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The Maths Behind the Heat Wave

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co-authored by John Norbury (Oxford University)

A heat wave in the UK is usually measured by temperatures reaching in excess of 30 degrees Celsius for more than a week. The current heat wave - the longest spell of hot weather in seven years - is welcomed by most after one of the most prolonged winters on record; but if temperatures remain high, health risks and environmental hazards, such as crop failure, wildfires, and water shortages, become a real concern. So how good are we at predicting heat waves? Are they related to global warming? And do we know when it will rain again?

Maths helps us to answer these questions. Forecasting a heat wave relies on an accurate prediction of the jet stream, and the attendant troughs of low pressure and ridges of high pressure. The present spell of hot weather in the UK is related to the dramatic shift of the jet stream in early July to a northerly position relative to the British Isles.

Jet streams were among the first meteorological phenomena to be understood and quantified by mathematics. The breakthrough was made by a Swedish meteorologist who moved to the United States in 1925 and subsequently changed the face of meteorology, in both the US and worldwide. His name was Carl-Gustaf Rossby, and his name is immortalized in the Rossby wave - a key mechanism in controlling extreme weather such as heat waves.

Rossby waves are similar to water waves, but instead of water moving up and down, the air stream undulates northwards and southwards. Rossby waves are revealed by distinctive patterns of the troughs and ridges. These mostly westward propagating waves are quirkily associated with eastward-moving weather. However, from time-to-time the waves come to a halt - remaining stationary over a particular region.

Forecasters call this "blocking", which is when a weather event such as a heat wave persists in an area for a long period of time. One undulation of a Rossby wave generally has an east-west extent of about five thousand kilometres (roughly across the northern Atlantic Ocean). By studying how the actual spacing of the troughs and ridges compares to an undulation calculated for the average winds, we estimate their subsequent motion.

Understanding how patterns in the jet streams change with the climate is critical. One of the major factors that influence the dynamics of a jet stream is the thermal wind equation. This equation states that the vertical rate of change of the horizontal wind is proportional to the horizontal rate of change of temperature. This means that the decrease in temperature towards the North Pole causes winds to develop a larger eastward component with altitude. The strong eastward moving jet streams are in part a consequence of the simple fact that the equator is warmer than the north and south poles. So, if the equator to pole temperature differences were to be modified as a consequence of climate change, this would most likely have a major impact on the behaviour of the jet streams.

Because jet streams are large-scale features of weather, forecasters can be reasonably confident in their ability to predict how the jets will change from day to day. The computer models used in forecasting divide the entire atmosphere into millions of 'weather pixels'. Each pixel covers a horizontal area roughly 15km by 15km, and there are about 70 vertical layers of these pixels throughout the depth of the atmosphere. One of the crucial factors in determining whether or not a weather system can be forecast on the computer accurately is the ratio of the size of the system to the size of a typical pixel: weather phenomenon that are many more times larger than a weather pixel, and will therefore be represented by perhaps many thousands of pixels, are likely to be forecast well. This is the case for the jet streams.

On the other hand we know that prolonged periods of hot weather invariably give rise to severe thunderstorms. Thunderstorms are usually very local phenomenon, and they are not well resolved by the current computer models. For this reason, the periods of hot weather can be forecast with considerable accuracy, while forecasters may only be able to indicate the likelihood of isolated, but often damaging, thunderstorms.

Ian Roulstone and John Norbury are authors of Invisible in the Storm: The Role of Mathematics in Understanding Weather (Princeton University Press, 2013).