Mathematics of climate change analysis

Summary: Mathematicians and scientists use sophisticated models to track and predict global climate change.

The term “climate change” refers to the changing distribution of weather patterns. Climate is considered to be the average of 30 years of weather. In other words, climate is the distribution from which weather is drawn. Global warming refers to the change in climate in such a way that warmer weather is increasingly likely. In fact, it is not just the warming itself that is of concern but also the rate of change of the warming process since ecological systems typically cannot adapt to a rapidly changing climate. According to the 2007 Synthesis Report by the Intergovernmental Panel on Climate Change (IPCC), “Warming of the climate system is unequivocal, as is now evident from observations of increases in global average air and ocean temperatures, widespread melting of snow and ice and rising global average sea level.” The main cause of changing climate is the increasing atmospheric concentrations of greenhouse gases (carbon dioxide, methane, and nitrous oxide), which effectively act as a blanket over the atmosphere.

The IPCC report noted, “There is very high confidence that the net effect of human activities since 1750 has been one of warming.” The evidence for the warming of the climate includes more than the measurement of global average temperatures, as physical evidence such as glacier melt also exists. Most predictions of global warming are based on data models, and mathematics is used extensively to measure and quantify atmospheric carbon dioxide and aerosols, which are believed to add to the problem. The National Oceanic and Atmospheric Administration (NOAA) and the U.S. National Aeronautics and Space Administration (NASA) are two large federal agencies that are involved in the collection, analysis, and dissemination of climate data. They employ a diverse range of mathematicians, statisticians, scientists, and others, and they have partnerships with many academic institutions, government agencies, and businesses around the world. Researchers do agree, however, that the current and future consequences of climate change disproportionately impact the world’s poor.

Climate as a Distribution

In order to understand global warming, it is important to understand that the term refers to a distribution. It is easy to dismiss the notion of global warming on a cold winter’s day, a mild summer day, or any day where the weather is cooler than expected. In fact, an unusually hot or cold event is not evidence for or against global warming. To aid in understanding climate as a distribution, consider a set of 20 cards numbered 1–20. One card will be drawn at a time with replacement. The value 10.5 is the average of these cards; a card selected below 10.5 will represent a below-average temperature for the day, and one selected above 10.5 will represent an above-average for the day. Further, the farther away the value of a card is from 10.5 will represent a larger deviation from average temperature. This example represents a stable climate. Some days are colder than average, and others are hotter than average. But, over time, roughly an equal number of colder days and hotter days occur. Moreover, if the value of the cards were unknown, basic statistical sampling ideas could be used to estimate the average.

To represent a changing climate, start with the same set of cards and consider values below or above 10.5 as a colder or hotter day. But this time, every time a card is drawn and replaced, the next higher card will be added to the set. For example, after the first card is drawn, a 21 will be added to the set, then a 22 will be added after the second draw, then a 23 after the third, and so on. At first, this change would be barely noticed if at all since the cards drawn will be roughly equal above and below 10.5. After some time, however, one would start to question the assumption that 10.5 is the average. In this case, if the values of the cards were unknown, basic statistical sampling techniques could not be used to estimate the average since the average is in fact changing. In this example, if 10.5 is taken as the average, then values below 10.5 still occur but are becoming less likely. In other words, record lows can still occur—and will still happen—even though climate is warming.

To complicate this example further, consider this same experiment being performed simultaneously by 2000 people to represent different locations around Earth. When the set of cards have values from 1 to 100, one individual would have only a 10% chance of drawing a card below 10.5, but it is expected that approximately 200 of the 2000 experiments will draw a card below 10.5. In other words, even though climate is warming, there will still be places that have colder than average days.

In terms of actual weather, consider Figure 1, which provides the average monthly temperature anomalies in degrees Celsius for December 2009 compared to the average from 1951 to 1980. The month of December was slightly colder for most of the United States. The overall average for the world was 0.60 degrees Celsius higher than the baseline years. One month, or even one year or a few years, of above average temperatures does not provide conclusive evidence for or against global warming as these abnormalities could be explained as normal variations in weather.

Evidence of Warming

Calculating global mean temperatures each year provides one form of evidence for global warming. For example, Figure 2 displays mean global temperature anomalies dating to 1850. Even though the overall trend is upward, variation from one year to the next can go in either direction. Gerald Meehl, who has a Ph.D. in climate dynamics and works at the National Center for Atmospheric Research (NCAR), collected information from 1800 weather stations across the United States that have been operating since 1950. He and his colleagues looked at the ratio of record highs to record lows and grouped the ratios by decade. From the 1950s through the 2000s, the ratio of record highs to lows was 1.09:1, 0.77:1, 0.78:1, 1.14:1, 1.36:1, 2.04:1. From the 1950s through the 1980s, the ratios might be considered to be in the range of normal variation for a stable climate. On the other hand, by the 2000s, it certainly appears that the observations no longer represent normal variation, and that the climate distribution is getting warmer.

