Vacuum Solution for Solov'ev's Equilibrium Configuration in Tokamaks*

In this work we have revised the Solov'ev analytical solution for tokamak equilibrium. We point out that this solution in the Solov'ev formulation is inapplicable in the vacuum, since it assumes a finite current density even outside the separatrix. The realistic vacuum should be current-free. To amend this vacuum solution problem, we use the Green function method to compute the vacuum fields caused by the plasma current, together with the fields corresponding to the homogeneous solution to the Grad-Shafranov equation. To construct the full solution, we match the vacuum field with the Solov'ev solution on the last closed flux surface. The total solution is then extended to the vacuum region to get the realistic vacuum solution. We find that the actual vacuum solution is vastly different from the Solov'ev solution, especially the X-point structure. The X-point solution is not of hyperbola type. Sometimes, even extra X points occur on the outboard midplane. The results are important for understanding the X point and separatrix structure.