ANALYSIS MHD SOLAR ACTIVITY USING ROBUST METHODS

Abstract

The aim of this paper is to analyze magnetohydrodynamic (MHD) solar activity on the basis of modeling and predicting the Wolf number dynamics. This study is based on the use of adaptive regression modeling approach implemented in the software package ARDM (adaptive regression dynamic modeling). The preliminary data was investigated for the presence of trend stability with fractal analysis. Currently, not only the experts in the field of solar physics, climatologists, and researchers in many fields related to the issue, but most of the world's population is conscious of the solar activity influence on many processes occurring in the world and directly defining the conditions of life and health. Most of the phenomena taking place on the Earth are indeed related to the solar activity either directly or indirectly. The solar activity is characterized by the number of spots produced on the sun surface. Their number is determined by a formula proposed by Rudolf Wolf. Variations of the solar activity are cyclic, with their minimum and maximum. Correspondingly, the effects of its influence are different. Thus, the ARDM software package is modified and the analysis, modeling and forecasting of the solar activity are carried out. The initial data includes two time series: the Wolf number over the period between 2000 and 2011 averaged monthly (124 cases) and the Wolf number over the same period averaged weekly (484 cases).

Headnote

ABSTRACT

The aim of this paper is to analyze magnetohydrodynamic (MHD) solar activity on the basis of modeling and predicting the Wolf number dynamics. This study is based on the use of adaptive regression modeling approach implemented in the software package ARDM (adaptive regression dynamic modeling). The preliminary data was investigated for the presence of trend stability with fractal analysis.

Currently, not only the experts in the field of solar physics, climatologists, and researchers in many fields related to the issue, but most of the world's population is conscious of the solar activity influence on many processes occurring in the world and directly defining the conditions of life and health. Most of the phenomena taking place on the Earth are indeed related to the solar activity either directly or indirectly. The solar activity is characterized by the number of spots produced on the sun surface. Their number is determined by a formula proposed by Rudolf Wolf. Variations of the solar activity are cyclic, with their minimum and maximum. Correspondingly, the effects of its influence are different.

Thus, the ARDM software package is modified and the analysis, modeling and forecasting of the solar activity are carried out.

The initial data includes two time series: the Wolf number over the period between 2000 and 2011 averaged monthly (124 cases) and the Wolf number over the same period averaged weekly (484 cases).

Keywords: MHD solar activity, Wolf number, regression analysis, fractal geometry

(ProQuest: ... denotes formulae omitted.)

INTRODUCTION

Despite the advances in modern ground-based and space observations of the Sun using the high spectral and spatial resolution devices for spectro and polarimetric data analysis and the progress made in studying the structure and the dynamics of sunspots, the problem of establishing a link between the global dynamo processes in the Sun and formation of sunspots is very complicated to solve, since spontaneous generation of self-organizing magnetic structures (e.g. sunspots and pores) presents a complex interaction between convection and magnetic fields at various scales. Thus, during a simulation of sunspots formation it is necessary to take into account a large number of multiparametric dynamic processes (e.g. influence from self-forming strong horizontal magnetic fields and self-organizing turbulent ascending and descending streams) [1]. At the same time, helioseismology studies of small magnetic sunspots self-organization into larger structures show the importance of taking into account turbulent convection. The recent achievements in numerical simulation of sunspots magnetohydrodynamics have allowed building realistic radioactive numerical models [2].

The solar magnetic activity cycles and large-scale structures of the solar magnetic field may be described by non-linear dynamo models in the convection zone [3]. The nonlinearity effects in solar magnetic activity appear for a reason. This is connected with the fact that the interaction between the magnetic field and the moving plasma is a nonlinear process due to the solar axial rotation. We should also note that oscillations of the Wolf daily number are of noise nature. The considered periods are of quasi-periodic nature with irregular change of phase and amplitude. It has been found previously the chaotic nature of solar activity processes manifests over relatively long periods of time [4]. Short-period processes (e.g. solar flashes) submit to the stochastic laws. It is noted that the methods of Fourier and wavelet analyses could be used for studying long-period change of solar activity [5]. In some works the multi-fractal nature of solar activity is found at different time scales [6].

In the present work, within the robust approach dynamic features of solar activity have been determined on the basis of the Wolf daily number analysis. It is well known that astrophysical objects are non-stationary open systems whose evolution appears to be very individual with inevitable intermittency effects manifesting [5]. In such a dynamics the components with various frequency ranges (specific - resonance and non-specific - chaotic) manifest. The feature of the robust approach is introducing informational parameters describing the dynamics of solar activity at various frequency ranges.

The present work has the following structure. In the first section the basic concepts and relations of the multifractal analysis of wolf number time series, which is a phenomenological approach to chaotic signal analysis allowing determining informational significance of low-frequency resonance and high-frequency components of the studied dynamics, are represented. The second section contains the results of constructed the adaptive regression dynamic models . The prospects of using robust analysis to study the processes of plasma convective motion in solar atmosphere are discussed in the last section.

SUBJECT & METHOD(S) OF RESEARCH: MULTIFRACTAL ANALYSIS OF WOLF NUMBER TIME SERIES

When using fractal analysis of two time series, it is revealed that the Hurst exponent has a value of about 0.96, which indicates a strong trend stability of the series. Multifractal analysis shows that the monthly averaging provides a better forecast than the weekly one; the monthly data provides more information and a greater pair of correlation integrals. The Holder exponents also reveal a high degree of trend stability for both of the series. When modeling the monthly Wolf number, the best model is the one that includes the centering data, polyharmonic trend, ARMA (0,6) and ARCH (1) models. The combined model is as follows:

... (1)

where e (t - 1), e (t - 2), ... are components of the ARMA (0,6); Y (t - 1), ... are terms of the ARCH model (1), s(t) is "white" noise.

