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In a warming climate, greenhouse gases modulate thermal cooling to space from the surface and atmosphere, which is a fundamental feedback process that affects climate sensitivity. Recent studies have found that when relative humidity (RH) is constant with global warming, Earth's clear‐sky longwave feedback would be dominated by surface cooling to space. Using a millennium‐length coupled general circulation model and accurate line‐by‐line radiative transfer calculations, here we show that the atmospheric cooling to space accounts for 12%–50% of the feedback parameter from poles to tropics. A simple yet comprehensive model is proposed here for explaining the atmospheric feedback process. It is found that when RH is held constant, the atmospheric feedback stabilizes the climate because (a) water vapor spectral lines are weakened by the collision‐broadening effect between water vapor and radiatively inert background gases, and (b) thermal emissions from other greenhouse gases increases due to enhanced Planck emission which is proportional to the surface warming. Each mechanism is responsible for half of the atmospheric feedback. We further elucidate that in hotter climates, the atmospheric feedback is more stabilizing because of (a) greater tropospheric opacity, and (b) more dramatic changes in air temperature with respect to transmission, owing to the pseudo‐adiabatic expansion of air with surface warming. The sum of surface and atmospheric feedback, the clear‐sky longwave feedback is accurately predicted by the simple model from the climate base state. Our study provides a theoretical way for assessing Earth's clear‐sky longwave feedback, with important implications for Earth‐like planets.

Observations and model simulations have shown that Earth maintains a stable longwave radiative feedback process. When the surface warms by 1 K, Earth allows for 1.7 to 2.0 Wm^{−2} of extra thermal cooling to escape to space in cloud‐free conditions. Recent studies have claimed that this enhanced thermal cooling to space can be explained by emissions from the surface passing through the atmosphere's infrared window. However, we find that a large portion of the stability actually results from enhanced atmospheric emission during global warming, which arises from the weakening of spectral lines broadening by radiatively inert gases (N_{2}, O_{2}, Ar) as the Earth warms. It is a well‐understood phenomenon in spectral physics but has been largely ignored in the feedback literature. As a result, the feedback responses from the thermal radiative effects of greenhouse gases tend to stabilize the climate, rather than initializing a runaway of thermal radiative energy. This study further proposes a simple theory for accurately predicting the clear‐sky longwave feedback from climate base states.

An atmospheric feedback process maintained by greenhouse gases crucially stabilizes Earth's climate under global warming

Earth's clear‐sky thermal energy budget is unlikely to runaway due to its stable atmospheric composition and thermodynamic structure

A simple, analytical model can accurately predict the state‐dependent clear‐sky longwave feedback spectrum

As a measure of habitability, the temperature of a planet is determined by the energy balance between the absorption of sunlight and the loss of thermal heat to space. Earth has been habitable for three or 4 billion years (Lepot et al., 2008; Nutman et al., 2016), owing to a relatively stable energy balance.

The thermal cooling to space is modulated by gases that are radiatively active in the longwave (thermal) spectra via the greenhouse effect. Simpson (1928) formulated a simple model to explain thermal cooling to space when water vapor is the only greenhouse gas (GHG) as a function of surface temperature, assuming constant longwave transmission per mass of water vapor. The same assumption was used in other conceptual models (Ingersoll, 1969; Nakajima et al., 1992), which we refer to as the “Simpsonian” model. It implies that once the atmosphere becomes opaque, the outgoing longwave radiation (OLR) cannot increase further when the surface warms. In this case, the planet's thermal budget would become unstable because, if the absorption of sunlight exceeds the threshold OLR, the ocean would evaporate uncontrollably, causing infinite warming and a runaway greenhouse effect (Goldblatt et al., 2013).

Observations and advanced Earth system models have shown that the Earth's climate system is relatively stable, despite longwave spectra being nearly opaque (only 17% of the global‐mean surface thermal emission is transmitted through the clear‐sky atmosphere to space (Costa & Shine, 2012; Huang & Huang, 2022, Figure S1). This stability is quantified by the clear‐sky longwave feedback parameter, defined as the change in OLR per degree of surface warming. It is well‐constrained to be −1.82 ± 0.12 Wm^{−2}K^{−1} (Flynn & Mauritsen, 2020) and is found to be relatively constant across a wide range of surface temperatures from the poles to the tropics based on observations and advanced Earth system models (Koll & Cronin, 2018; Raghuraman et al., 2019; Zhang et al., 2020).

Using a simple single‐column atmosphere with global‐mean atmospheric temperature and greenhouse gases and a global‐mean surface temperature of 288 K, it is possible to obtain clear‐sky feedback of 1.8 Wm^{−2}K^{−1} (34% of the increase in surface thermal emission per 1 K). The transmitted surface emission accounts for half of this value (0.9 Wm^{−2}K^{−1}), implying that the other half must emanate from the atmosphere. Thus, for this simple single‐column representation, the thermal energy balance is much more stable than the prediction of a Simpsonian model. In this paper, we show that this result largely holds for a more realistic representation of Earth's atmosphere.