In Figure 2, the baseline is the average from 1961 to 1990, with regression lines for different time periods from 1850 to 2009, 1910 to 2009, 1960 to 2009, and 1985 to 2009. The data are the HadCRUT3 data set provided by the University of East Anglia Climatic Research Unit.

Beyond data, a warming climate should present physical evidence in the form of melting ice. Figure 3 is one of a number of glacier image pairs, which are pictures of glaciers taken from the same vantage but 40–100 years apart. The change is striking. Where there was once ice, there is now ocean water with the glacier retreating about seven miles. In the foreground, thick vegetation exists where there was once rock. This change is because of microclimate changes since the ice is no longer cooling that area. Along with melting glaciers, Arctic sea ice is decreasing rapidly and permafrost is melting. In fact, the entire village of Newtok, Alaska, must be relocated because the loss of permafrost has allowed the banks of the Ninglick River to erode.

Melting ice is just one source of evidence of a changing climate. During most of the twentieth century, sea level was rising at a rate of 0.07 inches per year, but by the 1990s that rate increased to 0.12 inches per year. In 2006, the National Arbor Day Foundation updated its plant hardiness zone maps, and most of the zones shifted northward. In other words, many plants can now be grown where they could not before because of their cold hardiness. There have already been observed shifts in species ranges, a northward shift, as well as shifts in phenology (seasonal biological timing) toward events such as early blooming. In fact, many species have seasonal behavior that is occurring 15–20 days earlier than the behavior occurred in the mid-twentieth century.

The general trend of warming is only part of the story. If the planet warmed a degree or two over millions of years, then ecological processes could adapt and societies could migrate. Figure 2 has least squares regression lines calculated over the time periods of 1850–2009, 1910–2009, 1960–2009, and 1985–2009. The four regression lines are as follows:

In each case, the slope of the line, with units of degrees Celsius per year, is increasing as the time periods are shortened toward more recent years. More importantly, the 95% confidence intervals for the slopes are (0.00387, 0.00495), (0.00657, 0.00844), (0.01151, 0.01577), (0.01254, 0.02347), respectively. The first three intervals do not overlap, and so the slopes of the lines are significantly different. This provides evidence not only for overall warming but also that the rate of warming is increasing. Some species, trees for example, will simply not be able to adjust their ranges quickly enough to adapt to the warming climate.

Climate Science

Climate models, which incorporate mathematical topics such as dynamical systems, statistics, differential equations, and applied probability, are used to predict future global average temperature.

Mathematician Ka-Kit Tung, in his book Topics in Mathematical Modeling, provides a simple climate model. The model is

The left-hand side represents change in temperature. There are three basic terms on the right-hand side that contribute to temperature change. The first term has incoming solar radiation at the top of Earth’s atmosphere,

Q s(y),

where the s(y) term distributes the radiation differently depending on the latitudey=sin⊖ with ⊖ representing latitude. The term also takes into consideration how much radiation is absorbed

(1 - α(y))

where α(y) is the fraction reflected or albedo. The next term,

I(y)

represents outward radiation, and the last term,

D(y)

represents heat transportation from warmer latitudes to colder latitudes. In Tung’s textbook, this simplified model is analyzed to gain understanding of possible locations in ice lines.

The more complex computer simulations that model climate are built with assumptions related to population growth and societal choices, such as energy use or technological change. These assumptions are then used to predict how greenhouse gases will increase. The effect of increased greenhouse gases in trapping heat is well understood, and in terms of the simple climate model above, the increase in greenhouse gases decreases outward radiation. Beyond that, the increase in carbon levels itself is a problem as oceans work to absorb some of this carbon in the form of carbonic acid. The increase in carbonic acid in the oceans increases the acidity levels, which damages coral, crustaceans, sea urchins, and mollusks.

For each scenario, many different models are considered, and the predictions are averaged to produce the graph on the left side of Figure 4. The three higher curves illustrate the average warming. On the right side of the graph is a range based on the various models. A distribution has been created, and based on the graph, one could say that, by 2100, global mean temperatures will increase between approximately 1.5 degrees Celsius and 3.5 degrees Celsius, but the distribution around the three scenarios presented is from approximately 1 degree Celsius to 6 degrees Celsius. The right side of Figure 4 presents the predicted temperature changes as a distribution across Earth, and it is predicted that the Arctic region will warm more than the equatorial region.

