A Study of the Coefficient a in the Extended Nonlinear Guiding Center Theory

The diffusion mechanism of charged particles is one of the very im- portant problems in astrophysics. Recently, Matthaeus et al. (2003) developed a nonlinear guiding center theory (NLGC) of the perpendicular diffusion of en- ergetic charged particles with the parallel diffusion coefficient as an input. This theory agrees with the results of numerical simulations very well. In addition, Qin (2007) developed a nonlinear parallel diffusion theory (NLPA) following the idea of NLGC. Combining the NLGC with the NLPA, a new theory, NLGC-E was developed to solve the parallel and perpendicular diffusion coefficients simultaneously. In the NLGC theory, there is a coefficient a² which is selected to be 1/3 for the best agreement with the results of numerical simulations. In this work we tried the different values of a² for the NLGC-E theory to determine the optimal value which ?ts best the results of numerical simulations.     

Analytic forms of the perpendicular cosmic ray diffusion coefficient for an arbitrary turbulence spectrum and applications on transport of Galactic protons and acceleration at interplanetary shocks

We investigate cosmic ray scattering in the direction perpendicular to a mean magnetic field. Unlike in previous articles we employ a general form of the turbulence wave spectrum with arbitrary behavior in the energy range. By employing an improved version of the nonlinear guiding center theory we compute analytically the perpendicular mean free path. As shown, the energy range spectral index, has a strong influence on the perpendicular diffusion coefficient. If this parameter is larger than one we find for some cases a perpendicular diffusion coefficient that is independent of the parallel mean free path and particle energy. Two applications are considered, namely transport of Galactic protons in the solar system and diffusive particle acceleration at highly perpendicular interplanetary shock waves.     

Time-dependent transport of energetic particles in magnetic turbulence: computer simulations versus analytical theory

We explore numerically the transport of energetic particles in a turbulent magnetic field configuration. A test-particle code is employed to compute running diffusion coefficients as well as particle distribution functions in the different directions of space. Our numerical findings are compared with models commonly used in diffusion theory such as Gaussian distribution functions and solutions of the cosmic ray Fokker–Planck equation. Furthermore, we compare the running diffusion coefficients across the mean magnetic field with solutions obtained from the time-dependent version of the unified non-linear transport theory. In most cases we find that particle distribution functions are indeed of Gaussian form as long as a two-component turbulence model is employed. For turbulence setups with reduced dimensionality, however, the Gaussian distribution can no longer be obtained. It is also shown that the unified non-linear transport theory agrees with simulated perpendicular diffusion coefficients as long as the pure two-dimensional model is excluded.     

Quasi-linear perpendicular diffusion coefficients of charged cosmic rays calculated directly from the Newton–Lorentz equation

It is the subject of this article to determine diffusion coefficients of charged test particles perpendicular to a mean magnetic field analytically by starting directly from the Newton–Lorentz equation. We employ the so-called standard quasi-linear theory to test the well-known field line random walk limit. It is demonstrated in this article that there is an additional gyroresonant contribution which describes the scattering of the particle away from a single field line. For certain situations, this additional contribution is small and can be neglected. For other cases, we obtain new results such as a rigidity-dependent quasi-linear perpendicular mean free path. The results of this article are relevant for particles in the solar wind which move scatter free or nearly scatter free in the direction parallel to the magnetic field of the Sun.     

Space Flows and Disturbances Due to Bodies in Motion Through the Magnetoplasma

In this paper a method is concerned which makes it possible to describe numerically and analytically the most famous structures in the non-equilibrium ionosphere, such as stratified and yacht sail like structures, flute jets, wakes and clouds. These problems are of practical interest in space sciences, astrophysics and in turbulence theory, and also of fundamental interest since they enable one to concentrate on the effects of the ambient electric and magnetic fields. Disturbances of charged particle flows due to the ambient flow interactions with bodies are simulated with taking into account the ambient magnetic field effect. The effects of interactions between solid surfaces and the flows was simulated by making use of an original image method. The flow disturbances were described by the Boltzmann equation. In the case of the ambient homogeneous magnetic field the Boltzmann equation is solved analytically. The case of diffuse reflection of particles by surface is considered in detail. The disturbances of charged particle concentration are calculated in 3D space. The contours of constant particle concentration obtained from numerical simulations illustrate the dynamics of developing stratifications and flute structures in charged particle jets and feffect. The wakes under the ambient magnetic field effect. The basic goal of this paper is to present the method and to demonstate its possibility for simulations of turbulence, plasma jets, wakes and clouds inthe ionosphere and Space when effects of electric and magnetic fields are taken into account. This revised version was published online in July 2006 with corrections to the Cover Date.     

