The different transport regimes of pitch-angle scattering of energetic particles

Recently an advanced nonlinear diffusion theory for particle transport across the mean magnetic field has been developed. The method used in the derivation of the latter theory is based on the cosmic ray Fokker-Planck equation. In the present article we use the same approach to describe pitch-angle scattering and parallel spatial diffusion nonlinearly. Furthermore, we derive the quasilinear transport theory, the weakly nonlinear theory as well as the Bohm limit as special cases from our more general approach.     

The Balbus–Hawley instability in weakly ionized discs

MHD in protostellar discs is modified by the Hall current when the ambipolar diffusion approximation breaks down. Here I examine the Balbus–Hawley (magnetorotational) instability of a weak, vertical magnetic field within a weakly ionized disc. Vertical stratification is neglected, and a linear analysis is undertaken for the case in which the wavevector of the perturbation is parallel to the magnetic field.
The growth rate depends on whether the initial magnetic field is parallel or antiparallel to the angular momentum of the disc. The parallel case is less (more) unstable than the antiparallel case if the Hall current is dominated by negative (positive) species. The less-unstable orientation is stable for χ ≲0.5, where χ is the ratio of a generalized neutral–ion collision frequency to the Keplerian frequency. The other orientation has a formal growth rate of the order of the Keplerian angular frequency even in the limit χ →0! In this limit the wavelength of the fastest-growing mode tends to infinity, so the minimum level of ionization for instability is determined by the requirement that a wavelength fit within a disc scaleheight. In the ambipolar diffusion case, this requires χ > v _A c _s; in the Hall case this imposes a potentially much weaker limit,         

An explicit scheme for multifluid magnetohydrodynamics

When modelling astrophysical fluid flows, it is often appropriate to discard the canonical magnetohydrodynamic approximation, thereby freeing the magnetic field to diffuse with respect to the bulk velocity field. As a consequence, however, the induction equation can become problematic to solve via standard explicit techniques. In particular, the Hall diffusion term admits fast-moving whistler waves which can impose a vanishing time-step limit.
Within an explicit differencing framework, a multifluid scheme for weakly ionized plasmas is presented which relies upon a new approach to integrating the induction equation efficiently. The first component of this approach is a relatively unknown method of accelerating the integration of parabolic systems by enforcing stability over large compound time-steps rather than over each of the constituent substeps. This method, Super Time-Stepping, proves to be very effective in applying a part of the Hall term up to a known critical value. The excess of the Hall term above this critical value is then included via a new scheme for pure Hall diffusion.     

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.     

Numerical investigation of non-resonant and resonant scattering of charged particles with a spatially varying magnetic field

By integrating many charged particle trajectories in a magnetic field model consisting of a series of equally spaced field discontinuities with equal angular displacements, constant ¦B¦ and successive displacements oppositely directed, a parallel diffusion coefficient K _∥ is obtained. The particle gyroradius was kept sufficiently small for the interaction to be non-resonant. The diffusion coefficient is found to be in good agreement with that predicted by the known reflection properties for charged particles of individual discontinuities. However an attempt to reproduce the diffusion coefficient using the results of a recent study by Klimas and Sandri of a non-local diffusion equation applying to the non-resonant case lead to too low a value of K _∥. The computational approach was also applied to the case where the particle motion was in resonance with the wavelength of the train of discontinuities and a lower limit to K _∥ obtained. This lower limit exceeded the quasi-linear approximation value for K _∥ under resonant scattering conditions.     

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.     

Directional diffusion coefficients of solar protons inside and outside the bow shock

The directional diffusion coefficients of low-energy (? 0.3 MeV) solar protons inside and outside the bow shock are examined during the solar flare event of 24 January 1969. The data are derived from simultaneous observations obtained by Explorer 33 inside the magnetosheath and by Explorer 35 in the interplanetary medium. Although the gross properties of the spin-averaged intensities on a diffusion-type plot appear to be the same in both media, the directional intensities show significant variations. It is shown that directional intensities of low-energy protons can be described reasonably well by anisotropic diffusion with an associated diffusion coefficient. Directional diffusion coefficients are found to differ by a factor as much as three among different directions in space, and from the spin-averaged diffusion coefficient. This suggests that anisotropic diffusion does indeed take place and that so called ‘isotropic’ diffusion coefficients derived in the past from spin-averaged intensities may actually be directional diffusion coefficients in cases where substantial anisotropies (> 50 per cent) exist. The typical postulated ratio of field aligned to cross-field diffusion coefficients is $\text{κ⊥}\text{κ∥}\text{< 0.1}$. The present data would indicate a ratio of ?0.3. This value of the anisotropy is to be taken only as an upper limit of the ratio because of the limitations introduced by the wide field of view of the detectors (~90°) and the lack of directional measurements over the entire sphere. Comparison between directional diffusion coefficients in the interplanetary medium and magnetosheath derived from identical directions in space implies changes in the parameters of the interplanetary magnetic field as it interacts with the bow shock.     

