Please enjoy Gillen D’Arcy Wood discussing his new book TAMBORA: The Eruption That Changed the World, due out from Princeton University Press in May.
Gillen D’Arcy Wood discusses his new book TAMBORA: The Eruption That Changed the World
The Mystery of Snowfall Explained
The view from my backyard.
We’re still in the midst of a blizzard — indeed Princeton University Press is actually closed for the day as we huddle down at home with 810 inches and counting on the ground. But what better time to share this opinion piece from Ian Roulstone, meteorologist and author of Invisible in the Storm. While we’re waiting for an opportunity to get out there and shovel, we hope you will enjoy this article about the strange world of snowfall prediction and why it can’t be more accurate:
A snowcovered landscape is one of the classic images showcasing the beauty of weather on Earth. We are awed by the grandeur of whitecapped mountains and the almost magical quality of snowcovered trees. We are also frustrated when the tempests of winter reach far and wide, striking as they have done this year in America’s southern states.
When it comes to forecasting the likelihood of a blizzard, the weather anchors know what to say. But when asked to predict how much snow will actually accumulate, they will give estimates. Why?
Hurricane Sandy and Global Warnings, an original article by Ian Roulstone and John Norbury
Hurricane Sandy and Global Warnings
Ian Roulstone and John Norbury
There are many heroes in the story of Hurricane Sandy, but we arguably owe the greatest debt of gratitude to mathematicians who wrangle massive amounts of data to improve the accuracy of our weather predictions. Two devastating storms, decades apart, provide a fantastic snapshot of how weather prediction has improved thanks to the introduction of computational mathematics over the last century.
Just over 75 years ago, on September 9^{th} 1938 above the warm tropical waters near the Cape Verde islands, a storm gathered. As the weather system intensified, it was ushered westward by the prevailing largerscale ridge of high pressure over the Atlantic. By the 16^{th} the storm had become a hurricane, and the captain of a Brazilian freighter caught sight of the tempest northeast of Puerto Rico. He radioed the U.S. Weather Bureau to warn them of the impending danger – no satellites or sophisticated computer models to help the forecasters in those days.
A deep trough of low pressure over Appalachia forced the storm northward, avoiding the Bahamas and Florida, and towards the northeastern seaboard of the United States. The forecasters were relying on realtime reports of the storm’s progress, but it advanced at an incredible pace, moving northward at nearly 70mph. By the time the Weather Bureau realised it was on a collision course with Long Island it was too late. The death toll from the Great New England Hurricane approached 600, with over 700 injured, and the damage was estimated at $308 million – or around $4.8 billion at today’s prices.
History very nearly repeated itself on October 29 and 30^{th} last year, when Hurricane Sandy slammed into New Jersey. Meteorologists referred to Superstorm Sandy as a “multihazard event”, with major damage resulting from wind gusts, from high seas, from a tidal surge, from heavy rain, and even from driving snow. The number of fatalities in the U.S., attributed either directly or indirectly to Hurricane Sandy, were around 160: a tragedy, but mercifully fewer than the number killed by the Great Hurricane of 1938.
It is almost certain that timely warnings averted greater catastrophe last year. Unlike the storm of 1938, which caught forecasters by surprise, one of the most remarkable features of the forecast of Hurricane Sandy from the European Centre for MediumRange Weather Forecasts (ECMWF) was the prediction made on October 21^{st}, 36 hours before Sandy even formed, of a oneinfour chance of a severe storm, centred on New York, on October 30^{th}.
ECMWF routinely produce two types of forecast for 10 days ahead. As they state in a recent newsletter “The ECMWF global mediumrange forecast comprises a highresolution forecast (HRES) and an ensemble of lowerresolution forecasts (ENS)”, and it was the ENS that helped forewarn of Sandy.
To calculate a forecast we use supercomputers to solve seven equations for the seven basic variables that describe weather: wind speed and direction (3 variables), pressure, temperature, air density, and humidity. The equations governing weather are highly nonlinear. This means that the ‘cause and effect’ relationships between the basic variables can become ferociously complex. To deal with the potential loss of predictability, forecasters study not one, but many forecasts, called an ensemble. Each member of the ensemble is started from a slightly different initial state. These different initial states reflect our ignorance of exactly how weather systems form. If the forecasts predict similar outcomes, we can be reasonably confident, but if they produce very different scenarios, then the situation is more problematic.
In the figure below the ensemble of forecasts for Sandy, starting at midday on October 23^{rd} indicates the high probability of the ‘left turn’ and the most probable landfall – information that helped save lives. The inset at top right shows the strike probability chart that highlights the region around New York within which there is 25% chance of a severe storm by midnight on October 30^{th}. This forecast was computed from an earlier ensemble starting at midday on October 21^{st} and gave forecasters the vital “headsup” of severe weather striking a highly populated area.
