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Numerical simulations based on particle methods have been widely used in various fields including astrophysics. To date, simulation softwares have been developed by individual researchers or research groups in each field, with a huge amount of time and effort, even though numerical algorithms used are very similar. To improve the situation, we have developed a framework, called FDPS, which enables researchers to easily develop massively parallel particle simulation codes for arbitrary particle methods. Until version 3.0, FDPS have provided API only for C++ programing language. This limitation comes from the fact that FDPS is developed using the template feature in C++, which is essential to support arbitrary data types of particle. However, there are many researchers who use Fortran to develop their codes. Thus, the previous versions of FDPS require such people to invest much time to learn C++. This is inefficient. To cope with this problem, we newly developed a Fortran interface layer in FDPS, which provides API for Fortran. In order to support arbitrary data types of particle in Fortran, we design the Fortran interface layer as follows. Based on a given derived data type in Fortran representing particle, a Python script provided by us automatically generates a library that manipulates the C++ core part of FDPS. This library is seen as a Fortran module providing API of FDPS from the Fortran side and uses C programs internally to interoperate Fortran with C++. In this way, we have overcome several technical issues when emulating `template' in Fortran. By using the Fortran interface, users can develop all parts of their codes in Fortran. We show that the overhead of the Fortran interface part is sufficiently small and a code written in Fortran shows a performance practically identical to the one written in C++., Comment: 10 pages, 10 figures; accepted for publication in PASJ; a typo in author name is corrected

Numerical simulations based on particle methods have been widely used in various fields including astrophysics. To date, simulation softwares have been developed by individual researchers or research groups in each field, with a huge amount of time and effort, even though numerical algorithms used are very similar. To improve the situation, we have developed a framework, called FDPS, which enables researchers to easily develop massively parallel particle simulation codes for arbitrary particle methods. Until version 3.0, FDPS have provided API only for C++ programing language. This limitation comes from the fact that FDPS is developed using the template feature in C++, which is essential to support arbitrary data types of particle. However, there are many researchers who use Fortran to develop their codes. Thus, the previous versions of FDPS require such people to invest much time to learn C++. This is inefficient. To cope with this problem, we newly developed a Fortran interface layer in FDPS, which provides API for Fortran. In order to support arbitrary data types of particle in Fortran, we design the Fortran interface layer as follows. Based on a given derived data type in Fortran representing particle, a Python script provided by us automatically generates a library that manipulates the C++ core part of FDPS. This library is seen as a Fortran module providing API of FDPS from the Fortran side and uses C programs internally to interoperate Fortran with C++. In this way, we have overcome several technical issues when emulating `template' in Fortran. By using the Fortran interface, users can develop all parts of their codes in Fortran. We show that the overhead of the Fortran interface part is sufficiently small and a code written in Fortran shows a performance practically identical to the one written in C++., Comment: 10 pages, 10 figures; accepted for publication in PASJ; a typo in author name is corrected

Magnetorotational instability (MRI) has a potential to generate the vigorous turbulence in protoplanetary disks, although its turbulence strength and accretion stress remains debatable because of the uncertainty of MRI with low ionization fraction. We focus on the heating of electrons by strong electric fields which amplifies nonideal magnetohydrodynamic effects. The heated electrons frequently collide with and stick to dust grains, which in turn decreases the ionization fraction and is expected to weaken the turbulent motion driven by MRI. In order to quantitatively investigate the nonlinear evolution of MRI including the electron heating, we perform magnetohydrodynamical simulation with the unstratified shearing box. We introduce a simple analytic resistivity model depending on the current density by mimicking resistivity given by the calculation of ionization. Our simulation confirms that the electron heating suppresses magnetic turbulence when the electron heating occurs with low current density. We find a clear correlation between magnetic stress and its current density, which means that the magnetic stress is proportional to the squared current density. When the turbulent motion is completely suppressed, laminar accretion flow is caused by ordered magnetic field. We give an analytical description of the laminar state by using a solution of linear perturbation equations with resistivity. We also propose a formula that successfully predicts the accretion stress in the presence of the electron heating., Comment: 12 pages, 12 figures, accepted for publication in ApJ

