# A Sufficient Condition in Resistive MHD

Abstract: A sufficient stability condition with respect to purely growing modes is derived for resistive MHD. Though it may be, in general, violated, its ability to reduce in the appropriate limits to known necessary and sufficient stability conditions makes it instructive and conceptually useful.

Topics: Dissipative system (51%)

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Abstract: In a general stability condition obtained by the author in a previous work physically motivated test functions are introduced. This leads to simplified versions of the stability functional, which makes its evaluation and minimization more tractable. In the case of special force-free fields the simplified functional reduces to a good approximation of the exact stability functional derived by other means. It turns out that in this case the condition is sufficient for nonlinear stability also.

8 citations

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Abstract: The general sufficient condition obtained by the author in a previous work is analysed with respect to its “nearness” to necessity. It is found that for physically reasonable approximations the condition is in some sense necessary and sufficient for stability against all modes. This together with Hermiticity makes its analytical and numerical evaluation worth doing for the optimization of magnetic configurations.

5 citations

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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.

3 citations

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Abstract: A special Hopf bifurcation in resistive magnetohydrodynamics can easily be identified by starting from a particularly suitable form of the linearized equations which was previously introduced by the author. As usual, the bifurcation can lead to nonlinear periodic oscillations. These oscillations can eventually be destroyed, which raises the question of the nature of the ultimate attractors.

3 citations

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Abstract: A stability condition in resistive magnetohydrodynamics in the presence of equilibrium mass flow is obtained. This is compared with the stability condition in ideal magnetohydrodynamics in the presence of a resistive wall.

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Abstract: An energy principle for 2-d resistive instabilities has been found. It leads to a necessary and sufficient condition for stability allowing the use of test functions. One simple consequence is that the current density in a plasma with arbitrary cross section should not increase to the outside. Otherwise the plasma would be unstable against resistive instabilities.

20 citations

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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 .

15 citations

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Abstract: A general sufficient condition for stability against purely growing modes is derived. It is a quadratic symmetric form very suitable to numerical treatment and minimization. It has potential applications in plasma stability with non-ideal equations and real geometry.

5 citations