To calculate linear oscillations and waves in dynamics of gas and plasma one uses as a rule the old classical method of dispersion equation for complex frequencies $ω$ and wave numbers $k$: $ε(ω,k)=0$. This method appears to be inapplicable, f.e., in the case of waves in Maxwellian collisionless plasma when dispersion equation has no solutions. By means of some refined sophistication L.~Landau in 1946 has suggested in this case actually to replace the dispersion equation with another one, having a specific solution (``Landau damping'') and being now widely used in plasma physics. Recently we have suggested a quite new universal method of two-dimensional Laplace transformation (in coordinate $x$ and time $t$ for plane wave case), that allows to obtain asymptotical solutions of original Vlasov plasma equations as inseparable sets of coupled oscillatory modes (but not a single wave like $\exp(-iωt+i k x)$). The mode parameters are defined in this case by double-poles ($ω_n$,$k_n$) of Laplace image $E(ω_n,k_n)$ of electrical field $E(x,t)$. This method allows one to obtain the whole set of oscillatory modes for every concrete problem. It leads to some new ideology in the theory of plasma oscillations, which are considered as a combination of coupled oscillatory modes (characterized by pairs ($ω_n$,$k_n$) and amplitudes) and depend not only on the intrinsic plasma parameters, but also on mutually dependent self-consistent initial and boundary conditions and on method of plasma oscillations excitation.
The equivalence between the Laplace transform [Landau L., J. Phys. USSR, 10 (1946), 25] and Hermite transform [Zocco and Schekochihin, Phys. Plasmas, 18, 102309 (2011)] solutions of the linear collisionless Landau damping problem is proven.
A general solution to linearized Vlasov equation for an electron electrostatic wave in a homogeneous unmagnetized plasma is derived. The quasi-linear diffusion coefficient resulting from this solution is a continuous function of omega in contrast to that derived from the traditional Vlasov treatment. The general solution is also equivalent to the Landau treatment of the plasma normal oscillations, and hence leads to the well-known Landau damping.
Three-dimensional numerical simulations of decaying turbulence in a magnetized plasma are performed using a so-called FLR-Landau fluid model which incorporates linear Landau damping and finite Larmor radius (FLR) corrections. It is shown that compared to simulations of compressible Hall-MHD, linear Landau damping is responsible for significant damping of magnetosonic waves, which is consistent with the linear kinetic theory. Compressibility of the fluid and parallel energy cascade along the ambient magnetic field are also significantly inhibited when the beta parameter is not too small. In con…
Models of covariant linear electromagnetic constitutive relations are formulated that have wide applicability to the computation of susceptibility tensors for dispersive and inhomogeneous media. A perturbative framework is used to derive a linear constitutive relation for a globally neutral plasma enabling one to describe in this context a generalized Landau damping mechanism for non-stationary inhomogeneous plasma states.
Growth rate and gain of SBS considering the reduced Landau damping due to particle trapping has been proposed to predict the growth and average level of SBS reflectivity. Due to particle trapping, the reduced Landau damping has been taken used of to calculate the gain of SBS, which will make the simulation data of SBS average reflectivity be consistent to the Tang model better. This work will solve the pending questions in laser-plasma interaction and have wide applications in parametric instabilities.
