Editorial Type:
Article
Article Type:
Research Article

An Analytic Model for the Clear-Sky Longwave Feedback

Daniel D. B. Koll aLaboratory for Climate and Ocean-Atmosphere Studies, Department of Atmospheric and Oceanic Sciences, Peking University, Beijing, China

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https://orcid.org/0000-0002-9076-6901
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Nadir Jeevanjee bGeophysical Fluid Dynamics Laboratory, Princeton, New Jersey

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Nicholas J. Lutsko cScripps Institution of Oceanography, La Jolla, California

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Online Publication:
09 Aug 2023
Print Publication:
01 Aug 2023
DOI:
https://doi.org/10.1175/JAS-D-22-0178.1
Page(s):
1923–1951
Open access
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Abstract

Climate models and observations robustly agree that Earth’s clear-sky longwave feedback has a value of about −2 W m−2 K−1, suggesting that this feedback can be estimated from first principles. In this study, we derive an analytic model for Earth’s clear-sky longwave feedback. Our approach uses a novel spectral decomposition that splits the feedback into four components: a surface Planck feedback and three atmospheric feedbacks from CO2, H2O, and the H2O continuum. We obtain analytic expressions for each of these terms, and the model can also be framed in terms of Simpson’s law and deviations therefrom. We validate the model by comparing it against line-by-line radiative transfer calculations across a wide range of climates. Additionally, the model qualitatively matches the spatial feedback maps of a comprehensive climate model. For present-day Earth, our analysis shows that the clear-sky longwave feedback is dominated by the surface in the global mean and in the dry subtropics; meanwhile, atmospheric feedbacks from CO2 and H2O become important in the inner tropics. Together, these results show that a spectral view of Earth’s clear-sky longwave feedback elucidates not only its global-mean magnitude, but also its spatial pattern and its state dependence across past and future climates.

Significance Statement

The climate feedback determines how much our planet warms due to changes in radiative forcing. For more than 50 years scientists have been predicting this feedback using complex numerical models. Except for cloud effects the numerical models largely agree, lending confidence to global warming predictions, but nobody has yet derived the feedback from simpler considerations. We show that Earth’s clear-sky longwave feedback can be estimated using only pen and paper. Our results confirm that numerical climate models get the right number for the right reasons, and allow us to explain regional and state variations of Earth’s climate feedback. These variations are difficult to understand solely from numerical models but are crucial for past and future climates.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Daniel D. B. Koll, dkoll@pku.edu.cn

Abstract

Climate models and observations robustly agree that Earth’s clear-sky longwave feedback has a value of about −2 W m−2 K−1, suggesting that this feedback can be estimated from first principles. In this study, we derive an analytic model for Earth’s clear-sky longwave feedback. Our approach uses a novel spectral decomposition that splits the feedback into four components: a surface Planck feedback and three atmospheric feedbacks from CO2, H2O, and the H2O continuum. We obtain analytic expressions for each of these terms, and the model can also be framed in terms of Simpson’s law and deviations therefrom. We validate the model by comparing it against line-by-line radiative transfer calculations across a wide range of climates. Additionally, the model qualitatively matches the spatial feedback maps of a comprehensive climate model. For present-day Earth, our analysis shows that the clear-sky longwave feedback is dominated by the surface in the global mean and in the dry subtropics; meanwhile, atmospheric feedbacks from CO2 and H2O become important in the inner tropics. Together, these results show that a spectral view of Earth’s clear-sky longwave feedback elucidates not only its global-mean magnitude, but also its spatial pattern and its state dependence across past and future climates.

Significance Statement

The climate feedback determines how much our planet warms due to changes in radiative forcing. For more than 50 years scientists have been predicting this feedback using complex numerical models. Except for cloud effects the numerical models largely agree, lending confidence to global warming predictions, but nobody has yet derived the feedback from simpler considerations. We show that Earth’s clear-sky longwave feedback can be estimated using only pen and paper. Our results confirm that numerical climate models get the right number for the right reasons, and allow us to explain regional and state variations of Earth’s climate feedback. These variations are difficult to understand solely from numerical models but are crucial for past and future climates.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Daniel D. B. Koll, dkoll@pku.edu.cn
Keywords: Feedback; Longwave radiation; Radiation; Radiative transfer

1. Introduction

Earth’s climate sensitivity is a crucial factor in understanding and predicting climate change. While uncertainty in climate sensitivity is dominated by cloud feedbacks, the magnitude of climate sensitivity is largely set by the clear-sky longwave feedback λLW. Early studies estimated λLW to be −2.2 to −2.3 W m−2 K−1 (Manabe and Wetherald 1967; Budyko 1969). These estimates were impressively close to the current best estimates from climate models and observations, which agree on a fairly narrow range for λLW of about −1.8 to −2.2 W m−2 K−1 (Andrews et al. 2012; Chung et al. 2010; Kluft et al. 2019; Zhang et al. 2020; Zelinka et al. 2020). By contrast, the recent Sherwood et al. (2020) assessment estimated the total cloud feedback to be both smaller in magnitude and less certain at +0.45 ± 0.33 W m−2 K−1.

The robustness of the clear-sky longwave feedback suggests that one should be able to understand and describe its governing physics in fairly simple form. A simple model for λLW would provide definitive support for the value of −2 W m−2 K−1 derived from observations and climate models. It would also allow us to understand the state dependence of λLW: at warm enough temperatures Earth’s atmosphere transitions to a runaway state, in which λLW becomes zero or even changes sign, but it is unclear how λLW varies between today’s value and the runaway limit. Similarly, there is a long-standing interest in using paleoclimate proxies to constrain present-day climate sensitivity (Tierney et al. 2020), but this effort suffers from uncertainty regarding the state dependence of climate feedbacks (Meraner et al. 2013; Bloch-Johnson et al. 2015). Finally, geographic variation in feedbacks and their importance for the so-called pattern effect is an ongoing topic of research (Armour et al. 2013; Andrews et al. 2015, 2018), but if λLW has state dependence then that dependence should also influence the spatial pattern of λLW. For example, if the global-mean λLW was different in past climates due to changes in the global-mean surface temperature, then present-day λLW should show regional variation due to Earth’s surface temperature pattern, suggesting a close link between state dependence and spatial dependence of λLW.

One of the earliest models for λLW was proposed by Simpson (1928a), who found that an atmosphere that is optically thick due to water vapor would have a clear-sky longwave feedback that is approximately zero, suggesting Earth should be in a runaway greenhouse. Although this early model was abandoned by Simpson (1928b) as being overly simplistic, Ingram (2010) resolved the Simpsonian “paradox” by separating out the parts of Earth’s outgoing radiation spectrum that are optically thick due to water vapor (and for which λLW is approximately zero) from the optically thin “window” region. Koll and Cronin (2018) subsequently quantified Ingram’s argument: using fixed relative humidity (RH), single-column calculations they argued that for present-day Earth the clear-sky longwave feedback is dominated by the surface:
λLW≈λsurf.
Here λsurf is the surface Planck feedback, which is smaller than a blackbody’s feedback because greenhouse gases block the surface’s emission outside the spectral window. Meanwhile, the atmosphere itself contributes less to λLW in the present climate, and so to first order its contribution can be ignored. It follows that atmospheric feedback terms which are often the focus of climate model or observational analyses—the atmospheric component of the Planck feedback, the lapse rate feedback, and the water vapor feedback—roughly cancel (Koll and Cronin 2018; Jeevanjee et al. 2021a).

The match between λLW and the surface Planck feedback λsurf in Eq. (1) is not exact, however. Follow-up work found that λsurf only accounts for 50%–90% of λLW in different regions, with about 60% in the global mean (Raghuraman et al. 2019; Feng et al. 2023), implying a gap in the argument of Koll and Cronin (2018). Similarly, Seeley and Jeevanjee (2021) showed that in hot, high-CO2 climates λsurf becomes negligible yet λLW does not go to zero. As the surface warms the atmosphere is still able to increase its emission to space in spectral regions that are dominated by CO2. This emission mostly comes from the upper atmosphere, and gives rise to a spectral CO2 “radiator fin” feedback. The existence of a CO2 feedback means λLW must depend on CO2 concentration, and thus must have CO2 state dependence. Moreover, the CO2 feedback has to depend on the atmospheric lapse rate: if the atmosphere was isothermal with zero lapse rate, CO2’s forcing and feedback would both have to be zero, in line with previous work which tried to quantify the dependence of CO2 forcing on the lapse rate (Huang and Bani Shahabadi 2014; Dufresne et al. 2020), even if the details of the forcing mechanism are still disputed (Seeley 2018; Romps et al. 2022). So while the “surface-only” feedback picture from Koll and Cronin (2018) gives a reasonable first-order approximation to λLW, more terms are needed to describe λLW quantitatively.

In this study, we aim to derive a simple model of Earth’s feedback that can quantitatively capture the magnitude of λLW as well as its state dependence and regional variations. The model decomposes λLW into the surface Planck feedback (λsurf) plus three atmospheric terms: a CO2 band feedback (λCO2), a non-Simpsonian water vapor band feedback (λH2O), and a destabilizing water vapor continuum feedback (λcnt). Although these feedbacks are less familiar, they represent the different substances through which Earth gives longwave radiation off to space, and how each substance changes its emission under surface warming. As shown below, expressions can be derived for each spectral feedback term starting from the basic equations of radiative transfer. These expressions can be interpreted as a global-mean model for λLW or in terms of local feedbacks (Feldl and Roe 2013; Armour et al. 2013; Bloch-Johnson et al. 2020). That is, each atmospheric column is treated as an isolated 1D system whose longwave feedback depends on its local surface temperature. We validate the model (and the utility of the spectral decomposition) by comparing it against calculations with a line-by-line radiation code.

Our model of λLW is based on spectroscopic thinking and hence represents a different perspective than the conventional decomposition which breaks the clear-sky longwave feedback into Planck, lapse rate, and water vapor feedbacks (e.g., Soden et al. 2008; Sherwood et al. 2020; Zelinka et al. 2020). The conventional decomposition has been an important tool for understanding λLW and for diagnosing the physics governing outgoing longwave radiation in climate models. However, it also obscures large cancellations between the atmospheric part of the Planck feedback, the lapse rate feedback, and the water vapor feedback (Held and Shell 2012; Koll and Cronin 2018; Jeevanjee et al. 2021a). By obscuring these cancellations, the conventional decomposition can give a false impression of the uncertainty of climate models. The same cancellations also make it difficult to understand the state dependence of λLW—Planck, lapse rate, and water vapor feedbacks all increase in a warmer climate, but it is far from obvious how these changes add up to affect λLW (Meraner et al. 2013). Building on previous discussions of spectral feedbacks (e.g., Huang et al. 2010; Huang et al. 2014; Koll and Cronin 2018; Pan and Huang 2018; Seeley and Jeevanjee 2021; Jeevanjee et al. 2021a; Kluft et al. 2021; Feng et al. 2023), our goal in this paper is to show that the issues that arise in the conventional decomposition can be resolved by viewing λLW in terms of its spectral components instead.

The layout of the rest of this paper is as follows. Section 2 discusses several preliminaries which are necessary for the main derivations: an idealized Clausius–Clapeyron relation, an analytic approximation for moist lapse rates, and idealized band models for H2O and CO2 spectroscopy. Section 3 lays out our spectral framework and introduces the emission-level approximation, our spectral decomposition of λLW, and a description of the numerical line-by-line calculations. Section 4 derives analytic expressions for Earth’s emission temperature in different parts of the spectrum, which are then used in section 5 to derive analytic feedbacks. Our expressions compare favorably against the state dependence of λLW from line-by-line calculations. Next, section 6 uses these results to understand the spatial pattern of Earth’s clear-sky longwave feedback. We generate global maps of Earth’s clear-sky longwave feedback using a radiative kernel and climate model data. We then show that our analytic expressions recover qualitatively similar feedback patterns, which implies that the spatial pattern of λLW can be largely understood using our analytic model. Breaking λLW up into surface versus atmospheric terms, we find that the surface dominates λLW in the global mean as well as in the dry subtropics, with a spatial pattern set by the pattern of atmospheric relative humidity, while atmospheric feedbacks become significant in the inner tropics, with spatial patterns that are set by regional lapse rate changes under warming. The manuscript closes in section 7 with conclusions and broader discussion of the results.

