N. I. Grishanov

Dielectric permeability of a mirror-trapped plasma

Analytical expressions for the wave permeability tensor are derived for a two-dimensional plasma model of a straight axisymmetric mirror trap. The dielectric tensor components are found through a solution of the Vlasov equation, using the theory of Jacobian elliptic functions. The bounce-resonance effect of trapped particles on wave dissipation is analysed. It is shown that collisionless wave dissipation in the plasma with a mirror-trap configuration of a magnetic field can differ essentially from Landau damping in a plasma with straight uniform magnetic field lines. This dielectric tensor can be used in numerical calculations of Alfven and ion cyclotron heating of mirror-trap plasmas.

Wave dissipation by electron Landau damping in low aspect ratio tokamaks with elliptic magnetic surfaces

Longitudinal dielectric permittivity elements are derived for radio-frequency waves in an axisymmetric tokamak with elliptic magnetic surfaces, for arbitrary elongation and inverse aspect ratio. A collisionless plasma model is considered. Drift-kinetic equation is solved separately for untrapped (passing or circulating) and three groups of the trapped particles as a boundary-value problem. Bounce resonances are taken into account. A coordinate system with the “straight” magnetic field lines is used. Permittivity elements, evaluated in the paper, are suitable to estimate the wave dissipation by electron Landau damping (e.g., during the plasma heating and current drive generation) in the frequency range of Alfven, fast magnetosonic, and lower hybrid waves, for both the large and low aspect ratio tokamaks. The dissipated wave power is expressed by the summation of terms including the imaginary parts of both the diagonal and nondiagonal elements of the longitudinal permittivity.Longitudinal dielectric permittivity elements are derived for radio-frequency waves in an axisymmetric tokamak with elliptic magnetic surfaces, for arbitrary elongation and inverse aspect ratio. A collisionless plasma model is considered. Drift-kinetic equation is solved separately for untrapped (passing or circulating) and three groups of the trapped particles as a boundary-value problem. Bounce resonances are taken into account. A coordinate system with the “straight” magnetic field lines is used. Permittivity elements, evaluated in the paper, are suitable to estimate the wave dissipation by electron Landau damping (e.g., during the plasma heating and current drive generation) in the frequency range of Alfven, fast magnetosonic, and lower hybrid waves, for both the large and low aspect ratio tokamaks. The dissipated wave power is expressed by the summation of terms including the imaginary parts of both the diagonal and nondiagonal elements of the longitudinal permittivity.

Longitudinal permittivity of magnetospheric plasmas with dipole and circular magnetic field lines

Longitudinal permittivity elements of an axisymmetric magnetosphere are derived by solving the drift-kinetic equation for the trapped particles, in a two-dimensional plasma model with dipole and circular magnetic field lines. The drift-kinetic equation is solved for waves in the frequency range much larger than the drift frequencies. It is shown that the collisionless dissipation of such waves by the bounce-resonant interaction with the trapped particles differs substantially from one in a straight magnetic field. The estimations of the electron Landau damping for kinetic Alfven waves in magnetospheric plasmas are presented. Imaginary part of the longitudinal permittivity is analyzed numerically in the wide range of wave frequencies and plasma parameters.

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

The contributions of untrapped and three groups of trapped particles to the longitudinal permittivity of a tokamak plasma with elliptic magnetic surfaces are derived for radio frequency waves in a wide range of frequencies, mode number, and plasma parameters. The analytical expressions of the longitudinal permittivity elements are obtained by using the kinetic theory of dielectric tensor elements, where the drift kinetic equation is solved as a boundary-value problem. Considered is a collisionless plasma model of an axisymmetric tokamak with small ellipticity and a large aspect ratio. The limit to the known results for toroidal plasmas with the circular cross-section of the magnetic surfaces is shown.

