Nikolay Grishanov - Academia.edu (original) (raw)
Papers by Nikolay Grishanov
Plasma Physics Reports, Dec 1, 2013
The contribution of untrapped and two groups of trapped particles to the longitudinal (with respe... more The contribution of untrapped and two groups of trapped particles to the longitudinal (with respect to the magnetic field) elements of the dielectric susceptibility is determined by solving the drift-kinetic equations for such particles in axisymmetric tokamaks with Solov’ev equilibrium. The obtained dielectric characteristics are applicable for studying linear wave processes in the frequency range of Alfvén and fast magnetosonic waves in small- and large-aspect-ratio tokamaks with circular, elliptical, and D-shaped cross sections of magnetic surfaces. The high-frequency power absorbed in plasma via electron Landau damping is estimated by summing up terms containing the imaginary parts of both diagonal and non-diagonal elements of the longitudinal susceptibility. The imaginary part of the longitudinal susceptibility is calculated numerically for spherical tokamaks in a wide range of wave frequencies and magnetic surface radii.
Problems of Atomic Science and Technology
Kinetic models of magnetized current-carrying plasma have been developed to study the influence o... more Kinetic models of magnetized current-carrying plasma have been developed to study the influence of magnetic drift effects on the wave-particle interactions in tokamaks and cylindrical plasma columns. The drift-kinetic equations are derived for the perturbed distribution functions of trapped and untrapped particles in a two-dimensional axisymmetric toroidal plasma, taking into account their bounce oscillations and the finite orbit-widths of their banana trajectories.
Journal of Plasma Physics, 2016
Dispersion characteristics have been analysed for field-aligned electron–cyclotron waves (also kn... more Dispersion characteristics have been analysed for field-aligned electron–cyclotron waves (also known as right-hand polarized waves, extraordinary waves or whistlers) in a cylindrical magnetic mirror plasma including electrons with anisotropic temperature. It is shown that the instability of these waves is possible only in the range below the minimal electron–cyclotron frequency, which is much lower than the gyrotron frequency used for electron–cyclotron resonance power input into the plasma, under the condition where the perpendicular temperature of the resonant electrons is larger than their parallel temperature. The growth rates of whistler instability in the two magnetized plasma models, where the stationary magnetic field is either uniform or has a non-uniform magnetic mirror configuration, are compared.
Âëèÿíèå çàïåðòûõ è ïðîëåòíûõ...
The linearized Vlasov equation is rewritten for charged particles in the two-dimensional axisymme... more The linearized Vlasov equation is rewritten for charged particles in the two-dimensional axisymmetric plasma models using the cylindrical coordinates. There is described a method of its solution by the Fourier expansions of the perturbed distribution functions over the gyrophase angle in velocity space, conservation integrals of particle motion in the curvilinear magnetic field and smallness of the magnetization parameters. Such an approach allows us to evaluate the main contributions of untrapped and trapped particles to the transverse and longitudinal dielectric tensor components for electromagnetic waves in tokamaks, straight mirror-traps, laboratoty dipole magnetospheric plasmas and inner part of the Earth's magnetosphere. KEY WORDS: Vlasov equation, kinetic wave theory, tokamaks, mirror traps, magnitospheric plasmas. ... ... , 4,
arXiv: Plasma Physics, 2004
In this paper, we derive the dispersion equations for field-aligned cyclotron waves in an axisymm... more In this paper, we derive the dispersion equations for field-aligned cyclotron waves in an axisymmetric dipole magnetospheric plasmas with both the bi-Maxwellian and bi-Lorentzian distribution functions. To evaluate the contribution of trapped particles to the transverse current density components the Vlasov equation is solved using a standard method of switching to new variables associated with conservation integrals; new time-like variable is introduced (instead of the geomagnetic latitude angle) to describe the bounce-motion of trapped particles along the geomagnetic field; the perturbed electric field and current density components are Fourier-decomposed over the length of the geomagnetic field lines. As a result, the transverse permittivity elements are expressed by summation of bounce-resonant terms including the double integration in velocity space, the resonant denominators, and the corresponding phase coefficients. Due to geomagnetic field nonuniformity, the wave-particle re...
