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Register a new userCollisionsless amplifying of longitudinal electron waves in two-stream plasma
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V. N. Soshnikov
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To better understanding the principal features of collisionless damping/growing plasma waves we have implemented a demonstrative calculation for the simplest cases of electron waves in two-stream plasmas with the delta-function type electron velocity distribution function of each of the streams with velocities v(1) and v(2). The traditional dispersion equation is reduced to an algebraic 4th order equation, for which numerical solutions are presented for a variant of equal stream densities. In the case of uniform half-infinite slab one finds two dominant type solutions: non-damping forward waves and forward complex conjugated exponentially both damping and growing waves. Beside it in this case there is no necessity of calculation any logarithmically divergent indefinite integrals. The possibility of wave amplifying might be useful in practical applications.
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We have considered an expansion of solutions of the non-linear equations for both longitudinal and transverse waves in collisionless Maxwellian plasma in series of non-damping overtones of the field E(x,t) and electron velocity distribution function f=f(0) +f(1) where f(0) is background Maxwellian electron distribution function and f(1) is perturbation. The electrical field and perturbation f(1) are presented as a series of non-damping harmonics with increasing frequencies of the order n and the same propagation speed. It is shown presence of recurrent relations for arising overtones. Convergence of the series is provided by a power law parameter series convergence. There are proposed also successive procedures of cutting off the distribution function f(1) to the condition of positivity f near the singularity points where kinetic equation becomes inapplicable. In this case, at poles absence the solution reduces to non-damping Vlasov waves (oscillations). In the case of transverse waves, dispersion equation has two roots, corresponding to the branches of fast electromagnetic and slow electron waves. There is noted a possibility of experimental testing appearing exotic results with detecting frequencies and amplitudes of n-order overtones.
A dynamic mitigation mechanism of the two-stream instability is discussed based on a phase control for plasma and fluid instabilities. The basic idea for the dynamic mitigation mechanism by the phase control was proposed in the paper [Phys. Plasmas 19, 024503(2012)]. The mitigation method is applied to the two-stream instability in this paper. In general, instabilities appear from the perturbations, and normally the perturbation phase is unknown. Therefore, the instability growth rate is discussed in fluids and plasmas. However, if the perturbation phase is known, the instability growth can be controlled by a superimposition of perturbations imposed actively. For instance, a perturbed driver induces a perturbation to fluids or plasmas; if the perturbation induced by the perturbed driver is oscillated actively by the driver oscillation, the perturbation phase is known and the perturbation amplitude can be controlled, like a feedforward control. The application result shown in this paper demonstrates that the dynamic mitigation mechanism works well to smooth the non-uniformities and mitigate the instabilities in plasmas.
In this paper we have criticized the so-called Landau damping theory. We have analyzed solutions of the standard dispersion equations for longitudinal (electric) and transversal (electromagnetic and electron) waves in half-infinite slab of the uniform collisionless plasmas with non-Maxwellian and Maxwellian-like electron energy distribution functions. One considered the most typical cases of both the delta-function type distribution function (the plasma stream with monochromatic electrons) and distribution functions, different from Maxwellian ones as with a surplus as well as with a shortage in the Maxwellian distribution function tail. It is shown that there are present for the considered cases both collisionless damping and also non-damping electron waves even in the case of non-Maxwellian distribution function.
This paper presents a study of the two-stream instability of an electron beam propagating in a finite-size plasma placed between two electrodes. It is shown that the growth rate in such a system is much smaller than that of an infinite plasma or a finite size plasma with periodic boundary conditions. Even if the width of the plasma matches the resonance condition for a standing wave, a spatially growing wave is excited instead with the growth rate small compared to that of the standing wave in a periodic system. The approximate expression for this growth rate is $gamma approx (1/13)omega_{pe}(n_{b}/n_{p})(Lomega_{pe}/v_{b})ln (Lomega_{pe}/v_{b})[ 1-0.18cos ( Lomega_{pe}/v_{b}+{pi }/{2}) ]$, where $omega_{pe}$ is the electron plasma frequency, $n_{b}$ and $n_{p}$ are the beam and the plasma densities, respectively, $v_{b}$ is the beam velocity, and $L$ is the plasma width. The frequency, wave number and the spatial and temporal growth rates as functions of the plasma size exhibit band structure. The amplitude of saturation of the instability depends on the system length, not on the beam current. For short systems, the amplitude may exceed values predicted for infinite plasmas by more than an order of magnitude.
The dynamics of electron-plasma waves are described at arbitrary collisionality by considering the full Coulomb collision operator. The description is based on a Hermite-Laguerre decomposition of the velocity dependence of the electron distribution function. The damping rate, frequency, and eigenmode spectrum of electron-plasma waves are found as functions of the collision frequency and wavelength. A comparison is made between the collisionless Landau damping limit, the Lenard-Bernstein and Dougherty collision operators, and the electron-ion collision operator, finding large deviations in the damping rates and eigenmode spectra. A purely damped entropy mode, characteristic of a plasma where pitch-angle scattering effects are dominant with respect to collisionless effects, is shown to emerge numerically, and its dispersion relation is analytically derived. It is shown that such a mode is absent when simplified collision operators are used, and that like-particle collisions strongly influence the damping rate of the entropy mode.
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