Astron. Astrophys. 329, 1125-1137 (1998)

## 2. Description of the model

### 2.1. Assumptions

Prominences are formed of plasma sufficiently ionised and dense that we can consider the plasma (and the neutral elements) frozen in the magnetic field. Furthermore, their long-time existence and the slow motions of the plasma inside them, allows us to consider their global structure as quasi-static, in a first approximation. The plasma is typically of the order of 0.01-0.1, so the field can almost be considered as force-free. We choose below a linear force-free field because we want to have a large class of 3-D equilibrium to understand the presence of feet. As emphasized in Sect. 1, the main criterion for a magnetic configuration that can support a prominence is the existence of dips in the field lines. In the following we will suppose that cold material is present in such dips while we neglect its effect on the magnetic field.

### 2.2. Three-dimensional linear force-free field solutions

A linear force free field satisfies:

with constant. This leads to the differential equation:

We use a cartesian system of coordinates, where z refers to the height, y to the prominence axis, and x to the direction perpendicular to the prominence. The solution of Eq. (3) can be expressed in periodic harmonics for the field in the x and y directions, as shown in previous works (Nakagawa & Raadu 1972 and Démoulin et al. 1989). and are the periodicity of the field in the x and the y directions. We assume a dependence of the field in the vertical direction as , so that the magnetic field decreases with height. We avoid the solutions that give a dependence like , because they give non-physical results. For the vertical component of the field, we only keep the solution that behaves like , so this gives an odd solution in the x direction. Hence, the inversion line for the photospheric field stands at , which is easy to deal with. The harmonics for the magnetic field are then given by:

with is the amplitude of the harmonic , and

From Eq. (9) we see that has a maximum value (, ). For , becomes an oscillating solution with height, which is not physical. So we restrict ourselves to . In the following, the values given for are normalized to . So the value of in Mm will be:

### 2.3. The criterion for dips in the field lines

In this section we derive the expression of the curvature of a field line as a function of the magnetic-field spatial derivatives. Rather than using the properties of a force-free field, we use the general definition for the curvature of a field line. Let () be the local Frenet vector base associated to a field line path ( is the local tangent, and the local curvature direction). We note the curvature radius of the field line, and s the arc length on field lines. Then we can write:

The field line equation is given by:

Whatever is the variable , we can write that

Then, combining Eq. (12) and (13), we easily get to:

Introducing the result of Eq. (14) in Eq. (11) we reach:

In general, it is not easy to deal with this equation, but in our case, we are only interested in the curvature of the field lines where . In this condition, we finally obtain:

© European Southern Observatory (ESO) 1998

Online publication: December 16, 1997