Dynamic Stabilization of the Screw Pinch

Berge, Gerhard; Freidberg, J.P.

doi:10.1063/1.1693539

Cited by 2 publications

(4 citation statements)

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“…We denote the roots of Eq. P(bl)=O (bll __<bl2__< b13 ) through bl (i= 1,2,3). Since sn (u; k) is a periodic function of the variavle u with a period of 4K(k) (K(k) is a total elliptic integral of the first kind), from relation (2.18) follows a conclusion about a periodic nature of the energy transformation process.…”

Section: Initial Equation Nonlinear Interaction Of Two Modesmentioning

confidence: 84%

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To the theory of nonlinear mode coupling in dynamic stabilization of MHD modes

1973

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The nonlinear interaction of oscillation modes is investigated on the basis of Lagrangian formalism. Equations describing the changes of the bound mode amplitudes versus time, are obtained. It is shown that the energy transformation between different modes is of a periodic nature: if in the initial moment of time an appreciable part of the energy is contained, for example, in the m-th mode, then after aperiod of time T t (called a time of nonlinear interaction) the energy will be transformed to the n-th mode. Expressions for T t for cases with the interaction of two and three modes are obtained. As a particular case the process of nonlinear interaction of the electron "transverse" and "longitudinal" oscillations in the highfrequency hybrid resonance region of a "weakly" inhomogeneous plasma was investigated. IntroductionAt present the methods of plasma dynamic stabilization draw a great attention of both theoreticians and experimenters (see, e.g. [1-6] and vast References listed therein). This interest can be explained by the fact that such methods favour an efficient stabilization of the most dangerous from a standpoint of the plasma long confinement, "sausage-type" and "kink-type" hydrodynamic instabilities (unstable oscillations with m = (0,1), respectively).However, in the known theoretical papers the analyisys of the plasma configuration stability with respect to small perturbations is based on the known procedure of linearization of the MHD equations. Proceeding from the linearized equations one can determine conditions for plasma stability with respect to perturbations only with an infinitely small amplitude. What concerns perturbations with sufficiently large amplitudes, in this case the results of the linear stability theory can not be applied, even if for the reason, that in the nonlinear approximation the oscillation modes interact effectively. The nonlinear interaction of oscillations leads to the transformation of energy from one mode to another one. Under certain conditions this effect favours the stabilization of the unstable oscillation. In fact, assume for definiteness, that oscillation with m = 0 is unstable one (" sausage-type" instability) whereas oscillations with m_-__ 1 are stable ones. The linear theory defines such a state as "absolutely un-* On leave from the Atomic Energy State Committee, USSR.

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Section: Initial Equation Nonlinear Interaction Of Two Modesmentioning

confidence: 84%

“…[1][2][3][4][5][6] and vast References listed therein). This interest can be explained by the fact that such methods favour an efficient stabilization of the most dangerous from a standpoint of the plasma long confinement, "sausage-type" and "kink-type" hydrodynamic instabilities (unstable oscillations with m = (0,1), respectively).…”

Section: Introductionmentioning

confidence: 99%

To the theory of nonlinear mode coupling in dynamic stabilization of MHD modes

1973

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“…In the longitudinal scheme the oscillatory component in the magnetic field is by definition parallel to the main field and in the transverse scheme it is orthogonal to the main field. However, the usefulness of this distinction is somewhat doubtful because it appears that in many cases [35][36][37][38] a combination of the two schemes is preferable and necessary. We shall then turn to the discussion of more specific problems.…”

Section: Dynamic Stabilization Of Plasmasmentioning

confidence: 99%

“…In recent theoretical work by Berge [37], the possibility of producing a dynamic equilibrium in a toroidal geometry by oscillating currents in S. = 1 windings was studied. These currents produce forces which in the main make the plasma column move in a helical shape with constant cross-section.…”

Section: Bergementioning

confidence: 99%