Imagine that a highly-toxic pollutant is released in the middle of Manhattan on a windy day. What is the appropriate response? Evacuate one square block? Ten square blocks? The whole city? How much time do patients at a hospital five blocks north and 20 blocks east have to get out before the concentration is dangerously high?
|Manhattan. Mid and lower Manhattan as seen from the Empire State Building.
Image Credit: Marcela McGreal (CC BY 2.0)
These are complicated, high-stakes questions. The answer depends on how chemicals in the air spread through the grid of the city. Similar questions come up in many other contexts: How do chemicals spread throughout the bloodstream? In the water supply? In medical tests or research tools that send fluid through long, thin grooves?
In new research published in Physical Review Letters, two UK scientists show how the concentration of a suddenly released pollutant would change over time through Manhattan, or in any other rectangular network. Their work is based on a sophisticated mathematical theory and their technique can easily be applied to study networks of other shapes.
Before we get into the details, it helps to consider the basics of how substances spread through the air. Just as the scent of a burning candle diffuses into a room, chemicals released in the air diffuse into the surrounding environment. This happens because molecules have kinetic energy—they are constantly moving around and bumping into one another. This leads to mixing, as molecules in highly concentrated areas disperse into the surrounding air.
Molecules can also be carried along by the wind. When a fluid (like air or water) moves a substance from one place to another, the process is called advection (similar to the more familiar convection that helps cook your food). A pollutant released into the air on a windy day spreads by diffusion and advection.
The Manhattan scenario isn’t just a question of how a pollutant disperses in an open space, though. It is complicated by the fact that densely packed city streets act kind of like pipes, controlling how wind flows through the city. Diffusion, advection, and the layout of the streets all factor into how a pollutant spreads through the city.
Scientists usually approximate the way something spreads through a network with a Gaussian distribution. Picture a flat park with a hill in the middle. The top of the hill represents the high concentration of the solution at its center of mass. Move away from the center in any direction and the concentration decreases exponentially.
This type of approximation can work well if you are close to the center of mass of the pollutant, but it doesn’t work so well as you move away from it. However, predicting the concentration of a pollutant everywhere—including at the tail ends—is important if you’re talking about a toxic pollutant, especially if it’s harmful even in low concentrations. This motivated mathematicians Alexandra Tzella from the University of Birmingham and Jacques Vanneste from the University of Edinburgh to find a better approximation.
Their work started with a simple model of a rectangular network; imagine a piece of graph paper that stretches out to infinity in all directions. Fluid (air) can flow along the lines (streets) vertically or horizontally. Past research indicates that this is a good model for studying how pollution spreads through dense city centers.
The researchers then applied something called the theory of large deviations to describe dispersion in this simple model. The theory explores the probability of something happening far from the center of mass of a system. It led to an approximation that captures both the Gaussian core and what happens in the tails.
Their results imply that the geometry of the network complicates the way a pollutant spreads out, both at the core (where the concentration is highest) and at the tails (where the concentration is lowest). A moderate amount of time after a pollutant is released, the effects in the tails are especially significant and can be visualized by the diamond and triangular shapes depicted in the images below. In the absence of a network, the corresponding patches would be circular and, in the presence of a background wind, displaced.
|Simulations of the concentration of a pollutant released in a rectangular network,
obtained after a moderate amount of time in the absence of wind (left) and constant
wind (right). The colors go from high concentration (red) to low concentration (blue).
Image Credit: Alexandra Tzella/Physical Review Letters.
In other words, this new model gives a different and more accurate picture of how the pollutant spreads through the city over time. It holds for moderate times and a large range of distances from the center of mass.
This study focused on rectangular networks, but the researchers say their approach will work well for other layouts too. Understanding the intricacies of how advection, diffusion, and geometry affect the way a substance spreads through a fluid network is not just important for emergency planning. It has applications related to blood flowing through veins, water through pipes, and oil through cracks in a reservoir. That’s one of the great things about physics: a situation that looks oddly specific at first is often, at its core, just one application of common processes at work.