General Relativistic Machian Universe

The Machian Universe, is usually described with Newtonian Physics, We give an alternative General Relativistic picture for Mach’s Universe. As such, we show that, in the correct Machian limit, Schwarzschild’s metric is coherent with Robertson-Walker’s, on condition that there be a cosmological constant, or the Universe’s rotation—or both. It is now confirmed that the Universe is accelerating, so the former condition applies. The latter was also confirmed one more time with the recently discovered NASA space probes anomalies. From Kerr-Lense-Thirring solution, we find an inverse scale-factor dependent angular speed; we then, show that the cosmological “constant” may have Classically originated from a centrifugal acceleration field.     

Monte Carlo model of the highly anisotropic solar proton event of 20 April 1971

The anisotropy of the particle distribution and its variation with time at 1 AU early in a solar cosmic ray event can provide information on the pitch-angle scattering of the particles in the interplanetary medium. The proton event of 20 April 1971 is described in which the anisotropy of the 7.6–55 MeV energy channel remained large (? 100%) and field-aligned well into the decay phase of the event. A Monte Carlo technique, which gives the pitch-angle distribution, is employed to investigate two models put forward to explain this sustained anisotropy. It is shown that the observed event is consistent with one model in which the injection of particles at the Sun decayed with ane-folding time of 7 hr. In this model the parallel propagation is determined by small-angle scattering in a diverging field equivalent to a uniform diffusion coefficient of 2.1 × 10²² cm² s^?1 (the corresponding classical mean free path is 0.90 AU). A model with impulsive injection and in whichκ(r) increases strongly with distance from the Sun cannot satisfactorily explain the observations.     

Anisotropic diffusion of solar cosmic rays

A simple model is described for solar cosmic ray events which appears to be in reasonable accord with observations. The model is based partly on some earlier models, together with the assumption that the diffusion of particles is strongly anisotropic due to the presence of the interplanetary magnetic field. Some remarks concerning the limitations of the diffusion equation are included, and it is pointed out that the propagation of solar cosmic rays might be best described in terms of an analogy to electrical transmission lines rather than to the conduction of heat as is usually done.     

中尺度磁绳结构的行星际观测

由磁绳结构主导、平均尺度约二、三十个小时的行星际磁云是日冕物质抛射在行星际膨胀、传播的体现。最近，Moldwin等人报道在太阳风中还观测到一些尺度在几十分钟的小尺度磁绳结构，并认为太阳风中的磁绳结构在尺度分布上可能具有双峰特征，在全面检视了WIND卫星(1995年-2000年)和ACE卫星(1998年-2000年)的观测资料后，发现了在行星际太阳风中一些尺度为几个小时的中尺度磁绳结构，利用初步整理的其中28个中尺度磁绳结构事件，认为太阳风中的磁绳结构在尺度分布上可能是连续的，这对行星际太阳风中磁绳结构物理起源的研究可能提出重要的物理限制。     

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.     

Field-aligned suprathermal electron fluxes below 270 km in the auroral zone

Observations are reported of field aligned etectron fluxes in the energy range 50–500 eV at altitudes below 270 km from two rocket flights in the auroral zone. The regions of field aligned suprathermal electrons occurred in bursts of a few seconds duration, and in some instances the energy of the peak field aligned flux was in the range 100–500 eV. Theoretical calculations of the pitch angle distribution were made using the Monte Carlo technique for two model atmospheres having exospheric temperatures of 750 and 1500 K bracketing the expected auroral zone exospheric temperature. The calculations were made for the case of incident field aligned suprathermal fluxes with no local parallel electric field and also for the case of a local constant parallel electric field. Comparison of theoretical and experimental pitch angle distributions showed that in one case at 270 km a parallel electric field of 1–2 mV/m fitted the data whereas another burst at 210 km required a parallel electric field of about 10 mV/m to produce a field aligned distribution of 230 eV electrons as pronounced as was observed. Furthermore in this latter case the lack of strong field alignment at 500 eV pointed to localisation of the parallel electric field to an altitude range of 20–30 km about the rocket altitude.     

Magnetic Models of HD 115708 and HD 119419

The magnetic fields of the chemically peculiar stars HD 115708 and HD 119419 were modeled using observed curves of variation of the magnetic field with the phase of the rotational period. It turned out that the field of HD 115708 is described, in a first approximation, by a central dipole, while the field of HD 119419 is described by an off-center dipole. The main parameters of the magnetic fields of both stars and maps of the surface field-strength distribution were obtained. The dipole axis of the first star lies in the equatorial plane while that of the second is almost parallel to the axis of rotation.     

