X-point collapse and saturation in the nonlinear tearing mode reconnection

.D .D .G .Haines, W estudy thenonlinearevolution oftheresistivetearing m odein slab geom etry in two dim ensions. W e show that,in the strongly driven regim e (large (cid:1) 0 ),a collapse ofthe X -pointoccurs once the island width exceedsa certain criticalvalue (cid:24) 1 = (cid:1) 0 .A currentsheetisform ed and thereconnection is exponentialin tim e with a growth rate / (cid:17) 1 = 2 ,where (cid:17) is the resistivity. Ifthe aspect ratio of the currentsheetissu(cid:14) ciently large,the sheetcan itselfbecom e tearing-m ode unstable,giving rise to secondary islands,which then coalesce with the originalisland.The saturated state dependson the value of(cid:1) 0 .Forsm all(cid:1) 0 ,the saturation am plitude is / (cid:1) 0 and quantitatively agreeswith the theoreticalprediction.If(cid:1) 0 islarge enough forthe X -pointcollapse