Spiral Defect Chaos and the Skew-Varicose Instability inGeneralizations of the Swift-Hohenberg Equation

Mean flows are known to play an important role in the dynamics of the Spiral Defect Chaos state and in the existence of the skew-varicose instability in Rayleighâ��Bernard Convection. SDC only happens in large domains, so computations involving the full three-dimensional PDEs for convection are very time-consuming. We therefore explore the phenomena of Spiral Defect Chaos and the skew-varicose instability in Generalized Swiftâ��Hohenberg (GSH) models that include the effects of long-range mean flows. Our analysis is aimed at linking the two phenomena. We apply analytical and numerical methods to study the linear stability of stripe patterns in two generalizations of the two-dimensional Swiftâ��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. Close to the onset of pattern formation, the skew varicose instability has the same dependence on wave number as the Eckhaus instability provided the coupling to the mean flow is greater than a critical value. We 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 cross-over from the behaviour characteristic of no-slip to that of stress-free boundaries as the coupling to the mean flow increases or as the Prandtl number decreases. The region of stable stripes is completely eliminated by the cross-roll instability for large coupling to the mean flow or small Prandtl number. We characterize the nonlinear evolution of the modes that are responsible for the skew varicose instability in order to understand whether the bifurcation from stable stripes at the skew-varicose instability is supercritical or subcritical. The systems of ODEs, which are derived from the PDEs by selecting 3 relevant modes and truncating, show that the skew-varicose instability is supercritical whereas for an extension with 5 relevant modes shows the skew-varicose instability is subcritical. We solve the PDEs of one GSH model in spatially-extended domains for very long times, much longer than previous efforts in the literature. We are able to investigate the influence of domain size and other parameters much more systematically, and to develop a criterion for when the spiral defect chaos state could be expected to persist in the long time limit. The importance of the mean flow can be adjusted via the Prandtl number or parameter that accounts for the fluid boundary conditions on the horizontal surfaces in a convecting layer and hence we establish a relation between these parameters that preserves the same pattern. We further analyze the onset of chaotic state, and its dependence on the Prandtl number and the domain size. An outstanding issue in the understanding of SDC is that it exists at parameter values where simple straight roll convection is also stable, and the region of co-existence increases as the domain size increases. The results of our numerical simulations are coupled with the analysis of the skew-varicose instability of the straight-roll pattern in the Generalized Swiftâ��Hohenberg equation, allowing us to identify the role that skew-varicose events in local patches of stripes play in maintaining Spiral Defect Chaos.