6 Summary and Conclusions

Over the past few decades there has been a tremendous improvement of sunspot models, in part driven by improvements of instrumentation leading to unprecedented observational constraints on models, in part driven by the increase of computing power finally allowing radiative MHD simulations of entire sunspots. While for a long time penumbral fine structure was only accessible through simplified models (assuming certain field geometries or convective flow structures), in recent years numerical simulations evolved to a degree where they start capturing the essential elements of sunspot fine structure.

To explain different aspects, two “simple” model classes have been proposed for the penumbra: (a) Flux tube models, which assume that the magnetic field structure associated with penumbral filaments can be described as flux tubes located close to the τ = 1 surface. In these models flows channeled by magnetic fields account for the filamentation, the Evershed flow, and the line asymmetries. While these models are most successful in reproducing spectropolarimetric measurements they have difficulties to reproduce the overall downflow in the outer penumbra and to account for all of the energy transport needed to explain the penumbral brightness. (b) The gappy penumbra model, which makes the strong assumption that regions of strong magnetic field and convective energy transport are spatially separated. This model assumes field-free gaps in which energy transport by sufficiently deep reaching elongated convection cells. This model does not give an explanation for the Evershed flow (i.e., the model does not require the existence of strong horizontal flows), but such could be included. However, these flows would be essentially non-magnetic and it has not yet been demonstrated whether the gappy model can explain the observed spectropolarimetric signature. With respect to the energy transport this class of models differs fundamentally from flux tube models. While in the former only the footpoints provide hot plasma to the photosphere, in the latter upflows along the full length of the filaments are responsible for the energy supply.

Radiative MHD simulations reveal a full spectrum of magneto-convection regimes depending on the strength and inclination angle of the background field. Some of the convection patterns found in simulations have some resemblance of features found in simpler models. However, none of the simpler models can give a good characterization of magneto-convection in the penumbra as a whole. While umbral dots are essentially field-free upflow plumes (with much similarity to the gappy models), filaments in the inner penumbra remain magnetized with field strength around 1 kG. The outer penumbra with strong horizontal outflows is best characterized by anisotropic magneto-convection with no clear separation between convection and magnetic field. The manifestation of field inclination, field strength and horizontal flows close to photospheric layers show much similarity with flux tube models. However, the latter do not include overturning convective motions throughout the penumbra (except for the footpoints), which is the key element in radiative MHD simulations (as well as the gappy model).

Yet, at this point, it has not been fully investigated if radiative MHD simulations are consistent with all observational facts. In particular, it remains to be seen whether the flow pattern of convective rolls can be measured, and whether the observed penumbral line asymmetries in the Stokes parameters including the NCP can be reproduced by such models. Spectropolarimetric measurements need to have a spatial resolution of better than 0.1 arcsec to be comparable to the models. On the other hand, models need to be able to explain the substantial brightness of the penumbra, which has not been achieved by models not relying on overturning motions throughout the penumbra. Making an unambiguous connection between model results and observations requires the forward modeling of spectral lines and detailed comparison with observations, work in that direction is currently in progress.

Detailed modeling of formation, evolution, and decay of sunspots remains at this point an open problem. However, MHD simulations are now in the process of approaching the relevant scales and it can be expected that within the next decade substantial process will be made in this regard. Photospheric MHD simulations will reach domain sizes that can host entire active regions and domain depths, which allow coupling to flux emergence simulations in the lower convection zone.

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