of partial corotation near the first perihelion as evidence that PSP was already close tothe Alfv´en critical zone. Farther from the Sun, there is no apparent corotation of the solar wind (seeFigures 4 and 10) and such slippage becomes complete. However, near first perihelion the partialcorotation indicates partial slippage, and suggests that neighboring magnetic flux tubes could havesubstantially different V T values. In other words, in addition to the radial velocity shear suggestedin Figure 3 and Section 4.2, there could also be longitudinal velocity shear associated with partialcorotation.Intriguingly, in Figure 10(b) part of the V T distribution is actually faster than the corotationalspeed, which could be attributed to Kelvin-Helmholtz rollups in the mixing layer outside the Alfv´encritical zone. These interpretations of large reported V T need to be viewed as tentative, given thatmodeling has so far not been able to reproduce V T values as large as those discussed here.4.4. Domains and Anisotropy of Alfv´enic Fluctuations
Ever since the seminal work of Belcher & Davis Jr. (1971), it has been recognized that magneticand velocity fields in the solar wind tend to fluctuate together, which has been attributed to anAlfv´en mode v = ± b √ µ ρ , (3)where v ≡ V − V and b ≡ B − B , subtracting any large-scale (mean) fields V and B , ρ is themass density, and the right hand side is the magnetic fluctuation expressed in terms of the Alfv´enspeed. Even at large amplitudes such fluctuations are solutions of the incompressible MHD equations(Moffatt 1978). If there is a mean magnetic field B , then the the “+” sign indicates propagationalong − B and the “ − ” sign indicates propagation along B .For compressible MHD, large-amplitude propagating solutions exist for the Alfv´en mode so long asthe total field magnitude | B | = | ( B + b ) | is uniform (Goldstein et al. 1974; Barnes & Hollweg 1974;Barnes 1979). However, note that the divergence requirement on the magnetic field limits the spatialregion over which this “incompressible” mode can exist (Barnes 1979). From the work of Belcher &7 |B| (nT) P D F Mean: 50.65 (a) |B| (nT) P D F Mean: 93.90 (b) |B| (nT) P D F Mean: 53.28 (c)
Figure 11. (a) PDF of magnetic field magnitude | B | (in nT) as measured by PSP sampled at a cadence of1 NYs ( ≈ .
87 s) during 2018 Oct 31, 0800-1600 UT, about 6 days before first perihelion. Vertical dashedline indicates mean value. Clumps in the distribution of | B | correspond to local flux tubes. (b) Similar PDFfor 2018 Nov 6, 0000-0800 UT, including the time of first perihelion. The mean of | B | was generally largerwhen PSP was closer to the Sun. (c) Similar PDF for 2018 Nov 11, 0000-0800 UT, or 5 days after firstperihelion. At this location, the entire 8-h period effectively comprises a single domain.
Davis Jr. (1971) and many others, in situ measurements of solar wind fluctuations throughout theheliosphere have indicated that such Alfv´enic fluctuations are predominantly outward. In
PSP data,such fluctuations are common and frequently of large amplitude, with | b | ∼ | B | (see, e.g., Figure 2of Kasper et al. 2019). Parashar et al. (2020) and Horbury et al. (2020) recently described severalmeasures of Alfv´enicity as applied to the PSP
E1 data. Other aspects of Alfv´enic fluctuations havealso gained recent attention (Matteini et al. 2018, 2019; D’Amicis et al. 2020).The nearly constant magnetic field magnitude | B | (a distinctive property of Alfv´enic fluctuations)is often evident in PSP data as illustrated in Figure 5. For such cases, in terms of its components, thevector B is randomly walking on a sphere of nearly constant | B | , as described by Barnes (1981). Inturbulence, constant magnetic pressure may be associated with rapid, local relaxation processes thatalso favor patches of flow-field alignment, as in Eq. (3) (Matthaeus et al. 2008; Osman et al. 2011).Matteini et al. (2015) offer an alternative view of constancy of | B | , namely, that it is associated withconservation of ion kinetic energy in the reference frame of observed alpha particle motion.As can be seen from Figure 5, PSP data from the first encounter reveal Alfv´enic domains withnearly constant | B | that are often separated by sharp, major jumps in the components of B , asnecessary to preserve the divergence condition ∇ · B = 0 when ∇ · V (cid:54) = 0. Another indication of thedomain structure comes from probability distribution functions (PDFs) of | B | as shown in Figure 11.It is clear that eight hour samples near perihelion are likely to contain one to a few regions of nearlyconstant magnetic field. This is consistent with Alfv´enic turbulence, but in addition, it is consistentwith mixing layer dynamics, as we shall see below.At the interface between two plasma flows with relative shear, once the condition ∆ V > V A is met,Kelvin-Helmholtz dynamics are possible. This develops into a mixing layer that eventually includesrollups and magnetic polarity reversals (switchbacks) (Malagoli et al. 1996), and the mixing layeris expected to grow with distance along the flow, or in this case with distance from