by Judith Curry

The IPCC defines climate sensitivity as “a metric used to characterise the response of the global climate system to a given forcing. It is broadly defined as the equilibrium global mean surface temperature change following a doubling of atmospheric CO 2 concentration.”





The no feedback sensitivity is the direct response of the surface temperature to radiative forcing by the increased CO2, without any feedbacks. Why is this interesting/important? The no feedback sensitivity is in principle much easier to calculate (and can presumably be calculated with certainty) and it provides a reference point for assessing the sensitivities associated with climate feedbacks in the overall climate sensitivity to CO2.

The CO2 no feedback sensitivity is an idealized concept; we cannot observe it or conduct such an experiment in the atmosphere. Hence, the CO2 no feedback sensitivity can only be calculated using models. Determination of the no feedback sensitivity has two parts:

calculation of the direct radiative forcing associated with doubling CO2

determination of the equilibrium change of global mean surface temperature in response to the CO2 forcing

The IPCC TAR adopted the value of 3.7 W/m2 for the direct CO2 forcing, and I could not find an updated value from the AR4. This forcing translates into 1C of surface temperature change. These numbers do not seem to be disputed, even by most skeptics. Well, perhaps they should be disputed.

On the previous greenhouse thread, Jeff Id wrote :

I also wouldn’t be surprised to find out that warming from doubling of CO2 even with no feedback has been incorrectly estimated. I haven’t found a good reference for the 1C calculation as yet. The only stuff out there is too basic to correctly capture the magnitude IMO.

Fred Moolten replies:

Jeff, the 1C value for a forcing of 3.7 W/m^2 (the canonical value for doubled CO2 based on radiative transfer equations and spectroscopic data) is derived by differentiating the Stefan -Boltzmann equation that equates flux (F) to a constant (sigma) x the fourth power of temperature. We get dF/dT = 4 sigma T^3, and by inversion, dT/dF = 1/(4sigmaT^3). Substituting 3.7 for dF and 255 K for Earth’s radiating temperature, and assuming a linear lapse rate, dT becomes almost exactly 1 deg C. In fact, however, the models almost uniformly yield a slightly different value of about 1.2 C, based on variations that occur with latitude and season.

Peter317 then writes:

Yes, a body will radiate ~3.7 W/m2 more energy if its temperature is increased by ~1c. However, that does not mean that increasing the energy flow to the body by that amount is going to increase its temperature by ~1C. Any increase in the temperature has to increase the energy flow away from the body by all means it can, ie conduction, convection, evaporation as well as radiation. When the outflow of energy equals the inflow then thermal equilibrium exists with the body at a higher temperature. But that temperature cannot be as much as ~1C higher, as the radiative outflow has to be less than the total outflow.

I am siding with Jeff Id and Peter317 on this one, there is too much that is not clear on this subject, and translating 3.7 W/m2 into 1C temperature increase seems incorrect to me. To try to sort out this issue, I committed to spending 1 day preparing this post. Perhaps I am missing some crucial references. In any event, some rethinking and re-evaluation of this topic seems to be in order.

Direct CO2 radiative forcing

The concept of radiative forcing, as it is used by the IPCC, is clearly described in this Appendix to the TAR.

The direct CO2 radiative forcing is the change in infrared radiative fluxes for a doubling CO2 (typically from 287 to 574 ppm), without any feedback processes (e.g. from changing atmospheric water vapor amount or cloud characteristics.)

The IPCC defines radiative forcing as ‘the change in net (down minus up) irradiance (solar plus longwave; in W m–2) at the tropopause after allowing for stratospheric temperatures to readjust to radiative equilibrium, but with surface and tropospheric temperatures and state held fixed at the unperturbed values’.

When reading a paper about radiative forcing, be sure you check to see what level in the atmosphere they are referring to; in addition to the tropopause level, it might refer to to the top of the atmosphere (or top of model; this definition is used by Soden and Held) or to surface forcing. Note Collins et al. considers radiative forcing at all three levels.

Modeling the direct CO2 forcing

In the Best of the Greenhouse thread, we argued that the basic mechanism of the so-called Tyndall gas effect (e.g. the infrared radiative emission of gases) is well understood. In the thread on Confidence in Radiative Transfer Models, we argued that line-by-line radiative transfer codes and the best band models can accurately simulate clear sky (no clouds, aerosols) infrared radiation fluxes at the surface provided that the vertical profiles of atmospheric temperature and trace gas concentrations are specified accurately.

