Photospheric Electric Fields and Energy Fluxes in the Eruptive Active Region NOAA 11158
Maria D. Kazachenko, George H. Fisher, Brian T. Welsch, Yang Liu, Xudong Sun
PPhotospheric Electric Fields and Energy Fluxes in the EruptiveActive Region NOAA 11158
Maria D. Kazachenko , George H. Fisher , Brian T. Welsch , Yang Liu , Xudong Sun [email protected] Received ; accepted Space Sciences Laboratory, UC Berkeley, CA 94720, USA W. W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA94305, USA a r X i v : . [ a s t r o - ph . S R ] A ug ABSTRACT
How much electromagnetic energy crosses the photosphere in evolving solaractive regions? With the advent of high-cadence vector magnetic field observa-tions, addressing this fundamental question has become tractable. In this paper,we apply the “PTD-Doppler-FLCT-Ideal” (
PDFI ) electric field inversion tech-nique of Kazachenko et al. (2014) to a 6-day HMI/SDO vector magnetogramand Doppler velocity sequence, to find the electric field and Poynting flux evolu-tion in active region NOAA 11158, which produced an X2.2 flare early on 2011February 15. We find photospheric electric fields ranging up to 2 V/cm. ThePoynting fluxes range from [ − . . × ergs · cm − s − , mostly positive,with the largest contribution to the energy budget in the range of [10 –10 ]ergs · cm − s − . Integrating the instantaneous energy flux over space and time, wefind that the total magnetic energy accumulated above the photosphere from theinitial emergence to the moment before the X2.2 flare to be E = 10 . × ergs,which is partitioned as 2 . . × ergs, respectively, between free andpotential energies. Those estimates are consistent with estimates from preflarenon-linear force-free field (NLFFF) extrapolations and the Minimum CurrentCorona estimates (MCC), in spite of our very different approach. This studyof photospheric electric fields demonstrates the potential of the PDFI approachfor estimating Poynting fluxes and opens the door to more quantitative studiesof the solar photosphere and more realistic data-driven simulations of coronalmagnetic field evolution.
Subject headings:
Sun: magnetic field, Sun: flares, Sun: sunspots
Contents1 INTRODUCTION 12 METHODOLOGY:
PDFI
TECHNIQUE, POYNTING & HELICITYFLUXES 43 DATA REDUCTION: NOAA 11158 8 B . . . . . . . . . . . . . . . . . . . . . . . 83.2 Deriving Velocity Vector Fields: V . . . . . . . . . . . . . . . . . . . . . . . 103.3 How do Errors in Magnetic Field Measurements Affect our Electric FieldsEstimates? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 B preflare and B postflare . . . . . . . . . . . . . . . . 164.1.2 Velocity Field: V preflare and V postflare . . . . . . . . . . . . . . . . . 174.1.3 Electric Field: E preflare and E postflare . . . . . . . . . . . . . . . . . 194.1.4 Poynting Flux Vector: S preflare and S postflare . . . . . . . . . . . . . 204.2 Six-Day Evolution of Vertical Energy and Helicity Fluxes . . . . . . . . . . . 234.2.1 Evolution of Free, Potential, and Total Energy Fluxes . . . . . . . . . 244.2.2 Helicity Flux Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4 –
1. INTRODUCTION
The advent of high-cadence, large-scale vector magnetic field and Doppler velocitymeasurements from instruments such as the Helioseismic and Magnetic Imager (HMI,Schou et al. (2012)) on NASA’s Solar Dynamics Observatory (SDO) satellite (Pesnell et al.2012), the Spectropolarimeter instrument (SP; Lites et al. 2013) on the Solar OpticalTelescope (Tsuneta et al. 2008) aboard the
Hinode satellite (Kosugi et al. 2007), andimproved capabilities of ground-based instruments, such as SOLIS (e.g., Keller et al. 2003),make the estimation of electric fields in the solar photosphere possible. The calculationof the electric field from magnetic and Doppler data is critically important for variousquantitative studies of the solar atmosphere. First, if we know both electric and magneticfield vectors in the photosphere, we can estimate both the Poynting flux of magnetic energyand the flux of relative magnetic helicity entering the corona. Second, as demonstratedin a magneto-frictional model by Cheung and DeRosa (2012), the ability to compute theelectric field enables the driving of time-dependent simulations of the coronal magneticfield from photospheric magnetogram sequences. Combining electric field estimation witha magneto-frictional model of the evolving solar corona is the goal of the Coronal GlobalEvolutionary Model (CGEM) project (Fisher et al. 2015). Kazachenko et al. (2014) modified and extended the electric field inversion methodsintroduced by Fisher et al. (2010, 2012), to create a comprehensive technique forcalculating photospheric electric fields from vector magnetogram sequences. The newmethod, which we dubbed the
PDFI (an abbreviation for P oloidal-Toroidal Decomposition[PTD]- D oppler- F ourier Local Correlation Tracking [FLCT]- I deal) technique, incorporatesDoppler velocities from non-normal viewing angles (which are relevant to most solarobservations) and a faster and a more robust Poisson equation solver, for obtaining the http://cgem.stanford.edu PDFI technique, using synthetic data from anelastic MHD (
ANMHD ) simulations (Abbettet al. 2004), we found that the
PDFI method has less than 1% error in the total Poyntingflux and a 10% error in the helicity flux rate if we reconstruct it at the normal viewing angle( θ = 0) and less than 25% and 10% errors respectively at large viewing angles ( θ = 60 ◦ )(Kazachenko et al. 2014).In this paper, we take the next step forward, and apply the PDFI technique toobservations. The flare-productive active region (AR) NOAA 11158 was observed by HMInearly continuously for a six-day period over 10-16 February 2011, starting from its initialemergence near 14:00 UT on 10 February. We use the sequence of magnetic and Dopplerfield measurements of NOAA 11158 to derive the temporal evolution of electric field,Poynting, and helicity fluxes during these six days. The evolution of this AR included twolarge bipoles emerging in close proximity, with strong shearing motion between the centralsunspots (Schrijver et al. 2011; Sun et al. 2012). Over a six-day period, the AR hosted anX2.2 flare (with the GOES soft X-ray flux peaking at 01:56 UT on February 15) leadingto a pronounced halo CME, three M-class flares, and over twenty C-class flares. Since thisactive region was the first one for which the HMI vector magnetic field data were widelydistributed to the scientific community, its magnetic field has been thoroughly studied:the fast sunspot rotation from 20 hours before to 1 hour after the X2.2 flare (Jiang et al.2012; Vemareddy et al. 2012a), the flare related enhancement in the horizontal magneticfield along the magnetic polarity inversion line (PIL; Gosain (2012); Wang et al. (2012);Liu et al. (2012a)), abrupt changes in the vertical Lorentz force vectors (Petrie 2012;Alvarado-G´omez et al. 2012) and horizontal Lorentz forces (Petrie 2013; Wang et al. 2014),the evolution of relative and current magnetic helicities (Jing et al. 2012), the injection ofoppositely signed helicity through the photosphere (Dalmasse et al. 2013), the subsurfacethree-dimensional magnetic structure (Chintzoglou and Zhang 2013), the magnetic and 3 –velocity field transients driven by the flare (Maurya et al. 2012). Numerous approacheshave been used to calculate the energy associated with X2.2 flare and the AR as a whole:the DAVE4VM method (Liu and Schuck 2012; Tziotziou et al. 2013), non-linear forcefree extrapolation (Sun et al. 2012; Tziotziou et al. 2013), the Minimum Current CoronaModel (Tarr et al. 2013), and the coronal forward-fitting method (Aschwanden et al. 2014;Malanushenko et al. 2014). In this paper, we apply the
PDFI technique (Kazachenko et al.2014) to derive electric fields at the photosphere, and use those to estimate the photosphericenergy and helicities fluxes’ evolution in NOAA 11158.The paper is structured as follows. In Section 2 we briefly review the
PDFI method itselfand its recent improvements. In Section 3 we describe the observations of the emerging,flaring NOAA 11158, and quantify how the uncertainties in the HMI observations propagateinto derived electric field and Poynting fluxes. In Section 4, we describe derived electricfields, Poynting, and helicity fluxes in NOAA 11158, and in Section 5 we discuss resultsand draw conclusions. In addition, in Appendix A we show how our Mercator-reprojectedmagnetic fields in Cartesian coordinates can be scaled to apply to relatively small regionson the surface of the Sun to derive the electric fields in a local Cartesian coordinate system. 4 –
2. METHODOLOGY:
PDFI
TECHNIQUE, POYNTING & HELICITYFLUXES
The
PDFI technique uses the evolution of the vector magnetic field B and horizontaland Doppler velocities V to estimate electric fields in the solar photosphere. It is describedin detail in § PDFI method combines the inductive contribution to the electric field E from solution to Faraday’s law using the Poloidal-Toroidal-Decomposition (PTD) technique (Fisher et al. 2010, 2012), with non-inductive contributions from ( − V × B ). To find the velocity V , we use Doppler measurements andthe Fourier Local Correlation Tracking ( FLCT ) technique. The
