Nonlinear Wave-particle Interactions Research Articles

Toroidal Alfvén eigenmodes with nonlinear gyrokinetic and fluid hybrid models

Alfvén eigenmodes may be important in driving fast particle transport in magnetic confinement fusion devices, with potentially deleterious results. To explain and predict this behaviour, numerical simulations are necessary. In order to predict transport, modes must be simulated through to their nonlinear saturated state. In this work, the first simulations of non-linear wave-particle interaction between an energetic particle population and a Toroidal Alfvén Eigenmode are performed in which fluctuations responding self-consistently to modification of the fast particle profile are calculated with gyrokinetic treatment of all plasma species. Results from two such gyrokinetic codes are compared with new results from non-perturbative and perturbative fluid-gyrokinetic hybrid codes. There is a power-law relationship between the saturated magnetic perturbation amplitude, δB∕B0, and the linear mode growth rate, γL. All models show a transition from a higher to a lower exponent regime with increasing γL. Measured values of the higher exponent from different codes fall in a range between 1.45 and 1.79, while the lower exponent falls in a range between 0.47 and 0.79. There is a consistent difference of 1.0 between the higher and lower exponents independent of the model. The absolute level of saturated δB∕B0 is determined by the damping rate. In the fluid-gyrokinetic hybrid codes, an ad-hoc damping is applied, while in the gyrokinetic case the measured damping is consistent with the estimated rate of physical electron Landau damping.

Spectra of keV protons related to ion-cyclotron wave packets

We use the Fokker-Planck-Kolmogorov equation to study the statistical aspects of stochastic dynamics of the radiation belt (RB) protons driven by nonlinear electromagnetic ion-cyclotron (EMIC) wave packets. We obtain the spectra of keV protons scattered by these waves that show steeping near the gyroresonance, the signature of resonant wave-particle interaction that cannot be described by a simple power law. The most likely mechanism for proton precipitation events in RBs is shown to be nonlinear wave-particle interaction, namely, the scattering of RB protons into the loss cone by EMIC waves.

Kinetic equation for nonlinear resonant wave-particle interaction

We investigate the nonlinear resonant wave-particle interactions including the effects of particle (phase) trapping, detrapping, and scattering by high-amplitude coherent waves. After deriving the relationship between probability of trapping and velocity of particle drift induced by nonlinear scattering (phase bunching), we substitute this relation and other characteristic equations of wave-particle interaction into a kinetic equation for the particle distribution function. The final equation has the form of a Fokker-Planck equation with peculiar advection and collision terms. This equation fully describes the evolution of particle momentum distribution due to particle diffusion, nonlinear drift, and fast transport in phase-space via trapping. Solutions of the obtained kinetic equation are compared with results of test particle simulations.

Gyrophase bunched ions in the plasma sheet

Gyrophase bunched ions were first detected in the upstream region of the Earth’s bow shock in the early 1980s which is formed by the microphysical process associated with reflected solar wind ions at the bow shock. Inside the magnetosphere, the results of computer simulations demonstrated that nonlinear wave-particle interaction can also result in the gyrophase bunching of particles. However, to date direct observations barely exist regarding this issue occurred inside the magnetosphere. In this paper, we report for the first time an event of gyrophase bunched ions observed in the near-Earth plasma sheet. The nongyrotropic distributions of ions were closely accompanied with the electromagnetic waves at the oxygen cyclotron frequency. The phase of bunched ions and the phase of waves mainly have very narrow phase differences (<30°) when the O+ band waves are remarkably enhanced, which indicates that the wave and particle are closely corotating. The “electric phase bunching” is considered to be a possible mechanism for the formation of the gyrophase bunched distributions in this case. The MVA analysis suggests that the oxygen band waves possess left helicity with respect to the propagation direction, which agrees with the characteristic of electromagnetic ion cyclotron waves. The observation of O+ ions composition suggests that the oxygen band waves are excited due to the enhancements of the O+ ion density. This study suggests that the gyrophase bunching is a significant nonlinear effect that exists not only in the bow shock but also in the inner magnetosphere.

Two dimensional kinetic analysis of electrostatic harmonic plasma 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.

