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    (a) Graph of CO2 emission levels as given by Eq. (3) for qi = 280 ppmv and qf = 4qi. (b) As in (a), but from RFM calculations with CO2 only for our BASE atmosphere. RFM emission levels are diagnosed by the condition τν = τemCO2 = 0.5, and are geometrically averaged (coarse-grained) over 10 cm−1 bins. (c) As in (b), but pem 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 “pem 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 pem 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 Ts and Γstrat = 0 and (b) Ts = 300 K and variable Γstrat. Optimization of l in (a) yields l = 10.2 cm−1, and optimization of κ0 in (b) for −4 < Γstrat < 0 K km−1 yields κ0 = 50 m2 kg−1. With these parameter values, the good fit in these panels across a range of Ts and Γstrat provides a first validation of (7).

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    Maps of (a) CO2 forcing F4× with CO2 only from a LBL calculation; (b) as in (a), but using Eq. (7); (d) surface temperature Ts; and (e) stratospheric emission temperature Tstrat, 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 Ts map in (d) is almost identical to the F4× maps in (a) and (b), showing that the spatial variations in F4× in the CO2-only case stem almost entirely from Ts, with Tstrat variations playing a much smaller role. Accordingly, the strong meridional gradient in zonal-mean Ts matches that of F4× [(c) and (f)], while the meridional gradient in Tstrat 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 τν = 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 Ts = 300, no CO2, and varying RH. (c) Validation of the simple model (14) for F4× in the presence of H2O, as compared to F4× calculated by RFM. This comparison is made for idealized atmospheric columns with Ts = 300, qi = 280 ppmv, and varying RH. The simple expressions (8) and (12) predict Tem± very well except at low RH in the “+” region, leading to small (~0.5 W/m2) errors in F4× 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 F4×, 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 F4× for our idealized atmospheric columns with varying Ts and all other parameters fixed at the BASE values. The presence of H2O (blue) sets a limit on F4×(Ts) 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 pem 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 pem emitting downward toward the tropopause are shown in blue, and levels pem 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 F4×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 Ts 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 κtot, κlines, and κctm are calculated with RFM, whereas κest is given by Eqs. (10) with (κref,κref+) set to (κtot(Tref),κtot+(Tref+)), and where (Tref,Tref+)=(245,275)K. The estimated profile κ is a poor approximation to κtot, far from Tref due to our neglect of temperature scaling of line absorption, but is acceptably close within 20 K or so of Tref. 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 τem as a function of the parameter γ defined in (B2). At γ = 0, τem=eγEuler0.56, close to our values for τemCO2 and τemH2O. (b) Plot of normalized exact OLR from Eq. (B7), as well as normalized OLR from the emission level approximation (B3) as a function of τs and with γ = 0.4 (black dot in left panel; see text for discussion of this value). The EL approximation is reasonable, even near τs = 1, with maximum errors of roughly 15%.

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An Analytical Model for Spatially Varying Clear-Sky CO2 Forcing

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  • 1 a NOAA/Geophysical Fluid Dynamics Laboratory, Princeton, New Jersey
  • | 2 b Harvard University Center for the Environment, Cambridge, Massachusetts
  • | 3 c Geosciences Department, Princeton University, Princeton, New Jersey
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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 Ts, 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 Ts. 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 Ts 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 Ts 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.

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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

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 Ts, 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 Ts. 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 Ts 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 Ts 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.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

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

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):
κref(ν)=κ0exp(|νν0|l).
These are reference absorption coefficients evaluated at a fixed pressure and temperature, which we take to be pref = 100 hPa and Tref = 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 κref 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 τν(p) at a given wavenumber ν:
τν(p)=Dκref(ν)0pqpgprefdp=Dκref(ν)q2gp2pref.
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/pref 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 τem. (The emission level approximation is discussed further in appendix B.) Setting τν = τem in (2) and combining with Eq. (1) then yields the “emission pressure” pem(ν, q):
pem(ν,q)=2τemgprefDqκ0p0(q)exp(|νν0|2l).
The pressure p0(q) ≡ pem(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 τemCO2=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 νem emitting at a given p and q:
νem±(p,q)=ν0±lln(Dqκ0p22τemgpref).
Note the logarithmic dependence of νem on q in this equation.

