Quasilinear Term Research Articles

Nonlinear Hyperbolic Equations in Infinite Homogeneous Waveguides

ABSTRACT In this paper we prove global and almost global existence theorems for nonlinear wave equations with quadratic nonlinearities in infinite homogeneous waveguides. We can handle both the case of Dirichlet boundary conditions and Neumann boundary conditions. In the case of Neumann boundary conditions we need to assume a natural nonlinear Neumann condition on the quasilinear terms. The results that we obtain are sharp in terms of the assumptions on the dimensions for the global existence results and in terms of the lifespan for the almost global results. For nonlinear wave equations, in the case where the infinite part of the waveguide has spatial dimension three, the hypotheses in the theorem concern whether or not the Laplacian for the compact base of the waveguide has a zero mode or not.

Magnetic field influence on the low-temperature heat capacity of the CDW compound blue bronze [formula omitted

We have investigated the effect of magnetic field on the low-temperature heat capacity C p of the quasi-1 D compound blue bronze Rb 0.3 MoO 3 , characterized by a charge density wave (CDW) ground state. Below 1 K, C p is dominated by a quasi-linear term, γ T , characteristic of the disordered state of the CDW. Under applied field (between 0 and 2 T), there is an increase of the amplitude γ with H, and simultaneously a downward deviation from the linear regime occurs at an increasing crossover temperature. This could be due to the opening of a (pseudo)-gap in the corresponding spectral density of excitations, which develops with increasing magnetic field.

Computation of Fokker-Planck equation

In plasma physics, the interaction of radio-frequency waves with a plasma is described by a Fokker-Planck equation with an added quasilinear term. In nonlinear filtering with conditional probability density of the state x t {x_t} given the observation { y ( s ) : 0 ≤ s ≤ t } \left \{ {y\left ( s \right ):0 \le s \le t} \right \} is also described by a Fokker-Planck equation with an added first order term. Method for solving Fokker-Planck equation by means of ordinary differential equations is discussed.

A geometrical approach to front propagation problems in bounded domains with Neumann-type boundary conditions

We are interested in the asymptotic behavior of the solutions of scaled reaction-diffusion equations in bounded domains, associated with Neumann type boundary conditions, and more precisely in cases when such behavior is described in terms of moving interfaces. A typical example is the case of the Allen–Cahn equation associated with an oblique derivative boundary condition, where the generation of a front moving by mean curvature with an angle boundary condition is shown. In order to establish such results rigourously, we modify and adapt the “geometrical approach” introduced by P. E. Souganidis and the first author for solving problems in RN : we provide a new definition of weak solution for the global-in-time motion of fronts with curvature-dependent velocities and with angle boundary conditions, which turns out to be equivalent to the level-set approach when there is no fattening phenomenon. We use this definition to obtain the asymptotic behavior of the solutions of a large class of reaction-diffusion equations, including the case of quasilinear ones and (x, t)dependent reaction terms, but also with any, possibly nonlinear, Neumann boundary conditions.

Long-time heat release in the quasi-1D conductor (TMTSF)2PF6 at low temperatures

We have measured the heat capacity and the heat release of the quasi-1D conductor (TMTSF)2PF6 in its spin-density wave ground state. Below 1 K the presence of low-energy excitations is at the origin of slow heat release over time scales from about 1 s up to 105 s. In a first attempt, we analyzed this behaviour with a modified version of the standard tunneling model used in the case of amorphous solids, introducing a short-time cut-off in the distribution of the relaxation times of tunneling states. A remarkable feature of the heat release in (TMTSF)2PF6 is its high absolute value, which is 3–4 orders of magnitude larger than in structural glasses. The heat capacity contains a Schottky and a quasi-linear term.

Electron cyclotron resonance ion source one-dimensional fluid modeling

A one-dimensional (1D) fluid computer model for multiple ion species in an electron cyclotron resonance ion source (ECRIS) plasma has been developed. The ions species are assumed to be highly collisionally coupled and are treated using 1D fluid equations. The non-Maxwellian anisotropic electron distribution function is modeled by a 1D bounce-averaged Fokker–Planck code. ECR heating is included in the model as a quasilinear rf-diffusion term including relativistic detuning, rf pitch-angle scattering, and multiple resonance frequencies/locations. In a typical ECRIS, the electrons are very noncollisional and confined magnetically. The ions follow this electron confinement via the electrostatic potential. The 1D axial electrostatic potential profile predicted by the model shows an ion confining core electrostatic well as expected in ECRIS plasmas. Modeling results for the Argonne National Laboratory ECR-I ECRIS configuration are presented along with a discussion of the difficulties in benchmarking the model with Faraday cup measurements.

