Vacuum solution for Solov’ev’s equilibrium configuration in tokamaks

In this work, we construct a tokamak plasma equilibrium generated by a combination of currents flowing within the plasma and in distant external magnetic field coils. The plasma toroidal current density takes the simple form inside the plasma (where r and z are cylindrical coordinates, and a, b, and R are constants), and is zero in the surrounding vacuum. The Grad–Shafranov equation possesses a well-known exact analytic solution within the plasma due to Solov’ev. We use a Green’s function method to compute the poloidal magnetic flux generated by plasma currents, together with a parameterized homogeneous solution to the Grad–Shafranov equation (that is well-behaved at small r and z), to construct the vacuum solution. The vacuum solution is matched to the analytic solution on the last closed magnetic flux surface (LCFS) to determine the parameters in the homogeneous solution. This procedure is performed for both up–down-symmetric double-null and up–down-asymmetric single-null equilibria. We find that any magnetic X-points on the LCFS are distorted due to the fact that one quadrant is filled by a current-carrying plasma, whereas the other three are filled by a vacuum in which no current flows. In particular, the vacuum quadrant opposite the plasma-filled quadrant expands at the expense of the other three quadrants.