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 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.     

垂直激波条件下能量粒子加速机制的模拟研究

在具有湍动的磁场和垂直激波条件下对大量测试粒子的轨迹进行了数值计算,研究了激波强度和粒子初始能量对于粒子穿越激波的平均能量变化的影响,分析了漂移加速(SDA)在不同条件下对粒子加速的贡献,并给出了一个与数值结果相符合的漂移加速理论公式△E=amvivup(1-1/s).结果表明,加入磁场湍流后,垂直激波条件下粒子仍主要受到漂移加速作用,而基于粒子引导中心的耗散漂移加速理论在此条件下失效.      

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.     

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.     

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.     

A heuristic derivation of an improved analytical theory for perpendicular diffusion of charged particles

A heuristic derivation of an improved nonlinear guiding center theory for perpendicular diffusion of charged particles is presented. This new derivation complements previous work which is based on an indirect solution of the Fokker–Planck equation. The new derivation confirms the improved theory for diffusion of charged particles and makes the validity of the theory plausible.     

Dust in magnetised plasmas: Basic theory and some applications

In this paper we review the theory of charged test particle motion in magnetic fields. This theory is then extended to charged $\text{dust}$ particles, for which gravity and charge fluctuations play an important role. It is shown that systematic drifts perpendicular to the magnetic field and stochastic transport effects may then have to be considered - none of which occur in the case of atomic particles (with the exception of charge exchange reactions). Some applications of charged dust particle transport theory to planetary rings are then briefly discussed.     

存在初始引导场情况下的无碰撞磁场重联

采用二维三分量的全粒子模拟方法研究了不同初始引导场情况下的无碰撞磁场重联及初态为一维的Harris电流片.结果表明,Bz0＞0.5B0的强引导场不仅会显著改变粒子的运动轨迹,而且会改变重联区附近的电场和流场结构,从而影响重联率和电子加速.运用广义欧姆定律解释了不同引导场下电场的结构特征.另外,通过对扩散区附近束流电子的跟踪研究发现,在二维模型中,不论引导场强弱,位于扩散区中心垂直模拟平面的感应电场对电子加速起主要作用,而扩散区外平面电场的贡献很小.      

The Compton–Getting effect of energetic particles with an anisotropic pitch-angle distribution

This paper presents a simulation of anisotropy measurements by the low-energy charged particle (LECP) experiment on Voyager 1 for cases when the particle pitch-angle distribution function in the solar wind plasma reference frame is not isotropic. The model includes both the Compton–Getting anisotropy and perpendicular diffusion anisotropy that possibly exists in the upstream region of the termination shock. The results show that the Voyager 1 data cannot rule out either the model with zero solar wind speed or the one with a finite speed on qualitative basis. The determination of solar wind speed using the Compton–Getting effect is affected by the assumption of the magnetic field direction and perpendicular diffusion anisotropy. Because the pitch-angle distribution anisotropy is so large, a small uncertainty in the magnetic field direction can produce very different solar wind speeds ranging from ∼0 to >400 km/s. In fact, if the magnetic field is chosen to be in the Parker spiral direction, which is consistent with the magnetometer measurement on Voyager 1, the derived solar wind speed is still close to the supersonic value. Only the two lowest-energy channels of the LECP instrument may give a definitive answer to the solar wind speed. However, because these channels contain a very high level of cosmic ray background, an uncertainty of just a few percent in the background can entirely hamper the estimate of solar wind speed.     