Within the framework of the hypothesis offered by authors about complex-valued nature of physical quantities, the effect of the Landau damping has been explored with assumption that not only frequency can be a small imaginary component but also a wave vector. The numerical solution of the obtained dispersion equation testifies that uncollisional damping is accompanied in a certain region of space by antidumping of waves, and in particular situations antidumping may prevail over damping. It is possible that this effect may explain the experimental difficulties connected with inhibition of insta…
Separated spin-up and spin-down quantum kinetics is derived for more detailed research of the spin-electron acoustic waves. Kinetic theory allows to obtain spectrum of the spin-electron acoustic waves including effects of occupation of quantum states more accurately than quantum hydrodynamics. We apply quantum kinetic to calculate the Landau damping of the spin-electron acoustic waves. We have considered contribution of ions dynamics in the spin-electron acoustic wave spectrum. We obtain contribution of ions in the Landau damping in temperature regime of classic ions. Kinetic analysis for ion-…
The nonlinear theory of dust-acoustic waves (DAWs) with Landau damping is studied in an unmagnetized dusty negative-ion plasma in the extreme conditions when the free electrons are absent. The cold massive charged dusts are described by fluid equations, whereas the two-species of ions (positive and negative) are described by the kinetic Vlasov equations. A Korteweg de-Vries (KdV) equation with Landau damping, governing the dynamics of weakly nonlinear and weakly dispersive DAWs, is derived following Ott and Sudan [Phys. Fluids {\bf 12}, 2388 (1969)]. It is shown that for some typical laborator…
Starting from the Wigner-Moyal equation coupled to Poisson's equation, a simplified set of equations describing nonlinear Landau damping of Langmuir waves is derived. This system is studied numerically, with a particular focus on the transition from the classical to the quantum regime. In the quantum regime several new features are found. This includes a quantum modified bounce frequency, and the discovery that bounce-like amplitude oscillations can take place even in the absence of trapped particles. The implications of our results are discussed.
The nonlinear theory of Landau damping of electrostatic wave envelopes (WEs) is revisited in a quantum electron-positron (EP) pair plasma. Starting from a Wigner-Moyal equation coupled to the Poisson equation and applying the multiple scale technique, we derive a nonlinear Schr{ö}dinger (NLS) equation which governs the evolution of electrostatic WEs. It is shown that the coefficients of the NLS equation, including the nonlocal nonlinear term, which appears due to the resonant particles having group velocity of the WEs, are significantly modified by the particle dispersion. The effects of the q…
Kinetic treatments of drift-tearing modes that match an inner resonant layer solution to an external MHD region solution, characterised by $Δ^{\prime}$, fail to properly match the ideal MHD boundary condition on the parallel electric field, $E_{\parallel}.$ In this paper we demonstrate how consideration of ion sound and ion Landau damping effects achieves this and place the theory on a firm footing. As a consequence, these effects contribute quite significantly to the critical value of $Δ^{\prime}$ for instability of drift-tearing modes and play a key role in determining the minimum value for …
We perform fully kinetic simulations of flows known to produce dynamo in magnetohydrodynamics (MHD), considering scenarios with low Reynolds number and high magnetic Prandtl number, relevant for galaxy cluster scale fluctuation dynamos. We find that Landau damping on the electrons leads to a rapid decay of magnetic perturbations, impeding the dynamo. This collisionless damping process operates on spatial scales where electrons are nonmagnetized, reducing the range of scales where the magnetic field grows in high magnetic Prandtl number fluctuation dynamos. When electrons are not magnetized dow…
An exact solution of the collisionless time-dependent Vlasov equation is found for the first time. By means of this solution the behavior of the Langmuir waves in the nonlinear stage is considered. The analysis is restricted by the consideration of the first nonlinear approximation keeping the second power of the electric strength. It is shown that in general the waves with finite amplitudes are not subject to damping. Only in the linear approximation, when the wave amplitude is very small, are the waves experiencing damping. It is shown that with the definite resonance conditions imposed, the…
Bandyopadhyay and Das [Phys. Plasmas, 9, 465-473, 2002] have derived a nonlinear macroscopic evolution equation for ion acoustic wave in a magnetized plasma consisting of warm adiabatic ions and non-thermal electrons including the effect of Landau damping. In that paper they have also derived the corresponding nonlinear evolution equation when coefficient of the nonlinear term of the above mentioned macroscopic evolution equation vanishes, the nonlinear behaviour of the ion acoustic wave is described by a modified macroscopic evolution equation. But they have not considered the case when the c…
In this paper, we consider the nonlinear Vlasov-Poisson equations in a weakly collisional regime and study the linear Boltzmann collision operator. We prove that Landau damping still occurs in this case.
The fundamental problem of Farley-Buneman turbulence in the auroral $E$-region has been discussed and debated extensively in the past two decades. In the present paper we intend to clarify the different steps that the auroral $E$-region plasma has to undergo before reaching a steady state. The mode-coupling calculation, for Farley-Buneman turbulence, is developed in order to place it in perspective and to estimate its magnitude relative to the anomalous effects which arise through the nonlinear wave-particle interaction. This nonlinear effect, known as nonlinear ``Landau damping'' is due to th…
Landau damping of Langmuir waves is shown to have hydrodynamic roots, and, in principle, might have been predicted (along with Langmuir waves) several decades earlier, soon after Jeans (1902) paper appeared.
