Nonlinear Wave-particle Interactions Research Articles

Hele-Shaw beach creation by breaking waves: a mathematics-inspired experiment

Fundamentals of nonlinear wave-particle interactions are studied experimentally in a Hele-Shaw configuration with wave breaking and a dynamic bed. To design this configuration, we determine, mathematically, the gap width which allows inertial flows to survive the viscous damping due to the side walls. Damped wave sloshing experiments compared with simulations confirm that width-averaged potential-flow models with linear momentum damping are adequately capturing the large scale nonlinear wave motion. Subsequently, we show that the four types of wave breaking observed at real-world beaches also emerge on Hele-Shaw laboratory beaches, albeit in idealized forms. Finally, an experimental parameter study is undertaken to quantify the formation of quasi-steady beach morphologies due to nonlinear, breaking waves: berm or dune, beach and bar formation are all classified. Our research reveals that the Hele-Shaw beach configuration allows a wealth of experimental and modelling extensions, including benchmarking of forecast models used in the coastal engineering practice, especially for shingle beaches.

Landau damping and the onset of particle trapping in quantum plasmas

Using analytical theory and simulations, we assess the impact of quantum effects on non-linear wave-particle interactions in quantum plasmas. We more specifically focus on the resonant interaction between Langmuir waves and electrons, which, in classical plasmas, lead to particle trapping. Two regimes are identified depending on the difference between the time scale of oscillation tB(k)=m/eEk of a trapped electron and the quantum time scale tq(k)=2m/ℏk2 related to recoil effect, where E and k are the wave amplitude and wave vector. In the classical-like regime, tB(k) < tq(k), resonant electrons are trapped in the wave troughs and greatly affect the evolution of the system long before the wave has had time to Landau damp by a large amount according to linear theory. In the quantum regime, tB(k) > tq(k), particle trapping is hampered by the finite recoil imparted to resonant electrons in their interactions with plasmons.

Relativistic surfatron process for Landau resonant electrons in radiation belts

Abstract. Recent theoretical studies of the nonlinear wave-particle interactions for relativistic particles have shown that Landau resonant orbits could be efficiently accelerated along the mean background magnetic field for propagation angles θ in close proximity to a critical propagation θc associated with a Hopf–Hopf bifurcation condition. In this report, we extend previous studies to reach greater modeling capacities for the study of electrons in radiation belts by including longitudinal wave effects and inhomogeneous magnetic fields. We find that even though both effects can limit the surfatron acceleration of electrons in radiation belts, gains in energy of the order of 100 keV, taking place on one tenth of a millisecond, are sufficiently strong for the mechanism to be relevant to radiation belt dynamics.

Electrostatic instability in magnetically confined inhomogeneous plasma driven by nonlinear force

Abstract A theoretical investigation on amplification of electrostatic ion acoustic wave in magnetically confined plasma has been presented in this paper. This investigation considers nonlinear wave–particle interaction process, called plasma maser effect, in presence of drift wave turbulence supported by magnetically confined inhomogeneous plasma. The role of associated nonlinear dissipative force in this effect in a confined plasma has been analyzed. The nonlinear force, which arises as a result of resonant interaction between electrons and modulated fields, is shown to drive the instability. Using the ion fluid equation and the ion equation of continuity, the nonlinear dispersion relation of a test ion acoustic wave has been derived, and the growth rate of ion acoustic wave in presence of low frequency drift wave turbulence has been estimated using Helimak data.

Nanosecond Pulses in a THz Gyrotron Oscillator Operating in a Mode-Locked Self-ConsistentQ-Switch Regime

An experimental study of a nanosecond pulsed regime in a THz gyrotron oscillator operating in a self-consistent Q-switch regime has been carried out. The gyrotron is operated in the TE(7,2) transverse mode radiating at a frequency of 260.5 GHz. The 5 W nanosecond pulses are obtained in a self-consistent Q-switch regime in which the cavity diffraction quality factor dynamically varies by nearly 2 orders of magnitude on a subnanosecond time scale via the nonlinear interaction of different mode-locked frequency-equidistant sidebands. The experimental results are in good agreement with numerical simulations performed with the TWANG code based on a slow time scale formulation of the self-consistent time-dependent nonlinear wave particle interaction equations.

