N. N. Zol’nikova

Surfatron acceleration of protons by an electromagnetic wave at the heliosphere periphery

The trapping and subsequent efficient surfatron acceleration of weakly relativistic protons by an electromagnetic wave propagating across an external magnetic field in plasma at the heliosphere periphery is considered. The problem is reduced to analysis of a second-order time-dependent nonlinear equation for the wave phase on the particle trajectory. The conditions of proton trapping by the wave, the dynamics of the components of the particle momentum and velocity, the structure of the phase plane, the particle trajectories, and the dependence of the acceleration rate on initial parameters of the problem are analyzed. The asymptotic behavior of the characteristics of accelerated particles for the heliosphere parameters is investigated. The optimum conditions for surfatron acceleration of protons by an electromagnetic wave are discussed. It is demonstrated that the experimentally observed deviation of the spectra of cosmic-ray protons from standard power-law dependences can be caused by the surfatron mechanism. It is shown that protons with initial energies of several GeV can be additionally accelerated in the heliosphere (the region located between the shock front of the solar wind and the heliopause at distances of about 100 astronomical units (a.u.) from the Sun) up to energies on the order of several thousands of GeV. In order to explain the proton spectra in the energy range of ∼20–500 GeV, a two-component phenomenological model is proposed. The first component corresponds to the constant (in this energy range) galactic contribution, while the second (variable) component corresponds to the heliospheric contribution, which appears due to the additional acceleration of soft cosmic-ray protons at the heliosphere periphery. Variations in the proton spectra measured on different time scales between 1992 and 2008 in the energy range from several tens to several hundred GeV, as well as the dependence of these spectra on the heliospheric weather, can be explained by surfatron acceleration of protons in the heliosphere.

Exactly solvable model of resonance tunneling of an electromagnetic wave in plasma containing short-scale inhomogeneities

Different variants of resonance tunneling of a transverse electromagnetic wave through a plasma layer containing short-scale (subwavelength) inhomogeneities, including evanescence regions to which approximate methods are inapplicable, are analyzed in the framework of an exactly solvable one-dimensional model. Complex plasma density profiles described by a number of free parameters determining the permittivity modulation depth, the characteristic scale lengths of plasma structures, their number, and the thickness of the inhomogeneous plasma layer are considered. It is demonstrated that reflection-free propagation of the wave incident on the layer from vacuum (the effect of wave-barrier transillumination) can be achieved for various sets of such structures, including plasma density profiles containing a stochastic component. Taking into account cubic nonlinearity, it is also possible to obtain an exact solution to the one-dimensional problem on the nonlinear transillumination of nonuniform plasma. In this case, the thicknesses of the evanescence regions decrease appreciably. The problem of resonance tunneling of electromagnetic waves through such barriers is of interest for a number of practical applications.

Effect of the longitudinal momentum of electrons on their surfatron acceleration by an electromagnetic wave in space plasma

By numerically calculating the second-order nonlinear time-dependent equation for the wave phase on a particle trajectory, the effect of the longitudinal (with respect to the external magnetic field) momentum of electrons on the dynamics of their surfatron acceleration by an electromagnetic wave propagating across the external magnetic field in space plasma is analyzed. It is shown that, for strongly relativistic initial values of the longitudinal component of the electron momentum (the other parameters of the problem being fixed), the electrons are trapped into the ultrarelativistic regime of surfatron acceleration within a definite interval of the initial wave phase Ψ(0) on the particle trajectory. It was assumed in the calculations that Ψ(0) ≤ π. For the initial wave phases lying within the interval of 0 < Ψ(0) ≤ π, the electrons are immediately trapped by the wave, whereas at π ≤ Ψ(0) ≤ 0, no electron trapping is observed even at long computation times. This result substantially simplifies estimates of the wave damping caused by particle acceleration. The dynamics of the velocity components, momentum, and relativistic factor of electrons in the course of their ultrarelativistic acceleration are considered. The obtained results present interest for the development of modern concepts of the mechanisms for the generation of ultrarelativistic particles in space plasma, correct interpretation of experimental data on the flows of such particles, explanation of possible reasons for the deviation of the fast particle spectra observed in the heliosphere from the standard power-law scaling, and analysis of the relation between such deviations and the space weather.

