A 3-DIMENSIONAL MAGNETOHYDRODYNAMIC FORMALISM FOR CORONAL HELMET STREAMERS

A three-dimensional magnetohydrodynamic (MHD) formulation of the steady state coronal helmet-streamer problem is presented. It includes the simple azimuthally symmetric (partial derivative/partial derivative phi = 0) and two-dimensional (partial derivative/partial derivative phi = 0, B(phi) = 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, phi, theta), in which the problem is defined and boundary conditions established; (2) computationally convenient (nu, mu, xi), in which the entire space is represented by a rectangular box of sizes (1, 2, 1) (nu = r0/r, mu = cos theta, xi = phi/2pi), therefore allowing also the imposition of boundary conditions at infinity, nu = 0 (e.g., vanishing of the magnetic field components, etc.); (3) local Frenet's, (l, c, n), defined by the orthogonal unit vectors e(l) = B/B, e(c) = R(partial derivative e(l)/partial derivative l), and e(n) = e(l) x e(c) (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.