the Sun. Now,in our hypothesis (see Figure 3), there are numerous magnetic flux tubes in the nascent solar wind.Some of the interfaces between these should develop the Kelvin-Helmholtz instability and mixinglayers. These mixing layers should grow until they come into contact. When they do, it is possiblethat they merge in the sense that the shear-driven dynamics (i.e., flocculation) homogenizes themagnetic pressure within the merged region. We interpret the domains of Alfv´enic turbulence with8nearly constant | B | as such (possibly merged) mixing layers, and they exhibit sharp boundaries astopological defects across which the dynamics have not yet balanced the magnetic pressure. PSP dataalso provide some evidence that these domains become larger with increasing heliocentric distance r , as expected for mixing layers that grow and merge. In Figure 11(c) we see a case 5 days afterfirst perihelion when a very narrow distribution of nearly constant | B | was observed for an entire8-h period, in contrast with the 8-h period near first perihelion in which multiple distributions wereobserved (Figure 11(b)). As described earlier for V T , sub-distributions in the time series for | B | that last several hours are more common at increased r several days away from perihelion, while thevariation in PSP travel speed was relatively minor.In Figure 5 we see that at some times within a domain of nearly constant | B | , in association withparticularly strong magnetic fluctuations, | B | temporarily decreases for up to a few minutes beforereturning back to the same nearly constant level. At such times the magnetic pressure balance istemporarily disrupted within the domain. Sudden drops in | B | have previously been reported duringswitchbacks, i.e., reversals in B R (Bale et al. 2019; Kasper et al. 2019). Here we note that a temporarydecrease in | B | can occur together with strong fluctuations in any of the magnetic field components,e.g., in Figure 5 at hour 5.37 or 5.63, and are not particular to switchbacks. This is consistent withthe view that many switchbacks belong to a continuum of fluctuations that can occur in all fieldcomponents as part of in situ dynamics in the solar wind.To examine the Alfv´enic magnetic fluctuations in more detail, we calculated statistics of magneticincrements ∆B = B ( t + τ ) − B ( t ) for time lags τ of 1 s, 10 s, 1 min, 10 min, 1 h, and 6 h. (Notethat the FIELDS instrument typically samples the magnetic field at 299 Hz so even the 1-s lag ismuch longer than the instrumental resolution.) In order to study the fluctuation anisotropy, wedecomposed the magnetic increments into a parallel component ∆ B (cid:107) along the magnetic field B ( t )and two components along basis vectors perpendicular to B ( t ), in the R-T plane (∆ B , a roughlylongitudinal increment) and perpendicular to the R-T plane (∆ B , a roughly latitudinal increment).We calculated the variances (mean squares) of these quantities as a measure of the scale-dependentfluctuation energy in these components. To measure the conservation of | B | , we also calculated thevariance of increments in | B | .The variance ratio of the magnitude increment to the total increment, (cid:104) (∆ | B | ) (cid:105) / (cid:104) ( ∆B ) (cid:105) , usuallyremained below 0.05 throughout both the first and second PSP orbits, only rarely exceeding 0.2,for all values of τ . This confirms the basically Alfv´enic nature of the fluctuations. Throughoutboth orbits, there were some special time periods with an unusually low ratio, i.e., especially goodmagnetic pressure balance. It turns out that such special time periods occurred near both the firstand second perihelia, i.e., during 2018 Nov 3-10 and during 2019 Apr 3-6. We display results for thisratio and ratios between variances of increment components, as a function of time lag τ , for thesespecial times near the first and second perihelia and also for the entire data sets of the first orbit(2018 Oct 6 to 2018 Dec 19) and second orbit (2019 Feb 20 to 2019 May 15) in Figure 12.This figure shows that the variance ratio of the magnitude increment to the total increment (bluecurves) is indeed lower for the special periods near first perihelion (a) and second perihelion (c)compared with the full orbits (b and d). Yet even for data from the full orbits, the ratio for lags upto 1 h remains below 0.025, confirming the near constancy of | B | and magnetic pressure over suchtime scales. At τ = 6 h, the ratio increases, indicating that this time scale is frequently greater thanthe domain duration; even so the ratio remains below 0.06.9 < Δ | B | > / < Δ B >< Δ B ‖ > / < Δ B > , < Δ B > / < Δ B > Time lag (s)(a) 1st perihelion<Δ|B| >/< ΔB ><ΔB >/<ΔB ><ΔB ‖2 >/< ΔB > < Δ | B | > / < Δ B >< Δ B ‖ > / < Δ B > , < Δ B > / < Δ B > Time lag (s)(b) 1st orbit<Δ|B| >/< ΔB ><ΔB >/<ΔB ><ΔB ‖2 >/< ΔB > < Δ | B | > / < Δ B >< Δ B ‖ > / < Δ B > , < Δ B > / < Δ B > Time lag (s)(c) 2nd perihelion<Δ|B| >/< ΔB ><ΔB >/<ΔB ><ΔB ‖2 >/< ΔB > < Δ | B | > / < Δ B >< Δ B ‖ > / < Δ B > , < Δ B > / < Δ B > Time lag (s)(d) 2nd orbit<Δ|B| >/< ΔB ><ΔB >/<ΔB ><ΔB ‖2 >/< ΔB > Figure 12.
