Theory Of Parametric Instabilities Research Articles

Kinetic theory of parametric instabilities of lower hybrid waves in tokamaks in the electromagnetic framework

Parametric instabilities during the injection of lower hybrid waves into tokamak edge plasmas have received considerable attention for affecting the current drive at high density, mainly through spectral broadening. Here, kinetic theory of parametric instabilities was revisited in the electromagnetic framework. From Vlasov–Maxwell equations in the guiding center formalism, the theory of parametric instabilities in the electromagnetic framework is presented with appropriate nonlinearity treatment. Numerical calculations are performed under realistic lower hybrid current drive experimental parameters. Comparison between the electrostatic and electromagnetic results shows that the electromagnetic corrections have a stabilizing effect on the parametric process in all decay channels. The quantitative differences in growth rates, as well as in the most unstable regions, might have a non-negligible impact on the amplification factor of the parametric instabilities. However, with the dependence of parametric instabilities on the local plasma parameters, calculations show the same tendency between the electromagnetic and electrostatic model in a wide parameter range.

Linear theory of multibeam parametric instabilities in homogeneous plasmas

Linear theory of parametric instabilities resulting from N overlapping laser beams with an arbitrary beam configuration and polarization in a homogeneous plasma is presented. A general dispersion relation valid for both N-beam stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS) is derived. Analytically, we obtain the growth rate for a two-beam system and find the significance of two special geometries by sharing a common decay wave. By virtue of these geometries, the coefficient matrix of the general dispersion relation is decomposable and reducible in dimensions to give relatively simple formulae. Therefore, the most unstable mode is a trade-off among beam number, geometry, and polarization. We have also considered the spatio-temporal evolution of a highly symmetric N-beam shared scattered light (SL) mode, defined by N beams sharing a common scattered wave, and obtained convective gains for SRS and SBS. An estimation shows that both SRS and SBS in a N-beam SL mode may have significant impacts on inertial confinement fusion.

Nonlinearity in parametric instabilities during the injection of lower hybrid waves into tokamak plasmas

Nonlinear kinetic theory of parametric instabilities is developed and applied to study the parametric process during the injection of lower hybrid waves into tokamak plasmas. Four approaches containing different nonlinear terms are numerically solved and compared. For the resonant mode decay, the quasi-linear treatment is enough for the daughter branches. However, for the quasi-mode decay, the contribution from the nonlinear response of quasi-mode should be included in the governing equations of the sidebands. The results with correct nonlinear terms can recover those from the usually used fluid-kinetic hybrid approach in the electrostatic limit, but the kinetic theory and its simplification method developed here could be more generally applied to study parametric instabilities within the full electromagnetic framework.

Experiments and theory on parametric instabilities excited in HF heating experiments at HAARP

Parametric instabilities excited by O-mode HF heater and the induced ionospheric modification were explored via HAARP digisonde operated in a fast mode. The impact of excited Langmuir waves and upper hybrid waves on the ionosphere are manifested by bumps in the virtual spread, which expand the ionogram echoes upward as much as 140 km and the downward range spread of the sounding echoes, which exceeds 50 km over a significant frequency range. The theory of parametric instabilities is presented. The theory identifies the ionogram bump located between the 3.2 MHz heater frequency and the upper hybrid resonance frequency and the bump below the upper hybrid resonance frequency to be associated with the Langmuir and upper hybrid instabilities, respectively. The Langmuir bump is located close to the upper hybrid resonance frequency, rather than to the heater frequency, consistent with the theory. Each bump in the virtual height spread of the ionogram is similar to the cusp occurring in daytime ionograms at the E-F2 layer transition, indicating that there is a small ledge in the density profile similar to E-F2 layer transitions. The experimental results also show that the strong impact of the upper hybrid instability on the ionosphere can suppress the Langmuir instability.

PARAMETRIC INSTABILITY OF TWO-DISK ROTOR WITH TWO INERTIA ASYMMETRIES

Determination of operating conditions of parametric instability is crucial to the design and usage of the inertia asymmetric rotor. Current research mostly focused on the rotor with single inertia asymmetric disk. There are few studies on the multi-disk rotors with multiple inertia asymmetries. In fact, the interaction between the multiple parametric excitations with various phasing and amplitude, which are induced by the multiple unsymmetrical disks, would make the instability behavior of the system differ distinctly from that of the single-disk rotor system. Thus, the parametric instability of the two-disk rotor system with two inertia asymmetries is studied herein. Two important indicators for describing the unstable regions, namely the unstable rotating speed and width of the unstable region, are defined and derived using the parametric instability theory and Taylor expansion technique. For a practical used two-disk unsymmetrical rotor, three design parameters (inertia excitation phasing, relative position of the disk, and inertia ratio) are discussed in detail for their effects on the two indicators. It is shown from the results that the dynamic instability of the two-disk unsymmetrical rotor system indeed has some unique features that differ from that of the one-disk rotor system. The interaction of the two inertia parametric excitations could be utilized to control (or enhance) the unstable regions.

