## 2. Flux tube model and equilibriumAs a model for a coronal loop, we consider a deformed cylindrical
flux tube of length
The cross-sections of flux tubes are in general not constant along
the loop axis. For example, at the very base of coronal loops, i.e.,
in the convection layer, the magnetic flux is concentrated in several
narrow tubes. As the magnetic field lines penetrate the photosphere
and chromosphere they fan out due to a decrease of the confining
pressure. In our model, in which we focus on single flux tubes, the
variation of the cross-section is incorporated in the description of
the In the corona, the pressure scale height is of the order of 50 Mm
(Priest 1982). Because the height of most flux tubes is
much smaller, the influence of gravity can be neglected in the coronal
part. However, in the chromospheric and photospheric parts of the flux
tube, the scale height can be as small as a few hundred kilometers and
the influence of gravity cannot be neglected. In our model, its
effects are included by considering regions of length
near the bases of the flux tube in which the
gravity is non-zero. In these regions, the gravitational field is
aligned along the The magnetostatic equilibrium equations read: where the magnetic field, the current density, the pressure, and
the plasma density are denoted by ,
, where is the magnetic flux through a circle
with radius where is the Grad-Shafranov differential operator: From force balance it then follows that the basic equations may be written as (see Low 1975, and Beliïn et al. 1996b): where , , and are free equilibrium profiles. Although the shape of the flux tube is determined by pressure balance between the internal and external plasma, we here specify it a priori in order to avoid the complicated solution of a free boundary problem. Furthermore, boundary conditions at the ends of the flux tube ( and ) and at the outermost flux surface at should be imposed. We consider a class of equilibria for which the dependence of
on and This choice of equilibria still contains an infinite amount of
freedom through the profiles ,
, , and
. We distinguish the freedom in amplitudes from
the freedom in profile form. Two profiles of order unity, viz.,
where The other free equilibrium profiles, viz., , and are chosen as follows: where where In the next section, the continuum equations are derived for a flux coordinate system. Therefore, all equilibrium quantities should be expressed in terms of the flux coordinates , where is a longitudinal coordinate in the flux surfaces. Because Eq. (6) is initially solved on a structured grid, this implies that a grid inversion needs to be carried out: Once this inversion has been carried out all physical quantities can be expressed in terms of the flux geometry and the free equilibrium profiles. For example, the contra- and covariant components of the magnetic field are given by: where the Jacobian and is the poloidal magnetic field. For the solution of Eq. (6) we have used the numerical code PARIS described in Beliïn et al. (1996b)). This code exploits isoparametric bicubic Hermite elements. It provides a very accurate grid inversion and a very accurate solution for a moderate number of grid-points. © European Southern Observatory (ESO) 1997 Online publication: June 5, 1998 |