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Astron. Astrophys. 332, 1075-1081 (1998)

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1. Introduction

Sunspot penumbrae show structures in the form of alternating dark and bright fibrils extending in the radial direction across the penumbra. Observations by Beckers & Schröter (1969) have already suggested that the magnetic field inside the dark filaments might be close to horizontal whereas the field in the bright filaments is less inclined with respect to the vertical. This magnetic field structure has implications for the Evershed flow which was also suggested to be associated mainly with the dark filaments. The association of the Evershed flow with the dark penumbral fibrils has recently been confirmed by Shine et al. (1994) and Rimmele (1994, 1995b). In these observations, it was also found that the Evershed effect is a quasi-periodic time-dependent phenomenon which extends beyond the penumbral boundary (Rimmele 1995b). Furthermore, there is evidence (Rimmele 1995a) that the Evershed flow occurs in thin channels which are elevated above the continuum height for most parts of the penumbra. Very recently, Westendorp-Plaza et al. (1997) have actually for the first time identified the regions of return flux for the field lines carrying the Evershed flow. Their observations support the view that the Evershed flow is carried by low-lying field lines.

The suggested spatial variation of the inclination angle of the magnetic field with azimuth was also found by Lites et al. (1990) and Degenhardt & Wiehr (1991). Recently, very high resolution observations with the Lockheed tunable filtergraph on the Swedish Solar Observatory in La Palma (Title et al. 1993) showed that the magnetic field of a sunspot penumbra indeed varies between nearly horizontal in the dark penumbral filaments and much less inclined in the bright filaments. Similar results where obtained by Rimmele (1995a) and StanchfieldII et al. (1997). The observations of Title et al. (1993) show that the mean inclination of the magnetic field increases from [FORMULA] - [FORMULA] to [FORMULA] - [FORMULA] across the penumbra and that there is a rapid azimuthal variation of the inclination angle of about [FORMULA]. Little or no variation of the total field strength with azimuth is found by Title et al. (1993) and Rimmele (1995a) (see, however, the discussion in StanchfieldII et al. (1997)). On the basis of these observations, Title et al. (1993) proposed a tentative model for the magnetic field structure, which resembles that of laminated force-free fields (Low 1988a, 1988b). Since these fields only exist for Cartesian and spherical geometry, the model of Title et al. (1993) could so far not be backed up by a self-consistent calculation.

In a recent paper, Martens et al. (1996) presented a linear force-free model for a fluted sunspot. The model is able to explain some of the properties of fluted sunspots. Due to the high azimuthal wave number of the fluted force-free component, the scale height of this component is very small and the flutedness is confined to the chromosphere in this model. Furthermore, the magnetic loops corresponding to the dark filaments are very short in this model and the filaments do consist of a series of loops. This implies that the Evershed flow would have to consist of a phase-coordinated flow along many short loops. There is no explanation how this could be achieved. Therefore there is a clear need for improved models.

The fundamental problem posed by the observations is that a genuinely three-dimensional solution of the magnetohydrostatic equations is necessary. Though there has been some progress concerning analytical solutions in three dimensions recently (e.g. Neukirch 1995, 1997), these solutions do either not apply to the problem or they suffer from the same deficiencies as the linear force-free solutions. A different approach is therefore necessary.

In the present paper we present a method which allows us to find a self-consistent version of the model originally put forward by Title et al. (1993). The method is based on an asymptotic expansion of the three-dimensional force-free equations. The lowest order of this expansion has solutions which are formally equivalent to laminated force-free fields and a large number of different solutions can be found. These solutions can be used to model the magnetic field of the penumbra of a fluted sunspot.

In the following, we first briefly review the theory of laminated force-free fields and then present the expansion procedure. A specific solution which is useful as a model for a fluted penumbral magnetic field is presented and its properties are discussed especially in connection with a theoretical explanation of the Evershed flow.

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© European Southern Observatory (ESO) 1998

Online publication: March 30, 1998