Schematic diagram of the upper right quarter of the field and flow configuration. B and b indicate the x- and y-components of the magnetic field (dashed curves) and V and u the x- and y-components 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 slow-mode 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 (Sweet-Parker reconnection). Only the solution passing through the origin corresponds to a reconnection-type 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.
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 steady-state solution to an accuracy of 10−6.
Evolution of the reconnection rate, MA , for two different singular, steady-state 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 steady-state 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 steady-state solution with a uniform diffusivity profile and a uniform Ba . The unstable initial configuration does not reach a new equilibrium. Instead, it approaches the long-term decay behavior of the self-similar solution (dashed lines). Here, η 0 = 10−4.
These curves show the decay of the transverse magnetic field that occurs when a singular, steady-state 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 steady-state 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.
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 .
Diffusion region length α for different levels of approximation.
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