Stability of an array of deep cells in directional solidification

We develop an effective interface formalism to perform the linear stability analysis of long-wavelength perturbations of a periodic array of fingerlike cells in the small-Péclet-number limit of directional solidification. The growth rate Ω for perturbations of wave number q is found to have two branches. The first branch ${\mathrm{\Omega }}_1$(q) is stable for all wave numbers. The second branch is of the diffusive form ${\mathrm{\Omega }}_2$(q)=-${\mathit{Bq}}^2$, where B is proportional to the derivative dΔT/dΛ of the thermal undercooling ΔT with respect to the cell spacing Λ. The point of marginal array stability in this system therefore coincides with the point of minimum undercooling in an analogous way to lamellar eutectics.