Geodesic acoustic mode (GAM) is analyzed by using modified gyro-kinetic (MGK) equation applicable to low-frequency microinstabilities in a rotating axisymmetric plasma. Dispersion relation of GAM in the presence of arbitrary Mach number is analytically derived. Toroidal rotation plays the same effects on the GAM regardless of the orientation of equilibrium flow. It is shown that the toroidal Mach number $M$ increases the GAM frequency and dramatically decreases the Landau damping rate. The valid of classical gyro-kinetic (CGK) equation is also examined. For zero electron temperature, CGK is id…
Geodesic acoustic modes (GAMs) are oscillations of the electric field whose importance in tokamak plasmas is due to their role in the regulation of turbulence. The linear collisionless damping of GAMs is investigated here by means of analytical theory and numerical simulations with the global gyrokinetic particle-in-cell code ORB5. The combined effect of the phase mixing and Landau damping is found to quickly redistribute the GAM energy in phase-space, due to the synergy of the finite orbit width of the passing ions and the cascade in wave number given by the phase mixing. When plasma paramete…
The kinetic-MHD hybrid simulation approach for macroscopic instabilities in plasmas can be extended to include the kinetic effects of both thermal ions and energetic ions. The new coupling scheme includes synchronization of density and parallel velocity between thermal ions and MHD, in addition to pressure coupling, to ensure the quasineutrality condition and avoid numerical errors. The new approach has been implemented in the kinetic-MHD code M3D-C1-K, and was used to study the thermal ion kinetic effects and Landau damping in fishbone modes in both DIII-D and NSTX. It is found that the therm…
A Korteweg-de Vries (KdV) equation including the effect of Landau damping is derived to study the propagation of weakly nonlinear and weakly dispersive ion acoustic waves in a collisionless unmagnetized plasma consisting of warm adiabatic ions and two different species of electrons at different temperatures. The hotter energetic electron species follows the nonthermal velocity distribution of Cairns et al. [Geophys. Res. Lett. 22, 2709 (1995)] whereas the cooler electron species obeys the Boltzmann distribution. It is found that the coefficient of the nonlinear term of this KdV like evolution …
This paper addresses the linear propagation of an electron wave in a collisionless, inhomogeneous, nonstationary and anisotropic plasma. The plasma is characterized by its distribution function, $f_H$, at zero order in the wave amplitude. This distribution function, from which are derived all the hydrodynamical quantities, may be chosen arbitrarily, provided that it solves Vlasov equation. Then, from the linearized version of the electrons equation of motion, and from Gauss law, is derived an envelope equation for the wave amplitude, assumed to evolve over time and space scales much larger tha…
The linear destabilization and nonlinear saturation of energetic-particle driven Alfvénic instabilities in tokamaks strongly depend on the damping channels. In this work, the collisionless damping mechanisms of Alfvénic modes are investigated within a gyrokinetic framework, by means of global simulations with the particle-in-cell code ORB5, and compared with the eigenvalue code LIGKA and reduced models. In particular, the continuum damping and the Landau damping (of ions and electrons) are considered. The electron Landau damping is found to be dominant on the ion Landau damping for experimenta…
The effects of linear Landau damping on the nonlinear propagation of dust-acoustic solitary waves (DASWs) are studied in a collisionless unmagnetized dusty plasma with two species of positive ions. The extremely massive, micron-seized, cold and negatively charged dust particles are described by fluid equations, whereas the two species of positive ions, namely the cold (heavy) and hot (light) ions are described by the kinetic Vlasov equations. Following Ott and Sudan [Phys. Fluids {\bf 12}, 2388 (1969)], and by considering lower and higher-order perturbations, the evolution of DASWs with Landau…