The effect ofannealing conditions on the physical properties ofNd0.67Sr0.33MnO3-δ

A study of the influence of annealing in atmospheres with definedoxygen partial pressures (1.5×10-6 Pa ⩽ PO2 ⩽ 10+5 Pa) on the electrical resistivity ρ(T),magnetic susceptibility χ(T), and heat capacity cp(T) ofceramic samples of Nd0.67Sr0.33MnO3-δ is presented. ρ(T) at 1000 °C showsno appreciable change for 10+2 Pa ⩽ PO2 ⩽ 10+5 Pa. For samples annealed in air a sharp peak isfound in cp(T) and in ρ(T) at the ferromagnetic transition(TC≈238 K). The insulator-metal transitionis observed below TC as a broad peak in ρ(T).Annealing at lower PO2 leads to an increase of ρwhereas TC decreases only by ≈2 K.Annealing at PO2 ⩽ 6.3×10-2 Pa suppressesTC and results in spin canting and a vanishing of the peakin cp(T). The specific heat (2-300 K) for samples annealed inair and in Ar/5% H2 is determined and thermodynamicstandard values are calculated. The problem of the separation ofcp(T) into lattice, electronic, and Mn and Nd magneticcontributions is discussed. cp(T < 20 K) is dominated by aSchottky-like anomaly from the Zeeman-split ground-state doublet ofNd3+. A large quasi-linear term is mimicked by abroadening of the Schottky-like anomaly and by magneticcontributions from Mn in oxygen-deficient samples. The results arediscussed in the context of oxygen deficiency δ, carrierconcentration, defect chemistry, and disorder.

Neoclassical and anomalous transport in axisymmetric toroidal plasmas with electrostatic turbulence

Neoclassical and anomalous transport fluxes are determined for axisymmetric toroidal plasmas with weak electrostatic fluctuations. The neoclassical and anomalous fluxes are defined based on the ensemble-averaged kinetic equation with the statistically averaged nonlinear term. The anomalous forces derived from that quasilinear term induce the anomalous particle and heat fluxes. The neoclassical banana-plateau particle and heat fluxes and the bootstrap current are also affected by the fluctuations through the parallel anomalous forces and the modified parallel viscosities. The quasilinear term, the anomalous forces, and the anomalous particle and heat fluxes are evaluated from the fluctuating part of the drift kinetic equation. The averaged drift kinetic equation with the quasilinear term is solved for the plateau regime to derive the parallel viscosities modified by the fluctuations. The entropy production rate due to the anomalous transport processes is formulated and used to identify conjugate pairs of the anomalous fluxes and forces, which are connected by the matrix with the Onsager symmetry.

Minority-ion distribution function of a plasma under fundamental and second-harmonic ion-cyclotron heating

The distribution function of the minority ions during ion-cyclotron heating is calculated from a kinetic equation composed of a Landau collision term and a surface-averaged quasi-linear heating term. The kinetic equation is solved by a moment method in which the minority-ion distribution function is expanded in irreducible tensorial Hermite polynomials. The coefficients of the expansion are shown to be solutions of a system of coupled algebraic equations, and the effective minority-ion temperature is deduced from a compatibility constraint. The latter equation is in general too complicated to be solved analytically. The distribution function obtained here is therefore a semi-analytical result.

Calculation of Energetic Ion Tail from Ion Cyclotron Resonance Frequency Heating

The second harmonic frequency of hydrogen ion cyclotron resonance heating experiment on HT-6M tokamak was studied by adding the quasi-linear wave-ion interaction term in the two-dimensional (velocity space), time-dependent, nonlinear and multispecies Fokker-Planck equation. The temporal evolution of ion distribution function and relevant parameters were calculated and compared with experiment data. The calculation shows that the ion temperature increases, high-energy ion tail (above 5 keV) and anisotropy appear when the wave is injected to plasma. The simulations are in reasonable agreement with experiment data.