Ratios of variances of magnetic field increments measured by
PSP (a) near 1st perihelion, 2018Nov 3-10, (b) during 1st orbit, from 2018 Oct 6 to 2018 Dec 19, (c) near 2nd perihelion, 2019 Apr 3-6, and(d) during 2nd orbit, from 2019 Feb 20 to 2019 May 15. For varying time lags τ , vector magnetic increments ∆B = B ( t + τ ) − B ( t ) are calculated and decomposed into a component ∆ B (cid:107) parallel to B ( t ) (mostly radial)and two perpendicular components, ∆ B in the R-T plane (mostly longitudinal) and ∆ B out of the R-Tplane (mostly latitudinal). The magnitude increment (∆ | B | ) is also calculated. The low variance ratios ofthe magnitude increment and parallel increment to the total increment (blue and red curves, respectively)indicate the near constancy of | B | and magnetic pressure, a characteristic of Alfv´enic fluctuations. Thevariance ratio of longitudinal to latitudinal increments (black curves) is between 1.4 and 2 for the longest(6-h) increments but decreases to about 1 for the shortest (1-s) increments. The anisotropy of longer-timeincrements is unlikely to originate in or below the inner corona, and can be attributed to longitudinal velocityshear near the Alfv´en critical zone due to partial corotation, leading to perpendicular field increments thatare predominantly longitudinal over large scales and isotropize after a turbulent cascade to smaller scales. The ratio of the parallel increment variance to total increment variance (red curves) is also quitelow ( < .
1) for the 1-s time lags, and it grows larger for longer time lags according to the nearconstancy of | B | and the increase in the increment amplitude | ∆ B | for increasing lag τ . For a smallamplitude ( | ∆ B | (cid:28) | B | ), we would expect ∆ B (cid:107) ≈ ∆ | B | . However, for large-amplitude Alfv´en modefluctuations that maintain constant | B | , i.e., for spherical polarization in which B remains on a spherein its component space, ∆ B (cid:107) is directly related to the fluctuation amplitude. Here this geometric0effect dominates over the actual magnitude fluctuations, with (cid:104) (∆ B (cid:107) ) (cid:105) (cid:29) (cid:104) (∆ | B | ) (cid:105) even for oursmallest (1-s) lags.A surprising result from this analysis is an anisotropy between the two perpendicular components ofthe magnetic field increment. The variance ratio of roughly longitudinal to latitudinal perpendicularincrements, (cid:104) (∆ B ) (cid:105) / (cid:104) (∆ B ) (cid:105) , ranges from 1.4 to 2 for the longest (6-h) lags while decreasing toabout 1 for the shortest (1-s) lags. The transition seems to relate to the correlation time of severalminutes. This anisotropy for long lags persists throughout both orbits, though on average it isparticularly strong during time periods with better magnetic pressure balance such as the times closeto perihelia. As such it does not appear to be related to the direction of the PSP orbital motion,which varies strongly and systematically throughout the orbit.This anisotropy of magnetic increments for long τ is unlikely to originate in or below the innercorona, in which the latitudinal and longitudinal directions are not strongly distinguished. It can beunderstood in terms of velocity shear above the Alfv´en critical zone between flux tubes with varyingdegrees of longitudinal corotation, leading to perpendicular field increments that are predominantlylongitudinal over large scales and then isotropize after a turbulent cascade to smaller scales.4.5. Cross Helicity and Signatures of Velocity Shear
In terms of a volume average, here designated as (cid:104)· · ·(cid:105) , the cross helicity may be defined as H c ≡ (cid:104) v · b (cid:105) = 14 (cid:0) Z − Z − (cid:1) (4)where the fluctuations in magnetic field are computed in Alfv´en units as b = ( B − B ) / √ µ ρ , and theEls¨asser energies are Z = (cid:104)| v + b | (cid:105) and Z − = (cid:104)| v − b | (cid:105) . The traditional view is that the ± Els¨asserfluctuations z ± = v ± b comprise wave packets that propagate either along the B direction ( z − ), oropposed to it ( z + ). This definition corresponds to and generalizes the large amplitude eigenmodesdescribed in the previous section (cf. Eq. 3). H c is an ideal invariant of the incompressible MHDsystem and has significance whether or not a mean magnetic field B is present.It is well known that in the inner heliosphere, solar wind fluctuations have a strong cross-helicityin the sense that propagation is dominantly outward (Belcher & Davis Jr. 1971). In our simplecartoon (Figure 3), if all the magnetic field is outward and the fast outward streams are located inthe flux tubes with