2. Preliminaries

Our goal is to derive the longwave feedback of a cloud-free vertical column. The column’s state can be specified using five parameters: Ts, γlr, RH, qCO2, and Tstrat. Here Ts is the surface temperature, γlr ≡ dlnT/dlnP is the temperature lapse rate, RH is the relative humidity, qCO2 is the CO2 mass mixing ratio, and Tstrat is the stratospheric temperature. We idealize the state of the column by treating γlr, RH, and qCO2 as vertically uniform; all are defined more precisely below. Similarly, we approximate the stratosphere as isothermal.

a. Clausius–Clapeyron

The Clausius–Clapeyron relation governs the temperature dependence of the saturation vapor pressure e*(T) and is an essential element of our analytic model. The Clausius–Clapeyron relation is often solved by ignoring the temperature dependence of the latent heat of vaporization, dln⁡e*/dln⁡T=Lυ(T)/(RυT)≈Lυ(T0)/(RυT), which leads to the quasi-exponential approximation
e*≈e0*(T0)exp⁡[−Lυ(T0)Rυ(1T−1T0)].
This quasi-exponential form does not lead to closed-form analytic expressions in the equations of radiative transfer, however, so we require a simpler form of the Clausius–Clapeyron relation. We obtain this by approximating the Clausius–Clapeyron relation further as dln⁡e*/dln⁡T=Lυ(T)/(RυT)≈const, which leads to a simple power law between temperature and saturation vapor pressure (Koll and Cronin 2019),
e*≈e0*(T0)(TT0)γwv,
where
γwv≡Lυ(T0)RυT0.
Here T0 is an arbitrary reference temperature around which we are approximating the saturation vapor pressure as a power law. We emphasize that T0 is effectively a thermodynamic constant and does not change with surface warming. The nondimensional power-law exponent is large and reflects the steep rise of e* with temperature; at Earthlike temperatures, γwv ≈ 20. The fractional increase in saturation vapor pressure per unit warming is d​ln⁡e*/dT=γwv/T∼7%K−1, in line with other Clausius–Clapeyron approximations.

Figure 1 compares the approximations in Eqs. (2) and (3) against a fit based on experimental data (Huang 2018). Considering that a typical tropical atmospheric column spans the vertical temperature range 200–300 K, the quasi-exponential approximation is very accurate, whereas our power-law approximation only matches to roughly a factor of 2. Nevertheless, as shown below, this accuracy is good enough to match numerical calculations.

Fig. 1.
Fig. 1.

Different approximations to the Clausius–Clapeyron relation. The black curve is the fit based on experimental data (Huang 2018). The blue curve is the commonly used quasi-exponential approximation. The orange curve is the power-law approximation used in this work. The saturation vapor pressure is with respect to liquid water. In this plot (T0,e0*) are set equal to the triple-point values of H2O, so γwv = 19.8.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0178.1

b. Bulk moist lapse rate

The vertical temperature–pressure profile of an atmospheric column can be specified via the lapse rate exponent
γlr=d​ln⁡T/d​ln⁡p,
where p is pressure. For a dry adiabat the lapse rate exponent is vertically uniform, γlr = Rd/cp ≈ 2/7. For a moist atmosphere γlr varies both as a function of temperature and pressure, but due to the latent heat release in a convecting parcel it is generally smaller than the dry lapse rate: γlr ≤ Rd/cp.
To obtain analytically tractable expressions we would like to treat γlr as constant in the vertical even for a moist column, so we diagnose a bulk γlr using the surface and tropopause values of (T, p):
γlr≈ln⁡(Ttp/Ts)ln⁡(ptp/ps).
Assuming that the tropopause temperature stays constant in response to surface temperature changes, in accord with the FAT/FiTT hypothesis (Hartmann and Larson 2002; Seeley et al. 2019), then all that is needed is an expression for how ptp depends on Ts. We can derive such an expression by first obtaining an expression for the tropopause height ztp, following Romps (2016). From MSE conservation along an undilute moist adiabat between the surface and tropopause,
ztp≈1g[cp(Ts−Ttp)+Lυqs*],
where qs* is the mass mixing ratio of water at saturation, q*, evaluated at the surface and we neglect q* at the tropopause. The value for ptp can then be obtained as
ptp=pse−ztp/H,
where H is the scale height of pressure [=RdTav/g] and Tav ≡ (Ts + Ttp)/2. Plugging this into (6) yields
γlr≈RdTavln⁡(Ts/Ttp)cp(Ts−Ttp)+Lυqs*.
One can show that Eq. (9) correctly reduces to the dry lapse rate γlr = Rd/cp by setting qs*=0 and series expanding the logarithm, assuming Ts − Ttp ≪ Ttp. In practice the latter assumption is not strictly true but the resulting deviation from the dry adiabat is small even for a 100 K difference between surface and tropopause.
According to the bulk approximation, γlr is constant in the vertical and varies only in response to climatic changes (e.g., changes in surface temperature). One can then integrate Eq. (5) to solve for the column’s temperature–pressure profile. This leads to a power law similar to a dry adiabat,
T(p)=Ts(pps)γlr(Ts),
where the only difference to a dry adiabat is that now the lapse rate depends on surface temperature.

Figure 2 (left) compares profiles based on Eq. (10) to moist adiabatic profiles. The moist adiabats are obtained by numerically integrating a generalized form of the moist adiabat which does not approximate water vapor as a dilute substance and thus remains valid at high temperatures (Ding and Pierrehumbert 2016). In all cases, the tropopause temperature is assumed to be fixed and equal to Tstrat = 200 K. The analytic profiles given by Eq. (10) produce a reasonable fit to the moist adiabats, though at surface temperatures below 340 K they produce slightly colder tropospheres. The tropopause pressure is accurately reproduced, as the analytic profiles always reach the tropopause at roughly the same point as the moist adiabats.

Fig. 2.
Fig. 2.

Moist adiabatic lapse rates vs our analytic approximation. (left) Pressure–temperature profiles following a moist adiabat (solid) and following the bulk lapse rate approximation (dashed). (right) Adiabatic lapse rate γlr numerically computed at three fixed temperature levels inside the troposphere (light blue) compared with the bulk approximation in Eq. (9) (orange). Note that γlr(T) is undefined if T is larger than the surface temperature Ts. The average γlr (dark blue) is a mass-weighted mean of all numerical lapse rates inside the troposphere, 1/(ps−ptp)×∫ptppsγlrdp.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0178.1

Figure 2 (right) compares the Ts dependence of γlr. First, the moist adiabatic T(p) profiles shown in Fig. 2 (left) are used to numerically compute γlr at individual levels of the troposphere. Because our bulk expression for γlr only depends on temperature, and not pressure, the moist adiabatic values of γlr are similarly shown at fixed temperature levels. Additionally, for each adiabatic T(p) profile we compute the average moist lapse rate using a mass-weighted mean, 1/(ps−ptp)×∫ptppsγlrdp. Figure 2 shows that our analytic approximation captures the Ts dependence of the average moist lapse rate relatively well, though this general agreement can obscure significant differences at individual levels. For example, our analytic approximation of γlr deviates by more than a factor of 2 from the moist-adiabatic γlr at the T = 220 K level. We will show below that these details of atmospheric lapse rates do not have a major impact on Earth’s longwave feedback at low surface temperatures, but they become increasingly important above ∼300 K.

c. H2O and CO2 spectroscopy

The third ingredient for our derivations is a model of H2O and CO2 spectroscopy. We follow previous studies and model the absorption cross sections of H2O and CO2 as log-linear band shapes. Despite the simplicity of these models, they are able to explain numerous features of Earth’s climate, including the logarithmic nature of CO2 forcing, the temperature dependence of Earth’s surface feedback, and the vertical structure of radiative cooling (Crisp et al. 1986; Pierrehumbert 2010; Wilson and Gea-Banacloche 2012; Koll and Cronin 2018; Jeevanjee and Fueglistaler 2020; Romps et al. 2022). Because we explore feedbacks over a wide range of temperatures, we additionally need to account for the H2O continuum. We do so by approximating the continuum as a gray absorber.

For CO2, the absorption cross section is
κCO2=κ0(pp0)exp⁡(−|ν−ν0|lν),
where κ0 is the absorption cross section in the center of the band, p0 is a reference pressure, ν is wavenumber, ν0 the wavenumber of the center of the band, and lν the decay rate of the absorption cross section in wavenumber space. Previous work fit these parameters to the CO2 absorption spectrum at a reference pressure of p0 = 0.1 bar (Jeevanjee et al. 2021b). Because the choice of reference pressure is arbitrary, we here rescale the fits to the dry surface pressure in our calculations (i.e., the surface pressure excluding the contribution of water vapor), p0 = 1 bar. The resulting values are κ0 = 500 m2 kg−1, ν0 = 667.5 cm−1, and lν = 10.2 cm−1.
H2O band absorption can similarly be modeled using a log-linear shape, though one has to account for the fact that H2O has two bands which are relevant for Earth’s longwave feedback. The rotation band determines H2O absorption at wavenumbers less than 1000 cm−1 and the vibration–rotation band at wavenumbers larger than 1000 cm−1. We model these two bands as
κH2O,line=(pp0) max⁡[κrotexp⁡(−|ν−νrot|lrot),κv–rexp⁡(−|ν−νv–r|lv–r)].
The first term in the max(⋅⋅⋅) expression represents the rotation band, which dominates at low wavenumbers, while the second term represents the vibration–rotation band at high wavenumbers. The factor p/p0 in front of both H2O and CO2 cross sections reflects pressure broadening: under present-Earth conditions CO2 and H2O absorption lines become wider due to collisions of those molecules with the background air (N2 or O2). This has the overall effect that both gases become more efficient absorbers at higher pressure.
In contrast to the CO2 and H2O bands, the H2O continuum is dominated by self-broadening so the continuum cross section is independent of pressure and instead scales as ∝e=RHe*. Although continuum absorption is not uniform with respect to wavenumber, its spectral dependence is significantly weaker than the H2O or CO2 bands. We therefore approximate the continuum as a gray absorber and write
κH2O,cnt=κcntRHe*(T)e0*(TT0)−a,
where the dimensionless exponent a captures the direct temperature dependence which acts to weaken the continuum (Pierrehumbert 2010). The total H2O cross section is the sum of line and continuum absorption, κH2O=κH2O,line+κH2O,cnt. Because the line opacity decreases exponentially away from H2O band centers, the total opacity becomes largely dominated by the continuum in the window region around ∼1000 cm−1.

Our model of H2O spectroscopy has eight parameters: κrot, lrot, rot, κv–r, lv–r, νv–r, κcnt, a. We set νrot = 150 cm−1 and νv–r = 1500 cm−1, and fit the remaining parameters using the median-smoothed H2O cross sections shown in Fig. 3 across the wavenumber range 150 cm−1 ≤ ν ≤ 1500 cm−1. The results are sensitive to the smoothing procedure, that is whether one uses a geometric mean or a median. Because the average transmission across a spectral band tends to be dominated by the most optically thin frequencies (Pierrehumbert 2010), we use a median filter. To perform the fits we use the nonlinear least squares algorithm scipy.optimize.curve_fit, with a reference temperature of T0 = 300 K. We first fit the parameters κrot, lrot, κv–r, lv–r to H2O line opacities only, and then use these parameters to fit κcnt and a to H2O cross sections that include both line and continuum opacity. The resulting values are κrot = 165 m2 kg−1, lrot = 55 cm−1, κv–r = 15 m2 kg−1, lv–r = 38 cm−1, κcnt = 3 × 10−3 m2 kg−1, and a = 7, which broadly match the H2O fits previously reported in Jeevanjee and Fueglistaler (2020). Table 1 summarizes the thermodynamic and spectral parameters used in this paper.

Fig. 3.
Fig. 3.

Idealized band models compared against the absorption cross sections of (top) CO2 and (bottom) H2O. Gray envelopes show cross sections computed at line-by-line spectral resolution; solid lines are the cross sections smoothed by a median filter with width 25 cm−1. Dashed lines are the band models for CO2 and H2O bands (the sum of line and continuum absorption), while dotted lines show the gray H2O continuum model only. The CO2 band model assumes the absorption cross section is independent of temperature, so only one dashed line is shown in the top-right panel.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0178.1

Table 1.

List of parameters and, where applicable, assumed values.
Table 1.

Figure 3 compares the idealized band models with line-by-line absorption cross sections. Overall, the shape of the cross sections is captured fairly well. The median CO2 and H2O cross sections scale linearly with total pressure, as expected for pressure broadening. The increasing H2O absorption in response to warming around 1000 cm−1 is also qualitatively captured by our gray continuum model, even though the H2O continuum itself is actually not gray.

Figure 3 (right column) shows that the slopes of the CO2 and H2O bands flatten as temperature increases, with roughly constant opacity in the band centers but increasing opacity in the band wings. This behavior is not captured by our simple models. Physically, absorption band slopes can depend on temperature due to the shifting population of different molecular excitation states. For example, the wings of the 667 cm−1 CO2 band consist of multiple smaller bands that correspond to transitions between excited states of CO2 (so-called hot bands), while the center of the CO2 band is dominated by transitions to/from the ground state of CO2. As temperature rises more CO2 molecules leave the ground state and access excited states, which in turn preferentially increases the opacity in the wings of the CO2 band. To keep our parameterizations simple, however, we do not attempt to model the temperature dependence of the band slopes.

