Convection in binary fluid mixtures. II. Localized traveling waves.

Abstract

Nonlinear, spatially localized structures of traveling convection rolls that are surrounded by quiescent fluid in horizontal layers of binary fluids heated from below are investigated in quantitative detail as a function of Rayleigh number for two different Soret coupling strengths (separation ratios) with Lewis and Prandtl numbers characterizing ethanol-water mixtures. A finite-difference method was used to solve the full hydrodynamic field equations numerically in a vertical cross section perpendicular to the roll axes subject to realistic horizontal and laterally periodic boundary conditions with different periodicity lengths. Structure and dynamics of these localized traveling waves (LTW's) are dominated by the concentration field. As in the spatially extended convective states that are investigated in an accompanying paper, the Soret-induced concentration variations strongly influence, via density changes, the buoyancy forces that drive convection. The spatiotemporal properties of this feedback mechanism, involving boundary layers and concentration plumes, show that LTW's are strongly nonlinear states. Light intensity distributions are determined that can be observed in side-view shadowgraphs done with horizontal light along the roll axes. Detailed analyses of all fields are made using color-coded isoplots, among others. In the frame comoving with their drift velocity, LTW's display a nontrivial spatiotemporal symmetry consisting of time translation by one-half an oscillation period combined with vertical reflection through the horizontal midplane of the layer. A time-averaged concentration current is driven by a phase difference between the waves of concentration and vertical velocity in the bulk of the LTW state. The associated large-scale concentration redistribution stabilizes the LTW and controls its drift velocity into the quiescent fluid by generating a buoyancy-reducing concentration "barrier" ahead of the leading LTW front. All considered LTW's drift very slowly into the direction of the phase velocity of the pattern. For weak Soret coupling, $\ensuremath{\psi}=\ensuremath{-}0.08$, LTW's have a small selected width and exist in a narrow band of Rayleigh numbers above the stability threshold for growth of TW's. For stronger coupling, $\ensuremath{\psi}=\ensuremath{-}0.25$, LTW's exist below the bifurcation threshold for extended TW's in a narrow band of Rayleigh numbers. In its lower part, LTW's have a small selected width. For somewhat higher Rayleigh numbers, there exist two LTW attractors with two different widths. For yet higher Rayleigh numbers, there is again only one LTW attractor; however, with a broader width. Dynamical properties and the dependence on the system length are analyzed. Comparisons with experiments are presented.