Violation of adiabaticity in magnetic billiards due to separatrix crossings

We consider dynamics of magnetic billiards with curved boundaries and strong inhomogeneous magnetic field. We investigate a violation of adiabaticity of charged particle motion in this system. The destruction of adiabatic invariance is due to the change of type of the particle trajectory: particles can drift along the boundary reflecting from it or rotate around the magnetic field at some distance from the boundary without collisions with it. Trajectories of these two types are demarcated in the phase space by a separatrix. Crossings of the separatrix result in jumps of the adiabatic invariant. We derive an asymptotic formula for such a jump and demonstrate that an accumulation of these jumps leads to the destruction of the adiabatic invariance

Remarkable charged particle dynamics near magnetic field null lines

The study of charged-particle motion in electromagnetic fields is a rich source of problems, models, and new phenomena for nonlinear dynamics. The case of a strong magnetic field is well studied in the framework of a guiding center theory, which is based on conservation of an adiabatic invariant – the magnetic moment. This theory ceases to work near a line on which the magnetic field vanishes – the magnetic field null line. In this paper we show that the existence of these lines leads to remarkable phenomena which are new both for nonlinear dynamics in general and for the theory of charged-particle motion. We consider the planar motion of a charged particle in a strong stationary perpendicular magnetic field with a null line and a strong electric field. We show that particle dynamics switch between a slow guiding center motion and the fast traverse along a segment of the magnetic field null line. This segment is the same (in the principal approximation) for all particles with the same total energy. During the phase of a guiding center motion, the magnetic moment of particle’s Larmor rotation stays approximately constant, i.e., it is an adiabatic invariant. However, upon each traversing of the null-line, the magnetic moment changes in a random fashion, causing the particle choose a new trajectory of the guiding center motion. This results in a stationary distribution of the magnetic moment, which only depends on the particle’s total energy. The jumps in the adiabatic invariant are described by Painleve II equation

Charged particle nonlinear resonance with localized electrostatic wave-packets

A resonant wave-particle interaction, in particular a nonlinear resonance characterized by particle phase trapping, is an important process determining charged particle energization in many space and laboratory plasma systems. Although an individual charged particle motion in the nonlinear resonance is well described theoretically, the kinetic equation modeling the long-term evolution of the resonant particle ensemble has been developed only recently. This study is devoted to generalization of this equation for systems with localized wave packets propagating with the wave group velocity different from the wave phase velocity. We limit our consideration to the Landau resonance of electrons and waves propagating in an inhomogeneous magnetic field. Electrons resonate with the wave field-aligned electric fields associated with gradients of wave electrostatic potential or variations of the field-aligned component of the wave vector potential. We demonstrate how wave-packet properties determine the efficiency of resonant particle acceleration and derive the nonlocal integral operator acting on the resonant particle distribution. This operator describes particle distribution variations due to interaction with one wave-packet. We solve kinetic equation with this operator for many wave-packets and show that solutions coincide with the results of the numerical integration of test particle trajectories. To demonstrate the range of possible applications of the proposed approach, we consider the electron evolution induced by the Landau resonances with packets of kinetic Alfven waves, electron acoustic waves, and very oblique whistler waves in the near-Earth space plasma

Kinetic equation for nonlinear wave-particle interaction: solution properties and asymptotic dynamics

We consider a kinetic equation describing evolution of the particle distribution function in a system with nonlinear wave-particle interactions (trappings into resonance and nonlinear scatterings). We study properties of its solutions and show that the only stationary solution is a constant, and that all solutions with smooth initial conditions tend to a constant as time grows. The resulting flattening of the distribution function in the domain of nonlinear interactions is similar to one described by the quasi-linear plasma theory, but the distribution evolves much faster. The results are confirmed numerically for a model problem

Nonresonant charged particle acceleration by electrostatic wave propagating across fluctuating magnetic field

In this Letter, we demonstrate the effect of nonresonant charged-particle acceleration by an electrostatic wave propagating across the background magnetic field. We show that in the absence of resonance (i.e., when particle velocities are much smaller than the wave phase velocity) particles can be accelerated by electrostatic waves provided that the adiabaticity of particle motion is destroyed by magnetic field fluctuations. Thus, in a system with stochastic particle dynamics the electrostatic wave should be damped even in the absence of Landau resonance. The proposed mechanism is responsible for the acceleration of particles that cannot be accelerated via resonant wave-particle interactions. Simplicity of this straightforward acceleration scenario indicates a wide range of possible applications