Saturated magnetic field amplification at supernova shocks

Cosmic ray streaming instabilities at supernova shocks are discussed in the quasi-linear diffusion formalism which takes into account the feedback effect of wave growth on the cosmic ray streaming motion. In particular, the non-resonant instability that leads to magnetic field amplification in the short wavelength regime is considered. The linear growth rate is calculated using kinetic theory for a streaming distribution. We show that the non-resonant instability is actually driven by a compensating current in the background plasma. The non-resonant instability can develop into a non-linear regime generating turbulence. The saturation of the amplified magnetic fields due to particle diffusion in the turbulence is derived analytically. It is shown that the evolution of parallel and perpendicular cosmic ray pressures is predominantly determined by non-resonant diffusion. However, the saturation is determined by resonant diffusion which tends to reduce the streaming motion through pitch angle scattering. The saturated level can exceed the mean background magnetic field.     

Energetic particle acceleration in a three-dimensional magnetic field reconnection model: the role of magnetohydrodynamic turbulence

The role of magnetohydrodynamic (MHD) turbulence in the cosmic ray acceleration process in a volume with a reconnecting magnetic field is studied by means of Monte Carlo simulations. We performed modelling of proton acceleration, with the three-dimensional analytic model of stationary reconnection of Craig et al. providing the unperturbed background conditions. Perturbations of particle trajectories resulting from a turbulent magnetic field component were simulated using small-amplitude pitch-angle momentum scattering, enabling modelling of both small- and large-amplitude turbulence in a wide wavevector range. Within the approach, no second-order Fermi acceleration process is allowed. Comparison of the acceleration process in models involving particle trajectory perturbations with the unperturbed model reveals that the turbulence can substantially increase the acceleration efficiency, enabling much higher final particle energies and flat particle spectra.     

Simulation studies of electron acceleration by ion ring distributions in solar flares

Using a 2 1/2-D fully relativistic electromagnetic particle-in-cell code (PIC) we have investigated a potential electron acceleration mechanism in solar flares. The free energy is provided by ions which have a ring velocity distribution about the magnetic field direction. Ion rings may be produced by perpendicular shocks, which could in turn be generated by the super-Alfvénic motion of magnetic flux tubes emerging from the photosphere or by coronal mass ejections (CMEs). Such ion distributions are known to be unstable to the generation of lower hybrid waves, which have phase velocities in excess of the electron thermal speed parallel to the field and can, therefore, resonantly accelerate electrons in that direction. The simulations show the transfer of perpendicular ion energy to energetic electrons via lower hybrid wave turbulence. With plausible ion ring velocities, the process can account for the observationally inferred fluxes and energies of non-thermal electrons during the impulsive phase of flares. Our results also show electrostatic wave generation close to the plasma frequency: we suggest that this is due to a bump-in-tail instability of the electron distribution.     

Spontaneous magnetic reconnection

The present review concerns the relevance of collisionless reconnection in the astrophysical context. Emphasis is put on recent developments in theory obtained from collisionless numerical simulations in two and three dimensions. It is stressed that magnetic reconnection is a universal process of particular importance under collisionless conditions, when both collisional and anomalous dissipation are irrelevant. While collisional (resistive) reconnection is a slow, diffusive process, collisionless reconnection is spontaneous. On any astrophysical time scale, it is explosive. It sets on when electric current widths become comparable to the leptonic inertial length in the so-called lepton (electron/positron) “diffusion region”, where leptons de-magnetise. Here, the magnetic field contacts its oppositely directed partner and annihilates. Spontaneous reconnection breaks the original magnetic symmetry, violently releases the stored free energy of the electric current, and causes plasma heating and particle acceleration. Ultimately, the released energy is provided by mechanical motion of either the two colliding magnetised plasmas that generate the current sheet or the internal turbulence cascading down to lepton-scale current filaments. Spontaneous reconnection in such extended current sheets that separate two colliding plasmas results in the generation of many reconnection sites (tearing modes) distributed over the current surface, each consisting of lepton exhausts and jets which are separated by plasmoids. Volume-filling factors of reconnection sites are estimated to be as large as \({<}10^{-5}\) per current sheet. Lepton currents inside exhausts may be strong enough to excite Buneman and, for large thermal pressure anisotropy, also Weibel instabilities. They bifurcate and break off into many small-scale current filaments and magnetic flux ropes exhibiting turbulent magnetic power spectra of very flat power-law shape \(W_b\propto k^{-\alpha }\) in wavenumber k with power becoming as low as \(\alpha \approx 2\). Spontaneous reconnection generates small-scale turbulence. Imposed external turbulence tends to temporarily increase the reconnection rate. Reconnecting ultra-relativistic current sheets decay into large numbers of magnetic flux ropes composed of chains of plasmoids and lepton exhausts. They form highly structured current surfaces, “current carpets”. By including synchrotron radiation losses, one favours tearing-mode reconnection over the drift-kink deformation of the current sheet. Lepton acceleration occurs in the reconnection-electric field in multiple encounters with the exhausts and plasmoids. This is a Fermi-like process. It results in power-law tails on the lepton energy distribution. This effect becomes pronounced in ultra-relativistic reconnection where it yields extremely hard lepton power-law energy spectra approaching \(F(\gamma )\propto \gamma ^{-1}\), with \(\gamma \) the lepton energy. The synchrotron radiation limit becomes substantially exceeded. Relativistic reconnection is a probable generator of current and magnetic turbulence, and a mechanism that produces high-energy radiation. It is also identified as the ultimate dissipation mechanism of the mechanical energy in collisionless magnetohydrodynamic turbulent cascades via lepton-inertial-scale turbulent current filaments. In this case, the volume-filling factor is large. Magnetic turbulence causes strong plasma heating of the entire turbulent volume and violent acceleration via spontaneous lepton-scale reconnection. This may lead to high-energy particle populations filling the whole volume. In this case, it causes non-thermal radiation spectra that span the entire interval from radio waves to gamma rays.     

