Abstract
Theoretical studies of the plasmoid instability generally assume that the reconnecting magnetic fields are symmetric. We relax this assumption by performing twodimensional resistive magnetohydrodynamic simulations of the plasmoid instability during asymmetric inflow magnetic reconnection. Magnetic asymmetry modifies the onset, scaling, and dynamics of this instability. Magnetic islands develop preferentially into the weak magnetic field upstream region. Outflow jets from individual Xpoints impact plasmoids obliquely rather than directly as in the symmetric case. Consequently, deposition of momentum by the outflow jets into the plasmoids is less efficient, the plasmoids develop net vorticity, and shear flow slows down secondary merging between islands. Secondary merging events have asymmetry along both the inflow and outflow directions. Downstream plasma is more turbulent in cases with magnetic asymmetry because islands are able to roll around each other after exiting the current sheet. As in the symmetric case, plasmoid formation facilitates faster reconnection for at least small and moderate magnetic asymmetries. However, when the upstream magnetic field strengths differ by a factor of 4, the reconnection rate plateaus at a lower value than expected from scaling the symmetric results. We perform a parameter study to investigate the onset of the plasmoid instability as a function of magnetic asymmetry and domain size. There exist domain sizes for which symmetric simulations are stable but asymmetric simulations are unstable, suggesting that moderate magnetic asymmetry is somewhat destabilizing. We discuss the implications for plasmoid and flux rope formation in solar eruptions, laboratory reconnection experiments, and space plasmas. The differences between symmetric and asymmetric simulations provide some hints regarding the nature of the threedimensional plasmoid instability.
The authors thank A. Bhattacharjee, P. A. Cassak, T. G. Forbes, L. Guo, Y.M. Huang, K. E. Korreck, N. F. Loureiro, M. Oka, J. C. Raymond, K. K. Reeves, S. L. Savage, L. S. Shepherd, C. R. Sovinec, and H. D. Winter for useful discussions. The authors thank members of NIMROD Team for ongoing code development that helped make this work possible. In particular, we thank J. King for suggesting memory management strategies that allowed larger simulations to be performed. Resources supporting this work were provided by the NASA HighEnd Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center. This research has benefited greatly from the use of NASA's Astrophysics Data Service.
This research was supported by NASA Grants NNX09AB17G, NNX11AB61G, and NNX12AB25G; NASA contract NNM07AB07C; and NSF SHINE Grant AGS1156076 to the Smithsonian Astrophysical Observatory. A.K.Y. acknowledges support from the NSFREU solar physics program at the Center for Astrophysics, Grant No. ATM0851866.
I. INTRODUCTION
II. NUMERICAL METHOD AND PROBLEM SETUP
III. SIMULATION RESULTS
A. Nonlinear dynamics
B. Reconnection rate
IV. ONSET OF INSTABILITY
V. OBSERVATIONAL CONSEQUENCES
A. Solar atmosphere
B. Laboratory experiments
C. Space plasmas
VI. DISCUSSION
Key Topics
 Magnetic reconnection
 32.0
 Magnetic fields
 26.0
 Vortex dynamics
 17.0
 Magnetic islands
 15.0
 Plasma instabilities
 12.0
Figures
Simulation of the plasmoid instability with symmetric inflow ( ) at t = 324. Shown are contours of the (a) magnetic flux; (b) outofplane current density, Jy ; (c) plasma density, ρ; (d) plasma pressure, p; (e) the outflow component of velocity, Vx ; (f) the inflow component of velocity, Vz ; and (g) the vorticity, . Xpoints are denoted by “×” and Opoints are denoted by “ ”. The green dots in panels (e) and (f) represent flow stagnation points.
Simulation of the plasmoid instability with symmetric inflow ( ) at t = 324. Shown are contours of the (a) magnetic flux; (b) outofplane current density, Jy ; (c) plasma density, ρ; (d) plasma pressure, p; (e) the outflow component of velocity, Vx ; (f) the inflow component of velocity, Vz ; and (g) the vorticity, . Xpoints are denoted by “×” and Opoints are denoted by “ ”. The green dots in panels (e) and (f) represent flow stagnation points.
Simulation of the plasmoid instability with asymmetric inflow ( ) at t = 731. Shown are contours of the (a) magnetic flux; (b) outofplane current density, Jy ; (c) plasma density, ρ; (d) plasma pressure, p; (e) the outflow component of velocity, Vx ; (f) the inflow component of velocity, Vz ; and (g) the vorticity, . Xpoints are denoted by “×” and Opoints are denoted by “ ”. The green dots in panels (e) and (f) represent flow stagnation points.
Simulation of the plasmoid instability with asymmetric inflow ( ) at t = 731. Shown are contours of the (a) magnetic flux; (b) outofplane current density, Jy ; (c) plasma density, ρ; (d) plasma pressure, p; (e) the outflow component of velocity, Vx ; (f) the inflow component of velocity, Vz ; and (g) the vorticity, . Xpoints are denoted by “×” and Opoints are denoted by “ ”. The green dots in panels (e) and (f) represent flow stagnation points.
Secondary merging between two plasmoids at t = 818 for and . Shown are (a) magnetic flux contours and velocity vectors in the frame of the moving Xpoint, , and (b) outofplane current density, Jy . The local flow pattern is dominated by shear flow associated with island vorticity. The longest vector corresponds to a velocity of 0.17 while the characteristic Alfvén speed in the islands is ∼0.1. The Xpoint is denoted by “×.”
Secondary merging between two plasmoids at t = 818 for and . Shown are (a) magnetic flux contours and velocity vectors in the frame of the moving Xpoint, , and (b) outofplane current density, Jy . The local flow pattern is dominated by shear flow associated with island vorticity. The longest vector corresponds to a velocity of 0.17 while the characteristic Alfvén speed in the islands is ∼0.1. The Xpoint is denoted by “×.”
The reconnection rate as a function of time for simulations with different asymmetries and domain sizes. The reconnection rate is given by the maximum outofplane electric field among all of the Xpoints in a simulation. The horizontal line segments indicate the reconnection rate predicted by Eq. (18) . The predictions for are not shown, but are factors of greater than the predictions for .
The reconnection rate as a function of time for simulations with different asymmetries and domain sizes. The reconnection rate is given by the maximum outofplane electric field among all of the Xpoints in a simulation. The horizontal line segments indicate the reconnection rate predicted by Eq. (18) . The predictions for are not shown, but are factors of greater than the predictions for .
Results from a parameter study to test the dependence of the initial magnetic asymmetry ratio, , and the halfsize of the computational domain along the outflow direction, x max, on the onset of the plasmoid instability. Simulations are classified as unstable if new Xpoints form in the current sheet before . Blue triangles indicate instability, while red diamonds indicate stability. The gray dotted lines are contours of constant initial hybrid Lundquist number, .
Results from a parameter study to test the dependence of the initial magnetic asymmetry ratio, , and the halfsize of the computational domain along the outflow direction, x max, on the onset of the plasmoid instability. Simulations are classified as unstable if new Xpoints form in the current sheet before . Blue triangles indicate instability, while red diamonds indicate stability. The gray dotted lines are contours of constant initial hybrid Lundquist number, .
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