Magnetorotational instability (MRI) has a potential to generate the vigorous turbulence in protoplanetary disks, although its turbulence strength and accretion stress remains debatable because of the uncertainty of MRI with low ionization fraction. We focus on the heating of electrons by strong electric fields which amplifies nonideal magnetohydrodynamic effects. The heated electrons frequently collide with and stick to dust grains, which in turn decreases the ionization fraction and is expected to weaken the turbulent motion driven by MRI. In order to quantitatively investigate the nonlinear evolution of MRI including the electron heating, we perform magnetohydrodynamical simulation with the unstratified shearing box. We introduce a simple analytic resistivity model depending on the current density by mimicking resistivity given by the calculation of ionization. Our simulation confirms that the electron heating suppresses magnetic turbulence when the electron heating occurs with low current density. We find a clear correlation between magnetic stress and its current density, which means that the magnetic stress is proportional to the squared current density. When the turbulent motion is completely suppressed, laminar accretion flow is caused by ordered magnetic field. We give an analytical description of the laminar state by using a solution of linear perturbation equations with resistivity. We also propose a formula that successfully predicts the accretion stress in the presence of the electron heating., Comment: 12 pages, 12 figures, accepted for publication in ApJ

The giant impact (GI) is one of the most important hypotheses both in planetary science and geoscience, since it is related to the origin of the Moon and also the initial condition of the Earth. A number of numerical simulations have been done using the smoothed particle hydrodynamics (SPH) method. However, GI hypothesis is currently in a crisis. The "canonical" GI scenario failed to explain the identical isotope ratio between the Earth and the Moon. On the other hand, little has been known about the reliability of the result of GI simulations. In this paper, we discuss the effect of the resolution on the results of the GI simulations by varying the number of particles from $3 \times10^3$ to $10^8$. We found that the results does not converge, but shows oscillatory behaviour. We discuss the origin of this oscillatory behaviour., Comment: Accepted to PASJ, an animation is available at https://vimeo.com/194156367

We present the basic idea, implementation, measured performance and performance model of FDPS (Framework for developing particle simulators). FDPS is an application-development framework which helps the researchers to develop particle-based simulation programs for large-scale distributed-memory parallel supercomputers. A particle-based simulation program for distributed-memory parallel computers needs to perform domain decomposition, redistribution of particles, and gathering of particle information for interaction calculation. Also, even if distributed-memory parallel computers are not used, in order to reduce the amount of computation, algorithms such as Barnes-Hut tree method should be used for long-range interactions. For short-range interactions, some methods to limit the calculation to neighbor particles are necessary. FDPS provides all of these necessary functions for efficient parallel execution of particle-based simulations as "templates", which are independent of the actual data structure of particles and the functional form of the interaction. By using FDPS, researchers can write their programs with the amount of work necessary to write a simple, sequential and unoptimized program of O(N^2) calculation cost, and yet the program, once compiled with FDPS, will run efficiently on large-scale parallel supercomputers. A simple gravitational N-body program can be written in around 120 lines. We report the actual performance of these programs and the performance model. The weak scaling performance is very good, and almost linear speedup was obtained for up to the full system of K computer. The minimum calculation time per timestep is in the range of 30 ms (N=10^7) to 300 ms (N=10^9). These are currently limited by the time for the calculation of the domain decomposition and communication necessary for the interaction calculation. We discuss how we can overcome these bottlenecks., Comment: 22 pages, 27 figures,accepted for publication in PASJ. The FDPS package is here https://github.com/fdps/fdps