Parallel permittivity elements are derived for radio frequency waves in an axisymmetric D-shaped ... more Parallel permittivity elements are derived for radio frequency waves in an axisymmetric D-shaped tokamak with Soloviev type equilibrium under arbitrary aspect ratio, arbitrary elongation and moderate triangularity. The drift-kinetic equation is solved separately for untrapped (passing or circulating) and three groups of trapped particles as a boundary-value problem, accounting for the transit-time and bounce resonances, respectively. It is shown that the new two groups of trapped particles can appear only in tokamaks with large elongation and large central beta. Our dielectric characteristics can be applied for both the large and low aspect ratio tokamaks with circular, elliptic and D-shaped magnetic surfaces to study the wave propagation and wave dissipation by electron Landau damping during the plasma heating and current drive generation. The dissipated wave power is expressed by the summation of terms including the imaginary parts of both the diagonal and non-diagonal elements of...
Plasma and Fusion Research
The transverse dielectric susceptibility elements are derived for electromagnetic cyclotron waves... more The transverse dielectric susceptibility elements are derived for electromagnetic cyclotron waves in an axisymmetric laboratory dipole magnetosphere accounting for the cyclotron and bounce resonances of trapped and untrapped particles. A bi-Kappa (or bi-Lorentzian) distribution function is invoked to model the energetic particles with anisotropic temperature. The steady-state two-dimensional (2D) magnetic field is modeled by laboratory dipole approximation for a superconducting ring current of finite radius. Derived for field-aligned circularlypolarized waves the dispersion relations are suitable for analyzing both the whistler instability in the range below the electron-cyclotron frequency, and the proton-cyclotron instability in the range below the ion-cyclotron frequency. The instability growth rates in the 2D laboratory magnetosphere are defined by the contributions of energetic particles to the imaginary part of transverse susceptibility.
The Vlasov equation is solved for charged particles in two-dimensional axisymmetric plasma models... more The Vlasov equation is solved for charged particles in two-dimensional axisymmetric plasma models using: (i) convenient curvilinear (cylindrical, spherical, toroidal) coordinates, (ii) Fourier expansions of the perturbed distribution functions over the gyrophase angle in velocity space, (iii) smallness of the magnetization parameters, (iv) new variables by the conservation integrals to describe the bounce periodic motion of the trapped and untrapped particles along the equilibrium magnetic field line. An approach developed in the paper allows us to evaluate the main contribution of untrapped and trapped particles to the transverse and longitudinal dielectric permittivity elements for radio frequency waves in tokamaks, mirror-traps, and Earth’s magnetosphere.
Plasma Physics and Controlled Fusion, 2001
ABSTRACT In this paper the transverse and longitudinal dielectric tensor elements are evaluated f... more 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.
Arxiv preprint physics/ …, 2004
Levitated dipole configurations provide a promising approach [1, 2] for the magnetic confinement ... more Levitated dipole configurations provide a promising approach [1, 2] for the magnetic confinement of plasmas for fusion research and planetary plasmaspheric simulations. The point dipole magnetic field is a good approximation for magnetospheric plasmas of the Earth and other planets, ...
Arxiv preprint physics/ …, 2004
Abstract: Parallel permittivity elements are derived for RF waves in an axisymmetric D-shaped tok... more Abstract: Parallel permittivity elements are derived for RF waves in an axisymmetric D-shaped tokamak with Solov'ev type equilibrium. Drift-kinetic equation is solved separately for untrapped (passing or circulating) and three groups of trapped particles as a boundary-value ...
Arxiv preprint physics/ …, 2004
Abstract: Dispersion equations have been derived for field-aligned cyclotron waves in dipole magn... more Abstract: Dispersion equations have been derived for field-aligned cyclotron waves in dipole magnetospheric plasmas with anisotropic temperature. Vlasov equation is solved for trapped particles with both the bi-Maxwellian and bi-Lorentzian distribution functions accounting ...
Physics of Plasmas, 2002
ABSTRACT Longitudinal dielectric permittivity elements are derived for radio-frequency waves in a... more ABSTRACT 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 Alfvén, 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. © 2002 American Institute of Physics.
Toroidicity-induced Alfvin Eigenmodes (TAEs) are studied actively in the context of their possibl... more Toroidicity-induced Alfvin Eigenmodes (TAEs) are studied actively in the context of their possible destabilization by the fast ions. As is well known, conditions of TAEs excitation depend on the TAEs damping in the concrete tokamak. In this paper, collisionless TAEs dissipation due to untrapped and trapped electrons (electron Landau damping) is analyzed in a toroidal plasma with elliptic magnetic surfaces.
