Axisymmetric magnetohydrodynamic equations: Exact solutions for stationary incompressible flows

The paper describes a new set of exact analytical solutions of the axisymmetric magnetohydrodynamic (MHD) equations for stationary and incompressible flows. The magnetic and flow surfaces are assumed to form a regularly nested set to stress the physical character of the solutions. As a result, these show a self-similar radial behavior. A suitable choice of the reference system reduces the problem to a single differential equation equivalent to the generalized Grad–Shafranov equation. The proposed method leads to the explicit expressions of the magnetic and flow surfaces. The analytical form of all the physical quantities for each possible shape of the surfaces can then be obtained in terms of two arbitrary functions of the self-similar radial variable, one being the density. By comparison, standard treatments of the generalized Grad–Shafranov equation require the specification of up to five arbitrary functions.