Figure 1a plots pem(ν) from Eq. (3) for an initial CO2 concentration qi = 0.000 280 × 44/29 = 0.000 425 kg kg−1, and for a final CO2 concentration of qf = 4qi. Using a logarithmic axis for pem(ν) yields “emission pressure triangles” in the νp plane, with the triangle in the qf case being taller and wider than that from qi. 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.

Fig. 1.
Fig. 1.

(a) Graph of CO2 emission levels as given by Eq. (3) for qi = 280 ppmv and qf = 4qi. (b) As in (a), but from RFM calculations with CO2 only for our BASE atmosphere. RFM emission levels are diagnosed by the condition τν = τemCO2 = 0.5, and are geometrically averaged (coarse-grained) over 10 cm−1 bins. (c) As in (b), but pem 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 “pem 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 pem 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.

Citation: Journal of Climate 34, 23; 10.1175/JCLI-D-19-0756.1

But first, we should validate Eq. (3) and the associated triangle picture in Fig. 1a. To this end, we calculate pem(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 pem 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-cm1 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 pem there is roughly equal to p0 = 16 hPa, the sorted output shows that the most strongly absorbing wavenumbers have pem 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 qi and qf 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 qi curve has a corresponding point on the dashed orange qf 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 qi and qf 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 qf curve above p0(qi) (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
TstratT(p0(qi)p0(qf)),
where we take a geometric mean of p0(qi) and p0(qf). For given surface and stratospheric temperatures Ts and Tstrat, 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 qi to qf is given by
Δν=lln(qfqi).

As an aside, we note that the logarithmic dependence of Δν on q, which follows from (4), arises because τν~qe|νν0|/l. This implies that for fixed p and τν = τem, an arithmetic change in νem (which causes a uniform widening of the CO2 band) requires a geometric increase in q, because the ν dependence of τν 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 κref(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=2lln(qfqi)[πB(ν0,Ts)πB(ν0,Tstrat)](CO2-only).

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 Ts and Tstrat. This suggests that CO2 forcing is primarily governed by the surface–stratosphere temperature contrast TsTstrat, and that the tropospheric lapse rates emphasized by H16 are only a proxy for TsTstrat, insofar as their vertical integral determines TsTstrat. 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 Ts, a constant lapse rate of Γ ≡ −dT/dz = 7 K km−1 up to a tropopause at Ttp ≡ 200 K, with constant stratospheric lapse rate Γstrat 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 qi = 280 ppmv. For many calculations we will use a preferred BASE column with Ts = 300 K, tropospheric RH = 0.75, and Γstrat = 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 F4× from a quadrupling of CO2 for our idealized single columns with variable surface temperature Ts, isothermal stratosphere (Γstrat = 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 Tstrat, 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.

Fig. 2.
Fig. 2.

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

Citation: Journal of Climate 34, 23; 10.1175/JCLI-D-19-0756.1

Next we optimize κ0. We do this by considering the same columns as in the previous paragraph but with Ts = 300 K and with variable Γstrat. These more realistic, nonisothermal stratospheres now allow us to probe which κ0 value yields the most appropriate emission pressure p0 and hence Tstrat [cf. Eqs. (3) and (5)]. A comparison of F4x 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 < Γstrat < 0 K km−1, the value κ0 = 50 m2 kg−1 provides an excellent fit. Note, however, that larger errors appear for larger magnitude Γstrat, 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 Γstrat = −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.

Table 1.

4. Geographic distribution of F4× 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 qi = 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 F4× 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.

Fig. 3.
Fig. 3.

