## 4. ConclusionsIn this paper an MHD model for solar coronal plumes has been presented. Coronal plumes have been treated as stationary, axisymmetric structures and spherical coordinates have been employed. Since both observational evidence and theoretical investigations seem to agree about the intrinsic magnetic nature of coronal plumes, a linearisation with respect to the magnetic field has been used by assuming a low-beta coronal plasma. This method allows one to decouple the momentum equation components along and across the field lines and to tackle the problem in three distinct steps: - The zeroth order potential field is calculated assuming a background radial field and superimposing a non-radial contribution due to a given flux distribution at the plume base.
- A Bernoulli-type equation is solved for the density along the zeroth order magnetic field lines in the isothermal case. The transonic solution is imposed for the flow along each field line.
- The modification to the magnetic field, due to the unbalanced forces, is worked out by numerically solving a second order, Poisson-like PDE for the magnetic flux function (transfield or generalised Grad-Shafranov equation).
The method allows for the presence of three free functions, namely the radial field component at the plume base, the density at the plume base and the (constant) temperature along each field line. In the first part of the work, the plume structure has been considered to be purely radial in order to investigate easily the behaviour of the various physical quantities. The results are obviously what is expected for an isothermal, radial solar wind but with different conditions along each field line. For example, a plume which is hotter than the surroundings shows an increasing ratio of axis to background densities and higher flow speeds (the sonic point occurs closer to the Sun). An original contribution to our radial model is the calculation of the field line displacement due to the unbalanced pressure gradients. This is shown to follow closely the plasma beta behaviour, that is the angular displacement decreases until 2-3 and then it increases at larger distances, the only difference being due to the line-tying constraints at and . Obviously, our model retains its validity only until the plasma beta becomes comparable with unity, that is between 10 and 100 for typical coronal values, well beyond observational limits. In the second part the assumption of a purely radial background field has been relaxed by adding to it the contribution due to a flux concentration at the plume base. The resulting potential field shows similarities with that believed to lead to plume formation (closed bipolar loops interacting with a stronger open flux region). However, the main result of our non-radial analysis is the modelling of the observed super-radial expansion near the plume base, through a direct comparison with observational data. The good agreement between the theoretical model and the observations confirms that the plume structure is mainly determined by magnetic effects, whereas pressure and inertial forces only provide higher order perturbations. Another new feature is a slight enhancement in the flow speed (by a few kilometers per second) at the plume's axis and close to the coronal base, due to the concentration of the field lines; however this does not seem to affect the flow at larger distances (the position of the sonic points remains the same as in the radial case). Future efforts to improve this model will follow three directions: a better modelling of the coronal potential field, allowing for a non-radial plume axis (plumes far from solar poles appear to be bent towards the equator), a more realistic treatment of the plasma energetics, including heat deposition close to the plume base, and possibly the relaxation of the low-beta assumption, thus allowing one to model the behaviour of plumes at large distances from the Sun. © European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |