Alexei Vasiliev

Captures into resonance and scattering on resonance in dynamics of a charged relativistic particle in magnetic field and electrostatic wave

Abstract The dynamics of a relativistic charged particle in a uniform magnetic field and an electrostatic plane wave is studied. For a small-amplitude high-frequency wave, the system is reduced to a two-degree-of-freedom Hamiltonian system with slow and fast variables. In this system, the phenomena of capture into resonance and scattering on the resonance are described. These phenomena determine the long-term dynamics of the particle.

Nonlinear electron acceleration by oblique whistler waves: Landau resonance vs. cyclotron resonance

This paper is devoted to the study of the nonlinear interaction of relativistic electrons and high amplitude strongly oblique whistler waves in the Earths radiation belts. We consider electron trapping into Landau and fundamental cyclotron resonances in a simplified model of dipolar magnetic field. Trapping into the Landau resonance corresponds to a decrease of electron equatorial pitch-angles, while trapping into the first cyclotron resonance increases electron equatorial pitch-angles. For 100 keV electrons, the energy gained due to trapping is similar for both resonances. For electrons with smaller energy, acceleration is more effective when considering the Landau resonance. Moreover, trapping into the Landau resonance is accessible for a wider range of initial pitch-angles and initial energies in comparison with the fundamental resonance. Thus, we can conclude that for intense and strongly oblique waves propagating in the quasi-electrostatic mode, the Landau resonance is generally more important than the fundamental one.

Changes in the adiabatic invariant and streamline chaos in confined incompressible Stokes flow.

The steady incompressible flow in a unit sphere introduced by Bajer and Moffatt [J. Fluid Mech. 212, 337 (1990)] is discussed. The velocity field of this flow differs by a small perturbation from an integrable field whose streamlines are almost all closed. The unperturbed flow has two stationary saddle points (poles of the sphere) and a two-dimensional separatrix passing through them. The entire interior of the unit sphere becomes the domain of streamline chaos for an arbitrarily small perturbation. This phenomenon is explained by the nonconservation of a certain adiabatic invariant that undergoes a jump when a streamline crosses a small neighborhood of the separatrix of the unperturbed flow. An asymptotic formula is obtained for the jump in the adiabatic invariant. The accumulation of such jumps in the course of repeated crossings of the separatrix results in the complete breaking of adiabatic invariance and streamline chaos. (c) 1996 American Institute of Physics.

Fast transport of resonant electrons in phase space due to nonlinear trapping by whistler waves

We present an analytical, simplified formulation accounting for the fast transport of relativistic electrons in phase space due to wave-particle resonant interactions in the inhomogeneous magnetic field of Earths radiation belts. We show that the usual description of the evolution of the particle velocity distribution based on the Fokker-Planck equation can be modified to incorporate nonlinear processes of wave-particle interaction, including particle trapping. Such a modification consists in one additional operator describing fast particle jumps in phase space. The proposed, general approach is used to describe the acceleration of relativistic electrons by oblique whistler waves in the radiation belts. We demonstrate that for a wave power distribution with a hard enough power law tail P(Bw2)∝Bw−η such that η < 5/2, the efficiency of nonlinear acceleration could be more effective than the conventional quasi-linear acceleration for 100 keV electrons.

Probability of relativistic electron trapping by parallel and oblique whistler-mode waves in Earth's radiation belts

We investigate electron trapping by high-amplitude whistler-mode waves propagating at small as well as large angles relative to geomagnetic field lines. The inhomogeneity of the background magnetic field can result in an effective acceleration of trapped particles. Here, we derive useful analytical expressions for the probability of electron trapping by both parallel and oblique waves, paving the way for a full analytical description of trapping effects on the particle distribution. Numerical integrations of particle trajectories allow to demonstrate the accuracy of the derived analytical estimates. For realistic wave amplitudes, the levels of probabilities of trapping are generally comparable for oblique and parallel waves, but they turn out to be most efficient over complementary energy ranges. Trapping acceleration of 100 keV electrons.

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.

Chaotic advection in a cubic Stokes flow

Abstract A stationary incompressible Stokes flow in a sphere is considered. The flow was introduced by Stone et al. (1991) as a flow inside a neutrally buoyant spherical drop immersed in a linear flow. The velocity field of the flow is a result of a small perturbation of an integrable velocity field with almost all streamlines closed. Under arbitrarily small pertubation a large domain of chaotic advection within the sphere arises. This phenomenon is explained by quasirandom changes in the adiabatic invariant of the flow, which occur as a streamline crosses the two-dimensional separatrix of the unperturbed flow. Phase portraits of the averaged system are constructed. An asymptotic formula for the change in the adiabatic invariant due to the separatrix crossing is derived. The process of diffusion of the adiabatic invariant due to multiple separatrix crossings is described.

Phase change between separatrix crossings in slow-fast Hamiltonian systems

We consider a Hamiltonian system with slow and fast motions, one degree of freedom corresponding to fast motion, and the other degrees of freedom corresponding to slow motion. Suppose that at frozen values of the slow variables there is a non-degenerate saddle point and a separatrix on the phase plane of the fast variables. In the process of variation of the slow variables, the projection of a phase trajectory onto the phase plane of the fast variables may repeatedly cross the separatrix. These crossings are described by the crossing parameter called the pseudo-phase. We obtain an asymptotic formula for the pseudo-phase dependence on the initial conditions, and calculate the change of the pseudo-phase between two subsequent separatrix crossings.

Change of the adiabatic invariant at a separatrix in a volume-preserving 3D system

A 3D volume-preserving system is considered. The system differs by a small perturbation from an integrable one. In the phase space of the unperturbed system there are regions filled with closed phase trajectories, where the system has two independent first integrals. These regions are separated by a 2D separatrix passing through non-degenerate singular points. Far from the separatrix, the perturbed system has an adiabatic invariant. When a perturbed phase trajectory crosses the two-dimensional separatrix of the unperturbed system, this adiabatic invariant undergoes a quasi-random jump. The formula for this jump is obtained. If the geometry of the system allows for multiple separatrix crossings, the destruction of adiabatic invariance is possible, leading to chaotic behaviour in the system. An example of such a system is given.

Destruction of adiabatic invariance at resonances in slow¿fast Hamiltonian systems

There are many problems that lead to analysis of dynamical systems in which one can distinguish motions of two types: slow one and fast one. An averaging over fast motion is used for approximate description of the slow motion. First integrals of the averaged system are approximate first integrals of the exact system, i.e. adiabatic invariants. Resonant phenomena in fast motion (capture into resonance, scattering on resonance) lead to inapplicability of averaging, destruction of adiabatic invariance, dynamical chaos and transport in large domains in the phase space. In the paper perturbation theory methods for description of these phenomena are outlined. We also consider as an example the problem of surfatron acceleration of a relativistic charged particle.