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.     

On the effects of dynamical turbulence on the perpendicular diffusion of low-energy cosmic ray electrons

Diffusion perpendicular to the heliospheric magnetic field plays an integral role in the transport of charged particles in the heliosphere. In this study the perpendicular diffusion coefficient of low-energy cosmic ray electrons is calculated, using an equation derived from the random ballistic decorrelation interpretation of nonlinear guiding centre theory. An observationally motivated 2D turbulence power spectrum is assumed and the effects of various turbulence inputs on the resulting perpendicular diffusion coefficient are investigated. The perpendicular diffusion coefficients are first determined at 1 AU, for both magnetostatic and dynamical turbulence conditions. These solutions are also evaluated for radial distances of 0.1 AU to 10 AU to further investigate the values of the perpendicular diffusion coefficients in the very inner heliosphere. The results of this study show that the dissipation range of the turbulence power spectrum provides a negligible contribution towards the perpendicular diffusion coefficient, and that solutions derived using only the energy containing range serve as good approximations for solutions derived assuming the full 2D turbulence power spectrum. Finally, it is shown that the effects of dynamical turbulence, as considered in the present study, do not affect the perpendicular diffusion coefficients derived from the scattering theory considered here.     

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.     

Velocity correlation functions of charged particles derived from the Fokker–Planck equation

An important ingredient in theories for diffusion of charged particles across a mean magnetic field are velocity correlation functions along and across that field. In the current article we present an analytical study of these functions by investigating the two-dimensional Fokker–Planck equation. We show that for an isotropic pitch-angle Fokker–Planck coefficient, the parallel velocity correlation function is an exponential function in agreement with the standard model used previously. For other forms of the pitch-angle diffusion coefficient, however, we find non-exponential forms. Also a new, velocity correlation function based, approach for deriving the so-called Earl-relation is presented. This new derivation is more systematic and simpler than previous derivations. We also discuss higher-order velocity correlations and the applicability of the quasi-normal hypothesis in particle diffusion theory. Furthermore, we compute velocity correlation functions across the mean field and develop an alternative theory for perpendicular diffusion.     

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.     

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.     

Numerical integration of stochastic differential equations: A parallel cosmic ray modulation implementation on Africa’s fastest computer

Three-dimensional studies of the transport and modulation of cosmic ray particles in turbulent astrospheres require large-scale simulations using specialized scientific codes. Essentially, a multi-dimensional Fokker-Planck type equation (a parabolic diffusion equation) must be integrated numerically. One such approach is to convert the relevant transport equation into a set of stochastic differential equations (SDEs), with the latter much easier to handle numerically. Due to the growing demand for high performance computing resources, research into the application of effective and suitable numerical algorithms to solve such equations is needed. We present a case study of the performance of a custom-written FORTRAN SDE numerical solver on the CHPC (Centre for High Performance Computing) Lengau cluster in South Africa for a realistic test problem with different set-ups. It is shown that SDE codes can scale very well on large parallel computing platforms. Finally, we consider an extremely computationally expensive application of the SDE approach to cosmic ray modulation, studying the behaviour of galactic cosmic ray proton latitude gradients and relative amplitudes in a physics-first manner. This is done using a modulation code that employs diffusion coefficients derived from first principles, which in turn are functions of turbulence quantities in reasonable agreement with spacecraft observations and modelled using a two-component turbulence transport model (TTM). We show that this approach leads to reduced latitude gradients qualitatively in line with spacecraft observations of the same, without making ad hoc assumptions as to anisotropic perpendicular diffusion coefficients as are often made in many cosmic ray modulation studies.     

行星际结构与垂直无碰撞激波的相互作用

应用一维混合模拟方法数值研究了两种行星际结构──反向磁场和高密度等离子团与垂直无碰撞激波的相互作用．结果表明，随着激波上游区磁场的反向，下游区磁场将逐渐改变符号，且等离子体密度和速度分别呈现较强的湍动．激波上游和下游的物理量依然满足Rankine-Hugonoit关系．当高密度的等离子体团通过垂直无碰撞激波时，部分质子被激波反射，部分质子被加速并进入下游区域．由于质子速度分布为非Maxwell分布，在激波下游也激发出较强的湍动．     

Peculiarities of galactic cosmic ray Forbush effects during October–November 2003

Features of two successive Forbush effects of the galactic cosmic ray intensity in October–November 2003 have been studied based on the neutron monitors data. The rigidity spectrum of the galactic cosmic ray intensity in the course of the first Forbush effect (22–27 October) is gradually hardening, while the rigidity spectrum of the second Forbush effect (28 October–10 November) from the starting moment is very hard. As far, the energy range of the turbulence of the interplanetary magnetic field is in general responsible for the diffusion of galactic cosmic ray particles of the energy 5–50 GeV (to which neutron monitors are sensitive), we postulate that the gradually hardening (from day to day) of the rigidity spectrum of the first Forbush effect is associated with the enhancement of the power spectral density in the energy range of the interplanetary magnetic field turbulence caused by the large scale irregularities generated due to the interaction of the extending high speed disturbances with the background solar wind. The very hard rigidity spectrum (from the starting moment) of the second Forbush effect is generally associated with the well established new structure of the energy range of the interplanetary magnetic field turbulence enriched by the already created large scale irregularities. The gradually softening of the rigidity spectrum during the recovery phase of the second Forbush effect confirms that the disturbed interplanetary magnetic field turbulence step by step returns to the initial state.     