Nonthermal Radiation from Supernova Remnant Shocks

Most of high energy cosmic rays (CRs) are thought to be produced by diffusive shock acceleration (DSA) at supernova remnants (SNRs) within the Galaxy. Fortunately, nonthermal emissions from CR protons and electrons can provide direct observational evidence for such a model and place strong constraints on the complex nonlinear plasma processes in DSA theory. In this study we calculate the energy spectra of CR protons and electrons in Type Ia SNRs, using time-dependent DSA simulations that incorporate phenomenological models for some wave-particle interactions. We demonstrate that the time-dependent evolution of the self-amplified magnetic fields, Alfvnic drift, and escape of the highest energy particles affect the energy spectra of accelerated protons and electrons, and so resulting nonthermal radiation spectrum. Especially, the spectral cutoffs in X-ray and -ray emission spectra are regulated by the evolution of the highest energy particles, which are injected at the early phase of SNRs. Thus detailed understandings of nonlinear wave-particle interactions and time-dependent DSA simulations of SNRs are crucial in testing the SNR hypothesis for the origin of Galactic cosmic rays.

Heating of ions by high frequency electromagnetic waves in magnetized plasmas

The heating of ions by high frequency electrostatic waves in magnetically confined plasmas has been a paradigm for studying nonlinear wave-particle interactions. The frequency of the waves is assumed to be much higher than the ion cyclotron frequency and the waves are taken to propagate across the magnetic field. In fusion type plasmas, electrostatic waves, like the lower hybrid wave, cannot access the core of the plasma. That is a domain for high harmonic fast waves or electron cyclotron waves—these are primarily electromagnetic waves. Previous studies on heating of ions by two or more electrostatic waves are extended to two electromagnetic waves that propagate directly across the confining magnetic field. While the ratio of the frequency of each wave to the ion cyclotron frequency is large, the frequency difference is assumed to be near the ion cyclotron frequency. The nonlinear wave-particle interaction is studied analytically using a two time-scale canonical perturbation theory. The theory elucidates the effects of various parameters on the gain in energy by the ions—parameters such as the amplitudes and polarizations of the waves, the ratio of the wave frequencies to the cyclotron frequency, the difference in the frequency of the two waves, and the wave numbers associated with the waves. For example, the ratio of the phase velocity of the envelope formed by the two waves to the phase velocity of the carrier wave is important for energization of ions. For a positive ratio, the energy range is much larger than for a negative ratio. So waves like the lower hybrid waves will impart very little energy to ions. The theoretical results are found to be in good agreement with numerical simulations of the exact dynamical equations. The analytical results are used to construct mapping equations, simplifying the derivation of the motion of ions, which are, subsequently, used to follow the evolution of an ion distribution function. The heating of ions can then be properly quantified in terms of the wave parameters and can be conveniently used to find ideal conditions needed to heat ions by high frequency electromagnetic waves.

Nonlinear dynamical modelling of chaotic electrostatic ion cyclotron oscillations by jerk equations

Plasma being a nonlinear and complex system, is capable of sustaining a wide spectrum of waves, oscillations and instabilities. These fluctuations interact nonlinearly amongst themselves and also with particles: electrons/ions and thus lead to nonlinear wave-wave or wave-particle interaction. In the presence of coherent waves the particles are accelerated whereas irregular oscillations can give rise to particle heating which is also called stochastic heating. Particle orbits are known to be randomized by the wave fields such that their motion can also become stochastic. For fusion to be sustained one needs a very high temperature plasma for an extended duration. It quite common to deploy external waves like electron cyclotron waves or ion cyclotron waves for plasma heating and current drive. These external waves also work only in certain regimes. Conventional plasma techniques have been able to answer several of the observations of the above processes related to heating transport etc, but nonlinear dynamics as a tool has helped in comprehending the plasma oscillations better. We have for the first time obtained a Third Order nonlinear ordinary differential equation (TONLODE) also known as jerk equation to describe the electrostatic ion cyclotron plasma oscillations in a magnetic field. The interesting feature of this equation is that it does not require an external forcing term to obtain chaotic behaviour.

