On a general stability condition in resistive MHD
TL;DR: In this article, the general sufficient condition obtained by the author in a previous work is analyzed with respect to its "nearness" to necessity, and it is found that for physically reasonable approximations the condition is in some sense necessary and sufficient for stability against all modes.
About: This article is published in Physics Letters A.The article was published on 1991-12-30 and is currently open access. It has received 5 citations till now. The article focuses on the topics: Stability criterion.
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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
TL;DR: A special Hopf bifurcation in resistive magnetohydrodynamics can be identified by starting from a particularly suitable form of the linearized equations which was previously introduced by the author as mentioned in this paper.
3 citations
TL;DR: In this article, a stability condition in resistive magnetohydrodynamics in the presence of equilibrium mass flow is obtained, which is compared with the stability condition of ideal magnetodynamic systems in the case of a resistive wall.
TL;DR: In this paper, a general stability condition for a class of real systems, occurring especially in plasma physics, is proved to persist to second order, despite the addition of several antisymmetric operators of first order in the linearized stability equation.
Abstract: Two general problems related to resistive magnetohydrodynamic stability are addressed in this paper. First, a general stability condition previously derived by the author for a class of real systems, occurring especially in plasma physics, is proved to persist to second order, despite the addition of several antisymmetric operators of first order in the linearized stability equation. Second, for a special but representative choice of the stability operators, a nonperturbative analysis demonstrates the existence of a critical density for the appearance of an overstability and the connected Hopf bifurcation, as suggested in a previous paper [Phys. Lett. A 180, 257 (1993)].
01 Jan 1996
TL;DR: A brief summary of energy methods for linear stability in dissipative magnetohydrodynamics is given in this article, where the methods are equally efficient for fixed and free boundary problems.
Abstract: A brief summary of energy methods for linear stability in dissipative magnetohydrodynamics is given. In this case, the methods are equally efficient for fixed and free boundary problems. Linear asymptotic stability has implications in nonlinear stability, at least for a modest but finite level of perturbations.
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01 Jan 1985
TL;DR: In this paper, the authors consider the problem of finite element decomposition in the context of differential eigenvalue problems and apply it to test cases of continuous spectrum and non-conformity.
Abstract: 1. Finite Element Methods for the Discretization of Differential Eigenvalue Problems.- 1.1 A Classical Model Problem.- 1.1.1 Exact Problem.- 1.1.2 Approximate Problem.- 1.1.3 Questions on Numerical Stability.- 1.2 A Non-Standard Model Problem.- 1.2.1 Exact Problem.- 1.2.2 Conforming "Polluting" Approximations.- 1.2.3 "Non-Polluting" Conforming Approximation.- 1.2.4 Non-Conforming Approximation.- 1.3 Spectral Stability.- 1.3.1 General Considerations.- 1.3.2 Stability Conditions.- 1.3.3 Order of Convergence.- 1.4 Finite Elements of Order p.- 1.4.1 Discontinuous Finite Elements S0p.- 1.4.2 Continuous Finite Elements S1p (Lagrange Elements).- 1.4.3 C1-Finite Elements S2p (Hermite Elements).- 1.4.4 Application to the Model Problems.- 1.4.5 Non-Conformmg Lagrange Elements.- 1.4.6 Non-Conforming Hermite Elements with Collocation.- 1.5 Some Comments.- 2. The Ideal MHD Model.- 2.1 Basic Equations.- 2.2 Static Equilibrium.- 2.3 Linearized MHD Equations.- 2.4 Variational Formulation.- 2.5 Stability Considerations.- 2.6 Mechanical Analogon.- 3. Cylindrical Geometry.- 3.1 MHD Equations in Cylindrical Geometry.- 3.1.1 The AGV and Hain-Lust Equations.- 3.1.2 Continuous Spectrum.- 3.1.3 An Analytic Solution.- 3.2 Six Test Cases.- 3.2.1 Test Case A: Homogeneous Currentless Plasma Cylinder..