Spiral-defect chaos: Swift-Hohenberg model versus Boussinesq equations

Exploring spiral defect chaos in generalized Swift-Hohenberg models with mean flow

We explore the phenomenon of spiral defect chaos in two types of generalized Swift-Hohenberg model equations that include the effects of long-range drift velocity or mean flow. We use spatially extended domains and integrate the equations for very long times to study the pattern dynamics as the magnitude of the mean flow is varied. The magnitude of the mean flow is adjusted via a real and continuous parameter that accounts for the fluid boundary conditions on the horizontal surfaces in a convecting layer. For weak values of the mean flow, we find that the patterns exhibit a slow coarsening to a state dominated by large and very slowly moving target defects. For strong enough mean flow, we identify the existence of spatiotemporal chaos, which is indicated by a positive leading-order Lyapunov exponent. We compare the spatial features of the mean flow field with that of Rayleigh-Bénard convection and quantify their differences in the neighborhood of spiral defects.

Skew-varicose instability in two-dimensional generalized Swift-Hohenberg equations

We apply analytical and numerical methods to study the linear stability of stripe patterns in two generalizations of the two-dimensional Swift-Hohenberg equation that include coupling to a mean flow. A projection operator is included in our models to allow exact stripe solutions. In the generalized models, stripes become unstable to the skew-varicose, oscillatory skew-varicose, and cross-roll instabilities, in addition to the usual Eckhaus and zigzag instabilities. We analytically derive stability boundaries for the skew-varicose instability in various cases, including several asymptotic limits. We also use numerical techniques to determine eigenvalues and hence stability boundaries of other instabilities. We extend our analysis to both stress-free and no-slip boundary conditions and we note a crossover from the behavior characteristic of no-slip to that of stress-free boundaries as the coupling to the mean flow increases or as the Prandtl number decreases. Close to the critical value of the bifurcation parameter, the skew-varicose instability has the same curvature as the Eckhaus instability provided the coupling to the mean flow is greater than a critical value. The region of stable stripes is completely eliminated by the cross-roll instability for large coupling to the mean flow.

Characterization of the domain chaos convection state by the largest Lyapunov exponent

Using numerical integrations of the Boussinesq equations in rotating cylindrical domains with realistic boundary conditions, we have computed the value of the largest Lyapunov exponent lambda1 for a variety of aspect ratios and driving strengths. We study in particular the domain chaos state, which bifurcates supercritically from the conducting fluid state and involves extended propagating fronts as well as point defects. We compare our results with those from Egolf, [Nature 404, 733 (2000)], who suggested that the value of lambda1 for the spiral defect chaos state of a convecting fluid was determined primarily by bursts of instability arising from short-lived, spatially localized dislocation nucleation events. We also show that the quantity lambda1 is not intensive for aspect ratios Gamma over the range 20<Gamma<40 and that the scaling exponent of lambda1 near onset is consistent with the value predicted by the amplitude equation formalism.

Mean flow and spiral defect chaos in Rayleigh-Bénard convection

We describe a numerical procedure to construct a modified velocity field that does not have any mean flow. Using this procedure, we present two results. First, we show that, in the absence of the mean flow, spiral defect chaos collapses to a stationary pattern comprising textures of stripes with angular bends. The quenched patterns are characterized by mean wave numbers that approach those uniquely selected by focus-type singularities, which, in the absence of the mean flow, lie at the zigzag instability boundary. The quenched patterns also have larger correlation lengths and are comprised of rolls with less curvature. Secondly, we describe how the mean flow can contribute to the commonly observed phenomenon of rolls terminating perpendicularly into lateral walls. We show that, in the absence of the mean flow, rolls begin to terminate into lateral walls at an oblique angle. This obliqueness increases with the Rayleigh number.