Parametric excitation of oscillations in the hybrid resonance region

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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“…As shown in [13], the nonlinear longitudinal oscillations in a weakly inhomogeneous plasma are described by the following system of equations…”

Section: Rp-1/a13-all Qlmentioning

confidence: 99%

“…In paper [13] an assumption was made about the absence of internal resonance between the electron natural oscillations (co(+)4:nco(-), n is an integer). In this case the electron oscillations, excited at the frequency co(+) can not cause the build-up of oscillations at the frequency o9.…”

Section: Rp-1/a13-all Qlmentioning

confidence: 99%

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

1973

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“…In earlier papers [11,12] we investigated the longitudinal oscillations of an inhomogeneous magneto-active plasma in the regions of highfrequency and low-frequency hybrid resonances. Fundamentally, the following results were obtained: a) in the case of large amplitudes of electron and ion oscillations, the strongly nonlinear oscillations of an inhomogeneous plasma (its density n = n(r)) are no longer single-stream and laminar: the plasma motion becomes multistream (i. e. turbulent), b) the feasibility of a parametric build-up of oscillations at a basic (second) sub-harmonic of the external highfrequency was pointed out in Ref.…”

Section: Introductionmentioning

confidence: 99%

“…In these papers [11,12], however, it was overlooked that allowance for the_inhomogeneity of the external magnetic field (Bo = B 0 (r)) may also produce some specific effects. In particular, in this paper it is shown that the dependence Bo(?)…”

Section: Introductionmentioning

confidence: 99%

Parametric instability in the region of low-frequency hybrid resonances

Demchenko¹,

El-Naggar

1973

Nucl. Fusion

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The non-linear longitudinal oscillations of a homogeneous plasma embedded in a weakly inhomogeneous magnetic field in the presence of an external high frequency field are investigated. The non-linearity of oscillations is manifested in two ways: (a) allowance for the terms (V⃗∇)V⃗ in the equations of motion and (b) dependence of the plasma natural oscillation frequency on the displacement amplitude (or, otherwise, the loss of isochronity). It is demonstrated that the terms (V⃗∇)V⃗ do not restrict the amplitude increase in the regions of hybrid resonances. The non-linear terms arising in the equations of motion governed by the inhomogeneity of B⃗0, item (a), restrict the oscillation amplitude in the resonant areas and, item (b), lead to a parametric build-up of oscillations. The conditions for the parametric instability and the growth rates of unstable oscillations are determined.

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“…The excitation of ultra-and sub-harmonic oscillations, however, were not treated in the above papers. The possibility of parametric excitation of plasma oscillations^ with a frequency equal to the double hybrid frequency (a special case of ultraharmonic oscillations) was noted in Demchenko & El-Nagger (1971). Here the stability of longitudinal oscillations in a magnetoactive plasma is investigated.…”

Section: Introductionmentioning

confidence: 99%