On purely growing modes in real systems
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.
About: This article is published in Physics Letters A.The article was published on 1983-03-14. It has received 5 citations till now. The article focuses on the topics: Plasma stability & Quadratic equation.
Citations
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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, 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.
5 citations
TL;DR: In this paper, a formalism for the construction of energy principles for dissipative systems is presented, and the stability problem of resistive magnetohydrodynamic (MHD) is addressed: the energy principle for ideal MHD is generalized and the Stability criterion by Tasso [Phys. Lett. 147, 28 (1990)] is shown to be necessary in addition to sufficient for real growth rates.
Abstract: A formalism for the construction of energy principles for dissipative systems is presented. It is shown that dissipative systems satisfy a conservation law for the bilinear Hamiltonian provided the Lagrangian is time invariant. The energy on the other hand, differs from the Hamiltonian by being quadratic and by having a negative definite time derivative (positive power dissipation). The energy is a Lyapunov functional whose definiteness yields necessary and sufficient stability criteria. The stability problem of resistive magnetohydrodynamic (MHD) is addressed: the energy principle for ideal MHD is generalized and the stability criterion by Tasso [Phys. Lett. 147, 28 (1990)] is shown to be necessary in addition to sufficient for real growth rates. An energy principle is found for the inner layer equations that yields the resistive stability criterion D-R<0 in the incompressible limit, whereas the tearing mode criterion Delta'<0 is shown to result from the conservation law of the bilinear concomitant in the resistive layer. (C) 1997 American Institute of Physics.
4 citations
TL;DR: Three theorems presented in this paper show that under some conditions, Π>0 makes the mode stable, while Π<0 make the mode unstable.
Abstract: In this paper, we point out that for a static system the eigenvalue integral relation (iσ)2I+iσΦ+Σ=0 derived from an eigenequation\(\hat M\psi = [(i\sigma )^2 \hat L_2 + (i\sigma )\hat L_1 + \hat L_0 ]\psi (r) = 0\) only gives the sufficiency of condition gZ>0 for the stability of a certain mode, but it can not provide the necessity to the condition in general. Three theorems presented in this paper show that under some conditions, Π>0 makes the mode stable, while Π<0 makes the mode unstable.
2 citations
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)].
References
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TL;DR: In this article, it was shown that for a large class of Lagrangian systems, a stable configuration of the system which is linearly stable under a small variation of the steady state, to all orders of perturbation theory in the variation.
Abstract: It is shown for a large class of Lagrangian systems that a steady state configuration of the system which is linearly stable remains so under a small variation of the steady state, to all orders of perturbation theory in the variation. The principle is applied to two special systems: an incompressible ideal fluid, and a gas of charged particles interacting through their average fields.
33 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 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
TL;DR: In this article, the LRP 90 Reference CRPP-REPORT-1975-002 Record created on 2008-04-18, modified on 2017-05-12 was used.
13 citations