Elliptic Magnetic Surfaces Research Articles

Current density and poloidal magnetic field for toroidal elliptic plasmas with triangularity

Changes in the poloidal magnetic field around a tokamak magnetic surface due to different values of triangularity and ellipticity are analyzed in this paper. The treatment here presented allows the determination of the poloidal magnetic field from knowledge of the toroidal current density. Different profiles of these currents are studied. Improvements in the analytic forms of the magnetic surfaces have also been found. The treatment has been performed using a recent published system of coordinates. Suitable analytic equations have been used for the elliptic magnetic surfaces with triangularity and Shafranov shift.

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 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.

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

The parallel dielectric permittivity elements of a toroidal collisionless plasma with elliptic magnetic surfaces are analyzed for waves in the frequency range of the toroidicity-induced Alfvén eigenmodes (TAEs). The dielectric characteristics are derived by solving the drift-kinetic equation of the trapped and untapped particles in tokamaks with large aspect ratio and small elongation. The dissipated power of the Alfvén waves is expressed by the summation of terms including the non-diagonal elements of the parallel permittivity. Estimations of the TAEs electron Landau damping are carried out for the JET plasma parameters.

Collisional Neoclassical Transport for Elliptic Magnetic Surfaces with Triangularity and Grad–Shafranov Shift

An analysis of neoclassical transport for tokamaks in the collisional regime has been carried out for toroidal elliptic magnetic surfaces with triangularity and Grad–Shafranov shift. Several formulas for this general case have been obtained. Several numerical results are presented here with different ellipticity, triangularity and Grad–Shafranov shift. The elliptic elongation becomes an important parameter determining the diffusion of charged particles.

Kinetic Alfvén wave dissipation in tokamaks with elliptic magneticsurfaces

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 Alfvén 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 Alfvén 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 Alfvén waves in plasmas with high-temperature electrons, when the bounce resonant interaction between the wave and the trapped electrons becomes substantial.

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.

Pfirsch–Schlüter transport for toroidal elliptic magnetic surfaces in tokamak

Transport equations in the collisional regime have been found for very general geometries in tokamaks, using resistive magnetohydrodynamic equations, neglecting the toroidal field Eφ and assuming toroidal axisymmetry. Detailed analytic results have been obtained for families of elliptic magnetic surfaces. In the limit case of equal semiaxis and large aspect ratio, the well known Pfirsch–Schlüter result is obtained.

The perpendicular permittivity of magnetized toroidal plasmas with elliptic magnetic surfaces

The Vlasov equation for charged particles is analyzed in an axially symmetric toroidal plasma configuration with an elliptic cross section of magnetic surfaces. The asymptotic solution of the Vlasov equation is found. The analytical expressions for the perpendicular component of the dielectric permittivity tensor are obtained. These expressions are used for theoretical analyses of the trapped and untrapped ion influence on the collisionless cyclotron wave dissipation. The evaluated dielectric tensor components can be used for computer calculations of the radio frequency field structure and the collisionless dissipated power related to trapped and untrapped ions in tokamak plasmas. It is shown that ion cyclotron resonance dissipation in tokamaks depends on toroidicity and ellipticity parameters and does not depend on plasma temperature.

The parallel permittivity of magnetized toroidal plasmas with elliptic magnetic surfaces

The asymptotic solution of the Vlasov equation under the drift approximation, for an axially symmetric toroidal plasma configuration with an elliptic cross section of magnetic surfaces, is presented. The analytical expressions for the parallel component of the dielectric permittivity tensor are obtained. These expressions are used for theoretical analyses of the trapped and untrapped electron influence on the collisionless wave dissipation. The evaluated dielectric tensor components can be used for computer calculations of the radio frequency field structure and the collisionless dissipated power related to trapped and untrapped electrons in tokamak plasmas.

Reconnection of magnetic field-lines in an elongated toroidal plasma

A poloidal quadrupole field is applied to a tokamak discharge in order to study the magnetic field-line reconnection. Elongation of the poloidal cross-section of the plasma is observed when the quadrupole field strength is below a critical value. As the quadrupole field is increased above this critical field strength, reconnection occurs and the magnetic axis changes from being the center of elliptic magnetic surfaces (i.e., an ’’O’’ point) to the center of symmetry of hyperbolic surfaces (an ’’X’’ point). The reverse process is also observed.

Mercier's criterion for a class of M.H.D. Equilibria

For a class of axially symmetric equilibria, the terms in Mercier's criterion can be reduced to complete elliptic integrals with certain arguments and coefficients. For one particular choice of the equilibrium parameters the latter quantities can be evaluated explicitly without using any expansion. This case corresponds to “vertical” elliptic magnetic surfaces near the magnetica axis.