1 Introduction

Plasma flows in the solar photosphere are of great importance in understanding the dynamics in the solar photosphere and in studying connections of photospheric flow fields and solar flaring activities. Different methods have been developed to derive flow fields in the photosphere. Local correlation tracking, initially developed by November and Simon (1988), has been widely used to study velocities inside supergranules and active regions. Different types of feature-tracking methods (see, e.g., Hagenaar et al., 1999) were also developed to study various aspects of the solar dynamics, including magnetic-flux fragmentation and merging, determination of statistical parameters of solar dynamos, and others. Comparisons among different feature-tracking methods were also carried out (DeForest et al. 2007), and the results from these different methods were in reasonable agreement. Many authors also realized that the magnetic feature-tracking method might not be valid when magnetic-flux emergence or other complicated activities were occurring inside the studied active regions. Therefore, some new methods, with the magnetic induction equation taken into account, were developed to enhance the reliability of the inferred flow fields in such regions (Kusano et al. 2002; Longcope 2004; Welsch et al. 2004; Georgoulis and LaBonte 2006; Schuck 2008).

Meanwhile, with the rapid development of helioseismology, mapping subsurface plasma flows has become possible for ring-diagram analysis (see, e.g., Komm et al., 2005) and for time–distance helioseismology (see, e.g., Zhao et al., 2012). In particular, time–distance helioseismology provides a high spatial resolution inference of subsurface flow fields, which has allowed studies of supergranules and active regions. In fact, the subsurface flow fields obtained from time–distance measurements have also been compared with the results derived from local correlation tracking for the quiet Sun (De Rosa, Duvall, and Toomre 2000; Švanda, Zhao, and Kosovichev 2007), and reasonable agreements were reported. However, a systematic comparison of photospheric and subphotospheric flows derived for active regions has not yet been carried out.

In this article, we compare photospheric flows and subphotospheric flows inside two selected active regions, and try to understand the similarities and discrepancies of these results. The photospheric flow field is obtained from the Differential Affine Velocity Estimator for Vector Magnetograms (DAVE4VM: the Schuck 2008) using the data from the Helioseismic and Magnetic Imager (HMI)-observed vector magnetograms. The subphotospheric flow field is from the HMI time–distance analysis pipeline (Zhao et al. 2012) using the HMI Doppler velocity, but only results from the shallowest depth, i.e. a depth range of 0−1 Mm (hereafter, −0.5 Mm for simplicity), are used. It is very interesting to see how the results from these two very different analysis techniques using different data inputs agree or differ. Moreover, the differences in the flow fields between different depths can help us understand the rapidly developing complex active regions and flaring activities inside these regions. We introduce the procedure of data reduction in Section 2, and present our results in Section 3, together with a brief discussion. Our conclusions are given in Section 4.

2 Data Reduction

We use Doppler velocity and vector magnetic-field data taken by HMI (Scherrer et al. 2012; Schou et al. 2012) to derive the subsurface flow field and photospheric flow field, respectively. The HMI instrument is a filtergraph with a full-disk coverage of 4096×4096 pixels. The spatial resolution is about 1′′ with a 0.5′′ pixel size. The spectral line is the Fe i 6173 Å  absorption line formed in the photosphere (Norton et al. 2006). There are two CCD cameras in the instrument, the “front camera” and the “side camera.” The front camera acquires filtergrams at six wavelengths across the Fe i 6173 Å line in two polarization states with 3.75 seconds between the images. It takes 45 seconds to acquire a set of 12 filtergrams. This set of data is used to derive Dopplergrams and line-of-sight magnetograms (Couvidat et al. 2012). The side camera is dedicated to measuring the vector magnetic field. It takes 135 seconds to obtain the filtergrams in six polarization states at six wavelength positions. The Stokes parameters [I, Q, U, V ] are then computed from those measurements. In order to suppress the p-mode oscillations and increase the signal-to-noise ratio, the Stokes parameters are usually derived from the filtergrams averaged over a certain period of time. Currently an average is computed over 720 seconds. This average requires extra filtergrams before and after the nominal 720-second temporal window because the filtergrams must be interpolated onto a regular and finer grid before averaging. The averaging uses a cosine-apodized boxcar with a full width half maximum (FWHM) of 720 seconds. The tapered temporal window used is actually 1215 seconds long. The Stokes parameters are then inverted to produce the vector magnetic-field using the Very Fast Inversion of the Stokes Vector (VFISV) inversion algorithm, a Milne–Eddington (ME)-based approach (Borrero et al. 2011). The 180∘ ambiguity of the azimuth is solved based on the “minimum energy” algorithm (Metcalf 1994; Metcalf et al. 2006; Leka et al. 2009). Finally, the disambiguated vector magnetic-field data of active regions are deprojected into the heliographic coordinates.

