5. Application to some known exact force-free solutions
We now test the scheme presented in the previous section by running our code EXTRAPOL on some analytical and semi-numerical exact solutions of the non linear force-free equations Eq. (1)-(2). The boundary values of these exact solutions are used as simulated magnetograms. Hopefully, in these cases one knows the solution above in the domain too, and compare the reconstructed and the exact solutions (which is not the case for the actual corona!). There are only very few known exact solutions of the force-free equations. Let us presents the results obtained with our code on two cases that have been also used by other methods such as the VIM (Demoulin et al. 1992) for the first one and the Resistive Relaxation Method (Mikic & McClymont 1994) for the second one. Note that another class of related solutions that will not be tested here are those found by Cuperman & Ditkowski (1991).
5.1. The Low (1982) solution
Our first target is the well-known solution of Low (1982) for which the magnetic field is given by:
We choose for the function
We fix hereafter and
Our numerical box size corresponds to the choice , , . A non-uniform mesh with nodes was chosen as in Demoulin et al. (1992). The analytical solution is then computed on the mesh and in particular the values taken by and on the boundary provide boundary conditions for our force-free reconstruction procedure. We choose in Eq. (53) (i.e., the parameter necessary to fix the outer iteration corresponding to the injection of . Our method converges up to a Lorentz force of order for a number of inner Grad-Rubin iterations . The numerical error is defined as in Amari et al. (1998). We also found that choosing implies increasing Ngradrub up to about 12 to reach a Lorentz force of the same order. Fig. 1 shows some field lines of the exact solution (top) and the the corresponding field lines obtained from our computation.
Some discrepancies (up to few percents) between the exact and the computed solution are found in the domain, and these can reach almost for the field lines approaching the lateral boundaries of the box. These can be explained by our choice for the boundary condition () on these boundaries for the computed solution, while the exact solution does not decrease fast enough and even more pathologically in the horizontal plane (see Amari et al. 1998). Note that because applying this boundary condition results in a difference between the computed and exact solution, but still allows to reach a force-free equilibrium. However this equilibrium shows a different behaviour than the exact solution near the boundary, but there is no intrinsic instability as in the VIM (Wu 1990 and references therein, Cuperman et al. 1990a-b, Demoulin et al. 1992). It is worth noting that we have also performed some higher resolution run, with ,, which, unlike the VIM, gave even better results, allowing the boundaries to be pushed far away. Note that this `'robustness `' property (good behaviour while increasing spatial resolution) as well as the convergence of the method even for this type of lateral and top boundary conditions results from the well-posed formulation we have adopted, unlike for the VIM which is associated to an ill-posed mathematical problem (see Low & Lou 1991 and Amari et al. 1998). In this latter method errors increase exponentially with height (Demoulin et al. 1992) and this is a property intrinsic to the method (and not the numerical scheme used for the extrapolation), which implies that the computed solution will eventually diverge, while our solution never diverges for an arbitrarily large box. Actually the bigger is our box, the bigger the region of agreement between our solution and the exact one is, a property that we checked with the higher resolution run, pushing the lateral boundaries to , , .
5.2. Low & Lou's (1991) solution
The solution is supposed to be axi-symmetric and writes in spherical coordinates:
where is an a priori unknown function of , a solution of the nonlinear partial differential equation (see Low & Lou 1991). A family of solutions can be generated by choosing
for odd n, and a a real constant. P is then the solution of the following boundary-value problem:
We then solve numerically Eqs. (71)- (72). Usual transformations (Low & Lou 1991) then allow to get the solution in cartesian coordinates, in the upper half space.
Our numerical box is taken such that , , . A non-uniform mesh is generated with , , with most of the cells concentrated in the inner stronger field region. Once BVP (71)-(72) is solved, one deduces the corresponding three components (,,), the associated electric currents and on the same nodes of the mesh used by our force-free reconstruction code EXTRAPOL, and then computes the solution. One then use and (for the nodes in only) as boundary condition for the reconstruction procedure. We found that using and 4 inner iterations() allows to decrease the Lorentz force down to values of order .
Fig. 2 shows some field lines of the exact solution (top) and the corresponding field lines resulting from our reconstruction procedure. The errors, defined as for the previous case (Low's (1982) solution), are even less or of the order of in the larger part of the domain, except again near the lateral and top boundaries where the imposed boundary condition and the exact one disagree. Actually, those discrepancies are however smaller than those of the case of Low's (1982) model for the lateral boundaries because the magnetic field now decreases faster with distance. The case of the top boundary is different because of the existence, in the exact solution, of a pathological field-line in the center of the box that crosses almost vertically the top boundary while it has to match the applied boundary condition in the calculation, which will be difficult to fulfill, even with a large box. Note that despite the much better asymptotic behaviour of this force-free solution for the magnetic field the electric currents are distributed on a scale that is still large, which results in a configuration that does not quickly approach toward the potential field as it is often the case in the corona, outside regions of more localized electric currents.
© European Southern Observatory (ESO) 1999
Online publication: October 14, 1999