The Coulomb scattering effect on trapped particles bounce-resonance dissipation in magnetized toroidal plasmas

Abstract

The solution of the Vlasov equation with a simplified Fokker-Planck collision operator is presented for axially symmetric tokamak plasmas with a circular cross section of magnetic surfaces. The analytical expression for the parallel component of the dielectric permittivity tensor of trapped particles is obtained. This expression is used for theoretical analyses of the collision effect on the bounce-resonance wave dissipation. Conditions for a collisionless description of radio-frequency ͑rf͒ oscillations are found. This dielectric tensor components can be used for computer calculations of the rf field structure and the rf dissipated power related to trapped electrons in tokamak plasmas.

Geodesic mode spectrum modified by the energetic particles in tokamak plasmas

“…[16]. It is easy to perform the integration over the u-variable accounting bounce resonances that pro-…”

“…(1) changing the variable from λ to the new κ-variable for the untrapped κ 2 = 2ε/(1 + ε − λ) and trapped particles κ 2 = (1 + ε − λ)/2ε. In the untrapped equation, the Jacobi functions [15] sin ϑ/2 = sn(κ, x), cos ϑ/2 = cn(κ, x) with the Jacobi variable [16] ϑ/2 = am(κ, x) are introduced that trans-…”

“…A novel method of Jacobi functions [15] adopted for the calculation of the electron wave dissipation [16] in tokamaks is applied to solve the drift kinetic equation for the energetic particles (T h T i ) in the limit of the higher bounce frequency in comparison with the GAM one,…”

Collisionless Dissipation of the Toroidicity-induced Alfvén Eigenmodes in Elongated Tokamaks by the Trapped and Untrapped Electrons

“…[20] for short-wavelength magnetohydrodynamic (fast magnetosonic and Alfve¨n) waves and in Ref. [21] for Alfve¨n and lower hybrid waves. In particular, the solution of the Vlasov equation with a simpli¢ed Fokker-Planck collision operator was presented in [21] for a tokamak plasma model with circular magnetic surfaces.…”

“…[21] for Alfve¨n and lower hybrid waves. In particular, the solution of the Vlasov equation with a simpli¢ed Fokker-Planck collision operator was presented in [21] for a tokamak plasma model with circular magnetic surfaces. However, for fast Alfve¨n waves (with phase velocities of the order of the electron thermal velocity) [18,20,22] the main mechanism of the wave dissipation is the collisionless interaction with untrapped electrons.…”