MAGNETIC FLUX TRANSPORT AND THE LONG-TERM EVOLUTION OF SOLAR ACTIVE REGIONS

We characterize the evolution of the active regions using 304 Å light curves, which are good indicators of the general trend of emergence, growth, and decay in an active region (e.g., Ugarte-Urra & Warren 2012). The light curves are calculated by integrating the excess counts from a background image in all pixels within a 21° × 21° field of view around the center of each region in the Heliographic maps. The background image, which approximates the background of the quiet Sun pre-existent to the active region emergence, is estimated from the median intensity of the bottom 20% intensities in time at each pixel, including times before the emergence. Data is discarded if it lies beyond 0.9 R_⊙ of the center of the EUVI and AIA images to avoid the large interpolation edge effects near the solar limb. Elsewhere a correction is implemented based on a parameterization of the dependence of these effects with longitude at the latitude of interest (see Ugarte-Urra & Warren 2012).

The left panel in Figure 1 shows the light curves of the active regions in the data set with the times of peak in 304 Å intensity co-aligned. The times of peak intensity were determined by smoothing the light curves with a 24 hr time window, thus minimizing spikes due to transient activity like flares. To take into account different quiet Sun background levels, we have also subtracted the starting intensity from each light curve. The curves reflect the well known evolution of active region magnetic fields with emergence leading to a rapid rising phase followed by a longer decay period. In bipolar active regions, the rise time to maximum development (when emergence ceases) is typically shorter than five days (Harvey 1993), but it can be longer if multiple bipoles are involved and this is the case for some regions in our data set. The decay in most regions in our sample is close to monotonic, but there are regions, e.g., 11726, that exhibit large variations in 304 Å intensity which are due to interaction with shorter lived nearby flux.

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Also evident is that the brighter regions live longer. This is reminiscent of the observed property that active regions with larger fluxes and areas have longer lifetimes (e.g., Mosher 1977). As one will expect intuitively (and we will show quantitatively later in the paper) this is not coincidental, but it has its foundation in the relationship between magnetic flux and 304 Å intensity. An interesting property, that to our knowledge has not been reported before is that the evolution light curves are scalable. In other words, by knowing the peak intensity of a region in 304 Å, one can estimate its lifetime. This is demonstrated on the right panel of Figure 1. In this diagram we show all the light curves scaled to the active region 11158 peak intensity. That same scaling factor is used to scale the time, meaning that an active region twice as bright at peak, will live twice as long. In the figure, intensities have been also normalized by the AR 11158 peak intensity, and the times by the 11158 lifetime, set as the time when the 304 Å intensities reach the flat quiet Sun levels marginally above the original quiet Sun. That plateau is discussed by Sainz Dalda & Martinez Pillet (2008).

To understand the intensity decay, it is important to understand the relationship between the 304 Å maps and the magnetic field because it is the magnetic field evolution on the surface that drives the upper atmosphere response.

The relationship between the X-ray and EUV intensities and magnetic quantities, such as the total unsigned magnetic flux in an active region, has been firmly established (see Gurman et al. 1974; Schrijver 1987; Fisher et al. 1998; Pevtsov et al. 2003; van Driel-Gesztelyi et al. 2003; Fludra & Ireland 2008). Chromospheric EUV lines like He i 584.34 Å are no exception, as shown by CDS/SoHO measurements (Fludra et al. 2002; Fludra & Ireland 2008). We show in Figure 2 that this relationship also holds true for He ii 303.8 Å, the dominant spectral line in the 304 Å channel (O'Dwyer et al. 2010).

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In Figure 2 we plot the total number of AIA 304 Å counts per second in an active region as a function of total unsigned flux as measured from HMI magnetograms. The data points represent observations tracking each active region from limb to limb with a cadence of 1 hr. AIA 304 Å images were corrected for the time dependent degradation in sensitivity in the filter response, provided by the instrument team within the standard processing software. A threshold of 85 DN/s/pixel was used to discriminate between the pixels with active region contribution from the quiet Sun pixels. The field-of view ranged between 215''–325'' depending on the size of the region, with the exception of active region 11672 in which we used a 100'' × 110'' area. In the magnetograms, only field strengths above 20 Gauss, in absolute value, were considered.

The power-law relationship can be expressed as $\mathrm{log}({I}_{\mathrm{AIA}304})=A+B\mathrm{log}| \phi |$, where I_AIA304 is in DN/s, B = 0.71 ± 0.01 and A = −8.41 ± 0.15. This link between the 304 Å integrated intensity and the magnetic flux is important, not only for its implications in terms of how the EUV emission is generated, as the previous studies have argued, but also because it allows the 304 Å intensity to be used as a proxy of magnetic flux. This is particularly interesting now, as we do not currently have magnetic field measurements from outside the Earth's line of sight perspective, but we do have 304 Å measurements thanks to STEREO. The 304 Å emission is better than other lines at tracking flux evolution. It is formed closer to the solar surface than other EUV and X-ray lines, which extend over larger volumes in the atmosphere and are more sensitive to transient energetic events that generate significant density changes with their respective intensity fluctuations.

