Nonlinear theory for relativistic plasma wakefields in the blowout regime

We present a theory for nonlinear, multidimensional plasma waves with phase velocities near the speed of light. It is appropriate for describing plasma waves excited when all electrons are expelled out from a finite region by either the space charge of a short electron beam or the radiation pressure of a short intense laser. It works very well for the first bucket before phase mixing occurs. We separate the plasma response into a cavity or blowout region void of all electrons and a sheath of electrons just beyond the cavity. This simple model permits the derivation of a single equation for the boundary of the cavity. It works particularly well for narrow electron bunches and for short lasers with spot sizes matched to the radius of the cavity. It is also used to describe the structure of both the accelerating and focusing fields in the wake.     

On the low-threshold parametric mechanism of the anomalous power absorption in electron cyclotron resonance heating experiments in toroidal devices

The experimental conditions that facilitate the excitation of parametric decay instabilities upon the electron cyclotron resonance heating of a plasma at the second harmonic extraordinary wave in tokamaks and stellarators and, as a result, make anomalous absorption of microwave power possible have been analyzed. It has been shown that, in the case of a nonmonotonic radial profile of the plasma density, when the beam of electron cyclotron waves passes near the equatorial plane of a toroidal device, the parametric excitation of electron Bernstein waves, as well as the generation of ion Bernstein waves propagating from the parametric decay region to the nearest ion cyclotron harmonic, where they efficiently interact with ions, is possible. The proposed theoretical model can explain the anomalous generation of accelerated ions observed upon electron cyclotron heating in small and moderate toroidal facilities.     

Nonlinear resonance of plasma waves in the thermal irregularities of ionospheric plasma

We study the effect of striction plasma density disturbances on the generation intensity of longitudional cold and plasma oscillations due to polarization of the magnetic field-aligned ionospheric plasma irregularities with δN_o<0 by a powerful radio wave. It is assumed that the plasma density level inside the irregularity intersects the upper-hybrid resonance level, in the vicinity of which the cold oscillations excited directly by a powerful radio wave are transformed to shorter-wave plasma oscillations. We consider the short plasma wave limit to reduce the problem to a system of two coupled equations for the cold wave induction and plasma wave electric field. The first equation is supplemented by a local source equal to the integral of the plasma wave electric field in the resonance region. The second equation involves the cold wave induction at the resonance point and describes the electric field of interacting waves in the resonance vicinity. We use simplifications connected with the small absorption of plasma waves propagating inside the irregularity and weak radiation of these waves outside the irregularity. These conditions correspond to the generation of eigenmodes of plasma oscillations trapped in the irregularity. We have obtained a resonance-type nonlinear equation for the electric field intensity (or energy flux) of eigenmode plasma waves with allowance for striction disturbances of the plasma density profile in the resonance region. It is shown that the striction expulsion of plasma is responsible for the occurrence of coefficients describing the change in the intensity of excitation and radiation of plasma waves at the irregularity boundary. Such an expulsion leads to variations of the efficient generation band of plasma eigenmodes with the total phase increment of the wave in the irregularity. It also leads to a change in the phase shift of the plasma wave reflected from the resonance. These coefficients and the nonlinear phase shift are expressed in terms of real wave functions of the nonlinear Airy equation which describes the electric field of the excited waves in the resonance vicinity when the dissipation is absent. Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation, Russian Academy of Sciences, Troitsk, Moscow region, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 3, pp. 270–297, March, 1998.     

Direct observation of electron-Bernstein wave heating by O-X-B-mode conversion at low magnetic field in the WEGA stellarator

The ordinary-extraordinary-Bernstein-mode conversion process for overdense plasma heating with electron-Bernstein waves is demonstrated in the WEGA stellarator at low magnetic field (approximately 50 mT) at 2.45 GHz. For the first time the conversion from an O wave to an X wave is clearly demonstrated by probe measurements of amplitude and phase of the wave field in the conversion region and supported by two-dimensional full-wave calculations. The propagation and resonant absorption of the Bernstein wave is measured in fast power modulation experiments.     

