2. The problem
The set of equations that describe the equilibrium of the coronal magnetic field in the half-space , when plasma pressure and gravitational forces are neglected, are the well known force-free equations (Parker 1979):
in which as well as are unknowns.
The analysis of set of characteristics curves of this system, which is in general nonlinear (Grad and Rubin 1958, Parker 1995), shows that this system has a mixed elliptic-hyperbolic type structure. This complex structure of the problem (already known in fluid mechanics as the Beltrami field equations) makes this problem a formidable task to solve, and still makes it an open field of research in applied mathematics (Laurence & Avellaneda 1993), even in bounded domains. Moreover the astrophysical constraints, as seeking a solution in a domain that may be unbounded as add another non-trivial difficulty.
This mixed nature implies the requirement of two types of boundary conditions:
Because is unbounded, one may also require the asymptotic boundary condition:
© European Southern Observatory (ESO) 1999
Online publication: October 14, 1999