DRAFT VERSION DECEMBER 14, 2017Preprint typeset using LATEX style emulateapj v. 01/23/15

TRANSPORT OF INTERNETWORK MAGNETIC FLUX ELEMENTS IN THE SOLAR PHOTOSPHERE

PIYUSH AGRAWAL1,2 , MARK P. RAST1,2 , MILAN GOSIC3 , LUIS R. BELLOT RUBIO3 , AND MATTHIAS REMPEL4

1Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO 80309, USA; piyush.agrawal@colorado.edu2Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO 80303, USA; mark.rast@lasp.colorado.edu

3Instituto de Astrofısica de Andalucıa (CSIC), Apdo. de Correos 3004, E-18080 Granada, Spain and4High Altitude Observatory, National Center for Atmospheric Research, Boulder, CO 80307, USA

Draft version December 14, 2017

ABSTRACTThe motions of small-scale magnetic flux elements in the solar photosphere can provide some measure of theLagrangian properties of the convective flow. Measurements of these motions have been critical in estimatingthe turbulent diffusion coefficient in flux-transport dynamo models and in determining the Alfven wave ex-citation spectrum for coronal heating models. We examine the motions of internetwork flux elements in a 24hour long Hinode/NFI magnetogram sequence with 90 second cadence, and study both the scaling of their meansquared displacement and the shape of their displacement probability distribution as a function of time. We findthat the mean squared displacement scales super-diffusively with a slope of about 1.48. Super-diffusive scal-ing has been observed in other studies for temporal increments as small as 5 seconds, increments over whichballistic scaling would be expected. Using high-cadence MURaM simulations, we show that the observedsuper-diffusive scaling at short temporal increments is a consequence of random changes in the barycenterpositions due to flux evolution. We also find that for long temporal increments, beyond granular lifetimes, theobserved displacement distribution deviates from that expected for a diffusive process, evolving from Rayleighto Gaussian. This change in the distribution can be modeled analytically by accounting for supergranular ad-vection along with motions due to granulation. These results complicate the interpretation of magnetic elementmotions as strictly advective or diffusive on short and long timescales and suggest that measurements of mag-netic element motions must be used with caution in turbulent diffusion or wave excitation models. We proposethat passive trace motions in measured photospheric flows may yield more robust transport statistics.Keywords: Sun: photosphere — Sun: granulation

1. INTRODUCTION

The motions of small-scale magnetic flux elements in thesolar photosphere are largely determined by plasma flows.Studying these motions can contribute to our understandingof the Lagrangian dynamics in the radiative boundary layerof the highly turbulent solar convection zone. This in turncan inform models of coronal heating by Alfven waves, sincethe spectrum of those waves depends on the ‘footpoint’ mo-tions (e.g., Cranmer & van Ballegooijen 2005; van Ballegooi-jen et al. 2014; Van Kooten & Cranmer 2017). Additionally,flux-transport models of the solar dynamo rely on cross equa-torial reconnection of the opposite polarity field along withpoleward transport of the residual to reverse the sign of theglobal field every half cycle period (e.g., Babcock & Babcock1955; Wang et al. 1989; Dikpati & Gilman 2007; Jiang et al.2014). These processes are often modeled as due to the com-bined action of meridional flow and supergranular diffusion.

For low molecular diffusivities, turbulent transport in thecontinuum approximation can be described in terms of La-grangian parcel motions (e.g., Toschi & Bodenschatz 2009).This approach faces some challenges in the context of so-lar magnetic flux elements because the magnetic field motionis not strictly passive but back-reacts on the flow, the two-dimensional motions observed in the photosphere representsome unknown average of the flow over the range of depthsto which the field extends, and the elements themselves havefinite lifetimes. Of these, the first effect may be small be-cause the ratio of plasma to magnetic energy density in thequiet-sun photosphere is large, the second may contribute tothe observed field strength dependence of the element motions(e.g., Hagenaar et al. 1999; Yang et al. 2015), and the last may

make the interpretation of element transport as a diffusive pro-cess more challenging (e.g., Yuste et al. 2013), though this is-sue has not yet been studied in the solar context. While it isimportant that these difficulties be examined in future work,here, as a first approximation and in common with most previ-ous studies, we treat the magnetic flux elements as Lagrangiantracers to understand the implications of their motions underthat assumption.

In that context, transport is usually characterized by howthe mean squared displacement of the flux-weighted barycen-ter of magnetic elements scales with time (e.g., Lawrence &Schrijver 1993), 〈r2〉 ∝ tγ , where r is the Lagrangian dis-placement of each element over a temporal increment t, andγ is the inferred scaling exponent. The temporal increment,or time interval, refers to the time elapsed since any momentalong the trajectory of a flux element, not just the time sinceits emergence. Thus, for a given temporal increment, multi-ple displacement measurements are possible when the trajec-tory spans a length of time longer than the temporal incre-ment being considered. For very short time intervals, belowthe Lagrangian integral time (the autocorrelation time of thevelocity along a parcel trajectory), the motion of any singleflux element is expected to be highly correlated and the meansquared displacement of all the elements should scale ballis-tically (γ = 2). Subsequently, as the Lagrangian motionsde-correlate, both because the trajectories spatially sample awider range of the flow, which has a finite spatial correlationlength, and because the flow itself temporally de-correlates,an intermediate value of 1 < γ < 2 is expected. This scal-ing is sometimes referred to as super-diffusive. Finally, overintervals longer than the Eulerian integral time (the autocor-

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relation time of the velocity at a fixed point), the Lagrangianmotions are expected to be fully de-correlated, and the dis-placements of the flux elements should display a random walkwith diffusive scaling γ = 1. Once measured, γ can be used tocalculate an effective diffusion coefficient (Abramenko et al.2011), but, as we shall see, the physical interpretation of theprocess as diffusion may be problematic.

