Nonlinear Saturation Mechanism Research Articles

Theory and simulation of high-gain ion-ripple lasers.

Both the Raman, coherent Compton, and Compton regimes of high-gain ion-ripple Lasers (IRL's) are studied. The IRL works by coupling a negative-energy beam wave (or ponderomotive potential) to an electromagnetic wave by means of the ion ripple (backward Raman scattering or backward coherent Compton scattering) or by amplification of an em wave by negative Landau damping (backward Compton scattering). By employing fluid theory, the dispersion relation for wave coupling is derived and used to calculate the radiation frequency and linear growth rate. The nonlinear saturation mechanism is explored. A multidimensional (one dimension in space, three dimensions in momenta and fields) particle-in-cell simulation code was developed to verify the ideas, scaling laws, and nonlinear mechanisms. The effect of momentum spread is also studied; there is a slow decrease in the growth rate and efficiency as well as broadening of the radiation spectrum. This scheme may provide tunable sources of coherent high-power radiation. By proper choice of device parameters, sources of microwaves, optical, and perhaps even x rays can be made. An ion ripple in a plasma can provide a very short undulator wavelength (e.g., \ensuremath{\sim}${10}^{\mathrm{\ensuremath{-}}2}$ cm) and strong dc electric fields (e.g., \ensuremath{\sim}${10}^{10}$ V/m; equivalent to a magnetic field of 30 T). The plasma also produces an ion channel that provides guiding of electron and laser beams. The IRL's may generate high-power (e.g., g1 MW), coherent, short-wavelength photons with relatively low beam energy (e.g., \ensuremath{\sim}10 MeV) and possibly low beam quality requirement. The availability of tunable sources for wide wavelength regimes, coherence and high power, as well as lower cost and simplicity of equipment are emphasized.

A kinetic theory of trapped-electron-driven drift wave turbulence in a sheared magnetic field

A kinetic theory of collisionless and dissipative trapped-electron-driven drift wave turbulence in a sheared magnetic field is presented. Weak turbulence theory is employed to calculate the nonlinear electron and ion responses and to derive a wave kinetic equation that determines the nonlinear evolution of trapped-electron mode turbulence. The saturated fluctuation spectrum is calculated using the condition of nonlinear saturation. The turbulent transport coefficients (D, χi, χe), are, in turn, calculated using the saturated fluctuation spectrum. Because of the disparity in the three different radial scale lengths of the slab-like eigenmode: Δ (trapped-electron layer width), xt (turning point width), and xi (Landau damping point), Δ<xt<xi, it is found that ion Compton scattering rather than trapped-electron Compton scattering is the dominant nonlinear saturation mechanism. Ion Compton scattering transfers wave energy from short to long wavelengths where the wave energy is shear damped. As a consequence, a saturated fluctuation spectrum ‖φ‖2(kθ)∼k−αθ (α=2 and 3 for the dissipative and collisionless regimes, respectively) occurs for kθ ρs<1 and is heavily damped for kθ ρs>1. The predicted fluctuation level and transport coefficients are well below the ‘‘mixing length’’ estimate. This is due to the contribution of radial wave numbers x−1t<kr≤ρ−1i to the nonlinear couplings, the effect of radial localization of the trapped-electron response to a layer of width Δ, and the weak turbulence factor 〈γle/ωk〉k<1, which enters the saturation level.

Weak turbulence theory of ion temperature gradient modes for inverted density plasmas

Typical profiles measured in H-mode (‘‘high confinement’’) discharges from tokamaks such as JET [Proceedings of the 15th European Conference on Controlled Fusion and Plasma Heating (European Physical Society, Geneva, 1988), Vol. 12B, Part 1, p. 2239] and DIII-D [Phys. Fluids 31, 3738 (1988)] suggest that the ion temperature gradient instability threshold parameter ηi (≡d ln Ti/ d ln ni) could be negative in many cases. Previous linear theoretical calculations [Phys. Fluids B 1, 1185 (1989)] have established the onset conditions for these negative ηi modes and the fact that their growth rate is much smaller than their real frequency over a wide range of negative ηi values. This has motivated the present nonlinear weak turbulence analysis to assess the relevance of such instabilities for confinement in H-mode plasmas. The nonlinear eigenmode equation indicates that the three-wave coupling to shorter wavelength modes is the dominant nonlinear saturation mechanism. It is found that both the saturation level for these fluctuations and the magnitude of the associated ion thermal diffusivity are considerably smaller than the strong turbulence mixing-length-type estimates for the more conventional positive ηi instabilities.

Comment on “Generation of broadband noise in the magnetotail by the beam acoustic instability” by P. B. Dusenbery

It is shown here that of the three plasma models for the generation of broadband noise in the magnetotail proposed by Dusenbery and Lyons (1985), two result in the same types of instabilities that are also excited in the Grabbe and Eastman (GE, 1984) model, while the third model introduces other modes not present in the GE model. For the plasma parameters given in the GE model, the ion/ion acoustic instability is a nonresonant one, even though the phase velocity of the excited waves falls within the distribution function of the core ions. The nonlinear saturation mechanism of the ion/ion acoustic instability is the trapping of both the core and the beam ions. In a reply, Dusenbery addresses several ongoing controversies which have resulted from studies of wave particle interactions in the plasma sheet boundary layer.

