Plasma and Fusion Research: Rapid Communications

Volume 4, 049 (2009)First Demonstration of Collisionless Driven Reconnection
in a Multi-Hierarchy Simulation
Shunsuke USAMI1) , Hiroaki OHTANI1,2) , Ritoku HORIUCHI1,2) and Mitsue DEN3)
1)National Institute for Fusion Science, Toki 509-5292, JapanThe Graduate University for Advanced Studies (SOKENDAI), Toki 509-5292, Japan
3)
National Institute of Information and Communications Technology, Koganei 184-8795, Japan
2)(Received 31 August 2009 / Accepted 22 September 2009)
A multi-hierarchy simulation model for magnetic reconnection studies is developed in which macroscopic
and microscopic physics are expressed consistently and simultaneously. We are the first to have successfully
demonstrated collisionless driven reconnection in the framework of a multi-hierarchy model. Magnetic reconnection is found to occur in a micro-hierarchy upon plasma injection from a macro-hierarchy.
c 2009 The Japan Society of Plasma Science and Nuclear Fusion ResearchKeywords: multi-hierarchy simulation, magnetic reconnection, MHD, PIC
DOI: 10.1585/pfr.4.049
Magnetic reconnection is believed to play an important role in solar flares, geomagnetic substorms, and tokamak disruptions [1–3]. Furthermore, in recent years, magnetic reconnection has attracted considerable attention asa multi-hierarchy phenomenon. When magnetic reconnection occurs, the field topology changes on a macroscopic
scale and global plasma transport takes place. On the otherhand, an electric resistivity controlled by a microscopic
process is necessary as a trigger. Therefore, for complete understanding of magnetic reconnection [4], in themagnetic reconnection interlocked simulation (MARIS)
project, we develop a (three-dimensional) multi-hierarchysimulation model, which calculates both microscopic andmacroscopic physics consistently and simultaneously.
In the multi-hierarchy structure of magnetic reconnection, the characteristic space-time scale changes with distance from the neutral sheet. Dynamics within the ionmeandering orbit scale lmi is controlled by kinetic physics,
while plasma behaviors outside the ion skin depth di canbe expressed by a one-fluid model [5].
Based on the above features, we employ the domain
decomposition method for our multi-hierarchy model [6–8]. The domains diﬀer in algorithm. Physics in the domain where microscopic kinetic eﬀects play crucial roles
is solved by particle-in-cell (PIC) simulation method [9].
This domain is called PIC domain. On the other hand, dynamics on the periphery of the PIC domain is expressed bymagnetohydrodynamics (MHD) simulation method. We
shall refer to this domain as MHD domain. Between thePIC and MHD domains, an Interface domain with a finite
width is inserted. The physics in the Interface domain iscalculated by both the PIC and MHD algorithms. Macroscopic physical quantities in the Interface domain (for in-stance, magnetic field and fluid velocities) are obtained bya hand-shake scheme, QInterface = aQMHD + (1 − a)QPIC ,where QMHD and QPIC indicate the values of Q calculatedby the MHD and PIC algorithms, respectively. The parameter a is a function of the coordinates. Individual particle
velocities in the Interface domain are newly determined so
as to satisfy the (shifted) Maxwellian distribution using theobtained macroscopic quantities at every time step. ThePIC and MHD domains can be smoothly interlocked viathe Interface domain.
In previous work, as numerical tests, we simulated thepropagation of linear Alfvén waves in the multi-hierarchymodel [6, 7]. The model was not configured for reconnection. For instance, the external magnetic field was not
anti-parallel but was uniform (see Refs. [6, 7] for details).
Smooth wave propagation was observed.
In this paper, we connect the PIC and MHD domains
at the upstream boundary for simulations of collisionlessdriven reconnection. Figure 1 shows a schematic diagramof our multi-hierarchy model. The PIC domain covers
the central region close to the neutral sheet (|y/(c/ωce )| <11.875), and the MHD domains are outside the PIC domain (15.875 < |y/(c/ωce )| < 19.875), where c is the speedof light, and ωce is the electron gyrofrequency. The Interface domain is located between the PIC and MHD domains(11.875 < |y/(c/ωce )| < 15.875).
The initial condition is given by a one-dimensional
Harris-type equilibrium as Bx (y) = B0 tanh(y/L) for themagnetic field, where B0 is a constant, and L is the scaleheight along the y axis. The system is periodic in the xand z directions and is free in the y direction. An external
electric field Ezd (t) is applied at the entire outside boundary(y/(c/ωce ) = ±19.875) of the MHD domain. Plasmas are
injected inward in the y direction. The electric field Ezd (t)author’s e-mail: usami.shunsuke@nifs.ac.jp049-1

