Gopal Hazra1, Bidya Binay Karak2, and Arnab Rai Choudhuri1
1. Department of Physics, Indian Institute of Science, Bangalore 560012, India
2. Nordita KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91, Stockholm, Sweden
In recent years the flux-transport dynamo model has become popular due to its success in reproducing many important observational features of the solar cycle. However, in spite of this success, some recent observations raise doubts about the validity of the flux-transport dynamo model. The meridional circulation of the Sun is a crucial component of this model and some recent analyses of observational data suggest that this circulation may be quite different from what had been assumed in the early models of the flux-transport dynamo. We now show that the attractive features of the flux-transport dynamo can be retained even with meridional circulations more complicated than what had so far been assumed in theoretical models, provided some conditions are satisfied.
The majority of flux-transport dynamo models assume a single cell of meridional circulation encompassing one hemisphere of the convection zone with a poleward flow in the upper layers and an equatorward flow at the bottom of the convection zone. The equatorward flow at the bottom of the convection zone is essential to produce the equatorward shift of the sunspot belts with the progress of the sunspot cycle, leading to appropriate theoretical butterfly diagrams1. The poleward flow near the solar surface is also important for the propagation of the surface radial field poleward. We now basically find that a theoretical model can reproduce the basic patterns of observational data as long as we have an equatorward flow at the bottom of the convection zone, even if the meridional circulation pattern is more complicated within the body of the convection zone.
Until recently observational data existed only for the poleward meridional circulation in the upper layers of the convection zone. Our study was motivated by two recent independent measurements by Hathaway2, who found the evidence of return flow of meridional circulation at depths of 50–70 Mm only, and Zhao et al.3, who found the indication of two cells in radius. However, the later authors could not provide information about the flow below 0.75Rsun because of their limited data set.
To answer the question whether these results pose a threat to the flux-transport dynamo model, we have carried out some simulations with different possible multi-cell meridional circulations in the flux-transport dynamo model4. We first consider the case of a shallow meridional circulation confined only in the upper portions of the convection zone, with no flow underneath. This scenario cannot reproduce the correct behavior of the solar magnetic fields. Even on putting an additional cell of meridional circulation below the upper cell, we cannot match observations if the lower cell has a poleward flow at the bottom of the convection zone.
Figure 1 | a) Streamlines of meridional circulation with three radially stacked cells. (b) vθ as a function of r/R☉ at the mid-latitude of θ = 45°. (c) Butterfly diagram i.e, the time-latitude plot of the toroidal field at the bottom of the convection zone (r=0.70 R☉). (d) Time-latitude plot of the radial field at the surface of the Sun. All the toroidal and radial fields are in the unit of B0.
Only when we introduce multiple cells in such a way that there is an equatorward flow at the bottom of the convection zone, we do get appropriate butterfly diagrams. Figure 1a and 1b show results obtained with three cells of meridional circulation stacked one over the other in the radial direction—the upper two cells being in conformity with Zhao et al.’s findings, whereas the third cell provides a equatorward flow at the bottom of the convection zone. We find a solar-like butterfly diagram, presumably because the toroidal field generated in the tachocline is advected equatorward by the equatorward meridional flow there.
Finally we have considered some simulations with fairly complicated multi-cell meridional circulation in such a way that there is a significant amount of equatorward flow at the bottom of the convection zone at the low latitudes. Figure 2 shows a typical meridional circulation structure and the butterfly diagram using this flow.
We find a remarkably good solar-like equatorward propagation. Apart from equatorward meridional circulation at the bottom of the convection zone, we found that an equatorward latitudinal turbulent pumping of appropriate strength present there also can give rise to solar-like behavior. This is consistent to the earlier results of Guerrero \& de Gouveia Dal Pino5.
Figure 2 | Streamlines for complicated meridional circulation patterns. The blue contours imply counter-clockwise circulation, whereas the red contours imply clockwise circulation. (b) Butterfly diagram.
We have also checked whether the observed correlation between the polar field at the solar minimum and the strength of the next cycle can be reproduced with multicellular meridional flow. We have found that this correlation can be reproduced robustly in a high diffusivity dynamo model.
In summary, we conclude that the attractive features of the flux-transport dynamo can be retained even if the return flow of meridional circulation is at a shallow depth as suggested by recent observations, provided there are additional cells of meridional circulation below such that there is an equatorward flow near the bottom of the convection zone. If future observations show that an equatorward flow at the bottom of the convection zone does not exist, only then a drastic revision of the flux-transport dynamo will be needed.
 Choudhuri, A. R., Schussler, & Dikpati, M. 1995, A&A, 303, L29
 Hathaway, D.H. 2012, ApJ, 760, 84
 Zhao, J., Bogart, R. S., Kosovichev, A. G., Duvall, T. L., & Hartlep, T. 2013, ApJ, 774, L29
 Hazra, G., Karak, B. B., & Choudhuri, A. R. 2014, ApJ, 782, 93
 Guerrero, G., & de Gouveia Dal Pino, E. M. 2008, A&A, 485, 267