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

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1. Introduction

Sunspots are the sites of strong magnetic fields. They are generally associated with flux emergence at the solar surface and are the best observed forms of structured fields in the form of thick flux tubes. Flux tubes are widely regarded as the fundamental building block of the photospheric magnetic field. Despite several observational and theoretical advances in sunspot physics, the basic processes that lead to the coolness of spots, their formation and equilibrium structure are not fully understood. The development of sunspot models has been hampered by the complexity of the MHD equations, which has imposed restrictions on the analytic and numerical solutions that can be obtained.

Theoretical models developed so far treat the equilibrium of a sunspot as a 2-dimensional structure extending vertically through the photosphere. Some of the basic characteristics of the flux tube behavior are explained by such models, even though they unrealistically treat the momentum balance equation in isolation, without taking into account the energy balance. Another assumption that has often been invoked is the thin flux tube approximation (e.g., Defouw 1976; Roberts & Webb 1978), which works well for small-scale flux tubes, but is inappropriate for thick flux tubes, such as pores and sunspots. This is because the horizontal dimensions of these tubes are typically many times the atmospheric scale height. Solutions for thick flux have been obtained under various approximations.

The earliest quantitative model for sunspots was developed by Schlüter & Temesváry (1958), based upon a similarity assumption, in which the stratification and field geometry are specified. This work was extended by Deinzer (1965), Yun (1970, 1971), Solov'ev (1982, 1983), Jakimiec (1965), Jakimiec & Zabza (1966), Landman & Finn (1979), and Low (1980). Return flux models which allow the magnetic field lines to re-enter the solar surface just outside the spot are a further development of the similarity assumption (Skumanich & Osherovich 1981; Osherovich 1982; Osherovich & Flá 1983; Osherovich & Lawrence 1983). In recent years, the solution of the magnetostatic equations has broadly followed two approaches: direct solution of the partial differential equations, and free surface problem.

In the first class of solutions, the earlier models assumed a continuous variation of the magnetic field across the spot. Here the separation between the internal and external regions is not sharply defined; rather the magnetic field is assumed to fall smoothly from a maximum at the axis to zero at some large radial distance. Based upon this assumption, magnetostatic equilibria were constructed by Pizzo (1986). The inclusion of a sharp interface between the sunspot and ambient medium in the form of a current sheet was treated amongst others by Simon & Weiss (1970), Meyer et al. (1977), Simon et al. (1983), and Pizzo (1990).

In the second approach, the direct solution of the equilibrium force balance equation is replaced by a free surface problem over which the total pressure, which is the sum of the gas and magnetic pressures, is continuous across the current sheet. Such models have been constructed for example by Schmidt & Wegemann (1983), and Jahn (1989).

All the above mentioned models are somewhat restrictive and describe a subset of the family of equilibrium solutions. Furthermore, they assume a static situation from the beginning, which may be a limitation. Also, it is well known that sunspots are not truly static structures but evolve with time. In order to model this evolution, we need to solve the time dependent MHD equations. There are numerous time dependent studies of thick flux tubes such as those carried out in 2-D by Deinzer et al. (1984a,b), Grossmann Doerth et al. (1989, 1994), Knölker et al. (1991), Steiner et al. (1994) and in 3-D by Nordlund & Stein (1989, 1990). These elaborate studies have contributed significantly to understanding the nature of flux tubes and also to model the interaction of convection with magnetic fields. For example, the work by Nordlund & Stein (1989, 1990) considers the interaction of granulation with a strong vertical magnetic field and shows that the latter suppresses the convective transport of energy leading to a cooling of the atmosphere which may simulate umbra formation in a sunspot. In the present investigation we adopt a somewhat different approach. Starting from a potential field configuration, which is clearly not in dynamical equilibrium, we attempt to examine whether the temporal evolution of this state can lead to an equilibrium solution which for instance is similar to the solutions computed using the magnetostatic equations. Cooling effects due to the reduced convective transport in the sunspot will not be considered in this study and are deferred to a future investigation. The main focus of this work is to use a viable numerical technique for modeling dynamic phenomena in sunspots. In the first part of this study, we apply this method to examine the time dependent relaxation of a potential field. In subsequent papers, we hope to extend our calculations to study dynamic behaviour in sunspots associated with wave motions, Evershed flows and non-adiabatic effects involving radiative energy transport.

We present a numerical simulation for the equilibrium configuration of a sunspot starting from an arbitrary initial state using dynamic relaxation. For the purposes of this investigation, we work within the framework of a distributed magnetic field configuration, deferring the treatment of a current sheet to a subsequent paper. The computational method is based on ZEUS-2D, a code for solving the MHD equations in 2-D (Stone & Norman 1992a, 1992b). We present in Sect. 2 the MHD equations, written in a form which lends itself naturally to conservation of the physical variables. In Sect. 3, we discuss the initial state, which corresponds to a potential magnetic field over the computational domain. Next, we point out the initialization and boundary conditions for the MHD variables. The results of our numerical simulation are presented in Sect. 4. Finally, we discuss the significance of the results and outline future work Sect. 5.

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998