Relativistic warm plasma waveguide under the effects of plasma electrons and HF electrical field on Buneman instability

The stabilization effect of a strong HF (pump) electrical field and plasma electrons on a two-stream (Buneman) instability in a plane relativistic warm plasma waveguide is investigated; using the separation method to solve the two-fluid plasma model we separate the problem into two parts. The “temporal” (dynamical) part enables us to determine the frequencies and growth rates of unstable waves; this part within the redefinition of natural frequencies coincides with the system describing HF suppression of Buneman instability in uniform unbounded plasma. Natural frequencies of oscillations and spatial distribution of the amplitude of the self-consistent electrical field are determined from the solution of a boundary-value problem (“space part”) taking into account specific spatial distribution of plasma density. Plasma electrons are considered to have a relativistic thermal velocity. It is shown that the growth rate of instability in relativistic warm plasma is reduced compared to non-relativistic (cold or warm) plasma and relativistic cold plasma. In addition, it is found that the plasma electrons have no effect on the solution of the space part of the problem.     

基于范阿伦卫星观测的槽区嘶声波冷等离子体色散关系的适用性评估

等离子体层嘶声波对电子的散射损失是地球内外辐射带之间的槽区(1.8≤L≤3)形成的主要机制.冷等离子体色散关系被广泛地运用于量化嘶声波对高能电子的散射效应研究中,而在真实的磁层环境中,热等离子体的存在会修正嘶声波的色散特性.基于范阿伦双星的观测数据,对比了利用磁场观测数据得到的槽区嘶声波观测幅值和反演幅值,并研究了空间位置与地磁活动水平对嘶声波冷等离子体色散关系适用性的影响.结果表明,冷等离子体近似整体上高估了嘶声波的幅值,观测幅值与反演幅值的差异有着很强的日夜不对称性,而没有明显的地磁活动强度依赖性.此外发现,波动磁场的反演强度在低频(高频)处显著低于(高于)观测强度,意味着冷等离子体近似整体上高估(低估)了嘶声波对槽区较低(较高)能量电子的散射强度.研究证明,槽区嘶声波冷等离子体色散关系的适用范围有很强的空间区域与频率局限性,这对深入理解槽区电子的动态演化过程有非常重要的意义.     

Linear Theory of Radially Inhomogeneous, Unneutralized, Relativistic Electron Beams

A set of four, coupled, first-order differential equations describing the normal modes of a cylindrically symmetric, cold fluid, unneutralized, relativistic electron beam of arbitrary radial profile is derived. Effects of beam rotation and equilibrium fields are treated exactly. The differential equations are found to have singular points for a radially inhomogeneous beam wherever the eigenfrequency equals the cyclotron or Doppler resonance frequencies. The resulting branch cuts in the dispersion function give rise to secularly decaying contributions to the initial value problem. The rate of decay and the character of the eigenmode near the singularity are determined from the solution of the corresponding indicial equation. Discrete eigenmodes also exist and are obtained by numerical solution of the differential equations and boundary conditions. For realistic solid-beam equilibria, only one slow cyclotron-mode exists for any given pair of axial and azimuthal wavenumbers, and that mode is localized at the beam edge. Under identical conditions several slow space-charge modes exist and are not so distorted. However, even at the space-charge limit, the phase velocity of long-wavelength slow space-charge waves does not decrease to zero. These results are relevant to the Autoresonant and Converging Guide collective ion acceleration proposals.     

Dielectric Characteristics for Radio Frequency Waves in a Laboratory Dipole Plasma

Transverse and parallel dielectric permittivity elements have been derived for radio frequency waves in a laboratory dipole magnetic field plasma. Vlasov equation is resolved for both the trapped and untrapped particles as a boundary value problem to define their separate contributions to the dielectric tensor components. To estimate the wave power absorbed in the plasma volume the perturbed electric field and current density components are decomposed in a Fourier series over the poloidal angle. In this case, the dielectric characteristics can be analyzed independently of the solution of the Maxwell's equations. As usual, imaginary part of the parallel permittivity elements is necessary to estimate the electron Landau damping of radio frequency waves, whereas imaginary part of the transverse permittivity elements is important to estimate the wave dissipation by the cyclotron resonances. Computations of the imaginary part of the parallel permittivity elements are carried out in a wide range of the wave frequencies. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Electrostatic Waves in the Warm Magnetoplasma at the Cyclotron Harmonic Frequencies

Mode conversion and collisionless absorption of electromagnetic wave at the cyclotron harmonic frequencies in an inhomogeneous non-Maxwellian magnetoplasma have been studied. Under suitable energy transfer condition the converted electrostatic wave (plasma wave) either grows or damps. The expressions for the growth/damping rates of this wave have been derived and studied at the cyclotron harmonic frequencies. The effect of the temperature anisotropy on the growth/damping rate of the electrostatic wave at the second cyclotron harmonic frequency has been shown. Growth of such electrostatic waves at ionospheric heights may explain the observed upper hybrid resonance (UHR) echoes and noise bands at the second cyclotron harmonic frequency.     