Electron trapping and acceleration by kinetic Alfvén waves in solar flares

Context. Theoretical models and spacecraft observations of solar flares highlight the role of wave-particle interaction for non-local electron acceleration. In one scenario, the acceleration of a large electron population up to high energies is due to the transport of electromagnetic energy from the loop-top region down to the footpoints, which is then followed by the energy being released in dense plasma in the lower atmosphere.Aims. We consider one particular mechanism of non-linear electron acceleration by kinetic Alfven waves. Here, waves are generated by plasma flows in the energy release region near the loop top. We estimate the efficiency of this mechanism and the energies of accelerated electrons.Methods. We use analytical estimates and test-particle modelling to investigate the effects of electron trapping and acceleration by kinetic Alfven waves in the inhomogeneous plasma of the solar corona.Results. We demonstrate that, for realistic wave amplitudes, electrons can be accelerated up to 10−1000 keV during their propagation along magnetic field lines. Here the electric field that is parallel to the direction of the background magnetic field is about 10 to 103 times the amplitude of the Dreicer electric field. The acceleration mechanism strongly depends on electron scattering which is due to collisions that only take place near the loop footpoints.Conclusions. The non-linear wave-particle interaction can play an important role in the generation of relativistic electrons within flare loops. Electron trapping and coherent acceleration by kinetic Alfven waves represent the energy cascade from large-scale plasma flows that originate at the loop-top region down to the electron scale. The non-diffusive character of the non-linear electron acceleration may be responsible for the fast generation of high-energy particles.

Experimental characterization of nonlinear processes of whistler branch waves

Experiments in the Space Physics Simulation Chamber at the Naval Research Laboratory isolated and characterized important nonlinear wave-wave and wave-particle interactions that can occur in the Earth's Van Allen radiation belts by launching predominantly electrostatic waves in the intermediate frequency range with wave normal angle greater than 85° and measuring the nonlinearly generated electromagnetic scattered waves. The scattered waves have a perpendicular wavelength that is nearly an order of magnitude larger than that of the pump wave. Calculations of scattering efficiency from experimental measurements demonstrate that the scattering efficiency is inversely proportional to the damping rate and trends towards unity as the damping rate approaches zero. Signatures of both wave-wave and wave-particle scatterings are also observed in the triggered emission process in which a launched wave resonant with a counter-propagating electron beam generates a large amplitude chirped whistler wave. The possibility of nonlinear scattering or three wave decay as a saturation mechanism for the triggered emission is suggested. The laboratory experiment has inspired the search for scattering signatures in the in situ data of chorus emission in the radiation belts.

Mixed Diffusive-Convective Relaxation of a Warm Beam of Energetic Particles in Cold Plasma

This work addresses the features of fast particle transport in the bump-on-tail problem for varying the width of the fluctuation spectrum, in the view of possible applications to studies of energetic particle transport in fusion plasmas. Our analysis is built around the idea that strongly-shaped beams do not relax through diffusion only and that there exists an intermediate time scale where the relaxations are convective (ballistic-like). We cast this idea in the form of a self-consistent nonlinear dynamical model, which extends the classic equations of the quasi-linear theory to “broad” beams with internal structure. We also present numerical simulation results of the relaxation of a broad beam of energetic particles in cold plasma. These generally demonstrate the mixed diffusive-convective features of supra-thermal particle transport essentially depending on nonlinear wave-particle interactions and phase-space structures. Taking into account the modes of the stable linear spectrum is crucial for the self-consistent evolution of the distribution function and the fluctuation intensity spectrum.

Two-dimensional time evolution of beam-plasma instability in the presence of binary collisions

Energetic electrons produced during solar flares are known to be responsible for generating solar type III radio bursts. The radio emission is a byproduct of Langmuir wave generation via beam-plasma interaction and nonlinear wave-wave and wave-particle interaction processes. In addition to type III radio bursts, electrons traveling downwards toward the chromosphere lead to the hard X-ray emission via electron-ion collisions. Recently, the role of Langmuir waves on the X-ray-producing electrons has been identified as important, because Langmuir waves may alter the electron distribution, thereby affecting the X-ray profile. Both Coulomb collisions and wave-particle interactions lead electrons to scattering and energy exchange that necessitates considering the two-dimensional (2D) problem in velocity space. The present paper investigates the influence of binary collisions on the beam-plasma instability development in 2D in order to elucidate the nonlinear dynamics of Langmuir waves and binary collisions. The significance of the present findings in the context of solar physics is discussed.