Long-term evolution of electron distribution function due to nonlinear resonant interaction with whistler mode waves

Accurately modelling and forecasting of the dynamics of the Earth’s radiation belts with the available computer resources represents an important challenge that still requires significant advances in the theoretical plasma physics field of wave–particle resonant interaction. Energetic electron acceleration or scattering into the Earth’s atmosphere are essentially controlled by their resonances with electromagnetic whistler mode waves. The quasi-linear diffusion equation describes well this resonant interaction for low intensity waves. During the last decade, however, spacecraft observations in the radiation belts have revealed a large number of whistler mode waves with sufficiently high intensity to interact with electrons in the nonlinear regime. A kinetic equation including such nonlinear wave–particle interactions and describing the long-term evolution of the electron distribution is the focus of the present paper. Using the Hamiltonian theory of resonant phenomena, we describe individual electron resonance with an intense coherent whistler mode wave. The derived characteristics of such a resonance are incorporated into a generalized kinetic equation which includes non-local transport in energy space. This transport is produced by resonant electron trapping and nonlinear acceleration. We describe the methods allowing the construction of nonlinear resonant terms in the kinetic equation and discuss possible applications of this equation

Kinetic equation for nonlinear resonant wave-particle interaction

We investigate nonlinear resonant wave-particle interactions including effects of particle (phase) trapping, detrapping, and scattering by high-amplitude coherent waves. After deriving the relation 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

Charged particle dynamics in turbulent current sheet

We study dynamics of charged particle in current sheets with magnetic fluctuations. We use the adiabatic theory to describe the nonperturbed charged particle motion and show that magnetic field fluctuations destroy the adiabatic invariant. We demonstrate that the evolution of particle adiabatic invariant's distribution is described by a diffusion equation and derive analytical estimates of the rate of adiabatic invariant's diffusion. This rate is proportional to power density of magnetic field fluctuations. We compare analytical estimates with numerical simulations. We show that adiabatic invariant diffusion results in transient particles trapping in the current sheet. For magnetic field fluctuation amplitude few times larger than a normal magnetic field component, more than 50% of transient particles become trapped. We discuss the possible consequences of destruction of adiabaticity of the charged particle motion on the state of the current sheets

Probabilistic approach to nonlinear wave-particle resonant interaction

In this paper we provide a theoretical model describing the evolution of the charged particle distribution function in a system with nonlinear wave particle interactions. Considering a system with strong electrostatic waves propagating in an inhomogeneous magnetic field, we demonstrate that individual particle motion can be characterized by the probability of trapping into the resonance with the wave and by the efficiency of scattering at resonance. These characteristics, being derived for a particular plasma system, can be used to construct a kinetic equation (or generalized Fokker-Planck equation) modelling the long-term evolution of the particle distribution. In this equation, effects of charged particle trapping and transport in phase space are simulated with a nonlocal operator. We demonstrate that solutions of the derived kinetic equations agree with results of test particle tracing. The applicability of the proposed approach for the description of space and laboratory plasma systems is also discussed

Trapping (capture) into resonance and scattering on resonance: summary of results for space plasma systems

In the present review we survey space plasma systems where the nonlinear resonant interaction between charged particles and electromagnetic waves plays an important role. We focus on particle acceleration by strong electromagnetic waves. We start with presenting a general description of nonlinear resonant interaction based on the theory of slowfast Hamiltonian systems with resonances. Then we turn to several manifestations of the resonance effects in various space plasma systems. We describe a universal approach for evaluating main characteristics of the resonant particle dynamics: probability of trapping into resonance, energy change due to scattering and trapping. Then we demonstrate how effects of nonlinear resonant trapping and scattering can be combined in a generalized kinetic equation. We also discuss the stability of trapped motion and evolution of particle ensemble in systems with trapping. The main objective of this review is to provide a general approach for characterizing plasma systems with nonlinear resonant interactions