Nonlinear Wave-particle Interactions Research Articles

Backward test particle simulation of nonlinear cyclotron wave-particle interactions in the radiation belts

Wave-particle interaction plays a crucial role in the dynamics of the Earth’s radiation belts. Cyclotron resonance between coherent whistler mode electromagnetic waves and energetic electrons of the radiation belts is often called a coherent instability. Coherent instability leads to wave amplification/generation and particle acceleration/scattering. The effect of wave on particle’s distribution function is a key component of the instability. In general, whistler wave amplitude can grow over threshold of quasi-linear (linear) diffusion theory which analytically tracks the time-evolution of a particle distribution. Thus, a numerical approach is required to model the nonlinear wave induced perturbations on particle distribution function. A backward test particle model is used to determine the energetic electrons phase space dynamics as a result of coherent whistler wave instability. The results show the formation of a phase space features with much higher resolution than is available with forward scattering models. In the nonlinear regime the formation of electron phase space holes upstream of a monochromatic wave is observed. The results validate the nonlinear phase trapping mechanism that drives nonlinear whistler mode growth. The key differences in phase-space perturbations between the linear and nonlinear scenarios are also illustrated. For the linearized equations or for low (below threshold) wave amplitudes in the nonlinear case, there is no formation of a phase-space hole and both models show features that can be characterized as linear striations or ripples in phase-space.

A hybrid Eulerian-Lagrangian Vlasov method for nonlinear wave-particle interaction in weakly inhomogeneous magnetic field

We present a hybrid Eulerian-Lagrangian (HEL) Vlasov method for nonlinear resonant wave-particle interactions in weakly inhomogeneous magnetic field. The governing Vlasov equation is derived from a recently proposed resonance tracking Hamiltonian theory. It gives the evolution of the distribution function with a scale-separated Hamiltonian that contains the fast-varying coherent wave-particle interaction and slowly-varying motion about the resonance frame of reference. The hybrid scheme solves the fast-varying phase space evolution on Eulerian grid with an adaptive time step and then advances the slowly-varying dynamics by Lagrangian method along the resonance trajectory. We apply the HEL method to study the frequency chirping of whistler-mode chorus wave in the magnetosphere and the self-consistent simulations reproduce the chirping chorus wave and give high-resolution phase space dynamics of energetic particles at low computational cost. The scale-separated HEL approach could provide additional insights of the wave instabilities and wave-particle nonlinear coherence compared to the conventional Vlasov and particle-in-cell methods.

Electron cyclotron resonance during plasma initiation

Electron-cyclotron resonance heating (ECRH) is the main heating mechanism in the Wendelstein 7-X (W7-X) stellarator. Although second-harmonic ECRH (X2) has been used routinely for plasma startup, startup at third harmonic (X3) is known to be much more difficult. In this work, we investigate the energy gain of particles during nonlinear wave–particle interaction for conditions relevant to second- and third-harmonic startups in W7-X. We take into account both the beam and the ambient magnetic field inhomogeneities. The latter is shown to significantly increase the mean energy gain resulting from a single wave–particle resonant interaction. In W7-X-like conditions, the improvement in maximum gained energy is up to 4 times the analogous uniform magnetic field case. However, this improvement is not enough to ensure X3 startup. The optimal magnetic field inhomogeneity length scale for average energy gain and start up in W7-X-like conditions is found to be in the range of $1$ to $3\ {\rm km}^{-1}$ . A possibility of using multiple beams with neighbouring resonances is also considered. A considerable enhancement of the energy gain is demonstrated.

Nonlinear Landau damping of electron Bernstein waves in MAST-U

The Mega-Amp Spherical Tokamak Upgrade (MAST-U) is preparing to investigate microwave based current drive using an advanced wave coupling scheme known as O-X-B at high power. The main goal is to assert if this is an efficient method for current drive in the future fusion demonstration power plant STEP. A model for the nonlinear wave-particle interaction nonlinear Landau damping (NLD) is presented. A relativistic, kinetic and electrostatic description is used. For MAST-U, NLD could lead to heating and current drive at the upper hybrid (UH) layer or at the electron cyclotron (EC) resonance of a substantially downshifted daughter wave produced by the wave-particle interactions. It is found that the typical treatment used for nonlinear waveinteractions fails at two critical points in the vicinity of the UH layer.

