Asymptotic vacuum solution at tokamak X-point tip

The paper solves an outstanding problem of X point at tokamak edge that the sum of the angles of four quadrants bordering the X point is not 2 π, if a true vacuum surround plasma torus is considered. This is realized by chipping off a thin layer of plasma edge so that the sharp corner of the plasma-filled quadrant becomes a hyperbola. It is proved that with a hyperbolic plasma-vacuum boundary the asymptotic vacuum solution at the vertex of hyperbola can be obtained by the conformal transformation. The results show that a new X-shape configuration does appear. However, the new X point lies in the vacuum region bordering four vacuum quadrants, instead on the plasma-vacuum interface with one plasma-filled quadrant. In the new X-shape configuration, an acute angle constituted by the asymptotes of the hyperbolic boundary of plasma-filled quadrant leads to an obtuse angle of the opposite vacuum quadrant and vice verse. In the case with an acute angle by the hyperbola asymptotes, the new X point shifts toward the vertex of the hyperbolic plasma boundary, In the case with an obtuse angle, the new X point shifts outward the vertex of the hyperbolic plasma boundary. Nevertheless, the shift can be small if the layer chipped off is thin. The results help to understand the actual X-point structure at tokamak edge.