A three-dimensional magnetohydrodynamic (MHD) formulation of the steady state coronal helmet-streamer problem is presented. It includes the simple azimuthally symmetric (∂/∂φ = 0) and two-dimensional (∂/∂φ = 0, Bφ = 0) cases. The major mathematical difficulty - the correct, iterative calculation of the transverse electrical currents, which are the sources of the fields in Maxwell's equations - is eliminated. This is achieved by the elaboration of an algorithm connecting four different coordinate spaces: (1) spherical (r, φ, θ), in which the problem is defined and boundary conditions established; (2) computationally convenient (ν, μ, ξ), in which the entire space is represented by a rectangular box of sizes (1, 2, 1) (ν = ro/r, μ = cos θ, ξ = φ/2π), therefore allowing also the imposition of boundary conditions at infinity, ν = 0 (e.g., vanishing of the magnetic field components, etc.); (3) local Frenet's, (l, c, n), defined by the orthogonal unit vectors el = B/B, ec = R(∂el/∂l), and en = el x ec (tangent to the field line, pointing toward the center of curvature of the field line, and normal to the osculatory plane of the field, respectively; R is the curvature radius), required for the integration of the conductive, MHD equations along magnetic field lines; and (4) Cartesian (x, y, z), in which Frenet's unit vectors as well as the derivatives along their directions are defined. An analytical proof of the results for a particular two-dimensional model is presented.
- Sun: corona