This April, with headlines about climate change appearing almost daily, climate scientists and mathematicians got together for a unique workshop on how climate models might be improved. These models, although highly mathematical, are often designed, interpreted, and tweaked by people whose main expertise is not mathematics. Participants agreed that mathematicians can and should play a more active role in clarifying the meaning of climate models, figuring out how to aggregate the results of different models, and adding new components (such as models of sea ice) that are not treated adequately in the existing models.

The Symposium on Climate Change, sponsored by the Mathematical Sciences Research Institute
in Berkeley, featured two separate but intertwining events. On April 11, Congressman Jerry McNerney and California state legislator Ira Ruskin joined six climate and energy experts for a public forum in San Francisco that drew about 320 people. On the following two days, about 75 people met at MSRI for a scientific symposium directed more specifically to problems of climate modeling.

Climate change (which includes, but is not limited to, global warming) has finally begun to enter public discourse not as a political slogan, but as a reality that humans will have to deal with over the next century. Last fall, the British government issued the Stern Review, a first attempt to quantify the economic impact of climate change. This spring, the Intergovernmental Panel on Climate Change (IPCC) issued a series of assessments in which they concluded, among other things, that the climate is warming and that it is "very likely" that most of the warming is caused by humans. Even the U.S. Supreme Court got into the act, with an April 2 ruling that carbon dioxide emissions are subject to regulation as a pollutant.

Mathematicians have been largely absent from the public debate---even though many policy decisions will hinge on the interpretation of complicated, highly mathematical climate models. "I think that mathematicians as a community are reluctant to get involved in something that is politically so hot," says Christopher Jones of the University of North Carolina, an organizer of MSRI's scientific symposium. "But we can't afford not to get involved." The primary goal of the symposium was to explore ways to attract mathematicians to what is, for them, a largely unexplored branch of science. "We need a stamp of approval that tells mathematicians, especially young mathematicians, that this is an okay program of research," says Mary Lou Zeeman of Bowdoin University, who helped to organize the scientific symposium.

The Symposium on Climate Change was organized with unusual rapidity. The seeds were planted in November, when McNerney, who has a PhD in mathematics and worked for many years on alternative sources of energy, won election to Congress in California's 11th district (south of Berkeley). "We got interested in hosting an event with McNerney, and then many things began suggesting climate change as a subject," says David Eisenbud, the departing director of MSRI. Eisenbud and Jones, who was at MSRI for a semester-long program on dynamical systems, began inviting speakers in February, only two months before the event. "It was a miracle that it all came together, but that's a sign of how much people care about this topic," Jones says.

Eisenbud invited Inez Fung, co-director of UC Berkeley's Institute for the Environment, to be the keynote speaker at both the public forum and the scientific symposium. Fung presented a compelling scientific case that global warming is real and has been caused by human activity. The dossier of evidence, she said, contains at least five clues:

*the extremely tight correlation between historical carbon dioxide levels and historical climate, as determined from ice-core samples;

*the dramatic increase in measured concentrations of carbon dioxide (and the even more pronounced increase in methane concentrations, a greenhouse gas that is 20 times more potent than carbon dioxide);

*isotopic evidence that the additional carbon dioxide has come from fossil-fuel burning;
an observed increase in temperature over recent years, with 11 of the 12 warmest years on record occurring since 1995; and

*the unanimity of climate models in forecasting an increase in temperature over the next century (with the increases ranging from 1 to 12 degrees Centigrade).

The clarity of global warming dissolves into frustrating ambiguity when climate modelers are asked to predict the future, however. One speaker after another at the symposium lamented the use of climate models as crystal balls. "There are demands being made on these climate models that the models weren't constructed for," says Doug Nychka, a statistician at the National Center for Atmospheric Research.

For example, the IPCC report relies on 24 climate models, most of which were developed by various national weather services (NCAR being one of them). These models all share some common physics: the conservation of mass and energy in the atmosphere and in the ocean, radiative forcing from the sun, and so on. Nevertheless, they are very different in their detailed assumptions. Yet the IPCC report averages them all---a process comparable to averaging apples and oranges to determine what a generic fruit looks like.

To make things worse, every individual model contains vast uncertainties. No model includes a realistic description of sea ice, according to Cecilia Bitz of the University of Washington---a particularly troublesome problem, because the ice in the Arctic Ocean and Greenland is melting much more rapidly than the models predict. The descriptions of the ocean in most climate models are highly simplistic, and most ocean models treat the atmosphere equally roughly. And then there is the modelers' dirty secret: Every model has parameters that are "tuned" to make the output match observations more closely. In some cases, as pointed out by William Collins of UC Berkeley, the tuning causes the models to violate the very physics that has been so painstakingly included.

Finally, even a perfect model may still produce a range of possible answers, because of what Jim McWilliams of UCLA called "irreducible uncertainty." Weather models can project wind speeds a few hours into the future, but the wind speed over Scotland in 2050 may be essentially unknowable.

Most of the speakers expressed their belief that these problems can be addressed mathematically. Statisticians may turn up better ways to combine models than simple brute-force averaging. Tuning should be done openly, not clandestinely, with attention to defining the parameter space in which the values are being chosen. Above all, the uncertainties should be embraced, and not hidden. "Somehow, you have to shed the idea that the models represent an exact state of the environment," Jones says. "My suspicion," says Nychka, "is that there will be a continuum of models, and a person will have to make a choice, or else cast the results as a distribution."

The symposium also included some impressive examples of mathematical climate analysis. Ben Santer of Lawrence Livermore National Laboratory explained how to attribute climate changes to human or nonhuman causes (such as solar fluctuations or volcanoes), by combining several different climate variables into an "anthropogenic warming pattern." Max Aufhammer of UC Berkeley merged economics and climate science to quantify the effects of aerosols and temperature increases on rice harvests in Asia. (Both had a negative effect; the atmospheric haze, however, had a stronger effect than warming.)

Zeeman says that the symposium held two significant surprises for her. One is that climate modelers already have models of "intermediate complexity," which they use to build intuition but don't publish. "Those are the models dynamical systems people would love to get their hands on," she says. "Could we get the modelers to make them available?"

The second surprise, Zeeman says, is "the role that mathematicians can serve as translators." For example, mathematicians can help atmospheric scientists talk with oceanographers, or can help formulate climate models in terms of economic risk. According to Hans Kaper, a program director for applied mathematics at the National Science Foundation, economists and businesspeople must be brought into the study of climate change. "Their input will give you credibility to the outside world," Kaper says.

Readers who want to learn more about climate modeling won't have long to wait. Zeeman is organizing a SIAM minisymposium on climate change for next January's Joint Mathematics Meetings, where Fung is scheduled to give a plenary talk. Other workshops and summer schools will surely follow. (MSRI, the Institute for Mathematics and its Applications, and the Newton Institute for Mathematical Sciences have all expressed interest.) Says Nychka, "There's a lot of math to do here, and some of it, I think, has not been invented yet."

Dana Mackenzie writes from Santa Cruz, California.