· 04 November 2009 by Robert Matthews
WE'VE all watched those vast heaps of cotton woolfloat across the sky. Lofted and shaped by updrafts of warm air,cumulus clouds mesmerise with their constantly changing shape. Somegrow ever taller, while others wither and die before our eyes. All bearwitness to the ceaseless roiling of the ocean of air we call theatmosphere.
About 80 years ago, the British mathematician Lewis Fry Richardson waspondering the shapes of such clouds when a startling thought occurredto him: the laws that govern the atmosphere might actually be verysimple.
Even at the time, with scientific meteorology still in its infancy, the ideaseemed absurd: key equations governing the behaviour of the 5 millionbillion tonnes of air above us had already been identified - and theywere anything but simple.
No one was more aware of this than Richardson, who is recognised as one ofthe founders of modern weather forecasting. Even now, the world's mostpowerful computers are pushed to their limits extracting predictions offuture weather and climate from the equations he wrestled with usingpencil and paper.
Yet Richardson suspected that behind the mathematical complexity of theatmosphere lay a far simpler reality - if only we looked at it theright way.
Now an international team of researchers analysing signals from satellites,aircraft and ground-based stations have found clear evidence thatRichardson's intuition was right and that the complexity of theatmosphere could really be an illusion.
The results point to a new view of the atmosphere as a vast collection ofcascade-like processes, with large structures the size of continentsbreaking down to feed ever-smaller ones, right down to zephyrs of airno bigger than a fly.
The implications promise to transform the way we predict everything fromtomorrow's local weather to the changing climate of the entire planet."We may never be able to view the atmosphere and climate in the sameway again," says team member Shaun Lovejoyof McGill University in Montreal, Canada. "Rather than seeing them asso complex that only equally complex numerical models can make sense ofthem, we're seeing a kind of scale-by-scale simplicity."
Richardson had a reputation for having ideas decades ahead of his time. Hepioneered the study of fractal geometry - the study of patterns thatlook the same no matter how much you magnify them - though the word"fractal" had yet to be coined. Look at the honeycomb pattern in abeehive, say, and the hexagonal structure is only visible if you're nottoo close or too far away. But look at some kinds of plants and you'llsee their fronds are made up of ever-smaller versions of the overallleaf. This is known as scale invariance, and is a feature of fractals.Richardson noticed that coastlines have a similar property, theirjagged outlines appearing just as jagged as one zooms in toever-smaller scales.
Attempting to capture this mathematically, Richardson found the same behaviour insimple formulas called power laws, by which one quantity changesaccording to another raised to some power. Even something as simple astiling your bathroom wall follows a power law: reduce the length ofeach square tile by 1/l and you'll need l2 asmany tiles. Such laws also reproduce the scale invariance of objectslike ferns and coastlines, which retain the same basic form no matterhow big the change in scale.
It was while looking for other examples of self-similarity that Richardsoncame to ponder the skies above: he noticed how the shape of clouds isconstantly modified by the invisible whirls and eddies of turbulent airthat surround them.
To get some insight into the laws governing turbulent fluids, Richardsonperformed simple experiments in which he threw bits of parsnip into alake and watched how they moved apart under the action of the whirlsand eddies on the surface. As with coastlines, Richardson found that ascale-invariant power law seemed to apply - an observation thatinspired him to poetry: "Big whirls have little whirls that feed ontheir velocity, and little whirls have lesser whirls, and so on toviscosity" - a parody of Jonathan Swift's famous 18th-century doggerelabout fleas and the little fleas that bite 'em.
But behind the humour lay Richardson's growing conviction that theatmosphere is just a collection of cascade-like processes, with largestructures breaking down to feed ever-smaller ones, creating afractal-like structure which acted according to power laws.
As with his work on weather forecasting, Richardson could only dream of atime when his ideas could be properly investigated. That time seemed tocome in the 1980s, when fractals and scale invariance hit thescientific big time. Simple scaling laws were suddenly claimed tounderpin everything from the size and frequency of earthquakes andavalanches to the rise and fall of stock markets. So why didn't anyoneput Richardson's idea to the test and search for simple power lawsdescribing the entire atmosphere?
