"Simply, turbulence is very hard," says George Papanicolaou of Stanford University. "Every hard problem in classical physics finds itself embedded in turbulence. It is nonlinear, chaotic, stochastic."

   One of the words most frequently used to describe the state of turbulence is "ubiquitous." Turbulent flows are found not only throughout the world of nature, but throughout the world of research. The understanding of turbulence theory is essential in a range of scientific disciplines, including engineering, astrophysics, geology, and weather prediction. Turbulence is also, in the words of the legendary physicist Richard Feynman, "the last great unsolved problem of classical physics." Indeed, progress in the theory of turbulence has remained relatively stagnant for the past 30 years, while the exponential growth in computing power has allowed turbulence researchers to model turbulent flow to an extent never before dreamed possible.

   Mathematician George Papanicolaou, now at Stanford University, has been a major figure in the application of turbulence theory for two decades. In the mid-1970s, with Alain Bensoussan, now president of the French Institute National des Recherche Informatique et Automatique, and Jacques-Louis Lions of the College de France, Papanicolaou pioneered the mathematical study of the effective properties of materials. This concept, which has come to be known as "homogenization," has had applications in the study of composite materials and complex fluids. Papanicolaou, who describes himself as a "user of turbulence theory," has also done seminal work in the mathematical study of wave propagation in reflective seismograms and of the focusing of high intensity laser beams when they pass through inhomogenous media.

   Papanicolaou, 52, was born in Athens, Greece. He received his bachelor's degree in engineering at Union College in Schenectady, New York, and his master's degree and Ph.D. degree in mathematics at the Courant Institute at New York University, where he became a full professor in 1976 and, three years later, the director of the Division of Wave Propagation and Applied Mathematics. From 1990 to 1992, he was a visiting member of the Institute for Advanced Study in Princeton. In the summer of 1993, he moved to Stanford University, where he is a professor of mathematics. From his office at Stanford, Papanicolaou spoke with Science Watch correspondent Gary Taubes.

            It is said that turbulence theory has not changed much in 30 years. Why is that? Why did Feynman call it the last great unsolved problem?

   Papanicolaou: Simply, turbulence is very hard. Every hard problem in classical physics finds itself embedded in turbulence. It is nonlinear, chaotic, stochastic. And there is no separation of scales—you must deal with a very large number of scales of irregularities. It's just a mess. In most other physics problems, you can get control by reducing them to simpler problems that you can understand. You can separate scales, for instance, and determine that certain scales are not important. You can limit the phenomena. Or perhaps the inhomogeneity, the chaotic behavior, is not there all the time, so you can somehow approach it. In turbulence all these things happen at once, and you don't know how to separate them out.

            In spite of the difficulties that you mention, you describe yourself as a user of turbulence theory. What are some examples of how turbulence is used to understand problems?

   Papanicolaou: In communication theory, for instance, turbulence theory is used to understand how radio waves degrade by going through a turbulent atmosphere. We're interested in how radio waves are affected by the turbulent environment. The electromagnetic waves move so fast, we can assume that the turbulent motion is frozen. So the waves sample a frozen picture of the turbulent atmosphere. What will happen? If the waves are very long, they're not affected by the turbulence because the turbulence is on a finer scale. It's when the wavelengths are comparable to the scales of turbulence in the atmosphere that you get interactions. For the atmosphere, for example, it could be anywhere from one meter to 10 to 20 centimeters. And the shorter the waves get, the more prone they are to be scattered by inhomogeneities.
   Then there are diffusion problems—for example, pollution released into the atmosphere. If the atmosphere was quiet, then there would be very little diffusion. But because of the existence of turbulence, the observed diffusion is enormous. This kind of work really took off in the last 20 years, and we have a much better understanding of what's going on. Computational experimentation has helped a lot.
   Turbulence also arises in geophysical problems. The top 20 to 30 kilometers of the Earth's crust is extremely inhomogenous, and a lot of interesting things take place there. Wave propagation has to be understood—seismic waves, elastic waves propagating in the crust of the earth—and the extreme complexities are largely due to the inhomogenous structure of the earth. The same is true with the underground flow of water, the formation of aquifers, and the dispersion of pollutants underground. Those are problems that are mathematically very similar to propagation in atmospheric turbulence because you can pretty much assume that the inhomogenous structure is frozen in place as the dispersion process or the wave process takes place. A lot of this research was spurred by the study of secondary oil recovery, in which oil companies pump water down to drive oil out from this porous medium where it's pretty much embedded.

            Have computers helped make inroads into the understanding of turbulence?