weaker magnetic field, then the cross-helicity of long-wavelength fluctuations willbe negative and they will travel outward except in switchback regions. If the prevailing magneticpolarity is inward (as it is during E1 and E2), in the context of our cartoon the faster streamsshould still be in the weaker flux tubes, to give a positive cross helicity and the observed outwardpropagation. In a more complete description of the solar wind there is also likely to be a broadbandspectrum of more standard outward propagating Alfv´enic fluctuations.In PSP observations close to the Sun, the cross helicity measured by the ratio σ c = ( Z − Z − ) / ( Z + Z − ) is generally quite large, suggestive of a preponderance of outward traveling Alfv´en waves. Thereare departures from Alfv´enicity for increments at small lags, including reduced cross-helicity, asreported by Parashar et al. (2020). At the same time, our present analysis shows that the magneticmagnitude increment is quite small for small time lags (Figure 12), which is consistent with mostlyAlfv´en-mode fluctuations. This could be because above the conventional Alfv´en point velocity shearscan begin supplying turbulence energy (Zank et al. 1996; Breech et al. 2008) that remains nearlyincompressible but not entirely outward-directed.1 Figure 13. (Left) Vorticity in the z -direction and (right) magnetic field in the x -direction (shown by colorscales) from 3D compressible MHD simulation at t = 120. Vortex rollup, inhibited by the magnetic field,is just beginning to take effect. Initially flow reversal occurred across two thin planes at y = L y / L y / x -direction, stronger outside those two planesand weaker between them.5. RESULTS: 3D COMPRESSIBLE MHD SIMULATION OF MIXING LAYER DYNAMICSHaving examined several plasma and magnetic field diagnostics in the
PSP data, we now turn tothe results of more local compressible MHD simulations, emphasizing points of comparison with theobservations. We seek to examine further possible points of consistency with plasma dynamics drivenby nonlinear mixing layer dynamics as envisioned in § Compressible MHD simulation results
We carried out a number of compressible MHD runs using the approach outlined in § PSP ? In the actualsolar wind, as we have pointed out there is at least partial corotation in the longitudinal direction(see Section 4.3) and, we believe, also some slippage of individual flux tubes and longitudinal velocityshear as well (see Section 4.4).The baseline parameters corresponding to the results shown here were given in Table 1. Differentruns (not shown) were done in two dimensions and in three dimensions, with varying velocity contrast∆ U across the shear layers, and several values of uniform B x in the strong and weak magnetic field2 Figure 14. (Left) Vorticity in the z -direction and (right) magnetic field in the x -direction (shown by colorscales) from 3D compressible MHD simulation at a later time, t = 290. Vortex rollup is now well developed,producing two switchback regions, which recur intermittently throughout the simulation run. regions. Within the range of parameters that were varied, the results were all similar; therefore weshow just one case in the diagnostics here, as described in Table 1.Beginning from the initial state described above, the dynamics proceed along the lines of a hydro-dynamic mixing layer. The initially planar vorticity layers distort due to the early stages of vortexrollup. The magnetic field is too weak to prevent the distortion from reaching macroscopic dimen-sions. A snapshot of this state is shown in the two panels of Figure 13, where the breakup of thevortex layers begins along with large magnetic field directional deflections and small regions of weakfield in which the polarity reverses. Figure 14 describes the state of the system later, at simulationtime t = 290, when the shear-driven dynamics are more fully developed and clearly in a nonlinearstage. In particular, the phenomenon of rollups has noticeably emerged, where the vortex layers havefolded. Shocklets have formed at locations at which the flow direction change is relatively abrupt.Perhaps most importantly, there are now large transverse velocities in both the positive and nega-tive ˆ y (vertical) directions. The large deflected motions have carried along the magnetic field (rightpanel), which also exhibits large transverse deflections. Examination of the sign and magnitude ofthe streamwise ˆ x -direction magnetic field (indicated by the color legend) indicates the presence ofregions of polarity reversals, or “switchbacks”. Here these are entirely caused by nonlinear instabilitydriven by the initial shear layers.A complementary graph of the magnetic field components is shown in Figure 15 for time t = 290 ofthe MHD simulation. The spatial structure is sampled as a function of distance s along a trajectoryat an 18 ◦ angle relative to the axes of the box that threads through the (periodic) box several times,to produce a one-dimensional series that spans about ten correlation lengths, similar to the PSP datasample shown in Figure 5. For reasons to be discussed shortly, we associate the x -direction along themean field in the simulation with the − R direction for
