The magnetic field dominates most of the corona, and it is probably the origin of a large variety of structures and phenomena, such as flares, Coronal Mass Ejections, prominences and coronal heating (Priest 1982). Unfortunately the magnetic field is not yet observationally accessible in the tenuous and hot plasma that fills the corona (see Sakurai 1989, Amari & Demoulin 1992, and references therein). One possible familiar approach consists in solving the equations of a model (defined by some reasonable assumptions about the physical state of the corona) as a Boundary Value Problem (BVP), the boundary conditions being taken to be the measured values of the magnetic field in the denser and cooler photosphere: this is the so called Reconstruction problem of the coronal magnetic field. Many problems have been encountered since the early attempts of Schmidt (1964), as the observational problems to get rid of the ambiguity that remains in the transverse component of the photospheric magnetic field (Amari & Demoulin 1992, McClymont et al. 1997, and references therein), or the problems related to the choice of boundary conditions that make a well set BVP (Aly 1989)
In the simplest approximation the coronal magnetic field is current-free. This only requires the longitudinal component of the photospheric field as a boundary condition (Schmidt, 1964), and the solution can be computed using either a Green's function method or Laplace solver methods for the magnetic field or the vector potential. The mathematics of the various related BVPs (e.g., their well-(or ill-) posedness properties), are also known (Aly 1987, Amari et al. 1998).
In many active regions, where the magnetic configuration is known to have stored free energy, the current free assumption is not relevant. One can then introduced the so-called constant- force-free hypothesis, which allows for the presence of electric currents in the corona. The magnetic field is computed, for a given value of , from its longitudinal component by using either Fourier transform (Nakagawa et al. 1973, Alissandrakis 1981) or Green's function (Chiu & Hilton 1977, Semel 1988) techniques. Other spectral methods have been recently proposed (Boulmezaoud et al. 1998). It is also possible to solve it by regularizing an ill-posed BVP (in which the three components of the magnetic field are used) (Amari et al. 1998). However, the non-regularization or partial regularization of the so called Vertical Integration Method (VIM), leads in general to an amplification of errors (Wu et al. 1990 and referncess therein, Cuperman et al. 1990a-b, Cuperman et al. 1991, Demoulin et al. 1992). In addition the total energy of the linear force-free field in an unbounded domain such as the exterior of a star shaped domain is infinite, and is in general infinite in the case of the upper half space (except for some particular periodic solution satisfying some special conditions, Alissandrakis 1981, Aly 1992). Moreover the electric currents are uniformly distributed, while observations clearly show strong localized shear along the neutral line of many active region magnetic configurations (Hagyard 1988, Hofmann & Kalman 1991).
Modeling such strong localized electric currents needs to assume that the coronal magnetic configuration is in a non-linear, force-free state. In this case one can distinguish two types of methods associated to different classes of BVPs, Extrapolation Methods and Reconstruction Methods. In the first class of methods the three components of the magnetic field are used as boundary conditions. The equations are thus vertically integrated step by step, from the photosphere towards the corona, without incorporating any type of asymptotic boundary conditions. This give rise to the VIM (Wu et al. 1990 and references therein, Cuperman et al. 1990a-b, 1991, Demoulin et al. 1992). This method, associated to an ill-posed boundary value problem, has not yet been proved to be convincingly regularized, still ending with an exponential growing of the errors with height, prohibiting extrapolation up to reasonable heights. The second class of methods considers a BVP that only requires the normal component of the field on the boundary () and the normal component of the electric current say, where . Now the problem is considered in the whole domain and the solution is globally sought. It has been tackled by the use of iterative methods introduced by Grad and Rubin (Grad & Rubin 1958, Sakurai 1981, Sakurai et al. 1985) and by the Resistive MHD Relaxation Method (Mikic & McClymont 1994, Jiao et al. 1997). Roumeliotis (1997) presented a Relaxation Method in which the three components of the magnetic field are used at the photospheric level. Another method (see Amari & Demoulin 1992), is the Method of Weighted Residuals (Pridmore-Brown 1981). This method is based on the minimization of two residuals, one associated with the Laplace force that has to vanish for a force-free magnetic field, and the other one with the difference between the directions of the observed transverse photospheric magnetic field and of the computed one. However, some aspects, such as the choice of test functions to be used for scalar products, as well as some other points concerning the definite positiveness of one functional to be minimized, are not yet clear. Other computational schemes such as collocation or least square methods have also been proposed in Amari & Demoulin (1992), but they have not been tested so far.
Sakurai (1981, 1985) presented a Green's function approach of the Grad-Rubin formulation. Practically, the standard Green-Function formulation is however numerically expensive, since at each step of the iterative scheme one would need to compute an integral over the whole volume to get the value of the magnetic field at each point! An alternative approach proposed by Sakurai (1981, 1985) is to discretize the integral involving the Green's function by introducing "finite-element"-like dicretization for the electric currents. The process thus consists in starting from an initial current-free field line, putting current on it, and then retracing the correct perturbed field line carrying the electric current just put on. In this method, the field lines are discretized into a finite number of nodes (which define the degrees of freedom of the problem) and the nodes locations then become the unknowns of the problem for tracing the field lines. The latter are determined by solving a system of nonlinear algebraic equations, whose convergence is related in some sense to the absolute value of , and has not been proved to hold for large values of .
In this paper we consider another class of Grad-Rubin Methods that used the vector potential representation of the magnetic field. The paper is organized as follows. In Sect. 2 we present the general problem that is solved. In Sect. 3, we present the class of Grad-Rubin-like computational methods for solving the non-linear force-free case. We introduce in particular a new Vector Potential formulation in Sect. 4. We then present some results obtained with our method when applied to some particular known solutions in Sect. 5. Sect. 6 gathers concluding remarks.
It should be noted that a portion of the present study has been published in the proceeding of a conference (Amari et al. 1997).
© European Southern Observatory (ESO) 1999
Online publication: October 14, 1999