On the reconstruction of the nonlinear force-free coronal magnetic field from boundary data

Using a simple model in which the corona is represented by the half-space domain Ω = {z > 0} and the photosphere by the boundary plane ∂Ω = {z = 0}, we discuss some important aspects of the general problem of the reconstruction of the magnetic field B in a small isolated coronal region from the values of the vector B¦∂Ω measured by a magnetograph over its whole basis. Assuming B to be force-free in Ω: (i) we derive a series of relations which must be necessarily satisfied by the boundary field B¦∂Ω, and then by the magnetograph data if the force-free assumption is actually correct; (ii) we show how to extract directly from the measured B¦∂Ω some useful informations about the energy of B in Ω and the topological structure of its field lines; (iii) we present a critical discussion of the two methods which have been proposed so far for computing effectively B in Ω from B¦∂Ω.