DAVE4VM (Schuck 2008) is used to derive the vector velocity field in the photosphere from HMI time series deprojected, registered vector magnetic-field data. The cadence of the data is 720 seconds. The window size used in DAVE4VM is 19 pixels, which is determined by examining the slope, Pearson linear correlation coefficient, and Spearman rank order between ▽ h ⋅(v z B h −v h B z ) and δB z /δt, where v z and v h are vertical and horizontal velocities, and B z and B h are vertical and horizontal magnetic fields, as suggested in Schuck (2008).

The subphotospheric flow fields are obtained from the HMI time–distance data analysis pipeline, and all analysis details of the technique and an example of subphotospheric flow field of one active region can be found in Zhao et al. (2012). The data input for such a pipeline analysis is HMI-observed Dopplergrams with a sampling rate of 0.12∘ pixel−1. Although the pipeline provides subsurface flow fields down to 20 Mm in depth, here we only use the shallowest depth, i.e. −0.5 Mm, for a direct comparison with the photospheric flow fields.

We select two active regions to carry out this comparison. One is a mature yet simple active region, AR 11084; the other is an emerging, multipolar, and complicated active region, AR 11158. The time–distance helioseismology method requires eight-hour long Doppler velocities to derive the horizontal velocity field in the subsurface; thus, the velocity is an average during that eight-hour time interval. For AR 11084, we use data during the period of 2 July 2010 00:00 UT to 08:00 UT, and for AR 11158 the data used cover the period from 14 February 2011 06:00 UT to 14:00 UT. For the velocity in the photosphere, DAVE4VM can provide the velocity field at a cadence of 720 seconds, the same as that for HMI vector magnetic-field data. For comparison, we average eight derived photospheric velocity fields during that eight-hour time interval, one per hour, to obtain an eight-hour averaged photospheric velocity field. This velocity field is used for comparison with the velocity obtained for the −0.5 Mm layer.

3 Results and Discussion

3.1 AR 11084

AR 11084 is a mature simple active region with a stable and relatively round sunspot located inside it. Figure 1 shows the evolution of the magnetic field in AR 11084 in the eight hours during which the study is carried out; no significant evolution is seen during this eight-hour interval for the sunspot.

Figure 1
figure 1

Vertical magnetic field in active region AR 11084 at 00:00 UT, 02:00 UT, 06:00 UT, and 08:00 UT of 2 July 2010. Black and white refer to negative and positive fields, respectively.

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Figure 2 shows horizontal flows in this active region plotted on a continuum intensity image. The arrows represent the flows in the photosphere (green) and in the −0.5 Mm layer (red). Only velocities where the horizontal speed in the photosphere is greater than 0.07 km s−1 are plotted. Note that the scales are different in the two maps in order to better display the results. In general, both maps show very similar flow patterns: inward flows in the sunspot umbra and outward flows in the areas surrounding the sunspot. The inward- and outward-flow pattern in mature sunspots in the photosphere was previously reported by, e.g., Schröter (1962), Muller (1973), and Wang and Zirin (1992). Separation of the inward and outward flows takes place in the sunspot penumbra. But the boundaries between the two opposite-flow directions are different in these two layers. The area having inward flows at the −0.5 Mm layer is larger than that in the photosphere. In other words, the flows in the penumbra that have already become outflows in the photosphere are still inward flows at the −0.5 Mm layer. This inward flow may play a role in containing the sunspot (Zhao, Kosovichev, and Sekii 2010).