Table 1 summarizes the measured values of γ from re-cent studies, where the studies are limited to those of in-ternetwork elements, the focus of this paper. The valuesof γ vary depending on the data set examined and the fea-ture tracking algorithm employed, but what is peculiar is thatthe scaling, even at very short temporal increments, is super-diffusive (e.g., Abramenko et al. 2011; Chitta et al. 2012; Ja-farzadeh et al. 2014), even though one would expect theseincrements to be significantly shorter than both the Eulerianand Lagrangian integral times.

The scaling of the mean-squared displacement with timeprovides only limited information about the underlying flowswhich guide the motion of the flux elements. The probabilitydistribution of the Lagrangian displacements is more sensitiveto the flow properties and can capture some of the effects ofturbulent intermittency (Rast et al. 2016). For very short tem-poral increments, over which the mean squared displacementof the flux elements should scale ballistically, the probabilitydistribution of the displacements should reflect the underly-ing Lagrangian velocity distribution. As the motions of theflux elements de-correlate and the scaling becomes diffusive,the probability distribution should, for a two-dimensional mo-tion, approach a Rayleigh distribution (the distribution wouldbe Maxwellian for three-dimensional motions). We find thatdisplacement distributions of internetwork magnetic elementsdo not behave as expected.

In §3 we examine the displacement probability distributionof magnetic elements as a function of time. While the dis-tribution approaches Rayleigh as the trajectories de-correlateover granular lifetimes, at longest intervals (t & 2 hours), itsurprisingly becomes Gaussian. With the help of a simplecorrelated random walk model with drift motion we demon-strate that this is likely due to the presence of an underly-ing large-scale supergranular flow which dominates granu-lar motions over long timescales. We expect that transportby supergranular motions on timescales long compared totheir lifetimes would be similarly affected by the underlyingmeridional flow. The multiscale and intermittent nature of theflows leads to flux transport that cannot be described as eitherstrictly diffusive or purely advective, scaling is thus subdom-inant (Rast & Pinton 2011). In §4 we revisit the observeddiscrepancy in scaling at shortest times and, with the help ofa radiative magnetohydrodynamic simulation of solar granu-lation, show that the observed super-diffusive scaling is likelyan artifact of random changes in the barycenter positions ofthe magnetic elements, induced by rapid changes in their fluxcontent or configuration.

2. OBSERVATIONAL DATA AND RESULTS

To investigate the transport of small-scale internetworkmagnetic flux elements, we analyzed observations obtainedwith the Narrowband Filter Imager (NFI; Tsuneta et al. 2008)on-board the Hinode spacecraft (Kosugi et al. 2007). The datasequence is a part of the Hinode Operation Plan 151 (HOP151). The measurements are well suited to study the quiet-sun magnetic fields on temporal scales from minutes to daysbecause of their high spatial and temporal resolution, flux sen-

Table 1Recent work on the transport of internetwork magnetic flux elements.

Author Instrument Cadence (s) γa

Abramenko et al. (2011)b BBSO/NST 10 1.48-1.67Chitta et al. (2012)b SST/CRISP 5 1.59Giannattasio et al. (2014a)c Hinode/SOT 90 1.55Giannattasio et al. (2014b)c Hinode/SOT 90 1.44Jafarzadeh et al. (2014)b Sunrise/SuFI 3-12 1.69Manso Sainz et al. (2011)c Hinode/SOT 28 0.96, 1.70

This workc Hinode/NFI 90 1.48aScaling exponent γ computed using bmagnetic bright points andcmagnetic flux elements.

Figure 1. Sample Hinode/NFI magnetogram saturated at ±30 Gauss. Thered circle marks the boundary of the internetwork region. Colored curves arethe trajectories of individual flux elements.

sitivity, and long duration.The data are an uninterrupted time sequence of magne-

tograms, about 24 hours long, starting at 08:32:00 UT on 23November 2010, taken with 90 second cadence. They covera field of view of about 41 × 46 Mm2, with 0”.16 (116 km)pixel size, and were constructed using Stoke I and V mea-surements ±160 mA from the 589.6 nm Na I D1 line corecenter. Post processing of the data and removal of the p-modesignal by the application of a subsonic filter (Title et al. 1989;Straus et al. 1992) yields magnetograms with a noise level of4 Gauss, an estimate based on the standard-deviation of pix-els with no clear magnetic signal. More details regarding dataprocessing and calibration can be found in Gosic et al. (2014).

The magnetogram sequence employed in this study sam-ples the quiet sun at disk center, capturing the spatio-temporalevolution of a single supergranule and its surroundings. A sin-gle frame from the time series is shown in Figure 1. The redcircle, with radius 9.3 Mm (corresponding to 0.8 times thatof the supergranular cell (Gosic et al. 2016)), outlines the in-ternetwork region within which flux elements were identifiedand tracked. The multi-color curves indicate the trajectoriesof some representative flux elements.