Can we believe calculations from line-by-line radiative transfer codes for an atmosphere with doubled CO2? Well, the line-by-line codes and the RRTM band model have been shown to do a good job of capturing radiative flux variations in response to substantial (more than an order of magnitude) variation in water vapor content for a polar versus tropical atmosphere. Since the radiative properties of the CO2 molecule are simpler than those for the H20 molecule, we infer that the line-by-line codes and RRTM should perform accurate calculations for an atmosphere with doubled CO2. The best of the radiative transfer band models used in climate models (e.g. RRTM, GISS) should also be doing this correctly. Although Collins et al. does point out that many of the climate models radiative transfer codes do not compare well with the line by line models for CO2 doubling.

One of the nicest analyses I’ve seen that explains the latitudinal variations of CO2 forcing is given by Pielke Sr (see also this additional post), which provides total, troposphere, and surface forcing for tropical and subarctic summer, winter. The impacts of doubling CO2 are seen to vary strongly with latitude and season, owing the blocking effects by clouds and water vapor on the CO2 forcing. For example, the surface forcing is much larger in the subarctic winter than in the tropics, since the subarctic winter air is very dry and clouds are thin, and hence CO2 plays a bigger role.

Calculating the global CO2 forcing taking into account the geographical and seasonal variations has been done by Myhre and Stordahl 1997 (anyone locate a free online copy of this?) and also Soden and Held. To account for the variations in clouds, humidity and temperature, Myhre and Stordahl took the approach of using temperature and water vapor from the ECMWF analyses, climatological ozone data, and ISCCP cloud data; if I were designing this experiment, I would have made the same choices. The Myhre and Stordahl calculations are dated owing to deficiencies in the band model, and their line-by-line model was almost certainly not treating the far infrared correctly.

Note, Myhre and Stordahl found a 12.7% difference between CO2 radiative forcing with clouds versus clear sky. Radiative forcing with clouds (and aerosols) is a more complicated problem with greater uncertainties. But the calculations made by Stephens and Woods cited at Pielke Sr’s site are probably definitive in terms of radiative transfer in a cloudy atmosphere.

While the definitive global CO2 forcing calculations might not have been made yet with the currently best available radiative transfer codes, this is certainly something that it should be feasible to do with fairly high accuracy. And it is certainly something that should be done, rather than continuing to cite the 3.7 W/m2 value that was determined ca 2000 and does not use the latest radiative transfer codes that have been improved and validated. Note the relatively recent Collins et al. study only considers a midlatitude summer atmosphere.

Equilibrium surface temperature change

So once we calculate the direct no feedback CO2 forcing (to a doubling of CO2, it is often stated that all scientists agree that the Earth’s temperature would respond by increasing 1C.

From its heritage from 1D radiative-convective models of the earth’s atmosphere, the IPCC AR4 WGI Section 2.2 describes the concept of determining the change in equilibrium surface temperature from radiative forcing (RF):

Radiative forcing can be related through a linear relationship to the global mean equilibrium temperature change at the surface (ΔT s ): ΔT s = λRF, where λ is the climate sensitivity parameter. This equation, developed from these early climate studies, represents a linear view of global mean climate change between two equilibrium climate states. The TAR and other assessments have concluded that RF is a useful tool for estimating, to a first order, the relative global climate impacts of differing climate change mechanisms (Ramaswamy et al., 2001; Jacob et al., 2005). In particular, RF can be used to estimate the relative equilibrium globally averaged surface temperature change due to different forcing agents.

According to this simple model that relates radiative forcing at the tropopause to a surface temperature change, there is an equilibrium relationship between these two variables. The physical relationship between these two variables requires many many assumptions, including zero heat capacity of the surface and a convective link between the surface and the tropopause. Well, these kind of assumptions were arguably useful in 1967, at the time of the famous Manabe and Wetherald 1967 paper, but why are we still using such a vastly oversimplified model to assess climate sensitivity?

Surface radiative forcing – link to surface temperature

If our primary variable of interest is the surface temperature, it seems more reasonable to use the surface radiative forcing in a direct calculation of the surface temperature change, accounting for the different surface types (e.g. snow/ice, bare land, ocean) and different surface energy balance regimes.