QUASI-LINEAR ELASTODYNAMIC EQUATIONS FOR FINITE DIFFERENCE SOLUTIONS IN DISCONTINUOUS MEDIA

The linear elastodynamic equations are ill-posed for models which contain high contrast density discontinuities. This paper presents a quasi-linear superset of the linear equations that is well-posed for this situation. The extended system contains a conservation of mass equation and a quasi-linear convective term in the momentum equation. Density, momentum, and stress are the field variables in the quasi-linear system, which is cast in a first order form. Using a Lax–Wendroff finite difference approximation, the utility of the quasi-linear system is demonstrated by modeling underwater acoustic scattering from a truncated ice sheet. The model contains air, ice, and water with a density contrast between air and ice or water of O(103). Superlinear convergence of the Lax–Wendroff scheme is demonstrated for his heterogeneous medium problem.

Poloidal electric field due to electron-cyclotron resonance heating in tokamaks

The poloidal electric field generated by electron-cyclotron resonance heating is investigated for a tokamak plasma in the collisionless regime. This poloidal electric field is calculated by solving an adjoint equation to the linearized Fokker-Planck equation with a quasi-linear diffusion term. It is found from this calculation that the magnitude and the sign of the poloidal electric field depend strongly on the values of the inverse aspect ratio, the poloidal angle of the absorption point, the parallel velocity of resonant electrons normalized by the thermal velocity, and the strength of the relativistic correction to the resonance condition.

Effects of magnetized alpha particles on lower hybrid heating and current drive in a reactor grade plasma

The results of computer simulations of lower hybrid current drive (LHCD) for ITER parameters are presented. The damping of LH waves by alpha particles due to the interaction at high cyclotron resonances is obtained by solving the 2-D Fokker-Planck equation with the corresponding quasi-linear term (FPA code). This code is incorporated into a spatially 1-D LHCD code so as to obtain radial profiles of the radiofrequency (RF) power absorbed by electrons (1-D Fokker-Planck code) and alpha particles as well as the radial profile of the driven current. One-pass absorption is assumed and verified by a detailed analysis of LH rays for an ITER-like magnetic configuration. Noticeable absorption by alpha paraticles is observed only below ∼5 GHz of the generator frequency and is still acceptable down to ∼3 GHz (19% of the input power Pin, with a 16% reduction of the total driven current for Pin = 50 MW). Two competing quasi-linear effects influence the damping rates: the RF field reduces the slopes of the alpha particle distribution function but, since the absorption is accompanied by acceleration of groups of alpha particles above their initial velocity, the RF field also increases the number of resonant particles. Both the global results and the damping rates are compared with those obtained for unmagnetized alpha particles and with the linear approximation. For frequencies above ∼3 GHz, the latter approach was found to significantly overestimate the absorption by alpha particles. Some further mechanisms not included in the code but having possible effects on the interaction are also discussed.

Effect of friction on low frequency sound propagation in a gas-liquid foam

A model is proposed for propagation of low frequency acoustic disturbances in a gas-liquid foam with consideration of friction on interphase boundaries during liquid motion in a system of interconnected microcapillaries. A Burgers equation with quasilinear convolution-type term is obtained. Structure and dynamics of linear signals are studied over the range of applicability of the model.

Potential Enhancement Due to RF-Induced Parallel Velocity Space Diffusion in a Tandem Mirror

Potential enhancement due to RF-induced parallel velocity space diffusion in a tandem mirror is studied by using a one-dimensional model based on the linearized Fokker-Plank equation with a quasi-linear RF diffusion term. A modified Boltzmann relation between the plasma density and the confining potential is obtained as a function of the RF diffusion coefficient under the assumption of small deviation from the Maxwellian for distribution function. The relation shows that the confining potential increases with the increase of the RF diffusion coefficient. The confining potential is saturated for the flat distribution function in the limit of strong RF diffusion.