Parametric excitation of surface electron cyclotron X- and O- modes by alternating electric fields

Propagation of different electromagnetic surface waves (SW) is recognized as one of the most possible cause of undesirable plasma periphery heating, strengthening interaction between plasma column and walls of fusion device, impurity generation. This is found out clearly in experiments with ion cyclotron heating [1]. On the other hand in order to supply periodical operation of fusion devices, one can clean the devices inner surface. One of the prospective methods that can be used for this procedure is application both ion and electron cyclotron hardwares (without dismantling the whole device) for sustaining gas discharge, which can remove different contaminations from the chamber wall. The urgency of researches into parametric instability of SW at harmonics of cyclotron frequencies is caused as well by wide application of SW in various plasma technologies [2]. The present paper is devoted to development theory of parametric instability of slow surface Xand O-modes at electron cyclotron frequency for the model of semi-bounded plasma. Surface electron cyclotron O-modes (SECOM) were found theoretically [3] to propagate along plane plasmadielectric interface, when an external constant magnetic field is parallel to the plasma boundary and penetration depth of the modes is approximately equal to their wavelength. SECOM frequencies decrease if their wave number grows, their damping is determined by both collision (interaction between plasma particles) and kinetic (interaction between particles and plasma interface) mechanisms. Dispersion properties of surface electron cyclotron X-modes (SECXM) were studied theoretically in [4]. Unlike the SECOM they are unidirectional modes, their propagation coincides with direction of Larmor rotation of electrons on the plasma surface. They can propagate even along plasma-metal interface that makes them especially dangerous for destroying walls of fusion chambers. Sets of equations for these Xand O-modes are obtained by application of non-linear boundary conditions for their tangential magnetic field. Non-linear boundary conditions, which describe discontinuity of tangential component of their magnetic field are formulated by the aid of conception of existence of surface electric current, which is induced by the mentioned above external alternation electric field on the plasma interface. The reduced sets of equations for the Xand O-modes wave packets are obtained in approximation that the packets are composed by the main harmonic and two nearest satellite harmonics. EPJ Web of Conferences DOI: 10.1051/

Observation of three-mode parametric interactions in long optical cavities

We report the observation of three-mode optoacoustic parametric interactions of the type predicted to cause parametric instabilities in a $77\text{\ensuremath{-}}\mathrm{m}$-long, high-optical-power cavity that uses suspended sapphire mirrors. Resonant interaction occurs between two distinct optical modes and an acoustic mode of one mirror when the difference in frequency between the two optical cavity modes is close to the frequency of the acoustic mode. Experimental results validate the theory of parametric instability in high-power optical cavities, and demonstrate tunable parametric gain $\ensuremath{\sim}{10}^{\ensuremath{-}2}$ and more than $20\phantom{\rule{0.3em}{0ex}}\mathrm{dB}$ amplification of a high-order optical mode power generated by an applied acoustic signal.

The role of nonlinear beating currents in the theory of parametric instabilities

It is shown that there are two contributions to the nonlinear beating current density arising from the interaction of a high‐frequency pump wave with a Langmuir wave. One of these has its origin in the product of the charge density of the Langmuir wave with the velocity vector of an electron due to the uniform high‐frequency pump field. The other one has not been given in explicit form before; its direction is that of the wave vector of the Langmuir wave for the case of an unmagnetized plasma and a dipole pump. If the wave vector of the Langmuir wave is parallel to the pump electric field and if the frequency ω2 of the Langmuir wave is only slightly smaller than the pump frequency ω1, then the two contributions almost cancel each other. However, in the more complicated case of collisional and magnetized plasmas the wave vector of the Langmuir wave is generally not parallel to the pump electric field. The current result makes it plausible that the two contributions to the nonlinear beating current may not cancel each other even for ω2 = ω1.

