Oscillatory instability of deep cells in directional solidification

Coherent sidebranching of deep cells has been observed experimentally in the vicinity of the cell to dendrite transition. We propose that this phenomenon results from an oscillatory instability of deep cells, where cell widths and tip positions oscillate coherently in phase. We perform a linear-stability analysis of a periodic array of deep cells in the small-P\'eclet-number limit and derive the stability limits of the oscillatory mode.