Instability of Ion Cyclotron Waves (ICWS) at the Expense of Lower Hybrid Drift Waves (LHDWS) Turbulence Energy

Instability of ion cyclotron waves(ICWs) is investigated in presence of lower hybrid drift waves(LHDWs) turbulence. Plasma inhomogeneity in the Earth’s magnetopause region supports a range of low frequency drift wave turbulent fields due to gradients in density in different regions of the media. One of these drift phenomena is identified as lower hybrid drift waves (LHDWs) which satisfies resonant conditions ω − k · v = 0. We have considered a nonlinear wave-particle interaction model where the resonant wave that accelerates the particle in magnetopause may transfer its energy to ion cyclotron waves through a modulated field. In spite of the frequency gaps between the two waves, energy can be transferred nonlinearly to generate unstable ion cyclotron waves which always do not satisfy the resonant condition Ω−K · v ≠ 0 and the nonlinear scattering condition Ω − ω − (K − k) · v ̸= 0. Here, ω and Ω are frequencies of the resonant and the nonresonant waves respectively and k and K are the corresponding wave numbers. We have obtained a nonlinear dispersion relation for ion cyclotron waves(ICWs) in presence of lower hybrid drift waves(LHDWs)turbulence. The growth rate of the ion cyclotron waves using space observational data in the magnetopause region has been estimated.

Scattering of energetic electrons through nonlinear cyclotron resonance with coherent whistler-mode hiss emissions

Recent observations have revealed that plasmaspheric hiss consists of many discrete waves called “hiss elements.” However, the interaction of energetic electrons (10 keV to several MeV) with the plasmaspheric hiss has only been simulated by the quasilinear (QL) diffusion theory, which does not take the fine wave structure into account. The QL theory cannot address nonlinear particle motions determined by the inhomogeneity factor, which influences the scattering of electrons in pitch angle and energy. This study aims to identify differences between the nonlinear wave–particle interaction and QL theory for plasmaspheric hiss emissions. We conduct test particle simulations to demonstrate the nonlinear interactions between hiss waves and electrons. The nonlinear theory is used to model hiss elements consisting of discrete frequencies and continuous phases. Unlike the other theories, the frequency and amplitude variations in time of the hiss packet are taken into account. Frequencies of the packets are determined to satisfy the separability criterion; when the criterion is met, resonance overlapping is absent, and the electrons can generate each wave element independently. The realistic simulation model of hiss waves reproduces the scattering of electrons by both first- and second-order resonances. We also evaluate the efficiency of electron scattering by calculating nonlinear diffusion coefficients. The diffusion coefficient of equatorial pitch angle is of the same order of magnitude as those calculated by the QL diffusion theory, while we find the effective acceleration of resonant electrons by successive nonlinear trapping, which is not evaluated by the QL theory.

Nonlinear Electron Phase‐Space Dynamics in Spontaneous Excitation of Falling‐Tone Chorus

AbstractWhistler mode chorus, characterized by frequency chirping, is an important type of waves in planetary magnetospheres. To investigate the role of nonlinear wave particle interactions in excitation of chorus, analysis of electron phase space dynamics is required. While electron phase hole associated with rising tone chorus has been well studied, the phase space structure corresponding to falling tone chorus is less understood. Here, we investigate in detail the electron phase space dynamics in a spontaneously generated falling tone chorus with a parabolic type magnetic field, where field strength decreases away from the equator. We demonstrate the formation and evolution of electron phase space clump from downstream to upstream regions, and show that the variation of frequency chirping rate is caused by the movement of the source region. The results are consistent with the recently proposed Trap‐Release‐Amplify model, and should be useful to understand the mechanism of chorus frequency chirping.

Analytical results for phase bunching in the pendulum model of wave-particle interactions

Radiation belt electrons are strongly affected by resonant interactions with cyclotron-resonant waves. In the case of a particle passing through resonance with a single, coherent wave, a Hamiltonian formulation is advantageous. With certain approximations, the Hamiltonian has the same form as that for a plane pendulum, leading to estimates of the change at resonance of the first adiabatic invariant I, energy, and pitch angle. In the case of large wave amplitude (relative to the spatial variation of the background magnetic field), the resonant change in I and its conjugate phase angle ξ are not diffusive but determined by nonlinear dynamics. A general analytical treatment of slow separatrix crossing has long been available and can be used to give the changes in I associated with “phase bunching,” including the detailed dependence on ξ, in the nonlinear regime. Here we review this treatment, evaluate it numerically, and relate it to previous analytical results for nonlinear wave-particle interactions. “Positive phase bunching” can occur for some particles even in the pendulum Hamiltonian approximation, though the fraction of such particles may be exponentially small.

