FLARES, MAGNETIC FIELDS, AND SUBSURFACE VORTICITY: A SURVEY OF GONG AND MDI DATA

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1 The Astrophysical Journal, 645: , 2006 July 10 # The American Astronomical Society. All rights reserved. Printed in U.S.A. FLARES, MAGNETIC FIELDS, AND SUBSURFACE VORTICITY: A SURVEY OF GONG AND MDI DATA D. Mason, 1,2 R. Komm, 2 F. Hill, 2 R. Howe, 2 D. Haber, 3 and B. W. Hindman 3 Received 2005 October 7; accepted 2006 February 26 ABSTRACT We search for a relation between flows below active regions and flare events occurring in those active regions. For this purpose, we determine the subsurface flows from high-resolution Global Oscillation Network Group (GONG) and Michelson Doppler Imager ( MDI) Dynamics Program data using the ring-diagram technique. We then calculate the vorticity of the flows associated with active regions and compare it with a proxy of the total X-ray flare intensity of these regions using data from the Geostationary Operation Environmental Satellite (GOES ). We have analyzed 408 active regions with X-ray flare activity from GONG and 159 active regions from MDI data. Both data sets lead to similar results. The maximum unsigned zonal and meridional vorticity components of active regions are correlated with the total flare intensity; this behavior is most apparent at values greater than 3:2 ; 10 5 Wm 2. These vorticity components show a linear relation with the logarithm of the flare intensity that is dependent on the maximum unsigned magnetic flux; vorticity values are proportional to the product of total flare intensity and maximum unsigned magnetic flux for flux values greater than about 36 G. Active regions with strong flare intensity show a dipolar pattern in the zonal and meridional vorticity component that reverses at depths between 2 and 5 Mm. A measure of this pattern shows the same kind of relation with total flare intensity as the vorticity components. The vertical vorticity component shows no clear relation to flare activity. Subject headinggs: Sun: activity Sun: flares Sun: helioseismology Sun: magnetic fields Sun: photosphere 1. INTRODUCTION In this study, we search for a relation between flare events in active regions and the subsurface flows associated with these regions. Current theories suggest that magnetic flux tubes, created at the base of the convection zone, rise toward the surface where turbulent flows braid and intertwine them and magnetic energy is catastrophically released in flare events (see Priest & Forbes 2002 for a review). A possible connection between subsurface flows and flare events could be, for example, the so-called -effect (Longcope et al. 1998) where the twist of flux tubes results from kinetic helicity of turbulent flows. Some of the twist of flux tubes could also be the result of dynamo action (Choudhuri et al. 2004; Choudhuri 2003). Both magnetic field and photospheric plasma flow complexity contribute to the energy buildup (Romano et al. 2005; Maeshiro et al. 2005). The relation between surface motions and flare activity has been studied extensively. For example, correlation tracking studies have discovered a few cases of small-scale vorticity at the surface preceding a flaring event with timescales around an hour (Yang et al. 2004) and sunspot rotation in flaring active regions for a few cases more (Brown et al. 2003). Statistical properties of active regions and their relation to flare activity have also been studied (Lu 1995; Abramenko & Longcope 2005). The techniques of local helioseismology now make it possible to study subsurface flows associated with active regions in great detail. Previous studies have focused on large or small spatial scales relative to the size of active regions. For example, when averaged over a Carrington rotation, time-distance studies have shown convergence of subsurface flows toward active latitudes as well as increased vorticity (Zhao & Kosovichev 2004). On 1 University of Southern California, Los Angeles, CA National Solar Observatory, 950 North Cherry Avenue, Tucson, AZ JILA, University of Colorado, 440 UCB Boulder, CO smaller scales, time-distance studies have shown that magnetic flux and shear in near-surface layers can influence each other (Zhao et al. 2004) and that powerful downflows exist below the surface in sunspots (Zhao et al. 2001). Seismic holography studies have found complex subsurface flows in particular active regions ( Braun et al. 2004), and ring-diagram analysis has found similar results (Haber et al. 2003), particularly relating to vorticity (Komm et al. 2004a). Komm et al. (2005a) looked at the statistics of kinetic helicity density of subsurface flows in relation to flare activity but only for active regions of three Carrington rotations. While these studies have looked at properties that distinguish individual active regions, they have been limited to a few examples and lack a large statistical sample. This study focuses on individual active regions, their magnetic flux, and subsurface flows in synoptic maps, which represent an average over the disk passage of an active region. Systematization of the ring-diagram analysis has provided the tools and reliability needed for such a task (Haber et al. 2002; Hill et al. 2003), and the Global Oscillation Network Group (GONG) along with the MDI instrument on board the Solar and Heliospheric Observatory (SOHO) provide the data to measure subsurface flows. Since flares occur on shorter timescales and smaller spatial scales than can be analyzed with the current ring-diagram technique, we cannot resolve flare events. Instead, we look for systematic differences between active regions with and without pronounced flare activity. Expanding on the work by Komm et al. (2005a), we analyze nearly all MDI and GONG data processed with the ring-diagram technique and focus on vorticity to characterize the subsurface flows. The kinetic helicity, studied in Komm et al. (2005a), is a convenient scalar descriptor of flow topology. Komm et al. (2004b) noted that strong active regions show dipolar patterns in kinetic helicity maps and that these patterns are the result of similar ones in the corresponding vorticity maps. To study the topology and the dipolar patterns of

