Absorption spectra of selective absorbers, here CO2 and H2O, are taken from the catalog used for the Atmospheric Radiative Transfer Simulator, ARTS . ARTS includes treatments of line broadening – with the treatment of the foreign broadening appropriate for Earth's atmosphere and a representation of continuum absorption following the approach of as modified by . Other data sources include monthly mean, gridded (0.25∘×0.25∘) near-surface (2 m) air temperatures, and column water vapor for the 240 months between 2001 and 2021, and these are taken from reanalyses of meteorological data ERA5,. Cloud data are based on measurements using the AATSR instrument which flew on ENVISAT . The record extends from May 2002 through April 2012, and level-3 cloud top temperature and cloud fraction are used.

Concepts are developed for understanding the emission of terrestrial radiation, 99 % of which is emitted in the 50 to 2000 cm-1 wavenumber (denoted by ν) interval. This is sometimes referred to as the longwave or thermal infrared part of the electromagnetic spectrum.

We adopt terminology (see also Table ) that will be standard for many readers. The Planck source function is denoted by Bν and depends on wavenumber ν and temperature T. The spectral irradiance is denoted by Fν and, unless indicated otherwise, is assumed to describe the outgoing thermal irradiance at the top of the atmosphere. The mass absorption cross-section κν,x refers to a constituent x (either c for carbon dioxide or v for water vapor) whose density is denoted by ρx.

The optical depth between two heights z1 and z2 is denoted by τν,x(z1,z2) and defined as 1τν,x(z1,z2)=∫z1z2κν,xρxdz≈κ‾ν,xMx(z1,z2). The approximation defines the path-integrated mass burden of x, denoted Mx, and a mean mass absorption cross-section, denoted κ‾ν,x. Hereafter, we denote the partial water vapor column burden Mv by W and the partial CO2 burden Mc by C. W and C are equal to their respective column burdens when the path is taken to extend through the entirety of the atmosphere. The effective mass absorption coefficient includes the effects of continuum absorption and pressure broadening by adopting a single value at an effective pressure and temperature, (P,T)=(850hPa,280K). An exception is for the case of CO2, as used in estimates of the forcing, for which we adopt values (P,T)=(500hPa,255K) to be more representative of the levels where the forcing establishes itself. Reducing the effective pressure and temperature for H2O to (P,T)=(700hPa,270K) changes estimates of the radiative response by about 2 %.

The transmissivity through an absorber x is given as e-τν,x/μ, where μ is the diffusivity factor. It is introduced by taking an effective zenith angle θ to scale the path length by μ-1=(cos⁡θ)-1 through the medium and thereby apply an equation originally valid for radiances to irradiances. The value of θ depends on the optical depth , but a value of θ=53∘ roughly corresponds to the average for optical depths uniformly distributed between 0 and 1, resulting in the commonly adopted value of μ-1=1.6. Beer's law thus becomes 2Fν(z)=πe-τν,x(z,ze)/μBν(Te), where subscript e denotes the emission value of a particular variable, e.g., height ze or temperature Te.

We introduce the idea of spectral masking as a useful implication of combining Eq. () with our classification of emitters. To illustrate the idea, we consider the case where water vapor is the only atmospheric absorber so that Te,y=Tsfc.

Accepting, for the moment, our assertion that the water vapor emission temperature remains invariant, it then follows from Eq. () that 4δFν=πe-τν,v/μδBν,sfc, where δBν,sfc denotes changes from surface emissions at wavenumber ν. Equation () can be derived more formally see, e.g., Eq. 5 in the SI of, which motivates Eq. () as a formalization of our ideas instead of the simpler first-to-1 model. From Eq. (), at wavenumbers where water vapor is optically thick, δFν→0. This is what is meant by spectral masking. Put more generally, at wavenumbers where an invariant emitter dominates emissions, it masks the radiative response of underlying sensitive emitters to warming. call this “spectral cancellation of surface feedbacks”. We prefer the term masking because the surface still responds to warming, but, as viewed from space, the response is hidden or masked.