ON THE CONNECTION BETWEEN MICROBURSTS AND NONLINEAR ELECTRONIC STRUCTURES IN PLANETARY RADIATION BELTS

ABSTRACT Using a dynamical-system approach, we have investigated the efficiency of large-amplitude whistler waves for causing microburst precipitation in planetary radiation belts by modeling the microburst energy and particle fluxes produced as a result of nonlinear wave–particle interactions. We show that wave parameters, consistent with large-amplitude oblique whistlers, can commonly generate microbursts of electrons with hundreds of keV-energies as a result of Landau trapping. Relativistic microbursts (&gt;1 MeV) can also be generated by a similar mechanism, but require waves with large propagation angles &theta; kB &gt; 50 &deg; ?> and phase-speeds v &Phi; &ge; c / 9 ?> . Using our result for precipitating density and energy fluxes, we argue that holes in the distribution function of electrons near the magnetic mirror point can result in the generation of double layers and electron solitary holes consistent in scales (of the order of Debye lengths) to nonlinear structures observed in the radiation belts by the Van Allen Probes. Our results indicate a relationship between nonlinear electrostatic and electromagnetic structures in the dynamics of planetary radiation belts and their role in the cyclical production of energetic electrons ( E &ge; 100 ?> keV) on kinetic timescales, which is much faster than previously inferred.

Saturation mechanism in a two-stream free-electron laser

The effect of a guide field on the saturation mechanism in a two-stream free-electron laser (FEL) is verified. Two monoenergetic electron beams with a vanishing pitch-angle spread are considered. Nonlinear wave–particle interaction is described by a set of coupled differential equations in a 1-D approximation. Output power is presented as a function of the axial distance. It was found that through using a focusing mechanism, the two-stream FEL reached the saturation regime in a shorter axial distance in comparison with the case of no focusing mechanism.

PARTICLE ACCELERATION IN SUPERNOVA REMNANTS

Most high energy cosmic rays (CRs) are thought to be produced by diffusive shock acceleration (DSA) in supernova remnants (SNRs) within the Galaxy. Plasma and MHD simulations have shown that the self-excitation of MHD waves and amplification of magnetic fields via plasma instabilities are an integral part of DSA for strong collisionless shocks. In this study we explore how plasma processes such as plasma instabilities and wave-particle interactions can affect the energy spectra of CR protons and electrons, using time-dependent DSA simulations of SNR shocks. We demonstrate that the time-dependent evolution of the shock dynamics, the self-amplified magnetic fields and Alfvenic drift govern the highest energy end of the CR energy spectra. As a result, the spectral cutoffs in nonthermal X-ray and γ-ray radiation spectra are regulated by the evolution of the highest energy particles, which are injected at the early phase of SNRs. We also find that the maximum energy of CR protons can be boosted significantly only if the scale height of the magnetic field precursor is long enough to contain the diffusion lengths of the particles of interests. Thus, detailed understandings of nonlinear wave-particle interactions and time-dependent DSA simulations are crucial for understanding the nonthermal radiation from CR acceleration sources.

Laboratory studies of nonlinear whistler wave processes in the Van Allen radiation belts

Important nonlinear wave-wave and wave-particle interactions that occur in the Earth's Van Allen radiation belts are investigated in a laboratory experiment. Predominantly electrostatic waves in the whistler branch are launched that propagate near the resonance cone with measured wave normal angle greater than 85°. When the pump amplitude exceeds a threshold ∼5×10−6 times the background magnetic field, wave power at frequencies below the pump frequency is observed at wave normal angles (∼55°). The scattered wave has a perpendicular wavelength that is nearly an order of magnitude larger than that of the pump wave. Occasionally, the parametric decay of a lower hybrid wave into a magnetosonic wave and a whistler wave is simultaneously observed with a threshold of δB/B0∼7×10−7.

Kinetic theory of turbulence for parallel propagation revisited: Formal results

In a recent paper, Gaelzer et al. [Phys. Plasmas 22, 032310 (2015)] revisited the second-order nonlinear kinetic theory for turbulence propagating in directions parallel/anti-parallel to the ambient magnetic field. The original work was according to Yoon and Fang [Phys. Plasmas 15, 122312 (2008)], but Gaelzer et al. noted that the terms pertaining to discrete-particle effects in Yoon and Fang's theory did not enjoy proper dimensionality. The purpose of Gaelzer et al. was to restore the dimensional consistency associated with such terms. However, Gaelzer et al. was concerned only with linear wave-particle interaction terms. The present paper completes the analysis by considering the dimensional correction to nonlinear wave-particle interaction terms in the wave kinetic equation.