Nonlinear triggering process of whistler-mode emissions in a homogeneous magnetic field

We perform an electromagnetic particle simulation of triggered emissions in a uniform magnetic field for understanding of nonlinear wave–particle interaction in the vicinity of the magnetic equator. A finite length of a whistler-mode triggering wave packet with a constant frequency is injected by oscillating an external current at the equator. We find that the first subpacket of triggered emissions is generated in the homogeneous magnetic field. By analyzing resonant currents and resonant electron dynamics in the simulation, we find that the formation of an electron hole in a velocity phase space results in resonant currents, and the currents cause wave amplification and frequency increase. We obtain the interaction time of counter-streaming resonant electrons in a triggering wave packet with a finite width. By changing the duration time of the triggering pulse, we evaluate the interaction time necessary for formation of an electron hole. We conduct 4 runs with different duration times of the triggering pulse, 980, 230, 105, 40 Omega _e^{-1}, which correspond to cases with interaction times, 370%, 86%, 39%, and 15% of the nonlinear trapping period, respectively. We find generation of triggered emissions in the three cases of 370%, 86%, and 39%, which agrees with the conventional nonlinear model that the nonlinear transition time, which is necessary for formation of resonant currents, is about a quarter of the nonlinear trapping period.Graphic

Wave-particle interactions in quantum plasmas

Wave-particle interaction (WPI) is one of the most fundamental processes in plasma physics in which one most prominent example is the Landau damping. Owing to its excellent energy-exchange mechanism, the WPI has gained increasing interest not only from theoretical points of view, but also its many important applications including plasma heating and plasma acceleration. In this review work, we present theoretical backgrounds of linear and nonlinear wave-particle interactions in quantum plasmas. Specifically, we focus on the wave-particle interactions for homogeneous plasma waves (i.e., waves with infinite extent rather than a localized pulse) as well as for propagating electrostatic waves in the weak and strong quantum regimes to demonstrate the modifications of several classical features including those associated with resonant and trapped particles. Finally, the future perspectives of WPI in quantum plasmas are presented.

Interaction between energetic-ions and internal kink modes in a weak shear tokamak plasma

Based on the conventional tokamak HL-2A-like parameters and profiles, the linear properties and the nonlinear dynamics of non-resonant kink mode (NRK) and non-resonant fishbone instability (NRFB) in reversed shear tokamak plasmas are investigated by using the global hybrid kinetic-magnetohydrodynamic nonlinear code M3D-K. This work mainly focuses on the effect of passing energetic-ions on the NRK and NRFB instabilities, which is different from the previous works. It is demonstrated that the NRFB can be destabilized by the passing energetic-ions when the energetic-ion beta β h exceeds a critical value. The transition from NRK to NRFB occurs when the energetic-ion beta β h increases to above a critical value. The resonance condition responsible for the excitation of NRFB is interestingly found to be satisfied at ω t + ω p ≈ ω, where ω t is the toroidal motion frequency, ω p is the poloidal motion frequency and ω is the mode frequency. The nonlinear evolutions of NRFB’s mode structures and Poincaré plots are also analyzed in this work and it is found that the NRFB can induce evident energetic-ion loss/redistribution, which can degrade the performance of the plasmas. These findings are conducive to understanding the mechanisms of NRFB induced energetic-ion loss/redistribution through nonlinear wave-particle interaction.

Electrostatic weak turbulence theory for warm magnetized plasmas

Electrostatic weak turbulence theory for plasmas immersed in an ambient magnetic field is developed by employing a hybrid two-fluid and kinetic theories. The nonlinear susceptibility response function is calculated with the use of warm two-fluid equations. The linear dispersion relations for longitudinal electrostatic waves in magnetized plasmas are also obtained within the warm two-fluid theoretical scheme. However, dissipations that arise from linear and nonlinear wave–particle interactions cannot be discussed with the macroscopic two-fluid theory. To compute such collisionless dissipation effects, linearized kinetic theory is utilized. Moreover, a particle kinetic equation, which is necessary for a self-consistent description of the problem, is derived from the quasilinear kinetic theory. The final set of equations directly generalizes the electrostatic weak turbulence theory in unmagnetized plasmas, which could be applied for a variety of problems including the electron beam–plasma interactions in magnetized plasma environments.

