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.     

Radial and latitudinal gradients of galactic cosmic rays in the heliosphere at solar maximum

Our understanding of galactic cosmic ray (GCR) modulation has advanced greatly in the last three decades. However, we still need an appropriate knowledge of the GCR intensity gradient. Numerical simulations of the transport particle equation allow interpretation of cosmic ray intensities in the heliosphere. We use the numerical solution of the GCR transport equation during solar maximum epoch to compute the radial and latitudinal gradients. Our analysis indicates that adiabatic energy loss plays an important role in the radial distribution of GCR in the inner heliosphere, while in the outer region the diffusion and convection are the relevant processes. The latitudinal gradient is small.     

The FLUKA code for space applications: recent developments.

The FLUKA Monte Carlo transport code is widely used for fundamental research, radioprotection and dosimetry, hybrid nuclear energy system and cosmic ray calculations. The validity of its physical models has been benchmarked against a variety of experimental data over a wide range of energies, ranging from accelerator data to cosmic ray showers in the earth atmosphere. The code is presently undergoing several developments in order to better fit the needs of space applications. The generation of particle spectra according to up-to-date cosmic ray data as well as the effect of the solar and geomagnetic modulation have been implemented and already successfully applied to a variety of problems. The implementation of suitable models for heavy ion nuclear interactions has reached an operational stage. At medium/high energy FLUKA is using the DPMJET model. The major task of incorporating heavy ion interactions from a few GeV/n down to the threshold for inelastic collisions is also progressing and promising results have been obtained using a modified version of the RQMD-2.4 code. This interim solution is now fully operational, while waiting for the development of new models based on the FLUKA hadron-nucleus interaction code, a newly developed QMD code, and the implementation of the Boltzmann master equation theory for low energy ion interactions.     

Empirical model of cosmic ray spectrum in energy interval 1 MeV–100 GeV during 11-year solar cycle

The galactic cosmic rays (GCR) are the main ionization source at altitude of ∼3–35 km in the atmosphere. For high latitude anomalous cosmic ray (ACR) component has also a significant influence on the atmospheric ionization. We propose an empirical model for differential spectra D(E) of galactic and anomalous cosmic rays in energy interval 1 MeV–100 GeV during solar cycle. In the model data are used which cover three solar cycles: 20, 22 and 23. The LEAP87, IMAX92, CAPRICE94, AMS98 and BESS experimental spectra for protons and alpha particles are fitted to the proposed empirical model. The modulated GCR differential spectra are compared with force-field approximation to the one-dimensional transport equation and with solutions of two-dimensional cosmic ray transport equation. For experimental spectra, the calculation of the model parameters is performed by Levenberg–Marquardt algorithm, applied to the special case of least squares. Algorithm that combines the rapid local convergence of Newton–Raphson method with globally convergent method for non-linear systems of equations is applied for theoretically obtained differential spectra. The described programmes are realized in algorithmic language C++. The proposed model gives practical possibility for investigation of experimental data from measurements of galactic cosmic rays and their anomalous component.     

Cosmic ray modulation in the outer heliosphere: Predictions for cosmic ray intensities up to the heliopause along Voyager 1 and 2 trajectories

Time dependent cosmic ray modulation in the outer heliosphere is calculated and results are compared to Voyager 1 and 2 observations using a two-dimensional time-dependent cosmic ray transport model. We predict possible future 133–242 MeV proton observations along the Voyager 1 and 2 spacecraft trajectories. Recent theoretical advances in cosmic ray transport parameters are introduced in order to provide a time-dependence for the assumed transport parameters used in the model. This leads to results that are in general compatible with the spacecraft observations in the inner and outer heliosphere over multiple solar cycles. However, for the outer heliosphere, we find that the Voyager 1 and 2 spacecraft observations cannot be fitted with an identical set of parameters along both trajectories. This indicates a possible asymmetric heliosphere or a symmetric heliosphere but with different diffusion parameters in the northern and southern hemispheres, respectively. Furthermore, results indicate that Voyager 2 observations are still under the influence of solar cycle related changes because of the large modulation volume between the heliopause and spacecraft location in contrast to Voyager 1 which shows a steady increase in cosmic ray intensities.     