The formation of layered structures by parametric instability of finite-amplitude gravity waves

VHF Doppler radars occasionally indicate the presence of almost monochromatic long-period gravity waves in the mesosphere. The radar echo intensity occurs in the form of nearly horizontally extended layers which are assumed to be generated by enhanced turbulence although the Richardson numbers are significantly larger than unity, excluding both shear and static instabilities as primarymechanisms for transition to turbulence. In good agreement with these observations, the theory of parametric instability of finite-amplitude monochromatic gravity waves predicts fast growing anisotropic small-scale disturbances whose size is limited by molecular dissipation. The disturbances form turbulent layers moving with the phase velocity of the primary wave. They are “frozen” in the primary wave flow (Taylor's hypothesis) and thus confirm the general assumption that the fluid velocity can be obtained from the Doppler shift of backscattered radar echoes.

HF‐enhanced ion and plasma line spectra with two pumps

Spectra of the HF‐enhanced ion line and upshifted and downshifted plasma lines were obtained by a multiple‐pulse technique (9 pulses) and by a technique using pulse‐to‐pulse correlation (512 pulses) with a short (about 1 ms) interpulse period for high frequency resolution. Observations using the multiple‐pulse technique suggest the presence of two different types of spectral features. One type has a spectral width typically greater than 1 kHz, and another type a width much less than 400 Hz, the frequency resolution of the multiple‐pulse technique used. Observations using pulses with a short interpulse period show the width of the narrow spectral features to be in the 20‐ to 60‐Hz range. The results are in good qualitative agreement with predictions of the linear theory of parametric instabilities excited by two pumps. Observations of the narrow spectral features can be used to measure the line‐of‐sight component of ionospheric drift velocities of the electron gas with an accuracy of about 2 m/s.

Nonlinear mode coupling and saturation of decay instability in ion beam-plasma system

Properties of nonlinear mode coupling for the decay instability of the fast ion beams are experimentally clarified. Experimental results are qualitatively explained by parametric instability theory. Further, the nonlinear damping of the instability, due to the excitation of a new subharmonic wave by mixing with the pump, is observed to seriously affect its saturation.

On the MHD stability of a helical magnetic field of arbitrary amplitude

The authors have obtained a solution to the nonlinear MHD equations, which describes a helical magnetic field of arbitrarily large amplitude. The frequency and wave number of this field are not restricted by any particular relation. The stability properties of this large amplitude wave have been considered. This approach enables us to link together parametric instability theory with conventional MHD analysis of spatially varying magnetic fields.

General quasi-linear theory of parametric instabilities

The general form of the quasi‐linear equation is derived for plasmas in which there is a finite amplitude pump wave. Electrostatic turbulence is considered and the pump wave is treated in the dipole approximation. Appropriate number, momentum, and energy moment constraints are shown to be computable from the diffusion equation. The derivation of an H theorem indicates that the quasi‐linear diffusion of parametric instabilities always tends to an accessible asymptotic steady state.

Unified formalism of lower hybrid parametric instabilities in plasmas

A unified theory of parametric instabilities in the lower hybrid frequency region is formulated using the drift kinetic equation for electrons and the Vlasov equation for unmagnetized ions. The three wave decay process into various decay channels (viz., two lower hybrid waves, lower hybrid and ion acoustic waves) are examined and compared for various pump frequencies and powers. Nonlinear scattering rates by electrons and ions (i.e., nonlinear Landau damping) are evaluated and compared with previous calculations. It is seen that the dominant channel of decay is through ion acoustic and lower hybrid waves when Te/2Ti≳4 and through lower hybrid waves and lower hybrid quasi-modes when Te/2Ti<4.

Parametric Instabilities in Magnetized Plasma

The theory of parametric instabilities driven by a finite wavenumber general pump wave is developed for homogeneous magnetized plasma. The pump wave as well as the decay products can be electrostatic as well as electromagnetic. This formalism is applied to describe lower hybrid decay into electromagnetic modes. A separate treatment for the all electrostatic wave case is given using the particle drift equations. In particular the resonant decay instabilities and the oscillating two stream instabilities are studied.

