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Weakly nonlinear morphological instability of a cylindrical crystal growing from a pure undercooled melt.

We develop a weakly nonlinear morphological stability analysis for a sphere growing from its pure undercooled melt. For a sphere perturbed by a specific planform (a single spherical harmonic) we perform an expansion in the planform amplitude, A, to calculate the nonlinear critical radius (above which the chosen planform will be unstable for finite A), to lowest order in A, by setting the normal velocity corresponding to the fundamental perturbing mode to zero. We study the nonlinear critical radius as a function of the amplitude to identify the various bifurcations, which are transcritical (requiring an expansion to second order in A) for planforms generating physically distinct shapes for positive and negative amplitudes, and subcritical or supercritical (requiring an expansion to third order in A) for planforms for which the positive and negative amplitude shapes are related by rotation or translation. Bifurcations that are not transcritical are subcritical, except for a few harmonics of lower order or extreme ratios of the thermal conductivity in the solid to that in the liquid. We also treat the anomalous case of a perturbation by a first-order spherical harmonic (which is neutrally stable according to the linear theory, corresponding to a translation without any shape change) and observe that the second harmonic becomes unstable before the perturbing mode itself.