Abstract
We explain two puzzling aspects of Petschek's model for fast reconnection. One is its failure to occur in plasma simulations with uniform resistivity. The other is its inability to provide anything more than an upper limit for the reconnection rate. We have found that previously published analytical solutions based on Petschek's model are structurally unstable if the electrical resistivity is uniform. The structural instability is associated with the presence of an essential singularity at the Xline that is unphysical. By requiring that such a singularity does not exist, we obtain a formula that predicts a specific rate of reconnection. For uniform resistivity, reconnection can only occur at the slow, SweetParker rate. For nonuniform resistivity, reconnection can occur at a much faster rate provided that the resistivity profile is not too flat near the Xline. If this condition is satisfied, then the scale length of the nonuniformity determines the reconnection rate.
This work was supported by NSF Grants ATM0734032 and AGS0962698, NASA Grants NNX08AG44G and NNX10AC04G to the University of New Hampshire, and subcontract SVT7702 from the Smithsonian Astrophysical Observatory in support of their NASA Grants NNM07AA02C and NNM07AB07C. D. B. Seaton was supported by PRODEX Grant C90193 managed by the European Space Agency in collaboration with the Belgian Federal Science Policy Office, and by Grant FP7/20072013 from the European Commission's Seventh Framework Program under the agreement eHeroes (Project No. 284461). Additional support was provided by the Leverhulme Trust to E. R. Priest.
I. INTRODUCTION
II. AVERAGED RESISTIVEMHDEQUATIONS
III. STEADYSTATE SOLUTIONS
IV. TIMEDEPENDENT SOLUTIONS
V. IMPROVED DIFFUSION REGION MODEL
VI. CONCLUSIONS
Key Topics
 Electrical resistivity
 20.0
 Diffusion
 16.0
 Numerical solutions
 15.0
 Magnetohydrodynamics
 11.0
 Solution processes
 10.0
Figures
Schematic diagram of the upper right quarter of the field and flow configuration. B and b indicate the x and ycomponents of the magnetic field (dashed curves) and V and u the x and ycomponents of the bulk flow (hollow arrows). The shaded area indicates the expected current density distribution. The parameter α is the nominal length of the diffusion region beyond which the current density is bifurcated into a pair of slowmode shocks. L is the total length of the current layer including both diffusion and shock regions.
Schematic diagram of the upper right quarter of the field and flow configuration. B and b indicate the x and ycomponents of the magnetic field (dashed curves) and V and u the x and ycomponents of the bulk flow (hollow arrows). The shaded area indicates the expected current density distribution. The parameter α is the nominal length of the diffusion region beyond which the current density is bifurcated into a pair of slowmode shocks. L is the total length of the current layer including both diffusion and shock regions.
Outflow velocity V prescribed by (27) for different values of the integration constant. Solutions are shown for η = η 0 (uniform resistivity) and α = 1 (SweetParker reconnection). Only the solution passing through the origin corresponds to a reconnectiontype scenario.
Outflow velocity V prescribed by (27) for different values of the integration constant. Solutions are shown for η = η 0 (uniform resistivity) and α = 1 (SweetParker reconnection). Only the solution passing through the origin corresponds to a reconnectiontype scenario.
The outflow velocity V, current layer thickness a, and transverse field component b as functions of the distance, x, along the current layer. The neutral line is at x = 0 and the tip of the current layer (including slow shocks) is at x = 1. The vertical dashed line at x = α indicates the tip of the diffusion region. This particular solution is the only nonsingular solution of the nozzle Equation (27) when the magnetic diffusivity has a Gaussian profile with scale length lg = 0.1.
The outflow velocity V, current layer thickness a, and transverse field component b as functions of the distance, x, along the current layer. The neutral line is at x = 0 and the tip of the current layer (including slow shocks) is at x = 1. The vertical dashed line at x = α indicates the tip of the diffusion region. This particular solution is the only nonsingular solution of the nozzle Equation (27) when the magnetic diffusivity has a Gaussian profile with scale length lg = 0.1.
This figure shows the transition from a nonequilibrium initial state to the nonsingular steady state for a uniform external field (Ba = −1) and a Gaussian diffusivity profile with the scale length lg = 0.25 in normalized units. Panels (a), (b), and (c) show the layer thickness, a, the outflow velocity, V, and the transverse magnetic field, b, respectively, as functions of the distance, x, and the time, t. By t = 5 the numerical solution matches the nonsingular steadystate solution to an accuracy of 10−6.