3. Spectral framework

a. The emission-level approximation

To decompose the net longwave feedback into its spectral components we first need to consider the outgoing longwave flux (OLR) of a vertical column. At a spectral wavenumber ν, the column’s longwave flux varies vertically according to the monochromatic optical thickness τ* and the angle cos(θ) with which radiation propagates through the column. Assuming that the atmosphere’s longwave radiation follows a known angular distribution, e.g., isotropic, these quantities can be combined into the vertical coordinate τ=τ*/cos⁡(θ¯). Here cos⁡(θ¯) describes the average angle of propagation, and τ varies from τ = 0 at the TOA to τ = τsurf at the surface (e.g., Pierrehumbert 2010). The column’s OLR is then equal to
OLR=∫0∞πBν(Ts)e−τsurfdν+∫0∞∫0τsurfπBν[T(τ)]e−τdτdν.
The optical thicknesses τ and τsurf are functions of ν, so the order of integration cannot be switched. Physically, the first term corresponds to the surface’s emission to space, while the second term corresponds to an integral of the emission coming from each vertical level in the atmosphere.
The emission-level or radiating-level approximation states that the atmosphere’s emission to space [the second integral in Eq. (14)] originates from the vertical level at which optical thickness τ is order unity. The intuition behind the emission-level approximation is that levels of the atmosphere for which τ ≪ 1 are optically thin and do not contribute much to the TOA flux, while most emission from levels with τ ≫ 1 is absorbed by the overlying atmosphere and so its contribution to the TOA flux is also small. The emission level has been defined at slightly different values of τ, but all definitions agree on a value of order unity (Pierrehumbert 2010; Jeevanjee et al. 2021b). For simplicity, we define the emission level here as the level at which τ = 1. The temperature at this level is then the emission level temperature, Trad ≡ T(τ = 1), so
OLR≈∫0∞πBν(Ts)e−τsurf dν+∫0∞πBν[Trad(ν)] dν.
Given the emission-level approximation, the clear-sky longwave feedback is determined by how the surface emission and the atmospheric emission change in response to warming,
−λLW=dOLRdTs≈∫0∞πdBνdT|Tse−τsurfdν+∫0∞πdBνdT|TraddTraddTsdν.
The minus sign ensures consistency with the sign convention used in most climate studies: OLR typically increases in response to surface warming, so λLW < 0. Note that Eq. (16) does not contain any terms ∝ dτsurf/dTs because the resulting contribution to change in the surface emission decreases with warming at exactly the same rate as the atmospheric emission increases [this can be seen by differentiating Eq. (14) first before applying the emission-level approximation].

b. Spectral feedback decomposition

The net feedback in Eq. (16) can be decomposed into multiple spectral regions or bands. The surface term dominates in the window region where τsurf < 1 and the feedback is primarily a function of surface temperature Ts. The atmospheric emission dominates where τsurf > 1, and its magnitude primarily depends on the derivative dTrad/dTs. As we show below, dTrad/dTs differs depending on the opacity source at a given wavenumber. In this work we only consider Earth’s dominant greenhouse gases, CO2 and H2O, where H2O’s radiative effect additionally varies between the H2O bands and the H2O continuum, so we split the spectral integral into four terms:
−λLW=∫surfπdBνdT|Tse−τsurfdν+∫CO2πdBνdT|TCO2dTCO2dTsdν+∫H2OπdBνdT|TH2OdTH2OdTsdν+∫cntπdBνdT|TcntdTcntdTsdν=−(λsurf+λCO2+λH2O+λcnt),
where TCO2, TH2O, and Tcnt are the emission temperatures in the CO2 band, the H2O band, and the H2O continuum, respectively (the wavenumber range of each integral is discussed in section 5a). Based on the emitter, we refer to the four feedback terms as the surface feedback (λsurf), the CO2 band feedback (λCO2), the (non-Simpsonian) H2O band feedback (λH2O), and the H2O continuum feedback (λcnt).

Our spectral decomposition complements the conventional feedback decomposition which splits λLW into Planck, lapse rate, and water vapor (or relative humidity) feedbacks. The surface feedback λsurf measures the OLR increase due to surface warming while keeping the atmosphere fixed. This term is identical to the surface contribution of the Planck feedback, or “surface kernel,” in the conventional decomposition (Soden et al. 2008). As for the atmospheric feedback, Eq. (16) shows that it depends on the total derivative of Trad, that is, on dTrad/dTs. The conventional decomposition can be interpreted as splitting the total derivative dTrad/dTs up into various partial derivatives (uniform warming versus lapse rate versus water vapor changes), while using a single, spectrally averaged Trad. In contrast, our decomposition splits the atmosphere’s feedback into three different bands, but still retains the total derivative dTrad/dTs in each band. In principle our decomposition could be split further to recover the conventional decomposition. That is, one could further decompose dTrad/dTs in each band into partial derivatives of Trad that correspond to vertically uniform warming, lapse rate warming, and water vapor changes—see Jeevanjee et al. (2021a) for more details. Here, however, we do not pursue this approach because our analytic expressions are general enough to predict Trad and the total derivative dTrad/dTs.

We use relative humidity as the state variable throughout this paper, so the analytic results are compatible with papers that argue for the use of relative humidity in feedback decompositions instead of specific humidity (Held and Shell 2012; Jeevanjee et al. 2021a). In the fixed-RH framework the conventional water vapor feedback is replaced by a relative humidity feedback, which measures the clear-sky feedback due to RH changes. It is worth noting that the RH feedback is small in individual climate models, and its multimodel mean is close to zero (Zelinka et al. 2020). In the derivations below we therefore treat RH as an external parameter whose value is assumed constant under surface warming.

c. Line-by-line calculations

To calculate spectral feedbacks numerically we use a 1D line-by-line model, PyRADS (Koll and Cronin 2018). The model’s radiative transfer includes HITRAN2016 CO2 and H2O absorption data as well as the H2O component of the MTCKD continuum version 3.2 (Mlawer et al. 2012; Gordon et al. 2017). Calculations cover the spectral range 0.1–2500 cm−1 with a resolution of Δν = 0.01 cm−1, while the vertical resolution is 50 points in log pressure. In general the angular distribution of longwave radiation cos⁡(θ¯) varies in the vertical as well as across wavenumber (Li 2000; Feng and Huang 2019); however, a common approximation is to assume cos⁡(θ¯)=3/5 (Elsasser 1942), which is also used here.

The 1D calculations assume the atmosphere’s temperature profile follows either a moist adiabat or a power-law temperature–pressure profile that is consistent with our bulk lapse rate approximation. In both cases the troposphere is capped by a tropopause at 200 K, while the overlying stratosphere is isothermal at the same temperature. Relative humidity in the troposphere is vertically uniform while the H2O mass fraction in the stratosphere is set equal to its value at the tropopause. CO2 is treated as uniformly mixed in the vertical and fixed with respect to surface temperature. Because we are considering a wide range of surface temperatures, across which the tropopause pressure varies substantially, we vary the vertical grid spacing in PyRADS: for each surface temperature, the model-top pressure is set to a slightly lower value than the estimated tropopause pressure based on our bulk lapse rate formulation, which ensures the model’s top is always in the stratosphere and the tropopause is well resolved.

The spectrally resolved feedback is the difference in the spectrally resolved outgoing longwave flux, OLRν, between a base state and a perturbed state with warmed surface and atmosphere,
−λν=OLRν(Ts+ΔTs,T+ΔT)−OLRν(Ts,T)ΔTs.
We use ΔTs = 1 K, while ΔT denotes the atmospheric temperature perturbation caused by the surface warming ΔTs. Because relative humidity is kept fixed, the atmospheric warming T + ΔT also implies an increase in specific humidity.

Previous work has used various approaches to interpret line-by-line output. Seeley and Jeevanjee (2021) defined CO2 versus H2O bands based on the column-integrated, spectrally smoothed optical thickness of CO2 and H2O. However, the behavior of H2O differs strongly between the H2O bands and the H2O continuum, and it is difficult to distinguish these terms based on column-integrated optical thicknesses. For example, the H2O continuum might have a larger integrated optical thickness at some wavenumber than the H2O bands, but because continuum absorption decays more rapidly with altitude than band absorption [κcnt∝e*(T) versus κH2O∝p] the emission at the level where τ ∼ 1 could still be determined by the H2O bands.

Instead we first split the net feedback into its contributions from the surface versus atmosphere. The spectrally resolved surface feedback is the feedback in response to surface-only warming while keeping the atmosphere fixed,
−λsurfν=OLRν(Ts+ΔTs,T)−OLRν(Ts,T)ΔTs.
If we integrate λsurfν over all wavenumbers we get the surface feedback λsurf, equivalent to the surface kernel of Soden et al. (2008). The atmospheric feedback is equal to the difference between λν and λsurfν,
−λatmν=OLRν(Ts,T+ΔT)−OLRν(Ts,T)ΔTs.
We split λatmν into different bands based on the spectrally resolved emission pressures of CO2, H2O, and the H2O continuum. For each absorber PyRADS computes the optical thickness as a function of pressure and wavenumber, τ(p, ν). We define the CO2 emission pressure as the pressure at which the optical thickness of CO2 is equal to unity,
τCO2(prad,ν)=1,
which can be solved in each wavenumber bin to find prad(ν) (in practice we interpolate to find the pressure at which log[τ] = 0). The emission pressures of H2O and the H2O continuum are determined for each wavenumber bin in the same manner. The CO2 band feedback λCO2 is then the integral of λatmν over all wavenumbers at which CO2 has the smallest emission pressure, the H2O band feedback λH2O is the integral of λatmν over all wavenumbers at which H2O has the smallest emission pressure, and so on. The spectral decomposition is recomputed each time the atmosphere or surface state is varied, thereby allowing us to capture the state dependence of the longwave feedback not just due to changes in the atmosphere’s and surface’s emission but also due to changes in the width of spectral bands. We note that this approach is justified if one emitter clearly dominates the atmosphere’s emission at a given wavenumber, such that its emission pressure prad is much lower than that of any other emitters, but could be misleading if two emitters have very similar emission pressures. In practice, H2O and CO2 absorption cross sections decrease quasi exponentially away from their band centers (see section 2), which means the wavenumber range over which two absorbers can have a similar emission pressure is limited.

4. Emission temperatures

The feedbacks are set by the temperatures at the τ = 1 levels, so we seek analytic expressions for the emission temperatures TCO2, TH2O, and Tcnt. The optical thickness of a generic absorber is
τ=∫κqdpgcos⁡(θ¯),
where κ is the absorption cross section and q is the absorber’s mass-specific concentration. We use this equation to derive expressions for the emission temperatures by first writing the optical thickness in each band as a function of atmospheric temperature, then inverting these relations to find the emission temperature at the τ = 1 level.

a. CO2

CO2 is well mixed in the atmosphere so its mass-specific concentration qCO2 is vertically uniform. As discussed in section 2, its absorption cross section depends linearly on pressure due to pressure broadening and can be written as κCO2(ν,p)=κCO2*(ν)(p/p0), where κCO2* captures the wavenumber dependence of the CO2 absorption cross section, κCO2*∝exp⁡(−|ν−ν0|/lν), while p0 is an reference pressure. Because we previously chose p0 to be equal to the dry surface pressure, one can write κCO2*(ν)≈κCO2(ν,ps) (the approximation is due to neglecting the mass contribution of water vapor to ps). The optical thickness at a vertical level with temperature and pressure (T, p) is then
τCO2=∫0pκCO2*(p′ps)qCO2dp′gcos⁡(θ¯),=κCO2*2gcos⁡(θ¯)psqCO2p2,=κCO2*ps2gcos⁡(θ¯)qCO2(pps)2=κCO2*ps2gcos⁡(θ¯)qCO2(TTs)2/γlr≡τCO2*(ν) qCO2×(TTs)2/γlr,
where the fourth step uses the bulk lapse rate. Note that all spectroscopic parameters as well as ps and g are combined into a reference optical thickness, τCO2*(ν), which encapsulates how CO2 absorption varies with respect to wavenumber ν, surface pressure ps, and gravity g, but which can be treated as constant in response to warming.

b. Non-Simpsonian H2O

As for CO2, the absorption cross section of H2O scales linearly with pressure and can be written as κH2O(ν,p)=κH2O*(ν)(p/ps). We use the Clausius–Clapeyron power-law approximation to write the saturation specific humidity as q*≈Rd/Rυ×e0*/p×(T/T0)γwv and the specific humidity as q=RH×q*. The optical thickness of H2O at a level (T, p) is then
τ H 2 O = ∫ 0 p κ H 2 O * ( p ′ p s )q d p ′ gcos⁡( θ ¯ ) , ≈RH  κ H 2 O * e 0 * gcos⁡( θ ¯ ) R d R υ × ∫ 0 p ( p ′ p s ) ( T ′ T 0 ) γ wv d p ′ p ′ =RH  κ H 2 O * e 0 * gcos⁡( θ ¯ ) R d R υ × ∫ 0 T ( T ′ T s ) 1/ γ lr ( T ′ T 0 ) γ wv 1 γ lr d T ′ T ′ =RH  κ H 2 O * e 0 * gcos⁡( θ ¯ ) R d R υ 1 γ lr ( T 0 T s ) 1/ γ lr × ∫ 0 T ( T ′ T 0 ) γ wv +1/ γ lr d T ′ T ′ =RH  κ H 2 O * e 0 * gcos⁡( θ ¯ ) R d R υ 1 1+ γ wv γ lr × ( T T 0 ) ( 1+ γ wv γ lr ) / γ lr ( T 0 T s ) 1/ γ lr ≡RH  τ H 2 O * ( ν ) 1 1+ γ wv γ lr × ( T T 0 ) ( 1+ γ wv γ lr ) / γ lr ( T 0 T s ) 1/ γ lr ,
where the second step uses the Clausius–Clapeyron power law and also replaces the water vapor concentration in the stratosphere with the water vapor concentration of a moist adiabat that extends all the way to the top of atmosphere. We again define a reference optical thickness, τH2O*(ν), which encapsulates how H2O band absorption varies with respect to wavenumber ν, and gravity g, but which is independent of temperature.

