Let’s first remember that, as we previously talked about, Climate sensitivity corresponds to the amount of warming that will occur from a certain amount of “forcing”-change in the Earth’s radiative cooling that arises form something other than temperature changes themselves. In order for the Earth to maintain a constant temperature, the amount of radiation it receives from the sun must equal the amount that it reflects or re-emits back to space. Otherwise, because it is gaining or losing energy, it will warm or cool. Of course, warmer bodies emit more infrared radiation, so the Earth just needs to warm or cool enough to make the net radiation flux equal to zero again. The precise amount of temperature change necessary corresponds to the climate sensitivity. Of course, once again, there are caveats, which related to my previous post. But let’s forget those for a moment. Is it possible, in principle, to determine the climate sensitivity from looking at the Earth’s radiation budget and it’s response to temperature changes (feedback)? In fact, it is, in principle, possible, and one can even quantitatively describe how this can be done. However, keep in mind that it will only be possible to do so if the changes in the Earth’s radiation budget can be explicitly isolated as changes due to temperature, and changes due to other things, like CO2 (“forcing”).
This point is greatly emphasized by Roy Spencer, but our interest here is only in the Mathematics so this can be better described to people. I may not be the best person to do this, but it interests me and I’m going to try. The sensitivity of the climate relates to the theoretical response with no feedback as:
Where delta T corresponds to the response with feedback, delta T naught to the zero feedback response, and f to the feedback factor. The feedback factor in turn depends on the radiation flux caused by temperature change (the total flux minus the forcing flux) as:
Where ΔF/K is radiation flux per degree Kelvin, and 3.3 corresponds to the zero feedback flux per degree Kelvin, so for a feedback factor of 0, the ratio would be 3.3 W/m2 and if the feedback factor is one (corresponding to division by zero in our sensitivity formula), then no amount of change in K will create any change in ΔF, corresponding to infinite sensitivity. A feedback factor of -1, corresponding to half the zero feedback sensitivity, requires that the flux per Kelvin be double the no feedback value. Roy has been finding values of flux per Kelvin of ~6 W/m2, which corresponds to a feedback of ~-0.82, almost cutting the zero feedback sensitivity in half. The amount of temperature change to eliminate the flux from doubling CO2 (3.7 W/m2) would be a little over .6 degrees Kelvin, which is a very low sensitivity. Of course, determining the value of ΔF/K is difficult, as Spencer has himself pointed out. But the values found in climate models appear to be distinctly different from the observed fluxes.
Anyway, the main point of this post was to explain the math involved. I may revisit this later to investigate the feedbacks myself. It’s perhaps a topic for future investigation.
I would also like thank Roy for confirming my calculations.