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Self-similar evolution of fast magnetic reconnection in free space: a new model for astrophysical reconnection
en
新田 伸也
ニッタ シンヤ
NITTA Shin-ya
総研大甲第668号
We present a new model for time evolution of the fast magnetic reconnection in free space, which is characterized by self similarity. The possibility of this type of evolution is verified by numerical simulations. We also find an analytical solution which is consistent with the numerical result. In many cases of astrophysical problems, e.g., solar flares or geomagnetospheric substorms, the spatial scale of the reconnection system significantly expands as time proceeds. The focus of this work is on this expanding phase. The resultant spatial scale of the reconnection system is much larger than the initial scale (its dynamic range is 10 5-10 7 in order of magnitude). Thus, actual astrophysical reconnection must be treated as an evolutionary process in a free space which is free from any influence of external circumstances, at least in its expanding phase just after the onset. Eventually, the evolution will be strongly influenced by these external circumstances, and will settle into a final state. Even in this final state, we can expect the influence of the expanding phase will continue to affect the later evolution of the system. In spite of this, most previous numerical works focused on the character of evolution strongly affected by artificial boundary conditions on the simulation boundary. On the other hand, most analytical works focused on a stationary state of the reconnection as a boundary problem. However, we do not know how we should impose a well described boundary condition for these cases, because it is actually determined as a result of the evolutionary process of this expanding phase. Hence, the freely expanding phase is essential to our understanding of the properties of astrophysical magnetic reconnections. Our new model for magnetic reconnection alms to clarify a realistic evolution and spontaneous structure formation in free space. We assume the reconnection arising in an asymptotically uniform current sheet system (the Harris current sheet). The only fixed spatial scale in this system is the initial current sheet thickness, which is finite. Such a system probably has a self similar solution, because when the system sufficiently matures, there is no fixed proper spatial scale in the system other than the size of the expanding system itself. Thus, it is worthwhile to study the possibility of self-similar evolution of magnetic reconnection. We do this both numerically and analytically as outlined below. First we study it numerically, wishing to obtain evidence of self-similar evolution of the system. The reconnection is supposed to be triggered by artificially enhanced resistivity in the middle of the current sheet, which is held as a constant, independent of the time. This is a simplified model for anomalous resistivity. We were able to carry the computer simulation for the period while the system expanded by almost three orders of magnitude in the spatial scale and we succeeded in finding the expected self similar expansion of the system. The characteristic structure around the diffusion region is quite similar to the Petschek model which is characterized by a pair of slow-mode shocks and the fast-mode rarefaction-dominated inflow. In the outer region, a vortex-like return flow takes place driven by fast-mode compression caused by the piston effect of the reconnection jet takes place. The entire reconnection system expands self-similarly. However, owing to technical reasons in computer simulation, the dynamic range of the expansion in the spatial scale studied by our numerical simulation is not sufficient to constitute evidence that the obtained evolution is truly a self-similar one. In order to check this, we sought a self similar solution of the inflow region by an analytical study and compared the solution with our numerical result. By assuming that deviation owing to the reconnection from the initial equilibrium state is very small, we can analyze it with a perturbative method. This approximation is relevant for the inflow region to the original current sheet. We adopt a traditional mathematical method called the Grad-Shafranov approach. After a long derivation, we obtain several equations for the inflow region. One of them is a second order partial differential equation of the elliptical type for the magnetic flux function. We call it the Grad-Shafranov [GS] equation. Each of the other equations shows an algebraic relation between a physical quantity and the magnetic flux function. Thus, by solving the G S equation under a relevant boundary condition, we can obtain the distribution of magnetic flux function which shows the magnetic structure of the system. Once we obtain the magnetic flux function, we can easily derive the distributions of other quantities from other equations. The obtained solution for the inflow region is fairly consistent with our numerical solution. This analytical study confirms the existence of self-similar growth. On the other hand, numerical study by time-dependent computer simulation verifies the stability of the self-similar growth with respect to any MHD mode. Hence, these two approaches are complementary, and their results confirm the stable self-similar evolution of the fast magnetic reconnection system.
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