On the origin of a kink in the hardness spectra of cosmic-ray protons and helium nuclei in the vicinity of 230 GV

A three-component phenomenological model for the description of specific features of spectra of cosmic-ray protons and helium nuclei in the hardness range from 30 to 2 × 105 GV is proposed. The first component corresponds to the constant background; the second component, to a variable “soft” (30–500 GV) heliospheric source; and the third component, to a variable “hard” (0.5–200 TV) galactic source inside a local bubble. The corresponding “surfatron accelerators” are responsible for the existence and variability of both sources. In order for such accelerators to operate, there should be an extended area with a nearly uniform and constant (in both the magnitude and direction) magnetic field and electromagnetic waves propagating perpendicular (or obliquely) to it. The dimensions of each source determine the maximum energy to which cosmic rays can be accelerated. The soft source with a size of ~100 au lies at the periphery of the heliosphere, beyond the terminal shock, while the hard source with a size of >0.1 pc is located near the boundary of a local interstellar cloud at a distance of ~0.01 pc from the Sun. A kink in the hardness spectra of p and He (near the hardness of about 230 GV) is caused by the variability of physical conditions in the acceleration region and depends on the relation between the amplitudes and power-law indices of the background, the soft heliospheric source, and the nearby hard galactic source. Ultrarelativistic acceleration of p and He in space plasma by an electromagnetic wave propagating perpendicular to the external magnetic field is investigated using numerical calculations. The conditions for particle trapping by the wave, as well as the dynamics of the velocity and momentum components, are analyzed. The calculations show that, in contrast to electrons and positrons (e+), a trapped proton can escape from the effective potential well after a relatively short time, thereby terminating to accelerate. Such an effect gives rise to softer spectra of p and He sources as compared to those of e+. The possibility of deviation of the spectra of accelerated protons from standard power-law dependences due to the surfatron mechanism is discussed.

Surfatron accelerator in the local interstellar cloud

Taking into account results of numerous experiments, the variability of the energy spectra of cosmic rays (protons and helium nuclei) in the energy range of 10 GeV to ~107 GeV is explained on the basis of a hypothesis of the existence of two variable sources close to the Sun. The first (soft) surfatron source (with a size of ~100 AU) is located at the periphery of the heliosphere. The second (hard) surfatron source (with a size of ~1 pc) is situated in the Local Interstellar Cloud (LIC) at a distance of <1 pc. The constant background is described by a power-law spectrum with a slope of ~2.75. The variable heliospheric surfatron source is described by a power-law spectrum with a variable amplitude, slope, and cutoff energy, the maximum cutoff energy being in the range of EСН/Z < 1000 GeV. The variable surfatron source in the LIC is described by a power-law spectrum with a variable amplitude, slope, and cut-off energy, the maximum cut-off energy being EСL/Z ≤ 3 × 106 GeV. The proposed model is used to approximate data from several experiments performed at close times. The energy of each cosmic-ray component is calculated. The possibility of surfatron acceleration of Fe nuclei (Z = 26) in the LIC up to an energy of ECL ~ 1017 eV and electron and positrons to the “knee” in the energy spectrum is predicted. By numerically solving a system of nonlinear equations describing the interaction between an electromagnetic wave and a charged particle with an energy of up to E/Z ~ 3 × 106 GeV, the possibility of trapping, confinement, and acceleration of charged cosmic-ray particles by a quasi-longitudinal plasma wave is demonstrated.

On the reason for the kink in the rigidity spectra of cosmic-ray protons and helium nuclei near 230 GV

A three-component phenomenological model describing the specific features of the spectrum of cosmic-ray protons and helium nuclei in the rigidity range of 30–2×105 GV is proposed. The first component corresponds to the constant background; the second, to the variable “soft” (30–500 GV) heliospheric source; and the third, to the variable “hard” (0.5–200 TV) source located inside a local bubble. The existence and variability of both sources are provided by the corresponding “surfatron accelerators,” whose operation requires the presence of an extended region with an almost uniform (in both magnitude and direction) magnetic field, orthogonally (or obliquely) to which electromagnetic waves propagate. The maximum energy to which cosmic rays can be accelerated is determined by the source size. The soft source with a size of ∼100 AU is located at the periphery of the heliosphere, behind the front of the solar wind shock wave. The hard source with a size of >0.1 pc is located near the boundary of an interstellar cloud at a distance of ∼0.01 pc from the Sun. The presence of a kink in the rigidity spectra of p and He near 230 GV is related to the variability of the physical conditions in the acceleration region and depends on the relation between the amplitudes and power-law exponents in the dependences of the background, soft heliospheric source, and hard near galactic source. The ultrarelativistic acceleration of p and He by an electromagnetic wave propagating in space plasma across the external magnetic field is numerically analyzed. Conditions for particle trapping by the wave and the dynamics of the particle velocity and momentum components are considered. The calculations show that, in contrast to electrons and positrons (e+), the trapped protons relatively rapidly escape from the effective potential well and cease to accelerate. Due to this effect, the p and He spectra are softer than that of e+. The possibility that the spectra of accelerated protons deviate from standard power-law dependences due to the surfatron mechanism is discussed.