c. H2O continuum

Absorption by the H2O continuum strengthens in response to increasing water vapor concentrations and weakens in response to warming, κH2O,cnt=κcnt×RH e*(T)/e*(T0)×(T/T0)−a. The optical thickness of the continuum is then
τ cnt =RH ∫ 0 p κ cnt e * ( T ′ ) e * ( T 0 ) ( T ′ T 0 ) −a q d p ′ gcos⁡( θ ¯ ) , ≈ RH 2   κ cnt e 0 * gcos⁡( θ ¯ ) R d R υ × ∫ 0 T ( T ′ T 0 ) 2 γ wv −a d p ′ p , = RH 2   κ cnt e 0 * gcos⁡( θ ¯ ) R d R υ × ∫ 0 T ( T ′ T 0 ) 2 γ wv −a 1 γ lr d T ′ T ′ , = RH 2   κ cnt e 0 * gcos⁡( θ ¯ ) R d R υ 1 ( 2 γ wv −a ) γ lr × ( T T 0 ) 2 γ wv −a ≡ RH 2   τ cnt * 1 ( 2 γ wv −a ) γ lr × ( T T 0 ) 2 γ wv −a ,
where the second and third steps make the same assumptions as the derivation for the H2O band. Here the reference optical thickness τcnt* encapsulates how the H2O self-continuum varies with respect to gravity g but has no dependence on wavenumber or temperature.

d. Emission temperatures

By setting τ = 1 and inverting the above relations, we arrive at the emission temperatures in the CO2 band, the H2O band, and the H2O self-continuum:
TCO2=Ts[1τCO2*(ν)qCO2]γlr/2,
TH2O=T0[1+γwvγlrτH2O*(ν)RH]γlr/(1+γwvγlr)(TsT0)1/(1+γwvγlr),
Tcnt=T0[(2γwv−a)γlrτcnt*RH2]1/(2γwv−a).
To interpret these emission temperatures, consider whether a given emitter stabilizes or destabilizes Earth’s climate. For CO2 it is easy to see that the feedback is always stabilizing. Ignoring lapse rate changes, we have TCO2∝Ts, so dTCO2/dTs>0. More intuitively, the optical thickness of CO2 can be written as
τCO2∝(TTs)2/γlr=(pps)2.
The emission level of CO2 is therefore a fixed function of pressure at a given atmospheric CO2 concentration. Given that the atmosphere’s temperature at a fixed pressure level always increases in response to surface warming, TCO2 also has to increase under warming. This effect can be thought of as a spectral radiator fin, and is also valid if the lapse rate γlr varies under surface warming. It implies that even if the atmosphere stops emitting more at all other wavenumbers, so dTrad/dTs = 0 outside the CO2 band, the presence of CO2 still allows the atmosphere to shed more energy to space in response to surface warming (Seeley and Jeevanjee 2021).
Next, our expressions suggest that the feedback from H2O is small and, to first order, might be negligible. Equation (26b) shows TH2O∝Ts1/(1+γwvγlr), where representative values for Earth’s tropics are γwv ∼ 20 and γlr ∼ 1/7, so the H2O emission temperature only depends weakly on surface temperature, TH2O∝Ts1/4. This small exponent is closely related to Simpson’s “paradox” (Ingram 2010) or Simpson’s “law” (Jeevanjee et al. 2021a), which state that TH2O is approximately independent of surface temperature. In the limit γwvγlr=d​ln⁡e*/d​ln⁡p≫1, that is, if water vapor increases much faster in the vertical than the total atmospheric mass, then
TH2O≈T0[γwvγlrτH2O*(ν)RH]1/γwv,
and TH2O ceases to depend on Ts. If the lapse rate is also independent of Ts we recover Simpson’s law:
dTH2OdTs≈0.
In reality, however, water vapor dominates much of the spectrum so even minor deviations from Simpson’s law can have a notable impact on the longwave feedback. Deviations arise because the H2O optical thickness is sensitive to pressure broadening and because changes in γlr modify the total water vapor path inside the atmospheric column. For present-day Earth the net impact of these changes is to increase the H2O emission temperature under surface warming: since TH2O∝Ts1/4, it follows that dTH2O/dTs>0, which means the H2O bands tend to stabilize Earth’s climate.
Finally, Tcnt has no direct dependence on surface temperature, but is sensitive to lapse rate changes. If we take the continuum’s emission temperature [Eq. (26c)], and assume that the direct temperature dependence of the continuum a ∼ 7 is much smaller than its temperature dependence due to the Clausius–Clapeyron relation, 2γwv ∼ 40, we have
Tcnt∝T0×[γlr]1/(2γwv).
Because the lapse rate γlr decreases under surface warming we have dTcnt/dTs = dTcnt/dγlr × dγlr/dTs < 0. Physically, this effect can be understood by considering the impact of γlr on the atmosphere’s total water vapor path. If one decreases the lapse rate γlr while keeping Ts fixed, the atmospheric column warms and thus can store more water vapor. To still maintain an optical thickness of unity then requires that the continuum’s emission level moves to colder temperatures. Our expressions thus predict that the H2O continuum gives rise to a destabilizing feedback.

e. Comparison against LBL calculations

Equations (26a)–(26c) predict how Earth’s emission temperature varies in response to changes in Ts, qCO2, γlr, and RH. To test these equations, we perform four sets of numerical experiments with PyRADS in which we variously change Ts, qCO2, γlr, and RH while holding the other parameters fixed. The default values are Ts = 290 K, 400 ppm of CO2, γlr = 2/7, and RH = 0.8. To match our underlying assumptions, we assume a bulk tropospheric lapse rate, so T=Ts(p/ps)γlr, which means the temperature profile differs from an adiabat if γlr < 2/7. The troposphere is capped by an isothermal stratosphere which is kept fixed at Tstrat = 200 K. Note that in Eqs. (26a)–(26c) the dependence on wavenumber only enters through the reference optical thicknesses τCO2*, τH2O*, and τcnt*, which are evaluated using the cross sections from section 2. Because the cross sections were fit independently, the analytic Trad expressions do not contain any free tuning parameters.

To compare the analytic results against line-by-line calculations we first numerically compute the top-of-atmosphere spectral flux OLRν for a given set of (Ts, qCO2, γlr, RH). We then smooth OLRν with a median filter of width 50 cm−1, before inverting it using the Planck function to find the atmosphere’s emission temperature (also known as brightness temperature) at a given wavenumber. Finally, we combine our analytic expressions into a single emission temperature via
Trad=max⁡[Tstrat,min⁡[Ts,TCO2,TH2O,Tcnt]],
to compare directly with temperatures from line-by-line calculations.

Figure 4 demonstrates that the analytic results compare favorably against numerical calculations. Even though the analytic Trad shapes are idealized compared to the numerical calculations, the overall response of Trad to perturbations is captured well. First, increasing CO2 concentration lowers Trad around 667 cm−1, which corresponds to the wings of the CO2 band. This is simply a spectrally resolved view of how increasing CO2 acts as a radiative forcing (Jeevanjee et al. 2021b). Second, warming the surface while keeping all other parameters fixed has multiple effects. The main impact is to increase the emission temperature in the window region between ∼800 and 1200 cm−1. In addition, there are secondary impacts: surface warming also shrinks the width of the CO2 band and slightly increases the emission temperature in the H2O bands below 600 cm−1 and above 1300 cm−1 (this latter effect is hard to see in Fig. 4). The increased emission in the H2O bands shows that Simpson’s law in Eq. (29) is not exact, an effect that is captured by our analytic expressions. Third, reducing the lapse rate γlr preserves the width of the CO2 band, but it flattens the steepness of its slopes and increases the emission temperature in the center of the band. In the H2O bands, a smaller γlr while keeping Ts fixed also leads to a non-Simpsonian increase of the emission temperature in the H2O bands. In contrast to the H2O bands, the emission temperature of the H2O continuum around 1000 cm−1 decreases as γlr is reduced. As discussed above, this is because the atmospheric water path increases with a smaller γlr, which reduces Tcnt. The feedback of the H2O continuum therefore has the opposite sign as the H2O bands, in line with the analytic results. Finally, reducing the relative humidity increases Trad in all regions dominated by water vapor, both in the H2O bands below 600 cm−1 and above 1300 cm−1 and in the H2O continuum around 1000 cm−1, while the CO2 band is unaffected.

Fig. 4.
Fig. 4.

Analytic emission temperatures (dashed) compared against numerical line-by-line results smoothed with a median filter of width 50 cm−1. Large panels show the entire infrared spectrum; small panels are zoomed in on the CO2 band. The y axes are flipped so that emission temperature decreases going up, the same way temperature decreases with altitude in Earth’s atmosphere.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0178.1

Overall, Fig. 4 underlines that comparatively simple physics is sufficient to explain the spectrally resolved response of Trad to different climate perturbations. To connect Fig. 4 back to the total clear-sky longwave feedback we only need to consider how these changes in Trad play out once we average them into spectral bands, and how multiple bands add up to determine the net longwave feedback.

5. Analytic feedbacks

Having derived expressions for the emission temperature in different parts of the LW spectrum, and verified these expressions against line-by-line calculations, we can now derive analytic expressions for the four spectral feedbacks: λsurf, λCO2, λH2O, and λcnt. Above each spectral feedback was defined as an integral over a wavenumber range [Eq. (17)], but the wavenumber ranges were not further specified. We therefore first define and estimate the width of the different spectral bands.

a. Bandwidths

We define an absorption band as the spectral range in which a given absorber has the coldest emission temperature compared to all other absorbers (this is equivalent to the highest-altitude emission level), and thus dominates the column’s emission to space. For example, the CO2 band is defined as all wavenumbers in which TCO2<min⁡[TH2O,Tcnt,Ts], as illustrated in Fig. 5a. The width of the CO2 band can then be computed from the two wavenumbers νedge that define the edges of the CO2 band, which is where the emission temperature of CO2 is equal to the emission temperatures of its neighboring absorbers: TCO2(νedge)=min⁡[TH2O(νedge),Tcnt,Ts].

Fig. 5.
Fig. 5.

Illustration of spectral bandwidths. The emission temperature is equal to the emission temperature of whichever emitter is coldest, Trad=min⁡[TCO2,TH2O,Tcnt,Ts], or the stratospheric temperature. (left) Lines show the analytic Trad (solid) and surface temperature Ts (dashed), while colored regions illustrate which emitters dominate in which band. The calculation shown uses Ts = 260 K, RH = 0.8, and 400 ppm of CO2. (right) Bandwidths as a function of surface temperature, numerically calculated based on our emission temperature expressions. Here ΔνH2O refers only to the rotational band at wavenumbers lower than 1000 cm−1. The jumps at ∼280, ∼295, and ∼325 K occur when the H2O band starts intersecting the CO2 band, when the continuum becomes opaque, and when the continuum becomes opaque on the left side of the CO2 band, at wavenumbers less than about 600 cm−1, respectively.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0178.1

1) CO2 bandwidth

To estimate the width of the CO2 band we consider three situations: 1) the CO2 concentration qCO2 is so low that even in the center of the CO2 band the optical thickness is less than one, 2) a dry atmosphere in which there is no overlap between CO2 and H2O bands, and 3) a moist atmosphere in which there is some overlap between CO2 and H2O.

