Fractal Dimension and Maximum Sunspot Number in Solar Cycle R.-S. Kim 1,2, Y. Yi 1, S.-W. Kim 2, K.-S. Cho 2, Y.-J. Moon 2 1 Chungnam national University 2 Korea Astronomy and Space science Institute

Contents  Introduction  Prediction of solar activity cycle  Fractal dimension  Calculation of the fractal dimension  Cycle variation of the fractal dimension  Prediction of maximum sunspot number  Conclusion

Prediction of solar activity cycle  Prediction of solar activity cycle  When, how strong~?  Prediction method Period analysis: power spectrum Gleissberg cycle: long period (80 years) Precursor method: extended period

Fractal dimension  What is the Fractal?  Fractal has statistical self-similarity at all resolutions and is generated by an infinitely recursive process.  Fractal dimension  New parameter for quantitatively describing the characteristics of an irregular time series.

 Higuchi’s Method (1988)  The fractal dimension approaches 1, if a time series is regular  ‘Marginal fractal’ 2, if a time series is completely random  ‘Extreme fractal’  N: data number, m: initial time, k: interval time (resolution) Fractal dimension m

 The length of the curve for the time interval k,  The average value over k sets of L m (k) L m (k) = { ( ∑ | X(m+ik) - X(m+(i-1) k ) | ) × (N-1) /[(N-m)/k]k} × 1/k  k ↓, data number ↑ ; k ↑, data number ↓  if | X(m+ik) - X(m+(i-1) k ) | is constant, ~ k -2  Extreme fractal  If ~ k -D, the curve’s fractal dimension is D

 Cycle fractal dimension  Sunspot number : The daily relative sunspot numbers since 1850 from SIDC (Solar Influences Data analysis Center).  log = a – D log k(a = constant) Cycle variation of the fractal dimension

 Cycle variation  10 ~ 22 solar cycle Cycle variation of the fractal dimension

 Sunspot number

Cycle variation of the fractal dimension  Fractal dimension vs. Maximum sunspot number (r = -0.95)

 Fractal dimension vs. Maximum sunspot number  The coefficient of correlation for increasing phase Prediction of maximum sunspot number

 Observed vs. predicted maximum sunspot number  Using fractal dimension of increasing phase (4 years) Prediction of maximum sunspot number

Conclusion  Variation of the fractal dimension  Cycle fractal dimensions and maximum sunspot numbers have inverse relationship.  Prediction method  By using fractal dimension for increasing phase of solar cycle, we can predict maximum sunspot number  Observed and predicted sunspot numbers have good relationship (r = 0.90).