Nonlinear Landau Damping Research Articles

Nonlinear Landau damping and modulation of electrostatic waves in a nonextensive electron-positron-pair plasma.

The nonlinear theory of amplitude modulation of electrostatic wave envelopes in a collisionless electron-positron (EP) pair plasma is studied by using a set of Vlasov-Poisson equations in the context of Tsallis' q-nonextensive statistics. In particular, the previous linear theory of Langmuir oscillations in EP plasmas [Saberian and Esfandyari-Kalejahi, Phys. Rev. E 87, 053112 (2013)] is rectified and modified. Applying the multiple scale technique (MST), it is shown that the evolution of electrostatic wave envelopes is governed by a nonlinear Schrödinger (NLS) equation with a nonlocal nonlinear term ∝P∫|ϕ(ξ',τ)|(2)dξ'ϕ/(ξ-ξ') [where P denotes the Cauchy principal value, ϕ is the small-amplitude electrostatic (complex) potential, and ξ and τ are the stretched coordinates in MST], which appears due to the wave-particle resonance. It is found that a subregion 1/3<q≲3/5 of superextensivity (q<1) exists where the carrier-wave frequency can turn over with the group velocity going to zero and then to negative values. The effects of the nonlocal nonlinear term and the nonextensive parameter q are examined on the modulational instability of wave envelopes, as well as on the solitary wave solution of the NLS equation. It is found that the modulated wave packet is always unstable (nonlinear Landau damping) due to the nonlocal nonlinearity in the NLS equation. Furthermore, the effect of the nonlinear Landau damping is to slow down the amplitude of the wave envelope, and the corresponding decay rate can be faster the larger is the number of superthermal particles in pair plasmas.

Landau Damping in the Kuramoto Model

We consider the Kuramoto model of globally coupled phase oscillators in its continuum limit, with individual frequencies drawn from a distribution with density of class $${C^n}$$ ( $${n\geq 4}$$ ). A criterion for linear stability of the uniform stationary state is established which, for basic examples in the literature, is equivalent to the standard condition on the coupling strength. We prove that, under this criterion, the Kuramoto order parameter, when evolved under the full nonlinear dynamics, asymptotically vanishes (with polynomial rate n) for every trajectory issued from a sufficiently small $${C^n}$$ perturbation. The proof uses techniques from the Analysis of PDEs and closely follows recent proofs of the nonlinear Landau damping in the Vlasov equation and Vlasov-HMF model.

Explicit high-order non-canonical symplectic particle-in-cell algorithms for Vlasov-Maxwell systems

Explicit high-order non-canonical symplectic particle-in-cell algorithms for classical particle-field systems governed by the Vlasov-Maxwell equations are developed. The algorithms conserve a discrete non-canonical symplectic structure derived from the Lagrangian of the particle-field system, which is naturally discrete in particles. The electromagnetic field is spatially discretized using the method of discrete exterior calculus with high-order interpolating differential forms for a cubic grid. The resulting time-domain Lagrangian assumes a non-canonical symplectic structure. It is also gauge invariant and conserves charge. The system is then solved using a structure-preserving splitting method discovered by He et al. [preprint arXiv:1505.06076 (2015)], which produces five exactly soluble sub-systems, and high-order structure-preserving algorithms follow by combinations. The explicit, high-order, and conservative nature of the algorithms is especially suitable for long-term simulations of particle-field systems with extremely large number of degrees of freedom on massively parallel supercomputers. The algorithms have been tested and verified by the two physics problems, i.e., the nonlinear Landau damping and the electron Bernstein wave.

Landau Damping in Sobolev Spaces for the Vlasov-HMF Model

We consider the Vlasov-HMF (Hamiltonian Mean-Field) model. We consider solutions starting in a small Sobolev neighborhood of a spatially homogeneous state satisfying a linearized stability criterion (Penrose criterion). We prove that these solutions exhibit a scattering behavior to a modified state, which implies a nonlinear Landau damping effect with polynomial rate of damping.

