An Analytical Model for Spatially Varying Clear-Sky CO2 Forcing

Nadir Jeevanjee; Jacob T. Seeley; David Paynter; Stephan Fueglistaler

doi:10.1175/JCLI-D-19-0756.1

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(a) Graph of CO₂ emission levels as given by Eq. (3) for q_i = 280 ppmv and q_f = 4q_i. (b) As in (a), but from RFM calculations with CO₂ only for our BASE atmosphere. RFM emission levels are diagnosed by the condition τ_ν = $τ_{em}^{{CO}_{2}}$ = 0.5, and are geometrically averaged (coarse-grained) over 10 cm⁻¹ bins. (c) As in (b), but p_em values from the left (right) side of the 667-cm⁻¹ peak are separately sorted into descending (ascending) order rather than coarse-grained. The idealized “p_em triangles” in (a) roughly match the peak emission pressures in (b) and the blocking of surface emission seen in (c), but do not capture the smallest p_em values in (c). The green dashed lines at top in (a) and (b) depict the negative stratospheric contribution to the forcing, the orange lines in all panels depict the null tropospheric contribution to the forcing, and the red solid lines at bottom in (a) and (c) depict the positive surface contribution. Equation (7) quantifies these contributions.

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Comparison of (7) vs RFM for our idealized, CO₂-only single columns with (a) variable T_s and Γ_strat = 0 and (b) T_s = 300 K and variable Γ_strat. Optimization of l in (a) yields l = 10.2 cm⁻¹, and optimization of κ₀ in (b) for −4 < Γ_strat < 0 K km⁻¹ yields κ₀ = 50 m² kg⁻¹. With these parameter values, the good fit in these panels across a range of T_s and Γ_strat provides a first validation of (7).

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Maps of (a) CO₂ forcing $F_{4 \times}$ with CO₂ only from a LBL calculation; (b) as in (a), but using Eq. (7); (d) surface temperature T_s; and (e) stratospheric emission temperature T_strat, as diagnosed by Eq. (5). (c) Zonal means of (a) and (b); (f) zonal means of (d) and (e). The spatial variations in CO₂ forcing, and in particular the meridional gradient, are captured by the analytical model. Furthermore, the T_s map in (d) is almost identical to the $F_{4 \times}$ maps in (a) and (b), showing that the spatial variations in $F_{4 \times}$ in the CO₂-only case stem almost entirely from T_s, with T_strat variations playing a much smaller role. Accordingly, the strong meridional gradient in zonal-mean T_s matches that of $F_{4 \times}$ [(c) and (f)], while the meridional gradient in T_strat is weak.

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Zonal mean forcing for our GCM snapshot for both the CO₂-only and H₂O overlap cases, as computed with our global LBL code. H₂O strongly modulates the CO₂ forcing outside the dry polar regions, thus also modulating the meridional gradient in CO₂ forcing.

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(a),(b) As in Figs. 1a and 1b but with H₂O overlap, again for the BASE atmosphere. H₂O emission levels are shown in blue; in (a) they are given by Eqs. (12) while in (b) they are diagnosed directly from RFM by τ_ν = 0.5 and geometrically averaged over 10-cm⁻¹ bins, just as for CO₂. Note that (b) shows that the presence of H₂O implies that increasing CO₂ blocks tropospheric H₂O emission rather than surface emission. This is idealized in (a), which assumes a single emission level in each of two spectral regions, denoted “−” and “+” and spanning the wavenumber ranges 550–600 and 750–800 cm⁻¹, respectively.

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(a),(b) Validation of our simple expressions (8) and (12) for band-averaged H₂O emission temperatures, as compared to the band average of (13) from RFM. This comparison is made for idealized atmospheric columns with T_s = 300, no CO₂, and varying RH. (c) Validation of the simple model (14) for $F_{4 \times}$ in the presence of H₂O, as compared to $F_{4 \times}$ calculated by RFM. This comparison is made for idealized atmospheric columns with T_s = 300, q_i = 280 ppmv, and varying RH. The simple expressions (8) and (12) predict $T_{em}^{\pm}$ very well except at low RH in the “+” region, leading to small (~0.5 W/m²) errors in $F_{4 \times}$ at these RH values.

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(a)–(c) As in Figs. 3a–c, but now including H₂O overlap, where (b) is generated using Eq. (14). (d) The column RH field for this GCM snapshot, calculated as the tropospheric precipitable water divided by its saturation value. The color bar is adjusted for comparison with (a) and (b). Large RH variations in the tropics cause corresponding variations in $F_{4 \times}$ , which are captured by the analytical model. The effect of these RH variations is probed in (c), where we fix RH = 0.75 in the analytical model and find an almost uniform forcing in the tropics, consistent with the limit shown in Fig. 8.

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CO₂ forcing $F_{4 \times}$ for our idealized atmospheric columns with varying T_s and all other parameters fixed at the BASE values. The presence of H₂O (blue) sets a limit on $F_{4 \times}$ (T_s) which does not exist in the CO₂-only case (red). These behaviors are well captured by the analytical model [lines, given by Eqs. (14) and (7)]. The kinks in the blue lines are consequences of employing the emission level approximation [cf. Fig. B1].

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(a),(b) Emission levels $p_{em}^{↑ ↓}$ as seen from the tropopause as defined by Eq. (15), evaluated (a) analytically using Eq. (16) and (b) numerically from RFM output, for q = 280 (dashed) and 1120 ppmv (dashed) and the BASE atmosphere. Levels $p_{em}^{↓}$ emitting downward toward the tropopause are shown in blue, and levels $p_{em}^{↑}$ emitting upward toward the tropopause in red. (c),(d) As in (a) and (b), but for the corresponding upwelling and downwelling spectral fluxes. The comparison between theory and LBL output validates the theory, to first order. Furthermore, the change in upwelling flux [red lines in (c)] can be estimated as that given by Eq. (7), but there is an additional compensating change in downwelling radiance [blue lines in (c)], yielding the instantaneous tropopause forcing, (17).

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Spatial distributions of tropopause forcing $F_{4 \times}^{tp}$ , (top) with CO₂ only and (bottom) with H₂O overlap, from (left) our global LBL calculation and (center) Eq. (17), along with (right) zonal means. The excellent agreement between the LBL calculation and Eq. (17) validates the claim that instantaneous tropopause forcing is independent of stratospheric temperature. The spatial variations seen here are thus determined entirely by T_s with modulation by H₂O, even more so than for the TOA forcings of Fig. 3 and 7.

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Profiles of various contributions to spectrally averaged H₂O absorption coefficients in our BASE column for the wavenumber regions (a) 525–625 and (b) 725–825 cm⁻¹. The profiles of κ_tot, κ_lines, and κ_ctm are calculated with RFM, whereas κ_est is given by Eqs. (10) with $(κ_{ref}^{-}, κ_{ref}^{+})$ set to $(κ_{tot}^{-} (T_{ref}^{-}), κ_{tot}^{+} (T_{ref}^{+}))$ , and where $(T_{ref}^{-}, T_{ref}^{+}) = (245, 275) K$ . The estimated profile κ⁻ is a poor approximation to $κ_{tot}^{-}$ , far from $T_{ref}^{-}$ due to our neglect of temperature scaling of line absorption, but is acceptably close within 20 K or so of $T_{ref}^{-}$ . The horizontal axis in both panels is logarithmic, with the same geometric range (of 150) in each.

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(a) Plot of Eq. (B6) for emission levels τ_em as a function of the parameter γ defined in (B2). At γ = 0, $τ_{em} = e^{- γ_{Euler}} \approx 0.56$ , close to our values for $τ_{em}^{{CO}_{2}}$ and $τ_{em}^{H_{2} O}$ . (b) Plot of normalized exact OLR from Eq. (B7), as well as normalized OLR from the emission level approximation (B3) as a function of τ_s and with γ = 0.4 (black dot in left panel; see text for discussion of this value). The EL approximation is reasonable, even near τ_s = 1, with maximum errors of roughly 15%.

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Author: A. J. HENRY

No. III

BAROMETRIC PRESSURE

VERIFICATIONS

MISCELLANEOUS PHENOMENA

Editorial Type:: Article

Article Type:: Research Article

An Analytical Model for Spatially Varying Clear-Sky CO₂ Forcing

Nadir Jeevanjee¹, Jacob T. Seeley², David Paynter¹, and Stephan Fueglistaler³

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¹ ^aNOAA/Geophysical Fluid Dynamics Laboratory, Princeton, New Jersey
| ² ^bHarvard University Center for the Environment, Cambridge, Massachusetts
| ³ ^cGeosciences Department, Princeton University, Princeton, New Jersey

Published-online:: 09 Nov 2021

Print Publication:: 01 Dec 2021

DOI:: https://doi.org/10.1175/JCLI-D-19-0756.1

Page(s):: 9463–9480

Received:: 09 Oct 2019

Final Form:: 20 Aug 2021

Published Online:: 09 Nov 2021

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Abstract

Clear-sky CO₂ forcing is known to vary significantly over the globe, but the state dependence that controls this is not well understood. Here we extend the formalism of Wilson and Gea-Banacloche to obtain a quantitatively accurate analytical model for spatially varying instantaneous CO₂ forcing, which depends only on surface temperature T_s, stratospheric temperature, and column relative humidity (RH). This model shows that CO₂ forcing can be considered a swap of surface emission for stratospheric emission, and thus depends primarily on surface–stratosphere temperature contrast. The strong meridional gradient in CO₂ forcing is thus largely due to the strong meridional gradient in T_s. In the tropics and midlatitudes, however, the presence of H₂O modulates the forcing by replacing surface emission with RH-dependent atmospheric emission. This substantially reduces the forcing in the tropics, introduces forcing variations due to spatially varying RH, and sets an upper limit (with respect to T_s variations) on CO₂ forcing that is reached in the present-day tropics. In addition, we extend our analytical model to the instantaneous tropopause forcing, and find that this forcing depends on T_s only, with no dependence on stratospheric temperature. We also analyze the τ = 1 approximation for the emission level and derive an exact formula for the emission level, which yields values closer to τ = 1/2 than to τ = 1.

