The role of maths in weather and climate prediction
9 August 2013
In their new book, 'Invisible in the Storm: The Role of Mathematics in Understanding Weather', mathematicians Ian Roulstone (University of Surrey) and John Norbury (University of Oxford) explore how mathematics and meteorology come together to improve weather and climate prediction. Here they share their insight into maths, numerical weather prediction and climate models.
In 1904, the Norwegian physicist Vilhelm Bjerknes published a short paper entitled Weather Forecasting as a Problem in Mechanics and Physics, which heralded the beginning of a new era in meteorology and weather prediction. Bjerknes described a vision in which the laws of physics, and the equations that encapsulate them, were to become the cornerstone of a rational approach to forecasting. His ideas became the foundation of modern weather and climate prediction.
We had to wait nearly half a century before modern electronic computing made Bjerknes' vision a realistic prospect, and Vilhelm lived just long enough to see the beginnings of numerical weather prediction. But how would Bjerknes react if he could see a modern forecasting office? The breathtaking pace in the development of computers, satellites and associated technology such as weather radar, enable us to monitor weather around the world 24/7.
Since the 1950s these developments have improved the accuracy of weather forecasts. It would have been impossible for Bjerknes to anticipate such progress, and he would be astonished at what is now at the disposal of forecasters. But he might well ask if we get caught out by the occasional storm that wasn't predicted  and Bjerknes would almost certainly anticipate an affirmative answer.
So why do forecasters, over a century after Bjerknes' landmark paper, still find some situations particularly problematic? Today, as a matter of routine, we calculate subtle changes in the basic variables  wind speed and direction, temperature, pressure, density and humidity  at millions of data points in our atmosphere, using seven basic equations at each data point. This amounts to solving tens of millions of equations. Yet we can solve these equations in minutes, because we have computers capable of over one thousand billion calculations per second (a measure we call a petaflop), and databases hold information in multiple petabytes.
However, lurking in the fundamental equations is mathematics that makes forecasting so difficult, and makes weather so interesting. And Bjerknes understood the maths.
Behind the physics of the atmosphere is a phenomenon that challenges even the most powerful supercomputer simulations we use in forecasting today, and it will continue to challenge us in the future. It is called nonlinearity. This means that 'cause and effect' relationships between the basic variables become ferociously complex. We are familiar with the notion of chaos  that small changes in one variable can have a huge impact on other variables, as epitomised by Edward Lorenz in his famous allegory we know as the butterfly effect. Chaos in the weather is a consequence of the nonlinearity in the equations.
But while there's a devil in the mathematical detail, maths also offers us ways of dealing with these problems so that we can make useful predictions, and continue to improve our forecasts, even in the presence of chaos. By combining the basic variables we can understand what orchestrates the myriad of local interactions to produce coherent, swirling, cyclones, with warm and cold fronts.
For example, by combining equations describing heat and moisture with equations governing the wind and pressure, we can form a new variable called potential vorticity, or 'PV'. Vorticity is a measure of swirling motion. PV actually helps us to identify key mechanisms that are responsible for the development, the intensity, and the motion of weather systems  including superstorms such as Hurricane Sandy  because it encapsulates overarching physical principles that control the otherwise complicated 'cause and effect' relationships. These principles enable us to decide what is predictable amid the detailed interactions.
Much of the current research within the Met Office and universities is devoted to improving the numerical methods that lie at the heart of the gargantuan computer models. As more detail is incorporated into the simulations, so it becomes even more important to represent the overarching principles that govern largescale weather patterns even more accurately. Amazingly, the mathematics describing PV was developed initially by Vilhelm Bjerknes over a hundred years ago.
As we note in our new book, Invisible in the Storm, "There are many detailed interactions and there are degrees of unpredictability. But there are also many stabilizing mechanisms and, most importantly, for understanding and prediction, there is the maths to quantify the rules."
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