Nevertheless, recent studies have refined the Simpsonian model and explored its implication for understanding Earth's longwave feedback parameter (Ingram, 2010; Jeevanjee et al., 2021; Koll & Cronin, 2018). Ingram (2010) conducts a systematic examination of why the pure‐Simpsonian model failed to explain the climate of Earth. It suggests that the radiation sourced from water vapor follows the Simpsonian model, considering the relative humidity (RH) is near‐constant with surface warming (Held & Shell, 2012; Ingram, 2010; Raghuraman et al., 2019; Zhang et al., 2020), while the radiation sources from surface emission and other greenhouse gases largely follow the increase in Planck function of the atmosphere at the height of emission to space. Koll and Cronin (2018) then shows that the OLR is largely dominated by the Planck function of surface temperature that passes through water vapor window channels. Along the same line, Jeevanjee et al. (2021) further emphasizes that feedback due to air temperature and water vapor naturally cancels out in water vapor absorption channels when holding RH fixed. In this case, the longwave clear‐sky feedback is equivalent to surface cooling to space, which is referred to as the surface temperature (surface Planck) feedback (Jeevanjee et al., 2021; Koll & Cronin, 2018), defined as the change in clear‐sky OLR due to 1 K increase of Planck function of surface temperature. These studies expect the clear‐sky longwave feedback to be near‐zero in water vapor absorption channels and to be qualitatively explained by the surface temperature feedback in the water vapor window channels (Jeevanjee et al., 2021; Koll & Cronin, 2018) and are referred to as partly Simpsonian models.

The partly Simpsonian model makes a clear point that the fixed‐RH feedback analysis framework (Held & Shell, 2012; Ingram, 2013) naturally incorporates the compensation between temperature and water vapor feedback. The assumption of vertically uniform warming under fixed RH (Jeevanjee et al., 2021) is also consistent with comprehensive feedback analysis studies in that the global‐mean feedback caused by vertical warming structure or varying RH is small (Zelinka et al., 2020). However, the partly Simpsonian models do not explain the simulated or observed feedback parameter in a clear‐sky condition. First, one may note that the clear‐sky surface temperature feedback is commonly calculated with a clear‐sky radiative kernel (Y. Huang et al., 2017; Kramer et al., 2019; Soden et al., 2008; etc.), yielding a value of −1.2 Wm^{−2}K^{−1}. The same magnitude is found by Raghuraman et al. (2019) using partial radiative perturbation in global reanalyses. This −1.2 Wm^{−2}K^{−1} here is higher than the crude estimation of −0.9 Wm^{−2}K^{−1} (using global‐mean temperature) because the polar region, where the atmosphere is more transparent, contributes more than tropical regions to 1 K of global‐mean warming (Holland & Bitz, 2003). Second, it has been found that the spectrally resolved feedback, derived from GCMs and separated from space‐born satellite observations, spans over infrared channels, including both water vapor windows and absorption channels of all major greenhouse gases (Brindley & Bantges, 2016; X. Huang et al., 2014; Y. Huang & Ramaswamy, 2009). Third, idealized model simulation suggests that as surface temperature feedback vanishes to zero with increasing water vapor mass, that is, when the runaway greenhouse effect was expected to occur (Koll & Cronin, 2018), the feedback can become even more stable due to increased CO_{2} (Seeley & Jeevanjee, 2021). Hence, these multiple lines of evidence suggest that a large portion of Earth's stable clear‐sky longwave feedback cannot be explained by the surface cooling process in the water vapor window.

Building upon the comprehensive feedback analysis studies (Flynn & Mauritsen, 2020; Held & Shell, 2012; Sherwood et al., 2020; Soden et al., 2008; Zelinka et al., 2020), well‐developed radiative transfer theories (Goody & Yung, 1989), and (partly ) Simpsonian models (Ingram, 2010, 2013; Jeevanjee et al., 2021; Koll & Cronin, 2018), this study proposes a comprehensive yet simple model to explain the clear‐sky longwave feedback of Earth. In Section 2, we examine the atmospheric component of the radiative feedback by combining accurate line‐by‐line (LBL) radiative transfer calculations and GFDL's CM3 model (Donner et al., 2011; Griffies et al., 2011). Based on an analytical model that deconstructs the sources of radiative feedback in a transmission coordinate, Section 3 proposes an emission temperature shift theory, which solves the change of Planck function of temperature in transmission coordinate analytically, considering the overlap of major greenhouse gases in an idealized atmospheric condition. Section 4 further shows that the analytical model achieves quantitative accuracy in predicting the clear‐sky longwave feedback parameter of the millennium‐length coupled model from the base state of the climate for a wide range of surface temperatures. Our theory suggests that the conical clear‐sky longwave feedback of Earth at around −1.8 Wm^{−2}K^{−1} critically relies on three key conditions.

A relatively stable thermodynamic structure of the atmosphere with a near‐constant relative humidity, lapse rate, and tropopause with respect to temperature, to be described by the simplified atmospheric model proposed in Section 3. Under this condition, feedback in infrared window channels can be explained by surface temperature feedback alone; it spectrally integrates to −1.5 to −0.9 Wm^{−2}K^{−1}, from polar to tropical regions.

An atmospheric composition dominated by mass‐conserving radiatively inert background gas such as N_{2} and O_{2} , which weakens the foreign collision‐broadening of water vapor line absorption at a fixed saturation vapor pressure (Clough & Iacono, 1995; Goody & Yung, 1989; Paynter & Ramaswamy, 2011; Pierrehumbert, 2010). It contributes to the feedback in water vapor absorption channels, summing up to −0.1 to −0.4 Wm^{−2}K^{−1} when condition 1 is maintained.

A relatively constant column mass of non‐condensable GHGs is maintained in the feedback process, owing to physical and chemical processes within the atmosphere and its interaction with other components of the climate system (e.g., the global carbon and methane cycle) (Berner, 2003; Post et al., 1990; Wahlen, 1993). It contributes to feedback in absorption channels of carbon dioxide, ozone, methane, and nitrous oxide, summing up to −0.1 to −0.5 Wm^{−2}K^{−1} when conditions 1 and 2 are maintained.

