Solving the nonlinear force-free equations in full 3-D is extremely difficult. Configurations with one or two invariant coordinate(s) are more suitable for an analytic or semi-analytic treatment. Solutions in the form of an infinitely long cylinder with axial symmetry are the simplest cases, and two best known examples are Lundquist’s (1950) solution in terms of Bessel functions ( = constant), and a solution used by Gold and Hoyle (1960) in their flare model ( constant, all field lines have the same pitch in the direction of the axis). Low (1973) considered a 1D Cartesian (slab) geometry and analyzed slow time evolution of the force-free field with resistive diffusion.

In Cartesian 2D geometry with one ignorable coordinate in the horizontal (depth) direction, one ends up with a second-order partial differential equation, called the Grad–Shafranov equation in plasma physics. The force-free Grad–Shafranov equation is a special case of the Grad–Shafranov equation for magneto-static equilibria (see Grad and Rubin, 1958), which allow to compute plasma equilibria with one ignorable coordinate, e.g., a translational, rotational or helical symmetry. For an overview on how the Grad–Shafranov equation can be derived for arbitrary curvilinear coordinates with axisymmetry we refer to (Marsh, 1996, Section 3.2.). In the Cartesian case one finds (see, e.g., Sturrock, 1994, Section 13.4)

where the magnetic flux function depends only on two spatial coordinates and any choice of generates a solution of a magneto-static equilibrium with symmetry. For static equilibria with a vanishing plasma pressure gradient the method naturally provides us force-free configurations. A popular choice for the generating function is an exponential ansatz, see, e.g., Low (1977), Birn et al. (1978), and Priest and Milne (1980). The existence of solutions (sometimes multiple, sometimes none) and bifurcation of a solution sequence have been extensively investigated (e.g., Birn and Schindler, 1981). We will consider the Grad–Shafranov equation in spherical polar coordinates in the following.

Living Rev. Solar Phys. 9, (2012), 5
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