Abstract: The directional solidification in the undercooled pure melt influenced by a transverse far field flow was studied by using the multiple scale method. The result shows that in the boundary layer near the liquid-solid interface, when affected by a transverse far field flow, the temperature distribution in the direction of crystal growth presents an oscillatory and decay front in the side of liquid phase. The crucial distinguishing feature of a temperature pattern due to the transverse convection is the additional periodic modulation of the pattern in the growth direction. The wave number and eigenvalue that satisfy the Mullins-Sekerka dispersion relation are suppressed by the transverse far field flow.