First, at very low CO2 concentrations the bandwidth of CO2 is simply equal to zero. From the optical thickness of CO2 [Eq. (23)], the column-integrated optical thickness in the middle of the CO2 band is equal to τCO2(ν0,Ts)=qCO2τCO2*(ν0) so this occurs when
ΔνCO2=0,if qCO2τCO2*(ν0)<1.
As a representative value, we evaluate τCO2*(ν0) using κ0 = 500 m2 kg−1 from section 2c. We find that the middle of the CO2 band becomes optically thick above a CO2 concentration of ∼0.2 ppm. Note this value is only approximate, as our idealized band model deviates from real CO2 absorption cross section in the middle of the CO2 band (see Fig. 3).
Second, at nonnegligible CO2 concentrations and low water vapor concentrations, CO2–H2O overlap is negligible. Physically, this occurs either when the surface temperature is cold or the relative humidity is low; for simplicity we refer to this as the “cold” regime. In this regime the edge of the CO2 band can be defined as the wavenumber νcold at which TCO2 intersects with the surface temperature Ts, TCO2(νcold)=Ts. The emission temperature of CO2 is equal to TCO2=Ts×(τCO2*qCO2)−γlr/2 [Eq. (26a)], while our model of CO2 spectroscopy states τCO2(ν)*∝exp⁡(−|ν−ν0|/lν) [Eq. (11)]. Combining the two equations yields
νcold=ν0±lνlog⁡[qCO2τCO2*(ν0)],
where τCO2*(ν0)=κ0(ν0)ps/(2g) is the reference optical thickness in the center of the CO2 band. The overall width of the CO2 band in the cold regime is therefore
ΔνCO2cold=2lνlog⁡[qCO2τCO2*(ν0)].
To estimate the order of magnitude of ΔνCO2cold we again use κ0 = 500 m2 kg−1 and a qCO2 that corresponds to 400 ppm of CO2. The optical thickness in the center of the CO2 band is τCO2*(ν0)∼2600. This large optical thickness decreases exponentially with wavenumber away from ν0, so that TCO2=Ts only ∼80 cm−1 away from ν0. Because CO2’s band shape is symmetric about ν0, the present-day CO2 bandwidth is thus roughly 160 cm−1.
Third, at high water vapor concentrations, surface emission is replaced by H2O emission. Physically, this occurs either when the surface temperature is hot and/or relative humidity is high; for simplicity we refer to this as the “hot” regime. In this regime we solve the CO2 bandwidth as TCO2(νhot)=TH2O(νhot). Because the CO2 band decays much faster with wavenumber away from its band center than the H2O band does (lν ∼ 10 cm−1 versus lrot ∼ 55 cm−1; see Table 1), we further approximate TH2O as constant across the CO2 band and equal to its value in the CO2 band center TH2O(ν)≈TH2O(ν0). Combining the emission temperature of CO2 [Eq. (26a)] with our model of CO2 spectroscopy [Eq. (11)],
νhot=ν0±lνlog⁡{qCO2τCO2*(ν0)[TH2O(ν0)Ts]2/γlr},
where the emission temperature of H2O can be evaluated using Eq. (26b). Physically speaking, the H2O emission temperature is colder than the surface, TH2O(ν0)/Ts<1, so our model correctly captures the fact that H2O–CO2 overlap decreases the width of the CO2 band. Taking into account all three regimes, the overall width of the CO2 band is therefore
ΔνCO2={0,if qCO2τCO2*(ν0)<1,2×min⁡(νhot−ν0,νcold−ν0),if qCO2τCO2*(ν0)≥1.

2) H2O bandwidth

To determine the width of the H2O band the potential overlap with CO2 matters less because the CO2 band is too narrow to block a significant portion of the emission by H2O (at present-day CO2 concentrations). However, at high water vapor concentrations, competition between the H2O bands and the H2O continuum becomes important, so we again consider a “cold” and a “hot” regime. At low water vapor concentrations (physically, at cold temperature or low relative humidity) continuum absorption is negligible and we solve TH2O(νcold)=Ts. Combining the emission temperature of H2O [Eq. (26b)] with our H2O band model [Eq. (12)], this leads to
νLcold=νrot+lrot​log⁡[RHτrot*(νrot)1+γwvγlr(TsT0)γwv],
νRcold=νv–r−lv–r​log⁡[RHτv–r*(νv–r)1+γwvγlr(TsT0)γwv],
where νL is the left edge of the window below ∼1000 cm−1, and νR is the right edge of the window above ∼1000 cm−1 (see Fig. 5). The two H2O bands have different spectral slopes, and subscript “rot” denotes quantities that are related to the rotational H2O band at wavenumbers below 1000 cm−1 while subscript “v–r” denotes quantities related to the vibrational–rotational H2O band at wavenumbers above 1000 cm−1 (see section 2). At high water vapor concentrations, the continuum cuts off emission from the surface so the H2O band edge νhot is determined by TH2O(νhot)=Tcnt. Using the emission temperature of H2O [Eq. (26b)] and our H2O band model, we find
νLhot=νrot+lrot​log⁡[RHτrot*(νrot)1+γwvγlr(T0Ts)1/γlr(TcntT0)(1+γwvγlr)/γlr],
νRhot=νv–r−lv–r​log⁡[RHτv–r*(νv–r)1+γwvγlr(T0Ts)1/γlr(TcntT0)(1+γwvγlr)/γlr],
where the continuum emission temperature is given by Eq. (26c). Combining both regimes, the window width due to H2O absorption is therefore
Δνsurf(Ts,RH,γlr)=νR−νL=max⁡(νRcold,νRhot)−min⁡(νLcold,νLhot)⋅
Similar to the CO2 bandwidth, Eqs. (37) and (38) become invalid at very low RH or Ts because in those situations H2O ceases to be optically thick at all wavenumbers [mathematically, this happens when RH or Ts become small enough that the logarithms in Eqs. (37) and (38) change sign]. We do not consider the limit RH → 0 in this paper, but care should be taken when applying our results to extremely dry or cold atmospheres.
Finally, our feedback expression for the H2O band feedback requires us to separately specify the width of the rotational H2O band below 1000 cm−1. This width can be estimated by assuming that the rotational band always extends from 0 cm−1 to the left edge of the window region νL (see Fig. 5). Doing so presumes that H2O is always optically thick at low wavenumbers around ν = 0 cm−1. While this assumption again breaks down in very cold or dry climates (the maximum absorption in the rotational band occurs around ν ∼ 150 cm−1, not 0 cm−1, so low wavenumbers could become optically thin even if the band center is still optically thick), in those climates the H2O band feedback becomes negligible relative to the surface anyway. The width of the rotational H2O band is then
ΔνH2O(Ts,RH,γlr)≈νL−0=min⁡(νLcold,νLhot),
where the wavenumber νL denotes the left edge of the surface window (see above), as well as the right edge of the rotational H2O band.

b. Surface feedback

The surface feedback is given by
−λsurf=∫surfπdBνdT|Tse−τsurfdν.
The column-integrated optical thickness at a single frequency is the sum over all absorbers at that frequency, τsurf(ν)=τH2O(ν)+τCO2(ν)+τcnt. However, the optical thickness of H2O and CO2 drops off exponentially as a function of wavenumber away from their band centers. Thus, most frequencies are either so optically thick with respect to H2O and CO2 that all surface radiation is absorbed by the atmosphere (and hence does not contribute to the surface feedback), or so optically thin that we can ignore H2O and CO2. Inside the window we therefore only consider absorption by the gray continuum, τsurf ≈ τcnt, while the H2O and CO2 bands primarily set the width of the window.
To determine the width of the window we first consider an atmosphere without CO2. As discussed above, in this case the window region is set the H2O bands, with νL denoting the left window edge around ∼700 cm−1 and νR the right window edge around ∼1200 cm−1. The H2O continuum is gray and so can be taken out of the spectral integral,
−λsurf≈e−τcnt(Ts)∫νLνRπdBνdT|Tsdν.
We approximate the integral by treating the Planck function derivative as constant with respect to wavenumber, evaluated at the central wavenumber ν˜ of the window region, so ∫dBν/dTdν∝dBν˜/dT×Δν. In reality the Planck derivative is not constant with wavenumber, so our approximation should only be treated as a scaling which we account for by including a scaling constant csurf. The magnitude of csurf is further discussed below. The result is
−λsurf≈csurf×πdBν˜dT|Ts e−τcnt(Ts) Δνsurf,
where Δνsurf = νR − νL is the window region width due to H2O band absorption [see Eq. (39)], and we determine the central wavenumber of the window as ν˜=(νR+νL)/2.
Next, we add the effect of CO2-surface spectral blocking. Even if the atmosphere contained no water vapor whatsoever, part of the surface’s emission would still be absorbed by CO2 and thus have no effect on the TOA feedback. We account for the potential overlap between the surface and CO2 by simply subtracting the CO2 bandwidth from the H2O-only window width,
Δν˜surf=max⁡[0,Δνsurf(Ts,RH,γlr)−ΔνCO2(qCO2)],
where ΔνCO2 is defined above [Eq. (36)] and the tilde distinguishes the window width here from the H2O-only window width. Our final expression for the surface feedback is thus
−λsurf≈csurf×πdBν˜dT|Ts e−τcnt(Ts) Δν˜surf.

c. H2O band feedback

The H2O band feedback is given by
−λH2O=∫H2OπdBνdT|TH2OdTH2OdTsdν.
As sketched in Fig. 5, we consider the rotational H2O band as ranging from ν ≈ 0 to the left edge of the window, νL. We do not consider the potential feedback from the vibration–rotation band at wavenumbers higher than ∼1250 cm−1 and, for purposes of the H2O band feedback, also ignore CO2–H2O overlap effects.
The derivative of TH2O can be solved analytically. If water vapor behaved strictly according to Simpson’s law then dTH2O/dTs=0 and the H2O band feedback would be zero. Simpson’s law is only an approximation, however, so
dTH2OdTs=∂TH2O∂Ts+∂TH2O∂γlrdγlrdTs=11+γwvγlrTH2OTs+γwvγlr−γwvlog⁡(TsT0)+log⁡(1+γwvγlrRHτ0*)(1+γwvγlr)2TH2O×dγlrdTs.
One could also explicitly write out the lapse rate derivative dγlr/dTs, but the resulting expressions are long and do not lead to additional physical insight, so in practice we evaluate dγlr/dTs numerically. To estimate a typical value for dTH2O/dTs we ignore lapse rate changes, that is, the second term in Eq. (45). Assuming values representative of Earth’s tropics, 1 + γwvγlr = 1 + 1/7 × 20 ∼ 4, and representative temperatures TH2O∼240K (see Fig. 4) and Ts ∼ 300 K, a characteristic value for dTH2O/dTs is thus
dTH2OdTs∼14×240300=15,
in line with the numerical results of Jeevanjee et al. (2021a).
Next, we treat the H2O band feedback similar to the surface feedback. We assume the integrand of the spectral feedback integral is approximately constant with respect to wavenumber, and equal to its value at a central frequency ν˜. The feedback is then
−λH2O=∫0νLπdBνdT|TH2OdTH2OdTsdν≈cH2O×πdBν˜dT|TH2O(ν˜)×dTH2OdTs|ν˜×ΔνH2O,
where ΔνH2O=νL is the width of the H2O band, ν˜=νL/2 is the central wavenumber of the H2O band, and cH2O is again a scaling constant to account for the fact that we are replacing a spectral integral with simple multiplication.

d. H2O continuum feedback

The H2O continuum feedback is
−λcnt=∫cntπdBνdT|TcntdTcntdTsdν.
We apply the same logic as for the surface and H2O band feedbacks. The derivative dTcnt/dTs can be solved for analytically: Tcnt has no dependence on Ts other than through lapse rate changes, so
dTcntdTs=∂Tcnt∂γlrdγlrdTs=Tcntγlr(2γwv−a)dγlrdTs.
One important difference between the continuum and the other feedbacks is that the continuum is transparent across all wavenumbers at low surface temperatures, and only becomes optically thick at high surface temperatures. We approximate the continuum’s emissivity as 1−e−τcnt, which correctly captures the limiting behavior of an emitter at small and large optical thickness (τcnt ≪ 1 versus τcnt ≫ 1). The continuum can only dominate the atmosphere’s emission at wavenumbers at which CO2 and H2O absorption are weak, so we set the effective width of the continuum equal to the width of the window region Δν˜surf, defined above. The continuum feedback is then
−λcnt=∫cntπdBνdT|TcntdTcntdTsdν≈ccnt×πdBν˜dT|Tcnt×dTcntdTs×Δν˜surf(1−e−τcnt),
where ccnt is again a scaling constant. The sign of λcnt is positive because the bulk lapse rate decreases with warming, dγlr/dTs < 0. As discussed above, this means the H2O continuum acts as a positive/destabilizing feedback and has the opposite sign of the negative/stabilizing H2O band feedback.

e. CO2 band feedback

Next, we consider the CO2 feedback. Unlike the H2O band and continuum, however, the emission temperature of CO2 varies strongly with wavenumber, which makes it difficult to approximate the CO2 feedback integral via simple multiplication. Instead, we introduce an idealized CO2 “ditch” model, illustrated in Fig. 6. Our approach is closely related to the CO2 forcing models of Wilson and Gea-Banacloche (2012) and Jeevanjee et al. (2021b)—in appendix A we show that our ditch model can also be used to rederive the results of those previous studies, underlining the close relationship between forcing and feedbacks.

Fig. 6.
Fig. 6.

A CO2 “ditch” model: the CO2 band emits πBν(Tcold) in its center, its flanks emit πBν(Thot), and the slopes in-between are approximated as linear and symmetric. The shaded blue area is the OLR contribution from the CO2 band. (left) In cold climates or at high CO2 abundances, the CO2 band center radiates from the stratosphere. (right) In hot climates or at low CO2 abundances, the CO2 band center radiates from the troposphere.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0178.1

We approximate the CO2 band as symmetric around the central frequency ν0 = 667 cm−1. The center of the band emits πBν(Tcold) while outside the band the emission is πBν(Thot). Here Tcold and Thot are cold and hot emission temperatures, while νhot and νcold denote the edges of the CO2 ditch. At low and moderate surface temperatures the CO2 band center around 667 cm−1 radiates from the stratosphere, so Tcold is equal to the stratospheric temperature. However, this situation is no longer true at high surface temperatures. Physically, the tropopause rises as the surface warms, so if one warms the surface while holding CO2 concentration fixed (this is implicit in the definition of a climate feedback), parts of the CO2 band that were previously in the stratosphere have to start radiating from the troposphere. Eventually, even the CO2 band center radiates from the troposphere so the rectangular CO2 ditch turns into a triangular trough (see Fig. 6b). Here we leave our expressions general to allow for either situation.