PDFI technique is tested andits accuracy is characterized in detail in § PDFI technique.The fundamental idea of the
PTD part of the
PDFI method is that the magnetic field, B , defined on the photospheric surface, has a solenoidal nature and thus can be specifiedby two scalar functions B and J : B = ∇ × ∇ × B ˆz + ∇ × J ˆz , (1)where ˆ z points upward from the photosphere. Taking a partial time derivative ofEquation (1), and demanding that B obeys Faraday’s law, ∂ B ∂t = − ( ∇ × c E ) , (2)we find a solution for the inductive part of the electric field E P (where “P” stands forPTD), in terms of the partial time derivatives of the poloidal and toroidal potentials, ˙ B and˙ J respectively: c E P = −∇ × ˙ B ˆz − ˙ J ˆz . (3)As described in Kazachenko et al. (2014), solving two-dimensional Poisson equations in thedomain, where we observe the vector magnetic fields, we determine ˙ B , ˙ J and ∂ ˙ B ∂z . Note 5 –that the vector magnetic field data completely specify the source terms of these Poissonequations (see § E is a combination of the inductive and non-inductive parts, c E = −∇ × ˙ B ˆz − ˙ J ˆz − ∇ ψ ≡ c E P (cid:124)(cid:123)(cid:122)(cid:125) inductive − ∇ ψ. (cid:124)(cid:123)(cid:122)(cid:125) non-inductive (4)The non-inductive components to the scalar-potential part of the solution include threeseparate contributions: (1) − V × B from Doppler measurements (the “D” in PDFI ), (2) − V × B from Fourier Local Correlation Tracking results (the “F” in PDFI ), and (3) ascalar potential contribution added to impose the constraint E · B = , consistent with theideal MHD Ohm’s law (the “I” in PDFI ). When adding the − V × B contributions above,any inductive contributions from these terms are removed, since all inductive contributionsare already included in the E P solution. Our approach for handling these non-inductivecontributions is described in detail in Kazachenko et al. (2014).To calculate the flux of electromagnetic energy at the photosphere, given by thePoynting flux vector S = c π ( E × B ) , (5)we use the observed magnetic field vector and the electric field vector derived using the PDFI method. Since we are interested in the amount of energy flowing into and out of thecorona, we focus most of our attention on the vertical component of Poynting flux, S z = c π ( E x B y − E y B x ) . (6)This depends upon the horizontal components of both the electric field and the magneticfield. We further decompose S z into two contributions, the flux of potential-field energy,and the flux of free magnetic energy. The basic idea is that the horizontal magneticfield B h can be divided into a potential-field contribution B Ph , and a contribution, B fh , 6 –due to currents that flow into the atmosphere from the photosphere (Welsch 2006): S z = c π E h × (cid:16) B Ph + B fh (cid:17) . In this paper we use the Green’s function to find the potentialfield contribution, and subtract this contribution from the measured horizontal fields to find B fh . More discussion of the potential and free energy decomposition can be found in § (cid:18) dH R dt (cid:19) = − (cid:90) (cid:0) A P × E (cid:1) · ˆz da = − (cid:90) (cid:0) A Px E y − A Py E x (cid:1) da (cid:124) (cid:123)(cid:122) (cid:125) PDFI , (7)where A P = (cid:16) ∂ B P ∂y , − ∂ B P ∂x , (cid:17) = ∇ × B P ˆz is the vector potential that generates the potentialfield B P in volume V above the photosphere, which matches the photospheric normal field B z at z = 0. (Note that a similar expression for helicity, Equation (41) in Kazachenkoet al. (2014), contains a typographical error – E z should be replaced by E x .) Here B P canbe found by solving the Poisson equation A5 in Fisher et al. (2010). Adopting the namingconvention from § PDFI electric field E will be referred to as the PDFI helicity flux rate or (cid:0) dH R dt (cid:1) P DF I .If we have an ideal electric field c E = − V × B , then the helicity flux rate becomes(Berger 1984): (cid:18) dH R dt (cid:19) = − (cid:90) [( A P · V h ) B z − ( A P · B h ) V z ] da (cid:124) (cid:123)(cid:122) (cid:125) DFI . (8)For observations near disk center, where the line-of-sight direction approximates the verticaldirection, in Equation (8), we can make the assumption that V h can be determined withour FLCT flow estimates, and V z from our Doppler velocity measurements. Adopting thenaming convention from § DFI electric field solution ( D oppler F LCT I deal).When comparing helicity fluxes calculated using the PDFI and
DFI techniques, it is 7 –important to remember that
PDFI and
DFI methods are independent of each other, hencetheir results are not necessarily consistent. For more details on
PDFI and
DFI helicity fluxesand the quality of their reconstruction using
ANMHD simulations, see §
3. DATA REDUCTION: NOAA 11158
We derive the evolution of the magnetic field, electric field, Poynting, and helicityfluxes in NOAA 11158 using series of HMI vector magnetograms and the
PDFI method.A six-day, uninterrupted, 12-minute-cadence data set allowed us to study in detail bothlong-term, gradual evolution, as well as rapid changes centered around the region’s X-classflare. In this section we describe the data set and the coordinate system re-projection weuse. NOAA 11158 was the source of an X2.2 flare on 2011/02/15, starting at 01:44 UT,peaking at 01:56 UT, and ending at 02:06 UT. A front-side halo CME accompanied theflare (Schrijver et al. 2011). Prior to the X2.2 flare, the largest flare in this region was anM6.6 on 2011/02/13 at 17:28 UT, a little more than 30 hours before the X2.2 flare.HMI observed AR 11158 in great detail, routinely generating filtergrams in sixpolarization states at six wavelengths on the Fe I 617.3 nm spectral line. From thesefiltergrams, images for the Stokes parameters, I, Q, U, and V were derived which, usingthe Very Fast Inversion of the Stokes Algorithm (VFISV) code (Borrero et al. 2011), wereinverted into the magnetic field vector components. To resolve the 180 ◦ azimuthal fieldambiguity the “minimum energy” method (Metcalf et al. 1994; Leka et al. 2009) was used.In addition, we flipped the azimuths of the transverse magnetic field vectors in all pixelswhich exhibited single-frame fluctuations in azimuth of larger than 120 ◦ and for which suchflipping reduced time variation in the azimuth (Welsch et al. 2013). To study preflare photospheric magnetic evolution, and to baseline this evolutionagainst postflare evolution, we obtained 153 hours of 12-minute-cadence 0.5 (cid:48)(cid:48) -pixel HMI 9 –vector-magnetogram data, from the beginning of the active region emergence, about fourdays before the X2.2 flare, to two days after the flare: t start = February 10 2011 14:00 UT,with the active region centered at (S19, E50); and t end = February 16 2011 23:48 UT, withthe active region at (S21, W37).We rotated the active region to disk center and transformed it to a local Mercatorre-projected Cartesian coordinate system (Welsch et al. 2013). To do so, in the firststep, we re-projected the observed magnetic vectors’ components onto radial/horizontalcoordinate axes. We then converted the Cartesian output grid’s points into plane-of-sky(POS) coordinates, and interpolated the radial and horizontal components of the magneticfield, B r and B h , onto the remapped output grid points. Finally, to account for small,whole-frame shifts of the AR’s structure between successive measurements, we co-alignedthe data to sub-pixel scale. Note that since FLCT (Fisher and Welsch 2008), and indeedany method of estimating the optical flow (e.g., Schuck (2006)), depends upon imagestructure (e.g., gradients), conformal mappings are preferred since they are shape-preservingfor infinitesimally small objects. Accordingly, we use Mercator re-projection (Welschet al. 2009), with equally spaced grid points as an input for FLCT . After re-projection, topreserve physical quantities of magnetic fields and velocities, we corrected the flux densitiesfor the distortion of pixel scale introduced by re-projection; the details of the appliedcorrection-factors are given in Appendix A.For the minimum magnetic field to consider in the PTD, we chose a threshold of | B | = 250 Gauss, consistent with the upper limit of the uncertainty in the horizontal andvertical components of the magnetic field (Hoeksema et al. 2014). To avoid spurious signalsin electric fields, we apply a mask, where we set any pixel’s magnetic field components tozero if in any of three consecutive frames it has | B | <
250 Gauss. To increase the accuracyof the calculated electric fields we also added a boundary area of 55-pixels width/height 10 –padded with zeroes around the periphery of the magnetogram (see § dt = 720 sec) and has a field of view of 665 ×
645 pixels with a pixel sizeof 360 .