This power law can be used to translate the 304 Å light curves in Figure 1 into total unsigned magnetic flux curves for the full lifetime of the active regions. An intermediate step is necessary to go from the integrated intensities of the STEREO heliographic maps to the AIA intensities in the powerlaw. That relationship in our analysis is: log(I_AIA304) = C + D log(I_HG304), with C = −0.9 ± 0.3 and D = 1.11 ± 0.04.

To assess the importance of the chosen thresholds in the flux-luminosity relationship we repeated the analysis for all regions in different scenarios: AIA thresholds of 5, 85 and 190 DN/s and HMI thresholds of 20, 40, and 80 Gauss. While the slopes of the power law can change significantly in all those permutations due to the added or reduced 304 Å counts and magnetic fluxes, the correlation coefficients are larger than 0.90 in all of them. In the case we selected is ρろー = 0.96. The best fits correspond to an AIA threshold of 85 DN/s with marginal differences in the correlation and fit uncertainties between the 20 (our choice), 40, and 80 Gauss cases. The fit coefficients for the 40 and 80 Gauss cases are respectively A = −7.69, B = 0.68 and A = −6.83, B = 0.64.

While we cannot observe the evolution of the magnetic field when active regions are outside the field of view of available magnetometers such as MDI/SoHO or HMI/SDO (both observing from the Earth's perspective), there has been significant progress in recent years in simulating how magnetic field distributions evolve in time. The AFT model is able to evolve observed flux concentrations by simply applying a velocity field that realistically simulates the flows on the photosphere, making it possible to compare observed and simulated magnetic quantities for the complete history of an active region.

The AFT baseline model assimilates magnetic data from HMI magnetograms while the region is on the Earth side. Elsewhere it relies on the prescribed flows to allow the flux concentrations to evolve in time: moving, merging and canceling with other flux concentrations. We investigated the details of the active region evolution in the AFT model by comparing the total unsigned magnetic flux from HMI directly to the total unsigned magnetic flux from the AFT baseline model. It was initially found that the AFT model underestimated the total unsigned magnetic flux, in particular for active regions newly emerged on the western hemisphere. This may explain the flux discrepancy previously noted by Upton & Hathaway (2014a). These comparisons allowed us to make improvements to the original AFT baseline by optimizing the weighting used in the assimilation process such that the baseline total unsigned magnetic flux best matched the observed data, while continuing to mitigate the limb effects seen in the HMI data.

Figure 3 shows comparisons of the total unsigned magnetic flux $| {\phi }_{304}|$ inferred from the 304 Å light curves, as described earlier, to the one predicted by the AFT model $| {\phi }_{\mathrm{SFT}}|$ after integrating the flux density in the SFT maps for absolute values larger than 20 Gauss. The assimilated data is shown by the gray shaded areas in the Figure, while areas with the white background are being evolved solely by the AFT model. The close match of the curves during times of assimilation serves as an assurance that our 304 Å proxy approach works because the AFT model is nearly identical to the real HMI data during this time.

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We looked at SFT light curves for two different integration areas, shown with different colors in Figure 3, one matching the field of view of the integration area in 304 Å maps, the second one extending that area 25° to the East and West in order to assess the influence of differential rotation on the comparisons. While making the heliographic maps we tuned the rotation rate to the latitude of the region; the maps produced by the SFT model are made at the Carrington rotation rate, thus active regions can rotate out of a fixed longitude. The comparisons reveal that rotation of flux out of the box is not a major source of error. For NOAA 11672, the weakest region, enlarging the area increases the flux because the additional background quiet Sun flux has more weight than in stronger regions.

As illustrated by the 11158 curve, the AFT model can accurately reproduce the active region evolution when the active region is not being observed by HMI and the model is operating without assimilation. Additional flux may emerge, however, while the active region is on the far-side (e.g., 11150, 11272, and 11726). The AFT assimilation process corrects for this new emergence when this flux is observed by HMI and rotates over the East limb. In that respect, the 304 Å proxy is providing information not available to the model.

In the previous sections we established that the peak integrated intensity in 304 Å light curves can accurately determine the duration of the decay phase in an active region. We also showed that 304 Å can provide information about magnetic properties of an active region when magnetic field measurements are not available. Finally and complementary to the others, we have seen that the AFT model has the ability to make these properties evolve realistically in time. The next obvious question that can be raised is: can we combine these three pieces of information to provide predictions about the flux evolution over an active region lifetime?

