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Longitudinal Dynamics and Acceleration
Published in Volker Ziemann, ®, 2019
The left-hand plot in Figure 5.7 shows the average phase ⟨ϕ⟩ (top) and energy ⟨ϕ˙⟩ (bottom) for the ensemble of particles as function of time for the same starting conditions used in Figure 5.6. We observe that there is an apparent damping of the averages. This effect, often called Landau damping, is not real damping in the sense that energy is dissipated, but a loss of coherence in the ensemble of particles. We mentioned in the previous paragraph that the oscillation amplitude of each individual particle is unaffected, but the particles get out of step and the coherent motion of the average is lost. Instead, the distribution of the ensemble becomes wider. The right-hand side in Figure 5.7 shows the initial and final projections of the distribution of particles after 100Ts, which corresponds to the bottom right plot in Figure 5.6. Initially the distributions are well localized, as shown by the histograms in lighter gray. After 100Ts the projections of the smeared out ring, visible in Figure 5.6, appear as spread-out projection in Figure 5.7, for the phase in the top plot and the energy η in the bottom plot. Again, the loss of coherence leads to a widened final distribution. In order to maintain a small beam size after injecting into a ring one obviously has to pay special care to adjust the energy and also the timing, which would lead to the same widening of the distribution.
Permittivity and Wave Propagation in Plasmas
Published in V. L. Ginzburg, Oleg Glebov, Applications of Electrodynamics in Theoretical Physics and Astrophysics, 2017
It can be seen from this equation that even in the absence of collisions some absorption still takes place (the imaginary part of εij does not vanish), and this absorption is due only to those particles whose velocities satisfy the condition (12.33). But the condition (12.33) is nothing other than the Čerenkov radiation condition (see (6.58)). This immediately makes clear the physical cause of the absorption that occurs in the absence of collisions (at first glance, this result might seem paradoxical, for instance from the viewpoint of the elementary theory discussed above). Indeed, under the condition (12.33), a particle (e.g. an electron) emits Čerenkov waves (for these we have ω/k = vph= v cos θ, where θ is the angle between k and v). Any emission process can be reversed, however, and therefore under the same condition for ω and k the wave should produce the reverse effect, i.e. Čerenkov absorption, which involves the transfer of the appropriate energy and momentum from the wave to the particle. This is, of course, only a repetition of the interpretation of the collisionless damping discussed in Chapter 7. It is perhaps worthwhile emphasizing that when an “external” wave propagates in a plasma (in this case a plasma wave) we must consider its interaction with a “collective” of particles, rather than a single particle. Hence we must take into consideration, in addition to wave absorption, the stimulated emission of waves (this was also mentioned in Chapter 7). The collisionless absorption of plasma waves is often referred to as Landau damping since it was first analysed by Landau [153] when he solved the kinetic problem with initial conditions. Apart from the “Čerenkov interpretation”, collisionless damping can also be treated in a different way, and this can be especially simply done for cos θ=1, when ω/k = vph = v. Then the wave and the particle travel in the same direction, and those particles that slowly overtake the wave transfer their energy to it, while the particles that lag behind the wave obtain energy from it. The difference between these energies, proportional to (∂f00/∂v)v=vph, is what determines the collisionless damping.
Special Topics
Published in James J Y Hsu, Nanocomputing, 2017
Here, u≡k^·v⇀ $ u \equiv \hat{k} \cdot \overset{\rightharpoonup}{v} $ . Eq. (9.14) is identical to Eq. (9.8) if f0(u) is Maxwellian after an integration by parts. The Landau damping arises from ω = ku where the phase velocity of the wave equals the particle velocity. If there are more particles moving slower than the wave as in a Maxwellian distribution, the wave energy would be damped. On the other hand, when there is a bump on the tail of the electron distribution, so that there are more particles moving faster than the phase velocity of the wave at the rising slope of the velocity distribution, the wave can grow. By expanding Eq. (9.14) in ku/ω to yield 1≈ωp2/ω2(1+32k2υT2ωp2/ωp2)+...+iπωp2/k2(∂2f0/∂u)u=ω/k $ 1 \approx \omega^{2}_{p} /\omega^{2}(1 + \frac{3}{2}k^{2} \upsilon^{2}_{T} \omega^{2}_{p} /\omega^{2}_{p} ) + ... + i\pi \omega^{2}_{p} /k^{2} (\partial^{2} f0/\partial u)_{u = \omega /k} $ , the Bohm- Gross plasma dispersion relation can be found, ω2≈ωp2+32k2vT2 $ \omega^{2} \approx \omega_{p}^{2} + \frac{3}{2} k^2 v^2_T $ , as well as the damping rate of Eq. (9.11).
Electrostatic electron plasma wave envelope with nonlinear Landau damping in nonthermal plasmas
Published in Waves in Random and Complex Media, 2022
Debkumar Chakraborty, Samiran Ghosh
However, the dynamics of modulated electron plasma wave (EPW) in the presence of nonlinear Landau damping is investigated theoretically in the usual electron-ion plasma [59–61]. Later, based on the work of Ref. [59], different modulated plasma modes are investigated analytically in the presence of nonlinear Landau damping [53]. Physically, this nonlinear Landau damping occurs when the particle velocity approaches the group velocity of the modulated wave packet. In presence of such nonlinear effect the modulated wave is governed by a mNLSE which contains a nonlocal nonlinear term [59]. Furthermore, the usual NLSE type equations are solved numerically using finite difference, pseudo-spectral and split-step methods [24,62,63]. The mNLSE with nonlinear Landau damping in usual electron-ion plasma is simulated numerically with the finite difference scheme [61]. However, to the best of our knowledge, the modulated EPW dynamics with nonlinear Landau damping in presence of nonthermal (Cairns and Kappa) electrons is not well investigated.
Magnetoacoustic solitons in pair-ion fullerene plasma
Published in Waves in Random and Complex Media, 2020
Hafeez Ur-Rehman, S. Mahmood, S. Hussain
Magnetoacoustic wave is a magnetohydrodynamic (MHD) wave which is partially longitudinal (due to density compression) as well as partially transverse (due to compression of the magnetic field). It propagates in plasma in the perpendicular direction of the magnetic field. This MHD wave has properties in which both density and magnetic field compressions happen to be in phase and they play an important role for its propagation [19]. These waves are also named as fast MHD waves in plasma. In single fluid theory, the magnetoacoustic waves are nondispersive, however, in two fluid or multi-fluid approach these waves become dispersive and wave dispersion effects appear through electron inertial length in electron-ion plasma case [20, 21]. Magnetoacoustic waves are used for plasma heating through nonlinear ion Landau damping effects [22]. This wave can also derive plasma current in controlled thermo nuclear fusion devices [23–25].