Mathematics of clouds
The "Mathematics of Clouds" explores the intricate relationship between mathematics and atmospheric phenomena, particularly cloud formation and behavior. Understanding clouds involves identifying various mathematical patterns and models that describe their dynamics, structure, and the environmental conditions influencing them. For instance, the behavior of clouds can be quantified through parameters like motion, composition, and density, with significant contributions coming from organizations like NASA and NOAA.
Clouds are formed when water vapor condenses into droplets or ice crystals, a process influenced by air temperature and humidity levels. The concepts of saturation mixing ratio and dew point are crucial in determining when clouds will form based on the air's capacity to hold moisture. Atmospheric stability is another critical factor; it refers to how air parcels behave when lifted, leading to different cloud types based on conditions such as the dry and saturated adiabatic lapse rates.
Mathematically, phenomena such as fractal clouds and point clouds are also of interest, highlighting both their geometric properties and their aesthetic appeal. Collectively, these insights contribute to a deeper understanding of weather patterns, climate change, and the fundamental processes that govern the formation of clouds in our atmosphere.
Mathematics of clouds
Summary: The formation and behavior of clouds can be mathematically modeled and studied.
Mathematics has been called “the science of patterns.” In clouds and the atmosphere, generally there is no end to the patterns that may be observed, quantified, and more clearly understood using mathematics. Mathematicians have long modeled the behavior and structure of clouds.
Applied mathematicians continue to develop ways to detect clouds and quantify motion, composition, density, top altitude, and the distance between clouds, among other characteristics. In 1999, the U.S. National Aeronautics and Space Administration (NASA) launched the Multi-Angle Imaging SpectroRadiometer to measure environmental and climate data from nine different angles, including cloud data.
The U.S. National Oceanic and Atmospheric Administration (NOAA) is one of the largest organizations specializing in the study of the environment. In 2010, a NOAA team led by physicist Graham Feingold reported its findings that clouds form synchronous patterns, meaning that individual clouds in a group respond to signals from other clouds, an effect also observed in chirping crickets or flashing fireflies. This research has implications for interpreting climate change data. There are also mathematical objects such as point clouds that are of interest in geometry, imaging, and efficient distribution mining. Fractal clouds are appreciated for their mathematical properties and their artistic qualities.
Water in the Air
Air is composed primarily of nitrogen (78%) and oxygen (21%). Argon comprises nearly 1%, leaving little room for the remaining gasses, including carbon dioxide, ozone, and neon. This recitation, however, is for dry air. Water vapor, the invisible gas from which clouds are constructed, can account for 0% to 4% of any given parcel of air. In order to form a cloud, water vapor must change phase to either liquid water droplets or ice crystals.
The Transformation of Water into Clouds
The amount of water vapor that can be held in a parcel of air is determined primarily by the temperature of the air; warm air can hold more and cold air less. The amount of water vapor held in a parcel of air is identified by the mixing ratio:
The amount of water vapor a parcel of air can hold is called the “saturation mixing ratio”:
Relative humidity is a measure of how much vapor a parcel of air is holding compared to how much it could possibly hold and is expressed algebraically as
The dew point is the temperature at which a parcel of air becomes saturated. At this point, the saturation mixing ratio and the actual mixing ratio are equal to one another, and the relative humidity is therefore 100%. A further drop in temperature should produce condensation as water changes phase from vapor to liquid cloud droplets or solid ice crystals—a cloud is born.
The Unstable Atmosphere
Clouds are often the result of lifting in the atmosphere. When a parcel of air rises, it generally cools, and this cooling produces condensation. The way in which the lifting is accomplished can lead to dramatic differences in the appearance of the cloud. When whole layers of air are gently lifted in an atmosphere that is stable, stratus clouds are formed, whereas the more dramatic vertical structure of a cumulus cloud comes from runaway convection, a self-perpetuating process that can build clouds more than 12 kilometers (km) or 40,000 feet tall.
What is a stable atmosphere? Temperatures generally decrease with height. The rate of change is, of course, variable but it is referred to as the “lapse rate” (Γ) of the atmosphere. A parcel of air, distinct from the air that surrounds it, may be forced to rise or descend and will cool or warm as a result. Pressure generally decreases with height, and a parcel that rises into a zone of lower pressure will expand, doing work on the environment and therefore cooling. The rate at which a parcel of air cools as a result of this sort of ascension is known as the “dry adiabatic lapse rate” (Γd) which is approximately 10 degrees Celsius per km. When the dew point is reached in the parcel and condensation occurs, latent heat is released as a result of the phase change and the parcel is warmed.
The result is a lower lapse rate, the saturated adiabatic lapse rate (Γs). The saturated lapse rate depends on the amount of moisture being condensed but 6 degrees Celsius per km may be used as a rough estimate.
Now if Γ<Γd, the atmosphere is stable because unsaturated air that is made to rise will cool at approximately 10 degrees Celsius per km and will find itself in air that is increasingly warmer than itself. The greater the difference Γd-Γ, the greater the force restoring the parcel to its previous altitude. The force may be quantified as
where g is the gravitational constant, T is temperature, and δz is a small upward displacement of the parcel from its equilibrium level. Consider the implications of a temperature inversion in which temperature actually increases with height and Γ is a negative quantity. Now consider a situation in which the atmosphere cools strongly with height, that is Γ >Γd. Then, the restoring force becomes negative. Air that rises becomes warmer than its surroundings and so continues to rise. This leads to the runaway convection that builds the towering cumulonimbus clouds that can produce thunderstorms, lightning, and hail.
Bibliography
Adam, John. Mathematics in Nature: Modeling Patterns in the Natural World. Princeton, NJ: Princeton University Press, 2003.
Feingold, Graham, et al. “Precipitation-Generated Oscillations in Open Cellular Cloud Fields.” Nature 466, no. 12 (August 2010).
Wallace, John M., and Peter V. Hobbs. Atmospheric Science. Burlington, MA: Academic Press, 2006.