The mass absorption cross-sections of H2O and CO2 are presented in Fig. (). For W≈25 kg m-2, corresponding to the present-day globally averaged column burden, at wavenumbers where κν,v>0.04, the atmosphere is considered to be optically thick. This is satisfied over most of the thermal infrared, the exception being wavenumbers between 800 and 1200 cm-1, which define the atmospheric window, emphasizing that it depends on the value of W. Fig. () also shows that CO2, whose column burden C≈6 kg m-2, is the dominant absorber between 585 and 750 cm-1 and will need to be accounted for in any fuller treatment of the radiative response to warming.

Because W increases exponentially with Tsfc, the atmosphere will become opaque at lower values of κν,v as Tsfc rises, thus reducing its ability to transmit a radiative response to space. We quantify this effect through the introduction of a quantity, 5χ(T)=14σT3∫0∞dFνdTdν<1, which measures the broadband sensitivity of radiant energy to warming relative to that expected for a black body. introduce the same quantity (their Eq. 4) and call it the average transmission. We prefer to think of χ as the fractional (spectral) support for the radiant response, because this terminology aligns better with the more colorful way of thinking and the first-to-1 model that we keep in the back of our minds.

As an example, for the simple case of the water-vapor-only atmosphere δFν is given by Eq. () and τν,v(T)=κν,vW(T), such that 6χ(T)=π4σT3∫0∞e-κν,vW(T)μdBνdTdν. Rescaling ν by introducing the coordinate x(ν), such that 7dx=14σT3dBνdTdν, stretches the ν axis so that equally spaced x intervals carry equal amounts of the radiative emission response to warming. In terms of x, χ(T)=∫e-κx,vW(T)/μdx<1 is just the area under the curves in Fig. () and shows how an emission response is supported over some subset of x corresponding to wavenumbers where water vapor is optically thin or transparent, i.e., κx,v≪μ/W(T).

For the first-to-1 model, the curves in Fig. () would vary between zero and one. Intermediate values emerge both due to spectral averaging and from intermediate optical depths. They highlight the complexity of the line-by-line variability of the spectral transmissivity e-κx,vW(T)/μ (which the stroke width used to render the plot is too wide to fully resolve). The effects of the differences between near-line versus continuum (or far-line or dimer) absorption on χ can also be discerned from the way in which the window closes in Fig. (). The former is associated with a narrowing of the window (region of support) with temperature, while the latter is apparent from weaker support as W becomes large. Continuum emission is more broadband or gray, whereas line absorption, which more nearly results in e-κx,vW/μ∈{0,1}, remains more colorful and better aligns with first-to-1 thinking (i.e., τν,v is either 0 or much larger than 1) and the concept of masking.

Simpson's law provides the justification for idealizing water vapor in the troposphere as an invariant emitter and hence for Eq. (). It states that if the relative humidity R is fixed, W depends only on T. Modulo the effects of pressure broadening on κv, this means that τν,v likewise only depends on T, and hence, the emission temperature (effectively where T(τν,v≈1)) does not change with warming. This basic idea was developed and used by a number of investigators to study runaway greenhouse atmospheres before pointed out its earlier articulation by .

The heuristic formalized by Eq. () also helps understand how CO2 influences the radiative response to warming. If, in radiative equilibrium, the absorption of radiant energy is independent of T, then the emission must also be independent of T. This is a rough description of the stratosphere and means that, at wavelengths where CO2 is optically thick in the stratosphere, it behaves like an invariant emitter seefor a more thorough discussion of this point. This is not a consequence of Simpson's law, where concentrations adjust to temperature to maintain the same emission. In this case, temperatures adjust to concentrations to maintain the same emission.

Similar arguments could be applied to ozone, but its burden is more sensitive to temperature, and its influence is not considered here.

