The Dynamics of Patterns: A Survey

Abstract

A survey of the derivation and properties of the phase and amplitude equations in systems far from equilibrium is given. We study both the near onset and far from onset cases. In the former, both the amplitude and phase are governed by partial differential equations and, for real boundary conditions, the effects of large scale mean drift fields are minimal. The most interesting new developments concern the role that modulational instabilities play when the dominant microscopic structures are travelling or standing waves. In particular, it is shown that localized and strongly disordered behavior can occur. The application of these ideas in the context of convection in binary fluid mixtures is discussed. Far from onset amplitudes are slaved through algebraic equations to the modulus of the phase gradient, but mean drift effects become more important. I discuss the role that defects and focus singularities of the phase equation play in completing one’s understanding of the pattern dynamics. In particular, I suggest that the structure and dynamics of defects can be found from singular (particle-like) solutions of the phase equation and illustrate this idea in two cases. First, I find the shape of the dislocation solution in high Prandtl number fluids. Second, I compute solutions which describe the breaking of the circular symmetry about a focus singularity in which the focus (umbilicus) is shifted off-center by a dipole shaped mean drift. Finally, building on ideas first proposed by Gollub, McCarriar and Steinman, I suggest a specific mechanism for the onset of turbulence in convecting fluids of low to moderate Prandtl number at Rayleigh numbers of about 4.5 R_c. The transition involves two features. First, the wavenumber selected by circular patches about sidewall focus singularities approaches the skew varicose instability boundary and the pattern attempts to eliminate rolls through defect nucleation. Second, and crucial to sustaining the time dependence, I show that the advection of wavenumber by mean drift overcomes the stabilizing effect of diffusion and causes the focus singularities to act as sources of wavenumber. It is the continual production of wavenumber and the resulting defect nucleation initiated by skew varicose instabilities which together lead to the turbulent behavior of the pattern.