Langmuir Wave Dispersion Relation Research Articles

A new approach to the evaluation and solution of the relativistic kinetic dispersion relation and verification with continuum kinetic simulation

The present work describes a new approach to evaluation and root finding for the kinetic dispersion relation of Langmuir waves, which is central to the analytical understanding of collisionless damping in plasmas. The plasma dispersion function is solved to machine precision using direct integration in the complex plane in combination with an analytic evaluation of the residue to account for the deformation along the Landau contour. To efficiently attain machine precision, the contour is displaced in the complex plane prior to integration, and numerical subtleties related to the placement of the contour are discussed. The approach is generic in that it applies to arbitrary distribution functions, with the present manuscript focused on relativistic cases. Detailed verification of results via direct kinetic simulation in a variety of configuration space dimensions is also presented. Finally, the technique is applied to the challenging case of highly relativistic (i.e. extremely hot) plasmas. Here we show both qualitative agreement with prior work, as well as the disappearance of the Landau root which would have significant implication for real-life observation or experiment.

Linear pair-creation damping of high-frequency plasma oscillation

We have studied the linear dispersion relation for Langmuir waves in plasmas of very high density, based on the Dirac–Heisenberg–Wigner formalism. The vacuum contribution to the physical observables leads to ultraviolet divergences, which are removed by a charge renormalization. The remaining vacuum contribution is small and is in agreement with previously derived expressions for the time-dependent vacuum polarization. The main new feature of the theory is a damping mechanism similar to Landau damping, but where the plasmon energy gives rise to creation of electron–positron pairs. The dependence of the damping rate (pair-creation rate) on the wavenumber, temperature, and density is analyzed. Finally, the analytical results of linearized theory are compared with numerical solutions.

Kinetic-Simulation Study of Propagation of Langmuir-Like Ionic Waves in Dusty Plasma

The propagation of ionic perturbations in a dusty plasma is considered through a three-species kinetic simulation approach, in which the temporal evolution of all three elements, i.e., electrons, ions, and dust particles are followed based on the Vlasov equation coupled with the Poisson equation. Two cases are focused upon: first, a fully electron-depleted dusty plasma, i.e., a plasma consisting of ions and dust particles. The second case includes dusty plasmas with large electron-to-ion temperature ratios. The main features of the ionic waves in these two settings, including the dispersion relation and the Landau damping rate, are studied. It is shown that the dispersion relation of the ionic waves perfectly matches the dispersion relation of Langmuir waves and hence is called Langmuir-like ionic waves and can be considered as Langmuir-like ionic waves. These waves can be theoretically predicted by the dispersion relation of the dust-ion-acoustic waves. The transition of ionic waves from dust-ion-acoustic to Langmuir-like waves is shown to be sharp/smooth in first/second case. The Landau damping rates based on simulation results are presented and compared with theoretical predictions wherever possible.

Two particle-in-grid realisations on spacetrees

The present paper studies two particle management strategies for dynamically adaptive Cartesian grids at hands of a particle-in-cell code. One holds the particles within the grid cells, the other within the grid vertices. The fundamental challenge for the algorithmic strategies results from the fact that particles may run through the grid without velocity constraints. To facilitate this, we rely on multiscale grid representations. They allow us to lift and drop particles between different spatial resolutions. We call this cell-based strategy particle in tree (PIT). Our second approach assigns particles to vertices describing a dual grid (PIDT) and augments the lifts and drops with multiscale linked cells. Our experiments validate the two schemes at hands of an electrostatic particle-in-cell code by retrieving the dispersion relation of Langmuir waves in a thermal plasma. They reveal that different particle and grid characteristics favour different realisations. The possibility that particles can tunnel through an arbitrary number of grid cells implies that most data is exchanged between neighbouring ranks, while very few data is transferred non-locally. This constraints the scalability as the code potentially has to realise global communication. We show that the merger of an analysed tree grammar with PIDT allows us to predict particle movements among several levels and to skip parts of this global communication a priori. It is capable to outperform several established implementations based upon trees and/or space-filling curves.