This article has a companion article which can be found at http://journals.ametsoc.org/doi/abs/10.1175/JCLI-D-20-0396.1

Corresponding author: Nadir Jeevanjee, nadir.jeevanjee@noaa.gov

Abstract

This article has a companion article which can be found at http://journals.ametsoc.org/doi/abs/10.1175/JCLI-D-20-0396.1

Corresponding author: Nadir Jeevanjee, nadir.jeevanjee@noaa.gov

Keywords: Climate change; Greenhouse gases; Radiative forcing

1. Introduction

Changes in Earth’s CO₂ greenhouse effect (i.e., CO₂ radiative forcing) have been a primary driver of past and present climate changes, and are well simulated by state-of-the-art radiation codes (e.g., Mlynczak et al. 2016; Pincus et al. 2015; Oreopoulos et al. 2012; Forster et al. 2011). While this accuracy is critical for credible climate simulation and has thus been a priority for radiation research, less emphasis has been placed on an intuitive understanding of CO₂ forcing and its dependence on atmospheric state variables and hence geography or climate. For instance, zonally averaged clear-sky CO₂ forcing exhibits a marked meridional gradient (e.g., Huang et al. 2016), but what causes this? Answering such questions seems particularly worthwhile given the central role of CO₂ forcing in modern climate change.

While not very well understood, this dependence of CO₂ forcing on atmospheric state (and the ensuing spatial heterogeneity of CO₂ forcing) has been known for some time and has been variously attributed to heterogeneities in surface temperature, lapse rate, water vapor, and cloudiness (Zhang and Huang 2014; Byrne and Goldblatt 2014; Feldl and Roe 2013; Govindasamy and Caldeira 2000; Shine and Forster 1999; Myhre and Stordal 1997; Kiehl and Briegleb 1993). Such studies have typically still emphasized global mean forcing, however, and any attribution of the spatial structure has been only qualitative. Recently, however, Huang et al. (2016, hereafter H16) studied the spatial heterogeneity of CO₂ forcing, and developed a highly accurate multilinear regression model for CO₂ forcing that identified the lapse rate as the most important single predictor for clear-sky CO₂ forcing, followed by water vapor path. While these results point the way toward understanding, such regression models cannot tell us whether their predictors have a fundamental significance or are simply correlated with the state variables that really matter. Furthermore, such models offer limited mechanistic insight beyond that already required to sensibly choose predictors.

Here we attempt to push our understanding further by developing an analytical model based on first principles for spatially varying clear-sky CO₂ forcing. The analytical model builds on that of Wilson and Gea-Banacloche (2012) by accounting for water vapor (H₂O) overlap and nonisothermal stratospheres. It accurately emulates the global distribution of clear-sky radiative forcing produced by benchmark radiation codes, and its simplicity allows us to identify and understand the driving factors behind the geographical distribution of this forcing.

We begin in section 2 with a heuristic derivation of the analytical model, followed by validation against a line-by-line (LBL) benchmark in section 3. In section 4 we use the analytical model to compute the global distribution of CO₂ forcing in the absence of H₂O for a snapshot of GCM output, again comparing to a LBL benchmark. In this CO₂-only case, the analytical model shows that CO₂ forcing arises from surface–stratosphere temperature contrast, and thus that meridional gradients in CO₂ forcing are due almost entirely to the meridional surface temperature gradient. In section 5 we extend the analytical model to account for H₂O overlap. We again compute global forcing distributions using both the analytical model and the LBL benchmark, and find that H₂O overlap strongly modulates the meridional gradient in CO₂ forcing, by substantially reducing the forcing in the tropics as well as introducing variations from spatially varying column relative humidity. We also find that Simpson’s law, which says that H₂O emission temperatures at optically thick wavenumbers do not depend on surface temperature,^¹ implies an upper limit (with respect to spatial variations) on CO₂ forcing. Furthermore, this limit appears to be reached in the present-day tropics.

This work focuses primarily on the instantaneous, top-of atmosphere (TOA), clear-sky forcing. While clouds do not qualitatively change the meridional forcing gradient studied here (e.g., Fig. 1c of H16), they do reduce global mean CO₂ forcing by 20%–25% and also modulate its spatial pattern (Pincus et al. 2020; H16), so we consider possible extensions of this work to cloudy skies in the discussion. Our focus on instantaneous TOA forcing also means that we largely neglect the effects of stratospheric temperature adjustment due to increased CO₂ (H16; Hansen et al. 1997; IPCC 1994). This effect is nonnegligible, as the difference between instantaneous TOA forcing and stratosphere-adjusted forcing can be as large as 40% (H16; Zhang and Huang 2014).^² To address this, in section 7 we extend our formalism to the instantaneous tropopause forcing, which better approximates the stratosphere-adjusted forcing, and we argue that our conclusions should apply to stratosphere-adjusted forcing as well. Recent work has identified additional, smaller adjustments to radiative forcing, most notably tropospheric adjustments, but we do not consider these here (see, e.g., Sherwood et al. 2015; Ramaswamy et al. 2019).

2. Theory

In this section we heuristically derive an analytical model for (clear-sky, instantaneous, TOA) CO₂ forcing, where CO₂ is the only radiatively active species (i.e., “CO₂-only”) and we consider the 500–850-cm⁻¹ spectral region only. This spectral region corresponds to the CO₂ ν2 bending-vibration mode centered roughly around the 667-cm⁻¹ absorption peak, and we will refer to it heuristically as the 667-cm⁻¹ band, or simply the CO₂ band.

We begin with a piecewise-exponential parameterization of the spectrum of CO₂ mass absorption coefficients, following the approach of Jeevanjee and Fueglistaler (2020b), Wilson and Gea-Banacloche (2012), and Crisp et al. (1986):

κ_{ref} (ν) = κ_{0} \exp (- \frac{| ν - ν_{0} |}{l}) .

(1)

These are reference absorption coefficients evaluated at a fixed pressure and temperature, which we take to be p_ref = 100 hPa and T_ref = 250 K. Here ν denotes wavenumber (rather than frequency), ν₀ = 667.5 cm⁻¹, κ₀ = 50 m² kg⁻¹ is a representative mass absorption coefficient at ν₀ (discussed further below), and the “spectroscopic decay” parameter l = 10.2 cm⁻¹ sets the rate at which κ_ref declines exponentially away from band center. The parameters l and κ₀ may be obtained by fitting (1) to modeled absorption spectra, but the parameters turn out to depend somewhat on details of the fit (Jeevanjee and Fueglistaler 2020b; Wilson and Gea-Banacloche 2012). Instead, we opt to determine these parameters via optimization as described in section 3.

We now write down the optical depth τ_ν(p) at a given wavenumber ν:

τ_{ν} (p) = D κ_{ref} (ν) \int_{0}^{p} \frac{q p^{'}}{g p_{ref}} d p^{'} = \frac{D κ_{ref} (ν) q}{2 g} \frac{p^{2}}{p_{ref}} .

(2)

Here q is the CO₂ mass concentration (kg kg⁻¹), and D = 1.5 is a diffusivity factor required by the two-stream approximation (implicit in what follows), which truncates the 3D radiation field to upwelling and downwelling fluxes (Pierrehumbert 2010; Clough et al. 1992). The factor of

p^{'} / p_{ref}

in Eq. (2) accounts for pressure broadening, which causes absorption coefficients away from line centers to scale approximately linearly with pressure (Pierrehumbert 2010). We neglect temperature scaling of absorption coefficients.

Although one can use Eq. (2) to solve the radiative transfer equations explicitly, we instead employ the “emission level” approximation wherein we approximate the emission to space from CO₂ at a given wavenumber as occurring entirely at a certain emission level τ_em. (The emission level approximation is discussed further in appendix B.) Setting τ_ν = τ_em in (2) and combining with Eq. (1) then yields the “emission pressure” p_em(ν, q):

p_{em} (ν, q) = \underset{p_{0} (q)}{\underset{︸}{\sqrt{\frac{2 τ_{em} g p_{ref}}{D q κ_{0}}}}} \exp (- \frac{| ν - ν_{0} |}{2 l}) .