PSP measurements, and the y -direction withthe T direction, so the left panel uses B R = − B x and B T = B y . Note the region of approximately3 Bx0.300.250.200.150.100.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 s=0.0 ♦♦ ♦♦ s=5.5 s=14.0s=18.0
Figure 15. (left) Magnetic field components and magnitude from the 3D compressible MHD simulation attime t = 290. Note the presence of “switchbacks”, i.e., reversals of B R , as well as large symmetric fluctuationsof transverse B T . The magnetic field magnitude | B | is relatively constant within regions delineated by theproximity of switchbacks. This figure can be qualitatively compared with Figure 5. (right) The simulationplane and trajectory employed to obtain the data in the left panel. The trajectory is drawn, annotated withreference distances s . constant | B | and the presence of switchbacks in the simulation plot, in qualitative accord with the PSP data.We will now show several diagnostics that permit a quantitative comparison of several featuresof the simulation and the observations by
PSP . We do not expect precise correspondence becausethe simulation setup is a vast oversimplification of the interplanetary physical system. But if ourconjectures are correct concerning the basic physics that drives the evolution in the solar wind, thenwe might find some consistency in the comparisons.5.2.
Comparison with PSP data: Magnetic Field Magnitude
A special property of large amplitude Alfv´en waves, a constant magnitude | B | , is apparently afamiliar property in MHD turbulence. Constant magnitude patches or regions have also been observedin PSP data, as shown for the first
PSP encounter in Figures 5 and 11. An analysis of the shear-drivenMHD simulation results also shows a similar distribution of magnetic field magnitude, as illustratedin Figure 16. Note that at the earlier time, t = 120, the distribution has peaks associated with theinitial conditions, in which | B | was concentrated at two initial values, shown by vertical dot-dashedlines at 0.08 and 0.18. These peaks have smoothed somewhat by the later time t = 290, representinga more developed dynamical state that we consider comparable to the solar wind at initial PSP perihelia, somewhat downstream of the Alfv´en critical zone. Even at the later simulation time oneobserves in the PDF of | B | the presence of preferred values of | B | or sub-distributions as seen in the PSP data. 5.3.
Comparison with PSP data: Magnetic Switchbacks
Considerable attention has been given to the appearance of switchbacks in the
PSP data (Baleet al. 2019; Dudok de Wit et al. 2020) where, as mentioned earlier, they are seen more dramatically4
Figure 16.
PDF of magnetic field magnitude from the 3D compressible MHD simulation at simulationtimes t = 120 and t = 290. Initial magnetic field values are annotated. This figure can be qualitativelycompared with Figure 11(b). Figure 17. (Left) PDF of radial magnetic field from 8 hours of data that include
PSP ’s 1st perihelion;(right) PDF of “radial” x -component of magnetic field at two times from the shear-driven 3D compressibleMHD simulation. near perihelion than in the more distant solar wind. While many switchbacks are seen in the firstorbit, as in Figure 2, a close up look at one hour of data, as in Figure 5, shows that most of the solarwind is filled with unipolar negative radial field.For the present purposes it is of interest to compare the frequency of occurrence of reversed polaritymagnetic fields in the presence of a dominant magnetic polarity and strong shear. In this way one cancompare switchbacks from PSP with magnetic polarity reversals in our standard MHD simulation,noting that for most of E1 and E2 the large-scale magnetic field at
PSP was nearly in the − Rdirection, which we associate with the x -direction along the mean field in the simulation, whereas a PSP measurement of a transverse component such as T corresponds to the component along y , thedirection that cuts across shear layers in the simulation. To this end we compute the distribution of5 Figure 18.