Maps of (a) CO2 forcing F4× with CO2 only from a LBL calculation; (b) as in (a), but using Eq. (7); (d) surface temperature Ts; and (e) stratospheric emission temperature Tstrat, 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 Ts map in (d) is almost identical to the F4× maps in (a) and (b), showing that the spatial variations in F4× in the CO2-only case stem almost entirely from Ts, with Tstrat variations playing a much smaller role. Accordingly, the strong meridional gradient in zonal-mean Ts matches that of F4× [(c) and (f)], while the meridional gradient in Tstrat is weak.

Citation: Journal of Climate 34, 23; 10.1175/JCLI-D-19-0756.1

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 values6 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 F4×. The only spatially varying quantities in (7) are Ts and Tstrat, which are plotted in Figs. 3d–f. The Ts map is almost identical to the F4× maps, showing that the spatial variations in F4x in the CO2-only case stem almost entirely from Ts, with Tstrat variations playing a much smaller role (R2 = 0.961 between the maps of Ts and LBL F4×). Accordingly, the strong meridional gradient in zonal mean Ts matches that of F4×, while the meridional gradient in Tstrat is weak (Figs. 3c,f) With such weak Tstrat gradients, both the large-scale meridional gradient in F4× as well as the regional features mentioned above can then be understood simply as consequences of variations in surface temperature. (In particular, the negative F4× values over Antarctica occur because there we find Ts < Tstrat.) 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 F4× including H2O overlap

a. Heuristics

We now consider overlap8 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 F4× 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 F4×, which appear related to synoptic-scale weather.

Fig. 4.
Fig. 4.

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.

Citation: Journal of Climate 34, 23; 10.1175/JCLI-D-19-0756.1

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 pem(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 F4× seen in Figs. 4 and 7a.

Fig. 5.
Fig. 5.

(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 τν = 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.

Citation: Journal of Climate 34, 23; 10.1175/JCLI-D-19-0756.1

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 τH2O to find T(τ±=τemH2O). We then invoke Eq. (B4), which says that the emission temperatures may be approximated as the minima of Ts and T(τ±=τemH2O):
Tem±min[Ts,T(τ±=τemH2O)].
Appendix B derives and validates this emission level approximation, and also derives a value of τemH2O=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
T¯emTem++Tem2,
which can be substituted into (7) for Ts, 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±=τemH2O), 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-broadening9 only:
κ=κrefppref,
κ+=κref+RHRHrefe(α0σ)(TTref+).
The reference absorption coefficients κref± are evaluated at distinct reference pressures and temperatures (pref±,Tref±), and κ+ also requires a reference relative humidity RHref. The constant α0L/(RυTref+2) results from linearization of the exponent in Clausius–Clapeyron, and eα0(TTref+) combines with the RH/RHref 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 Ttp 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=DκrefpprefWVP0exp(LRυT),
where WVP0=(Ts+Ttp)RHpυ/(2ΓL) depends on RH and has units of water vapor path, pυ=2.5×1011Pa, the saturation vapor pressure pυ*(T)=pυexp(L/RυT), and all other symbols have their usual meaning.
For τ+, the self-broadening scaling, (10b), makes for a different calculation. Denoting vapor density by ρυ [kg m−3; saturation value is denoted with an asterisk (*)] and noting that ρυ(T)ρυ(Tref+)exp[α0(TTref+)], we have
τ+=DTtpTκ+ρυdTΓDRH2RHrefρυ*(Tref+)TtpTκref+eα(TTref+)dTΓ,=DRH2ρυ*(Tref+)κref+RHrefΓαeα(TTref+),whereα2α0σ.
Inverting Eqs. (11) at τ±=τemH2O then yields [employing the Lambert W function that satisfies W(xex) = x]
T(τ=τemH2O)=T*W[T*Tref(DWVP0κref/τem)RdΓ/g],whereT*LRdΓgRυ
T(τ+=τemH2O)=Tref++1αln[τemΓαRHrefDRH2ρυ*(Tref+)κref+].
Note the dependence of T(τ+=τemH2O) 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±=τemH2O) and hence Tem±, 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(τem) for that wavenumber is fixed, and hence does not depend on Ts; it is this Ts 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 Ts 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 Ts variations, which appears to be reached in the present-day tropics.