2 1544 MASON ET AL. Vol. 645 subsurface flows associated with active regions, we decided to focus here on the vorticity vector. This allows us to perform an unprecedented survey containing 784 active regions identified by flare production and plasma-flow topology observed by GONG and 364 active regions observed by MDI, of which about half show X-ray flare intensity. With the scale of our study, we can relate subsurface flows unambiguously with flare productivity and establish a quantitative relation between flow vorticity, magnetic flux, and flare intensity of active regions. This shows that a study of the diagnostics of subsurface flows is complementary to magnetic measurements. 2. DATA AND METHOD 2.1. Data Source We analyze observations obtained during 43 consecutive Carrington rotations CR (2001 July 27 to 2004 November 8) for which we have high-resolution full-disk Doppler data from the GONG+ network. In addition, we analyze Dynamics Program data from the MDI instrument on board SOHO covering 20 Carrington rotations during CR (1996 May 23 to 2001 May 20) and the five consecutive rotations CR (2002 January 11 to 2002 May 21) that overlap with data from the GONG+ network. We determine the horizontal components of solar subsurface flows with a ring-diagram analysis using the same technique as described by Haber et al. (2002) for the dense-pack analysis of MDI Dynamics Program data. The dense-pack analysis has been implemented as the GONG ring-diagram pipeline (Corbard et al. 2003; Hill et al. 2003), while the MDI data are analyzed with the dense-pack pipeline devised by Haber et al. (2002). For consistency, we use only MDI Dynamics Program data obtained before 2003, which have been analyzed with the same inversion technique as the GONG data, the so-called regularized least-squares inversion technique (Thompson et al. 1996; Haber et al. 2002). MDI data obtained after 2002 have been analyzed with the optimally located averages inversion technique ( Haber et al. 2004). We derive daily flow maps of horizontal velocities from the dense-pack analysis and combine them to form synoptic flow maps on a grid with centers spaced by 7N5 in latitude and longitude ranging from 52N5 in latitude at 16 depths from 0.6 to 16 Mm. We also estimate the vertical velocity component as described in Komm et al. (2004b). To focus on the variation of the flows, we remove the large-scale trends in latitude of the horizontal flows and calculate residual synoptic flow maps, as described in Komm et al. (2004b). More details of the analysis can be found in Komm et al. (2005a, 2005b). As a measure of solar activity, we use the NSO Kitt Peak synoptic charts. 4 We convert the magnetogram data to absolute values and bin them into circular areas with 15 diameter centered on a grid with 7N5 spacing in latitude and longitude to match the dense-pack mosaic. We identify the active regions in Carrington rotations analyzed here using the NOAA active region numbers listed in the Active Region Monitor (ARM) at NASA Goddard Space Flight Center s Solar Data Analysis Center (SDAC). 5 To identify each active region, we create a circular mask in heliographic coordinates that fully encloses the active region. A consequence of this method is that neighboring quiet regions might be included in the active region-specific data, an effect that 4 Available at 5 Available at Fig. 1. Synoptic maps of CR Top: Magnetic flux. Second: Binned unsigned magnetic flux. Third: Zonal vorticity component! x. Fourth: Meridional vorticity component! y. Bottom: Vertical vorticity component! z. Active regions of FI above 3:2 ; 10 5 Wm 2 are circled. grows with irregularly shaped active regions. Despite this concern, our method adequately achieves the goal of isolating values related to one active region from another in addition to isolating the active regions from the quiet Sun. Moreover, extrema in the magnetic flux and vorticity consistently concentrate in the active region proper so that a loose mask produces the highest chance of registering these extrema. We determine a proxy of the total X-ray flare activity of these active regions using the sum of peak flare intensities (FIs, measured in W m 2 ) over an active region s lifetime reported in data from the Geostationary Operational Environmental Satellite (GOES). 6 A proxy calculated 6 Available at

3 No. 2, 2006 SURVEY OF GONG AND MDI DATA 1545 Fig. 3. Log-log scatter plot of flare intensity FI against B max. large-scale components of the horizontal flows have been removed. In this study, we derive only the vorticity of the residual flows, as defined by equations (1a) (1d): w ¼ :<v;! x z ;! y x ; ð1aþ ð1bþ ð1cþ Fig. 2. Zonal vorticity,! x, at 300 longitude in CR 1993 as a function of latitude and depth. Top: Gross magnetic flux (solid line) and binned over 15 (dotted line). Second: The x-component of vorticity (! x ). Third: Signal-to-error ratio. Bottom: Idealized schematic for flows below a strong active region (with arbitrary amplitudes). The arrows in the second panel represent the meridional and vertical velocity components with the vertical one increased by a factor of 10 for visibility. from integrated intensity values, also provided by GOES, is linearly related to FI on a log-log scale and would thus lead to similar results. In this study, we analyze the subset of active regions for which GOES records X-ray flare intensity. From a total of 784 active regions in the GONG and 364 regions in the MDI data, there are 408 active regions with X-ray flare intensity from GONG and 159 regions from MDI Vorticity and Magnetic Flux In this paper, we examine vorticity, w, which is a vector quantity that corresponds to changing orientation in space of fluid particles and is thus a quantity associated with mixing and turbulence (e.g., Lesieur 1987). The measured velocity components