Study on longitudinal dispersion relation in one-dimensional relativistic plasma: Linear theory and Vlasov simulation

Zhang, H.; Wu, S. Z.; Zhou, C. T.; Zhu, Shun‐Peng; He, X. T.

doi:10.1063/1.4821606

Cited by 6 publications

(12 citation statements)

References 34 publications

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“…It has to be highlighted that the Zhang dispersion relation (Zhang et al. 2013) described by the green line coincides with the red line until the effects of the real dispersion relations, which include the effects of the magnetic field and the - dimension case, are present. The Zhang dispersion relation is considered for a maximum wavenumber , arguing that it is sufficient for typical relativistic plasma oscillations and not because a cutoff is predicted as happens in our approach.…”

Section: Relativistic Dispersion Relation For the Longitudinal Casementioning

confidence: 95%

“…The Fourier component of the -current vector is By using (3.5), we arrive at where and and where (see appendix B) where are is Kelvin function of order (Synge 1956) and It has to be noted that other works constrain the problem to one dimension and, consequently, they use the one-dimensional Jüttner distribution function (Zhang et al. 2013).…”

Section: Dispersion Relationsmentioning

confidence: 99%

“…2018) and the one-dimensional linear longitudinal case (Zhang et al. 2013). Particularly, the work done by Zhang et al.…”

Section: Introductionmentioning

confidence: 99%

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Modified Vlasov equation and dispersion relations for a relativistic radiating electron plasma

2019

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A modified Vlasov equation is obtained by developing a covariant statistical mechanics for a system of electrons without considering the effects of the ions and including the Landau–Lifshitz equation of motion. General dispersion relations for the transverse and longitudinal modes for any temperature are expressed. The results are similar to those found by Hakim & Mangeney (Phys. Fluids, vol. 14, 1971, pp. 2751–2781) for both the modified Vlasov equation and the dispersion relations. However, for the longitudinal mode, unlike the development of Hakim and Mangeney, correct expansions are done in order to give a numerical approach to obtain the longitudinal relativistic dispersion relations for any value of the wavenumber. Accordingly, new loop solutions, with turning points, crossing the super-luminous region and the super-thermal region are found. Although the expressions for the Landau damping and the damping due to the radiation reaction force coincide with the Hakim and Mangeney results for some particular cases, in general they are different. A Landau anti-damping appears in the second branch of the loop in a small region between the cutoff point and the intersection with the super-thermal line. The analysis of this effect leads us to a kind of wave pulse. We will call them bipolar waves. The treatment contains the relativistic interactions between all the electrons in the system with retarded effects. This explain the differences with Zhang’s recent work (Phys. Plasmas, vol. 20, 2013, 092112–092132). It is shown that for low densities, the cutoff of the wave is due to the dispersion relations and not due to the radiation reaction force damping. While for both high densities and temperatures, the damping due to the radiation reaction force is important.

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Section: Relativistic Dispersion Relation For the Longitudinal Casementioning

confidence: 95%

Section: Dispersion Relationsmentioning

confidence: 99%

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Modified Vlasov equation and dispersion relations for a relativistic radiating electron plasma

2019

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“…All computations are performed in normalized variables: the plasma frequency is use to set temporal and spatial scales; momenta are normalized to mc; the electric field is normalized to mcx p =e; and the background density n 0 scales out. Oscillations are excited by initializing the distribution function with a spatial modulation of wave number k f ðt ¼ 0; x; pÞ ¼ n 0 ð1 þ A cos kxÞf eq ðpÞ (22) with k ¼ 2pn=L, where n is an integer. We identify f ð1Þ ¼ A n 0 cosðkxÞ f eq ðpÞ.…”

Section: Examplesmentioning

confidence: 99%

A time-implicit numerical method and benchmarks for the relativistic Vlasov–Ampere equations

Carrie

Shadwick

2016

Physics of Plasmas

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We present a time-implicit numerical method to solve the relativistic Vlasov-Ampere system of equations on a two dimensional phase space grid. The time-splitting algorithm we use allows the generalization of the work presented here to higher dimensions keeping the linear aspect of the resulting discrete set of equations. The implicit method is benchmarked against linear theory results for the relativistic Landau damping for which analytical expressions using the Maxwell-J€ uttner distribution function are derived. We note that, independently from the shape of the distribution function, the relativistic treatment features collective behaviours that do not exist in the nonrelativistic case. The numerical study of the relativistic two-stream instability completes the set of benchmarking tests. V

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“…For high-frequency electrostatic (plasma waves) and electromagnetic modes recent work has been developed in Refs. (Fichtner & Schlickeiser, 1995;Schlickeiser & Kneller, 1997;Melrose, 1999;Bergman & Eliasson, 2001;Podesta, 2008;Bers et al, 2009;Schlickeiser, 2010;Zhang et al, 2013;López et al;2014) in the context of laser thermonuclear fusion. Our work is therefore an extension of these works to low-frequency spectrum.…”

Section: Introductionmentioning

confidence: 99%

Dispersion relation of low-frequency electrostatic waves in plasmas with relativistic electrons

et al. 2016

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The dispersion relation of electrostatic waves with phase velocities smaller than the electron thermal velocity is investigated in relativistic temperature plasmas. The model equations are the electron relativistic collisionless hydrodynamic equations and the ion non-relativistic Vlasov equation, coupled to the Poisson equation. The complex frequency of electrostatic modes are calculated numerically as a function of the relevant parameters kλ De and ZT e /T i where k is the wavenumber, λ De , the electron Debye length, T e and T i the electron and ion temperature, and Z, the ion charge number. Useful analytic expressions of the real and imaginary parts of frequency are also proposed. The non-relativistic results established in the literature from the kinetic theory are recovered and the role of the relativistic effects on the dispersion and the damping rate of electrostatic modes is discussed. In particular, it is shown that in highly relativistic regime the electrostatic waves are strongly damped.

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Study on longitudinal dispersion relation in one-dimensional relativistic plasma: Linear theory and Vlasov simulation

Cited by 6 publications

References 34 publications

Modified Vlasov equation and dispersion relations for a relativistic radiating electron plasma

Modified Vlasov equation and dispersion relations for a relativistic radiating electron plasma

A time-implicit numerical method and benchmarks for the relativistic Vlasov–Ampere equations

Dispersion relation of low-frequency electrostatic waves in plasmas with relativistic electrons

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