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Multidimensional electron beam-plasma instabilities in the relativistic regime
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Figures

Image of FIG. 1.

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FIG. 1.

Number of citations per year received by Fried’s (Ref. 5) and Weibel’s (Ref. 6) 1959 articles until 2009 (from ISI Web of Knowledge).

Image of FIG. 2.

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FIG. 2.

Schematic representation of the FIS and the collisionless shock context. (a) A petawatt laser generates a relativistic electron beam which then deposits its energy near the pellet center. (b) A collisionless shock travels through the interstellar medium. After particles undergo first-order Fermi acceleration (dashed line), some escape upstream (plain line) and trigger turbulence through beam-plasma instabilities. The typical size of the system is indicated in each case.

Image of FIG. 3.

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FIG. 3.

Basic principle of the particle-in-cell simulation technique.

Image of FIG. 4.

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FIG. 4.

Sketch of the system considered in the present review. “RC” here stands for “return current.” Ions are fixed.

Image of FIG. 5.

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FIG. 5.

Schematic representations of the distribution functions considered.

Image of FIG. 6.

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FIG. 6.

Growth rate maps ( units) in the cold-limit for and varying beam densities: (a) and (b) .

Image of FIG. 7.

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FIG. 7.

Factor determining the cold filamentation growth rate for both and (Table I), in terms of the beam Lorentz factor . The factor peaks for .

Image of FIG. 8.

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FIG. 8.

Full spectrum largest growth rate in terms of . Without any free parameter left, this graph is universal. The largest growth rate the system can experience is for and .

Image of FIG. 9.

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FIG. 9.

(a) A group of monokinetic electrons initially in phase with a growing wave remains so during one -folding time. (b) A thermal velocity spread produces a spatial spread at , where is the growth rate. If this spread is much smaller than the wavelength, the interaction is quasimonochromatic.

Image of FIG. 10.

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FIG. 10.

(a) Growth rate map ( units) with the waterbag model for , , , and . (b) Vector field representation of the corresponding electric fluctuations. The flow is along the axis.

Image of FIG. 11.

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FIG. 11.

Growth rate map ( units) for . Other parameters are those of Fig. 10. The flow is along the axis.

Image of FIG. 12.

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FIG. 12.

(a) Momentum spread ( units) stabilizing the filamentation instability vs the beam relativistic factor in waterbag model (29) for two beam-to-plasma density ratios. (b) Maximum growth rate ( units) as a function of the beam thermal spread ( units) in the waterbag model for two beam-to-plasma density ratios. The beam Lorentz factor is . Parallel and transverse beam spreads are set equal in both cases.

Image of FIG. 13.

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FIG. 13.

Waterbag spectra ( units) for a beam parallel spread of (a) and (b). Density ratio is , beam Lorentz factor , and plasma temperatures . The beam transverse spread is in both cases.

Image of FIG. 14.

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FIG. 14.

Growth rate map ( units) with the Maxwell–Jüttner model for , , , and . The flow is along the axis.

Image of FIG. 15.

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FIG. 15.

Typical distribution functions subject to the Weibel and the filamentation instabilities. An anisotropic Weibel-unstable hot plasma can be approximated by a cold filamentation-unstable two-beam system.

Image of FIG. 16.

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FIG. 16.

Schematic representation of various settings involving the Weibel and the filamentation instability. (a) The plasma is Weibel stable (WS), the beam is filamentation unstable (FU). (b) The plasma is Weibel stable, the beam is filamentation stable. (c) The plasma is Weibel unstable (WU), the beam is filamentation stable (FS). (d) The plasma is Weibel unstable, the beam is filamentation unstable, and the two instabilities interact.

Image of FIG. 17.

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FIG. 17.

Phase velocity diagrams for a hot relativistic beam passing through a 5 keV plasma. Parameters are and . Beam temperatures are 5 keV (a), 50 keV (b) and 1 MeV (c). Upper plots: growth rate maps ( units). Lower plots: phase velocity diagrams. The beam (red) and plasma (blue) velocity distributions formally extend all over the domain . The contours shown are isocontours of the distribution functions enclosing 99% of the particles. For , the contour appears like a line.

Image of FIG. 18.

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FIG. 18.

Hierarchy map in the cold-limit in terms of the beam Lorentz factor and the beam to plasma density ratio .

Image of FIG. 19.

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FIG. 19.

Behavior of in terms of , where is the frontier density ratio.

Image of FIG. 20.

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FIG. 20.

