4.1. Comparison with the linear force-free model
There 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 z. The variation of the inclination angle with height is completely determined by the function f we choose, but in any case it will have a much weaker dependence on z than the linear force-free model. In the linear force-free model, the magnetic field drops off exponentially and the scale height of the fluted part is very small. This is actually one of the shortcomings of that model.
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
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 approximation
An 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 r and z for . The plots for the locations of minimum and maximum inclination do not differ very much from this plot. It can be seen that the residual force is small close to the origin of the transformed radial coordinate system and then rises almost linearly in r away from that origin. This can be understood by an inspection of Eq. (58). Most of the variation of the residual force expression obviously comes from the factor . The factor just seems to be a minor modulation of the first factor. The plot also shows the limits of the approximation scheme. At the boundaries (in r) the dimensionless residual force has reached a value of almost 0.6 which is at the very limit of what can be considered as acceptable for a small parameter. On the other hand, we have only considered the lowest order of the expansion scheme and higher order corrections could make the representation of the field even better. One should also keep in mind that by an expansion procedure like this one can usually not expect to get a convergent series but only an asymptotic series (though in a mathematically rigorous sense we would still need to prove that the series is indeed asymptotic).
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 a. This would give a kind of optimum value for the location of the origin of the transformed r -coordinate. Since we expect that the increase in accuracy achieved by such a procedure will be small we have not carried this out here.
© European Southern Observatory (ESO) 1998
Online publication: March 30, 1998