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Manuscript Title: MACH: a computer code for solution of the poloidal asymmetry eigenvalue problem in tokamaks.
Authors: S.E. Attenberger, S.P. Hirshman, W.A. Houlberg
Program title: MACH
Catalogue identifier: ACPY_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 79(1994)341
Programming language: Fortran.
Keywords: General purpose, Matrix, Equilibrium, Plasma physics, Poloidal rotation, Mach number, Mathieu equation, Momentum balance, Eigenvalue, Recursive method.
Classification: 4.8, 19.6.

Nature of problem:
The ions in a tokamak plasma gyrate around and stream along magnetic field lines that encircle the torus in both the poloidal (short- circumference) and toroidal (long circumference) directions. In the infinite aspect ratio limit, the geometry is cylindrical and the plasma ion density is poloidally symmetric, but at finite aspect ratios the symmetry is broken. Given a poloidal "Mach number" that is related to the plasma rotation velocity, the poloidal variation of the plasma density is determined from the steady state momentum balance equation. A "shock" is formed when the rotation velocity is near a critical value (Mach 1), giving a nearly discontinuous poloidal variation of the plasma density.

Solution method:
The momentum balance equation is transformed into an eigenvalue equation of the Mathieu type with an extra damping term. Only the eigenvalue associated with the lowest order eigenfunction corresponds to a physical solution, i.e., one where the solution is nonzero over the range eta= [0,2phi], where eta is a phase-shifted poloidal angle. The solution is represented in terms of a Fourier expansion, where the Fourier coefficients depend on the eigenvalue. A recursion procedure has been found that permits determination of the correct eigenvalue in much less time than the usual matrix determinant method. It is therefore appropriate for use in a tokamak transport code where the momentum equation must be solved repeatedly.

The reduction of the momentum balance equation to a Mathieu form assumes a large (but finite) aspect ratio torus.