Directional solidification into static stability

Consider the directional solidification of a binary alloy rejecting a heavy solute as it solidifies upward. If the solidification front is planar, the fluid melt ahead of the front is stably stratified and convection is not expected. In this paper we analyse the linear stability of planar solidification asymptotically in the limit of large solutal Rayleigh number, R. Three distinct linear modes are found which correspond to internal waves,buoyancy edge waves, or morphological modes. Of these three modes, only the morphological modes are subject to an instability. We find that for large Rayleigh number this instability first occurs at long wavelengths with wavenumbers that scale on R(-1/14). The scalings derived from the linear analysis are used to construct a nonlinear theory for the morphological instability in the large Rayleigh number limit. Similarity solutions are found which describe steadily convecting, non-planar growth reminiscent of an observed phenomenon known as steepling.