Elsevier

Journal of Computational Physics

Volume 227, Issue 2, 10 December 2007, Pages 1340-1352
Journal of Computational Physics

Multi-scale plasma simulation by the interlocking of magnetohydrodynamic model and particle-in-cell kinetic model

https://doi.org/10.1016/j.jcp.2007.09.011Get rights and content

Abstract

Many kinds of simulation models have been developed to understand the complex plasma systems. However, these simulation models have been separately performed because the fundamental assumption of each model is different and restricts the physical processes in each spatial and temporal scales. On the other hand, it is well known that the interactions among the multiple scales may play crucial roles in the plasma phenomena (e.g. magnetic reconnection, collisionless shock), where the kinetic processes in the micro-scale may interact with the global structure in the fluid dynamics. To take self-consistently into account such multi-scale phenomena, we have developed a new simulation model by directly interlocking the fluid simulation of the magnetohyrdodynamics (MHD) model and the kinetic simulation of the particle-in-cell (PIC) model. The PIC domain is embedded in a small part of MHD domain. The both simulations are performed simultaneously in each domain and the bounded data are frequently exchanged each other to keep the consistency between the models. We have applied our new interlocked simulation to Alfvén wave propagation problem as a benchmark test and confirmed that the waves can propagate smoothly through the boundaries of each domain.

Introduction

Plasma consists essentially of a multi-scale system, that is, various structures and waves are spontaneously formed and excited in widely different scales. The hierarchy of spatial and temporal structure is attributed to the kinetic of particles and the fluid dynamics. Therefore, to capture accurately the property in each scale, numerical plasma simulation has to be performed by an appropriate model. Magneteohydrodynamic (MHD) simulation is one of the most widely utilized tools in plasma research, and it has given substantial contribution to the study of global scale dynamics in space, astrophysical, and experimental plasmas. The MHD equation is derived from the approximation to large scales in space and time, so that it cannot deal with any kinetic process. On the other hand, the particle-in-cell (PIC), Vlasov and Fokker–Planck simulations are performed in order to handle the micro-scale processes. Since these kinetic simulations essentially contain any plasma processes in their system, the system size is severely restricted in small and short range. Therefore, we have used some idealized (and sometimes impractical) settings for the initial and boundary conditions.

These simulation models in macro- and micro-scales have been separately developed so far. This is because the approximation in fundamental theory in each scale is restricted to specific scale. However, the mutual interaction between the macro-scale dynamics and the micro-scale kinetics plays important roles for various multi-scale plasma phenomena. For instance, the non-gyrotropic particle orbit and the micro-scale instability growing on magnetic neutral lines are widely believed to play crucial roles for the creation of electric field, which drives magnetic reconnection on a thin current sheet [2]. Nevertheless, since there is no adequate method to treat the micro-scale kinetics in the macro-scale system, we have had to treat all the microscopic processes as an artificial model parameter so-called ”anomalous resistivity” in the large scale MHD simulation. Particle acceleration on large scale collisionless shocks is also the typical multi-scale plasma phenomena, in which reflected and accelerated particles may modify the large scale plasma flow.

Hence, a new methodology to connect consistently the dynamics of the macro-scale and micro-scale is greatly required to be developed. Once this type of method is established, it could help our understanding of complicated plasma processes, and may improve the predictability of the processes. The multi-scale simulation is an issue not only in plasma physics but also in any research fields of sciences and technologies, and the new algorithm and the mathematical framework are demanded to overcome the challenging problems for interconnecting physical processes in vastly different scales. The objective of this paper is to propose a new type of multi-scale plasma simulation algorithm, whereby the mutual interaction between large scale MHD process and micro-scale plasma kinetics is able to be taken into account directly. The strategy in our model is that, only in a limited region where the micro-process is crucial for macro-scale dynamics, the plasma kinetics are calculated with keeping the consistency with the large scale dynamics. In order to accomplish the consistency, we have developed the way to connect the MHD simulation and the kinetic simulation. This new model is an application of the macro–micro interlocked (MMI) simulation which has been recently proposed by [4], for plasma.

In this paper, we adopt the particle-in-cell (PIC) model for the kinetic simulation, and the conventional finite difference method for the MHD simulation, respectively. Therefore, our MHD and PIC interlocked model is resembled to the hybrid continuum-atomistic simulation, which is quickly grown for the study of multi-scale hydrodynamics. For instance, [7] proposed the connection of the conventional fluid model and the particle-based Direct Simulation Monte–Carlo (DSMC) model. However, we should mention the peculiarity in plasma physics in contrast to the hydrodynamics simulations. Since the several characteristics contained in the particle-based (PIC) model are negated in the continuum (MHD) model, some sophisticated filter, which can pass only the proper components for each the macro- and micro-scale models, has to be developed. Therefore, the method proposed here is not a simple application of the hydrodynamic connection model.

This paper is organized as follows: In Section 2, the simulation models are described both for the MHD and the kinetic simulation, and then, the interlocking procedure is explained both for the spatial and temporal connections, respectively. In Section 3, the interlocked model are examined based on the benchmark test for the Alfvén wave propagation problem. Finally, the prospects of the new model are discussed in Section 4.

Section snippets

Simulation model

The basic configuration of our MMI plasma simulation is schematically illustrated in Fig. 1. The system consists of two kinds of bounded areas, and in each area the MHD and PIC simulations are performed, respectively. PIC simulations are embedded in the MHD simulation, and the mutual interactions are managed through the boundary among the MHD and PIC domains. The main advantage of our MMI plasma simulation is that both the MHD and PIC simulations are performed simultaneously and the kinetic

Test on Alfvén wave propagation

The interlocked simulations between PIC and MHD are applied on the Alfvén wave propagation. The simulations are one-dimensional in space of X-axis, but fully three-dimensional in velocity. In the present PIC simulation code, we have used Kyoto university one-dimensional ElectroMagnetic Particle cOde (KEMPO1) [3]. Hereafter, magnetic field and density are normalized by the background value B0 and N0. Velocity, time, and length, are normalized by the Alfvén velocity VA, the inverse of proton

Summary and discussion

We have explained the algorithm of the MHD and PIC interlocked simulation. This new algorithm is applied to Alfvén wave propagation problem in one-dimensional system. The wave smoothly propagates from MHD into PIC domain and ejected again into MHD. While the wave is propagating in the PIC domain, the wave almost keeps its form and amplitude, even though some high-frequency fluctuations are excited. Their power is smaller than that of the Alfvén wave.

Here, we discuss about a wave shift

Acknowledgements

The authors thank Prof. T.Sato, and the member of Holistic Simulation Research Program in the Earth Simulator Center, Japan Agency for Marine-Earth Science and Technology (JAMSTEC). This research is one of the results from the program of “Development of Multi-Scale Coupled Simulation Algorithm” in the Earth Simulator Center. This work was also supported by the Grant-in-Aid for Creative Scientific Research “The Basic Study of Space Weather Prediction” (17GS0208, Head Investigator: K. Shibata)

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