Current sheet with medium scale developed turbulence and the formation of the plasma sheet of Earth''s magnetosphere and solar prominences

A current sheet model with developed medium scale turbulence has been constructed. It is suggested that regular plasma flow in the current sheet is compensated by diffusive flux and plasma mixing, leading to temperature equalization. The analyzed turbulence has the form of electrostatic vortices in which electrons and ions move with the same velocities and hence does not lead to anomalous resistivity and current dissipation. It is possible to determine the plasma pressure dependence on magnetic vector potential and to find the Grad—Shafranov equation solutions. The theory is used to explain the Earth's magnetosphere plasma sheet characteristics. It is taken into account that experimentally observed plasma velocity fluctuations in the Earth's plasma sheet and quiescent prominences are much higher than regular plasma flow velocities. The analysis of turbulent current sheet dynamics after the regular motion weakening allows to construct the prominence formation theory. The decreasing of plasma pressure in the sheet due to diffusion leads to field-aligned plasma flow and plasma tube filling by cold chromospheric plasma by the action of siphon mechanism.     

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.     

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.     

磁扰期间的三维漂移壳分离

基于磁层粒子动力学理论,首先对比了计算漂移壳分离的引导中心法和磁力线追踪法,计算表明两种方法的计算结果一致.然后分别采用T89c和T96磁层磁场模式,用磁力线追踪法数值计算了不同初始位置(≤9Re)、不同初始投掷角、不同Kp指数和不同太阳风压力下,带电粒子的漂移壳分离.计算结果揭示了漂移壳分离随初始位置、投掷角、Kp指数和太阳风压力的变化.其具体特征如下. (1)随着径向距离的增大,漂移壳分离效应愈加显著,由正午出发的粒子将被稳定捕获,而午夜出发的径向距离≥7Re的部分大投掷角粒子将沿磁层顶逃逸. (2)正午出发的粒子,漂移到午夜时其漂移壳随投掷角减小向外排列;午夜出发的粒子,漂移到正午时其漂移壳随投掷角增大排列; 90°投掷角粒子在磁赤道面的漂移壳沿着磁场等值线排列. (3)漂移壳分离随Kp指数和太阳风压力增大变得显著,且随这两种扰动参数的变化特征和趋势是基本相似的.      

Simulation study of motion of charged particles trapped in Earth’s magnetosphere

This article aims to understand the motion of the charged particles trapped in the Earth’s inner magnetosphere. The emphasis is on identifying the numerical scheme, which is appropriate to characterize the trajectories of the charged particles of different energies that enter the Earth’s magnetosphere and get trap along the magnetic field lines. These particles perform three different periodic motions, namely: gyration, bounce, and azimuthal drift. However, often, the gyration of the particle is ignored, and only the guiding center of the particle is traced to reduce the computational time. It is because the simulation of all three motions (gyro, bounce, and drift) together needed a robust numerical scheme, which has less numerical dissipation. We have developed a three-dimensional test particle simulation model in which the relativistic equation of motion is solved numerically using the fourth and sixth-order Runge-Kutta methods. The stability of the simulation model is verified by checking the conservation of total kinetic energy and adiabatic invariants linked with each type of motion. We found that the sixth-order Runge-Kutta method is suitable to trace the complete trajectories of both proton and electron of a wide energy range, 5 keV to 250 MeV for L = 2 – 6. We have estimated the bounce and drift periods from the simulations, and they are found to be in good agreement with the theory. The study implies that a simulation model with sixth-order Runge-Kutta method can be applied to the time-vary, non-analytical form of magnetic configuration in future studies to understand the dynamics of charged particles trapped in Earth’s magnetosphere.     

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.     

Wobble of a spin stabilized satellite with cross products of inertia and magnetic attitude control

Asymmetrical spin stabilized satellite dynamics in the vicinity of the required motion is considered. The principal axis of the maximum moment of inertia slightly deviates from its assumed direction in the satellite reference frame. This is formalized in the cross products of inertia. This inertial uncertainty results in a wobble, that is undesired angular velocity components perpendicular to the rotation axis, and oscillations of this axis near the required direction. The torque-free motion is investigated first. Expressions that explicitly relate satellite inertia parameters to wobble are provided. Wobble evolution under the action of magnetic damping control is analyzed next. Its amplitude approximate exponential decay behavior and residual unavoidable wobble level are derived. These expressions are compared with numerical simulation results of nonlinear equations of motion including various disturbance sources.     

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.