16 km, which at disk center is equivalent to the original 0.5” size of HMI’s pixels.Further detail on the data cube preparation and calibration, for a shorter time period, canbe found in Welsch et al. (2013).Figure 1 shows the final vertical magnetic field in a subregion of the full-disk data arrayafter re-projection in the beginning (Panel A ), middle ( B, C ) and the postflare ( D ) timesof the magnetogram sequence. Note that the positive and negative vertical magnetic fluxes, B z ( t ), shown on the right panel, nearly balance each other; the signed magnetic fluxes growfrom essentially zero to roughly 1 . × Mx ( − . × Mx) at the time of the flare(vertical dashed line). The flux emerges in two phases – an initial, gradual phase is followedby a much more rapid phase, a pattern seen in the emergence of many active regions (Fuand Welsch 2015).
To derive the three-component velocity vector of the magnetized plasma, we usedthe following two methods: the method of Welsch et al. (2013) for calibrated line-of-sight(LOS) Doppler velocity component V LOS , and
FLCT for the horizontal velocities V h . Toderive and calibrate the Doppler velocity for instrumental effects and convective blueshift,we used three successive vector magnetograms and one Dopplergram coincident with thecentral magnetogram. Following the idea of Welsch et al. (2013), that the Doppler shiftsmeasured along polarity inversion lines (PILs) of the LOS magnetic field determine onecomponent of the velocity perpendicular to the magnetic field, we calibrated the quantity V LOS by subtracting the median Doppler velocity among all pixels on LOS PILs. To find 11 –Fig. 1.—
Panels A-D:
HMI vertical magnetic field ( B z ) maps at 4 different times of NOAA11158 evolution. Panel E : evolution of the positive and negative vertical magnetic fluxesof the 6-day interval, with the diamonds indicating the times of the four images (A-D) onthe left. An X-class flare occurred at the time corresponding to the vertical dashed line.The black box in
Panel D indicates the field of view of Figures 4–8, however all energy andhelicity estimates described further are for the entire field of view.the horizontal velocity V h , we determined local displacements of magnetic flux between twosuccessive images in the neighborhood of each pixel, employing the following three steps.First, we masked the initial and final images with a Gaussian windowing function with an e -folding width of σ F LCT = 5 pixels; second, we cross-correlated the two masked images;finally, we found the peak of the cross correlation function. The vector displacement of thispeak from zero is the inferred spatial displacement of the pattern in the neighborhood ofthe windowing function’s center.To calculate electric fields, and hence Poynting and helicity fluxes (Equations (6)and (7)), we use Doppler and
FLCT velocities as an input into the
PDFI inversion. Forcomparison, apart from the
PDFI , we also use Doppler and
FLCT velocities on their own asan independent estimate for the helicity flux rate (see Equation (8)).To summarize, as a result of the data reduction we obtained a six-day data cubeconsisting of 768 frames (two frames less than the original dataset to obtain
FLCT velocities), 12 –each of which contains data for three components of the velocity field and three componentsof the magnetic field with a field of view of 665 ×
645 pixels, a pixel size of 360 .
16 km, anda time step of dt = 720 sec. 13 – When testing the
PDFI electric field inversion technique using MHD simulations(Kazachenko et al. 2014), we have the advantage of knowing that the errors in the inputdata are zero, but such is not the case with the HMI data (Liu et al. 2012b). Hence toestimate the uncertainties in the derived electric field and Poynting fluxes, we have toaccount for uncertainties in the HMI input data, which arise primarily from estimation andinversion of the Stokes profiles (Liu et al. 2012b; Hoeksema et al. 2014).Fitting the core of magnetic-field-values distribution in the weak-field regions with aGaussian and assuming that its width indicates the noise level in the measurements, weestimate the errors in B x , B y and B z in AR 11158 dataset during six days of AR evolution(see Figure 2). The fluctuation of the error in B x , B y and B z varies within 100 Gauss forthe horizontal magnetic field and within 30 Gauss for vertical magnetic field due largelyto effects arising from SDO’s 24-hour orbital period (Hoeksema et al. 2014). We use thosevalues, i.e., [100 , ,
30] Gauss, respectively, as noise thresholds for magnetic field values.
Stdev for B x , B y , B z
13 14 15 16Time, days020406080100120 S t de v , G au ss Stdev Bx Stdev By Stdev Bz Fig. 2.— Estimated noise levels in B x , B y and B z , as functions of time in February 2011.Note the periodic variation, believed to be a function of orbital phase of the SDO satellite. 14 –Fig. 3.— Pixel-by-pixel scatter plots showing the ensemble of perturbed electric field solu-tions as a function of the unperturbed electric field solution, for all three components of E ,for one particular realization of pseudo-random noise.We use Monte Carlo simulations to estimate the errors in the PDFI electric fields causedby the uncertainty in the HMI data. The sensitivity of
PDFI electric field solutions to vectormagnetogram noise is exacerbated by the fact that source terms of the Poisson equationsthat are solved as part of the electric field inversions involve temporal and/or spatialderivatives, which greatly amplify the noise. This is, however, ameliorated by the fact thatsolutions to Poisson’s equation tend to smooth out the effect of noisy source terms. Sincethe entire inversion procedure is quite complex, our approach is to start from a given setof input magnetic field data, and then add pseudo-random, Gaussian perturbations to thedata, consistent with the noise thresholds given above ([100 , ,
30] Gauss). By comparingthe ensemble of electric field solutions that are perturbed about our initial solution, we cancharacterize the resulting errors of the electric field inversion. Figure 3 shows three scatterplots between the original and the perturbed the electric fields for one particular realizationof pseudo-random noise, one plot for each component.The resulting uncertainties in E x and E y are smaller than those for E z , reflecting thefact that the inductive horizontal electric field from PTD, E Ph , depends on B z , which has 15 –much smaller intrinsic errors than B x and B y , while the inductive part of the vertical electricfield, E Pz , depends on the more-noisy B x and B y . Thus our study not only shows that theelectric field inversion errors are not too large, but the distribution of these errors about thediagonal line provides quantitative estimates for the random errors, and deviations fromunity slope provide some information about more systematic errors. To scale the deviationsrelative to the unperturbed electric fields, we fit the difference between the electric fieldfrom the perturbed magnetic field and the electric field from the unperturbed magneticfield with a Gaussian and then normalize it by dividing it by the standard deviation of theunperturbed data. We find the standard deviation σ = [0 . , . , . , .
