Volume 576, April 2015
|Number of page(s)||12|
|Published online||13 March 2015|
When the general theory of reconnection was developed, an elegant derivation of the reconnection rate was presented that makes use of an Euler potential representation of the magnetic field (Hesse & Schindler 1988; Schindler 2007). Here we present an alternative derivation, based on Cartesian tensors, for resistive MHD. Consider magnetic field lines passing through a non-ideal region . Inside this region, the resistivity η ≠ 0. Outside is an ideal plasma, where η = 0. Consider a closed path that passes though parallel to a magnetic field line, as shown in Fig. A.1.
Magnetic field lines passing through a non-ideal region . A closed path, shown in red, defines the boundary of a surface . In , the path is parallel to the magnetic field. The section of the path inside is shown in green. In the text, this is referred to as Γ.
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Let denote the surface bounded by the closed path. The rate of reconnection is the rate of change of flux Φ through the surface where n is the unit normal to the surface . In order to differentiate the integral with respect to time, one must move from an Eulerian frame to a Lagrangian one. This can be achieved via an application of Nanson’s formula to give Here, geometric quantities that are now written in captials are relative to a Lagrangian frame with surface , e.g. n is Eulerian and N is Lagrangian. F = ∂x/∂X is the deformation gradient, a second order Cartesian tensor relating the Lagrangian and Eulerian frames. Associated with this is J = det(F). In the last integral, dots over terms represent differentiation with respect to time. It can be shown that where L = ∂u/∂x. Here, u is the Eulerian velocity, making L an entirely Eulerian tensor. It can also be shown that by differentiating the expression FF-1 = I. Collecting these results together, it follows that By expressing the integrand in terms of vectors and making use of the resistive induction equation, one finds where η = η(x) is the resistivity. By an application of Stokes’ theorem, one can show that Taking the dot product of a unit vector with the resistive Ohm’s law gives where η = (μ0σ)-1 with conductivity σ = σ(x) and (constant) magnetic permeability μ0. As η = 0 outside , the integration only gives a non-zero value along the section of the path inside (the green path in Fig. A.1). If this section is labelled Γ, it follows that It then follows that The choice of field line, and hence path, taken through was arbitrary. Therefore, it is common to choose the field line that returns the largest magnitude. Since, in this work, we are integrating along field lines that pass through points connecting all four flux systems (four-colour points), the reconnection rate ℛ(t) is taken to be Here the integration is along the field line that gives the largest magnitude for the integrated parallel electric field. is the set of field lines that pass through four-colour points, and varies in time. Γ is the path along the field line in the current sheet.
© ESO, 2015
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