Mathematical analysis of hurricanes and tornadoes

Summary: Mathematical analysis and modeling have been used to attempt to predict and simulate hurricanes and tornadoes.

Hurricanes and tornadoes are both potentially catastrophic types of storms that cause billions of dollars in damage and claim many lives each year. Predicting major weather events of these types is difficult, though mathematical modeling and computer power have allowed mathematicians and scientists to make advances in storm science. The term “cyclone” is often erroneously applied to tornadoes; it properly refers to the class of storms originating over water that includes hurricanes, typhoons, and tropical cyclones. Tornadoes and cyclones are characterized by revolving forms and high winds, but tornadoes are typically smaller, faster spawning, shorter lived, and their damage is usually more focused. Mathematical analysis and modeling of storms draws from many fields.

For example, vector calculus plays a substantial role in analyzing and modeling these storms, since both pressure and humidity can be represented as scalar fields and wind as a vector field. Theories and equations from physics for conservation of mass and energy, along with angular momentum and shear, are also quite important. Historically, challenges in storm description, prediction, modeling, and simulation have often been related to data collection and computing power. One of the earliest systematic data collection and prediction efforts was conducted in the 1880s by John Finley of the U.S. Army Signal Corps, but for a variety of sociopolitical reasons, federal research lagged until about World War II. The emergence of Doppler radar advanced storm science, as did computers in the 1970s that were capable of generating three-dimensional models. However, even in the twenty-first century, no one can perfectly predict the emergence, path, strength, or damage of a hurricane or tornado. Even with multiple stations and satellites, data are still sometimes sparse or difficult to integrate across sources, and this type of research raises theoretical questions about the limits of predictability. At the same time, early warning systems that give even a few hours of notice regarding approaching storms are widely considered to be beneficial, and mathematicians continue to contribute to this area. Actuaries are also involved in calculating the costs of these storms, in terms of both money and lives.

A tornado is a rotating column of air that is in contact with both the ground and a cloud. Tornadoes are generally spawned by thunderstorms. The United States has the highest incidence of tornadoes of any country in the world, in part because of the confluence of cold air from Canada, warm, moist air from the Gulf of Mexico, and dry air from the Southwest. A related phenomenon is water spouts, which are essentially tornadoes that form over water, especially in tropical areas. A hurricane is a powerful, spiraling storm that begins over a warm sea, near the equator. “Hurricane” is, in fact, just one name for the kind of storm scientists refer to as a “strong tropical cyclone.” Depending on where they occur, hurricanes are given a different label. If they begin over the Atlantic Basin (Atlantic Ocean north of the equator, the Caribbean Sea, the Gulf of Mexico) or the Northeast Pacific Ocean, they are called “hurricanes.”

When the same kind of storm occurs in the western North Pacific Ocean, it is called a “typhoon.” In the southwest Pacific Ocean and the Indian Ocean, the storms are referred to as “cyclones.” No matter what it is called when a hurricane, typhoon, or cyclone hits land, it can do great damage through fierce winds, torrential rains, inland flooding, and huge waves crashing ashore. A powerful hurricane can kill more people and destroy more property than any other natural disaster. Hurricanes and other cyclones form in the tropics during summer and fall.

Predicting Major Storms

A few very important characteristics of hurricane are as follows:

• Hurricanes form under weak, high-altitude winds
• Hurricanes have no fronts
• Hurricanes main energy source is the latent heat of condensation
• The center of a storm is warmer than the surrounding air
• Hurricane winds weaken with height
• Strongest winds are near the Earth’s surface
• Hurricanes weaken rapidly over land

As global weather patterns become more erratic as evidenced in the early twenty-first century, it has become difficult to accurately forecast hurricanes. However, mathematics allows forecasters a thorough insight into the mechanisms of weather features, including large-amplitude water waves and sustained winds cloud structure. Moreover, statistical models built from historical data perform with greater precision. Also, scientists use high-quality time series data along with less precise time series data using a Bayesian approach, which does not require data to have uniform precision. This way, scientists have been able to forecast U.S. hurricanes six months in advance.

Wind engineer Herbert Saffir and meteorologist Robert Simpson introduced the very popular Saffir–Simpson wind scale, which is a 1–5 categorization based on the hurricane’s intensity at the indicated time. This scale is an excellent tool for alerting the public about the possible impacts of various-intensity hurricanes. However, the scale does not address the potential for other hurricane-related impacts, such as storm surges, rainfall-induced floods, and tornadoes.

The estimation of hurricane-generated waves and surges in coastal waters is of critical importance to the timely evacuation of coastal residents and the assessment of damage to coastal property in the event that a storm makes landfall. Tornado wind speed or intensity is rated using the Fujita scale, named for Tetsuya Theodore Fujita. It is based on the subjective assessment of the damage caused to human and vegetation structures by the tornado. Its original development was linked to the Beaufort wind force scale, named for Francis Beaufort. Ratings range from a minimum of “F0” to a maximum of “F6.” It is also sometimes called the Fujita–Pearson scale to recognize contributions of Allen Pearson, who was director of the National Severe Storms Forecast Center at the time. The scale has since been revised by data gathered from structural engineers and others that suggested that the original wind speeds were too high for categories F3 and above.

To provide accurate estimates for wave height, scientists use Wave Model (WAM). WAM is built around the solution to the action balance equation in terms of an action density function. With the aid of FORTRAN and other programming languages today, WAM is an extremely efficient model.

Hurricane size (extent of hurricane-force winds), local bathymetry (depth of near-shore waters), topography, the hurricane’s forward speed, and its angle to the coast are all factors that affect the surge that is produced. Mathematicians and scientists are working hard to develop a reliable technique for prediction of storm surges. The capability for prediction of hurricane surges is based primarily on the use of analytic and mathematical models, which estimate the interactions between winds and ocean, also taking into account numerous other factors. One of the models used for storm-surge modeling is known as the Advanced Circulation Model (ADCIRC). This is a finite-element circulation model based on the two-dimensional, depth-integrated shallow-water equations representing the conservation laws for mass and momentum. The momentum equations are combined with the continuity equation and result in the generalized wave continuity equation. ADCIRC is implemented in spherical coordinates for this application. As expected, many parameters can be set to optimize running the model for specific applications and locations.

Bibliography

Adam, John. Mathematics in Nature: Modeling Patterns in the Natural World. Princeton, NJ: Princeton University Press, 2003.

Elsner, J. B., et al. “Bayesian Analysis of U.S. Hurricane Climate.” Journal of Climate 14, no 23 (2001).