29/10/2013 05:48 GMT | Updated 23/01/2014 18:58 GMT

Invisible in St Jude's Storm: The Role of Maths in Getting the Forecast Right

Co-authored by John Norbury (Oxford University)

Miranda Prynne in The Daily Telegraphreported "The Met Office's £30million supercomputer was able to predict the size and path of the St Jude's Day storm four days before it had formed by using calculations from millions of sites around the world to simulate the weather." The forecast of the St Jude's Day storm was good and, as Prynne highlights, it was picked up by the Met Office before it had actually formed. However, if all the forecasters had to rely on was the mind-boggling computational power of their supercomputer, then most of the forecasts would be rubbish. The reason this forecast was very good - like so many these days - is that our ever-growing knowledge of how the atmosphere works has been extremely carefully incorporated into the computer algorithms using state-of-the-art mathematics.

The atmosphere is a complicated, often chaotic, physical system. The laws that govern the wind, the warmth, the humidity and the air pressure contain many subtle and important feedbacks, and number-crunching alone is insufficient to calculate the weather accurately.

Consider a simple desk-top toy that adorns many executive offices. The double pendulum consists of two rods: the first swings from a pivot attached to a table, and is joined by another pivot to a second freely-swinging rod. When the rods are set in motion and the amplitude of the to-ing and fro-ing is not too large, the motion is predictable. But if the second rod is released from a point above the table, the motion of one rod can affect the other drastically, and predicting the subsequent motion of the rods rapidly becomes impossible - no matter how much computational power we have to calculate that motion.

So chaos is possible in this desk-top toy with only two components: in our computer models of the atmosphere, forecasters are dealing with a problem that has several billion components. The reason we are able to forecast the weather with increasing accuracy is that while there are many detailed interactions, together with degrees of unpredictability, there are also many stabilising mechanisms and, most importantly, for understanding and prediction, there is the maths to quantify the rules.

If we combine the basic weather variables of wind speed and direction, air pressure, density, temperature and humidity, we can then study other quantities that reveal the coherence of entire weather systems. First we conserve air mass and thermal energy, as the air motion compresses and distorts the moving air parcels. Secondly, and more tellingly for swirling storms, we also subtly conserve potential vorticity (or PV for short). This is a measure of swirl of the wind combined with conservation of thermal energy. The PV equation tells us that PV does not change every time variables such as the wind or pressure change. Rather we need to change the energy of the swirling mass of air, which processes such as water vapour condensation do. Cloud condensation powers storms and hurricanes. So PV encapsulates, in a single variable, much of what we need to know about the choreography of entire swirling weather systems.

The PV of a rotating weather system is to a meteorologist what angular momentum is to a spinning ice-skater. Just as an ice-skater uses the conservation of angular momentum to spin faster by drawing their arms towards their body, weather forecasters monitor PV to see how hurricanes and middle latitude storms will strengthen or weaken.

By using maths, forecasters incorporate in the computer programs the knowledge we gain by using new variables such as PV. Predicting the likely development of the St Jude's day storm was possible thanks to the maths that informed the development of new algorithms in the supercomputers of today.

Twelve months ago this week, the forecasts of Superstorm Sandy were also remarkably accurate and, as we explained in our article in Scientific American, maths played a crucial role in getting the predictions right, which helped save lives and livelihoods.

Prynne concludes by reporting that the next Met Office supercomputer will cost £100 million, which will be money well spent. And there is a new generation of satellites observing the atmosphere and the oceans ever-more closely. To get the best value from these investments, it is imperative that we continue to use maths to transform our understanding of the atmosphere into ever-better algorithms that keep us ahead of the game in the enduring challenge of weather forecasting.

Ian Roulstone and John Norbury are authors of Invisible in the Storm: the role of mathematics in understanding weather.