At wavenumbers on the shoulders of its central absorption feature (band), near 600 and 733 cm-1, CO2 is less absorbing but still absorbing enough to become optically thick within the troposphere. At these wavenumbers, CO2 behaves like a sensitive emitter. In doing so, it competes with H2O (more so at wavenumbers on the low-energy side of the absorption band, where H2O is more absorbing, e.g., Fig. ) for control of emission to space. At wavenumbers where CO2 wins the battle by becoming optically thick above the emission height of water vapor, it re-establishes a radiative response to warming that H2O would have otherwise masked. Where CO2 emits at heights below the water vapor emission, its radiative response to warming is masked. The lack of concentration gradients in CO2 complicates the picture as they contribute to a more graduated change in τν,c than in τν,v, which defocuses the emission height and hence the idea of a single or dominant emitter.

Notwithstanding the difficulties of treating the overlap between CO2 and water vapor at wavenumbers where both have intermediate optical depths, Eq. () helps understand the basic physics of the radiant energy exchange and anticipate effects that gray thinking would obscure. Specifically, to account for CO2, the dominant emitters in Eq. () are chosen based on whether or not an atmospheric absorber is optically thick at a particular value of ν. When τν of one of the absorbers exceeds unity, its emission height and temperature are set to the height where τν=1. When both absorbers are optically thick, the dominant absorber is the first to 1 (lowest emission temperature), and surface emissions (in that case, third to 1) are neglected. By fixing the temperature of the stratosphere to Tcp, we effectively account for stratospheric adjustment and hence for the differentiated response of stratospheric versus tropospheric CO2 to δTsfc.

Figure  shows the fractional (spectral) support of the response χ calculated using this model. In contrast to Fig. , which was calculated for water vapor alone, the spectral support for the radiative response vanishes in the vicinity of the central CO2 absorption feature at 667 cm-1 and is re-established on its shoulders. Figure  highlights the dual role of CO2 in modulating the radiative response to warming. On the one hand, it masks surface emissions. On the other hand, it re-establishes a radiative response over parts of the spectrum that would otherwise be masked by water vapor. These effects depend on Tsfc. The masking by stratospheric CO2 becomes more important at colder temperatures, where the stratosphere is more massive and where the troposphere contains less water vapor. The re-establishment of the radiative response on the shoulder of the CO2 absorption band becomes more prominent at larger Tsfc and is essential for maintaining some support for the radiative response at very warm temperatures. On balance, the presence of CO2 moderates the dependence of χ on temperature see

From our understanding of the temperature influence on the emission of thermal radiation, for small changes in Tsfc, we expect 9δF=λδTsfc, which introduces the proportionality constant λ as the radiative response parameter. It is closely related to the radiative feedback parameter, which is often denoted by the same symbol using the same expression, modulo a change in the sign convention to allow an increase in F with T to be associated with λ<0, as expected for the net feedback in a stable system. In what follows, we decompose λ into a part that comes from changes in longwave and shortwave radiant energy transfer, such that λ=λ(lw)+λ(sw).

In clear skies, the longwave radiative response to a change in Tsfc, as predicted by Eq. () with the first-to-1 approximation, is given by 10Λ(T)≡π∫e-(τν,v/μ)dBνdTdν=χ(T)4σT3, where we distinguish the radiative response estimated heuristically, which we denote with Λ, from the true value of the clear-sky radiative response, which we denote with λcs(lw). For the case of a pure-water-vapor atmosphere, and modulo ambiguity in how W is defined to vary with T, Eq. () is identical to Eq. (3) in . It yields the expectation that 11λcs(lw)≈Λ(Tsfc)=χ(Tsfc)4σTsfc3.

For Tsfc=288K and R=0.8, Λ=1.9 Wm-2 K-1 (Fig. ), which is indistinguishable from the estimate for λcs(lw) under similar conditions. The estimate of is slightly larger at λcs(lw)≈2.3 Wm-2 K-1, but this is consistent with their calculations having been based on a much drier atmosphere. Figure  demonstrates that Λ also captures the sensitivity of λcs(lw) to temperature, humidity, and the presence of CO2, all forms of state dependence that have been identified and explored in a number of recent studies .

