Quadrupolar magnetic reconnection in solar flares. I. Three-dimensional geometry inferred from Yohkoh observations

We analyze the three-dimensional geometry of solar flares that show so-call ed interacting flare loops in soft X-ray, hard X-ray, and radio emission, a s previously identified by Hanaoka and Nishio. The two flare loops that app ear brightest after the flare are assumed to represent the outcome of a qua drupolar magnetic reconnection process, during which the connectivity of ma gnetic polarities is exchanged between the four loop footpoints. We paramet erize the three-dimensional geometry of the four involved magnetic field li nes with circular segments, additionally constrained by the geometric condi tion that the two pre-reconnection held lines have to intersect each other at the onset of the reconnection process, leading to a 10 parameter model. We fit this 10 parameter model to Yohkoh Soft and Hard X-Ray Telescopes (SX T and HXT) data of 10 solar flares and determine in this way the loop sizes and relative orientation of interacting held lines before and after reconn ection. We apply a flare model by Melrose to calculate the magnetic flux tr ansfer and energy released when two current-carrying field lines reconnect to form a new current-carrying system in a quadrupolar geometry. The findin gs and conclusions are the following. (1) The pre-reconnection held lines a lways show a strong asymmetry in size, consistent with the scenario of newl y emerging small-scale loops that reconnect with preexisting large-scale lo ops. (2) The relative angle between reconnecting field lines is nearly coll inear in half of the cases, and nearly perpendicular in the other half, con trary to the antiparallel configuration that is considered to be most effic ient for magnetic reconnection. (3) The angle between interacting field lin es is reduced by approximate to 10 degrees-50 degrees after quadrupolar rec onnection. (4) The small-scale flare loop experiences a shrinkage by a fact or of 1.31 +/- 0.44, which is consistent with the scaling law found from pr evious electron time-of-flight measurements, suggesting that electron accel eration occurs near the cusp of quadrupolar configurations. (5) The large-s cale loop is found to dominate the total induction between current-carrying loops, providing a simple estimate of the maximum magnetic energy availabl e for flare energy release because of current transfer, which scales as Del ta E-I approximate to 10(29.63)(r(2)/10(9) cm)(I-2/10(11) A)(2) (with r(2) the curvature radius and I-2 the current of the large-scale loop) and is fo und to correlate with observed flare energies deduced from soft X-ray and h ard X-ray fluxes. Most of the energy is transferred to small-scale loops th at have one-half of the large-scale current (I-1 = I-2/2). (6) The quadrupo lar reconnection geometry provides also a solution of Canfield's dilemma of the offset between the maximum of vertical currents and the HXR flare loop footpoints. (7) The quadrupolar geometry provides not only a framework for interacting double-loop flares, but it can also be considered as a general ized version of (cusp-shaped) single-loop flares.