Iterative solution of the Grad–Shafranov equation in symmetric magnetic coordinates

Abstract

The inverse Grad–Shafranov equation for axisymmetric magnetohydrodynamic equilibria is reformulated in symmetric magnetic coordinates (in which magnetic field lines look “straight,” and the geometric toroidal angle is one of the coordinates). The poloidally averaged part of the equilibrium condition and Ampère law takes the form of two first-order ordinary differential equations, with the two arbitrary flux functions, pressure and force-free part of the current density, as sources. The condition for the coordinates to be flux coordinates, and the poloidally varying part of the equilibrium equation are similarly transformed into a set of first-order ordinary differential equations, with coefficients depending on the metric, and explicitly solved for the radial derivatives of the coefficients of the Fourier representation of the Cartesian coordinates in the poloidal angle. The derivation exploits the existence of Boozer–White coordinates, but does not require to find these coordinates explicitly; on the other hand, it offers a simple recipe to perform the transformation to Boozer–White coordinates, if required. Use of symmetric flux coordinates is advantageous for the formulation of many problems of equilibrium, stability, and wave propagation in tokamak plasmas, since these coordinates have the simplest metric of their class. It is also shown that in symmetric flux coordinates the Lagrangian equations of the drift motion of charged particles are automatically solved for the time derivatives, with right-hand sides closely related to the coefficients of the inverse Grad–Shafranov equation.