If the energy available at the earth’s surface increases by 1 W/m2, how does the surface temperature change? The simple Planck calculation works for a surface absorbing/emitting as a black body, with zero heat capacity and change in the surface turbulent heat loss. Such a situation does not occur on the planet earth, although land surfaces are a closer approximation to this ideal than the ocean or snow/ice. Consider the Arctic Ocean sea ice during summer when it is melting; the surface temperature remains near 0C as the ice melts, with the heating (including solar heating) melting the ice rather than raising the surface temperature. Consider the tropical ocean, where the surface heating is mixed over a layer of order 100 m in depth, and likely contributes to an increase in evaporative cooling.

So understanding how the earth’s surfaces accommodates an increased flux of heat is key to understanding climate sensitivity IMO, and using the surface forcing seems to be the logical way to approach this. The IPCC cautions that:

It should be noted that a perturbation to the surface energy budget involves sensible and latent heat fluxes besides solar and longwave irradiance; therefore, it can quantitatively be very different from the RF[tropopause], which is calculated at the tropopause, and thus is not representative of the energy balance perturbation to the surface-troposphere (climate) system. While the surface forcing adds to the overall description of the total perturbation brought about by an agent, the RF and surface forcing should not be directly compared nor should the surface forcing be considered in isolation for evaluating the climate response (see, e.g., the caveats expressed in Manabe and Wetherald, 1967; Ramanathan, 1981).

I checked the “caveats” in Manabe and Wetherald 1967, p 250: The basic shortcoming of this line of argument may be that it is based upon the heat balance only of the earth’s surface, instead of that of the atmosphere as a whole. This argument didn’t make much sense to me, but then I didn’t spend much time on it. The Ramanathan 1981 paper is interesting (I don’t recall reading it before.) Ramanthan emphasizes that it is not just the radiative forcing at the surface, but also the atmospheric warming from CO2 that amplifies the surface warming. This is simple enough to include. Ramanathan concludes that it is possible to formulate the climate sensitivity problem from the viewpoint of the surface energy balance approach.

Redesigning the CO2 no feedback sensitivity analysis

I think the correct way to do this problem is to use the surface energy balance approach, as broadly outlined by Ramanathan. I would design the analysis in the following way:

1. Compute the surface radiative forcing and its amplification by the atmospheric warming in a manner following Myhre and Stordal 1997, using gridded global fields of of the input variables obtained from observations (e.g. the ECMWF reanalysis, ISCCP clouds, satellite ozone, some sort of aerosol optical depth from satellite. Conduct the calculations daily over two different annual cycles (say 1 El Nino and 1 La Nina year). These two different years provide an estimate of the uncertainty in the sensitivity associated with the base state of the atmosphere. Note, each annual forcing dataset will need to be run repetitively for maybe up to a decade to get equilibrium for the ocean and sea ice models. A grid resolution of 2.5 degrees should be fine.

2. Use the calculated fluxes to force the surface component of a climate model (without the atmosphere), including the ocean, sea ice, and land subsystem models, for the baseline (preindustrial) and the doubled CO2 forcing. Conduct two calculations for both the baseline and perturbed cases:

keep the the (turbulent) sensible and latent fluxes for the perturbed case the same as for the baseline case determine the perturbed surface temperatures by calculating the turbulent sensible and latent heat fluxes using the perturbed surface temperatures

Note, these two different ways of treating the sensible and latent heat fluxes tell you different things about sensitivity (without allowing the evaporative flux in #2 to change the radiative flux).

This is how I would do the analysis to determine the CO2 no feedback sensitivity. The number would almost certainly be less than 1C.

Conclusions

My 1 day time limit is up, I feel like this needs more thought, but this is what I came up with, look forward to your feedback. The simplified radiative forcing model that relates the tropospheric radiative flux to the surface temperature requires far too many unwarranted assumptions (my concerns with the equation ΔT s = λRF will be the subject of a future post.)

I’ve mentioned these general ideas a number of times before, including my opinion that the equation ΔT s = λRF was not carved in stone on Mount Sinai. I’m not sure if I have an solutions here, but there are a whole lot of questions on this subject that require much more substantive answers. I look forward to your ideas on this.