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Astron. Astrophys. 327, 786-794 (1997)

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5. Discussion

We should point out that the aim of this study is not to present a realistic model for a sunspot, but rather to demonstrate that a sunspot-like configuration can develop through the dynamic relaxation of a potential field configuration under the action of pressure forces. Earlier work by Pizzo (1986) had indicated that one can iteratively construct a magnetostatic equilibrium solution, by relaxing a potential field configuration through a sequence of equilibria. There are, however, two assumptions which are inherent in this procedure which need to be checked, viz. whether the different equilibrium solutions are in fact connected and secondly whether flows can be neglected. These assumptions can be verified by solving the the time dependent equations which permit one to see the dynamical evolution of the potential field solution. Our simulation indeed shows, that a quasi-equilibrium state can be achieved, which is roughly similar to the configuration computed by Pizzo (1986) and one in which flows are small in the asymptotic time limit. An interesting feature of our simulation is the development of transient flows, which have a peak value of about 1.5 km s-1. These flow gradually diminish and after long enough time become vanishingly small. This asymptotic state can be regarded as the final equilibrium solution, in which all quantities are constant with time.

In the present calculation we have assumed a smooth radial distribution of the magnetic field. Observations of pores (e.g., Steshenko 1967) and the similarity in flux distribution of most umbrae (Gokhale & Zwaan 1972) support the current sheet models, which allow for a sharp transition between the sunspot and quiet photosphere. Although this feature is absent in our model, we doubt whether its presence would change the essential nature of our results. Furthermore, the inclusion of a current sheet would involve major modifications to the ZEUS-2D package, which we defer to a later calculation.

In our analysis we have considered the lower boundary above the superadiabatic temperature layer. The reason for this is not because we believe that the influence of this layer is unimportant, but rather we expect that if the magnetic field is sufficiently strong, which is the case in these calculations, then the convective instability associated with the superadiabatic layer would be suppressed by the strong magnetic field. Indeed earlier studies on magneto-convection based on the Boussinesq approximation by Galloway & Moore (1979), and Weiss (1981 a,b) (additional references can be found for example in the review by Proctor 1992) have shown that convection in the presence of a vertical magnetic field leads to a sweeping of the field to the cell boundaries and to a realignment of the field lines in such a way as to minimize their interference to the convective motions. Thus, the spot can be regarded as the region characterized by a strong vertical field and with significantly reduced convection. It is, however, well known that in certain cases oscillatory convection with motions essentially aligned with the vertical field can continue to exist, no matter now strong the field (Syrovatskii & Zhughzda 1967). For further details the reader can consult the papers by Proctor & Weiss (1982), Knobloch & Proctor (1981) and Hurlburt et al. (1989). Our aim in this work is to examine the dynamical motions that arise in the photospheric and chromospheric regions of sunspots where nonlinear effects may become important. We expect to incorporate, in subsequent papers, the oscillatory motions present in the convection zone through their buffeting action on the lower boundary in our problem.

A more important physical effect that has been neglected in the present work is the inclusion of a realistic energy equation which takes into account radiative and convective transport. It is well known (e.g., Spruit 1977) that radiation plays a significant role in the energy balance of flux tubes, particularly in the photospheric layers. Furthermore, the suppression of convection by the strong magnetic field and its effect on the thermodynamic structure of a spot also has not been treated by us, but which has been taken into account for instance by Nordlund & Stein (1989, 1990). However, we expect to consider this refinement in a subsequent paper of this series by incorporating an energy equation in the analysis.

This study marks the beginning of an investigation into various time dependent processes in thick flux tubes, such as sunspots, pores etc. It is well known from observations that such flux tubes are not static, but evolve and decay in time. Furthermore, they support a variety of wave motions, which are likely to be nonlinear in the upper layers of the solar atmosphere. In order to model these phenomena, an effective method is needed for solving the nonlinear time dependent MHD equations in multi-dimensions. We have developed such a method, based upon the ZEUS-2D algorithm, for modeling dynamical phenomena in sunspot-like configurations. The results indicate that our approach is robust and can be successfully applied to simulate the temporal evolution of a thick stratified flux tube. In subsequent papers, we shall enlarge the scope of the study to treat oscillations and their interaction with external wave modes, radiative transfer and their effect on the thermal structure, current sheets and ultimately the birth and decay of spots.

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998