2.1. Tracking AlgorithmWe used a semi-automatic procedure to determine the tra-

jectories of the magnetic flux elements. Identification and

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tracking were carried out automatically, but the results wererevised manually at each time-step to verify the performanceof the code. In cases of element misidentification or interac-tion (merging, cancellation or fragmentation), the code out-put was corrected and the tracking continued from that stepuntil precise trajectories were derived. The feature identifi-cation algorithm employed the clumping method of Parnellet al. (2009), with a minimum unsigned flux density of 12Gauss (3 times the noise level) and a minimum element areaof 4 pixels. This yielded elements with a mean unsigned fluxdensity of about 23 Gauss. To identify individual elementsduring interactions we used the downhill method of Welsch& Longcope (2003), which allowed us to maintain their iden-tification over long periods of time. Our tracking approachthus overcomes some of the difficulties faced by standard al-gorithms during element interactions (e.g., Gosic et al. 2014,2016). The tracking was repeated twice to check for consis-tency, and nearly identical results were obtained each time.Further details can be found in Gosic (2012)).

For the analysis that follows, we restrict identification toelements whose trajectories begin within the internetwork re-gion defined by the red circle in Figure 1. Further, elementlifetimes ranged from 1.5 minutes to 5.5 hours, and we re-tained for analysis only those with lifetimes ≥ 4.5 minutes.This yields a total of 6463 unique magnetic flux element tra-jectories over the 958 magnetogram sequence. The changein flux-weighted barycenter positions of these elements as afunction of time forms the basis for the displacement statis-tics analyzed below.

2.2. Mean Squared DisplacementThe mean squared displacement of the flux elements is plot-

ted as a function of temporal increment in Figure 2 (red stars).It approximates a power law with a super-diffusive scalingexponent, γ = 1.48. This is consistent with previous mea-surements (Table 1), the value falling in-between ballistic anddiffusive values. There is some deviation from the power-lawbehavior for time intervals greater than about 2 hours whichexceeds the displacement variance. In §3.2 we suggest thatthis deviation at long times reflects advection by the underly-ing supergranular flow, for which there is strong evidence inthe displacement distributions.

2.3. Displacement Probability DistributionsWhile the mean squared displacement of the flux elements

closely follows a power-law for time intervals below 2 hours(Figure 2), the underlying probability distribution of the dis-placements evolves over this range of increments. The leftcolumn of Figure 3 shows the observed distributions for fourdifferent temporal increments, t = 1.5, 6, 45 and 120 min-utes, along with the best-fit Rayleigh and Gaussian functions(blue (dashed) and green (solid) curves, respectively). Forshort increments, t = 1.5 minutes (Figure 3a), the distri-bution is neither Rayleigh nor Gaussian, but shows an el-evated probability for large displacements. In §4 we arguethat this is likely due to the apparent motion of the magneticelement barycenters when they are subject to flux evolutionand element identification uncertainties. For intermediate in-crements, t = 6 minutes (Figure 3b), the distribution ap-proximates a Rayleigh distribution because the motions havelargely de-correlated. Somewhat surprisingly, for longer tem-poral increments the distribution does not remain Rayleigh(t = 45 minutes, Figure 3c) but instead becomes nearly Gaus-sian (t = 120 minutes, Figure 3d). We show below that a

Figure 2. Mean squared displacement as a function of temporal incrementfor the magnetic flux elements in Hinode/NFI data (red stars) and for therandom walk model (blue squares) of §3.2. The best-fit slope for the Hin-ode/NFI data is 1.48 (red dashed line), computed from fitting the data below2 hours. The scaling for the model is 1.97 and 1.96 (blue dashed lines) atthe shortest and the longest temporal increments, respectively. The verticalblack dashed lines correspond to the increments for which the displacementprobability distributions are shown in Figure 3. For both observations and themodel, the variance of the squared displacement at each temporal incrementdoes not exceed the size of the plotting symbol.larger scale drift component (likely originating with super-granulation in the observations) superimposed on the granularmotion can explain both this change and the deviation frompower-law scaling at long intervals.

3. MODEL: CORRELATED RANDOM WALK WITH DRIFT

3.1. Model DefinitionUnder the assumption that magnetic flux elements are ad-

vected passively by the underlying plasma flow, we model theeffects of granular and supergranular flows on magnetic ele-ment motions as random walk and drift contributions to themotion of Lagrangian ‘walkers’. The effect of granular flowis modeled as a correlated random walk, a random walk withan imposed correlation in the step direction. The random walksteps are of size vg ∆t, where both vg and the step interval ∆tare prescribed constants. The direction of travel of a walker istaken to be θ + ∆θ, where θ is the direction of motion duringthe previous step and ∆θ, computed anew for each time-step,is a uniformly sampled random variable between −π/C andπ/C. The parameter C constrains the step direction (inducesmemory) and thus controls the time it takes for the motionsto de-correlate. For C = 1, ∆θ ∈ [−π, π) and the motion isdelta-correlated (step direction de-correlates after one time-step). If C is large, the change in direction of the walker (∆θ)at each time-step is small, and the motion stays correlated fora longer time. When interpreting observations in light of thismodel, the parameter C is chosen so that the random walkcomponent of the motion de-correlates over a granular life-time.

To keep the number of free model parameters to a mini-mum, the supergranular contribution to the magnetic elementmotions is modeled as a uniform constant drift in a single di-rection. This approximation is reasonable as long as the ele-ments being tracked are internetwork elements (as is the casefor our data) and the supergranular flow is approximately spa-tially uniform and steady over any individual granulation in-duced random walk trajectory. This is true if the supergranularmotions are directed approximately radially away from their

4 AGRAWAL ET AL.

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Figure 3. Displacement probability distributions for Hinode/NFI data (left column) and for the random walk model (right column) of §3.2, at different temporalincrements, are shown here. The red stars are the distribution values, the blue (dashed) and the green (solid) curves refer to the best-fit Rayleigh and Gaussiancurves, respectively. Note that only ∼ 2% of the flux elements survive to contribute to the Hinode/NFI data distribution at 120 minutes, and since the bin size isheld constant at about 0.13 Mm, the histogram is noisy.

