Proceedings of ITC/ISHW2007

Intermittent dynamics of nonlinear resistive tearing modes at
extremely high magnetic Reynolds number
Takahiro MIYOSHI1) , Masahiro BECCHAKU1) and Kanya KUSANO2)
1)

2)

Hiroshima University, Higashi-hiroshima 739-8526, JapanThe Earth Simulator Center, Japan Agency for Marine-Earth Science and Technology, Yokohama 236-0001, Japan(Received / Accepted )Nonlinear dynamics of the resistive tearing instability in high magnetic Reynolds number (R m ) plasmas
is studied by newly developing an accurate and robust resistive magnetohydrodynamic (MHD) scheme. Theresults show that reconnection processes strongly depend on R m . Particularly, in a high R m case, small-scale
plasmoids induced by a secondary instability are intermittently generated and ejected accompanied by fast shocks.
According to the intermittent processes, the reconnection rate increases intermittently at a later nonlinear stage.
Keywords: tearing instability, high magnetic Reynolds number, MHD, simulation, secondary instability
DOI: 10.1585/pfr.1.001

The tearing instability is one of the most basic andimportant mechanism of plasma dynamics both in fusion
and astrophysical processes. Particularly, resistive tearing
modes have been well investigated theoretically and numerically as the most basic modes of the tearing instability[1, 2, 3]. However, our understanding of the nonlinear dynamics of resistive tearing instability in case of magnetic
Reynolds numbers, R m , as high as the practical systems isstill severely limited. A major reason of that is attributed to
the fact that the numerical resolution of simulations is toostrongly restricted to resolve a thin current sheet, which isbelieved to be formed in the realistic systems. Therefore,
in this study, an accurate and robust resistive MHD scheme
is developed, and nonlinear simulations at the highest-everresolution are carried out in order to find a new dynamic
regime of the resistive tearing instability.
First, we develop an accurate, eﬃcient, and robust numerical solver for resistive compressible magnetohydrodynamics (MHD). The Harten-Lax-van Leer-Discontinuities(HLLD) approximate Riemann solver [4], which is one ofthe promissing shock capturing solver for ideal MHD fromthe viewpoint of its resolution, robustness, and eﬃciency,
is applied to ideal terms of resistive MHD. Also, higherorder accuracy is achieved by the MUSCL method withlimiters. As a divergence cleaning method, hyperbolic divergence cleaning method [5] is adopted. Resistive terms,
on the other hand, are calculated by a classical centeredfinite diﬀerence method. In order to confirm the applicability of the present strategy to resistive MHD, severanl numerical tests are performed. As a first test, nonlinear simulations for the resistive tearing mode with a quite small
apmplitude are compared with the linear theory [1]. WhenRm is large enough (about more than 10 5 ), the results ofboth are almost corresponding. We also perform anotherauthor’s e-mail: miyoshi@sci.hiroshima-u.ac.jp

test of which both the present scheme and the (fully) centered finite diﬀerence scheme apply the nonlinear simulation of the tearing instability for a linear force free magnetic field. The results of both are almost the same at aninitial stage, while the centered finite diﬀerence scheme isbrokendown at a later stage. Thus, it is concluded that thepresent scheme achieves a high degree of the numerical accuracy and robustness even for resistive MHD processes.
Subsequently, nonlinear simulations of resistive tearing modes are performed in a simple 2D slab geometry
with uniform resistivity η (≡ R −1m ). The initial condition isgiven by the Harris equiribrium, B 0x = tanh(y/δ), where
the thickness of the initial current sheet δ is set to 0.5.
The simulation box is such that −12.8 ≤ x ≤ 12.8 and−6.4 ≤ y ≤ 6.4. The periodic boundary condition isapplied to x direction, while the symmetric condition isadopted for y boundary. In order to resolve the resistivelayer suﬃciently, non-uniform grid is adopted for y direction. The finest grid spacing is about 0.0102 for x direction and 0.0005 for y direction. In this paper, two casesof relatively low R m , Rm = 103 , and relatively high R m ,Rm = 104 , are presented in particular. The nonlinear tearing mode at R m = 103 steadily grows and is almost saturated as expected from the previous works. On the otherhand, it is found that at a high R m , Rm = 104 , a new dynamics arises after the formation of a thin current sheet atthe initial nonlinear stage. Fig. 1 show the current density and mass density distribution around the current sheet.
We find that many secondary plasmoids are intermittently
created in the thin current sheet and interacted with eachother. Through the nonlinear interaction of the plasmoids,
various fine structures associated with fast shocks are generated even in the uniform η model. Fig. 2 show the temporal evolution of the maximum electric field in the current
sheet induced by the resistivity at both R m cases. ThoughProceedings of ITC/ISHW2007

Fig. 1 Distributions of (a) the current density and (b) the density
at Rm = 104

Fig. 2 Time evolution of the maximum resistive electric field in
the current sheet at Rm = 103 (green line) and 104 (redline).

multiple X points are advected at R m = 104 , the resistive
electric field is almost considered as the reconnection rate.
It is found that the reconnection rate is much enhanced intermittently at Rm = 104 even though the linear growthrate of the resistive tearing instability is reduced with anincreasing of R m .
The results indicate that the nonlinear dynamics at ahigh Rm is much diﬀerent from our conventional understanding based on the linear theory and the simulations atmodest value of R m . It strongly suggests that the realistic
MHD dynamics at extremely high R m (e.g., more than 10 14in the solar corona!) is still veiled, and it is likely that somehierarchical MHD dynamics is involved to connect macroscale plasma evolution and micro-scale kinetic processes.
[1] H. P. Furth, J. Killeen, and M. N. Rosenbluth, Phys. Fluids
6, 459 (1963).
[2] P. H. Rutherford, Phys. Fluids 16, 1903 (1973).
[3] L.-N. Hau and S.-W. Chiou, J. Geophys. Res. 106, 8371
(2001).
[4] T. Miyoshi and K. Kusano, J. Comput. Phys. 208, 315(2005).
[5] A. Dedner, F. Kemm, D. Kröner, C.-D. Munz