   Papanicolaou: Most of the interesting things that have happened in turbulence are empirical models that are checked computationally. Of course, the big progress since the 1960s is the ability to do numerical simulations with a computer. That's an enormous improvement. Nowadays, the kind of tedious analytical work that was being done 30 years ago has almost disappeared. Instead, people do bigger and bigger numerical simulations. A huge bank of data for turbulent flows has been generated, and people use it to test their empirical theories.

            Has computer modeling actually allowed for any true understanding of turbulence theory, or has it been simply a tool to help model empirical problems?

   Papanicolaou: Here's a striking thing: In 1941, Kolmogorov, the great Soviet mathematician, wrote a two-page paper on turbulence, using very exciting physical reasoning to produce a heuristic law that tells how energy is distributed among different scales in turbulence. In the atmosphere, for instance, turbulence manifests itself on scales from millimeters and centimeters to meters and maybe even tens of meters. So how much energy does each scale carry in a typical developed turbulence? How does the energy distribute itself among the various scales? That's a very essential question. Is there a universal law that describes this?
   Kolmogorov did a simple dimensional analysis and came out with the so-called "five-thirds law," which says that there is an intermediate range of scales—in which, as the length of the scales decreases, the energy decreases—and that this cascade of energy has a universal character. The 1950s and 1960s saw a flurry of physical experiments in which discrepancies in this law were found. For the past 20 years, people have been simulating the problems on the computer to see if Kolmogorov was right, or if the researchers proposing heuristic improvements based on experiment were right. That has been extremely fruitful research. And it's now a benchmark problem. Every time a new supercomputer comes out, people immediately test what it will do for the turbulence problem.

            For the past 15 years, we've heard about chaos theory as the potential answer to the problem of understanding turbulence. What is the relationship between the two?

   Papanicolaou: Chaos theory is related to the subject of the onset of turbulence, when you take a fluid which is at first quiet, then drive it harder and harder until the flow becomes turbulent. I, on the other hand, am talking about fully developed turbulence, which is thoroughly distinct in terms of the mathematical technology involved. For instance, turbulence in the atmosphere is there and never ceases to be there—it's fully developed. So the question is, how do you describe developed turbulence? It's well past the point of its onset and has spread everywhere. The tools needed are traditionally very different from those pertaining to the onset of turbulence.

            What about advances in pure mathematical theory? Have those led to any true understanding of turbulence?

   Papanicolaou: There really hasn't been much progress. There was the so-called renormalization group method that was started in the 1970s by physicists, primarily Kenneth Wilson of Cornell. His idea was that when you're dealing with problems in which many scales interact simultaneously, you take a few scales at a time and calculate the interactions between them and then put the whole thing together in a self-consistent way and come up with a global theory. It's a nice idea, but it has been understood mathematically only for toy problems. So far it hasn't really played any role that we would have wanted it to play—for example, in actually generating equations that drive interactions between the scales so that we can really solve problems.
   Take, for example, turbulent flow around an airplane. To this day, when people want to study how airplanes behave when they fly through turbulence, they resort to entirely empirical heuristic theories, with a lot of adjustable parameters that can be tuned.

            What is the ultimate problem that you would like to see solved in your field?

   Papanicolaou: One really interesting problem is to think of clever ways to make numerical calculations that really straddle many scales. So far the numerical calculations have been rather straightforward—direct numerical calculation: write down the equations, put them on the computer, solve them. There have to be more intelligent ways of approaching this, to put more insight into the computer modeling. In the next 10 or 20 years, that's what's going to happen. The computational schemes are going to become increasingly intelligent, more adaptive. We are going to put into computer code the ability to recognize its environment and adapt, to become more efficient, and to be guided by the theory.The scant theory that exists right now is not employed in any intrinsic way when you use a computer to help make the problem more efficient. For turbulence it would be enormously important to be able to do that.

            Are you satisfied with the progress you've made in the past 25 years? What's on your agenda for upcoming research?

   Papanicolaou: Of course I'm not satisfied. There are two things I still hope I'll be able to do. One is to find a way to create numerical computational methods that really use theoretical insight. This is the Holy Grail among a small group of people who really understand the mathematics: to find a way to really make the computer leverage your insight. I'm trying to do that now. As far as analytical things, I underestimated the importance in the last 20-some years of chaos theory. I felt it was overplayed and that the results were too qualitative and too thin for my tastes. But now I'm beginning to change my mind. Some substance is really coming out of chaos theory and the subject is solidifying, and I'd like to understand it. The glitz is wearing off, and now there seems to be some interesting work going on.