On Climate Sensitivity and why it is probably small

What is climate sensitivity?

The equilibrium climate sensitivity refers to the equilibrium change in average global surface air temperature following a unit change in the radiative forcing. This sensitivity, denoted here as λ, therefore has units of °K / (W/m²).

(Note that occasionally, the sensitivity, λ is defined as the inverse of the above definition, that is, the radiative forcing required to change the temperature by one degree. This definition is used in the Cess et al. figure below. When in doubt, look at the units!)

Instead of the above definition of λ, the global climate sensitivity can also be expressed as the temperature change ΔT_x2, following a doubling of the atmospheric CO₂ content. Such an increase is equivalent to a radiative forcing of 3.8 W/m². Hence, ΔT_x2=3.8 W/m² λ.

The actual value of the climate sensitivity is a trillion dollar question (the cost of implementing Kyoto-like measures). If it is high, it would imply that anthropogenic greenhouse gases could significantly offset the global equilibrium temperature, while if it is small, the anthropogenic effect cannot be large, and any measures we take will not have a noticeable effect. Hence, its value is very important.

The IPCC states (in fact, a dozen times in their third scientific report) that the climate sensitivity is likely to be in the range of ΔT_x2 = 1.5 to 4.5°K. Below, we'll try to understand where this number comes from, why it is uncertain (at least for IPCC climatologists who rely on global circulation models) and what we can do about it.

The sensitivity of a Black Body Earth

Let us try to estimate the sensitivity of a Black Body Earth. That is, suppose Earth would have been a perfect absorber of visible and infrared radiation (and therefore also a perfect emitter of those wavelengths). What would its temperature be?

In equilibrium, each unit area facing the sun receives a total radiative flux of $$ F_0 \approx 1366 W/m^2 .$$ Thus, the total radiation absorbed by Earth is $$ P_{in} =\pi R^2 F_0,$$ which is the flux times the cross-section of the surface pointing to the sun. The total radiated flux is: $$ P_{out} = 4 \pi R^2 \sigma T^4 ,$$ which is the total surface area of Earth times the flux emitted per unit surface area. σ is the Stephan-Boltzmann constant. In equilibrium, the total absorbed flux equals the radiation emitted, hence, $$ P_{out} = P_{in} ~~\Rightarrow ~~ T = \left(F_0 / 4 \over \sigma\right)^{1/4} = \left(S \over \sigma\right)^{1/4} ,$$ where S = F₀/4 is the average flux received by a unit surface area on Earth.

We can now obtain the climate sensitivity, if we differentiate the black body equilibrium temperature of Earth: $$ \lambda = {dT \over dS} = {1 \over 4} {T \over S} = 0.21^\circ K/(W/m^2) .$$ This is the temperature change that would follow a unit change in the radiative flux reaching a unit area on Earth, on average.

The sensitivity of a Gray-Body Earth

In reality, Earth is not a perfect absorber, nor is it a perfect emitter.

Because its albedo is not zero, part of the solar radiation is reflected back to space without participating in the radiative equilibrium. Thus, the energy flux absorbed is: $$ P_{in} = \pi R^2 (1-\alpha) F_0,$$ where $\alpha$ is the albedo.

Moreover, because Earth is not a perfect black body, it is not a perfect emitter. Its emissivity ε is less than unity, and the total energy output is: $$ P_{out} = 4 \pi R^2 \epsilon \sigma T^4. $$ The equilibrium now gives: T = \left((1-\alpha) F_0 / 4 \over \epsilon \sigma\right)^{1/4} = \left(S \over \epsilon \sigma\right)^{1/4}, where the flux reaching a unit area (while excluding the flux reflected back to space) is $$ S=(1-\alpha) F_0/4 \approx (1-0.3) 1366/4~W/(m^2)\approx 240~W/(m^2) .$$ The sensitivity now obtained is given by: $$ \lambda = {dT \over dS} = {1 \over 4} {T \over S} = 0.30^\circ K/(W/m^2) .$$ Note the following:

• This is the often quoted "Black-Body" sensitivity, whereas in fact it is the sensitivity obtained for a Gray Earth, namely, after the albedo is taken into consideration.
• This sensitivity implicitly assumes that by changing the global temperature we don't change the albedo nor the emissivity (aka, no "feedbacks"). We assumed so because when we carried out the differentiation, we didn't differentiate $(1-\alpha)$ nor $\epsilon$.
• This sensitivity translates to an equilibrium CO₂ doubling temperature of about 1.2°K.

The effect of feedbacks

The climate system is more complicated than a black body, or a gray body for that matter. In reality, changing the temperature would necessarily change the albedo and also the emissivity.

For example, suppose we impose a positive radiative forcing (e.g., we double the atmospheric CO₂ content). As a result, the global temperature will increase. A higher global temperature would then imply that there is more water vapor in the atmosphere. However, water vapor is an excellent greenhouse gas, so we would indirectly be forcing more positive forcing which would tend to further increase the temperature (i.e., "a positive feedback").

