Direct Measurement of the Solar-Wind Taylor Microscale using MMS Turbulence Campaign Data
Riddhi Bandyopadhyay, William H. Matthaeus, Alexandros Chasapis, Christopher T. Russell, Robert J. Strangeway, Roy B. Torbert, Barbara L. Giles, Daniel J. Gershman, Craig J. Pollock, James L. Burch
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Direct Measurement of the Solar-Wind Taylor Microscale using MMS Turbulence Campaign Data
Riddhi Bandyopadhyay, ∗ William H. Matthaeus,
1, 2
Alexandros Chasapis, Christopher T. Russell, Robert J. Strangeway, Roy B. Torbert, Barbara L. Giles, Daniel J. Gershman, Craig J. Pollock, andJames L. Burch Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA Bartol Research Institute, University of Delaware, Newark, DE 19716, USA Laboratory for Atmospheric and Space Physics, University of Colorado Boulder, Boulder, Colorado, USA University of California, Los Angeles, California 90095-1567, USA University of New Hampshire, Durham, New Hampshire 03824, USA NASA Goddard Space Flight Center, Greenbelt, Maryland 20771, USA Denali Scientific, Fairbanks, Alaska 99709, USA Southwest Research Institute, San Antonio, Texas 78238-5166, USA (Received; Revised; Accepted)
ABSTRACTUsing the novel Magnetospheric Multiscale (MMS) mission data accumulated during the 2019 MMSSolar Wind Turbulence Campaign, we calculate the Taylor microscale ( λ T ) of the turbulent magneticfield in the solar wind. The Taylor microscale represents the onset of dissipative processes in classicalturbulence theory. An accurate estimation of Taylor scale from spacecraft data is, however, usuallydifficult due to low time cadence, the effect of time decorrelation, and other factors. Previous reportswere based either entirely on the Taylor frozen-in approximation, which conflates time dependence,or that were obtained using multiple datasets, which introduces sample-to-sample variation of plasmaparameters, or where inter-spacecraft distance were larger than the present study. The unique config-uration of linear formation with logarithmic spacing of the 4 MMS spacecraft, during the campaign,enables a direct evaluation of the λ T from a single dataset, independent of the Taylor frozen-in ap-proximation. A value of λ T ≈ Keywords: turbulence, plasmas, solar wind INTRODUCTION: TURBULENCE SCALESTurbulence is a multi-scale phenomena. The turbulentsolar wind possesses structures and processes with broadrange of length scales (Verscharen et al. 2019). The dif-ferent characteristic length scales enter into the dynam-ics in various ways. For example, the correlation scalerepresents the sizes of the most energetic eddies (Smithet al. 2001). The mean-free path between collisionsdetermine the collisionality of the plasma. Proton ki-netic physics dominates near the proton inertial lengthand gyro-radius (Leamon et al. 1998); similarly elec-tron physics becomes important at the electron inertial [email protected] ∗ Now at: Department of Astrophysical Sciences, Princeton Uni-versity, Princeton, NJ 08544, USA length and gyro-radius (Alexandrova et al. 2012). Thesedifferent characteristic scales can provide useful infor-mation regarding the propagation of energetic particles,such as cosmic rays in the solar wind (Jokipii 1973).Of these various scales there are several related di-rectly to fundamental turbulence properties, and under-standing these in various space and astrophysical venuescontributes in the understanding of physical effects rang-ing from reconnection to particle heating and scattering.For an initial orientation, We can appeal to analogieswith hydrodynamics, to outline relationships that existamong these scales in classical turbulence. Accordingly,we use as a reference point the case in which the dissipa-tion is controlled by a simple scalar kinematic viscosity ν . We may begin with the scale at which the bulk ofturbulence energy resides, or is injected; we call this theenergy-containing