Numerical study of solar flare particle propagation in the heliosphere

Numerical solutions are presented for the propagation of solar cosmic rays interplanetary space, including the effects of pitch-angle scattering and adiabatic focusing. The intensity-time profiles can be well fitted by a simple radial spatial diffusion equation with scattering mean-free path λ_fit. For low-rigidity particles the radial mean-free path so obtained is significantly larger than the mean-free path calculated from the scattering coefficient due to the inapplicability of the diffusive approximation early in the event. The well-known discrepency between λ_fit and the theoretical predictions may be resolved by these calculations.     

Effects of solar modulation on the cosmic ray positron fraction

We implemented a 2D Monte Carlo model to simulate the solar modulation of galactic cosmic rays. The model is based on the Parker’s transport equation which contains diffusion, convection, particle drift and energy loss. Following the evolution in time of the solar activity, we are able to modulate a local interstellar spectrum (LIS), that we assumed isotropic beyond the termination shock, down to the Earth position inside the heliosphere. In this work we focused our attention to the cosmic ray positron fraction at energy below ∼10 GeV, showing how the particle drift processes could explain different results for AMS-01 and PAMELA. We compare our modulated spectra with observations at Earth, and then make a prediction of the cosmic ray positron fraction for the AMS-02 experiment.     

Radial distribution of galactic cosmic rays at solar maximum

In this paper we analyze the spatial distribution of galactic cosmic rays during periods of maximum solar activity of the cycles 21, 22 and 23. We have used a two dimensional model to solve the cosmic ray transport equation. This model includes all relevant physical processes: diffusion, convection, drift and shock effects on cosmic ray propagation inside the heliosphere. We focused on the study of the radial distribution of galactic cosmic rays, and compare our results with the spacecraft observations for two energies (175 MeV H and 265 MeV/n He). Although the radial intensities of galactic cosmic rays can be explained qualitatively with all three local interstellar spectra (LISs) used in this work, we applied a reduced chi-squared analysis to investigate the best LIS that could fit the data.     

Extragalactic origin of antiprotons

We have studied the effect of Galactic modulation on cosmic rays entering the Galaxy from outside for two different models for the confinement of cosmic rays, using one dimensional transport equation. From this study, the role of extragalactic cosmic rays has been examined critically in the context of the recent data on antiprotons. We have arrived at the conclusion that they are not a significant source of cosmic ray antiprotons. However, determination of the energy spectrum of Ps at least up to a few tens of GeV would provide information on the modulation of cosmic rays, while entering the Galaxy from outside.     

The physics of the FLUKA code: Recent developments

FLUKA is a Monte-Carlo code able to simulate interaction and transport of hadrons, heavy ions and electromagnetic particles from few keV (or thermal neutron) to cosmic ray energies in whichever material. The highest priority in the design and development of the code has always been the implementation and improvement of sound and modern physical models. A summary of the FLUKA physical models is given, while recent developments are described in detail: among the others, extensions of the intermediate energy hadronic interaction generator, refinements in photon cross sections and interaction models, analytical on-line evolution of radio-activation and remnant dose. In particular, new developments in the nucleus–nucleus interaction models are discussed. Comparisons with experimental data and examples of applications of relevance for space radiation are also provided.     

Non-linear diffusion of cosmic rays

The propagation of cosmic rays in the interstellar medium after their release from the sources – supernova remnants – can be attended by the development of streaming instability. The instability creates MHD turbulence that changes the conditions of particle transport and leads to a non-linear diffusion of cosmic rays. We present a self-similar solution of the equation of non-linear diffusion for particles ejected from a SNR and discuss how obtained results may change the physical picture of cosmic ray propagation in the Galaxy.     