14] for E x , E y , E z and S z , respectively.To summarize, the levels of the errors in the HMI data lead to errors in the electric fieldup to 14% and 13% in E x and E y and 18% in E z . The error in the vertical Poynting flux, S z , is 14%. If we add these uncertainties from the data (in quadrature) to the errors in thetotal Poynting flux that we found when testing the PDFI method (where the discrepancy, δS z , ranged from 1% to 25% for viewing angles ranging from 0 to 60 ◦ , see Figure 10 ofKazachenko et al. (2014)), then we end up with the error ranging from 14% to 29% in thevertical Poynting flux. 16 –
4. RESULTS
To describe the photospheric electric fields E and energy fluxes S in evolving NOAA11158, we use two approaches. In Section 4.1 we show the spatial distribution of B , V , E and S at two times, before and after the X2.2 flare. In Section 4.2 we analyze their temporalspatially integrated evolution over the six days of observations. These approaches allow usto capture both the spatial structure at quiet times before the flare with the changes duringthe flare, and the overall long-term behavior of the active region. As an example of the spatial distribution of electric fields and Poynting fluxes in NOAA11158, we selected two instances before T pre = 01 : 36 UT and after T post = 02 : 12 UT theX2.2 flare. These instances are of particular interest since the photospheric vector magneticfield changed abruptly during the flare (Petrie 2012; Wang et al. 2014), and various flaresignatures, such as flare ribbons, have been observed during this time frame. Here weinvestigate the changes in velocities, electric fields, and Poynting fluxes associated withthese changes. We note that the 12-minute-cadence data shown here are derived from atapered temporal average that is performed every 720 seconds using observations collectedover 22 . B preflare and B postflare Figure 4 shows the vector magnetic fields B centered at NOAA 11158 before (preflare, T pre = 01 : 36 UT) and the first time step after the onset of the flare (postflare, 17 – T post = 02 : 12 UT) and also the difference image between the two (right panel). Asseen from the difference image on the right, the horizontal magnetic field close to PILincreased by over 300 G during the flare (see arrows), while the vertical magnetic fieldremained nearly constant. This horizontal-field increase is consistent with the magneticfield contraction scenario of Hudson et al. (2008) and is described in detail by Petrie(2012); Wang et al. (2014) and Sun et al. (2012). On the difference image we also noticethe two circular patterns, directed counter-clockwise in negative (pixel coordinates [20,90])and clockwise in positive (pixel coordinates [120,80]) polarities, meaning that the fieldconnecting positive and negative polarities becomes more left-handed. Since both preflareand postflare magnetic fields have a right-handed twist (i.e. horizontal magnetic fieldshave clockwise orientation in the negative polarity and counter-clockwise orientation inthe positive polarity), the observed change, dB , decreases the twist present in the preflaremagnetogram – direct evidence of an abrupt magnetic twist decrease in both sunspotsduring the flare, which contradicts arguments (Melrose 1995) that the vertical currentdensity through the photosphere should not change on the flare time scale. The suddenchange in twist might arise from removal of magnetic helicity from the active-region’smagnetic field by a coronal mass ejection (Longcope and Welsch 2000; Petrie 2012). V preflare and V postflare For completeness, in Figure 5 we show the three components of the velocity fieldat the times before and after the flare as well as the difference image between the two.The Doppler velocity has a strong (1 km s − ) upflow close to PIL and in the sunspot’sumbra. During the flare the Doppler speed does not exhibit any prominent changes (see thebackground of the right panel), although such changes have been reported in other flares(Deng et al. 2006). However the horizontal velocity field does change: there is an apparent 18 –Fig. 4.— Horizontal (arrows) and vertical (grayscale background) components of the mag-netic field in NOAA 11158 at preflare ( top left ) and postflare times ( bottom left ), and thedifference horizontal field between the two, postflare minus preflare ( right panel). The back-ground white and black colors represent positive and negative vertical magnetic fields, re-spectively. The blue and red colors correspond to horizontal field in areas of positive andnegative values of the background vertical magnetic field B z . The white and black contoursoutline the positive and negative vertical magnetic fields at B z = ±
400 Gauss. The arrowsin the right bottom corners show scales for horizontal magnetic field vectors, B h (left panels)or its change, dB h (right panel). ∼ . top left ) and postflare times ( bottom left ), and the difference image between thetwo ( right panel). The background white and black colors represent positive and negativeDoppler velocities, respectively, where positive is toward the viewer (opposite to the usualastrophysical convention). The blue and red colors correspond to V h originating in areas ofpositive and negative values of background vertical velocity. The white and black contoursoutline the positive and negative vertical magnetic fields at B z = ±
400 Gauss. The arrowsin the right bottom corners show scales for horizontal velocities. E preflare and E postflare Finally, in Figure 6 we show the vector electric field maps (“vector electrograms”)calculated using the
PDFI method before and after the flare, and the difference betweenthe two. The horizontal electric fields E h range from 0 to 1 . E h at the PIL is a consequence of thesteady velocity upflows and large horizontal magnetic fields along the PIL that lead to anon-inductive, horizontal electric field, E Dh — the “D” in PDFI — oriented in a perpendiculardirection to the PIL. Given the flare-driven increase in the horizontal magnetic field atand near the flaring PIL, this E Dh and hence E h also increase, by up to 0 . E z varies within [ − . , . E z ≈ − . E Pz , is related to the time derivative of vertical current density via ∂ t J z = c ( ∇ cE Pz ) / (4 π ). Hence, the presence of a nonzero E z is related to changes in thevertical current, which is mostly concentrated close to the main PIL (Petrie 2013; Janvieret al. 2014). S preflare and S postflare Figure 7 shows snapshots of the three components of the Poynting flux vector, S = c π E × B , before and after the flare and also the difference image between the two.The vertical Poynting varies within [ − . , . × ergs · cm − · s − . The maximumvalue of this range is around three times smaller than the steady-state photospheric energyflux estimated from the Stefan-Boltzmann law, given the temperature T photosphere = 5500 K , f ∼ × erg · cm − · s − . By analyzing the distribution of the vertical Poynting flux,we find that more than half of the spatially integrated signed Poynting flux both at preflareand postflare instants is injected in the range of S z = [10 , ] ergs · cm − · s − , whilethe rest of the energy is injected in the range of S z = [10 , ] ergs · cm − · s − with the 21 –Fig. 6.— “Electrogram”: Horizontal (arrows) and vertical (background) electric field compo-nents in NOAA 11158 at preflare and postflare times, and the difference image between thetwo. The blue and red colors correspond to E h in areas of positive and negative background E z . The white and black contours outline the positive and negative vertical magnetic fieldsat B z = ±
400 Gauss. The arrows in the right bottom corners show scales of horizontalvectors.positive flux. We find the strongest vertical Poynting fluxes close to the main PIL: duringthe flare the mean values of positive and negative S z contributions within a white dashedbox, shown on the right panel of Figure 7, change from (1 . . × ergs · cm − · s − and from ( − . − . × ergs · cm − · s − respectively, or spatially integrating over thesame box in terms of energy injection rate, from 1 . × ergs s − to 4 . × ergs s − (see white enhancement on the right panel). Summed over the 22 minutes of the GOESflare time for this event, the energy crossing the photosphere during the flare within the 22 –box, would amount to about 4 . × erg, that is very small compared to typical energiesreleased in large flares such as this (10 erg or more). We also note that the absolute valueof the spatially integrated positive Poynting flux (over NOAA 11158 ) is more that twotimes larger than the absolute value of the spatially integrated negative Poynting flux –more magnetic energy is transported through the photosphere upward into the corona thandownward.Similar to the evolution in E h , the magnitude of the horizontal component of thePoynting flux, S h , increases near the PIL by up to 8 × ergs · cm − · s − . As noted above,the combination of steady upflows found in the Dopplergrams at and near the PIL with thesystematic increase in the transverse magnetic fields along the PIL leads to an increase inthe non-inductive, horizontal electric field, E Dh , which in turn produces an increase in thePoynting flux there.Summarizing some of the changes in the photosphere during the flare, in Figure 8, weshow the vertical and horizontal magnetic fields, horizontal electric field and the verticalPoynting flux before and after the flare and also the scans along a fixed x value (or columnin the 2D image) through the center of the active region across the PIL at those twomoments (right column). From the right panel we find that while the vertical magnetic fielddoes not exhibit any significant changes (see B z -panel, first row), the horizontal magneticfield increases at the PIL by over 300 Gauss (see B h -panel, second row). This leads to anincrease in the horizontal Doppler electric field by up to 0 . E h close tovertical dotted lines, third row) and an increase in the vertical Poynting flux close to PILby up to 10 ergs · cm − s − , from [1 .
