On the multistream approach of relativistic Weibel instability. I. Linear analysis and specific illustrations

Abstract

Équipe 107 : Physique des plasmas chaudsInternational audienceA one-dimensional multistream formalism is extended for the study of temperature anisotropy driven Weibel-type instabilities in collisionless and relativistic plasma. The formulation is based on a Hamiltonian reduction technique using the invariance of generalized canonical momentum in transverse direction. The Vlasov-Maxwell model is expressed in terms of an ensemble of one-dimensional Vlasov-type equations, coupled together with the Maxwell equations in a self-consistent way. Although the model is fundamentally nonlinear, this first of three companion papers focuses on the linear aspect. Dispersion relations of the Weibel instability are derived in the linear regime for different kinds of polarization of the electromagnetic potential vector. The model allows new unexpected insights on the instability: enhanced growth rates for the Weibel instability are predicted when a dissymmetric distribution is considered in p(perpendicular to). In the case of a circular polarization, a simplification of the linear analysis can be obtained by the introduction of the ``multiring'' approach allowing to extend the analytical model of Yoon and Davidson [Phys. Rev. A 35, 2718 (1987)]. Applications of this model are left to the other two papers of the series where specific problems are addressed pertaining to the nonlinear and relativistic dynamics of magnetically trapped particles met in the saturation regime of the Weibel instability

Fluid description of Weibel-type instabilities via full pressure tensor dynamics

“…However, some of the conclusions drawn within these cold models, such as the complete de-coupling between the CFI purely electromagnetic (e.m.) and electrostatic features [10] in some limits, have proven not to survive in a full kinetic treatment with non-null beam temperatures [11][12][13]. Kinetic features have been instead successfully described by means of kinetic reduced models based on specific classes of electron distribution functions, such as waterbags [14][15][16][17], multi-stream distributions [18], combinations of the two [19] or some of their special cases, such as "multi-ring" distributions [20][21][22]. While on the one hand such special classes of particle distributions 45001-p1 are useful both for the numerical initialisation of these instabilities, especially in the relativistic regime, and for their numerical simulation at a lower computational cost, on the other hand the exact analytical calculations they typically allow provide a valuable insight into the physics of Weibel-type modes.…”

Vlasov models for kinetic Weibel-type instabilities

“…Relying on the similarity between WI and CFI instabilities, we have recently built a new theoretical tool starting from the possibility of representing the temperature anisotropy by means of a finite number of counter-streaming streams (see Inglebert et al 2012;Ghizzo & Bertrand 2013;Ghizzo 2013a). This model, which we have called the multi-stream model, turns out to be of more general importance and not limited to the Weibel instability.…”

A pressure tensor description for the time-resonant Weibel instability

“…y,α completely determines that of Π (1) xx,α and Π (1) zz,α via plasma compressibility effects, but these Π α components are not linearly involved in the fluid velocity dynamics. Notice that the Π xz,α component remains zero during the whole linear evolution and that the two other non-diagonal components can evolve even if the equilibrium pressure is isotropic due to the first r.h.s term in equations (16) and (17). The role of an initial pressure anisotropy on the magnetic field generation clearly appears in the second r.h.s.…”

“…Reduced kinetic models, instead, such as those based on the evolution of waterbag or multi-stream distributions [19], successfully describe these kinetic features. The multistream model, in particular, deals with Fried's analogy in depth by sampling the distribution function with a bunch of parallel beams, taking advantage of the potential cyclic variables of the problem ( [20], [16], [17]). The fluid model we here consider, extended to include the full pressure tensor dynamics, can be compared to a reduced kinetic model.…”