From Held and Zhao 2011, a simulation with an atmospheric model of the change in the number of tropical cyclones that form over each hemisphere and over the globe when sea surface temperatures (SSTs) are raised uniformly by 2C (labelled P2K), when the CO2 is doubled with fixed SSTs, and when SSTs and CO2 are increased together.
Suppose that we have a model of the climatic response to gradually increasing CO2, and we examine the globally-averaged incoming top-of-atmosphere flux, , as a function of time (using a large ensemble of runs of the model to average out internal variability). Letting refer to the difference between two climate states, for example the difference between the climates of 2100 and 2000 in a particular model, we end up looking at an expression like
where is the global mean surface temperature and refers to all of the other things on which depends. Here is the CO2 concentration, or, to the extend the useful range of this linearization, log(CO2). The forcing might be defined as . We typically go a step further and write so that we can think of this last term as a feedback, modifying the radiative restoring strength,
i.e, so that . While this is a formal manipulation that you can always perform if you want to, it is obviously more useful when is actually more or less proportional to . Ideally, there is a causal chain: => => . But what if the change in due to an increase in CO2 results from some other causal chain that doesn’t pass through the warming of the surface (or the warming of the strongly coupled surface-troposphere system)?