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centers and, over the lifetime of an individual flux element,the granular motions do not induce a significant deviationfrom that radial direction. We can determine if our model isself-consistent by checking if these conditions are met for theparameters that approximate granular and supergranular con-tributions. For these (see §3.2), we find that the average devi-ation of a random walk trajectory from the direction of driftis about ±2.3◦ over the time it takes for the drift componentto carry the walkers 30 Mm (the characteristic supergranularlength scale, Rieutord & Rincon (2010)). Thus, motion due togranulation approximately samples only a small region aboutthe mean supergranular drift direction, though radial gradientsin the supergranular flow may pose difficulties for modelingvery long-lived element displacements that traverse the fullsupergranular extent. The same drift can be applied to eachrandom walk realization, even if in reality the supergranularflow carries each magnetic element in a radially different di-rection, because the granular component carries the individualwalkers in all directions, and only the distance traveled, notthe direction of travel, is of interest to the displacement statis-tics. In practice, when constructing the two-dimensional cor-related random walk model, we choose the drift to act in thepositive x-direction and vectorially add vsg ∆t to the corre-lated random walk component vg ∆t at each time-step, wherevsg is the drift velocity.

3.2. Model Displacement Probability DistributionsTo qualitatively compare model results with observations,

we take C = 7, vg = 5.5 km/s and vsg = 0.5 km/s. Whenscaled with ∆t = 0.4 seconds, the de-correlation time of ran-dom walk component is on average about 6 minutes. Thesevalues have not been fine tuned, but are reasonably represen-tative of granular and supergranular horizontal flow veloci-ties and typical granule lifetimes (e.g., Rast 2003; Nordlundet al. 2009; Rieutord & Rincon 2010). Using these param-eters, we compute the trajectories of the walkers and deter-mine their displacements as a function of temporal increment.The resulting displacement distributions (Figure 3e-h) quali-tatively agree with the observations for all but the shortest in-crements. As mentioned previously, at these shortest times theobserved distribution reflects the underlying Lagrangian ve-locity distribution and any pathologies associated with track-ing the barycenter of magnetic elements. The later issue doesnot enter this simplified correlated random walk model, andwe discuss its role in the observations in more detail in §4. Itis worth noting here, that even in the absence of the barycen-ter complications, the observed distribution at early times isunlikely to be captured by this simplified model since un-like for the observations, the Lagrangian velocity distributionin the model is a delta function. What is important is thatthis simple model captures the evolution of the distributionat longer times. As is the case for magnetic elements in theHinode/NFI data, the distribution first becomes Rayleigh, asthe motions de-correlate over temporal increments longer thanthe Lagrangian autocorrelation time, and then, for still longerincrements, approaches an offset Gaussian, as the slow driftcomes to dominate the displacement.

This dependence of the shape of the probability distributionfunction on temporal increment can be expressed analytically.For time intervals long compared to the granular correlationtime, the walker motions combine a random walk with a drift.The former yields a two-dimensional Gaussian spread of thewalkers positions about the origin while the latter advects theorigin downstream. By bivariate transformation of random

variables (e.g., Casella & Berger 2002; Hogg & Tanis 2006;Rast & Pinton 2009, and this paper’s Appendix A), for spatialoffsets in x and y which are Gaussianly distributed with equalvariances σ about x0 and y0, the radial distance r from a fixedorigin at (0, 0) is distributed as

P (r) =r

σ2e−(r−r0)2/2σ2

e−r0r/σ2

I0(r0r/σ2) , (1)

where r20 = x20 + y20 and I0 is the lowest order modi-fied Bessel function of the first kind (Abramowitz & Ste-gun 1972). In our model of the Lagrangian displace-ment of the walkers, both r0 and σ increase with tem-poral increment due to the drift and random walk com-ponents of the motion, respectively. For small values ofr0 compared to σ, the product of the last two terms inEquation (1) approaches one and the distribution becomesRayleigh, as expected for a two-dimensional random walkcentered at origin. For large r0 and small σ, I0(r0r/σ

2) ∼exp(r0r/σ

2)/√

2πr0r/σ2 (Abramowitz & Stegun 1972),and the distribution becomes

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(r/r0)

2πσ2e−(r−r0)2/2σ2

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Thus for large r0 and small σ, the distribution is nearly Gaus-sian around r = r0, with only slight distortion by the r/r0pre-factor to ensure that P (r) = 0 at r = 0. Note that for arandom walk without drift, x0 and y0 are typically taken to bezero and the resulting distribution is Rayleigh for all times, butfor our case with finite drift contribution r0 = x0 = vsgt, thedistribution evolves from Rayleigh to Gaussian at long times(Figure 3e-h).

