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Astron. Astrophys. 350, 1051-1059 (1999)

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2. The problem

The set of equations that describe the equilibrium of the coronal magnetic field in the half-space [FORMULA], when plasma pressure and gravitational forces are neglected, are the well known force-free equations (Parker 1979):

[EQUATION]

in which [FORMULA] as well as [FORMULA] 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 [FORMULA] add another non-trivial difficulty.

This mixed nature implies the requirement of two types of boundary conditions:

  • First of all, the elliptic part, resulting from the assumption that the RHS of Eq. (1) is given (the electric current), is rather well known, since it is nothing else than the Biot and Savart law, and just requires the value of [FORMULA] on [FORMULA] to compute [FORMULA] in the whole domain, as expected for any elliptic problem:

    [EQUATION]

    where [FORMULA] is a given regular function.
  • Then from Eqs. (1)-(2) one gets a hyperbolic equation for [FORMULA] (for [FORMULA] given):

    [EQUATION]

    and therefore one may give the value of [FORMULA] in the part [FORMULA] of [FORMULA] where [FORMULA], say:

    [EQUATION]

    where [FORMULA] is a given regular function. Note that this type of boundary condition is sufficient if one reasonably assumes that every field line of the coronal magnetic field has its two footpoints connected to the boundary [FORMULA]. Configurations having non-connected field lines (magnetic islands) would otherwise lead to the impossibility of transporting information from the boundary [FORMULA] (Aly 1988).

Because [FORMULA] is unbounded, one may also require the asymptotic boundary condition:

[EQUATION]

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© European Southern Observatory (ESO) 1999

Online publication: October 14, 1999
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