Quasi-linear Analysis Research Articles

Cosmic ray current driven turbulence in shocks with efficient particle acceleration: the oblique, long-wavelength mode instability

In order for diffusive shock acceleration (DSA) to accelerate particles to high energies, the energetic particles must be able to interact with magnetic turbulence over a broad wavelength range. The weakly anisotropic distribution of accelerated particles, i.e., cosmic rays (CRs), is believed capable of producing this turbulence in a symbiotic relationship where the magnetic turbulence required to accelerate the CRs is created by the accelerated CRs themselves. In efficient DSA, this wave-particle interaction can be strongly nonlinear where CRs modify the plasma flow and the specific mechanisms of magnetic field amplification. Resonant interactions have long been known to amplify magnetic fluctuations on the scale of the CR gyroradius, and Bell (2004) showed that the CR current can efficiently amplify magnetic fluctuations with scales smaller than the CR gyroradius. Here, we show with a multi-scale, quasi-linear analysis that the presence of turbulence with scales shorter than the CR gyroradius enhances the growth of modes with scales longer than the gyroradius, at least for particular polarizations. We use a mean-field approach to average the equation of motion and the induction equation over the ensemble of magnetic field oscillations accounting for the anisotropy of relativistic particles on the background plasma. We derive the response of the magnetized CR current on magnetic field fluctuations and show that, in the presence of short-scale, Bell-type turbulence, long wavelength modes are amplified. The polarization, helicity, and angular dependence of the growth rates are calculated for obliquely propagating modes for wavelengths both below and above the CR mean free path. The long wavelength growth rates we estimate for typical supernova remnant parameters are sufficiently fast to suggest a fundamental increase in the maximum CR energy a given shock can produce.

Experimental Electron Temperature Gradient Dependence of Heavy Impurity Transport in Fusion Devices

The turbulent impurity (nickel) transport dependence on the normalized electron temperature gradient has been analyzed in sawtooth-free electron cyclotron wave heated Tore Supra plasmas. In the core, our experimental analysis shows that the lower R/L((T)(e)), the lower the nickel diffusion coefficient. The latter decreases until the instability threshold is reached. The experimental threshold is in agreement with the one computed by a gyrokinetic model. Further out, R/L((T)(e)) plays no role in the impurity diffusion. This set of experimental results is consistent with a quasilinear gyrokinetic analysis.

Mach number dependence of electron heating in high Mach number quasiperpendicular shocks

The efficiency of electron heating through microinstabilities generated in the transition region of a quasiperpendicular shock for a wide range of Mach numbers is investigated by utilizing particle-in-cell (PIC) simulation and model analyses. In the model analyses saturation levels of effective electron temperature as a result of microinstabilities are estimated from an extended quasilinear (trapping) analysis for relatively low (high) Mach number shocks. Here, modified two-stream instability (MTSI) is assumed to become dominant in low Mach number regime, while Buneman instability (BI) is assumed to become dominant in high Mach number regime. It is revealed that Mach number dependence of the effective electron temperature in the MTSI dominant case is essentially different from that in the BI dominant case. The effective electron temperature through the MTSI does not depend much on the Mach number, although that through the BI increases with the Mach number as in the past studies. The results are confirmed to be consistent with the PIC simulations both in qualitative and quantitative levels. The model analyses predict that a critical Mach number, above which a steep rise in electron heating rate occurs, may arise at the Mach number of a few tens.

The role of ion and electron electrostatic turbulence in characterizing stationary particle transport in the core of tokamak plasmas

A general feature of particle transport in the core of tokamak plasmas is that when core particle sources are small, a stationary peaked density profile is provided by a balance of outward diffusion and inward convection, driven by either neoclassical or turbulent mechanisms. The turbulent contribution to the off-diagonal elements of the transport matrix is very sensitive to the type of dominant instability of the background turbulence. We present here a detailed quasi-linear gyrokinetic analysis of stationary turbulent particle transport by means of analytical and numerical calculations to show how the actual parametric dependence of the stationary normalized density gradient can strongly vary between an ion temperature gradient (ITG) dominated turbulence and a trapped electron mode dominated turbulence regime. It is also shown how the maximal achievable normalized density gradient is reached when the turbulence regime is in a mixed state. This result is interpreted as the interplay of different physical mechanisms arising from (linear) wave–particle resonances. The results presented here are addressed to interpret some of the still unresolved issues in interpreting known experimental results.

