Contribution of trapped and untrapped particles to the longitudinal permeability of a toroidal plasma

Abstract

The contributions of trapped and untrapped particles to the longitudinal permeability of a toroidal plasma are analyzed for radio frequency waves in a wide range of frequencies, mode numbers, and plasma parameters. A collisionless plasma model of an axisymmetric tokamak with circular magnetic surfaces and large aspect ratio is considered. The analytical expressions of the longitudinal permeability elements are obtained by using the kinetic theory of dielectric tensor components, where the Vlasov equation is solved as a boundary-value problem. The radial structures of the imaginary part of the longitudinal permeability (for both the untrapped and trapped particles) are studied for parabolic profiles of plasma density, temperature, and tokamak safety factor. The results of the paper are interesting for the problems of Alfvén (and lower-hybrid) resonance heating and current drive by the Čerenkov dissipation mechanism of the waves in toroidal plasmas.

Longitudinal permittivity of a tokamak plasma with elliptic and circular magnetic surfaces

“…The expressions of the longitudinal permittivity elements obtained in the paper have a quite simple form in order to calculate the contributions of the trapped and untrapped particles to Re⑀ ʈ m,mЈ and Im⑀ ʈ m,mЈ ; that was demonstrated in Ref. 11 for tokamaks with circular magnetic surfaces by using the usual PC-computer. However, the solution of twodimensional Maxwell's equations with the ''nonlocal'' dielectric tensor needs the additional intellectual and technical efforts.…”

“…If aϭb ͑or ϭ0) in Eq. ͑4͒, this equation becomes the initial kinetic equation to evaluate the longitudinal permittivity elements in a tokamak plasma with circular magnetic surfaces, when 11 the equilibrium magnetic field is approximated by H(r,) ϭH 0 (r)(1Ϫ⑀ cos ).…”

“…And really, assuming ϭ0 ͑and/or ␤ϭ0) in Eqs. ͑20͒-͑24͒, the contributions of untrapped and trapped particles to the longitudinal permittivity can be described as 11, where they were used to analyze numerically the wave dissipation in tokamaks with both uniform and nonuniform plasmas in the frequency range of Alfvén and lower-hybrid waves. In particular, some concrete results were obtained there for a toroidal plasma model with the main plasma/antenna parameters of the ''Tokamak Chauffage Alfvén'' ͑TCA/BR͒ tokamak 16 with the Alfvén wave heating, assembled at the University of São Paulo ͑Brazil͒.…”

“…In particular, the quite simple ͑and the quite correct͒ expressions of the longitudinal permittivity elements in a tokamak with circular magnetic surfaces were presented in Ref. 11. There it was demonstrated that the numerical analysis of both the real and imaginary parts of this plasma characteristic is possible without problems for both the untrapped and trapped particles.…”

Toroidicity effects on the absorption of IC waves by electrons

“…It has been known for some time that toroidicity influences the absorption of the compressional wave (also known as a magnetosonic, or fast wave (FW)) by electrons in the ion cyclotron (IC) and magnetohydrodynamic (MHD) range of frequencies [1][2][3][4][5][6][7]. These authors have pointed out that the poloidal modulation of the parallel velocity of charged particles modifies the longitudinal permeability of the plasma (whose imaginary part describes electron Landau damping of electrostatic waves).…”