Exact magnetohydrodynamic equilibria with flow and effects on the Shafranov shift

Abstract

Exact solutions of the equation governing the equilibrium magnetohydrodynamic states of an axisymmetric plasma with incompressible flows of arbitrary direction [Phys. Pasmas 5 (1998) 2378] are constructed for toroidal current density profiles peaked on the magnetic axis in connection with the ansatz S=− ku, where S= d/ du[ ϱ( dΦ/ du) 2] ( k is a parameter, u labels the magnetic surfaces; ϱ( u) and Φ( u) are the density and the electrostatic potential, respectively). They pertain to either unbounded plasmas of astrophysical concern or bounded plasmas of arbitrary aspect ratio. For k=0, a case which includes flows parallel to the magnetic field, the solutions are expressed in terms of Kummer functions while for k≠0 in terms of Airy functions. On the basis of a tokamak solution with k≠0 describing a plasma surrounded by a perfectly conducted boundary of rectangular cross-section it turns out that the Shafranov shift is a decreasing function which can vanish for a positive value of k. This value is larger the smaller the aspect ratio of the configuration.