(3)

The pressure p₀(q) ≡ p_em(ν₀, q) is an effective emission pressure at the center of the CO₂ band. We show in appendix B that a suitable CO₂ emission level for our purposes is

τ_{em}^{{CO}_{2}} = 0.5

. With this input, and for q = 280 ppmv, we find p₀ = 16 hPa, well into the stratosphere.

Equation (3) can also be inverted for the wavenumbers ν_em emitting at a given p and q:

ν_{em}^{\pm} (p, q) = ν_{0} \pm l \ln (\frac{D q κ_{0} p^{2}}{2 τ_{em} g p_{ref}}) .

(4)

Note the logarithmic dependence of ν_em on q in this equation.

Figure 1a plots p_em(ν) from Eq. (3) for an initial CO₂ concentration q_i = 0.000 280 × 44/29 = 0.000 425 kg kg⁻¹, and for a final CO₂ concentration of q_f = 4q_i. Using a logarithmic axis for p_em(ν) yields “emission pressure triangles” in the ν–p plane, with the triangle in the q_f case being taller and wider than that from q_i. Crucially, this growth in the triangle means that as q increases, some surface emission is blocked (red line segments) and new stratospheric emission is added (green line segments). In a moment, we will use this insight to write down an analytical expression for CO₂ forcing.

But first, we should validate Eq. (3) and the associated triangle picture in Fig. 1a. To this end, we calculate p_em(ν) with a benchmark line-by-line code (see calculation details in section 3). The line-by-line output has too much fine-scale spectral variation to make a useful comparison to the idealized p_em triangles, however, so we must smooth it somehow. This can be done by either coarse-graining (Fig. 1b), or sorting into descending (ascending) order on the left (right) side of the 667-cm⁻¹ peak (Fig. 1c).

These panels reveal both strengths and weaknesses of the triangle picture. The peak coarse-grained emission pressures in Fig. 1b quantitatively match those of Fig. 1a, thus confirming the increase in stratospheric emission (green dashed lines). Also, the blocking of surface emission depicted in Fig. 1a is also seen in the sorted output in Fig. 1c (red solid lines; these are not seen in Fig. 1b due to the coarse-graining). At the same time, however, the absorption coefficients and hence emission pressures near the center of the CO₂ band (the Q-branch) exhibit an extreme wavenumber dependence (Coakley and Yang 2014), so that although the coarse-grained p_em there is roughly equal to p₀ = 16 hPa, the sorted output shows that the most strongly absorbing wavenumbers have p_em much less than 16 hPa.^³ Thus while the LBL calculation supports the main features of the triangle picture, the notions of a “representative” peak absorption coefficient κ₀ or an “effective” peak emission pressure p₀ are imperfect idealizations, whose limitations we will encounter below.

We now proceed to a heuristic estimate of the CO₂ forcing

F

, defined as the difference in outgoing longwave between the q_i and q_f cases. As hinted at above, the key insight is to assess the contributions to the forcing at each height, rather than each wavenumber as is customary (e.g., Dufresne et al. 2020). In Fig. 1a, each orange point on the solid q_i curve has a corresponding point on the dashed orange q_f curve at the same height, and thus both points have the same temperature and thus emission to space (neglecting variations in Planck function across these small spectral intervals). The orange segments along the q_i and q_f curves thus make identical contributions to the outgoing longwave, and thus can be neglected in calculating

F

.^⁴ There are thus only two contributions to

F

: the new stratospheric emission from the q_f curve above p₀(q_i) (dashed green), and the blocked surface emission at wavelengths that were previously optically thin (solid red). In other words, the forcing is simply a swap of surface emission for stratospheric emission. This new stratospheric emission is of course what cools the stratosphere in response to increased CO₂ (Wang and Huang 2020), and it emanates from a characteristic stratospheric temperature

T_{strat} \equiv T (\sqrt{p_{0} (q_{i}) p_{0} (q_{f})}),

(5)

where we take a geometric mean of p₀(q_i) and p₀(q_f). For given surface and stratospheric temperatures T_s and T_strat, then, their contributions to the forcing can be estimated once we know the spectral width Δν over which these contributions are made (Fig. 1a). Using (4), we find that this effective widening of the CO₂ band from changing q_i to q_f is given by

Δ ν = l \ln (\frac{q_{f}}{q_{i}}) .

(6)

As an aside, we note that the logarithmic dependence of Δν on q, which follows from (4), arises because $τ_{ν} ~ q e^{- | ν - ν_{0} | / l}$ . This implies that for fixed p and τ_ν = τ_em, an arithmetic change in ν_em (which causes a uniform widening of the CO₂ band) requires a geometric increase in q, because the ν dependence of τ_ν is exponential. Since the forcing is proportional to Δν (Fig. 1a), this is then the origin of the logarithmic scaling of CO₂ forcing [as understood heuristically in, e.g., Pierrehumbert (2010); see also chapter 2 of Seeley (2018)]. Also note that the overall scale of Δν is governed by the spectroscopic decay parameter l, which also governs the exponential decay of κ_ref(ν) in Eq. (1).^⁵

Returning to our derivation, if we denote the hemispherically integrated Planck function by πB(ν, T) (units of W m⁻² cm⁻¹), and if we approximate the average Planck function across the CO₂ band by evaluating it at ν₀, we can write

F

in this CO₂-only case as

F = 2 l \ln (\frac{q_{f}}{q_{i}}) [π B (ν_{0}, T_{s}) - π B (ν_{0}, T_{strat})] ({CO}_{2} - only) .

(7)

This expression is equivalent to Eq. (25) of Wilson and Gea-Banacloche (2012). Note that besides the initial and final CO₂ concentrations, the only atmospheric state variables appearing in Eq. (7) are T_s and T_strat. This suggests that CO₂ forcing is primarily governed by the surface–stratosphere temperature contrast T_s − T_strat, and that the tropospheric lapse rates emphasized by H16 are only a proxy for T_s − T_strat, insofar as their vertical integral determines T_s − T_strat. Further physical implications of Eq. (7) will be discussed when we study spatial variations of CO₂ forcing in section 4.

3. Line-by-line calculations and parameter optimization

In the remainder of this paper we will test Eq. (7), as well as its extension to account for H₂O overlap, using line-by-line radiative transfer calculations, applied to both idealized single columns and GCM output. This section details those calculations, and uses them to optimize the parameters κ₀ and l appearing in Eqs. (3) and (7), respectively.

a. Line-by-line calculations

Our idealized single column calculations use the Reference Forward Model (Dudhia 2017) for both line-by-line spectroscopy and radiative transfer. We use HITRAN 2016 spectroscopic data for all available spectral lines of H₂O and CO₂ within 500–850 cm⁻¹, for only the most common isotopologue of both gases. We consider highly idealized atmospheric profiles with variable T_s, a constant lapse rate of Γ ≡ −dT/dz = 7 K km⁻¹ up to a tropopause at T_tp ≡ 200 K, with constant stratospheric lapse rate Γ_strat above. We take relative humidity (RH) to be uniform in the troposphere, while specific humidity is uniform in the stratosphere and equal to the tropopause value. Our baseline CO₂ concentration is q_i = 280 ppmv. For many calculations we will use a preferred BASE column with T_s = 300 K, tropospheric RH = 0.75, and Γ_strat = 0. We run RFM at a spectral resolution of 0.1 cm⁻¹ (forcing values accurate to within 0.3% relative to benchmark calculations at 10⁻³ cm⁻¹; not shown) and on a vertical grid with uniform spacing of 100 m up to model top at 50 km. Calculations include H₂O continuum effects (unless otherwise noted), which are parameterized using RFM’s implementation of the MT_CKD2.5 continuum (Mlawer et al. 2012). CO₂ line shapes include line mixing corrections following Strow et al. (1994). We neglect the 1000-cm⁻¹ CO₂ band for the sake of a clean comparison with Eq. (7), but for a quadrupling to 1120 ppmv this band can contribute an additional ~1 W m⁻² of forcing in a global average, a roughly 10% effect (e.g., Zhao et al. 2018). See Zhong and Haigh (2013) for further analyses of the contributions from additional CO₂ bands, and their effect on the logarithmic scaling of CO₂ forcing.

The “global” LBL calculations (i.e., parallelized calculations on GCM output) shown below follow those of Paynter and Ramaswamy (2012) at a resolution of 0.01 cm⁻¹, using RFM to produce optical depth profiles and then solving the radiative transfer equations with four quadrature points per flux calculation, following the method of Clough et al. (1992). For simplicity the surface emissivity is set to 1 at all locations in all calculations, although the central role we find for surface emission means that uncertainties in surface emissivity (e.g., Feldman et al. 2014) may be relevant for uncertainties in CO₂ forcing.

b. Parameter optimization

We begin with preliminary calculations that we use to set the parameters l and κ₀, and which also serve as a first, idealized test of (7). We start by calculating the instantaneous TOA forcing $F_{4 \times}$ from a quadrupling of CO₂ for our idealized single columns with variable surface temperature T_s, isothermal stratosphere (Γ_strat = 0), and for CO₂ as the only radiatively active species (CO₂ only). Because these stratospheres are isothermal the parameter κ₀ is not needed to determine T_strat, so these calculations can be used to set l without compensating errors from κ₀ optimization.