Generation of fast charged particles in plasma by a wave with a stochastically jumping phase

The interaction between charged plasma particles and an electromagnetic wave with a stochastic jumping phase is analyzed by numerical simulations. It is demonstrated that, in the course of interaction, the particle energy can increase by more than one order of magnitude. Optimal conditions for efficient interaction of charged plasma particles with a wave having a stochastically jumping phase are determined. According to the simulation results, substantial acceleration of charged plasma particles by a wave with a stochastically jumping phase takes place both at fixed time intervals between phase jumps and when these intervals are random. The influence of the wave parameters, such as the wave amplitude, the characteristic time interval between phase jumps, and the characteristic magnitude of these jumps, on the acceleration dynamics is analyzed.

Waveguide propagation of intense electromagnetic radiation in slightly inhomogeneous nonlinear media

The propagation of self-localizing beams of electromagnetic waves in the form of nonlinear waveguides in a slightly inhomogeneous medium is studied analytically and numerically. The trajectories of the axial ray are studied as a function of its direction and the field strength at the initial point on the basis of a nonlinear scalar Helmholtz equation. Analytic expressions are derived. The longitudinal refractive index, the field intensity, and the waveguide radius are plotted as functions of the instantaneous position of the point on the axial ray. Deep penetration of the beam into the opaque region and the position of the screening surface are studied as functions of the parameters of the beam and the medium. A steady-state 3D problem is analyzed for a power-law nonlinearity with an arbitrary power. A 2D problem is analyzed for the case of a ponderomotive nonlinearity with saturation.

On the cause of variability of the cosmic ray spectrum in the knee region

Cosmic ray (CR) energy spectra for H, He, Si, and Fe nuclei with energy-to-charge number ratios ℰ/Z in the range from 10 to 5 × 107 GeV are studied using observational data obtained at different times in different energy ranges: AMS-02, CREAM, Tibet ASγ, Tibet (hybrid), GRAPES-3, KASCADE, and KASCADE-Grande. Comparison of the H and He CR fluxes according to the KASCADE and KASCADE-Grande data (for different models of deconvolving CR spectra) with the Tibet ASγ and Tibet (hybrid) data obtained at another time in the range of ℰ/Z ∼ 3 × 106 GeV demonstrates space weather-caused variability of the CR flux. This feature of CR energy spectra in the Tibet ASγ data is most clearly observed in the spectra of heavier nuclei (Si and Fe) according to the KASCADE-Grande and GRAPES-3 data. The variability in the energy spectra of all CRs in the vicinity of the “knee” is shown in the data of Yakutsk EAS, CASA-BLANCA, and Tibet-III experiments. The variability of the CR flux on a time scale on the order of several years exists only if the source corresponding to the peak in the energy spectrum is situated at a distance of no more than 1 pc from the Sun. Rapid surfatron acceleration of CRs may result from colliding interstellar clouds nearest to the Sun (LIC and G). This acceleration mechanism allows one to explain the variability of the CR spectrum in the range 103 GeV < ℰ/Z < 108 GeV. Conditions for the trapping of strongly relativistic Fe nuclei by an electromagnetic wave, the dynamics of the components of the particle velocity and momentum, and the dependence of the particle acceleration rate on the initial parameters of the problem are analyzed using numerical calculations. The structure of the phase plane of the accelerated Fe nuclei is examined. Optimal conditions for the implementation of ultrarelativistic surfatron acceleration of Fe nuclei by an electromagnetic wave are formulated.

Analysis of the dependence of surfatron acceleration of electrons by an electromagnetic wave in space plasma on the particle momentum along the wave front

Based on the numerical solution of the nonlinear nonstationary second-order equation for the wave phase on the particle trajectory, the dynamics of surfatron acceleration of electrons by an electromagnetic wave propagating across the external magnetic field in space plasma is analyzed as a function of the electron momentum along the wave front. Numerical calculations show that, for strongly relativistic initial values of the electron momentum component along the wave front gy(0) (the other parameters of the problem being the same), electrons are trapped into the regime of ultrarelativistic surfatron acceleration within a certain interval of the initial wave phase Ψ(0) on the particle trajectory. It is assumed in the calculations that |Ψ(0)| ≤ π. For strongly relativistic values of gy(0), electrons are immediately trapped by the wave for 19% of the initial values of the phase Ψ(0) (favorable phases). For the rest of the values of Ψ(0), trapping does not occur even at long times. This circumstance substantially simplifies estimations of the wave damping due to particle acceleration in subsequent calculations. The dynamics of the relativistic factor and the components of the electron velocity and momentum under surfatron acceleration is also analyzed. The obtained results are of interest for the development of modern concepts of possible mechanisms of generation of ultrarelativistic particle fluxes in relatively calm space plasma, as well as for correct interpretation of observational data on the fluxes of such particles and explanation of possible reasons for the deviation of ultrarelativistic particle spectra detected in the heliosphere from the standard power-law scalings and the relation of these variations to space weather and large-scale atmospheric processes similar to tropical cyclones.