This section analytically derives the atmospheric and surface contribution to the clear‐sky longwave feedback parameter. By definition, the clear‐sky longwave feedback, *α*, is the change of OLR per degree of surface warming in cloud‐ and aerosol‐free conditions. At a surface temperature, *T*_{s}, spectrally resolved OLR (*R*, Wm^{−2} cm^{−1}) can be decomposed as a sum of transmitted thermal emissions from the surface, troposphere, and stratosphere as (Goody & Yung, 1989; Wallace & Hobbs, 2006):*T* is the atmospheric temperature and *B* is the Planck function of a given temperature. At a frequency *υ*, *υ* − *δυ*/2 and *υ* + *δυ*/2 (*δυ* = 1 cm^{−1} here) from a layer to space. Here we assume azimuthal homogeneity in spectral radiances by adopting a diffusivity factor (Armstrong, 1968; Li, 2000) of *R*_{trop} and *R*_{strat} are used to represent the sum of tropospheric and stratospheric contribution to *R*, respectively. Because temperature tends to decrease monotonically with height while transmission monotonically increases with heights from the surface to the tropopause, we can express temperature as a function of transmission in the form of

The clear‐sky longwave feedback *α* (Wm^{−2}K^{−1}) is defined as the change in clear‐sky OLR per degree of surface warming. *α* can be expressed as a sum of four terms, including the surface (*α*_{srf}), the troposphere (*α*_{trop}), the stratosphere (−*∂R*_{strat}/*∂T*_{s}), and the change of transmission (

We find that the feedback due to the increased opacity, *∂R*_{strat}/*∂T*_{s}, is also found to be small (see Figure B2a) and that it tends to offset *α*_{trop} is a reasonable approximation for atmospheric feedback (as a sum of *α*_{trop}, *∂R*_{strat}/*∂T*_{s}), and is referred to as *α*_{Atm} hereinafter. Hence, the clear‐sky longwave feedback can be expressed as:

To examine the relative importance of *α*_{srf} and *α*_{Atm}, a set of line‐by‐line (LBL) calculations are conducted using temperature, humidity, and pressure levels from an equilibrium climate sensitivity experiment with a coupled global circulation model, GFDL's CM3 (Donner et al., 2011; Griffies et al., 2011; Paynter et al., 2018) (described in Appendix A). Figure 2 shows *α*, *α*_{srf}, and *α*_{Atm} for every 5‐K bin of surface temperature from 252.5 to 302.5 K (covering 89% of model grids). The *α*_{srf} is equivalent to the surface temperature kernel in a clear‐sky condition. Soden et al. (2008) shows that this term increases (less stable) from −1.5 Wm^{−2}K^{−1} in polar regions to −0.8 Wm^{−2}K^{−1} in the deep tropics, which is consistent with our result using accurate LBL computations (Figure 2a yellow curve). Here we further show that the magnitude of *α*_{srf} is determined by two factors: the derivative of the Planck function (*∂B*(*T*_{s})/*∂T*_{s}) and the vertically integrated transmission of the GHGs (*T*_{s}, transmission through water vapor decays with *T*_{s}. With only line absorption (dotted curves in Figure 2a), *α*_{srf} is almost constant with *T*_{s}, indicating that the *α*_{srf} is nearly constant. However, as continuum absorption increases with specific humidity, we show that including continuum absorption would dramatically decrease transmission when *T*_{s} is higher than 280 K (solid curves in Figure 2a), so that the actual *α*_{srf} increases (less negative) with *T*_{s}.

In contrast, the clear‐sky longwave feedback, *α*, tends to slightly decrease with *T*_{s} rather than increase with it, because *α*_{Atm} becomes more important in regions warmer than 280 K. It is critical, considering the present‐day global surface temperature is at 288 K. Regardless of the water vapor continuum, *α*_{Atm} takes about −0.2 to −0.9 Wm^{−2}K^{−1} of the feedback parameter. Similar statistics have been noted in Raghuraman et al. (2019) based on reanalysis of present‐day Earth and can be inferred as the discrepancy between clear‐sky longwave feedback evaluated for GCMs (Flynn & Mauritsen, 2020; Sherwood et al., 2020; Zhang et al., 2020) and the surface temperature kernel. Thus it would appear that a partly Simpsonian model (Jeevanjee et al., 2021; Koll & Cronin, 2018), which only considers *α*_{srf}, underestimates the stability of *α* at high *T*_{s} by ignoring the contribution of *α*_{Atm}. In particular, it seems to underestimate the stability of *α* at high *T*_{s}.

Furthermore, the spectrally resolved global‐mean feedback parameter suggests that *α*_{Atm} is significant across infrared channels (Figure 2b), in line with spectrally resolved feedback obtained from GCMs and observations (Brindley & Bantges, 2016; X. Huang et al., 2014; Y. Huang et al., 2010) and idealized simulations (Seeley & Jeevanjee, 2021, for CO_{2}). Figures 2c and 2d shows *α*_{Atm} with water vapor but no other GHGs in blue, and *α*_{Atm} with well‐mixed GHGs and O3 but no water vapor in red (experiments described in Appendix A). While existing literature (Ingram, 2010; Jeevanjee et al., 2021; Seeley & Jeevanjee, 2021) expects well‐mixed GHGs and O3 to be major sources of the atmospheric feedback, only half of the *α*_{Atm} is explained by spectral ranges sensitive to these GHGs (the red‐shaded area in Figure 2). Water vapor absorption accounts for the other half of *α*_{Atm} between 255 and 300 K *T*, potentially due to the foreign collision‐broadening effect (Clough & Iacono, 1995; Goody & Yung, 1989; Ingram, 2010; Paynter & Ramaswamy, 2011; Pierrehumbert, 2010). *α*_{Atm} in absorption channels of these major greenhouses gases is further explained in Section 3 by solving the change of temperature‐transmission relation with surface warming.

