Is the Solar Cycle Chaotic?

Posted by: chaamjamal on: August 26, 2018

In: Uncategorized
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FIGURE 1: ORW SHAPE: Optimal Regular Wave Function

solarcycleorwf

FIGURE 2: ORW RESIDUAL ANALYSIS AND THE HURST EXPONENT

solarcycle-hurst-exponents

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Sunspots are evanescent dark spots on the sun ranging in size from megameters to gigameters in diameter. Their life span varies from two days to two weeks and their number at any given time from a few to more than a hundred. The total number of sunspots is known to serve as a measure of the level of solar activity (Rogers, 2006) (Yan, 2013). The time series of the number of sunspots appears to form a cyclical pattern with a period of about eleven-years. However, both the period and the amplitude of this cycle are variable and irregular apparently containing long run and irregular cycles of their own (Hathaway, 1994) (Rogers, 2006) (Miletsky, 2014).
Sunspot cycles have presented researchers with an enigmatic and vexing quandary for centuries because their irregular nature is not well understood (Hathaway, The Solar Cycle, 2010). Sunspot counts have been recorded and studied since 1610 and all aspects of the patterns in the time series have been subjected to intense scrutiny and interpretation in terms of solar phenomena and their effects on earth (Usoskin, A solar cycle lost, 2009) (Usoskin, Grand minima and maxima of solar activity, 2007) (Hathaway, The Solar Cycle, 2010).
The interest in sunspot numbers has grown in the climate change era due to advances in satellite measurements of solar activity and also because of the possible effects of changes in solar irradiance on climate (Weart, 2003) (Haigh, 2007) (Fox, 2004). However, certain issues in the utility of sunspot count time series data remain unresolved, the most prominent being the irregular nature of both the short term solar cycle and the long wave of its amplitude.
In this short post we show that these issues may be addressed by separating the regular cyclical components of the solar cycle from the irregular and by describing the system as a sum of two independent phenomena – one cyclical and the other random and chaotic. The random component is shown to be a non-Gaussian Hurst process with dependence and persistence, properties known to create irregular patterns from randomness (Hurst, 1951) (Koutsoyiannis D. , 2002) (Mandelbrot B. , 1972) (Kim, 2006) (Watari, 1995) (Zhou, 2014).
A hurdle to the analysis and understanding of the cyclical behavior of sunspot counts has been the irregular nature of these cycles. We therefore propose that the phenomenon is best described as the sum of two components – one regular and cyclical and the other irregular and random. For mean monthly sunspot counts in the sample period 1/1818-11/2015 we show that the regular and cyclical component of the phenomenon consists of two superimposed wave functions, one a short wave and the other a long wave. FIGURE 1.
The short wave is identified as a 131-month asymmetric and triangular cycle of sunspot counts with a 50-month rising leg and an 81-month falling leg. The long wave is found to be a 100-year symmetric triangular wave in which the amplitude of the short wave fluctuates between 100 and 230 sunspots (FIGURE 1). This optimal regular component of sunspot number behavior is constructed by minimizing the residuals of the compound wave function.
It is shown that these residuals are not random Gaussian but that they tend to form patterns (FIGURE 2). Rescaled range analysis of the residuals shows that they contain the Hurst effect of memory and persistence and it is proposed that the patterns in the residuals may be explained in terms of the non-linear dynamics and chaos. It is proposed therefore that the behavior of sunspot cycles may be understood in terms of its regular and cyclical components overlaid with a chaotic process.