25 to 2 . × ergs · cm − s − . 23 –Fig. 7.— Horizontal (arrows) and vertical (background) Poynting vector field componentsin NOAA 11158 at preflare and postflare times, and the difference image between the two.The blue and red colors correspond to S h in positive and negative areas of the background S z . The white and black contours outline the positive and negative vertical magnetic fieldsat B z = ±
400 Gauss. The arrows in the right bottom corners show scales of horizontalvectors. The range of the background S z is [ − , × ergs cm − s − in the left panelsand [ − . , . × erg cm − s − in the right panel. The white dashed box in the centerof the right panel indicates the field of view where energy injection rates are calculated (seeSection 4.1.4). To quantify the long-term evolution of the energy and helicity fluxes in NOAA 11158,we integrate the photospheric Poynting and helicity flux maps over the AR’s field of view 24 –(665 × Panels (A-F) of Figure 9 show the six-day time evolution of area-integrated potentialand free vertical energy fluxes and their sum, the vertical Poynting flux: [ S z,p , S z,f , S z ]. Thepotential component, S z,p is calculated by taking a time derivative of the coronal potentialenergy, computed as a surface integral over the photosphere, with the potential functioncomputed using a Green’s function technique. The free component is the difference between S z and S z,p : S z,f = S z − S z,p . In the three insets on the left panels we also show evolution of S z from one hour before to four hours after the flare. In panels A and C, large fluctuationsare present in S z,p and S z (which were calculated independently), with timescales rangingfrom about 4 −
24 hr. In panel B, S z,f does not exhibit such strong fluctuations at shortertimescales, which evidently cancel out in the difference S z - S z,p ; a residual 24-hr. periodicitycan still be seen, however. These fluctuations are related to orbital motions of the SDOsatellite (Liu et al. 2012b) and are present over the whole field of view. Unfortunately, thesefluctuations are internal to the HMI polarization measurements, hence their removal is atime-consuming and complicated task. Efforts to address these artifacts are underway, butthey have not been fully addressed at the time this article was written. Fortunately, whenthese fluctuations are integrated over several hours, they do not cause large fluctuations inthe total cumulative fluxes (see panel E).How do S z,p , S z,f , and S z change during the flare? Insets of Panels A-C of Figure 9show that S z,f and S z increase during the flare, after which they return to nearly preflarevalues. We can understand this change in the following way. Since the change in themagnetic field close to the PIL is practically a step function (Petrie 2012), the value ofthe Poynting flux, which involves the temporal derivative of magnetic field, should indeed 25 –appear like a delta function in time. We conclude that while the transients in the verticalPoynting flux include spurious signals due to flare-induced effects on the HMI spectral line(Maurya et al. 2012), the observed peak in the Poynting flux around the flare time is real.(The spatial distributions of the Poynting flux before and after the flare are shown in thebottom row of Figure 8.)Panel D shows a steady growth in the region’s total unsigned flux after February 14,after the bulk of the region’s flux has emerged. The growth at the time of the flare isconsistent with the steady upflows seen along the main PIL, which we expect to carrymagnetic fields upward from the solar interior, across the photosphere. A slight 24-hrperiodicity is discernible from 14 February onward.As shown in panel E, if we integrate S z,f , S z,p and S z in time, the fluctuationsseen in panels A-C largely disappear. Notably, no contribution from the flare transientis obvious. We also observe that, for a time, before February 14 12:00 UT, the activeregion appears to possess negative free energy, a spurious result. A probable explanationfor this is that our Poynting flux estimates are affected by the orbital motions, while thepotential energy that depends only on B z suffers differently from this systematic error. Analternative approach that might ameliorate the negative free energy budget is modificationof boundary conditions used to compute the potential field – if instead of the Neumannboundary condition, given by B z , we had used a “hybrid” Dirichlet-Neumann boundarycondition (Welsch and Fisher 2015), which uses both ( ∇ h · B h ) and B z from the HMI data,our estimate of the potential energy would be E p,hybrid = 4 . × erg by the flare time (cf. E p = 8 . × erg ) implying three times more free energy E f,hybrid = 6 . × erg.To conclude, taking into account the 29% uncertainty in S z , due to uncertainties inthe HMI data and the PDFI method (see Section 3.3), we find the total energy that enteredthe photosphere by the flare time to be E = [10 . ± . × erg, which consists of 26 – E p = 8 . × erg of potential energy and E f = 2 . × erg of free energy. Figure 10 shows time evolution of area integrated helicity flux rates dH R dt (left) andhelicity fluxes H R (right) in NOAA 11158 calculated using the PDFI electric fields (
PDFI )and the velocity field alone (
DFI ). The apparent large-scale 24-hour periodicity is mostlikely due to signal contamination from the satellite motion (Liu et al. 2012b). Until 12:00UT on February 14 2011, the two helicity flux rates match each other, but at later timesthey diverge: dH R dt from the velocity field ( DFI , black) becomes significantly larger than thatfrom the
PDFI electric field (green). What could be the cause of this difference? Note thatthe primary extra input to the
PDFI results compared to the
DFI results is informationabout ˙ J z , which sets the inductive E z ; this, in turn, is coupled to E h through the idealOhm’s constraint, E · B = 0. To investigate the role that the ideal Ohm’s constraint playsin the PDFI helicity estimate, we calculated the helicity flux rate from the non-ideal
PDF (i.e.PTD-Doppler-FLCT) electric fields (see blue curve on Figure 10). We find that the
PDF and the
DFI helicity flux rates have a similar time evolution, although the
PDF helicity fluxrate is somewhat smaller than the
DFI rate. The discrepancy between
PDFI and
DFI helicityflux rates could arise because the
DFI helicity flux rate is only sensitive to emergence andshearing. Shearing motions however, can then induce the photospheric field to unwind viathe propagation of torsional Alfv´en waves into the interior, along with the accompanying˙ J z and E z signatures, that only PDFI captures (cf., the transient twist variations modeledby Longcope and Welsch 2000). The constraint E · B = 0 then couples the resulting changein E z (and ˙ J z ) to a change in E h , thereby further reducing the PDFI helicity flux rate.Integrating over time we find that the total accumulated helicity flux injected through thephotosphere from the start until the flare time (right panel) is H R,DF I = 7 × Mx , 27 –which is similar to the value from PDFI : H R,P DF I = 8 . × Mx . 28 –Fig. 8.— Left and Middle columns:
Vertical and horizontal magnetic fields, horizontal electricfield, and the vertical Poynting flux before and after the flare.
