RUDI GAELZER

In this paper, we derive the dielectric tensor for a plasma containing particles described by an
anisotropic superthermal (bi-kappa) velocity distribution function. The tensor components are
written in terms of the two-variables kappa plasma special functions, recently defined by Gaelzer
and Ziebell [Phys. Plasmas 23, 022110 (2016)]. We also obtain various new mathematical properties for these functions, which are useful for the analytical treatment, numerical implementation,
and evaluation of the functions and, consequently, of the dielectric tensor. The formalism developed
here and in the previous paper provides a mathematical framework for the study of electromagnetic
waves propagating at arbitrary angles and polarizations in a superthermal plasma.

We study the dispersion relation for low frequency waves in the whistler mode propagating along
the ambient magnetic field, considering ions and electrons with product-bi-kappa (PBK) velocity
distributions and taking into account the presence of a population of dust particles. The results
obtained by numerical analysis of the dispersion relation show that the decrease in the j indexes in
the ion PBK distribution contributes to the increase in magnitude of the growth rates of the ion
firehose instability and the size of the region in wave number space where the instability occurs. It
is also shown that the decrease in the j indexes in the electron PBK distribution contribute to
decrease in the growth rates of instability, despite the fact that the instability occurs due to the
anisotropy in the ion distribution function. For most of the interval of j values which has been
investigated, the ability of the non-thermal ions to increase the instability overcomes the tendency
of decrease due to the non-thermal electron distribution, but for very small values of the kappa
indexes the deleterious effect of the non-thermal electrons tends to overcome the effect due to the
non-thermal ion distribution.

Velocity distribution functions (VDFs) that exhibit a power-law dependence on the high-energy tail have been the subject of intense research by the plasma physics community. Such functions, known as kappa or superthermal distributions, have been found to provide a better fitting to the VDFs measured by spacecraft in the solar wind. One of the problems that is being addressed on this new light is the temperature anisotropy of solar wind protons and electrons. In the literature, the general treatment for waves excited by (bi-)Maxwellian plasmas is well-established. However, for kappa distributions, the wave characteristics have been studied mostly for the limiting cases of purely parallel or perpendicular propagation, relative to the ambient magnetic field. Contributions to the general case of obliquely-propagating electromagnetic waves have been scarcely reported so far. The absence of a general treatment prevents a complete analysis of the wave-particle interaction in kappa plasmas, since some instabilities can operate simultaneously both in the parallel and oblique directions. In a recent work, Gaelzer and Ziebell [J. Geophys. Res. 119, 9334 (2014)] obtained expressions for the dielectric tensor and dispersion relations for the low-frequency, quasi-perpendicular dispersive Alfvén waves resulting from a kappa VDF. In the present work, the formalism introduced by Ref. 1 is generalized for the general case of electrostatic and/or electromagnetic waves propagating in a kappa plasma in any frequency range and for arbitrary angles. An isotropic distribution is considered, but the methods used here can be easily applied to more general anisotropic distributions, such as the bi-kappa or product-bi-kappa.

The radiation emission mechanism responsible for both type-II and type-III solar radio bursts is commonly
accepted as plasma emission. Recently Ganse et al. suggested that type-II radio bursts may be enhanced when the electron foreshock geometry of a coronal mass ejection contains a double hump structure. They reasoned that the counter-streaming electron beams that exist between the double shocks may enhance the nonlinear coalescence interaction, thereby giving rise to more efficient generation of radiation. Ganse et al. employed a particle-in-cell simulation to study such a scenario. The present paper revisits the same problem with EM weak turbulence theory, and show that the fundamental (F) emission is not greatly affected by the presence of counter-streaming beams, but the harmonic (H) emission becomes somewhat more effective when the two beams are present. The present finding is thus complementary to the work by Ganse et al.
Key words: plasmas – radiation mechanisms: non-thermal – solar wind – Sun: radio radiation – turbulence – waves

Electrostatic harmonic Langmuir waves are virtual modes excited in weakly turbulent plasmas, first observed in early laboratory beam-plasma experiments as well as in rocket-borne active experiments in space. However, their unequivocal presence was confirmed through computer simulated experiments and subsequently theoretically explained. The peculiarity of harmonic Langmuir waves is that while their existence requires nonlinear response, their excitation mechanism and subsequent early time evolution are governed by essentially linear process. One of the unresolved theoretical issues regards the role of nonlinear wave-particle interaction process over longer evolution time period. Another outstanding issue is that existing theories for these modes are limited to one-dimensional space. The present paper carries out two dimensional theoretical analysis of fundamental and (first) harmonic Langmuir waves for the first time. The result shows that harmonic Langmuir wave is essentially governed by (quasi)linear process and that nonlinear wave-particle interaction plays no significant role in the time evolution of the wave spectrum. The numerical solutions of the two-dimensional wave spectra for fundamental and harmonic Langmuir waves are also found to be consistent with those obtained by direct particle-in-cell simulation method reported in the literature.