represent the average flow in a volume element defined by the horizontal size of each dense pack and the depth extent of the inversion kernels; the resulting velocities represent a spatial average. Since we cannot directly measure the velocity gradients, we derive them by numerical differentiation using three-point Lagrangian interpolation. To remove the effect of differential rotation, average meridional flow, and their average depth variation, and to emphasize the influence of magnetic activity, we calculate the vorticity from the residual velocities where the! z y ; ð1dþ with v as residual velocity and x indicating zonal (east-west), y meridional (north-south), and z vertical direction. In general, we ignore the vorticity values at the two most poleward latitudes of 45 and 52N5. The errors of the measured horizontal velocities increase with increasing distance from disk center by about a factor of 2 from equator to 52N5 latitude (Komm et al. 2005b), and as a consequence, the error of the derived vertical velocity component increases as well. This center-to-limb variation is most likely due to foreshortening that gets stronger with increasing distance from disk center. The additional numerical differentiation to calculate the vorticity tends to further enhance any noise present in the data. However, this is not much of a concern for this study, since active regions are located at lower latitudes. To represent the subsurface vorticity of an active region, we choose the absolute maxima of the three components of vorticity within the active region mask, represented by! x; max,! y; max, and! z;max. These values adequately reflect the extent of fluid twist in an active region and tend to have the highest signal-to-noise ratio (S/ N) values. Similarly, to represent the magnetic activity of an active region, we choose the absolute maximum of the binned magnetic flux within the active region mask, B max. In maps of flow vorticity, many active regions show dipolar structures that are most pronounced in! x and! y (see x 3.1). To measure this dipolar pattern, we define a structural component measure! s. For a circled active region, a swath covering two dense-pack regions is defined along the anticipated vorticity dipole for the x- and y-components. A width of two dense-pack

4 1546 MASON ET AL. Vol. 645 Fig. 4. Maximum vorticity of active regions as a function of their maximum magnetic flux at four different depths. The columns plot! x;max,! z;max,! s, and! a against B max. The rows plot 1.2, 3.1, 8.5, and 14.3 Mm. patches ensures that the maximum value of the vorticity pattern is included. A single-patch swath might miss it since active regions are not necessarily centered on the dense-pack grid. This swath is then divided in half to isolate the ends of the dipoles. Corresponding to the findings in x 3.1, for! x, the minimum value from the southern half of the swath is subtracted from the maximum value in the northern half. For! y, the minimum value from the eastern half is subtracted from the maximum value in the western. The resulting differences in! x and! y are added to obtain! s. If the flows within an active region show a dipolar structure of this orientation, the value of! s will be of the same order as the sum of! x;max and! y;max. If the direction of the dipole is reversed,! s will return negative values of similar size. If the flows do not lead to dipoles in! x and! y,! s will approach zero. To provide a control quantity for the effects of random fluctuations in the vorticity field, we have also created an antistructural component of vorticity! a in which the masking swaths are rotated 90 counterclockwise. We do not include the vertical vorticity component because it is generally about 1 order of magnitude smaller than the other two components when calculated on the dense-pack grid and the dipolar structures are less pronounced. 3. RESULTS 3.1. Example of Vorticity Maps Figure 1 shows, as an example, synoptic maps for CR 1993 of the surface magnetic flux and the three vorticity components at a depth of 11.6 Mm. The circles identify seven active regions that produce more than 3:2 ; 10 5 Wm 2 in FI (see x 3.3 for this value). The binned unsigned magnetic flux in the second panel of Figure 1 shows that the dense-pack grid is adequate for identifying individual active regions. As these maps show, flare-producing active regions also produce distinct dipole signatures in! x and! y. A cursory visual inspection leads to the impression that! x shows a stronger presence of these dipoles than! y in some active regions. While showing a general positive inclination for flareproducing active regions,! z has difficulty distinguishing them from the quiet Sun. Because the dipoles of the x-component of vorticity are orthogonal to the ones of the y-component, the phenomenon suggests an axisymmetrical flow whose central axis lies in the radial direction and pierces the center of the active region. Specifically,! x exhibits a positive pole directly to the north of the active region center and a negative pole directly to the south. Correspondingly,! y exhibits a positive pole directly to the west and a negative pole directly to the east. To provide some perspective on the variation with depth of! x, we include a depth-latitude slice of active region AR located at 300 longitude and 7 latitude in CR In Figure 2, the dipoles are readily apparent, along with a reversal at depths between 2 and 7 Mm. The signal-to-error plot in the third panel indicates that the dipoles and their reversal are clearly significant. An idealized schematic of the kind of flows that would accompany this kind of dipolar structure in vorticity is presented in the bottom panel Magnetic Flux For comparison, we first provide correlations between magnetic flux and flare activity. In Figure 3 we plot the unsigned magnetic maximum (B max ) for each active region against the sum of peak flare intensities for that active region (FI). Despite a large amount of scatter, a significant positive correlation is apparent.