Section 2 shows that a partly Simpsonian model proposed by existing literature underestimates the magnitude of clear‐sky longwave feedback, *α*, nor its dependence upon surface temperatures. It is found that the remaining feedback from the atmosphere, *α*_{Atm}, is sourced majorly from changes in the Planck function of temperature with respect to the transmission of the base‐state troposphere. In the following context of this section, radiative and atmospheric processes that control *α*_{Atm} are analytically derived.

Here we use *T*_{e} to denote the brightness temperature derived from OLR (*R*), and express *R*_{trop} in Equation 1 alternatively as*α*_{Atm} in Equation 3 then becomes:*r* is introduced to quantify the temperature shift in transmission coordinate by assuming it is vertically uniform (Figure 3):

Note that although *B*(*rT*_{e}) − *B*(*T*_{e}) is used to quantify changes in tropospheric contribution to OLR, *rTe* does not necessarily equal to the brightness temperature of OLR in the warm state.

A linearization of Equation 4 yields:

Therefore, the magnitude of *α*_{Atm} is controlled by *R*_{trop}, as given by radiative transfer at the base state, and *r* − 1 per degree of warming, which also depends on changes in radiative transfer with surface warming.

The emission temperature shift ratio, *r*, is jointly determined by layer‐by‐layer air temperature, partial pressure of background gases (foreign pressure), and partial pressure of every GHG in the base state atmosphere, as well as their impacts on the layer‐by‐layer transmission spectra with surface warming. Despite the complexity, *r* can be inferred from the base state if the change of these properties with surface warming follows a predictable pattern. And it does, as a consequence of basic thermodynamic relations.

First, temperature and pressure are linked via the temperature lapse rate, Γ, under the hydrostatic balance, based on the barometric formula (Wallace & Hobbs, 2006):*i* denotes a discrete atmospheric layer, *g* = 9.80 m/s^{2} is gravity and ^{−1} K^{−1} is the specific gas content of the air. *p*_{s} is surface pressure and is considered to be constant with warming because the dry air mass, which consists of more than 98% of the atmosphere, is conserved.

Second, when the surface warms from *T*_{s} to *T*_{s} to the tropopause changes little in this process (see Figure 3a) (Ingram, 2010), we may infer the air pressure at *T*_{i} changes from *p*_{i} to _{eff}, is used to describe the change of air pressure at a given *T*_{i} with warming. Assuming the bottom of the atmosphere expands pseudo‐adiabatically under fixed RH with the surface warming, Γ_{eff} is then the pseudo‐adiabatic lapse rate from *T*_{s} to

Lastly, the partial pressure of water vapor and other GHGs are physically linked to air temperature and air pressure, respectively. While the partial pressure of well‐mixed GHGs, by definition, is fixed at pressure levels, we may treat tropospheric O_{3} similarly as well‐mixed. It gives the partial pressure of these gases being proportional to the air pressure:

The partial pressure of water vapor, *k*_{CC} to describe saturation vapor pressure and to represent

*k*_{CC} can be derived by taking a derivative of the widely used August–Roche–Magnus formula (Alduchov & Eskridge, 1996) over temperature *T*_{i}, as

Therefore, the change of air temperature, foreign pressure, and partial pressure of every GHG with surface warming can be inferred from their base states. By adopting a strong‐line approximation (Goody & Yung, 1989; Pierrehumbert, 2010), transmission caused by a saturated line over a spectral interval (i.e., 1 cm^{−1}) can be explicitly expressed as a function of the three variables (see discussion in Appendix B):*υ* − *δυ*/2 and *υ* + *δυ*/2. *G*. This approximation is obtained by integrating over the far‐tail of Lorentz profile (Goody & Yung, 1989; Pierrehumbert, 2010). Here we assume that the width of the Lorentz profile collision broadening is proportional to *p*_{i} and independent of *T*_{i} and the line intensity is proportional to

First, we derive *r* in idealized conditions. Take the 280‐K surface temperature bin as an example, the mean tropospheric lapse rate Γ is close to the pseudo‐adiabatic lapse rate Γ_{eff} = 6 K km^{−1}. For Γ ≈ Γ_{eff} and absorptions of well‐mixed greenhouse gases, we expect *T*′, with _{3} (red‐shaded area in Figure 2), Figure 3b depicts that the actual temperature‐transmission profile of the warm state is very close to *r* is the same as the change in air temperature in the pressure coordinate. As a result, *α*_{Atm} comes from the change in Planck emission in the pressure coordinate and is referred to as a “Planckian” feedback response.

Then, for water vapor line absorption, the saturation vapor pressure (*k*_{CC}) retains the transmission‐temperature relation close to the base state (*r* = 1), while the foreign‐collision broadening effect, as a function of air pressure (linked to temperature with *A*(Γ)), shifts the transmission‐temperature relation at a rate close to *r* can be solved as (Appendix B2):*k*_{CC} = 0.09 (at *T*_{e} = 250K), Γ = 6 Kkm^{−1}, and *A*(Γ) = 5.7 at *T*_{s} = 280 K, this equation expects *r* − 1 to be dampened to roughly 15% compared to *r* is weaker but proportional to the change in air temperature in the pressure coordinate, leading to a ’dampened Planckian’ feedback response.