Effects of Spatial Variation of Thermal Electrons on Whistler-Mode Waves in Magnetosphere

A ray-tracing method is developed to evaluate the wave growth/damping and specifically propagation trajectories of the magnetospherically reflected Whistler-mode waves. The methodology is valid for weak wave growth/damping when plasma is comprised of a cold electron population and a hot electron population, together with background neutralizing ions, e.g. protons. The effect of anisotropic thermal electrons on the propagation of Whistler-mode waves is studied in detail. Numerical results are obtained for a realistic spatial variation model of plasma population, including the cold electron density distribution, and the thermal electron density and temperature distribution. It is found that, analogous to the case of the typical cold plasma approximation, the overall ray path of Whistler-mode waves is insensitive to the thermal electron density and temperature anisotropy, and the ray path reflects where wave frequency is below or comparable to the local lower hybrid resonance frequency flhr. However, the wave growth is expected to be influenced by the thermal electron population. The results present a first detailed verification for the validity of the typical cold plasma approximation for the propagation of Whistler-mode waves and may account for the observation that the Whistler-mode waves tend to propagate on a particular magnetic shell L where the wave frequency is comparable to fthe.     

Propagation of acoustic waves in a medium with the Rayleigh energy release mechanism

This research continues theoretical studies of propagation of acoustic waves in a plasma considering it in the context of a Rayleigh medium. For the first time, the solution to the problem with the boundary and not the initial conditions is examined. It is shown that for small values of the parameter characterizing the energy input in the plasma, the amplification coefficients of a harmonic acoustic wave in the problem of propagation of the initial perturbation and in the problem with the boundary conditions are close. However, if the energy input increases, the amplification of the wave propagating from the source is larger than in the problem of the initial perturbation propagation. The same concerns the amplification of waves with different frequencies for fixed parameters of the plasma; i.e., the difference between the amplification coefficients is larger, the lower the wave frequency. The resultant analytic dependences make it possible to determine exactly which of the problems (with the initial or boundary conditions) should be solved to compute the amplification coefficient of acoustic waves under specific experimental conditions.     

Plasma-sheath transition in the magnetized plasma-wall problem for collisionless ions

A model for the collisionless plasma-wall problem under the action of an applied magnetic field is developed. The behavior of its solution is examined and found to be qualitatively consistent with experiment. The plasma and the sheath are then modeled separately to obtain the position of the quasi-neutral plasma boundary and the position of the edge of the electron-free sheath. It is shown that the plasma boundary can be specified as the point where the component of the ion velocity normal to the wall reaches the ion sound speed (Bohm criterion), and the sheath edge is specified as the point corresponding to Godyak's condition for the electric field. Studying the behavior near the plasma boundary and the sheath edge, the plasma solution and the solution of the space charge region are patched together to approximate the solution of the plasma-wall problem.     

On exterior boundary-value problems of the Helmholtz equation not deduced from waves

A boundary integral equation is applied to describe a special kind of exterior Helmholtz boundary-value problem that is not deduced from waves. Then the asymptotic property of O(r ^–2) decay at infinity and the uniqueness of the solution as well as its finite energy property are discussed.     

Generation and Radiation of Waves at Combined Frequencies in a Warm Inhomogeneous Plasma Layer

We investigate generation and radiation of waves at combined frequencies from an arbitrary inhomogeneous, isotropic plasma layer, when electromagnetic waves are obliquely incident on it to interact with a surface wave at the plasma boundary. This interaction has been described for both S- and P-polarized waves. We consider a warm plasma layer with thickness very small as compared to the wavelengths of oscillations. It is shown that generated waves are strongly amplified, compared to cold plasma, when phase velocities of generated waves approaches the electron thermal velocity. Waves are not emitted when P-polarized waves are incident perpendicular onto the plasma layer.     