The results of this calculation, using both RFM as well as (7), are shown in Fig. 2a for various values of l. The value l = 10.2 cm⁻¹ minimizes the errors in this comparison and yields an excellent fit, and will be used henceforth. Note that this value is close to the l = 11–11.5-cm⁻¹ range reported in Jeevanjee and Fueglistaler (2020b) and Wilson and Gea-Banacloche (2012) from direct fits to the spectroscopy, and that all values in this range yield a reasonable fit in Fig. 2a.

Next we optimize κ₀. We do this by considering the same columns as in the previous paragraph but with T_s = 300 K and with variable Γ_strat. These more realistic, nonisothermal stratospheres now allow us to probe which κ₀ value yields the most appropriate emission pressure p₀ and hence T_strat [cf. Eqs. (3) and (5)]. A comparison of $F_{4 x}$ as computed by RFM and (7) for these columns and for various values of κ₀ is shown in Fig. 2b. This panel shows that for typical values of −4 < Γ_strat < 0 K km⁻¹, the value κ₀ = 50 m² kg⁻¹ provides an excellent fit. Note, however, that larger errors appear for larger magnitude Γ_strat, showing the limitations of using a single idealized emission pressure p₀ to represent emission near band center; in these extreme cases there are unrealistically warm temperatures near model top (~400 K for Γ_strat = −6 K km⁻¹) that are probed by only the most absorbent wavenumbers (Fig. 1c), and such wavenumbers are not well represented by the coarse-grained average (Figs. 1a,b).

The values of l and κ₀ determined here, as well as other parameter values used in this paper, are tabulated in Table 1.

Table 1.

Parameters for the simple model of CO₂ forcing. See referenced sections for details.

4. Geographic distribution of $F_{4 \times}$ with CO₂ only

Now we apply Eq. (7) along with (5) to more realistic atmospheric columns to obtain a geographical distribution of CO₂ forcing. We continue to consider the CO₂-only case, postponing an analysis of the effects of H₂O overlap to sections 5 and 6. We also only consider forcings relative to a uniform, preindustrial values of q_i = 280 ppmv.

We take as atmospheric data a 22 March 1981 snapshot from a historical run of GFDL’s AM3 (Donner et al. 2011). This equinoctial snapshot has meridional temperature gradients typical of the annual mean, but also exhibits zonal variations due to synoptic-scale weather, which provides a more stringent test of our simple model than annual mean fields. We calculate the forcing $F_{4 \times}$ from a quadrupling of CO₂ for each column using our global LBL code as well as Eq. (7), with the results in Figs. 3a–c. Despite its simplicity, Eq. (7) captures the spatial pattern and overall magnitude of CO₂ forcing as calculated by the global LBL, in both the zonal mean and fully spatially resolved (R² = 0.994 for the latter). The most conspicuous errors are a small overall positive bias, as well as a larger overestimate of the zonal mean forcing near 50°N. Both of these errors appear to be due to the idealization of a single emission pressure p₀ at the center of the CO₂ band; in section 7 we evaluate the tropopause forcing that does not depend on p₀, and these errors disappear (Fig. 10). The larger error near 50°N seems related to a vertical minimum in stratospheric temperatures near p₀ at those latitudes, which biases our estimate of stratospheric emission.

Several other features of Figs. 3a–c deserve mention. As pointed out in the introduction and also found in previous studies (which typically include H₂O and clouds; e.g., H16; Byrne and Goldblatt 2014; Myhre and Stordal 1997), there is a strong meridional gradient in CO₂ forcing, with large values in the tropics and values close to zero or even negative near the poles. [The potentially surprising negative values^⁶ over Antarctica were emphasized by Schmithüsen et al. (2015), but subsequently put into context by Smith et al. (2018), Flanner et al. (2018), and Freese and Cronin (2021).^⁷] There are also several small-scale regions of enhanced forcing throughout the tropics, as well as a diminished forcing over the Tibetan Plateau.

The simplicity of (7) allows us to identify the origin of these and other spatial variations in $F_{4 \times}$ . The only spatially varying quantities in (7) are T_s and T_strat, which are plotted in Figs. 3d–f. The T_s map is almost identical to the $F_{4 \times}$ maps, showing that the spatial variations in $F_{4 x}$ in the CO₂-only case stem almost entirely from T_s, with T_strat variations playing a much smaller role (R² = 0.961 between the maps of T_s and LBL $F_{4 \times}$ ). Accordingly, the strong meridional gradient in zonal mean T_s matches that of $F_{4 \times}$ , while the meridional gradient in T_strat is weak (Figs. 3c,f) With such weak T_strat gradients, both the large-scale meridional gradient in $F_{4 \times}$ as well as the regional features mentioned above can then be understood simply as consequences of variations in surface temperature. (In particular, the negative $F_{4 \times}$ values over Antarctica occur because there we find T_s < T_strat.) Physically, surface temperatures are critical because they dictate the strength of the emission blocked by the widened CO₂ band (red lines in Fig. 1a).

5. Theory for $F_{4 \times}$ including H₂O overlap

a. Heuristics

We now consider overlap^⁸ between the 667 cm⁻¹ CO₂ band and the H₂O rotational band and continuum. To get a feel for the impact of H₂O overlap, Fig. 4 shows the zonal mean forcing for our GCM snapshot for both the CO₂-only and H₂O overlap cases, as computed with our global LBL code. It is immediately apparent that H₂O overlap significantly modulates the meridional gradient in CO₂ forcing from the CO₂-only case, by significantly reducing $F_{4 \times}$ in the tropics (H₂O overlap makes little difference in the very dry regions poleward of roughly ±65°). A map of this forcing (Fig. 7a, presented in the next section) also shows zonal asymmetries in tropical $F_{4 \times}$ , which appear related to synoptic-scale weather.

To understand these features, we must understand how H₂O changes the heuristic picture of CO₂ forcing in Fig. 1. Returning to our idealized single-column calculations, Fig. 5b shows p_em(ν) as calculated by RFM for q = 0, 280, and 1120 ppmv in our BASE atmospheric column but now in the presence of H₂O. We see that the surface emission from Fig. 1 is replaced by tropospheric emission from H₂O. This should indeed reduce the forcing relative to the CO₂-only case, as increasing CO₂ will now displace H₂O emission from the atmosphere rather than warmer surface emission. Furthermore, this displaced H₂O emission will itself depend on relative humidity RH, as drier areas will emit from closer to the surface and hence at warmer temperatures, yielding a stronger forcing; this potentially explains the meridional gradient and zonal asymmetries in tropical $F_{4 \times}$ seen in Figs. 4 and 7a.

To construct an analog to Fig. 1a, we first assume that the H₂O emission on each side of the CO₂ band has an (RH-dependent) emission temperature (continuing to make the emission level approximation), and that under an increase in CO₂ it is this emission that will be replaced by stratospheric emission. This idealization is depicted in Fig. 5a. We take the 550–600 cm⁻¹ spectral interval to be the low wavenumber side of the CO₂ band, and quantities averaged over or pertaining to this interval will be signified with a minus sign (−); similarly, we take 750–800 cm⁻¹ as the high-wavenumber side, and quantities averaged over or pertaining to this interval will be signified with a plus sign (+).

To turn the heuristic picture of Fig. 5a into a formula that generalizes (7), we will estimate spectrally averaged H₂O optical depths τ^±, which we can combine with an emission level τ^H2O to find

T (τ^{\pm} = τ_{em}^{H_{2} O})

. We then invoke Eq. (B4), which says that the emission temperatures may be approximated as the minima of T_s and

T (τ^{\pm} = τ_{em}^{H_{2} O})

T_{em}^{\pm} \equiv \min [T_{s}, T (τ^{\pm} = τ_{em}^{H_{2} O})] .

(8)

Appendix B derives and validates this emission level approximation, and also derives a value of

τ_{em}^{H_{2} O} = 0.6

. This derivation holds only for a single wavenumber, however, and there is an implicit but strong assumption in Eq. (8) that in spectrally averaging Eq. (B4), we may commute the “min” function with the spectral averaging. The limitations of this assumption will become evident below. Regardless, with (8) in hand we may then construct a mean H₂O emission temperature

{\bar{T}}_{em} \equiv \frac{T_{em}^{+} + T_{em}^{-}}{2},

(9)

which can be substituted into (7) for T_s, in line with the heuristic picture in Fig. 5a.

b. Theory

Now we proceed with the quantitative details. The reader uninterested in the following details of H₂O radiative transfer may skip to Eq. (12), which give the desired expressions for $T (τ^{\pm} = τ_{em}^{H_{2} O})$ , and proceed from there.

Since optical depth is a vertical integral of absorber density times absorption coefficient, a prerequisite for calculating τ^± is to obtain estimated, spectrally averaged H₂O absorption coefficients κ^±. A complication, however, is that κ⁻ is dominated by line absorption, whereas κ⁺ is dominated by continuum absorption (Shine et al. 2012; see also appendix A). Accordingly, we approximate κ⁻ as scaling with foreign pressure broadening only (Pierrehumbert 2010), while κ⁺ scales with self-broadening^⁹ only:

κ^{-} = κ_{ref}^{-} \frac{p}{p_{ref}^{-}},

(10a)

κ^{+} = κ_{ref}^{+} \frac{RH}{{RH}_{ref}} e^{(α_{0} - σ) (T - T_{ref}^{+})} .