The dispersion relation for parallel propagating waves in the ion-cyclotron branch is investigated
numerically by considering that the velocity distribution of the ion population is a function of type
product-bi-kappa. We investigate the effects of the non-thermal features and of the anisotropy associated with this type of distribution on the ion-cyclotron instability, as well as the influence of different forms of the electron distribution, by considering Maxwellian distributions, bi-kappa distributions, and product-bi-kappa distributions. The cases of ions described by either Maxwellian or bi-kappa distributions are also considered, for comparison. The results of the numerical analysis show that the increase in the non-thermal character associated with the anisotropic kappa distributions for ions contributes to enhance the instability as compared to that obtained in the Maxwellian case, in magnitude and in wave number range, with more significant enhancement for the case of ion product-bi-kappa distributions than for the case of ion bi-kappa distributions. It is also shown
that the ion-cyclotron instability is decreased if the electrons are described by product-bi-kappa distributions, while electrons described by bi-kappa distributions lead to growth rates which are very
similar to those obtained considering a Maxwellian distribution for the electron population.

Yoon and Fang [Phys. Plasmas 15, 122312 (2008)] formulated a second-order nonlinear kinetic theory that describes the turbulence propagating in directions parallel/anti-parallel to the ambient magnetic field. Their theory also includes discrete-particle effects, or the effects due to spontaneously emitted thermal fluctuations. However, terms associated with the spontaneous fluctuations in particle and wave kinetic equations in their theory contain proper dimensionality only for an artificial one-dimensional situation. The present paper extends the analysis and re-derives the dimensionally correct kinetic equations for three-dimensional case. The new formalism properly describes the effects of spontaneous fluctuations emitted in three-dimensional space, while the collectively emitted turbulence propagates predominantly in directions parallel/anti-parallel to the ambient magnetic field. As a first step, the present investigation focuses on linear wave-particle interaction terms only. A subsequent paper will include the dimensionally correct nonlinear wave-particle interaction terms.

The plasma emission, or electromagnetic (EM) radiation at the plasma frequency and/or its harmonic(s), is generally accepted as the radiation mechanism responsible for solar type II and III radio bursts. Identification and characterization of these solar radio burst phenomena were done in the 1950s. Despite many decades of theoretical research since then, a rigorous demonstration of the plasma emission process based upon first principles was not available until recently, when, in a recent Letter, Ziebell et al. reported the first complete numerical solution of EM weak turbulence equations; thus, quantitatively analyzing the plasma emission process starting from the initial electron beam and the associated beam-plasma (or Langmuir wave) instability, as well as the subsequent nonlinear conversion of electrostatic Langmuir turbulence into EM radiation. In the present paper, the same problem is revisited in order to elucidate the detailed physical mechanisms that could not be reported in the brief Letter format. Findings from the present paper may be useful for interpreting observations and full-particle numerical simulations.

We investigate the dispersion relation for low frequency electromagnetic waves propagating parallel to the ambient magnetic field, considering that the velocity distributions of ions and electrons can be either bi-Maxwellian of product bi-kappa distributions. The effect of the anisotropy and non-thermal features associated to the product-bi-kappa distributions on the firehose instability are numerically investigated. The general conclusion to be drawn from the results obtained is that the increase in non-thermal features which is consequence of the decrease of the κ indexes in the ion distribution contributes to increase the instability in magnitude and wave number range, in comparison with bi-Maxwellian distributions with similar temperature anisotropy, and that the increase of non-thermal features in the electron distribution contributes to the quenching of the instability, which is nevertheless driven by the anisotropy in the ion distribution. Significant differences between results obtained either considering product-bi-kappa distributions or bi-kappa distributions are also reported.

Known radiation emission mechanisms in plasmas include bremmstrahlung (or free-free emission), gyro- and synchrotron radiation, cyclotron maser, and plasma emission. For unmagnetized plasmas, only bremmstrahlung and plasma emissions are viable. Of these, bremmstrahlung becomes inoperative in the absence of collisions, and the plasma emission requires the presence of electron beam, followed by various scattering and conversion processes. The present Letter proposes a new type of radiation emission process for plasmas in a state of thermodynamic quasi-equilibrium between particles and enhanced Langmuir turbulence. The radiation emission mechanism proposed in the present Letter is not predicted by the linear theory of thermal plasmas, but it relies on nonlinear wave-particle resonance processes. The electromagnetic particle-in-cell numerical simulation supports the new mechanism.

