Perpendicular transport of charged particles: Results for the unified nonlinear transport theory derived from the Newton–Lorentz equation

The perpendicular diffusion coefficient is calculated by combining a recently developed Unified Nonlinear Transport (UNLT) theory with the Newton–Lorentz equation. The total perpendicular mean free path can be described as a combination of a guiding center contribution and a microscopic contribution. It is shown that the total mean free path depends strongly on the energy range of the turbulence power spectrum and on particle energy. Further, a slab/2D composite model is used to investigate the influence of each contribution to the total mean free path for a quasi-3D turbulence model. For pure 2D turbulence the UNLT reduces to the NLGC-theory. For pure slab turbulence the guiding center contribution is subdiffusive in accordance with simulations and the theorem on reduced dimensionality. Conversely, the microscopic contribution is non-zero, which has to be interpreted as normal diffusion.     

Numerical investigation of the influence of large turbulence scales on the parallel and perpendicular transport of cosmic rays

In recent analytical investigations it has been demonstrated that the turbulence behavior at large scales has a very strong influence on the perpendicular diffusion coefficient of charged particles. In the present paper we use computer simulations to investigate numerically cross field transport and particle propagation along the mean magnetic field for different turbulence models at large scales. Our results are compared with quasilinear theory and nonlinear diffusion theories. We show that for different forms of the turbulence spectrum at large scales, the perpendicular mean free paths obtained numerically are in agreement with recent predictions made by analytical theory. It is also shown that the parallel diffusion coefficient contains always a strong nonlinear contribution which is, however, independent of the assumed spectrum at large scales.     

On the universality of asymptotic limits in the theory of field line diffusion and perpendicular transport of energetic particles

We discuss the random walk of magnetic field lines in astrophysical plasmas. Based on the standard theory of field line diffusion we show that there are two asymptotic limits. In these limits field line wandering is universal because in both regimes the field line diffusion coefficient depends only on fundamental length scales and absolute magnetic field strengths. As examples we discuss the field line diffusion coefficient for different prominent turbulence models namely the slab model, the two-dimensional model, and the Goldreich–Sridhar model. We show that the field line diffusion coefficient for the latter model agrees with the results obtained for slab and two-dimensional turbulence in limiting cases. We also discuss the transport of energetic particles perpendicular with respect to the mean magnetic field. Based on the unified nonlinear transport theory we consider again asymptotic limits. It is shown that one can identify four different regimes in which the transport is again universal. In all four cases perpendicular transport only depends on fundamental length scales of turbulence, magnetic field values, and the parallel diffusion coefficient.     

The transport of galactic and jovian cosmic ray electrons in the heliosphere

An overview is given on what we know about the cosmic ray diffusion process from the modelling of low-energy (MeV) electron transport in the heliosphere. For energies below ∼300 MeV, these electrons give a direct indication of the average mean free paths because they do not experience large adiabatic energy changes and their modulation is largely unaffected by global gradient and curvature drifts. Apart from galactic cosmic ray electrons, the jovian magnetosphere at ∼5 AU in the ecliptic plane is also a relatively strong source of MeV electrons, with energies up to ∼30 MeV. Therefore, when modelling the transport of these particles in the inner heliosphere, a three-dimensional treatment is essential. By comparing these models to observations from the Ulysses, Pioneer and Voyager missions, important conclusions can be made on e.g., the relative contributions of the galactic and jovian electrons to the total electron intensity, the magnitude of the parallel and perpendicular transport coefficients, and the time dependant treatment thereof.     

