A chart appears on page 45 of the 2007 Synthesis Report of the Intergovernmental Panel on Climate Change (IPCC), laying out projections for what global temperature and sea level should look like by the end of this century. Both are projected to rise, which will come as no surprise to anyone who's been paying even the slightest attention to the headlines over the past decade or so. In both cases, however, the projections span a wide range of possibilities. The temperature, for example, is likely to rise anywhere from 1.8 C to 6.4 C (3.2 F to 11.5 F), while sea level could increase by as little as 7 inches or by as much as 23 — or anywhere in between.

It all sounds appallingly vague, and the fact that it's all based on computer models probably doesn't reassure the general public all that much. For many people, "model" is just another way of saying "not the real world." In fairness, the wide range of possibilities in part reflects uncertainty about human behavior: The chart lays out different possible scenarios based on how much CO2 and other greenhouse gases humans might emit over the coming century. Whether the world adopts strict emissions controls or decides to ignore the climate problem entirely will make a huge difference to how much warming is likely to happen.

But even when you factor out the vagaries of politics and economics, and assume future emissions are known perfectly, the projections from climate models still cover a range of temperatures, sea levels, and other manifestations of climate change. And while there's just one climate, there's more than one way to simulate it. The IPCC's numbers come from averaging nearly two dozen individual models produced by institutions including the National Center for Atmospheric Research (NCAR), the Geophysical Fluid Dynamics Laboratory (GFDL), the U.K.'s Met Office, and more. All of these models have features in common, but they're constructed differently — and all of them leave some potentially important climate processes out entirely. So the question remains: How much can we really trust climate models to tell us about the future?

The answer, says Keith Dixon, a modeler at GFDL, is that it all depends on questions you're asking. "If you want to know 'is climate change something that should be on my radar screen?'" he says, "then you end up with some very solid results. The climate is warming, and we can say why. Looking to the 21st century, all reasonable projections of what humans will be doing suggest that not only will the climate continue to warm, you have a good chance of it accelerating. Those are global-scale issues, and they're very solid."

The reason they're solid is that, right from the emergence of the first crude versions back in the 1960s, models have been at their heart a series of equations that describe airflow, radiation and energy balance as the Sun warms the Earth and the Earth sends some of that warmth back out into space. "It literally comes down to mathematics," says Peter Gleckler, a research scientist with the Program for Climate Model Diagnosis and Intercomparison at Livermore National Laboratory, and the basic equations are identical from one model to another. "Global climate models," he says, echoing Dixon, "are designed to deal with large-scale flow of the atmosphere, and they do very well with that."

The problem is that warming causes all sorts of changes — in the amount of ice in the Arctic, in the kind of vegetation on land, in ocean currents, in permafrost and cloud cover and more — that in turn can either cause more warming, or cool things off. To model the climate accurately, you have to account for all of these factors. Unfortunately, says James Hurrell, who led the NCAR's most recent effort to upgrade its own climate model, you can't. "Sometimes you don't include processes simply because you don't understand them well enough," he says. "Sometimes it's because they haven't even been discovered yet."

A good example of the former, says Dixon, is the global carbon cycle — the complex interchange of carbon between oceans, atmosphere, and biosphere. Since atmospheric carbon dioxide is driving climate change, it's obviously important, but until about 15 years ago, it was too poorly understood to be included in the models. "Now," says Dixon, "we're including it — we're simulating life, not just physics." Equations representing ocean dynamics and sea ice also have been added to climate models as scientists have understood these crucial processes better.

Other important phenomena, such as changes in clouds, are still too complex to model accurately. "We can't simulate individual cumulus clouds," says Dixon, because they're much smaller than the 200-kilometer grid boxes that make up climate models' representation of the world. The same applies to aerosols — tiny particles, including natural dust and manmade soot — that float around in the atmosphere and can cool or warm the planet, depending on their size and composition.

But there's no one right way to model these small-scale phenomena. "We don't have the observations and don't have the theory," says Gleckler. The best they can do on this point is to simulate the net effect of all the clouds or aerosols in a grid box, a process known as "parameterization." Different modeling centers go about it in different ways, which, unsurprisingly, leads to varying results. "It's not a science for which everything is known, by definition," says Gleckler. "Many groups around the world are pursuing their own research pathways to develop improved models." If the past is any guide, modelers will be able to abandon parameterizations one by one, replacing them with mathematical representations of real physical processes.

