Nonlinear physics and energetic particle transport features of the beam–plasma instability

Abstract

In this paper we study transport features of a one-dimensional beam-plasma system in the presence of multiple resonances. As a model description of the general problem of a warm energetic particle beam, we assume n cold supra-thermal beams and investigate the self-consistent evolution in the presence of the complete spectrum of nearly degenerate Langmuir modes. A qualitative transport estimation is obtained by computing the Lagrangian Coherent Structures of the system on given temporal scales. This leads to the splitting of the phase space into regions where the local transport processes are relatively faster. The general theoretical framework is applied to the case of the nonlinear dynamics of two cold beams, for which numerical simulation results are illustrated and analysed.

Resonance overlap and nonlinear features of the beam–plasma system

“…As a result, a large number of resonances can be easily described, up to dealing with the so-called quasi-linear limit (Berk et al 1995a;Volokitin & Krafft 2012). The conceptual framework that justifies the implementation of a Hamiltonian approach is the so-called notion of 'quasi-stationary states' (Berk et al 1995b;Elskens & Escande 2003;Carlevaro et al 2014), i.e. transient states of the discrete system, in which a time is spent by the dynamics proportional to the number of considered particles (in practice, charged macroparticles).…”

“…In this work, following O' Neil &Malmberg (1968) andO'Neil et al (1971) (see also Carlevaro et al (2015Carlevaro et al ( , 2016a for other reference details), the plasma is addressed as a cold linear dielectric medium represented by longitudinal electrostatic Langmuir waves, whose density n p is assumed much greater that the beam density n B . In this respect, we define η ≡ n B /n p as one of the fundamental parameters of the model.…”

“…A detailed description of the phase-space dynamics is also provided by means of the Lagrangian coherent structure (LCS) technique (see Haller (2015) for a complete review). This approach is particularly suited as a post-processing technique for describing particle motion in phase space in the presence of a time-dependent scalar potential and it has been already applied to the BPS in Carlevaro et al (2015). The value of calculating LCSs in this work stems from the fact that they generalize dynamical structures observed in autonomous and periodic systems, e.g.…”

Mixed Diffusive-Convective Relaxation of a Warm Beam of Energetic Particles in Cold Plasma

“…In particular, we adopt the Hamiltonian formulation of the problem described in Ref. [51], where the broad supra-thermal particle beam is discretized as superposition of n ≫ 1 cold beams self-consistently evolving in the presence of m n modes nearly degenerate with Langmuir waves, i.e., with frequency ω k j ≃ ω p for j = 1, ..., m. This ensures that the dielectric function of the cold background plasma is nearly vanishing and allows casting the Poisson equation for each plasma oscillation into the form of a simple evolution equation, while particles trajectories are solved from the equations of motion (2) [9].…”

“…In the following, we first briefly summarize the Hamiltonian formulation of the broad beam relaxation in cold plasma derived in Ref. [51] (Sec.4.1). Then, we discuss the dimensionless parameters that are used to characterize the different nonlinear dynamic regimes (Sec.4.2), which will be studied numerically with four different set-ups of initial conditions (Sec.4.3).…”