Competition between stimulated Raman scattering and two-plasmon decay in inhomogeneous plasma

We demonstrate competitions between stimulated Raman scattering (SRS) and two-plasmon decay (TPD) in the laser polarization plane in inhomogeneous near quarter-critical density plasma by using linear convective gain analysis and two-dimensional (2D) particle-in-cell (PIC) simulations. Linear theoretical analysis implies that convective SRS occurs in a wider and lower density region than absolute SRS and has a shared occurrence region with convective TPD. This convective SRS prefers a parameter space with the laser intensity larger than the order of 1015 W/cm2 and the density scale length about several hundreds microns, which may be common in large scale direct-drive scheme, shock ignition scheme, and hybrid-drive scheme. A convective nature and saturation mechanism under these parameter regions are identified to be Langmuir decay instability and strong pump depletion. The significance of this convective SRS is shown in our 2D PIC simulations that hot electrons are reduced through suppressing the electron staged acceleration by TPD in the lower density region due to its high phase velocity. Temperature induced competitions are also studied using a relativistic modification to the Langmuir wave dispersion relation when Te>5 keV. Both absolute and convective SRS are observed to be dominant in the simulations when the temperature is as high as 10 keV.

Bump-on-tail instability in space plasmas

Bump-on-tail instability of Langmuir waves propagating in unmagnetized Lorentzian plasma modeled by a Kappa velocity distribution with spectral index κ has been investigated in this paper. Growth rate of this microinstability has been determined analytically from the Langmuir wave dispersion relation. Change in the maximum growth rate with increasing κ has been numerically estimated for different number density and temperature ratios using solar wind data.

Influence of quantum energy equation on electronic plasma oscillations

Five-moment approximation in hydrodynamics includes not only a continuity equation and a momentum balance equation, but also an energy equation. A set of hydrodynamic equations in this approximation is presented for a system of nonrelativistic quantum particles with the Coulomb interaction. This set of equations is linearized, and dispersion relation for Langmuir waves in quasi-neutral system of electrons and immobile ions is obtained. The plasma waves suffer damping in this case, and also there is another damping branch.

Hydrodynamic limit of Wigner-Poisson kinetic theory: Revisited

In this paper, we revisit the hydrodynamic limit of the Langmuir wave dispersion relation based on the Wigner-Poisson model in connection with that obtained directly from the original Lindhard dielectric function based on the random-phase-approximation. It is observed that the (fourth-order) expansion of the exact Lindhard dielectric constant correctly reduces to the hydrodynamic dispersion relation with an additional term of fourth-order, beside that caused by the quantum diffraction effect. It is also revealed that the generalized Lindhard dielectric theory accounts for the recently discovered Shukla-Eliasson attractive potential (SEAP). However, the expansion of the exact Lindhard static dielectric function leads to a k4 term of different magnitude than that obtained from the linearized quantum hydrodynamics model. It is shown that a correction factor of 1/9 should be included in the term arising from the quantum Bohm potential of the momentum balance equation in fluid model in order for a correct plasma dielectric response treatment. Finally, it is observed that the long-range oscillatory screening potential (Friedel oscillations) of type cos(2kFr)/r3, which is a consequence of the divergence of the dielectric function at point k = 2kF in a quantum plasma, arises due to the finiteness of the Fermi-wavenumber and is smeared out in the limit of very high electron number-densities, typical of white dwarfs and neutron stars. In the very low electron number-density regime, typical of semiconductors and metals, where the Friedel oscillation wavelength becomes much larger compared to the interparticle distances, the SEAP appears with a much deeper potential valley. It is remarked that the fourth-order approximate Lindhard dielectric constant approaches that of the linearized quantum hydrodynamic in the limit if very high electron number-density. By evaluation of the imaginary part of the Lindhard dielectric function, it is shown that the Landau-damping region in ω-k plane increases dramatically by increase of the electron number-density.