Acceleration of particles by black hole with gravitomagnetic charge immersed in magnetic field

The collision of test charged particles in the vicinity of an event horizon of a weakly magnetized non-rotating black hole with gravitomagnetic charge has been studied. The presence of the external magnetic field decreases the innermost stable circular orbits (ISCO) radii of charged particles. The opposite mechanism occurs when there is nonvanishing gravitomagnetic charge. For a collision of charged particle moving at ISCO and the neutral particle falling from infinity the maximal collision energy can be decreased by gravitomagnetic charge in the presence of external asymptotically uniform magnetic field.     

Diffusion and dispersion in magnetohydrodynamic turbulence: The influence of mean magnetic fields

The properties of magnetohydrodynamic (MHD) turbulence under the influence of a strong mean magnetic field are investigated from the Lagrangian viewpoint by tracking fluid particles in direct numerical simulations. The particle trajectories show characteristic bends near vortex sheets. A strong mean magnetic field leads to preferential diffusion parallel to the mean magnetic field. The two‐particle relative dispersion process shows a dependence on the orientation of the initial separation vector. The relative dispersion is slowed down for initial separation vectors aligned with the mean magnetic field. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variations of cosmic-ray energy in interplanetary space

A consistent theory of energy exchange between high-energy charged cosmic-ray particles and the random inhomogeneities of a magnetic field frozen in the moving solar wind plasma is developed. It is shown that the mode of the particle energy variations at a given law of plasma velocity variation in space is determined by the specific form of the particle distribution function. The equation for the density of cosmic-ray energy is obtained. Consideration is given to the generation of a charged particle energy spectrum in the course of multiple scatterings by the random inhomogeneities of the magnetic field.     

Propagation and absorption of electron-cyclotron maser radiation during solar flares

The electron-cyclotron maser is believed to be the source of microwave spike bursts often observed during solar and stellar flares. Partial absorption of this radiation as it propagates through the corona can produce plasma heating and soft X-ray emission over an extended region. In this paper, the propagation and absorption of the maser radiation during solar flares are examined through linear theory and electro-magnetic particle simulations. It is shown using linear theory that strong absorption of the radiation should occur as it propagates towards the second harmonic layer where the magnetic field is half as strong as in the emission region. Only radiation propagating nearly parallel to the magnetic field in a low-temperature plasma may be able to escape under certain, limited conditions. Finite temperature effects can cause radiation propagating nearly perpendicular to the magnetic field to refract, causing enhanced absorption. Particle simulations are then used to evaluate the nonlinear response of the plasma as the maser radiation propagates through the absorption layer. It is shown that some of the maser radiation is able to escape through a process of absorption below the second harmonic of the local gyrofrequency and re-emission above it. The fraction able to escape is much higher than that predicted by linear theory, although the amount of escaping energy is only a small fraction of the incident energy. The bulk of incident energy goes into the perpendicular heating of the ambient electrons, with the rate of energy absorption showing no signs of leveling off during the simulations. This indicates that the absorption layer does not become optically thin after continuous heating by the maser radiation. A few electrons are accelerated to several tens of keVs as a result of the heating.     