Turbulence is thought to play a role in the heating of the solar wind plasma, though many questions remain to be solved regarding the exact nature of the mechanisms driving this process in the heliosphere. In particular, the physics of the collisionless interactions between particles and turbulent electromagnetic fields in the kinetic dissipation range of the turbulent cascade remains incompletely understood. A recent analysis of an interval of Magnetosphere Multiscale (MMS) observations has used the field-particle correlation technique to demonstrate that electron Landau damping is involved i…
Incorporation of kinetic effects such as Landau damping into a fluid framework was pioneered by Hammett and Perkins PRL 1990, by obtaining closures of the fluid hierarchy, where the gyrotropic heat flux fluctuations or the deviation of the 4th-order gyrotropic fluid moment, are expressed through lower-order fluid moments. To obtain a closure of a fluid model expanded around a bi-Maxwellian distribution function, the usual plasma dispersion function $Z(ζ)$ that appears in kinetic theory or the associated plasma response function $R(ζ)=1 + ζZ(ζ)$, have to be approximated with a suitable Padé app…
We show that the linearized Vlasov-Poisson equations around self-similar non-homogeneous states near zero contain the full plasma echo mechanism, yielding Gevrey 3 as a critical stability class. Moreover, here Landau damping may persist despite blow-up: We construct a critical Gevrey regularity class in which the force field converges in $L^2$. Thus, on the one hand, the physical phenomenon of Landau damping holds. On the other hand, the density diverges to infinity in Sobolev regularity. Hence, ``strong damping'' cannot hold.
The pondermotive potential in the X-ray Raman compression can generate an electron band gap which suppresses the Landau damping. The regime is identified where a Langmuir wave can be driven without damping in the stimulated Raman compression. It is shown that the partial wave breaking and the frequency detuning due to the trapped particles would be greatly reduced.
The linear and nonlinear theories of electron-acoustic waves (EAWs) are studied in a partially degenerate quantum plasma with two-temperature electrons and stationary ions. The initial equilibrium of electrons is assumed to be given by the Fermi-Dirac distribution at finite temperature. By employing the multi-scale asymptotic expansion technique to the one-dimensional Wigner-Moyal and Poisson equations, it is shown that the effects of multi-plasmon resonances lead to a modified complex Korteweg-de Vries (KdV) equation with a new nonlocal nonlinearity. Besides giving rise to a nonlocal nonlinea…
This paper addresses the linear and nonlinear three-dimensional propagation of an electron wave in a collisionless plasma that may be inhomogeneous, nonstationary, anisotropic and even weakly magnetized. The wave amplitude, together with any hydrodynamic quantity characterizing the plasma (density, temperature,...) are supposed to vary very little within one wavelength or one wave period. Hence, the geometrical optics limit is assumed, and the wave propagation is described by a first order differential equation. This equation explicitly accounts for three-dimensional effects, plasma inhomogene…
We study the nonlinear propagation of ion-acoustic waves (IAWs) in an unmagnetized collisionless plasma with the effects of electron and ion Landau damping in the weak quantum (semiclassical) regime, i.e., when the typical ion-acoustic (IA) length scale is larger than the thermal de Broglie wavelength. Starting from a set of classical and semiclassical Vlasov equations for ions and electrons, coupled to the Poisson equation, we derive a modified (by the particle dispersion) Korteweg-de Vries (KdV) equation which governs the evolution of IAWs with the effects of wave-particle resonance. It is f…
To address the problem of Landau damping in kinetic turbulence, the forcing of the linearized Vlasov equation by a stationary random source is considered. It is found that the time-asymptotic density response is dominated by resonant particle interactions that are synchronized with the source. The energy consumption of this response is calculated, implying an effective damping rate, which is the main result of this paper. Evaluating several cases, it is found that the effective damping rate can differ from the Landau damping rate in magnitude and also, remarkably, in sign. A limit is demonstra…
