Simon Wing¹, Jay R. Johnson², and Angelos Vourlidas¹
1.The Johns Hopkins University, Applied Physics Laboratory, Laurel, Maryland, USA
2.Andrews University, Berrien Springs, Michigan, USA

The solar dynamo is not fully understood. Predicting sunspot number (SSN) is still a challenge. A key question is how much information is contained in the past history of the observations and how much of that information is lost as the system evolves. In this work, we look at the flow of information in solar activity cycles using transfer entropy to examine the relationship between various observables that play key roles in the solar dynamo, namely the polar field, meridional flow, and SSN. The transfer entropy (TE) from variable x to y, TE(x → y), gives a measure of information transfer from variable x to y, given that all past values of y are known. Unlike the correlation function, TE(x → y) is usually not the same as TE(y → x).

To illustrate the power of this technique, we examine the information transfer between aa index and SSN. We show that the time-shifted cross correlations, namely, corr(aa index(t), SSN(t + τ)) and corr(SSN(t), aa index(t + τ)), have roughly same amplitudes. However, TE(SSN → aa index) is larger than TE(aa index → SSN), suggesting that more information is transferred from SSN to aa index than the other way around (see Figure 3 in Ref. 1). This information cannot be discerned from the standard correlational analysis.

Polar field and SSN

We examine the information transfer between the polar field and SSN. The transfer of information from the polar field to SSN peaks at lag time (τ) ~30–40 months (see Figure 4 in Ref. 1). The lag time can be interpreted as the average transport time of fluxes from the polar region to low latitude photosphere in the period analyzed (1967–2014).

Figure 1|The long-term effect of the polar faculae (proxy for polar fields) on sunspot production. TE(polar faculae → SSN) and TE(SSN → polar faculae) are plotted in blue and red curves, respectively, for the period 1906–2014. The solid and dashed green curves show the mean and 3σ of the noise. There is also a long-term effect of the SSN on polar faculae.

To investigate the long-term effect of the polar field on SSN, we use polar faculae, which have been suggested as a good proxy for polar fields. The polar faculae data are available since 1906. Figure 1 plots TE(polar faculae → SSN) and TE(SSN → polar faculae) for the period 1906–2014. The transfer of information from the polar faculae to the SSN peaks at τ ~ 30–40 months (similar to that for the polar field), but thereafter it remains at a persistent low level for at least 400 months (~ 3 solar cycles). The latter may indicate the persistency of the polar fields from cycle to cycle. It may also be consistent with the idea that the polar fields from the last 3 or more solar cycles can affect the production of SSN of the subsequent cycle^[2]. Interestingly, the information transfer from the polar faculae to SSN has minima at lags of one and two solar cycle periods. Figure 1 also suggests that there is information transfer from SSN to the polar field. The amount of flux emergence, which can be proxied by SSN, provides the seeds of the polar fields several years later^[3]. The multiple peaks with small time separations in the red curve may result from the episodic nature of the flux transfer to the poles. For example, it is well-known that flux is transported to the poles in surges^[4].

Parameters that affect the polar field

Figure 2|TE(meridional flow → polar field) and TE(SSN → polar field) are plotted in blue and red curves, respectively, for the period 1986–2012. The curves are noisy because of the limited availability of the meridional flow data. The solid and dashed green curves show the mean and 3σ of the noise.

In some models, the polar field is mainly determined by the meridional-flow speed^[5] while in others the polar field may have been determined to a large extent by the flux emergence^[3]. We examine the contributions of the meridional flow and flux emergence (proxied by SSN) to the polar field. Figure 2 plots TE(meridional flow → polar field) and TE(SSN → polar field) for 1986–2012. It turns out that both the meridional flow speed and flux emergence (proxied by the SSN) transfer information to the polar field, but one transfers more information than the other, depending on the lag times. The meridional-flow speed transfers more information to the polar field than SSN at τ ~28–30 and ~90–110 months, which may be consistent with some models. However, the flux emergence transfers more information to the polar field than the meridional flow at τ ~60–80 months, which may be consistent with a recently developed surface flux transport model.

Closing remarks

We have also examined the information transfer from the meridional flow to SSN. The transfer of information from the meridional flow to SSN peaks at τ ~110–120 months, suggesting that the meridional flow can be used to predict SSN about one cycle ahead. This result is presented in our recently published paper^[1]. It is hoped that our analysis can provide observational constraints to solar cycle models and theories.

References:
[1] Wing, S., Johnson,J., & Vourlidas, A., 2018, ApJ, 854, 85
[2] Charbonneau, P., Dikpati, M., 2000, ApJ, 543, 1027-1043
[3] Upton, L., Hathaway, D. H., 2014, ApJ, 792, 142
[4] Wang, Y.-M., Nash, A. G., & Sheeley, N. R., Jr., 1989, ApJ, 347, 529–539
[5] Choudhuri, A. R., Chatterjee, P., & Jiang, J. 2007, Physical Review Letters, 98, 131103