DISCUSSIONS & RESULTS: THE RESULTS OF CONSTRUCTED THE ARDM MODELS

The standard deviation (SD) of the model is 10.09, the external SD is 20.00. The forecast for the period between November 2010 and April 2012 is constructed. A comparison of the forecast built in ARDM with the one built in the Solar Influences Data Analysis Center (Belgium) (http://sidc.oma.be/html/wolfimms.html) is carried out as well.

The curves (Fig.1 and Fig.2) show that the model constructed in the ARDM (Fig.1) provides a closer approximation to the original data compared to the information provided by the Solar Influences Data Analysis Center (Belgium) (Fig.2). The forecast obtained by the modeling on the basis of the ARM-approach is closer to the observational results obtained over the period between November 2010 and April 2011.

A combined model for weekly data includes centering data, polyharmonic trend, ARMA (0,6) model and ARCH (1):

...

SD is 16.91, an external SD - 22.77.

Forecast for weekly data is presented in Fig. 3. Fig. 4 shows stretched part of Fig. 3, including the time interval from 2010 to April 2012.

From the curves of weekly averaged forecasts demonstrated in Fig. 1,3,4, two serially increasing peaks of the solar activity occurring between middle of 2011 and the first quarter of 2012 are more noticeable compared with monthly averaged forecasts. However, this feature is not reflected in thcurves constructed on the basis of the the Belgian center which does not take into account the periodic variations of the Wolf number.

CONCLUSION

As results, we should say the following. The main mechanism of solar cycle is hydromagnetic dynamo operating in the convective zone [6]. However, the existence of magnetic field in convective zone means there is magnetic field in internal zone of radiation equilibrium. Considering the fact that the Sun was formed as a result of compression of a gas-dust cloud containing magnetic field, it is believed that there might be some residual magnetic field of cosmogonic origin independent of dynamo in radiative equilibrium zone. In this respect, differential rotation of the radiative zone may be a non-stationary transition phenomenon related to internal distribution of perturbations from the convective zone due to magnetic tension [6]. Thus, formation of the solar dynamic magnetic field is based on evolution of small-scale magnetic helicity and of a complex non-linear structure [7]. On the other hand, coronal mass ejection (CME) is closely linked to the processes of magnetic energy transformation into kinetic and heat energies (magnetic reconnection) in the Sun's corona and solar flares. Usually, CME and solar flares arise is solar active areas where there is strong magnetic field. Thus, since the evolution of Wolf numbers is based on formation mechanism of magnetic spots as a result of negative effective instability of magnetic pressure, largescale magnetic field in the convective zone is subjected to strong turbulent diffusion and the dynamics of CME and solar flares are the main mechanisms of solar energy ejection, all these processes should be analyzed as a single multi-parametric correlation process. This is also confirmed by the results of the present work. In particular, it follows from the structure of time dependence of non-stationarity factor as a precursor of the moments when dynamic solar activity is increased as well as correlation between Wolf numbers and amount of solar energy ejected [8]. The correlations and anticorrelations found in cross-correlations depend on the Sun's 27-days period of rotational dynamics. This is a confirmation of periodic dynamic processes in the convective and radiative equilibrium zones. Peng Zou, et. al. (2014) [9] determined from the analysis of correlation measurements of solar filaments' positive entropy relative to solar dynamo and dynamics of sunspots that coronal activity of the Sun's upper layer described by the number of solar filaments was chaotic and had a very complex behavior, but still the connection between solar filaments and sunspots remained unclear. Nevertheless, according to section 4 of the present article, some correlations between Wolf numbers and the Sun's energy ejection may be found. And if the numerical simulation carried out by Peng Zou, et. al. (2014) [9] cannot directly serve as a tool of predicting the Sun's activity, then in the future such a method might be developed on the basis of robust analysis of solar activity.

Thus, robust analysis allows detecting the moments of most significant restructuring occurring in the solar atmosphere which are revealed from maximums of the nonstationarity factor. Depending on duration of the averaging period, the non-stationarity effects related to both the well-known 11-years periods and the processes within each of such cycles are revealed. The robust approach determines the information of the dynamics of correlations between the signals simultaneously measured - dynamic variables of solar activity. To consider cross-correlations taking place between various solar activity parameters, the well-known set of Wolf numbers and the consequence of values of total energy emitted by the Sun's corona are used. The three-dimensional representations of cross-correlation functions enabled to determine individual features of the Sun's evolution.

In conclusion, it should be noted that the study of magnetized plasma's selforganization and the formation of solar stable magnetic structures are still some of the most complicated problems of modern astrophysics [10]. The Sun is a non-stationary open system whose dynamics appears to be highly individual with unavoidable manifestation of intermittency effects [11]. In these dynamics the contributions to various frequency ranges are manifested: specific - resonance components with wide sets of frequencies and non-specific - chaotic components [12]. The information on such contributions to the studied signals may be obtained within the robust analysis whose specifics lies in introducing informational parameters characterizing components of the investigated signals in various frequency ranges [13]. The results of the work can be used to process positional observations also [14, 15].

ACKNOWLEDGEMENTS

The work is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University. We would also like to especially acknowledge the contributions of S.G.Valeev for the materials were provided for the article. This work was partially supported by scholarship of the President of the Russian Federation to young scientists and post-graduate students number СП-3225.2018.3. This work was partially supported by the Russian Foundation for Basic Research, grant nos. 16-3260071-mol_a_dk, 18-32-00895 mol_a and Russian Science Foundation 18-72-10037.

References

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