Simplified versions of a stability condition in resistive MHD
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.
About: This article is published in Physics Letters A.The article was published on 1992-10-05. It has received 8 citations till now. The article focuses on the topics: Stability criterion & Stability (learning theory).
Citations
More filters
TL;DR: In this paper, the relation between the helical bifurcation of a Taylor relaxed state (a Beltrami equilibrium) and a tearing mode was analyzed in a Hamiltonian framework.
Abstract: The relation between the helical bifurcation of a Taylor relaxed state (a Beltrami equilibrium) and a tearing mode is analyzed in a Hamiltonian framework. Invoking an Eulerian representation of the Hamiltonian, the symplectic operator (defining a Poisson bracket) becomes non-canonical, i.e. the symplectic operator has a nontrivial cokernel (dual to its nullspace), foliating the phase space into level sets of Casimir invariants. A Taylor relaxed state is an equilibrium point on a Casimir (helicity) leaf. Changing the helicity, equilibrium points may bifurcate to produce helical relaxed states; a necessary and sufficient condition for bifurcation is derived. Tearing yields a helical perturbation on an unstable equilibrium, producing a helical structure approximately similar to a helical relaxed state. A slight discrepancy found between the helically bifurcated relaxed state and the linear tearing mode viewed as a perturbed, singular equilibrium state is attributed to a Casimir element (named ?helical flux?) pertinent to a ?resonance singularity? of the non-canonical symplectic operator. While the helical bifurcation can occur at discrete eigenvalues of the Beltrami parameter, the tearing mode, being a singular eigenfunction, exists for an arbitrary Beltrami parameter. Bifurcated Beltrami equilibria appearing on the same helicity leaf are isolated by the helical-flux Casimir foliation. The obstacle preventing the tearing mode to develop in the ideal limit turns out to be the shielding current sheet on the resonant surface, preventing the release of the ?potential energy?. When this current is dissipated by resistivity, reconnection is allowed and tearing instability occurs. The ?? criterion for linear tearing instability of Beltrami equilibria is shown to be directly related to the spectrum of the curl operator.
31 citations
TL;DR: In this paper, a generalization of the theory of singular differential equations is presented, where singular Casimir elements stemming from this singularity are unearthed using a generalisation of the functional derivative that occurs in the Poisson bracket.
10 citations
Posted Content•
TL;DR: In this article, a physical interpretation of Casimir elements as adiabatic invariants was proposed, based on coarse-grained microscopic angle variables, and a macroscopic hierarchy on which the separated action variables become adiabilistic invariants.
Abstract: The infinite-dimensional mechanics of fluids and plasmas can be formulated as "noncanonical" Hamiltonian systems on a phase space of Eulerian variables. Singularities of the Poisson bracket operator produce singular Casimir elements that foliate the phase space, imposing topological constraints on the dynamics. Here we proffer a physical interpretation of Casimir elements as \emph{adiabatic invariants} ---upon coarse graining microscopic angle variables, we obtain a macroscopic hierarchy on which the separated action variables become adiabatic invariants. On reflection, a Casimir element may be \emph{unfrozen} by recovering a corresponding angle variable; such an increase in the number of degrees of freedom is, then, formulated as a \emph{singular perturbation}. As an example, we propose a canonization of the resonant-singularity of the Poisson bracket operator of the linearized magnetohydrodynamics equations, by which the ideal obstacle (resonant Casimir element) constraining the dynamics is unfrozen, giving rise to a tearing-mode instability.
10 citations
Additional excerpts
...Evidently, μ ≥ λ1 destroys the coercivity of 〈Hμũ, ũ〉 with respect to the norm ‖ũ‖ (2), violating the sufficient condition of stability [24] (see also [9, 8, 14, 19, 18])....
[...]
TL;DR: In this article, the stability of generalized Trkal flows is investigated and a sufficient condition for nonlinear stability is derived for low velocities, which reduces essentially to the nonlinear sufficient stability condition for force free fields, previously found by the author.
Abstract: Force free fields can be generalized to Trkal flows in magnetohydrodynamics if a special velocity field parallel to the magnetic field is introduced. Such flows decay exponentially in time in case a constant viscosity and a constant resistivity are added. The stability of generalized Trkal flows is investigated and a sufficient condition for nonlinear stability is derived. For low velocities, this condition reduces essentially to the nonlinear sufficient stability condition for force free fields, previously found by the author.
9 citations
10 Feb 2014
TL;DR: In this article, a physical interpretation of Casimir elements as adiabatic invariants is proposed, based on coarse graining microscopic angle variables, and a macroscopic hierarchy on which the separated action variables become adiabel invariants.
Abstract: The infinite-dimensional mechanics of fluids and plasmas can be formulated as “noncanonical” Hamiltonian systems on a phase space of Eulerian variables. Singularities of the Poisson bracket operator produce singular Casimir elements that foliate the phase space, imposing topological constraints on the dynamics. Here we proffer a physical interpretation of Casimir elements as adiabatic invariants —upon coarse graining microscopic angle variables, we obtain a macroscopic hierarchy on which the separated action variables become adiabatic invariants. On reflection, a Casimir element may be unfrozen by recovering a corresponding angle variable; such an increase in the number of degrees of freedom is, then, formulated as a singular perturbation. As an example, we propose a canonization of the resonant-singularity of the Poisson bracket operator of the linearized magnetohydrodynamics equations, by which the ideal obstacle (resonant Casimir element) constraining the dynamics is unfrozen, giving rise to a tearing-mode instability.
7 citations
References
More filters
TL;DR: In this paper, it was shown that force-free magnetic fields in resistive magnetohydrostatics are those for which α is constant in space and time, where α is defined by ▽ × B = α B.
35 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, 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