scale λ c . For a turbulence amplitude a r X i v : . [ phy s i c s . s p ace - ph ] J un Bandyopadhyay et al. Z , with units of speed, one finds immediately a nonlin-ear time scale, or the eddy turnover time τ nl = λ c /Z .Smaller scale structures will have faster time scalesthat depend on their characteristics speeds. Using Kol-mogorov’s famous similarity hypothesis as a guide (Kol-mogorov 1941), we may estimate the speed of structuresat smaller scales (cid:96) to be Z (cid:96) ∼ (cid:15) / (cid:96) / = τ nl ( (cid:96)/L ) / ,where we also use the de K´arm´an & Howarth (1938)estimate of the decay rate (cid:15) ∼ Z /λ c . probing at stillsmaller scales, very much smaller than λ c , eventuallydissipative processes of viscous origin become important.As a first approximation one may estimate the time scalefor dissipation of a structure (e.g., a vortex) at scale (cid:96) tobe τ diss = (cid:96) /ν , using standard viscous dissipation as amodel. A reasonable way to estimate the characteristicscale at which dissipation becomes dominant is to askwhen the eddy or structure at scale (cid:96) become criticallydamped. This occurs when the intrinsic nonlinear timebalances the local-in-scale dissipative time. For this,we solve τ nl ( η ) = τ diss ( η ) finding η = λ c ( ν/Zλ c ) / = λ c Re − / . The scale η is often called the Kolmogorovdissipation scale, and we recognize standard definition ofthe large scale Reynolds number Re ≡ Zλ c /ν . Note thatif the critically damped scale is known or estimated, asit might be in a plasma identified with ion inertial scalefor example, then the effective Reynolds number may bedefined as R eff = ( λ c /η ) / , or R eff = ( λ c /d i ) / if dissi-pation is presumed to become dominant over nonlineareffects at scales comparable to the ion inertial length d i .Yet another scale, generally intermediate to λ c and η may be defined by equating the large scale eddy turnovertime to the scale-dependent dissipative time. Thus, τ nl = τ diss ( (cid:96) ) is solved by a particular value (cid:96) = λ T . Thislength scale is the Taylor microscale , the subject of thepresent paper. The first of its several equivalent defini-tions highlights a particular physical property, namelythat it is critically damped at the large scale nonlin-ear time. Before turning to its evaluation in the MMSTurbulence Campaign, we introduce and discuss severaladditional properties of the Taylor scale λ T . TAYLOR MICROSCALELike the majority of concepts in plasma turbulence,the Taylor scale is also borrowed from hydrodynamicturbulence research. The Taylor scale can be viewed asthe measure of curvature of the autocorrelation function( R ( r ) = (cid:104) F ( x ) · F ( x + r ) (cid:105) ) at the origin; for isotropy, λ = R (0) R (cid:48)(cid:48) (0) , (1)where F is the fluctuating field of interest, e.g., veloc-ity field ( v ) in hydrodynamic turbulence, or magnetic field ( b ) in magnetohydrodynamic (MHD) and plasmaturbulence. Here, we consider only the magnetic fieldfluctuations. For small lags r , using R ( r ) = R ( − r ) , arequirement of statistical homogeneity, the autocorrela-tion function near the origin can be Taylor expanded,assuming isotropy, as R ( r ) = 1 − r λ + · · · , (2)Another physically revealing way to view the Taylormicroscale is obtained by noting that for viscous-like ν dissipation in an incompressible medium, the Taylorscale is also related to dissipation, in thatd (cid:104)| b | (cid:105) d t = ν (cid:104)|∇ × b | (cid:105) = ν (cid:104)| b | (cid:105) λ . (3)In this sense, the Taylor scale is the “equivalent dis-sipation scale,” so that, at any instant of time, the dis-sipation rate is the same if all the energy were at theTaylor scale. This is, in fact, the physical basis uponwhich Taylor (Taylor 1935) first formulated the ideaof this particular length scale. Some older turbulencetexts (Hinze 1975) refer to the Taylor scale as “the dis-sipation