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Researchers confirm 125 year-old theorem 

By Dawn Wiseman

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Next month, George Vatistas, with fellow professor Kamran Siddiqui and graduate student Hamid Ait Abderrahmane (Department of Mechanical and Industrial Engineering), will be published in the Physical Review Letters, the world's foremost physics letters journal. Their contribution will focus on recent experiments that confirm for the first time J.J. Thomson’s 125 year-old theorem on the stability of vortex rings.

In 1989, Vatistas was exploring the properties of water vortices by spinning up liquid in tall slender cylinders. Instead of the relatively smooth-sided funnel he was expecting, large waves undulated up the length of his whirlpool, obstructing the view of the central core he wanted to study.

In an attempt to minimize this visual noise, Vatistas lowered the liquid height to work with a much shallower pool. When he spun up the liquid again, the core, when viewed from above, eventually resolved into a polygonal triangular shape. He remembers his first impression of this geometric beauty as, “Wow.”

A review of the literature sent Vatistas back more than 100 years to an 1883 treatise by physicist J.J. Thomson. Thomson, who won the 1906 Nobel Prize in Physics for his identification of the electron, built on work by Lord Kelvin (and others including Kirchoff, Gröbli and Poincaré) to mathematically demonstrate what should occur in systems of point vortices or masses.

To understand Thomson’s conclusions, picture a satellite image of a hurricane. The eye is the core of a huge whirling mass of air which extends out from the centre of the storm. Pressure differences at the edge between the air inside and outside the eye can actually spin off other vortices in the form of tornadoes, which orbit the eye until such time as they dissipate.

Thomson said that under ideal circumstances, naturally occurring stable systems of up to six vortices (and potentially seven) could rotate around the core of a larger spinning mass. His theory also implied that the radius from the core of the parent vortex to the cores of its daughter vortices would be half that of the total radius of the parent vortex.

Vatistas suspected that the triangular polygon he produced in 1989 was indeed a manifestation of Thomson’s treatise, where three daughter vortices were equally distributed around the parent’s central core. He even produced the square, pentagonal, and hexagonal cross-sectional cores predicted for four, five and six daughter vortices, but then the system appeared to give over to turbulence.

Nineteen years later, Vatistas, Siddiqui and PhD student Abderrahmane, decided to leverage advances in image processing and the availability of new experimental equipment to revisit the problem. This time, they were able to “see” further.

They reproduced the earlier results and established their stability. As Vatistas explained, each equilibrium state is produced within a range of rotational velocities. While more daughter vortices are produced as the velocity increases, the range of velocities for which the system is stable decreases.

He added, “Even now we are not able to produce the seven-vortex system. But this is critically stable and likely cannot be formed under real conditions. It is like trying to balance a pencil on its tip. In theory, you should be able to do it, but in reality it cannot be done.”

The team also confirmed the key relationship between the radii. “This is where we have clearly established a first,” said Vatistas; in 2002, a group at the University of California (Berkeley) came close, but did not verify the radial relationship.

Why is this important? The confirmation of Thomson’s treatise suggests that it can be used with confidence to describe the almost endless naturally occurring phenomena which appear to be analogous to vortex systems.

Some, like the formation of spiral galaxies are too far away for us to study without this type of connection. Others are much closer to home: tornadoes, hurricanes and specific atmospheric conditions at the poles. As the authors underlined in their letter, the close resemblance of their results to satellite imagery of weather patterns in Antartica “is intriguing.”

For Vatistas, the possibilities for new research “are very exciting. I’m fascinated by galaxies and the classical turbulence we seem to see there.” He smiled. “I like to think beyond the conventional, but I have to keep my feet on the ground. I am a scientist.”