The Astrophysical Journal, 720:717–722, 2010 September 1 doi:10.1088/0004-637X/720/1/717C© 2010. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

MAGNETIC ENERGY SPECTRA IN SOLAR ACTIVE REGIONS

Valentyna Abramenko and Vasyl Yurchyshyn

Big Bear Solar Observatory, 40386 N. Shore Lane, Big Bear City, CA 92314, USAReceived 2010 April 21; accepted 2010 July 14; published 2010 August 12

ABSTRACT

Line-of-sight magnetograms for 217 active regions (ARs) with different flare rates observed at the solar disk centerfrom 1997 January until 2006 December are utilized to study the turbulence regime and its relationship to flareproductivity. Data from the SOHO/MDI instrument recorded in the high-resolution mode and data from the BBSOmagnetograph were used. The turbulence regime was probed via magnetic energy spectra and magnetic dissipationspectra. We found steeper energy spectra for ARs with higher flare productivity. We also report that both the powerindex, α, of the energy spectrum, E(k) ∼ k−α , and the total spectral energy, W = ∫

E(k)dk, are comparablycorrelated with the flare index, A, of an AR. The correlations are found to be stronger than those found between theflare index and the total unsigned flux. The flare index for an AR can be estimated based on measurements of α andW as A = 10b(αW )c, with b = −7.92 ± 0.58 and c = 1.85 ± 0.13. We found that the regime of the fully developedturbulence occurs in decaying ARs and in emerging ARs (at the very early stage of emergence). Well-developedARs display underdeveloped turbulence with strong magnetic dissipation at all scales.

Key words: Sun: flares – Sun: photosphere – Sun: surface magnetism – turbulence

Online-only material: color figures

1. INTRODUCTION

Existing methods for predicting the flare activity of activeregions (ARs) are not numerous (e.g., Abramenko et al. 2002;Falconer et al. 2002, 2003, 2006; Leka & Barnes 2003, 2007;McAteer et al. 2005; Schrijver 2007; Georgoulis & Rust 2007;Barnes & Leka 2008; see review by McAteer et al. 2009) andthe majority of them are based on first-order statistical momentsof the magnetic field (test parameters are derived from B). Atthe same time, Leka & Barnes (2007) noted that the explorationof higher order statistical moments seems beneficial given thenonlinear nature of a flaring process.

Earlier we presented results of a small statistical study(Abramenko 2005) focused on the relationship between theflare productivity of solar ARs and the power-law index, α,of the magnetic energy spectrum, E(k) ∼ k−α . This magneticenergy spectrum technique is based on the second-order statis-tical moment of a two-dimensional field and shows the distri-bution of energy over spatial scales. The study was based on16 ARs observed predominantly during 2000–2003 with theSOHO/MDI instrument (Scherrer et al. 1995) performing inthe high-resolution (HR) mode. The findings were promising:flaring ARs were reported to display steeper power spectra withthe power index exceeding a magnitude of 2. Flare-quiet ARsexhibited a Kolmogorov-type spectrum with α close to 5/3(Kolmogorov 1991, hereafter K41).

Power spectra calculations, based on the technique describedin Abramenko (2005), will be a part of the pipeline system de-signed to process real-time data flowing from the Helioseismicand Magnetic Imager1 that operates on board the Solar Dy-namics Observatory.2 Here, we evaluate the performance of themethod based on a large data set. We report that both the mag-netic energy stored in large-scale structures and the magneticenergy cascaded by turbulence at small-scale structures (be-low 10 Mm) are comparably correlated with flare productivity

1 http://hmi.stanford.edu/2 http://sdo.gsfc.nasa.gov/

(Section 3). From the magnetic energy spectra, we also deter-mined at what spatial scales the bulk of the magnetic energydissipation occurs (Section 4). This allowed us to make an in-ference about the characteristics of the turbulent regime in ARs,which may be useful as a constraint criterion for the MHDmodeling of ARs.

2. DATA

We selected 217 ARs measured near the solar disk center(no further than 20◦ away from the central meridian), so thatthe projection effect can be neglected. The set covers theperiod between 1997 January and 2006 December. All the ARsdisplayed nonzero flare activity, i.e., at least one GOES flarewas produced by an AR during its passage across the solardisk. Given the typical average flux densities in ARs, smoothpower spectra are not usually obtained for ARs with unsignedmagnetic flux less than 1022 Mx. Therefore, we required thateach AR should possess unsigned magnetic flux that exceedsthis threshold value.