Figure 4a-d displays the positions of the random walk-ers (black points) in our correlated random walk model attimes corresponding to the displacement distributions shownin Figure 3e-h. Over very short time intervals, the trajec-tories are radially ballistic, directed away from the origin.Over longer intervals, walkers’ positions are distributed asa two-dimensional Gaussian ‘cloud’ about the drift positionr0 = vsgt, and the displacement distribution about that posi-tion is Rayleigh. As r0 is small compared to σ, the peak of theGaussian cloud is still close to the origin, and the displace-ments, when computed from the origin, are approximatelyRayleigh distributed (see Figure 4b and its correspondingdisplacement distribution in Figure 3f). Since the standard-deviation σ of the random walk component increases as t1/2,while r0 due to the drift increases as t, after sufficiently longtimes the Gaussian cloud drifts away from the origin and theshape of the displacement distribution, as computed from theorigin, changes from Rayleigh to nearly Gaussian (see Fig-ure 4d and its corresponding displacement distribution in Fig-ure 3h). That change is apparent with increasing time in boththe model distributions (right column of Figure 3) and thosederived from the observations (left column of Figure 3).

The change is also reflected in the mean squared displace-ment vs. time curve in Figure 2, in which the curve forwalkers (blue squares) is plotted along with measurementsfrom Hinode data. For the shortest temporal increments, thewalker motions are highly correlated and the scaling is bal-listic. For longer increments the walker motions de-correlateand the scaling becomes super-diffusive, flattening towardswhat would, in the non-drifting case, become the diffusivevalue. With drift, this is interrupted, and the scaling revertsto ballistic as drift contribution to the displacement comes

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to dominate the motion. Power-law fits to the model dataat the shortest and longest times have indices of 1.97 and1.96, respectively. This reversion to ballistic scaling occursfor temporal increments longer than (vg/vsg)

2 ∗ τg if the driftis steady, where τg is a typical granule correlation time. If thecorrelation time of the random component and the timescaleon which drift dominates the motions are sufficiently wellseparated, then a diffusive scaling between the two ballisticregimes can be achieved. This is not the case here becausethe ratio of the granular random walk velocity to the super-granular drift velocity is insufficiently large. For temporal in-crements greater than 2 hours, the Hinode data may show thebeginning of the change in scaling. Unfortunately, the fluxelements do not have sufficiently long lifetimes to recover thefull reversion to a ballistic scaling.

4. FLUX ELEMENTS AND PASSIVE TRACERS IN MURAMSIMULATIONS

As noted previously, the model of §3, based on a correlatedrandom walk with drift, does not capture the observed meansquared displacement scaling for magnetic elements at short-est time intervals. The observed scaling at shortest time inter-vals is super-diffusive, rather than ballistic, as in the simplifiedmodel. This is true for element trajectories in our Hinode ob-servations and for trajectories determined using data with ca-dences as high as 5 seconds (Chitta et al. 2012, Table 1). Oneexpects these time intervals to be well below the Lagrangianand Eulerian integral times, and under the assumption thatflux elements are advected passively, the element displace-ments should thus scale ballistically.

To uncover the origin of this discrepancy, we analyzeda small-scale dynamo quiet-sun simulation using a modi-fied version of the MURaM radiative magnetohydrodynamicscode (Vogler et al. 2005). The simulation is similar to the runO16b described in Rempel (2014), but with the vertical do-main size extended to 1.7 Mm above the photosphere and theradiative transfer computed using four opacity bins. The over-all domain size is about 6 × 6 × 4 Mm3, with a uniform gridspacing of 16 km. The simulation has no imposed mean mag-netic field, but the mixed field is generated and maintained bya small-scale dynamo. The average unsigned vertical mag-netic flux density at optical depth unity has a value of about80 Gauss, which is quite representative of the quiet-sun mag-netism (see e.g., Orozco Suarez et al. 2007; Danilovic et al.2016). The simulation spans about one hour with 2.0625 sec-ond cadence (1801 snapshots). We note that the simulation

captures granular flows only, not those at larger scales, sothe contribution of supergranulation, critical to transport overlonger time intervals (§3.2 above), cannot be examined. Inthis section, we focus on time intervals below 100 secondsfor which we expect the scaling to be independent of super-granular motions, and are particularly interested in the short-est temporal increments for which ballistic scaling is expectedbut not seen in the observations.

We identified and tracked flux elements in the simulationsusing vertical magnetic flux density images at optical depthunity, employing a minimum unsigned flux density thresholdof three times the standard-deviation above the mean. Thethreshold was computed independently for each image. At thenative resolution of the simulation, this corresponds to pixelswith unsigned flux density in excess of about 470 Gauss. Wenote that the threshold values differ for the simulation and theobservational data (§2.1) due to differing image resolution anddifferences in the optical depth surface on which the elementsare tracked. Only magnetic elements with areas greater than20 pixels and lifetimes in excess of 10 time-steps (about 20seconds) were included in the study of the displacement statis-tics in the simulations. We discarded elements that split ormerged over the 100 second interval in order to minimize un-certainty in the displacement measurement, which was basedon changes in the flux-weighted barycenter positions.

Figure 5 plots the mean squared displacement as a func-tion of temporal increment for the magnetic elements trackedin the MURaM simulation (red stars). The curve approx-imates a power-law with slope 1.69, similar to the scalingfound using observations of magnetic bright points with sim-ilar cadence (e.g., Chitta et al. 2012; Abramenko et al. 2011,Table 1). Moreover, the shape of the curve (deviating froma power-law in detail) is similar to that seen in observa-tions (compare with Figure 8 in Chitta et al. (2012) and Fig-ure 5 in Abramenko et al. (2011)). Both the observations andthe simulations show a shallower slope for shortest temporalincrements, with the scaling exponent in the MURaM simu-lation at shortest increments having a value of 1.21.