5 No. 2, 2006 SURVEY OF GONG AND MDI DATA 1547 Fig. 5. Maximum vorticity of active regions as a function of their flare intensity at four different depths. The columns plot! x;max,! z;max,! s, and! a against FI. The rows plot 1.2, 3.1, 8.5, and 14.3 Mm. Colors indicate ranges of maximum magnetic flux (B max ) of the active regions: purple (10 30 G), blue (30 70 G), green (70 90 G ), and red ( G). Black points do not fall into any of the above ranges. We calculate linear correlation coefficients (also called Pearson) and Spearman rank-order correlation coefficients. As a nonparametric correlation, the rank correlation is more robust than linear correlations with regard to outliers in the data (e.g., Press et al. 1992). The Spearman rank-order correlation coefficient between FI and B max is 0.514, while the Pearson correlation coefficient between log FI and B 2 max is Since B2 max correlates with log FI better than any other form of B max, we use B 2 max for comparisons throughout the rest of this paper. Because we bin down magnetic flux to the resolution of our vorticity maps, we lose information on small spatial scales. One might imagine that the correlation between the unbinned magnetic flux and flare activity would be higher than the binned magnetic flux. This, however, is not the case. When using the fullresolution maps from Kitt Peak, we find a Spearman rank correlation between unbinned B max and FI of 0.437, while the Pearson correlation coefficient between unbinned B 2 max and log FI is In Figure 4, we plot! x; max,! z; max,! s, and! a against B max for four representative depths. The quantities! x;max and! s show a general positive correlation with B max,while! z;max shows a small positive correlation and! a shows none. The meridional component! y;max (not shown) shows a similar behavior to! x;max.in this scatter plot and all others that follow, error bars are not plotted to aid clarity. When these scatter plots are restricted to vorticity values that are at least 3 times larger than the corresponding errors, we find that the near-horizontal cloud in the lower left-hand corner thins out but none of the plots change in character. This is probably because flow maxima in active regions tend to have high S/N values Vorticity Measures and Flare Activity Figure 5 shows semilog scatter plots of! x;max,! z;max,! s, and! a against FI at the same representative depths of 1.2, 3.1, 8.5, and 14.3 Mm as in the previous section. The meridional component! y;max shows similar results to the zonal one,! x;max, and is not included in the following sections in order to reduce redundancy. Each active region data point is sorted by B max into five ranges: G, G, G, G, and active regions below 20 G or above 160 G. For each magnetic range, we have 61, 187, 82, 70, and 8 active regions, respectively. The range of values in! x; max and! s are a factor of roughly 3.3 larger at 1.2 Mm than at the lower depths, which is not so for! z; max. This indicates that the vertical gradients of the horizontal velocities change more rapidly close to the surface than at deeper layers. For FI values greater than about 3:2 ; 10 5 Wm 2,the values of! x; max and! s are clearly related to FI, while for smaller values, the values of! x; max and! s are small and their relation with FI is not as apparent. The data points with high B max tend to separate from the rest of the points, a fact that is best exemplified in the plots of! x; max at 8.5 and 14.3 Mm. Each magnetic range appears to have its own slope directly proportional to the values of B max in that range. Results for! s duplicate those for! x;max at all depths except at 3.1 Mm, where positive correlations reverse to negative correlations. To quantify this relation, we calculate Pearson correlation coefficients between! x;max and the logarithm of FI as a function of depth for the same four B max ranges, as shown in the top left panel of Figure 6. At all depths, the correlation values vary

6 1548 MASON ET AL. Vol. 645 Fig. 6. Correlations for! x;max (left) and! z;max (right) with FI and B max as a function of depth. Top: Pearson correlations of! with log FI. Second: Pearson correlations of! with B 2 max. Symbols indicate four ranges of maximum magnetic flux B max of the active regions: triangles (10 30 G), diamonds (30 70 G), squares (70 90 G ), and crosses ( G). The dashed line represents the Pearson correlation between B 2 max and log FI over all magnetic ranges. Bottom: Partial correlations between!, B2 max, and FI for the highest magnetic range ( G). The symbols indicate which variable s influence is removed: triangles ( FI), diamonds (!), and squares (B 2 max ). systematically with magnetic activity so that active regions in the highest magnetic range demonstrate the highest correlation values. This separation grows wider at depths below about 7 Mm. Also of interest are two local minima that occur in all samples at 1.5 and 5.8 Mm, which correspond to depths with low S/N in Figure 2. Spearman rank-order correlations for all depths and ranges (not shown) nearly duplicate the Pearson semilog correlations, indicating that a semilog representation is appropriate. Below about 7 Mm, the highest magnetic range sample exceeds the correlation between B 2 max and log FI. The correlation between! x; max and B 2 max (Fig. 6, middle left panel ) shows a similar stratification but far less definitely. The fact that active regions with high magnetic activity record similar Pearson correlations to those between! x; max and log FI suggests that the three quantities are highly intercorrelated. Partial correlations measure the direct influence of an independent variable on a dependent one (Neter et al. 1992). In the case of! x;max, B 2 max, and log FI, three partial correlations can be derived by measuring the correlation between each pair of variables with the influence of the third one taken out. The bottom left panel of Figure 6 shows these partial correlations for the subset with the highest magnetic activity. Clearly, at all depths except for 1.5 and 5.8 Mm, the correlation between! x;max and log FI is substantially larger than the correlation between B 2 max and log FI and somewhat higher than the correlation between B 2 max and! x; max. The exceptions coincide with the local minima observed in the top left panel. The right column of Figure 6 shows the results of the same analysis for! z; max. The correlations between log FI and! z; max (top right panel ) remain always close to zero, but there is a general increasing trend below a depth of about 4 Mm. This trend is most striking in the high-activity subset where the correlation reaches a significant value of 0.42 at 15.8 Mm. The correlations between! z; max and B 2 max (middle right panel ) do not differ much from zero, and there is again a trend of increasing correlations with greater depth, albeit less pronounced. The partial correlations show that the strongest connection exists between log FI and B 2 max. Figure 7 repeats the correlation analysis of Figure 6 for! s and! a. The magnitude of Pearson correlation coefficients between! s and log FI (top left panel ) is slightly smaller than those between! x; max and log FI for the two highest magnetic ranges across all depths. The two lowest magnetic ranges exhibit somewhat smaller correlation coefficients than in Figure 6. Between about 2 and 5 Mm, the sign of the correlations are negative. This range of anticorrelation coincides with the hump observed between the local minima in the top left panel of Figure 6. The middle left panel of Figure 7 shows generally low but stratified correlations between! s and B 2 max except for the highest magnetic range. Here the valley of negative correlations is obvious along with high positive correlations below a depth of about 7 Mm.