The temperature sensitivity of Λ is interesting in its own right as it explains a state dependence of the climate sensitivity see also; here, it is also highlighted because it will influence interpretations of cloud effects on the radiative response to warming. From Fig. , three temperature regimes can be identified: a cold, T<275K, “Budyko” regime where Λ increases only slightly (dΛ/dT≈0.004 Wm-2 K-2) and can hence be well approximated as constant, and a warm regime, 285K<T<305K, over which the radiative response to warming reduces sharply (dΛ/dT≈-0.08 Wm-2 K-2) with temperature. This is due to closing the atmospheric window due to continuum absorption from water vapor (compare the solid and dotted lines for χ in Fig. ), and Λ is thus sensitive to the humidity model WR versus Wsfc see alsoon this point. A third regime emerges at very warm temperatures, T>305 K. Here, Λ is roughly constant but small (Λ≈0.25 Wm-2 K-1). In this, regime CO2 plays an important role in maintaining a radiative response (compare solid teal and black lines in Fig. ) in an atmosphere that is optically thick in water vapor across the thermal infrared .

The moderating effects of CO2 on the temperature dependence of Λ reduce its maximum value from 2.55 to 2.17 Wm-2 K-1 and increase its minimum value from 0.05 to 0.26 Wm-2 K-1. The former effect arises from spectral masking at wavenumbers where CO2 is optically thick within the stratosphere and is more important in cold and dry atmospheres where surface emissions would otherwise dominate. The latter effect comes from CO2 wing absorption reclaiming spectral emissions from water vapor at warm temperatures (Fig. ). The moderating effect of CO2 on Λ is somewhat smaller than the warm-regime limit of λcs(lw)≈1 Wm-2 K-1, as estimated by and . Some of the difference can be explained by the use of an unrealistically cold stratosphere in those studies – decreasing Tcp to 150 K increases the asymptotic value of Λ to 0.44 Wm-2 K-1. The remaining difference likely reflects the crude treatment of emissions at intermediate optical depths by our model.

To the extent that λcs(lw) can be usefully approximated by Λ(Tsfc), it demonstrates that this response is something that is quite easy to understand and, given knowledge of the H2O and CO2 absorption spectra, to quantify. Moreover, because the dual effects of CO2 appear to approximately compensate one another at Earth-like temperatures (see Fig. ), Λ≈Λv. This indicates that the reduction in λcs(lw) from what would be expected from a black body largely measures how effective water vapor is at controlling emission to space and thereby masking the spectral response of emissions to surface warming, an idea that seems to have been the first to appreciate. It also explains why simply approximating 12Λ≈π∫8001200dBνdTdν, as proposed by and as might be justified by the first-to-1 model, provides such a reasonable estimate of λcs(lw).

Application of Eq. () yields a model of CO2 forcing similar to that first proposed by and developed later, in more detail, by . The starting point is to describe the irradiance as a function of the CO2 burden C, its spectral mass absorption coefficient κν,c, and the limiting temperatures Tcp and T⋆, such that 13F(C)=π∫0∞[e-Cκν,c/μBν(T⋆)+1-e-Cκν,c/μBν(Tcp)]dν, with T⋆=min⁡(Tsfc,T*) where WR(T*)=κν,v-1. This defines T⋆ as the temperature at which the W, distributed with Tsfc following WR, would attain an optical thickness of 1 or Tsfc, whichever is smaller. Through its dependence on κν,v, it will vary with ν. The choice of a fixed stratospheric CO2 emission temperature set to the cold point (Fig. b) provides a simple way to account for stratospheric adjustment by ensuring that the emission temperature of stratospheric CO2 remains invariant. As such, it anticipates our interest in the radiative response to changing CO2, i.e., the forcing.

