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There is increasing interest in finding the relation between the sunspot number (SSN) and solar polar field. In general, fractal properties may be observed in the time series of the dynamics of complex systems, such as solar activity and climate. This study investigated the relations between the SSN and solar polar field by performing a multifractal analysis. To investigate the change in multifractality, we applied a wavelet transform to time series. When the SSN was maximum and minimum, the SSN showed monofractality or weak multifractality. The solar polar field showed weak multifractality when that was maximum and minimum. When the SSN became maximum, the fractality of the SSN changed from multifractality to monofractality. The multifractality of SSN became large before two years of SSN maximum, then that of the solar polar field became large and changed largely. It was found that the change in SSN triggered the change in the solar polar field. Hence, the SSN and solar polar field were closely correlated from the view point of fractals. When the maximum solar polar field before the maximum SSN was larger, the maximum SSN of the next cycle was larger. The formation of the magnetic field of the sunspots was correlated with the solar polar field.

Various objects in nature exhibit the so-called self-similarity or fractal property. Nature is full of fractals, for instance, trees, rivers, coastlines, mountains, clouds, and seashells. Monofractal signals are homogeneous in that they have the same scaling properties and are characterized by a fractal dimension. On the other hand, multifractal signals are nonuniform, more complex and can be decomposed into many subsets characterized by the different dimensions of the fractal. Fractal properties can also be observed in a time series representing the dynamics of complex systems. A change in fractality can appear with a phase transition or change of state. As an example, the multifractal properties of daily rainfall were studied in an East Asian monsoon climate with extreme rainfall and in a temperate climate with moderate rainfall [

The sunspot number (SSN) has an 11-year cycle and the solar magnetic field has a 22-year cycle, exactly twice that of the sunspot cycle, because the polarity of the field returns to its original value every two sunspot cycles. During the past ~120 years, Earth’s surface temperature has been correlated with both the decadal averages and solar cycle minimum values of the geomagnetic aa index [

The aim of our work is to investigate the relations between the SSN and solar polar field from the view point of fractals. To examine the changes in multifractality, we performed a multifractal analysis on the SSN, and solar polar field by using the wavelet transform. The wavelet transform can perform reliable multifractal analysis [

We used several solar activity indices to see the solar activity in a multifaceted manner. We used the monthly SSN provided by Solar Influences Data Analysis Center (sidc.oma.be) and the solar polar field throughout the solar sunspot cycle provided by the Wilcox Solar Observatory (http://wso.stanford.edu/).

For the analysis, we used the Daubechies wavelet, which is widely used in solving a broad range of problems, e.g., self-similarity properties of a signal or fractal problems and signal discontinuities. We used a discrete signal which was fitted the Daubechies mother wavelet with the capacity of accurate inverse transformation. Thus, we can precisely calculate the following optimum τ(q), which can be regarded as a characteristic function of the fractal behavior. We can define the τ(q) from the power-law behavior of the partition function, as shown in Equation (2). We then calculated the scaling of the partition function Z_{q}(a), which is defined as the sum of the q-th powers of the modulus of the wavelet transform coefficients at scale a, where q is the q-th moment. In our calculation, the wavelet-transform coefficients did not become zero. Thus, for an accurate calculation, the summation was considered for the whole set. Muzy et al. [_{q}(a) as the sum of the q-th powers of the local maxima of the modulus to avoid dividing by zero. We obtained the following partition function Z_{q}(a):

Z q ( a ) = ∑ | W φ [ f ] ( a , b ) | q , (1)

where W φ [ f ] ( a , b ) , a, and b are the wavelet coefficient of function f, a scale parameter, and a space parameter, respectively. W ϕ [ f ] ( a , b ) is defined as below.

