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Manuscript Title: A Numerical Code for the Calculation of Relativistic Electron Cyclotron Damping with an Arbitrary Distribution Function at an Arbitrary Harmonic
Authors: I. Shkarofsky, M. Shoucri
Program title: DAMPING
Catalogue identifier: AEIS_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 182(2011)1507
Programming language: FORTRAN.
Computer: Any computer or work station.
Operating system: Any with a Fortran compiler.
RAM: 8 Mbytes, using double precision arithmetic, with 400x200 grid points, on an ALPHA server.
Keywords: relativistic cyclotron harmonic damping, relativistic Landau damping, anti-Hermititian tensor, arbitrary distribution function.
PACS: 52.25.Os, 52.35.Hr, 52.40.Db.
Classification: 19.8.

External routines: Requires a link to the NAG libraries for the Bessel functions subroutines.

Nature of problem:
The computer program calculates generalized expressions for the anti-Hermitian part of the relativistic dielectric tensor in a plasma for an arbitrary distribution function, with comparison to the analytical expressions derived in [1], associated with a relativistic Maxwellian distribution, for any n-th cyclotron harmonic and for the relativistic Landau damping case when n =0. The anti-Hermitian parts of the dielectric tensor are important to know, since they determine the damping or emission of an electromagnetic wave within the plasma medium. New analytical expressions are derived for the relativistic Landau damping and for the electron cyclotron damping including the case n|| = 1, for the case of an arbitrary distribution function. This is important since studies of the RF heating of tokamak plasmas in a fully self-consistent way involves the simultaneous description of the temporal evolution of the heated species velocity distribution function using a Fokker-Planck code, coupled to the damping of the wave on the evolving distribution function. We can evaluate numerically the degree by which a non-Maxwellian nature of the relativistic distribution function changes the Landau damping and the electron cyclotron damping, with respect to the Maxwellian case. In addition this is also important in the study of electron cyclotron emission, which can be a sensitive indicator of non-thermal electron distributions.

Solution method:
The relativistic expressions for the anti-Hermitian parts of the dielectric tensor elements can be expressed as a single integral over the parallel momentum variable [1]. This single integral is calculated along the resonance curve, for an arbitrary distribution function calculated from a relativistic Fokker-Planck code. Since Fokker-Planck codes usually evolve the distribution functions using a spherical coordinate system in momentum space, the present code has to first transform the distribution function calculated by the Fokker-Planck code from a spherical coordinate system to a cylindrical coordinate system in momentum space, where the single integral over the parallel momentum variable can be calculated along the resonance curves. In the evaluation of the term in Eq.(8), cubic spline are used for the calculation of the derivatives.

Running time:
An execution on an ALPHA server 4000 from Digital requires less than 10 sec CPU time on a single processor for 400x200 grid points.

References:
[1] I.P. Shkarofsky , J. Plasma Phys. 61 (1999) 107