Energy Principle for 3-d Resistive Instabilities in Shaped-cross-section Tokamaks
TL;DR: In this paper, an energy principle for "helical" incompressible perturbations in shaped cross-section plasmas is derived in the Tokamak scaling (epsilon identical to ka approximately=Bperpendicular to /Bz<<1).
Abstract: An energy principle for 'helical' incompressible perturbations in shaped cross-section plasmas is derived in the Tokamak scaling ( epsilon identical to ka approximately=Bperpendicular to /Bz<<1). Two models for the resistivity are used. The resistivity is assumed to be transported either by the fluid or by the magnetic surfaces. In the first case generalized rippling and tearing modes are discovered, while in the latter case the rippling is cancelled in a self-consistent way. The Euler equation for the tearing modes generalizes the previously derived equation for two-dimensional perturbations. It is pointed out that the energy principle cannot be extended to higher orders in epsilon .
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TL;DR: The nonsymmetric eigenvalue problem Ax = λBx is discussed with special emphasis on linear algebra theory, on algorithms implemented for solving large-scale systems, and on interpreting complex spectra obtained in applications from physics and engineering.
99 citations
TL;DR: In this article, the resistive MHD equations are linearized around an equilibrium with cylindrical symmetry and solved numerically as a complex eigenvalue problem, which allows one to solve for very small resistivity γ≈10 -10.
54 citations
TL;DR: In this article, a dispersion relation for the resistive tearing mode is derived in a closed analytical form, where generalized models of the resistivity are considered, and a further transport equation for the electron temperature is assumed in the presented theory.
Abstract: In collision‐dominated plasmas, as well as in plasma systems with anomalous dissipation, the resistivity, in general, depends on the local plasma parameters. Except for some special applications, this dependence is usually not taken into account for the resistive tearing mode. In this paper, a dispersion relation for the resistive tearing mode is derived in a closed analytical form, where generalized models for the resistivity are considered. Because of the dependence on electron temperature, e.g., for the Spitzer resistivity, a further transport equation for the electron temperature is assumed in the presented theory. For applications to more realistic systems, the equilibrium configuration is assumed to be weakly two dimensional. Thus the obtained dispersion relation represents a generalization, which includes a number of results of previous authors. In order to determine this dispersion relation, a constant Ψ approximation is not applied such that the parameter regime of maximum growth rate is covered by the presented theory.
38 citations
TL;DR: In this paper, the authors evaluated the linear tearing mode stability for 1D equilibria in cylindrical geometry with arbitrary current profiles and showed that for hollow currents with two singular q surfaces extremely accurate results showing the coupling influence of both singularities are obtained.
Abstract: The linear tearing mode stability is evaluated for 1D equilibria in cylindrical geometry with arbitrary current profiles. This treatment combines the energy method and the Ritz-Galerkin procedure using finite elements. Particularly for hollow currents with two singular q surfaces extremely accurate results showing the coupling influence of both singularities are obtained.
16 citations
TL;DR: In this article, physically motivated test functions are introduced to simplify the stability functional, which makes its evaluation and minimization more tractable, and the simplified functional reduces to a good approximation of the exact stability functional.
8 citations
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TL;DR: In this article, the stability and radial distribution of linear tearing modes in cylindrical tokamaks of various radial profiles were analyzed and a double tearing mode with two points of discontinuity in the radial magnetic field perturbation was found.
Abstract: Detailed computational results are presented on the stability and radial distribution of linear tearing modes in cylindrical tokamaks of various radial profiles. In the case of a skin‐current profile, a “double tearing mode”, with two points of discontinuity in the radial magnetic field perturbation is found. An analytical method is also derived for comparison of the stability of different radial profiles. It is further shown that the tearing mode can be driven by finite electron viscosity, as well as by the usual finite resistivity mechanism.
509 citations
TL;DR: In this paper, the magnetohydrodynamic stability of a straight plasma column with elliptic cross section, carrying a uniform axial current, is investigated by extremizing the Lagrangian of the system using a natural coordinate system based on the magnetic field lines.
Abstract: The magnetohydrodynamic stability of a straight plasma column with elliptic cross section, carrying a uniform axial current, is investigated by extremizing the Lagrangian of the system using a natural coordinate system based on the magnetic field lines. Stability criteria are derived and growth rates are obtained analytically for systems with a uniform mass density inside the plasma. It is shown that the coupling between kink modes and Alfven waves produced by noncircularity is a destabilizing effect. A technique for solving the problem numerically is also discussed and used to demonstrate the effect of a spatially varying plasma density on the growth rate.
78 citations
TL;DR: Optimized current profiles are shown to be capable of providing simultaneous stability against all resistive kink modes in the tokamak as discussed by the authors. But their performance is limited by the complexity of the current profiles.
Abstract: Optimized current profiles are shown to be capable of providing simultaneous stability against all resistive kink modes in the tokamak.
64 citations
TL;DR: In this article, an energy principle for 2-d resistive instabilities has been found, which leads to a necessary and sufficient condition for stability allowing the use of test functions.
Abstract: An energy principle for 2-d resistive instabilities has been found. It leads to a necessary and sufficient condition for stability allowing the use of test functions. One simple consequence is that the current density in a plasma with arbitrary cross section should not increase to the outside. Otherwise the plasma would be unstable against resistive instabilities.
20 citations
TL;DR: In this paper, an energy principle for the dissipative two-fluid theory in Lagrangian form is given, which represents a necessary and sufficient condition for stability allowing the use of test functions.
Abstract: Abstract An energy principle for the dissipative two-fluid theory in Lagrangian form is given. It represents a necessary and sufficient condition for stability allowing the use of test functions. It is exact for two-dimensional disturbances but is still correct in terms of perturbation theory for long wavelengths along the magnetic field. This may well find application in tokamak plasmas. A discussion of the general case and its relation to the stability of flows in hydrodynamics is given. This energy principle may be applied for estimating the magnitude of residual tearing modes in tokamaks.
15 citations