Plasma Physics Reports, Dec 1, 2013
The contribution of untrapped and two groups of trapped particles to the longitudinal (with respe... more The contribution of untrapped and two groups of trapped particles to the longitudinal (with respect to the magnetic field) elements of the dielectric susceptibility is determined by solving the drift-kinetic equations for such particles in axisymmetric tokamaks with Solov’ev equilibrium. The obtained dielectric characteristics are applicable for studying linear wave processes in the frequency range of Alfvén and fast magnetosonic waves in small- and large-aspect-ratio tokamaks with circular, elliptical, and D-shaped cross sections of magnetic surfaces. The high-frequency power absorbed in plasma via electron Landau damping is estimated by summing up terms containing the imaginary parts of both diagonal and non-diagonal elements of the longitudinal susceptibility. The imaginary part of the longitudinal susceptibility is calculated numerically for spherical tokamaks in a wide range of wave frequencies and magnetic surface radii.
Problems of Atomic Science and Technology
Kinetic models of magnetized current-carrying plasma have been developed to study the influence o... more Kinetic models of magnetized current-carrying plasma have been developed to study the influence of magnetic drift effects on the wave-particle interactions in tokamaks and cylindrical plasma columns. The drift-kinetic equations are derived for the perturbed distribution functions of trapped and untrapped particles in a two-dimensional axisymmetric toroidal plasma, taking into account their bounce oscillations and the finite orbit-widths of their banana trajectories.
Journal of Plasma Physics, 2016
Dispersion characteristics have been analysed for field-aligned electron–cyclotron waves (also kn... more Dispersion characteristics have been analysed for field-aligned electron–cyclotron waves (also known as right-hand polarized waves, extraordinary waves or whistlers) in a cylindrical magnetic mirror plasma including electrons with anisotropic temperature. It is shown that the instability of these waves is possible only in the range below the minimal electron–cyclotron frequency, which is much lower than the gyrotron frequency used for electron–cyclotron resonance power input into the plasma, under the condition where the perpendicular temperature of the resonant electrons is larger than their parallel temperature. The growth rates of whistler instability in the two magnetized plasma models, where the stationary magnetic field is either uniform or has a non-uniform magnetic mirror configuration, are compared.
Âëèÿíèå çàïåðòûõ è ïðîëåòíûõ...
The linearized Vlasov equation is rewritten for charged particles in the two-dimensional axisymme... more The linearized Vlasov equation is rewritten for charged particles in the two-dimensional axisymmetric plasma models using the cylindrical coordinates. There is described a method of its solution by the Fourier expansions of the perturbed distribution functions over the gyrophase angle in velocity space, conservation integrals of particle motion in the curvilinear magnetic field and smallness of the magnetization parameters. Such an approach allows us to evaluate the main contributions of untrapped and trapped particles to the transverse and longitudinal dielectric tensor components for electromagnetic waves in tokamaks, straight mirror-traps, laboratoty dipole magnetospheric plasmas and inner part of the Earth's magnetosphere. KEY WORDS: Vlasov equation, kinetic wave theory, tokamaks, mirror traps, magnitospheric plasmas. ... ... , 4,
arXiv: Plasma Physics, 2004
In this paper, we derive the dispersion equations for field-aligned cyclotron waves in an axisymm... more In this paper, we derive the dispersion equations for field-aligned cyclotron waves in an axisymmetric dipole magnetospheric plasmas with both the bi-Maxwellian and bi-Lorentzian distribution functions. To evaluate the contribution of trapped particles to the transverse current density components the Vlasov equation is solved using a standard method of switching to new variables associated with conservation integrals; new time-like variable is introduced (instead of the geomagnetic latitude angle) to describe the bounce-motion of trapped particles along the geomagnetic field; the perturbed electric field and current density components are Fourier-decomposed over the length of the geomagnetic field lines. As a result, the transverse permittivity elements are expressed by summation of bounce-resonant terms including the double integration in velocity space, the resonant denominators, and the corresponding phase coefficients. Due to geomagnetic field nonuniformity, the wave-particle re...