Finally, the effects of GHGs overlap and pseudo‐adiabatic upward shift (characterized via Γ_{eff}) is incorporated comprehensively in Appendix B. We derive that *f*_{G}, *f*_{ql}, and *f*_{qc} are unitless coefficients that represent contributions from each absorption mechanism for transmission at every wavenumber. They are estimated as regression coefficients based on Equation B9 from LBL calculations performed at a reference state and are included in the supplementary materials. *k*_{l} is an unitless line intensity parameter for CO_{2}, which is set to 0.02 in this study.

In principle, from Equation 10 and Equation 5, we expect *α*_{Atm} to be largely Planckian in channels more sensitive to well‐mixed greenhouse gases or O_{3} (*f*_{ql} ≪ *f*_{G}), with the *r* approaching to a larger end following Equation 8. In channels more sensitive to water vapor line absorption (*f*_{ql} ≫ *f*_{G}), we expect *α*_{Atm} to be a dampened Planckian response, with *r* approaching to a smaller end following Equation 9, at a rate controlled by the Clausius‐Clapeyron relation. Furthermore, *α*_{Atm} is expected to be more stabilizing when the effective lapse rate (Γ_{eff}) is steeper than the base state lapse rate (*Γ*). The impacts of each term on Earth's atmosphere are examined and elucidated in Section 4.

By assuming *α*_{Atm}, which are further summed with the surface temperature feedback, *α*_{srf}, for the total feedback, *α*. For each composited 5‐K surface temperature bin, only the temperature, partial pressure of gases, and radiative fluxes at the base state are used, in addition to the LBL‐derived regression coefficients at a reference state.

The results match well with LBL for a wide range of surface temperature from 255 to 300 K, as presented in Figure 4a. Figure 4b further shows the spectrally resolved *α*_{Atm} at 280 K surface temperature as an example. There is negative bias in [800 1,000] cm^{−1} caused by the neglected surface transmission change (^{−1} in the center of CO_{2} absorption channel caused by the neglected stratospheric feedback (

For well‐mixed GHGs and O_{3} (the red‐shaded area in Figures 4b and 4d), *α*_{Atm} is caused by temperature increase at the same value of integrated column mass from the top of the troposphere. In the absence of water vapor, this feedback process is straightforward to be understood and has been viewed as the ’Planckian‐like’ feedback by Ingram (2010), with *r* being approximately *r* on *α*_{Atm}, we take a broadband approximation of Equation 4 by linearizing it over *r* following the Stefan‐Boltzmann law, with Δ*T*_{s} = 1K:*T*_{s} = 280K and *R*_{trop} = 190 Wm^{−2} (integrate over the red curve in Figure 4d), *r* equals to 1.0036, Equation 11 estimates *α*_{Atm} roughly as −2.7 Wm^{−2}K^{−1}. With *α*_{srf} = −1.2 Wm^{−2}K^{−1}, Planck feedback is roughly −3.9 Wm^{−2}K^{−1}, which is close to existing literature (Flynn & Mauritsen, 2020; Sherwood et al., 2020). Moreover, we note that *α*_{Atm} in O_{3} absorption spectrum is well predicted by using the base state transmittance through vertically resolving O_{3} profiles but assuming in Equation 10 that (a) no changes in stratospheric emission nor transmission and (b) changes in the tropospheric emission comes from vertically uniform tropospheric O_{3}. While O_{3} increases the opacity of the water vapor window around 1,080 cm^{−1} to result in less negative *α*_{srf}, our results suggest that the atmospheric feedback due to thermal emissions of stratospheric O_{3} is negligible and that the role of tropospheric O_{3} is similar to well‐mixed GHGs in stabilizing the clear‐sky longwave feedback (see Figure B2d compared to Figure 4b).

*α*_{Atm} in water vapor absorption channels (the blue‐shaded area in Figure 4) is substantial and is spectrally integrated into half of the *α*_{Atm}. This result is counter‐intuitive, as partly Simpsonian models expected zero emission temperature shift from the exponential CC relation, which was considered to outweigh the foreign pressure‐broadening effect. Here we show that in water vapor absorption channels, the magnitude of *r* − 1 is reduced to not zero, but roughly 15% of *T*_{s} = 280K, *R*_{trop} in water vapor absorption channels is roughly 105 Wm^{−2} (integrated over the blue‐shaded area of the red curve in Figure 4d), *r* = 1.0006 leads to *α*_{Atm} around −0.25 Wm^{−2}K^{−1} following Equation 11, which is consistent with the blue curve in Figure 2c at 280 K *T*.

Outside of strong water vapor absorption channels, the overlap between water vapor line and continuum absorption with other gases control the spectral shape of *r* and hence *α*_{Atm} (Figures 4b and 4d). Contributions from water vapor absorption decreases *r* significantly in absorption channel of CO_{2}, CH_{4}, N_{2}O, and O_{3} via *f*_{ql} and *f*_{qc} in Equation 10. Although *r* in water vapor absorption channels is smaller compared to other GHGs, greater *R*_{trop} is emitted from water vapor channels (105 Wm^{−2} compared to 85 Wm^{−2}K^{−1}) because (a) water vapor absorption is strong in the troposphere to mask over surface emissions but weaker in the stratosphere to transmit tropospheric emissions, and (b) Planck function at tropospheric temperature peaks within the water vapor rational‐vibrational spectrum. Thus water vapor can contribute half of *α*_{Atm} owing to the compensation from greater tropospheric emission.

Furthermore, we find that the clear‐sky longwave feedback, *α*, remains relatively constant (between −1.7 and −2.0 Wm^{−2}K^{−1}) because *α*_{Atm} becomes more negative with *T*_{s} to compensate for vanishing *α*_{srf}, rather than being explained by *α*_{srf} alone (Koll & Cronin, 2018). It is especially important, considering the present‐day global‐mean surface temperature is at 288 K, where *α*_{Atm} has become critical. Figure B2 further shows that this enhancement of *α*_{Atm} with surface temperature is robust regardless of the combination of GHGs in radiative calculations.

