1. The bioconvection of swimming microorganisms in a large shallow fluid layer. Click on the image for a larger version. A horizontal cross-section is shown at the top surface at an instant in time during the pattern evolution. The colors represent cell concentration where red is large and blue is small concentration. Red regions are the locations of falling plumes of microorganisms that have gathered at the surface and exceeded a critical density. The spatial structure of the pattern is very complex and the result of nonlinear interactions. (To reference this figure and for more details see Paul and Karimi, Physical Review E, 87, 053016, 2013 available here).
2. The long-time pattern after 106 time units from a numerical solution of the Generalized Swift-Hohenberg equation with the inclusion of mean flow. The color contours can be thought of as representing the temperature field of a fluid convection experiment where dark would be hot rising fluid and light cool falling. Click on the image for a larger version. For this simulation we have used Manneville’s model with c2=0.7, the order parameter ε = 0.7, the Prandtl number σ = 0.2, and the mean flow coupling strength gm = 50. This geometry would be equivalent to an aspect ratio Γ = 128 Rayleigh-Benard convection domain. This illustrates one great advantage of the Swift-Hohenberg system — it is still far beyond reach to simulate a Rayleigh-Benard convection domain with an aspect ratio of 128 for one million nondimensional time units! (To reference this figure and for more details see Karimi, Huang, and Paul, Physical Review E, 84, 046215, 2011 available here).
3. The spatial variation of the leading order Lyapunov vector can yield insights into local regions that are contributing significantly to the disorder present in the pattern dynamics. In this figure the time average of the magnitude of the leading order Lyapunov vector is shown as color contours for chaotic Rayleigh-Benard convection in a cylindrical domain. Red is large magnitude and blue is small magnitude. Click on the image for a larger version. For this simulation the aspect ratio Γ = 30, the Rayleigh number R = 2580, and the Prandtl number σ = 1. (To reference this figure and for more details see Karimi and Paul, Physical Review E, 85, 046201, 2012 available here).