For the majority of ARs (215), MDI/HR magnetograms(pixel size of 0.′′6) were utilized. For two extremely flare-productive ARs only Big Bear Solar Observatory (BBSO)data, obtained at very good seeing conditions with the DigitalMagnetograph (DMG; Spirock 2005; pixel size of 0.′′6), wereavailable. As we have shown earlier (Abramenko et al. 2001),magnetic energy spectra calculated for the same AR fromMDI/HR and BBSO/DMG magnetograms agree very wellat scales larger than 3 Mm. This allows us to use differentdata without the risk of skewing the correlation. From MDIfull disk magnetograms, we determined the trend in the totalunsigned flux in each AR during a three-day time intervalcentered at the time of the MDI/HR magnetogram acquisition.When AR flux variations did not exceed ±10% of the meanvalue, we classified the AR as a stable, well-developed AR. ARsdisplaying larger monotonous changes in the flux were classifiedas emerging or decaying. Unipolar sunspots were detected byvisual assessment.

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The flare productivity of an AR was measured by the flareindex, A, introduced in Abramenko (2005). Since the X-rayclassification of solar flares (X, M, C, and B) is based on adenary logarithmic scale, we can define the flare index as

A = (100S(X) + 10S(M) + S(C) + 0.1S(B))/t. (1)

Here, S(j ) is the sum of all GOES flare magnitudes in the jthX-ray class:

S(j ) =Nj∑i=1

I(j )i , (2)

where Nj = NX,NM,NC , and NB are the numbers of flares ofX, M, C, and B classes, respectively, that occurred in a given ARduring its passage across the solar disk, which is represented bythe time interval t measured in days. I (j )

i = I(X)i , I

(M)i , I

(C)i , and

I(B)i are the GOES magnitudes of X, M, C, and B flares. The

interval t was taken to be 27/2 days for the majority of the ARswith the exception of the emerging ones. In general, those ARsthat produced only C-class flares have flare indices smaller than2, whereas several X-class flares will result in the flare indexexceeding 100. The highest flare index of 584 was registeredfor NOAA AR 10486, which, unfortunately, was not includedin the analysis because it was outside of the MDI/HR field ofview.

3. MAGNETIC ENERGY SPECTRA

Photospheric plasma is thought to be in a turbulent state,subject to the continuous sub-photospheric convection. It can beanalyzed with the energy spectrum (which is frequently referredto as a power spectrum). The energy spectrum is based onsecond-order statistical moments of a field and is very useful forprobing the inter-scale energy cascade and dissipation regimes(see, e.g., Monin & Yaglom 1975; Biskamp 1993).

Our method for calculating the one-dimensional angle-integrated energy spectrum, E(k), of a two-dimensional struc-ture (e.g., a line-of-sight magnetogram, Bz(x, y)) was describedin Abramenko et al. (2001) and later improved in Abramenko(2005). The most reliable scale interval for deriving the energycascade in the spectrum is 3–10 Mm when utilizing MDI/HRand BBSO data (Abramenko et al. 2001). At scales smaller than2–3 Mm, the influence of insufficient resolution is substantial,while the contribution from large sunspots at scales exceeding10 Mm contaminates the slope of the turbulent cascade in thespectrum. For all the ARs, we used the above linear interval tomeasure the power index, α, of the magnetic energy spectrum,E(k) ∼ k−α , where k is a wavenumber inversely proportional tothe spatial scale, r = 2π/k. The index was computed from thebest linear fit in the double logarithmic plot via the IDL/LINFITroutine by minimizing the χ2 error statistic. Figure 1 shows theenergy spectra with three ARs of different flare productivity,which is evident from the different steepness of the slopes. Thewell-defined power-law energy cascade interval is observed at3–10 Mm in all the cases.

The relationship between the power and flare indices is shownin Figure 2(a). Positive correlation between these parameters(ρ = 0.57 with a 95% confidence interval of 0.478–0.66according to the Fisher’s Z-transformation statistical test ofsignificance3) allows us to conclude that, in general, steeperspectra are associated with a higher flaring rate. Unipolar ARs

3 http://icp.giss.nasa.gov/education/statistics/page3.html

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Figure 1. Magnetic energy spectra, E(k), shown for three ARs with differentflare indices, A. Vertical dotted segments mark the linear interval between 3and 10 Mm, where the best linear fits (thin lines) were calculated. The powerindex, α, defined as E(k) ∼ k−α and measured as a slope of the best fit for eachspectrum is shown.

(A color version of this figure is available in the online journal.)

(green circles) form a separate low-flaring/shallow-spectrumsubset. Emerging, decaying, and stable ARs are distributed moreor less uniformly over the diagram.

