A numerical code for the calculation of relativistic electron cyclotron damping with an arbitrary distribution function at an arbitrary harmonic

Abstract

The relativistic expressions for the anti-Hermitian parts of the relativistic dielectric tensor elements can be expressed as a single integral over the parallel momentum variable, allowing an arbitrary electron distribution function. A computer program has been written for the calculation of this single integral. The numerical results are tested for a relativistic Maxwellian distribution function and agree with analytical expressions for this case. The numerical code is therefore an essential element in a more general validation, evaluation and demonstration of powerful analytical results presented. The computer program is then applied to the calculation of relativistic electron cyclotron harmonic damping at any arbitrary harmonic, for any distorted electron distribution function, distorted for example by an electric field or by RF power sources, as for instance by both electron cyclotron and lower-hybrid waves, as calculated from a relativistic Fokker–Planck code. For generality, we also include the case of relativistic Landau damping, also used to check the code. Program summaryProgram title: DAMPINGCatalogue identifier: AEIS_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEIS_v1_0.htmlProgram obtainable from: CPC Program Library, Queenʼs University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 84 108No. of bytes in distributed program, including test data, etc.: 2 581 423Distribution format: tar.gzProgramming language: FORTRANComputer: Any computer or work stationOperating system: Any with a Fortran compilerRAM: 8 Mbytes, using double precision arithmetic, with 400×200 grid points, on an ALPHA serverClassification: 19.8External 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 involve 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 s CPU time on a single processor for 400×200 grid points.