The problem, says Lovejoy, lies with the word "simple". When fractals beganmaking headlines, researchers raced to find the power laws behind ahost of natural phenomena. In particular, they sought the value of the"exponent" in these power laws, the one number that governed the extentto which the phenomenon in question changed with scale. (In thebathroom tile example, the exponent is 2.) But they soon ran intotrouble. "They found that this single-exponent approach didn't alwayswork," says Lovejoy. "Many phenomena failed to obey power laws with oneexponent, and people started to give up on them, saying the idea wasoverly simplistic and had been oversold."
Among the ideas abandoned was Richardson's claim that the atmosphere is ruledby power laws. But in their race to move on, many researchers hadoverlooked the possibility that describing the atmosphere might be atad more complex than describing a coastline - and so might have neededa slightly more sophisticated approach.
Take air pressure, for example. The familiar isobars on weather chartsdefine regions of equal pressure, similar to the elevation contours ona map. Indeed, an isobar can be thought of as a kind of "coastline",described by its own fractal law. But there's a key difference: acoastline's shape is defined only at one specific value of height - sealevel. In contrast, the isobars of air pressure form a whole array ofshapes at different heights, like Russian dolls nested within eachother. Air pressure is what mathematicians call a multifractal field,described by a whole set of power laws, rather than just one.
The failure of researchers to find simple power laws for the entireatmosphere says more about their naivety than about Richardson's idea,but it cast a long shadow over attempts to apply fractals tometeorology, as Lovejoy himself discovered early in his research career.Inspired by the work of the French mathematician Benoît Mandelbrot, whocoined the term "fractal" in the 1970s, Lovejoy devoted half of hisdoctoral thesis to evidence for power laws governing rainfall. "I wasgetting ready to move on to my postdoctoral research when I learnedthat my thesis had been rejected," he recalls. "The examiner couldn'tsee any connection between fractals and rainfall and I was advised toremove all references to it."
Not wanting to risk a final rejection, Lovejoy did as he was told andresolved to publish the excised findings. They appeared in Science in 1982 (vol 216, p 185), along with Lovejoy's tentative claim that they might just be linked to Richardson's outlandish idea.
In the following years, Lovejoy teamed up with Daniel Schertzer,now at the University of Paris East, France, and set about searchingfor evidence for multifractal power laws lurking in weather data.
They focused on rainfall: a meteorological phenomenon whose familiaritymasks the complexity of its origins. Triggered by a delicate balance ofatmospheric factors, rainfall is tough to model even using the mostpowerful supercomputers. Yet by analysing data from the rain-detectingradar network around Montreal, Schertzer and Lovejoy found evidence foran underlying simplicity to the process.
The radar data allowed them to plot the amount of rainfall in an area, asthey zoomed in and out at different scales. The researchers found theirplots could be described by power laws with different exponents - astrong hint that rainfall is a multifractal process, with theunderlying physics cascading down to ever-smaller scales.
While intriguing, the discovery was far from compelling. For a start, thedata only allowed Schertzer and Lovejoy to extract power laws spanningscales between about 100 kilometres and 1 kilometre. To properlysupport their theory that the atmosphere is multifractal, they wouldhave to show the scaling laws still held out to scales of tens ofthousands of kilometres - the size of the entire planet.
The team realised that one source of meteorological data was up to the job: Orbiting satellites. Scanning the planet evenly and in great detail,they build up a consistent picture at scales ranging from a fewkilometres to the whole planet. And Schertzer and Lovejoy realised thata satellite launched in 1997 by NASA and the Japanese space agency JAXAcould allow them to put the multifractal theory to a truly global test.
Orbiting the planet every 90 minutes, the Tropical Rainfall Measuring Mission (TRMM)peers down on a broad swathe of the Earth with sensors that detect thetelltale signs of rainfall on scales down to a few kilometres. Togetherwith their colleagues at McGill University and the University of ParisEast, Schertzer and Lovejoy analysed 1200 consecutive orbits of TRMM,looking for signs of multifractal behaviour in the atmosphere.