Views from two different angles of the surface boundaries delimiting the domains governed by distinct instability classes in the parameter space for a 5 keV plasma. The color code refers to the maximum growth rate in units. The surface approximately parallel to the plane defines the two-stream/oblique frontier (two-stream prevails in the low limit). The second surface defines the filamentation/oblique frontier (filamentation prevails in the high limit).

Image of FIG. 21.

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FIG. 21.

Theoretical estimate of the saturated magnetic energy density normalized to the total electron energy density (solid) and corresponding growth rate (dashed). A Maxwell–Jüttner model is assumed with , , , and .

Image of FIG. 22.

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FIG. 22.

Predicted saturated magnetic energy density as a function of the beam temperature . A Maxwell–Jüttner model is assumed with , , and .

Image of FIG. 23.

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FIG. 23.

Normalized box-averaged field energy densities in the 1D simulation: That of driven by the two-stream instability is . The energy densities of the magnetic and electrostatic component of the filamentation instability are denoted as and .

Image of FIG. 24.

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FIG. 24.

(a) displays the electrostatic computed by the 1D simulation of the two-stream instability in units of . (b) shows the amplitude modulus in the wavenumber (solid curve) and in (dashed curve).

Image of FIG. 25.

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FIG. 25.

Ten-logarithmic electron phase space distributions computed by the 1D simulation of the two-stream instability at the times (a), 74 (b), 83 (c), and 133 (d). (enhanced online). [URL: http://dx.doi.org/10.1063/1.3514586.1]10.1063/1.3514586.1

Image of FIG. 26.

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FIG. 26.

The fields normalized with computed by the 1D simulation: (a) shows , (b) shows and (c) . The normalized magnetic pressure (solid black curve) is compared to at the time in (d). The filaments merging process are clearly evidenced in (a).

Image of FIG. 27.

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FIG. 27.

The electrostatic (blue dashed curve) is compared in panel (a) to (solid black curve) at . In (b), the time-evolution of these amplitudes are compared at the position , where both are maximally positive. All curves are expressed in units of .

Image of FIG. 28.

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FIG. 28.

The ten-logarithmic electron phase space distributions. The distributions for the momentum component along the beam direction are shown in (a) and (b) at the time , while (c) displays the total distribution for the momentum component along the simulation direction. The total distribution for the momentum component along the beam (d) and in the simulation direction are shown in (e) at . (enhanced online). [URL: http://dx.doi.org/10.1063/1.3514586.2]10.1063/1.3514586.2

Image of FIG. 29.

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FIG. 29.

The electromagnetic fields at in the 2D simulation, which resolves the plane, and has the beam direction aligned with : The distributions [(a)–(c)] display , , and , normalized to . The normalized field energies of (dashed blue curve) and of are plotted in (d) in units of the initial electron kinetic energy density. (enhanced online). [URL: http://dx.doi.org/10.1063/1.3514586.3]10.1063/1.3514586.3

Image of FIG. 30.

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FIG. 30.

The electromagnetic fields at in the 2D simulation, which resolves the plane, and has the beam direction aligned with : The distributions [(a)–(c)] display , , and , normalized to . The box-averaged field energy densities (dashed blue curve) and are plotted in (d) in units of the initial electron kinetic energy density. (enhanced online). [URL: http://dx.doi.org/10.1063/1.3514586.4]10.1063/1.3514586.4

Image of FIG. 31.

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FIG. 31.

2D PIC simulation of a waterbag beam-plasma system with , , , : (a) beam and (b) plasma density profiles at the end of the linear phase; simulated growth rate maps calculated from (c) and (d) spectra over . The beam flows along the -axis.

Image of FIG. 32.

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FIG. 32.

phase spaces of (a) beam and (b) plasma electrons close to the field saturation time. The parameters are those of Fig. 31.

Image of FIG. 33.

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FIG. 33.

Time histories of beam (red) and plasma (blue) kinetic energies for (a) (a) and (b). The other parameters are , , . All energies are normalized to the initial beam energy. The gray dashed line indicates the saturation time of the fields following the linear phase.

Image of FIG. 34.

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FIG. 34.

Vector field representation of the simulated electric fluctuations at . The parameters are those of Fig. 31. The color code refers to the simulated growth rate.

Image of FIG. 35.

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FIG. 35.

2D PIC simulation of a waterbag beam-plasma system with , , , : simulated growth rate maps calculated from (a) and spectra (b) over and , respectively. The beam drift is along the -axis.

Image of FIG. 36.

