In this paper, we have presented a numerical method for reconstructing the coronal magnetic field as a force-free magnetic field from its value given on the boundary of the domain. Let us summarize here the main points we have discussed and our main results:
(a) The boundary-value problem is formulated such that it corresponds to a well-posed mathematical problem: the normal component of the magnetic field is imposed on the boundary of the domain, and only on that part of this boundary where . We impose on the lateral boundaries so that does not need to be specified on these boundaries, provided that these boundaries are put far enough to mimic the behaviour of the solution at infinity.
(b) We have derived a Grad Rubin Vector Potential formulation of this BVP to ensure up to machine roundoff numerical errors. We have shown that this problem may be equivalent to solve a sequence of linear elliptic boundary-value problems for the Vector Potential, and hyperbolic ones for . This current formulation is relevant for seeking regular enough solutions but not equilibria having current sheets. Weak formulations of these methods are however currently under study and would be reported in a next Paper of the series (Amari & Boulmezaoud 1999, in preparation). We have implemented this method in our computational code EXTRAPOL, in a relatively efficient numerical way. Other mathematical approaches allowing the existence of critical points in the configuration are also currently studied.
(c) We have successively applied our method to theoretical magnetograms obtained from two exact known solutions, the solutions of Low (1982) and Low & Lou (1991). The method converges up to a small residual Lorentz force, in a reasonable number of iterations. Some discrepancies between the exact solution and the reconstructed one occurred near the top or lateral boundaries of the computational box, and have been explained by the relatively bad asymptotic behaviour of Low's (1982) solution, or the existence of an almost vertical pathological field line in the solution of Low & Lou (1991), which makes difficult to match our applied boundary conditions on these boundaries (). Other approaches involving the assumption of potential field near the boundaries, or approximation of the Green formulae that can explicitly give the normal component at those boundaries are currently under development. Another approach could be to map the infinite upper half-plane onto the bounded square box by using a class of mappings that represent the generalization of conformal mappings used in two dimensions.
(d) Our formulation is better than the (VIM) (Wu 1985, Cuperman et al. 1990a-b, Demoulin et al. 1992) since it corresponds to a mathematically well-posed boundary value problem. Although it may exhibit some residual discrepancies with the exact known solution, errors never increase exponentially up to blowing up as in the VIM. Moreover as it was shown by Bineau (1972) another consequence is that the solution is expected to be continuous respect to boundary conditions, at least for not to large (Amari et al. 1998).
(e) Our method is different from Sakurai's (1981) approach in which, instead of solving an elliptic problem for , he uses a more local approach where the location of the nodes that discretize a given field line are computed once some electric currents () are injected in this field line, as the solution of non-linear system of equations that does not take into account the contribution of the whole computational domain (as one would expect in an elliptic problem). This approach allows a fast enough computation, which might be useful for some very concentrated (almost thin isolated) tube-like configurations, but it is not yet clear how this truncation procedure (by solving a single problem for each field line) may be involved in the numerical instabilities encountered in solving the nonlinear system for cases corresponding to large values of . Indeed Sakurai's (1981) approach might be considered as a Lagrangian discretization method while we have presented a Eulerian type discretization that would be more suited to highly sheared magnetic configurations. The two methods should be worth to be kept and used for different types of data and configurations. The results presented in this Paper seem to be optimistic as regards the application of the method to simulated magnetograms. The next step currently under development is the application of this method to various sets of data provided by vector magnetographs. However there are several important points that need to be emphasized, and that make actual data much more difficult to handle than exact force-free solutions:
i) First of all data are much more noisy, because of the errors on the transverse magnetic field measurements that are larger than on the longitudinal one (Amari & Demoulin 1992, Klimchuk & Canfield 1993, McClymont et al. 1997). Other errors may also arise after the resolution of the ambiguity that exists on the transverse component. These errors depend on the method that is used (Mikic & Amari 1999, in preparation). Eventually the non-force-free character of the photosphere (Aly 1989) may be taken into account. Actually from point (b) above, the well-posedness of our formulation (for at least not too large), would make the solution not very sensitive to errors expected on the photospheric measurements. We are currently working on the project of simulating the error effects (Amari et al. 1999, in preparation) of these instrumental errors on the transverse field components, by introducing some random noise in the simulated data obtained from some highly sheared force-free solutions obtained by a relaxation code (Yang et al. 1986, Klimchuk & Sturrock 1992), and then reconstructing them with our method.
ii) Related to these errors, one may also find that, unlike theoretical magnetograms, actual data are far from smooth. This implies that any reconstruction method should be either robust or one will have to smooth the data prior to reconstruction, which may introduce possible added deviations from the sought solution, since there is no unique way of smoothing.
iii) One non negligible difficulty that has to be taken into account is the needs for computing from the photospheric normal components of the magnetic field and of the electric current. Weak field regions cannot be ruled out in a straightforward way since high shear can be localized near the neutral line (Hagyard 1988).
iv) One final point is that unlike for theoretical magnetograms, one never knows a priori the solution in the corona in order to check the reconstructed one. However an alternative can be the use of YOHKOH or SOHO/EIT data (for different heights). These data would have to be used a posteriori to check if the computed structures has such loops or `'footpoints" that match the coronal observed ones, but not use these data set to fix a remaining free parameter such as in linear force-free constant- extrapolations!
© European Southern Observatory (ESO) 1999
Online publication: October 14, 1999