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Manuscript Title: ECMC, a portable two-dimensional code for plasma equilibrium computation on coaxial-multiple-coil systems.
Authors: N.O. Fuentes, H.O. Gavarini
Program title: ECMC
Catalogue identifier: ADBB_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 90(1995)169
Programming language: Fortran.
Computer: DEC-VAX 11/750.
Operating system: VMS - V4.6, UNIX - V4, MS-DOS 3.0 and above.
RAM: 1850K words
Word size: 32
Keywords: Plasma physics, Equilibrium, Compact torus, Field reversed, Configuration, Two-dimensional, Coaxial systems, Multiple coil systems, Finite difference, Successive over Relaxation, Portable code, Interactive code.
Classification: 19.6.

Nature of problem:
This two-dimensional code computes field reversed configuration (FRC) equilibria for any particular experimental arrangement. The real geometry of a coaxial multiple-coil system can be reproduced from the input data of each coil position in labortory coordinates. The respective currents are used to satisfy Ampere's law for the derivatives of poloidal magnetic flux function, Psi. With this condition it is not necessary to assume a particular shape for Psi, so the outer boundary of the computational region does not need to be delimited by the coils. Both, open and closed magnetic field regions are included in the code simulation. Users must give a plasma trapped current value as constraint parameter. They may also provide the dependence of plasma pressure as a function of Psi, changing that given by default.

Solution method:
Plasma equilibrium states are found by solving finite difference approximations to the Grad-Shafranov equation rewritten using the successive over relaxation (SOR) method. The poloidal magnetic flux function, Psi, is given by the contribution of two terms: an homogeneous part, Psih, and a particular solution, Psis. The latter takes into account the existence of the source term due to plasma current density, and Psih is obtained as summation of N-terms of the form (CmPsim), each one associated to each coil. Here, Psim is an homogeneous solution that takes the value 1 on m-coil and 0 on the other coils. Coefficients Cm are derived from the system of equations that results when writing the Ampere's law over a contour encircling each m-coil. To do that, magnetic field components are replaced by derivatives of Psi and known coil currents, given as input parameters, are used. Computation begins by calculating coil positions on the grid and coefficients of SOR representation, all depending only on the geometry. Then, Psih and Psis are determined from given initial approximating functions and improved with the SOR method before constructing Psi. The calculation proceeds by updating Psi with SOR until convergence requirement is achieved. In order to fulfill the global constraint imposed on plasma current, Itheta=const., plasma pressure is recalculated as the computation goes on. All variables are used in a dimensionless form by the code, but Gaussian units are used in input-output data with exception of currents which are expressed in Amperes.

Restrictions:
The model used here neglects toroidal component of magnetic field and plasma rotation. Mirror symmetry about the midplane z=0, and axial symmetry are considered. The poloidal magnetic flux function must be Psi=0 for r=0, r -> infinity, z -> infinity.

Unusual features:
The code allows the computational region to be divided into four zones in order to include infinite spatial coordinates (r,z). When going to infinity each coordinate is transformed into its reciprocal. Adopting this option to run the code, only one of the zones contains the total number of coils and the plasma. Plasma pressure dependence as a function of Psi may be changed when desired simply by changing that given by default with another routine provided by the user. Users may change running parameters (coil positions and currents, plasma trapped currents, data to be written in output files, convergence criteria) interactively.

Running time:
For the corresponding test cases 55 min., 59 min. and 40 min. on an Intel 80486 PC.

References:
[1] R.L. Spencer and D.W. Hewett, Phys. Fluids 25(1982)1365.
[2] D.W. Hewett and R.L. Spencer, Phys. Fluids 26(1983)1299.
[3] M. Kako, T. Ishimura and T. Amano, J. Phys. Soc. Jpn. 52(1983)3056.