Suppression of the large‐scale Lorentz force by turbulence

The components of the total stress tensor (Reynolds stress plus Maxwell stress) are computed within the quasilinear approximation for a driven turbulence influenced by a large‐scale magnetic background field. The conducting fluid has an arbitrary magnetic Prandtl number and the turbulence without the background field is assumed as homogeneous and isotropic with a free Strouhal number St. The total large‐scale magnetic tension is always reduced by the turbulence with the possibility of a ‘catastrophic quenching’ for large magnetic Reynolds number Rm so that even its sign is reversed. The total magnetic pressure is enhanced by turbulence in the high‐conductivity limit but it is reduced in the low‐conductivity limit. Also in this case the sign of the total pressure may reverse but only for special turbulences with sufficiently large St > 1. The turbulence‐induced terms of the stress tensor are suppressed by strong magnetic fields. For the tension term this quenching grows with the square of the Hartmann number of the magnetic field. For microscopic (i.e. small) diffusivity values the magnetic tension term becomes thus highly quenched even for field amplitudes much smaller than their equipartition value. In the opposite case of large‐eddy simulations the magnetic quenching is only mild but then also the turbulence‐induced Maxwell tensor components for weak fields remain rather small (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effective potential energy for relativistic particles in the field of inclined rotating magnetized sphere

The dynamics of a charged relativistic particle in electromagnetic field of a rotating magnetized celestial body with the magnetic axis inclined to the axis of rotation is studied. The covariant Lagrangian function in the rotating reference frame is found. Effective potential energy is defined on the base of the first integral of motion. The structure of the equipotential surfaces for a relativistic charged particle is studied and depicted for different values of the dipole moment. It is shown that there are trapping regions for the particles of definite energies.     

Proton activity of the Sun in current solar cycle 24

We present a study of seven large solar proton events in the current solar cycle 24(from 2009 January up to the current date). They were recorded by the GOES spacecraft with the highest proton fluxes being over 200 pfu for energies 10 Me V. In situ particle measurements show that:(1) The profiles of the proton fluxes are highly dependent on the locations of their solar sources, namely flares or coronal mass ejections(CMEs), which confirms the "heliolongitude rules" associated with solar energetic particle fluxes;(2) The solar particle release(SPR) times fall in the decay phase of the flare emission, and are in accordance with the times when the CMEs travel to an average height of 7.9 solar radii; and(3) The time differences between the SPR and the flare peak are also dependent on the locations of the solar active regions. The results tend to support the scenario of proton acceleration by the CME-driven shock,even though there exists a possibility of particle acceleration at the flare site, with subsequent perpendicular diffusion of accelerated particles in the interplanetary magnetic field. We derive the integral time-of-maximum spectra of solar protons in two forms: a single power-law distribution and a power law roll-over with an exponential tail. It is found that the unique ground level enhancement that occurred in the event on 2012 May 17 displays the hardest spectrum and the largest roll-over energy which may explain why this event could extend to relativistic energies.     

From Small-Scale Dynamo to Isotropic MHD Turbulence

We consider the problem of incompressible, forced, nonhelical, homogeneous, isotropic MHD turbulence with no mean magnetic field. This problem is essentially different from the case with externally imposed uniform mean field. There is no scale-by-scale equipartition between magnetic and kinetic energies as would be the case for the Alfvén-wave turbulence. The isotropic MHD turbulence is the end state of the turbulent dynamo which generates folded fields with small-scale direction reversals. We propose that the statistics seen in numerical simulations of isotropic MHD turbulence could be explained as a superposition of these folded fields and Alfvén-like waves that propagate along the folds.     

Diffusion of cosmic rays in a multiphase interstellar medium swept-up by a supernova remnant blast wave

Supernova remnants (SNRs) are one of the most energetic astrophysical events and are thought to be the dominant source of Galactic cosmic rays (CRs). A recent report on observations from the Fermi satellite has shown a signature of pion decay in the gamma-ray spectra of SNRs. This provides strong evidence that high-energy protons are accelerated in SNRs. The actual gamma-ray emission from pion decay should depend on the diffusion of CRs in the interstellar medium. In order to quantitatively analyse the diffusion of high-energy CRs from acceleration sites, we have performed test particle numerical simulations of CR protons using a three-dimensional magnetohydrodynamics (MHD) simulation of an interstellar medium swept-up by a blast wave. We analyse the diffusion of CRs at a length scale of order a few pc in our simulated SNR, and find the diffusion of CRs is precisely described by a Bohm diffusion, which is required for efficient acceleration at least for particles with energies above 30 TeV for a realistic interstellar medium. Although we find the possibility of a superdiffusive process (travel distance ∝ t^0.75) in our simulations, its effect on CR diffusion at the length scale of the turbulence in the SNR is limited.