Ion Heating from Nonlinear Landau Damping of High Mode Number Toroidal Alfvén Eigenmodes

Bulk ion heating rate from nonlinear Landau damping of high mode number Toroidal Alfvén Eigenmodes (TAEs) is calculated in the frame work of weak turbulence theory. The heating rate is lower than the nonlinear spectral transfer rate to more stable modes, but relatively insensitive to the details of linear damping mechanisms.

Particle-in-cell simulations of parametric decay instability of radiofrequency wave in the ion cyclotron range of frequency in an inhomogeneous plasma

Particle-in-cell simulations have been conducted to investigate the parametric decay processes of radiofrequency waves in the ion cyclotron range of frequency in inhomogeneous plasmas. By choosing the parameters close to those in the ion Bernstein wave heating experiments on the HT-7 tokamak (Li et al 2001 Plasma Phys. Control. Fusion 43 1227–38), decay channels similar to experimental observations have been found in simulations. It is shown that due to strong inhomogeneity of tokamak edge plasma, layers may exist where selection rules of the mode–mode coupling are satisfied, and resonant parametric decays can occur if the input wave power is sufficiently high. As a result, daughter waves at half-, second- and third-harmonic frequencies of the pump wave can be excited. The nonresonant parametric decay is also observed, and ions are heated via nonlinear Landau damping.

THE EFFECT OF NONLINEAR LANDAU DAMPING ON ULTRARELATIVISTIC BEAM PLASMA INSTABILITIES

Very-high energy gamma-rays from extragalactic sources pair-produce off of the extragalactic background light, yielding an electron-positron pair beam. This pair beam is unstable to various plasma instabilities, especially the "oblique" instability, which can be the dominant cooling mechanism for the beam. However, recently, it has been claimed that nonlinear Landau damping renders it physically irrelevant by reducing the effective damping rate to a low level. Here, we show with numerical calculations that the effective damping rate is $8\times 10^{-4}$ of the growth rate of the linear instability, which is sufficient for the "oblique" instability to be the dominant cooling mechanism of these pair beams. In particular, we show that previous estimates of this rate ignored the exponential cutoff in the scattering amplitude at large wavenumber and assumed that the damping of scattered waves entirely depends on collisions, ignoring collisionless processes. We find that the total wave energy eventually grows to approximate equipartition with the beam by increasingly depositing energy into long wavelength modes. As we have not included the effect of nonlinear wave-wave interactions on these long wavelength modes, this scenario represents the "worst-case" scenario for the oblique instability. As it continues to drain energy from the beam at a faster rate than other processes, we conclude that the "oblique" instability is sufficiently strong to make it the physically dominant cooling mechanism for high-energy pair beams in the intergalactic medium.

Finite Amplitude Effects on Landau Damping and Diminished Transportation of Trapped Electrons

We studied the effect of finite amplitude, which was ignored in Landau’s linearized theory and also studied a transition criterion between linear and nonlinear Landau damping. To adjust the finite amplitude of wave and the initial trapped electron flux JxT0, we used particle-in-cell simulation that can uniquely modulate these parameters. We predicted on the finite amplitude of wave potential and kinetic energies; these matched the simulation exactly. Oscillation and damping rates in Landau damping agreed remarkably well with those of theory and differed substantially from theory beyond certain range of amplitude. Either large amplitude or large JxT0 increased degree of damping; this increase diminished the transportation of trapped electrons. More accurate description of linear Landau damping coupled with the finite amplitude effect was possible. Although previous theories predict nonlinear waves in large amplitude, we did not observe the expected nonlinear wave because of diminished JxT0 at some conditions.

Laser light triggers increased Raman amplification in the regime of nonlinear Landau damping.

Stimulated Raman backscattering (SRS) has many unwanted effects in megajoule-scale inertially confined fusion (ICF) plasmas. Moreover, attempts to harness SRS to amplify short laser pulses through backward Raman amplification have achieved limited success. In high-temperature fusion plasmas, SRS usually occurs in a kinetic regime where the nonlinear response of the Langmuir wave to the laser drive and its host of complicating factors make it difficult to predict the degree of amplification that can be achieved under given experimental conditions. Here we present experimental evidence of reduced Landau damping with increasing Langmuir wave amplitude and determine its effects on Raman amplification. The threshold for trapping effects to influence the amplification is shown to be very low. Above threshold, the complex SRS dynamics results in increased amplification factors, which partly explains previous ICF experiments. These insights could aid the development of more efficient backward Raman amplification schemes in this regime.