(10b)

The reference absorption coefficients

κ_{ref}^{\pm}

are evaluated at distinct reference pressures and temperatures

(p_{ref}^{\pm}, T_{ref}^{\pm})

, and κ⁺ also requires a reference relative humidity RH_ref. The constant

α_{0} \equiv L / (R_{υ} T_{ref}^{+ 2})

results from linearization of the exponent in Clausius–Clapeyron, and

e^{α_{0} (T - T_{ref}^{+})}

combines with the RH/RH_ref factor to give the required vapor pressure scaling [see also Eq. (A1)]. The constant σ = 0.02 K⁻¹ is an explicit temperature scaling coefficient. Equation (10b) and the parameter values therein are derived in detail and evaluated in appendix A. Parameter values are recorded in Table 1.

The approximations (10) then allow for an analytical evaluation of τ^±, as follows. We integrate using temperature as our dummy integration variable, and set the lower bound of the integral to the cold-point tropopause temperature T_tp whose H₂O concentrations are assumed negligible (here and below we take the cold point as the tropopause). For τ⁻, which we model as being due to line absorption, such a calculation was already performed in Jeevanjee and Fueglistaler (2020b), so we simply quote their Eq. (12):

τ^{-} = D κ_{ref}^{-} \frac{p}{p_{ref}^{-}} {WVP}_{0} \exp (- \frac{L}{R_{υ} T}),

(11a)

where

{WVP}_{0} = (T_{s} + T_{tp}) RH p_{υ}^{\infty} / (2 Γ L)

depends on RH and has units of water vapor path,

p_{υ}^{\infty} = 2.5 \times 10^{11} Pa

, the saturation vapor pressure

p_{υ}^{*} (T) = p_{υ}^{\infty} \exp (- L / R_{υ} T)

, and all other symbols have their usual meaning.

For τ⁺, the self-broadening scaling, (10b), makes for a different calculation. Denoting vapor density by ρ_υ [kg m⁻³; saturation value is denoted with an asterisk (*)] and noting that

ρ_{υ} (T) \approx ρ_{υ} (T_{ref}^{+}) \exp [α_{0} (T - T_{ref}^{+})]

, we have

\begin{array}{l} τ^{+} = D \int_{T_{tp}}^{T} κ^{+} ρ_{υ} \frac{d T^{'}}{Γ} \\ \approx D \frac{{RH}^{2}}{{RH}_{ref}} ρ_{υ}^{*} (T_{ref}^{+}) \int_{T_{tp}}^{T} κ_{ref}^{+} e^{α (T^{'} - T_{ref}^{+})} \frac{d T^{'}}{Γ}, \\ = D \frac{{RH}^{2} ρ_{υ}^{*} (T_{ref}^{+}) κ_{ref}^{+}}{{RH}_{ref} Γ α} e^{α (T - T_{ref}^{+})}, \\ where α \equiv 2 α_{0} - σ . \end{array}

(11b)

Inverting Eqs. (11) at

τ^{\pm} = τ_{em}^{H_{2} O}

then yields [employing the Lambert W function that satisfies W(xe^x) = x]

T (τ^{-} = τ_{em}^{H_{2} O}) = \frac{T *}{W [\frac{T *}{T_{ref}^{-}} {(D {WVP}_{0} κ_{ref}^{-} / τ_{em})}^{R_{d} Γ / g}]}, where T * \equiv \frac{L R_{d} Γ}{g R_{υ}}

(12a)

T (τ^{+} = τ_{em}^{H_{2} O}) = T_{ref}^{+} + \frac{1}{α} \ln [\frac{τ_{em} Γ α {RH}_{ref}}{D {RH}^{2} ρ_{υ}^{*} (T_{ref}^{+}) κ_{ref}^{+}}] .

(12b)

Note the dependence of

T (τ^{+} = τ_{em}^{H_{2} O})

on RH² in (12b), characteristic of the continuum. Equation (12) provides the expressions we seek, and will be combined below with Eqs. (8) and (9) to yield a generalization of (7) valid in the presence of H₂O.

Before validating these expressions for $T (τ^{\pm} = τ_{em}^{H_{2} O})$ and hence $T_{em}^{\pm}$ , we return to the topic of Simpson’s law. As first noted by Simpson (1928), H₂O optical depth at a given wavenumber and at fixed RH is to a first approximation a function of temperature only, due to the dominant influence of Clausius-Clapeyron scaling. This means that T(τ_em) for that wavenumber is fixed, and hence does not depend on T_s; it is this T_s invariance of H₂O emission temperatures that we refer to as Simpson’s law (Jeevanjee et al. 2021). Indeed, Simpson’s law can be seen in Eqs. (11) and (12), which do not exhibit any explicit T_s dependence. While Simpson’s law is known to have various implications for other aspects of climate,^¹⁰ we will see that for CO₂ forcing it leads to an upper limit on CO₂ forcing with respect to T_s variations, which appears to be reached in the present-day tropics.

c. Validation

We validate the expressions (8) and (12) for

T_{em}^{\pm}

by comparing them to the spectral average of

T_{em} (ν) \equiv \min [T_{s}, T (τ_{ν} = τ_{em})]

(13)

as calculated from RFM output for our single columns with T_s = 300 K, no CO₂, and with varying RH. The ground truth

\int T_{em} (ν) d ν

for

T_{em}^{\pm}

is compared to our estimates from (8) and (12) in Figs. 6a and 6b, which show that Eqs. (8) and (12) do an excellent job of capturing the variation of

T_{em}^{-}

with RH, and do a good job with

T_{em}^{+}

down to RH values near 0.25, around which a significant fraction of wavenumbers in the “+” spectral region become optically thin and thus have T_em(ν) = T_s. In this case the “min” function in (13) does not commute with the spectral averaging, violating the assumption behind (8).

With some confidence in our estimate of

T_{em}^{\pm}

, we now substitute

{\bar{T}}_{em}

from Eq. (9) into Eq. (7) to obtain an expression for CO₂ forcing in the presence of H₂O overlap:

F = 2 l \ln (\frac{q_{f}}{q_{i}}) [π B (ν_{0}, {\bar{T}}_{em}) - π B (ν_{0}, T_{strat})] (w / H_{2} O overlap) .

(14)

Note that as RH → 0,

{\bar{T}}_{em} \to T_{s}

so this equation indeed generalizes (7).

As a preliminary test of (14) we take our single-column, T_s = 300 K, variable RH calculations (with q_i = 280 ppmv) and compare $F_{4 \times}$ as calculated from RFM with $F_{4 \times}$ calculated from (14) and (12). The result is shown in Fig. 6c and shows quite good agreement between the two, although the errors in $T_{em}^{+}$ at low RH discussed above do lead to small (~0.5 W m⁻²) errors in $F_{4 \times}$ .

6. Geographic distribution of $F_{4 \times}$ with H₂O overlap

We now estimate $F_{4 \times}$ with H₂O overlap for our GCM snapshot using Eqs. (8), (9), (12), and (14), where Γ in Eq. (12) is diagnosed for each column as a mass-weighted tropospheric average,^¹¹ and column RH is diagnosed for each GCM column as the precipitable water in the troposphere divided by its saturation value. The results of this computation are shown in Figs. 7b and 7c, and show that Eqs. (7) and (12) indeed capture the spatial distribution and overall magnitude of $F_{4 \times}$ with H₂O overlap, with similar agreement to the CO₂-only case. This supports the heuristic picture of Fig. 5a, namely that the effect of H₂O on CO₂ forcing can be thought of as simply a change in the intensity of the emission (i.e., the T_em) blocked by CO₂.

We now return to the T_s invariance of T(τ^± = τ_em) in Eq. (12). One consequence of this T_s invariance, in combination with Eq. (14), is that while CO₂-only forcing (at fixed q_i) grows with increasing T_s (Fig. 2a), forcing with H₂O overlap should asymptote to a constant value of (14) evaluated on the average of the temperatures in Eq. (12) (assuming fixed RH and T_strat). We confirm this in Fig. 8, which shows $F_{4 \times}$ calculated from both RFM as well as Eqs. (7) and (12), for a series of our idealized atmospheric columns with variable T_s and fixed q_i = 280 ppmv and RH = 0, 0.75. These plots confirm that the presence of H₂O sets an upper limit on $F_{4 \times}$ with respect to T_s which is well captured by our analytical model. Physically, as T_s increases so does the water vapor path and hence the H₂O optical thickness at all wavenumbers. There is thus a transition in the origin of the emission blocked by increasing CO₂, from surface emission to emission from H₂O, the latter of which is T_s invariant. In reality this occurs at different water vapor paths for different wavenumbers, and thus in the spectral integral this transition is smooth and begins even at T_s = 250 K (blue dots in Fig. 8). In our analytical model, however, this transition can only occur separately for the “−” and “+” regions [Eq. (12)] so this transition is more abrupt (blue curve in Fig. 8). Indeed, the kinks in the blue curve in Fig. 8 arise precisely from the kink in the emission level approximation shown in the right panel of Fig. B1.