Enhanced pinch effect due to the electrostatic potential

An inward particle pinch appears necessary to explain experimental results in tokamaks. Neither the neoclassical pinch effect, which is too small, nor the off-diagonal quasilinear term, which is usually outward in the trapped particle regime, can account for the observations. A mechanism for an enhanced inward pinch is proposed, based on results for an asymmetric magnetic field bump [Phys. Fluids 15, 2211 (1972)]. Because turbulent fluctuations also break toroidal symmetry, an enhanced inward pinch driven by the fluctuations and the Ohmic inductive field E is expected. To demonstrate this effect, an inward particle flux is calculated for a model tokamak configuration that has an electrostatic potential bump Φ0 at toroidal angle ζ=2π. For the parameter regime r/R&amp;lt;eΦ0/Te&amp;lt;1, the flux is found to be Γ=−4.47K(q)(r/R)L(Φ0)(vte/Rνe) 1/2cNE/B, where r(R) is minor (major) radius, B is the magnetic field strength, vte is the electron thermal speed, νe is the electron–ion collision frequency, q is the safety factor, and K(q) and L(Φ0) are functions of q and Φ0, respectively. The results are also applicable to an asymmetric potential bump created externally to enhance the inward pinch flux of high energy, collisionless particles.

Local full-wave energy and quasilinear analysis in nonuniform plasmas

The subject of local wave energy in plasmas is treated via quasilinear theory from the dual perspectives of the action-angle formalism and gyrokinetic analysis. This work presents an extension to all orders in the gyroradius of the self-consistent wave-propagation/quasilinear-absorption problem using gyrokinetics. Questions of when and under what conditions local energy should be of definite sign are answered using the action-angle formalism. An important result is that the 'dielectric operators' of the linearized wave equation and of the local energy are not the same, a fact which is obscured when the eikonal or WKB assumption is invoked. This study was originally motivated by concern over the question of local energy for RF-heating of plasmas, where in certain instances, full-wave effects such as refraction, strong absorption, and mode conversion are of primary importance. Fundamentally, the RF-absorption must equate with the energy moment of the quasilinear term to achieve a correct energy balance.

ECRH current drive in Tokamak plasmas

The current drive by electron cyclotron resonance heating (ECRH) is investigated in a Tokamak with arbitrary aspect ratio. The r.f. power is assumed to be weak and the wave-induced current is calculated by using an adjoint equation to the steady-state linearized Fokker-Planck equation with a quasi-linear diffusion term. The trapped electrons and the modification of electron cyclotron resonance condition by the relativistic mass increase are shown to have significant effects on the efficiency of this current drive. The efficiency strongly depends on the values of the parallel velocity u0 of resonant electrons, the inverse aspect ratio in , the poloidal angle theta 0 of the absorption point, and the relativistic parameter S which represents the strength of the relativistic correction to the resonance condition. It is also pointed out that the electron-electron collisions enhance the trapping effects compared with those of the Lorentz gas model.

Neutral particle emission in JET during radiofrequency heating experiments

The paper deals with the simulation of neutral particle spectra obtained in JET with an array of four passive neutral particle analysers during discharges with applied stationary or modulated radiofrequency power in the ion cyclotron range of frequencies for a deuterium plasma with hydrogen minority. The distribution function of the hydrogen ions has been computed using a one-dimensional Fokker-Planck equation describing wave-particle interaction by a quasi-linear diffusion term in velocity space. Both stationary and oscillating solutions of the Fokker-Planck equation have been worked out and the computed hydrogen fluxes have been found to be in very good agreement with the measured ones. In this context, the problems arising in the deduction of the power deposition profiles in modulated discharges are discussed. The profiles of power delivered to the electrons and the bulk ions have been obtained. It is shown that no adjustable parameters need to be introduced and that the main source of error lies in the uncertainty in the knowledge of the ion density.

Absorption of radio-frequency waves

The authors discuss two aspects of the theory of radio frequency heating of tokamaks. The first is the development of a simplified mathematical treatment of ion cyclotron absorption and its application to a study of the effect of a small concentration of high energy fusion products. The second is an investigation of the interaction of high amplitude electron cyclotron waves with a plasma. The nonlinear behaviour of electrons in such a wave is studied, with a view to improving on the usual quasilinear diffusion term used to describe wave-particle interactions.