Next, the higher water content would imply that more clouds are formed. Clouds have two effects, that of a blanket (i.e., reducing the emissivity) and hence increasing the temperature (i.e., more positive feedback). But clouds are white, and thus increase the reflectivity of Earth (increase the albedo). This of course tends to reduce the temperature (i.e., a negative feedback).

Other feedbacks include those of ice/albedo, dust, lapse rate, and even different feedbacks through the ecological system (e.g., see the Daisy World for a nice theoretical example).

Because such feedbacks exist, climate sensitivity is more complicated. Because climate variables (such as the amount of water vapor) should depend on the temperature as the basic variable and not the radiative flux, let us look at the inverse of the sensitivity, and how it depends on temperature. If we differentiate the definition of sensitivity, we find: $$ \lambda^{-1} = {dS \over dT} = {\partial S \over \partial T} + {\partial S \over \partial \alpha} {d\alpha \over dT} + {\partial S \over \partial \epsilon} {d\epsilon \over dT}. $$ Namely, the change in the radiative flux is the direct change associated with a change in temperature (while keeping the albedo and emissivity constant), plus the changes associated with the changed albedo and a changed emissivity. Plugging in eq. (1), we have: $$ \lambda^{-1} = {1 \over 4}{T \over S} + { S \over (1- \alpha)} {d\alpha \over dT} + {S \over \epsilon} {d\epsilon \over dT} $$ Thus, if we wish to estimate the sensitivity of the global climate, we need to know how much the albedo changes if we change the temperature (contributing to $d\alpha / dT$) and how much the emissivity changes as a function of temperature.

For example, more water vapor at higher temperatures implies that the emissivity decreases with temperature (water vapor blocks infrared from escaping), so $d\epsilon /dT$ is negative. This implies that $\lambda^{-1}$ decreases and the sensitivity $\lambda$ increases.

If there are several feedback mechanisms, we can write that each one contributes to a changed sensitivity (both through albedo and through a changed emissivity): where $dQ_i$ is the effective change to the energy budget through feedback i, following a change in temperature $dT$.

The sensitivity can be obtained through two conceptually different methods: Using computer simulations (i.e., global circulation models) and empirically.

Climate Sensitivity from Global Circulation Models

The standard way to obtain the climate sensitivity is to carry out a computer simulation of the global climate, namely, to use a global circulation model (GCM). Specifically, the global climate can be simulated under two conditions and compared. For example, one can simulate the global climate under some baseline conditions and then simulate the climate when some additional radiative forcing is present (e.g., with a doubled atmospheric content of CO₂). The results of the two simulations can then be used to study the effects of the applied radiative forcing. For example, one can estimate the effect on the average global temperature, or look for particular fingerprints of the particular radiative forcing applied.

Global circulation models are very powerful tools. Because in principle all the different aspects of the simulations can be controlled and studied, they can serve as detailed "climate laboratories", and thus have notable advantages. Specifically,

• GCMs can be used to analyze the effect of different components in the climate system. For example, one can separate the behavior of different feedbacks (e.g., water vapor, ice, etc.).
• GCMs can be used to estimate the effects of different types of forcing. This is because different geographic distribution of forcings can in principle cause different regional variations and in principle even different global temperature variations (even if the net change to the radiative budget is the same!)

There is however one HUGE drawback, because of which GCMs are not suited for predicting future change in the global temperature. The sensitivity obtained by running different GCMs can vary by more than a factor of 3 between different climate models!

The above figure explains why this large uncertainty exists. Plotted are the sensitivities obtained in different GCMs (in 1989, but the situations today is very similar), as a function of the contribution of the changed cloud cover to the energy budget, as quantified using ΔQ_cloud/ΔT.

One can clearly see from fig. 1 that the cloud cover contribution is the primary variable which determines the overall sensitivity of the models. Moreover, because the value of this feedback mechanism varies from model to model, so does the prediction of the overall climate sensitivity. Clearly, if we were to know ΔQ_cloud/ΔT to higher accuracy, the sensitivity would have been known much better. But this is not the case.

The problem with clouds is really an Achilles heel for GCMs. The reason is that cloud physics takes place on relatively small spatial and temporal scales (km's and mins), and thus cannot be resolved by GCMs. This implies that clouds in GCMs are parameterized and dealt with empirically, that is, with recipes for how their average characteristics depend on the local temperature and water vapor content. Different recipes give different cloud cover feedbacks and consequently different overall climate sensitivities.

The bottom line, GCMs cannot be used to predict future global warming, and this will remain the case unless we better understand the different effects of clouds and learn how to quantify them.

Empirical determinations of Climate Sensitivity

Instead of trying to simulate climate variations, it is possible to use past climate variations to empirically estimate the global climate sensitivity. That is, look for past variations in the energy budget and compare those with actual temperature variations.

Suppose for example that over a given time span, conditions on Earth varied such that the energy budget changes. These variations could arise from different atmospheric content (e.g., CO₂), different surface albedo (e.g., ice cover variations, vegetation) or other factors. This implies that over the time span, the energy budget could have changed by some ΔS, which can be estimated.