The limit seen in Fig. 8 is reached beginning at roughly T_s ≈ 300 K, a typical T_s of the present-day tropics. This suggests that the forcing curves in the tropics in Fig. 7c may be thought of as having attained a global maximum (for RH ≈ 0.75), with further local maxima in the subtropics arising only from the low RH values there. Indeed, recalculating $F_{4 \times}$ for our GCM snapshot using (14) and (12) but fixing RH = 0.75 yields the dashed red curve, which varies very little across the tropics. Comparison of the $F_{4 \times}$ and RH maps in Fig. 7 shows that the zonal asymmetries in tropical $F_{4 \times}$ are also due to zonal asymmetries in RH, due to the intrusion of deep tropical moisture filaments into the subtropics (e.g., Pierrehumbert and Roca 1998; Pierrehumbert 1998).

It is important to note that this upper limit on CO₂ forcing is with respect to T_s variations only, and assumes a fixed preindustrial baseline concentration q_i as well as a fixed T_strat. Increasing q_i much beyond preindustrial values brings secondary CO₂ bands with much weaker H₂O overlap into play (Zhong and Haigh 2013), and this limit then no longer applies.

7. Tropopause forcing

So far we have focused solely on instantaneous TOA forcing, as it is the simplest version of CO₂ forcing to compute numerically. But as discussed in the introduction, the stratosphere-adjusted forcing has long been recognized to be more directly related to surface warming (e.g., Hansen et al. 1997; IPCC 1994; Rind and Lacis 1993). While calculating stratospheric adjustments is outside the scope of this paper, we can improve upon the instantaneous TOA forcing by considering the instantaneous tropopause forcing $F^{tp}$ , which is well known to be a better approximation to the stratosphere-adjusted forcing. In this section we develop an analytical model for $F^{tp}$ analogous to Eq. (7), and ask whether our conclusions from previous sections hold for $F^{tp}$ as well.

To describe

F^{tp}

we must estimate both upwelling and downwelling fluxes at the tropopause. For this we need the corresponding emission pressures

p_{em}^{↑}

and

p_{em}^{↓}

, which lie at an optical distance of τ_em below and above the tropopause, respectively (in height). These are determined by the equations

τ (p_{em}^{↑}) = τ (p_{tp}) + τ_{em},

(15a)

τ (p_{em}^{↓}) = τ (p_{tp}) - τ_{em},

(15b)

where p_tp is the tropopause pressure. These equations can be solved numerically from RFM output at each wavenumber, or evaluated analytically using (2), the latter of which yields

p_{em}^{↑} = \sqrt{p_{tp}^{2} + \frac{2 g p_{ref} τ_{em}}{D κ (ν) q}},

(16a)

p_{em}^{↓} = \sqrt{p_{tp}^{2} - \frac{2 g p_{ref} τ_{em}}{D κ (ν) q}} .

(16b)

These numerical and analytical emission pressures are plotted in Figs. 9a and 9b for the tropopause pressure p_tp = 130 hPa from our BASE profile and for CO₂ only, continuing to use

τ_{em}^{{CO}_{2}} = 0.5

. The corresponding fluxes at the tropopause are shown in Figs. 9c and 9d, where the analytical fluxes are computed as

π B [ν_{0}, T (p_{em}^{↑ ↓})]

, from both RFM output and the analytical expressions (16).

Figure 9 serves two purposes. First, it shows that our analytical formalism captures the first order behavior of the tropopause emission pressures and hence fluxes, just as it does at the TOA. Second, it allows us to heuristically derive an expression for

F^{tp}

, as follows. From Fig. 9c, the forcing from the upwelling (red lines) is given by Eq. (7); indeed, this figure gives another heuristic derivation of (7). But, the downwelling contribution to the forcing in Fig. 9c cancels the stratospheric term in (7), leaving only the surface term. This argument extends straightforwardly to the case with H₂O overlap, where T_s must be replaced by

{\bar{T}}_{em}

but the cancellation of the stratospheric term still holds. The tropopause forcing is thus

F^{tp} = 2 l \ln (q_{f} / q_{i}) π B (ν_{0}, {\bar{T}}_{em}) .

(17)

Equation (17) is quite striking, in that it says that the instantaneous tropopause forcing is independent of stratospheric temperatures. Physically, this arises because the additional (in the same sense as Fig. 1) upwelling and downwelling shown in Fig. 9 both originate from very near the tropopause, and thus have emission temperatures nearly equal to the tropopause temperature and thus cancel.

To test Eq. (17), we repeat the global calculations of $F_{4 \times}$ shown in Figs. 3 and 7, but now for the tropopause forcing $F_{4 \times}^{tp}$ where we take the tropopause in each column to be the cold point. The result is shown in Fig. 10. The agreement in both the CO₂-only and H₂O overlap cases is excellent. Interestingly, the accuracy of the analytical model is better for $F_{4 \times}^{tp}$ than $F_{4 \times}$ (cf. Fig. 3), because there is no stratospheric term in (17) and thus no errors arising from the idealization of a single “peak” emission pressure p₀ [cf. Eqs. (3) and (5)].

Although the instantaneous tropopause forcing $F^{tp}$ is independent of stratospheric temperature, the adjusted tropopause forcing will of course depend on the stratospheric temperature adjustment, as this will change the downwelling but not the upwelling at the tropopause. This upsets the cancellation of the added upwelling and downwelling described above. But, it turns out that the adjustment to tropopause forcing from changes in lower stratospheric downwelling is much smaller (10%–15%) than the adjustment to TOA forcing from changes in upper stratospheric upwelling (30%–40%) (e.g., Richardson et al. 2019). This is presumably because the stratospheric temperature adjustment is much more pronounced in the upper rather than lower stratosphere (Wang and Huang 2020). Thus $F^{tp}$ is a good (i.e., to within 15%) estimate of the adjusted forcing, and the stratospheric adjustment can be considered a relatively minor correction. Furthermore, since $F^{tp}$ depends solely on T_s and not on T_strat, the claim that the meridional gradient in CO₂ forcing is largely governed by the meridional T_s gradient is even more accurate for $F^{tp}$ than $F$ . All of this suggests that our conclusion that spatial variations in CO₂ forcing are due predominantly to surface temperature variations, with modulation by H₂O, should hold for stratosphere adjusted forcing as well as the instantaneous forcings considered here.

8. Summary and discussion

We summarize our main results as follows:

Clear-sky TOA CO₂ forcing $F$ in the absence of H₂O can be viewed as a swap of surface emission for stratospheric emission [Fig. 1a, Eq. (7)]. Thus, $F$ is governed by surface–stratosphere temperature contrast, and the strong meridional gradient in $F$ can be attributed largely to the meridional gradient in surface temperature (Fig. 3).
The meridional forcing gradient is significantly modulated by the presence of H₂O (Fig. 4), where H₂O replaces surface emission at the edges of the CO₂ band with colder atmospheric emission (Fig. 5).
The T_s invariance of H₂O emission temperatures $T_{em}^{\pm}$ implies an upper limit (at fixed RH and with respect to T_s variations) on CO₂ forcing (Fig. 8). This limit is likely reached in the present-day tropics (Fig. 7).

We also considered the instantaneous tropopause forcing $F^{tp}$ and found that it depends on T_s only, so the above conclusions also hold (perhaps even more so) for $F^{tp}$ . However, it would still be useful in future work to calculate the full stratosphere-adjusted rather than instantaneous forcings, using perhaps a simple method for the stratospheric adjustment such as fixed dynamical heating (Fels et al. 1980).

Another extension of this work would be to generalize Eq. (7) to cloudy columns, and hence to compute all-sky forcing. This might be accomplished by replacing T_s with a diagnosed cloud-top temperature, just as we replaced T_s by T_em in the presence of H₂O. Clouds, like H₂O, should simply change the upwelling radiation that is blocked by additional CO₂. This is already well known in the literature as the “cloud masking” of CO₂ forcing (e.g., H16), but might be succinctly and quantitatively described by the substitution of cloud-top temperature for T_s in (7). Such an approach, applied to feedbacks rather than forcing, was recently taken in McKim et al. (2021).

Although this work focuses on the spatial variations of CO₂ forcing, the physics of these variations is simply the atmospheric state dependence of CO₂ forcing, which also has implications for CO₂ forcing as a function of base climate. For instance, a very cold Snowball Earth climate (e.g., Hoffman et al. 2017) will have negligible H₂O and a much smaller surface–stratosphere temperature contrast, which would lead to much reduced CO₂ forcings relative to the present day. This fact and its implications for exiting the Snowball Earth state were noted by Pierrehumbert (2004), but Eq. (7) makes this precise and allows for quantitative estimates of this effect.