Parametric instabilities generated by an energetic electron beam

The large amplitude electron plasma oscillations produced by a low density energetic electron beam can parametrically decay into growing non-resonant electron and ion waves. A similar parametric instability occurs in a plasma driven by a sufficiently strong applied electric field whose frequency is near the electron plasma frequency. A two-specie one-dimensional particle simulation model was used to study the beam-generated parametric instability and the results are compared with corresponding one-specie beam and external field simulations. For low beam densities the individual non-resonant modes grow approximately at the rate predicted by parametric instability theory. The growth of these modes eventually causes the resonant mode energy to decay from its saturation value at a rate proportional to the original parametric growth rate. The electron distribution function fe(v) develops suprathermal tails which cause enhanced non-thermal electric field energy fluctuations whose spectrum W(k<or=0.1ke) approximately k-2.

Thresholds of parametric instabilities near the lower-hybrid frequency

Resonant decay instabilities of a pump wave with frequency ω0 near the lower-hybrid frequency ϑlh are analyzed with respect to the wavenumber k of the decay waves and the ratio ω0/ωlh to determine the decay process with the minimum threshold. It was found that the lowest thresholds are for decay into an electron plasma (lower-hybrid) wave plus either a backward ion-cyclotron wave, an ion Bernstein wave, or a low-frequency sound wave. For ω0&amp;lt; (2ωlh)1/2, it was found that these decay processes can occur and have faster growth than ion quasi-modes provided the drift velocity (cE0/B0) is much less than the sound speed. In many cases of interest, electromagnetic corrections to the lower-hybrid wave rule out decay into all but short wavelength (kρi ≳1) waves. The experimental results are consistent with the linear theory of parametric instabilities in a homogeneous plasma.

Threshold of Lower Hybrid Parametric Instability in a Modulated Electron-Beam–Plasma System

An experimental study of parametric instability driven by a beam modulation near the lower hybrid frequency is presented. If the modulation voltage exceeds a threshold value, the pump wave appears to decay into an ion acoustic wave and a lower hybrid wave. The threshold electron drift speed is found to be of the order of the ion acoustic speed as predicted by a theory of parametric instability in an inhomogeneous plasma.

Parametric instabifity of transverse and Langmuir waves in a plasma

The parametric excitation of transverse and Langmuir waves by an externally-driven electromagnetic field of frequency (ω0 &gt; 2ωp) in a warm and collisional plasma is studied, using the fluid equations. By an application of the multiple- time-scale perturbation method, the threshold intensity and the growth rate above threshold are obtained. The results are compared with those of Goldman (1969) and Prasad (1968), both of whom worked with a kinetic model.The theory of parametric instabilities in plasmas has been the subject of numerous investigations in recent years. Broadly speaking, the instabilities can be grouped into two categories: those for which the excited waves are purely electrostatic (see e.g. DuBois &amp; Goldman 1965, 1967; Silin 1965; Lee &amp; Su 1966; Jackson 1967; Nishikawa 1968; Kaw &amp; Dawson 1969; Tzoar 1969; Sanmartin 1970; McBride 1970; Perkins &amp; Flick 1971; Fejer &amp; Leer 1972a, b; Bezzerides &amp; Weinstock 1972; DuBois &amp; Goldman 1972), and those for which one of the excited waves is electromagnetic (see e.g. Goldman &amp; Dubois 1965; Montgomery &amp; Alexeff 1966; Chen &amp; Lewak 1970; Bodner &amp; Eddleman 1972; Fejer &amp; Leer 1972b; Lee &amp; Kaw 1972; Forslund et al. 1972).

Arecibo heating experiments

Enhancements of various features of the incoherent scatter spectrum are observed when the ionosphere is illuminated with powerful, high frequency radio waves. The radio waves excite plasma instabilities producing lines or more complex spectral features near the local plasma frequency, at the local ion acoustic frequency, near the local gyrofrequency, and near twice the gyrofrequency. The enhancements occur in a thin slab as observed by the incoherent scatter radar and at both upshifted and downshifted frequencies with respect to the probing radar frequency. The enhancements are observed to vary with time when the excitation is held constant and is turned on or off.The high power radio waves are produced by a 160 kw transmitter feeding a log‐periodic pair of curtains mounted at the focus of the 1000‐ft reflector and covering the frequency range from 5 to 12 MHz. The effects are observed with the incoherent scatter radar using the same reflector and with ionosondes and photometers.The frequencies of the enhanced plasma line and the ion line and their relation to the pump (high frequency radio wave) frequency are predictable from available parametric instability theory. Other spectral features are being explained as the theory develops with the help of the observations. There remain some discrepancies, in particular the asymmetries in intensity, width, and fluctuations of the upshifted compared to the downshifted plasma lines.