The CO2 band is relatively narrow, so we can neglect the wavenumber dependence of the Planck function and evaluate it at the center of the CO2 band, πBν(T)≈πBν0(T). Treating the slopes of the CO2 ditch as piecewise linear, the OLR from the CO2 band is then simply the blue area under the ditch in Fig. 6a,
OLRCO2=2∫ν0νhotπBν0(TCO2)dν=[πBν0(Thot)+πBν0(Tcold)](νhot−νcold)+2πBν0(Tcold)(νcold−ν0)⋅
The OLR change in response to some climate perturbation is
ΔOLRCO2=OLRCO2′−OLRCO2=[πBν0(Thot′)+πBν0(Tcold′)](νhot′−νcold′)−[πBν0(Thot)+πBν0(Tcold)](νhot−νcold)+2πBν0(Tcold′)(νcold′−ν0)−2πBν0(Tcold)(νcold−ν0),
where primes indicate perturbed variables. For the CO2 band feedback, the relevant perturbation is a change in surface temperature ΔTs, while for the forcing the relevant perturbation is a change in qCO2 (see appendix A). If ΔTs is small enough, we can series expand and drop higher-order terms. For example, the perturbation of the emission at the CO2 band edge is
πBν0(Thot′)=πBν0(Thot)+πdBν0dT|ThotdThotdTsΔTs,
with similar expressions for Tcold′, ν′hot, and ν′cold. Plugging back into Eq. (52), the feedback of the CO2 ditch is
−λCO2=lim⁡ΔTs→0ΔOLRCO2ΔTs=[πdBν0dT|ThotdThotdTs+πdBν0dT|TcolddTcolddTs](νhot−νcold)+[πBν0(Thot)+πBν0(Tcold)](dνhotdTs−dνcolddTs)+2πdBν0dT|TcolddTcolddTs(νcold−ν0)+2Bν0(Tcold)dνcolddTs.
Equation (53) gives the most general expression for the feedback of the CO2 ditch. Geometrically, the blue area under the CO2 ditch changes if the flanks and center rise while the edges remain fixed (terms proportional to dThot/dTs and dTcold/dTs), or if the edges move while the flanks and center of the ditch remain fixed (terms proportional to dνhot/dTs and dνcold/dTs). To evaluate Eq. (53) we thus need to specify how the parameters Thot, Tcold, νhot, and νcold vary as a function of surface temperature.
At cold surface temperatures we again ignore H2O absorption around the CO2 band so Thot = Ts. Similarly, the tropopause is low and the CO2 band center radiates from the stratosphere, so Tcold = Tstrat and dTcold/dTs = 0. As in section 5a, we find the band edges νhot and νcold by solving TCO2(νhot)=Ts and TCO2(νcold)=Tstrat. The results are νhot=ν0+lνlog⁡[τCO2*(ν0)qCO2], and νcold=ν0+lνlog⁡[τCO2*(ν0)qCO2(Tstrat/Ts)2/γlr]. We can see that the hot CO2 band edge does not change under surface warming, dνhot/dTs = 0, while the sensitivity of the cold or stratospheric band edge to surface warming is
dνcolddTs=∂νcold∂Ts|γlr+∂νcold∂γlr|TsdγlrdTs=−2lνγlrTs+2lνγlr2log⁡(TsTstrat)dγlrdTs.
The lapse rate change dγlr/dTs is always negative, so the portion of the CO2 band inside the stratosphere shrinks, dνcold/dTs = 0. Geometrically, since νhot stays fixed while νcold moves toward the center of the CO2 band, the CO2 band slope becomes shallower and the blue area under the CO2 ditch increases—an OLR increase, or a stabilizing feedback. Physically, this is a simple consequence of a rising tropopause. As the surface warms, the tropopause moves to lower pressures, thus moving more of CO2’s emission from the cold stratosphere into the warmer tropopause. Plugging back into Eq. (53), the CO2 band feedback at cold surface temperatures is
−λCO2cool=πdBν0dT|Ts2γlrlog⁡(TsTstrat)+[πBν0(Ts)−πBν0(Tstrat)]×[2lνγlrTs−2lνγlr2log⁡(TsTstrat)dγlrdTs].
At high surface temperatures the CO2 band center moves into the tropopause and the rectangular ditch turns into a triangle (see lower left in Fig. 4, and sketch in Fig. 6b). We set νcold = ν0, where the central wavenumber ν0 is set by the spectroscopic properties of CO2 and so is fixed under surface warming (dνcold/dTs = 0). The emission temperature in the center of the CO2 band is now Tcold=TCO2(ν0), where TCO2 is the emission temperature of CO2 [Eq. (26a)]. The crucial difference between high and low surface temperatures is that once the CO2 band center moves into the tropopause Tcold is no longer constant,
dTCO2(ν0)dTs=∂TCO2(ν0)∂Ts|γlr+∂TCO2(ν0)∂γlr|TsdγlrdTs=TCO2(ν0)Ts−TCO2(ν0)2log⁡[qCO2τCO2*(ν0)] dγlrdTs.
The outer edges of the CO2 band at high temperatures are set by water vapor absorption, Thot=min⁡[TH2O(ν0),Tcnt]. We treat H2O as Simpsonian, so dThot/dTs ≈ 0, and also ignore non-Simpsonian shifts in the outer CO2 band edge, dνhot/dTs ≈ 0. Plugging back into Eq. (53), the feedback at high surface temperatures is then
−λCO2hot=πdBν0dT|TcolddTcolddTs(νhot−νcold)=πdBν0dT|TcolddTcolddTs lνlog⁡[τCO2*(ν0)qCO2(ThotTs)2/γlr].
Geometrically, the behavior of the CO2 band at high temperatures is dictated by the rise in the center of the band, dTcold/dTs. Since the band center emits more in response to surface warming, dTcold/dTs > 0, the blue area under the triangular ditch goes up—again, an OLR increase, which leads to a stabilizing feedback. Physically, once the center of the CO2 band radiates from inside the troposphere, we have dTcold/dTs ∝ −dγlr/dTs, which means the rate at which emission increases is highly sensitive to the rate at which the upper atmosphere warms via the changing lapse rate.

Finally, when does the CO2 band center change from a stratospheric radiator at low Ts to a tropospheric radiator at high Ts, which also determines the transition between λCO2cool and λCO2hot? Based on line-by-line calculations with 400 ppm of CO2, appendix B shows that the smoothed emission temperature in the CO2 band center moves out of the stratosphere at surface temperatures above 310 K. We therefore identify 310 K as the transition point between the low-temperature and high-temperature CO2 feedback regimes. Note, however, that this value also depends on CO2 concentration.

Multiplying the low-temperature regime with a scaling constant cCO2, similar to our other spectral feedbacks, the overall CO2 band feedback is thus
λCO2={cCO2×λCO2cool,if Ts≤310 K,λCO2hot+b,if Ts>310 K,
where we choose the constant b to ensure that λCO2 remains continuous at 310 K (in practice b is always of order unity, b ∼ 0.5).

f. Validation against LBL calculations

To test our analytic feedback expressions, we again use 1D calculations with PyRADS. One potential issue is that our derivations use the bulk lapse rate approximation, and so might differ from realistic feedbacks. Figure 7 compares feedbacks calculated with a moist adiabat to feedbacks with bulk lapse rate profiles. Overall, the bulk lapse rate approximation only introduces minor errors in λLW over the temperature range 250–320 K. We therefore consider the bulk lapse rate approximation sufficiently accurate below 320 K, while care should be taken when applying our analytic expressions to extremely hot climates. To better match the derivations, the PyRADS calculations here also use vertical profiles with a bulk lapse rate, so T=Ts(p/ps)γlr. We explore the surface temperature dependence of spectral feedbacks at high and low relative humidity (RH = 0.8 and RH = 0.1), without CO2 and with 400 ppm of CO2, for four sets of calculations in total.

Fig. 7.
Fig. 7.

The impact of the bulk lapse rate approximation on longwave feedbacks is modest below ∼320 K, but becomes significant at high temperatures. Solid lines are numerical feedbacks calculated assuming the atmosphere follows a moist adiabatic profile; dashed lines are numerical feedbacks calculated assuming the atmosphere follows our bulk lapse rate approximation.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0178.1

To compare our analytic expressions against the 1D calculations we need to specify the scaling constants csurf,cH2O,ccnt, and cCO2. We pick these constants to match the 1D calculations at RH = 0.8 and 400 ppm of CO2. The temperature dependence varies significantly between different feedbacks, so we choose csurf to match λsurf at low temperatures (Ts = 250 K), ccnt to match λcnt at high temperatures (Ts = 330 K), and cH2O and cCO2 to match λH2O and λCO2 around Earth’s present-day mean temperature (Ts = 290 K). Table 1 gives the resulting values for the above 1D calculations with bulk lapse rates, and for another set of 1D calculations with moist lapse rates. In agreement with Fig. 7, the scaling constants vary little between the two sets of calculations. In this section we choose the scaling constants to match the idealized 1D calculations with bulk lapse rates, while section 6 considers a feedback calculation specifically for present-day Earth, and so uses the scaling constants that match the moist adiabatic calculations. Regardless of the exact values, the scaling constants are always of order unity.

Figure 8 shows that our analytic expressions successfully capture the basic state dependence of λLW as well as of its spectral constituents. The longwave feedback λLW is sensitive to changes in surface temperature, but it also varies in response to humidity and CO2 changes. Comparing the left and right columns in Fig. 8, λLW becomes larger with decreasing relative humidity (also see McKim et al. 2021). Comparing the top and bottom rows, adding CO2 to an atmosphere without any CO2 evens out the temperature dependence of λLW, by decreasing λLW at cold temperatures and increasing λLW at high temperatures. Importantly, the analytic expressions capture most of the variation in λLW, including its state dependence.

Fig. 8.
Fig. 8.

Spectral feedbacks calculated using PyRADS and assuming a bulk lapse rate (symbols) compared against the analytic scalings (lines). (top) Calculations without CO2 and (bottom) with 400 ppm of CO2. The large panels show feedbacks, while small panels show the corresponding analytic emission temperatures.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0178.1

To understand the behavior of λLW we turn to the individual spectral feedbacks. The surface feedback λsurf is generally the dominant term in the spectral decomposition. Without CO2, λsurf makes up at least 90% of λLW below 300 K. The presence of CO2 decreases λsurf but even in this case λsurf makes up at least 60% of λLW below 300 K. Our analytic expressions thus agree with previous studies which showed that Earth’s longwave feedback is dominated by the surface feedback (Koll and Cronin 2018; Raghuraman et al. 2019). This situation changes at high temperatures, however, once the surface window closes, at which point λLW becomes dominated by atmospheric feedbacks.

In line with section 4, the CO2 band feedback acts to stabilize Earth’s climate in warm climates, and its importance increases with surface temperature. Below 300 K, λCO2 contributes less than 20% of the total feedback, but its magnitude grows rapidly with surface temperature such that at 330 K and high relative humidity λCO2 makes up almost 70% of λLW. Interestingly, for large RH λCO2 becomes equal to λsurf at surface temperatures around ∼305 K. Extrapolating from these 1D calculations to Earth’s spatial feedback pattern, we can expect that Earth’s feedback is dominated by the surface in most regions, but that atmospheric feedbacks become important in the inner tropics—an issue explored in detail in section 6.

Finally, again in line with our analytic results, the two water vapor feedbacks λH2O and λcnt have opposing signs. At high relative humidity λH2O and λcnt partially cancel. In contrast, at low relative humidity λcnt becomes negligible while λH2O only changes moderately—a non-Simpsonian effect. The different sensitivity to RH arises because the continuum’s optical thickness scales as τcnt ∝ RH2, whereas the optical thickness in the water vapor bands only scales as τH2O∝RH. Decreases in relative humidity therefore increase λLW both by increasing the surface feedback λsurf and by reducing λcnt, so that H2O acts as a net stabilizing feedback. Comparing λH2O and λCO2 at present-day CO2 levels, we see that the two feedbacks are roughly equal in magnitude. Non-Simpsonian H2O effects are thus about as important as the CO2 band for Earth’s current longwave feedback.

6. The spatial pattern of λLW

In the previous two sections we demonstrated that the analytic expressions summarized in Table 2 accurately capture the behavior of Earth’s emission temperature Trad as well as the state dependence of λLW. These feedback expressions can be interpreted as either a model for the global-mean feedback or as a model for the local feedback of an isolated atmospheric column, so the state dependence of λLW shown in Fig. 8 should also appear as a spatial dependence in Earth’s clear-sky longwave feedback.

Table 2.

Summary of main theoretical results.
Table 2.

