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Center for Nonlinear Studies |
Maxwell's equations imply that exponentially smaller non-ideal effects than commonly assumed can give rapid magnetic reconnection in space and astrophysical plasmas. An integration of the electric field parallel to a magnetic field line--from one connection to another--gives the field-line voltage V. When V=0 for every line, Faraday's law requires that magnetic field lines act as stretchable strings, which can become ever more entangled but cannot be cut. However, a high entanglement makes the field line connections exponentially sensitive to a small voltage V. The cause is well known in popular culture as the butterfly effect and in the theory of deterministic dynamical systems as a sensitive dependence on initial conditions. Nevertheless, the importance to magnetic reconnection is not generally recognized. Two-coordinate models are too constrained geometrically for the required entanglement. Otherwise, the effect is general and can be studied in a simple model that is periodic in two coordinates and bounded in the third. If a magnetic field is embedded in a plasma that is subjected to a force, the natural evolution of the field lines is to increasing entanglement when the voltage V can be approximated as zero. Increasing entanglement occurs for an arbitrarily smooth initial state and forcing provided the forcing is (1) sufficiently strong and (2) depends in a non-trivial way on all three spatial coordinates. The increasing field line entanglement makes the field line connections exponentially more sensitive to a non-zero voltage V until the rate of reconnection balances the rate of increasing entanglement. Even the smallest non-ideal effect, electron inertia, gives sufficient voltage for the required reconnection. The complicated spatial variation in the exponential sensitivity to non-ideal effects localizes the reconnection rather than the spatial variation in V. The energy density of the magnetic field is enhanced by the entanglement of the field lines, and this energy can be released at speeds up to the Alfvén speed once magnetic connections are broken by an exponentially small field-line voltage V. Physics of Plasmas 20, 032903 (2013) gives additional information and references. This work was supported in part by the U.S. Department of Energy Grant ER5433 to Columbia University.
Host: Xianzhu Tang, T-5