Solid red curve shows the PDF of the transverse velocity normalized to the 1-hour runningaverage of the Alfv´en speed during the first
PSP encounter. Dash-dotted orange curve shows the PDF of thetransverse velocity from the shear-driven 3D compressible MHD simulation, normalized to the local Alfv´enspeed. Compare with Figures 8 and 9. the radial magnetic field component B R from 8 hours of PSP data near first perihelion (2018 Nov 6,0000-0800 UT) and compare this to the distribution of B R ≡ − B x from the simulation at two times.This comparison is presented in the two panels of Figure 17.The qualitative features of these distributions are quite similar: The PSP data show one strongpeak at a dominant negative polarity, with a shelf-like distribution that extends to positive polarityvalues, indicating switchbacks. In the simulation there are two preferred values of dominant polarityat the earlier time shown, and a single strong peak at the later time that we believe better representsthe more developed dynamics downstream of the Alfv´en critical zone. At both times the simulationshows a relatively flat, low-level, shelf-like distribution of reversed polarity, much like the observeddistribution in the left panel.The simulation data in Figure 17, as well as analysis of several other simulations we have carriedout, demonstrate that switchbacks occur with similar frequencies in MHD mixing layer dynamics andin the
PSP data near first perihelion.5.4.
Comparison with PSP data: Transverse Velocities
The
PSP results shown above demonstrate that the transverse velocity components near first
PSP perihelion are essentially bounded by the local or neighborhood value of the Alfv´en speed. It is alsointeresting, using the same normalization, to compare the distribution of the V T component of velocityin the PSP data with the corresponding “non-radial” (transverse) component V y of the plasma velocityin the MHD simulation data. One can see in Figure 18 that the local Alfv´en speed represents anapproximate limit that constrains the dynamics in both the simulation and PSP observations, whichis consistent with the reasoning above (see Section 4.3). DISCUSSION AND CONCLUSIONSMotivated by
STEREO observations (DeForest et al. 2016), we have examined the possibility thatshear driven dynamics drive the transition from elongated, or striated, structures in the lower coronato more isotropic, or flocculated, structures above the the Alfv´en critical zone. The associated6release of energy in the sheared flows represents an additional source of energy over and above thepreexisting Alfv´enic turbulence originating at lower coronal altitudes. If the above hypothesis iscorrect, this transition signals an enhancement in outer scale turbulence energy that persists to muchlarger heliocentric distances. This hypothesis has been examined here beginning with clues fromimaging, and further motivated by the large turbulence amplitudes seen in global MHD simulations.
Parker Solar Probe provides an opportunity to begin detailed analysis of the consequences of theshear driving hypothesis. The first results of this analysis have been presented in some detail here.The basic picture is that of a magnetized parallel mixing layer in which the velocity contrasts arelarge enough to cause significant deflections and even reversals of the magnetic field. The salientfeatures of the mixing layer are well known in hydrodynamic, engineering and practical applications;a common example is shown in Figure 19. The magnetic field parallel to the flow presents a compli-cation in that it resists deflection. However, as anticipated in theory (Chandrasekhar 1981; Miura& Pritchett 1982) and demonstrated in simulations (Goldstein et al. 1989b; Roberts et al. 1992;Malagoli et al. 1996; Landi et al. 2006), a sufficiently strong shear in comparison with the ambientAlfv´en speed will produce the typical Kelvin-Helmholtz-type rollups. We show that, in principle,episodic switchbacks can be accounted for by in situ solar wind dynamics, not requiring nonradialinputs from the Sun or its inner corona.The heliospheric simulations that confirm the likelihood of such conditions in the solar wind areconsistent with a number of previous observations (Borovsky 2016; Usmanov et al. 2018; Horburyet al. 2018; Lockwood et al. 2019). Indeed the expected amplitude of turbulent fluctuations inferredfrom turbulence modeling suggests that large amplitude departures from the laminar state, includingswitchbacks, may be anticipated as
PSP perihelia migrate inward towards the Alfv´en critical zone.It is also important to recognize that the fluctuations that propagate and convect outward from thelower corona and into the critical zone from below may not be uniform or homogeneous. In fact, itis well established in observations that the vertical velocity of fluctuations may vary considerably,with typical variations due to type II spicules (De Pontieu et al. 2009) in the range of 50-100 kms − . Similar contrasts in radial velocity are seen throughout the inner corona in analysis of deepexposure STEREO-A /COR2 coronagraph images (DeForest et al. 2018). These types of fluctuationsmay also cause magnetic reversals (Samanta et al. 2019) that might propagate upwards and possiblysurvive to the Alfv´en critical zone. If they do survive to the Alfv´en critical zone, these fluctuationswould be expected, under the right detailed conditions, to contribute to driving of turbulence throughnonlinear instability.
