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 Psi have the form

A = A(0)(R)sinphi + A(1)(R)z, psi = psi(0)(R)sinphi + psi(1)(R)z,

in terms of cylindrical polar coordinates (R,phi,z). First of all, two non-linear fourth-order equations for A(1) and Psi(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 Psi(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 gamma 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,phi,z ) = (1,pi/2,0), say.