On the linear stability of needle crystals : evolution of a Zel'dovich localized front deformation

Publisher Website
Google Scholar
Abstract
We derive from the linearized version of the dynamical integro-differential equation satisfied by small deformations of a 2-D needle-crystal the equations of evolution of a Zel'dovich wavepacket (a localized deformation starting from the tip region with a characteristic wavevector in the unstable region of the local planar spectrum). We define the conditions under which the « locally planar » type of description put forward by previous heuristic theories is valid. We find that the time evolution of the wavevector and spatial width of the deformation is primarily governed, not by the Zel'dovich stretching effect, but by a « differential amplification » effect due to spatial localization, which was up to now ignored. We show that, due to the parabolic shape of the basic front profile, the corresponding equations have an « asymptotically marginal » solution (the wavevector locks on the local marginal value, the amplitude saturates), thus strengthening the plausibility of the side-branching scenario based on transient amplification