I study the dynamics of turbulent Plane Couette flow, the motion of fluid in a rectangular box like the one above. The box has solid walls on the top and bottom. These walls slide at constant speeds in opposite directions: the top wall moves away from the viewer, the bottom towards. This shearing motion drives the fluid. If the walls move fast enough, the flow becomes turbulent.
The color in the picture above indicates the speed of the fluid in the direction of the moving walls. Red indicates fluid being dragged away from the viewer by the top wall; blue, towards. The top wall and the upper half of the fluid are cut away to show what happens inside.
Just above the onset of turbulence, the flow follows characteristic patterns of behavior. The most important patterns are counter-rotating rolls that stretch along the direction of the wall motion, and whose circular faces are visible in the front plane. The rolls circulate high- and low-speed flow from the walls towards the center, causing alternating, wobbly streaks of high (red) and low (blue) fluid. As time progresses, the roll-streak structures appear (in isolation or in roughly regular arrays), wobble, 'burst' into finer-scale turbulence, and then reform and repeat –never quite the same way, but in recognizable patterns. You can see this in many of my plane Couette movies.
The main point of my research is to find a way to speak about and understand these patterns of behavior mathematically. We know the equations of motion for fluids (the Navier-Stokes equations) and we can use them to simulate fluid flows on computers. But the computer simulation tells us how every individual arrow moves from instant to instant, not why they align and move in the patterns that we observe. We can predict all the details but we don't understand the self-organization of the whole flow.
Our plan for understanding the whole is to find the set of exact solutions of Navier-Stokes that organize the flow's dynamics. We build up a repertoire of known, exact patterns and then use this to analyze what we see in general: always something familiar, never exactly the same thing twice. The movies of periodic orbits show a few examples of such patterns.
The last paragraph is (in a very general way) the idea behind Periodic Orbit Theory. This has been successfully applied to problems in low-dimensional nonlinear dynamical systems and nonlinear quantum mechanics. We're hoping it's key to turbulence, too.