Tiny particles high in our atmosphere play a key role in our climate, but deciphering the mathematics to calculate their impact on global warming has been a 15-year labour of love for Macquarie University mathematician Associate Professor Stuart Hawkins.
Tenacity: Associate Professor Stuart Hawkins, pictured, and his collaborators focussed on solving a complex climate maths problem for 15 years until they made the breakthrough, which has just been published in the Journal of Quantatative Spectroscopy and Radiative Transfer.
When microscopic particles - like mineral dust, or soot from fires - drift into the upper atmosphere, they change how the sun’s light reaches the Earth below.
But because each particle has a unique, irregular shape, it’s extremely difficult to determine their overall impact.
“These particles might cool the Earth by reflecting light back into space, or warm it by reflecting light to the ground or warm the surrounding air by absorbing the light,” says Associate Professor Hawkins, a lecturer in computational mathematics at Macquarie University’s School of Mathematical and Physical Sciences.
“Scientists couldn’t accurately calculate their effect because the mathematical tools available only worked well for nearly spherical particles; but these particles are mostly not spherical.”
Associate Professor Hawkins first learned of this conundrum in 2008, when the atmospheric physicist, Associate Professor Michael Box, told a lecture audience that without a suitable model to show the effect of these particles, the calculations predicting how the climate would behave in future had a significant gap.
This revelation was to shape the next 15 years of Associate Professor Hawkins’ career; and it has finally led to an important breakthrough.
Cracking the code
In a new paper published in the Journal of Quantitative Spectroscopy and Radiative Transfer, Associate Professor Hawkins and his collaborators have unveiled an elegant solution: a method to accurately calculate how light scatters off irregularly shaped particles, including long, thin particles – something previous methods couldn't achieve.
Desert particles: Sand storms like this one, pictured, release mineral particles into the upper atmosphere that can either cool Earth by reflecting sunlight into space or warm it by reflecting light to the ground.
Their calculations remain accurate even for particles 15 times longer than they are wide, far exceeding what current methods can handle.
The work builds on another crucial paper published just weeks earlier in Advances in Computational Mathematics, which provided the mathematical foundation needed to make the breakthrough possible.
This supporting paper, representing 15 years of collaboration, developed a new system of equations that remains stable at all frequencies - a critical requirement for reliable calculations.
“Scientists have been able to solve the equations for spherical particles for about 100 years," says Associate Professor Hawkins.
“However, critical particles such as the minerals that originate in Australia’s deserts and end up in the atmosphere are non-spherical, so we could not accurately simulate light scattering from this important family of particles.”
Learning from sound and light
One of the keys lay in Associate Professor Hawkins’ early work as a postdoctoral researcher, as he tried to adapt an algorithm designed for acoustic (sound) waves to work for light.
Light scatter: The sun's rays passing through our atmosphere, pictured, behave differently when they encounter tiny non-spherical particles, and new research can now calculate their effect.
“Acoustic and light waves behave differently, but their underlying mathematical equations are deeply connected, so I thought it would be an easy conversion.
“But after six months of checking and debugging, I reached a standstill,” says Associate Professor Hawkins.
This early setback eventually led to the breakthrough that made the new work possible two decades later.
Associate Professor Hawkins and his collaborators were finally able to combine two significant advances: a new set of mathematical equations describing how light interacts with particles, and an efficient way to solve these equations using computers.
The method is particularly notable for its stability. Unlike previous approaches that often failed when dealing with certain particle shapes or sizes, this new method remains reliable across a much wider range of scenarios.
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The result is what mathematicians call a T-matrix – a grid of numbers containing all the information about how a particle scatters light at a particular frequency.
This matrix can help climate scientists calculate how light passes through a column of air containing these particles, which is critical information for climate models.
Associate Professor Hawkins has also created a clever open-access computer program to smooth the way, called T-Mat Solver.
A matrix approach
“The program allows users to find the shape that fits their data, then apply the T-matrix to predict behaviour at all other wavelengths of interest,” says Associate Professor Hawkins.
“Previously, scientists had to assume these particles were close to spherical, which most are not.
“Now we can work with differently shaped spheroids, or average results over a range of random-shaped particles.”
The program’s usefulness extends beyond climate science. The same mathematical principles could apply to any process involving objects that scatter light or sound, from medical imaging to the design of new materials.
“These materials might arrange the particles in specific ways to slow light down, stop it altogether, or reflect it,” Associate Professor Hawkins says.
“The T-matrix is the perfect tool for this because it tells you what one individual particle does, and can account for the interactions and reflections between particles, while delivering stable and reliable results across a wide range of particle shapes and sizes.”
Stuart Hawkins in an Associate Professor in the School of Mathematical and Physical Sciences at Macquarie University.