Automated forecasts serve important role in space weather science, by providing statistical insights to flare-trigger mechanisms, and by enabling tailor-made forecasts and high-frequency forecasts. Only by realtime forecast we can experimentally measure the performance of flare-forecasting methods while confidently avoiding overlearning. We have been operating unmanned flare forecast service since August, 2015 that provides 24-hour-ahead forecast of solar flares, every 12 minutes. We report the method and prediction results of the system., Comment: 6 pages, 4 figures

The giant impact (GI) is one of the most important hypotheses both in planetary science and geoscience, since it is related to the origin of the Moon and also the initial condition of the Earth. A number of numerical simulations have been done using the smoothed particle hydrodynamics (SPH) method. However, GI hypothesis is currently in a crisis. The "canonical" GI scenario failed to explain the identical isotope ratio between the Earth and the Moon. On the other hand, little has been known about the reliability of the result of GI simulations. In this paper, we discuss the effect of the resolution on the results of the GI simulations by varying the number of particles from $3 \times10^3$ to $10^8$. We found that the results does not converge, but shows oscillatory behaviour. We discuss the origin of this oscillatory behaviour., Comment: Accepted to PASJ, an animation is available at https://vimeo.com/194156367

Automated forecasts serve important role in space weather science, by providing statistical insights to flare-trigger mechanisms, and by enabling tailor-made forecasts and high-frequency forecasts. Only by realtime forecast we can experimentally measure the performance of flare-forecasting methods while confidently avoiding overlearning. We have been operating unmanned flare forecast service since August, 2015 that provides 24-hour-ahead forecast of solar flares, every 12 minutes. We report the method and prediction results of the system., Comment: 6 pages, 4 figures

We present the basic idea, implementation, measured performance and performance model of FDPS (Framework for developing particle simulators). FDPS is an application-development framework which helps the researchers to develop particle-based simulation programs for large-scale distributed-memory parallel supercomputers. A particle-based simulation program for distributed-memory parallel computers needs to perform domain decomposition, redistribution of particles, and gathering of particle information for interaction calculation. Also, even if distributed-memory parallel computers are not used, in order to reduce the amount of computation, algorithms such as Barnes-Hut tree method should be used for long-range interactions. For short-range interactions, some methods to limit the calculation to neighbor particles are necessary. FDPS provides all of these necessary functions for efficient parallel execution of particle-based simulations as "templates", which are independent of the actual data structure of particles and the functional form of the interaction. By using FDPS, researchers can write their programs with the amount of work necessary to write a simple, sequential and unoptimized program of O(N^2) calculation cost, and yet the program, once compiled with FDPS, will run efficiently on large-scale parallel supercomputers. A simple gravitational N-body program can be written in around 120 lines. We report the actual performance of these programs and the performance model. The weak scaling performance is very good, and almost linear speedup was obtained for up to the full system of K computer. The minimum calculation time per timestep is in the range of 30 ms (N=10^7) to 300 ms (N=10^9). These are currently limited by the time for the calculation of the domain decomposition and communication necessary for the interaction calculation. We discuss how we can overcome these bottlenecks., Comment: 22 pages, 27 figures,accepted for publication in PASJ. The FDPS package is here https://github.com/fdps/fdps

In this paper, we propose observational methods for detecting lightning in protoplanetary disks. We do so by calculating the critical electric field strength in the lightning matrix gas (LMG), the parts of the disk where the electric field is strong enough to cause lightning. That electric field accelerates multiple positive ion species to characteristic terminal velocities. In this paper, we present three distinct discharge models, with corresponding critical electric fields. We simulate the position-velocity diagrams and the integrated emission maps for the models. We calculate the measure of sensitivity values for detection of the models, and for distinguishing between the models. At the distance of TW-Hya (54pc), LMG that occupies $2\pi$ in azimuth and $25 \mathrm{au}

