Li, Gao and Zhan (Solar Physics 255, 289, 2009) showed that the monthly sunspot number (SN) and sunspot area (SA) series over 1874-2007 are phase-synchronized only within a narrow low-frequency band of roughly 7-12 years (the Schwabe-cycle band), display strong phase mixing at higher frequencies, and exhibit a persistent but time-varying lag of SN behind SA, with a reported extreme asynchrony near 35 months around 1928. The open problem — documented by that paper but left unexplained there and in the subsequent literature — is whether solar dynamo theory can predict or explain these detailed phase relations: (a) why coherent phase variables between the two indices exist only near the Schwabe band, (b) what produces the strong phase mixing at higher frequencies, and (c) the origin of the persistent, time-variable lag. Later literature has generalized the phenomenology to many other pairs of activity indices and argued that the large apparent SN-SA offsets are largely artifacts of the sunspot-number definition, but no dynamo-theoretic (or other physical) account reproducing the documented frequency-resolved phase statistics has been published, so the explanatory question remains open.

Background

Sunspot activity is quantified by two canonical indices derived from the same photospheric spots. The sunspot number (SN, also called the Wolf or international sunspot number) is a count-based index R=k(10g+f)R = k(10g + f), where gg is the number of spot groups, ff the number of individual spots, and kk an observatory-dependent calibration factor; the sunspot area (SA) is the total projected area of all visible spots on the solar disk, compiled since 1874 by the Royal Greenwich Observatory and its successors, conventionally expressed in millionths of the solar hemisphere. Both indices vary with the approximately 11-year Schwabe cycle, the magnetic activity cycle produced by the solar dynamo — the self-exciting magnetohydrodynamic process, driven by the Sun's differential rotation, convection, and meridional circulation, that maintains and periodically reverses the large-scale solar magnetic field. In the prevailing Babcock-Leighton flux-transport picture, differential rotation shears poloidal field into toroidal field, buoyant magnetic flux tubes then emerge through the convection zone to form spot groups at low latitudes, and surface flux-transport processes (turbulent diffusion plus poleward meridional flow) recycle the decayed flux into the new poloidal field. Dynamo models of this type are routinely tuned to reproduce the amplitudes and periods of activity indices, but they predict large-scale field evolution, not the individual spot groups from which SN and SA are built. Li, Gao and Zhan (2009) analyzed the phase relation of the smoothed monthly mean SN and SA series from November 1874 to September 2007 with linear spectral methods (fast Fourier transform) and nonlinear time-series tools: the cross-wavelet transform and wavelet coherence (time-frequency maps that expose locally coherent oscillations and their relative phase), and the cross-recurrence plot with its line of synchronization (LOS) — a phase-space technique in which two series are embedded in a common state space and the LOS traces how far one series must be shifted at each epoch to match the other, giving a time-resolved measure of which index leads and by how much. They found a high level of phase synchronization confined to time scales of about 7 to 12 years, noisy behavior with strong phase mixing (random distribution of relative phases) at higher frequencies, a reported global correlation of about 0.985 at zero shift, and an LOS indicating that SN lags SA over most of the record, with a reported largest asynchrony of roughly 35 months around 1928 and greater asynchrony near cycle minima and maxima than on the rising and falling branches. Follow-up work generalized the phenomenology rather than explaining it: Zhang, Le and Zhang (2012) found the same band-confined in-phase structure and high-frequency phase mixing for the sunspot number versus the 10.7 cm radio flux (the solar radio emission at 10.7 cm wavelength, a standard indicator of chromospheric and coronal activity); Tang and Deng (2013) and Deng, Li, Zheng and Cheng (2013) documented phase asynchrony of the flare index (a daily aggregate measure of the number and importance of solar flares) and of the radio flux relative to sunspot activity, interpreting chromospheric and coronal lags through a heuristic integral-response model; Xu, Xie and Qu (2017) and Li, Xu, Yin, Xie and Feng (2020) extended phase-relation studies to total solar irradiance and to thirteen monthly activity indices. Crucially, Ramesh and Vasantharaju (2014) analyzed cycles 21-23 and argued that the temporal offsets between SN and SA are artificial rather than real: both indices derive from the same observed spots, their overall cross-correlation lag is only about 1-2 months, and large peak-occurrence offsets (up to 23 months) arise from underestimation of the sunspot number near maximum caused by the equal weighting of spot groups in the Wolf definition. Li, Li, Zhang and Feng (2016) likewise showed that the SA-SN relation is statistically nonlinear (clearest at cycle maximum) and suggested the correction factor in the SN definition as a contributing cause. The International Space Science Institute (ISSI) review by Ermolli, Shibasaki, Tlatov and van Driel-Gesztelyi (2014) recounts the Li et al. (2009) phase results descriptively, with no dynamo-theoretical explanation reported. What remains missing is a physical account — dynamo-theoretic, or explicitly demonstrating the contrary — of the full frequency-resolved phase structure between the two most fundamental solar activity indices.

Problem Statement

Can the documented phase relations between the monthly mean sunspot number (SN) and sunspot area (SA) series over 1874-2007 be derived from solar dynamo theory — or shown to arise entirely from the construction of the two indices — and specifically: (a) why do coherent, phase-locked variables between SN and SA exist only in the narrow low-frequency band of roughly 7-12 years around the Schwabe cycle, with no coherent phase structure outside it; (b) what physical mechanism produces the strong phase mixing (apparently random distribution of relative phases) of the high-frequency components of the two series; and (c) what is the origin of the persistent but time-variable lag of SN behind SA seen in the cross-recurrence line of synchronization, including large excursions such as the reported roughly 35-month asynchrony around 1928 and the greater asynchrony near cycle minima and maxima? A satisfactory answer gives a physical account of this frequency-resolved phase structure — from a dynamo-theoretic model of the dynamo and flux emergence mapped through both index definitions, or by demonstrating that the structure follows entirely from index construction and measurement — and should also take into account the later cross-correlation evidence that overall SN-SA lags are small and that large peak offsets may be artifacts of the Wolf-number definition.

References

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  2. XueFeng Zhang, GuiMing Le, YanXia Zhang, 'Phase relationship between the relative sunspot number and solar 10.7 cm flux', Chinese Science Bulletin 57, 2078-2082 (2012), DOI: 10.1007/s11434-012-5104-4, https://doi.org/10.1007/s11434-012-5104-4
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