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

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##### Citation

Brambilla, M. (2003). Iterative solution of the Grad–Shafranov equation in symmetric
magnetic coordinates.* Physics of Plasmas,* *10*(9), 3674-3683.
doi:10.1063/1.1600736.

Cite as: https://hdl.handle.net/11858/00-001M-0000-0027-3ABB-E

##### 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.