Right column:
Vertical scansfor B z , B h , E h and S z before (blue) and after (red) the flare at the locations shown withblue and red lines in the two left columns. Horizontal dotted lines on left and middle panelscorrespond to vertical dotted lines on the right panel. 29 –Fig. 9.— Evolution of magnetic and Poynting fluxes in NOAA 11158 during six days: (A-C)Area-integrated potential (A), free (B) and total (C) Poynting fluxes. Corner insets showevolution of the same quantities one hour before and 2 hours after the flare; the X-axis is inhours; (D) Area integrated unsigned magnetic flux, (E) Area and time integrated free andpotential components of the vertical Poynting flux and their sum, (F) GOES 5-minutes softX-ray light curve (1-8 ˚A channel). The vertical dotted line indicates the GOES flare peaktime at 01:56 UT. All the quantities were calculated within the FOV shown in A-D Panelsin Figure 1. 30 –Fig. 10.— Evolution of helicity flux rates and helicity fluxes in NOAA 11158 during sixdays: Left:
Spatially integrated helicity flux rates dH R dt calculated from velocity field ( DFI ,black) and
PDFI (green) and
PDF (blue) electric fields;
Right : Total helicity fluxes H R , i.e.quantities on the left panel integrated in time. 31 –
5. DISCUSSION
Here, we discuss how electric fields, energy fluxes (Poynting fluxes), and helicity fluxesestimated with the
PDFI method compare to earlier results.Using the
PDFI method, we find photospheric electric field components in NOAA 11158whose amplitude varies from − . . . E z ≈ − . E z to be E z ≈ [0 . − .
2] V/cm, i.e. significantly smaller than our estimates.Other attempts have focused on estimating the electric field in the corona inside thereconnecting current sheet (RCS) (Poletto and Kopp 1986; Wang et al. 2003, 2004; Qiuet al. 2002, 2004; Jing et al. 2005). For example, using a relationship between the electricfield along the current sheet and the observable velocity and magnetic field (Priest andForbes 2000), Poletto and Kopp (1986) derived the maximum value of electric field in theRCS to be 2 V/cm. In a similar way, Wang et al. (2003) found the coronal electric fieldduring the two-ribbon flare occurring in two stages: the coronal electric field remained near1 V/cm averaged over 20 minutes during the first stage, and was followed by values of 0 . . − S z ≈ × ergs · cm − s − , i.e. several times smaller than thesteady state solar luminosity, 6 × ergs · cm − s − , but of the same order of magnitude.If we look at the values of S z across the AR, where most of the magnetic energy is comingfrom, we find even smaller values, ranging from 10 to 10 ergs · cm − s − . A questionnaturally follows this calculation: Is such a Poynting flux consistent with the amount ofmagnetic energy stored in the coronal part of the active region? This is the question weaddress in Table 1.In Table 1, we compare estimates for coronal energy and helicity for NOAA 11158 thatwe derive in this paper with results from other papers. In the first four rows, we showresults from Evolutionary estimates , i.e. works by Tarr et al. (2013); Liu and Schuck (2012);Tziotziou et al. (2013); Vemareddy et al. (2012b) and Jing et al. (2012), where energy orhelicity or both are found from the evolution of photospheric magnetic fields, by summingthe energy or helicity flux rates injected through the photosphere from the beginning of themagnetogram sequence until the moment before the flare. In the last five rows, we showresults from
Instantaneous estimates , i.e. works by Malanushenko et al. (2014); Aschwandenet al. (2014); Sun et al. (2012); Tziotziou et al. (2013) and Jing et al. (2012), where a singleclose to the flare time magnetogram or EUV image is used to estimate the energy or helicityof the corona. The “Method” column shows the type of method used to find the energy 33 –and helicity estimates on the right. In some papers, other methods were used to calculatethe potential energy; they are indicated with a letter next to the estimate. In addition, weindicate the type of input data used in the calculations. This helps to explain the differencein results between some papers that used the same methods, but applied them to differentinput data, e.g. differences in estimates of the potential field energy, E P , by Tarr et al.(2013) and Sun et al. (2012), where B LOS and B z have been used respectively.The total magnetic energy from different models, shown in Table 1, ranges from6 × ergs (Malanushenko et al. 2014) to 12 × ergs (Tziotziou et al. 2013). Wenoticed that both coronal NLFFF methods, which use the EUV coronal loops instead of thetransverse magnetic field as a constraint for the NLFFF extrapolation (Malanushenko et al.2014; Aschwanden et al. 2014), derive total energies that are the smallest of all total energyestimates: 6 × ergs and 8 . × ergs respectively. In contrast, the photosphericNLFFF methods, which use a vector magnetogram for extrapolation (Sun et al. 2012;Tziotziou et al. 2013), yield the largest estimates for the total energy: 10 . × ergsand 12 × ergs respectively. The evolutionary estimates, that derive the total energyby integrating the energy flux, inferred from the photospheric velocity or electric fields,yield total energies in between the two. For example, using the MCC method, Tarr et al.(2013) derive E = 8 . × ergs. Using DAVE4VM approach, Liu et al. (2012b) andTziotziou et al. (2013) yield similar estimates of E = 8 . × ergs and E = 8 . × ergs, respectively. Finally, in this paper using the PDFI method we find the total energyof E = [10 . ± . × ergs. To conclude, taking the PDFI and HMI uncertainties intoaccount, we find that the total energy, estimated right before the flare, is consistent with E from DAVE4VM, MCC and NLFFF and is slightly larger than the coronal NLFFFestimates.If we look at the temporal evolution of energy E , we notice that E derived from PDFI
34 –(Figure 9) is consistent with E from DAVE4VM (Figure 14 in Liu et al. (2012b)) andNLFFF (Figure 4 in Sun et al. (2012)). In fact, the energies E derived from the PDFI andDAVE4VM are almost identical until February 14 18:00, and then start diverging severalhours before the flare. By the end of magnetogram sequence, at 18:00 UT on 16 February,the E from DAVE4VM and PDFI are 12 × ergs and 14 × ergs, respectively.Still, this discrepancy lies within the uncertainty of the total energy estimate (29%, seeSection 3.3). Comparing the temporal evolution of E from PDFI and NLFFFs, we againfind that they are consistent with each other, with less than 15% differences between thetwo, which is within our uncertainty (29%, see Section 3.3).The potential field energy from different models, also shown in Table 1, ranges from4 . × ergs (Malanushenko et al. 2014) to 8 . × ergs (this paper). Similar to ourapproach, Sun et al. (2012) used the Green’s function and B z to estimate potential fieldenergy of 8 . × ergs, that, within the HMI uncertainty, is consistent with our estimate.The potential field energies calculated by Tarr et al. (2013); Malanushenko et al. (2014)and Aschwanden et al. (2014) are smaller than E p from this paper and from Sun et al.(2012). This difference might be due to the fact, that in contrast to this paper and Sunet al. (2012), where B z is used to calculate E p , Tarr et al. (2013); Malanushenko et al.(2014) and Aschwanden et al. (2014) used B LOS that tend to underestimate the field at B LOS > . − . E p in Table 1.The free magnetic energy from different models, also shown in Table 1, ranges from1 . × ergs (Aschwanden et al. 2014) to 2 . × ergs (Tarr et al. 2013). Using theMCC model and the flare ribbon locations, which allow one to derive the footprint of the 35 –reconnecting magnetic fields, Tarr et al. (2013) find the initial pre-flare free energy in thecorona to be E f = 2 . × ergs, consistent with estimates of Sun et al. (2012) andthe results of this paper. Tarr et al. (2013) also find that more than 50% of this energy, dE = 1 . × ergs, is released during the flare. Coronal NLFFF methods (Aschwandenet al. 2014; Malanushenko et al. 2014) find that 60% to 80% of the free energy is releasedduring the flare, but the values of the pre-flare free energy E f that they find are roughlyfactors of two (or more) times smaller than the values from Tarr et al. (2013); Sun et al.(2012), and this paper.Another important fact one must keep in mind when comparing cumulative freePoynting fluxes and coronal free energies, is that the total Poynting flux only gives usinformation about the total energy that entered the corona from the photosphere. Whatwe do not know is how much of this total energy leaves the corona from above into theheliosphere during eruptive flares prior the X2.2 flare. For this reason, our PDFI freeenergy estimate is the upper limit of the energy available in the corona. To summarize, thefree magnetic energy, which we find from a difference of integrated Poynting flux and thepotential energy, is up to two times larger than the free energy estimated from the coronalNLFFF codes, and 20 −
30% smaller than free-energy estimates from the MCC modeland photospheric NLFFF extrapolations (Tarr et al. 2013; Sun et al. 2012). The temporalevolution of the
PDFI free energy is within 20 −
30% of E f ( t ) from the photospheric NLFFFcode.Finally, in the last column of Table 1, we compare the total relative magnetic helicitiesbefore the X2.2 flare. The time integrated total helicity flux we find with DFI and
PDFI techniques is H R,DF I = 7 × M x and H R,P DF I = 8 . × M x respectively. Thedifference between the two is consistent with the results we found for the ANMHD testcase, where the DFI method reconstructed the total helicity around 10% more accurately 36 –than the
PDFI method (see Table 3 in Kazachenko et al. (2014)). For comparison, usingDAVE4VM, Liu et al. (2012b) and Tziotziou et al. (2013) found that the total amountof helicity injected into corona is 6 . × Mx and 8 . × Mx respectively. Theseresults are consistent with our PDFI estimate, given the differences between the DAVE4VMand the
PDFI accuracies (Kazachenko et al. 2014) – the ratio between the total
PDFI -and DAVE4VM-reconstructed helicity fluxes and the actual helicity flux are 0 .
