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Very few people disagree with the basic fact that the greenhouse gas CO2 warms the climate, but without some kind of positive feedback mechanism, it doesn’t add very much: around 1°C-1.2°C per doubling of CO2.

The statement is correct, but very misleading. In the absence of feedbacks, the climate sensitivity is in fact 1 to 1.2 deg C per the IPCC. However, the impact of feed backs is actually greater than the direct impact of increased CO2. This can be clearly concluded without relying on climate models at all, by looking at historical ice ages. These cycles are caused by variations in the Earth's orbit, which produce a small change in incident radiation. See here for example. Yet it would take an enormous change in solar output or CO2 concentration to directly cause the observed change in temperature. here is a good explanation of feedback effects, which states

To put this in perspective, it would take about five doublings of CO2 or a 7% increase in the total solar radiation hitting the Earth to produce the magnitude of climate change typical of glacial-to-interglacial transitions

Thus it is clear - without relying on models - that the feedback effect must be greater than the direct effect.

The phrase "without some kind of positive feedback mechanism" gives the impression that scientists have postulated some undefined mechanism in order to drive projected temperatures higher. Climate sensitivity without feedbacks can be calculated fairly accurately on a single sheet of paper, (see here for example) and the result will be approximately 1 deg C. But the result is not relevant to anything, because the real world has many feedbacks, so the focus in climate science over the past several decades has been on quantifying their effects. At this point, most of them are well understood, their net effect is overwhelmingly positive, and they are all solidly based on physics. Reference 2 is a very good explanation.

According to James Hansen, the best source of information about feedbacks is not models, but paleoclimate data:

Models are imperfect and we will never be sure that they include all important processes. Fortunately, Earth's history provides a remarkably rich record of how our planet responded to climate forcings in the past. Paleoclimate records yield, by far, our most accurate assessment of climate sensitivity and climate feedbacks.

In the same paper, Hansen calculates the climate sensitivity in various units, and expresses it as 2 to 4 degrees C for a doubling of CO2:

The empirical fast-feedback climate sensitivity that we infer from the LGM-Holocene comparison is thus 5°C/6.5 W/m2 ~ 3⁄4 ± 1⁄4 °C per W/m2 or 3 ± 1°C for doubled CO2. The fact that ice sheet and GHG boundary conditions are actually slow climate feedbacks is irrelevant for the purpose of evaluating the fast-feedback climate sensitivity. 7 This empirical climate sensitivity incorporates all fast response feedbacks in the real- world climate system, including changes of water vapor, clouds, aerosols, aerosol effects on clouds, and sea ice. In contrast to climate models, which can only approximate the physical processes and may exclude important processes, the empirical result includes all processes that exist in the real world – and the physics is exact.

As an example of feedbacks, water vapor is a very powerful greenhouse gas, and warmer air holds more water vapor. This is a scientific principal that has been well-understood for over a hundred years. If temperature increases because of higher CO2, the atmosphere will hold more water vapor, which itself will cause a greater greenhouse effect. Calculating the impact of the CO2 while disregarding the water vapor is arbitrarily (or perhaps intentionally) ignoring a significant part of the question. Other feedbacks have similar effects, such as changes in reflectivity due to melting of snow and ice, release of CO2 from ocean water because of increasing ocean temperatures, and release of methane from melting permafrost.