Effects of the position of the solar wind termination shock and the heliopause on the heliospheric modulation of cosmic rays

The effects of changing the position of the solar wind termination shock and the position of the heliopause, and therefore the extent of the heliosheath, on the modulation of cosmic ray protons are illustrated. An improved numerical model with diffusive termination shock acceleration, a heliosheath and drifts is used. The modulation is computed in the equatorial plane and at 35 heliolatitude using recently derived diffusion coefficients applicable to a number of cosmic ray species during both magnetic polarity cycles of the Sun. It was found that qualitatively the modulation results for the different heliopause positions are similar although they differ quantitatively, e.g., clearly different radial gradients are predicted for the regions beyond the termination shock compared to inside the shock. The difference between the modulation for the two solar polarity cycles are less significant at a heliolatitude of 35° than in the equatorial plane. We found that moving the termination shock from 90 to 100 AU, with the heliopause fixed at 120 AU, caused only quantitative differences so that the exact position of the TS in the outer heliosphere seems not crucially important to global modulation. Moving the heliopause outwards, to represent the modulation in the tail region of the heliosphere, causes overall decreases in the cosmic ray intensities but not linearly as a function of energy, e.g., at 1 GeV the effect is insignificant. We conclude from this modelling that the modulation of protons in the heliospheric nose and tail regions are qualitatively similar although, clear quantitative and interesting differences occur.     

基于ACE飞船观测的银河宇宙线与太阳风变化的统计研究

基于ACE飞船的资料,通过时序迭加方法统计分析了最近两个太阳活动极小年时期（2007.0-2009.0和2016.5-2019.0年）的宇宙线计数与太阳风参数的关系.结果表明,宇宙线的计数受太阳风共转流相互作用区的强烈影响,宇宙线计数变化与快慢太阳风流界面的位置密切相关,例如流界面的穿越通常伴随着宇宙线计数的下降.分析表明,第一时段的具有“雪犁”效应的宇宙线计数下降对应于流界面附近的扩散系数急剧下降,而第二时段的非“雪犁”效应的计数下降可能是由穿越流界面后的扩散系数增大引起的.日球层电流片也与宇宙线计数变化存在一定的相关性,宇宙线粒子在日球层电流片附近存在一定程度的堆积.太阳风对宇宙线的作用机制表明,宇宙线的漂移和扩散效应决定了其在1AU附近的分布变化.      

Modelling of galactic Carbon in an asymmetrical heliosphere: Effects of asymmetrical modulation conditions

Observations of galactic cosmic rays (GCRs) from the two Voyager spacecraft inside the heliosheath indicate significant differences between them, suggesting that in addition to a possible global asymmetry in the north–south dimensions (meridional plane) of the heliosphere, it is also possible that different modulation (turbulence) conditions could exist between the two hemispheres of the heliosphere. We focus on illustrating the effects on GCR Carbon of asymmetrical modulation conditions combined with a heliosheath thickness that has a significant dependence on heliolatitude. To reflect different modulation conditions between the two heliospheric hemispheres in our numerical model, the enhancement of both polar and radial perpendicular diffusion off the ecliptic plane is assumed to differ from heliographic pole to pole. The computed radial GCR intensities at polar angles of 55° (approximating the Voyager 1 direction) and 125° (approximating the Voyager 2 direction) are compared at different energies and for both particle drift cycles. This is done in the context of illustrating how different values of the enhancement of both polar and radial perpendicular diffusion between the two hemispheres contribute to causing differences in radial intensities during solar minimum and moderate maximum conditions. We find that in the A > 0 cycle these differences between 55° and 125° change both quantitatively and qualitatively for the assumed asymmetrical modulation condition as reflected by polar diffusion, while in the A < 0 cycle, minute quantitative differences are obtained. However, when both polar and radial perpendicular diffusion have significant latitude dependences, major differences in radial intensities between the two polar angles are obtained in both polarity cycles. Furthermore, significant differences in radial intensity gradients obtained in the heliosheath at lower energies may suggest that the solar wind turbulence at and beyond the solar wind termination shock must have a larger latitudinal dependence.     

Particle acceleration and transport at an oblique CME-driven shock

In gradual solar energetic particle (SEP) events, protons and heavy ions are often accelerated to >100 MeV/nucleon at a CME-driven shock. In this work, we study particle acceleration at an oblique shock by extending our earlier particle acceleration and transport in heliosphere (PATH) code to include shocks with arbitrary θ_BN, where θ_BN is the angle between the upstream magnetic field and the shock normal. Instantaneous particle spectra at the shock front are obtained by solving the transport equation using the total diffusion coefficient κ, which is a function of the parallel diffusion coefficient κ_∥ and the perpendicular diffusion coefficient κ_⊥. In computing κ_∥ and κ_⊥, we use analytic expressions derived previously. The particle maximum energy at the shock front as a function of time, the time intensity profiles and particle spectra at 1 AU for five θ_BN’s are calculated for an example shock.     