Polarization vector formalism of plasma weak turbulence

This paper formulates the plasma weak turbulent theory based on the unit electric field polarization vector. This concept is not intrinsically new, and partial formulations of weak turbulence processes based on the polarization vector approach are found in the literature. However, the present paper applies such a method uniformly to all the relevant processes for the first time, thus unifying the existing formalisms. The present result potentially provides many advantages including the fact that it facilitates the complex manipulations of various tensor coupling coefficients that dictate the wave–wave and nonlinear wave–particle interactions. To demonstrate its validity, the limit of unmagnetized plasmas is considered, and it is shown that the known results are recovered. The present formalism can be extended to more complex situations including magnetized plasmas, which is a subject of future work.

The in-situ exploration of Jupiter’s radiation belts

Jupiter has the most complex and energetic radiation belts in our Solar System and one of the most challenging space environments to measure and characterize in-depth. Their hazardous environment is also a reason why so many spacecraft avoid flying directly through their most intense regions, thus explaining how Jupiter’s radiation belts have kept many of their secrets so well hidden, despite having been studied for decades. In this paper we argue why these secrets are worth unveiling. Jupiter’s radiation belts and the vast magnetosphere that encloses them constitute an unprecedented physical laboratory, suitable for interdisciplinary and novel scientific investigations: from studying fundamental high energy plasma physics processes which operate throughout the Universe, such as adiabatic charged particle acceleration and nonlinear wave-particle interactions, to exploiting the astrobiological consequences of energetic particle radiation. The in-situ exploration of the uninviting environment of Jupiter’s radiation belts presents us with many challenges in mission design, science planning, instrumentation, and technology. We address these challenges by reviewing the different options that exist for direct and indirect observations of this unique system. We stress the need for new instruments, the value of synergistic Earth and Jupiter-based remote sensing and in-situ investigations, and the vital importance of multi-spacecraft in-situ measurements. While simultaneous, multi-point in-situ observations have long become the standard for exploring electromagnetic interactions in the inner Solar System, they have never taken place at Jupiter or any strongly magnetized planet besides Earth. We conclude that a dedicated multi-spacecraft mission to Jupiter is an essential and obvious way forward for exploring the planet’s radiation belts. Besides guaranteeing numerous discoveries and huge leaps in our understanding of radiation belt systems, such a mission would also enable us to view Jupiter, its extended magnetosphere, moons, and rings under new light, with great benefits for space, planetary, and astrophysical sciences. For all these reasons, in-situ investigations of Jupiter’s radiation belts deserve to be given a high priority in the future exploration of our Solar System. This article is based on a White Paper submitted in response to the European Space Agency’s call for science themes for its Voyage 2050 programme.

A Statistical Analysis of Duration and Frequency Chirping Rate of Falling Tone Chorus

AbstractThe duration (τ) and chirping rate (Γ) of whistler mode chorus waves are two of the most important properties to understand chorus generation mechanism and to quantify effects of nonlinear wave particle interactions on radiation belt electron acceleration. In this study, we perform the first statistical analysis of the duration and chirping rate of falling tone chorus elements using Van Allen Probes data. We found that τ increases and Γ decreases with increasing L‐shell, although the dependence is weak. The duration from dawnside and dayside have similar distributions, which is a bit longer than those from duskside and nightside. However, Γ has little dependence on MLT. The relation between τ and Γ shows that τ scales with Γ as , supporting one of the previous theoretical models of rising tone chorus in Teng et al. (2017). Our results should provide important insights to deepen our understanding of falling tone chorus excitation.

UBER v1.0: a universal kinetic equation solver for radiation belts

Abstract. Recent proceedings in radiation belt studies have proposed new requirements for numerical methods to solve the kinetic equations involved. In this article, we present a numerical solver that can solve the general form of the radiation belt Fokker–Planck equation and Boltzmann equation in arbitrarily provided coordinate systems and with user-specified boundary geometry, boundary conditions, and equation terms. The solver is based upon the mathematical theory of stochastic differential equations, whose computational accuracy and efficiency are greatly enhanced by specially designed adaptive algorithms and a variance reduction technique. The versatility and robustness of the solver are exhibited in four example problems. The solver applies to a wide spectrum of radiation belt modeling problems, including the ones featuring non-diffusive particle transport such as that arising from nonlinear wave–particle interactions.

