While it is true that the sciences already comprise some of the most mathematically intense disciplines in existence, it is also true that most scientists are not mathematicians. In most cases this difference is of little concern, but it does hold the potential to create significant problems, particularly when it comes to the design and interpretation of the computational models that comprise an ever more significant part of the scientific toolbox. This problem is particularly acute when it becomes necessary to extend an existing model or make comparisons between two different simulations, both tasks that have become central to understanding the consequences of global climate change. Until recently, however, the mathematicians whose services are so desperately needed in this effort have been reluctant to join it - the politics have just been too hot. But as this month's issue of the SIAM News (June 2007, 40/5) highlights in two separate stories, all of that is starting to change.

Perhaps the most significant recent event signaling this awakening was the Symposium on Climate Change this past April at the Mathematical Sciences Research Institute in Berkeley, California ("Mathematicians Confront Climate Change" by Dana Mackenzie). The symposium garnered a high degree of attention for an event organized with such an extraordinary degree of rapidity (only five month elapsed between its conception and the actual event), with about 320 people attending a public forum on April 11 and about 75 meeting over the next two days to discuss more technical issues. In a move many outside the mathematical community may be inclined to misinterpret, the symposium began by criticizing the way in which climate modeling is currently handled.

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.

Speakers also pointed out that current climatic models contain significant and troubling simplifications, and that all contained "tuning" parameters designed to adjust the model's behavior to better match historic results - even when they cause "the models to violate the very physics that [have] been so painstakingly included."

It is important to realize that these criticism are not an attack on the more general predictions that most climatic models have arrived at - there is significant agreement about these results at this point. Instead, the message is that climate models are not being used to their full potential; much more detailed and reliable results are possible. But to reach this goal will require a more rigorous approach to modeling than is currently employed - and reaching that level of rigor will require the involvement of dedicated mathematicians.

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, not hidden.

The symposium was not focused entirely on criticizing current climate modeling practice, however.

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 volcanos), by combining several different climate variables into an "anthropogenic warming pattern."

Efforts to model the impact of climate change on specific economic activities where also showcased.

Models related to climate change - and human efforts to cope with it - were also showcased at the eighth conference of the Society for Industrial and Applied Mathematics Activity Group on Geosciences (SIAM/GS) held the month before ("Geosciences Conference Tackles Global Issues" by Barry A. Cipra). The two topics perhaps most interesting to readers of Wordlchanging are recent efforts to model CO₂ sequestration and to better understand the effects of hurricanes on surface waters and costal areas.

The idea of sequestering CO₂ by injecting it into deep aquifers and hoping that it stays put is tempting - deep aquifers are estimated to have a carbon storage capacity near 500 gigatons (about 20 years worth of anthropogenic CO₂ at current production rates) - but begs the question about whether it actually will stay put. Several of the talks at the SIAM/GS conference were focused on modeling this process, with mixed results. Current models indicate that proposed injection techniques are feasible, though all of the speakers emphasized the need for more complete models than are currently used. Unfortunately, human activity may already be making this form of carbon sequestration obsolete before it has even begun.

One of the easiest ways for the CO₂ in [an injection] plume to get back into the atmosphere is through an abandoned well. Even a filled-in or capped well might provide a direct route upward. Texas alone has more than a million such manmade fissures. The Alberta Basin in Canada is another densely welled area. Michael Celia of Princeton and Jan Nordbotten of the University of Bergen, Norway, described... models and... simulations of the potential for CO₂ leakage in a square region 50 kilometers on a side near Wabamim Lake, southwest of Edmonton, which is dotted with oil wells... To a depth of nearly 3000 meters there are a dozen sandstone and limestone aquifers, separated by shale and other aquitards. The locations and depths of the wells are known (most are between one and two kilometers deep). What's uncertain is the condition of the wells, in particular properties like their effective permeability.

...The researchers produced a probability distribution for the wells' permeability, then ran Monte Carlo simulations for 32 years worth of plume migration... They conclude that existing wells (and more are being drilled every year) are potentially important pathways [for CO₂ escape]...

Nordbotten stresses the need for more empirical data, a sentiment echoed by many of the other speakers. At this point too little is known about actual "on-the-ground" conditions to effectively evaluate the potential of carbon sequestration via deep well injection.

While carbon sequestration is a long-term program, hurricanes are a more immediate threat, and a session at SIAM/GS also focused on modeling these monster storms.

...Joannes Westernick of the University of Notre Dame presented an analysis of storm surges from hurricanes Katrina and Rita. Westernick's group combined detailed topographic models of the Gulf Coast, from Sabine Lake on the Texas-Louisiana border to Mobile Bay, Alabama, with extensive measurements form the hurricanes themselves... The model... includes information on the frictional resistance to storm surges arising from vegetation (e.g., march, cypress forest) and other land usage...

The "hindcast" results are encouraging. Using one-second timesteps and algorithms tailored to high velocities and large spatial gradients, the model, which computes a day's worth of surge in just under an hour, postdicts water levels to within about 10% of measured values.

The session also offered some surprises, particularly when it came to the implications of recent efforts to model hurricanes over the open ocean.

Winds and tides are the usual suspects in ocean energetics, [William] Dewar [of Florida State University, who gave a presentation on ocean dynamics] says, but it's possible that ocean life - especially zooplankton, which exist in huge quantities - also plays an important role in turbulent mixing.

"Surprisingly little power is involved in mixing the modern ocean," Dewar says. A single, hundred-watt kitchen blender would suffice to mic a cubic kilometer at observed rates. The incessant motion of myriad critters, such as salps (filter feeders similar to jellyfish, but with the beginnings of a backbone), which propel themselves vertically by enveloping and ejecting tiny quantities of water - not to mention the titanic battles between sperm whales and giant squid (of which there may be a billion worldwide) - contribute terawatts of power. "Clouds" of zooplankton, for example, migrate daily from the surface to as far as a kilometer below, the equivalent of a 400-gigawatt blender. All this bioenergetics raises an interesting question: Could overfishing be yet another human mechanism of climate change?

Results such as these highlight the power of mathematical models to not only elaborate and refine our current understanding of the world, but also to uncover potential interactions between humans and the environment that we might not otherwise expect. As scientists produce more data and develop more detailed models, the involvement of mathematicians will also become more necessary to ensure that the potential of their work is fully realized.

(The stories referenced in this article should become available in the SIAM News Archives sometime in the next two months. Image of Hurricane Katrina courtesy of the Wikimedia Commons.)