A recent paper [Ziebell et al., Phys. Plasmas 21, 010701 (2014)] discusses a new type of radiation emission process for plasmas in a state of quasi-equilibrium between the particles and enhanced Langmuir turbulence. Such a system may be an example of the so-called “turbulent quasi-equilibrium.” In the present paper, it is shown on the basis of electromagnetic weak turbulence theory that an initial thermal equilibrium state (i.e., only electrostatic fluctuations and Maxwellian particle distributions) transitions toward the turbulent quasi-equilibrium state with enhanced electromagnetic radiation spectrum, thus demonstrating that the turbulent quasi-equilibrium discussed in the above paper correctly describes the weakly turbulent plasma dynamically interacting with electromagnetic fluctuations, while maintaining a dynamical steady-state in the average sense.

The effects of velocity distribution functions (VDF) that exhibit a power-law dependence on the high-energy tail have been the subject of intense research by the space plasma community. Such functions, known as superthermal or kappa distributions, have been found to provide a better fitting to the VDF measured by several spacecraft in the plasma environment of the solar wind. In the literature, the general treatment for waves excited by (bi-)Maxwellian plasmas is well-established. However, for kappa distributions, either isotropic or anisotropic, the wave characteristics have been studied mostly for the limiting cases of purely parallel or perpendicular propagation. Contributions for the general case of obliquely-propagating waves have been scarcely reported so far. In this work we introduce a mathematical formalism that provides expressions for the dielectric tensor components and subsequent dispersion relations for oblique propagating dispersive Alfvén waves (DAW) resulting from a kappa VDF. We employ an isotropic distribution, but the methods used here can be easily applied to more general anisotropic distributions, such as the bi-kappa or product-bi-kappa. The effect of the kappa index and thermal corrections on the dispersion relations of DAW is discussed.

The plasma emission is the radiation mechanism responsible for solar type II and type III radio bursts. The first theory of plasma emission was put forth in the 1950s, but the rigorous demonstration of the process based upon first principles had been lacking. The present Letter reports the first complete numerical solution of electromagnetic weak turbulence equations. It is shown that the fundamental emission is dominant and unless the beam speed is substantially higher than the electron thermal speed, the harmonic emission is not likely to be generated. The present findings may be useful for validating reduced models and for interpreting particle-in-cell simulations.

In this paper an electromagnetic particle code is used to investigate the spontaneous thermal emission. Specifically we perform particle-in-cell simulations employing a non-relativistic isotropic Maxwellian particle distribution to show that thermal fluctuations are related to the origin of spontaneous magnetic field fluctuation. These thermal fluctuations can become seed for further amplification mechanisms and thus be considered at the origin of the cosmological magnetic field, at microgauss levels. Our numerical results are in accordance with theoretical results presented in the literature.

This Letter reports on the results of numerical simulations which may provide a possible explanation for the strahl broadening during quiet solar conditions. The relevant processes involved in the broadening are due to kinetic quasi-linear wave-particle interaction. Making use of static analytical electron distribution in an inhomogeneous field, it is found that self-generated electrostatic waves at the plasma frequency, i.e., Langmuir waves, are capable of scattering the strahl component, resulting in energy and pitch-angle diffusion that broadens its velocity distribution significantly. The present theoretical results provide an alternative or complementary explanation to the usual whistler diffusion scenario, suggesting that self-induced electrostatic waves at the plasma frequency might play a key role in broadening the solar wind strahl during quiet solar conditions.

The solar wind electrons observed at 1 AU are characterized by velocity distribution functions (VDF) that deviate from the Maxwellian form in a high energy regime. Such a feature is often modeled by a kappa distribution. In the present paper a self-consistent theory of quiet-time solar wind electrons that contain a power-law tail component, f ∝ v-α is discussed. These electrons are assumed to be in dynamic equilibrium with enhanced electrostatic fluctuations with peak frequency near the plasma frequency (i.e., the Langmuir turbulence). In order to verify the theoretical prediction, the solar wind electrons in the high-energy range known as the super-halo distribution detected by WIND and STEREO spacecraft are compared against the theoretical model where it was found that the theoretical power-law index is intermittent with regard to the observed range of indices, thus indicating that the turbulent equilibrium model of suprathermal solar wind electrons may be valid.

The statistical mechanical reformulation of weak turbulence theory for unmagnetized plasmas including fully electromagnetic effects was carried out by Yoon [Phys. Plasmas 13, 022302 (2006)]. However, the wave kinetic equation for the transverse wave ignores the nonlinear three-wave interaction that involves two transverse waves and a Langmuir wave, the incoherent analogue of the so-called Raman scattering process, which may account for the third and higher-harmonic plasma emissions. The present paper extends the previous formalism by including such a term.