Time simulation of the nonlinear wave–particle interaction in meters-long traveling-wave tubes

We propose a multi-particle self-consistent Hamiltonian (derived from an N-body description) that is applicable for periodic structures, such as traveling-wave tubes (TWTs), gyrotrons, free-electron lasers, or particle accelerators. We build a one-dimension symplectic multi-particle algorithm to simulate the nonlinear wave–particle interaction in the time domain occurring in an experimental 3-m-long helix TWT. Our algorithm is efficient thanks to a drastic reduction model. A 3D helix version of our reduction model is provided. Finally, we establish an explicit expression of the electromagnetic power in the time domain and in non-monochromatic (non-“continuous waveform”) regime.

A “Trap‐Release‐Amplify” Model of Chorus Waves

AbstractWhistler mode chorus waves are quasi‐coherent electromagnetic emissions with frequency chirping. Various models have been proposed to understand the chirping mechanism, which is a long‐standing problem in space plasmas. Based on analysis of effective wave growth rate and electron phase space dynamics in a self‐consistent particle simulation, we propose a phenomenological model called the “Trap‐Release‐Amplify” (TaRA) model for chorus. In this model, phase space structures of correlated electrons are formed by nonlinear wave particle interactions, which mainly occur in the downstream of equator. When released from the wave packet in the upstream, these electrons lead to selective amplification of new emissions which satisfy the phase‐locking condition to maximize wave power transfer, resulting in frequency chirping. The phase‐locking condition at the release point gives a chirping rate that is, fully consistent with the one by Helliwell in case of a nonuniform background magnetic field. The nonlinear wave particle interaction part of the TaRA model results in a chirping rate that is, proportional to wave amplitude, a conclusion originally reached by Vomvoridis et al. Therefore, the TaRA model unifies two different results from seemingly unrelated studies. Furthermore, the TaRA model naturally explains fine structures of chorus waves, including subpackets and bandwidth, and their evolution through dynamics of phase‐trapped electrons. Finally, we suggest that this model could be applied to explain other related phenomena, including frequency chirping of chorus in a uniform background magnetic field and of electromagnetic ion cyclotron waves in the magnetosphere.

Linear and Nonlinear Effects of Proton Temperature Anisotropy on Proton-beam Instability in the Solar Wind

Abstract Solar wind observations have shown that the drift velocity of the proton beam relative to the background proton is of the order of the local Alfvén velocity. The proton-beam instability has been suggested to play an important role in decelerating proton beams. Most existing studies examined the kinetic properties of the proton-beam instability in simple Maxwellian or drift-Maxwellian plasmas. In this paper, we systemically investigate the effects of the proton temperature anisotropy on the growth rates of proton-beam instabilities using linear Vlasov equations and the nonlinear evolution of these instabilities at different propagation angles using 1D and 2D hybrid simulations. The results show that the real frequencies, growth rates, and threshold conditions strongly depend on the proton temperature anisotropy T i ⊥ / T i ∣ ∣ and the parallel proton beta β i ∣ ∣ . The parallel magnetosonic/whistler (M/W), backward M/W, and oblique Alfvén/ion-cyclotron (A/IC) instabilities are likely to grow in the regime of T i ⊥ / T i ∣ ∣ < 1 , whereas the parallel M/W, oblique A/IC, parallel A/IC, and mirror instabilities tend to be excited in the regime of T i ⊥ / T i ∣ ∣ > 1 . Moreover, the oblique A/IC instability with a linear wave growth phase can quickly reduce the proton-beam drift velocity when T i ⊥ / T i ∣ ∣ < 1 , while the nonlinear wave–particle interaction for the parallel A/IC and mirror instabilities tends to gradually reduce the drift velocity when T i ⊥ / T i ∣ ∣ > 1 . The linear and nonlinear evolution of the proton-beam instability associated with the proton temperature anisotropy can be used to explain the observed dependence of the proton-beam drift velocity on the plasma beta and temperature anisotropy in the solar wind.

A Study on Instabilities and Nonlinear Wave Energy Exchange Process in Terrestrial Ionospheric Plasma

From ground and satellite based observatories, various forms of nonlinear effects and plasma waves instabilities are observed in the Earth’s ionosphere. In this paper we present a brief review on nonlinear wave-particle interaction energy exchange mode plasma maser instability and on probable growth rate of high frequency non-resonant waves at different regions of the Earth’s ionosphere.