MHD study for the adiabatic expansion acceleration process associated with the spontaneous fast magnetic reconnection is reported. When the fast reconnection process steadily generates a plasmoid in the downstream, the adiabatic expansion acceleration region appears between the reconnection jet and plasmoid. It is pointed out that the appearance of the acceleration region is required to steadily keep the reconnection process. The reconnection jet and plasmoid is generally high beta but the plasma pressure in the acceleration region is extremely low, when the reconnection jet is supersonic. This feature may become a signature to detect where the Petschek reconnection is steadily caused in the current sheet of the geomagnetotail. (c) 2005 COSPAR. Published by Elsevier Ltd. All rights reserved.