Calculation of the vacuum Green’s function valid even for high toroidal mode numbers in tokamaks

The present evaluation of the Green’s function used for the magnetic scalar potential in vacuum calculations for axisymmetric geometry has been found to be deficient even for moderately high, n, the toroidal mode number. This is relevant to the edge localized peeling-ballooning modes calculated by gato, pest and other mhd stability codes. The deficiency was due to the loss of numerical precision arising from the upward recursion relation used for generating the functions from the values at n = 0 from the complete elliptic integrals of the first and second kinds. To ameliorate this, a direct integration of the integral representation of the function is crafted to achieve the necessary high accuracy for moderately high mode numbers, with due consideration to the singular behavior of the integrand involved. At higher mode numbers the loss of numerical precision due to cancellations from the oscillatory behavior of the integrand is further avoided by judiciously deforming the integration contour into the complex plane to obtain a new integral representation for the Green’s function. Near machine precision, roughly 12–16 digits, can be achieved by using a combination of these techniques. The relation to the associated Legendre functions, as well as a novel integral representation of these are also described.