We have developed UFCORIN, a platform for studying and automating space weather prediction. Using our system we have tested 6,160 different combinations of SDO/HMI data as input data, and simulated the prediction of GOES X-ray flux for 2 years (2011-2012) with one-hour cadence. We have found that direct comparison of the true skill statistics (TSS) from small cross-validation sets is ill-posed, and used the standard scores ($z$) of the TSS to compare the performance of the various prediction strategies. The $z$ of a strategy is a stochastic variable of the stochastically-chosen cross-validation dataset, and the $z$ for the three strategies best at predicting X, $\geq$M and $\geq$C class flares are better than the average $z$ of the 6,160 strategies by 2.3$\sigma$, 2.1$\sigma$, 3.8$\sigma$ confidence levels, respectively. The best three TSS values were $0.75\pm0.07$, $0.48\pm0.02$, and $0.56\pm0.04$, respectively., Comment: 47 pages, 11 figures, accepted for publication in Space Weather: http://onlinelibrary.wiley.com/doi/10.1002/2015SW001257/full

In this paper, we propose observational methods for detecting lightning in protoplanetary disks. We do so by calculating the critical electric field strength in the lightning matrix gas (LMG), the parts of the disk where the electric field is strong enough to cause lightning. That electric field accelerates multiple positive ion species to characteristic terminal velocities. In this paper, we present three distinct discharge models, with corresponding critical electric fields. We simulate the position-velocity diagrams and the integrated emission maps for the models. We calculate the measure of sensitivity values for detection of the models, and for distinguishing between the models. At the distance of TW-Hya (54pc), LMG that occupies $2\pi$ in azimuth and $25 \mathrm{au}

We have developed UFCORIN, a platform for studying and automating space weather prediction. Using our system we have tested 6,160 different combinations of SDO/HMI data as input data, and simulated the prediction of GOES X-ray flux for 2 years (2011-2012) with one-hour cadence. We have found that direct comparison of the true skill statistics (TSS) from small cross-validation sets is ill-posed, and used the standard scores ($z$) of the TSS to compare the performance of the various prediction strategies. The $z$ of a strategy is a stochastic variable of the stochastically-chosen cross-validation dataset, and the $z$ for the three strategies best at predicting X, $\geq$M and $\geq$C class flares are better than the average $z$ of the 6,160 strategies by 2.3$\sigma$, 2.1$\sigma$, 3.8$\sigma$ confidence levels, respectively. The best three TSS values were $0.75\pm0.07$, $0.48\pm0.02$, and $0.56\pm0.04$, respectively., Comment: 47 pages, 11 figures, accepted for publication in Space Weather: http://onlinelibrary.wiley.com/doi/10.1002/2015SW001257/full

Massive (hypermassive and supramassive) neutron stars are likely to be often formed after the merger of binary neutron stars. We explore the evolution process of the remnant massive neutron stars and gravitational waves emitted by them, based on numerical-relativity simulations for binary neutron star mergers employing a variety of equations of state and choosing a plausible range of the neutron-star mass of binaries. We show that the lifetime of remnant hypermassive neutron stars depends strongly on the total binary mass and also on the equations of state. Gravitational waves emitted by the remnant massive neutron stars universally have a quasiperiodic nature of an approximately constant frequency although the frequency varies with time. We also show that the frequency and time-variation feature of gravitational waves depend strongly on the equations of state. We derive a fitting formula for the quasiperiodic gravitational waveforms, which may be used for the data analysis of a gravitational-wave signal., Comment: 33 pages, 18 figures. Accepted for publication in Physical Review D

Massive (hypermassive and supramassive) neutron stars are likely to be often formed after the merger of binary neutron stars. We explore the evolution process of the remnant massive neutron stars and gravitational waves emitted by them, based on numerical-relativity simulations for binary neutron star mergers employing a variety of equations of state and choosing a plausible range of the neutron-star mass of binaries. We show that the lifetime of remnant hypermassive neutron stars depends strongly on the total binary mass and also on the equations of state. Gravitational waves emitted by the remnant massive neutron stars universally have a quasiperiodic nature of an approximately constant frequency although the frequency varies with time. We also show that the frequency and time-variation feature of gravitational waves depend strongly on the equations of state. We derive a fitting formula for the quasiperiodic gravitational waveforms, which may be used for the data analysis of a gravitational-wave signal., Comment: 33 pages, 18 figures. Accepted for publication in Physical Review D

