H. Ohtani

M. Den

R. Horiuchi

S. Usami

National Institute for Fusion Science (NIFS-Repository)

 Collisionless magnetic reconnection is one of the fundamental processes in which energy is converted from magnetic field energy to kinetic energy. It plays an essential role in the rapid energy release in laboratory fusion device and astrophysical plasmas. Furthermore, magnetic reconnection has an aspect of cross-hierarchy phenomenon. For comprehension of magnetic reconnection process as a multi-hierarchy phenomenon, we have developed the multi-hierarchy simulation model which solves macroscopic and microscopic physics simultaneously and self-consistently. Our multi-hierarchy model is based on the domain decomposition method. The simulation domain is divided into macro- and micro-hierarchies, and different algorithms are used in the different domains. The physics in the macro-hierarchy (MHD domain) is calculated by the MHD algorithm, and the dynamics in the micro-hierarchy (PIC domain) is expressed by the PIC algorithm. Between two hierarchies, the interface domain with a finite width is inserted, where physical quantities in the macro-hierarchy and micro-hierarchy are exchanged. We call this method “hand-shaking scheme”. In order to examine validity of hand-shaking scheme between two hierarchies in our multi-hierarchy model, we simulated the propagation of linear Alfvén wave in the uniform magnetic field. Waves were observed to propagate smoothly in the multi-hierarchy simulation box [1]. Furthermore, multi-hierarchy simulations of plasma injection were performed. It was confirmed that plasma continuously and smoothly flowed from the MHD to PIC domains across the magnetic field [2, 3]. In 2009, using two-hierarchy-connected model in the upstream direction with periodic condition in the downstream direction, we succeeded in the first demonstration of multi-hierarchy simulation on magnetic reconnection (In Fig. 1, we show the magnetic field lines in the (x, y) plane) [4]. Recently we have improved our multi-hierarchy model where open boundary condition is applied in the downstream direction. Magnetic reconnection is driven in the same way as the first model. Figure 2 displays the magnetic field lines in the (x, y) plane. Plasma flows come from MHD to PIC domains and drive magnetic reconnection at the center in the PIC domain. The magnetic field lines are smoothly connected between two hierarchies. We confirmed that the magnetic reconnection phenomena found in our multi-hierarchy model exhibit true physics, by comparing them with pure PIC simulation results.  Fig. 1: Magnetic field lines in multi-hierarchy simulation for a periodic boundary condition in the downstream direction (x).  Fig. 2: Magnetic field lines in multi-hierarchy simulation for an open boundary condition in the downstream direction (x).  1) S. Usami, H. Ohtani, R. Horiuchi, and M. Den,  Comm. in Comput. Phys. 4 (2008) 537. 2) S. Usami, H. Ohtani, R. Horiuchi, and M. Den, J. Plasma Fusion Res. 85 (2009) 585. 3) S. Usami, H. Ohtani, R. Horiuchi, and M. Den,  Comm. in Comput. Phys. in press. 4) S. Usami, H. Ohtani, R. Horiuchi, and M. Den, Plasma Fusion Res. 4 (2009) 049. Reﬂective ﬂakes (mica or aluminum ﬂakes, commer-cial products such as Kalliroscope, etc) are widely usedto visualize ﬂuid motions in the laboratory. An exampleis shown in Fig. 1(a), which is a ﬂake visualization of asteady ﬂow in a precessing sphere (Fig. 2). In this exper-iment, a small amount (about 13 ppm) of TiO2-coatedmica, whose size is about (10 µm)2 × 0.1 µm, is seeded tothe water conﬁned in the sphere. The incident rays area thin laser sheet which runs through the centre of thesphere perpendicularly to the spin axis (see Fig. 2). Re-ﬂected light by the ﬂakes is recorded by a digital camerafrom the perspective parallel to the spin axis. Since thevisualized pattern is stationary, we may suppose that theﬂow is steady. However, can we extract the informationof ﬂow structures from the visualized pattern?This fundamental problem has been investigated byquite a few authors (see the references in Ref. 1), butthere is no clear answer. In order to investigate themechanism of ﬂake visualizations, we derive1), under theassumption that ﬂakes are inﬁnitely thin elliptic diskswithout inertia, the governing equations of ﬂakes:dXdt= u�X(t), t�(1)for the translational motion of the position vector X(t),anddNdt= −�N ×∇��N · u�×N (2)for the rotational motion of the normal vector N(t).Here, u(x, t) denotes the ﬂuid velocity at position xand time t. Note that (2) indicates that the temporalevolution of ﬂake orientations is identical to that of theinﬁnitesimal material surface elements.Then, we may numerically reproduce the labora-tory visualization (Fig. 1a) by tracking ﬂakes accordingto (1) and (2) in the steady ﬂow which is simulated bya spectral method in the precisely same ﬂow conditionsas in the experiment. Since (1) indicates that the spa-tial distribution of ﬂakes in an incompressible ﬂuid isuniform if it is initially so, the pattern observed in ﬂakevisualizations stems from their non-uniform orientations.Therefore, ﬂake orientations at each point on the lasersheet are numerically determined to simulate the inten-sity of the observed light based on the probability forﬂakes to reﬂect the incident rays to the observer. Ob-tained result is shown in Fig. 1(b), which is in excellentagreement with the laboratory result in Fig. 1(a). Wemay, therefore, conclude that the motion of ﬂakes is welldescribed by (1) and (2).By further detailed numerical investigations, it isshown that ﬂake orientations are isotropic on the threeconcentric bright circles in Fig. 1, whereas those oﬀ thecircles are strongly anisotropic. Hence, the three circlesare always bright irrespective of the angles of incidentrays and perspective, but the observed light intensity inthe latter regions depends on the angles. It is conﬁrmedboth by laboratory experiments and numerical simula-tions that observed patterns are indeed diﬀerent for dif-ferent angles of the incident rays. This result implies thata visualized image by a single pair of incident rays andperspective cannot distinguish these essentially diﬀerentregions, and we need to use multiple light sources or per-spectives to determine ﬂake orientations. The possibilityof the identiﬁcation of ﬂow structures, especially coher-ent structures in turbulence, based on thus determinedﬂake orientations is under investigation.(a)(b)Fig. 1: Flake visualization of a steady ﬂow in the pre-cessing sphere (Fig. 2). (a) Laboratory experiment. (b)Numerical simulation based on (1) and (2).ΩsSpin axisΩp Precession axisFig. 2: Precessing sphere.1) Goto, S., Kida, S., and Fujiwara, S.: J. Fluid Mech.(2011) in press.375§3. Multi-Hierarchy Simulation of Collisionless Driven Reconnection  in an Open SystemUsami, S., Ohtani, H., Horiuchi, R.,  Den, M. (NICT)

Multi-Hierarchy Simulation of Collisionless Driven Reconnection in an Open System

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Multi-Hierarchy Simulation of Collisionless Driven Reconnection in an Open System

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