A comparison of the spectra in Figure 1 also suggests thatnot only the slope but also the amplitudes of a spectrum may berelated to flare productivity. Thus, we calculated two additionalparameters from the spectra. The first parameter, 〈E(klow)〉,characterizes the amplitude of a spectrum at low wavenumbersand is derived as E(k) averaged over the interval ranging fromthe smallest k to k = 2π/10 Mm−1. The correlation coefficientbetween this parameter and the flare index A is 0.65 with a 95%confidence interval of 0.57–0.72. The second parameter is thetotal spectral energy, W = ∫

E(k)dk, where the integration isperformed over all the wavenumbers where E(k) is nonzero. Onaverage, 85% ± 5% of the total spectral energy is concentratedat scales larger than 10 Mm, so that W may be considered ameasure of the energy accumulated in large-scale structures ofan AR. The correlation between W and the flare index A (seeFigure 2(b)) was found to be somewhat higher: ρ = 0.68 witha 95% confidence interval of 0.60–0.75.

Note that W can also be computed directly from B2z according

to Parseval’s theorem (e.g., Kammler 2000, p. 74). However,computing W via integration of a Fourier transform has anadvantage because the integration over an annulus also actsas a noise filter by leaving out the high-frequency cornersin a two-dimensional wavenumber box. When analyzed dataare obtained from MDI/HR, the difference in derived valuesis less than 0.5%. However, the difference increases whenground-based data are analyzed. When a mixed data set is used(space- and ground-based observations), the values of W derivedvia integration of the Fourier transform are more consistent.Therefore, the total spectral energy used here was calculatedusing the integration approach.

So, the power index of the spectrum, the total spectralenergy, and the averaged large-scale amplitude are comparablycorrelated with the flare index. We may conclude that both

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Figure 2. Flare index, A, plotted vs. (a) the power index, α, of the magnetic energy spectrum (the vertical dashed segment marks the power index of the fully developedKolmogorov-type turbulence of α = 5/3), (b) the total spectral energy, W, (c) the power index weighted by the total spectral energy, W, and (d) the total unsignedflux. The solid lines show the best linear fits to the data points. The correlation coefficients, ρ, are shown. Black, red, blue, and green circles mark well-developed,emerging, and decaying ARs, and unipolar sunspots, respectively.

(A color version of this figure is available in the online journal.)

the turbulent energy cascade and the presence of large-scalestructures in an AR are relevant to flare activity.

For the purpose of prediction of an AR’s flare productivity, weweighted the total spectral energy, W, and the power index, α,to capture the contribution of both small- and large-scale effects(Figure 2(c)). The correlation coefficient in this case increasedto 0.71 with a 95% confidence interval of 0.63–0.77. The flareindex can then be fitted assuming

A = 10b(αW )c, (3)

where b = −7.92±0.58 and c = 1.85±0.13, with the reducedχ2 = 0.29.

In general, the most powerful flares occur in strong, largeARs with a considerable amount of magnetic flux (e.g., Barnes& Leka 2008). And yet, the total unsigned flux is only veryweakly correlated with the flare index (see Figure 2(d), thecorrelation coefficient is 0.37 with a 95% confidence intervalof 0.25–0.49). This is understandable when we recall previousstudies showing that not only the magnetic flux but rather thecomplexity of the magnetic structure is relevant to the flaringrate (Sammis et al. 2000; Abramenko et al. 2002; Falconer et al.2002, 2003, 2006; Leka & Barnes 2003, 2007; McAteer et al.2005; Schrijver 2007; Georgoulis & Rust 2007).

4. MAGNETIC DISSIPATION SPECTRA

As it follows from Figure 2(a), the majority of ARs (especiallythose of high flare activity) display energy spectra steeper thanthe Kolmogorov-type spectrum. To infer a physical meaning of

this result, we will analyze the magnetic energy dissipation ratesand magnetic dissipation spectra.

The magnetic energy dissipation rate is related to the pres-ence of electric currents, i.e., 〈ε〉 ∼ η〈j2〉 (e.g., Biskamp 1993).In MHD models of turbulence, dissipative structures are visu-alized via (squared) currents (e.g., Biskamp & Welter 1989;Biskamp 1996; Schaffenberger et al. 2006; Pietarila Grahamet al. 2009; Servidio et al. 2009), which in two-dimensional im-ages appear to be predominantly located along magnetic fielddiscontinuities, frequently referred to as current sheets. Fromtwo-dimensional MHD modeling, Biskamp & Welter (1989)found that when the magnetic Reynolds number (which is theratio of the characteristic values of the advection terms to themagnetic diffusivity, and quantifies the strength of advectionrelative to magnetic diffusion) is low, these current sheets areextended and rare, and they become shorter and more numerousas the Reynolds number increases. Thus, the magnetic dissipa-tion spectrum represents the distribution of dissipative structuresover many spatial scales, and is a reasonable proxy for statisticsof current structures in an AR.

