Contributions to the linear and nonlinear theory of the beam–plasma interaction

Abstract

We focus our attention on some relevant aspects of the beam–plasma instability in order to refine some features of the linear and nonlinear dynamics. After a re-analysis of the Poisson equation and of the assumption dealing with the background plasma in the form of a linear dielectric, we study the non-perturbative properties of the linear dispersion relation, showing the necessity for a better characterization of the mode growth rate in those flat regions of the distribution function where the Landau formula is no longer predictive. We then upgrade the original $N$ -body approach in O'Neil et al. (Phys. Fluids, vol. 14, 1971, pp. 1204–1212), in order to include a return current in the background plasma. This correction term is responsible for smaller saturation levels and growth rates of the Langmuir modes, as result of the energy density transferred to the plasma via the return current. Finally, we include friction effects, as those due to the collective influence of all the plasma charges on the motion of the beam particles. The resulting force induces a progressive resonance detuning, because particles are losing energy and decreasing their velocity. This friction phenomenon gives rise to a deformation of the distribution function, associated with a significant growth of the less energetic particle population. The merit of this work is to show how a fine analysis of the beam–plasma instability outlines a number of subtleties about the linear, intermediate and late dynamics which can be of relevance when such a system is addressed as a paradigm to describe relevant nonlinear wave–particle phenomena (Chen & Zonca, Rev. Mod. Phys., vol. 88, 2016, 015008).

One dimensional reduced model for ITER relevant energetic particle transport

“…The instability features of a single resonant mode, can be analyzed by explicitly expressing ω → ω + iγ D , where now ω can include a real frequency shift with respect to the Langmuir mode frequency ω p and we recall that γD is the linear drive. We can write the normalized dispersion relation for a give mode as [49] 2…”

Cosmological Vlasov–Poisson equations for dark matter

“…In the so-called single-wave model thereof, certain assumptions about the involved velocities are employed such that the plasma response is effectively non-resonant; therefore, the plasma can be treated as a bulk with respect to the propagation of the electron beam, while the plasma response can be incorporated in a form of a real dielectric function (see, e.g. Carlevaro et al 2014Carlevaro et al , 2020. Similarly as in the cosmological case, also here the Vlasov-Poisson equations are directly related to a Hamiltonian principle (Tennyson et al 1994;Antoniazzi et al 2006), and thus can be reduced to Newtonian-type equation of motion coupled to a Poisson equation.…”

“…wave-particle interaction. Furthermore, it is assumed that the electron beam frequency is (roughly) equal to the plasma frequency, which that the dielectric of the plasma is nearly vanishing (Carlevaro et al 2020), thereby justifying linear temporal expansions applied to the dielectric. This allows to incorporate the effect of the plasma density in the Poisson equation by a linear dielectric, and at the same time, cast the Poisson equation into a simple evolution equation (O 'Neil and Malmberg 1968).…”

Cosmological Vlasov-Poisson equations for dark matter: Recent developments and connections to selected plasma problems

“…In the so-called single-wave model thereof, certain assumptions about the involved velocities are employed such that the plasma response is effectively non-resonant; therefore, the plasma can be treated as a bulk with respect to the propagation of the electron beam, while the plasma response can be incorporated in a form of a real dielectric function (see e.g. [214,215]). Similarly as in the cosmological case, also here the Vlasov-Poisson equations are directly related to a Hamiltonian principle [217,218], and thus can be reduced to Newtonian-type equation of motion coupled to a Poisson equation.…”

“…It is usually assumed that 𝛼 := 𝜌 b /𝜌 p 1 is a perturbatively small control parameter of the kinetic wave-particle interaction. Furthermore, it is assumed that the electron beam frequency is (roughly) equal to the plasma frequency, which ensures that the dielectric of the plasma is nearly vanishing [215], thereby justifying linear temporal expansions applied to the dielectric. This allows to incorporate the effect of the plasma density in the Poisson Fig.…”