G. Guerrero^{1}, P. K. Smolarkiewicz^{2}, A. G. Kosovichev^{3} & N. N. Mansour^{4}

1. Physics Department, Universidade Federal de Minas Gerais, Av. Presidente Antônio Carlos, 6627, Belo Horizonte – MG, Brazil， 31270-901

2. European Centre for Medium-Range Weather Forecasts, Reading RG2 9AX, UK

3. Big Bear Solar Observatory, NJIT, Big Bear City, CA 92314-9672, USA

4. NASA Ames Research Center, Moffett Field, CA 94040, USA

The spatial and temporal evolution of sunspots is the surface manifestation of a rather complex magnetic process happening in the solar interior. Unfortunately, the weak helioseismic signature of the magnetic field makes it difficult to obtain some information about its dynamics at subsurface levels. Over the past 60 years, theorists have tried to develop a consistent physical model to explain this process using a few observational hints, numerical models, and a lot of imagination. It has been a very dynamic field of research in which many questions are still open. Here I briefly describe two different approaches to model the physics of solar cycles.

The solar magnetic cycle is thought to be hydromagnetic in nature and is called dynamo. Its study has been guided by the observations of the large scale, azimuthally averaged, plasma motions, namely differential rotation and meridional flow. Thus, modelers need to worry about the explanation for both, the fluid motions and the magnetic field dynamics. However, these two aspects of the problem have the same origin: the small-scale, turbulent, and helical motions of the fluid resulting from the convective instability in the upper 30% of the solar radius.

An elegant framework to describe these phenomena is the mean-field approximation where the quantities in the magnetohydrodynamics (MHD) equations are separated in their small (turbulent) and their large (Reynolds averaged) scales. When the mean-field approximation is applied to the momentum equation, the result is that the large-scale motions depend mainly on the contribution of the turbulence via the so called Reynolds stresses. They are capable of efficiently transporting angular momentum in radius and latitude. Hydrodynamic (HD) mean-field models nicely reproduce the solar differential rotation pattern. For the meridional circulation, most of the models result in a flow consistent with one big circulation cell per hemisphere, spanning from top to bottom of the convection zone^{[1]}. This is intriguingly in contradiction with recent helioseismology results indicating that the meridional flow might be consistent with several circulation cells in radius and latitude^{[2]}.

Figure 1 |Differential rotation parameter, χ_{Ω}(it measures the difference in angular velocity between the equator and 60 degrees latitude) as a function of the Rossby number. Negative (positive) values of χ_{Ω}indicate anti-solar (solar-like) differential rotation. The transition between the two regimes occurs in a very sharp range of Ro (note that this might not be the case in MHD models). There is few observational evidence of anti-solar rotation for cool stars in the literature, new observations will let us know whether the results obtained by simulations are realistic in the near future.

As for the magnetic part, the evolution of the large-scale magnetic field depends not only on the profile of differential rotation but also on the turbulent motions via the electromotive force (emf). Unfortunately this emf is not well known, and even when most of the observational constraints are used, a wide range of dynamo solutions is still allowed. One class of mean-field models, more phenomenological in its assumptions, is the so-called flux-transport dynamo model^{[3]}. It has been successful in reproducing the observed magnetic features and has been thoroughly used in forecasting studies. Nevertheless, the flux-transport model requires either a single circulation cell or then a coherent equatorward flow at the base of the convection zone, neither of which has been observed so far. It also assumes the buoyancy of thin magnetic flux tubes, from bottom to top of the CZ, and the existence of these rising tubes is still a matter of debate.

Figure 2 |Differential rotation profile, meridional circulation and angular momentum fluxes for models, F_{ω}^{RS}, F_{ω}^{MC}, F_{z}^{RS}and F_{z}^{MC}(from left to right) for simulations with rotation period a) 112, b) 56, c) 28 and d) 14 days. In the meridional flow plots (second column), dashed (solid) lines indicate counterclockwise (clockwise) circulation. The model corresponding to the Sun (model c) reproduces fairly well the latitudinal differential rotation and the tachocline, however the iso-rotation contours are aligned along cylindrical shapes. In the Sun these contours have conical shape. This is one of the most important issues to be solved in global solar simulations.

Starting with the seminal works of P. Gilman in the late 70s, another approach is possible and consists of solving together all the HD or MHD equations in spherical geometry and with the appropriate boundary conditions. This is called global modeling. Several groups around the world are tackling the solar dynamo problem under this framework by considering different numerical techniques in their simulations. With respect to both the plasma motions and the magnetic field, the results seem to be converging. Here I describe the results recently obtained by Guerrero et al.^{[4]} for HD simulations using numerical code EULAG.

Figure 3 |Time-latitude, butterfly diagram, showing the three components of the magnetic field. The color bar indicates the magnitude of the field in Tesla. In this particular simulations the cycle period is short, ~2.5-years, and the solution is symmetric around the equator (see upper panel corresponding to the toroidal component). This picture will appear in a forthcoming publication.

The establishment of mean flows depends on the delicate balance between buoyant and Coriolis forces (this balance can be controlled through a non-dimensional quantity called Rossby number, see Figure 1). For larger Rossby numbers (i.e., dominant convective motions), the differential rotation profile is anti-solar, meaning rapid poles and slower equator. The collective action of small scale motions in the meridional direction creates a coherent counterclockwise (clockwise) meridional circulation in the northern (southern) hemisphere. These circulation cells transport angular momentum toward the higher latitudes (Panels a and b of Figure 2). For smaller Rossby numbers (dominant rotation), the equator rotates faster than the poles such as observed in the Sun. The meridional flow exhibits a complicated multicellular patter consistent with the helioseismic results^{[2]} (Panels c and d of Figure 2). The decline of the angular velocity in the upper part of the convection zone (also called near-surface shear layer), might happen because the surface fluid motions, granulation and supergranulation, evolve on a time-scale much shorter than the rotation time-scale^{[4]}. The poleward migration observed in the plasma flow at all latitudes could be a consequence of this negative near-surface shear via a mechanism called gyroscopic pumping.

Obtaining large-scale dynamo solutions with global simulations has been very common in recent years. The solutions go from toroidal wreathes of inverse polarity around the equator, for models where convection dominates over rotation, to magnetic cycles with different configurations and periods (see, e.g., in Figure 3), for rotation dominated models^{[5]}. Although up to the date there is not a model able to reproduce all the solar features, the recent results are exciting and promise a new era of self-consistent simulations of solar and stellar interiors.

### References

[1] Kitchatinov, L. & Rudiger, G. 2005, *AN*, **326**, 379

[2] Zhao, J, Bogart, R. S., Kosovichev, A. G., Duvall, T. L., Jr., & Hartlep, T., 2013, *ApJL*, **774**, L29

[3] Guerrero, G. & de Gouveia Dal Pino, E. M., 2008, *A&A*, **485**, 267

[4] Guerrero, G., Smoloarkiewicz P. K., Kosovichev, A. G., & Mansour, N., 2013, *ApJ*, **779**, 176

[5] Ghizaru, M., Charbonneau, P., & Smolarkiewicz, P. K., 2010, *ApJL*, **715**, L133