7 No. 2, 2006 SURVEY OF GONG AND MDI DATA 1549 Fig. 7. Same as Fig. 6, but for! s and! a. The partial correlations of! s in the bottom left panel mimic the partial correlation with! x; max at depths below about 7 Mm, while at depths between about 2 and 7 Mm the ones that exclude B 2 max or FI show negative values, as expected from correlations with! s shown in the top left and middle left panels of Figure 7. As described in x 2.2, we have created an antistructure component,! a, as a control value to! s. The scatter plot between! a and log FI exhibit horizontal clouds at all depths (right column of Fig. 7);! a shows practically no correlation with log FI. The partial correlation plot in the bottom right panel of Figure 7 shows that any correlation between the three values! a, log FI, and B 2 max exists only between B 2 max and log FI MDI Results To compare MDI and GONG data, we select the common data set that spans five Carrington rotations (CR ) and includes 134 active regions. We divide the data into two categories depending on whether they lie within or outside the active region masks. The two subsets indicate high and low S/N areas. For! z, GONG and MDI data are significantly correlated (with values greater than 0.7) at all depths for both subsets. For! x, the correlation between GONG and MDI data of the active region subset is also very high with similar values to! z except within 2 Mm of the surface, while the correlation of the quiet-region subset shows values of about 0.4 on average. This simple test shows that we can overall expect that MDI and GONG data lead to similar results. Figure 8 shows! x;max,! z;max,! s, and! a derived from MDI data as a function of log FI. We sort the data according to the same magnetic ranges as in x 3.3. For each magnetic range from lowest to highest, we have 21, 82, 35, and 19 active regions. Only two active regions fall outside these ranges. Three features are immediately present. First, there are far fewer data points, especially for active regions with FIs above 3:2 ; 10 5 Wm 2 compared to the GONG data. Second, the vorticity values derived from MDI data are smaller than the GONG vorticity values at a given flare activity. Third, visual correlations between vorticity values and flare activity are less apparent, possibly because of the lack of extreme values. As a consequence, correlations will not be very significant. To examine whether general features in the GONG data are repeated in the MDI data, we provide the correlations between! s,! a, and log FI in Figure 9. The top left and middle left panels show the same valleys at the same depths as in Figure 7, although the dependence on depth is less clear. The bottom left panel shows that below about 7 Mm, partial correlations between! s and log FI exceed those between B 2 max and log FI, exhibiting very similar behavior to the GONG data in Figure 7. However, the partial correlation coefficients between! s and B 2 max slightly exceed those between! s and log FI, the opposite of that observed in the GONG data. The results of! a in MDI data (Fig. 9, top and middle right panels) agree with the GONG data by exhibiting small correlations between! a and log FI and! a and B 2 max. Although the MDI

8 1550 MASON ET AL. Vol. 645 Fig. 8. Vorticity components of active regions in the MDI survey as a function of their FI at four different depths. The columns plot! x;max,! z;max,! s,and! a and the rows plot 1.2, 3.1, 8.5, and 14.3 Mm. Colors are the same as in Fig. 5. data tend to give larger absolute values to the correlation coefficients, the significance of these correlations is low, since we have so few data points. For example, the lowest magnetic range reaches a relatively high correlation coefficient of 0.37 at 4.4 Mm between! a and log FI, but the confidence level is only 90%. The partial correlations (bottom right panel ) are highest between B 2 max and log FI also agreeing with the GONG data. While the differences in partial correlations disappear at depths below 10 Mm, the confidence interval also lowers to around 80% Quantitative Relation between! s,b max, and log FI The stratification by magnetic range observed in the correlations between! x;max or! s and log FI in both the GONG and MDI data suggests a continuous relationship between the regression slope for! x;max or! s and log FI and the magnetic range chosen for the data points that are sampled. To derive this relation, we divide the data into subsets containing active regions with B max values within a 24 G span, where each subsequent subset is shifted by 3 G. The lowest subset contains active regions with B max ranging from 0 to 24 G, while the highest subset ranges from 121 to 145 G. As a consequence, neighboring ranges overlap, and the resulting slopes are not independent of neighboring slopes. While a shorter magnetic span will reduce overlap, it will also reduce the number of active regions included and make the estimate of the slope less reliable. On the other hand, a larger magnetic span will increase the number of data points in a subset, but it will also lose some detail in the variation of the slopes. Tests with different magnetic spans suggest that a 24 G span provides a good compromise. Most importantly, this span permits us to extend our analysis