Figure 2
figure 2

Horizontal flows in AR 11084 in the photosphere (green arrows) and in the −0.5 Mm layer (red arrows) overplotted on a continuum intensity image taken at 2 July 2010 04:00 UT. The photospheric flow is an average of eight velocity maps at a cadence of one hour from 00:00 UT to 08:00 UT derived by DAVE4VM. The flow in the −0.5 Mm layer is derived by a time–distance helioseismology method applied to eight-hour Dopplergrams observed from 2 July 2010 00:00 UT to 08:00 UT. Only velocities at locations where the horizontal velocity in the photosphere is greater than 0.07 km s−1 are plotted.

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To better illustrate the difference of the flows in these two layers, we mark with asterisks in Figure 3 the locations where the angle between the two flows is greater than 90∘. This shows that the major differences between these two sets of results are inside the sunspot penumbra and other areas far away from the sunspot, where the method DAVE4VM becomes insensitive due to weak magnetic field strength.

Figure 3
figure 3

Background image is the 2 July 2010 04:00 UT continuum intensity data. Asterisks correspond to the areas where the angle between the two flows in the photosphere and the −0.5 Mm layer is greater than 90∘.

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While there are a few areas that show different flows in the two layers, most of the areas actually show very similar flow patterns in both layers. Figure 4 shows scatter plots of the velocities in the two layers. In the top panels are scatter plots of v x (right) and v y (left). The magnitude (\(\vert v\vert = \sqrt{v_{x}^{2} + v_{y}^{2}} \)) and azimuthal angle [θ] of the horizontal velocity are plotted in the bottom panels. The azimuthal angle ranges from −180∘ to 180∘. The horizontal axis represents velocity in the −0.5 Mm layer from the time–distance helioseismology technique, and the vertical axis velocity in the photosphere from the DAVE4VM method. The Pearson correlation coefficient is 0.89 for v x , 0.76 for v y , 0.48 for |v|, and 0.64 for θ. v x has a better correlation than v y , and this was also reported by Švanda, Zhao, and Kosovichev (2007) for the quiet-Sun study. Poor correlation in |v| implies a gradient between the two layers. We also calculate the vector correlation coefficient and the Cauchy–Schwarz inequality (Schrijver et al. 2006). The vector correlation coefficient [C vec] is defined as

$$ C_{\mathrm{vec}} = \frac{\sum_{i} \mathbf{v}_i\cdot {\mathbf{u}_i}}{(\sum_{i} \mathbf{v}_i^2\sum_{i} \mathbf{u}_i^2)^{1/2}}, $$
(1)

where v i and u i are velocities in two layers at pixel i. The Cauchy–Schwarz inequality [C cs] is defined as

$$ C_{\mathrm{cs}} = \frac{1}{M}\sum_{i} \frac{{\mathbf{v}_i}\cdot {\mathbf{u}_i}}{|{\mathbf{v}_i}||{\mathbf{u}_i}|}, $$
(2)

where M is the total number of pixels in the region studied. Here we use only two components of the vector velocity field, i.e. the horizontal velocity, to compute the coefficients. The vector correlation coefficient is 0.80, and the Cauchy–Schwarz inequality is 0.76. This indicates that the flows in the two layers are very similar.

Figure 4
figure 4

Scatter plots between the velocities of two layers for v x (top left), v y (top right), their magnitudes (|v|; bottom left), and azimuthal angles (θ; bottom right). The horizontal axis represents velocity in the −0.5 Mm layer from the time–distance helioseismology method, the vertical axis velocity in the photosphere from the DAVE4VM. The Pearson correlation coefficient is 0.89 for v x , 0.76 for v y , 0.48 for |v|, and 0.64 for θ.