scale,” although later Kolomogrov (Kolmogorov1941) introduced the similarity variable ( η ), now knownas the Kolmogorov length scale, to denote the scale atwhich eddies become critically damped. Notionally, theTaylor microscale represents the largest eddies in thedissipation range, or equivalently, the smallest eddies inthe inertial range. The interpretation, of course, maynot be so straightforward in plasma turbulence. How-ever, one may draw some conclusions by analogy withhydrodynamics.Keeping this parallel in mind, we recall that, indeed,space plasma observations show that the transition ofKolmogorov -5/3 spectra to a steeper slope occur atsomewhat larger scales than ion-inertial scale or protongyro radius (Leamon et al. 2000). In classical hydrody-namic turbulence, the Taylor scale is greater than theKolmogorov length scale ( η ). Therefore, if one treats theion-inertial length or the proton gyro radius as equiva-lent to the classical-turbulence Kolmogorov scale, wheredissipative (or kinetic) processes become dominant, theTaylor scale provides a natural descriptor of the slightsteepening of spectra, and the onset of dissipation, priorto dissipation becoming dominant at still smaller scales.This also fits well with the idea, from reconnection stud-ies, that intense kinetic activity in current sheets is ini-tiated at some multiple of the ion scales (Shay et al.1998) LIMITATIONS aylor Microscale: MMS Turbulence Campaign λ T , requires measurementof the curvature of the correlation function near theorigin , precisely demanding such information regard-ing small spatial range fluctuations. But simultaneousmulti-point data, especially at small separation, havegenerally not been available. Even when multi-pointmeasurements have been obtained, those intervals aretypically not very long (e.g., Roberts et al. 2015; Chas-apis et al. 2017). For statistical studies of turbulencein the solar wind, continuous intervals of duration ofat least a few hours, corresponding to at least a fewspacecraft-frame correlation times, are desirable (Isaacset al. 2015). Consequently, previous studies were forcedto perform analyses compiled from a number of differentintervals. At that point, additional uncertainty is intro-duced by variation of plasma conditions from interval-to-interval. The first MMS mini campaign, named the MMS Solar-Wind Turbulence Campaign , explicitly over-comes these limitations, as discussed in the next section.However, we note that the data studied are also limitedin that there is one sampling direction i.e., all spacecraftare in a line. Although four-point measurements providemany advantages, this and other related limitations areintrinsic to four-point measurements. MMS SOLAR-WIND TURBULENCE CAMPAIGN
Magnetopause Bow shock MMSlocation
Figure 1.
Top: Configuration of the 4 MMS spacecraft andthe flow speed direction during the MMS turbulence cam-paign. Bottom: Location of the MMS, along with the nomi-nal magnetopause and bow-shock locations. The GeocentricSolar Ecliptic (GSE) (Franz & Harper 2002) coordinate sys-tem is used, in which the XY-plane is defined by the Earthmean ecliptic and the +X-axis is defined by the Earth-Sunvector.
The Magnetospheric Multiscale (MMS) mission, waslaunched in 2015 with the primary goal of studying mag-netic reconnection a process responsible for releasingmagnetic energy into flows and internal energy. Thefour MMS spacecraft are equipped with state-of-the-art instruments with unprecedented resolution. DuringFebruary 2019, the MMS apogee was raised to ∼
27 R E on Earth’s dayside of the magnetosphere and outside theion foreshock region. This orbit allowed the spacecraftto sample the pristine solar wind, outside the Earthsmagnetosheath and far from the bow shock, for extendedperiods of time (see Fig. 1).During the first mini campaign, the four MMS space-craft were arranged in a “string of pearls” or “beads on Bandyopadhyay et al.
Figure 2.
