4. Magnetic fields from localised sources
The photospheric distribution of the flux function 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
Here, and L are free constants defining the strength and the linear extent of the magnetic region, respectively. The sign of determines whether the region is a local source () or sink () of coronal magnetic field lines.
Substituting Eq. into Eq. , we first get the expression for ,
which can also be expressed (Gradshteyn & Ryzhik 1980) in series form as
Obviously, one can also use Eq. (9) or (10) instead of to derive alternative expressions for and .
This expression shows that magnetic field lines are straight lines described by . In addition, in the limit Eqs. reduce to
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. 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. with the source width. From Eqs. and , the vertical magnetic field component has the following photospheric distribution,
This expression shows that decreases symmetrically about the source centre, , from the maximum value as increases. We now define the width of the photospheric magnetic source as the distance between the points about at which 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 , 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 (). Given that the magnetic structure is y-invariant, it is represented by a cut through a plane in this figure and the subsequent ones of the same kind. Fig. 1a, with Mm, resembles the magnetic field produced by an infinitely thin line source (Eq. ), while the structure shown in Fig. 1b is generated by a broad source of half-width Mm. It is apparent the difference between the two magnetic field configurations not too far from the source.
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 is then a superposition of flux functions of each of the N individual unipolar magnetic regions,
where the are given by Eq. ,
with , and the parameters describing the strength, width and position of the n-th magnetic element.
4.2. Uniform background magnetic field
where, again, the global magnetic field lines are given as contours .
© European Southern Observatory (ESO) 1999
Online publication: November 3, 1999