Double Power Law: Variation over One Solar Cycle

A more in depth understanding of the double power law model can be obtained by investigating the temporal variation of the slope parameters of the separate power laws. Figure 9 shows how the fitted parameters of the smooth double power law model vary over one solar cycle.

Figure 9: Temporal variation of the parameters of the smooth double power law model over the solar cycle. Panel (a) shows αα, panel (b) shows ββ, panel (c) shows 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 panel (d) shows σσ._

Panel (a) shows the temporal variation of α_α _, the slope of the left hand power law. Typically, the value of α_α _ lies between −1.7 and −1.8 but there are plenty of upward fluctuations in the curve. There is much more consistency in the value at times of reduced solar activity (from 2017 onwards reaching solar minimum in late 2019). Although the fluctuations might simply be a reflection of statistical variations in the data, one other possible explanation could be that the decay of large scale active regions does contribute to the small scale flux distribution over shorter time scales. This would explain why these fluctuations seem to occur mainly during enhanced solar activity.

Panel (b) shows the temporal variation of β_β _, the slope of the right hand power law. There is a noticeable trend: the slope (magnitude of β_β _) becomes steeper at solar minimum and flatter at solar maximum. Again, there are a number of outliers in the graph. Interestingly, the spikes all point towards more negative values, whereas the spikes in the α_α _ curve all point towards less negative values. A value of −5 was set as the lower limit in the maximum likelihood estimation optimisation algorithm, so it is possible that the true value of β_β _ in some magnetograms is less than −5. However, due to numerical limitation we have to compromise slightly on accuracy. For the most part, the value of β_β _ is greater than −5. The upper envelope of the curve seems to vary roughly between −3.0 (low activity) and −2.0 (high activity)

The temporal variation of the ‘kink location’ (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}) is shown in panel (c). Once again, the data is spiky but typically the value lies between 1019\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$10^{19}$\end{document} and 1020Mx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$10^{20}~\mathrm{Mx}$\end{document}. The values seem to be slightly higher at solar minimum than at solar maximum but whether this is a real feature of the distribution or the result of statistical fluctuations is unclear. This question could be investigated further in the future.

Finally, in panel (d) we see the temporal variation of the smoothing parameter (σ_σ _). A larger value of σ_σ _ means the transition from slope α_α _ to slope β_β _ is sharper. Low values of σ_σ _ means that the transition takes place over a wider range. Typically, the value lies between 2 and 4, however there are a number of spikes in either direction (both toward a value of 0 and toward a value of 15). A value of 15 was set as an upper limit for the maximum likelihood estimation optimisation algorithm. The spikes toward the highest value of σ_σ _ could be even higher, but we argue that a value of 15 is already sharp enough to convey the required information. In addition, computational limitations, again, mean we have to make a compromise on accuracy in these situations.

We see that there is much more variation in the β_β _-parameter than the α_α _-parameter. This tells us that the slope of the small-scale flux distribution is much more stable over a solar cycle than that of the large-scale flux distribution. The right hand slope (β_β _) varies with changing solar activity; typically having a shallower slope at solar maximum and a steeper drop-off at solar minimum. This is to be expected because the right hand power law slope (β_β _) is affected by the presence of large-scale active regions.

The fact that the power law index of the small-scale flux distribution appears to remain unchanged on average regardless of the level of solar activity could be interpreted as an indication that the long-term behaviour of the small-scale flux distribution is not dominated by the fragmentation of larger features, but largely due to a mechanism which perpetually generates new, small-scale features throughout the entire solar cycle.

We note that our results are qualitatively consistent with those of Song et al. (2024). Although we did not assume from the outset that a double power law distribution is the correct fit for the data, we found using statistical goodness of fit tests that over a full solar cycle a smooth double power law is the best fitting distribution function amongst those we tested. This very similar to the “two-segment” power law distribution fitted by Song et al. (2024), although this would correspond to our sharp double power law and not to the smooth double power law we found to be the best fit over the full solar cycle. Quantitatively, we found that our power law index for the small scale flux part (α_α _) is roughly between −1.7 and −1.8 with more upward fluctuations during solar maximum, whereas Song et al. (2024) quote a slightly larger average value of −1.64. We note that it is possible that when averaging over the whole cycle we would arrive at a higher value due to the upward fluctuations we find. However, given the nature of the variations of α_α _ as shown in panel (a) of Figure 9 we do not believe that an average value would necessarily be meaningful. Our value of the power law index for the large scale flux part (β_β ) is roughly in the same range, but at some times considerably lower than the corresponding values given by Song et al. (2024), which vary between −1.92 at solar maximum and −2.27 at solar minimum. We would suggest that this difference could be due to the difference between the sharp double power law (“two-segment” power law) used by Song et al. (2024) and the smooth double power law which we use. This is also reflected by the fact that Song et al. (2024) fix the transition point between the two segments of their double power law to 5.5×1018Mx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$5.5 \times 10^{18}~\mathrm{Mx}$\end{document}, whereas we have a transition region whose position (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 width (σ_σ _) is part of the fitting process. Despite these differences which we consider to be relatively minor, we find it very encouraging that our findings using a statistical goodness of fit approach seem to give results that are largely consistent with the findings by Song et al. (2024).