A key complication in climate modeling is the existence of feedback loops. A feedback loop is created when a change in one factor causes a change in a second factor that then either reinforces or diminishes the change in the first factor. While each scenario sets out greenhouse gas levels, the models must then attempt to take into account how warming may, in fact, increase warming or decrease warming. For example, one positive feedback loop involves melting ice. As ice melts, the Earth’s albedo (reflectivity) changes so that less solar radiation is reflected out to space. In the climate model above, the σ(y) term is decreased so that more solar radiation is absorbed. In other words, as the planet warms, ice melts. However, there are now fewer reflective white surfaces and more dark surfaces, which will then absorb even more solar radiation and increase the planet’s warming. Another potential positive feedback loop arises from melting permafrost. As the permafrost melts, partially decomposed organic matter will decompose more fully and release carbon into the atmosphere. Even more uncertainty arises with the effect of clouds. Low clouds tend to cool by reflecting more energy than they trap, while the reverse is true for high clouds. As surface temperature increases, there is increased evaporation from the oceans, creating more water vapor and hence clouds. But the type of clouds that arise will depend on whether this is a positive or negative feedback loop.

Of course, to many people, an increase of a few degrees Celsius does not seem to be drastic enough to impact life on Earth significantly. But consider that during the twentieth century, global average temperatures increased by less than 1 degree Celsius. Nevertheless, there has already been observed disappearing glaciers, loss of Arctic sea ice, changing species habitat and phenology, and a new plant hardiness map. In fact, a difference of approximately 0.2 degrees Celsius was the difference between the Medieval Warm Period (c. 950–1250) and the cooling period (c. 1400–1700). The warm period led to the Norse migrating to Greenland and bountiful harvests and population increases in Europe. This period was followed by a cooling period that led to the collapse of the Norse Greenland society and starvation in Europe.

Impacts of Climate Change

The general consensus in the scientific community in 2010 is that warming has occurred and will continue to take place even with changes. Debate continues on precisely how much warming will occur and the exact nature of the ramifications. The questions are by how much, and what should people expect to happen? Species ranges are already changing, and, in some cases, species ranges are disappearing as appears to be the case for polar bears. Unfortunately, the speed of warming will lead to some species not being able to change their range quickly enough, resulting in extinction. The changing phenology is already causing ecological disruption. Some plants are blooming earlier, but the species that feed on them are not arriving earlier, leading to decreased food supply.

As Figure 3 shows, ice is melting and more of that is expected. The loss of Arctic ice will decrease polar bear populations. The melting glaciers of the Tibetan Plateau are of particular importance. These glaciers are responsible for supplying water to about 2 billion people, and data suggests that the Tibetan Plateau is warming twice as fast as the global average. Once these glaciers are gone, so is the water supply. The melting of glaciers and land ice, along with the thermal expansion of water, will raise sea levels. One example is Bangladesh, which faces severe threats from sea level increases since millions of people live along a coastline that may be underwater in the future.

There are additional predictions as of 2010, based on models and scientific expertise. An increase of 2 degrees Celsius from pre-industrial levels would lead to a fall in agricultural yields in the developed world, a 97% loss of coral reefs, and 16% of global ecosystems transformed. With an increase of 3 degrees Celsius, few ecosystems could adapt and an additional 25–40 million people would be displaced from the coasts because of sea level rise. If global average temperatures rose to 4 degrees Celsius above pre-industrial levels, entire regions would be out of agricultural production, including Australia.

Climate Change and Societies

The joint science academies’ statement on sustainability, energy efficiency, and climate protection issued in 2007 by the G8 nations and Brazil, China, India, Mexico, and South Africa, said that, “Many of the world’s poorest people, who lack the resources to respond to the impacts of climate change, are likely to suffer the most.” The warming of the planet will have some advantages and disadvantages, although there will be more disadvantages. Some warmer climate species will have expanded ranges and be able to thrive, while arctic species may lose their entire ecosystem. Some countries will be impacted more than others, and the wealthier countries will have a better ability to adapt. The examples that have been given here of societies that already have been or will likely be impacted are all examples of poorer societies.

The people of Newtok, Alaska, are poor; in 2010, Bangladesh ranked 183 in the world in terms of GDP per capita; and there is considerable poverty in regions in and around the Tibetan Plateau. Part of the tragedy is that these are not the people who are largely responsible for increasing greenhouse gases. China and the United States are the largest emitters of carbon dioxide, but on a per capita basis, the United States far exceeds China. In general, it is the industrialized nations that contribute the most to greenhouse gases. Figure 4 provides different models for future climate change, and these are primarily based on the models that predict future greenhouse gas emissions, and it is the more industrialized nations that have the resources to make reductions in these emissions.

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