Time series of solar wind observations from16:39:00 - 21:41:00 UTC on 24 Feb 2019. Top panel: GSEcomponents ofthe MMS1/FGM magnetic field, second panel:Proton number density measured by MMS1 and Wind, thirdpanel: X component of the proton velocity, in GSE coordi-nate system, measured by MMS1 and Wind, bottom panel:Proton temperature measured by MMS1 and Wind. string” formation instead of the usual tetrahedral forma-tion. With the spacecraft baseline almost perpendicularto the solar wind flow, the spacecraft were separatedby logarithmic distances ranging from 25 to 200 km,and the baseline separations remain unchanged within10%. This configuration allows direct investigation ofthe scale-dependent nature of the solar wind structuresnear proton scales. Laboratory experiments have uti-lized such formations (Cartagena-Sanchez et al. 2019),but this kind of data are novel in observations. Thiswork is the first of several studies undertaken to takeadvantage of this unique configuration (see also Chas-apis et al. (2020)). Although not relevant for this study,the spacecraft spin-axis were tilted about 15 ◦ to obtainimproved electric field measurements in the solar wind.A schematic configuration of the four MMS spacecraftin the solar wind, during the Turbulence Campaign, isprovided in the top panel of Fig. 1. The orbital con-text plot showing MMS location relative to the nominalmagnetopause (Shue et al. 1998) and bow shock (Farris& Russell 1994), is illustrated in the bottom panel ofFig. 1. MMS DATADuring the three-week long mini campaign, a num-ber of useful solar wind and foreshock intervals were se-lected. The longest of the selected pristine solar wind in-tervals, a continuous interval of five hours of burst-modedata, on 24 February 2019, from 16:39:00 to 21:41:00UTC, is analyzed in this paper. No signature of reflectedions from the bow shock is found. For this interval, we
Table 1.
Parameters for MMS interval on 24 February 2019,from 16:00 to 21:00 UTC (5 hours). Quantities with an as-terisks ( ∗ ) have been estimated using Wind data, and forthose, the MMS estimates are given in parenthesis.Solar-wind speed V SW = 322 km s − Correlation Length λ c = 3 . × kmIon inertial length d i = 91 kmIon gyroradius ρ i = 64 ∗ (150) kmElectron inertial length d e = 2.3 kmDebye length λ D = 10 mProton beta β i = 0.5 ∗ (2.5)Magnetic field B = |(cid:104) B (cid:105)| = 3.4 nTMagnetic-field fluctuation B rms /B = 0.72Proton density (cid:104) N i (cid:105) = 6.2 cm − Proton temperature (cid:104) T i (cid:105) = 2.5 ∗ (12.4) eV did not detect any high-frequency waves characteristic ofthe foreshock. We note, however, that the other 5-hourinterval (17 February 2019, from 11:24:00 to 16:24:00UTC) chosen as a part of the turbulence campaign, hasforeshock signatures, and consequently that interval wasnot considered for this analysis.To evaluate the magnetic field Taylor microscale fromtwo-spacecraft correlation data, We employ data fromthe Fluxgate Magnetometer (FGM) aboard each each ofthe four MMS spacecraft(Russell et al. 2016). The toppanel of Fig. 2 shows the three Cartesian componentsof magnetic field in the GSE coordinate system (Franz& Harper 2002), recorded in this period by the FGMonboard MMS1. It is apparent that this interval is richin structures, including numerous current sheets, fluxtubes, and broad band random fluctuations - taken to-gether these represent a fairly typical sample of solarwind turbulence (Bruno & Carbone 2005).The Fast Plasma Investigation (FPI) (Pollock et al.2016) instrument measures proton and electron distri-bution functions and moments every 150ms and 30ms,respectively. Due to the limitations of the FPI instru-ments in the solar wind (Bandyopadhyay et al. 2018a),some systematic uncertainties remain in the moments,and more so in the higher-order moments. Therefore,we cross-check the proton moments in the selected in-terval with Wind (Ogilvie et al. 1995; Lepping et al.1995) data, time-shifted to the bow-shock nose. TheMMS and Wind estimates of proton density, velocity(X GSE component), and temperature are shown in thebottom three panels in Fig. 2. The density and velocityare in adequately close agreement, but significant dis-crepancies exist in the proton temperature values. TheFPI estimates of temperature are significantly greaterthan the Wind values. Given the known limitations of aylor Microscale: MMS Turbulence Campaign Figure 3.
