We examine the effect of weak, small-scale magnetic field structure on the rate of reconnection in a strongly magnetized plasma. This affects the rate of reconnection by reducing the transverse scale for reconnection flows and by allowing many independent flux reconnection events to occur simultaneously. Allowing only for the first effect and using Goldreich & Sridhar's model of strong turbulence in a magnetized plasma with negligible intermittency, we find a lower limit for the reconnection speed ~VA-3/16L3/4, where VA is the Alfven speed, L is the Lundquist number, and is the large-scale magnetic Mach number of the turbulence. We derive an upper limit of ~VA2 by invoking both effects. We argue that generic reconnection in turbulent plasmas will normally occur at close to this upper limit. The fraction of magnetic energy that goes directly into electron heating scales as -2/5L8/5, and the thickness of the current sheet scales as -3/5L-2/5. A significant fraction of the magnetic energy goes into high-frequency Alfven waves. The angle between adjacent field lines on the same side of the reconnection layer is ~-1/5L6/5 on the scale of the current sheet thickness. We claim that the qualitative sense of these conclusions, that reconnection is fast even though current sheets are narrow, is almost independent of the local physics of reconnection and the nature of the turbulent cascade. As the consequence of this the Galactic and solar dynamos are generically fast, i.e., do not depend on the plasma resistivity.

https://iopscience.iop.org/article/10.1086/307233/pdf

Reconnection in a weakly stochastic field

Model of the High-Energy Phase of Solar Flares

Current sheets formed in magnetic reconnection events are found to be unstable to high-wavenumber perturbations. The instability is very fast: its maximum growth rate scales as S1∕4vA∕LCS, where LCS is the length of the sheet, vA the Alfven speed, and S the Lundquist number. As a result, a chain of plasmoids (secondary islands) is formed, whose number scales as S3∕8.

Instability of current sheets and formation of plasmoid chains

A general picture of magnetic reconnection in the framework of 2‐D incompressible resistive magnetohydrodynamic theory is presented. Numerical studies of (quasi‐) steady‐state driven reconnection reveal current sheet formation for Mach numbers M=u/vA exceeding the Sweet–Parker reconnection rate MSP=(η/LvA)1/2. Since the thickness δ of the current sheet is found to be invariant to a change of the resistivity η, its length Δ increases rapidly with decreasing η or increasing M, which can be written in the form Δ∼(M/MSP)4, so that Δ reaches the global system size L within a short range of the parameter M/MSP. The results are rather insensitive to the particular choice of boundary conditions. Because of the presence of a current sheet, the overall reconnection process is quite slow. This picture essentially agrees with Syrovatsky’s [Sov. Phys. JETP 33, 933 (1971)] theory and disproves Petschek’s [AAS/NASA Symposium on the Physics of Solar Flares, (NASA, Washington, DC, 1964) p. 425] mechanism of fast magnetic reconnection. A theory of the solution in the external and in the diffusion region is developed and analytical expressions in agreement with the simulation results are obtained by means of a variational principle. For sufficiently long current sheets the tearing mode becomes unstable in spite of the stabilizing effect of the inhomogeneous flow. The tearing mode contributes to the overall reconnection process, but a general assessment of this effect in the asymptotic regime of almost vanishing η is difficult.

Magnetic Reconnection via Current Sheets

This paper provides a short review of some of the basic concepts related to the origin of coronal mass ejections (CMEs). The various ideas which have been put forward to explain the initiation of CMEs are categorized in terms of whether they are force-free or non-force-free and ideal or nonideal. A few representative models of each category are examined to illustrate the principles involved. At the present time there is no model which is sufficiently developed to aid forecasters in their efforts to predict CMEs, but given the current pace of research, this situation could improve dramatically in the near future.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2000JA000005

A review on the genesis of coronal mass ejections

The stability of a plane current layer is analyzed in the hydromagnetic approximation, allowing for finite isotropic resistivity. The effect of a small layer curvature is simulated by a gravitational field. In an incompressible fluid, there can be three basic types of ``resistive'' instability: a long‐wave ``tearing'' mode, corresponding to breakup of the layer along current‐flow lines; a short‐wave ``rippling'' mode, due to the flow of current across the resistivity gradients of the layer; and a low‐g gravitational interchange mode that grows in spite of finite magnetic shear. The time scale is set by the resistive diffusion time τR and the hydromagnetic transit time τH of the layer. For large S = τR/τH, the growth rate of the ``tearing'' and ``rippling'' modes is of order τR−3/5τH−2/5, and that of the gravitational mode is of order τR−1/3τH−2/3. As S → ∞, the gravitational effect dominates and may be used to stabilize the two nongravitational modes. If the zero‐order configuration is in equilibrium, the...

Finite‐Resistivity Instabilities of a Sheet Pinch

The distortion of a magnetic well in a minimum‐B system by a plasma is calculated. The equations of hydromagnetic equilibria with a tensor pressure are used. The pressure tensor is assumed to be a function of B only. The axisymmetric case is considered in detail by using a stream function for the magnetic field. The resulting partial differential equation is solved by finite‐difference methods. Numerical solutions are presented for the stuffed‐cusp configuration. The effect of increasing the magnitude of the plasma pressure is discussed.

Computation of Hydromagnetic Equilibria with Finite Pressure in Minimum-B Systems

A study is made of various processes that occur prior to the pinch effect when an electric field is applied to a deuterium gas. The variables of the problem are the percentage of ionization, the electron and ion temperatures, the resistivity of the gas, and the current density. A one‐dimensional problem is considered in which the above variables are determined as functions of one‐space dimension and the time. The equations determining these variables are: the heat developed as the current flows through the gas equals the rate of increase of the internal energy of the plasma, the equation describing the rate of ionization, the equation describing the rate of transfer of energy from electrons to ions and the electromagnetic field equations. These equations are a generalization of the plasma equations solved by Wyld and Watson in that spatial dependence is included, i.e., a current layer is calculated instead of assuming a constant current. The equations are solved numerically using an IBM 704 computer.

/pdf/boundary-layer-formation-in-the-pinch-3376kj7q46.pdf

John Killeen

Papers

Finite‐Resistivity Instabilities of a Sheet Pinch

Computation of Hydromagnetic Equilibria with Finite Pressure in Minimum-B Systems

Boundary‐Layer Formation in the Pinch