The theory of plasma waves and Landau damping in Maxwellian plasmas, Landau's ``rule of pass around poles'' include doubtful statements, particularly related to an artificial ``constructing'' of the dispersion equation, what should allow the possibility of its solution otherwise not existing at all, and the possibility of analytical continuations of corresponding very specific ruptured functions in the one-dimensional Laplace transformation, used by Landau, what is the base of his theory. We represent, as an accessible variant, a more general alternative theory based on a two-dimensional Lapla…
A Neural Operator Closure for Landau Damping in Electrostatic Plasma
Queen Mary University of London（伦敦大学玛丽女王学院）；University of Colorado, Boulder（科罗拉多大学博尔德分校）；Institute for Plasma Science and Technology, Consiglio Nazionale delle Ricerche（意大利国家研究委员会等离子科学与技术研究所）
We prove Landau damping for the collisionless Vlasov equation with a class of $L^1$ interaction potentials (including the physical case of screened Coulomb interactions) on $\mathbb R^3_x \times \mathbb R^3_v$ for localized disturbances of an infinite, homogeneous background. Unlike the confined case $\mathbb T^3_x \times \mathbb R_v^3$, results are obtained for initial data in Sobolev spaces (as well as Gevrey and analytic classes). For spatial frequencies bounded away from zero, the Landau damping of the density is similar to the confined case. The finite regularity is possible due to an add…
We present a study of the standard plasma physics test, Landau damping, using the Particle-In-Cell (PIC) algorithm. The Landau damping phenomenon consists of the damping of small oscillations in plasmas without collisions. In the PIC method, a hybrid discretization is constructed with a grid of finitely supported basis functions to represent the electric, magnetic and/or gravitational fields, and a distribution of delta functions to represent the particle field. Approximations to the dispersion relation are found to be inadequate in accurately calculating values for the electric field frequenc…
It is found that the presence of the so-called non-axisymmetric resonances of wave-particle interaction in stellarators [which are associated with the lack of axial symmetry of the magnetic configuration, Kolesnichenko et al., Phys. Plasmas 9 (2002) 517] may have a strong stabilizing influence through Landau mechanism on the Toroidicity-induced Alfvén Eigenmodes (TAE) and isomon modes (Alfvénic modes with equal poloidal and toroidal mode numbers and frequencies in the continuum region) destabilized by the energetic ions. These resonances involve largest harmonics of the equilibrium magnetic fi…
In this paper, we study Landau damping in the weakly collisional limit of a Vlasov-Fokker-Planck equation with nonlinear collisions in the phase-space $(x,v) \in \mathbb T_x^n \times \mathbb R^n_v$. The goal is four-fold: (A) to understand how collisions suppress plasma echoes and enable Landau damping in agreement with linearized theory in Sobolev spaces, (B) to understand how phase mixing accelerates collisional relaxation, (C) to understand better how the plasma returns to global equilibrium during Landau damping, and (D) to rule out that collision-driven nonlinear instabilities dominate. W…
In this note we present the main results from the recent work hal-00376547/arXiv:0904.2760, which for the first time establish Landau damping in a nonlinear context.
This paper brings further insight into the recently published N-body description of Debye shielding and Landau damping [Escande D F, Elskens Y and Doveil F 2014 Plasma Phys. Control. Fusion 57 025017]. Its fundamental equation for the electrostatic potential is derived in a simpler and more rigorous way. Various physical consequences of the new approach are discussed, and this approach is compared with the seminal one by Pines and Bohm [Pines D and Bohm D 1952 Phys. Rev. 85 338--353].
