DIRECTIONAL SOLIDIFICATION AT HIGH-SPEED .1. SECONDARY INSTABILITIES

We present an extensive analytical and numerical analysis of secondary instabilities in directional solidification in the limit of high spee d, which is by now accessible in real experiments. The important featu re in this regime is that front dynamics are quasilocal. From symmetry and scaling arguments, we write down the general form of the nonlinea r equation for the interface, an equation to which the present study p ertains. In order to determine the values of the coefficients, a deriv ation from the fully nonlocal model was performed. We consider the gen eral case where mass diffusion is allowed in both phases, and its spec ial restrictions to the one-sided model (appropriate for regular mater ials), and the symmetric one (appropriate for liquid crystals). We fir st focus on the appearance of cellular structures (primary instability ). In the symmetric case the structures are rather shallow, in accord with experiments. In the one-sided model, the front generically develo ps, in a certain region of parameter space, cusp singularities. These can be avoided by allowing a small amount of diffusion in the growing phase; the front then reaches a stationary state. Stationary states ar e in turn subject to instabilities (secondary instabilities). Besides the Eckhaus instability, we find parity-breaking (PB), vacillating-bre athing (VB), and period-halving (PH) bifurcations, regardless of the d etails of the model, a fact which points to their genericity. Another line of research developed in this paper is the analytical analysis of these bifurcations. The PB and PH bifurcations are analyzed close to the codimension-two bifurcation point where the first and second harmo nics are dangerous. The results emerging from this analysis are suppor ted by the full numerics. The VB mode is analyzed analytically by mean s of an analogy with the problem of a quasi-free-electron in a crystal . Finally we discuss some questions beyond secondary instabilities. We find that this system exhibits an anomalous growth mode, observed in many systems. Among other pertinent features, we find that the broken- parity (BP) state is subject to a long-wavelength instability, causing a fragmentation of the extended state. This provides a signature of i ts ''solitarylike'' persistence observed in many experiments. Another important dynamical characteristic is that on increase of the growth s peed the VB mode suffers a PB instability, and acquires a quasiperiodi c motion (mixture of BP and VB modes which are incommensurate), which constitutes a prelude to a chaotic regime, discussed in the companion paper.