Spectral power density of magnetic field mea-sured by MMS1. Kolmogorov scaling ∼ f − / is shown forreference. The vertical lines represent the correlation length( kλ c = 1), the ion-inertial length ( kd i = 1), and the ion gyro-radius ( kρ i = 1), with wavenumber k (cid:39) (2 πf ) / |(cid:104) V (cid:105)| . Thepart of the spectrum where the signal-to-noise ratio decreasesbelow ∼
5, is grey-shaded, to indicate that this region is noisedominated. Note that the flattening in the high-frequencyrange ( f (cid:38) FPI in the solar wind, we use the Wind measurements oftemperature to evaluate proton beta and other relevantparameters. The average values of the plasma parame-ters are reported in Table 1.Fig. 3 shows spacecraft-frame frequency spectrum ofthe magnetic field during this period. A clear Kol-mogorov scaling ( ∼ f − / ) can be seen at scales smallerthan the correlation length, λ c (inferred from the Tay-lor hypothesis). A break in spectral slope from ∼ f − / to ∼ f − / is observed near (inferred) kinetic scales ( d i or ρ i ). Often these scales are associated with the dis-sipation scale ( λ diss ) in collisionless plasmas, equivalentto Kolmogorov scale ( η ) in classical turbulence. Kineticdissipative processes, such as wave damping, are effec-tive in these small plasma kinetic scales. For example,Leamon et al. (2000) and Wang et al. (2018) argued thatthe ion inertial scale controls the spectral break and on-set of strong dissipation, while Bruno & Trenchi (2014)suggested the break frequency is associated with the res-onance condition for parallel propagating Alfv´en wave.Another possibility is that the largest of the proton ki-netic scales terminates the inertial range and controlsthe spectral break(Chen et al. 2014). The flatteningnear f (cid:38) TAYLOR MICROSCALE: RESULTS
Figure 4.
Magnetic-field correlation function based onfrozen-in approximation (green, solid line) and obtained fromtwo-spacecraft evaluation (red, cross symbols). An exponen-tial fit (blue, dashed line) to the single-spacecraft measure-ment is used to obtain the correlation length. A quadratic fit(black, thin line) to the multi-spacecraft points estimates theTaylor scale. The inset plots part of the correlation functionenlarged near the origin to clearly show the multi-spacecraftpoints and the parabolic fit.
To estimate the Taylor scale from this interval, werecall the approximation near the origin: R ( r ) ≈ − r λ , (4)where the higher-order terms are neglected. Therefore,one may obtain the Taylor microscale by fitting the au-tocorrelation function R ( r ) to a parabolic curve at theorigin. Clearly, the quadratic approximation holds bet-ter as one asymptotically approaches smaller values of r . Previous multi-spacecraft estimates (Matthaeus et al.2005) were evaluated with the Cluster spacecraft, withseparations in the range 150 km ≤ r ≤
270 km. Here,we extend that analysis by approaching the origin closerby about an order of magnitude, with 25 km ≤ r ≤ λ T by fitting the the six availabletwo-point correlation function to a parabolic curve. Theresulting value of the Taylor scale is λ T = 6933km. Forcomparison, we also show the single-spacecraft, frozen-in hypothesis based evaluation of the correlation func-tion. Evidently, the single-spacecraft estimate decaysmuch rapidly closer to the origin, presumably due totime decorrelation of the solar wind fluctuations in thosescales (Matthaeus et al. 2010). At large lags, however, Bandyopadhyay et al. the frozen-in based correlation function exhibits approx-imately exponential decay and provides a satisfactory es-timate of the correlation length, about 320 , λ T since thecurvature at the origin is undefined. Solar wind viscosity – An accurate estimation of theTaylor scale also permits an evaluation of an effectiveviscosity (or, turbulent viscosity/resisivity) of the solarwind, according to the expression, ν = (cid:15) E λ T , (5)where (cid:15) is the cascade rate ≈ − and E is thefluctuation energy per unit mass. The cascade rate canbe obtained, for example, from the third order law orother estimates, see e.g., Verma et al. (1995); Sorriso-Valvo et al. (2007); MacBride et al. (2008); Bandyopad-hyay et al. (2018b). Putting in the rest of the values: λ T = 6933 km and 2 E = (cid:104)| b | (cid:105) = 324 km s − , we ob-tain ν ≈
