Barriers in the transition to global chaos in collisionless magnetic reconnection. I. Ridges of the finite time Lyapunov exponent field

Abstract

The transitional phase from local to global chaos in the magnetic field of a reconnecting current layer is investigated. Regions where the magnetic field is stochastic exist next to regions where the field is more regular. In regions between stochastic layers and between a stochastic layer and an island structure, the field of the finite time Lyapunov exponent (FTLE) shows a structure with ridges. These ridges, which are special gradient lines that are transverse to the direction of minimum curvature of this field, are approximate Lagrangian coherent structures (LCS) that act as barriers for the transport of field lines.

Coherent structures and the saturation of a nonlinear dynamo

“…Therefore, the maximum FTLE provides a measure of the exponential separation between two neighboring field lines after a finite field-line-time τ , i. e., after a finite distance along the field line. homoclinic and heteroclinic crossings of attracting and repelling LCSs, where a lobe dynamics mechanism takes place (Grasso et al 2010;Borgogno et al 2011;Yeates & Hornig 2011;Rempel et al 2012). Both FTLE fields are obtained by fixing the evolution (dynamic) time (t 0 = 100 for the left panel and t 0 = 1700 for the right panel) and setting τ = 9/B rms , where B rms = 0.014 for t 0 = 100 and B rms = 0.37 for t 0 = 1700.…”

“…6). If one applies the Lagrangian techniques discussed in the previous section to the magnetic field, the identification of magnetic LCSs provides the main barriers to the transport of field lines, a topic of great interest in magnetic reconnection studies (Evans et al 2004;Grasso et al 2010;Borgogno et al 2011;Yeates & Hornig 2011). To obtain the magnetic LCSs, the magnetic field at a fixed dynamic time t 0 is used and the maximum FTLE field is computed by integrating…”

“…where the parameter (position) s along the field line is seen as an effective time, or fieldline-time (Borgogno et al 2011). The flow map for B is defined as φ s0+τ s0 : x(s 0 ) → x(s 0 + τ ).…”

Barriers in the transition to global chaos in collisionless magnetic reconnection. II. Field line spectroscopy

“…However, no information can be extracted on the field-line-time a field line spends around these stochastic regions or on the field-line-time it takes to cross them when transition to global chaos occurs and no magnetic barriers exist anymore. In the first of these companion papers 1 we have shown that in regions between stochastic layers the field of the finite time Lyapunov exponent (FTLE) shows a structure with ridges. 1 There the ridges have been identified with barriers to field line motion by means of the invariant stable and unstable manifolds associated with distinguished hyperbolic trajectories.…”

“…In the first of these companion papers 1 we have shown that in regions between stochastic layers the field of the finite time Lyapunov exponent (FTLE) shows a structure with ridges. 1 There the ridges have been identified with barriers to field line motion by means of the invariant stable and unstable manifolds associated with distinguished hyperbolic trajectories. 2 Since the FTLE is a field line quantity, the ridges are approximate Lagrangian coherent structures (LCS's).…”