2. Flux tube model and equilibrium
As a model for a coronal loop, we consider a deformed cylindrical flux tube of length L and radius R, which is threaded by a magnetic field and embedded in a field-free medium. The curvature of the magnetic axis of the tube is neglected. A cylindrical coordinate system is adopted, with the z -axis along the axis of the tube. The flux tube is axisymmetric and, hence, none of the static equilibrium quantities depends on the azimuthal coordinate . The ends of the tube represent the chromosphere/photosphere while the mid-plane of the tube represents the apex of the loop. This model is described in Beliïn et al. (1996b) and the geometry is illustrated in Fig. 1.
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 z -dependence of the tube radius . We consider expanded tubes only, i.e., tubes having larger cross-sections at the top than at the bottom.
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 z -axis and it is directed towards the ends of the tube. In the coronal part of the flux tube gravity is neglected.
The magnetostatic equilibrium equations read:
where the magnetic field, the current density, the pressure, and the plasma density are denoted by , , P, and , respectively. The magnetic field in axisymmetric cylindrical geometries may be written as:
where is the Grad-Shafranov differential operator:
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 z is separable, i.e., . Furthermore, we consider equilibrium classes for which so that is separable also:
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., F and , are introduced such that
where H is the pressure scale height determined by the strength of gravity, is the normalization of the density. The profile can be used to model an additional stratification of the density in the region where :
where µ is the ratio between the density at and , and determines how fast the transition from to takes place.
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