The science of weather and climate prediction was utterly transformed in the second half of the 20^{th} Century by highperformance computing. But in order to fully exploit the computational power, and the information gathered by weather satellites and weather radar, we need mathematics. As we explained in our article in Scientific American [hyperlink] math quantified the choreography of Hurricane Sandy. And to account for the everpresent uncertainties in the science of weather forecasting, math delivers the tools to analyse the predictions and to highlight the dangers.
Lives were saved because of the quality of our weather forecasts, which are made possible by an international group of mathematicians and weather prediction centers. The math that helps us quantify uncertainty in weather forecasting is being used to quantify uncertainty in climate prediction. It is easy to underestimate the value of this research, but investing in this science is vital if we are to stave off future billions in damages.
For further insights into the math behind weather and climate prediction, see Roulstone and Norbury’s new book Invisible in the Storm: The Role of Mathematics in Understanding Weather.
Climate Change: a Movie and the Math by Ian Roulstone and John Norbury
Climate Change: a Movie and the Math
By Ian Roulstone and John Norbury
Next week the Intergovernmental Panel on Climate Change (IPCC) will release the first of three reports that constitute their Fifth Assessment Report on climate change. This first report, The Physical Science Basis, will cover a huge range of topics from the carbon cycle to extreme weather. But climate prediction also relies heavily on mathematics, which is used to quantify uncertainties and improve the models.
The role of math is illustrated by a remarkable video of our everchanging weather. Last month the National Oceanic and Atmospheric Administration (NOAA) decommissioned Geostationary Operational Environmental Satellite 12 (GOES12), which monitored our weather for the past 10 years from its isolated vantage point 36,000 kilometers above America and the Atlantic Ocean.
GOES12 had seen it all – from wildfires, volcanic ash, and landscape parched by drought, to Hurricanes Ike, Katrina and Sandy, and the blizzards that gripped the central United States in the winter of 200910. NOAA created a video – 187 seconds and 3641 images – one snapshot from each day of its operational life, which amounts to 10 years’ weather flashing before our eyes in just over 3 minutes. It’s dramatic and amazing:
In Scientific American, Evelyn Lamb commented on how this video highlights “a tension between the unpredictability of the weather and its repetitiveness”. Even after a few seconds it becomes clear that the patterns revealed by clouds differ from one part of the globe to another. Great towering cumulonimbus bubble up and unleash thunderstorms in tropical regions every day, while in more temperate midlatitudes, the ubiquitous low pressure systems whirl across the Atlantic carrying their warm and cold fronts to Europe. The occasional hurricane, spawned in the tropics, careers towards the United States (Hurricane Sandy can be seen at about 2’50’’). But the mayhem is orchestrated: the cyclones almost seem like a train of ripples or waves, following preferred tracks, and the towering storms are confined largely to the tropics.

In fact, this movie is affording us a glimpse of a remarkable world – it is a rollercoaster ride on the ‘weather attractor’.
An ‘attractor’ is a mathematician’s way of representing recurring behavior in complex systems, such as our atmosphere. A familiar illustration of an attractor can be seen in the figure below, and it is named after one of the fathers of chaos, Edward Lorenz.

It is impossible to illustrate the weather attractor for the atmosphere in terms of a simple threedimensional image: Lorenz’s very simple model of a circulating cell had only three variables. Our modern computer models used in climate prediction have around 100 million variables, so the attractor resides in a space we cannot even begin to visualise. And this is why the movie created by NOAA is so valuable: it gives us a vivid impression of the repetitiveness emerging from otherwise complex, chaotic behaviour.
Weather forecasters try to predict how our atmosphere evolves and how it moves around the attractor – a hugely difficult task that requires us to explore many possible outcomes (called an ensemble of forecasts) when trying to estimate the weather several days ahead. But climate scientists are faced with a very different problem: instead of trying to figure out which point on the 100 milliondimensional attractor represents the weather 100 years from now, they are trying to figure out whether the shape of the attractor is changing. In other words, are the butterfly wings ‘folding’ as the average weather changes? This is a mathematician’s way of quantifying climate change.
If 100 years from now, when a distant successor of GOES12 is retired, our descendants create a movie of this future weather, will they see the same patterns of recurring behaviour, or will there be more hurricanes? Will the waves of cyclones follow different tracks? And will tropical storms be more intense? Math enables us to “capture the pattern” even though chaos stops us from saying exactly what will happen, and to calculate answers to these questions we have to calculate how the weather attractor is changing.
This article is crossposted with the Huffington Post: http://www.huffingtonpost.com/ianroulstone/climatepredictionmathematics_b_3961853.html
For further insights into the math behind weather and climate prediction, see Roulstone and Norbury’s new book Invisible in the Storm: The Role of Mathematics in Understanding Weather.