The maser synchrotron instability in an inhomogeneous medium: Determination of the spectral intensity of auroral kilometric radiation

The auroral kilometric radiation (AKR), which emanates from high‐altitude auroral regions of the earth, is mainly emitted in cold plasma depleted regions on the right‐handed X mode. The maser synchrotron instability (MSI) is likely to be responsible for the generation of this emission, accounting for the low density of the cold plasma at the source and the small instantaneous bandwidth of AKR. As the emitted frequency lies very close to the local X mode cutoff frequency, the existence of a gradient in the earth's magnetic field dramatically influences the wave propagation at the source, and thus the resonant wave‐particle process leading to the MSI. A fully inhomogeneous treatment is therefore required. In particular, the magnetic field inhomogeneity quenches the usual nonlinear saturation mechanisms and limits the spatial extension of the source region. It becomes then plausible to determine directly the AKR intensity through a linear transfer equation, with a source term linked to the spontaneous X mode emission. Using the evaluation of the gain carried out in a previous work, we solve this linear transfer equation and determine the AKR spectral intensity. The computed AKR fluxes are consistent with those observed, provided that nonthermal features of the electronic distribution function are present at sufficient energies in the source region. It is found that no nonlinear effect occurs for most AKR events, while the saturation level linked to the trapping of resonant electrons in the wave electric field corresponds well to the maximum observed AKR spectral intensity. Finally, a scenario is proposed for the generation of AKR X mode.

The evolution of nonlinear Alfvén waves subject to growth and damping

Results of a numerical study of Alfvén waves are presented subject to nonlinearity, dispersion, growth, and damping. The model presented is the derivative nonlinear Schrödinger equation, modified to include linear growth and damping processes. The processes that are considered are wave amplification by streaming particle distributions, and damping resulting from ion-cyclotron resonance absorption. These growth and damping mechanisms are dominant in different portions of wavenumber space. The primary role of nonlinearity is the transfer of wave energy from growing or amplified wavenumbers to those which are damped. A nonlinear saturation mechanism thereby results, in which instability of low wavenumber modes may be quenched. A simple phenomenological model is developed, which accounts for many of the salient features of the numerical calculations. The application of these results to observations of Alfvén waves upstream of the Earth’s bow shock is briefly considered. It is suggested that the short wavelength ‘‘shocklet’’ structures resemble the soliton-like pulses that emerge from the driven derivative nonlinear Schrödinger equation. However, the nonlinear effects discussed in this paper do not seem responsible for limiting the amplitude of the ‘‘low-frequency’’ waves in the foreshock region.

Theory of resistivity-gradient-driven turbulence

A theory of the nonlinear evolution and saturation of resistivity-driven turbulence, which evolves from linear rippling instabilities, is presented. The nonlinear saturation mechanism is identified both analytically and numerically. Saturation occurs when the turbulent diffusion of the resistivity is large enough so that dissipation caused by parallel electron thermal conduction balances the nonlinearly modified resistivity gradient driving term. The levels of potential, resistivity, and density fluctuations at saturation are calculated. A combination of computational modeling and analytic treatment is used in this investigation.

Experimental evidence on the dependence of 140‐megahertz radar auroral backscatter characteristics on ionospheric conductivity

From combined STARE and magnetometer observations during a moderately disturbed period, it is deduced that the conductivity rather than the electric field is the prime agent responsible for the long‐period variations of both the ground magnetic signatures and the backscatter cross section. At times, there is an almost linear dependence between backscatter amplitude and conductivity suggesting that, within the framework of a nonlinear saturation mechanism, it is the relative amplitude Δn/n of the plasma wave irregularities rather than the absolute amplitude Δn that must be limited. In addition to backscatter amplitude modulation, there is also a modulation observed in the E field direction which is interpreted as being due to secondary east‐west polarization electric fields controlled by conductivity modifications in the region.

Nonlinear saturation of drift waves in sheared magnetic field

A nonlinear saturation mechanism of electrostatic drift waves in a sheared magnetic field is presented. The basic physical mechanism is the coupling of the unstable, lowest radial mode to higher modes which are damped because of their short radial wavelengths. Stationary spectra are obtained analytically and applied to the current-driven drift wave instability.

On nonlinear water waves under a light wind and Landau type equations near the stability threshold

Various mechanisms of nonlinear saturation of water wave growth under the action of a light wind are discussed. The unstable wind may be saturated by nonlinear dissipation due to the energy transfer to the damping harmonics of the wave. Other nonlinear saturation mechanisms: nonlinear frequency shift, self-modulation or self-focusing of a wave packet may be effective in certain wavenumber regions. In case the wind speed is close to the critical one, an equation is derived for the complex wave amplitude. This equation describes all these nonlinear effects in near-critical systems. In the one-dimensional case this is the nonlinear Shrodinger equation with complex coefficients. Its solutions under various conditions are discussed.

