scholarly journals An Energy Principle for Ideal MHD Equilibria with Flows

2013 ◽  
Author(s):  
Yao Zhou and Hong Qin
Keyword(s):  
Energy Principle ◽  
Ideal Mhd ◽  
Mhd Equilibria

Localized Ideal and Resistive Instabilities in 3-dimensional MHD Equilibria with Closed Field Lines

1990 ◽  
Vol 45 (9-10) ◽  
pp. 1074-1076
Author(s):  
Dario Correa-Restrepo
Keyword(s):  
Radial Direction ◽  
Mhd Equations ◽  
Closed Field ◽  
Stability Criteria ◽  
Energy Principle ◽  
3 Dimensional ◽  
Pressure Surface ◽  
Mhd Equilibria ◽  
Field Lines ◽  
Low Shear

Abstract A class of stability criteria is derived for MHD equilibria with closed field lines. The destabilizing perturbations have finite gradients along the field and are localized around a field line, the localiza-tion being stronger on the pressure surface than in the radial direction. By contrast, in sheared configurations the localization is comparable in both directions. The derived stability criteria are less stringent than those obtained for MHD equilibria with shear for similarly localized perturbations in the limit of low shear. These results, obtained from the energy principle, are a particular case of those obtained by solving the linearized resistive MHD equations with an appropriate ansatz and subsequently taking the limit of vanishing shear and resistivity.

Ballooning Modes in Three-dimensional MHD Equilibria with Shear

1978 ◽  
Vol 33 (7) ◽  
pp. 789-791 ◽  
Author(s):  
D. Correa-Restrepo
Keyword(s):  
Magnetic Field ◽  
Eigenvalue Problem ◽  
Field Line ◽  
Magnetic Field Line ◽  
Three Dimensional ◽  
One Dimensional ◽  
Ideal Mhd ◽  
Mhd Equilibria

Stability with respect to ballooning modes in arbitrary, three-dimensional, ideal MHD equilibria with shear is studied. The destabilizing perturbations considered here have finite gradients along the field and are localized around a closed magnetic field line, the localization being weaker on the surface than transversally to it. This kind of localization allows the problem of stability to be reduced to the solution of a one-dimensional eigenvalue problem.

The only three classes of symmetric MHD equilibria

1980 ◽  
Vol 24 (3) ◽  
pp. 515-518 ◽  
Author(s):  
J. W. Edenstrasser
Keyword(s):  
Space Variable ◽  
Magnetic Axis ◽  
Ideal Mhd ◽  
Mhd Equilibria

Ideal MHD equilibria with an ignorable space variable are investigated. It is shown that only three classes of these symmetric equilibria exist: the systems with a straight, a (cylindric) helical, and a circular magnetic axis.

Computation of Tokamak Equilibria With Steady Flow

1987 ◽  
Vol 42 (10) ◽  
pp. 1154-1166 ◽  
Author(s):  
W. Kerner ◽  
S. Tokuda
Keyword(s):  
Steady Flow ◽  
Stationary Flow ◽  
Plasma Boundary ◽  
Ideal Mhd ◽  
Mhd Equilibria ◽  
Free Plasma

The equations for ideal MHD equilibria with stationary flow are re-examined and addressed as numerically applied to tokamak configurations with a free plasma boundary. Both the isothermal (purely toroidal flow) and the poloidal flow cases are treated. Experiment-relevant states with steady flow (so far only in the toroidal direction) are computed by the modified SELENE40 code.

