Equilibrium and stability of MHD-fluids by dynamic techniques

Berge, Gerhard

doi:10.1088/0029-5515/12/1/010

Cited by 28 publications

(7 citation statements)

References 57 publications

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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: 95%

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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: 95%

“…[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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“…Section II below reviews the basics of this theory. There are, however, a few cases of vibrational stabilization of distributed parameter systems (Berge, 1972;Chelomey, 1956;Meerkov and Shapiro, 1976;Osovets, 1974;and Wolf, 1969); these cases are discussed in Section III.…”

Section: I;mentioning

confidence: 99%

Method of vibrational control in the problem of stabilization of chemical reactors. Progress report, July 1, 1981-June 30, 1982

Meerkov¹

1982

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A problem of vibrational stabilization of systems described by partial differential equations is considered. After reviewing the vibrational control theory for lumped parameter systems, a few cases of vibrational stabilization of distributed parameter systems are discussed. A condition of vibrational stabilizabi li ty for a class of linear partial differential equations is given and an example of vibrational stabilization in a nonlinear situation (Benard instability) is presented.

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“…Methods of dynamic and feedback stabilization [ 1J of such instabilities have, therefore, become very important, in the last few years. Berge [ 2] has given an excellent review of dynamic stabilization techniques involving the use of oscillating magnetic fields in the axial (or toroidal) and/or poloidal components.…”

Section: Introductionmentioning

confidence: 99%

“…Most of the papers reviewed by Berge [ 2] consider oscillating fields with frequencies much larger than the ion-acoustic frequency kc s . In this case, there is hardly any coupling of energy to the natural ion-acoustic waves in the plasma.…”

Section: Introductionmentioning

confidence: 99%

Excitation and suppression of kink modes by coupling to ion-acoustic waves

1975

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The stability of kink modes in a cylindrical plasma column under the influence of an oscillating magnetic field applied in the azimuthal direction is studied. The frequency of the applied field is chosen close to the ion-acoustic frequency. It is found that coupling to damped ion-acoustic waves induces a small real frequency to the kink modes which can significantly affect their growth rates. Conditions for parametrically suppressing (or exciting) these modified kink modes are specified.

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Equilibrium and stability of MHD-fluids by dynamic techniques

Cited by 28 publications

References 57 publications

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

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

Method of vibrational control in the problem of stabilization of chemical reactors. Progress report, July 1, 1981-June 30, 1982

Excitation and suppression of kink modes by coupling to ion-acoustic waves

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