Summary and Conclusions

In this paper we have presented the analysis of 260 full disk line-of-sight HMI magnetograms over an 11 year period from May 2010 to March 2021. We used the modified clumping algorithm to identify individual magnetic flux features and generated size distributions of these features. We then used the maximum likelihood method to obtain fits to these observational size distributions by nine different probablity density functions: (single) power law PDF, expontial PDF, lognormal PDF, Weibull PDF, truncated Weibull PDF, sharp double power law PDF, smooth double power law PDF, Weibull-lognormal PDF, and truncated Weibull-lognormal PDF.

Using a variety of statistical goodness-of-fit criteria we established that over a full solar cycle the smooth double power law seems to represent the best fit to the observations. We analysed how the parameters of the smooth double power law change over the solar cycle. On the basis of this analysis we found that the power law index for the small flux features (α_α _) shows no long-term solar cycle variation, whereas the power law index for the large flux features (β_β ) shows a variation which is in line with the decrease of large scale active region flux during solar minimum. The two other parameters of the smooth double power law PDF are the location of the transition between the two power laws (xc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x{c}$\end{document}) and the width of the transition (σ_σ _). On the basis of our analysis we could not identify any significant long-term variation of these two parameters over the full solar cycle.

The smooth double power law is an example of a bi-modal distribution. Bi-modal distributions are often taken to be an indication that the two different parts of the distribution could be generated by different physical processes. If interpreted in such a way our result could be viewed as being supportive of processes keeping the small-scale flux distribution largely time-invariant over a solar cycle, for example small-scale dynamo processes as discussed in Rempel et al. (2023), while the solar cycle variation of the large-scale flux distribution is generated by a large-scale dynamo. However, one has to be cautious not to overinterpret our results for a number of reasons. Firstly, the analysis presented in this paper is based on a single solar cycle only. It will need to be confirmed over other solar cycles, which clearly is a long-term project. Secondly, our results do not exclude the possibility that a single physical mechanism could be responsible for both the relative temporal stability of the flux distribution at small scales while also generating the variation of the large-scale distribution over the solar cycle. A solid theoretical understanding of the various physical processes is necessary to be able to distinguish between the different options. Finally, we note that the power indices as well as the other parameters of the flux distribution are subject to statistical fluctuations. While the overall long-term trend in the (large-scale) power law index β_β _ is relatively obvious, we cannot exclude the possibilty that the statistical fluctuations could prevent us from detecting much smaller systematic variations in α_α _, for example. In this context, we mention the work by Korpi-Lagg et al. (2022) who found no signficant solar cycle dependence of the quiet sun internetwork magnetic field fluctuations when using 1∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1^{\circ}$\end{document} data patches, but clear solar cycle dependence of network and internetwork magnetic field fluctuations when using larger patches (15∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$15^{ \circ}$\end{document}). Although the paper by Korpi-Lagg et al. (2022) studies a different quantity than we did in the current paper, it shows that the question of solar cycle dependence of small scale magnetism is a subtle one. Clearly, further studies of this important problem are warranted, both on the observational/data analysis side and on the theoretical side.