Figure 5 elucidates that *α*_{Atm} enhances across all absorption channels with surface temperature because of two factors. First, in warmer regions, more tropospheric emission (*R*_{trop}) is radiated to TOA in weak absorption channels (Figure 5f near the edge of shaded areas), where OLR is more sensitive to emissions from the lower troposphere. It partly explains *α*_{Atm} in radiative fins of GHGs (Figure 5c compared to d). Second, the emission temperature shift increases substantially with surface temperature (*r*, Figure 5e). It is responsible for more negative *α*_{Atm} at 300 K than at 255 K across all absorption channels. This is because *r* accounts for different tropospheric warming structures from the poles to the tropics. If the troposphere warms more than the surface, as in the tropical region (at 300 K in Figure 5), there would be a greater temperature shift in pressure coordinate (smaller Γ_{eff} and larger *A*(Γ_{eff})) and hence in the transmission coordinate (larger *r*), thus amplifying *α*_{Atm}. Such effect is characterized by the pseudo‐adiabatic thermal expansion using Γ_{eff} in this study via the term *A*(Γ_{eff}) − *A*(Γ) in Equation 10. This crude approximation generally captures the warming structure in the upper and middle troposphere (Figures 5a and 5b) and facilitates an accurate feedback prediction across different base states (surface temperatures) without knowing actual temperature profiles in the warm states. Our result supports the conclusion that upper‐tropospheric warming can amplify the negative feedback in the radiative fins of CO_{2} in a hot climate, as found in Seeley and Jeevanjee (2021) via a mechanism‐denial experiment using idealized simulations. With an analytical, spectral explanation, the enhancing *α*_{Atm} with *T*_{s} caused by *A*(Γ_{eff}) − *A*(Γ) is consistent with the conventional understanding of the lapse rate feedback, which is known to contribute to destabilizing feedback process in the polar regions but stabilizing process in the tropical regions (Colman & Soden, 2021; Manabe & Wetherald, 1975; Soden et al., 2008).

We note that although the impact of RH is implicit in Equation 3 or Equation 10, the column‐mean RH, as well as the vertical RH structure, can be important for the state‐dependent clear‐sky longwave feedback parameters. While the column‐mean RH affects the *α*_{srf} via the vertically integrated transmission of the base state atmosphere *α*_{Atm} (Bourdin et al., 2021) via the contribution from water vapor (*f*_{ql} and *f*_{qc}, which depend on vapor pressure in Equation B9) and the OLR across infrared spectra. A spectrally varying effective RH, determined from the vertical levels where *T*_{e} is located, is used to produce Figures 3 and 4. Therefore, RH controls these base‐state quantities (*p*_{q}, *R*_{trop}, and *T*_{e}) and should be treated carefully.

Based on line‐by‐line radiative transfer calculations and a millennium‐length coupled general circulation model, this study presents a novel, simple theory to explain the effect of greenhouse gases (GHGs) on outgoing longwave radiation (OLR) for quantitatively evaluating clear‐sky longwave feedback. This theory proposes that the complex clear‐sky longwave feedback can be viewed as a sum of two processes (Equation 3): (a) feedback due to surface cooling to space, *α*_{srf}, which only depends on surface temperature and the total transmission through the atmosphere; and (b) feedback due to atmospheric cooling to space, *α*_{Atm}, which depends on the thermodynamic structure and gas composition within the atmosphere. We further show that *α*_{Atm} sources from increased emission temperatures with warming caused by the well‐understood collision‐broadening effect and the presence of well‐mixed GHGs and O_{3}. The *α*_{Atm} decreases from −0.2 Wm^{−2}K^{−1} at 255 K to −0.9 Wm^{−2}K^{−1} at 300 K, the magnitude of which is quantitatively predicted by the emission temperature shift theory via pseudo‐adiabatic lapse rates (Equation 10). In the absence of *α*_{Atm}, the clear‐sky longwave feedback would increase from −1.5 Wm^{−2}K^{−1} at 255 K to −0.9 Wm^{−2}K^{−1} at 300 K because water vapor continuum absorption increases the *α*_{srf} (Figure 2), rather than being relatively constant at around −1.8 Wm^{−2}K^{−1}. Thus, without *α*_{Atm}, the clear‐sky longwave feedback parameter would be only half as stable as it is, which is an important source of the paradox found in Simpson (1928) (while Simpson (1928) also overestimates the atmospheric opacity when it adopts a gray atmosphere assumption). We conclude that GHGs induce an atmospheric feedback process that critically stabilizes Earth's climate.

The analytical decomposition of the clear‐sky longwave feedback into *α*_{srf} and *α*_{Atm} is consistent with feedback analysis studies that built upon radiative kernel methods (Soden et al., 2008; X. Huang et al., 2014; Y. Huang et al., 2017; Kramer et al., 2019). These studies decompose the clear‐sky longwave radiative processes into contributions from surface temperature, layer‐by‐layer air temperature, and water vapor. Our result shows the two atmospheric terms in the kernel method (air temperature and water vapor) compensate for each other but not fully, due to the foreign collision broadening effect: in strong water vapor absorption channels, one may expect roughly 15% of the atmospheric feedback to remain when relative humidity, as opposed to specific humidity, is fixed (as inferred from Equation 8 compared to Equation 9, Figure 3b compared to Figure 3c, and Figure 4d). Further studies will be conducted to link the analytical framework proposed here with conventional feedback analysis.