PSP now has made several passes through the outer parts of this region, and provides substantialdata relevant to the present suggestions. Here we have examined the time series, and the distribu-tions of magnetic field magnitude, radial magnetic field and transverse velocities. All of these appearto be consistent with expectations and simulation results for a shear-driven dynamics scenario. Fur-thermore, the domains of Alfv´enic fluctuations are consistent with mixing layers that grow and/ormerge with distance from the Alfv´en critical zone, and we find anisotropy among the perpendicularmagnetic increments with stronger longitudinal increments at large scales, which seems unlikely toarise from deep in the solar corona but could be explained in terms of longitudinal velocity shearassociated with partial corotation.As an additional step to test this hypothesis, we carried out 3D high Mach number compressibleMHD simulations as driven by an initial planar velocity shear layer with a parallel sheared mag-7
Figure 19.
Images of plumes above smokestacks in Newark, Delaware, USA with escaping vapors exhibitinga sequence of changes analogous to what is envisioned for the shear driven dynamics of the young solar wind.Upon escape from the constraining smokestack, the plume is initially well-collimated. Roll-ups are initiatednear the edges due to shear. At greater distances the plumes become wider and more isotropic. netic field and a velocity contrast of three times the Alfv´en speed. Vortex rollup and nonlinearKelvin-Helmholtz activity is anticipated and observed. As expected, the magnetic field is deflected,sometimes through large angles. Reversals of the field direction, corresponding to the phenomenonof “switchbacks” in the
PSP observations, are seen with similar frequency in the simulations and inthe observations. The distributions of transverse velocity, radial magnetic field, and magnetic fieldmagnitude all show similarities between the simulation results and
PSP observations from the firstencounter.Several features of the
PSP observations are related to the familiar appearance of high cross-helicitystates (Alfv´enicity) in the inner heliosphere (Belcher & Davis Jr. 1971; Bruno & Carbone 2013), as wellas their familiar sense of polarization associated with a predominantly outward propagating character.Large-amplitude Alfv´en waves of pure polarization are required to be “spherically polarized” with themagnetic amplitude wandering on a constant magnitude surface (Barnes & Hollweg 1974). Ordinarilyone would associate such states with incompressibility. Indeed, even though the magnetosonic Machnumbers exceed unity here, the density fluctuations are observed to be small (Krupar et al. 2020),presumably because the Alfv´encity is large enough to prohibit proliferation of compressive modes.Furthermore we observed that the magnitude of the magnetic field is relatively constant in patches.This may be an additional indicator of large scale flux tube structures originating in the lower coronaand subsequent mixing layers, where transverse pressure balance is approximately realized due toquasi-two-dimensional turbulent relaxation (Servidio et al. 2008).Another interesting feature of the observations relates to the outward-propagating polarization ofthe cross-helicity, viewed in the context of the idealized configuration illustrated in Figure 3. Belowthe Alfv´en critical zone, substantial cross-helicity can be present in the vertical (axial) magnetic fieldof the flux tubes and the excess radial (axial) velocity found in some magnetic flux tubes. Both of8these features are inferred from coronagraph observations (DeForest et al. 2018). It is interesting thatto maintain a sense of outward propagation at the scale of the (model) flux tubes in a unipolar region,the faster flowing flux tubes must coexist with weaker axial magnetic fields, while flux tubes withslower radial speeds would have stronger radial magnetic fields. This sense of correlation remainsthe same whether the unipolar region has positive or negative radial magnetic field. This sense maydominate for other reasons; for example, strong closed fields may inhibit acceleration of the nascentsolar wind. Apparently the regions with opposite sense of polarization, i.e., those that correspond toinward propagation, are not present at significant levels during the
PSP first perihelion, which liesoutside the Alfv´en critical zone. This situation may change when