We study how the magnetorotational instability (MRI) in protoplanetary disks is affected by the electric discharge caused by the electric field in the resistive magnetohydrodynamic. We performed three-dimensional shearing box simulations with various values of plasma beta and electrical breakdown models. We find that the MRI is self-sustaining in spite of the high resistivity. The instability gives rise to the large electric field that causes the electrical breakdown, and the breakdown maintains the high degree of ionization required for the instability. The condition for this self-sustained MRI is set by the balance between the energy supply from the shearing motion and the energy consumed by ohmic dissipation. We apply the condition to various disk models and study where the active, self-sustained, and dead zones of MRI are located. In the fiducial minimum-mass solar-nebula model, the newly found sustained zone occupies only a limited volume of the disk. In the late-phase gas-depleted disk models, however, the sustained zone occupies a larger volume of the disk.

We study how the magnetorotational instability (MRI) in protoplanetary disks is affected by the electric discharge caused by the electric field in the resistive MHD. We have performed three-dimensional shearing box simulations with various values of plasma beta and electrical breakdown models. We find the self-sustainment of the MRI in spite of the high resistivity. The instability gives rise to the large electric field that causes the electrical breakdown, and the breakdown maintains the high ionization degree required for the instability. The condition for this self-sustained MRI is set by the balance between the energy supply from the shearing motion and the energy consumed by the Ohmic dissipation. We apply the condition to various disk models and study where the active, self-sustained, and dead zones of MRI are. In the fiducial minimum-mass solar nebula (MMSN) model, the newly-found sustained zone occupies only the limited volume of the disk. In the late-phase gas-depleted disk models, however, the sustained zone occupies larger volume of the disk., Comment: 23 pages, 9 figures,accepted to The Astrophysical Journal, Fix Figure 6

We present a data format for the output of general N-body simulations, allowing the presence of individual time steps. By specifying a standard, different N-body integrators and different visualization and analysis programs can all share the simulation data, independent of the type of programs used to produce the data. Our Particle Stream Data Format, PSDF, is specified in YAML, based on the same approach as XML but with a simpler syntax. Together with a specification of PSDF, we provide background and motivation, as well as specific examples in a variety of computer languages. We also offer a web site from which these examples can be retrieved, in order to make it easy to augment existing codes in order to give them the option to produce PSDF output., Comment: 5 pages; submitted to New Astronomy

We study how the magnetorotational instability (MRI) in protoplanetary disks is affected by the electric discharge caused by the electric field in the resistive MHD. We have performed three-dimensional shearing box simulations with various values of plasma beta and electrical breakdown models. We find the self-sustainment of the MRI in spite of the high resistivity. The instability gives rise to the large electric field that causes the electrical breakdown, and the breakdown maintains the high ionization degree required for the instability. The condition for this self-sustained MRI is set by the balance between the energy supply from the shearing motion and the energy consumed by the Ohmic dissipation. We apply the condition to various disk models and study where the active, self-sustained, and dead zones of MRI are. In the fiducial minimum-mass solar nebula (MMSN) model, the newly-found sustained zone occupies only the limited volume of the disk. In the late-phase gas-depleted disk models, however, the sustained zone occupies larger volume of the disk., Comment: 23 pages, 9 figures,accepted to The Astrophysical Journal, Fix Figure 6