This figure shows the transition from a nonequilibrium initial state to the nonsingular steady state for a uniform external field (Ba = −1) and a Gaussian diffusivity profile with the scale length lg = 0.25 in normalized units. Panels (a), (b), and (c) show the layer thickness, a, the outflow velocity, V, and the transverse magnetic field, b, respectively, as functions of the distance, x, and the time, t. By t = 5 the numerical solution matches the nonsingular steadystate solution to an accuracy of 10−6.
Evolution of the reconnection rate, MA , for two different singular, steadystate solutions. For these solutions, the diffusivity is nonuniform and has a Gaussian profile with a scale length lg = 0.1. The time t is normalized to the Alfvén scale time. Both solutions rapidly evolve towards the reconnection rate of MA = 0.0279 of the nonsingular steadystate solution (dashed line). Here, η 0 = 10−4 and Ba = −1.
Evolution of the reconnection rate, MA , for two different singular, steadystate solutions. For these solutions, the diffusivity is nonuniform and has a Gaussian profile with a scale length lg = 0.1. The time t is normalized to the Alfvén scale time. Both solutions rapidly evolve towards the reconnection rate of MA = 0.0279 of the nonsingular steadystate solution (dashed line). Here, η 0 = 10−4 and Ba = −1.
Evolution of the reconnection rate, MA , and the diffusion region length, α, for a steadystate solution with a uniform diffusivity profile and a uniform Ba . The unstable initial configuration does not reach a new equilibrium. Instead, it approaches the longterm decay behavior of the selfsimilar solution (dashed lines). Here, η 0 = 10−4.
Evolution of the reconnection rate, MA , and the diffusion region length, α, for a steadystate solution with a uniform diffusivity profile and a uniform Ba . The unstable initial configuration does not reach a new equilibrium. Instead, it approaches the longterm decay behavior of the selfsimilar solution (dashed lines). Here, η 0 = 10−4.
These curves show the decay of the transverse magnetic field that occurs when a singular, steadystate equilibrium with uniform resistivity is smoothed. The parameter b 1 is the spatial gradient of the transverse magnetic field at x = 0. The dashed curve labeled a 2 = 0 corresponds to the case considered by Malyshkin and Kulsrud where the infinite series expansion for the solution is truncated by assuming that ∂ 2 a/∂x 2 = 0 at x = 0. The solid curve labeled a 10 = 0 shows the decay when the initial state is truncated at the 10th order by setting ∂ 10 a/∂x 10 = 0 at x = 0. For both cases, η 0 = 10−4 and Ba = −1.
These curves show the decay of the transverse magnetic field that occurs when a singular, steadystate equilibrium with uniform resistivity is smoothed. The parameter b 1 is the spatial gradient of the transverse magnetic field at x = 0. The dashed curve labeled a 2 = 0 corresponds to the case considered by Malyshkin and Kulsrud where the infinite series expansion for the solution is truncated by assuming that ∂ 2 a/∂x 2 = 0 at x = 0. The solid curve labeled a 10 = 0 shows the decay when the initial state is truncated at the 10th order by setting ∂ 10 a/∂x 10 = 0 at x = 0. For both cases, η 0 = 10−4 and Ba = −1.
Characteristic paths in the center of the diffusion region for uniform η and Ba . Only the single characteristic at x = 0 connects the steadystate solution at t = ∞ with its initial conditions.
Characteristic paths in the center of the diffusion region for uniform η and Ba . Only the single characteristic at x = 0 connects the steadystate solution at t = ∞ with its initial conditions.
Plot of solutions to (52) assuming uniform resistivity. Solutions are shown for α = 1. The critical point for the flow to escape at the Alfvén speed is located at x = 0.505 and V = 0.236. The solution passing through the critical point is structurally unstable.
Plot of solutions to (52) assuming uniform resistivity. Solutions are shown for α = 1. The critical point for the flow to escape at the Alfvén speed is located at x = 0.505 and V = 0.236. The solution passing through the critical point is structurally unstable.
Comparison of the iterated external fields and (solid curves) with the initial and the initial solution for (dashed curves). The dotted curve shows Somov–Titov analytical approximation for . The values of η 0, lg , and MA are the same as for Figure 3 .
Comparison of the iterated external fields and (solid curves) with the initial and the initial solution for (dashed curves). The dotted curve shows Somov–Titov analytical approximation for . The values of η 0, lg , and MA are the same as for Figure 3 .
Tables
Diffusion region length α for different levels of approximation.
Diffusion region length α for different levels of approximation.
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