Oxygen-ion-beam-driven electrostatic ion cyclotron instability of hydrogen plasma

The electrostatic ion cyclotron instability of hydrogen plasma driven by an oxygen ion beam and resulting turbulent heating of both ion species is investigated. The instability growth rate exceeds the oxygen ion gyrofrequency, so that the oxygen ions may be considered as unmagnetized during the process of waves growth. As a result the instability is developed due to inverse Landau damping of the ion cyclotron waves caused by thermal motion of oxygen ions across the magnetic field. The quasilinear analysis of the turbulent heating of both ion species resulted from their interactions with ion cyclotron turbulence indicates that this instability may be responsible for the observed anisotropic heating of auroral outflowing oxygen O+ ions in the ionosphere.

3D PiC code simulations for a laboratory experimental investigation of Auroral Kilometric Radiation mechanisms

Auroral Kilometric Radiation (AKR), occurs naturally in the polar regions of the Earth's magnetosphere where electrons are accelerated by electric fields into the increasing planetary magnetic dipole. Here conservation of the magnetic moment converts axial to rotational momentum forming a horseshoe distribution in velocity phase space. This distribution is unstable to cyclotron emission with radiation emitted in the X-mode. In a scaled laboratory reproduction of this process, a 75–85 keV electron beam of 5–40 A was magnetically compressed by a system of solenoids and emissions were observed for cyclotron frequencies of 4.42 GHz and 11.7 GHz resonating with near cut-off TE0,1 and TE0,3 modes, respectively. Here we compare these measurements with numerical predictions from the 3D PiC code KARAT. The 3D simulations accurately predicted the radiation modes and frequencies produced by the experiment. The predicted conversion efficiency between electron kinetic and wave field energy of around 1% is close to the experimental measurements and broadly consistent with quasi-linear theoretical analysis and geophysical observations.

Weak turbulence theory and simulation of the gyro-water-bag model

The thermal confinement time of a magnetized fusion plasma is essentially determined by turbulent heat conduction across the equilibrium magnetic field. To achieve the study of turbulent thermal diffusivities, Vlasov gyrokinetic description of the magnetically confined plasmas is now commonly adopted, and offers the advantage over fluid models (MHD, gyrofluid) to take into account nonlinear resonant wave-particle interactions which may impact significantly the predicted turbulent transport. Nevertheless kinetic codes require a huge amount of computer resources and this constitutes the main drawback of this approach. A unifying approach is to consider the water-bag representation of the statistical distribution function because it allows us to keep the underlying kinetic features of the problem, while reducing the Vlasov kinetic model into a set of hydrodynamic equations, resulting in a numerical cost comparable to that needed for solving multifluid models. The present paper addresses the gyro-water-bag model derived as a water-bag-like weak solution of the Vlasov gyrokinetic models. We propose a quasilinear analysis of this model to retrieve transport coefficients allowing us to estimate turbulent thermal diffusivities without computing the full fluctuations. We next derive another self-consistent quasilinear model, suitable for numerical simulation, that we approximate by means of discontinuous Galerkin methods.

Quasi-linear analysis of whistler waves driven by relativistic runaway beams in tokamaks

The presence of a whistler wave instability (WWI) excited by runaway electrons may be the reason for the observation that the number of runaway electrons produced during disruptions in large tokamaks depends sensitively on the magnetic field strength. Previous work has shown that the linear growth rates of these waves are such that they are stable for high magnetic field (so the runaway beam can form) but unstable for low magnetic field. Here, it is shown that the quasi-linear diffusion process due to the WWI represents a very efficient pitch-angle scattering mechanism for runaways and consequently may stop runaway beam formation in large tokamak disruptions.

Semiclassical analysis for Hartree equation

We justify WKB analysis for Hartree equation in space dimension at least three, in a regime which is supercritical as far as semiclassical analysis is concerned. The main technical remark is that the nonlinear Hartree term can be considered as a semilinear perturbation. This is in contrast with the case of the nonlinear Schrodinger equation with a local nonlinearity, where quasilinear analysis is needed to treat the nonlinearity.