Influence of trapped electrons on the dielectric properties of the Earth’s radiation belts

The dielectric tensor of an axisymmetric magnetosphere is derived by solving the Vlasov equations for trapped particles for a two‐dimensional plasma model with circular magnetic field lines. Because of the nonuniformity of the geomagnetic field any Fourier harmonic of the perturbed current density (after its Fourier expansion over the geomagnetic latitude) is proportional to the sum over all harmonics of the oscillating electric field; that leads to additional wave dissipation effects. The resonant conditions for wave–particle interactions are discussed taking into account the cyclotron and bounce resonances; these conditions in magnetospheric plasmas are entirely different from the Landau damping resonant factor in the uniform magnetic field. The numerical calculations of the longitudinal permeability in the Earth’s radiation belts (the equatorial distance of the magnetic field line is five Earth’s radii) are carried out for short‐period oscillations with the frequency ω∼2 s−1 and longitudinal wave numbe...

Contribution of trapped particles to the dielectric tensor of magnetospheric plasmas

Analytical expressions for the components of the dielectric permeability operator tensor of plasmas confined in planetary dipolar magnetospheres are derived. The kinetic Vlasovs equation of a two-dimensional plasma model, confined by an axisymmetric mirror trap, with circular magnetic field lines has been used. The conditions for single-particle resonances of trapped particles are discussed, taking into account the bounce resonances. It is shown that the collisionless Cherenkov damping of the Alfven and magnetosonic waves differs substantially from the Cherenkov damping of these modes in the straight confining magnetic field case; this is true due to the presence of the trapped particles. Estimations of the damping coefficients for Langmuir and kinetic Alfven waves are made, and some observational consequences of this study are discussed.

Kinetic Alfvén wave dissipation in tokamaks with elliptic magnetic surfaces

The longitudinal permittivity elements of a toroidal plasma with elliptic magnetic surfaces are evaluated by solving the drift kinetic equation as a boundary-value problem for waves in the frequency range much more than the drift frequency. The linearized drift kinetic equation is solved separately for untrapped, and three groups of trapped particles. The quadratic corrections over the inverse tokamak aspect ratio are taken into account to approximate the equilibrium magnetic field. Some physical features related to the absorption of monochromatic kinetic Alfven waves in collisionless elongated tokamaks are analysed for the case when only usual untrapped and trapped particles-electrons can exist in the plasma volume. The radial structure of the imaginary part of the longitudinal permittivity is calculated for a toroidal plasma having the main Joint European Torus (JET) parameters. For simplicity, the parabolic profiles are considered for plasma density and temperature. It is shown, that the kinetic Alfven wave dissipation, in cold/warm plasmas, occurs mainly by the effective interaction with untrapped electrons. There is another situation and other dissipation mechanisms for the slow Alfven waves in plasmas with high-temperature electrons, when the bounce resonant interaction between the wave and the trapped electrons becomes substantial.

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

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 Alfven (and lower-hybrid) resonance heating and current drive by the Cerenkov dissipation mechanism of the waves in toroidal plasmas.

Parallel Permittivity Elements for Radio Frequency Waves in Elongated D-shaped Tokamaks

Parallel permittivity elements are derived for radio-frequency waves in an axisymmetric tokamak with D-shaped transverse cross-sections of the magnetic surfaces under arbitrary aspect ratio, arbitrary elongation and small triangularity. The bounce resonances are taken into account for untrapped (passing or circulating) and three groups of trapped particles. The corresponding limits for the simpler plasma models are considered. Our dielectric characteristics are suitable to estimate the wave dissipation by electron Landau damping during the plasma heating and current drive generation in the frequency range of Alfv ´ en and fast magnetosonic waves, for both the large and low aspect ratio tokamaks with circular, elliptic and D-shaped magnetic surfaces. The dissipated wave power is expressed by the summation of terms including the imaginary parts of both the diagonal and non-diagonal elements of the parallel permittivity.

Radio Frequency Dielectric Characteristics of a Collisionless Laboratory Dipole Plasma

Transverse and longitudinal dielectric permittivity elements have been derived for radio‐frequency waves in a laboratory dipole plasma model by solving the Vlasov equation for trapped and untrapped particles accounting for the cyclotron and bounce resonances.