The state dependence of CO₂ forcing may also be relevant to the spread in CO₂ forcing among GCMs (e.g., Soden et al. 2018; Chung and Soden 2015a,b; Zhang and Huang 2014). This spread is often attributed to parameterization error in GCM broadband radiation schemes, but may also have a contribution from spread in GCM base states. Equation (14) is computationally inexpensive to evaluate (no spectral or vertical integration required) and thus might be applied to GCM output to estimate this contribution. Indeed, one can simply differentiate (7) with respect to T_s and evaluate at T_s = 288 K, obtaining 2l(ln2)π(∂B/∂T)(ν₀, 288 K) = 0.070 W m⁻² K⁻¹ for CO₂ doubling. Thus, biases of 2 K in T_s (Flato et al. 2013) should bias $F_{2 \times}$ by roughly 0.14 W m⁻². One can also consider T_strat biases, which by a similar differentiation of (7) but with respect to T_strat and evaluated at T_strat = 220 K yields a sensitivity of −0.04 W m⁻² K⁻¹. Biases of 4–5 K in T_strat (Butchart et al. 2011) would thus similarly bias $F_{2 \times}$ by 0.1–0.2 W m⁻². Note that these sensitivities to T_s and T_strat mean that $F$ is not entirely independent of the warming it produces, thus measuring the degree to which the usual forcing-feedback framework is only an approximation.

Finally, it is worth noting that our analytical model can explain empirically determined features of the linear regression model of H16. For example, p₀ (280 ppmv) = 16 hPa from (3) is close to the empirically determined 10-hPa value used in H16 to evaluate stratospheric temperatures. As another example, consider H16’s T_s regression coefficient of 0.066 W m⁻² K⁻¹ for CO₂ doubling. According to our model, this coefficient should simply be the 0.070 W m⁻² K⁻¹ calculated in the previous paragraph, a close numerical agreement.

Acknowledgments

NJ thanks Yi Huang for discussions, Robert Pincus and V. Ramaswamy for encouragement, and W. Happer for discussion of CO₂ spectroscopy and for pointing out the Wilson and Gea-Banacloche (2012) reference. The authors thank five anonymous reviewers as well as Pu Lin and Mike Winton for detailed feedback on the manuscript at various stages. NJ was supported by a Harry Hess post-doctoral fellowship from Princeton Geosciences, JS is supported by a HUCE fellowship, and SF acknowledges support from NSF Grants 1660538 and 1733818.

APPENDIX A

Estimate for H₂O Self-Broadened Absorption Coefficient

This appendix discusses our determination of the spectrally averaged reference absorption coefficients $κ_{ref}^{\pm}$ appearing in (10), and also derives the expression (10b) for the self-broadened absorption coefficient profile κ⁺.

Self-broadened continuum H₂O absorption coefficients exhibit both an explicit temperature scaling and pressure broadening, the latter of which scales linearly with vapor pressure p_υ rather than the dry air pressure p (Pierrehumbert 2010). These scalings are thus relative to a reference temperature and reference vapor pressure, the latter of which can be written in terms of the saturation vapor pressure

p_{υ}^{*}

and reference RH as

p_{υ, ref} = {RH}_{ref} p_{υ}^{*} (T_{ref})

. The vapor pressure scaling can then be written as

\frac{p_{υ}}{p_{υ, ref}} = \frac{RH p_{υ}^{*} (T)}{{RH}_{ref} p_{υ}^{*} (T_{ref})} \approx \frac{RH}{{RH}_{ref}} e^{α_{0} (T - T_{ref})}, where α_{0} \equiv \frac{L}{R_{υ} T_{ref}^{2}} .

(A1)

As for the explicit temperature scaling, this takes the form

e^{σ (T_{ref} - T)}

(Mlawer et al. 2012).

Since the “+” wavenumber region is dominated by continuum absorption (as we will see), we will adopt the above vapor pressure scaling for κ⁺, as well as the explicit temperature scaling coefficient σ = 0.021 K⁻¹ relevant for this wavenumber region (Mlawer et al. 2012). We specify reference values RH_ref = 0.75 and

T_{ref}^{+} = 275 K

for κ⁺, and

(p_{ref}^{-}, T_{ref}^{-}) = (370 hPa, 245 K)

for κ⁻, which will scale with the dry air pressure [Eq. (10a)]. These reference pressures and temperatures are in principle arbitrary, but values near the emission pressures and temperatures can be expected to minimize errors from our various approximations. We can now write down κ⁺ as

κ^{+} = κ_{ref}^{+} \frac{RH}{{RH}_{ref}} e^{(α_{0} - σ) (T - T_{ref}^{+})} .

(A2)

This is Eq. (10b) in the main text. Equation (10a) is standard and can be found in textbooks (e.g., Pierrehumbert 2010), although it neglects temperature scaling of line absorption, an issue to which we return below.

To gauge the accuracy of Eq. (10), Fig. A1 shows profiles of spectrally averaged total absorption coefficient κ_tot, lines-only contribution κ_lines, and the difference κ_ctm, which we can ascribe to the continuum, for both the “+” and “−” wavenumber regions. These profiles are calculated via RFM for our BASE column, where κ_lines is calculating by running RFM without continuum effects, and all spectral averages are performed geometrically rather than arithmetically. Figure 8 shows that for our BASE column the continuum contribution κ_ctm dominates in the “+” region but not in the “−” region, justifying our use of continuum scalings for the “+” region only. This figure also shows our estimates (10), with $κ_{ref}^{\pm}$ taken to be equal to $κ_{tot}^{\pm} (p_{ref}^{\pm}, T_{ref}^{\pm}, {RH}_{ref})$ , yielding $κ_{ref}^{-} = 0.1 m^{2} {kg}^{- 1}$ and $κ_{ref}^{+} = 0.025 m^{2} {kg}^{- 1}$ . Our estimates (10) thus agree with $κ_{tot}^{\pm}$ at $(p_{ref}^{\pm}, T_{ref}^{\pm}, {RH}_{ref})$ by construction, but due to the many approximations we have made do not have the same logarithmic slope (i.e., scaling) as κ_tot. However, because H₂O optical depth is an integral of κ_tot weighted by p_υ, Clausius–Clapeyron scaling means it is only important for κ to have the right order of magnitude in the range of (T_em − 20 K, T_em) or so, within which our estimates are accurate to roughly a factor of 2 (by our choice of $T_{ref}^{\pm}$ ).

It is interesting to note that the logarithmic slopes of κ_lines and κ_etm are comparable for a given wavenumber range, despite the naive expectation that κ_lines scales with p (which varies by a factor of 5 over the vertical range shown in Fig. A1) and κ_ctm scales with p_υ (which varies by a factor of 700). However, κ_lines also exhibits a temperature scaling, which we ignore and which accounts for much of the error in the slope of κ in Fig. A1a. At the same time, κ_ctm also exhibits a temperature scaling but with opposite sign, which weakens its Clausius–Clapeyron scaling [Eq. (A2)]. These opposing temperature scalings for κ_lines and κ_ctm modify our naive expectations, and seem to conspire to produce surprisingly similar overall logarithmic slopes. Whether or not this is a coincidence, or is related to the hypothesis that continuum absorption is simply due to far-wing line absorption (e.g., Ma et al. 2008), could be investigated further.

APPENDIX B

On the Emission Level Approximation and the Choice of τ_em

In sections 2 and 5 we made the “emission level” approximation that emission to space can be regarded as originating from a single level. We take this level to be the surface when the atmosphere is sufficiently optically thin, and we set these levels as $τ_{em}^{{CO}_{2}} = 0.5$ and $τ_{em}^{H_{2} O} = 0.6$ when the surface optical depth is greater than those values. This appendix discusses this approximation, and justifies these choices of τ_em for our applications. Other values of τ_em may be required for other applications. Note also that the emission level we define here is distinct from the more general emission level defined in Dufresne et al. (2020); their emission level characterizes atmospheric emission even in the optically thin limit. Furthermore, our emission level τ_em need not coincide with the maximum of the “weighting function” or the cooling-to-space profile, though these also occur at τ ~ O(1) (Jeevanjee and Fueglistaler 2020a).

It will be convenient to use the framework and notation of Jeevanjee and Fueglistaler (2020a), which considers a gray gas with idealized optical depth, temperature, and source function profiles

τ = τ_{s} {(\frac{p}{p_{s}})}^{β}, T = T_{s} {(\frac{p}{p_{s}})}^{R_{d} Γ / g}, B = B_{s} {(\frac{T}{T_{s}})}^{α},

where subscript s denotes the surface value of a quantity and B has units of W m⁻². These profiles combine to yield

B (τ) = B_{s} {(\frac{τ}{τ_{s}})}^{γ}, where

(B1)

γ \equiv \frac{d \ln B}{d \ln τ} = α \frac{R_{d} Γ}{g} \frac{1}{β} .

(B2)

Now, the emission level (EL) approximation simply says that

OLR \approx {\begin{matrix} B_{s} & if τ_{s} < τ_{em} \\ B (τ_{em}) & if τ_{s} \geq τ_{em} \end{matrix} (EL approximation)

(B3)

for some emission level optical depth τ_em, which may depend on the parameters introduced above. This τ_em may be thought of as characterizing the transition between surface and atmospheric emission, or equivalently between optically thin and optically thick regimes. As such, we expect τ_em ~ O(1) (Jeevanjee and Fueglistaler 2020b; Petty 2006; Wallace and Hobbs 2006), as we will indeed find below. Note that in terms of an effective emission temperature T_em which satisfies OLR ≈ B(T_em), the EL approximation can be rewritten as

T_{em} = \min [T_{s}, T (τ_{em})],

(B4)

which is the form used in the main text [e.g., Eq. (8)].