Not only is the super-diffusive scaling at such short incre-ments surprising in itself, but the increase in the scaling ex-ponent with increasing increment in the absence of any large-scale flow (or in situations where the contribution from thelarge-scale flow is negligible, as is the case for observationsat these increments) is also hard to understand. Without alarge-scale flow component, the scaling exponent should de-

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(M

m2)

Figure 5. Mean squared displacement vs. temporal increment for flux ele-ments in MURaM simulations tracked at native resolution (red stars), trackedat degraded 116 km resolution (green diamonds), passive tracers (black dots)and the model presented in §4 (blue squares). The best-fit slope at the short-est temporal increments for flux elements tracked at native resolution is 1.21(red dashed line) and that for passive tracers is 1.99 (black dashed line). Notethat the plots for black dots and green diamonds are shifted vertically upwardto better compare their shape with other curves at long increments. The vari-ance of the mean squared displacements for all the curves do not exceed thesize of the plotting symbol.

20 40 60 80v (km/s)

0.001

0.010

0.100

P (

v)

2 4 6 8 10

0.001

0.010

0.100

Figure 6. Lagrangian velocity distribution of the flux elements (red) and thepassive tracers (black) tracked in the MURaM simulations. Plot in the insetshows the distribution for velocities less than 10 km/s.

crease with increasing increment as the motions de-correlatewith time. Moreover, oddities are found in the Lagrangianvelocity distribution of the magnetic elements in the simu-lation. The red curve in Figure 6 plots the Lagrangian ve-locity distribution determined from the flux element barycen-ter displacements after one time-step. It shows unrealisticallyhigh velocity values, well in excess of the photospheric soundspeed (∼ 7 km/s, e.g., Nordlund et al. (2009)). To check thatthis is not an artifact of our tracking algorithm, we analyzedmagnetic bright point displacements in the photospheric con-tinuum intensity images of the MURaM solution. The track-ing was done by Samuel Van Kooten using the algorithm de-scribed in Van Kooten & Cranmer (2017). That algorithm isindependent of the one we employed, yet we found the veloc-ity and displacement statistics in agreement with those of themagnetic flux elements presented in this work.

We suggest that both the non-ballistic scaling of the ele-ment displacements at short time intervals and spurious su-personic Lagrangian velocities deduced from the element mo-tions in the simulations are a consequence of using magneticelements’ barycenters in determining their positions and mis-interpreting all changes in those positions as true motions.Changes in the magnetic flux content of an element subjectsits barycenter to random changes in position. This intro-duces jitter that is erroneously interpreted as motion. Figure 7presents an illustrative example. The outline of the magneticelement at two consecutive time-steps is shown in red andblue, with blue being the later time. The red arrow indi-cates the displacement (scaled for illustration), as computedfrom the shift in element’s barycenter. As the magnetic ele-ment evolves, the disappearance of flux displaces its barycen-ter with a magnitude and direction drastically different fromwhat would be achieved by passive advection due to under-lying plasma flow. The black arrow in the figure (scaled asthe red arrow) indicates the displacement computed using thevector average of the plasma velocities over all grid-pointsconstituting the element at the initial time-step. Magneticflux evolution thus introduces a random component to the de-duced motions of flux element barycenters which dominatesthe scaling at shortest time intervals and can lead to unrealis-tically large Lagrangian velocity values. This is true irrespec-tive of whether the barycenter definition is flux-weighted orposition-weighted.

To further assess the impact of this jitter, we compared mag-netic element motions with passive tracer motions in the sim-ulation. The motions of passive tracers were evolved using thephotospheric horizontal plasma velocity at their locations, andare independent of any contribution from flux evolution. Forpoint-like passive tracers seeded at random locations, the scal-ing for the shortest temporal increments is ballistic. To ensurethat the flux element positions are not biased, we separatelyseeded the passive tracers co-spatially with the flux elements,so that they initially occupied the same area (the same grid-point locations) as the corresponding elements. The passivetracer ‘element’ positions were then evolved using the hori-zontal velocity field averaged over the area they spanned. Thistraces the motions of their position-weighted barycenter andallows direct comparison between the flux element and pas-sive tracer statistics if the field distribution over the elementsis nearly uniform. This approximation holds for the mag-netic elements identified using the employed three standard-deviation threshold, and for these elements we found consis-tent velocity and displacement statistics independent of thebarycenter definition.

The barycenter velocity distribution of the passive tracer‘elements’ shows no pathologically high values (black curvesin Figure 6), and for short temporal increments, the meansquared displacement of the ‘elements’ scales ballistically(for black dots in Figure 5), in contrast to the super-diffusivescaling for magnetic elements (for red stars in Figure 5). Forlonger temporal increments, the mean squared displacementscaling of the flux elements and passive tracers agree, sug-gesting that over longer timescales the passive advection offlux elements due to the plasma flow dominates the randombarycenter jitter. For sufficiently long increments (not shownin the plot), and in the absence of any large-scale flow (dis-cussed in §3.2), the slope for both curves should approach thediffusive value, though direct comparison between the flux el-ement and passive tracer statistics makes the most sense overshort temporal increments over which the passive tracer and

8 AGRAWAL ET AL.

0.0 0.1 0.2 0.3 0.4 0.5 0.6x (Mm)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

y (M

m)

−5

0

5

10

Flu

x d

en

sity (

in u

nits o

f 1

00

Ga

uss)

Figure 7. Red and blue contours mark the boundary of the area coveredby a flux element at two consecutive time-steps (blue being the later time).Red and black arrows are scaled with respect to each other and signify thediffering magnitude and direction of the displacement of magnetic elementbarycenter due to flux evolution and plasma flow, respectively.

element locations remain contiguous.The scaling behavior of the magnetic flux elements in MU-

