Exploring Quaternion Flipping of the Sun’s Magnetic Field in the Centennial Sunspot Record

Abstract

Sunspots are a major proxy for measuring the solar input into the Earth’s atmospheric dynamics. This paper explores whether sunspot evolution is underpinned by quaternion flipping in the 19th and 20th century sunspot histories. Quaternion flipping occurs when there is self-similarity in an R-matrix of sunspot cycles, where successive cycles evolve by rotating 1800 in a plane and the subsequent conjugation of subsequent cycles is in the reverse order. The historical sunspot record is partitioned according to quadratic clusters and significant periodicities in the spectral analysis, such as the 65.8 year signal. The flipping is then simulated by reversing the time axis between two adjoining ~ 66 year cycles in the sunspot record, producing an R3R2=-R4 quaternion flip, namely, between 1822–1888 and a reversed time series of 1976 − 1910. The Hausdorff dimensions and multi-taper spectral results for each flipped period are calculated in their entirety, where there was a 94.2% common spectral signal and 98.1% self-similarity. The R3R2=-R4 flip maintained a Hausdorff dimension of between 1.080 ± 0.025. The Hausdorff dimension of 1.081 is significant, since it generates a quadratic Julia set envelope of z2+c where c = ¼. For any values greater than 1.081, the Julia magnetic loops collapse and the quadratic point at z = ½ bifurcates into a pair of repelling fixed points off the real axis (that is, magnetic dipoles). This may explain the flipping in polarity after sunspot maximums and the self-similarity in frequencies between sunspot numbers and solar emissions. Further, there is a possibility that this centennial flipping can be traced from the Little Ice Age and the next phase is a centennial maximum mid-century. This research shows a fundamental relationship in a developing ‘fractal mechanics’ of solar emissions and the nature of its periodic input into the Earth’s atmospheric dynamics.