Grad–Shafranov reconstruction of the magnetic configuration in the reconnection X-point vicinity in compressible plasma

The reconstruction problem for steady symmetrical two-dimensional magnetic reconnection is addressed in the frame of a two-fluid approximationwith neglected ion current. This approach yields Poisson’s equation for the magnetic potential of the in-plane magnetic field, wherethe right-hand side contains the out-of-plane electron current density with the reversed sign. In the simplest case of uniform electron temperatureand number density and neglecting the electron inertia, Poisson’s equation turns to the Grad–Shafranov one. With boundary conditionsfixed at any unclosed curve (the satellite trajectory), both equations result in an ill-posed problem. Since the magnetic configuration inthe reconnection region is highly stretched, one can make use of the boundary layer approximation; hence, the problem becomes well-posed.The described approach is generalized for the case of nonuniform electron temperature and number density. The benchmark reconstructionof the PIC simulations data has shown that the main contribution for inaccuracy arises from replacing Poisson’s equation by the equation ofGrad–Shafranov. Under this substitution, the reachable cross-size of the reconstructed region is shrinking down to fractions of the protoninertial length. Artificial smoothing, demanded by solving the ill-posed problem, and boundary layer approximation represent two alternativemethods of problem regularization. In terms of the reconstruction error, they perform nearly the same; the second method benefits from thecomparative simplicity and less restrictions imposed on the boundary shape.