银河宇宙线的起源和初级质子谱

本文采用无界银河系的爆发点源各向同性弥散的模型研究了初级质子的输运方程.用加速器实验得到的P-P碰撞截面的最新资料和一个合理的扩散系数, 得到了该方程的解, 进而讨论了该解的一些特点以及诸多类似点源对地球附近质子谱的可能贡献.该模型所预期的质子谱能较好地说明实验结果.      

Acceleration at the termination shock: Anisotropies and spectra

Low-energy termination shock particle populations observed by the Voyagers upstream of the shock exhibited strong field-aligned beaming with anisotropies of the order of unity. The Parker transport equation is valid only for nearly isotropic phase space distributions and is inapplicable to these highly beamed populations. The usual approach is to revert to the more general focused transport equation retaining pitch-angle information. We developed a complimentary technique employing a three-moment expansion of the Skilling equation using Legendre polynomials. We investigate the effects of adiabatic focusing and reflection on the diffusive acceleration process at oblique shock waves. It is shown that low-energy particle intensities are discontinuous and sharply peaked at the shock, consistent with the observations. Particle spectra are not only harder than the power laws predicted from diffusive transport theory, but also exhibit spectral gaps near the low-energy acceleration threshold due to more efficient acceleration by scattering and mirroring. Our model also predicts upstream anisotropies as high as 100% for highly oblique shocks whereas downstream distributions are nearly isotropic.     

超高能宇宙线电子在星际介质中的传播

本文研究能量高达1012eV以上宇宙线电子在星际介质中的传播特征, 得到了一些有趣的结果。作为一种自然的猜测, 宇宙线电子高能成份的区域性特征, 可能是导致银河系γ射线辐射的非均匀成团状分布结构的直接原因。      

Prediction of expected global climate change by forecasting of galactic cosmic ray intensity time variation in near future based on solar magnetic field data

A method of prediction of expected part of global climate change caused by cosmic ray (CR) by forecasting of galactic cosmic ray intensity time variation in near future based on solar activity data prediction and determined parameters of convection-diffusion and drift mechanisms is presented. This gave possibility to make prediction of expected part of global climate change, caused by long-term cosmic ray intensity variation. In this paper, we use the model of cosmic ray modulation in the Heliosphere, which considers a relation between long-term cosmic ray variations with parameters of the solar magnetic field. The later now can be predicted with good accuracy. By using this prediction, the expected cosmic ray variations in the near Earth space also can be estimated with a good accuracy. It is shown that there are two possibilities: (1) to predict cosmic ray intensity for 1–6 months by using a delay of long-term cosmic ray variations relatively to effects of the solar activity and (2) to predict cosmic ray intensity for the next solar cycle. For the second case, the prediction of the global solar magnetic field characteristics is crucial. For both cases, reliable long-term cosmic ray and solar activity data as well as solar magnetic field are necessary. For solar magnetic field, we used results of two magnetographs (from Stanford and Kitt Peak Observatories). The obtained forecasting of long-term cosmic ray intensity variation we use for estimation of the part of global climate change caused by cosmic ray intensity changing (influenced on global cloudiness covering).     

Modeling the acceleration and modulation of anomalous cosmic ray oxygen

After the solar wind termination shock crossings of the Voyager spacecraft, the acceleration of anomalous cosmic rays has become a very contentious subject. In this paper we examine several topics pertinent to anomalous cosmic ray oxygen acceleration and transport using a numerical cosmic ray modulation model. These include the effects of drifts on a purely Fermi I accelerated spectra, the effects of introducing higher charge states of oxygen into the modulation model, examining the viability of momentum diffusion as a re-acceleration process in the heliosheath and examining energy spectra, and intensity gradients, in the inner heliosphere during consecutive drift cycles.     

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.     

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.