Q&A with Ian Roulstone and John Norbury, authors of Invisible in the Storm
We are publishing Invisible in the Storm by Ian Roulstone and John Norbury next month. The book explains how mathematics and meteorology come together to predict the weekly weather, prepare us for incredible weather events like Hurricane Sandy, and contribute to our understanding of climate change. They kindly answered a few of my questions for the Princeton University Press Blog:
1. I’ll start with the thing everyone is talking about. It seems like extreme weather more prevalent in recent years. With Hurricane Sandy and the recent unprecedented Nor’Easter behind us (ed. note: I’m writing from NJ), it bears asking whether the future holds more extreme weather? Can mathematics help answer this question?
Mathematicians think about weather and climate in an unusual way. Our everchanging weather can be visualized as a curve meandering through an abstract mathematical space of logically possible weather. Any one point of the curve corresponds to a particular state of the weather. The surprise is that the curve does not wander around randomly–patterns emerge. One part of the pattern may correspond to ‘warm and dry’ and another part to ‘cold and wet’. Predicting changes in the weather for the week ahead involves working out if the curve will drift from one part of the pattern to another. Understanding climate involves working out how the pattern itself will change.
2. So, is the pattern changing toward more extreme weather or can we not answer this question yet?
If we compare the results from different climate models (from different research institutions and weather bureaus around the world), then they show an increase in global average temperature over the next century. However, this could lead to quite different conditions in different parts of the world. For example, if the Gulf Stream was weakened, Europe could experience colder weather. However, we know our models are not perfect, and mathematics is helping us to understand the errors that are inherent in the compuergenerated simulations. This work is important as it will help us to estimate the likely extremes in weather and climate with greater confidence.
3. To return to the end of your first answer, how can mathematics detect climate change?
Climate depends on many factors: the atmosphere, the oceans, the icecaps, land usage, and life in all its forms. Not only are there many interconnections between these systems, the timescales over which changes occur vary enormously: trees can be felled in a few hours or days–changing the character of the local landscape quickly–but carbon stocks in soil vary much more slowly, perhaps over several millennia. To predict future climate we have to account for the short and longtimescale effects, and this can pose subtle problems. Mathematics helps us to quantify how the different timescales of the changes in the components of the Earth system impact on predictions of climate change. Using mathematics, we calculate how cloud patterns change over the next five days, and how the Arctic icesheet changes over the next five years.
4. How does mathematics help forecasters predict the weather for the week ahead?
One of the main sources of information for a new forecast is yesterday’s forecast. New generations of satellites gather more, and more accurate, readings, ranging from the sea surface temperature to the state of the stratosphere. Data is exchanged freely around the world among weather bureaus; global weather prediction relies upon this protocol. However, we will never have perfect, complete weather data, and this is why we need mathematical techniques to combine the new information with the old.
5. Years ago, it seems like weather was much simpler — will it rain, snow, sleet, or be sunny? These days, mathematics enables weather forecasters to forecast more than rain or shine: the computer simulations are useful for predicting everything from pollen levels and pollution to flood risk and forest fires. Can you explain how mathematics is part of this?
Mathematics is the language we use to describe the world around us in a way that facilitates predictions of the future. Even though hay fever and floods are very different natural phenomena, predictions of their occurrence can be made using mathematical models. Weather forecasters are actively engaged in combining their predictions with models that help us forecast weatherrelated phenomena.
6. It sounds like a oneway street — mathematics helps us understand meteorology — but you note in the book that the relationship is more reciprocal. Can you elaborate?
To most of us, meteorology and mathematics are a world apart: why should calculus tell us anything about the formation of snowflakes? But mathematics has played an evergrowing and crucial role in the development of meteorology and weather forecasting over the past two centuries. Our story explains how mathematics that was originally developed for very different purposes, such as studying the ether or the dynamics of the solar system, is now helping us to understand the dynamics of the atmosphere and oceans, and the changes in our climate. And it is a twoway process: the diversity of phenomena we seek to quantify means we have to describe them using new mathematical ideas that capture the rapid changes, the slow changes, the randomness, and the order, we observe.
7. Is there a current area of mathematical/meteorological research that you are particularly excited about? Ie – what’s next?
There’s one particular subject that’s attracting a lot of attention right now: it is called data assimilation. This is the part of the forecasting process where new observational information about the state of the atmosphere is combined with the previous forecast, to give us the starting conditions for the next forecast. Improvements to this part of the forecasting process nearly always lead to better forecasts. And the technology applies to modelling the climate too. In this case, we’re not so much interested in whether we have the correct starting conditions, but whether we have used the correct values of the parameters that define the processes and physical phenomena which affect climate–for example, the carbon cycle, from leaves to biomass and carbon dioxide. One of the reasons this is a really exciting area for mathematicians is that we need some new mathematical ideas to analyse these problem. At the moment we rely heavily on math that was developed over 50 years ago–and it works very well–but as we strive to increase the detail we want to represent in our weather and climate models, we have to unravel the Gordian knot that ties together the many different parts of the Earth system we have to represent.