c. Validation

We validate the expressions (8) and (12) for Tem± by comparing them to the spectral average of
Tem(ν)min[Ts,T(τν=τem)]
as calculated from RFM output for our single columns with Ts = 300 K, no CO2, and with varying RH. The ground truth Tem(ν)dν for Tem± 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 Tem with RH, and do a good job with Tem+ down to RH values near 0.25, around which a significant fraction of wavenumbers in the “+” spectral region become optically thin and thus have Tem(ν) = Ts. In this case the “min” function in (13) does not commute with the spectral averaging, violating the assumption behind (8).
Fig. 6.
Fig. 6.

(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 Ts = 300, no CO2, and varying RH. (c) Validation of the simple model (14) for F4× in the presence of H2O, as compared to F4× calculated by RFM. This comparison is made for idealized atmospheric columns with Ts = 300, qi = 280 ppmv, and varying RH. The simple expressions (8) and (12) predict Tem± very well except at low RH in the “+” region, leading to small (~0.5 W/m2) errors in F4× at these RH values.

Citation: Journal of Climate 34, 23; 10.1175/JCLI-D-19-0756.1

With some confidence in our estimate of Tem±, we now substitute T¯em from Eq. (9) into Eq. (7) to obtain an expression for CO2 forcing in the presence of H2O overlap:
F=2lln(qfqi)[πB(ν0,T¯em)πB(ν0,Tstrat)](w/H2Ooverlap).
Note that as RH → 0, T¯emTs so this equation indeed generalizes (7).

As a preliminary test of (14) we take our single-column, Ts = 300 K, variable RH calculations (with qi = 280 ppmv) and compare F4× as calculated from RFM with F4× calculated from (14) and (12). The result is shown in Fig. 6c and shows quite good agreement between the two, although the errors in Tem+ at low RH discussed above do lead to small (~0.5 W m−2) errors in F4×.

6. Geographic distribution of F4× with H2O overlap

We now estimate F4× 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 F4× 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 Tem) blocked by CO2.

Fig. 7.
Fig. 7.

(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 F4×, 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.

Citation: Journal of Climate 34, 23; 10.1175/JCLI-D-19-0756.1

We now return to the Ts invariance of T(τ± = τem) in Eq. (12). One consequence of this Ts invariance, in combination with Eq. (14), is that while CO2-only forcing (at fixed qi) grows with increasing Ts (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 Tstrat). We confirm this in Fig. 8, which shows F4× calculated from both RFM as well as Eqs. (7) and (12), for a series of our idealized atmospheric columns with variable Ts and fixed qi = 280 ppmv and RH = 0, 0.75. These plots confirm that the presence of H2O sets an upper limit on F4× with respect to Ts which is well captured by our analytical model. Physically, as Ts 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 Ts 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 Ts = 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.

Fig. 8.
Fig. 8.

CO2 forcing F4× for our idealized atmospheric columns with varying Ts and all other parameters fixed at the BASE values. The presence of H2O (blue) sets a limit on F4×(Ts) 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].

Citation: Journal of Climate 34, 23; 10.1175/JCLI-D-19-0756.1

The limit seen in Fig. 8 is reached beginning at roughly Ts ≈ 300 K, a typical Ts 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 F4× 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 F4× and RH maps in Fig. 7 shows that the zonal asymmetries in tropical F4× 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 Ts variations only, and assumes a fixed preindustrial baseline concentration qi as well as a fixed Tstrat. Increasing qi 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 Ftp, which is well known to be a better approximation to the stratosphere-adjusted forcing. In this section we develop an analytical model for Ftp analogous to Eq. (7), and ask whether our conclusions from previous sections hold for Ftp as well.

