Dielectric characteristics of axisymmetric low aspect ratio tokamak plasmas

Abstract

In this paper the transverse and longitudinal dielectric tensor elements are evaluated for radio-frequency waves in an axisymmetric tokamak with circular magnetic surfaces, without the smallness assumption of the inverse aspect ratio and poloidal magnetic field. A collisionless plasma model is considered. The Vlasov equation is solved separately for untrapped and trapped particles as a boundary-value problem. The cyclotron and bounce resonances are accounted. The coordinate system with the 'straight' magnetic field lines was used. The separate contributions of untrapped and trapped particles to the dielectric tensor components are expressed by the summation of bounce resonant terms, which include the double integration in velocity space, the resonant denominators, the phase coefficients, the standard elementary and elliptic functions. The permittivity elements presented in the paper are suitable to analyse the spatial structure of the electromagnetic fields and their interaction with charged particles (e.g. during the plasma heating and current drive generation) in the frequency range of Alfvén, fast magnetosonic, ion-cyclotron and lower hybrid waves, for both the low and large aspect ratio tokamaks.

Parallel permittivity elements for radio frequency waves in elongated D-shaped tokamaks

“…The same comments should be addressed to calculate the contribution of ions and electrons to the total current density components using Eq. (12).…”

“…[10][11][12] for tokamaks with circular transverse cross-sections of the magnetic surfaces). In both the large and low aspect ratio tokamaks with circular magnetic surfaces, the equilibrium magnetic field has only one minimum (or three extremums, with respect to the poloidal angle) in an equatorial plane at the external part of the magnetic surfaces.…”

“…In this paper, the parallel dielectric permittivity elements are derived for radio-frequency waves in an axisymmetric toroidal collisionless plasma with D-shaped magnetic surfaces under arbitrary aspect ratio, arbitrary elongation, and small triangularity. The drift-kinetic equation is solved separately for untrapped and three groups of trapped particles as a boundary-value problem, using an approach developed for low aspect ratio tokamaks with circular [12], elliptic [16] and D-shaped [17] magnetic surfaces. The limits for the more simple plasma models are considered.…”

“…(21, 49), can be reduced to i) the third kind elliptic integrals in low ( < 1) aspect ratio tokamaks [12], or ii) the first kind elliptic integrals in large ( << 1) aspect ratio tokamaks [9] As was mentioned above, the d-trapped particles are absent in a tokamak with circular magnetic surfaces, therefore m,m ,d…”

“…As a result, in tokamaks with circular magnetic surfaces, e.g., such as TCABR (Tokamak Chauffage Alfvén Brésilien) at the São Paulo University, the expressions (24)- (26), (34)- (36) can be reduced to the corresponding results in Ref. [12]. In particular, if b = a and λ → 0, d → 0, the contribution of untrapped and t-trapped particles to the parallel permittivity elements are…”