to the higher magnetic ranges without losing too much accuracy. The top panel of Figure 10 shows the number of active regions as a function of the central value of each activity range. Here we focus on! s and calculate the linear regression slope between! s and ðlog FIÞfor each subset of activity. Since we find the clearest stratification at a depth of 11.6 Mm (see Fig. 7, top left panel ), we use it as our sampling depth. The middle panel of Figure 10 shows the regression slope as a function of magnetic range. There is a strong positive correlation between the slope of! s and log FI and the magnetic range. Errors, represented by dotted lines above and below the solid line, increase at low and high magnetic ranges. This occurs because fewer points are sampled at these ranges as indicated by the top panel. We repeat the same analysis for the MDI data. The relatively small number of active regions in each magnetic range (Fig. 10, top panel ) results in a loss in accuracy, as indicated by increased errors of the slopes. We restrict the magnetic ranges because there are so few active regions at extreme magnetic ranges. The MDI results, shown in the bottom panel of Figure 10, show a linear trend with a zero-slope intercept at around 30 G just as the GONG results. The measured slopes are smaller than the GONG ones, a consequence of smaller vorticity values (see x 3.4). These findings suggest that an equation can be created to link! s, B max, and log FI for B max greater than about 35 G. Since the GONG results are derived from a larger sample of active regions and extend to higher magnetic flux values than the MDI results, we focus on the GONG results. Figure 5 shows a linear relation between! s and log FI and suggests that regression lines for

9 No. 2, 2006 SURVEY OF GONG AND MDI DATA 1551 Fig. 9. Same as Fig. 7, but for! s and! a derived from the MDI survey. different magnetic ranges intersect at a point around 10 6 Wm 2. Figure 10 shows that its slope is a linear function of B max for B max values greater than 36 G. This suggests that! s is related to FI and B max in the following way:! s ¼ a þ (b þ cb max )(log FI log d ); ð2þ with a ¼ ð2:64 0:41Þ; 10 6 s 1, b ¼ ( 1:40 0:16) ; 10 6 m 2 Ws 1, c ¼ ð3:88 0:27Þ ; 10 8 m 2 WGs 1,and log d ¼ ð1:39 0:02Þ ; 10 1.Theparametersband c represent the y-intercept and the slope of the linear regression between! s and log FI as a function of B max (see Fig. 10). The parameters a and d represent the! s and FI values at the point where the regression lines for different magnetic ranges intersect (see Fig. 5). Because the intersection point is to the left of log FI ¼ 0, and the slope between! s and log FI for all magnetic ranges changes with the central value B max,they-intercepts show a linear relation with B max as represented in the term (b þ cb max ). To calculate d, we determine the y-intercepts of the regression lines as a function of magnetic range and minimize the factor in front of B max by varying log d. When this factor approaches zero, then d represents FI at the intersection point. The minimizing value of log d (d ¼ 0:92 ; 10 6 Wm 2 ) leads to a factor close to zero that is 3 orders of magnitude smaller than its error. From the factor error, we estimate the error of log d by finding the parameter values above and below the minimizing value that produce a factor equivalent to its error. Once log d is determined, determining the value for a is trivial, since it is merely the height of the y-intercept versus B max relation for a zero slope. When applying this analysis to! x;max, and! y;max, we get similar results. However, the errors on the parameters in equation (2) are lowest for! s. We can reformulate equation (2) to express flare activity as a function of! s and B max for B max values greater than 36 G with the same four constants: log FI ¼! s a b þ cb max þ log d: 4. SUMMARY AND DISCUSSION We have calculated the vorticity components of residual subsurface flows for 43 consecutive Carrington rotations of GONG data and 23 Carrington rotations of MDI data where the contributions of the average zonal and meridional flow have been removed. The largest vorticity values and fluctuations are observed in layers close to the surface. At least part of these fluctuations might be artifacts due to the difficulty in accurately measuring flows near the surface and the resulting error propagation, which is compounded in the case of the GONG instruments which must observe through the Earth s atmosphere. In this study, we have shown that flare activity is intrinsically linked to subsurface phenomena on timescales and spatial scales comparable to the lifetime and size of active regions. Active ð3þ