The answer to the question is affirmative. The importance of the peak 304 Å intensity is that it is directly linked to maximum flux in the region (Figure 2), which coincides with the stoppage of flux emergence. From that moment, the evolution of the flux is fundamentally governed by the diffusive motions of convection, along with the differential rotation and meridional flows (e.g., see review Jiang et al. 2014). Transport by diffusion has been known for many decades (Leighton 1964). In the case of a bipole, it has been shown (Mosher 1977) that the flux decay in one polarity can be expressed as

$$\begin{eqnarray}&&{\phi }_{+}={\phi }_{o}\mathrm{erf}\left(\displaystyle \frac{{\rm{a}}}{\sqrt{4\mathrm{Dt}}}\right)\end{eqnarray} \tag{ 1 }$$

where ϕ_o is the initial flux, 2a is the separation between polarities, and D is the diffusion coefficient. The expression illustrates that provided we know the diffusion that affects the field, we only need to know the strength and the separation of the polarities at the start of the process, to predict the decay over time. Simple as this is, however, it is insufficient to fit the slope of the light curves in Figure 1. The limiting factor is that it describes ideal conditions and it does not take into consideration the flux lost under the sensitivity threshold of the instrument or the effect of the convective motions concentrating the magnetic field into the magnetic network. This can be addressed numerically in a model like the AFT model.

In Figure 4, we show comparisons between $| {\phi }_{304}|$ and $| {\phi }_{\mathrm{SFT}}|$ with the restriction that the assimilation of new magnetic field data is stopped when the 304 Å light curve reaches its peak intensity. From that moment onward, the total flux curve is from a field distribution created purely by the modeled transport. The plots show that the AFT model does very well in reproducing the amount of flux decayed over one solar rotation.

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In most cases the active region evolution is well matched for the entire period. In cases like 11158, observations and model diverge as the region leaves the Earth view for a second rotation. This may be caused by interactions with nearby flux that has not been assimilated into the model. In fact, two small regions emerged to the East on March 19 and 26, interacted and disappeared, coincidental with drops in the observed total flux. In cases like 11259 and 11672, a rapid decay is observed immediately after the assimilation process is halted. This is likely caused by stochastic differences between the actual and simulated convection patterns. As the active region flux adjusts to the new network pattern, leading and following polarity flux are more prone to interact and cancel with one another. After this brief adjustent period, the decay process stabilizes and mirrors the evolution of the 304 Å flux. We note that this effect may be more pronounced for active regions with a smaller separation distance between the leading and following polarity. For example, while 11259 and 11765 had similar peak intensities, 11765 had about twice separation distance and did not experience a substantial adjustment period. It should also be noted that most of these discrepancies are mitigated by continued assimilation of magnetograms when they are available.

The bottom three panels in Figure 4 (NOAA 11272, 11484, and 11726) are a special subset in the predictions because the peak in 304 Å is reached while on the back side of the Sun and therefore we do not have information about the flux distribution when the decay starts. This makes them excellent cases to test how 304 Å images and light curves serve as the single source of information for a new emergence and its future decay. For these cases, we manually insert a bipolar active region in the AFT maps when the 304 Å has reached its maximum value. In the case of 11484 and 11726, the assimilation is stopped before the actual emergence of the region.

The bipolar strength is chosen from the total unsigned magnetic flux inferred from 304 Å, split between the two polarities. The separation is derived from the weighted standard deviation from the centroid of the 304 Å intensity distribution over a threshold of 170 DN/s. That distance is shown as a blue circle in Figure 5. We determined the relationship between this standard deviation and the distance between polarities for 19 active regions near central meridian and used it to estimate the polarity separation for the three regions with emergences on the back side of the Sun where we only have information about 304 Å. While it is possible in some cases to also estimate the tilt angle for the two polarities based on a 2D Gaussian fit of the 304 Å intensity distribution (green elipses in the figure), the method proved unreliable in general without further constraints. The tilt angle (γがんま) was, therefore, simply estimated from Joy's law for a given latitude L, $\gamma (L)=0.51| L| -0\buildrel{\circ}\over{.} 8$. This can be a crude approximation because it is well known that there is a wide scatter around the latitude average (Wang et al. 1989).

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How reliable are these predictions? The flux can be consistently predicted within a factor of 2. Figure 6 shows a comparison of 304 Å and SFT fluxes for times when the curves reach a fraction (0.8, 0.6, 0.4, and 0.2) of the peak in the $| {\phi }_{304}|$ smoothed light curve. When plotted as a function of time since peak intensity, precision remains rather constant, with only a few points suggesting that it may slowly worsen with time. This does not necessarily imply that the modeling of the flux decay is inaccurate at longer times. Given that many points lie well within that factor of 2 range, it can just be a reflection of the increased chance of other contributions not considered in the model, such as new emergences, affecting the light curves.

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These results confirm that, despite the approximations, 304 Å measurements can be sufficient when coupled with a SFT model, to establish how the total unsigned flux of an active region changes over long timescales. Furthermore, a SFT model that combines assimilation of nearside data with far side data from 304 Å measurements should be able to provide a more complete picture of the magnetic field configuration of the entire Sun. This has significant implications for space weather predictions, such as solar irradiance, solar wind, and coronal field models.