Particle systems and nonlinear Landau damping

Some works dealing with the long-time behavior of interacting particle systems are reviewed and put into perspective, with focus on the classical Kolmogorov–Arnold–Moser theory and recent results of Landau damping in the nonlinear perturbative regime, obtained in collaboration with Clément Mouhot. Analogies are discussed, as well as new qualitative insights in the theory. Finally, the connection with a more recent work on the inviscid Landau damping near the Couette shear flow, by Bedrossian and Masmoudi, is briefly discussed.

Ion acceleration by parallel propagating nonlinear Alfvén wave packets in a radially expanding plasma

Abstract. The numerical simulation of the nonlinear evolution of the parallel propagating Alfvén waves in a radially expanding plasma is performed by using a kinetic-fluid model (the Vlasov–MHD model). In our study, both the nonlinear evolution of the Alfvén waves and the radial evolution of the velocity distribution function (VDF) are treated simultaneously. On the other hand, important ion kinetic effects such as ion cyclotron damping and instabilities driven by the non-equilibrium ion velocity distributions are not included in the present model. The results indicate that the steepened Alfvén wave packets outwardly accelerate ions, which can be observed as the beam components in the interplanetary space. The energy of imposed Alfvén waves is converted into the longitudinal fluctuations by the nonlinear steepening and the nonlinear Landau damping. The wave shoaling due to the inhomogeneity of the phase velocity is also observed.

Nonlinear Landau damping of high frequency waves in non-Maxwellian plasmas

Space plasmas often possess non-Maxwellian distribution functions which have a significant effect on the plasma waves. When a laser or electron beam passes through a dense plasma, hot low density electron populations can be generated to alter the wave damping/growth rate. In this paper, we present theoretical analysis of the nonlinear Landau damping for Langmuir waves in a plasma where two electron populations are found. The results show a marked difference between the Maxwellian and non-Maxwellian instantaneous damping rates when we employ a non-Maxwellian distribution function called the generalized (r, q) distribution function, which is the generalized form of the kappa and Maxwellian distribution functions. In the limiting case of r = 0 and q → ∞, it reduces to the classical Maxwellian distribution function, and when r = 0 and q → κ + 1, it reduces to the kappa distribution function.

Exact evaluation of the rates of electrostatic decay and scattering off thermal ions for an unmagnetized Maxwellian plasma

Electrostatic decay of Langmuir waves into Langmuir and ion sound waves (L→L′+S) and scattering of Langmuir waves off thermal ions (L+i→L′+i′, also called “nonlinear Landau damping”) are important nonlinear weak-turbulence processes. The rates for these processes depend on the quadratic longitudinal response function α(2) (or, equivalently, the quadratic longitudinal susceptibility χ(2)), which describes the second-order response of a plasma to electrostatic wave fields. Previous calculations of these rates for an unmagnetized Maxwellian plasma have relied upon an approximate form for α(2) that is valid where two of the wave fields are fast (i.e., vϕ=ω/k≫Ve where ω is the angular frequency, k is the wavenumber, and Ve is the electron thermal speed) and one is slow (vϕ≪Ve). Recently, an exact expression was derived for α(2) that is valid for any phase speeds of the three waves in an unmagnetized Maxwellian plasma. Here, this exact α(2) is applied to the calculation of the three-dimensional rates for electrostatic decay and scattering off thermal ions, and the resulting exact rates are compared with the approximate rates. The calculations are performed using previously derived three-dimensional rates for electrostatic decay given in terms of a general α(2), and newly derived three-dimensional rates for scattering off thermal ions; the scattering rate is derived assuming a Maxwellian ion distribution, and both rates are derived assuming arc distributions for the wave spectra. For most space plasma conditions, the approximate rate is found to be accurate to better than 20%; however, for sufficiently low Langmuir phase speeds (vϕ/Ve≈3) appropriate to some spatial domains of the foreshock regions of planetary bow shocks and type II solar radio bursts, the use of the exact rate may be necessary for accurate calculations. The relative rates of electrostatic decay and scattering off thermal ions are calculated for a range of parameters using the exact expressions for the rates; electrostatic decay is found to have the larger growth rate over the whole range of parameters, consistent with previous approximate calculations.

