Introduction

Improving our understanding of the spatiotemporal behaviour of the distribution of magnetic flux on the solar surface is important for several areas of solar physics. On the one hand, the characteristics of the solar surface magnetic field can provide insights into the dynamo processes taking place in the Sun’s interior (e.g. Brandenburg, Sokoloff, and Subramanian 2012; Charbonneau 2014, 2020; Brun and Browning 2017; Rempel et al. 2023). On the other hand, with accurate measurements of the solar magnetic field currently only possible at photospheric levels (e.g. Bellot Rubio and Orozco Suárez 2019; Wiegelmann and Sakurai 2021), the structure and dynamics of the photospheric magnetic flux also provides a crucial input into our understanding of the structure of the coronal magnetic field and phenomena like coronal heating and magnetic activity processes (e.g. Mackay and Yeates 2012; Petrie 2013; Pevtsov et al. 2021).

Because it involves the largest scales, the most obvious manifestation of the processes generating the solar magnetic field in the solar interior is the 11-year cycle in sunspot numbers, which is a 22-year cycle if the polarity of magnetic fields is taken into account (e.g. Solanki, Inhester, and Schüssler 2006; Hathaway 2015). While sunspots are observed in the active-region bands between ±40∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\pm 40^{\circ}$\end{document} latitude, magnetic features of all sizes can be observed on the solar surface and smaller magnetic features seem to be distributed across the entire solar surface (e.g. Solanki, Inhester, and Schüssler 2006).

Due to their general importance, different aspects of the solar surface magnetic field and its characteristics have been investigated in the past. Numerous studies focus on larger magnetic features such as pores, sunspot umbrae, sunspots and sunspot groups. Observations have shown a strong correlation between the magnetic flux and area of sunspots, as well as active regions in general, (e.g. Nicholson 1933; Houtgast and van Sluiters 1948; Pevtsov et al. 2014; Nagovitsyn, Pevtsov, and Osipova 2017; Pevtsov et al. 2021; Wang, Jiang, and Luo 2023) and hence the distribution of both of these quantities, as well as active regions in general, has been investigated intensively (e.g. Kuklin 1980; Bogdan et al. 1988; Baumann and Solanki 2005; Zharkov, Zharkova, and Ipson 2005; Schad and Penn 2010; Jiang et al. 2011; Nagovitsyn, Pevtsov, and Livingston 2012; Cho et al. 2015; Muñoz-Jaramillo et al. 2015; Kostyuchenko 2017; Nikbakhsh et al. 2019; Tlatov, Riehokainen, and Tlatova 2019; Nagovitsyn and Pevtsov 2021; Sakurai and Toriumi 2023; Kumar, Kumar, and Vats 2025). Other studies have focussed on bipolar magnetic regions (e.g. Tang, Howard, and Adkins 1984; Harvey and Zwaan 1993; Zhang, Wang, and Liu 2010), the quiet network magnetic flux (e.g. Schrijver et al. 1997), ephemeral regions (e.g. Parnell 2002), or emerging flux features (e.g. Thornton and Parnell 2011).

Instead of considering either larger scale features or a specific set of features there have also been a number of studies of the distribution of features across all scales. For example, Meunier (2003) used SOHO/MDI full-disk magnetograms to examine the statistical distribution of the magnetic flux and the variation of this distribution, which they found to be a power law, with the solar cycle. Parnell et al. (2009) combined SOHO/MDI data with observations by Hinode/SOT, which allowed them to cover five orders of magnitude in flux. These authors also found that the magnetic flux distribution is represented by a single power law. A more recent study of a similar type, but concentrating on quiet sun flux elements, has been undertaken by Javaherian et al. (2017) on the basis of SDO/HMI magnetograms, with the authors finding a broken (or double) power law distribution function. Depending on the data sets and features studied, very different forms for the distribution functions have been put forward, for example exponential distributions (e.g. Tang, Howard, and Adkins 1984; Schrijver et al. 1997), log-normal distributions (e.g. Bogdan et al. 1988; Baumann and Solanki 2005; Zhang, Wang, and Liu 2010; Schad and Penn 2010), double log-normal distributions (Kuklin 1980; Nagovitsyn, Pevtsov, and Livingston 2012; Nagovitsyn and Pevtsov 2021, e.g.), or, as already mentioned above, power law distributions (e.g. Zharkov, Zharkova, and Ipson 2005; Meunier 2003; Parnell et al. 2009; Thornton and Parnell 2011; Shapoval et al. 2018). Other forms of suggested distribution functions are polynomials (e.g. Harvey and Zwaan 1993), the Weibull distribution (e.g. Parnell 2002), broken (or double) power law distributions (e.g. Javaherian et al. 2017; Song et al. 2024), and bi-modal log-normal functions (e.g. Cho et al. 2015), as well as combinations of distribution functions such as power law and log-normal (e.g. Jiang et al. 2011) and Weibull and log-normal (e.g. Muñoz-Jaramillo et al. 2015). The distribution functions mentioned above can be split into to different classes: unimodal distributions such as, for example a power-law distribution, and bimodal distributions like the hybrid distributions, broken/double power laws or double log-normal distributions. Bimodal distributions are often regarded as being indicative of two different physical processes generating the distinct parts of the distribution (e.g. Nagovitsyn, Pevtsov, and Livingston 2012; Muñoz-Jaramillo et al. 2015; Song et al. 2024) or that it could be representing different stages in the evolution of certain magnetic features (e.g. Tlatov, Riehokainen, and Tlatova 2019). On the other hand, unimodal distributions (e.g. Schrijver et al. 1997; Parnell et al. 2009) could be indicative of a single physical process operating over all scales of the distribution.

In this paper we carry out a statistical analysis of photospheric magnetic flux features over a full solar cycle using data collected by the Helioseismic and Magnetic Imager (HMI) instrument on the Solar Dynamics Observatory (SDO). The methods and data we use are very similar in general to Song et al. (2024). However, a major difference is that our study tests a number of different distributions against our data set (more details in Section 2) to determine the best fitting distribution over a full solar cycle.

In Section 2 we introduce the data set and our method of analysis before highlighting our results in Section 3. We discuss the implications of our findings in Section 4 and provide some concluding remarks in Section 5.