RaM (for red stars in Figure 5) can be reproduced using asimple two-component model of their motion: motion due toa correlated random walk (corresponding to passive advectionby granular flows) and motion due to delta-correlated jitter(mimicking random barycenter motion due to flux evolution).The correlated motion follows the same formulation as thatin §3.1 without the supergranular drift component (C = 8.4,vg = 2.0 km/s, vsg = 0 km/s). The parameter C is chosenso that, when scaled with MURaM time-step ∆t = 2.0625seconds, the motion de-correlates over a typical granular life-time (about 6 minutes). The granular velocity vg is chosento be the mean of the passive tracer barycenter velocity mag-nitudes when the tracers are seeded co-spatially with the fluxelement locations (as described above). We note that vg hereis distinct from that used in §3.2. Here, since it is based onpassive tracers in the plasma velocity field, it does not includethe barycenter jitter due to flux element evolution. The jittercomponent is instead explicitly modeled as a vector of con-stant magnitude with random orientation and is added to thecorrelated granular motions at each time-step. Plotted in Fig-ure 5 with blue squares is the model displacement when usingthe values above, along with a jitter magnitude 3.5 km/s. Thevalue of 3.5 km/s is arbitrary but demonstrates that this modelcan capture the scaling observed for the magnetic flux ele-ments over short temporal increments. The real distributionof jitter velocity magnitudes in the simulation can be deter-mined by the difference between the measured flux elementbarycenter values (red distributions in Figure 6) and their pas-sive tracer counterparts (black distributions in Figure 6). It isbroad with a mean value of 2.3 km/s and a standard-deviationof 3.8 km/s. What is important is that this highly simplifiedmodel captures the effect of both the correlated granular flow(which would alone yield ballistic scaling) and the randomapparent motions due to flux element evolution (which wouldalone yield diffusive scaling). Together these two componentsexplain the super-diffusive scaling at the shortest time inter-vals and the increase in scaling exponent as the temporal in-crement increases.

Table 2Scaling vs. image resolution, computed for flux elements tracked in

MURaM simulations

Pixel resolution Flux density thresholda γ0b σvc

(km) (Gauss) (km/s)16 470 1.21 3.3927 412 1.32 2.7438.5 377 1.44 2.4347 354 1.50 2.25100 254 1.73 1.62116 234 1.75 1.58

aaverage of three times the standard-deviation values for all magnetograms,bscaling at the shortest temporal increments, cstandard-deviation of theLagrangian velocity distribution of the flux elements.

4.1. Magnetic element displacement vs. image resolutionThe barycenter jitter contribution to magnetic element mo-

tion is sensitive to the physical processes governing flux evo-lution, image resolution and cadence, and the feature track-ing algorithm and identification parameters. Green diamondsin Figure 5 illustrate the effect of image resolution on themean squared displacement. Flux elements were identifiedand tracked after convolving the MURaM images with a two-dimensional Gaussian kernel, degrading the MURaM imageresolution to∼116 km (Gaussian full-width at half maximumequal to the ratio of target pixel size and MURaM pixel size).The scaling exponent obtained (for green diamonds) has alarger value at the shortest increments than that obtained whentracking elements at the native resolution (for red stars). Ta-ble 2 shows the monotonic increase in the short incrementscaling exponent γ0 with decreasing resolution. While atthese timescales the flow contribution to the displacement isballistic, the barycenter jitter contribution to the scaling de-creases with decreasing resolution. This is because the fluxelements being tracked in a degraded image are on averagelarger so their barycenter positions are less sensitive to smallchanges in the field configuration, the probability for largechanges in flux density is reduced by the reduction in resolu-tion, and the shape of the elements in a low-resolution imageare less irregular making the barycenter definition more robustand its position less sensitive to barycenter jitter.

It is important to also note that the plot for the low-resolution results for flux elements (green diamonds) in Fig-ure 5 were shifted vertically upward to better compare itsshape with the native resolution (red star) displacements atlongest increments. The magnitude of the displacement de-pends on the Lagrangian velocity distribution which decreasesin width with decreasing resolution (σv in Table 2). The ve-locity distribution captures both the advective and the jittercontributions, both of which are reduced when the image isdegraded. This leads to smaller mean squared displacementsfor the same temporal increment. Thus, both the scaling coef-ficient at the shortest time intervals and the magnitude of themean squared displacements are not robust and depend on theimage properties and the feature tracking parameters.

5. SUMMARY AND CONCLUSION

We examined the transport of internetwork magnetic fluxelements in Hinode/NFI data and found, as in previous stud-ies (e.g., Chitta et al. 2012; Abramenko et al. 2011; Jafarzadehet al. 2014), that their mean squared displacement scalessuper-diffusively with time even for the shortest temporal in-crements. In addition, the shape of the underlying displace-ment probability distribution evolves from Rayleigh to Gaus-sian as the increment increases. Using a correlated random

9

walk model with a drift component, we have demonstratedthat this is likely due to supergranular motions dominatinggranular motions for time intervals long compared to the gran-ular correlation time. We suggest that over intervals longerthan supergranular correlation times, the distributions wouldbe similarly affected by the underlying meridional flow. Thus,the interpretation of flux element motion as a strictly diffusiveprocess is likely incorrect.