- 3.2.2 Test Case B: Continuous Spectrum.- 3.2.3 Test Case C: Particular Free Boundary Mode.- 3.2.4 Test Case D: Unstable Region for k = -0.2, m = 1.- 3.2.5 Test Case E: Unstable Region for k = -0.2, m = 2.- 3.2.6 Test Case F: Internal Kink Mode.- 3.3 Approximations.- 3.3.1 Conforming Finite Elements.- 3.3.2 Non-Conforming Finite Elements.- 3.4 Polluting Finite Elements.- 3.4.1 Hat Function Elements.- 3.4.2 Application to Test Case A.- 3.5 Conforming Non-Polluting Finite Elements.- 3.5.1 Linear Elements.- 3.5.2 Quadratic Elements.- 3.5.3 Third-Order Lagrange Elements.- 3.5.4 Cubic Hermite Elements.- 3.6 Non-Conforming Non-Polluting Elements.- 3.6.1 Linear Elements.- 3.6.2 Quadratic Elements.- 3.6.3 Lagrange Cubic Elements.- 3.6.4 Hermite Cubic Elements with Collocation.- 3.7 Applications and Comparison Studies (with M.-A. Secretan).- 3.7.1 Application to Test Case A.- 3.7.2 Application to Test Case B.- 3.7.3 Application to Test Case C.- 3.7.4 Application to Test Case F.- 3.8 Discussion and Conclusion.- 4. Two-Dimensional Finite Elements Applied to Cylindrical Geometry.- 4.1 Conforming Finite Elements.- 4.1.1 Conforming Triangular Finite Elements.- 4.1.2 Conforming Lowest-Order Quadrangular Finite Elements.- 4.2 Non-Conforming, Finite Hybrid Elements.- 4.2.1 Finite Hybrid Elements Formulation.- 4.2.2 Lowest-Order Finite Hybrid Elements.- 4.2.3 Application to the Test Cases.- 4.2.4 Explanation of the Spectral Shift.- 4.2.5 Convergence Properties.- 4.3 Discussion.- 5. ERATO: Application to Toroidal Geometry.- 5.1 Static Equilibrium.- 5.1.1 Grad-Schluter-Shafranov Equation.- 5.1.2 Weak Formulation.- 5.2 Mapping of (?, ?) into (?, ?) Coordinates in ?p.- 5.3 Variational Formulation of the Potential and Kinetic Energies..- 5.4 Variational Formulation of the Vacuum Energy.- 5.5 Finite Hybrid Elements.- 5.6 Extraction of the Rapid Angular Variation.- 5.7 Calculation of ?-Limits (with F. Troyon).- 6. HERA: Application to Helical Geometry (Peter Merkel, IPP Garching).- 6.1 Equilibrium.- 6.2 Variational Formulation of the Stability Problem.- 6.3 Applications.- 6.3.1 Straight Heliac.- 6.3.2 Straight Heliotron Equilibria.- 6.3.3 Large-k Ballooning Modes.- 6.3.4 Conclusion.- 7. Similar Problems.- 7.1 Similar Problems in Plasma Physics.- 7.1.1 Resistive Spectrum in a Cylinder.- 7.1.2 Non-Linear Plasma Wave Equation (with M. C. Festeau-Barrioz).- 7.1.3 Alfven and ICRF Heating in a Tokamak (with K. Appert, T. Hellsten, J. Vaclavik, and L. Villard).- 7.2 Similar Problems in Other Domains.- 7.2.1 Stability of a Compressible Gas in a Rotating Cylinder..- 7.2.2 Normal Modes in the Oceans.- Appendices.- A: Variational Formulation of the Ballooning Mode Criterion.- B.1 The Problem.- B.2 Two Numberings of the Components.- B.3 Resolution for Numbering (D1).- B.4 Resolution for Numbering (D2).- B.5 Higher Order Finite Elements.- C: Organization of ERATO.- D: Listing of ERATO 3 (with R. Iacono).- References.
90 citations
TL;DR: In this paper, the stability of multihelical tearing modes in tokamaks with shaped cross sections is determined numerically, and the method allows inclusion of a large number of singular surfaces resolved with high accuracy.
Abstract: The stability of multihelical tearing modes in tokamaks with shaped cross sections is determined numerically. The method allows inclusion of a large number of singular surfaces resolved with high accuracy. Poloidal and radial couplings are discussed and the convergence is well understood. Modes of high poloidal m number are found to be unstable for typical equilibria. Completely stable current distributions have been constructed for D-shaped plasmas.
19 citations
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 .
17 citations
TL;DR: In this article, a sufficient stability condition with respect to purely growing modes is derived for resistive MHDs, which is a necessary and sufficient condition for purely growing MHD.
7 citations
TL;DR: In this article, a general sufficient condition for stability against purely growing modes is derived, which is a quadratic symmetric form very suitable to numerical treatment and minimization, and has potential applications in plasma stability with non-ideal equations and real geometry.
5 citations