Vlasov multi-dimensional model dispersion relation

A hybrid model of the Vlasov equation in multiple spatial dimension D > 1 [H. A. Rose and W. Daughton, Phys. Plasmas 18, 122109 (2011)], the Vlasov multi dimensional model (VMD), consists of standard Vlasov dynamics along a preferred direction, the z direction, and N flows. At each z, these flows are in the plane perpendicular to the z axis. They satisfy Eulerian-type hydrodynamics with coupling by self-consistent electric and magnetic fields. Every solution of the VMD is an exact solution of the original Vlasov equation. We show approximate convergence of the VMD Langmuir wave dispersion relation in thermal plasma to that of Vlasov-Landau as N increases. Departure from strict rotational invariance about the z axis for small perpendicular wavenumber Langmuir fluctuations in 3D goes to zero like θN, where θ is the polar angle and flows are arranged uniformly over the azimuthal angle.

Excitation of flow instabilities due to nonlinear scale invariance

A novel route to instabilities and turbulence in fluid and plasma flows is presented in kinetic Vlasov-Maxwell model. New kind of flow instabilities is shown to arise due to the availability of new kinetic energy sources which are absent in conventional treatments. The present approach is based on a scale invariant nonlinear analytic formalism developed to address irregular motions on a chaotic attractor or in turbulence in a more coherent manner. We have studied two specific applications of this turbulence generating mechanism. The warm plasma Langmuir wave dispersion relation is shown to become unstable in the presence of these multifractal measures. In the second application, these multifractal measures are shown to induce naturally non-Gaussian, i.e., a stretched, Gaussian distribution and anomalous transport for tracer particles from the turbulent advection-diffusion transport equation in a Vlasov plasma flow.

Landau damping of Langmuir waves in non-Maxwellian plasmas

As free electrons move in the nearest neighbour ion’s potential well, the equilibrium velocity departs from Maxwell distribution. The effect of the non-Maxwellian velocity distribution function (NMVDF) on many properties of the plasma such as the transport coefficients, the kinetic energy, and the degree of ionization is found to be noticeable. A correction to the Langmuir wave dispersion relation is proved to arise due to the NMVDF as well [Phys. Plasmas 17, 052105 (2010)]. The study is extended hereafter to include the effect of NMVDF on the Landau damping of Langmuir wave.

Langmuir wave dispersion relation in non-Maxwellian plasmas

The Langmuir wave dispersion relation is derived in partially ionized plasmas, where free electrons are confined to move in a nearest neighbor ions’ potential well. The equilibrium velocity distribution function experiences then, a departure from Maxwell distribution function. The effect of the non-Maxwellian character of the distribution function on the Langmuir phase and group velocities as well as the phase matching conditions and the nonlinear growth rate of decay instability is investigated. The proposed Langmuir wave dispersion relation is relevant to dense and cryogenic plasmas.

Kinetic dispersion of Langmuir waves. I. The Langmuir decay instability

We derive a fully kinetic, three-dimensional dispersion relation for Langmuir waves with a focus on the Langmuir decay instability (LDI). The kinetic dispersion is compared to the standard fluid dispersion found with an equation of state (EOS) closure. The EOS closure fails to capture the intricacies of the nonlinear pressure when high frequency electron plasma waves (EPWs) couple to low frequency ion acoustic waves (IAWs). In particular, we find discrepancies in the kλd scaling of the LDI growth rate, where k is the wavenumber of the incident EPW and λd is the Debye length. As a result, the kinetic dispersion relation for LDI results in instability thresholds that can be in excess of twice those predicted by the fluid theory. Both the fluid and kinetic dispersion relations predict a nonlinear frequency shift due to the beating of the pump and scattered EPWs, but again the kλd scaling of these frequency shifts differ. In addition, the kinetic dispersion predicts a nonlinear reduction in the IAW damping from the three-wave interaction.

Plasma equations in general relativity

Vlasov’s equation and the ideal multifluid equations are considered in manifestly covariant form. In the latter case, a thermodynamic closure (locally the first law of thermodynamics) leads to a generalized Kelvin/Helmholtz theorem. In the former case, the local dispersion relation for Langmuir waves in a strong gravitational field is derived and solved.