A Green's Function Method for the Vacuum Contribution to the MHD Stability of Helically Symmetric Equilibria

The HERA stability code [1] computes the spectrum of the ideal MHD stability operator of helically symmetric equilibria. HERA is a modification of the ERATO code [2], which treats the stability of axisymmetric equilibria. In both codes two different procedures are provided for calculating the potential energy of a vacuum region bounded by the plasma surface and a surrounding conducting wall. The first approach treats the vacuum formally in the same way as the plasma by considering it as a shearless and pressureless plasma. The second method uses a Green's function technique. The problem is reduced to a coupled set of integral equations on the boundaries of the vacuum region. While the former approach has the advantage that vacuum and plasma region are treated in one step, the Green's function technique allows the case of a conducting wall at infinity to be easily solved. In the present paper the application of the Green's function method to the helical case is described. In helical geometry the main obstacles are to find a sufficiently fast and accurate algorithm for computing the Fourier-transformed Green's function and its normal derivative and to treat properly the logarithmic singularities of the Green's function and its derivative. For the axisymmetric case in ERATO the singularities of the normal derivative are eliminated from the integral equations by subtracting known integrals with the same singularities [3], [4]. Here the Green's function and its