In this section we therefore analyze the spatial pattern of λLW for Earth’s present-day climate. First, we generate a map of λLW using the radiative kernel technique (Soden et al. 2008). Next, we generate a map of λLW using our analytic expressions. The radiative kernel technique cannot be used to determine the feedback contributions of individual gases and our analytic expressions only account for the feedback from Earth’s dominant greenhouse gases, H2O and CO2, whereas the radiative kernel includes additional greenhouse gases such as O3 and CH4. We therefore split λLW into only two terms, namely, the surface feedback λsurf and the atmospheric feedback λatm = λLW − λsurf. Despite the idealizations in our analytic approach compared to a full radiative kernel, we find that the resulting feedback maps are in qualitative agreement. This allows us to attribute the spatial pattern of λLW, as deduced from the radiative kernel, to geographic variations in the inputs of our analytic model.

a. Inputs for feedback maps

For the kernel calculation, we use the HadGEM2 radiative kernel. For consistency with the analytic model (which assumes the stratosphere is isothermal and at a fixed temperature), we set the kernel to zero in the stratosphere. The tropopause is defined as in Soden et al. (2008): the tropopause pressure ptp increases linearly with latitude, from 0.1 bar at the equator to 0.3 bar at the poles. The analytic model also assumes RH stays fixed under surface warming, so we do not include RH changes in the kernel calculation. Doing so is justified because the RH feedback only makes a minor contribution to λLW in individual climate models, and it moreover tends to cancel in the multimodel mean (Zelinka et al. 2020). To compute the forced response, we use HadGEM2 climatologies from the CMIP5 archive for a preindustrial control simulation and an abrupt-4xCO2 simulation, where the climatologies are 50-yr averages (for 4xCO2, years 100–150 after increasing CO2). Multiplying the kernel with the forced response gives a map of the change in top-of-atmosphere (TOA) radiation (Soden et al. 2008). To compute a feedback, one additionally needs to normalize the change in TOA radiation by a change in surface temperature. Consistent with our assumption of an isolated atmospheric column we compute local–local feedback maps, that is, we divide the local change in OLR deduced from the kernel by the local change in surface temperature (Feldl and Roe 2013; Armour et al. 2013; Bloch-Johnson et al. 2020). To distinguish between surface and atmospheric feedbacks in the kernel method we compute the clear-sky longwave feedback λLW and the surface feedback λsurf, where the second is equal to the surface kernel; the atmospheric feedback is then computed as the residual λatm = λLW − λsurf.

We compare the kernel-derived feedback maps against maps from our analytic expressions. The surface feedback λsurf is the same as in section 5, while the atmospheric feedback is the sum over all atmospheric terms λatm=λCO2+λH2O+λcnt. The analytic expressions require six input parameters: CO2 concentration, surface temperature Ts, stratosphere temperature Tstrat, relative humidity RH, temperature lapse rate γlr, and the change in lapse rate under surface warming dγlr/dTs. Except for the lapse rate change dγlr/dTs, all these inputs can be obtained from a single climate state (here, the HadGEM2 preindustrial state) and do not require knowledge of the climate’s forced response. CO2 is set to be spatially uniform at 400 ppm (results are highly similar if using a preindustrial 285 ppm); the surface temperature Ts is taken as the air temperature at 2 m; and the stratospheric temperature Tstrat is set equal to the temperature at the tropopause pressure level, Tstrat = T(ptp), where ptp is defined using via the above tropopause definition of Soden et al. (2008). The relative humidity RH is set equal to the column relative humidity, defined as the ratio between the atmospheric column’s water vapor path and its water vapor path at saturation (e.g., Bretherton et al. 2005),
RH=WVP WVP*
=∫ptppsqdp/g ∫ptppsq*dp/g .
Here the vertical integral is taken from the tropopause ptp down to the surface to exclude the strongly subsaturated stratosphere. One could in principle also approximate RH using other measures of atmospheric humidity; however, the column relative humidity is a natural choice because it correctly captures the atmosphere’s total water vapor path, which in turn determines the width of the window region and λsurf.
Next, the lapse rate γlr = dlnT/dlnp varies strongly in the vertical. We compute a bulk lapse rate using a mass-weighted vertical average,
γlr=1p1−ptp∫ptpp1pTdTdpdp,
where the average is taken from the tropopause ptp down to a near-surface pressure p1. Some polar regions have such strong surface inversions that the inferred bulk lapse rate becomes negative, whereas our derivations break down if γlr < 0. At the same time, the map of γlr should reflect near-surface inversions over subtropical eastern ocean basins and deep boundary layers over tropical land, discussed below. We therefore define p1 similar to ptp, as varying linearly in latitude from p1 = 1 bar at the equator to p1 = 0.85 bar at the poles. One could also evaluate γlr using the bulk lapse rate definition from Eq. (6) in combination with a tropopause definition; however, this approach makes the inferred lapse rates quite sensitive to the tropopause definition, which we sidestep by using the mass-weighted average in Eq. (61) instead. Finally, the only input in our analytic expressions that requires information about the climate’s forced response is the change in lapse rate dγlr/dTs, which is computed using the difference in γlr between the HadGEM2 4xCO2 and preindustrial simulations.

Figure 9 shows maps of the input data from HadGEM2, which we use below to evaluate the analytic expressions. In the top two rows, large variations are notable in the maps of surface temperature Ts, column relative humidity RH, and bulk lapse rate γlr. In contrast, apart from minor stationary wave patterns in the northern midlatitudes, the stratospheric temperature Tstrat is zonally fairly uniform and varies by only about 20 K between the equator and poles. The bottom row shows the normalized bulk lapse change, dln(γlr)/dTs = 1/(Δγlr) × (Δγlr/ΔTs), computed using the bulk lapse rate difference Δγlr between 4xCO2 and preindustrial simulations. The bulk lapse rate change shows an equator–pole contrast, with a decrease in γlr at low and midlatitudes and an increase in γlr at high latitudes. This contrast is in line with previous studies—for a moist adiabat the atmospheric temperature–pressure profile becomes less steep under warming, so γlr decreases in the tropics, while the opposite occurs at high latitudes (e.g., Payne et al. 2015; Cronin and Jansen 2016; Stuecker et al. 2018). There is also a noticeable tropical land–ocean contrast in the bulk lapse rate change, with tropical land areas showing near-zero lapse rate change. This is likely due to compensation between moist-adiabatic warming aloft, which is uniform across the tropics and tends to decrease γlr, and amplified land surface warming, which increases γlr (Byrne and O’Gorman 2013). Conversely, subtropical eastern ocean basins have the same moist adiabatic warming aloft but suppressed surface warming, both of which contribute to strong decreases in γlr.

Fig. 9.
Fig. 9.

Input data used to evaluate the analytic feedback maps in Fig. 10. (top),(middle) Fields from a HadGEM2 preindustrial simulation. (bottom right) The normalized bulk lapse rate change dIn(γlr)/dTs computed using the HadGEM2 4xCO2 and preindustrial simulations.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0178.1

b. Feedback maps

Figure 10 shows the feedback maps resulting from kernel and analytic calculations. Overall, we find good qualitative agreement between kernel-derived feedbacks and our analytic approximations. The global pattern of λLW in both maps shows clear contrasts between the high latitudes, subtropics, and inner tropics (Fig. 10, top row). The value of λLW is smallest in the inner tropics, especially in the intertropical convergence zone (ITCZ), while it is largest in the subtropics, especially over eastern ocean basins. The agreement is less good at small scales, with the analytic map of λLW showing less regional structure and deviating from the kernel-derived map in continental interiors and over the Southern Ocean. This is plausible given the idealizations in our derivations, such as representing realistic vertical temperature profiles by a smooth power law. However, small-scale differences tend to cancel when taking a zonal or global mean. The zonal mean of λLW in our analytic estimate agrees with the zonal mean of the kernel λLW to within 11% at each latitude. The global-mean values of λLW are almost identical, with −2.15 W m−2 K−1 for the kernel calculation and −2.16 W m−2 K−1 for the analytic estimate. Note that these global mean averages are weighted by the HadGEM2 pattern of surface warming, which is required to convert a local–local feedback map into a global mean (Feldl and Roe 2013; Armour et al. 2013).

Fig. 10.
Fig. 10.

Feedback maps showing feedbacks computed (left) with a radiative kernel and (center) with our analytic expressions. The analytic maps are calculated from the inputs shown in Fig. 9. (top) The net longwave clear-sky feedback λLW, (middle) the surface component λsurf, and (bottom) the atmospheric component λatm. Means above each panel are area-weighted global means that are weighted by the pattern of surface warming. (right) Zonal mean values.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0178.1

The qualitative agreement between the λLW maps also holds separately for surface and atmospheric feedbacks, though differences are larger here. The kernel-derived map of λsurf is almost uniform at high latitudes, large in magnitude over subtropical desert regions, and small in magnitude over the ITCZ. The analytic map of λsurf qualitatively matches this pattern, though it overpredicts the magnitude of λsurf in the global mean by 0.18 W m−2 K−1, or 13%. Conversely, the analytic estimate underpredicts λatm relative to the kernel-derived map in the global mean by 0.17 W m−2 K−1, or 22%. In addition, the analytic λatm map predicts that the atmospheric feedback goes almost to zero at the poles, whereas the kernel-derived λatm map shows a small but clearly nonzero feedback. The strong differences at the poles again presumably arise because our derivations fail to capture the atmospheric feedback response in areas with inversions and other complex temperature–pressure profiles.

In addition to an overall spatial agreement, both kernel and analytic feedback calculations agree that the surface dominates the net longwave feedback. Figure 11 shows that the surface’s contribution to the total feedback is about 50% at low latitudes and increases toward the poles, reaching about 75% in the kernel maps and over 90% in the analytic maps. One plausible reason why the analytic maps tend to overestimate λsurf/λLW at high latitudes is that our expressions do not include minor greenhouse gases such as ozone or methane. Any additional atmospheric absorption from such gases reduces the window width via Δνsurf×e−τcnt and thus also the surface feedback λsurf (also see Feng et al. 2023). This effect should be most clearly visible at high latitudes, where water vapor concentrations are low and Δνsurf is large, while at low latitudes Δνsurf×e−τcnt is already small due to the water vapor continuum, leaving less room for other greenhouse gases to affect λsurf. Nevertheless, in line with the results from section 5, both kernel and analytic maps show that λLW is dominated by λsurf across most of the globe. In contrast, atmospheric feedbacks only start to rival λsurf in the inner tropics and particularly inside the ITCZ (Fig. 10). Our finding agrees with other published estimates: the simple area-weighted global mean of λsurf/λLW is 60% in our kernel calculation and 67% in our analytic estimate, well in line with the results of Raghuraman et al. (2019), who deduced 63% using a different methodology. Similarly, Feng et al. (2023) found that λsurf/λLW varies between 88% at the poles to 50% in the tropics, in good agreement with Fig. 11. We conclude that our analytic model of λLW has notable biases at regional scales but it is sufficient to understand the factors that underlie the large-scale pattern of λLW, which we turn to next.

Fig. 11.
Fig. 11.

Zonal mean fraction of the surface feedback to the net feedback, λsurf/λLW, based on the radiative kernel (solid) and our analytic expressions (dashed).

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0178.1

c. What controls the large-scale pattern of λLW?

The match between our analytic model and the kernel calculation implies that one can explain much of the spatial structure of λLW in terms of the analytic model’s input parameters. We do this by calculating correlations between λsurf and λatm from the kernel-derived feedback maps against the analytic model’s five main inputs: surface temperature Ts, column relative humidity RH, stratospheric temperature Tstrat, bulk lapse rate γlr, and the change in bulk lapse rate under warming dγlr/dTs. Spatial CO2 contrasts are small (e.g., Fraser et al. 1983), and so do not need to be considered here.

Figure 12 shows the resulting spatial correlations between the kernel-derived feedback maps (left column of Fig. 10) and the five inputs from HadGEM2 (Fig. 9). Because the feedback maps differ strongly between tropics and extratropics in terms of zonal variation and magnitude, we compute correlations separately in these two regions (data are split based on being equatorward or poleward of 30° latitude). Based on the inherent correlations between the five input maps, we consider a correlation significant if its coefficient exceeds |r| ≥ 0.75 (the largest intrainput correlations are r = −0.71 between Ts and Tstrat in the tropics, and r = −0.76 between Ts and dγlr/dTs in the extratropics; not shown).

Fig. 12.
Fig. 12.

Spatial correlation between the kernel-derived feedback maps of λsurf and λatm (Fig. 10, left column), and the inputs to our analytic model (Fig. 9). (top) Correlations between inputs and λsurf; (bottom) correlations between inputs and λatm. (left) Correlations inside the tropics; (right) correlations in the extratropics. Dark colors highlight particularly strong correlations (|r| ≥ 0.75), while the tropics and extratropics are defined as all points equatorward and poleward of 30° latitude, respectively.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0178.1

In line with our analytic model, we find that the kernel-derived λsurf is strongly correlated with column RH in the tropics (r = 0.83), while it does not show strong correlation with any inputs in the extratropics (|r| < 0.4). This underlines the importance of the subtropical dry radiator fin regions for λsurf, which are clearly visible as the dark blue regions in Fig. 9 (top right) and the yellow regions in Fig. 10 (center left). As expected, the sign of the correlation is positive which means λsurf becomes less negative, or less stabilizing, as column RH increases.