The magnetic dissipation spectrum allows us to probe the stateof the turbulence. For fully developed turbulence (K41; highReynolds number), the bulk of the magnetic energy dissipationoccurs at small scales, kd, whereas the energy input occursat large scales, ke (Figure 3), and the energy cascades fromlarge to small scales without any losses. When the energy inputinterval and the dissipation interval overlap, dissipation occurs atintermediate scales along the cascade. This condition occurs inthe state of underdeveloped turbulence (low Reynolds number),when large-scale structures might interfere with the turbulent

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Figure 3. MDI/HR magnetogram for a quiet-Sun area recorded on 2001 June 5/13:07 UT (left) and corresponding spectra (right): the energy spectrum E(k) (bluecurve, right axis) and dissipation spectrum, k2E(k) (red curve, left axis). The image size is 266 × 202 arcsec. The magnetogram is scaled from −150 G to 150 G.The arrows ke and kd mark the maxima of the energy (ke) and dissipation (kd) spectra. Positions of the maxima were derived by five-point box car averaging (blackcurves). The maxima of the energy and dissipation spectra are distinctly separated in the wavenumber space.

(A color version of this figure is available in the online journal.)

cascade at small scales. It is a challenge to model such a fieldbecause no K41 simplifications are applicable.

The magnetic energy dissipation spectra are defined as(Monin & Yaglom 1975; Biskamp 1993)

Edis(k) = 2ηk2E(k), (4)

where η is the magnetic diffusivity coefficient. (Note that Eand Edis in Equation (4) have different dimensions.) Then therate of magnetic energy dissipation normalized by the magneticdiffusivity can be derived as (Biskamp 1993)

〈ε〉/η = 2∫ ∞

0k2E(k)dk. (5)

From observations we can derive the function k2E(k), whichis proportional to the dissipation spectrum under an assumptionthat η is uniform over the AR area. In our case, both k2E(k) and〈ε〉/η are associated with the dissipation of the Bz componentonly.

We calculated k2E(k) spectra for all ARs in our data set.Typical examples are shown in Figure 4. At the early stagesof development of the emerging ARs, the separation distance(kd − ke) is the largest, which is similar to the fully developedturbulence conditions seen in the quiet Sun (see Figure 3).Later on this distance decreases as kd shifts toward smallerwavenumbers (larger scales), so that the intervals of energyand dissipation become exceedingly overlapping. This impliesthe formation of large-scale dissipative structures. Decayingmagnetic complexes show quite opposite behavior (Figure 4,middle row). Well-developed ARs (bottom row in Figure 4)show a significant overlap of the energy and dissipation intervalssuggesting that, to the contrary of the fully developed turbulencephenomenology, significant dissipation takes place at all spatialscales. Thus, for the majority of well-developed ARs, one shouldexpect a state of underdeveloped turbulence in the photospherewith the dissipation of the magnetic energy at all observablespatial scales.

We then compared the magnitudes of 〈ε〉/η to the flare index,A. Their correlation turned out to be positive with ρ = 0.53with a 95% confidence interval of 0.43–0.62. This indicates thatthe rate of magnetic energy dissipation in the photosphere isrelevant to the flare activity.

5. CONCLUSIONS AND DISCUSSION

In this study, we analyzed second-order statistical momentsof solar magnetic fields of 217 ARs observed with the MDI

instrument in the HR mode during the 23rd solar cycle. Theangle-integrated magnetic energy spectra of solar ARs displaya well-defined power-law region, which indicates the presenceof a turbulent nonlinear energy cascade. The power index, α,measured at 3–10 Mm scale range was found to be correlatedwell with the flare index, A (correlation coefficient ρ = 0.57).This result further supports our previous findings based on only16 ARs (Abramenko 2005). The power indices range between1.3 and 3.0, with the majority of ARs having the power index inthe range of 1.6–2.3. No particular preference for the classical5/3 index was found. These values are surprisingly in agreementwith recent numerical simulations of decaying MHD turbulence(Lee et al. 2010). The model results showed that in equivalentinitial magnetic configurations different types of spectra (fromk−3/2 to k−2) may emerge depending on the intrinsic nonlineardynamics of the flow.

