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-P\'eclet-number limit of directional solidification. The growth rate {Ω} for perturbations of wave number q is found to have two branches. The first branch Ω₁(q) is stable for all wave numbers. The second branch is of the diffusive form Ω₂(q)=-Bq², where B is proportional to the derivative d{Δ}T/d{Λ} of the thermal undercooling {Δ}T 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.