Two dimensional ideal incompressible MHD turbulence is studied in a Lagrangian framework in which the non-local pressure forces are approximated to be locally isotropic. Within this model current sheet formation at null points and along field lines is observed. In the case of field lines, this behavior is found to be robust under addition of random pressure fluctuations from isotropy. The value of a quantity called the fluid balance of strain is found to be important in predicting where and when current sheet formation occurs.