c 2009 The Japan Society of Plasma
Science and Nuclear Fusion Research

Plasma and Fusion Research: Rapid Communications

Volume 4, 049 (2009)

Fig. 1 Schematic diagram of the multi-hierarchy simulation
box. The simulation domain is divided into PIC, Interface, and MHD domains.

Fig. 3 Spatial profiles of various terms in the out-of-plane components of the ion momentum equation along the y axis
passing the X point at ωce t = 1323.

main. An X point exists at the center. We can see that inflows come inward from the MHD domain and drive magnetic reconnection at the center in the PIC domain. Notethat magnetic reconnection does not completely reach asteady state. Unfortunately, if the simulation continues fora longer time, two outflows would collide with each otherbecause the x direction is taken to be a periodic boundarycondition.
Here, we need to examine whether the collisionlessdriven reconnection found in our multi-hierarchy simulation model represents a true physical phenomenon.
Ishizawa and Horiuchi [5] studied the violation mechanismof the frozen-in condition in magnetic reconnection with aPIC simulation. They found that ion nongyrotropic pressure causes violation of the ion frozen-in constraint bycomparing terms in the momentum equation of ions,Fig. 2 Magnetic lines of force (top panel) and fluid velocity vectors (second and third panels) at ωce t = 1323. The fluidvelocities v x (solid line) and vy (dash-dot line) along the yaxis passing the X point are shown in the right side of thesecond panel. The third small panel is an enlarged viewof the red rectangular region in the second panel.

E + ui /c × B  −(1/en)∇ · Pi + (mi /e)ui · ∇ui , (1)where E is the electric field, and e, n, mi , ui , and Pi represent the charge, number density, mass, fluid velocity,and pressure tensor of ions, respectively. In Fig. 3, weshow the spatial profiles of the out-of-plane componentsof terms in Eq. (1) along the y axis passing the X point.
< 5( l ), theWe can see that in the region |y/(c/ωce )|∼mipressure tensor term is dominant, and it leads to breaking the frozen-in constraint. On the other hand, the pressure tensor almost cancels out the inertia term in the region< |y/(c/ω )| < 10( d )), and the relation E+u /c×B| = 0(5∼ce ∼iizholds approximately. These results are consistent withthose obtained in Ref. [5]. Therefore, our multi-hierarchyin the MHD domain is programmed to evolve from zero
to a constant value E0 , which corresponds to −0.04B0 with
the PIC unit system.
Figure 2 displays the magnetic lines of force (toppanel) and fluid velocity vectors (second and third panels)in the (x, y) plane in the whole multi-hierarchy simulation
domain at ωce t = 1323. (An enlarged view of the red rectangular region is also presented in the third panel.) Thelines of force and the fluid velocity are smoothly connected
between the MHD and PIC domains via the Interface do049-2

Plasma and Fusion Research: Rapid Communications

Volume 4, 049 (2009)

model exhibits a true physical phenomenon.
This work was done as a first step in the multihierarchy simulation of magnetic reconnection. Next, wechange the boundary conditions in the x direction from periodic to open. This enables us to continue simulations
for a much longer time, until a steady state of magnetic
reconnection. In the near future, we would also like toconsider hierarchy connections at the reconnection downstream boundary.
This work was partially supported by the Research
Cooperation Program on Hierarchy and Holism in Natural Sciences 2 at the National Institutes of Natural Sciences and the General Coordinated Research at the National Institute for Fusion Science (NIFS08KTAL015,NIFS09KNXN174).

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