To describe Ftp we must estimate both upwelling and downwelling fluxes at the tropopause. For this we need the corresponding emission pressures pem and pem, which lie at an optical distance of τem below and above the tropopause, respectively (in height). These are determined by the equations
τ(pem)=τ(ptp)+τem,
τ(pem)=τ(ptp)τem,
where ptp 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
pem=ptp2+2gprefτemDκ(ν)q,
pem=ptp22gprefτemDκ(ν)q.
These numerical and analytical emission pressures are plotted in Figs. 9a and 9b for the tropopause pressure ptp = 130 hPa from our BASE profile and for CO2 only, continuing to use τemCO2=0.5. The corresponding fluxes at the tropopause are shown in Figs. 9c and 9d, where the analytical fluxes are computed as πB[ν0,T(pem)], from both RFM output and the analytical expressions (16).
Fig. 9.
Fig. 9.

(a),(b) Emission levels pem 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 pem emitting downward toward the tropopause are shown in blue, and levels pem 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).

Citation: Journal of Climate 34, 23; 10.1175/JCLI-D-19-0756.1

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 Ftp, 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 Ts must be replaced by T¯em but the cancellation of the stratospheric term still holds. The tropopause forcing is thus
Ftp=2lln(qf/qi)πB(ν0,T¯em).
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 F4× shown in Figs. 3 and 7, but now for the tropopause forcing F4×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 F4×tp than F4× (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)].

Fig. 10.
Fig. 10.

Spatial distributions of tropopause forcing F4×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 Ts with modulation by H2O, even more so than for the TOA forcings of Fig. 3 and 7.

Citation: Journal of Climate 34, 23; 10.1175/JCLI-D-19-0756.1

Although the instantaneous tropopause forcing Ftp 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 Ftp 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 Ftp depends solely on Ts and not on Tstrat, the claim that the meridional gradient in CO2 forcing is largely governed by the meridional Ts gradient is even more accurate for Ftp 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 Ts invariance of H2O emission temperatures Tem± implies an upper limit (at fixed RH and with respect to Ts 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 Ftp and found that it depends on Ts only, so the above conclusions also hold (perhaps even more so) for Ftp. 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 Ts with a diagnosed cloud-top temperature, just as we replaced Ts by Tem 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 Ts 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 Ts and evaluate at Ts = 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 Ts (Flato et al. 2013) should bias F2× by roughly 0.14 W m−2. One can also consider Tstrat biases, which by a similar differentiation of (7) but with respect to Tstrat and evaluated at Tstrat = 220 K yields a sensitivity of −0.04 W m−2 K−1. Biases of 4–5 K in Tstrat (Butchart et al. 2011) would thus similarly bias F2× by 0.1–0.2 W m−2. Note that these sensitivities to Ts and Tstrat 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 Ts 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 κref± 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 pυ 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 pυ* and reference RH as pυ,ref=RHrefpυ*(Tref). The vapor pressure scaling can then be written as
pυpυ,ref=RHpυ*(T)RHrefpυ*(Tref)RHRHrefeα0(TTref),whereα0LRυTref2.
As for the explicit temperature scaling, this takes the form eσ(TrefT) (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 RHref = 0.75 and Tref+=275K for κ+, and (pref,Tref)=(370hPa,245K) 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+=κref+RHRHrefe(α0σ)(TTref+).
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 κtot, lines-only contribution κlines, and the difference κctm, which we can ascribe to the continuum, for both the “+” and “−” wavenumber regions. These profiles are calculated via RFM for our BASE column, where κlines 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 κctm 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 κref± taken to be equal to κtot±(pref±,Tref±,RHref), yielding κref=0.1m2kg1 and κref+=0.025m2kg1. Our estimates (10) thus agree with κtot± at (pref±,Tref±,RHref) by construction, but due to the many approximations we have made do not have the same logarithmic slope (i.e., scaling) as κtot. However, because H2O optical depth is an integral of κtot weighted by pυ, Clausius–Clapeyron scaling means it is only important for κ to have the right order of magnitude in the range of (Tem − 20 K, Tem) or so, within which our estimates are accurate to roughly a factor of 2 (by our choice of Tref±).

Fig. A1.
Fig. A1.

