Fast, purely growing collisionless reconnection as an eigenfunction problem related to but not involving linear whistler waves
If either finite electron inertia or finite resistivity is included in 2D magnetic reconnection, the two-fluid equations become a pair of second-order differential equations coupling the out-of-plane magnetic field and vector potential to each other to form a fourth-order system. The coupling at an X-point is such that out-of-plane even-parity electric and odd-parity magnetic fields feed off each other to produce instability if the scale length on which the equilibrium magnetic field changes is less than the ion skin depth. The instability growth rate is given by an eigenvalue of the fourth-order system determined by boundary and symmetry conditions. The instability is a purely growing mode, not a wave, and has growth rate of the order of the whistler frequency. The spatial profile of both the out-of-plane electric and magnetic eigenfunctions consists of an inner concave region having extent of the order of the electron skin depth, an intermediate convex region having extent of the order of the equilibrium magnetic field scale length, and a concave outer exponentially decaying region. If finite electron inertia and resistivity are not included, the inner concave region does not exist and the coupled pair of equations reduces to a second-order differential equation having non-physical solutions at an X-point.
© 2014 AIP Publishing LLC. Received 3 August 2014; accepted 23 September 2014; published online 10 October 2014. This material is based upon work supported by the U.S. Department of Energy Office of Science, Office of Fusion Energy Sciences under Award Nos. DE-FG02-04ER54755 and DE-SC0010471, by the National Science Foundation under Award No. 1059519, and by the Air Force Office of Scientific Research under Award No. FA9550-11-1-0184.
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