94 and 1 . . × Mx and 5 . × Mx respectively. Asshown by Schuck (2008) for the ANMHD test case, combining DAVE flows with ANMHD’svertical velocity overestimates the total helicity flux by at least 40%, hence the disagreementbetween DAVE and PDFI results for NOAA 11158 is not surprising. Finally, using differentNLFFF approaches, Tziotziou et al. (2013) and Jing et al. (2012) find H R = 13 × Mx and H R = 5 . × Mx respectively. To summarize, using the PDFI method we find therelative magnetic helicity consistent with DAVE4VM estimates (Liu et al. 2012b; Tziotziouet al. 2013), but very different from (and in between) the NLFFF estimates (Tziotziou et al.2013; Jing et al. 2012).
6. CONCLUSION
The electric field on the Sun plays an important role in transporting energy, heatingplasma, and accelerating and transporting charged particles. Estimates of photosphericelectric and magnetic field vectors make the estimation of Poynting flux of electromagneticenergy crossing the photosphere and the flux of relative magnetic helicity straightforward.Taking advantage of the newly released high temporal and spatial resolution HMI vectormagnetograms (Schou et al. 2012), and the recently developed
PDFI electric-field inversion 37 –Table 1: Summary Table: Comparison of coronal energy and helicity in NOAA 11158 wherean X2.2 flare occurred on February 15 2011 01:35 UT: dE - change in the coronal freemagnetic energy during the flare, [ E f , E p , E ] - free, potential and total magnetic energiesin the corona before the flare at 01:35 UT, H R – relative magnetic helicity in the coronabefore the flare at 01:35 UT. In Evolutionary or Photospheric estimates, coronal energy andhelicity are calculated cumulatively from tracking magnetic field evolution, i.e. by integratingphotospheric energy and helicity flux rates in time. In
Instantaneous or Coronal Estimatesenergy and helicity are calculated instantaneously, using extrapolation of the photosphericvector magnetic fields. All quantities have been calculated using the indicated
Method , unlessa more specific method is used, like in the case of potential energies (see letters ( d ) − ( f ) ). Inphotospheric estimates we use a start time of 14:11 UT on 10 February. Paper Method Data dE E f E p E H R ergs 10 Mx Evolutionary EstimatesThis paper
PDFI
Method B , V z – . . ( d ) . . ... DFI method B , V z – – – – 7 . B LOS . . . ( d ) . B , V z – – – 8 . B , V z – – – 8 . . B – – – – 6 . B – – – – 5 . Instantaneous Estimates
Malanushenko et al. (2014) Coronal NLFFF ( a ) B LOS . . ( e ) ( a ) B LOS . . . ( f ) . ( b ) B . . . ( d ) . ( c ) B . ( b ) B – – – – 5 . NLFFF method : ( a ) – EUV loops instead of B t are used as a constraint, ( b ) – Wiegelmann (2004), ( c ) – Georgoulis et al. (2012). Potential field methods: ( d ) – Green’s function, Sakurai (1989), ( e ) – Green’s function, Chiu and Hilton (1977), ( f ) – Aschwanden and Sandman (2010).
38 –method (Kazachenko et al. 2014), we apply the method to a six-day sequence of vectormagnetic field measurements of NOAA 11158, from 10 to 16 February 2011. From these12-minute cadence measurements, we derive the evolution of electric fields, the Poyntingflux, and the helicity flux. During the interval of study, an X2.2 flare occurred, along with35 M- and C-class flares.We analyze the spatial distribution of the derived electric field and Poynting fluxmaps, their temporal evolution, and changes during the X2.2 flare. We compare derived
PDFI electric fields with various estimates of a typical coronal and photospheric electricfields made to date,
PDFI energies and helicities with those previously reported in theliterature for this active region (NLFFF, MCC and DAVE4VM estimates). The results arethe following:1. We find the photospheric electric field vector, which typically ranges from − . − . × erg · cm − s − with majority of the energy flux moving upward into corona and more than half of thetotal energy input rate injected from within the range of [10 to 10 ] erg · cm − s − .The largest vertical Poynting flux is concentrated at the PIL.3. Integrating the Poynting flux in time we find the total magnetic energy before theflare, E = [10 . ± . × ergs. In spite of a very different approach, it is consistentwithin the uncertainty with the total energies from DAVE4VM, MCC and NLFFFmethods’ estimates and larger than the coronal NLFFF estimates. 39 –4. The potential field magnetic energy before the flare, estimated via the Green’sfunction, E p = 8 . × ergs, is consistent with E p = 8 . × ergs (Sun et al. 2012).Tarr et al. (2013); Malanushenko et al. (2014) and Aschwanden et al. (2014) derivea similar or somewhat smaller E p , using different computation methods and B LOS instead of B z that is prone to being underestimated in the strong field regions due tolimitation of the LOS pipeline algorithm (Hoeksema et al. 2014).5. The free magnetic energy before the flare from the PDFI method, E f = 2 . × ergs, is up to two times larger than the free energy estimated by coronal NLFFFcodes (Aschwanden et al. 2014; Malanushenko et al. 2014), and 20 −
30% smaller thanthe free energy estimates from MCC model and photospheric NLFFF extrapolations(Tarr et al. 2013; Sun et al. 2012).6. Analyzing the temporal evolution of cumulative energy, E , from PDFI , NLFFFs andDAVE4VM, we find less than 15% differences between the three. The temporalevolution of the
PDFI E f is less than 20 −
30% different from the E f from thephotospheric NLFFF codes and several times larger than E f from the coronal NLFFFcodes (Aschwanden et al. 2014; Malanushenko et al. 2014).7. We find the relative magnetic helicity to be consistent with DAVE4VM estimates(Liu et al. 2012b; Tziotziou et al. 2013), but very different from the NLFFF estimates(Tziotziou et al. 2013; Jing et al. 2012).8. Using Monte-Carlo simulations, we find that the levels of the errors in the HMI datalead to uncertainties in the horizontal electric field of 13% to 18% that result inerrors in the vertical Poynting flux of around 14%. If we add those uncertainties tothe errors that we found when testing the PDFI method ( <
25% in S z , Kazachenkoet al. (2014)), then we end up with 14% to 29% errors in the vertical Poynting fluxdepending on the LOS angle. Also, since some free energy might have been released 40 –by flares prior to the X2.2 flare, we view the PDFI estimates of the free energy injectedthrough the photosphere to be upper limits on the total free energy available beforethe flare.This study is the first application of the
PDFI electric field inversion technique tophotospheric vector magnetic field and Doppler measurements. We find that the totalamount of energy and helicity injected through the photosphere before the flare estimatedby the
PDFI method is consistent with estimates from other approaches, in spite of differingtechniques. This agreement is very promising, implying that the
PDFI technique is notonly capable of describing the coronal energy and helicity budget, but can also provideinstantaneous estimates of energy and helicity transferred through the photosphere.We believe that both the derived dataset of
PDFI electric fields and the