Comparison of formulas for resonant interactions between energetic electrons and oblique whistler-mode waves

Test particle simulation is a useful method for studying both linear and nonlinear wave-particle interactions in the magnetosphere. The gyro-averaged equations of particle motion for first-order and other cyclotron harmonic resonances with oblique whistler-mode waves were first derived by Bell [J. Geophys. Res. 89, 905 (1984)] and the most recent relativistic form was given by Ginet and Albert [Phys. Fluids B 3, 2994 (1991)], and Bortnik [Ph.D. thesis (Stanford University, 2004), p. 40]. However, recently we found there was a (−1)l−1 term difference between their formulas of perpendicular motion for the lth-order resonance. This article presents the detailed derivation process of the generalized resonance formulas, and suggests a check of the signs for self-consistency, which is independent of the choice of conventions, that is, the energy variation equation resulting from the momentum equations should not contain any wave magnetic components, simply because the magnetic field does not contribute to changes of particle energy. In addition, we show that the wave centripetal force, which was considered small and was neglect in previous studies of nonlinear interactions, has a profound time derivative and can significantly enhance electron phase trapping especially in high frequency waves. This force can also bounce the low pitch angle particles out of the loss cone. We justify both the sign problem and the missing wave centripetal force by demonstrating wave-particle interaction examples, and comparing the gyro-averaged particle motion to the full particle motion under the Lorentz force.

On the dimensionally correct kinetic theory of turbulence for parallel propagation

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.

Wave-particle interactions in the outer radiation belts

Data from the Van Allen Probes have provided the first extensive evidence of non-linear (as opposed to quasi-linear) wave-particle interactions in space with the associated rapid (fraction of a bounce period) electron acceleration to hundreds of keV by Landau resonance in the parallel electric fields of time domain structures (TDS) and very oblique chorus waves. The experimental evidence, simulations, and theories of these processes are discussed. {\bf Key words:} the radiation belts, wave-particle interaction, plasma waves and instabilities

Electron scattering and nonlinear trapping by oblique whistler waves: The critical wave intensity for nonlinear effects

In this paper, we consider high-energy electron scattering and nonlinear trapping by oblique whistler waves via the Landau resonance. We use recent spacecraft observations in the radiation belts to construct the whistler wave model. The main purpose of the paper is to provide an estimate of the critical wave amplitude for which the nonlinear wave-particle resonant interaction becomes more important than particle scattering. To this aim, we derive an analytical expression describing the particle scattering by large amplitude whistler waves and compare the corresponding effect with the nonlinear particle acceleration due to trapping. The latter is much more rare but the corresponding change of energy is substantially larger than energy jumps due to scattering. We show that for reasonable wave amplitudes ∼10–100 mV/m of strong whistlers, the nonlinear effects are more important than the linear and nonlinear scattering for electrons with energies ∼10–50 keV. We test the dependencies of the critical wave amplitude on system parameters (background plasma density, wave frequency, etc.). We discuss the role of obtained results for the theoretical description of the nonlinear wave amplification in radiation belts.

Non-linear wave-particle interactions and fast ion loss induced by multiple Alfvén eigenmodes in the DIII-D tokamak

A new non-linear feature has been observed in fast-ion loss from tokamak plasmas in the form of oscillations at the sum, difference and second harmonic frequencies of two independent Alfvén eigenmodes (AEs). Full orbit calculations and analytic theory indicate this non-linearity is due to coupling of fast-ion orbital response as it passes through each AE—a change in wave-particle phase k · r by one mode alters the force exerted by the next. The loss measurement is of barely confined, non-resonant particles, while similar non-linear interactions can occur between well-confined particles and multiple AEs leading to enhanced fast-ion transport.

Latitudinal dependence of nonlinear interaction between electromagnetic ion cyclotron wave and terrestrial ring current ions

Electromagnetic ion cyclotron (EMIC) waves can lead to the rapid decay (on a timescale of hours) of the terrestrial ring current. Such decay process is usually investigated in the framework of quasi-linear theory. Here, both theoretical analysis and test-particle simulation are performed to understand the nonlinear interaction between ring current ions and EMIC waves. In particular, the dependence of the nonlinear wave-particle interaction processes on the ion initial latitude is investigated in detail. These nonlinear processes are classified into the phase trapping and phase bunching, and the phase bunching is further divided into the channel and cluster effects. Compared to the prediction of the quasi-linear theory, the ring current decay rate can be reduced by the phase trapping, increased by the channel effect phase bunching, but non-deterministically influenced by the cluster effect phase bunching. The ion initial latitude changes the occurrence of the phase trapping, modulates the transport direction and strength of the cluster effect phase bunching, and only slightly affects the channel effect phase bunching. The current results suggest that the latitudinal dependence of these nonlinear processes should be considered in the evaluation of the ring current decay induced by EMIC waves.