Sometimes, modelers don't understand a process well enough to include it at all, even if they know it could be important. One example is a caveat that appears on that 2007 IPCC chart. The projected range of sea-level rise, it warns, explicitly excludes "future rapid dynamical changes in ice flow." In other words, if land-based ice in Greenland and Antarctica starts moving more quickly toward the sea than it has in the past — something glaciologists knew was possible, but hadn't yet been documented — these estimates would be incorrect. And sure enough, satellites have now detected such movements. "The last generation of NCAR models," says Hurrell, "had no ice sheet dynamics at all. The model we just released last summer does, but the representation is relatively crude. In a year or two, we'll have a more sophisticated update."

Sophistication only counts, however, if the models end up doing a reasonable job of representing the real world. It's not especially useful to wait until 2100 to find out, so modelers do the next best thing: They perform "hindcasts," which are the inverse of forecasts. "We start the models from the middle of the 1800s," says Dixon, "and let them run through the present." If a model reproduces the overall characteristics of the real-world climate record reasonably well, that's a good sign.

What the models don't try to do is to match the timing of short-term climate variations we've experienced. A model might produce a Dust Bowl like that of the 1930s, but in the model it might happen in the 1950s. It should produce the ups and downs of El Niño and La Niña currents in the Pacific with about the right frequency and intensity, but not necessarily at the same times as they happen in the real Pacific. Models should show slowdowns and accelerations in the overall warming trend, the result of natural fluctuations, at about the rate they happen in the real climate. But they won't necessarily show the specific flattening of global warming we've observed during the past decade — a temporary slowdown that had skeptics declaring the end of climate change.

It's also important to realize that climate represents what modelers call a boundary condition. Blizzards in the Sahara are outside the boundaries of our current climate, and so are stands of palm trees in Greenland next year. But within those boundaries, things can bounce around a great deal from year to year or decade to decade. What modelers aim to produce is a virtual climate that resembles the real one in a statistical sense, with El Niños, say, appearing about as often as they do in reality, or hundred-year storms coming once every hundred years or so.

This is one essential difference between weather forecasting and climate projection. Both use computer models, and in some cases, even the very same models. But weather forecasts start out with the observed state of the atmosphere and oceans at this very moment, then project it forward. It's not useful for our day-to-day lives to know that September has this average high or that average low; we want to know what the actual temperature will be tomorrow, and the day after, and next week. Because the atmosphere is chaotic, anything less than perfect knowledge of today's conditions (which is impossible, given that observations are always imperfect) will make the forecast useless after about two weeks.

Since climate projections go out not days or weeks, but decades, modelers don't even try to make specific forecasts. Instead, they look for changes in averages — in boundary conditions. They want to know if Septembers in 2050 will be generally warmer than Septembers in 2010, or whether extreme weather events — droughts, torrential rains, floods — will become more or less frequent. Indeed, that's the definition of climate: the average conditions in a particular place.

"Because models are put together by different scientists using different codes, each one has its strengths and weaknesses," says Dixon. "Sometimes one [modeling] group ends up with too much or too little sea ice but does very well with El Niño and precipitation in the continental U.S., for example," while another nails the ice but falls down on sea-level rise. When you average many models together, however, the errors tend to cancel.

Even when models reproduce the past reasonably well, however, it doesn't guarantee that they're equally reliable at projecting the future. That's in part because some changes in climate are non-linear, which is to say that a small nudge can produce an unexpectedly large result. Again, ice sheets are a good example: If you look at melting alone, it's pretty straightforward to calculate how much extra water will enter the sea for every degree of temperature rise. But because meltwater can percolate down to lubricate the undersides of glaciers, and because warmer oceans can lift the ends of glaciers up off the sea floor and remove a natural brake, the ice itself can end up getting dumped into the sea, unmelted. A relatively small temperature rise can thus lead to an unexpectedly large increase in sea level. That particular non-linearity was already suspected, if not fully understood, but there could be others lurking in the climate system.

Beyond that, says Dixon, if three-fourths of the models project that the Sahel (the area just south of the Sahara) will get wetter, for example, and a fourth says it will dry out, "there's a tendency to go with the majority. But we can't rule out without a whole lot of investigation whether the minority is doing something right. Maybe they have a better representation of rainfall patterns." Even so, he says, if you have the vast majority coming up with similar results, and you go back to the underlying theory, and it makes physical sense, that tends to give you more confidence they're right. The best confidence-builder of all, of course, is when a trend projected by models shows up in observations — warmer springs and earlier snowmelt in the Western U.S., for example, which not only makes physical sense in a warming world, but which is clearly happening.