So Random

Fall 2011

so-randomFluid dynamics, the branch of physics concerned with the behavior of liquids and gases in motion, can explain what happens when cream swirls in a cup of coffee, when wisps of smoke rise from an extinguished campfire, when water burbles down a creek bed. Drop a grain of sand into that creek, for example, and there is an equation that will describe the movement of the sand and pinpoint the one single place where the sand will end up.

But if a storm rolls in and the water starts frothing, all bets are off, according to Gregory Eyink, a professor in the Department of Applied Mathematics and Statistics at the Whiting School. Using a virtual-reality computer simulation, Eyink has shown that if two identical particles are dropped into a turbulent flow-a flow that is fast and irregular-at the same time and same place, they will end up in completely random spots. “The theories that assume there’s one solution assume smooth velocity,” says Eyink. “But it turns out that in turbulent flow, things are not sufficiently smooth and one solution is no longer true. If you put a particle in it and let it go, it’s totally unpredictable. Where it goes becomes random,” says Eyink, who reported his findings in the May 27, 2011, issue of Physical Review E.

Eyink’s work builds on conjectures about particle separation in turbulent fluids made in the 1920s by the physicist (and meteorologist) Louis Fry Richardson. In the mid-1990s, researchers from Europe confirmed Richardson’s conjecture when they studied particle separation in a mathematical model of turbulence. Based on this model, they concluded that in real fluids, particles will end up in totally random places, a physical phenomenon known as “spontaneous stochasticity.”

Eyink wanted empirical proof. He used a database of computer-generated “virtual reality” turbulence that he created with mechanical engineer Charles Meneveau and computer scientist Randal Burns, both of the Whiting School, and physicist Alexander Szalay of the Krieger School of Arts and Sciences. This database is “real” in the sense that the fluid velocities in it satisfy the same equations of motion as real physical fluids do. Eyink dropped identical virtual particles into the database at the same spot and then tracked their movement. Along the way, he gave the particle random “kicks.” Predictably, the particles followed different paths. However, even as the kicks got weaker and their effects lessened, the particles still went to random places.

Moreover, Eyink’s results were contrary to a basic principle established by Nobel Prize-winning physicist Hannes Alfven that holds that certain kinds of objects-magnetic field lines (like magnetic lines of force seen with iron fillings)–are carried by a fluid as if they were light threads cast into the flow. This phenomenon, known as “flux freezing,” has long been a problematic theory: “It explains many different processes, including how stars form,” says Eyink. “But it has also been observed that many times it doesn’t work-for example, in solar flares.” In Eyink’s simulation, the “threads” of magnetic field-lines did not end up at one location but instead moved randomly. “With this technique it showed up really clearly,” said Eyink. “The results were amazingly good.

“Stochastic magnetic flux freezing in turbulent flows should help us to understand violent events in the sun such as solar flares, which can disrupt power and communication networks on Earth, and the generation of Earth’s own magnetic field, which helps to shield us from some harmful radiations and which is presently decreasing, about a 10-15 percent decline over the last 150 years,” says Eyink.