倒V电子沉降通量能谱的理论计算

沿极光区磁力线大约在2000公里到8000公里的高度范围内,存在着一个等离子体湍流和大尺度平行电场的加速区。沿磁力线运动的等离子体片中的电子通过此加速区时,受到等离子体湍流和平行电场的共同作用,形成电子沉降的倒V结构。从一维准线性的动力学方程出发,导出了沉降电子通量的能谱方程,得出了电子通量能谱的理论公式。对等离子体湍流和平行电场对沉降电子能谱的影响作了分析和讨论。本文所提出的理论可以解释目前观测到的某些基本现象。      

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.     

Plasma sheet coefficient of diffusion: Predictions and observations

The theory of a plasma sheet with medium scale developed turbulence predicts a value for the plasma sheet diffusion coefficient in the Z direction. Its value becomes very near to the diffusion coefficient calculated from the assumption of isotropic turbulence on the basis of ISEE-2 velocity fluctuations in the X and Y directions. INTERBALL/Tail CORALL bulk velocity measurements make it possible to determine velocity fluctuations in the Z direction and calculate the diffusion coefficient in this Z direction. It is shown that INTERBALL/Tail observations are in very good agreement with theory predictions.     

Direct and indirect estimates of turbulence around the turbopause

Direct and indirect estimates of turbulence parameters at altitudes 80–120 km are reviewed. It is shown that there are contradictions in data on absolute value, shape, seasonal variations etc. of eddy diffusion coefficient K_t revealed or accepted by different authors. The strongest contradiction is in the views on the seasonal variations of turbulence. Possible explanation may be connected with the correct account for mean motions. Data on turbopause height obtained from neutral composition data (Ar, N₂) are discussed. The revealed reverse connection of this height with the temperature at 120 km is discussed in terms of K_t dependence on temperature gradient.     

Solar flare particle injection spectra and spectra of flux maxima at the point of observation

Numerical models of impulsive solar flare particle events usually assume the radial diffusion coefficient to be independent of energy per nucleon, T, although the observations indicate a T^0.5 dependence (constant mean free path). The assumption of a constant diffusion coefficient results in a preservation of a power law injection spectrum at all radial distances throughout the event. We investigate the effect of an energy dependent diffusion coefficient on the spectrum of flux maxima at a fixed point in interplanetary space. This spectrum is harder than that of initial differential number densities close to the sun. Furthermore, the spectrum hardens with increasing radial distance which seems to be at variance with observations.     

Cosmic ray acceleration by supernova shocks

We analyse the results of recent measurements of nonthermal emission from individual supernova remnants (SNRs) and their correspondence to the nonlinear kinetic theory of cosmic ray (CR) acceleration in SNRs. It is shown that the theory fits these data in a satisfactory way and provides the strong evidences for the efficient CR production in SNRs accompanied by significant magnetic field amplification. Magnetic field amplification leads to considerable increase of CR maximum energy so that the spectrum of CRs accelerated in SNRs is consistent with the requirements for the formation of Galactic CR spectrum up to the energy ∼10¹⁷ eV.     

用GAO-YONG湍流模式对分离流动的数值模拟

采用GAO-YONG湍流模式对二维管道内三角形钝体绕流问题以及槽道内后台阶分离流动进行了数值模拟.求解程序基于开源数值计算平台OpenFOAM (Open Field Operation and Manipulation),数值模拟结果很好地预测了钝体绕流问题中漩涡脱落的尾流的流动趋势以及后台阶流动中分离再附的流动结构,同时分析了速度分布以及摩擦系数等参数并与实验值进行了对比,结果符合很好.这表明GAO-YONG湍流模式对大分离流动有较好的预测能力,对工程实践具有指导性作用.     

利用谱方法数值求解三维球坐标系下的Navier-Stokes方程

从三维球坐标系下的Navier-Stokes方程出发,采用谱配置方法建立了一个三维球坐标系下中层大气波动传播的全非线性动力学数值模式;构造了合理的纬向边界,并利用初始给定的重力波波包对该模式进行了数值检验．计算结果展示了重力波波包在大气中传播的三维图像,波包整体向上传播,波相关扰动速度随高度的增加而增长．     

Transport of ions in presence of induced electric field and electrostatic turbulence: Source of ions injected into ring current

The transport of ions from the polar ionosphere to the inner magnetosphere during stormtime conditions has been computed using a Monte Carlo diffusion code. The effect of the electrostatic turbulence assumed to be present during the substorm expansion phase was simulated by a process that accelerated the ions stochastically perpendicular to the magnetic field with a diffusion coefficient proportional to the energization rate of the ions by the induced electric field. This diffusion process was continued as the ions were convected from the plasma sheet boundary layer to the double-spiral injection boundary. Inward of the injection boundary, the ions were convected adiabatically. By using as input an O⁺ flux of 2.8 × 10⁸ cm^?2 s^?1 (w > 10 eV) and an H⁺ flux of 5.5 × 10⁸ cm^?2 s^?1 (w > .63 eV), the computed distribution functions of the ions in the ring current were found to be in good agreement, over a wide range in L (4 to 8), with measurements made with the ISEE-1 satellite during a storm. This O⁺ flux and a large part of the H⁺ flux are consistent with the DE satellite measurements of the polar ionospheric outflow during disturbed times.