Charged particle acceleration takes place ubiquitously in the Universe including the near-Earth heliospheric environment. Typical in situ spacecraft measurements made in the solar wind show that the charged particle velocity distribution contains energetic components with quasi scale-free power-law velocity dependence, f ∼ v − α , for high velocity range. In this Review a theory of quiet-time solar-wind electrons that contain a suprathermal component is discussed, in which these electrons are taken to be in dynamical equilibrium with Langmuir turbulence. This Review includes an overview of the Langmuir turbulence theory, as well as a discussion on asymptotic equilibrium solution of Langmuir turbulence/suprathermal electron system. Theoretical predictions of high-energy electron velocity power-law distribution index is then compared against the recent observations of the superhalo electron velocity distribution made by instruments onboard WIND and STEREO spacecraft. It is shown that the theoretical prediction of velocity power-law index is intermediate to the observed range.

In a pair of recent papers (Ziebell et al 2008 Phys. Plasmas 15 032303, 2008 Plasma Phys. Control. Fusion 50 085011) it was shown, within the context of weak turbulence theory, that the Langmuir turbulence generated by the bump-in-tail instability does not lead to Langmuir condensation (or accumulation of wave energy and momentum in the long-wavelength regime) in two dimensions. The present analysis finds that it is important to include the spontaneous scattering off Langmuir turbulence of the electrons, which is ignored in the customary literature when compared with a similar process involving ions, in order to recover the condensation of Langmuir waves in two dimensions.

We utilize a kinetic description to study the dispersion relation of Alfvén waves propagating parallelly to the ambient magnetic field in a dusty plasma, taking into account the fluctuation of the charge of the dust particles, which is due to inelastic collisions with electrons and ions. We consider a plasma in which the velocity distribution functions of the plasma particles are modelled as anisotropic kappa distributions, study the dispersion relation for several combinations of the parameters κ∥ and κ⊥, and emphasize the effect of the anisotropy of the distributions on the mode coupling which occurs in a dusty plasma, between waves in the branch of circularly polarized waves and waves in the whistler branch.

This article demonstrates the generation of enhanced ion-acoustic waves by beam-plasma instability in a density cavity. The self-consistent equations of weak turbulence theory that include quasi-linear, decay, and scattering processes as well as convective and dispersive effects are numerically solved for a one-dimensional density cavity. It is shown that significant enhancements of ion-acoustic waves occur in the presence of counterstreaming electron beams and that the enhanced ion-acoustic waves are initially localized near the center of the density cavity at large wavelengths. Later in the evolution, the enhancement in the spectrum of ion-acoustic waves spreads out toward the edges of the cavity, with a shift to smaller wavelengths, while the enhancement near the center of the cavity tends to decrease in magnitude. The significance of the present findings is discussed.

The problem of electron-beam propagation in inhomogeneous solar wind is intimately related to the solar type II and/or type III radio bursts. Many scientists have addressed this issue in the past by means of quasi-linear theory, but in order to fully characterize the nonlinear dynamics, one must employ weak-turbulence theory. Available numerical solutions of the weak-turbulence theory either rely on only one nonlinear process (either decay or scattering), or when both nonlinear terms are included, the inhomogeneity effect is generally ignored. The present paper reports the full solution of weak-turbulence theory that includes both decay and scattering processes, and also incorporating the effects of density gradient. It is found that the quasi-linear effect sufficiently accounts for the primary Langmuir waves, but to properly characterize the back-scattered Langmuir wave, which is important for eventual radiation generation, it is found that both nonlinear decay and scattering processes make comparable contributions. Such a finding may be important in the quantitative analysis of the plasma emission process with application to solar type II and/or type III radio bursts.

We present general expressions for the components of the dielectric tensor of magnetized dusty plasmas, valid for arbitrary direction of propagation and for situations in which populations of dust particles of different sizes are present in the plasma. These expressions are derived using a kinetic approach which takes into account the variation of the charge of the dust particles due to inelastic collisions with electrons and ions, and features the components of the dielectric tensor in terms of a finite and an infinite series, containing all effects of harmonics and Larmor radius, and is valid for the whole range of frequencies above the plasma frequency of the dust particles, which are assumed to be motionless. The integrals in velocity space which appear in the dielectric tensor are solved assuming that the electron and ion populations are described by anisotropic non-thermal distributions characterized by parameters κ ∥ and κ ⊥ , featuring the Maxwellian as a limiting case. These integrals can be written in terms of generalized dispersion functions, which can be expressed in terms of hypergeometric functions. The formulation therefore becomes specially suitable for numerical analysis.

Beam–plasma instability of inhomogeneous media is an important problem associated with practical applications in space and laboratory plasmas. Previous investigations of the spatially inhomogeneous beam–plasma instability problem assumed one-dimensional velocity and wave number space. This paper extends the available theory to two-dimensional velocity and wave number space. A fully self-consistent set of quasilinear particle and wave kinetic equations is solved for two-dimensional velocity and wave number space in both time and one-dimensional spatial inhomogeneity scale length. The analytical equations include induced and spontaneous emission processes, as well as convective and dispersive effects.