Parallel permittivity elements are derived for radio frequency waves in an axisymmetric D-shaped ... more Parallel permittivity elements are derived for radio frequency waves in an axisymmetric D-shaped tokamak with Soloviev type equilibrium under arbitrary aspect ratio, arbitrary elongation and moderate triangularity. The drift-kinetic equation is solved separately for untrapped (passing or circulating) and three groups of trapped particles as a boundary-value problem, accounting for the transit-time and bounce resonances, respectively. It is shown that the new two groups of trapped particles can appear only in tokamaks with large elongation and large central beta. Our dielectric characteristics can be applied for both the large and low aspect ratio tokamaks with circular, elliptic and D-shaped magnetic surfaces to study the wave propagation and wave dissipation by electron Landau damping during the plasma heating and current drive generation. The dissipated wave power is expressed by the summation of terms including the imaginary parts of both the diagonal and non-diagonal elements of...
Plasma and Fusion Research
The transverse dielectric susceptibility elements are derived for electromagnetic cyclotron waves... more The transverse dielectric susceptibility elements are derived for electromagnetic cyclotron waves in an axisymmetric laboratory dipole magnetosphere accounting for the cyclotron and bounce resonances of trapped and untrapped particles. A bi-Kappa (or bi-Lorentzian) distribution function is invoked to model the energetic particles with anisotropic temperature. The steady-state two-dimensional (2D) magnetic field is modeled by laboratory dipole approximation for a superconducting ring current of finite radius. Derived for field-aligned circularlypolarized waves the dispersion relations are suitable for analyzing both the whistler instability in the range below the electron-cyclotron frequency, and the proton-cyclotron instability in the range below the ion-cyclotron frequency. The instability growth rates in the 2D laboratory magnetosphere are defined by the contributions of energetic particles to the imaginary part of transverse susceptibility.
The Vlasov equation is solved for charged particles in two-dimensional axisymmetric plasma models... more The Vlasov equation is solved for charged particles in two-dimensional axisymmetric plasma models using: (i) convenient curvilinear (cylindrical, spherical, toroidal) coordinates, (ii) Fourier expansions of the perturbed distribution functions over the gyrophase angle in velocity space, (iii) smallness of the magnetization parameters, (iv) new variables by the conservation integrals to describe the bounce periodic motion of the trapped and untrapped particles along the equilibrium magnetic field line. An approach developed in the paper allows us to evaluate the main contribution of untrapped and trapped particles to the transverse and longitudinal dielectric permittivity elements for radio frequency waves in tokamaks, mirror-traps, and Earth’s magnetosphere.
Plasma Physics and Controlled Fusion, 2001
ABSTRACT In this paper the transverse and longitudinal dielectric tensor elements are evaluated f... more 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.
Arxiv preprint physics/ …, 2004
Levitated dipole configurations provide a promising approach [1, 2] for the magnetic confinement ... more Levitated dipole configurations provide a promising approach [1, 2] for the magnetic confinement of plasmas for fusion research and planetary plasmaspheric simulations. The point dipole magnetic field is a good approximation for magnetospheric plasmas of the Earth and other planets, ...
Arxiv preprint physics/ …, 2004
Abstract: Parallel permittivity elements are derived for RF waves in an axisymmetric D-shaped tok... more Abstract: Parallel permittivity elements are derived for RF waves in an axisymmetric D-shaped tokamak with Solov'ev type equilibrium. Drift-kinetic equation is solved separately for untrapped (passing or circulating) and three groups of trapped particles as a boundary-value ...
Arxiv preprint physics/ …, 2004
Abstract: Dispersion equations have been derived for field-aligned cyclotron waves in dipole magn... more Abstract: Dispersion equations have been derived for field-aligned cyclotron waves in dipole magnetospheric plasmas with anisotropic temperature. Vlasov equation is solved for trapped particles with both the bi-Maxwellian and bi-Lorentzian distribution functions accounting ...
Physics of Plasmas, 2002
ABSTRACT Longitudinal dielectric permittivity elements are derived for radio-frequency waves in a... more ABSTRACT 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 Alfvén, 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. © 2002 American Institute of Physics.
Toroidicity-induced Alfvin Eigenmodes (TAEs) are studied actively in the context of their possibl... more Toroidicity-induced Alfvin Eigenmodes (TAEs) are studied actively in the context of their possible destabilization by the fast ions. As is well known, conditions of TAEs excitation depend on the TAEs damping in the concrete tokamak. In this paper, collisionless TAEs dissipation due to untrapped and trapped electrons (electron Landau damping) is analyzed in a toroidal plasma with elliptic magnetic surfaces.