## 2 Linear Force-Free Fields

Linear force-free fields are characterized by

where the force-free parameter is constant. Taking the curl of Equation (15) and using the solenoidal
condition (16) we derive a vector Helmholtz equation:
which can be solved by a separation ansatz, a Green’s function method (Chiu and Hilton, 1977) or a Fourier
method (Alissandrakis, 1981). These methods can also be used to compute a potential field by choosing
.
For computing the solar magnetic field in the corona with the linear force-free model one needs
only measurements of the LOS photospheric magnetic field. The force-free parameter is a
priori unknown and we will discuss later how can be approximated from observations.
Seehafer (1978) derived solutions of the linear force-free equations (local Cartesian geometry
with in the photosphere and is the height from the Sun’s surface) in the form:

with and .
As the boundary condition, the method uses the distribution of on the photosphere .
The coefficients can be obtained by comparing Equation (20) for with the magnetogram
data. In practice, Seehafer’s (1978) method is used for calculating the linear force-free field (or potential
field for ) for a given magnetogram (e.g., MDI on SOHO) and a given value of as follows. The
observed magnetogram which covers a rectangular region extending from 0 to in and 0 to in
is artificially extended onto a rectangular region covering to and to by
taking an antisymmetric mirror image of the original magnetogram in the extended region, i.e.,

This makes the total magnetic flux in the whole extended region to be zero. (Alternatively one
may pad the extended region with zeros, although in this case the total magnetic flux may
be non-zero.) The coefficients are derived from this enlarged magnetogram with the
help of a Fast Fourier Transform. In order for to be real and positive so that solutions
(18) – (20) do not diverge at infinity, should not exceed the maximum value for given and
,
Usually, is normalized by the harmonic mean of and defined by

For we have . With this normalization the values of fall into the range
.