## 4. Discussion## 4.1. Comparison with the linear force-free modelThere are clear differences between this model and the linear
force-free model of Martens et al. (1996). In the present model the
field strength drops off with height more or less like
for large Another difference is that the parts of the penumbra where the field is nearly horizontal do not form rather short loops but quite extended radial loops which can more easily account for the observed Evershed flow. Whereas in the linear force-free model the direction and the amplitude of the current density are determined by the magnetic field because is a constant, in the present model is a function of the azimuth. The relation between and can be calculated either from Eq. (26) or Eq. (28). We get With the form of given in Eq. (55), we obtain Since the cosine has its extrema where the sine has its zeros and vice versa, we have the situation that the maximum current density flows along field lines having the average inclination, whereas the current density vanishes along field lines having the maximum or minimum inclination. Furthermore the direction of the current flow changes its sign every half wave length in azimuth. This coincides exactly with the discussion given in Title et al. (1993) and sketched in their Fig. 17. This vindicates our approach because it was our aim to come up with a self-consistent version of their schematic model. ## 4.2. Quality of the approximationAn important point to investigate is the quality of the approximation scheme that we have presented. A good way to do this is to investigate the magnitude of the residual force due to the approximation. To be able to judge the quality of the approximation, we need to compare the residual force to a quantity of the same dimension. A convenient measure for the strength of the force is . We then obtain Note that the residual force has only a poloidal component. One can immediately see that the residual force vanishes at the origin of the transformed coordinate system (). In Fig. 4 we show a surface plot of the residual force as
function of
We mention as a possibility that in principle, one could integrate
the amplitude of the force over the volume under consideration and
minimise this integral with respect to the parameter © European Southern Observatory (ESO) 1998 Online publication: March 30, 1998 |