## 4. Magnetic fields from localised sourcesThe 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, 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 regionWe model a unipolar magnetic region that is localised around by a boundary condition of the following form Here, and Substituting Eq. into Eq. , we first get the expression for , Next, Eqs. and yield the related solution for the flux function , which can also be expressed (Gradshteyn & Ryzhik 1980) in series form as or, using the so-called Polygamma functions, as Now, the magnetic field components , can be obtained from Eqs. and , Obviously, one can also use Eq. (9) or (10) instead of to derive alternative expressions for and . At this stage, we consider the limit and use Eq. to obtain 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
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 Fig. 1 shows typical examples of two sources of different extents
centred about the origin (). Given
that the magnetic structure is
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 where the are given by Eq. , with ,
and
the parameters describing the
strength, width and position of the ## 4.2. Uniform background magnetic fieldA large scale, uniform, horizontal background field () has a flux function clearly given by The total flux function of a group of magnetic elements in such a background field is then a superposition of expressions and , where, again, the global magnetic field lines are given as contours . © European Southern Observatory (ESO) 1999 Online publication: November 3, 1999 |