Effect of the background magnetic field inhomogeneity on generation processes of whistler‐mode chorus and broadband hiss‐like emissions

By a series of self‐consistent electron hybrid code simulations, we study the effect of the background magnetic field inhomogeneity on the generation process of whistler‐mode chorus emissions. Chorus with rising tones are generated through nonlinear wave‐particle interactions occurring around the magnetic equator. The mirror force plays an important role in the nonlinear interactions, and the spatial inhomogeneity of the background magnetic field is a key parameter of the chorus generation process. We have conducted numerical experiments with different spatial inhomogeneities to understand properties of the chorus generation process. We assume the same initial condition of energetic electrons at the magnetic equator in all simulation runs. The simulation results reveal that the spectral characteristics of chorus significantly vary depending on the magnetic field inhomogeneity. Whistler‐mode emissions are generated and propagate away from the equator in all simulation runs, but distinct chorus elements with rising tones are only reproduced in the cases of small inhomogeneities. In the simulation that had the smallest inhomogeneity, we find excitation of broadband hiss‐like emission (BHE) whose amplitudes are comparable to discrete chorus elements found in other simulation runs. The BHE consists of many wave elements with rising tones nonlinearly triggered in the region close to the magnetic equator. We show that the small spatial inhomogeneity of the background magnetic field results in the small threshold amplitude for the nonlinear wave growth and allows the triggering process of rising tone elements to emerge easily in the equatorial region of the magnetosphere.

Effects of collisions on energetic particle-driven chirping bursts

In the presence of an energetic particle population in a dissipative plasma, self-trapped structures in phase-space (holes and clumps) emerge from nonlinear wave-particle interactions. Their dynamics can lead to a nonlinear continuous shifting of the wave frequency (chirping). The effects of collisions on chirping characteristics are investigated, with a one-dimensional kinetic model. Existing analytic theory is extended to account for Krook-like collisions, which quantitatively explains a significant departure from widely accepted square-root time dependency. Relaxation oscillations, associated with chirping bursts, are investigated in the presence of dynamical friction and velocity-diffusion. The period increases with decreasing drag and weakly increases with decreasing diffusion. The mechanism is clarified with a simple semi-analytic model of hole/clump pair, which satisfies a Fokker-Planck equation. The model shows that the linear growth rate cannot be obtained simply by fitting an exponential to the amplitude time-series.

Significance of Wave-Particle Interaction Analyzer for direct measurements of nonlinear wave-particle interactions

Abstract. In the upcoming JAXA/ERG satellite mission, Wave Particle Interaction Analyzer (WPIA) will be installed as an onboard software function. We study the statistical significance of the WPIA for measurement of the energy transfer process between energetic electrons and whistler-mode chorus emissions in the Earth's inner magnetosphere. The WPIA measures a relative phase angle between the wave vector E and velocity vector v of each electron and computes their inner product W, where W is the time variation of the kinetic energy of energetic electrons interacting with plasma waves. We evaluate the feasibility by applying the WPIA analysis to the simulation results of whistler-mode chorus generation. We compute W using both a wave electric field vector observed at a fixed point in the simulation system and a velocity vector of each energetic electron passing through this point. By summing up Wi of an individual particle i to give Wint, we obtain significant values of Wint as expected from the evolution of chorus emissions in the simulation result. We can discuss the efficiency of the energy exchange through wave-particle interactions by selecting the range of the kinetic energy and pitch angle of the electrons used in the computation of Wint. The statistical significance of the obtained Wint is evaluated by calculating the standard deviation σW of Wint. In the results of the analysis, positive or negative Wint is obtained at the different regions of velocity phase space, while at the specific regions the obtained Wint values are significantly greater than σW, indicating efficient wave-particle interactions. The present study demonstrates the feasibility of using the WPIA, which will be on board the upcoming ERG satellite, for direct measurement of wave-particle interactions.

Monte-Carlo model for nonlinear interactions of Alfvén eigenmodes with energetic ions

A Monte-Carlo model for interactions between a single Alfvén eigenmode and energetic ions in a tokamak is presented. A phenomenological decorrelation of the wave-particle phase is introduced to mimic decorrelation by collisions or by other waves. Analysis is dedicated to how the strength of the phase decorrelation affects the nonlinear wave-particle interactions. Several of the phenomena that have been observed in some earlier models describing the nonlinear dynamics of Alfvén eigenmodes have been verified, such as the growth and saturation of the wave mode amplitude giving rise to a localized flattening of the distribution function, as well as the generation of coherent structures in the distribution function. The degree of phase decorrelation is shown to have a strong effect on the dynamics of the Alfvén eigenmode excitation.

