The discrete charm of rain

Rain is made of drops. That statement may seem trivial, but the consequences of the discrete nature of rain are hardly insignificant. For example, radar measurements of storm intensity, often presented on maps as shown in figure 1, only work because rain falls as drops. Radar sensors—on ground-based instruments and from satellites—estimate the amount of precipitation from the intensity of echoes reflected off the raindrops. Because of their reliance on an indirect measurement, radar measurements can be inaccurate. Nonetheless, the technique allows scientists to study spatial variability over a range of about 500 km; for larger distances, the return echo is too noisy. In contrast, the collection area of rain gauges and the disdrometers that measure the size and speed of falling drops is roughly 50–1300 cm².

The big-picture view of radar helps in many hydrological applications. For example, radar measurements of the spatial distribution of rainfall in an entire river basin can enable hydrologists to better predict floods, the most common natural disaster in the US. The danger arises not only from storms that last a long time or cover a wide area but also from short, localized, intense storms that lead to flash floods.

A knowledge of the drop-size distribution—that is, the probability of having a drop of a given diameter—is essential to obtain an empirical relationship that gives the rainfall rate as a function of the measured echo intensity. After all, the intensity varies as the sixth power of the drop diameter, and the rainfall rate depends on the volume of the drop and the speed at which it falls, which is a function of size. These days, to improve radar estimates of rainfall rates, researchers are using disdrometers to measure the drop-size distributions and are exploring how those distributions vary not only inside a single shower but also under such different meteorological conditions as thunderstorms, midlatitude showers, and monsoons.

Measurement and models

Early investigators of rainfall’s discrete properties exposed flour-covered sheets or absorbent papers for a couple of minutes or so and measured the size of drops by the dough pellets or stains they left. However, the modern rain gauges introduced at the beginning of the 20th century allowed for prolonged and precise measurements of the rainfall rate, defined as the total volume of water passing through the instrument collecting area in a given interval of time. As a consequence, the rainfall rate (or equivalently, the flux, which divides out the measuring area) is by far the most common variable used to describe rainfall, and investigators often view rainfall as continuous. The discrete-drop perspective, however, can be quite different.

Our principal line of research is to mathematically model rainfall. But why is that important? Rainfall, together with other forms of precipitation, is an integral part of the global water cycle. As such, it plays an important role in how energy is exchanged among Earth, atmosphere, and space; those exchanges, in turn, influence atmospheric and oceanic circulations. (See the article by Tristan L’Ecuyer and Jonathan Jiang, PHYSICS TODAY, July 2010, page 36 .) A proper rainfall model would allow scientists to improve the parameterization of global circulation models and to better address a number of questions regarding processes driven by rainfall; relevant issues include soil infiltration, runoff production, the nitrogen content of soil minerals, vegetation growth, and the spread of infectious diseases. In all those applications, the variable describing rainfall is typically the flux, which is clearly linked to drop diameters and the rate at which drops hit the ground. That link, however, is often disregarded in the statistical modeling of rainfall.

A not-so-simple question

To illustrate the importance of a proper theoretical framework for rainfall, let us ask if global warming has increased the duration of rainstorms. The question presumes a well-defined notion of a rain event. Sometimes it is obvious whether two raindrops belong to the same storm. If they are, for example, separated by a number of days, they certainly belong to different meteorological perturbations. But consider a day during which it rains on and off. Perhaps it rains steadily for 30 minutes, pauses or drizzles for an hour, and then rains hard for another 30 minutes. Should the two spells of intense rain be regarded as part of an extended event or as two separate events? Does the amount of rain precipitated in the second 30 minutes depend on what happened in the first 30 minutes? The answer to that question is crucially important for those of us who model rainfall. Finally, consider that measurement instruments are fixed on the ground while rain-producing clouds move through the sky. What seemed like an uncomplicated question is quickly revealed to be not so straightforward.

Scientists have adapted many different frameworks to address our deceptively simple question. The various definitions in a given framework affect both the derived duration of a rain event and the amount of water it precipitates. A statistical model that is reasonable in one framework may be completely inappropriate in another. As a result, comparisons among different studies are problematic. To properly address the question requires a detailed analysis of the interdrop time intervals and drop diameters (see figure 2). In fact, only with such studies can one investigate the time scales relevant to the dynamics between storms and the far more complicated intrastorm dynamics responsible for the pattern of intense and sparse precipitation inside a single meteorological perturbation.

Universal aspects

Some features of a rain model need to take location into account and so are different for midlatitude versus tropical applications. Others are adjusted, for example, when modeling a thunderstorm. But some properties hold everywhere: A rain shower in Rome has much in common with one in New York. Theorists who model rain as a continuous phenomenon have proposed random cascade models and self-organized criticality as mechanisms for enforcing universal properties. Both those models are stationary; in other words, they assume that the fluctuations in the rainfall rate are compatible with a rule that is invariant in time, space, or both. Such models naturally explain why, no matter where you look, rainfall rates at different time resolutions appear to be statistically self-similar: In the jargon of the field, the rainfall rate is said to exhibit scaling behavior.

The discrete view of rainfall offers a different perspective. The self-similarity of the rainfall rate need not be generated from a stationary statistical model; it could also result from a nonstationary one. Such a model would be compatible with the observation that drop-size distributions vary in time rather than being unique. Its universal character derives from the ability to ascribe the variability of drop-size distributions to changes in a few parameters that give a distribution its specific form. By way of analogy, think of distributions of sizes, weights, reading ability, and so forth in a population. All of those are derived from the same basic Gaussian shape by changing the mean value and variance.

In the field of complexity science, scaling or self-similarity is considered to be a manifestation of self-emergent properties and long-range interactions among a system’s constituents. And indeed, those features are part and parcel of random cascade models, self-organized criticality, and other widely applicable mechanisms. On the other hand, nonstationary models, though they give the impression of statistical self-similarity when tested with current standard techniques, do not imply any particular emergent property.

Is rain a complex emergent phenomenon, or is it merely complicated? That is just one of many questions about rainfall and its modeling that await resolution. We think the investigation of the droplike structure of rain is essential if clear, satisfying answers are to emerge.