Pattern selection in directional solidification

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Abstract
The equations of motion for a slightly deformed planar solidification front advancing with a steady velocity and stabilized by a thermal gradient are integrated numerically. The regime where the interface develops a cellular pattern is analyzed. We study the case where the system is subjected to Gaussian white noise. The wave number is found to change in a discontinuous way, through strongly nonlinear events which either create or destroy one cell at a time. Ultimately, the process appears to lead toward a unique final wave number. The structure of the equations is examined with respect to the wave-number dependence and the selection phenomenon observed in the computer simulations is understood phenomenologically on the basis of a phase-space type of argument.