Random walk of magnetic field lines: Subdiffusive,diffusive, and superdiffusive regimes

One- and two-dimensional models of magnetic field fluctuations and turbulence are widely used in space-, astrophysical, and laboratory contexts. In the present article we use a generalized form of the turbulence wave spectrum to calculate field line diffusion coefficients analytically and numerically. General conditions are derived for which field line wandering behaves subdiffusively, diffusively, and superdiffusively.     

太阳风中动力论Alfven波的湍流谱(a)朗道衰减

提出一个太阳风中Alfven脉动湍流的新模式,动力论Alfven波是Alfven波和离子声波非解耦的新波模。由太阳向外传播的各种波长的动力Alfven波的非线性相互作用推导出动力论Alfven脉动湍流功率谱P_k,在Alfven半径以外,P_k∝k-3/2,而在Alfven半径以内,由太阳附近的P_k∝k-1变化成P_k∝k-3/2动力论Alfven脉动在Alfven半径以内完成朗道衰减。新模式克服了以前理论模式遇到的困难。      

Modulation of cosmic rays: Perpendicular diffusion and drifts in a heliosphere with a solar wind termination shock

In this study the roles of polar perpendicular diffusion and drifts are illustrated in a model containing a heliosheath and diffusive shock acceleration as applied to the solar wind termination shock. Of particular interest is the relation of polar perpendicular diffusion to particle drifts and how the effectiveness of the termination shock acceleration of galactic and anomalous protons is influenced by this relation. We found that drifts have a more prominent effect than the polar enhancement of perpendicular diffusion so that its omission from termination shock models would produce unrealistically large shock acceleration and consequently also larger modulation effects throughout the heliosphere. The computed spectra at a heliolatitude of 35° are almost similar for the two polarity epochs indicating that the two Voyager spacecraft might not observe differences between the two cycles in future.     

The role of radial perpendicular diffusion and latitude dependent acceleration along the solar wind termination shock

A numerical model, based on Parker’s transport equation, describing the modulation of anomalous cosmic rays and containing diffusive shock acceleration is applied. The role of radial perpendicular diffusion at the solar wind termination shock, and as the dominant diffusion coefficient in the outer heliosphere, is studied, in particular the role it plays in the effectiveness of the acceleration of anomalous protons and helium when its latitude dependence is changed. It is found that the latitudinal enhancement of radial perpendicular diffusion towards the heliospheric poles and along the termination shock has a prominent effect on the acceleration of these particles. It results in a ‘break’ in the energy spectrum for anomalous protons at ∼6.0 MeV, causing the spectral index to change from E^−1.38 to E^−2.23, but for anomalous helium at ∼3.0 MeV, changing the spectral index from E^−1.38 to E^−2.30. When approaching the simulated TS, the changes in the modulated spectra as they unfold to a ‘steady’ power law shape at energies below 50 MeV are much less prominent as a function of radial distances when radial perpendicular diffusion is increased with heliolatitude.     

Modelling of low-energy galactic electrons in the heliosheath

The modulation of cosmic ray electrons in the heliosphere plays an important role in improving our understanding and assessment of the processes applicable to low-energy galactic electrons. A full three-dimensional numerical model based on Parker’s transport equation is used to study the modulation of 10 MeV galactic electrons, in particular inside the heliosheath. The emphasis is placed on the role that perpendicular diffusion plays in causing the extraordinary large increase in the observed intensities of these electrons in the heliosheath. The modelling is compared with observations of 6–14 MeV electrons from the Voyager 1 mission. Results are shown for the radial intensity profiles of these electrons, as well as the modulation effects of varying the extent of the heliosheath by changing the location of the termination shock and the heliopause and the value of the local interstellar spectrum. We confirm that the heliosheath acts as a modulation ‘barrier’ for low-energy galactic electrons. The significance of this result depends on how wide the inner heliosheath is; on how high the very local interstellar spectrum is at these low energies (E < 100 MeV) and on how small perpendicular diffusion is inside the inner heliosheath.     