Seed magnetic Fields Generated by Primordial Supernova Explosions

The origin of the magnetic field in galaxies is an open question in astrophysics. Several mechanisms have been proposed related, in general, to the generation of small seed fields amplified by a dynamo mechanism. In general, these mechanisms have difficulty in satisfying both the requirements of a sufficiently high strength for the magnetic field and the necessary large coherent scales. We show that the formation of dense and turbulent shells of matter, in the multiple explosion scenario of Miranda &38; Opher for the formation of the large-scale structures of the Universe, can naturally act as a seed for the generation of a magnetic field. During the collapse and explosion of Population III objects, a temperature gradient not parallel to a density gradient can naturally be established, producing a seed magnetic field through the Biermann battery mechanism. We show that seed magnetic fields ∼ 10⁻¹²–10⁻¹⁴ G can be produced in this multiple explosion scenario on scales of the order of clusters of galaxies (with coherence length L  ∼ 1.8 Mpc) and up to ∼ 4.5 × 10⁻¹⁰ G on scales of galaxies ( L  ∼ 100 kpc).     

An upper limit to the density inhomogeneity in the universe

An upper limit to the amplitude of the overall density fluctuation has been found by means of the gravitational lensing effect of the density inhomogeneity on the luminosities of quasars with larger redshifts. The observed differences of luminosities of quasars located at different directions are partially given by the lensing effect, therefore, a useful upper limit to the inhomogeneity can be derived if the luminosity distribution of quasars is uniform enough. We obtain that, in the case of the density parameter of the Universe =1, the overall matter should be less clustered than the luminous matter by a factor of at least 3. This result may not favour the biased dark matter scenario for the formation of large-scale structure in the Universe.     

Kolmogorov dissipation scales in weakly ionized plasmas

In a weakly ionized plasma, the evolution of the magnetic field is described by a 'generalized Ohm's law' that includes the Hall effect and the ambipolar diffusion terms. These terms introduce additional spatial and time-scales which play a decisive role in the cascading and the dissipation mechanisms in magnetohydrodynamic turbulence. We determine the Kolmogorov dissipation scales for the viscous, the resistive and the ambipolar dissipation mechanisms. The plasma, depending on its properties and the energy injection rate, may preferentially select one of these dissipation scales, thus determining the shortest spatial scale of the supposedly self-similar spectral distribution of the magnetic field. The results are illustrated taking the partially ionized part of the solar atmosphere as an example. Thus, the shortest spatial scale of the supposedly self-similar spectral distribution of the solar magnetic field is determined by any of the four dissipation scales given by the viscosity, the Spitzer resistivity (electron–ion collisions), the resistivity due to electron–neutral collisions and the ambipolar diffusivity. It is found that the ambipolar diffusion dominates for reasonably large energy injection rate. The robustness of the magnetic helicity in the partially ionized solar atmosphere would facilitate the formation of self-organized vortical structures.     

Strong cosmic-ray scattering in an anisotropic random magnetic field

We calculate the kinetic coefficients and the transport mean free paths of high-energy particles parallel to the regular magnetic field in the approximation of a large-scale anisotropic random magnetic field by using a nonlinear collision integral, i.e., by taking into account the processes of strong random scattering. We consider the diffusion of solar and Galactic cosmic rays by two-dimensional turbulence. Strong random scattering by two-dimensional turbulence is shown to reduce the parallel transport mean free path several fold. The momentum dependence of the parallel mean free path does not change, Λ_∥ ∝ p^2?v. In the case of strong random scattering by turbulence formed by several modes, the parallel transport mean free path is Λ_∥ ∝ p. We show that two-dimensional turbulence can make a major contribution to the parallel transport mean free paths of cosmic rays in the heliosphere and the interstellar medium.     

Probability distribution functions for the Sun's magnetic field

Magnetoconvection structures the Sun's magnetic field cover a vast range of scales, down to the magnetic diffusion scale that is orders of magnitude smaller than the resolution of current telescopes. The statistical properties of this structuring are governed by probability density functions (PDFs) for the flux densities and by the angular distribution functions for the orientations of the field vectors. The magnetic structuring on sub‐pixel scales greatly affects the field properties averaged over the resolution element due to the non‐linear relation between polarization and magnetic field. Here we use a Hinode SOT/SP data set for the quiet Sun disk center to explore the complex behavior of the 6301–6302 Å Stokes line profile system and identify the observables that allow us to determine the distribution functions in the most robust and least model dependent way. The angular distribution is found to be strongly peaked around the vertical direction for large flux densities but widens with decreasing flux density to become isotropic in the limit of zero flux density. The noise‐corrected PDFs for the vertical, horizontal, and total flux densities all have a narrowly peaked maximum at zero flux density that can be fitted with a stretched exponential, while the extended wings decline quadratically (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)