No abstract

It is well known that absolute instabilities can be located by prescribed mappings from the complex frequency plane to the wave-number plane through the dispersion relation D(w, k) = 0. However, in many systems of physical interest the dispersion relation is polynominal in w while transcendental in k, and the implementation of this mapping procedure is particularly difficult. If one maps consecutive deformations of the Fourier integral path (originally along the real k-axis) into the w-plane, points having (8D/8k) = 0 are readily detected by the distinctive feature of their local maps. It is shown that a simple topological relationship between these points and the image of the real k-axis determines the stability characteristics of the system, without mapping from the w-plane back into the k-plane.

This chapter looks at the evolution of linear instabilities, which results from small-amplitude perturbations of equilibria. The study is critical in testing stability/instability of equilibria in nonlinear dynamics, and entails a study of the evolutions of instabilities in space and time. The starting point is based upon the linearized equations for small-amplitude perturbations superimposed on the equilibrium being tested. This entails finding the linear natural modes within space and time dependence. The instability evolution from localized perturbations is given by the Green's function for the linearly-unstable medium, leading to distinguishing between absolute and convective instability evolutions. The former indicates an unstable normal mode which eventually grows in time at every point in space, while the latter entails that the instability growth convects away from the originally localized perturbations, and eventually, the localized excitation region reverts back to the original equilibrium.

A numerical integration of Eq. (5.16) of Ref. 5 gives M = 0.78 [(S/Sir)/(X 2 /A)] for X = 2931 nm, L = 2 cm, and F~ 1, but this 0.78 correction has not been applied to any value of TR quoted herein. n M. Gross, Co Fabre, P. Pillet, and S. Haroche, Phys. Rev. Lett. 36, 1035 (1976); A. Flusberg, T. Mossberg, and S. R. Hartmann, Phys. Lett. 58A, 373 (1976). 12 Without a magnetic field several sublevels of 7.P 3 / 2 are excited leading to beats in the SF output: Q. H. F. Vrehen, H. M. J. Hikspoors, and H. M. Gibbs, Phys. Rev. Lett. 38, 764 (1977). 13 The dipole moments (Ref. 2) for the strongest transitions are, in 10" 18 esu cm, d SF~1 2, d purn -0.5, d lp _$ D = 1.75, and d ls _Qp=4A.

1-D Eulerian Vlasov-Maxwell simulations are presented which show kinetic enhancement of stimulated Raman backscatter (SRBS) due to electron trapping in regimes of heavy linear Landau damping. The conventional Raman Langmuir wave is transformed into a set of beam acoustic modes [L. Yin et al., Phys. Rev. E 73, 025401 (2006)]. For the first time, a low phase velocity electron acoustic wave (EAW) is seen developing from the self-consistent Raman physics. Backscatter of the pump laser off the EAW fluctuations is reported and referred to as electron acoustic Thomson scatter. This light is similar in wavelength to, although much lower in amplitude than, the reflected light between the pump and SRBS wavelengths observed in single hot spot experiments, and previously interpreted as stimulated electron acoustic scatter [D. S. Montgomery et al., Phys. Rev. Lett. 87, 155001 (2001)]. The EAW observed in our simulations is strongest well below the phase-matched frequency for electron acoustic scatter, and therefore the EAW is not produced by it. The beating of different beam acoustic modes is proposed as the EAW excitation mechanism, and is called beam acoustic decay. Supporting evidence for this process, including bispectral analysis, is presented. The linear electrostatic modes, found by projecting the numerical distribution function onto a GaussHermite basis, include beam acoustic modes (some of which are unstable even without parametric coupling to light waves) and a strongly-damped EAW similar to the observed one. This linear EAW results from non-Maxwellian features in the electron distribution, rather than nonlinearity due to electron trapping.

Substantial radio-frequency power in the ion-cyclotron range of frequencies can be effectively coupled to a tokamak plasma from poloidal current strap antennas at the plasma edge. If there exists an ion–ion hybrid resonance inside the plasma, then some of the power from the antenna, delivered into the plasma by fast Alfvén waves, can be mode converted to ion-Bernstein waves. In tokamak confinement fields the mode-converted ion-Bernstein waves can damp effectively and locally on electrons [A. K. Ram and A. Bers, Phys. Fluids B 3, 1059 (1991)]. The usual mode-conversion analysis that studies the propagation of fast Alfvén waves in the immediate vicinity of the ion–ion hybrid resonance is extended to include the propagation and reflection of the fast Alfvén waves on the high magnetic-field side of the ion–ion hybrid resonance. It is shown that there exist plasma conditions for which the entire fast Alfvén wave power incident on the ion–ion hybrid resonance can be converted to ion-Bernstein waves. In this extended analysis of the mode conversion process, the fast Alfvén waves can be envisioned as being coupled to an internal plasma resonator. This resonator extends from the low magnetic-field cutoff near the ion–ion hybrid resonance to the high magnetic-field cutoff. The condition for 100% mode conversion corresponds to a critical coupling of the fast Alfvén waves to this internal resonator. As an example, the appropriate plasma conditions for 100% mode conversion are determined for the Tokamak Fusion Test Reactor (TFTR) [R. Majeski et al., Proceedings of the 11th Topical Conference on RF Power in Plasmas, Palm Springs (American Institute of Physics, New York, 1995), Vol. 355, p. 63] experimental parameters.

Mode conversion of the fast Alfvén wave (FAW) at the ion-hybrid frequency in the ion cyclotron range of frequencies (ICRF) is studied in the presence of ion cyclotron absorption and direct electron damping in a tokamak plasma. The usual Budden model is extended to include the effect of electron damping and of the high-field-side cutoff, and is solved analytically and numerically. The mode-conversion efficiency is given as a function of the Budden transmission coefficient and of a phase integral, which describes interference between the incoming and outgoing waves. In incidence from the low-field side, a discrete spectrum of phases exists for which complete absorption (i.e., combined mode conversion and direct electron damping) of the FAW for a single transit of the resonance region can be achieved. This permits efficient electron heating and/or current drive via mode conversion of FAWs.

A new phenomenon of coherent acceleration of ions by a discrete spectrum of electrostatic waves propagating perpendicularly to a uniform magnetic field is described. It allows the energization of ions whose initial energies correspond to a region of phase space which is below the chaotic domain. The ion orbits below the chaotic domain are described very accurately using a perturbation analysis to second order in the wave amplitudes. This analysis shows that the coherent acceleration takes place only when the wave spectrum contains at least two waves whose frequencies are separated by an amount close to an integer multiple of the cyclotron frequency. The way the ion energization depends on the wavenumbers and wave amplitudes is also presented in detail using the results of the perturbation analysis.