Probing heliospheric diffusion coefficients with solar energetic particles

The dynamics of solar particle events provide a direct link to the understanding of properties of wave–particle interactions, and to the nature of the solar wind fluctuations. Depending on their energy, the often simultaneously observed electrons, protons and ions interact with different wavenumber ranges of the fluctuations, and are sensitive to various aspects of the dynamical nature of the solar wind turbulence. In general, the evolution of particle events is also sensitive to the spatial variation of the transport parameters between the Sun and a few AU. Together with in situ plasma and magnetic field observations this information can be used to extrapolate the properties of transport parameters into the more distant Heliosphere. Recent developments in the theory of parallel transport of energetic particles, and examples for the modelling of solar particle events and the derivation of transport parameters are considered. A dynamical quasi-linear theory is presented which gives special emphasis to the geometry and dynamic nature of the fluctuations, and which is able to provide particle mean free paths solely from observed plasma parameters, in good agreement with those derived by the modelling. Possibilities to apply the above results to the study of other energetic particle processes in the Heliosphere are discussed.     

Energy dependence of the rigidity spectrum of Forbush decrease of galactic cosmic ray intensity

We show that rigidity spectrum of Forbush decrease (Fd) of galactic cosmic ray (GCR) intensity in September 9–23, 2005 clearly depends on energy. We calculated rigidity spectrum of the Fd based on the neutron monitors and Nagoya muon telescope channels’ data divided in three groups according to their cut off rigidities. We found that temporal changes of rigidity spectrum exponent γ are approximately similar for all cut off rigidity groups, but γ values are the larger the higher are cut off rigidities. We conclude that rigidity spectrum of Fd is hard for lower energy range and is soft for the higher energy range. We believe that an energy dependence of the power law rigidity spectrum of Fd is observed owing to the preferential convection–diffusion mechanism during Fd in September 9–23, 2005. It is a reflection of an influence of the temporal changes of the structure of the interplanetary magnetic field (IMF) turbulence in different range of frequency f during Fd. Particularly, a decisive role in formation of the character of the rigidity spectrum belongs to the changes of the exponent ν of the power spectral density (PSD) of the IMF turbulence (PSD ∝ f^−ν). The exponent ν is greater for high frequency region of the IMF turbulence (responsible for scattering of low rigidity particles of GCR), than for low frequency region of the IMF turbulence (being responsible for scattering of higher rigidity particles). Also, we challenge to estimate an existence of slab/2D structure of solar wind turbulence during the Fd in September 9–23, 2005 based on the distribution of average turbulence energy among the IMF’s components.     

Comparison of the effects of two models for perpendicular diffusion on cosmic-ray latitudinal gradients

We compare the effects of two different models for perpendicular diffusion on the latitudinal gradients of galactic cosmic ray protons during solar minimum conditions. These two models correspond to the newly developed non-linear guiding center theory [Matthaeus, W.H., Qin, G., Bieber, J.W., Zank, G.P. Nonlinear collisionless perpendicular diffusion of charged particles. Astrophys. J. Lett., 590 (1), L53–L56, 2003] and the theory based on a velocity correlation function approach [Bieber, J.W., Matthaeus, W.H. Perpendicular diffusion and drift at intermediate cosmic-ray energies. Astrophys. J., 485 (2) 655–659, 1997]. In this ab initio study a steady-state two-dimensional numerical modulation model is used which incorporates a state-of-the-art turbulence model. We show that the non-linear guiding center theory predicts a mean free path that has a rigidity dependence that better accounts for the latitudinal gradients measured by Ulysses during its first fast latitude scan in 1994/1995.     

Turbulence dissipation scale in the inner heliosphere

Small scale turbulence in the solar corona and the solar wind is considered. The estimates of dissipation scale in the inner heliosphere are obtained in the assumption that the initial source of turbulence is located near the chromosphere-corona transition layer. Theoretical results are compared with radiooccultation data.