A strong limitation on the rapidity of flux-pile-up reconnection

The reconnection rate which can be achieved in the steady-state flux-pile-up regime is severely limited by the gas pressure of the plasma. Using the family of solutions obtained previously by Priest and Forbes, we show that the Alfven Mach number of the plasma flowing towards the reconnection site cannot exceed pi beta(e)/(8 ln R(me)), where beta(e) and R(me) are the plasma beta and magnetic Reynolds numbers at large distance. This limit corresponds to a very weak flux-pile-up, and it is a factor of beta(e) slower than the maximum Petschek rate. Thus the maximum flux-pile-up reconnection rate in the corona is at least two orders of magnitude smaller than the rate implied by hare observations.