Smooth Turbulence

Water rushing through pipes, currents in the ocean, weather patterns, airplanes taking off, blood running in veins, chemicals churning in a mixer, even milk being stirred into coffee: all are governed by a single phenomenon – turbulence. Scientists and mathematicians have been studying turbulence since the legendary 18th-century mathematician Leonhard Euler formulated the first mathematical equations for describing liquid flow in non-linear patterns. This formula was refined in the next century by Claude-Louis Navier and George Gabriel Stokes, who noted that viscosity, even when minimal, influences fluid motion and incorporated terms to reflect its effect. The Navier-Stokes equations, as well as a number of equations derived from them, are still used today in scientific fields ranging from chemical and aerospace engineering to climate modeling.

 

Nonetheless, these useful equations still hold some puzzles for mathematicians, and the prestigious Clay Institute in Cambridge, Mass., has included the Navier-Stokes equations in a short list of the seven most challenging mathematical problems for the new millenium. Whoever solves any one of them is promised a $1 million prize. The burning question for the Navier-Stokes equations is whether the solutions to them “break” or “blow up” within a finite time period or whether they remain “smooth” ad infinitum. By “breaking,” mathematicians mean that as the solutions progress in time, some of the quantities might become infinite and the mathematical equation would no longer be a valid model for the physical process. In fact, some mathematicians believe that turbulence is linked to an infinite number of solution breaks, which can occur at any moment. However, proof for Navier-Stokes solutions breaking or remaining smooth has never been established – hence the challenge.

 

Prof. Edriss Titi of the Computer Science and Applied Mathematics Department is fascinated by the ways in which mathematics contributes to solving real-world problems: “Physicists ask: ‘What are the mechanisms that underlie this phenomenon?’ Engineers ask: ‘How can I control this phenomenon?’ But it is mathematicians who study the observations of the other two, develop a proper mathematical framework (and once in a while a completely new theory) and apply rigorous methods, to simplify and answer the others’ questions.”

 

Titi has shown that several equations, variations on Navier-Stokes that apply to specific patterns of turbulent flow, are able to produce smooth solutions at all times, with no breaks. The first, which he proved while at Cornell University, was for helical flows. As the name implies, in helical flows symmetrical lines of flow swirl around a central vortex. But this kind of flow, says Titi, is not truly three-dimensional. One can treat the flow lines as though they lie on two-dimensional surfaces, and this reduces the complexity of the problem. Titi and his colleagues proved that helical flows are invariant – once they start they’ll go on forever without breaking, according to the formula.

 

The second set of equations is truly three-dimensional, making it more complex, and Titi’s proof for smoothness at all times in these equations settles a major open problem in the mathematical theory of geophysical fluid dynamics and adds significantly to the under-standing of the field.

 

The equations were originally formulated in the 1920s, for weather prediction, by Lewis Fry Richardson – one of the first to try modeling the complex movement of air around the globe. In doing so, he had to deal with movement along an enormously wide, thin curving layer of fluid. (If the earth were a very large apple, the atmosphere would be the thin peel.) Taking advantage of the shallow atmosphere, Richardson distorted the Navier-Stokes equations, replacing the equation for vertical motion with the so-called “hydrostatic balance” equation, which is based on a simplified balance between the rate of change in pressure with respect to depth and the buoyancy forces. 

 

“Richardson’s ‘Primitive Equations of Large-Scale Ocean and Atmosphere Dynamics’ ruined the elegant symmetry of Navier-Stokes,” says Titi. Mathematically, these equations appear to be more difficult to approach than the original Navier-Stokes, and it was only last year that mathematicians succeeded in proving they possess smooth solutions for even a short period. So it was quite a surprise when Titi and his student were able to prove that these equations are eternally smooth.

 

Titi: “The charm of challenging mathematical problems such as Navier-Stokes is that they can be simply formulated and explained, yet they keep us busy trying to solve them for decades, centuries – sometimes millennia.” 

 

 

Prof. Edriss S. Titi was born and raised in an Arab family in the Old City of Akko. His parents barely finished grade school, but they insisted on providing their four children with the best education available, at Akko’s private Franciscan Terra Sancta school. There, with the help of teachers he’s in touch with to this day, Titi excelled in mathematics and physics. In 1974, he began studies at the Technion, in Haifa. 

 

After receiving an M.Sc. in theoretical mathematics from the Technion, Titi switched to applied mathematics, completing his Ph.D. at Indiana University and postdoctoral research at the University of Chicago. He was a lecturer for two years at Cornell University, then moved to the University of California at Irvine, where he became a full professor in 1989. Titi first came to the Weizmann Institute as a visiting scientist in 1999; he joined the Mathematics and Computer Science Faculty in 2003.