Earlier this year, they published their findings in Geophysical Research Letters (vol 36, p L01801)- and they were simply stunning. The satellite data generated abeautiful collection of fractals and followed power laws on scales fromtens of thousands of kilometres down to about 10 kilometres.
"It's rare that fundamental theories that have been marginalised for 80 yearsare suddenly and decisively proven," says Lovejoy. "Yet this is what webelieve we have done for Richardson's idea that atmospheric dynamicsare cascade processes."
-- It's rare that fundamental theories that have been marginalised for 80 years are suddenly and decisively proven--
Danny McKenna of the US National Center for Atmospheric Researchin Boulder, Colorado, says the evidence for power laws is convincing.He believes they could prove vital in tackling one of the mostnotorious problems in modelling the atmosphere.
Going off grid
Today's computer models represent the atmosphere as a vast grid-like pattern ofcells, whose meteorological properties are calculated using the complexequations formulated by Richardson and his successors. The finer thegrid, the better the simulation, but even the world's fastestsupercomputers can't cope when the grid is made up of cells smallerthan about 100 square kilometres.
To get around the problem, modellers have come up with estimates of whathappens inside the cells, called parameterisations. The problem withsuch parameterisations is that they can fall victim to the notoriousbutterfly effect, by which even small inaccuracies in the initialconditions can be magnified to huge size by the non-linear nature ofthe processes underlying the weather. This can lead to unreliableforecasts.
McKenna believes that the discovery of scaling laws could transform thesituation by providing insights into phenomena that take place onscales smaller than 100 kilometres. Robin Hogan of the University ofReading, UK, agrees that they could be a big improvement on existingtechniques. "Although we won't know what individual eddies are doing atthis sub-grid scale, their net ability to, say, transport heatvertically could be estimated," he says.
Now Lovejoy's team is keen to see cascades extend the reach and reliabilityof current models. While the existing models cannot handle structuresmuch smaller than 100 kilometres across, the cascades may continue downto scales smaller than a millimetre. "Cascades could help fill in thatmissing factor of 100 million or so," says Lovejoy.
To find out, he and his colleagues are now working with researchers at the US National Oceanic and Atmospheric Administrationin Boulder, Colorado, on incorporating multifractal techniques intolive computer models of the atmosphere. Their aim is to make bothweather and climate models reliable at the finest scales possible. It'sa challenging goal, but one that Lovejoy believes is achievable."Obviously there are many issues to be resolved," he cautions, "and itmay be some years before the techniques are implemented."
Nevertheless,it seems we are closing in on a new era in our understanding of theatmosphere, one in which computer models finally get to grips with itsfull complexity in all its beautiful simplicity. And with the need forreliable predictions of the future climate more pressing than ever,Richardson's genius may have cut through the clouds of complexity inthe nick of time.
"Thehistory of science shows that complex phenomena usually give way tounderlying simplicity," says Lovejoy. "And simplicity points the way tothe future".
A Reality check for climate models
It's not just more reliable weather forecasts we can expect by swappingcomplex numerical models for the simpler ones advocated by Britishmathematician Lewis Fry Richardson. Robin Hogan at the University of Reading,UK, believes that such power laws could also act as a vital realitycheck on climate models. Put simply, if a given model doesn't reproducethe real atmosphere's multifractal behaviour and its power laws,something must be missing.
Which raises an obvious question: how well do current models of theatmosphere perform? After all, if there's any truth in Richardson'sidea, their computational complexity should give rise to cascadingsimplicity. Jonathan Stolle at McGill University in Montreal, Canada,has teamed up with his colleague Shaun Lovejoy, and David Schertzer atthe University of Paris East, France, to examine this issue - and sofar the results are encouraging.
"We'verecently demonstrated that the top traditional numerical models havevirtually perfect cascade structures from around 10,000 kilometres downto 100 kilometres," says Lovejoy. Power laws may be able to extend themodels' reach - and accuracy - even further.