Variational principles for equilibrium states with plasma flow

Hameiri, Eliezer

doi:10.1063/1.872995

Cited by 70 publications

(75 citation statements)

References 46 publications

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“…The two last integrals were deduced for the particular case of incompressible flows in the papers [8,9], cf. also papers [18,19,20] where the vorticity and helicity analogs were obtained for the MHD flows with the specific spatial symmetry.…”

Section: Variational Principle and Canonical Variablesmentioning

confidence: 99%

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Canonical description of ideal magnetohydrodynamic flows and integrals of motion

Kats

2004

Phys. Rev. E

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In the framework of the variational principle the canonical variables describing magnetohydrodynamic (MHD) flows of general type (i.e., with spatially varying entropy and nonzero values of all topological invariants) are introduced. It is shown that the velocity representation of the Clebsch type following from the variational principle with constraints is equivalent to that resulting from the generalization of the Weber transformation performed in the paper for the case of arbitrary MHD flows. Using such complete velocity representation enables us not only to describe the general type flows in terms of single-valued functions, but also to solve the intriguing problem of the "missing" MHD integrals of motion. The set of hitherto known MHD local invariants and integrals of motion appears to be incomplete: for the vanishing magnetic field it does not reduce to the set of the conventional hydrodynamic invariants. And if the analogs of the vorticity and helicity were discussed earlier for the particular cases, the analog of Ertel invariant has been so far unknown. It is shown that all "missing" invariants are expressed in terms of the decomposition of the velocity representation into the "hydrodynamic" and "magnetic" parts. In spite of the nonunique character of such representation it is shown that there exists a natural restriction of the gauge transformations set allowing one to make the invariants gauge independent. It is found that on the basis of the new invariants introduced a wide set of high-order invariants can be constructed. The new invariants are relevant both for the deeper insight into the problem of the topological structure of the MHD flows as a whole and for the examination of the stability problems. The additional advantage of the proposed approach is that it enables one to deal with discontinuous flows, including all types of possible breaks.

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Section: Variational Principle and Canonical Variablesmentioning

confidence: 99%

“…The related quantities were mentioned for the specific cases of symmetric flows in the works [18,19,20], the vorticity and helicity invariants for the incompressible flows have been obtained recently in Refs. [8,9].…”

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confidence: 99%

Canonical description of ideal magnetohydrodynamic flows and integrals of motion

Kats

2004

Phys. Rev. E

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“…[13] including a couple of additional issues: the impact of flow on the axial current density and the role of the individual terms of a quantity A involved in the stability condition [Eqs. (14)(15)(16)(17)(18)(19) of Sec. III].…”

Section: Introductionmentioning

confidence: 99%

Magnetohydrodynamic counter-rotating vortices and synergetic stabilizing effects of magnetic field and plasma flow

Throumoulopoulos

Tasso

2010

Physics of Plasmas

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A nonlinear two-dimensional steady state solution in the framework of hydrodynamics describing a row of periodic counter-rotating vortices is extended to the magnetohydrodynamic (MHD) equilibrium equation with incompressible flow of arbitrary direction. The extended solution covers a variety of equilibria because four surface quantities remain free. Similar to the case of the MHD cat-eyes equilibrium [Throumoulopoulos et al., J. Phys. A: Math. Theor. 42, 335501 (2009)] and unlike linear equilibria, the flow has a strong impact on isobaric surfaces by forming pressure islands located within the counter-rotating vortices even for values of β (defined as the ratio of the thermal pressure over the external axial magnetic-field pressure) on the order of 0.01. Also, the axial current density is appreciably modified by the flow. Furthermore, a magnetic-field-aligned flow of experimental fusion relevance, i.e for Alfvén Mach numbers of the order of 0.01, and the flow shear in combination with the variation of the magnetic field perpendicular to the magnetic surfaces have significant stabilizing effects potentially related to the equilibrium nonlinearity. The stable region is enhanced by an external axial magnetic field.

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“…The asymptotic boundary condition v ∞ B ∞ enhances the probability that the stationary flows are stable, at least in the ideal region (see, e.g. Hameiri, 1998). The assumption that E z = 0, or better to say E z → 0, does not alter the results.…”

Section: The Basic Incompressible Resistive Mhd Equations and Assumptmentioning

confidence: 99%

Topological skeleton of the 2-D slightly non-ideal MHD system close to X-type magnetic null points – an analysis of the general solution for the generic case

2012

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Abstract. The appearance of eruptive space plasma processes, e.g. in eruptive flares as observed in the solar atmosphere, is usually assumed to be caused by magnetic reconnection, often connected with singular points of the magnetic field.We are interested in the general relation between the eigenvalues of the Jacobians of the plasma velocity and the magnetic field and their relation to the shape of a spatially varying, localized non-idealness or resistivity, i.e. we are searching for the general solution. We perform a local analysis of almost all regular, generic, structurally stable non-ideal or resistive MHD solutions. Therefore we use Taylor expansions of the magnetic field, the velocity field and all other physical quantities, including the non-idealness, and with the method of comparison of coefficients, the non-linear resistive MHD system is solved analytically, locally in a close vicinity of the null point.We get a parameterised general solution that provides us with the topological and geometrical skeleton of the resistive MHD fields. These local solutions provide us with the "roots" of field and streamlines around the null points of basically all possible 2-D reconnection solutions. We prove mathematically that necessarily, the flow close to the magnetic X-point must also be of X-point type to guarantee positive dissipation of energy and annihilation of magnetic flux. We also prove that, if the non-idealness has only a onedimensional, sheet-like structure, only one separatrix line can be crossed by the plasma flow, similar to known reconnective annihilation solutions.

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Variational principles for equilibrium states with plasma flow

Cited by 70 publications

References 46 publications

Canonical description of ideal magnetohydrodynamic flows and integrals of motion

Canonical description of ideal magnetohydrodynamic flows and integrals of motion

Magnetohydrodynamic counter-rotating vortices and synergetic stabilizing effects of magnetic field and plasma flow

Topological skeleton of the 2-D slightly non-ideal MHD system close to X-type magnetic null points – an analysis of the general solution for the generic case

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