Over the same time span, over which the radiation budget varied, the global temperature would have varied as well, giving rise to a ΔT which in principle can be estimated as well.

If we compare the two, we obtain an estimate for the climate sensitivity: Note that this assumes that the climate had a long enough time to reach equilibrium (otherwise we are not estimating the equilibrium climate sensitivity). If the time scale is shorter than the time it takes to reach equilibrium (which is several millennia, the time it takes the ice-caps to adjust to any change), then where d is the damping expected on the given time scale (e.g., over a century, we expect a climate response which is only about 80% of the equilibrium response).

This method has several noteworthy advantages over the usage of GCMs:

• Even if we don't understand the climate system (and in particular, the effects of clouds), we are measuring Earth's actual behavior, not its simulated one (which of course is only as good as the physical ingredients we use to simulate).
• Since we can use different time scales, we could in principle obtain several independent estimates for the sensitivity, that is, we have an internal consistency check. This cannot be said about GCMs, since plugging in the wrong physics to different GCMs would consistently give the wrong results in all the simulations, with no way of knowing it.

But there are also several major drawbacks:

• Using paleoclimatic data to reconstruct past climate variations (radiative budget and temperature) is tricky.
• We cannot separate the effects of different components in the climate system (e.g., we cannot single out and quantify just the effect of clouds for example, since we measure the overall sensitivity).
• We cannot distinguish between different climate forcing, since we implicitly assume a one to one relation between radiative forcing and global temperature change.
• It is very hard to analyze regional variations.

Nevertheless, because this methodology is orthogonal to that of using GCMs, it is well worth pursuing.

Different empirical analyses have been previously carried out on different time bases (ranging from the 20th century global warming to the cooling from the mid Cretaceous).

In my own research, I added a few more time scale (e.g., the 11 year solar cycle or the Phanerozoic as a whole), but I also included the previous analyses as well. This was done for two reasons. First, by comparing different time scales, it is possible to check the consistency of the empirical approach. Second, unlike the previous analyses, I included the radiative forcing associated with the cosmic ray flux / cloud cover variations, that is, estimate the sensitivity based on: where ΔS_CRF term included is the forcing associated with the CRF variations on the given time scale. Different time scales include different CRF variations, such that this correction is different for case to case.

The different time scales included in my analysis are:

• The 11-year solar cycle (averaged over the past 300 years).
• Warming over the 20th century
• Warming since the last glacial maximum (i.e, 20,000 years ago)
• Cooling from the Eocene to the present epoch
• Cooling from the mid-Cretaceous
• Comparison between the Phanerozoic temperature variations (over the past 550 Million years) to different the CO₂ reconstructions
• Comparison between the Phanerozoic temperature variations and the cosmic ray flux reaching the Earth (as reconstructed using Iron-Meteorites and astronomical data, e.g., read this).

The results are summarized in the figure below. The left panel excludes the effect of cosmic rays while that on the right includes.

It was found that if the cosmic ray flux climate link is ignored and one averages the different empirical estimates for the sensitivity, one obtains that λ=0.54±0.12°K/(W m^-2). This corresponds to CO₂ doubling temperature of ΔT_x2=2.0±0.5°K.

If the cosmic ray flux climate link is included in the radiation budget, averaging the different estimates for the sensitivity give a somewhat lower result, namely, that λ=0.35±0.09°K/(W m^-2). (Corresponding to ΔT_x2=1.3±0.4°). Interestingly, this result is quite similar to the so called "black body" (i.e., corresponding to a climate system with feedbacks that tend to cancel each other).

One of the most interesting points evident when comparing the two panels of fig. 2 is the fact that once the CRF is included in the sensitivity estimate, the scatter in the different sensitivity estimates is much smaller. This is strong empirical evidence suggesting that the CRF is indeed affecting the climate. The reason is that if the CRF would not have had any climatic effect, the CRF/climate corrections ΔS_CRF would have corresponded to adding random numbers to the values on the left (i.e, to ΔS₀, without the CRF). This would have tended to increase the scatter even more. However, once these "random" numbers were added, the scatter was significantly decreased instead, as you would expect if the corrections are real.

Summary

• Earth's climate sensitivity is not expected to be that of a "black body" because of different feedbacks known to exist in the climate system.
• Although Global Circulation models are excellent tools for studying some questions, they are very bad at predicting the global climate sensitivity because the cloud feedback is essentially unknown. It is the main reason why the sensitivity is (not) predicted this way with an uncertainty of a factor of 3!
• Climate Sensitivity can be estimated empirically. A relatively low value (one which corresponds to net cancelation of the feedbacks) is obtained.
• Empirical Climate sensitivities obtained on different time scales are significantly more consistent with each other if the Cosmic Ray flux / Climate link is included. This is yet another indication that this link is real.

Bibliography

Nir J. Shaviv, "On Climate Response to Changes in the Cosmic Ray Flux and Radiative Budget", JGR-Space, vol. 110, A08105 (Abstract, PDF).