Exact solutions for reconnective magnetic annihilation

A family of exact solutions of the steady resistive nonlinear magnetohydrod ynamic equations in two dimensions (x, y) is presented for reconnective ann ihilation, in which the magnetic field is advected across one pair of separ atrices and diffuses across the other pair. They represent a two-fold gener alization of the previous Craig-Henton solution, since a dimensionless free parameter (gamma) in the new solutions equals unity in the previous soluti ons and the components (v(xe),v(ye)) and (B-xe, B-ye) Of plasma velocity an d magnetic field at a fixed external point (x, y) = (1, 0), say, may all be imposed, whereas only three of these four components are free in the previ ous solutions. The solutions have the exact forms A = A(0)(x) + A(1)(x)y, phi = phi(0)(x) + phi(1)(x)y for the magnetic flux function (A) and stream function (phi), so that the e lectric current is no longer purely a function of z as it was previously. T he origin (0,0) represents both a stagnation point and a magnetic null poin t, where the plasma velocity (v = del x phi (z) over cap) and magnetic fiel d (B = del x A (z) over cap) both vanish. A current sheet extends along the y-axis. The nonlinear fourth-order equations for A(1) and phi(1) are solved in the limit of small dimensionless resistivity (large magnetic Reynolds number) u sing the method of matched asymptotic expansions. Although the solution has a weak boundary layer near z = 0, we show that a composite asymptotic repr esentation on 0 less than or equal to x less than or equal to 1 is given by the leading-order outer solution, which has a simple closed-form structure . This enables the equations for A(0) and phi(0) to be solved explicitly, f rom which their representation for small resistivity is obtained. The effec t of the five parameters (v(xe), v(ye), B-xe, B-ye, gamma) On the solutions is determined, including their influence on the width of the diffusion reg ion and the inclinations of the streamlines and magnetic field lines at the origin. Several possibilities for generalizing these solutions for asymmetric recon nective annihilation in two and three dimensions are also presented.