Three-halves law in sunspot cycle shape

Abstract

Annual mean sunspot numbers R(t) since 1700 show evidence of a non-linear effect, first evidenced by the detection of a third harmonic in R±(t), the alternating representation of the magnetic (22 yr) cycle of solar activity. The form of the non-linearity proves to be a three-halves law R(t) = 100[|Rlin(t)|/83]3/2, where Rlin(t), also an alternating quantity, is a presumed underlying or ‘linearized’ sunspot number. The non-linearity is of such a nature as to cause strong semicycles to be sharper than sinusoidal and to produce the inflexion in R±(t) noted at sunspot minimum. The difference R(t)−|Rlin(t)| is sufficiently like a third harmonic, for semicycles of average strength, to explain the band around 22/3 yr which is noticeable in the spectrum of R±(t). However, the third harmonic alone is not sufficient to account for the observed dependence of semicycle shape on amplitude, whereas the three-halves law accounts economically for a range of effects. A search for a physical explanation of a three-halves law reveals that just such a law results because large sunspot groups, such as occur around strong sunspot maxima, enter into the sunspot number, as conventionally defined, over more days than small groups, simply because large groups last longer. Semicycle asymmetry, which cannot result from a simple non-linear law, is here ascribed to magnetic buoyancy acting preferentially on the anti-nodal layers of a travelling wave. Profiles for semicycles of different strengths have been constructed on the assumption that the underlying influence is sinusoidal. Each sinusoid is distorted by the three-halves law, and then made unsymmetrical by applying a buoyancy theory for magnetized plasma rising against viscous drag. The majority of past semicycles, including those of the eighteenth century, can be matched quite well by the artificial profiles, a conclusion that supports the idea of an underlying influence that is sinusoidal, and the hypothesis that sunspots result from an upward travelling wave from a submerged 22 yr oscillator.