Kinetic theory of the diochotron instability

Nocentini, A.; Berk, H. L.; Sudan, R. N.

doi:10.1017/s0022377800003858

Cited by 21 publications

(10 citation statements)

References 8 publications

(3 reference statements)

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“…Before coming to the stability problem, it should be pointed out that the zero radial thickness assumption for equilibria in I and here, is a reasonable modelling of a ring whose z-extension 2z 0 is much larger than its r-extension 2r 0 . In particular, there is no contradiction with the result of Nocentini, Berk & Sudan (1968) of a minimum possible non-zero ring thickness. This is because there are ions as well as electrons in the present problem, and because the ^-velocity of the electrons here is relativistic.…”

Section: Introductionmentioning

confidence: 60%

Linear stability analysis of a class of solutions of the filament model for a stationary field electron ring accelerator

Chen

Ehrman

1976

J. Plasma Phys.

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The stability of two of the short time scale solutions obtained in the filament model for a stationary field electron ring accelerator is studied by means of a linearized perturbation on the electron and ion distribution functions. The Vlasov equations for the electrons and ions are coupled. The coupling of these equations results in a Fredholm integral equation which can be converted into an infinite system of linear equations. A stability criterion is then obtained from the zeros of the truncated determinants of the coefficients of the infinite system. One of the equilibrium solutions, for which the ion density in phase space is a monotonic non-increasing function of the unperturbed ion Hamiltonian, is entirely stable. The other solution, for which this monotonicity does not hold, may be unstable, but turns out to be stable provided that both the non-monotonicity parameter ∈ and the ratio of the ion period to the electron period are not too large. In the nonmonotonic case, the ion density in physical space also does not drop off to zero monotonically from the centre of the ring. For the unstable cases, the instability is driven by a ‘spike’ in the phase-space ion density located at the maximum value of the unperturbed ion Hamiltonian.

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Section: Introductionmentioning

confidence: 60%

Linear stability analysis of a class of solutions of the filament model for a stationary field electron ring accelerator

Chen

Ehrman

1976

J. Plasma Phys.

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“…Summarizing, the hydrodynamic Jeans-type instability arises because of a`thermodynamic non-uniformity' of the stellar disc ± the system is not sufficiently`hot' in the equatorial plane, equation (43). (According to Griv & Peter 1996a, in plasma physics an instability of non-axisymmetric perturbations of the Jeans type is known as the negative-mass instability of a relativistic charged particle ring or the diocotron instability of a non-relativistic ring that caused azimuthal clumping of beams in synchrotrons, betatrons and mirror machines; see Nocentini, Berk &Sudan 1968 andDavidson 1992 for an explanation of these instabilities of systems of electrically charged particles.) Griv & Peter (1996a) have pointed out that the hypothesis that density waves are excited by the classical Jeans instability in a given disc of stars during many rotations of the system encounters a severe challenge.…”

Section: ã Jkj Fxxmentioning

confidence: 99%

Small-amplitude density waves in galactic discs with radial gradients

Griv¹,

Yuan²,

Gedalin³

1999

Monthly Notices of the Royal Astronomical Society

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Stellar discs of highly flattened giant galaxies, including that of the Milky Way, are studied by linear theory to determine the stability of such discs against small‐amplitude gravity perturbations. In order to understand the physics of the problem better, the simplest theoretical model is applied. That is, the local disc is studied by employing the method of particle orbit theory. In this purely Lagrangian method, an approximate solution of the Newtonian equations of the motion of stars is obtained using a general technique based upon the perturbation method. In the second order of Lindblad’s epicyclic theory, expressions are found for the unperturbed motions of stars in a stationary system with an axially symmetric mass distribution. Then, expressions are found for the perturbed motions of stars when the small non‐axisymmetric gravity perturbation is additionally taken into account. The perturbed terms are obtained as second‐order oscillations. To describe the ordered behaviour of a medium near its quasi‐equilibrium state, these equations for the trajectories of stars are used to obtain the dispersion relation that connects the frequency of excited collective oscillations with the wavenumber throughout the disc, including resonant regions. Using the dispersion relation, a new class of gradient microinstabilities of a non‐uniformly rotating disc inherent in an inhomogeneous system is discussed. The Landau mechanism of excitation of spiral density waves works at the corotation resonance between stars and hydrodynamically (Jeans) stable perturbations (e.g. those produced by a bar‐like structure, a spontaneous perturbation and/or a companion galaxy). A physical aetiology of the gradient microinstabilities of collisionless stellar discs is explained. Such instabilities can develop only if the inhomogeneous and non‐uniformly rotating disc of stars is Jeans‐stable. Certain astronomical implications of the theory for actual galaxies are explored as well. In particular, the development of these instabilities of a stellar disc can result directly in the formation of different observable structural features, e.g. spiral arms and collisionless dynamical relaxation of the system on the Hubble time‐scale.

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“…(19), (20), (24) For two-dimensional spatial fb(xIk)=fb(HIy variations, the perturbed distribution function and electrostatic potential are xpressed as 1 6 fb (xk't)=6^fb(X'yk )exP (-iwt), Making use of the method of characteristics, we integrate the linearized Vlasov equation from t'=--to t'=t. Neglecting initial perturbations, the formal solution for 6 fb can be expressed as 1 7…”

Section: )mentioning

confidence: 99%