The total spectral energy, W = ∫E(k)dk, is found to be

correlated well with the flare index (ρ = 0.68), while spectralenergy weighted by the power index shows the strongestcorrelation with the flare index (ρ = 0.71), which allowedus to determine an empirical description of this relationship:A = 10b(αW )c, where b = −7.92±0.58 and C = 1.85 ± 0.13.

Combined analysis of magnetic energy and magnetic dissi-pation spectra showed that in the majority of well-developedARs, the turbulent energy cascade is augmented by magneticenergy dissipation at all scales. We thus argue that a state ofunderdeveloped turbulence exists in the photosphere of matureARs.

The magnetic energy dissipation rate, 〈ε〉/η, correlates withflare productivity in the same degree as the power index does(ρ = 0.53). As long as the energy dissipation rate is proportionalto the square of the electric currents, we argue that the presenceof the currents is relevant to flare productivity. Also, goodcorrelation between the energy dissipation rate and the flaringrate is in agreement with earlier reports by Schrijver et al. (2005).

It is known from direct calculations based on vector magne-tograms that electric currents are ubiquitous in ARs (see, e.g.,Abramenko et al. 1991, 1996; Leka et al. 1993, 1996; Pevtsovet al. 1994; Wheatland 2000; Zhang 2002; Schrijver et al. 2005,Schrijver et al. 2008; Leka & Barnes 2007; Schrijver 2009). Wefound here that photospheric magnetic fields are in a state of un-derdeveloped turbulence when both the energy cascade and theenergy dissipation at all scales are present in the system. We,therefore, arrive at the conclusion that both large- and small-scale dissipative structures (currents) are relevant to flaring.

On the other hand, Fisher et al. (1998) found no correlationbetween photospheric currents and the soft X-ray luminosity of

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Figure 4. Top: energy spectra, E(k) (blue lines), and dissipation spectra, k2E(k) (double red lines), plotted for an emerging AR. Panels (a), (b), and (c) correspond tothree consecutive moments during the emergence. Vertical blue (red) bars mark the maximum of the energy (dissipation) spectrum. The blue bars correspond to ke andthe red bars correspond to kd . As the AR emerges, kd shifts toward the smaller wavenumbers. Middle: energy and dissipation spectra for a decaying magnetic complexNOAA ARs 9682 and 9712. The right panel shows a superposition of the dissipation spectra at the well-developed (double red line) and decaying (solid green line)state of the magnetic complex. As the magnetic fields decay, kd shifts toward small scales (larger wavenumbers). Bottom: energy spectra and dissipation spectra forthree well-developed ARs. The spectra are overlapped for each case.

(A color version of this figure is available in the online journal.)

ARs. This was also noted, but not discussed, by Schrijver et al.(2005). We suggest that this apparent controversy is due to thedifference in the nature of flares and the soft X-ray emission.We may consider flares as sporadic explosive events caused bystrongly nonlinear dynamics relevant to large- and small-scalemagnetic discontinuities (Falconer et al. 2002, 2003, 2006; seealso Schrijver 2009), whereas soft X-ray emission reflects amore stationary and homogeneous process of coronal heatingrather related to ubiquitous small-scale discontinuities formedin situ (Klimchuk 2006).

We would also like to note that the power law of the magneticenergy spectra, explored in this paper, should not be confusedwith the power law found in the distribution of the magneticflux in flux concentrations recently reported by Parnell et al.(2009). At first sight, both of them characterize the structureof the magnetic field. However, they address different physi-cal consequences of the magnetic field structuring. The powerlaw of the magnetic energy spectrum represents the distribu-tion of magnetic energy, B2, over spatial scales and quantifiesturbulence in an AR. Here, the smallest magnetic elements arerepresented by the tail of the spectrum usually associated withlow spectrum amplitudes. The power law found in the distri-bution of magnetic flux represents the frequency (abundance)

of magnetic elements of different sizes and implies a uniquemechanism of the formation of magnetic flux concentrations(say, fragmentation process; see Abramenko & Longcope 2005for more discussion). The smallest magnetic elements are themost frequent and are represented by the highest amplitudes ofthe distribution.

Modern computational capabilities allow us to develop MHDmodels that take into account the turbulent regime and turbulentdissipation (e.g., Lionello et al. 2010; Klimchuk et al. 2010).Therefore, diagnostics of turbulence derived from a largeuniform data set is essential for constructing and restrainingthese models.

We are grateful to the anonymous referee for constructivecriticism and useful comments, thus allowing us to improve themanuscript substantially. SOHO is a project of international co-operation between ESA and NASA. This work was supported bythe NSF grant ATM-0716512 and NASA grants NNX08AJ20Gand NNX08AQ89G.

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