Landau Damping and the Velocity Fourier Transform of the Initial Perturbation

Abstract This research was stimulated by the recent studies of damping solutions in dynamically stable spherical stellar systems. Using the simplest model of the homogeneous stellar medium, we discuss nontrivial features of stellar systems. Taking them into account will make it possible to correctly interpret the results obtained earlier and will help to set up decisive numerical experiments in the future. In particular, we compare the initial value problem versus the eigenvalue problem. It turns out that in the unstable regime, the Landau-damped waves can be represented as a superposition of van Kampen modes plus a discrete damped mode, usually ignored in the stability study. This mode is a solution complex conjugate to the unstable Jeans mode. In contrast, the Landau-damped waves are not genuine modes: in modes, eigenfunctions depend on time as exp ( − iωt), while the waves do not have eigenfunctions on the real v-axis at all. However, ‘eigenfunctions’ on the complex v-contours do exist. Deviations from the Landau damping are common and can be due to singularities or cut-off of the initial perturbation above some fixed value in the velocity space.

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Reflection and absorption of ordinary waves in an inhomogeneous plasma

Journal of Plasma Physics ◽

10.1017/s0022377800026945 ◽

1993 ◽

Vol 50 (1) ◽

pp. 109-123 ◽

Cited By ~ 1

Author(s):

R. Croci

Keyword(s):

Magnetic Field ◽

Fourier Transform ◽

Electric Field ◽

Maxwell Equations ◽

Inhomogeneous Plasma ◽

Landau Damping ◽

Equilibrium Plasma ◽

Homogeneous Plasma ◽

The Fourier Transform ◽

Uniform Equilibrium

This study treats the system of Vlasov and Maxwell equations for the Fourier transform in space and time of a plasma referred to Cartesian co-ordinates with the z co-ordinate parallel to the uniform equilibrium magnetic field; the equilibrium plasma density depends on ηχ, where η is a parameter, and goes to zero as │χ│ → ∞. The component ky of the wave vector is taken to be zero, whereas kz is non-zero, thereby allowing for Landau damping. Coupling of the components of the electric field parallel and perpendicular to the equilibrium magnetic field (the ordinary and extraordinary waves in a homogeneous plasma) is neglected.

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Collisionless Plasma Oscillations, Their Residual Presentation and the Mechanism of Landau Damping

Zeitschrift für Naturforschung A ◽

10.1515/zna-1974-1225 ◽

1974 ◽

Vol 29 (12) ◽

pp. 1863-1873

Author(s):

G. Ecker ◽

G. Frömling

Keyword(s):

Initial Phase ◽

Initial Perturbation ◽

Collisionless Plasma ◽

Time Development ◽

Landau Damping ◽

Velocity Spread ◽

Particle Beams ◽

Plasma Oscillations ◽

Phase Mixing ◽

Analytic Expressions

The description of the electron oscillations of a collisionless plasma by the usual residual presentation is insufficient in the initial phase of time development and for perturbations of small velocity spread. We derived criteria for the number of residual terms which have to be taken into account and obtain analytic expressions for the remaining integral. Decomposing the initial perturbation into velocity beams we show that Landau damping is due to phase mixing caused by free streaming of particle beams modified through the response of the main plasma body.

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Comparison of Calculated and Recorded Electron Energy-Loss Spectra of Aluminium and Carbon

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100133916 ◽

1990 ◽

Vol 48 (2) ◽

pp. 64-65

Author(s):

L. Reimer ◽

R. Oelgeklaus

Keyword(s):

Fourier Transform ◽

Energy Loss ◽

Electron Energy ◽

Rotational Symmetry ◽

Mean Free Path ◽

Single Scattering ◽

Specimen Thickness ◽

Two Dimensional ◽

Image Position ◽

Electron Energy Loss

Quantitative electron energy-loss spectroscopy (EELS) needs a correction for the limited collection aperture α and a deconvolution of recorded spectra for eliminating the influence of multiple inelastic scattering. Reversely, it is of interest to calculate the influence of multiple scattering on EELS. The distribution f(w,θ,z) of scattered electrons as a function of energy loss w, scattering angle θ and reduced specimen thickness z=t/Λ (Λ=total mean-free-path) can either be recorded by angular-resolved EELS or calculated by a convolution of a normalized single-scattering function ϕ(w,θ). For rotational symmetry in angle (amorphous or polycrystalline specimens) this can be realised by the following sequence of operations :(1)where the two-dimensional distribution in angle is reduced to a one-dimensional function by a projection P, T is a two-dimensional Fourier transform in angle θ and energy loss w and the exponent -1 indicates a deprojection and inverse Fourier transform, respectively.

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FR-IR Molecular Microanalysis System

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100134958 ◽

1990 ◽

Vol 48 (2) ◽

pp. 270-271

Author(s):

John A. Reffner ◽

William T. Wihlborg

Keyword(s):

Fourier Transform ◽

Fourier Transform Infrared ◽

Integrated System ◽

Morphological Examination ◽

Fully Integrated ◽

Ir Microscopy ◽

Normal Functions ◽

Infrared Microspectroscopy ◽

Infrared Spectral ◽

Ft Ir

The IRμs™ is the first fully integrated system for Fourier transform infrared (FT-IR) microscopy. FT-IR microscopy combines light microscopy for morphological examination with infrared spectroscopy for chemical identification of microscopic samples or domains. Because the IRμs system is a new tool for molecular microanalysis, its optical, mechanical and system design are described to illustrate the state of development of molecular microanalysis. Applications of infrared microspectroscopy are reviewed by Messerschmidt and Harthcock.Infrared spectral analysis of microscopic samples is not a new idea, it dates back to 1949, with the first commercial instrument being offered by Perkin-Elmer Co. Inc. in 1953. These early efforts showed promise but failed the test of practically. It was not until the advances in computer science were applied did infrared microspectroscopy emerge as a useful technique. Microscopes designed as accessories for Fourier transform infrared spectrometers have been commercially available since 1983. These accessory microscopes provide the best means for analytical spectroscopists to analyze microscopic samples, while not interfering with the FT-IR spectrometer’s normal functions.

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The extended Fourier algorithm: Application in discrete optics and electron holography

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100172322 ◽

1994 ◽

Vol 52 ◽

pp. 918-919

Author(s):

E. Voelkl ◽

L. F. Allard

Keyword(s):

Fourier Transform ◽

Fourier Transforms ◽

Sampling Rate ◽

Electron Holography ◽

Optical Bench ◽

Fourier Space ◽

Ccd Cameras ◽

Simulated Image ◽

Initial Sampling ◽

Fourier Algorithm

The conventional discrete Fourier transform can be extended to a discrete Extended Fourier transform (EFT). The EFT allows to work with discrete data in close analogy to the optical bench, where continuous data are processed. The EFT includes a capability to increase or decrease the resolution in Fourier space (thus the argument that CCD cameras with a higher number of pixels to increase the resolution in Fourier space is no longer valid). Fourier transforms may also be shifted with arbitrary increments, which is important in electron holography. Still, the analogy between the optical bench and discrete optics on a computer is limited by the Nyquist limit. In this abstract we discuss the capability with the EFT to change the initial sampling rate si of a recorded or simulated image to any other(final) sampling rate sf.

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