Exact Solutions for Spine Reconnective Magnetic Annihilation

Solutions for spine reconnective annihilation are presented which satisfy exactly the three-dimensional equations of steady-state resistive incompressible magnetohydrodynamics (MHD). The magnetic flux function ( A ) and stream function have the form $$A = A_{0}(R) \sin \phi + A_{1}(R)z, \qquad \Psi = \Psi _{0}(R) \sin \phi + \Psi _{1}(R)z,$$ in terms of cylindrical polar coordinates ( R , { , z ). First of all, two non-linear fourth-order equations for A 1 and 1 are solved by the method of matched asymptotic expansions when the magnetic Reynolds number is much larger than unity. The solution, for which a composite asymptotic expansion is given in closed form, possesses a weak boundary layer near the spine ( R = 0). These solutions are used to solve the remaining two equations for A 0 and 0 . Physically, the magnetic field is advected across the fan separatrix surface and diffuses across the spine curve. Different members of a family of solutions are determined by values of a free parameter n and the components ( B Re , B ze ) and ( v Re , v ze ) of the magnetic field and plasma velocity at a fixed external point ( R , { , z ) = (1, ~ /2,0), say.