To determine τ_em, we first analytically compute the OLR for our idealized gray gas,^{^B1} using Eq. (B1) and assuming τ_s ≫ 1:

\begin{array}{l} OLR = \int_{0}^{\infty} B_{s} {(τ / τ_{s})}^{γ} e^{- τ} d τ \\ = \frac{B_{s}}{τ_{s}^{γ}} \tilde{Γ} (1 + γ), \end{array}

(B5)

where

\tilde{Γ} (γ + 1) \equiv \int_{0}^{\infty} x^{γ} e^{- x} d x

denotes Euler’s gamma function, and the tilde is introduced to distinguish it from the atmospheric lapse rate. We may then combine Eqs. (B1), (B3), and (B5) and solve for τ_em, obtaining

τ_{em} = {[\tilde{Γ} (1 + γ)]}^{1 / γ} .

(B6)

A plot of this curve is shown in Fig. B1a. To determine τ_em, then, we simply need appropriate values for γ for CO₂ and H₂O emission. For CO₂, τ_em only enters our theory quantitatively in determining p₀(q) [Eq. (3)], which lies in the stratosphere where Γ ≈ −2 K km⁻¹. Using this value for Γ and also setting β = 2 [Eq. (2)] and α = 4 (Jeevanjee and Fueglistaler 2020a), Eq. (B2) then yields

γ^{{CO}_{2}} = - 0.1

. Plugging this into (B6) yields

τ_{em}^{{CO}_{2}} \approx 0.5

(Fig. B1a, red dot).

For H₂O, we are interested in tropospheric emission (Γ ≈ 7 K km⁻¹) in the neighborhood of the CO₂ band (α = 4). Jeevanjee and Fueglistaler (2020b) found β = 5.5 for line absorption,^{^B2} thus yielding $γ^{H_{2} O} = 0.15$ and hence $τ_{em}^{H_{2} O} = 0.6$ (Fig. B1a, blue dot).

With Eq. (B6) in hand, we may also explicitly evaluate the accuracy of the EL approximation (B3) across a range of τ_s values. To do this we need an exact expression for the OLR, generalizing (B5) and valid for all τ_s:

\begin{array}{l} OLR = B_{s} e^{- τ_{s}} + \int_{0}^{τ_{s}} B_{s} {(τ / τ_{s})}^{γ} e^{- τ} d τ \\ = B_{s} e^{- τ_{s}} + \frac{B_{s}}{τ_{s}^{γ}} \tilde{Γ} (1 + γ, τ_{s}), \end{array}

(B7)

where

\tilde{Γ} (1 + γ, τ_{s}) \equiv \int_{0}^{τ_{s}} x^{γ} e^{- x} d x

is now the (lower) incomplete gamma function, which differs from Euler’s gamma function only in the upper limit of the integral. We compare Eq. (B7) to the EL approximation (B3) in Fig. B1b. We choose γ = 0.4 (black dot in Fig. B1a), as it is appropriate for tropospheric CO₂ (Jeevanjee and Fueglistaler 2020a) and yields larger errors than the tropospheric H₂O value of γ = 0.15, allowing for a more conservative error assessment. Even with this conservative value of γ, the EL approximation is quite reasonable: errors never exceed 15% or so. For γ = 0.15, the errors do not exceed 7%. For greenhouses gases besides H₂O and CO₂, however, the value of γ may vary beyond the range considered here; see Jeevanjee and Fueglistaler (2020a) for further discussion.

Finally, we note that the γ parameter of Eq. (B2) was also found by Jeevanjee and Fueglistaler (2020a) to determine the validity of the cooling-to-space approximation, which holds when

γ ≪ 1

. In this limit, we may Taylor-expand the

\tilde{Γ}

function in (B6) and invoke the fact that

{[d \tilde{Γ} (x) / d x] |}_{x = 1} = - γ_{Euler}

, where γ_Euler is the Euler-Mascheroni constant (yet another gamma). A little calculation then shows that

\lim_{γ \to 0} τ_{em} = e^{- γ_{Euler}} = 0.56 .

(B8)

This gives a preferred value for τ_em when |γ| ≪ 1 (Fig. B1a, dotted horizontal line), and indeed this value is very close to both

τ_{em}^{CO 2}

and

τ_{em}^{H 2 O}

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Simpson (1928), Ingram (2010), Jeevanjee et al. (2021), and references therein.

Consequently, the global mean instantaneous TOA forcing values shown here will be significantly lower than the standard stratosphere-adjusted value of 3.7 W m⁻² per doubling (Ramaswamy et al. 2001; Myhre et al. 1998).

Indeed, CO₂ is known to emit strongly from not only the stratosphere, but the mesosphere and thermosphere as well (e.g. Curtis and Goody 1956; Mlynczak et al. 2010).

⁴

To the extent that the cooling-to-space approximation holds (Jeevanjee and Fueglistaler 2020a), this claim of unchanged cooling-to-space is consistent with a negligible change in tropospheric heating rate for the CO₂-only case, e.g. Fig. 6c of Sejas et al. (2016).

⁵

The spectroscopic decay parameter also plays a key role in setting the magnitude of radiative cooling (Jeevanjee and Fueglistaler 2020b).

⁶

Note that this negative CO₂ forcing is related to, but distinct from, the negative climatological greenhouse effect discussed in, e.g., Sejas et al. (2018).

⁷

In particular, a negative instantaneous forcing can still lead to a positive surface temperature perturbation, because of stratospheric adjustment as well as surface-troposphere decoupling.

⁸

Strictly speaking, “overlap” refers to absorption and emission by multiple gases at a given wavenumber, and this is indeed what is modeled in our LBL calculations. The simple model developed below, however, treats wavenumbers as either entirely H₂O or CO₂-dominated, so the simple model idealizes overlap as the presence of wavenumbers that transition from H₂O to CO₂-dominated as CO₂ concentrations are increased.

⁹

We neglect here the foreign-broadened component of the continuum, which is weaker in the moister, tropical columns where continuum absorption is significant (Shine et al. 2012).

¹⁰

For example, there are far-reaching implications for the runaway greenhouse (Nakajima et al. 1992), outgoing longwave radiation (Koll and Cronin 2018), radiative cooling and precipitation (Jeevanjee and Romps 2018), and the water vapor feedback (Ingram 2010; Jeevanjee 2018; Jeevanjee et al. 2021).

¹¹

More precisely, Γ is obtained as a mass-weighted average between the cold-point tropopause and either 1) the surface or 2) the highest temperature inversion below the cold point (e.g. a trade inversion). The sensitivity to Γ is minor and almost identical results can be obtained with a uniform Γ = 6.5 K km⁻¹.

^B1

We continue to employ a two-stream approximation and assume here that τ implicitly contains a diffusivity factor.

^B2

A β value appropriate for continuum absorption can be read off from Eq. (13) of Jeevanjee and Fueglistaler (2020b) by doubling the contribution from Clausius-Clapeyron scaling due to the quadratic dependence of continuum absorption on vapor pressure, which roughly doubles β to 10. This halves $γ^{H 2 O}$ to 0.08, but yields negligible changes in $τ_{em}^{H 2 O}$ (Fig. B1a).

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An Analytical Model for Spatially Varying Clear-Sky CO₂ Forcing

Abstract

Abstract

1. Introduction

2. Theory

3. Line-by-line calculations and parameter optimization

a. Line-by-line calculations

b. Parameter optimization

4. Geographic distribution of $F_{4 \times}$ with CO₂ only

5. Theory for $F_{4 \times}$ including H₂O overlap

a. Heuristics

b. Theory

c. Validation

6. Geographic distribution of $F_{4 \times}$ with H₂O overlap

7. Tropopause forcing

8. Summary and discussion

Acknowledgments

APPENDIX A

Estimate for H₂O Self-Broadened Absorption Coefficient

APPENDIX B

On the Emission Level Approximation and the Choice of τ_em

REFERENCES

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GENERAL CONDITIONS

No. III

BAROMETRIC PRESSURE

VERIFICATIONS

MISCELLANEOUS PHENOMENA

An Analytical Model for Spatially Varying Clear-Sky CO2 Forcing

Abstract

Abstract

1. Introduction

2. Theory

3. Line-by-line calculations and parameter optimization

a. Line-by-line calculations

b. Parameter optimization

4. Geographic distribution of F4× with CO2 only

5. Theory for F4× including H2O overlap

a. Heuristics

b. Theory

c. Validation

6. Geographic distribution of F4× with H2O overlap

7. Tropopause forcing

8. Summary and discussion

Acknowledgments

APPENDIX A

Estimate for H2O Self-Broadened Absorption Coefficient

APPENDIX B

On the Emission Level Approximation and the Choice of τem

REFERENCES

Share Link

AMS Publications

Get Involved with AMS

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An Analytical Model for Spatially Varying Clear-Sky CO₂ Forcing

4. Geographic distribution of $F_{4 \times}$ with CO₂ only

5. Theory for $F_{4 \times}$ including H₂O overlap

6. Geographic distribution of $F_{4 \times}$ with H₂O overlap

Estimate for H₂O Self-Broadened Absorption Coefficient

On the Emission Level Approximation and the Choice of τ_em