Computation of ideal MHD equilibria

1976 ◽  
Vol 12 (1) ◽  
pp. 33-44 ◽  
Author(s):  
K. Lackner
Keyword(s):  
Ideal Mhd ◽  
Mhd Equilibria

scholarly journals Magnetohydrodynamic stability of plasmas with ideal and relaxed regions

2009 ◽  
Vol 75 (5) ◽  
pp. 637-659 ◽  
Author(s):  
R. L. MILLS ◽  
M. J. HOLE ◽  
R. L. DEWAR
Keyword(s):  
Lagrange Equation ◽  
Beltrami Equation ◽  
Limit Problem ◽  
Cylindrical Geometry ◽  
Large Solution ◽  
Small Solution ◽  
Energy Principle ◽  
Single Interface ◽  
Ideal Mhd ◽  
The Difference

AbstractA unified energy principle approach is presented for analysing the magnetohydrodynamic (MHD) stability of plasmas consisting of multiple ideal and relaxed regions. The gauge a = ξ × B for the vector potential, a, of linearized perturbations is used, with the equilibrium magnetic field B obeying a Beltrami equation, ∇ × B = αB, in relaxed regions. In a region with such a force-free equilibrium Beltrami field we show that ξ obeys the same Euler–Lagrange equation whether ideal or relaxed MHD is used for perturbations, except in the neighbourhood of the magnetic surfaces where B · ∇ is singular. The difference at singular surfaces is analysed in cylindrical geometry: in ideal MHD only Newcomb's small solutions are allowed, whereas in relaxed MHD only the odd-parity large solution and even-parity small solution are allowed. A procedure for constructing global multi-region solutions in cylindrical geometry is presented. Focusing on the limit where the two interfaces approach each other arbitrarily closely, it is shown that the singular-limit problem encountered previously by Hole et al. in multi-region relaxed MHD is stabilized if the relaxed-MHD region between the coalescing interfaces is replaced by an ideal-MHD region. We then present a stable (k, pressure) phase-space plot, which allows us to determine the form a stable pressure and field profile must take in the region between the interfaces. From this knowledge, we conclude that there exists a class of single-interface plasmas that were found to be stable by Kaiser and Uecker, but are shown to be unstable when the interface is resolved.

An energy criterion for the linear stability of conservative flows

2000 ◽  
Vol 402 ◽  
pp. 329-348 ◽  
Author(s):  
P. A. DAVIDSON
Keyword(s):  
Variational Principle ◽  
Linear Stability ◽  
Energy Criterion ◽  
Conservative System ◽  
Base Flow ◽  
Recirculating Flow ◽  
Ideal Mhd ◽  
Mhd Equilibria ◽  
Euler Flows ◽  
The Stability

We investigate the linear stability of inviscid flows which are subject to a conservative body force. This includes a broad range of familiar conservative systems, such as ideal MHD, natural convection, flows driven by electrostatic forces and axisymmetric, swirling, recirculating flow. We provide a simple, unified, linear stability criterion valid for any conservative system. In particular, we establish a principle of maximum action of the formformula herewhere η is the Lagrangian displacement,e is a measure of the disturbance energy, T and V are the kinetic and potential energies, and L is the Lagrangian. Here d represents a variation of the type normally associated with Hamilton's principle, in which the particle trajectories are perturbed in such a way that the time of flight for each particle remains the same. (In practice this may be achieved by advecting the streamlines of the base flow in a frozen-in manner.) A simple test for stability is that e is positive definite and this is achieved if L(η) is a maximum at equilibrium. This captures many familiar criteria, such as Rayleigh's circulation criterion, the Rayleigh–Taylor criterion for stratified fluids, Bernstein's principle for magnetostatics, Frieman & Rotenberg's stability test for ideal MHD equilibria, and Arnold's variational principle applied to Euler flows and to ideal MHD. There are three advantages to our test: (i) d2T(η) has a particularly simple quadratic form so the test is easy to apply; (ii) the test is universal and applies to any conservative system; and (iii) unlike other energy principles, such as the energy-Casimir method or the Kelvin–Arnold variational principle, there is no need to identify all of the integral invariants of the flow as a precursor to performing the stability analysis. We end by looking at the particular case of MHD equilibria. Here we note that when u and B are co-linear there exists a broad range of stable steady flows. Moreover, their stability may be assessed by examining the stability of an equivalent magnetostatic equilibrium. When u and B are non-parallel, however, the flow invariably violates the energy criterion and so could, but need not, be unstable. In such cases we identify one mode in which the Lagrangian displacement grows linearly in time. This is reminiscent of the short-wavelength instability of non-Beltrami Euler flows.

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