We carry out a systematic study of the dispersion relation for linear electrostatic waves in an arbitrarily degenerate quantum electron plasma. We solve for the complex frequency spectrum for arbitrary values of wavenumber $k$ and level of degeneracy $μ$. Our finding is that for large $k$ and high $μ$ the real part of the frequency $ω_{r}$ grows linearly with $k$ and scales with $μ$ only because of the scaling of the Fermi energy. In this regime the relative Landau damping rate $γ/ω_{r}$ becomes independent of $k$ and varies inversly with $μ$. Thus, damping is weak but finite at moderate level…
We give a new, simpler, proof of nonlinear Landau damping on T^d in Gevrey-1/s regularity (s > 1/3) which matches the regularity requirement predicted by the formal analysis of Mouhot and Villani in the original proof of Landau damping [Acta Mathematica 2011]. Our proof combines in a novel way ideas from the original proof of Landau damping and the proof of inviscid damping in 2D Euler [arXiv:1306.5028]. As in the work on 2D Euler, we use paraproduct decompositions and controlled regularity loss to replace the Newton iteration scheme employed in the original proof. We perform time-response est…
The derivation of Debye shielding and Landau damping from the $N$-body description of plasmas is performed directly by using Newton's second law for the $N$-body system. This is done in a few steps with elementary calculations using standard tools of calculus, and no probabilistic setting. Unexpectedly, Debye shielding is encountered together with Landau damping. This approach is shown to be justified in the one-dimensional case when the number of particles in a Debye sphere becomes large. The theory is extended to accommodate a correct description of trapping and chaos due to Langmuir waves. …
Using an exact law for incompressible Hall magnetohydrodynamics (HMHD) turbulence, the energy cascade rate is computed from three-dimensional HMHD-CGL (bi-adiabatic ions and isothermal electrons) and Landau fluid (LF) numerical simulations that feature different intensities of Landau damping over a broad range of wavenumbers, typically $0.05\lesssim k_\perp d_i \lesssim100$. Using three sets of cross-scale simulations where turbulence is initiated at large, medium and small scales, the ability of the fluid energy cascade to "sense" the kinetic Landau damping at different scales is tested. The …
We establish the possibility of Landau damping for gravitational scalar waves which propagate in a non-collisional gas of particles. In particular, under the hypothesis of homogeneity and isotropy, we describe the medium at the equilibrium with a Jüttner-Maxwell distribution, and we analytically determine the damping rate from the Vlasov equation. We find that damping occurs only if the phase velocity of the wave is subluminal throughout the propagation within the medium. Finally, we investigate relativistic media in cosmological settings by adopting numerical techniques.
Effects of non-thermal high-energy electrons on Langmuir wave-particle interaction are investigated by an initial value approach. A Vlasov-Poisson simulation is employed which is based on the splitting scheme by Cheng and Knorr [Cheng, C.Z. and G. Knorr, 1976: J. Comput. Phys. 22, 330-351.]. The kappa distribution function is taken as an example of non-thermal electrons. The modification is manifested as an increase in the Landau damping rate and a decrease in the real frequency for a long wavelength limit. A part of the analyses by the modified plasma dispersion function [Summers, D. and R.M.…
Landau damping is the tendency of solutions to the Vlasov equation towards spatially homogeneous distribution functions. The distribution functions however approach the spatially homogeneous manifold only weakly, and Boltzmann entropy is not changed by Vlasov equation. On the other hand, density and kinetic energy density, which are integrals of the distribution function, approach spatially homogeneous states strongly, which is accompanied by growth of the hydrodynamic entropy. Such a behavior can be seen when Vlasov equation is reduced to the evolution equations for density and kinetic energy…
The Caldeira-Leggett Hamiltonian (Eq. (1) below) describes the interaction of a discrete harmonic oscillator with a continuous bath of harmonic oscillators. This system is a standard model of dissipation in macroscopic low temperature physics, and has applications to superconductors, quantum computing, and macroscopic quantum tunneling. The similarities between the Caldeira-Leggett model and the linearized Vlasov-Poisson equation are analyzed, and it is shown that the damping in the Caldeira-Leggett model is analogous to that of Landau damping in plasmas [1]. An invertible linear transformatio…
How turbulent energy is dissipated in weakly collisional space and astrophysical plasmas is a major open question. Here, we present the application of a field-particle correlation technique to directly measure the transfer of energy between the turbulent electromagnetic field and electrons in the Earth's magnetosheath, the region of solar wind downstream of the Earth's bow shock. The measurement of the secular energy transfer from the parallel electric field as a function of electron velocity shows a signature consistent with Landau damping. This signature is coherent over time, close to the p…
Heliospheric plasma turbulence plays a key role in transferring the energy of large-scale magnetic field and plasma flow fluctuations to smaller scales where the energy can be dissipated, ultimately leading to plasma heating. High-quality measurements of electromagnetic fields and electron velocity distributions by the Magnetospheric Multiscale (MMS) mission in Earth's magnetosheath present a unique opportunity to characterize plasma turbulence and to determine the mechanisms responsible for its dissipation. We apply the field-particle correlation technique to a set of twenty MMS magnetosheath…
