Explaining the ultimate example of chaos theory

Turbulence is seen as the ultimate example of chaos theory: the way a butterfly flaps its wings in Australia could be linked to whether a hurricane forms over the Caribbean Sea or not. 

But proving that mathematically is fiendishly difficult – so difficult that The Clay Institute has put up a million-dollar-prize for solutions of the Navier-Stokes equations that underpin fluid mechanics.

A group of mathematicians from the University of Maryland in the US has developed the tools to take it on, however, in the process proving one of the underlying theories of turbulence – Batchelor’s Law.

The new proof could lead to more exact modelling of turbulence in many realms, from the aerodynamics of cars to the formation of cyclones.

Batchelor’s Law describes the size and distribution of the swirls and eddies that form as fluids mix, in the way that a drop of milk spreads through tea. If the tea is stirred, swirls of many sizes form: some are the size of the teaspoon, but on those swirls there are smaller swirls, each with smaller swirls, forming a complex structure similar to a fractal. 

But unlike a fractal, the smaller swirls are not exact replicas of the large ones, so George Batchelor’s 1959 prediction was approximate – as a physicist he could only draw from what was observed in nature (for example for salt mixing through seawater) or in controlled lab experiments: observations that could not track every molecule’s movement.

However, the Maryland researchers were able to develop a rigorous proof – so far, only for a limited set of circumstances that don’t qualify them for the million dollars, but their new techniques will enable a slew of new proofs in the coming years, hopes one of the authors, Jacob Bedrossian.

“This is opening a door to understanding turbulence at a broader level,” he says. “There is a lot of work to be done, a lot of new mathematical ideas need to be invented, but the playing field is open – the game is on.”

Bedrossian, Samuel Punshon-Smith (now at Brown University) and Alex Blumenthal unveiled their techniques in a series of three talks at the Society for Industrial and Applied Mathematics Conference on Analysis of Partial Differential Equations. 

The breakthrough was made possible by the mathematicians bridging the gap between their three different sets of expertise.

Bedrossian, who studies fluid flow using partial differential equations, sat down to explore the problem of turbulent mixing with Punshon-Smith, who uses partial differential equations to study probability in stochastic systems – systems with some noise or randomness in them. 

The pair explored papers by physicists, which, although expressed in different language to their native mathematical tongue, led them to realise they needed to understand the trajectories of the particles in the fluid, which behave chaotically.

“We had a vision that maybe it could be done, but it was near impossible,” Bedrossian says. “If you had asked me three years ago if we could solve this, I would have said ‘check back in 100 years’.” 

At this point, the pair spoke to Blumenthal, who worked on dynamical systems and ergodic theory, a branch of mathematics that includes chaos theory. To him the proof seemed quite achievable – simplistic models of the kind of system Bedrossian and Punshon-Smith were working on had been developed in previous decades, but never applied to the real world.

As the three set to work, Blumenthal realised what he had bitten off. 

“Everything in partial differential equations is harder than you expect. Working in infinite dimensions is like stepping in quicksand,” he said.

However, the group persevered and developed tools which they say “bridge the canyon between fluid mechanics [Bedrossian’s field] and dynamical systems [Blumenthal’s field]”.

“Sam’s work in probability is the common thread that helps connect what Alex and I do,” Bedrossian says. “No single person would have had enough expertise in all the fields to build the bridge.”

To document the techniques needed four papers: one describing how particles that are initially close – such as drops of milk in tea – end up widely separated; a second and third that deal with the speed and mixing of the fluids; and the last to convert the findings into statements that amount to a rigorous proof of Batchelor’s Law.

Navid Constantinou, an ocean and atmospheric modeller with the Australian National University who was not involved in the research, says that the rigorous nature of the proof would help with developing more accurate climate models, because of the wide-ranging applicability of Batchelor’s Law.

“Now they have proved how the energy is distributed in each spatial scale, it is useful for climate models,” he says.

“When you are running a global model, you have limited computer resources that can at best resolve down to a few kilometres, but this gives you a way to include anything smaller than that, we know now we can trust Batchelor’s Law.” 

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