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3.2 AR 11158

AR 11158 is a multipolar, complex active region, producing several major flares during its disk passage (see, e.g., Sun et al. 2012). The flows studied here in the photosphere are an average of eight one-hour cadence flow maps derived by DAVE4VM for the period of 14 February 2011 06:00 UT to 14:00 UT, during which magnetic flux was emerging in this region. This is demonstrated by the temporal profiles of magnetic-flux evolution in the region (see Figure 5). The two vertical dotted lines mark the time interval during which the flows are calculated. Note also that the positive and negative fluxes are well balanced. Figure 6 shows the vertical magnetic field in AR 11158 in this eight-hour time interval. The negative field patches N1 and N2 (see the bottom right panel) rotate counterclockwise, while the positive field patch P1 (leading polarity) undergoes separation, moving toward the Northwest. Shear motion along the polarity inversion line between N1 and P2 is also clearly seen. These flow patterns are confirmed by the photospheric flow map by DAVE4VM (see Figure 7).

Figure 5
figure 5

Temporal profiles of magnetic flux in AR 11158 on 14 February 2011. The thick solid line represents unsigned flux, while the solid and dashed lines are positive flux and absolute value of negative flux. Vertical magnetic fields greater than 100 Gauss, the noise level of the magnetic field strength, are used to measure the fluxes. Two vertical dotted lines mark the time interval from 06:00 UT to 14:00 UT in which the flows are analyzed here.

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Figure 6
figure 6

Evolution of magnetic field in AR 11158 during 06:00 UT – 14:00 UT 14 February 2011. White and black represent positive and negative vertical magnetic fields, respectively. P1 and P2 in the bottom right panel denote positive polarity patches; N1 and N2 denote negative field patches.

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Figure 7
figure 7

Horizontal velocity in the photosphere in AR 11158 derived by DAVE4VM. The image shows the vertical magnetic field with the positive field in white and negative field in black. The arrows represent horizontal velocity. The image is an average of eight velocity maps at a cadence of one hour from 06:00 UT – 14:00 UT 14 February 2011. Only velocities greater than 0.07 km s−1 are plotted. Black (white) arrows indicate that the vertical magnetic fields in the pixels are positive (negative).

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The flows in AR 11158 in the photosphere and at the −0.5 Mm layer are compared in Figure 8. Outward flows are clearly seen in the areas surrounding the sunspots in both layers, as also seen in AR 11084. However, the well-documented flux-emergence-related surface flows in the photosphere (see, e.g., Brown et al., 2003, Zhang, Li, and Song, 2007, Schmieder et al., 1994, Deng et al., 2006) – the rotation of negative field patches N1 and N2, the separation motion of positive field patch P1, and the shear motion along the polarity inversion line between P2 and N1 – do not have counterparts in the −0.5 Mm layer. Instead, significant inward flows in the sunspots are seen (e.g. in N1, N2, and P1). Figure 9 shows the flow fields in these patches in detail. Figure 10 is another way to show the difference of the flows in the two layers: the asterisks mark areas where the angle between the two flows is greater than 90∘. A significant difference occurs in the rotational sunspots, N1 and N2, and in the separation-motion sunspot P1, indicating that the surface flows in the photosphere that are related to flux emergence do not extend very deeply into the subsurface. Those flows are probably caused by the magnetic field, which is emerging and expanding into the solar corona. Gradients in magnetic field due to its rapid emergence and expansion drive motions in the photosphere, as demonstrated in magnetohydrodynamic (MHD) simulations (see, e.g., Magara and Longcope, 2003; Manchester et al., 2004; Fan, 2009). This has less effect in the subphotosphere.

Figure 8
figure 8

Vertical magnetic field in AR 11158 (image) overplotted by the horizontal velocities in the photosphere (green arrows) and in the −0.5 Mm layer (red arrows). Black and white in the image refer to negative and positive fields, respectively. The photospheric velocity is the same as in Figure 7. The flow in the −0.5 Mm layer is derived by a time–distance helioseismology method applied to eight-hour Dopplergrams observed from 06:00 UT – 14:00 UT 14 February 2011. Only velocities at locations where the horizontal velocity in the photosphere is greater than 0.07 km s−1 are plotted.