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FIG. 36.

phase spaces of (a) beam and (b) plasma electrons at the field saturation time. The parameters are those of Fig. 35.

Image of FIG. 37.

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FIG. 37.

2D PIC simulations of Maxwell–Jüttner beam-plasma systems: beam [(a), (c), and (e)] and plasma [(b), (d), and (f)] density profiles at the end of the linear phase with [(a) and (b)] , , , [(c) and (d)] , , , [(e) and (f)] , , . In all cases, and the beam flows along the -axis.

Image of FIG. 38.

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FIG. 38.

2D PIC simulations of a Maxwell–Jüttner beam-plasma system with , , , and : (a) beam and (b) plasma density profiles; growth rate maps obtained from (c) and (d) spectra.

Image of FIG. 39.

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FIG. 39.

Comparison between 1D and 2D simulations in the oblique regime: time evolutions of the normalized electric energy (a) and kinetic beam energy (b). The spatial axis resolved in the 1D simulations makes a varying angle (from 0° to 65°) with the beam axis.

Image of FIG. 40.

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FIG. 40.

3D PIC simulation of a Maxwell–Jüttner beam-plasma system with , , , , and : isosurfaces of the beam (upper plots) and plasma (lower plots) density profiles at successive times. The beam flows rightward.

Image of FIG. 41.

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FIG. 41.

(a) Theoretical growth rate map with the parameters of Fig. 40. (b) Growth rate map for the parameters best fitting the simulated beam-plasma at : , , , and .

Image of FIG. 42.

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FIG. 42.

Superimposed iso-surfaces of beam (red) and plasma (blue) density profiles at .

Image of FIG. 43.

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FIG. 43.

Growth rate map ( units) of the electron beam-plasma system with a guiding magnetic field. Parameters are , , and . Filamentation is stabilized, and some wave vectors support more than one unstable mode. Ions are fixed. The flow is along the axis.

Image of FIG. 44.

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FIG. 44.

Growth rate of the flow aligned unstable modes for a cold proton beam with interacting with a cold plasma 100 times denser. A guiding magnetic field is accounted for such as . Blue (thin plain): resonant Alfvén unstable modes. Purple (thin dashed): nonresonant Bell’s unstable modes. Yellow (bold plain): Two-stream instability. Green (bold dashed): Buneman instability. The proton to electron mass ratio is 1836.

Tables

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Table I.

Analytical expressions of the maximum growth rate and associated wave vector in the cold-limit for each instability class. For , see Ref. 7 for two-stream, Ref. 133 for filamentation, and Ref. 8 for oblique. For , there is no oblique extremum. See Refs. 134 and 28 for two-stream and filamentation in this case.

Generic image for table

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Table II.

Kinetic scalings of the maximum filamentation, oblique and two-stream growth rates in the high - and -limits (Ref. 110). For the cold-fluid scalings, see Table I.

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/content/aip/journal/pop/17/12/10.1063/1.3514586
2010-12-28
2014-04-17

Abstract

The interest in relativistic beam-plasma instabilities has been greatly rejuvenated over the past two decades by novel concepts in laboratory and space plasmas. Recent advances in this long-standing field are here reviewed from both theoretical and numerical points of view. The primary focus is on the two-dimensional spectrum of unstable electromagnetic waves growing within relativistic, unmagnetized, and uniform electron beam-plasma systems. Although the goal is to provide a unified picture of all instability classes at play, emphasis is put on the potentially dominant waves propagating obliquely to the beam direction, which have received little attention over the years. First, the basic derivation of the general dielectric function of a kinetic relativistic plasma is recalled. Next, an overview of two-dimensional unstable spectra associated with various beam-plasma distribution functions is given. Both cold-fluid and kinetic linear theory results are reported, the latter being based on waterbag and Maxwell–Jüttner model distributions. The main properties of the competing modes (developing parallel, transverse, and oblique to the beam) are given, and their respective region of dominance in the system parameter space is explained. Later sections address particle-in-cell numerical simulations and the nonlinear evolution of multidimensional beam-plasma systems. The elementary structures generated by the various instability classes are first discussed in the case of reduced-geometry systems. Validation of linear theory is then illustrated in detail for large-scale systems, as is the multistaged character of the nonlinear phase. Finally, a collection of closely related beam-plasma problems involving additional physical effects is presented, and worthwhile directions of future research are outlined.

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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Multidimensional electron beam-plasma instabilities in the relativistic regime
http://aip.metastore.ingenta.com/content/aip/journal/pop/17/12/10.1063/1.3514586
10.1063/1.3514586
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