Destabilization of magnetosonic-whistler waves by a relativistic runaway beam

Magnetosonic-whistler waves may be destabilized by runaway electrons both in fusion and astrophysical plasmas. A linear instability growth rate of these waves in the presence of a runaway avalanche is calculated both perturbatively and by numerical solution of the full dispersion equation. The local threshold of the instability depends on the fraction of runaways, the magnetic field, and the temperature of the background plasma. The quasilinear analysis shows that the main result of the instability is the scattering of the electrons in pitch-angle. It appears possible that this instability could explain why the number of runaway electrons generated in tokamak disruptions depends on the strength of the magnetic field.

High-order electromechanical /spl Sigma//spl Delta/ modulation in micromachined inertial sensors

Analysis of second-order electromechanical sigma-delta (/spl Sigma//spl Delta/) inertial sensors shows that in-band quantization error introduces a resolution penalty, which cannot be eliminated by oversampling. In addition, a tradeoff between resolution and phase compensation forces such systems to operate with reduced phase margin. This paper introduces high-order electromechanical /spl Sigma//spl Delta/ modulation as an approach, which eliminates the quantization noise overhead and allows for increased phase compensation without degrading the resolution. Quasi-linear analysis is used to evaluate the contribution of the individual noise sources to the output of the system and to examine the effect of noise interaction on the behavior of electromechanical /spl Sigma//spl Delta/ modulators.

First observation of the B̃A11 state of SiH2 and SiD2 radicals by optical-optical double resonance spectroscopy

The B1A1 state of SiH2 and SiD2 was observed by the optical-optical double resonance technique for the first time. The electronic band origin of the B state of SiD2 was determined to be 27 214.11 cm(-1). A very clear exclusive behavior depending on the even/odd value of the bending vibrational quantum number was observed in the spectra, representing a quasilinear behavior of the B state. The barrier height to linearity was estimated to be approximately 125 cm(-1) by the quasilinear analysis of the bending vibrational level structure of SiD2.

Coupling of Sloshing and Ship Motions

The coupled effect between ship motions and sloshing is studied. Two-dimensional experiments of a hull section containing tanks filled with different levels of water excited in sway by regular waves have been conducted. Steady-state results are obtained for the sway amplitude. Even if violent sloshing occurs in the tanks, the steady-state motion is almost linear and sinusoidal with the frequency of the linear incident waves. This implies that higher-order harmonics of the sloshing force are filtered out by the system. Simulations of the modeled case are performed using a linear and a nonlinear sloshing model and mainly assuming linear external flow. For steady-state motion, a convolution formulation does not improve the results relative to using constant coefficients in the equation of motion. However, in order to properly model the transient behavior in an irregular sea, a convolution formulation must be included. The treatment of the retardation function for the external problem is discussed in detail. A good agreement between experiments and computations is reported. The calculated coupled motion is sensitive to the damping of the sloshing motion in a certain frequency range where the coupled sloshing and ship motions cause resonant ship motions. A quasilinear frequency domain analysis is used to explain this by introducing the sloshing loads as a frequency dependent spring.

Unsaturated quasi-linear flow analysis in V-shaped domains

An explicit, essentially 2D solution for steady unsaturated seepage flow from infinity to a corner, with boundaries kept at constant suction, is obtained for the Kirchhoff potential by the method of separation of variables. A system of coordinates coinciding with the corner boundaries is selected. Distributions of the pressure, stream lines and velocities are derived. Non-existence of steady flows at certain corner orientations, deflection of the incident flow by slanted boundaries and inflection points on the stream lines close to the vertex are discussed. One-dimensional limit for zones far from the trough is examined. Refraction and further collimation of the upper 2D flow in the second underlying porous medium with implications to geotechnical capillary barriers is studied. A vadose zone originated accretion is matched with a saturated ‘wing’, which appears on a slanted bedrock. Mathematically, this matching is done by linking the Dupuit–Forchheimer and quasi-linear models. A tilted water table is shown to entrain the recharging moisture with curvilinear stream lines in the unsaturated zone.

Transport of megaelectron volt protons for fast ignition

Quasi-linear analysis demonstrates that intense and nonrelativistic proton beams do not lose collectively their kinetic energy through transverse Weibel electromagnetic instabilities when interacting with supercompressed plasmas of inertial confinement interest.