Next, we find λatm is most strongly correlated with γlr and dγlr/dTs in the tropics (r = 0.75 for both), and with dγlr/dTs in the extratropics (r = 0.88). Of the two parameters that show strong correlations with λatm in the tropics, γlr and dγlr/dTs, which one is more important? We performed a test with the analytical model in which we set dγlr/dTs = 0 (not shown). Doing so eliminates most tropical structure in the map of λatm, which indicates that λatm is largely determined by dγlr/dTs, not γlr. The correlation between λatm and dγlr/dTs is positive, which is intuitive: λatm becomes more negative if the upper atmosphere warms more relative to the surface, i.e., if γlr decreases. The spatial variability of λatm is largest in the tropics, and can be can be understood in terms of the map of dγlr/dTs already discussed in section 6a: tropical λatm is large over subtropical eastern ocean basins due to suppressed surface warming, and small over land due to enhanced surface warming, where these warming patterns are relative to the approximately uniform warming of the tropical free troposphere (Byrne and O’Gorman 2013).

The correlations shown in Fig. 12 are between fields derived from two independent methods, and so are nontrivial. Appendix C shows that the same analysis performed with λsurf and λatm from our analytic feedback maps identifies the same dominant relations (e.g., λsurf is most strongly correlated with column RH in the tropics), though most correlation coefficients are unsurprisingly even larger (e.g., r = 0.93 for the analytic λsurf and tropical column RH). Our results thus underline that the spatial pattern of λLW can be understood, at least in rough terms and on large spatial scales, by Earth’s spatial pattern of relative humidity and lapse rate changes. Relative humidity and lapse rate changes dominate the pattern of λLW in the tropics, where they control λsurf and λatm, respectively, while lapse rate changes dominate the pattern of λatm in the extratropics.

7. Discussion and conclusions

In this paper we have presented a novel decomposition of Earth’s clear-sky longwave feedback λLW into four spectral components, namely, a surface Planck feedback (λsurf) and three atmospheric feedbacks: a CO2 band feedback (λCO2), a (non-Simpsonian) water vapor band feedback (λH2O), and a destabilizing water vapor continuum feedback (λcnt). We have derived simple analytic expressions for each of these spectral feedbacks, which accurately reproduce the results of line-by-line calculations and qualitatively match the feedback map computed from a radiative kernel. In principle one could extend this approach even further to account for additional complicating factors, such as the effect of additional greenhouse gases or a more realistic stratosphere. However, our results already show that from a radiative perspective the factors determining λLW can be understood fairly easily, adding further support to the close agreement between observations and climate models.

The picture of Earth’s clear-sky longwave feedback that emerges from this perspective is relatively simple, consisting of a surface feedback plus atmospheric feedbacks from CO2 and H2O. At present the surface feedback λsurf is the most important contributor in the global mean and at most latitudes, with its spatial pattern determined by the distribution of atmospheric water vapor. λsurf is largest in the dry subtropics, consistent with the view that these are the locus of Earth’s stabilizing longwave feedback (Pierrehumbert 1995; McKim et al. 2021), and smallest in the inner tropics, where the surface’s emission is blocked by the H2O continuum. The atmospheric feedbacks from the CO2 and H2O bands play a supporting role to λsurf at mid- and high latitudes, but they rival the surface feedback in the inner tropics, with the global pattern of λatm largely determined by the pattern of the atmospheric lapse rate change dγlr/dTs. The H2O continuum provides a negligible feedback below ∼310 K (see section 5), but the continuum itself is still important through its influence on λsurf.

This spectral picture is arguably a more intuitive starting point for reasoning about different climates than the conventional decomposition of λLW into Planck, lapse rate, and water vapor feedbacks. As discussed by Cronin and Dutta (2023), it is nontrivial to accurately estimate the supposedly simple Planck feedback from first principles. Similarly, one can qualitatively reason that lapse rate and water vapor feedbacks both increase in magnitude under global warming, but these are large and of opposite sign, so it is difficult to predict their net change and, by extension, the Ts dependence of λLW, in the conventional decomposition without resorting to numerical models. The strong cancellations between Planck, lapse rate, and water vapor feedbacks can be alleviated by considering conventional feedbacks in a fixed relative humidity framework (Ingram 2010; Held and Shell 2012), but this comes at the cost that the state dependence of the Planck feedback is no longer trivial to understand at fixed RH.

In contrast, the state dependence of λLW is fairly straightforward to understand from a spectral perspective, at least in broad brushstrokes. For present-day Earth the Ts dependence of λLW is dominated by the surface in most regions. If relative humidity is fixed, λsurf increases at very cold temperatures, peaks around 260–290 K depending on RH, and then decreases again (see section 5). The decrease is rapid at high RH due to the H2O continuum, but much slower at low RH. Atmospheric feedbacks also have state dependence. All of them increase in magnitude as the atmosphere warms, and are further amplified by a weakening lapse rate. In the tropics the state dependence of λLW is thus set by the interplay between a decreasing surface feedback and increasing atmospheric feedbacks. This can lead to surprising dynamics—at high RH, λsurf decreases in magnitude more rapidly with warming than the atmospheric feedbacks from λCO2 and λH2O increase. As a result, λLW becomes nonmonotonic with warming and develops a local minimum around ∼310 K, which leads to a local maximum in climate sensitivity (Seeley and Jeevanjee 2021).

The state dependence of λLW at temperatures far above ∼310 K is beyond the scope of this paper, but a spectral perspective points to the importance of stabilizing H2O and CO2 bands versus the destabilizing H2O continuum as Earth approaches the runaway greenhouse. The main caveat here is that Earth’s net feedback does not necessarily stay dominated by λLW at very high surface temperatures, and atmospheric feedbacks are also complicated at high temperatures by effects such as nondilute thermodynamics and surface pressure changes (Goldblatt et al. 2013; Ramirez et al. 2014).

There are several remaining shortcomings in our analysis of λLW that are beyond the scope of this paper. A major one is our assumption that the atmosphere can be described by a single bulk lapse rate, such that temperature has to monotonically decrease with altitude. In the real world inversions are common, particularly in polar regions and over subtropical oceans. Comparable to the long-standing discussion about how to interpret the lapse rate feedback at high latitudes in the conventional decomposition (e.g., Cai and Lu 2009; Payne et al. 2015; Stuecker et al. 2018; Boeke et al. 2021; Henry et al. 2021), we therefore expect that our approach here only provides a first step toward understanding the processes which shape λLW in inversion regions.

Another assumption is that we ignore stratospheric changes, even though stratospheric cooling induced by rising CO2 levels is a major and robust signal of anthropogenic warming (e.g., Vallis et al. 2014). It is notable that the radiative changes due to stratospheric cooling are also hard to intuitively explain using conventional feedbacks. Climate model analyses typically treat the stratosphere’s fast radiative adjustment to CO2 changes as distinct from Planck, lapse rate, and water vapor feedbacks. Our derivations here sidestep this issue and treat Tstrat as a fixed parameter. Similarly, our derivations ignore the potential feedback from relative humidity changes. In reality there is no guarantee that relative humidity will remain constant under global warming, let alone would have been similar in past climates. In principle our analysis starting from the emission level approximation can be extended to estimate the feedbacks associated with changes in either RH or Tstrat; RH changes would lead to a feedback term proportional to ∂Trad/∂RH, while stratospheric changes would lead to a feedback term proportional to ∂Trad/∂Tstrat.

Acknowledgments.

D.D.B.K. thanks Jeevanjee Gardens in Nairobi. N.J.L. was supported by the NOAA Climate Program Office’s Modeling, Analysis, Predictions, and Projections program through Grant NA20OAR4310387.

Data availability statement.

HadGEM2 GCM data are publicly available in CMIP data archives. The HadGEM2 radiative kernel is available at https://archive.researchdata.leeds.ac.uk/382. Scripts to compute analytical feedbacks are available at https://github.com/danielkoll/spectral_feedbacks.

APPENDIX A

CO2 Forcing

The CO2 ditch model can be used to explain the CO2 forcing in addition to the CO2 band feedback. This section rederives the CO2 forcing expressions from Wilson and Gea-Banacloche (2012) and Jeevanjee et al. (2021b), which are valid as long as the CO2 band center radiates from the stratosphere. Note that our CO2 band feedback model only considers OLR changes inside the CO2 band (see Fig. 6). This is because the effect of CO2 on λH2O or λsurf is separately considered in the derivation of those feedbacks. Forcing is defined as the OLR change integrated across all wavenumbers, however, so here we need to consider the expanded shaded region shown in Fig. A1. The OLR integrated across this expanded region, OLR+, is
OLR+=2∫ν0νν∞πBν0(Trad)dν=[πBν0(Thot)+πBν0(Tcold)](νhot−νcold)+2πBν0(Tcold)(νcold−ν0)+2πBν0(Thot)(ν∞−νhot).
The forcing from a doubling of CO2 is then
FCO22x=−dOLR+d​log⁡2(qCO2)=−ln⁡(2)dOLR+d​ln⁡qCO2=−ln⁡(2){[πBν0(Thot)+πBν0(Tcold)](dνhotd​ln⁡qCO2−dνcoldd​ln⁡qCO2)+2πBν0(Tcold)dνcoldd​ln⁡qCO2−2πBν0(Thot)dνhotd​ln⁡qCO2}.
The minus sign in the first line ensures that forcing is positive when OLR decreases, while the base-2 logarithm is necessary because forcing is defined with respect to a CO2 doubling. In the second step we then change the logarithm’s base to the natural logarithm, while in the third step we treat the emission temperatures Thot and Tcold as constant. This is valid because the derivative of OLR with respect to qCO2 is taken at fixed Ts (i.e., at fixed surface temperature, the temperature outside the CO2 band and in the stratosphere are both independent of CO2 concentration).
Fig. A1.
Fig. A1.

CO2 ditch model for the CO2 forcing. The shaded blue area is the OLR contribution from the CO2 band as well as neighboring spectral regions. The band edges νhot and νcold vary in response to CO2 concentration qCO2, while ν∞ is sufficiently far away from the CO2 band to be constant with respect to qCO2.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0178.1

The CO2 band edges are defined by TCO2(νhot)=Thot and TCO2(cold)=Tstrat. Solving for νhot and νcold we find
νhot=ν0+lν​log⁡[qCO2τCO2*(ν0)(ThotTs)2/γlr],
νcold=ν0+lν​log⁡[qCO2τCO2*(ν0)(TstratTs)2/γlr].
We can see that the CO2 band edges shift equally in response to a CO2 increase:
dνhotd​ln⁡qCO2=dνcoldd​ln⁡qCO2=lν.
It follows that the first term proportional to dνhot/d​ln⁡qCO2−dνcold/d​ln⁡qCO2 in Eq. (A2) is zero. The CO2 forcing is thus
FCO22x=2ln⁡(2)lν[πBν0(Thot)−πBν0(Tcold)],
which is identical to the analytic CO2 forcing model in Jeevanjee et al. [2021b, their Eqs. (7) and (14)].

APPENDIX B

Transition from Stratospheric to Tropospheric CO2 Radiator Fin

At high surface temperatures the CO2 band center transitions from mainly radiating from the stratosphere to mainly radiating from the troposphere. Figure B1 shows smoothed brightness temperatures Tb computed from the 1D line-by-line calculations described in section 5, with a CO2 volume-mixing ratio of 400 ppm. In the middle of the CO2 band, at about 667 cm−1, CO2 radiates from the troposphere at surface temperatures above ∼310 K. In rough agreement with the line-by-line results, our analytic CO2 brightness temperatures predict this transition happens at a surface temperature of ∼320 K (dashed lines in Fig. B1). In practice we therefore use a transition temperature of Ts,0 = 310 K for 400 ppm of CO2 to determine when CO2 changes from a stratospheric to a tropospheric radiator.

Fig. B1.
Fig. B1.

Brightness temperatures computed from line-by-line calculations and smoothed with a 50 cm−1 median filter (solid) vs analytic emission temperatures (dashed). (top) Calculations use a bulk lapse rate profile, T(p)=Ts(p/ps)γlr (bottom) Calculations use a moist adiabat.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0178.1

APPENDIX C

Spatial Correlations in Analytic Feedback Maps

Figure C1 repeats the same analysis as in Fig. 12, but using the analytic feedback maps of λsurf and λatm. Given that the analytic model is computed using the input fields from Fig. 9, it is not surprising that most correlations between inputs and feedback maps are even higher than in Fig. 12. With the exception of λatm in the tropics, for which the correlation between the analytic λatm and γlr is slightly lower than between kernel-derived λatm and γlr, Fig. C1 identifies the same strong correlations as Fig. 12.

Fig. C1.
Fig. C1.

Spatial correlation between the analytic feedback maps of λsurf and λatm (Fig. 10, right column) and the inputs to our analytic model (Fig. 9). (top) Correlations between inputs and λsurf; (bottom) correlations between inputs and λatm. (left) Correlations inside the tropics; (right) correlations in the extratropics. Dark colors highlight particularly strong correlations (|r| ≥ 0.8).

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0178.1

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