150 km s − . This value is considerably largerthan the one obtained using Braginskii (Braginskii 1965)formalism, which is based on simple particle-particlecollision (Montgomery 1983). Our result also pro-vides improvement on earlier indirect estimates, basedon turbulence-cascade phenomenology (Coleman 1968;Verma 1996). DISCUSSION AND CONCLUSIONSIn this paper, MMS data accumulated during the tur-bulence campaign have been used to evaluate the Taylormicroscale of magnetic field fluctuations using a multi-spacecraft technique, and taking advantage of a uniquebeads-on-a-string flight formation. The previous esti-mate by Matthaeus et al. (2005), using Cluster data,is λ T = 2478 km, which is about 3 times smaller thanthe present evaluation. The deviation is possibly dueto the relatively larger spacecraft separation used in theCluster data set, comparatively shorter intervals, andmixing of different solar wind intervals. It is also possi-ble that this level of variability is intrinsic to the solarwind for a variety of reasons including 1 /f noise, streamstructure (Matthaeus & Goldstein 1986). Another possi-bility is that the differences may be attributable to thedifferences in the formation of the Cluster and MMSspacecraft. The Cluster spacecraft pair baselines arein a tetrahedron, introducing anisotropy effects, whichare not present in the MMS linear formation analyzedhere. These limitations are inherent to four-point mea-surements, and can be overcome by large constellations of simultaneous, in-situ measurements (Matthaeus et al.2019; Klein et al. 2019; TenBarge et al. 2019).We note here that the two-spacecraft data points covera very small range, in contrast with the frozen-in basedcorrelation function (compare the inset in Fig. 4 to themain plot). We, however, do not expect any weaknessin the analysis due to this point. The estimation of cor-relation length by an exponential approximation is onlyvalid at long lag, while the quadratic approximation tothe expansion of the correlation function is expected tohold only near the origin. Therefore, the small cover-age of scale is not expected to hinder the Taylor-scaleestimation.We find that the break frequency, in the magnetic-fieldspectrum, is situated between the Taylor scale ( λ T ) anddissipative scales ( d i , ρ i ). Wang et al. (2018), showedthat for β ∼ d i than ρ i , and it is insensitiveto β -values. In general, λ T /λ diss > d i or ion gyro-radius ρ i with the dissipation scale, then wefind that λ T /λ diss ≈
70. The relationship of Taylor scaleto ion kinetic scales in the solar wind however, appearsto be much more variable than it is in hydrodynamics,and in particular the relationship has been found to de-pend on the turbulence cascade rate (Matthaeus et al.2008).The present paper is a step in a broad progressionof interplanetary measurements of fundamental plasmaturbulence properties. The 1980 NASA Plasma Turbu-lence Explorer Panel emphasized the need for simulta-neous multi-point measurements, in particular, plasmaand magnetic field measurements, to make progress inthis area (Montgomery & et al. aylor Microscale: MMS Turbulence Campaign A. MEASURE OF CONFIDENCE IN THE TWO-SPACECRAFT ANALYSESAs can be seen in Fig. 4, the signals between spacecraft are strongly correlated; however, the differences are verysmall between them. The very small variation in the 2-spacecraft correlations, between 1 and 0 . b , R ( r ) = (cid:104) b ( x ) · b ( x + r ) (cid:105) . (A1)Keeping in mind that the uncertainty in FGM magnetic field measurements are less than δb = 0 . b ∼ b ( x ) · b ( x + r ), is less than 2 δb/b ∼ .
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