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 κtot, κlines, and κctm are calculated with RFM, whereas κest is given by Eqs. (10) with (κref,κref+) set to (κtot(Tref),κtot+(Tref+)), and where (Tref,Tref+)=(245,275)K. The estimated profile κ is a poor approximation to κtot, far from Tref due to our neglect of temperature scaling of line absorption, but is acceptably close within 20 K or so of Tref. The horizontal axis in both panels is logarithmic, with the same geometric range (of 150) in each.

Citation: Journal of Climate 34, 23; 10.1175/JCLI-D-19-0756.1

It is interesting to note that the logarithmic slopes of κlines and κetm are comparable for a given wavenumber range, despite the naive expectation that κlines scales with p (which varies by a factor of 5 over the vertical range shown in Fig. A1) and κctm scales with pυ (which varies by a factor of 700). However, κlines 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, κctm also exhibits a temperature scaling but with opposite sign, which weakens its Clausius–Clapeyron scaling [Eq. (A2)]. These opposing temperature scalings for κlines and κctm 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 τem

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 τemCO2=0.5 and τemH2O=0.6 when the surface optical depth is greater than those values. This appendix discusses this approximation, and justifies these choices of τem for our applications. Other values of τem 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 τem 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=τs(pps)β,T=Ts(pps)RdΓ/g,B=Bs(TTs)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)=Bs(ττs)γ,where
γdlnBdlnτ=αRdΓg1β.
Now, the emission level (EL) approximation simply says that
OLR{Bsifτs<τemB(τem)ifτsτem(EL approximation)
for some emission level optical depth τem, which may depend on the parameters introduced above. This τem 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 τem ~ 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 Tem which satisfies OLR ≈ B(Tem), the EL approximation can be rewritten as
Tem=min[Ts,T(τem)],
which is the form used in the main text [e.g., Eq. (8)].
To determine τem, we first analytically compute the OLR for our idealized gray gas,B1 using Eq. (B1) and assuming τs ≫ 1:
OLR=0Bs(τ/τs)γeτdτ=BsτsγΓ˜(1+γ),
where Γ˜(γ+1)0xγexdx 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 τem, obtaining
τem=[Γ˜(1+γ)]1/γ.
A plot of this curve is shown in Fig. B1a. To determine τem, then, we simply need appropriate values for γ for CO2 and H2O emission. For CO2, τem 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 γCO2=0.1. Plugging this into (B6) yields τemCO20.5 (Fig. B1a, red dot).
Fig. B1.
Fig. B1.

(a) Plot of Eq. (B6) for emission levels τem as a function of the parameter γ defined in (B2). At γ = 0, τem=eγEuler0.56, close to our values for τemCO2 and τemH2O. (b) Plot of normalized exact OLR from Eq. (B7), as well as normalized OLR from the emission level approximation (B3) as a function of τs and with γ = 0.4 (black dot in left panel; see text for discussion of this value). The EL approximation is reasonable, even near τs = 1, with maximum errors of roughly 15%.

Citation: Journal of Climate 34, 23; 10.1175/JCLI-D-19-0756.1

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 γH2O=0.15 and hence τemH2O=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 τs values. To do this we need an exact expression for the OLR, generalizing (B5) and valid for all τs:
OLR=Bseτs+0τsBs(τ/τs)γeτdτ=Bseτs+BsτsγΓ˜(1+γ,τs),
where Γ˜(1+γ,τs)0τsxγexdx 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 γ1. In this limit, we may Taylor-expand the Γ˜ function in (B6) and invoke the fact that [dΓ˜(x)/dx]|x=1=γEuler, where γEuler is the Euler-Mascheroni constant (yet another gamma). A little calculation then shows that
limγ0τem=eγEuler=0.56.
This gives a preferred value for τem when |γ| ≪ 1 (Fig. B1a, dotted horizontal line), and indeed this value is very close to both τemCO2 and τemH2O.

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2

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).

3

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 γH2O to 0.08, but yields negligible changes in τemH2O (Fig. B1a).

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