Exact solutions of the equation governing the equilibrium magetohydrodynamic states of an axisymmetric plasma with incompressible flows of arbitrary direction [H. Tasso and G.N.Throumoulopoulos, Phys. Pasmas { \bf 5}, 2378 (1998)] are constructed for toroidal current density profiles peaked on the magnetic axis in connection with the ansatz $S=-ku$, where $S=d/d u [\varrho (d\Phi/du)^2]$ ($k$ is a parameter, $u$ labels the magnetic surfaces; $\varrho(u)$ and $\Phi(u)$ 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 $k\neq 0$ in terms of Airy functions. On the basis of a tokamak solution with $k\neq 0$ 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.
Categories: physics.plasm-ph