The Weibel instability in relativistic plasmas

Achterberg, A.; Wiersma, J.; Norman, C.

doi:10.1051/0004-6361:20065366

Cited by 80 publications

(65 citation statements)

References 29 publications

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“…A first discussion of these stabilization mechanisms can be found in Wiersma & Achterberg (2004). The companion paper (Achterberg et al 2007) discusses them in more detail.…”

Section: Discussionmentioning

confidence: 99%

“…Starting from first principles, we present a detailed analysis of the dispersion relation that determines the growth rateσ, both in a fluid description of the plasma and in a kinetic waterbag description. These results are important for the analysis of the non-linear stage of the relativistic Weibel instability, which is sensitive to the dispersion properties of the plasma; see the companion paper (Achterberg et al 2007) for details. From the dispersion relation we derive expressions for the growth rateσ using various approximations: non-relativistic and ultrarelativistic; with and without a background magnetic field.…”

Section: Introductionmentioning

confidence: 97%

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The Weibel instability in relativistic plasmas

Achterberg¹,

Wiersma²

2007

A&A

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105

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Aims. We discuss the linear theory of the Weibel instability in a relativistic plasma driven by ultra-relativistic beams, describing the physics of the generation of magnetic fields in the ultra-relativistic shocks associated with Gamma Ray Bursts (GRBs). We perform a detailed analysis of the linear dispersion relation for the benefit of non-linear calculations that we discuss in the companion paper. Methods. We use a covariant approach, where the linear response of the beam-plasma system is determined from the polarization tensor. This tensor relates the four-current density to the four-potential of the electromagnetic field. Showing that two approaches, one based on a fluid model and one on a kinetic description that uses a waterbag distribution for the phase-space density of the beam particles, yield essentially the same result, we compare our results to those obtained by other approaches. We mainly consider the symmetric case of two counterstreaming (but otherwise identical) beams. Results. We show that the effect of an asymmetry in the beam densities is small for typical parameters, and briefly discuss the effect of an ambient magnetic field. The dispersion relation of the Weibel instability driven by ultra-relativistic beams is rather insensitive to the model used to describe the plasma. The properties of the instability, such as the growth rate and the range of unstable wavelengths, are governed by only two parameters: the ratio of the plasma frequency squared of the beam and hot background plasma, and a "Mach number", which is essentially the ratio of the beam momentum and the momentum associated with thermal velocity (∼sound speed) in the beam plasma. We also show that, at least for the parameters associated with the ultra-relativistic shocks in GRBs, the influence of the magnetic field is small, and the results for an unmagnetized plasma can be used. Conclusions.

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“…A first discussion of these stabilization mechanisms can be found in Wiersma & Achterberg (2004). The companion paper (Achterberg et al 2007) discusses them in more detail.…”

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

The Weibel instability in relativistic plasmas

Achterberg¹,

Wiersma²

2007

A&A

Self Cite

105

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“…Another way of evaluating the field at saturation is to write that as it grows, particles start oscillating in the field of the fastest growing k m at frequency [37,38],…”

Section: Field At Saturationmentioning

confidence: 99%

Collisionless shock formation, spontaneous electromagnetic fluctuations, and streaming instabilities

Bret

Stockem²,

Fiúza³

et al. 2013

Physics of Plasmas

105

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Collisionless shocks are ubiquitous in astrophysics and in the lab. Recent numerical simulations and experiments have shown how they can arise from the encounter of two collisionless plasma shells. When the shells interpenetrate, the overlapping region turns unstable, triggering the shock formation. As a first step towards a microscopic understanding of the process, we analyze here in detail the initial instability phase. On the one hand, 2D relativistic PIC simulations are performed where two symmetric initially cold pair plasmas collide. On the other hand, the instabilities at work are analyzed, as well as the field at saturation and the seed field which gets amplified. For mildly relativistic motions and onward, Weibel modes govern the linear phase. We derive an expression for the duration of the linear phase in good agreement with the simulations. This saturation time constitutes indeed a lower-bound for the shock formation time.

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“…Achterberg & Norman (1980) made ad hoc modifications to the simple result for stationary planar shocks to include many corrections and predicted the ubiquitous power-law spectrum of diffusive shock acceleration. Lee & Fisk (1982) considered particle acceleration by a selfsimilar blast wave propagating into a cold stationary medium where the radial diffusion coefficient is independent of energy and of self-similar form.…”

Section: Introductionmentioning

confidence: 99%

Coupled Hydromagnetic Wave Excitation and Ion Acceleration at an Evolving Coronal/Interplanetary Shock

Lee

2005

ASTROPHYS J SUPPL S

242

177

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An analytical quasilinear theory is presented for the evolution of a ''gradual'' event consisting of solar energetic particles (SEPs) accelerated at an evolving coronal/interplanetary shock. The upstream ion transport is described by the two-stream moments of the focused transport equation, which accommodate the large streaming anisotropies observed near event onset. The proton transport equations and a wave kinetic equation are solved together for the coupled behavior of the hydromagnetic waves and the energetic protons. The theory includes diffusive shock acceleration, ion advection with the solar wind, spatial diffusion upstream of the shock, magnetic focusing, wave excitation by the energetic protons, and minor ions as test particles. A number of approximations are made for analytical tractability. The predictions reproduce the observed phases of most gradual SEP events: onset, a ''plateau'' with large streaming anisotropy, an ''energetic storm particle'' (ESP) enhancement prior to shock passage, and the decaying ''invariant spectra'' after shock passage. The theory treats naturally the transition from a scatter-dominated sheath adjacent to the shock where the wave intensity is enhanced to the nearly scatter-free ion transport in interplanetary space. The plateau is formed by ions that are extracted from the outer edge of the scatter-dominated sheath by magnetic focusing and escape into interplanetary space; it corresponds quantitatively to the ''streaming limit'' identified and interpreted in gradual events by D. V. Reames and C. K. Ng. The ion energy spectra at the shock have the standard power-law form dependent on shock strength, which is expected for diffusive shock acceleration, with a high-energy cutoff whose form is determined self-consistently by the ion escape rate. The increased shock strength, magnetic field magnitude, and injection energies close to the Sun account for the observed predominance of high-energy ions early in the event. The downstream ion transport is determined under two extreme assumptions: (i) vanishing diffusive transport and (ii) effective diffusive transport leading to small ion spatial gradients. The latter assumption reproduces the invariant spectra, spatial gradients, and exponential temporal decay observed in the late phase of many events. The minor ion distributions exhibit fractionation due to rigidity-dependent transport and acceleration. However, their energy spectra, spatial gradients, and high-energy cutoffs do not reproduce observed forms and lead to excessive fractionation. The origin of these discrepancies is probably the neglect of nonlinear processes. Although not easily incorporated in the theory, these processes could substantially modify the predicted wave intensity. An illustrative calculation assuming an arbitrary power-law form for the wave intensity demonstrates the sensitive dependence of ion fractionation on the power-law index. Subject headingg s: acceleration of particles -interplanetary medium -MHD -shock wavesSun: particle emission

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The Weibel instability in relativistic plasmas

Cited by 80 publications

References 29 publications

The Weibel instability in relativistic plasmas

The Weibel instability in relativistic plasmas

Collisionless shock formation, spontaneous electromagnetic fluctuations, and streaming instabilities

Coupled Hydromagnetic Wave Excitation and Ion Acceleration at an Evolving Coronal/Interplanetary Shock

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