Wave Trajectory and Electron Cyclotron Heating in Tokamak Plasmas

Wave trajectories in high density tokamak plasmas are studied numerically. Results show that the ordinary wave injected at an appropriate incident angle can propagate into the dense plasmas and is mode-converted to the extraordinary wave at the plasma cutoff, is further converted to the electron Bernstein wave during passing a loop or a folded curve near the upper hybrid resonance (UHR) layer, and is cyclotron damped away, resulting in local electron heating before arriving at the cyclotron resonance layer. Similar trajectory and damping are obtained when a microwave in a form of extraordinary wave is injected quasi-perpendicularly in the direction of decreasing toroidal field.     

On the Symmetrical Modes in a Cylindrical Nonisothermal Plasma Waveguide in a Finite External Magnetic Field

A MHD, symmetrical, nonpotential waves in nonuniform, cylindrical nonisothermal plasma waveguide placed in a finite external magnetic field H ₀ directed along the axis of the waveguide are considered. The thermal velocity of the electrons ν_Te = (kT_e/m_e)^1/2 (k — Boltzman's constant, T_e — temperature, m_e — mass of the electron) is not neglected. As a consequence we obtain one more solution in the plasma region and respectively an equation of third power on χ² — the transverse wave number. As far as we know only equations of second power on χ² describing non-potential waves are investigated. The dispersion equations are written solving the boundary-value problem. Our results coincide with well known ones when the electron thermal velocity tends to zero.     

Effect on Landau damping rates for a non-Maxwellian distribution function consisting of two electron populations

In many physical situations where a laser or electron beam passes through a dense plasma,hot low-density electron populations can be generated,resulting in a particle distribution function consisting of a dense cold population and a small hot population.Presence of such low-density electron distributions can alter the wave damping rate.A kinetic model is employed to study the Landau damping of Langmuir waves when a small hot electron population is present in the dense cold electron population with non-Maxwellian distribution functions.Departure of plasma from Maxwellian distributions significantly alters the damping rates as compared to the Maxwellian plasma.Strong damping is found for highly nonMaxwellian distributions as well as plasmas with a higher density and hot electron population.Existence of weak damping is also established when the distribution contains broadened flat tops at the low energies or tends to be Maxwellian.These results may be applied in both experimental and space physics regimes.     

Effect on Landau damping rates for non-Maxwellian distribution function consisting of two electron populations

In many physical situations where a laser or electron beam passes through a dense plasma, hot low-density electron populations can be generated, resulting in a particle distribution function consisting of a dense cold population and a small hot population. Presence of such low-density electron distributions can alter the wave damping rate. Kinetic model is employed to study the Landau damping of Langmuir waves when a small hot electron population is present in the dense cold electron population with non-Maxwellian distribution functions. Departure of plasma from Maxwellian distributions significantly alters the damping rates as compared to the Maxwellian plasma. Strong damping is found for highly non-Maxwellian distributions as well as plasmas with higher dense and hot electron population. Existence of weak damping is also established when the distribution contains broadened flat tops at the low energies or tends to be Maxwellian. These results may be applied in both experimental and space physics regimes.     

Electron cyclotron resonance heating in a short cylindrical plasma system

Electron cyclotron resonance (ECR) plasma is produced and studied in a small cylindrical system. Microwave power is delivered by a CW magnetron at 2.45 GHz in TE¹⁰ mode and launched radially to have extraordinary (X) wave in plasma. The axial magnetic field required for ECR in the system is such that the first two ECR surfaces (B = 875.0 G andB = 437.5 G) reside in the system. ECR plasma is produced with hydrogen with typical plasma density n^e as 3.2 × 10¹⁰ cm^-3 and plasma temperature T^e between 9 and 15 eV. Various cut-off and resonance positions are identified in the plasma system. ECR heating (ECRH) of the plasma is observed experimentally. This heating is because of the mode conversion of X-wave to electron Bernstein wave (EBW) at the upper hybrid resonance (UHR) layer. The power mode conversion efficiency is estimated to be 0.85 for this system. The experimental results are presented in this paper.