We propose Paraiso, a domain specific language embedded in functional programming language Haskell, for automated tuning of explicit solvers of partial differential equations (PDEs) on GPUs as well as multicore CPUs. In Paraiso, one can describe PDE solving algorithms succinctly using tensor equations notation. Hydrodynamic properties, interpolation methods and other building blocks are described in abstract, modular, re-usable and combinable forms, which lets us generate versatile solvers from little set of Paraiso source codes. We demonstrate Paraiso by implementing a compressive hydrodynamics solver. A single source code less than 500 lines can be used to generate solvers of arbitrary dimensions, for both multicore CPUs and GPUs. We demonstrate both manual annotation based tuning and evolutionary computing based automated tuning of the program., Comment: 52 pages, 14 figures, accepted for publications in Computational Science and Discovery

We present a data format for the output of general N-body simulations, allowing the presence of individual time steps. By specifying a standard, different N-body integrators and different visualization and analysis programs can all share the simulation data, independent of the type of programs used to produce the data. Our Particle Stream Data Format, PSDF, is specified in YAML, based on the same approach as XML but with a simpler syntax. Together with a specification of PSDF, we provide background and motivation, as well as specific examples in a variety of computer languages. We also offer a web site from which these examples can be retrieved, in order to make it easy to augment existing codes in order to give them the option to produce PSDF output., Comment: 5 pages; submitted to New Astronomy

We propose Paraiso, a domain specific language embedded in functional programming language Haskell, for automated tuning of explicit solvers of partial differential equations (PDEs) on GPUs as well as multicore CPUs. In Paraiso, one can describe PDE solving algorithms succinctly using tensor equations notation. Hydrodynamic properties, interpolation methods and other building blocks are described in abstract, modular, re-usable and combinable forms, which lets us generate versatile solvers from little set of Paraiso source codes. We demonstrate Paraiso by implementing a compressive hydrodynamics solver. A single source code less than 500 lines can be used to generate solvers of arbitrary dimensions, for both multicore CPUs and GPUs. We demonstrate both manual annotation based tuning and evolutionary computing based automated tuning of the program., Comment: 52 pages, 14 figures, accepted for publications in Computational Science and Discovery

We study the role of dust-dust collisional charging in protoplanetary discs. We show that dust-dust collisional charging becomes an important process in determining the charge state of dust and gas, if there is dust enhancement and/or dust is fluffy, so that dust surface area per disc volume is locally increased. We solve the charge equilibrium equations for various disc environments and dust number density $\eta$, using general purpose graphic processors (GPGPU) and {\sc cuda} programming language. We found that as dust number density $\eta$ increases, the charge distribution experience four phases. In one of these phases the electrostatic field $E$ caused by dust motion increases as $E \propto \eta^4$. As a result, macroscopic electric discharge takes place, for example at $\eta = 70$ (in units of minimum-mass solar nebula (MMSN) values, considering two groups of fluffy dust with radii $10^{-2}\unit{cm}$, $10^{2}\unit{cm}$). We present a model that describes the charge exchange processes in the discs as an electric circuit. We derive analytical formulae of critical dust number density for lightning, as functions of dust parameters. We estimate the total energy, intensity and event ratio of such discharges (`lightning'). We discuss the possibility of observing lightning and sprite discharges in protoplanetary discs by Astronomically Low Frequency ({\em ALF}) waves, {\em IR} images, {\em UV} lines, and high energy gamma rays. We also discuss the effects of lightning on chondrule heating, planetesimal growth and magnetorotational instability of the disc., Comment: 27 pages, 38 figures, Accepted to MNRAS on 2009 October 8. Received 2009 October 5; in original form 2009 August 11