Nonlinear Frequency Oscillation of Alfvén Eigenmodes in Fusion Plasmas

A nonlinear oscillation of frequency and amplitude is found by massively parallel gyrokinetic simulations of Alfvén eigenmodes excited by energetic particles in toroidal plasmas. The fast and repetitive frequency chirping is induced by the evolution of coherent structures in the phase space. The dynamics of the coherent structures is controlled by the competition between the phase-space island formation due to the nonlinear particle trapping and the island destruction due to the free streaming. The chirping dynamics provides a conceptual framework for understanding nonlinear wave-particle interactions underlying the transport process in collisionless plasmas.

Dynamical-systems approach to relativistic nonlinear wave-particle interaction in collisionless plasmas

In this report, we present a dynamical systems approach to study the exact nonlinear wave-particle interaction in relativistic regime. We give particular attention to the effect of wave obliquity on the dynamics of the orbits by studying the specific cases of parallel (θ = 0) and perpendicular (θ = -π/2) propagations in comparison to the general case of oblique propagation θ =]-π/2,0[. We found that the fixed points of the system correspond to Landau resonance and that the dynamics can evolve from trapping to surfatron acceleration for propagation angles obeying a Hopf bifurcations condition. Cyclotron-resonant particles are also studied by the construction of a pseudo-potential structure in the Lorentz factor γ. We derived a condition for which Arnold diffusion results in relativistic stochastic acceleration. Hence, two general conclusions are drawn: (1) The propagation angle θ can significantly alter the dynamics of the orbits at both Landau and cyclotron-resonances. (2) Considering the short-time scales upon which the particles are accelerated, these two mechanisms for Landau and cyclotron resonant orbits could become potential candidates for problems of particle energization in collisionless space and cosmic plasmas.

On unstable electromagnetic radiation through nonlinear wave–particle interactions in presence of drift wave turbulence

AbstractA study on the generation of unstable electromagnetic wave through nonlinear wave–particle interactions in an inhomogeneous plasma has been presented. Drift wave turbulence, which is one of the common features of inhomogeneous plasma, is found to be strongly in phase relation with thermal particles. The plasma particles, accelerated by drift wave turbulence field, may transfer their energy to electromagnetic O-mode through a modulated field. This process has been described by Vlasov Maxwell system of equations. From the nonlinear dispersion relation for O-mode the growth rate has been estimated. It has been observed that growth of unstable O-mode may predict magnetospheric electromagnetic radiation phenomena.

Electron energy diffusion and advection due to non-linear electron-chorus wave interactions

Non-linear plasma waves effects such as phase bunching and phase trapping can cause drastic changes (tens keVs) in the energies of electrons in the magnetosphere at the timescale of a few seconds. Previous studies of non-linear wave effects mainly focused on the energy changes of an electron passing through a plasma wave region, thus covering only a fraction of the electron's single-trip trajectory between its two mirror points. In this study, we use a test-particle method to study the cumulative non-linear wave effects over multiple electron bounce periods. From our simulation results and analyses, we find that non-linear wave-particle interactions are rather common with the presence of chorus waves. The non-linear wave effects take the form of energy advection, which can accelerate electrons rather efficiently. However, energy advection is often accompanied by electron pitch angle advection, causing electron pitch angles to decrease and constraining electron energy advection. The test particle approach we present here can be used to estimate nonlinear transport coefficients, which need to be incorporated into global radiation belt models.