10 1552 MASON ET AL. Vol. 645 Fig. 10. Linear regression slopes for! s with log FI as a function of the central value of magnetic range for active regions present in the GONG and MDI surveys. Top: Number of active regions for each central value in the GONG data (dark line) and the MDI data (light line). Middle: Linearregression slope derived from GONG data. Bottom: Linear regression slope derived from MDI data. Dotted lines indicate the slope error. regions with pronounced flare activity have large values in zonal and meridional vorticity components,! x and! y, with a repeated dipole pattern in both. Large flare activity will generally accompany large magnetic flux and large vorticity. Low flare activity can accompany a wide range of magnetic flux, but the associated flows can only have small vorticity. The separation of these two ranges is visible as a knee at a total flare intensity (FI) value of 3:2 ; 10 5 Wm 2. Above this value, large vorticity values will accompany large magnetic flux for a given FI, and larger flare activity will accompany lower magnetic flux for a given vorticity value. We quantify the relation between! s, B 2 max, and log FI for flux values greater than about 36 G at a depth of 11.6 Mm where the relation is most apparent. These three quantities are clearly interdependent, and the correlation analysis shows that for large values of magnetic flux, vorticity measurements can be a better indicator of flare activity than magnetic flux. The MDI data show a similar result within the limit of a smaller sample of active regions above the threshold. This shows the importance of continuous coverage, which will be possible with the Helioseismic and Magnetic Imager instrument on board the Solar Dynamics Observatory and the ongoing operation of GONG. The dipolar pattern of zonal and meridional vorticity strongly suggest an axisymmetric flow pattern associated with flaring active regions. The reversal of this pattern at depths between 2 and 6 Mm, together with the fact that these vorticity components reflect mainly the vertical gradients of the horizontal velocity components ( Komm et al. 2005c), imply that the vertical gradients change sign at these depths. Previous studies show that horizontal flows converge toward active regions at depths less than 10 Mm, implying downflows, and diverge from active regions at greater depths, implying upflows ( Komm et al. 2005b; Haber et al. 2004; Zhao & Kosovichev 2004). Together this implies a strong flow convergence (or large downflows) near 6 Mm and a weak one near 2 Mm, as depicted in the schematic in the bottom panel of Figure 2. Numerical studies by Schüssler & Rempel (2005) show that surface cooling of emerging flux tubes leads to downflows at shallower depths. Together with pressure buildup by upflows in flux tubes at greater depth, this provides a mechanism to dynamically disconnect magnetic fields from their magnetic roots at depths above 10 Mm, surprisingly close to the observed transition from downflow to upflow. Interestingly, the depths of 2 and 10 Mm coincide with observations of soundspeed anomalies found under active regions ( Hughes et al. 2005) and suggest that below-average sound speeds coincide with small downflows close to the surface and above-average sound speeds coincide with strong downflows at greater depth. While we cannot rule out that the differences between near-surface and somewhat deeper layers are due to artifacts, they might suggest that the layers within 2 Mm of the surface indicate the boundary layer between photosphere and convection zone. We find a positive correlation between magnetic flux and total flare activity that is in agreement with other observational studies attempting to correlate energy or area of active regions with flare activity (Mayfield & Lawrence 1985; Kucera et al. 1997). The fact that binning the magnetic flux from magnetogram to densepack resolution slightly improves correlations between B max and FI might simply reflect that flare intensity is measured as an integral quantity of an active region and is not localized within a region. The area of active regions might also be correlated with the vorticity of associated subsurface flows. However, when we use the circular thresholds, defined to identify the active regions, as an estimate of their area, we find generally poor correlations between area estimates and flare activity or vorticity. Our result suggests that it might be possible to predict the overall flare activity of an active region from input magnetic flux and subsurface vorticity. Encouraged by the results, we plan to investigate the level of accuracy that is possible with a relation such as equation (3), given the large scatter of the data. However, to be useful as a prediction tool, the analysis would have to be extended to address temporal changes and active region evolution. We plan to explore other research interests motivated by this study. For example, since the dipolar structure in zonal and meridional vorticity is related to the change with depth of the divergence of the horizontal flows, we plan to examine the divergence patterns in active regions. In addition, using larger dense-pack patches, González Hernández et al. (2006) are able to explore greater depths than with the standard ring-diagram analysis. This promises to extend some of our observations, such as the increasing correlation between! z;max and flare activity with depth, which suggests structural twists in active regions as discussed by Zhao & Kosovichev (2003). While our work provides novel statistical information on spatial scales comparable to the sizes of active regions, it would be useful to apply this analysis to higher resolution flow maps as well. This might be possible with the inclusion of time-distance analysis in the GONG pipeline (Hill et al. 2004) in the near future.