We study the role of dust-dust collisional charging in protoplanetary discs. We show that dust-dust collisional charging becomes an important process in determining the charge state of dust and gas, if there is dust enhancement and/or dust is fluffy, so that dust surface area per disc volume is locally increased. We solve the charge equilibrium equations for various disc environments and dust number density $\eta$, using general purpose graphic processors (GPGPU) and {\sc cuda} programming language. We found that as dust number density $\eta$ increases, the charge distribution experience four phases. In one of these phases the electrostatic field $E$ caused by dust motion increases as $E \propto \eta^4$. As a result, macroscopic electric discharge takes place, for example at $\eta = 70$ (in units of minimum-mass solar nebula (MMSN) values, considering two groups of fluffy dust with radii $10^{-2}\unit{cm}$, $10^{2}\unit{cm}$). We present a model that describes the charge exchange processes in the discs as an electric circuit. We derive analytical formulae of critical dust number density for lightning, as functions of dust parameters. We estimate the total energy, intensity and event ratio of such discharges (`lightning'). We discuss the possibility of observing lightning and sprite discharges in protoplanetary discs by Astronomically Low Frequency ({\em ALF}) waves, {\em IR} images, {\em UV} lines, and high energy gamma rays. We also discuss the effects of lightning on chondrule heating, planetesimal growth and magnetorotational instability of the disc., Comment: 27 pages, 38 figures, Accepted to MNRAS on 2009 October 8. Received 2009 October 5; in original form 2009 August 11

We simulate the acceleration processes of collisionless particles in a shock structure with magnetohydrodynamical (MHD) fluctuations. The electromagnetic field is represented as a sum of MHD shock solution ($\Mag_0, \Ele_0$) and torsional Alfven modes spectra ($\delta \Mag, \delta \Ele $). We represent fluctuation modes in logarithmic wavenumber space. Since the electromagnetic fields are represented analytically, our simulations can easily cover as large as eight orders of magnitude in resonant frequency, and do not suffer from spatial limitations of box size or grid spacing. We deterministically calculate the particle trajectories under the Lorenz force for time interval of up to ten years, with a time step of $\sim 0.5 \sec$. This is sufficient to resolve Larmor frequencies without a stochastic treatment. Simulations show that the efficiency of the first order Fermi acceleration can be parametrized by the fluctuation amplitude $\eta \equiv < \delta B^2 > ^{\frac 1 2} {B_0}^{-1}$ . Convergence of the numerical results is shown by increasing the number of wave modes in Fourier space while fixing $\eta$. Efficiency of the first order Fermi acceleration has a maximum at $ \eta \simeq 10^1$. The acceleration rate depends on the angle between the shock normal and $\Mag_0$, and is highest when the angle is zero. Our method will provide a convenient tool for comparing collisionless turbulence theories with, for example, observations of bipolar structure of super nova remnants (SNRs) and shell-like synchrotron-radiating structure., Comment: 11 pages, 4 figures, accepted for publication in The Astrophysical Journal letters

We simulate the acceleration processes of collisionless particles in a shock structure with magnetohydrodynamical (MHD) fluctuations. The electromagnetic field is represented as a sum of MHD shock solution ($\Mag_0, \Ele_0$) and torsional Alfven modes spectra ($\delta \Mag, \delta \Ele $). We represent fluctuation modes in logarithmic wavenumber space. Since the electromagnetic fields are represented analytically, our simulations can easily cover as large as eight orders of magnitude in resonant frequency, and do not suffer from spatial limitations of box size or grid spacing. We deterministically calculate the particle trajectories under the Lorenz force for time interval of up to ten years, with a time step of $\sim 0.5 \sec$. This is sufficient to resolve Larmor frequencies without a stochastic treatment. Simulations show that the efficiency of the first order Fermi acceleration can be parametrized by the fluctuation amplitude $\eta \equiv < \delta B^2 > ^{\frac 1 2} {B_0}^{-1}$ . Convergence of the numerical results is shown by increasing the number of wave modes in Fourier space while fixing $\eta$. Efficiency of the first order Fermi acceleration has a maximum at $ \eta \simeq 10^1$. The acceleration rate depends on the angle between the shock normal and $\Mag_0$, and is highest when the angle is zero. Our method will provide a convenient tool for comparing collisionless turbulence theories with, for example, observations of bipolar structure of super nova remnants (SNRs) and shell-like synchrotron-radiating structure., Comment: 11 pages, 4 figures, accepted for publication in The Astrophysical Journal letters