Mathematics and lightning
Mathematics plays a crucial role in understanding and predicting lightning, a natural electrical phenomenon observed globally for millennia. Lightning occurs primarily as cloud-to-ground strikes, where a massive discharge of electricity travels from thunderclouds to the Earth. The voltage of a lightning strike can reach up to one billion volts, vastly exceeding that of standard electrical outlets, and the heat generated can exceed 30,000 kelvins, resulting in the explosive sound of thunder. Mathematical techniques are employed to model lightning behavior, analyze its patterns, and predict strikes based on various factors like weather conditions, geography, and historical data. For instance, mathematicians use probabilistic models to determine areas more prone to lightning, aiding in fire prevention strategies through optimized fire break placements. The study of fractals has also revealed that lightning follows non-linear paths, allowing researchers to apply fractal geometry in their analyses. Innovations like the Geostationary Lightning Mapper and machine-learning algorithms are enhancing the ability to forecast lightning events, potentially providing advanced warnings. Ultimately, the intersection of mathematics and meteorology deepens our understanding of this dynamic natural phenomenon and its impact on the environment and human safety.
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Mathematics and lightning
SUMMARY: Lightning is studied, modeled, and predicted using mathematical techniques.
Lightning is an electrical phenomenon of nature that has been observed by people around the world for thousands of years. Thunder is the sound of lightning, created by the intense heat of a lightning bolt. Many people may have learned as children a simple calculation for estimating the distance of lightning based on the sound of thunder. Since thunder travels about one mile in five seconds, a fifteen-second delay between the time lightning is seen and the time the thunder is heard indicates that the lightning strike was about three miles away.
Lightning strikes occur frequently around the globe, with an estimated 25 million cloud-to-ground strikes per year in the United States alone. Lightning has many religious associations, and it is often used as a metaphor for sudden insight or inspiration. Mathematician Carl Friedrich Gauss is reported to have said, regarding a problem he had been working on, “Like a sudden flash of lightning, the riddle was solved.” Lightning is studied by mathematicians, often in collaboration with scientists in other fields, to better understand the various facets of this complex phenomenon.
Among the several types of lightning that occur, the most seen and the most dangerous is cloud-to-ground lightning, caused by the discharge of electrons into the Earth from thunderclouds in the atmosphere. The voltage released by a bolt of cloud-to-ground lightning is on the order of 1 million times the voltage in a standard electrical outlet.
The excess of electrons at the base of a thundercloud repels electrons on the ground deep into the Earth, inducing a strong positive charge on the ground below. While air usually acts as an insulator, preventing the flow of electric current, the strong electric field between a storm cloud and the Earth can reach tens of thousands of volts per inch, pulling air molecules apart into negatively charged electrons and positive ions. This creates pathways of ionized air known as “streamers.” The freely moving charges in the ionized air allow electric current to flow through it.
A lightning strike occurs when a streamer carrying electrons from the cloud toward the Earth meets a shorter, positively charged airstream reaching up from an object on the Earth. This creates a complete conductive pathway between the cloud and the ground and a sudden and massive discharge of electrons into the Earth.
Between an average thundercloud and the Earth, there are an estimated 108 volts, reaching 109 (1 billion) volts in more-intense strikes. For perspective, one may compare 1.2 × 108 volts between a thundercloud and the Earth to the 120 volts delivered by a standard electrical outlet in the United States:
Voltage between cloud and ground
= 1.2 × 108 volts
= 1.2 × 102 × 106 volts
= 120 volts × 106
= Voltage in standard electrical outlet × 1 million.
The heat created by the electric current in a bolt of lightning reaches temperatures up to 30,000 kelvins (K), more than five times the temperature of the surface of the sun and hot enough to melt rock and fuse soil and sand into glass. The temperature on the Kelvin scale is the temperature in degrees Celsius plus 273.15. The intense heat in a channel of lightning causes the air within the channel to expand rapidly, sending out a shock wave that weakens into the acoustic wave of thunder. The electric current and heat of a lightning strike can start forest fires, damage property, destroy electrical equipment, and cause serious or fatal injuries to people and animals. According to estimates by the National Weather Service, lightning causes on average about 60 deaths and 300 injuries in the United States each year.
Statistics collected by NASA satellites have found that most of the eastern half of the United States sustains about eight flashes of lightning per square mile per year (decreasing to less than one per square mile per year toward the West Coast). Since 1 mi2 = 640 acres, this translates to eight flashes per 640 acres per year, or one flash per 80 acres per year. Accordingly, a one-acre lot in this region would be struck by lightning on average once every 80 years.
Mathematical research can help to predict the behavior of lightning strikes based on weather patterns and other variables; for example, by modeling probabilistic distributions of lightning strikes according to factors such as time, geography, and strength. The mathematical theory of highly optimized tolerance (HOT) is useful in controlling forest fires caused by lightning. This theory suggests optimal placement of fire breaks: if data or other evidence suggests that lightning strikes some areas of a forest more frequently than others, then large fires can best be prevented by purposefully cutting fire breaks that create sections whose sizes are inversely proportional to the rate at which lightning strikes. Other mathematicians are interested in studying the patterns and geometry of lightning. Mathematician Benoit Mandelbrot, known for his study of fractal patterns, noted that lightning does not travel in a straight line but rather in patterns reminiscent of fractals. Techniques of fractal modeling are used to study fractal patterns in the ionized plasma structures of lightning streamers. Morphological filtering and gradient detection can be used to help visualize lightning in satellite imagery and separate it from other visible effects, such as city lights.
In 2016, the National Oceanic and Atmospheric Administration (NOAA) launched a satellite called the Geostationary Lightning Mapper (GLM) with more advanced sensors to capture severe weather events such as lightning strikes. The utility of the GLM was made more important as its data is used to populate databases. These, in turn, allow Artificial Intelligence (AI) technologies to use this recorded data to more accurately predict lightning strikes. Machine-learning algorithms are being developed to more accurately forecast lightning strikes, with some that can provide up to an hour of advanced warning. One hybird effort combines machine-learning algorithm with weather forecasts. A computer can then determine relationships between different weather variables and incidents of lightning to make its predictions.
Bibliography
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