PSP passes through or below thecritical zone. For example, the inward-type modes may build up in the critical zone due to sharpAlfv´en speed gradients or due to stagnation.It appears that velocity differences between flux tubes may be available, under the right conditions,to drive large amplitude fluctuations that are responsible for the transition between striation in thesub-Alfv´enic inner corona and flocculation in the super-Alfv´enic outer corona. This phenomenonmay be characteristic of what we may reasonably call the “young solar wind”. The region in whichthis appears to occur is outside the Alfv´en critical zone and near the first β = 1 zone. This iswhere pressure fluctuations become large enough to overcome the rigidity of the magnetic field.Vortex rollups and large deflections or switchbacks of the magnetic field become possible. From thefirst detailed examination of the relevant evidence presented here, it appears that this hypothesis isreasonable, or at least not ruled out.The injection of additional turbulence energy due to shear-induced rollup may set the scale ofthe energy-containing eddies in the region of injection, thus determining the turbulence correlationlength observed from about 40 R (cid:12) outward to 1 au and beyond. In this regard the potential for asignificant additional injection of energy outside the Alfv´en critical zone may act as an “afterburner”that further boosts heating and acceleration.While the evidence summarized above appears to support the mixing layer hypothesis, it does notdiminish the potential importance of large amplitude fluctuations that originate in the lower corona,propagate outward, and survive into the region where mixing layer dynamics occurs. Such fluctuationscould be generated by field line stirring and reconnection in the photospheric “furnace” that producesbraiding of field lines, nanoflares, and a turbulent cascade that is probably responsible for heatingthe corona and accelerating the wind to supersonic and super-Alfv´enic speeds (McKenzie et al. 1995;Axford et al. 1999; Matthaeus et al. 1999; Cranmer et al. 2007; Verdini et al. 2010). There could alsobe ejecta from large scale interchange reconnection (Fisk & Kasper 2020) or wavelike fluctuationslaunched from spicules (Samanta et al. 2019). Some fluctuations of these types originating from loweraltitudes may also produce local large-angle magnetic deflections. We suspect that it will be difficultto rule out contributions to large angular deflections due to several potential sources. In any case,such fluctuations, upon arrival in the Alfv´en critical zone, would contribute to the perturbations thatunleash the nonlinear magnetized mixing layer phenomena that we describe here. Predictions for future perihelia.
As perihelia move closer and then enter the Alfv´en critical zone, weexpect to observe a further increase in both the mean magnetic field and the amplitude of broadbandturbulence. As we move closer to the region of “striations”, the more random fluctuations seen dueto rollups should give way to more organized patterns of near-radially aligned flux tubes. Thesestriations contain the velocity shears and magnetic shears that provide the energy for the rollups9further along. Approaching these more organized magnetic structures, we expect the frequency ofswitchbacks to decrease, and the sharpness of velocity jumps to increase as the sub-Alfv´enic region isapproached, assuring greater confinement, and the suppression of Kelvin-Helmholtz-like mixing layerdynamics. The amplitude and frequency of occurrence of large fluctuating tangential velocity shoulddecrease, while periods of corotation should become more frequent as the plasma at lower altitudescomes under increasing control of the more rigid lower coronal magnetic field. In this same region
PSP may begin to see other signatures, for example, component reconnection between adjacent striatedflux tubes, or indications of helical field lines within them. It is possible that if
PSP perihelia liedeep enough in the corona, the striated flux tubes may display properties such as Beltrami, Alfv´enic,and force-free signatures that are indicative of approach to a generalized relaxed state of turbulence(Servidio et al. 2008). This is directly analogous to our compressible MHD simulation results at veryearly times and the behavior of familiar smoke plumes near the confining smoke pipes as seen inFigure 19.In this paper we have presented a hypothesis regarding the role of shear driven dynamics in theregion currently explored by
PSP as well as supporting evidence. As a logical consequence, we developa set of expectations for what
PSP will observe as its perihelia explore deeper in the Alfv´en criticalzone and below. These predictions will soon be tested. In any case, approaching these regions for first-ever in situ observation,
PSP is expected to reveal important features of the plasma, electromagnetic,and energetic particle environment in the solar corona that shape the entire heliosphere.ACKNOWLEDGMENTSThe authors are grateful to Wiwithawin Charoenngam for plotting assistance. This work uti-lizes data produced collaboratively between AFRL/ADAPT and NSO/NISP. This research has beensupported in part by grant RTA6280002 from Thailand Science Research and Innovation and theParker Solar Probe mission under the ISOIS project (contract NNN06AA01C) and a subcontractto University of Delaware from Princeton University (SUB0000165). M.L.G. acknowledges supportfrom the Parker Solar Probe FIELDS MAG team. Y.Y. and M.W. acknowledge support from NSFC(Grants 11672123,11902138, and 91752201). Additional support is acknowledged from the NASALWS program (NNX17AB79G) and the HSR program (80NSSC18K1210 & 80NSSC18K1648).REFERENCES
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