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Astron. Astrophys. 351, 733-740 (1999)

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4. Magnetic fields from localised sources

The photospheric distribution of the flux function [FORMULA] has so far been arbitrary. In what follows, we shall introduce the basic element of this paper, a finite width, y-invariant source of magnetic field, which can be used to generalise the infinitely thin line current source used by other authors (e.g. Priest et al. 1994; Mackay & Priest 1996).

Suitable combinations of unipolar sources can then be used to model more complex photospheric structures and the related coronal magnetic field configurations.

4.1. Unipolar magnetic region

We model a unipolar magnetic region that is localised around [FORMULA] by a boundary condition [FORMULA] of the following form

[EQUATION]

Here, [FORMULA] and L are free constants defining the strength and the linear extent of the magnetic region, respectively. The sign of [FORMULA] determines whether the region is a local source ([FORMULA]) or sink ([FORMULA]) of coronal magnetic field lines.

Substituting Eq. [FORMULA] into Eq. [FORMULA], we first get the expression for [FORMULA],

[EQUATION]

Next, Eqs. [FORMULA] and [FORMULA] yield the related solution for the flux function [FORMULA],

[EQUATION]

which can also be expressed (Gradshteyn & Ryzhik 1980) in series form as

[EQUATION]

or, using the so-called Polygamma [FORMULA] functions, as

[EQUATION]

Now, the magnetic field components [FORMULA], [FORMULA] can be obtained from Eqs. [FORMULA] and [FORMULA],

[EQUATION]

Obviously, one can also use Eq. (9) or (10) instead of [FORMULA] to derive alternative expressions for [FORMULA] and [FORMULA].

At this stage, we consider the limit [FORMULA] and use Eq. [FORMULA] to obtain

[EQUATION]

This expression shows that magnetic field lines are straight lines described by [FORMULA]. In addition, in the limit [FORMULA] Eqs. [FORMULA] reduce to

[EQUATION]

[EQUATION]

These are the magnetic field components of an infinitely thin and long line source (compare with Eq. [2.1.2] in Mackay & Priest 1996). Therefore, despite using Eq. [FORMULA] to (arbitrarily) define the internal structure of a photospheric source, when its size is reduced to zero the coronal magnetic field becomes that of a line source, as expected.

In addition, it can be of interest to link the scaling parameter L in Eq. [FORMULA] with the source width. From Eqs. [FORMULA] and [FORMULA], the vertical magnetic field component has the following photospheric distribution,

[EQUATION]

This expression shows that [FORMULA] decreases symmetrically about the source centre, [FORMULA], from the maximum value [FORMULA] as [FORMULA] increases. We now define the width of the photospheric magnetic source as the distance between the points about [FORMULA] at which [FORMULA] has decreased by a factor e with respect to the central value. It turns out that, with this definition, the width of the photospheric source is about [FORMULA], which gives a clear physical interpretation of the parameter L.

Fig. 1 shows typical examples of two sources of different extents centred about the origin ([FORMULA]). Given that the magnetic structure is y-invariant, it is represented by a cut through a [FORMULA] plane in this figure and the subsequent ones of the same kind. Fig. 1a, with [FORMULA] Mm, resembles the magnetic field produced by an infinitely thin line source (Eq. [12]), while the structure shown in Fig. 1b is generated by a broad source of half-width [FORMULA] Mm. It is apparent the difference between the two magnetic field configurations not too far from the source.

[FIGURE] Fig. 1a and b. Examples of the coronal magnetic field pattern from a single source with half-width a  L=0.1 Mm, and b  L=10 Mm.

Having obtained the solution for an individual, unipolar magnetic element, we can immediately generalise the previous results to the case of an arbitrary number of such localised sources or sinks placed on the photosphere. The related flux function [FORMULA] is then a superposition of flux functions [FORMULA] of each of the N individual unipolar magnetic regions,

[EQUATION]

where the [FORMULA] are given by Eq. [FORMULA],

[EQUATION]

with [FORMULA], [FORMULA] and [FORMULA] the parameters describing the strength, width and position of the n-th magnetic element.

4.2. Uniform background magnetic field

A large scale, uniform, horizontal background field ([FORMULA]) has a flux function [FORMULA] clearly given by

[EQUATION]

The total flux function of a group of magnetic elements in such a background field is then a superposition of expressions [FORMULA] and [FORMULA],

[EQUATION]

where, again, the global magnetic field lines are given as contours [FORMULA].

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© European Southern Observatory (ESO) 1999

Online publication: November 3, 1999
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