Super-diffusive scaling is found in studies using observa-tional data with cadences as short as 5 seconds (e.g., Chittaet al. 2012), much shorter than the expected Lagrangianand Eulerian integral times of the flow and thus capturingtimescales over which the displacement scaling should be bal-listic. We investigated the underlying causes for this discrep-ancy by tracking flux elements in a MURaM simulation with 2second cadence and found similar super-diffusive scaling forshort temporal increments. Comparison between the flux el-ement and passive tracer statistics in the simulation suggeststhat the super-diffusive scaling over short temporal intervalsis a consequence of misinterpreting flux element barycentermotion as strictly due to plasma flows. In addition to mo-tions induced by the underlying flow, barycenter positions aresubject to jitter induced by magnetic flux evolution. This im-parts a random component to the measured motions and con-tributes strongly to the scaling at the shortest temporal incre-ments and results in the observed super-diffusive scaling. Themeasured Lagrangian velocity distribution reflects these spu-rious motions as well, showing values well in excess of thephotospheric sound speed. Moreover, the jitter contributiondepends on the underlying physical processes governing fluxevolution, image resolution and cadence, and the magnetic el-ement identification scheme employed.

These results suggest that using displacement measure-ments of flux element barycenters to directly determinediffusion coefficients or wave forcing by magnetic elementmotions in the solar photosphere may be problematic.However, the artifacts identified may be partially overcomeby employing passive tracers as a proxy, as their motion isindependent of flux evolution. Rather than tracking magneticelements over long periods of time, it may be preferable tocompute the photospheric horizontal plasma velocity usingmethods such as structure or correlation tracking (e.g., Simonet al. 1988; Roudier et al. 1999; Potts et al. 2004; Roudieret al. 2012; Attie et al. 2016). The velocity field could then beused to compute passive tracer trajectories and displacementstatistics. In addition to likely being more robust, thiswould avoid the ubiquitous difficulties associated with fluxelement sparsity, identification and evolution, and allow theinvestigation of displacement along many more trajectoriesthan is usually possible, with each lasting longer than typicalmagnetic element lifetimes. From the trajectories, one couldmeasure both the drift contribution of larger scale flows andthe effective diffusive component of the random motions byfitting the observed displacement probability distribution withthe analytic function of Equation 1. The fit would yield boththe width of the displacement distribution σ(t), a measureof the diffusive component of the motion, and the averagedisplacement due to the drift motion r0(t). The successof this method would rely on the accuracy of the derivedphotospheric plasma velocity field and the ability to separatethe diffusive and the drift components if the large-scale flowis not uniform in space and time, but from this work, weexpect the method to yield a more authentic measure of themagnetic flux element motion.

This paper is based on the data acquired during Hinode Op-eration Plan 151. We thank the Hinode Chief Observers fortheir efforts in executing this plan. Hinode was developedand launched by ISAS/JAXA with NAOJ as a domestic part-ner and NASA and STFC (UK) as international partners. Itis operated by these agencies in cooperation with ESA andNSC (Norway). This work has been partially funded by theSpanish Ministerio de Economıa y Competitividad throughprojects ESP2013-47349-C6-1-R and ESP2016-77548-C5-1-R including European FEDER funds. The research has madeuse of NASA’s Astrophysics Data System Bibliographic Ser-vices. NCAR is supported by the National Science Foun-dation. The authors thank Samuel Van Kooten for mag-netic bright points tracking. MPR was partially supported byNASA award NNX12AB35G. P. Agrawal acknowledges thesupport of the University of Colorado’s George Ellery HaleGraduate Student Fellowship.

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APPENDIX

BIVARIATE TRANSFORMATION OF RANDOM VARIABLES

To determine the probability density of a function u = f(x, y) of independent random variables x and y, with individualprobability densities P (x) and P (y) and joint probability density Pxy(x, y) = P (x)P (y), define a second function v = g(x, y)chosen so that its probability density can be integrated out of the joint probability density, leaving that of the target functionbehind. For u and v with inverses x = h1(u, v) and y = h2(u, v), the joint probability density of u and v is (e.g., Casella &Berger 2002; Hogg & Tanis 2006)

Puv(u, v) = Pxy(h1, h2)

∣∣∣∣∣∣∂h1

∂u∂h1

∂v

∂h2

∂u∂h2

∂v

∣∣∣∣∣∣ (A1)

andP (u) =

∫P (u, v) dv . (A2)

To derive Equation 1 in Section 3.2, consider P (r) with r =√x2 + y2, where as in §3.2, x and y are independent random

spatial offsets Gaussianly distributed about x0 and y0 with equal variance σ. By bivariate transformation, the joint probabilitydensity

Pxy(x, y) =1

2πσ2e−[(x−x0)

2+(y−y0)2]/2σ2

. (A3)

can be written in terms of r (u in the general notation above) and θ = tan−1(y/x) (v in the general notation above), withinverse functions x = r cos θ (h1 in the general notation above) and y = r sin θ (h2 in the general notation above), by evaluatingEquation A1,

Prθ(r, θ) =r

2πσ2e−[r2+r20−2r0r cos(θ−φ)]/2σ2

=r

2πσ2e−(r−r0)2/2σ2

e−r0r/σ2

er0rcos(θ−φ)/σ2

, (A4)

with φ = tan−1(y0/x0) and r0 =√x20 + y20 . Integrating over all θ then yields the probability density of r (Equation 1 in the

main text),

P (r) =r

2πσ2e−(r−r0)2/2σ2

e−r0r/σ2

∫ 2π

0

er0rcos(θ−φ)/σ2

dθ =r

σ2e−(r−r0)2/2σ2

e−r0r/σ2

I0(r0r/σ2) , (A5)

where, I0 is the lowest order modified Bessel function of the first kind (Abramowitz & Stegun 1972).