Considering the global‐mean feedback parameter is a weighted sum of relatively constant feedback (with respect to base state *T*_{s}, Figure 2) over a surface warming response Δ*T*_{s}, our result suggests that the global‐mean clear‐sky longwave feedback parameter is not extremely sensitive to the pattern of Δ*T*_{s}, under the condition that the atmospheric warming follows a constant relative humidity and uniform upward shift of temperature (see assumptions made in Section 3). Therefore, the state‐dependency of the climate sensitivity on Δ*T*_{s} pattern, as disclosed in Andrews et al. (2015), may come from regions with varying relative humidity (Pan & Huang, 2018), such as over the tropical ocean, and regions where the atmospheric warming is insensitive to surface warming (e.g., temperature inversion layer (Andrews & Webb, 2018)), for which the assumption that tropospheric warming is strictly coupled with *T*_{s} (Equation 6) does not apply. Feedback processes in those regions are found to be affected by remote responses to Δ*T*_{s} patterns (Zhou et al., 2017), rather than by local Δ*T*_{s} alone. Considering these uncertainties sources majorly from the tropical ocean (Andrews et al., 2015; Dong et al., 2020) where the radiative contribution of *T*_{s} (*α*_{srf}) is small (−0.9 Wm^{−2}K^{−1} in Soden et al. (2008) and Figure 2a at 300 K), our results are consistent with Dessler et al. (2018) who finds that the radiation balances under the impact of internal variability are more related to atmospheric temperature rather than surface temperature. Note that the discussion here does not involve clear‐sky shortwave processes nor cloud radiative feedback, which has been found to be an important source of the state‐dependency (Andrews et al., 2015; Zhou et al., 2015, 2017).

In a climate hotter than Earth's tropics (i.e., 300 K in Figures 5b and 5d), our study suggests that *α*_{Atm} alone explains the clear‐sky longwave feedback since *α*_{srf} would vanish to zero (Koll & Cronin, 2018; Seeley & Jeevanjee, 2021). Following Figure 5 and the discussion in Section 4, *α*_{Atm} would become more negative (stable) than the −0.9 Wm^{−2}K^{−1} because (a) tropospheric cooling to space (*R*_{trop}) increases as the troposphere becomes thicker and more opaque with *T*_{s} and (b) the emission temperature shift (*r*) is enhanced by a steeper pseudo‐adiabatic lapse rate (Γ_{eff}), which gives rise to stronger upper‐tropospheric warming than the surface. The sensitivity to the upper troposphere can be further amplified if CO_{2} mass increases with surface warming, as shown in Seeley and Jeevanjee (2021). While Kluft et al. (2021) has questioned the effectiveness of CO_{2} on the feedback process, it is evident in our study that the presence of CO_{2} is not required for the negative atmospheric feedback process at all, because the negative feedback process is sufficiently maintained by water vapor via the collision‐broadening with nitrogen and oxygen in the absence of any other GHGs (Figures 1d, and Figures B2e and B2f compared to Figures B2c and B2d).

Importantly, the stability and predictability of the clear‐sky longwave feedback rely on the robust, near‐constant relative humidity, lapse rate, and tropopause at temperature levels with surface warming (Held & Shell, 2012; Ingram, 2010). In this process, the mass of non‐condensable gases is well‐maintained by the atmosphere and other components of the climate system. As long as a similar evolving thermodynamic pattern is exhibited, the theory presented in this study is generalizable to past, present, and future climates of Earth, as well as other planets. At the time of longwave saturation, any breakdown of this pattern might trigger a runaway greenhouse effect, either locally, seasonally, or globally. Thus, our study suggests that the runaway greenhouse effect occurs conditionally, rather than unconditionally (Ingersoll, 1969; Nakajima et al., 1992). While Earth‐like climate may become unstable given sufficiently high radiation disruptions (e.g., from insolation or anthropogenic emissions) in simulations with idealized thermodynamic pattern (Goldblatt et al., 2013), our results indicate that such runaway might initiate from a surface temperature much higher than present‐day Earth (i.e., at and beyond the boiling point) so that the foreign pressure‐broadening effect becomes weak enough (high saturation vapor pressure vs. conserved background gases) to be overcome by positive shortwave feedback from clouds, albedo, and water vapor. These conditions should be examined with care in future studies when addressing the emergence of the runaway greenhouse effect on Earth and other Earth‐like planets.

Two experiments are conducted with the Geophysical Fluid Dynamics Laboratory (GFDL)'s CM3 (Donner et al., 2011; Griffies et al., 2011) in Paynter et al. (2018). The first is a control run, where CO_{2} is fixed at a pre‐industrial level, and the second is an experiment run, where CO_{2} increases by 1% per year until reaching a doubling and then CO_{2} is held constant until equilibrium with the control run is reached. We evaluate these two runs at the equilibrium state (approximately 4.8 K of warming, 4,800 years after CO_{2} doubling).

Every grid point from the control run is composited into every 5‐K bin of surface temperature. Figures in this study show bins from 252.5 to 302.5 K, covering 89% of these grids. The mean profiles of each bin are obtained from both the control and the experimental run. Using these composited profiles, a set of radiative transfer calculations is conducted. Spectrally resolved gas optical depths are calculated using a new benchmark line‐by‐line model, pyLBL (

Four experiments are conducted to decompose the longwave radiative feedback.

Atmospheric profiles and surface temperature from the control run.

Atmospheric profiles and surface temperature from the experimental run.

Temperature profiles and surface temperature from the experimental state while holding relative humidity fixed at the control run.

Temperature profiles and surface temperature from the experimental state while holding profiles above the cold‐point tropopause of the control run fixed.