An N-fold increase in the burden gives rise to F, given by the change in the irradiance (Eq. ) at the new burden; in clear skies, this becomes 14Fcs(N)=F(NC)-F(C)=π∫0∞e-Cκν,c/μ-e-NCκν,c/μBν(T⋆)-Bν(Tcp)dν. As climate sensitivity is usually referred to as the temperature response to a doubling of atmospheric CO2, in the remainder of the paper, we equate Fcs with Fcs(2). With Tcp=200K ranging from 194 to 204 K , Fcs varies from 4.55 to 4.22 Wm-2. These values compare favorably with estimates of the adjusted clear-sky flux in the literature, which range from 4.3 to 4.9 Wm-2 . The model is not only qualitatively informative; it also has quantitative fidelity, as demonstrated by is ability to capture the sensitivities to various quantities seen in more complex calculations, e.g., as in .

Following and subsequent studies e.g.,, two approximations make it possible to cast Eq. () into an even simpler form. The first is to replace the Planck source function with its band-averaged or band-centered values. This is justified because the difference between the CO2 transmissivities vanishes for τν,c≪1 and for τν,c≫1 so that Bν only contributes to the integral in the vicinity of νc. This allows it to be approximated by its central value and for T⋆ to be approximated by a band-averaged (567.5 to 767.5 cm-1) value: T⋆‾≡1200cm-1∫567.5767.5T⋆dν≈282.13K, where T⋆ is defined as previously described. The second approximation is justified graphically in Fig. , which shows that the envelope of the CO2 absorption spectrum falls off exponentially with ν as αe-‖ν-νc‖/l. This implies that, for a CO2 burden of C, τν,c>1 for νc-lln⁡(αC)<ν<νc+lln⁡(αC). It follows that, for a burden of NC, the atmosphere becomes optically thick for the larger interval – larger by the amount 2lln⁡(N). With these simplifications, Eq. () is simplified to 15Fcs(N)≈2πlln⁡NBνc(min⁡(T⋆‾,Tsfc))-Bνc(Tcp). For the same range of Tcp (194 to 204 K), Fcs varies from 4.3 to 4.0 Wm-2, comparable to estimates from the direct integration of Eq. ().

Dividing the estimate of the forcing from Eq. () by the radiative response from Eq. () gives an expression for the clear-sky climate sensitivity Scs: 16Scs=∫0∞e-κν,cCμ-e-2κν,cCμBν(T⋆)-Bν(Tcp)dν∫0∞e-κν,vWμdBνdTdν=2.3K, where Tcp is taken as the average across the stated range, and the net effect of CO2 on the radiative response to surface warming is assumed to be negligible. The additional simplifications of Eq. () for forcing and Eq. () for the radiative response yield a simpler expression in that it no longer depends explicitly on the absorption spectra of CO2 and H2O. With these approximations, 17Scs≈Bνc(min⁡(T⋆‾,Tsfc))-Bνc(Tcp)2σ∫8001200dBνdTdνlln⁡2=2.4K. By virtue of assuming a fixed window, Eq. () will not, however, generalize as well as Eq. () to warmer temperatures.

As a comparison, for radiative convective equilibrium, estimate Scs=2.1K, albeit for a drier atmosphere. The ability to derive Eq. () from the simple heuristic and its interpretation and/or simplification in the form of Eq. () illustrate how the value of the clear-sky climate sensitivity and its dependence on quantities like surface and tropopause temperature (Tcp) are quite easy to understand and predict. This understanding, as we show next, provides a different, and we believe better, basis for quantifying the effect of clouds.

From a radiant energy transfer perspective, one important distinction between clouds and water vapor is that clouds are neither colorful nor necessarily Simpsonian. Their grayness makes them effective in modifying both the clear-sky forcing and the clear-sky radiative response to warming. Some of these effects are well known, but others are only beginning to be appreciated or have been overlooked entirely.