RF and IF ports matching circuit synthesis for a simultaneous conjugate-matched mixer using quasi-linear analysis

A quasi-linear two-port approach between RF and IF ports to design a simultaneous conjugate-matched mixer is presented in this paper. Conventionally, mixer design is treated as a nonlinear three-port device problem. Nonetheless, with the exception of the large-signal local oscillator (LO) that exists at the LO port, the input RF and output IF signals that exist at the RF and IF ports, respectively, are small signals. Consequently, mixers can be approximated as bilateral quasi-linear two-port circuits with a time-variant transfer function between the RF and IF ports, in which the LO port of the mixer is treated as part of the two-port network. With this approximation, it can be shown mathematically that the optimum source and load matching networks required for attaining simultaneous conjugate match at the RF and IF ports are actually time invariant, thus implying that it is possible to synthesize these optimum impedance values. This proposed mixer design technique, together with the equations derived, are verified with block-diagram simulation and experimental measurements of two 2.4-GHz RF/420-MHz IF double-balanced diode mixers.

The Dominance of Dynamic Barlike Instabilities in the Evolution of a Massive Stellar Core Collapse That “Fizzles”

Core collapse in a massive rotating star may halt at subnuclear density if the core contains angular momentum J 1049 g cm2 s-1. An aborted collapse can lead to the formation of a rapidly rotating equilibrium object, which, because of its high electron fraction, Ye > 0.4, and high entropy per baryon, Sb/k ≈ 1-2, is secularly and dynamically stable. The further evolution of such a is driven by deleptonization and cooling of the hot, dense material. These processes cause the fizzler both to contract toward neutron star densities and to spin up, driving it toward instability points of the barlike modes. Using linear stability analyses to study the latter case, we find that the stability properties of fizzlers are similar to those of Maclaurin spheroids and polytropes despite the nonpolytropic nature and extreme compressibility of the fizzler equation of state. For fizzlers with the specific angular momentum distribution of the Maclaurin spheroids, secular and dynamic barlike instabilities set in at T/|W| ≈ 0.14 and 0.27, respectively, where T is the rotational kinetic energy and W is the gravitational energy of the fizzler, the same limits as found for Maclaurin spheroids. For fizzlers in which angular momentum is more concentrated toward the equator, the secular stability limits drop dramatically. For the most extreme angular momentum distribution we consider, the secular stability limit for the barlike modes falls to T/|W| ≈ 0.038, compared with T/|W| ≈ 0.09-0.10 for the most extreme polytropic cases known previously (Imamura et al.). For fixed equation-of-state parameters, the secular and dynamic stability limits occur at roughly constant mass over the range of typical fizzler central densities. Deleptonization and cooling decrease the limiting masses on timescales shorter than the growth time for secular instability. Consequently, unless an evolving fizzler reaches neutron star densities first, it will always encounter dynamic barlike instabilities before secular instabilities have time to grow. Quasi-linear analysis shows that the angular momentum loss during the early nonlinear evolution of the dynamic barlike instability is dominated by Newtonian self-interaction gravitational torques rather than by the emission of gravitational wave (GW) radiation. GW emission may dominate after the initial dynamic evolutionary phase ends. Nonlinear hydrodynamics simulations with a proper equation of state will be required to determine the ultimate outcome of such evolutions and to refine predictions of GW production by barlike instabilities.

Nonlinear cyclotron resonant wave-particle interaction in a nonuniform magnetic field

Quasilinear analysis of wave-particle interactions is presented for plasma flowing in a weakly nonuniform magnetic field configuration. The method presented is based on a scale separation between the length scales of quasilinear relaxation and the magnetic field inhomogeneity, allowing one to obtain large scale solutions for both particle distribution functions and wave spectra, without going into the details of the small scale quasilinear relaxation. The numerical example shows the existence of a secondary instability for an initially stable particle distribution function.

Interaction between lower hybrid waves and energetic ions in a tokamak system

This paper presents a numerical quasilinear analysis of the interaction between lower hybrid waves and energetic ions created by radio wave heating in a tokamak environment, discussing conditions which are favourable to significant interaction, and the consequent radial location of energy deposition.

Quasilinear evolution of the weakly relativistic electron cyclotron maser instability

This paper presents a quasilinear analysis of the relativistic electron cyclotron maser instability in which the self-consistent set of equations governing the evolution of the particle distribution function and the energy spectra of unstable waves is numerically solved for parameters typical of the Earth’s auroral zone plasma, taking into account both resonant and non-resonant diffusions. The results obtained show that only 0.1%∼0.2% of the particle energy is converted into wave energy by the loss cone instability, and also show that the saturation amplitude for the extraordinary mode increases in proportion to the ratio between electron cyclotron frequency and electron plasma frequency, in agreement with previous results obtained with numerical simulations.