W ϕ [ f ] ( a , b ) = 1 | a | ∫ − ∞ + ∞ f ( t ) φ * ( t − b a ) d t (2)

where f ( t ) is data and φ is wavelet function. For small scales, we expect

Z q ( a ) ~ a τ ( q ) . (3)

First, we investigated the changes in Z_{q}(a) in the time series at a different scale a for each moment q. We plotted the logarithm of Z_{q}(a) against that of time scale a. Here τ(q) is the slope of the fitted straight line for each q. Next, we plotted τ(q) versus q, which is the multifractal spectrum. The time window was advanced by one year, which was repeated. Monofractal and multifractal signals were defined as follows: A monofractal signal corresponds to a straight line for τ(q), while a multifractal signal τ(q) is nonlinear [^{2} value, which is the coefficient of determination, for the fitted straight line. If R^{2} ≥ 0.98, the time series is monofractal; if 0.98 > R^{2}, it is multifractal.

A time window was fixed to 6 years for the following reasons. We calculated the wavelets with time windows of 10, 6, and 4 years. Initially, when a time window was 10 years, a fractality changed slowly. By integrating the wavelet coefficient in a wide range, small changes were canceled. Thus, this case was inappropriate to find a fast change of climatic regime shift. Next, when the time window was 4 years, the fractality changed quickly. The overlap of the first and following data was 3 years, which was shorter than the 9 years when the time window was 10 years, and the change of fractality was large. Thus, this case was also inappropriate. Finally, when the time window was 6 years, a moderate change in fractality was observed. Hence, the time window was fixed to 6 years.

We calculated the multifractal spectrum τ(q) of the SSN between 1910 and 2010. The multifractal spectrum τ(q) between 1967 and 1979 is shown in

We plotted the τ(−6) of each index, where q = −6 is the appropriate number for showing a change in τ. The large negative values of τ(−6) show large

multifractality. Thus, τ(−6) is not always equal to the fractality gained from the R^{2} value.

We investigated the change in the SSN. To detect the changes in multifractality, we examined the multifractal analysis of the SSN. The τ(−6) of the SSN and SSN as well as the solar cycle number of the SSN are shown in

Fractal properties were regularly changing. When the SSN was maximum (minimum), the τ(−6) of the SSN became maximum, as shown by 1 (3) in

The solar polar field is shown in

Next, we show the τ(−6) of the solar polar field and the solar polar field in

became maximum, the τ(−6) of the solar polar field became maximum, as shown by 5 in

We show the τ(−6) of the solar polar field and SSN between 1980 and 2015 in

minimum in cycle 23. Those results coincided with the result of

When the SSN and solar polar field were maximum and minimum, the τ(−6) of the SSN and solar polar field strength became maximum. The fractal dimensions decrease with increasing mean magnetic field, implying that the magnetic field distribution is more regular in active regions [

When the change in the solar polar field was small, the multifractality was weak. The solar polar field showed weak multifractality when the solar polar field was maximum and minimum. We compared the τ(−6) of SSN with the solar polar field. In

comparing with 5 (the solar field maximum) and 3 (the SSN minimum) shown in

The solar polar field makes the magnetic field of the sunspots when the solar cycle is the minimum [

In this study, we investigated the relation between the SSN and the solar polar field by performing a multifractal analysis. To detect the changes in multifractality, we performed a multifractal analysis on the SSN, and the solar polar field using the wavelet transform.

1) When the SSN was maximum and minimum, the SSN showed monofractality or weak multifractality. The solar polar field showed weak multifractality when that was maximum and minimum.

2) When the SSN became maximum, the fractality of the SSN changed from multifractality to monofractality.

3) The multifractality of SSN became large before two years of SSN maximum, then that of the solar polar field became large and changed largely. It was found that the change in SSN triggered the change in the solar polar field. Hence, the SSN and solar polar field were closely correlated from the view point of fractals.

4) When the maximum solar polar field before the maximum SSN was larger, the maximum SSN of the next cycle was larger. The formation of the magnetic field of the sunspots was correlated with the solar polar field.

The author declares no conflicts of interest regarding the publication of this paper.

Maruyama, F. (2019) Relationship between the Sunspot Number and Solar Polar Field by Wavelet-Based Multifractal Analysis. Journal of Applied Mathematics and Physics, 7, 1043-1051. https://doi.org/10.4236/jamp.2019.75070