The above analysis identifies ways in which the amount and distribution of clouds influences estimates of climate sensitivity even if the coverage, albedo, and temperature of the clouds do not change. It also identifies δTcld as a bit of a joker through its ability to substantially increase or decrease the radiative response. Below, we work through a few examples to illustrate these effects.

Returning to Eq. () and introducing λcld to represent the (long- and shortwave) radiative response to changes in the coverage (or albedo) of clouds and (1-fα)λcs(sw) to represent the all-sky changes in shortwave radiation with warming, 25S=(1-fh)Fcs(1-ηf)λcs(lw)-λcld-(1-fα)λcs(sw). By writing the surface albedo changes in terms of their clear-sky value λcs(sw), we account for cloud masking through fα so that Eq. () explicitly accounts for the varied cloud effects on climate sensitivity (see also Table ). The contribution of cloud coverage (or albedo) changes on the radiative response λcld is usually associated with net albedo changes and historically has been the main focus of cloud feedback studies; the other terms are mixed together with the clear-sky response. To the extent that cloud coverage or albedo changes are correlated with surface albedo changes, λcld and fα will not be independent. On a more detailed level, subtleties will arise due to differences in cloud albedo and cloud coverage; for instance, ambiguity among the terms may arise as clouds shift in location, thereby changing the planetary albedo and their cloud top temperature while maintaining a fixed coverage.

Above, it was shown that, for δTcld≈12δTsfc, we expect ηf≈fh. For clouds to maintain a neutral effect on the climate sensitivity in the presence of cloud coverage changes would, from Eq. () with ηf≈fh+λcld/Λ(Tsfc), require δTcld=f-fh+λcld/ΛTsfcfδTsfc. For λcld≈-0.2 Wm-2 K-1, as assessed by , δTcld≈914δTsfc. Recent work suggesting that λcld may be even smaller motivates us to adopt this, admittedly crude, approximation. This amounts to approximating 26(1-fh)Fcs(1-ηf)λcs(lw)-λcld≈Fcsλcs(lw)=Scs. It then follows that 27S=Scs1-(1-fα)λcs(sw)(1-fh)λcs(lw)-1≈43Scs. The 43 adjustment to the clear-sky climate sensitivity from surface albedo changes is estimated using the previously cited values of fh=0.25, with fα=0.5, and λcs(sw)=0.7 from . Because the ice margins are cloudier than the Earth as a whole, one might expect fα>f; however, the complete masking of surface changes only arises for clouds with an optical thickness much greater than 1. With Scs=2.3K, this implies S≈3.07 K. Equations () and () point out how a reasonably physical and quantitatively accurate estimate of Earth's equilibrium climate sensitivity can be obtained by assuming that the main effect of clouds is to mask surface albedo changes and how, in this case, the climate sensitivity can be reasonably estimated given knowledge of the H2O and CO2 spectroscopy, which determines Scs, the total cloud fraction f (as an approximation for fα), and an estimate of the surface albedo changes with warming.

For a planet without clouds but with the same λcs(sw), S≈3.7 K, which is considerably larger. Turning the argument around, for a given δTcld, this quantifies how large λcld would need to be for clouds to make our planet more rather than less sensitive to forcing.

While an estimated climate sensitivity of about 3 K will not raise any eyebrows, the way it was arrived at provides a new and hopefully fertile approach to thinking about clouds. Traditional feedback analysis adopts a gray perspective and attempts to explain sources of differences in estimates of λ(lw) due to changes in quantities such as the lapse rate or in humidity. This fails to adequately separate cloud from clear-sky effects and obscures the essential question as to what controls the emission temperature of clouds and how their present-day distributions mask well understood clear-sky effects.

To better link the contributions of the radiative response to the physics of radiant energy transfer, a different research program is needed. Such a program would employ first-principle models of radiant energy transfer and observations to

quantify Scs as the clear-sky Simpsonian response to warming, including the effects of CO2 and other long-lived greenhouse gases (sensitive emitters)