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

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2. Basic theory

2.1. Non-existence of laminated force-free fields in cylindrical coordinates

We assume that the magnetic field of the sunspot penumbra is force-free and therefore obeys the equations

[EQUATION]

From Eq. (1) we conclude that

[EQUATION]

where we have absorbed [FORMULA] into [FORMULA].

The observations (Degenhardt & Wiehr 1991; Title et al. 1993; Rimmele 1995a) show that simple sunspots may have a fluted penumbra with rapid variations of the inclination of the magnetic field with azimuth ([FORMULA]), however with almost constant magnetic pressure at constant radius. Since the azimuthal component of the magnetic field is small compared to the other components and the sunspot field can be considered as being force free, also the electric currents seem to flow mostly in the meridional planes ([FORMULA] constant). The appropriate model to describe such a magnetic field would be given by laminated non-linear force-free fields (Low 1988a, 1988b). However, these fields do only exist in Cartesian and spherical coordinates; in cylindrical coordinates appropriate for the description of sunspots such fields do not exist. This can be most easily seen as follows. The basic assumption is

[EQUATION]

Then Eq. (5) is automatically fulfilled. If we take the curl of Eq. (2) and multiply the result by [FORMULA] we get

[EQUATION]

On the other hand

[EQUATION]

It follows that [FORMULA] may not depend on [FORMULA] which precludes flutedness for force-free fields in cylindrical coordinates. This means that only rotationally symmetric equilibria of this type are possible and therefore it is impossible to find an exact force-free solution appropriate for fluted sunspots.

2.2. Asymptotic expansion method

However, if we are mainly interested in the penumbral structure, we could try to find an approximate solution with the desired properties, based on laminated equilibria. We start by writing Eqs. (4) and (5) together with Eq. (3) in cylindrical coordinates:

[EQUATION]

We now introduce a new radial coordinate

[EQUATION]

where a is a yet unspecified radius located somewhere inside the penumbra (actually a could be treated as a free parameter which can be used later to minimise the residual force resulting from the approximation). If the radial extent of the penumbra is [FORMULA], we now expand the equations above in the parameter

[EQUATION]

which we assume to be smaller than one (this is a kind of large-aspect-ratio expansion with [FORMULA] being the aspect ratio). This type of expansion procedure together with the coordinate transformation Eq. (15) has been introduced by Kiessling (1995) in the framework of toroidally confined plasma equilibria.

With [FORMULA] we get

[EQUATION]

We now introduce the following expansion scheme in [FORMULA]

[EQUATION]

Note that [FORMULA] is of order [FORMULA] whereas all other quantities are of order [FORMULA]. To lowest order in [FORMULA] we get:

[EQUATION]

Eqs. (26) - (30) are completely equivalent to the equations for laminated force-free fields in Cartesian geometry (Low 1988a) with the replacements [FORMULA] and [FORMULA].

Multiplying Eq. (26) by [FORMULA] and Eq. (28) by [FORMULA] and adding the two equations, we get:

[EQUATION]

Eq. (29) can be solved by

[EQUATION]

Eq. (30) is solved by introducing a flux function [FORMULA]:

[EQUATION]

Inserting Eq. (33) into Eq. (27) we obtain

[EQUATION]

Eqs. (34) and (31) allow solutions of the form (Low 1988a)

[EQUATION]

where [FORMULA]. So in this approximation we have a field without a [FORMULA] component that is potential in the [FORMULA] -z -plane with an arbitrary modulation in the [FORMULA] -direction.

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© European Southern Observatory (ESO) 1998

Online publication: March 30, 1998
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