11 No. 2, 2006 SURVEY OF GONG AND MDI DATA 1553 This work was supported by NASA grants NAG and NNG 05HL41 to the National Solar Observatory and NAG , NAG , and by NSF grant ATM to the University of Colorado. This work utilizes data obtained by the Global Oscillation Network Group (GONG) project, managed by the National Solar Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA), Inc., under a cooperative agreement with the National Science Foundation. The data were acquired by instruments operated by the Big Bear Solar Observatory, High Altitude Observatory, Learmonth Solar Observatory, Udaipur Solar Observatory, Instituto de Astrofísica de Canarias, and Cerro Tololo Inter- American Observatory. SOHO is a mission of international cooperation between ESA and NASA. NSO/ Kitt Peak data used here are produced cooperatively by NSF/ NSO, NASA GSFC, and NOAA/SEC. The ring-fitting analysis is based on algorithms developed by Haber, Hindman, and Larsen with support from NASA and Stanford University. This work is carried out through the National Solar Observatory Research Experiences for Undergraduate (REU) site program, which is cofunded by the Department of Defense in partnership with the National Science Foundation REU Program. Abramenko, V. I., & Longcope, D. W. 2005, ApJ, 619, 1160 Braun, D. C., Birch, A. C., & Lindsey, C. 2004, in Proc. SOHO 14/GONG 2004, Helio- and Asteroseismology: Toward a Golden Future, ed. D. Danesy ( ESA SP-559; Noordwijk: ESA), 337 Brown, D. S., Nightingale, R. W., Alexander, D., Schrijver, C. J., Metcalf, T. R., Shine, R. A., Title, A. M., & Wolfson, C. J. 2003, Sol. Phys., 216, 79 Choudhuri, A. R. 2003, Sol. Phys., 215, 31 Choudhuri, A. R., Chatterjee, P., & Nandy, D. 2004, ApJ, 615, L57 Corbard, T., Toner, C., Hill, F., Hanna, K. D., Haber, D. A., Hindman, B. W., & Bogart, R. S. 2003, in Local and Global Helioseismology: The Present and Future, ed. H. Sawaya-Lacoste ( ESA SP-517; Noordwijk: ESA), 255 González Hernández, I., Komm, R., Hill, F., Howe, R., Corbard, T., & Haber, D. A. 2006, ApJ, 638, 576 Haber, D. A., Hindman, B. W., & Toomre, J. 2003, in Proc. SOHO 12/GONG+ 2002: Local and Global Helioseismology: The Present and Future, ed. H. Sawaya-Lacoste (ESA SP-517; Noordwijk: ESA), 103 Haber, D. A., Hindman, B. W., Toomre, J., Bogart, R. S., Larsen, R. M., & Hill, F. 2002, ApJ, 570, 855 Haber, D. A., Hindman, B. W., Toomre, J., & Thompson, M. J. 2004, Sol. Phys., 220, 371 Hill, F., et al. 2003, in Proc. SOHO 12/GONG+ 2002: Local and Global Helioseismology: The Present and Future, ed. H. Sawaya-Lacoste ( ESA SP- 517; Noordwijk: ESA), , in Proc. SOHO 14/GONG 2004, Helio- and Asteroseismology: Towards a Golden Future, ed. D. Danesy (ESA SP-559; Noordwijk: ESA), 128 Hughes, S. J., Rajaguru, S.P, & Thompson, M. J. 2005, ApJ, 627, 1040 Komm, R. W., Corbard, T., Durney, B. R., González Hernández, I., Hill, F., Howe, R., & Toner, C. 2004a, ApJ, 605, 554 REFERENCES Komm, R. W., Howe, R., González Hernández, I., Hill, F., Haber, D., Hindman, B., & Corbard, T. 2004b, in Proc. SOHO 14/GONG 2004, Helioand Asteroseismology: Towards a Golden Future, ed. D. Danesy ( ESA SP- 559; Noordwijk: ESA), 520 Komm, R. W., Howe, R., Hill, F., González Hernández, I., & Toner, C. 2005a, ApJ, 630, 1184 Komm, R. W., Howe, R., Hill, F., González Hernández, I., Toner, C., & Corbard, T. 2005b, ApJ, 631, 636 Komm, R. W., Howe, R., Hill, F., Haber, D. A., & González Hernández, I. 2005c, Eos. Trans. AGU, 86(18), Jt. Assem. Suppl., abstract SP43B-04 Kucera, T. A., Dennis, B. R., Schwartz, R. A., & Shaw, D. 1997, ApJ, 475, 338 Lesieur, M. 1987, Turbulence in Fluids ( Dordrecht: Kluwer) Longcope, D. W., Fisher, G. H., & Pevtsov, A. A. 1998, ApJ, 507, 417 Lu, E. T. 1995, ApJ, 446, L109 Maeshiro, T., Kusano, K., Yokoyama, T., & Sakurai, T. 2005, ApJ, 620, 1069 Mayfield, E. B., & Laurence, J. K. 1985, Sol. Phys., 96, 293 Neter, J., Wasserman, W., & Whitmore, G. A. 1992, Applied Statistics ( Newton: Allyn & Bacon) Press, W. H., Flannery, B. P., Teukolsky, S. A., & Vetterling, W. T. 1992, Numerical Recipes in C (Cambridge: Cambridge Univ. Press) Priest, E. R., & Forbes, T. G. 2002, Astron. Astrophys. Rev., 10, 313 Romano, P., Contarino, L., & Zuccarello, F. 2005, A&A, 433, 683 Schüssler, M., & Rempel, M. 2005, A&A, 441, 337 Thompson, M. J., et al. 1996, Science, 272, 1300 Yang, G., Xu, Y., Cao, W., Wang, H., Denker, C., & Rimmele, T. R. 2004, ApJ, 617, L151 Zhao, J., & Kosovichev, A. G. 2003, ApJ, 591, , ApJ, 603, 776 Zhao, J., Kosovichev, A. G., & Duvall, T. L. Jr., 2001, ApJ, 557, , ApJ, 607, L135

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