Cosmic ray streaming in clusters of galaxies

The observed bimodality in radio luminosity in galaxy clusters is puzzling. We investigate the possibility that cosmic-ray (CR) streaming in the intra-cluster medium can 'switch off' hadronically induced radio and gamma-ray emission. For self-confined CRs, this depends on the source of MHD wave damping: if only non-linear Landau damping operates, then CRs stream on the slow Alfvenic timescale, but if turbulent wave damping operates, super-Alfvenic streaming is possible. As turbulence increases, it promotes outward streaming more than it enables inward turbulent advection. Curiously, the CR flux is independent of $\nabla f$ (as long as it is non-zero) and depends only on plasma parameters; this enables radio halos with flat inferred CR profiles to turn off. We perform 1D time-dependent calculations of a radio mini-halo (Perseus) and giant radio halo (Coma) and find that both diminish in radio luminosity by an order of magnitude in several hundred Myr, given plausible estimates for the magnetic field in the outskirts of the cluster. Due to the energy dependence of CR streaming, spectral curvature develops, and radio halos turn off more slowly at low frequencies -- properties consistent with observations. Similarly, CR streaming rapidly turns off gamma-ray emission at the high-energies probed by Cherenkov telescopes, but not at the low energies probed by Fermi. CR mediated wave-heating of the ICM is unaffected, as it is dominated by ~GeV CRs which stream Alfvenically.

Simulation of linear and nonlinear Landau damping of lower hybrid waves

The linear physics of lower hybrid waves (LHWs) and their nonlinear interaction with particles through Landau damping are studied with the gyrokinetic electron and fully kinetic ion (GeFi) particle simulation model in the electrostatic limit. Unlike most other wave modes, the LHWs can resonantly interact with both electrons and ions, with the former being highly magnetized and latter nearly unmagnetized around the lower hybrid frequency. Direct interactions of LHWs with electrons and/or ions are investigated for cases with various k∥/k,Ti/Te, and wave amplitudes. In the linear electron Landau damping (ELD), the dispersion relation and the linear damping rate obtained from our simulation agree well with the analytical linear theory. As the wave amplitude increases, the nonlinear Landau effects are present, and a transition from strong decay at smaller amplitudes to weak decay at larger amplitudes is observed. In the nonlinear stage, the LHWs in the long time evolution finally exhibit a steady Bernstein-Greene-Kruskal mode, in which the wave amplitude is saturated above the noise level. While the resonant electrons are trapped in the wave field in the nonlinear ELD, the resonant ions are untrapped in the LHW time scales. The ion Landau damping is thus predominantly in a linear fashion, leading to a wave saturation level significantly lower than that in the ELD. On the long time scales, however, the ions are still weakly trapped. The results show a coupling between the LHW frequency and the ion cyclotron frequency during the long-time LHW evolution.

Highly efficient numerical algorithm based on random trees for accelerating parallel Vlasov–Poisson simulations

An efficient numerical method based on a probabilistic representation for the Vlasov–Poisson system of equations in the Fourier space has been derived. This has been done theoretically for arbitrary dimensional problems, and particularized to unidimensional problems for numerical purposes. Such a representation has been validated theoretically in the linear regime comparing the solution obtained with the classical results of the linear Landau damping.The numerical strategy followed requires generating suitable random trees combined with a Padé approximant for approximating accurately a given divergent series. Such series are obtained by summing the partial contributions to the solution coming from trees with arbitrary number of branches. These contributions, coming in general from multi-dimensional definite integrals, are efficiently computed by a quasi-Monte Carlo method. It is shown how the accuracy of the method can be effectively increased by considering more terms of the series.The new representation was used successfully to develop a Probabilistic Domain Decomposition method suited for massively parallel computers, which improves the scalability found in classical methods. Finally, a few numerical examples based on classical phenomena such as the non-linear Landau damping, and the two streaming instability are given, illustrating the remarkable performance of the algorithm, when compared the results with those obtained using a classical method.