Experiment a is used as the “base state” and experiment b is used as the “warm state” in this study. The clear‐sky longwave feedback is estimated from the difference in OLR between the a and b, as shown in Figure 4. We find that the feedback estimated from c to a is similar to b to a, confirming that RH feedback is small from a global‐mean perspective (Held & Shell, 2012). Experiment *d* is used in Appendix C for validating Equation 3, in which feedback sourced from the stratosphere is manually neglected.

These LBL calculations are driven by three combinations of greenhouse gases and an experiment that excludes water vapor continuum absorption.

“All gases”: water vapor and O_{3} profiles and well‐mixed CO_{2}, N_{2}O, CH_{4} at the pre‐industrial gas level.

“WGHGO3”: O_{3} profile and well‐mixed CO_{2}, N_{2}O, and CH_{4} (without water vapor).

“H2O”: water vapor line and continuum absorption (without other GHGs).

“noctm”: water vapor line absorptions, O_{3} profile and well‐mixed CO_{2}, N_{2}O, CH_{4} at the pre‐industrial gas level.

Background gases, including N_{2} and O_{2} , are held constant at fixed numbers of molecules, regardless of the combinations of greenhouse gases.

Following Equation 4.15 in Goody and Yung (1989) (Goody & Yung, 1989) and Equation. 4.69 in Pierrehumbert (2010) (Pierrehumbert, 2010), the averaged transmission between *υ* − *δυ*/2 and *υ* + *δυ*/2 due to a strong gas line is proportional to the square root of the prod of *i* denotes a discrete atmospheric layer, *υ* − *δυ*/2 and *υ* + *δυ*/2 from this layer to space. *p*_{i} and *T*_{i} are the air pressure and temperature. *G*. This approximation is obtained by integrating over the far‐tail of Lorentz profile (Goody & Yung, 1989; Pierrehumbert, 2010). Here we assume that the width of the Lorentz profile collision broadening is proportional to *p*_{i} and independent of *T*_{i} and the line intensity is proportional to *k*_{l} is taken to be 0.02 for CO_{2} but zero otherwise due to (a) a low concentration of other well‐mixed gases on Earth, and (b) a much stronger impact from the temperature dependence of saturation vapor pressure. The validity of Equation B1 is examined in Figure B1 using water vapor absorption as an example. It shows that *k*_{l} is taken to be zero.

In the following derivation, we treat ^{−1} resolution outputted from line‐by‐line calculation at a reference state (280K surface temperature in this study and as provided in the supplementary).

For well‐mixed gas with constant volume‐mixing ratio *n*, *p*_{gas,i} ≡ *np*_{i}. If surface warms from *T*_{s} to *T*′, when *k*_{l} = 0:*T*_{s}. Hence, the air temperature that contributes the same weight to TOA systematically increases by *r*.

This approximation of *k*_{l} = 0 holds well for CH_{4}, N_{2} O, and tropospheric O_{3}, due to their low concentration in Earth's atmosphere. The coefficient for temperature‐dependent line intensity, *k*_{l}, is taken to be 0.02 for CO_{2}.

The partial pressure of water vapor *p*_{q}, unlike well‐mixed gases, is determined by saturation vapor pressure and relative humidity. *ql* denotes water vapor line absorption. *k*_{CC} is a linear coefficient to approximate the Clausius–Clapeyron equation.

We can solve for the *r* required to reach the same transmission by taking a logarithm of Equation B7 and applying a first‐order Taylor expansion:

Self‐continuum absorption of water vapor depends only on temperature that determines the vapor pressure (Paynter & Ramaswamy, 2011). Although the strong‐line approximation does not strictly work for self‐continuum absorption, a similar approximation can be applied to account for the impact of vapor pressure on the averaged transmission between *υ* − *δυ*/2 and *υ* + *δυ*/2 due to self‐continuum absorption:*qc* denoting water vapor self‐continuum absorption. This approximation neglects the effect of temperature on continuum absorption and partly leads to bias in Figure 4 at high surface temperature.

A log of averaged transmission between *υ* − *δυ*/2 and *υ* + *δυ*/2 is taken:*G* here denotes for major greenhouse gases other than water vapor, including CO_{2}, CH_{4}, N_{2}O, and O_{3}.

By taking a first‐order approximation of the logarithm of this equation, we can solve for *r* as:*f*_{CC} represents the response from exponential dependence of saturation vapor pressure on air temperature, *f*_{BM} represents the response from foreign pressure that is regulated by lapse rate under the hydrostatic balance, *f*_{Sum} represents the response from the effect of temperature in mass density. The dependence of line intensity of CO_{2} on temperature is included using *k*_{l} = 0.02. These coefficients can be estimated from radiative transfer calculations performed at the base state, by treating *n* = 0.5:^{−1} and for the entire infrared from 20 to 3,250 cm^{−1}). Note here coefficients of O_{3} are treated as well‐mixed gases because tropospheric O_{3} does not strongly vary with height (or air temperature). Figure B2 shows the predicted feedback parameters with different mixtures of greenhouse gases, in comparison with LBL results which exclude changes in the stratosphere.

We acknowledge GFDL resources made available for this research. We thank three anonymous reviewers for their constructive comments. Nadir Jeevanjee and Shiv Priyam Raghuraman are acknowledged for their comments and suggestions on an internal review of the manuscript. Jing Feng is supported by the NOAA Climate Program Office under Grant U8R1ES2/P01.

The line‐by‐line derived regression coefficients used in Equation 10 and an example code of the analytical model is accessible at (Feng, 2023). The benchmark line‐by‐line model, pyLBL, is available at