Andrés Muñoz Jaramillo1-3
1. Department of Physics, Montana State University, Bozeman, MT 59717, USA
2. Space Sciences Laboratory, University of California, Berkeley, CA 94720, USA
3. W.W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305, USA
From clouds to river networks, from blood vessels to ferns and threes, self-similarity is a common theme in nature, and the Sun is no exception. Driven by its interaction with turbulent convection, the solar magnetic field arranges itself into structures, remarkably like each other, spanning a wide range of spatial scales (See Figure 1). This is evidence that there is a common generative set of mechanisms at play behind them.
Figure 1. Example of self-similarity in SDO/HMI (a, b, c) and Hinode/SOT (d) magnetograms. Chosen bipolar structures span roughly four orders of magnitude in area (two orders of magnitude in length-scale). Day of observation is 05-Jun-2012.
Talking about self-similarity is not possible without the concept of scale. For this reason, understanding the relative abundance of structures of different sizes is a powerful tool for gaining insight into their generative processes. This task has received continuous attention during the last thirty years[1-4], but no consensus has been reached regarding the probability distribution best characterizes solar data.
Taking advantage of 11 different databases (constructed using data from 9 different instruments), we tested which, out of the distributions that have been proposed in so far, is the one that fits all sets best. Furthermore, by using three types of observations (sunspot group area, sunspot area, and active region flux) we explored why, if all these quantities are related, different studies have reached different conclusions.
Figure 2. Empirical size distribution of photospheric magnetic structures. Sunspot group data measured by the Royal Greenwich Observatory (RGO) (shown in gray) is used as reference. Each color indicates the empirical distribution of a different type of data and instrument: (b) blue HMI sunspot group area, (c) red sunspot area measured by the San Fernando Observatory (SFO), and (d) green unsigned MDI active region flux. Data types are reconciled using proportionality constants. The same composite distribution (shown as a solid dark red line) is overplotted over all sets. It is constructed using a linear combination of Weibull (dashed blue line), and log-normal (dotted yellow line) distributions.
As shown in Figure 2, we found that data are best fitted by a composite distribution (made from a combination of Weibull and Log-Normal distributions). Additionally, we found that the apparent discrepancy between previous studies arises from the fact that different data types sample different sections of this composite distribution (see the relative position of each data type with respect to the reference Royal Greenwich Observatory data in Figure 2).
Based on the remarkable coincidence of the log-normal component of the composite distribution and the characteristics of bipolar active region data (see Figure 2-d), we propose that this is evidence of two separate mechanisms giving rise to visible structures on the photosphere: one directly connected to the global component of the dynamo (and the generation of bipolar active regions), and the other with the small-scale component of the dynamo (and the fragmentation of magnetic structures due to their interaction with turbulent convection). The transition between these two kinds of structures occurs between 1021 Mx and 1022 Mx (see Figure 3-a).
Although our databases do not contain structures smaller than a pore, we can extrapolate the composite distribution into scales smaller than those visible in our data (see Figure 3-b). We find that, for these scales, the Weibull distribution shows the log-linear behavior expected of a power-law distribution. This makes our result consistent with a recent study that found a power-law distribution spanning five orders of magnitude (by applying six different detection algorithms to SOHO/MDI and Hinode/SOT data)[4].
Figure 3. (a) Relative contribution of the Weibull and log-normal components to the composite distribution. (b) Extrapolation of the composite distribution towards smaller domains showing behavior consistent with the log-linearity of a power-law.
The agreement we found between our difference databases, and the fact that it allows us to reconcile the previous work that has been done in the past, gives us confidence that we are on the right track towards the definite characterization of the size distribution of photospheric magnetic features. Our next task will be to take advantage of the high cadence, high duty cycle, high resolution, and full-disk coverage that HMI provides, to perform a characterization of photospheric magnetism of unprecedented detail and open a new chapter in our understanding of the mechanisms that give rise to the photospheric magnetic field.
More details about this work can be found in [5], which has been recently accepted for publication in ApJ. In the meantime, the body of the paper can be downloaded from the ArXiv: http://arxiv.org/abs/1410.6281
References
[1] Harvey, K.L. & Zwaan, C. 1993, Sol. Phys., 148, 85.
[2] Schrijver, C.J., Title, A.M., van Ballegooijen, A.A., Hagenaar, H.J., & Shine, R.A. 1997, ApJ, 487, 424.
[3] Baumann, I. & Solanki, S.K. 2005, A&A, 443, 1061
[4] Parnell, C.E., DeForest, C.E., Hagenaar, H.J., Johnston, B.A., Lamb, D.A., & Welsch, B.T. 2009, ApJ, 698, 75.
[5] Muñoz-Jaramillo, A., Senkpeil, R.R., Windmueller, J.C., Amouzou, E.C.; Longcope, D.W., Tlatov, A.G., Nagovitsyn, Y.A., Pevtsov, A.A., Chapman, G.A., Cookson, A.M., Yeates, A.R., Watson, F.T., Balmaceda, L.A., DeLuca, E.E., & Martens, P.C.H. 2014