RELAXATION OF BLAZAR-INDUCED PAIR BEAMS IN COSMIC VOIDS

The stability properties of a low density ultra relativistic pair beam produced in the intergalactic medium by multi-TeV gamma-ray photons from blazars are analyzed. The problem is relevant for probes of magnetic field in cosmic voids through gamma-ray observations. In addition, dissipation of such beams could affect considerably the thermal history of the intergalactic medium and structure formation. We use a Monte Carlo method to quantify the properties of the blazar induced electromagnetic shower, in particular the bulk Lorentz factor and the angular spread of the pair beam generated by the shower, as a function of distance from the blazar itself. We then use linear and nonlinear kinetic theory to study the stability of the pair beam against the growth of electrostatic plasma waves, employing the Monte Carlo results for our quantitative estimates. We find that the fastest growing mode, like any perturbation mode with even a very modest component perpendicular to the beam direction, cannot be described in the reactive regime. Due to the effect of non-linear Landau damping, which suppresses the growth of plasma oscillations, the beam relaxation timescale is found significantly longer than the inverse Compton loss time. Finally, density inhomogeneities associated with cosmic structure induce loss of resonance between the beam particles and plasma oscillations, strongly inhibiting their growth. We conclude that relativistic pair beams produced by blazars in the intergalactic medium are stable on timescales long compared to the electromagnetic cascade's. There appears to be little or no effect of pair-beams on the intergalactic medium.

Nonlinear Landau damping and formation of Bernstein-Greene-Kruskal structures for plasmas with q-nonextensive velocity distributions

In the past, long-time evolution of an initial perturbation in collisionless Maxwellian plasma (q = 1) has been simulated numerically. The controversy over the nonlinear fate of such electrostatic perturbations was resolved by Manfredi [Phys. Rev. Lett. 79, 2815–2818 (1997)] using long-time simulations up to t=1600ωp−1. The oscillations were found to continue indefinitely leading to Bernstein-Greene-Kruskal (BGK)-like phase-space vortices (from here on referred as “BGK structures”). Using a newly developed, high resolution 1D Vlasov-Poisson solver based on piecewise-parabolic method (PPM) advection scheme, we investigate the nonlinear Landau damping in 1D plasma described by toy q-distributions for long times, up to t=3000ωp−1. We show that BGK structures are found only for a certain range of q-values around q = 1. Beyond this window, for the generic parameters, no BGK structures were observed. We observe that for values of q&amp;lt;1 where velocity distributions have long tails, strong Landau damping inhibits the formation of BGK structures. On the other hand, for q&amp;gt;1 where distribution has a sharp fall in velocity, the formation of BGK structures is rendered difficult due to high wave number damping imposed by the steep velocity profile, which had not been previously reported. Wherever relevant, we compare our results with past work.

Small BGK waves and nonlinear Landau damping (higher dimensions)

Small BGK waves and nonlinear Landau damping (higher dimensions)

Decays of electron Bernstein waves near plasma edge

Nonlinear wave-wave couplings near the upper hybrid resonance are studied via particle-in-cell simulations. It is found that the decay of an electron Bernstein wave (EBW) depends on the ratio of the incident frequency and electron cyclotron frequency. For ratios less than two, parametric decay into a lower hybrid wave (or an ion Bernstein wave) and EBWs at a lower frequency is observed. For ratios larger than two, the daughter waves could be an electron cyclotron quasi-mode and another EBW or an ion wave and EBW. For sufficiently high incident power, the former process may dominate. Because of the electron cyclotron quasi-mode, electrons can be strongly heated by nonlinear Landau damping. As a result, the bulk of the incident power can be absorbed near plasma edge at high power. The increase in number of decay channels with frequency implies that the allowable power into the plasma must decrease with frequency.