Ion-beam-driven lower-hybrid instability and resultant anomalous beam slowing

A lower-hybrid instability with ion cyclotron harmonics is observed to be driven by an ion beam injected obliquely to the magnetic field confining the isothermal plasma of the Q-1 double plasma device. The instability occurs with the injection of a low-density, low-velocity beam and propagates normal to the field with phase velocity ω/k⊥ ≅ ub⊥, the perpendicular velocity component of the spiraling ions. The frequency spectrum, propagation, and growth rate are all in good agreement with a numerical calculation based on linear kinetic theory. Pulsed beams are used to follow the instability from the linearly growing stage to non-linear saturation. The anomalous perpendicular momentum loss of the beam is examined by both direct energy analysis and by measurements of the resultant beam orbit modifications. By varying the beam parameters, a transition of the non-linear saturation mechanism from the quasi-linear to the trapping regime is demonstrated.

Turbulence theory for the dissipative trapped electron instability

Drift orbit diffusion induced by turbulence acting on trapped electrons is shown to reduce and broaden the magnetic drift resonance and produce the dominant nonlinear saturation mechanism for the dissipative trapped electron instability. The fluctuation level obtained from such a theory is found to be consistent with present experimental observations.

High frequency parametric wave phenomena and plasma heating: A review

Abstract A survey of parametric instabilities in plasma, and associated particle heating, is presented. A brief summary of linear theory is given. The physical mechanism of decay instability, the purely growing mode (oscillating two-stream instability) and soliton and density cavity formation is presented. Effects of density gradients are discussed. Possible nonlinear saturation mechanisms are pointed out. Experimental evidence for the existence of parametric instabilities in both unmagnetized and magnetized plasmas is reviewed in some detail. Experimental observation of plasma heating associated with the presence of parametric instabilities is demonstrated by a number of examples. Possible application of these phenomena to heating of pellets by lasers and heating of magnetically confined fusion plasmas by high power microwave sources is discussed.

Parametrische Instabilitäten in Plasmen

AbstractLinear and nonlinear theoretical aspects of parametric instabilities in plasmas are reviewed. Applications in laser‐fusion and heating of toroidally confined plasmas are considered. We start with the electrostatic decay and modulational instabilities in homogeneous, unmagnetized plasmas. The basic equations are derived in a physically simple manner. Electromagnetic modes are included using a wave description for the scattering instabilities (Raman and Brillouin scattering). Kinetic limits where quasi‐modes are involved (Compton scattering) and the corresponding instabilities in magnetic plasmas are discussed. Using a WKB‐analysis the effects of density, temperature, and expansion velocity gradients as well as turbulence, finite laser bandwidth, boundaries, and oblique incidence are estimated. The calculations are extended to include such phenomena as sidescattering and decay near the quartercritical and critical density where the WKB approximation breaks down. Nonlinear saturation mechanisms are discussed for Raman, Brillouin, decay, and modulational instabilities. Special attention is given to pump depletion and particle trapping. The existence of solitary waves and the stability of the latter are investigated. The conclusions are compared with results of experiments as well as computer simulations.

Nonlinear saturation and angular rotation of instabilities in the E layer

Two plasma instabilities have been identified as causing small‐scale irregularities in the equatorial E layer. These instabilities are referred to as the two‐stream and gradient drift instabilities. Here a unified calculation is made of the saturated density amplitudes of these instabilities. The present nonlinear saturation mechanism is referred to as perturbed orbits or diffusing orbits. The density fluctuations required for this saturation are roughly 10% for the gradient drift instability and 1% for the two‐stream instability. This agrees with rocket measurements over India. In addition, a calculation is made of the frequency shift of two‐stream waves. This shift is a nonlinear effect due to the slowing down of electrons by the unstable waves, and could be viewed as an enhanced friction. It is shown that this frequency shift will cause the angular spectrum of the two‐stream waves to be rotated upwards or downwards, depending on the direction of the electrojet. Such rotations have been observed by radar.

Nonlinear Saturation and Harmonic Generation of an Electron Plasma Wave

The spatial growing plasma wave excited by an electron beam been studied. The gradual change of the dispersion roots from the cold beam type to the hot beam type was observed and it was in good agreement with the theoretical prediction by O'Neil and Malmberg. The nonlinear saturation mechanism of the growing plasma wave was also interpreted by the change of the dispersion roots which was caused by the feedback effect of the wave on the beam particles. the harmonic spectrum near onset of the saturation did not show the power spectrum of ω -5 or ω -2.5 . The termination of the saturation level of the unstable plasma wave was observed for the case of the high beam current, but the mechanism is not clear.