# Going with the Flowby Mary Hrovat

Imagine the Gulf Stream, with blue-green water flowing smoothly in something like straight lines. The wind kicks up. Some of the water begins to move faster than the rest. Imagine the wind curving some of the straight lines, so all of the water is no longer going in the same direction at the same speed. Eventually, some water even starts to move backward relative to the rest before flowing forward again. Small circles form on one side and break free, spinning away from the current.

These circular motions are eddies, which can cover tens or even hundreds of miles of the ocean’s surface. On Shouhong Wang’s computer screen, the bucolic blue-green flow with its swirling eddies is a numerical simulation, representing a powerful blend of physical data and mathematical analysis. Wang, a professor of mathematics at IU Bloomington, is using this simulated stream to help resolve some long-standing problems in fluid dynamics, the scientific study of fluids (gases and liquids) in motion.

There is currently no theory that can predict where eddies will form and break away. Turbulent flow at the boundary layer—the line along the surface of the water where eddies develop and split off from the rest of the current—involves rapidly changing velocities as well as circular motion, and it is difficult to describe and interpret mathematically. Wang is working on a new mathematical theory that will increase understanding of the dynamics of ocean flows and how a flow separates from the boundary layer.

Wang points out that the boundary separation of fluid flows plays a role in many physical problems. The new mathematics he is developing can be applied, he says, in any situation where there is an incompressible flow. “Incompressible” means that water doesn’t get more dense, or compress, when you squeeze it. Most gases, including the gases in the atmosphere, do get more dense when the pressure increases. In some cases, though, they can be treated as incompressible, so the work that Wang is doing could help in developing more precise climate models.

 A computer simulation illustrating an eddy's velocity. Courtesy Shouhong Wang.

Wang and several collaborators are analyzing large amounts of data collected by the U.S. Office of Naval Research. The ONR placed floats, or drifters, in the Atlantic Ocean and let them drift with the motion of the water, taking regular measurements of how fast and how far the drifters moved. To mathematically interpret the data, Wang and his colleagues first created snapshots of conditions at each instant that data were recorded, then used the physical data to develop numerical models of the formation of the eddies. Based on the numerical models, Wang and postdoctoral fellow Cheng Wang are creating computer simulations to show the flow structure after eddies form and break away from the boundary layer.

The more difficult part of the work is to explain what is happening mathematically as the system changes over time. Mathematicians have two ways of studying the changes in fluid flow. One approach relies on what are known as the Navier-Stokes equations to describe fluid flows in terms of pressure, density, and velocity. In the area of topology—the study of particular figures and shapes—these equations help to classify the structure of the flow patterns, relating topological concepts (like centers) to observable physical phenomena (like circulating areas). The other tool mathematicians use to describe fluid flows is Lagrangian dynamics, which studies the motions of individual cells of fluid in their physical context.Working with Professor Michael Ghil of UCLA and Professor Tian Ma of Sichuan University, Wang is pursuing connections between these two approaches. So far, he says, they have been able to give “the first rigorous characterization” of the boundary layer separation, when eddies begin to spin off. As his work continues, Wang hopes researchers will gain new tools to use in addressing some of the more complicated problems in oceanic and atmospheric physics.