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Figure 9
figure 9

Enlarged sub-areas of Figure 8 to show details in the three interesting regions that include patch N2 (top), patches N1 and P2 (middle), and patch P1 (bottom) marked in Figure 6. Both vertical and horizontal axes are in megameters.

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Figure 10
figure 10

Background image is the vertical magnetic field in AR 11158. Asterisks represent the areas where the angle between the horizontal velocities in the photosphere and the −0.5 Mm layer is greater than 90∘. White (black) asterisks correspond to the areas where the vertical field is negative (positive).

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The scatter plots of velocities in two layers in AR 11158 are presented in Figure 11. As expected, the correlation coefficient is lower than that for AR 11084. Once again, the v x has a better correlation than v y .

Figure 11
figure 11

Scatter plots between the velocities of two layers for v x (top left), v y (top right), their magnitudes (|v|; bottom left), and azimuthal angles (θ; bottom right), for AR 11158. The horizontal axis represents velocity in the −0.5 Mm layer from the time–distance helioseismology method, while the vertical axis represents velocity in the photosphere from the DAVE4VM. The Pearson correlation coefficient is 0.78 for v x , 0.70 for v y , 0.56 for |v|, and 0.43 for θ. The vector correlation coefficient for the two components (horizontal velocity) is 0.73, and the Cauchy–Schwarz inequality is 0.59.

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3.3 Discussion

For both active regions, we also extend the subsurface flows, obtained from the time–distance data analysis pipeline, down to −6 Mm in depth to compare with the −0.5 Mm flow fields. For depths down to −3 Mm, the flow patterns look quite similar to that at −0.5 Mm, indicating that similar flow structures extend to a depth of −3 Mm or so. Below −3 Mm, the outflow structure beyond the sunspot penumbra remains largely unchanged, but the inflow structure inside the penumbra has changed to outflows. The depth variation of subsurface flow fields obtained from time–distance analysis has been discussed by various authors (e.g. Zhao, Kosovichev, and Sekii, 2010). These large-scale outflows in sunspots have been reproduced by MHD numerical simulations. For example, Rempel (2011) showed that the large-scale outflows, extending from the photosphere to the deep interior, were associated with the sunspot penumbra. His simulation also showed converging flows in a sunspot without a penumbra. However, for a sunspot with a well-developed penumbra, no obvious converging flows were found in that simulation, which is not consistent with our results here, i.e. that the converging flows are in the umbra and part of the penumbra for both layers. This discrepancy deserves more study.

We also compared the vertical flows derived from both techniques, but found poor correlation between the two. The photospheric vertical flows in sunspots derived from DAVE4VM are dominated by upward flows, consistent with the strong outflows in the penumbra known as the Evershed effect. However, the subsurface vertical flows derived from time–distance analysis are dominated by downward flows, as discussed by Hindman, Haber, and Toomre (2009) and Zhao, Kosovichev, and Sekii (2010).

4 Conclusions

We compare flows in the photosphere and in the −0.5 Mm layer in two active regions. One is a mature simple active region; the other is an emerging, complex active region. We found that for the mature simple active region the flows in both layers show very similar patterns: inward flow in the sunspot umbra and outward flow in the areas surrounding the sunspot. The boundary separating these two types of flows occurs in the sunspot penumbra, but the location of separation is slightly different in the two layers. The inward-flow area in the sunspot is larger in the −0.5 Mm layer than that in the photosphere. In other words, the flows in these areas where there are inward flows in the −0.5 Mm layer have become outward flow in the photosphere. Inward flow in the sunspots in the subsurface is suggested to play a role in containing the sunspots (Zhao, Kosovichev, and Sekii 2010).

For the emerging, complex active region, the flows in the two layers show both great similarities and significant differences. Although both layers show similar outward flows in the areas surrounding the sunspots, the well-documented flux-emergence-related surface flows seen in the photosphere, such as separation motion of leading and following polarity patches, fast rotation of sunspots, and apparent shear motion along the polarity inversion lines, do not have counterparts in the −0.5 Mm layer. This implies that the cause of these flows, which is probably the gradient of the magnetic field due to its rapid emergence and expansion into the corona, has less effect in the subsurface.