GLOBAL WEAK SOLUTIONS FOR THE RELATIVISTIC WATERBAG CONTINUUM

In this paper we consider the relativistic waterbag continuum which is a useful PDE for collisionless kinetic plasma modeling recently developed in Ref. 11. The waterbag representation of the statistical distribution function of particles can be viewed as a special class of exact weak solution of the Vlasov equation, allowing to reduce this latter into a set of hydrodynamic equations (with the complexity of a multi-fluid model) while keeping its kinetic features (Landau damping and nonlinear resonant wave-particle interaction). These models are very promising because they are very useful for analytical theory and numerical simulations of laser-plasma and gyrokinetic physics.10–16, 56, 57The relativistic waterbag continuum is derived from two phase-space variable reductions of the relativistic Vlasov–Maxwell equations through the existence of two underlying exact invariants, one coming from physics properties of the dynamics is the canonical transverse momentum, and the second, named the "water-bag" and coming from geometric property of the phase-space is just the direct consequence of the Liouville Theorem. In this paper we prove the existence and uniqueness of global weak entropy solutions of the relativistic waterbag continuum. Existence is based on vanishing viscosity method and bounded variations (BV) estimates to get compactness while proof of uniqueness relies on kinetic formulation of the relativistic waterbag continuum and the associated kinetic entropy defect measure.

Amplitude dependence of frequency sweep rates of whistler mode chorus emissions

[1] The frequency and amplitude characteristics of chorus emissions are studied by performing electron hybrid code simulations with different initial number densities of energetic electrons. Chorus emissions with rising tones are generated in all simulation runs except for the simulation assuming the lowest number density. The frequency sweep rates of reproduced chorus vary depending on the variation of the wave amplitude of respective chorus elements. We find that the theoretically estimated frequency sweep rates are consistent with the simulation results. The simulation results reveal that the characteristic frequency variation of chorus elements showing rising tones are formed at the region very close to the magnetic equator while the wave amplitudes of chorus elements are significantly amplified through their propagation away from the equator. The spatial scale of the region where the observed explosive wave growth varies in each simulation run, corresponding to the difference of the wave amplitudes of reproduced chorus elements. We estimate the spatial scale hc, a measure of the spatial extent of the generation region derived from the nonlinear wave growth theory, using parameters of the simulation runs and find its consistency with the simulation results. We also analyze the energy spectra of reproduced chorus elements in the simulation results and find that the spectra of chorus are essentially different from those predicted by the linear growth rates in the frequency domain. These results clearly demonstrate that the nonlinear wave-particle interaction governs the chorus generation mechanism.

Amplification of ion acoustic wave in an inhomogeneous plasma through nonlinear wave-particle interaction with drift wave turbulence

A study on the amplification of ion acoustic wave in an inhomogeneous plasma has been made on the basis of a nonlinear wave-particle interaction process called plasma maser effect. The drift wave instability, which is a universal phenomenon of an inhomogeneous confined plasma system, is found to be strongly in phase relation with thermal particles and may transfer its wave energy nonlinearly through a modulated field to ion acoustic wave. Considering a Maxwellian distribution function model for inhomogeneous plasmas under the standard local approximation, we have estimated the growth rate for ion acoustic wave, which is obtained by using the nonlinear dispersion relation. It has been found that amplification of ion acoustic wave is possible at the expense of drift wave turbulent energy. This result may be particularly important for stability of various drift modes in magnetically confined plasma as well as for transport of momentum and energy in such inhomogeneous systems.

Nonlinear turbulence theory and simulation of Buneman instability

In the present paper, the weak turbulence theory for reactive instabilities, formulated in a companion paper [P. H. Yoon, Phys. Plasmas 17, 112316 (2010)], is applied to the strong electron-ion two-stream (or Buneman) instability. The self-consistent theory involves quasilinear velocity space diffusion equation for the particles and nonlinear wave kinetic equation that includes quasilinear (or induced emission) term as well as nonlinear wave-particle interaction term (or a term that represents an induced scattering off ions). We have also performed one-dimensional electrostatic Vlasov simulation in order to benchmark the theoretical analysis. Under the assumption of self-similar drifting Gaussian distribution function for the electrons it is shown that the current reduction and the accompanying electron heating as well as electric field turbulence generation can be discussed in a self-consistent manner. Upon comparison with the Vlasov simulation result it is found that quasilinear wave kinetic equation alone is insufficient to account for the final saturation amplitude. Upon including the nonlinear scattering term in the wave kinetic equation, however, we find that a qualitative agreement with the simulation is recovered. From this, we conclude that the combined quasilinear particle diffusion plus induced emission and scattering (off ions) processes adequately account for the nonlinear development of the Buneman instability.