PDFI method itself will be useful to the science community for analysis of the evolution andspatial distribution of the photospheric electric fields, fluxes of energy and helicity, andtheir relationships with flare activity. In addition,
PDFI electric fields can be used astime-dependent boundary conditions for data-driven models of coronal magnetic fieldevolution (Fisher et al. 2015). The dataset for NOAA 11158 is available for downloading onour website . http://cgem.stanford.edu
41 –We thank the US taxpayers for providing the funding that made this research possible.We thank the anonymous referee for thoughtful input that have improved the manuscript.We acknowledge funding from the Coronal Global Evolutionary Model (CGEM) award NSFAGS 1321474 (MDK, BTW, GHF), Coronal Global Evolutionary Model (CGEM) awardNASA Award NNX13AK39G (XS, YL), NSF Award AGS-1048318 (GHF), NASA AwardNNX13AK54G (MDK), NSF SHINE Postdoc Award 1027296 (MDK), NSF’s NationalSpace Weather Program AGS-1024862 (BTW), the NASA Living-With-a-Star TR&TProgram NNX11AQ56G (MDK, BTW, GHF), and the NASA Heliophysics Theory ProgramNNX11AJ65G (GHF, BTW). 42 –
A. SCALING B AND E TO CARTESIAN MERCATOR MESH
The data we analyze have been transformed from plane-of-sky to Mercator projectionwith a local, Cartesian coordinate system centered on NOAA 11158. The distortion of pixelscales in this transformation has implications for inferring electric fields from magneticevolution that we describe in detail here.When observed magnetic fields are reprojected from the Sun’s observed plane-of-skysurface onto a plane using a conformal (angle-preserving) mapping, pixel areas in differentregions of the new coordinate system generally do not correspond to the same physical areason the Sun. Denoting the area of a pixel ij on the Sun as A Sij , and in the mapped coordinatesas A Mij , A Mij ≈ A Sij near the center of the projection, but A Mij (cid:54) = A Sij far from center. Incontrast, the reprojection does not directly alter the magnetic field values themselves: thefields are interpolated onto the new grid point, but the interpolation attempts to faithfullyrepresent measured field values at each point. While observed magnetic field vector datahave units of field strength, i.e. Gauss, they are more accurately described as measurementsof pixel-averaged flux densities, Mx/cm . This is because the flux in pixel ij could beconfined within a subregion of the pixel with fraction area f ij A Sij , where the fill fraction f ij obeys 0 < f ij <
1, implying a true field strength B true = B app /f larger that the apparentfield strength (“pixel-averaged flux density”) B app measured by the instrument.The question, however, arises: Do we need to modify the field strengths in the newprojection – call the original field strengths B S , and the reprojected field strengths B M –to account for the distorted areas? If so, how? Do vertical and horizontal magnetic fieldcomponents need to be compensated in the same way? How should velocities and electricfields be modified to compensate for this distortion?Below we consider the transformation used in this paper, from the plane-of-sky toMercator projection (Welsch et al. 2009). Briefly, the distortion of scale in cylindrical 43 –projections such as Mercator is a function of latitude alone. The horizontal Mercatorcoordinate, x , is mapped one-to-one with the heliocentric longitude, φ , and is independent oflatitude. Because the distance between lines of constant longitude decreases with increasinglatitude, the projection’s scale (the distance of the sphere corresponding to a fixed distancein the projected image) must decrease (fewer solar Mm per Mercator pixel) with increasinglatitude. As lines of constant longitude converge towards the poles on a spherical surface,the physical distance between such lines goes to zero. In Mercator coordinates, however, thedistance between two lines of constant longitude ∆ x , is fixed and is independent of latitude θ . Effectively, this means the projection magnifies distances towards the poles. (Thiseffect is easily seen on Mercator projections of the Earth’s surface, on which Antarcticaand Greenland, for example, appear too large relative to landmasses at lower latitudes.)Consequently, displacements in the vertical Mercator coordinate, dy , corresponding to afixed latitudinal displacement dθ , should increase towards the poles. Because the physicallength corresponding to a fixed dy shrinks towards the poles as 1 /cos ( θ ), and dx scales inthe same way, pixel areas in the reprojected system scale as cos ( θ ) compared to areas onthe Sun.Consider two reprojected pixels with the same normal magnetic field, one at a highlatitude and one at a low latitude, denoting these B Hn = B Ln respectively. These have thesame fluxes in the projection, Φ M,H ≈ Φ M,L . The flux in each reprojected pixel, however,corresponds to different fluxes on the Sun: Φ
M,L ≈ Φ S,L , but Φ
S,H < Φ M,H since the actualsolar area corresponding to the high-latitude pixels is smaller. We choose to handle this bymultiplying B n in each reprojected pixel by cos ( θ ), to compensate for distortion of its areaas a function of latitude; at the same time, we do not rescale pixel lengths.When flux emerges into a pixel, v n transports horizontal field B h along a pixel edge oflength L over a time interval ∆ t , meaning Φ em = v n LB h ∆ t has emerged (see discussion in 44 –Section 2, and Figure 3 of Welsch et al. (2013)). This means that the normal flux in thepixels that share this edge must change by Φ em , to account for the emerged flux. We haveassumed that flux in each pixel sharing L has been compensated by a factor of cos ( θ ). Forthe emerged horizontal flux to match the changes in the vertical flux, two factors of cos ( θ )must be present in the product v n B h , since we are not rescaling L . We can scale v n by cos α ( θ ) and B h by cos β ( θ ), which yields α + β = 2 . (A1)When flux is horizontally transported from one pixel to another, v h transports thevertical field B n across a pixel edge of length L over a time interval ∆ t , meaning a fluxΦ xport = v h LB n ∆ t has been moved. This means the normal fluxes in the pixels that sharethis edge must each change by Φ xport , to account for the transported flux. Because B n hasalready been rescaled by cos θ , v h in the product v h LB n does not need any scaling.The pixel-integrated vertical Poynting flux S z = ( E h × B h ) at high latitude must bescaled by cos ( θ ), to account for the area distortion. Recall that cE h is proportional to thesum of v n B h , scaled by cos α + β ( θ ), and v h B n , are already scaled by cos ( θ ). Because S z mustscale as cos ( θ ) for any v , we can consider the special case v h = 0, implying S z = v n B h , so α + 2 β = 2 . (A2)Comparing equations A1 and A2 yields β = 0, so B h is not changed, but α = 2, so v n is scaled by cos ( θ ). Physically, this implies that the vertical transport of emerging fluxat high latitudes is scaled to make sure a “scaled amount” of flux emerges. This scalingimplies that E h is automatically and implicitly scaled (via scaling applied to B h and v n ) by cos ( θ ), and that E z = v h × B h is unscaled. 45 –Summarizing the above, we scale B n , v n , E h by cos ( θ ) and do not scale B h , v h , E n . 46 – REFERENCES
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