Exploring the role of long-range coupling on chaotic fluid flows using Lyapunov vectors*

We study how weak long-range spatial coupling affects the chaotic dynamics of fluid flows. We use the Generalized Swift-Hohenberg (GSH) equation to quantify the role of a weak mean flow, or wind, on high-dimensional chaotic dynamics. The GSH equation is a canonical pattern forming system that has provided important insights into the spiral defect chaos state of Rayleigh-B\'{e}nard convection. It has been shown that spiral defect chaos requires a mean flow, yet the physical mechanisms behind this interesting result are not completely understood. A central feature of the GSH equation is a mean flow whose magnitude can be continuously varied. We investigate how the mean flow magnitude affects the chaotic dynamics by computing Lyapunov vectors and Lyapunov exponents which describe the growth or decay of small perturbations to the dynamics. Our study is motivated, in part, by our recent findings of the significant influence of spatial coupling on the chaotic dynamics of lattices of coupled maps. We are interested in building upon our physical understanding of the dynamical influence of the mean flow using the Lyapunov based diagnostics over a broad range of mean flow magnitudes.